Apostol, Tom M. (Editor)
1991-01-01
In this 'Project Mathematics! series, sponsored by California Institute for Technology (CalTech), the mathematical concept of polynomials in rectangular coordinate (x, y) systems are explored. sing film footage of real life applications and computer animation sequences, the history of, the application of, and the different linear coordinate systems for quadratic, cubic, intersecting, and higher degree of polynomials are discussed.
Tunable grating with active feedback
Rosset, Samuel; O'Brien, Benjamin M.; Gisby, Todd; Xu, Daniel; Shea, Herbert R.; Anderson, Iain A.
2013-04-01
We report on the use of capacitive self-sensing to operate a DEA-based tunable grating in closed-loop mode. Due to their large strain capabilities, DEAs are key candidates for tunable optics applications. However, the viscoelasticity of elastomers is detrimental for applications that require long-term stability, such as tunable gratings and lenses. We show that capacitive sensing of the electrode strain can be used to suppress the strain drift and increase the response speed of silicone-based actuators. On the other hand, VHB actuators exhibit a time-dependent permittivity, which causes a drift between the device capacitance and its strain.
Gratings in passive and active optical waveguides
DEFF Research Database (Denmark)
Berendt, Martin Ole
1999-01-01
mode losses confirmed. An elaborated grating model, including the detailed shape of the index modulation, has been developed. This model improves the interpretation of grating growth dynamic, which is of value to both; analysis of the UV imprinting set-ups, and to the investigation of photosensitivity...... 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...
Active resonant subwavelength grating for scannerless range imaging sensors.
Energy Technology Data Exchange (ETDEWEB)
Kemme, Shanalyn A.; Nellums, Robert O.; Boye, Robert R.; Peters, David William
2006-11-01
In this late-start LDRD, we will present a design for a wavelength-agile, high-speed modulator that enables a long-term vision for the THz Scannerless Range Imaging (SRI) sensor. It takes the place of the currently-utilized SRI micro-channel plate which is limited to photocathode sensitive wavelengths (primarily in the visible and near-IR regimes). Two of Sandia's successful technologies--subwavelength diffractive optics and THz sources and detectors--are poised to extend the capabilities of the SRI sensor. The goal is to drastically broaden the SRI's sensing waveband--all the way to the THz regime--so the sensor can see through image-obscuring, scattering environments like smoke and dust. Surface properties, such as reflectivity, emissivity, and scattering roughness, vary greatly with the illuminating wavelength. Thus, objects that are difficult to image at the SRI sensor's present near-IR wavelengths may be imaged more easily at the considerably longer THz wavelengths (0.1 to 1mm). The proposed component is an active Resonant Subwavelength Grating (RSG). Sandia invested considerable effort on a passive RSG two years ago, which resulted in a highly-efficient (reflectivity greater than gold), wavelength-specific reflector. For this late-start LDRD proposal, we will transform the passive RSG design into an active laser-line reflector.
Gain-phase grating based on spatial modulation of active Raman gain in cold atoms
International Nuclear Information System (INIS)
Kuang Shangqi; Jin Chunshui; Li Chun
2011-01-01
In order to obtain an atomic grating which can diffract light into the high-order directions more efficiently, a gain-phase grating (GPG) based on the spatial modulation of active Raman gain is theoretically presented. This grating is induced by a pump field and a standing wave in ultracold atoms, and it not only diffracts a weak probe field propagating along a direction normal to the standing wave into the high-order directions, but also amplifies the amplitude of the zero-order diffraction. In contrast with electromagnetically induced grating or electromagnetically induced phase grating, the GPG has larger diffraction efficiencies in the high-order directions. Hence it is more suitable to be utilized as an all-optical router in optical networking and communication.
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)
Alessi, David A; Rosso, Paul A; Nguyen, Hoang T; Aasen, Michael D; Britten, Jerald A; Haefner, Constantin
2016-12-26
Laser energy absorption and subsequent heat removal from diffraction gratings in chirped pulse compressors poses a significant challenge in high repetition rate, high peak power laser development. In order to understand the average power limitations, we have modeled the time-resolved thermo-mechanical properties of current and advanced diffraction gratings. We have also developed and demonstrated a technique of actively cooling Petawatt scale, gold compressor gratings to operate at 600W of average power - a 15x increase over the highest average power petawatt laser currently in operation. Combining this technique with low absorption multilayer dielectric gratings developed in our group would enable pulse compressors for petawatt peak power lasers operating at average powers well above 40kW.
Mason, JC
2002-01-01
Chebyshev polynomials crop up in virtually every area of numerical analysis, and they hold particular importance in recent advances in subjects such as orthogonal polynomials, polynomial approximation, numerical integration, and spectral methods. Yet no book dedicated to Chebyshev polynomials has been published since 1990, and even that work focused primarily on the theoretical aspects. A broad, up-to-date treatment is long overdue.Providing highly readable exposition on the subject''s state of the art, Chebyshev Polynomials is just such a treatment. It includes rigorous yet down-to-earth coverage of the theory along with an in-depth look at the properties of all four kinds of Chebyshev polynomials-properties that lead to a range of results in areas such as approximation, series expansions, interpolation, quadrature, and integral equations. Problems in each chapter, ranging in difficulty from elementary to quite advanced, reinforce the concepts and methods presented.Far from being an esoteric subject, Chebysh...
Dobbs, David E.
2010-01-01
This note develops and implements the theory of polynomial asymptotes to (graphs of) rational functions, as a generalization of the classical topics of horizontal asymptotes and oblique/slant asymptotes. Applications are given to hyperbolic asymptotes. Prerequisites include the division algorithm for polynomials with coefficients in the field of…
Identification of time-varying neural dynamics from spiking activities using Chebyshev polynomials.
Song Xu; Yang Li; Xudong Wang; Chan, Rosa H M
2016-08-01
Neural plasticity, elicited by processes such as development and learning, is an important biological attribute which can be viewed as a time-varying property of the nervous system. In this paper, we investigated the novel use of Chebyshev polynomials to estimate the changes in model parameters efficiently for time-varying dynamical systems with binary inputs and outputs. A forward orthogonal least square (FOLS) algorithm selected the significant model terms. Extensive simulations showed that the proposed algorithm identified the system changes more accurately in comparison with adaptive filter. This approach can be applied to identify not only gradual but also abrupt temporal evolutions of neural dynamics underlying nervous system activity with high sensitivity and accuracy by observing input and output spike trains only.
Svebak, Sven
2016-08-01
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.
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.
The Trigonometric Polynomial Like Bernstein Polynomial
2014-01-01
A symmetric basis of trigonometric polynomial space is presented. Based on the basis, symmetric trigonometric polynomial approximants like Bernstein polynomials are constructed. Two kinds of nodes are given to show that the trigonometric polynomial sequence is uniformly convergent. The convergence of the derivative of the trigonometric polynomials is shown. Trigonometric quasi-interpolants of reproducing one degree of trigonometric polynomials are constructed. Some interesting properties of the trigonometric polynomials are given. PMID:25276845
The Trigonometric Polynomial Like Bernstein Polynomial
Directory of Open Access Journals (Sweden)
Xuli Han
2014-01-01
Full Text Available A symmetric basis of trigonometric polynomial space is presented. Based on the basis, symmetric trigonometric polynomial approximants like Bernstein polynomials are constructed. Two kinds of nodes are given to show that the trigonometric polynomial sequence is uniformly convergent. The convergence of the derivative of the trigonometric polynomials is shown. Trigonometric quasi-interpolants of reproducing one degree of trigonometric polynomials are constructed. Some interesting properties of the trigonometric polynomials are given.
Directory of Open Access Journals (Sweden)
Sven Svebak
2016-08-01
Full Text Available 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.
Szegő, G
1939-01-01
The general theory of orthogonal polynomials was developed in the late 19th century from a study of continued fractions by P. L. Chebyshev, even though special cases were introduced earlier by Legendre, Hermite, Jacobi, Laguerre, and Chebyshev himself. It was further developed by A. A. Markov, T. J. Stieltjes, and many other mathematicians. The book by Szegő, originally published in 1939, is the first monograph devoted to the theory of orthogonal polynomials and its applications in many areas, including analysis, differential equations, probability and mathematical physics. Even after all the
El-Guindy, Ahmad; Ismail, Mourad E. H.
2013-01-01
We identify the Atkin polynomials in terms of associated Jacobi polynomials. Our identificationthen takes advantage of the theory of orthogonal polynomials and their asymptotics to establish many new properties of the Atkin polynomials. This shows that co-recursive polynomials may lead to interesting sets of orthogonal polynomials.
Irreducible multivariate polynomials obtained from polynomials in ...
Indian Academy of Sciences (India)
over K(X)? We provided some methods to construct irreducible multivariate polynomials over an arbi- trary field, starting from arbitrary irreducible polynomials in fewer variables, of which we mention the following two results: Theorem A. If we write an irreducible polynomial f ∈ K[X] as a sum of polynomials a0,..., an ∈ K[X] ...
Irreducible multivariate polynomials obtained from polynomials in ...
Indian Academy of Sciences (India)
We provided some methods to construct irreducible multivariate polynomials over an arbi- trary field, starting from arbitrary irreducible polynomials in fewer variables, of which we mention the following two results: Theorem A. If we write an irreducible polynomial f ∈ K[X] as a sum of polynomials a0,..., an ∈ K[X] with deg a0 ...
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.
Improved polynomial remainder sequences for Ore polynomials.
Jaroschek, Maximilian
2013-11-01
Polynomial remainder sequences contain the intermediate results of the Euclidean algorithm when applied to (non-)commutative polynomials. The running time of the algorithm is dependent on the size of the coefficients of the remainders. Different ways have been studied to make these as small as possible. The subresultant sequence of two polynomials is a polynomial remainder sequence in which the size of the coefficients is optimal in the generic case, but when taking the input from applications, the coefficients are often larger than necessary. We generalize two improvements of the subresultant sequence to Ore polynomials and derive a new bound for the minimal coefficient size. Our approach also yields a new proof for the results in the commutative case, providing a new point of view on the origin of the extraneous factors of the coefficients.
Chaos, Fractals, and Polynomials.
Tylee, J. Louis; Tylee, Thomas B.
1996-01-01
Discusses chaos theory; linear algebraic equations and the numerical solution of polynomials, including the use of the Newton-Raphson technique to find polynomial roots; fractals; search region and coordinate systems; convergence; and generating color fractals on a computer. (LRW)
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....
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....
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 ...
Poly-Frobenius-Euler polynomials
Kurt, Burak
2017-07-01
Hamahata [3] defined poly-Euler polynomials and the generalized poly-Euler polynomials. He proved some relations and closed formulas for the poly-Euler polynomials. By this motivation, we define poly-Frobenius-Euler polynomials. We give some relations for this polynomials. Also, we prove the relationships between poly-Frobenius-Euler polynomials and Stirling numbers of the second kind.
Coherent orthogonal polynomials
International Nuclear Information System (INIS)
Celeghini, E.; Olmo, M.A. del
2013-01-01
We discuss a fundamental characteristic of orthogonal polynomials, like the existence of a Lie algebra behind them, which can be added to their other relevant aspects. At the basis of the complete framework for orthogonal polynomials we include thus–in addition to differential equations, recurrence relations, Hilbert spaces and square integrable functions–Lie algebra theory. We start here from the square integrable functions on the open connected subset of the real line whose bases are related to orthogonal polynomials. All these one-dimensional continuous spaces allow, besides the standard uncountable basis (|x〉), for an alternative countable basis (|n〉). The matrix elements that relate these two bases are essentially the orthogonal polynomials: Hermite polynomials for the line and Laguerre and Legendre polynomials for the half-line and the line interval, respectively. Differential recurrence relations of orthogonal polynomials allow us to realize that they determine an infinite-dimensional irreducible representation of a non-compact Lie algebra, whose second order Casimir C gives rise to the second order differential equation that defines the corresponding family of orthogonal polynomials. Thus, the Weyl–Heisenberg algebra h(1) with C=0 for Hermite polynomials and su(1,1) with C=−1/4 for Laguerre and Legendre polynomials are obtained. Starting from the orthogonal polynomials the Lie algebra is extended both to the whole space of the L 2 functions and to the corresponding Universal Enveloping Algebra and transformation group. Generalized coherent states from each vector in the space L 2 and, in particular, generalized coherent polynomials are thus obtained. -- Highlights: •Fundamental characteristic of orthogonal polynomials (OP): existence of a Lie algebra. •Differential recurrence relations of OP determine a unitary representation of a non-compact Lie group. •2nd order Casimir originates a 2nd order differential equation that defines the
Incomplete q-Chebyshev Polynomials
Ercan, Elif; Firengiz, Mirac Cetin; Tuglu, Naim
2016-01-01
In this paper, we get the generating functions of q-Chebyshev polynomials using operator. Also considering explicit formulas of q-Chebyshev polynomials, we give new generalizations of q-Chebyshev polynomials called incomplete q-Chebyshev polynomials of the first and second kind. We obtain recurrence relations and several properties of these polynomials. We show that there are connections between incomplete q-Chebyshev polynomials and the some well-known polynomials. Keywords: q-Chebyshev poly...
Polynomial Graphs and Symmetry
Goehle, Geoff; Kobayashi, Mitsuo
2013-01-01
Most quadratic functions are not even, but every parabola has symmetry with respect to some vertical line. Similarly, every cubic has rotational symmetry with respect to some point, though most cubics are not odd. We show that every polynomial has at most one point of symmetry and give conditions under which the polynomial has rotational or…
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...
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.
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.
Laser-induced gratings in the gas phase excited via Raman-active transitions
Energy Technology Data Exchange (ETDEWEB)
Kozlov, D.N. [General Physics Inst., Russian Academy of Sciences, Moscow (Russian Federation); Bombach, R.; Hemmerling, B.; Hubschmid, W. [Paul Scherrer Inst. (PSI), Villigen (Switzerland)
1997-06-01
We report on a new time resolved coherent Raman technique that is based on the generation of thermal gratings following a population change among molecular levels induced by stimulated Raman pumping. This is achieved by spatially and temporally overlapping intensity interference patterns generated independently by two lasers. When this technique is used in carbon dioxide, employing transitions which belong to the Q-branches of the {nu}{sub 1}/2{nu}{sub 2} Fermi dyad, it is possible to investigate molecular energy transfer processes. (author) 2 figs., 10 refs.
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 ...
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.
El-Khoury, O.; Kim, C.; Shafieezadeh, A.; Hur, J. E.; Heo, G. H.
2015-06-01
This study performs a series of numerical simulations and shake-table experiments to design and assess the performance of a nonlinear clipped feedback control algorithm based on optimal polynomial control (OPC) to mitigate the response of a two-span bridge equipped with a magnetorheological (MR) damper. As an extended conventional linear quadratic regulator, OPC provides more flexibility in the control design and further enhances system performance. The challenges encountered in this case are (1) the linearization of the nonlinear behavior of various components and (2) the selection of the weighting matrices in the objective function of OPC. The first challenge is addressed by using stochastic linearization which replaces the nonlinear portion of the system behavior with an equivalent linear time-invariant model considering the stochasticity in the excitation. Furthermore, a genetic algorithm is employed to find optimal weighting matrices for the control design. The input current to the MR damper installed between adjacent spans is determined using a clipped stochastic optimal polynomial control algorithm. The performance of the controlled system is assessed through a set of shake-table experiments for far-field and near-field ground motions. The proposed method showed considerable improvements over passive cases especially for the far-field ground motion.
International Nuclear Information System (INIS)
El-Khoury, O; Shafieezadeh, A; Hur, J E; Kim, C; Heo, G H
2015-01-01
This study performs a series of numerical simulations and shake-table experiments to design and assess the performance of a nonlinear clipped feedback control algorithm based on optimal polynomial control (OPC) to mitigate the response of a two-span bridge equipped with a magnetorheological (MR) damper. As an extended conventional linear quadratic regulator, OPC provides more flexibility in the control design and further enhances system performance. The challenges encountered in this case are (1) the linearization of the nonlinear behavior of various components and (2) the selection of the weighting matrices in the objective function of OPC. The first challenge is addressed by using stochastic linearization which replaces the nonlinear portion of the system behavior with an equivalent linear time-invariant model considering the stochasticity in the excitation. Furthermore, a genetic algorithm is employed to find optimal weighting matrices for the control design. The input current to the MR damper installed between adjacent spans is determined using a clipped stochastic optimal polynomial control algorithm. The performance of the controlled system is assessed through a set of shake-table experiments for far-field and near-field ground motions. The proposed method showed considerable improvements over passive cases especially for the far-field ground motion. (paper)
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...
Darboux polynomials and first integrals of natural polynomial Hamiltonian systems
International Nuclear Information System (INIS)
Maciejewski, Andrzej J.; Przybylska, Maria
2004-01-01
We show that for a natural polynomial Hamiltonian system the existence of a single Darboux polynomial (a partial polynomial first integral) is equivalent to the existence of an additional first integral functionally independent with the Hamiltonian function. Moreover, we show that, in a case when the degree of potential is odd, the system does not admit any proper Darboux polynomial, i.e., the only Darboux polynomials are first integrals
Context-free grammars for several polynomials associated with Eulerian polynomials
Ma, Shi-Mei; Ma, Jun; Yeh, Yeong-Nan; Zhu, Bao-Xuan
2016-01-01
In this paper, we present grammatical descriptions of several polynomials associated with Eulerian polynomials, including q-Eulerian polynomials, alternating run polynomials and derangement polynomials. As applications, we get several convolution formulas involving these polynomials.
Classifying polynomials and identity testing
Indian Academy of Sciences (India)
m = nO(log n). [2]. This provides a excellent classification of easy polynomials. Hence, any polynomial that cannot be written as a determinant of a small sized matrix is not easy. A simple counting argument shows that there exist many such polynomials. However, prov- ing an explicitly given polynomial to be hard has.
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...
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...
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
Fast methods for resumming matrix polynomials and Chebyshev matrix polynomials
International Nuclear Information System (INIS)
Liang Wanzen; Baer, Roi; Saravanan, Chandra; Shao Yihan; Bell, Alexis T.; Head-Gordon, Martin
2004-01-01
Fast and effective algorithms are discussed for resumming matrix polynomials and Chebyshev matrix polynomials. These algorithms lead to a significant speed-up in computer time by reducing the number of matrix multiplications required to roughly twice the square root of the degree of the polynomial. A few numerical tests are presented, showing that evaluation of matrix functions via polynomial expansions can be preferable when the matrix is sparse and these fast resummation algorithms are employed
Approximations of orthogonal polynomials in terms of Hermite polynomials
J.L. López; N.M. Temme (Nico)
1999-01-01
textabstractSeveral orthogonal polynomials have limit forms in which Hermite polynomials show up. Examples are limits with respect to certain parameters of the Jacobi and Laguerre polynomials. In this paper we are interested in more details of these limits and we give asymptotic representations of
Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials
Koornwinder, T.H.; Bouzeffour, F.
2011-01-01
Nonsymmetric Askey-Wilson polynomials are usually written as Laurent polynomials. We write them equivalently as 2-vector-valued symmetric Laurent polynomials. Then the Dunkl-Cherednik operator of which they are eigenfunctions, is represented as a 2 × 2 matrix-valued operator. As a new result made
Cyclotomy and Cyclotomic Polynomials
Indian Academy of Sciences (India)
Home; Journals; Resonance – Journal of Science Education; Volume 4; Issue 12. Cyclotomy and Cyclotomic Polynomials - The Story of how Gauss Narrowly Missed Becoming a Philologist. B Sury. General Article Volume 4 Issue 12 December 1999 pp 41-53 ...
International Nuclear Information System (INIS)
Li, Weijie; Ho, Siu Chun Michael; Song, Gangbing
2016-01-01
Steel reinforcement corrosion is one of the dominant causes for structural deterioration for reinforced concrete structures. This paper presents a novel corrosion detection technique using an active thermal probe. The technique takes advantage of the fact that corrosion products have poor thermal conductivity, which will impede heat propagation generated from the active thermal probe. At the same time, the active thermal probe records the temperature response. The presence of corrosion products can thus be detected by analyzing the temperature response after the injection of heat at the reinforcement-concrete interface. The feasibility of the proposed technique was firstly analyzed through analytical modeling and finite element simulation. The active thermal probe consisted of carbon fiber strands to generate heat and a fiber optic Bragg grating (FBG) temperature sensor. Carbon fiber strands are used due to their corrosion resistance. Wet-dry cycle accelerated corrosion experiments were performed to study the effect of corrosion products on the temperature response of the reinforced concrete sample. Results suggest a high correlation between corrosion severity and magnitude of the temperature response. The technique has the merits of high accuracy, high efficiency in measurement and excellent embeddability. (paper)
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....
Interpolation and Polynomial Curve Fitting
Yang, Yajun; Gordon, Sheldon P.
2014-01-01
Two points determine a line. Three noncollinear points determine a quadratic function. Four points that do not lie on a lower-degree polynomial curve determine a cubic function. In general, n + 1 points uniquely determine a polynomial of degree n, presuming that they do not fall onto a polynomial of lower degree. The process of finding such a…
Fourier series in orthogonal polynomials
Osilenker, Boris
1999-01-01
This book presents a systematic course on general orthogonal polynomials and Fourier series in orthogonal polynomials. It consists of six chapters. Chapter 1 deals in essence with standard results from the university course on the function theory of a real variable and on functional analysis. Chapter 2 contains the classical results about the orthogonal polynomials (some properties, classical Jacobi polynomials and the criteria of boundedness).The main subject of the book is Fourier series in general orthogonal polynomials. Chapters 3 and 4 are devoted to some results in this topic (classical
Generalized q-Hermite polynomials
International Nuclear Information System (INIS)
Berg, C.; Ruffing, A.
2001-01-01
We consider two operators A and A + in a Hilbert space of functions on the exponential lattice {q n ,-q n vertical stroke n element of Z}, where 0 0 and that A and A + are ladder operators for the sequence ψ n =(A + ) n ψ 0 . The sequence (ψ n /ψ 0 ) is shown to be a family of orthogonal polynomials, which we identify as symmetrized q-Laguerre polynomials. We obtain in this way a new proof of the orthogonality for these polynomials. When γ=0 the polynomials are the discrete q-Hermite polynomials of type II, studied in several papers on q-quantum mechanics. (Orig.) (orig.)
Complex Polynomial Vector Fields
DEFF Research Database (Denmark)
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...
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 ...
A Characterization of Polynomials
DEFF Research Database (Denmark)
Andersen, Kurt Munk
1996-01-01
Given the problem:which functions f(x) are characterized by a relation of the form:f[x1,x2,...,xn]=h(x1+x2+...+xn), where n>1 and h(x) is a given function? Here f[x1,x2,...,xn] denotes the divided difference on n points x1,x2,...,xn of the function f(x).The answer is: f(x) is a polynomial of degr...
Energy Technology Data Exchange (ETDEWEB)
Milks, Matthew M; Guise, Hubert de [Department of Physics, Lakehead University, Thunder Bay, ON, P7B 5E1 (Canada)
2005-12-01
The construction of su(2) intelligent states is simplified using a polynomial representation of su(2). The cornerstone of the new construction is the diagonalization of a 2 x 2 matrix. The method is sufficiently simple to be easily extended to su(3), where one is required to diagonalize a single 3 x 3 matrix. For two perfectly general su(3) operators, this diagonalization is technically possible but the procedure loses much of its simplicity owing to the algebraic form of the roots of a cubic equation. Simplified expressions can be obtained by specializing the choice of su(3) operators. This simpler construction will be discussed in detail.
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....... In this thesis we study the real roots of the chromatic polynomial, termed chromatic roots, and focus on how certain properties of a graph affect the location of its chromatic roots. Firstly, we investigate how the presence of a certain spanning tree in a graph affects its chromatic roots. In particular we prove...... of certain minor-closed families of graphs. Later, we study the Tutte polynomial of a graph, which contains the chromatic polynomial as a specialisation. We discuss a technique of Thomassen using which it is possible to deduce that the roots of the chromatic polynomial are dense in certain intervals. We...
Laksâ, Arne
2015-11-01
B-splines are the de facto industrial standard for surface modelling in Computer Aided design. It is comparable to bend flexible rods of wood or metal. A flexible rod minimize the energy when bending, a third degree polynomial spline curve minimize the second derivatives. B-spline is a nice way of representing polynomial splines, it connect polynomial splines to corner cutting techniques, which induces many nice and useful properties. However, the B-spline representation can be expanded to something we can call general B-splines, i.e. both polynomial and non-polynomial splines. We will show how this expansion can be done, and the properties it induces, and examples of non-polynomial B-spline.
Classifying polynomials and identity testing
Indian Academy of Sciences (India)
hard to compute [3,4]! Therefore, the solution to. PIT problem has a key role in our attempt to com- putationally classify polynomials. In this article, we will focus on this connection between PIT and polynomial classification. We now formally define arithmetic circuits and the identity testing problem. 1.1 Problem definition.
Polynomial methods in combinatorics
Guth, Larry
2016-01-01
This book explains some recent applications of the theory of polynomials and algebraic geometry to combinatorics and other areas of mathematics. One of the first results in this story is a short elegant solution of the Kakeya problem for finite fields, which was considered a deep and difficult problem in combinatorial geometry. The author also discusses in detail various problems in incidence geometry associated to Paul Erdős's famous distinct distances problem in the plane from the 1940s. The proof techniques are also connected to error-correcting codes, Fourier analysis, number theory, and differential geometry. Although the mathematics discussed in the book is deep and far-reaching, it should be accessible to first- and second-year graduate students and advanced undergraduates. The book contains approximately 100 exercises that further the reader's understanding of the main themes of the book. Some of the greatest advances in geometric combinatorics and harmonic analysis in recent years have been accompl...
Complex Polynomial Vector Fields
DEFF Research Database (Denmark)
Dias, Kealey
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...... or meromorphic (allowing poles as singularities) functions. There already exists a well-developed theory for iterative holomorphic dynamical systems, and successful relations found between iteration theory and flows of vector fields have been one of the main motivations for the recent interest in holomorphic...... 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...
On the limit from q-Racah polynomials to big q-Jacobi polynomials
Koornwinder, T.H.
2011-01-01
A limit formula from q-Racah polynomials to big q-Jacobi polynomials is given which can be considered as a limit formula for orthogonal polynomials. This is extended to a multi-parameter limit with 3 parameters, also involving (q-)Hahn polynomials, little q-Jacobi polynomials and Jacobi polynomials.
Application of a composition of generating functions for obtaining explicit formulas of polynomials
Kruchinin, Vladimir; Kruchinin, Dmitry
2012-01-01
Using notions of composita and composition of generating functions we obtain explicit formulas for Chebyshev polynomials, Legendre polynomials, Gegenbauer polynomials, Associated Laguerre polynomials, Stirling polynomials, Abel polynomials, Bernoulli Polynomials of the Second Kind, Generalized Bernoulli polynomials, Euler Polynomials, Peters polynomials, Narumi polynomials, Humbert polynomials, Lerch polynomials and Mahler polynomials.
Czech Academy of Sciences Publication Activity Database
Krčmařík, David; Slavík, Radan; Kulishov, M.; Karásek, Miroslav
2009-01-01
Roč. 27, č. 13 (2009), s. 2335-2342 ISSN 0733-8724 R&D Projects: GA ČR(CZ) GA102/07/0999; GA AV ČR KJB200670601 Institutional research plan: CEZ:AV0Z20670512 Keywords : optical communications * optical fibre * fibre gratings Subject RIV: JA - Electronics ; Optoelectronics, Electrical Engineering Impact factor: 2.185, year: 2009
Iwata, Fujio
2001-06-01
Toppan Printing Co., Ltd. originated the name of 'grating image'. It means an image that consists of diffraction grating dots that look similar to the halftone dots of conventional printing. We proposed this new display method using simple gratings in order to enhance the visual effects when illumination is made by a fluorescent lamp. We considered the use of simple gratings as elemental dots, and used a number of elemental dots to display a 2D image. This method produces an effect something like the halftone dots of printing. The grating image technology grows from its starting to become able to produce 3D images and a 3D-video system using an electron beam grating-writing system.
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
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....
Sum rules for zeros of polynomials and generalized Lucas polynomials
Ricci, Paolo Emilio
1993-10-01
A representation formula in terms of generalized Lucas polynomials of the second kind [see formula (4.3)], for the sum rules Js(i) introduced by Case [J. Math. Phys. 21, 702 (1980)] and studied by Dehesa et al. [J. Math. Phys. 26, 1547 (1985); 26, 2729 (1985)] in order to obtain information about the zeros' distribution of eigenfunctions of a class of ordinary polynomial differential operators, is derived.
Polynomial Regressions and Nonsense Inference
Directory of Open Access Journals (Sweden)
Daniel Ventosa-Santaulària
2013-11-01
Full Text Available Polynomial specifications are widely used, not only in applied economics, but also in epidemiology, physics, political analysis and psychology, just to mention a few examples. In many cases, the data employed to estimate such specifications are time series that may exhibit stochastic nonstationary behavior. We extend Phillips’ results (Phillips, P. Understanding spurious regressions in econometrics. J. Econom. 1986, 33, 311–340. by proving that an inference drawn from polynomial specifications, under stochastic nonstationarity, is misleading unless the variables cointegrate. We use a generalized polynomial specification as a vehicle to study its asymptotic and finite-sample properties. Our results, therefore, lead to a call to be cautious whenever practitioners estimate polynomial regressions.
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
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....
Stochastic Estimation via Polynomial Chaos
2015-10-01
processes. As originally developed by Norbert Weiner, a polynomial chaos represents key properties of a stochastic process through the application of...polynomial chaos method for representing the properties of second order stochastic processes. As originally developed by Norbert Weiner, a...any real or complex functional ][xF in 2L converges to ][xF in the 2L sense with Wiener measure.[3] In the same reference, the orthogonality of
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...... function. We give a detailed study of the case when m has exactly two distinct prime factors, and classify the minimum possible degree for a symmetric representing polynomial....
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
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.
A 'missing' family of classical orthogonal polynomials
International Nuclear Information System (INIS)
Vinet, Luc; Zhedanov, Alexei
2011-01-01
We study a family of 'classical' orthogonal polynomials which satisfy (apart from a three-term recurrence relation) an eigenvalue problem with a differential operator of Dunkl type. These polynomials can be obtained from the little q-Jacobi polynomials in the limit q = -1. We also show that these polynomials provide a nontrivial realization of the Askey-Wilson algebra for q = -1.
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
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...
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.
Polynomial invariants to quantify Four-body Correlations
Sharma, Santosh Shelly; Sharma, Naresh Kumar
2013-03-01
Local unitary invariance and notion of negativity fonts are used as the principle tools to construct four qubit polynomial invariants of degree 8, 12, and 24. Determinants of negativity fonts are linked to matrices obtained from state operator through selective partial transposition. Our general aim is to construct the polynomial invariants that quantify entanglement due to K - body correlations in an N-qubit (N ? K) pure state. This is done by constructing N-qubit invariants from multivariate forms with (K - 1)-qubit invariants as coefficients. In particular, the invariant that quantifies entanglement due to N-body correlations is obtained from a biform having as coefficients the N - 1 qubit invariants. A polynomial invariant that is non-zero on four qubit pure states with four-body correlations and zero on all other states, is identified. Classification of four qubit states into seven major classes, using criterion based on the nature of correlations, is discussed. We gratefully acknowledge financial support from CNPq Brazil and Faep, UEL, Brazil.
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.
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.
Aluminum nitride grating couplers.
Ghosh, Siddhartha; Doerr, Christopher R; Piazza, Gianluca
2012-06-10
Grating couplers in sputtered aluminum nitride, a piezoelectric material with low loss in the C band, are demonstrated. Gratings and a waveguide micromachined on a silicon wafer with 600 nm minimum feature size were defined in a single lithography step without partial etching. Silicon dioxide (SiO(2)) was used for cladding layers. Peak coupling efficiency of -6.6 dB and a 1 dB bandwidth of 60 nm have been measured. This demonstration of wire waveguides and wideband grating couplers in a material that also has piezoelectric and elasto-optic properties will enable new functions for integrated photonics and optomechanics.
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.
Plain Polynomial Arithmetic on GPU
International Nuclear Information System (INIS)
Haque, Sardar Anisul; Maza, Marc Moreno
2012-01-01
As for serial code on CPUs, parallel code on GPUs for dense polynomial arithmetic relies on a combination of asymptotically fast and plain algorithms. Those are employed for data of large and small size, respectively. Parallelizing both types of algorithms is required in order to achieve peak performances. In this paper, we show that the plain dense polynomial multiplication can be efficiently parallelized on GPUs. Remarkably, it outperforms (highly optimized) FFT-based multiplication up to degree 2 12 while on CPU the same threshold is usually at 2 6 . We also report on a GPU implementation of the Euclidean Algorithm which is both work-efficient and runs in linear time for input polynomials up to degree 2 18 thus showing the performance of the GCD algorithm based on systolic arrays.
Orthogonal Polynomials and their Applications
Dehesa, Jesús; Marcellan, Francisco; Francia, José; Vinuesa, Jaime
1988-01-01
The Segovia meeting set out to stimulate an intensive exchange of ideas between experts in the area of orthogonal polynomials and its applications, to present recent research results and to reinforce the scientific and human relations among the increasingly international community working in orthogonal polynomials. This volume contains original research papers as well as survey papers about fundamental questions in the field (Nevai, Rakhmanov & López) and its relationship with other fields such as group theory (Koornwinder), Padé approximation (Brezinski), differential equations (Krall, Littlejohn) and numerical methods (Rivlin).
Recursive formula to compute Zernike radial polynomials.
Honarvar Shakibaei, Barmak; Paramesran, Raveendran
2013-07-15
In optics, Zernike polynomials are widely used in testing, wavefront sensing, and aberration theory. This unique set of radial polynomials is orthogonal over the unit circle and finite on its boundary. This Letter presents a recursive formula to compute Zernike radial polynomials using a relationship between radial polynomials and Chebyshev polynomials of the second kind. Unlike the previous algorithms, the derived recurrence relation depends neither on the degree nor on the azimuthal order of the radial polynomials. This leads to a reduction in the computational complexity.
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...
Polynomial Regressions and Nonsense Inference
DEFF Research Database (Denmark)
Ventosa-Santaulària, Daniel; Rodríguez-Caballero, Carlos Vladimir
Polynomial specifications are widely used, not only in applied economics, but also in epidemiology, physics, political analysis, and psychology, just to mention a few examples. In many cases, the data employed to estimate such estimations are time series that may exhibit stochastic nonstationary ...
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)
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.
Aspects of the Tutte polynomial
DEFF Research Database (Denmark)
Ok, Seongmin
This thesis studies various aspects of the Tutte polynomial, especially focusing on the Merino-Welsh conjecture. We write T(G;x,y) for the Tutte polynomial of a graph G with variables x and y. In 1999, Merino and Welsh conjectured that if G is a loopless 2-connected graph, then T(G;1,1) ≤ max{T(G;2......,0), T(G;0,2)}. The three numbers, T(G;1,1), T(G;2,0) and T(G;0,2) are respectively the numbers of spanning trees, acyclic orientations and totally cyclic orientations of G. First, I extend Negami's splitting formula to the multivariate Tutte polynomial. Using the splitting formula, Thomassen and I found......-Welsh conjecture has a stronger version claiming that the Tutte polynomial is convex on the line segment between (2,0) and (0,2) for loopless 2-connected graphs. Chavez-Lomeli et al. proved that this holds for coloopless paving matroids, and I provide a shorter proof of their theorem. I also prove it for minimally...
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.
Tables of properties of airfoil polynomials
Desmarais, Robert N.; Bland, Samuel R.
1995-01-01
This monograph provides an extensive list of formulas for airfoil polynomials. These polynomials provide convenient expansion functions for the description of the downwash and pressure distributions of linear theory for airfoils in both steady and unsteady subsonic flow.
Random walk polynomials and random walk measures
van Doorn, Erik A.; Schrijner, Pauline
1993-01-01
Random walk polynomials and random walk measures play a prominent role in the analysis of a class of Markov chains called random walks. Without any reference to random walks, however, a random walk polynomial sequence can be defined (and will be defined in this paper) as a polynomial sequence{Pn(x)}
Positive trigonometric polynomials and signal processing applications
Dumitrescu, Bogdan
2007-01-01
Presents the results on positive trigonometric polynomials within a unitary framework; the theoretical results obtained partly from the general theory of real polynomials, partly from self-sustained developments. This book provides information on the theory of sum-of-squares trigonometric polynomials in two parts: theory and applications.
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
Sibling curves of polynomials | Wiggins | Quaestiones Mathematicae
African Journals Online (AJOL)
Sibling curves were demonstrated in papers [2, 3] as a novel way to visualize the zeros of complex valued functions. In this paper, we continue the work done in those papers by focusing solely on polynomials. We proceed to prove that the number of sibling curves of a polynomial is the degree of the polynomial. Keywords: ...
Restricted Schur polynomials and finite N counting
International Nuclear Information System (INIS)
Collins, Storm
2009-01-01
Restricted Schur polynomials have been posited as orthonormal operators for the change of basis from N=4 SYM to type IIB string theory. In this paper we briefly expound the relationship between the restricted Schur polynomials and the operators forwarded by Brown, Heslop, and Ramgoolam. We then briefly examine the finite N counting of the restricted Schur polynomials.
On polynomial and multiplicative radicals | Tumurbat | Quaestiones ...
African Journals Online (AJOL)
We show that polynomial and multiplicative radicals in [1] are special cases of radicals defined by means of elements. We scrutinize the way of defining a radical γG by a subset G of polynomials in noncommuting indeterminates. Defining polynomial radicals, Drazin and Roberts [1] required that the set G be closed
J.L. López; N.M. Temme (Nico)
1999-01-01
textabstractThis is a second paper on finite exact representations of certain polynomials in terms of Hermite polynomials. The representations have asymptotic properties and include new limits of the polynomials, again in terms of Hermite polynomials. This time we consider the generalized Bernoulli,
Special polynomials associated with rational solutions of some hierarchies
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
Polynomial Mappings with Small Degree
Directory of Open Access Journals (Sweden)
Jelonek Zbigniew
2015-06-01
Full Text Available Let Xn be an affine variety of dimension n and Yn be a quasi-projective variety of the same dimension. We prove that for a quasi-finite polynomial mapping ƒ : Xn → Yn, every non-empty component of the set Yn\\ ƒ (Xn is closed and it has dimension greater or equal to n μ(ƒ, where μ(ƒ is a geometric degree of ƒ. Moreover, we prove that generally, if ƒ : Xn → Yn is any polynomial mapping, then either every non-empty component of the set is of dimension ≥ n μ(ƒ or ƒ contracts a subvariety of dimension ≥ n μ(ƒ + 1.
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.
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...
Poly-Cauchy and Peters mixed-type polynomials
Kim, Dae San; Kim, Taekyun
2013-01-01
The Peters polynomials are a generalization of Boole polynomials. In this paper, we consider Peters and poly-Cauchy mixed type polynomials and investigate the properties of those polynomials which are derived from umbral calculus. Finally, we give various identities of those polynomials associated with special polynomials.
Energy Technology Data Exchange (ETDEWEB)
Nunes, J.A.; Tong, W.G. [San Diego State Univ., CA (United States). Dept. of Chemistry; Chandler, D.W.; Rahn, L.A. [Sandia National Lab., Livermore, CA (United States). Combustion Research Facility
1995-04-01
A novel four-wave mixing technique for the detection of circular dichroism in optically active liquid samples is demonstrated. When two cross-polarized laser beams are crossed at a small angle in a circular dichroic liquid a weak thermal grating is produced with a phase depending on the sign of the circular dichroism. The authors show that the polarization of one of the beams can be modified to allow coherent interference with an intensity-grating induced thermal grating. A probe beam scattering from the composite grating results in a signal that reveals the sign and magnitude of the circular dichroism. The use of this technique to optimize the signal-to-noise ratio in the presence of scattered light and laser intensity noise is discussed.
Evolutionary Design of Polynomial Controller
R. Matousek; S. Lang; P. Minar; P. Pivonka
2011-01-01
In the control theory one attempts to find a controller that provides the best possible performance with respect to some given measures of performance. There are many sorts of controllers e.g. a typical PID controller, LQR controller, Fuzzy controller etc. In the paper will be introduced polynomial controller with novel tuning method which is based on the special pole placement encoding scheme and optimization by Genetic Algorithms (GA). The examples will show the perform...
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
Codimensions of generalized polynomial identities
International Nuclear Information System (INIS)
Gordienko, Aleksei S
2010-01-01
It is proved that for every finite-dimensional associative algebra A over a field of characteristic zero there are numbers C element of Q + and t element of Z + such that gc n (A)∼Cn t d n as n→∞, where d=PI exp(A) element of Z + . Thus, Amitsur's and Regev's conjectures hold for the codimensions gc n (A) of the generalized polynomial identities. Bibliography: 6 titles.
International Nuclear Information System (INIS)
Shu, X. W.; Wang, J. X.; Yaqoob, T.
2010-01-01
We extend the study of the core of the Fe Kα emission line at ∼6.4 keV in Seyfert galaxies reported by Yaqoob and Padmanabhan using a larger sample observed by the Chandra high-energy grating (HEG). The sample consists of 82 observations of 36 unique sources with z H 23 cm -2 ) Seyfert galaxies to date. From an empirical and uniform analysis, we present measurements of the Fe Kα line centroid energy, flux, equivalent width (EW), and intrinsic width (FWHM). The Fe Kα line is detected in 33 sources, and its centroid energy is constrained in 32 sources. In 27 sources, the statistical quality of the data is good enough to yield measurements of the FWHM. We find that the distribution in the line centroid energy is strongly peaked around the value for neutral Fe, with over 80% of the observations giving values in the range 6.38-6.43 keV. Including statistical errors, 30 out of 32 sources (∼94%) have a line centroid energy in the range 6.35-6.47 keV. The mean EW, among the observations in which a non-zero lower limit could be measured, was 53 ± 3 eV. The mean FWHM from the subsample of 27 sources was 2060 ± 230 km s -1 . The mean EW and FWHM are somewhat higher when multiple observations for a given source are averaged. From a comparison with the Hβ optical emission-line widths (or, for one source, Brα), we find that there is no universal location of the Fe Kα line-emitting region relative to the optical broad-line region (BLR). In general, a given source may have contributions to the Fe Kα line flux from parsec-scale distances from the putative black hole, down to matter a factor ∼2 closer to the black hole than the BLR. We confirm the presence of the X-ray Baldwin effect, an anti-correlation between the Fe Kα line EW and X-ray continuum luminosity. The HEG data have enabled isolation of this effect to the narrow core of the Fe Kα line.
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.
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 Chebyshev Polynomials of Matrices
Czech Academy of Sciences Publication Activity Database
Faber, V.; Liesen, J.; Tichý, Petr
2010-01-01
Roč. 31, č. 4 (2010), s. 2205-2221 ISSN 0895-4798 R&D Projects: GA AV ČR IAA100300802 Grant - others:GA AV ČR(CZ) M100300901 Institutional research plan: CEZ:AV0Z10300504 Source of funding: I - inštitucionálna podpora na rozvoj VO Keywords : matrix approximation problems * Chebyshev polynomials * complex approximation theory * Krylov subspace methods * Arnoldi's method Subject RIV: BA - General Mathematics Impact factor: 1.725, year: 2010
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.
On classical orthogonal polynomials and differential operators
International Nuclear Information System (INIS)
Miranian, L
2005-01-01
It is well known that the four families of classical orthogonal polynomials (Jacobi, Bessel, Hermite and Laguerre) each satisfy an equation FP n (x) = λ n P n (x), n ≥ 0, for an appropriate second-order differential operator F. In this paper it is shown that any linear differential operator U which has the Jacobi, Bessel, Hermite or Laguerre polynomials as eigenfunctions has to be a polynomial with constant coefficients in the classical second-order operator F
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.
Model selection for univariable fractional polynomials.
Royston, Patrick
2017-07-01
Since Royston and Altman's 1994 publication ( Journal of the Royal Statistical Society, Series C 43: 429-467), fractional polynomials have steadily gained popularity as a tool for flexible parametric modeling of regression relationships. In this article, I present fp_select, a postestimation tool for fp that allows the user to select a parsimonious fractional polynomial model according to a closed test procedure called the fractional polynomial selection procedure or function selection procedure. I also give a brief introduction to fractional polynomial models and provide examples of using fp and fp_select to select such models with real data.
Acceleration of computation of φ-polynomials.
Kaya, Ilhan; Rolland, Jannick
2013-11-18
The benefits of making an effective use of impressive computational power offered by multi-core platforms are investigated for the computation of φ-polynomials used in the description of freeform surfaces. Specifically, we devise parallel algorithms based upon the recurrence relations of both Zernike polynomials and gradient orthogonal Q-polynomials and implement these parallel algorithms on Graphical Processing Units (GPUs) respectively. The results show that more than an order of magnitude improvement is achieved in computational time over a sequential implementation if these recurrence-based parallel algorithms are adopted in the computation of the φ-polynomials.
The q-Laguerre matrix polynomials.
Salem, Ahmed
2016-01-01
The Laguerre polynomials have been extended to Laguerre matrix polynomials by means of studying certain second-order matrix differential equation. In this paper, certain second-order matrix q-difference equation is investigated and solved. Its solution gives a generalized of the q-Laguerre polynomials in matrix variable. Four generating functions of this matrix polynomials are investigated. Two slightly different explicit forms are introduced. Three-term recurrence relation, Rodrigues-type formula and the q-orthogonality property are given.
Control to Facet for Polynomial Systems
DEFF Research Database (Denmark)
Sloth, Christoffer; Wisniewski, Rafael
2014-01-01
This paper presents a solution to the control to facet problem for arbitrary polynomial vector fields defined on simplices. The novelty of the work is to use Bernstein coefficients of polynomials for determining certificates of positivity. Specifically, the constraints that are set up...... for the controller design are solved by searching for polynomials in Bernstein form. This allows the controller design problem to be formulated as a linear programming problem. Examples are provided that demonstrate the efficiency of the method for designing controls for polynomial systems....
Orthonormal polynomials in wavefront analysis: analytical solution.
Mahajan, Virendra N; Dai, Guang-ming
2007-09-01
Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. In recent papers, we derived closed-form polynomials that are orthonormal over a hexagonal pupil, such as the hexagonal segments of a large mirror. We extend our work to elliptical, rectangular, and square pupils. Using the circle polynomials as the basis functions for their orthogonalization over such pupils, we derive closed-form polynomials that are orthonormal over them. These polynomials are unique in that they are not only orthogonal across such pupils, but also represent balanced classical aberrations, just as the Zernike circle polynomials are unique in these respects for circular pupils. The polynomials are given in terms of the circle polynomials as well as in polar and Cartesian coordinates. Relationships between the orthonormal coefficients and the corresponding Zernike coefficients for a given pupil are also obtained. The orthonormal polynomials for a one-dimensional slit pupil are obtained as a limiting case of a rectangular pupil.
Directory of Open Access Journals (Sweden)
Ryoo CS
2010-01-01
Full Text Available The purpose of this paper is to give some properties of several Bernstein type polynomials to represent the fermionic -adic integral on . From these properties, we derive some interesting identities on the Euler numbers and polynomials.
International Nuclear Information System (INIS)
Dubetsky, B.; Berman, P.R.; Sleator, T.
1992-01-01
A theory of a grating simulated echo (GTE) is developed. The GSE involves the sequential excitation of atoms by two counterpropagating traveling waves, a standing wave, and a third traveling wave. It is shown that the echo signal is very sensitive to small changes in atomic velocity, much more sensitive than the normal stimulated echo. Use of the GSE as a collisional probe or accelerometer is discussed
Orthogonal polynomials from Hermitian matrices. II
Odake, Satoru; Sasaki, Ryu
2018-01-01
This is the second part of the project "unified theory of classical orthogonal polynomials of a discrete variable derived from the eigenvalue problems of Hermitian matrices." In a previous paper, orthogonal polynomials having Jackson integral measures were not included, since such measures cannot be obtained from single infinite dimensional Hermitian matrices. Here we show that Jackson integral measures for the polynomials of the big q-Jacobi family are the consequence of the recovery of self-adjointness of the unbounded Jacobi matrices governing the difference equations of these polynomials. The recovery of self-adjointness is achieved in an extended ℓ2 Hilbert space on which a direct sum of two unbounded Jacobi matrices acts as a Hamiltonian or a difference Schrödinger operator for an infinite dimensional eigenvalue problem. The polynomial appearing in the upper/lower end of the Jackson integral constitutes the eigenvector of each of the two-unbounded Jacobi matrix of the direct sum. We also point out that the orthogonal vectors involving the q-Meixner (q-Charlier) polynomials do not form a complete basis of the ℓ2 Hilbert space, based on the fact that the dual q-Meixner polynomials introduced in a previous paper fail to satisfy the orthogonality relation. The complete set of eigenvectors involving the q-Meixner polynomials is obtained by constructing the duals of the dual q-Meixner polynomials which require the two-component Hamiltonian formulation. An alternative solution method based on the closure relation, the Heisenberg operator solution, is applied to the polynomials of the big q-Jacobi family and their duals and q-Meixner (q-Charlier) 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
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.
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)
Integral inequalities for self-reciprocal polynomials
Indian Academy of Sciences (India)
that is, the coefficients satisfy ak = an−k for k = 0, 1,...,n. We denote the class of self-reciprocal polynomials of order n by Pn. The properties of self-reciprocal polynomials have been studied by several mathemati- cians. We refer to [1–5] and the references given therein. Here, we are concerned with the following interesting ...
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.
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 ...
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
Exceptional polynomials and SUSY quantum mechanics
Indian Academy of Sciences (India)
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 potential.
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)
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
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
Block orthogonal polynomials: I. Definitions and properties
International Nuclear Information System (INIS)
Normand, Jean-Marie
2007-01-01
Constrained orthogonal polynomials have been recently introduced in the study of the Hohenberg-Kohn functional to provide basis functions satisfying particle number conservation for an expansion of the particle density. More generally, we define block orthogonal (BO) polynomials which are orthogonal, with respect to a first Euclidean inner product, to a given i-dimensional subspace E i of polynomials associated with the constraints. In addition, they are mutually orthogonal with respect to a second Euclidean inner product. We recast the determination of these polynomials into a general problem of finding particular orthogonal bases in an Euclidean vector space endowed with distinct inner products. An explicit two step Gram-Schmidt orthogonalization (G-SO) process to determine these bases is given. By definition, the standard block orthogonal (SBO) polynomials are associated with a choice of E i equal to the subspace of polynomials of degree less than i. We investigate their properties, emphasizing similarities to and differences from the standard orthogonal polynomials. Applications to classical orthogonal polynomials will be given in forthcoming papers
The Medusa Algorithm for Polynomial Matings
DEFF Research Database (Denmark)
Boyd, Suzanne Hruska; Henriksen, Christian
2012-01-01
The Medusa algorithm takes as input two postcritically finite quadratic polynomials and outputs the quadratic rational map which is the mating of the two polynomials (if it exists). Specifically, the output is a sequence of approximations for the parameters of the rational map, as well as an image...
Indian Academy of Sciences (India)
For a polynomial of degree , we have obtained an upper bound involving coefficients of the polynomial, for moduli of its zeros of smallest moduli, and then a refinement of the well-known Eneström–Kakeya theorem (under certain conditions).
Polynomial approximation approach to transient heat conduction ...
African Journals Online (AJOL)
This work reports polynomial approximation approach to transient heat conduction in a long slab, long cylinder and sphere with linear internal heat generation. It has been shown that the polynomial approximation method is able to calculate average temperature as a function of time for higher value of Biot numbers.
Orthonormal polynomials in wavefront analysis: error analysis.
Dai, Guang-Ming; Mahajan, Virendra N
2008-07-01
Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. However, they are not appropriate for noncircular pupils, such as annular, hexagonal, elliptical, rectangular, and square pupils, due to their lack of orthogonality over such pupils. We emphasize the use of orthonormal polynomials for such pupils, but we show how to obtain the Zernike coefficients correctly. We illustrate that the wavefront fitting with a set of orthonormal polynomials is identical to the fitting with a corresponding set of Zernike polynomials. This is a consequence of the fact that each orthonormal polynomial is a linear combination of the Zernike polynomials. However, since the Zernike polynomials do not represent balanced aberrations for a noncircular pupil, the Zernike coefficients lack the physical significance that the orthonormal coefficients provide. We also analyze the error that arises if Zernike polynomials are used for noncircular pupils by treating them as circular pupils and illustrate it with numerical examples.
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.
Generating polynomials using function composition | Aichinger ...
African Journals Online (AJOL)
We characterize the integral polynomials that can be obtained from x and x2 (or from 1, x, and x3) by using +;-, and function composition. Keywords: Near-rings, integer polynomials, subnear-ring membership problem. Quaestiones Mathematicae 36(2013), 39-46 ...
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 ...
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
Polynomials formalism of quantum numbers
International Nuclear Information System (INIS)
Kazakov, K.V.
2005-01-01
Theoretical aspects of the recently suggested perturbation formalism based on the method of quantum number polynomials are considered in the context of the general anharmonicity problem. Using a biatomic molecule by way of example, it is demonstrated how the theory can be extrapolated to the case of vibrational-rotational interactions. As a result, an exact expression for the first coefficient of the Herman-Wallis factor is derived. In addition, the basic notions of the formalism are phenomenologically generalized and expanded to the problem of spin interaction. The concept of magneto-optical anharmonicity is introduced. As a consequence, an exact analogy is drawn with the well-known electro-optical theory of molecules, and a nonlinear dependence of the magnetic dipole moment of the system on the spin and wave variables is established [ru
Fuzzy Morphological Polynomial Image Representation
Directory of Open Access Journals (Sweden)
Chin-Pan Huang
2010-01-01
Full Text Available A novel signal representation using fuzzy mathematical morphology is developed. We take advantage of the optimum fuzzy fitting and the efficient implementation of morphological operators to extract geometric information from signals. The new representation provides results analogous to those given by the polynomial transform. Geometrical decomposition of a signal is achieved by windowing and applying sequentially fuzzy morphological opening with structuring functions. The resulting representation is made to resemble an orthogonal expansion by constraining the results of opening to equate adapted structuring functions. Properties of the geometric decomposition are considered and used to calculate the adaptation parameters. Our procedure provides an efficient and flexible representation which can be efficiently implemented in parallel. The application of the representation is illustrated in data compression and fractal dimension estimation temporal signals and images.
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...... that are subcodes of the binary Reed-Muller codes and can be very simply instrumented, 3) a new class of constacyclic codes that are subcodes of thep-ary "Reed-Muller codes," 4) two new classes of binary convolutional codes with large "free distance" derived from known binary cyclic codes, 5) two new classes...... 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....
Differential expansion for link polynomials
Bai, C.; Jiang, J.; Liang, J.; Mironov, A.; Morozov, A.; Morozov, An.; Sleptsov, A.
2018-03-01
The differential expansion is one of the key structures reflecting group theory properties of colored knot polynomials, which also becomes an important tool for evaluation of non-trivial Racah matrices. This makes highly desirable its extension from knots to links, which, however, requires knowledge of the 6j-symbols, at least, for the simplest triples of non-coincident representations. Based on the recent achievements in this direction, we conjecture a shape of the differential expansion for symmetrically-colored links and provide a set of examples. Within this study, we use a special framing that is an unusual extension of the topological framing from knots to links. In the particular cases of Whitehead and Borromean rings links, the differential expansions are different from the previously discovered.
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.
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.
Tutte polynomial in functional magnetic resonance imaging
García-Castillón, Marlly V.
2015-09-01
Methods of graph theory are applied to the processing of functional magnetic resonance images. Specifically the Tutte polynomial is used to analyze such kind of images. Functional Magnetic Resonance Imaging provide us connectivity networks in the brain which are represented by graphs and the Tutte polynomial will be applied. The problem of computing the Tutte polynomial for a given graph is #P-hard even for planar graphs. For a practical application the maple packages "GraphTheory" and "SpecialGraphs" will be used. We will consider certain diagram which is depicting functional connectivity, specifically between frontal and posterior areas, in autism during an inferential text comprehension task. The Tutte polynomial for the resulting neural networks will be computed and some numerical invariants for such network will be obtained. Our results show that the Tutte polynomial is a powerful tool to analyze and characterize the networks obtained from functional magnetic resonance imaging.
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)
Improved polynomial remainder sequences for Ore polynomials☆
Jaroschek, Maximilian
2013-01-01
Polynomial remainder sequences contain the intermediate results of the Euclidean algorithm when applied to (non-)commutative polynomials. The running time of the algorithm is dependent on the size of the coefficients of the remainders. Different ways have been studied to make these as small as possible. The subresultant sequence of two polynomials is a polynomial remainder sequence in which the size of the coefficients is optimal in the generic case, but when taking the input from applications, the coefficients are often larger than necessary. We generalize two improvements of the subresultant sequence to Ore polynomials and derive a new bound for the minimal coefficient size. Our approach also yields a new proof for the results in the commutative case, providing a new point of view on the origin of the extraneous factors of the coefficients. PMID:26523087
Resolving capacity of the circular Zernike polynomials.
Svechnikov, M V; Chkhalo, N I; Toropov, M N; Salashchenko, N N
2015-06-01
Circular Zernike polynomials are often used for approximation and analysis of optical surfaces. In this paper, we analyse their lateral resolving capacity, illustrating the effects of a lack of approximation by a finite set of polynomials and answering the following questions: What is the minimum number of polynomials that is necessary to describe a local deformation of a certain size? What is the relationship between the number of approximating polynomials and the spatial spectrum of the approximation? What is the connection between the mean-square error of approximation and the number of polynomials? The main results of this work are the formulas for calculating the error of fitting the relief and the connection between the width of the spatial spectrum and the order of approximation.
Block orthogonal polynomials: II. Hermite and Laguerre standard block orthogonal polynomials
International Nuclear Information System (INIS)
Normand, Jean-Marie
2007-01-01
The standard block orthogonal (SBO) polynomials P i;n (x), 0 ≤ i ≤ n are real polynomials of degree n which are orthogonal with respect to a first Euclidean scalar product to polynomials of degree less than i. In addition, they are mutually orthogonal with respect to a second Euclidean scalar product. Applying the general results obtained in a previous paper, we determine and investigate these polynomials when the first scalar product corresponds to Hermite (resp. Laguerre) polynomials. These new sets of polynomials, which we call Hermite (resp. Laguerre) SBO polynomials, provide a basis of functional spaces well suited for some applications requiring to take into account special linear constraints which can be recast into an Euclidean orthogonality relation
Recurrence relations on the generelizad poly-Genocchi polynomials
Kurt, Burak; Bilgic, Secil
2017-07-01
In this work, we introduce and investigate poly-Genocchi numbers and polynomials. We give recurrence relations for the poly-Genocchi numbers. Also, we prove a relation between multi-poly-Genocchi polynomial and multi-poly-Bernoulli polynomial.
Exact multi-restricted Schur polynomial correlators
International Nuclear Information System (INIS)
Bhattacharyya, Rajsekhar; Koch, Robert de Mello; Stephanou, Michael
2008-01-01
We derive a product rule satisfied by restricted Schur polynomials. We focus mostly on the case that the restricted Schur polynomial is built using two matrices, although our analysis easily extends to more than two matrices. This product rule allows us to compute exact multi-point correlation functions of restricted Schur polynomials, in the free field theory limit. As an example of the use of our formulas, we compute two point functions of certain single trace operators built using two matrices and three point functions of certain restricted Schur polynomials, exactly, in the free field theory limit. Our results suggest that gravitons become strongly coupled at sufficiently high energy, while the restricted Schur polynomials for totally antisymmetric representations remain weakly interacting at these energies. This is in perfect accord with the half-BPS (single matrix) results of hep-th/0512312. Finally, by studying the interaction of two restricted Schur polynomials we suggest a physical interpretation for the labels of the restricted Schur polynomial: the composite operator χ R,(r n ,r m ) (Z,X) is constructed from the half BPS 'partons' χ r n (Z) and χ r m (X).
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.
More on rotations as spin matrix polynomials
International Nuclear Information System (INIS)
Curtright, Thomas L.
2015-01-01
Any nonsingular function of spin j matrices always reduces to a matrix polynomial of order 2j. The challenge is to find a convenient form for the coefficients of the matrix polynomial. The theory of biorthogonal systems is a useful framework to meet this challenge. Central factorial numbers play a key role in the theoretical development. Explicit polynomial coefficients for rotations expressed either as exponentials or as rational Cayley transforms are considered here. Structural features of the results are discussed and compared, and large j limits of the coefficients are examined
The Translated Dowling Polynomials and Numbers.
Mangontarum, Mahid M; Macodi-Ringia, Amila P; Abdulcarim, Normalah S
2014-01-01
More properties for the translated Whitney numbers of the second kind such as horizontal generating function, explicit formula, and exponential generating function are proposed. Using the translated Whitney numbers of the second kind, we will define the translated Dowling polynomials and numbers. Basic properties such as exponential generating functions and explicit formula for the translated Dowling polynomials and numbers are obtained. Convexity, integral representation, and other interesting identities are also investigated and presented. We show that the properties obtained are generalizations of some of the known results involving the classical Bell polynomials and numbers. Lastly, we established the Hankel transform of the translated Dowling numbers.
Small oscillations, Sturm sequences, and orthogonal polynomials
International Nuclear Information System (INIS)
Baake, M.
1986-07-01
The relation between small oscillations of one-dimensional mechanical systems and the theory of orthogonal polynomials is investigated. It is shown how the polynomials provide a natural tool to determine the eigenfrequencies and eigencoordinates completely, where the existence of a certain two-termed recurrence formula is essential. Physical and mathematical statements are formulated in terms of the recursion coefficients which can directly be obtained from the corresponding secular equation. Several known as well as new results on Sturm sequences and orthogonal polynomials are presented with respect to the treatment of small oscillations. (orig.)
Zhao, Chunyu; Burge, James H
2013-12-16
Zernike polynomials are an orthonormal set of scalar functions over a circular domain, and are commonly used to represent wavefront phase or surface irregularity. In optical testing, slope or curvature of a surface or wavefront is sometimes measured instead, from which the surface or wavefront map is obtained. Previously we derived an orthonormal set of vector polynomials that fit to slope measurement data and yield the surface or wavefront map represented by Zernike polynomials. Here we define a 3-element curvature vector used to represent the second derivatives of a continuous surface, and derive a set of orthonormal curvature basis functions that are written in terms of Zernike polynomials. We call the new curvature functions the C polynomials. Closed form relations for the complete basis set are provided, and we show how to determine Zernike surface coefficients from the curvature data as represented by the C polynomials.
Adapted polynomial chaos expansion for failure detection
International Nuclear Information System (INIS)
Paffrath, M.; Wever, U.
2007-01-01
In this paper, we consider two methods of computation of failure probabilities by adapted polynomial chaos expansions. The performance of the two methods is demonstrated by a predator-prey model and a chemical reaction problem
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...
Twisted Polynomials and Forgery Attacks on GCM
DEFF Research Database (Denmark)
Abdelraheem, Mohamed Ahmed A. M. A.; Beelen, Peter; Bogdanov, Andrey
2015-01-01
nonce misuse resistance, such as POET. The algebraic structure of polynomial hashing has given rise to security concerns: At CRYPTO 2008, Handschuh and Preneel describe key recovery attacks, and at FSE 2013, Procter and Cid provide a comprehensive framework for forgery attacks. Both approaches rely...... heavily on the ability to construct forgery polynomials having disjoint sets of roots, with many roots (“weak keys”) each. Constructing such polynomials beyond naïve approaches is crucial for these attacks, but still an open problem. In this paper, we comprehensively address this issue. We propose to use...... 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...
Some results for extended Jacobi polynomials
International Nuclear Information System (INIS)
Khan, I.A.
1991-06-01
In this paper, some contiguous function relations and generating functions have been established for extended Jacobi polynomials. The results obtained here, are of very general nature. (author). 5 refs
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.)
Some generating functions of Laguerre polynomials
Directory of Open Access Journals (Sweden)
Bidyut Kumar Guha Thakurta
1987-01-01
Full Text Available In this note a class of interesting generating relation, which is stated in the form of theorem, involving Laguerre polynomials is derived. Some applications of the theorem are also given here.
On linear equations with general polynomial solutions
Laradji, A.
2018-04-01
We provide necessary and sufficient conditions for which an nth-order linear differential equation has a general polynomial solution. We also give necessary conditions that can directly be ascertained from the coefficient functions of the equation.
Handbook on semidefinite, conic and polynomial optimization
Anjos, Miguel F
2012-01-01
This book offers the reader a snapshot of the state-of-the-art in the growing and mutually enriching areas of semidefinite optimization, conic optimization and polynomial optimization. It covers theory, algorithms, software and applications.
Solving Bivariate Polynomial Systems on a GPU
International Nuclear Information System (INIS)
Moreno Maza, Marc; Pan Wei
2012-01-01
We present a CUDA implementation of dense multivariate polynomial arithmetic based on Fast Fourier Transforms over finite fields. Our core routine computes on the device (GPU) the subresultant chain of two polynomials with respect to a given variable. This subresultant chain is encoded by values on a FFT grid and is manipulated from the host (CPU) in higher-level procedures. We have realized a bivariate polynomial system solver supported by our GPU code. Our experimental results (including detailed profiling information and benchmarks against a serial polynomial system solver implementing the same algorithm) demonstrate that our strategy is well suited for GPU implementation and provides large speedup factors with respect to pure CPU code.
Polynomial analysis of ambulatory blood pressure measurements
Zwinderman, A. H.; Cleophas, T. A.; Cleophas, T. J.; van der Wall, E. E.
2001-01-01
In normotensive subjects blood pressures follow a circadian rhythm. A circadian rhythm in hypertensive patients is less well established, and may be clinically important, particularly with rigorous treatments of daytime blood pressures. Polynomial analysis of ambulatory blood pressure monitoring
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
A Polynomial Estimate of Railway Line Delay
DEFF Research Database (Denmark)
Cerreto, Fabrizio; Harrod, Steven; Nielsen, Otto Anker
2017-01-01
by numerical analysis with simulation. This paper proposes an analytical estimate of aggregate delay with a polynomial form. The function returns the aggregate delay of a railway line resulting from an initial, primary, delay. Analysis of the function demonstrates that there should be a balance between the two...... remedial measures, supplement and buffer. Numerical analysis of a Copenhagen Sbane line shows that the polynomial function is valid even when theoretical assumptions are violated....
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.
Integral inequalities for self-reciprocal polynomials
Indian Academy of Sciences (India)
Integral inequalities for self-reciprocal polynomials. HORST ALZER. Morsbacher Str. 10, D-51545 Waldbröl, Germany. E-mail: H.Alzer@gmx.de. MS received 14 November 2007. Abstract. Let n ≥ 1 be an integer and let Pn be the class of polynomials P of degree at most n satisfying znP(1/z) = P (z) for all z ∈ C. Moreover, ...
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.
Local fibred right adjoints are polynomial
DEFF Research Database (Denmark)
Kock, Anders; Kock, Joachim
2013-01-01
For any locally cartesian closed category E, we prove that a local fibred right adjoint between slices of E is given by a polynomial. The slices in question are taken in a well known fibred sense......For any locally cartesian closed category E, we prove that a local fibred right adjoint between slices of E is given by a polynomial. The slices in question are taken in a well known fibred sense...
Fast polynomial multiplication on a GPU
International Nuclear Information System (INIS)
Maza, Marc Moreno; Pan Wei
2010-01-01
We present CUDA implementations of Fast Fourier Transforms over finite fields. This allows us to develop GPU support for dense univariate polynomial multiplication leading to speedup factors in the range 21 - 37 with respect to the best serial C-code available to us, for our largest input data sets. Since dense univariate polynomial multiplication is a core routine in symbolic computation, this is promising result for the integration of GPU support into computer algebra systems.
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)
Vector-Valued Jack Polynomials from Scratch
Directory of Open Access Journals (Sweden)
Jean-Gabriel Luque
2011-03-01
Full Text Available Vector-valued Jack polynomials associated to the symmetric group S_N are polynomials with multiplicities in an irreducible module of S_N and which are simultaneous eigenfunctions of the Cherednik-Dunkl operators with some additional properties concerning the leading monomial. These polynomials were introduced by Griffeth in the general setting of the complex reflections groups G(r,p,N and studied by one of the authors (C. Dunkl in the specialization r=p=1 (i.e. for the symmetric group. By adapting a construction due to Lascoux, we describe an algorithm allowing us to compute explicitly the Jack polynomials following a Yang-Baxter graph. We recover some properties already studied by C. Dunkl and restate them in terms of graphs together with additional new results. In particular, we investigate normalization, symmetrization and antisymmetrization, polynomials with minimal degree, restriction etc. We give also a shifted version of the construction and we discuss vanishing properties of the associated polynomials.
Strong asymptotics for extremal polynomials associated with weights on ℝ
Lubinsky, Doron S
1988-01-01
0. The results are consequences of a strengthened form of the following assertion: Given 0 1. Auxiliary results include inequalities for weighted polynomials, and zeros of extremal polynomials. The monograph is fairly self-contained, with proofs involving elementary complex analysis, and the theory of orthogonal and extremal polynomials. It should be of interest to research workers in approximation theory and orthogonal polynomials.
Fabrication of Polymer Optical Fibre (POF) Gratings.
Luo, Yanhua; Yan, Binbin; Zhang, Qijin; Peng, Gang-Ding; Wen, Jianxiang; Zhang, Jianzhong
2017-03-04
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 Connection Coefficients of the Chebyshev-Boubaker Polynomials
Directory of Open Access Journals (Sweden)
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.
On the connection coefficients of the Chebyshev-Boubaker polynomials.
Barry, Paul
2013-01-01
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.
Beta-integrals and finite orthogonal systems of Wilson polynomials
International Nuclear Information System (INIS)
Neretin, Yu A
2002-01-01
The integral is calculated and the system of orthogonal polynomials with weight equal to the corresponding integrand is constructed. This weight decreases polynomially, therefore only finitely many of its moments converge. As a result the system of orthogonal polynomials is finite. Systems of orthogonal polynomials related to 5 H 5 -Dougall's formula and the Askey integral is also constructed. All the three systems consist of Wilson polynomials outside the domain of positiveness of the usual weight
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.
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.
Fiber Grating Environmental Sensing System
Schulz, Whitten L.; Udd, Eric
2003-07-29
Fiber grating environmental measurement systems are comprised of sensors that are configured to respond to changes in moisture or chemical content of the surrounding medium through the action of coatings and plates inducing strain that is measured. These sensors can also be used to monitor the interior of bonds for degradation due to aging, cracking, or chemical attack. Means to multiplex these sensors at high speed and with high sensitivity can be accomplished by using spectral filters placed to correspond to each fiber grating environmental sensor. By forming networks of spectral elements and using wavelength division multiplexing arrays of fiber grating sensors may be processed in a single fiber line allowing distributed high sensitivity, high bandwidth fiber optic grating environmental sensor systems to be realized.
Diffraction-grating neutron interferometers
International Nuclear Information System (INIS)
Ioffe, A.I.
1988-01-01
Aberration distortions of wavefronts in a very cold neutron interferometer using diffraction gratings are analyzed. Aberrations that considerably reduce the efficiency of a two-grating interferometer are shown to be fully compensable by adding a third diffraction grating, which also permits the interferometer to operate with a non-collimated and non-monochromatized illuminating beam thereby raising its efficiency. A fourth diffraction grating additionally permits compensation of effects of the terrestrial rotation that affect performance of a large interferometer in which the spatial separation of beams can be of the order of a few meters. It is demonstrated to be practically possible to implement an interferometer for neutrons having a wavelength λ = 20 A and to use it in experiments aimed at finding the electric charge of the neutron at the level of 10 -23 to 10 -22 of the electronic charge. (orig.)
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.
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....
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.
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
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.].
Extending a Property of Cubic Polynomials to Higher-Degree Polynomials
Miller, David A.; Moseley, James
2012-01-01
In this paper, the authors examine a property that holds for all cubic polynomials given two zeros. This property is discovered after reviewing a variety of ways to determine the equation of a cubic polynomial given specific conditions through algebra and calculus. At the end of the article, they will connect the property to a very famous method…
Ethical concerns related to grateful patient philanthropy: the physician's perspective.
Wright, Scott M; Wolfe, Leah; Stewart, Rosalyn; Flynn, John A; Paisner, Richard; Rum, Steve; Parson, Gregory; Carrese, Joseph
2013-05-01
Philanthropic contributions to academic medical centers from grateful patients support research, patient care, education, and capital projects. The goal of this study was to identify the ethical concerns associated with philanthropic gifts from grateful patients. A qualitative study design was selected. Investigators conducted in-depth semi-structured interviews with 20 Department of Medicine physicians at Johns Hopkins who were identified by Development Office staff as experienced and successful in this realm-those having relationships with multiple patients who have made philanthropic contributions. Interview transcripts were independently coded by two investigators. Content analysis identified several themes related to ethical concerns. Eighteen informants (90 %) were Associate Professors or Professors; two (10 %) were females. Four thematic domains emerged related to ethical concerns associated with philanthropy from grateful patients: (i) impact of gift on the doctor-patient relationship; (ii) gift acquisition considered beyond the physician's professional role; (iii) justice and fairness; and (iv) vulnerability of patients. Despite acknowledging at least one of the aforementioned concerns, eleven physician informants (55 %) expressed the view that there were no ethical issues involved with grateful patient philanthropy. In this paper, we report that physicians involved in grateful patient philanthropy are aware of, and in some cases troubled by, the ethical concerns related to this activity. Further studies could examine how best to prepare faculty for the challenges that may accompany these gifts so as to help them maintain expected professional and ethical standards when accepting grateful patient philanthropy.
Time-dependent generalized polynomial chaos
International Nuclear Information System (INIS)
Gerritsma, Marc; Steen, Jan-Bart van der; Vos, Peter; Karniadakis, George
2010-01-01
Generalized polynomial chaos (gPC) has non-uniform convergence and tends to break down for long-time integration. The reason is that the probability density distribution (PDF) of the solution evolves as a function of time. The set of orthogonal polynomials associated with the initial distribution will therefore not be optimal at later times, thus causing the reduced efficiency of the method for long-time integration. Adaptation of the set of orthogonal polynomials with respect to the changing PDF removes the error with respect to long-time integration. In this method new stochastic variables and orthogonal polynomials are constructed as time progresses. In the new stochastic variable the solution can be represented exactly by linear functions. This allows the method to use only low order polynomial approximations with high accuracy. The method is illustrated with a simple decay model for which an analytic solution is available and subsequently applied to the three mode Kraichnan-Orszag problem with favorable results.
Chemical Reaction Networks for Computing Polynomials.
Salehi, Sayed Ahmad; Parhi, Keshab K; Riedel, Marc D
2017-01-20
Chemical reaction networks (CRNs) provide a fundamental model in the study of molecular systems. Widely used as formalism for the analysis of chemical and biochemical systems, CRNs have received renewed attention as a model for molecular computation. This paper demonstrates that, with a new encoding, CRNs can compute any set of polynomial functions subject only to the limitation that these functions must map the unit interval to itself. These polynomials can be expressed as linear combinations of Bernstein basis polynomials with positive coefficients less than or equal to 1. In the proposed encoding approach, each variable is represented using two molecular types: a type-0 and a type-1. The value is the ratio of the concentration of type-1 molecules to the sum of the concentrations of type-0 and type-1 molecules. The proposed encoding naturally exploits the expansion of a power-form polynomial into a Bernstein polynomial. Molecular encoders for converting any input in a standard representation to the fractional representation as well as decoders for converting the computed output from the fractional to a standard representation are presented. The method is illustrated first for generic CRNs; then chemical reactions designed for an example are mapped to DNA strand-displacement reactions.
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.
Quantum chaotic dynamics and random polynomials
International Nuclear Information System (INIS)
Bogomolny, E.; Bohigas, O.; Leboeuf, P.
1995-11-01
The distribution of roots of polynomials of high degree with random coefficients is investigated which, among others, appear naturally in the context of 'quantum chaotic dynamics'. It is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane. In order to further increase this tendency, the particular case of self-inverse random polynomials is studied, and it is shown that for them a finite portion of all roots lies exactly on the unit circle. Correlation functions of these roots are also computed analytically, and compared to the correlations of eigenvalues of random matrices. The problem of ergodicity of chaotic wavefunctions is also considered. Special attention is devoted to the role of symmetries in the distribution of roots of random polynomials. (author)
Markov and Bernstein type inequalities for polynomials
Directory of Open Access Journals (Sweden)
Mohapatra RN
1999-01-01
Full Text Available In an answer to a question raised by chemist Mendeleev, A. Markov proved that if is a real polynomial of degree , then The above inequality which is known as Markov's Inequality is best possible and becomes equality for the Chebyshev polynomial . Few years later, Serge Bernstein needed the analogue of this result for the unit disk in the complex plane instead of the interval and the following is known as Bernstein's Inequality. If is a polynomial of degree then This inequality is also best possible and is attained for , being a complex number. The above two inequalities have been the starting point of a considerable literature in Mathematics and in this article we discuss some of the research centered around these inequalities.
Dominating Sets and Domination Polynomials of Paths
Directory of Open Access Journals (Sweden)
Saeid Alikhani
2009-01-01
Full Text Available Let G=(V,E be a simple graph. A set S⊆V is a dominating set of G, if every vertex in V\\S is adjacent to at least one vertex in S. Let 𝒫ni be the family of all dominating sets of a path Pn with cardinality i, and let d(Pn,j=|𝒫nj|. In this paper, we construct 𝒫ni, and obtain a recursive formula for d(Pn,i. Using this recursive formula, we consider the polynomial D(Pn,x=∑i=⌈n/3⌉nd(Pn,ixi, which we call domination polynomial of paths and obtain some properties of this polynomial.
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.
Some Inequalities for the Derivative of Polynomials
Directory of Open Access Journals (Sweden)
Sunil Hans
2014-01-01
Full Text Available If pz=∑υ=0ncυzυ is a polynomial of degree n, having no zeros in z<1, then Aziz (1989 proved maxz=1p′z≤n/2Mα2+Mα+π21/2, where Mα=max1≤k≤npeiα+2kπ/n. In this paper, we consider a class of polynomial Pnμ of degree n, defined as pz=a0+∑υ=μnaυzυ and present certain generalizations of above inequality and some other well-known results.
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....
Some Relationships between the Analogs of Euler Numbers and Polynomials
Directory of Open Access Journals (Sweden)
Kim T
2007-01-01
Full Text Available We construct new twisted Euler polynomials and numbers. We also study the generating functions of the twisted Euler numbers and polynomials associated with their interpolation functions. Next we construct twisted Euler zeta function, twisted Hurwitz zeta function, twisted Dirichlet -Euler numbers and twisted Euler polynomials at non-positive integers, respectively. Furthermore, we find distribution relations of generalized twisted Euler numbers and polynomials. By numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the twisted -Euler polynomials. Finally, we give a table for the solutions of the twisted -Euler polynomials.
Some Relationships between the Analogs of Euler Numbers and Polynomials
Directory of Open Access Journals (Sweden)
C. S. Ryoo
2007-10-01
Full Text Available We construct new twisted Euler polynomials and numbers. We also study the generating functions of the twisted Euler numbers and polynomials associated with their interpolation functions. Next we construct twisted Euler zeta function, twisted Hurwitz zeta function, twisted Dirichlet l-Euler numbers and twisted Euler polynomials at non-positive integers, respectively. Furthermore, we find distribution relations of generalized twisted Euler numbers and polynomials. By numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the twisted q-Euler polynomials. Finally, we give a table for the solutions of the twisted q-Euler polynomials.
Equivalences of the multi-indexed orthogonal polynomials
International Nuclear Information System (INIS)
Odake, Satoru
2014-01-01
Multi-indexed orthogonal polynomials describe eigenfunctions of exactly solvable shape-invariant quantum mechanical systems in one dimension obtained by the method of virtual states deletion. Multi-indexed orthogonal polynomials are labeled by a set of degrees of polynomial parts of virtual state wavefunctions. For multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson, and Askey-Wilson types, two different index sets may give equivalent multi-indexed orthogonal polynomials. We clarify these equivalences. Multi-indexed orthogonal polynomials with both type I and II indices are proportional to those of type I indices only (or type II indices only) with shifted parameters
Location of zeros of Wiener and distance polynomials.
Dehmer, Matthias; Ilić, Aleksandar
2012-01-01
The geometry of polynomials explores geometrical relationships between the zeros and the coefficients of a polynomial. A classical problem in this theory is to locate the zeros of a given polynomial by determining disks in the complex plane in which all its zeros are situated. In this paper, we infer bounds for general polynomials and apply classical and new results to graph polynomials namely Wiener and distance polynomials whose zeros have not been yet investigated. Also, we examine the quality of such bounds by considering four graph classes and interpret the results.
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)
Orthogonal polynomials of compact simple Lie groups: branching rules for polynomials
International Nuclear Information System (INIS)
Nesterenko, M; Patera, J; Szajewska, M; Tereszkiewicz, A
2010-01-01
Polynomials in this paper are defined starting from a compact semisimple Lie group. A known classification of maximal, semisimple subgroups of simple Lie groups is used to select the cases to be considered here. A general method is presented and all the cases of rank ≤3 are explicitly studied. We derive the polynomials of simple Lie groups B 3 and C 3 as they are not available elsewhere. The results point to far reaching Lie theoretical connections to the theory of multivariable orthogonal polynomials.
Airy polynomials, three-variable Hermite polynomials and the paraxial wave equation
International Nuclear Information System (INIS)
Torre, A
2012-01-01
Within the context of an investigation of the potential of functions, which directly or indirectly relate to the Airy functions, to yield solutions of the 2D paraxial wave equation, we consider here the Airy polynomials. We see that appropriate Gaussian-modulated versions of such polynomials, which could in a sense parallel the Hermite–Gaussian wavefunctions of both elegant and standard type, yield, when assumed as initial functions, solutions of the 2D PWE shaping in the form of a complex Gaussian modulation of three-variable Hermite polynomials. A basic characterization of these wavefunctions is provided, having as a touchstone the Hermite–Gaussian wavefunctions. (paper)
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
Inequalities for a Polynomial and its Derivative
Indian Academy of Sciences (India)
Annual Meetings · Mid Year Meetings · Discussion Meetings · Public Lectures · Lecture Workshops · Refresher Courses · Symposia · Live Streaming. Home; Journals; Proceedings – Mathematical Sciences; Volume 110; Issue 2. Inequalities for a Polynomial and its Derivative. V K Jain. Volume 110 Issue 2 May 2000 pp 137- ...
Integral Inequalities for Self-Reciprocal Polynomials
Indian Academy of Sciences (India)
Annual Meetings · Mid Year Meetings · Discussion Meetings · Public Lectures · Lecture Workshops · Refresher Courses · Symposia · Live Streaming. Home; Journals; Proceedings – Mathematical Sciences; Volume 120; Issue 2. Integral Inequalities for Self-Reciprocal Polynomials. Horst Alzer. Volume 120 Issue 2 April 2010 ...
Integral inequalities for self-reciprocal polynomials
Indian Academy of Sciences (India)
Annual Meetings · Mid Year Meetings · Discussion Meetings · Public Lectures · Lecture Workshops · Refresher Courses · Symposia · Live Streaming. Home; Journals; Proceedings – Mathematical Sciences; Volume 120; Issue 2. Integral Inequalities for Self-Reciprocal Polynomials. Horst Alzer. Volume 120 Issue 2 April 2010 ...
Polynomial stabilization of some dissipative hyperbolic systems
Czech Academy of Sciences Publication Activity Database
Ammari, K.; Feireisl, Eduard; Nicaise, S.
2014-01-01
Roč. 34, č. 11 (2014), s. 4371-4388 ISSN 1078-0947 R&D Projects: GA ČR GA201/09/0917 Institutional support: RVO:67985840 Keywords : exponential stability * polynomial stability * observability inequality Subject RIV: BA - General Mathematics Impact factor: 0.826, year: 2014 http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=9924
Bernoulli Polynomials, Fourier Series and Zeta Numbers
DEFF Research Database (Denmark)
Scheufens, Ernst E
2013-01-01
Fourier series for Bernoulli polynomials are used to obtain information about values of the Riemann zeta function for integer arguments greater than one. If the argument is even we recover the well-known exact values, if the argument is odd we find integral representations and rapidly convergent...
Euler Polynomials, Fourier Series and Zeta Numbers
DEFF Research Database (Denmark)
Scheufens, Ernst E
2012-01-01
Fourier series for Euler polynomials is used to obtain information about values of the Riemann zeta function for integer arguments greater than one. If the argument is even we recover the well-known exact values, if the argument is odd we find integral representations and rapidly convergent series....
D-stability of polynomial matrices
Czech Academy of Sciences Publication Activity Database
Henrion, Didier; Bachelier, O.; Šebek, M.
2001-01-01
Roč. 74, č. 8 (2001), s. 845-856 ISSN 0020-7179 R&D Projects: GA AV ČR KSK1019101 Institutional research plan: AV0Z1075907 Keywords : polynomial methods * robust control * linear matrix inequalities Subject RIV: BC - Control Systems Theory Impact factor: 0.736, year: 2001
Odd Degree Polynomials on Real Banach Spaces
Czech Academy of Sciences Publication Activity Database
Aron, R. M.; Hájek, Petr Pavel
2007-01-01
Roč. 11, č. 1 (2007), s. 143-153 ISSN 1385-1292 R&D Projects: GA AV ČR IAA100190502; GA ČR GA201/04/0090 Institutional research plan: CEZ:AV0Z10190503 Keywords : odd degree polynomials * zero sets Subject RIV: BA - General Mathematics Impact factor: 0.356, year: 2007
Quantum Hilbert matrices and orthogonal polynomials
DEFF Research Database (Denmark)
Andersen, Jørgen Ellegaard; Berg, Christian
2009-01-01
Using the notion of quantum integers associated with a complex number q≠0 , we define the quantum Hilbert matrix and various extensions. They are Hankel matrices corresponding to certain little q -Jacobi polynomials when |q|... of reciprocal Fibonacci numbers called Filbert matrices. We find a formula for the entries of the inverse quantum Hilbert matrix....
Characteristic polynomials of linear polyacenes and their ...
Indian Academy of Sciences (India)
Administrator
Abstract. Coefficients of characteristic polynomials (CP) of linear polyacenes (LP) have been shown to be obtainable from Pascal's triangle by using a graph factorisation and squaring technique. Strong subspectrality existing among the members of the linear polyacene series has been shown from the derivation of the CP's.
Polynomial computation of Hankel singular values
Kwakernaak, H.
1992-01-01
A revised and improved version of a polynomial algorithm is presented. It was published by N.J. Young (1990) for the computation of the singular values and vectors of the Hankel operator defined by a linear time-invariant system with a rotational transfer matrix. Tentative numerical experiments
Characteristic polynomials of linear polyacenes and their ...
Indian Academy of Sciences (India)
Coefficients of characteristic polynomials (CP) of linear polyacenes (LP) have been shown to be obtainable from Pascal's triangle by using a graph factorisation and squaring technique. Strong subspectrality existing among the members of the linear polyacene series has been shown from the derivation of the CP's. Thus it ...
Coherent states for polynomial su(2) algebra
International Nuclear Information System (INIS)
Sadiq, Muhammad; Inomata, Akira
2007-01-01
A class of generalized coherent states is constructed for a polynomial su(2) algebra in a group-free manner. As a special case, the coherent states for the cubic su(2) algebra are discussed. The states so constructed reduce to the usual SU(2) coherent states in the linear limit
On zeros of certain Dirichlet polynomials
BLLACA, KAJTAZ H.
2015-01-01
In this article we establish the zero-free region of certain Dirichlet polynomials $L_{F, X}$ arising in approximate functional equation for functions in the Selberg class and we prove an asymptotic formula for the number of zeros of $L_{F, X}$.
Exceptional polynomials and SUSY quantum mechanics
Indian Academy of Sciences (India)
Exceptional polynomials and SUSY quantum mechanics. K V S SHIV CHAITANYA1,∗, S SREE RANJANI2,. PRASANTA K PANIGRAHI3, R RADHAKRISHNAN4 and V SRINIVASAN4. 1BITS Pilani, Hyderabad Campus, Jawahar Nagar, Shameerpet Mandal, Hyderabad 500 078, India. 2Faculty of Science and Technology, ...
Deformation Prediction Using Linear Polynomial Functions ...
African Journals Online (AJOL)
The predictions are compared with measured data reported in literature and the results are discussed. The computational aspects of implementation of the model are also discussed briefly. Keywords: Linear Polynomial, Structural Deformation, Prediction Journal of the Nigerian Association of Mathematical Physics, Volume ...
Subintegrality, invertible modules and Laurent polynomial extensions
Indian Academy of Sciences (India)
Indian Acad. Sci. (Math. Sci.) Vol. 125, No. 2, May 2015, pp. 149–160. c Indian Academy of Sciences. Subintegrality, invertible modules and Laurent polynomial extensions. VIVEK SADHU. Department of Mathematics ...... comments which have improved the exposition. Further, he would like to thank CSIR,. India for financial ...
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.
Error Minimization of Polynomial Approximation of Delta
Indian Academy of Sciences (India)
2016-01-27
Jan 27, 2016 ... The difference between Universal time (UT) and Dynamical time (TD), known as Delta ( ) is tabulated for the first day of each year in the Astronomical Almanac. During the last four centuries it is found that there are large differences between its values for two consecutive years. Polynomial ...
On Arithmetic-Geometric-Mean Polynomials
Griffiths, Martin; MacHale, Des
2017-01-01
We study here an aspect of an infinite set "P" of multivariate polynomials, the elements of which are associated with the arithmetic-geometric-mean inequality. In particular, we show in this article that there exist infinite subsets of probability "P" for which every element may be expressed as a finite sum of squares of real…
On the zeros of the classical polynomials
International Nuclear Information System (INIS)
Calogero, F.
1977-01-01
Some theorems and conjectures concerning the zeros of the classical polynomials (of Hermite, Laguerre and Jacobi) are announced. The theorems, whose proof is not reported, have been obtained from the relationship between the motion of the zeros of special solutions of simple linear partial differential equations (in x and t), and one-dimensional many-body problems. (author)
Hermite polynomials and quasi-classical asymptotics
Czech Academy of Sciences Publication Activity Database
Ali, S. T.; Engliš, Miroslav
2014-01-01
Roč. 55, č. 4 (2014), 042102 ISSN 0022-2488 R&D Projects: GA ČR GA201/09/0473 Institutional support: RVO:67985840 Keywords : quantization * Hermite polynomials * reproducing kernel Subject RIV: BA - General Mathematics Impact factor: 1.243, year: 2014 http://scitation.aip.org/content/aip/journal/jmp/55/4/10.1063/1.4869325
Polynomial Asymptotes of the Second Kind
Dobbs, David E.
2011-01-01
This note uses the analytic notion of asymptotic functions to study when a function is asymptotic to a polynomial function. Along with associated existence and uniqueness results, this kind of asymptotic behaviour is related to the type of asymptote that was recently defined in a more geometric way. Applications are given to rational functions and…
The 6 Vertex Model and Schubert Polynomials
Directory of Open Access Journals (Sweden)
Alain Lascoux
2007-02-01
Full Text Available We enumerate staircases with fixed left and right columns. These objects correspond to ice-configurations, or alternating sign matrices, with fixed top and bottom parts. The resulting partition functions are equal, up to a normalization factor, to some Schubert polynomials.
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....
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.
Indecomposability of polynomials via Jacobian matrix
International Nuclear Information System (INIS)
Cheze, G.; Najib, S.
2007-12-01
Uni-multivariate decomposition of polynomials is a special case of absolute factorization. Recently, thanks to the Ruppert's matrix some effective results about absolute factorization have been improved. Here we show that with a jacobian matrix we can get sharper bounds for the special case of uni-multivariate decomposition. (author)
transformation of independent variables in polynomial regression ...
African Journals Online (AJOL)
Ada
preferable when possible to work with a simple functional form in transformed variables rather than with a more complicated form in the original variables. In this paper, it is shown that linear transformations applied to independent variables in polynomial regression models affect the t ratio and hence the statistical ...
Neutron interferometers with diffraction gratings
International Nuclear Information System (INIS)
Ioffe, A.I.
1983-01-01
A neutron interferometer is described in which the amplitude coherent division of the wave fronts is realized by means of neutron diffraction gratings. Photolithographic gratings on glass with a rectangular surface relief profile with a 58 Ni sprayed layer 2000 A thick are used as gratings. In contrast to perfect-crystal neutron interferometers the designed interferometer is capable of operating in the longwave neutron spectrum region. Variation of the value of spatial division of the interfering beams (up to 50 cm) and rather a high efficiency of the amergent beam together with the elemination of neutron beam passage through the interferometer coherent divosor material in such an interferometer permit to use it for solving problems of the solid-state physics and nuclear physics, for example, foA searching for the Yang Mills long-range field
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...... wavelength. It is shown that it is possible to tune and modulate a DFB fiber laser with both strain from a piezoelectric transducer and by temperature through resistive heating of a methal film. Both a chemical deposited silver layer and an electron-beam evaporation technique has been investigated, to find...
Engineered plasmon focusing on functional gratings
Offerhaus, Herman L.; van den Bergen, B; van Hulst, N.F.
2005-01-01
We report on the engineering of plasmon propagation and focusing by dedicated curved gratings and noncollinear phasematching. Gratings were created on gold by focused ion beam milling and plasmons were measured using phase sensitive PSTM.
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....
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.
On the Symmetric Properties for the Generalized Twisted Bernoulli Polynomials
Directory of Open Access Journals (Sweden)
Kim Taekyun
2009-01-01
Full Text Available We study the symmetry for the generalized twisted Bernoulli polynomials and numbers. We give some interesting identities of the power sums and the generalized twisted Bernoulli polynomials using the symmetric properties for the -adic invariant integral.
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.
On a Family of Multivariate Modified Humbert Polynomials
Aktaş, Rabia; Erkuş-Duman, Esra
2013-01-01
This paper attempts to present a multivariable extension of generalized Humbert polynomials. The results obtained here include various families of multilinear and multilateral generating functions, miscellaneous properties, and also some special cases for these multivariable polynomials. PMID:23935411
Irreducibility results for compositions of polynomials in several ...
Indian Academy of Sciences (India)
We obtain explicit upper bounds for the number of irreducible factors for a class of compositions of polynomials in several variables over a given field. In particular, some irreducibility criteria are given for this class of compositions of polynomials.
Recurrence relations for the Cartesian derivatives of the Zernike polynomials.
Stephenson, Philip C L
2014-04-01
A recurrence relation for the first-order Cartesian derivatives of the Zernike polynomials is derived. This relation is used with the Clenshaw method to determine an efficient method for calculating the derivatives of any linear series of Zernike polynomials.
Fabrication of Polymer Optical Fibre (POF Gratings
Directory of Open Access Journals (Sweden)
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.
Khan, Waseem A; Haroon, Hiba
2016-01-01
In 2008, Liu and Wang established various symmetric identities for Bernoulli, Euler and Genocchi polynomials. In this paper, we extend these identities in a unified and generalized form to families of Hermite-Bernoulli, Euler and Genocchi polynomials. The procedure followed is that of generating functions. Some relevant connections of the general theory developed here with the results obtained earlier by Pathan and Khan are also pointed out.
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
Weighted inequalities for generalized polynomials with doubling weights.
Joung, Haewon
2017-01-01
Many weighted polynomial inequalities, such as the Bernstein, Marcinkiewicz, Schur, Remez, Nikolskii inequalities, with doubling weights were proved by Mastroianni and Totik for the case [Formula: see text], and by Tamás Erdélyi for [Formula: see text]. In this paper we extend such polynomial inequalities to those for generalized trigonometric polynomials. We also prove the large sieve for generalized trigonometric polynomials with doubling weights.
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 ...
Holographic Gratings for Slow-Neutron Optics
Klepp, Juergen; Pruner, Christian; Tomita, Yasuo; Geltenbort, Peter; Drevenšek-Olenik, Irena; Gyergyek, Saso; Kohlbrecher, Joachim; Fally, Martin
2012-01-01
Recent progress in the development of holographic gratings for neutron-optics applications is reviewed. We summarize the properties of gratings recorded in deuterated (poly)methylmethacrylate, holographic polymer-dispersed liquid crystals and nanoparticle-polymer composites revealed by diffraction experiments with slow neutrons. Existing and anticipated neutron-optical instrumentations based on holographic gratings are discussed.
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…
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
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
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.
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.
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.
The quality of zero bounds for complex polynomials.
Dehmer, Matthias; Tsoy, Yury Robertovich
2012-01-01
In this paper, we evaluate the quality of zero bounds on the moduli of univariate complex polynomials. We select classical and recently developed bounds and evaluate their quality by using several sets of complex polynomials. As the quality of priori bounds has not been investigated thoroughly, our results can be useful to find optimal bounds to locate the zeros of complex polynomials.
Lowering and raising operators for some special orthogonal polynomials
Koornwinder, T.H.
2006-01-01
Abstract: This paper discusses operators lowering or raising the degree but preserving the parameters of special orthogonal polynomials. Results for one-variable classical (q-)orthogonal polynomials are surveyed. For Jacobi polynomials associated with root system BC2 a new pair of lowering and
Multiple Representations and the Understanding of Taylor Polynomials
Habre, Samer
2009-01-01
The study of Maclaurin and Taylor polynomials entails the comprehension of various new mathematical ideas. Those polynomials are initially discussed at the college level in a calculus class and then again in a course on numerical methods. This article investigates the understanding of these polynomials by students taking a numerical methods class…
The Gibbs Phenomenon for Series of Orthogonal Polynomials
Fay, T. H.; Kloppers, P. Hendrik
2006-01-01
This note considers the four classes of orthogonal polynomials--Chebyshev, Hermite, Laguerre, Legendre--and investigates the Gibbs phenomenon at a jump discontinuity for the corresponding orthogonal polynomial series expansions. The perhaps unexpected thing is that the Gibbs constant that arises for each class of polynomials appears to be the same…
Root and critical point behaviors of certain sums of polynomials
Indian Academy of Sciences (India)
Seon-Hong Kim
2018-04-24
Apr 24, 2018 ... Introduction. There is an extensive literature concerning roots of sums of polynomials. Many authors. [5–7] 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?' ...
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 ...
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.
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
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.
Modelling Trends in Ordered Correspondence Analysis Using Orthogonal Polynomials.
Lombardo, Rosaria; Beh, Eric J; Kroonenberg, Pieter M
2016-06-01
The core of the paper consists of the treatment of two special decompositions for correspondence analysis of two-way ordered contingency tables: the bivariate moment decomposition and the hybrid decomposition, both using orthogonal polynomials rather than the commonly used singular vectors. To this end, we will detail and explain the basic characteristics of a particular set of orthogonal polynomials, called Emerson polynomials. It is shown that such polynomials, when used as bases for the row and/or column spaces, can enhance the interpretations via linear, quadratic and higher-order moments of the ordered categories. To aid such interpretations, we propose a new type of graphical display-the polynomial biplot.
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...... 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...
Planar Harmonic and Monogenic Polynomials of Type A
Directory of Open Access Journals (Sweden)
Charles F. Dunkl
2016-10-01
Full Text Available Harmonic polynomials of type A are polynomials annihilated by the Dunkl Laplacian associated to the symmetric group acting as a reflection group on R N . The Dunkl operators are denoted by T j for 1 ≤ j ≤ N , and the Laplacian Δ κ = ∑ j = 1 N T j 2 . This paper finds the homogeneous harmonic polynomials annihilated by all T j for j > 2 . The structure constants with respect to the Gaussian and sphere inner products are computed. These harmonic polynomials are used to produce monogenic polynomials, those annihilated by a Dirac-type operator.
Zernike polynomials for photometric characterization of LEDs
International Nuclear Information System (INIS)
Velázquez, J L; Ferrero, A; Pons, A; Campos, J; Hernanz, M L
2016-01-01
We propose a method based on Zernike polynomials to characterize photometric quantities and descriptors of light emitting diodes (LEDs) from measurements of the angular distribution of the luminous intensity, such as total luminous flux, BA, inhomogeneity, anisotropy, direction of the optical axis and Lambertianity of the source. The performance of this method was experimentally tested for 18 high-power LEDs from different manufacturers and with different photometric characteristics. A small set of Zernike coefficients can be used to calculate all the mentioned photometric quantities and descriptors. For applications not requiring a great accuracy such as those of lighting design, the angular distribution of the luminous intensity of most of the studied LEDs can be interpolated with only two Zernike polynomials. (paper)
Orthogonal polynomials in neutron transport theory
Energy Technology Data Exchange (ETDEWEB)
Dehesa, J.S. (Granada Univ. (Spain). Facultad de Ciencias)
1982-01-01
The asymptotic average properties of zeros of the polynomials gsub(k)sup(m) (x), which play a fundamental role in neutron transport and radiative transfer theories, are investigated analytically in terms of the angular expansion coefficients wsub(k) of the scattering kernel for three wide classes of scattering models. In particular it is found that the scattering models of Eccleston-McCormick (J. Nucl. Energy.; 24:23 (1970)), Shultis et al (Nucl. Sci. Eng.; 59:53 (1976)) and Henyey-Greenstein (Astrophys. J.; 93:70 (1941)) belong in one of the above-mentioned classes, and their associated polynomials gsub(k)sup(m) (x) have the same asymptotic density of zeros.
Polynomial chaos representation of databases on manifolds
Energy Technology Data Exchange (ETDEWEB)
Soize, C., E-mail: christian.soize@univ-paris-est.fr [Université Paris-Est, Laboratoire Modélisation et Simulation Multi-Echelle, MSME UMR 8208 CNRS, 5 bd Descartes, 77454 Marne-La-Vallée, Cedex 2 (France); Ghanem, R., E-mail: ghanem@usc.edu [University of Southern California, 210 KAP Hall, Los Angeles, CA 90089 (United States)
2017-04-15
Characterizing the polynomial chaos expansion (PCE) of a vector-valued random variable with probability distribution concentrated on a manifold is a relevant problem in data-driven settings. The probability distribution of such random vectors is multimodal in general, leading to potentially very slow convergence of the PCE. In this paper, we build on a recent development for estimating and sampling from probabilities concentrated on a diffusion manifold. The proposed methodology constructs a PCE of the random vector together with an associated generator that samples from the target probability distribution which is estimated from data concentrated in the neighborhood of the manifold. The method is robust and remains efficient for high dimension and large datasets. The resulting polynomial chaos construction on manifolds permits the adaptation of many uncertainty quantification and statistical tools to emerging questions motivated by data-driven queries.
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.
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.
A Deterministic and Polynomial Modified Perceptron Algorithm
Directory of Open Access Journals (Sweden)
Olof Barr
2006-01-01
Full Text Available We construct a modified perceptron algorithm that is deterministic, polynomial and also as fast as previous known algorithms. The algorithm runs in time O(mn3lognlog(1/ρ, where m is the number of examples, n the number of dimensions and ρ is approximately the size of the margin. We also construct a non-deterministic modified perceptron algorithm running in timeO(mn2lognlog(1/ρ.
Polynomials and identities on real Banach spaces
Czech Academy of Sciences Publication Activity Database
Hájek, Petr Pavel; Kraus, M.
2012-01-01
Roč. 385, č. 2 (2012), s. 1015-1026 ISSN 0022-247X R&D Projects: GA ČR(CZ) GAP201/11/0345 Institutional research plan: CEZ:AV0Z10190503 Keywords : Polynomials on Banach spaces Subject RIV: BA - General Mathematics Impact factor: 1.050, year: 2012 http://www.sciencedirect.com/science/article/pii/S0022247X11006743
Piecewise polynomial solutions to linear inverse problems
DEFF Research Database (Denmark)
Hansen, Per Christian; Mosegaard, K.
1996-01-01
We have presented a new algorithm PP-TSVD that computes piecewise polynomial solutions to ill-posed problems, without a priori knowledge about the positions of the break points. In particular, we can compute piecewise constant functions that describe layered models. Such solutions are useful, e.g.......g., in seismological problems, and the algorithm can also be used as a preprocessor for other methods where break points/discontinuities must be incorporated explicitly....
Polynomial Variables and the Jacobian Problem
Indian Academy of Sciences (India)
Most readers would be familiar with the technique of shifting the origin to simplify the equation -of a locus. This process is the same as choosing new coordinate axes, ..... 1982 by H Bass, E H Connell and D Wright. Points at Infinity. Let f be a nonzero polynomial in X, Y, and let d = deg (f). Then f can be written in the form f =.
Moments, positive polynomials and their applications
Lasserre, Jean Bernard
2009-01-01
Many important applications in global optimization, algebra, probability and statistics, applied mathematics, control theory, financial mathematics, inverse problems, etc. can be modeled as a particular instance of the Generalized Moment Problem (GMP) . This book introduces a new general methodology to solve the GMP when its data are polynomials and basic semi-algebraic sets. This methodology combines semidefinite programming with recent results from real algebraic geometry to provide a hierarchy of semidefinite relaxations converging to the desired optimal value. Applied on appropriate cones,
Completeness of the ring of polynomials
DEFF Research Database (Denmark)
Thorup, Anders
2015-01-01
Consider the polynomial ring R:=k[X1,…,Xn]R:=k[X1,…,Xn] in n≥2n≥2 variables over an uncountable field k. We prove that R is complete in its adic topology, that is, the translation invariant topology in which the non-zero ideals form a fundamental system of neighborhoods of 0. In addition we prove...
Zero sets of polynomials in several variables
Czech Academy of Sciences Publication Activity Database
Aron, R. M.; Hájek, Petr Pavel
2006-01-01
Roč. 86, č. 6 (2006), s. 561-568 ISSN 0003-889X R&D Projects: GA ČR(CZ) GA201/01/1198; GA ČR(CZ) GA201/04/0090; GA AV ČR(CZ) IAA1019205 Institutional research plan: CEZ:AV0Z10190503 Keywords : polynomial * zero set Subject RIV: BA - General Mathematics Impact factor: 0.341, year: 2006
Trigonometric Polynomials For Estimation Of Spectra
Greenhall, Charles A.
1990-01-01
Orthogonal sets of trigonometric polynomials used as suboptimal substitutes for discrete prolate-spheroidal "windows" of Thomson method of estimation of spectra. As used here, "windows" denotes weighting functions used in sampling time series to obtain their power spectra within specified frequency bands. Simplified windows designed to require less computation than do discrete prolate-spheroidal windows, albeit at price of some loss of accuracy.
Detecting Prime Numbers via Roots of Polynomials
Dobbs, David E.
2012-01-01
It is proved that an integer n [greater than or equal] 2 is a prime (resp., composite) number if and only if there exists exactly one (resp., more than one) nth-degree monic polynomial f with coefficients in Z[subscript n], the ring of integers modulo n, such that each element of Z[subscript n] is a root of f. This classroom note could find use in…
Multivariate Krawtchouk Polynomials and Composition Birth and Death Processes
Directory of Open Access Journals (Sweden)
Robert Griffiths
2016-05-01
Full Text Available This paper defines the multivariate Krawtchouk polynomials, orthogonal on the multinomial distribution, and summarizes their properties as a review. The multivariate Krawtchouk polynomials are symmetric functions of orthogonal sets of functions defined on each of N multinomial trials. The dual multivariate Krawtchouk polynomials, which also have a polynomial structure, are seen to occur naturally as spectral orthogonal polynomials in a Karlin and McGregor spectral representation of transition functions in a composition birth and death process. In this Markov composition process in continuous time, there are N independent and identically distributed birth and death processes each with support 0 , 1 , … . The state space in the composition process is the number of processes in the different states 0 , 1 , … . Dealing with the spectral representation requires new extensions of the multivariate Krawtchouk polynomials to orthogonal polynomials on a multinomial distribution with a countable infinity of states.
Doubling (Dual) Hahn Polynomials: Classification and Applications
Oste, Roy; Van der Jeugt, Joris
2016-01-01
We classify all pairs of recurrence relations in which two Hahn or dual Hahn polynomials with different parameters appear. Such couples are referred to as (dual) Hahn doubles. The idea and interest comes from an example appearing in a finite oscillator model [Jafarov E.I., Stoilova N.I., Van der Jeugt J., J. Phys. A: Math. Theor. 44 (2011), 265203, 15 pages, arXiv:1101.5310]. Our classification shows there exist three dual Hahn doubles and four Hahn doubles. The same technique is then applied to Racah polynomials, yielding also four doubles. Each dual Hahn (Hahn, Racah) double gives rise to an explicit new set of symmetric orthogonal polynomials related to the Christoffel and Geronimus transformations. For each case, we also have an interesting class of two-diagonal matrices with closed form expressions for the eigenvalues. This extends the class of Sylvester-Kac matrices by remarkable new test matrices. We examine also the algebraic relations underlying the dual Hahn doubles, and discuss their usefulness for the construction of new finite oscillator models.
Soft x-ray blazed transmission grating spectrometer with high resolving power and extended bandpass
Heilmann, Ralf K.; Bruccoleri, Alexander Robert; Schattenburg, Mark
2016-04-01
A number of high priority questions in astrophysics can be addressed by a state-of-the-art soft x-ray grating spectrometer, such as the role of Active Galactic Nuclei in galaxy and star formation, characterization of the Warm-Hot Intergalactic Medium and the “missing baryon” problem, characterization of halos around the Milky Way and nearby galaxies, as well as stellar coronae and surrounding winds and disks. An Explorer-scale, large-area (> 1,000 cm2), high resolving power (R = λ/Δλ > 3,000) soft x-ray grating spectrometer is highly feasible based on Critical-Angle Transmission (CAT) grating technology. Still significantly higher performance can be provided by a CAT grating spectrometer on an X-ray-Surveyor-type mission. CAT gratings combine the advantages of blazed reflection gratings (high efficiency, use of higher diffraction orders) with those of conventional transmission gratings (low mass, relaxed alignment tolerances and temperature requirements, transparent at higher energies) with minimal mission resource requirements. They are high-efficiency blazed transmission gratings that consist of freestanding, ultra-high aspect-ratio grating bars fabricated from silicon-on-insulator (SOI) wafers using advanced anisotropic dry and wet etch techniques. Blazing is achieved through grazing-incidence reflection off the smooth grating bar sidewalls. The reflection properties of silicon are well matched to the soft x-ray band. Nevertheless, CAT gratings with sidewalls made of higher atomic number elements allow extension of the CAT grating principle to higher energies and larger dispersion angles. We show x-ray data from metal-coated CAT gratings and demonstrate efficient blazing to higher energies and larger blaze angles than possible with silicon alone. We also report on measurements of the resolving power of a breadboard CAT grating spectrometer consisting of a Wolter-I slumped-glass focusing mirror pair from Goddard Space Flight Center and CAT gratings, to be
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.
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 gr...
Improved light harvest in diffraction grating-embedded TiO2 nanoparticle film
Lee, Jeeyoung; Lee, Myeongkyu
2017-12-01
We show that a high-efficiency diffraction grating can be embedded into nanoparticulate TiO2 film via imprinting combined with TiCl4 treatment. The grating-embedded film consists of two layers in intimate contact. A thin TiO2 layer was first patterned on a glass substrate by imprinting. The patterned layer was TiCl4-treated with a higher concentration than the over-coated thicker layer, so that it diffracts incident light as a refractive-index grating. Gratings with a period scaled down to 1 µm could be embedded into the film. Diffraction efficiency increased with an increasing grating height and an efficiency over 80% was achieved in the near-ultraviolet and visible range. Dye-sensitized solar cells fabricated using a grating-embedded TiO2 photoanode exhibited much better photovoltaic performance than those without a grating. It was also found that the incorporation of a diffraction grating greatly enhances the photocatalytic activity of nanoparticulate TiO2 film. All these are attributed to improved light harvest.
From $r$-Linearized Polynomial Equations to $r^m$-Linearized Polynomial Equations
Fernando, Neranga; Hou, Xiang-dong
2015-01-01
Let $r$ be a prime power and $q=r^m$. For $0\\le i\\le m-1$, let $f_i\\in \\mathbb{F}_r[x]$ be $q$-linearized and $a_i\\in \\mathbb{F}_q$. Assume that $z\\in \\mathbb{\\bar{F}}_r$ satisfies the equation $\\sum_{i=0}^{m-1}a_if_i(z)^{r^i}=0$, where $\\sum_{i=0}^{m-1}a_if_i^{r_i}\\in \\mathbb{F}_q[x]$ is an $r$-linearized polynomial. It is shown that $z$ satisfies a $q$-linearized polynomial equation with coefficients in $\\mathbb{F}_r$. This result provides an explanation for numerous permutation polynomials...
Heilmann, Ralf K.; Bruccoleri, Alexander R.; Kolodziejczak, Jeffery; Gaskin, Jessica A.; O'Dell, Stephen L.; Bhatia, Ritwik; Schattenburg, Mark L.
2016-07-01
A number of high priority subjects in astrophysics can be addressed by a state-of-the-art soft x-ray grating spectrometer, such as the role of Active Galactic Nuclei in galaxy and star formation, characterization of the Warm-Hot Intergalactic Medium and the missing baryon problem, characterization of halos around the Milky Way and nearby galaxies, as well as stellar coronae and surrounding winds and disks. An Explorer-scale, largearea (> 1,000 cm2), high resolving power (R =λ/Δλ > 3,000) soft x-ray grating spectrometer is highly feasible based on Critical-Angle Transmission (CAT) grating technology, even for telescopes with angular resolution of 5-10 arcsec. Still, significantly higher performance can be provided by a CAT grating spectrometer on an X-ray- Surveyor-type mission. CAT gratings combine the advantages of blazed reflection gratings (high efficiency, use of higher diffraction orders) with those of conventional transmission gratings (lowmass, relaxed alignment tolerances and temperature requirements, transparent at higher energies) with minimalmission resource requirements. They are high-efficiency blazed transmission gratings that consist of freestanding, ultra-high aspect-ratio grating bars fabricated from silicon-on-insulator (SOI) wafers using advanced anisotropic dry and wet etch techniques. Blazing is achieved through grazing-incidence reflection off the smooth grating bar sidewalls. The reflection properties of silicon are well matched to the soft x-ray band, and existing silicon CAT gratings can exceed 30% absolute diffraction efficiency, with clear paths for further improvement. Nevertheless, CAT gratings with sidewalls made of higher atomic number elements allow extension of the CAT grating principle to higher energies and larger dispersion angles, thus enabling higher resolving power at shorter wavelengths. We show x-ray data from CAT gratings coated with a thin layer of platinum using atomic layer deposition, and demonstrate efficient
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.
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.
Exponential operators, generalized polynomials and evolution problems
International Nuclear Information System (INIS)
Dattoli, G.; Mancho, A.M.; Quattromini, M.; Torre, A.
2001-01-01
The operator (d/dx) χ d/dx plays a central role in the theory of operational calculus. Its exponential form is crucial in problems relevant to solutions of Fokker-Planck and Schroedinger equations. We explore the formal properties of the evolution operators associated to (d/dx) χ d/dx, discuss its link to special forms of Laguerre polynomials and Laguerre-based functions. The obtained results are finally applied to specific problems concerning the solution of Fokker-Planck equations relevant to the beam lifetime in storage rings
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
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.
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.
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
Explicitly integrable polynomial Hamiltonians and evaluation of Lie transformations
International Nuclear Information System (INIS)
Shi, J.; Yan, Y.T.
1993-01-01
We have found that any homogeneous polynomial can be written as a sum of integrable polynomials of the same degree, with which each associated polynomial Hamiltonian is integrable, and the associated Lie transformation can be evaluated exactly. An integrable polynomial factorization has thus been developed to convert a sympletic map in the form of a Dragt-Finn factorization into a product of exactly evaluable Lie transformations associated with integrable polynomials. Having a small number of factorization bases of integrable polynomials enables one to consider a factorization with the use of high-order symplectic integrators so that a symplectic map can always be evaluated with the desired accuracy. The results are significant for studying the long-term stability of beams in accelerators
Comparative assessment of freeform polynomials as optical surface descriptions.
Kaya, Ilhan; Thompson, Kevin P; Rolland, Jannick P
2012-09-24
Slow-servo single-point diamond turning as well as advances in computer controlled small lap polishing enables the fabrication of freeform optics, or more specifically, optical surfaces for imaging applications that are not rotationally symmetric. Various forms of polynomials for describing freeform optical surfaces exist in optical design and to support fabrication. A popular method is to add orthogonal polynomials onto a conic section. In this paper, recently introduced gradient-orthogonal polynomials are investigated in a comparative manner with the widely known Zernike polynomials. In order to achieve numerical robustness when higher-order polynomials are required to describe freeform surfaces, recurrence relations are a key enabler. Results in this paper establish the equivalence of both polynomial sets in accurately describing freeform surfaces under stringent conditions. Quantifying the accuracy of these two freeform surface descriptions is a critical step in the future application of these tools in both advanced optical system design and optical fabrication.
Polynomials for torus links from Chern-Simons gauge theories
International Nuclear Information System (INIS)
Isidro, J.M.; Labastida, J.M.F.; Ramallo, A.V.
1993-01-01
Invariant polynomials for torus links are obtained in the framework of the Chern-Simons topological gauge theory. The polynomials are computed as vacuum expectation values on the three-sphere of Wilson line operators representing the Verlinde algebra of the corresponding rational conformal field theory. In the case of the SU(2) gauge theory our results provide explicit expressions for the Jones polynomial as well as for the polynomials associated to the N-state (N>2) vertex models (Akutsu-Wadati polynomials). By means of the Chern-Simons coset construction, the minimal unitary models are analyzed, showing that the corresponding link invariants factorize into two SU(2) polynomials. A method to obtain skein rules from the Chern-Simons knot operators is developed. This procedure yields the eigenvalues of the braiding matrix of the corresponding conformal field theory. (orig.)
Polarization Measurements on SUMI's TVLS Gratings
Kobayashi, K.; West, E. A.; Davis, J. M.; Gary, G. A.
2007-01-01
We present measurements of toroidal variable-line-space (TVLS) gratings for the Solar Ultraviolet Magnetograph Investigation (SUMI), currently being developed at the National Space Science and Technology Center (NSSTC). SUMI is a spectro-polarimeter designed to measure magnetic fields in the solar chromosphere by observing two UV emission lines sensitive to magnetic fields, the CIY line at 155nm and the MgII line at 280nm. The instrument uses a pair of TVLS gratings, to observe both linear polarizations simultaneously. Efficiency measurements were done on bare aluminum gratings and aluminum/MgF2 coated gratings, at both linear polarizations.
Polarization Measurements on SUMI's TVLS Gratings
Kobayashi, K.; West, E. A.; Davis, J. M.; Gary, G. A.
2007-01-01
We present measurements of toroidal variable-line-space (TVLS) gratings for the Solar Ultraviolet Magnetograph Investigation (SUMI), currently being developed an the National Space Science and Technology Center (NSSTC). SUMI zs a spectro-polarimeter designed no measure magnetic fields in the solar chromosphere by observing two UV emission lines sensitive to magnetic fields, the C-IV line at 155nm and the Mg-II line at 280nm. The instrument uses a pair of TVLS gratings, to observe both linear polarizations simultaneously. Efficiency measurements were done on bare aluminum gratings and MgF2 coated gratings, at both linear polarizations.
Astronomical large Ge immersion grating by Canon
Sukegawa, Takashi; Suzuki, Takeshi; Kitamura, Tsuyoshi
2016-07-01
Immersion grating is a powerful optical device for thee infrared high-resolution spectroscope. Germanium (GGe) is the best material for a mid-infrared immersion grating because of Ge has very large reflective index (n=4.0). On the other hands, there is no practical Ge immersion grating under 5umm use. It was very difficult for a fragile IR crystal to manufacture a diffraction grating precisely. Our original free-forming machine has accuracy of a few nano-meter in positioning and stability. We already fabricated the large CdZnTe immersion grating. (Sukegawa et al. (2012), Ikeda et al. (2015)) Wee are developing Ge immersion grating that can be a good solution for high-resolution infrared spectroscopy with the large ground-based/space telescopes. We succeeded practical Ge immersion grating with the grooved area off 75mm (ruled direction) x 119mm (grove width) and the blaze angle of 75 degrees. Our astronomical large Ge immersion grating has the grooved area of 155mm (ruled direction) x 41mmm (groove width) and groove pitch off 91.74um. We also report optical performance of astronomical large Ge immersion grating with a metal coating on the diffraction surface.
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
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.
Recurrence relations and weight functions for approximation by orthonormal polynomials
International Nuclear Information System (INIS)
Bisoi, A.K.
1989-12-01
The popular three-term recurrence relation is found to be a numerically unstable algorithm for generation of higher order orthogonal polynomials. For stability, a k-term recurrence relation for generating the k th order polynomial is suggested. Instead of assigning equal weight to data points with unspecified errors, the use of the Jacobian weight function is suggested for generating the orthogonal polynomials to effect a closer representation. (author). 6 refs, 1 tab
Vector polynomials for direct analysis of circular wavefront slope data.
Mahajan, Virendra N; Acosta, Eva
2017-10-01
In the aberration analysis of a circular wavefront, Zernike circle polynomials are used to obtain its wave aberration coefficients. To obtain these coefficients from the wavefront slope data, we need vector functions that are orthogonal to the gradients of the Zernike polynomials, and are irrotational so as to propagate minimum uncorrelated random noise from the data to the coefficients. In this paper, we derive such vector functions, which happen to be polynomials.
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 ...
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.
An Analytic Formula for the A_2 Jack Polynomials
Directory of Open Access Journals (Sweden)
Vladimir V. Mangazeev
2007-01-01
Full Text Available In this letter I shall review my joint results with Vadim Kuznetsov and Evgeny Sklyanin [Indag. Math. 14 (2003, 451-482] on separation of variables (SoV for the $A_n$ Jack polynomials. This approach originated from the work [RIMS Kokyuroku 919 (1995, 27-34] where the integral representations for the $A_2$ Jack polynomials was derived. Using special polynomial bases I shall obtain a more explicit expression for the $A_2$ Jack polynomials in terms of generalised hypergeometric functions.
Tabulating knot polynomials for arborescent knots
International Nuclear Information System (INIS)
Mironov, A; Morozov, A; Morozov, A; Sleptsov, A; Ramadevi, P; Singh, Vivek Kumar
2017-01-01
Arborescent knots are those which can be represented in terms of double fat graphs or equivalently as tree Feynman diagrams. This is the class of knots for which the present knowledge is sufficient for lifting topological description to the level of effective analytical formulas. The paper describes the origin and structure of the new tables of colored knot polynomials, which will be posted at the dedicated site (http://knotebook.org). Even if formal expressions are known in terms of modular transformation matrices, the computation in finite time requires additional ideas. We use the ‘family’ approach, suggested in Mironov and Morozov (2015 Nucl. Phys . B 899 395–413), and apply it to arborescent knots in the Rolfsen table by developing a Feynman diagram technique, associated with an auxiliary matrix model field theory. Gauge invariance in this theory helps to provide meaning to Racah matrices in the case of non-trivial multiplicities and explains the need for peculiar sign prescriptions in the calculation of [21]-colored HOMFLY-PT polynomials. (paper)
Positive trigonometric polynomials and signal processing applications
Dumitrescu, Bogdan
2017-01-01
This revised edition is made up of two parts: theory and applications. Though many of the fundamental results are still valid and used, new and revised material is woven throughout the text. As with the original book, the theory of sum-of-squares trigonometric polynomials is presented unitarily based on the concept of Gram matrix (extended to Gram pair or Gram set). The programming environment has also evolved, and the books examples are changed accordingly. The applications section is organized as a collection of related problems that use systematically the theoretical results. All the problems are brought to a semi-definite programming form, ready to be solved with algorithms freely available, like those from the libraries SeDuMi, CVX and Pos3Poly. A new chapter discusses applications in super-resolution theory, where Bounded Real Lemma for trigonometric polynomials is an important tool. This revision is written to be more appealing and easier to use for new readers. < Features updated information on LMI...
Tabulating knot polynomials for arborescent knots
Mironov, A.; Morozov, A.; Morozov, A.; Ramadevi, P.; Singh, Vivek Kumar; Sleptsov, A.
2017-02-01
Arborescent knots are those which can be represented in terms of double fat graphs or equivalently as tree Feynman diagrams. This is the class of knots for which the present knowledge is sufficient for lifting topological description to the level of effective analytical formulas. The paper describes the origin and structure of the new tables of colored knot polynomials, which will be posted at the dedicated site (http://knotebook.org). Even if formal expressions are known in terms of modular transformation matrices, the computation in finite time requires additional ideas. We use the ‘family’ approach, suggested in Mironov and Morozov (2015 Nucl. Phys. B 899 395-413), and apply it to arborescent knots in the Rolfsen table by developing a Feynman diagram technique, associated with an auxiliary matrix model field theory. Gauge invariance in this theory helps to provide meaning to Racah matrices in the case of non-trivial multiplicities and explains the need for peculiar sign prescriptions in the calculation of [21]-colored HOMFLY-PT polynomials.
Simulation of aspheric tolerance with polynomial fitting
Li, Jing; Cen, Zhaofeng; Li, Xiaotong
2018-01-01
The shape of the aspheric lens changes caused by machining errors, resulting in a change in the optical transfer function, which affects the image quality. At present, there is no universally recognized tolerance criterion standard for aspheric surface. To study the influence of aspheric tolerances on the optical transfer function, the tolerances of polynomial fitting are allocated on the aspheric surface, and the imaging simulation is carried out by optical imaging software. Analysis is based on a set of aspheric imaging system. The error is generated in the range of a certain PV value, and expressed as a form of Zernike polynomial, which is added to the aspheric surface as a tolerance term. Through optical software analysis, the MTF of optical system can be obtained and used as the main evaluation index. Evaluate whether the effect of the added error on the MTF of the system meets the requirements of the current PV value. Change the PV value and repeat the operation until the acceptable maximum allowable PV value is obtained. According to the actual processing technology, consider the error of various shapes, such as M type, W type, random type error. The new method will provide a certain development for the actual free surface processing technology the reference value.
Ye, Jingfei; Gao, Zhishan; Wang, Shuai; Cheng, Jinlong; Wang, Wei; Sun, Wenqing
2014-10-01
Four orthogonal polynomials for reconstructing a wavefront over a square aperture based on the modal method are currently available, namely, the 2D Chebyshev polynomials, 2D Legendre polynomials, Zernike square polynomials and Numerical polynomials. They are all orthogonal over the full unit square domain. 2D Chebyshev polynomials are defined by the product of Chebyshev polynomials in x and y variables, as are 2D Legendre polynomials. Zernike square polynomials are derived by the Gram-Schmidt orthogonalization process, where the integration region across the full unit square is circumscribed outside the unit circle. Numerical polynomials are obtained by numerical calculation. The presented study is to compare these four orthogonal polynomials by theoretical analysis and numerical experiments from the aspects of reconstruction accuracy, remaining errors, and robustness. Results show that the Numerical orthogonal polynomial is superior to the other three polynomials because of its high accuracy and robustness even in the case of a wavefront with incomplete data.
Fiber Bragg Grating Based Thermometry.
Ahmed, Zeeshan; Filla, James; Guthrie, William; Quintavalle, John
In recent years there has been considerable interest in developing photonic temperature sensors such as the Fiber Bragg gratings (FBG) as an alternative to resistance thermometry. In this study we examine the thermal response of FBGs over the temperature range of 233 K to 393 K. We demonstrate, in hermetically sealed dry Argon environment, FBG devices show a quadratic dependence on temperature with expanded uncertainties (k=2) of ≈500 mK. Our measurements indicate that the combined measurement uncertainty is dominated by uncertainty in determining peak center fitting and thermal ageing of polyimide coated fibers.
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
Measurement of grating visibility of a fiber Bragg grating based on bent-spectral analysis.
Gunawardena, Dinusha S; Lai, Man-Hong; Lim, Kok-Sing; Ali, Muhammad M; Ahmad, Harith
2015-02-10
In this study, a technique for measuring the grating visibility of the fiber Bragg grating (FBG) based on bent-spectral analysis is proposed. From varying ac and dc coupling coefficients at different bending radii, the grating visibility is estimated with the aid of a simple mathematical model. The investigation begins with the estimation of the grating visibility from the transmission spectra of the FBG during the inscription process. After that, the FBGs are subjected to a bending test with reducing radii, and again the transmission spectra are recorded. It is shown that the estimated grating visibility is in agreement with the result determined from the earlier inscription process.
Overview on grating developments at ESA
Guldimann, B.; Deep, A.; Vink, R.; Harnisch, B.; Kraft, S.; Sierk, B.; Bazalgette, G.; Bézy, J.-L.
2017-11-01
In the frame of recent studies and missions, ESA has been performing various pre-developments of optical gratings for instruments operating at wavelengths from the UV up to the SWIR. The instrument requirements of Sentinel-4, Sentinel-5, CarbonSat and FLEX are driving the need for advanced designs and technologies leading to gratings with high efficiency, high spectral resolution, low stray light and low polarization sensitivities. Typical ESA instruments (e.g. Sciamachy, GOME, MERIS, OLCI, NIRSpec) were and are based on ruled gratings or gratings manufactured with one holographic photoresist mask layer which is transferred to an optical substrate (e.g. glass, glass ceramic) with dry etching methods and subsequently either coated with a reflective coating or used as a mold for replication. These manufacturing methods lead to blazed grating profiles with a metallic reflective surface. The vast majority of spectrometers on ground are still based on such gratings. In general, gratings based on grooved metallic surfaces tend for instance to polarize the incoming light significantly and are therefore not always suitable for ESA's needs of today. Gratings made for space therefore evolved to many other designs and concepts which will be reported in this paper.
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%.
New generation DWDM fibre grating devices
Zervas, M.N.
2000-01-01
Using a recently developed inverse scattering layer-peeling algorithm and a modified stroboscopic grating writing technique, we have designed and successfully demonstrated novel grating devices, such as 50GHz-bandwidth dispersion compensators and square dispersionless filters, suitable for future high performance DWDM optical systems.
High order Bragg grating microfluidic dye laser
DEFF Research Database (Denmark)
Balslev, Søren; Kristensen, Anders
2004-01-01
We demonstrate a single mode distributed feedback liquid dye laser, based on a short 133 'rd order Bragg grating defined in a single polymer layer between two glass substrates.......We demonstrate a single mode distributed feedback liquid dye laser, based on a short 133 'rd order Bragg grating defined in a single polymer layer between two glass substrates....
Straw combustion on slow-moving grates
DEFF Research Database (Denmark)
Kær, Søren Knudsen
2005-01-01
Combustion of straw in grate-based boilers is often associated with high emission levels and relatively poor fuel burnout. A numerical grate combustion model was developed to assist in improving the combustion performance of these boilers. The model is based on a one-dimensional ‘‘walking...
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 Continued Fraction Expansion of Real Roots of Polynomial Systems
DEFF Research Database (Denmark)
Mantzaflaris, Angelos; Mourrain, Bernard; Tsigaridas, Elias
2011-01-01
We elaborate on a correspondence between the coefficients of a multivariate polynomial represented in the Bernstein basis and in a tensor-monomial basis, which leads to homography representations of polynomial functions that use only integer arithmetic (in contrast to the Bernstein basis) and are...
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.
Optimal Approximation of Biquartic Polynomials by Bicubic Splines
Directory of Open Access Journals (Sweden)
Kačala Viliam
2018-01-01
The goal of this paper is to resolve this problem. Unlike the spline curves, in the case of spline surfaces it is insufficient to suppose that the grid should be uniform and the spline derivatives computed from a biquartic polynomial. We show that the biquartic polynomial coefficients have to satisfy some additional constraints to achieve optimal approximation by bicubic splines.
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 top...
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.
Orthonormal polynomials for elliptical wavefronts with an arbitrary orientation.
Díaz, José A; Navarro, Rafael
2014-04-01
We generalize the analytical form of the orthonormal elliptical polynomials for any arbitrary aspect ratio to arbitrary orientation and give expression for them up to the 4th order. The utility of the polynomials is demonstrated by obtaining the expansion up to the 8th order in two examples of an off-axis wavefront exiting from an optical system with a vignetted pupil.
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…
On peculiar properties of generating functions of some orthogonal polynomials
International Nuclear Information System (INIS)
Szabłowski, Paweł J
2012-01-01
We prove that for |x| ⩽ |t| ≥( t i )/(q) i h n+i ( x|q) =h n (x|t,q) Σ i≥0 (t i )/(q) i h i (x|q), where h n (x|q) and h n (x|t, q) are respectively the so-called q-Hermite and the big q-Hermite polynomials, and (q) n denotes the so-called q-Pochhammer symbol. We prove similar equalities involving big q-Hermite and Al-Salam–Chihara polynomials, and Al-Salam–Chihara and the so-called continuous dual q-Hahn polynomials. Moreover, we are able to relate in this way some other ‘ordinary’ orthogonal polynomials such as, e.g., Hermite, Chebyshev or Laguerre. These equalities give a new interpretation of the polynomials involved and moreover can give rise to a simple method of generating more and more general (i.e. involving more and more parameters) families of orthogonal polynomials. We pose some conjectures concerning Askey–Wilson polynomials and their possible generalizations. We prove that these conjectures are true for the cases q = 1 (classical case) and q = 0 (free case), thus paving the way to generalization of Askey–Wilson polynomials at least in these two cases. (paper)
On an inequality concerning the polar derivative of a polynomial
Indian Academy of Sciences (India)
We also extend Zygmund's inequality to the polar derivative of a polynomial. Keywords. Zygmund's inequality; polar derivative; Lp-norm inequalities. 1. Introduction and statement of results. Let P (z) be a polynomial of degreen and let P (z) be its derivative. Then according to the famous result known as Bernstein's inequality ...
Numerical Performances of Two Orthogonal Polynomials in the Tau ...
African Journals Online (AJOL)
In this work, efforts are made at comparing the numerical effectiveness of the two most accurate orthogonal polynomials; Chebyshev and Legendre polynomials. Although the two have different weight functions, but they most time give close results especially when they are considered in the same interval. This work has ...
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 ...
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.
Zeros and uniqueness of Q-difference polynomials of meromorphic ...
Indian Academy of Sciences (India)
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 [ f n ( z ) f ( q z + c ) ] ( k ) and [ f n ( z ) ( f ( q z + c ) − f ( z ) ) ] ( k ) , where () is a transcendental function of zero order.
Conditional Density Approximations with Mixtures of Polynomials
DEFF Research Database (Denmark)
Varando, Gherardo; López-Cruz, Pedro L.; Nielsen, Thomas Dyhre
2015-01-01
Mixtures of polynomials (MoPs) are a non-parametric density estimation technique especially designed for hybrid Bayesian networks with continuous and discrete variables. Algorithms to learn one- and multi-dimensional (marginal) MoPs from data have recently been proposed. In this paper we introduce...... is found. We illustrate and study the methods using data sampled from known parametric distributions, and we demonstrate their applicability by learning models based on real neuroscience data. Finally, we compare the performance of the proposed methods with an approach for learning mixtures of truncated...... basis functions (MoTBFs). The empirical results show that the proposed methods generally yield models that are comparable to or significantly better than those found using the MoTBF-based method....
Kinetic term anarchy for polynomial chaotic inflation
Nakayama, Kazunori; Takahashi, Fuminobu; Yanagida, Tsutomu T.
2014-09-01
We argue that there may arise a relatively flat inflaton potential over super-Planckian field values with an approximate shift symmetry, if the coefficients of the kinetic terms for many singlet scalars are subject to a certain random distribution. The inflation takes place along the flat direction with a super-Planckian length, whereas the other light directions can be stabilized by the Hubble-induced mass. The inflaton potential generically contains various shift-symmetry breaking terms, leading to a possibly large deviation of the predicted values of the spectral index and tensor-to-scalar ratio from those of the simple quadratic chaotic inflation. We revisit a polynomial chaotic inflation in supergravity as such.
Dynamic normal forms and dynamic characteristic polynomial
DEFF Research Database (Denmark)
Frandsen, Gudmund Skovbjerg; Sankowski, Piotr
2011-01-01
We present the first fully dynamic algorithm for computing the characteristic polynomial of a matrix. In the generic symmetric case, our algorithm supports rank-one updates in O(n2logn) randomized time and queries in constant time, whereas in the general case the algorithm works in O(n2klogn......) randomized time, where k is the number of invariant factors of the matrix. The algorithm is based on the first dynamic algorithm for computing normal forms of a matrix such as the Frobenius normal form or the tridiagonal symmetric form. The algorithm can be extended to solve the matrix eigenproblem...... with relative error 2−b in additional O(nlog2nlogb) time. Furthermore, it can be used to dynamically maintain the singular value decomposition (SVD) of a generic matrix. Together with the algorithm, the hardness of the problem is studied. For the symmetric case, we present an Ω(n2) lower bound for rank...
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.
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.
Structured matrix based methods for approximate polynomial GCD
Boito, Paola
2011-01-01
Defining and computing a greatest common divisor of two polynomials with inexact coefficients is a classical problem in symbolic-numeric computation. The first part of this book reviews the main results that have been proposed so far in the literature. As usual with polynomial computations, the polynomial GCD problem can be expressed in matrix form: the second part of the book focuses on this point of view and analyses the structure of the relevant matrices, such as Toeplitz, Toepliz-block and displacement structures. New algorithms for the computation of approximate polynomial GCD are presented, along with extensive numerical tests. The use of matrix structure allows, in particular, to lower the asymptotic computational cost from cubic to quadratic order with respect to polynomial degree. .
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)
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.
Jack polynomial fractional quantum Hall states and their generalizations
International Nuclear Information System (INIS)
Baratta, Wendy; Forrester, Peter J.
2011-01-01
In the study of fractional quantum Hall states, a certain clustering condition involving up to four integers has been identified. We give a simple proof that particular Jack polynomials with α=-(r-1)/(k+1), (r-1) and (k+1) relatively prime, and with partition given in terms of its frequencies by [n 0 0 (r-1)s k0 r-1 k0 r-1 k...0 r-1 m] satisfy this clustering condition. Our proof makes essential use of the fact that these Jack polynomials are translationally invariant. We also consider nonsymmetric Jack polynomials, symmetric and nonsymmetric generalized Hermite and Laguerre polynomials, and Macdonald polynomials from the viewpoint of the clustering.
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.
Mobile device camera design with Q-type polynomials to achieve higher production yield.
Ma, Bin; Sharma, Katelynn; Thompson, Kevin P; Rolland, Jannick P
2013-07-29
The camera lenses that are built into the current generation of mobile devices are extremely stressed by the excessively tight packaging requirements, particularly the length. As a result, the aspheric departures and slopes on the lens surfaces, when designed with conventional power series based aspheres, are well beyond those encountered in most optical systems. When the as-manufactured performance is considered, the excessive aspheric slopes result in unusually high sensitivity to tilt and decenter and even despace resulting in unusually low manufacturing yield. Q(bfs) polynomials, a new formulation for nonspherical optical surfaces introduced by Forbes, not only build on orthogonal polynomials, but their unique normalization provides direct access to the RMS slope of the aspheric departure during optimization. Using surface shapes with this description in optimization results in equivalent performance with reduced alignment sensitivity and higher yield. As an additional approach to increasing yield, mechanically imposed external pivot points, introduced by Bottema, can be used as a design technique to further reduce alignment sensitivity and increase yield. In this paper, the Q-type polynomials and external pivot points were applied to a mobile device camera lens designed using an active RMS slope constraint that was then compared to a design developed using conventional power series surface descriptions. Results show that slope constrained Q-type polynomial description together with external pivot points lead directly to solutions with significantly higher manufacturing yield.
Yamaguchi, Kenzo; Fujii, Masamitsu
2015-12-01
The enhanced optical properties of metallic subwavelength gratings with very narrow slits have recently been extensively studied in the field of plasmonics. The optical transmission and reflection of such nanostructures, which act as nano-electro-mechanical systems (NEMS) actuators, can be electrically controlled by varying their geometrical parameters, giving them great flexibility for numerous applications in photonics, opto-electronics, and sensing. The previous challenges in controlling the optical properties were overcome by forming a metallic subwavelength grating with an NEMS actuator in mid-air, allowing the grating to be physically moved with the bias voltage. The device can shift the plasmon resonance wavelength with an electrical signal. The resonance wavelength for Wood's anomaly at the infrared region is predicted through simulations to shift by approximately 150 nm. We discuss the effect of polarization on the optical properties and grating mechanism. The reported effect may be used to achieve active spectral tuning and switching in a wide range of applications.
A neural network model for the self-organization of cortical grating cells.
Bauer, C; Burger, T; Stetter, M; Lang, E W
2000-01-01
A neural network model with incremental Hebbian learning of afferent and lateral synaptic couplings is proposed,which simulates the activity-dependent self-organization of grating cells in upper layers of striate cortex. These cells, found in areas V1 and V2 of the visual cortex of monkeys, respond vigorously and exclusively to bar gratings of a preferred orientation and periodicity. Response behavior to varying contrast and to an increasing number of bars in the grating show threshold and saturation effects. Their location with respect to the underlying orientation map and their nonlinear response behavior are investigated. The number of emerging grating cells is controlled in the model by the range and strength of the lateral coupling structure.
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....
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.}
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.
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.
Speed enhancement in VCSELs employing grating mirrors
DEFF Research Database (Denmark)
Chung, Il-Sug; Mørk, Jesper
2013-01-01
In recent years, various approaches to improve the speed of directly modulated vertical-cavity surface-emitting lasers (VCSELs) have been reported and demonstrated good improvement. In this paper, we propose and numerically investigate a new possibility of using high-index-contrast grating (HCG......) as mirror for VCSELs. By changing the grating design, one can control the reflection delay of the grating mirror, enabling the control of cavity photon lifetime. On the other hand, short energy penetration depth of the HCG results in smaller modal volume, compared to DBR VCSELs. An example structure shows...... that the HCG VCSEL has a 30-% higher 3-dB bandwidth than the DBR VCSEL....
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.
Production of diffraction gratings using holographic interferometry
International Nuclear Information System (INIS)
Ecevit, F.N.; Guven, H.; Aydin, R.
1989-09-01
Holographic transmission gratings are produced using low power He-Ne laser and the 488-nm Ar-ion laser line. From the observed data of the Hg spectrum and the 488.0-nm, 514.5-nm and 632.8-nm laser lines the fringe spacings of the gratings are calculated. Using the gratings produced with the He-Ne laser the Rydberg constant is determined by measuring the diffraction angles of the Balmer series in the H-atomic spectrum. (author). 12 refs, 4 figs, 1 tab
Heilmann, Ralf K.; Bruccoleri, Alexander; Schattenburg, Mark; Kolodziejczak, jeffery; Gaskin, Jessica; O'Dell, Stephen L.
2017-01-01
A number of high priority subjects in astrophysics are addressed by a state-of-the-art soft x-ray grating spectrometer, e.g. the role of Active Galactic Nuclei in galaxy and star formation, characterization of the WHIM and the “missing baryon” problem, characterization of halos around the Milky Way and nearby galaxies, and stellar coronae and surrounding winds and disks. An Explorer-scale, large-area (A > 1,000 cm2), high resolving power (R > 3,000) soft x-ray grating spectrometer is highly feasible based on Critical-Angle Transmission (CAT) grating technology, even for telescopes with angular resolution of 5-10 arcsec. Significantly higher performance could be provided by a CAT grating spectrometer on an X-ray-Surveyor-type mission (A > 4,000 cm2, R > 5,000). CAT gratings combine advantages of blazed reflection gratings (high efficiency, use of higher orders) with those of transmission gratings (low mass, relaxed alignment tolerances and temperature requirements, transparent at higher energies) with minimal mission resource requirements. Blazing is achieved through grazing-incidence reflection off the smooth silicon grating bar sidewalls. Silicon is well matched to the soft x-ray band, and 30% absolute diffraction efficiency has been acheived with clear paths for further improvement. CAT gratings with sidewalls made of high-Z elements allow extension of blazing to higher energies and larger dispersion angles, enabling higher resolving power at shorter wavelengths. X-ray data from CAT gratings coated with a thin layer of platinum using atomic layer deposition demonstrate efficient blazing to higher energies and much larger blaze angles than possible with silicon alone. Measurements of the resolving power of a breadboard CAT grating spectrometer consisting of a Wolter-I slumped-glass focusing optic from GSFC and CAT gratings, taken at the MSFC Stray Light Facility, have demonstrated resolving power > 10,000. Thus currently fabricated CAT gratings are compatible
vs. a polynomial chaos-based MCMC
Siripatana, Adil
2014-08-01
Bayesian Inference of Manning\\'s n coefficient in a Storm Surge Model Framework: comparison between Kalman lter and polynomial based method Adil Siripatana Conventional coastal ocean models solve the shallow water equations, which describe the conservation of mass and momentum when the horizontal length scale is much greater than the vertical length scale. In this case vertical pressure gradients in the momentum equations are nearly hydrostatic. The outputs of coastal ocean models are thus sensitive to the bottom stress terms de ned through the formulation of Manning\\'s n coefficients. This thesis considers the Bayesian inference problem of the Manning\\'s n coefficient in the context of storm surge based on the coastal ocean ADCIRC model. In the first part of the thesis, we apply an ensemble-based Kalman filter, the singular evolutive interpolated Kalman (SEIK) filter to estimate both a constant Manning\\'s n coefficient and a 2-D parameterized Manning\\'s coefficient on one ideal and one of more realistic domain using observation system simulation experiments (OSSEs). We study the sensitivity of the system to the ensemble size. we also access the benefits from using an in ation factor on the filter performance. To study the limitation of the Guassian restricted assumption on the SEIK lter, 5 we also implemented in the second part of this thesis a Markov Chain Monte Carlo (MCMC) method based on a Generalized Polynomial chaos (gPc) approach for the estimation of the 1-D and 2-D Mannning\\'s n coe cient. The gPc is used to build a surrogate model that imitate the ADCIRC model in order to make the computational cost of implementing the MCMC with the ADCIRC model reasonable. We evaluate the performance of the MCMC-gPc approach and study its robustness to di erent OSSEs scenario. we also compare its estimates with those resulting from SEIK in term of parameter estimates and full distributions. we present a full analysis of the solution of these two methods, of the
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.
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.
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
Coherent states and Hermite polynomials on quaternionic Hilbert spaces
Energy Technology Data Exchange (ETDEWEB)
Thirulogasanthar, K; Krzyzak, A [Department of Computer Science and Software Engineering, Concordia University, 1455 de Maisonneuve Blvd. W, Montreal, Quebec H3G 1M8 (Canada); Honnouvo, G, E-mail: santhar@cs.concordia.c, E-mail: krzyzak@cs.concordia.c, E-mail: g-honnouvo@yahoo.f [Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, Quebec H3A 2K6 (Canada)
2010-09-24
Coherent states, similar to the canonical coherent states of the harmonic oscillator, labeled by quaternions are established on the right and left quaternionic Hilbert spaces. On the left quaternionic Hilbert space reproducing kernels are established. As was claimed by Adler and Millard (J. Math. Phys. 1997 38 2117-26) it is proved that these coherent states cannot be realized as a group action on a quaternionic Hilbert space. As an extension of the complex Hermite polynomials, quaternionic Hermite polynomials are defined and these polynomials are associated with an operator as eigenfunctions.
Local polynomial Whittle estimation covering non-stationary fractional processes
DEFF Research Database (Denmark)
Nielsen, Frank
This paper extends the local polynomial Whittle estimator of Andrews & Sun (2004) to fractionally integrated processes covering stationary and non-stationary regions. We utilize the notion of the extended discrete Fourier transform and periodogram to extend the local polynomial Whittle estimator...... to the non-stationary region. By approximating the short-run component of the spectrum by a polynomial, instead of a constant, in a shrinking neighborhood of zero we alleviate some of the bias that the classical local Whittle estimators is prone to. This bias reduction comes at a cost as the variance is in...
Chromatic Polynomials Of Some (m,l-Hyperwheels
Directory of Open Access Journals (Sweden)
Julian A. Allagan
2014-03-01
Full Text Available In this paper, using a standard method of computing the chromatic polynomial of hypergraphs, we introduce a new reduction theorem which allows us to find explicit formulae for the chromatic polynomials of some (complete non-uniform $(m,l-$hyperwheels and non-uniform $(m,l-$hyperfans. These hypergraphs, constructed through a ``join" graph operation, are some generalizations of the well-known wheel and fan graphs, respectively. Further, we revisit some results concerning these graphs and present their chromatic polynomials in a standard form that involves the Stirling numbers of the second kind.
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.
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...
Asymptotic distribution of zeros of polynomials satisfying difference equations
Krasovsky, I. V.
2003-01-01
We propose a way to find the asymptotic distribution of zeros of orthogonal polynomials pn(x) satisfying a difference equation of the formB(x)pn(x+[delta])-C(x,n)pn(x)+D(x)pn(x-[delta])=0.We calculate the asymptotic distribution of zeros and asymptotics of extreme zeros of the Meixner and Meixner-Pollaczek polynomials. The distribution of zeros of Meixner polynomials shows some delicate features. We indicate the relation of our technique to the approach based on the Nevai-Dehesa-Ullman distribution.
Xu, Xiangru; Huang, Wei; Xu, Mingfei
2015-10-19
Optical lithography has approached a regime of high numerical aperture and wide field, where the impact of polarization aberration on imaging quality turns to be serious. Most of the existing studies focused on the distribution rule of polarization aberration on the pupil, and little attention had been paid to the field. In this paper, a new orthonormal set of polynomials is established to describe the polarization aberration of rotationally symmetric optical systems. The polynomials can simultaneously reveal the distribution rules of polarization aberration on the exit pupil and the field. Two examples are given to verify the polynomials.
Robust stability of fractional order polynomials with complicated uncertainty structure.
Matušů, Radek; Şenol, Bilal; Pekař, Libor
2017-01-01
The main aim of this article is to present a graphical approach to robust stability analysis for families of fractional order (quasi-)polynomials with complicated uncertainty structure. More specifically, the work emphasizes the multilinear, polynomial and general structures of uncertainty and, moreover, the retarded quasi-polynomials with parametric uncertainty are studied. Since the families with these complex uncertainty structures suffer from the lack of analytical tools, their robust stability is investigated by numerical calculation and depiction of the value sets and subsequent application of the zero exclusion condition.
On an Ordering-Dependent Generalization of the Tutte Polynomial
Geloun, Joseph Ben; Caravelli, Francesco
2017-09-01
A generalization of the Tutte polynomial involved in the evaluation of the moments of the integrated geometric Brownian in the Itô formalism is discussed. The new combinatorial invariant depends on the order in which the sequence of contraction-deletions have been performed on the graph. Thus, this work provides a motivation for studying an order-dependent Tutte polynomial in the context of stochastic differential equations. We show that in the limit of the control parameters encoding the ordering going to zero, the multivariate Tutte-Fortuin-Kasteleyn polynomial is recovered.
Polynomial Chaos Surrogates for Bayesian Inference
Le Maitre, Olivier
2016-01-06
The Bayesian inference is a popular probabilistic method to solve inverse problems, such as the identification of field parameter in a PDE model. The inference rely on the Bayes rule to update the prior density of the sought field, from observations, and derive its posterior distribution. In most cases the posterior distribution has no explicit form and has to be sampled, for instance using a Markov-Chain Monte Carlo method. In practice the prior field parameter is decomposed and truncated (e.g. by means of Karhunen- Lo´eve decomposition) to recast the inference problem into the inference of a finite number of coordinates. Although proved effective in many situations, the Bayesian inference as sketched above faces several difficulties requiring improvements. First, sampling the posterior can be a extremely costly task as it requires multiple resolutions of the PDE model for different values of the field parameter. Second, when the observations are not very much informative, the inferred parameter field can highly depends on its prior which can be somehow arbitrary. These issues have motivated the introduction of reduced modeling or surrogates for the (approximate) determination of the parametrized PDE solution and hyperparameters in the description of the prior field. Our contribution focuses on recent developments in these two directions: the acceleration of the posterior sampling by means of Polynomial Chaos expansions and the efficient treatment of parametrized covariance functions for the prior field. We also discuss the possibility of making such approach adaptive to further improve its efficiency.
Dynamics of polynomial Chaplygin gas warm inflation
Energy Technology Data Exchange (ETDEWEB)
Jawad, Abdul [COMSATS Institute of Information Technology, Department of Mathematics, Lahore (Pakistan); Chaudhary, Shahid [Sharif College of Engineering and Technology, Department of Mathematics, Lahore (Pakistan); Videla, Nelson [Pontificia Universidad Catolica de Valparaiso, Instituto de Fisica, Valparaiso (Chile)
2017-11-15
In the present work, we study the consequences of a recently proposed polynomial inflationary potential in the context of the generalized, modified, and generalized cosmic Chaplygin gas models. In addition, we consider dissipative effects by coupling the inflation field to radiation, i.e., the inflationary dynamics is studied in the warm inflation scenario. We take into account a general parametrization of the dissipative coefficient Γ for describing the decay of the inflaton field into radiation. By studying the background and perturbative dynamics in the weak and strong dissipative regimes of warm inflation separately for the positive and negative quadratic and quartic potentials, we obtain expressions for the most relevant inflationary observables as the scalar power spectrum, the scalar spectral, and the tensor-to-scalar ratio. We construct the trajectories in the n{sub s}-r plane for several expressions of the dissipative coefficient and compare with the two-dimensional marginalized contours for (n{sub s}, r) from the latest Planck data. We find that our results are in agreement with WMAP9 and Planck 2015 data. (orig.)
Liquid crystal on subwavelength metal gratings
Energy Technology Data Exchange (ETDEWEB)
Palto, S. P.; Barnik, M. I.; Artemov, V. V.; Shtykov, N. M.; Geivandov, A. R.; Yudin, S. G.; Gorkunov, M. V. [Shubnikov Institute of Crystallography of Russian Academy of Sciences, Leninsky pr. 59, 119333 Moscow (Russian Federation)
2015-06-14
Optical and electrooptical properties of a system consisting of subwavelength metal gratings and nematic liquid crystal layer are studied. Aluminium gratings that also act as interdigitated electrodes are produced by focused ion beam lithography. It is found that a liquid crystal layer strongly influences both the resonance and light polarization properties characteristic of the gratings. Enhanced transmittance is observed not only for the TM-polarized light in the near infrared spectral range but also for the TE-polarized light in the visible range. Although the electrodes are separated by nanosized slits, and the electric field is strongly localized near the surface, a pronounced electrooptical effect is registered. The effect is explained in terms of local reorientation of liquid crystal molecules at the grating surface and propagation of the orientational deformation from the surface into the bulk of the liquid crystal layer.
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.
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.
A summation procedure for expansions in orthogonal polynomials
International Nuclear Information System (INIS)
Garibotti, C.R.; Grinstein, F.F.
1977-01-01
Approximants to functions defined by formal series expansions in orthogonal polynomials are introduced. They are shown to be convergent even out of the elliptical domain where the original expansion converges
Force prediction in cold rolling mills by polynomial methods
Directory of Open Access Journals (Sweden)
Nicu ROMAN
2007-12-01
Full Text Available A method for steel and aluminium strip thickness control is provided including a new technique for predictive rolling force estimation method by statistic model based on polynomial techniques.
on the performance of Autoregressive Moving Average Polynomial
African Journals Online (AJOL)
Timothy Ademakinwa
Using numerical example, DL, PDL, ARPDL and. ARMAPDL models were fitted. Autoregressive Moving Average Polynomial Distributed Lag Model. (ARMAPDL) model performed better than the other models. Keywords: Distributed Lag Model, Selection Criterion, Parameter Estimation, Residual Variance. ABSTRACT. 247.
Almost normed spaces of functions with polynomial asymptotic behaviour
International Nuclear Information System (INIS)
Kudryavtsev, L D
2003-01-01
We consider a linear space of functions asymptotically approaching polynomials of degree not higher than a fixed one as the independent variable approaches infinity. Such a space cannot be normed if the functions in it possess a certain smoothness. For this reason the concept of almost normed space is introduced and the spaces in question, namely, spaces of functions asymptotically or strongly asymptotically approaching polynomials, are shown to be almost normed. The completeness of these spaces in the metric generated by their almost norm is also proved, the connection between the asymptotic approach and the strong asymptotic approach of functions to polynomials is studied, and a new (and shorter) proof of the criterion for the asymptotic approach of functions to polynomials is presented
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...
A general polynomial solution to convection–dispersion equation ...
Indian Academy of Sciences (India)
Jiao Wang
s12040-017-0820-4. A general polynomial solution to convection–dispersion equation using ... water pollution of groundwater, numerical models are increasingly used in .... to convective transport by water flow is negligi- ble. Equation (4) is ...
An Elementary Proof of the Polynomial Matrix Spectral Factorization Theorem
Ephremidze, Lasha
2010-01-01
A very simple and short proof of the polynomial matrix spectral factorization theorem (on the unit circle as well as on the real line) is presented, which relies on elementary complex analysis and linear algebra.
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.
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 ...
Design of robust differential microphone arrays with orthogonal polynomials.
Pan, Chao; Benesty, Jacob; Chen, Jingdong
2015-08-01
Differential microphone arrays have the potential to be widely deployed in hands-free communication systems thanks to their frequency-invariant beampatterns, high directivity factors, and small apertures. Traditionally, they are designed and implemented in a multistage way with uniform linear geometries. This paper presents an approach to the design of differential microphone arrays with orthogonal polynomials, more specifically with Jacobi polynomials. It first shows how to express the beampatterns as a function of orthogonal polynomials. Then several differential beamformers are derived and their performance depends on the parameters of the Jacobi polynomials. Simulations show the great flexibility of the proposed method in terms of designing any order differential microphone arrays with different beampatterns and controlling white noise gain.
Orthogonal sets of data windows constructed from trigonometric polynomials
Greenhall, Charles A.
1990-01-01
Suboptimal, easily computable substitutes for the discrete prolate-spheroidal windows used by Thomson for spectral estimation are given. Trigonometric coefficients and energy leakages of the window polynomials are tabulated.
Cubic Polynomials with Rational Roots and Critical Points
Gupta, Shiv K.; Szymanski, Waclaw
2010-01-01
If you want your students to graph a cubic polynomial, it is best to give them one with rational roots and critical points. In this paper, we describe completely all such cubics and explain how to generate them.
Fitting discrete aspherical surface sag data using orthonormal polynomials.
Hilbig, David; Ceyhan, Ufuk; Henning, Thomas; Fleischmann, Friedrich; Knipp, Dietmar
2015-08-24
Characterizing real-life optical surfaces usually involves finding the best-fit of an appropriate surface model to a set of discrete measurement data. This process can be greatly simplified by choosing orthonormal polynomials for the surface description. In case of rotationally symmetric aspherical surfaces, new sets of orthogonal polynomials were introduced by Forbes to replace the numerical unstable standard description. From these, for the application of surface retrieval using experimental ray tracing, the sag orthogonal Q(con)-polynomials are of particular interest. However, these are by definition orthogonal over continuous data and may not be orthogonal for discrete data. In this case, the simplified solution is not valid. Hence, a Gram-Schmidt orthonormalization of these polynomials over the discrete data set is proposed to solve this problem. The resulting difference will be presented by a performance analysis and comparison to the direct matrix inversion method.
Log-concavity of the genus polynomials of Ringel Ladders
Directory of Open Access Journals (Sweden)
Jonathan L Gross
2015-10-01
Full Text Available A Ringel ladder can be formed by a self-bar-amalgamation operation on a symmetric ladder, that is, by joining the root vertices on its end-rungs. The present authors have previously derived criteria under which linear chains of copies of one or more graphs have log-concave genus polyno- mials. Herein we establish Ringel ladders as the first significant non-linear infinite family of graphs known to have log-concave genus polynomials. We construct an algebraic representation of self-bar-amalgamation as a matrix operation, to be applied to a vector representation of the partitioned genus distribution of a symmetric ladder. Analysis of the resulting genus polynomial involves the use of Chebyshev polynomials. This paper continues our quest to affirm the quarter-century-old conjecture that all graphs have log-concave genus polynomials.
Constructing the Lyapunov Function through Solving Positive Dimensional Polynomial System
Directory of Open Access Journals (Sweden)
Zhenyi Ji
2013-01-01
Full Text Available We propose an approach for constructing Lyapunov function in quadratic form of a differential system. First, positive polynomial system is obtained via the local property of the Lyapunov function as well as its derivative. Then, the positive polynomial system is converted into an equation system by adding some variables. Finally, numerical technique is applied to solve the equation system. Some experiments show the efficiency of our new algorithm.
Irreducibility results for compositions of polynomials in several ...
Indian Academy of Sciences (India)
R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22
such that the corresponding polynomial f (X, Y ) is reducible over Q(X). Indeed, one may choose for instance a0(X), a1(X), . . . , am−2(X) to be any polynomials with coefficients in Q, with a0(X) = 0, of degrees less than or equal to d, and define am−1(X) by the equality am−1(X) = −Xd − 5X − 5 −. ∑. 0≤i≤m−2 ai(X).
Commutators with idempotent values on multilinear polynomials in ...
Indian Academy of Sciences (India)
ring of R and C = Z(Q), the center of Q, which is called the extended centroid of R. We also make frequent use of the theory of generalized polynomial identities and differential identities. In particular, we need to recall that when R is a prime ring and I an ideal of R, then R, I and Q satisfy the same generalized polynomial ...
Interpretation of stream programs: characterizing type 2 polynomial time complexity
Férée , Hugo; Hainry , Emmanuel; Hoyrup , Mathieu; Péchoux , Romain
2010-01-01
International audience; We study polynomial time complexity of type 2 functionals. For that purpose, we introduce a first order functional stream language. We give criteria, named well-founded, on such programs relying on second order interpretation that characterize two variants of type 2 polynomial complexity including the Basic Feasible Functions (BFF). These charac- terizations provide a new insight on the complexity of stream programs. Finally, we adapt these results to functions over th...
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. .
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
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.
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.
Multi-variable Gould-Hopper and Laguerre polynomials
Directory of Open Access Journals (Sweden)
Caterina Cassisa
2007-12-01
Full Text Available The monomiality principle was introduced by G. Dattoli, in order to derive the properties of special or generalized polynomials starting from the corresponding ones of monomials. In this article we show a general technique to extend themonomiality approach tomulti-index polynomials in several variables. Application to the case of Hermite, Laguerre-type and mixed-type (i.e. between Laguerre and Hermite are derived.
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...
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....
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
Fast parallel computation of polynomials using few processors
DEFF Research Database (Denmark)
Valiant, Leslie; Skyum, Sven
1981-01-01
It is shown that any multivariate polynomial that can be computed sequentially in C steps and has degree d can be computed in parallel in 0((log d) (log C + log d)) steps using only (Cd)0(1) processors.......It is shown that any multivariate polynomial that can be computed sequentially in C steps and has degree d can be computed in parallel in 0((log d) (log C + log d)) steps using only (Cd)0(1) processors....
Fast Parallel Computation of Polynomials Using Few Processors
DEFF Research Database (Denmark)
Valiant, Leslie G.; Skyum, Sven; Berkowitz, S.
1983-01-01
It is shown that any multivariate polynomial of degree $d$ that can be computed sequentially in $C$ steps can be computed in parallel in $O((\\log d)(\\log C + \\log d))$ steps using only $(Cd)^{O(1)} $ processors.......It is shown that any multivariate polynomial of degree $d$ that can be computed sequentially in $C$ steps can be computed in parallel in $O((\\log d)(\\log C + \\log d))$ steps using only $(Cd)^{O(1)} $ processors....
Optimal Approximation of Biquartic Polynomials by Bicubic Splines
Kačala, Viliam; Török, Csaba
2018-02-01
Recently an unexpected approximation property between polynomials of degree three and four was revealed within the framework of two-part approximation models in 2-norm, Chebyshev norm and Holladay seminorm. Namely, it was proved that if a two-component cubic Hermite spline's first derivative at the shared knot is computed from the first derivative of a quartic polynomial, then the spline is a clamped spline of class C2 and also the best approximant to the polynomial. Although it was known that a 2 × 2 component uniform bicubic Hermite spline is a clamped spline of class C2 if the derivatives at the shared knots are given by the first derivatives of a biquartic polynomial, the optimality of such approximation remained an open question. The goal of this paper is to resolve this problem. Unlike the spline curves, in the case of spline surfaces it is insufficient to suppose that the grid should be uniform and the spline derivatives computed from a biquartic polynomial. We show that the biquartic polynomial coefficients have to satisfy some additional constraints to achieve optimal approximation by bicubic splines.
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.
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.
Super-A-polynomial for knots and BPS states
International Nuclear Information System (INIS)
Fuji, Hiroyuki; Gukov, Sergei; Sułkowski, Piotr
2013-01-01
We introduce and compute a 2-parameter family deformation of the A-polynomial that encodes the color dependence of the superpolynomial and that, in suitable limits, reduces to various deformations of the A-polynomial studied in the literature. These special limits include the t-deformation which leads to the “refined A-polynomial” introduced in the previous work of the authors and the Q-deformation which leads, by the conjecture of Aganagic and Vafa, to the augmentation polynomial of knot contact homology. We also introduce and compute the quantum version of the super-A-polynomial, an operator that encodes recursion relations for S r -colored HOMFLY homology. Much like its predecessor, the super-A-polynomial admits a simple physical interpretation as the defining equation for the space of SUSY vacua (= critical points of the twisted superpotential) in a circle compactification of the effective 3d N=2 theory associated to a knot or, more generally, to a 3-manifold M. Equivalently, the algebraic curve defined by the zero locus of the super-A-polynomial can be thought of as the space of open string moduli in a brane system associated with M. As an inherent outcome of this work, we provide new interesting formulas for colored superpolynomials for the trefoil and the figure-eight knot.
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...
Quantum Interference, Graphs, Walks, and Polynomials.
Tsuji, Yuta; Estrada, Ernesto; Movassagh, Ramis; Hoffmann, Roald
2018-04-09
In this paper, we explore quantum interference (QI) in molecular conductance from the point of view of graph theory and walks on lattices. By virtue of the Cayley-Hamilton theorem for characteristic polynomials and the Coulson-Rushbrooke pairing theorem for alternant hydrocarbons, it is possible to derive a finite series expansion of the Green's function for electron transmission in terms of the odd powers of the vertex adjacency matrix or Hückel matrix. This means that only odd-length walks on a molecular graph contribute to the conductivity through a molecule. Thus, if there are only even-length walks between two atoms, quantum interference is expected to occur in the electron transport between them. However, even if there are only odd-length walks between two atoms, a situation may come about where the contributions to the QI of some odd-length walks are canceled by others, leading to another class of quantum interference. For nonalternant hydrocarbons, the finite Green's function expansion may include both even and odd powers. Nevertheless, QI can in some circumstances come about for nonalternants from cancellation of odd- and even-length walk terms. We report some progress, but not a complete resolution, of the problem of understanding the coefficients in the expansion of the Green's function in a power series of the adjacency matrix, these coefficients being behind the cancellations that we have mentioned. Furthermore, we introduce a perturbation theory for transmission as well as some potentially useful infinite power series expansions of the Green's function.
One-parameter extension of the Doi-Peliti formalism and its relation with orthogonal polynomials.
Ohkubo, Jun
2012-10-01
An extension of the Doi-Peliti formalism for stochastic chemical kinetics is proposed. Using the extension, path-integral expressions consistent with previous studies are obtained. In addition, the extended formalism is naturally connected to orthogonal polynomials. We show that two different orthogonal polynomials, i.e., Charlier polynomials and Hermite polynomials, can be used to express the Doi-Peliti formalism explicitly.
Orthonormal polynomials for annular pupil including a cross-shaped obstruction.
Dai, Fengzhao; Wang, Xiangzhao; Sasaki, Osami
2015-04-01
By nonrecursive matrix method using the Zernike circle polynomials as the basis functions, we derived a set of polynomials up to fourth order which is approximately orthonormal for optical systems with an annular pupil having a cross-shaped obstruction. The performance of the polynomials is compared with the strictly orthonormal polynomials with some numerical examples.
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
Orthonormal aberration polynomials for optical systems with circular and annular sector pupils.
Díaz, José Antonio; Mahajan, Virendra N
2013-02-20
Using the Zernike circle polynomials as the basis functions, we obtain the orthonormal polynomials for optical systems with circular and annular sector pupils by the Gram-Schmidt orthogonalization process. These polynomials represent balanced aberrations yielding minimum variance of the classical aberrations of rotationally symmetric systems. Use of the polynomials obtained is illustrated with numerical examples.
International Nuclear Information System (INIS)
Daskaloyannis, C.
2000-01-01
The integrals of motion of the classical two-dimensional superintegrable systems close in a restrained polynomial Poisson algebra, whose general form is discussed. Each classical superintegrable problem has a quantum counterpart, a quantum superintegrable system. The polynomial Poisson algebra is deformed to a polynomial associative algebra, the finite-dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique. It is conjectured that the finite-dimensional representations of the polynomial algebra are determined by the energy eigenvalues of the superintegrable system. The calculation of energy eigenvalues is reduced to the solution of algebraic equations, which are universal for a large number of two-dimensional superintegrable systems. (author)
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.
International Nuclear Information System (INIS)
Koike, Masato; Namioka, Takeshi
1997-01-01
Comparative studies have been made on the holographic plane grating and the ruled varied-line-spacing (VLS) plane grating designed for two kinds of objective Monk - Gillieson type high-resolution grazing incidence monochromator, I and II. The ray-traced performance of monochromator types I and II on a synchrotron radiation beam line was evaluated in terms of resolving power and spectral purity by the introduction of new concepts of effective Gaussian line and purity profiles. The resolving power defined on the basis of the effective Gaussian profile is consistent with the spectral purity of the beam emerging from the exit slit and is more realistic as compared with those defined in the conventional manner, especially when spectral images have asymmetric profiles. It is concluded that holographic plane gratings recorded with a spherical and an aspheric wave front are capable of providing high resolution with high spectral purity and are fully interchangeable with the corresponding ruled VLS plane gratings. This interchangeability provides more flexibility for users in choosing a proper grating for a high-resolution grazing incidence monochromator of the Monk - Gillieson type. copyright 1997 Optical Society of America
Low aberration monolithic diffraction gratings for high performance optical spectrometers
Triebel, Peter; Moeller, Tobias; Diehl, Torsten; Gatto, Alexandre; Pesch, Alexander; Erdmann, Lars E.; Burkhardt, Matthias; Kalies, Alexander
2017-09-01
Gratings are the core element of the spectrometer. For imaging spectrometers beside the polarization sensitivity and efficiency the imaging quality of the diffraction grating is essential. Lenses and mirrors can be produced with lowest wavefront aberrations. Low aberration imaging quality of the grating is required not to limit the overall imaging quality of the instrument. Different types of spectrometers will lead to different requirements on the wavefront aberrations for their specific diffraction gratings. The wavefront aberration of an optical grating is a combination of the substrate wavefront and the grating wavefront. During the manufacturing process of the grating substrate different processes can be applied in order to minimize the wavefront aberrations. The imaging performance of the grating is also optimized due to the recording setup of the holography. This technology of holographically manufactured gratings is used for transmission and reflection gratings on different types of substrates like prisms, convex and concave spherical and aspherical surface shapes, free-form elements. All the manufactured gratings are monolithic and can be coated with high reflection and anti-reflection coatings. Prism substrates were used to manufacture monolithic GRISM elements for the UV to IR spectral range preferably working in transmission. Besides of transmission gratings, numerous spectrometer setups (e.g. Offner, Rowland circle, Czerny-Turner system layout) working on the optical design principles of reflection gratings. The present approach can be applied to manufacture high quality reflection gratings for the EUV to the IR. In this paper we report our latest results on manufacturing lowest wavefront aberration gratings based on holographic processes in order to enable at least diffraction limited complex spectrometric setups over certain wavelength ranges. Beside the results of low aberration gratings the latest achievements on improving efficiency together with
On equiconvergence of ultraspherical polynomials solution of one-speed neutron transport equation
International Nuclear Information System (INIS)
Yilmazer, Ayhan; Tombakoglu, Mehmet
2006-01-01
Ultraspherical polynomial approximation is used in slab criticality calculations for strongly anisotropic scattering. A unique and general formulation is developed for slab criticality condition whose sub-cases are spherical harmonics approximation, Chebyshev polynomial approximation of first and second kind. Since Legendre polynomials, Chebyshev Polynomials of first and second kinds are special cases of ultraspherical polynomials; our formulation inherently covers all these approximations and lets one to employ any other ultraspherical polynomial approximation in the solution of one-speed neutron transport equation. Our calculations showed that solution of one-speed neutron transport equation for various degrees of anisotropy and cross-section parameters is almost insensitive to the choice of ultraspherical polynomial with the present days' computing capabilities. In other words, as much as high order ultraspherical polynomial approximation is used the solution converges to the same value for a specified problem regardless the type of the ultraspherical polynomial assigned in the solution as equiconvergence theorem of Jacobi polynomials states
Polymer planar Bragg grating for sensing applications
Rosenberger, M.; Hartlaub, N.; Koller, G.; Belle, S.; Schmauss, B.; Hellmann, R.
2013-05-01
Bragg gratings have become indispensable as optical sensing elements and are already used for a variety of technical applications. Mainly silica fiber Bragg gratings (FBGs) have been extensively studied over the last decades and are nowadays commercially available. Bragg grating sensors consisting of other materials like polymers, however, have only recently come into the focus of fundamental and applied research. Polymers exhibit significantly different properties advantageous for many sensing applications and therefore provide a good alternative to silica based devices. In addition, polymer materials are inexpensive, simple to handle as well as available in various forms like liquid resists or bulk material. Accordingly, polymer integrated optics attract increasing interest and can serve as a substitute for optical fibers. We report on the fabrication of a planar Bragg grating sensor in bulk Polymethylmethacrylate (PMMA). The sensor consists of an optical waveguide and a Bragg grating, both written simultaneously into a PMMA chip by a single writing step, for which a phase mask covered by an amplitude mask is placed on top of the PMMA and exposed to the UV radiation of a KrF excimer laser. Depending on the phase mask period, different Bragg gratings reflecting in the telecommunication wavelength range are fabricated and characterized. Reflection and transmission measurements show a narrow reflection band and a high reflectivity of the polymer planar Bragg grating (PPBG). After connecting to a single mode fiber, the portable PPBG based sensor was evaluated for different measurands like humidity and strain. The sensor performance was compared to already existing sensing systems. Due to the obtained results as well as the rapid and cheap fabrication of the sensor chip, the PPBG qualifies for a low cost sensing element.
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.
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
Moniem, T. A.
2016-05-01
This article presents a methodology for an integrated Bragg grating using an alloy of GaAs, AlGaAs, and InGaAs with a controllable refractive index to obtain an adaptive Bragg grating suitable for many applications on optical processing and adaptive control systems, such as limitation and filtering. The refractive index of a Bragg grating is controlled by using an external electric field for controlling periodic modulation of the refractive index of the active waveguide region. The designed Bragg grating has refractive indices programmed by using that external electric field. This article presents two approaches for designing the controllable refractive indices active region of a Bragg grating. The first approach is based on the modification of a planar micro-strip structure of the iGaAs traveling wave as the active region, and the second is based on the modification of self-assembled InAs/GaAs quantum dots of an alloy from GaAs and InGaAs with a GaP traveling wave. The overall design and results are discussed through numerical simulation by using the finite-difference time-domain, plane wave expansion, and opto-wave simulation methods to confirm its operation and feasibility.
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.
Fast Minimum Variance Beamforming Based on Legendre Polynomials.
Bae, MooHo; Park, Sung Bae; Kwon, Sung Jae
2016-09-01
Currently, minimum variance beamforming (MV) is actively investigated as a method that can improve the performance of an ultrasound beamformer, in terms of the lateral and contrast resolution. However, this method has the disadvantage of excessive computational complexity since the inverse spatial covariance matrix must be calculated. Some noteworthy methods among various attempts to solve this problem include beam space adaptive beamforming methods and the fast MV method based on principal component analysis, which are similar in that the original signal in the element space is transformed to another domain using an orthonormal basis matrix and the dimension of the covariance matrix is reduced by approximating the matrix only with important components of the matrix, hence making the inversion of the matrix very simple. Recently, we proposed a new method with further reduced computational demand that uses Legendre polynomials as the basis matrix for such a transformation. In this paper, we verify the efficacy of the proposed method through Field II simulations as well as in vitro and in vivo experiments. The results show that the approximation error of this method is less than or similar to those of the above-mentioned methods and that the lateral response of point targets and the contrast-to-speckle noise in anechoic cysts are also better than or similar to those methods when the dimensionality of the covariance matrices is reduced to the same dimension.
International Nuclear Information System (INIS)
Odake, Satoru; Sasaki, Ryu
2009-01-01
Two sets of infinitely many exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials are presented. They are derived as the eigenfunctions of shape invariant and thus exactly solvable quantum mechanical Hamiltonians, which are deformations of those for the Wilson and Askey-Wilson polynomials in terms of a degree l (l=1,2,...) eigenpolynomial. These polynomials are exceptional in the sense that they start from degree l≥1 and thus not constrained by any generalisation of Bochner's theorem.
Hyperspectral grating optimization and manufacturing considerations
Ziph-Schatzberg, Leah; Swartz, Barry; Warren, Chris; Santman, Jeff; Saleh, Mohammad; Wiggins, Richard; Crifasi, Joe; Comstock, Lovell; Taylor, Kevan
2015-06-01
Hyperspectral imaging systems are finding broader applications in both the commercial and aerospace markets. It is becoming clear that to optimize the performance of these systems, their instrument transfer function needs to be tailored for each application. Vis-SWIR systems in the full 400nm to 2500nm waveband present particular design and manufacturing challenges. A single blazed grating is inadequate for a system operating in the full vis-SWIR wavelength range. In addition, optical materials and broad band coatings present a challenge for non-reflective systems. An understanding of the application and wavelengths of interest, combined with a judicious choice of a focal plane array, can then lead to an optimized system for the specific application. The ability to tailor the grating and manufacture a wide variety of grating profiles and substrate shapes becomes a significant performance enabler. This paper will discuss how the use of optical, coating, and grating design/analysis software, combined with grating manufacturing techniques assure meeting high performance requirements for different applications.
Kim, Jongyul; Lee, Kye Hong; Lim, Chang Hwy; Kim, Taejoo; Ahn, Chi Won; Cho, Gyuseong; Lee, Seung Wook
2013-06-01
The fabrication of gratings including metal deposition processes for highly neutron absorbing lines is a critical issue to achieve a good visibility of the grating-based phase imaging system. The source grating for a neutron Talbot-Lau interferometer is an array of Gadolinium (Gd) structures that are generally made by sputtering, photo-lithography, and chemical wet etching. However, it is very challenging to fabricate a Gd structure with sufficient neutron attenuation of approximately more than 20 μm using a conventional metal deposition method because of the slow Gd deposition rate, film stress, high material cost, and so on. In this article, we fabricated the source gratings for neutron Talbot-Lau interferometers by filling the silicon structure with Gadox particles. The new fabrication method allowed us a very stable and efficient way to achieve a much higher Gadox filled structure than a Gd film structure, and is even more suitable for thermal polychromatic neutrons, which are more difficult to stop than cold neutrons. The newly fabricated source gratings were tested at the polychromatic thermal neutron grating interferometer system of HANARO at the Korea Atomic Energy Research Institute, and the visibilities and images from the neutron phase imaging system with the new source gratings were compared with those fabricated by a Gd deposition method.
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
Ladder Operators for Lamé Spheroconal Harmonic Polynomials
Directory of Open Access Journals (Sweden)
Ricardo Méndez-Fragoso
2012-10-01
Full Text Available Three sets of ladder operators in spheroconal coordinates and their respective actions on Lamé spheroconal harmonic polynomials are presented in this article. The polynomials are common eigenfunctions of the square of the angular momentum operator and of the asymmetry distribution Hamiltonian for the rotations of asymmetric molecules, in the body-fixed frame with principal axes. The first set of operators for Lamé polynomials of a given species and a fixed value of the square of the angular momentum raise and lower and lower and raise in complementary ways the quantum numbers $n_1$ and $n_2$ counting the respective nodal elliptical cones. The second set of operators consisting of the cartesian components $hat L_x$, $hat L_y$, $hat L_z$ of the angular momentum connect pairs of the four species of polynomials of a chosen kind and angular momentum. The third set of operators, the cartesian components $hat p_x$, $hat p_y$, $hat p_z$ of the linear momentum, connect pairs of the polynomials differing in one unit in their angular momentum and in their parities. Relationships among spheroconal harmonics at the levels of the three sets of operators are illustrated.
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.
High-index-contrast subwavelength grating VCSEL
DEFF Research Database (Denmark)
Gilet, Philippe; Olivier, Nicolas; Grosse, Philippe
2010-01-01
In this article, we report our results on 980nm high-index-contrast subwavelength grating (HCG) VCSELs for optical interconnection applications. In our structure, a thin undoped HCG layer replaces a thick p-type Bragg mirror. The HCG mirror can feasibly achieve polarization-selective reflectivities...... close to 100%. The investigated structure consists of a HCG mirror with an underneath /4-thick oxide gap, four p-type GaAlAs/GaAs pairs for current spreading, three InGaAs/GaAs quantum wells, and an n-type GaAlAs/GaAs Bragg mirror. The HCG structure was defined by e-beam lithography and dry etching....... The current oxide aperture and the oxide gap underneath the HCG were simultaneously formed by the selective wet oxidation process. Compared to air-gap high contrast grating mirrors demonstrated elsewhere, our grating mirrors are particular since they are supported by thinner /4 aluminium oxide layer, and thus...
Optically tunable chirped fiber Bragg grating.
Li, Zhen; Chen, Zhe; Hsiao, V K S; Tang, Jie-Yuan; Zhao, Fuli; Jiang, Shao-Ji
2012-05-07
This work presents an optically tunable chirped fiber Bragg grating (CFBG). The CFBG is obtained by a side-polished fiber Bragg grating (SPFBG) whose thickness of the residual cladding layer in the polished area (D(RC)) varies with position along the length of the grating, which is coated with a photoresponsive liquid crystal (LC) overlay. The reflection spectrum of the CFBG is tuned by refractive index (RI) modulation, which comes from the phase transition of the overlaid photoresponsive LC under ultraviolet (UV) light irradiation. The broadening in the reflection spectrum and corresponding shift in the central wavelength are observed with UV light irradiation density of 0.64mW/mm. During the phase transition of the photoresponsive LC, the RI increase of the overlaid LC leads to the change of the CFBG reflection spectrum and the change is reversible and repeatable. The optically tunable CFBGs have potential use in optical DWDM system and an all-fiber telecommunication system.
Plasmonic Transmission Gratings – Fabrication and Characterization
DEFF Research Database (Denmark)
Sierant, Aleksandra; Jany, Benedykt; Bartoszek-Bober, Dobrosława
realization is given by the use of a metallic diffraction grating, where the diffracted light couples to the SPP. Here, we propose metallic periodic transmission gratings, processed onto a glass substrate, with various periods and fill factors. The gratings are milled in a plain gold layer with a focused ion......Surface plasmon polaritons (SPPs) are collective electron oscillations, confined at metal-dielectric interfaces. Coupling incident photons to SPPs may lead to spectrally broad field enhancement and confinement below the diffraction limit [1]. This phenomenon facilitates various applications......) Simulations. [1] W. L. Barnes, A. Dereux, T. W. Ebbesen, Nature 424, 824–830 (2003) [2] X. D. Hoa, A. G. Kirk, M. Tabrizian, Biosensors and Bioelectronics, 23, 2, 151-160 (2007) [3] T. Kawalec, et al., Opt. Lett. 39, 2932 (2014)...
Bragg Grating Optical Filters by UV Nanoimprinting
Directory of Open Access Journals (Sweden)
M. Casalboni
2012-01-01
Full Text Available Results on an optical waveguide filter operating in the near IR region are reported. The device consists of a hybrid sol-gel -based grating loaded waveguide, obtained through the merging of conventional photolithography and UV-nanoimprinting. Starting from submicrometric gratings, fabricated by electron beam lithography, a soft mould has been produced and the original structures were replicated onto sol-gel photosensitive films. A final photolithographic step allowed the production of grating-loaded channel waveguides. The devices were optically characterized by transmission measurements in the telecom range 1450–1590 nm. The filter extinction ratio is −11 dB and the bandwidth is 1.7 nm.
Ultrabroadband TM reflection from high contrast grating: why?
Gushchin, I.; Tishchenko, A.V.; Parriaux, O.; Hoekstra, Hugo
2009-01-01
A grating mode analysis of the unusually broadband TM reflection from a high contrast binary grating sheds light on the origin of this effect. This interpretation will be submitted to the workshop attendance.
High Efficiency Low Scatter Echelle Grating, Phase I
National Aeronautics and Space Administration — A high efficiency low scatter echelle grating will be developed using a novel technique of multiple diamond shaving cuts. The grating will have mirror surfaces on...
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.
Reflectivity-modulated grating-mirror
DEFF Research Database (Denmark)
2012-01-01
region in a layer structure comprising a p- and a n-doped semiconductor layer with an electrooptic material layer (12) arranged there between. The grating region comprises a grating structure formed by periodic perforations to change the refractive index periodically in directions normal...... a reflectivity with little or no out coupling and a reflectivity with normal out coupling, wherein lasing in the VCL is supported at both the first and the second reflectivity. As the out coupling mirror modulates the output, the lasing does not need to be modulated, and the invention provides the advantage...
Application of spherical gratings in synchrotron radiation spectroscopy
International Nuclear Information System (INIS)
Hogrefe, H.; Howells, M.R.; Hoyer, E.
1986-05-01
The recent development in gracing incidence grating monochromator design is discussed and the performance limiting for such instruments are examined. Especially the aberrations of toroidal and spherical gratings are investigated using the optical path function concept. It is shown that large radius spherical gratings, which can be produced with better slope tolerances than aspherics, also yield smaller overall line curvature than toroids. Therefore, a new simple spherical grating monochromator design is proposed and its performance is analyzed
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.
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
Dispersion engineering for vertical microcavities using subwavelength gratings.
Wang, Zhaorong; Zhang, Bo; Deng, Hui
2015-02-20
We show that the energy-momentum dispersion of a vertical semiconductor microcavity can be modified by design using a high-index-contrast subwavelength grating (SWG) as a cavity mirror. We analyze the angular dependence of the reflection phase of the SWG to illustrate the principles of dispersion engineering. We show examples of engineered dispersions such as ones with much reduced or increased energy density of states and one with a double-well-shaped dispersion. This method of dispersion engineering is compatible with maintaining a high cavity quality factor and incorporating fully protected active media inside the cavity, thus enabling the creation of new types of cavity quantum electrodynamics systems.
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.
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....... Furthermore, an empirical investigation of the 30 DJIA stocks shows that this estimator indicates stronger persistence in volatility than the standard local Whittle estimator....
Recurrence Relations for Orthogonal Polynomials on Triangular Domains
Directory of Open Access Journals (Sweden)
Abedallah Rababah
2016-04-01
Full Text Available In Farouki et al, 2003, Legendre-weighted orthogonal polynomials P n , r ( u , v , w , r = 0 , 1 , … , n , n ≥ 0 on the triangular domain T = { ( u , v , w : u , v , w ≥ 0 , u + v + w = 1 } are constructed, where u , v , w are the barycentric coordinates. Unfortunately, evaluating the explicit formulas requires many operations and is not very practical from an algorithmic point of view. Hence, there is a need for a more efficient alternative. A very convenient method for computing orthogonal polynomials is based on recurrence relations. Such recurrence relations are described in this paper for the triangular orthogonal polynomials, providing a simple and fast algorithm for their evaluation.
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)
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.
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.
Design and development of long-period grating sensors for ...
Indian Academy of Sciences (India)
Abstract. Long Period Gratings (LPGs) have been developed using carbon diox- ide laser in a standard optical fibre. LPGs with a periodicity of 600 μm and grating length of 24 mm have been inscribed on standard single mode fibre. Such gratings have been used in designing temperature sensors and temperature is ...
75 FR 41889 - Certain Steel Grating From China
2010-07-19
... COMMISSION Certain Steel Grating From China Determination On the basis of the record \\1\\ developed in the... steel grating from China, provided for in subheading 7308.90.70 of the Harmonized Tariff Schedule of the... imports of certain steel gratings from China were being subsidized within the meaning of section 703(b) of...
75 FR 8746 - Certain Steel Grating From China
2010-02-25
... COMMISSION Certain Steel Grating From China AGENCY: United States International Trade Commission. ACTION... retarded, by reason of subsidized and less-than-fair-value imports from China of certain steel gratings..., producers, or exporters ] in China of certain steel gratings, and that such products are being sold in the...
A Root Isolation Algorithm for Sparse Univariate Polynomials
Alonso, Maria Emilia; Galligo, André
2012-01-01
8 double pages.; International audience; We consider a univariate polynomial f with real coefficients having a high degree $N$ but a rather small number $d+1$ of monomials, with $d\\ll N$. Such a sparse polynomial has a number of real root smaller or equal to $d$. Our target is to find for each real root of $f$ an interval isolating this root from the others. The usual subdivision methods, relying either on Sturm sequences or Moebius transform followed by Descartes's rule of sign, destruct the...
Polynomials-Based Terminal Control of Affine Systems
Directory of Open Access Journals (Sweden)
A. E. Golubev
2015-01-01
Full Text Available One of the approaches to solving terminal control problems for affine dynamical systems is based on the use of polynomials of degree 2n − 1, where n is the order of the system in question. In this paper, we investigate the terminal control problem for which the final state of the system coincides with the origin in the phase space. We seek a set of initial states such that the solution of the terminal control problem can be constructed by using a polynomial of degree 2n − 2.Note that solution of the terminal control problem in question can be used to solve the problem of stabilizing the zero equilibrium in a finite time.For the second-order systems we prove the necessary and sufficient conditions for existence of the polynomial of the second degree which determines the solution of the terminal problem. The solutions of the terminal control problem based on the polynomials of second and third degree are given. As an example, the terminal control problem is considered for the simple pendulum.We also discuss solution of the terminal problem for affine systems of the third order, based on the use of the fourth and fifth degree polynomials. The necessary and sufficient conditions for existence of the fourth-degree polynomial such that its phase graph connects an arbitrary initial state of the system and the origin are obtained.For systems of arbitrary order n we obtain the necessary and sufficient conditions for existence of a solution of the terminal problem using the polynomial of degree 2n − 2. We also give the solution of the problem by means of the polynomial of degree 2n − 1.Further research can be focused on extending the results obtained in this note to terminal control problems where the desired final state of the system is not necessarily the origin.One of the potential application areas for the obtained theoretical results is automatic control of technical plants like unmanned aerial vehicles and mobile robots.
The Steenrod problem of realizing polynomial cohomology rings
Andersen, Kasper K. S.; Grodal, Jesper
2007-01-01
In this paper we completely classify which graded polynomial R-algebras in finitely many even degree variables can occur as the singular cohomology of a space with coefficients in R, a 1960 question of N. E. Steenrod, for a commutative ring R satisfying mild conditions. In the fundamental case R = Z, our result states that the only polynomial cohomology rings over Z which can occur, are tensor products of copies of H^*(CP^\\infty;Z) = Z[x_2], H^*(BSU(n);Z) = Z[x_4,x_6,...,x_{2n}], and H^*(BSp(...
Tricomplex Dynamical Systems Generated by Polynomials of Odd Degree
Parisé, Pierre-Olivier; Rochon, Dominic
In this paper, we give the exact interval of the cross section of the Multibrot sets generated by the polynomial zp + c where z and c are complex numbers and p > 2 is an odd integer. Furthermore, we show that the same Multibrots defined on the hyperbolic numbers are always squares. Moreover, we give a generalized 3D version of the hyperbolic Multibrot set and prove that our generalization is an octahedron for a specific 3D slice of the dynamical system generated by the tricomplex polynomial ηp + c where p > 2 is an odd integer.
Polynomial Approximation Techniques for Differential Equations in Electrochemical Problems
1981-01-15
LEYE L I oFFIPE-W NAVAL RESEARCH ConracEi NP014-8--0107 Task No. NR 359-718 1>. -/ TECHNICAL REP&T O. 4 Polynomial Approximation Techniques for...If the number of points xn is increased, a new polynomial may thus be found which de - scribes O(x) at these new points. Therefore, in the limit, O...has been used to simulate O(x) in the interval [xlXn1. Certain of these quadrature formulas lead to the well known Newton -Cotes, trapezoidal, and
The matrix and polynomial approaches to Lanczos-type algorithms
Brezinski, C.; Redivo-Zaglia, M.; Sadok, H.
2000-11-01
Lanczos method for solving a system of linear equations can be derived by using formal orthogonal polynomials. It can be implemented by several recurrence relationships, thus leading to several algorithms. In this paper, the Lanczos/Orthodir algorithm will be derived in two different ways. The first one is based on a matrix approach and on the recursive computation of two successive regular matrices. We will show that it can be directly obtained from the orthogonality conditions and the fact that Lanczos method is a Krylov subspace method. The second approach is based on formal orthogonal polynomials. The case of breakdowns will be treated similarly.
Orthogonal polynomials on the unit circle part 2 spectral theory
Simon, Barry
2013-01-01
This two-part book is a comprehensive overview of the theory of probability measures on the unit circle, viewed especially in terms of the orthogonal polynomials defined by those measures. A major theme involves the connections between the Verblunsky coefficients (the coefficients of the recurrence equation for the orthogonal polynomials) and the measures, an analog of the spectral theory of one-dimensional Schrödinger operators. Among the topics discussed along the way are the asymptotics of Toeplitz determinants (Szegő's theorems), limit theorems for the density of the zeros of orthogonal po
Orthogonal polynomials on the unit circle part 1 classical theory
2009-01-01
This two-part book is a comprehensive overview of the theory of probability measures on the unit circle, viewed especially in terms of the orthogonal polynomials defined by those measures. A major theme involves the connections between the Verblunsky coefficients (the coefficients of the recurrence equation for the orthogonal polynomials) and the measures, an analog of the spectral theory of one-dimensional Schrodinger operators. Among the topics discussed along the way are the asymptotics of Toeplitz determinants (SzegÅ‘'s theorems), limit theorems for the density of the zeros of orthogonal po
Gaussian polynomials and content ideal in trivial extensions
International Nuclear Information System (INIS)
Bakkari, C.; Mahdou, N.
2006-12-01
The goal of this paper is to exhibit a class of Gaussian non-coherent rings R (with zero-divisors) such that wdim(R) = ∞ and fPdim(R) is always at most one and also exhibits a new class of rings (with zerodivisors) which are neither locally Noetherian nor locally domain where Gaussian polynomials have a locally principal content. For this purpose, we study the possible transfer of the 'Gaussian' property and the property 'the content ideal of a Gaussian polynomial is locally principal' to various trivial extension contexts. This article includes a brief discussion of the scopes and limits of our result. (author)
Discrete-Time Filter Synthesis using Product of Gegenbauer Polynomials
Directory of Open Access Journals (Sweden)
N. Stojanovic
2016-09-01
Full Text Available 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 approximations(Chebyshev, Legendre, Butterworth are shown to be a special cases of proposed approximation method.
Fitting by Orthonormal Polynomials of Silver Nanoparticles Spectroscopic Data
Bogdanova, Nina; Koleva, Mihaela
2018-02-01
Our original Orthonormal Polynomial Expansion Method (OPEM) in one-dimensional version is applied for first time to describe the silver nanoparticles (NPs) spectroscopic data. The weights for approximation include experimental errors in variables. In this way we construct orthonormal polynomial expansion for approximating the curve on a non equidistant point grid. The corridors of given data and criteria define the optimal behavior of searched curve. The most important subinterval of spectra data is investigated, where the minimum (surface plasmon resonance absorption) is looking for. This study describes the Ag nanoparticles produced by laser approach in a ZnO medium forming a AgNPs/ZnO nanocomposite heterostructure.
The Laplace transform and polynomial approximation in L2
DEFF Research Database (Denmark)
Labouriau, Rodrigo
2016-01-01
This short note gives a sufficient condition for having the class of polynomials dense in the space of square integrable functions with respect to a finite measure dominated by the Lebesgue measure in the real line, here denoted by L2. It is shown that if the Laplace transform of the measure...... concerning the polynomial approximation is original, even thought the proof is relatively simple. Additionally, an alternative stronger condition (easier to be verified) not involving the calculation of the Laplace transform is given. The condition essentially says that the density of the measure should have...
On the Lojasiewicz exponent at infinity of real polynomials
International Nuclear Information System (INIS)
Ha Huy Vui; Pham Tien Son
2007-07-01
Let f : R n → R be a nonconstant polynomial function. In this paper, using the information from 'the curve of tangency' of f, we provide a method to determine the Lojasiewicz exponent at infinity of f. As a corollary, we give a computational criterion to decide if the Lojasiewicz exponent at infinity is finite or not. Then, we obtain a formula to calculate the set of points at which the polynomial f is not proper. Moreover, a relation between the Lojasiewicz exponent at infinity of f with the problem of computing the global optimum of f is also established. (author)
Popov form computation for matrices of Ore polynomials
DEFF Research Database (Denmark)
Khochtali, Mohamed; Né Nielsen, Johan Rosenkilde; Storjohann, Arne
2017-01-01
Let F[∂; s, d] be a ring of Ore polynomials over a field. We give a new deterministic algorithm for computing the Popov form P of a non-singular matrix A ∈ F[∂; s, d]n×n. Our main focus is to ensure controlled growth in the size of coefficients from F in the case F = k(z), and even k = Q. Our....... The resulting bit-complexity for the differential and shift polynomial case over ℚ(z) improves upon the previous best....
Riemann hypothesis for period polynomials of modular forms.
Jin, Seokho; Ma, Wenjun; Ono, Ken; Soundararajan, Kannan
2016-03-08
The period polynomial r(f)(z) for an even weight k≥4 newform f∈S(k)(Γ(0)((N)) is the generating function for the critical values of L(f,s) . It has a functional equation relating r(f)(z) to r(f)(-1/Nz). We prove the Riemann hypothesis for these polynomials: that the zeros of r(f)(z) lie on the circle |z|=1/√N . We prove that these zeros are equidistributed when either k or N is large.
Representation of moduli spaces of curves and calculation of extremal polynomials
International Nuclear Information System (INIS)
Bogatyrev, A B
2003-01-01
The classical Chebyshev and Zolotarev polynomials are the first ranks of the hierarchy of extremal polynomials, which are typical solutions of problems on the conditional minimization of the uniform norm over a space of polynomials. In the general case such polynomials are connected with hyperelliptic curves the genus of which labels the ranks of the hierarchy. Representations of the moduli spaces of such curves are considered in this paper with applications to the calculation of extremal polynomials. Uniformizing curves by special Schottky groups one obtains effectively computable parametric expressions for extremal polynomials in terms of linear series of Poincare
Undergraduate Experiment with Fractal Diffraction Gratings
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…
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.
Large-aperture subwavelength grating couplers.
Qi, Fan; Ma, Qingyan; Wang, Yufei; Zheng, Wanhua
2016-04-10
Subwavelength nanostructure grating couplers fabricated on silicon-on-insulator substrates are used to simplify the fabrication process while maintaining high coupling efficiency. The main obstacle for their application in photonic integrated circuits is the small aperture size of the nanostructure when TE polarization is involved, since they are difficult to achieve with 193 nm deep-ultraviolet lithography and cause problems in inductively coupled plasma etching. A larger lateral period has been used to increase the aperture size. Here, we propose that decreasing the effective index of the nanostructure can also enlarge the aperture size. We analyze the two methods in detail with a rectangle-hole nanostructure and 220 nm thick waveguide layer, aiming at TE polarization centered at 1560 nm. We find performance degenerations for large lateral periods, and this can be simply compensated by adjusting the width of the rectangle hole. The minimum linewidth of the nanostructure can reach 240 nm, while the coupling efficiency is just slightly decreased. The backreflections of a large-aperture grating increase but stay in the same order with ordinary ones, and we also show that this can be overcome by apodizing the grating structure. Finally, we experimentally demonstrate the designed large-aperture grating couplers and the coupling efficiencies are higher than 35%, and reach a rectangle-hole width.
Disorder effects in subwavelength grating metamaterial waveguides
Czech Academy of Sciences Publication Activity Database
Ortega-Moñux, A.; Čtyroký, Jiří; Cheben, P.; Schmid, J. H.; Wang, S.; Molina-Fernández, I.; Halíř, R.
2017-01-01
Roč. 25, č. 11 (2017), s. 12222-12236 ISSN 1094-4087 R&D Projects: GA ČR(CZ) GA16-00329S Institutional support: RVO:67985882 Keywords : Subwavelength grating * Integrated photonics * Diffraction effects Subject RIV: BH - Optics, Masers, Lasers OBOR OECD: Optics (including laser optics and quantum optics) Impact factor: 3.307, year: 2016
Smart photogalvanic running-grating interferometer
DEFF Research Database (Denmark)
Kukhtarev, N. V.; Kukhtareva, T.; Edwards, M. E.
2005-01-01
Photogalvanic effect produces actuation of periodic motion of macroscopic LiNbO3 crystal. This effect was applied to the development of an all-optical moving-grating interferometer usable for optical trapping and transport of algae chlorella microorganisms diluted in water with a concentration...
Use of savart plates in grating interferometers.
Peek, T H
1971-05-01
An analysis is given of Savart plates for arbitrary angles between the optic axis and the plate normal. Conoscopic interference patterns of thin Savart plates cut nearly parallel to the optic axis are shown and the use of such plates combined with diffraction gratings is discussed.
130-nm tunable grating-mirror VCSEL
DEFF Research Database (Denmark)
Chung, Il-Sug; Mørk, Jesper
2014-01-01
We have reported that a combination of the high-index-contrast grating (HCG) mirror as movable mirror and the extended cavity configuration with an antireflection layer can provide a tuning wavelength range of 100 nm for tunable VCSELs. Here, we report that using the air-coupled cavity...
Disorder effects in subwavelength grating metamaterial waveguides
Czech Academy of Sciences Publication Activity Database
Ortega-Moñux, A.; Čtyroký, Jiří; Cheben, P.; Schmid, J. H.; Wang, S.; Molina-Fernández, I.; Halíř, R.
2017-01-01
Roč. 25, č. 11 (2017), s. 12222-12236 ISSN 1094-4087 R&D Projects: GA ČR(CZ) GA16-00329S Institutional support: RVO:67985882 Keywords : Subwavelength grating * Integrated photonics * Diffraction effect s Subject RIV: BH - Optics, Masers, Lasers OBOR OECD: Optics (including laser optics and quantum optics) Impact factor: 3.307, year: 2016
Surface Fluctuation Scattering using Grating Heterodyne Spectroscopy
DEFF Research Database (Denmark)
Edwards, R. V.; Sirohi, R. S.; Mann, J. A.
1982-01-01
Heterodyne photon spectroscopy is used for the study of the viscoelastic properties of the liquid interface by studying light scattered from thermally generated surface fluctuations. A theory of a heterodyne apparatus based on a grating is presented, and the heterodyne condition is given in terms...
Diffusion of solid fuelon a vibrating grate
DEFF Research Database (Denmark)
Sabelström, Hanna Katarina
to introduce a varying velocity depending on the position on the grate, a modification of the model is necessary where also the density will vary as a consequence of the continuity equation. The definition of the density will thereby change from being the particle density to be the cell density, i.e. a measure...
Fibre Bragg Grating and Long Period Grating Sensors in Polymer Optical Fibres
Bundalo, Ivan-Lazar; Bang, Ole; Nielsen, Kristian
2017-01-01
The work presented in this thesis focuses on improving the fabrication of Fibre Bragg Gratings (FBGs) and Long Period Gratings (LPGs) in microstructure polymer optical fibres (mPOF). It also focuses on exploring new options for biomedical and acoustic sensing with the purpose of expanding the range of applications and pushing the limits. The first part of the work focuses on the fabrication of FBGs in polymer optical fibres. FBGs are a periodic perturbation of the refractive index of the opti...
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
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.
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.
Recent developments of Bragg gratings in PMMA and TOPAS polymer optical fibers
DEFF Research Database (Denmark)
Webb, David; Kyriacos, Kalli; Carroll, Karen
temperature to the glass transition temperature, and this permanent change is affected by the thermal history of the gratings. We also report the first FBG inscription in microstructured polymer optical fibres fabricated from Topas. This material is fully polymerised and has a very low moisture absorption......, leading to very good fibre drawing properties. Furthermore, although Topas is chemically inert and biomolecules do not readily bind to its surface, treatment with Antraquinon and subsequent UV activation allows sensing molecules to be deposited in well defined spatial locations. When combined with grating...
Terahertz amplification in RTD-gated HEMTs with a grating-gate wave coupling topology
Energy Technology Data Exchange (ETDEWEB)
Condori Quispe, Hugo O.; Sensale-Rodriguez, Berardi [The University of Utah, Salt Lake City, Utah 84112 (United States); Encomendero-Risco, Jimy J.; Xing, Huili Grace [University of Notre Dame, Notre Dame, Indiana 46556 (United States); Cornell University, Ithaca, New York 14853 (United States)
2016-08-08
We theoretically analyze the operation of a terahertz amplifier consisting of a resonant-tunneling-diode gated high-electron-mobility transistor (RTD-gated HEMT) in a grating-gate topology. In these devices, the key element enabling substantial power gain is the efficient coupling of terahertz waves into and out of plasmons in the RTD-gated HEMT channel, i.e., the gain medium, via the grating-gate itself, part of the active device, rather than by an external antenna structure as discussed in previous works, therefore potentially enabling terahertz amplification with associated power gains >40 dB.
Mirror symmetry, toric branes and topological string amplitudes as polynomials
International Nuclear Information System (INIS)
Alim, Murad
2009-01-01
The central theme of this thesis is the extension and application of mirror symmetry of topological string theory. The contribution of this work on the mathematical side is given by interpreting the calculated partition functions as generating functions for mathematical invariants which are extracted in various examples. Furthermore the extension of the variation of the vacuum bundle to include D-branes on compact geometries is studied. Based on previous work for non-compact geometries a system of differential equations is derived which allows to extend the mirror map to the deformation spaces of the D-Branes. Furthermore, these equations allow the computation of the full quantum corrected superpotentials which are induced by the D-branes. Based on the holomorphic anomaly equation, which describes the background dependence of topological string theory relating recursively loop amplitudes, this work generalizes a polynomial construction of the loop amplitudes, which was found for manifolds with a one dimensional space of deformations, to arbitrary target manifolds with arbitrary dimension of the deformation space. The polynomial generators are determined and it is proven that the higher loop amplitudes are polynomials of a certain degree in the generators. Furthermore, the polynomial construction is generalized to solve the extension of the holomorphic anomaly equation to D-branes without deformation space. This method is applied to calculate higher loop amplitudes in numerous examples and the mathematical invariants are extracted. (orig.)
Polynomials constant on a hyperplane and CR maps of hyperquadrics
Lebl, J.; Peters, H.
2011-01-01
We prove a sharp degree bound for polynomials constant on a hyperplane with a fixed number of distinct monomials for dimensions 2 and 3. We study the connection with monomial CR maps of hyperquadrics and prove similar bounds in this setup with emphasis on the case of spheres. The results support
A reduced polynomial chaos expansion method for the stochastic ...
Indian Academy of Sciences (India)
... numerical examples, namely, bending of a stochastic beam, ﬂow through porous media with stochastic permeability and transverse bending of a plate with stochastic properties. The results obtained from the proposed method are compared with classical polynomial chaos and direct Monte Carlo simulation results.
Polynomial skew-products with wandering Fatou-disks
Peters, H.; Vivas, L.R.
2016-01-01
Until recently, little was known about the existence of wandering Fatou components for rational maps in more than one complex variables. In 2014, examples of wandering Fatou components were constructed in Astorg et al. [1] for polynomial skew-products with an invariant parabolic fiber. In 2004 Lilov
Some Polynomials Associated with the r-Whitney Numbers
Indian Academy of Sciences (India)
26
k=0. Wm,r(n, k)xk. They found some combinatorial identities by means of Riordan arrays. In this section we use a technique from linear algebra introduced by Rota [44] to give an alternative proof of some results of Cheon and Jung. Theorem 1. The exponential generating function for the r-Dowling polynomials is. ∞. ∑ n=0.