Sample records for active polynomial grating

  1. Design and fabrication of an active polynomial grating for soft-X-ray monochromators and spectrometers

    Chen, S J; Perng, S Y; Kuan, C K; Tseng, T C; Wang, D J


    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.

  2. Numerical scheme for the modal method based on subsectional Gegenbauer polynomial expansion: application to biperiodic binary grating.

    Edee, K; Plumey, J P


    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

  3. A Novel Active Grating Monochromator - Active Grating Spectrometer Beamline System for Inelastic Soft-X-ray Scattering Experiments

    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

  4. 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)

  5. Polynomial functors and polynomial monads

    Gambino, Nicola


    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.

  6. Orthogonal polynomials

    Freud, Géza


    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

  7. 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)


    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.

  8. Chebyshev polynomials

    Mason, JC


    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...

  9. Consequences of Laughter Upon Trunk Compression and Cortical Activation: Linear and Polynomial Relations

    Svebak, Sven


    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

  10. 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


    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

  11. Hydrodynamics-based functional forms of activity metabolism: a case for the power-law polynomial function in animal swimming energetics.

    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.

  12. Hydrodynamics-Based Functional Forms of Activity Metabolism: A Case for the Power-Law Polynomial Function in Animal Swimming Energetics

    Papadopoulos, Anthony


    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...

  13. Polynomially Bounded Sequences and Polynomial Sequences

    Okazaki Hiroyuki


    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].

  14. Factoring multivariate integral polynomials.

    Lenstra, A.K.


    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.

  15. New classes of test polynomials of polynomial algebras

    冯克勤; 余解台


    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.

  16. Generalized bivariate Fibonacci polynomials

    Catalani, Mario


    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.

  17. Factoring Polynomials and Fibonacci.

    Schwartzman, Steven


    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)

  18. Branched polynomial covering maps

    Hansen, Vagn Lundsgaard


    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....

  19. Branched polynomial covering maps

    Hansen, Vagn Lundsgaard


    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....

  20. Bernstein polynomials on Simplex

    Bayad, A.; Kim, T.; Rim, S. -H.


    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

  1. 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

  2. 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


    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.

  3. Generalized Fibonacci-Lucas Polynomials

    Mamta Singh


    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

  4. Weierstrass polynomials for links

    Hansen, Vagn Lundsgaard


    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....

  5. New results on permutation polynomials over finite fields

    Ma, Jingxue; Zhang, Tao; Feng, Tao; Ge, Gennian


    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...

  6. Quantum Grothendieck polynomials

    Kirillov, Anatol N.


    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.

  7. Weighted lattice polynomials

    Marichal, Jean-Luc


    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.

  8. Feynman Graph Polynomials

    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.

  9. Nonnegativity of uncertain polynomials

    Šiljak Dragoslav D.


    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.

  10. Jack polynomials in superspace

    Desrosiers, P; Mathieu, P


    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...

  11. On universal knot polynomials

    Mironov, A; Morozov, A


    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.

  12. Computing Modular Polynomials

    Charles, Denis; Lauter, Kristin


    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...

  13. 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)


    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.

  14. Reflectivity-modulated grating-mirror


    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...

  15. Additive and polynomial representations

    Krantz, David H; Suppes, Patrick


    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


    Zhiqiang LI; Yupeng QIAO; Hongsheng QI; Daizhan CHENG


    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.

  17. Complexity of Ising Polynomials

    Kotek, Tomer


    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 ...

  18. 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

  19. Nonconventional Polynomial CLT

    Hafouta, Y.; Kifer, Y.


    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.

  20. Functional composition of polynomials: indecomposability, Diophantine equations and lacunary polynomials

    Kreso, Dijana; Tichy, Robert F.


    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).

  1. Computing the Alexander Polynomial Numerically

    Hansen, Mikael Sonne


    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....

  2. Chromatic polynomials for simplicial complexes

    Møller, Jesper Michael; Nord, Gesche


    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...

  3. On Q-derived polynomials

    R.J. Stroeker (Roel)


    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

  4. Corrosion detection of steel reinforced concrete using combined carbon fiber and fiber Bragg grating active thermal probe

    Li, Weijie; Ho, Siu Chun Michael; Song, Gangbing


    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.

  5. 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....

  6. Oblivious Polynomial Evaluation

    Hong-Da Li; Dong-Yao Ji; Deng-Guo Feng; Bao Li


    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.

  7. Polynomial Learning of Distribution Families

    Belkin, Mikhail


    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...

  8. Symmetric functions and Hall polynomials

    MacDonald, Ian Grant


    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...

  9. Polynomial Regression on Riemannian Manifolds

    Hinkle, Jacob; Fletcher, P Thomas; Joshi, Sarang


    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.

  10. Deformed Mittag-Leffler Polynomials

    Miomir S. Stankovic; Marinkovic, Sladjana D.; Rajkovic, Predrag M.


    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.

  11. Witt Rings and Permutation Polynomials

    Qifan Zhang


    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.

  12. Polynomial Bell Inequalities

    Chaves, Rafael


    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.

  13. 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...

  14. Polynomial harmonic GMDH learning networks for time series modeling.

    Nikolaev, Nikolay Y; Iba, Hitoshi


    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

  15. Polynomial weights and code constructions

    Massey, J; Costello, D; Justesen, Jørn


    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...

  16. 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) ...

  17. Multilayer diffraction grating

    Barbee, Jr., Troy W.


    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.

  18. Cyclotomy and permutation polynomials of large indices

    WANG Qiang


    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.

  19. Difference equations of q-Appell polynomials

    Mahmudov, Nazim I.


    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

  20. Complex Roots of Quaternion Polynomials

    Dospra, Petroula; Poulakis, Dimitrios


    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...

  1. Orthogonal polynomials and deformed oscillators

    Borzov, V. V.; Damaskinsky, E. V.


    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.

  2. Polynomials Associated with Dihedral Groups

    Charles F. Dunkl


    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.

  3. An introduction to orthogonal polynomials

    Chihara, Theodore S


    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

  4. Evaluations of topological Tutte polynomials

    Ellis-Monaghan, Joanna A


    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.

  5. Active Q-switching of a fiber laser using a modulated fiber Fabry-Perot filter and a fiber Bragg grating

    Martínez Manuel, Rodolfo; Kaboko, J. J. M.; Shlyagin, M. G.


    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.

  6. Multilayer dielectric diffraction gratings

    Perry, Michael D.; Britten, Jerald A.; Nguyen, Hoang T.; Boyd, Robert; Shore, Bruce W.


    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.

  7. Uniqueness and Zeros of -Shift Difference Polynomials

    Kai Liu; Xin-Ling Liu; Ting-Bin Cao


    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.

  8. On the Irreducibility of Some Composite Polynomials

    M. Alizadeh


    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

  9. Bannai-Ito polynomials and dressing chains

    Derevyagin, Maxim; Tsujimoto, Satoshi; Vinet, Luc; Zhedanov, Alexei


    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.

  10. Ehrhart polynomials of matroid polytopes and polymatroids

    De Loera, Jesús A.; Haws, David C.; Köppe, Matthias


    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...

  11. 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.


    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.

  12. q-Bernstein polynomials, q-Stirling numbers and q-Bernoulli polynomials

    Kim, T.


    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.

  13. Modular forms and period polynomials

    Pasol, Vicentiu


    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.

  14. 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.

  15. Orthogonal Polynomials and their Applications

    Dehesa, Jesús; Marcellan, Francisco; Francia, José; Vinuesa, Jaime


    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).

  16. Symbolic computation of Appell polynomials using Maple

    H. Alkahby


    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.

  17. Diffraction by m-bonacci gratings

    Monsoriu, Juan A.; Giménez, Marcos H.; Furlan, Walter D.; Barreiro, Juan C.; Saavedra, Genaro


    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.

  18. Two polynomial division inequalities in

    Goetgheluck P


    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.

  19. 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.

  20. 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...

  1. Sheffer and Non-Sheffer Polynomial Families

    G. Dattoli


    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.

  2. Quantum F-polynomials in Classical Types

    Tran, Thao


    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.

  3. Bidirectional grating compressors

    Wang, Cheng; Li, Zhaoyang; Li, Shuai; Liu, Yanqi; Leng, Yuxin; Li, Ruxin


    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.

  4. Fractal Diffraction Grating

    Bak, Dongsu; Kim, Sang Pyo; Kim, Sung Ku; Soh, Kwang-Sup; Yee, Jae Hyung


    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.

  5. Application of Chebyshev Polynomial to simulated modeling

    CHI Hai-hong; LI Dian-pu


    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.

  6. An Improved Volumetric Estimation Using Polynomial Regression

    Noraini Abdullah; Amran Ahmed; Zainodin Hj. Jubok


    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...

  7. Computing the zeros of quaternion polynomials

    Serôdio, R.; Pereira, E.; Vitória, J.


    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...

  8. A Class of Binomial Permutation Polynomials

    Tu, Ziran; Zeng, Xiangyong; Hu, Lei; Li, Chunlei


    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.

  9. Positive trigonometric polynomials and signal processing applications

    Dumitrescu, Bogdan


    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.

  10. s-Numbers sequences for homogeneous polynomials

    Caliskan, Erhan; Rueda, Pilar


    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.

  11. Graph colorings, flows and arithmetic Tutte polynomial

    D'Adderio, Michele; Moci, Luca


    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.

  12. Lattice Platonic Solids and their Ehrhart polynomial

    E. J. Ionascu


    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.

  13. Lattice Platonic Solids and their Ehrhart polynomial

    Ionascu, Eugen J


    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.

  14. Discriminants of Polynomials Related to Chebyshev Polynomials: The 'Mutt and Jeff' Syndrome

    Tran, Khang


    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.

  15. Complete Bell polynomials and new generalized identities for polynomials of higher order

    Rubinstein, Boris Y


    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.

  16. Boundary integral equation Neumann-to-Dirichlet map method for gratings in conical diffraction.

    Wu, Yumao; Lu, Ya Yan


    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

  17. Normal BGG solutions and polynomials

    Cap, A; Hammerl, M


    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 ...

  18. BSDEs with polynomial growth generators

    Philippe Briand


    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.

  19. Twisted Polynomials and Forgery Attacks on GCM

    Abdelraheem, Mohamed Ahmed A. M. A.; Beelen, Peter; Bogdanov, Andrey;


    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...

  20. Space complexity in polynomial calculus

    Filmus, Y.; Lauria, M.; Nordström, J.; Ron-Zewi, N.; Thapen, Neil


    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

  1. Simplifying Tensor Polynomials with Indices

    Balfagón, A


    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//

  2. Roots of Quaternion Standard Polynomials

    Chapman, Adam


    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.

  3. Pattern Matching under Polynomial Transformation

    Butman, Ayelet; Clifford, Raphael; Jalsenius, Markus; Lewenstein, Noa; Porat, Benny; Porat, Ely; Sach, Benjamin


    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...

  4. Hybrid grating reflector with high reflectivity and broad bandwidth

    Taghizadeh, Alireza; Park, Gyeong Cheol; Mørk, Jesper; Chung, Il-Sug


    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...

  5. Polynomial approximation, local polynomial convexity, and degenerate CR singularities -- II

    Bharali, Gautam


    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...

  6. A point-wise fiber Bragg grating displacement sensing system and its application for active vibration suppression of a smart cantilever beam subjected to multiple impact loadings

    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)

  7. Modeling Component-based Bragg gratings Application: tunable lasers

    Hedara Rachida


    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.

  8. Chromatic Polynomials of Mixed Hypercycles

    Allagan Julian A.


    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

  9. Zeroes of random Reinhardt polynomials

    Karami, Arash


    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...

  10. Global Polynomial Kernel Hazard Estimation

    Hiabu, Munir; Miranda, Maria Dolores Martínez; Nielsen, Jens Perch;


    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...

  11. Nonnegative Polynomials and Sums of Squares

    Blekherman, Grigoriy


    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.

  12. An Improved Volumetric Estimation Using Polynomial Regression

    Noraini Abdullah


    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.

  13. Exceptional polynomials and SUSY quantum mechanics

    K V S Shiv Chaitanya; S Sree Ranjani; Prasanta K Panigrahi; R Radhakrishnan; V Srinivasan


    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.

  14. A new Arnoldi approach for polynomial eigenproblems

    Raeven, F.A.


    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.

  15. Landau and Kolmogoroff type polynomial inequalities

    Alves Claudia RR; Dimitrov Dimitar K


    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...

  16. Haglund's conjecture on 3-column Macdonald polynomials

    Blasiak, Jonah


    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...

  17. Accelerated graph-based spectral polynomial filters

    Knyazev, Andrew; Malyshev, Alexander,


    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...

  18. Quantum Schubert polynomials and quantum Schur functions

    Kirillov, Anatol N.


    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...

  19. Generalizations of Bernoulli numbers and polynomials

    Qiu-Ming Luo; Bai-Ni Guo; Feng Qi; Lokenath Debnath


    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.

  20. About polynomials related to a quadratic equation

    Groux, Roland


    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.

  1. Discrete least squares approximation with polynomial vectors

    Van Barel, Marc; Bultheel, Adhemar


    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...

  2. On permutation polynomials over finite fields

    C. Small; R. A. Mollin


    A polynomial f over a finite field F is called a permutation polynomial if the mapping F→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...

  3. The q-Laguerre matrix polynomials.

    Salem, Ahmed


    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

  4. Control to Facet for Polynomial Systems

    Sloth, Christoffer; Wisniewski, Rafael


    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....

  5. Polynomial chaotic inflation in supergravity revisited

    Kazunori Nakayama


    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.

  6. Multi-indexed (q)-Racah Polynomials

    Odake, Satoru


    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.

  7. Macdonald Polynomials and Multivariable Basic Hypergeometric Series

    Michael J. Schlosser


    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.

  8. On the Fermionic -adic Integral Representation of Bernstein Polynomials Associated with Euler Numbers and Polynomials

    Ryoo CS


    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.

  9. Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams

    Assaf, Sami; Searles, Dominic


    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...

  10. Dynamic Sensing Performance of a Point-Wise Fiber Bragg Grating Displacement Measurement System Integrated in an Active Structural Control System

    Chien-Ching Ma


    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.

  11. Tutorial: Applications of Fibre Gratings

    Hwayaw; Tam; Bai; ou; Guan; Shunyee; Liu


    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.

  12. Phase gratings for plasmon focusing

    Offerhaus, H.L.; Bergen, van den, GJA Gino; Hulst, van, N.F.


    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).

  13. Integral Method for Gratings

    Maystre, Daniel


    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.

  14. Ultra-High Temperature Gratings

    John Canning; Somnath Bandyopadhyay; Michael Stevenson; Kevin Cook


    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.

  15. Polynomial invariants of quantum codes

    Rains, E M


    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]).

  16. Algebras, dialgebras, and polynomial identities

    Bremner, Murray R


    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.

  17. Fuzzy Morphological Polynomial Image Representation

    Chin-Pan Huang


    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.

  18. Elementary combinatorics of the HOMFLYPT polynomial

    Chmutov, Sergei; Polyak, Michael


    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.

  19. Application of polynomial preconditioners to conservation laws

    Geurts, Bernard J.; Buuren, van René; Lu, Hao


    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

  20. An inequality for polynomials with elliptic majorant

    Nikolov Geno


    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 .

  1. 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

  2. Sums of Powers of Fibonacci Polynomials

    Helmut Prodinger


    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.

  3. The weighted lattice polynomials as aggregation functions

    Marichal, Jean-Luc


    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.

  4. A Note on Solvable Polynomial Algebras

    Huishi Li


    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.

  5. On the Zeros of a Polynomial

    V K Jain


    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).

  6. A generalization of the Bernoulli polynomials

    Pierpaolo Natalini; Angela Bernardini


    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).

  7. A generalization of the Bernoulli polynomials

    Pierpaolo Natalini


    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.

  8. Quantum Search for Zeros of Polynomials

    Weigert, S


    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.

  9. A quantum search for zeros of polynomials

    Weigert, Stefan [HuMP-Hull Mathematical Physics, Department of Mathematics, University of Hull, Hull HU6 7RX (United Kingdom)


    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.

  10. Point vortex equilibria related to Bessel polynomials

    O'Neil, Kevin A.


    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.

  11. Large degree asymptotics of generalized Bessel polynomials

    López, J.L.; Temme, N.M.


    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

  12. A $(p,q)$-Analogue of Poly-Euler Polynomials and Some Related Polynomials

    Komatsu, Takao; Ramírez, José L.; Sirvent, Víctor F.


    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.

  13. Sobolev orthogonal polynomials on a simplex

    Aktas, Rabia


    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.

  14. Orthogonal Polynomials from Hermitian Matrices II

    Odake, Satoru


    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...

  15. 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)

  16. Tutte polynomial in functional magnetic resonance imaging

    García-Castillón, Marlly V.


    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.

  17. Pure Bending Characteristic of Tilted Fiber Bragg Grating

    Bo Liu; Yin-Ping Miao; Hai-Bin Zhou; Qi-Da Zhao


    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.

  18. On some properties on bivariate Fibonacci and Lucas polynomials

    Belbachir, Hacéne; Bencherif, Farid


    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.

  19. Bernoulli-like polynomials associated with Stirling Numbers

    Bender, Carl M; Brody, Dorje C.; BERNHARD K. MEISTER


    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.

  20. Bragg grating rogue wave

    Degasperis, Antonio; Aceves, Alejandro B


    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.

  1. Silicon graphene Bragg gratings

    Capmany, Jose; Domenech, David; Munoz, Pascual


    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.

  2. Bragg grating rogue wave

    Degasperis, Antonio [Dipartimento di Fisica, “Sapienza” Università di Roma, P.le A. Moro 2, 00185 Roma (Italy); Wabnitz, Stefan, E-mail: [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)


    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.

  3. Electrically-programmable diffraction grating

    Ricco, A.J.; Butler, M.A.; Sinclair, M.B.; Senturia, S.D.


    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.

  4. Polynomial Interpolation in the Elliptic Curve Cryptosystem

    Liew K. Jie


    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.

  5. More on rotations as spin matrix polynomials

    Curtright, Thomas L. [Department of Physics, University of Miami, Coral Gables, Florida 33124-8046 (United States)


    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.

  6. Cycles are determined by their domination polynomials

    Akbari, Saieed; Oboudi, Mohammad Reza


    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.

  7. On function compositions that are polynomials

    Aichinger, Erhard


    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.

  8. On Combinatorial Formulas for Macdonald Polynomials

    Lenart, Cristian


    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...

  9. On Calculation of Adomian Polynomials by MATLAB



    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.

  10. Tutte Polynomial of Scale-Free Networks

    Chen, Hanlin; Deng, Hanyuan


    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.

  11. 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...

  12. Characteristic Polynomials of Sample Covariance Matrices

    Kösters, Holger


    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.

  13. 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


    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...

  14. 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.

  15. Superconformal minimal models and admissible Jack polynomials

    Blondeau-Fournier, Olivier; Ridout, David; Wood, Simon


    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.

  16. Inequalities for a polynomial and its derivative

    Chanam, Barchand; Dewan, K. K.


    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].

  17. Bergman orthogonal polynomials and the Grunsky matrix

    Beckermann, Bernhard; Stylianopoulos, Nikos


    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,...

  18. Equivalence of polynomial conjectures in additive combinatorics

    Lovett, Shachar


    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.

  19. Stochastic processes with orthogonal polynomial eigenfunctions

    Griffiths, Bob


    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.

  20. Ferrers Matrices Characterized by the Rook Polynomials

    MAHai-cheng; HUSheng-biao


    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.

  1. 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.)

  2. On Sharing, Memoization, and Polynomial Time

    Avanzini, Martin; Dal Lago, Ugo


    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.

  3. Laguerre polynomials method in the valon model

    Boroun, G R


    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.

  4. 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.

  5. Polynomial Subtraction Method for Disconnected Quark Loops

    Liu, Quan; Morgan, Ron


    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.

  6. A Polynomial Preconditioner for the CMRH Algorithm

    Shiji Xu; Jiangzhou Lai; Linzhang Lu


    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...

  7. Blind Signature Scheme Based on Chebyshev Polynomials

    Maheswara Rao Valluri


    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.

  8. Nonsymmetric Askey-Wilson polynomials and $Q$-polynomial distance-regular graphs

    Lee, Jae-Ho


    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...

  9. Vector-Valued Jack Polynomials from Scratch

    Jean-Gabriel Luque


    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.

  10. 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)

  11. Reliable Computational Predictions by Modeling Uncertainties Using Arbitrary Polynomial Chaos

    Witteveen, J.A.S.; Bijl, H


    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 ...

  12. Characterization of pinhole transmission gratings.

    Eidmann, K; Kühne, M; Müller, P; Tsakiris, G D


    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

  13. On the Connection Coefficients of the Chebyshev-Boubaker Polynomials

    Paul Barry


    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.

  14. Generalized Narayana Polynomials, Riordan Arrays and Lattice Paths

    Barry, Paul; Hennessy, Aoife


    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...

  15. A new class of generalized polynomials associated with Hermite and Bernoulli polynomials

    M. A. Pathan


    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

  16. 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

  17. Supersymmetric Bragg gratings

    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)

  18. 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.

  19. Hierarchical polynomial network approach to automated target recognition

    Kim, Richard Y.; Drake, Keith C.; Kim, Tony Y.


    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.

  20. Optical Fiber Grating based Sensors

    Michelsen, Susanne


    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...

  1. NMR Quantum Calculations of the Jones Polynomial

    Marx, Raimund; Kauffman, Louis; Lomonaco, Samuel; Spörl, Andreas; Pomplun, Nikolas; Myers, John; Glaser, Steffen J


    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.

  2. 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)

  3. Minimal residual method stronger than polynomial preconditioning

    Faber, V.; Joubert, W.; Knill, E. [Los Alamos National Lab., NM (United States)] [and others


    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.

  4. 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

  5. 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....

  6. Sparse DOA estimation with polynomial rooting

    Xenaki, Angeliki; Gerstoft, Peter; Fernandez Grande, Efren


    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...

  7. Incomplete Bivariate Fibonacci and Lucas -Polynomials

    Dursun Tasci


    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.

  8. On the Waring problem for polynomial rings

    Fröberg, Ralf; Shapiro, Boris


    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.

  9. Error Minimization of Polynomial Approximation of Delta

    Islam Sana; Sadiq Muhammad; Qureshi Muhammad Shahid


    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.

  10. Large Degree Asymptotics of Generalized Bessel Polynomials

    López, J. L.; Temme, Nico


    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...

  11. The chromatic polynomial and list colorings

    Thomassen, Carsten


    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....

  12. Five Constructions of Permutation Polynomials over $\\gf(q^2)$

    Ding, Cunsheng; Yuan, Pingzhi


    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...

  13. Perturbations around the zeros of classical orthogonal polynomials

    Sasaki, Ryu


    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.

  14. Perturbations around the zeros of classical orthogonal polynomials

    Sasaki, Ryu


    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.

  15. Inclusion-exclusion polynomials with large coefficients

    Bzdega, Bartlomiej


    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

  16. Scalar Field Theories with Polynomial Shift Symmetries

    Griffin, Tom; Horava, Petr; Yan, Ziqi


    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...

  17. Algebraic differential equations associated to some polynomials

    Barlet, Daniel


    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.

  18. Nondimensional Simplification of Tensor Polynomials with Indices

    Jaén, X


    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:

  19. Quantum Hilbert matrices and orthogonal polynomials

    Andersen, Jørgen Ellegaard; Berg, Christian


    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....

  20. Z-polynomials and ring commutativity

    Buckley, S.M.; McHale, D.


    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.

  1. 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.


    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.

  2. Interpolation of Shifted-Lacunary Polynomials

    Giesbrecht, Mark


    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<=e1polynomial in the (sparse) representation size, log(alpha)+ sum_i(log|c_i|+log(e_i)) and in particular is logarithmic in deg(f). Our method combines previous celebrated results on sparse interpolation and computing sparsest shifts, and provides a way to handle polynomials with extremely high degree which are, in some sense, sparse in information.

  3. Cumulants, lattice paths, and orthogonal polynomials

    Lehner, Franz


    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.

  4. Algebraic polynomial system solving and applications

    Bleylevens, I.W.M.


    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

  5. Polynomial stabilization of some dissipative hyperbolic systems

    Ammari, K.; Feireisl, Eduard; Nicaise, S.


    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

  6. 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)

  7. Ideals in Polynomial Near-rings

    Mark Farag


    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.

  8. 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)

  9. On an Inequality Concerning the Polar Derivative of a Polynomial

    A Aziz; N A Rather


    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.

  10. Representations of Knot Groups and Twisted Alexander Polynomials

    Xiao Song LIN


    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.

  11. Self-dual Koornwinder-MacDonald polynomials

    Van Diejen, J F


    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.

  12. Irreducibility Results for Compositions of Polynomials in Several Variables

    Anca Iuliana Bonciocat; Alexandru Zaharescu


    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.

  13. Remarks on Homogeneous Al-Salam and Carlitz Polynomials

    Jian-Ping Fang


    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.

  14. Identities involving Bessel polynomials arising from linear differential equations

    Kim, Taekyun; Kim, Dae San


    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.

  15. Best polynomial degree reduction on q-lattices with applications to q-orthogonal polynomials

    Ait-Haddou, Rachid


    We show that a weighted least squares approximation of q-Bézier coefficients provides the best polynomial degree reduction in the q-L2-norm. We also provide a finite analogue of this result with respect to finite q-lattices and we present applications of these results to q-orthogonal polynomials. © 2015 Elsevier Inc. All rights reserved.

  16. MEMS Bragg grating force sensor

    Reck, Kasper; Thomsen, Erik Vilain; Hansen, Ole


    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....

  17. Some Systems of Multivariable Orthogonal q-Racah polynomials

    Gasper, George; Rahman, Mizan


    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.

  18. On the Lorentz degree of a product of polynomials

    Ait-Haddou, Rachid


    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.

  19. Further Results on Permutation Polynomials over Finite Fields

    Yuan, Pingzhi; Ding, Cunsheng


    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 $\\gf_q$.

  20. Universality for polynomial invariants on ribbon graphs with flags

    Avohou, Remi C.; Geloun, Joseph Ben; Hounkonnou , Mahouton N.


    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.

  1. On conformal measures for infinitely renormalizable quadratic polynomials

    HUANG Zhiyong; JIANG Yunping; WANG Yuefei


    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.

  2. Probabilistic aspects of Al-Salam-Chihara polynomials

    Bryc, Wlodzimierz; Matysiak, Wojciech; Szablowski, Pawel J.


    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.

  3. Moments for Generating Functions of Al-Salam-Carlitz Polynomials

    Jian Cao


    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.

  4. Some advances in tensor analysis and polynomial optimization

    Li, Zhening; Ling, Chen; Wang, Yiju; Yang, Qingzhi


    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'...

  5. A Determinant Expression for the Generalized Bessel Polynomials

    Sheng-liang Yang; Sai-nan Zheng


    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.

  6. The Gibbs Phenomenon for Series of Orthogonal Polynomials

    Fay, T. H.; Kloppers, P. Hendrik


    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…

  7. Stretchable diffraction gratings for spectrometry

    Simonov, Aleksey N.; Grabarnik, Semen; Vdovin, Gleb


    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.

  8. Aptamer functionalized lipid multilayer gratings for label free detection of specific analytes

    Prommapan, Plengchart; Lowry, Troy W.; van Winkle, David; Lenhert, Steven


    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.

  9. High Resolution of the ECG Signal by Polynomial Approximation

    G. Rozinaj


    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.

  10. Theoretical analysis of novel fiber grating pair

    Wang, Liao; Jia, Hongzhi; Fang, Liang; You, Bei


    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.