Chen, S J; Perng, S Y; Kuan, C K; Tseng, T C; Wang, D J
2001-01-01
An active polynomial grating has been designed for use in synchrotron radiation soft-X-ray monochromators and spectrometers. The grating can be dynamically adjusted to obtain the third-order-polynomial surface needed to eliminate the defocus and coma aberrations at any photon energy. Ray-tracing results confirm that a monochromator or spectrometer based on this active grating has nearly no aberration limit to the overall spectral resolution in the entire soft-X-ray region. The grating substrate is made of a precisely milled 17-4 PH stainless steel parallel plate, which is joined to a flexure-hinge bender shaped by wire electrical discharge machining. The substrate is grounded into a concave cylindrical shape with a nominal radius and then polished to achieve a roughness of 0.45 nm and a slope error of 1.2 mu rad rms. The long trace profiler measurements show that the active grating can reach the desired third-order polynomial with a high degree of figure accuracy.
Polynomial modal analysis of lamellar diffraction gratings in conical mounting.
Randriamihaja, Manjakavola Honore; Granet, Gérard; Edee, Kofi; Raniriharinosy, Karyl
2016-09-01
An efficient numerical modal method for modeling a lamellar grating in conical mounting is presented. Within each region of the grating, the electromagnetic field is expanded onto Legendre polynomials, which allows us to enforce in an exact manner the boundary conditions that determine the eigensolutions. Our code is successfully validated by comparison with results obtained with the analytical modal method.
Study of physiology of visual cortex activated by rotating grating with functional MRI
International Nuclear Information System (INIS)
Liang Ping; Shao Qing; Zhang Zhiqiang; Lu Guangming
2004-01-01
Objective: To research the physiology of visual cortex activated by rotating grating with functional-MRI (fMRI), and to identify the components of the activation. Methods: Functional MRI was performed in 9 healthy volunteers by using GRE-EPI sequences on a 1.5 T MR scanner. In the block designing, rotating grating, static grating, and luminance were plotted as task states, while static grating, luminance, and darkness were set as control states. The stimuli tasks included six steps. Imaging processing and statistical analysis was carried out off-line using SPM99 in single-subject method. Results: Some respective areas of visual cortex were activated by the various stimuli information supplied by rotating grating. The strong activation in the middle of occipital lobe located at primary vision area was related to the stimuli of white luminance. Its average maximum points were at 13, -98, -2 and 11, -100, -41 The bilateral activations of Brodmann 19th area located at MT area were related to visual motion perception. Its average maximum points were at 46, -72, -2 and -44, -74, 0. The mild activation in the middle of occipital lobe was related to form perception. Its average maximum points were at -12, -98, -6 and -16, -96, -6. Conclusion: The plotting of control state is important in bock design. The effective visual information of rotating grating includes components of luminance, visual motion perception, and form perception. FMRI has potential as a tool for studying the basic physiology of visual cortex. (authors)
Papadopoulos, Anthony
2009-01-01
The first-degree power-law polynomial function is frequently used to describe activity metabolism for steady swimming animals. This function has been used in hydrodynamics-based metabolic studies to evaluate important parameters of energetic costs, such as the standard metabolic rate and the drag power indices. In theory, however, the power-law polynomial function of any degree greater than one can be used to describe activity metabolism for steady swimming animals. In fact, activity metabolism has been described by the conventional exponential function and the cubic polynomial function, although only the power-law polynomial function models drag power since it conforms to hydrodynamic laws. Consequently, the first-degree power-law polynomial function yields incorrect parameter values of energetic costs if activity metabolism is governed by the power-law polynomial function of any degree greater than one. This issue is important in bioenergetics because correct comparisons of energetic costs among different steady swimming animals cannot be made unless the degree of the power-law polynomial function derives from activity metabolism. In other words, a hydrodynamics-based functional form of activity metabolism is a power-law polynomial function of any degree greater than or equal to one. Therefore, the degree of the power-law polynomial function should be treated as a parameter, not as a constant. This new treatment not only conforms to hydrodynamic laws, but also ensures correct comparisons of energetic costs among different steady swimming animals. Furthermore, the exponential power-law function, which is a new hydrodynamics-based functional form of activity metabolism, is a special case of the power-law polynomial function. Hence, the link between the hydrodynamics of steady swimming and the exponential-based metabolic model is defined.
Directory of Open Access Journals (Sweden)
Anthony Papadopoulos
Full Text Available The first-degree power-law polynomial function is frequently used to describe activity metabolism for steady swimming animals. This function has been used in hydrodynamics-based metabolic studies to evaluate important parameters of energetic costs, such as the standard metabolic rate and the drag power indices. In theory, however, the power-law polynomial function of any degree greater than one can be used to describe activity metabolism for steady swimming animals. In fact, activity metabolism has been described by the conventional exponential function and the cubic polynomial function, although only the power-law polynomial function models drag power since it conforms to hydrodynamic laws. Consequently, the first-degree power-law polynomial function yields incorrect parameter values of energetic costs if activity metabolism is governed by the power-law polynomial function of any degree greater than one. This issue is important in bioenergetics because correct comparisons of energetic costs among different steady swimming animals cannot be made unless the degree of the power-law polynomial function derives from activity metabolism. In other words, a hydrodynamics-based functional form of activity metabolism is a power-law polynomial function of any degree greater than or equal to one. Therefore, the degree of the power-law polynomial function should be treated as a parameter, not as a constant. This new treatment not only conforms to hydrodynamic laws, but also ensures correct comparisons of energetic costs among different steady swimming animals. Furthermore, the exponential power-law function, which is a new hydrodynamics-based functional form of activity metabolism, is a special case of the power-law polynomial function. Hence, the link between the hydrodynamics of steady swimming and the exponential-based metabolic model is defined.
Active phase correction of high resolution silicon photonic arrayed waveguide gratings.
Gehl, M; Trotter, D; Starbuck, A; Pomerene, A; Lentine, A L; DeRose, C
2017-03-20
Arrayed waveguide gratings provide flexible spectral filtering functionality for integrated photonic applications. Achieving narrow channel spacing requires long optical path lengths which can greatly increase the footprint of devices. High index contrast waveguides, such as those fabricated in silicon-on-insulator wafers, allow tight waveguide bends which can be used to create much more compact designs. Both the long optical path lengths and the high index contrast contribute to significant optical phase error as light propagates through the device. Therefore, silicon photonic arrayed waveguide gratings require active or passive phase correction following fabrication. Here we present the design and fabrication of compact silicon photonic arrayed waveguide gratings with channel spacings of 50, 10 and 1 GHz. The largest device, with 11 channels of 1 GHz spacing, has a footprint of only 1.1 cm2. Using integrated thermo-optic phase shifters, the phase error is actively corrected. We present two methods of phase error correction and demonstrate state-of-the-art cross-talk performance for high index contrast arrayed waveguide gratings. As a demonstration of possible applications, we perform RF channelization with 1 GHz resolution. Additionally, we generate unique spectral filters by applying non-zero phase offsets calculated by the Gerchberg Saxton algorithm.
Superiority of legendre polynomials to Chebyshev polynomial in ...
African Journals Online (AJOL)
In this paper, we proved the superiority of Legendre polynomial to Chebyshev polynomial in solving first order ordinary differential equation with rational coefficient. We generated shifted polynomial of Chebyshev, Legendre and Canonical polynomials which deal with solving differential equation by first choosing Chebyshev ...
Freud, Géza
1971-01-01
Orthogonal Polynomials contains an up-to-date survey of the general theory of orthogonal polynomials. It deals with the problem of polynomials and reveals that the sequence of these polynomials forms an orthogonal system with respect to a non-negative m-distribution defined on the real numerical axis. Comprised of five chapters, the book begins with the fundamental properties of orthogonal polynomials. After discussing the momentum problem, it then explains the quadrature procedure, the convergence theory, and G. Szegő's theory. This book is useful for those who intend to use it as referenc
Birefringence Bragg Binary (3B) grating, quasi-Bragg grating and immersion gratings
Ebizuka, Noboru; Morita, Shin-ya; Yamagata, Yutaka; Sasaki, Minoru; Bianco, Andorea; Tanabe, Ayano; Hashimoto, Nobuyuki; Hirahara, Yasuhiro; Aoki, Wako
2014-07-01
A volume phase holographic (VPH) grating achieves high angular dispersion and very high diffraction efficiency for the first diffraction order and for S or P polarization. However the VPH grating could not achieve high diffraction efficiency for non-polarized light at a large diffraction angle because properties of diffraction efficiencies for S and P polarizations are different. Furthermore diffraction efficiency of the VPH grating extinguishes toward a higher diffraction order. A birefringence binary Bragg (3B) grating is a thick transmission grating with optically anisotropic material such as lithium niobate or liquid crystal. The 3B grating achieves diffraction efficiency up to 100% for non-polarized light by tuning of refractive indices for S and P polarizations, even in higher diffraction orders. We fabricated 3B grating with liquid crystal and evaluated the performance of the liquid crystal grating. A quasi-Bragg (QB) grating, which consists long rectangle mirrors aligned in parallel precisely such as a window shade, also achieves high diffraction efficiency toward higher orders. We fabricated QB grating by laminating of silica glass substrates and glued by pressure fusion of gold films. A quasi-Bragg immersion (QBI) grating has smooth mirror hypotenuse and reflector array inside the hypotenuse, instead of step-like grooves of a conventional immersion grating. An incident beam of the QBI grating reflects obliquely at a reflector, then reflects vertically at the mirror surface and reflects again at the same reflector. We are going to fabricate QBI gratings by laminating of mirror plates as similar to fabrication of the QB grating. We will also fabricate silicon and germanium immersion gratings with conventional step-like grooves by means of the latest diamond machining methods. We introduce characteristics and performance of these gratings.
Fabrication update on critical-angle transmission gratings for soft x-ray grating spectrometers
Heilmann, Ralf K.; Bruccoleri, Alex; Mukherjee, Pran; Yam, Jonathan; Schattenburg, Mark L.
2011-09-01
Diffraction grating-based, wavelength dispersive high-resolution soft x-ray spectroscopy of celestial sources promises to reveal crucial data for the study of the Warm-Hot Intergalactic Medium, the Interstellar Medium, warm absorption and outflows in Active Galactic Nuclei, coronal emission from stars, and other areas of interest to the astrophysics community. Our recently developed critical-angle transmission (CAT) gratings combine the advantages of the Chandra high and medium energy transmission gratings (low mass, high tolerance of misalignments and figure errors, polarization insensitivity) with those of blazed reflection gratings (high broad band diffraction efficiency, high resolution through use of higher diffraction orders) such as the ones on XMM-Newton. Extensive instrument and system configuration studies have shown that a CAT grating-based spectrometer is an outstanding instrument capable of delivering resolving power on the order of 5,000 and high effective area, even with a telescope point-spread function on the order of many arc-seconds. We have fabricated freestanding, ultra-high aspect-ratio CAT grating bars from silicon-on-insulator wafers using both wet and dry etch processes. The 200 nm-period grating bars are supported by an integrated Level 1 support mesh, and a coarser external Level 2 support mesh. The resulting grating membrane is mounted to a frame, resulting in a grating facet. Many such facets comprise a grating array that provides light-weight coverage of large-area telescope apertures. Here we present fabrication results on the integration of CAT gratings and the different high-throughput support mesh levels and on membrane-frame bonding. We also summarize recent x-ray data analysis of 3 and 6 micron deep wet-etched CAT grating prototypes.
Irreducible multivariate polynomials obtained from polynomials in ...
Indian Academy of Sciences (India)
Hall, 1409 W. Green Street, Urbana, IL 61801, USA. E-mail: Nicolae. ... Theorem A. If we write an irreducible polynomial f ∈ K[X] as a sum of polynomials a0,..., an ..... This shows us that deg ai = (n − i) deg f2 for each i = 0,..., n, so min k>0.
Branched polynomial covering maps
DEFF Research Database (Denmark)
Hansen, Vagn Lundsgaard
2002-01-01
A Weierstrass polynomial with multiple roots in certain points leads to a branched covering map. With this as the guiding example, we formally define and study the notion of a branched polynomial covering map. We shall prove that many finite covering maps are polynomial outside a discrete branch ...... set. Particular studies are made of branched polynomial covering maps arising from Riemann surfaces and from knots in the 3-sphere. (C) 2001 Elsevier Science B.V. All rights reserved.......A Weierstrass polynomial with multiple roots in certain points leads to a branched covering map. With this as the guiding example, we formally define and study the notion of a branched polynomial covering map. We shall prove that many finite covering maps are polynomial outside a discrete branch...
Discrete-time state estimation for stochastic polynomial systems over polynomial observations
Hernandez-Gonzalez, M.; Basin, M.; Stepanov, O.
2018-07-01
This paper presents a solution to the mean-square state estimation problem for stochastic nonlinear polynomial systems over polynomial observations confused with additive white Gaussian noises. The solution is given in two steps: (a) computing the time-update equations and (b) computing the measurement-update equations for the state estimate and error covariance matrix. A closed form of this filter is obtained by expressing conditional expectations of polynomial terms as functions of the state estimate and error covariance. As a particular case, the mean-square filtering equations are derived for a third-degree polynomial system with second-degree polynomial measurements. Numerical simulations show effectiveness of the proposed filter compared to the extended Kalman filter.
Stability analysis of polynomial fuzzy models via polynomial fuzzy Lyapunov functions
Bernal Reza, Miguel Ángel; Sala, Antonio; JAADARI, ABDELHAFIDH; Guerra, Thierry-Marie
2011-01-01
In this paper, the stability of continuous-time polynomial fuzzy models by means of a polynomial generalization of fuzzy Lyapunov functions is studied. Fuzzy Lyapunov functions have been fruitfully used in the literature for local analysis of Takagi-Sugeno models, a particular class of the polynomial fuzzy ones. Based on a recent Taylor-series approach which allows a polynomial fuzzy model to exactly represent a nonlinear model in a compact set of the state space, it is shown that a refinemen...
A new class of generalized polynomials associated with Hermite and Bernoulli polynomials
Directory of Open Access Journals (Sweden)
M. A. Pathan
2015-05-01
Full Text Available In this paper, we introduce a new class of generalized polynomials associated with the modified Milne-Thomson's polynomials Φ_{n}^{(α}(x,ν of degree n and order α introduced by Derre and Simsek.The concepts of Bernoulli numbers B_n, Bernoulli polynomials B_n(x, generalized Bernoulli numbers B_n(a,b, generalized Bernoulli polynomials B_n(x;a,b,c of Luo et al, Hermite-Bernoulli polynomials {_HB}_n(x,y of Dattoli et al and {_HB}_n^{(α} (x,y of Pathan are generalized to the one {_HB}_n^{(α}(x,y,a,b,c which is called the generalized polynomial depending on three positive real parameters. Numerous properties of these polynomials and some relationships between B_n, B_n(x, B_n(a,b, B_n(x;a,b,c and {}_HB_n^{(α}(x,y;a,b,c are established. Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating functions. These results extend some known summations and identities of generalized Bernoulli numbers and polynomials
Optical fiber Bragg gratings. Part II. Modeling of finite-length gratings and grating arrays.
Passaro, Vittorio M N; Diana, Roberto; Armenise, Mario N
2002-09-01
A model of both uniform finite-length optical fiber Bragg gratings and grating arrays is presented. The model is based on the Floquet-Bloch formalism and allows rigorous investigation of all the physical aspects in either single- or multiple-periodic structures realized on the core of a monomodal fiber. Analytical expressions of reflectivity and transmittivity for both single gratings and grating arrays are derived. The influence of the grating length and the index modulation amplitude on the reflected and transmitted optical power for both sinusoidal and rectangular profiles is evaluated. Good agreement between our method and the well-known coupled-mode theory (CMT) approach has been observed for both single gratings and grating arrays only in the case of weak index perturbation. Significant discrepancies exist there in cases of strong index contrast because of the increasing approximation of the CMT approach. The effects of intragrating phase shift are also shown and discussed.
Liu, Xudong; Chen, Xuequan; Parrott, Edward P. J.; Han, Chunrui; Humbert, Georges; Crunteanu, Aurelian; Pickwell-MacPherson, Emma
2018-05-01
Active broadband terahertz (THz) polarization manipulation devices are challenging to realize, but also of great demand in broadband terahertz systems. Vanadium dioxide (VO2) shows a promising phase transition for active control of THz waves and provides broadband polarization characteristics when integrated within grating-type structures. We creatively combine a VO2-based grating structure with a total internal reflection (TIR) geometry providing a novel interaction mechanism between the electromagnetic waves and the device, to realize a powerful active broadband THz polarization-controlling device. The device is based on a Si-substrate coated with a VO2 layer and a metal grating structure on top, attached to a prism for generating the TIR condition on the Si-VO2-grating interface. The grating is connected to electrodes for electrically switching the VO2 between its insulating and conducting phases. By properly selecting the incident angle of the THz waves, the grating direction, and the incident polarization state, we first achieved a broadband intensity modulator under a fused silica prism with an average modulation depth of 99.75% in the 0.2-1.1 THz region. Additionally, we realized an active ultra-broadband quarter-wave converter under a Si prism that can be switched between a 45° linear rotator and a quarter wave converter in the 0.8-1.5 THz region. This is the first demonstration of an active quarter-wave converter with ultra-broad bandwidth performance. Our work shows a highly flexible and multifunctional polarization-controlling device for broadband THz applications.
Branched polynomial covering maps
DEFF Research Database (Denmark)
Hansen, Vagn Lundsgaard
1999-01-01
A Weierstrass polynomial with multiple roots in certain points leads to a branched covering map. With this as the guiding example, we formally define and study the notion of a branched polynomial covering map. We shall prove that many finite covering maps are polynomial outside a discrete branch...... set. Particular studies are made of branched polynomial covering maps arising from Riemann surfaces and from knots in the 3-sphere....
On the number of polynomial solutions of Bernoulli and Abel polynomial differential equations
Cima, A.; Gasull, A.; Mañosas, F.
2017-12-01
In this paper we determine the maximum number of polynomial solutions of Bernoulli differential equations and of some integrable polynomial Abel differential equations. As far as we know, the tools used to prove our results have not been utilized before for studying this type of questions. We show that the addressed problems can be reduced to know the number of polynomial solutions of a related polynomial equation of arbitrary degree. Then we approach to these equations either applying several tools developed to study extended Fermat problems for polynomial equations, or reducing the question to the computation of the genus of some associated planar algebraic curves.
On generalized Fibonacci and Lucas polynomials
Energy Technology Data Exchange (ETDEWEB)
Nalli, Ayse [Department of Mathematics, Faculty of Sciences, Selcuk University, 42075 Campus-Konya (Turkey)], E-mail: aysenalli@yahoo.com; Haukkanen, Pentti [Department of Mathematics, Statistics and Philosophy, 33014 University of Tampere (Finland)], E-mail: mapehau@uta.fi
2009-12-15
Let h(x) be a polynomial with real coefficients. We introduce h(x)-Fibonacci polynomials that generalize both Catalan's Fibonacci polynomials and Byrd's Fibonacci polynomials and also the k-Fibonacci numbers, and we provide properties for these h(x)-Fibonacci polynomials. We also introduce h(x)-Lucas polynomials that generalize the Lucas polynomials and present properties of these polynomials. In the last section we introduce the matrix Q{sub h}(x) that generalizes the Q-matrix whose powers generate the Fibonacci numbers.
Stabilisation of discrete-time polynomial fuzzy systems via a polynomial lyapunov approach
Nasiri, Alireza; Nguang, Sing Kiong; Swain, Akshya; Almakhles, Dhafer
2018-02-01
This paper deals with the problem of designing a controller for a class of discrete-time nonlinear systems which is represented by discrete-time polynomial fuzzy model. Most of the existing control design methods for discrete-time fuzzy polynomial systems cannot guarantee their Lyapunov function to be a radially unbounded polynomial function, hence the global stability cannot be assured. The proposed control design in this paper guarantees a radially unbounded polynomial Lyapunov functions which ensures global stability. In the proposed design, state feedback structure is considered and non-convexity problem is solved by incorporating an integrator into the controller. Sufficient conditions of stability are derived in terms of polynomial matrix inequalities which are solved via SOSTOOLS in MATLAB. A numerical example is presented to illustrate the effectiveness of the proposed controller.
Bai , Shi; Bouvier , Cyril; Kruppa , Alexander; Zimmermann , Paul
2016-01-01
International audience; The general number field sieve (GNFS) is the most efficient algo-rithm known for factoring large integers. It consists of several stages, the first one being polynomial selection. The quality of the selected polynomials can be modelled in terms of size and root properties. We propose a new kind of polynomials for GNFS: with a new degree of freedom, we further improve the size property. We demonstrate the efficiency of our algorithm by exhibiting a better polynomial tha...
Nonnegativity of uncertain polynomials
Directory of Open Access Journals (Sweden)
iljak Dragoslav D.
1998-01-01
Full Text Available The purpose of this paper is to derive tests for robust nonnegativity of scalar and matrix polynomials, which are algebraic, recursive, and can be completed in finite number of steps. Polytopic families of polynomials are considered with various characterizations of parameter uncertainty including affine, multilinear, and polynomic structures. The zero exclusion condition for polynomial positivity is also proposed for general parameter dependencies. By reformulating the robust stability problem of complex polynomials as positivity of real polynomials, we obtain new sufficient conditions for robust stability involving multilinear structures, which can be tested using only real arithmetic. The obtained results are applied to robust matrix factorization, strict positive realness, and absolute stability of multivariable systems involving parameter dependent transfer function matrices.
Diffraction by m-bonacci gratings
International Nuclear Information System (INIS)
Monsoriu, Juan A; Giménez, Marcos H; Furlan, Walter D; Barreiro, Juan C; Saavedra, Genaro
2015-01-01
We present a simple diffraction experiment with m-bonacci gratings as a new interesting generalization of the Fibonacci ones. Diffraction by these non-conventional structures is proposed as a motivational strategy to introduce students to basic research activities. The Fraunhofer diffraction patterns are obtained with the standard equipment present in most undergraduate physics labs and are compared with those obtained with regular periodic gratings. We show that m-bonacci gratings produce discrete Fraunhofer patterns characterized by a set of diffraction peaks which positions are related to the concept of a generalized golden mean. A very good agreement is obtained between experimental and numerical results and the students’ feedback is discussed. (paper)
On Multiple Polynomials of Capelli Type
Directory of Open Access Journals (Sweden)
S.Y. Antonov
2016-03-01
Full Text Available This paper deals with the class of Capelli polynomials in free associative algebra F{Z} (where F is an arbitrary field, Z is a countable set generalizing the construction of multiple Capelli polynomials. The fundamental properties of the introduced Capelli polynomials are provided. In particular, decomposition of the Capelli polynomials by means of the same type of polynomials is shown. Furthermore, some relations between their T -ideals are revealed. A connection between double Capelli polynomials and Capelli quasi-polynomials is established.
Chromatic polynomials for simplicial complexes
DEFF Research Database (Denmark)
Møller, Jesper Michael; Nord, Gesche
2016-01-01
In this note we consider s s -chromatic polynomials for finite simplicial complexes. When s=1 s=1 , the 1 1 -chromatic polynomial is just the usual graph chromatic polynomial of the 1 1 -skeleton. In general, the s s -chromatic polynomial depends on the s s -skeleton and its value at r...
Roots of the Chromatic Polynomial
DEFF Research Database (Denmark)
Perrett, Thomas
The chromatic polynomial of a graph G is a univariate polynomial whose evaluation at any positive integer q enumerates the proper q-colourings of G. It was introduced in connection with the famous four colour theorem but has recently found other applications in the field of statistical physics...... extend Thomassen’s technique to the Tutte polynomial and as a consequence, deduce a density result for roots of the Tutte polynomial. This partially answers a conjecture of Jackson and Sokal. Finally, we refocus our attention on the chromatic polynomial and investigate the density of chromatic roots...
General Reducibility and Solvability of Polynomial Equations ...
African Journals Online (AJOL)
General Reducibility and Solvability of Polynomial Equations. ... Unlike quadratic, cubic, and quartic polynomials, the general quintic and higher degree polynomials cannot be solved algebraically in terms of finite number of additions, ... Galois Theory, Solving Polynomial Systems, Polynomial factorization, Polynomial Ring ...
Certain non-linear differential polynomials sharing a non zero polynomial
Directory of Open Access Journals (Sweden)
Majumder Sujoy
2015-10-01
functions sharing a nonzero polynomial and obtain two results which improves and generalizes the results due to L. Liu [Uniqueness of meromorphic functions and differential polynomials, Comput. Math. Appl., 56 (2008, 3236-3245.] and P. Sahoo [Uniqueness and weighted value sharing of meromorphic functions, Applied. Math. E-Notes., 11 (2011, 23-32.].
Polynomial Heisenberg algebras
International Nuclear Information System (INIS)
Carballo, Juan M; C, David J Fernandez; Negro, Javier; Nieto, Luis M
2004-01-01
Polynomial deformations of the Heisenberg algebra are studied in detail. Some of their natural realizations are given by the higher order susy partners (and not only by those of first order, as is already known) of the harmonic oscillator for even-order polynomials. Here, it is shown that the susy partners of the radial oscillator play a similar role when the order of the polynomial is odd. Moreover, it will be proved that the general systems ruled by such kinds of algebras, in the quadratic and cubic cases, involve Painleve transcendents of types IV and V, respectively
Polynomial optimization : Error analysis and applications
Sun, Zhao
2015-01-01
Polynomial optimization is the problem of minimizing a polynomial function subject to polynomial inequality constraints. In this thesis we investigate several hierarchies of relaxations for polynomial optimization problems. Our main interest lies in understanding their performance, in particular how
Birth-death processes and associated polynomials
van Doorn, Erik A.
2003-01-01
We consider birth-death processes on the nonnegative integers and the corresponding sequences of orthogonal polynomials called birth-death polynomials. The sequence of associated polynomials linked with a sequence of birth-death polynomials and its orthogonalizing measure can be used in the analysis
Extended biorthogonal matrix polynomials
Directory of Open Access Journals (Sweden)
Ayman Shehata
2017-01-01
Full Text Available The pair of biorthogonal matrix polynomials for commutative matrices were first introduced by Varma and Tasdelen in [22]. The main aim of this paper is to extend the properties of the pair of biorthogonal matrix polynomials of Varma and Tasdelen and certain generating matrix functions, finite series, some matrix recurrence relations, several important properties of matrix differential recurrence relations, biorthogonality relations and matrix differential equation for the pair of biorthogonal matrix polynomials J(A,B n (x, k and K(A,B n (x, k are discussed. For the matrix polynomials J(A,B n (x, k, various families of bilinear and bilateral generating matrix functions are constructed in the sequel.
Bannai-Ito polynomials and dressing chains
Derevyagin, Maxim; Tsujimoto, Satoshi; Vinet, Luc; Zhedanov, Alexei
2012-01-01
Schur-Delsarte-Genin (SDG) maps and Bannai-Ito polynomials are studied. SDG maps are related to dressing chains determined by quadratic algebras. The Bannai-Ito polynomials and their kernel polynomials -- the complementary Bannai-Ito polynomials -- are shown to arise in the framework of the SDG maps.
Grate-firing of biomass for heat and power production
DEFF Research Database (Denmark)
Yin, Chungen; Rosendahl, Lasse; Kær, Søren Knudsen
2008-01-01
bed on the grate, and the advanced secondary air supply (a real breakthrough in this technology) are highlighted for grate-firing systems. Amongst all the issues or problems associated with grate-fired boilers burning biomass, primary pollutant formation and control, deposition formation and corrosion......As a renewable and environmentally friendly energy source, biomass (i.e., any organic non-fossil fuel) and its utilization are gaining an increasingly important role worldwide Grate-firing is one of the main competing technologies in biomass combustion for heat and power production, because it can...... combustion mechanism, the recent breakthrough in the technology, the most pressing issues, the current research and development activities, and the critical future problems to be resolved. The grate assembly (the most characteristic element in grate-fired boilers), the key combustion mechanism in the fuel...
Constructing general partial differential equations using polynomial and neural networks.
Zjavka, Ladislav; Pedrycz, Witold
2016-01-01
Sum fraction terms can approximate multi-variable functions on the basis of discrete observations, replacing a partial differential equation definition with polynomial elementary data relation descriptions. Artificial neural networks commonly transform the weighted sum of inputs to describe overall similarity relationships of trained and new testing input patterns. Differential polynomial neural networks form a new class of neural networks, which construct and solve an unknown general partial differential equation of a function of interest with selected substitution relative terms using non-linear multi-variable composite polynomials. The layers of the network generate simple and composite relative substitution terms whose convergent series combinations can describe partial dependent derivative changes of the input variables. This regression is based on trained generalized partial derivative data relations, decomposed into a multi-layer polynomial network structure. The sigmoidal function, commonly used as a nonlinear activation of artificial neurons, may transform some polynomial items together with the parameters with the aim to improve the polynomial derivative term series ability to approximate complicated periodic functions, as simple low order polynomials are not able to fully make up for the complete cycles. The similarity analysis facilitates substitutions for differential equations or can form dimensional units from data samples to describe real-world problems. Copyright © 2015 Elsevier Ltd. All rights reserved.
Point-by-point written fiber-Bragg gratings and their application in complex grating designs.
Marshall, Graham D; Williams, Robert J; Jovanovic, Nemanja; Steel, M J; Withford, Michael J
2010-09-13
The point-by-point technique of fabricating fibre-Bragg gratings using an ultrafast laser enables complete control of the position of each index modification that comprises the grating. By tailoring the local phase, amplitude and spacing of the grating's refractive index modulations it is possible to create gratings with complex transmission and reflection spectra. We report a series of grating structures that were realized by exploiting these flexibilities. Such structures include gratings with controlled bandwidth, and amplitude- and phase-modulated sampled (or superstructured) gratings. A model based on coupled-mode theory provides important insights into the manufacture of such gratings. Our approach offers a quick and easy method of producing complex, non-uniform grating structures in both fibres and other mono-mode waveguiding structures.
Vortices and polynomials: non-uniqueness of the Adler–Moser polynomials for the Tkachenko equation
International Nuclear Information System (INIS)
Demina, Maria V; Kudryashov, Nikolai A
2012-01-01
Stationary and translating relative equilibria of point vortices in the plane are studied. It is shown that stationary equilibria of any system containing point vortices with arbitrary choice of circulations can be described with the help of the Tkachenko equation. It is also obtained that translating relative equilibria of point vortices with arbitrary circulations can be constructed using a generalization of the Tkachenko equation. Roots of any pair of polynomials solving the Tkachenko equation and the generalized Tkachenko equation are proved to give positions of point vortices in stationary and translating relative equilibria accordingly. These results are valid even if the polynomials in a pair have multiple or common roots. It is obtained that the Adler–Moser polynomial provides non-unique polynomial solutions of the Tkachenko equation. It is shown that the generalized Tkachenko equation possesses polynomial solutions with degrees that are not triangular numbers. (paper)
Generalizations of orthogonal polynomials
Bultheel, A.; Cuyt, A.; van Assche, W.; van Barel, M.; Verdonk, B.
2005-07-01
We give a survey of recent generalizations of orthogonal polynomials. That includes multidimensional (matrix and vector orthogonal polynomials) and multivariate versions, multipole (orthogonal rational functions) variants, and extensions of the orthogonality conditions (multiple orthogonality). Most of these generalizations are inspired by the applications in which they are applied. We also give a glimpse of these applications, which are usually generalizations of applications where classical orthogonal polynomials also play a fundamental role: moment problems, numerical quadrature, rational approximation, linear algebra, recurrence relations, and random matrices.
Stochastic Estimation via Polynomial Chaos
2015-10-01
AFRL-RW-EG-TR-2015-108 Stochastic Estimation via Polynomial Chaos Douglas V. Nance Air Force Research...COVERED (From - To) 20-04-2015 – 07-08-2015 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER Stochastic Estimation via Polynomial Chaos ...This expository report discusses fundamental aspects of the polynomial chaos method for representing the properties of second order stochastic
The spectral combination characteristic of grating and the bi-grating diffraction imaging effect
Institute of Scientific and Technical Information of China (English)
2007-01-01
This paper reports on a new property of grating, namely spectral combination, and on bi-grating diffraction imaging that is based on spectral combination. The spectral combination characteristic of a grating is the capability of combining multiple light beams of different wavelengths incident from specific angles into a single beam. The bi-grating diffraction imaging is the formation of the image of an object with two gratings: the first grating disperses the multi-color light beams from the object and the second combines the dispersed light beams to form the image. We gave the conditions necessary for obtaining the spectral combination. We also presented the equations that relate the two gratings’ spatial frequencies, diffraction orders and positions necessary for obtaining the bi-grating diffraction imaging.
A New Generalisation of Macdonald Polynomials
Garbali, Alexandr; de Gier, Jan; Wheeler, Michael
2017-06-01
We introduce a new family of symmetric multivariate polynomials, whose coefficients are meromorphic functions of two parameters ( q, t) and polynomial in a further two parameters ( u, v). We evaluate these polynomials explicitly as a matrix product. At u = v = 0 they reduce to Macdonald polynomials, while at q = 0, u = v = s they recover a family of inhomogeneous symmetric functions originally introduced by Borodin.
Special polynomials associated with some hierarchies
International Nuclear Information System (INIS)
Kudryashov, Nikolai A.
2008-01-01
Special polynomials associated with rational solutions of a hierarchy of equations of Painleve type are introduced. The hierarchy arises by similarity reduction from the Fordy-Gibbons hierarchy of partial differential equations. Some relations for these special polynomials are given. Differential-difference hierarchies for finding special polynomials are presented. These formulae allow us to obtain special polynomials associated with the hierarchy studied. It is shown that rational solutions of members of the Schwarz-Sawada-Kotera, the Schwarz-Kaup-Kupershmidt, the Fordy-Gibbons, the Sawada-Kotera and the Kaup-Kupershmidt hierarchies can be expressed through special polynomials of the hierarchy studied
Spherical grating spectrometers
O'Donoghue, Darragh; Clemens, J. Christopher
2014-07-01
We describe designs for spectrometers employing convex dispersers. The Offner spectrometer was the first such instrument; it has almost exclusively been employed on satellite platforms, and has had little impact on ground-based instruments. We have learned how to fabricate curved Volume Phase Holographic (VPH) gratings and, in contrast to the planar gratings of traditional spectrometers, describe how such devices can be used in optical/infrared spectrometers designed specifically for curved diffraction gratings. Volume Phase Holographic gratings are highly efficient compared to conventional surface relief gratings; they have become the disperser of choice in optical / NIR spectrometers. The advantage of spectrometers with curved VPH dispersers is the very small number of optical elements used (the simplest comprising a grating and a spherical mirror), as well as illumination of mirrors off axis, resulting in greater efficiency and reduction in size. We describe a "Half Offner" spectrometer, an even simpler version of the Offner spectrometer. We present an entirely novel design, the Spherical Transmission Grating Spectrometer (STGS), and discuss exemplary applications, including a design for a double-beam spectrometer without any requirement for a dichroic. This paradigm change in spectrometer design offers an alternative to all-refractive astronomical spectrometer designs, using expensive, fragile lens elements fabricated from CaF2 or even more exotic materials. The unobscured mirror layout avoids a major drawback of the previous generation of catadioptric spectrometer designs. We describe laboratory measurements of the efficiency and image quality of a curved VPH grating in a STGS design, demonstrating, simultaneously, efficiency comparable to planar VPH gratings along with good image quality. The stage is now set for construction of a prototype instrument with impressive performance.
A Summation Formula for Macdonald Polynomials
de Gier, Jan; Wheeler, Michael
2016-03-01
We derive an explicit sum formula for symmetric Macdonald polynomials. Our expression contains multiple sums over the symmetric group and uses the action of Hecke generators on the ring of polynomials. In the special cases {t = 1} and {q = 0}, we recover known expressions for the monomial symmetric and Hall-Littlewood polynomials, respectively. Other specializations of our formula give new expressions for the Jack and q-Whittaker polynomials.
Weierstrass polynomials for links
DEFF Research Database (Denmark)
Hansen, Vagn Lundsgaard
1997-01-01
There is a natural way of identifying links in3-space with polynomial covering spaces over thecircle. Thereby any link in 3-space can be definedby a Weierstrass polynomial over the circle. Theequivalence relation for covering spaces over thecircle is, however, completely different from...
Golden, Ryan; Cho, Ilwoo
2015-01-01
In this paper, we study structure theorems of algebras of symmetric functions. Based on a certain relation on elementary symmetric polynomials generating such algebras, we consider perturbation in the algebras. In particular, we understand generators of the algebras as perturbations. From such perturbations, define injective maps on generators, which induce algebra-monomorphisms (or embeddings) on the algebras. They provide inductive structure theorems on algebras of symmetric polynomials. As...
Kim, Youngju; Kim, Jongyul; Kim, Daeseung; Hussey, Daniel. S.; Lee, Seung Wook
2018-03-01
We introduce an analyzer grating based on a structured scintillator fabricated by a gadolinium oxysulfide powder filling method for a symmetric Talbot-Lau neutron grating interferometer. This is an alternative way to analyze the Talbot self-image of a grating interferometer without using an absorption grating to block neutrons. Since the structured scintillator analyzer grating itself generates the signal for neutron detection, we do not need an additional scintillator screen as an absorption analyzer grating. We have developed and tested an analyzer grating based on a structured scintillator in our symmetric Talbot-Lau neutron grating interferometer to produce high fidelity absorption, differential phase, and dark-field contrast images. The acquired images have been compared to results of a grating interferometer utilizing a typical absorption analyzer grating with two commercial scintillation screens. The analyzer grating based on the structured scintillator enhances interference fringe visibility and shows a great potential for economical fabrication, compact system design, and so on. We report the performance of the analyzer grating based on a structured scintillator and evaluate its feasibility for the neutron grating interferometer.
Associated polynomials and birth-death processes
van Doorn, Erik A.
2001-01-01
We consider sequences of orthogonal polynomials with positive zeros, and pursue the question of how (partial) knowledge of the orthogonalizing measure for the {\\it associated polynomials} can lead to information about the orthogonalizing measure for the original polynomials, with a view to
National Research Council Canada - National Science Library
Battiato, James
1998-01-01
Coupled mode theory was used to model reflection fiber gratings. The effects of experimental parameters on grating characteristics were modeled for both uniform and non-uniform grating profiles using this approach...
Scattering theory and orthogonal polynomials
International Nuclear Information System (INIS)
Geronimo, J.S.
1977-01-01
The application of the techniques of scattering theory to the study of polynomials orthogonal on the unit circle and a finite segment of the real line is considered. The starting point is the recurrence relations satisfied by the polynomials instead of the orthogonality condition. A set of two two terms recurrence relations for polynomials orthogonal on the real line is presented and used. These recurrence relations play roles analogous to those satisfied by polynomials orthogonal on unit circle. With these recurrence formulas a Wronskian theorem is proved and the Christoffel-Darboux formula is derived. In scattering theory a fundamental role is played by the Jost function. An analogy is deferred of this function and its analytic properties and the locations of its zeros investigated. The role of the analog Jost function in various properties of these orthogonal polynomials is investigated. The techniques of inverse scattering theory are also used. The discrete analogues of the Gelfand-Levitan and Marchenko equations are derived and solved. These techniques are used to calculate asymptotic formulas for the orthogonal polynomials. Finally Szego's theorem on toeplitz and Hankel determinants is proved using the recurrence formulas and some properties of the Jost function. The techniques of inverse scattering theory are used to calculate the correction terms
Kessler, Terrance J [Mendon, NY; Bunkenburg, Joachim [Victor, NY; Huang, Hu [Pittsford, NY
2007-02-13
A plurality of gratings (G1, G2) are arranged together with a wavefront sensor, actuators, and feedback system to align the gratings in such a manner, that they operate like a single, large, monolithic grating. Sub-wavelength-scale movements in the mechanical mounting, due to environmental influences, are monitored by an interferometer (28), and compensated by precision actuators (16, 18, 20) that maintain the coherently additive mode. The actuators define the grating plane, and are positioned in response to the wavefronts from the gratings and a reference flat, thus producing the interferogram that contains the alignment information. Movement of the actuators is also in response to a diffraction-limited spot on the CCD (36) to which light diffracted from the gratings is focused. The actuator geometry is implemented to take advantage of the compensating nature of the degrees of freedom between gratings, reducing the number of necessary control variables.
Fermionic formula for double Kostka polynomials
Liu, Shiyuan
2016-01-01
The $X=M$ conjecture asserts that the $1D$ sum and the fermionic formula coincide up to some constant power. In the case of type $A,$ both the $1D$ sum and the fermionic formula are closely related to Kostka polynomials. Double Kostka polynomials $K_{\\Bla,\\Bmu}(t),$ indexed by two double partitions $\\Bla,\\Bmu,$ are polynomials in $t$ introduced as a generalization of Kostka polynomials. In the present paper, we consider $K_{\\Bla,\\Bmu}(t)$ in the special case where $\\Bmu=(-,\\mu'').$ We formula...
Relations between Möbius and coboundary polynomials
Jurrius, R.P.M.J.
2012-01-01
It is known that, in general, the coboundary polynomial and the Möbius polynomial of a matroid do not determine each other. Less is known about more specific cases. In this paper, we will investigate if it is possible that the Möbius polynomial of a matroid, together with the Möbius polynomial of
Matrix product formula for Macdonald polynomials
Cantini, Luigi; de Gier, Jan; Wheeler, Michael
2015-09-01
We derive a matrix product formula for symmetric Macdonald polynomials. Our results are obtained by constructing polynomial solutions of deformed Knizhnik-Zamolodchikov equations, which arise by considering representations of the Zamolodchikov-Faddeev and Yang-Baxter algebras in terms of t-deformed bosonic operators. These solutions are generalized probabilities for particle configurations of the multi-species asymmetric exclusion process, and form a basis of the ring of polynomials in n variables whose elements are indexed by compositions. For weakly increasing compositions (anti-dominant weights), these basis elements coincide with non-symmetric Macdonald polynomials. Our formulas imply a natural combinatorial interpretation in terms of solvable lattice models. They also imply that normalizations of stationary states of multi-species exclusion processes are obtained as Macdonald polynomials at q = 1.
Matrix product formula for Macdonald polynomials
International Nuclear Information System (INIS)
Cantini, Luigi; Gier, Jan de; Michael Wheeler
2015-01-01
We derive a matrix product formula for symmetric Macdonald polynomials. Our results are obtained by constructing polynomial solutions of deformed Knizhnik–Zamolodchikov equations, which arise by considering representations of the Zamolodchikov–Faddeev and Yang–Baxter algebras in terms of t-deformed bosonic operators. These solutions are generalized probabilities for particle configurations of the multi-species asymmetric exclusion process, and form a basis of the ring of polynomials in n variables whose elements are indexed by compositions. For weakly increasing compositions (anti-dominant weights), these basis elements coincide with non-symmetric Macdonald polynomials. Our formulas imply a natural combinatorial interpretation in terms of solvable lattice models. They also imply that normalizations of stationary states of multi-species exclusion processes are obtained as Macdonald polynomials at q = 1. (paper)
Arabic text classification using Polynomial Networks
Directory of Open Access Journals (Sweden)
Mayy M. Al-Tahrawi
2015-10-01
Full Text Available In this paper, an Arabic statistical learning-based text classification system has been developed using Polynomial Neural Networks. Polynomial Networks have been recently applied to English text classification, but they were never used for Arabic text classification. In this research, we investigate the performance of Polynomial Networks in classifying Arabic texts. Experiments are conducted on a widely used Arabic dataset in text classification: Al-Jazeera News dataset. We chose this dataset to enable direct comparisons of the performance of Polynomial Networks classifier versus other well-known classifiers on this dataset in the literature of Arabic text classification. Results of experiments show that Polynomial Networks classifier is a competitive algorithm to the state-of-the-art ones in the field of Arabic text classification.
Radiative properties tailoring of grating by comb-drive microactuator
International Nuclear Information System (INIS)
Jiao, Y.; Liu, L.H.; Liu, L.J.; Hsu, P.-F.
2014-01-01
Micro-scale grating structures are widely researched in recent years. Although micro-scale fabrication technology is highly advanced today, with grating aspect ratio greater than 25:1 being achievable some fabrication requirements, such as fine groove processing, are still challenging. Comb-drive microactuator is proposed in this paper to be utilized on simple binary grating structures for tailoring or modulating spectral radiation properties by active adjustment. The rigorous coupled-wave analysis (RCWA) is used to calculate the absorptance of proposed structures and to investigate the impacts brought by the geometry and displacement of comb-drive microactuator. The results show that the utilization of comb-drive microactuator on grating improves the absorptance of simple binary grating while avoiding the difficulty fine groove processing. Spectral radiation property tailoring after gratings are fabricated becomes possible with the comb-drive microactuator structure. - Highlights: • A microscale grating structure with comb-driven microactuator is proposed. • The movement of microactuator changes peak absorptance resonance wavelength. • Geometric and displacement effects of comb finger on absorptance are investigated. • Both RCWA and LC circuit models are developed to predict the resonance wavelength. • Resonance frequency equations of LC circuits allow quick design analysis
Many-body orthogonal polynomial systems
International Nuclear Information System (INIS)
Witte, N.S.
1997-03-01
The fundamental methods employed in the moment problem, involving orthogonal polynomial systems, the Lanczos algorithm, continued fraction analysis and Pade approximants has been combined with a cumulant approach and applied to the extensive many-body problem in physics. This has yielded many new exact results for many-body systems in the thermodynamic limit - for the ground state energy, for excited state gaps, for arbitrary ground state avenges - and are of a nonperturbative nature. These results flow from a confluence property of the three-term recurrence coefficients arising and define a general class of many-body orthogonal polynomials. These theorems constitute an analytical solution to the Lanczos algorithm in that they are expressed in terms of the three-term recurrence coefficients α and β. These results can also be applied approximately for non-solvable models in the form of an expansion, in a descending series of the system size. The zeroth order order this expansion is just the manifestation of the central limit theorem in which a Gaussian measure and hermite polynomials arise. The first order represents the first non-trivial order, in which classical distribution functions like the binomial distributions arise and the associated class of orthogonal polynomials are Meixner polynomials. Amongst examples of systems which have infinite order in the expansion are q-orthogonal polynomials where q depends on the system size in a particular way. (author)
Vertex models, TASEP and Grothendieck polynomials
International Nuclear Information System (INIS)
Motegi, Kohei; Sakai, Kazumitsu
2013-01-01
We examine the wavefunctions and their scalar products of a one-parameter family of integrable five-vertex models. At a special point of the parameter, the model investigated is related to an irreversible interacting stochastic particle system—the so-called totally asymmetric simple exclusion process (TASEP). By combining the quantum inverse scattering method with a matrix product representation of the wavefunctions, the on-/off-shell wavefunctions of the five-vertex models are represented as a certain determinant form. Up to some normalization factors, we find that the wavefunctions are given by Grothendieck polynomials, which are a one-parameter deformation of Schur polynomials. Introducing a dual version of the Grothendieck polynomials, and utilizing the determinant representation for the scalar products of the wavefunctions, we derive a generalized Cauchy identity satisfied by the Grothendieck polynomials and their duals. Several representation theoretical formulae for the Grothendieck polynomials are also presented. As a byproduct, the relaxation dynamics such as Green functions for the periodic TASEP are found to be described in terms of the Grothendieck polynomials. (paper)
Electro-optic diffraction grating tuned laser
International Nuclear Information System (INIS)
Hughes, R.S.
1975-01-01
An electro-optic diffraction grating tuned laser comprising a laser medium, output mirror, retro-reflective grating and an electro-optic diffraction grating beam deflector positioned between the laser medium and the reflective diffraction grating is described. An optional angle multiplier may be used between the electro-optic diffraction grating and the reflective grating. (auth)
On the Laurent polynomial rings
International Nuclear Information System (INIS)
Stefanescu, D.
1985-02-01
We describe some properties of the Laurent polynomial rings in a finite number of indeterminates over a commutative unitary ring. We study some subrings of the Laurent polynomial rings. We finally obtain two cancellation properties. (author)
M-Polynomials and Topological Indices of Titania Nanotubes
Directory of Open Access Journals (Sweden)
Mobeen Munir
2016-10-01
Full Text Available Titania is one of the most comprehensively studied nanostructures due to their widespread applications in the production of catalytic, gas sensing, and corrosion-resistant materials. M-polynomial of nanotubes has been vastly investigated, as it produces many degree-based topological indices, which are numerical parameters capturing structural and chemical properties. These indices are used in the development of quantitative structure-activity relationships (QSARs in which the biological activity and other properties of molecules, such as boiling point, stability, strain energy, etc., are correlated with their structure. In this report, we provide M-polynomials of single-walled titania (SW TiO2 nanotubes and recover important topological degree-based indices to theoretically judge these nanotubes. We also plot surfaces associated to single-walled titania (SW TiO2 nanotubes.
Best polynomial degree reduction on q-lattices with applications to q-orthogonal polynomials
Ait-Haddou, Rachid
2015-06-07
We show that a weighted least squares approximation of q-Bézier coefficients provides the best polynomial degree reduction in the q-L2-norm. We also provide a finite analogue of this result with respect to finite q-lattices and we present applications of these results to q-orthogonal polynomials. © 2015 Elsevier Inc. All rights reserved.
Best polynomial degree reduction on q-lattices with applications to q-orthogonal polynomials
Ait-Haddou, Rachid; Goldman, Ron
2015-01-01
We show that a weighted least squares approximation of q-Bézier coefficients provides the best polynomial degree reduction in the q-L2-norm. We also provide a finite analogue of this result with respect to finite q-lattices and we present applications of these results to q-orthogonal polynomials. © 2015 Elsevier Inc. All rights reserved.
Computing the Alexander Polynomial Numerically
DEFF Research Database (Denmark)
Hansen, Mikael Sonne
2006-01-01
Explains how to construct the Alexander Matrix and how this can be used to compute the Alexander polynomial numerically.......Explains how to construct the Alexander Matrix and how this can be used to compute the Alexander polynomial numerically....
Near-infrared light-controlled tunable grating based on graphene/elastomer composites
Wang, Fei; Jia, Shuhai; Wang, Yonglin; Tang, Zhenhua
2018-02-01
A near-infrared (nIR) light actuated tunable transmission optical grating based on graphene nanoplatelet (GNP)/polydimethylsiloxane (PDMS) and PDMS is proposed. A simple fabrication protocol is studied that allows integration of the grating with the actuation mechanism; both components are made from soft elastomers, and this ensure the tunability and the light-driven operation of the grating. The resulting grating structure demonstrates continuous period tunability of 2.7% under an actuation power density of 220 mW cm-2 within a period of 3 s and also demonstrates a time-independent characteristic. The proposed infrared activated grating can be developed for wireless remote light splitting in bio/chemical sensing and optical telecommunications applications.
Density of Real Zeros of the Tutte Polynomial
DEFF Research Database (Denmark)
Ok, Seongmin; Perrett, Thomas
2018-01-01
The Tutte polynomial of a graph is a two-variable polynomial whose zeros and evaluations encode many interesting properties of the graph. In this article we investigate the real zeros of the Tutte polynomials of graphs, and show that they form a dense subset of certain regions of the plane. This ....... This is the first density result for the real zeros of the Tutte polynomial in a region of positive volume. Our result almost confirms a conjecture of Jackson and Sokal except for one region which is related to an open problem on flow polynomials.......The Tutte polynomial of a graph is a two-variable polynomial whose zeros and evaluations encode many interesting properties of the graph. In this article we investigate the real zeros of the Tutte polynomials of graphs, and show that they form a dense subset of certain regions of the plane...
Density of Real Zeros of the Tutte Polynomial
DEFF Research Database (Denmark)
Ok, Seongmin; Perrett, Thomas
2017-01-01
The Tutte polynomial of a graph is a two-variable polynomial whose zeros and evaluations encode many interesting properties of the graph. In this article we investigate the real zeros of the Tutte polynomials of graphs, and show that they form a dense subset of certain regions of the plane. This ....... This is the first density result for the real zeros of the Tutte polynomial in a region of positive volume. Our result almost confirms a conjecture of Jackson and Sokal except for one region which is related to an open problem on flow polynomials.......The Tutte polynomial of a graph is a two-variable polynomial whose zeros and evaluations encode many interesting properties of the graph. In this article we investigate the real zeros of the Tutte polynomials of graphs, and show that they form a dense subset of certain regions of the plane...
Parallel Construction of Irreducible Polynomials
DEFF Research Database (Denmark)
Frandsen, Gudmund Skovbjerg
Let arithmetic pseudo-NC^k denote the problems that can be solved by log space uniform arithmetic circuits over the finite prime field GF(p) of depth O(log^k (n + p)) and size polynomial in (n + p). We show that the problem of constructing an irreducible polynomial of specified degree over GF(p) ...... of polynomials is in arithmetic NC^3. Our algorithm works over any field and compared to other known algorithms it does not assume the ability to take p'th roots when the field has characteristic p....
DesAutels, G Logan; Powers, Peter; Brewer, Chris; Walker, Mark; Burky, Mark; Anderson, Gregg
2008-07-20
An optical temperature sensor was created using a femtosecond micromachined diffraction grating inside transparent bulk 6H-SiC, and to the best of our knowledge, this is a novel technique of measuring temperature. Other methods of measuring temperature using fiber Bragg gratings have been devised by other groups such as Zhang and Kahrizi [in MEMS, NANO, and Smart Systems (IEEE, 2005)]. This temperature sensor was, to the best of our knowledge, also used for a novel method of measuring the linear and nonlinear coefficients of the thermal expansion of transparent and nontransparent materials by means of the grating first-order diffracted beam. Furthermore the coefficient of thermal expansion of 6H-SiC was measured using this new technique. A He-Ne laser beam was used with the SiC grating to produce a first-order diffracted beam where the change in deflection height was measured as a function of temperature. The grating was micromachined with a 20 microm spacing and has dimensions of approximately 500 microm x 500 microm (l x w) and is roughly 0.5 microm deep into the 6H-SiC bulk. A minimum temperature of 26.7 degrees C and a maximum temperature of 399 degrees C were measured, which gives a DeltaT of 372.3 degrees C. The sensitivity of the technique is DeltaT=5 degrees C. A maximum deflection angle of 1.81 degrees was measured in the first-order diffracted beam. The trend of the deflection with increasing temperature is a nonlinear polynomial of the second-order. This optical SiC thermal sensor has many high-temperature electronic applications such as aircraft turbine and gas tank monitoring for commercial and military applications.
Zhao, Dan; Wang, Xiao; Mu, Jie; Li, Zhilin; Zuo, Yanlei; Zhou, Song; Zhou, Kainan; Zeng, Xiaoming; Su, Jingqin; Zhu, Qihua
2017-02-01
The grating tiling technology is one of the most effective means to increase the aperture of the gratings. The line-density error (LDE) between sub-gratings will degrade the performance of the tiling gratings, high accuracy measurement and compensation of the LDE are of significance to improve the output pulses characteristics of the tiled-grating compressor. In this paper, the influence of LDE on the output pulses of the tiled-grating compressor is quantitatively analyzed by means of numerical simulation, the output beams drift and output pulses broadening resulting from the LDE are presented. Based on the numerical results we propose a compensation method to reduce the degradations of the tiled grating compressor by applying angular tilt error and longitudinal piston error at the same time. Moreover, a monitoring system is setup to measure the LDE between sub-gratings accurately and the dispersion variation due to the LDE is also demonstrated based on spatial-spectral interference. In this way, we can realize high-accuracy measurement and compensation of the LDE, and this would provide an efficient way to guide the adjustment of the tiling gratings.
Optimization over polynomials : Selected topics
Laurent, M.; Jang, Sun Young; Kim, Young Rock; Lee, Dae-Woong; Yie, Ikkwon
2014-01-01
Minimizing a polynomial function over a region defined by polynomial inequalities models broad classes of hard problems from combinatorics, geometry and optimization. New algorithmic approaches have emerged recently for computing the global minimum, by combining tools from real algebra (sums of
Parallel multigrid smoothing: polynomial versus Gauss-Seidel
International Nuclear Information System (INIS)
Adams, Mark; Brezina, Marian; Hu, Jonathan; Tuminaro, Ray
2003-01-01
Gauss-Seidel is often the smoother of choice within multigrid applications. In the context of unstructured meshes, however, maintaining good parallel efficiency is difficult with multiplicative iterative methods such as Gauss-Seidel. This leads us to consider alternative smoothers. We discuss the computational advantages of polynomial smoothers within parallel multigrid algorithms for positive definite symmetric systems. Two particular polynomials are considered: Chebyshev and a multilevel specific polynomial. The advantages of polynomial smoothing over traditional smoothers such as Gauss-Seidel are illustrated on several applications: Poisson's equation, thin-body elasticity, and eddy current approximations to Maxwell's equations. While parallelizing the Gauss-Seidel method typically involves a compromise between a scalable convergence rate and maintaining high flop rates, polynomial smoothers achieve parallel scalable multigrid convergence rates without sacrificing flop rates. We show that, although parallel computers are the main motivation, polynomial smoothers are often surprisingly competitive with Gauss-Seidel smoothers on serial machines
Parallel multigrid smoothing: polynomial versus Gauss-Seidel
Adams, Mark; Brezina, Marian; Hu, Jonathan; Tuminaro, Ray
2003-07-01
Gauss-Seidel is often the smoother of choice within multigrid applications. In the context of unstructured meshes, however, maintaining good parallel efficiency is difficult with multiplicative iterative methods such as Gauss-Seidel. This leads us to consider alternative smoothers. We discuss the computational advantages of polynomial smoothers within parallel multigrid algorithms for positive definite symmetric systems. Two particular polynomials are considered: Chebyshev and a multilevel specific polynomial. The advantages of polynomial smoothing over traditional smoothers such as Gauss-Seidel are illustrated on several applications: Poisson's equation, thin-body elasticity, and eddy current approximations to Maxwell's equations. While parallelizing the Gauss-Seidel method typically involves a compromise between a scalable convergence rate and maintaining high flop rates, polynomial smoothers achieve parallel scalable multigrid convergence rates without sacrificing flop rates. We show that, although parallel computers are the main motivation, polynomial smoothers are often surprisingly competitive with Gauss-Seidel smoothers on serial machines.
Efficient computation of Laguerre polynomials
A. Gil (Amparo); J. Segura (Javier); N.M. Temme (Nico)
2017-01-01
textabstractAn efficient algorithm and a Fortran 90 module (LaguerrePol) for computing Laguerre polynomials . Ln(α)(z) are presented. The standard three-term recurrence relation satisfied by the polynomials and different types of asymptotic expansions valid for . n large and . α small, are used
Orthogonal polynomials in transport theories
International Nuclear Information System (INIS)
Dehesa, J.S.
1981-01-01
The asymptotical (k→infinity) behaviour of zeros of the polynomials gsub(k)sup((m)(ν)) encountered in the treatment of direct and inverse problems of scattering in neutron transport as well as radiative transfer theories is investigated in terms of the amplitude antiwsub(k) of the kth Legendre polynomial needed in the expansion of the scattering function. The parameters antiwsub(k) describe the anisotropy of scattering of the medium considered. In particular, it is shown that the asymptotical density of zeros of the polynomials gsub(k)sup(m)(ν) is an inverted semicircle for the anisotropic non-multiplying scattering medium
Chromatic polynomials of random graphs
International Nuclear Information System (INIS)
Van Bussel, Frank; Fliegner, Denny; Timme, Marc; Ehrlich, Christoph; Stolzenberg, Sebastian
2010-01-01
Chromatic polynomials and related graph invariants are central objects in both graph theory and statistical physics. Computational difficulties, however, have so far restricted studies of such polynomials to graphs that were either very small, very sparse or highly structured. Recent algorithmic advances (Timme et al 2009 New J. Phys. 11 023001) now make it possible to compute chromatic polynomials for moderately sized graphs of arbitrary structure and number of edges. Here we present chromatic polynomials of ensembles of random graphs with up to 30 vertices, over the entire range of edge density. We specifically focus on the locations of the zeros of the polynomial in the complex plane. The results indicate that the chromatic zeros of random graphs have a very consistent layout. In particular, the crossing point, the point at which the chromatic zeros with non-zero imaginary part approach the real axis, scales linearly with the average degree over most of the density range. While the scaling laws obtained are purely empirical, if they continue to hold in general there are significant implications: the crossing points of chromatic zeros in the thermodynamic limit separate systems with zero ground state entropy from systems with positive ground state entropy, the latter an exception to the third law of thermodynamics.
New polynomial-based molecular descriptors with low degeneracy.
Directory of Open Access Journals (Sweden)
Matthias Dehmer
Full Text Available In this paper, we introduce a novel graph polynomial called the 'information polynomial' of a graph. This graph polynomial can be derived by using a probability distribution of the vertex set. By using the zeros of the obtained polynomial, we additionally define some novel spectral descriptors. Compared with those based on computing the ordinary characteristic polynomial of a graph, we perform a numerical study using real chemical databases. We obtain that the novel descriptors do have a high discrimination power.
Need for higher order polynomial basis for polynomial nodal methods employed in LWR calculations
International Nuclear Information System (INIS)
Taiwo, T.A.; Palmiotti, G.
1997-01-01
The paper evaluates the accuracy and efficiency of sixth order polynomial solutions and the use of one radial node per core assembly for pressurized water reactor (PWR) core power distributions and reactivities. The computer code VARIANT was modified to calculate sixth order polynomial solutions for a hot zero power benchmark problem in which a control assembly along a core axis is assumed to be out of the core. Results are presented for the VARIANT, DIF3D-NODAL, and DIF3D-finite difference codes. The VARIANT results indicate that second order expansion of the within-node source and linear representation of the node surface currents are adequate for this problem. The results also demonstrate the improvement in the VARIANT solution when the order of the polynomial expansion of the within-node flux is increased from fourth to sixth order. There is a substantial saving in computational time for using one radial node per assembly with the sixth order expansion compared to using four or more nodes per assembly and fourth order polynomial solutions. 11 refs., 1 tab
Low crosstalk Arrayed Waveguide Grating with Cascaded Waveguide Grating Filter
International Nuclear Information System (INIS)
Deng Yang; Liu Yuan; Gao Dingshan
2011-01-01
We propose a highly compact and low crosstalk arrayed waveguide grating (AWG) with cascaded waveguide grating (CWGF). The side lobes of the silicon nanowire AWG, which are normally introduced by fabrication errors, can be effectively suppressed by the CWGF. And the crosstalk can be improved about 15dB.
M-Polynomials and Topological Indices of Dominating David Derived Networks
Directory of Open Access Journals (Sweden)
Kang Shin Min
2018-03-01
Full Text Available There is a strong relationship between the chemical characteristics of chemical compounds and their molecular structures. Topological indices are numerical values associated with the chemical molecular graphs that help to understand the physical features, chemical reactivity, and biological activity of chemical compound. Thus, the study of the topological indices is important. M-polynomial helps to recover many degree-based topological indices for example Zagreb indices, Randic index, symmetric division idex, inverse sum index etc. In this article we compute M-polynomials of dominating David derived networks of the first type, second type and third type of dimension n and find some topological properties by using these M-polynomials. The results are plotted using Maple to see the dependence of topological indices on the involved parameters.
An ultra-high-vacuum multiple grating chamber and scan drive with improved grating change
International Nuclear Information System (INIS)
Hulbert, S.L.; Holly, D.J.; Middleton, F.H.; Wallace, D.J.; Wisconsin Univ., Stoughton, WI; Wisconsin Univ., Stoughton, WI
1989-01-01
We describe a new grating chamber and scan drive which has been designed, built, and tested by Physical Sciences Laboratory of the University of Wisconsin for the new high flux, high-resolution spectroscopy branch line of the TOK hybrid wiggler/undulator on the NSLS VUV ring. The chamber will contain spherical gratings to be used in the Spherical Grating Monochromator (SGM) configuration introduced by Chen and Sette. The grating chamber houses five 180 mm x 35 mm x 30 mm gratings capable of scanning a range of 12 degree (-14 degree to +8 degree with respect to the incoming beam direction) for VUV and soft X-ray diffraction. The gratings can be switched and precisely indexed while under ultra-high vacuum (UHV) at any scan angle and are mechanically isolated from the vacuum chamber to prevent inaccuracies due to chamber distortions. The gratings can separately be adjusted for height, yaw, pitch, and roll, with the latter three performed while in vacuo. The scan drive provides a resolution of 0.03 arc sec with linearity over the 12 degree range of ∼1.5 arc sec and absolute reproducibility of 1 arc sec. 5 refs., 5 figs
Sheffer and Non-Sheffer Polynomial Families
Directory of Open Access Journals (Sweden)
G. Dattoli
2012-01-01
Full Text Available By using the integral transform method, we introduce some non-Sheffer polynomial sets. Furthermore, we show how to compute the connection coefficients for particular expressions of Appell polynomials.
Off-plane x-ray reflection grating fabrication
Peterson, Thomas J.; DeRoo, Casey T.; Marlowe, Hannah; McEntaffer, Randall L.; Miles, Drew M.; Tutt, James H.; Schultz, Ted B.
2015-09-01
Off-plane X-ray diffraction gratings with precision groove profiles at the submicron scale will be used in next generation X-ray spectrometers. Such gratings will be used on a current NASA suborbital rocket mission, the Off-plane Grating Rocket Experiment (OGRE), and have application for future grating missions. The fabrication of these gratings does not come without challenges. High performance off-plane gratings must be fabricated with precise radial grating patterns, optically at surfaces, and specific facet angles. Such gratings can be made using a series of common micro-fabrication techniques. The resulting process is highly customizable, making it useful for a variety of different mission architectures. In this paper, we detail the fabrication method used to produce high performance off-plane gratings and report the results of a preliminary qualification test of a grating fabricated in this manner. The grating was tested in the off-plane `Littrow' configuration, for which the grating is most efficient for a given diffraction order, and found to achieve 42% relative efficiency in the blaze order with respect to all diffracted light.
Generalized Pseudospectral Method and Zeros of Orthogonal Polynomials
Directory of Open Access Journals (Sweden)
Oksana Bihun
2018-01-01
Full Text Available Via a generalization of the pseudospectral method for numerical solution of differential equations, a family of nonlinear algebraic identities satisfied by the zeros of a wide class of orthogonal polynomials is derived. The generalization is based on a modification of pseudospectral matrix representations of linear differential operators proposed in the paper, which allows these representations to depend on two, rather than one, sets of interpolation nodes. The identities hold for every polynomial family pνxν=0∞ orthogonal with respect to a measure supported on the real line that satisfies some standard assumptions, as long as the polynomials in the family satisfy differential equations Apν(x=qν(xpν(x, where A is a linear differential operator and each qν(x is a polynomial of degree at most n0∈N; n0 does not depend on ν. The proposed identities generalize known identities for classical and Krall orthogonal polynomials, to the case of the nonclassical orthogonal polynomials that belong to the class described above. The generalized pseudospectral representations of the differential operator A for the case of the Sonin-Markov orthogonal polynomials, also known as generalized Hermite polynomials, are presented. The general result is illustrated by new algebraic relations satisfied by the zeros of the Sonin-Markov polynomials.
On the Connection Coefficients of the Chebyshev-Boubaker Polynomials
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Paul Barry
2013-01-01
Full Text Available The Chebyshev-Boubaker polynomials are the orthogonal polynomials whose coefficient arrays are defined by ordinary Riordan arrays. Examples include the Chebyshev polynomials of the second kind and the Boubaker polynomials. We study the connection coefficients of this class of orthogonal polynomials, indicating how Riordan array techniques can lead to closed-form expressions for these connection coefficients as well as recurrence relations that define them.
Flexible PCPDTBT:PCBM solar cells with integrated grating structures
DEFF Research Database (Denmark)
Oliveira Hansen, Roana Melina de; Liu, Yinghui; Madsen, Morten
2013-01-01
We report on development of flexible PCPDTBT:PCBM solar cells with integrated diffraction gratings on the bottom electrodes. The presented results address PCPDTBT:PCBM solar cells in an inverted geometry, which contains implemented grating structures whose pitch is tuned to match the absorption...... spectra of the active layer. This optimized solar cell structure leads to an enhanced absorption in the active layer and thus improved short-circuit currents and power conversion efficiencies in the fabricated devices. Fabrication of the solar cells on thin polyimide substrates which are compatible...
Polynomial sequences generated by infinite Hessenberg matrices
Directory of Open Access Journals (Sweden)
Verde-Star Luis
2017-01-01
Full Text Available We show that an infinite lower Hessenberg matrix generates polynomial sequences that correspond to the rows of infinite lower triangular invertible matrices. Orthogonal polynomial sequences are obtained when the Hessenberg matrix is tridiagonal. We study properties of the polynomial sequences and their corresponding matrices which are related to recurrence relations, companion matrices, matrix similarity, construction algorithms, and generating functions. When the Hessenberg matrix is also Toeplitz the polynomial sequences turn out to be of interpolatory type and we obtain additional results. For example, we show that every nonderogative finite square matrix is similar to a unique Toeplitz-Hessenberg matrix.
Special polynomials associated with rational solutions of some hierarchies
International Nuclear Information System (INIS)
Kudryashov, Nikolai A.
2009-01-01
New special polynomials associated with rational solutions of the Painleve hierarchies are introduced. The Hirota relations for these special polynomials are found. Differential-difference hierarchies to find special polynomials are presented. These formulae allow us to search special polynomials associated with the hierarchies. It is shown that rational solutions of the Caudrey-Dodd-Gibbon, the Kaup-Kupershmidt and the modified hierarchy for these ones can be obtained using new special polynomials.
Deep-etched sinusoidal polarizing beam splitter grating.
Feng, Jijun; Zhou, Changhe; Cao, Hongchao; Lv, Peng
2010-04-01
A sinusoidal-shaped fused-silica grating as a highly efficient polarizing beam splitter (PBS) is investigated based on the simplified modal method. The grating structure depends mainly on the ratio of groove depth to grating period and the ratio of incident wavelength to grating period. These ratios can be used as a guideline for the grating design at different wavelengths. A sinusoidal-groove PBS grating is designed at a wavelength of 1310 nm under Littrow mounting, and the transmitted TM and TE polarized waves are mainly diffracted into the zeroth order and the -1st order, respectively. The grating profile is optimized by using rigorous coupled-wave analysis. The designed PBS grating is highly efficient (>95.98%) over the O-band wavelength range (1260-1360 nm) for both TE and TM polarizations. The sinusoidal grating can exhibit higher diffraction efficiency, larger extinction ratio, and less reflection loss than the rectangular-groove PBS grating. By applying wet etching technology on the rectangular grating, which was manufactured by holographic recording and inductively coupled plasma etching technology, the sinusoidal grating can be approximately fabricated. Experimental results are in agreement with theoretical values.
DEFF Research Database (Denmark)
Taghizadeh, Alireza
This thesis deals with theoretical investigations of a newly proposed grating structure, referred to as hybrid grating (HG) as well as vertical cavity lasers based on the grating reflectors. The HG consists of a near-subwavelength grating layer and an unpatterned high-refractive-index cap layer...... directions, which is analogous to electronic quantum wells in conduction or valence bands. Several interesting configurations of heterostructures have been investigated and their potential in fundamental physics study and applications are discussed. For numerical and theoretical studies, a three...... feasibility than the HCG-based ones. Furthermore, the concept of cavity dispersion in vertical cavities is introduced and its importance in the modal properties is numerically investigated. The dispersion curvature of a cavity mode is interpreted as the effective photon mass of the cavity mode. In a vertical...
M-Polynomial and Degree-Based Topological Indices of Polyhex Nanotubes
Directory of Open Access Journals (Sweden)
Mobeen Munir
2016-12-01
Full Text Available The discovery of new nanomaterials adds new dimensions to industry, electronics, and pharmaceutical and biological therapeutics. In this article, we first find closed forms of M-polynomials of polyhex nanotubes. We also compute closed forms of various degree-based topological indices of these tubes. These indices are numerical tendencies that often depict quantitative structural activity/property/toxicity relationships and correlate certain physico-chemical properties, such as boiling point, stability, and strain energy, of respective nanomaterial. To conclude, we plot surfaces associated to M-polynomials and characterize some facts about these tubes.
Cosmographic analysis with Chebyshev polynomials
Capozziello, Salvatore; D'Agostino, Rocco; Luongo, Orlando
2018-05-01
The limits of standard cosmography are here revised addressing the problem of error propagation during statistical analyses. To do so, we propose the use of Chebyshev polynomials to parametrize cosmic distances. In particular, we demonstrate that building up rational Chebyshev polynomials significantly reduces error propagations with respect to standard Taylor series. This technique provides unbiased estimations of the cosmographic parameters and performs significatively better than previous numerical approximations. To figure this out, we compare rational Chebyshev polynomials with Padé series. In addition, we theoretically evaluate the convergence radius of (1,1) Chebyshev rational polynomial and we compare it with the convergence radii of Taylor and Padé approximations. We thus focus on regions in which convergence of Chebyshev rational functions is better than standard approaches. With this recipe, as high-redshift data are employed, rational Chebyshev polynomials remain highly stable and enable one to derive highly accurate analytical approximations of Hubble's rate in terms of the cosmographic series. Finally, we check our theoretical predictions by setting bounds on cosmographic parameters through Monte Carlo integration techniques, based on the Metropolis-Hastings algorithm. We apply our technique to high-redshift cosmic data, using the Joint Light-curve Analysis supernovae sample and the most recent versions of Hubble parameter and baryon acoustic oscillation measurements. We find that cosmography with Taylor series fails to be predictive with the aforementioned data sets, while turns out to be much more stable using the Chebyshev approach.
Curved VPH gratings for novel spectrographs
Clemens, J. Christopher; O'Donoghue, Darragh; Dunlap, Bart H.
2014-07-01
The introduction of volume phase holographic (VPH) gratings into astronomy over a decade ago opened new possibilities for instrument designers. In this paper we describe an extension of VPH grating technology that will have applications in astronomy and beyond: curved VPH gratings. These devices can disperse light while simultaneously correcting aberrations. We have designed and manufactured two different kinds of convex VPH grating prototypes for use in off-axis reflecting spectrographs. One type functions in transmission and the other in reflection, enabling Offnerstyle spectrographs with the high-efficiency and low-cost advantages of VPH gratings. We will discuss the design process and the tools required for modelling these gratings along with the recording layout and process steps required to fabricate them. We will present performance data for the first convex VPH grating produced for an astronomical spectrograph.
Multilevel weighted least squares polynomial approximation
Haji-Ali, Abdul-Lateef
2017-06-30
Weighted least squares polynomial approximation uses random samples to determine projections of functions onto spaces of polynomials. It has been shown that, using an optimal distribution of sample locations, the number of samples required to achieve quasi-optimal approximation in a given polynomial subspace scales, up to a logarithmic factor, linearly in the dimension of this space. However, in many applications, the computation of samples includes a numerical discretization error. Thus, obtaining polynomial approximations with a single level method can become prohibitively expensive, as it requires a sufficiently large number of samples, each computed with a sufficiently small discretization error. As a solution to this problem, we propose a multilevel method that utilizes samples computed with different accuracies and is able to match the accuracy of single-level approximations with reduced computational cost. We derive complexity bounds under certain assumptions about polynomial approximability and sample work. Furthermore, we propose an adaptive algorithm for situations where such assumptions cannot be verified a priori. Finally, we provide an efficient algorithm for the sampling from optimal distributions and an analysis of computationally favorable alternative distributions. Numerical experiments underscore the practical applicability of our method.
High Resolution of the ECG Signal by Polynomial Approximation
Directory of Open Access Journals (Sweden)
G. Rozinaj
2006-04-01
Full Text Available Averaging techniques as temporal averaging and space averaging have been successfully used in many applications for attenuating interference [6], [7], [8], [9], [10]. In this paper we introduce interference removing of the ECG signal by polynomial approximation, with smoothing discrete dependencies, to make up for averaging methods. The method is suitable for low-level signals of the electrical activity of the heart often less than 10 m V. Most low-level signals arising from PR, ST and TP segments which can be detected eventually and their physiologic meaning can be appreciated. Of special importance for the diagnostic of the electrical activity of the heart is the activity bundle of His between P and R waveforms. We have established an artificial sine wave to ECG signal between P and R wave. The aim focus is to verify the smoothing method by polynomial approximation if the SNR (signal-to-noise ratio is negative (i.e. a signal is lower than noise.
Relations between zeros of special polynomials associated with the Painleve equations
International Nuclear Information System (INIS)
Kudryashov, Nikolai A.; Demina, Maria V.
2007-01-01
A method for finding relations of roots of polynomials is presented. Our approach allows us to get a number of relations between the zeros of the classical polynomials as well as the roots of special polynomials associated with rational solutions of the Painleve equations. We apply the method to obtain the relations for the zeros of several polynomials. These are: the Hermite polynomials, the Laguerre polynomials, the Yablonskii-Vorob'ev polynomials, the generalized Okamoto polynomials, and the generalized Hermite polynomials. All the relations found can be considered as analogues of generalized Stieltjes relations
On polynomial solutions of the Heun equation
International Nuclear Information System (INIS)
Gurappa, N; Panigrahi, Prasanta K
2004-01-01
By making use of a recently developed method to solve linear differential equations of arbitrary order, we find a wide class of polynomial solutions to the Heun equation. We construct the series solution to the Heun equation before identifying the polynomial solutions. The Heun equation extended by the addition of a term, -σ/x, is also amenable for polynomial solutions. (letter to the editor)
A new Arnoldi approach for polynomial eigenproblems
Energy Technology Data Exchange (ETDEWEB)
Raeven, F.A.
1996-12-31
In this paper we introduce a new generalization of the method of Arnoldi for matrix polynomials. The new approach is compared with the approach of rewriting the polynomial problem into a linear eigenproblem and applying the standard method of Arnoldi to the linearised problem. The algorithm that can be applied directly to the polynomial eigenproblem turns out to be more efficient, both in storage and in computation.
Orthogonal Polynomials and Special Functions
Assche, Walter
2003-01-01
The set of lectures from the Summer School held in Leuven in 2002 provide an up-to-date account of recent developments in orthogonal polynomials and special functions, in particular for algorithms for computer algebra packages, 3nj-symbols in representation theory of Lie groups, enumeration, multivariable special functions and Dunkl operators, asymptotics via the Riemann-Hilbert method, exponential asymptotics and the Stokes phenomenon. The volume aims at graduate students and post-docs working in the field of orthogonal polynomials and special functions, and in related fields interacting with orthogonal polynomials, such as combinatorics, computer algebra, asymptotics, representation theory, harmonic analysis, differential equations, physics. The lectures are self-contained requiring only a basic knowledge of analysis and algebra, and each includes many exercises.
Multiple Meixner polynomials and non-Hermitian oscillator Hamiltonians
International Nuclear Information System (INIS)
Ndayiragije, F; Van Assche, W
2013-01-01
Multiple Meixner polynomials are polynomials in one variable which satisfy orthogonality relations with respect to r > 1 different negative binomial distributions (Pascal distributions). There are two kinds of multiple Meixner polynomials, depending on the selection of the parameters in the negative binomial distribution. We recall their definition and some formulas and give generating functions and explicit expressions for the coefficients in the nearest neighbor recurrence relation. Following a recent construction of Miki, Tsujimoto, Vinet and Zhedanov (for multiple Meixner polynomials of the first kind), we construct r > 1 non-Hermitian oscillator Hamiltonians in r dimensions which are simultaneously diagonalizable and for which the common eigenstates are expressed in terms of multiple Meixner polynomials of the second kind. (paper)
Colouring and knot polynomials
International Nuclear Information System (INIS)
Welsh, D.J.A.
1991-01-01
These lectures will attempt to explain a connection between the recent advances in knot theory using the Jones and related knot polynomials with classical problems in combinatorics and statistical mechanics. The difficulty of some of these problems will be analysed in the context of their computational complexity. In particular we shall discuss colourings and groups valued flows in graphs, knots and the Jones and Kauffman polynomials, the Ising, Potts and percolation problems of statistical physics, computational complexity of the above problems. (author). 20 refs, 9 figs
Spherical grating based x-ray Talbot interferometry
Energy Technology Data Exchange (ETDEWEB)
Cong, Wenxiang, E-mail: congw@rpi.edu, E-mail: xiy2@rpi.edu, E-mail: wangg6@rpi.edu; Xi, Yan, E-mail: congw@rpi.edu, E-mail: xiy2@rpi.edu, E-mail: wangg6@rpi.edu; Wang, Ge, E-mail: congw@rpi.edu, E-mail: xiy2@rpi.edu, E-mail: wangg6@rpi.edu [Biomedical Imaging Center, Rensselaer Polytechnic Institute, Troy, New York 12180 (United States)
2015-11-15
Purpose: Grating interferometry is a state-of-the-art x-ray imaging approach, which can acquire information on x-ray attenuation, phase shift, and small-angle scattering simultaneously. Phase-contrast imaging and dark-field imaging are very sensitive to microstructural variation and offers superior contrast resolution for biological soft tissues. However, a common x-ray tube is a point-like source. As a result, the popular planar grating imaging configuration seriously restricts the flux of photons and decreases the visibility of signals, yielding a limited field of view. The purpose of this study is to extend the planar x-ray grating imaging theory and methods to a spherical grating scheme for a wider range of preclinical and clinical applications. Methods: A spherical grating matches the wave front of a point x-ray source very well, allowing the perpendicular incidence of x-rays on the grating to achieve a higher visibility over a larger field of view than the planer grating counterpart. A theoretical analysis of the Talbot effect for spherical grating imaging is proposed to establish a basic foundation for x-ray spherical gratings interferometry. An efficient method of spherical grating imaging is also presented to extract attenuation, differential phase, and dark-field images in the x-ray spherical grating interferometer. Results: Talbot self-imaging with spherical gratings is analyzed based on the Rayleigh–Sommerfeld diffraction formula, featuring a periodic angular distribution in a polar coordinate system. The Talbot distance is derived to reveal the Talbot self-imaging pattern. Numerical simulation results show the self-imaging phenomenon of a spherical grating interferometer, which is in agreement with the theoretical prediction. Conclusions: X-ray Talbot interferometry with spherical gratings has a significant practical promise. Relative to planar grating imaging, spherical grating based x-ray Talbot interferometry has a larger field of view and
Spherical grating based x-ray Talbot interferometry
International Nuclear Information System (INIS)
Cong, Wenxiang; Xi, Yan; Wang, Ge
2015-01-01
Purpose: Grating interferometry is a state-of-the-art x-ray imaging approach, which can acquire information on x-ray attenuation, phase shift, and small-angle scattering simultaneously. Phase-contrast imaging and dark-field imaging are very sensitive to microstructural variation and offers superior contrast resolution for biological soft tissues. However, a common x-ray tube is a point-like source. As a result, the popular planar grating imaging configuration seriously restricts the flux of photons and decreases the visibility of signals, yielding a limited field of view. The purpose of this study is to extend the planar x-ray grating imaging theory and methods to a spherical grating scheme for a wider range of preclinical and clinical applications. Methods: A spherical grating matches the wave front of a point x-ray source very well, allowing the perpendicular incidence of x-rays on the grating to achieve a higher visibility over a larger field of view than the planer grating counterpart. A theoretical analysis of the Talbot effect for spherical grating imaging is proposed to establish a basic foundation for x-ray spherical gratings interferometry. An efficient method of spherical grating imaging is also presented to extract attenuation, differential phase, and dark-field images in the x-ray spherical grating interferometer. Results: Talbot self-imaging with spherical gratings is analyzed based on the Rayleigh–Sommerfeld diffraction formula, featuring a periodic angular distribution in a polar coordinate system. The Talbot distance is derived to reveal the Talbot self-imaging pattern. Numerical simulation results show the self-imaging phenomenon of a spherical grating interferometer, which is in agreement with the theoretical prediction. Conclusions: X-ray Talbot interferometry with spherical gratings has a significant practical promise. Relative to planar grating imaging, spherical grating based x-ray Talbot interferometry has a larger field of view and
Uniqueness and zeros of q-shift difference polynomials
Indian Academy of Sciences (India)
In this paper, we consider the zero distributions of -shift difference polynomials of meromorphic functions with zero order, and obtain two theorems that extend the classical Hayman results on the zeros of differential polynomials to -shift difference polynomials. We also investigate the uniqueness problem of -shift ...
The Flexibility of Pusher Furnace Grate
Directory of Open Access Journals (Sweden)
Słowik J.A.
2016-12-01
Full Text Available The lifetime of guide grates in pusher furnaces for heat treatment could be increased by raising the flexibility of their structure through, for example, the replacement of straight ribs, parallel to the direction of grate movement, with more flexible segments. The deformability of grates with flexible segments arranged in two orientations, i.e. crosswise (perpendicular to the direction of compression and lengthwise (parallel to the direction of compression, was examined. The compression process was simulated using SolidWorks Simulation program. Relevant regression equations were also derived describing the dependence of force inducing the grate deformation by 0.25 mm ‒ modulus of grate elasticity ‒ on the number of flexible segments in established orientations. These calculations were made in Statistica and Scilab programs. It has been demonstrated that, with the same number of segments, the crosswise orientation of flexible segments increases the grate structure flexibility in a more efficient way than the lengthwise orientation. It has also been proved that a crucial effect on the grate flexibility has only the quantity and orientation of segments (crosswise / lengthwise, while the exact position of segments changes the grate flexibility by less than 1%.
Factoring polynomials over arbitrary finite fields
Lange, T.; Winterhof, A.
2000-01-01
We analyse an extension of Shoup's (Inform. Process. Lett. 33 (1990) 261–267) deterministic algorithm for factoring polynomials over finite prime fields to arbitrary finite fields. In particular, we prove the existence of a deterministic algorithm which completely factors all monic polynomials of
Additive and polynomial representations
Krantz, David H; Suppes, Patrick
1971-01-01
Additive and Polynomial Representations deals with major representation theorems in which the qualitative structure is reflected as some polynomial function of one or more numerical functions defined on the basic entities. Examples are additive expressions of a single measure (such as the probability of disjoint events being the sum of their probabilities), and additive expressions of two measures (such as the logarithm of momentum being the sum of log mass and log velocity terms). The book describes the three basic procedures of fundamental measurement as the mathematical pivot, as the utiliz
The grating as an accelerating structure
International Nuclear Information System (INIS)
Fernow, R.C.
1991-02-01
This report considers the use of a diffraction grating as an accelerating structure for charged particle beams. We examine the functional dependence of the electromagnetic fields above the surface of a grating. Calculations are made of the strength of the accelerating modes for structures with π and 2π phase advance per period and for incident waves polarized with either the E or H vector along the grooves of the grating. We consider examples of using gratings in a laser linac and in a grating lens. We also briefly examine previous results published about this subject. 36 refs
Enhanced Raman scattering in porous silicon grating.
Wang, Jiajia; Jia, Zhenhong; Lv, Changwu
2018-03-19
The enhancement of Raman signal on monocrystalline silicon gratings with varying groove depths and on porous silicon grating were studied for a highly sensitive surface enhanced Raman scattering (SERS) response. In the experiment conducted, porous silicon gratings were fabricated. Silver nanoparticles (Ag NPs) were then deposited on the porous silicon grating to enhance the Raman signal of the detective objects. Results show that the enhancement of Raman signal on silicon grating improved when groove depth increased. The enhanced performance of Raman signal on porous silicon grating was also further improved. The Rhodamine SERS response based on Ag NPs/ porous silicon grating substrates was enhanced relative to the SERS response on Ag NPs/ porous silicon substrates. Ag NPs / porous silicon grating SERS substrate system achieved a highly sensitive SERS response due to the coupling of various Raman enhancement factors.
Varied line-space gratings and applications
International Nuclear Information System (INIS)
McKinney, W.R.
1991-01-01
This paper presents a straightforward analytical and numerical method for the design of a specific type of varied line-space grating system. The mathematical development will assume plane or nearly-plane spherical gratings which are illuminated by convergent light, which covers many interesting cases for synchrotron radiation. The gratings discussed will have straight grooves whose spacing varies across the principal plane of the grating. Focal relationships and formulae for the optical grating-pole-to-exist-slit distance and grating radius previously presented by other authors will be derived with a symbolic algebra system. It is intended to provide the optical designer with the tools necessary to design such a system properly. Finally, some possible advantages and disadvantages for application to synchrotron to synchrotron radiation beamlines will be discussed
A Determinant Expression for the Generalized Bessel Polynomials
Directory of Open Access Journals (Sweden)
Sheng-liang Yang
2013-01-01
Full Text Available Using the exponential Riordan arrays, we show that a variation of the generalized Bessel polynomial sequence is of Sheffer type, and we obtain a determinant formula for the generalized Bessel polynomials. As a result, the Bessel polynomial is represented as determinant the entries of which involve Catalan numbers.
A generalization of the Bernoulli polynomials
Directory of Open Access Journals (Sweden)
Pierpaolo Natalini
2003-01-01
Full Text Available A generalization of the Bernoulli polynomials and, consequently, of the Bernoulli numbers, is defined starting from suitable generating functions. Furthermore, the differential equations of these new classes of polynomials are derived by means of the factorization method introduced by Infeld and Hull (1951.
Information-theoretic lengths of Jacobi polynomials
Energy Technology Data Exchange (ETDEWEB)
Guerrero, A; Dehesa, J S [Departamento de Fisica Atomica, Molecular y Nuclear, Universidad de Granada, Granada (Spain); Sanchez-Moreno, P, E-mail: agmartinez@ugr.e, E-mail: pablos@ugr.e, E-mail: dehesa@ugr.e [Instituto ' Carlos I' de Fisica Teorica y Computacional, Universidad de Granada, Granada (Spain)
2010-07-30
The information-theoretic lengths of the Jacobi polynomials P{sup ({alpha}, {beta})}{sub n}(x), which are information-theoretic measures (Renyi, Shannon and Fisher) of their associated Rakhmanov probability density, are investigated. They quantify the spreading of the polynomials along the orthogonality interval [- 1, 1] in a complementary but different way as the root-mean-square or standard deviation because, contrary to this measure, they do not refer to any specific point of the interval. The explicit expressions of the Fisher length are given. The Renyi lengths are found by the use of the combinatorial multivariable Bell polynomials in terms of the polynomial degree n and the parameters ({alpha}, {beta}). The Shannon length, which cannot be exactly calculated because of its logarithmic functional form, is bounded from below by using sharp upper bounds to general densities on [- 1, +1] given in terms of various expectation values; moreover, its asymptotics is also pointed out. Finally, several computational issues relative to these three quantities are carefully analyzed.
Transversals of Complex Polynomial Vector Fields
DEFF Research Database (Denmark)
Dias, Kealey
Vector fields in the complex plane are defined by assigning the vector determined by the value P(z) to each point z in the complex plane, where P is a polynomial of one complex variable. We consider special families of so-called rotated vector fields that are determined by a polynomial multiplied...... by rotational constants. Transversals are a certain class of curves for such a family of vector fields that represent the bifurcation states for this family of vector fields. More specifically, transversals are curves that coincide with a homoclinic separatrix for some rotation of the vector field. Given...... a concrete polynomial, it seems to take quite a bit of work to prove that it is generic, i.e. structurally stable. This has been done for a special class of degree d polynomial vector fields having simple equilibrium points at the d roots of unity, d odd. In proving that such vector fields are generic...
On Multiple Interpolation Functions of the -Genocchi Polynomials
Directory of Open Access Journals (Sweden)
Jin Jeong-Hee
2010-01-01
Full Text Available Abstract Recently, many mathematicians have studied various kinds of the -analogue of Genocchi numbers and polynomials. In the work (New approach to q-Euler, Genocchi numbers and their interpolation functions, "Advanced Studies in Contemporary Mathematics, vol. 18, no. 2, pp. 105–112, 2009.", Kim defined new generating functions of -Genocchi, -Euler polynomials, and their interpolation functions. In this paper, we give another definition of the multiple Hurwitz type -zeta function. This function interpolates -Genocchi polynomials at negative integers. Finally, we also give some identities related to these polynomials.
Polynomial regression analysis and significance test of the regression function
International Nuclear Information System (INIS)
Gao Zhengming; Zhao Juan; He Shengping
2012-01-01
In order to analyze the decay heating power of a certain radioactive isotope per kilogram with polynomial regression method, the paper firstly demonstrated the broad usage of polynomial function and deduced its parameters with ordinary least squares estimate. Then significance test method of polynomial regression function is derived considering the similarity between the polynomial regression model and the multivariable linear regression model. Finally, polynomial regression analysis and significance test of the polynomial function are done to the decay heating power of the iso tope per kilogram in accord with the authors' real work. (authors)
The modified Gauss diagonalization of polynomial matrices
International Nuclear Information System (INIS)
Saeed, K.
1982-10-01
The Gauss algorithm for diagonalization of constant matrices is modified for application to polynomial matrices. Due to this modification the diagonal elements become pure polynomials rather than rational functions. (author)
Approximating Exponential and Logarithmic Functions Using Polynomial Interpolation
Gordon, Sheldon P.; Yang, Yajun
2017-01-01
This article takes a closer look at the problem of approximating the exponential and logarithmic functions using polynomials. Either as an alternative to or a precursor to Taylor polynomial approximations at the precalculus level, interpolating polynomials are considered. A measure of error is given and the behaviour of the error function is…
Talbot Carpet Simulation for X-ray grating interferometer
Energy Technology Data Exchange (ETDEWEB)
Kim, Youngju; Oh, Ohsung; Jeong, Hanseong; Kim, Jeongho; Lee, Seung Wook [Pusan National University, Busan (Korea, Republic of); Kim, Jongyul; Moon, Myungkook [Korea Atomic Energy Research Institute, Daejeon (Korea, Republic of)
2015-05-15
In this study, Talbot carpet simulator has been developed to visualize the X-ray grating interference patterns in grating interferometer. We have simulated X-ray interference for a variety of simulations and demonstrated a few examples in this summary. Grating interferometer produces interference of X-ray called Talbot pattern with gratings manufactured in micro scale. Talbot pattern is self-images of phase grating which develops interference as beam splitter that is one of gratings consisted of interferometer. As the other gratings, there are source grating makes coherence and analyze grating is used to analyze interference onto detector. Talbot carpet has been studied as the beam behavior which is distinguished with common X-ray imaging systems. It is helpful to understand grating interferometer and possible to expect beams' oscillation for designing theoretically. We confirm pattern has periodicity produced by interference after pi and pi/2 phase grating and changes in the perpendicular direction to entrance face according to phase objects.
Trimborn, Barbara; Meyer, Pascal; Kunka, Danays; Zuber, Marcus; Albrecht, Frederic; Kreuer, Sascha; Volk, Thomas; Baumbach, Tilo; Koenig, Thomas
2016-01-21
X-ray grating interferometry is one among various methods that allow extracting the so-called phase and visibility contrasts in addition to the well-known transmission images. Crucial to achieving a high image quality are the absorption gratings employed. Here, we present an in-depth analysis of how the grating type and lamella heights influence the final images. Benchmarking gratings of two different designs, we show that a frequently used proxy for image quality, a grating's so-called visibility, is insufficient to predict contrast-to-noise ratios (CNRs). Presenting scans from an excised rat lung, we demonstrate that the CNRs obtained for transmission and visibility images anti-correlate. This is explained by the stronger attenuation implied by gratings that are engineered to provide high visibilities by means of an increased lamella height. We show that even the visibility contrast can suffer from this effect when the associated reduced photon flux on the detector is not outweighed by a corresponding gain in visibility. Resulting in an inevitable trade-off between the quality of the two contrasts, the question of how an optimal grating should be designed can hence only be answered in terms of Pareto optimality.
Numerical Simulation of Polynomial-Speed Convergence Phenomenon
Li, Yao; Xu, Hui
2017-11-01
We provide a hybrid method that captures the polynomial speed of convergence and polynomial speed of mixing for Markov processes. The hybrid method that we introduce is based on the coupling technique and renewal theory. We propose to replace some estimates in classical results about the ergodicity of Markov processes by numerical simulations when the corresponding analytical proof is difficult. After that, all remaining conclusions can be derived from rigorous analysis. Then we apply our results to seek numerical justification for the ergodicity of two 1D microscopic heat conduction models. The mixing rate of these two models are expected to be polynomial but very difficult to prove. In both examples, our numerical results match the expected polynomial mixing rate well.
Exceptional polynomials and SUSY quantum mechanics
Indian Academy of Sciences (India)
Abstract. We show that for the quantum mechanical problem which admit classical Laguerre/. Jacobi polynomials as solutions for the Schrödinger equations (SE), will also admit exceptional. Laguerre/Jacobi polynomials as solutions having the same eigenvalues but with the ground state missing after a modification of the ...
A companion matrix for 2-D polynomials
International Nuclear Information System (INIS)
Boudellioua, M.S.
1995-08-01
In this paper, a matrix form analogous to the companion matrix which is often encountered in the theory of one dimensional (1-D) linear systems is suggested for a class of polynomials in two indeterminates and real coefficients, here referred to as two dimensional (2-D) polynomials. These polynomials arise in the context of 2-D linear systems theory. Necessary and sufficient conditions are also presented under which a matrix is equivalent to this companion form. (author). 6 refs
Global sensitivity analysis by polynomial dimensional decomposition
Energy Technology Data Exchange (ETDEWEB)
Rahman, Sharif, E-mail: rahman@engineering.uiowa.ed [College of Engineering, The University of Iowa, Iowa City, IA 52242 (United States)
2011-07-15
This paper presents a polynomial dimensional decomposition (PDD) method for global sensitivity analysis of stochastic systems subject to independent random input following arbitrary probability distributions. The method involves Fourier-polynomial expansions of lower-variate component functions of a stochastic response by measure-consistent orthonormal polynomial bases, analytical formulae for calculating the global sensitivity indices in terms of the expansion coefficients, and dimension-reduction integration for estimating the expansion coefficients. Due to identical dimensional structures of PDD and analysis-of-variance decomposition, the proposed method facilitates simple and direct calculation of the global sensitivity indices. Numerical results of the global sensitivity indices computed for smooth systems reveal significantly higher convergence rates of the PDD approximation than those from existing methods, including polynomial chaos expansion, random balance design, state-dependent parameter, improved Sobol's method, and sampling-based methods. However, for non-smooth functions, the convergence properties of the PDD solution deteriorate to a great extent, warranting further improvements. The computational complexity of the PDD method is polynomial, as opposed to exponential, thereby alleviating the curse of dimensionality to some extent.
Polynomial asymptotic stability of damped stochastic differential equations
Directory of Open Access Journals (Sweden)
John Appleby
2004-08-01
Full Text Available The paper studies the polynomial convergence of solutions of a scalar nonlinear It\\^{o} stochastic differential equation\\[dX(t = -f(X(t\\,dt + \\sigma(t\\,dB(t\\] where it is known, {\\it a priori}, that $\\lim_{t\\rightarrow\\infty} X(t=0$, a.s. The intensity of the stochastic perturbation $\\sigma$ is a deterministic, continuous and square integrable function, which tends to zero more quickly than a polynomially decaying function. The function $f$ obeys $\\lim_{x\\rightarrow 0}\\mbox{sgn}(xf(x/|x|^\\beta = a$, for some $\\beta>1$, and $a>0$.We study two asymptotic regimes: when $\\sigma$ tends to zero sufficiently quickly the polynomial decay rate of solutions is the same as for the deterministic equation (when $\\sigma\\equiv0$. When $\\sigma$ decays more slowly, a weaker almost sure polynomial upper bound on the decay rate of solutions is established. Results which establish the necessity for $\\sigma$ to decay polynomially in order to guarantee the almost sure polynomial decay of solutions are also proven.
International Nuclear Information System (INIS)
Konakli, Katerina; Sudret, Bruno
2016-01-01
The growing need for uncertainty analysis of complex computational models has led to an expanding use of meta-models across engineering and sciences. The efficiency of meta-modeling techniques relies on their ability to provide statistically-equivalent analytical representations based on relatively few evaluations of the original model. Polynomial chaos expansions (PCE) have proven a powerful tool for developing meta-models in a wide range of applications; the key idea thereof is to expand the model response onto a basis made of multivariate polynomials obtained as tensor products of appropriate univariate polynomials. The classical PCE approach nevertheless faces the “curse of dimensionality”, namely the exponential increase of the basis size with increasing input dimension. To address this limitation, the sparse PCE technique has been proposed, in which the expansion is carried out on only a few relevant basis terms that are automatically selected by a suitable algorithm. An alternative for developing meta-models with polynomial functions in high-dimensional problems is offered by the newly emerged low-rank approximations (LRA) approach. By exploiting the tensor–product structure of the multivariate basis, LRA can provide polynomial representations in highly compressed formats. Through extensive numerical investigations, we herein first shed light on issues relating to the construction of canonical LRA with a particular greedy algorithm involving a sequential updating of the polynomial coefficients along separate dimensions. Specifically, we examine the selection of optimal rank, stopping criteria in the updating of the polynomial coefficients and error estimation. In the sequel, we confront canonical LRA to sparse PCE in structural-mechanics and heat-conduction applications based on finite-element solutions. Canonical LRA exhibit smaller errors than sparse PCE in cases when the number of available model evaluations is small with respect to the input
Energy Technology Data Exchange (ETDEWEB)
Konakli, Katerina, E-mail: konakli@ibk.baug.ethz.ch; Sudret, Bruno
2016-09-15
The growing need for uncertainty analysis of complex computational models has led to an expanding use of meta-models across engineering and sciences. The efficiency of meta-modeling techniques relies on their ability to provide statistically-equivalent analytical representations based on relatively few evaluations of the original model. Polynomial chaos expansions (PCE) have proven a powerful tool for developing meta-models in a wide range of applications; the key idea thereof is to expand the model response onto a basis made of multivariate polynomials obtained as tensor products of appropriate univariate polynomials. The classical PCE approach nevertheless faces the “curse of dimensionality”, namely the exponential increase of the basis size with increasing input dimension. To address this limitation, the sparse PCE technique has been proposed, in which the expansion is carried out on only a few relevant basis terms that are automatically selected by a suitable algorithm. An alternative for developing meta-models with polynomial functions in high-dimensional problems is offered by the newly emerged low-rank approximations (LRA) approach. By exploiting the tensor–product structure of the multivariate basis, LRA can provide polynomial representations in highly compressed formats. Through extensive numerical investigations, we herein first shed light on issues relating to the construction of canonical LRA with a particular greedy algorithm involving a sequential updating of the polynomial coefficients along separate dimensions. Specifically, we examine the selection of optimal rank, stopping criteria in the updating of the polynomial coefficients and error estimation. In the sequel, we confront canonical LRA to sparse PCE in structural-mechanics and heat-conduction applications based on finite-element solutions. Canonical LRA exhibit smaller errors than sparse PCE in cases when the number of available model evaluations is small with respect to the input
Optical Fiber Grating based Sensors
DEFF Research Database (Denmark)
Michelsen, Susanne
2003-01-01
In this thesis differenct optical fiber gratings are used for sensor purposes. If a fiber with a core concentricity error (CCE) is used, a directional dependent bend sensor can be produced. The CCE direction can be determined by means of diffraction. This makes it possible to produce long......-period gratings in a fiber with a CCE direction parallel or perpendicular to the writing direction. The maximal bending sensitivity is independent on the writing direction, but the detailed bending response is different in the two cases. A temperature and strain sensor, based on a long-period grating and two...... sampled gratings, was produced and investigated. It is based on the different temperature and strain response of these gratings. Both a transfer matrix method and an overlap calculation is performed to explain the sensor response. Another type of sensor is based on tuning and modulation of a laser...
Degenerate r-Stirling Numbers and r-Bell Polynomials
Kim, T.; Yao, Y.; Kim, D. S.; Jang, G.-W.
2018-01-01
The purpose of this paper is to exploit umbral calculus in order to derive some properties, recurrence relations, and identities related to the degenerate r-Stirling numbers of the second kind and the degenerate r-Bell polynomials. Especially, we will express the degenerate r-Bell polynomials as linear combinations of many well-known families of special polynomials.
Commutators with idempotent values on multilinear polynomials in ...
Indian Academy of Sciences (India)
Multilinear polynomial; derivations; generalized polynomial identity; prime ring; right ideal. Abstract. Let R be a prime ring of characteristic different from 2, C its extended centroid, d a nonzero derivation of R , f ( x 1 , … , x n ) a multilinear polynomial over C , ϱ a nonzero right ideal of R and m > 1 a fixed integer such that.
Polynomial weights and code constructions
DEFF Research Database (Denmark)
Massey, J; Costello, D; Justesen, Jørn
1973-01-01
polynomial included. This fundamental property is then used as the key to a variety of code constructions including 1) a simplified derivation of the binary Reed-Muller codes and, for any primepgreater than 2, a new extensive class ofp-ary "Reed-Muller codes," 2) a new class of "repeated-root" cyclic codes...... of long constraint length binary convolutional codes derived from2^r-ary Reed-Solomon codes, and 6) a new class ofq-ary "repeated-root" constacyclic codes with an algebraic decoding algorithm.......For any nonzero elementcof a general finite fieldGF(q), it is shown that the polynomials(x - c)^i, i = 0,1,2,cdots, have the "weight-retaining" property that any linear combination of these polynomials with coefficients inGF(q)has Hamming weight at least as great as that of the minimum degree...
The generalized Yablonskii-Vorob'ev polynomials and their properties
International Nuclear Information System (INIS)
Kudryashov, Nikolai A.; Demina, Maria V.
2008-01-01
Rational solutions of the generalized second Painleve hierarchy are classified. Representation of the rational solutions in terms of special polynomials, the generalized Yablonskii-Vorob'ev polynomials, is introduced. Differential-difference relations satisfied by the polynomials are found. Hierarchies of differential equations related to the generalized second Painleve hierarchy are derived. One of these hierarchies is a sequence of differential equations satisfied by the generalized Yablonskii-Vorob'ev polynomials
2-variable Laguerre matrix polynomials and Lie-algebraic techniques
International Nuclear Information System (INIS)
Khan, Subuhi; Hassan, Nader Ali Makboul
2010-01-01
The authors introduce 2-variable forms of Laguerre and modified Laguerre matrix polynomials and derive their special properties. Further, the representations of the special linear Lie algebra sl(2) and the harmonic oscillator Lie algebra G(0,1) are used to derive certain results involving these polynomials. Furthermore, the generating relations for the ordinary as well as matrix polynomials related to these matrix polynomials are derived as applications.
Describing Quadratic Cremer Point Polynomials by Parabolic Perturbations
DEFF Research Database (Denmark)
Sørensen, Dan Erik Krarup
1996-01-01
We describe two infinite order parabolic perturbation proceduresyielding quadratic polynomials having a Cremer fixed point. The main ideais to obtain the polynomial as the limit of repeated parabolic perturbations.The basic tool at each step is to control the behaviour of certain externalrays.......Polynomials of the Cremer type correspond to parameters at the boundary of ahyperbolic component of the Mandelbrot set. In this paper we concentrate onthe main cardioid component. We investigate the differences between two-sided(i.e. alternating) and one-sided parabolic perturbations.In the two-sided case, we prove...... the existence of polynomials having an explicitlygiven external ray accumulating both at the Cremer point and at its non-periodicpreimage. We think of the Julia set as containing a "topologists double comb".In the one-sided case we prove a weaker result: the existence of polynomials havingan explicitly given...
Orthogonal polynomials derived from the tridiagonal representation approach
Alhaidari, A. D.
2018-01-01
The tridiagonal representation approach is an algebraic method for solving second order differential wave equations. Using this approach in the solution of quantum mechanical problems, we encounter two new classes of orthogonal polynomials whose properties give the structure and dynamics of the corresponding physical system. For a certain range of parameters, one of these polynomials has a mix of continuous and discrete spectra making it suitable for describing physical systems with both scattering and bound states. In this work, we define these polynomials by their recursion relations and highlight some of their properties using numerical means. Due to the prime significance of these polynomials in physics, we hope that our short expose will encourage experts in the field of orthogonal polynomials to study them and derive their properties (weight functions, generating functions, asymptotics, orthogonality relations, zeros, etc.) analytically.
A note on some identities of derangement polynomials.
Kim, Taekyun; Kim, Dae San; Jang, Gwan-Woo; Kwon, Jongkyum
2018-01-01
The problem of counting derangements was initiated by Pierre Rémond de Montmort in 1708 (see Carlitz in Fibonacci Q. 16(3):255-258, 1978, Clarke and Sved in Math. Mag. 66(5):299-303, 1993, Kim, Kim and Kwon in Adv. Stud. Contemp. Math. (Kyungshang) 28(1):1-11 2018. A derangement is a permutation that has no fixed points, and the derangement number [Formula: see text] is the number of fixed-point-free permutations on an n element set. In this paper, we study the derangement polynomials and investigate some interesting properties which are related to derangement numbers. Also, we study two generalizations of derangement polynomials, namely higher-order and r -derangement polynomials, and show some relations between them. In addition, we express several special polynomials in terms of the higher-order derangement polynomials by using umbral calculus.
Topological quantum information, virtual Jones polynomials and Khovanov homology
International Nuclear Information System (INIS)
Kauffman, Louis H
2011-01-01
In this paper, we give a quantum statistical interpretation of the bracket polynomial state sum 〈K〉, the Jones polynomial V K (t) and virtual knot theory versions of the Jones polynomial, including the arrow polynomial. We use these quantum mechanical interpretations to give new quantum algorithms for these Jones polynomials. In those cases where the Khovanov homology is defined, the Hilbert space C(K) of our model is isomorphic with the chain complex for Khovanov homology with coefficients in the complex numbers. There is a natural unitary transformation U:C(K) → C(K) such that 〈K〉 = Trace(U), where 〈K〉 denotes the evaluation of the state sum model for the corresponding polynomial. We show that for the Khovanov boundary operator ∂:C(K) → C(K), we have the relationship ∂U + U∂ = 0. Consequently, the operator U acts on the Khovanov homology, and we obtain a direct relationship between the Khovanov homology and this quantum algorithm for the Jones polynomial. (paper)
Polynomial solutions of the Monge-Ampère equation
Energy Technology Data Exchange (ETDEWEB)
Aminov, Yu A [B.Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, Khar' kov (Ukraine)
2014-11-30
The question of the existence of polynomial solutions to the Monge-Ampère equation z{sub xx}z{sub yy}−z{sub xy}{sup 2}=f(x,y) is considered in the case when f(x,y) is a polynomial. It is proved that if f is a polynomial of the second degree, which is positive for all values of its arguments and has a positive squared part, then no polynomial solution exists. On the other hand, a solution which is not polynomial but is analytic in the whole of the x, y-plane is produced. Necessary and sufficient conditions for the existence of polynomial solutions of degree up to 4 are found and methods for the construction of such solutions are indicated. An approximation theorem is proved. Bibliography: 10 titles.
Apodized grating coupler using fully-etched nanostructures
International Nuclear Information System (INIS)
Wu Hua; Li Chong; Guo Xia; Li Zhi-Yong
2016-01-01
A two-dimensional apodized grating coupler for interfacing between single-mode fiber and photonic circuit is demonstrated in order to bridge the mode gap between the grating coupler and optical fiber. The grating grooves of the grating couplers are realized by columns of fully etched nanostructures, which are utilized to digitally tailor the effective refractive index of each groove in order to obtain the Gaussian-like output diffractive mode and then enhance the coupling efficiency. Compared with that of the uniform grating coupler, the coupling efficiency of the apodized grating coupler is increased by 4.3% and 5.7%, respectively, for the nanoholes and nanorectangles as refractive index tunes layer. (paper)
DEFF Research Database (Denmark)
Marckmann, Carl Johan
2003-01-01
Research Center (MIC) at the Technical University of Denmark. The Bragg gratings were fabricated at COM using UV irradiation of the planar waveguides using the phase mask method. The induction of a frozen-in DC electric field into the samples was performed by thermal poling of the Bragg gratings...... layers, it becam possible to investigate the symmetry properties of the third-order nonlinearities. Contrary to the expectations for an amorphous material, the measurements indicated an almost polarization independent third-order nonlinearity - the most probable explanation being electrostriction......The subject of this ph.d. thesis was the development of an electrically switchable Bragg grating made in an optical waveguide using thermal poling to be applied within optical telecommunication systems. The planar waveguides used in this thesis were fabricated at the Micro- and Nanotechnology...
Nanoporous Polymeric Grating-Based Biosensors
Gao, Tieyu; Hsiao, Vincent; Zheng, Yue Bing; Huang, Tony Jun
2012-01-01
We demonstrate the utilization of an interferometrically created nanoporous polymeric gratings as a platform for biosensing applications. Aminopropyltriethoxysilane (APTES)-functionalized nanoporous polymeric gratings was fabricated by combining holographic interference patterning and APTES-functionalization of pre-polymer syrup. The successful detection of multiple biomolecules indicates that the biofunctionalized nanoporous polymeric gratings can act as biosensing platforms which are label-free, inexpensive, and applicable as high-throughput assays. Copyright © 2010 by ASME.
Nanoporous Polymeric Grating-Based Biosensors
Gao, Tieyu
2012-05-02
We demonstrate the utilization of an interferometrically created nanoporous polymeric gratings as a platform for biosensing applications. Aminopropyltriethoxysilane (APTES)-functionalized nanoporous polymeric gratings was fabricated by combining holographic interference patterning and APTES-functionalization of pre-polymer syrup. The successful detection of multiple biomolecules indicates that the biofunctionalized nanoporous polymeric gratings can act as biosensing platforms which are label-free, inexpensive, and applicable as high-throughput assays. Copyright © 2010 by ASME.
Zeros and uniqueness of Q-difference polynomials of meromorphic ...
Indian Academy of Sciences (India)
Meromorphic functions; Nevanlinna theory; logarithmic order; uniqueness problem; difference-differential polynomial. Abstract. In this paper, we investigate the value distribution of -difference polynomials of meromorphic function of finite logarithmic order, and study the zero distribution of difference-differential polynomials ...
Laguerre polynomials by a harmonic oscillator
Baykal, Melek; Baykal, Ahmet
2014-09-01
The study of an isotropic harmonic oscillator, using the factorization method given in Ohanian's textbook on quantum mechanics, is refined and some collateral extensions of the method related to the ladder operators and the associated Laguerre polynomials are presented. In particular, some analytical properties of the associated Laguerre polynomials are derived using the ladder operators.
Encapsulation process for diffraction gratings.
Ratzsch, Stephan; Kley, Ernst-Bernhard; Tünnermann, Andreas; Szeghalmi, Adriana
2015-07-13
Encapsulation of grating structures facilitates an improvement of the optical functionality and/or adds mechanical stability to the fragile structure. Here, we introduce novel encapsulation process of nanoscale patterns based on atomic layer deposition and micro structuring. The overall size of the encapsulated structured surface area is only restricted by the size of the available microstructuring and coating devices; thus, overcoming inherent limitations of existing bonding processes concerning cleanliness, roughness, and curvature of the components. Finally, the process is demonstrated for a transmission grating. The encapsulated grating has 97.5% transmission efficiency in the -1st diffraction order for TM-polarized light, and is being limited by the experimental grating parameters as confirmed by rigorous coupled wave analysis.
Julia Sets of Orthogonal Polynomials
DEFF Research Database (Denmark)
Christiansen, Jacob Stordal; Henriksen, Christian; Petersen, Henrik Laurberg
2018-01-01
For a probability measure with compact and non-polar support in the complex plane we relate dynamical properties of the associated sequence of orthogonal polynomials fPng to properties of the support. More precisely we relate the Julia set of Pn to the outer boundary of the support, the lled Julia...... set to the polynomial convex hull K of the support, and the Green's function associated with Pn to the Green's function for the complement of K....
An introduction to orthogonal polynomials
Chihara, Theodore S
1978-01-01
Assuming no further prerequisites than a first undergraduate course in real analysis, this concise introduction covers general elementary theory related to orthogonal polynomials. It includes necessary background material of the type not usually found in the standard mathematics curriculum. Suitable for advanced undergraduate and graduate courses, it is also appropriate for independent study. Topics include the representation theorem and distribution functions, continued fractions and chain sequences, the recurrence formula and properties of orthogonal polynomials, special functions, and some
Imaging characteristics of Zernike and annular polynomial aberrations.
Mahajan, Virendra N; Díaz, José Antonio
2013-04-01
The general equations for the point-spread function (PSF) and optical transfer function (OTF) are given for any pupil shape, and they are applied to optical imaging systems with circular and annular pupils. The symmetry properties of the PSF, the real and imaginary parts of the OTF, and the modulation transfer function (MTF) of a system with a circular pupil aberrated by a Zernike circle polynomial aberration are derived. The interferograms and PSFs are illustrated for some typical polynomial aberrations with a sigma value of one wave, and 3D PSFs and MTFs are shown for 0.1 wave. The Strehl ratio is also calculated for polynomial aberrations with a sigma value of 0.1 wave, and shown to be well estimated from the sigma value. The numerical results are compared with the corresponding results in the literature. Because of the same angular dependence of the corresponding annular and circle polynomial aberrations, the symmetry properties of systems with annular pupils aberrated by an annular polynomial aberration are the same as those for a circular pupil aberrated by a corresponding circle polynomial aberration. They are also illustrated with numerical examples.
An elastomeric grating coupler
Kocabas, A.; Ay, F.; Dana, A.; Aydinli, A.
We report on a novel nondestructive and reversible method for coupling free space light to planar optical waveguides. In this method, an elastomeric grating is used to produce an effective refractive index modulation on the surface of the optical waveguide. The external elastomeric grating binds to
DEFF Research Database (Denmark)
Zhang, C.; Webb, D.J.; Kalli, K.
We report for the first time fibre Bragg grating inscription in microstructured optical fibre fabricated from Topas® cyclic olefin copolymer. The temperature sensitivity of the grating was studied revealing a positive Bragg wavelength shift of approximately 0.8 nmK-1,the largest sensitivity yet...
Polynomial selection in number field sieve for integer factorization
Directory of Open Access Journals (Sweden)
Gireesh Pandey
2016-09-01
Full Text Available The general number field sieve (GNFS is the fastest algorithm for factoring large composite integers which is made up by two prime numbers. Polynomial selection is an important step of GNFS. The asymptotic runtime depends on choice of good polynomial pairs. In this paper, we present polynomial selection algorithm that will be modelled with size and root properties. The correlations between polynomial coefficient and number of relations have been explored with experimental findings.
Polynomial solutions of nonlinear integral equations
International Nuclear Information System (INIS)
Dominici, Diego
2009-01-01
We analyze the polynomial solutions of a nonlinear integral equation, generalizing the work of Bender and Ben-Naim (2007 J. Phys. A: Math. Theor. 40 F9, 2008 J. Nonlinear Math. Phys. 15 (Suppl. 3) 73). We show that, in some cases, an orthogonal solution exists and we give its general form in terms of kernel polynomials
Polynomial solutions of nonlinear integral equations
Energy Technology Data Exchange (ETDEWEB)
Dominici, Diego [Department of Mathematics, State University of New York at New Paltz, 1 Hawk Dr. Suite 9, New Paltz, NY 12561-2443 (United States)], E-mail: dominicd@newpaltz.edu
2009-05-22
We analyze the polynomial solutions of a nonlinear integral equation, generalizing the work of Bender and Ben-Naim (2007 J. Phys. A: Math. Theor. 40 F9, 2008 J. Nonlinear Math. Phys. 15 (Suppl. 3) 73). We show that, in some cases, an orthogonal solution exists and we give its general form in terms of kernel polynomials.
Laguerre polynomials by a harmonic oscillator
International Nuclear Information System (INIS)
Baykal, Melek; Baykal, Ahmet
2014-01-01
The study of an isotropic harmonic oscillator, using the factorization method given in Ohanian's textbook on quantum mechanics, is refined and some collateral extensions of the method related to the ladder operators and the associated Laguerre polynomials are presented. In particular, some analytical properties of the associated Laguerre polynomials are derived using the ladder operators. (paper)
Remarks on determinants and the classical polynomials
International Nuclear Information System (INIS)
Henning, J.J.; Kranold, H.U.; Louw, D.F.B.
1986-01-01
As motivation for this formal analysis the problem of Landau damping of Bernstein modes is discussed. It is shown that in the case of a weak but finite constant external magnetic field, the analytical structure of the dispersion relations is of such a nature that longitudinal waves propagating orthogonal to the external magnetic field are also damped, contrary to normal belief. In the treatment of the linearized Vlasov equation it is found convenient to generate certain polynomials by the problem at hand and to explicitly write down expressions for these polynomials. In the course of this study methods are used that relate to elementary but fairly unknown functional relationships between power sums and coefficients of polynomials. These relationships, also called Waring functions, are derived. They are then used in other applications to give explicit expressions for the generalized Laguerre polynomials in terms of determinant functions. The properties of polynomials generated by a wide class of generating functions are investigated. These relationships are also used to obtain explicit forms for the cumulants of a distribution in terms of its moments. It is pointed out that cumulants (or moments, for that matter) do not determine a distribution function
Thermal and Structural Analysis of FIMS Grating
Directory of Open Access Journals (Sweden)
K.-I. Seon
2001-06-01
Full Text Available Far ultraviolet IMaging Spectrograph (FIMS should be designed to maintain its structural stability and to minimize optical performance degradation in launch and in operation enviroments. The structural and thermal analyzes of grating and grating mount system, which are directly related to FIMS optical performance, was performed using finite element method. The grating mount was made to keep the grating stress down, while keeping the natural frequency of the grating mount higher than 100 Hz. Transient and static thermal analyzes were also performed and the results shows that the thermal stress on the grating can be attenuated sufficiently The optical performance variation due to temperature variation was within the allowed range.
General quantum polynomials: irreducible modules and Morita equivalence
International Nuclear Information System (INIS)
Artamonov, V A
1999-01-01
In this paper we continue the investigation of the structure of finitely generated modules over rings of general quantum (Laurent) polynomials. We obtain a description of the lattice of submodules of periodic finitely generated modules and describe the irreducible modules. We investigate the problem of Morita equivalence of rings of general quantum polynomials, consider properties of division rings of fractions, and solve Zariski's problem for quantum polynomials
Multiple Meixner polynomials and non-Hermitian oscillator Hamiltonians
Ndayiragije, François; Van Assche, Walter
2013-01-01
Multiple Meixner polynomials are polynomials in one variable which satisfy orthogonality relations with respect to $r>1$ different negative binomial distributions (Pascal distributions). There are two kinds of multiple Meixner polynomials, depending on the selection of the parameters in the negative binomial distribution. We recall their definition and some formulas and give generating functions and explicit expressions for the coefficients in the nearest neighbor recurrence relation. Followi...
Multivariable biorthogonal continuous--discrete Wilson and Racah polynomials
International Nuclear Information System (INIS)
Tratnik, M.V.
1990-01-01
Several families of multivariable, biorthogonal, partly continuous and partly discrete, Wilson polynomials are presented. These yield limit cases that are purely continuous in some of the variables and purely discrete in the others, or purely discrete in all the variables. The latter are referred to as the multivariable biorthogonal Racah polynomials. Interesting further limit cases include the multivariable biorthogonal Hahn and dual Hahn polynomials
Embedded high-contrast distributed grating structures
Zubrzycki, Walter J.; Vawter, Gregory A.; Allerman, Andrew A.
2002-01-01
A new class of fabrication methods for embedded distributed grating structures is claimed, together with optical devices which include such structures. These new methods are the only known approach to making defect-free high-dielectric contrast grating structures, which are smaller and more efficient than are conventional grating structures.
Primitive polynomials selection method for pseudo-random number generator
Anikin, I. V.; Alnajjar, Kh
2018-01-01
In this paper we suggested the method for primitive polynomials selection of special type. This kind of polynomials can be efficiently used as a characteristic polynomials for linear feedback shift registers in pseudo-random number generators. The proposed method consists of two basic steps: finding minimum-cost irreducible polynomials of the desired degree and applying primitivity tests to get the primitive ones. Finally two primitive polynomials, which was found by the proposed method, used in pseudorandom number generator based on fuzzy logic (FRNG) which had been suggested before by the authors. The sequences generated by new version of FRNG have low correlation magnitude, high linear complexity, less power consumption, is more balanced and have better statistical properties.
Neck curve polynomials in neck rupture model
International Nuclear Information System (INIS)
Kurniadi, Rizal; Perkasa, Yudha S.; Waris, Abdul
2012-01-01
The Neck Rupture Model is a model that explains the scission process which has smallest radius in liquid drop at certain position. Old fashion of rupture position is determined randomly so that has been called as Random Neck Rupture Model (RNRM). The neck curve polynomials have been employed in the Neck Rupture Model for calculation the fission yield of neutron induced fission reaction of 280 X 90 with changing of order of polynomials as well as temperature. The neck curve polynomials approximation shows the important effects in shaping of fission yield curve.
The finite Fourier transform of classical polynomials
Dixit, Atul; Jiu, Lin; Moll, Victor H.; Vignat, Christophe
2014-01-01
The finite Fourier transform of a family of orthogonal polynomials $A_{n}(x)$, is the usual transform of the polynomial extended by $0$ outside their natural domain. Explicit expressions are given for the Legendre, Jacobi, Gegenbauer and Chebyshev families.
Algebraic limit cycles in polynomial systems of differential equations
International Nuclear Information System (INIS)
Llibre, Jaume; Zhao Yulin
2007-01-01
Using elementary tools we construct cubic polynomial systems of differential equations with algebraic limit cycles of degrees 4, 5 and 6. We also construct a cubic polynomial system of differential equations having an algebraic homoclinic loop of degree 3. Moreover, we show that there are polynomial systems of differential equations of arbitrary degree that have algebraic limit cycles of degree 3, as well as give an example of a cubic polynomial system of differential equations with two algebraic limit cycles of degree 4
From sequences to polynomials and back, via operator orderings
Energy Technology Data Exchange (ETDEWEB)
Amdeberhan, Tewodros, E-mail: tamdeber@tulane.edu; Dixit, Atul, E-mail: adixit@tulane.edu; Moll, Victor H., E-mail: vhm@tulane.edu [Department of Mathematics, Tulane University, New Orleans, Louisiana 70118 (United States); De Angelis, Valerio, E-mail: vdeangel@xula.edu [Department of Mathematics, Xavier University of Louisiana, New Orleans, Louisiana 70125 (United States); Vignat, Christophe, E-mail: vignat@tulane.edu [Department of Mathematics, Tulane University, New Orleans, Louisiana 70118, USA and L.S.S. Supelec, Universite d' Orsay (France)
2013-12-15
Bender and Dunne [“Polynomials and operator orderings,” J. Math. Phys. 29, 1727–1731 (1988)] showed that linear combinations of words q{sup k}p{sup n}q{sup n−k}, where p and q are subject to the relation qp − pq = ı, may be expressed as a polynomial in the symbol z=1/2 (qp+pq). Relations between such polynomials and linear combinations of the transformed coefficients are explored. In particular, examples yielding orthogonal polynomials are provided.
Connection coefficients between Boas-Buck polynomial sets
Cheikh, Y. Ben; Chaggara, H.
2006-07-01
In this paper, a general method to express explicitly connection coefficients between two Boas-Buck polynomial sets is presented. As application, we consider some generalized hypergeometric polynomials, from which we derive some well-known results including duplication and inversion formulas.
Geometrical optics modeling of the grating-slit test.
Liang, Chao-Wen; Sasian, Jose
2007-02-19
A novel optical testing method termed the grating-slit test is discussed. This test uses a grating and a slit, as in the Ronchi test, but the grating-slit test is different in that the grating is used as the incoherent illuminating object instead of the spatial filter. The slit is located at the plane of the image of a sinusoidal intensity grating. An insightful geometrical-optics model for the grating-slit test is presented and the fringe contrast ratio with respect to the slit width and object-grating period is obtained. The concept of spatial bucket integration is used to obtain the fringe contrast ratio.
Least squares orthogonal polynomial approximation in several independent variables
International Nuclear Information System (INIS)
Caprari, R.S.
1992-06-01
This paper begins with an exposition of a systematic technique for generating orthonormal polynomials in two independent variables by application of the Gram-Schmidt orthogonalization procedure of linear algebra. It is then demonstrated how a linear least squares approximation for experimental data or an arbitrary function can be generated from these polynomials. The least squares coefficients are computed without recourse to matrix arithmetic, which ensures both numerical stability and simplicity of implementation as a self contained numerical algorithm. The Gram-Schmidt procedure is then utilised to generate a complete set of orthogonal polynomials of fourth degree. A theory for the transformation of the polynomial representation from an arbitrary basis into the familiar sum of products form is presented, together with a specific implementation for fourth degree polynomials. Finally, the computational integrity of this algorithm is verified by reconstructing arbitrary fourth degree polynomials from their values at randomly chosen points in their domain. 13 refs., 1 tab
International Nuclear Information System (INIS)
Yasa, F.; Anli, F.; Guengoer, S.
2007-01-01
We present analytical calculations of spherically symmetric radioactive transfer and neutron transport using a hypothesis of P1 and T1 low order polynomial approximation for diffusion coefficient D. Transport equation in spherical geometry is considered as the pseudo slab equation. The validity of polynomial expansionion in transport theory is investigated through a comparison with classic diffusion theory. It is found that for causes when the fluctuation of the scattering cross section dominates, the quantitative difference between the polynomial approximation and diffusion results was physically acceptable in general
On Roots of Polynomials and Algebraically Closed Fields
Directory of Open Access Journals (Sweden)
Schwarzweller Christoph
2017-10-01
Full Text Available In this article we further extend the algebraic theory of polynomial rings in Mizar [1, 2, 3]. We deal with roots and multiple roots of polynomials and show that both the real numbers and finite domains are not algebraically closed [5, 7]. We also prove the identity theorem for polynomials and that the number of multiple roots is bounded by the polynomial’s degree [4, 6].
Topological string partition functions as polynomials
International Nuclear Information System (INIS)
Yamaguchi, Satoshi; Yau Shingtung
2004-01-01
We investigate the structure of the higher genus topological string amplitudes on the quintic hypersurface. It is shown that the partition functions of the higher genus than one can be expressed as polynomials of five generators. We also compute the explicit polynomial forms of the partition functions for genus 2, 3, and 4. Moreover, some coefficients are written down for all genus. (author)
Rotation of 2D orthogonal polynomials
Czech Academy of Sciences Publication Activity Database
Yang, B.; Flusser, Jan; Kautský, J.
2018-01-01
Roč. 102, č. 1 (2018), s. 44-49 ISSN 0167-8655 R&D Projects: GA ČR GA15-16928S Institutional support: RVO:67985556 Keywords : Rotation invariants * Orthogonal polynomials * Recurrent relation * Hermite-like polynomials * Hermite moments Subject RIV: JD - Computer Applications, Robotics Impact factor: 1.995, year: 2016 http://library.utia.cas.cz/separaty/2017/ZOI/flusser-0483250.pdf
q-analogue of the Krawtchouk and Meixner orthogonal polynomials
International Nuclear Information System (INIS)
Campigotto, C.; Smirnov, Yu.F.; Enikeev, S.G.
1993-06-01
The comparative analysis of Krawtchouk polynomials on a uniform grid with Wigner D-functions for the SU(2) group is presented. As a result the partnership between corresponding properties of the polynomials and D-functions is established giving the group-theoretical interpretation of the Krawtchouk polynomials properties. In order to extend such an analysis on the quantum groups SU q (2) and SU q (1,1), q-analogues of Krawtchouk and Meixner polynomials of a discrete variable are studied. The total set of characteristics of these polynomials is calculated, including the orthogonality condition, normalization factor, recurrent relation, the explicit analytic expression, the Rodrigues formula, the difference derivative formula and various particular cases and values. (R.P.) 22 refs.; 2 tabs
Skew-orthogonal polynomials and random matrix theory
Ghosh, Saugata
2009-01-01
Orthogonal polynomials satisfy a three-term recursion relation irrespective of the weight function with respect to which they are defined. This gives a simple formula for the kernel function, known in the literature as the Christoffel-Darboux sum. The availability of asymptotic results of orthogonal polynomials and the simple structure of the Christoffel-Darboux sum make the study of unitary ensembles of random matrices relatively straightforward. In this book, the author develops the theory of skew-orthogonal polynomials and obtains recursion relations which, unlike orthogonal polynomials, depend on weight functions. After deriving reduced expressions, called the generalized Christoffel-Darboux formulas (GCD), he obtains universal correlation functions and non-universal level densities for a wide class of random matrix ensembles using the GCD. The author also shows that once questions about higher order effects are considered (questions that are relevant in different branches of physics and mathematics) the ...
Some properties of generalized self-reciprocal polynomials over finite fields
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Ryul Kim
2014-07-01
Full Text Available Numerous results on self-reciprocal polynomials over finite fields have been studied. In this paper we generalize some of these to a-self reciprocal polynomials defined in [4]. We consider some properties of the divisibility of a-reciprocal polynomials and characterize the parity of the number of irreducible factors for a-self reciprocal polynomials over finite fields of odd characteristic.
Speed and the coherence of superimposed chromatic gratings.
Bosten, J M; Smith, L; Mollon, J D
2016-05-01
On the basis of measurements of the perceived coherence of superimposed drifting gratings, Krauskopf and Farell (1990) proposed that motion is analysed independently in different chromatic channels. They found that two gratings appeared to slip if each modulated one of the two 'cardinal' color mechanisms S/(L+M) and L/(L+M). If the gratings were defined along intermediate color directions, observers reported a plaid, moving coherently. We hypothesised that slippage might occur in chromatic gratings if the motion signal from the S/(L+M) channel is weak and equivalent to a lower speed. We asked observers to judge coherence in two conditions. In one, S/(L+M) and L/(L+M) gratings were physically the same speed. In the other, the two gratings had perceptually matched speeds. We found that the relative incoherence of cardinal gratings is the same whether gratings are physically or perceptually matched in speed. Thus our hypothesis was firmly contradicted. In a control condition, observers were asked to judge the coherence of stationary gratings. Interestingly, the difference in judged coherence between cardinal and intermediate gratings remained as strong as it was when the gratings moved. Our results suggest a possible alternative interpretation of Krauskopf and Farell's result: the processes of object segregation may precede the analysis of the motion of chromatic gratings, and the same grouping signals may prompt object segregation in the stationary and moving cases. Copyright © 2016 Elsevier Ltd. All rights reserved.
Performance testing of an off-plane reflection grating and silicon pore optic spectrograph at PANTER
Marlowe, Hannah; McEntaffer, Randall L.; Allured, Ryan; DeRoo, Casey T.; Donovan, Benjamin D.; Miles, Drew M.; Tutt, James H.; Burwitz, Vadim; Menz, Benedikt; Hartner, Gisela D.; Smith, Randall K.; Cheimets, Peter; Hertz, Edward; Bookbinder, Jay A.; Günther, Ramses; Yanson, Alex; Vacanti, Giuseppe; Ackermann, Marcelo
2015-10-01
An x-ray spectrograph consisting of aligned, radially ruled off-plane reflection gratings and silicon pore optics (SPO) was tested at the Max Planck Institute for Extraterrestrial Physics PANTER x-ray test facility. SPO is a test module for the proposed Arcus mission, which will also feature aligned off-plane reflection gratings. This test is the first time two off-plane gratings were actively aligned to each other and with an SPO to produce an overlapped spectrum. We report the performance of the complete spectrograph utilizing the aligned gratings module and plans for future development.
Varied line-space gratings: past, present and future
International Nuclear Information System (INIS)
Hettrick, M.C.
1985-08-01
A classically ruled diffraction grating consists of grooves which are equidistant, straight and parallel. Conversely, the so-called ''holographic'' grating (formed by the interfering waves of coherent visible light), although severely constrained by the recording wavelength and recording geometry, has grooves which are typically neither equidistant, straight nor parallel. In contrast, a varied line-space (VLS) grating, in common nomenclature, is a design in which the groove positions are relatively unconstrained yet possess sufficient symmetry to permit mechanical ruling. Such seemingly exotic gratings are no longer only a theoretical curiosity, but have been ruled and used in a wide variety of applications. These include: (1) aberration-corrected normal incidence concave gratings for Seya-Namioka monochromators and optical de-multiplexers, (2) flat-field grazing incidence concave gratings for plasma diagnostics, (3) aberration-corrected grazing incidence plane gratings for space-borne spectrometers, (4) focusing grazing incidence plane grating for synchrotron radiation monochromators, and (5) wavefront generators for visible interferometry of optical surfaces (particularly aspheres). Future prospects of VLS gratings as dispersing elements, wavefront correctors and beamsplitters appear promising. The author discusses the history of VLS gratings, their present applications, and their potential in the future. 61 refs., 24 figs
M-Polynomials and topological indices of V-Phenylenic Nanotubes and Nanotori.
Kwun, Young Chel; Munir, Mobeen; Nazeer, Waqas; Rafique, Shazia; Min Kang, Shin
2017-08-18
V-Phenylenic nanotubes and nanotori are most comprehensively studied nanostructures due to widespread applications in the production of catalytic, gas-sensing and corrosion-resistant materials. Representing chemical compounds with M-polynomial is a recent idea and it produces nice formulas of degree-based topological indices which correlate chemical properties of the material under investigation. These indices are used in the development of quantitative structure-activity relationships (QSARs) in which the biological activity and other properties of molecules like boiling point, stability, strain energy etc. are correlated with their structures. In this paper, we determine general closed formulae for M-polynomials of V-Phylenic nanotubes and nanotori. We recover important topological degree-based indices. We also give different graphs of topological indices and their relations with the parameters of structures.
Fabrication of Polymer Optical Fibre (POF Gratings
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Yanhua Luo
2017-03-01
Full Text Available Gratings inscribed in polymer optical fibre (POF have attracted remarkable interest for many potential applications due to their distinctive properties. This paper overviews the current state of fabrication of POF gratings since their first demonstration in 1999. In particular we summarize and discuss POF materials, POF photosensitivity, techniques and issues of fabricating POF gratings, as well as various types of POF gratings.
on the performance of Autoregressive Moving Average Polynomial
African Journals Online (AJOL)
Timothy Ademakinwa
Distributed Lag (PDL) model, Autoregressive Polynomial Distributed Lag ... Moving Average Polynomial Distributed Lag (ARMAPDL) model. ..... Global Journal of Mathematics and Statistics. Vol. 1. ... Business and Economic Research Center.
Application of polynomial preconditioners to conservation laws
Geurts, Bernardus J.; van Buuren, R.; Lu, H.
2000-01-01
Polynomial preconditioners which are suitable in implicit time-stepping methods for conservation laws are reviewed and analyzed. The preconditioners considered are either based on a truncation of a Neumann series or on Chebyshev polynomials for the inverse of the system-matrix. The latter class of
Symmetric functions and orthogonal polynomials
Macdonald, I G
1997-01-01
One of the most classical areas of algebra, the theory of symmetric functions and orthogonal polynomials has long been known to be connected to combinatorics, representation theory, and other branches of mathematics. Written by perhaps the most famous author on the topic, this volume explains some of the current developments regarding these connections. It is based on lectures presented by the author at Rutgers University. Specifically, he gives recent results on orthogonal polynomials associated with affine Hecke algebras, surveying the proofs of certain famous combinatorial conjectures.
Applications of polynomial optimization in financial risk investment
Zeng, Meilan; Fu, Hongwei
2017-09-01
Recently, polynomial optimization has many important applications in optimization, financial economics and eigenvalues of tensor, etc. This paper studies the applications of polynomial optimization in financial risk investment. We consider the standard mean-variance risk measurement model and the mean-variance risk measurement model with transaction costs. We use Lasserre's hierarchy of semidefinite programming (SDP) relaxations to solve the specific cases. The results show that polynomial optimization is effective for some financial optimization problems.
Polynomially Riesz elements | Živković-Zlatanović | Quaestiones ...
African Journals Online (AJOL)
A Banach algebra element ɑ ∈ A is said to be "polynomially Riesz", relative to the homomorphism T : A → B, if there exists a nonzero complex polynomial p(z) such that the image Tp ∈ B is quasinilpotent. Keywords: Homomorphism of Banach algebras, polynomially Riesz element, Fredholm spectrum, Browder element, ...
Fiber facet gratings for high power fiber lasers
Vanek, Martin; Vanis, Jan; Baravets, Yauhen; Todorov, Filip; Ctyroky, Jiri; Honzatko, Pavel
2017-12-01
We numerically investigated the properties of diffraction gratings designated for fabrication on the facet of an optical fiber. The gratings are intended to be used in high-power fiber lasers as mirrors either with a low or high reflectivity. The modal reflectance of low reflectivity polarizing grating has a value close to 3% for TE mode while it is significantly suppressed for TM mode. Such a grating can be fabricated on laser output fiber facet. The polarizing grating with high modal reflectance is designed as a leaky-mode resonant diffraction grating. The grating can be etched in a thin layer of high index dielectric which is sputtered on fiber facet. We used refractive index of Ta2O5 for such a layer. We found that modal reflectance can be close to 0.95 for TE polarization and polarization extinction ratio achieves 18 dB. Rigorous coupled wave analysis was used for fast optimization of grating parameters while aperiodic rigorous coupled wave analysis, Fourier modal method and finite difference time domain method were compared and used to compute modal reflectance of designed gratings.
Symmetric integrable-polynomial factorization for symplectic one-turn-map tracking
International Nuclear Information System (INIS)
Shi, Jicong
1993-01-01
It was found that any homogeneous polynomial can be written as a sum of integrable polynomials of the same degree which Lie transformations can be evaluated exactly. By utilizing symplectic integrators, an integrable-polynomial factorization is developed to convert a symplectic map in the form of Dragt-Finn factorization into a product of Lie transformations associated with integrable polynomials. A small number of factorization bases of integrable polynomials enable one to use high order symplectic integrators so that the high-order spurious terms can be greatly suppressed. A symplectic map can thus be evaluated with desired accuracy
Polynomial fuzzy model-based approach for underactuated surface vessels
DEFF Research Database (Denmark)
Khooban, Mohammad Hassan; Vafamand, Navid; Dragicevic, Tomislav
2018-01-01
The main goal of this study is to introduce a new polynomial fuzzy model-based structure for a class of marine systems with non-linear and polynomial dynamics. The suggested technique relies on a polynomial Takagi–Sugeno (T–S) fuzzy modelling, a polynomial dynamic parallel distributed compensation...... surface vessel (USV). Additionally, in order to overcome the USV control challenges, including the USV un-modelled dynamics, complex nonlinear dynamics, external disturbances and parameter uncertainties, the polynomial fuzzy model representation is adopted. Moreover, the USV-based control structure...... and a sum-of-squares (SOS) decomposition. The new proposed approach is a generalisation of the standard T–S fuzzy models and linear matrix inequality which indicated its effectiveness in decreasing the tracking time and increasing the efficiency of the robust tracking control problem for an underactuated...
Connections between the matching and chromatic polynomials
Directory of Open Access Journals (Sweden)
E. J. Farrell
1992-01-01
Full Text Available The main results established are (i a connection between the matching and chromatic polynomials and (ii a formula for the matching polynomial of a general complement of a subgraph of a graph. Some deductions on matching and chromatic equivalence and uniqueness are made.
On Generalisation of Polynomials in Complex Plane
Directory of Open Access Journals (Sweden)
Maslina Darus
2010-01-01
Full Text Available The generalised Bell and Laguerre polynomials of fractional-order in complex z-plane are defined. Some properties are studied. Moreover, we proved that these polynomials are univalent solutions for second order differential equations. Also, the Laguerre-type of some special functions are introduced.
Technique for image interpolation using polynomial transforms
Escalante Ramírez, B.; Martens, J.B.; Haskell, G.G.; Hang, H.M.
1993-01-01
We present a new technique for image interpolation based on polynomial transforms. This is an image representation model that analyzes an image by locally expanding it into a weighted sum of orthogonal polynomials. In the discrete case, the image segment within every window of analysis is
Okounkov's BC-Type Interpolation Macdonald Polynomials and Their q=1 Limit
Koornwinder, T.H.
2015-01-01
This paper surveys eight classes of polynomials associated with A-type and BC-type root systems: Jack, Jacobi, Macdonald and Koornwinder polynomials and interpolation (or shifted) Jack and Macdonald polynomials and their BC-type extensions. Among these the BC-type interpolation Jack polynomials were
Complex Polynomial Vector Fields
DEFF Research Database (Denmark)
Dias, Kealey
vector fields. Since the class of complex polynomial vector fields in the plane is natural to consider, it is remarkable that its study has only begun very recently. There are numerous fundamental questions that are still open, both in the general classification of these vector fields, the decomposition...... of parameter spaces into structurally stable domains, and a description of the bifurcations. For this reason, the talk will focus on these questions for complex polynomial vector fields.......The two branches of dynamical systems, continuous and discrete, correspond to the study of differential equations (vector fields) and iteration of mappings respectively. In holomorphic dynamics, the systems studied are restricted to those described by holomorphic (complex analytic) functions...
Interlacing of zeros of quasi-orthogonal meixner polynomials | Driver ...
African Journals Online (AJOL)
... interlacing of zeros of quasi-orthogonal Meixner polynomials Mn(x;β; c) with the zeros of their nearest orthogonal counterparts Mt(x;β + k; c), l; n ∈ ℕ, k ∈ {1; 2}; is also discussed. Mathematics Subject Classication (2010): 33C45, 42C05. Key words: Discrete orthogonal polynomials, quasi-orthogonal polynomials, Meixner
Discriminants and functional equations for polynomials orthogonal on the unit circle
International Nuclear Information System (INIS)
Ismail, M.E.H.; Witte, N.S.
2000-01-01
We derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and q-difference equations for these polynomials. A general functional equation is found which allows one to relate the zeros of the orthogonal polynomials to the stationary values of an explicit quasi-energy and implies recurrences on the orthogonal polynomial coefficients. We also evaluate the discriminants and quantized discriminants of polynomials orthogonal on the unit circle
Contributions to fuzzy polynomial techniques for stability analysis and control
Pitarch Pérez, José Luis
2014-01-01
The present thesis employs fuzzy-polynomial control techniques in order to improve the stability analysis and control of nonlinear systems. Initially, it reviews the more extended techniques in the field of Takagi-Sugeno fuzzy systems, such as the more relevant results about polynomial and fuzzy polynomial systems. The basic framework uses fuzzy polynomial models by Taylor series and sum-of-squares techniques (semidefinite programming) in order to obtain stability guarantees...
Metrology measurements for large-aperture VPH gratings
Zheng, Jessica R.; Gers, Luke; Heijmans, Jeroen
2013-09-01
The High Efficiency and Resolution Multi Element Spectrograph (HERMES) for the Australian Astronomical Observatory (AAO) uses four large aperture, high angle of incidence volume phase holographic gratings (VPHG) for high resolution `Galactic archaeology' spectroscopy. The large clear aperture, the high diffraction efficiency, the line frequency homogeneity, and mosaic alignment made manufacturing and testing challenging. We developed new metrology systems at the AAO to verify the performance of these VPH gratings. The measured diffraction efficiencies and line frequency of the VPH gratings received so far meet the vendor's provided data. The wavefront quality for the Blue VPH grating is good but the Green and Red VPH gratings need to be post polishing.
On the Lorentz degree of a product of polynomials
Ait-Haddou, Rachid
2015-01-01
In this note, we negatively answer two questions of T. Erdélyi (1991, 2010) on possible lower bounds on the Lorentz degree of product of two polynomials. We show that the correctness of one question for degree two polynomials is a direct consequence of a result of Barnard et al. (1991) on polynomials with nonnegative coefficients.
Strong result for real zeros of random algebraic polynomials
Directory of Open Access Journals (Sweden)
T. Uno
2001-01-01
Full Text Available An estimate is given for the lower bound of real zeros of random algebraic polynomials whose coefficients are non-identically distributed dependent Gaussian random variables. Moreover, our estimated measure of the exceptional set, which is independent of the degree of the polynomials, tends to zero as the degree of the polynomial tends to infinity.
Large degree asymptotics of generalized Bessel polynomials
J.L. López; N.M. Temme (Nico)
2011-01-01
textabstractAsymptotic expansions are given for large values of $n$ of the generalized Bessel polynomials $Y_n^\\mu(z)$. The analysis is based on integrals that follow from the generating functions of the polynomials. A new simple expansion is given that is valid outside a compact neighborhood of the
Waveguide silicon nitride grating coupler
Litvik, Jan; Dolnak, Ivan; Dado, Milan
2016-12-01
Grating couplers are one of the most used elements for coupling of light between optical fibers and photonic integrated components. Silicon-on-insulator platform provides strong confinement of light and allows high integration. In this work, using simulations we have designed a broadband silicon nitride surface grating coupler. The Fourier-eigenmode expansion and finite difference time domain methods are utilized in design optimization of grating coupler structure. The fully, single etch step grating coupler is based on a standard silicon-on-insulator wafer with 0.55 μm waveguide Si3N4 layer. The optimized structure at 1550 nm wavelength yields a peak coupling efficiency -2.6635 dB (54.16%) with a 1-dB bandwidth up to 80 nm. It is promising way for low-cost fabrication using complementary metal-oxide- semiconductor fabrication process.
Linear operator pencils on Lie algebras and Laurent biorthogonal polynomials
International Nuclear Information System (INIS)
Gruenbaum, F A; Vinet, Luc; Zhedanov, Alexei
2004-01-01
We study operator pencils on generators of the Lie algebras sl 2 and the oscillator algebra. These pencils are linear in a spectral parameter λ. The corresponding generalized eigenvalue problem gives rise to some sets of orthogonal polynomials and Laurent biorthogonal polynomials (LBP) expressed in terms of the Gauss 2 F 1 and degenerate 1 F 1 hypergeometric functions. For special choices of the parameters of the pencils, we identify the resulting polynomials with the Hendriksen-van Rossum LBP which are widely believed to be the biorthogonal analogues of the classical orthogonal polynomials. This places these examples under the umbrella of the generalized bispectral problem which is considered here. Other (non-bispectral) cases give rise to some 'nonclassical' orthogonal polynomials including Tricomi-Carlitz and random-walk polynomials. An application to solutions of relativistic Toda chain is considered
Papatryfonos, Konstantinos; Saladukha, Dzianis; Merghem, Kamel; Joshi, Siddharth; Lelarge, Francois; Bouchoule, Sophie; Kazazis, Dimitrios; Guilet, Stephane; Le Gratiet, Luc; Ochalski, Tomasz J.; Huyet, Guillaume; Martinez, Anthony; Ramdane, Abderrahim
2017-02-01
Single-mode diode lasers on an InP(001) substrate have been developed using InAs/In0.53Ga0.47As quantum dash (Qdash) active regions and etched lateral Bragg gratings. The lasers have been designed to operate at wavelengths near 2 μm and exhibit a threshold current of 65 mA for a 600 μm long cavity, and a room temperature continuous wave output power per facet >5 mW. Using our novel growth approach based on the low ternary In0.53Ga0.47As barriers, we also demonstrate ridge-waveguide lasers emitting up to 2.1 μm and underline the possibilities for further pushing the emission wavelength out towards longer wavelengths with this material system. By introducing experimentally the concept of high-duty-cycle lateral Bragg gratings, a side mode suppression ratio of >37 dB has been achieved, owing to an appreciably increased grating coupling coefficient of κ ˜ 40 cm-1. These laterally coupled distributed feedback (LC-DFB) lasers combine the advantage of high and well-controlled coupling coefficients achieved in conventional DFB lasers, with the regrowth-free fabrication process of lateral gratings, and exhibit substantially lower optical losses compared to the conventional metal-based LC-DFB lasers.
Higher order branching of periodic orbits from polynomial isochrones
Directory of Open Access Journals (Sweden)
B. Toni
1999-09-01
Full Text Available We discuss the higher order local bifurcations of limit cycles from polynomial isochrones (linearizable centers when the linearizing transformation is explicitly known and yields a polynomial perturbation one-form. Using a method based on the relative cohomology decomposition of polynomial one-forms complemented with a step reduction process, we give an explicit formula for the overall upper bound of branch points of limit cycles in an arbitrary $n$ degree polynomial perturbation of the linear isochrone, and provide an algorithmic procedure to compute the upper bound at successive orders. We derive a complete analysis of the nonlinear cubic Hamiltonian isochrone and show that at most nine branch points of limit cycles can bifurcate in a cubic polynomial perturbation. Moreover, perturbations with exactly two, three, four, six, and nine local families of limit cycles may be constructed.
Access Platforms for Offshore Wind Turbines Using Gratings
DEFF Research Database (Denmark)
Andersen, Thomas Lykke; Rasmussen, Michael R.
2008-01-01
The paper deals with forces generated by a stationary jet on different types of gratings and a solid plate. The force reduction factors for the different gratings compared to the solid plate mainly depend on the porosity of the gratings, but the geometry of the grating is also of some importance........ The derived reduction factors are expected to be applicable to design of offshore wind turbine access platforms with gratings where slamming also is an important factor....
On the estimation of the degree of regression polynomial
International Nuclear Information System (INIS)
Toeroek, Cs.
1997-01-01
The mathematical functions most commonly used to model curvature in plots are polynomials. Generally, the higher the degree of the polynomial, the more complex is the trend that its graph can represent. We propose a new statistical-graphical approach based on the discrete projective transformation (DPT) to estimating the degree of polynomial that adequately describes the trend in the plot
Precise rotational alignment of x-ray transmission diffraction gratings
International Nuclear Information System (INIS)
Hill, S.L.
1988-01-01
Gold transmission diffraction gratings used for x-ray spectroscopy must sometimes be rotationally aligned to the axis of a diagnostic instrument to within sub-milliradian accuracy. We have fabricated transmission diffraction gratings with high line-densities (grating period of 200 and 300 nm) using uv holographic and x-ray lithography. Since the submicron features of the gratings are not optically visible, precision alignment is time consuming and difficult to verify in situ. We have developed a technique to write an optically visible alignment pattern onto these gratings using a scanning electron microscope (SEM). At high magnification (15000 X) several submicron lines of the grating are observable in the SEM, making it possible to write an alignment pattern parallel to the grating lines in an electron-beam-sensitive coating that overlays the grating. We create an alignment pattern by following a 1-cm-long grating line using the SEM's joystick-controlled translation stage. By following the same grating line we are assured the traveled direction of the SEM electron beam is parallel to the grating to better than 10 μradian. The electron-beam-exposed line-width can be large (5 to 15 μm wide) depending on the SEM magnification, and is therefore optically visible. The exposed pattern is eventually made a permanent feature of the grating by ion beam etching or gold electroplating. The pattern can be used to accurately align the grating to the axis of a diagnostic instrument. More importantly, the alignment of the grating can be quickly verified in situ
On Some Extensions of Szasz Operators Including Boas-Buck-Type Polynomials
Directory of Open Access Journals (Sweden)
Sezgin Sucu
2012-01-01
Full Text Available This paper is concerned with a new sequence of linear positive operators which generalize Szasz operators including Boas-Buck-type polynomials. We establish a convergence theorem for these operators and give the quantitative estimation of the approximation process by using a classical approach and the second modulus of continuity. Some explicit examples of our operators involving Laguerre polynomials, Charlier polynomials, and Gould-Hopper polynomials are given. Moreover, a Voronovskaya-type result is obtained for the operators containing Gould-Hopper polynomials.
On associated polynomials and decay rates for birth-death processes
van Doorn, Erik A.
2001-01-01
We consider sequences of orthogonal polynomials and pursue the question of how (partial) knowledge of the orthogonalizing measure for the {\\it associated polynomials} can lead to information about the orthogonalizing measure for the original polynomials. In particular, we relate the supports of the
On associated polynomials and decay rates for birth-death processes
van Doorn, Erik A.
2003-01-01
We consider sequences of orthogonal polynomials and pursue the question of how (partial) knowledge of the orthogonalizing measure for the associated polynomials can lead to information about the orthogonalizing measure for the original polynomials. In particular, we relate the supports of the two
Energy Technology Data Exchange (ETDEWEB)
Alessi, D. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
2016-11-01
Pulse compressors for ultrafast lasers have been identified as a technology gap in the push towards high peak power systems with high average powers for industrial and scientific applications. Gratings for ultrashort (sub-150fs) pulse compressors are metallic and can absorb a significant percentage of laser energy resulting in up to 40% loss as well as thermal issues which degrade on-target performance. We have developed a next generation gold grating technology which we have scaled to the petawatt-size. This resulted in improvements in efficiency, uniformity and processing as compared to previous substrate etched gratings for high average power. This new design has a deposited dielectric material for the grating ridge rather than etching directly into the glass substrate. It has been observed that average powers as low as 1W in a compressor can cause distortions in the on-target beam. We have developed and tested a method of actively cooling diffraction gratings which, in the case of gold gratings, can support a petawatt peak power laser with up to 600W average power. We demonstrated thermo-mechanical modeling of a grating in its use environment and benchmarked with experimental measurement. Multilayer dielectric (MLD) gratings are not yet used for these high peak power, ultrashort pulse durations due to their design challenges. We have designed and fabricated broad bandwidth, low dispersion MLD gratings suitable for delivering 30 fs pulses at high average power. This new grating design requires the use of a novel Out Of Plane (OOP) compressor, which we have modeled, designed, built and tested. This prototype compressor yielded a transmission of 90% for a pulse with 45 nm bandwidth, and free of spatial and angular chirp. In order to evaluate gratings and compressors built in this project we have commissioned a joule-class ultrafast Ti:Sapphire laser system. Combining the grating cooling and MLD technologies developed here could enable petawatt laser systems to
Some Polynomials Associated with the r-Whitney Numbers
Indian Academy of Sciences (India)
26
Abstract. In the present article we study three families of polynomials associated with ... [29, 39] for their relations with the Bernoulli and generalized Bernoulli polynomials and ... generating functions in a similar way as in the classical cases.
The Bessel polynomials and their differential operators
International Nuclear Information System (INIS)
Onyango Otieno, V.P.
1987-10-01
Differential operators associated with the ordinary and the generalized Bessel polynomials are defined. In each case the commutator bracket is constructed and shows that the differential operators associated with the Bessel polynomials and their generalized form are not commutative. Some applications of these operators to linear differential equations are also discussed. (author). 4 refs
High-mechanical-strength single-pulse draw tower gratings
Rothhardt, Manfred W.; Chojetzki, Christoph; Mueller, Hans Rainer
2004-11-01
The inscription of fiber Bragg gratings during the drawing process is a very useful method to realize sensor arrays with high numbers of gratings and excellent mechanical strength and also type II gratings with high temperature stability. Results of single pulse grating arrays with numbers up to 100 and definite wavelengths and positions for sensor applications were achieved at 1550 nm and 830 nm using new photosensitive fibers developed in IPHT. Single pulse type I gratings at 1550 nm with more than 30% reflectivity were shown first time to our knowledge. The mechanical strength of this fiber with an Ormocer coating with those single pulse gratings is the same like standard telecom fibers. Weibull plots of fiber tests will be shown. At 830 nm we reached more than 10% reflectivity with single pulse writing during the fiber drawing in photosensitive fibers with less than 16 dB/km transmission loss. These gratings are useful for stress and vibration sensing applications. Type II gratings with reflectivity near 100% and smooth spectral shape and spectral width of about 1 nm are temperature stable up to 1200 K for short time. They are also realized in the fiber drawing process. These gratings are useful for temperature sensor applications.
Conference on Commutative rings, integer-valued polynomials and polynomial functions
Frisch, Sophie; Glaz, Sarah; Commutative Algebra : Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions
2014-01-01
This volume presents a multi-dimensional collection of articles highlighting recent developments in commutative algebra. It also includes an extensive bibliography and lists a substantial number of open problems that point to future directions of research in the represented subfields. The contributions cover areas in commutative algebra that have flourished in the last few decades and are not yet well represented in book form. Highlighted topics and research methods include Noetherian and non- Noetherian ring theory as well as integer-valued polynomials and functions. Specific topics include: · Homological dimensions of Prüfer-like rings · Quasi complete rings · Total graphs of rings · Properties of prime ideals over various rings · Bases for integer-valued polynomials · Boolean subrings · The portable property of domains · Probabilistic topics in Intn(D) · Closure operations in Zariski-Riemann spaces of valuation domains · Stability of do...
VCSELs and silicon light sources exploiting SOI grating mirrors
DEFF Research Database (Denmark)
Chung, Il-Sug; Mørk, Jesper
2012-01-01
In this talk, novel vertical-cavity laser structure consisting of a dielectric Bragg reflector, a III-V active region, and a high-index-contrast grating made in the Si layer of a silicon-on-insulator (SOI) wafer will be presented. In the Si light source version of this laser structure, the SOI...... the Bragg reflector. Numerical simulations show that both the silicon light source and the VCSEL exploiting SOI grating mirrors have superior performances, compared to existing silicon light sources and long wavelength VCSELs. These devices are highly adequate for chip-level optical interconnects as well...
Fundamental limit of light trapping in grating structures
Yu, Zongfu
2010-08-11
We use a rigorous electromagnetic approach to analyze the fundamental limit of light-trapping enhancement in grating structures. This limit can exceed the bulk limit of 4n 2, but has significant angular dependency. We explicitly show that 2D gratings provide more enhancement than 1D gratings. We also show the effects of the grating profile’s symmetry on the absorption enhancement limit. Numerical simulations are applied to support the theory. Our findings provide general guidance for the design of grating structures for light-trapping solar cells.
Sibling curves of quadratic polynomials | Wiggins | Quaestiones ...
African Journals Online (AJOL)
Sibling curves were demonstrated in [1, 2] as a novel way to visualize the zeroes of real valued functions. In [3] it was shown that a polynomial of degree n has n sibling curves. This paper focuses on the algebraic and geometric properites of the sibling curves of real and complex quadratic polynomials. Key words: Quadratic ...
Dual exponential polynomials and linear differential equations
Wen, Zhi-Tao; Gundersen, Gary G.; Heittokangas, Janne
2018-01-01
We study linear differential equations with exponential polynomial coefficients, where exactly one coefficient is of order greater than all the others. The main result shows that a nontrivial exponential polynomial solution of such an equation has a certain dual relationship with the maximum order coefficient. Several examples illustrate our results and exhibit possibilities that can occur.
EUV properties of two diffraction gratings
International Nuclear Information System (INIS)
Cotton, D.; Chakrabarti, S.; Edelstein, J.; Pranke, J.; Christensen, A.B.
1988-01-01
The efficiency and scattering characteristics of a mechanically ruled grating (MRG) and a holographically ruled grating (HRG) are presented. One of these gratings will be employed in the Extreme Ultraviolet Spectrometer, an instrument of the Remote Atmospheric and Ionospheric Detector System to be flown aboard a TIROS satellite in 1991. The HRG showed much less Lyman alpha scattering, while the MRG had the better efficiency over most of the spectral range covered. 8 refs
Hybrid grating reflectors: Origin of ultrabroad stopband
DEFF Research Database (Denmark)
Park, Gyeong Cheol; Taghizadeh, Alireza; Chung, Il-Sug
2016-01-01
Hybrid grating (HG) reflectors with a high-refractive-index cap layer added onto a high contrast grating (HCG) provide a high reflectance close to 100% over a broader wavelength range than HCGs. The combination of a cap layer and a grating layer brings a strong Fabry-Perot (FP) resonance as well ...
Generalized Freud's equation and level densities with polynomial
Indian Academy of Sciences (India)
Home; Journals; Pramana – Journal of Physics; Volume 81; Issue 2. Generalized Freud's equation and level densities with polynomial potential. Akshat Boobna Saugata Ghosh. Research Articles Volume 81 ... Keywords. Orthogonal polynomial; Freud's equation; Dyson–Mehta method; methods of resolvents; level density.
Polynomial fuzzy observer designs: a sum-of-squares approach.
Tanaka, Kazuo; Ohtake, Hiroshi; Seo, Toshiaki; Tanaka, Motoyasu; Wang, Hua O
2012-10-01
This paper presents a sum-of-squares (SOS) approach to polynomial fuzzy observer designs for three classes of polynomial fuzzy systems. The proposed SOS-based framework provides a number of innovations and improvements over the existing linear matrix inequality (LMI)-based approaches to Takagi-Sugeno (T-S) fuzzy controller and observer designs. First, we briefly summarize previous results with respect to a polynomial fuzzy system that is a more general representation of the well-known T-S fuzzy system. Next, we propose polynomial fuzzy observers to estimate states in three classes of polynomial fuzzy systems and derive SOS conditions to design polynomial fuzzy controllers and observers. A remarkable feature of the SOS design conditions for the first two classes (Classes I and II) is that they realize the so-called separation principle, i.e., the polynomial fuzzy controller and observer for each class can be separately designed without lack of guaranteeing the stability of the overall control system in addition to converging state-estimation error (via the observer) to zero. Although, for the last class (Class III), the separation principle does not hold, we propose an algorithm to design polynomial fuzzy controller and observer satisfying the stability of the overall control system in addition to converging state-estimation error (via the observer) to zero. All the design conditions in the proposed approach can be represented in terms of SOS and are symbolically and numerically solved via the recently developed SOSTOOLS and a semidefinite-program solver, respectively. To illustrate the validity and applicability of the proposed approach, three design examples are provided. The examples demonstrate the advantages of the SOS-based approaches for the existing LMI approaches to T-S fuzzy observer designs.
Nanostructure Diffraction Gratings for Integrated Spectroscopy and Sensing
Guo, Junpeng (Inventor)
2016-01-01
The present disclosure pertains to metal or dielectric nanostructures of the subwavelength scale within the grating lines of optical diffraction gratings. The nanostructures have surface plasmon resonances or non-plasmon optical resonances. A linear photodetector array is used to capture the resonance spectra from one of the diffraction orders. The combined nanostructure super-grating and photodetector array eliminates the use of external optical spectrometers for measuring surface plasmon or optical resonance frequency shift caused by the presence of chemical and biological agents. The nanostructure super-gratings can be used for building integrated surface enhanced Raman scattering (SERS) spectrometers. The nanostructures within the diffraction grating lines enhance Raman scattering signal light while the diffraction grating pattern of the nanostructures diffracts Raman scattering light to different directions of propagation according to their wavelengths. Therefore, the nanostructure super-gratings allows for the use of a photodetector array to capture the surface enhanced Raman scattering spectra.
Solutions of interval type-2 fuzzy polynomials using a new ranking method
Rahman, Nurhakimah Ab.; Abdullah, Lazim; Ghani, Ahmad Termimi Ab.; Ahmad, Noor'Ani
2015-10-01
A few years ago, a ranking method have been introduced in the fuzzy polynomial equations. Concept of the ranking method is proposed to find actual roots of fuzzy polynomials (if exists). Fuzzy polynomials are transformed to system of crisp polynomials, performed by using ranking method based on three parameters namely, Value, Ambiguity and Fuzziness. However, it was found that solutions based on these three parameters are quite inefficient to produce answers. Therefore in this study a new ranking method have been developed with the aim to overcome the inherent weakness. The new ranking method which have four parameters are then applied in the interval type-2 fuzzy polynomials, covering the interval type-2 of fuzzy polynomial equation, dual fuzzy polynomial equations and system of fuzzy polynomials. The efficiency of the new ranking method then numerically considered in the triangular fuzzy numbers and the trapezoidal fuzzy numbers. Finally, the approximate solutions produced from the numerical examples indicate that the new ranking method successfully produced actual roots for the interval type-2 fuzzy polynomials.
Extended asymmetric-cut multilayer X-ray gratings.
Prasciolu, Mauro; Haase, Anton; Scholze, Frank; Chapman, Henry N; Bajt, Saša
2015-06-15
The fabrication and characterization of a large-area high-dispersion blazed grating for soft X-rays based on an asymmetric-cut multilayer structure is reported. An asymmetric-cut multilayer structure acts as a perfect blazed grating of high efficiency that exhibits a single diffracted order, as described by dynamical diffraction throughout the depth of the layered structure. The maximum number of grating periods created by cutting a multilayer deposited on a flat substrate is equal to the number of layers deposited, which limits the size of the grating. The size limitation was overcome by depositing the multilayer onto a substrate which itself is a coarse blazed grating and then polish it flat to reveal the uniformly spaced layers of the multilayer. The number of deposited layers required is such that the multilayer thickness exceeds the step height of the substrate structure. The method is demonstrated by fabricating a 27,060 line pairs per mm blazed grating (36.95 nm period) that is repeated every 3,200 periods by the 120-μm period substrate structure. This preparation technique also relaxes the requirements on stress control and interface roughness of the multilayer film. The dispersion and efficiency of the grating is demonstrated for soft X-rays of 13.2 nm wavelength.
Adaptable Diffraction Gratings With Wavefront Transformation
Iazikov, Dmitri; Mossberg, Thomas W.; Greiner, Christoph M.
2010-01-01
Diffraction gratings are optical components with regular patterns of grooves, which angularly disperse incoming light by wavelength. Traditional diffraction gratings have static planar, concave, or convex surfaces. However, if they could be made so that they can change the surface curvature at will, then they would be able to focus on particular segments, self-calibrate, or perform fine adjustments. This innovation creates a diffraction grating on a deformable surface. This surface could be bent at will, resulting in a dynamic wavefront transformation. This allows for self-calibration, compensation for aberrations, enhancing image resolution in a particular area, or performing multiple scans using different wavelengths. A dynamic grating gives scientists a new ability to explore wavefronts from a variety of viewpoints.
An overview on polynomial approximation of NP-hard problems
Directory of Open Access Journals (Sweden)
Paschos Vangelis Th.
2009-01-01
Full Text Available The fact that polynomial time algorithm is very unlikely to be devised for an optimal solving of the NP-hard problems strongly motivates both the researchers and the practitioners to try to solve such problems heuristically, by making a trade-off between computational time and solution's quality. In other words, heuristic computation consists of trying to find not the best solution but one solution which is 'close to' the optimal one in reasonable time. Among the classes of heuristic methods for NP-hard problems, the polynomial approximation algorithms aim at solving a given NP-hard problem in poly-nomial time by computing feasible solutions that are, under some predefined criterion, as near to the optimal ones as possible. The polynomial approximation theory deals with the study of such algorithms. This survey first presents and analyzes time approximation algorithms for some classical examples of NP-hard problems. Secondly, it shows how classical notions and tools of complexity theory, such as polynomial reductions, can be matched with polynomial approximation in order to devise structural results for NP-hard optimization problems. Finally, it presents a quick description of what is commonly called inapproximability results. Such results provide limits on the approximability of the problems tackled.
Diffraction efficiency of radially-profiled off-plane reflection gratings
Miles, Drew M.; Tutt, James H.; DeRoo, Casey T.; Marlowe, Hannah; Peterson, Thomas J.; McEntaffer, Randall L.; Menz, Benedikt; Burwitz, Vadim; Hartner, Gisela; Laubis, Christian; Scholze, Frank
2015-09-01
Future X-ray missions will require gratings with high throughput and high spectral resolution. Blazed off-plane reflection gratings are capable of meeting these demands. A blazed grating profile optimizes grating efficiency, providing higher throughput to one side of zero-order on the arc of diffraction. This paper presents efficiency measurements made in the 0.3 - 1.5 keV energy band at the Physikalisch-Technische Bundesanstalt (PTB) BESSY II facility for three holographically-ruled gratings, two of which are blazed. Each blazed grating was tested in both the Littrow configuration and anti-Littrow configuration in order to test the alignment sensitivity of these gratings with regard to throughput. This paper outlines the procedure of the grating experiment performed at BESSY II and discuss the resulting efficiency measurements across various energies. Experimental results are generally consistent with theory and demonstrate that the blaze does increase throughput to one side of zero-order. However, the total efficiency of the non-blazed, sinusoidal grating is greater than that of the blazed gratings, which suggests that the method of manufacturing these blazed profiles fails to produce facets with the desired level of precision. Finally, evidence of a successful blaze implementation from first diffraction results of prototype blazed gratings produce via a new fabrication technique at the University of Iowa are presented.
Marlowe, Hannah; McEntaffer, Randall L.; Allured, Ryan; DeRoo, Casey; Miles, Drew M.; Donovan, Benjamin D.; Tutt, James H.; Burwitz, Vadim; Menz, Benedikt; Hartner, Gisela D.; Smith, Randall K.; Günther, Ramses; Yanson, Alex; Vacanti, Giuseppe; Ackermann, Marcelo
2015-05-01
An X-ray spectrograph consisting of aligned, radially ruled off-plane reflection gratings and silicon pore optics (SPO) was tested at the Max Planck Institute for extraterrestrial Physics PANTER X-ray test facility. The SPO is a test module for the proposed Arcus mission, which will also feature aligned off-plane reflection gratings. This test is the first time two off-plane gratings were actively aligned to each other and with a SPO to produce an overlapped spectrum. We report the performance of the complete spectrograph utilizing the aligned gratings module and plans for future development.
A note on the zeros of Freud-Sobolev orthogonal polynomials
Moreno-Balcazar, Juan J.
2007-10-01
We prove that the zeros of a certain family of Sobolev orthogonal polynomials involving the Freud weight function e-x4 on are real, simple, and interlace with the zeros of the Freud polynomials, i.e., those polynomials orthogonal with respect to the weight function e-x4. Some numerical examples are shown.
About the solvability of matrix polynomial equations
Netzer, Tim; Thom, Andreas
2016-01-01
We study self-adjoint matrix polynomial equations in a single variable and prove existence of self-adjoint solutions under some assumptions on the leading form. Our main result is that any self-adjoint matrix polynomial equation of odd degree with non-degenerate leading form can be solved in self-adjoint matrices. We also study equations of even degree and equations in many variables.
An X-ray grazing incidence phase multilayer grating
Chernov, V A; Mytnichenko, S V
2001-01-01
An X-ray grazing incidence phase multilayer grating, representing a thin grating placed on a multilayer mirror, is proposed. A high efficiency of grating diffraction can be obtained by the possibility of changing the phase shift of the wave diffracted from the multilayer under the Bragg and total external reflection conditions. A grazing incidence phase multilayer grating consisting of Pt grating stripes on a Ni/C multilayer and optimized for the hard X-ray range was fabricated. Its diffraction properties were studied at photon energies of 7 and 8 keV. The obtained maximum value of the diffraction efficiency of the +1 grating order was 9% at 7 keV and 6.5% at 8 keV. The data obtained are in a rather good accordance with the theory.
Towards freeform curved blazed gratings using diamond machining
Bourgenot, C.; Robertson, D. J.; Stelter, D.; Eikenberry, S.
2016-07-01
Concave blazed gratings greatly simplify the architecture of spectrographs by reducing the number of optical components. The production of these gratings using diamond-machining offers practically no limits in the design of the grating substrate shape, with the possibility of making large sag freeform surfaces unlike the alternative and traditional method of holography and ion etching. In this paper, we report on the technological challenges and progress in the making of these curved blazed gratings using an ultra-high precision 5 axes Moore-Nanotech machine. We describe their implementation in an integral field unit prototype called IGIS (Integrated Grating Imaging Spectrograph) where freeform curved gratings are used as pupil mirrors. The goal is to develop the technologies for the production of the next generation of low-cost, compact, high performance integral field unit spectrometers.
Apodized grating coupler using fully-etched nanostructures
Wu, Hua; Li, Chong; Li, Zhi-Yong; Guo, Xia
2016-08-01
A two-dimensional apodized grating coupler for interfacing between single-mode fiber and photonic circuit is demonstrated in order to bridge the mode gap between the grating coupler and optical fiber. The grating grooves of the grating couplers are realized by columns of fully etched nanostructures, which are utilized to digitally tailor the effective refractive index of each groove in order to obtain the Gaussian-like output diffractive mode and then enhance the coupling efficiency. Compared with that of the uniform grating coupler, the coupling efficiency of the apodized grating coupler is increased by 4.3% and 5.7%, respectively, for the nanoholes and nanorectangles as refractive index tunes layer. Project supported by the National Natural Science Foundation of China (Grant Nos. 61222501, 61335004, and 61505003), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20111103110019), the Postdoctoral Science Foundation of Beijing Funded Project, China (Grant No. Q6002012201502), and the Science and Technology Research Project of Jiangxi Provincial Education Department, China (Grant No. GJJ150998).
Two polynomial representations of experimental design
Notari, Roberto; Riccomagno, Eva; Rogantin, Maria-Piera
2007-01-01
In the context of algebraic statistics an experimental design is described by a set of polynomials called the design ideal. This, in turn, is generated by finite sets of polynomials. Two types of generating sets are mostly used in the literature: Groebner bases and indicator functions. We briefly describe them both, how they are used in the analysis and planning of a design and how to switch between them. Examples include fractions of full factorial designs and designs for mixture experiments.
Stable piecewise polynomial vector fields
Directory of Open Access Journals (Sweden)
Claudio Pessoa
2012-09-01
Full Text Available Let $N={y>0}$ and $S={y<0}$ be the semi-planes of $mathbb{R}^2$ having as common boundary the line $D={y=0}$. Let $X$ and $Y$ be polynomial vector fields defined in $N$ and $S$, respectively, leading to a discontinuous piecewise polynomial vector field $Z=(X,Y$. This work pursues the stability and the transition analysis of solutions of $Z$ between $N$ and $S$, started by Filippov (1988 and Kozlova (1984 and reformulated by Sotomayor-Teixeira (1995 in terms of the regularization method. This method consists in analyzing a one parameter family of continuous vector fields $Z_{epsilon}$, defined by averaging $X$ and $Y$. This family approaches $Z$ when the parameter goes to zero. The results of Sotomayor-Teixeira and Sotomayor-Machado (2002 providing conditions on $(X,Y$ for the regularized vector fields to be structurally stable on planar compact connected regions are extended to discontinuous piecewise polynomial vector fields on $mathbb{R}^2$. Pertinent genericity results for vector fields satisfying the above stability conditions are also extended to the present case. A procedure for the study of discontinuous piecewise vector fields at infinity through a compactification is proposed here.
Second-harmonic generation in second-harmonic fiber Bragg gratings.
Steel, M J; de Sterke, C M
1996-06-20
We consider the production of second-harmonic light in gratings resonant with the generated field, through a Green's function approach. We recover some standard results and obtain new limits for the uniform grating case. With the extension to nonuniform gratings, we find the Green's function for the second harmonic in a grating with an arbitrary phase shift at some point. We then obtain closed form approximate expressions for the generated light for phase shifts close to π/2 and at the center of the grating. Finally, comparing the uniform and phase-shifted gratings with homogeneous materials, we discuss the enhancement in generated light and the bandwidth over which it occurs, and the consequences for second-harmonic generation in optical fiber Bragg gratings.
q-Bernoulli numbers and q-Bernoulli polynomials revisited
Directory of Open Access Journals (Sweden)
Kim Taekyun
2011-01-01
Full Text Available Abstract This paper performs a further investigation on the q-Bernoulli numbers and q-Bernoulli polynomials given by Acikgöz et al. (Adv Differ Equ, Article ID 951764, 9, 2010, some incorrect properties are revised. It is point out that the generating function for the q-Bernoulli numbers and polynomials is unreasonable. By using the theorem of Kim (Kyushu J Math 48, 73-86, 1994 (see Equation 9, some new generating functions for the q-Bernoulli numbers and polynomials are shown. Mathematics Subject Classification (2000 11B68, 11S40, 11S80
Simple design of slanted grating with simplified modal method.
Li, Shubin; Zhou, Changhe; Cao, Hongchao; Wu, Jun
2014-02-15
A simplified modal method (SMM) is presented that offers a clear physical image for subwavelength slanted grating. The diffraction characteristic of the slanted grating under Littrow configuration is revealed by the SMM as an equivalent rectangular grating, which is in good agreement with rigorous coupled-wave analysis. Based on the equivalence, we obtained an effective analytic solution for simplifying the design and optimization of a slanted grating. It offers a new approach for design of the slanted grating, e.g., a 1×2 beam splitter can be easily designed. This method should be helpful for designing various new slanted grating devices.
Defect grating modes as superimposed grating states
van Groesen, Embrecht W.C.; Sopaheluwakan, A.; Andonowati, A.; de Ridder, R.M; de Ridder, R.M.; Altena, G; Altena, G.; Geuzebroek, D.H.; Geuzenboek, D.; Dekker, R.; Dekker, R
2003-01-01
For a symmetric grating structure with a defect, we show that a fully transmitted defect mode in the band gap can be obtained as a superposition of two steady states: an amplified and an attenuated defect state. Without scanning the whole band gap by transmission calculations, this simplifies the
Computing Galois Groups of Eisenstein Polynomials Over P-adic Fields
Milstead, Jonathan
The most efficient algorithms for computing Galois groups of polynomials over global fields are based on Stauduhar's relative resolvent method. These methods are not directly generalizable to the local field case, since they require a field that contains the global field in which all roots of the polynomial can be approximated. We present splitting field-independent methods for computing the Galois group of an Eisenstein polynomial over a p-adic field. Our approach is to combine information from different disciplines. We primarily, make use of the ramification polygon of the polynomial, which is the Newton polygon of a related polynomial. This allows us to quickly calculate several invariants that serve to reduce the number of possible Galois groups. Algorithms by Greve and Pauli very efficiently return the Galois group of polynomials where the ramification polygon consists of one segment as well as information about the subfields of the stem field. Second, we look at the factorization of linear absolute resolvents to further narrow the pool of possible groups.
Fast beampattern evaluation by polynomial rooting
Häcker, P.; Uhlich, S.; Yang, B.
2011-07-01
Current automotive radar systems measure the distance, the relative velocity and the direction of objects in their environment. This information enables the car to support the driver. The direction estimation capabilities of a sensor array depend on its beampattern. To find the array configuration leading to the best angle estimation by a global optimization algorithm, a huge amount of beampatterns have to be calculated to detect their maxima. In this paper, a novel algorithm is proposed to find all maxima of an array's beampattern fast and reliably, leading to accelerated array optimizations. The algorithm works for arrays having the sensors on a uniformly spaced grid. We use a general version of the gcd (greatest common divisor) function in order to write the problem as a polynomial. We differentiate and root the polynomial to get the extrema of the beampattern. In addition, we show a method to reduce the computational burden even more by decreasing the order of the polynomial.
Response of fiber Bragg gratings to longitudinal ultrasonic waves.
Minardo, Aldo; Cusano, Andrea; Bernini, Romeo; Zeni, Luigi; Giordano, Michele
2005-02-01
In the last years, fiber optic sensors have been widely exploited for several sensing applications, including static and dynamic strain measurements up to acoustic detection. Among these, fiber Bragg grating sensors have been indicated as the ideal candidate for practical structural health monitoring in light of their unique advantages over conventional sensing devices. Although this class of sensors has been successfully tested for static and low-frequency measurements, the identification of sensor performances for high-frequency detection, including acoustic emission and ultrasonic investigations, is required. To this aim, the analysis of feasibilty on the use of fiber Bragg grating sensors as ultrasonic detectors has been carried out. In particular, the response of fiber Bragg gratings subjected to the longitudinal ultrasonic (US) field has been theoretically and numerically investigated. Ultrasonic field interaction has been modeled, taking into account the direct deformation of the grating pitch combined with changes in local refractive index due to the elasto-optic effect. Numerical results, obtained for both uniform and Gaussian-apodized fiber Bragg gratings, show that the grating spectrum is strongly influenced by the US field in terms of shape and central wavelength. In particular, a key parameter affecting the grating response is the ratio between the US wavelength and the grating length. Normal operation characterized by changes in wavelength of undistorted Bragg peak is possible only for US wavelengths longer than the grating length. For US wavelengths approaching the grating length, the wavelength change is accompanied by subpeaks formation and main peak amplitude modulation. This effect can be attributed to the nonuniformity of the US perturbation along the grating length. At very high US frequencies, the grating is not sensitive any longer. The results of this analysis provide useful tools for the design of grating-based ultrasound sensors for
Undergraduate experiment with fractal diffraction gratings
International Nuclear Information System (INIS)
Monsoriu, Juan A; Furlan, Walter D; Pons, Amparo; Barreiro, Juan C; Gimenez, Marcos H
2011-01-01
We present a simple diffraction experiment with fractal gratings based on the triadic Cantor set. Diffraction by fractals is proposed as a motivating strategy for students of optics in the potential applications of optical processing. Fraunhofer diffraction patterns are obtained using standard equipment present in most undergraduate physics laboratories and compared with those obtained with conventional periodic gratings. It is shown that fractal gratings produce self-similar diffraction patterns which can be evaluated analytically. Good agreement is obtained between experimental and numerical results.
Bragg Fibers with Soliton-like Grating Profiles
Directory of Open Access Journals (Sweden)
Bugaychuk S.
2016-01-01
Full Text Available Nonlinear dynamical system corresponding to the optical holography in a nonlocal nonlinear medium with dissipation contains stable localized spatio-temporal states, namely the grid dissipative solitons. These solitons display a non-uniform profile of the grating amplitude, which has the form of the dark soliton in the reflection geometry. The transformation of the grating amplitude gives rise many new atypical effects for the beams diffracted on such grating, and they are very suitable for the fiber Brass gratings. The damped nonlinear Schrodinger equation is derived that describes the properties of the grid dissipative soliton.
Bich Do, Danh; Lin, Jian Hung; Diep Lai, Ngoc; Kan, Hung-Chih; Hsu, Chia Chen
2011-08-01
We demonstrate the fabrication of a three-dimensional (3D) polymer quadratic nonlinear (χ(2)) grating structure. By performing layer-by-layer direct laser writing (DLW) and spin-coating approaches, desired photobleached grating patterns were embedded in the guest--host dispersed-red-1/poly(methylmethacrylate) (DR1/PMMA) active layers of an active-passive alternative multilayer structure through photobleaching of DR1 molecules. Polyvinyl-alcohol and SU8 thin films were deposited between DR1/PMMA layers serving as a passive layer to separate DR1/PMMA active layers. After applying the corona electric field poling to the multilayer structure, nonbleached DR1 molecules in the active layers formed polar distribution, and a 3D χ(2) grating structure was obtained. The χ(2) grating structures at different DR1/PMMA nonlinear layers were mapped by laser scanning second harmonic (SH) microscopy, and no cross talk was observed between SH images obtained from neighboring nonlinear layers. The layer-by-layer DLW technique is favorable to fabricating hierarchical 3D polymer nonlinear structures for optoelectronic applications with flexible structural design.
Neutron diffraction from holographic gratings in PMMA
International Nuclear Information System (INIS)
Havermeyer, F.; Kraetzig, E.; Rupp, R.A.; Schubert, D.W.
1999-01-01
Complete text of publication follows. By definition photorefractive materials change the refractive index for light under the action of light. Using the spatially modulated light intensity pattern from the interference of two plane waves, volume phase gratings with accurately defined spacings can be produced. Depending on the material there are many physical origins for these gratings, but in most cases they are linked to a density modulation and, consequently, to a refractive index grating for neutrons. By diffraction of light or neutrons from such gratings even small refractive index changes down to Δn ∼ 10 -7 - 10 -9 can be measured. In our photopolymer system PMMA/MMA (poly(methyl methacrylate) with a content of 10-20% of the residual monomer methyl methacrylate) inhomogeneous illumination leads to local post-polymerisation processes of the residual monomer. The resulting light-optical refractive index grating is caused by the modulation of the monomer/polymer ratio as well as by the modulation of the total density. Only by the unique combination of methods for light and neutron diffraction, available at HOLONS (Holography and Neutron Scattering, instrument at the GKSS research centre), both contributions can be separated. We discuss the angular dependence of the neutron diffraction efficiency for weakly and strongly (efficiencies up to 60% have been achieved) modulated gratings and propose a simple model for the evaluation of the gratings. (author)
Guts of surfaces and the colored Jones polynomial
Futer, David; Purcell, Jessica
2013-01-01
This monograph derives direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses, we prove that the growth of the degree of the colored Jones polynomials is a boundary slope of an essential surface in the knot complement. We show that certain coefficients of the polynomial measure how far this surface is from being a fiber for the knot; in particular, the surface is a fiber if and only if a particular coefficient vanishes. We also relate hyperbolic volume to colored Jones polynomials. Our method is to generalize the checkerboard decompositions of alternating knots. Under mild diagrammatic hypotheses, we show that these surfaces are essential, and obtain an ideal polyhedral decomposition of their complement. We use normal surface theory to relate the pieces of the JSJ decomposition of the complement to the combinatorics of certain surface spines (state graphs). Since state graphs have p...
Computing Tutte polynomials of contact networks in classrooms
Hincapié, Doracelly; Ospina, Juan
2013-05-01
Objective: The topological complexity of contact networks in classrooms and the potential transmission of an infectious disease were analyzed by sex and age. Methods: The Tutte polynomials, some topological properties and the number of spanning trees were used to algebraically compute the topological complexity. Computations were made with the Maple package GraphTheory. Published data of mutually reported social contacts within a classroom taken from primary school, consisting of children in the age ranges of 4-5, 7-8 and 10-11, were used. Results: The algebraic complexity of the Tutte polynomial and the probability of disease transmission increases with age. The contact networks are not bipartite graphs, gender segregation was observed especially in younger children. Conclusion: Tutte polynomials are tools to understand the topology of the contact networks and to derive numerical indexes of such topologies. It is possible to establish relationships between the Tutte polynomial of a given contact network and the potential transmission of an infectious disease within such network
Directory of Open Access Journals (Sweden)
Bangyong Sun
2014-01-01
Full Text Available The polynomial regression method is employed to calculate the relationship of device color space and CIE color space for color characterization, and the performance of different expressions with specific parameters is evaluated. Firstly, the polynomial equation for color conversion is established and the computation of polynomial coefficients is analysed. And then different forms of polynomial equations are used to calculate the RGB and CMYK’s CIE color values, while the corresponding color errors are compared. At last, an optimal polynomial expression is obtained by analysing several related parameters during color conversion, including polynomial numbers, the degree of polynomial terms, the selection of CIE visual spaces, and the linearization.
Exponential time paradigms through the polynomial time lens
Drucker, A.; Nederlof, J.; Santhanam, R.; Sankowski, P.; Zaroliagis, C.
2016-01-01
We propose a general approach to modelling algorithmic paradigms for the exact solution of NP-hard problems. Our approach is based on polynomial time reductions to succinct versions of problems solvable in polynomial time. We use this viewpoint to explore and compare the power of paradigms such as
Immersion Gratings for Infrared High-resolution Spectroscopy
Sarugaku, Yuki; Ikeda, Yuji; Kobayashi, Naoto; Kaji, Sayumi; Sukegawa, Takashi; Sugiyama, Shigeru; Nakagawa, Takao; Arasaki, Takayuki; Kondo, Sohei; Nakanishi, Kenshi; Yasui, Chikako; Kawakita, Hideyo
2016-10-01
High-resolution spectroscopy in the infrared wavelength range is essential for observations of minor isotopologues, such as HDO for water, and prebiotic organic molecules like hydrocarbons/P-bearing molecules because numerous vibrational molecular bands (including non-polar molecules) are located in this wavelength range. High spectral resolution enables us to detect weak lines without spectral line confusion. This technique has been widely used in planetary sciences, e.g., cometary coma (H2O, CO, and organic molecules), the martian atmosphere (CH4, CO2, H2O and HDO), and the upper atmosphere of gas giants (H3+ and organic molecules such as C2H6). Spectrographs with higher resolution (and higher sensitivity) still have a potential to provide a plenty of findings. However, because the size of spectrographs scales with the spectral resolution, it is difficult to realize it.Immersion grating (IG), which is a diffraction grating wherein the diffraction surface is immersed in a material with a high refractive index (n > 2), provides n times higher spectral resolution compared to a reflective grating of the same size. Because IG reduces the size of spectrograph to 1/n compared to the spectrograph with the same spectral resolution using a conventional reflective grating, it is widely acknowledged as a key optical device to realize compact spectrographs with high spectral resolution.Recently, we succeeded in fabricating a CdZnTe immersion grating with the theoretically predicted diffraction efficiency by machining process using an ultrahigh-precision five-axis processing machine developed by Canon Inc. Using the same technique, we completed a practical germanium (Ge) immersion grating with both a reflection coating on the grating surface and the an AR coating on the entrance surface. It is noteworthy that the wide wavelength range from 2 to 20 um can be covered by the two immersion gratings.In this paper, we present the performances and the applications of the immersion
Grism and immersion grating for space telescope
Ebizuka, Noboru; Oka, Kiko; Yamada, Akiko; Ishikawa, Mami; Kashiwagi, Masako; Kodate, Kashiko; Hirahara, Yasuhiro; Sato, Shuji; Kawabata, Koji S.; Wakaki, Moriaki; Morita, Shin-ya; Simizu, Tomoyuki; Yin, Shaohui; Omori, Hitoshi; Iye, Masanori
2017-11-01
The grism is a versatile dispersion element for an astronomical instrument ranging from ultraviolet to infrared. Major benefit of using a grism in a space application, instead of a reflection grating, is the size reduction of optical system because collimator and following optical elements could locate near by the grism. The surface relief (SR) grism is consisted a transmission grating and a prism, vertex angle of which is adjusted to redirect the diffracted beam straight along the direct vision direction at a specific order and wavelength. The volume phase holographic (VPH) grism consists a thick VPH grating sandwiched between two prisms, as specific order and wavelength is aligned the direct vision direction. The VPH grating inheres ideal diffraction efficiency on a higher dispersion application. On the other hand, the SR grating could achieve high diffraction efficiency on a lower dispersion application. Five grisms among eleven for the Faint Object Camera And Spectrograph (FOCAS) of the 8.2m Subaru Telescope with the resolving power from 250 to 3,000 are SR grisms fabricated by a replication method. Six additional grisms of FOCAS with the resolving power from 3,000 to 7,000 are VPH grisms. We propose "Quasi-Bragg grism" for a high dispersion spectroscopy with wide wavelength range. The germanium immersion grating for instance could reduce 1/64 as the total volume of a spectrograph with a conventional reflection grating since refractive index of germanium is over 4.0 from 1.6 to 20 μm. The prototype immersion gratings for the mid-InfraRed High dispersion Spectrograph (IRHS) are successfully fabricated by a nano-precision machine and grinding cup of cast iron with electrolytic dressing method.
Root and Critical Point Behaviors of Certain Sums of Polynomials
Indian Academy of Sciences (India)
13
There is an extensive literature concerning roots of sums of polynomials. Many papers and books([5], [6],. [7]) have written about these polynomials. Perhaps the most immediate question of sums of polynomials,. A + B = C, is “given bounds for the roots of A and B, what bounds can be given for the roots of C?” By. Fell [3], if ...
Fibre gratings for high temperature sensor applications
Canning, J.; Sommer, K.; Englund, M.
2001-07-01
Phosphosilicate fibre gratings can be stabilized at temperatures in excess of 500 °C for sensor applications by optimizing thermal and UV presensitization recipes. Furthermore, the use of 193 nm presensitization prevents the formation of OH absorption bands, extending the use of fibre gratings across the entire wavelength spectrum. Gratings for operation at 700 °C retaining up to 70% reflectivity after 30 min are demonstrated.
Polarization sensitivity testing of off-plane reflection gratings
Marlowe, Hannah; McEntaffer, Randal L.; DeRoo, Casey T.; Miles, Drew M.; Tutt, James H.; Laubis, Christian; Soltwisch, Victor
2015-09-01
Off-Plane reflection gratings were previously predicted to have different efficiencies when the incident light is polarized in the transverse-magnetic (TM) versus transverse-electric (TE) orientations with respect to the grating grooves. However, more recent theoretical calculations which rigorously account for finitely conducting, rather than perfectly conducting, grating materials no longer predict significant polarization sensitivity. We present the first empirical results for radially ruled, laminar groove profile gratings in the off-plane mount which demonstrate no difference in TM versus TE efficiency across our entire 300-1500 eV bandpass. These measurements together with the recent theoretical results confirm that grazing incidence off-plane reflection gratings using real, not perfectly conducting, materials are not polarization sensitive.
The chromatic polynomial and list colorings
DEFF Research Database (Denmark)
Thomassen, Carsten
2009-01-01
We prove that, if a graph has a list of k available colors at every vertex, then the number of list-colorings is at least the chromatic polynomial evaluated at k when k is sufficiently large compared to the number of vertices of the graph.......We prove that, if a graph has a list of k available colors at every vertex, then the number of list-colorings is at least the chromatic polynomial evaluated at k when k is sufficiently large compared to the number of vertices of the graph....
Polymer optical fiber bragg grating sensors
DEFF Research Database (Denmark)
Stefani, Alessio; Yuan, Scott Wu; Andresen, Søren
2010-01-01
Fiber-optical accelerometers based on polymer optical fiber Bragg gratings are reported. We have written fiber Bragg gratings for 1550 nm and 850 nm operations, characterized their temperature and strain response, and tested their performance in a prototype accelerometer....
BSDEs with polynomial growth generators
Directory of Open Access Journals (Sweden)
Philippe Briand
2000-01-01
Full Text Available In this paper, we give existence and uniqueness results for backward stochastic differential equations when the generator has a polynomial growth in the state variable. We deal with the case of a fixed terminal time, as well as the case of random terminal time. The need for this type of extension of the classical existence and uniqueness results comes from the desire to provide a probabilistic representation of the solutions of semilinear partial differential equations in the spirit of a nonlinear Feynman-Kac formula. Indeed, in many applications of interest, the nonlinearity is polynomial, e.g, the Allen-Cahn equation or the standard nonlinear heat and Schrödinger equations.
Overview of diffraction gratings technologies for space-flight satellites and astronomy
Cotel, Arnaud; Liard, Audrey; Desserouer, Frédéric; Bonnemason, Francis; Pichon, Pierre
2014-09-01
The diffraction gratings are widely used in Space-flight satellites for spectrograph instruments or in ground-based telescopes in astronomy. The diffraction gratings are one of the key optical components of such systems and have to exhibit very high optical performances. HORIBA Jobin Yvon S.A.S. (part of HORIBA Group) is in the forefront of such gratings development for more than 40 years. During the past decades, HORIBA Jobin Yvon (HJY) has developed a unique expertise in diffraction grating design and manufacturing processes for holographic, ruled or etched gratings. We will present in this paper an overview of diffraction grating technologies especially designed for space and astronomy applications. We will firstly review the heritage of the company in this field with the space qualification of different grating types. Then, we will describe several key grating technologies developed for specific space or astronomy projects: ruled blazed low groove density plane reflection grating, holographic blazed replica plane grating, high-groove density holographic toroidal and spherical grating and transmission Fused Silica Etched (FSE) grismassembled grating.
Polynomial chaos functions and stochastic differential equations
International Nuclear Information System (INIS)
Williams, M.M.R.
2006-01-01
The Karhunen-Loeve procedure and the associated polynomial chaos expansion have been employed to solve a simple first order stochastic differential equation which is typical of transport problems. Because the equation has an analytical solution, it provides a useful test of the efficacy of polynomial chaos. We find that the convergence is very rapid in some cases but that the increased complexity associated with many random variables can lead to very long computational times. The work is illustrated by exact and approximate solutions for the mean, variance and the probability distribution itself. The usefulness of a white noise approximation is also assessed. Extensive numerical results are given which highlight the weaknesses and strengths of polynomial chaos. The general conclusion is that the method is promising but requires further detailed study by application to a practical problem in transport theory
Minimal residual method stronger than polynomial preconditioning
Energy Technology Data Exchange (ETDEWEB)
Faber, V.; Joubert, W.; Knill, E. [Los Alamos National Lab., NM (United States)] [and others
1994-12-31
Two popular methods for solving symmetric and nonsymmetric systems of equations are the minimal residual method, implemented by algorithms such as GMRES, and polynomial preconditioning methods. In this study results are given on the convergence rates of these methods for various classes of matrices. It is shown that for some matrices, such as normal matrices, the convergence rates for GMRES and for the optimal polynomial preconditioning are the same, and for other matrices such as the upper triangular Toeplitz matrices, it is at least assured that if one method converges then the other must converge. On the other hand, it is shown that matrices exist for which restarted GMRES always converges but any polynomial preconditioning of corresponding degree makes no progress toward the solution for some initial error. The implications of these results for these and other iterative methods are discussed.
Bernoulli numbers and polynomials from a more general point of view
International Nuclear Information System (INIS)
Dattoli, G.; Cesarano, C.; Lorenzutta, S.
2000-01-01
In this work it is applied the method of generating function, to introduce new forms of Bernoulli numbers and polynomials, which are exploited to derive further classes of partial sums involving generalized many index many variable polynomials. Analogous considerations are developed for the Euler numbers and polynomials [it
Generalizations of an integral for Legendre polynomials by Persson and Strang
Diekema, E.; Koornwinder, T.H.
2012-01-01
Persson and Strang (2003) evaluated the integral over [−1,1] of a squared odd degree Legendre polynomial divided by x2 as being equal to 2. We consider a similar integral for orthogonal polynomials with respect to a general even orthogonality measure, with Gegenbauer and Hermite polynomials as
Gratings in passive and active optical waveguides
DEFF Research Database (Denmark)
Berendt, Martin Ole
1999-01-01
will not only couple to the backward propagating fundamental mode, but also to cladding modes. Cladding modes are strongly bound, but slightly leaky, higher-order modes in the core-cladding-air index structure. If the waveguide is not surrounded by air, but by a recoating the cladding modes become highly...... attenuated. In either case the cladding mode coupling gives loss on the short wavelength side of the reflection band. The cladding mode coupling loss is a major problem for the utilization of fiber Bragg gratings in wavelength division multiplexed (WDM) system. In this project, a numerical model for cladding...... mode coupling has been developed. The model can predict the spectral location and size of coupling, for various fiber designs. By the aid of this modeling tool, a fiber has been optimized to give low cladding-mode losses. The optimized fiber has been produced and the predicted reduction of cladding...
Eye aberration analysis with Zernike polynomials
Molebny, Vasyl V.; Chyzh, Igor H.; Sokurenko, Vyacheslav M.; Pallikaris, Ioannis G.; Naoumidis, Leonidas P.
1998-06-01
New horizons for accurate photorefractive sight correction, afforded by novel flying spot technologies, require adequate measurements of photorefractive properties of an eye. Proposed techniques of eye refraction mapping present results of measurements for finite number of points of eye aperture, requiring to approximate these data by 3D surface. A technique of wave front approximation with Zernike polynomials is described, using optimization of the number of polynomial coefficients. Criterion of optimization is the nearest proximity of the resulted continuous surface to the values calculated for given discrete points. Methodology includes statistical evaluation of minimal root mean square deviation (RMSD) of transverse aberrations, in particular, varying consecutively the values of maximal coefficient indices of Zernike polynomials, recalculating the coefficients, and computing the value of RMSD. Optimization is finished at minimal value of RMSD. Formulas are given for computing ametropia, size of the spot of light on retina, caused by spherical aberration, coma, and astigmatism. Results are illustrated by experimental data, that could be of interest for other applications, where detailed evaluation of eye parameters is needed.
Animating Nested Taylor Polynomials to Approximate a Function
Mazzone, Eric F.; Piper, Bruce R.
2010-01-01
The way that Taylor polynomials approximate functions can be demonstrated by moving the center point while keeping the degree fixed. These animations are particularly nice when the Taylor polynomials do not intersect and form a nested family. We prove a result that shows when this nesting occurs. The animations can be shown in class or…
Siewert, F; Löchel, B; Buchheim, J; Eggenstein, F; Firsov, A; Gwalt, G; Kutz, O; Lemke, St; Nelles, B; Rudolph, I; Schäfers, F; Seliger, T; Senf, F; Sokolov, A; Waberski, Ch; Wolf, J; Zeschke, T; Zizak, I; Follath, R; Arnold, T; Frost, F; Pietag, F; Erko, A
2018-01-01
Blazed gratings are of dedicated interest for the monochromatization of synchrotron radiation when a high photon flux is required, such as, for example, in resonant inelastic X-ray scattering experiments or when the use of laminar gratings is excluded due to too high flux densities and expected damage, for example at free-electron laser beamlines. Their availability became a bottleneck since the decommissioning of the grating manufacture facility at Carl Zeiss in Oberkochen. To resolve this situation a new technological laboratory was established at the Helmholtz Zentrum Berlin, including instrumentation from Carl Zeiss. Besides the upgraded ZEISS equipment, an advanced grating production line has been developed, including a new ultra-precise ruling machine, ion etching technology as well as laser interference lithography. While the old ZEISS ruling machine GTM-6 allows ruling for a grating length up to 170 mm, the new GTM-24 will have the capacity for 600 mm (24 inch) gratings with groove densities between 50 lines mm -1 and 1200 lines mm -1 . A new ion etching machine with a scanning radiofrequency excited ion beam (HF) source allows gratings to be etched into substrates of up to 500 mm length. For a final at-wavelength characterization, a new reflectometer at a new Optics beamline at the BESSY-II storage ring is under operation. This paper reports on the status of the grating fabrication, the measured quality of fabricated items by ex situ and in situ metrology, and future development goals.
Learning Read-constant Polynomials of Constant Degree modulo Composites
DEFF Research Database (Denmark)
Chattopadhyay, Arkadev; Gavaldá, Richard; Hansen, Kristoffer Arnsfelt
2011-01-01
Boolean functions that have constant degree polynomial representation over a fixed finite ring form a natural and strict subclass of the complexity class \\textACC0ACC0. They are also precisely the functions computable efficiently by programs over fixed and finite nilpotent groups. This class...... is not known to be learnable in any reasonable learning model. In this paper, we provide a deterministic polynomial time algorithm for learning Boolean functions represented by polynomials of constant degree over arbitrary finite rings from membership queries, with the additional constraint that each variable...
A comparison of high-order polynomial and wave-based methods for Helmholtz problems
Lieu, Alice; Gabard, Gwénaël; Bériot, Hadrien
2016-09-01
The application of computational modelling to wave propagation problems is hindered by the dispersion error introduced by the discretisation. Two common strategies to address this issue are to use high-order polynomial shape functions (e.g. hp-FEM), or to use physics-based, or Trefftz, methods where the shape functions are local solutions of the problem (typically plane waves). Both strategies have been actively developed over the past decades and both have demonstrated their benefits compared to conventional finite-element methods, but they have yet to be compared. In this paper a high-order polynomial method (p-FEM with Lobatto polynomials) and the wave-based discontinuous Galerkin method are compared for two-dimensional Helmholtz problems. A number of different benchmark problems are used to perform a detailed and systematic assessment of the relative merits of these two methods in terms of interpolation properties, performance and conditioning. It is generally assumed that a wave-based method naturally provides better accuracy compared to polynomial methods since the plane waves or Bessel functions used in these methods are exact solutions of the Helmholtz equation. Results indicate that this expectation does not necessarily translate into a clear benefit, and that the differences in performance, accuracy and conditioning are more nuanced than generally assumed. The high-order polynomial method can in fact deliver comparable, and in some cases superior, performance compared to the wave-based DGM. In addition to benchmarking the intrinsic computational performance of these methods, a number of practical issues associated with realistic applications are also discussed.
Corrugated grating on organic multilayer Bragg reflector
Jaquet, Sylvain; Scharf, Toralf; Herzig, Hans Peter
2007-08-01
Polymeric multilayer Bragg structures are combined with diffractive gratings to produce artificial visual color effects. A particular effect is expected due to the angular reflection dependence of the multilayer Bragg structure and the dispersion caused by the grating. The combined effects can also be used to design particular filter functions and various resonant structures. The multilayer Bragg structure is fabricated by spin-coating of two different low-cost polymer materials in solution on a cleaned glass substrate. These polymers have a refractive index difference of about 0.15 and permit multilayer coatings without interlayer problems. Master gratings of different periods are realized by laser beam interference and replicated gratings are superimposed on the multilayer structure by soft embossing in a UV curing glue. The fabrication process requires only polymer materials. The obtained devices are stable and robust. Angular dependent reflection spectrums for the visible are measured. These results show that it is possible to obtain unexpected reflection effects. A rich variety of color spectra can be generated, which is not possible with a single grating. This can be explained by the coupling of transmission of grating orders and the Bragg reflection band. A simple model permits to explain some of the spectral vs angular dependence of reflected light.
Dynamic optical coupled system employing Dammann gratings
Di, Caihui; Zhou, Changhe; Ru, Huayi
2004-10-01
With the increasing of the number of users in optical fiber communications, fiber-to-home project has a larger market value. Then the need of dynamic optical couplers, especially of N broad-band couplers, becomes greater. Though some advanced fiber fusion techniques have been developed, they still have many shortcomings. In this paper we propose a dynamic optical coupled system employing even-numbered Dammann gratings, which have the characteristic that the phase distribution in the first half-period accurately equals to that in the second-period with π phase inversion. In our experiment, we divide a conventional even-numbered Dammann grating into two identical gratings. The system can achieve the beam splitter and combiner as the switch between them according to the relative shift between two complementary gratings. When there is no shift between the gratings, the demonstrated 1×8 dynamic optical coupler achieves good uniformity of 0.06 and insertion loss of around 10.8 dB for each channel as a splitter. When the two gratings have an accurate shift of a half-period between them, our system has a low insertion loss of 0.46 dB as a combiner at a wavelength of 1550 nm.
Fabrication of high edge-definition steel-tape gratings for optical encoders
Ye, Guoyong; Liu, Hongzhong; Yan, Jiawei; Ban, Yaowen; Fan, Shanjin; Shi, Yongsheng; Yin, Lei
2017-10-01
High edge definition of a scale grating is the basic prerequisite for high measurement accuracy of optical encoders. This paper presents a novel fabrication method of steel tape gratings using graphene oxide nanoparticles as anti-reflective grating strips. Roll-to-roll nanoimprint lithography is adopted to manufacture the steel tape with hydrophobic and hydrophilic pattern arrays. Self-assembly technology is employed to obtain anti-reflective grating strips by depositing the graphene oxide nanoparticles on hydrophobic regions. A thin SiO2 coating is deposited on the grating to protect the grating strips. Experimental results confirm that the proposed fabrication process enables a higher edge definition in making steel-tape gratings, and the new steel tape gratings offer better performance than conventional gratings.
Svebak, Sven
2016-01-01
Results from two studies of biological consequences of laughter are reported. A proposed inhibitory brain mechanism was tested in Study 1. It aims to protect against trunk compression that can cause health hazards during vigorous laughter. Compression may be maximal during moderate durations and, for protective reasons, moderate in enduring vigorous laughs. Twenty-five university students volunteered to see a candid camera film. Laughter responses (LR) and the superimposed ha-responses were operationally assessed by mercury-filled strain gauges strapped around the trunk. On average, the thorax compression amplitudes exceeded those of the abdomen, and greater amplitudes were seen in the males than in the females after correction for resting trunk circumference. Regression analyses supported polynomial relations because medium LR durations were associated with particularly high thorax amplitudes. In Study 2, power changes were computed in the beta and alpha EEG frequency bands of the parietal cortex from before to after exposure to the comedy “Dinner for one” in 56 university students. Highly significant linear relations were calculated between the number of laughs and post-exposure cortical activation (increase of beta, decrease of alpha) due to high activation after frequent laughter. The results from Study 1 supported the hypothesis of a protective brain mechanism that is activated during long LRs to reduce the risk of harm to vital organs in the trunk cavity. The results in Study 2 supported a linear cortical activation and, thus, provided evidence for a biological correlate to the subjective experience of mental refreshment after laughter. PMID:27547260
Suppressing Ghost Diffraction in E-Beam-Written Gratings
Wilson, Daniel; Backlund, Johan
2009-01-01
A modified scheme for electron-beam (E-beam) writing used in the fabrication of convex or concave diffraction gratings makes it possible to suppress the ghost diffraction heretofore exhibited by such gratings. Ghost diffraction is a spurious component of diffraction caused by a spurious component of grating periodicity as described below. The ghost diffraction orders appear between the main diffraction orders and are typically more intense than is the diffuse scattering from the grating. At such high intensity, ghost diffraction is the dominant source of degradation of grating performance. The pattern of a convex or concave grating is established by electron-beam writing in a resist material coating a substrate that has the desired convex or concave shape. Unfortunately, as a result of the characteristics of electrostatic deflectors used to control the electron beam, it is possible to expose only a small field - typically between 0.5 and 1.0 mm wide - at a given fixed position of the electron gun relative to the substrate. To make a grating larger than the field size, it is necessary to move the substrate to make it possible to write fields centered at different positions, so that the larger area is synthesized by "stitching" the exposed fields.
Complex centers of polynomial differential equations
Directory of Open Access Journals (Sweden)
Mohamad Ali M. Alwash
2007-07-01
Full Text Available We present some results on the existence and nonexistence of centers for polynomial first order ordinary differential equations with complex coefficients. In particular, we show that binomial differential equations without linear terms do not have complex centers. Classes of polynomial differential equations, with more than two terms, are presented that do not have complex centers. We also study the relation between complex centers and the Pugh problem. An algorithm is described to solve the Pugh problem for equations without complex centers. The method of proof involves phase plane analysis of the polar equations and a local study of periodic solutions.
Differential recurrence formulae for orthogonal polynomials
Directory of Open Access Journals (Sweden)
Anton L. W. von Bachhaus
1995-11-01
Full Text Available Part I - By combining a general 2nd-order linear homogeneous ordinary differential equation with the three-term recurrence relation possessed by all orthogonal polynomials, it is shown that sequences of orthogonal polynomials which satisfy a differential equation of the above mentioned type necessarily have a differentiation formula of the type: gn(xY'n(x=fn(xYn(x+Yn-1(x. Part II - A recurrence formula of the form: rn(xY'n(x+sn(xY'n+1(x+tn(xY'n-1(x=0, is derived using the result of Part I.
Considering a non-polynomial basis for local kernel regression problem
Silalahi, Divo Dharma; Midi, Habshah
2017-01-01
A common used as solution for local kernel nonparametric regression problem is given using polynomial regression. In this study, we demonstrated the estimator and properties using maximum likelihood estimator for a non-polynomial basis such B-spline to replacing the polynomial basis. This estimator allows for flexibility in the selection of a bandwidth and a knot. The best estimator was selected by finding an optimal bandwidth and knot through minimizing the famous generalized validation function.
Open Problems Related to the Hurwitz Stability of Polynomials Segments
Directory of Open Access Journals (Sweden)
Baltazar Aguirre-Hernández
2018-01-01
Full Text Available In the framework of robust stability analysis of linear systems, the development of techniques and methods that help to obtain necessary and sufficient conditions to determine stability of convex combinations of polynomials is paramount. In this paper, knowing that Hurwitz polynomials set is not a convex set, a brief overview of some results and open problems concerning the stability of the convex combinations of Hurwitz polynomials is then provided.
Perturbative approach to continuum generation in a fiber Bragg grating.
Westbrook, P S; Nicholson, J W
2006-08-21
We derive a perturbative solution to the nonlinear Schrödinger equation to include the effect of a fiber Bragg grating whose bandgap is much smaller than the pulse bandwidth. The grating generates a slow dispersive wave which may be computed from an integral over the unperturbed solution if nonlinear interaction between the grating and unperturbed waves is negligible. Our approach allows rapid estimation of large grating continuum enhancement peaks from a single nonlinear simulation of the waveguide without grating. We apply our method to uniform and sampled gratings, finding good agreement with full nonlinear simulations, and qualitatively reproducing experimental results.
The computation of bond percolation critical polynomials by the deletion–contraction algorithm
International Nuclear Information System (INIS)
Scullard, Christian R
2012-01-01
Although every exactly known bond percolation critical threshold is the root in [0,1] of a lattice-dependent polynomial, it has recently been shown that the notion of a critical polynomial can be extended to any periodic lattice. The polynomial is computed on a finite subgraph, called the base, of an infinite lattice. For any problem with exactly known solution, the prediction of the bond threshold is always correct for any base containing an arbitrary number of unit cells. For unsolved problems, the polynomial is referred to as the generalized critical polynomial and provides an approximation that becomes more accurate with increasing number of bonds in the base, appearing to approach the exact answer. The polynomials are computed using the deletion–contraction algorithm, which quickly becomes intractable by hand for more than about 18 bonds. Here, I present generalized critical polynomials calculated with a computer program for bases of up to 36 bonds for all the unsolved Archimedean lattices, except the kagome lattice, which was considered in an earlier work. The polynomial estimates are generally within 10 −5 –10 −7 of the numerical values, but the prediction for the (4,8 2 ) lattice, though not exact, is not ruled out by simulations. (paper)
Barkhouser, Robert H.; Arns, James; Gunn, James E.
2014-08-01
The Prime Focus Spectrograph (PFS) is a major instrument under development for the 8.2 m Subaru telescope on Mauna Kea. Four identical, fixed spectrograph modules are located in a room above one Nasmyth focus. A 55 m fiber optic cable feeds light into the spectrographs from a robotic fiber positioner mounted at the telescope prime focus, behind the wide field corrector developed for Hyper Suprime-Cam. The positioner contains 2400 fibers and covers a 1.3 degree hexagonal field of view. Each spectrograph module will be capable of simultaneously acquiring 600 spectra. The spectrograph optical design consists of a Schmidt collimator, two dichroic beamsplitters to separate the light into three channels, and for each channel a volume phase holographic (VPH) grating and a dual- corrector, modified Schmidt reimaging camera. This design provides a 275 mm collimated beam diameter, wide simultaneous wavelength coverage from 380 nm to 1.26 µm, and good imaging performance at the fast f/1.1 focal ratio required from the cameras to avoid oversampling the fibers. The three channels are designated as the blue, red, and near-infrared (NIR), and cover the bandpasses 380-650 nm (blue), 630-970 nm (red), and 0.94-1.26 µm (NIR). A mosaic of two Hamamatsu 2k×4k, 15 µm pixel CCDs records the spectra in the blue and red channels, while the NIR channel employs a 4k×4k, substrate-removed HAWAII-4RG array from Teledyne, with 15 µm pixels and a 1.7 µm wavelength cutoff. VPH gratings have become the dispersing element of choice for moderate-resolution astronomical spectro- graphs due their potential for very high diffraction efficiency, low scattered light, and the more compact instru- ment designs offered by transmissive dispersers. High quality VPH gratings are now routinely being produced in the sizes required for instruments on large telescopes. These factors made VPH gratings an obvious choice for PFS. In order to reduce risk to the project, as well as fully exploit the performance
Miller, W., Jr.; Li, Q.
2015-04-01
The Wilson and Racah polynomials can be characterized as basis functions for irreducible representations of the quadratic symmetry algebra of the quantum superintegrable system on the 2-sphere, HΨ = EΨ, with generic 3-parameter potential. Clearly, the polynomials are expansion coefficients for one eigenbasis of a symmetry operator L2 of H in terms of an eigenbasis of another symmetry operator L1, but the exact relationship appears not to have been made explicit. We work out the details of the expansion to show, explicitly, how the polynomials arise and how the principal properties of these functions: the measure, 3-term recurrence relation, 2nd order difference equation, duality of these relations, permutation symmetry, intertwining operators and an alternate derivation of Wilson functions - follow from the symmetry of this quantum system. This paper is an exercise to show that quantum mechancal concepts and recurrence relations for Gausian hypergeometrc functions alone suffice to explain these properties; we make no assumptions about the structure of Wilson polynomial/functions, but derive them from quantum principles. There is active interest in the relation between multivariable Wilson polynomials and the quantum superintegrable system on the n-sphere with generic potential, and these results should aid in the generalization. Contracting function space realizations of irreducible representations of this quadratic algebra to the other superintegrable systems one can obtain the full Askey scheme of orthogonal hypergeometric polynomials. All of these contractions of superintegrable systems with potential are uniquely induced by Wigner Lie algebra contractions of so(3, C) and e(2,C). All of the polynomials produced are interpretable as quantum expansion coefficients. It is important to extend this process to higher dimensions.
International Nuclear Information System (INIS)
Miller, W Jr; Li, Q
2015-01-01
The Wilson and Racah polynomials can be characterized as basis functions for irreducible representations of the quadratic symmetry algebra of the quantum superintegrable system on the 2-sphere, HΨ = EΨ, with generic 3-parameter potential. Clearly, the polynomials are expansion coefficients for one eigenbasis of a symmetry operator L 2 of H in terms of an eigenbasis of another symmetry operator L 1 , but the exact relationship appears not to have been made explicit. We work out the details of the expansion to show, explicitly, how the polynomials arise and how the principal properties of these functions: the measure, 3-term recurrence relation, 2nd order difference equation, duality of these relations, permutation symmetry, intertwining operators and an alternate derivation of Wilson functions - follow from the symmetry of this quantum system. This paper is an exercise to show that quantum mechancal concepts and recurrence relations for Gausian hypergeometrc functions alone suffice to explain these properties; we make no assumptions about the structure of Wilson polynomial/functions, but derive them from quantum principles. There is active interest in the relation between multivariable Wilson polynomials and the quantum superintegrable system on the n-sphere with generic potential, and these results should aid in the generalization. Contracting function space realizations of irreducible representations of this quadratic algebra to the other superintegrable systems one can obtain the full Askey scheme of orthogonal hypergeometric polynomials. All of these contractions of superintegrable systems with potential are uniquely induced by Wigner Lie algebra contractions of so(3, C) and e(2,C). All of the polynomials produced are interpretable as quantum expansion coefficients. It is important to extend this process to higher dimensions. (paper)
Exploiting a Transmission Grating Spectrometer
Energy Technology Data Exchange (ETDEWEB)
Ronald E. Bell
2004-12-08
The availability of compact transmission grating spectrometers now allows an attractive and economical alternative to the more familiar Czerny-Turner configuration for many high-temperature plasma applications. Higher throughput is obtained with short focal length refractive optics and stigmatic imaging. Many more spectra can be obtained with a single spectrometer since smaller, more densely packed optical input fibers can be used. Multiple input slits, along with a bandpass filter, can be used to maximize the number of spectra per detector, providing further economy. Curved slits can correct for the strong image curvature of the short focal length optics. Presented here are the governing grating equations for both standard and high-dispersion transmission gratings, defining dispersion, image curvature, and desired slit curvature, that can be used in the design of improved plasma diagnostics.
Exploiting a Transmission Grating Spectrometer
International Nuclear Information System (INIS)
Bell, Ronald E.
2004-01-01
The availability of compact transmission grating spectrometers now allows an attractive and economical alternative to the more familiar Czerny-Turner configuration for many high-temperature plasma applications. Higher throughput is obtained with short focal length refractive optics and stigmatic imaging. Many more spectra can be obtained with a single spectrometer since smaller, more densely packed optical input fibers can be used. Multiple input slits, along with a bandpass filter, can be used to maximize the number of spectra per detector, providing further economy. Curved slits can correct for the strong image curvature of the short focal length optics. Presented here are the governing grating equations for both standard and high-dispersion transmission gratings, defining dispersion, image curvature, and desired slit curvature, that can be used in the design of improved plasma diagnostics
Solving the interval type-2 fuzzy polynomial equation using the ranking method
Rahman, Nurhakimah Ab.; Abdullah, Lazim
2014-07-01
Polynomial equations with trapezoidal and triangular fuzzy numbers have attracted some interest among researchers in mathematics, engineering and social sciences. There are some methods that have been developed in order to solve these equations. In this study we are interested in introducing the interval type-2 fuzzy polynomial equation and solving it using the ranking method of fuzzy numbers. The ranking method concept was firstly proposed to find real roots of fuzzy polynomial equation. Therefore, the ranking method is applied to find real roots of the interval type-2 fuzzy polynomial equation. We transform the interval type-2 fuzzy polynomial equation to a system of crisp interval type-2 fuzzy polynomial equation. This transformation is performed using the ranking method of fuzzy numbers based on three parameters, namely value, ambiguity and fuzziness. Finally, we illustrate our approach by numerical example.
Grating-Coupled Waveguide Cloaking
International Nuclear Information System (INIS)
Wang Jia-Fu; Qu Shao-Bo; Ma Hua; Wang Cong-Min; Wang Xin-Hua; Zhou Hang; Xu Zhuo; Xia Song
2012-01-01
Based on the concept of a grating-coupled waveguide (GCW), a new strategy for realizing EM cloaking is presented. Using metallic grating, incident waves are firstly coupled into the effective waveguide and then decoupled into free space behind, enabling EM waves to pass around the obstacle. Phase compensation in the waveguide keeps the wave-front shape behind the obstacle unchanged. Circular, rectangular and triangular cloaks are presented to verify the robustness of the GCW cloaking. Electric field animations and radar cross section (RCS) comparisons convincingly demonstrate the cloaking effect
A high-order q-difference equation for q-Hahn multiple orthogonal polynomials
DEFF Research Database (Denmark)
Arvesú, J.; Esposito, Chiara
2012-01-01
A high-order linear q-difference equation with polynomial coefficients having q-Hahn multiple orthogonal polynomials as eigenfunctions is given. The order of the equation coincides with the number of orthogonality conditions that these polynomials satisfy. Some limiting situations when are studie....... Indeed, the difference equation for Hahn multiple orthogonal polynomials given in Lee [J. Approx. Theory (2007), ), doi: 10.1016/j.jat.2007.06.002] is obtained as a limiting case....
On the Lorentz degree of a product of polynomials
Ait-Haddou, Rachid
2015-01-01
In this note, we negatively answer two questions of T. Erdélyi (1991, 2010) on possible lower bounds on the Lorentz degree of product of two polynomials. We show that the correctness of one question for degree two polynomials is a direct consequence
Generalized Freud's equation and level densities with polynomial potential
Boobna, Akshat; Ghosh, Saugata
2013-08-01
We study orthogonal polynomials with weight $\\exp[-NV(x)]$, where $V(x)=\\sum_{k=1}^{d}a_{2k}x^{2k}/2k$ is a polynomial of order 2d. We derive the generalised Freud's equations for $d=3$, 4 and 5 and using this obtain $R_{\\mu}=h_{\\mu}/h_{\\mu -1}$, where $h_{\\mu}$ is the normalization constant for the corresponding orthogonal polynomials. Moments of the density functions, expressed in terms of $R_{\\mu}$, are obtained using Freud's equation and using this, explicit results of level densities as $N\\rightarrow\\infty$ are derived.
Dual-function beam splitter of a subwavelength fused-silica grating.
Feng, Jijun; Zhou, Changhe; Zheng, Jiangjun; Cao, Hongchao; Lv, Peng
2009-05-10
We present the design and fabrication of a novel dual-function subwavelength fused-silica grating that can be used as a polarization-selective beam splitter. For TM polarization, the grating can be used as a two-port beam splitter at a wavelength of 1550 nm with a total diffraction efficiency of 98%. For TE polarization, the grating can function as a high-efficiency grating, and the diffraction efficiency of the -1st order is 95% under Littrow mounting. This dual-function grating design is based on a simplified modal method. By using the rigorous coupled-wave analysis, the optimum grating parameters can be determined. Holographic recording technology and inductively coupled plasma etching are used to manufacture the fused-silica grating. Experimental results are in agreement with the theoretical values.
Electromagnetically induced two-dimensional grating assisted by incoherent pump
Energy Technology Data Exchange (ETDEWEB)
Chen, Yu-Yuan; Liu, Zhuan-Zhuan; Wan, Ren-Gang, E-mail: wrg@snnu.edu.cn
2017-04-25
We propose a scheme for realizing electromagnetically induced two-dimensional grating in a double-Λ system driven simultaneously by a coherent field and an incoherent pump field. In such an atomic configuration, the absorption is suppressed owing to the incoherent pumping process and the probe can be even amplified, while the refractivity is mainly attributed to the dynamically induced coherence. With the help of a standing-wave pattern coherent field, we obtain periodically modulated refractive index without or with gain, and therefore phase grating or gain-phase grating which diffracts a probe light into high-order direction efficiently can be formed in the medium via appropriate manipulation of the system parameters. The diffraction efficiency attainable by the present gratings can be controlled by tuning the coherent field intensity or the interaction length. Hence, the two-dimensional grating can be utilized as all-optical splitter or router in optical networking and communication. - Highlights: • Two-dimensional grating is coherently induced in four-level atoms. • Phase and gain-phase gratings are obtained assisted by incoherent pump. • The diffraction power is improved due to the enhanced refraction modulation. • The gratings can be utilized as multi-channel all-optical splitter and router.
H∞ Control of Polynomial Fuzzy Systems: A Sum of Squares Approach
Directory of Open Access Journals (Sweden)
Bomo W. Sanjaya
2014-07-01
Full Text Available This paper proposes the control design ofa nonlinear polynomial fuzzy system with H∞ performance objective using a sum of squares (SOS approach. Fuzzy model and controller are represented by a polynomial fuzzy model and controller. The design condition is obtained by using polynomial Lyapunov functions that not only guarantee stability but also satisfy the H∞ performance objective. The design condition is represented in terms of an SOS that can be numerically solved via the SOSTOOLS. A simulation study is presented to show the effectiveness of the SOS-based H∞ control designfor nonlinear polynomial fuzzy systems.
Zeros and logarithmic asymptotics of Sobolev orthogonal polynomials for exponential weights
Díaz Mendoza, C.; Orive, R.; Pijeira Cabrera, H.
2009-12-01
We obtain the (contracted) weak zero asymptotics for orthogonal polynomials with respect to Sobolev inner products with exponential weights in the real semiaxis, of the form , with [gamma]>0, which include as particular cases the counterparts of the so-called Freud (i.e., when [phi] has a polynomial growth at infinity) and Erdös (when [phi] grows faster than any polynomial at infinity) weights. In addition, the boundness of the distance of the zeros of these Sobolev orthogonal polynomials to the convex hull of the support and, as a consequence, a result on logarithmic asymptotics are derived.
Structural Design of a Compact in-Plane Nano-Grating Accelerometer
International Nuclear Information System (INIS)
Yao Bao-Yin; Zhou Zhen; Feng Li-Shuang; Wang Wen-Pu; Wang Xiao
2012-01-01
A combination of large mass, weak spring and nano-grating is the key for a nano-grating accelerometer to measure nano-G acceleration. A novel compact nano-grating accelerometer integrating a large mass with nano-grating is proposed. First, the numbers of diffraction orders are calculated. Then, structure parameters are optimized by finite element analysis to achieve a high sensitivity in an ideal vibration mode. Finally, we design the fabrication method to form such a compact nano-grating accelerometer and successfully fabricate the uniform and well-designed nano-gratings with a period of 847 nm, crater of 451 nm by an FIB/SEM dual beam system. Based on the ANSYS simulation, a nano-grating accelerometer is predicted to work in the first modal and enables the accelerometer to have displacement sensitivity at 197 nm/G with a measurement range of ±1 G, corresponding to zeroth diffraction beam optical sensitivity 1%/mG. The nano-gratings fabricated are very close to those designed ones within experimental error to lay the foundation for the sequent fabrication. These results provide a theoretical basis for the design and fabrication of nano-grating accelerometers
Transparent Electrochemical Gratings from a Patterned Bistable Silver Mirror.
Park, Chihyun; Na, Jongbeom; Han, Minsu; Kim, Eunkyoung
2017-07-25
Silver mirror patterns were formed reversibly on a polystyrene (PS)-patterned electrode to produce gratings through the electrochemical reduction of silver ions. The electrochemical gratings exhibited high transparency (T > 95%), similar to a see-through window, by matching the refractive index of the grating pattern with the surrounding medium. The gratings switch to a diffractive state upon the formation of a mirror pattern (T modulation, NIR light reflection, and on-demand heat transfer.
Some Results on the Independence Polynomial of Unicyclic Graphs
Directory of Open Access Journals (Sweden)
Oboudi Mohammad Reza
2018-05-01
Full Text Available Let G be a simple graph on n vertices. An independent set in a graph is a set of pairwise non-adjacent vertices. The independence polynomial of G is the polynomial I(G,x=∑k=0ns(G,kxk$I(G,x = \\sum\
Limit cycles bifurcating from the periodic annulus of cubic homogeneous polynomial centers
Directory of Open Access Journals (Sweden)
Jaume Llibre
2015-10-01
Full Text Available We obtain an explicit polynomial whose simple positive real roots provide the limit cycles which bifurcate from the periodic orbits of any cubic homogeneous polynomial center when it is perturbed inside the class of all polynomial differential systems of degree n.
Liquid filling of photonic crystal fibres for grating writing
DEFF Research Database (Denmark)
Sørensen, Henrik Rokkjær; Canning, John; Lægsgaard, Jesper
2007-01-01
liquid filling of photonic crystal fibres reduces the scattering from air–glass interfaces during Bragg grating writing in many layered photonic crystal fibres. Within experimental uncertainty, the grating index modulation of a grating written in germanium-doped photonic crystal fibre with 10 rings...
Polynomial Poisson algebras: Gel'fand-Kirillov problem and Poisson spectra
Lecoutre, César
2014-01-01
We study the fields of fractions and the Poisson spectra of polynomial Poisson algebras.\\ud \\ud First we investigate a Poisson birational equivalence problem for polynomial Poisson algebras over a field of arbitrary characteristic. Namely, the quadratic Poisson Gel'fand-Kirillov problem asks whether the field of fractions of a Poisson algebra is isomorphic to the field of fractions of a Poisson affine space, i.e. a polynomial algebra such that the Poisson bracket of two generators is equal to...
21 CFR 133.146 - Grated cheeses.
2010-04-01
... Products § 133.146 Grated cheeses. (a) Description. Grated cheeses is the class of foods prepared by..., and skim milk cheese for manufacturing may not be used. All cheese ingredients used are either made... ___ cheese”, the name of the cheese filling the blank. (ii) If only parmesan and romano cheeses are used and...
A MANUALLY OPERATED CASSAVA GRATING MACHINE
African Journals Online (AJOL)
Dr Obe
1984-09-01
Sep 1, 1984 ... substantial losses arising from the inability of the person to hold small pieces of cassava roots for grating. Happily, there now exist various. Versions of mechanical graters which are driven by electric motors or small internal combustion engines. In fact, it may be said that cassava grating has been effectively.
On an Inequality Concerning the Polar Derivative of a Polynomial
Indian Academy of Sciences (India)
Abstract. In this paper, we present a correct proof of an -inequality concerning the polar derivative of a polynomial with restricted zeros. We also extend Zygmund's inequality to the polar derivative of a polynomial.
The potential of diffraction grating for spatial applications
Jourlin, Y.; Parriaux, O.; Pigeon, F.; Tischenko, A. V.
2017-11-01
Diffraction gratings are know, and have been fabricated for more than one century. They are now making a come back for two reasons: first, because they are now better understood which leads to the efficient exploitation of what was then called their "anomalies"; secondly, because they are now fabricable by means of the modern manufacturing potential of planar technologies. Novel grating can now perform better than conventional gratings, and address new application fields which were not expected to be theirs. This is the case of spatial applications where they can offer multiple optical functions, low size, low weight and mechanical robustness. The proposed contribution will briefly discuss the use of gratings for spatial applications. One of the most important applications is in the measurement of displacement. Usual translation and rotation sensors are bulky devices, which impose a system breakdown leading to cumbersome and heavy assemblies. We are proposing a miniaturized version of the traditional moving grating technique using submicron gratings and a specific OptoASIC which enables the measurement function to be non-obtrusively inserted into light and compact electro-mechanical systems. Nanometer resolution is possible with no compromise on the length of the measurement range. Another family of spatial application is in the field of spectrometers where new grating types allow a more flexible processing of the optical spectrum. Another family of applications addresses the question of inter-satellite communications: the introduction of gratings in laser cavities or in the laser mirrors enables the stabilization of the emitted polarization, the stabilization of the frequency as well as wide range frequency sweeping without mobile parts.
Classification of complex polynomial vector fields in one complex variable
DEFF Research Database (Denmark)
Branner, Bodil; Dias, Kealey
2010-01-01
This paper classifies the global structure of monic and centred one-variable complex polynomial vector fields. The classification is achieved by means of combinatorial and analytic data. More specifically, given a polynomial vector field, we construct a combinatorial invariant, describing...... the topology, and a set of analytic invariants, describing the geometry. Conversely, given admissible combinatorial and analytic data sets, we show using surgery the existence of a unique monic and centred polynomial vector field realizing the given invariants. This is the content of the Structure Theorem......, the main result of the paper. This result is an extension and refinement of Douady et al. (Champs de vecteurs polynomiaux sur C. Unpublished manuscript) classification of the structurally stable polynomial vector fields. We further review some general concepts for completeness and show that vector fields...
Design of compressors for FEL pulses using deformable gratings
Bonora, Stefano; Fabris, Nicola; Frassetto, Fabio; Giovine, Ennio; Miotti, Paolo; Quintavalla, Martino; Poletto, Luca
2017-06-01
We present the optical layout of soft X-rays compressors using reflective grating specifically designed to give both positive or negative group-delay dispersion (GDD). They are tailored for chirped-pulse-amplification experiments with FEL sources. The optical design originates from an existing compressor with plane gratings already realized and tested at FERMI, that has been demonstrated capable to introduce tunable negative GDD. Here, we discuss two novel designs for compressors using deformable gratings capable to give both negative and positive GDD. Two novel designs are discussed: 1) a design with two deformable gratings and an intermediate focus between the twos, that is demonstrated capable to introduce positive GDD; 2) a design with one deformable grating giving an intermediate focus, followed by a concave mirror and a plane grating, that is capable to give both positive and negative GDD depending on the distance between the second mirror and the second grating. Both the designs are tunable in wavelength and GDD, by acting on the deformable gratings, that are rotated to tune the wavelength and the GDD and deformed to introduce the radius required to keep the spectral focus. The deformable gratings have a laminar profile and are ruled on a thin silicon plane substrate. A piezoelectric actuator is glued on the back of the substrate and is actuated to give a radius of curvature that is varying from infinite (plane) to few meters. The ruling procedure, the piezoelectric actuator and the efficiency measurements in the soft X-rays will be presented. Some test cases are discussed for wavelengths shorter than 12 nm.
Grating-assisted surface acoustic wave directional couplers
Golan, G.; Griffel, G.; Seidman, A.; Croitoru, N.
1991-07-01
Physical properties of novel grating-assisted Y directional couplers are examined using the coupled-mode theory. A general formalism for the analysis of the lateral perturbed directional coupler properties is presented. Explicit expressions for waveguide key parameters such as coupling length, grating period, and other structural characterizations, are obtained. The influence of other physical properties such as time and frequency response or cutoff conditions are also analyzed. A plane grating-assisted directional coupler is presented and examined as a basic component in the integrated acoustic technology.
Sensitive visual test for concave diffraction gratings.
Bruner, E. C., Jr.
1972-01-01
A simple visual test for the evaluation of concave diffraction gratings is described. It is twice as sensitive as the Foucault knife edge test, from which it is derived, and has the advantage that the images are straight and free of astigmatism. It is particularly useful for grating with high ruling frequency where the above image faults limit the utility of the Foucault test. The test can be interpreted quantitatively and can detect zonal grating space errors of as little as 0.1 A.
Holographic grating relaxation technique for soft matter science
Energy Technology Data Exchange (ETDEWEB)
Lesnichii, Vasilii, E-mail: vasilii.lesnichii@physchem.uni-freiburg.de [Institute of Physical Chemistry, Albertstraße 21, Institute of Macromolecular Chemistry, Stefan-Meier-Str. 31, Albert-Ludwigs Universität, Freiburg im Breisgau 79104 (Germany); ITMO University, Kronverksky prospekt 49, Saint-Petersburg 197101 (Russian Federation); Kiessling, Andy [Institute of Physical Chemistry, Albertstraße 21, Institute of Macromolecular Chemistry, Stefan-Meier-Str. 31, Albert-Ludwigs Universität, Freiburg im Breisgau 79104 (Germany); Current address: Illinois Institute of Technology, 10 West 33rd Street, Chicago,IL60616 (United States); Bartsch, Eckhard [Institute of Physical Chemistry, Albertstraße 21, Institute of Macromolecular Chemistry, Stefan-Meier-Str. 31, Albert-Ludwigs Universität, Freiburg im Breisgau 79104 (Germany); Veniaminov, Andrey, E-mail: veniaminov@phoi.ifmo.ru [ITMO University, Kronverksky prospekt 49, Saint-Petersburg 197101 (Russian Federation)
2016-06-17
The holographic grating relaxation technique also known as forced Rayleigh scattering consists basically in writing a holographic grating in the specimen of interest and monitoring its diffraction efficiency as a function of time, from which valuable information on mass or heat transfer and photoinduced transformations can be extracted. In a more detailed view, the shape of the relaxation curve and the relaxation rate as a function of the grating period were found to be affected by the architecture of diffusing species (molecular probes) that constitute the grating, as well as that of the environment they diffuse in, thus making it possible to access and study spatial heterogeneity of materials and different modes of e.g., polymer motion. Minimum displacements and spatial domains approachable by the technique are in nanometer range, well below spatial periods of holographic gratings. In the present paper, several cases of holographic relaxation in heterogeneous media and complex motions are exemplified. Nano- to micro-structures or inhomogeneities comparable in spatial scale with holographic gratings manifest themselves in relaxation experiments via non-exponential decay (stepwise or stretched), spatial-period-dependent apparent diffusion coefficient, or unusual dependence of diffusion coefficient on molecular volume of diffusing probes.
International Nuclear Information System (INIS)
Brasil, D A; Alves, J A P; Pekelsky, J R
2015-01-01
This work describes the development of a grating diffratometer to provide traceable calibration of grating pitch in range 20 μm to 350 nm. The approach is based on the Littrow configuration in which a laser beam is directed onto the grating which is mounted on a rotary table and can be turned so that each selected diffraction order is retro-reflected in the laser incidence direction. A beamsplitter and a lens direct the reflected diffraction order to form a small image spot on a CCD camera and the spot centering is used to adjust to rotation angle, thereby giving the diffraction angle. Knowing the diffraction angle for several orders and the wavelength of the laser, the average grating pitch can be determined to an uncertainty the order of 14 pm. (paper)
Self Referencing Heterodyne Transient Grating Spectroscopy with Short Wavelength
Directory of Open Access Journals (Sweden)
Jakob Grilj
2015-04-01
Full Text Available Heterodyning by a phase stable reference electric field is a well known technique to amplify weak nonlinear signals. For short wavelength, the generation of a reference field in front of the sample is challenging because of a lack of suitable beamsplitters. Here, we use a permanent grating which matches the line spacing of the transient grating for the creation of a phase stable reference field. The relative phase among the two can be changed by a relative translation of the permanent and transient gratings in direction orthogonal to the grating lines. We demonstrate the technique for a transient grating on a VO2 thin film and observe constructive as well as destructive interference signals.
Unified design of sinusoidal-groove fused-silica grating.
Feng, Jijun; Zhou, Changhe; Cao, Hongchao; Lu, Peng
2010-10-20
A general design rule of deep-etched subwavelength sinusoidal-groove fused-silica grating as a highly efficient polarization-independent or polarization-selective device is studied based on the simplified modal method, which shows that the device structure depends little on the incident wavelength, but mainly on the ratio of groove depth to incident wavelength and the ratio of wavelength to grating period. These two ratios could be used as the design guidelines for wavelength-independent structure from deep ultraviolet to far infrared. The optimized grating profile with a different function as a polarizing beam splitter, a polarization-independent two-port beam splitter, or a polarization-independent grating with high efficiency of -1st order is obtained at a wavelength of 1064 nm, and verified by using the rigorous coupled-wave analysis. The performance of the sinusoidal grating is better than a conventional rectangular one, which could be useful for practical applications.
Random polynomials and expected complexity of bisection methods for real solving
DEFF Research Database (Denmark)
Emiris, Ioannis Z.; Galligo, André; Tsigaridas, Elias
2010-01-01
, and by Edelman and Kostlan in order to estimate the real root separation of degree d polynomials with i.i.d. coefficients that follow two zero-mean normal distributions: for SO(2) polynomials, the i-th coefficient has variance (d/i), whereas for Weyl polynomials its variance is 1/i!. By applying results from....... The second part of the paper shows that the expected number of real roots of a degree d polynomial in the Bernstein basis is √2d ± O(1), when the coefficients are i.i.d. variables with moderate standard deviation. Our paper concludes with experimental results which corroborate our analysis....
O(N) symmetries, sum rules for generalized Hermite polynomials and squeezed states
International Nuclear Information System (INIS)
Daboul, Jamil; Mizrahi, Salomon S
2005-01-01
Quantum optics has been dealing with coherent states, squeezed states and many other non-classical states. The associated mathematical framework makes use of special functions as Hermite polynomials, Laguerre polynomials and others. In this connection we here present some formal results that follow directly from the group O(N) of complex transformations. Motivated by the squeezed states structure, we introduce the generalized Hermite polynomials (GHP), which include as particular cases, the Hermite polynomials as well as the heat polynomials. Using generalized raising operators, we derive new sum rules for the GHP, which are covariant under O(N) transformations. The GHP and the associated sum rules become useful for evaluating Wigner functions in a straightforward manner. As a byproduct, we use one of these sum rules, on the operator level, to obtain raising and lowering operators for the Laguerre polynomials and show that they generate an sl(2, R) ≅ su(1, 1) algebra
Euler polynomials and identities for non-commutative operators
De Angelis, Valerio; Vignat, Christophe
2015-12-01
Three kinds of identities involving non-commutating operators and Euler and Bernoulli polynomials are studied. The first identity, as given by Bender and Bettencourt [Phys. Rev. D 54(12), 7710-7723 (1996)], expresses the nested commutator of the Hamiltonian and momentum operators as the commutator of the momentum and the shifted Euler polynomial of the Hamiltonian. The second one, by Pain [J. Phys. A: Math. Theor. 46, 035304 (2013)], links the commutators and anti-commutators of the monomials of the position and momentum operators. The third appears in a work by Figuieira de Morisson and Fring [J. Phys. A: Math. Gen. 39, 9269 (2006)] in the context of non-Hermitian Hamiltonian systems. In each case, we provide several proofs and extensions of these identities that highlight the role of Euler and Bernoulli polynomials.
On integral and finite Fourier transforms of continuous q-Hermite polynomials
International Nuclear Information System (INIS)
Atakishiyeva, M. K.; Atakishiyev, N. M.
2009-01-01
We give an overview of the remarkably simple transformation properties of the continuous q-Hermite polynomials H n (x vertical bar q) of Rogers with respect to the classical Fourier integral transform. The behavior of the q-Hermite polynomials under the finite Fourier transform and an explicit form of the q-extended eigenfunctions of the finite Fourier transform, defined in terms of these polynomials, are also discussed.
On polynomial selection for the general number field sieve
Kleinjung, Thorsten
2006-12-01
The general number field sieve (GNFS) is the asymptotically fastest algorithm for factoring large integers. Its runtime depends on a good choice of a polynomial pair. In this article we present an improvement of the polynomial selection method of Montgomery and Murphy which has been used in recent GNFS records.
Grateful Med: getting started.
Shearer, B; McCann, L; Crump, W J
1990-01-01
When a local medical library is not available, it is often necessary for physicians to discover alternate ways to receive medical information. Rural physicians, particularly, can make use of a computer program called Grateful Med that provides access to the same literature available to physicians in large cities. This program permits the user to perform database searches on the National Library of Medicine database (MEDLINE), corresponding to the primary index to medical literature, Index Medicus. In this article, we give the procedure for procuring a National Library of Medicine password and for making efficient use of the Grateful Med program.
Automorphisms of Algebras and Bochner's Property for Vector Orthogonal Polynomials
Horozov, Emil
2016-05-01
We construct new families of vector orthogonal polynomials that have the property to be eigenfunctions of some differential operator. They are extensions of the Hermite and Laguerre polynomial systems. A third family, whose first member has been found by Y. Ben Cheikh and K. Douak is also constructed. The ideas behind our approach lie in the studies of bispectral operators. We exploit automorphisms of associative algebras which transform elementary vector orthogonal polynomial systems which are eigenfunctions of a differential operator into other systems of this type.
Iridescence in Meat Caused by Surface Gratings
Directory of Open Access Journals (Sweden)
Ali Kemal Yetisen
2013-11-01
Full Text Available The photonic structure of cut muscle tissues reveals that the well-ordered gratings diffract light, producing iridescent colours. Cut fibrils protruding from the muscle surface create a two-dimensional periodic array, which diffract light at specific wavelengths upon illumination. However, this photonic effect misleads consumers in a negative way to relate the optical phenomenon with the quality of the product. Here we discuss the fundamentals of this optical phenomenon and demonstrate a methodology for quantitatively measuring iridescence caused by diffraction gratings of muscle tissue surface of pork (Sus scrofa domesticus using reflection spectrophotometry. Iridescence was discussed theoretically as a light phenomenon and spectral measurements were taken from the gratings and monitored in real time during controlled drying. The findings show that the intensity of diffraction diminishes as the surface grating was dried with an air flow at 50 °C for 2 min while the diffracted light wavelength was at 585 ± 9 nm. Our findings indicate that the diffraction may be caused by a blazed surface grating. The implications of the study include providing guidelines to minimise the iridescence by altering the surface microstructure, and in consequence, removing the optical effect.
Development of a segmented grating mount system for FIREX-1
International Nuclear Information System (INIS)
Ezaki, Y; Tabata, M; Kihara, M; Horiuchi, Y; Endo, M; Jitsuno, T
2008-01-01
A mount system for segmented meter-sized gratings has been developed, which has a high precision grating support mechanism and drive mechanism to minimize both deformation of the optical surfaces and misalignments in setting a segmented grating for obtaining sufficient performance of the pulse compressor. From analytical calculations, deformation of the grating surface is less than 1/20 lambda RMS and the estimated drive resolution for piston and tilt drive of the segmented grating is 1/20 lambda, which are both compliant with the requirements for the rear-end subsystem of FIREX-1
Holographic diffraction gratings as laser radiation protection filters
International Nuclear Information System (INIS)
Pantelic, D.; Pantelic, G.
2006-01-01
Holographic volume diffraction gratings are used as attenuation filters, due to their selective spectral transmission. They can be tailored to reflect or transmit narrow spectral ranges by adjusting spatial frequency of Bragg grating in carefully chosen photosensitive materials, like silver-halide emulsion or di-chromated gelatin layers. If properly recorded and chemically processed, resulting gratings can significantly attenuate light at wavelengths corresponding to various laser spectral lines. Thus, they can be used as filters in laser protection goggles. We analyze the characteristics of Bragg gratings necessary to obtain high attenuation coefficients. Also, their angular selectivity is taken into account and corresponding experimental conditions are investigated. Although di-chromated gelatin seems to be almost ideal material, due to its almost 100% diffraction efficiency, environmental stability is poor (degradation under humid environment), thus making its practical usage difficult. Thus, we have analyzed alternative materials like di-chromated pullulan, which is stable under normal environmental conditions (without drop in diffraction efficiency after prolonged exposure to humidity). Pullulan is polymer (polysaccharide) of biologic origin produced by certain bacteria. If doped with chromium ions it becomes photosensitive, enabling recording of diffraction gratings with spatial frequency of more than 3000 lines/mm. Material is chemically processed by mixture of isopropyl alcohol and water. Both thick and thin layers can be produced by gravity settling. Spectral properties of resulting gratings are analyzed, showing that they can significantly attenuate laser light of particular wavelength, depending of grating period and its slant angle. (authors)
Model based control of grate combustion; Modellbaserad roststyrning
Energy Technology Data Exchange (ETDEWEB)
Broden, Henrik; Kjellstroem, Bjoern; Niklasson, Fredrik; Boecher Poulsen, Kristian
2006-12-15
An existing dynamic model for grate combustion has been further developed. The model has been used for studies of possible advantages that can be gained from utilisation of measurements of grate temperatures and fuel bed height for control of a boiler after disturbances caused by varying fuel moisture and fuel feeding. The objective was to asses the possibilities to develop a control system that would adjust for such disturbances quicker than measurements of steam output and oxygen in the exhaust. The model is based on dividing the fuel bed into three layers, where the different layers include fuel being dried, fuel being pyrolysed and char reacting with oxygen. The grate below the fuel bed is also considered. A mass balance, an energy balance and a volume balance is considered for each layer in 22 cells along the grate. The energy balances give the temperature distribution and the volume balances the bed height. The earlier version of the model could not handle layers that are consumed. This weakness has now been eliminated. Comparisons between predicted grate temperatures and measurements in a 25 MW boiler fuelled with biofuel have been used for validation of the model. The comparisons include effects of variations in primary air temperature, fuel moisture and output power. The model shows good agreement with observations for changes in the air temperature but the ability of the model to predict effects of changed fuel moisture is difficult to judge since the steam dome pressure control caused simultaneous changes of the primary air flow, which probably had a larger influence on the grate temperature. A linearised, tuned and reduced version of the model was used for design of a linear quadratic controller. This was used for studies of advantages of using measurements of grate temperatures and bed height for control of pusher velocity, grate speed, primary air flow and air temperature after disturbances of fuel moisture and fuel flow. Measurements of the grate
Thermal annealing of tilted fiber Bragg gratings
González-Vila, Á.; Rodríguez-Cobo, L.; Mégret, P.; Caucheteur, C.; López-Higuera, J. M.
2016-05-01
We report a practical study of the thermal decay of cladding mode resonances in tilted fiber Bragg gratings, establishing an analogy with the "power law" evolution previously observed on uniform gratings. We examine how this process contributes to a great thermal stability, even improving it by means of a second cycle slightly increasing the annealing temperature. In addition, we show an improvement of the grating spectrum after annealing, with respect to the one just after inscription, which suggests the application of this method to be employed to improve saturation issues during the photo-inscription process.
Families of superintegrable Hamiltonians constructed from exceptional polynomials
International Nuclear Information System (INIS)
Post, Sarah; Tsujimoto, Satoshi; Vinet, Luc
2012-01-01
We introduce a family of exactly-solvable two-dimensional Hamiltonians whose wave functions are given in terms of Laguerre and exceptional Jacobi polynomials. The Hamiltonians contain purely quantum terms which vanish in the classical limit leaving only a previously known family of superintegrable systems. Additional, higher-order integrals of motion are constructed from ladder operators for the considered orthogonal polynomials proving the quantum system to be superintegrable. (paper)
Raising and Lowering Operators for Askey-Wilson Polynomials
Directory of Open Access Journals (Sweden)
Siddhartha Sahi
2007-01-01
Full Text Available In this paper we describe two pairs of raising/lowering operators for Askey-Wilson polynomials, which result from constructions involving very different techniques. The first technique is quite elementary, and depends only on the ''classical'' properties of these polynomials, viz. the q-difference equation and the three term recurrence. The second technique is less elementary, and involves the one-variable version of the double affine Hecke algebra.
Echelle grating multi-order imaging spectrometer utilizing a catadioptric lens
Chrisp, Michael P; Bowers, Joel M
2014-05-27
A cryogenically cooled imaging spectrometer that includes a spectrometer housing having a first side and a second side opposite the first side. An entrance slit is on the first side of the spectrometer housing and directs light to a cross-disperser grating. An echelle immersions grating and a catadioptric lens are positioned in the housing to receive the light. A cryogenically cooled detector is located in the housing on the second side of the spectrometer housing. Light from the entrance slit is directed to the cross-disperser grating. The light is directed from the cross-disperser grating to the echelle immersions grating. The light is directed from the echelle immersions grating to the cryogenically cooled detector on the second side of the spectrometer housing.
Phasor analysis of binary diffraction gratings with different fill factors
International Nuclear Information System (INIS)
MartInez, Antonio; Sanchez-Lopez, Ma del Mar; Moreno, Ignacio
2007-01-01
In this work, we present a simple analysis of binary diffraction gratings with different slit widths relative to the grating period. The analysis is based on a simple phasor technique directly derived from the Huygens principle. By introducing a slit phasor and a grating phasor, the intensity of the diffracted orders and the grating's resolving power can be easily obtained without applying the usual Fourier transform operations required for these calculations. The proposed phasor technique is mathematically equivalent to the Fourier transform calculation of the diffraction order amplitude, and it can be useful to explain binary diffraction gratings in a simple manner in introductory physics courses. This theoretical analysis is illustrated with experimental results using a liquid crystal device to display diffraction gratings with different fill factors
Díaz Mendoza, C.; Orive, R.; Pijeira Cabrera, H.
2008-10-01
We study the asymptotic behavior of the zeros of a sequence of polynomials whose weighted norms, with respect to a sequence of weight functions, have the same nth root asymptotic behavior as the weighted norms of certain extremal polynomials. This result is applied to obtain the (contracted) weak zero distribution for orthogonal polynomials with respect to a Sobolev inner product with exponential weights of the form e-[phi](x), giving a unified treatment for the so-called Freud (i.e., when [phi] has polynomial growth at infinity) and Erdös (when [phi] grows faster than any polynomial at infinity) cases. In addition, we provide a new proof for the bound of the distance of the zeros to the convex hull of the support for these Sobolev orthogonal polynomials.
Mathematical Use Of Polynomials Of Different End Periods Of ...
African Journals Online (AJOL)
This paper focused on how polynomials of different end period of random numbers can be used in the application of encryption and decryption of a message. Eight steps were used in generating information on how polynomials of different end periods of random numbers in the application of encryption and decryption of a ...
Overview of diffraction gratings technologies for spaceflight satellites and ground-based telescopes
Cotel, A.; Liard, A.; Desserouer, F.; Pichon, P.
2017-11-01
The diffraction gratings are widely used in Space-flight satellites for spectrograph instruments or in ground-based telescopes in astronomy. The diffraction gratings are one of the key optical components of such systems and have to exhibit very high optical performances. HORIBA Jobin Yvon S.A.S. (part of HORIBA Group) is in the forefront of such gratings development for more than 40 years. During the past decades, HORIBA Jobin Yvon (HJY) has developed a unique expertise in diffraction grating design and manufacturing processes for holographic, ruled or etched gratings. We will present in this paper an overview of diffraction grating technologies especially designed for space and astronomy applications. We will firstly review the heritage of the company in this field with the space qualification of different grating types. Then, we will describe several key grating technologies developed for specific space or astronomy projects: ruled blazed low groove density plane reflection grating, high-groove density holographic toroidal and spherical grating, and finally transmission Fused Silica Etched (FSE) grism-assembled grating. We will not present the Volume Phase Holographic (VPHG) grating type which is used in Astronomy.
Computing derivative-based global sensitivity measures using polynomial chaos expansions
International Nuclear Information System (INIS)
Sudret, B.; Mai, C.V.
2015-01-01
In the field of computer experiments sensitivity analysis aims at quantifying the relative importance of each input parameter (or combinations thereof) of a computational model with respect to the model output uncertainty. Variance decomposition methods leading to the well-known Sobol' indices are recognized as accurate techniques, at a rather high computational cost though. The use of polynomial chaos expansions (PCE) to compute Sobol' indices has allowed to alleviate the computational burden though. However, when dealing with large dimensional input vectors, it is good practice to first use screening methods in order to discard unimportant variables. The derivative-based global sensitivity measures (DGSMs) have been developed recently in this respect. In this paper we show how polynomial chaos expansions may be used to compute analytically DGSMs as a mere post-processing. This requires the analytical derivation of derivatives of the orthonormal polynomials which enter PC expansions. Closed-form expressions for Hermite, Legendre and Laguerre polynomial expansions are given. The efficiency of the approach is illustrated on two well-known benchmark problems in sensitivity analysis. - Highlights: • Derivative-based global sensitivity measures (DGSM) have been developed for screening purpose. • Polynomial chaos expansions (PC) are used as a surrogate model of the original computational model. • From a PC expansion the DGSM can be computed analytically. • The paper provides the derivatives of Hermite, Legendre and Laguerre polynomials for this purpose
Exact polynomial solutions of second order differential equations and their applications
International Nuclear Information System (INIS)
Zhang Yaozhong
2012-01-01
We find all polynomials Z(z) such that the differential equation where X(z), Y(z), Z(z) are polynomials of degree at most 4, 3, 2, respectively, has polynomial solutions S(z) = ∏ n i=1 (z − z i ) of degree n with distinct roots z i . We derive a set of n algebraic equations which determine these roots. We also find all polynomials Z(z) which give polynomial solutions to the differential equation when the coefficients of X(z) and Y(z) are algebraically dependent. As applications to our general results, we obtain the exact (closed-form) solutions of the Schrödinger-type differential equations describing: (1) two Coulombically repelling electrons on a sphere; (2) Schrödinger equation from the kink stability analysis of φ 6 -type field theory; (3) static perturbations for the non-extremal Reissner–Nordström solution; (4) planar Dirac electron in Coulomb and magnetic fields; and (5) O(N) invariant decatic anharmonic oscillator. (paper)
The Jones polynomial as a new invariant of topological fluid dynamics
International Nuclear Information System (INIS)
Ricca, Renzo L; Liu, Xin
2014-01-01
A new method based on the use of the Jones polynomial, a well-known topological invariant of knot theory, is introduced to tackle and quantify topological aspects of structural complexity of vortex tangles in ideal fluids. By re-writing the Jones polynomial in terms of helicity, the resulting polynomial becomes then function of knot topology and vortex circulation, providing thus a new invariant of topological fluid dynamics. Explicit computations of the Jones polynomial for some standard configurations, including the Whitehead link and the Borromean rings (whose linking numbers are zero), are presented for illustration. In the case of a homogeneous, isotropic tangle of vortex filaments with same circulation, the new Jones polynomial reduces to some simple algebraic expression, that can be easily computed by numerical methods. This shows that this technique may offer a new setting and a powerful tool to detect and compute topological complexity and to investigate relations with energy, by tackling fundamental aspects of turbulence research. (paper)
PLOTNFIT.4TH, Data Plotting and Curve Fitting by Polynomials
International Nuclear Information System (INIS)
Schiffgens, J.O.
1990-01-01
1 - Description of program or function: PLOTnFIT is used for plotting and analyzing data by fitting nth degree polynomials of basis functions to the data interactively and printing graphs of the data and the polynomial functions. It can be used to generate linear, semi-log, and log-log graphs and can automatically scale the coordinate axes to suit the data. Multiple data sets may be plotted on a single graph. An auxiliary program, READ1ST, is included which produces an on-line summary of the information contained in the PLOTnFIT reference report. 2 - Method of solution: PLOTnFIT uses the least squares method to calculate the coefficients of nth-degree (up to 10. degree) polynomials of 11 selected basis functions such that each polynomial fits the data in a least squares sense. The procedure incorporated in the code uses a linear combination of orthogonal polynomials to avoid 'i11-conditioning' and to perform the curve fitting task with single-precision arithmetic. 3 - Restrictions on the complexity of the problem - Maxima of: 225 data points per job (or graph) including all data sets 8 data sets (or tasks) per job (or graph)
Multivariate Local Polynomial Regression with Application to Shenzhen Component Index
Directory of Open Access Journals (Sweden)
Liyun Su
2011-01-01
Full Text Available This study attempts to characterize and predict stock index series in Shenzhen stock market using the concepts of multivariate local polynomial regression. Based on nonlinearity and chaos of the stock index time series, multivariate local polynomial prediction methods and univariate local polynomial prediction method, all of which use the concept of phase space reconstruction according to Takens' Theorem, are considered. To fit the stock index series, the single series changes into bivariate series. To evaluate the results, the multivariate predictor for bivariate time series based on multivariate local polynomial model is compared with univariate predictor with the same Shenzhen stock index data. The numerical results obtained by Shenzhen component index show that the prediction mean squared error of the multivariate predictor is much smaller than the univariate one and is much better than the existed three methods. Even if the last half of the training data are used in the multivariate predictor, the prediction mean squared error is smaller than the univariate predictor. Multivariate local polynomial prediction model for nonsingle time series is a useful tool for stock market price prediction.
Twisted Polynomials and Forgery Attacks on GCM
DEFF Research Database (Denmark)
Abdelraheem, Mohamed Ahmed A. M. A.; Beelen, Peter; Bogdanov, Andrey
2015-01-01
Polynomial hashing as an instantiation of universal hashing is a widely employed method for the construction of MACs and authenticated encryption (AE) schemes, the ubiquitous GCM being a prominent example. It is also used in recent AE proposals within the CAESAR competition which aim at providing...... in an improved key recovery algorithm. As cryptanalytic applications of our twisted polynomials, we develop the first universal forgery attacks on GCM in the weak-key model that do not require nonce reuse. Moreover, we present universal weak-key forgeries for the nonce-misuse resistant AE scheme POET, which...
A polynomial based model for cell fate prediction in human diseases.
Ma, Lichun; Zheng, Jie
2017-12-21
Cell fate regulation directly affects tissue homeostasis and human health. Research on cell fate decision sheds light on key regulators, facilitates understanding the mechanisms, and suggests novel strategies to treat human diseases that are related to abnormal cell development. In this study, we proposed a polynomial based model to predict cell fate. This model was derived from Taylor series. As a case study, gene expression data of pancreatic cells were adopted to test and verify the model. As numerous features (genes) are available, we employed two kinds of feature selection methods, i.e. correlation based and apoptosis pathway based. Then polynomials of different degrees were used to refine the cell fate prediction function. 10-fold cross-validation was carried out to evaluate the performance of our model. In addition, we analyzed the stability of the resultant cell fate prediction model by evaluating the ranges of the parameters, as well as assessing the variances of the predicted values at randomly selected points. Results show that, within both the two considered gene selection methods, the prediction accuracies of polynomials of different degrees show little differences. Interestingly, the linear polynomial (degree 1 polynomial) is more stable than others. When comparing the linear polynomials based on the two gene selection methods, it shows that although the accuracy of the linear polynomial that uses correlation analysis outcomes is a little higher (achieves 86.62%), the one within genes of the apoptosis pathway is much more stable. Considering both the prediction accuracy and the stability of polynomial models of different degrees, the linear model is a preferred choice for cell fate prediction with gene expression data of pancreatic cells. The presented cell fate prediction model can be extended to other cells, which may be important for basic research as well as clinical study of cell development related diseases.
Transmission Grating and Optics Technology Development for the Arcus Explorer Mission
Heilmann, Ralf; Arcus Team
2018-01-01
Arcus is a high-resolution x-ray spectroscopy MIDEX mission selected for a Phase A concept study. It is designed to explore structure formation through measurements of hot baryon distributions, feedback from black holes, and the formation and evolution of stars, disks, and exoplanet atmospheres. The design provides unprecedented sensitivity in the 1.2-5 nm wavelength band with effective area above 450 sqcm and spectral resolution R > 2500. The Arcus technology is based on 12 m-focal length silicon pore optics (SPO) developed for the European Athena mission, and critical-angle transmission (CAT) x-ray diffraction gratings and x-ray CCDs developed at MIT. The modular design consists of four parallel channels, each channel holding an optics petal, followed by a grating petal. CAT gratings are lightweight, alignment insensitive, high-efficiency x-ray transmission gratings that blaze into high diffraction orders, leading to high spectral resolution. Each optics petal represents an azimuthal sub-aperture of a full Wolter optic. The sub-aperturing effect increases spectral resolving power further. Two CCD readout strips receive photons from each channel, including higher-energy photons in 0th order. Each optics petal holds 34 SPO modules. Each grating petal holds 34 grating windows, and each window holds 4-6 grating facets. A grating facet consists of a silicon grating membrane, bonded to a flexure frame that interfaces with the grating window. We report on a sequence of tests with increasing complexity that systematically increase the Technology Readiness Level (TRL) for the combination of CAT gratings and SPOs towards TLR 6. CAT gratings have been evaluated in x rays for diffraction efficiency (> 30% at 2.5 nm) and for resolving power (R> 10,000). A CAT grating/SPO combination was measured at R ~ 3100 at blaze angles smaller than design values, exceeding Arcus requirements. Efficiency and resolving power were not impacted by vibration and thermal testing of gratings. A
Calculation of Smith-Purcell radiation from a volume strip grating
International Nuclear Information System (INIS)
Kube, G.
2005-01-01
Smith-Purcell radiation is generated by a charged particle beam passing close to the surface of a diffraction grating. Experimental investigations show a strong dependency of the emitted radiation intensity on the form of the grating profile. This influence is expressed by the radiation factor which is a measure of the grating efficiency, in close analogy to reflection coefficients of optical grating theories. The radiation factor depends on beam energy and observation geometry. Up to now calculations for radiation factors exist for lamellar, sinusoidal and echelette-type grating profiles. In this paper, calculations of Smith-Purcell radiation factors for volume strip gratings which are separated by vacuum gaps are presented. They are based on the modal expansion method and restricted to perfectly conducting grating surfaces and to electron trajectories perpendicular to the grating grooves. An infinite system of coupled linear algebraic equations for the scattered and the transmitted wave amplitudes is derived by imposing the continuity condition at the open end of the grooves, and by the boundary conditions at the remaining part of the interface. Numerical results are presented and discussed in view of using Smith-Purcell radiation for particle beam diagnostic purposes
Analysis of surface absorbed dose in X-ray grating interferometry
Energy Technology Data Exchange (ETDEWEB)
Wang, Zhili, E-mail: wangnsrl@ustc.edu.cn [National Synchrotron Radiation Laboratory, University of Science and Technology of China, Hefei 230026 (China); Wu, Zhao; Gao, Kun; Wang, Dajiang; Chen, Heng; Wang, Shenghao [National Synchrotron Radiation Laboratory, University of Science and Technology of China, Hefei 230026 (China); Wu, Ziyu, E-mail: wuzy@ustc.edu.cn [National Synchrotron Radiation Laboratory, University of Science and Technology of China, Hefei 230026 (China); Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049 (China)
2014-10-15
Highlights: • Theoretical framework for dose estimation in X-ray grating interferometry. • Potential dose reduction of X-ray grating interferometry compared to conventional radiography. • Guidelines for optimization of X-ray grating interferometry for dose-sensitive applications. • Measure to compare various existing X-ray phase contrast imaging techniques. - Abstract: X-ray phase contrast imaging using grating interferometry has shown increased contrast over conventional absorption imaging, and therefore the great potential of dose reduction. The extent of the dose reduction depends on the geometry of grating interferometry, the photon energy, the properties of the sample under investigation and the utilized detector. These factors also determine the capability of grating interferometry to distinguish between different tissues with a specified statistical certainty in a single raw image. In this contribution, the required photon number for imaging and the resulting surface absorbed dose are determined in X-ray grating interferometry, using a two-component imaging object model. The presented results confirm that compared to conventional radiography, phase contrast imaging using grating interferometry indeed has the potential of dose reduction. And the extent of dose reduction is strongly dependent on the imaging conditions. Those results provide a theoretical framework for dose estimation under given imaging conditions before experimental trials, and general guidelines for optimization of grating interferometry for those dose-sensitive applications.
Analysis of surface absorbed dose in X-ray grating interferometry
International Nuclear Information System (INIS)
Wang, Zhili; Wu, Zhao; Gao, Kun; Wang, Dajiang; Chen, Heng; Wang, Shenghao; Wu, Ziyu
2014-01-01
Highlights: • Theoretical framework for dose estimation in X-ray grating interferometry. • Potential dose reduction of X-ray grating interferometry compared to conventional radiography. • Guidelines for optimization of X-ray grating interferometry for dose-sensitive applications. • Measure to compare various existing X-ray phase contrast imaging techniques. - Abstract: X-ray phase contrast imaging using grating interferometry has shown increased contrast over conventional absorption imaging, and therefore the great potential of dose reduction. The extent of the dose reduction depends on the geometry of grating interferometry, the photon energy, the properties of the sample under investigation and the utilized detector. These factors also determine the capability of grating interferometry to distinguish between different tissues with a specified statistical certainty in a single raw image. In this contribution, the required photon number for imaging and the resulting surface absorbed dose are determined in X-ray grating interferometry, using a two-component imaging object model. The presented results confirm that compared to conventional radiography, phase contrast imaging using grating interferometry indeed has the potential of dose reduction. And the extent of dose reduction is strongly dependent on the imaging conditions. Those results provide a theoretical framework for dose estimation under given imaging conditions before experimental trials, and general guidelines for optimization of grating interferometry for those dose-sensitive applications
Grating geophone signal processing based on wavelet transform
Li, Shuqing; Zhang, Huan; Tao, Zhifei
2008-12-01
Grating digital geophone is designed based on grating measurement technique benefiting averaging-error effect and wide dynamic range to improve weak signal detected precision. This paper introduced the principle of grating digital geophone and its post signal processing system. The signal acquisition circuit use Atmega 32 chip as core part and display the waveform on the Labwindows through the RS232 data link. Wavelet transform is adopted this paper to filter the grating digital geophone' output signal since the signal is unstable. This data processing method is compared with the FIR filter that widespread use in current domestic. The result indicates that the wavelet algorithm has more advantages and the SNR of seismic signal improve obviously.
Fractional order differentiation by integration with Jacobi polynomials
Liu, Dayan
2012-12-01
The differentiation by integration method with Jacobi polynomials was originally introduced by Mboup, Join and Fliess [22], [23]. This paper generalizes this method from the integer order to the fractional order for estimating the fractional order derivatives of noisy signals. The proposed fractional order differentiator is deduced from the Jacobi orthogonal polynomial filter and the Riemann-Liouville fractional order derivative definition. Exact and simple formula for this differentiator is given where an integral formula involving Jacobi polynomials and the noisy signal is used without complex mathematical deduction. Hence, it can be used both for continuous-time and discrete-time models. The comparison between our differentiator and the recently introduced digital fractional order Savitzky-Golay differentiator is given in numerical simulations so as to show its accuracy and robustness with respect to corrupting noises. © 2012 IEEE.
Synchronization of generalized Henon map using polynomial controller
International Nuclear Information System (INIS)
Lam, H.K.
2010-01-01
This Letter presents the chaos synchronization of two discrete-time generalized Henon map, namely the drive and response systems. A polynomial controller is proposed to drive the system states of the response system to follow those of the drive system. The system stability of the error system formed by the drive and response systems and the synthesis of the polynomial controller are investigated using the sum-of-squares (SOS) technique. Based on the Lyapunov stability theory, stability conditions in terms of SOS are derived to guarantee the system stability and facilitate the controller synthesis. By satisfying the SOS-based stability conditions, chaotic synchronization is achieved. The solution of the SOS-based stability conditions can be found numerically using the third-party Matlab toolbox SOSTOOLS. A simulation example is given to illustrate the merits of the proposed polynomial control approach.
Fractional order differentiation by integration with Jacobi polynomials
Liu, Dayan; Gibaru, O.; Perruquetti, Wilfrid; Laleg-Kirati, Taous-Meriem
2012-01-01
The differentiation by integration method with Jacobi polynomials was originally introduced by Mboup, Join and Fliess [22], [23]. This paper generalizes this method from the integer order to the fractional order for estimating the fractional order derivatives of noisy signals. The proposed fractional order differentiator is deduced from the Jacobi orthogonal polynomial filter and the Riemann-Liouville fractional order derivative definition. Exact and simple formula for this differentiator is given where an integral formula involving Jacobi polynomials and the noisy signal is used without complex mathematical deduction. Hence, it can be used both for continuous-time and discrete-time models. The comparison between our differentiator and the recently introduced digital fractional order Savitzky-Golay differentiator is given in numerical simulations so as to show its accuracy and robustness with respect to corrupting noises. © 2012 IEEE.
Smith-Purcell radiation from concave dotted gratings
Sergeeva, D. Yu.; Tishchenko, A. A.; Aryshev, A. S.; Strikhanov, M. N.
2018-02-01
We present the first-principles theory of Smith-Purcell effect from the concave dotted grating consisting of bent chains of separated micro- or nanoparticles. The numerical analysis demonstrates that the obtained spectral-angular distributions change significantly depending on the structure of the grating.
Three-port beam splitter of a binary fused-silica grating.
Feng, Jijun; Zhou, Changhe; Wang, Bo; Zheng, Jiangjun; Jia, Wei; Cao, Hongchao; Lv, Peng
2008-12-10
A deep-etched polarization-independent binary fused-silica phase grating as a three-port beam splitter is designed and manufactured. The grating profile is optimized by use of the rigorous coupled-wave analysis around the 785 nm wavelength. The physical explanation of the grating is illustrated by the modal method. Simple analytical expressions of the diffraction efficiencies and modal guidelines for the three-port beam splitter grating design are given. Holographic recording technology and inductively coupled plasma etching are used to manufacture the fused-silica grating. Experimental results are in good agreement with the theoretical values.
Diffraction Efficiency Testing of Sinusoidal and Blazed Off-Plane Reflection Gratings
Tutt, James H.; McEntaffer, Randall L.; Marlowe, Hannah; Miles, Drew M.; Peterson, Thomas J.; Deroo, Casey T.; Scholze, Frank; Laubis, Christian
2016-09-01
Reflection gratings in the off-plane mount have the potential to enhance the performance of future high resolution soft X-ray spectrometers. Diffraction efficiency can be optimized through the use of blazed grating facets, achieving high-throughput on one side of zero-order. This paper presents the results from a comparison between a grating with a sinusoidally grooved profile and two gratings that have been blazed. The results show that the blaze does increase throughput to one side of zero-order; however, the total throughput of the sinusoidal gratings is greater than the blazed gratings, suggesting the method of manufacturing the blazed gratings does not produce precise facets. The blazed gratings were also tested in their Littrow and anti-Littrow configurations to quantify diffraction efficiency sensitivity to rotations about the grating normal. Only a small difference in the energy at which efficiency is maximized between the Littrow and anti-Littrow configurations is seen with a small shift in peak efficiency towards higher energies in the anti-Littrow case. This is due to a decrease in the effective blaze angle in the anti-Littrow mounting. This is supported by PCGrate-SX V6.1 modeling carried out for each blazed grating which predicts similar response trends in the Littrow and anti-Littrow orientations.
Phasor analysis of binary diffraction gratings with different fill factors
Energy Technology Data Exchange (ETDEWEB)
MartInez, Antonio [Departamento de Ciencia de Materiales, Optica y TecnologIa Electronica, Universidad Miguel Hernandez, 03202 Elche (Spain); Sanchez-Lopez, Ma del Mar [Instituto de BioingenierIa y Departamento de Fisica y Arquitectura de Computadores, Universidad Miguel Hernandez, 03202 Elche (Spain); Moreno, Ignacio [Departamento de Ciencia de Materiales, Optica y TecnologIa Electronica, Universidad Miguel Hernandez, 03202 Elche (Spain)
2007-09-11
In this work, we present a simple analysis of binary diffraction gratings with different slit widths relative to the grating period. The analysis is based on a simple phasor technique directly derived from the Huygens principle. By introducing a slit phasor and a grating phasor, the intensity of the diffracted orders and the grating's resolving power can be easily obtained without applying the usual Fourier transform operations required for these calculations. The proposed phasor technique is mathematically equivalent to the Fourier transform calculation of the diffraction order amplitude, and it can be useful to explain binary diffraction gratings in a simple manner in introductory physics courses. This theoretical analysis is illustrated with experimental results using a liquid crystal device to display diffraction gratings with different fill factors.
Non-existence criteria for Laurent polynomial first integrals
Directory of Open Access Journals (Sweden)
Shaoyun Shi
2003-01-01
Full Text Available In this paper we derived some simple criteria for non-existence and partial non-existence Laurent polynomial first integrals for a general nonlinear systems of ordinary differential equations $\\dot x = f(x$, $x \\in \\mathbb{R}^n$ with $f(0 = 0$. We show that if the eigenvalues of the Jacobi matrix of the vector field $f(x$ are $\\mathbb{Z}$-independent, then the system has no nontrivial Laurent polynomial integrals.
Vanishing of Littlewood-Richardson polynomials is in P
Adve, Anshul; Robichaux, Colleen; Yong, Alexander
2017-01-01
J. DeLoera-T. McAllister and K. D. Mulmuley-H. Narayanan-M. Sohoni independently proved that determining the vanishing of Littlewood-Richardson coefficients has strongly polynomial time computational complexity. Viewing these as Schubert calculus numbers, we prove the generalization to the Littlewood-Richardson polynomials that control equivariant cohomology of Grassmannians. We construct a polytope using the edge-labeled tableau rule of H. Thomas-A. Yong. Our proof then combines a saturation...
Recurrence coefficients for discrete orthonormal polynomials and the Painlevé equations
International Nuclear Information System (INIS)
Clarkson, Peter A
2013-01-01
We investigate semi-classical generalizations of the Charlier and Meixner polynomials, which are discrete orthogonal polynomials that satisfy three-term recurrence relations. It is shown that the coefficients in these recurrence relations can be expressed in terms of Wronskians of modified Bessel functions and confluent hypergeometric functions, respectively for the generalized Charlier and generalized Meixner polynomials. These Wronskians arise in the description of special function solutions of the third and fifth Painlevé equations. (paper)
MEMS Bragg grating force sensor
DEFF Research Database (Denmark)
Reck, Kasper; Thomsen, Erik Vilain; Hansen, Ole
2011-01-01
We present modeling, design, fabrication and characterization of a new type of all-optical frequency modulated MEMS force sensor based on a mechanically amplified double clamped waveguide beam structure with integrated Bragg grating. The sensor is ideally suited for force measurements in harsh...... environments and for remote and distributed sensing and has a measured sensitivity of -14 nm/N, which is several times higher than what is obtained in conventional fiber Bragg grating force sensors. © 2011 Optical Society of America....
Optically controlled tunable dispersion compensators based on pumped fiber gratings.
Shu, Xuewen; Sugden, Kate; Bennion, Ian
2011-08-01
We demonstrate optically tunable dispersion compensators based on pumping fiber Bragg gratings made in Er/Yb codoped fiber. The tunable dispersion for a chirped grating and also a uniform-period grating was successfully demonstrated in the experiment. The dispersion of the chirped grating was tuned from 900 to 1990 ps/nm and also from -600 to -950 ps/nm in the experiment. © 2011 Optical Society of America
Design and Use of a Learning Object for Finding Complex Polynomial Roots
Benitez, Julio; Gimenez, Marcos H.; Hueso, Jose L.; Martinez, Eulalia; Riera, Jaime
2013-01-01
Complex numbers are essential in many fields of engineering, but students often fail to have a natural insight of them. We present a learning object for the study of complex polynomials that graphically shows that any complex polynomials has a root and, furthermore, is useful to find the approximate roots of a complex polynomial. Moreover, we…
Review of High-Speed Fiber Optic Grating Sensors Systems
Energy Technology Data Exchange (ETDEWEB)
Udd, E; Benterou, J; May, C; Mihailov, S J; Lu, P
2010-03-24
Fiber grating sensors can be used to support a wide variety of high speed measurement applications. This includes measurements of vibrations on bridges, traffic monitoring on freeways, ultrasonic detection to support non-destructive tests on metal plates and providing details of detonation events. This paper provides a brief overview of some of the techniques that have been used to support high speed measurements using fiber grating sensors over frequency ranges from 10s of kHz, to MHZ and finally toward frequencies approaching the GHz regime. Very early in the development of fiber grating sensor systems it was realized that a high speed fiber grating sensor system could be realized by placing an optical filter that might be a fiber grating in front of a detector so that spectral changes in the reflection from a fiber grating were amplitude modulated. In principal the only limitation on this type of system involved the speed of the output detector which with the development of high speed communication links moved from the regime of 10s of MHz toward 10s of GHz. The earliest deployed systems involved civil structures including measurements of the strain fields on composite utility poles and missile bodies during break tests, bridges and freeways. This was followed by a series of developments that included high speed fiber grating sensors to support nondestructive testing via ultrasonic wave detection, high speed machining and monitoring ship hulls. Each of these applications involved monitoring mechanical motion of structures and thus interest was in speeds up to a few 10s of MHz. Most recently there has been interest in using fiber grating to monitor the very high speed events such as detonations and this has led to utilization of fiber gratings that are consumed during an event that may require detection speeds of hundreds of MHz and in the future multiple GHz.
International Nuclear Information System (INIS)
Cavallo, A.; Hsuan, H.; Boyd, D.; Grek, B.; Johnson, D.; Kritz, A.; Mikkelsen, D.; LeBlanc, B.; Takahashi, H.
1984-10-01
During first harmonic electron cyclotron heating (ECH) on the Princeton Divertor Experiment (PDX) (R 0 = 137 cm, a = 40 cm), electron temperature was monitored using a grating polychromator which measured second harmonic electron cyclotron emission from the low field side of the tokamak. Interference from the high power heating pulse on the broadband detectors in the grating instrument was eliminated by using a waveguide filter in the transmission line which brought the emission signal to the grating instrument. Off-axis (approx. 4 cm) location of the resonance zone resulted in heating without sawtooth or m = 1 activity. However, heating with the resonance zone at the plasma center caused very large amplitude sawteeth accompanied by strong m = 1 activity: ΔT/T/sub MAX/ approx. = 0.41, sawtooth period approx. = 4 msec, m = 1 period approx. = 90 μ sec, (11 kHz). This is the first time such intense MHD activity driven by ECH has been observed. (For both cases there was no sawtooth activity in the ohmic phase of the discharge before ECH.) At very low densities there is a clear indication that a superthermal electron population is created during ECH
Optical Fiber Grating Hydrogen Sensors: A Review.
Dai, Jixiang; Zhu, Li; Wang, Gaopeng; Xiang, Feng; Qin, Yuhuan; Wang, Min; Yang, Minghong
2017-03-12
In terms of hydrogen sensing and detection, optical fiber hydrogen sensors have been a research issue due to their intrinsic safety and good anti-electromagnetic interference. Among these sensors, hydrogen sensors consisting of fiber grating coated with sensitive materials have attracted intensive research interests due to their good reliability and distributed measurements. This review paper mainly focuses on optical fiber hydrogen sensors associated with fiber gratings and various materials. Their configurations and sensing performances proposed by different groups worldwide are reviewed, compared and discussed in this paper. Meanwhile, the challenges for fiber grating hydrogen sensors are also addressed.
Sub-wavelength grating structure on the planar waveguide (Conference Presentation)
Qing-Song, Zhu; Sheng-Hui, Chen
2016-10-01
Making progress in recent years, with the technology of the grating, the grating period can be reduced to shrink the size of the light coupler on a waveguide. The working wavelength of the light coupler can be in the range from the near-infrared to visible. In this study , we used E-gun evaporation system with ion-beam-assisted deposition system to fabricate bottom cladding (SiO2), guiding layer (Ta2O5) and Distributed Bragg Reflector(DBR) of the waveguide on the silicon substrate. Electron-beam lithography is used to make sub-wavelength gratings and reflector grating on the planar waveguide which is a coupling device on the guiding layer. The best fabrication parameters were analyzed to deposit the film. The exposure and development times also influenced to fabricate the grating quality. The purpose is to reduce the device size and enhance coupling efficiency which maintain normal incidence of the light . We designed and developed the device using the Finite-Difference Time-Domain (FDTD) method. The grating period, depth, fill factor, film thickness, Distributed Bragg Reflector(DBR) numbers and reflector grating period have been discussed to enhance coupling efficiency and maintained normal incidence of the light. According to the simulation results, when the wavelength is 1300 nm, the coupling grating period is 720 nm and the Ta2O5 film is 460 nm with 360 nm of reflector grating period and 2 layers of Distributed Bragg Reflector, which had the optimum coupling efficiency and normal incidence angle. In the measurement, We successfully measured the TE wave coupling efficiency of the photoresist grating coupling device.
Talbot effect of the defective grating in deep Fresnel region
Teng, Shuyun; Wang, Junhong; Zhang, Wei; Cui, Yuwei
2015-02-01
Talbot effect of the grating with different defect is studied theoretically and experimentally in this paper. The defects of grating include the loss of the diffraction unit, the dislocation of the diffraction unit and the modulation of the unit separation. The exact diffraction distributions of three kinds of defective gratings are obtained according to the finite-difference time-domain (FDTD) method. The calculation results show the image of the missing or dislocating unit appears at the Talbot distance (as mentioned in K. Patorski Prog. Opt., 27, 1989, pp.1-108). This is the so-called self-repair ability of grating imaging. In addition, some more phenomena are discovered. The loss or the dislocation of diffraction unit causes the diffraction distortion within a certain radial angle. The regular modulation of unit separation changes the original diffraction, but the new periodicity of the diffraction distribution rebuilds. The self-imaging of grating with smaller random modulation still keeps the partial self-repair ability, and yet this characteristic depends on the modulation degree of defective grating. These diffraction phenomena of the defective gratings are explained by use of the diffraction theory of grating. The practical experiment is also performed and the experimental results confirm the theoretic predictions.
Two polynomial division inequalities in
Directory of Open Access Journals (Sweden)
Goetgheluck P
1998-01-01
Full Text Available This paper is a first attempt to give numerical values for constants and , in classical estimates and where is an algebraic polynomial of degree at most and denotes the -metric on . The basic tools are Markov and Bernstein inequalities.
The cross waveguide grating: proposal, theory and applications.
Muñoz, Pascual; Pastor, Daniel; Capmany, José
2005-04-18
In this paper a novel grating-like integrated optics device is proposed, the Cross Waveguide Grating (XWG). The device is based upon a modified configuration of a traditional Arrayed Waveguide Grating (AWG). The Arrayed Waveguides part is changed, as detailed along this document, giving the device both the ability of multi/demultiplexing and power splitting/coupling. Design examples and transfer function simulations show good agreement with the presented theory. Finally, some of the envisaged applications are outlined.
Asymptotics for the ratio and the zeros of multiple Charlier polynomials
Ndayiragije, François; Van Assche, Walter
2011-01-01
We investigate multiple Charlier polynomials and in particular we will use the (nearest neighbor) recurrence relation to find the asymptotic behavior of the ratio of two multiple Charlier polynomials. This result is then used to obtain the asymptotic distribution of the zeros, which is uniform on an interval. We also deal with the case where one of the parameters of the various Poisson distributions depend on the degree of the polynomial, in which case we obtain another asymptotic distributio...
H∞ Control of Polynomial Fuzzy Systems: A Sum of Squares Approach
Bomo W. Sanjaya; Bambang Riyanto Trilaksono; Arief Syaichu-Rohman
2014-01-01
This paper proposes the control design ofa nonlinear polynomial fuzzy system with H∞ performance objective using a sum of squares (SOS) approach. Fuzzy model and controller are represented by a polynomial fuzzy model and controller. The design condition is obtained by using polynomial Lyapunov functions that not only guarantee stability but also satisfy the H∞ performance objective. The design condition is represented in terms of an SOS that can be numerically solved via the SOSTOOLS. A simul...
Euler Polynomials and Identities for Non-Commutative Operators
De Angelis, V.; Vignat, C.
2015-01-01
Three kinds of identities involving non-commutating operators and Euler and Bernoulli polynomials are studied. The first identity, as given by Bender and Bettencourt, expresses the nested commutator of the Hamiltonian and momentum operators as the commutator of the momentum and the shifted Euler polynomial of the Hamiltonian. The second one, due to J.-C. Pain, links the commutators and anti-commutators of the monomials of the position and momentum operators. The third appears in a work by Fig...
Multicore optical fiber grating array fabrication for medical sensing applications
Westbrook, Paul S.; Feder, K. S.; Kremp, T.; Taunay, T. F.; Monberg, E.; Puc, G.; Ortiz, R.
2015-03-01
In this work we report on a fiber grating fabrication platform suitable for parallel fabrication of Bragg grating arrays over arbitrary lengths of multicore optical fiber. Our system exploits UV transparent coatings and has precision fiber translation that allows for quasi-continuous grating fabrication. Our system is capable of both uniform and chirped fiber grating array spectra that can meet the demands of medical sensors including high speed, accuracy, robustness and small form factor.
Hybrid grating reflectors: Origin of ultrabroad stopband
Energy Technology Data Exchange (ETDEWEB)
Park, Gyeong Cheol; Taghizadeh, Alireza; Chung, Il-Sug, E-mail: ilch@fotonik.dtu.dk [DTU Fotonik, Department of Photonics Engineering, Technical University of Denmark, DK-2800 Kgs. Lyngby (Denmark)
2016-04-04
Hybrid grating (HG) reflectors with a high-refractive-index cap layer added onto a high contrast grating (HCG) provide a high reflectance close to 100% over a broader wavelength range than HCGs. The combination of a cap layer and a grating layer brings a strong Fabry-Perot (FP) resonance as well as a weak guided mode (GM) resonance. Most of the reflected power results from the FP resonance, while the GM resonance plays a key role in achieving a reflectance close to 100% as well as broadening the stopband. An HG sample with 7 InGaAlAs quantum wells included in the cap layer has been fabricated by directly wafer-bonding a III-V cap layer onto a Si grating layer. Its reflection property has been characterized. This heterogeneously integrated HG reflector may allow for a hybrid III-V on Si laser to be thermally efficient, which has promising prospects for silicon photonics light sources and high-speed operation.
Local polynomial Whittle estimation of perturbed fractional processes
DEFF Research Database (Denmark)
Frederiksen, Per; Nielsen, Frank; Nielsen, Morten Ørregaard
We propose a semiparametric local polynomial Whittle with noise (LPWN) estimator of the memory parameter in long memory time series perturbed by a noise term which may be serially correlated. The estimator approximates the spectrum of the perturbation as well as that of the short-memory component...... of the signal by two separate polynomials. Including these polynomials we obtain a reduction in the order of magnitude of the bias, but also in‡ate the asymptotic variance of the long memory estimate by a multiplicative constant. We show that the estimator is consistent for d 2 (0; 1), asymptotically normal...... for d ε (0, 3/4), and if the spectral density is infinitely smooth near frequency zero, the rate of convergence can become arbitrarily close to the parametric rate, pn. A Monte Carlo study reveals that the LPWN estimator performs well in the presence of a serially correlated perturbation term...
Polynomial algebra of discrete models in systems biology.
Veliz-Cuba, Alan; Jarrah, Abdul Salam; Laubenbacher, Reinhard
2010-07-01
An increasing number of discrete mathematical models are being published in Systems Biology, ranging from Boolean network models to logical models and Petri nets. They are used to model a variety of biochemical networks, such as metabolic networks, gene regulatory networks and signal transduction networks. There is increasing evidence that such models can capture key dynamic features of biological networks and can be used successfully for hypothesis generation. This article provides a unified framework that can aid the mathematical analysis of Boolean network models, logical models and Petri nets. They can be represented as polynomial dynamical systems, which allows the use of a variety of mathematical tools from computer algebra for their analysis. Algorithms are presented for the translation into polynomial dynamical systems. Examples are given of how polynomial algebra can be used for the model analysis. alanavc@vt.edu Supplementary data are available at Bioinformatics online.
Polynomial chaos expansion with random and fuzzy variables
Jacquelin, E.; Friswell, M. I.; Adhikari, S.; Dessombz, O.; Sinou, J.-J.
2016-06-01
A dynamical uncertain system is studied in this paper. Two kinds of uncertainties are addressed, where the uncertain parameters are described through random variables and/or fuzzy variables. A general framework is proposed to deal with both kinds of uncertainty using a polynomial chaos expansion (PCE). It is shown that fuzzy variables may be expanded in terms of polynomial chaos when Legendre polynomials are used. The components of the PCE are a solution of an equation that does not depend on the nature of uncertainty. Once this equation is solved, the post-processing of the data gives the moments of the random response when the uncertainties are random or gives the response interval when the variables are fuzzy. With the PCE approach, it is also possible to deal with mixed uncertainty, when some parameters are random and others are fuzzy. The results provide a fuzzy description of the response statistical moments.
Polynomial Vector Fields in One Complex Variable
DEFF Research Database (Denmark)
Branner, Bodil
In recent years Adrien Douady was interested in polynomial vector fields, both in relation to iteration theory and as a topic on their own. This talk is based on his work with Pierrette Sentenac, work of Xavier Buff and Tan Lei, and my own joint work with Kealey Dias.......In recent years Adrien Douady was interested in polynomial vector fields, both in relation to iteration theory and as a topic on their own. This talk is based on his work with Pierrette Sentenac, work of Xavier Buff and Tan Lei, and my own joint work with Kealey Dias....
Time-domain Brillouin scattering assisted by diffraction gratings
Matsuda, Osamu; Pezeril, Thomas; Chaban, Ievgeniia; Fujita, Kentaro; Gusev, Vitalyi
2018-02-01
Absorption of ultrashort laser pulses in a metallic grating deposited on a transparent sample launches coherent compression/dilatation acoustic pulses in directions of different orders of acoustic diffraction. Their propagation is detected by delayed laser pulses, which are also diffracted by the metallic grating, through the measurement of the transient intensity change of the first-order diffracted light. The obtained data contain multiple frequency components, which are interpreted by considering all possible angles for the Brillouin scattering of light achieved through multiplexing of the propagation directions of light and coherent sound by the metallic grating. The emitted acoustic field can be equivalently presented as a superposition of plane inhomogeneous acoustic waves, which constitute an acoustic diffraction grating for the probe light. Thus the obtained results can also be interpreted as a consequence of probe light diffraction by both metallic and acoustic gratings. The realized scheme of time-domain Brillouin scattering with metallic gratings operating in reflection mode provides access to wide range of acoustic frequencies from minimal to maximal possible values in a single experimental optical configuration for the directions of probe light incidence and scattered light detection. This is achieved by monitoring the backward and forward Brillouin scattering processes in parallel. Potential applications include measurements of the acoustic dispersion, simultaneous determination of sound velocity and optical refractive index, and evaluation of samples with a single direction of possible optical access.
[Diffraction gratings used in x-ray spectroscopy]: Final report
International Nuclear Information System (INIS)
Smith, H.I.
1988-01-01
This subcontract was initiated in order to facilitate the development at MIT of technologies for fabricating the very fine diffraction grating required in x-ray spectroscopy at Lawrence Livermore Laboratory (LLL). These gratings are generally gold transmission gratings with spatial periods of 200 nm or less. The major focus of our efforts was to develop a means of fabricating gratings of 100 nm period. We explored two approaches: e-beam fabrication of x-ray lithography masks, and achromatic holographic lithography. This work was pursued by Erik Anderson as a major component of his Ph.D. thesis. Erik was successful in both the e-beam and holographic approaches. However, the e-beam method proved to be highly impractical: exposure times of about 115 days would be required to cover an area of 1 cm 2 . The achromatic holography, on the other hand, should be capable of exposing areas well in excess of 1 cm 2 in times under 1 hour. Moreover, 100 nm-period gratings produced by achromatic holography are coherent over their entire area whereas gratings produced by e-beam lithography are coherent only over areas /approximately/100 μm. The remainder of this report consists of portions excerpted from Erik Anderson's thesis. These contain all the details of our work on 100 nm period gratings. 26 refs., 17 figs
The influence of grating shape formation fluctuation on DFB laser diode threshold condition
Bao, Shiwei; Song, Qinghai; Xie, Chunmei
2018-03-01
Not only the grating material refractive index itself but also the Bragg grating physical shape formation affects the coupling strength greatly. The Bragg grating shape includes three factors, namely grating depth, duty ratio and grating angle. During the lithography and wet etching process, there always will be some fluctuation between the target and real grating shape formation after fabrication process. This grating shape fluctuation will affect the DFB coupling coefficient κ , and then consequently threshold current and corresponding wavelength. This paper studied the grating shape formation fluctuation influence to improve the DFB fabrication yield. A truncated normal random distribution fluctuation is considered in this paper. The simulation results conclude that it is better to choose relative thicker grating depth with lower refractive index to obtain a better fabrication tolerance, while not quite necessary to spend too much effort on improving lithography and wet etching process to get a precisely grating duty ratio and grating angle.
Directory of Open Access Journals (Sweden)
Hjalmar Rosengren
2006-12-01
Full Text Available We study multivariable Christoffel-Darboux kernels, which may be viewed as reproducing kernels for antisymmetric orthogonal polynomials, and also as correlation functions for products of characteristic polynomials of random Hermitian matrices. Using their interpretation as reproducing kernels, we obtain simple proofs of Pfaffian and determinant formulas, as well as Schur polynomial expansions, for such kernels. In subsequent work, these results are applied in combinatorics (enumeration of marked shifted tableaux and number theory (representation of integers as sums of squares.
Measurement of a discontinuous object based on a dual-frequency grating
Institute of Scientific and Technical Information of China (English)
Qiao Nao-Sheng; Cai Xin-Hua; Yao Chun-Mei
2009-01-01
The dual-frequency grating measurement theory is proposed in order to carry out the measurement of a discontinuous object. Firstly, the reason why frequency spectra are produced by low frequency gratings and high frequency gratings in the field of frequency is analysed, and the relationship between the wrapped-phase and the unwrappingphase is discussed. Secondly, a method to combine the advantages of the two kinds of gratings is proposed: one stripe is produced in the mutation part of the object measured by a suitable low frequency grating designed by MATLAB, then the phase produced by the low frequency grating need not be unfolded. The integer series of stripes is produced by a high frequency grating designed by MATLAB based on the frequency ratio of the two kinds of gratings and the high frequency wrapped-phase, and the high frequency unwrapping-phase is then obtained. In order to verify the correctness of the theoretical analysis, a steep discontinuous object of 600×600 pixels and 10.00 mm in height is simulated and a discontinuous object of ladder shape which is 32.00 mm in height is used in experiment. Both the simulation and the experiment can restore the discontinuous object height accurately by using the dual-frequency grating measurement theory.
Szász-Durrmeyer operators involving Boas-Buck polynomials of blending type.
Sidharth, Manjari; Agrawal, P N; Araci, Serkan
2017-01-01
The present paper introduces the Szász-Durrmeyer type operators based on Boas-Buck type polynomials which include Brenke type polynomials, Sheffer polynomials and Appell polynomials considered by Sucu et al. (Abstr. Appl. Anal. 2012:680340, 2012). We establish the moments of the operator and a Voronvskaja type asymptotic theorem and then proceed to studying the convergence of the operators with the help of Lipschitz type space and weighted modulus of continuity. Next, we obtain a direct approximation theorem with the aid of unified Ditzian-Totik modulus of smoothness. Furthermore, we study the approximation of functions whose derivatives are locally of bounded variation.
Szász-Durrmeyer operators involving Boas-Buck polynomials of blending type
Directory of Open Access Journals (Sweden)
Manjari Sidharth
2017-05-01
Full Text Available Abstract The present paper introduces the Szász-Durrmeyer type operators based on Boas-Buck type polynomials which include Brenke type polynomials, Sheffer polynomials and Appell polynomials considered by Sucu et al. (Abstr. Appl. Anal. 2012:680340, 2012. We establish the moments of the operator and a Voronvskaja type asymptotic theorem and then proceed to studying the convergence of the operators with the help of Lipschitz type space and weighted modulus of continuity. Next, we obtain a direct approximation theorem with the aid of unified Ditzian-Totik modulus of smoothness. Furthermore, we study the approximation of functions whose derivatives are locally of bounded variation.
Nanoscale freestanding gratings for ultraviolet blocking filters
Energy Technology Data Exchange (ETDEWEB)
van Beek, J.T.; Fleming, R.C.; Hindle, P.S.; Prentiss, J.D.; Schattenburg, M.L. [Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (United States); Ritzau, S. [Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States)
1998-11-01
Ultraviolet (UV) blocking filters are needed for atomic flux imaging in environments where high levels of ultraviolet radiation are present. Freestanding gratings are a promising candidate for UV filtering. They have a high aspect ratio ({approximately}13), narrow ({approximately}40 nm) slots, and effectively block UV radiation. The grating fabrication process makes use of several etching, electroplating, and lithographic steps and includes an optional step to plug pinholes induced by particles during processing. Gratings were successfully manufactured and tested. Measured UV transmissions of {approximately}10{sup {minus}5} and particle transmissions of {approximately}10{percent} are in agreement with theoretical predictions. {copyright} {ital 1998 American Vacuum Society.}
An extension of Krawtchouk\\'s polynomials to the contstruction of ...
African Journals Online (AJOL)
A simple method is described for the construction of a set of orthogonal polynomials for any case where the proportions of observations follow a binomial distribution. The least squares equation which fits the data is determined using the properties of orthogonal polynomials and the analysis of variance technique.
Geometry of polynomials and root-finding via path-lifting
Kim, Myong-Hi; Martens, Marco; Sutherland, Scott
2018-02-01
Using the interplay between topological, combinatorial, and geometric properties of polynomials and analytic results (primarily the covering structure and distortion estimates), we analyze a path-lifting method for finding approximate zeros, similar to those studied by Smale, Shub, Kim, and others. Given any polynomial, this simple algorithm always converges to a root, except on a finite set of initial points lying on a circle of a given radius. Specifically, the algorithm we analyze consists of iterating where the t k form a decreasing sequence of real numbers and z 0 is chosen on a circle containing all the roots. We show that the number of iterates required to locate an approximate zero of a polynomial f depends only on log\\vert f(z_0)/ρ_\\zeta\\vert (where ρ_\\zeta is the radius of convergence of the branch of f-1 taking 0 to a root ζ) and the logarithm of the angle between f(z_0) and certain critical values. Previous complexity results for related algorithms depend linearly on the reciprocals of these angles. Note that the complexity of the algorithm does not depend directly on the degree of f, but only on the geometry of the critical values. Furthermore, for any polynomial f with distinct roots, the average number of steps required over all starting points taken on a circle containing all the roots is bounded by a constant times the average of log(1/ρ_\\zeta) . The average of log(1/ρ_\\zeta) over all polynomials f with d roots in the unit disk is \
The Combinatorial Rigidity Conjecture is False for Cubic Polynomials
DEFF Research Database (Denmark)
Henriksen, Christian
2003-01-01
We show that there exist two cubic polynomials with connected Julia sets which are combinatorially equivalent but not topologically conjugate on their Julia sets. This disproves a conjecture by McMullen from 1995.......We show that there exist two cubic polynomials with connected Julia sets which are combinatorially equivalent but not topologically conjugate on their Julia sets. This disproves a conjecture by McMullen from 1995....
Ratio asymptotics of Hermite-Pade polynomials for Nikishin systems
International Nuclear Information System (INIS)
Aptekarev, A I; Lopez, Guillermo L; Rocha, I A
2005-01-01
The existence of ratio asymptotics is proved for a sequence of multiple orthogonal polynomials with orthogonality relations distributed among a system of m finite Borel measures with support on a bounded interval of the real line which form a so-called Nikishin system. For m=1 this result reduces to Rakhmanov's celebrated theorem on the ratio asymptotics for orthogonal polynomials on the real line.
International Nuclear Information System (INIS)
Calogero, F.
1978-01-01
Let zsub(j)(α, β) be the jth zero of the Jacobi polynomial J sub(n)sup(α,β)(z), and xsub(j) the jth zero of the Hermite polynomial Hsub(n)(x). Then, as t→infinity, zsub(j)(at,bt)=(b-a)/(b+a)+t sup(-1/2)c x sub(j)+t -1 4/3(n+1/2+xsub(j) 2 )(a-b)/(a+b) 2 +0(t sup(-3/2)), with c=(ab)sup(1/2) [(a+b)/2]sup(-3/2) a>0, b>0. This formula implies the limit relation n exclamation mark lim sub(t→infinity) [t sup(-n/2)J sub(n)sup(at,bt) ((b-a)/(b+a)+t sup(-1/2)x)] = [(a+b)c/4]sup(n) Hsub(n)(chi/c). (author)
Dynamic population gratings in rare-earth-doped optical fibres
Energy Technology Data Exchange (ETDEWEB)
Stepanov, Serguei [Optics Department, CICESE, km.107 carr. Tijuana-Ensenada, Ensenada, 22860, BC (Mexico)], E-mail: steps@cicese.mx
2008-11-21
Dynamic Bragg gratings can be recorded in rare-earth (e.g. Er, Yb) doped optical fibres by two counter-propagating mutually coherent laser waves via local saturation of the fibre optical absorption or gain (in optically pumped fibres). Typical recording cw light power needed for efficient grating formation is of sub-mW-mW scale which results in characteristic recording/erasure times of 10-0.1 ms. This review paper discusses fundamental aspects of the population grating formation, their basic properties, relating wave-mixing processes and also considers different applications of these dynamic gratings in single-frequency fibre lasers, tunable filters, optical fibre sensors and adaptive interferometry.
Dynamic population gratings in rare-earth-doped optical fibres
International Nuclear Information System (INIS)
Stepanov, Serguei
2008-01-01
Dynamic Bragg gratings can be recorded in rare-earth (e.g. Er, Yb) doped optical fibres by two counter-propagating mutually coherent laser waves via local saturation of the fibre optical absorption or gain (in optically pumped fibres). Typical recording cw light power needed for efficient grating formation is of sub-mW-mW scale which results in characteristic recording/erasure times of 10-0.1 ms. This review paper discusses fundamental aspects of the population grating formation, their basic properties, relating wave-mixing processes and also considers different applications of these dynamic gratings in single-frequency fibre lasers, tunable filters, optical fibre sensors and adaptive interferometry.
Congruences concerning Legendre polynomials III
Sun, Zhi-Hong
2010-01-01
Let $p>3$ be a prime, and let $R_p$ be the set of rational numbers whose denominator is coprime to $p$. Let $\\{P_n(x)\\}$ be the Legendre polynomials. In this paper we mainly show that for $m,n,t\\in R_p$ with $m\
Global Polynomial Kernel Hazard Estimation
DEFF Research Database (Denmark)
Hiabu, Munir; Miranda, Maria Dolores Martínez; Nielsen, Jens Perch
2015-01-01
This paper introduces a new bias reducing method for kernel hazard estimation. The method is called global polynomial adjustment (GPA). It is a global correction which is applicable to any kernel hazard estimator. The estimator works well from a theoretical point of view as it asymptotically redu...
Optimal Conformal Polynomial Projections for Croatia According to the Airy/Jordan Criterion
Directory of Open Access Journals (Sweden)
Dražen Tutić
2009-05-01
Full Text Available The paper describes optimal conformal polynomial projections for Croatia according to the Airy/Jordan criterion. A brief introduction of history and theory of conformal mapping is followed by descriptions of conformal polynomial projections and their current application. The paper considers polynomials of degrees 1 to 10. Since there are conditions in which the 1st degree polynomial becomes the famous Mercator projection, it was not considered specifically for Croatian territory. The area of Croatia was defined as a union of national territory and the continental shelf. Area definition data were taken from the Euro Global Map 1:1 000 000 for Croatia, as well as from two maritime delimitation treaties. Such an irregular area was approximated with a regular grid consisting of 11 934 ellipsoidal trapezoids 2' large. The Airy/Jordan criterion for the optimal projection is defined as minimum of weighted mean of Airy/Jordan measure of distortion in points. The value of the Airy/Jordan criterion is calculated from all 11 934 centres of ellipsoidal trapezoids, while the weights are equal to areas of corresponding ellipsoidal trapezoids. The minimum is obtained by Nelder and Mead’s method, as implemented in the fminsearch function of the MATLAB package. Maps of Croatia representing the distribution of distortions are given for polynomial degrees 2 to 6 and 10. Increasing the polynomial degree results in better projections considering the criterion, and the 6th degree polynomial provides a good ratio of formula complexity and criterion value.
Polynomial realization of the Uq (sl(3)) Gel'fand-(Weyl)-Zetlin basis
International Nuclear Information System (INIS)
Dobrev, V.K.; Truini, P.
1996-01-01
We give an explicit realization of the U ≡ U q (sl(3)) Gel'fand-(Weyl)-Zetlin (GWZ) basis as polynomial functions in three variables. This realization is obtained in two complementary ways. First we establish a 1-to-1 correspondence between the abstract GWZ basis and explicit polynomials in the quantum subgroup U + of the raising generators. We then use an explicit construction of arbitrary lowest weight (holomorphic) representations of U in terms of three variables on which the generators of U are realized as q-difference operators. Applying the GWZ corresponding polynomials in this realization to the lowest weight vector (the function 1) produces one realization of our GWZ basis. Another realization of the GWZ polynomial basis is found by the explicit diagonalization of the operators of isospin I-circumflex 2 , third component of isospin I-circumflex z , and hypercharge Y-circumflex, in the same realization as q-difference operators. The result is that the eigenvectors can be written in terms of q-hypergeometric polynomials in our three variables. Finally we construct an explicit scalar product (adapting the Shapovalov form to our setting). Using it we prove the orthogonality of our GWZ polynomials for which we use both realizations. This provides a polynomial construction for the orthonormal GWZ basis. We work for generic q, leaving the root of unity case for a following paper. It seems that our results are new also in the classical situation (q=1). (author). 20 refs
Real zeros of classes of random algebraic polynomials
Directory of Open Access Journals (Sweden)
K. Farahmand
2003-01-01
Full Text Available There are many known asymptotic estimates for the expected number of real zeros of an algebraic polynomial a0+a1x+a2x2+⋯+an−1xn−1 with identically distributed random coefficients. Under different assumptions for the distribution of the coefficients {aj}j=0n−1 it is shown that the above expected number is asymptotic to O(logn. This order for the expected number of zeros remains valid for the case when the coefficients are grouped into two, each group with a different variance. However, it was recently shown that if the coefficients are non-identically distributed such that the variance of the jth term is (nj the expected number of zeros of the polynomial increases to O(n. The present paper provides the value for this asymptotic formula for the polynomials with the latter variances when they are grouped into three with different patterns for their variances.
a Unified Matrix Polynomial Approach to Modal Identification
Allemang, R. J.; Brown, D. L.
1998-04-01
One important current focus of modal identification is a reformulation of modal parameter estimation algorithms into a single, consistent mathematical formulation with a corresponding set of definitions and unifying concepts. Particularly, a matrix polynomial approach is used to unify the presentation with respect to current algorithms such as the least-squares complex exponential (LSCE), the polyreference time domain (PTD), Ibrahim time domain (ITD), eigensystem realization algorithm (ERA), rational fraction polynomial (RFP), polyreference frequency domain (PFD) and the complex mode indication function (CMIF) methods. Using this unified matrix polynomial approach (UMPA) allows a discussion of the similarities and differences of the commonly used methods. the use of least squares (LS), total least squares (TLS), double least squares (DLS) and singular value decomposition (SVD) methods is discussed in order to take advantage of redundant measurement data. Eigenvalue and SVD transformation methods are utilized to reduce the effective size of the resulting eigenvalue-eigenvector problem as well.
Czech Academy of Sciences Publication Activity Database
Knížek, J.; Tichý, Petr; Beránek, L.; Šindelář, Jan; Vojtěšek, B.; Bouchal, P.; Nenutil, R.; Dedík, O.
2010-01-01
Roč. 7, č. 10 (2010), s. 48-60 ISSN 0974-5718 Grant - others:GA MZd(CZ) NS9812; GA ČR(CZ) GAP304/10/0868 Institutional research plan: CEZ:AV0Z10300504; CEZ:AV0Z10750506 Keywords : polynomial regression * orthogonalization * numerical methods * markers * biomarkers Subject RIV: BA - General Mathematics
Quantum entanglement via nilpotent polynomials
International Nuclear Information System (INIS)
Mandilara, Aikaterini; Akulin, Vladimir M.; Smilga, Andrei V.; Viola, Lorenza
2006-01-01
We propose a general method for introducing extensive characteristics of quantum entanglement. The method relies on polynomials of nilpotent raising operators that create entangled states acting on a reference vacuum state. By introducing the notion of tanglemeter, the logarithm of the state vector represented in a special canonical form and expressed via polynomials of nilpotent variables, we show how this description provides a simple criterion for entanglement as well as a universal method for constructing the invariants characterizing entanglement. We compare the existing measures and classes of entanglement with those emerging from our approach. We derive the equation of motion for the tanglemeter and, in representative examples of up to four-qubit systems, show how the known classes appear in a natural way within our framework. We extend our approach to qutrits and higher-dimensional systems, and make contact with the recently introduced idea of generalized entanglement. Possible future developments and applications of the method are discussed
Experimental observation of acoustic sub-harmonic diffraction by a grating
Energy Technology Data Exchange (ETDEWEB)
Liu, Jingfei, E-mail: benjamin.jf.liu@gatech.edu; Declercq, Nico F., E-mail: declercqdepatin@gatech.edu [Laboratory for Ultrasonic Nondestructive Evaluation “LUNE,” Georgia Tech Lorraine, Georgia Tech-CNRS UMI2958, Georgia Institute of Technology, 2, rue Marconi, Metz 57070 (France)
2014-06-28
A diffraction grating is a spatial filter causing sound waves or optical waves to reflect in directions determined by the frequency of the waves and the period of the grating. The classical grating equation is the governing principle that has successfully described the diffraction phenomena caused by gratings. However, in this work, we show experimental observation of the so-called sub-harmonic diffraction in acoustics that cannot be explained by the classical grating equation. Experiments indicate two physical phenomena causing the effect: internal scattering effects within the corrugation causing a phase shift and nonlinear acoustic effects generating new frequencies. This discovery expands our current understanding of the diffraction phenomenon, and it also makes it possible to better design spatial diffraction spectra, such as a rainbow effect in optics with a more complicated color spectrum than a traditional rainbow. The discovery reveals also a possibly new technique to study nonlinear acoustics by exploitation of the natural spatial filtering effect inherent to an acoustic diffraction grating.
Vacuum Predisperser For A Large Plane-Grating Spectrograph
Engleman, R.; Palmer, B. A.; Steinhaus, D. W.
1980-11-01
A plane grating predisperser has been constructed which acts as an "order-sorter" for a large plane-grating spectrograph. This combination can photograph relatively wide regions of spectra in a single exposure with no loss of resolution.
Nuclear-magnetic-resonance quantum calculations of the Jones polynomial
International Nuclear Information System (INIS)
Marx, Raimund; Spoerl, Andreas; Pomplun, Nikolas; Schulte-Herbrueggen, Thomas; Glaser, Steffen J.; Fahmy, Amr; Kauffman, Louis; Lomonaco, Samuel; Myers, John M.
2010-01-01
The repertoire of problems theoretically solvable by a quantum computer recently expanded to include the approximate evaluation of knot invariants, specifically the Jones polynomial. The experimental implementation of this evaluation, however, involves many known experimental challenges. Here we present experimental results for a small-scale approximate evaluation of the Jones polynomial by nuclear magnetic resonance (NMR); in addition, we show how to escape from the limitations of NMR approaches that employ pseudopure states. Specifically, we use two spin-1/2 nuclei of natural abundance chloroform and apply a sequence of unitary transforms representing the trefoil knot, the figure-eight knot, and the Borromean rings. After measuring the nuclear spin state of the molecule in each case, we are able to estimate the value of the Jones polynomial for each of the knots.
All-silicon nanorod-based Dammann gratings.
Li, Zile; Zheng, Guoxing; He, Ping'An; Li, Song; Deng, Qiling; Zhao, Jiangnan; Ai, Yong
2015-09-15
Established diffractive optical elements (DOEs), such as Dammann gratings, whose phase profile is controlled by etching different depths into a transparent dielectric substrate, suffer from a contradiction between the complexity of fabrication procedures and the performance of such gratings. In this Letter, we combine the concept of geometric phase and phase modulation in depth, and prove by theoretical analysis and numerical simulation that nanorod arrays etched on a silicon substrate have a characteristic of strong polarization conversion between two circularly polarized states and can act as a highly efficient half-wave plate. More importantly, only by changing the orientation angles of each nanorod can the arrays control the phase of a circularly polarized light, cell by cell. With the above principle, we report the realization of nanorod-based Dammann gratings reaching diffraction efficiencies of 50%-52% in the C-band fiber telecommunications window (1530-1565 nm). In this design, uniform 4×4 spot arrays with an extending angle of 59°×59° can be obtained in the far field. Because of these advantages of the single-step fabrication procedure, accurate phase controlling, and strong polarization conversion, nanorod-based Dammann gratings could be utilized for various practical applications in a range of fields.
A novel method for length of chirped fiber Bragg grating sensor
Li, Zhenwei; Wei, Peng; Liu, Taolin
2018-03-01
Length of chirped fiber Bragg grating sensor is very important for detonation velocity. Different from other ways, we proposed a novel method based on the optical frequency domain reflection theory to measure the length of chirped fiber grating sensor in non-contact condition. This method adopts a tunable laser source to provide wavelength scanning laser, which covers the Full Width at Half Maximum of spectrum of the chirped fiber Bragg grating sensor. A Michelson interferometer is used to produce optical interference signal. Finally, the grating's length is attainable by distance domain signal. In theory, length resolution of chirped fiber Bragg grating sensor could be 0.02 mm. We perform a series of length measurement experiments for chirped fiber grating sensor, including comparison experiments with hot-tip method. And the experiment results show that the novel method could accurately measure the length of chirped fiber Bragg grating sensors, and the length differences between the optical frequency domain reflection method and the hot-tip probe method are very small.
Fast tunable blazed MEMS grating for external cavity lasers
Tormen, Maurizio; Niedermann, Philippe; Hoogerwerf, Arno; Shea, Herbert; Stanley, Ross
2017-11-01
Diffractive MEMS are interesting for a wide range of applications, including displays, scanners or switching elements. Their advantages are compactness, potentially high actuation speed and in the ability to deflect light at large angles. We have designed and fabricated deformable diffractive MEMS grating to be used as tuning elements for external cavity lasers. The resulting device is compact, has wide tunability and a high operating speed. The initial design is a planar grating where the beams are free-standing and attached to each other using leaf springs. Actuation is achieved through two electrostatic comb drives at either end of the grating. To prevent deformation of the free-standing grating, the device is 10 μm thick made from a Silicon on Insulator (SOI) wafer in a single mask process. At 100V a periodicity tuning of 3% has been measured. The first resonant mode of the grating is measured at 13.8 kHz, allowing high speed actuation. This combination of wide tunability and high operating speed represents state of the art in the domain of tunable MEMS filters. In order to improve diffraction efficiency and to expand the usable wavelength range, a blazed version of the deformable MEMS grating has been designed. A key issue is maintaining the mechanical properties of the original device while providing optically smooth blazed beams. Using a process based on anisotropic KOH etching, blazed gratings have been obtained and preliminary characterization is promising.
Two-dimensional grating guided-mode resonance tunable filter.
Kuo, Wen-Kai; Hsu, Che-Jung
2017-11-27
A two-dimensional (2D) grating guided-mode resonance (GMR) tunable filter is experimentally demonstrated using a low-cost two-step nanoimprinting technology with a one-dimensional (1D) grating polydimethylsiloxane mold. For the first nanoimprinting, we precisely control the UV LED irradiation dosage and demold the device when the UV glue is partially cured and the 1D grating mold is then rotated by three different angles, 30°, 60°, and 90°, for the second nanoimprinting to obtain 2D grating structures with different crossing angles. A high-refractive-index film ZnO is then coated on the surface of the grating structure to form the GMR filter devices. The simulation and experimental results demonstrate that the passband central wavelength of the filter can be tuned by rotating the device to change azimuth angle of the incident light. We compare these three 2D GMR filters with differential crossing angles and find that the filter device with a crossing angle of 60° exhibits the best performance. The tunable range of its central wavelength is 668-742 nm when the azimuth angle varies from 30° to 90°.
Weierstrass method for quaternionic polynomial root-finding
Falcão, M. Irene; Miranda, Fernando; Severino, Ricardo; Soares, M. Joana
2018-01-01
Quaternions, introduced by Hamilton in 1843 as a generalization of complex numbers, have found, in more recent years, a wealth of applications in a number of different areas which motivated the design of efficient methods for numerically approximating the zeros of quaternionic polynomials. In fact, one can find in the literature recent contributions to this subject based on the use of complex techniques, but numerical methods relying on quaternion arithmetic remain scarce. In this paper we propose a Weierstrass-like method for finding simultaneously {\\sl all} the zeros of unilateral quaternionic polynomials. The convergence analysis and several numerical examples illustrating the performance of the method are also presented.
A set of sums for continuous dual q-2-Hahn polynomials
International Nuclear Information System (INIS)
Gade, R. M.
2009-01-01
An infinite set {τ (l) (y;r,z)} r,lisanelementofN 0 of linear sums of continuous dual q -2 -Hahn polynomials with prefactors depending on a complex parameter z is studied. The sums τ (l) (y;r,z) have an interpretation in context with tensor product representations of the quantum affine algebra U q ' (sl(2)) involving both a positive and a negative discrete series representation. For each l>0, the sum τ (l) (y;r,z) can be expressed in terms of the sum τ (0) (y;r,z), continuous dual q 2 -Hahn polynomials, and their associated polynomials. The sum τ (0) (y;r,z) is obtained as a combination of eight basic hypergeometric series. Moreover, an integral representation is provided for the sums τ (l) (y;r,z) with the complex parameter restricted by |zq| -2 -Hahn polynomials.
Fibonacci-like Differential Equations with a Polynomial Non-Homogeneous Part
Asveld, P.R.J.
1989-01-01
We investigate non-homogeneous linear differential equations of the form $x''(t) + x'(t) - x(t) = p(t)$ where $p(t)$ is either a polynomial or a factorial polynomial in $t$. We express the solution of these differential equations in terms of the coefficients of $p(t)$, in the initial conditions, and
System Construction for the Measurement of Bragg Grating Characteristics in Optical Fibers
West, Douglas P.
1995-01-01
Bragg gratings are used to measure strain in optical fibers. To measure strain they are sometimes used as a smart structure. They must be characterized after they are written to determine their spectral response. This paper deals with the test setup to characterize Bragg grating spectral responses.Bragg gratings are a photo-induced phenomena in optical fibers. The gratings can be used to measure strain by measuring the shift in wavelength. They placed the fibers into a smart structure to measure the stress and strain produced on support columns placed in bridges. As the cable is subjected to strain the grating causes a shift to a longer wavelength if the fiber is stretched and a shift to a shorter wavelength shift if the fiber is compacted. Our applications involve using the fibers to measure stress and strain on airborne systems. There are many ways to write Bragg gratings into optical fibers. Our focus is on side writing the grating. Our capabilities are limited in the production rate of the gratings. The Bragg grating is written into a fiber and becomes a permanent fixture. We are writing the grating to be centered at 1300 nm because that is the standard phase mask wavelength.
Sparse DOA estimation with polynomial rooting
DEFF Research Database (Denmark)
Xenaki, Angeliki; Gerstoft, Peter; Fernandez Grande, Efren
2015-01-01
Direction-of-arrival (DOA) estimation involves the localization of a few sources from a limited number of observations on an array of sensors. Thus, DOA estimation can be formulated as a sparse signal reconstruction problem and solved efficiently with compressive sensing (CS) to achieve highresol......Direction-of-arrival (DOA) estimation involves the localization of a few sources from a limited number of observations on an array of sensors. Thus, DOA estimation can be formulated as a sparse signal reconstruction problem and solved efficiently with compressive sensing (CS) to achieve...... highresolution imaging. Utilizing the dual optimal variables of the CS optimization problem, it is shown with Monte Carlo simulations that the DOAs are accurately reconstructed through polynomial rooting (Root-CS). Polynomial rooting is known to improve the resolution in several other DOA estimation methods...
Cylinder and metal grating polarization beam splitter
Yang, Junbo; Xu, Suzhi
2017-08-01
We propose a novel and compact metal grating polarization beam splitter (PBS) based on its different reflected and transmitted orders. The metal grating exhibits a broadband high reflectivity and polarization dependence. The rigorous coupled wave analysis is used to calculate the reflectivity and the transmitting spectra and optimize the structure parameters to realize the broadband PBS. The finite-element method is used to calculate the field distribution. The characteristics of the broadband high reflectivity, transmitting and the polarization dependence are investigated including wavelength, period, refractive index and the radius of circle grating. When grating period d = 400 nm, incident wavelength λ = 441 nm, incident angle θ = 60° and radius of circle d/5, then the zeroth reflection order R0 = 0.35 and the transmission zeroth order T0 = 0.08 for TE polarization, however, T0 = 0.34 and R0 = 0.01 for TM mode. The simple fabrication method involves only single etch step and good compatibility with complementary metal oxide semiconductor technology. PBS designed here is particularly suited for optical communication and optical information processing.
Discrete dipole approximation simulation of bead enhanced diffraction grating biosensor
International Nuclear Information System (INIS)
Arif, Khalid Mahmood
2016-01-01
We present the discrete dipole approximation simulation of light scattering from bead enhanced diffraction biosensor and report the effect of bead material, number of beads forming the grating and spatial randomness on the diffraction intensities of 1st and 0th orders. The dipole models of gratings are formed by volume slicing and image processing while the spatial locations of the beads on the substrate surface are randomly computed using discrete probability distribution. The effect of beads reduction on far-field scattering of 632.8 nm incident field, from fully occupied gratings to very coarse gratings, is studied for various bead materials. Our findings give insight into many difficult or experimentally impossible aspects of this genre of biosensors and establish that bead enhanced grating may be used for rapid and precise detection of small amounts of biomolecules. The results of simulations also show excellent qualitative similarities with experimental observations. - Highlights: • DDA was used to study the relationship between the number of beads forming gratings and ratio of first and zeroth order diffraction intensities. • A very flexible modeling program was developed to design complicated objects for DDA. • Material and spatial effects of bead distribution on surfaces were studied. • It has been shown that bead enhanced grating biosensor can be useful for fast detection of small amounts of biomolecules. • Experimental results qualitatively support the simulations and thus open a way to optimize the grating biosensors.
Fiber Bragg Grating Sensors for Harsh Environments
Directory of Open Access Journals (Sweden)
Stephen J. Mihailov
2012-02-01
Full Text Available Because of their small size, passive nature, immunity to electromagnetic interference, and capability to directly measure physical parameters such as temperature and strain, fiber Bragg grating sensors have developed beyond a laboratory curiosity and are becoming a mainstream sensing technology. Recently, high temperature stable gratings based on regeneration techniques and femtosecond infrared laser processing have shown promise for use in extreme environments such as high temperature, pressure or ionizing radiation. Such gratings are ideally suited for energy production applications where there is a requirement for advanced energy system instrumentation and controls that are operable in harsh environments. This paper will present a review of some of the more recent developments.
Pemodelan Tapis Fabry-perot pada Serat Optik dengan Menggunakan Fiber Bragg Grating
Pramuliawati, Septi; ', Saktioto; ', Defrianto
2015-01-01
Fabry-perot filter was successfully developed by a uniform Fiber Bragg Grating in fiber optic. A characterization of Bragg Grating was analyzed by using computational model with second-order of Transfer Matrix Method based on Coupled Mode Theory. The reflectivity, length of grating, and bandwidth were parametrics to determine the performance of single Bragg Grating. The transmission spectrum showed the longer grating is designed, the larger the reflectivity was produced, so that the transmiss...
Uniquely identifiable tamper-evident device using coupling between subwavelength gratings
Fievre, Ange Marie Patricia
Reliability and sensitive information protection are critical aspects of integrated circuits. A novel technique using near-field evanescent wave coupling from two subwavelength gratings (SWGs), with the input laser source delivered through an optical fiber is presented for tamper evidence of electronic components. The first grating of the pair of coupled subwavelength gratings (CSWGs) was milled directly on the output facet of the silica fiber using focused ion beam (FIB) etching. The second grating was patterned using e-beam lithography and etched into a glass substrate using reactive ion etching (RIE). The slightest intrusion attempt would separate the CSWGs and eliminate near-field coupling between the gratings. Tampering, therefore, would become evident. Computer simulations guided the design for optimal operation of the security solution. The physical dimensions of the SWGs, i.e. period and thickness, were optimized, for a 650 nm illuminating wavelength. The optimal dimensions resulted in a 560 nm grating period for the first grating etched in the silica optical fiber and 420 nm for the second grating etched in borosilicate glass. The incident light beam had a half-width at half-maximum (HWHM) of at least 7 microm to allow discernible higher transmission orders, and a HWHM of 28 microm for minimum noise. The minimum number of individual grating lines present on the optical fiber facet was identified as 15 lines. Grating rotation due to the cylindrical geometry of the fiber resulted in a rotation of the far-field pattern, corresponding to the rotation angle of moire fringes. With the goal of later adding authentication to tamper evidence, the concept of CSWGs signature was also modeled by introducing random and planned variations in the glass grating. The fiber was placed on a stage supported by a nanomanipulator, which permitted three-dimensional displacement while maintaining the fiber tip normal to the surface of the glass substrate. A 650 nm diode laser was
The effect of aberrated recording beams on reflecting Bragg gratings
SeGall, Marc; Ott, Daniel; Divliansky, Ivan; Glebov, Leonid B.
2013-03-01
The effect of aberrations present in the recording beams of a holographic setup is discussed regarding the period and spectral response of a reflecting volume Bragg grating. Imperfect recording beams result in spatially varying resonant wavelengths and the side lobes of the spectrum are washed out. Asymmetrical spectra, spectral broadening, and a reduction in peak diffraction efficiency may also be present, though these effects are less significant for gratings with wider spectral widths. Reflecting Bragg gratings (RBGs) are used as elements in a variety of applications including spectral beam combining1,2, mode locking3,4, longitudinal and transverse mode selection in lasers5,6, and sensing7,8. For applications requiring narrow spectral selectivity9, or large apertures10, these gratings must have a uniform period throughout the length of the recording medium, which may be on the order of millimeters. However, when using typical recording techniques such as two-beam interference for large aperture gratings and phase-mask recording of fiber gratings, aberrations from the optical elements in the system result in an imperfect grating structure11-13. In this paper we consider the effects of aberrations on large aperture gratings recorded in thick media using the two-beam interference technique. Previous works in analyzing the effects of aberrations have considered the effects of aberrations in a single recording plane where the beams perfectly overlap. Such an approach is valid for thin media (on the order of tens of microns), but for thick recording media (on the order of several millimeters) there will be a significant shift in the positions of the beams relative to each other as they traverse the recording medium. Therefore, the fringe pattern produced will not be constant throughout the grating if one or both beams have a non-uniform wavefront. Such non-uniform gratings may have a wider spectral width, a shifted resonant wavelength, or other problems. It is
High-order diffraction gratings for high-power semiconductor lasers
International Nuclear Information System (INIS)
Vasil’eva, V. V.; Vinokurov, D. A.; Zolotarev, V. V.; Leshko, A. Yu.; Petrunov, A. N.; Pikhtin, N. A.; Rastegaeva, M. G.; Sokolova, Z. N.; Shashkin, I. S.; Tarasov, I. S.
2012-01-01
A deep diffraction grating with a large period (∼2 μm) within one of the cladding layers is proposed for the implementation of selective feedback in a semiconductor laser. Frequency dependences of reflectance in the 12th diffraction order for rectangular, triangular, and trapezoidal diffraction gratings are calculated. It is shown that the maximum reflectance of the waveguide mode is attained using a rectangular or trapezoidal grating ∼2 μm deep in the laser structure. Deep trapezoidal diffraction gratings with large periods are fabricated in the Al 0.3 Ga 0.7 As cladding layer of a GaAs/AlGaAs laser structure using photolithography and reactive ion etching.
Measurement of chloride-ion concentration with long-period grating technology
Tang, Jaw-Luen; Wang, Jian-Neng
2007-06-01
A simple and low-cost long-period fiber grating (LPG) sensor suited for chloride-ion concentration measurement is presented. The LPG sensor is found to be sensitive to the refractive index of the medium around the cladding surface of the sensing grating, thus offering the prospect of development of practical sensors such as an ambient index sensor or a chemical concentration indicator with high stability and reliability. We measured chloride ions in a typical concrete sample immersed in salt water solutions with different weight concentrations ranging from 0% to 25%. Results show that the LPG sensor exhibited a linear decrease in the transmission loss and resonance wavelength shift when the concentration increased. The measurement accuracy for the concentration of salt in water solution is estimated to be 0.6% and the limit of detection for chloride ions is about 0.04%. To further enhance its sensitivity for chloride concentrations, we coated a monolayer of colloidal gold nanoparticles as the active material on the grating surface of the LPG sensor. The operating principle of sensing is based on the sensitivity of localized surface plasmon resonance of self-assembled gold colloids on the grating section of the LPG. With this method, a factor of two increase in the sensitivity of detecting chemical solution concentrations was obtained. The advantages of this type of fiber-optic sensor are that it is compact, relatively simple to construct and easy to use. Moreover, the sensor has the potential capability for on-site, in vivo and remote sensing, and it has potential use as a disposable sensor.
Femtosecond laser pulse written Volume Bragg Gratings
Directory of Open Access Journals (Sweden)
Richter Daniel
2013-11-01
Full Text Available Femtosecond laser pulses can be applied for structuring a wide range of ransparent materials. Here we want to show how to use this ability to realize Volume-Bragg-Gratings in various- mainly non-photosensitive - glasses. We will further present the characteristics of the realized gratings and a few elected applications that have been realized.
Polynomials in finite geometries and combinatorics
Blokhuis, A.; Walker, K.
1993-01-01
It is illustrated how elementary properties of polynomials can be used to attack extremal problems in finite and euclidean geometry, and in combinatorics. Also a new result, related to the problem of neighbourly cylinders is presented.
Dirichlet polynomials, majorization, and trumping
International Nuclear Information System (INIS)
Pereira, Rajesh; Plosker, Sarah
2013-01-01
Majorization and trumping are two partial orders which have proved useful in quantum information theory. We show some relations between these two partial orders and generalized Dirichlet polynomials, Mellin transforms, and completely monotone functions. These relations are used to prove a succinct generalization of Turgut’s characterization of trumping. (paper)
The neighbourhood polynomial of some families of dendrimers
Nazri Husin, Mohamad; Hasni, Roslan
2018-04-01
The neighbourhood polynomial N(G,x) is generating function for the number of faces of each cardinality in the neighbourhood complex of a graph and it is defined as (G,x)={\\sum }U\\in N(G){x}|U|, where N(G) is neighbourhood complex of a graph, whose vertices of the graph and faces are subsets of vertices that have a common neighbour. A dendrimers is an artificially manufactured or synthesized molecule built up from branched units called monomers. In this paper, we compute this polynomial for some families of dendrimer.
Grating-based tomography applications in biomedical engineering
Schulz, Georg; Thalmann, Peter; Khimchenko, Anna; Müller, Bert
2017-10-01
For the investigation of soft tissues or tissues consisting of soft and hard tissues on the microscopic level, hard X-ray phase tomography has become one of the most suitable imaging techniques. Besides other phase contrast methods grating interferometry has the advantage of higher sensitivity than inline methods and the quantitative results. One disadvantage of the conventional double-grating setup (XDGI) compared to inline methods is the limitation of the spatial resolution. This limitation can be overcome by removing the analyser grating resulting in a single-grating setup (XSGI). In order to verify the performance of XSGI concerning contrast and spatial resolution, a quantitative comparison of XSGI and XDGI tomograms of a human nerve was performed. Both techniques provide sufficient contrast to allow for the distinction of tissue types. The spatial resolution of the two-fold binned XSGI data set is improved by a factor of two in comparison to XDGI which underlies its performance in tomography of soft tissues. Another application for grating-based X-ray phase tomography is the simultaneous visualization of soft and hard tissues of a plaque-containing coronary artery. The simultaneous visualization of both tissues is important for the segmentation of the lumen. The segmented data can be used for flow simulations in order to obtain information about the three-dimensional wall shear stress distribution needed for the optimization of mechano-sensitive nanocontainers used for drug delivery.
Mechanism of optical unidirectional transmission in subwavelength dual-metal gratings
Gao, H.; Zheng, Z. Y.; Hao, H. Y.; Dong, A. G.; Fan, Z. J.; Liu, D. H.
2014-03-01
The mechanism of optical unidirectional (OUD) transmission in parallel subwavelength dual-metal gratings was investigated. It was found that this kind of OUD phenomenon originates from the coupling of the surface plasmon polaritons (SPPs) between the front grating and a layer of metal film which replaces the rear grating. The higher the intensity of the coupled SPPs at the entrances of the rear grating, the higher the transmittance can be achieved. Basing on this property, an effective OUD example was achieved by exploring the intensity difference at the entrances of the rear gratings between the two incidences of opposite directions. In this kind of OUD, the positive transmittance can exceed 80 % and the difference between the transmittances of the two opposite directions can be as large as 63 %. The detailed design process was also presented.
Diffraction efficiency calculations of polarization diffraction gratings with surface relief
Nazarova, D.; Sharlandjiev, P.; Berberova, N.; Blagoeva, B.; Stoykova, E.; Nedelchev, L.
2018-03-01
In this paper, we evaluate the optical response of a stack of two diffraction gratings of equal one-dimensional periodicity. The first one is a surface-relief grating structure; the second, a volume polarization grating. This model is based on our experimental results from polarization holographic recordings in azopolymer films. We used films of commercially available azopolymer (poly[1-[4-(3-carboxy-4-hydroxyphenylazo) benzenesulfonamido]-1,2-ethanediyl, sodium salt]), shortly denoted as PAZO. During the recording process, a polarization grating in the volume of the material and a relief grating on the film surface are formed simultaneously. In order to evaluate numerically the optical response of this “hybrid” diffraction structure, we used the rigorous coupled-wave approach (RCWA). It yields stable numerical solutions of Maxwell’s vector equations using the algebraic eigenvalue method.
A new derivation of the highest-weight polynomial of a unitary lie algebra
International Nuclear Information System (INIS)
P Chau, Huu-Tai; P Van, Isacker
2000-01-01
A new method is presented to derive the expression of the highest-weight polynomial used to build the basis of an irreducible representation (IR) of the unitary algebra U(2J+1). After a brief reminder of Moshinsky's method to arrive at the set of equations defining the highest-weight polynomial of U(2J+1), an alternative derivation of the polynomial from these equations is presented. The method is less general than the one proposed by Moshinsky but has the advantage that the determinantal expression of the highest-weight polynomial is arrived at in a direct way using matrix inversions. (authors)
A probabilistic approach of sum rules for heat polynomials
International Nuclear Information System (INIS)
Vignat, C; Lévêque, O
2012-01-01
In this paper, we show that the sum rules for generalized Hermite polynomials derived by Daboul and Mizrahi (2005 J. Phys. A: Math. Gen. http://dx.doi.org/10.1088/0305-4470/38/2/010) and by Graczyk and Nowak (2004 C. R. Acad. Sci., Ser. 1 338 849) can be interpreted and easily recovered using a probabilistic moment representation of these polynomials. The covariance property of the raising operator of the harmonic oscillator, which is at the origin of the identities proved in Daboul and Mizrahi and the dimension reduction effect expressed in the main result of Graczyk and Nowak are both interpreted in terms of the rotational invariance of the Gaussian distributions. As an application of these results, we uncover a probabilistic moment interpretation of two classical integrals of the Wigner function that involve the associated Laguerre polynomials. (paper)
Observation of narrowband intrinsic spectra of Brillouin dynamic gratings.
Song, Kwang Yong; Yoon, Hyuk Jin
2010-09-01
We experimentally demonstrate that the reflection spectrum of a Brillouin dynamic grating in a polarization-maintaining fiber can be much narrower than the intrinsic linewidth of the stimulated Brillouin scattering, matching well with the theory of a fiber Bragg grating in terms of the linewidth and the reflectivity. A 3 dB bandwidth as narrow as 10.5 MHz is observed with the Brillouin dynamic grating generated in a 9 m uniform fiber.
A Combinatorial Proof of a Result on Generalized Lucas Polynomials
Directory of Open Access Journals (Sweden)
Laugier Alexandre
2016-09-01
Full Text Available We give a combinatorial proof of an elementary property of generalized Lucas polynomials, inspired by [1]. These polynomials in s and t are defined by the recurrence relation 〈n〉 = s〈n-1〉+t〈n-2〉 for n ≥ 2. The initial values are 〈0〉 = 2; 〈1〉= s, respectively.
A Kantorovich Type of Szasz Operators Including Brenke-Type Polynomials
Directory of Open Access Journals (Sweden)
Fatma Taşdelen
2012-01-01
convergence properties of these operators by using Korovkin's theorem. We also present the order of convergence with the help of a classical approach, the second modulus of continuity, and Peetre's -functional. Furthermore, an example of Kantorovich type of the operators including Gould-Hopper polynomials is presented and Voronovskaya-type result is given for these operators including Gould-Hopper polynomials.
Lower bounds for the circuit size of partially homogeneous polynomials
Czech Academy of Sciences Publication Activity Database
Le, Hong-Van
2017-01-01
Roč. 225, č. 4 (2017), s. 639-657 ISSN 1072-3374 Institutional support: RVO:67985840 Keywords : partially homogeneous polynomials * polynomials Subject RIV: BA - General Mathematics OBOR OECD: Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8) https://link.springer.com/article/10.1007/s10958-017-3483-4
Generalized catalan numbers, sequences and polynomials
KOÇ, Cemal; GÜLOĞLU, İsmail; ESİN, Songül
2010-01-01
In this paper we present an algebraic interpretation for generalized Catalan numbers. We describe them as dimensions of certain subspaces of multilinear polynomials. This description is of utmost importance in the investigation of annihilators in exterior algebras.
Multilevel weighted least squares polynomial approximation
Haji-Ali, Abdul-Lateef; Nobile, Fabio; Tempone, Raul; Wolfers, Sö ren
2017-01-01
, obtaining polynomial approximations with a single level method can become prohibitively expensive, as it requires a sufficiently large number of samples, each computed with a sufficiently small discretization error. As a solution to this problem, we propose
An algorithmic approach to solving polynomial equations associated with quantum circuits
International Nuclear Information System (INIS)
Gerdt, V.P.; Zinin, M.V.
2009-01-01
In this paper we present two algorithms for reducing systems of multivariate polynomial equations over the finite field F 2 to the canonical triangular form called lexicographical Groebner basis. This triangular form is the most appropriate for finding solutions of the system. On the other hand, the system of polynomials over F 2 whose variables also take values in F 2 (Boolean polynomials) completely describes the unitary matrix generated by a quantum circuit. In particular, the matrix itself can be computed by counting the number of solutions (roots) of the associated polynomial system. Thereby, efficient construction of the lexicographical Groebner bases over F 2 associated with quantum circuits gives a method for computing their circuit matrices that is alternative to the direct numerical method based on linear algebra. We compare our implementation of both algorithms with some other software packages available for computing Groebner bases over F 2
Recurrence approach and higher order polynomial algebras for superintegrable monopole systems
Hoque, Md Fazlul; Marquette, Ian; Zhang, Yao-Zhong
2018-05-01
We revisit the MIC-harmonic oscillator in flat space with monopole interaction and derive the polynomial algebra satisfied by the integrals of motion and its energy spectrum using the ad hoc recurrence approach. We introduce a superintegrable monopole system in a generalized Taub-Newman-Unti-Tamburino (NUT) space. The Schrödinger equation of this model is solved in spherical coordinates in the framework of Stäckel transformation. It is shown that wave functions of the quantum system can be expressed in terms of the product of Laguerre and Jacobi polynomials. We construct ladder and shift operators based on the corresponding wave functions and obtain the recurrence formulas. By applying these recurrence relations, we construct higher order algebraically independent integrals of motion. We show that the integrals form a polynomial algebra. We construct the structure functions of the polynomial algebra and obtain the degenerate energy spectra of the model.
Optical and x-ray alignment approaches for off-plane reflection gratings
Allured, Ryan; Donovan, Benjamin D.; DeRoo, Casey T.; Marlowe, Hannah R.; McEntaffer, Randall L.; Tutt, James H.; Cheimets, Peter N.; Hertz, Edward; Smith, Randall K.; Burwitz, Vadim; Hartner, Gisela; Menz, Benedikt
2015-09-01
Off-plane reflection gratings offer the potential for high-resolution, high-throughput X-ray spectroscopy on future missions. Typically, the gratings are placed in the path of a converging beam from an X-ray telescope. In the off-plane reflection grating case, these gratings must be co-aligned such that their diffracted spectra overlap at the focal plane. Misalignments degrade spectral resolution and effective area. In-situ X-ray alignment of a pair of off-plane reflection gratings in the path of a silicon pore optics module has been performed at the MPE PANTER beamline in Germany. However, in-situ X-ray alignment may not be feasible when assembling all of the gratings required for a satellite mission. In that event, optical methods must be developed to achieve spectral alignment. We have developed an alignment approach utilizing a Shack-Hartmann wavefront sensor and diffraction of an ultraviolet laser. We are fabricating the necessary hardware, and will be taking a prototype grating module to an X-ray beamline for performance testing following assembly and alignment.
Canonical basis for type A4 (II) - Polynomial elements in one variable
International Nuclear Information System (INIS)
Hu Yuwang; Ye Jiachen
2003-12-01
All the 62 monomial elements in the canonical basis B of the quantized enveloping algebra for type A 4 have been determined. According to Lusztig's idea, the elements in the canonical basis B consist of monomials and linear combinations of monomials (for convenience, we call them polynomials). In this note, we compute all the 144 polynomial elements in one variable in the canonical basis B of the quantized enveloping algebra for type A 4 based on our joint note. We conjecture that there are other polynomial elements in two or three variables in the canonical basis B, which include independent variables and dependent variables. Moreover, it is conjectured that there are no polynomial elements in the canonical basis B with four or more variables. (author)
Polarizing beam splitter of deep-etched triangular-groove fused-silica gratings.
Zheng, Jiangjun; Zhou, Changhe; Feng, Jijun; Wang, Bo
2008-07-15
We investigated the use of a deep-etched fused-silica grating with triangular-shaped grooves as a highly efficient polarizing beam splitter (PBS). A triangular-groove PBS grating is designed at a wavelength of 1550 nm to be used in optical communication. When it is illuminated in Littrow mounting, the transmitted TE- and TM-polarized waves are mainly diffracted in the minus-first and zeroth orders, respectively. The design condition is based on the average differences of the grating mode indices, which is verified by using rigorous coupled-wave analysis. The designed PBS grating is highly efficient over the C+L band range for both TE and TM polarizations (>97.68%). It is shown that such a triangular-groove PBS grating can exhibit a higher diffraction efficiency, a larger extinction ratio, and less reflection loss than the binary-phase fused-silica PBS grating.
Discrete-Time Filter Synthesis using Product of Gegenbauer Polynomials
N. Stojanovic; N. Stamenkovic; I. Krstic
2016-01-01
A new approximation to design continuoustime and discrete-time low-pass filters, presented in this paper, based on the product of Gegenbauer polynomials, provides the ability of more flexible adjustment of passband and stopband responses. The design is achieved taking into account a prescribed specification, leading to a better trade-off among the magnitude and group delay responses. Many well-known continuous-time and discrete-time transitional filter based on the classical polynomial approx...
Bounds and asymptotics for orthogonal polynomials for varying weights
Levin, Eli
2018-01-01
This book establishes bounds and asymptotics under almost minimal conditions on the varying weights, and applies them to universality limits and entropy integrals. Orthogonal polynomials associated with varying weights play a key role in analyzing random matrices and other topics. This book will be of use to a wide community of mathematicians, physicists, and statisticians dealing with techniques of potential theory, orthogonal polynomials, approximation theory, as well as random matrices. .
On Linear Combinations of Two Orthogonal Polynomial Sequences on the Unit Circle
Directory of Open Access Journals (Sweden)
Suárez C
2010-01-01
Full Text Available Let be a monic orthogonal polynomial sequence on the unit circle. We define recursively a new sequence of polynomials by the following linear combination: , , . In this paper, we give necessary and sufficient conditions in order to make be an orthogonal polynomial sequence too. Moreover, we obtain an explicit representation for the Verblunsky coefficients and in terms of and . Finally, we show the relation between their corresponding Carathéodory functions and their associated linear functionals.
On the existence of polynomial Lyapunov functions for rationally stable vector fields
DEFF Research Database (Denmark)
Leth, Tobias; Wisniewski, Rafal; Sloth, Christoffer
2018-01-01
This paper proves the existence of polynomial Lyapunov functions for rationally stable vector fields. For practical purposes the existence of polynomial Lyapunov functions plays a significant role since polynomial Lyapunov functions can be found algorithmically. The paper extents an existing result...... on exponentially stable vector fields to the case of rational stability. For asymptotically stable vector fields a known counter example is investigated to exhibit the mechanisms responsible for the inability to extend the result further....
Wolf, Alexey; Dostovalov, Alexandr; Skvortsov, Mikhail; Raspopin, Kirill; Parygin, Alexandr; Babin, Sergey
2018-05-01
In this work, long high-quality fiber Bragg gratings with phase shifts in the structure are inscribed directly in the optical fiber by point-by-point technique using femtosecond laser pulses. Phase shifts are introduced during the inscription process with a piezoelectric actuator, which rapidly shifts the fiber along the direction of its movement in a chosen point of the grating with a chosen shift value. As examples, single and double π phase shifts are introduced in fiber Bragg gratings with a length up to 34 mm in passive fibers, which provide corresponding transmission peaks with bandwidth less than 1 pm. It is shown that 37 mm π -phase-shifted grating inscribed in an active Er-doped fiber forms high-quality DFB laser cavity generating single-frequency radiation at 1550 nm with bandwidth of 20 kHz and signal-to-noise ratio of >70 dB. The inscription technique has a high degree of performance and flexibility and can be easily implemented in fibers of various types.
On Modular Counting with Polynomials
DEFF Research Database (Denmark)
Hansen, Kristoffer Arnsfelt
2006-01-01
For any integers m and l, where m has r sufficiently large (depending on l) factors, that are powers of r distinct primes, we give a construction of a (symmetric) polynomial over Z_m of degree O(\\sqrt n) that is a generalized representation (commonly also called weak representation) of the MODl f...
Using InTeGrate materials to develop interdisciplinary thinking for a sustainable future
Awad, A. A.; Gilbert, L.; Iverson, E. A. R.; Manduca, C. A.; Steer, D. N.
2017-12-01
InTeGrate materials focus on societal grand challenges, sustainability, and interdisciplinary problems through developing geoscientific habits of mind, the use of credible data, and systems thinking. The materials are freely available 2-3 week modules and courses that allow instructors to focus on a wide variety of topics from regulating carbon emissions, changing biosphere, and storms and community resilience to environmental justice, ocean sustainability, and humans' dependence on mineral resources, integrating a variety of relevant interdisciplinary activities throughout. Presented with interdisciplinary approaches, students learn with tools to integrate engineering, policy, economics, and social aspects with the science to address the challenges. Students' ability to apply interdisciplinary approaches to address sustainability problems is made visible through the essays they write as a part of the materials assessment. InTeGrate modules have been adopted and implemented by faculty members interested in sustainability themes and innovative pedagogy, and have reached more than 50,000 students in all 50 states, Puerto Rico, India, and Micronesia. Student data were collected from 533 assessment essays in 57 undergraduate classes. The essays required students to describe a global challenge in an interdisciplinary manner through identifying scientific implications, and connecting it to economic, social and policy decisions. Students also completed a second essay assessing their systems thinking ability, a geoscience literacy exam (GLE), and demographic and attitudinal surveys. Scores for students enrolled in classes using InTeGrate materials were compared to scores from students in similar classes that did not use InteGrate materials. The InTeGrate and control groups had equivalent GLE scores and demographic characteristics. Essay scores for students in InTeGrate introductory or majors courses outperformed students in comparable level control courses as measured by
Numerical modelling of a straw-fired grate boiler
DEFF Research Database (Denmark)
Kær, Søren Knudsen
2004-01-01
The paper presents a computational fluid dynamics (CFD) analysis of a 33 MW straw-fired grate boiler. Combustion on the grate plays akey-role in the analysis of these boilers and in this work a stand-alone code was used to provide inlet conditions for the CFD analysis. Modelpredictions were...... compared with available gas temperature and species concentration measurements showing good agreement. Combustionof biomass in grate-based boilers is often associated with high emission levels and relatively high amounts of unburnt carbon in the fly ash.Based on the CFD analysis, it is suggested that poor...
Poirot, Jordan; De Luna, Paolo; Rainer, Gregor
2016-04-01
We comprehensively characterize spiking and visual evoked potential (VEP) activity in tree shrew V1 and V2 using Cartesian, hyperbolic, and polar gratings. Neural selectivity to structure of Cartesian gratings was higher than other grating classes in both visual areas. From V1 to V2, structure selectivity of spiking activity increased, whereas corresponding VEP values tended to decrease, suggesting that single-neuron coding of Cartesian grating attributes improved while the cortical columnar organization of these neurons became less precise from V1 to V2. We observed that neurons in V2 generally exhibited similar selectivity for polar and Cartesian gratings, suggesting that structure of polar-like stimuli might be encoded as early as in V2. This hypothesis is supported by the preference shift from V1 to V2 toward polar gratings of higher spatial frequency, consistent with the notion that V2 neurons encode visual scene borders and contours. Neural sensitivity to modulations of polarity of hyperbolic gratings was highest among all grating classes and closely related to the visual receptive field (RF) organization of ON- and OFF-dominated subregions. We show that spatial RF reconstructions depend strongly on grating class, suggesting that intracortical contributions to RF structure are strongest for Cartesian and polar gratings. Hyperbolic gratings tend to recruit least cortical elaboration such that the RF maps are similar to those generated by sparse noise, which most closely approximate feedforward inputs. Our findings complement previous literature in primates, rodents, and carnivores and highlight novel aspects of shape representation and coding occurring in mammalian early visual cortex. Copyright © 2016 the American Physiological Society.
Bernoulli numbers and polynomials from a more general point of view
Energy Technology Data Exchange (ETDEWEB)
Dattoli, G. [ENEA, Centro Ricerche Frascati, Frascati, RM(Italy). Div. Fisica Applicata; Cesarano, C. [Ulm Univ., Ulm (Germany). Dept. of Mathematics; Lonzellutta, S. [ENEA, Centro Ricerche E. Clementel, Bologna (Italy). Div. Fisica Applicata
2000-07-01
In this work it is applied the method of generating function, to introduce new forms of Bernoulli numbers and polynomials, which are exploited to derive further classes of partial sums involving generalized many index many variable polynomials. Analogous considerations are developed for the Euler numbers and polynomials. [Italian] Si applica il metodo della funzione generatrice per introdurre nuove forme di numeri e polinomi di Bernoulli che vengono utilizzati per sviluppare e per calcolare somme parziali che coinvolgono polinomi a piu' indici ed a piu' variabili. Si sviluppano considerazioni analoghe per i polinomi ed i numeri di Eulero.
Quantum Hurwitz numbers and Macdonald polynomials
Harnad, J.
2016-11-01
Parametric families in the center Z(C[Sn]) of the group algebra of the symmetric group are obtained by identifying the indeterminates in the generating function for Macdonald polynomials as commuting Jucys-Murphy elements. Their eigenvalues provide coefficients in the double Schur function expansion of 2D Toda τ-functions of hypergeometric type. Expressing these in the basis of products of power sum symmetric functions, the coefficients may be interpreted geometrically as parametric families of quantum Hurwitz numbers, enumerating weighted branched coverings of the Riemann sphere. Combinatorially, they give quantum weighted sums over paths in the Cayley graph of Sn generated by transpositions. Dual pairs of bases for the algebra of symmetric functions with respect to the scalar product in which the Macdonald polynomials are orthogonal provide both the geometrical and combinatorial significance of these quantum weighted enumerative invariants.
Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies
International Nuclear Information System (INIS)
Hampton, Jerrad; Doostan, Alireza
2015-01-01
Sampling orthogonal polynomial bases via Monte Carlo is of interest for uncertainty quantification of models with random inputs, using Polynomial Chaos (PC) expansions. It is known that bounding a probabilistic parameter, referred to as coherence, yields a bound on the number of samples necessary to identify coefficients in a sparse PC expansion via solution to an ℓ 1 -minimization problem. Utilizing results for orthogonal polynomials, we bound the coherence parameter for polynomials of Hermite and Legendre type under their respective natural sampling distribution. In both polynomial bases we identify an importance sampling distribution which yields a bound with weaker dependence on the order of the approximation. For more general orthonormal bases, we propose the coherence-optimal sampling: a Markov Chain Monte Carlo sampling, which directly uses the basis functions under consideration to achieve a statistical optimality among all sampling schemes with identical support. We demonstrate these different sampling strategies numerically in both high-order and high-dimensional, manufactured PC expansions. In addition, the quality of each sampling method is compared in the identification of solutions to two differential equations, one with a high-dimensional random input and the other with a high-order PC expansion. In both cases, the coherence-optimal sampling scheme leads to similar or considerably improved accuracy
Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies
Hampton, Jerrad; Doostan, Alireza
2015-01-01
Sampling orthogonal polynomial bases via Monte Carlo is of interest for uncertainty quantification of models with random inputs, using Polynomial Chaos (PC) expansions. It is known that bounding a probabilistic parameter, referred to as coherence, yields a bound on the number of samples necessary to identify coefficients in a sparse PC expansion via solution to an ℓ1-minimization problem. Utilizing results for orthogonal polynomials, we bound the coherence parameter for polynomials of Hermite and Legendre type under their respective natural sampling distribution. In both polynomial bases we identify an importance sampling distribution which yields a bound with weaker dependence on the order of the approximation. For more general orthonormal bases, we propose the coherence-optimal sampling: a Markov Chain Monte Carlo sampling, which directly uses the basis functions under consideration to achieve a statistical optimality among all sampling schemes with identical support. We demonstrate these different sampling strategies numerically in both high-order and high-dimensional, manufactured PC expansions. In addition, the quality of each sampling method is compared in the identification of solutions to two differential equations, one with a high-dimensional random input and the other with a high-order PC expansion. In both cases, the coherence-optimal sampling scheme leads to similar or considerably improved accuracy.
Välimäki, Vesa; Pekonen, Jussi; Nam, Juhan
2012-01-01
Digital subtractive synthesis is a popular music synthesis method, which requires oscillators that are aliasing-free in a perceptual sense. It is a research challenge to find computationally efficient waveform generation algorithms that produce similar-sounding signals to analog music synthesizers but which are free from audible aliasing. A technique for approximately bandlimited waveform generation is considered that is based on a polynomial correction function, which is defined as the difference of a non-bandlimited step function and a polynomial approximation of the ideal bandlimited step function. It is shown that the ideal bandlimited step function is equivalent to the sine integral, and that integrated polynomial interpolation methods can successfully approximate it. Integrated Lagrange interpolation and B-spline basis functions are considered for polynomial approximation. The polynomial correction function can be added onto samples around each discontinuity in a non-bandlimited waveform to suppress aliasing. Comparison against previously known methods shows that the proposed technique yields the best tradeoff between computational cost and sound quality. The superior method amongst those considered in this study is the integrated third-order B-spline correction function, which offers perceptually aliasing-free sawtooth emulation up to the fundamental frequency of 7.8 kHz at the sample rate of 44.1 kHz. © 2012 Acoustical Society of America.
Two-port connecting-layer-based sandwiched grating by a polarization-independent design.
Li, Hongtao; Wang, Bo
2017-05-02
In this paper, a two-port connecting-layer-based sandwiched beam splitter grating with polarization-independent property is reported and designed. Such the grating can separate the transmission polarized light into two diffraction orders with equal energies, which can realize the nearly 50/50 output with good uniformity. For the given wavelength of 800 nm and period of 780 nm, a simplified modal method can design a optimal duty cycle and the estimation value of the grating depth can be calculated based on it. In order to obtain the precise grating parameters, a rigorous coupled-wave analysis can be employed to optimize grating parameters by seeking for the precise grating depth and the thickness of connecting layer. Based on the optimized design, a high-efficiency two-port output grating with the wideband performances can be gained. Even more important, diffraction efficiencies are calculated by using two analytical methods, which are proved to be coincided well with each other. Therefore, the grating is significant for practical optical photonic element in engineering.
Tensor calculus in polar coordinates using Jacobi polynomials
Vasil, Geoffrey M.; Burns, Keaton J.; Lecoanet, Daniel; Olver, Sheehan; Brown, Benjamin P.; Oishi, Jeffrey S.
2016-11-01
Spectral methods are an efficient way to solve partial differential equations on domains possessing certain symmetries. The utility of a method depends strongly on the choice of spectral basis. In this paper we describe a set of bases built out of Jacobi polynomials, and associated operators for solving scalar, vector, and tensor partial differential equations in polar coordinates on a unit disk. By construction, the bases satisfy regularity conditions at r = 0 for any tensorial field. The coordinate singularity in a disk is a prototypical case for many coordinate singularities. The work presented here extends to other geometries. The operators represent covariant derivatives, multiplication by azimuthally symmetric functions, and the tensorial relationship between fields. These arise naturally from relations between classical orthogonal polynomials, and form a Heisenberg algebra. Other past work uses more specific polynomial bases for solving equations in polar coordinates. The main innovation in this paper is to use a larger set of possible bases to achieve maximum bandedness of linear operations. We provide a series of applications of the methods, illustrating their ease-of-use and accuracy.
Real-root property of the spectral polynomial of the Treibich-Verdier potential and related problems
Chen, Zhijie; Kuo, Ting-Jung; Lin, Chang-Shou; Takemura, Kouichi
2018-04-01
We study the spectral polynomial of the Treibich-Verdier potential. Such spectral polynomial, which is a generalization of the classical Lamé polynomial, plays fundamental roles in both the finite-gap theory and the ODE theory of Heun's equation. In this paper, we prove that all the roots of such spectral polynomial are real and distinct under some assumptions. The proof uses the classical concept of Sturm sequence and isomonodromic theories. We also prove an analogous result for a polynomial associated with a generalized Lamé equation, where we apply a new approach based on the viewpoint of the monodromy data.
On Sequences of Numbers and Polynomials Defined by Linear Recurrence Relations of Order 2
Directory of Open Access Journals (Sweden)
Tian-Xiao He
2009-01-01
Full Text Available Here we present a new method to construct the explicit formula of a sequence of numbers and polynomials generated by a linear recurrence relation of order 2. The applications of the method to the Fibonacci and Lucas numbers, Chebyshev polynomials, the generalized Gegenbauer-Humbert polynomials are also discussed. The derived idea provides a general method to construct identities of number or polynomial sequences defined by linear recurrence relations. The applications using the method to solve some algebraic and ordinary differential equations are presented.
Numerical Solutions for Convection-Diffusion Equation through Non-Polynomial Spline
Directory of Open Access Journals (Sweden)
Ravi Kanth A.S.V.
2016-01-01
Full Text Available In this paper, numerical solutions for convection-diffusion equation via non-polynomial splines are studied. We purpose an implicit method based on non-polynomial spline functions for solving the convection-diffusion equation. The method is proven to be unconditionally stable by using Von Neumann technique. Numerical results are illustrated to demonstrate the efficiency and stability of the purposed method.
Directory of Open Access Journals (Sweden)
S. Vukotic
2016-08-01
Full Text Available Digital polynomial-based interpolation filters implemented using the Farrow structure are used in Digital Signal Processing (DSP to calculate the signal between its discrete samples. The two basic design parameters for these filters are number of polynomial-segments defining the finite length of impulse response, and order of polynomials in each polynomial segment. The complexity of the implementation structure and the frequency domain performance depend on these two parameters. This contribution presents estimation formulae for length and polynomial order of polynomial-based filters for various types of requirements including attenuation in stopband, width of transitions band, deviation in passband, weighting in passband/stopband.
Modeling, simulation, and design of SAW grating filters
Schwelb, Otto; Adler, E. L.; Slaboszewicz, J. K.
1990-05-01
A systematic procedure for modeling, simulating, and designing SAW (surface acoustic wave) grating filters, taking losses into account, is described. Grating structures and IDTs (interdigital transducers) coupling to SAWs are defined by cascadable transmission-matrix building blocks. Driving point and transfer characteristics (immittances) of complex architectures consisting of gratings, transducers, and coupling networks are obtained by chain-multiplying building-block matrices. This modular approach to resonator filter analysis and design combines the elements of lossy filter synthesis with the transmission-matrix description of SAW components. A multipole filter design procedure based on a lumped-element-model approximation of one-pole two-port resonator building blocks is given and the range of validity of this model examined. The software for simulating the performance of SAW grating devices based on this matrix approach is described, and its performance, when linked to the design procedure to form a CAD/CAA (computer-aided design and analysis) multiple-filter design package, is illustrated with a resonator filter design example.
Invariant hyperplanes and Darboux integrability of polynomial vector fields
International Nuclear Information System (INIS)
Zhang Xiang
2002-01-01
This paper is composed of two parts. In the first part, we provide an upper bound for the number of invariant hyperplanes of the polynomial vector fields in n variables. This result generalizes those given in Artes et al (1998 Pac. J. Math. 184 207-30) and Llibre and Rodriguez (2000 Bull. Sci. Math. 124 599-619). The second part gives an extension of the Darboux theory of integrability to polynomial vector fields on algebraic varieties
A Formally Verified Conflict Detection Algorithm for Polynomial Trajectories
Narkawicz, Anthony; Munoz, Cesar
2015-01-01
In air traffic management, conflict detection algorithms are used to determine whether or not aircraft are predicted to lose horizontal and vertical separation minima within a time interval assuming a trajectory model. In the case of linear trajectories, conflict detection algorithms have been proposed that are both sound, i.e., they detect all conflicts, and complete, i.e., they do not present false alarms. In general, for arbitrary nonlinear trajectory models, it is possible to define detection algorithms that are either sound or complete, but not both. This paper considers the case of nonlinear aircraft trajectory models based on polynomial functions. In particular, it proposes a conflict detection algorithm that precisely determines whether, given a lookahead time, two aircraft flying polynomial trajectories are in conflict. That is, it has been formally verified that, assuming that the aircraft trajectories are modeled as polynomial functions, the proposed algorithm is both sound and complete.
On selfadjoint functors satisfying polynomial relations
DEFF Research Database (Denmark)
Agerholm, Troels; Mazorchuk, Volodomyr
2011-01-01
We study selfadjoint functors acting on categories of finite dimen- sional modules over finite dimensional algebras with an emphasis on functors satisfying some polynomial relations. Selfadjoint func- tors satisfying several easy relations, in particular, idempotents and square roots of a sum...
SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos
Energy Technology Data Exchange (ETDEWEB)
Ahlfeld, R., E-mail: r.ahlfeld14@imperial.ac.uk; Belkouchi, B.; Montomoli, F.
2016-09-01
A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrix is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the Hankel matrix of moments. It can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions. The extension to more input variables suggested in this work, is an anisotropic and adaptive version of Smolyak's algorithm that is solely based on the moments of the input probability distributions. It is referred to as SAMBA (PC), which is short for Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos. It is illustrated that for moderately high-dimensional problems (up to 20 different input variables or histograms) SAMBA can significantly simplify the calculation of sparse Gaussian quadrature rules. SAMBA's efficiency for multivariate functions with regard to data availability is further demonstrated by analysing higher order convergence and accuracy for a set of nonlinear test functions with 2, 5
SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos
International Nuclear Information System (INIS)
Ahlfeld, R.; Belkouchi, B.; Montomoli, F.
2016-01-01
A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrix is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the Hankel matrix of moments. It can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions. The extension to more input variables suggested in this work, is an anisotropic and adaptive version of Smolyak's algorithm that is solely based on the moments of the input probability distributions. It is referred to as SAMBA (PC), which is short for Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos. It is illustrated that for moderately high-dimensional problems (up to 20 different input variables or histograms) SAMBA can significantly simplify the calculation of sparse Gaussian quadrature rules. SAMBA's efficiency for multivariate functions with regard to data availability is further demonstrated by analysing higher order convergence and accuracy for a set of nonlinear test functions with 2, 5 and 10
Bayer Demosaicking with Polynomial Interpolation.
Wu, Jiaji; Anisetti, Marco; Wu, Wei; Damiani, Ernesto; Jeon, Gwanggil
2016-08-30
Demosaicking is a digital image process to reconstruct full color digital images from incomplete color samples from an image sensor. It is an unavoidable process for many devices incorporating camera sensor (e.g. mobile phones, tablet, etc.). In this paper, we introduce a new demosaicking algorithm based on polynomial interpolation-based demosaicking (PID). Our method makes three contributions: calculation of error predictors, edge classification based on color differences, and a refinement stage using a weighted sum strategy. Our new predictors are generated on the basis of on the polynomial interpolation, and can be used as a sound alternative to other predictors obtained by bilinear or Laplacian interpolation. In this paper we show how our predictors can be combined according to the proposed edge classifier. After populating three color channels, a refinement stage is applied to enhance the image quality and reduce demosaicking artifacts. Our experimental results show that the proposed method substantially improves over existing demosaicking methods in terms of objective performance (CPSNR, S-CIELAB E, and FSIM), and visual performance.
Polynomials in algebraic analysis
Multarzyński, Piotr
2012-01-01
The concept of polynomials in the sense of algebraic analysis, for a single right invertible linear operator, was introduced and studied originally by D. Przeworska-Rolewicz \\cite{DPR}. One of the elegant results corresponding with that notion is a purely algebraic version of the Taylor formula, being a generalization of its usual counterpart, well known for functions of one variable. In quantum calculus there are some specific discrete derivations analyzed, which are right invertible linear ...
International Nuclear Information System (INIS)
van Diejen, J.F.
1997-01-01
Two families (type A and type B) of confluent hypergeometric polynomials in several variables are studied. We describe the orthogonality properties, differential equations, and Pieri-type recurrence formulas for these families. In the one-variable case, the polynomials in question reduce to the Hermite polynomials (type A) and the Laguerre polynomials (type B), respectively. The multivariable confluent hypergeometric families considered here may be used to diagonalize the rational quantum Calogero models with harmonic confinement (for the classical root systems) and are closely connected to the (symmetric) generalized spherical harmonics investigated by Dunkl. (orig.)
Global sensitivity analysis using sparse grid interpolation and polynomial chaos
International Nuclear Information System (INIS)
Buzzard, Gregery T.
2012-01-01
Sparse grid interpolation is widely used to provide good approximations to smooth functions in high dimensions based on relatively few function evaluations. By using an efficient conversion from the interpolating polynomial provided by evaluations on a sparse grid to a representation in terms of orthogonal polynomials (gPC representation), we show how to use these relatively few function evaluations to estimate several types of sensitivity coefficients and to provide estimates on local minima and maxima. First, we provide a good estimate of the variance-based sensitivity coefficients of Sobol' (1990) [1] and then use the gradient of the gPC representation to give good approximations to the derivative-based sensitivity coefficients described by Kucherenko and Sobol' (2009) [2]. Finally, we use the package HOM4PS-2.0 given in Lee et al. (2008) [3] to determine the critical points of the interpolating polynomial and use these to determine the local minima and maxima of this polynomial. - Highlights: ► Efficient estimation of variance-based sensitivity coefficients. ► Efficient estimation of derivative-based sensitivity coefficients. ► Use of homotopy methods for approximation of local maxima and minima.
Modern Theory of Gratings Resonant Scattering: Analysis Techniques and Phenomena
Sirenko, Yuriy K
2010-01-01
Diffraction gratings are one of the most popular objects of analysis in electromagnetic theory. The requirements of applied optics and microwave engineering lead to many new problems and challenges for the theory of diffraction gratings, which force us to search for new methods and tools for their resolution. In Modern Theory of Gratings, the authors present results of the electromagnetic theory of diffraction gratings that will constitute the base of further development of this theory, which meet the challenges provided by modern requirements of fundamental and applied science. This volume covers: spectral theory of gratings (Chapter 1) giving reliable grounds for physical analysis of space-frequency and space-time transformations of the electromagnetic field in open periodic resonators and waveguides; authentic analytic regularization procedures (Chapter 2) that, in contradistinction to the traditional frequency-domain approaches, fit perfectly for the analysis of resonant wave scattering processes; paramet...
Application of Chybeshev Polynomials in Factorizations of Balancing and Lucas-Balancing Numbers
Directory of Open Access Journals (Sweden)
Prasanta Kumar Ray
2012-01-01
Full Text Available In this paper, with the help of orthogonal polynomial especially Chybeshev polynomials of first and second kind, number theory and linear algebra intertwined to yield factorization of the balancing and Lucas-balancing numbers.
Bandwidth-Tunable Fiber Bragg Gratings Based on UV Glue Technique
Fu, Ming-Yue; Liu, Wen-Feng; Chen, Hsin-Tsang; Chuang, Chia-Wei; Bor, Sheau-Shong; Tien, Chuen-Lin
2007-07-01
In this study, we have demonstrated that a uniform fiber Bragg grating (FBG) can be transformed into a chirped fiber grating by a simple UV glue adhesive technique without shifting the reflection band with respect to the center wavelength of the FBG. The technique is based on the induced strain of an FBG due to the UV glue adhesive force on the fiber surface that causes a grating period variation and an effective index change. This technique can provide a fast and simple method of obtaining the required chirp value of a grating for applications in the dispersion compensators, gain flattening in erbium-doped fiber amplifiers (EDFAs) or optical filters.
STABILITY SYSTEMS VIA HURWITZ POLYNOMIALS
Directory of Open Access Journals (Sweden)
BALTAZAR AGUIRRE HERNÁNDEZ
2017-01-01
Full Text Available To analyze the stability of a linear system of differential equations ẋ = Ax we can study the location of the roots of the characteristic polynomial pA(t associated with the matrix A. We present various criteria - algebraic and geometric - that help us to determine where the roots are located without calculating them directly.
Space complexity in polynomial calculus
Czech Academy of Sciences Publication Activity Database
Filmus, Y.; Lauria, M.; Nordström, J.; Ron-Zewi, N.; Thapen, Neil
2015-01-01
Roč. 44, č. 4 (2015), s. 1119-1153 ISSN 0097-5397 R&D Projects: GA AV ČR IAA100190902; GA ČR GBP202/12/G061 Institutional support: RVO:67985840 Keywords : proof complexity * polynomial calculus * lower bounds Subject RIV: BA - General Mathematics Impact factor: 0.841, year: 2015 http://epubs.siam.org/doi/10.1137/120895950
Directory of Open Access Journals (Sweden)
Hamed Kharrati
2012-01-01
Full Text Available This study presents an improved model and controller for nonlinear plants using polynomial fuzzy model-based (FMB systems. To minimize mismatch between the polynomial fuzzy model and nonlinear plant, the suitable parameters of membership functions are determined in a systematic way. Defining an appropriate fitness function and utilizing Taylor series expansion, a genetic algorithm (GA is used to form the shape of membership functions in polynomial forms, which are afterwards used in fuzzy modeling. To validate the model, a controller based on proposed polynomial fuzzy systems is designed and then applied to both original nonlinear plant and fuzzy model for comparison. Additionally, stability analysis for the proposed polynomial FMB control system is investigated employing Lyapunov theory and a sum of squares (SOS approach. Moreover, the form of the membership functions is considered in stability analysis. The SOS-based stability conditions are attained using SOSTOOLS. Simulation results are also given to demonstrate the effectiveness of the proposed method.
Lu, Wenlong; Xie, Junwei; Wang, Heming; Sheng, Chuan
2016-01-01
Inspired by track-before-detection technology in radar, a novel time-frequency transform, namely polynomial chirping Fourier transform (PCFT), is exploited to extract components from noisy multicomponent signal. The PCFT combines advantages of Fourier transform and polynomial chirplet transform to accumulate component energy along a polynomial chirping curve in the time-frequency plane. The particle swarm optimization algorithm is employed to search optimal polynomial parameters with which the PCFT will achieve a most concentrated energy ridge in the time-frequency plane for the target component. The component can be well separated in the polynomial chirping Fourier domain with a narrow-band filter and then reconstructed by inverse PCFT. Furthermore, an iterative procedure, involving parameter estimation, PCFT, filtering and recovery, is introduced to extract components from a noisy multicomponent signal successively. The Simulations and experiments show that the proposed method has better performance in component extraction from noisy multicomponent signal as well as provides more time-frequency details about the analyzed signal than conventional methods.
Lowry, Troy Warren
The dynamic self-organization of lipids in biological systems is a highly regulated process that enables the compartmentalization of living systems at microscopic and nanoscopic levels. Exploiting the self-organization and innate biofunctionality of lyotropic liquid crystalline phospholipids, a novel nanofabrication process called "nanointaglio" was invented in order to rapidly and scalably integrate lipid nanopatterns onto the surface. The work presented here focuses on using nanointaglio fabricated lipid diffraction micro- and nanopatterns for the development of new sensing and bioactivity studies. The lipids are patterned as diffraction gratings for sensor functionality. The lipid multilayer gratings operate as nanomechanical sensor elements that are capable of transducing molecular binding to fluid lipid multilayers into optical signals in a label free manner due to shape changes in the lipid nanostructures. To demonstrate the label free detection capabilities, lipid nanopatterns are shown to be suitable for the integration of chemically different lipid multilayer gratings into a sensor array capable of distinguishing vapors by means of an optical nose. Sensor arrays composed of six different lipid formulations are integrated onto a surface and their optical response to three different vapors (water, ethanol and acetone) in air as well as pH under water is monitored as a function of time. Principal component analysis of the array response results in distinct clustering, indicating the suitability of the arrays for distinguishing these analytes. Importantly, the nanointaglio process used is capable of producing lipid gratings out of different materials with sufficiently uniform heights for the fabrication of an optical nose. A second main application is demonstrated for the study of membrane binding proteins. Although in vitro methods for assaying the catalytic activity of individual enzymes are well established, quantitative methods for assaying the kinetics of
Effective length of short Fabry-Perot cavity formed by uniform fiber Bragg gratings.
Barmenkov, Yuri O; Zalvidea, Dobryna; Torres-Peiró, Salvador; Cruz, Jose L; Andrés, Miguel V
2006-07-10
In this paper, we describe the properties of Fabry-Perot fiber cavity formed by two fiber Bragg gratings in terms of the grating effective length. We show that the grating effective length is determined by the group delay of the grating, which depends on its diffraction efficiency and physical length. We present a simple analytical formula for calculation of the effective length of the uniform fiber Bragg grating and the frequency separation between consecutive resonances of a Fabry-Perot cavity. Experimental results on the cavity transmission spectra for different values of the gratings' reflectivity support the presented theory.
Lusine Poghosyan
2014-01-01
The paper considers convergence acceleration of the quasi-periodic and the quasi-periodic-rational interpolations by application of polynomial corrections. We investigate convergence of the resultant quasi-periodic-polynomial and quasi-periodic-rational-polynomial interpolations and derive exact constants of the main terms of asymptotic errors in the regions away from the endpoints. Results of numerical experiments clarify behavior of the corresponding interpolations for moderate number of in...
Zernike polynomial based Rayleigh-Ritz model of a piezoelectric unimorph deformable mirror
CSIR Research Space (South Africa)
Long, CS
2012-04-01
Full Text Available , are routinely and conveniently described using Zernike polynomials. A Rayleigh-Ritz structural model, which uses Zernike polynomials directly to describe the displacements, is proposed in this paper. The proposed formulation produces a numerically inexpensive...
Cascading metallic gratings for broadband absorption enhancement in ultrathin plasmonic solar cells
International Nuclear Information System (INIS)
Wen, Long; Sun, Fuhe; Chen, Qin
2014-01-01
The incorporation of plasmonic nanostructures in the thin-film solar cells (TFSCs) is a promising route to harvest light into the nanoscale active layer. However, the light trapping scheme based on the plasmonic effects intrinsically presents narrow-band resonant enhancement of light absorption. Here we demonstrate that by cascading metal nanogratings with different sizes atop the TFSCs, broadband absorption enhancement can be realized by simultaneously exciting multiple localized surface plasmon resonances and inducing strong coupling between the plasmonic modes and photonic modes. As a proof of concept, we demonstrate of 66.5% in the photocurrent in an ultrathin amorphous silicon TFSC with two-dimensional cascaded gratings over the reference cell without gratings
Polynomial structures in one-loop amplitudes
International Nuclear Information System (INIS)
Britto, Ruth; Feng Bo; Yang Gang
2008-01-01
A general one-loop scattering amplitude may be expanded in terms of master integrals. The coefficients of the master integrals can be obtained from tree-level input in a two-step process. First, use known formulas to write the coefficients of (4-2ε)-dimensional master integrals; these formulas depend on an additional variable, u, which encodes the dimensional shift. Second, convert the u-dependent coefficients of (4-2ε)-dimensional master integrals to explicit coefficients of dimensionally shifted master integrals. This procedure requires the initial formulas for coefficients to have polynomial dependence on u. Here, we give a proof of this property in the case of massless propagators. The proof is constructive. Thus, as a byproduct, we produce different algebraic expressions for the scalar integral coefficients, in which the polynomial property is apparent. In these formulas, the box and pentagon contributions are separated explicitly.
Optimization of polynomials in non-commuting variables
Burgdorf, Sabine; Povh, Janez
2016-01-01
This book presents recent results on positivity and optimization of polynomials in non-commuting variables. Researchers in non-commutative algebraic geometry, control theory, system engineering, optimization, quantum physics and information science will find the unified notation and mixture of algebraic geometry and mathematical programming useful. Theoretical results are matched with algorithmic considerations; several examples and information on how to use NCSOStools open source package to obtain the results provided. Results are presented on detecting the eigenvalue and trace positivity of polynomials in non-commuting variables using Newton chip method and Newton cyclic chip method, relaxations for constrained and unconstrained optimization problems, semidefinite programming formulations of the relaxations and finite convergence of the hierarchies of these relaxations, and the practical efficiency of algorithms.
A general theory of interference fringes in x-ray phase grating imaging
International Nuclear Information System (INIS)
Yan, Aimin; Wu, Xizeng; Liu, Hong
2015-01-01
Purpose: The authors note that the concept of the Talbot self-image distance in x-ray phase grating interferometry is indeed not well defined for polychromatic x-rays, because both the grating phase shift and the fractional Talbot distances are all x-ray wavelength-dependent. For x-ray interferometry optimization, there is a need for a quantitative theory that is able to predict if a good intensity modulation is attainable at a given grating-to-detector distance. In this work, the authors set out to meet this need. Methods: In order to apply Fourier analysis directly to the intensity fringe patterns of two-dimensional and one-dimensional phase grating interferometers, the authors start their derivation from a general phase space theory of x-ray phase-contrast imaging. Unlike previous Fourier analyses, the authors evolved the Wigner distribution to obtain closed-form expressions of the Fourier coefficients of the intensity fringes for any grating-to-detector distance, even if it is not a fractional Talbot distance. Results: The developed theory determines the visibility of any diffraction order as a function of the grating-to-detector distance, the phase shift of the grating, and the x-ray spectrum. The authors demonstrate that the visibilities of diffraction orders can serve as the indicators of the underlying interference intensity modulation. Applying the theory to the conventional and inverse geometry configurations of single-grating interferometers, the authors demonstrated that the proposed theory provides a quantitative tool for the grating interferometer optimization with or without the Talbot-distance constraints. Conclusions: In this work, the authors developed a novel theory of the interference intensity fringes in phase grating x-ray interferometry. This theory provides a quantitative tool in design optimization of phase grating x-ray interferometers
A general theory of interference fringes in x-ray phase grating imaging.
Yan, Aimin; Wu, Xizeng; Liu, Hong
2015-06-01
The authors note that the concept of the Talbot self-image distance in x-ray phase grating interferometry is indeed not well defined for polychromatic x-rays, because both the grating phase shift and the fractional Talbot distances are all x-ray wavelength-dependent. For x-ray interferometry optimization, there is a need for a quantitative theory that is able to predict if a good intensity modulation is attainable at a given grating-to-detector distance. In this work, the authors set out to meet this need. In order to apply Fourier analysis directly to the intensity fringe patterns of two-dimensional and one-dimensional phase grating interferometers, the authors start their derivation from a general phase space theory of x-ray phase-contrast imaging. Unlike previous Fourier analyses, the authors evolved the Wigner distribution to obtain closed-form expressions of the Fourier coefficients of the intensity fringes for any grating-to-detector distance, even if it is not a fractional Talbot distance. The developed theory determines the visibility of any diffraction order as a function of the grating-to-detector distance, the phase shift of the grating, and the x-ray spectrum. The authors demonstrate that the visibilities of diffraction orders can serve as the indicators of the underlying interference intensity modulation. Applying the theory to the conventional and inverse geometry configurations of single-grating interferometers, the authors demonstrated that the proposed theory provides a quantitative tool for the grating interferometer optimization with or without the Talbot-distance constraints. In this work, the authors developed a novel theory of the interference intensity fringes in phase grating x-ray interferometry. This theory provides a quantitative tool in design optimization of phase grating x-ray interferometers.
Measurement system for diffraction efficiency of convex gratings
Liu, Peng; Chen, Xin-hua; Zhou, Jian-kang; Zhao, Zhi-cheng; Liu, Quan; Luo, Chao; Wang, Xiao-feng; Tang, Min-xue; Shen, Wei-min
2017-08-01
A measurement system for diffraction efficiency of convex gratings is designed. The measurement system mainly includes four components as a light source, a front system, a dispersing system that contains a convex grating, and a detector. Based on the definition and measuring principle of diffraction efficiency, the optical scheme of the measurement system is analyzed and the design result is given. Then, in order to validate the feasibility of the designed system, the measurement system is set up and the diffraction efficiency of a convex grating with the aperture of 35 mm, the curvature-radius of 72mm, the blazed angle of 6.4°, the grating period of 2.5μm and the working waveband of 400nm-900nm is tested. Based on GUM (Guide to the Expression of Uncertainty in Measurement), the uncertainties in the measuring results are evaluated. The measured diffraction efficiency data are compared to the theoretical ones, which are calculated based on the grating groove parameters got by an atomic force microscope and Rigorous Couple Wave Analysis, and the reliability of the measurement system is illustrated. Finally, the measurement performance of the system is analyzed and tested. The results show that, the testing accuracy, the testing stability and the testing repeatability are 2.5%, 0.085% and 3.5% , respectively.
High performance Si immersion gratings patterned with electron beam lithography
Gully-Santiago, Michael A.; Jaffe, Daniel T.; Brooks, Cynthia B.; Wilson, Daniel W.; Muller, Richard E.
2014-07-01
Infrared spectrographs employing silicon immersion gratings can be significantly more compact than spectro- graphs using front-surface gratings. The Si gratings can also offer continuous wavelength coverage at high spectral resolution. The grooves in Si gratings are made with semiconductor lithography techniques, to date almost entirely using contact mask photolithography. Planned near-infrared astronomical spectrographs require either finer groove pitches or higher positional accuracy than standard UV contact mask photolithography can reach. A collaboration between the University of Texas at Austin Silicon Diffractive Optics Group and the Jet Propulsion Laboratory Microdevices Laboratory has experimented with direct writing silicon immersion grating grooves with electron beam lithography. The patterning process involves depositing positive e-beam resist on 1 to 30 mm thick, 100 mm diameter monolithic crystalline silicon substrates. We then use the facility JEOL 9300FS e-beam writer at JPL to produce the linear pattern that defines the gratings. There are three key challenges to produce high-performance e-beam written silicon immersion gratings. (1) E- beam field and subfield stitching boundaries cause periodic cross-hatch structures along the grating grooves. The structures manifest themselves as spectral and spatial dimension ghosts in the diffraction limited point spread function (PSF) of the diffraction grating. In this paper, we show that the effects of e-beam field boundaries must be mitigated. We have significantly reduced ghost power with only minor increases in write time by using four or more field sizes of less than 500 μm. (2) The finite e-beam stage drift and run-out error cause large-scale structure in the wavefront error. We deal with this problem by applying a mark detection loop to check for and correct out minuscule stage drifts. We measure the level and direction of stage drift and show that mark detection reduces peak-to-valley wavefront error
Simulation Studies of the Dielectric Grating as an Accelerating and Focusing Structure
International Nuclear Information System (INIS)
Soong, Ken; Peralta, E.A.; Byer, R.L.; Colby, E.
2011-01-01
A grating-based design is a promising candidate for a laser-driven dielectric accelerator. Through simulations, we show the merits of a readily fabricated grating structure as an accelerating component. Additionally, we show that with a small design perturbation, the accelerating component can be converted into a focusing structure. The understanding of these two components is critical in the successful development of any complete accelerator. The concept of accelerating electrons with the tremendous electric fields found in lasers has been proposed for decades. However, until recently the realization of such an accelerator was not technologically feasible. Recent advances in the semiconductor industry, as well as advances in laser technology, have now made laser-driven dielectric accelerators imminent. The grating-based accelerator is one proposed design for a dielectric laser-driven accelerator. This design, which was introduced by Plettner, consists of a pair of opposing transparent binary gratings, illustrated in Fig. 1. The teeth of the gratings serve as a phase mask, ensuring a phase synchronicity between the electromagnetic field and the moving particles. The current grating accelerator design has the drive laser incident perpendicular to the substrate, which poses a laser-structure alignment complication. The next iteration of grating structure fabrication seeks to monolithically create an array of grating structures by etching the grating's vacuum channel into a fused silica wafer. With this method it is possible to have the drive laser confined to the plane of the wafer, thus ensuring alignment of the laser-and-structure, the two grating halves, and subsequent accelerator components. There has been previous work using 2-dimensional finite difference time domain (2D-FDTD) calculations to evaluate the performance of the grating accelerator structure. However, this work approximates the grating as an infinite structure and does not accurately model a
Computation of the Likelihood in Biallelic Diffusion Models Using Orthogonal Polynomials
Directory of Open Access Journals (Sweden)
Claus Vogl
2014-11-01
Full Text Available In population genetics, parameters describing forces such as mutation, migration and drift are generally inferred from molecular data. Lately, approximate methods based on simulations and summary statistics have been widely applied for such inference, even though these methods waste information. In contrast, probabilistic methods of inference can be shown to be optimal, if their assumptions are met. In genomic regions where recombination rates are high relative to mutation rates, polymorphic nucleotide sites can be assumed to evolve independently from each other. The distribution of allele frequencies at a large number of such sites has been called “allele-frequency spectrum” or “site-frequency spectrum” (SFS. Conditional on the allelic proportions, the likelihoods of such data can be modeled as binomial. A simple model representing the evolution of allelic proportions is the biallelic mutation-drift or mutation-directional selection-drift diffusion model. With series of orthogonal polynomials, specifically Jacobi and Gegenbauer polynomials, or the related spheroidal wave function, the diffusion equations can be solved efficiently. In the neutral case, the product of the binomial likelihoods with the sum of such polynomials leads to finite series of polynomials, i.e., relatively simple equations, from which the exact likelihoods can be calculated. In this article, the use of orthogonal polynomials for inferring population genetic parameters is investigated.
Generating the patterns of variation with GeoGebra: the case of polynomial approximations
Attorps, Iiris; Björk, Kjell; Radic, Mirko
2016-01-01
In this paper, we report a teaching experiment regarding the theory of polynomial approximations at the university mathematics teaching in Sweden. The experiment was designed by applying Variation theory and by using the free dynamic mathematics software GeoGebra. The aim of this study was to investigate if the technology-assisted teaching of Taylor polynomials compared with traditional way of work at the university level can support the teaching and learning of mathematical concepts and ideas. An engineering student group (n = 19) was taught Taylor polynomials with the assistance of GeoGebra while a control group (n = 18) was taught in a traditional way. The data were gathered by video recording of the lectures, by doing a post-test concerning Taylor polynomials in both groups and by giving one question regarding Taylor polynomials at the final exam for the course in Real Analysis in one variable. In the analysis of the lectures, we found Variation theory combined with GeoGebra to be a potentially powerful tool for revealing some critical aspects of Taylor Polynomials. Furthermore, the research results indicated that applying Variation theory, when planning the technology-assisted teaching, supported and enriched students' learning opportunities in the study group compared with the control group.
Non-Abelian integrable hierarchies: matrix biorthogonal polynomials and perturbations
Ariznabarreta, Gerardo; García-Ardila, Juan C.; Mañas, Manuel; Marcellán, Francisco
2018-05-01
In this paper, Geronimus–Uvarov perturbations for matrix orthogonal polynomials on the real line are studied and then applied to the analysis of non-Abelian integrable hierarchies. The orthogonality is understood in full generality, i.e. in terms of a nondegenerate continuous sesquilinear form, determined by a quasidefinite matrix of bivariate generalized functions with a well-defined support. We derive Christoffel-type formulas that give the perturbed matrix biorthogonal polynomials and their norms in terms of the original ones. The keystone for this finding is the Gauss–Borel factorization of the Gram matrix. Geronimus–Uvarov transformations are considered in the context of the 2D non-Abelian Toda lattice and noncommutative KP hierarchies. The interplay between transformations and integrable flows is discussed. Miwa shifts, τ-ratio matrix functions and Sato formulas are given. Bilinear identities, involving Geronimus–Uvarov transformations, first for the Baker functions, then secondly for the biorthogonal polynomials and its second kind functions, and finally for the τ-ratio matrix functions, are found.
A Non-Polynomial Gravity Formulation for Loop Quantum Cosmology Bounce
Directory of Open Access Journals (Sweden)
Stefano Chinaglia
2017-09-01
Full Text Available Recently the so-called mimetic gravity approach has been used to obtain corrections to the Friedmann equation of General Relativity similar to the ones present in loop quantum cosmology. In this paper, we propose an alternative way to derive this modified Friedmann equation via the so-called non-polynomial gravity approach, which consists of adding geometric non-polynomial higher derivative terms to Hilbert–Einstein action, which are nonetheless polynomials and lead to a second-order differential equation in Friedmann–Lemaître–Robertson–Walker space-times. Our explicit action turns out to be a realization of the Helling proposal of effective action with an infinite number of terms. The model is also investigated in the presence of a non-vanishing cosmological constant, and a new exact bounce solution is found and studied.
Magneto-Optic Fiber Gratings Useful for Dynamic Dispersion Management and Tunable Comb Filtering
International Nuclear Information System (INIS)
Bao-Jian, Wu; Xin, Lu; Kun, Qiu
2010-01-01
Intelligent control of dispersion management and tunable comb filtering in optical network applications can be performed by using magneto-optic fiber Bragg gratings (MFBGs). When a nonuniform magnetic field is applied to the MFBG with a constant grating period, the resulting grating response is equivalent to that of a conventional chirped grating. Under a linearly nonuniform magnetic field along the grating, a linear dispersion is achieved in the grating bandgap and the maximal dispersion slope can come to 1260 ps/nm 2 for a 10-mm-long fiber grating at 1550 nm window. Similarly, a Gaussian-apodizing sampled MFBG is also useful for magnetically tunable comb filtering, with potential application to clock recovery from return-to-zero optical signals and optical carrier tracking. (condensed matter: electronic structure, electrical, magnetic, and optical properties)
Nonclassical Orthogonal Polynomials and Corresponding Quadratures
Fukuda, H; Alt, E O; Matveenko, A V
2004-01-01
We construct nonclassical orthogonal polynomials and calculate abscissas and weights of Gaussian quadrature for arbitrary weight and interval. The program is written by Mathematica and it works if moment integrals are given analytically. The result is a FORTRAN subroutine ready to utilize the quadrature.
Hamed Kharrati; Sohrab Khanmohammadi; Witold Pedrycz; Ghasem Alizadeh
2012-01-01
This study presents an improved model and controller for nonlinear plants using polynomial fuzzy model-based (FMB) systems. To minimize mismatch between the polynomial fuzzy model and nonlinear plant, the suitable parameters of membership functions are determined in a systematic way. Defining an appropriate fitness function and utilizing Taylor series expansion, a genetic algorithm (GA) is used to form the shape of membership functions in polynomial forms, which are afterwards used in fuzzy m...
Multiwavelength optical scatterometry of dielectric gratings
Yashina, Nataliya P.
2012-08-01
Modern scatterometry problems arising in the lithography production of periodic gratings are in the focus of the work. The performance capabilities of a novel theoretical and numerical modeling oriented to these problems are considered. The approach is based on rigorous solutions of 2-D initial boundary value problems of the gratings theory. The quintessence and advantage of the method is the possibility to perform an efficient analysis simultaneously and interactively both for steady state and transient processes of the resonant scattering of electromagnetic waves by the infinite and compact periodic structures. © 2012 IEEE.
DEFF Research Database (Denmark)
Ravn, Jesper N.
1992-01-01
A simple approach for the calculation of self-diffraction in a thin combined phase and amplitude grating is presented. The third order nonlinearity, the electron-hole recombination time, and the ambipolar diffusion coefficient in a ZnO crystal are measured by means of laser-induced self-diffracti......A simple approach for the calculation of self-diffraction in a thin combined phase and amplitude grating is presented. The third order nonlinearity, the electron-hole recombination time, and the ambipolar diffusion coefficient in a ZnO crystal are measured by means of laser-induced self...
Reduction of Bragg-grating-induced coupling to cladding modes
DEFF Research Database (Denmark)
Berendt, Martin Ole; Bjarklev, Anders Overgaard; Soccolich, C.E.
1999-01-01
gratings in a depressed-cladding fiber are compared with simulations. The model gives good agreement with the measured transmission spectrum and accounts for the pronounced coupling to asymmetrical cladding modes, even when the grating is written with the smallest possible blaze. The asymmetry causing...... this is accounted for by the unavoidable attenuation of the UV light. It is found for the considered fiber designs that a high numerical-aperture fiber increases the spectral separation between the Bragg resonance and the onset of cladding-mode losses. A depressed-cladding fiber reduces the coupling strength......We discuss fiber designs that have been suggested for the reduction of Bragg-grating induced coupling to cladding modes. The discussion is based on a theoretical approach that includes the effect of asymmetry in the UV-induced index grating, made by UV-side writing. Experimental results from...
A general U-block model-based design procedure for nonlinear polynomial control systems
Zhu, Q. M.; Zhao, D. Y.; Zhang, Jianhua
2016-10-01
The proposition of U-model concept (in terms of 'providing concise and applicable solutions for complex problems') and a corresponding basic U-control design algorithm was originated in the first author's PhD thesis. The term of U-model appeared (not rigorously defined) for the first time in the first author's other journal paper, which established a framework for using linear polynomial control system design approaches to design nonlinear polynomial control systems (in brief, linear polynomial approaches → nonlinear polynomial plants). This paper represents the next milestone work - using linear state-space approaches to design nonlinear polynomial control systems (in brief, linear state-space approaches → nonlinear polynomial plants). The overall aim of the study is to establish a framework, defined as the U-block model, which provides a generic prototype for using linear state-space-based approaches to design the control systems with smooth nonlinear plants/processes described by polynomial models. For analysing the feasibility and effectiveness, sliding mode control design approach is selected as an exemplary case study. Numerical simulation studies provide a user-friendly step-by-step procedure for the readers/users with interest in their ad hoc applications. In formality, this is the first paper to present the U-model-oriented control system design in a formal way and to study the associated properties and theorems. The previous publications, in the main, have been algorithm-based studies and simulation demonstrations. In some sense, this paper can be treated as a landmark for the U-model-based research from intuitive/heuristic stage to rigour/formal/comprehensive studies.
Schur Stability Regions for Complex Quadratic Polynomials
Cheng, Sui Sun; Huang, Shao Yuan
2010-01-01
Given a quadratic polynomial with complex coefficients, necessary and sufficient conditions are found in terms of the coefficients such that all its roots have absolute values less than 1. (Contains 3 figures.)