Chen, S J; Perng, S Y; Kuan, C K; Tseng, T C; Wang, D J
2001-01-01
An active polynomial grating has been designed for use in synchrotron radiation soft-X-ray monochromators and spectrometers. The grating can be dynamically adjusted to obtain the third-order-polynomial surface needed to eliminate the defocus and coma aberrations at any photon energy. Ray-tracing results confirm that a monochromator or spectrometer based on this active grating has nearly no aberration limit to the overall spectral resolution in the entire soft-X-ray region. The grating substrate is made of a precisely milled 17-4 PH stainless steel parallel plate, which is joined to a flexure-hinge bender shaped by wire electrical discharge machining. The substrate is grounded into a concave cylindrical shape with a nominal radius and then polished to achieve a roughness of 0.45 nm and a slope error of 1.2 mu rad rms. The long trace profiler measurements show that the active grating can reach the desired third-order polynomial with a high degree of figure accuracy.
Edee, K; Plumey, J P
2015-03-01
The modal method based on Gegenbauer polynomials (MMGE) is extended to the case of bidimensional binary gratings. A new concept of modified polynomials is introduced in order to take into account boundary conditions and also to make the method more flexible in use. In the previous versions of MMGE, an undersized matrix relation is obtained by solving Maxwell's equations, and the boundary conditions complement this undersized system. In the current work, contrary to this previous version of the MMGE, boundary conditions are incorporated into the definition of a new basis of polynomial functions, which are adapted to the boundary value problem of interest. Results are successfully compared for both metallic and dielectric structures to those obtained from the modal method based on Fourier expansion (MMFE) and MMFE with adaptative spatial resolution. PMID:26366651
By using two aspherical variable-line-space active gratings and applying the energy compensation principle, we have designed a very efficient active grating monochromator -- active grating spectrometer (AGM-AGS) beamline system for the photon demanding inelastic soft-x-ray scattering experiments. During the energy scan, the defocus and coma aberrations of the AGM can be completely eliminated to make the focal point fixed at the sample position and to maintain high spectral resolution for the entire spectral range. The AGS, which has an optical system identical to that of the AGM, but positioned reversely along the optical path, collects the photons emitted from the sample with a nearly identical energy spread as the AGM and focus them onto a position sensitive detector located at the exit slit position. The ray tracing results show that the efficiency of the AGM-AGS is two orders of magnitudes higher than that of conventional design while maintaining a very high spectral resolution
Mechanics of dielectric elastomer-activated deformable transmission grating
Laminating a thin layer of elastomeric grating on the surface of a prestretched dielectric elastomer (DE) membrane forms a basic design of electrically tunable transmission grating. We analyze the inhomogeneous deformation of a circular multiple-region configuration. Variation of the geometric and material parameters, as well as of the critical condition determined by loss of tension instability, is probed to aid the design of a DE-based deformable grating. The predicted changes in the grating period agree substantially with the experimental results reported by Aschwanden et al (Aschwanden et al 2007 IEEE Photon. Technol. Lett. 19 1090). (paper)
Polynomial functors and polynomial monads
Gambino, Nicola
2009-01-01
We study polynomial functors over locally cartesian closed categories. After setting up the basic theory, we show how polynomial functors assemble into a double category, in fact a framed bicategory. We show that the free monad on a polynomial endofunctor is polynomial. The relationship with operads and other related notions is explored.
Freud, Géza
1971-01-01
Orthogonal Polynomials contains an up-to-date survey of the general theory of orthogonal polynomials. It deals with the problem of polynomials and reveals that the sequence of these polynomials forms an orthogonal system with respect to a non-negative m-distribution defined on the real numerical axis. Comprised of five chapters, the book begins with the fundamental properties of orthogonal polynomials. After discussing the momentum problem, it then explains the quadrature procedure, the convergence theory, and G. Szegő's theory. This book is useful for those who intend to use it as referenc
Active temperature compensation design of sensor with fiber gratings
Xingfa Dong(董兴法); Yonglin Huang(黄勇林); Li Jiang(姜莉); Guiyun Kai(开桂云); Xiaoyi Dong(董孝义)
2004-01-01
A technique for compensation of temperature effects in fiber grating sensors is reported. For strain sensors and other sensors related to strain such as electromagnetic sensors, a novel structure is designed, which uses two fiber Bragg gratings (FBGs) as strain differential sensor and has temperature effects cancelled. Using this technique, the stress sensitivity has been amplified and gets up to 0.226 nm/N, the total variation in wavelength difference within the range of 3-45 ℃ is 0.03 nm, 1/14 of the uncompensated FBG.The structure can be used in the temperature-insensitive static strain measurement and minor-vibration measurement.
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...
Svebak, Sven
2016-01-01
Results from two studies of biological consequences of laughter are reported. A proposed inhibitory brain mechanism was tested in Study 1. It aims to protect against trunk compression that can cause health hazards during vigorous laughter. Compression may be maximal during moderate durations and, for protective reasons, moderate in enduring vigorous laughs. Twenty-five university students volunteered to see a candid camera film. Laughter responses (LR) and the superimposed ha-responses were operationally assessed by mercury-filled strain gauges strapped around the trunk. On average, the thorax compression amplitudes exceeded those of the abdomen, and greater amplitudes were seen in the males than in the females after correction for resting trunk circumference. Regression analyses supported polynomial relations because medium LR durations were associated with particularly high thorax amplitudes. In Study 2, power changes were computed in the beta and alpha EEG frequency bands of the parietal cortex from before to after exposure to the comedy “Dinner for one” in 56 university students. Highly significant linear relations were calculated between the number of laughs and post-exposure cortical activation (increase of beta, decrease of alpha) due to high activation after frequent laughter. The results from Study 1 supported the hypothesis of a protective brain mechanism that is activated during long LRs to reduce the risk of harm to vital organs in the trunk cavity. The results in Study 2 supported a linear cortical activation and, thus, provided evidence for a biological correlate to the subjective experience of mental refreshment after laughter. PMID:27547260
Active terahertz beam steering by photo-generated graded index gratings in thin semiconductor films.
Steinbusch, T P; Tyagi, H K; Schaafsma, M C; Georgiou, G; Gómez Rivas, J
2014-11-01
We demonstrate active beam steering of terahertz radiation using a photo-excited thin layer of gallium arsenide. A constant gradient of phase discontinuity along the interface is introduced by an spatially inhomogeneous density of free charge carriers that are photo-generated in the GaAs with an optical pump. The optical pump has been spatially modulated to form the shape of a planar blazed grating. The phase gradient leads to an asymmetry between the +1 and -1 transmission diffracted orders of more than a factor two. Optimization of the grating structure can lead to an asymmetry of more than one order of magnitude. Similar to metasurfaces made of plasmonic antennas, the photo-generated grating is a planar structure that can achieve large beam steering efficiency. Moreover, the photo-generation of such structures provides a platform for active THz beam steering. PMID:25401807
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.
Papadopoulos, Anthony
2009-01-01
The first-degree power-law polynomial function is frequently used to describe activity metabolism for steady swimming animals. This function has been used in hydrodynamics-based metabolic studies to evaluate important parameters of energetic costs, such as the standard metabolic rate and the drag power indices. In theory, however, the power-law polynomial function of any degree greater than one can be used to describe activity metabolism for steady swimming animals. In fact, activity metaboli...
Polynomially Bounded Sequences and Polynomial Sequences
Okazaki Hiroyuki
2015-09-01
Full Text Available In this article, we formalize polynomially bounded sequences that plays an important role in computational complexity theory. Class P is a fundamental computational complexity class that contains all polynomial-time decision problems [11], [12]. It takes polynomially bounded amount of computation time to solve polynomial-time decision problems by the deterministic Turing machine. Moreover we formalize polynomial sequences [5].
Factoring multivariate integral polynomials.
Lenstra, A.K.
1983-01-01
An algorithm is presented to factorize polynomials in several variables with integral coefficients that is polynomial-time in the degrees of the polynomial to be factored, for any fixed number of variables. The algorithm generalizes the algorithm presented by A. K. Lenstra et al. to factorize integral polynomials in one variable.
New classes of test polynomials of polynomial algebras
冯克勤; 余解台
1999-01-01
A polynomial p in a polynomial algebra over a field is called a test polynomial if any endomorphism of the polynomial algebra that fixes p is an automorphism. some classes of new test polynomials recognizing nonlinear automorphisms of polynomial algebras are given. In the odd prime characteristic case, test polynomials recognizing non-semisimple automorphisms are also constructed.
Generalized bivariate Fibonacci polynomials
Catalani, Mario
2002-01-01
We define generalized bivariate polynomials, from which upon specification of initial conditions the bivariate Fibonacci and Lucas polynomials are obtained. Using essentially a matrix approach we derive identities and inequalities that in most cases generalize known results.
Factoring Polynomials and Fibonacci.
Schwartzman, Steven
1986-01-01
Discusses the factoring of polynomials and Fibonacci numbers, offering several challenges teachers can give students. For example, they can give students a polynomial containing large numbers and challenge them to factor it. (JN)
Branched polynomial covering maps
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
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....
Bernstein polynomials on Simplex
Bayad, A.; Kim, T.; Rim, S. -H.
2011-01-01
We prove two identities for multivariate Bernstein polynomials on simplex, which are considered on a pointwise. In this paper, we study good approximations of Bernstein polynomials for every continuous functions on simplex and the higher dimensional q-analogues of Bernstein polynomials on simplex
Coherent orthogonal polynomials
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 L2 functions and to the corresponding Universal Enveloping Algebra and transformation group. Generalized coherent states from each vector in the space L2 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
Fiber Bragg grating strain sensors to monitor and study active volcanoes
Sorrentino, Fiodor; Beverini, Nicolò; Carbone, Daniele; Carelli, Giorgio; Francesconi, Francesco; Gambino, Salvo; Giacomelli, Umberto; Grassi, Renzo; Maccioni, Enrico; Morganti, Mauro
2016-04-01
Stress and strain changes are among the best indicators of impending volcanic activity. In volcano geodesy, borehole volumetric strain-meters are mostly utilized. However, they are not easy to install and involve high implementation costs. Advancements in opto-electronics have allowed the development of low-cost sensors, reliable, rugged and compact, thus particularly suitable for field application. In the framework of the EC FP7 MED-SUV project, we have developed strain sensors based on the fiber Bragg grating (FBG) technology. In comparison with previous implementation of the FBG technology to study rock deformations, we have designed a system that is expected to offer a significantly higher resolution and accuracy in static measurements and a smooth dynamic response up to 100 Hz, implying the possibility to observe seismic waves. The system performances are tailored to suit the requirements of volcano monitoring, with special attention to power consumption and to the trade-off between performance and cost. Preliminary field campaigns were carried out on Mt. Etna (Italy) using a prototypal single-axis FBG strain sensor, to check the system performances in out-of-the-lab conditions and in the harsh volcanic environment (lack of mains electricity for power, strong diurnal temperature changes, strong wind, erosive ash, snow and ice during the winter time). We also designed and built a FBG strain sensor featuring a multi-axial configuration which was tested and calibrated in the laboratory. This instrument is suitable for borehole installation and will be tested on Etna soon.
Generalized Fibonacci-Lucas Polynomials
Mamta Singh
2013-12-01
Full Text Available Various sequences of polynomials by the names of Fibonacci and Lucas polynomials occur in the literature over a century. The Fibonacci polynomials and Lucas polynomials are famous for possessing wonderful and amazing properties and identities. In this paper, Generalized Fibonacci-Lucas Polynomials are introduced and defined by the recurrence relation with and . Some basic identities of Generalized Fibonacci-Lucas Polynomials are obtained by method of generating function. Keywords: Fibonacci polynomials, Lucas polynomials, Generalized Fibonacci polynomials, Generalized Fibonacci-Lucas polynomials.
Weierstrass polynomials for links
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 that for...... links in 3-space. This paper initiates a study of the connections between polynomial covering spaces over the circle and links in 3-space....
New results on permutation polynomials over finite fields
Ma, Jingxue; Zhang, Tao; Feng, Tao; Ge, Gennian
2015-01-01
Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, four classes of monomial complete permutation polynomials and one class of trinomial complete permutation polynomials are presented, one of which confirms a conjecture proposed by Wu et al. (Sci. China Math., to appear. Doi: 10.1007/s11425-014-4964-2). Furthermore, we give two classes of trinomial permutation polynomials, and make some pr...
Quantum Grothendieck polynomials
Kirillov, Anatol N.
1996-01-01
We study the algebraic aspects of (small) quantum equivariant $K$-theory of flag manifold. Lascoux-Sch\\"utzenberger's type formula for quantum double and quantum double dual Grothendieck polynomials and the quantum Cauchy identity for quantum Grothendieck polynomials are obtained.
Marichal, Jean-Luc
2007-01-01
We define the concept of weighted lattice polynomial functions as lattice polynomial functions constructed from both variables and parameters. We provide equivalent forms of these functions in an arbitrary bounded distributive lattice. We also show that these functions include the class of discrete Sugeno integrals and that they are characterized by a median based decomposition formula.
Bogner, Christian; Weinzierl, Stefan
The integrand of any multiloop integral is characterized after Feynman parametrization by two polynomials. In this review we summarize the properties of these polynomials. Topics covered in this paper include among others: spanning trees and spanning forests, the all-minors matrix-tree theorem, recursion relations due to contraction and deletion of edges, Dodgson's identity and matroids.
Nonnegativity of uncertain polynomials
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.
Jack polynomials in superspace
Desrosiers, P; Mathieu, P
2003-01-01
This work initiates the study of {\\it orthogonal} symmetric polynomials in superspace. Here we present two approaches leading to a family of orthogonal polynomials in superspace that generalize the Jack polynomials. The first approach relies on previous work by the authors in which eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland Hamiltonian were constructed. Orthogonal eigenfunctions are now obtained by diagonalizing the first nontrivial element of a bosonic tower of commuting conserved charges not containing this Hamiltonian. Quite remarkably, the expansion coefficients of these orthogonal eigenfunctions in the supermonomial basis are stable with respect to the number of variables. The second and more direct approach amounts to symmetrize products of non-symmetric Jack polynomials with monomials in the fermionic variables. This time, the orthogonality is inherited from the orthogonality of the non-symmetric Jack polynomials, and the value of the norm is given exp...
Mironov, A; Morozov, A
2015-01-01
We present a universal knot polynomials for 2- and 3-strand torus knots in adjoint representation, by universalization of appropriate Rosso-Jones formula. According to universality, these polynomials coincide with adjoined colored HOMFLY and Kauffman polynomials at SL and SO/Sp lines on Vogel's plane, and give their exceptional group's counterparts on exceptional line. We demonstrate that [m,n]=[n,m] topological invariance, when applicable, take place on the entire Vogel's plane. We also suggest the universal form of invariant of figure eight knot in adjoint representation, and suggest existence of such universalization for any knot in adjoint and its descendant representation. Properties of universal polynomials and applications of these results are discussed.
Charles, Denis; Lauter, Kristin
2004-01-01
We present a new probabilistic algorithm to compute modular polynomials modulo a prime. Modular polynomials parameterize pairs of isogenous elliptic curves and are useful in many aspects of computational number theory and cryptography. Our algorithm has the distinguishing feature that it does not involve the computation of Fourier coefficients of modular forms. We avoid computing the exponentially large integral coefficients by working directly modulo a prime and computing isogenies between e...
Laser-induced gratings in the gas phase excited via Raman-active transitions
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.
Reflectivity-modulated grating-mirror
2012-01-01
The invention relates to vertical cavity lasers (VCL) incorporating a reflectivity-modulated grating mirror (1) for modulating the laser output. A cavity is formed by a bottom mirror (4), an active region (3), and an outcoupling top grating mirror (1) formed by a periodic refractive index grating...... advantage of lower power consumption at high modulation speeds.......The invention relates to vertical cavity lasers (VCL) incorporating a reflectivity-modulated grating mirror (1) for modulating the laser output. A cavity is formed by a bottom mirror (4), an active region (3), and an outcoupling top grating mirror (1) formed by a periodic refractive index grating...... 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 to the...
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
STABILITY OF SWITCHED POLYNOMIAL SYSTEMS
Zhiqiang LI; Yupeng QIAO; Hongsheng QI; Daizhan CHENG
2008-01-01
This paper investigates the stability of (switched) polynomial systems. Using semi-tensor product of matrices, the paper develops two tools for testing the stability of a (switched) polynomial system. One is to convert a product of multi-variable polynomials into a canonical form, and the other is an easily verifiable sufficient condition to justify whether a multi-variable polynomial is positive definite. Using these two tools, the authors construct a polynomial function as a candidate Lyapunov function and via testing its derivative the authors provide some sufficient conditions for the global stability of polynomial systems.
Complexity of Ising Polynomials
Kotek, Tomer
2011-01-01
This paper deals with the partition function of the Ising model from statistical mechanics, which is used to study phase transitions in physical systems. A special case of interest is that of the Ising model with constant energies and external field. One may consider such an Ising system as a simple graph together with vertex and edge weight values. When these weights are considered indeterminates, the partition function for the constant case is a trivariate polynomial Z(G;x,y,z). This polynomial was studied with respect to its approximability by L. A. Goldberg, M. Jerrum and M. Patersonin 2003. Z(G;x,y,z) generalizes a bivariate polynomial Z(G;t,y), which was studied in by D. Andr\\'{e}n and K. Markstr\\"{o}m in 2009. We consider the complexity of Z(G;t,y) and Z(G;x,y,z) in comparison to that of the Tutte polynomial, which is well-known to be closely related to the Potts model in the absence of an external field. We show that Z(G;\\x,\\y,\\z) is #P-hard to evaluate at all points in $mathbb{Q}^3$, except those in ...
Densification via polynomial extensions
Galatos, N.; Horčík, Rostislav
Vienna: Vienna University of Technology, 2014 - (Baaz, M.; Ciabattoni, A.; Hetzl, S.). s. 179-182 [LATD 2014. Logic, Algebra and Truth Degrees. 16.07.2014-19.07.2014, Vienna] Institutional support: RVO:67985807 Keywords : densification * commutative ordered monoid * commutative residuated chain * idempotent semiring * polynomial extension Subject RIV: BA - General Mathematics
Nonconventional Polynomial CLT
Hafouta, Y.; Kifer, Y.
2015-01-01
We obtain a functional central limit theorem (CLT) for sums of the form $\\xi_N(t)=\\frac1{\\sqrt N}\\sum_{n=1}^{[Nt]}\\big(F(X(q_1(n)),...,X(q_\\ell(n)))-\\bar F\\big)$ where $q_1,...,q_\\ell$ are polynomials.
Kreso, Dijana; Tichy, Robert F.
2015-01-01
Starting from Ritt's classical theorems, we give a survey of results in functional decomposition of polynomials and of applications in Diophantine equations. This includes sufficient conditions for the indecomposability of polynomials, the study of decompositions of lacunary polynomials and the finiteness criterion for the equations of type f(x) = g(y).
Computing the Alexander Polynomial Numerically
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....
Chromatic polynomials for simplicial complexes
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 r is the n...
R.J. Stroeker (Roel)
2002-01-01
textabstractA Q-derived polynomial is a univariate polynomial, defined over the rationals, with the property that its zeros, and those of all its derivatives are rational numbers. There is a conjecture that says that Q-derived polynomials of degree 4 with distinct roots for themselves and all their
Li, Weijie; Ho, Siu Chun Michael; Song, Gangbing
2016-04-01
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.
Complex Polynomial Vector Fields
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...... 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....
Oblivious Polynomial Evaluation
Hong-Da Li; Dong-Yao Ji; Deng-Guo Feng; Bao Li
2004-01-01
The problem of two-party oblivious polynomial evaluation(OPE)is studied,where one party(Alice)has a polynomial P(x)and the other party(Bob)with an input x wants to learn P(x)in such an oblivious way that Bob obtains P(x)without learning any additional information about P except what is implied by P(x)and Alice does not know Bob's input x.The former OPE protocols are based on an intractability assumption except for OT protocols.In fact,evaluating P(x)is equivalent to computing the product of the coefficient vectors(a0,...,an)and(1,...,xn).Using this idea,an efficient scale product protocol of two vectors is proposed first and then two OPE protocols are presented which do not need any other cryptographic assumption except for OT protocol.Compared with the existing OPE protocol,another characteristic of the proposed protocols is the degree of the polynomial is private.Another OPE protocol works in case of existence of untrusted third party.
Polynomial Learning of Distribution Families
Belkin, Mikhail
2010-01-01
The question of polynomial learnability of probability distributions, particularly Gaussian mixture distributions, has recently received significant attention in theoretical computer science and machine learning. However, despite major progress, the general question of polynomial learnability of Gaussian mixture distributions still remained open. The current work resolves the question of polynomial learnability for Gaussian mixtures in high dimension with an arbitrary fixed number of components. The result on learning Gaussian mixtures relies on an analysis of distributions belonging to what we call "polynomial families" in low dimension. These families are characterized by their moments being polynomial in parameters and include almost all common probability distributions as well as their mixtures and products. Using tools from real algebraic geometry, we show that parameters of any distribution belonging to such a family can be learned in polynomial time and using a polynomial number of sample points. The r...
Symmetric functions and Hall polynomials
MacDonald, Ian Grant
1998-01-01
This reissued classic text is the acclaimed second edition of Professor Ian Macdonald's groundbreaking monograph on symmetric functions and Hall polynomials. The first edition was published in 1979, before being significantly expanded into the present edition in 1995. This text is widely regarded as the best source of information on Hall polynomials and what have come to be known as Macdonald polynomials, central to a number of key developments in mathematics and mathematical physics in the 21st century Macdonald polynomials gave rise to the subject of double affine Hecke algebras (or Cherednik algebras) important in representation theory. String theorists use Macdonald polynomials to attack the so-called AGT conjectures. Macdonald polynomials have been recently used to construct knot invariants. They are also a central tool for a theory of integrable stochastic models that have found a number of applications in probability, such as random matrices, directed polymers in random media, driven lattice gases, and...
Polynomial Regression on Riemannian Manifolds
Hinkle, Jacob; Fletcher, P Thomas; Joshi, Sarang
2012-01-01
In this paper we develop the theory of parametric polynomial regression in Riemannian manifolds and Lie groups. We show application of Riemannian polynomial regression to shape analysis in Kendall shape space. Results are presented, showing the power of polynomial regression on the classic rat skull growth data of Bookstein as well as the analysis of the shape changes associated with aging of the corpus callosum from the OASIS Alzheimer's study.
Deformed Mittag-Leffler Polynomials
Miomir S. Stankovic; Marinkovic, Sladjana D.; Rajkovic, Predrag M.
2010-01-01
The starting point of this paper are the Mittag-Leffler polynomials introduced by H. Bateman [1]. Based on generalized integer powers of real numbers and deformed exponential function, we introduce deformed Mittag-Leffler polynomials defined by appropriate generating function. We investigate their recurrence relations, differential properties and orthogonality. Since they have all zeros on imaginary axes, we also consider real polynomials with real zeros associated to them.
Witt Rings and Permutation Polynomials
Qifan Zhang
2005-01-01
Let p be a prime number. In this paper, the author sets up a canonical correspondence between polynomial functions over Z/p2Z and 3-tuples of polynomial functions over Z/pZ. Based on this correspondence, he proves and reproves some fundamental results on permutation polynomials mod pl. The main new result is the characterization of strong orthogonal systems over Z/p1Z.
Chaves, Rafael
2016-01-01
It is a recent realization that many of the concepts and tools of causal discovery in machine learning are highly relevant to problems in quantum information, in particular quantum nonlocality. The crucial ingredient in the connection between both fields is the mathematical theory of causality, allowing for the representation of arbitrary causal structures and providing a rigorous tool to reason about probabilistic causation. Indeed, Bell's theorem concerns a very particular kind of causal structure and Bell inequalities are a special case of linear constraints following from such models. It is thus natural to look for generalizations involving more complex Bell scenarios. The problem, however, relies on the fact that such generalized scenarios are characterized by polynomial Bell inequalities and no current method is available to derive them beyond very simple cases. In this work, we make a significant step in that direction, providing a new, general, and conceptually clear method for the derivation of polynomial Bell inequalities in a wide class of scenarios. We also show how our construction can be used to allow for relaxations of causal constraints and naturally gives rise to a notion of nonsignaling in generalized Bell networks.
Complex Polynomial Vector Fields
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...
Polynomial harmonic GMDH learning networks for time series modeling.
Nikolaev, Nikolay Y; Iba, Hitoshi
2003-12-01
This paper presents a constructive approach to neural network modeling of polynomial harmonic functions. This is an approach to growing higher-order networks like these build by the multilayer GMDH algorithm using activation polynomials. Two contributions for enhancement of the neural network learning are offered: (1) extending the expressive power of the network representation with another compositional scheme for combining polynomial terms and harmonics obtained analytically from the data; (2) space improving the higher-order network performance with a backpropagation algorithm for further gradient descent learning of the weights, initialized by least squares fitting during the growing phase. Empirical results show that the polynomial harmonic version phGMDH outperforms the previous GMDH, a Neurofuzzy GMDH and traditional MLP neural networks on time series modeling tasks. Applying next backpropagation training helps to achieve superior polynomial network performances. PMID:14622880
Polynomial weights and code constructions
Massey, J; Costello, D; Justesen, Jørn
1973-01-01
For any nonzero elementcof a general finite fieldGF(q), it is shown that the polynomials(x - c)^i, i = 0,1,2,cdots, have the "weight-retaining" property that any linear combination of these polynomials with coefficients inGF(q)has Hamming weight at least as great as that of the minimum degree...
Parallel Construction of Irreducible Polynomials
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) ...
Multilayer diffraction grating
Barbee, Jr., Troy W.
1990-01-01
This invention is for a reflection diffraction grating that functions at X-ray to VUV wavelengths and at normal angles of incidence. The novel grating is comprised of a laminar grating of period D with flat-topped grating bars. A multiplicity of layered synthetic microstructures, of period d and comprised of alternating flat layers of two different materials, are disposed on the tops of the grating bars of the laminar grating. In another embodiment of the grating, a second multiplicity of layered synthetic microstructures are also disposed on the flat faces, of the base of the grating, between the bars. D is in the approximate range from 3,000 to 50,000 Angstroms, but d is in the approximate range from 10 to 400 Angstroms. The laminar grating and the layered microstructures cooperatively interact to provide many novel and beneficial instrumentational advantages.
Cyclotomy and permutation polynomials of large indices
WANG Qiang
2012-01-01
We use cyclotomy to design new classes of permutation polynomials over finite fields. This allows us to generate many classes of permutation polynomials in an algorithmic way. Many of them are permutation polynomials of large indices.
Difference equations of q-Appell polynomials
Mahmudov, Nazim I.
2014-01-01
In this paper, we study some properties of the q-Appell polynomials, including the recurrence relations and the q-difference equations which extend some known calssical (q=1) results. We also provide the recurrence relations and the q-difference equations for q-Bernoulli polynomials, q-Euler polynomials, q-Genocchi polynomials and for newly defined q-Hermite polynomials, as special cases of q-Appell polynomials
Complex Roots of Quaternion Polynomials
Dospra, Petroula; Poulakis, Dimitrios
2015-01-01
The polynomials with quaternion coefficients have two kind of roots: isolated and spherical. A spherical root generates a class of roots which contains only one complex number $z$ and its conjugate $\\bar{z}$, and this class can be determined by $z$. In this paper, we deal with the complex roots of quaternion polynomials. More precisely, using B\\'{e}zout matrices, we give necessary and sufficient conditions, for a quaternion polynomial to have a complex root, a spherical root, and a complex is...
Orthogonal polynomials and deformed oscillators
Borzov, V. V.; Damaskinsky, E. V.
2015-10-01
In the example of the Fibonacci oscillator, we discuss the construction of oscillator-like systems associated with orthogonal polynomials. We also consider the question of the dimensions of the corresponding Lie algebras.
Polynomials Associated with Dihedral Groups
Charles F. Dunkl
2007-03-01
Full Text Available There is a commutative algebra of differential-difference operators, with two parameters, associated to any dihedral group with an even number of reflections. The intertwining operator relates this algebra to the algebra of partial derivatives. This paper presents an explicit form of the action of the intertwining operator on polynomials by use of harmonic and Jacobi polynomials. The last section of the paper deals with parameter values for which the formulae have singularities.
An introduction to orthogonal polynomials
Chihara, Theodore S
2011-01-01
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
Evaluations of topological Tutte polynomials
Ellis-Monaghan, Joanna A
2011-01-01
We find a number of new combinatorial identities for, and interpretations of evaluations of, the topological Tutte polynomials of Las Vergnas, $L(G)$, and of and Bollob\\'as and Riordan, $R(G)$, as well as for the classical Tutte polynomial $T(G)$. For example, we express $R(G)$ and $T(G)$ as a sum of chromatic polynomials, show that $R(G)$ counts non-crossing graph states and $k$-valuations, and reformulate the Four Colour Theorem in terms of $R(G)$. Our main approach is to apply identities for the topological transition polynomial, one involving twisted duals, and one involving doubling the edges of a graph. These identities for the transition polynomial allow us to show that the Penrose polynomial $P(G)$ can be recovered from $R(G)$, a fact that we use to obtain identities and interpretations for $R(G)$. We also consider enumeration of circuits in medial graphs and use this to relate $R(G)$ and $L(G)$ for graphs embedded in low genus surfaces.
Martínez Manuel, Rodolfo; Kaboko, J. J. M.; Shlyagin, M. G.
2016-02-01
We propose and demonstrate a simple and robust actively Q-switched erbium-doped fiber ring cavity laser. The Q-switching is based on dynamic spectral overlapping of two filters, namely a fiber Bragg grating-based filter and a fiber Fabry-Perot tunable filter. Using 3.5 m of erbium-doped fiber and a pump power of only 60 mW, Q-switched pulses with a peak power of 9.7 W and a pulse duration of 500 ns were obtained. A pulse repetition rate can be continuously varied from a single shot to a few KHz.
Multilayer dielectric diffraction gratings
Perry, Michael D.; Britten, Jerald A.; Nguyen, Hoang T.; Boyd, Robert; Shore, Bruce W.
1999-01-01
The design and fabrication of dielectric grating structures with high diffraction efficiency used in reflection or transmission is described. By forming a multilayer structure of alternating index dielectric materials and placing a grating structure on top of the multilayer, a diffraction grating of adjustable efficiency, and variable optical bandwidth can be obtained. Diffraction efficiency into the first order in reflection varying between 1 and 98 percent has been achieved by controlling the design of the multilayer and the depth, shape, and material comprising the grooves of the grating structure. Methods for fabricating these gratings without the use of ion etching techniques are described.
Uniqueness and Zeros of -Shift Difference Polynomials
Kai Liu; Xin-Ling Liu; Ting-Bin Cao
2011-08-01
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 difference polynomials that share a common value.
On the Irreducibility of Some Composite Polynomials
M. Alizadeh
2012-06-01
Full Text Available . In this paper we study the irreducibility of some composite polynomials, constructed by a polynomial composition method over finite fields. Finally, a recurrent method for constructing families of irreducible polynomials of higher degree from given irreducible polynomials over finite fields is given
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.
Ehrhart polynomials of matroid polytopes and polymatroids
De Loera, Jesús A.; Haws, David C.; Köppe, Matthias
2007-01-01
We investigate properties of Ehrhart polynomials for matroid polytopes, independence matroid polytopes, and polymatroids. In the first half of the paper we prove that for fixed rank their Ehrhart polynomials are computable in polynomial time. The proof relies on the geometry of these polytopes as well as a new refined analysis of the evaluation of Todd polynomials. In the second half we discuss two conjectures about the h^*-vector and the coefficients of Ehrhart polynomials of matroid polytop...
Fabrication update on critical-angle transmission gratings for soft x-ray grating spectrometers
Heilmann, Ralf K.; Bruccoleri, Alex; Mukherjee, Pran; Yam, Jonathan; Schattenburg, Mark L.
2011-09-01
Diffraction grating-based, wavelength dispersive high-resolution soft x-ray spectroscopy of celestial sources promises to reveal crucial data for the study of the Warm-Hot Intergalactic Medium, the Interstellar Medium, warm absorption and outflows in Active Galactic Nuclei, coronal emission from stars, and other areas of interest to the astrophysics community. Our recently developed critical-angle transmission (CAT) gratings combine the advantages of the Chandra high and medium energy transmission gratings (low mass, high tolerance of misalignments and figure errors, polarization insensitivity) with those of blazed reflection gratings (high broad band diffraction efficiency, high resolution through use of higher diffraction orders) such as the ones on XMM-Newton. Extensive instrument and system configuration studies have shown that a CAT grating-based spectrometer is an outstanding instrument capable of delivering resolving power on the order of 5,000 and high effective area, even with a telescope point-spread function on the order of many arc-seconds. We have fabricated freestanding, ultra-high aspect-ratio CAT grating bars from silicon-on-insulator wafers using both wet and dry etch processes. The 200 nm-period grating bars are supported by an integrated Level 1 support mesh, and a coarser external Level 2 support mesh. The resulting grating membrane is mounted to a frame, resulting in a grating facet. Many such facets comprise a grating array that provides light-weight coverage of large-area telescope apertures. Here we present fabrication results on the integration of CAT gratings and the different high-throughput support mesh levels and on membrane-frame bonding. We also summarize recent x-ray data analysis of 3 and 6 micron deep wet-etched CAT grating prototypes.
q-Bernstein polynomials, q-Stirling numbers and q-Bernoulli polynomials
Kim, T.
2010-01-01
In this paper, we give new identities involving Phillips q-Bernstein polynomials and we derive some interesting properties of q-Berstein polynomials associated with q-Stirling numbers and q-Bernoulli polynomials.
Modular forms and period polynomials
Pasol, Vicentiu
2012-01-01
We study the space of period polynomials associated with modular forms for finite index subgroups of the modular group. For the full modular group, this space is endowed with a pairing, corresponding to the Petersson inner product on modular forms via a formula of Haberland, and with an action of Hecke operators, defined algebraically by Zagier. We extend Haberland's formula to arbitrary modular forms for finite index subgroups of the modular group, and we show that it conceals two stronger formulas. We extend the action of Hecke operators to \\Gamma_0(N) and \\Gamma_1(N), and we prove algebraically that the pairing on period polynomials appearing in Haberland's formula is Hecke equivariant. Two indefinite theta series identities follow from this proof. We give two ways of determining the extra relations satisfied by the even and odd parts of period polynomials associated with cusp forms, which are independent of the period relations.
Plain Polynomial Arithmetic on GPU
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 212 while on CPU the same threshold is usually at 26. 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 218 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).
Symbolic computation of Appell polynomials using Maple
H. Alkahby
2001-07-01
Full Text Available This work focuses on the symbolic computation of Appell polynomials using the computer algebra system Maple. After describing the traditional approach of constructing Appell polynomials, the paper examines the operator method of constructing the same Appell polynomials. The operator approach enables us to express the Appell polynomial as Bessel function whose coefficients are Euler and Bernuolli numbers. We have also constructed algorithms using Maple to compute Appell polynomials based on the methods we have described. The achievement is the construction of Appell polynomials for any function of bounded variation.
Diffraction by m-bonacci gratings
Monsoriu, Juan A.; Giménez, Marcos H.; Furlan, Walter D.; Barreiro, Juan C.; Saavedra, Genaro
2015-11-01
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.
Two polynomial division inequalities in
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.
Entanglement conditions and polynomial identities
We develop a rather general approach to entanglement characterization based on convexity properties and polynomial identities. This approach is applied to obtain simple and efficient entanglement conditions that work equally well in both discrete as well as continuous-variable environments. Examples of violations of our conditions are presented.
On Modular Counting with Polynomials
Hansen, Kristoffer Arnsfelt
For any integers m and l, where m has r sufficiently large (depending on l) factors, that are powers of r distinct primes, we give a construction of a (symmetric) polynomial over Z_m of degree O(\\sqrt n) that is a generalized representation (commonly also called weak representation) of the MODl f...
Sheffer and Non-Sheffer Polynomial Families
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.
Quantum F-polynomials in Classical Types
Tran, Thao
2009-01-01
In their "Cluster Algebras IV" paper, Fomin and Zelevinsky defined F-polynomials and g-vectors, and they showed that the cluster variables in any cluster algebra can be expressed in a formula involving the appropriate F-polynomial and g-vector. In "F-polynomials in Quantum Cluster Algebras," the predecessor to this paper, we defined and proved the existence of quantum F-polynomials, which are analogs of F-polynomials in quantum cluster algebras in the sense that cluster variables in any quantum cluster algebra can be expressed in a similar formula in terms of quantum F-polynomials and g-vectors. In this paper, we give formulas for both F-polynomials and quantum F-polynomials for cluster algebras of classical type when the initial exchange matrix is acyclic.
Bidirectional grating compressors
Wang, Cheng; Li, Zhaoyang; Li, Shuai; Liu, Yanqi; Leng, Yuxin; Li, Ruxin
2016-07-01
A bidirectional grating compressor for chirped pulse amplifiers is presented. It compresses a laser beam simultaneously in two opposite directions. The pulse compressor is shown to promote chirped pulse amplifiers' output energy without grating damages. To verify the practicability, an experiment is carried out. In addition, a crosscorrelation instrument is designed and set up to test the time synchronization between these two femtosecond pulses.
Bak, Dongsu; Kim, Sang Pyo; Kim, Sung Ku; Soh, Kwang-Sup; Yee, Jae Hyung
1998-01-01
We consider an optical diffraction grating in which the spatial distribution of open slits forms a fractal set. The Fraunhofer diffraction patterns through the fractal grating are obtained analytically for the simplest triad Cantor type and its generalized version. The resulting interference patterns exhibit characteristics of the original fractals and their scaling properties.
Application of Chebyshev Polynomial to simulated modeling
CHI Hai-hong; LI Dian-pu
2006-01-01
Chebyshev polynomial is widely used in many fields, and used usually as function approximation in numerical calculation. In this paper, Chebyshev polynomial expression of the propeller properties across four quadrants is given at first, then the expression of Chebyshev polynomial is transformed to ordinary polynomial for the need of simulation of propeller dynamics. On the basis of it,the dynamical models of propeller across four quadrants are given. The simulation results show the efficiency of mathematical model.
An Improved Volumetric Estimation Using Polynomial Regression
Noraini Abdullah; Amran Ahmed; Zainodin Hj. Jubok
2011-01-01
The polynomial regression (PR) technique is used to estimate the parameters of the dependent variable having a polynomial relationship with the independent variable. Normality and nonlinearity exhibit polynomial characterization of power terms greater than 2. Polynomial Regression models (PRM) with the auxiliary variables are considered up to their third order interactions. Preliminary, multicollinearity between the independent variables is minimized and statistical tests involving the Global...
Computing the zeros of quaternion polynomials
Serôdio, R.; Pereira, E.; Vitória, J.
2001-01-01
A method is developed to compute the zeros of a quaternion polynomial with all terms of the form qkXk. This method is based essentially in Niven's algorithm [1], which consists of dividing the polynomial by a characteristic polynomial associated to a zero. The information about the trace and the norm of the zero is obtained by an original idea which requires the companion matrix associated to the polynomial. The companion matrix is represented by a matrix with complex entries. Three numerical...
A Class of Binomial Permutation Polynomials
Tu, Ziran; Zeng, Xiangyong; Hu, Lei; Li, Chunlei
2013-01-01
In this note, a criterion for a class of binomials to be permutation polynomials is proposed. As a consequence, many classes of binomial permutation polynomials and monomial complete permutation polynomials are obtained. The exponents in these monomials are of Niho type.
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.
s-Numbers sequences for homogeneous polynomials
Caliskan, Erhan; Rueda, Pilar
2015-01-01
We extend the well known theory of $s$-numbers of linear operators to homogeneous polynomials defined between Banach spaces. Approximation, Kolmogorov and Gelfand numbers of polynomials are introduced and some well-known results of the linear and multilinear settings are obtained for homogeneous polynomials.
Graph colorings, flows and arithmetic Tutte polynomial
D'Adderio, Michele; Moci, Luca
2011-01-01
We introduce the notions of arithmetic colorings and arithmetic flows over a graph with labelled edges, which generalize the notions of colorings and flows over a graph. We show that the corresponding arithmetic chromatic polynomial and arithmetic flow polynomial are given by suitable specializations of the associated arithmetic Tutte polynomial, generalizing classical results of Tutte.
Lattice Platonic Solids and their Ehrhart polynomial
E. J. Ionascu
2013-01-01
Full Text Available First, we calculate the Ehrhart polynomial associated to an arbitrary cube with integer coordinates for its vertices. Then, we use this result to derive relationships between the Ehrhart polynomials for regular lattice tetrahedra and those for regular lattice octahedra. These relations allow one to reduce the calculation of these polynomials to only one coefficient.
Lattice Platonic Solids and their Ehrhart polynomial
Ionascu, Eugen J
2011-01-01
First, we calculate the Ehrhart polynomial associated to an arbitrary cube with integer coordinates for its vertices. Then, we use this result to derive relationships between the Ehrhart polynomials for regular lattice tetrahedrons and those for regular lattice octahedrons. These relations allow one to reduce the calculation of these polynomials to only one coefficient.
Discriminants of Polynomials Related to Chebyshev Polynomials: The 'Mutt and Jeff' Syndrome
Tran, Khang
2016-01-01
The discriminants of certain polynomials related to Chebyshev polynomials factor into the product of two polynomials, one of which has coefficients that are much larger than the other's. Remarkably, these polynomials of dissimilar size have "almost" the same roots, and their discriminants involve exactly the same prime factors.
Complete Bell polynomials and new generalized identities for polynomials of higher order
Rubinstein, Boris Y
2009-01-01
The relations between the Bernoulli and Eulerian polynomials of higher order and the complete Bell polynomials are found that lead to new identities for the Bernoulli and Eulerian polynomials and numbers of higher order. General form of these identities is considered and generating function for polynomials satisfying this general identity is found.
Boundary integral equation Neumann-to-Dirichlet map method for gratings in conical diffraction.
Wu, Yumao; Lu, Ya Yan
2011-06-01
Boundary integral equation methods for diffraction gratings are particularly suitable for gratings with complicated material interfaces but are difficult to implement due to the quasi-periodic Green's function and the singular integrals at the corners. In this paper, the boundary integral equation Neumann-to-Dirichlet map method for in-plane diffraction problems of gratings [Y. Wu and Y. Y. Lu, J. Opt. Soc. Am. A26, 2444 (2009)] is extended to conical diffraction problems. The method uses boundary integral equations to calculate the so-called Neumann-to-Dirichlet maps for homogeneous subdomains of the grating, so that the quasi-periodic Green's functions can be avoided. Since wave field components are coupled on material interfaces with the involvement of tangential derivatives, a least squares polynomial approximation technique is developed to evaluate tangential derivatives along these interfaces for conical diffraction problems. Numerical examples indicate that the method performs equally well for dielectric or metallic gratings. PMID:21643404
Normal BGG solutions and polynomials
Cap, A; Hammerl, M
2012-01-01
First BGG operators are a large class of overdetermined linear differential operators intrinsically associated to a parabolic geometry on a manifold. The corresponding equations include those controlling infinitesimal automorphisms, higher symmetries, and many other widely studied PDE of geometric origin. The machinery of BGG sequences also singles out a subclass of solutions called normal solutions. These correspond to parallel tractor fields and hence to (certain) holonomy reductions of the canonical normal Cartan connection. Using the normal Cartan connection, we define a special class of local frames for any natural vector bundle associated to a parabolic geometry. We then prove that the coefficient functions of any normal solution of a first BGG operator with respect to such a frame are polynomials in the normal coordinates of the parabolic geometry. A bound on the degree of these polynomials in terms of representation theory data is derived. For geometries locally isomorphic to the homogeneous model of ...
BSDEs with polynomial growth generators
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.
Twisted Polynomials and Forgery Attacks on GCM
Abdelraheem, Mohamed Ahmed A. M. A.; Beelen, Peter; Bogdanov, Andrey;
2015-01-01
twisted polynomials from Ore rings as forgery polynomials. We show how to construct sparse forgery polynomials with full control over the sets of roots. We also achieve complete and explicit disjoint coverage of the key space by these polynomials. We furthermore leverage this new construction 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 is a...
Space complexity in polynomial calculus
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: 0.741, year: 2014 http://epubs.siam.org/doi/10.1137/120895950
Simplifying Tensor Polynomials with Indices
Balfagón, A
1998-01-01
We are presenting an algorithm capable of simplifying tensor polynomials with indices when the building tensors have index symmetry properties. These properties include simple symmetry, cyclicity and those due to the presence of partial and covariant derivatives. We are also including some examples using the Riemann tensor as a paradigm. The algorithm is part of a Mathematica package called Tools of Tensor Calculus (TTC) [web address: http//baldufa.upc.es/ttc
Roots of Quaternion Standard Polynomials
Chapman, Adam
2011-01-01
Here we present a reduction of any quaternion standard polynomial equation into an equation with two central variables and quaternion coefficients. If only pure imaginary roots are in demand, then the equation is with one central variable. As a result of this reduction we obtain formulas for the solutions of quadratic equations. Another result is a routine for analytically solving cubic quaternion equations assuming they have at least one pure imaginary root.
Pattern Matching under Polynomial Transformation
Butman, Ayelet; Clifford, Raphael; Jalsenius, Markus; Lewenstein, Noa; Porat, Benny; Porat, Ely; Sach, Benjamin
2011-01-01
We consider a class of pattern matching problems where a polynomial transformation can be applied to the pattern at every alignment. Given a pattern of length m and a longer text of length n where both are assumed to contain integer values only, we show O(n log m) algorithms for pattern matching under linear transformations even when wildcard symbols can occur in the input. We then show how to extend the technique to polynomial transformations of arbitrary degree. Next we consider the problem of finding the minimum Hamming distance under polynomial transformation. We show that, for any epsilon > 0, there cannot exist an O(nm^(1-epsilon)) algorithm for additive and linear transformations conditional on the hardness of the classic 3SUM problem. Finally, we consider a version of the Hamming distance problem under additive transformations with a bound k on the maximum distance that need be reported. We give a deterministic O(nk log k) time solution which we then improve by careful use of randomisation to O(n sqrt...
Hybrid grating reflector with high reflectivity and broad bandwidth
Taghizadeh, Alireza; Park, Gyeong Cheol; Mørk, Jesper; Chung, Il-Sug
2014-01-01
We suggest a new type of grating reflector denoted hybrid grating (HG) which shows large reflectivity in a broad wavelength range and has a structure suitable for realizing a vertical cavity laser with ultra-small modal volume. The properties of the grating reflector are investigated numerically......). By using an active III-V layer, a laser can be realized where the gain region is integrated into the mirror itself...
Polynomial approximation, local polynomial convexity, and degenerate CR singularities -- II
Bharali, Gautam
2010-01-01
We provide some conditions for the graph of a Hoelder-continuous function on \\bar{D}, where \\bar{D} is a closed disc in the complex plane, to be polynomially convex. Almost all sufficient conditions known to date --- provided the function (say F) is smooth --- arise from versions of the Weierstrass Approximation Theorem on \\bar{D}. These conditions often fail to yield any conclusion if rank_R(DF) is not maximal on a sufficiently large subset of \\bar{D}. We bypass this difficulty by introducin...
In this work, active vibration suppression of a smart cantilever beam subjected to disturbances from multiple impact loadings is investigated with a point-wise fiber Bragg grating (FBG) displacement sensing system. An FBG demodulator is employed in the proposed fiber sensing system to dynamically demodulate the responses obtained by the FBG displacement sensor with high sensitivity. To investigate the ability of the proposed FBG displacement sensor as a feedback sensor, velocity feedback control and delay control are employed to suppress the vibrations of the first three bending modes of the smart cantilever beam. To improve the control performance for the first bending mode when the cantilever beam is subjected to an impact loading, we improve the conventional velocity feedback controller by tuning the control gain online with the aid of information from a higher vibration mode. Finally, active control of vibrations induced by multiple impact loadings due to a plastic ball is performed with the improved velocity feedback control. The experimental results show that active vibration control of smart structures subjected to disturbances such as impact loadings can be achieved by employing the proposed FBG sensing system to feed back out-of-plane point-wise displacement responses with high sensitivity. (paper)
Modeling Component-based Bragg gratings Application: tunable lasers
Hedara Rachida
2011-09-01
Full Text Available The principal function of a grating Bragg is filtering, which can be used in optical fibers based component and active or passive semi conductors based component, as well as telecommunication systems. Their ideal use is with lasers with fiber, amplifiers with fiber or Laser diodes. In this work, we are going to show the principal results obtained during the analysis of various types of grating Bragg by the method of the coupled modes. We then present the operation of DBR are tunable. The use of Bragg gratings in a laser provides single-mode sources, agile wavelength. The use of sampled grating increases the tuning range.
Chromatic Polynomials of Mixed Hypercycles
Allagan Julian A.
2014-08-01
Full Text Available We color the vertices of each of the edges of a C-hypergraph (or cohypergraph in such a way that at least two vertices receive the same color and in every proper coloring of a B-hypergraph (or bihypergraph, we forbid the cases when the vertices of any of its edges are colored with the same color (monochromatic or when they are all colored with distinct colors (rainbow. In this paper, we determined explicit formulae for the chromatic polynomials of C-hypercycles and B-hypercycles
Zeroes of random Reinhardt polynomials
Karami, Arash
2012-01-01
For a Reinhardt domain $\\Omega$ with the smooth boundary in $\\mathbb{C}^{m+1}$ and a positive smooth measure $\\mu$ on the boundary of $\\Omega$, we consider the ensemble $P_{N}$ of polynomials of degree $N$ with the Gaussian probability measure $\\gamma_{N}$ which is induced by $L^{2}(\\partial\\Omega,d\\mu)$. Our aim is to compute scaling limit distribution function and scaling limit pair correlation function between zeros when $z\\in\\partial\\Omega$. First of all we apply stationary phase method t...
Global Polynomial Kernel Hazard Estimation
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...... reduces bias with unchanged variance. A simulation study investigates the finite-sample properties of GPA. The method is tested on local constant and local linear estimators. From the simulation experiment we conclude that the global estimator improves the goodness-of-fit. An especially encouraging result...
Nonnegative Polynomials and Sums of Squares
Blekherman, Grigoriy
2010-01-01
In the smallest cases where there exist nonnegative polynomials that are not sums of squares we present a complete classification of the differences between these sets. We show that in these cases the fundamental reason that the set of sums of squares is smaller than the set of nonnegative polynomials is that polynomials of degree d satisfy certain linear relations known as the Cayley-Bacharach relations, which are not satisfied by polynomials of full degree 2d. For any nonnegative polynomial that is not a sum of squares we can write down a linear inequality coming from a Cayley-Bacharach relation that certifies that the polynomial is not a sum of squares. We also present structure results on the strictly positive sums of squares that lie on the boundary of the cone of sums of squares and results on extreme rays of the cone dual to the cone of sums of squares.
An Improved Volumetric Estimation Using Polynomial Regression
Noraini Abdullah
2011-12-01
Full Text Available The polynomial regression (PR technique is used to estimate the parameters of the dependent variable having a polynomial relationship with the independent variable. Normality and nonlinearity exhibit polynomial characterization of power terms greater than 2. Polynomial Regression models (PRM with the auxiliary variables are considered up to their third order interactions. Preliminary, multicollinearity between the independent variables is minimized and statistical tests involving the Global, Correlation Coefficient, Wald, and Goodness-of-Fit tests, are carried out to select significant variables with their possible interactions. Comparisons between the polynomial regression models (PRM are made using the eight selection criteria (8SC. The best regression model is identified based on the minimum value of the eight selection criteria (8SC. The use of an appropriate transformation will increase in the degree of a statistically valid polynomial, hence, providing a better estimation for the model.
Exceptional polynomials and SUSY quantum mechanics
K V S Shiv Chaitanya; S Sree Ranjani; Prasanta K Panigrahi; R Radhakrishnan; V Srinivasan
2015-07-01
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. Then, we claim that the existence of these exceptional polynomials leads to the presence of non-trivial supersymmetry.
A new Arnoldi approach for polynomial eigenproblems
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.
Landau and Kolmogoroff type polynomial inequalities
Alves Claudia RR; Dimitrov Dimitar K
1999-01-01
Let be integers. Denote by the norm . For various positive values of and we establish Kolmogoroff type inequalities with certain constants , which hold for every ( denotes the space of real algebraic polynomials of degree not exceeding ). For the particular case and , we provide a complete characterisation of the positive constants and , for which the corresponding Landau type polynomial inequalities hold. In each case we determine the corresponding extremal polynomials for which e...
Haglund's conjecture on 3-column Macdonald polynomials
Blasiak, Jonah
2014-01-01
We prove a positive combinatorial formula for the Schur expansion of LLT polynomials indexed by a 3-tuple of skew shapes. This verifies a conjecture of Haglund. The proof requires expressing a noncommutative Schur function as a positive sum of monomials in Lam's algebra of ribbon Schur operators. Combining this result with the expression of Haglund, Haiman, and Loehr for transformed Macdonald polynomials in terms of LLT polynomials then yields a positive combinatorial rule for transformed Mac...
Accelerated graph-based spectral polynomial filters
Knyazev, Andrew; Malyshev, Alexander,
2015-01-01
Graph-based spectral denoising is a low-pass filtering using the eigendecomposition of the graph Laplacian matrix of a noisy signal. Polynomial filtering avoids costly computation of the eigendecomposition by projections onto suitable Krylov subspaces. Polynomial filters can be based, e.g., on the bilateral and guided filters. We propose constructing accelerated polynomial filters by running flexible Krylov subspace based linear and eigenvalue solvers such as the Block Locally Optimal Precond...
Quantum Schubert polynomials and quantum Schur functions
Kirillov, Anatol N.
1997-01-01
We introduce the quantum multi-Schur functions, quantum factorial Schur functions and quantum Macdonald polynomials. We prove that for restricted vexillary permutations the quantum double Schubert polynomial coincides with some quantum multi-Schur function and prove a quantum analog of the Nagelsbach-Kostka and Jacobi-Trudi formulae for the quantum double Schubert polynomials in the case of Grassmannian permutations. We prove, also, an analog of the Billey-Jockusch-Stanley formula for quantum...
Generalizations of Bernoulli numbers and polynomials
Qiu-Ming Luo; Bai-Ni Guo; Feng Qi; Lokenath Debnath
2003-01-01
The concepts of Bernoulli numbers Bn, Bernoulli polynomials Bn(x), and the generalized Bernoulli numbers Bn(a,b) are generalized to the one Bn(x;a,b,c) which is called the generalized Bernoulli polynomials depending on three positive real parameters. Numerous properties of these polynomials and some relationships between Bn, Bn(x), Bn(a,b), and Bn(x;a,b,c) are established.
About polynomials related to a quadratic equation
Groux, Roland
2011-01-01
We consider here a particular quadratic equation linking two elements of a C-Algebra. By analysing powers of the unknowns, it appears a double sequence of polynomials related to classical Bernoulli polynomials. We get the generating functions, integral forms and explicit formulas for the coefficients involving cosecant and tangent numbers. We also study the use of these polynomials for the calculation of some integral transforms.
Discrete least squares approximation with polynomial vectors
Van Barel, Marc; Bultheel, Adhemar
1993-01-01
We give a solution of a discrete least squares approximation problem in terms of orthogonal polynomial vectors. The degrees of the polynomial elements of these vectors can be different. An algorithm is constructed computing the coefficients of recurrence relations for the orthogonal polynomial vectors. In case the function values are prescribed in points on the real line or on the unit circle variants of the original algorithm can be designed which are an order of magnitude more efficient. Al...
On permutation polynomials over finite fields
C. Small; R. A. Mollin
1987-01-01
A polynomial f over a finite field F is called a permutation polynomial if the mapping FÃ¢Â†Â’F defined by f is one-to-one. In this paper we consider the problem of characterizing permutation polynomials; that is, we seek conditions on the coefficients of a polynomial which are necessary and sufficient for it to represent a permutation. We also give some results bearing on a conjecture of Carlitz which says essentially that for any even integer m, the cardinality of finite fields admitting pe...
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. PMID:27190749
Control to Facet for Polynomial Systems
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....
Polynomial chaotic inflation in supergravity revisited
Kazunori Nakayama
2014-10-01
Full Text Available We revisit a polynomial chaotic inflation model in supergravity which we proposed soon after the Planck first data release. Recently some issues have been raised in Ref. [12], concerning the validity of our polynomial chaotic inflation model. We study the inflaton dynamics in detail, and confirm that the inflaton potential is very well approximated by a polynomial potential for the parameters of our interest in any practical sense, and in particular, the spectral index and the tensor-to-scalar ratio can be estimated by single-field approximation. This justifies our analysis of the polynomial chaotic inflation in supergravity.
Multi-indexed (q)-Racah Polynomials
Odake, Satoru
2012-01-01
As the second stage of the project $multi-indexed orthogonal polynomials$, we present, in the framework of `discrete quantum mechanics' with real shifts in one dimension, the multi-indexed (q)-Racah polynomials. They are obtained from the (q)-Racah polynomials by multiple application of the discrete analogue of the Darboux transformations or the Crum-Krein-Adler deletion of `virtual state' vectors of type I and II, in a similar way to the multi-indexed Laguerre and Jacobi polynomials reported earlier. The virtual state vectors are the `solutions' of the matrix Schr\\"odinger equation with negative `eigenvalues', except for one of the two boundary points.
Macdonald Polynomials and Multivariable Basic Hypergeometric Series
Michael J. Schlosser
2007-03-01
Full Text Available We study Macdonald polynomials from a basic hypergeometric series point of view. In particular, we show that the Pieri formula for Macdonald polynomials and its recently discovered inverse, a recursion formula for Macdonald polynomials, both represent multivariable extensions of the terminating very-well-poised ${}_6phi_5$ summation formula. We derive several new related identities including multivariate extensions of Jackson's very-well-poised ${}_8phi_7$ summation. Motivated by our basic hypergeometric analysis, we propose an extension of Macdonald polynomials to Macdonald symmetric functions indexed by partitions with complex parts. These appear to possess nice properties.
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.
Assaf, Sami; Searles, Dominic
2016-01-01
We introduce two new bases for polynomials that lift monomial and fundamental quasisymmetric functions to the full polynomial ring. By defining a new condition on pipe dreams, called quasi-Yamanouchi, we give a positive combinatorial rule for expanding Schubert polynomials into these new bases that parallels the expansion of Schur functions into fundamental quasisymmetric functions. As a result, we obtain a refinement of the stable limits of Schubert polynomials to Stanley symmetric functions...
Chien-Ching Ma
2011-12-01
Full Text Available In this work, a fiber Bragg grating (FBG sensing system which can measure the transient response of out-of-plane point-wise displacement responses is set up on a smart cantilever beam and the feasibility of its use as a feedback sensor in an active structural control system is studied experimentally. An FBG filter is employed in the proposed fiber sensing system to dynamically demodulate the responses obtained by the FBG displacement sensor with high sensitivity. For comparison, a laser Doppler vibrometer (LDV is utilized simultaneously to verify displacement detection ability of the FBG sensing system. An optical full-field measurement technique called amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI is used to provide full-field vibration mode shapes and resonant frequencies. To verify the dynamic demodulation performance of the FBG filter, a traditional FBG strain sensor calibrated with a strain gauge is first employed to measure the dynamic strain of impact-induced vibrations. Then, system identification of the smart cantilever beam is performed by FBG strain and displacement sensors. Finally, by employing a velocity feedback control algorithm, the feasibility of integrating the proposed FBG displacement sensing system in a collocated feedback system is investigated and excellent dynamic feedback performance is demonstrated. In conclusion, our experiments show that the FBG sensor is capable of performing dynamic displacement feedback and/or strain measurements with high sensitivity and resolution.
Tutorial: Applications of Fibre Gratings
Hwayaw; Tam; Bai; ou; Guan; Shunyee; Liu
2003-01-01
Fibre grating is an important enabling technology that has found numerous applications in both telecommunications and sensor systems. This tutorial describes the basic characteristics of fibre gratings and gives examples of where they are being employed.
Phase gratings for plasmon focusing
Offerhaus, H.L.; Bergen, van den, GJA Gino; Hulst, van, N.F.
2005-01-01
We report gratings structures realized for the creation of focused plasmons through noncollinear phasematching. The gratings are created on gold by focused ion beam milling and the plasmons were measured using phase sensitive photon scanning tunneling microscope (PSTM).
Maystre, Daniel
2014-01-01
The chapter contains a detailed presentation of the surface integral theory for modelling light diffraction by surface-relief diffraction gratings having a one-dimensional periodicity. Several different approaches are presented, leading either to a single integral equation, or to a system of coupled integral equations. Special attention is paid to the singularities of the kernels, and to different techniques to accelerate the convergence of the numerical computations. The theory is applied to gratings having different profiles with or without edges, to real metal and dielectrics, and to perfectly conducting substrates.
Ultra-High Temperature Gratings
John Canning; Somnath Bandyopadhyay; Michael Stevenson; Kevin Cook
2008-01-01
Regenerated gratings seeded by type-Ⅰ gratings are shown to withstand temperatures beyond 1000 ℃. The method of regeneration offers a new approach to increasing temperature resistance of stable fibre Bragg and other gratings. These ultra-high temperature (UHT) gratings extend the applicability of silicate based components to high temperature applications such as monitoring of smelters and vehicle and aircraft engines to high power fibre lasers.
Polynomial invariants of quantum codes
Rains, E M
1997-01-01
The weight enumerators (quant-ph/9610040) of a quantum code are quite powerful tools for exploring its structure. As the weight enumerators are quadratic invariants of the code, this suggests the consideration of higher-degree polynomial invariants. We show that the space of degree k invariants of a code of length n is spanned by a set of basic invariants in one-to-one correspondence with S_k^n. We then present a number of equations and inequalities in these invariants; in particular, we give a higher-order generalization of the shadow enumerator of a code, and prove that its coefficients are nonnegative. We also prove that the quartic invariants of a ((4,4,2)) are uniquely determined, an important step in a proof that any ((4,4,2)) is additive ([2]).
Algebras, dialgebras, and polynomial identities
Bremner, Murray R
2012-01-01
This is a survey of some recent developments in the theory of associative and nonassociative dialgebras, with an emphasis on polynomial identities and multilinear operations. We discuss associative, Lie, Jordan, and alternative algebras, and the corresponding dialgebras; the KP algorithm for converting identities for algebras into identities for dialgebras; the BSO algorithm for converting operations in algebras into operations in dialgebras; Lie and Jordan triple systems, and the corresponding disystems; and a noncommutative version of Lie triple systems based on the trilinear operation abc-bca. The paper concludes with a conjecture relating the KP and BSO algorithms, and some suggestions for further research. Most of the original results are joint work with Raul Felipe, Luiz A. Peresi, and Juana Sanchez-Ortega.
Fuzzy Morphological Polynomial Image Representation
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.
Elementary combinatorics of the HOMFLYPT polynomial
Chmutov, Sergei; Polyak, Michael
2008-01-01
We explore Jaeger's state model for the HOMFLYPT polynomial. We reformulate this model in the language of Gauss diagrams and use it to obtain Gauss diagram formulas for a two-parameter family of Vassiliev invariants coming from the HOMFLYPT polynomial. These formulas are new already for invariants of degree 3.
Application of polynomial preconditioners to conservation laws
Geurts, Bernard J.; Buuren, van René; Lu, Hao
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 p
An inequality for polynomials with elliptic majorant
Nikolov Geno
1999-01-01
Let be the transformed Chebyshev polynomial of the first kind, where . We show here that has the greatest uniform norm in of its -th derivative among all algebraic polynomials of degree not exceeding , which vanish at and satisfy the inequality at the points .
The Bessel polynomials and their differential operators
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
Sums of Powers of Fibonacci Polynomials
Helmut Prodinger
2009-11-01
Using the explicit (Binet) formula for the Fibonacci polynomials, a summation formula for powers of Fibonacci polynomials is derived straightforwardly, which generalizes a recent result for squares that appeared in Proc. Ind. Acad. Sci. (Math. Sci.) 118 (2008) 27--41.
The weighted lattice polynomials as aggregation functions
Marichal, Jean-Luc
2006-01-01
We define the concept of weighted lattice polynomials as lattice polynomials constructed from both variables and parameters. We provide equivalent forms of these functions in an arbitrary bounded distributive lattice. We also show that these functions include the class of discrete Sugeno integrals and that they are characterized by a remarkable median based decomposition formula.
A Note on Solvable Polynomial Algebras
Huishi Li
2014-03-01
Full Text Available In terms of their defining relations, solvable polynomial algebras introduced by Kandri-Rody and Weispfenning [J. Symbolic Comput., 9(1990] are characterized by employing Gr\\"obner bases of ideals in free algebras, thereby solvable polynomial algebras are completely determinable and constructible in a computational way.
V K Jain
2009-02-01
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).
A generalization of the Bernoulli polynomials
Pierpaolo Natalini; Angela Bernardini
2003-01-01
A generalization of the Bernoulli polynomials and, consequently, of the Bernoulli numbers, is defined starting from suitable generating functions. Furthermore, the differential equations of these new classes of polynomials are derived by means of the factorization method introduced by Infeld and Hull (1951).
A generalization of the Bernoulli polynomials
Pierpaolo Natalini
2003-01-01
Full Text Available A generalization of the Bernoulli polynomials and, consequently, of the Bernoulli numbers, is defined starting from suitable generating functions. Furthermore, the differential equations of these new classes of polynomials are derived by means of the factorization method introduced by Infeld and Hull (1951.
Quantum Search for Zeros of Polynomials
Weigert, S
2003-01-01
A quantum mechanical search procedure to determine the real zeros of a polynomial is introduced. It is based on the construction of a spin observable whose eigenvalues coincide with the zeros of the polynomial. Subsequent quantum mechanical measurements of the observable output directly the numerical values of the zeros. Performing the measurements is the only computational resource involved.
A quantum search for zeros of polynomials
Weigert, Stefan [HuMP-Hull Mathematical Physics, Department of Mathematics, University of Hull, Hull HU6 7RX (United Kingdom)
2003-12-01
A quantum mechanical search procedure to determine the real zeros of a polynomial is introduced. It is based on the construction of a spin observable whose eigenvalues coincide with the zeros of the polynomial. Subsequent quantum mechanical measurements of the observable output directly the numerical values of the zeros. Performing the measurements is the only computational resource involved.
Point vortex equilibria related to Bessel polynomials
O'Neil, Kevin A.
2016-05-01
The method of polynomials is used to construct two families of stationary point vortex configurations. The vortices are placed at the reciprocals of the zeroes of Bessel polynomials. Configurations that translate uniformly, and configurations that are completely stationary, are obtained in this way.
Large degree asymptotics of generalized Bessel polynomials
López, J.L.; Temme, N.M.
2011-01-01
Asymptotic 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 origin in t
A $(p,q)$-Analogue of Poly-Euler Polynomials and Some Related Polynomials
Komatsu, Takao; Ramírez, José L.; Sirvent, Víctor F.
2016-01-01
In the present article, we introduce a $(p,q)$-analogue of the poly-Euler polynomials and numbers by using the $(p,q)$-polylogarithm function. These new sequences are generalizations of the poly-Euler numbers and polynomials. We give several combinatorial identities and properties of these new polynomials. Moreover, we show some relations with the $(p,q)$-poly-Bernoulli polynomials and $(p,q)$-poly-Cauchy polynomials. The $(p,q)$-analogues generalize the well-known concept of the $q$-analogue.
Sobolev orthogonal polynomials on a simplex
Aktas, Rabia
2011-01-01
The Jacobi polynomials on the simplex are orthogonal polynomials with respect to the weight function $W_\\bg(x) = x_1^{\\g_1} ... x_d^{\\g_d} (1- |x|)^{\\g_{d+1}}$ when all $\\g_i > -1$ and they are eigenfunctions of a second order partial differential operator $L_\\bg$. The singular cases that some, or all, $\\g_1,...,\\g_{d+1}$ are -1 are studied in this paper. Firstly a complete basis of polynomials that are eigenfunctions of $L_\\bg$ in each singular case is found. Secondly, these polynomials are shown to be orthogonal with respect to an inner product which is explicitly determined. This inner product involves derivatives of the functions, hence the name Sobolev orthogonal polynomials.
Orthogonal Polynomials from Hermitian Matrices II
Odake, Satoru
2016-01-01
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 $\\ell^2$ Hilbert space on which a direct sum of two unbounded Jacobi matrices acts as a Hamiltonian or a difference Schr\\"odinger operator for an infinite dimensional eigenvalue problem. The polynomial appearing in the upper/lower end of Jackson integral constitutes the eigenvector of each of the two unbounded Jacobi matrix of the direct sum. We also point out...
Matrix product formula for Macdonald polynomials
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)
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.
Pure Bending Characteristic of Tilted Fiber Bragg Grating
Bo Liu; Yin-Ping Miao; Hai-Bin Zhou; Qi-Da Zhao
2008-01-01
a novel structure of the pure macro-bending sensor based on the tilted fiber Bragg grating (TFBG) is proposed. The TFBG located in the half circle with the different diameters is bent at a constant angle with respect to the tilted grating planes. With the variations of the curvature, the core-mode resonance is unchanged and the transmission power of cladding modes detected by the photodiodes varies linearly with curvature, while the ghost mode changes by the form of two-order polynomial. So we can use the transmission power of ghost mode or other cladding modes to detect bending curvature as shape sensor. From a practical point of view, the sensor proposed here is simple, low cost and easy to implement. Moreover, it is possible to make a temperature-insensitive shape sensor due to the same temperature characteristic between the core mode and the cladding modes.
On some properties on bivariate Fibonacci and Lucas polynomials
Belbachir, Hacéne; Bencherif, Farid
2007-01-01
In this paper we generalize to bivariate polynomials of Fibonacci and Lucas, properties obtained for Chebyshev polynomials. We prove that the coordinates of the bivariate polynomials over appropriate basis are families of integers satisfying remarkable recurrence relations.
Bernoulli-like polynomials associated with Stirling Numbers
Bender, Carl M; Brody, Dorje C.; BERNHARD K. MEISTER
2005-01-01
The Stirling numbers of the first kind can be represented in terms of a new class of polynomials that are closely related to the Bernoulli polynomials. Recursion relations for these polynomials are given.
Degasperis, Antonio; Aceves, Alejandro B
2015-01-01
We derive the rogue wave solution of the classical massive Thirring model, that describes nonlinear optical pulse propagation in Bragg gratings. Combining electromagnetically induced transparency with Bragg scattering four-wave mixing, may lead to extreme waves at extremely low powers.
Silicon graphene Bragg gratings
Capmany, Jose; Domenech, David; Munoz, Pascual
2013-01-01
We propose the use of interleaved graphene sections on top of a silicon waveguide to implement tunable Bragg gratings. The filter central wavelength and bandwidth can be controlled changing the chemical potential of the graphene sections. Apodization techniques are also presented.
Degasperis, Antonio [Dipartimento di Fisica, “Sapienza” Università di Roma, P.le A. Moro 2, 00185 Roma (Italy); Wabnitz, Stefan, E-mail: stefan.wabnitz@unibs.it [Dipartimento di Ingegneria dell' Informazione, Università degli Studi di Brescia and INO-CNR, via Branze 38, 25123 Brescia (Italy); Aceves, Alejandro B. [Southern Methodist University, Dallas (United States)
2015-06-12
We derive the rogue wave solution of the classical massive Thirring model, that describes nonlinear optical pulse propagation in Bragg gratings. Combining electromagnetically induced transparency with Bragg scattering four-wave mixing may lead to extreme waves at extremely low powers.
Electrically-programmable diffraction grating
Ricco, A.J.; Butler, M.A.; Sinclair, M.B.; Senturia, S.D.
1998-05-26
An electrically-programmable diffraction grating is disclosed. The programmable grating includes a substrate having a plurality of electrodes formed thereon and a moveable grating element above each of the electrodes. The grating elements are electrostatically programmable to form a diffraction grating for diffracting an incident beam of light as it is reflected from the upper surfaces of the grating elements. The programmable diffraction grating, formed by a micromachining process, has applications for optical information processing (e.g. optical correlators and computers), for multiplexing and demultiplexing a plurality of light beams of different wavelengths (e.g. for optical fiber communications), and for forming spectrometers (e.g. correlation and scanning spectrometers). 14 figs.
Polynomial Interpolation in the Elliptic Curve Cryptosystem
Liew K. Jie
2011-01-01
Full Text Available Problem statement: In this research, we incorporate the polynomial interpolation method in the discrete logarithm problem based cryptosystem which is the elliptic curve cryptosystem. Approach: In this study, the polynomial interpolation method to be focused is the Lagrange polynomial interpolation which is the simplest polynomial interpolation method. This method will be incorporated in the encryption algorithm of the elliptic curve ElGamal cryptosystem. Results: The scheme modifies the elliptic curve ElGamal cryptosystem by adding few steps in the encryption algorithm. Two polynomials are constructed based on the encrypted points using Lagrange polynomial interpolation and encrypted for the second time using the proposed encryption method. We believe it is safe from the theoretical side as it still relies on the discrete logarithm problem of the elliptic curve. Conclusion/Recommendations: The modified scheme is expected to be more secure than the existing scheme as it offers double encryption techniques. On top of the existing encryption algorithm, we managed to encrypt one more time using the polynomial interpolation method. We also have provided detail examples based on the described algorithm.
More on rotations as spin matrix polynomials
Curtright, Thomas L. [Department of Physics, University of Miami, Coral Gables, Florida 33124-8046 (United States)
2015-09-15
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.
Cycles are determined by their domination polynomials
Akbari, Saieed; Oboudi, Mohammad Reza
2009-01-01
Let $G$ be a simple graph of order $n$. A dominating set of $G$ is a set $S$ of vertices of $G$ so that every vertex of $G$ is either in $S$ or adjacent to a vertex in $S$. The domination polynomial of $G$ is the polynomial $D(G,x)=\\sum_{i=1}^{n} d(G,i) x^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. In this paper we show that cycles are determined by their domination polynomials.
On function compositions that are polynomials
Aichinger, Erhard
2015-01-01
For a polynomial map $\\tupBold{f} : k^n \\to k^m$ ($k$ a field), we investigate those polynomials $g \\in k[t_1,\\ldots, t_n]$ that can be written as a composition $g = h \\circ \\tupBold{f}$, where $h: k^m \\to k$ is an arbitrary function. In the case that $k$ is algebraically closed of characteristic~$0$ and $\\tupBold{f}$ is surjective, we will show that $g = h \\circ \\tupBold{f}$ implies that $h$ is a polynomial.
On Combinatorial Formulas for Macdonald Polynomials
Lenart, Cristian
2008-01-01
A recent breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of a pair of statistics on fillings of Young diagrams. Ram and Yip gave a formula for the Macdonald polynomials of arbitrary type in terms of so-called alcove walks; these originate in the work of Gaussent-Littelmann and of the author with Postnikov on discrete counterparts to the Littelmann path model. In this paper, w...
On Calculation of Adomian Polynomials by MATLAB
Hossein ABOLGHASEMI
2011-01-01
Full Text Available Adomian Decomposition Method (ADM is an elegant technique to handle an extensive class of linear or nonlinear differential and integral equations. However, in case of nonlinear equations, ADM demands a special representation of each nonlinear term, namely, Adomian polynomials. The present paper introduces a novel MATLAB code which computes Adomian polynomials associated with several types of nonlinearities. The code exploits symbolic programming incorporated with a recently proposed alternative scheme to be straightforward and fast. For the sake of exemplification, Adomian polynomials of famous nonlinear operators, computed by the code, are given.
Tutte Polynomial of Scale-Free Networks
Chen, Hanlin; Deng, Hanyuan
2016-05-01
The Tutte polynomial of a graph, or equivalently the q-state Potts model partition function, is a two-variable polynomial graph invariant of considerable importance in both statistical physics and combinatorics. The computation of this invariant for a graph is NP-hard in general. In this paper, we focus on two iteratively growing scale-free networks, which are ubiquitous in real-life systems. Based on their self-similar structures, we mainly obtain recursive formulas for the Tutte polynomials of two scale-free networks (lattices), one is fractal and "large world", while the other is non-fractal but possess the small-world property. Furthermore, we give some exact analytical expressions of the Tutte polynomial for several special points at ( x, y)-plane, such as, the number of spanning trees, the number of acyclic orientations, etc.
Transversals of Complex Polynomial Vector Fields
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...
Characteristic Polynomials of Sample Covariance Matrices
Kösters, Holger
2009-01-01
We investigate the second-order correlation function of the characteristic polynomial of a sample covariance matrix. Starting from an explicit formula for the generating function, we re-obtain several well-known kernels from random matrix theory.
Thermodynamic characterization of networks using graph polynomials
Ye, Cheng; Peron, Thomas K DM; Silva, Filipi N; Rodrigues, Francisco A; Costa, Luciano da F; Torsello, Andrea; Hancock, Edwin R
2015-01-01
In this paper, we present a method for characterizing the evolution of time-varying complex networks by adopting a thermodynamic representation of network structure computed from a polynomial (or algebraic) characterization of graph structure. Commencing from a representation of graph structure based on a characteristic polynomial computed from the normalized Laplacian matrix, we show how the polynomial is linked to the Boltzmann partition function of a network. This allows us to compute a number of thermodynamic quantities for the network, including the average energy and entropy. Assuming that the system does not change volume, we can also compute the temperature, defined as the rate of change of entropy with energy. All three thermodynamic variables can be approximated using low-order Taylor series that can be computed using the traces of powers of the Laplacian matrix, avoiding explicit computation of the normalized Laplacian spectrum. These polynomial approximations allow a smoothed representation of the...
Solving Bivariate Polynomial Systems on a GPU
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.
Superconformal minimal models and admissible Jack polynomials
Blondeau-Fournier, Olivier; Ridout, David; Wood, Simon
2016-01-01
We give new proofs of the rationality of the N=1 superconformal minimal model vertex operator superalgebras and of the classification of their modules in both the Neveu-Schwarz and Ramond sectors. For this, we combine the standard free field realisation with the theory of Jack symmetric functions. A key role is played by Jack symmetric polynomials with a certain negative parameter that are labelled by admissible partitions. These polynomials are shown to describe free fermion correlators, suitably dressed by a symmetrising factor. The classification proofs concentrate on explicitly identifying Zhu's algebra and its twisted analogue. Interestingly, these identifications do not use an explicit expression for the non-trivial vacuum singular vector. While the latter is known to be expressible in terms of an Uglov symmetric polynomial or a linear combination of Jack superpolynomials, it turns out that standard Jack polynomials (and functions) suffice to prove the classification.
Inequalities for a polynomial and its derivative
Chanam, Barchand; Dewan, K. K.
2007-12-01
Let , 1[less-than-or-equals, slant][mu][less-than-or-equals, slant]n, be a polynomial of degree n such that p(z)[not equal to]0 in z0, then for 0Yadav and Pukhta [K.K. Dewan, R.S. Yadav, M.S. Pukhta, Inequalities for a polynomial and its derivative, Math. Inequal. Appl. 2 (2) (1999) 203-205] proved Equality holds for the polynomial where n is a multiple of [mu]E In this paper, we obtain an improvement of the above inequality by involving some of the coefficients. As an application of our result, we further improve upon a result recently proved by Aziz and Shah [A. Aziz, W.M. Shah, Inequalities for a polynomial and its derivative, Math. Inequal. Appl. 7 (3) (2004) 379-391].
Bergman orthogonal polynomials and the Grunsky matrix
Beckermann, Bernhard; Stylianopoulos, Nikos
2016-01-01
By exploiting a link between Bergman orthogonal polynomials and the Grunsky matrix, probably first observed by Kühnau in 1985, we improve some recent results on strong asymptotics of Bergman polynomials outside the domain G of orthogonality, and on entries of the Bergman shift operator. In our proofs we suggest a new matrix approach involving the Grunsky matrix, and use well-established results in the literature relating properties of the Grunsky matrix to the regularity of the boundary of G,...
Equivalence of polynomial conjectures in additive combinatorics
Lovett, Shachar
2010-01-01
We study two conjectures in additive combinatorics. The first is the polynomial Freiman-Ruzsa conjecture, which relates to the structure of sets with small doubling. The second is the inverse Gowers conjecture for $U^3$, which relates to functions which locally look like quadratics. In both cases a weak form, with exponential decay of parameters is known, and a strong form with only a polynomial loss of parameters is conjectured. Our main result is that the two conjectures are in fact equivalent.
Stochastic processes with orthogonal polynomial eigenfunctions
Griffiths, Bob
2009-12-01
Markov processes which are reversible with either Gamma, Normal, Poisson or Negative Binomial stationary distributions in the Meixner class and have orthogonal polynomial eigenfunctions are characterized as being processes subordinated to well-known diffusion processes for the Gamma and Normal, and birth and death processes for the Poisson and Negative Binomial. A characterization of Markov processes with Beta stationary distributions and Jacobi polynomial eigenvalues is also discussed.
Ferrers Matrices Characterized by the Rook Polynomials
MAHai-cheng; HUSheng-biao
2003-01-01
In this paper,we show that there exist precisely W(A) Ferrers matrices F(C1,C2,…,cm)such that the rook polynomials is equal to the rook polynomial of Ferrers matrix F(b1,b2,…,bm), where A={b1,b2-1,…,bm-m+1} is a repeated set,W(A) is weight of A.
Quantum group invariants and link polynomials
A general method is developed for constructing quantum group invariants and determining their eigenvalues. Applied to the universal R-matrix this method leads to the construction of a closed formula for link polynomials. To illustrate the application of this formula, the quantum groups Uq(E8), Uq(so(2m+1)) and Uq(gl(m)) are considered as examples, and corresponding link polynomials are obtained. (orig.)
On Sharing, Memoization, and Polynomial Time
Avanzini, Martin; Dal Lago, Ugo
2015-01-01
We study how the adoption of an evaluation mechanism with sharing and memoization impacts the class of functions which can be computed in polynomial time. We first show how a natural cost model in which lookup for an already computed result has no cost is indeed invariant. As a corollary, we then prove that the most general notion of ramified recurrence is sound for polynomial time, this way settling an open problem in implicit computational complexity.
Laguerre polynomials method in the valon model
Boroun, G R
2014-01-01
We used the Laguerre polynomials method for determination of the proton structure function in the valon model. We have examined the applicability of the valon model with respect to a very elegant method, where the structure of the proton is determined by expanding valon distributions and valon structure functions on Laguerre polynomials. We compared our results with the experimental data, GJR parameterization and DL model. Having checked, this method gives a good description for the proton structure function in valon model.
Positive maps, positive polynomials and entanglement witnesses
We link the study of positive quantum maps, block positive operators and entanglement witnesses with problems related to multivariate polynomials. For instance, we show how indecomposable block positive operators relate to biquadratic forms that are not sums of squares. Although the general problem of describing the set of positive maps remains open, in some particular cases we solve the corresponding polynomial inequalities and obtain explicit conditions for positivity.
Polynomial Subtraction Method for Disconnected Quark Loops
Liu, Quan; Morgan, Ron
2014-01-01
The polynomial subtraction method, a new numerical approach for reducing the noise variance of Lattice QCD disconnected matrix elements calculation, is introduced in this paper. We use the MinRes polynomial expansion of the QCD matrix as the approximation to the matrix inverse and get a significant reduction in the variance calculation. We compare our results with that of the perturbative subtraction and find that the new strategy yields a faster decrease in variance which increases with quark mass.
A Polynomial Preconditioner for the CMRH Algorithm
Shiji Xu; Jiangzhou Lai; Linzhang Lu
2011-01-01
Many large and sparse linear systems can be solved efficiently by restarted GMRES and CMRH methods Sadok 1999. The CMRH(m) method is less expensive and requires slightly less storage than GMRES(m). But like GMRES, the restarted CMRH method may not converge. In order to remedy this defect, this paper presents a polynomial preconditioner for CMRH-based algorithm. Numerical experiments are given to show that the polynomial preconditioner is quite simple and easily constructed and the preconditio...
Blind Signature Scheme Based on Chebyshev Polynomials
Maheswara Rao Valluri
2011-12-01
Full Text Available A blind signature scheme is a cryptographic protocol to obtain a valid signature for a message from a signer such that signer’s view of the protocol can’t be linked to the resulting message signature pair. This paper presents blind signature scheme using Chebyshev polynomials. The security of the given scheme depends upon the intractability of the integer factorization problem and discrete logarithms ofChebyshev polynomials.
Nonsymmetric Askey-Wilson polynomials and $Q$-polynomial distance-regular graphs
Lee, Jae-Ho
2015-01-01
In his famous theorem (1982), Douglas Leonard characterized the $q$-Racah polynomials and their relatives in the Askey scheme from the duality property of $Q$-polynomial distance-regular graphs. In this paper we consider a nonsymmetric (or Laurent) version of the $q$-Racah polynomials in the above situation. Let $\\Gamma$ denote a $Q$-polynomial distance-regular graph that contains a Delsarte clique $C$. Assume that $\\Gamma$ has $q$-Racah type. Fix a vertex $x \\in C$. We partition the vertex s...
Vector-Valued Jack Polynomials from Scratch
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.
Diffusion tensor image registration using polynomial expansion
In this paper, we present a deformable registration framework for the diffusion tensor image (DTI) using polynomial expansion. The use of polynomial expansion in image registration has previously been shown to be beneficial due to fast convergence and high accuracy. However, earlier work was developed only for 3D scalar medical image registration. In this work, it is shown how polynomial expansion can be applied to DTI registration. A new measurement is proposed for DTI registration evaluation, which seems to be robust and sensitive in evaluating the result of DTI registration. We present the algorithms for DTI registration using polynomial expansion by the fractional anisotropy image, and an explicit tensor reorientation strategy is inherent to the registration process. Analytic transforms with high accuracy are derived from polynomial expansion and used for transforming the tensor's orientation. Three measurements for DTI registration evaluation are presented and compared in experimental results. The experiments for algorithm validation are designed from simple affine deformation to nonlinear deformation cases, and the algorithms using polynomial expansion give a good performance in both cases. Inter-subject DTI registration results are presented showing the utility of the proposed method. (paper)
Reliable Computational Predictions by Modeling Uncertainties Using Arbitrary Polynomial Chaos
Witteveen, J.A.S.; Bijl, H
2006-01-01
Inherent physical uncertainties can have a significant influence on computational predictions. It is therefore important to take physical uncertainties into account to obtain more reliable computational predictions. The Galerkin polynomial chaos method is a commonly applied uncertainty quantification method. However, the polynomial chaos expansion has some limitations. Firstly, the polynomial chaos expansion based on classical polynomials can achieve exponential convergence for a limited set ...
Characterization of pinhole transmission gratings.
Eidmann, K; Kühne, M; Müller, P; Tsakiris, G D
1990-01-01
Gold pinhole transmission gratings fabricated by Heidenhain GmbH primarily for the purpose of studying the radiation of intense soft x-ray sources have been tested with the synchrotron radiation of BESSY. Typical results for the spectral dependence of the grating efficiency into the various diffraction orders are presented in a wavelength region ranging from 4 to 20 nm. Also the influence of grating irregularities has been studied. With appropriate grating parameters quite good agreement between the experimental results and theoretical Calculations is Obtained. PMID:21307429
On the Connection Coefficients of the Chebyshev-Boubaker Polynomials
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.
Generalized Narayana Polynomials, Riordan Arrays and Lattice Paths
Barry, Paul; Hennessy, Aoife
2012-01-01
We study a family of polynomials in two variables, identifying them as the moments of a two-parameter family of orthogonal polynomials. The coefficient array of these orthogonal polynomials is shown to be an ordinary Riordan array. We express the generating function of the sequence of polynomials under study as a continued fraction, and determine the corresponding Hankel transform. An alternative characterization of the polynomials in terms of a related Riordan array is also given. This Riord...
A new class of generalized polynomials associated with Hermite and Bernoulli polynomials
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
Radiative properties tailoring of grating by comb-drive microactuator
Micro-scale grating structures are widely researched in recent years. Although micro-scale fabrication technology is highly advanced today, with grating aspect ratio greater than 25:1 being achievable some fabrication requirements, such as fine groove processing, are still challenging. Comb-drive microactuator is proposed in this paper to be utilized on simple binary grating structures for tailoring or modulating spectral radiation properties by active adjustment. The rigorous coupled-wave analysis (RCWA) is used to calculate the absorptance of proposed structures and to investigate the impacts brought by the geometry and displacement of comb-drive microactuator. The results show that the utilization of comb-drive microactuator on grating improves the absorptance of simple binary grating while avoiding the difficulty fine groove processing. Spectral radiation property tailoring after gratings are fabricated becomes possible with the comb-drive microactuator structure. - Highlights: • A microscale grating structure with comb-driven microactuator is proposed. • The movement of microactuator changes peak absorptance resonance wavelength. • Geometric and displacement effects of comb finger on absorptance are investigated. • Both RCWA and LC circuit models are developed to predict the resonance wavelength. • Resonance frequency equations of LC circuits allow quick design analysis
The supersymmetric (SUSY) structure of coupled-mode equations that describe scattering of optical waves in one-dimensional Bragg gratings is highlighted. This property can find applications to the synthesis of special Bragg filters and distributed-feedback (DFB) optical cavities. In particular, multiple SUSY (Darboux–Crum) transformations can be used to synthesize DFB filters with any desired number of resonances at target frequencies. As an example, we describe the design of a DFB structure with a set of equally-spaced resonances, i.e. a frequency comb transmission filter. (paper)
Time-dependent generalized polynomial chaos
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.
Hierarchical polynomial network approach to automated target recognition
Kim, Richard Y.; Drake, Keith C.; Kim, Tony Y.
1994-02-01
A hierarchical recognition methodology using abductive networks at several levels of object recognition is presented. Abductive networks--an innovative numeric modeling technology using networks of polynomial nodes--results from nearly three decades of application research and development in areas including statistical modeling, uncertainty management, genetic algorithms, and traditional neural networks. The systems uses pixel-registered multisensor target imagery provided by the Tri-Service Laser Radar sensor. Several levels of recognition are performed using detection, classification, and identification, each providing more detailed object information. Advanced feature extraction algorithms are applied at each recognition level for target characterization. Abductive polynomial networks process feature information and situational data at each recognition level, providing input for the next level of processing. An expert system coordinates the activities of individual recognition modules and enables employment of heuristic knowledge to overcome the limitations provided by a purely numeric processing approach. The approach can potentially overcome limitations of current systems such as catastrophic degradation during unanticipated operating conditions while meeting strict processing requirements. These benefits result from implementation of robust feature extraction algorithms that do not take explicit advantage of peculiar characteristics of the sensor imagery, and the compact, real-time processing capability provided by abductive polynomial networks.
Optical Fiber Grating based Sensors
Michelsen, Susanne
2003-01-01
In this thesis differenct optical fiber gratings are used for sensor purposes. If a fiber with a core concentricity error (CCE) is used, a directional dependent bend sensor can be produced. The CCE direction can be determined by means of diffraction. This makes it possible to produce long......-period gratings in a fiber with a CCE direction parallel or perpendicular to the writing direction. The maximal bending sensitivity is independent on the writing direction, but the detailed bending response is different in the two cases. A temperature and strain sensor, based on a long-period grating and two...... sampled gratings, was produced and investigated. It is based on the different temperature and strain response of these gratings. Both a transfer matrix method and an overlap calculation is performed to explain the sensor response. Another type of sensor is based on tuning and modulation of a laser...
NMR Quantum Calculations of the Jones Polynomial
Marx, Raimund; Kauffman, Louis; Lomonaco, Samuel; Spörl, Andreas; Pomplun, Nikolas; Myers, John; Glaser, Steffen J
2009-01-01
The repertoire of problems theoretically solvable by a quantum computer recently expanded to include the approximate evaluation of knot invariants, specifically the Jones polynomial. The experimental implementation of this evaluation, however, involves many known experimental challenges. Here we present experimental results for a small-scale approximate evaluation of the Jones Polynomial by nuclear-magnetic resonance (NMR), in addition we show how to escape from the limitations of NMR approaches that employ pseudo pure states. Specifically, we use two spin 1/2 nuclei of natural abundance chloroform and apply a sequence of unitary transforms representing the Trefoil Knot, the Figure Eight Knot and the Borromean Rings. After measuring the state of the molecule in each case, we are able to estimate the value of the Jones Polynomial for each of the knots.
Quantum chaotic dynamics and random polynomials
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)
Minimal residual method stronger than polynomial preconditioning
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.
Polynomial chaos functions and stochastic differential equations
The Karhunen-Loeve procedure and the associated polynomial chaos expansion have been employed to solve a simple first order stochastic differential equation which is typical of transport problems. Because the equation has an analytical solution, it provides a useful test of the efficacy of polynomial chaos. We find that the convergence is very rapid in some cases but that the increased complexity associated with many random variables can lead to very long computational times. The work is illustrated by exact and approximate solutions for the mean, variance and the probability distribution itself. The usefulness of a white noise approximation is also assessed. Extensive numerical results are given which highlight the weaknesses and strengths of polynomial chaos. The general conclusion is that the method is promising but requires further detailed study by application to a practical problem in transport theory
Polynomial Vector Fields in One Complex Variable
Branner, Bodil
In recent years Adrien Douady was interested in polynomial vector fields, both in relation to iteration theory and as a topic on their own. This talk is based on his work with Pierrette Sentenac, work of Xavier Buff and Tan Lei, and my own joint work with Kealey Dias.......In recent years Adrien Douady was interested in polynomial vector fields, both in relation to iteration theory and as a topic on their own. This talk is based on his work with Pierrette Sentenac, work of Xavier Buff and Tan Lei, and my own joint work with Kealey Dias....
Sparse DOA estimation with polynomial rooting
Xenaki, Angeliki; Gerstoft, Peter; Fernandez Grande, Efren
2015-01-01
Direction-of-arrival (DOA) estimation involves the localization of a few sources from a limited number of observations on an array of sensors. Thus, DOA estimation can be formulated as a sparse signal reconstruction problem and solved efficiently with compressive sensing (CS) to achieve...... highresolution imaging. Utilizing the dual optimal variables of the CS optimization problem, it is shown with Monte Carlo simulations that the DOAs are accurately reconstructed through polynomial rooting (Root-CS). Polynomial rooting is known to improve the resolution in several other DOA estimation methods...
Incomplete Bivariate Fibonacci and Lucas -Polynomials
Dursun Tasci
2012-01-01
Full Text Available We define the incomplete bivariate Fibonacci and Lucas -polynomials. In the case =1, =1, we obtain the incomplete Fibonacci and Lucas -numbers. If =2, =1, we have the incomplete Pell and Pell-Lucas -numbers. On choosing =1, =2, we get the incomplete generalized Jacobsthal number and besides for =1 the incomplete generalized Jacobsthal-Lucas numbers. In the case =1, =1, =1, we have the incomplete Fibonacci and Lucas numbers. If =1, =1, =1, =⌊(−1/(+1⌋, we obtain the Fibonacci and Lucas numbers. Also generating function and properties of the incomplete bivariate Fibonacci and Lucas -polynomials are given.
On the Waring problem for polynomial rings
Fröberg, Ralf; Shapiro, Boris
2011-01-01
In this note we discuss an analog of the classical Waring problem for C[x_0, x_1,...,x_n]. Namely, we show that a general homogeneous polynomial p \\in C[x_0,x_1,...,x_n] of degree divisible by k\\ge 2 can be represented as a sum of at most k^n k-th powers of homogeneous polynomials in C[x_0, x_1,...,x_n]. Noticeably, k^n coincides with the number obtained by naive dimension count.
Error Minimization of Polynomial Approximation of Delta
Islam Sana; Sadiq Muhammad; Qureshi Muhammad Shahid
2008-09-01
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 approximations have been developed to obtain the values of for any time of a year for the period AD 1620 to AD 2000 (Meeu 2000) as no dynamical theories describe the variations in . In this work, a new set of polynomials for is obtained for the period AD 1620 to AD 2007 that is found to produce better results compared to previous attempts.
Large Degree Asymptotics of Generalized Bessel Polynomials
López, J. L.; Temme, Nico
2011-01-01
Asymptotic 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 origin in the $z-$plane. New forms of expansions in terms of elementary functions valid in sectors not containing the turning points $z=\\pm i/n$ are derived, and a new expansion in terms of modified Bessel fu...
The chromatic polynomial and list colorings
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....
Five Constructions of Permutation Polynomials over $\\gf(q^2)$
Ding, Cunsheng; Yuan, Pingzhi
2015-01-01
Four recursive constructions of permutation polynomials over $\\gf(q^2)$ with those over $\\gf(q)$ are developed and applied to a few famous classes of permutation polynomials. They produce infinitely many new permutation polynomials over $\\gf(q^{2^\\ell})$ for any positive integer $\\ell$ with any given permutation polynomial over $\\gf(q)$. A generic construction of permutation polynomials over $\\gf(2^{2m})$ with o-polynomials over $\\gf(2^m)$ is also presented, and a number of new classes of per...
Perturbations around the zeros of classical orthogonal polynomials
Sasaki, Ryu
2014-01-01
Starting from degree N solutions of a time dependent Schroedinger-like equation for classical orthogonal polynomials, a linear matrix equation describing perturbations around the N zeros of the polynomial is derived. The matrix has remarkable Diophantine properties. Its eigenvalues are independent of the zeros. The corresponding eigenvectors provide the representations of the lower degree (0,1,...,N-1) polynomials in terms of the zeros of the degree N polynomial. The results are valid universally for all the classical orthogonal polynomials, including the Askey scheme of hypergeometric orthogonal polynomials and its q-analogues.
Perturbations around the zeros of classical orthogonal polynomials
Sasaki, Ryu
2015-04-01
Starting from degree N solutions of a time dependent Schrödinger-like equation for classical orthogonal polynomials, a linear matrix equation describing perturbations around the N zeros of the polynomial is derived. The matrix has remarkable Diophantine properties. Its eigenvalues are independent of the zeros. The corresponding eigenvectors provide the representations of the lower degree ( 0 , 1 , … , N - 1 ) polynomials in terms of the zeros of the degree N polynomial. The results are valid universally for all the classical orthogonal polynomials, including the Askey scheme of hypergeometric orthogonal polynomials and its q-analogues.
Inclusion-exclusion polynomials with large coefficients
Bzdega, Bartlomiej
2012-01-01
We prove that for every positive integer $k$ there exist an inclusion-exclusion polynomial $Q_{\\{q_1,q_2,...,q_k\\}}$ with the height at least $c^{2^k}\\prod_{j=1}^{k-2}q_j^{2^{k-j-1}-1}$, where $c$ is a positive constant and $q_1
Scalar Field Theories with Polynomial Shift Symmetries
Griffin, Tom; Horava, Petr; Yan, Ziqi
2014-01-01
We continue our study of naturalness in nonrelativistic QFTs of the Lifshitz type, focusing on scalar fields that can play the role of Nambu-Goldstone (NG) modes associated with spontaneous symmetry breaking. Such systems allow for an extension of the constant shift symmetry to a shift by a polynomial of degree $P$ in spatial coordinates. These "polynomial shift symmetries" in turn protect the technical naturalness of modes with a higher-order dispersion relation, and lead to a refinement of the proposed classification of infrared Gaussian fixed points available to describe NG modes in nonrelativistic theories. Generic interactions in such theories break the polynomial shift symmetry explicitly to the constant shift. It is thus natural to ask: Given a Gaussian fixed point with polynomial shift symmetry of degree $P$, what are the lowest-dimension operators that preserve this symmetry, and deform the theory into a self-interacting scalar field theory with the shift symmetry of degree $P$? To answer this (essen...
Algebraic differential equations associated to some polynomials
Barlet, Daniel
2013-01-01
We compute the Gauss-Manin differential equation for any period of a polynomial in \\ $\\C[x_{0},\\dots, x_{n}]$ \\ with \\ $(n+2)$ \\ monomials. We give two general factorizations theorem in the algebra \\ $\\C$ \\ for such a differential equations.
Nondimensional Simplification of Tensor Polynomials with Indices
Jaén, X
1999-01-01
We are presenting an algorithm capable of simplifying tensor polynomials withindices when the building tensors have index symmetry properties. Theseproperties include simple symmetry, cyclicity and those due to the presence ofcovariant derivatives. The algorithm is part of a Mathematica package calledTools of Tensor Calculus (TTC) [web address: http://baldufa.upc.es/ttc
Quantum Hilbert matrices and orthogonal polynomials
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....
Z-polynomials and ring commutativity
Buckley, S.M.; McHale, D.
2012-01-01
We characterise polynomials f with integer coefficients such that a ring with unity R is necessarily commutative if f(x) is central for all x Ɛ R. We also solve the corresponding problem without the assumption that the ring has a unity.
Thermodynamic characterization of networks using graph polynomials
Ye, Cheng; Comin, César H.; Peron, Thomas K. DM.; Silva, Filipi N.; Rodrigues, Francisco A.; Costa, Luciano da F.; Torsello, Andrea; Hancock, Edwin R.
2015-09-01
In this paper, we present a method for characterizing the evolution of time-varying complex networks by adopting a thermodynamic representation of network structure computed from a polynomial (or algebraic) characterization of graph structure. Commencing from a representation of graph structure based on a characteristic polynomial computed from the normalized Laplacian matrix, we show how the polynomial is linked to the Boltzmann partition function of a network. This allows us to compute a number of thermodynamic quantities for the network, including the average energy and entropy. Assuming that the system does not change volume, we can also compute the temperature, defined as the rate of change of entropy with energy. All three thermodynamic variables can be approximated using low-order Taylor series that can be computed using the traces of powers of the Laplacian matrix, avoiding explicit computation of the normalized Laplacian spectrum. These polynomial approximations allow a smoothed representation of the evolution of networks to be constructed in the thermodynamic space spanned by entropy, energy, and temperature. We show how these thermodynamic variables can be computed in terms of simple network characteristics, e.g., the total number of nodes and node degree statistics for nodes connected by edges. We apply the resulting thermodynamic characterization to real-world time-varying networks representing complex systems in the financial and biological domains. The study demonstrates that the method provides an efficient tool for detecting abrupt changes and characterizing different stages in network evolution.
Interpolation of Shifted-Lacunary Polynomials
Giesbrecht, Mark
2008-01-01
Given a "black box" function to evaluate an unknown rational polynomial f in Q[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity t, the shift alpha, the exponents 0<=e1
Cumulants, lattice paths, and orthogonal polynomials
Lehner, Franz
2001-01-01
A formula expressing free cumulants in terms of the Jacobi parameters of the corresponding orthogonal polynomials is derived. It combines Flajolet's theory of continued fractions and Lagrange inversion. For the converse we discuss Gessel-Viennot theory to express Hankel determinants in terms of various cumulants.
Algebraic polynomial system solving and applications
Bleylevens, I.W.M.
2010-01-01
The problem of computing the solutions of a system of multivariate polynomial equations can be approached by the Stetter-Möller matrix method which casts the problem into a large eigenvalue problem. This Stetter-Möller matrix method forms the starting point for the development of computational proce
Polynomial stabilization of some dissipative hyperbolic systems
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
Indecomposability of polynomials via Jacobian matrix
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)
Ideals in Polynomial Near-rings
Mark Farag
2002-01-01
In this paper, we further explore the relationship between the ideals of N and those of N[x], where N is a zero-symmetric right near-ring with identity and N[x] is the polynomial near-ring introduced by Bagley in 1993.
Function approximation with polynomial regression slines
Principles of the polynomial regression splines as well as algorithms and programs for their computation are presented. The programs prepared using software package MATLAB are generally intended for approximation of the X-ray spectra and can be applied in the multivariate calibration of radiometric gauges. (author)
On an Inequality Concerning the Polar Derivative of a Polynomial
A Aziz; N A Rather
2007-08-01
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.
Representations of Knot Groups and Twisted Alexander Polynomials
Xiao Song LIN
2001-01-01
We present a twisted version of the Alexander polynomial associated with a matrix representation of the knot group. Examples of two knots with the same Alexander module but differenttwisted Alexander polynomials are given.
Self-dual Koornwinder-MacDonald polynomials
Van Diejen, J F
1995-01-01
We prove certain duality properties and present recurrence relations for a four-parameter family of self-dual Koornwinder-Macdonald polynomials. The recurrence relations are used to verify Macdonald's normalization conjectures for these polynomials.
Irreducibility Results for Compositions of Polynomials in Several Variables
Anca Iuliana Bonciocat; Alexandru Zaharescu
2005-05-01
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.
Remarks on Homogeneous Al-Salam and Carlitz Polynomials
Jian-Ping Fang
2014-01-01
Several multilinear generating functions of the homogeneous Al-Salam and Carlitz polynomials are derived from q-operator. In addition, two interesting relationships of product of this kind of polynomials are obtained.
Identities involving Bessel polynomials arising from linear differential equations
Kim, Taekyun; Kim, Dae San
2016-01-01
In this paper, we study linear di?erential equations arising from Bessel polynomials and their applications. From these linear differential equations, we give some new and explicit identities for Bessel polynomials.
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-L
MEMS Bragg grating force sensor
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....
Some Systems of Multivariable Orthogonal q-Racah polynomials
Gasper, George; Rahman, Mizan
2004-01-01
In 1991 Tratnik derived two systems of multivariable orthogonal Racah polynomials and considered their limit cases. q-Extensions of these systems are derived, yielding systems of multivariable orthogonal q-Racah polynomials, from which systems of multivariable orthogonal q-Hahn, dual q-Hahn, q-Krawtchouk, q-Meixner, and q-Charlier polynomials follow as special or limit cases.
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.
Further Results on Permutation Polynomials over Finite Fields
Yuan, Pingzhi; Ding, Cunsheng
2013-01-01
Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we develop general theorems on permutation polynomials over finite fields. As a demonstration of the theorems, we present a number of classes of explicit permutation polynomials on $\\gf_q$.
Universality for polynomial invariants on ribbon graphs with flags
Avohou, Remi C.; Geloun, Joseph Ben; Hounkonnou , Mahouton N.
2013-01-01
In this paper, we analyze the Bollobas and Riordan polynomial for ribbon graphs with flags introduced in arXiv:1301.1987 and prove its universality. We also show that this polynomial can be defined on some equivalence classes of ribbon graphs involving flag moves and that the new polynomial is still universal on these classes.
On conformal measures for infinitely renormalizable quadratic polynomials
HUANG Zhiyong; JIANG Yunping; WANG Yuefei
2005-01-01
We study a conformal measure for an infinitely renormalizable quadratic polynomial. We prove that the conformal measure is ergodic if the polynomial is unbranched and has complex bounds. The main technique we use in the proof is the three-dimensional puzzle for an infinitely renormalizable quadratic polynomial.
Probabilistic aspects of Al-Salam-Chihara polynomials
Bryc, Wlodzimierz; Matysiak, Wojciech; Szablowski, Pawel J.
2003-01-01
We solve the connection coefficient problem between the Al-Salam-Chihara polynomials and the q-Hermite polynomials, and we use the resulting identity to answer a question from probability theory. We also derive the distribution of some Al-Salam-Chihara polynomials, and compute determinants of related Hankel matrices.
Moments for Generating Functions of Al-Salam-Carlitz Polynomials
Jian Cao
2012-01-01
We employ the moment representations for Al-Salam-Carlitz polynomials and show how to deduce bilinear, trilinear, and multilinear generating functions for Al-Salam-Carlitz polynomials. Moreover, we obtain two terminating generating functions for Al-Salam-Carlitz polynomials by the method of moments.
Some advances in tensor analysis and polynomial optimization
Li, Zhening; Ling, Chen; Wang, Yiju; Yang, Qingzhi
2014-01-01
Tensor analysis (also called as numerical multilinear algebra) mainly includes tensor decomposition, tensor eigenvalue theory and relevant algorithms. Polynomial optimization mainly includes theory and algorithms for solving optimization problems with polynomial objects functions under polynomial constrains. This survey covers the most of recent advances in these two fields. For tensor analysis, we introduce some properties and algorithms concerning the spectral radius of nonnegative tensors'...
A Determinant Expression for the Generalized Bessel Polynomials
Sheng-liang Yang; Sai-nan Zheng
2013-01-01
Using the exponential Riordan arrays, we show that a variation of the generalized Bessel polynomial sequence is of Sheffer type, and we obtain a determinant formula for the generalized Bessel polynomials. As a result, the Bessel polynomial is represented as determinant the entries of which involve Catalan numbers.
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…
Stretchable diffraction gratings for spectrometry
Simonov, Aleksey N.; Grabarnik, Semen; Vdovin, Gleb
2007-07-01
We have investigated the possibility of using transparent stretchable diffraction gratings for spectrometric applications. The gratings were fabricated by replication of a triangular-groove master into a transparent viscoelastic. The sample length, and hence the spatial period, can be reversibly changed by mechanical stretching. When used in a monochromator with two slits, the stretchable grating permits scanning the spectral components over the output slit, converting the monochromator into a scanning spectrometer. The spectral resolution of such a spectrometer was found to be limited mainly by the wave-front aberrations due to the grating deformation. A model relating the deformation-induced aberrations in different diffraction orders is presented. In the experiments, a 12-mm long viscoelastic grating with a spatial frequency of 600 line pairs/mm provided a full-width at half-maximum resolution of up to ~1.2 nm in the 580-680 nm spectral range when slowly stretched by a micrometer screw and ~3 nm when repeatedly stretched by a voice coil at 15 Hz. Comparison of aberrations in transmitted and diffracted beams measured by a Shack- Hartmann wave-front sensor showed that astigmatisms caused by stretch-dependent wedge deformation are the main factors limiting the resolution of the viscoelastic-grating-based spectrometer.
Aptamer functionalized lipid multilayer gratings for label free detection of specific analytes
Prommapan, Plengchart; Lowry, Troy W.; van Winkle, David; Lenhert, Steven
2015-03-01
Lipid multilayer gratings have been formed on surfaces with a period of 700 nm. When illuminated with white light incident at about 50°, these gratings diffract green light perpendicular to their surface. We demonstrate the potential of these gratings as sensors for analytes by monitoring changes in the diffracted light due to the changes in the size and shape of the grating in response to analyte binding. To demonstrate this potential application, a lipid multilayer grating was functionalized with a thrombin binding aptamer. The selectivity of our aptamer functionalized lipid gratings was confirmed both by monitoring the diffracted light intensity and by fluorescence microscopy. Furthermore, the results show that the binding activity between the aptamer and thrombin depends on the relative composition of a zwitterionic lipid (DOPC) and a cationic lipid (DOTAP). This work shows that nanostructured lipid multilayers on surfaces are a promising nanomaterial for label-free bio-sensing applications.
High Resolution of the ECG Signal by Polynomial Approximation
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.
Theoretical analysis of novel fiber grating pair
Wang, Liao; Jia, Hongzhi; Fang, Liang; You, Bei
2016-06-01
A novel fiber grating pair that consists of a conventional long-period fiber grating and a fiber Bragg cladding grating (FBCG) is proposed. The FBCG is a new type of fiber grating in which refractive index modulation is formed in the cladding. Through the coupled-mode theory, we accurately calculate the coupling coefficients between modes supported in the fibers. And some other mode coupling features in the fiber cladding gratings are analyzed in detail. The calculation of the modes involved in this paper is based on a model of three-layer step-index fiber geometry. Then, we have investigated the sensitivity characteristics for variation of the modulation strengths of the fiber Bragg cladding gratings' resonance peaks and the long-period cladding gratings' (LPCGs) dual resonant peaks. Finally, the modulation strength sensitivity of the grating pair's three resonant peaks is demonstrated, and the results indicate that these grating pairs may find potential applications in optical fiber sensing.
Polynomial threshold functions and Boolean threshold circuits
Hansen, Kristoffer Arnsfelt; Podolskii, Vladimir V.
2013-01-01
secondary interest. We show that PTFs on general Boolean domains are tightly connected to depth two threshold circuits. Our main results in regard to this connection are: PTFs of polynomial length and polynomial degree compute exactly the functions computed by THRMAJ circuits. An exponential length lower...... bound for PTFs that holds regardless of degree, thereby extending known lower bounds for THRMAJ circuits. We generalize two-party unbounded error communication complexity to the multi-party number-on-the-forehead setting, and show that communication lower bounds for 3-player protocols would yield size...... lower bounds for THRTHR circuits. We obtain several other results about PTFs. These include relationships between weight and degree of PTFs, and a degree lower bound for PTFs of constant length. We also consider a variant of PTFs over the max-plus algebra. We show that they are connected to PTFs over...
Venereau polynomials and related fiber bundles
Kaliman, Shulim; ZAIDENBERG, MIKHAIL
2003-01-01
The Venereau polynomials v-n:=y+x^n(xz+y(yu+z^2)), n>= 1, on A4 have all fibers isomorphic to the affine space A3. Moreover, for all n>= 1 the map (v-n, x) : A4 -> A2 yields a flat family of affine planes over A2. In the present note we show that over the punctured plane A2\\0, this family is a fiber bundle. This bundle is trivial if and only if v-n is a variable of the ring C[x][y,z,u] over C[x]. It is an open question whether v1 and v2 are variables of the polynomial ring C[x,y,z,u]. S. Vene...
Tabulating knot polynomials for arborescent knots
Mironov, A; Morozov, An; Sleptsov, A; Ramadevi, P; Singh, Vivek Kumar
2016-01-01
Arborescent knots are the ones 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 enough 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. 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, and apply it to arborescent knots in 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 polynomials.
On computing factors of cyclotomic polynomials
Brent, Richard P.
1993-07-01
For odd square-free n > 1 the cyclotomic polynomial {Φ_n}(x) satisfies the identity of Gauss, 4{Φ_n}(x) = A_n^2 - {( - 1)^{(n - 1)/2}}nB_n^2. A similar identity of Aurifeuille, Le Lasseur, and Lucas is {Φ_n}({( - 1)^{(n - 1)/2}}x) = C_n^2 - nxD_n^2 or, in the case that n is even and square-free, ± {Φ_{n/2}}( - {x^2}) = C_n^2 - nxD_n^2. Here, {A_n}(x), ldots ,{D_n}(x) are polynomials with integer coefficients. We show how these coefficients can be computed by simple algorithms which require O({n^2}) arithmetic operations and work over the integers. We also give explicit formulae and generating functions for {A_n}(x), ldots ,{D_n}(x) , and illustrate the application to integer factorization with some numerical examples.
Zernike polynomials for photometric characterization of LEDs
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)
Polynomial Operators on Classes of Regular Languages
Klíma, Ondřej; Polák, Libor
We assign to each positive variety mathcal V and each natural number k the class of all (positive) Boolean combinations of the restricted polynomials, i.e. the languages of the form L_0a_1 L_1a_2dots a_ell L_ell, text{ where } ell≤ k, a 1,...,a ℓ are letters and L 0,...,L ℓ are languages from the variety mathcal V. For this polynomial operator we give a certain algebraic counterpart which works with identities satisfied by syntactic (ordered) monoids of languages considered. We also characterize the property that a variety of languages is generated by a finite number of languages. We apply our constructions to particular examples of varieties of languages which are crucial for a certain famous open problem concerning concatenation hierarchies.
On Polynomial Sized MDP Succinct Policies
Liberatore, P
2011-01-01
Policies of Markov Decision Processes (MDPs) determine the next action to execute from the current state and, possibly, the history (the past states). When the number of states is large, succinct representations are often used to compactly represent both the MDPs and the policies in a reduced amount of space. In this paper, some problems related to the size of succinctly represented policies are analyzed. Namely, it is shown that some MDPs have policies that can only be represented in space super-polynomial in the size of the MDP, unless the polynomial hierarchy collapses. This fact motivates the study of the problem of deciding whether a given MDP has a policy of a given size and reward. Since some algorithms for MDPs work by finding a succinct representation of the value function, the problem of deciding the existence of a succinct representation of a value function of a given size and reward is also considered.
General Linearized Polynomial Interpolation and Its Applications
Xie, Hongmei; Suter, Bruce W
2011-01-01
In this paper, we first propose a general interpolation algorithm in a free module of a linearized polynomial ring, and then apply this algorithm to decode several important families of codes, Gabidulin codes, KK codes and MV codes. Our decoding algorithm for Gabidulin codes is different from the polynomial reconstruction algorithm by Loidreau. When applied to decode KK codes, our interpolation algorithm is equivalent to the Sudan-style list-1 decoding algorithm proposed by K/"otter and Kschischang for KK codes. The general interpolation approach is also capable of solving the interpolation problem for the list decoding of MV codes proposed by Mahdavifar and Vardy, and has a lower complexity than solving linear equations.
Line Complexity Asymptotics of Polynomial Cellular Automata
Stone, Bertrand
2016-01-01
Cellular automata are discrete dynamical systems that consist of patterns of symbols on a grid, which change according to a locally determined transition rule. In this paper, we will consider cellular automata that arise from polynomial transition rules, where the symbols in the automaton are integers modulo some prime $p$. We are principally concerned with the asymptotic behavior of the line complexity sequence $a_T(k)$, which counts, for each $k$, the number of coefficient strings of length...
Block Toeplitz methods in polynomial matrix computations
Zuniga, J. C.; Henrion, Didier
Leuven: Kathlolieke Universiteit, 2004 - (de Moor, B.; Motmans, B.; Willems, J.), s. 1-7 ISBN 90-5682-517-8. [MTNS 2004 /16./. Leuven (BE), 05.07.2004-09.07.2004] R&D Projects: GA ČR GA102/02/0709 Institutional research plan: CEZ:AV0Z1075907 Keywords : polynomial matrices * numerical linear algebra * computer - aided control system design Subject RIV: BC - Control Systems Theory
Pure Imaginary Roots of Quaternion Standard Polynomials
Chapman, Adam
2011-01-01
In this paper, we present a new method for solving standard quaternion equations. Using this method we reobtain the known formulas for the solution of a quadratic quaternion equation, and provide an explicit solution for the cubic quaternion equation, as long as the equation has at least one pure imaginary root. We also discuss the number of essential pure imaginary roots of a two-sided quaternion polynomial.
Polynomial approximation of functions in Sobolev spaces
Dupont, T.; Scott, R.
1980-01-01
Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hilbert are presented. Using an averaged Taylor series, we represent a function as a polynomial plus a remainder. The remainder can be manipulated in many ways to give different types of bounds. Approximation of functions in fractional order Sobolev spaces is treated as well as the usual integer order spaces and several nonstandard Sobolev-like spaces.
Real meromorphic functions and linear differential polynomials
LANGLEY; J.; K.
2010-01-01
We determine all real meromorphic functions f in the plane such that f has finitely many zeros, the poles of f have bounded multiplicities, and f and F have finitely many non-real zeros, where F is a linear differential polynomial given by F = f (k) +Σk-1j=0ajf(j) , in which k≥2 and the coefficients aj are real numbers with a0≠0.
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
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 pro...
Products of Random Matrices from Polynomial Ensembles
Kieburg, Mario; Kösters, Holger
2016-01-01
Very recently we have shown that the spherical transform is a convenient tool for studying the relation between the joint density of the singular values and that of the eigenvalues for bi-unitarily invariant random matrices. In the present work we discuss the implications of these results for products of random matrices. In particular, we derive a transformation formula for the joint densities of a product of two independent bi-unitarily invariant random matrices, the first from a polynomial ...
Reverse-engineering of polynomial dynamical systems
Jarrah, Abdul Salam; Laubenbacher, Reinhard; Stigler, Brandilyn; Stillman, Michael
2006-01-01
Multivariate polynomial dynamical systems over finite fields have been studied in several contexts, including engineering and mathematical biology. An important problem is to construct models of such systems from a partial specification of dynamic properties, e.g., from a collection of state transition measurements. Here, we consider static models, which are directed graphs that represent the causal relationships between system variables, so-called wiring diagrams. This paper contains an algo...
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…
Polynomial chaos representation of a stochastic preconditioner
Desceliers, Christophe; Ghanem, R; Soize, Christian
2005-01-01
A method is developed in this paper to accelerate the convergence in computing the solution of stochastic algebraic systems of equations. The method is based on computing, via statistical sampling, a polynomial chaos decomposition of a stochastic preconditioner to the system of equations. This preconditioner can subsequently be used in conjunction with either chaos representations of the solution or with approaches based on Monte Carlo sampling. In addition to presenting the supporting theory...
A Deterministic and Polynomial Modified Perceptron Algorithm
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.
Polynomial sequences for bond percolation critical thresholds
In this paper, I compute the inhomogeneous (multi-probability) bond critical surfaces for the (4, 6, 12) and (34, 6) lattices using the linearity approximation described in Scullard and Ziff (2010 J. Stat. Mech. P03021), implemented as a branching process of lattices. I find the estimates for the bond percolation thresholds, pc(4, 6, 12) = 0.693 778 49... and pc(34, 6) = 0.434 370 77..., to be compared with Parviainen's numerical results of pc≈0.693 733 83 and 0.434 306 21 (Parviainen, 2007 J. Phys. A: Math. Theor. 40 9253). These deviations are of the order of 10−5, as is standard for this method, although they are larger than Parviainen's typical standard error of 10−7. Deriving thresholds in this way for a given lattice leads to a polynomial with integer coefficients, whose root in [0, 1] gives the estimate for the bond threshold. I show how the method can be refined, leading to a sequence of higher-order polynomials giving predictions that probably converge to the exact answer. Finally, I discuss how this fact hints that for certain graphs, such as the kagome lattice, the exact bond threshold may not be the root of any polynomial with integer coefficients
A homological study of Green polynomials
Kato, Syu
2011-01-01
We interpret the orthogonality relation of Kostka polynomials arising from complex reflection groups (c.f. [Shoji, Invent. Math. 74 (1983), J. Algebra 245 (2001)] and [Lusztig, Adv. Math. 61 (1986)]) in terms of homological algebra. This leads us to the notion of Kostka system, which can be seen as a categorical counter-part of Kostka polynomials. Then, we show that every generalized Springer correspondence (in good characteristic) (c.f. [Lusztig, Invent. Math. 75 (1984)]) gives rise to a Kostka system. This enables us to see the top-term generation property of the homology of generalized Springer fibers, and the transition formula of Kostka polynomials between two generalized Springer correspondences of type $\\mathsf{BC}$. The latter enhances one of the main results from [Ciubotaru-Kato-K, Invent. Math., to appear] to its graded version. In the appendix, we present a purely algebraic proof that a Kostka system exists for type $\\mathsf{A}$, and therefore one can skip geometric sections \\S 3--5 to see the key ...
The bivariate Rogers-Szegoe polynomials
We present an operator approach to deriving Mehler's formula and the Rogers formula for the bivariate Rogers-Szegoe polynomials hn(x, y vertical bar q). The proof of Mehler's formula can be considered as a new approach to the nonsymmetric Poisson kernel formula for the continuous big q-Hermite polynomials Hn(x; a vertical bar q) due to Askey, Rahman and Suslov. Mehler's formula for hn(x, y vertical bar q) involves a 3Φ2 sum and the Rogers formula involves a 2Φ1 sum. The proofs of these results are based on parameter augmentation with respect to the q-exponential operator and the homogeneous q-shift operator in two variables. By extending recent results on the Rogers-Szegoe polynomials hn(x vertical bar q) due to Hou, Lascoux and Mu, we obtain another Rogers-type formula for hn(x, y vertical bar q). Finally, we give a change of base formula for Hn(x; a vertical bar q) which can be used to evaluate some integrals by using the Askey-Wilson integral
Characteristic polynomials of pseudo-Anosov maps
Birman, Joan; Kawamuro, Keiko
2010-01-01
We study the relationship between three different approaches to the action of a pseudo-Anosov mapping class $[F]$ on a surface: the original theorem of Thurston, its algorithmic proof by Bestvina-Handel, and related investigations of Penner-Harer. Bestvina and Handel represent $[F]$ as a suitably chosen homotopy equivalence $f: G\\to G$ of a finite graph, with an associated transition matrix $T$ whose largest eigenvalue is the dilatation of $[F]$. Extending a skew-symmetric form introduced by Penner and Harer to the setting of Bestvina and Handel, we show that the characteristic polynomial of $T$ is a monic and palindromic or anti-palindromic polynomial, possibly multiplied by a power of $x$. Moreover, it factors as a product of three polynomials. One of them reflects the action of $[F]$ on a certain symplectic space, the second one relates to the degeneracies of the skew-symmetric form, and the third one reflects the restriction of $f$ to the vertices of $G$. We give an application to the problem of deciding ...
Role of discriminantly separable polynomials in integrable dynamical systems
Dragović, Vladimir; Kukić, Katarina
2014-11-01
Discriminantly separable polynomials of degree two in each of the three variables are considered. Those polynomials are by definition polynomials which discriminants are factorized as the products of the polynomials in one variable. Motivating example for introducing such polynomials is the famous Kowalevski top. Motivated by the role of such polynomials in the Kowalevski top, we generalize Kowalevski's integration procedure on a whole class of systems basically obtained by replacing so called the Kowalevski's fundamental equation by some other instance of the discriminantly separable polynomial. We present also the role of the discriminantly separable polynomils in twowell-known examples: the case of Kirchhoff elasticae and the Sokolov's case of a rigid body in an ideal fluid.
Analysis of cubic permutation polynomials for turbo codes
Trifina, Lucian
2011-01-01
Quadratic permutation polynomials (QPPs) have been widely studied and used as interleavers in turbo codes. However, less attention has been given to cubic permutation polynomials (CPPs). This paper proves a theorem which states sufficient and necessary conditions for a cubic permutation polynomial to be a null permutation polynomial. The result is used to reduce the search complexity of CPP interleavers for short lengths (multiples of 8, between 40 and 256), by improving the distance spectrum over the set of polynomials with the largest spreading factor. The comparison with QPP interleavers is made in terms of search complexity and upper bounds of the bit error rate (BER) and frame error rate (FER) for AWGN channel. Cubic permutation polynomials leading to better performance than quadratic permutation polynomials are found for some lengths.
Near-perfect diffraction grating rhomb
Wantuck, Paul J.
1990-01-01
A near-perfect grating rhomb enables an output beam to be diffracted to an angle offset from the input beam. The correcting grating is tipped relative to the dispersing grating to provide the offset angle. The correcting grating is further provided with a groove spacing which differs from the dispersing grating groove space by an amount effective to substantially remove angular dispersion in the output beam. A near-perfect grating rhomb has the capability for selective placement in a FEL to suppress sideband instabilities arising from the FEL.
Flexible PCPDTBT:PCBM solar cells with integrated grating structures
Oliveira Hansen, Roana Melina de; Liu, Yinghui; Madsen, Morten;
2013-01-01
spectra of the active layer. This optimized solar cell structure leads to an enhanced absorption in the active layer and thus improved short-circuit currents and power conversion efficiencies in the fabricated devices. Fabrication of the solar cells on thin polyimide substrates which are compatible......We report on development of flexible PCPDTBT:PCBM solar cells with integrated diffraction gratings on the bottom electrodes. The presented results address PCPDTBT:PCBM solar cells in an inverted geometry, which contains implemented grating structures whose pitch is tuned to match the absorption...
High Efficiency Low Scatter Echelle Grating Project
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...
Polymer optical fiber bragg grating sensors
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....
Grating-Coupled Waveguide Cloaking
WANG Jia-Fu; QU Shao-Bo; XU Zhuo; MA Hua; WANG Cong-Min; XIA Song; WANG Xin-Hua; ZHOU Hang
2012-01-01
Based on the concept of a grating-coupled waveguide (GCW),a new strategy for realizing EM cloaking is presented.Using metallic grating,incident waves are firstly coupled into the effective waveguide and then decoupled into free space behind,enabling EM waves to pass around the obstacle.Phase compensation in the waveguide keeps the wave-front shape behind the obstacle unchanged.Circular,rectangular and triangular cloaks are presented to verify the robustness of the GCW cloaking.Electric field animations and radar cross section (RCS)comparisons convincingly demonstrate the cloaking effect.
Calculation of thermal noise in grating reflectors
Heinert, Daniel; Kroker, Stefanie; Friedrich, Daniel; HILD, Stefan; Kley, Ernst-Bernhard; Leavey, Sean; Martin, Iain W.; Nawrodt, Ronny; Tünnermann, Andreas; Vyatchanin, Sergey P.; YAMAMOTO Kazuhiro
2013-01-01
Grating reflectors have been repeatedly discussed to improve the noise performance of metrological applications due to the reduction or absence of any coating material. So far, however, no quantitative estimate on the thermal noise of these reflective structures exists. In this work we present a theoretical calculation of a grating reflector's noise. We further apply it to a proposed 3rd generation gravitational wave detector. Depending on the grating geometry, the grating material and the te...
Efficient iterative technique for designing bragg gratings
Plougmann, Nikolai; Kristensen, Martin
2004-01-01
We present a new iterative method for designing Bragg gratings based on the Levenberg-Marquardt method of minimizing a chi-squared merit function. It is effective for designing both weak and strong gratings and is particularly well suited for unchirped gratings....
Tunable Fiber Gratings and Their Applications
Z.; Fang; L.; Zhao; L.; Li; K.; Gao; Y.; Zhou; J.; Geng; R.; Qu; G.; Chen
2003-01-01
Some practical research topics on tunable fiber gratings in author's group are presented, including tuning speed, tuning range, tuning characteristics of gratings in HB fiber, and the tunability of the line-width. The applications of fiber gratings in communication and sensing are also discussed.
Maximally positive polynomial systems supported on circuits
Bihan, Frédéric
2013-01-01
A real polynomial system with support $\\calW \\subset \\Z^n$ is called {\\it maximally positive} if all its complex solutions are positive solutions. A support $\\calW$ having $n+2$ elements is called a circuit. We previously showed that the number of non-degenerate positive solutions of a system supported on a circuit $\\calW \\subset\\Z^n$ is at most $m(\\calW)+1$, where $m(\\calW) \\leq n$ is the degeneracy index of $\\calW$. We prove that if a circuit $\\calW \\subset \\Z^n$ supports a maximally positi...
Pseudorandom Generators for Polynomial Threshold Functions
Meka, Raghu; Zuckerman, David
2009-01-01
We study the natural question of constructing pseudorandom generators (PRGs) for low-degree polynomial threshold functions (PTFs). We give a PRG with seed-length log n/eps^{O(d)} fooling degree d PTFs with error at most eps. Previously, no nontrivial constructions were known even for quadratic threshold functions and constant error eps. For the class of degree 1 threshold functions or halfspaces, we construct PRGs with much better dependence on the error parameter eps and obtain a PRG with se...
Softness, Polynomial Boundedness and Amplitudes' Positivity
Bai, Dong
2016-01-01
In this note, we study the connection between infrared (IR) and ultraviolet (UV) behaviors of scattering amplitudes of massless channels by exploiting dispersion relations and positivity bounds. Given forward scattering amplitudes which scale as $\\mathcal{A}(s)\\sim s^M$ in the IR ($s\\to0$) and could be embedded into UV completions satisfying unitarity, analyticity, crossing symmetry and polynomial boundedness $|\\mathcal{A}(s)|< c\\, |s|^N$ ($|s|\\to\\infty$), with $M$ and $N$ integers, we show that the inequality $2\\ceil*{\\frac{N}{2}}\\ge M \\ge 0$ must hold, where $\\ceil*{x}$ is the smallest integer greater than or equal to $x$.
Conditional Density Approximations with Mixtures of Polynomials
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...... two methods for learning MoP approximations of conditional densities from data. Both approaches are based on learning MoP approximations of the joint density and the marginal density of the conditioning variables, but they differ as to how the MoP approximation of the quotient of the two densities is...
A Matricial Algorithm for Polynomial Refinement
King, Emily J
2011-01-01
In order to have a multiresolution analysis, the scaling function must be refinable. That is, it must be the linear combination of 2-dilation, $\\mathbb{Z}$-translates of itself. Refinable functions used in connection with wavelets are typically compactly supported. In 2002, David Larson posed the question, "Are all polynomials (of a single variable) finitely refinable?" That summer the author proved that the answer indeed was true using basic linear algebra. The result was presented in a number of talks but had not been typed up until now. The purpose of this short note is to record that particular proof.
Time-reversal symmetry and random polynomials
Braun, D; Zyczkowski, K
1996-01-01
We analyze the density of roots of random polynomials where each complex coefficient is constructed of a random modulus and a fixed, deterministic phase. The density of roots is shown to possess a singular component only in the case for which the phases increase linearly with the index of coefficients. This means that, contrary to earlier belief, eigenvectors of a typical quantum chaotic system with some antiunitary symmetry will not display a clustering curve in the stellar representation. Moreover, a class of time-reverse invariant quantum systems is shown, for which spectra display fluctuations characteristic of orthogonal ensemble, while eigenvectors confer to predictions of unitary ensemble.
Inverting Onto Functions and Polynomial Hierarchy
Buhrman, H.; Fortnow, L.; Koucký, Michal; Rogers, J.D.; Vereshchagin, N.K.
Berlin : Springer-Verlag, 2007 - (Diekert, V.; Volkov, M.; Voronkov, A.), s. 92-103 ISBN 978-3-540-74509-9. - (Lecture Notes in Computer Science. 4649). [International Computer Science Symposium in Russia, CSR 2007. Jekaterinburg (RU), 03.09.2007-07.09.2007] R&D Projects: GA ČR GA201/05/0124; GA ČR GP201/07/P276 Institutional research plan: CEZ:AV0Z10190503 Keywords : one-way functions * polynomial hierarchy * Kolmogorov generic oracle s Subject RIV: BA - General Mathematics
On difference equations for orthogonal polynomials on nonuniform lattices
By the study of various properties of some divided-difference equations, we simplify the definition of classical orthogonal polynomials given by Atakishiyev, Rahman and Suslov (1995), then prove that orthogonal polynomials obtained by some modifications of the classical orthogonal polynomials on nonuniform lattices satisfy a single fourth-order linear homogeneous divided-difference equation with polynomial coefficients. Moreover, we factorize and solve explicitly these divided-difference equations. Also, we prove that the product of two functions, each of which satisfying a second-order linear homogeneous divided-difference equation is a solution of a fourth-order linear homogeneous divided-difference equation. This result holds in particular when the divided-difference operator is carefully replaced by the Askey-Wilson operator Dq, following pioneer work by Alphonse Magnus (1988) connecting Dq and divided-difference operators. Finally, we propose a method to look for polynomial solutions of linear divided-difference equations with polynomial coefficients. (author)
VCSELs and silicon light sources exploiting SOI grating mirrors
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...... Bragg reflector. Numerical simulations show that both the silicon light source and the VCSEL exploiting SOI grating mirrors have superior performances, compared to existing silicon light sources and long wavelength VCSELs. These devices are highly adequate for chip-level optical interconnects as well as...
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 ostatní: 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
High-temperature diffraction gratings for synchrotron radiation
SiC-based mechanically ruled master gratings and replicas are developed for synchrotron radiation instruments. An SiC-based gold replica grating without any thermal deformation due to active cooling is used in a high-photon-flux-soft x-ray monochromator that is installed in a bending magnet beamline. An SiC-based gold master grating is used in a vacuum ultraviolet/soft x-ray monochromator installed in an undulator beamline with slight groove shape deformation. This deformation is caused by the thermal change of the gold film occurring at higher than 250--300 degree C. A method for cleaning carbon-contaminated synchrotron radiation optics is tested. The ultraviolet ozone ashing method effectively cleans carbon contamination on the optics and is useful for extending the lifetime of synchrotron radiation optics
Darboux polynomials for Lotka-Volterra systems in three dimensions
Christodoulides, Yiannis T
2008-01-01
We consider Lotka-Volterra systems in three dimensions depending on three real parameters. By using elementary algebraic methods we classify the Darboux polynomials (also known as second integrals) for such systems for various values of the parameters, and give the explicit form of the corresponding cofactors. More precisely, we show that a Darboux polynomial of degree greater than one is reducible. In fact, it is a product of linear Darboux polynomials and first integrals.
Polynomial force approximations and multifrequency atomic force microscopy
Daniel Platz; Daniel Forchheimer; Tholén, Erik A; David B. Haviland
2013-01-01
We present polynomial force reconstruction from experimental intermodulation atomic force microscopy (ImAFM) data. We study the tip–surface force during a slow surface approach and compare the results with amplitude-dependence force spectroscopy (ADFS). Based on polynomial force reconstruction we generate high-resolution surface-property maps of polymer blend samples. The polynomial method is described as a special example of a more general approximative force reconstruction, where the aim is...
A Laguerre Polynomial Orthogonality and the Hydrogen Atom
Dunkl, Charles F.
2000-01-01
The radial part of the wave function of an electron in a Coulomb potential is the product of a Laguerre polynomial and an exponential with the variable scaled by a factor depending on the degree. This note presents an elementary proof of the orthogonality of wave functions with differing energy levels. It is also shown that this is the only other natural orthogonality for Laguerre polynomials. By expanding in terms of the usual Laguerre polynomial basis an analogous strange orthogonality is o...
Optimization of coefficients of lists of polynomials by evolutionary algorithms
Sendra Pons, Juan Rafael; Winkler, Stephan M.
2015-01-01
We here discuss the optimization of coefficients of lists of polynomials using evolutionary computation. The given polynomials have 5 variables, namely t, a1, a2, a3, a4, and integer coefficients. The goal is to find integer values i, with i 2 {1, 2, 3, 4}, substituting ai such that, after crossing out the gcd (greatest common divisor) of all coefficients of the polynomials, the resulting integers are minimized in absolute value. Evolution strategies, a special class of heu...
Quantization of gauge fields, graph polynomials and graph cohomology
Kreimer, Dirk; Sars, Matthias; van Suijlekom, Walter D.
2012-01-01
We review quantization of gauge fields using algebraic properties of 3-regular graphs. We derive the Feynman integrand at n loops for a non-abelian gauge theory quantized in a covariant gauge from scalar integrands for connected 3-regular graphs, obtained from the two Symanzik polynomials. The transition to the full gauge theory amplitude is obtained by the use of a third, new, graph polynomial, the corolla polynomial. This implies effectively a covariant quantization without ghosts, where al...
Characteristic and Counting Polynomials: Modelling Nonane Isomers Properties
Jäntschi, Lorentz; BOLBOACA, Sorana D.; FURDUI, Cristina Maria
2009-01-01
Abstract The major goal of this study was to investigate the broad application of graph polynomials to the analysis of Henry?s law constants (solubility) of nonane isomers. In this context, Henry?s law constants of nonane isomers were modelled using characteristic and counting polynomials. The characteristic and counting polynomials on the distance matrix, on the maximal fragments matrix, on the complement of maximal fragments matrix, and on the Szeged matrix were calculated for ea...
LINX-1: a code for linking polynomial cross section files
The capabilities of the LINX-1 code are described. It was developed for the purpose of linking seperate fuel assembly and reflector node polynomial cross section files, obtained by the POLX-1 code, together into a single reactor polynomial cross section library. The output of the polynomial cross section library can be in either binary or fixed (BCD) format. Input data requirements and the format of the output file generated by LINX-1 are also described. 2 refs
Differentiation by integration with Jacobi polynomials
Liu, Da-Yan; Perruquetti, Wilfrid
2010-01-01
In this paper, the numerical differentiation by integration method based on Jacobi polynomials originally introduced by Mboup, Fliess and Join is revisited in the central case where the used integration window is centered. Such method based on Jacobi polynomials was introduced through an algebraic approach and extends the numerical differentiation by integration method introduced by Lanczos. The here proposed method is used to estimate the $n^{th}$ ($n \\in \\mathbb{N}$) order derivative from noisy data of a smooth function belonging to at least $C^{n+1+q}$ $(q \\in \\mathbb{N})$. In the recent paper of Mboup, Fliess and Join, where the causal and anti-causal case were investigated, the mismodelling due to the truncation of the Taylor expansion was investigated and improved allowing a small time-delay in the derivative estimation. Here, for the central case, we show that the bias error is $O(h^{q+2})$ where $h$ is the integration window length for $f\\in C^{n+q+2}$ in the noise free case and the corresponding conv...
On Factorization of Generalized Macdonald Polynomials
Kononov, Ya
2016-01-01
A remarkable feature of Schur functions -- the common eigenfunctions of cut-and-join operators from $W_\\infty$ -- is that they factorize at the peculiar two-parametric topological locus in the space of time-variables, what is known as the hook formula for quantum dimensions of representations of $U_q(SL_N)$ and plays a big role in various applications. This factorization survives at the level of Macdonald polynomials. We look for its further generalization to generalized Macdonald polynomials (GMP), associated in the same way with the toroidal Ding-Iohara-Miki algebras, which play the central role in modern studies in Seiberg-Witten-Nekrasov theory. In the simplest case of the first-coproduct eigenfunctions, where GMP depend on just two sets of time-variables, we discover a weak factorization -- on a one- (rather than four-) parametric slice of the topological locus, what is already a very non-trivial property, calling for proof and better understanding.
ASYMPTOTIC EXPANSIONS OF ZEROS FOR KRAWTCHOUK POLYNOMIALS WITH ERROR BOUNDS
ZHU Xiao-feng; LI Xiu-chun
2006-01-01
Krawtchouk polynomials are frequently applied in modern physics. Based on the results which were educed by Li and Wong, the asymptotic expansions of Krawtchouk polynomials are improved by using Airy function, and uniform asymptotic expansions are got. Furthermore, the asymptotic expansions of the zeros for Krawtchouk polynomials are again deduced by using the property of the zeros of Airy function, and their corresponding error bounds are discussed. The obtained results give the asymptotic property of Krawtchouk polynomials with their zeros, which are better than the results educed by Li and Wong.
Quantum Schubert polynomials and the Vafa-Intriligator formula
Kirillov, A N; Kirillov, Anatol N.; Maeno, Toshiaki
1996-01-01
We introduce a quantization map and study the quantization of Schubert and Grothendieck polynomials, monomials, elementary and complete polynomials. Our construction is based on the quantum Cauchy identity. As a corollary, we prove the Lascoux-Schützenberger type formula for quantum Schubert polynomials of the flag manifold. Our formula gives a simple method for computation of quantum Schubert polynomials. We also prove the higher genus analog of Vafa-Intriligator's formula for the flag manifold. We introduce the Extended Ehresman-Bruhat order on the symmetric group and prove the equivariant quantum Pieri formula.
An Analytic Formula for the A_2 Jack Polynomials
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.
Unimodularity of zeros of self-inversive polynomials
Lalin, Matilde N
2012-01-01
We generalise a necessary and sufficient condition given by Cohn for all the zeros of a self-inversive polynomial to be on the unit circle. Our theorem implies some sufficient conditions found by Lakatos, Losonczi and Schinzel. We apply our result to the study of a polynomial family closely related to Ramanujan polynomials, recently introduced by Gun, Murty and Rath, and studied by Murty, Smyth and Wang and Lal\\'in and Rogers. We prove that all polynomials in this family have their zeros on the unit circle, a result conjectured by Lal\\'in and Rogers on computational evidence.
Multi-indexed Wilson and Askey-Wilson Polynomials
Odake, Satoru
2013-01-01
As the third stage of the project multi-indexed orthogonal polynomials, we present, in the framework of 'discrete quantum mechanics' with pure imaginary shifts in one dimension, the multi-indexed Wilson and Askey-Wilson polynomials. They are obtained from the original Wilson and Askey-Wilson polynomials by multiple application of the discrete analogue of the Darboux transformations or the Crum-Krein-Adler deletion of 'virtual state solutions' of type I and II, in a similar way to the multi-indexed Laguerre, Jacobi and (q-)Racah polynomials reported earlier.
Quantum algorithms for virtual Jones polynomials via Thistlethwaite theorems
Vélez, Mario; Ospina, Juan
2010-04-01
Recently a quantum algorithm for the Jones polynomial of virtual links was proposed by Kauffman and Dye via the implementation of the virtual braid group in anyonic topological quantum computation when the virtual crossings are considered as generalized swap gates. Also recently, a mathematical method for the computation of the Jones polynomial of a given virtual link in terms of the relative Tuttle polynomial of its face (Tait) graph with some suitable variable substitutions was proposed by Diao and Hetyei. The method of Diao and Hetyei is offered as an alternative to the ribbon graph approach according to which the Tutte polynomial of a given virtual link is computed in terms of the Bollobás- Riordan polynomial of the corresponding ribbon graph. The method of Diao and Hetyei can be considered as an extension of the celebrated Thistlethwaite theorem according to which invariant polynomials for knots and links are derived from invariant polynomials for graphs. Starting from these ideas we propose a quantum algorithm for the Jones polynomial of a given virtual link in terms of the generalized Tutte polynomials by exploiting the Thistlethwaite theorem and the Kauffman algorithm . Our method is claimed as the quantum version of the Diao-Hetyei method. Possible supersymmetric implementations of our algortihm are discussed jointly with its formulations using topological quantum lambda calculus.
A Bivariate Analogue to the Composed Product of Polynomials
Donald Mills; Kent M. Neuerburg
2003-01-01
The concept of a composed product for univariate polynomials has been explored extensively by Brawley, Brown, Carlitz, Gao,Mills, et al. Starting with these fundamental ideas andutilizing fractional power series representation(in particular, the Puiseux expansion) of bivariate polynomials, we generalize the univariate results. We define a bivariate composed sum,composed multiplication,and composed product (based on function composition). Further, we investigate the algebraic structure of certain classes of bivariate polynomials under these operations. We also generalize a result of Brawley and Carlitz concerningthe decomposition of polynomials into irreducibles.
Compact Imaging Spectrometer Utilizing Immersed Gratings
Chrisp, Michael P. (Danville, CA); Lerner, Scott A. (Corvallis, OR); Kuzmenko, Paul J. (Livermore, CA); Bennett, Charles L. (Livermore, CA)
2006-03-21
A compact imaging spectrometer with an immersive diffraction grating that compensates optical distortions. The imaging spectrometer comprises an entrance slit for transmitting light, a system for receiving the light and directing the light, an immersion grating, and a detector array. The entrance slit, the system for receiving the light, the immersion grating, and the detector array are positioned wherein the entrance slit transmits light to the system for receiving the light and the system for receiving the light directs the light to the immersion grating and the immersion grating receives the light and directs the light through an optical element to the detector array.
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
Determinantal and permanental representation of generalized bivariate Fibonacci p-polynomials
Kaygisiz, Kenan; Sahin, Adem
2011-01-01
In this paper, we give some determinantal and permanental representations of generalized bivariate Fibonacci p-polynomials by using various Hessenberg matrices. The results that we obtained are important since generalized bivariate Fibonacci p-polynomials are general form of, for example, bivariate Fibonacci and Pell p-polynomials, second kind Chebyshev polynomials, bivariate Jacobsthal polynomials etc.
Inequalities for a Polynomial and its Derivative
V K Jain
2000-05-01
For an arbitrary entire function and any > 0, let $M(f, r):=\\max_{|z|=r}|f(z)|$. It is known that if is a polynomial of degree having no zeros in the open unit disc, and $m:=\\min_{|z|=1}|p(z)|$, then $$M(p',1)≤\\frac{n}{2}\\{M(p,1)-m\\},$$ $$M(p, R)≤\\left(\\frac{R^n+1}{2}\\right)M(p, 1)-m\\left(\\frac{R^n-1}{2}\\right), R>> 1.$$ It is also known that if has all its zeros in the closed unit disc, then $$M(p', 1)≥\\frac{n}{2}\\{M(p, 1)+m\\}.$$ The present paper contains certain generalizations of these inequalities
Orthogonal polynomials for refinable linear functionals
Laurie, Dirk; de Villiers, Johan
2006-12-01
A refinable linear functional is one that can be expressed as a convex combination and defined by a finite number of mask coefficients of certain stretched and shifted replicas of itself. The notion generalizes an integral weighted by a refinable function. The key to calculating a Gaussian quadrature formula for such a functional is to find the three-term recursion coefficients for the polynomials orthogonal with respect to that functional. We show how to obtain the recursion coefficients by using only the mask coefficients, and without the aid of modified moments. Our result implies the existence of the corresponding refinable functional whenever the mask coefficients are nonnegative, even when the same mask does not define a refinable function. The algorithm requires O(n^2) rational operations and, thus, can in principle deliver exact results. Numerical evidence suggests that it is also effective in floating-point arithmetic.
Study on the Grey Polynomial Geometric Programming
LUODang
2005-01-01
In the model of geometric programming, values of parameters cannot be gotten owing to data fluctuation and incompletion. But reasonable bounds of these parameters can be attained. This is to say, parameters of this model can be regarded as interval grey numbers. When the model contains grey numbers, it is hard for common programming method to solve them. By combining the common programming model with the grey system theory,and using some analysis strategies, a model of grey polynomial geometric programming, a model of 8 positioned geometric programming and their quasi-optimum solution or optimum solution are put forward. At the same time, we also developed an algorithm for the problem.This approach brings a new way for the application research of geometric programming. An example at the end of this paper shows the rationality and feasibility of the algorithm.
Dynamic normal forms and dynamic characteristic polynomial
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...
Digital terrain modeling with the Chebyshev polynomials
Florinsky, I V
2015-01-01
Mathematical problems of digital terrain analysis include interpolation of digital elevation models (DEMs), DEM generalization and denoising, and computation of morphometric variables by calculation of partial derivatives of elevation. Traditionally, these procedures are based on numerical treatments of two-variable discrete functions of elevation. We developed a spectral analytical method and algorithm based on high-order orthogonal expansions using the Chebyshev polynomials of the first kind with the subsequent Fejer summation. The method and algorithm are intended for DEM analytical treatment, such as, DEM global approximation, denoising, and generalization as well as computation of morphometric variables by analytical calculation of partial derivatives. To test the method and algorithm, we used a DEM of the Northern Andes including 230,880 points (the elevation matrix 480 $\\times$ 481). DEMs were reconstructed with 480, 240, 120, 60, and 30 expansion coefficients. The first and second partial derivatives ...
Polynomial Approximations of Electronic Wave Functions
Panin, Andrej I
2010-01-01
This work completes the construction of purely algebraic version of the theory of non-linear quantum chemistry methods. It is shown that at the heart of these methods there lie certain algebras close in their definition to the well-known Clifford algebra but quite different in their properties. The most important for quantum chemistry property of these algebras is the following : for a fixed number of electrons the corresponding sector of the Fock space becomes a commutative algebra and its ideals are determined by the order of excitations from the Hartree-Fock reference state. Quotients of this algebra can also be endowed with commutative algebra structures and quotient Schr{\\"o}dinger equations are exactly the couple cluster type equations. Possible computer implementation of multiplication in the aforementioned algebras is described. Quality of different polynomial approximations of configuration interaction wave functions is illustrated with concrete examples. Embedding of algebras of infinitely separated...
The Polynomially Exponential Time Restrained Analytical Hierarchy
眭跃飞
1991-01-01
A polynomially exponential time restrained analytical hierarchy is introduced with the basic properties of the hierarchy followed.And it will be shown that there is a recursive set A such that A does not belong to any level of the p-arithmetical hierarchies.Then we shall prove that there are recursive sets A and B such that the different levels of the analytical hierarchy relative to A are different and for some n every level higher than n of the analytical hierarchy relative to B is the same as the n-th level.And whether the higher levels are collapsed into some lower level is neither provable nor disprovable in set theory and several other results.
On Nilpotent Elements of Skew Polynomial Rings
J. Esmaeili
2012-09-01
Full Text Available We study the structure of the set of nilpotent elements in skew polynomial ring R[x; α], when R is an α-Armendariz ring. We prove that if R is a nil α-Armendariz ring and α t = IR, then the set of nilpotent elements of R is an α-compatible subrng of R. Also, it is shown that if R is an α-Armendariz ring and α t = IR, then R is nil α-Armendariz. We give some examples of non α-Armendariz rings which are nil α-Armendariz. Moreover, we show that if α t = IR for some positive integer t and R is a nil α-Armendariz ring and nil(R[x][y; α] = nil(R[x][y], then R[x] is nil α-Armendariz. Some results of [3] follow as consequences of our results
Properties of the corolla polynomial of a 3-regular graph
Kreimer, Dirk; Yeats, Karen
2012-01-01
We investigate combinatorial properties of a graph polynomial indexed by half-edges of a graph which was introduced recently to understand the connection between Feynman rules for scalar field theory and Feynman rules for gauge theory. We investigate the new graph polynomial as a stand-alone object.
Zonal polynomials and hypergeometric functions of quaternion matrix argument
Li, Fei; Xue, Yifeng
2009-01-01
We define zonal polynomials of quaternion matrix argument and deduce some important formulae of zonal polynomials and hypergeometric functions of quaternion matrix argument. As an application, we give the distributions of the largest and smallest eigenvalues of a quaternion central Wishart matrix $W\\sim\\mathbb{Q}W(n,\\Sigma)$, respectively.
On the Structure of Cubic and Quartic Polynomials
Haramaty, Elad
2009-01-01
In this paper we study the structure of polynomials of degree three and four that have high bias or high Gowers norm, over arbitrary prime fields. In particular we obtain the following results. 1. We give a canonical representation for degree three or four polynomials that have a significant bias (i.e. they are not equidistributed). This result generalizes the corresponding results from the theory of quadratic forms. It also significantly improves the results of Green and Tao and Kaufman and Lovett for such polynomials. 2. For the case of degree four polynomials with high Gowers norm we show that (a subspace of co-dimension O(1) of) F^n can be partitioned to subspaces of dimension Omega(n) such that on each of the subspaces the polynomial is equal to some degree three polynomial. It was shown by Green and Tao and by Lovett, Meshulam and Samorodnitsky that a quartic polynomial with a high Gowers norm is not necessarily correlated with any cubic polynomial. Our result shows that a slightly weaker statement does...
On fully split lacunary polynomials in finite fields
Bibak, Khodakhast; Shparlinski, Igor E.
2011-01-01
We estimate the number of possible types degree patterns of $k$-lacunary polynomials of degree $t < p$ which split completely modulo $p$. The result is based on a combination of a bound on the number of zeros of lacunary polynomials with some graph theory arguments.
Learning Read-constant Polynomials of Constant Degree modulo Composites
Chattopadhyay, Arkadev; Gavaldá, Richard; Hansen, Kristoffer Arnsfelt;
2011-01-01
known to be learnable in any reasonable learning model. In this paper, we provide a deterministic polynomial time algorithm for learning Boolean functions represented by polynomials of constant degree over arbitrary finite rings from membership queries, with the additional constraint that each variable...
A generating algorithm for ribbon tableaux and spin polynomials
Francois Descouens
2006-01-01
We describe a general algorithm for generating various families of ribbon tableaux and computing their spin polynomials. This algorithm is derived from a new matricial coding. An advantage of this new notation lies in the fact that it permits one to generate ribbon tableaux with skew shapes. This algorithm permits us to compute quickly big LLT polynomials in MuPAD-Combinat.
On peculiar properties of generating functions of some orthogonal polynomials
We prove that for |x| ⩽ |t| ≥(ti)/(q)ihn+i( x|q) =hn(x|t,q) Σi≥0(ti)/(q)ihi(x|q), where hn(x|q) and hn(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)
Extended Fibonacci numbers and polynomials with probability applications
Demetrios L. Antzoulakos
2004-10-01
Full Text Available The extended Fibonacci sequence of numbers and polynomials is introduced and studied. The generating function, recurrence relations, an expansion in terms of multinomial coefficients, and several properties of the extended Fibonacci numbers and polynomials are obtained. Interesting relations between them and probability problems which take into account lengths of success and failure runs are also established.
On linear operators preserving the set of positive polynomials
Guterman, Alexander; Shapiro, Boris
2008-01-01
Following the classical approach of P´olya-Schur theory [14] we initiate in this paper the study of linear operators acting on R[x] and preserving either the set of positive univariate polynomials or similar sets of non-negative and elliptic polynomials.
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…
Generalization of the Macdonald formula for Hall-Littlewood polynomials
Klostermann, Inka
2011-01-01
We study the Gaussent-Littelmann formula for Hall-Littlewood polynomials and we develop combinatorial tools to describe the formula in a purely combinatorial way for type A_n. Furthermore, we show by using these tools that the Gaussent-Littelmann formula and the well-known Macdonald formula for Hall-Littlewood polynomials for type A_n are the same.
A Parallel Algorithm for Finding Roots of a Complex Polynomial
程锦松
1990-01-01
A distribution theory of the roots of a polynomial and a parallel algorithm for finding roots of a complex polynomial based on that theory are developed in this paper.With high parallelism,the algorithm is an improvement over the Wilf algorithm[3].
Approximation to Continuous Functions by a Kind of Interpolation Polynomials
Yuan Xue-gang; Wang De-hui
2001-01-01
In this paper, an interpolation polynomial operator Fn (f; l, x ) is constructed based on the zeros of a kind of Jacobi polynomials as the interpolation nodes. For any continuous function f(x)∈ Cb[1,1] (0≤b≤l) Fn(f; l,x) converges to f(x) uniformly, where l is an odd number.
On -Euler Numbers Related to the Modified -Bernstein Polynomials
Min-Soo Kim; Daeyeoul Kim; Taekyun Kim
2010-01-01
We consider q-Euler numbers, polynomials, and q-Stirling numbers of first and second kinds. Finally, we investigate some interesting properties of the modified q-Bernstein polynomials related to q-Euler numbers and q-Stirling numbers by using fermionic p-adic integrals on ℤp.
A new two-variable generalization of the Jones polynomial
Goundaroulis, Dimos; Lambropoulou, Sofia
2016-01-01
We present a new 2-variable generalization of the Jones polynomial that can be defined through the skein relation of the Jones polynomial. The well-definedness of this new generalization is proved algebraically. We also give a closed combinatorial formula for this new classical link invariant.
Polynomial perturbations of hermitian linear functionals and difference equations
Cantero, M J; Velázquez, L
2009-01-01
This paper is devoted to the study of general (Laurent) polynomial modifications of moment functionals on the unit circle, i.e., associated with hermitian Toeplitz matrices. We present a new approach which allows us to study polynomial modifications of arbitrary degree. The main objective is the characterization of the quasi-definiteness of the functionals involved in the problem in terms of a difference equation relating the corresponding Schur parameters. The results are presented in the general framework of (non necessarily quasi-definite) hermitian functionals, so that the maximum number of orthogonal polynomials is characterized by the number of consistent steps of an algorithm based on the referred recurrence for the Schur parameters. Some concrete applications to the study of orthogonal polynomials on the unit circle show the effectiveness of this new approach: an exhaustive and instructive analysis of the functionals coming from a general inverse polynomial perturbation of degree one for the Lebesgue ...
Multiple Meixner polynomials and non-Hermitian oscillator Hamiltonians
Ndayiragije, F.; Van Assche, W.
2013-12-01
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.
Low degree polynomial equations arithmetic, geometry and topology
Kollár, J
1996-01-01
These are the notes of my lectures at the 1996 European Congress of Mathematicians. Polynomials appear in mathematics frequently, and we all know from experience that low degree polynomials are easier to deal with than high degree ones. It is, however, not clear that there is a well defined class of ``low degree" polynomials. For many questions, polynomials behave well if their degree is low enough, but the precise bound on the degree depends on the concrete problem. It turns out that there is a collection of basic questions in arithmetic, algebraic geometry and topology all of which give the same class of ``low degree" polynomials. The aim of this lecture is to explain these properties and to provide a survey of the known results.
Approximating smooth functions using algebraic-trigonometric polynomials
Sharapudinov, Idris I.
2011-01-01
The problem under consideration is that of approximating classes of smooth functions by algebraic-trigonometric polynomials of the form p_n(t)+\\tau_m(t), where p_n(t) is an algebraic polynomial of degree n and \\tau_m(t)=a_0+\\sum_{k=1}^ma_k\\cos k\\pi t+b_k\\sin k\\pi t is a trigonometric polynomial of order m. The precise order of approximation by such polynomials in the classes W^r_\\infty(M) and an upper bound for similar approximations in the class W^r_p(M) with \\frac43 are found. The proof of these estimates uses mixed series in Legendre polynomials which the author has introduced and investigated previously. Bibliography: 13 titles.
Approximating smooth functions using algebraic-trigonometric polynomials
The problem under consideration is that of approximating classes of smooth functions by algebraic-trigonometric polynomials of the form pn(t)+τm(t), where pn(t) is an algebraic polynomial of degree n and τm(t)=a0+Σk=1mak cos kπt + bk sin kπt is a trigonometric polynomial of order m. The precise order of approximation by such polynomials in the classes Wr∞(M) and an upper bound for similar approximations in the class Wrp(M) with 4/3< p<4 are found. The proof of these estimates uses mixed series in Legendre polynomials which the author has introduced and investigated previously. Bibliography: 13 titles.
Properties of the zeros of generalized basic hypergeometric polynomials
Bihun, Oksana; Calogero, Francesco
2015-11-01
We define the generalized basic hypergeometric polynomial of degree N in terms of the generalized basic hypergeometric function, by choosing one of its parameters to allow the termination of the series after a finite number of summands. In this paper, we obtain a set of nonlinear algebraic equations satisfied by the N zeros of the polynomial. Moreover, we obtain an N × N matrix M defined in terms of the zeros of the polynomial, which, in turn, depend on the parameters of the polynomial. The eigenvalues of this remarkable matrix M are given by neat expressions that depend only on some of the parameters of the polynomial; that is, the matrix M is isospectral. Moreover, in case the parameters that appear in the expressions for the eigenvalues of M are rational, the matrix M has rational eigenvalues, a Diophantine property.
On Polynomial Kernels for Structural Parameterizations of Odd Cycle Transversal
Jansen, Bart M P
2011-01-01
The Odd Cycle Transversal problem (OCT) asks whether a given graph can be made bipartite (i.e., 2-colorable) by deleting at most l vertices. We study structural parameterizations of OCT with respect to their polynomial kernelizability, i.e., whether instances can be efficiently reduced to a size polynomial in the chosen parameter. It is a major open problem in parameterized complexity whether Odd Cycle Transversal admits a polynomial kernel when parameterized by l. On the positive side, we show a polynomial kernel for OCT when parameterized by the vertex deletion distance to the class of bipartite graphs of treewidth at most w (for any constant w); this generalizes the parameter feedback vertex set number (i.e., the distance to a forest). Complementing this, we exclude polynomial kernels for OCT parameterized by the distance to outerplanar graphs, conditioned on the assumption that NP \
Higher order branching of periodic orbits from polynomial isochrones
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.
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. PMID:25791164
Fiber Bragg Grating Based Thermometry
Ahmed, Zeeshan; Filla, James; Guthrie, William; Quintavalle, John
2016-01-01
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 a hermetically sealed dry Argon environment, that FBG devices show a quadratic dependence on temperature with expanded uncertainties (k = 2) of ~500 mK. Our measurements indicate that the combined m...
Nested long period grating interferometers
Murphy, Richard P.; James, Stephen W.; Tatam, Ralph P.
2007-01-01
The concept of nested fibre optic long period grating (LPG) based interferometers is introduced. A number of in-series, identical LPGs may be used to form a set of nested, multiplexed Mach-Zehnder interferometers that may demodulated and demultiplexed by virtue of a Fourier analysis of the optical spectrum. The concept is demonstrated by the use of three LPGs to form a nested set of interferometers.
Diffraction gratings for lighting applications
Cornelissen, Hugo J.; de Boer, Dick K. G.; Tukker, Teus
2013-09-01
Sub-micron diffraction gratings have been used for two LED illumination applications. One is to create a transparent see through luminaire which can be used to illuminate and read a paper document or e-book. A second is a light sensor that can be used in a feedback loop to control a multicolor LED lamp. Optical design and experimental proof-of-principle are presented.
Fabrication of large-area and low mass critical-angle x-ray transmission gratings
Heilmann, Ralf K.; Bruccoleri, Alex R.; Guan, Dong; Schattenburg, Mark L.
2014-07-01
Soft x-ray spectroscopy of celestial sources with high resolving power R = E/ΔE and large collecting area addresses important science listed in the Astro2010 Decadal Survey New Worlds New Horizons, such as the growth of the large scale structure of the universe and its interaction with active galactic nuclei, the kinematics of galactic outflows, as well as coronal emission from stars and other topics. Numerous studies have shown that a transmission grating spectrometer based on lightweight critical-angle transmission (CAT) gratings can deliver R = 3000-5000 and large collecting area with high efficiency and minimal resource requirements, providing spectroscopic figures of merit at least an order of magnitude better than grating spectrometers on Chandra and XMM-Newton, as well as future calorimeter-based missions. The recently developed CAT gratings combine the advantages of transmission gratings (low mass, relaxed figure and alignment tolerances) and blazed reflection gratings (high broad band diffraction efficiency, utilization of higher diffraction orders). Their working principle based on blazing through reflection off the smooth, ultra-high aspect ratio grating bar sidewalls has previously been demonstrated on small samples with x rays. For larger gratings (area greater than 1 inch square) we developed a fabrication process for grating membranes with a hierarchy of integrated low-obscuration supports. The fabrication involves a combination of advanced lithography and highly anisotropic dry and wet etching techniques. We report on the latest fabrication results of free-standing, large-area CAT gratings with polished sidewalls and preliminary x-ray tests.
Varied line-space gratings and applications
This paper presents a straightforward analytical and numerical method for the design of a specific type of varied line-space grating system. The mathematical development will assume plane or nearly-plane spherical gratings which are illuminated by convergent light, which covers many interesting cases for synchrotron radiation. The gratings discussed will have straight grooves whose spacing varies across the principal plane of the grating. Focal relationships and formulae for the optical grating-pole-to-exist-slit distance and grating radius previously presented by other authors will be derived with a symbolic algebra system. It is intended to provide the optical designer with the tools necessary to design such a system properly. Finally, some possible advantages and disadvantages for application to synchrotron to synchrotron radiation beamlines will be discussed
Self-imaging by a volume grating
Forte, Gustavo; Lencina, Alberto; Tebaldi, Myrian; Bolognini, Néstor
2011-05-01
The self-image phenomenon by a volume grating is proposed and theoretically analyzed. A theoretical model based on a path integral formulation to describe wave propagation through the grating inhomogeneous medium is applied. A modified version of the scalar diffraction theory Fresnel propagator is obtained which allows calculating the diffracted field amplitude by the grating. The proposed model is applied to amplitude and/or phase volume gratings. Remarkable features appear, in particular at the fractional Talbot distance 0.125 zT. In this case, if an in-phase real and imaginary grating modulation is considered a self-image intensity profile is observed for determined values of the absorptive and refractive parameters. On the other hand, a spatial comb intensity profile for a near half period shift between the real and imaginary grating modulations is found.
On a class of polynomials associated with the Cliques in a graph and its applications
E. J. Farrell
1989-03-01
Full Text Available The clique polynomial of a graph is defined. An explicit formula is then derived for the clique polynomial of the complete graph. A fundamental theorem and a reduction process is then given for clique polynomials. Basic properties of the polynomial are also given. It is shown that the number theoretic functions defined by Menon are related to clique polynomials. This establishes a connection between the clique polynomial and decompositions of finite sets, symmetric groups and analysis.
A link polynomial via a vertex-edge-face state model
Fiedler, Thomas
2007-01-01
We construct a 2-variable link polynomial, called $W_L$, for classical links by considering simultaneously the Kauffman state models for the Alexander and for the Jones polynomials. We conjecture that this polynomial is the product of two 1-variable polynomials, one of which is the Alexander polynomial. We refine $W_L$ to an ordered set of 3-variable polynomials for those links in 3-space which contain a Hopf link as a sublink.
Advanced grating laser designs for microwave generation
Ibsen, M.; Ronnekleiv, E.; Hadeler, O.; Cowle, G.J.; Laming, R.I.; Zervas, M. N.
2000-01-01
Fibre Bragg gratings have over a relatively short evolutionary process matured from prototypes in laboratory environments to commercial products in real world applications. This short process has been driven by a combined effort from many research groups throughout the scientific communities in the development and refinement of grating fabrication techniques as well as a number of potential applications. One field that has attracted attention from many obvious applications of Bragg gratings i...
Hurvitz, G.; Ehrlich, Y.; Strum, G.; Shpilman, Z.; Levy, I.; Fraenkel, M.
2012-01-01
A novel fabrication method for soft x-ray transmission grating and other optical elements is presented. The method uses Focused-Ion-Beam (FIB) technology to fabricate high-quality free standing grating bars on Transmission Electron Microscopy grids (TEM-grid). High quality transmission gratings are obtained with superb accuracy and versatility. Using these gratings and back-illuminated CCD camera, absolutely calibrated x-ray spectra can be acquired for soft x-ray source diagnostics in the 100...
Multilayer dielectric gratings for tiled-gratings compression of petawatt pulses.
Cotel, Arnaud
2007-01-01
Pulse compression diffraction gratings represent currently an important bottleneck for the development of energetic high-intensity Petawatt laser. Indeed, the laser-induced damage threshold of standard gold-coated gratings and the diffraction efficiency are limited. That's why we have developed a new generation of diffraction gratings: ! multilayer dielectric (MLD) gratings. Studies of MLD g! ratings applied to the Pico2000 laser project are the first part of my thesis work. On the other hand...
Perfect crystallike gratings for cold neutrons
We report on significant improvements of the performance of thick diffraction gratings for cold neutrons. The basis material for the production of holographic gratings by optical means is photosensitized deuterated poly(methyl methacrylate) (D-PMMA). The properties of these gratings now approach those of perfect monochromator crystals for neutrons of shorter wavelength: for cold neutrons with 1.0 nm wavelength the gratings exhibit a reflectivity in the percent range which makes them suitable for a wide range of applications in neutron optics
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
Conformal Laplace superintegrable systems in 2D: polynomial invariant subspaces
Escobar-Ruiz, M. A.; Miller, Willard, Jr.
2016-07-01
2nd-order conformal superintegrable systems in n dimensions are Laplace equations on a manifold with an added scalar potential and 2n-1 independent 2nd order conformal symmetry operators. They encode all the information about Helmholtz (eigenvalue) superintegrable systems in an efficient manner: there is a 1-1 correspondence between Laplace superintegrable systems and Stäckel equivalence classes of Helmholtz superintegrable systems. In this paper we focus on superintegrable systems in two-dimensions, n = 2, where there are 44 Helmholtz systems, corresponding to 12 Laplace systems. For each Laplace equation we determine the possible two-variate polynomial subspaces that are invariant under the action of the Laplace operator, thus leading to families of polynomial eigenfunctions. We also study the behavior of the polynomial invariant subspaces under a Stäckel transform. The principal new results are the details of the polynomial variables and the conditions on parameters of the potential corresponding to polynomial solutions. The hidden gl 3-algebraic structure is exhibited for the exact and quasi-exact systems. For physically meaningful solutions, the orthogonality properties and normalizability of the polynomials are presented as well. Finally, for all Helmholtz superintegrable solvable systems we give a unified construction of one-dimensional (1D) and two-dimensional (2D) quasi-exactly solvable potentials possessing polynomial solutions, and a construction of new 2D PT-symmetric potentials is established.
Recent developments of Bragg gratings in PMMA and TOPAS polymer optical fibers
Webb, David; Kyriacos, Kalli; Carroll, Karen; Zhang, Chi; Komodromos, Michael Frederick; Argyros, Alex; Large, Maryanne; Emiliyanov, Grigoriy Andreev; Bang, Ole; Kjær, Erik Michael
We report on the temperature response of FBGs recorded in pure PMMA and TOPAS holey fibers. The gratings are fabricated in the near IR using a cw He-Cd laser operating at 325nm. The room temperature grating response is non-linear and characterised by quadratic behaviour for temperatures from room......, 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...
Continuously tunable laser based on polarization gratings in azobenzene-containing material
Two simple procedures to manufacture continuously tunable miniature lasers in azobenzene-containing material were developed. Both types of lasers rely on the distributed feedback provided by polarization gratings. In a first approach tuning is achieved by changing the thickness of the active wave guiding layer by means of a wedge layer and in a second approach by gradually changing the spatial frequency of the refractive index grating obtained by modification of the Lloyd interferometer set-up used for holographic inscription of the gratings. A continuous tuning range of up to 35 nm has been demonstrated. (paper)
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.
Ladder operators and recursion relations for the associated Bessel polynomials
Introducing the associated Bessel polynomials in terms of two non-negative integers, and under an integrability condition we simultaneously factorize their corresponding differential equation into a product of the ladder operators by four different ways as shape invariance symmetry equations. This procedure gives four different pairs of recursion relations on the associated Bessel polynomials. In spite of description of Bessel and Laguerre polynomials in terms of each other, we show that the associated Bessel differential equation is factorized in four different ways whereas for Laguerre one we have three different ways
Ladder operators and recursion relations for the associated Bessel polynomials
Fakhri, H.; Chenaghlou, A.
2006-10-01
Introducing the associated Bessel polynomials in terms of two non-negative integers, and under an integrability condition we simultaneously factorize their corresponding differential equation into a product of the ladder operators by four different ways as shape invariance symmetry equations. This procedure gives four different pairs of recursion relations on the associated Bessel polynomials. In spite of description of Bessel and Laguerre polynomials in terms of each other, we show that the associated Bessel differential equation is factorized in four different ways whereas for Laguerre one we have three different ways.
Ladder operators and recursion relations for the associated Bessel polynomials
Fakhri, H. [Institute for Studies in Theoretical Physics and Mathematics (IPM), PO Box 19395-5531, Tehran (Iran, Islamic Republic of) and Department of Theoretical Physics and Astrophysics, Physics Faculty, Tabriz University, PO Box 51666-16471, Tabriz (Iran, Islamic Republic of)]. E-mail: hfakhri@ipm.ir; Chenaghlou, A. [Institute for Studies in Theoretical Physics and Mathematics (IPM), PO Box 19395-5531, Tehran (Iran, Islamic Republic of) and Physics Department, Faculty of Science, Sahand University of Technology, PO Box 51335-1996, Tabriz (Iran, Islamic Republic of)]. E-mail: a.chenaghlou@sut.ac.ir
2006-10-30
Introducing the associated Bessel polynomials in terms of two non-negative integers, and under an integrability condition we simultaneously factorize their corresponding differential equation into a product of the ladder operators by four different ways as shape invariance symmetry equations. This procedure gives four different pairs of recursion relations on the associated Bessel polynomials. In spite of description of Bessel and Laguerre polynomials in terms of each other, we show that the associated Bessel differential equation is factorized in four different ways whereas for Laguerre one we have three different ways.
Polynomial system solving for decoding linear codes and algebraic cryptanalysis
Bulygin, Stanislav
2009-01-01
This thesis is devoted to applying symbolic methods to the problems of decoding linear codes and of algebraic cryptanalysis. The paradigm we employ here is as follows. We reformulate the initial problem in terms of systems of polynomial equations over a finite field. The solution(s) of such systems should yield a way to solve the initial problem. Our main tools for handling polynomials and polynomial systems in such a paradigm is the technique of Gröbner bases and normal form reductions. The ...
Stable radial distortion calibration by polynomial matrix inequalities programming
Heller, Jan; Pajdla, Tomas
2014-01-01
Polynomial and rational functions are the number one choice when it comes to modeling of radial distortion of lenses. However, several extrapolation and numerical issues may arise while using these functions that have not been covered by the literature much so far. In this paper, we identify these problems and show how to deal with them by enforcing nonnegativity of certain polynomials. Further, we show how to model these nonnegativities using polynomial matrix inequalities (PMI) and how to estimate the radial distortion parameters subject to PMI constraints using semidefinite programming (SDP). Finally, we suggest several approaches on how to incorporate the proposed method into the overall camera calibration procedure.
Ratio Monotonicity of Polynomials Derived from Nondecreasing Sequences
Chen, William Y C; Zhou, Elaine L F
2010-01-01
The ratio monotonicity of a polynomial is a stronger property than log-concavity. Let P(x) be a polynomial with nonnegative and nondecreasing coefficients. We prove the ratio monotone property of P(x+1), which leads to the log-concavity of P(x+c) for any $c\\geq 1$ due to Llamas and Mart\\'{\\i}nez-Bernal. As a consequence, we obtain the ratio monotonicity of the Boros-Moll polynomials obtained by Chen and Xia without resorting to the recurrence relations of the coefficients.
Transfer matrix computation of generalised critical polynomials in percolation
Scullard, Christian R.; Jacobsen, Jesper Lykke
2012-01-01
Percolation thresholds have recently been studied by means of a graph polynomial $P_B(p)$, henceforth referred to as the critical polynomial, that may be defined on any periodic lattice. The polynomial depends on a finite subgraph $B$, called the basis, and the way in which the basis is tiled to form the lattice. The unique root of $P_B(p)$ in $[0,1]$ either gives the exact percolation threshold for the lattice, or provides an approximation that becomes more accurate with appropriately increa...
Separation of variables in the A$_{2}$ type Jack polynomials
Kuznetsov, V B
1995-01-01
An integral operator M is constructed performing a separation of variables for the 3-particle quantum Calogero-Sutherland (CS) model. Under the action of M the CS eigenfunctions (Jack polynomials for the root system A_2) are transformed to the factorized form \\phi(y_1)\\phi(y_2), where \\phi(y) is a trigonometric polynomial of one variable expressed in terms of the {}_3F_2 hypergeometric series. The inversion of M produces a new integral representation for the A_2 Jack polynomials.
Combinatorial theory of Macdonald polynomials I: Proof of Haglund's formula
Haglund, J.; Haiman, M.; Loehr, N.
2005-01-01
Haglund recently proposed a combinatorial interpretation of the modified Macdonald polynomials H̃μ. We give a combinatorial proof of this conjecture, which establishes the existence and integrality of H̃μ. As corollaries, we obtain the cocharge formula of Lascoux and Schützenberger for Hall–Littlewood polynomials, a formula of Sahi and Knop for Jack's symmetric functions, a generalization of this result to the integral Macdonald polynomials Jμ, a formula for H̃μ in terms of Lascoux–Leclerc–Th...
The F-pure threshold of quasi-homogeneous polynomials
Müller, Susanne
2016-01-01
Inspired by the work of Bhatt and Singh (see: arXiv:1307.1171) we compute the $F$-pure threshold of quasi-homogeneous polynomials. We first consider the case of a curve given by a quasi-homogeneous polynomial $f$ in three variables $x,y,z$ of degree equal to the degree of $xyz$ and then we proceed with the general case of a Calabi-Yau hypersurface, i.e. a hypersurface given by a quasi-homogeneous polynomial $f$ in $n+1$ variables $x_0, \\ldots, x_n$ of degree equal to the degree of $x_0 \\cdots...
Generalizations of Poly-Bernoulli numbers and polynomials
Jolany, Hassan; Darafsheh, M. R.; Alikelaye, R. Eizadi
2012-01-01
The Concepts of poly-Bernoulli numbers $B_n^{(k)}$, poly-Bernoulli polynomials $B_n^{k}{(t)}$ and the generalized poly-bernoulli numbers $B_{n}^{(k)}(a,b)$ are generalized to $B_{n}^{(k)}(t,a,b,c)$ which is called the generalized poly-Bernoulli polynomials depending on real parameters \\textit{a,b,c}. Some properties of these polynomials and some relationships between $B_n^{k}$, $B_n^{(k)}(t)$, $B_{n}^{(k)}(a,b)$ and $B_{n}^{(k)}(t,a,b,c)$ are established
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.
Uniform asymptotics of the coefficients of unitary moment polynomials
Hiary, Ghaith A
2010-01-01
Keating and Snaith showed that the $2k^{th}$ absolute moment of the characteristic polynomial of a random unitary matrix evaluated on the unit circle is given by a polynomial of degree $k^2$. In this article, uniform asymptotics for the coefficients of that polynomial are derived, and a maximal coefficient is located. Some of the asymptotics are given in explicit form. Numerical data to support these calculations are presented. Some apparent connections between random matrix theory and the Riemann zeta function are discussed.
q-Bernoulli numbers and q-Bernoulli polynomials revisited
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
Integral Inequalities for Self-Reciprocal Polynomials
Horst Alzer
2010-04-01
Let $n≥ 1$ be an integer and let $\\mathcal{P}_n$ be the class of polynomials of degree at most satisfying $z^nP(1/z)=P(z)$ for all $z\\in C$. Moreover, let be an integer with $1≤ r≤ n$. Then we have for all $P\\in\\mathcal{P}_n$: $$_n(r)\\int^{2}_0|P(e^{it})|^2dt≤\\int^{2}_0|P^{(r)}(e^{it})|^2dt≤_n(r)\\int^{2}_0|P(e^{it})|^2dt$$ with the best possible factors \\begin{equation*}_n(r)=\\begin{cases}\\prod^{r-1}_{j=0}\\left(\\frac{n}{2}-j\\right)^2, < \\text{if is even},\\\\ \\frac{1}{2}\\left[\\prod^{r-1}_{j=0}\\left(\\frac{n+1}{2}-j\\right)^2+\\prod^{r-1}_{j=0}\\left(\\frac{n-1}{2}-j\\right)^2\\right], < \\text{if is odd},\\end{cases}\\end{equation*} and \\begin{equation*}_n(r)=\\frac{1}{2}\\prod\\limits^{r-1}_{j=0}(n-j)^2.\\end{equation*} This refines and extends a result due to Aziz and Zargar (1997).
Constructing Polynomial Spectral Models for Stars
Rix, Hans-Walter; Conroy, Charlie; Hogg, David W
2016-01-01
Stellar spectra depend on the stellar parameters and on dozens of photospheric elemental abundances. Simultaneous fitting of these $\\mathcal{N}\\sim \\,$10-40 model labels to observed spectra has been deemed unfeasible, because the number of ab initio spectral model grid calculations scales exponentially with $\\mathcal{N}$. We suggest instead the construction of a polynomial spectral model (PSM) of order $\\mathcal{O}$ for the model flux at each wavelength. Building this approximation requires a minimum of only ${\\mathcal{N}+\\mathcal{O}\\choose\\mathcal{O}}$ calculations: e.g. a quadratic spectral model ($\\mathcal{O}=\\,$2), which can then fit $\\mathcal{N}=\\,$20 labels simultaneously, can be constructed from as few as 231 ab initio spectral model calculations; in practice, a somewhat larger number ($\\sim\\,$300-1000) of randomly chosen models lead to a better performing PSM. Such a PSM can be a good approximation to ab initio spectral models only over a limited portion of label space, which will vary case by case. Y...
Polynomial Method for PLL Controller Optimization
Tsung-Yu Chiou
2011-06-01
Full Text Available The Phase-Locked Loop (PLL is a key component of modern electronic communication and control systems. PLL is designed to extract signals from transmission channels. It plays an important role in systems where it is required to estimate the phase of a received signal, such as carrier tracking from global positioning system satellites. In order to robustly provide centimeter-level accuracy, it is crucial for the PLL to estimate the instantaneous phase of an incoming signal which is usually buried in random noise or some type of interference. This paper presents an approach that utilizes the recent development in the semi-definite programming and sum-of-squares field. A Lyapunov function will be searched as the certificate of the pull-in range of the PLL system. Moreover, a polynomial design procedure is proposed to further refine the controller parameters for system response away from the equilibrium point. Several simulation results as well as an experiment result are provided to show the effectiveness of this approach.
Polynomial Time Algorithms for Minimum Energy Scheduling
Baptiste, Philippe; Durr, Christoph
2009-01-01
The aim of power management policies is to reduce the amount of energy consumed by computer systems while maintaining satisfactory level of performance. One common method for saving energy is to simply suspend the system during the idle times. No energy is consumed in the suspend mode. However, the process of waking up the system itself requires a certain fixed amount of energy, and thus suspending the system is beneficial only if the idle time is long enough to compensate for this additional energy expenditure. In the specific problem studied in the paper, we have a set of jobs with release times and deadlines that need to be executed on a single processor. Preemptions are allowed. The processor requires energy L to be woken up and, when it is on, it uses one unit of energy per one unit of time. It has been an open problem whether a schedule minimizing the overall energy consumption can be computed in polynomial time. We solve this problem in positive, by providing an O(n^5)-time algorithm. In addition we pr...
Polynomial super-gl(n) algebras
We introduce a class of finite-dimensional nonlinear superalgebras L L0-bar + L1-bar providing gradings of L0-bar = gl(n) ≅ sl(n) + gl(1). Odd generators close by anticommutation on polynomials (of degree >1) in the gl(n) generators. Specifically, we investigate 'type I' super-gl(n) algebras, having odd generators transforming in a single irreducible representation of gl(n) together with its contragredient. Admissible structure constants are discussed in terms of available gl(n) couplings, and various special cases and candidate superalgebras are identified and exemplified via concrete oscillator constructions. For the case of the n-dimensional defining representation, with odd generators Qa, Q-barb and even generators Eab, a, b = 1, ..., n, a three-parameter family of quadratic super-gl(n) algebras (deformations of sl(n/1)) is defined. In general, additional covariant Serre-type conditions are imposed in order that the Jacobi identities are fulfilled. For these quadratic super-gl(n) algebras, the construction of Kac modules and conditions for atypicality are briefly considered. Applications in quantum field theory, including Hamiltonian lattice QCD and spacetime supersymmetry, are discussed
Superradiance in volume diffraction grating
Baryshevsky, V. G.; Batrakov, K. G.
2001-01-01
Volume Free Electron Laser (VFEL) was proposed in [1-4]. It can operate in the wide spectral range from microwaves to X-rays. To simulate the processes which take place in VFEL the superradiance from a short electron pulse moving in a volume diffraction grating is studied in wavelength range ~4 mm. It is supposed that Bragg condition for emitted photons is fulfilled and dynamical diffraction takes place. The spectral-angular distributions for transmitted and diffracted waves are derived. It i...
Fiber Bragg Grating Based Thermometry
Ahmed, Zeeshan; Guthrie, William; Quintavalle, John
2016-01-01
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 a hermetically sealed dry Argon environment, that 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 the peak center fitting and by thermal aging of polyimide coated fibers.
Fiber optic diffraction grating maker
Deason, Vance A.; Ward, Michael B.
1991-01-01
A compact and portable diffraction grating maker comprised of a laser beam, optical and fiber optics devices coupling the beam to one or more evanescent beam splitters, and collimating lenses or mirrors directing the split beam at an appropriate photosensitive material. The collimating optics, the output ends of the fiber optic coupler and the photosensitive plate holder are all mounted on an articulated framework so that the angle of intersection of the beams can be altered at will without disturbing the spatial filter, collimation or beam quality, and assuring that the beams will always intersect at the position of the plate.
Alshal, Mohamed A.; Allam, Nageh K.
2016-07-01
A design for a highly efficient modified grating crystalline silicon (c-Si) thin film solar cell is demonstrated and analyzed using the two-dimensional (2-D) finite element method. The suggested grating has a double-sided pyramidal structure. The incorporation of the modified grating in a c-Si thin film solar cell offers a promising route to harvest light into the few micrometers active layer. Furthermore, a layer of silicon nitride is used as an antireflection coating (ARC). Additionally, the light trapping through the suggested design is significantly enhanced by the asymmetry of the top and bottom pyramids. The effects of the thickness of the active layer and facet angle of the pyramid on the spectral absorption, ultimate efficiency (η), and short-circuit current density (J sc) are investigated. The numerical results showed 87.9% efficiency improvement over the conventional thin film c-Si solar cell counterpart without gratings.
Tilt sensitivity of the two-grating interferometer
Anderson, Christopher N.; Naulleau, Patrick P.
2008-01-30
Fringe formation in the two-grating interferometer is analyzed in the presence of a small parallelism error between the diffraction gratings assumed in the direction of grating shear. Our analysis shows that with partially coherent illumination, fringe contrast in the interference plane is reduced in the presence of nonzero grating tilt with the effect proportional to the grating tilt angle and the grating spatial frequencies. Our analysis also shows that for a given angle between the gratings there is an angle between the final grating and the interference plane that optimizes fringe contrast across the field.
Inverse Scattering for Gratings and Wave Guides
Eskin, Gregory; Ralston, James; Yamamoto, Masahiro
2007-01-01
We consider the problem of unique identification of dielectric coefficients for gratings and sound speeds for wave guides from scattering data. We prove that the "propagating modes" given for all frequencies uniquely determine these coefficients. The gratings may contain conductors as well as dielectrics and the boundaries of the conductors are also determined by the propagating modes.
Femtosecond laser pulse written Volume Bragg Gratings
Richter Daniel
2013-11-01
Full Text Available Femtosecond laser pulses can be applied for structuring a wide range of ransparent materials. Here we want to show how to use this ability to realize Volume-Bragg-Gratings in various- mainly non-photosensitive - glasses. We will further present the characteristics of the realized gratings and a few elected applications that have been realized.
Antireflective characteristics of hemispherical grid grating
REN Zhibin; JIANG Huilin; LIU Guojun; SUN Qiang
2005-01-01
In this paper, the optical characteristics of new type hemispherical grid subwavelength grating are studied by using multi-level column structure approximation and rigorous coupled-wave analysis. This kind of grating could be fabricated by chemical methods, thus simplifying the fabrication technology of subwavelength gratings for visible light. By computer simulation and calculation, the hemispherical grid subwavelength gratings are proved to have antireflective characteristics. Two design schemes of this kind of grating are presented. In the first scheme, the grating could achieve a reflectivity as low as 3.4416×10-7, which can be adapted to 0.46―0.7 μm of visible waveband and ±12° incident angle field. In the second scheme, the grating can achieve a reflectivity as low as 3.112×10-4 and adapted to the whole visible waveband and ±23° incident angle field. The application field of the latter scheme is wider than that of the former. The results of this paper could provide reference for the applications of the hemispherical grid subwavelength gratings for the visible waveband.
Straw combustion on slow-moving grates
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...
High order Bragg grating microfluidic dye laser
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....
Quantization of gauge fields, graph polynomials and graph homology
We review quantization of gauge fields using algebraic properties of 3-regular graphs. We derive the Feynman integrand at n loops for a non-abelian gauge theory quantized in a covariant gauge from scalar integrands for connected 3-regular graphs, obtained from the two Symanzik polynomials. The transition to the full gauge theory amplitude is obtained by the use of a third, new, graph polynomial, the corolla polynomial. This implies effectively a covariant quantization without ghosts, where all the relevant signs of the ghost sector are incorporated in a double complex furnished by the corolla polynomial–we call it cycle homology–and by graph homology. -- Highlights: •We derive gauge theory Feynman from scalar field theory with 3-valent vertices. •We clarify the role of graph homology and cycle homology. •We use parametric renormalization and the new corolla polynomial
On polynomial mappings from the plane to the plane
Krzyżanowska, Iwona; Szafraniec, Zbigniew
2012-01-01
Let $f:{\\mathbb R}^2\\longrightarrow {\\mathbb R}^2$ be a generic polynomial mapping. There are constructed quadratic forms whose signatures determine the number of positive and negative cusps of $f$.
Complete permutation polynomials over finite fields of odd characteristic
Xu, Guangkui; Cao, Xiwang; Tu, Ziran; Zeng, Xiangyong; Hu, Lei
2013-01-01
In this paper, we present three classes of complete permutation monomials over finite fields of odd characteristic. Meanwhile, the compositional inverses of these complete permutation polynomials are also proposed.
Polynomials on the space of ω-ultradifferentiable functions
Katarzyna Grasela
2007-01-01
The space of polynomials on the the space \\(D_{\\omega}\\) of \\(\\omega\\)-ultradifferentiable functions is represented as the direct sum of completions of symmetric tensor powers of \\(D^{\\prime}_{\\omega}\\).
APPROXIMATION BY GENERALIZED MKZ-OPERATORS IN POLYNOMIAL WEIGHTED SPACES
Lucyna Rempulska; Mariola Skorupka
2007-01-01
We prove some approximation properties of generalized Meyer-K(o)nig and Zeller operators for differentiable functions in polynomial weighted spaces. The results extend some results proved in [ 1-3,7-16].
Inner approximations for polynomial matrix inequalities and robust stability regions
Henrion, Didier
2011-01-01
Following a polynomial approach, many robust fixed-order controller design problems can be formulated as optimization problems whose set of feasible solutions is modelled by parametrized polynomial matrix inequalities (PMI). These feasibility sets are typically nonconvex. Given a parametrized PMI set, we provide a hierarchy of linear matrix inequality (LMI) problems whose optimal solutions generate inner approximations modelled by a single polynomial sublevel set. Those inner approximations converge in a strong analytic sense to the nonconvex original feasible set, with asymptotically vanishing conservatism. One may also impose the hierarchy of inner approximations to be nested or convex. In the latter case they do not converge any more to the feasible set, but they can be used in a convex optimization framework at the price of some conservatism. Finally, we show that the specific geometry of nonconvex polynomial stability regions can be exploited to improve convergence of the hierarchy of inner approximation...
On the one dimensional polynomial and regular images of Rn
Fernando Galván, José Francisco
2014-01-01
his work we present a full geometric characterization of the 1-dimensional polynomial and regular images S of Rn and we compute for all of them the invariants ppSq and rpSq, already introduced in [FG2].
Stability of the Bose-Einstein condensate under polynomial perturbations
Gielerak, R.; Damek, J.
2002-01-01
The problem of the Bose-Einstein condensate preservation under thermofield and standard gauge-invariant perturbations is discussed. A new result on stability of the condensate under thermofield perturbations of a polynomial type is presented.
Some functional inequalities on polynomial volume growth Lie groups
Chamorro, Diego
2010-01-01
We study in this article some Sobolev-type inequalities on polynomial volume growth Lie groups. We show in particular that improved Sobolev inequalities can be extended without the use of the Littlewood-Paley decomposition to this general framework.
Force prediction in cold rolling mills by polynomial methods
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.
Maximum likelihood polynomial regression for robust speech recognition
LU Yong; WU Zhenyang
2011-01-01
The linear hypothesis is the main disadvantage of maximum likelihood linear re- gression （MLLR）. This paper applies the polynomial regression method to model adaptation and establishes a nonlinear model adaptation algorithm using maximum likelihood polyno
Polynomial form of the Hilbert-Einstein action
Katanaev, M. O.
2005-01-01
Configuration space of general relativity is extended by inclusion of the determinant of the metric as a new independent variable. As the consequence the Hilbert-Einstein action takes a polynomial form.
Quantization of gauge fields, graph polynomials and graph homology
Kreimer, Dirk, E-mail: kreimer@physik.hu-berlin.de [Humboldt University, 10099 Berlin (Germany); Sars, Matthias [Humboldt University, 10099 Berlin (Germany); Suijlekom, Walter D. van [Radboud University Nijmegen, 6525 AJ Nijmegen (Netherlands)
2013-09-15
We review quantization of gauge fields using algebraic properties of 3-regular graphs. We derive the Feynman integrand at n loops for a non-abelian gauge theory quantized in a covariant gauge from scalar integrands for connected 3-regular graphs, obtained from the two Symanzik polynomials. The transition to the full gauge theory amplitude is obtained by the use of a third, new, graph polynomial, the corolla polynomial. This implies effectively a covariant quantization without ghosts, where all the relevant signs of the ghost sector are incorporated in a double complex furnished by the corolla polynomial–we call it cycle homology–and by graph homology. -- Highlights: •We derive gauge theory Feynman from scalar field theory with 3-valent vertices. •We clarify the role of graph homology and cycle homology. •We use parametric renormalization and the new corolla polynomial.
Cauchy-Kowalevski and polynomial ordinary differential equations
Roger J. Thelwell
2012-01-01
Full Text Available The Cauchy-Kowalevski Theorem is the foremost result guaranteeing existence and uniqueness of local solutions for analytic quasilinear partial differential equations with Cauchy initial data. The techniques of Cauchy-Kowalevski may also be applied to initial-value ordinary differential equations. These techniques, when applied in the polynomial ordinary differential equation setting, lead one naturally to a method in which coefficients of the series solution are easily computed in a recursive manner, and an explicit majorization admits a clear a priori error bound. The error bound depends only on immediately observable quantities of the polynomial system; coefficients, initial conditions, and polynomial degree. The numerous benefits of the polynomial system are shown for a specific example.
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...
Planar-grating klystron experiment
This paper reports on a coherent radiation source which uses an electron beam to drive a resonator which consists of two short sections of metal grating embedded in a parallel-plate waveguide structure that has been operated in the millimeter-wavelength regime. The fields in the first of the grating sections imparts velocity modulation to the beam and the second extracts power from a bunched beam. Thus, the device functions like a two-cavity klystron. However, the open quasi-optical coupling structure can be utilized at wavelengths which are shorter than those that are practicable in a conventional closed cavity klystron design. The electron beam energy and current employed in these experiments are modest (10's of kv and 1-2 A) and the primary motivation for the work is to develop convenient moderate power sources for various applications of millimeter- and submillimeter-wavelength radiation. It is interesting to note, however, that the topology of the resonator in very-high-power radiation sources at centimeter-millimeter wavelengths
Performance comparison of polynomial representations for optimizing optical freeform systems
Brömel, A.; Gross, H.; Ochse, D.; Lippmann, U.; Ma, C.; Zhong, Y.; Oleszko, M.
2015-09-01
Optical systems can benefit strongly from freeform surfaces, however the choice of the right representation isn`t an easy one. Classical representations like X-Y-polynomials, as well as Zernike-polynomials are often used for such systems, but should have some disadvantage regarding their orthogonality, resulting in worse convergence and reduced quality in final results compared to newer representations like the Q-polynomials by Forbes. Additionally the supported aperture is a circle, which can be a huge drawback in case of optical systems with rectangular aperture. In this case other representations like Chebyshev-or Legendre-polynomials come into focus. There are a larger number of possibilities; however the experience with these newer representations is rather limited. Therefore in this work the focus is on investigating the performance of four widely used representations in optimizing two ambitious systems with very different properties: Three-Mirror-Anastigmat and an anamorphic System. The chosen surface descriptions offer support for circular or rectangular aperture, as well as different grades of departure from rotational symmetry. The basic shapes are for example a conic or best-fit-sphere and the polynomial set is non-, spatial or slope-orthogonal. These surface representations were chosen to evaluate the impact of these aspects on the performance optimization of the two example systems. Freeform descriptions investigated here were XY-polynomials, Zernike in Fringe representation, Q-polynomials by Forbes, as well as 2-dimensional Chebyshev-polynomials. As a result recommendations for the right choice of freeform surface representations for practical issues in the optimization of optical systems can be given.
Asymptotic analysis of the Nörlund and Stirling polynomials
Mark Daniel Ward
2012-04-01
Full Text Available We provide a full asymptotic analysis of the N{\\"o}rlund polynomials and Stirling polynomials. We give a general asymptotic expansion---to any desired degree of accuracy---when the parameter is not an integer. We use singularity analysis, Hankel contours, and transfer theory. This investigation was motivated by a need for such a complete asymptotic description, with parameter 1/2, during this author's recent solution of Wilf's 3rd (previously Unsolved Problem.
Polynomials whose reducibility is related to the Goldbach conjecture
Borwein, Peter B.; Choi, Stephen K. K.; Martin, Greg; Samuels, Charles L.
2014-01-01
We introduce a collection of polynomials $F_N$, associated to each positive integer $N$, whose divisibility properties yield a reformulation of the Goldbach conjecture. While this reformulation certainly does not lead to a resolution of the conjecture, it does suggest two natural generalizations for which we provide some numerical evidence. As these polynomials $F_N$ are independently interesting, we further explore their basic properties, giving, among other things, asymptotic estimates on t...
Finding low-weight polynomial multiples using discrete logarithm
Didier, Frédéric; Laigle-Chapuy, Yann
2007-01-01
Finding low-weight multiples of a binary polynomial is a difficult problem arising in the context of stream ciphers cryptanalysis. The classical algorithm to solve this problem is based on a time memory trade-off. We will present an improvement to this approach using discrete logarithm rather than a direct representation of the involved polynomials. This gives an algorithm which improves the theoretical complexity, and is also very flexible in practice.
Exploiting symmetries in SDP-relaxations for polynomial optimization
Riener, Cordian; Theobald, Thorsten; Andrén, Lina Jansson; Lasserre, Jean B.
2011-01-01
In this paper we study various approaches for exploiting symmetries in polynomial optimization problems within the framework of semi definite programming relaxations. Our special focus is on constrained problems especially when the symmetric group is acting on the variables. In particular, we investigate the concept of block decomposition within the framework of constrained polynomial optimization problems, show how the degree principle for the symmetric group can be computationally exploited...
Modeling Microwave Structures in Time Domain Using Laguerre Polynomials
Z. Raida; Lacik, J.
2006-01-01
The paper is focused on time domain modeling of microwave structures by the method of moments. Two alternative schemes with weighted Laguerre polynomials are presented. Thanks to their properties, these schemes are free of late time oscillations. Further, the paper is aimed to effective and accurate evaluation of Green's functions integrals within these schemes. For this evaluation, a first- and second-order polynomial approximation is developed. The last part of the paper deals with mode...
Auslander-Reiten conjecture for symmetric algebras of polynomial growth
Zhou, Guodong; Zimmermann, Alexander
2010-01-01
This paper studies self-injective algebras of polynomial growth. We prove that the derived equivalence classification of weakly symmetric algebras of domestic type coincides with the classification up to stable equivalences (of Morita type). As for weakly symmetric non-domestic algebras of polynomial growth, up to some scalar problems, the derived equivalence classification coincides with the classification up to stable equivalences of Morita type. As a consequence, we get the validity of the...
On Sharing, Memoization, and Polynomial Time (Long Version)
Avanzini, Martin; Lago, Ugo Dal
2015-01-01
We study how the adoption of an evaluation mechanism with sharing and memoization impacts the class of functions which can be computed in polynomial time. We first show how a natural cost model in which lookup for an already computed value has no cost is indeed invariant. As a corollary, we then prove that the most general notion of ramified recurrence is sound for polynomial time, this way settling an open problem in implicit computational complexity.
Exact Bivariate Polynomial Factorization in Q by Approximation of Roots
Feng, Yong; Wu, Wenyuan; Zhang, Jingzhong
2010-01-01
Factorization of polynomials is one of the foundations of symbolic computation. Its applications arise in numerous branches of mathematics and other sciences. However, the present advanced programming languages such as C++ and J++, do not support symbolic computation directly. Hence, it leads to difficulties in applying factorization in engineering fields. In this paper, we present an algorithm which use numerical method to obtain exact factors of a bivariate polynomial with rational coeffici...
Strong and ratio asymptotics for Laguerre polynomials revisited
Deaño, Alfredo; Huertas, Edmundo J.; Marcellán, Francisco
2013-01-01
In this paper we consider the strong asymptotic behavior of Laguerre polynomials in the complex plane. The leading behavior is well known from Perron and Mehler-Heine formulas, but higher order coefficients, which are important in the context of Krall-Laguerre or Laguerre-Sobolev-type orthogonal polynomials, are notoriously difficult to compute. In this paper, we propose the use of an alternative expansion, due to Buchholz, in terms of Bessel functions of the first kind. The coefficients in t...
General Convolution Identities for Bernoulli and Euler Polynomials
Dilcher, K.; Vignat , C.
2015-01-01
Using general identities for difference operators, as well as a technique of symbolic computation and tools from probability theory, we derive very general kth order (k \\ge 2) convolution identities for Bernoulli and Euler polynomials. This is achieved by use of an elementary result on uniformly distributed random variables. These identities depend on k positive real parameters, and as special cases we obtain numerous known and new identities for these polynomials. In particular we show that ...
Invariant hyperplanes and Darboux integrability of polynomial vector fields
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
Zeros of differential polynomials in real meromorphic functions
Bergweiler, W.; Eremenko, A.; Langley, J.
2004-01-01
We show that for a real transcendental meromorphic function f, the differential polynomial f'+f^m with m > 4 has infinitely many non-real zeros. Similar results are obtained for differential polynomials f'f^m-1. We specially investigate the case of meromorphic functions with finitely many poles. We show by examples the precision of our results. One of our main tools is the Fatou theorem from complex dynamics.
CONVERGENCE ARTE FOR INTERATES OF q-BERNSTEIN POLYNOMIALS
无
2007-01-01
Recently, q-Bernstein polynomials have been intensively investigated by a number of authors. Their results show that for q ≠ 1, q-Bernstein polynomials possess of many interesting properties. In this paper, the convergence rate for iterates of both q-Bernstein when n →∞ and convergence rate of Bn(f,q;x) when f ∈ Cn-1[0, 1], q →∞ are also presented.
Fast parallel computation of polynomials using few processors
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
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....
Software for Exact Integration of Polynomials over Polyhedra
De Loera, Jesus; Koeppe, Matthias; Moreinis, Stanislav; Pinto, Gregory; Wu, Jianqiu
2011-01-01
We are interested in the fast computation of the exact value of integrals of polynomial functions over convex polyhedra. We present speed ups and extensions of the algorithms presented in previous work. We present the new software implementation and provide benchmark computations. The computation of integrals of polynomials over polyhedral regions has many applications; here we demonstrate our algorithmic tools solving a challenge from combinatorial voting theory.
Sparse polynomial interpolation and the fast Euclidean algorithm
Go, Soo
2012-01-01
We introduce an algorithm to interpolate sparse multivariate polynomials with integer coefficients. Our algorithm modifies Ben-Or and Tiwari's deterministic algorithm for interpolating over rings of characteristic zero to work modulo p, a smooth prime of our choice. We present benchmarks comparing our algorithm to Zippel's probabilistic sparse interpolation algorithm, demonstrating that our algorithm makes fewer probes for sparse polynomials. Our interpolation algorithm requires finding roo...
Nonstandard decision methods for the solvability of real polynomial equations
曾广兴
1999-01-01
For a multivariate polynomial equation with coefficients in a computable ordered field, two criteria of this equation having real solutions are given. Based on the criteria, decision methods for the existence of real zeros and the semidefiniteness of binary polynomials are provided. With the aid of computers, these methods are used to solve several examples. The technique is to extend the original field involved in the question to a computable non-Archimedean ordered field containing infinitesimal elements.
Twisted exponential sums of polynomials in one variable
无
2010-01-01
The twisted T-adic exponential sums associated to a polynomial in one variable are studied.An explicit arithmetic polygon in terms of the highest two exponents of the polynomial is proved to be a lower bound of the Newton polygon of the C-function of the twisted T-adic exponential sums.This bound gives lower bounds for the Newton polygon of the L-function of twisted p-power order exponential sums.
First extension groups of Verma modules and $R$-polynomials
Abe, Noriyuki
2010-01-01
We study the first extension groups between Verma modules. There was a conjecture which claims that the dimensions of the higher extension groups between Verma modules are the coefficients of $R$-polynomials defined by Kazhdan-Lusztig. This conjecture was known as the Gabber-Joseph conjecture (although Gebber and Joseph did not state.) However, Boe gives a counterexample to this conjecture. In this paper, we study how far are the dimensions of extension groups from the coefficients of $R$-polynomials.
Compact imaging spectrometer utilizing immersed gratings
Chrisp, Michael P. (Danville, CA); Lerner, Scott A. (Corvallis, OR); Kuzmenko, Paul J. (Livermore, CA); Bennett, Charles L. (Livermore, CA)
2007-07-03
A compact imaging spectrometer with an immersive diffraction grating that compensates optical distortions. The imaging spectrometer comprises an entrance slit for transmitting light, means for receiving the light and directing the light, an immersion grating, and a detector array. The entrance slit, the means for receiving the light, the immersion grating, and the detector array are positioned wherein the entrance slit transmits light to the means for receiving the light and the means for receiving the light directs the light to the immersion grating and the immersion grating receives the light and directs the light to the means for receiving the light, and the means for receiving the light directs the light to the detector array.
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...
Traversa, Fabio Lorenzo; Ramella, Chiara; Bonani, Fabrizio; Di Ventra, Massimiliano
2015-07-01
Memcomputing is a novel non-Turing paradigm of computation that uses interacting memory cells (memprocessors for short) to store and process information on the same physical platform. It was recently proven mathematically that memcomputing machines have the same computational power of nondeterministic Turing machines. Therefore, they can solve NP-complete problems in polynomial time and, using the appropriate architecture, with resources that only grow polynomially with the input size. The reason for this computational power stems from properties inspired by the brain and shared by any universal memcomputing machine, in particular intrinsic parallelism and information overhead, namely, the capability of compressing information in the collective state of the memprocessor network. We show an experimental demonstration of an actual memcomputing architecture that solves the NP-complete version of the subset sum problem in only one step and is composed of a number of memprocessors that scales linearly with the size of the problem. We have fabricated this architecture using standard microelectronic technology so that it can be easily realized in any laboratory setting. Although the particular machine presented here is eventually limited by noise-and will thus require error-correcting codes to scale to an arbitrary number of memprocessors-it represents the first proof of concept of a machine capable of working with the collective state of interacting memory cells, unlike the present-day single-state machines built using the von Neumann architecture. PMID:26601208
Factorization of colored knot polynomials at roots of unity
Ya. Kononov
2015-07-01
Full Text Available HOMFLY polynomials are the Wilson-loop averages in Chern–Simons theory and depend on four variables: the closed line (knot in 3d space–time, representation R of the gauge group SU(N and exponentiated coupling constant q. From analysis of a big variety of different knots we conclude that at q, which is a 2m-th root of unity, q2m=1, HOMFLY polynomials in symmetric representations [r] satisfy recursion identity: Hr+m=Hr⋅Hm for any A=qN, which is a generalization of the property Hr=H1r for special polynomials at m=1. We conjecture a further generalization to arbitrary representation R, which, however, is checked only for torus knots. Next, Kashaev polynomial, which arises from HR at q2=e2πi/|R|, turns equal to the special polynomial with A substituted by A|R|, provided R is a single-hook representations (including arbitrary symmetric – what provides a q−A dual to the similar property of Alexander polynomial. All this implies non-trivial relations for the coefficients of the differential expansions, which are believed to provide reasonable coordinates in the space of knots – existence of such universal relations means that these variables are still not unconstrained.
An overview on polynomial approximation of NP-hard problems
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.
Characterizing Polynomial Time Computability of Rational and Real Functions
Walid Gomaa
2009-11-01
Full Text Available Recursive analysis was introduced by A. Turing [1936], A. Grzegorczyk [1955], and D. Lacombe [1955]. It is based on a discrete mechanical framework that can be used to model computation over the real numbers. In this context the computational complexity of real functions defined over compact domains has been extensively studied. However, much less have been done for other kinds of real functions. This article is divided into two main parts. The first part investigates polynomial time computability of rational functions and the role of continuity in such computation. On the one hand this is interesting for its own sake. On the other hand it provides insights into polynomial time computability of real functions for the latter, in the sense of recursive analysis, is modeled as approximations of rational computations. The main conclusion of this part is that continuity does not play any role in the efficiency of computing rational functions. The second part defines polynomial time computability of arbitrary real functions, characterizes it, and compares it with the corresponding notion over rational functions. Assuming continuity, the main conclusion is that there is a conceptual difference between polynomial time computation over the rationals and the reals manifested by the fact that there are polynomial time computable rational functions whose extensions to the reals are not polynomial time computable and vice versa.
Factorization of colored knot polynomials at roots of unity
Kononov, Ya.; Morozov, A.
2015-07-01
HOMFLY polynomials are the Wilson-loop averages in Chern-Simons theory and depend on four variables: the closed line (knot) in 3d space-time, representation R of the gauge group SU (N) and exponentiated coupling constant q. From analysis of a big variety of different knots we conclude that at q, which is a 2m-th root of unity, q2m = 1, HOMFLY polynomials in symmetric representations [ r ] satisfy recursion identity: Hr+m =Hr ṡHm for any A =qN, which is a generalization of the property Hr = H1r for special polynomials at m = 1. We conjecture a further generalization to arbitrary representation R, which, however, is checked only for torus knots. Next, Kashaev polynomial, which arises from HR at q2 = e 2 πi / | R |, turns equal to the special polynomial with A substituted by A| R |, provided R is a single-hook representations (including arbitrary symmetric) - what provides a q - A dual to the similar property of Alexander polynomial. All this implies non-trivial relations for the coefficients of the differential expansions, which are believed to provide reasonable coordinates in the space of knots - existence of such universal relations means that these variables are still not unconstrained.
A new spectrometer with no moving parts uses a two-dimensional Si-based charge-coupled diode (CCD) array detector and an integrated grating consisting of three subgratings. The effective spectral range imaged on the detector is magnified threefold. The digitized spectral image in the 200-1000 nm wavelength range can be measured quickly. The nonlinear relationship between CCD pixel position and wavelength is corrected with multiple polynomial functions in the calibration procedure, which fits the data using a mathematical pattern-analysis method. The instrument can be applied for rapid spectroscopic data analyses in many types of photoelectronic experiments and routine testing
On the values of independence and domination polynomials at specific points
Saeid Alikhani
2012-06-01
Full Text Available Let G be a simple graph of order n. We consider the independence polynomial and the domination polynomial of a graph G. The value of a graph polynomial at a specific point can give sometimes a very surprising information about the structure of the graph. In this paper we investigate independence and domination polynomial at -1 and 1.
Soft x-ray transmission gratings
A technique was developed for producing transmission diffraction gratings suitable for use in the soft x-ray region. Thin self-supporting films of a transparent material are overlaid with several thousand opaque metallic strips per mm. Gratings with 2100, 2400, and 5600 1/mm have been produced and tested. Representative spectra over the wavelength range from 17.2 to 40.0 nm are given for a grating consisting of a 120-nm-thick Al support layer overlaid with 2400, 34-nm-thick, Ag strips/mm. The absolute transmittance is approx. 13% at 30 nm, and the efficiency in the first order is approx. 16%. The observed resolution of approx. 2A is acceptable for many of the potential applications. These gratings have several advantages over the two presently available alternatives in the soft x-ray region (i.e., reflection gratings used at grazing incidence and free-standing metallic wire transmission gratings). Fabrication is relatively quick, simple, and cheap. The support layer can also serve as a filter and help conduct excessive heat away. Higher line densities and hence higher resolutions are possible, and when used at normal incidence the spectra are aberration free. Suitable materials, component thicknesses, and line densities can be chosen to produce a grating of optimum characteristics for a particular application
Permutation Polynomials of Degree 6 or 7 over Finite Fields of Characteristic 2
Li, Jiyou; Chandler, David B.; Xiang, Qing
2010-01-01
In \\cite{D1}, Dickson listed all permutation polynomials up to degree 5 over an arbitrary finite field, and all permutation polynomials of degree 6 over finite fields of odd characteristic. The classification of degree 6 permutation polynomials over finite fields of characteristic 2 was left incomplete. In this paper we complete the classification of permutation polynomials of degree 6 over finite fields of characteristic 2. In addition, all permutation polynomials of degree 7 over finite fie...
Variations of the Ramanujan polynomials and remarks on $\\zeta(2j+1)/\\pi^{2j+1}$
Lalin, Matilde
2011-01-01
We observe that five polynomial families have all of their zeros on the unit circle. We prove the statements explicitly for four of the polynomial families. The polynomials have coefficients which involve Bernoulli numbers, Euler numbers, and the odd values of the Riemann zeta function. These polynomials are closely related to the Ramanujan polynomials, which were recently introduced by Murty, Smyth and Wang. Our proofs rely upon theorems of Schinzel, and Lakatos and Losonczi and some generalizations.
Discriminants and functional equations for polynomials orthogonal on the unit circle
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
Permutation Polynomials of Degree 6 or 7 over Finite Fields of Characteristic 2
Li, Jiyou; Xiang, Qing
2010-01-01
In \\cite{D1}, Dickson listed all permutation polynomials up to degree 5 over an arbitrary finite field, and all permutation polynomials of degree 6 over finite fields of odd characteristic. The classification of degree 6 permutation polynomials over finite fields of characteristic 2 was left incomplete. In this paper we complete the classification of permutation polynomials of degree 6 over finite fields of characteristic 2. In addition, all permutation polynomials of degree 7 over finite fields of characteristic 2 are classified.
Gratings in passive and active optical waveguides
Berendt, Martin Ole
1999-01-01
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...
Fiber Bragg Grating Sensors for Harsh Environments
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.
An electromagnetically induced grating by microwave modulation
Xiao, Zhi-Hong; Shin, Sung Guk; Kim, Kisik, E-mail: kisik@inha.ac.k [Department of Physics, Inha University, Incheon 402-751 (Korea, Republic of)
2010-08-28
We study the phenomenon of an electromagnetically induced phase grating in a double-dark state system of {sup 87}Rb atoms, the two closely placed lower fold levels of which are coupled by a weak microwave field. Owing to the existence of the weak microwave field, the efficiency of the phase grating is strikingly improved, and an efficiency of approximately 33% can be achieved. Under the action of the weak standing wave field, the high efficiency of the phase grating can be maintained by modulating the strength and detuning of the weak microwave field, increasing the strength of the standing wave field. (fast track communication)
Hurvitz, G; Strum, G; Shpilman, Z; Levy, I; Fraenkel, M
2012-01-01
A novel fabrication method for soft x-ray transmission grating and other optical elements is presented. The method uses Focused-Ion-Beam (FIB) technology to fabricate high-quality free standing grating bars on Transmission Electron Microscopy grids (TEM-grid). High quality transmission gratings are obtained with superb accuracy and versatility. Using these gratings and back-illuminated CCD camera, absolutely calibrated x-ray spectra can be acquired for soft x-ray source diagnostics in the 100-3000 eV spectral range. Double grating combinations of identical or different parameters are easily fabricated, allowing advanced one-shot application of transmission grating spectroscopy. These applications include spectroscopy with different spectral resolutions, bandwidths, dynamic ranges, and may serve for identification of high-order contribution, and spectral calibrations of various x-ray optical elements.
Hurvitz, G.; Ehrlich, Y.; Strum, G.; Shpilman, Z.; Levy, I.; Fraenkel, M.
2012-08-01
A novel fabrication method for soft x-ray transmission grating and other optical elements is presented. The method uses focused-ion-beam technology to fabricate high-quality free standing grating bars on transmission electron microscopy grids. High quality transmission gratings are obtained with superb accuracy and versatility. Using these gratings and back-illuminated CCD camera, absolutely calibrated x-ray spectra can be acquired for soft x-ray source diagnostics in the 100-3000 eV spectral range. Double grating combinations of identical or different parameters are easily fabricated, allowing advanced one-shot application of transmission grating spectroscopy. These applications include spectroscopy with different spectral resolutions, bandwidths, dynamic ranges, and may serve for identification of high-order contribution, and spectral calibrations of various x-ray optical elements.
Okayama, Hideaki; Onawa, Yosuke; Shimura, Daisuke; Yaegashi, Hiroki; Sasaki, Hironori
2016-08-01
We describe a Bragg grating with a phase shift section and a sampled grating scheme that converts input polarization to orthogonal polarization. A very narrow polarization-independent wavelength peak can be generated by phase shift structures and polarization-independent multiple diffraction peaks by sampled gratings. The characteristics of the device were examined by transfer matrix and finite-difference time-domain methods.
On Different Classes of Algebraic Polynomials with Random Coefficients
K. Farahmand
2008-01-01
Full Text Available The expected number of real zeros of the polynomial of the form a0+a1x+a2x2+⋯+anxn, where a0,a1,a2,…,an is a sequence of standard Gaussian random variables, is known. For n large it is shown that this expected number in (−∞,∞ is asymptotic to (2/πlogn. In this paper, we show that this asymptotic value increases significantly to n+1 when we consider a polynomial in the form a0(n01/2x/1+a1(n11/2x2/2+a2(n21/2x3/3+⋯+an(nn1/2xn+1/n+1 instead. We give the motivation for our choice of polynomial and also obtain some other characteristics for the polynomial, such as the expected number of level crossings or maxima. We note, and present, a small modification to the definition of our polynomial which improves our result from the above asymptotic relation to the equality.
Ladder Operators for Lamé Spheroconal Harmonic Polynomials
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
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)
Efficient computer algebra algorithms for polynomial matrices in control design
Baras, J. S.; Macenany, D. C.; Munach, R.
1989-01-01
The theory of polynomial matrices plays a key role in the design and analysis of multi-input multi-output control and communications systems using frequency domain methods. Examples include coprime factorizations of transfer functions, cannonical realizations from matrix fraction descriptions, and the transfer function design of feedback compensators. Typically, such problems abstract in a natural way to the need to solve systems of Diophantine equations or systems of linear equations over polynomials. These and other problems involving polynomial matrices can in turn be reduced to polynomial matrix triangularization procedures, a result which is not surprising given the importance of matrix triangularization techniques in numerical linear algebra. Matrices with entries from a field and Gaussian elimination play a fundamental role in understanding the triangularization process. In the case of polynomial matrices, matrices with entries from a ring for which Gaussian elimination is not defined and triangularization is accomplished by what is quite properly called Euclidean elimination. Unfortunately, the numerical stability and sensitivity issues which accompany floating point approaches to Euclidean elimination are not very well understood. New algorithms are presented which circumvent entirely such numerical issues through the use of exact, symbolic methods in computer algebra. The use of such error-free algorithms guarantees that the results are accurate to within the precision of the model data--the best that can be hoped for. Care must be taken in the design of such algorithms due to the phenomenon of intermediate expressions swell.
Orbifold E-functions of dual invertible polynomials
Ebeling, Wolfgang; Gusein-Zade, Sabir M.; Takahashi, Atsushi
2016-08-01
An invertible polynomial is a weighted homogeneous polynomial with the number of monomials coinciding with the number of variables and such that the weights of the variables and the quasi-degree are well defined. In the framework of the search for mirror symmetric orbifold Landau-Ginzburg models, P. Berglund and M. Henningson considered a pair (f , G) consisting of an invertible polynomial f and an abelian group G of its symmetries together with a dual pair (f ˜ , G ˜) . We consider the so-called orbifold E-function of such a pair (f , G) which is a generating function for the exponents of the monodromy action on an orbifold version of the mixed Hodge structure on the Milnor fibre of f. We prove that the orbifold E-functions of Berglund-Henningson dual pairs coincide up to a sign depending on the number of variables and a simple change of variables. The proof is based on a relation between monomials (say, elements of a monomial basis of the Milnor algebra of an invertible polynomial) and elements of the whole symmetry group of the dual polynomial.
Access Platforms for Offshore Wind Turbines Using Gratings
Andersen, Thomas Lykke; Rasmussen, Michael R.
2008-01-01
The paper deals with forces generated by a stationary jet on different types of gratings and a solid plate. The force reduction factors for the different gratings compared to the solid plate mainly depend on the porosity of the gratings, but the geometry of the grating is also of some importance....... The derived reduction factors are expected to be applicable to design of offshore wind turbine access platforms with gratings where slamming also is an important factor....
Hydraulic Capacity of an ADA Compliant Street Drain Grate
Lottes, Steven A. [Argonne National Lab. (ANL), Argonne, IL (United States); Bojanowski, Cezary [Argonne National Lab. (ANL), Argonne, IL (United States)
2015-09-01
Resurfacing of urban roads with concurrent repairs and replacement of sections of curb and sidewalk may require pedestrian ramps that are compliant with the American Disabilities Act (ADA), and when street drains are in close proximity to the walkway, ADA compliant street grates may also be required. The Minnesota Department of Transportation ADA Operations Unit identified a foundry with an available grate that meets ADA requirements. Argonne National Laboratory’s Transportation Research and Analysis Computing Center used full scale three dimensional computational fluid dynamics to determine the performance of the ADA compliant grate and compared it to that of a standard vane grate. Analysis of a parametric set of cases was carried out, including variation in longitudinal, gutter, and cross street slopes and the water spread from the curb. The performance of the grates was characterized by the fraction of the total volume flow approaching the grate from the upstream that was captured by the grate and diverted into the catch basin. The fraction of the total flow entering over the grate from the side and the fraction of flow directly over a grate diverted into the catch basin were also quantities of interest that aid in understanding the differences in performance of the grates. The ADA compliant grate performance lagged that of the vane grate, increasingly so as upstream Reynolds number increased. The major factor leading to the performance difference between the two grates was the fraction of flow directly over the grates that is captured by the grates.
Sampled phase-shift fiber Bragg gratings
Xu Wang(王旭); Chongxiu Yu(余重秀); Zhihui Yu(于志辉); Qiang Wu(吴强)
2004-01-01
A phase-shift fiber Bragg grating (FBG) with sampling is proposed to generate a multi-channel bandpass filter in the background of multi-channel stopbands. The sampled noire fiber gratings are analyzed by Fourier transform theory first, and then simulation and experiment are performed, the results show that transmission peaks are opened in every reflective channel, the spectrum shape of every channel is identical.It can be used to fabricate multi-wavelength distributed feedback (DFB) fiber laser.
Gratings in plasmonic V-groove waveguides
Smith, Cameron; Cuesta, Irene Fernandez; Kristensen, Anders
2011-01-01
We introduce visible light optical gratings to surface plasmon V-groove waveguides. Gradient e-beam dosage onto silicon stamp enables structuring V-grooves of varying depth. Nanoimprint lithography maintains a Λ=265 nm corrugation for gold surface devices.......We introduce visible light optical gratings to surface plasmon V-groove waveguides. Gradient e-beam dosage onto silicon stamp enables structuring V-grooves of varying depth. Nanoimprint lithography maintains a Λ=265 nm corrugation for gold surface devices....
The problem of creating a stray radiation grating which has as large a compartment ratio as possible, without the Bucky factor (ratio of the total incident radiation to the total radiation passing through) rising to an impermissible degree and which can be suitably made, particularly as a focussing stray radiation grating, is solved by the radiation absorbing layer being a pigment/binder layer and by the strips carrying a further layer, which is electrically conducting. (orig./HP)
Decompositions of Trigonometric Polynomials with Applications to Multivariate Subdivision Schemes
Dyn, Nira
2009-01-01
We study multivariate trigonometric polynomials, satisfying a set of constraints close to the known Strung-Fix conditions. Based on the polyphase representation of these polynomials relative to a general dilation matrix, we develop a simple constructive method for a special type of decomposition of such polynomials. These decompositions are of interest to the analysis of convergence and smoothness of multivariate subdivision schemes associated with general dilation matrices. We apply these decompositions, by verifying sufficient conditions for the convergence and smoothness of multivariate scalar subdivision schemes, proved here. For the convergence analysis our sufficient conditions apply to arbitrary dilation matrices, while the previously known necessary and sufficient conditions are relevant only in case of dilation matrices with a self similar tiling. For the analysis of smoothness, we state and prove two theorems on multivariate matrix subdivision schemes, which lead to sufficient conditions for C^1 lim...
Equations on knot polynomials and 3d/5d duality
Mironov, A
2012-01-01
We briefly review the current situation with various relations between knot/braid polynomials (Chern-Simons correlation functions), ordinary and extended, considered as functions of the representation and of the knot topology. These include linear skein relations, quadratic Plucker relations, as well as "differential" and (quantum) A-polynomial structures. We pay a special attention to identity between the A-polynomial equations for knots and Baxter equations for quantum relativistic integrable systems, related through Seiberg-Witten theory to 5d super-Yang-Mills models and through the AGT relation to the q-Virasoro algebra. This identity is an important ingredient of emerging a 3d-5d generalization of the AGT relation. The shape of the Baxter equation (including the values of coefficients) depend on the choice of the knot/braid. Thus, like the case of KP integrability, where (some, so far torus) knots parameterize particular points of the Universal Grassmannian, in this relation they parameterize particular ...
Polynomial quasisolutions of linear differential-difference equations
Valery B. Cherepennikov
2006-01-01
Full Text Available The paper discusses a linear differential-difference equation of neutral type with linear coefficients, when at the initial time moment \\(t=0\\ the value of the desired function \\(x(t\\ is known. The authors are not familiar with any results which would state the solvability conditions for the given problem in the class of analytical functions. A polynomial of some degree \\(N\\ is introduced into the investigation. Then the term "polynomial quasisolution" (PQ-solution is understood in the sense of appearance of the residual \\(\\Delta (t=O(t^N\\, when this polynomial is substituted into the initial problem. The paper is devoted to finding PQ-solutions for the initial-value problem under analysis.
Polynomial parametrization of Pythagorean quadruples, quintuples and sextuples
Frisch, Sophie
2011-01-01
A Pythagorean n-tuple is an integer solution of x_1^2+...+x_{n-1}^2=x_n^2. For n=4 and n=6, the Pythagorean n-tuples admit a parametrization by a single n-tuple of polynomials with integer coefficients (which is impossible for n=3). For n=5, there is an integer-valued polynomial Pythagorean 5-tuple which parametrizes Pythagorean quintuples (similar to the case n=3). Pythagorean quadruples are closely related to (integer) Descartes quadruples (solutions of 2(b_1^2+b_2^2+b_3^2+b_4^2) = (b_1+b_2+b_3+b_4)^2), which we also parametrize by a Descartes quadruple of polynomials with integer coefficients.
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.
The Kauffman bracket and the Jones polynomial in quantum gravity
In the loop representation the quantum states of gravity are given by knot invariants. From general arguments concerning the loop transform of the exponential of the Chern-Simons form, a certain expansion of the Kauffman bracket knot polynomial can be formally viewed as a solution of the Hamiltonian constraint with a cosmological constant in the loop representation. The Kauffman bracket is closely related to the Jones polynomial. In this paper the operation of the Hamiltonian on the power expansions of the Kauffman bracket and Jones polynomials is analyzed. It is explicitly shown that the Kauffman bracket is a formal solution of the Hamiltonian constraint to third order in the cosmological constant. We make use of the extended loop representation of quantum gravity where the analytic calculation can be thoroughly accomplished. Some peculiarities of the extended loop calculus are considered and the significance of the results to the case of the conventional loop representation is discussed. (orig.)
Polynomial chaos expansion with random and fuzzy variables
Jacquelin, E.; Friswell, M. I.; Adhikari, S.; Dessombz, O.; Sinou, J.-J.
2016-06-01
A dynamical uncertain system is studied in this paper. Two kinds of uncertainties are addressed, where the uncertain parameters are described through random variables and/or fuzzy variables. A general framework is proposed to deal with both kinds of uncertainty using a polynomial chaos expansion (PCE). It is shown that fuzzy variables may be expanded in terms of polynomial chaos when Legendre polynomials are used. The components of the PCE are a solution of an equation that does not depend on the nature of uncertainty. Once this equation is solved, the post-processing of the data gives the moments of the random response when the uncertainties are random or gives the response interval when the variables are fuzzy. With the PCE approach, it is also possible to deal with mixed uncertainty, when some parameters are random and others are fuzzy. The results provide a fuzzy description of the response statistical moments.
Basic polynomial invariants, fundamental representations and the Chern class map
Baek, Sanghoon; Zainoulline, Kirill
2011-01-01
Consider a crystallographic root system together with its Weyl group $W$ acting on the weight lattice $M$. Let $Z[M]^W$ and $S^*(M)^W$ be the $W$-invariant subrings of the integral group ring $Z[M]$ and the symmetric algebra $S^*(M)$ respectively. A celebrated theorem of Chevalley says that $Z[M]^W$ is a polynomial ring over $Z$ in classes of fundamental representations $w_1,...,w_n$ and $S^*(M)^{W}$ over rational numbers is a polynomial ring in basic polynomial invariants $q_1,...,q_n$, where $n$ is the rank. In the present paper we establish and investigate the relationship between $w_i$'s and $q_i$'s over the integers.
Characterizing Polynomial Time Computability of Rational and Real Functions
Gomaa, Walid
2009-01-01
Recursive analysis was introduced by A. Turing [1936], A. Grzegorczyk [1955], and D. Lacombe [1955]. It is based on a discrete mechanical framework that can be used to model computation over the real numbers. In this context the computational complexity of real functions defined over compact domains has been extensively studied. However, much less have been done for other kinds of real functions. This article is divided into two main parts. The first part investigates polynomial time computability of rational functions and the role of continuity in such computation. On the one hand this is interesting for its own sake. On the other hand it provides insights into polynomial time computability of real functions for the latter, in the sense of recursive analysis, is modeled as approximations of rational computations. The main conclusion of this part is that continuity does not play any role in the efficiency of computing rational functions. The second part defines polynomial time computability of arbitrary real ...
Applying polynomial filtering to mass preconditioned Hybrid Monte Carlo
Haar, Taylor; Zanotti, James; Nakamura, Yoshifumi
2016-01-01
The use of mass preconditioning or Hasenbusch filtering in modern Hybrid Monte Carlo simulations is common. At light quark masses, multiple filters (three or more) are typically used to reduce the cost of generating dynamical gauge fields; however, the task of tuning a large number of Hasenbusch mass terms is non-trivial. The use of short polynomial approximations to the inverse has been shown to provide an effective UV filter for HMC simulations. In this work we investigate the application of polynomial filtering to the mass preconditioned Hybrid Monte Carlo algorithm as a means of introducing many time scales into the molecular dynamics integration with a simplified parameter tuning process. A generalized multi-scale integration scheme that permits arbitrary step- sizes and can be applied to Omelyan-style integrators is also introduced. We find that polynomial-filtered mass-preconditioning (PF-MP) performs as well as or better than standard mass preconditioning, with significantly less fine tuning required.
Multivariate polynomial interpolation and sampling in Paley-Wiener spaces
Bailey, B A
2010-01-01
In this paper, an equivalence between existence of particular exponential Riesz bases for multivariate bandlimited functions and existence of certain polynomial interpolants for these bandlimited functions is given. For certain classes of unequally spaced data nodes and corresponding $\\ell_2$ data, the existence of these polynomial interpolants allows for a simple recovery formula for multivariate bandlimited functions which demonstrates $L_2$ and uniform convergence on $\\mathbb{R}^d$. A simpler computational version of this recovery formula is also given, at the cost of replacing $L_2$ and uniform convergence on $\\mathbb{R}^d$ with $L_2$ and uniform convergence on increasingly large subsets of $\\mathbb{R}^d$. As a special case, the polynomial interpolants of given $\\ell_2$ data converge in the same fashion to the multivariate bandlimited interpolant of that same data. Concrete examples of pertinant Riesz bases and unequally spaced data nodes are also given.
Describing Quadratic Cremer Point Polynomials by Parabolic Perturbations
Sørensen, Dan Erik Krarup
1996-01-01
.Polynomials of the Cremer type correspond to parameters at the boundary of ahyperbolic component of the Mandelbrot set. In this paper we concentrate onthe main cardioid component. We investigate the differences between two-sided(i.e. alternating) and one-sided parabolic perturbations.In the two-sided case, we...... prove the existence of polynomials having an explicitlygiven external ray accumulating both at the Cremer point and at its non-periodicpreimage. We think of the Julia set as containing a "topologists double comb".In the one-sided case we prove a weaker result: the existence of polynomials havingan...... explicitly given external ray accumulating the Cremer point, but having in itsimpression both the Cremer point and its other preimage. We think of the Julia setas containing a "topologists single comb".By tuning, similar results hold on the boundary of any hyperbolic component of theMandelbrot set....
On adaptive weighted polynomial preconditioning for Hermitian positive definite matrices
Fischer, Bernd; Freund, Roland W.
1992-01-01
The conjugate gradient algorithm for solving Hermitian positive definite linear systems is usually combined with preconditioning in order to speed up convergence. In recent years, there has been a revival of polynomial preconditioning, motivated by the attractive features of the method on modern architectures. Standard techniques for choosing the preconditioning polynomial are based only on bounds for the extreme eigenvalues. Here a different approach is proposed, which aims at adapting the preconditioner to the eigenvalue distribution of the coefficient matrix. The technique is based on the observation that good estimates for the eigenvalue distribution can be derived after only a few steps of the Lanczos process. This information is then used to construct a weight function for a suitable Chebyshev approximation problem. The solution of this problem yields the polynomial preconditioner. In particular, we investigate the use of Bernstein-Szego weights.
Integrability of the dynamical systems with polynomial Hamiltonians
The paper is devoted to an exhaustive study of the integrability of dynamical systems described by polynomial Hamiltonians. After a general analysis of the conditions in which the integration of the equation of motion is possible, we determine the classes of polynomial potentials satisfying these conditions. We effectively study two models admitting both exact solutions and chaotic ones. Despite the fact that the maximal degree of the potential is different in the two cases, an interesting connection can be established between the two models. This connection is given by studying the similar classes of periodical solutions admitted by the two models. The paper will end with a study referring to a non-polynomial potential of the Thomas-Fermi type. (author)
Tensor calculus in polar coordinates using Jacobi polynomials
Vasil, Geoffrey M; Lecoanet, Daniel; Olver, Sheehan; Brown, Benjamin P; Oishi, Jeffrey S
2015-01-01
Spectral methods are an efficient way to solve partial differential equations on domains possessing certain symmetries. The utility of a method depends strongly on the choice of spectral basis. In this paper we describe a set of bases built out of Jacobi polynomials, and associated operators for solving scalar, vector, and tensor partial differential equations in polar coordinates on a unit disk. By construction, the bases satisfy regularity conditions at r=0 for any tensorial field. The coordinate singularity in a disk is a prototypical case for many coordinate singularities. The work presented here extends to other geometries. The operators represent covariant derivatives, multiplication by azimuthally symmetric functions, and the tensorial relationship between fields. These arise naturally from relations between classical orthogonal polynomials, and form a Heisenberg algebra. Other past work uses more specific polynomial bases for solving equations in polar coordinates. The main innovation in this paper is...
Dynamic optical coupled system employing Dammann gratings
Di, Caihui; Zhou, Changhe; Ru, Huayi
2004-10-01
With the increasing of the number of users in optical fiber communications, fiber-to-home project has a larger market value. Then the need of dynamic optical couplers, especially of N broad-band couplers, becomes greater. Though some advanced fiber fusion techniques have been developed, they still have many shortcomings. In this paper we propose a dynamic optical coupled system employing even-numbered Dammann gratings, which have the characteristic that the phase distribution in the first half-period accurately equals to that in the second-period with π phase inversion. In our experiment, we divide a conventional even-numbered Dammann grating into two identical gratings. The system can achieve the beam splitter and combiner as the switch between them according to the relative shift between two complementary gratings. When there is no shift between the gratings, the demonstrated 1×8 dynamic optical coupler achieves good uniformity of 0.06 and insertion loss of around 10.8 dB for each channel as a splitter. When the two gratings have an accurate shift of a half-period between them, our system has a low insertion loss of 0.46 dB as a combiner at a wavelength of 1550 nm.
Skew-orthogonal polynomials, differential systems and random matrix theory
Ghosh, Saugata [Abdus Salam ICTP, Strada Costiera 11, 34100, Trieste (Italy)
2007-01-26
We study skew-orthogonal polynomials with respect to the weight function exp [ - 2V(x)], with V(x) = {sigma}{sup 2d}{sub K=1}(u{sub K}/K)x{sup K}, u{sub 2d} > 0, d > 0. A finite subsequence of such skew-orthogonal polynomials arising in the study of orthogonal and symplectic ensembles of random matrices satisfies a system of differential-difference-deformation equation. The vectors formed by such subsequence have the rank equal to the degree of the potential in the quaternion sense. These solutions satisfy certain compatibility condition and hence admit a simultaneous fundamental system of solutions.
The Laplace transform and polynomial approximation in L2
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...
Connection preserving deformations and q-semi-classical orthogonal polynomials
Ormerod, Christopher M.; Witte, N. S.; Forrester, Peter J.
2011-09-01
We present a framework for the study of q-difference equations satisfied by q-semi-classical orthogonal systems. As an example, we identify the q-difference equation satisfied by a deformed version of the little q-Jacobi polynomials as a gauge transformation of a special case of the associated linear problem for q-PVI. We obtain a parametrization of the associated linear problem in terms of orthogonal polynomial variables and find the relation between this parametrization and that of Jimbo and Sakai.
Algorithm that Solves 3-SAT in Polynomial Time
Steinmetz, Jason W
2011-01-01
The question of whether the complexity class P is equal to the complexity class NP has been a seemingly intractable problem for over 4 decades. It has been clear that if an algorithm existed that would solve the problems in the NP class in polynomial time then P would equal NP. However, no one has yet been able to create that algorithm or to successfully prove that such an algorithm cannot exist. The algorithm that will be presented in this paper runs in polynomial time and solves the 3-satisfiability or 3-SAT problem, which has been proven to be NP-complete, thus indicating that P = NP.
Polynomial birth-death distribution approximation in Wasserstein distance
Xia, Aihua; Zhang, Fuxi
2008-01-01
The polynomial birth-death distribution (abbr. as PBD) on $\\ci=\\{0,1,2, >...\\}$ or $\\ci=\\{0,1,2, ..., m\\}$ for some finite $m$ introduced in Brown & Xia (2001) is the equilibrium distribution of the birth-death process with birth rates $\\{\\alpha_i\\}$ and death rates $\\{\\beta_i\\}$, where $\\a_i\\ge0$ and $\\b_i\\ge0$ are polynomial functions of $i\\in\\ci$. The family includes Poisson, negative binomial, binomial and hypergeometric distributions. In this paper, we give probabilistic proofs of variou...
Modeling Microwave Structures in Time Domain Using Laguerre Polynomials
Z. Raida
2006-09-01
Full Text Available The paper is focused on time domain modeling of microwave structures by the method of moments. Two alternative schemes with weighted Laguerre polynomials are presented. Thanks to their properties, these schemes are free of late time oscillations. Further, the paper is aimed to effective and accurate evaluation of Green's functions integrals within these schemes. For this evaluation, a first- and second-order polynomial approximation is developed. The last part of the paper deals with modeling microstrip structures in the time domain. Conditions of impedance matching are derived, and the proposed approach is verified by modeling a microstrip filter.
Exact Bivariate Polynomial Factorization in Q by Approximation of Roots
Feng, Yong; Zhang, Jingzhong
2010-01-01
Factorization of polynomials is one of the foundations of symbolic computation. Its applications arise in numerous branches of mathematics and other sciences. However, the present advanced programming languages such as C++ and J++, do not support symbolic computation directly. Hence, it leads to difficulties in applying factorization in engineering fields. In this paper, we present an algorithm which use numerical method to obtain exact factors of a bivariate polynomial with rational coefficients. Our method can be directly implemented in efficient programming language such C++ together with the GNU Multiple-Precision Library. In addition, the numerical computation part often only requires double precision and is easily parallelizable.
Multi-mode entangled states represented as Grassmannian polynomials
Maleki, Y.
2016-06-01
We introduce generalized Grassmannian representatives of multi-mode state vectors. By implementing the fundamental properties of Grassmann coherent states, we map the Hilbert space of the finite-dimensional multi-mode states to the space of some Grassmannian polynomial functions. These Grassmannian polynomials form a well-defined space in the framework of Grassmann variables; namely Grassmannian representative space. Therefore, a quantum state can be uniquely defined and determined by an element of Grassmannian representative space. Furthermore, the Grassmannian representatives of some maximally entangled states are considered, and it is shown that there is a tight connection between the entanglement of the states and their Grassmannian representatives.
On Star-Varieties with Almost Polynomial Growth
A. Giambruno; S. Mishchenko
2001-01-01
Let 2 be a variety of algebras with involution * over a field of characteristic zero and cn(V, *) the corresponding sequence of *-codimensions. Here,we characterize those varieties V such that cn (V, *) is polynomially bounded. We prove that V is such a variety if and only if G2,M V, where G2 and M are two explicit finite-dimensional algebras with involution previously constructed. It follows that G2 and M generate the only two varieties of algebras with involution with almost polynomial growth and there is no variety with intermediate growth.
Bispectral commuting difference operators for multivariable Askey-Wilson polynomials
Iliev, Plamen
2008-01-01
We construct a commutative algebra A_z, generated by d algebraically independent q-difference operators acting on variables z_1, z_2,..., z_d, which is diagonalized by the multivariable Askey-Wilson polynomials P_n(z) considered by Gasper and Rahman [6]. Iterating Sears' transformation formula, we show that the polynomials P_n(z) possess a certain duality between z and n. Analytic continuation allows us to obtain another commutative algebra A_n, generated by d algebraically independent differ...
Local polynomial Whittle estimation covering non-stationary fractional processes
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...... in.ated by a multiplicative constant. We show consistency and asymptotic normality for d ε (-1/2), and if the spectral density of the short-run component is in.nitely smooth near frequency zero we obtain an optimal rate of convergence for this setting, i.e. convergence arbitrarily close to the...
Method of resolution of 3SAT in polynomial time
Salemi, Luigi
2009-01-01
Presentation of a Method for determining whether a problem 3Sat has solution, and if yes to find one, in time max O(n^15). Is thus proved that the problem 3Sat is fully resolved in polynomial time and therefore that it is in P, by the work of Cook and Levin, and can transform a SAT problem in a 3Sat in polynomial time (ref. Karp), it follows that P = NP. Open Source program is available at http://www.visainformatica.it/3sat
Skew-orthogonal polynomials, differential systems and random matrix theory
We study skew-orthogonal polynomials with respect to the weight function exp[-2V (x)], with V (x) = ΣK=12d (uK/K)xK, u2d > 0, d > 0. A finite subsequence of such skew-orthogonal polynomials arising in the study of Orthogonal and Symplectic ensembles of random matrices, satisfy a system of differential-difference-deformation equation. The vectors formed by such subsequence has the rank equal to the degree of the potential in the quaternion sense. These solutions satisfy certain compatibility condition and hence admit a simultaneous fundamental system of solutions. (author)
Gaussian polynomials and content ideal in trivial extensions
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)
Evolution method and HOMFLY polynomials for virtual knots
Bishler, Ludmila; Morozov, Andrey; Morozov, Anton
2014-01-01
Following the suggestion of arXiv:1407.6319 to lift the knot polynomials for virtual knots and links from Jones to HOMFLY, we apply the evolution method to calculate them for an infinite series of twist-like virtual knots and antiparallel 2-strand links. Within this family one can check topological invariance and understand how differential hierarchy is modified in virtual case. This opens a way towards a definition of colored (not only cabled) knot polynomials, though problems still persist beyond the first symmetric representation.
On the Action of Steenrod Operations on Polynomial Algebras
KARACA, İsmet
1998-01-01
Let \\( \\bba \\) be the mod-\\( p \\) Steenrod Algebra. Let \\( p \\) be an odd prime number and \\( Zp = Z/pZ \\). Let \\( Ps = Zp [x1,x2,\\ldots,xs]. \\) A polynomial \\( N \\in Ps \\) is said to be hit if it is in the image of the action \\( A \\otimes Ps \\ra Ps. \\) In [10] for \\( p=2, \\) Wood showed that if \\( \\a(d+s) > s \\) then every polynomial of degree \\( d \\) in \\( Ps \\) is hit where \\( \\a(d+s) \\) denotes the number of ones in the binary expansion of \\( d+s \\). Latter in [6] Monks extended a resu...
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...
Koch, Frieder J; Detlefs, Carsten; Schröter, Tobias J; Kunka, Danays; Last, Arndt; Mohr, Jürgen
2016-05-01
Refractive X-ray lenses are in use at a large number of synchrotron experiments. Several materials and fabrication techniques are available for their production, each having their own strengths and drawbacks. We present a grating interferometer for the quantitative analysis of single refractive X-ray lenses and employ it for the study of a beryllium point focus lens and a polymer line focus lens, highlighting the differences in the outcome of the fabrication methods. The residuals of a line fit to the phase gradient are used to quantify local lens defects, while shape aberrations are quantified by the decomposition of the retrieved wavefront phase profile into either Zernike or Legendre polynomials, depending on the focus and aperture shape. While the polymer lens shows better material homogeneity, the beryllium lens shows higher shape accuracy. PMID:27137533
On lifting $q$-difference operators for a chain of basic hypergeometric polynomials
Atakishiyeva, Mesuma; Atakishiyev, Natig
2011-01-01
We construct lifting $q$-difference operators for a chain of basic hypergeometric polynomials, which composed of the continuous $q$-Hermite polynomials $H_{\\,n}(x\\,|\\,q)$ of Rogers, the continuous big $q$-Hermite polynomials $H_n(x;a|\\,q)$, the Al-Salam-Chihara polynomials $Q_n(x;a,b\\,|\\,q)$, the continuous dual $q$-Hahn polynomials $p_n(x;a,b,c\\,|\\,q)$ and, finally, the Askey-Wilson polynomials $p_n(x;a,b,c,d\\,|\\,q)$ on the five different levels within the Askey $q$-scheme....
Two-variable orthogonal polynomials of big q-Jacobi type
Lewanowicz, Stanislaw; Wozny, Pawel
2010-01-01
A four-parameter family of orthogonal polynomials in two discrete variables is defined for a weight function of basic hypergeometric type. The polynomials, which are expressed in terms of univariate big q-Jacobi polynomials, form an extension of Dunkl's bivariate (little) q-Jacobi polynomials [C.F. Dunkl, Orthogonal polynomials in two variables of q-Hahn and q-Jacobi type, SIAM J. Algebr. Discrete Methods 1 (1980) 137-151]. We prove orthogonality property of the new polynomials, and show that they satisfy a three-term relation in a vector-matrix notation, as well as a second-order partial q-difference equation.
Analytical and numerical study on grating depth effects in grating coupled waveguide sensors
Horvath, R.; Wilcox, L.C.; Pedersen, H.C.; Skivesen, N.; Hesthaven, J.S.; Johansen, P.M.
2005-01-01
The in-coupling process for grating-coupled planar optical waveguide sensors is investigated in the case of TE waves. A simple analytical model based on the Rayleigh-Fourier-Kiselev method is applied to take into account the depth of the grating coupler, which is usually neglected in the modeling...
Efficient modeling of photonic crystals with local Hermite polynomials
Boucher, C. R.; Li, Zehao; Albrecht, J. D.; Ram-Mohan, L. R.
2014-04-01
Developing compact algorithms for accurate electrodynamic calculations with minimal computational cost is an active area of research given the increasing complexity in the design of electromagnetic composite structures such as photonic crystals, metamaterials, optical interconnects, and on-chip routing. We show that electric and magnetic (EM) fields can be calculated using scalar Hermite interpolation polynomials as the numerical basis functions without having to invoke edge-based vector finite elements to suppress spurious solutions or to satisfy boundary conditions. This approach offers several fundamental advantages as evidenced through band structure solutions for periodic systems and through waveguide analysis. Compared with reciprocal space (plane wave expansion) methods for periodic systems, advantages are shown in computational costs, the ability to capture spatial complexity in the dielectric distributions, the demonstration of numerical convergence with scaling, and variational eigenfunctions free of numerical artifacts that arise from mixed-order real space basis sets or the inherent aberrations from transforming reciprocal space solutions of finite expansions. The photonic band structure of a simple crystal is used as a benchmark comparison and the ability to capture the effects of spatially complex dielectric distributions is treated using a complex pattern with highly irregular features that would stress spatial transform limits. This general method is applicable to a broad class of physical systems, e.g., to semiconducting lasers which require simultaneous modeling of transitions in quantum wells or dots together with EM cavity calculations, to modeling plasmonic structures in the presence of EM field emissions, and to on-chip propagation within monolithic integrated circuits.
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.
Trends and future of fiber Bragg grating sensing technologies: tailored draw tower gratings (DTGs)
Lindner, E.; Hartung, A.; Hoh, D.; Chojetzki, C.; Schuster, K.; Bierlich, J.; Rothhardt, M.
2014-05-01
Today fiber Bragg gratings are commonly used in sensing technology as well as in telecommunications. Numerous requirements must be satisfied for their application as a sensor such as the number of sensors per system, the measurement resolution and repeatability, the sensor reusability as well as the sensor costs. In addition current challenges need to be met in the near future for sensing fibers to keep and extend their marketability such as the suitability for sterilization, hydrogen darkening or the separation of strain and temperature (or pressure and temperature). In this contribution we will give an outlook about trends and future of the fiber Bragg gratings in sensing technologies. Specifically, we will discuss how the use of draw tower grating technology enables the production of tailored Bragg grating sensing fibers, and we will present a method of separating strain and temperature by the use of a single Bragg grating only, avoiding the need for additional sensors to realize the commonly applied temperature compensation.
Polynomial matrix equation design of reduced dimension observers
Kraffer, Ferdinand
Kusadasi: IEEE, 2004, s. 1-6. [IEEE Mediterranean Conference on Control and Automation /12./. Kusadasi (TR), 06.06.2004-09.06.2004] R&D Projects: GA ČR GA102/01/0608 Institutional research plan: CEZ:AV0Z1075907 Keywords : linear systems * reduced dimension observers * polynomial equations Subject RIV: BC - Control Systems Theory