Chen, S J; Perng, S Y; Kuan, C K; Tseng, T C; Wang, D J
2001-01-01
An active polynomial grating has been designed for use in synchrotron radiation soft-X-ray monochromators and spectrometers. The grating can be dynamically adjusted to obtain the third-order-polynomial surface needed to eliminate the defocus and coma aberrations at any photon energy. Ray-tracing results confirm that a monochromator or spectrometer based on this active grating has nearly no aberration limit to the overall spectral resolution in the entire soft-X-ray region. The grating substrate is made of a precisely milled 17-4 PH stainless steel parallel plate, which is joined to a flexure-hinge bender shaped by wire electrical discharge machining. The substrate is grounded into a concave cylindrical shape with a nominal radius and then polished to achieve a roughness of 0.45 nm and a slope error of 1.2 mu rad rms. The long trace profiler measurements show that the active grating can reach the desired third-order polynomial with a high degree of figure accuracy.
Constructive feedforward neural networks using hermite polynomial activation functions.
Ma, Liying; Khorasani, K
2005-07-01
In this paper, a constructive one-hidden-layer network is introduced where each hidden unit employs a polynomial function for its activation function that is different from other units. Specifically, both a structure level as well as a function level adaptation methodologies are utilized in constructing the network. The functional level adaptation scheme ensures that the "growing" or constructive network has different activation functions for each neuron such that the network may be able to capture the underlying input-output map more effectively. The activation functions considered consist of orthonormal Hermite polynomials. It is shown through extensive simulations that the proposed network yields improved performance when compared to networks having identical sigmoidal activation functions.
Gratings in passive and active optical waveguides
DEFF Research Database (Denmark)
Berendt, Martin Ole
1999-01-01
This research project has focused on gratings in optical waveguides. These gratings may be produced by UV photon imprinting in optical fibers or planar technology waveguides. The gratings are optical waveguide equivalents of bulk dielectric mirrors or diffraction grating. For a grating in a waveg......This research project has focused on gratings in optical waveguides. These gratings may be produced by UV photon imprinting in optical fibers or planar technology waveguides. The gratings are optical waveguide equivalents of bulk dielectric mirrors or diffraction grating. For a grating...... mode losses confirmed. An elaborated grating model, including the detailed shape of the index modulation, has been developed. This model improves the interpretation of grating growth dynamic, which is of value to both; analysis of the UV imprinting set-ups, and to the investigation of photosensitivity...... mechanisms in silica. An other important part of the project aimed at perfection of distributed feed back DFB fiber lasers. At the outset of this project, DFB fiber lasers had been demonstrated, now the DFB fiber lasers are in commercial production. One of the problems that had to be overcome was to secure...
International Nuclear Information System (INIS)
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
Freud, Géza
1971-01-01
Orthogonal Polynomials contains an up-to-date survey of the general theory of orthogonal polynomials. It deals with the problem of polynomials and reveals that the sequence of these polynomials forms an orthogonal system with respect to a non-negative m-distribution defined on the real numerical axis. Comprised of five chapters, the book begins with the fundamental properties of orthogonal polynomials. After discussing the momentum problem, it then explains the quadrature procedure, the convergence theory, and G. Szegő's theory. This book is useful for those who intend to use it as referenc
Active temperature compensation design of sensor with fiber gratings
Institute of Scientific and Technical Information of China (English)
Xingfa Dong(董兴法); Yonglin Huang(黄勇林); Li Jiang(姜莉); Guiyun Kai(开桂云); Xiaoyi Dong(董孝义)
2004-01-01
A technique for compensation of temperature effects in fiber grating sensors is reported. For strain sensors and other sensors related to strain such as electromagnetic sensors, a novel structure is designed, which uses two fiber Bragg gratings (FBGs) as strain differential sensor and has temperature effects cancelled. Using this technique, the stress sensitivity has been amplified and gets up to 0.226 nm/N, the total variation in wavelength difference within the range of 3-45 ℃ is 0.03 nm, 1/14 of the uncompensated FBG.The structure can be used in the temperature-insensitive static strain measurement and minor-vibration measurement.
Mason, JC
2002-01-01
Chebyshev polynomials crop up in virtually every area of numerical analysis, and they hold particular importance in recent advances in subjects such as orthogonal polynomials, polynomial approximation, numerical integration, and spectral methods. Yet no book dedicated to Chebyshev polynomials has been published since 1990, and even that work focused primarily on the theoretical aspects. A broad, up-to-date treatment is long overdue.Providing highly readable exposition on the subject''s state of the art, Chebyshev Polynomials is just such a treatment. It includes rigorous yet down-to-earth coverage of the theory along with an in-depth look at the properties of all four kinds of Chebyshev polynomials-properties that lead to a range of results in areas such as approximation, series expansions, interpolation, quadrature, and integral equations. Problems in each chapter, ranging in difficulty from elementary to quite advanced, reinforce the concepts and methods presented.Far from being an esoteric subject, Chebysh...
Energy Technology Data Exchange (ETDEWEB)
Blackett, S.A. [Univ. of Auckland (New Zealand). Dept of Engineering Science
1996-02-01
Numerical analysis is an important part of Engineering. Frequently relationships are not adequately understood, or too complicated to be represented by theoretical formulae. Instead, empirical approximations based on observed relationships can be used for simple fast and accurate evaluations. Historically, storage of data has been a large constraint on approximately methods. So the challenge is to find a sufficiently accurate representation of data which is valid over as large a range as possible while requiring the storage of only a few numerical values. Polynomials, popular as approximation functions because of their simplicity, can be used to represent simple data. Equation 1.1 shows a simple 3rd order polynomial approximation. However, just increasing the order and number of terms included in a polynomial approximation does not improve the overall result. Although the function may fit exactly to observed data, between these points it is likely that the approximation is increasingly less smooth and probably inadequate. An alternative to adding further terms to the approximation is to make the approximation rational. Equation 1.2 shows a rational polynomial, 3rd order in the numerator and denominator. A rational polynomial approximation allows poles and this can greatly enhance an approximation. In Sections 2 and 3 two different methods for fitting rational polynomials to a given data set are detailed. In Section 4, consideration is given to different rational polynomials used on adjacent regions. Section 5 shows the performance of the rational polynomial algorithms. Conclusions are presented in Section 6.
Dobbs, David E.
2010-01-01
This note develops and implements the theory of polynomial asymptotes to (graphs of) rational functions, as a generalization of the classical topics of horizontal asymptotes and oblique/slant asymptotes. Applications are given to hyperbolic asymptotes. Prerequisites include the division algorithm for polynomials with coefficients in the field of…
Directory of Open Access Journals (Sweden)
Anthony Papadopoulos
Full Text Available The first-degree power-law polynomial function is frequently used to describe activity metabolism for steady swimming animals. This function has been used in hydrodynamics-based metabolic studies to evaluate important parameters of energetic costs, such as the standard metabolic rate and the drag power indices. In theory, however, the power-law polynomial function of any degree greater than one can be used to describe activity metabolism for steady swimming animals. In fact, activity metabolism has been described by the conventional exponential function and the cubic polynomial function, although only the power-law polynomial function models drag power since it conforms to hydrodynamic laws. Consequently, the first-degree power-law polynomial function yields incorrect parameter values of energetic costs if activity metabolism is governed by the power-law polynomial function of any degree greater than one. This issue is important in bioenergetics because correct comparisons of energetic costs among different steady swimming animals cannot be made unless the degree of the power-law polynomial function derives from activity metabolism. In other words, a hydrodynamics-based functional form of activity metabolism is a power-law polynomial function of any degree greater than or equal to one. Therefore, the degree of the power-law polynomial function should be treated as a parameter, not as a constant. This new treatment not only conforms to hydrodynamic laws, but also ensures correct comparisons of energetic costs among different steady swimming animals. Furthermore, the exponential power-law function, which is a new hydrodynamics-based functional form of activity metabolism, is a special case of the power-law polynomial function. Hence, the link between the hydrodynamics of steady swimming and the exponential-based metabolic model is defined.
Svebak, Sven
2016-01-01
Results from two studies of biological consequences of laughter are reported. A proposed inhibitory brain mechanism was tested in Study 1. It aims to protect against trunk compression that can cause health hazards during vigorous laughter. Compression may be maximal during moderate durations and, for protective reasons, moderate in enduring vigorous laughs. Twenty-five university students volunteered to see a candid camera film. Laughter responses (LR) and the superimposed ha-responses were operationally assessed by mercury-filled strain gauges strapped around the trunk. On average, the thorax compression amplitudes exceeded those of the abdomen, and greater amplitudes were seen in the males than in the females after correction for resting trunk circumference. Regression analyses supported polynomial relations because medium LR durations were associated with particularly high thorax amplitudes. In Study 2, power changes were computed in the beta and alpha EEG frequency bands of the parietal cortex from before to after exposure to the comedy “Dinner for one” in 56 university students. Highly significant linear relations were calculated between the number of laughs and post-exposure cortical activation (increase of beta, decrease of alpha) due to high activation after frequent laughter. The results from Study 1 supported the hypothesis of a protective brain mechanism that is activated during long LRs to reduce the risk of harm to vital organs in the trunk cavity. The results in Study 2 supported a linear cortical activation and, thus, provided evidence for a biological correlate to the subjective experience of mental refreshment after laughter. PMID:27547260
Institute of Scientific and Technical Information of China (English)
LI Chunxiang; ZHOU Dai
2004-01-01
The polynomial matrix using the block coefficient matrix representation auto-regressive moving average (referred to as the PM-ARMA) model is constructed in this paper for actively controlled multi-degree-of-freedom (MDOF) structures with time-delay through equivalently transforming the preliminary state space realization into the new state space realization. The PM-ARMA model is a more general formulation with respect to the polynomial using the coefficient representation auto-regressive moving average (ARMA) model due to its capability to cope with actively controlled structures with any given structural degrees of freedom and any chosen number of sensors and actuators. (The sensors and actuators are required to maintain the identical number.) under any dimensional stationary stochastic excitation.
Polynomially Bounded Sequences and Polynomial Sequences
Directory of Open Access Journals (Sweden)
Okazaki Hiroyuki
2015-09-01
Full Text Available In this article, we formalize polynomially bounded sequences that plays an important role in computational complexity theory. Class P is a fundamental computational complexity class that contains all polynomial-time decision problems [11], [12]. It takes polynomially bounded amount of computation time to solve polynomial-time decision problems by the deterministic Turing machine. Moreover we formalize polynomial sequences [5].
Institute of Scientific and Technical Information of China (English)
王雷
2008-01-01
<正>Polynomial functions are among the sim- plest expressions in algebra.They are easy to evaluate:only addition and repeated multipli- cation are required.Because of this,they are often used to approximate other more compli-
Slow light via a tapered grating: transfer matrix approach
Valkering, Theo P.
2006-01-01
Reflection properties of a nonuniform apodized Bragg grating connecting a homogeneous medium with a (half-infinite) uniform grating are investigated for system parameters such that the group velocity in the uniform grating is low. In particular a smooth polynomial transition of the index profile to
Factoring multivariate integral polynomials.
Lenstra, A.K.
1983-01-01
An algorithm is presented to factorize polynomials in several variables with integral coefficients that is polynomial-time in the degrees of the polynomial to be factored, for any fixed number of variables. The algorithm generalizes the algorithm presented by A. K. Lenstra et al. to factorize integral polynomials in one variable.
New classes of test polynomials of polynomial algebras
Institute of Scientific and Technical Information of China (English)
冯克勤; 余解台
1999-01-01
A polynomial p in a polynomial algebra over a field is called a test polynomial if any endomorphism of the polynomial algebra that fixes p is an automorphism. some classes of new test polynomials recognizing nonlinear automorphisms of polynomial algebras are given. In the odd prime characteristic case, test polynomials recognizing non-semisimple automorphisms are also constructed.
Polynomial selections and separation by polynomials
Wasowicz, Szymon
2008-01-01
Necessary and sufficient conditions under which two real functions defined on the real interval can be separated by a polynomial are given. An immediate consequence of the main result is the existence of the polynomial separation of convex functions of higher order. Another application is some Hyers-Ulam-stability-type result.
Factoring Polynomials and Fibonacci.
Schwartzman, Steven
1986-01-01
Discusses the factoring of polynomials and Fibonacci numbers, offering several challenges teachers can give students. For example, they can give students a polynomial containing large numbers and challenge them to factor it. (JN)
Branched polynomial covering maps
DEFF Research Database (Denmark)
Hansen, Vagn Lundsgaard
2002-01-01
A Weierstrass polynomial with multiple roots in certain points leads to a branched covering map. With this as the guiding example, we formally define and study the notion of a branched polynomial covering map. We shall prove that many finite covering maps are polynomial outside a discrete branch...... set. Particular studies are made of branched polynomial covering maps arising from Riemann surfaces and from knots in the 3-sphere. (C) 2001 Elsevier Science B.V. All rights reserved....
Fiber Bragg grating strain sensors to monitor and study active volcanoes
Sorrentino, Fiodor; Beverini, Nicolò; Carbone, Daniele; Carelli, Giorgio; Francesconi, Francesco; Gambino, Salvo; Giacomelli, Umberto; Grassi, Renzo; Maccioni, Enrico; Morganti, Mauro
2016-04-01
Stress and strain changes are among the best indicators of impending volcanic activity. In volcano geodesy, borehole volumetric strain-meters are mostly utilized. However, they are not easy to install and involve high implementation costs. Advancements in opto-electronics have allowed the development of low-cost sensors, reliable, rugged and compact, thus particularly suitable for field application. In the framework of the EC FP7 MED-SUV project, we have developed strain sensors based on the fiber Bragg grating (FBG) technology. In comparison with previous implementation of the FBG technology to study rock deformations, we have designed a system that is expected to offer a significantly higher resolution and accuracy in static measurements and a smooth dynamic response up to 100 Hz, implying the possibility to observe seismic waves. The system performances are tailored to suit the requirements of volcano monitoring, with special attention to power consumption and to the trade-off between performance and cost. Preliminary field campaigns were carried out on Mt. Etna (Italy) using a prototypal single-axis FBG strain sensor, to check the system performances in out-of-the-lab conditions and in the harsh volcanic environment (lack of mains electricity for power, strong diurnal temperature changes, strong wind, erosive ash, snow and ice during the winter time). We also designed and built a FBG strain sensor featuring a multi-axial configuration which was tested and calibrated in the laboratory. This instrument is suitable for borehole installation and will be tested on Etna soon.
Generalized Fibonacci-Lucas Polynomials
Directory of Open Access Journals (Sweden)
Mamta Singh
2013-12-01
Full Text Available Various sequences of polynomials by the names of Fibonacci and Lucas polynomials occur in the literature over a century. The Fibonacci polynomials and Lucas polynomials are famous for possessing wonderful and amazing properties and identities. In this paper, Generalized Fibonacci-Lucas Polynomials are introduced and defined by the recurrence relation with and . Some basic identities of Generalized Fibonacci-Lucas Polynomials are obtained by method of generating function. Keywords: Fibonacci polynomials, Lucas polynomials, Generalized Fibonacci polynomials, Generalized Fibonacci-Lucas polynomials.
Quantum Grothendieck polynomials
Kirillov, Anatol N.
1996-01-01
We study the algebraic aspects of (small) quantum equivariant $K$-theory of flag manifold. Lascoux-Sch\\"utzenberger's type formula for quantum double and quantum double dual Grothendieck polynomials and the quantum Cauchy identity for quantum Grothendieck polynomials are obtained.
Polynomial Graphs and Symmetry
Goehle, Geoff; Kobayashi, Mitsuo
2013-01-01
Most quadratic functions are not even, but every parabola has symmetry with respect to some vertical line. Similarly, every cubic has rotational symmetry with respect to some point, though most cubics are not odd. We show that every polynomial has at most one point of symmetry and give conditions under which the polynomial has rotational or…
Jack polynomials in superspace
Desrosiers, P; Mathieu, P
2003-01-01
This work initiates the study of {\\it orthogonal} symmetric polynomials in superspace. Here we present two approaches leading to a family of orthogonal polynomials in superspace that generalize the Jack polynomials. The first approach relies on previous work by the authors in which eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland Hamiltonian were constructed. Orthogonal eigenfunctions are now obtained by diagonalizing the first nontrivial element of a bosonic tower of commuting conserved charges not containing this Hamiltonian. Quite remarkably, the expansion coefficients of these orthogonal eigenfunctions in the supermonomial basis are stable with respect to the number of variables. The second and more direct approach amounts to symmetrize products of non-symmetric Jack polynomials with monomials in the fermionic variables. This time, the orthogonality is inherited from the orthogonality of the non-symmetric Jack polynomials, and the value of the norm is given exp...
Nonnegativity of uncertain polynomials
Directory of Open Access Journals (Sweden)
iljak Dragoslav D.
1998-01-01
Full Text Available The purpose of this paper is to derive tests for robust nonnegativity of scalar and matrix polynomials, which are algebraic, recursive, and can be completed in finite number of steps. Polytopic families of polynomials are considered with various characterizations of parameter uncertainty including affine, multilinear, and polynomic structures. The zero exclusion condition for polynomial positivity is also proposed for general parameter dependencies. By reformulating the robust stability problem of complex polynomials as positivity of real polynomials, we obtain new sufficient conditions for robust stability involving multilinear structures, which can be tested using only real arithmetic. The obtained results are applied to robust matrix factorization, strict positive realness, and absolute stability of multivariable systems involving parameter dependent transfer function matrices.
Mironov, A; Morozov, A
2015-01-01
We present a universal knot polynomials for 2- and 3-strand torus knots in adjoint representation, by universalization of appropriate Rosso-Jones formula. According to universality, these polynomials coincide with adjoined colored HOMFLY and Kauffman polynomials at SL and SO/Sp lines on Vogel's plane, and give their exceptional group's counterparts on exceptional line. We demonstrate that [m,n]=[n,m] topological invariance, when applicable, take place on the entire Vogel's plane. We also suggest the universal form of invariant of figure eight knot in adjoint representation, and suggest existence of such universalization for any knot in adjoint and its descendant representation. Properties of universal polynomials and applications of these results are discussed.
Yu, Jiun-Hung
2012-01-01
Polynomial remainder codes are a large class of codes derived from the Chinese remainder theorem that includes Reed-Solomon codes as a special case. In this paper, we revisit these codes and study them more carefully than in previous work. We explicitly allow the code symbols to be polynomials of different degrees, which leads to two different notions of weight and distance. Algebraic decoding is studied in detail. If the moduli are not irreducible, the notion of an error locator polynomial is replaced by an error factor polynomial. We then obtain a collection of gcd-based decoding algorithms, some of which are not quite standard even when specialized to Reed-Solomon codes.
Laser-induced gratings in the gas phase excited via Raman-active transitions
Energy Technology Data Exchange (ETDEWEB)
Kozlov, D.N. [General Physics Inst., Russian Academy of Sciences, Moscow (Russian Federation); Bombach, R.; Hemmerling, B.; Hubschmid, W. [Paul Scherrer Inst. (PSI), Villigen (Switzerland)
1997-06-01
We report on a new time resolved coherent Raman technique that is based on the generation of thermal gratings following a population change among molecular levels induced by stimulated Raman pumping. This is achieved by spatially and temporally overlapping intensity interference patterns generated independently by two lasers. When this technique is used in carbon dioxide, employing transitions which belong to the Q-branches of the {nu}{sub 1}/2{nu}{sub 2} Fermi dyad, it is possible to investigate molecular energy transfer processes. (author) 2 figs., 10 refs.
Additive and polynomial representations
Krantz, David H; Suppes, Patrick
1971-01-01
Additive and Polynomial Representations deals with major representation theorems in which the qualitative structure is reflected as some polynomial function of one or more numerical functions defined on the basic entities. Examples are additive expressions of a single measure (such as the probability of disjoint events being the sum of their probabilities), and additive expressions of two measures (such as the logarithm of momentum being the sum of log mass and log velocity terms). The book describes the three basic procedures of fundamental measurement as the mathematical pivot, as the utiliz
STABILITY OF SWITCHED POLYNOMIAL SYSTEMS
Institute of Scientific and Technical Information of China (English)
Zhiqiang LI; Yupeng QIAO; Hongsheng QI; Daizhan CHENG
2008-01-01
This paper investigates the stability of (switched) polynomial systems. Using semi-tensor product of matrices, the paper develops two tools for testing the stability of a (switched) polynomial system. One is to convert a product of multi-variable polynomials into a canonical form, and the other is an easily verifiable sufficient condition to justify whether a multi-variable polynomial is positive definite. Using these two tools, the authors construct a polynomial function as a candidate Lyapunov function and via testing its derivative the authors provide some sufficient conditions for the global stability of polynomial systems.
Kreso, Dijana; Tichy, Robert F.
2015-01-01
Starting from Ritt's classical theorems, we give a survey of results in functional decomposition of polynomials and of applications in Diophantine equations. This includes sufficient conditions for the indecomposability of polynomials, the study of decompositions of lacunary polynomials and the finiteness criterion for the equations of type f(x) = g(y).
Calculators and Polynomial Evaluation.
Weaver, J. F.
The intent of this paper is to suggest and illustrate how electronic hand-held calculators, especially non-programmable ones with limited data-storage capacity, can be used to advantage by students in one particular aspect of work with polynomial functions. The basic mathematical background upon which calculator application is built is summarized.…
Hetyei, Gábor
2010-01-01
We introduce the short toric polynomial associated to a graded Eulerian poset. This polynomial contains the same information as the two toric polynomials introduced by Stanley, but allows different algebraic manipulations. The intertwined recurrence defining Stanley's toric polynomials may be replaced by a single recurrence, in which the degree of the discarded terms is independent of the rank. A short toric variant of the formula by Bayer and Ehrenborg, expressing the toric $h$-vector in terms of the $cd$-index, may be stated in a rank-independent form, and it may be shown using weighted lattice path enumeration and the reflection principle. We use our techniques to derive a formula expressing the toric $h$-vector of a dual simplicial Eulerian poset in terms of its $f$-vector. This formula implies Gessel's formula for the toric $h$-vector of a cube, and may be used to prove that the nonnegativity of the toric $h$-vector of a simple polytope is a consequence of the Generalized Lower Bound Theorem holding for ...
Computing the Alexander Polynomial Numerically
DEFF Research Database (Denmark)
Hansen, Mikael Sonne
2006-01-01
Explains how to construct the Alexander Matrix and how this can be used to compute the Alexander polynomial numerically.......Explains how to construct the Alexander Matrix and how this can be used to compute the Alexander polynomial numerically....
ON PROPERTIES OF DIFFERENCE POLYNOMIALS
Institute of Scientific and Technical Information of China (English)
Chen Zongxuan; Huang Zhibo; Zheng Xiumin
2011-01-01
We study the value distribution of difference polynomials of meromorphic functions, and extend classical theorems of Tumura-Clunie type to difference polynomials. We also consider the value distribution of f(z)f(z+c).
Chromatic polynomials for simplicial complexes
DEFF Research Database (Denmark)
Møller, Jesper Michael; Nord, Gesche
2016-01-01
In this note we consider s s -chromatic polynomials for finite simplicial complexes. When s=1 s=1 , the 1 1 -chromatic polynomial is just the usual graph chromatic polynomial of the 1 1 -skeleton. In general, the s s -chromatic polynomial depends on the s s -skeleton and its value at r r is the n...
Kuipers, J.
2012-06-01
New features of the symbolic algebra package Form 4 are discussed. Most importantly, these features include polynomial factorization and polynomial gcd computation. Examples of their use are shown. One of them is an exact version of Mincer which gives answers in terms of rational polynomials and 5 master integrals.
R.J. Stroeker (Roel)
2002-01-01
textabstractA Q-derived polynomial is a univariate polynomial, defined over the rationals, with the property that its zeros, and those of all its derivatives are rational numbers. There is a conjecture that says that Q-derived polynomials of degree 4 with distinct roots for themselves and all their
Li, Weijie; Ho, Siu Chun Michael; Song, Gangbing
2016-04-01
Steel reinforcement corrosion is one of the dominant causes for structural deterioration for reinforced concrete structures. This paper presents a novel corrosion detection technique using an active thermal probe. The technique takes advantage of the fact that corrosion products have poor thermal conductivity, which will impede heat propagation generated from the active thermal probe. At the same time, the active thermal probe records the temperature response. The presence of corrosion products can thus be detected by analyzing the temperature response after the injection of heat at the reinforcement-concrete interface. The feasibility of the proposed technique was firstly analyzed through analytical modeling and finite element simulation. The active thermal probe consisted of carbon fiber strands to generate heat and a fiber optic Bragg grating (FBG) temperature sensor. Carbon fiber strands are used due to their corrosion resistance. Wet-dry cycle accelerated corrosion experiments were performed to study the effect of corrosion products on the temperature response of the reinforced concrete sample. Results suggest a high correlation between corrosion severity and magnitude of the temperature response. The technique has the merits of high accuracy, high efficiency in measurement and excellent embeddability.
Complex Polynomial Vector Fields
DEFF Research Database (Denmark)
Dias, Kealey
The two branches of dynamical systems, continuous and discrete, correspond to the study of differential equations (vector fields) and iteration of mappings respectively. In holomorphic dynamics, the systems studied are restricted to those described by holomorphic (complex analytic) functions...... 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....
A Characterization of Polynomials
DEFF Research Database (Denmark)
Andersen, Kurt Munk
1996-01-01
Given the problem:which functions f(x) are characterized by a relation of the form:f[x1,x2,...,xn]=h(x1+x2+...+xn), where n>1 and h(x) is a given function? Here f[x1,x2,...,xn] denotes the divided difference on n points x1,x2,...,xn of the function f(x).The answer is: f(x) is a polynomial of degree...
Some discrete multiple orthogonal polynomials
Arvesú, J.; Coussement, J.; van Assche, W.
2003-04-01
In this paper, we extend the theory of discrete orthogonal polynomials (on a linear lattice) to polynomials satisfying orthogonality conditions with respect to r positive discrete measures. First we recall the known results of the classical orthogonal polynomials of Charlier, Meixner, Kravchuk and Hahn (T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978; R. Koekoek and R.F. Swarttouw, Reports of the Faculty of Technical Mathematics and Informatics No. 98-17, Delft, 1998; A.F. Nikiforov et al., Classical Orthogonal Polynomials of a Discrete Variable, Springer, Berlin, 1991). These polynomials have a lowering and raising operator, which give rise to a Rodrigues formula, a second order difference equation, and an explicit expression from which the coefficients of the three-term recurrence relation can be obtained. Then we consider r positive discrete measures and define two types of multiple orthogonal polynomials. The continuous case (Jacobi, Laguerre, Hermite, etc.) was studied by Van Assche and Coussement (J. Comput. Appl. Math. 127 (2001) 317-347) and Aptekarev et al. (Multiple orthogonal polynomials for classical weights, manuscript). The families of multiple orthogonal polynomials (of type II) that we will study have a raising operator and hence a Rodrigues formula. This will give us an explicit formula for the polynomials. Finally, there also exists a recurrence relation of order r+1 for these multiple orthogonal polynomials of type II. We compute the coefficients of the recurrence relation explicitly when r=2.
Oblivious Polynomial Evaluation
Institute of Scientific and Technical Information of China (English)
Hong-Da Li; Dong-Yao Ji; Deng-Guo Feng; Bao Li
2004-01-01
The problem of two-party oblivious polynomial evaluation(OPE)is studied,where one party(Alice)has a polynomial P(x)and the other party(Bob)with an input x wants to learn P(x)in such an oblivious way that Bob obtains P(x)without learning any additional information about P except what is implied by P(x)and Alice does not know Bob's input x.The former OPE protocols are based on an intractability assumption except for OT protocols.In fact,evaluating P(x)is equivalent to computing the product of the coefficient vectors(a0,...,an)and(1,...,xn).Using this idea,an efficient scale product protocol of two vectors is proposed first and then two OPE protocols are presented which do not need any other cryptographic assumption except for OT protocol.Compared with the existing OPE protocol,another characteristic of the proposed protocols is the degree of the polynomial is private.Another OPE protocol works in case of existence of untrusted third party.
Polynomial Learning of Distribution Families
Belkin, Mikhail
2010-01-01
The question of polynomial learnability of probability distributions, particularly Gaussian mixture distributions, has recently received significant attention in theoretical computer science and machine learning. However, despite major progress, the general question of polynomial learnability of Gaussian mixture distributions still remained open. The current work resolves the question of polynomial learnability for Gaussian mixtures in high dimension with an arbitrary fixed number of components. The result on learning Gaussian mixtures relies on an analysis of distributions belonging to what we call "polynomial families" in low dimension. These families are characterized by their moments being polynomial in parameters and include almost all common probability distributions as well as their mixtures and products. Using tools from real algebraic geometry, we show that parameters of any distribution belonging to such a family can be learned in polynomial time and using a polynomial number of sample points. The r...
Symmetric functions and Hall polynomials
MacDonald, Ian Grant
1998-01-01
This reissued classic text is the acclaimed second edition of Professor Ian Macdonald's groundbreaking monograph on symmetric functions and Hall polynomials. The first edition was published in 1979, before being significantly expanded into the present edition in 1995. This text is widely regarded as the best source of information on Hall polynomials and what have come to be known as Macdonald polynomials, central to a number of key developments in mathematics and mathematical physics in the 21st century Macdonald polynomials gave rise to the subject of double affine Hecke algebras (or Cherednik algebras) important in representation theory. String theorists use Macdonald polynomials to attack the so-called AGT conjectures. Macdonald polynomials have been recently used to construct knot invariants. They are also a central tool for a theory of integrable stochastic models that have found a number of applications in probability, such as random matrices, directed polymers in random media, driven lattice gases, and...
Witt Rings and Permutation Polynomials
Institute of Scientific and Technical Information of China (English)
Qifan Zhang
2005-01-01
Let p be a prime number. In this paper, the author sets up a canonical correspondence between polynomial functions over Z/p2Z and 3-tuples of polynomial functions over Z/pZ. Based on this correspondence, he proves and reproves some fundamental results on permutation polynomials mod pl. The main new result is the characterization of strong orthogonal systems over Z/p1Z.
Deformed Mittag-Leffler Polynomials
Miomir S. Stankovic; Marinkovic, Sladjana D.; Rajkovic, Predrag M.
2010-01-01
The starting point of this paper are the Mittag-Leffler polynomials introduced by H. Bateman [1]. Based on generalized integer powers of real numbers and deformed exponential function, we introduce deformed Mittag-Leffler polynomials defined by appropriate generating function. We investigate their recurrence relations, differential properties and orthogonality. Since they have all zeros on imaginary axes, we also consider real polynomials with real zeros associated to them.
Polynomial Regression on Riemannian Manifolds
Hinkle, Jacob; Fletcher, P Thomas; Joshi, Sarang
2012-01-01
In this paper we develop the theory of parametric polynomial regression in Riemannian manifolds and Lie groups. We show application of Riemannian polynomial regression to shape analysis in Kendall shape space. Results are presented, showing the power of polynomial regression on the classic rat skull growth data of Bookstein as well as the analysis of the shape changes associated with aging of the corpus callosum from the OASIS Alzheimer's study.
Chaves, Rafael
2016-01-01
It is a recent realization that many of the concepts and tools of causal discovery in machine learning are highly relevant to problems in quantum information, in particular quantum nonlocality. The crucial ingredient in the connection between both fields is the mathematical theory of causality, allowing for the representation of arbitrary causal structures and providing a rigorous tool to reason about probabilistic causation. Indeed, Bell's theorem concerns a very particular kind of causal structure and Bell inequalities are a special case of linear constraints following from such models. It is thus natural to look for generalizations involving more complex Bell scenarios. The problem, however, relies on the fact that such generalized scenarios are characterized by polynomial Bell inequalities and no current method is available to derive them beyond very simple cases. In this work, we make a significant step in that direction, providing a new, general, and conceptually clear method for the derivation of polynomial Bell inequalities in a wide class of scenarios. We also show how our construction can be used to allow for relaxations of causal constraints and naturally gives rise to a notion of nonsignaling in generalized Bell networks.
Derivations and identities for Kravchuk polynomials
Bedratyuk, Leonid
2012-01-01
We introduce the notion of Kravchuk derivations of the polynomial algebra. We prove that any element of the kernel of the derivation gives a polynomial identity satisfied by the Kravchuk polynomials. Also, we prove that any kernel element of the basic Weitzenb\\"ok derivations yields a polynomial identity satisfied by the Kravchuk polynomials. We describe the corresponding intertwining maps.
Parallel Construction of Irreducible Polynomials
DEFF Research Database (Denmark)
Frandsen, Gudmund Skovbjerg
Let arithmetic pseudo-NC^k denote the problems that can be solved by log space uniform arithmetic circuits over the finite prime field GF(p) of depth O(log^k (n + p)) and size polynomial in (n + p). We show that the problem of constructing an irreducible polynomial of specified degree over GF(p) ...
Polynomial Regressions and Nonsense Inference
DEFF Research Database (Denmark)
Ventosa-Santaulària, Daniel; Rodríguez-Caballero, Carlos Vladimir
behavior. We extend Phillips’ (1986) results by proving an inference drawn from polynomial specifications, under stochastic nonstationarity, is misleading unless the variables cointegrate. We use a generalized polynomial specification as a vehicle to study its asymptotic and finite-sample properties. Our...
Complex Polynomial Vector Fields
DEFF Research Database (Denmark)
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...
Befriending Askey-Wilson polynomials
Szabłowski, Paweł J
2011-01-01
Although our main interest is with the Askey-Wilson (AW) polynomials we recall and review four other families of the so-called Askey-Wilson scheme of polynomials. We do this for completeness as well as for better exposition of AW properties. Our main results concentrate on the complex parameters case, revealing new fascinating symmetries between the variables and some of the parameters. In particular we express Askey-Wilson polynomials as linear combinations of Al-Salam--Chihara (ASC) polynomials which together with the obtained earlier expansion of the Askey-Wilson density forms complete generalization of the situation met in the case of Al-Salam--Chihara and q-Hermite polynomials and the Poisson-Mehler expansion formula. As a by-product we get useful identities involving ASC polynomials. Finally by certain re-scaling of variables and parameters we arrive to AW polynomials and AW densities that have clear probabilistic interpretation. We recall some known and present some believed to be unknown identities an...
Martínez Manuel, Rodolfo; Kaboko, J. J. M.; Shlyagin, M. G.
2016-02-01
We propose and demonstrate a simple and robust actively Q-switched erbium-doped fiber ring cavity laser. The Q-switching is based on dynamic spectral overlapping of two filters, namely a fiber Bragg grating-based filter and a fiber Fabry-Perot tunable filter. Using 3.5 m of erbium-doped fiber and a pump power of only 60 mW, Q-switched pulses with a peak power of 9.7 W and a pulse duration of 500 ns were obtained. A pulse repetition rate can be continuously varied from a single shot to a few KHz.
Orthogonal polynomials and deformed oscillators
Borzov, V. V.; Damaskinsky, E. V.
2015-10-01
In the example of the Fibonacci oscillator, we discuss the construction of oscillator-like systems associated with orthogonal polynomials. We also consider the question of the dimensions of the corresponding Lie algebras.
An introduction to orthogonal polynomials
Chihara, Theodore S
2011-01-01
Assuming no further prerequisites than a first undergraduate course in real analysis, this concise introduction covers general elementary theory related to orthogonal polynomials. It includes necessary background material of the type not usually found in the standard mathematics curriculum. Suitable for advanced undergraduate and graduate courses, it is also appropriate for independent study. Topics include the representation theorem and distribution functions, continued fractions and chain sequences, the recurrence formula and properties of orthogonal polynomials, special functions, and some
Constructing general partial differential equations using polynomial and neural networks.
Zjavka, Ladislav; Pedrycz, Witold
2016-01-01
Sum fraction terms can approximate multi-variable functions on the basis of discrete observations, replacing a partial differential equation definition with polynomial elementary data relation descriptions. Artificial neural networks commonly transform the weighted sum of inputs to describe overall similarity relationships of trained and new testing input patterns. Differential polynomial neural networks form a new class of neural networks, which construct and solve an unknown general partial differential equation of a function of interest with selected substitution relative terms using non-linear multi-variable composite polynomials. The layers of the network generate simple and composite relative substitution terms whose convergent series combinations can describe partial dependent derivative changes of the input variables. This regression is based on trained generalized partial derivative data relations, decomposed into a multi-layer polynomial network structure. The sigmoidal function, commonly used as a nonlinear activation of artificial neurons, may transform some polynomial items together with the parameters with the aim to improve the polynomial derivative term series ability to approximate complicated periodic functions, as simple low order polynomials are not able to fully make up for the complete cycles. The similarity analysis facilitates substitutions for differential equations or can form dimensional units from data samples to describe real-world problems.
Evaluations of topological Tutte polynomials
Ellis-Monaghan, Joanna A
2011-01-01
We find a number of new combinatorial identities for, and interpretations of evaluations of, the topological Tutte polynomials of Las Vergnas, $L(G)$, and of and Bollob\\'as and Riordan, $R(G)$, as well as for the classical Tutte polynomial $T(G)$. For example, we express $R(G)$ and $T(G)$ as a sum of chromatic polynomials, show that $R(G)$ counts non-crossing graph states and $k$-valuations, and reformulate the Four Colour Theorem in terms of $R(G)$. Our main approach is to apply identities for the topological transition polynomial, one involving twisted duals, and one involving doubling the edges of a graph. These identities for the transition polynomial allow us to show that the Penrose polynomial $P(G)$ can be recovered from $R(G)$, a fact that we use to obtain identities and interpretations for $R(G)$. We also consider enumeration of circuits in medial graphs and use this to relate $R(G)$ and $L(G)$ for graphs embedded in low genus surfaces.
Bannai-Ito polynomials and dressing chains
Derevyagin, Maxim; Tsujimoto, Satoshi; Vinet, Luc; Zhedanov, Alexei
2012-01-01
Schur-Delsarte-Genin (SDG) maps and Bannai-Ito polynomials are studied. SDG maps are related to dressing chains determined by quadratic algebras. The Bannai-Ito polynomials and their kernel polynomials -- the complementary Bannai-Ito polynomials -- are shown to arise in the framework of the SDG maps.
Uniqueness and Zeros of -Shift Difference Polynomials
Indian Academy of Sciences (India)
Kai Liu; Xin-Ling Liu; Ting-Bin Cao
2011-08-01
In this paper, we consider the zero distributions of -shift difference polynomials of meromorphic functions with zero order, and obtain two theorems that extend the classical Hayman results on the zeros of differential polynomials to -shift difference polynomials. We also investigate the uniqueness problem of -shift difference polynomials that share a common value.
Birth-death processes and associated polynomials
Doorn, van Erik A.
2003-01-01
We consider birth-death processes on the nonnegative integers and the corresponding sequences of orthogonal polynomials called birth-death polynomials. The sequence of associated polynomials linked with a sequence of birth-death polynomials and its orthogonalizing measure can be used in the analysis
On the Irreducibility of Some Composite Polynomials
Directory of Open Access Journals (Sweden)
M. Alizadeh
2012-06-01
Full Text Available . In this paper we study the irreducibility of some composite polynomials, constructed by a polynomial composition method over finite fields. Finally, a recurrent method for constructing families of irreducible polynomials of higher degree from given irreducible polynomials over finite fields is given
Fabrication update on critical-angle transmission gratings for soft x-ray grating spectrometers
Heilmann, Ralf K.; Bruccoleri, Alex; Mukherjee, Pran; Yam, Jonathan; Schattenburg, Mark L.
2011-09-01
Diffraction grating-based, wavelength dispersive high-resolution soft x-ray spectroscopy of celestial sources promises to reveal crucial data for the study of the Warm-Hot Intergalactic Medium, the Interstellar Medium, warm absorption and outflows in Active Galactic Nuclei, coronal emission from stars, and other areas of interest to the astrophysics community. Our recently developed critical-angle transmission (CAT) gratings combine the advantages of the Chandra high and medium energy transmission gratings (low mass, high tolerance of misalignments and figure errors, polarization insensitivity) with those of blazed reflection gratings (high broad band diffraction efficiency, high resolution through use of higher diffraction orders) such as the ones on XMM-Newton. Extensive instrument and system configuration studies have shown that a CAT grating-based spectrometer is an outstanding instrument capable of delivering resolving power on the order of 5,000 and high effective area, even with a telescope point-spread function on the order of many arc-seconds. We have fabricated freestanding, ultra-high aspect-ratio CAT grating bars from silicon-on-insulator wafers using both wet and dry etch processes. The 200 nm-period grating bars are supported by an integrated Level 1 support mesh, and a coarser external Level 2 support mesh. The resulting grating membrane is mounted to a frame, resulting in a grating facet. Many such facets comprise a grating array that provides light-weight coverage of large-area telescope apertures. Here we present fabrication results on the integration of CAT gratings and the different high-throughput support mesh levels and on membrane-frame bonding. We also summarize recent x-ray data analysis of 3 and 6 micron deep wet-etched CAT grating prototypes.
Iwata, Fujio
2001-06-01
Toppan Printing Co., Ltd. originated the name of 'grating image'. It means an image that consists of diffraction grating dots that look similar to the halftone dots of conventional printing. We proposed this new display method using simple gratings in order to enhance the visual effects when illumination is made by a fluorescent lamp. We considered the use of simple gratings as elemental dots, and used a number of elemental dots to display a 2D image. This method produces an effect something like the halftone dots of printing. The grating image technology grows from its starting to become able to produce 3D images and a 3D-video system using an electron beam grating-writing system.
Iwata, Fujio
1995-07-01
The word 'grating image' was first named by Toppan Printing Company, Ltd. It means that an image consists of grating dots. In 1988, we presented this new technology at the Optical Security Systems Symposium, in Switzerland. Then it was improved and applied in display application. Recently, it was further applied in 3D video systems. In this report, the development history and the recent situations of grating image technology are described.
Diffraction by m-bonacci gratings
Monsoriu, Juan A.; Giménez, Marcos H.; Furlan, Walter D.; Barreiro, Juan C.; Saavedra, Genaro
2015-11-01
We present a simple diffraction experiment with m-bonacci gratings as a new interesting generalization of the Fibonacci ones. Diffraction by these non-conventional structures is proposed as a motivational strategy to introduce students to basic research activities. The Fraunhofer diffraction patterns are obtained with the standard equipment present in most undergraduate physics labs and are compared with those obtained with regular periodic gratings. We show that m-bonacci gratings produce discrete Fraunhofer patterns characterized by a set of diffraction peaks which positions are related to the concept of a generalized golden mean. A very good agreement is obtained between experimental and numerical results and the students’ feedback is discussed.
Orthogonal Polynomials and Special Functions
Assche, Walter
2003-01-01
The set of lectures from the Summer School held in Leuven in 2002 provide an up-to-date account of recent developments in orthogonal polynomials and special functions, in particular for algorithms for computer algebra packages, 3nj-symbols in representation theory of Lie groups, enumeration, multivariable special functions and Dunkl operators, asymptotics via the Riemann-Hilbert method, exponential asymptotics and the Stokes phenomenon. The volume aims at graduate students and post-docs working in the field of orthogonal polynomials and special functions, and in related fields interacting with orthogonal polynomials, such as combinatorics, computer algebra, asymptotics, representation theory, harmonic analysis, differential equations, physics. The lectures are self-contained requiring only a basic knowledge of analysis and algebra, and each includes many exercises.
Plain Polynomial Arithmetic on GPU
Anisul Haque, Sardar; Moreno Maza, Marc
2012-10-01
As for serial code on CPUs, parallel code on GPUs for dense polynomial arithmetic relies on a combination of asymptotically fast and plain algorithms. Those are employed for data of large and small size, respectively. Parallelizing both types of algorithms is required in order to achieve peak performances. In this paper, we show that the plain dense polynomial multiplication can be efficiently parallelized on GPUs. Remarkably, it outperforms (highly optimized) FFT-based multiplication up to degree 212 while on CPU the same threshold is usually at 26. We also report on a GPU implementation of the Euclidean Algorithm which is both work-efficient and runs in linear time for input polynomials up to degree 218 thus showing the performance of the GCD algorithm based on systolic arrays.
Orthogonal Polynomials and their Applications
Dehesa, Jesús; Marcellan, Francisco; Francia, José; Vinuesa, Jaime
1988-01-01
The Segovia meeting set out to stimulate an intensive exchange of ideas between experts in the area of orthogonal polynomials and its applications, to present recent research results and to reinforce the scientific and human relations among the increasingly international community working in orthogonal polynomials. This volume contains original research papers as well as survey papers about fundamental questions in the field (Nevai, Rakhmanov & López) and its relationship with other fields such as group theory (Koornwinder), Padé approximation (Brezinski), differential equations (Krall, Littlejohn) and numerical methods (Rivlin).
On Polynomial Solutions of Linear Differential Equations with Polynomial Coefficients
Si, Do Tan
1977-01-01
Demonstrates a method for solving linear differential equations with polynomial coefficients based on the fact that the operators z and D + d/dz are known to be Hermitian conjugates with respect to the Bargman and Louck-Galbraith scalar products. (MLH)
Symbolic computation of Appell polynomials using Maple
Directory of Open Access Journals (Sweden)
H. Alkahby
2001-07-01
Full Text Available This work focuses on the symbolic computation of Appell polynomials using the computer algebra system Maple. After describing the traditional approach of constructing Appell polynomials, the paper examines the operator method of constructing the same Appell polynomials. The operator approach enables us to express the Appell polynomial as Bessel function whose coefficients are Euler and Bernuolli numbers. We have also constructed algorithms using Maple to compute Appell polynomials based on the methods we have described. The achievement is the construction of Appell polynomials for any function of bounded variation.
Cheng, L.K.; Nieuwland, R.A.; Toet, P.M.; Agovic, K.
2010-01-01
A brief overview of fiber Bragg grating based sensor technology from sensor head, read out unit and commercial applications is given. Fiber Bragg grating based sensor systems are becoming mature rapidly. Components for commercial pressure sensors and temperature sensors are available and slowly gett
DEFF Research Database (Denmark)
Zhang, C.; Webb, D.J.; Kalli, K.;
We report for the first time fibre Bragg grating inscription in microstructured optical fibre fabricated from Topas® cyclic olefin copolymer. The temperature sensitivity of the grating was studied revealing a positive Bragg wavelength shift of approximately 0.8 nmK-1,the largest sensitivity yet...
Bidirectional grating compressors
Wang, Cheng; Li, Zhaoyang; Li, Shuai; Liu, Yanqi; Leng, Yuxin; Li, Ruxin
2016-07-01
A bidirectional grating compressor for chirped pulse amplifiers is presented. It compresses a laser beam simultaneously in two opposite directions. The pulse compressor is shown to promote chirped pulse amplifiers' output energy without grating damages. To verify the practicability, an experiment is carried out. In addition, a crosscorrelation instrument is designed and set up to test the time synchronization between these two femtosecond pulses.
Bak, Dongsu; Kim, Sang Pyo; Kim, Sung Ku; Soh, Kwang-Sup; Yee, Jae Hyung
1998-01-01
We consider an optical diffraction grating in which the spatial distribution of open slits forms a fractal set. The Fraunhofer diffraction patterns through the fractal grating are obtained analytically for the simplest triad Cantor type and its generalized version. The resulting interference patterns exhibit characteristics of the original fractals and their scaling properties.
On Modular Counting with Polynomials
DEFF Research Database (Denmark)
Hansen, Kristoffer Arnsfelt
2006-01-01
For any integers m and l, where m has r sufficiently large (depending on l) factors, that are powers of r distinct primes, we give a construction of a (symmetric) polynomial over Z_m of degree O(\\sqrt n) that is a generalized representation (commonly also called weak representation) of the MODl...
Two polynomial division inequalities in
Directory of Open Access Journals (Sweden)
Goetgheluck P
1998-01-01
Full Text Available This paper is a first attempt to give numerical values for constants and , in classical estimates and where is an algebraic polynomial of degree at most and denotes the -metric on . The basic tools are Markov and Bernstein inequalities.
Global Polynomial Kernel Hazard Estimation
DEFF Research Database (Denmark)
Hiabu, Munir; Miranda, Maria Dolores Martínez; Nielsen, Jens Perch;
2015-01-01
This paper introduces a new bias reducing method for kernel hazard estimation. The method is called global polynomial adjustment (GPA). It is a global correction which is applicable to any kernel hazard estimator. The estimator works well from a theoretical point of view as it asymptotically redu...
Sheffer and Non-Sheffer Polynomial Families
Directory of Open Access Journals (Sweden)
G. Dattoli
2012-01-01
Full Text Available By using the integral transform method, we introduce some non-Sheffer polynomial sets. Furthermore, we show how to compute the connection coefficients for particular expressions of Appell polynomials.
Boundary integral equation Neumann-to-Dirichlet map method for gratings in conical diffraction.
Wu, Yumao; Lu, Ya Yan
2011-06-01
Boundary integral equation methods for diffraction gratings are particularly suitable for gratings with complicated material interfaces but are difficult to implement due to the quasi-periodic Green's function and the singular integrals at the corners. In this paper, the boundary integral equation Neumann-to-Dirichlet map method for in-plane diffraction problems of gratings [Y. Wu and Y. Y. Lu, J. Opt. Soc. Am. A26, 2444 (2009)] is extended to conical diffraction problems. The method uses boundary integral equations to calculate the so-called Neumann-to-Dirichlet maps for homogeneous subdomains of the grating, so that the quasi-periodic Green's functions can be avoided. Since wave field components are coupled on material interfaces with the involvement of tangential derivatives, a least squares polynomial approximation technique is developed to evaluate tangential derivatives along these interfaces for conical diffraction problems. Numerical examples indicate that the method performs equally well for dielectric or metallic gratings. PMID:21643404
Exceptional Polynomials and SUSY Quantum Mechanics
Chaitanya, K V S Shiv; Panigrahi, Prasanta K; Radhakrishnan, R; Srinivasan, V
2011-01-01
We show that the existence of exceptional polynomials leads to the presence of non-trivial supersymmetry. The existence of these polynomials leads to two distinct potentials in the Schr\\"odinger equation having the same eigenvalues. We describe two ways of finding new exactly solvable (ES) potentials. All Schr\\"odinger equations having the Laguerre and the Jacobi polynomials as solutions, have non-trivial supersymmetric partners with corresponding exceptional polynomials as solutions.
Application of Chebyshev Polynomial to simulated modeling
Institute of Scientific and Technical Information of China (English)
CHI Hai-hong; LI Dian-pu
2006-01-01
Chebyshev polynomial is widely used in many fields, and used usually as function approximation in numerical calculation. In this paper, Chebyshev polynomial expression of the propeller properties across four quadrants is given at first, then the expression of Chebyshev polynomial is transformed to ordinary polynomial for the need of simulation of propeller dynamics. On the basis of it,the dynamical models of propeller across four quadrants are given. The simulation results show the efficiency of mathematical model.
An Improved Volumetric Estimation Using Polynomial Regression
Noraini Abdullah; Amran Ahmed; Zainodin Hj. Jubok
2011-01-01
The polynomial regression (PR) technique is used to estimate the parameters of the dependent variable having a polynomial relationship with the independent variable. Normality and nonlinearity exhibit polynomial characterization of power terms greater than 2. Polynomial Regression models (PRM) with the auxiliary variables are considered up to their third order interactions. Preliminary, multicollinearity between the independent variables is minimized and statistical tests involving the Global...
Positive trigonometric polynomials and signal processing applications
Dumitrescu, Bogdan
2007-01-01
Presents the results on positive trigonometric polynomials within a unitary framework; the theoretical results obtained partly from the general theory of real polynomials, partly from self-sustained developments. This book provides information on the theory of sum-of-squares trigonometric polynomials in two parts: theory and applications.
Lattice Platonic Solids and their Ehrhart polynomial
Directory of Open Access Journals (Sweden)
E. J. Ionascu
2013-01-01
Full Text Available First, we calculate the Ehrhart polynomial associated to an arbitrary cube with integer coordinates for its vertices. Then, we use this result to derive relationships between the Ehrhart polynomials for regular lattice tetrahedra and those for regular lattice octahedra. These relations allow one to reduce the calculation of these polynomials to only one coefficient.
General Eulerian Numbers and Eulerian Polynomials
Directory of Open Access Journals (Sweden)
Tingyao Xiong
2013-01-01
Full Text Available We will generalize the definitions of Eulerian numbers and Eulerian polynomials to general arithmetic progressions. Under the new definitions, we have been successful in extending several well-known properties of traditional Eulerian numbers and polynomials to the general Eulerian polynomials and numbers.
s-Numbers sequences for homogeneous polynomials
Caliskan, Erhan; Rueda, Pilar
2015-01-01
We extend the well known theory of $s$-numbers of linear operators to homogeneous polynomials defined between Banach spaces. Approximation, Kolmogorov and Gelfand numbers of polynomials are introduced and some well-known results of the linear and multilinear settings are obtained for homogeneous polynomials.
Lattice Platonic Solids and their Ehrhart polynomial
Ionascu, Eugen J
2011-01-01
First, we calculate the Ehrhart polynomial associated to an arbitrary cube with integer coordinates for its vertices. Then, we use this result to derive relationships between the Ehrhart polynomials for regular lattice tetrahedrons and those for regular lattice octahedrons. These relations allow one to reduce the calculation of these polynomials to only one coefficient.
Normal BGG solutions and polynomials
Cap, A; Hammerl, M
2012-01-01
First BGG operators are a large class of overdetermined linear differential operators intrinsically associated to a parabolic geometry on a manifold. The corresponding equations include those controlling infinitesimal automorphisms, higher symmetries, and many other widely studied PDE of geometric origin. The machinery of BGG sequences also singles out a subclass of solutions called normal solutions. These correspond to parallel tractor fields and hence to (certain) holonomy reductions of the canonical normal Cartan connection. Using the normal Cartan connection, we define a special class of local frames for any natural vector bundle associated to a parabolic geometry. We then prove that the coefficient functions of any normal solution of a first BGG operator with respect to such a frame are polynomials in the normal coordinates of the parabolic geometry. A bound on the degree of these polynomials in terms of representation theory data is derived. For geometries locally isomorphic to the homogeneous model of ...
BSDEs with polynomial growth generators
Directory of Open Access Journals (Sweden)
Philippe Briand
2000-01-01
Full Text Available In this paper, we give existence and uniqueness results for backward stochastic differential equations when the generator has a polynomial growth in the state variable. We deal with the case of a fixed terminal time, as well as the case of random terminal time. The need for this type of extension of the classical existence and uniqueness results comes from the desire to provide a probabilistic representation of the solutions of semilinear partial differential equations in the spirit of a nonlinear Feynman-Kac formula. Indeed, in many applications of interest, the nonlinearity is polynomial, e.g, the Allen-Cahn equation or the standard nonlinear heat and Schrödinger equations.
Leont'ev, V. K.
2015-11-01
A pseudo-Boolean function is an arbitrary mapping of the set of binary n-tuples to the real line. Such functions are a natural generalization of classical Boolean functions and find numerous applications in various applied studies. Specifically, the Fourier transform of a Boolean function is a pseudo-Boolean function. A number of facts associated with pseudo-Boolean polynomials are presented, and their applications to well-known discrete optimization problems are described.
Stable piecewise polynomial vector fields
Directory of Open Access Journals (Sweden)
Claudio Pessoa
2012-09-01
Full Text Available Let $N={y>0}$ and $S={y<0}$ be the semi-planes of $mathbb{R}^2$ having as common boundary the line $D={y=0}$. Let $X$ and $Y$ be polynomial vector fields defined in $N$ and $S$, respectively, leading to a discontinuous piecewise polynomial vector field $Z=(X,Y$. This work pursues the stability and the transition analysis of solutions of $Z$ between $N$ and $S$, started by Filippov (1988 and Kozlova (1984 and reformulated by Sotomayor-Teixeira (1995 in terms of the regularization method. This method consists in analyzing a one parameter family of continuous vector fields $Z_{epsilon}$, defined by averaging $X$ and $Y$. This family approaches $Z$ when the parameter goes to zero. The results of Sotomayor-Teixeira and Sotomayor-Machado (2002 providing conditions on $(X,Y$ for the regularized vector fields to be structurally stable on planar compact connected regions are extended to discontinuous piecewise polynomial vector fields on $mathbb{R}^2$. Pertinent genericity results for vector fields satisfying the above stability conditions are also extended to the present case. A procedure for the study of discontinuous piecewise vector fields at infinity through a compactification is proposed here.
Pattern Matching under Polynomial Transformation
Butman, Ayelet; Clifford, Raphael; Jalsenius, Markus; Lewenstein, Noa; Porat, Benny; Porat, Ely; Sach, Benjamin
2011-01-01
We consider a class of pattern matching problems where a polynomial transformation can be applied to the pattern at every alignment. Given a pattern of length m and a longer text of length n where both are assumed to contain integer values only, we show O(n log m) algorithms for pattern matching under linear transformations even when wildcard symbols can occur in the input. We then show how to extend the technique to polynomial transformations of arbitrary degree. Next we consider the problem of finding the minimum Hamming distance under polynomial transformation. We show that, for any epsilon > 0, there cannot exist an O(nm^(1-epsilon)) algorithm for additive and linear transformations conditional on the hardness of the classic 3SUM problem. Finally, we consider a version of the Hamming distance problem under additive transformations with a bound k on the maximum distance that need be reported. We give a deterministic O(nk log k) time solution which we then improve by careful use of randomisation to O(n sqrt...
On Ternary Inclusion-Exclusion Polynomials
Bachman, Gennady
2010-01-01
Taking a combinatorial point of view on cyclotomic polynomials leads to a larger class of polynomials we shall call the inclusion-exclusion polynomials. This gives a more appropriate setting for certain types of questions about the coefficients of these polynomials. After establishing some basic properties of inclusion-exclusion polynomials we turn to a detailed study of the structure of ternary inclusion-exclusion polynomials. The latter subclass is exemplified by cyclotomic polynomials $\\Phi_{pqr}$, where $p
International Nuclear Information System (INIS)
In this work, active vibration suppression of a smart cantilever beam subjected to disturbances from multiple impact loadings is investigated with a point-wise fiber Bragg grating (FBG) displacement sensing system. An FBG demodulator is employed in the proposed fiber sensing system to dynamically demodulate the responses obtained by the FBG displacement sensor with high sensitivity. To investigate the ability of the proposed FBG displacement sensor as a feedback sensor, velocity feedback control and delay control are employed to suppress the vibrations of the first three bending modes of the smart cantilever beam. To improve the control performance for the first bending mode when the cantilever beam is subjected to an impact loading, we improve the conventional velocity feedback controller by tuning the control gain online with the aid of information from a higher vibration mode. Finally, active control of vibrations induced by multiple impact loadings due to a plastic ball is performed with the improved velocity feedback control. The experimental results show that active vibration control of smart structures subjected to disturbances such as impact loadings can be achieved by employing the proposed FBG sensing system to feed back out-of-plane point-wise displacement responses with high sensitivity. (paper)
Modeling Component-based Bragg gratings Application: tunable lasers
Directory of Open Access Journals (Sweden)
Hedara Rachida
2011-09-01
Full Text Available The principal function of a grating Bragg is filtering, which can be used in optical fibers based component and active or passive semi conductors based component, as well as telecommunication systems. Their ideal use is with lasers with fiber, amplifiers with fiber or Laser diodes. In this work, we are going to show the principal results obtained during the analysis of various types of grating Bragg by the method of the coupled modes. We then present the operation of DBR are tunable. The use of Bragg gratings in a laser provides single-mode sources, agile wavelength. The use of sampled grating increases the tuning range.
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
In this paper, we mainly study the relation of two cyclically reduced words w and w' on the condition they have the same trace polynomial (i.e., tr w= tr w' ). By defining an equivalence relation through such operators on words as inverse, cyclically left shift, and mirror, it is straightforward to get that w ～ w' implies tr w = tr w'. We show by a counter example that tr w = tr w' does not imply w ～ w'. And in two special cases, we prove that tr w = tr w' if and only if w ～ w'.
Zeroes of random Reinhardt polynomials
Karami, Arash
2012-01-01
For a Reinhardt domain $\\Omega$ with the smooth boundary in $\\mathbb{C}^{m+1}$ and a positive smooth measure $\\mu$ on the boundary of $\\Omega$, we consider the ensemble $P_{N}$ of polynomials of degree $N$ with the Gaussian probability measure $\\gamma_{N}$ which is induced by $L^{2}(\\partial\\Omega,d\\mu)$. Our aim is to compute scaling limit distribution function and scaling limit pair correlation function between zeros when $z\\in\\partial\\Omega$. First of all we apply stationary phase method t...
Chromatic Polynomials of Mixed Hypercycles
Directory of Open Access Journals (Sweden)
Allagan Julian A.
2014-08-01
Full Text Available We color the vertices of each of the edges of a C-hypergraph (or cohypergraph in such a way that at least two vertices receive the same color and in every proper coloring of a B-hypergraph (or bihypergraph, we forbid the cases when the vertices of any of its edges are colored with the same color (monochromatic or when they are all colored with distinct colors (rainbow. In this paper, we determined explicit formulae for the chromatic polynomials of C-hypercycles and B-hypercycles
An Improved Volumetric Estimation Using Polynomial Regression
Directory of Open Access Journals (Sweden)
Noraini Abdullah
2011-12-01
Full Text Available The polynomial regression (PR technique is used to estimate the parameters of the dependent variable having a polynomial relationship with the independent variable. Normality and nonlinearity exhibit polynomial characterization of power terms greater than 2. Polynomial Regression models (PRM with the auxiliary variables are considered up to their third order interactions. Preliminary, multicollinearity between the independent variables is minimized and statistical tests involving the Global, Correlation Coefficient, Wald, and Goodness-of-Fit tests, are carried out to select significant variables with their possible interactions. Comparisons between the polynomial regression models (PRM are made using the eight selection criteria (8SC. The best regression model is identified based on the minimum value of the eight selection criteria (8SC. The use of an appropriate transformation will increase in the degree of a statistically valid polynomial, hence, providing a better estimation for the model.
Control to Facet for Polynomial Systems
DEFF Research Database (Denmark)
Sloth, Christoffer; Wisniewski, Rafael
2014-01-01
This paper presents a solution to the control to facet problem for arbitrary polynomial vector fields defined on simplices. The novelty of the work is to use Bernstein coefficients of polynomials for determining certificates of positivity. Specifically, the constraints that are set up for the con......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....
Exceptional polynomials and SUSY quantum mechanics
Indian Academy of Sciences (India)
K V S Shiv Chaitanya; S Sree Ranjani; Prasanta K Panigrahi; R Radhakrishnan; V Srinivasan
2015-07-01
We show that for the quantum mechanical problem which admit classical Laguerre/Jacobi polynomials as solutions for the Schrödinger equations (SE), will also admit exceptional Laguerre/Jacobi polynomials as solutions having the same eigenvalues but with the ground state missing after a modification of the potential. Then, we claim that the existence of these exceptional polynomials leads to the presence of non-trivial supersymmetry.
A new Arnoldi approach for polynomial eigenproblems
Energy Technology Data Exchange (ETDEWEB)
Raeven, F.A.
1996-12-31
In this paper we introduce a new generalization of the method of Arnoldi for matrix polynomials. The new approach is compared with the approach of rewriting the polynomial problem into a linear eigenproblem and applying the standard method of Arnoldi to the linearised problem. The algorithm that can be applied directly to the polynomial eigenproblem turns out to be more efficient, both in storage and in computation.
Landau and Kolmogoroff type polynomial inequalities
Alves Claudia RR; Dimitrov Dimitar K
1999-01-01
Let be integers. Denote by the norm . For various positive values of and we establish Kolmogoroff type inequalities with certain constants , which hold for every ( denotes the space of real algebraic polynomials of degree not exceeding ). For the particular case and , we provide a complete characterisation of the positive constants and , for which the corresponding Landau type polynomial inequalities hold. In each case we determine the corresponding extremal polynomials for which e...
Quantum Schubert polynomials and quantum Schur functions
Kirillov, Anatol N.
1997-01-01
We introduce the quantum multi-Schur functions, quantum factorial Schur functions and quantum Macdonald polynomials. We prove that for restricted vexillary permutations the quantum double Schubert polynomial coincides with some quantum multi-Schur function and prove a quantum analog of the Nagelsbach-Kostka and Jacobi-Trudi formulae for the quantum double Schubert polynomials in the case of Grassmannian permutations. We prove, also, an analog of the Billey-Jockusch-Stanley formula for quantum...
Accelerated graph-based spectral polynomial filters
Knyazev, Andrew; Malyshev, Alexander,
2015-01-01
Graph-based spectral denoising is a low-pass filtering using the eigendecomposition of the graph Laplacian matrix of a noisy signal. Polynomial filtering avoids costly computation of the eigendecomposition by projections onto suitable Krylov subspaces. Polynomial filters can be based, e.g., on the bilateral and guided filters. We propose constructing accelerated polynomial filters by running flexible Krylov subspace based linear and eigenvalue solvers such as the Block Locally Optimal Precond...
Generalizations of Bernoulli numbers and polynomials
Qiu-Ming Luo; Bai-Ni Guo; Feng Qi; Lokenath Debnath
2003-01-01
The concepts of Bernoulli numbers Bn, Bernoulli polynomials Bn(x), and the generalized Bernoulli numbers Bn(a,b) are generalized to the one Bn(x;a,b,c) which is called the generalized Bernoulli polynomials depending on three positive real parameters. Numerous properties of these polynomials and some relationships between Bn, Bn(x), Bn(a,b), and Bn(x;a,b,c) are established.
Discrete least squares approximation with polynomial vectors
Van Barel, Marc; Bultheel, Adhemar
1993-01-01
We give a solution of a discrete least squares approximation problem in terms of orthogonal polynomial vectors. The degrees of the polynomial elements of these vectors can be different. An algorithm is constructed computing the coefficients of recurrence relations for the orthogonal polynomial vectors. In case the function values are prescribed in points on the real line or on the unit circle variants of the original algorithm can be designed which are an order of magnitude more efficient. Al...
Tutorial: Applications of Fibre Gratings
Institute of Scientific and Technical Information of China (English)
Hwayaw; Tam; Bai; ou; Guan; Shunyee; Liu
2003-01-01
Fibre grating is an important enabling technology that has found numerous applications in both telecommunications and sensor systems. This tutorial describes the basic characteristics of fibre gratings and gives examples of where they are being employed.
Polynomial chaotic inflation in supergravity revisited
Directory of Open Access Journals (Sweden)
Kazunori Nakayama
2014-10-01
Full Text Available We revisit a polynomial chaotic inflation model in supergravity which we proposed soon after the Planck first data release. Recently some issues have been raised in Ref. [12], concerning the validity of our polynomial chaotic inflation model. We study the inflaton dynamics in detail, and confirm that the inflaton potential is very well approximated by a polynomial potential for the parameters of our interest in any practical sense, and in particular, the spectral index and the tensor-to-scalar ratio can be estimated by single-field approximation. This justifies our analysis of the polynomial chaotic inflation in supergravity.
Asymptotics for a generalization of Hermite polynomials
Alfaro, M; Peña, A; Rezola, M L
2009-01-01
We consider a generalization of the classical Hermite polynomials by the addition of terms involving derivatives in the inner product. This type of generalization has been studied in the literature from the point of view of the algebraic properties. Thus, our aim is to study the asymptotics of this sequence of nonstandard orthogonal polynomials. In fact, we obtain Mehler--Heine type formulas for these polynomials and, as a consequence, we prove that there exists an acceleration of the convergence of the smallest positive zeros of these generalized Hermite polynomials towards the origin.
Relative risk regression models with inverse polynomials.
Ning, Yang; Woodward, Mark
2013-08-30
The proportional hazards model assumes that the log hazard ratio is a linear function of parameters. In the current paper, we model the log relative risk as an inverse polynomial, which is particularly suitable for modeling bounded and asymmetric functions. The parameters estimated by maximizing the partial likelihood are consistent and asymptotically normal. The advantages of the inverse polynomial model over the ordinary polynomial model and the fractional polynomial model for fitting various asymmetric log relative risk functions are shown by simulation. The utility of the method is further supported by analyzing two real data sets, addressing the specific question of the location of the minimum risk threshold.
Multi-indexed (q)-Racah Polynomials
Odake, Satoru
2012-01-01
As the second stage of the project $multi-indexed orthogonal polynomials$, we present, in the framework of `discrete quantum mechanics' with real shifts in one dimension, the multi-indexed (q)-Racah polynomials. They are obtained from the (q)-Racah polynomials by multiple application of the discrete analogue of the Darboux transformations or the Crum-Krein-Adler deletion of `virtual state' vectors of type I and II, in a similar way to the multi-indexed Laguerre and Jacobi polynomials reported earlier. The virtual state vectors are the `solutions' of the matrix Schr\\"odinger equation with negative `eigenvalues', except for one of the two boundary points.
On the verification of polynomial system solvers
Institute of Scientific and Technical Information of China (English)
Changbo CHEN; Marc MORENO MAZA; Wei PAN; Yuzhen XI
2008-01-01
We discuss the verification of mathematical software solving polynomial systems symbolically by way of triangular decomposition. Standard verification techniques are highly resource consuming and apply only to polynomial systems which are easy to solve. We exhibit a new approach which manipulates constructible sets represented by regular systems. We provide comparative benchmarks of different verification procedures applied to four solvers on a large set of well-known polynomial systems. Our experimental results illustrate the high effi-ciency of our new approach. In particular, we are able to verify triangular decompositions of polynomial systems which are not easy to solve.
Directory of Open Access Journals (Sweden)
Ryoo CS
2010-01-01
Full Text Available The purpose of this paper is to give some properties of several Bernstein type polynomials to represent the fermionic -adic integral on . From these properties, we derive some interesting identities on the Euler numbers and polynomials.
Genus expansion of HOMFLY polynomials
Mironov, A; Sleptsov, A
2013-01-01
In the planar limit of the 't Hooft expansion, the Wilson-loop average in 3d Chern-Simons theory (i.e. the HOMFLY polynomial) depends in a very simple way on representation (the Young diagram), so that the (knot-dependent) Ooguri-Vafa partition function becomes a trivial KP tau-function. We study higher genus corrections to this formula in the form of expansion in powers of z = q-q^{-1}. Expansion coefficients are expressed through the eigenvalues of the cut-and-join operators, i.e. symmetric group characters. Moreover, the z-expansion is naturally exponentiated. Representation through cut-and-join operators makes contact with Hurwitz theory and its sophisticated integrability properties. Our formulas describe the shape of genus expansion for the HOMFLY polynomials, which for their matrix model counterparts is usually controlled by Virasoro like constraints and AMM/EO topological recursion. The genus expansion differs from the better studied weak coupling expansion at finite number of colors N, which is descr...
Algorithms in Solving Polynomial Inequalities
Directory of Open Access Journals (Sweden)
Christopher M. Cordero
2015-11-01
Full Text Available A new method to solve the solution set of polynomial inequalities was conducted. When −1 −2 >0 ℎ 1,2∈ ℝ 10 if n is even. Then, the solution set is ∈ ℝ ∈ −∞,1 ∪ ,+∞ ∪ ,+1 : }. Thus, when −1−2…−≥0, the solution is ∈ ℝ ∈−∞, 1∪ ,+∞∪, +1: }. If is odd, then the solution set is ∈ ℝ ∈ ,+∞ ∪ ,+1 : }. Thus, when −1 −2…−≥0, the solution set is ∈ ℝ ∈ ,+∞∪, +1: }. Let −1−2…−<0 if n is even. Then, the solution set is ∈ ℝ ∈ ,+1 ∶ }. Thus, when −1 −2…−≤0, then the solution set is ∈ ℝ ∈, +1: }. If is an odd, then the solution set is ∈ ℝ ∈ −∞,1 ∪ ,+1 : }. Thus, when −1 −2 … − ≤0, the solution set is ∈ ℝ ∈ −∞,1 ∪ ,+1 : }. This research provides a novel method in solving the solution set of polynomial inequalities, in addition to other existing methods.
Directory of Open Access Journals (Sweden)
Chien-Ching Ma
2011-12-01
Full Text Available In this work, a fiber Bragg grating (FBG sensing system which can measure the transient response of out-of-plane point-wise displacement responses is set up on a smart cantilever beam and the feasibility of its use as a feedback sensor in an active structural control system is studied experimentally. An FBG filter is employed in the proposed fiber sensing system to dynamically demodulate the responses obtained by the FBG displacement sensor with high sensitivity. For comparison, a laser Doppler vibrometer (LDV is utilized simultaneously to verify displacement detection ability of the FBG sensing system. An optical full-field measurement technique called amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI is used to provide full-field vibration mode shapes and resonant frequencies. To verify the dynamic demodulation performance of the FBG filter, a traditional FBG strain sensor calibrated with a strain gauge is first employed to measure the dynamic strain of impact-induced vibrations. Then, system identification of the smart cantilever beam is performed by FBG strain and displacement sensors. Finally, by employing a velocity feedback control algorithm, the feasibility of integrating the proposed FBG displacement sensing system in a collocated feedback system is investigated and excellent dynamic feedback performance is demonstrated. In conclusion, our experiments show that the FBG sensor is capable of performing dynamic displacement feedback and/or strain measurements with high sensitivity and resolution.
Maystre, Daniel
2014-01-01
The chapter contains a detailed presentation of the surface integral theory for modelling light diffraction by surface-relief diffraction gratings having a one-dimensional periodicity. Several different approaches are presented, leading either to a single integral equation, or to a system of coupled integral equations. Special attention is paid to the singularities of the kernels, and to different techniques to accelerate the convergence of the numerical computations. The theory is applied to gratings having different profiles with or without edges, to real metal and dielectrics, and to perfectly conducting substrates.
Ultra-High Temperature Gratings
Institute of Scientific and Technical Information of China (English)
John Canning; Somnath Bandyopadhyay; Michael Stevenson; Kevin Cook
2008-01-01
Regenerated gratings seeded by type-Ⅰ gratings are shown to withstand temperatures beyond 1000 ℃. The method of regeneration offers a new approach to increasing temperature resistance of stable fibre Bragg and other gratings. These ultra-high temperature (UHT) gratings extend the applicability of silicate based components to high temperature applications such as monitoring of smelters and vehicle and aircraft engines to high power fibre lasers.
Assaf, Sami; Searles, Dominic
2016-01-01
We introduce two new bases for polynomials that lift monomial and fundamental quasisymmetric functions to the full polynomial ring. By defining a new condition on pipe dreams, called quasi-Yamanouchi, we give a positive combinatorial rule for expanding Schubert polynomials into these new bases that parallels the expansion of Schur functions into fundamental quasisymmetric functions. As a result, we obtain a refinement of the stable limits of Schubert polynomials to Stanley symmetric functions...
Pure Bending Characteristic of Tilted Fiber Bragg Grating
Institute of Scientific and Technical Information of China (English)
Bo Liu; Yin-Ping Miao; Hai-Bin Zhou; Qi-Da Zhao
2008-01-01
a novel structure of the pure macro-bending sensor based on the tilted fiber Bragg grating (TFBG) is proposed. The TFBG located in the half circle with the different diameters is bent at a constant angle with respect to the tilted grating planes. With the variations of the curvature, the core-mode resonance is unchanged and the transmission power of cladding modes detected by the photodiodes varies linearly with curvature, while the ghost mode changes by the form of two-order polynomial. So we can use the transmission power of ghost mode or other cladding modes to detect bending curvature as shape sensor. From a practical point of view, the sensor proposed here is simple, low cost and easy to implement. Moreover, it is possible to make a temperature-insensitive shape sensor due to the same temperature characteristic between the core mode and the cladding modes.
Degasperis, Antonio; Aceves, Alejandro B
2015-01-01
We derive the rogue wave solution of the classical massive Thirring model, that describes nonlinear optical pulse propagation in Bragg gratings. Combining electromagnetically induced transparency with Bragg scattering four-wave mixing, may lead to extreme waves at extremely low powers.
Energy Technology Data Exchange (ETDEWEB)
Degasperis, Antonio [Dipartimento di Fisica, “Sapienza” Università di Roma, P.le A. Moro 2, 00185 Roma (Italy); Wabnitz, Stefan, E-mail: stefan.wabnitz@unibs.it [Dipartimento di Ingegneria dell' Informazione, Università degli Studi di Brescia and INO-CNR, via Branze 38, 25123 Brescia (Italy); Aceves, Alejandro B. [Southern Methodist University, Dallas (United States)
2015-06-12
We derive the rogue wave solution of the classical massive Thirring model, that describes nonlinear optical pulse propagation in Bragg gratings. Combining electromagnetically induced transparency with Bragg scattering four-wave mixing may lead to extreme waves at extremely low powers.
BOUNDS FOR THE ZEROS OF POLYNOMIALS
Institute of Scientific and Technical Information of China (English)
W. M. Shah; A.Liman
2004-01-01
Let P(z) =n∑j=0 ajzj be a polynomial of degree n. In this paper we prove a more general result which interalia improves upon the bounds of a class of polynomials. We also prove a result which includes some extensions and generalizations of Enestrom-Kakeya theorem.
An inequality for polynomials with elliptic majorant
Nikolov Geno
1999-01-01
Let be the transformed Chebyshev polynomial of the first kind, where . We show here that has the greatest uniform norm in of its -th derivative among all algebraic polynomials of degree not exceeding , which vanish at and satisfy the inequality at the points .
Distortion control of conjugacies between quadratic polynomials
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
We use a new type of distortion control of univalent functions to give an alternative proof of Douady-Hubbard’s ray-landing theorem for quadratic Misiurewicz polynomials. The univalent maps arise from Thurston’s iterated algorithm on perturbation of such polynomials.
Uniqueness of meromorphic functions concerning differential polynomials
Institute of Scientific and Technical Information of China (English)
QIAO Lei
2007-01-01
Based on a unicity theorem for entire funcitions concerning differential polynomials proposed by M. L. Fang and W. Hong, we studied the uniqueness problem of two meromorphic functions whose differential polynomials share the same 1-point by proving two theorems and their related lemmas. The results extend and improve given by Fang and Hong's theorem.
Fostering Connections between Classes of Polynomial Functions.
Buck, Judy Curran
The typical path of instruction in high school algebra courses for the study of polynomial functions has been from linear functions, to quadratic functions, to polynomial functions of degree greater than two. This paper reports results of clinical interviews with an Algebra II student. The interviews were used to probe into the student's…
On Polynomial Functions over Finite Commutative Rings
Institute of Scientific and Technical Information of China (English)
Jian Jun JIANG; Guo Hua PENG; Qi SUN; Qi Fan ZHANG
2006-01-01
Let R be an arbitrary finite commutative local ring. In this paper, we obtain a necessary and sufficient condition for a function over R to be a polynomial function. Before this paper, necessary and sufficient conditions for a function to be a polynomial function over some special finite commutative local rings were obtained.
Point vortex equilibria related to Bessel polynomials
O'Neil, Kevin A.
2016-05-01
The method of polynomials is used to construct two families of stationary point vortex configurations. The vortices are placed at the reciprocals of the zeroes of Bessel polynomials. Configurations that translate uniformly, and configurations that are completely stationary, are obtained in this way.
Large degree asymptotics of generalized Bessel polynomials
López, J.L.; Temme, N.M.
2011-01-01
Asymptotic expansions are given for large values of $n$ of the generalized Bessel polynomials $Y_n^\\mu(z)$. The analysis is based on integrals that follow from the generating functions of the polynomials. A new simple expansion is given that is valid outside a compact neighborhood of the origin in t
The Approach of Moments for Polynomial Equations
Laurent, M.; Rostalski, P.
2010-01-01
In this article we present the moment based approach for computing all real solutions of a given system of polynomial equations. This approach builds upon a lifting method for constructing semidefinite relaxations of several nonconvex optimization problems, using sums of squares of polynomials and t
A generalization of the Bernoulli polynomials
Pierpaolo Natalini; Angela Bernardini
2003-01-01
A generalization of the Bernoulli polynomials and, consequently, of the Bernoulli numbers, is defined starting from suitable generating functions. Furthermore, the differential equations of these new classes of polynomials are derived by means of the factorization method introduced by Infeld and Hull (1951).
A generalization of the Bernoulli polynomials
Directory of Open Access Journals (Sweden)
Pierpaolo Natalini
2003-01-01
Full Text Available A generalization of the Bernoulli polynomials and, consequently, of the Bernoulli numbers, is defined starting from suitable generating functions. Furthermore, the differential equations of these new classes of polynomials are derived by means of the factorization method introduced by Infeld and Hull (1951.
Reliability polynomials crossing more than twice
Brown, J.I.; Koç, Y.; Kooij, R.E.
2011-01-01
In this paper we study all-terminal reliability polynomials of networks having the same number of nodes and the same number of links. First we show that the smallest possible size for a pair of networks that allows for two crossings of their reliability polynomials have seven nodes and fifteen edges
Application of polynomial preconditioners to conservation laws
Geurts, Bernard J.; Buuren, van René; Lu, Hao
2000-01-01
Polynomial preconditioners which are suitable in implicit time-stepping methods for conservation laws are reviewed and analyzed. The preconditioners considered are either based on a truncation of a Neumann series or on Chebyshev polynomials for the inverse of the system-matrix. The latter class of p
Indian Academy of Sciences (India)
V K Jain
2009-02-01
For a polynomial of degree , we have obtained an upper bound involving coefficients of the polynomial, for moduli of its zeros of smallest moduli, and then a refinement of the well-known Eneström–Kakeya theorem (under certain conditions).
Sums of Powers of Fibonacci Polynomials
Indian Academy of Sciences (India)
Helmut Prodinger
2009-11-01
Using the explicit (Binet) formula for the Fibonacci polynomials, a summation formula for powers of Fibonacci polynomials is derived straightforwardly, which generalizes a recent result for squares that appeared in Proc. Ind. Acad. Sci. (Math. Sci.) 118 (2008) 27--41.
A Note on Solvable Polynomial Algebras
Directory of Open Access Journals (Sweden)
Huishi Li
2014-03-01
Full Text Available In terms of their defining relations, solvable polynomial algebras introduced by Kandri-Rody and Weispfenning [J. Symbolic Comput., 9(1990] are characterized by employing Gr\\"obner bases of ideals in free algebras, thereby solvable polynomial algebras are completely determinable and constructible in a computational way.
Polynomial invariants of quantum codes
Rains, E M
1997-01-01
The weight enumerators (quant-ph/9610040) of a quantum code are quite powerful tools for exploring its structure. As the weight enumerators are quadratic invariants of the code, this suggests the consideration of higher-degree polynomial invariants. We show that the space of degree k invariants of a code of length n is spanned by a set of basic invariants in one-to-one correspondence with S_k^n. We then present a number of equations and inequalities in these invariants; in particular, we give a higher-order generalization of the shadow enumerator of a code, and prove that its coefficients are nonnegative. We also prove that the quartic invariants of a ((4,4,2)) are uniquely determined, an important step in a proof that any ((4,4,2)) is additive ([2]).
Algebras, dialgebras, and polynomial identities
Bremner, Murray R
2012-01-01
This is a survey of some recent developments in the theory of associative and nonassociative dialgebras, with an emphasis on polynomial identities and multilinear operations. We discuss associative, Lie, Jordan, and alternative algebras, and the corresponding dialgebras; the KP algorithm for converting identities for algebras into identities for dialgebras; the BSO algorithm for converting operations in algebras into operations in dialgebras; Lie and Jordan triple systems, and the corresponding disystems; and a noncommutative version of Lie triple systems based on the trilinear operation abc-bca. The paper concludes with a conjecture relating the KP and BSO algorithms, and some suggestions for further research. Most of the original results are joint work with Raul Felipe, Luiz A. Peresi, and Juana Sanchez-Ortega.
Fuzzy Morphological Polynomial Image Representation
Directory of Open Access Journals (Sweden)
Chin-Pan Huang
2010-01-01
Full Text Available A novel signal representation using fuzzy mathematical morphology is developed. We take advantage of the optimum fuzzy fitting and the efficient implementation of morphological operators to extract geometric information from signals. The new representation provides results analogous to those given by the polynomial transform. Geometrical decomposition of a signal is achieved by windowing and applying sequentially fuzzy morphological opening with structuring functions. The resulting representation is made to resemble an orthogonal expansion by constraining the results of opening to equate adapted structuring functions. Properties of the geometric decomposition are considered and used to calculate the adaptation parameters. Our procedure provides an efficient and flexible representation which can be efficiently implemented in parallel. The application of the representation is illustrated in data compression and fractal dimension estimation temporal signals and images.
Sobolev orthogonal polynomials on a simplex
Aktas, Rabia
2011-01-01
The Jacobi polynomials on the simplex are orthogonal polynomials with respect to the weight function $W_\\bg(x) = x_1^{\\g_1} ... x_d^{\\g_d} (1- |x|)^{\\g_{d+1}}$ when all $\\g_i > -1$ and they are eigenfunctions of a second order partial differential operator $L_\\bg$. The singular cases that some, or all, $\\g_1,...,\\g_{d+1}$ are -1 are studied in this paper. Firstly a complete basis of polynomials that are eigenfunctions of $L_\\bg$ in each singular case is found. Secondly, these polynomials are shown to be orthogonal with respect to an inner product which is explicitly determined. This inner product involves derivatives of the functions, hence the name Sobolev orthogonal polynomials.
Polynomials with Palindromic and Unimodal Coeﬃ cients
Institute of Scientific and Technical Information of China (English)
Hua SUN; Yi WANG; Hai Xia ZHANG
2015-01-01
Let f(q) = arqr +· · ·+asqs, with ar = 0 and as = 0, be a real polynomial. It is a palindromic polynomial of darga n if r+s = n and ar+i = as−i for all i. Polynomials of darga n form a linear subspace Pn(q) of R(q)n+1 of dimension ? n2 ?+1. We give transition matrices between two bases ?qj(1+q+· · ·+qn−2j)? , ?qj(1+q)n−2j? and the standard basis ?qj(1+qn−2j)? of Pn(q). We present some characterizations and sufficcient conditions for palindromic polynomials that can be expressed in terms of these two bases with nonnegative coefficients. We also point out the link between such polynomials and rank-generating functions of posets.
Tutte polynomial in functional magnetic resonance imaging
García-Castillón, Marlly V.
2015-09-01
Methods of graph theory are applied to the processing of functional magnetic resonance images. Specifically the Tutte polynomial is used to analyze such kind of images. Functional Magnetic Resonance Imaging provide us connectivity networks in the brain which are represented by graphs and the Tutte polynomial will be applied. The problem of computing the Tutte polynomial for a given graph is #P-hard even for planar graphs. For a practical application the maple packages "GraphTheory" and "SpecialGraphs" will be used. We will consider certain diagram which is depicting functional connectivity, specifically between frontal and posterior areas, in autism during an inferential text comprehension task. The Tutte polynomial for the resulting neural networks will be computed and some numerical invariants for such network will be obtained. Our results show that the Tutte polynomial is a powerful tool to analyze and characterize the networks obtained from functional magnetic resonance imaging.
Orthogonal Polynomials from Hermitian Matrices II
Odake, Satoru
2016-01-01
This is the second part of the project `unified theory of classical orthogonal polynomials of a discrete variable derived from the eigenvalue problems of hermitian matrices.' In a previous paper, orthogonal polynomials having Jackson integral measures were not included, since such measures cannot be obtained from single infinite dimensional hermitian matrices. Here we show that Jackson integral measures for the polynomials of the big $q$-Jacobi family are the consequence of the recovery of self-adjointness of the unbounded Jacobi matrices governing the difference equations of these polynomials. The recovery of self-adjointness is achieved in an extended $\\ell^2$ Hilbert space on which a direct sum of two unbounded Jacobi matrices acts as a Hamiltonian or a difference Schr\\"odinger operator for an infinite dimensional eigenvalue problem. The polynomial appearing in the upper/lower end of Jackson integral constitutes the eigenvector of each of the two unbounded Jacobi matrix of the direct sum. We also point out...
Matrix product formula for Macdonald polynomials
International Nuclear Information System (INIS)
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)
DEFF Research Database (Denmark)
Kaiser, W.; Bach, L.; Reithmaier, J. P.;
2003-01-01
37 GHz direct-modulation bandwidth could be obtained by a multi-section design with an integrated weakly coupled DBR grating. The laser shows side mode suppression ratios of 45 dB and output powers exceeding 20 mW.......37 GHz direct-modulation bandwidth could be obtained by a multi-section design with an integrated weakly coupled DBR grating. The laser shows side mode suppression ratios of 45 dB and output powers exceeding 20 mW....
Bernoulli-like polynomials associated with Stirling Numbers
Bender, Carl M; Brody, Dorje C.; BERNHARD K. MEISTER
2005-01-01
The Stirling numbers of the first kind can be represented in terms of a new class of polynomials that are closely related to the Bernoulli polynomials. Recursion relations for these polynomials are given.
HIGHER ORDER MULTIVARIABLE NORLUND EULER-BERNOULLI POLYNOMIALS
Institute of Scientific and Technical Information of China (English)
刘国栋
2002-01-01
The definitions of higher order multivariable Norlund Euler polynomials and Norlund Bernoulli polynomials are presented and some of their important properties are expounded. Some identities involving recurrence sequences and higher order multivariable Norlund Euler-Bernoulli polynomials are established.
Polynomial Interpolation in the Elliptic Curve Cryptosystem
Directory of Open Access Journals (Sweden)
Liew K. Jie
2011-01-01
Full Text Available Problem statement: In this research, we incorporate the polynomial interpolation method in the discrete logarithm problem based cryptosystem which is the elliptic curve cryptosystem. Approach: In this study, the polynomial interpolation method to be focused is the Lagrange polynomial interpolation which is the simplest polynomial interpolation method. This method will be incorporated in the encryption algorithm of the elliptic curve ElGamal cryptosystem. Results: The scheme modifies the elliptic curve ElGamal cryptosystem by adding few steps in the encryption algorithm. Two polynomials are constructed based on the encrypted points using Lagrange polynomial interpolation and encrypted for the second time using the proposed encryption method. We believe it is safe from the theoretical side as it still relies on the discrete logarithm problem of the elliptic curve. Conclusion/Recommendations: The modified scheme is expected to be more secure than the existing scheme as it offers double encryption techniques. On top of the existing encryption algorithm, we managed to encrypt one more time using the polynomial interpolation method. We also have provided detail examples based on the described algorithm.
Polynomial threshold functions and Boolean threshold circuits
DEFF Research Database (Denmark)
Hansen, Kristoffer Arnsfelt; Podolskii, Vladimir V.
2013-01-01
We study the complexity of computing Boolean functions on general Boolean domains by polynomial threshold functions (PTFs). A typical example of a general Boolean domain is 12n . We are mainly interested in the length (the number of monomials) of PTFs, with their degree and weight being...... of secondary interest. We show that PTFs on general Boolean domains are tightly connected to depth two threshold circuits. Our main results in regard to this connection are: PTFs of polynomial length and polynomial degree compute exactly the functions computed by THRMAJ circuits. An exponential length lower...
Exponential Polynomial Approximation with Unrestricted Upper Density
Institute of Scientific and Technical Information of China (English)
Xiang Dong YANG
2011-01-01
We take a new approach to obtaining necessary and sufficient conditions for the incompleteness of exponential polynomials in Lp/α, where Lp/α is the weighted Banach space of complex continuous functions f defined on the real axis (R)satisfying (∫+∞/-∞|f(t)|pe-α(t)dt)1/p, 1 < p < ∞, and α(t) is a nonnegative continuous function defined on the real axis (R). In this paper, the upper density of the sequence which forms the exponential polynomials is not required to be finite. In the study of weighted polynomial approximation, consideration of the case is new.
On function compositions that are polynomials
Aichinger, Erhard
2015-01-01
For a polynomial map $\\tupBold{f} : k^n \\to k^m$ ($k$ a field), we investigate those polynomials $g \\in k[t_1,\\ldots, t_n]$ that can be written as a composition $g = h \\circ \\tupBold{f}$, where $h: k^m \\to k$ is an arbitrary function. In the case that $k$ is algebraically closed of characteristic~$0$ and $\\tupBold{f}$ is surjective, we will show that $g = h \\circ \\tupBold{f}$ implies that $h$ is a polynomial.
More on rotations as spin matrix polynomials
Energy Technology Data Exchange (ETDEWEB)
Curtright, Thomas L. [Department of Physics, University of Miami, Coral Gables, Florida 33124-8046 (United States)
2015-09-15
Any nonsingular function of spin j matrices always reduces to a matrix polynomial of order 2j. The challenge is to find a convenient form for the coefficients of the matrix polynomial. The theory of biorthogonal systems is a useful framework to meet this challenge. Central factorial numbers play a key role in the theoretical development. Explicit polynomial coefficients for rotations expressed either as exponentials or as rational Cayley transforms are considered here. Structural features of the results are discussed and compared, and large j limits of the coefficients are examined.
Modular polynomials via isogeny volcanoes
Broker, Reinier; Sutherland, Andrew V
2010-01-01
We present a new algorithm to compute the classical modular polynomial Phi_n in the rings Z[X,Y] and (Z/mZ)[X,Y], for a prime n and any positive integer m. Our approach uses the graph of n-isogenies to efficiently compute Phi_n mod p for many primes p of a suitable form, and then applies the Chinese Remainder Theorem (CRT). Under the Generalized Riemann Hypothesis (GRH), we achieve an expected running time of O(n^3 (log n)^3 log log n), and compute Phi_n mod m using O(n^2 (log n)^2 + n^2 log m) space. We have used the new algorithm to compute Phi_n with n over 5000, and Phi_n mod m with n over 20000. We also consider several modular functions g for which Phi_n^g is smaller than Phi_n, allowing us to handle n over 60000.
Superconformal minimal models and admissible Jack polynomials
Blondeau-Fournier, Olivier; Ridout, David; Wood, Simon
2016-01-01
We give new proofs of the rationality of the N=1 superconformal minimal model vertex operator superalgebras and of the classification of their modules in both the Neveu-Schwarz and Ramond sectors. For this, we combine the standard free field realisation with the theory of Jack symmetric functions. A key role is played by Jack symmetric polynomials with a certain negative parameter that are labelled by admissible partitions. These polynomials are shown to describe free fermion correlators, suitably dressed by a symmetrising factor. The classification proofs concentrate on explicitly identifying Zhu's algebra and its twisted analogue. Interestingly, these identifications do not use an explicit expression for the non-trivial vacuum singular vector. While the latter is known to be expressible in terms of an Uglov symmetric polynomial or a linear combination of Jack superpolynomials, it turns out that standard Jack polynomials (and functions) suffice to prove the classification.
Thermodynamic characterization of networks using graph polynomials
Ye, Cheng; Peron, Thomas K DM; Silva, Filipi N; Rodrigues, Francisco A; Costa, Luciano da F; Torsello, Andrea; Hancock, Edwin R
2015-01-01
In this paper, we present a method for characterizing the evolution of time-varying complex networks by adopting a thermodynamic representation of network structure computed from a polynomial (or algebraic) characterization of graph structure. Commencing from a representation of graph structure based on a characteristic polynomial computed from the normalized Laplacian matrix, we show how the polynomial is linked to the Boltzmann partition function of a network. This allows us to compute a number of thermodynamic quantities for the network, including the average energy and entropy. Assuming that the system does not change volume, we can also compute the temperature, defined as the rate of change of entropy with energy. All three thermodynamic variables can be approximated using low-order Taylor series that can be computed using the traces of powers of the Laplacian matrix, avoiding explicit computation of the normalized Laplacian spectrum. These polynomial approximations allow a smoothed representation of the...
Inequalities for a polynomial and its derivative
Chanam, Barchand; Dewan, K. K.
2007-12-01
Let , 1[less-than-or-equals, slant][mu][less-than-or-equals, slant]n, be a polynomial of degree n such that p(z)[not equal to]0 in z0, then for 0Yadav and Pukhta [K.K. Dewan, R.S. Yadav, M.S. Pukhta, Inequalities for a polynomial and its derivative, Math. Inequal. Appl. 2 (2) (1999) 203-205] proved Equality holds for the polynomial where n is a multiple of [mu]E In this paper, we obtain an improvement of the above inequality by involving some of the coefficients. As an application of our result, we further improve upon a result recently proved by Aziz and Shah [A. Aziz, W.M. Shah, Inequalities for a polynomial and its derivative, Math. Inequal. Appl. 7 (3) (2004) 379-391].
Tutte Polynomial of Scale-Free Networks
Chen, Hanlin; Deng, Hanyuan
2016-05-01
The Tutte polynomial of a graph, or equivalently the q-state Potts model partition function, is a two-variable polynomial graph invariant of considerable importance in both statistical physics and combinatorics. The computation of this invariant for a graph is NP-hard in general. In this paper, we focus on two iteratively growing scale-free networks, which are ubiquitous in real-life systems. Based on their self-similar structures, we mainly obtain recursive formulas for the Tutte polynomials of two scale-free networks (lattices), one is fractal and "large world", while the other is non-fractal but possess the small-world property. Furthermore, we give some exact analytical expressions of the Tutte polynomial for several special points at ( x, y)-plane, such as, the number of spanning trees, the number of acyclic orientations, etc.
Twisted Polynomials and Forgery Attacks on GCM
DEFF Research Database (Denmark)
Abdelraheem, Mohamed Ahmed A. M. A.; Beelen, Peter; Bogdanov, Andrey;
2015-01-01
nonce misuse resistance, such as POET. The algebraic structure of polynomial hashing has given rise to security concerns: At CRYPTO 2008, Handschuh and Preneel describe key recovery attacks, and at FSE 2013, Procter and Cid provide a comprehensive framework for forgery attacks. Both approaches rely...... heavily on the ability to construct forgery polynomials having disjoint sets of roots, with many roots (“weak keys”) each. Constructing such polynomials beyond naïve approaches is crucial for these attacks, but still an open problem. In this paper, we comprehensively address this issue. We propose to use...... in an improved key recovery algorithm. As cryptanalytic applications of our twisted polynomials, we develop the first universal forgery attacks on GCM in the weak-key model that do not require nonce reuse. Moreover, we present universal weak-key forgeries for the nonce-misuse resistant AE scheme POET, which...
Transversals of Complex Polynomial Vector Fields
DEFF Research Database (Denmark)
Dias, Kealey
Vector fields in the complex plane are defined by assigning the vector determined by the value P(z) to each point z in the complex plane, where P is a polynomial of one complex variable. We consider special families of so-called rotated vector fields that are determined by a polynomial multiplied...... by rotational constants. Transversals are a certain class of curves for such a family of vector fields that represent the bifurcation states for this family of vector fields. More specifically, transversals are curves that coincide with a homoclinic separatrix for some rotation of the vector field. Given...... a concrete polynomial, it seems to take quite a bit of work to prove that it is generic, i.e. structurally stable. This has been done for a special class of degree d polynomial vector fields having simple equilibrium points at the d roots of unity, d odd. In proving that such vector fields are generic...
Characteristic Polynomials of Sample Covariance Matrices
Kösters, Holger
2009-01-01
We investigate the second-order correlation function of the characteristic polynomial of a sample covariance matrix. Starting from an explicit formula for the generating function, we re-obtain several well-known kernels from random matrix theory.
Local Polynomial Estimation of Distribution Functions
Institute of Scientific and Technical Information of China (English)
LI Yong-hong; ZENG Xia
2007-01-01
Under the condition that the total distribution function is continuous and bounded on (-∞,∞), we constructed estimations for distribution and hazard functions with local polynomial method, and obtained the rate of strong convergence of the estimations.
Concentration for noncommutative polynomials in random matrices
Meckes, Mark W.; Szarek, Stanislaw J.
2011-01-01
We present a concentration inequality for linear functionals of noncommutative polynomials in random matrices. Our hypotheses cover most standard ensembles, including Gaussian matrices, matrices with independent uniformly bounded entries and unitary or orthogonal matrices.
Characterization of pinhole transmission gratings.
Eidmann, K; Kühne, M; Müller, P; Tsakiris, G D
1990-01-01
Gold pinhole transmission gratings fabricated by Heidenhain GmbH primarily for the purpose of studying the radiation of intense soft x-ray sources have been tested with the synchrotron radiation of BESSY. Typical results for the spectral dependence of the grating efficiency into the various diffraction orders are presented in a wavelength region ranging from 4 to 20 nm. Also the influence of grating irregularities has been studied. With appropriate grating parameters quite good agreement between the experimental results and theoretical Calculations is Obtained. PMID:21307429
Blind Signature Scheme Based on Chebyshev Polynomials
Directory of Open Access Journals (Sweden)
Maheswara Rao Valluri
2011-12-01
Full Text Available A blind signature scheme is a cryptographic protocol to obtain a valid signature for a message from a signer such that signer’s view of the protocol can’t be linked to the resulting message signature pair. This paper presents blind signature scheme using Chebyshev polynomials. The security of the given scheme depends upon the intractability of the integer factorization problem and discrete logarithms ofChebyshev polynomials.
Bergman orthogonal polynomials and the Grunsky matrix
Beckermann, Bernhard; Stylianopoulos, Nikos
2016-01-01
By exploiting a link between Bergman orthogonal polynomials and the Grunsky matrix, probably first observed by Kühnau in 1985, we improve some recent results on strong asymptotics of Bergman polynomials outside the domain G of orthogonality, and on entries of the Bergman shift operator. In our proofs we suggest a new matrix approach involving the Grunsky matrix, and use well-established results in the literature relating properties of the Grunsky matrix to the regularity of the boundary of G,...
Positive maps, positive polynomials and entanglement witnesses
Skowronek, Lukasz
2009-01-01
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.
Positive maps, positive polynomials and entanglement witnesses
Energy Technology Data Exchange (ETDEWEB)
Skowronek, Lukasz; Zyczkowski, Karol [Institute of Physics, Jagiellonian University, Krakow (Poland)], E-mail: lukasz.skowronek@uj.edu.pl, E-mail: karol@tatry.if.uj.edu.pl
2009-08-14
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.
Equivalence of polynomial conjectures in additive combinatorics
Lovett, Shachar
2010-01-01
We study two conjectures in additive combinatorics. The first is the polynomial Freiman-Ruzsa conjecture, which relates to the structure of sets with small doubling. The second is the inverse Gowers conjecture for $U^3$, which relates to functions which locally look like quadratics. In both cases a weak form, with exponential decay of parameters is known, and a strong form with only a polynomial loss of parameters is conjectured. Our main result is that the two conjectures are in fact equivalent.
On Sharing, Memoization, and Polynomial Time
Avanzini, Martin; Dal Lago, Ugo
2015-01-01
We study how the adoption of an evaluation mechanism with sharing and memoization impacts the class of functions which can be computed in polynomial time. We first show how a natural cost model in which lookup for an already computed result has no cost is indeed invariant. As a corollary, we then prove that the most general notion of ramified recurrence is sound for polynomial time, this way settling an open problem in implicit computational complexity.
Stochastic processes with orthogonal polynomial eigenfunctions
Griffiths, Bob
2009-12-01
Markov processes which are reversible with either Gamma, Normal, Poisson or Negative Binomial stationary distributions in the Meixner class and have orthogonal polynomial eigenfunctions are characterized as being processes subordinated to well-known diffusion processes for the Gamma and Normal, and birth and death processes for the Poisson and Negative Binomial. A characterization of Markov processes with Beta stationary distributions and Jacobi polynomial eigenvalues is also discussed.
Quantum group invariants and link polynomials
International Nuclear Information System (INIS)
A general method is developed for constructing quantum group invariants and determining their eigenvalues. Applied to the universal R-matrix this method leads to the construction of a closed formula for link polynomials. To illustrate the application of this formula, the quantum groups Uq(E8), Uq(so(2m+1)) and Uq(gl(m)) are considered as examples, and corresponding link polynomials are obtained. (orig.)
ON ABEL-GONTSCHAROFF-GOULD'S POLYNOMIALS
Institute of Scientific and Technical Information of China (English)
He Tianxiao; Leetsch C. Hsu; Peter J. S. Shiue
2003-01-01
In this paper a connective study of Gould's annihilation coefficients and Abel-Gontscharoff polynomials is presented. It is shown that Gould's annihilation coefficients and Abel-Gontscharoff polynomials are actually equivalent to each other under certain linear substitutions for the variables. Moreover, a pair of related expansion formulas involving Gontscharoff's remainder and a new form of it are demonstrated, and also illustrated with several examples.
Local fibred right adjoints are polynomial
DEFF Research Database (Denmark)
Kock, Anders; Kock, Joachim
2013-01-01
For any locally cartesian closed category E, we prove that a local fibred right adjoint between slices of E is given by a polynomial. The slices in question are taken in a well known fibred sense......For any locally cartesian closed category E, we prove that a local fibred right adjoint between slices of E is given by a polynomial. The slices in question are taken in a well known fibred sense...
Ferrers Matrices Characterized by the Rook Polynomials
Institute of Scientific and Technical Information of China (English)
MAHai-cheng; HUSheng-biao
2003-01-01
In this paper,we show that there exist precisely W(A) Ferrers matrices F(C1,C2,…,cm)such that the rook polynomials is equal to the rook polynomial of Ferrers matrix F(b1,b2,…,bm), where A={b1,b2-1,…,bm-m+1} is a repeated set,W(A) is weight of A.
Laguerre polynomials method in the valon model
Boroun, G R
2014-01-01
We used the Laguerre polynomials method for determination of the proton structure function in the valon model. We have examined the applicability of the valon model with respect to a very elegant method, where the structure of the proton is determined by expanding valon distributions and valon structure functions on Laguerre polynomials. We compared our results with the experimental data, GJR parameterization and DL model. Having checked, this method gives a good description for the proton structure function in valon model.
Vector-Valued Jack Polynomials from Scratch
Directory of Open Access Journals (Sweden)
Jean-Gabriel Luque
2011-03-01
Full Text Available Vector-valued Jack polynomials associated to the symmetric group S_N are polynomials with multiplicities in an irreducible module of S_N and which are simultaneous eigenfunctions of the Cherednik-Dunkl operators with some additional properties concerning the leading monomial. These polynomials were introduced by Griffeth in the general setting of the complex reflections groups G(r,p,N and studied by one of the authors (C. Dunkl in the specialization r=p=1 (i.e. for the symmetric group. By adapting a construction due to Lascoux, we describe an algorithm allowing us to compute explicitly the Jack polynomials following a Yang-Baxter graph. We recover some properties already studied by C. Dunkl and restate them in terms of graphs together with additional new results. In particular, we investigate normalization, symmetrization and antisymmetrization, polynomials with minimal degree, restriction etc. We give also a shifted version of the construction and we discuss vanishing properties of the associated polynomials.
Diffusion tensor image registration using polynomial expansion
International Nuclear Information System (INIS)
In this paper, we present a deformable registration framework for the diffusion tensor image (DTI) using polynomial expansion. The use of polynomial expansion in image registration has previously been shown to be beneficial due to fast convergence and high accuracy. However, earlier work was developed only for 3D scalar medical image registration. In this work, it is shown how polynomial expansion can be applied to DTI registration. A new measurement is proposed for DTI registration evaluation, which seems to be robust and sensitive in evaluating the result of DTI registration. We present the algorithms for DTI registration using polynomial expansion by the fractional anisotropy image, and an explicit tensor reorientation strategy is inherent to the registration process. Analytic transforms with high accuracy are derived from polynomial expansion and used for transforming the tensor's orientation. Three measurements for DTI registration evaluation are presented and compared in experimental results. The experiments for algorithm validation are designed from simple affine deformation to nonlinear deformation cases, and the algorithms using polynomial expansion give a good performance in both cases. Inter-subject DTI registration results are presented showing the utility of the proposed method. (paper)
Reliable Computational Predictions by Modeling Uncertainties Using Arbitrary Polynomial Chaos
Witteveen, J.A.S.; Bijl, H
2006-01-01
Inherent physical uncertainties can have a significant influence on computational predictions. It is therefore important to take physical uncertainties into account to obtain more reliable computational predictions. The Galerkin polynomial chaos method is a commonly applied uncertainty quantification method. However, the polynomial chaos expansion has some limitations. Firstly, the polynomial chaos expansion based on classical polynomials can achieve exponential convergence for a limited set ...
Longhi, Stefano
2015-01-01
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.
International Nuclear Information System (INIS)
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)
On the Connection Coefficients of the Chebyshev-Boubaker Polynomials
Directory of Open Access Journals (Sweden)
Paul Barry
2013-01-01
Full Text Available The Chebyshev-Boubaker polynomials are the orthogonal polynomials whose coefficient arrays are defined by ordinary Riordan arrays. Examples include the Chebyshev polynomials of the second kind and the Boubaker polynomials. We study the connection coefficients of this class of orthogonal polynomials, indicating how Riordan array techniques can lead to closed-form expressions for these connection coefficients as well as recurrence relations that define them.
Explicit classes of permutation polynomials of F33m
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
Permutation polynomials have been an interesting subject of study for a long time and have applications in many areas of mathematics and engineering. However, only a small number of specific classes of permutation polynomials are known so far. In this paper, six classes of linearized permutation polynomials and six classes of nonlinearized permutation polynomials over F33m are presented. These polynomials have simple shapes, and they are related to planar functions.
Explicit classes of permutation polynomials of F33m
Institute of Scientific and Technical Information of China (English)
DING CunSheng; XIANG Qing; YUAN Jin; YUAN PingZhi
2009-01-01
Permutation polynomials have been an interesting subject of study for a long time and have applications in many areas of mathematics and engineering. However, only a small number of specific classes of permutation polynomials are known so far. In this paper, six classes of linearized permutation polynomials and six classes of nonlinearized permutation polynomials over F33 are pre-sented. These polynomials have simple shapes, and they are related to planar functions.
Picosecond Holographic-Grating Spectroscopy
Duppen, K.
1987-01-01
Interfering light waves produce an optical interference pattern in any medium that interacts with light. This modulation of some physical parameter of the system acts as a classical holographic grating for optical radiation. When such a grating is produced through interaction of pulsed light waves w
Optical Fiber Grating based Sensors
DEFF Research Database (Denmark)
Michelsen, Susanne
2003-01-01
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...
Stretchable diffraction gratings for spectrometry
Simonov, A.N.; Grabarnik, S.; Vdovine, G.V
2007-01-01
We have investigated the possibility of using transparent stretchable diffraction gratings for spectrometric applications. The gratings were fabricated by replication of a triangular-groove master into a transparent viscoelastic. The sample length, and hence the spatial period, can be reversibly cha
Brunner, T. A.; Ausschnitt, C. P.
2007-03-01
Despite the increasing use of advanced imaging methods to pattern chip features, process windows continue to shrink with decreasing critical dimensions. Controlling the manufacturing process within these shrinking windows requires monitor structures designed to maximize both sensitivity and robustness. In particular, monitor structures must exhibit a large, measurable response to dose and focus changes over the entire range of the critical features process window. Any process variations present fundamental challenges to the effectiveness of OPC methods, since the shape compensation assumes a repeatable process. One particular process parameter which is under increasing scrutiny is focus blur, e.g. from finite laser bandwidth, which can cause such OPC instability, and thereby damage pattern fidelity. We introduce a new type of test target called the Process Monitor Grating (PMG) which is designed for extreme sensitivity to process variation. The PMG design principle is to use assist features to zero out higher diffraction orders. We show via simulation and experiment that such structures are indeed very sensitive to process variation. In addition, PMG targets have other desirable attributes such as mask manufacturability, robustness to pattern collapse, and compatibility with standard CD metrology methods such as scatterometry. PMG targets are applicable to the accurate determination of dose and focus deviations, and in combination with an isofocal grating target, allow the accurate determination of focus blur. The methods shown in this paper are broadly applicable to the characterization of process deviations using test wafers or to the control of product using kerf structures.
Certain non-linear differential polynomials sharing a non zero polynomial
Directory of Open Access Journals (Sweden)
Majumder Sujoy
2015-10-01
functions sharing a nonzero polynomial and obtain two results which improves and generalizes the results due to L. Liu [Uniqueness of meromorphic functions and differential polynomials, Comput. Math. Appl., 56 (2008, 3236-3245.] and P. Sahoo [Uniqueness and weighted value sharing of meromorphic functions, Applied. Math. E-Notes., 11 (2011, 23-32.].
A new class of generalized polynomials associated with Hermite and Bernoulli polynomials
Directory of Open Access Journals (Sweden)
M. A. Pathan
2015-05-01
Full Text Available In this paper, we introduce a new class of generalized polynomials associated with the modified Milne-Thomson's polynomials Φ_{n}^{(α}(x,ν of degree n and order α introduced by Derre and Simsek.The concepts of Bernoulli numbers B_n, Bernoulli polynomials B_n(x, generalized Bernoulli numbers B_n(a,b, generalized Bernoulli polynomials B_n(x;a,b,c of Luo et al, Hermite-Bernoulli polynomials {_HB}_n(x,y of Dattoli et al and {_HB}_n^{(α} (x,y of Pathan are generalized to the one {_HB}_n^{(α}(x,y,a,b,c which is called the generalized polynomial depending on three positive real parameters. Numerous properties of these polynomials and some relationships between B_n, B_n(x, B_n(a,b, B_n(x;a,b,c and {}_HB_n^{(α}(x,y;a,b,c are established. Some implicit summation formulae and general symmetry identities are derived by using different analytical means and applying generating functions. These results extend some known summations and identities of generalized Bernoulli numbers and polynomials
Twisted Alexander polynomials of hyperbolic knots
Dunfield, Nathan M; Jackson, Nicholas
2011-01-01
We study a twisted Alexander polynomial naturally associated to a hyperbolic knot in an integer homology 3-sphere via a lift of the holonomy representation to SL(2, C). It is an unambiguous symmetric Laurent polynomial whose coefficients lie in the trace field of the knot. It contains information about genus, fibering, and chirality, and moreover is powerful enough to sometimes detect mutation. We calculated this invariant numerically for all 313,209 hyperbolic knots in S^3 with at most 15 crossings, and found that in all cases it gave a sharp bound on the genus of the knot and determined both fibering and chirality. We also study how such twisted Alexander polynomials vary as one moves around in an irreducible component X_0 of the SL(2, C)-character variety of the knot group. We show how to understand all of these polynomials at once in terms of a polynomial whose coefficients lie in the function field of X_0. We use this to help explain some of the patterns observed for knots in S^3, and explore a potential...
Algorithms for Testing Monomials in Multivariate Polynomials
Chen, Zhixiang; Liu, Yang; Schweller, Robert
2010-01-01
This paper is our second step towards developing a theory of testing monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by an arithmetic circuit has some types of monomials in its sum-product expansion. The complexity aspects of this problem and its variants have been investigated in our first paper by Chen and Fu (2010), laying a foundation for further study. In this paper, we present two pairs of algorithms. First, we prove that there is a randomized $O^*(p^k)$ time algorithm for testing $p$-monomials in an $n$-variate polynomial of degree $k$ represented by an arithmetic circuit, while a deterministic $O^*(6.4^k + p^k)$ time algorithm is devised when the circuit is a formula, here $p$ is a given prime number. Second, we present a deterministic $O^*(2^k)$ time algorithm for testing multilinear monomials in $\\Pi_m\\Sigma_2\\Pi_t\\times \\Pi_k\\Pi_3$ polynomials, while a randomized $O^*(1.5^k)$ algorithm is given for these polynomials. The first algorithm extends...
New development in theory of Laguerre polynomials
Guseinov, I I
2012-01-01
The new complete orthonormal sets of -Laguerre type polynomials (-LTP,) are suggested. Using Schr\\"odinger equation for complete orthonormal sets of -exponential type orbitals (-ETO) introduced by the author, it is shown that the origin of these polynomials is the centrally symmetric potential which contains the core attraction potential and the quantum frictional potential of the field produced by the particle itself. The quantum frictional forces are the analog of radiation damping or frictional forces suggested by Lorentz in classical electrodynamics. The new -LTP are complete without the inclusion of the continuum states of hydrogen like atoms. It is shown that the nonstandard and standard conventions of -LTP and their weight functions are the same. As an application, the sets of infinite expansion formulas in terms of -LTP and L-Generalized Laguerre polynomials (L-GLP) for atomic nuclear attraction integrals of Slater type orbitals (STO) and Coulomb-Yukawa like correlated interaction potentials (CIP) wit...
Quantum chaotic dynamics and random polynomials
Energy Technology Data Exchange (ETDEWEB)
Bogomolny, E.; Bohigas, O.; Leboeuf, P.
1995-11-01
The distribution of roots of polynomials of high degree with random coefficients is investigated which, among others, appear naturally in the context of `quantum chaotic dynamics`. It is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane. In order to further increase this tendency, the particular case of self-inverse random polynomials is studied, and it is shown that for them a finite portion of all roots lies exactly on the unit circle. Correlation functions of these roots are also computed analytically, and compared to the correlations of eigenvalues of random matrices. The problem of ergodicity of chaotic wavefunctions is also considered. Special attention is devoted to the role of symmetries in the distribution of roots of random polynomials. (author). 32 refs.
Uniform trigonometric polynomial B-spline curves
Institute of Scientific and Technical Information of China (English)
吕勇刚; 汪国昭; 杨勋年
2002-01-01
This paper presents a new kind of uniform spline curve, named trigonometric polynomialB-splines, over space Ω = span{sint, cost, tk-3,tk-4,…,t,1} of which k is an arbitrary integerlarger than or equal to 3. We show that trigonometric polynomial B-spline curves have many similarV properties to traditional B-splines. Based on the explicit representation of the curve we have also presented the subdivision formulae for this new kind of curve. Since the new spline can include both polynomial curves and trigonometric curves as special cases without rational form, it can be used as an efficient new model for geometric design in the fields of CAD/CAM.
A complete discrimination system for polynomials
Institute of Scientific and Technical Information of China (English)
杨路; 侯晓荣; 曾振柄
1996-01-01
Given a polynomial with symbolic/literal coefficients,a complete discrimination system is a set of explicit expressions in terms of the coefficients,which is sufficient for determining the numbers and multiplicities of the real and imaginary roots.Though it is of great significance,such a criterion for root-classification has never been given for polynomials with degrees greater than 4.The lack of efficient tools in this aspect extremely prevents computer implementations for Tarski’s and other methods in automated theorem proving.To remedy this defect,a generic algorithm is proposed to produce a complete discrimination system for a polynomial with any degrees.This result has extensive applications in various fields,and its efficiency was demonstrated by computer implementations.
Fast beampattern evaluation by polynomial rooting
Häcker, P.; Uhlich, S.; Yang, B.
2011-07-01
Current automotive radar systems measure the distance, the relative velocity and the direction of objects in their environment. This information enables the car to support the driver. The direction estimation capabilities of a sensor array depend on its beampattern. To find the array configuration leading to the best angle estimation by a global optimization algorithm, a huge amount of beampatterns have to be calculated to detect their maxima. In this paper, a novel algorithm is proposed to find all maxima of an array's beampattern fast and reliably, leading to accelerated array optimizations. The algorithm works for arrays having the sensors on a uniformly spaced grid. We use a general version of the gcd (greatest common divisor) function in order to write the problem as a polynomial. We differentiate and root the polynomial to get the extrema of the beampattern. In addition, we show a method to reduce the computational burden even more by decreasing the order of the polynomial.
Dominating Sets and Domination Polynomials of Paths
Directory of Open Access Journals (Sweden)
Saeid Alikhani
2009-01-01
Full Text Available Let G=(V,E be a simple graph. A set S⊆V is a dominating set of G, if every vertex in V\\S is adjacent to at least one vertex in S. Let 𝒫ni be the family of all dominating sets of a path Pn with cardinality i, and let d(Pn,j=|𝒫nj|. In this paper, we construct 𝒫ni, and obtain a recursive formula for d(Pn,i. Using this recursive formula, we consider the polynomial D(Pn,x=∑i=⌈n/3⌉nd(Pn,ixi, which we call domination polynomial of paths and obtain some properties of this polynomial.
Minimal residual method stronger than polynomial preconditioning
Energy Technology Data Exchange (ETDEWEB)
Faber, V.; Joubert, W.; Knill, E. [Los Alamos National Lab., NM (United States)] [and others
1994-12-31
Two popular methods for solving symmetric and nonsymmetric systems of equations are the minimal residual method, implemented by algorithms such as GMRES, and polynomial preconditioning methods. In this study results are given on the convergence rates of these methods for various classes of matrices. It is shown that for some matrices, such as normal matrices, the convergence rates for GMRES and for the optimal polynomial preconditioning are the same, and for other matrices such as the upper triangular Toeplitz matrices, it is at least assured that if one method converges then the other must converge. On the other hand, it is shown that matrices exist for which restarted GMRES always converges but any polynomial preconditioning of corresponding degree makes no progress toward the solution for some initial error. The implications of these results for these and other iterative methods are discussed.
Voltage-controlled compression for period tuning of optical surface relief gratings.
Kollosche, Matthias; Döring, Sebastian; Stumpe, Joachim; Kofod, Guggi
2011-04-15
This Letter reports on new methods and a consistent model for voltage tunable optical transmission gratings. Elastomeric gratings were molded from holographically written surface relief gratings in an azobenzene sol-gel material. These were placed on top of a transparent electroactive elastomeric substrate. Two different electro-active substrate elastomers were employed, with a large range of prestretches. A novel finite-deformation theory was found to match the device response excellently, without fitting parameters. The results clearly show that the grating underwent pure-shear deformation, and more surprisingly, that the mechanical properties of the electro-active substrate did not affect device actuation.
Error Minimization of Polynomial Approximation of Delta
Indian Academy of Sciences (India)
Islam Sana; Sadiq Muhammad; Qureshi Muhammad Shahid
2008-09-01
The difference between Universal time (UT) and Dynamical time (TD), known as Delta ( ) is tabulated for the first day of each year in the Astronomical Almanac. During the last four centuries it is found that there are large differences between its values for two consecutive years. Polynomial approximations have been developed to obtain the values of for any time of a year for the period AD 1620 to AD 2000 (Meeu 2000) as no dynamical theories describe the variations in . In this work, a new set of polynomials for is obtained for the period AD 1620 to AD 2007 that is found to produce better results compared to previous attempts.
Large Degree Asymptotics of Generalized Bessel Polynomials
López, J. L.; Temme, Nico
2011-01-01
Asymptotic expansions are given for large values of $n$ of the generalized Bessel polynomials $Y_n^\\mu(z)$. The analysis is based on integrals that follow from the generating functions of the polynomials. A new simple expansion is given that is valid outside a compact neighborhood of the origin in the $z-$plane. New forms of expansions in terms of elementary functions valid in sectors not containing the turning points $z=\\pm i/n$ are derived, and a new expansion in terms of modified Bessel fu...
Polynomial Vector Fields in One Complex Variable
DEFF Research Database (Denmark)
Branner, Bodil
In recent years Adrien Douady was interested in polynomial vector fields, both in relation to iteration theory and as a topic on their own. This talk is based on his work with Pierrette Sentenac, work of Xavier Buff and Tan Lei, and my own joint work with Kealey Dias.......In recent years Adrien Douady was interested in polynomial vector fields, both in relation to iteration theory and as a topic on their own. This talk is based on his work with Pierrette Sentenac, work of Xavier Buff and Tan Lei, and my own joint work with Kealey Dias....
The chromatic polynomial and list colorings
DEFF Research Database (Denmark)
Thomassen, Carsten
2009-01-01
We prove that, if a graph has a list of k available colors at every vertex, then the number of list-colorings is at least the chromatic polynomial evaluated at k when k is sufficiently large compared to the number of vertices of the graph.......We prove that, if a graph has a list of k available colors at every vertex, then the number of list-colorings is at least the chromatic polynomial evaluated at k when k is sufficiently large compared to the number of vertices of the graph....
Incomplete Bivariate Fibonacci and Lucas -Polynomials
Directory of Open Access Journals (Sweden)
Dursun Tasci
2012-01-01
Full Text Available We define the incomplete bivariate Fibonacci and Lucas -polynomials. In the case =1, =1, we obtain the incomplete Fibonacci and Lucas -numbers. If =2, =1, we have the incomplete Pell and Pell-Lucas -numbers. On choosing =1, =2, we get the incomplete generalized Jacobsthal number and besides for =1 the incomplete generalized Jacobsthal-Lucas numbers. In the case =1, =1, =1, we have the incomplete Fibonacci and Lucas numbers. If =1, =1, =1, =⌊(−1/(+1⌋, we obtain the Fibonacci and Lucas numbers. Also generating function and properties of the incomplete bivariate Fibonacci and Lucas -polynomials are given.
Genus Polynomials of Cycles with Double Edges
Institute of Scientific and Technical Information of China (English)
Eunyoung BAEK; Jongyook PARK
2011-01-01
Two cellular embeddings i : G → S and j : G → S of a connected graph G into a closed orientable surface S are equivalent if there is an orientation-preserving surface homeomorphism h: S → S such that hi = j. The genus polynomial of a graph G is defined by g[G](x)=∑∞g=0agxg,where ag is the number of equivalence classes of embeddings of G into the orientable surface Sg with g genera.In this paper, we compute the genus polynomial of a graph obtained from a cycle by replacing each edge by two multiple edges.
The classification of polynomial basins of infinity
DeMarco, Laura
2011-01-01
We consider the problem of classifying the dynamics of complex polynomials $f: \\mathbb{C} \\to \\mathbb{C}$ restricted to their basins of infinity. We synthesize existing combinatorial tools --- tableaux, trees, and laminations --- into a new invariant of basin dynamics we call the pictograph. For polynomials with all critical points escaping to infinity, we obtain a complete description of the set of topological conjugacy classes. We give an algorithm for constructing abstract pictographs, and we provide an inductive algorithm for counting topological conjugacy classes with a given pictograph.
Perturbations around the zeros of classical orthogonal polynomials
Sasaki, Ryu
2014-01-01
Starting from degree N solutions of a time dependent Schroedinger-like equation for classical orthogonal polynomials, a linear matrix equation describing perturbations around the N zeros of the polynomial is derived. The matrix has remarkable Diophantine properties. Its eigenvalues are independent of the zeros. The corresponding eigenvectors provide the representations of the lower degree (0,1,...,N-1) polynomials in terms of the zeros of the degree N polynomial. The results are valid universally for all the classical orthogonal polynomials, including the Askey scheme of hypergeometric orthogonal polynomials and its q-analogues.
Perturbations around the zeros of classical orthogonal polynomials
Sasaki, Ryu
2015-04-01
Starting from degree N solutions of a time dependent Schrödinger-like equation for classical orthogonal polynomials, a linear matrix equation describing perturbations around the N zeros of the polynomial is derived. The matrix has remarkable Diophantine properties. Its eigenvalues are independent of the zeros. The corresponding eigenvectors provide the representations of the lower degree ( 0 , 1 , … , N - 1 ) polynomials in terms of the zeros of the degree N polynomial. The results are valid universally for all the classical orthogonal polynomials, including the Askey scheme of hypergeometric orthogonal polynomials and its q-analogues.
Institute of Scientific and Technical Information of China (English)
Tan Xiaogang; Wei Ping; Li Liping
2009-01-01
To detect higher order polynomial phase signals (HOPPSs), the smoothed-pseudo polynomial Wigner-Ville distribution (SP-PWVD), an improved version of the polynomial Wigner-Ville distribution (PWVD), is pre-sented using a separable kernel. By adjusting the lengths of the functions in the kernel, the balance between resolution retaining and interference suppressing can be adjusted conveniently. The proposed method with merits of interference terms reduction and noise suppression can provide time frequency representation of better readability and more accurate instantaneous frequency (IF) estimation with higher order SP-PWVD. The performance of the SP-PWVD is verified by computer simulations.
Extreme Silica Optical Fibre Gratings
Directory of Open Access Journals (Sweden)
Kevin Cook
2008-10-01
Full Text Available A regenerated optical fibre Bragg grating that survives temperature cycling up to 1,295Ã‚Â°C is demonstrated. A model based on seeded crystallisation or amorphisation is proposed.
MEMS Bragg grating force sensor
DEFF Research Database (Denmark)
Reck, Kasper; Thomsen, Erik Vilain; Hansen, Ole
2011-01-01
We present modeling, design, fabrication and characterization of a new type of all-optical frequency modulated MEMS force sensor based on a mechanically amplified double clamped waveguide beam structure with integrated Bragg grating. The sensor is ideally suited for force measurements in harsh...... environments and for remote and distributed sensing and has a measured sensitivity of -14 nm/N, which is several times higher than what is obtained in conventional fiber Bragg grating force sensors. © 2011 Optical Society of America....
Quantum Hilbert matrices and orthogonal polynomials
DEFF Research Database (Denmark)
Andersen, Jørgen Ellegaard; Berg, Christian
2009-01-01
Using the notion of quantum integers associated with a complex number q≠0 , we define the quantum Hilbert matrix and various extensions. They are Hankel matrices corresponding to certain little q -Jacobi polynomials when |q|... of reciprocal Fibonacci numbers called Filbert matrices. We find a formula for the entries of the inverse quantum Hilbert matrix....
On "the matching polynomial of a polygraph"
Broersma, H.J.; Li, Xueliang
1993-01-01
In this note we give an explanation for two phenomena mentioned in the concluding remarks of “The matching polynomial of a polygraph” by Babić et al. The following results are obtained: 1. Although three matrices for given polygraphs defined in the above article in general have different orders, th
Optimization of Cubic Polynomial Functions without Calculus
Taylor, Ronald D., Jr.; Hansen, Ryan
2008-01-01
In algebra and precalculus courses, students are often asked to find extreme values of polynomial functions in the context of solving an applied problem; but without the notion of derivative, something is lost. Either the functions are reduced to quadratics, since students know the formula for the vertex of a parabola, or solutions are…
Bernoulli Polynomials, Fourier Series and Zeta Numbers
DEFF Research Database (Denmark)
Scheufens, Ernst E
2013-01-01
Fourier series for Bernoulli polynomials are used to obtain information about values of the Riemann zeta function for integer arguments greater than one. If the argument is even we recover the well-known exact values, if the argument is odd we find integral representations and rapidly convergent...
Piecewise polynomial representations of genomic tracks.
Tarabichi, Maxime; Detours, Vincent; Konopka, Tomasz
2012-01-01
Genomic data from micro-array and sequencing projects consist of associations of measured values to chromosomal coordinates. These associations can be thought of as functions in one dimension and can thus be stored, analyzed, and interpreted as piecewise-polynomial curves. We present a general framework for building piecewise polynomial representations of genome-scale signals and illustrate some of its applications via examples. We show that piecewise constant segmentation, a typical step in copy-number analyses, can be carried out within this framework for both array and (DNA) sequencing data offering advantages over existing methods in each case. Higher-order polynomial curves can be used, for example, to detect trends and/or discontinuities in transcription levels from RNA-seq data. We give a concrete application of piecewise linear functions to diagnose and quantify alignment quality at exon borders (splice sites). Our software (source and object code) for building piecewise polynomial models is available at http://sourceforge.net/projects/locsmoc/.
Ideals in Polynomial Near-rings
Institute of Scientific and Technical Information of China (English)
Mark Farag
2002-01-01
In this paper, we further explore the relationship between the ideals of N and those of N[x], where N is a zero-symmetric right near-ring with identity and N[x] is the polynomial near-ring introduced by Bagley in 1993.
Miscellaneous identities of generalized Hermite polynomials
Directory of Open Access Journals (Sweden)
G. Dattoli
1997-11-01
Full Text Available We extend a number of identities valid for the ordinary case to generalized Hermite polynomials with two indices and two variables. These identities, new to the authors knowledge, are obtained by using an operatorial procedure based on the properties of the Weyl group.
Polynomial computation of Hankel singular values
Kwakernaak, Huibert
1992-01-01
A revised and improved version of a polynomial algorithm is presented. It was published by N.J. Young (1990) for the computation of the singular values and vectors of the Hankel operator defined by a linear time-invariant system with a rotational transfer matrix. Tentative numerical experiments indi
Euler Polynomials, Fourier Series and Zeta Numbers
DEFF Research Database (Denmark)
Scheufens, Ernst E
2012-01-01
Fourier series for Euler polynomials is used to obtain information about values of the Riemann zeta function for integer arguments greater than one. If the argument is even we recover the well-known exact values, if the argument is odd we find integral representations and rapidly convergent series....
Thermodynamic characterization of networks using graph polynomials
Ye, Cheng; Comin, César H.; Peron, Thomas K. DM.; Silva, Filipi N.; Rodrigues, Francisco A.; Costa, Luciano da F.; Torsello, Andrea; Hancock, Edwin R.
2015-09-01
In this paper, we present a method for characterizing the evolution of time-varying complex networks by adopting a thermodynamic representation of network structure computed from a polynomial (or algebraic) characterization of graph structure. Commencing from a representation of graph structure based on a characteristic polynomial computed from the normalized Laplacian matrix, we show how the polynomial is linked to the Boltzmann partition function of a network. This allows us to compute a number of thermodynamic quantities for the network, including the average energy and entropy. Assuming that the system does not change volume, we can also compute the temperature, defined as the rate of change of entropy with energy. All three thermodynamic variables can be approximated using low-order Taylor series that can be computed using the traces of powers of the Laplacian matrix, avoiding explicit computation of the normalized Laplacian spectrum. These polynomial approximations allow a smoothed representation of the evolution of networks to be constructed in the thermodynamic space spanned by entropy, energy, and temperature. We show how these thermodynamic variables can be computed in terms of simple network characteristics, e.g., the total number of nodes and node degree statistics for nodes connected by edges. We apply the resulting thermodynamic characterization to real-world time-varying networks representing complex systems in the financial and biological domains. The study demonstrates that the method provides an efficient tool for detecting abrupt changes and characterizing different stages in network evolution.
On the Schinzel Identity of Cyclotomic Polynomial
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
@@For integer n>0, let n(x) denote the nth cyclotomic polynomial n(x)=tackrel{01 be an odd square-free number.Aurifeuille and Le Lasseur［1］ proved thatequationn(x)=An2(x)-(-1)n-12)nxBn2(x).equation
Algebraic polynomial system solving and applications
Bleylevens, I.W.M.
2010-01-01
The problem of computing the solutions of a system of multivariate polynomial equations can be approached by the Stetter-Möller matrix method which casts the problem into a large eigenvalue problem. This Stetter-Möller matrix method forms the starting point for the development of computational proce
The GCD property and irreduciable quadratic polynomials
Directory of Open Access Journals (Sweden)
Saroj Malik
1986-01-01
Full Text Available The proof of the following theorem is presented: If D is, respectively, a Krull domain, a Dedekind domain, or a Prüfer domain, then D is correspondingly a UFD, a PID, or a Bezout domain if and only if every irreducible quadratic polynomial in D[X] is a prime element.
UNIQUENESS OF DIFFERENCE POLYNOMIALS OF MEROMORPHIC FUNCTIONS
Institute of Scientific and Technical Information of China (English)
刘永; 祁晓光
2014-01-01
In this article, we investigate the uniqueness problems of difference polynomials of meromorphic functions and obtain some results which can be viewed as discrete analogues of the results given by Shibazaki. Some examples are given to show the results in this article are best possible.
Polynomial Asymptotes of the Second Kind
Dobbs, David E.
2011-01-01
This note uses the analytic notion of asymptotic functions to study when a function is asymptotic to a polynomial function. Along with associated existence and uniqueness results, this kind of asymptotic behaviour is related to the type of asymptote that was recently defined in a more geometric way. Applications are given to rational functions and…
Scalar Field Theories with Polynomial Shift Symmetries
Griffin, Tom; Horava, Petr; Yan, Ziqi
2014-01-01
We continue our study of naturalness in nonrelativistic QFTs of the Lifshitz type, focusing on scalar fields that can play the role of Nambu-Goldstone (NG) modes associated with spontaneous symmetry breaking. Such systems allow for an extension of the constant shift symmetry to a shift by a polynomial of degree $P$ in spatial coordinates. These "polynomial shift symmetries" in turn protect the technical naturalness of modes with a higher-order dispersion relation, and lead to a refinement of the proposed classification of infrared Gaussian fixed points available to describe NG modes in nonrelativistic theories. Generic interactions in such theories break the polynomial shift symmetry explicitly to the constant shift. It is thus natural to ask: Given a Gaussian fixed point with polynomial shift symmetry of degree $P$, what are the lowest-dimension operators that preserve this symmetry, and deform the theory into a self-interacting scalar field theory with the shift symmetry of degree $P$? To answer this (essen...
Dynamic system uncertainty propagation using polynomial chaos
Institute of Scientific and Technical Information of China (English)
Xiong Fenfen; Chen Shishi; Xiong Ying
2014-01-01
The classic polynomial chaos method (PCM), characterized as an intrusive methodology, has been applied to uncertainty propagation (UP) in many dynamic systems. However, the intrusive polynomial chaos method (IPCM) requires tedious modification of the governing equations, which might introduce errors and can be impractical. Alternative to IPCM, the non-intrusive polynomial chaos method (NIPCM) that avoids such modifications has been developed. In spite of the frequent application to dynamic problems, almost all the existing works about NIPCM for dynamic UP fail to elaborate the implementation process in a straightforward way, which is important to readers who are unfamiliar with the mathematics of the polynomial chaos theory. Meanwhile, very few works have compared NIPCM to IPCM in terms of their merits and applicability. Therefore, the mathematic procedure of dynamic UP via both methods considering parametric and initial condition uncertainties are comparatively discussed and studied in the present paper. Comparison of accuracy and efficiency in statistic moment estimation is made by applying the two methods to several dynamic UP problems. The relative merits of both approaches are discussed and summarized. The detailed description and insights gained with the two methods through this work are expected to be helpful to engineering designers in solving dynamic UP problems.
Irreducibility Results for Compositions of Polynomials in Several Variables
Indian Academy of Sciences (India)
Anca Iuliana Bonciocat; Alexandru Zaharescu
2005-05-01
We obtain explicit upper bounds for the number of irreducible factors for a class of compositions of polynomials in several variables over a given field. In particular, some irreducibility criteria are given for this class of compositions of polynomials.
Combinatorial polynomials as moments, Hankel transforms and exponential Riordan arrays
Barry, Paul
2011-01-01
In the case of two combinatorial polynomials, we show that they can exhibited as moments of paramaterized families of orthogonal polynomials, and hence derive their Hankel transforms. Exponential Riordan arrays are the main vehicles used for this.
Representations of Knot Groups and Twisted Alexander Polynomials
Institute of Scientific and Technical Information of China (English)
Xiao Song LIN
2001-01-01
We present a twisted version of the Alexander polynomial associated with a matrix representation of the knot group. Examples of two knots with the same Alexander module but differenttwisted Alexander polynomials are given.
Identities involving Bessel polynomials arising from linear differential equations
Kim, Taekyun; Kim, Dae San
2016-01-01
In this paper, we study linear di?erential equations arising from Bessel polynomials and their applications. From these linear differential equations, we give some new and explicit identities for Bessel polynomials.
On an Inequality Concerning the Polar Derivative of a Polynomial
Indian Academy of Sciences (India)
A Aziz; N A Rather
2007-08-01
In this paper, we present a correct proof of an -inequality concerning the polar derivative of a polynomial with restricted zeros. We also extend Zygmund’s inequality to the polar derivative of a polynomial.
Best polynomial degree reduction on q-lattices with applications to q-orthogonal polynomials
Ait-Haddou, Rachid
2015-06-07
We show that a weighted least squares approximation of q-Bézier coefficients provides the best polynomial degree reduction in the q-L
Characterization and Enumeration of Good Punctured Polynomials over Finite Fields
Directory of Open Access Journals (Sweden)
Somphong Jitman
2016-01-01
Full Text Available A family of good punctured polynomials is introduced. The complete characterization and enumeration of such polynomials are given over the binary field F2. Over a nonbinary finite field Fq, the set of good punctured polynomials of degree less than or equal to 2 are completely determined. For n≥3, constructive lower bounds of the number of good punctured polynomials of degree n over Fq are given.
Theoretical analysis of novel fiber grating pair
Wang, Liao; Jia, Hongzhi; Fang, Liang; You, Bei
2016-06-01
A novel fiber grating pair that consists of a conventional long-period fiber grating and a fiber Bragg cladding grating (FBCG) is proposed. The FBCG is a new type of fiber grating in which refractive index modulation is formed in the cladding. Through the coupled-mode theory, we accurately calculate the coupling coefficients between modes supported in the fibers. And some other mode coupling features in the fiber cladding gratings are analyzed in detail. The calculation of the modes involved in this paper is based on a model of three-layer step-index fiber geometry. Then, we have investigated the sensitivity characteristics for variation of the modulation strengths of the fiber Bragg cladding gratings' resonance peaks and the long-period cladding gratings' (LPCGs) dual resonant peaks. Finally, the modulation strength sensitivity of the grating pair's three resonant peaks is demonstrated, and the results indicate that these grating pairs may find potential applications in optical fiber sensing.
On the Lorentz degree of a product of polynomials
Ait-Haddou, Rachid
2015-01-01
In this note, we negatively answer two questions of T. Erdélyi (1991, 2010) on possible lower bounds on the Lorentz degree of product of two polynomials. We show that the correctness of one question for degree two polynomials is a direct consequence of a result of Barnard et al. (1991) on polynomials with nonnegative coefficients.
Does the polynomial hierarchy collapse if onto functions are invertible?
H. Buhrman; L. Fortnow; M. Koucký; J.D. Rogers; N. Vereshchagin
2010-01-01
The class TFNP, defined by Megiddo and Papadimitriou, consists of multivalued functions with values that are polynomially verifiable and guaranteed to exist. Do we have evidence that such functions are hard, for example, if TFNP is computable in polynomial-time does this imply the polynomial-time hi
A Determinant Expression for the Generalized Bessel Polynomials
Sheng-liang Yang; Sai-nan Zheng
2013-01-01
Using the exponential Riordan arrays, we show that a variation of the generalized Bessel polynomial sequence is of Sheffer type, and we obtain a determinant formula for the generalized Bessel polynomials. As a result, the Bessel polynomial is represented as determinant the entries of which involve Catalan numbers.
On conformal measures for infinitely renormalizable quadratic polynomials
Institute of Scientific and Technical Information of China (English)
HUANG Zhiyong; JIANG Yunping; WANG Yuefei
2005-01-01
We study a conformal measure for an infinitely renormalizable quadratic polynomial. We prove that the conformal measure is ergodic if the polynomial is unbranched and has complex bounds. The main technique we use in the proof is the three-dimensional puzzle for an infinitely renormalizable quadratic polynomial.
Some advances in tensor analysis and polynomial optimization
Li, Zhening; Ling, Chen; Wang, Yiju; Yang, Qingzhi
2014-01-01
Tensor analysis (also called as numerical multilinear algebra) mainly includes tensor decomposition, tensor eigenvalue theory and relevant algorithms. Polynomial optimization mainly includes theory and algorithms for solving optimization problems with polynomial objects functions under polynomial constrains. This survey covers the most of recent advances in these two fields. For tensor analysis, we introduce some properties and algorithms concerning the spectral radius of nonnegative tensors'...
Canonical representation for approximating solution of fuzzy polynomial equations
Directory of Open Access Journals (Sweden)
M. Salehnegad
2010-06-01
Full Text Available In this paper, the concept of canonical representation is proposed to find fuzzy roots of fuzzy polynomial equations. We transform fuzzy polynomial equations to system of crisp polynomial equations, this transformation is perform by using canonical representation based on three parameters Value, Ambiguity and Fuzziness.
Universality for polynomial invariants on ribbon graphs with flags
Avohou, Remi C.; Geloun, Joseph Ben; Hounkonnou , Mahouton N.
2013-01-01
In this paper, we analyze the Bollobas and Riordan polynomial for ribbon graphs with flags introduced in arXiv:1301.1987 and prove its universality. We also show that this polynomial can be defined on some equivalence classes of ribbon graphs involving flag moves and that the new polynomial is still universal on these classes.
High Resolution of the ECG Signal by Polynomial Approximation
Directory of Open Access Journals (Sweden)
G. Rozinaj
2006-04-01
Full Text Available Averaging techniques as temporal averaging and space averaging have been successfully used in many applications for attenuating interference [6], [7], [8], [9], [10]. In this paper we introduce interference removing of the ECG signal by polynomial approximation, with smoothing discrete dependencies, to make up for averaging methods. The method is suitable for low-level signals of the electrical activity of the heart often less than 10 m V. Most low-level signals arising from PR, ST and TP segments which can be detected eventually and their physiologic meaning can be appreciated. Of special importance for the diagnostic of the electrical activity of the heart is the activity bundle of His between P and R waveforms. We have established an artificial sine wave to ECG signal between P and R wave. The aim focus is to verify the smoothing method by polynomial approximation if the SNR (signal-to-noise ratio is negative (i.e. a signal is lower than noise.
Discrete Fourier Analysis and Chebyshev Polynomials with G2 Group
Directory of Open Access Journals (Sweden)
Huiyuan Li
2012-10-01
Full Text Available The discrete Fourier analysis on the 30°-60°-90° triangle is deduced from the corresponding results on the regular hexagon by considering functions invariant under the group G2, which leads to the definition of four families generalized Chebyshev polynomials. The study of these polynomials leads to a Sturm-Liouville eigenvalue problem that contains two parameters, whose solutions are analogues of the Jacobi polynomials. Under a concept of m-degree and by introducing a new ordering among monomials, these polynomials are shown to share properties of the ordinary orthogonal polynomials. In particular, their common zeros generate cubature rules of Gauss type.
Flexible PCPDTBT:PCBM solar cells with integrated grating structures
DEFF Research Database (Denmark)
Oliveira Hansen, Roana Melina de; Liu, Yinghui; Madsen, Morten;
2013-01-01
We report on development of flexible PCPDTBT:PCBM solar cells with integrated diffraction gratings on the bottom electrodes. The presented results address PCPDTBT:PCBM solar cells in an inverted geometry, which contains implemented grating structures whose pitch is tuned to match the absorption...... spectra of the active layer. This optimized solar cell structure leads to an enhanced absorption in the active layer and thus improved short-circuit currents and power conversion efficiencies in the fabricated devices. Fabrication of the solar cells on thin polyimide substrates which are compatible...
Polynomial Operators on Classes of Regular Languages
Klíma, Ondřej; Polák, Libor
We assign to each positive variety mathcal V and each natural number k the class of all (positive) Boolean combinations of the restricted polynomials, i.e. the languages of the form L_0a_1 L_1a_2dots a_ell L_ell, text{ where } ell≤ k, a 1,...,a ℓ are letters and L 0,...,L ℓ are languages from the variety mathcal V. For this polynomial operator we give a certain algebraic counterpart which works with identities satisfied by syntactic (ordered) monoids of languages considered. We also characterize the property that a variety of languages is generated by a finite number of languages. We apply our constructions to particular examples of varieties of languages which are crucial for a certain famous open problem concerning concatenation hierarchies.
Eigenvalue conjecture and colored Alexander polynomials
Mironov, A
2016-01-01
We connect two important conjectures in the theory of knot polynomials. The first one is the property Al_R(q) = Al_{[1]}(q^{|R|}) for all single hook Young diagrams R, which is known to hold for all knots. The second conjecture claims that all the mixing matrices U_{i} in the relation {\\cal R}_i = U_i{\\cal R}_1U_i^{-1} between the i-th and the first generators {\\cal R}_i of the braid group are universally expressible through the eigenvalues of {\\cal R}_1. Since the above property of Alexander polynomials is very well tested, this relation provides a new support to the eigenvalue conjecture, especially for i>2, when its direct check by evaluation of the Racah matrices and their convolutions is technically difficult.
On Polynomial Sized MDP Succinct Policies
Liberatore, P
2011-01-01
Policies of Markov Decision Processes (MDPs) determine the next action to execute from the current state and, possibly, the history (the past states). When the number of states is large, succinct representations are often used to compactly represent both the MDPs and the policies in a reduced amount of space. In this paper, some problems related to the size of succinctly represented policies are analyzed. Namely, it is shown that some MDPs have policies that can only be represented in space super-polynomial in the size of the MDP, unless the polynomial hierarchy collapses. This fact motivates the study of the problem of deciding whether a given MDP has a policy of a given size and reward. Since some algorithms for MDPs work by finding a succinct representation of the value function, the problem of deciding the existence of a succinct representation of a value function of a given size and reward is also considered.
The Medusa Algorithm for Polynomial Matings
DEFF Research Database (Denmark)
Boyd, Suzanne Hruska; Henriksen, Christian
2012-01-01
The Medusa algorithm takes as input two postcritically finite quadratic polynomials and outputs the quadratic rational map which is the mating of the two polynomials (if it exists). Specifically, the output is a sequence of approximations for the parameters of the rational map, as well as an image...... of its Julia set. Whether these approximations converge is answered using Thurston's topological characterization of rational maps. This algorithm was designed by John Hamal Hubbard, and implemented in 1998 by Christian Henriksen and REU students David Farris and Kuon Ju Liu. In this paper we describe...... the algorithm and its implementation, discuss some output from the program (including many pictures) and related questions. Specifically, we include images and a discussion for some shared matings, Lattès examples, and tuning sequences of matings....
Venereau polynomials and related fiber bundles
Kaliman, Shulim; ZAIDENBERG, MIKHAIL
2003-01-01
The Venereau polynomials v-n:=y+x^n(xz+y(yu+z^2)), n>= 1, on A4 have all fibers isomorphic to the affine space A3. Moreover, for all n>= 1 the map (v-n, x) : A4 -> A2 yields a flat family of affine planes over A2. In the present note we show that over the punctured plane A2\\0, this family is a fiber bundle. This bundle is trivial if and only if v-n is a variable of the ring C[x][y,z,u] over C[x]. It is an open question whether v1 and v2 are variables of the polynomial ring C[x,y,z,u]. S. Vene...
On computing factors of cyclotomic polynomials
Brent, Richard P.
1993-07-01
For odd square-free n > 1 the cyclotomic polynomial {Φ_n}(x) satisfies the identity of Gauss, 4{Φ_n}(x) = A_n^2 - {( - 1)^{(n - 1)/2}}nB_n^2. A similar identity of Aurifeuille, Le Lasseur, and Lucas is {Φ_n}({( - 1)^{(n - 1)/2}}x) = C_n^2 - nxD_n^2 or, in the case that n is even and square-free, ± {Φ_{n/2}}( - {x^2}) = C_n^2 - nxD_n^2. Here, {A_n}(x), ldots ,{D_n}(x) are polynomials with integer coefficients. We show how these coefficients can be computed by simple algorithms which require O({n^2}) arithmetic operations and work over the integers. We also give explicit formulae and generating functions for {A_n}(x), ldots ,{D_n}(x) , and illustrate the application to integer factorization with some numerical examples.
Polynomial threshold functions and Boolean threshold circuits
DEFF Research Database (Denmark)
Hansen, Kristoffer Arnsfelt; Podolskii, Vladimir V.
2013-01-01
secondary interest. We show that PTFs on general Boolean domains are tightly connected to depth two threshold circuits. Our main results in regard to this connection are: PTFs of polynomial length and polynomial degree compute exactly the functions computed by THRMAJ circuits. An exponential length lower...... bound for PTFs that holds regardless of degree, thereby extending known lower bounds for THRMAJ circuits. We generalize two-party unbounded error communication complexity to the multi-party number-on-the-forehead setting, and show that communication lower bounds for 3-player protocols would yield size...... lower bounds for THRTHR circuits. We obtain several other results about PTFs. These include relationships between weight and degree of PTFs, and a degree lower bound for PTFs of constant length. We also consider a variant of PTFs over the max-plus algebra. We show that they are connected to PTFs over...
Zernike polynomials for photometric characterization of LEDs
International Nuclear Information System (INIS)
We propose a method based on Zernike polynomials to characterize photometric quantities and descriptors of light emitting diodes (LEDs) from measurements of the angular distribution of the luminous intensity, such as total luminous flux, BA, inhomogeneity, anisotropy, direction of the optical axis and Lambertianity of the source. The performance of this method was experimentally tested for 18 high-power LEDs from different manufacturers and with different photometric characteristics. A small set of Zernike coefficients can be used to calculate all the mentioned photometric quantities and descriptors. For applications not requiring a great accuracy such as those of lighting design, the angular distribution of the luminous intensity of most of the studied LEDs can be interpolated with only two Zernike polynomials. (paper)
Tabulating knot polynomials for arborescent knots
Mironov, A; Morozov, An; Sleptsov, A; Ramadevi, P; Singh, Vivek Kumar
2016-01-01
Arborescent knots are the ones which can be represented in terms of double fat graphs or equivalently as tree Feynman diagrams. This is the class of knots for which the present knowledge is enough for lifting topological description to the level of effective analytical formulas. The paper describes the origin and structure of the new tables of colored knot polynomials, which will be posted at the dedicated site. Even if formal expressions are known in terms of modular transformation matrices, the computation in finite time requires additional ideas. We use the "family" approach, and apply it to arborescent knots in Rolfsen table by developing a Feynman diagram technique, associated with an auxiliary matrix model field theory. Gauge invariance in this theory helps to provide meaning to Racah matrices in the case of non-trivial multiplicities and explains the need for peculiar sign prescriptions in the calculation of [21]-colored HOMFLY polynomials.
Circular β ensembles, CMV representation, characteristic polynomials
Institute of Scientific and Technical Information of China (English)
SU ZhongGen
2009-01-01
In this note we first briefly review some recent progress in the study of the circular β ensemble on the unit circle, where 0 > 0 is a model parameter. In the special cases β = 1,2 and 4, this ensemble describes the joint probability density of eigenvalues of random orthogonal, unitary and sympletic matrices, respectively. For general β, Killip and Nenciu discovered a five-diagonal sparse matrix model, the CMV representation. This representation is new even in the case β = 2; and it has become a powerful tool for studying the circular β ensemble. We then give an elegant derivation for the moment identities of characteristic polynomials via the link with orthogonal polynomials on the unit circle.
Identification of Microcalcifications with Orthogonal Polynomials Model
R. Krishnamoorthy,; N. Amudhavalli,; M.K. Sivakolundu
2010-01-01
Breast cancer is one of the leading causes of women mortality in the world. Mammography has been proved to be the primary radiologic procedure for the early detection of breast cancer. The two main indicators of breast cancer in a mammogram are the presence of a tumor mass and the presence of clustered microcalcifications (MCs). In order to aid radiologists several researchers have developed computer aided diagnosis (CAD) schemes. In this paper a new scheme based on orthogonal polynomials mod...
Moments, positive polynomials and their applications
Lasserre, Jean Bernard
2009-01-01
Many important applications in global optimization, algebra, probability and statistics, applied mathematics, control theory, financial mathematics, inverse problems, etc. can be modeled as a particular instance of the Generalized Moment Problem (GMP) . This book introduces a new general methodology to solve the GMP when its data are polynomials and basic semi-algebraic sets. This methodology combines semidefinite programming with recent results from real algebraic geometry to provide a hierarchy of semidefinite relaxations converging to the desired optimal value. Applied on appropriate cones,
Piecewise polynomial solutions to linear inverse problems
DEFF Research Database (Denmark)
Hansen, Per Christian; Mosegaard, K.
1996-01-01
We have presented a new algorithm PP-TSVD that computes piecewise polynomial solutions to ill-posed problems, without a priori knowledge about the positions of the break points. In particular, we can compute piecewise constant functions that describe layered models. Such solutions are useful, e.......g., in seismological problems, and the algorithm can also be used as a preprocessor for other methods where break points/discontinuities must be incorporated explicitly....
Polynomial approximation of functions in Sobolev spaces
Dupont, T.; Scott, R.
1980-01-01
Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hilbert are presented. Using an averaged Taylor series, we represent a function as a polynomial plus a remainder. The remainder can be manipulated in many ways to give different types of bounds. Approximation of functions in fractional order Sobolev spaces is treated as well as the usual integer order spaces and several nonstandard Sobolev-like spaces.
Normality and shared values concerning differential polynomials
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
Let F be a family of functions meromorphic in a domain D, let P be a polynomial with either deg P≥3 or deg P = 2 and P having only one distinct zero, and let b be a finite nonzero complex number. If, each pair of functions f and g in F, P (f)f and P (g)g share b in D, then F is normal in D.
Conditional Density Approximations with Mixtures of Polynomials
DEFF Research Database (Denmark)
Varando, Gherardo; López-Cruz, Pedro L.; Nielsen, Thomas Dyhre;
2015-01-01
Mixtures of polynomials (MoPs) are a non-parametric density estimation technique especially designed for hybrid Bayesian networks with continuous and discrete variables. Algorithms to learn one- and multi-dimensional (marginal) MoPs from data have recently been proposed. In this paper we introduce...... basis functions (MoTBFs). The empirical results show that the proposed methods generally yield models that are comparable to or significantly better than those found using the MoTBF-based method....
A Deterministic and Polynomial Modified Perceptron Algorithm
Directory of Open Access Journals (Sweden)
Olof Barr
2006-01-01
Full Text Available We construct a modified perceptron algorithm that is deterministic, polynomial and also as fast as previous known algorithms. The algorithm runs in time O(mn3lognlog(1/ρ, where m is the number of examples, n the number of dimensions and ρ is approximately the size of the margin. We also construct a non-deterministic modified perceptron algorithm running in timeO(mn2lognlog(1/ρ.
Reverse-engineering of polynomial dynamical systems
Jarrah, Abdul Salam; Laubenbacher, Reinhard; Stigler, Brandilyn; Stillman, Michael
2006-01-01
Multivariate polynomial dynamical systems over finite fields have been studied in several contexts, including engineering and mathematical biology. An important problem is to construct models of such systems from a partial specification of dynamic properties, e.g., from a collection of state transition measurements. Here, we consider static models, which are directed graphs that represent the causal relationships between system variables, so-called wiring diagrams. This paper contains an algo...
Detecting Prime Numbers via Roots of Polynomials
Dobbs, David E.
2012-01-01
It is proved that an integer n [greater than or equal] 2 is a prime (resp., composite) number if and only if there exists exactly one (resp., more than one) nth-degree monic polynomial f with coefficients in Z[subscript n], the ring of integers modulo n, such that each element of Z[subscript n] is a root of f. This classroom note could find use in…
Time-reversal symmetry and random polynomials
Braun, D; Kus, M.; Zyczkowski, K.
1996-01-01
We analyze the density of roots of random polynomials where each complex coefficient is constructed of a random modulus and a fixed, deterministic phase. The density of roots is shown to possess a singular component only in the case for which the phases increase linearly with the index of coefficients. This means that, contrary to earlier belief, eigenvectors of a typical quantum chaotic system with some antiunitary symmetry will not display a clustering curve in the stellar representation. M...
Georeferencing CAMS data: Polynomial rectification and beyond
Yang, Xinghe
The Calibrated Airborne Multispectral Scanner (CAMS) is a sensor used in the commercial remote sensing program at NASA Stennis Space Center. In geographic applications of the CAMS data, accurate geometric rectification is essential for the analysis of the remotely sensed data and for the integration of the data into Geographic Information Systems (GIS). The commonly used rectification techniques such as the polynomial transformation and ortho rectification have been very successful in the field of remote sensing and GIS for most remote sensing data such as Landsat imagery, SPOT imagery and aerial photos. However, due to the geometric nature of the airborne line scanner which has high spatial frequency distortions, the polynomial model and the ortho rectification technique in current commercial software packages such as Erdas Imagine are not adequate for obtaining sufficient geometric accuracy. In this research, the geometric nature, especially the major distortions, of the CAMS data has been described. An analytical step-by-step geometric preprocessing has been utilized to deal with the potential high frequency distortions of the CAMS data. A generic sensor-independent photogrammetric model has been developed for the ortho-rectification of the CAMS data. Three generalized kernel classes and directional elliptical basis have been formulated into a rectification model of summation of multisurface functions, which is a significant extension to the traditional radial basis functions. The preprocessing mechanism has been fully incorporated into the polynomial, the triangle-based finite element analysis as well as the summation of multisurface functions. While the multisurface functions and the finite element analysis have the characteristics of localization, piecewise logic has been applied to the polynomial and photogrammetric methods, which can produce significant accuracy improvement over the global approach. A software module has been implemented with full
Products of Random Matrices from Polynomial Ensembles
Kieburg, Mario; Kösters, Holger
2016-01-01
Very recently we have shown that the spherical transform is a convenient tool for studying the relation between the joint density of the singular values and that of the eigenvalues for bi-unitarily invariant random matrices. In the present work we discuss the implications of these results for products of random matrices. In particular, we derive a transformation formula for the joint densities of a product of two independent bi-unitarily invariant random matrices, the first from a polynomial ...
Completeness of the ring of polynomials
DEFF Research Database (Denmark)
Thorup, Anders
2015-01-01
Consider the polynomial ring R:=k[X1,…,Xn]R:=k[X1,…,Xn] in n≥2n≥2 variables over an uncountable field k. We prove that R is complete in its adic topology, that is, the translation invariant topology in which the non-zero ideals form a fundamental system of neighborhoods of 0. In addition we pro...
Some Orthogonal Polynomials in Four Variables
Directory of Open Access Journals (Sweden)
Charles F. Dunkl
2008-11-01
Full Text Available The symmetric group on 4 letters has the reflection group $D_3$ as an isomorphic image. This fact follows from the coincidence of the root systems $A_3$ and $D_3$. The isomorphism is used to construct an orthogonal basis of polynomials of 4 variables with 2 parameters. There is an associated quantum Calogero-Sutherland model of 4 identical particles on the line.
Algebraic polynomials and moments of stochastic integrals
Langovoy, Mikhail A
2011-01-01
We propose an algebraic method for proving estimates on moments of stochastic integrals. The method uses qualitative properties of roots of algebraic polynomials from certain general classes. As an application, we give a new proof of a variation of the Burkholder-Davis-Gundy inequality for the case of stochastic integrals with respect to real locally square integrable martingales. Further possible applications and extensions of the method are outlined.
Real meromorphic functions and linear differential polynomials
Institute of Scientific and Technical Information of China (English)
LANGLEY; J.; K.
2010-01-01
We determine all real meromorphic functions f in the plane such that f has finitely many zeros, the poles of f have bounded multiplicities, and f and F have finitely many non-real zeros, where F is a linear differential polynomial given by F = f (k) +Σk-1j=0ajf(j) , in which k≥2 and the coefficients aj are real numbers with a0≠0.
The bivariate Rogers Szegö polynomials
Chen, William Y. C.; Saad, Husam L.; Sun, Lisa H.
2007-06-01
We present an operator approach to deriving Mehler's formula and the Rogers formula for the bivariate Rogers-Szegö polynomials hn(x, y|q). The proof of Mehler's formula can be considered as a new approach to the nonsymmetric Poisson kernel formula for the continuous big q-Hermite polynomials Hn(x; a|q) due to Askey, Rahman and Suslov. Mehler's formula for hn(x, y|q) involves a 3phi2 sum and the Rogers formula involves a 2phi1 sum. The proofs of these results are based on parameter augmentation with respect to the q-exponential operator and the homogeneous q-shift operator in two variables. By extending recent results on the Rogers-Szegö polynomials hn(x|q) due to Hou, Lascoux and Mu, we obtain another Rogers-type formula for hn(x, y|q). Finally, we give a change of base formula for Hn(x; a|q) which can be used to evaluate some integrals by using the Askey-Wilson integral.
The bivariate Rogers-Szegoe polynomials
Energy Technology Data Exchange (ETDEWEB)
Chen, William Y C [Center for Combinatorics, LPMC, Nankai University, Tianjin 300071 (China); Saad, Husam L [Center for Combinatorics, LPMC, Nankai University, Tianjin 300071 (China); Sun, Lisa H [Center for Combinatorics, LPMC, Nankai University, Tianjin 300071 (China)
2007-06-08
We present an operator approach to deriving Mehler's formula and the Rogers formula for the bivariate Rogers-Szegoe polynomials h{sub n}(x, y vertical bar q). The proof of Mehler's formula can be considered as a new approach to the nonsymmetric Poisson kernel formula for the continuous big q-Hermite polynomials H{sub n}(x; a vertical bar q) due to Askey, Rahman and Suslov. Mehler's formula for h{sub n}(x, y vertical bar q) involves a {sub 3}{phi}{sub 2} sum and the Rogers formula involves a {sub 2}{phi}{sub 1} sum. The proofs of these results are based on parameter augmentation with respect to the q-exponential operator and the homogeneous q-shift operator in two variables. By extending recent results on the Rogers-Szegoe polynomials h{sub n}(x vertical bar q) due to Hou, Lascoux and Mu, we obtain another Rogers-type formula for h{sub n}(x, y vertical bar q). Finally, we give a change of base formula for H{sub n}(x; a vertical bar q) which can be used to evaluate some integrals by using the Askey-Wilson integral.
Nested Canalyzing, Unate Cascade, and Polynomial Functions.
Jarrah, Abdul Salam; Raposa, Blessilda; Laubenbacher, Reinhard
2007-09-15
This paper focuses on the study of certain classes of Boolean functions that have appeared in several different contexts. Nested canalyzing functions have been studied recently in the context of Boolean network models of gene regulatory networks. In the same context, polynomial functions over finite fields have been used to develop network inference methods for gene regulatory networks. Finally, unate cascade functions have been studied in the design of logic circuits and binary decision diagrams. This paper shows that the class of nested canalyzing functions is equal to that of unate cascade functions. Furthermore, it provides a description of nested canalyzing functions as a certain type of Boolean polynomial function. Using the polynomial framework one can show that the class of nested canalyzing functions, or, equivalently, the class of unate cascade functions, forms an algebraic variety which makes their analysis amenable to the use of techniques from algebraic geometry and computational algebra. As a corollary of the functional equivalence derived here, a formula in the literature for the number of unate cascade functions provides such a formula for the number of nested canalyzing functions.
Characteristic polynomials of pseudo-Anosov maps
Birman, Joan; Kawamuro, Keiko
2010-01-01
We study the relationship between three different approaches to the action of a pseudo-Anosov mapping class $[F]$ on a surface: the original theorem of Thurston, its algorithmic proof by Bestvina-Handel, and related investigations of Penner-Harer. Bestvina and Handel represent $[F]$ as a suitably chosen homotopy equivalence $f: G\\to G$ of a finite graph, with an associated transition matrix $T$ whose largest eigenvalue is the dilatation of $[F]$. Extending a skew-symmetric form introduced by Penner and Harer to the setting of Bestvina and Handel, we show that the characteristic polynomial of $T$ is a monic and palindromic or anti-palindromic polynomial, possibly multiplied by a power of $x$. Moreover, it factors as a product of three polynomials. One of them reflects the action of $[F]$ on a certain symplectic space, the second one relates to the degeneracies of the skew-symmetric form, and the third one reflects the restriction of $f$ to the vertices of $G$. We give an application to the problem of deciding ...
Role of discriminantly separable polynomials in integrable dynamical systems
Dragović, Vladimir; Kukić, Katarina
2014-11-01
Discriminantly separable polynomials of degree two in each of the three variables are considered. Those polynomials are by definition polynomials which discriminants are factorized as the products of the polynomials in one variable. Motivating example for introducing such polynomials is the famous Kowalevski top. Motivated by the role of such polynomials in the Kowalevski top, we generalize Kowalevski's integration procedure on a whole class of systems basically obtained by replacing so called the Kowalevski's fundamental equation by some other instance of the discriminantly separable polynomial. We present also the role of the discriminantly separable polynomils in twowell-known examples: the case of Kirchhoff elasticae and the Sokolov's case of a rigid body in an ideal fluid.
Novel Bandwidth Sensor Based Fiber Grating
Institute of Scientific and Technical Information of China (English)
无
2003-01-01
In this paper, the basic principle and the design method of the bandwidth sensing of fiber grating are expounded, respectively. Several novel bandwidth sensor based fiber grating are analyzed and discussed.
High Efficiency Low Scatter Echelle Grating Project
National Aeronautics and Space Administration — A high efficiency low scatter echelle grating will be developed using a novel technique of multiple diamond shaving cuts. The grating will have mirror surfaces on...
Polymer optical fiber bragg grating sensors
DEFF Research Database (Denmark)
Stefani, Alessio; Yuan, Scott Wu; Andresen, Søren;
2010-01-01
Fiber-optical accelerometers based on polymer optical fiber Bragg gratings are reported. We have written fiber Bragg gratings for 1550 nm and 850 nm operations, characterized their temperature and strain response, and tested their performance in a prototype accelerometer....
Grating-Coupled Waveguide Cloaking
Institute of Scientific and Technical Information of China (English)
WANG Jia-Fu; QU Shao-Bo; XU Zhuo; MA Hua; WANG Cong-Min; XIA Song; WANG Xin-Hua; ZHOU Hang
2012-01-01
Based on the concept of a grating-coupled waveguide (GCW),a new strategy for realizing EM cloaking is presented.Using metallic grating,incident waves are firstly coupled into the effective waveguide and then decoupled into free space behind,enabling EM waves to pass around the obstacle.Phase compensation in the waveguide keeps the wave-front shape behind the obstacle unchanged.Circular,rectangular and triangular cloaks are presented to verify the robustness of the GCW cloaking.Electric field animations and radar cross section (RCS)comparisons convincingly demonstrate the cloaking effect.
Efficient iterative technique for designing bragg gratings
DEFF Research Database (Denmark)
Plougmann, Nikolai; Kristensen, Martin
2004-01-01
We present a new iterative method for designing Bragg gratings based on the Levenberg-Marquardt method of minimizing a chi-squared merit function. It is effective for designing both weak and strong gratings and is particularly well suited for unchirped gratings....
Hybrid grating reflectors: Origin of ultrabroad stopband
DEFF Research Database (Denmark)
Park, Gyeong Cheol; Taghizadeh, Alireza; Chung, Il-Sug
2016-01-01
Hybrid grating (HG) reflectors with a high-refractive-index cap layer added onto a high contrast grating (HCG) provide a high reflectance close to 100% over a broader wavelength range than HCGs. The combination of a cap layer and a grating layer brings a strong Fabry-Perot (FP) resonance as well...
Tunable Fiber Gratings and Their Applications
Institute of Scientific and Technical Information of China (English)
Z.; Fang; L.; Zhao; L.; Li; K.; Gao; Y.; Zhou; J.; Geng; R.; Qu; G.; Chen
2003-01-01
Some practical research topics on tunable fiber gratings in author's group are presented, including tuning speed, tuning range, tuning characteristics of gratings in HB fiber, and the tunability of the line-width. The applications of fiber gratings in communication and sensing are also discussed.
VCSELs and silicon light sources exploiting SOI grating mirrors
DEFF Research Database (Denmark)
Chung, Il-Sug; Mørk, Jesper
2012-01-01
In this talk, novel vertical-cavity laser structure consisting of a dielectric Bragg reflector, a III-V active region, and a high-index-contrast grating made in the Si layer of a silicon-on-insulator (SOI) wafer will be presented. In the Si light source version of this laser structure, the SOI...... the Bragg reflector. Numerical simulations show that both the silicon light source and the VCSEL exploiting SOI grating mirrors have superior performances, compared to existing silicon light sources and long wavelength VCSELs. These devices are highly adequate for chip-level optical interconnects as well...
Compact Imaging Spectrometer Utilizing Immersed Gratings
Energy Technology Data Exchange (ETDEWEB)
Chrisp, Michael P. (Danville, CA); Lerner, Scott A. (Corvallis, OR); Kuzmenko, Paul J. (Livermore, CA); Bennett, Charles L. (Livermore, CA)
2006-03-21
A compact imaging spectrometer with an immersive diffraction grating that compensates optical distortions. The imaging spectrometer comprises an entrance slit for transmitting light, a system for receiving the light and directing the light, an immersion grating, and a detector array. The entrance slit, the system for receiving the light, the immersion grating, and the detector array are positioned wherein the entrance slit transmits light to the system for receiving the light and the system for receiving the light directs the light to the immersion grating and the immersion grating receives the light and directs the light through an optical element to the detector array.
Calculation of thermal noise in grating reflectors
Heinert, Daniel; Friedrich, Daniel; Hild, Stefan; Kley, Ernst-Bernhard; Leavey, Sean; Martin, Iain W; Nawrodt, Ronny; Tünnermann, Andreas; Vyatchanin, Sergey P; Yamamoto, Kazuhiro
2013-01-01
Grating reflectors have been repeatedly discussed to improve the noise performance of metrological applications due to the reduction or absence of any coating material. So far, however, no quantitative estimate on the thermal noise of these reflective structures exists. In this work we present a theoretical calculation of a grating reflector's noise. We further apply it to a proposed 3rd generation gravitational wave detector. Depending on the grating geometry, the grating material and the temperature we obtain a thermal noise decrease by up to a factor of ten compared to conventional dielectric mirrors. Thus the use of grating reflectors can substantially improve the noise performance in metrological applications.
Time-reversal symmetry and random polynomials
Braun, D; Zyczkowski, K
1996-01-01
We analyze the density of roots of random polynomials where each complex coefficient is constructed of a random modulus and a fixed, deterministic phase. The density of roots is shown to possess a singular component only in the case for which the phases increase linearly with the index of coefficients. This means that, contrary to earlier belief, eigenvectors of a typical quantum chaotic system with some antiunitary symmetry will not display a clustering curve in the stellar representation. Moreover, a class of time-reverse invariant quantum systems is shown, for which spectra display fluctuations characteristic of orthogonal ensemble, while eigenvectors confer to predictions of unitary ensemble.
Softness, Polynomial Boundedness and Amplitudes' Positivity
Bai, Dong
2016-01-01
In this note, we study the connection between infrared (IR) and ultraviolet (UV) behaviors of scattering amplitudes of massless channels by exploiting dispersion relations and positivity bounds. Given forward scattering amplitudes which scale as $\\mathcal{A}(s)\\sim s^M$ in the IR ($s\\to0$) and could be embedded into UV completions satisfying unitarity, analyticity, crossing symmetry and polynomial boundedness $|\\mathcal{A}(s)|< c\\, |s|^N$ ($|s|\\to\\infty$), with $M$ and $N$ integers, we show that the inequality $2\\ceil*{\\frac{N}{2}}\\ge M \\ge 0$ must hold, where $\\ceil*{x}$ is the smallest integer greater than or equal to $x$.
Maximally positive polynomial systems supported on circuits
Bihan, Frédéric
2013-01-01
A real polynomial system with support $\\calW \\subset \\Z^n$ is called {\\it maximally positive} if all its complex solutions are positive solutions. A support $\\calW$ having $n+2$ elements is called a circuit. We previously showed that the number of non-degenerate positive solutions of a system supported on a circuit $\\calW \\subset\\Z^n$ is at most $m(\\calW)+1$, where $m(\\calW) \\leq n$ is the degeneracy index of $\\calW$. We prove that if a circuit $\\calW \\subset \\Z^n$ supports a maximally positi...
Computing the Tutte Polynomial in Vertex-Exponential Time
DEFF Research Database (Denmark)
Björklund, Andreas; Husfeldt, Thore; Kaski, Petteri;
2008-01-01
The deletion–contraction algorithm is perhapsthe most popular method for computing a host of fundamental graph invariants such as the chromatic, flow, and reliability polynomials in graph theory, the Jones polynomial of an alternating link in knot theory, and the partition functions of the models...... algorithm that computes the Tutte polynomial—and hence, all the aforementioned invariants and more—of an arbitrary graph in time within a polynomial factor of the number of connected vertex sets. The algorithm actually evaluates a multivariate generalization of the Tutte polynomial by making use...... of an identity due to Fortuin and Kasteleyn. We also provide a polynomial-space variant of the algorithm and give an analogous result for Chung and Graham's cover polynomial....
Soft x-ray blazed transmission grating spectrometer with high resolving power and extended bandpass
Heilmann, Ralf K.; Bruccoleri, Alexander Robert; Schattenburg, Mark
2016-04-01
A number of high priority questions in astrophysics can be addressed by a state-of-the-art soft x-ray grating spectrometer, such as the role of Active Galactic Nuclei in galaxy and star formation, characterization of the Warm-Hot Intergalactic Medium and the “missing baryon” problem, characterization of halos around the Milky Way and nearby galaxies, as well as stellar coronae and surrounding winds and disks. An Explorer-scale, large-area (> 1,000 cm2), high resolving power (R = λ/Δλ > 3,000) soft x-ray grating spectrometer is highly feasible based on Critical-Angle Transmission (CAT) grating technology. Still significantly higher performance can be provided by a CAT grating spectrometer on an X-ray-Surveyor-type mission. CAT gratings combine the advantages of blazed reflection gratings (high efficiency, use of higher diffraction orders) with those of conventional transmission gratings (low mass, relaxed alignment tolerances and temperature requirements, transparent at higher energies) with minimal mission resource requirements. They are high-efficiency blazed transmission gratings that consist of freestanding, ultra-high aspect-ratio grating bars fabricated from silicon-on-insulator (SOI) wafers using advanced anisotropic dry and wet etch techniques. Blazing is achieved through grazing-incidence reflection off the smooth grating bar sidewalls. The reflection properties of silicon are well matched to the soft x-ray band. Nevertheless, CAT gratings with sidewalls made of higher atomic number elements allow extension of the CAT grating principle to higher energies and larger dispersion angles. We show x-ray data from metal-coated CAT gratings and demonstrate efficient blazing to higher energies and larger blaze angles than possible with silicon alone. We also report on measurements of the resolving power of a breadboard CAT grating spectrometer consisting of a Wolter-I slumped-glass focusing mirror pair from Goddard Space Flight Center and CAT gratings, to be
On an Inequality of Pual Turan Concerning Polynomials-II
Institute of Scientific and Technical Information of China (English)
Abdullah Mir∗
2015-01-01
Let P(z) be a polynomial of degree n and for any complex number α, let DαP(z)=nP(z)+(α−z)P′(z) denote the polar derivative of the polynomial P(z) with respect to α. In this paper, we obtain inequalities for the polar derivative of a poly-nomial having all zeros inside a circle. Our results shall generalize and sharpen some well-known results of Turan, Govil, Dewan et al. and others.
Some Results on the Polar Derivative of a Polynomial
Institute of Scientific and Technical Information of China (English)
Abdullah Mir; Bilal Dar
2014-01-01
Let P(z) be a polynomial of degree n and for any complex number α, let DαP(z)=nP(z)+(α-z)P′(z) denote the polar derivative of P(z) with respect toα. In this paper, we obtain certain inequalities for the polar derivative of a polynomial with restricted zeros. Our results generalize and sharpen some well-known polynomial inequalities.
Conventional modeling of the multilayer perceptron using polynomial basis functions
Chen, Mu-Song; Manry, Michael T.
1993-01-01
A technique for modeling the multilayer perceptron (MLP) neural network, in which input and hidden units are represented by polynomial basis functions (PBFs), is presented. The MLP output is expressed as a linear combination of the PBFs and can therefore be expressed as a polynomial function of its inputs. Thus, the MLP is isomorphic to conventional polynomial discriminant classifiers or Volterra filters. The modeling technique was successfully applied to several trained MLP networks.
THE λ-GR(O)BNER BASES UNDER POLYNOMIAL COMPOSITION
Institute of Scientific and Technical Information of China (English)
Jinwang LIU; Dongmei LI; Xiaosong CHEN
2007-01-01
Polynomial composition is the operation of replacing variables in a polynomial with other polynomials. λ-Gr(o)bner basis is an especial Gr(o)bner basis. The main problem in the paper is: when does composition commute with λ-Gr(o)bner basis computation? We shall answer better the above question. This has a natural application in the computation of λ-Gr(o)bner bases.
Multiple Twisted -Euler Numbers and Polynomials Associated with -Adic -Integrals
Directory of Open Access Journals (Sweden)
Jang Lee-Chae
2008-01-01
Full Text Available By using -adic -integrals on , we define multiple twisted -Euler numbers and polynomials. We also find Witt's type formula for multiple twisted -Euler numbers and discuss some characterizations of multiple twisted -Euler Zeta functions. In particular, we construct multiple twisted Barnes' type -Euler polynomials and multiple twisted Barnes' type -Euler Zeta functions. Finally, we define multiple twisted Dirichlet's type -Euler numbers and polynomials, and give Witt's type formula for them.
Quantization of gauge fields, graph polynomials and graph cohomology
Kreimer, Dirk; Sars, Matthias; van Suijlekom, Walter D.
2012-01-01
We review quantization of gauge fields using algebraic properties of 3-regular graphs. We derive the Feynman integrand at n loops for a non-abelian gauge theory quantized in a covariant gauge from scalar integrands for connected 3-regular graphs, obtained from the two Symanzik polynomials. The transition to the full gauge theory amplitude is obtained by the use of a third, new, graph polynomial, the corolla polynomial. This implies effectively a covariant quantization without ghosts, where al...
Darboux polynomials for Lotka-Volterra systems in three dimensions
Christodoulides, Yiannis T
2008-01-01
We consider Lotka-Volterra systems in three dimensions depending on three real parameters. By using elementary algebraic methods we classify the Darboux polynomials (also known as second integrals) for such systems for various values of the parameters, and give the explicit form of the corresponding cofactors. More precisely, we show that a Darboux polynomial of degree greater than one is reducible. In fact, it is a product of linear Darboux polynomials and first integrals.
Characteristic and Counting Polynomials: Modelling Nonane Isomers Properties
Jäntschi, Lorentz; BOLBOACA, Sorana D.; FURDUI, Cristina Maria
2009-01-01
Abstract The major goal of this study was to investigate the broad application of graph polynomials to the analysis of Henry?s law constants (solubility) of nonane isomers. In this context, Henry?s law constants of nonane isomers were modelled using characteristic and counting polynomials. The characteristic and counting polynomials on the distance matrix, on the maximal fragments matrix, on the complement of maximal fragments matrix, and on the Szeged matrix were calculated for ea...
Quantum Schubert polynomials and the Vafa-Intriligator formula
Kirillov, A N; Kirillov, Anatol N.; Maeno, Toshiaki
1996-01-01
We introduce a quantization map and study the quantization of Schubert and Grothendieck polynomials, monomials, elementary and complete polynomials. Our construction is based on the quantum Cauchy identity. As a corollary, we prove the Lascoux-Schützenberger type formula for quantum Schubert polynomials of the flag manifold. Our formula gives a simple method for computation of quantum Schubert polynomials. We also prove the higher genus analog of Vafa-Intriligator's formula for the flag manifold. We introduce the Extended Ehresman-Bruhat order on the symmetric group and prove the equivariant quantum Pieri formula.
An Analytic Formula for the A_2 Jack Polynomials
Directory of Open Access Journals (Sweden)
Vladimir V. Mangazeev
2007-01-01
Full Text Available In this letter I shall review my joint results with Vadim Kuznetsov and Evgeny Sklyanin [Indag. Math. 14 (2003, 451-482] on separation of variables (SoV for the $A_n$ Jack polynomials. This approach originated from the work [RIMS Kokyuroku 919 (1995, 27-34] where the integral representations for the $A_2$ Jack polynomials was derived. Using special polynomial bases I shall obtain a more explicit expression for the $A_2$ Jack polynomials in terms of generalised hypergeometric functions.
Structure relations for monic orthogonal polynomials in two discrete variables
Rodal, J.; Area, I.; Godoy, E.
2008-04-01
In this paper, extensions of several relations linking differences of bivariate discrete orthogonal polynomials and polynomials themselves are given, by using an appropriate vector-matrix notation. Three-term recurrence relations are presented for the partial differences of the monic polynomial solutions of admissible second order partial difference equation of hypergeometric type. Structure relations, difference representations as well as lowering and raising operators are obtained. Finally, expressions for all matrix coefficients appearing in these finite-type relations are explicitly presented for a finite set of Hahn and Kravchuk orthogonal polynomials.
The distribution of zeros of general q-polynomials
Álvarez-Nodarse, R.; Buendía, E.; Dehesa, J. S.
1997-10-01
A general system of q-orthogonal polynomials is defined by means of its three-term recurrence relation. This system encompasses many of the known families of q-polynomials, among them the q-analogue of the classical orthogonal polynomials. The asymptotic density of zeros of the system is shown to be a simple and compact expression of the parameters which characterize the asymptotic behaviour of the coefficients of the recurrence relation. This result is applied to specific classes of polynomials known by the names q-Hahn, q-Kravchuk, q-Racah, q-Askey and Wilson, Al Salam - Carlitz and the celebrated little and big q-Jacobi.
Multi-indexed Jacobi polynomials and Maya diagrams
Takemura, Kouichi
2014-11-01
Multi-indexed Jacobi polynomials are defined by the Wronskian of four types of eigenfunctions of the Pöschl-Teller Hamiltonian. We give a correspondence between multi-indexed Jacobi polynomials and pairs of Maya diagrams, and we show that any multi-indexed Jacobi polynomial is essentially equal to some multi-indexed Jacobi polynomial of two types of eigenfunction. As an application, we show a Wronskian-type formula of some special eigenstates of the deformed Pöschl-Teller Hamiltonian.
Polynomial analysis of canopy spectra and biochemical component content inversion
Institute of Scientific and Technical Information of China (English)
YAN Chunyan; LIU Qiang; NIU Zheng; WANG Jihua; HUANG Wenjiang; LIU Liangyun
2005-01-01
A polynomial expression model was developed in this paper to describe directional canopy spectra, and the decomposition of the polynomial expression was used as a tool for retrieving biochemical component content from canopy multi-angle spectra. First, the basic formula of the polynomial expression was introduced and the physical meaning of its terms and coefficients was discussed. Based on this analysis, a complete polynomial expression model and its decomposition method were given. By decomposing the canopy spectra simulated with SAILH model, it shows that the polynomial expression can not only fit well the canopy spectra, but also show the contribution of every order scattering to the whole reflectance. Taking the first scattering coefficients a10 and a01 for example, the test results show that the polynomial coefficients reflect very well the hot spot phenomenon and the effects of viewing angles, LAI and leaf inclination angle on canopy spectra. By coupling the polynomial expression with leaf model PROSPECT, a canopy biochemical component content inversion model was given. In the simulated test, the canopy multi-angle spectra were simulated by two different models, SAILH and 4-SCALE respectively, then the biochemical component content was retrieved by inverting the coupled polynomial expression + PROSPECT model. Results of the simulated test are promising, and when applying the algorithm to measured corn canopy multi-angle spectra, we also get relatively accurate chlorophyll content. It shows that the polynomial analysis provides a new method to get biochemical component content independent of any specific canopy model.
Multi-indexed Wilson and Askey-Wilson Polynomials
Odake, Satoru
2013-01-01
As the third stage of the project multi-indexed orthogonal polynomials, we present, in the framework of 'discrete quantum mechanics' with pure imaginary shifts in one dimension, the multi-indexed Wilson and Askey-Wilson polynomials. They are obtained from the original Wilson and Askey-Wilson polynomials by multiple application of the discrete analogue of the Darboux transformations or the Crum-Krein-Adler deletion of 'virtual state solutions' of type I and II, in a similar way to the multi-indexed Laguerre, Jacobi and (q-)Racah polynomials reported earlier.
ASYMPTOTIC EXPANSIONS OF ZEROS FOR KRAWTCHOUK POLYNOMIALS WITH ERROR BOUNDS
Institute of Scientific and Technical Information of China (English)
ZHU Xiao-feng; LI Xiu-chun
2006-01-01
Krawtchouk polynomials are frequently applied in modern physics. Based on the results which were educed by Li and Wong, the asymptotic expansions of Krawtchouk polynomials are improved by using Airy function, and uniform asymptotic expansions are got. Furthermore, the asymptotic expansions of the zeros for Krawtchouk polynomials are again deduced by using the property of the zeros of Airy function, and their corresponding error bounds are discussed. The obtained results give the asymptotic property of Krawtchouk polynomials with their zeros, which are better than the results educed by Li and Wong.
A Bivariate Analogue to the Composed Product of Polynomials
Institute of Scientific and Technical Information of China (English)
Donald Mills; Kent M. Neuerburg
2003-01-01
The concept of a composed product for univariate polynomials has been explored extensively by Brawley, Brown, Carlitz, Gao,Mills, et al. Starting with these fundamental ideas andutilizing fractional power series representation(in particular, the Puiseux expansion) of bivariate polynomials, we generalize the univariate results. We define a bivariate composed sum,composed multiplication,and composed product (based on function composition). Further, we investigate the algebraic structure of certain classes of bivariate polynomials under these operations. We also generalize a result of Brawley and Carlitz concerningthe decomposition of polynomials into irreducibles.
On factorization of generalized Macdonald polynomials
Kononov, Ya.; Morozov, A.
2016-08-01
A remarkable feature of Schur functions—the common eigenfunctions of cut-and-join operators from W_∞ —is that they factorize at the peculiar two-parametric topological locus in the space of time variables, which is known as the hook formula for quantum dimensions of representations of U_q(SL_N) and which plays a big role in various applications. This factorization survives at the level of Macdonald polynomials. We look for its further generalization to generalized Macdonald polynomials (GMPs), associated in the same way with the toroidal Ding-Iohara-Miki algebras, which play the central role in modern studies in Seiberg-Witten-Nekrasov theory. In the simplest case of the first-coproduct eigenfunctions, where GMP depend on just two sets of time variables, we discover a weak factorization—on a one- (rather than four-) parametric slice of the topological locus, which is already a very non-trivial property, calling for proof and better understanding.
On Factorization of Generalized Macdonald Polynomials
Kononov, Ya
2016-01-01
A remarkable feature of Schur functions -- the common eigenfunctions of cut-and-join operators from $W_\\infty$ -- is that they factorize at the peculiar two-parametric topological locus in the space of time-variables, what is known as the hook formula for quantum dimensions of representations of $U_q(SL_N)$ and plays a big role in various applications. This factorization survives at the level of Macdonald polynomials. We look for its further generalization to generalized Macdonald polynomials (GMP), associated in the same way with the toroidal Ding-Iohara-Miki algebras, which play the central role in modern studies in Seiberg-Witten-Nekrasov theory. In the simplest case of the first-coproduct eigenfunctions, where GMP depend on just two sets of time-variables, we discover a weak factorization -- on a one- (rather than four-) parametric slice of the topological locus, what is already a very non-trivial property, calling for proof and better understanding.
Effective grating theory for resonance domain surface-relief diffraction gratings.
Golub, Michael A; Friesem, Asher A
2005-06-01
An effective grating model, which generalizes effective-medium theory to the case of resonance domain surface-relief gratings, is presented. In addition to the zero order, it takes into account the first diffraction order, which obeys the Bragg condition. Modeling the surface-relief grating as an effective grating with two diffraction orders provides closed-form analytical relationships between efficiency and grating parameters. The aspect ratio, the grating period, and the required incidence angle that would lead to high diffraction efficiencies are predicted for TE and TM polarization and verified by rigorous numerical calculations.
Polymer lasers assembled by suspending membranes on a distributed feedback grating.
Zhai, Tianrui; Tong, Fei; Wang, Yonglu; Wu, Xiaofeng; Li, Songtao; Wang, Meng; Zhang, Xinping
2016-09-19
Polymer lasers are fabricated by an assembly method. A polymer membrane is directly attached on the one- or two- dimensional grating. The suspended membrane acts as an active waveguide, which is supported by the grating ridge, leaving air gaps in the grating valley. Most of the radiation is effectively confined within the active waveguide due to the strong reflection at the membrane/air interfaces. So, low threshold lasing can be achieved when the sample is optically pumped. This fabrication method provides an alternative to investigate low-threshold polymer lasers. PMID:27661937
Fiber Bragg Grating Based Thermometry
Ahmed, Zeeshan; Filla, James; Guthrie, William; Quintavalle, John
2016-01-01
In recent years there has been considerable interest in developing photonic temperature sensors such as the Fiber Bragg gratings (FBG) as an alternative to resistance thermometry. In this study we examine the thermal response of FBGs over the temperature range of 233 K to 393 K. We demonstrate, in a hermetically sealed dry Argon environment, that FBG devices show a quadratic dependence on temperature with expanded uncertainties (k = 2) of ~500 mK. Our measurements indicate that the combined m...
Diffraction gratings for lighting applications
Cornelissen, Hugo J.; de Boer, Dick K. G.; Tukker, Teus
2013-09-01
Sub-micron diffraction gratings have been used for two LED illumination applications. One is to create a transparent see through luminaire which can be used to illuminate and read a paper document or e-book. A second is a light sensor that can be used in a feedback loop to control a multicolor LED lamp. Optical design and experimental proof-of-principle are presented.
Fabrication of large-area and low mass critical-angle x-ray transmission gratings
Heilmann, Ralf K.; Bruccoleri, Alex R.; Guan, Dong; Schattenburg, Mark L.
2014-07-01
Soft x-ray spectroscopy of celestial sources with high resolving power R = E/ΔE and large collecting area addresses important science listed in the Astro2010 Decadal Survey New Worlds New Horizons, such as the growth of the large scale structure of the universe and its interaction with active galactic nuclei, the kinematics of galactic outflows, as well as coronal emission from stars and other topics. Numerous studies have shown that a transmission grating spectrometer based on lightweight critical-angle transmission (CAT) gratings can deliver R = 3000-5000 and large collecting area with high efficiency and minimal resource requirements, providing spectroscopic figures of merit at least an order of magnitude better than grating spectrometers on Chandra and XMM-Newton, as well as future calorimeter-based missions. The recently developed CAT gratings combine the advantages of transmission gratings (low mass, relaxed figure and alignment tolerances) and blazed reflection gratings (high broad band diffraction efficiency, utilization of higher diffraction orders). Their working principle based on blazing through reflection off the smooth, ultra-high aspect ratio grating bar sidewalls has previously been demonstrated on small samples with x rays. For larger gratings (area greater than 1 inch square) we developed a fabrication process for grating membranes with a hierarchy of integrated low-obscuration supports. The fabrication involves a combination of advanced lithography and highly anisotropic dry and wet etching techniques. We report on the latest fabrication results of free-standing, large-area CAT gratings with polished sidewalls and preliminary x-ray tests.
A new two-variable generalization of the Jones polynomial
Goundaroulis, Dimos; Lambropoulou, Sofia
2016-01-01
We present a new 2-variable generalization of the Jones polynomial that can be defined through the skein relation of the Jones polynomial. The well-definedness of this new generalization is proved algebraically. We also give a closed combinatorial formula for this new classical link invariant.
Learning Read-constant Polynomials of Constant Degree modulo Composites
DEFF Research Database (Denmark)
Chattopadhyay, Arkadev; Gavaldá, Richard; Hansen, Kristoffer Arnsfelt;
2011-01-01
is not known to be learnable in any reasonable learning model. In this paper, we provide a deterministic polynomial time algorithm for learning Boolean functions represented by polynomials of constant degree over arbitrary finite rings from membership queries, with the additional constraint that each variable...
Mutations of Laurent Polynomials and Flat Families with Toric Fibers
Ilten, Nathan Owen
2012-01-01
We give a general criterion for two toric varieties to appear as fibers in a flat family over the projective line. We apply this to show that certain birational transformations mapping a Laurent polynomial to another Laurent polynomial correspond to deformations between the associated toric varieties.
Animating Nested Taylor Polynomials to Approximate a Function
Mazzone, Eric F.; Piper, Bruce R.
2010-01-01
The way that Taylor polynomials approximate functions can be demonstrated by moving the center point while keeping the degree fixed. These animations are particularly nice when the Taylor polynomials do not intersect and form a nested family. We prove a result that shows when this nesting occurs. The animations can be shown in class or…
Pointwise Approximation Theorems for Combinations and Derivatives of Bernstein Polynomials
Institute of Scientific and Technical Information of China (English)
Lin Sen XIE
2005-01-01
We establish the pointwise approximation theorems for the combinations of Bernstein polynomials by the rth Ditzian-Totik modulus of smoothness ωγφ(f, t) where φ is an admissible step-weight function. An equivalence relation between the derivatives of these polynomials and the smoothness of functions is also obtained.
Lm Extremal Polynomials Associated with Generalized Jacobi Weights
Institute of Scientific and Technical Information of China (English)
Ying-guang Shi
2003-01-01
Asymptotic estimations of the Christoffel type functions for Lm extremal polynomials with an even integer m associated with generalized Jacobi weights are established. Also, asymptotic behavior of the zeros of the Lm extremal polynomials and the Cotes numbers of the corresponding Turan quadrature formula is given.
On -Euler Numbers Related to the Modified -Bernstein Polynomials
Min-Soo Kim; Daeyeoul Kim; Taekyun Kim
2010-01-01
We consider q-Euler numbers, polynomials, and q-Stirling numbers of first and second kinds. Finally, we investigate some interesting properties of the modified q-Bernstein polynomials related to q-Euler numbers and q-Stirling numbers by using fermionic p-adic integrals on ℤp.
Classification of complex polynomial vector fields in one complex variable
DEFF Research Database (Denmark)
Branner, Bodil; Dias, Kealey
2010-01-01
This paper classifies the global structure of monic and centred one-variable complex polynomial vector fields. The classification is achieved by means of combinatorial and analytic data. More specifically, given a polynomial vector field, we construct a combinatorial invariant, describing...
A Parallel Algorithm for Finding Roots of a Complex Polynomial
Institute of Scientific and Technical Information of China (English)
程锦松
1990-01-01
A distribution theory of the roots of a polynomial and a parallel algorithm for finding roots of a complex polynomial based on that theory are developed in this paper.With high parallelism,the algorithm is an improvement over the Wilf algorithm[3].
On the Structure of Cubic and Quartic Polynomials
Haramaty, Elad
2009-01-01
In this paper we study the structure of polynomials of degree three and four that have high bias or high Gowers norm, over arbitrary prime fields. In particular we obtain the following results. 1. We give a canonical representation for degree three or four polynomials that have a significant bias (i.e. they are not equidistributed). This result generalizes the corresponding results from the theory of quadratic forms. It also significantly improves the results of Green and Tao and Kaufman and Lovett for such polynomials. 2. For the case of degree four polynomials with high Gowers norm we show that (a subspace of co-dimension O(1) of) F^n can be partitioned to subspaces of dimension Omega(n) such that on each of the subspaces the polynomial is equal to some degree three polynomial. It was shown by Green and Tao and by Lovett, Meshulam and Samorodnitsky that a quartic polynomial with a high Gowers norm is not necessarily correlated with any cubic polynomial. Our result shows that a slightly weaker statement does...
Approximation to Continuous Functions by a Kind of Interpolation Polynomials
Institute of Scientific and Technical Information of China (English)
Yuan Xue-gang; Wang De-hui
2001-01-01
In this paper, an interpolation polynomial operator Fn (f; l, x ) is constructed based on the zeros of a kind of Jacobi polynomials as the interpolation nodes. For any continuous function f(x)∈ Cb[1,1] (0≤b≤l) Fn(f; l,x) converges to f(x) uniformly, where l is an odd number.
POLYNOMIAL RECURRENCE FOR L（E）VY PROCESSES
Institute of Scientific and Technical Information of China (English)
ZHAO MINZHI; YING JIANGANG
2004-01-01
In this paper, the authors study the ω-transience and ω-recurrence for Lévy processes with any weight function ω, give a relation between ω-recurrence and the last exit times. As a special case, the polynomial recurrence and polynomial transience are also studied.
Varied line-space gratings and applications
International Nuclear Information System (INIS)
This paper presents a straightforward analytical and numerical method for the design of a specific type of varied line-space grating system. The mathematical development will assume plane or nearly-plane spherical gratings which are illuminated by convergent light, which covers many interesting cases for synchrotron radiation. The gratings discussed will have straight grooves whose spacing varies across the principal plane of the grating. Focal relationships and formulae for the optical grating-pole-to-exist-slit distance and grating radius previously presented by other authors will be derived with a symbolic algebra system. It is intended to provide the optical designer with the tools necessary to design such a system properly. Finally, some possible advantages and disadvantages for application to synchrotron to synchrotron radiation beamlines will be discussed
The spectral combination characteristic of grating and the bi-grating diffraction imaging effect
Institute of Scientific and Technical Information of China (English)
ZHANG WeiPing; HE XiaoRong
2007-01-01
This paper reports on a new property of grating,namely spectral combination,and on bi-grating diffraction imaging that is based on spectral combination.The spectral combination characteristic of a grating is the capability of combining multiple light beams of different wavelengths incident from specific angles into a single beam.The bi-grating diffraction imaging is the formation of the image of an object with two gratings: the first grating disperses the multi-color light beams from the object and the second combines the dispersed light beams to form the image.We gave the conditions necessary for obtaining the spectral combination.We also presented the equations that relate the two gratings' spatial frequencies,diffraction orders and positions necessary for obtaining the bi-grating diffraction imaging.
Identification of Microcalcifications with Orthogonal Polynomials Model
Directory of Open Access Journals (Sweden)
R. Krishnamoorthy,
2010-05-01
Full Text Available Breast cancer is one of the leading causes of women mortality in the world. Mammography has been proved to be the primary radiologic procedure for the early detection of breast cancer. The two main indicators of breast cancer in a mammogram are the presence of a tumor mass and the presence of clustered microcalcifications (MCs. In order to aid radiologists several researchers have developed computer aided diagnosis (CAD schemes. In this paper a new scheme based on orthogonal polynomials model (OPM has been proposed and same scheme is applied for the identification of MCs in x-ray mammograms. Experimental results on both digitized mammogram images from the Mammographic Image Analysis Society (MIAS database and digital mammograms from local database are presented to demonstrate the effectiveness of the proposed method.
Digital terrain modeling with the Chebyshev polynomials
Florinsky, I V
2015-01-01
Mathematical problems of digital terrain analysis include interpolation of digital elevation models (DEMs), DEM generalization and denoising, and computation of morphometric variables by calculation of partial derivatives of elevation. Traditionally, these procedures are based on numerical treatments of two-variable discrete functions of elevation. We developed a spectral analytical method and algorithm based on high-order orthogonal expansions using the Chebyshev polynomials of the first kind with the subsequent Fejer summation. The method and algorithm are intended for DEM analytical treatment, such as, DEM global approximation, denoising, and generalization as well as computation of morphometric variables by analytical calculation of partial derivatives. To test the method and algorithm, we used a DEM of the Northern Andes including 230,880 points (the elevation matrix 480 $\\times$ 481). DEMs were reconstructed with 480, 240, 120, 60, and 30 expansion coefficients. The first and second partial derivatives ...
Inequalities for a Polynomial and its Derivative
Indian Academy of Sciences (India)
V K Jain
2000-05-01
For an arbitrary entire function and any > 0, let $M(f, r):=\\max_{|z|=r}|f(z)|$. It is known that if is a polynomial of degree having no zeros in the open unit disc, and $m:=\\min_{|z|=1}|p(z)|$, then $$M(p',1)≤\\frac{n}{2}\\{M(p,1)-m\\},$$ $$M(p, R)≤\\left(\\frac{R^n+1}{2}\\right)M(p, 1)-m\\left(\\frac{R^n-1}{2}\\right), R>> 1.$$ It is also known that if has all its zeros in the closed unit disc, then $$M(p', 1)≥\\frac{n}{2}\\{M(p, 1)+m\\}.$$ The present paper contains certain generalizations of these inequalities
Study on the Grey Polynomial Geometric Programming
Institute of Scientific and Technical Information of China (English)
LUODang
2005-01-01
In the model of geometric programming, values of parameters cannot be gotten owing to data fluctuation and incompletion. But reasonable bounds of these parameters can be attained. This is to say, parameters of this model can be regarded as interval grey numbers. When the model contains grey numbers, it is hard for common programming method to solve them. By combining the common programming model with the grey system theory,and using some analysis strategies, a model of grey polynomial geometric programming, a model of 8 positioned geometric programming and their quasi-optimum solution or optimum solution are put forward. At the same time, we also developed an algorithm for the problem.This approach brings a new way for the application research of geometric programming. An example at the end of this paper shows the rationality and feasibility of the algorithm.
The Polynomially Exponential Time Restrained Analytical Hierarchy
Institute of Scientific and Technical Information of China (English)
眭跃飞
1991-01-01
A polynomially exponential time restrained analytical hierarchy is introduced with the basic properties of the hierarchy followed.And it will be shown that there is a recursive set A such that A does not belong to any level of the p-arithmetical hierarchies.Then we shall prove that there are recursive sets A and B such that the different levels of the analytical hierarchy relative to A are different and for some n every level higher than n of the analytical hierarchy relative to B is the same as the n-th level.And whether the higher levels are collapsed into some lower level is neither provable nor disprovable in set theory and several other results.
Approximate protein structural alignment in polynomial time.
Kolodny, Rachel; Linial, Nathan
2004-08-17
Alignment of protein structures is a fundamental task in computational molecular biology. Good structural alignments can help detect distant evolutionary relationships that are hard or impossible to discern from protein sequences alone. Here, we study the structural alignment problem as a family of optimization problems and develop an approximate polynomial-time algorithm to solve them. For a commonly used scoring function, the algorithm runs in O(n(10)/epsilon(6)) time, for globular protein of length n, and it detects alignments that score within an additive error of epsilon from all optima. Thus, we prove that this task is computationally feasible, although the method that we introduce is too slow to be a useful everyday tool. We argue that such approximate solutions are, in fact, of greater interest than exact ones because of the noisy nature of experimentally determined protein coordinates. The measurement of similarity between a pair of protein structures used by our algorithm involves the Euclidean distance between the structures (appropriately rigidly transformed). We show that an alternative approach, which relies on internal distance matrices, must incorporate sophisticated geometric ingredients if it is to guarantee optimality and run in polynomial time. We use these observations to visualize the scoring function for several real instances of the problem. Our investigations yield insights on the computational complexity of protein alignment under various scoring functions. These insights can be used in the design of scoring functions for which the optimum can be approximated efficiently and perhaps in the development of efficient algorithms for the multiple structural alignment problem. PMID:15304646
Higher order branching of periodic orbits from polynomial isochrones
Directory of Open Access Journals (Sweden)
B. Toni
1999-09-01
Full Text Available We discuss the higher order local bifurcations of limit cycles from polynomial isochrones (linearizable centers when the linearizing transformation is explicitly known and yields a polynomial perturbation one-form. Using a method based on the relative cohomology decomposition of polynomial one-forms complemented with a step reduction process, we give an explicit formula for the overall upper bound of branch points of limit cycles in an arbitrary $n$ degree polynomial perturbation of the linear isochrone, and provide an algorithmic procedure to compute the upper bound at successive orders. We derive a complete analysis of the nonlinear cubic Hamiltonian isochrone and show that at most nine branch points of limit cycles can bifurcate in a cubic polynomial perturbation. Moreover, perturbations with exactly two, three, four, six, and nine local families of limit cycles may be constructed.
Asymptotic behaviour of zeros of exceptional Jacobi and Laguerre polynomials
Gómez-Ullate, David; Milson, Robert
2012-01-01
The location and asymptotic behaviour for large n of the zeros of exceptional Jacobi and Laguerre polynomials are discussed. The zeros of exceptional polynomials fall into two classes: the regular zeros, which lie in the interval of orthogonality and the exceptional zeros, which lie outside that interval. We show that the regular zeros have two interlacing properties: one is the natural interlacing between consecutive polynomials as a consequence of their Sturm-Liouville character, while the other one shows interlacing between the zeros of exceptional and classical polynomials. A generalization of the classical Heine-Mehler formula is provided for the exceptional polynomials, which allows to derive the asymptotic behaviour of their regular zeros. We also describe the location and the asymptotic behaviour of the exceptional zeros, which converge for large n to fixed values.
Properties of the zeros of generalized basic hypergeometric polynomials
Bihun, Oksana; Calogero, Francesco
2015-11-01
We define the generalized basic hypergeometric polynomial of degree N in terms of the generalized basic hypergeometric function, by choosing one of its parameters to allow the termination of the series after a finite number of summands. In this paper, we obtain a set of nonlinear algebraic equations satisfied by the N zeros of the polynomial. Moreover, we obtain an N × N matrix M defined in terms of the zeros of the polynomial, which, in turn, depend on the parameters of the polynomial. The eigenvalues of this remarkable matrix M are given by neat expressions that depend only on some of the parameters of the polynomial; that is, the matrix M is isospectral. Moreover, in case the parameters that appear in the expressions for the eigenvalues of M are rational, the matrix M has rational eigenvalues, a Diophantine property.
Multiple Meixner polynomials and non-Hermitian oscillator Hamiltonians
Ndayiragije, F.; Van Assche, W.
2013-12-01
Multiple Meixner polynomials are polynomials in one variable which satisfy orthogonality relations with respect to r > 1 different negative binomial distributions (Pascal distributions). There are two kinds of multiple Meixner polynomials, depending on the selection of the parameters in the negative binomial distribution. We recall their definition and some formulas and give generating functions and explicit expressions for the coefficients in the nearest neighbor recurrence relation. Following a recent construction of Miki, Tsujimoto, Vinet and Zhedanov (for multiple Meixner polynomials of the first kind), we construct r > 1 non-Hermitian oscillator Hamiltonians in r dimensions which are simultaneously diagonalizable and for which the common eigenstates are expressed in terms of multiple Meixner polynomials of the second kind.
Modelling Trends in Ordered Correspondence Analysis Using Orthogonal Polynomials.
Lombardo, Rosaria; Beh, Eric J; Kroonenberg, Pieter M
2016-06-01
The core of the paper consists of the treatment of two special decompositions for correspondence analysis of two-way ordered contingency tables: the bivariate moment decomposition and the hybrid decomposition, both using orthogonal polynomials rather than the commonly used singular vectors. To this end, we will detail and explain the basic characteristics of a particular set of orthogonal polynomials, called Emerson polynomials. It is shown that such polynomials, when used as bases for the row and/or column spaces, can enhance the interpretations via linear, quadratic and higher-order moments of the ordered categories. To aid such interpretations, we propose a new type of graphical display-the polynomial biplot. PMID:25791164
Structured matrix based methods for approximate polynomial GCD
Boito, Paola
2011-01-01
Defining and computing a greatest common divisor of two polynomials with inexact coefficients is a classical problem in symbolic-numeric computation. The first part of this book reviews the main results that have been proposed so far in the literature. As usual with polynomial computations, the polynomial GCD problem can be expressed in matrix form: the second part of the book focuses on this point of view and analyses the structure of the relevant matrices, such as Toeplitz, Toepliz-block and displacement structures. New algorithms for the computation of approximate polynomial GCD are presented, along with extensive numerical tests. The use of matrix structure allows, in particular, to lower the asymptotic computational cost from cubic to quadratic order with respect to polynomial degree. .
Hurvitz, G.; Ehrlich, Y.; Strum, G.; Shpilman, Z.; Levy, I.; Fraenkel, M.
2012-01-01
A novel fabrication method for soft x-ray transmission grating and other optical elements is presented. The method uses Focused-Ion-Beam (FIB) technology to fabricate high-quality free standing grating bars on Transmission Electron Microscopy grids (TEM-grid). High quality transmission gratings are obtained with superb accuracy and versatility. Using these gratings and back-illuminated CCD camera, absolutely calibrated x-ray spectra can be acquired for soft x-ray source diagnostics in the 100...
Multilayer dielectric gratings for tiled-gratings compression of petawatt pulses.
Cotel, Arnaud
2007-01-01
Pulse compression diffraction gratings represent currently an important bottleneck for the development of energetic high-intensity Petawatt laser. Indeed, the laser-induced damage threshold of standard gold-coated gratings and the diffraction efficiency are limited. That's why we have developed a new generation of diffraction gratings: ! multilayer dielectric (MLD) gratings. Studies of MLD g! ratings applied to the Pico2000 laser project are the first part of my thesis work. On the other hand...
Perfect crystallike gratings for cold neutrons
International Nuclear Information System (INIS)
We report on significant improvements of the performance of thick diffraction gratings for cold neutrons. The basis material for the production of holographic gratings by optical means is photosensitized deuterated poly(methyl methacrylate) (D-PMMA). The properties of these gratings now approach those of perfect monochromator crystals for neutrons of shorter wavelength: for cold neutrons with 1.0 nm wavelength the gratings exhibit a reflectivity in the percent range which makes them suitable for a wide range of applications in neutron optics
Etessami, Kousha; Yannakakis, Mihalis
2012-01-01
We show that one can approximate the least fixed point solution for a multivariate system of monotone probabilistic max(min) polynomial equations, referred to as maxPPSs (and minPPSs, respectively), in time polynomial in both the encoding size of the system of equations and in log(1/epsilon), where epsilon > 0 is the desired additive error bound of the solution. (The model of computation is the standard Turing machine model.) We establish this result using a generalization of Newton's method which applies to maxPPSs and minPPSs, even though the underlying functions are only piecewise-differentiable. This generalizes our recent work which provided a P-time algorithm for purely probabilistic PPSs. These equations form the Bellman optimality equations for several important classes of infinite-state Markov Decision Processes (MDPs). Thus, as a corollary, we obtain the first polynomial time algorithms for computing to within arbitrary desired precision the optimal value vector for several classes of infinite-state...
Monolithic integrated optic fiber Bragg grating sensor interrogator
Mendoza, Edgar A.; Esterkin, Yan; Kempen, Cornelia; Sun, Songjian
2010-04-01
Fiber Bragg gratings (FBGs) are a mature sensing technology that has gained rapid acceptance in civil, aerospace, chemical and petrochemical, medicine, aviation and automotive industries. Fiber Bragg grating sensors can be use for a variety of measurements including strain, stress, vibration, acoustics, acceleration, pressure, temperature, moisture, and corrosion distributed at multiple locations within the structure using a single fiber element. The most prominent advantages of FBGs are: small size and light weight, multiple FBG transducers on a single fiber, and immunity to radio frequency interference. A major disadvantage of FBG technology is that conventional state-of-the-art fiber Bragg grating interrogation systems are typically bulky, heavy, and costly bench top instruments that are assembled from off-the-shelf fiber optic and optical components integrated with a signal electronics board into an instrument console. Based on the need for a compact FBG interrogation system, this paper describes recent progress towards the development of a miniature fiber Bragg grating sensor interrogator (FBG-TransceiverTM) system based on multi-channel monolithic integrated optic sensor microchip technology. The integrated optic microchip technology enables the monolithic integration of all of the functionalities, both passive and active, of conventional bench top FBG sensor interrogators systems, packaged in a miniaturized, low power operation, 2-cm x 5-cm small form factor (SFF) package suitable for the long-term structural health monitoring in applications where size, weight, and power are critical for operation.
A link polynomial via a vertex-edge-face state model
Fiedler, Thomas
2007-01-01
We construct a 2-variable link polynomial, called $W_L$, for classical links by considering simultaneously the Kauffman state models for the Alexander and for the Jones polynomials. We conjecture that this polynomial is the product of two 1-variable polynomials, one of which is the Alexander polynomial. We refine $W_L$ to an ordered set of 3-variable polynomials for those links in 3-space which contain a Hopf link as a sublink.
Fiber Bragg Grating Based Thermometry
Ahmed, Zeeshan; Guthrie, William; Quintavalle, John
2016-01-01
In recent years there has been considerable interest in developing photonic temperature sensors such as the Fiber Bragg gratings (FBG) as an alternative to resistance thermometry. In this study we examine the thermal response of FBGs over the temperature range of 233 K to 393 K. We demonstrate, in a hermetically sealed dry Argon environment, that FBG devices show a quadratic dependence on temperature with expanded uncertainties (k = 2) of ~500 mK. Our measurements indicate that the combined measurement uncertainty is dominated by uncertainty in determining the peak center fitting and by thermal aging of polyimide coated fibers.
Analytical and numerical study on grating depth effects in grating coupled waveguide sensors
DEFF Research Database (Denmark)
Horvath, R.; Wilcox, L.C.; Pedersen, H.C.;
2005-01-01
. Analytical expressions are derived both for the position and width of the in-coupling peaks to illustrate the effects of grating depth on the guided mode resonances in grating coupled waveguide sensors. Numerical computations verify the model for shallow gratings both in terms of peak shape and position......The in-coupling process for grating-coupled planar optical waveguide sensors is investigated in the case of TE waves. A simple analytical model based on the Rayleigh-Fourier-Kiselev method is applied to take into account the depth of the grating coupler, which is usually neglected in the modeling...
Conformal Laplace superintegrable systems in 2D: polynomial invariant subspaces
Escobar-Ruiz, M. A.; Miller, Willard, Jr.
2016-07-01
2nd-order conformal superintegrable systems in n dimensions are Laplace equations on a manifold with an added scalar potential and 2n-1 independent 2nd order conformal symmetry operators. They encode all the information about Helmholtz (eigenvalue) superintegrable systems in an efficient manner: there is a 1-1 correspondence between Laplace superintegrable systems and Stäckel equivalence classes of Helmholtz superintegrable systems. In this paper we focus on superintegrable systems in two-dimensions, n = 2, where there are 44 Helmholtz systems, corresponding to 12 Laplace systems. For each Laplace equation we determine the possible two-variate polynomial subspaces that are invariant under the action of the Laplace operator, thus leading to families of polynomial eigenfunctions. We also study the behavior of the polynomial invariant subspaces under a Stäckel transform. The principal new results are the details of the polynomial variables and the conditions on parameters of the potential corresponding to polynomial solutions. The hidden gl 3-algebraic structure is exhibited for the exact and quasi-exact systems. For physically meaningful solutions, the orthogonality properties and normalizability of the polynomials are presented as well. Finally, for all Helmholtz superintegrable solvable systems we give a unified construction of one-dimensional (1D) and two-dimensional (2D) quasi-exactly solvable potentials possessing polynomial solutions, and a construction of new 2D PT-symmetric potentials is established.
Alshal, Mohamed A.; Allam, Nageh K.
2016-07-01
A design for a highly efficient modified grating crystalline silicon (c-Si) thin film solar cell is demonstrated and analyzed using the two-dimensional (2-D) finite element method. The suggested grating has a double-sided pyramidal structure. The incorporation of the modified grating in a c-Si thin film solar cell offers a promising route to harvest light into the few micrometers active layer. Furthermore, a layer of silicon nitride is used as an antireflection coating (ARC). Additionally, the light trapping through the suggested design is significantly enhanced by the asymmetry of the top and bottom pyramids. The effects of the thickness of the active layer and facet angle of the pyramid on the spectral absorption, ultimate efficiency (η), and short-circuit current density (J sc) are investigated. The numerical results showed 87.9% efficiency improvement over the conventional thin film c-Si solar cell counterpart without gratings.
vs. a polynomial chaos-based MCMC
Siripatana, Adil
2014-08-01
Bayesian Inference of Manning\\'s n coefficient in a Storm Surge Model Framework: comparison between Kalman lter and polynomial based method Adil Siripatana Conventional coastal ocean models solve the shallow water equations, which describe the conservation of mass and momentum when the horizontal length scale is much greater than the vertical length scale. In this case vertical pressure gradients in the momentum equations are nearly hydrostatic. The outputs of coastal ocean models are thus sensitive to the bottom stress terms de ned through the formulation of Manning\\'s n coefficients. This thesis considers the Bayesian inference problem of the Manning\\'s n coefficient in the context of storm surge based on the coastal ocean ADCIRC model. In the first part of the thesis, we apply an ensemble-based Kalman filter, the singular evolutive interpolated Kalman (SEIK) filter to estimate both a constant Manning\\'s n coefficient and a 2-D parameterized Manning\\'s coefficient on one ideal and one of more realistic domain using observation system simulation experiments (OSSEs). We study the sensitivity of the system to the ensemble size. we also access the benefits from using an in ation factor on the filter performance. To study the limitation of the Guassian restricted assumption on the SEIK lter, 5 we also implemented in the second part of this thesis a Markov Chain Monte Carlo (MCMC) method based on a Generalized Polynomial chaos (gPc) approach for the estimation of the 1-D and 2-D Mannning\\'s n coe cient. The gPc is used to build a surrogate model that imitate the ADCIRC model in order to make the computational cost of implementing the MCMC with the ADCIRC model reasonable. We evaluate the performance of the MCMC-gPc approach and study its robustness to di erent OSSEs scenario. we also compare its estimates with those resulting from SEIK in term of parameter estimates and full distributions. we present a full analysis of the solution of these two methods, of the
Tilt sensitivity of the two-grating interferometer
Energy Technology Data Exchange (ETDEWEB)
Anderson, Christopher N.; Naulleau, Patrick P.
2008-01-30
Fringe formation in the two-grating interferometer is analyzed in the presence of a small parallelism error between the diffraction gratings assumed in the direction of grating shear. Our analysis shows that with partially coherent illumination, fringe contrast in the interference plane is reduced in the presence of nonzero grating tilt with the effect proportional to the grating tilt angle and the grating spatial frequencies. Our analysis also shows that for a given angle between the gratings there is an angle between the final grating and the interference plane that optimizes fringe contrast across the field.
A comparison of high-order polynomial and wave-based methods for Helmholtz problems
Lieu, Alice; Gabard, Gwénaël; Bériot, Hadrien
2016-09-01
The application of computational modelling to wave propagation problems is hindered by the dispersion error introduced by the discretisation. Two common strategies to address this issue are to use high-order polynomial shape functions (e.g. hp-FEM), or to use physics-based, or Trefftz, methods where the shape functions are local solutions of the problem (typically plane waves). Both strategies have been actively developed over the past decades and both have demonstrated their benefits compared to conventional finite-element methods, but they have yet to be compared. In this paper a high-order polynomial method (p-FEM with Lobatto polynomials) and the wave-based discontinuous Galerkin method are compared for two-dimensional Helmholtz problems. A number of different benchmark problems are used to perform a detailed and systematic assessment of the relative merits of these two methods in terms of interpolation properties, performance and conditioning. It is generally assumed that a wave-based method naturally provides better accuracy compared to polynomial methods since the plane waves or Bessel functions used in these methods are exact solutions of the Helmholtz equation. Results indicate that this expectation does not necessarily translate into a clear benefit, and that the differences in performance, accuracy and conditioning are more nuanced than generally assumed. The high-order polynomial method can in fact deliver comparable, and in some cases superior, performance compared to the wave-based DGM. In addition to benchmarking the intrinsic computational performance of these methods, a number of practical issues associated with realistic applications are also discussed.
The spectral combination characteristic of grating and the bi-grating diffraction imaging effect
Institute of Scientific and Technical Information of China (English)
2007-01-01
This paper reports on a new property of grating, namely spectral combination, and on bi-grating diffraction imaging that is based on spectral combination. The spectral combination characteristic of a grating is the capability of combining multiple light beams of different wavelengths incident from specific angles into a single beam. The bi-grating diffraction imaging is the formation of the image of an object with two gratings: the first grating disperses the multi-color light beams from the object and the second combines the dispersed light beams to form the image. We gave the conditions necessary for obtaining the spectral combination. We also presented the equations that relate the two gratings’ spatial frequencies, diffraction orders and positions necessary for obtaining the bi-grating diffraction imaging.
Inverse Scattering for Gratings and Wave Guides
Eskin, Gregory; Ralston, James; Yamamoto, Masahiro
2007-01-01
We consider the problem of unique identification of dielectric coefficients for gratings and sound speeds for wave guides from scattering data. We prove that the "propagating modes" given for all frequencies uniquely determine these coefficients. The gratings may contain conductors as well as dielectrics and the boundaries of the conductors are also determined by the propagating modes.
Straw combustion on slow-moving grates
DEFF Research Database (Denmark)
Kær, Søren Knudsen
2005-01-01
Combustion of straw in grate-based boilers is often associated with high emission levels and relatively poor fuel burnout. A numerical grate combustion model was developed to assist in improving the combustion performance of these boilers. The model is based on a one-dimensional ‘‘walking...
Reflectivity-modulated grating-mirror
DEFF Research Database (Denmark)
2012-01-01
to the oscillation axis. A modulated voltage (91) is applied in reverse bias between the n- and p-doped layers to modulate the refractive index of the electrooptic material layer (12) and thereby the reflectivity spectrum of the grating mirror (1). The reflectivity of the grating mirror (1) can be modulated between...
Antireflective characteristics of hemispherical grid grating
Institute of Scientific and Technical Information of China (English)
REN Zhibin; JIANG Huilin; LIU Guojun; SUN Qiang
2005-01-01
In this paper, the optical characteristics of new type hemispherical grid subwavelength grating are studied by using multi-level column structure approximation and rigorous coupled-wave analysis. This kind of grating could be fabricated by chemical methods, thus simplifying the fabrication technology of subwavelength gratings for visible light. By computer simulation and calculation, the hemispherical grid subwavelength gratings are proved to have antireflective characteristics. Two design schemes of this kind of grating are presented. In the first scheme, the grating could achieve a reflectivity as low as 3.4416×10-7, which can be adapted to 0.46―0.7 μm of visible waveband and ±12° incident angle field. In the second scheme, the grating can achieve a reflectivity as low as 3.112×10-4 and adapted to the whole visible waveband and ±23° incident angle field. The application field of the latter scheme is wider than that of the former. The results of this paper could provide reference for the applications of the hemispherical grid subwavelength gratings for the visible waveband.
Ladder operators and recursion relations for the associated Bessel polynomials
International Nuclear Information System (INIS)
Introducing the associated Bessel polynomials in terms of two non-negative integers, and under an integrability condition we simultaneously factorize their corresponding differential equation into a product of the ladder operators by four different ways as shape invariance symmetry equations. This procedure gives four different pairs of recursion relations on the associated Bessel polynomials. In spite of description of Bessel and Laguerre polynomials in terms of each other, we show that the associated Bessel differential equation is factorized in four different ways whereas for Laguerre one we have three different ways
Ladder operators and recursion relations for the associated Bessel polynomials
Fakhri, H.; Chenaghlou, A.
2006-10-01
Introducing the associated Bessel polynomials in terms of two non-negative integers, and under an integrability condition we simultaneously factorize their corresponding differential equation into a product of the ladder operators by four different ways as shape invariance symmetry equations. This procedure gives four different pairs of recursion relations on the associated Bessel polynomials. In spite of description of Bessel and Laguerre polynomials in terms of each other, we show that the associated Bessel differential equation is factorized in four different ways whereas for Laguerre one we have three different ways.
Ladder operators and recursion relations for the associated Bessel polynomials
Energy Technology Data Exchange (ETDEWEB)
Fakhri, H. [Institute for Studies in Theoretical Physics and Mathematics (IPM), PO Box 19395-5531, Tehran (Iran, Islamic Republic of) and Department of Theoretical Physics and Astrophysics, Physics Faculty, Tabriz University, PO Box 51666-16471, Tabriz (Iran, Islamic Republic of)]. E-mail: hfakhri@ipm.ir; Chenaghlou, A. [Institute for Studies in Theoretical Physics and Mathematics (IPM), PO Box 19395-5531, Tehran (Iran, Islamic Republic of) and Physics Department, Faculty of Science, Sahand University of Technology, PO Box 51335-1996, Tabriz (Iran, Islamic Republic of)]. E-mail: a.chenaghlou@sut.ac.ir
2006-10-30
Introducing the associated Bessel polynomials in terms of two non-negative integers, and under an integrability condition we simultaneously factorize their corresponding differential equation into a product of the ladder operators by four different ways as shape invariance symmetry equations. This procedure gives four different pairs of recursion relations on the associated Bessel polynomials. In spite of description of Bessel and Laguerre polynomials in terms of each other, we show that the associated Bessel differential equation is factorized in four different ways whereas for Laguerre one we have three different ways.
Stable radial distortion calibration by polynomial matrix inequalities programming
Heller, Jan; Pajdla, Tomas
2014-01-01
Polynomial and rational functions are the number one choice when it comes to modeling of radial distortion of lenses. However, several extrapolation and numerical issues may arise while using these functions that have not been covered by the literature much so far. In this paper, we identify these problems and show how to deal with them by enforcing nonnegativity of certain polynomials. Further, we show how to model these nonnegativities using polynomial matrix inequalities (PMI) and how to estimate the radial distortion parameters subject to PMI constraints using semidefinite programming (SDP). Finally, we suggest several approaches on how to incorporate the proposed method into the overall camera calibration procedure.
Polynomial complexity algorithm for Max-Cut problem
Katkov, Mikhail
2010-01-01
The standard NP-complete max-cut problem is reformulated as a binary quadratic program xQx s.t x^2=1. This problem is further reformulated as global minimum of quartic polynomial (xQ'x - z)^2 + \\sum_i (x_i^2-1)^2+ \\alpha z^2, for some \\alpha. The global minimum is found by polynomial complexity semi-definite program. Numerical examples and code is provided. The resulting algorithm solves arbitrary max-cut problem in polynomial time, therefore P=NP.
Representations of non-negative polynomials via critical ideals
Hiep, Dang Tuan
2011-01-01
This paper studies the representations of a non-negative polynomial $f$ on a non-compact semi-algebraic set $K$ modulo its critical ideal. Under the assumptions that the semi-algebraic set $K$ is regular and $f$ satisfies the boundary Hessian conditions (BHC) at each zero of $f$ in $K$, we show that $f$ can be represented as a sum of squares (SOS) of real polynomials modulo its critical ideal if $f\\ge 0$ on $K$. In particular, we focus on the polynomial ring $\\mathbb R[x]$.
q-Bernoulli numbers and q-Bernoulli polynomials revisited
Directory of Open Access Journals (Sweden)
Kim Taekyun
2011-01-01
Full Text Available Abstract This paper performs a further investigation on the q-Bernoulli numbers and q-Bernoulli polynomials given by Acikgöz et al. (Adv Differ Equ, Article ID 951764, 9, 2010, some incorrect properties are revised. It is point out that the generating function for the q-Bernoulli numbers and polynomials is unreasonable. By using the theorem of Kim (Kyushu J Math 48, 73-86, 1994 (see Equation 9, some new generating functions for the q-Bernoulli numbers and polynomials are shown. Mathematics Subject Classification (2000 11B68, 11S40, 11S80
Generalized Freud's equation and level densities with polynomial potential
Boobna, Akshat; Ghosh, Saugata
2013-08-01
We study orthogonal polynomials with weight $\\exp[-NV(x)]$, where $V(x)=\\sum_{k=1}^{d}a_{2k}x^{2k}/2k$ is a polynomial of order 2d. We derive the generalised Freud's equations for $d=3$, 4 and 5 and using this obtain $R_{\\mu}=h_{\\mu}/h_{\\mu -1}$, where $h_{\\mu}$ is the normalization constant for the corresponding orthogonal polynomials. Moments of the density functions, expressed in terms of $R_{\\mu}$, are obtained using Freud's equation and using this, explicit results of level densities as $N\\rightarrow\\infty$ are derived.
Describing Quadratic Cremer Point Polynomials by Parabolic Perturbations
DEFF Research Database (Denmark)
Sørensen, Dan Erik Krarup
1996-01-01
We describe two infinite order parabolic perturbation proceduresyielding quadratic polynomials having a Cremer fixed point. The main ideais to obtain the polynomial as the limit of repeated parabolic perturbations.The basic tool at each step is to control the behaviour of certain externalrays.......Polynomials of the Cremer type correspond to parameters at the boundary of ahyperbolic component of the Mandelbrot set. In this paper we concentrate onthe main cardioid component. We investigate the differences between two-sided(i.e. alternating) and one-sided parabolic perturbations.In the two-sided case, we prove...
Generalizations of Poly-Bernoulli numbers and polynomials
Jolany, Hassan; Darafsheh, M. R.; Alikelaye, R. Eizadi
2012-01-01
The Concepts of poly-Bernoulli numbers $B_n^{(k)}$, poly-Bernoulli polynomials $B_n^{k}{(t)}$ and the generalized poly-bernoulli numbers $B_{n}^{(k)}(a,b)$ are generalized to $B_{n}^{(k)}(t,a,b,c)$ which is called the generalized poly-Bernoulli polynomials depending on real parameters \\textit{a,b,c}. Some properties of these polynomials and some relationships between $B_n^{k}$, $B_n^{(k)}(t)$, $B_{n}^{(k)}(a,b)$ and $B_{n}^{(k)}(t,a,b,c)$ are established
Alignment free characterization of 2D gratings
Madsen, Morten Hannibal; Hansen, Poul-Erik; Jørgensen, Jan Friis
2015-01-01
Fast characterization of 2-dimensional gratings is demonstrated using a Fourier lens optical system and a differential optimization algorithm. It is shown that both the grating specific parameters such as the basis vectors and the angle between them and the alignment of the sample, such as the rotation of the sample around the x-, y-, and z-axis, can be deduced from a single measurement. More specifically, the lattice vectors and the angle between them have been measured, while the corrections of the alignment parameters are used to improve the quality of the measurement, and hence reduce the measurement uncertainty. Alignment free characterization is demonstrated on both a 2D hexagonal grating with a period of 700 nm and a checkerboard grating with a pitch of 3000 nm. The method can also be used for both automatic alignment and in-line characterization of gratings.
Compact imaging spectrometer utilizing immersed gratings
Energy Technology Data Exchange (ETDEWEB)
Chrisp, Michael P. (Danville, CA); Lerner, Scott A. (Corvallis, OR); Kuzmenko, Paul J. (Livermore, CA); Bennett, Charles L. (Livermore, CA)
2007-07-03
A compact imaging spectrometer with an immersive diffraction grating that compensates optical distortions. The imaging spectrometer comprises an entrance slit for transmitting light, means for receiving the light and directing the light, an immersion grating, and a detector array. The entrance slit, the means for receiving the light, the immersion grating, and the detector array are positioned wherein the entrance slit transmits light to the means for receiving the light and the means for receiving the light directs the light to the immersion grating and the immersion grating receives the light and directs the light to the means for receiving the light, and the means for receiving the light directs the light to the detector array.
Constructing Polynomial Spectral Models for Stars
Rix, Hans-Walter; Conroy, Charlie; Hogg, David W
2016-01-01
Stellar spectra depend on the stellar parameters and on dozens of photospheric elemental abundances. Simultaneous fitting of these $\\mathcal{N}\\sim \\,$10-40 model labels to observed spectra has been deemed unfeasible, because the number of ab initio spectral model grid calculations scales exponentially with $\\mathcal{N}$. We suggest instead the construction of a polynomial spectral model (PSM) of order $\\mathcal{O}$ for the model flux at each wavelength. Building this approximation requires a minimum of only ${\\mathcal{N}+\\mathcal{O}\\choose\\mathcal{O}}$ calculations: e.g. a quadratic spectral model ($\\mathcal{O}=\\,$2), which can then fit $\\mathcal{N}=\\,$20 labels simultaneously, can be constructed from as few as 231 ab initio spectral model calculations; in practice, a somewhat larger number ($\\sim\\,$300-1000) of randomly chosen models lead to a better performing PSM. Such a PSM can be a good approximation to ab initio spectral models only over a limited portion of label space, which will vary case by case. Y...
Lower bounds for polynomials using geometric programming
Ghasemi, Mehdi
2011-01-01
We make use of a result of Hurwitz and Reznick, and a consequence of this result due to Fidalgo and Kovacec, to determine a new sufficient condition for a polynomial $f\\in\\mathbb{R}[X_1,...,X_n]$ of even degree to be a sum of squares. This result generalizes a result of Lasserre and a result of Fidalgo and Kovacec, and it also generalizes the improvements of these results given in [6]. We apply this result to obtain a new lower bound $f_{gp}$ for $f$, and we explain how $f_{gp}$ can be computed using geometric programming. The lower bound $f_{gp}$ is generally not as good as the lower bound $f_{sos}$ introduced by Lasserre and Parrilo and Sturmfels, which is computed using semidefinite programming, but a run time comparison shows that, in practice, the computation of $f_{gp}$ is much faster. The computation is simplest when the highest degree term of $f$ has the form $\\sum_{i=1}^n a_iX_i^{2d}$, $a_i>0$, $i=1,...,n$. The lower bounds for $f$ established in [6] are obtained by evaluating the objective function ...
Polynomial super-gl(n) algebras
International Nuclear Information System (INIS)
We introduce a class of finite-dimensional nonlinear superalgebras L L0-bar + L1-bar providing gradings of L0-bar = gl(n) ≅ sl(n) + gl(1). Odd generators close by anticommutation on polynomials (of degree >1) in the gl(n) generators. Specifically, we investigate 'type I' super-gl(n) algebras, having odd generators transforming in a single irreducible representation of gl(n) together with its contragredient. Admissible structure constants are discussed in terms of available gl(n) couplings, and various special cases and candidate superalgebras are identified and exemplified via concrete oscillator constructions. For the case of the n-dimensional defining representation, with odd generators Qa, Q-barb and even generators Eab, a, b = 1, ..., n, a three-parameter family of quadratic super-gl(n) algebras (deformations of sl(n/1)) is defined. In general, additional covariant Serre-type conditions are imposed in order that the Jacobi identities are fulfilled. For these quadratic super-gl(n) algebras, the construction of Kac modules and conditions for atypicality are briefly considered. Applications in quantum field theory, including Hamiltonian lattice QCD and spacetime supersymmetry, are discussed
Polynomial Time Algorithms for Minimum Energy Scheduling
Baptiste, Philippe; Durr, Christoph
2009-01-01
The aim of power management policies is to reduce the amount of energy consumed by computer systems while maintaining satisfactory level of performance. One common method for saving energy is to simply suspend the system during the idle times. No energy is consumed in the suspend mode. However, the process of waking up the system itself requires a certain fixed amount of energy, and thus suspending the system is beneficial only if the idle time is long enough to compensate for this additional energy expenditure. In the specific problem studied in the paper, we have a set of jobs with release times and deadlines that need to be executed on a single processor. Preemptions are allowed. The processor requires energy L to be woken up and, when it is on, it uses one unit of energy per one unit of time. It has been an open problem whether a schedule minimizing the overall energy consumption can be computed in polynomial time. We solve this problem in positive, by providing an O(n^5)-time algorithm. In addition we pr...
New Polynomial Law of Hadron Mass
Bottini, S
1998-01-01
A simple and general law of mass, intrinsically with zero freely-adaptable parameters, is shown to be valid for all the hadrons with one or more flavored (s, c, or b) quarks/antiquarks, both baryons and mesons. It establishes correspondences H-->{P_i} between these hadrons with at least one flavored constituent, H, and specific sets of lighter particles, {P_i}, in which the total baryon and lepton numbers are conserved. The law is polynomial with a varying degree that univocally depends only on the particles involved. Its statement may be given straightforwardly in terms of the binomial series converging to 1/sqrt{1-beta}. It asserts that, for each hadron H, there exists a certain set of lighter particles {P_i}, together with a specific reference hadron h_0 that fixes the mass scale, such that, for a value of the sum of this power series equal to the mass of H, a definite partial sum of the series equals the total mass of particles {P_i}. The starting, independent data in the rule are the masses of the hadron...
Polynomial Method for PLL Controller Optimization
Directory of Open Access Journals (Sweden)
Tsung-Yu Chiou
2011-06-01
Full Text Available The Phase-Locked Loop (PLL is a key component of modern electronic communication and control systems. PLL is designed to extract signals from transmission channels. It plays an important role in systems where it is required to estimate the phase of a received signal, such as carrier tracking from global positioning system satellites. In order to robustly provide centimeter-level accuracy, it is crucial for the PLL to estimate the instantaneous phase of an incoming signal which is usually buried in random noise or some type of interference. This paper presents an approach that utilizes the recent development in the semi-definite programming and sum-of-squares field. A Lyapunov function will be searched as the certificate of the pull-in range of the PLL system. Moreover, a polynomial design procedure is proposed to further refine the controller parameters for system response away from the equilibrium point. Several simulation results as well as an experiment result are provided to show the effectiveness of this approach.
International Nuclear Information System (INIS)
A new spectrometer with no moving parts uses a two-dimensional Si-based charge-coupled diode (CCD) array detector and an integrated grating consisting of three subgratings. The effective spectral range imaged on the detector is magnified threefold. The digitized spectral image in the 200-1000 nm wavelength range can be measured quickly. The nonlinear relationship between CCD pixel position and wavelength is corrected with multiple polynomial functions in the calibration procedure, which fits the data using a mathematical pattern-analysis method. The instrument can be applied for rapid spectroscopic data analyses in many types of photoelectronic experiments and routine testing
Certified counting of roots of random univariate polynomials
Cleveland, Joseph; Hauenstein, Jonathan D; Haywood, Ian; Mehta, Dhagash; Morse, Anthony; Robol, Leonardo; Schlenk, Taylor
2014-01-01
A challenging problem in computational mathematics is to compute roots of a high-degree univariate random polynomial. We combine an efficient multiprecision implementation for solving high-degree random polynomials with two certification methods, namely Smale's $\\alpha$-theory and one based on Gerschgorin's theorem, for showing that a given numerical approximation is in the quadratic convergence region of Newton's method of some exact solution. With this combination, we can certifiably count the number of real roots of random polynomials. We quantify the difference between the two certification procedures and list the salient features of both of them. After benchmarking on random polynomials where the coefficients are drawn from the Gaussian distribution, we obtain novel experimental results for the Cauchy distribution case.
Quantization of gauge fields, graph polynomials and graph homology
Energy Technology Data Exchange (ETDEWEB)
Kreimer, Dirk, E-mail: kreimer@physik.hu-berlin.de [Humboldt University, 10099 Berlin (Germany); Sars, Matthias [Humboldt University, 10099 Berlin (Germany); Suijlekom, Walter D. van [Radboud University Nijmegen, 6525 AJ Nijmegen (Netherlands)
2013-09-15
We review quantization of gauge fields using algebraic properties of 3-regular graphs. We derive the Feynman integrand at n loops for a non-abelian gauge theory quantized in a covariant gauge from scalar integrands for connected 3-regular graphs, obtained from the two Symanzik polynomials. The transition to the full gauge theory amplitude is obtained by the use of a third, new, graph polynomial, the corolla polynomial. This implies effectively a covariant quantization without ghosts, where all the relevant signs of the ghost sector are incorporated in a double complex furnished by the corolla polynomial–we call it cycle homology–and by graph homology. -- Highlights: •We derive gauge theory Feynman from scalar field theory with 3-valent vertices. •We clarify the role of graph homology and cycle homology. •We use parametric renormalization and the new corolla polynomial.
On polynomial mappings from the plane to the plane
Krzyżanowska, Iwona; Szafraniec, Zbigniew
2012-01-01
Let $f:{\\mathbb R}^2\\longrightarrow {\\mathbb R}^2$ be a generic polynomial mapping. There are constructed quadratic forms whose signatures determine the number of positive and negative cusps of $f$.
Inner approximations for polynomial matrix inequalities and robust stability regions
Henrion, Didier
2011-01-01
Following a polynomial approach, many robust fixed-order controller design problems can be formulated as optimization problems whose set of feasible solutions is modelled by parametrized polynomial matrix inequalities (PMI). These feasibility sets are typically nonconvex. Given a parametrized PMI set, we provide a hierarchy of linear matrix inequality (LMI) problems whose optimal solutions generate inner approximations modelled by a single polynomial sublevel set. Those inner approximations converge in a strong analytic sense to the nonconvex original feasible set, with asymptotically vanishing conservatism. One may also impose the hierarchy of inner approximations to be nested or convex. In the latter case they do not converge any more to the feasible set, but they can be used in a convex optimization framework at the price of some conservatism. Finally, we show that the specific geometry of nonconvex polynomial stability regions can be exploited to improve convergence of the hierarchy of inner approximation...
Symmetric polynomials in information theory: Entropy and subentropy
Energy Technology Data Exchange (ETDEWEB)
Jozsa, Richard; Mitchison, Graeme [DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA (United Kingdom)
2015-06-15
Entropy and other fundamental quantities of information theory are customarily expressed and manipulated as functions of probabilities. Here we study the entropy H and subentropy Q as functions of the elementary symmetric polynomials in the probabilities and reveal a series of remarkable properties. Derivatives of all orders are shown to satisfy a complete monotonicity property. H and Q themselves become multivariate Bernstein functions and we derive the density functions of their Levy-Khintchine representations. We also show that H and Q are Pick functions in each symmetric polynomial variable separately. Furthermore, we see that H and the intrinsically quantum informational quantity Q become surprisingly closely related in functional form, suggesting a special significance for the symmetric polynomials in quantum information theory. Using the symmetric polynomials, we also derive a series of further properties of H and Q.
An operator approach to the Al-Salam-Carlitz polynomials
Chen, William Y. C.; Saad, Husam L.; Sun, Lisa H.
2010-04-01
We present an operator approach to Rogers-type formulas and Mehler's formula for the Al-Salam-Carlitz polynomials Un(x,y,a;q). By using the q-exponential operator, we obtain a Rogers-type formula, which leads to a linearization formula. With the aid of a bivariate augmentation operator, we get a simple derivation of Mehler's formula due to Al-Salam and Carlitz ["Some orthogonal q-polynomials," Math. Nachr. 30, 47 (1965)]. By means of the Cauchy companion augmentation operator, we obtain an equivalent form of Mehler's formula. We also give several identities on the generating functions for products of the Al-Salam-Carlitz polynomials, which are extensions of the formulas for the Rogers-Szegö polynomials.
APPROXIMATION BY GENERALIZED MKZ-OPERATORS IN POLYNOMIAL WEIGHTED SPACES
Institute of Scientific and Technical Information of China (English)
Lucyna Rempulska; Mariola Skorupka
2007-01-01
We prove some approximation properties of generalized Meyer-K(o)nig and Zeller operators for differentiable functions in polynomial weighted spaces. The results extend some results proved in [ 1-3,7-16].
Discrete Darboux transformation for discrete polynomials of hypergeometric type
Bangerezako, Gaspard
1998-03-01
The Darboux transformation, well known in second-order differential operator theory, is applied to the difference equations satisfied by the discrete hypergeometric polynomials (Charlier, Meixner-Kravchuk, Hahn).
SOME EXTREMAL PROPERTIES OF THE INTEGRAL OF LEGENDRE POLYNOMIALS
Institute of Scientific and Technical Information of China (English)
史应光; 王子玉
2001-01-01
Some extremal properties of the integral of Legendre polynomials are given, which are of independent interest. Meanwhile they show that a conjecture of P. Erdos[1] is plausible and maybe provides some means to prove this conjecture.
An Elementary Proof of the Polynomial Matrix Spectral Factorization Theorem
Ephremidze, Lasha
2010-01-01
A very simple and short proof of the polynomial matrix spectral factorization theorem (on the unit circle as well as on the real line) is presented, which relies on elementary complex analysis and linear algebra.
Kauffman polynomials of some links and invariants of 3-manifolds
Institute of Scientific and Technical Information of China (English)
李起升
2002-01-01
Kauffman bracket polynomials of the so-called generalized tree-like links are studied. An algorithm of Witten type invariants, which was defined by Blanchet and Habegger et al. of more general 3-manifolds is given.
Force prediction in cold rolling mills by polynomial methods
Directory of Open Access Journals (Sweden)
Nicu ROMAN
2007-12-01
Full Text Available A method for steel and aluminium strip thickness control is provided including a new technique for predictive rolling force estimation method by statistic model based on polynomial techniques.
Stability of the Bose-Einstein condensate under polynomial perturbations
Gielerak, R.; Damek, J.
2002-01-01
The problem of the Bose-Einstein condensate preservation under thermofield and standard gauge-invariant perturbations is discussed. A new result on stability of the condensate under thermofield perturbations of a polynomial type is presented.
Maximum likelihood polynomial regression for robust speech recognition
Institute of Scientific and Technical Information of China (English)
LU Yong; WU Zhenyang
2011-01-01
The linear hypothesis is the main disadvantage of maximum likelihood linear re- gression （MLLR）. This paper applies the polynomial regression method to model adaptation and establishes a nonlinear model adaptation algorithm using maximum likelihood polyno
The Evolution of GDP in USA Using Polynomial Regression Analysis
Directory of Open Access Journals (Sweden)
Catalin Angelo Ioan
2013-10-01
Full Text Available The paper deals with the problem of statistical forecasts in terms of polynomial regression. Thus, it compares actual results with predicted variables using data sets sequentially go through all the set initially.
The Evolution of GDP in USA Using Polynomial Regression Analysis
Catalin Angelo Ioan; Gina Ioan
2013-01-01
The paper deals with the problem of statistical forecasts in terms of polynomial regression. Thus, it compares actual results with predicted variables using data sets sequentially go through all the set initially.
Cauchy-Kowalevski and polynomial ordinary differential equations
Directory of Open Access Journals (Sweden)
Roger J. Thelwell
2012-01-01
Full Text Available The Cauchy-Kowalevski Theorem is the foremost result guaranteeing existence and uniqueness of local solutions for analytic quasilinear partial differential equations with Cauchy initial data. The techniques of Cauchy-Kowalevski may also be applied to initial-value ordinary differential equations. These techniques, when applied in the polynomial ordinary differential equation setting, lead one naturally to a method in which coefficients of the series solution are easily computed in a recursive manner, and an explicit majorization admits a clear a priori error bound. The error bound depends only on immediately observable quantities of the polynomial system; coefficients, initial conditions, and polynomial degree. The numerous benefits of the polynomial system are shown for a specific example.
Nonstandard decision methods for the solvability of real polynomial equations
Institute of Scientific and Technical Information of China (English)
曾广兴
1999-01-01
For a multivariate polynomial equation with coefficients in a computable ordered field, two criteria of this equation having real solutions are given. Based on the criteria, decision methods for the existence of real zeros and the semidefiniteness of binary polynomials are provided. With the aid of computers, these methods are used to solve several examples. The technique is to extend the original field involved in the question to a computable non-Archimedean ordered field containing infinitesimal elements.
Performance comparison of polynomial representations for optimizing optical freeform systems
Brömel, A.; Gross, H.; Ochse, D.; Lippmann, U.; Ma, C.; Zhong, Y.; Oleszko, M.
2015-09-01
Optical systems can benefit strongly from freeform surfaces, however the choice of the right representation isn`t an easy one. Classical representations like X-Y-polynomials, as well as Zernike-polynomials are often used for such systems, but should have some disadvantage regarding their orthogonality, resulting in worse convergence and reduced quality in final results compared to newer representations like the Q-polynomials by Forbes. Additionally the supported aperture is a circle, which can be a huge drawback in case of optical systems with rectangular aperture. In this case other representations like Chebyshev-or Legendre-polynomials come into focus. There are a larger number of possibilities; however the experience with these newer representations is rather limited. Therefore in this work the focus is on investigating the performance of four widely used representations in optimizing two ambitious systems with very different properties: Three-Mirror-Anastigmat and an anamorphic System. The chosen surface descriptions offer support for circular or rectangular aperture, as well as different grades of departure from rotational symmetry. The basic shapes are for example a conic or best-fit-sphere and the polynomial set is non-, spatial or slope-orthogonal. These surface representations were chosen to evaluate the impact of these aspects on the performance optimization of the two example systems. Freeform descriptions investigated here were XY-polynomials, Zernike in Fringe representation, Q-polynomials by Forbes, as well as 2-dimensional Chebyshev-polynomials. As a result recommendations for the right choice of freeform surface representations for practical issues in the optimization of optical systems can be given.
On Some Inequalities Concerning Rate of Growth of Polynomials
Institute of Scientific and Technical Information of China (English)
Abdullah Mir; Imtiaz Hussain; Q. M. Dawood
2014-01-01
In this paper we consider a class of polynomials P(z)=a0+∑nv=t avzv, t≥1, not vanishing in|z|
ON THE COEFFICIENTS OF A POLYNOMIAL WITH RESTRICTED ZEROS
Institute of Scientific and Technical Information of China (English)
A.Aziz; W.M.Shah
2009-01-01
LetP(Z)=αn Zn + αn-1Zn-1 +…+α0 be a complex polynomial of degree n. There is a close connection between the coefficients and the zeros of P(z). In this paper we prove some sharp inequalities concerning the coeffi-cients of the polynomial P(z) with restricted zeros. We also establish a sufficient condition for the separation of zeros of P(z).
Solving Heat and Wave-Like Equations Using He's Polynomials
Directory of Open Access Journals (Sweden)
Syed Tauseef Mohyud-Din
2009-01-01
Full Text Available We use He's polynomials which are calculated form homotopy perturbation method (HPM for solving heat and wave-like equations. The proposed iterative scheme finds the solution without any discretization, linearization, or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that suggested technique solves nonlinear problems without using Adomian's polynomials is a clear advantage of this algorithm over the decomposition method.
Auslander-Reiten conjecture for symmetric algebras of polynomial growth
Zhou, Guodong; Zimmermann, Alexander
2010-01-01
This paper studies self-injective algebras of polynomial growth. We prove that the derived equivalence classification of weakly symmetric algebras of domestic type coincides with the classification up to stable equivalences (of Morita type). As for weakly symmetric non-domestic algebras of polynomial growth, up to some scalar problems, the derived equivalence classification coincides with the classification up to stable equivalences of Morita type. As a consequence, we get the validity of the...
Inequalities between gamma-polynomials of graph-associahedra
Aisbett, Natalie
2012-01-01
We prove a conjecture of Postnikov, Reiner and Williams by defining a partial order on the set of tree graphs with $n$ vertices that induces inequalities between the $\\gamma$-polynomials of their associated graph-associahedra. The partial order is given by relating trees that can be obtained from one another by operations called tree shifts. We also show that tree shifts lower the $\\gamma$-polynomials of graphs that are not trees, as do the flossing moves of Babson and Reiner.
Twisted exponential sums of polynomials in one variable
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
The twisted T-adic exponential sums associated to a polynomial in one variable are studied.An explicit arithmetic polygon in terms of the highest two exponents of the polynomial is proved to be a lower bound of the Newton polygon of the C-function of the twisted T-adic exponential sums.This bound gives lower bounds for the Newton polygon of the L-function of twisted p-power order exponential sums.
Polynomial Estimates for c-functions on Reductive Symmetric Spaces
DEFF Research Database (Denmark)
van den Ban, Erik; Schlichtkrull, Henrik
2012-01-01
The c-functions, related to a reductive symmetric space G/H and a fixed representation τ of a maximal compact subgroup K of G, are shown to satisfy polynomial bounds in imaginary directions.......The c-functions, related to a reductive symmetric space G/H and a fixed representation τ of a maximal compact subgroup K of G, are shown to satisfy polynomial bounds in imaginary directions....
JACOBI POLYNOMIALS USED TO INVERT THE LAPLACE TRANSFORM
Institute of Scientific and Technical Information of China (English)
A.Al-Shuaibi; F.Al-Rawjih
2004-01-01
Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find an approximation to f(t) by the use of the classical Jacobi polynomials. The main contribution of our work is the development of a new and very effective method to determine the coefficients in the finite series expansion that approximation f(t) in terms of Jacobi polynomials. Some numerical examples are illustrated.
Difference oscillator in terms of the Meixner polynomials
Atakishiyev, Natig M.; Jafarov, Elchin I.; Nagiyev, Shakir M.; Wolf, Kurt B.
1998-07-01
We discuss a difference model of the linear harmonic oscillator based on the Meixner polynomials. As limit and special cases, it contains difference oscillator models in terms of the Kravchuk and Charlier polynomials, as well as the wavefunctions of the linear harmonic oscillator in quantum mechanics. We show that the dynamical group is SU(1,1) and construct explicitly the corresponding coherent state. The reproducing kernel for the wavefunctions of the Meixner model is also found.
Polynomials whose reducibility is related to the Goldbach conjecture
Borwein, Peter B.; Choi, Stephen K. K.; Martin, Greg; Samuels, Charles L.
2014-01-01
We introduce a collection of polynomials $F_N$, associated to each positive integer $N$, whose divisibility properties yield a reformulation of the Goldbach conjecture. While this reformulation certainly does not lead to a resolution of the conjecture, it does suggest two natural generalizations for which we provide some numerical evidence. As these polynomials $F_N$ are independently interesting, we further explore their basic properties, giving, among other things, asymptotic estimates on t...
Finding low-weight polynomial multiples using discrete logarithm
Didier, Frédéric; Laigle-Chapuy, Yann
2007-01-01
Finding low-weight multiples of a binary polynomial is a difficult problem arising in the context of stream ciphers cryptanalysis. The classical algorithm to solve this problem is based on a time memory trade-off. We will present an improvement to this approach using discrete logarithm rather than a direct representation of the involved polynomials. This gives an algorithm which improves the theoretical complexity, and is also very flexible in practice.
Asymptotic analysis of the Nörlund and Stirling polynomials
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Mark Daniel Ward
2012-04-01
Full Text Available We provide a full asymptotic analysis of the N{\\"o}rlund polynomials and Stirling polynomials. We give a general asymptotic expansion---to any desired degree of accuracy---when the parameter is not an integer. We use singularity analysis, Hankel contours, and transfer theory. This investigation was motivated by a need for such a complete asymptotic description, with parameter 1/2, during this author's recent solution of Wilf's 3rd (previously Unsolved Problem.
The Terminal Hosoya Polynomial of Some Families of Composite Graphs
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Emeric Deutsch
2014-01-01
Full Text Available Let G be a connected graph and let Ω(G be the set of pendent vertices of G. The terminal Hosoya polynomial of G is defined as TH(G,t∶=∑x,y∈Ω(G:x≠ytdG(x,y, where dG(x,y denotes the distance between the pendent vertices x and y. In this note paper we obtain closed formulae for the terminal Hosoya polynomial of rooted product graphs and corona product graphs.
On Sharing, Memoization, and Polynomial Time (Long Version)
Avanzini, Martin; Lago, Ugo Dal
2015-01-01
We study how the adoption of an evaluation mechanism with sharing and memoization impacts the class of functions which can be computed in polynomial time. We first show how a natural cost model in which lookup for an already computed value has no cost is indeed invariant. As a corollary, we then prove that the most general notion of ramified recurrence is sound for polynomial time, this way settling an open problem in implicit computational complexity.
The Tutte polynomial of some self-similar graphs
Donno, Alfredo
2010-01-01
We study the Tutte polynomial of two infinite families of finite graphs: the Sierpi\\'{n}ski graphs, which are finite approximations of the well-known Sierpi\\'{n}ski gasket, and the Schreier graphs of the Hanoi Towers group $H^{(3)}$ acting on the rooted ternary tree. For both of them, we recursively describe the Tutte polynomial and we compute several special evaluations of it, giving interesting results about the combinatorial structure of these graphs.
Exploiting symmetries in SDP-relaxations for polynomial optimization
Riener, Cordian; Theobald, Thorsten; Andrén, Lina Jansson; Lasserre, Jean B.
2011-01-01
In this paper we study various approaches for exploiting symmetries in polynomial optimization problems within the framework of semi definite programming relaxations. Our special focus is on constrained problems especially when the symmetric group is acting on the variables. In particular, we investigate the concept of block decomposition within the framework of constrained polynomial optimization problems, show how the degree principle for the symmetric group can be computationally exploited...
Modeling Microwave Structures in Time Domain Using Laguerre Polynomials
Z. Raida; Lacik, J.
2006-01-01
The paper is focused on time domain modeling of microwave structures by the method of moments. Two alternative schemes with weighted Laguerre polynomials are presented. Thanks to their properties, these schemes are free of late time oscillations. Further, the paper is aimed to effective and accurate evaluation of Green's functions integrals within these schemes. For this evaluation, a first- and second-order polynomial approximation is developed. The last part of the paper deals with mode...
Fast parallel computation of polynomials using few processors
DEFF Research Database (Denmark)
Valiant, Leslie; Skyum, Sven
1981-01-01
It is shown that any multivariate polynomial that can be computed sequentially in C steps and has degree d can be computed in parallel in 0((log d) (log C + log d)) steps using only (Cd)0(1) processors.......It is shown that any multivariate polynomial that can be computed sequentially in C steps and has degree d can be computed in parallel in 0((log d) (log C + log d)) steps using only (Cd)0(1) processors....
Fast Parallel Computation of Polynomials Using Few Processors
DEFF Research Database (Denmark)
Valiant, Leslie G.; Skyum, Sven; Berkowitz, S.;
1983-01-01
It is shown that any multivariate polynomial of degree $d$ that can be computed sequentially in $C$ steps can be computed in parallel in $O((\\log d)(\\log C + \\log d))$ steps using only $(Cd)^{O(1)} $ processors.......It is shown that any multivariate polynomial of degree $d$ that can be computed sequentially in $C$ steps can be computed in parallel in $O((\\log d)(\\log C + \\log d))$ steps using only $(Cd)^{O(1)} $ processors....
Invariant hyperplanes and Darboux integrability of polynomial vector fields
International Nuclear Information System (INIS)
This paper is composed of two parts. In the first part, we provide an upper bound for the number of invariant hyperplanes of the polynomial vector fields in n variables. This result generalizes those given in Artes et al (1998 Pac. J. Math. 184 207-30) and Llibre and Rodriguez (2000 Bull. Sci. Math. 124 599-619). The second part gives an extension of the Darboux theory of integrability to polynomial vector fields on algebraic varieties
Zeros of differential polynomials in real meromorphic functions
Bergweiler, W.; Eremenko, A.; Langley, J.
2004-01-01
We show that for a real transcendental meromorphic function f, the differential polynomial f'+f^m with m > 4 has infinitely many non-real zeros. Similar results are obtained for differential polynomials f'f^m-1. We specially investigate the case of meromorphic functions with finitely many poles. We show by examples the precision of our results. One of our main tools is the Fatou theorem from complex dynamics.
CONVERGENCE ARTE FOR INTERATES OF q-BERNSTEIN POLYNOMIALS
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
Recently, q-Bernstein polynomials have been intensively investigated by a number of authors. Their results show that for q ≠ 1, q-Bernstein polynomials possess of many interesting properties. In this paper, the convergence rate for iterates of both q-Bernstein when n →∞ and convergence rate of Bn(f,q;x) when f ∈ Cn-1[0, 1], q →∞ are also presented.
Weyl and Lidski(i) Inequalities for General Hyperbolic Polynomials
Institute of Scientific and Technical Information of China (English)
Denis SERRE
2009-01-01
The roots of hyperbolic polynomials satisfy the linear inequalities that were previously established for the eigenvalues of Hermitian matrices, after a conjecture by A. Horn. Among them are the so-called Weyl and Lidski(i) inequalities. An elementary proof of the latter for hyperbolic polynomials is given. This proof follows an idea from H. Weinberger and is free from representation theory and Schubert calculus arguments, as well as from hyperbolic partial differential equations theory.
First extension groups of Verma modules and $R$-polynomials
Abe, Noriyuki
2010-01-01
We study the first extension groups between Verma modules. There was a conjecture which claims that the dimensions of the higher extension groups between Verma modules are the coefficients of $R$-polynomials defined by Kazhdan-Lusztig. This conjecture was known as the Gabber-Joseph conjecture (although Gebber and Joseph did not state.) However, Boe gives a counterexample to this conjecture. In this paper, we study how far are the dimensions of extension groups from the coefficients of $R$-polynomials.
Constructing Polynomial Spectral Models for Stars
Rix, Hans-Walter; Ting, Yuan-Sen; Conroy, Charlie; Hogg, David W.
2016-08-01
Stellar spectra depend on the stellar parameters and on dozens of photospheric elemental abundances. Simultaneous fitting of these { N } ˜ 10-40 model labels to observed spectra has been deemed unfeasible because the number of ab initio spectral model grid calculations scales exponentially with { N }. We suggest instead the construction of a polynomial spectral model (PSM) of order { O } for the model flux at each wavelength. Building this approximation requires a minimum of only ≤ft(≥nfrac{}{}{0em}{}{{ N }+{ O }}{{ O }}\\right) calculations: e.g., a quadratic spectral model ({ O }=2) to fit { N }=20 labels simultaneously can be constructed from as few as 231 ab initio spectral model calculations; in practice, a somewhat larger number (˜300-1000) of randomly chosen models lead to a better performing PSM. Such a PSM can be a good approximation only over a portion of label space, which will vary case-by-case. Yet, taking the APOGEE survey as an example, a single quadratic PSM provides a remarkably good approximation to the exact ab initio spectral models across much of this survey: for random labels within that survey the PSM approximates the flux to within 10-3 and recovers the abundances to within ˜0.02 dex rms of the exact models. This enormous speed-up enables the simultaneous many-label fitting of spectra with computationally expensive ab initio models for stellar spectra, such as non-LTE models. A PSM also enables the simultaneous fitting of observational parameters, such as the spectrum’s continuum or line-spread function.
Factorization of colored knot polynomials at roots of unity
Kononov, Ya.; Morozov, A.
2015-07-01
HOMFLY polynomials are the Wilson-loop averages in Chern-Simons theory and depend on four variables: the closed line (knot) in 3d space-time, representation R of the gauge group SU (N) and exponentiated coupling constant q. From analysis of a big variety of different knots we conclude that at q, which is a 2m-th root of unity, q2m = 1, HOMFLY polynomials in symmetric representations [ r ] satisfy recursion identity: Hr+m =Hr ṡHm for any A =qN, which is a generalization of the property Hr = H1r for special polynomials at m = 1. We conjecture a further generalization to arbitrary representation R, which, however, is checked only for torus knots. Next, Kashaev polynomial, which arises from HR at q2 = e 2 πi / | R |, turns equal to the special polynomial with A substituted by A| R |, provided R is a single-hook representations (including arbitrary symmetric) - what provides a q - A dual to the similar property of Alexander polynomial. All this implies non-trivial relations for the coefficients of the differential expansions, which are believed to provide reasonable coordinates in the space of knots - existence of such universal relations means that these variables are still not unconstrained.
Factorization of colored knot polynomials at roots of unity
Directory of Open Access Journals (Sweden)
Ya. Kononov
2015-07-01
Full Text Available HOMFLY polynomials are the Wilson-loop averages in Chern–Simons theory and depend on four variables: the closed line (knot in 3d space–time, representation R of the gauge group SU(N and exponentiated coupling constant q. From analysis of a big variety of different knots we conclude that at q, which is a 2m-th root of unity, q2m=1, HOMFLY polynomials in symmetric representations [r] satisfy recursion identity: Hr+m=Hr⋅Hm for any A=qN, which is a generalization of the property Hr=H1r for special polynomials at m=1. We conjecture a further generalization to arbitrary representation R, which, however, is checked only for torus knots. Next, Kashaev polynomial, which arises from HR at q2=e2πi/|R|, turns equal to the special polynomial with A substituted by A|R|, provided R is a single-hook representations (including arbitrary symmetric – what provides a q−A dual to the similar property of Alexander polynomial. All this implies non-trivial relations for the coefficients of the differential expansions, which are believed to provide reasonable coordinates in the space of knots – existence of such universal relations means that these variables are still not unconstrained.
Characterizing Polynomial Time Computability of Rational and Real Functions
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Walid Gomaa
2009-11-01
Full Text Available Recursive analysis was introduced by A. Turing [1936], A. Grzegorczyk [1955], and D. Lacombe [1955]. It is based on a discrete mechanical framework that can be used to model computation over the real numbers. In this context the computational complexity of real functions defined over compact domains has been extensively studied. However, much less have been done for other kinds of real functions. This article is divided into two main parts. The first part investigates polynomial time computability of rational functions and the role of continuity in such computation. On the one hand this is interesting for its own sake. On the other hand it provides insights into polynomial time computability of real functions for the latter, in the sense of recursive analysis, is modeled as approximations of rational computations. The main conclusion of this part is that continuity does not play any role in the efficiency of computing rational functions. The second part defines polynomial time computability of arbitrary real functions, characterizes it, and compares it with the corresponding notion over rational functions. Assuming continuity, the main conclusion is that there is a conceptual difference between polynomial time computation over the rationals and the reals manifested by the fact that there are polynomial time computable rational functions whose extensions to the reals are not polynomial time computable and vice versa.
Orthogonality of Hermite polynomials in superspace and Mehler type formulae
Coulembier, Kevin; Sommen, Frank
2010-01-01
In this paper, Hermite polynomials related to quantum systems with orthogonal O(m)-symmetry, finite reflection group symmetry G < O(m), symplectic symmetry Sp(2n) and superspace symmetry O(m) x Sp(2n) are considered. After an overview of the results for O(m) and \\cG, the orthogonality of the Hermite polynomials related to Sp(2n) is obtained with respect to the Berezin integral. As a consequence, an extension of the Mehler formula for the classical Hermite polynomials to Grassmann algebras is proven. Next, Hermite polynomials in a full superspace with O(m) x Sp(2n) symmetry are considered. It is shown that they are not orthogonal with respect to the canonically defined inner product. However, a new inner product is introduced which behaves correctly with respect to the structure of harmonic polynomials on superspace. This inner product allows to restore the orthogonality of the Hermite polynomials and also restores the hermiticity of a class of Schroedinger operators in superspace. Subsequently, a Mehler fo...
An overview on polynomial approximation of NP-hard problems
Directory of Open Access Journals (Sweden)
Paschos Vangelis Th.
2009-01-01
Full Text Available The fact that polynomial time algorithm is very unlikely to be devised for an optimal solving of the NP-hard problems strongly motivates both the researchers and the practitioners to try to solve such problems heuristically, by making a trade-off between computational time and solution's quality. In other words, heuristic computation consists of trying to find not the best solution but one solution which is 'close to' the optimal one in reasonable time. Among the classes of heuristic methods for NP-hard problems, the polynomial approximation algorithms aim at solving a given NP-hard problem in poly-nomial time by computing feasible solutions that are, under some predefined criterion, as near to the optimal ones as possible. The polynomial approximation theory deals with the study of such algorithms. This survey first presents and analyzes time approximation algorithms for some classical examples of NP-hard problems. Secondly, it shows how classical notions and tools of complexity theory, such as polynomial reductions, can be matched with polynomial approximation in order to devise structural results for NP-hard optimization problems. Finally, it presents a quick description of what is commonly called inapproximability results. Such results provide limits on the approximability of the problems tackled.
Conference on Commutative rings, integer-valued polynomials and polynomial functions
Frisch, Sophie; Glaz, Sarah; Commutative Algebra : Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions
2014-01-01
This volume presents a multi-dimensional collection of articles highlighting recent developments in commutative algebra. It also includes an extensive bibliography and lists a substantial number of open problems that point to future directions of research in the represented subfields. The contributions cover areas in commutative algebra that have flourished in the last few decades and are not yet well represented in book form. Highlighted topics and research methods include Noetherian and non- Noetherian ring theory as well as integer-valued polynomials and functions. Specific topics include: · Homological dimensions of Prüfer-like rings · Quasi complete rings · Total graphs of rings · Properties of prime ideals over various rings · Bases for integer-valued polynomials · Boolean subrings · The portable property of domains · Probabilistic topics in Intn(D) · Closure operations in Zariski-Riemann spaces of valuation domains · Stability of do...
Traversa, Fabio Lorenzo; Ramella, Chiara; Bonani, Fabrizio; Di Ventra, Massimiliano
2015-07-01
Memcomputing is a novel non-Turing paradigm of computation that uses interacting memory cells (memprocessors for short) to store and process information on the same physical platform. It was recently proven mathematically that memcomputing machines have the same computational power of nondeterministic Turing machines. Therefore, they can solve NP-complete problems in polynomial time and, using the appropriate architecture, with resources that only grow polynomially with the input size. The reason for this computational power stems from properties inspired by the brain and shared by any universal memcomputing machine, in particular intrinsic parallelism and information overhead, namely, the capability of compressing information in the collective state of the memprocessor network. We show an experimental demonstration of an actual memcomputing architecture that solves the NP-complete version of the subset sum problem in only one step and is composed of a number of memprocessors that scales linearly with the size of the problem. We have fabricated this architecture using standard microelectronic technology so that it can be easily realized in any laboratory setting. Although the particular machine presented here is eventually limited by noise-and will thus require error-correcting codes to scale to an arbitrary number of memprocessors-it represents the first proof of concept of a machine capable of working with the collective state of interacting memory cells, unlike the present-day single-state machines built using the von Neumann architecture. PMID:26601208
On the values of independence and domination polynomials at specific points
Directory of Open Access Journals (Sweden)
Saeid Alikhani
2012-06-01
Full Text Available Let G be a simple graph of order n. We consider the independence polynomial and the domination polynomial of a graph G. The value of a graph polynomial at a specific point can give sometimes a very surprising information about the structure of the graph. In this paper we investigate independence and domination polynomial at -1 and 1.
Fiber Bragg Grating Sensors for Harsh Environments
Directory of Open Access Journals (Sweden)
Stephen J. Mihailov
2012-02-01
Full Text Available Because of their small size, passive nature, immunity to electromagnetic interference, and capability to directly measure physical parameters such as temperature and strain, fiber Bragg grating sensors have developed beyond a laboratory curiosity and are becoming a mainstream sensing technology. Recently, high temperature stable gratings based on regeneration techniques and femtosecond infrared laser processing have shown promise for use in extreme environments such as high temperature, pressure or ionizing radiation. Such gratings are ideally suited for energy production applications where there is a requirement for advanced energy system instrumentation and controls that are operable in harsh environments. This paper will present a review of some of the more recent developments.
Thermal annealing of tilted fiber Bragg gratings
González-Vila, Á.; Rodríguez-Cobo, L.; Mégret, P.; Caucheteur, C.; López-Higuera, J. M.
2016-05-01
We report a practical study of the thermal decay of cladding mode resonances in tilted fiber Bragg gratings, establishing an analogy with the "power law" evolution previously observed on uniform gratings. We examine how this process contributes to a great thermal stability, even improving it by means of a second cycle slightly increasing the annealing temperature. In addition, we show an improvement of the grating spectrum after annealing, with respect to the one just after inscription, which suggests the application of this method to be employed to improve saturation issues during the photo-inscription process.
An electromagnetically induced grating by microwave modulation
Energy Technology Data Exchange (ETDEWEB)
Xiao, Zhi-Hong; Shin, Sung Guk; Kim, Kisik, E-mail: kisik@inha.ac.k [Department of Physics, Inha University, Incheon 402-751 (Korea, Republic of)
2010-08-28
We study the phenomenon of an electromagnetically induced phase grating in a double-dark state system of {sup 87}Rb atoms, the two closely placed lower fold levels of which are coupled by a weak microwave field. Owing to the existence of the weak microwave field, the efficiency of the phase grating is strikingly improved, and an efficiency of approximately 33% can be achieved. Under the action of the weak standing wave field, the high efficiency of the phase grating can be maintained by modulating the strength and detuning of the weak microwave field, increasing the strength of the standing wave field. (fast track communication)
Influence of grating thickness in low-contrast subwavelength grating concentrating lenses
Ye, Mao; Yi, Ya Sha
2016-07-01
Conventional subwavelength grating concentrating lenses are designed based on calculated phase overlap, wherein the phase change is fixed by the grating thickness, bar-width, and airgap, and therefore the focus. We found that certain concentration effects can still be maintained by changing the grating thickness with the same bar-widths and airgap dimensions. Following that, we discovered the existence of the grating thickness threshold; light concentration intensity spikes upon exceeding this limit. However, the light concentration property does not change continuously with respect to a steady increase in grating thickness. This observation indicates that there exists a concentration mode self-interference effect along the light propagation direction inside the gratings. Our results may provide guidance in designing and fabricating microlenses in a potentially more easy and controllable manner. Such approaches can be utilized in various integrated nanophotonics applications ranging from optical cavities and read/write heads to concentrating photovoltaics.
Hurvitz, G; Strum, G; Shpilman, Z; Levy, I; Fraenkel, M
2012-01-01
A novel fabrication method for soft x-ray transmission grating and other optical elements is presented. The method uses Focused-Ion-Beam (FIB) technology to fabricate high-quality free standing grating bars on Transmission Electron Microscopy grids (TEM-grid). High quality transmission gratings are obtained with superb accuracy and versatility. Using these gratings and back-illuminated CCD camera, absolutely calibrated x-ray spectra can be acquired for soft x-ray source diagnostics in the 100-3000 eV spectral range. Double grating combinations of identical or different parameters are easily fabricated, allowing advanced one-shot application of transmission grating spectroscopy. These applications include spectroscopy with different spectral resolutions, bandwidths, dynamic ranges, and may serve for identification of high-order contribution, and spectral calibrations of various x-ray optical elements.
Hurvitz, G.; Ehrlich, Y.; Strum, G.; Shpilman, Z.; Levy, I.; Fraenkel, M.
2012-08-01
A novel fabrication method for soft x-ray transmission grating and other optical elements is presented. The method uses focused-ion-beam technology to fabricate high-quality free standing grating bars on transmission electron microscopy grids. High quality transmission gratings are obtained with superb accuracy and versatility. Using these gratings and back-illuminated CCD camera, absolutely calibrated x-ray spectra can be acquired for soft x-ray source diagnostics in the 100-3000 eV spectral range. Double grating combinations of identical or different parameters are easily fabricated, allowing advanced one-shot application of transmission grating spectroscopy. These applications include spectroscopy with different spectral resolutions, bandwidths, dynamic ranges, and may serve for identification of high-order contribution, and spectral calibrations of various x-ray optical elements.
Okayama, Hideaki; Onawa, Yosuke; Shimura, Daisuke; Yaegashi, Hiroki; Sasaki, Hironori
2016-08-01
We describe a Bragg grating with a phase shift section and a sampled grating scheme that converts input polarization to orthogonal polarization. A very narrow polarization-independent wavelength peak can be generated by phase shift structures and polarization-independent multiple diffraction peaks by sampled gratings. The characteristics of the device were examined by transfer matrix and finite-difference time-domain methods.
Karp, Dmitry
2012-01-01
In this note we prove positivity of Maclaurin coefficients of polynomials written in terms of rising factorials and arbitrary log-concave sequences. These polynomials arise naturally when studying log-concavity of rising factorial series. We propose several conjectures concerning zeros and coefficients of a generalized form of those polynomials. We also consider polynomials whose generating functions are higher order Toeplitz determinants formed by rising factorial series. We make three conjectures about these polynomials. All proposed conjectures are supported by numerical evidence.
Discriminants and functional equations for polynomials orthogonal on the unit circle
International Nuclear Information System (INIS)
We derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and q-difference equations for these polynomials. A general functional equation is found which allows one to relate the zeros of the orthogonal polynomials to the stationary values of an explicit quasi-energy and implies recurrences on the orthogonal polynomial coefficients. We also evaluate the discriminants and quantized discriminants of polynomials orthogonal on the unit circle
Access Platforms for Offshore Wind Turbines Using Gratings
DEFF Research Database (Denmark)
Andersen, Thomas Lykke; Rasmussen, Michael R.
2008-01-01
The paper deals with forces generated by a stationary jet on different types of gratings and a solid plate. The force reduction factors for the different gratings compared to the solid plate mainly depend on the porosity of the gratings, but the geometry of the grating is also of some importance........ The derived reduction factors are expected to be applicable to design of offshore wind turbine access platforms with gratings where slamming also is an important factor....
Long period fiber gratings induced by mechanical resonance
Shahal, Shir; Duadi, Hamootal; Fridman, Moti
2015-01-01
We present a simple, and robust method for writing long period fiber gratings with low polarization dependent losses. Our method is based on utilizing mechanical vibrations of the tapered fiber while pooling it. Our method enables real-time tunability of the periodicity, efficiency and length of the grating. We also demonstrate complex grating by writing multiple gratings simultaneously. Finally, we utilized the formation of the gratings in different fiber diameters to investigate the Young's modulus of the fiber.
Theory of photorefractive dynamic grating formulation
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
The photorefractive holographic recording and two-beam coupling are both dynamic grating formulation process. The interference light intensity of the two coherent beams induces a phase grating though photo-induced refractive index variation and the phase grating changing the intensities of the two beams through beam-coupling take place at the same time. By solving simultaneously the band transport equations and wave-coupled equations, and using the light intensity modulation as the main variable, the analytic solution is obtained, which is valid for any light intensity modulation and constant light excitation efficiency. Here all the mechanics of drift, diffusion and photovoltaic effect are considered. The result shows that the modulation of the dynamic grating varies more slowly compared with that of the linear modulation approximation.
Hydraulic Capacity of an ADA Compliant Street Drain Grate
Energy Technology Data Exchange (ETDEWEB)
Lottes, Steven A. [Argonne National Lab. (ANL), Argonne, IL (United States); Bojanowski, Cezary [Argonne National Lab. (ANL), Argonne, IL (United States)
2015-09-01
Resurfacing of urban roads with concurrent repairs and replacement of sections of curb and sidewalk may require pedestrian ramps that are compliant with the American Disabilities Act (ADA), and when street drains are in close proximity to the walkway, ADA compliant street grates may also be required. The Minnesota Department of Transportation ADA Operations Unit identified a foundry with an available grate that meets ADA requirements. Argonne National Laboratory’s Transportation Research and Analysis Computing Center used full scale three dimensional computational fluid dynamics to determine the performance of the ADA compliant grate and compared it to that of a standard vane grate. Analysis of a parametric set of cases was carried out, including variation in longitudinal, gutter, and cross street slopes and the water spread from the curb. The performance of the grates was characterized by the fraction of the total volume flow approaching the grate from the upstream that was captured by the grate and diverted into the catch basin. The fraction of the total flow entering over the grate from the side and the fraction of flow directly over a grate diverted into the catch basin were also quantities of interest that aid in understanding the differences in performance of the grates. The ADA compliant grate performance lagged that of the vane grate, increasingly so as upstream Reynolds number increased. The major factor leading to the performance difference between the two grates was the fraction of flow directly over the grates that is captured by the grates.
Sampled phase-shift fiber Bragg gratings
Institute of Scientific and Technical Information of China (English)
Xu Wang(王旭); Chongxiu Yu(余重秀); Zhihui Yu(于志辉); Qiang Wu(吴强)
2004-01-01
A phase-shift fiber Bragg grating (FBG) with sampling is proposed to generate a multi-channel bandpass filter in the background of multi-channel stopbands. The sampled noire fiber gratings are analyzed by Fourier transform theory first, and then simulation and experiment are performed, the results show that transmission peaks are opened in every reflective channel, the spectrum shape of every channel is identical.It can be used to fabricate multi-wavelength distributed feedback (DFB) fiber laser.
Dynamic optical coupled system employing Dammann gratings
Di, Caihui; Zhou, Changhe; Ru, Huayi
2004-10-01
With the increasing of the number of users in optical fiber communications, fiber-to-home project has a larger market value. Then the need of dynamic optical couplers, especially of N broad-band couplers, becomes greater. Though some advanced fiber fusion techniques have been developed, they still have many shortcomings. In this paper we propose a dynamic optical coupled system employing even-numbered Dammann gratings, which have the characteristic that the phase distribution in the first half-period accurately equals to that in the second-period with π phase inversion. In our experiment, we divide a conventional even-numbered Dammann grating into two identical gratings. The system can achieve the beam splitter and combiner as the switch between them according to the relative shift between two complementary gratings. When there is no shift between the gratings, the demonstrated 1×8 dynamic optical coupler achieves good uniformity of 0.06 and insertion loss of around 10.8 dB for each channel as a splitter. When the two gratings have an accurate shift of a half-period between them, our system has a low insertion loss of 0.46 dB as a combiner at a wavelength of 1550 nm.
Orbifold E-functions of dual invertible polynomials
Ebeling, Wolfgang; Gusein-Zade, Sabir M.; Takahashi, Atsushi
2016-08-01
An invertible polynomial is a weighted homogeneous polynomial with the number of monomials coinciding with the number of variables and such that the weights of the variables and the quasi-degree are well defined. In the framework of the search for mirror symmetric orbifold Landau-Ginzburg models, P. Berglund and M. Henningson considered a pair (f , G) consisting of an invertible polynomial f and an abelian group G of its symmetries together with a dual pair (f ˜ , G ˜) . We consider the so-called orbifold E-function of such a pair (f , G) which is a generating function for the exponents of the monodromy action on an orbifold version of the mixed Hodge structure on the Milnor fibre of f. We prove that the orbifold E-functions of Berglund-Henningson dual pairs coincide up to a sign depending on the number of variables and a simple change of variables. The proof is based on a relation between monomials (say, elements of a monomial basis of the Milnor algebra of an invertible polynomial) and elements of the whole symmetry group of the dual polynomial.
Efficient computer algebra algorithms for polynomial matrices in control design
Baras, J. S.; Macenany, D. C.; Munach, R.
1989-01-01
The theory of polynomial matrices plays a key role in the design and analysis of multi-input multi-output control and communications systems using frequency domain methods. Examples include coprime factorizations of transfer functions, cannonical realizations from matrix fraction descriptions, and the transfer function design of feedback compensators. Typically, such problems abstract in a natural way to the need to solve systems of Diophantine equations or systems of linear equations over polynomials. These and other problems involving polynomial matrices can in turn be reduced to polynomial matrix triangularization procedures, a result which is not surprising given the importance of matrix triangularization techniques in numerical linear algebra. Matrices with entries from a field and Gaussian elimination play a fundamental role in understanding the triangularization process. In the case of polynomial matrices, matrices with entries from a ring for which Gaussian elimination is not defined and triangularization is accomplished by what is quite properly called Euclidean elimination. Unfortunately, the numerical stability and sensitivity issues which accompany floating point approaches to Euclidean elimination are not very well understood. New algorithms are presented which circumvent entirely such numerical issues through the use of exact, symbolic methods in computer algebra. The use of such error-free algorithms guarantees that the results are accurate to within the precision of the model data--the best that can be hoped for. Care must be taken in the design of such algorithms due to the phenomenon of intermediate expressions swell.
Ladder Operators for Lamé Spheroconal Harmonic Polynomials
Directory of Open Access Journals (Sweden)
Ricardo Méndez-Fragoso
2012-10-01
Full Text Available Three sets of ladder operators in spheroconal coordinates and their respective actions on Lamé spheroconal harmonic polynomials are presented in this article. The polynomials are common eigenfunctions of the square of the angular momentum operator and of the asymmetry distribution Hamiltonian for the rotations of asymmetric molecules, in the body-fixed frame with principal axes. The first set of operators for Lamé polynomials of a given species and a fixed value of the square of the angular momentum raise and lower and lower and raise in complementary ways the quantum numbers $n_1$ and $n_2$ counting the respective nodal elliptical cones. The second set of operators consisting of the cartesian components $hat L_x$, $hat L_y$, $hat L_z$ of the angular momentum connect pairs of the four species of polynomials of a chosen kind and angular momentum. The third set of operators, the cartesian components $hat p_x$, $hat p_y$, $hat p_z$ of the linear momentum, connect pairs of the polynomials differing in one unit in their angular momentum and in their parities. Relationships among spheroconal harmonics at the levels of the three sets of operators are illustrated.
A Faster Algorithm for Quasi-convex Integer Polynomial Optimization
Hildebrand, Robert
2010-01-01
We present a faster exponential-time algorithm for integer optimization over quasi-convex polynomials. We study the minimization of a quasi-convex polynomial subject to s quasi-convex polynomial constraints and integrality constraints for all variables. The new algorithm is an improvement upon the best known algorithm due to Heinz (Journal of Complexity, 2005). A lower time complexity is reached through applying a stronger ellipsoid rounding method and applying a recent advancement in the shortest vector problem to give a smaller exponential-time complexity of a Lenstra-type algorithm. For the bounded case, our algorithm attains a time-complexity of s (r l M d)^{O(1)} 2^{2n\\log_2(n) + O(n)} when M is a bound on the number of monomials in each polynomial and r is the binary encoding length of a bound on the feasible region. In the general case, s l^{O(1)} d^{O(n)} 2^{2n\\log_2(n)}. In each we assume d>=2 is a bound on the total degree of the polynomials and l bounds the maximum binary encoding size of the input...
Universal Racah matrices and adjoint knot polynomials. I. Arborescent knots
Mironov, A
2015-01-01
By now it is well established that the quantum dimensions of descendants of the adjoint representation can be described in a universal form, independent of a particular family of simple Lie algebras. The Rosso-Jones formula then implies a universal description of the adjoint knot polynomials for torus knots, which in particular unifies the HOMFLY (SU_N) and Kauffman (SO_N) polynomials. For E_8 the adjoint representation is also fundamental. We suggest to extend the universality from the dimensions to the Racah matrices and this immediately produces a unified description of the adjoint knot polynomials for all arborescent (double-fat) knots, including twist, 2-bridge and pretzel. Technically we develop together the universality and the "eigenvalue conjecture", which expresses the Racah and mixing matrices through the eigenvalues of the quantum R-matrix, and for dealing with the adjoint polynomials one has to extend it to the previously unknown 6x6 case. The adjoint polynomials do not distinguish between mutant...
Non-Symmetric Jack Polynomials and Integral Kernels
Baker, T H
1996-01-01
We investigate some properties of non-symmetric Jack, Hermite and Laguerre polynomials which occur as the polynomial part of the eigenfunctions for certain Calogero-Sutherland models with exchange terms. For the non-symmetric Jack polynomials, the constant term normalization ${\\cal N}_\\eta$ is evaluated using recurrence relations, and ${\\cal N}_\\eta$ is related to the norm for the non-symmetric analogue of the power-sum inner product. Our results for the non-symmetric Hermite and Laguerre polynomials allow the explicit determination of the integral kernels which occur in Dunkl's theory of integral transforms based on reflection groups of type $A$ and $B$, and enable many analogues of properties of the classical Fourier, Laplace and Hankel transforms to be derived. The kernels are given as generalized hypergeometric functions based on non-symmetric Jack polynomials. Central to our calculations is the construction of operators $\\widehat{\\Phi}$ and $\\widehat{\\Psi}$, which act as lowering-type operators for the n...
Decompositions of Trigonometric Polynomials with Applications to Multivariate Subdivision Schemes
Dyn, Nira
2009-01-01
We study multivariate trigonometric polynomials, satisfying a set of constraints close to the known Strung-Fix conditions. Based on the polyphase representation of these polynomials relative to a general dilation matrix, we develop a simple constructive method for a special type of decomposition of such polynomials. These decompositions are of interest to the analysis of convergence and smoothness of multivariate subdivision schemes associated with general dilation matrices. We apply these decompositions, by verifying sufficient conditions for the convergence and smoothness of multivariate scalar subdivision schemes, proved here. For the convergence analysis our sufficient conditions apply to arbitrary dilation matrices, while the previously known necessary and sufficient conditions are relevant only in case of dilation matrices with a self similar tiling. For the analysis of smoothness, we state and prove two theorems on multivariate matrix subdivision schemes, which lead to sufficient conditions for C^1 lim...
Characterizing Polynomial Time Computability of Rational and Real Functions
Gomaa, Walid
2009-01-01
Recursive analysis was introduced by A. Turing [1936], A. Grzegorczyk [1955], and D. Lacombe [1955]. It is based on a discrete mechanical framework that can be used to model computation over the real numbers. In this context the computational complexity of real functions defined over compact domains has been extensively studied. However, much less have been done for other kinds of real functions. This article is divided into two main parts. The first part investigates polynomial time computability of rational functions and the role of continuity in such computation. On the one hand this is interesting for its own sake. On the other hand it provides insights into polynomial time computability of real functions for the latter, in the sense of recursive analysis, is modeled as approximations of rational computations. The main conclusion of this part is that continuity does not play any role in the efficiency of computing rational functions. The second part defines polynomial time computability of arbitrary real ...
An Operator Approach to the Al-Salam-Carlitz Polynomials
Chen, William Y C; Sun, Lisa H
2009-01-01
We present an operator approach to Rogers-type formulas and Mehler's formulas for the Al-Salam-Carlitz polynomials $U_n(x,y,a;q)$. By using the q-exponential operator, we obtain a Rogers-type formula which leads to a linearization formula. With the aid of a bivariate augmentation operator, we get a simple derivation of Mehler's formula due to by Al-Salam and Carlitz, which requires a terminating condition on a ${}_3\\phi_2$ series. By means of the Cauchy companion augmentation operator, we obtain Mehler's formula in a similar form, but it does not need the terminating condition. We also give several identities on the generating functions for products of the Al-Salam-Carlitz polynomials which are extensions of formulas for Rogers-Szeg\\"o polynomials.
Double scaling limit for modified Jacobi-Angelesco polynomials
Deschout, Klaas
2011-01-01
We consider multiple orthogonal polynomials with respect to two modified Jacobi weights on touching intervals [a,0] and [0,1], with a < 0, and study a transition that occurs at a = -1. The transition is studied in a double scaling limit, where we let the degree n of the polynomial tend to infinity while the parameter a tends to -1 at a rate of O(n^{-1/2}). We obtain a Mehler-Heine type asymptotic formula for the polynomials in this regime. The method used to analyze the problem is the steepest descent technique for Riemann-Hilbert problems. A key point in the analysis is the construction of a new local parametrix.
The Laplace transform and polynomial approximation in L2
DEFF Research Database (Denmark)
Labouriau, Rodrigo
2016-01-01
This short note gives a sufficient condition for having the class of polynomials dense in the space of square integrable functions with respect to a finite measure dominated by the Lebesgue measure in the real line, here denoted by L2. It is shown that if the Laplace transform of the measure...... in play is bounded in a neighbourhood of the origin, then the moments of all order are finite and the class of polynomials is dense in L2. The existence of the moments of all orders is well known for the case where the measure is concentrated in the positive real line (see Feller, 1966), but the result...... concerning the polynomial approximation is original, even thought the proof is relatively simple. Additionally, an alternative stronger condition (easier to be verified) not involving the calculation of the Laplace transform is given. The condition essentially says that the density of the measure should have...
Polynomial quasisolutions of linear differential-difference equations
Directory of Open Access Journals (Sweden)
Valery B. Cherepennikov
2006-01-01
Full Text Available The paper discusses a linear differential-difference equation of neutral type with linear coefficients, when at the initial time moment \\(t=0\\ the value of the desired function \\(x(t\\ is known. The authors are not familiar with any results which would state the solvability conditions for the given problem in the class of analytical functions. A polynomial of some degree \\(N\\ is introduced into the investigation. Then the term "polynomial quasisolution" (PQ-solution is understood in the sense of appearance of the residual \\(\\Delta (t=O(t^N\\, when this polynomial is substituted into the initial problem. The paper is devoted to finding PQ-solutions for the initial-value problem under analysis.
Local polynomial Whittle estimation of perturbed fractional processes
DEFF Research Database (Denmark)
Frederiksen, Per; Nielsen, Frank; Nielsen, Morten Ørregaard
We propose a semiparametric local polynomial Whittle with noise (LPWN) estimator of the memory parameter in long memory time series perturbed by a noise term which may be serially correlated. The estimator approximates the spectrum of the perturbation as well as that of the short-memory component...... of the signal by two separate polynomials. Including these polynomials we obtain a reduction in the order of magnitude of the bias, but also in‡ate the asymptotic variance of the long memory estimate by a multiplicative constant. We show that the estimator is consistent for d 2 (0; 1), asymptotically normal...... for d ε (0, 3/4), and if the spectral density is infinitely smooth near frequency zero, the rate of convergence can become arbitrarily close to the parametric rate, pn. A Monte Carlo study reveals that the LPWN estimator performs well in the presence of a serially correlated perturbation term...
Polynomial chaos expansion with random and fuzzy variables
Jacquelin, E.; Friswell, M. I.; Adhikari, S.; Dessombz, O.; Sinou, J.-J.
2016-06-01
A dynamical uncertain system is studied in this paper. Two kinds of uncertainties are addressed, where the uncertain parameters are described through random variables and/or fuzzy variables. A general framework is proposed to deal with both kinds of uncertainty using a polynomial chaos expansion (PCE). It is shown that fuzzy variables may be expanded in terms of polynomial chaos when Legendre polynomials are used. The components of the PCE are a solution of an equation that does not depend on the nature of uncertainty. Once this equation is solved, the post-processing of the data gives the moments of the random response when the uncertainties are random or gives the response interval when the variables are fuzzy. With the PCE approach, it is also possible to deal with mixed uncertainty, when some parameters are random and others are fuzzy. The results provide a fuzzy description of the response statistical moments.
Basic polynomial invariants, fundamental representations and the Chern class map
Baek, Sanghoon; Zainoulline, Kirill
2011-01-01
Consider a crystallographic root system together with its Weyl group $W$ acting on the weight lattice $M$. Let $Z[M]^W$ and $S^*(M)^W$ be the $W$-invariant subrings of the integral group ring $Z[M]$ and the symmetric algebra $S^*(M)$ respectively. A celebrated theorem of Chevalley says that $Z[M]^W$ is a polynomial ring over $Z$ in classes of fundamental representations $w_1,...,w_n$ and $S^*(M)^{W}$ over rational numbers is a polynomial ring in basic polynomial invariants $q_1,...,q_n$, where $n$ is the rank. In the present paper we establish and investigate the relationship between $w_i$'s and $q_i$'s over the integers.
Integrability of the dynamical systems with polynomial Hamiltonians
International Nuclear Information System (INIS)
The paper is devoted to an exhaustive study of the integrability of dynamical systems described by polynomial Hamiltonians. After a general analysis of the conditions in which the integration of the equation of motion is possible, we determine the classes of polynomial potentials satisfying these conditions. We effectively study two models admitting both exact solutions and chaotic ones. Despite the fact that the maximal degree of the potential is different in the two cases, an interesting connection can be established between the two models. This connection is given by studying the similar classes of periodical solutions admitted by the two models. The paper will end with a study referring to a non-polynomial potential of the Thomas-Fermi type. (author)
Fractional order differentiation by integration with Jacobi polynomials
Liu, Dayan
2012-12-01
The differentiation by integration method with Jacobi polynomials was originally introduced by Mboup, Join and Fliess [22], [23]. This paper generalizes this method from the integer order to the fractional order for estimating the fractional order derivatives of noisy signals. The proposed fractional order differentiator is deduced from the Jacobi orthogonal polynomial filter and the Riemann-Liouville fractional order derivative definition. Exact and simple formula for this differentiator is given where an integral formula involving Jacobi polynomials and the noisy signal is used without complex mathematical deduction. Hence, it can be used both for continuous-time and discrete-time models. The comparison between our differentiator and the recently introduced digital fractional order Savitzky-Golay differentiator is given in numerical simulations so as to show its accuracy and robustness with respect to corrupting noises. © 2012 IEEE.
Factorization of Polynomials and GCD Computations for Finding Universal Denominators
Abramov, S. A.; Gheffar, A.; Khmelnov, D. E.
We discuss the algorithms which, given a linear difference equation with rational function coefficients over a field k of characteristic 0, compute a polynomial U(x) ∈ k[x] (a universal denominator) such that the denominator of each of rational solutions (if exist) of the given equation divides U(x). We consider two types of such algorithms. One of them is based on constructing a set of irreducible polynomials that are candidates for divisors of denominators of rational solutions, and on finding a bound for the exponent of each of these candidates (the full factorization of polynomials is used). The second one is related to earlier algorithms for finding universal denominators, where the computation of gcd was used instead of the full factorization. The algorithms are applicable to scalar equations of arbitrary orders as well as to systems of first-order equations.
Multivariate polynomial interpolation and sampling in Paley-Wiener spaces
Bailey, B A
2010-01-01
In this paper, an equivalence between existence of particular exponential Riesz bases for multivariate bandlimited functions and existence of certain polynomial interpolants for these bandlimited functions is given. For certain classes of unequally spaced data nodes and corresponding $\\ell_2$ data, the existence of these polynomial interpolants allows for a simple recovery formula for multivariate bandlimited functions which demonstrates $L_2$ and uniform convergence on $\\mathbb{R}^d$. A simpler computational version of this recovery formula is also given, at the cost of replacing $L_2$ and uniform convergence on $\\mathbb{R}^d$ with $L_2$ and uniform convergence on increasingly large subsets of $\\mathbb{R}^d$. As a special case, the polynomial interpolants of given $\\ell_2$ data converge in the same fashion to the multivariate bandlimited interpolant of that same data. Concrete examples of pertinant Riesz bases and unequally spaced data nodes are also given.
Applying polynomial filtering to mass preconditioned Hybrid Monte Carlo
Haar, Taylor; Zanotti, James; Nakamura, Yoshifumi
2016-01-01
The use of mass preconditioning or Hasenbusch filtering in modern Hybrid Monte Carlo simulations is common. At light quark masses, multiple filters (three or more) are typically used to reduce the cost of generating dynamical gauge fields; however, the task of tuning a large number of Hasenbusch mass terms is non-trivial. The use of short polynomial approximations to the inverse has been shown to provide an effective UV filter for HMC simulations. In this work we investigate the application of polynomial filtering to the mass preconditioned Hybrid Monte Carlo algorithm as a means of introducing many time scales into the molecular dynamics integration with a simplified parameter tuning process. A generalized multi-scale integration scheme that permits arbitrary step- sizes and can be applied to Omelyan-style integrators is also introduced. We find that polynomial-filtered mass-preconditioning (PF-MP) performs as well as or better than standard mass preconditioning, with significantly less fine tuning required.
Tensor calculus in polar coordinates using Jacobi polynomials
Vasil, Geoffrey M; Lecoanet, Daniel; Olver, Sheehan; Brown, Benjamin P; Oishi, Jeffrey S
2015-01-01
Spectral methods are an efficient way to solve partial differential equations on domains possessing certain symmetries. The utility of a method depends strongly on the choice of spectral basis. In this paper we describe a set of bases built out of Jacobi polynomials, and associated operators for solving scalar, vector, and tensor partial differential equations in polar coordinates on a unit disk. By construction, the bases satisfy regularity conditions at r=0 for any tensorial field. The coordinate singularity in a disk is a prototypical case for many coordinate singularities. The work presented here extends to other geometries. The operators represent covariant derivatives, multiplication by azimuthally symmetric functions, and the tensorial relationship between fields. These arise naturally from relations between classical orthogonal polynomials, and form a Heisenberg algebra. Other past work uses more specific polynomial bases for solving equations in polar coordinates. The main innovation in this paper is...
A novel single-order diffraction grating: Random position rectangle grating
Zuhua, Yang; Qiangqiang, Zhang; Jing, Wang; Quanping, Fan; Yuwei, Liu; Lai, Wei; Leifeng, Cao
2016-05-01
Spectral diagnosis of radiation from laser plasma interaction and monochromation of radiation source are hot and important topics recently. Grating is one of the primary optical elements to do this job. Although easy to fabricate, traditional diffraction grating suffers from multi-order diffraction contamination. On the other hand, sinusoidal amplitude grating has the nonharmonic diffraction property, but it is too difficult to fabricate, especially for x-ray application. A novel nonharmonic diffraction grating named random position rectangle grating (RPRG) is proposed in this paper. Theoretical analysis and experiment results show that the RPRG is both higher order diffraction suppressing and not difficult to fabricate. Additionally, it is highly efficient; its first order absolute theoretical diffraction efficiency reaches 4.1%. Our result shows that RPRG is a novel tool for radiation diagnosis and monochromation. Project supported by the National Natural Science Foundation of China (Grant No. 11375160) and the National Science Instruments Major Project of China (Grant No. 2012YQ130125).
Trends and future of fiber Bragg grating sensing technologies: tailored draw tower gratings (DTGs)
Lindner, E.; Hartung, A.; Hoh, D.; Chojetzki, C.; Schuster, K.; Bierlich, J.; Rothhardt, M.
2014-05-01
Today fiber Bragg gratings are commonly used in sensing technology as well as in telecommunications. Numerous requirements must be satisfied for their application as a sensor such as the number of sensors per system, the measurement resolution and repeatability, the sensor reusability as well as the sensor costs. In addition current challenges need to be met in the near future for sensing fibers to keep and extend their marketability such as the suitability for sterilization, hydrogen darkening or the separation of strain and temperature (or pressure and temperature). In this contribution we will give an outlook about trends and future of the fiber Bragg gratings in sensing technologies. Specifically, we will discuss how the use of draw tower grating technology enables the production of tailored Bragg grating sensing fibers, and we will present a method of separating strain and temperature by the use of a single Bragg grating only, avoiding the need for additional sensors to realize the commonly applied temperature compensation.
Moniem, T. A.
2016-05-01
This article presents a methodology for an integrated Bragg grating using an alloy of GaAs, AlGaAs, and InGaAs with a controllable refractive index to obtain an adaptive Bragg grating suitable for many applications on optical processing and adaptive control systems, such as limitation and filtering. The refractive index of a Bragg grating is controlled by using an external electric field for controlling periodic modulation of the refractive index of the active waveguide region. The designed Bragg grating has refractive indices programmed by using that external electric field. This article presents two approaches for designing the controllable refractive indices active region of a Bragg grating. The first approach is based on the modification of a planar micro-strip structure of the iGaAs traveling wave as the active region, and the second is based on the modification of self-assembled InAs/GaAs quantum dots of an alloy from GaAs and InGaAs with a GaP traveling wave. The overall design and results are discussed through numerical simulation by using the finite-difference time-domain, plane wave expansion, and opto-wave simulation methods to confirm its operation and feasibility.
On the Action of Steenrod Operations on Polynomial Algebras
KARACA, İsmet
1998-01-01
Let \\( \\bba \\) be the mod-\\( p \\) Steenrod Algebra. Let \\( p \\) be an odd prime number and \\( Zp = Z/pZ \\). Let \\( Ps = Zp [x1,x2,\\ldots,xs]. \\) A polynomial \\( N \\in Ps \\) is said to be hit if it is in the image of the action \\( A \\otimes Ps \\ra Ps. \\) In [10] for \\( p=2, \\) Wood showed that if \\( \\a(d+s) > s \\) then every polynomial of degree \\( d \\) in \\( Ps \\) is hit where \\( \\a(d+s) \\) denotes the number of ones in the binary expansion of \\( d+s \\). Latter in [6] Monks extended a resu...
On Continued Fraction Expansion of Real Roots of Polynomial Systems
DEFF Research Database (Denmark)
Mantzaflaris, Angelos; Mourrain, Bernard; Tsigaridas, Elias
2011-01-01
polynomials. A partial extension of Vincent’s Theorem for multivariate polynomials is presented, which allows us to prove the termination of the algorithm. Bounding functions, projection and preconditioning are employed to speed up the scheme. The resulting isolation boxes have optimized rational coordinates......, corresponding to the first terms of the continued fraction expansion of the real roots. Finally, we present new complexity bounds for a simplified version of the algorithm in the bit complexity model, and also bounds in the real RAM model for a family of subdivision algorithms in terms of the real condition...
Method of resolution of 3SAT in polynomial time
Salemi, Luigi
2009-01-01
Presentation of a Method for determining whether a problem 3Sat has solution, and if yes to find one, in time max O(n11). Is thus proved (if I am not mistaken yet) that the problem 3Sat is fully resolved in polynomial time and therefore that it is in P, by the work of Cook and Levin, and can transform a SAT problem in a 3Sat in polynomial time (ref. Karp), it follows that P = NP. Open Source program is available at http://www.visainformatica.it/3sat
The colored HOMFLY polynomial is q-holonomic
Garoufalidis, Stavros
2012-01-01
We prove that the colored HOMFLY polynomial of a link, colored by symmetric or exterior powers of the fundamental representation, is q-holonomic with respect to the color parameters. As a result, we obtain the existence of an (a,q) super-polynomial of all knots in 3-space. Our result has implications on the quantization of the SL(2,C) character variety of knots using ideal triangulations or the topological recursion, and motivates questions on the web approach to representation theory.
Multi-mode entangled states represented as Grassmannian polynomials
Maleki, Y.
2016-06-01
We introduce generalized Grassmannian representatives of multi-mode state vectors. By implementing the fundamental properties of Grassmann coherent states, we map the Hilbert space of the finite-dimensional multi-mode states to the space of some Grassmannian polynomial functions. These Grassmannian polynomials form a well-defined space in the framework of Grassmann variables; namely Grassmannian representative space. Therefore, a quantum state can be uniquely defined and determined by an element of Grassmannian representative space. Furthermore, the Grassmannian representatives of some maximally entangled states are considered, and it is shown that there is a tight connection between the entanglement of the states and their Grassmannian representatives.
Evolution method and HOMFLY polynomials for virtual knots
Bishler, Ludmila; Morozov, Andrey; Morozov, Anton
2014-01-01
Following the suggestion of arXiv:1407.6319 to lift the knot polynomials for virtual knots and links from Jones to HOMFLY, we apply the evolution method to calculate them for an infinite series of twist-like virtual knots and antiparallel 2-strand links. Within this family one can check topological invariance and understand how differential hierarchy is modified in virtual case. This opens a way towards a definition of colored (not only cabled) knot polynomials, though problems still persist beyond the first symmetric representation.
On Star-Varieties with Almost Polynomial Growth
Institute of Scientific and Technical Information of China (English)
A. Giambruno; S. Mishchenko
2001-01-01
Let 2 be a variety of algebras with involution * over a field of characteristic zero and cn(V, *) the corresponding sequence of *-codimensions. Here,we characterize those varieties V such that cn (V, *) is polynomially bounded. We prove that V is such a variety if and only if G2,M V, where G2 and M are two explicit finite-dimensional algebras with involution previously constructed. It follows that G2 and M generate the only two varieties of algebras with involution with almost polynomial growth and there is no variety with intermediate growth.
Polynomial birth-death distribution approximation in Wasserstein distance
Xia, Aihua; Zhang, Fuxi
2008-01-01
The polynomial birth-death distribution (abbr. as PBD) on $\\ci=\\{0,1,2, >...\\}$ or $\\ci=\\{0,1,2, ..., m\\}$ for some finite $m$ introduced in Brown & Xia (2001) is the equilibrium distribution of the birth-death process with birth rates $\\{\\alpha_i\\}$ and death rates $\\{\\beta_i\\}$, where $\\a_i\\ge0$ and $\\b_i\\ge0$ are polynomial functions of $i\\in\\ci$. The family includes Poisson, negative binomial, binomial and hypergeometric distributions. In this paper, we give probabilistic proofs of variou...
Modeling Microwave Structures in Time Domain Using Laguerre Polynomials
Directory of Open Access Journals (Sweden)
Z. Raida
2006-09-01
Full Text Available The paper is focused on time domain modeling of microwave structures by the method of moments. Two alternative schemes with weighted Laguerre polynomials are presented. Thanks to their properties, these schemes are free of late time oscillations. Further, the paper is aimed to effective and accurate evaluation of Green's functions integrals within these schemes. For this evaluation, a first- and second-order polynomial approximation is developed. The last part of the paper deals with modeling microstrip structures in the time domain. Conditions of impedance matching are derived, and the proposed approach is verified by modeling a microstrip filter.
Multi-mode entangled states represented as Grassmannian polynomials
Maleki, Y.
2016-09-01
We introduce generalized Grassmannian representatives of multi-mode state vectors. By implementing the fundamental properties of Grassmann coherent states, we map the Hilbert space of the finite-dimensional multi-mode states to the space of some Grassmannian polynomial functions. These Grassmannian polynomials form a well-defined space in the framework of Grassmann variables; namely Grassmannian representative space. Therefore, a quantum state can be uniquely defined and determined by an element of Grassmannian representative space. Furthermore, the Grassmannian representatives of some maximally entangled states are considered, and it is shown that there is a tight connection between the entanglement of the states and their Grassmannian representatives.
On a Family of 2-Variable Orthogonal Krawtchouk Polynomials
Directory of Open Access Journals (Sweden)
F. Alberto Grünbaum
2010-12-01
Full Text Available We give a hypergeometric proof involving a family of 2-variable Krawtchouk polynomials that were obtained earlier by Hoare and Rahman [SIGMA 4 (2008, 089, 18 pages] as a limit of the 9−j symbols of quantum angular momentum theory, and shown to be eigenfunctions of the transition probability kernel corresponding to a ''poker dice'' type probability model. The proof in this paper derives and makes use of the necessary and sufficient conditions of orthogonality in establishing orthogonality as well as indicating their geometrical significance. We also derive a 5-term recurrence relation satisfied by these polynomials.
Algorithms for Computing Triangular Decompositions of Polynomial Systems
Chen, Changbo
2011-01-01
We propose new algorithms for computing triangular decompositions of polynomial systems incrementally. With respect to previous works, our improvements are based on a {\\em weakened} notion of a polynomial GCD modulo a regular chain, which permits to greatly simplify and optimize the sub-algorithms. Extracting common work from similar expensive computations is also a key feature of our algorithms. In our experimental results the implementation of our new algorithms, realized with the {\\RegularChains} library in {\\Maple}, outperforms solvers with similar specifications by several orders of magnitude on sufficiently difficult problems.
Algorithm that Solves 3-SAT in Polynomial Time
Steinmetz, Jason W
2011-01-01
The question of whether the complexity class P is equal to the complexity class NP has been a seemingly intractable problem for over 4 decades. It has been clear that if an algorithm existed that would solve the problems in the NP class in polynomial time then P would equal NP. However, no one has yet been able to create that algorithm or to successfully prove that such an algorithm cannot exist. The algorithm that will be presented in this paper runs in polynomial time and solves the 3-satisfiability or 3-SAT problem, which has been proven to be NP-complete, thus indicating that P = NP.
Skew-orthogonal polynomials, differential systems and random matrix theory
International Nuclear Information System (INIS)
We study skew-orthogonal polynomials with respect to the weight function exp[-2V (x)], with V (x) = ΣK=12d (uK/K)xK, u2d > 0, d > 0. A finite subsequence of such skew-orthogonal polynomials arising in the study of Orthogonal and Symplectic ensembles of random matrices, satisfy a system of differential-difference-deformation equation. The vectors formed by such subsequence has the rank equal to the degree of the potential in the quaternion sense. These solutions satisfy certain compatibility condition and hence admit a simultaneous fundamental system of solutions. (author)
M-Polynomial and Related Topological Indices of Nanostar Dendrimers
Directory of Open Access Journals (Sweden)
Mobeen Munir
2016-09-01
Full Text Available Dendrimers are highly branched organic macromolecules with successive layers of branch units surrounding a central core. The M-polynomial of nanotubes has been vastly investigated as it produces many degree-based topological indices. These indices are invariants of the topology of graphs associated with molecular structure of nanomaterials to correlate certain physicochemical properties like boiling point, stability, strain energy, etc. of chemical compounds. In this paper, we first determine M-polynomials of some nanostar dendrimers and then recover many degree-based topological indices.
An orthogonal basis for the hyperbolic hybrid polynomial space
Institute of Scientific and Technical Information of China (English)
HUANG Yu; WANG GuoZhao
2007-01-01
Motivated by the wide usage of the Tchebyshev basis and Legendre basis in the algebra polynomial space, we construct an orthogonal basis with the properties of the H-Bézier basis in the hyperbolic hybrid polynomial space, which is similar to the Legendre basis and holds remarkable properties. Moreover, we derive the transformation matrices that map the H-Bézier basis and the orthogonal basis forms into each other. An example for approximating the degree reduction of the H- Bézier curves is sketched to illustrate the utility of the orthogonal basis.
Bispectral commuting difference operators for multivariable Askey-Wilson polynomials
Iliev, Plamen
2008-01-01
We construct a commutative algebra A_z, generated by d algebraically independent q-difference operators acting on variables z_1, z_2,..., z_d, which is diagonalized by the multivariable Askey-Wilson polynomials P_n(z) considered by Gasper and Rahman [6]. Iterating Sears' transformation formula, we show that the polynomials P_n(z) possess a certain duality between z and n. Analytic continuation allows us to obtain another commutative algebra A_n, generated by d algebraically independent differ...
Local polynomial Whittle estimation covering non-stationary fractional processes
DEFF Research Database (Denmark)
Nielsen, Frank
This paper extends the local polynomial Whittle estimator of Andrews & Sun (2004) to fractionally integrated processes covering stationary and non-stationary regions. We utilize the notion of the extended discrete Fourier transform and periodogram to extend the local polynomial Whittle estimator to...... in.ated by a multiplicative constant. We show consistency and asymptotic normality for d ε (-1/2), and if the spectral density of the short-run component is in.nitely smooth near frequency zero we obtain an optimal rate of convergence for this setting, i.e. convergence arbitrarily close to the...
A novel computational approach to approximate fuzzy interpolation polynomials.
Jafarian, Ahmad; Jafari, Raheleh; Mohamed Al Qurashi, Maysaa; Baleanu, Dumitru
2016-01-01
This paper build a structure of fuzzy neural network, which is well sufficient to gain a fuzzy interpolation polynomial of the form [Formula: see text] where [Formula: see text] is crisp number (for [Formula: see text], which interpolates the fuzzy data [Formula: see text]. Thus, a gradient descent algorithm is constructed to train the neural network in such a way that the unknown coefficients of fuzzy polynomial are estimated by the neural network. The numeral experimentations portray that the present interpolation methodology is reliable and efficient. PMID:27625982
A Root Isolation Algorithm for Sparse Univariate Polynomials
Alonso, Maria Emilia; Galligo, André
2012-01-01
8 double pages. International audience We consider a univariate polynomial f with real coefficients having a high degree $N$ but a rather small number $d+1$ of monomials, with $d\\ll N$. Such a sparse polynomial has a number of real root smaller or equal to $d$. Our target is to find for each real root of $f$ an interval isolating this root from the others. The usual subdivision methods, relying either on Sturm sequences or Moebius transform followed by Descartes's rule of sign, destruct...
ON COEFFICIENT POLYNOMIALS OF CUBIC HERMITE-PAD(E) APPROXIMATIONS TO THE EXPONENTIAL FUNCTION
Institute of Scientific and Technical Information of China (English)
Cheng-de Zheng; Guo-can Wang; Zhi-bin Li
2005-01-01
The polynomials related with cubic Hermite-Pade approximation to the exponential function are investigated which have degrees at most n, m, s respectively. A connection is given between the coefficients of each of the polynomials and certain hypergeometric functions, which leads to a simple expression for a polynomial in a special case. Contour integral representations of the polynomials are given. By using of the saddle point method the exact asymptotics of the polynomials are derived as n, m, s tend to infinity through certain ray sequence. Some further uniform asymptotic aspects of the polynomials are also discussed.
Complex and distributional weights for sieved ultraspherical polynomials
Directory of Open Access Journals (Sweden)
Jairo A. Charris
1996-01-01
Full Text Available Contour integral and distributional orthogonality of sieved ultraspherical polynomials are established for values of the parameters outside the natural range of orthogonality by positive measures on the real line. A general representation theorem for moment functionals is included.
On Polynomials of Prescribed Height in Finite Fields
Shparlinskiĭ, I. E.
1989-02-01
This paper deals with the set \\mathfrak{M}(B) of monic polynomials of degree n with integral coefficients belonging to a given n-dimensional cube B with side h. An asymptotic formula is obtained for the number of polynomials in \\mathfrak{M}(B) having a specific type of decomposition into irreducible factors modulo some prime p, and an asymptotic formula for the number of primitive polynomials modulo p in \\mathfrak{M}(B), which translates when n=1 into known results of I. M. Vinogradov on the distribution of primitive roots. These asymptotic formulas are nontrivial when h\\geq p^{n/(n+1)+\\varepsilon} for any \\varepsilon>0.Moreover, an asymptotic formula is obtained for the average value of the number of divisors modulo p of polynomials in \\mathfrak{M}(B), a result that is nontrivial when h\\geq\\max(p^{1-2/n}\\ln p,\\,p^{1/2}\\ln p).Bibliography: 11 titles.
Differentiation by integration using orthogonal polynomials, a survey
E. Diekema; T.H. Koornwinder
2012-01-01
This survey paper discusses the history of approximation formulas for n-th order derivatives by integrals involving orthogonal polynomials. There is a large but rather disconnected corpus of literature on such formulas. We give some results in greater generality than in the literature. Notably we un