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.
1980-08-01
a C) - I Fl C.) -KV II 0 LrJ -I- C) y j1 I U I - C) -7 4 2 *2 - -* -i ! 3 C ;4.~ ’I WALSH PREPROCESSO M OE. 2001A ,OLT" OUT C~M04N& FALM aLMs t Disa ...others being blanked. ’Fihe seql ency spect rum processor operates as follows: 1) Examines all NTI sampleJAs of’ thle Forwalrd ValshI ’ra sfor-mer...ichl sets up the processin g t hreshold. a. ’T’his analyvsis is based ((il a known di flerene in sCetuency spect rumn bet weeti noise onlIy cotidit ions
Matrix-valued polynomials in Lanczos type methods
Energy Technology Data Exchange (ETDEWEB)
Simoncini, V. [Universita di Padova (Italy); Gallopoulos, E. [Univ. of Illinois, Urbana, IL (United States)
1994-12-31
It is well known that convergence properties of iterative methods can be derived by studying the behavior of the residual polynomial over a suitable domain of the complex plane. Block Krylov subspace methods for the solution of linear systems A[x{sub 1},{hor_ellipsis}, x{sub s}] = [b{sub 1},{hor_ellipsis}, b{sub s}] lead to the generation of residual polynomials {phi}{sub m} {element_of} {bar P}{sub m,s} where {bar P}{sub m,s} is the subset of matrix-valued polynomials of maximum degree m and size s such that {phi}{sub m}(0) = I{sub s}, R{sub m} := B - AX{sub m} = {phi}{sub m}(A) {circ} R{sub 0}, where {phi}{sub m}(A) {circ} R{sub 0} := R{sub 0} - A{summation}{sub j=0}{sup m-1} A{sup j}R{sub 0}{xi}{sub j}, {xi}{sub j} {element_of} R{sup sxs}. An effective method has to balance adequate approximation with economical computation of iterates defined by the polynomial. Matrix valued polynomials can be used to improve the performance of block methods. Another approach is to solve for a single right-hand side at a time and use the generated information in order to update the approximations of the remaining systems. In light of this, a more general scheme is as follows: A subset of residuals (seeds) is selected and a block short term recurrence method is used to compute approximate solutions for the corresponding systems. At the same time the generated matrix valued polynomial is implicitly applied to the remaining residuals. Subsequently a new set of seeds is selected and the process is continued as above, till convergence of all right-hand sides. The use of a quasi-minimization technique ensures a smooth convergence behavior for all systems. In this talk the authors discuss the implementation of this class of algorithms and formulate strategies for the selection of parameters involved in the computation. Experiments and comparisons with other methods will be presented.
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.
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...
Optimal stability polynomials for numerical integration of initial value problems
Ketcheson, David I.
2013-01-08
We consider the problem of finding optimally stable polynomial approximations to the exponential for application to one-step integration of initial value ordinary and partial differential equations. The objective is to find the largest stable step size and corresponding method for a given problem when the spectrum of the initial value problem is known. The problem is expressed in terms of a general least deviation feasibility problem. Its solution is obtained by a new fast, accurate, and robust algorithm based on convex optimization techniques. Global convergence of the algorithm is proven in the case that the order of approximation is one and in the case that the spectrum encloses a starlike region. Examples demonstrate the effectiveness of the proposed algorithm even when these conditions are not satisfied.
Prime valued polynomials and class numbers of quadratic fields
Directory of Open Access Journals (Sweden)
Richard A. Mollin
1990-01-01
Full Text Available It is the purpose of this paper to give a survey of the relationship between the class number one problem for real quadratic fields and prime-producing quadratic polynomials; culminating in an overview of the recent solution to the class number one problem for real quadratic fields of Richaud-Degert type. We conclude with new conjectures, questions and directions.
Prime valued polynomials and class numbers of quadratic fields
Mollin, Richard A.
1990-01-01
It is the purpose of this paper to give a survey of the relationship between the class number one problem for real quadratic fields and prime-producing quadratic polynomials; culminating in an overview of the recent solution to the class number one problem for real quadratic fields of Richaud-Degert type. We conclude with new conjectures, questions and directions.
Note on the Solution of Transport Equation by Tau Method and Walsh Functions
Directory of Open Access Journals (Sweden)
Abdelouahab Kadem
2010-01-01
Full Text Available We consider the combined Walsh function for the three-dimensional case. A method for the solution of the neutron transport equation in three-dimensional case by using the Walsh function, Chebyshev polynomials, and the Legendre polynomials are considered. We also present Tau method, and it was proved that it is a good approximate to exact solutions. This method is based on expansion of the angular flux in a truncated series of Walsh function in the angular variable. The main characteristic of this technique is that it reduces the problems to those of solving a system of algebraic equations; thus, it is greatly simplifying the problem.
Approximating the Value of a Concurrent Reachability Game in the Polynomial Time Hierarchy
DEFF Research Database (Denmark)
Frederiksen, Søren Kristoffer Stiil; Miltersen, Peter Bro
2013-01-01
We show that the value of a finite-state concurrent reachability game can be approximated to arbitrary precision in TFNP[NP], that is, in the polynomial time hierarchy. Previously, no better bound than PSPACE was known for this problem. The proof is based on formulating a variant of the state...
Feature Detection and Curve Fitting Using Fast Walsh Transforms for Shock Tracking: Applications
Gnoffo, Peter A.
2017-01-01
Walsh functions form an orthonormal basis set consisting of square waves. Square waves make the system well suited for detecting and representing functions with discontinuities. Given a uniform distribution of 2p cells on a one-dimensional element, it has been proven that the inner product of the Walsh Root function for group p with every polynomial of degree shock wave form after the diaphragm bursts.
Far-field Diffraction Properties of Annular Walsh Filters
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Pubali Mukherjee
2013-01-01
Full Text Available Annular Walsh filters are derived from the rotationally symmetric annular Walsh functions which form a complete set of orthogonal functions that take on values either +1 or −1 over the domain specified by the inner and outer radii of the annulus. The value of any annular Walsh function is taken as zero from the centre of the circular aperture to the inner radius of the annulus. The three values 0, +1, and −1 in an annular Walsh function can be realized in a corresponding annular Walsh filter by using transmission values of zero amplitude (i.e., an obscuration, unity amplitude and zero phase, and unity amplitude and phase, respectively. Not only the order of the Walsh filter but also the size of the inner radius of the annulus provides an additional degree of freedom in tailoring of point spread function by using these filters for pupil plane filtering in imaging systems. In this report, we present the far-field amplitude characteristics of some of these filters to underscore their potential for effective use in several demanding applications like high-resolution microscopy, optical data storage, microlithography, optical encryption, and optical micromanipulation.
Kang, Seung-Wan; Byun, Gukdo; Park, Hun-Joon
2014-12-01
This paper presents empirical research into the relationship between leader-follower value congruence in social responsibility and the level of ethical satisfaction for employees in the workplace. 163 dyads were analyzed, each consisting of a team leader and an employee working at a large manufacturing company in South Korea. Following current methodological recommendations for congruence research, polynomial regression and response surface modeling methodologies were used to determine the effects of value congruence. Results indicate that leader-follower value congruence in social responsibility was positively related to the ethical satisfaction of employees. Furthermore, employees' ethical satisfaction was stronger when aligned with a leader with high social responsibility. The theoretical and practical implications are discussed.
A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus
Dunkl, Charles F.
2017-06-01
For each irreducible module of the symmetric group S_{N} there is a set of parametrized nonsymmetric Jack polynomials in N variables taking values in the module. These polynomials are simultaneous eigenfunctions of a commutative set of operators, self-adjoint with respect to two Hermitian forms, one called the contravariant form and the other is with respect to a matrix-valued measure on the N-torus. The latter is valid for the parameter lying in an interval about zero which depends on the module. The author in a previous paper [SIGMA 12 (2016), 033, 27 pages] proved the existence of the measure and that its absolutely continuous part satisfies a system of linear differential equations. In this paper the system is analyzed in detail. The N-torus is divided into (N-1)! connected components by the hyperplanes x_{i}=x_{j}, i
Speech/Nonspeech Detection Using Minimal Walsh Basis Functions
Directory of Open Access Journals (Sweden)
Moe Pwint
2006-10-01
Full Text Available This paper presents a new method to detect speech/nonspeech components of a given noisy signal. Employing the combination of binary Walsh basis functions and an analysis-synthesis scheme, the original noisy speech signal is modified first. From the modified signals, the speech components are distinguished from the nonspeech components by using a simple decision scheme. Minimal number of Walsh basis functions to be applied is determined using singular value decomposition (SVD. The main advantages of the proposed method are low computational complexity, less parameters to be adjusted, and simple implementation. It is observed that the use of Walsh basis functions makes the proposed algorithm efficiently applicable in real-world situations where processing time is crucial. Simulation results indicate that the proposed algorithm achieves high-speech and nonspeech detection rates while maintaining a low error rate for different noisy conditions.
Speech/Nonspeech Detection Using Minimal Walsh Basis Functions
Directory of Open Access Journals (Sweden)
Pwint Moe
2007-01-01
Full Text Available This paper presents a new method to detect speech/nonspeech components of a given noisy signal. Employing the combination of binary Walsh basis functions and an analysis-synthesis scheme, the original noisy speech signal is modified first. From the modified signals, the speech components are distinguished from the nonspeech components by using a simple decision scheme. Minimal number of Walsh basis functions to be applied is determined using singular value decomposition (SVD. The main advantages of the proposed method are low computational complexity, less parameters to be adjusted, and simple implementation. It is observed that the use of Walsh basis functions makes the proposed algorithm efficiently applicable in real-world situations where processing time is crucial. Simulation results indicate that the proposed algorithm achieves high-speech and nonspeech detection rates while maintaining a low error rate for different noisy conditions.
Power electronic systems Walsh analysis with Matlab
Deb, Anish
2014-01-01
A Totally Different Outlook on Power Electronic System AnalysisPower Electronic Systems: Walsh Analysis with MATLAB® builds a case for Walsh analysis as a powerful tool in the study of power electronic systems. It considers the application of Walsh functions in analyzing power electronic systems, and the advantages offered by Walsh domain analysis of power electronic systems. Solves Power Electronic Systems in an Unconventional WayThis book successfully integrates power electronics as well as systems and control. Incorporating a complete orthonormal function set very much unlike the sine-cosin
Directory of Open Access Journals (Sweden)
W. M. Abd-Elhameed
2015-01-01
Full Text Available The main aim of this research article is to develop two new algorithms for handling linear and nonlinear third-order boundary value problems. For this purpose, a novel operational matrix of derivatives of certain nonsymmetric generalized Jacobi polynomials is established. The suggested algorithms are built on utilizing the Galerkin and collocation spectral methods. Moreover, the principle idea behind these algorithms is based on converting the boundary value problems governed by their boundary conditions into systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable solvers. We support our algorithms by a careful investigation of the convergence analysis of the suggested nonsymmetric generalized Jacobi expansion. Some illustrative examples are given for the sake of indicating the high accuracy and efficiency of the two proposed algorithms.
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
Chen, Sheng; Hong, Xia; Khalaf, Emad F; Alsaadi, Fuad E; Harris, Chris J
2017-12-01
Complex-valued (CV) B-spline neural network approach offers a highly effective means for identifying and inverting practical Hammerstein systems. Compared with its conventional CV polynomial-based counterpart, a CV B-spline neural network has superior performance in identifying and inverting CV Hammerstein systems, while imposing a similar complexity. This paper reviews the optimality of the CV B-spline neural network approach. Advantages of B-spline neural network approach as compared with the polynomial based modeling approach are extensively discussed, and the effectiveness of the CV neural network-based approach is demonstrated in a real-world application. More specifically, we evaluate the comparative performance of the CV B-spline and polynomial-based approaches for the nonlinear iterative frequency-domain decision feedback equalization (NIFDDFE) of single-carrier Hammerstein channels. Our results confirm the superior performance of the CV B-spline-based NIFDDFE over its CV polynomial-based counterpart.
Spectrums Transform Operators in Bases of Fourier and Walsh Functions
Directory of Open Access Journals (Sweden)
V. V. Syuzev
2017-01-01
Full Text Available The problems of synthesis of the efficient algorithms for digital processing of discrete signals require transforming the signal spectra from one basis system into other. The rational solution to this problem is to construct the Fourier kernel, which is a spectrum of some basis functions, according to the system of functions of the other basis. However, Fourier kernel properties are not equally studied and described for all basis systems of practical importance. The article sets a task and presents an original way to solve the problem of mutual transformation of trigonometric Fourier spectrum into Walsh spectrum of different basis systems.The relevance of this theoretical and applied problem is stipulated, on the one hand, by the prevalence of trigonometric Fourier basis for harmonic representation of digital signals, and, on the other hand, by the fact that Walsh basis systems allow us to have efficient algorithms to simulate signals. The problem solution is achieved through building the Fourier kernel of a special structure that allows us to establish independent groups of Fourier and Walsh spectrum coefficients for further reducing the computational complexity of the transform algorithms.The article analyzes the properties of the system of trigonometric Fourier functions and shows its completeness. Considers the Walsh function basis systems in three versions, namely those of Hadamard, Paley, and Hartmut giving different ordering and analytical descriptions of the functions that make up the basis. Proves a completeness of these systems.Sequentially, for each of the three Walsh systems the analytical curves for the Fourier kernel components are obtained, and Fourier kernel themselves are built with binary rational number of samples of basis functions. The kernels are presented in matrix form and, as an example, recorded for a particular value of the discrete interval of N, equal to 8. The analysis spectral coefficients of the Fourier kernel
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...
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...
1981-08-01
Bo) YoYo = 0 (27) Equation (27) can be satisfied identically if the end conditions of the adjoint system are proportional to the end conditions of the...at the near end and carry on in one direction. Thus Eq. (35) is simplified to - tb - 6J(6y) - -( Yoyo +oyo’) 6 yo + fto 6yQdt tb- -f [(cyiBy’)6y + (yy+cy...the final values of the problem. 18 NUMERICAL RESULTS AND DISCUSSION The analysis presented in previous sections will now be tested by way of some
El novelazo diferido de Rodolfo Walsh1
Luppi, Juan Pablo
2016-01-01
En 1969, Rodolfo Walsh (1927-1977) desplaza propósitos estéticos que había priorizado cinco años atrás (cuando dejó el periodismo para escribir ficción) y trastoca la figuración autoral afirmada previamente. El acercamiento a la militancia en el vértigo cultural y vital de fines de los 60 (en el semanario del sindicalismo opositor en 1968-69, desde 1970 en agrupaciones armadas) oblitera el uso subjetivo del tiempo y desacomoda la autoexigencia de morosidad, pero no interrumpe tareas de escrit...
Szegő, G
1939-01-01
The general theory of orthogonal polynomials was developed in the late 19th century from a study of continued fractions by P. L. Chebyshev, even though special cases were introduced earlier by Legendre, Hermite, Jacobi, Laguerre, and Chebyshev himself. It was further developed by A. A. Markov, T. J. Stieltjes, and many other mathematicians. The book by Szegő, originally published in 1939, is the first monograph devoted to the theory of orthogonal polynomials and its applications in many areas, including analysis, differential equations, probability and mathematical physics. Even after all the
Polynomially Bounded Sequences and Polynomial Sequences
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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].
Dyadic Walsh analysis from 1924 onwards Walsh-Gibbs-Butzer dyadic differentiation in science
Stankovic, Radomir S; Schipp, Ferenc; Wade, William R; Su, Weiyi; Endow, Yasushi; Fridli, Sandor; Golubov, Boris I; Pichler, Franz
2015-01-01
Dyadic (Walsh) analysis emerged as a new research area in applied mathematics and engineering in early seventies within attempts to provide answers to demands from practice related to application of spectral analysis of different classes of signals, including audio, video, sonar, and radar signals. In the meantime, it evolved in a mature mathematical discipline with fundamental results and important features providing basis for various applications. The book will provide fundamentals of the area through reprinting carefully selected earlier publications followed by overview of recent results concerning particular subjects in the area written by experts, most of them being founders of the field, and some of their followers. In this way, this first volume of the two volume book offers a rather complete coverage of the development of dyadic Walsh analysis, and provides a deep insight into its mathematical foundations necessary for consideration of generalizations and applications that are the subject of the se...
Irreducible multivariate polynomials obtained from polynomials in ...
Indian Academy of Sciences (India)
over K(X)? We provided some methods to construct irreducible multivariate polynomials over an arbi- trary field, starting from arbitrary irreducible polynomials in fewer variables, of which we mention the following two results: Theorem A. If we write an irreducible polynomial f ∈ K[X] as a sum of polynomials a0,..., an ∈ K[X] ...
Sibling curves of polynomials | Wiggins | Quaestiones Mathematicae
African Journals Online (AJOL)
Sibling curves were demonstrated in papers [2, 3] as a novel way to visualize the zeros of complex valued functions. In this paper, we continue the work done in those papers by focusing solely on polynomials. We proceed to prove that the number of sibling curves of a polynomial is the degree of the polynomial. Keywords: ...
Koornwinder, T.H.
2012-01-01
Askey-Wilson polynomial refers to a four-parameter family of q-hypergeometric orthogonal polynomials which contains all families of classical orthogonal polynomials (in the wide sense) as special or limit cases.
Towards holographic "brain" memory based on randomization and Walsh-Hadamard transformation
Dolev, S.; Frenkel, S.; Hanemann, A.
2013-01-01
The holographic conceptual approach for cognitive processes in human brain is investigated by neuroscientists due to the ability of holography to describe sophisticated phenomena of human perception and cognition. In this work we suggest a new mathematical description for Pribram's holographic or "holonomic" representation approach for the mind. Namely, we consider: (i) randomization of information, and (ii) Walsh-Hadamard spectral representation of holograms, rather than the well-known Fourier transform representation. The randomization reflects the belief that perceptual processes are not direct, but depend on the perceiver's expectations and previous knowledge as well as the information available in the stimulus itself. The use of Fourier transform and in our case Walsh-Hadamard transform reflects the possibility that each neuron or group of neurons encode some information about the entire image rather than the whole information about a part of the image. We demonstrate that the Walsh-Hadamard transform has benefits over the general Fourier transform. The encoding is performed on randomized information that is then represented by a set of spectral Wash-Hadamard coefficients that have holographic properties. Namely, any portion of the set of coefficients defines a "blurry image" of the original data. The values of the coefficients of the Walsh-Hadamard transformation are distributed approximately normally when the information is randomized, ensuring, with high probability, that growing sets of coefficients implies a monotonic gain of information. Moreover the randomization of the original information yields robust code that is able to cope with missing coefficients. The bridge between the randomization and holographic encoding with the well-known holographic human brain assumption may bring an interesting interpretation of the perception phenomena. In particular, holographic encoding fits the mystery of the human memory encoding, where damage of portions leaves a
Directory of Open Access Journals (Sweden)
Erdoğan Şen
2013-01-01
Full Text Available We consider the following boundary-value problem of nonlinear fractional differential equation with p-Laplacian operator D0+β(ϕp(D0+αu(t+a(tf(u=0, 01, ϕp-1=ϕq, 1/p+1/q=1,0⩽γ0 are parameters, a:(0,1→[0,+∞, and f:[0,+∞→[0,+∞ are continuous. By the properties of Green function and Schauder fixed point theorem, several existence and nonexistence results for positive solutions, in terms of the parameters λ and μ are obtained. The uniqueness of positive solution on the parameters λ and μ is also studied. In the final section of this paper, we derive not only new but also interesting identities related special polynomials by which Caputo fractional derivative.
Vivaldi, F
2002-01-01
We introduce a class of dynamical systems of algebraic origin, consisting of self-interacting irreducible polynomials over a field. A polynomial $f$ is made to act on a polynomial $g$ by mapping the roots of the latter. This action identifies a new polynomial $h$, as the minimal polynomial of the displaced roots. By allowing several polynomials to act on one another, we obtain a self-interacting system with a rich dynamics and strong collective features, which affords a fresh viewpoint on some algebraic dynamical constructs. We identify the basic dynamical invariants and begin the study of periodic behaviour, organizing the polynomials into an oriented graph.
On L1-convergence of Walsh-Fourier series
Directory of Open Access Journals (Sweden)
C. W. Onneweer
1978-01-01
Full Text Available Let G denote the dyadic group, which has as its dual group the Walsh(-Paley functions. In this paper we formulate a condition for functions in L1(G which implies that their Walsh-Fourier series converges in L1(G-norm. As a corollary we obtain a Dini-Lipschitz-type theorem for L1(G convergence and we prove that the assumption on the L1(G modulus of continuity in this theorem cannot be weakened. Similar results also hold for functions on the circle group T and their (trigonometric Fourier series.
Reddy, A. Satyanarayana; Mehta, Shashank K
2011-01-01
A graph $X$ is said to be a pattern polynomial graph if its adjacency algebra is a coherent algebra. In this study we will find a necessary and sufficient condition for a graph to be a pattern polynomial graph. Some of the properties of the graphs which are polynomials in the pattern polynomial graph have been studied. We also identify known graph classes which are pattern polynomial graphs.
48 CFR 52.222-20 - Walsh-Healey Public Contracts Act.
2010-10-01
... Clauses 52.222-20 Walsh-Healey Public Contracts Act. As prescribed in 22.610, insert the following clause in solicitations and contracts covered by the Act: Walsh-Healey Public Contracts Act (OCT 2010) If... amount that exceeds or may exceed $15,000, and is subject to the Walsh-Healey Public Contracts Act, as...
29 CFR 785.4 - Application to Walsh-Healey Public Contracts Act.
2010-07-01
... 29 Labor 3 2010-07-01 2010-07-01 false Application to Walsh-Healey Public Contracts Act. 785.4... Considerations § 785.4 Application to Walsh-Healey Public Contracts Act. The principles set forth in this part... employees performing work subject to the provisions of the Walsh-Healey Public Contracts Act. ...
41 CFR 50-201.1 - The Walsh-Healey Public Contracts Act.
2010-07-01
... 41 Public Contracts and Property Management 1 2010-07-01 2010-07-01 true The Walsh-Healey Public Contracts Act. 50-201.1 Section 50-201.1 Public Contracts and Property Management Other Provisions Relating... Walsh-Healey Public Contracts Act. The Walsh-Healey Public Contracts Act, as amended (41 U.S.C. 35-45...
The Walsh Family Resilience Questionnaire: the Italian version
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Rocchi S
2017-12-01
Full Text Available Silvana Rocchi,1 Claudio Ghidelli,2 Roberto Burro,3 Michele Vitacca,4 Simonetta Scalvini,5 Anna Maria Della Vedova,6 Gianmarco Roselli,7 Jean-Pierre Ramponi,8 Giorgio Bertolotti9 1Psychology Service, ICS Maugeri Spa SB, Institute of Lumezzane, 2Psychological Counselling Service, Università Cattolica del Sacro Cuore, Brescia, 3Department of Human Sciences, University of Verona, Verona, 4Respiratory Rehabilitation Division, 5Cardiac Rehabilitation Division, ICS Maugeri Spa SB, Institute of Lumezzane, 6Department of Clinical and Experimental Sciences, University of Brescia, 7Department of Mental Health, Spedali Civili, 8ICS Maugeri Spa SB, Institute of Lumezzane, Brescia, 9Psychology Unit, ICS Maugeri Spa SB, Institute of Tradate, Varese, Italy Background: Resilience focuses on strength under stress, in the context of adversity. Walsh’s theoretical model identifies relational processes that allow families to tackle and overcome critical situations, dividing them into three domains of family function. The aim of this study was to assess resilience in families of patients with a chronic disease by adapting and validating the Italian version of the Walsh Family Resilience Questionnaire (Walsh-IT. Patients and methods: An Italian adult sample of 421 participants (patients and relatives was collected with the aim to assess the reliability and validity of the Walsh-IT. Concurrent validity was carried out by comparing this instrument with the Family Adaptability and Cohesion Evaluation Scale III (FACES III administered at the same time as the Walsh-IT. Results: Reliability showed high correlation between repeated measurements. The alpha coefficient was 0.946. Both parallel analysis and minimum average partial criteria suggested that the best number of domains is equal to 3, explaining 50.4% of the total variance. Based on the results obtained from the Rasch analysis, items 10, 11, 16, 22, and 23 have been removed resulting in a short-form questionnaire
Branched polynomial covering maps
DEFF Research Database (Denmark)
Hansen, Vagn Lundsgaard
1999-01-01
A Weierstrass polynomial with multiple roots in certain points leads to a branched covering map. With this as the guiding example, we formally define and study the notion of a branched polynomial covering map. We shall prove that many finite covering maps are polynomial outside a discrete branch...... set. Particular studies are made of branched polynomial covering maps arising from Riemann surfaces and from knots in the 3-sphere....
Branched polynomial covering maps
DEFF Research Database (Denmark)
Hansen, Vagn Lundsgaard
2002-01-01
A Weierstrass polynomial with multiple roots in certain points leads to a branched covering map. With this as the guiding example, we formally define and study the notion of a branched polynomial covering map. We shall prove that many finite covering maps are polynomial outside a discrete branch...... 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....
Sibling curves of quadratic polynomials | Wiggins | Quaestiones ...
African Journals Online (AJOL)
Sibling curves were demonstrated in [1, 2] as a novel way to visualize the zeroes of real valued functions. In [3] it was shown that a polynomial of degree n has n sibling curves. This paper focuses on the algebraic and geometric properites of the sibling curves of real and complex quadratic polynomials. Key words: Quadratic ...
Polynomial approximation approach to transient heat conduction ...
African Journals Online (AJOL)
This work reports polynomial approximation approach to transient heat conduction in a long slab, long cylinder and sphere with linear internal heat generation. It has been shown that the polynomial approximation method is able to calculate average temperature as a function of time for higher value of Biot numbers.
Coherent orthogonal polynomials
Energy Technology Data Exchange (ETDEWEB)
Celeghini, E., E-mail: celeghini@fi.infn.it [Dipartimento di Fisica, Università di Firenze and INFN–Sezione di Firenze, I50019 Sesto Fiorentino, Firenze (Italy); Olmo, M.A. del, E-mail: olmo@fta.uva.es [Departamento de Física Teórica and IMUVA, Universidad de Valladolid, E-47005, Valladolid (Spain)
2013-08-15
We discuss a fundamental characteristic of orthogonal polynomials, like the existence of a Lie algebra behind them, which can be added to their other relevant aspects. At the basis of the complete framework for orthogonal polynomials we include thus–in addition to differential equations, recurrence relations, Hilbert spaces and square integrable functions–Lie algebra theory. We start here from the square integrable functions on the open connected subset of the real line whose bases are related to orthogonal polynomials. All these one-dimensional continuous spaces allow, besides the standard uncountable basis (|x〉), for an alternative countable basis (|n〉). The matrix elements that relate these two bases are essentially the orthogonal polynomials: Hermite polynomials for the line and Laguerre and Legendre polynomials for the half-line and the line interval, respectively. Differential recurrence relations of orthogonal polynomials allow us to realize that they determine an infinite-dimensional irreducible representation of a non-compact Lie algebra, whose second order Casimir C gives rise to the second order differential equation that defines the corresponding family of orthogonal polynomials. Thus, the Weyl–Heisenberg algebra h(1) with C=0 for Hermite polynomials and su(1,1) with C=−1/4 for Laguerre and Legendre polynomials are obtained. Starting from the orthogonal polynomials the Lie algebra is extended both to the whole space of the L{sup 2} functions and to the corresponding Universal Enveloping Algebra and transformation group. Generalized coherent states from each vector in the space L{sup 2} and, in particular, generalized coherent polynomials are thus obtained. -- Highlights: •Fundamental characteristic of orthogonal polynomials (OP): existence of a Lie algebra. •Differential recurrence relations of OP determine a unitary representation of a non-compact Lie group. •2nd order Casimir originates a 2nd order differential equation that defines
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.
Approximation by weighted means of Walsh-Fourier series
Directory of Open Access Journals (Sweden)
F. Móricz
1996-01-01
Full Text Available We study the rate of approximation to functions in Lp and, in particular, in Lip(α,p by weighted means of their Walsh-Fourier series, where α>0 and 1≤p≤∞. For the case p=∞, Lp is interpreted to be CW, the collection of uniformly W continuous functions over the unit interval [0,1. We also note that the weighted mean kernel is quasi-positive, under fairly general conditions.
Herman's Condition and Siegel Disks of Bi-Critical Polynomials
Chéritat, Arnaud; Roesch, Pascale
2016-06-01
We extend a theorem of Herman from the case of unicritical polynomials to the case of polynomials with two finite critical values. This theorem states that Siegel disks of such polynomials, under a diophantine condition (called Herman's condition) on the rotation number, must have a critical point on their boundaries.
Quantum algorithms for the Jones polynomial and Khovanov homology
Kauffman, Louis H.; Lomonaco, Samuel J., Jr.
2012-06-01
This paper generalizes the AJL algorithm for quantum computation of the Jones polynomial to continuous ranges of values on the unit circle for the Jones parameter and shows that the Kauffman-Lomonaco 3-strand algorithm for the Jones polynomial is a special case of this generalization. We then describe a quantum algorithm for the Jones polynomial that is related to Khovanov homology.
Polynomial Optimization Methods
P. van Eeghen (Piet)
2013-01-01
htmlabstractThis thesis is an exposition of ideas and methods that help un- derstanding the problem of minimizing a polynomial over a basic closed semi-algebraic set. After the introduction of some the- ory on mathematical tools such as sums of squares, nonnegative polynomials and moment matrices,
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.
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.].
Signless Laplacian Polynomial and Characteristic Polynomial of a Graph
Directory of Open Access Journals (Sweden)
Harishchandra S. Ramane
2013-01-01
Full Text Available The signless Laplacian polynomial of a graph G is the characteristic polynomial of the matrix Q(G=D(G+A(G, where D(G is the diagonal degree matrix and A(G is the adjacency matrix of G. In this paper we express the signless Laplacian polynomial in terms of the characteristic polynomial of the induced subgraphs, and, for regular graph, the signless Laplacian polynomial is expressed in terms of the derivatives of the characteristic polynomial. Using this we obtain the characteristic polynomial of line graph and subdivision graph in terms of the characteristic polynomial of induced subgraphs.
Solution of nonlinear Volterra-Hammerstein integral equations via single-term Walsh series method
Directory of Open Access Journals (Sweden)
Sepehrian B.
2005-01-01
Full Text Available Single-term Walsh series are developed to approximate the solutions of nonlinear Volterra-Hammerstein integral equations. Properties of single-term Walsh series are presented and are utilized to reduce the computation of integral equations to some algebraic equations. The method is computationally attractive, and applications are demonstrated through illustrative examples.
Directory of Open Access Journals (Sweden)
K N Pushpalatha
2017-05-01
Full Text Available In an era of advanced computer technology world where innumerable services such as access to bank accounts, or access to secured data or entry to some national important organizations require authentication of genuine individual. Among all biometric personal identification systems, fingerprint recognition system is most accurate and economical technology. In this paper we have proposed fingerprint recognition system using Local Walsh Hadamard Transform (LWHT with Phase Magnitude Histograms (PMHs for feature extraction. Fingerprints display oriented texture-like patterns. Gabor filters have the property of capturing global and local texture information from blur or unclear images and filter bank provides the orientation features which are robust to image distortion and rotation. The LWHT algorithm is compared with other two approaches viz., Gabor Coefficients and Directional Features. The three methods are compared using FVC 2006 Finger print database images. It is found from the observation that the values of TSR, FAR and FRR have improved results compared to existing algorithm.
Superiority of legendre polynomials to Chebyshev polynomial in ...
African Journals Online (AJOL)
In this paper, we proved the superiority of Legendre polynomial to Chebyshev polynomial in solving first order ordinary differential equation with rational coefficient. We generated shifted polynomial of Chebyshev, Legendre and Canonical polynomials which deal with solving differential equation by first choosing Chebyshev ...
Igenberlina, Alua; Matin, Dauren; Turgumbayev, Mendybay
2017-09-01
In this paper, deviations of the partial sums of a multiple Fourier-Walsh series of a function in the metric L1(Qk) on a dyadic group are investigated. This estimate plays an important role in the study of equivalent normalizations in this space by means of a difference, oscillation, and best approximation by polynomials in the Walsh system. The classical classical Besov space and its equivalent normalizations are set forth in the well-known monographs of Nikolsky S.M., Besov O.V., Ilyin V.P., Triebel H.; in the works of Kazakh scientists such as Amanov T.I., Mynbaev K.T., Otelbaev M.O., Smailov E.S.. The Besov spaces on the dyadic group and the Vilenkin groups in the one-dimensional case are considered in works by Ombe H., Bloom Walter R, Fournier J., Onneweer C.W., Weyi S., Jun Tateoka.
Minkowski Polynomials and Mutations
Directory of Open Access Journals (Sweden)
Mohammad Akhtar
2012-12-01
Full Text Available Given a Laurent polynomial f, one can form the period of f: this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds. Mutations are a particular class of birational transformations acting on Laurent polynomials in two variables; they preserve the period and are closely connected with cluster algebras. We propose a higher-dimensional analog of mutation acting on Laurent polynomials f in n variables. In particular we give a combinatorial description of mutation acting on the Newton polytope P of f, and use this to establish many basic facts about mutations. Mutations can be understood combinatorially in terms of Minkowski rearrangements of slices of P, or in terms of piecewise-linear transformations acting on the dual polytope P* (much like cluster transformations. Mutations map Fano polytopes to Fano polytopes, preserve the Ehrhart series of the dual polytope, and preserve the period of f. Finally we use our results to show that Minkowski polynomials, which are a family of Laurent polynomials that give mirror partners to many three-dimensional Fano manifolds, are connected by a sequence of mutations if and only if they have the same period.
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
Directory of Open Access Journals (Sweden)
Bangyong Sun
2014-01-01
Full Text Available The polynomial regression method is employed to calculate the relationship of device color space and CIE color space for color characterization, and the performance of different expressions with specific parameters is evaluated. Firstly, the polynomial equation for color conversion is established and the computation of polynomial coefficients is analysed. And then different forms of polynomial equations are used to calculate the RGB and CMYK’s CIE color values, while the corresponding color errors are compared. At last, an optimal polynomial expression is obtained by analysing several related parameters during color conversion, including polynomial numbers, the degree of polynomial terms, the selection of CIE visual spaces, and the linearization.
Algebraic polynomials with random coefficients
Directory of Open Access Journals (Sweden)
K. Farahmand
2002-01-01
Full Text Available This paper provides an asymptotic value for the mathematical expected number of points of inflections of a random polynomial of the form a0(ω+a1(ω(n11/2x+a2(ω(n21/2x2+…an(ω(nn1/2xn when n is large. The coefficients {aj(w}j=0n, w∈Ω are assumed to be a sequence of independent normally distributed random variables with means zero and variance one, each defined on a fixed probability space (A,Ω,Pr. A special case of dependent coefficients is also studied.
Bernoulli Polynomials, Fourier Series and Zeta Numbers
DEFF Research Database (Denmark)
Scheufens, Ernst E
2013-01-01
Fourier series for Bernoulli polynomials are used to obtain information about values of the Riemann zeta function for integer arguments greater than one. If the argument is even we recover the well-known exact values, if the argument is odd we find integral representations and rapidly convergent...
Euler Polynomials, Fourier Series and Zeta Numbers
DEFF Research Database (Denmark)
Scheufens, Ernst E
2012-01-01
Fourier series for Euler polynomials is used to obtain information about values of the Riemann zeta function for integer arguments greater than one. If the argument is even we recover the well-known exact values, if the argument is odd we find integral representations and rapidly convergent series....
Wilson's disease: the 60th anniversary of Walshe's article on treatment with penicillamine.
Teive, Hélio A G; Barbosa, Egberto Reis; Lees, Andrew J
2017-01-01
This historical review describes Professor Walshe's seminal contribution to the treatment of Wilson's disease on the 60th anniversary of his pioneering article on penicillamine, the first effective treatment for the condition.
A Demonstration of the Use of Walsh Functions for Multiplexed Imaging
1990-12-01
IMAGING WITH WALSH FUNCTIONS . . . . 31 A. OVERVIEW .......................31 B. RADIOMETRY...................32 C. MULTIPLEXING VERSUS CONVENTIONAL...are credited with such discoveries as the unique ring system around Uranus in 1977, a ring of dust emission about the galactic center, an understanding...MULTIPLEXED IMAGING WITH WALSH FUNCTIONS A. OVERVIEW The reader should recall that the primary objective of this thesis project has been to demonstrate that
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
R.J. Stroeker (Roel)
2006-01-01
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
Galerkin orthogonal polynomials
Livermore, Philip W.
2010-03-01
The Galerkin method offers a powerful tool in the solution of differential equations and function approximation on the real interval [-1, 1]. By expanding the unknown function in appropriately chosen global basis functions, each of which explicitly satisfies the given boundary conditions, in general this scheme converges exponentially fast and almost always supplies the most terse representation of a smooth solution. To date, typical schemes have been defined in terms of a linear combination of two Jacobi polynomials. However, the resulting functions do not inherit the expedient properties of the Jacobi polynomials themselves and the basis set will not only be non-orthogonal but may, in fact, be poorly conditioned. Using a Gram-Schmidt procedure, it is possible to construct, in an incremental fashion, polynomial basis sets that not only satisfy any linear homogeneous boundary conditions but are also orthogonal with respect to the general weighting function (1-x)α(1+x)β. However, as it stands, this method is not only cumbersome but does not provide the structure for general index n of the functions and obscures their dependence on the parameters (α,β). In this paper, it is shown that each of these Galerkin basis functions, as calculated by the Gram-Schmidt procedure, may be written as a linear combination of a small number of Jacobi polynomials with coefficients that can be determined. Moreover, this terse analytic representation reveals that, for large index, the basis functions behave asymptotically like the single Jacobi polynomial Pn(α,β)(x). This new result shows that such Galerkin bases not only retain exponential convergence but expedient function-fitting properties too, in much the same way as the Jacobi polynomials themselves. This powerful methodology of constructing Galerkin basis sets is illustrated by many examples, and it is shown how the results extend to polar geometries. In exploring more generalised definitions of orthogonality involving
Piecewise polynomial functions
Energy Technology Data Exchange (ETDEWEB)
Carlson, R. E.
1976-01-01
This report contains a series of talks presented during the summer of 1975 at LLL on the general topic of piecewise polynomial functions. Such functions offer simple, accurate, economical, and flexible procedures for solving a wide variety of problems. The emphasis is on curve fitting, surface fitting, and applications to finite element approximations. This discussion is not intended as an all-inclusive, general presentation of piecewise polynomials. Rather, attention is directed to the basic of cubic (and bicubic) interpolation and approximation, since they represent a good first choice for many problems when increased accuracy over linear (or bilinear) approximations is desired. The text is divided into sections on univariate functions, cubic splines, bivariate functions, blending functions, and nonrectangular domains. 10 figures, 1 table. (RWR)
Multivariate piecewise polynomials
de Boor, C.
This article was supposed to be on `multivariate splines". An informal survey, taken recently by asking various people in Approximation Theory what they consider to be a `multivariate spline', resulted in the answer that a multivariate spline is a possibly smooth piecewise polynomial function of several arguments. In particular the potentially very useful thin-plate spline was thought to belong more to the subject of radial basis funtions than in the present article. This is all the more surprising to me since I am convinced that the variational approach to splines will play a much greater role in multivariate spline theory than it did or should have in the univariate theory. Still, as there is more than enough material for a survey of multivariate piecewise polynomials, this article is restricted to this topic, as is indicated by the (changed) title.
Mating Siegel Quadratic Polynomials
Yampolsky, Michael; Zakeri, Saeed
1998-01-01
Let F be a quadratic rational map of the sphere which has two fixed Siegel disks with bounded type rotation numbers theta and nu. Using a new degree 3 Blaschke product model for the dynamics of F and an adaptation of complex a priori bounds for renormalization of critical circle maps, we prove that F can be realized as the mating of two Siegel quadratic polynomials with the corresponding rotation numbers theta and nu.
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...
Fourier series in orthogonal polynomials
Osilenker, Boris
1999-01-01
This book presents a systematic course on general orthogonal polynomials and Fourier series in orthogonal polynomials. It consists of six chapters. Chapter 1 deals in essence with standard results from the university course on the function theory of a real variable and on functional analysis. Chapter 2 contains the classical results about the orthogonal polynomials (some properties, classical Jacobi polynomials and the criteria of boundedness).The main subject of the book is Fourier series in general orthogonal polynomials. Chapters 3 and 4 are devoted to some results in this topic (classical
A Combinatorial Proof of a Result on Generalized Lucas Polynomials
Laugier Alexandre; Saikia Manjil P.
2016-01-01
We give a combinatorial proof of an elementary property of generalized Lucas polynomials, inspired by [1]. These polynomials in s and t are defined by the recurrence relation 〈n〉 = s〈n-1〉+t〈n-2〉 for n ≥ 2. The initial values are 〈0〉 = 2; 〈1〉= s, respectively.
Chemical Reaction Networks for Computing Polynomials.
Salehi, Sayed Ahmad; Parhi, Keshab K; Riedel, Marc D
2017-01-20
Chemical reaction networks (CRNs) provide a fundamental model in the study of molecular systems. Widely used as formalism for the analysis of chemical and biochemical systems, CRNs have received renewed attention as a model for molecular computation. This paper demonstrates that, with a new encoding, CRNs can compute any set of polynomial functions subject only to the limitation that these functions must map the unit interval to itself. These polynomials can be expressed as linear combinations of Bernstein basis polynomials with positive coefficients less than or equal to 1. In the proposed encoding approach, each variable is represented using two molecular types: a type-0 and a type-1. The value is the ratio of the concentration of type-1 molecules to the sum of the concentrations of type-0 and type-1 molecules. The proposed encoding naturally exploits the expansion of a power-form polynomial into a Bernstein polynomial. Molecular encoders for converting any input in a standard representation to the fractional representation as well as decoders for converting the computed output from the fractional to a standard representation are presented. The method is illustrated first for generic CRNs; then chemical reactions designed for an example are mapped to DNA strand-displacement reactions.
Large level crossings of a random polynomial
Directory of Open Access Journals (Sweden)
Kambiz Farahmand
1988-01-01
Full Text Available We know the expected number of times that a polynomial of degree n with independent random real coefficients asymptotically crosses the level K, when K is any real value such that (K2/n→0 as n→∞. The present paper shows that, when K is allowed to be large, this expected number of crossings reduces to only one. The coefficients of the polynomial are assumed to be normally distributed. It is shown that it is sufficient to let K≥exp(nf where f is any function of n such that f→∞ as n→∞.
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…
Error Minimization of Polynomial Approximation of Delta
Indian Academy of Sciences (India)
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 ...
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...... examples of rotated families to argue this. There will be discussed several open questions concerning the number of transversals that can appear for a certain degree d of a polynomial vector field, and furthermore how transversals are analyzed with respect to bifurcations around multiple equilibrium points....
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...
A multivariate fast discrete Walsh transform with an application to function interpolation
Liu, Kwong-Ip; Dick, Josef; Hickernell, Fred J.
2009-09-01
For high dimensional problems, such as approximation and integration, one cannot afford to sample on a grid because of the curse of dimensionality. An attractive alternative is to sample on a low discrepancy set, such as an integration lattice or a digital net. This article introduces a multivariate fast discrete Walsh transform for data sampled on a digital net that requires only mathcal{O}(N log N) operations, where N is the number of data points. This algorithm and its inverse are digital analogs of multivariate fast Fourier transforms. This fast discrete Walsh transform and its inverse may be used to approximate the Walsh coefficients of a function and then construct a spline interpolant of the function. This interpolant may then be used to estimate the function's effective dimension, an important concept in the theory of numerical multivariate integration. Numerical results for various functions are presented.
Polynomial methods in combinatorics
Guth, Larry
2016-01-01
This book explains some recent applications of the theory of polynomials and algebraic geometry to combinatorics and other areas of mathematics. One of the first results in this story is a short elegant solution of the Kakeya problem for finite fields, which was considered a deep and difficult problem in combinatorial geometry. The author also discusses in detail various problems in incidence geometry associated to Paul Erdős's famous distinct distances problem in the plane from the 1940s. The proof techniques are also connected to error-correcting codes, Fourier analysis, number theory, and differential geometry. Although the mathematics discussed in the book is deep and far-reaching, it should be accessible to first- and second-year graduate students and advanced undergraduates. The book contains approximately 100 exercises that further the reader's understanding of the main themes of the book. Some of the greatest advances in geometric combinatorics and harmonic analysis in recent years have been accompl...
Complex Polynomial Vector Fields
DEFF Research Database (Denmark)
Dias, Kealey
The two branches of dynamical systems, continuous and discrete, correspond to the study of differential equations (vector fields) and iteration of mappings respectively. In holomorphic dynamics, the systems studied are restricted to those described by holomorphic (complex analytic) functions...... or meromorphic (allowing poles as singularities) functions. There already exists a well-developed theory for iterative holomorphic dynamical systems, and successful relations found between iteration theory and flows of vector fields have been one of the main motivations for the recent interest in holomorphic...... vector fields. Since the class of complex polynomial vector fields in the plane is natural to consider, it is remarkable that its study has only begun very recently. There are numerous fundamental questions that are still open, both in the general classification of these vector fields, the decomposition...
Polynomial Arithmetic and Contour Construction,
1977-10-01
the Root of an Equation. M. Dowell , and P. Jarratt , BIT, 12, 503 (1972) 19. A Machine Method for Solving Polynomial Equations. D. H. Lehmer, Journal...Computer. P. Jarratt , and D. Nudds, The Computer Journal, 8, 62 (1965) 22. A Three Stage Variable Shift Iteration for Polynomial Zeros and its Relation
Polynomial weights and code constructions
DEFF Research Database (Denmark)
Massey, J; Costello, D; Justesen, Jørn
1973-01-01
For any nonzero elementcof a general finite fieldGF(q), it is shown that the polynomials(x - c)^i, i = 0,1,2,cdots, have the "weight-retaining" property that any linear combination of these polynomials with coefficients inGF(q)has Hamming weight at least as great as that of the minimum degree...
Parallel Construction of Irreducible Polynomials
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) ...
Graphical Solution of Polynomial Equations
Grishin, Anatole
2009-01-01
Graphing utilities, such as the ubiquitous graphing calculator, are often used in finding the approximate real roots of polynomial equations. In this paper the author offers a simple graphing technique that allows one to find all solutions of a polynomial equation (1) of arbitrary degree; (2) with real or complex coefficients; and (3) possessing…
Modular polynomials for genus 2
Broker, Reinier; Lauter, Kristin
2008-01-01
Modular polynomials are an important tool in many algorithms involving elliptic curves. In this article we investigate their generalization to the genus 2 case following pioneering work by Gaudry and Dupont. We prove various properties of these genus 2 modular polynomials and give an improved way to explicitly compute them.
Optimization over polynomials : Selected topics
Laurent, M.; Jang, Sun Young; Kim, Young Rock; Lee, Dae-Woong; Yie, Ikkwon
2014-01-01
Minimizing a polynomial function over a region defined by polynomial inequalities models broad classes of hard problems from combinatorics, geometry and optimization. New algorithmic approaches have emerged recently for computing the global minimum, by combining tools from real algebra (sums of
Optimization over polynomials: Selected topics
M. Laurent (Monique); S.Y. Jang; Y.R. Kim; D.-W. Lee; I. Yie
2014-01-01
htmlabstractMinimizing a polynomial function over a region defined by polynomial inequalities models broad classes of hard problems from combinatorics, geometry and optimization. New algorithmic approaches have emerged recently for computing the global minimum, by combining tools from real algebra
Indecomposable polynomials and their spectrum
Bodin, Arnaud; Debes, Pierre; Najib, Salah
We address some questions concerning indecomposable polynomials and their spectrum. How does the spectrum behave via reduction or specialization, or via a more general ring morphism? Are the indecomposability properties equivalent over a field and over its algebraic closure? How many polynomials are decomposable over a finite field?
Information-theoretic lengths of Jacobi polynomials
Energy Technology Data Exchange (ETDEWEB)
Guerrero, A; Dehesa, J S [Departamento de Fisica Atomica, Molecular y Nuclear, Universidad de Granada, Granada (Spain); Sanchez-Moreno, P, E-mail: agmartinez@ugr.e, E-mail: pablos@ugr.e, E-mail: dehesa@ugr.e [Instituto ' Carlos I' de Fisica Teorica y Computacional, Universidad de Granada, Granada (Spain)
2010-07-30
The information-theoretic lengths of the Jacobi polynomials P{sup ({alpha}, {beta})}{sub n}(x), which are information-theoretic measures (Renyi, Shannon and Fisher) of their associated Rakhmanov probability density, are investigated. They quantify the spreading of the polynomials along the orthogonality interval [- 1, 1] in a complementary but different way as the root-mean-square or standard deviation because, contrary to this measure, they do not refer to any specific point of the interval. The explicit expressions of the Fisher length are given. The Renyi lengths are found by the use of the combinatorial multivariable Bell polynomials in terms of the polynomial degree n and the parameters ({alpha}, {beta}). The Shannon length, which cannot be exactly calculated because of its logarithmic functional form, is bounded from below by using sharp upper bounds to general densities on [- 1, +1] given in terms of various expectation values; moreover, its asymptotics is also pointed out. Finally, several computational issues relative to these three quantities are carefully analyzed.
Some New Formulae for Genocchi Numbers and Polynomials Involving Bernoulli and Euler Polynomials
Directory of Open Access Journals (Sweden)
Serkan Araci
2014-01-01
Full Text Available We give some new formulae for product of two Genocchi polynomials including Euler polynomials and Bernoulli polynomials. Moreover, we derive some applications for Genocchi polynomials to study a matrix formulation.
The number of polynomial solutions of polynomial Riccati equations
Gasull, Armengol; Torregrosa, Joan; Zhang, Xiang
2016-11-01
Consider real or complex polynomial Riccati differential equations a (x) y ˙ =b0 (x) +b1 (x) y +b2 (x)y2 with all the involved functions being polynomials of degree at most η. We prove that the maximum number of polynomial solutions is η + 1 (resp. 2) when η ≥ 1 (resp. η = 0) and that these bounds are sharp. For real trigonometric polynomial Riccati differential equations with all the functions being trigonometric polynomials of degree at most η ≥ 1 we prove a similar result. In this case, the maximum number of trigonometric polynomial solutions is 2η (resp. 3) when η ≥ 2 (resp. η = 1) and, again, these bounds are sharp. Although the proof of both results has the same starting point, the classical result that asserts that the cross ratio of four different solutions of a Riccati differential equation is constant, the trigonometric case is much more involved. The main reason is that the ring of trigonometric polynomials is not a unique factorization domain.
Gender Identity, Interactionism, and Politics: A Reply to Rogers and Walsh.
DeBold, Joseph F.; Luria, Zella
1983-01-01
Discusses the reasoning and strategies of the biological approach to sex differences research. Refutes Rogers' and Walsh's (1982) perspective that, because biological and environmental factors are intertwined in the development of the nervous system, the biological contribution cannot be studied profitably. (CMG)
Representation of American versus non-American fans in Baillie Walsh's Springsteen & I
Wilkinson, M.C.
2015-01-01
This article explores the representation of American versus non-American fans in Baillie Walsh's 2013 crowd/fan-sourced documentary Springsteen & I. The film—as much as it was fed by a wide and international range of fan-produced material—ultimately produced and presented one particular type of fan
Quadratic-like dynamics of cubic polynomials
Blokh, Alexander; Oversteegen, Lex; Ptacek, Ross; Timorin, Vladlen
2013-01-01
A small perturbation of a quadratic polynomial with a non-repelling fixed point gives a polynomial with an attracting fixed point and a Jordan curve Julia set, on which the perturbed polynomial acts like angle doubling. However, there are cubic polynomials with a non-repelling fixed point, for which no perturbation results into a polynomial with Jordan curve Julia set. Motivated by the study of the closure of the Cubic Principal Hyperbolic Domain, we describe such polynomials in terms of thei...
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,
Robust stability of fractional order polynomials with complicated uncertainty structure
Şenol, Bilal; Pekař, Libor
2017-01-01
The main aim of this article is to present a graphical approach to robust stability analysis for families of fractional order (quasi-)polynomials with complicated uncertainty structure. More specifically, the work emphasizes the multilinear, polynomial and general structures of uncertainty and, moreover, the retarded quasi-polynomials with parametric uncertainty are studied. Since the families with these complex uncertainty structures suffer from the lack of analytical tools, their robust stability is investigated by numerical calculation and depiction of the value sets and subsequent application of the zero exclusion condition. PMID:28662173
Cantor polynomials for semigroup sectors
Nathanson, Melvyn B.
2013-01-01
A packing function on a set Omega in R^n is a one-to-one correspondence between the set of lattice points in Omega and the set N_0 of nonnegative integers. It is proved that if r and s are relatively prime positive integers such that r divides s-1, then there exist two distinct quadratic packing polynomials on the sector {(x,y) \\in \\R^2 : 0 \\leq y \\leq rx/s}. For the rational numbers 1/s, these are the unique quadratic packing polynomials. Moreover, quadratic quasi-polynomial packing function...
Some discrete multiple orthogonal polynomials
Arvesú, J.; Coussement, J.; Van Assche, W.
2003-01-01
27 pages, no figures.-- MSC2000 codes: 33C45, 33C10, 42C05, 41A28.-- Issue title: "Proceedings of the 6th International Symposium on Orthogonal Polynomials, Special Functions and their Applications" (OPSFA-VI, Rome, Italy, 18-22 June 2001). MR#: MR1985676 (2004g:33015) Zbl#: Zbl 1021.33006 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 w...
Polynomial weights and code constructions.
Massey, J. L.; Costello, D. J., Jr.; Justesen, J.
1973-01-01
Study of certain polynomials with the 'weight-retaining' property that any linear combination of these polynomials with coefficients in a general finite field has Hamming weight at least as great as that of the minimum-degree polynomial included. This fundamental property is used in applications to Reed-Muller codes, a new class of 'repeated-root' binary cyclic codes, two new classes of binary convolutional codes derived from binary cyclic codes, and two new classes of binary convolutional codes derived from Reed-Solomon codes.
Sum rules for zeros of polynomials and generalized Lucas polynomials
Ricci, Paolo Emilio
1993-10-01
A representation formula in terms of generalized Lucas polynomials of the second kind [see formula (4.3)], for the sum rules Js(i) introduced by Case [J. Math. Phys. 21, 702 (1980)] and studied by Dehesa et al. [J. Math. Phys. 26, 1547 (1985); 26, 2729 (1985)] in order to obtain information about the zeros' distribution of eigenfunctions of a class of ordinary polynomial differential operators, is derived.
Roots of the Chromatic Polynomial
DEFF Research Database (Denmark)
Perrett, Thomas
. In this thesis we study the real roots of the chromatic polynomial, termed chromatic roots, and focus on how certain properties of a graph affect the location of its chromatic roots. Firstly, we investigate how the presence of a certain spanning tree in a graph affects its chromatic roots. In particular we prove...... a tight lower bound on the smallest non-trivial chromatic root of a graph admitting a spanning tree with at most three leaves. Here, non-trivial means different from 0 or 1. This extends a theorem of Thomassen on graphs with Hamiltonian paths. We also prove similar lower bounds on the chromatic roots...... extend Thomassen’s technique to the Tutte polynomial and as a consequence, deduce a density result for roots of the Tutte polynomial. This partially answers a conjecture of Jackson and Sokal. Finally, we refocus our attention on the chromatic polynomial and investigate the density of chromatic roots...
Polynomial Regressions and Nonsense Inference
Directory of Open Access Journals (Sweden)
Daniel Ventosa-Santaulària
2013-11-01
Full Text Available Polynomial specifications are widely used, not only in applied economics, but also in epidemiology, physics, political analysis and psychology, just to mention a few examples. In many cases, the data employed to estimate such specifications are time series that may exhibit stochastic nonstationary behavior. We extend Phillips’ results (Phillips, P. Understanding spurious regressions in econometrics. J. Econom. 1986, 33, 311–340. by proving that an inference drawn from polynomial specifications, under stochastic nonstationarity, is misleading unless the variables cointegrate. We use a generalized polynomial specification as a vehicle to study its asymptotic and finite-sample properties. Our results, therefore, lead to a call to be cautious whenever practitioners estimate polynomial regressions.
An introduction to orthogonal polynomials
Chihara, Theodore S
1978-01-01
Assuming no further prerequisites than a first undergraduate course in real analysis, this concise introduction covers general elementary theory related to orthogonal polynomials. It includes necessary background material of the type not usually found in the standard mathematics curriculum. Suitable for advanced undergraduate and graduate courses, it is also appropriate for independent study. Topics include the representation theorem and distribution functions, continued fractions and chain sequences, the recurrence formula and properties of orthogonal polynomials, special functions, and some
Polynomial chaos representation of databases on manifolds
Soize, C.; Ghanem, R.
2017-04-01
Characterizing the polynomial chaos expansion (PCE) of a vector-valued random variable with probability distribution concentrated on a manifold is a relevant problem in data-driven settings. The probability distribution of such random vectors is multimodal in general, leading to potentially very slow convergence of the PCE. In this paper, we build on a recent development for estimating and sampling from probabilities concentrated on a diffusion manifold. The proposed methodology constructs a PCE of the random vector together with an associated generator that samples from the target probability distribution which is estimated from data concentrated in the neighborhood of the manifold. The method is robust and remains efficient for high dimension and large datasets. The resulting polynomial chaos construction on manifolds permits the adaptation of many uncertainty quantification and statistical tools to emerging questions motivated by data-driven queries.
Polynomial chaos representation of databases on manifolds
Energy Technology Data Exchange (ETDEWEB)
Soize, C., E-mail: christian.soize@univ-paris-est.fr [Université Paris-Est, Laboratoire Modélisation et Simulation Multi-Echelle, MSME UMR 8208 CNRS, 5 bd Descartes, 77454 Marne-La-Vallée, Cedex 2 (France); Ghanem, R., E-mail: ghanem@usc.edu [University of Southern California, 210 KAP Hall, Los Angeles, CA 90089 (United States)
2017-04-15
Characterizing the polynomial chaos expansion (PCE) of a vector-valued random variable with probability distribution concentrated on a manifold is a relevant problem in data-driven settings. The probability distribution of such random vectors is multimodal in general, leading to potentially very slow convergence of the PCE. In this paper, we build on a recent development for estimating and sampling from probabilities concentrated on a diffusion manifold. The proposed methodology constructs a PCE of the random vector together with an associated generator that samples from the target probability distribution which is estimated from data concentrated in the neighborhood of the manifold. The method is robust and remains efficient for high dimension and large datasets. The resulting polynomial chaos construction on manifolds permits the adaptation of many uncertainty quantification and statistical tools to emerging questions motivated by data-driven queries.
Accurate estimation of solvation free energy using polynomial fitting techniques.
Shyu, Conrad; Ytreberg, F Marty
2011-01-15
This report details an approach to improve the accuracy of free energy difference estimates using thermodynamic integration data (slope of the free energy with respect to the switching variable λ) and its application to calculating solvation free energy. The central idea is to utilize polynomial fitting schemes to approximate the thermodynamic integration data to improve the accuracy of the free energy difference estimates. Previously, we introduced the use of polynomial regression technique to fit thermodynamic integration data (Shyu and Ytreberg, J Comput Chem, 2009, 30, 2297). In this report we introduce polynomial and spline interpolation techniques. Two systems with analytically solvable relative free energies are used to test the accuracy of the interpolation approach. We also use both interpolation and regression methods to determine a small molecule solvation free energy. Our simulations show that, using such polynomial techniques and nonequidistant λ values, the solvation free energy can be estimated with high accuracy without using soft-core scaling and separate simulations for Lennard-Jones and partial charges. The results from our study suggest that these polynomial techniques, especially with use of nonequidistant λ values, improve the accuracy for ΔF estimates without demanding additional simulations. We also provide general guidelines for use of polynomial fitting to estimate free energy. To allow researchers to immediately utilize these methods, free software and documentation is provided via http://www.phys.uidaho.edu/ytreberg/software. Copyright © 2010 Wiley Periodicals, Inc.
Chromatic polynomials of random graphs
Energy Technology Data Exchange (ETDEWEB)
Van Bussel, Frank; Fliegner, Denny; Timme, Marc [Max Planck Institute for Dynamics and Self-Organization (MPIDS), Goettingen (Germany); Ehrlich, Christoph [Department of Physics, Technical University of Dresden, Dresden (Germany); Stolzenberg, Sebastian, E-mail: fvb@nld.ds.mpg.d, E-mail: timme@nld.ds.mpg.d [Department of Physics, Ithaca and Weill Cornell Medical College, New York City, Cornell University, NY (United States)
2010-04-30
Chromatic polynomials and related graph invariants are central objects in both graph theory and statistical physics. Computational difficulties, however, have so far restricted studies of such polynomials to graphs that were either very small, very sparse or highly structured. Recent algorithmic advances (Timme et al 2009 New J. Phys. 11 023001) now make it possible to compute chromatic polynomials for moderately sized graphs of arbitrary structure and number of edges. Here we present chromatic polynomials of ensembles of random graphs with up to 30 vertices, over the entire range of edge density. We specifically focus on the locations of the zeros of the polynomial in the complex plane. The results indicate that the chromatic zeros of random graphs have a very consistent layout. In particular, the crossing point, the point at which the chromatic zeros with non-zero imaginary part approach the real axis, scales linearly with the average degree over most of the density range. While the scaling laws obtained are purely empirical, if they continue to hold in general there are significant implications: the crossing points of chromatic zeros in the thermodynamic limit separate systems with zero ground state entropy from systems with positive ground state entropy, the latter an exception to the third law of thermodynamics.
Chromatic polynomials of random graphs
Van Bussel, Frank; Ehrlich, Christoph; Fliegner, Denny; Stolzenberg, Sebastian; Timme, Marc
2010-04-01
Chromatic polynomials and related graph invariants are central objects in both graph theory and statistical physics. Computational difficulties, however, have so far restricted studies of such polynomials to graphs that were either very small, very sparse or highly structured. Recent algorithmic advances (Timme et al 2009 New J. Phys. 11 023001) now make it possible to compute chromatic polynomials for moderately sized graphs of arbitrary structure and number of edges. Here we present chromatic polynomials of ensembles of random graphs with up to 30 vertices, over the entire range of edge density. We specifically focus on the locations of the zeros of the polynomial in the complex plane. The results indicate that the chromatic zeros of random graphs have a very consistent layout. In particular, the crossing point, the point at which the chromatic zeros with non-zero imaginary part approach the real axis, scales linearly with the average degree over most of the density range. While the scaling laws obtained are purely empirical, if they continue to hold in general there are significant implications: the crossing points of chromatic zeros in the thermodynamic limit separate systems with zero ground state entropy from systems with positive ground state entropy, the latter an exception to the third law of thermodynamics.
Directory of Open Access Journals (Sweden)
Saifulnizam Bin Abdul Khalid
2013-01-01
Full Text Available This paper presents an improved Walsh function IWF algorithms as an alternative approach for active and reactive powers measurement in linear and nonlinear, balanced and unbalanced sinusoidal three-phase load system. It takes advantage of Walsh function unified approach, simple algorithm and its intrinsic high level of accuracy as a result of coefficient characteristics and energy behaviour representation. The developed algorithm was modeled on the Matlab Simulink software; different types of load, linear and nonlinear, were also modeled based on practical voltage and current waveforms and tested with the proposed improved Walsh algorithms. The IEEE standard 1459–2000 which is based on fast Fourier transform FFT approach was used as benchmark for the linear load system. The data obtained from laboratory experiment to determine power components in harmonic load systems using Fluke 435 power quality analyzer PQA which complies with IEC/EN61010-1-2001 standard was modeled and used to validate the improved algorithm for nonlinear load measurement. The results showed that the algorithm has the potential to effectively measure three-phase power components under different load conditions.
Simulation of aspheric tolerance with polynomial fitting
Li, Jing; Cen, Zhaofeng; Li, Xiaotong
2018-01-01
The shape of the aspheric lens changes caused by machining errors, resulting in a change in the optical transfer function, which affects the image quality. At present, there is no universally recognized tolerance criterion standard for aspheric surface. To study the influence of aspheric tolerances on the optical transfer function, the tolerances of polynomial fitting are allocated on the aspheric surface, and the imaging simulation is carried out by optical imaging software. Analysis is based on a set of aspheric imaging system. The error is generated in the range of a certain PV value, and expressed as a form of Zernike polynomial, which is added to the aspheric surface as a tolerance term. Through optical software analysis, the MTF of optical system can be obtained and used as the main evaluation index. Evaluate whether the effect of the added error on the MTF of the system meets the requirements of the current PV value. Change the PV value and repeat the operation until the acceptable maximum allowable PV value is obtained. According to the actual processing technology, consider the error of various shapes, such as M type, W type, random type error. The new method will provide a certain development for the actual free surface processing technology the reference value.
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.
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).
African Journals Online (AJOL)
Ignatius & Ebimene
0(1)( − 1), are finite real constants and is the order of the boundary value problem. This type of problem is a frequent occurrence in the mathematical modeling of viscoelastic flow, stability and instability convection analysis of heat transfer, and in other fields of science and technology. Hence, obtaining the solution of ...
Global Polynomial Kernel Hazard Estimation
DEFF Research Database (Denmark)
Hiabu, Munir; Miranda, Maria Dolores Martínez; Nielsen, Jens Perch
2015-01-01
This paper introduces a new bias reducing method for kernel hazard estimation. The method is called global polynomial adjustment (GPA). It is a global correction which is applicable to any kernel hazard estimator. The estimator works well from a theoretical point of view as it asymptotically redu...
Polynomial Regressions and Nonsense Inference
DEFF Research Database (Denmark)
Ventosa-Santaulària, Daniel; Rodríguez-Caballero, Carlos Vladimir
Polynomial specifications are widely used, not only in applied economics, but also in epidemiology, physics, political analysis, and psychology, just to mention a few examples. In many cases, the data employed to estimate such estimations are time series that may exhibit stochastic nonstationary ...
Stochastic Estimation via Polynomial Chaos
2015-10-01
Combinations Distribution Function Orthogonal Polynomial Set Gaussian Hermite Uniform Legendre Gamma Laguerre Beta Jacobi Poisson Charlier...E-9) By using the transformation of variables yx 2= along with the trigonometric identity ...2(CL 2 1)4(CL 4 1cos2ln 22 0 θθ θ −−=∫ dxx (E-14) The Clausen identity
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.
Romanovski polynomials in selected physics problems
Raposo, A. P.; Weber, H. J.; Alvarez-Castillo, D.; Kirchbach, M.
2007-01-01
We briefly review the five possible real polynomial solutions of hypergeometric differential equations. Three of them are the well known classical orthogonal polynomials, but the other two are different with respect to their orthogonality properties. We then focus on the family of polynomials which exhibits a finite orthogonality. This family, to be referred to as the Romanovski polynomials, is required in exact solutions of several physics problems ranging from quantum mechanics and quark ph...
A generalization of weight polynomials to matroids
Johnsen, Trygve; Roksvold, Jan Nyquist; Verdure, Hugues
2013-01-01
Generalizing polynomials previously studied in the context of linear codes, we define weight polynomials and an enumerator for a matroid $M$. Our main result is that these polynomials are determined by Betti numbers associated with graded minimal free resolutions of the Stanley-Reisner ideals of $M$ and so-called elongations of $M$. Generalizing Greene's theorem from coding theory, we show that the enumerator of a matroid is equivalent to its Tutte polynomial.
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.
Modular polynomial arithmetic in partial fraction decomposition
Abdali, S. K.; Caviness, B. F.; Pridor, A.
1977-01-01
Algorithms for general partial fraction decomposition are obtained by using modular polynomial arithmetic. An algorithm is presented to compute inverses modulo a power of a polynomial in terms of inverses modulo that polynomial. This algorithm is used to make an improvement in the Kung-Tong partial fraction decomposition algorithm.
A multidimensional systems approach to polynomial optimization
Bleylevens, I.W.M.; Hanzon, B.; Peeters, R.L.M.
2004-01-01
With any system of multivariate polynomial equations we can associate a system of multidimensional difference equations by interpreting the variables in the polynomial equations as shift operators working on a multidimensional time series. If the solution set of the system of multivariate polynomial
Directory of Open Access Journals (Sweden)
Stephen Z. Pinter
2007-02-01
Full Text Available Transmitter nonlinearity has been a major issue in many scenarios: cellular wireless systems have high power RF amplifier (HPA nonlinearity at the base station; satellite downlinks have nonlinear TWT amplifiers in the satellite transponder and multipath conditions in the ground station; and radio-over-fiber (ROF systems consist of a nonlinear optical link followed by a wireless channel. In many cases, the nonlinearity is simply ignored if there is no out-of-band emission. This results in poor BER performance. In this paper we propose a new technique to estimate the linear part of the wireless downlink in the presence of a nonlinearity using Walsh codes; Walsh codes are commonly used in CDMA downlinks. Furthermore, we show that equalizer performance is significantly improved by taking into account the presence of the nonlinearity during channel estimation. This is shown by using a regular decision feedback equalizer (DFE with both wireless and RF amplifier noise. We perform estimation in a multiuser CDMA communication system where all users transmit their signal simultaneously. Correlation analysis is applied to identify the channel impulse response (CIR and the derivation of key correlation relationships is shown. A difficulty with using Walsh codes in terms of their correlations (compared to PN sequences is then presented, as well as a discussion on how to overcome it. Numerical evaluations show a good estimation of the linear system with 54 users in the downlink and a signal-to-noise ratio (SNR of 25 dB. Bit error rate (BER simulations of the proposed identification and equalization algorithms show a BER of 10Ã¢ÂˆÂ’6 achieved at an SNR of Ã¢ÂˆÂ¼25 dB.
Directory of Open Access Journals (Sweden)
Pinter Stephen Z
2007-01-01
Full Text Available Transmitter nonlinearity has been a major issue in many scenarios: cellular wireless systems have high power RF amplifier (HPA nonlinearity at the base station; satellite downlinks have nonlinear TWT amplifiers in the satellite transponder and multipath conditions in the ground station; and radio-over-fiber (ROF systems consist of a nonlinear optical link followed by a wireless channel. In many cases, the nonlinearity is simply ignored if there is no out-of-band emission. This results in poor BER performance. In this paper we propose a new technique to estimate the linear part of the wireless downlink in the presence of a nonlinearity using Walsh codes; Walsh codes are commonly used in CDMA downlinks. Furthermore, we show that equalizer performance is significantly improved by taking into account the presence of the nonlinearity during channel estimation. This is shown by using a regular decision feedback equalizer (DFE with both wireless and RF amplifier noise. We perform estimation in a multiuser CDMA communication system where all users transmit their signal simultaneously. Correlation analysis is applied to identify the channel impulse response (CIR and the derivation of key correlation relationships is shown. A difficulty with using Walsh codes in terms of their correlations (compared to PN sequences is then presented, as well as a discussion on how to overcome it. Numerical evaluations show a good estimation of the linear system with 54 users in the downlink and a signal-to-noise ratio (SNR of 25 dB. Bit error rate (BER simulations of the proposed identification and equalization algorithms show a BER of achieved at an SNR of dB.
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.
Space complexity in polynomial calculus
Czech Academy of Sciences Publication Activity Database
Filmus, Y.; Lauria, M.; Nordström, J.; Ron-Zewi, N.; Thapen, Neil
2015-01-01
Roč. 44, č. 4 (2015), s. 1119-1153 ISSN 0097-5397 R&D Projects: GA AV ČR IAA100190902; GA ČR GBP202/12/G061 Institutional support: RVO:67985840 Keywords : proof complexity * polynomial calculus * lower bounds Subject RIV: BA - General Mathematics Impact factor: 0.841, year: 2015 http://epubs.siam.org/doi/10.1137/120895950
Aspects of the Tutte polynomial
DEFF Research Database (Denmark)
Ok, Seongmin
), T(G;0,2)}. The three numbers, T(G;1,1), T(G;2,0) and T(G;0,2) are respectively the numbers of spanning trees, acyclic orientations and totally cyclic orientations of G. First, I extend Negami's splitting formula to the multivariate Tutte polynomial. Using the splitting formula, Thomassen and I found......-connected then T(G;1,1) ≤ T(G;0,2). Strengthening Thomassen's idea that acyclic orientations dominate spanning trees in sparse graphs, I conjecture that the ratio T(G;2,0)/T(G;1,1) increases as G gets sparser. To support this conjecture, I prove a variant of the conjecture for series-parallel graphs. The Merino......This thesis studies various aspects of the Tutte polynomial, especially focusing on the Merino-Welsh conjecture. We write T(G;x,y) for the Tutte polynomial of a graph G with variables x and y. In 1999, Merino and Welsh conjectured that if G is a loopless 2-connected graph, then T(G;1,1) ≤ max{T(G;2,0...
Stable piecewise polynomial vector fields
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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.
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...... twisted polynomials from Ore rings as forgery polynomials. We show how to construct sparse forgery polynomials with full control over the sets of roots. We also achieve complete and explicit disjoint coverage of the key space by these polynomials. We furthermore leverage this new construction...
Ladder Operators for Lamé Spheroconal Harmonic Polynomials
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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.
Rotated Walsh-Hadamard Spreading with Robust Channel Estimation for a Coded MC-CDMA System
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Raulefs Ronald
2004-01-01
Full Text Available We investigate rotated Walsh-Hadamard spreading matrices for a broadband MC-CDMA system with robust channel estimation in the synchronous downlink. The similarities between rotated spreading and signal space diversity are outlined. In a multiuser MC-CDMA system, possible performance improvements are based on the chosen detector, the channel code, and its Hamming distance. By applying rotated spreading in comparison to a standard Walsh-Hadamard spreading code, a higher throughput can be achieved. As combining the channel code and the spreading code forms a concatenated code, the overall minimum Hamming distance of the concatenated code increases. This asymptotically results in an improvement of the bit error rate for high signal-to-noise ratio. Higher convolutional channel code rates are mostly generated by puncturing good low-rate channel codes. The overall Hamming distance decreases significantly for the punctured channel codes. Higher channel code rates are favorable for MC-CDMA, as MC-CDMA utilizes diversity more efficiently compared to pure OFDMA. The application of rotated spreading in an MC-CDMA system allows exploiting diversity even further. We demonstrate that the rotated spreading gain is still present for a robust pilot-aided channel estimator. In a well-designed system, rotated spreading extends the performance by using a maximum likelihood detector with robust channel estimation at the receiver by about 1 dB.
Energy Technology Data Exchange (ETDEWEB)
Basu, T.K.; Bhattacharya, T.K. (Indian Inst. of Tech., Kharagpur (IN). Dept. of Electrical Engineering); Purkayastha, P. (Oil India Ltd., Duliajan (IN))
1991-08-01
Medium range forecasts of hourly load demand for a span of 24 h to 168h are required for the preparation of short term maintenance schedules of unit auxiliaries and peaking stations apart from maintaining security constraints and minimizing operational costs. Forecasts of the daily load for seven days by the available multiplicative SARIMA model suffer from divergent error levels of multistep ahead forecasts, and to take all the features of any particular day, the order of such models will be prohibitively large. In the present paper a unique approach has been made to the time series analysis of stochastic processes: multigrade periodicities present in the daily load curve have been gainfully exploited by grouping the data in seven subgroups characterizing each week day separately and then each group is replaced by an orthogonal transform known as a Walsh transform. The amplitudes of the transformed spectrum have been time series modelled by the Box-Jenkins approach and forecasts are made by taking inverse (Walsh) transforms. (author).
Real zeros of classes of random algebraic polynomials
Directory of Open Access Journals (Sweden)
K. Farahmand
2003-01-01
Full Text Available There are many known asymptotic estimates for the expected number of real zeros of an algebraic polynomial a0+a1x+a2x2+⋯+an−1xn−1 with identically distributed random coefficients. Under different assumptions for the distribution of the coefficients {aj}j=0n−1 it is shown that the above expected number is asymptotic to O(logn. This order for the expected number of zeros remains valid for the case when the coefficients are grouped into two, each group with a different variance. However, it was recently shown that if the coefficients are non-identically distributed such that the variance of the jth term is (nj the expected number of zeros of the polynomial increases to O(n. The present paper provides the value for this asymptotic formula for the polynomials with the latter variances when they are grouped into three with different patterns for their variances.
2010-07-01
...-Healey Public Contracts Act of truck drivers employed by oil dealers. 50-210.1 Section 50-210.1 Public...-210.1 Coverage under the Walsh-Healey Public Contracts Act of truck drivers employed by oil dealers... Interpretations No. 2 1 with respect to coverage under the Walsh-Healey Public Contracts Act of truck drivers...
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
Efficient detection of symmetries polynomially parametrized curves
Alcázar Arribas, Juan Gerardo
2014-01-01
We present efficient algorithms for detecting central and mirror symmetry for the case of algebraic curves defined by means of polynomial parametrizations. The algorithms are based on an algebraic relationship between proper parametrizations of a same curve, which leads to a triangular polynomial system that can be solved in a very fast way; in particular, curves parametrized by polynomials of serious degrees/coefficients can be analyzed in a few seconds. In our analysis we provide a good num...
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.
29 CFR 1926.15 - Relationship to the Service Contract Act; Walsh-Healey Public Contracts Act.
2010-07-01
... Public Contracts Act. 1926.15 Section 1926.15 Labor Regulations Relating to Labor (Continued...-Healey Public Contracts Act. (a) A contract for “construction” is one for nonpersonal service. See, e.g... Walsh-Healey Public Contracts Act to certain particular working conditions. None of the described...
Connell, Mary
2012-01-01
Cross, Fine, Jones, and Walsh (2012) provided a thoughtful review and critique of a book chapter describing the interview process at Child Advocacy Centers. They observed some of the ways that concerns raised in that chapter are being addressed and described revised guidelines that further clarify issues. Ongoing research and examination of the…
Peach, Prunus persica (L.) Batsch (Rosaceae), has been reported to be a host of the apple maggot fly, Rhagoletis pomonella (Walsh), 1867 (Diptera: Tephritidae), an important quarantine pest of apple (Malus domestica Borkhausen) (Rosaceae) in the western U.S.A. However, all reports of peach as a hos...
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.
Computationally Efficient Robust Color Image Watermarking Using Fast Walsh Hadamard Transform
Directory of Open Access Journals (Sweden)
Suja Kalarikkal Pullayikodi
2017-10-01
Full Text Available Watermark is the copy deterrence mechanism used in the multimedia signal that is to be protected from hacking and piracy such a way that it can later be extracted from the watermarked signal by the decoder. Watermarking can be used in various applications such as authentication, video indexing, copyright protection and access control. In this paper a new CDMA (Code Division Multiple Access based robust watermarking algorithm using customized 8 × 8 Walsh Hadamard Transform, is proposed for the color images and detailed performance and robustness analysis have been performed. The paper studies in detail the effect of spreading code length, number of spreading codes and type of spreading codes on the performance of the watermarking system. Compared to the existing techniques the proposed scheme is computationally more efficient and consumes much less time for execution. Furthermore, the proposed scheme is robust and survives most of the common signal processing and geometric attacks.
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.
Application of polynomial preconditioners to conservation laws
Geurts, Bernardus J.; van Buuren, R.; Lu, H.
2000-01-01
Polynomial preconditioners which are suitable in implicit time-stepping methods for conservation laws are reviewed and analyzed. The preconditioners considered are either based on a truncation of a Neumann series or on Chebyshev polynomials for the inverse of the system-matrix. The latter class of
Connections between the matching and chromatic polynomials
Directory of Open Access Journals (Sweden)
E. J. Farrell
1992-01-01
Full Text Available The main results established are (i a connection between the matching and chromatic polynomials and (ii a formula for the matching polynomial of a general complement of a subgraph of a graph. Some deductions on matching and chromatic equivalence and uniqueness are made.
Multiplicative polynomial mappings on commutative Banach algebras
Directory of Open Access Journals (Sweden)
Labachuk O.V.
2012-12-01
Full Text Available We consider the multiplicative polynomial mappings on commutativealgebras in this work. We call a multiplicative polynomialtrivial, if it can be represented as a product of characters. Inthe paper we investigate the following question: does there existsa nontrivial multiplicative polynomial functional on a commutativealgebra?
General Reducibility and Solvability of Polynomial Equations ...
African Journals Online (AJOL)
A complete work on general reducibility and solvability of polynomial equations by algebraic meansradicals is developed. These equations called, reanegbèd and vic-emmeous are designed by using simple algebraic principles on how systems of equations and polynomials behave. Reanegbèd equations are capable of ...
A generalization of the Bernoulli polynomials
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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.
New pole placement algorithm - Polynomial matrix approach
Shafai, B.; Keel, L. H.
1990-01-01
A simple and direct pole-placement algorithm is introduced for dynamical systems having a block companion matrix A. The algorithm utilizes well-established properties of matrix polynomials. Pole placement is achieved by appropriately assigning coefficient matrices of the corresponding matrix polynomial. This involves only matrix additions and multiplications without requiring matrix inversion. A numerical example is given for the purpose of illustration.
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 imag...
Multivariate Krawtchouk Polynomials and Composition Birth and Death Processes
National Research Council Canada - National Science Library
Robert Griffiths
2016-01-01
.... The dual multivariate Krawtchouk polynomials, which also have a polynomial structure, are seen to occur naturally as spectral orthogonal polynomials in a Karlin and McGregor spectral representation...
M-Interval Orthogonal Polynomial Estimators with Applications
Jaroszewicz, Boguslaw Emanuel
In this dissertation, adaptive estimators of various statistical nonlinearities are constructed and evaluated. The estimators are based on classical orthogonal polynomials which allows an exact computation of convergence rates. The first part of the dissertation is devoted to the estimation of one- and multi-dimensional probability density functions. The most attractive computationally is the Legendre estimator, which corresponds to the mean square segmented polynomial approximation of a pdf. Exact bounds for two components of the estimation error--deterministic bias and random error--are derived for all the polynomial estimators. The bounds on the bias are functions of the "smoothness" of the estimated pdf as measured by the number of continuous derivatives the pdf possesses. Adaptively estimated the optimum number of orthonormal polynomials minimizes the total error. In the second part, the theory of polynomial estimators is applied to the estimation of derivatives of pdf and regression functions. The optimum detectors for small signals in nongaussian noise, as well as any kind of statistical filtering involving likelihood function, are based on the nonlinearity which is a ratio of the derivative of the pdf and the pdf itself. Several different polynomial estimators of this nonlinearity are developed and compared. The theory of estimation is then extended to the multivariable case. The partial derivative nonlinearity is used for detection of signals in dependent noise. When the dimensionality of the nonlinearity is very large, the transformed Hermite estimators are particularly useful. The estimators can be viewed as two-stage filters: the first stage is a pre -whitening filter optimum in gaussian noise and the second stage is a nonlinear filter, which improves performance in nongaussian noise. Filtering of this type can be applied to predictive coding, nonlinear identification and other estimation problems involving a conditional expected value. In the third
On the Polynomial Basis of GF(2n) Having a Small Number of Trace-One Elements
Jiantao Wang; Dong Zheng; Zheng Huang
2016-01-01
In the Galois fields GF(2n), a polynomial basis with a small number of trace-one elements is desirable for its convenience in computing. To find new irreducible polynomials g(x) over GF(2) with this property, we research into the auxiliary polynomial f(x)=(x+1)g(x) with roots {1,α1,α2,…,αn}, such that the symmetric polynomials sk=1+α1k+α2k+⋯+αnk are relative to the symmetric polynomials of g(x). We introduce a new class of polynomials with the number “1” occupying most of the values in its sk...
Zero points of general quaternionic polynomials
Janovská, Drahoslava; Opfer, Gerhard
2013-10-01
We consider quaternionic polynomials in the most general form. More precisely, polynomials are arbitrary, finite, sums of monomials, where monomials of degree j have the form a0ṡxṡa1ṡxṡa2ṡx⋯xṡaj-1ṡxṡaj, with a variable j. This allows also several monomials of the same degree. Our aim is to apply Newton's method to compute zero points of these polynomials, namely to calculate the Jacobi matrix needed in Newton's method.
On Calculation of Adomian Polynomials by MATLAB
Directory of Open Access Journals (Sweden)
Hossein ABOLGHASEMI
2011-01-01
Full Text Available Adomian Decomposition Method (ADM is an elegant technique to handle an extensive class of linear or nonlinear differential and integral equations. However, in case of nonlinear equations, ADM demands a special representation of each nonlinear term, namely, Adomian polynomials. The present paper introduces a novel MATLAB code which computes Adomian polynomials associated with several types of nonlinearities. The code exploits symbolic programming incorporated with a recently proposed alternative scheme to be straightforward and fast. For the sake of exemplification, Adomian polynomials of famous nonlinear operators, computed by the code, are given.
Empowering Polynomial Theory Conjectures with Spreadsheets
Directory of Open Access Journals (Sweden)
Chris Petersdinh
2017-06-01
Full Text Available Polynomial functions and their properties are fundamental to algebra, calculus, and mathematical modeling. Students who do not have a strong understanding of the relationship between factoring and solving equations can have difficulty with optimization problems in calculus and solving application problems in any field. Understanding function transformations is important in trigonometry, the idea of the general antiderivative, and describing the geometry of a problem mathematically. This paper presents spreadsheet activities designed to bolster students' conceptualization of the factorization theorem for polynomials, complex zeros of polynomials, and function transformations. These activities were designed to use a constructivist approach involving student experimentation and conjectures.
Polynomial entropies for Bott nondegenerate Hamiltonian systems
Labrousse, Clémence; Marco, Jean-Pierre
2012-01-01
In this paper, we study the entropy of a Hamiltonian flow in restriction to an enregy level where it admits a first integral which is nondegenerate in the Bott sense. It is easy to see that for such a flow, the topological entropy vanishes. We focus on the polynomial and the weak polynomial entropies. We prove that, under conditions on the critical level of the Bott first integral and dynamical conditions on the hamiltonian function, the weak polynomial entropy belongs to {0,1} and the polyno...
Homfly Polynomials of Generalized Hopf Links
Morton, Hugh R.; Hadji, Richard J.
2001-01-01
Following the recent work by T.-H. Chan in [HOMFLY polynomial of some generalized Hopf links, J. Knot Theory Ramif. 9 (2000) 865--883] on reverse string parallels of the Hopf link we give an alternative approach to finding the Homfly polynomials of these links, based on the Homfly skein of the annulus. We establish that two natural skein maps have distinct eigenvalues, answering a question raised by Chan, and use this result to calculate the Homfly polynomial of some more general reverse stri...
Representation of American versus non-American fans in Baillie Walsh's Springsteen & I
Directory of Open Access Journals (Sweden)
Maryn Claire Wilkinson
2015-06-01
Full Text Available This article explores the representation of American versus non-American fans in Baillie Walsh's 2013 crowd/fan-sourced documentary Springsteen & I. The film—as much as it was fed by a wide and international range of fan-produced material—ultimately produced and presented one particular type of fan as privileged and appropriate to speak for all: the American Springsteen fan. The film does so in three main ways: by presenting American fans as more authentically connected to Springsteen's language and lyrics (they truly "understand" his work; by showing that American fans relate to Springsteen's world and themes in more authentic ways; and by presenting American fans as experiencing a better, closer, more authentic Springsteen when seeing him perform live. The essay thus reexamines fan stereotyping from the perspective of national identity. It aims to rearticulate the necessity of the vigilance and scrutiny of crowd-sourced fan texts because they have profound effects on how fans are taught to view themselves in and by the media.
Harmonic polynomials, hyperspherical harmonics, and atomic spectra
Avery, John Scales
2010-01-01
The properties of monomials, homogeneous polynomials and harmonic polynomials in d-dimensional spaces are discussed. The properties are shown to lead to formulas for the canonical decomposition of homogeneous polynomials and formulas for harmonic projection. Many important properties of spherical harmonics, Gegenbauer polynomials and hyperspherical harmonics follow from these formulas. Harmonic projection also provides alternative ways of treating angular momentum and generalised angular momentum. Several powerful theorems for angular integration and hyperangular integration can be derived in this way. These purely mathematical considerations have important physical applications because hyperspherical harmonics are related to Coulomb Sturmians through the Fock projection, and because both Sturmians and generalised Sturmians have shown themselves to be extremely useful in the quantum theory of atoms and molecules.
Handbook on semidefinite, conic and polynomial optimization
Anjos, Miguel F
2012-01-01
This book offers the reader a snapshot of the state-of-the-art in the growing and mutually enriching areas of semidefinite optimization, conic optimization and polynomial optimization. It covers theory, algorithms, software and applications.
Quantum Communication and Quantum Multivariate Polynomial Interpolation
Diep, Do Ngoc; Giang, Do Hoang
2017-09-01
The paper is devoted to the problem of multivariate polynomial interpolation and its application to quantum secret sharing. We show that using quantum Fourier transform one can produce the protocol for quantum secret sharing distribution.
Energy Technology Data Exchange (ETDEWEB)
Schulze-Halberg, Axel, E-mail: axgeschu@iun.edu [Department of Mathematics and Actuarial Science, Indiana University Northwest, 3400 Broadway, Gary IN 46408 (United States); Department of Physics, Indiana University Northwest, 3400 Broadway, Gary IN 46408 (United States); Roy, Pinaki, E-mail: pinaki@isical.ac.in [Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108 (India)
2017-03-15
We construct energy-dependent potentials for which the Schrödinger equations admit solutions in terms of exceptional orthogonal polynomials. Our method of construction is based on certain point transformations, applied to the equations of exceptional Hermite, Jacobi and Laguerre polynomials. We present several examples of boundary-value problems with energy-dependent potentials that admit a discrete spectrum and the corresponding normalizable solutions in closed form.
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...
Rotation of 2D orthogonal polynomials
Czech Academy of Sciences Publication Activity Database
Yang, B.; Flusser, Jan; Kautský, J.
2018-01-01
Roč. 102, č. 1 (2018), s. 44-49 ISSN 0167-8655 R&D Projects: GA ČR GA15-16928S Institutional support: RVO:67985556 Keywords : Rotation invariants * Orthogonal polynomials * Recurrent relation * Hermite-like polynomials * Hermite moments Subject RIV: JD - Computer Applications, Robotics Impact factor: 1.995, year: 2016 http:// library .utia.cas.cz/separaty/2017/ZOI/flusser-0483250.pdf
Polynomial Chaos Surrogates for Bayesian Inference
Le Maitre, Olivier
2016-01-06
The Bayesian inference is a popular probabilistic method to solve inverse problems, such as the identification of field parameter in a PDE model. The inference rely on the Bayes rule to update the prior density of the sought field, from observations, and derive its posterior distribution. In most cases the posterior distribution has no explicit form and has to be sampled, for instance using a Markov-Chain Monte Carlo method. In practice the prior field parameter is decomposed and truncated (e.g. by means of Karhunen- Lo´eve decomposition) to recast the inference problem into the inference of a finite number of coordinates. Although proved effective in many situations, the Bayesian inference as sketched above faces several difficulties requiring improvements. First, sampling the posterior can be a extremely costly task as it requires multiple resolutions of the PDE model for different values of the field parameter. Second, when the observations are not very much informative, the inferred parameter field can highly depends on its prior which can be somehow arbitrary. These issues have motivated the introduction of reduced modeling or surrogates for the (approximate) determination of the parametrized PDE solution and hyperparameters in the description of the prior field. Our contribution focuses on recent developments in these two directions: the acceleration of the posterior sampling by means of Polynomial Chaos expansions and the efficient treatment of parametrized covariance functions for the prior field. We also discuss the possibility of making such approach adaptive to further improve its efficiency.
Flocke, N
2009-08-14
In this paper it is shown that shifted Jacobi polynomials G(n)(p,q,x) can be used in connection with the Gaussian quadrature modified moment technique to greatly enhance the accuracy of evaluation of Rys roots and weights used in Gaussian integral evaluation in quantum chemistry. A general four-term inhomogeneous recurrence relation is derived for the shifted Jacobi polynomial modified moments over the Rys weight function e(-Tx)/square root x. It is shown that for q=1/2 this general four-term inhomogeneous recurrence relation reduces to a three-term p-dependent inhomogeneous recurrence relation. Adjusting p to proper values depending on the Rys exponential parameter T, the method is capable of delivering highly accurate results for large number of roots and weights in the most difficult to treat intermediate T range. Examples are shown, and detailed formulas together with practical suggestions for their efficient implementation are also provided.
A fixed point method to compute solvents of matrix polynomials
Marcos, Fernando; Pereira, Edgar
2009-01-01
Matrix polynomials play an important role in the theory of matrix differential equations. We develop a fixed point method to compute solutions of matrix polynomials equations, where the matricial elements of the matrix polynomial are considered separately as complex polynomials. Numerical examples illustrate the method presented.
Strong asymptotics for extremal polynomials associated with weights on ℝ
Lubinsky, Doron S
1988-01-01
0. The results are consequences of a strengthened form of the following assertion: Given 0 1. Auxiliary results include inequalities for weighted polynomials, and zeros of extremal polynomials. The monograph is fairly self-contained, with proofs involving elementary complex analysis, and the theory of orthogonal and extremal polynomials. It should be of interest to research workers in approximation theory and orthogonal polynomials.
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.
Time-delay polynomial networks and rates of approximation
Directory of Open Access Journals (Sweden)
Irwin W. Sandberg
1998-01-01
Full Text Available We consider a large family of finite memory causal time-invariant maps G from an input set S to a set of ℝ-valued functions, with the members of both sets of functions defined on the nonnegative integers, and we give an upper bound on the error in approximating a G using a two-stage structure consisting of a tapped delay line and a static polynomial network N . This upper bound depends on the degree of the multivariable polynomial that characterizes N. Also given is a lower bound on the worst-case error in approximating a G using polynomials of a fixed maximum degree. These upper and lower bounds differ only by a multiplicative constant. We also give a corresponding result for the approximation of not-necessarily-causal input–output maps with inputs and outputs that may depend on more than one variable. This result is of interest, for example, in connection with image processing.
Background Subtraction of Raman Spectra Based on Iterative Polynomial Smoothing.
Wang, Tuo; Dai, Liankui
2017-06-01
In this paper, a novel background subtraction algorithm is presented that can automatically recover Raman signal. This algorithm is based on an iterative polynomial smoothing method that highly reduces the need for experience and a priori knowledge. First, a polynomial filter is applied to smooth the input spectrum (the input spectrum is just an original spectrum at the first iteration). The output curve of the filter divides the original spectrum into two parts, top and bottom. Second, a proportion is calculated between the lowest point of the signal in the bottom part and the highest point of the signal in the top part. The proportion is a key index that decides whether to go into a new iteration. If a new iteration is needed, the minimum value between the output curve and the original spectrum forms a new curve that goes into the same filter in the first step and continues as another iteration until no more iteration is needed to finally get the background of the original spectrum. Results from the simulation experiments not only show that the iterative polynomial smoothing algorithm achieves good performance, processing time, cost, and accuracy of recovery, but also prove that the algorithm adapts to different background types and a large signal-to-noise ratio range. Furthermore, real measured Raman spectra of organic mixtures and non-organic samples are used to demonstrate the application of the algorithm.
Evaluation of Piecewise Polynomial Equations for Two Types of Thermocouples
Directory of Open Access Journals (Sweden)
Andrew Chen
2013-12-01
Full Text Available Thermocouples are the most frequently used sensors for temperature measurement because of their wide applicability, long-term stability and high reliability. However, one of the major utilization problems is the linearization of the transfer relation between temperature and output voltage of thermocouples. The linear calibration equation and its modules could be improved by using regression analysis to help solve this problem. In this study, two types of thermocouple and five temperature ranges were selected to evaluate the fitting agreement of different-order polynomial equations. Two quantitative criteria, the average of the absolute error values |e|ave and the standard deviation of calibration equation estd, were used to evaluate the accuracy and precision of these calibrations equations. The optimal order of polynomial equations differed with the temperature range. The accuracy and precision of the calibration equation could be improved significantly with an adequate higher degree polynomial equation. The technique could be applied with hardware modules to serve as an intelligent sensor for temperature measurement.
Extending a Property of Cubic Polynomials to Higher-Degree Polynomials
Miller, David A.; Moseley, James
2012-01-01
In this paper, the authors examine a property that holds for all cubic polynomials given two zeros. This property is discovered after reviewing a variety of ways to determine the equation of a cubic polynomial given specific conditions through algebra and calculus. At the end of the article, they will connect the property to a very famous method…
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
Best polynomial degree reduction on q-lattices with applications to q-orthogonal polynomials
Ait-Haddou, Rachid
2015-06-07
We show that a weighted least squares approximation of q-Bézier coefficients provides the best polynomial degree reduction in the q-L2-norm. We also provide a finite analogue of this result with respect to finite q-lattices and we present applications of these results to q-orthogonal polynomials. © 2015 Elsevier Inc. All rights reserved.
Multilevel weighted least squares polynomial approximation
Haji-Ali, Abdul-Lateef
2017-06-30
Weighted least squares polynomial approximation uses random samples to determine projections of functions onto spaces of polynomials. It has been shown that, using an optimal distribution of sample locations, the number of samples required to achieve quasi-optimal approximation in a given polynomial subspace scales, up to a logarithmic factor, linearly in the dimension of this space. However, in many applications, the computation of samples includes a numerical discretization error. Thus, obtaining polynomial approximations with a single level method can become prohibitively expensive, as it requires a sufficiently large number of samples, each computed with a sufficiently small discretization error. As a solution to this problem, we propose a multilevel method that utilizes samples computed with different accuracies and is able to match the accuracy of single-level approximations with reduced computational cost. We derive complexity bounds under certain assumptions about polynomial approximability and sample work. Furthermore, we propose an adaptive algorithm for situations where such assumptions cannot be verified a priori. Finally, we provide an efficient algorithm for the sampling from optimal distributions and an analysis of computationally favorable alternative distributions. Numerical experiments underscore the practical applicability of our method.
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.
A series transformation formula and related polynomials
Directory of Open Access Journals (Sweden)
Khristo N. Boyadzhiev
2005-12-01
Full Text Available We present a formula that turns power series into series of functions. This formula serves two purposes: first, it helps to evaluate some power series in a closed form; second, it transforms certain power series into asymptotic series. For example, we find the asymptotic expansions for ÃŽÂ»>0 of the incomplete gamma function ÃŽÂ³(ÃŽÂ»,x and of the Lerch transcendent ÃŽÂ¦(x,s,ÃŽÂ». In one particular case, our formula reduces to a series transformation formula which appears in the works of Ramanujan and is related to the exponential (or Bell polynomials. Another particular case, based on the geometric series, gives rise to a new class of polynomials called geometric polynomials.
Sparse DOA estimation with polynomial rooting
DEFF Research Database (Denmark)
Xenaki, Angeliki; Gerstoft, Peter; Fernandez Grande, Efren
2015-01-01
Direction-of-arrival (DOA) estimation involves the localization of a few sources from a limited number of observations on an array of sensors. Thus, DOA estimation can be formulated as a sparse signal reconstruction problem and solved efficiently with compressive sensing (CS) to achieve...... highresolution imaging. Utilizing the dual optimal variables of the CS optimization problem, it is shown with Monte Carlo simulations that the DOAs are accurately reconstructed through polynomial rooting (Root-CS). Polynomial rooting is known to improve the resolution in several other DOA estimation methods...
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 second......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...
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.
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....
Dr. H. B. Kekre; Dr. Sudeep D. Thepade; Adib Parkar
2011-01-01
The storage of colour information in a greyscale image is not a new idea. Various techniques have been proposed using different colour spaces including the standard RGB colour space, the YUV colour space, and the YCbCr colour space. This paper extends the results described in [1] and [2]. While [1] describes the storage of colour information in a greyscale image using Haar wavelets, and [2] adds a comparison with Kekre’s wavelets, this paper adds a third transform – the Walsh transform and pr...
A polynomial based model for cell fate prediction in human diseases.
Ma, Lichun; Zheng, Jie
2017-12-21
Cell fate regulation directly affects tissue homeostasis and human health. Research on cell fate decision sheds light on key regulators, facilitates understanding the mechanisms, and suggests novel strategies to treat human diseases that are related to abnormal cell development. In this study, we proposed a polynomial based model to predict cell fate. This model was derived from Taylor series. As a case study, gene expression data of pancreatic cells were adopted to test and verify the model. As numerous features (genes) are available, we employed two kinds of feature selection methods, i.e. correlation based and apoptosis pathway based. Then polynomials of different degrees were used to refine the cell fate prediction function. 10-fold cross-validation was carried out to evaluate the performance of our model. In addition, we analyzed the stability of the resultant cell fate prediction model by evaluating the ranges of the parameters, as well as assessing the variances of the predicted values at randomly selected points. Results show that, within both the two considered gene selection methods, the prediction accuracies of polynomials of different degrees show little differences. Interestingly, the linear polynomial (degree 1 polynomial) is more stable than others. When comparing the linear polynomials based on the two gene selection methods, it shows that although the accuracy of the linear polynomial that uses correlation analysis outcomes is a little higher (achieves 86.62%), the one within genes of the apoptosis pathway is much more stable. Considering both the prediction accuracy and the stability of polynomial models of different degrees, the linear model is a preferred choice for cell fate prediction with gene expression data of pancreatic cells. The presented cell fate prediction model can be extended to other cells, which may be important for basic research as well as clinical study of cell development related diseases.
Polynomial Representations for a Wavelet Model of Interest Rates
Directory of Open Access Journals (Sweden)
Dennis G. Llemit
2015-12-01
Full Text Available In this paper, we approximate a non – polynomial function which promises to be an essential tool in interest rates forecasting in the Philippines. We provide two numerical schemes in order to generate polynomial functions that approximate a new wavelet which is a modification of Morlet and Mexican Hat wavelets. The first is the Polynomial Least Squares method which approximates the underlying wavelet according to desired numerical errors. The second is the Chebyshev Polynomial approximation which generates the required function through a sequence of recursive and orthogonal polynomial functions. We seek to determine the lowest order polynomial representations of this wavelet corresponding to a set of error thresholds.
Some Relationships between the Analogs of Euler Numbers and Polynomials
Directory of Open Access Journals (Sweden)
Kim T
2007-01-01
Full Text Available We construct new twisted Euler polynomials and numbers. We also study the generating functions of the twisted Euler numbers and polynomials associated with their interpolation functions. Next we construct twisted Euler zeta function, twisted Hurwitz zeta function, twisted Dirichlet -Euler numbers and twisted Euler polynomials at non-positive integers, respectively. Furthermore, we find distribution relations of generalized twisted Euler numbers and polynomials. By numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the twisted -Euler polynomials. Finally, we give a table for the solutions of the twisted -Euler polynomials.
Some Relationships between the Analogs of Euler Numbers and Polynomials
Directory of Open Access Journals (Sweden)
C. S. Ryoo
2007-10-01
Full Text Available We construct new twisted Euler polynomials and numbers. We also study the generating functions of the twisted Euler numbers and polynomials associated with their interpolation functions. Next we construct twisted Euler zeta function, twisted Hurwitz zeta function, twisted Dirichlet l-Euler numbers and twisted Euler polynomials at non-positive integers, respectively. Furthermore, we find distribution relations of generalized twisted Euler numbers and polynomials. By numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the twisted q-Euler polynomials. Finally, we give a table for the solutions of the twisted q-Euler polynomials.
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 Arithmetic-Geometric-Mean Polynomials
Griffiths, Martin; MacHale, Des
2017-01-01
We study here an aspect of an infinite set "P" of multivariate polynomials, the elements of which are associated with the arithmetic-geometric-mean inequality. In particular, we show in this article that there exist infinite subsets of probability "P" for which every element may be expressed as a finite sum of squares of real…
Exceptional polynomials and SUSY quantum mechanics
Indian Academy of Sciences (India)
2Faculty of Science and Technology, ICFAI Foundation for Higher Education, Donthanapally,. Hyderabad 501 203, India ... classical orthogonal polynomial systems (OPS) such as, the Hermite, Laguerre and the. Jacobi satisfy eq. (1). Note that y0 is a constant ..... Ltd., Singapore, 2001). [11] C Quesne, SIGMA 5, 084 (2009), ...
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
The 6 Vertex Model and Schubert Polynomials
Directory of Open Access Journals (Sweden)
Alain Lascoux
2007-02-01
Full Text Available We enumerate staircases with fixed left and right columns. These objects correspond to ice-configurations, or alternating sign matrices, with fixed top and bottom parts. The resulting partition functions are equal, up to a normalization factor, to some Schubert polynomials.
Modeling Power Amplifiers using Memory Polynomials
Kokkeler, Andre B.J.
2005-01-01
In this paper we present measured in- and output data of a power amplifier (PA). We compare this data with an AM-AM and AM-PM model. We conclude that a more sophisticated PA model is needed to cope with severe memory effects. We suggest to use memory polynomials and introduce two approaches to
Integral inequalities for self-reciprocal polynomials
Indian Academy of Sciences (India)
Annual Meetings · Mid Year Meetings · Discussion Meetings · Public Lectures · Lecture Workshops · Refresher Courses · Symposia · Live Streaming. Home; Journals; Proceedings – Mathematical Sciences; Volume 120; Issue 2. Integral Inequalities for Self-Reciprocal Polynomials. Horst Alzer. Volume 120 Issue 2 April 2010 ...
Deformation Prediction Using Linear Polynomial Functions ...
African Journals Online (AJOL)
The predictions are compared with measured data reported in literature and the results are discussed. The computational aspects of implementation of the model are also discussed briefly. Keywords: Linear Polynomial, Structural Deformation, Prediction Journal of the Nigerian Association of Mathematical Physics, Volume ...
Dynamic normal forms and dynamic characteristic polynomial
DEFF Research Database (Denmark)
Frandsen, Gudmund Skovbjerg; Sankowski, Piotr
2011-01-01
We present the first fully dynamic algorithm for computing the characteristic polynomial of a matrix. In the generic symmetric case, our algorithm supports rank-one updates in O(n2logn) randomized time and queries in constant time, whereas in the general case the algorithm works in O(n2klogn...
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....
Duality of orthogonal polynomials on a finite set
Borodin, Alexei
2001-01-01
We prove a certain duality relation for orthogonal polynomials defined on a finite set. The result is used in a direct proof of the equivalence of two different ways of computing the correlation functions of a discrete orthogonal polynomial ensemble.
On an inequality concerning the polar derivative of a polynomial
Indian Academy of Sciences (India)
Abstract. In this paper, we present a correct proof of an -inequality concerning the polar derivative of a polynomial with restricted zeros. We also extend Zygmund's inequality to the polar derivative of a polynomial.
Commutators with idempotent values on multilinear polynomials in ...
Indian Academy of Sciences (India)
Proceedings – Mathematical Sciences. Current Issue : Vol. 127, Issue 5. Current Issue Volume 127 | Issue 5. November 2017. Home · Volumes & Issues · Special Issues · Forthcoming Articles · Search · Editorial Board · Information for Authors · Subscription ...
Commutators with idempotent values on multilinear polynomials in ...
Indian Academy of Sciences (India)
Introduction and statements of the result. In mid-forties, after the development of the general structure theory for rings, a great deal of work was done that showed that under certain types of hypothesis, rings had to be commutative or almost commutative. A classical result of ring theory established by Jacobson generalizes at ...
Characterization and Enumeration of Good Punctured Polynomials over Finite Fields
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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.
Irreducibility results for compositions of polynomials in several ...
Indian Academy of Sciences (India)
R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22
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. Keywords. Composition of polynomials; irreducibility results. 1. Introduction.
On some properties of generalized Fibonacci and Lucas polynomials
Directory of Open Access Journals (Sweden)
Sümeyra Uçar
2017-07-01
Full Text Available In this paper we investigate some properties of generalized Fibonacci and Lucas polynomials. We give some new identities using matrices and Laplace expansion for the generalized Fibonacci and Lucas polynomials. Also, we introduce new families of tridiagonal matrices whose successive determinants generate any subsequence of these polynomials.
Approximating Exponential and Logarithmic Functions Using Polynomial Interpolation
Gordon, Sheldon P.; Yang, Yajun
2017-01-01
This article takes a closer look at the problem of approximating the exponential and logarithmic functions using polynomials. Either as an alternative to or a precursor to Taylor polynomial approximations at the precalculus level, interpolating polynomials are considered. A measure of error is given and the behaviour of the error function is…
On the 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.
On Multiple Generalized w-Genocchi Polynomials and Their Applications
Directory of Open Access Journals (Sweden)
Lee-Chae Jang
2010-01-01
Full Text Available We define the multiple generalized w-Genocchi polynomials. By using fermionic p-adic invariant integrals, we derive some identities on these generalized w-Genocchi polynomials, for example, fermionic p-adic integral representation, Witt's type formula, explicit formula, multiplication formula, and recurrence formula for these w-Genocchi polynomials.
Model-robust experimental designs for the fractional polynomial ...
African Journals Online (AJOL)
Fractional polynomial response surface models are polynomial models whose powers are restricted to a small predefined set of rational numbers. Very often these models can give a good a fit to the data and much more plausible behavior between design points than the polynomial models. In this paper, we propose a ...
Polynomial approximations of the Normal toWeibull Distribution transformation
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Andrés Feijóo
2014-09-01
Full Text Available Some of the tools that are generally employed in power system analysis need to use approaches based on statistical distributions for simulating the cumulative behavior of the different system devices. For example, the probabilistic load flow. The presence of wind farms in power systems has increased the use of Weibull and Rayleigh distributions among them. Not only the distributions themselves, but also satisfying certain constraints such as correlation between series of data or even autocorrelation can be of importance in the simulation. Correlated Weibull or Rayleigh distributions can be obtained by transforming correlated Normal distributions, and it can be observed that certain statistical values such as the means and the standard deviations tend to be retained when operating such transformations, although why this happens is not evident. The objective of this paper is to analyse the consequences of using such transformations. The methodology consists of comparing the results obtained by means of a direct transformation and those obtained by means of approximations based on the use of first and second degree polynomials. Simulations have been carried out with series of data which can be interpreted as wind speeds. The use of polynomial approximations gives accurate results in comparison with direct transformations and provides an approach that helps explain why the statistical values are retained during the transformations.
Polynomial Chaos Based Acoustic Uncertainty Predictions from Ocean Forecast Ensembles
Dennis, S.
2016-02-01
Most significant ocean acoustic propagation occurs at tens of kilometers, at scales small compared basin and to most fine scale ocean modeling. To address the increased emphasis on uncertainty quantification, for example transmission loss (TL) probability density functions (PDF) within some radius, a polynomial chaos (PC) based method is utilized. In order to capture uncertainty in ocean modeling, Navy Coastal Ocean Model (NCOM) now includes ensembles distributed to reflect the ocean analysis statistics. Since the ensembles are included in the data assimilation for the new forecast ensembles, the acoustic modeling uses the ensemble predictions in a similar fashion for creating sound speed distribution over an acoustically relevant domain. Within an acoustic domain, singular value decomposition over the combined time-space structure of the sound speeds can be used to create Karhunen-Loève expansions of sound speed, subject to multivariate normality testing. These sound speed expansions serve as a basis for Hermite polynomial chaos expansions of derived quantities, in particular TL. The PC expansion coefficients result from so-called non-intrusive methods, involving evaluation of TL at multi-dimensional Gauss-Hermite quadrature collocation points. Traditional TL calculation from standard acoustic propagation modeling could be prohibitively time consuming at all multi-dimensional collocation points. This method employs Smolyak order and gridding methods to allow adaptive sub-sampling of the collocation points to determine only the most significant PC expansion coefficients to within a preset tolerance. Practically, the Smolyak order and grid sizes grow only polynomially in the number of Karhunen-Loève terms, alleviating the curse of dimensionality. The resulting TL PC coefficients allow the determination of TL PDF normality and its mean and standard deviation. In the non-normal case, PC Monte Carlo methods are used to rapidly establish the PDF. This work was
Systematic comparison of the use of annular and Zernike circle polynomials for annular wavefronts.
Mahajan, Virendra N; Aftab, Maham
2010-11-20
The theory of wavefront analysis of a noncircular wavefront is given and applied for a systematic comparison of the use of annular and Zernike circle polynomials for the analysis of an annular wavefront. It is shown that, unlike the annular coefficients, the circle coefficients generally change as the number of polynomials used in the expansion changes. Although the wavefront fit with a certain number of circle polynomials is identically the same as that with the corresponding annular polynomials, the piston circle coefficient does not represent the mean value of the aberration function, and the sum of the squares of the other coefficients does not yield its variance. The interferometer setting errors of tip, tilt, and defocus from a four-circle-polynomial expansion are the same as those from the annular-polynomial expansion. However, if these errors are obtained from, say, an 11-circle-polynomial expansion, and are removed from the aberration function, wrong polishing will result by zeroing out the residual aberration function. If the common practice of defining the center of an interferogram and drawing a circle around it is followed, then the circle coefficients of a noncircular interferogram do not yield a correct representation of the aberration function. Moreover, in this case, some of the higher-order coefficients of aberrations that are nonexistent in the aberration function are also nonzero. Finally, the circle coefficients, however obtained, do not represent coefficients of the balanced aberrations for an annular pupil. The various results are illustrated analytically and numerically by considering an annular Seidel aberration function.
Orthogonal polynomials in neutron transport theory
Energy Technology Data Exchange (ETDEWEB)
Dehesa, J.S. (Granada Univ. (Spain). Facultad de Ciencias)
1982-01-01
The asymptotic average properties of zeros of the polynomials gsub(k)sup(m) (x), which play a fundamental role in neutron transport and radiative transfer theories, are investigated analytically in terms of the angular expansion coefficients wsub(k) of the scattering kernel for three wide classes of scattering models. In particular it is found that the scattering models of Eccleston-McCormick (J. Nucl. Energy.; 24:23 (1970)), Shultis et al (Nucl. Sci. Eng.; 59:53 (1976)) and Henyey-Greenstein (Astrophys. J.; 93:70 (1941)) belong in one of the above-mentioned classes, and their associated polynomials gsub(k)sup(m) (x) have the same asymptotic density of zeros.
Piecewise polynomial solutions to linear inverse problems
DEFF Research Database (Denmark)
Hansen, Per Christian; Mosegaard, K.
1996-01-01
We have presented a new algorithm PP-TSVD that computes piecewise polynomial solutions to ill-posed problems, without a priori knowledge about the positions of the break points. In particular, we can compute piecewise constant functions that describe layered models. Such solutions are useful, e.g.......g., in seismological problems, and the algorithm can also be used as a preprocessor for other methods where break points/discontinuities must be incorporated explicitly....
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.
Discrete Fourier Analysis and Chebyshev Polynomials with G2 Group
Li, Huiyuan; Sun, Jiachang; Xu, Yuan
2012-10-01
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.
Planar Harmonic and Monogenic Polynomials of Type A
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Charles F. Dunkl
2016-10-01
Full Text Available Harmonic polynomials of type A are polynomials annihilated by the Dunkl Laplacian associated to the symmetric group acting as a reflection group on R N . The Dunkl operators are denoted by T j for 1 ≤ j ≤ N , and the Laplacian Δ κ = ∑ j = 1 N T j 2 . This paper finds the homogeneous harmonic polynomials annihilated by all T j for j > 2 . The structure constants with respect to the Gaussian and sphere inner products are computed. These harmonic polynomials are used to produce monogenic polynomials, those annihilated by a Dirac-type operator.
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.
Multivariate Krawtchouk Polynomials and Composition Birth and Death Processes
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Robert Griffiths
2016-05-01
Full Text Available This paper defines the multivariate Krawtchouk polynomials, orthogonal on the multinomial distribution, and summarizes their properties as a review. The multivariate Krawtchouk polynomials are symmetric functions of orthogonal sets of functions defined on each of N multinomial trials. The dual multivariate Krawtchouk polynomials, which also have a polynomial structure, are seen to occur naturally as spectral orthogonal polynomials in a Karlin and McGregor spectral representation of transition functions in a composition birth and death process. In this Markov composition process in continuous time, there are N independent and identically distributed birth and death processes each with support 0 , 1 , … . The state space in the composition process is the number of processes in the different states 0 , 1 , … . Dealing with the spectral representation requires new extensions of the multivariate Krawtchouk polynomials to orthogonal polynomials on a multinomial distribution with a countable infinity of states.
Predicting Physical Time Series Using Dynamic Ridge Polynomial Neural Networks
Al-Jumeily, Dhiya; Ghazali, Rozaida; Hussain, Abir
2014-01-01
Forecasting naturally occurring phenomena is a common problem in many domains of science, and this has been addressed and investigated by many scientists. The importance of time series prediction stems from the fact that it has wide range of applications, including control systems, engineering processes, environmental systems and economics. From the knowledge of some aspects of the previous behaviour of the system, the aim of the prediction process is to determine or predict its future behaviour. In this paper, we consider a novel application of a higher order polynomial neural network architecture called Dynamic Ridge Polynomial Neural Network that combines the properties of higher order and recurrent neural networks for the prediction of physical time series. In this study, four types of signals have been used, which are; The Lorenz attractor, mean value of the AE index, sunspot number, and heat wave temperature. The simulation results showed good improvements in terms of the signal to noise ratio in comparison to a number of higher order and feedforward neural networks in comparison to the benchmarked techniques. PMID:25157950
On the minimum of a polynomial function on a basic closed semialgebraic set and applications
DEFF Research Database (Denmark)
Jeronimo, Gabriela; Perrucci, Daniel; Tsigaridas, Elias
2013-01-01
We give an explicit upper bound for the algebraic degree and an explicit lower bound for the absolute value of the minimum of a polynomial function on a compact connected component of a basic closed semialgebraic set when this minimum is not zero. As an application, we obtain a lower bound...
On the minimum of a polynomial function on a basic closed semialgebraic set and applications
DEFF Research Database (Denmark)
Jeronimo, Gabriela; Perrucci, Daniel; Tsigaridas, Elias
We give an explicit upper bound for the algebraic degree and an explicit lower bound for the absolute value of the minimum of a polynomial function on a compact connected component of a basic closed semialgebraic set when this minimum is not zero. As an application, we obtain a lower bound...
Uniform approximations of Bernoulli and Euler polynomials in terms of hyperbolic functions
J.L. López; N.M. Temme (Nico)
1998-01-01
textabstractBernoulli and Euler polynomials are considered for large values of the order. Convergent expansions are obtained for $B_n(nz+1/2)$ and $E_n(nz+1/2)$ in powers of $n^{-1$, with coefficients being rational functions of $z$ and hyperbolic functions of argument $1/2z$. These expansions are
Results on the deficiencies of some differential-difference polynomials of meromorphic functions
Directory of Open Access Journals (Sweden)
Zheng Xiu-Min
2016-01-01
and c be a non-zero complex constant, then δ(∞, f(zm f(z+cf′(z≥δ(∞, f′ and δ(∞,f(z+cnf′(z≥ δ(∞, f′. We also investigate the value distribution of some differential-difference polynomials taking small function a(z with respect to f(z.
Numerical solutions of fifth order boundary value problems using ...
African Journals Online (AJOL)
Mamadu-Njoseh polynomials are polynomials constructed in the interval [-1,1] with respect to the weight function () = 2 + 1. This paper aims at applying these polynomials, as trial functions satisfying the boundary conditions, in a numerical approach for the solution of fifth order boundary value problems. For this, these ...
Mahmood, Zahid; Ning, Huansheng; Ghafoor, AtaUllah
2017-03-24
Wireless Sensor Networks (WSNs) consist of lightweight devices to measure sensitive data that are highly vulnerable to security attacks due to their constrained resources. In a similar manner, the internet-based lightweight devices used in the Internet of Things (IoT) are facing severe security and privacy issues because of the direct accessibility of devices due to their connection to the internet. Complex and resource-intensive security schemes are infeasible and reduce the network lifetime. In this regard, we have explored the polynomial distribution-based key establishment schemes and identified an issue that the resultant polynomial value is either storage intensive or infeasible when large values are multiplied. It becomes more costly when these polynomials are regenerated dynamically after each node join or leave operation and whenever key is refreshed. To reduce the computation, we have proposed an Efficient Key Management (EKM) scheme for multiparty communication-based scenarios. The proposed session key management protocol is established by applying a symmetric polynomial for group members, and the group head acts as a responsible node. The polynomial generation method uses security credentials and secure hash function. Symmetric cryptographic parameters are efficient in computation, communication, and the storage required. The security justification of the proposed scheme has been completed by using Rubin logic, which guarantees that the protocol attains mutual validation and session key agreement property strongly among the participating entities. Simulation scenarios are performed using NS 2.35 to validate the results for storage, communication, latency, energy, and polynomial calculation costs during authentication, session key generation, node migration, secure joining, and leaving phases. EKM is efficient regarding storage, computation, and communication overhead and can protect WSN-based IoT infrastructure.
Dynamics of iterative schemes for quadratic polynomial
Goyal, Komal; Prasad, Bhagwati
2017-10-01
Most of the fractals are generated by applying a map recursively to an initial point or a set of the space such as complex plane. An orbit of a point is a sequence of iteratively obtained points from it and the orbit is said to be diverging when its points grow unbounded. A set of all the points whose orbits are not diverging may be termed as a fractal. The fractal dynamics of the orbits depends on the rule of iteration also. In this paper, we study the dynamics of iterations for a number of iterative schemes. The results regarding the escape criteria for quadratic complex polynomials under various iteration procedures are established.
Integral inequalities for self-reciprocal polynomials
Indian Academy of Sciences (India)
Self-reciprocal polynomials. 133 where xj = (N − k)/(N − j) ∈ (0, 1) for j = 0, 1,...,r − 1. Let Qr be the expression on the right-hand side of (2.4). We have. Q1 = 2x2. 0 + 2 ≥ 2. And, if Qr ≥ 2, then. Qr+1 = (1 − xr)2Qr + 4xr r−1. ∏ j=0. (1 + xj )2. ≥ 2(1 − xr)2 + 4xr = 2 + 2x2 r ≥ 2. By induction, we conclude that (2.4) holds for all r ...
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...... two methods for learning MoP approximations of conditional densities from data. Both approaches are based on learning MoP approximations of the joint density and the marginal density of the conditioning variables, but they differ as to how the MoP approximation of the quotient of the two densities...
Orthogonal polynomial approximation in higher dimensions: Applications in astrodynamics
Bani Younes, Ahmad Hani Abd Alqader
We propose novel methods to utilize orthogonal polynomial approximation in higher dimension spaces, which enable us to modify classical differential equation solvers to perform high precision, long-term orbit propagation. These methods have immediate application to efficient propagation of catalogs of Resident Space Objects (RSOs) and improved accounting for the uncertainty in the ephemeris of these objects. More fundamentally, the methodology promises to be of broad utility in solving initial and two point boundary value problems from a wide class of mathematical representations of problems arising in engineering, optimal control, physical sciences and applied mathematics. We unify and extend classical results from function approximation theory and consider their utility in astrodynamics. Least square approximation, using the classical Chebyshev polynomials as basis functions, is reviewed for discrete samples of the to-be-approximated function. We extend the orthogonal approximation ideas to n-dimensions in a novel way, through the use of array algebra and Kronecker operations. Approximation of test functions illustrates the resulting algorithms and provides insight into the errors of approximation, as well as the associated errors arising when the approximations are differentiated or integrated. Two sets of applications are considered that are challenges in astrodynamics. The first application addresses local approximation of high degree and order geopotential models, replacing the global spherical harmonic series by a family of locally precise orthogonal polynomial approximations for efficient computation. A method is introduced which adapts the approximation degree radially, compatible with the truth that the highest degree approximations (to ensure maximum acceleration error < 10-9 ms-2, globally) are required near the Earths surface, whereas lower degree approximations are required as radius increases. We show that a four order of magnitude speedup is
QAPV: a polynomial invariant for graph isomorphism testing
Directory of Open Access Journals (Sweden)
Valdir Agustinho de Melo
2013-08-01
Full Text Available To each instance of the Quadratic Assignment Problem (QAP a relaxed instance can be associated. Both variances of their solution values can be calculated in polynomial time. The graph isomorphism problem (GIP can be modeled as a QAP, associating its pair of data matrices with a pair of graphs of the same order and size. We look for invariant edge weight functions for the graphs composing the instances in order to try to find quantitative differences between variances that could be associated with the absence of isomorphism. This technique is sensitive enough to show the effect of a single edge exchange between two regular graphs of up to 3,000 vertices and 300,000 edges with degrees up to 200. Planar graph pairs from a dense family up to 300,000 vertices were also discriminated. We conjecture the existence of functions able to discriminate non-isomorphic pairs for every instance of the problem.
Zernike expansion of derivatives and Laplacians of the Zernike circle polynomials.
Janssen, A J E M
2014-07-01
The partial derivatives and Laplacians of the Zernike circle polynomials occur in various places in the literature on computational optics. In a number of cases, the expansion of these derivatives and Laplacians in the circle polynomials are required. For the first-order partial derivatives, analytic results are scattered in the literature. Results start as early as 1942 in Nijboer's thesis and continue until present day, with some emphasis on recursive computation schemes. A brief historic account of these results is given in the present paper. By choosing the unnormalized version of the circle polynomials, with exponential rather than trigonometric azimuthal dependence, and by a proper combination of the two partial derivatives, a concise form of the expressions emerges. This form is appropriate for the formulation and solution of a model wavefront sensing problem of reconstructing a wavefront on the level of its expansion coefficients from (measurements of the expansion coefficients of) the partial derivatives. It turns out that the least-squares estimation problem arising here decouples per azimuthal order m, and per m the generalized inverse solution assumes a concise analytic form so that singular value decompositions are avoided. The preferred version of the circle polynomials, with proper combination of the partial derivatives, also leads to a concise analytic result for the Zernike expansion of the Laplacian of the circle polynomials. From these expansions, the properties of the Laplacian as a mapping from the space of circle polynomials of maximal degree N, as required in the study of the Neumann problem associated with the transport-of-intensity equation, can be read off within a single glance. Furthermore, the inverse of the Laplacian on this space is shown to have a concise analytic form.
Chkifa, Abdellah
2015-04-08
Motivated by the numerical treatment of parametric and stochastic PDEs, we analyze the least-squares method for polynomial approximation of multivariate functions based on random sampling according to a given probability measure. Recent work has shown that in the univariate case, the least-squares method is quasi-optimal in expectation in [A. Cohen, M A. Davenport and D. Leviatan. Found. Comput. Math. 13 (2013) 819–834] and in probability in [G. Migliorati, F. Nobile, E. von Schwerin, R. Tempone, Found. Comput. Math. 14 (2014) 419–456], under suitable conditions that relate the number of samples with respect to the dimension of the polynomial space. Here “quasi-optimal” means that the accuracy of the least-squares approximation is comparable with that of the best approximation in the given polynomial space. In this paper, we discuss the quasi-optimality of the polynomial least-squares method in arbitrary dimension. Our analysis applies to any arbitrary multivariate polynomial space (including tensor product, total degree or hyperbolic crosses), under the minimal requirement that its associated index set is downward closed. The optimality criterion only involves the relation between the number of samples and the dimension of the polynomial space, independently of the anisotropic shape and of the number of variables. We extend our results to the approximation of Hilbert space-valued functions in order to apply them to the approximation of parametric and stochastic elliptic PDEs. As a particular case, we discuss “inclusion type” elliptic PDE models, and derive an exponential convergence estimate for the least-squares method. Numerical results confirm our estimate, yet pointing out a gap between the condition necessary to achieve optimality in the theory, and the condition that in practice yields the optimal convergence rate.
Positive trigonometric polynomials and signal processing applications
Dumitrescu, Bogdan
2017-01-01
This revised edition is made up of two parts: theory and applications. Though many of the fundamental results are still valid and used, new and revised material is woven throughout the text. As with the original book, the theory of sum-of-squares trigonometric polynomials is presented unitarily based on the concept of Gram matrix (extended to Gram pair or Gram set). The programming environment has also evolved, and the books examples are changed accordingly. The applications section is organized as a collection of related problems that use systematically the theoretical results. All the problems are brought to a semi-definite programming form, ready to be solved with algorithms freely available, like those from the libraries SeDuMi, CVX and Pos3Poly. A new chapter discusses applications in super-resolution theory, where Bounded Real Lemma for trigonometric polynomials is an important tool. This revision is written to be more appealing and easier to use for new readers. < Features updated information on LMI...
On factorization of generalized Macdonald polynomials
Energy Technology Data Exchange (ETDEWEB)
Kononov, Ya. [Landau Institute for Theoretical Physics, Chernogolovka (Russian Federation); HSE, Math Department, Moscow (Russian Federation); Morozov, A. [ITEP, Moscow (Russian Federation); Institute for Information Transmission Problems, Moscow (Russian Federation); National Research Nuclear University MEPhI, Moscow (Russian Federation)
2016-08-15
A remarkable feature of Schur functions - the common eigenfunctions of cut-and-join operators from W{sub ∞} - 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{sub q}(SL{sub 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. (orig.)
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 ...... study illustrates the performance of the proposed estimator compared to the classical local Whittle estimator and the local polynomial Whittle estimator. The empirical justi.cation of the proposed estimator is shown through an analysis of credit spreads....
Structural Reliability Analysis Using Orthogonalizable Power Polynomial Basis Vector
Li Yejun; Huang Bin
2017-01-01
A new method for structural reliability analysis using orthogonalizable power polynomial basis vector is presented. Firstly, a power polynomial basis vector is adopted to express the initial series solution of structural response, which is determined by a series of deterministic recursive equation based on perturbation technique, and then transferred to be a set of orthogonalizable power polynomial basis vector using the orthogonalization technique. By conducting Garlekin projection, an accel...
On Roots of Polynomials and Algebraically Closed Fields
Directory of Open Access Journals (Sweden)
Schwarzweller Christoph
2017-10-01
Full Text Available In this article we further extend the algebraic theory of polynomial rings in Mizar [1, 2, 3]. We deal with roots and multiple roots of polynomials and show that both the real numbers and finite domains are not algebraically closed [5, 7]. We also prove the identity theorem for polynomials and that the number of multiple roots is bounded by the polynomial’s degree [4, 6].
Representations of the Schroedinger group and matrix orthogonal polynomials
Energy Technology Data Exchange (ETDEWEB)
Vinet, Luc [Centre de recherches mathematiques, Universite de Montreal, CP 6128, succ. Centre-ville, Montreal, QC H3C 3J7 (Canada); Zhedanov, Alexei, E-mail: luc.vinet@umontreal.ca, E-mail: zhedanov@fti.dn.ua [Donetsk Institute for Physics and Technology, Donetsk 83114 (Ukraine)
2011-09-02
The representations of the Schroedinger group in one space dimension are explicitly constructed in the basis of the harmonic oscillator states. These representations are seen to involve matrix orthogonal polynomials in a discrete variable that have Charlier and Meixner polynomials as building blocks. The underlying Lie-theoretic framework allows for a systematic derivation of the structural formulas (recurrence relations, difference equations, Rodrigues' formula, etc) that these matrix orthogonal polynomials satisfy. (paper)
Applications of polynomial optimization in financial risk investment
Zeng, Meilan; Fu, Hongwei
2017-09-01
Recently, polynomial optimization has many important applications in optimization, financial economics and eigenvalues of tensor, etc. This paper studies the applications of polynomial optimization in financial risk investment. We consider the standard mean-variance risk measurement model and the mean-variance risk measurement model with transaction costs. We use Lasserre's hierarchy of semidefinite programming (SDP) relaxations to solve the specific cases. The results show that polynomial optimization is effective for some financial optimization problems.
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.
Using Tutte polynomials to characterize sexual contact networks
Cadavid Muñoz, Juan José
2014-06-01
Tutte polynomials are used to characterize the dynamic and topology of the sexual contact networks, in which pathogens are transmitted as an epidemic. Tutte polynomials provide an algebraic characterization of the sexual contact networks and allow the projection of spread control strategies for sexual transmission diseases. With the usage of Tutte polynomials, it allows obtaining algebraic expressions for the basic reproductive number of different pathogenic agents. Computations are done using the computer algebra software Maple, and it's GraphTheory Package. The topological complexity of a contact network is represented by the algebraic complexity of the correspondent polynomial. The change in the topology of the contact network is represented as a change in the algebraic form of the associated polynomial. With the usage of the Tutte polynomials, the number of spanning trees for each contact network can be obtained. From the obtained results in the polynomial form, it can be said that Tutte polynomials are of great importance for designing and implementing control measures for slowing down the propagation of sexual transmitted pathologies. As a future research line, the analysis of weighted sexual contact networks using weighted Tutte polynomials is considered.
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.
"Stuff From Back Home": Enda Walsh's The Walworth FarceDOI:10.5007/2175-8026.2010n58p333
Directory of Open Access Journals (Sweden)
Eamonn Jordan
2010-01-01
Full Text Available Since its first performance in 2006 by Druid Theatre Company, Enda Walsh's award-winning The Walworth Farce has toured Ireland, Britain, America, Canada, New Zealand and Australia to great acclaim, with the brilliance of the directing, design and acting engaging with the intelligence and theatricality of Walsh's script. The play deals with a family, who as part of a daily enforced ritual, re-enact a farce, written and directed by Dinny, the patriarch, a story which accounts for their exile in London, and away from their family home in Cork. Their enactment is an attempt to create a false memory, for their performance is very much at odds with the real events which provoked their exile.
Solution of inverse heat conduction equation with the use of Chebyshev polynomials
Directory of Open Access Journals (Sweden)
Joachimiak Magda
2016-12-01
Full Text Available A direct problem and an inverse problem for the Laplace’s equation was solved in this paper. Solution to the direct problem in a rectangle was sought in a form of finite linear combinations of Chebyshev polynomials. Calculations were made for a grid consisting of Chebyshev nodes, what allows us to use orthogonal properties of Chebyshev polynomials. Temperature distributions on the boundary for the inverse problem were determined using minimization of the functional being the measure of the difference between the measured and calculated values of temperature (boundary inverse problem. For the quasi-Cauchy problem, the distance between set values of temperature and heat flux on the boundary was minimized using the least square method. Influence of the value of random disturbance to the temperature measurement, of measurement points (distance from the boundary, where the temperature is not known arrangement as well as of the thermocouple installation error on the stability of the inverse problem was analyzed.
Polynomial description of inhomogeneous topological superconducting wires
Pérez, Marcos; Martínez, Gerardo
2017-11-01
We present the universal features of the topological invariant for p-wave superconducting wires after the inclusion of spatial inhomogeneities. Three classes of distributed potentials are studied, a single-defect, a commensurate and an incommensurate model, using periodic site modulations. An analytic polynomial description is achieved by splitting the topological invariant into two parts; one part depends on the chemical potential and the other does not. For the homogeneous case, an elliptical region is found where the topological invariant oscillates. The zeros of these oscillations occur at points where the fermion parity switches for finite wires. The increase of these oscillations with the inhomogeneity strength leads to new isolated non-topological phases. We characterize these new phases according to each class of spatial distributions. Such phases could also be observed in the XY model, to which our model is dual.
Polynomial Apodizers for Centrally Obscured Vortex Coronagraphs
Fogarty, Kevin; Pueyo, Laurent; Mazoyer, Johan; N'Diaye, Mamadou
2017-12-01
Several coronagraph designs have been proposed over the last two decades to directly image exoplanets. Among these designs, vector vortex coronagraphs provide theoretically perfect starlight cancellation along with small inner working angles when deployed on telescopes with unobstructed pupils. However, current and planned space missions and ground-based extremely large telescopes present complex pupil geometries, including large central obscurations caused by secondary mirrors, which prevent vortex coronagraphs from rejecting on-axis sources entirely. Recent solutions combining the vortex phase mask with a ring-apodized pupil have been proposed to circumvent this issue, but provide a limited throughput for vortex charges > 2. We present pupil plane apodizations for charge 2, 4, and 6 vector vortex coronagraphs that compensate for pupil geometries with circularly symmetric central obstructions caused by on-axis secondary mirrors. These apodizations are derived analytically and allow vortex coronagraphs to retain theoretically perfect nulling in the presence of obstructed pupils. For a charge 4 vortex, we design polynomial apodization functions assuming a grayscale apodizing filter that represent a substantial gain in throughput over the ring-apodized vortex coronagraph design, while for a charge 6 vortex, we design polynomial apodized vortex coronagraphs that have ≳ 70 % total energy throughput for the entire range of central obscuration sizes studied. We propose methods for optimizing apodizations produced with either grayscale apodizing filters or shaped mirrors. We conclude by demonstrating how this design may be combined with apodizations numerically optimized for struts and primary mirror segment gaps to design terrestrial exoplanet imagers for complex pupils.
Geometry of polynomials and root-finding via path-lifting
Kim, Myong-Hi; Martens, Marco; Sutherland, Scott
2018-02-01
Using the interplay between topological, combinatorial, and geometric properties of polynomials and analytic results (primarily the covering structure and distortion estimates), we analyze a path-lifting method for finding approximate zeros, similar to those studied by Smale, Shub, Kim, and others. Given any polynomial, this simple algorithm always converges to a root, except on a finite set of initial points lying on a circle of a given radius. Specifically, the algorithm we analyze consists of iterating where the t k form a decreasing sequence of real numbers and z 0 is chosen on a circle containing all the roots. We show that the number of iterates required to locate an approximate zero of a polynomial f depends only on log\\vert f(z_0)/ρ_\\zeta\\vert (where ρ_\\zeta is the radius of convergence of the branch of f-1 taking 0 to a root ζ) and the logarithm of the angle between f(z_0) and certain critical values. Previous complexity results for related algorithms depend linearly on the reciprocals of these angles. Note that the complexity of the algorithm does not depend directly on the degree of f, but only on the geometry of the critical values. Furthermore, for any polynomial f with distinct roots, the average number of steps required over all starting points taken on a circle containing all the roots is bounded by a constant times the average of log(1/ρ_\\zeta) . The average of log(1/ρ_\\zeta) over all polynomials f with d roots in the unit disk is \
Mathematical Use Of Polynomials Of Different End Periods Of ...
African Journals Online (AJOL)
This will tremendously reduce the danger of information intercept. The study discover how well to protect secret information through polynomials of different end periods of random numbers. Keywords: polynomials, end periods, random numbers, encryption, decryption, message. Journal of Technology and Education in ...
Polynomial expansions for solution of wave equation in quantum calculus
Directory of Open Access Journals (Sweden)
Akram Nemri
2010-12-01
Full Text Available In this paper, using the q^2 -Laplace transform early introduced by Abdi [1], we study q-Wave polynomials related with the q-difference operator ∆q,x . We show in particular that they are linked to the q-little Jacobi polynomials p_n (x; α, β | q^2 .
Optimum short-time polynomial regression for signal analysis
Indian Academy of Sciences (India)
A Sreenivasa Murthy
Taylor series representation, according to which smooth functions can be expressed point-wise as polynomials ... in the two approaches is that the Taylor series representation is a point-wise polynomial approximation, ...... In the literature, a dyadic set of windows has been used. [8, 9, 23], but by choosing a set of arithmetic ...
Optimum short-time polynomial regression for signal analysis
Indian Academy of Sciences (India)
We propose a short-time polynomial regression (STPR) for time-varying signal analysis. The advantage of using polynomials is that the notion of a spectrum is not needed and the signals can be analyzed in the time domain over short durations. In the presence of noise, such modeling becomes important, because the ...
Studying the multivariable Alexander polynomial by means of Seifert surfaces
Cimasoni, David
2004-01-01
We show how Seifert surfaces, so useful for the understanding of the Alexander polynomial \\Delta_L(t), can be generalized in order to study the multivariable Alexander polynomial \\Delta_L(t_1,...,t_\\mu). In particular, we give an elementary and geometric proof of the Torres formula.
On Period of the Sequence of Fibonacci Polynomials Modulo
Directory of Open Access Journals (Sweden)
İnci Gültekin
2013-01-01
Full Text Available It is shown that the sequence obtained by reducing modulo coefficient and exponent of each Fibonacci polynomials term is periodic. Also if is prime, then sequences of Fibonacci polynomial are compared with Wall numbers of Fibonacci sequences according to modulo . It is found that order of cyclic group generated with matrix is equal to the period of these sequences.
The truncated exponential polynomials, the associated Hermite forms and applications
Dattoli, G.; Migliorati, M.
2006-01-01
We discuss the properties of the truncated exponential polynomials and develop the theory of new form of Hermite polynomials, which can be constructed using the truncated exponential as a generating function. We derive their explicit forms and comment on their usefulness in applications, with particular reference to the theory of flattened beams, used in optics.
A reduced polynomial chaos expansion method for the stochastic ...
Indian Academy of Sciences (India)
a basis of polynomials orthogonal with respect to a given pdf, and the associated vectors of coef- ficients multiplying each polynomial are obtained using a Galerkin type of error minimization approach. Here an alternative approach is investigated. The solution is firstly projected into a finite dimensional orthonormal vector ...
2001-09-01
diagnosis natural language understanding circuit fault diagnosis pattern recognition machine vision nancial auditing map learning sensor...de nition of piecewise polynomial function shown in section provides the foundation for this type of approximation Schumaker proved... ACCA ACCB A ights degree of command and control FCC value is assumed to be the average of all the ACC values of the aircraft in the
Directory of Open Access Journals (Sweden)
Xiang Shen
2017-02-01
Full Text Available The Rational Function Model (RFM is a widely used generic sensor model for georeferencing satellite images. Owing to inaccurate measurement of satellite orbit and attitude, the Rational Polynomial Coefficients (RPCs provided by image vendors are commonly biased and cannot be directly used for high-precision remote-sensing applications. In this paper, we propose a new method for the bias compensation of RPCs using local polynomial models (including the local affine model and the local quadratic model, which provides the ability to correct non-rigid RPC deformations. Performance of the proposed approach was evaluated using a stereo triplet of ZY-3 satellite images and compared with conventional global-polynomial-based models (including the global affine model and the global quadratic model. The experimental results show that, when the same polynomial form was used, the correction residuals of the local model could be notably smaller than those of the global model, which indicates that the new method has great ability to remove complex errors existed in vendor-provided RPCs. In the experiments of this study, the accuracy of the local affine model was nearly 15% better than that of the global affine model. Performance of the local quadratic model was not as good as the local affine model when the number of Ground Control Points (GCPs was less than 10, but it improved rapidly with an increase in the number of redundant observations. In the test scenario with 15 GCPs, the accuracy of the local quadratic model was about 9% and 27% better than those of the local affine model and the global quadratic model, respectively.
Learning Read-constant Polynomials of Constant Degree modulo Composites
DEFF Research Database (Denmark)
Chattopadhyay, Arkadev; Gavaldá, Richard; Hansen, Kristoffer Arnsfelt
2011-01-01
Boolean functions that have constant degree polynomial representation over a fixed finite ring form a natural and strict subclass of the complexity class \\textACC0ACC0. They are also precisely the functions computable efficiently by programs over fixed and finite nilpotent groups. This class...... is not known to be learnable in any reasonable learning model. In this paper, we provide a deterministic polynomial time algorithm for learning Boolean functions represented by polynomials of constant degree over arbitrary finite rings from membership queries, with the additional constraint that each variable...... in the target polynomial appears in a constant number of monomials. Our algorithm extends to superconstant but low degree polynomials and still runs in quasipolynomial time....
77 FR 33790 - Hennion & Walsh, Inc. and Smart Trust; Notice of Application
2012-06-07
... and conditions of the application. 2. Applicants request relief to permit a Series to invest in... and Closed-end Funds will be deposited in a Series at prices which are based on the market value of... relief will provide investors with a practical, cost-efficient means of investing in a professionally...
Harmonic sums and polylogarithms generated by cyclotomic polynomials
Energy Technology Data Exchange (ETDEWEB)
Ablinger, Jakob; Schneider, Carsten [Johannes Kepler Univ., Linz (Austria). Research Inst. for Symbolic Computation; Bluemlein, Johannes [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany)
2011-05-15
The computation of Feynman integrals in massive higher order perturbative calculations in renormalizable Quantum Field Theories requires extensions of multiply nested harmonic sums, which can be generated as real representations by Mellin transforms of Poincare-iterated integrals including denominators of higher cyclotomic polynomials. We derive the cyclotomic harmonic polylogarithms and harmonic sums and study their algebraic and structural relations. The analytic continuation of cyclotomic harmonic sums to complex values of N is performed using analytic representations. We also consider special values of the cyclotomic harmonic polylogarithms at argument x=1, resp., for the cyclotomic harmonic sums at N{yields}{infinity}, which are related to colored multiple zeta values, deriving various of their relations, based on the stuffle and shuffle algebras and three multiple argument relations. We also consider infinite generalized nested harmonic sums at roots of unity which are related to the infinite cyclotomic harmonic sums. Basis representations are derived for weight w=1,2 sums up to cyclotomy l=20. (orig.)
An extended class of orthogonal polynomials defined by a Sturm–Liouville problem
National Research Council Canada - National Science Library
Gómez-Ullate, David; Kamran, Niky; Milson, Robert
2009-01-01
... eigenfunctions of a SturmâLiouville problem. As opposed to the classical orthogonal polynomial systems, these sequences start with a polynomial of degree one. We denote these polynomials as X 1 -...
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
Van Assche, W.; Yáñez, R. J.; Dehesa, J. S.
1995-08-01
The information entropy of the harmonic oscillator potential V(x)=1/2λx2 in both position and momentum spaces can be expressed in terms of the so-called ``entropy of Hermite polynomials,'' i.e., the quantity Sn(H):= -∫-∞+∞H2n(x)log H2n(x) e-x2dx. These polynomials are instances of the polynomials orthogonal with respect to the Freud weights w(x)=exp(-||x||m), m≳0. Here, a very precise and general result of the entropy of Freud polynomials recently established by Aptekarev et al. [J. Math. Phys. 35, 4423-4428 (1994)], specialized to the Hermite kernel (case m=2), leads to an important refined asymptotic expression for the information entropies of very excited states (i.e., for large n) in both position and momentum spaces, to be denoted by Sρ and Sγ, respectively. Briefly, it is shown that, for large values of n, Sρ+1/2logλ≂log(π√2n/e)+o(1) and Sγ-1/2log λ≂log(π√2n/e)+o(1), so that Sρ+Sγ≂log(2π2n/e2)+o(1) in agreement with the generalized indetermination relation of Byalinicki-Birula and Mycielski [Commun. Math. Phys. 44, 129-132 (1975)]. Finally, the rate of convergence of these two information entropies is numerically analyzed. In addition, using a Rakhmanov result, we describe a totally new proof of the leading term of the entropy of Freud polynomials which, naturally, is just a weak version of the aforementioned general result.
Papadopoulos, Anthony
2009-01-01
The first-degree power-law polynomial function is frequently used to describe activity metabolism for steady swimming animals. This function has been used in hydrodynamics-based metabolic studies to evaluate important parameters of energetic costs, such as the standard metabolic rate and the drag power indices. In theory, however, the power-law polynomial function of any degree greater than one can be used to describe activity metabolism for steady swimming animals. In fact, activity metabolism has been described by the conventional exponential function and the cubic polynomial function, although only the power-law polynomial function models drag power since it conforms to hydrodynamic laws. Consequently, the first-degree power-law polynomial function yields incorrect parameter values of energetic costs if activity metabolism is governed by the power-law polynomial function of any degree greater than one. This issue is important in bioenergetics because correct comparisons of energetic costs among different steady swimming animals cannot be made unless the degree of the power-law polynomial function derives from activity metabolism. In other words, a hydrodynamics-based functional form of activity metabolism is a power-law polynomial function of any degree greater than or equal to one. Therefore, the degree of the power-law polynomial function should be treated as a parameter, not as a constant. This new treatment not only conforms to hydrodynamic laws, but also ensures correct comparisons of energetic costs among different steady swimming animals. Furthermore, the exponential power-law function, which is a new hydrodynamics-based functional form of activity metabolism, is a special case of the power-law polynomial function. Hence, the link between the hydrodynamics of steady swimming and the exponential-based metabolic model is defined.
Directory of Open Access Journals (Sweden)
Anthony Papadopoulos
Full Text Available The first-degree power-law polynomial function is frequently used to describe activity metabolism for steady swimming animals. This function has been used in hydrodynamics-based metabolic studies to evaluate important parameters of energetic costs, such as the standard metabolic rate and the drag power indices. In theory, however, the power-law polynomial function of any degree greater than one can be used to describe activity metabolism for steady swimming animals. In fact, activity metabolism has been described by the conventional exponential function and the cubic polynomial function, although only the power-law polynomial function models drag power since it conforms to hydrodynamic laws. Consequently, the first-degree power-law polynomial function yields incorrect parameter values of energetic costs if activity metabolism is governed by the power-law polynomial function of any degree greater than one. This issue is important in bioenergetics because correct comparisons of energetic costs among different steady swimming animals cannot be made unless the degree of the power-law polynomial function derives from activity metabolism. In other words, a hydrodynamics-based functional form of activity metabolism is a power-law polynomial function of any degree greater than or equal to one. Therefore, the degree of the power-law polynomial function should be treated as a parameter, not as a constant. This new treatment not only conforms to hydrodynamic laws, but also ensures correct comparisons of energetic costs among different steady swimming animals. Furthermore, the exponential power-law function, which is a new hydrodynamics-based functional form of activity metabolism, is a special case of the power-law polynomial function. Hence, the link between the hydrodynamics of steady swimming and the exponential-based metabolic model is defined.
Constructing general partial differential equations using polynomial and neural networks.
Zjavka, Ladislav; Pedrycz, Witold
2016-01-01
Sum fraction terms can approximate multi-variable functions on the basis of discrete observations, replacing a partial differential equation definition with polynomial elementary data relation descriptions. Artificial neural networks commonly transform the weighted sum of inputs to describe overall similarity relationships of trained and new testing input patterns. Differential polynomial neural networks form a new class of neural networks, which construct and solve an unknown general partial differential equation of a function of interest with selected substitution relative terms using non-linear multi-variable composite polynomials. The layers of the network generate simple and composite relative substitution terms whose convergent series combinations can describe partial dependent derivative changes of the input variables. This regression is based on trained generalized partial derivative data relations, decomposed into a multi-layer polynomial network structure. The sigmoidal function, commonly used as a nonlinear activation of artificial neurons, may transform some polynomial items together with the parameters with the aim to improve the polynomial derivative term series ability to approximate complicated periodic functions, as simple low order polynomials are not able to fully make up for the complete cycles. The similarity analysis facilitates substitutions for differential equations or can form dimensional units from data samples to describe real-world problems. Copyright © 2015 Elsevier Ltd. All rights reserved.
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.
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
Polynomials in Control Theory Parametrized by Their Roots
Directory of Open Access Journals (Sweden)
Baltazar Aguirre-Hernández
2012-01-01
Full Text Available The aim of this paper is to introduce the space of roots to study the topological properties of the spaces of polynomials. Instead of identifying a monic complex polynomial with the vector of its coefficients, we identify it with the set of its roots. Viète's map gives a homeomorphism between the space of roots and the space of coefficients and it gives an explicit formula to relate both spaces. Using this viewpoint we establish that the space of monic (Schur or Hurwitz aperiodic polynomials is contractible. Additionally we obtain a Boundary Theorem.
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.
Asymptotic distribution of zeros of polynomials satisfying difference equations
Krasovsky, I. V.
2003-01-01
We propose a way to find the asymptotic distribution of zeros of orthogonal polynomials pn(x) satisfying a difference equation of the formB(x)pn(x+[delta])-C(x,n)pn(x)+D(x)pn(x-[delta])=0.We calculate the asymptotic distribution of zeros and asymptotics of extreme zeros of the Meixner and Meixner-Pollaczek polynomials. The distribution of zeros of Meixner polynomials shows some delicate features. We indicate the relation of our technique to the approach based on the Nevai-Dehesa-Ullman distribution.
Chromatic Polynomials Of Some (m,l-Hyperwheels
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Julian A. Allagan
2014-03-01
Full Text Available In this paper, using a standard method of computing the chromatic polynomial of hypergraphs, we introduce a new reduction theorem which allows us to find explicit formulae for the chromatic polynomials of some (complete non-uniform $(m,l-$hyperwheels and non-uniform $(m,l-$hyperfans. These hypergraphs, constructed through a ``join" graph operation, are some generalizations of the well-known wheel and fan graphs, respectively. Further, we revisit some results concerning these graphs and present their chromatic polynomials in a standard form that involves the Stirling numbers of the second kind.
On an Ordering-Dependent Generalization of the Tutte Polynomial
Geloun, Joseph Ben; Caravelli, Francesco
2017-09-01
A generalization of the Tutte polynomial involved in the evaluation of the moments of the integrated geometric Brownian in the Itô formalism is discussed. The new combinatorial invariant depends on the order in which the sequence of contraction-deletions have been performed on the graph. Thus, this work provides a motivation for studying an order-dependent Tutte polynomial in the context of stochastic differential equations. We show that in the limit of the control parameters encoding the ordering going to zero, the multivariate Tutte-Fortuin-Kasteleyn polynomial is recovered.
Directory of Open Access Journals (Sweden)
Darío Dawyd
2012-06-01
Full Text Available En el presente artículo se reconstruye la investigación que Rodolfo Walsh realizó sobre el enfrentamiento que costó la vida de Rosendo García, Domingo Blajaquis y Juan Zalazar y que publicó en siete notas en el semanario CGT entre el 16 de mayo y el 27 de junio de 1968. Aquellas notas son contrastadas con su publicación posterior en formato de libro como ¿Quien mató a Rosendo? en mayo de 1969 y se busca destacar las diferencias entre ambos formatos y contextos. Estas diferencias permiten relevar el contexto político acelerado en que se produjeron y la gran distancia entre un momento y otro a pesar de que transcurrió solo un año entre ambos. Con ello se pretende revisitar una de las investigaciones menos abordadas de Walsh y ponerla en contexto al interior de una etapa político-sindical poco estudiada, especialmente los significados de la división de la CGT y el peronismo, en tanto Walsh había retomado su militancia política y estrechado sus primeros vínculos con una de las fracciones del peronismo en disputa.In the present article we reconstructed Rodolfo Walsh's research about the death of Rosendo García, Domingo Blajaquis and Juan Zalazar that he published in seven notes in the weekly CGT between May and June in 1968. Those notes are compare with his later publication of the research as a the book ¿Quien mató a Rosendo? in May, 1969. We try to emphasize the differences between both formats and contexts because those differences allow as to relieve the critical political context in which they took place. The article underline of the worst approached investigations of Walsh and put it in its context of the meanings of the division of the CGT and the Peronism, considering that Walsh had taken again his political militancy and had make his first links with one of the fractions of the Peronism in dispute.
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....
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 Ising, Potts, and Fortuin–Kasteleyn in statistical physics. Prior to this work, deletion–contraction was also the fastest known general-purpose algorithm for these invariants, running in time roughly proportional to the number of spanning trees in the input graph.Here, we give a substantially faster...
Dynamics of quadratic polynomials: Complex bounds for real maps
Lyubich, Mikhail; Yampolsky, Michael
1995-01-01
We extend Sullivan's complex a priori bounds to real quadratic polynomials with essentially bounded combinatorics. Combined with the previous results of the first author, this yields complex bounds for all real quadratics. Local connectivity of the corresponding Julia sets follows.
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.
Gravity Anomaly of Polyhedral Bodies Having a Polynomial Density Contrast
D'Urso, M. G.; Trotta, S.
2017-07-01
We analytically evaluate the gravity anomaly associated with a polyhedral body having an arbitrary geometrical shape and a polynomial density contrast in both the horizontal and vertical directions. The gravity anomaly is evaluated at an arbitrary point that does not necessarily coincide with the origin of the reference frame in which the density function is assigned. Density contrast is assumed to be a third-order polynomial as a maximum but the general approach exploited in the paper can be easily extended to higher-order polynomial functions. Invoking recent results of potential theory, the solution derived in the paper is shown to be singularity-free and is expressed as a sum of algebraic quantities that only depend upon the 3D coordinates of the polyhedron vertices and upon the polynomial density function. The accuracy, robustness and effectiveness of the proposed approach are illustrated by numerical comparisons with examples derived from the existing literature.
Skew-orthogonal polynomials and random matrix theory
Ghosh, Saugata
2009-01-01
Orthogonal polynomials satisfy a three-term recursion relation irrespective of the weight function with respect to which they are defined. This gives a simple formula for the kernel function, known in the literature as the Christoffel-Darboux sum. The availability of asymptotic results of orthogonal polynomials and the simple structure of the Christoffel-Darboux sum make the study of unitary ensembles of random matrices relatively straightforward. In this book, the author develops the theory of skew-orthogonal polynomials and obtains recursion relations which, unlike orthogonal polynomials, depend on weight functions. After deriving reduced expressions, called the generalized Christoffel-Darboux formulas (GCD), he obtains universal correlation functions and non-universal level densities for a wide class of random matrix ensembles using the GCD. The author also shows that once questions about higher order effects are considered (questions that are relevant in different branches of physics and mathematics) the ...
Structural Reliability Analysis Using Orthogonalizable Power Polynomial Basis Vector
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Li Yejun
2017-01-01
Full Text Available A new method for structural reliability analysis using orthogonalizable power polynomial basis vector is presented. Firstly, a power polynomial basis vector is adopted to express the initial series solution of structural response, which is determined by a series of deterministic recursive equation based on perturbation technique, and then transferred to be a set of orthogonalizable power polynomial basis vector using the orthogonalization technique. By conducting Garlekin projection, an accelerating factor vector of the orthogonalizable power polynomial expansion is determined by solving small scale algebraic equations. Numerical results of a continuous bridge structure on reliability analysis shows that the proposed method can achieve the accuracy of the Direct Monte Carlo method and can save a lot of computation time at the same time, it is both accurate and efficient, and is very competitive to be used in structural reliability analysis.
Numerical Simulation of Polynomial-Speed Convergence Phenomenon
Li, Yao; Xu, Hui
2017-10-01
We provide a hybrid method that captures the polynomial speed of convergence and polynomial speed of mixing for Markov processes. The hybrid method that we introduce is based on the coupling technique and renewal theory. We propose to replace some estimates in classical results about the ergodicity of Markov processes by numerical simulations when the corresponding analytical proof is difficult. After that, all remaining conclusions can be derived from rigorous analysis. Then we apply our results to seek numerical justification for the ergodicity of two 1D microscopic heat conduction models. The mixing rate of these two models are expected to be polynomial but very difficult to prove. In both examples, our numerical results match the expected polynomial mixing rate well.
Classification of complex polynomial vector fields in one complex variable
DEFF Research Database (Denmark)
Branner, Bodil; Dias, Kealey
2010-01-01
This paper classifies the global structure of monic and centred one-variable complex polynomial vector fields. The classification is achieved by means of combinatorial and analytic data. More specifically, given a polynomial vector field, we construct a combinatorial invariant, describing...... the topology, and a set of analytic invariants, describing the geometry. Conversely, given admissible combinatorial and analytic data sets, we show using surgery the existence of a unique monic and centred polynomial vector field realizing the given invariants. This is the content of the Structure Theorem......, the main result of the paper. This result is an extension and refinement of Douady et al. (Champs de vecteurs polynomiaux sur C. Unpublished manuscript) classification of the structurally stable polynomial vector fields. We further review some general concepts for completeness and show that vector fields...
High-accuracy polynomial solutions of the classical Stefan problem
Kot, V. A.
2017-09-01
High-accuracy polynomial solutions of the Stefan problem for a semi-infinite medium with Dirichlet and Neumann boundary conditions and boundary conditions of general form are presented. The initial temperature of the medium was assumed to be equal to its phase-transition temperature. With the use of the integral method of boundary characteristics, based on the multiple integration of the heat-conduction equation, sequences of identical equalities with different boundary conditions were obtained. On the basis of these equalities, polynomial solutions of different degrees were constructed. High efficiency of the approach proposed was demonstrated by different examples. The polynomial solutions of the second and third degrees surpass in approximation accuracy the analogous known solutions. The accuracy of the calculations of the interphase boundary with the use of the fourth- and fifth-degree polynomials is higher by several orders of magnitude then that of numerical methods.
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.
Guts of surfaces and the colored Jones polynomial
Futer, David; Purcell, Jessica
2013-01-01
This monograph derives direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses, we prove that the growth of the degree of the colored Jones polynomials is a boundary slope of an essential surface in the knot complement. We show that certain coefficients of the polynomial measure how far this surface is from being a fiber for the knot; in particular, the surface is a fiber if and only if a particular coefficient vanishes. We also relate hyperbolic volume to colored Jones polynomials. Our method is to generalize the checkerboard decompositions of alternating knots. Under mild diagrammatic hypotheses, we show that these surfaces are essential, and obtain an ideal polyhedral decomposition of their complement. We use normal surface theory to relate the pieces of the JSJ decomposition of the complement to the combinatorics of certain surface spines (state graphs). Since state graphs have p...
Using Tutte polynomials to analyze the structure of the benzodiazepines
Cadavid Muñoz, Juan José
2014-05-01
Graph theory in general and Tutte polynomials in particular, are implemented for analyzing the chemical structure of the benzodiazepines. Similarity analysis are used with the Tutte polynomials for finding other molecules that are similar to the benzodiazepines and therefore that might show similar psycho-active actions for medical purpose, in order to evade the drawbacks associated to the benzodiazepines based medicine. For each type of benzodiazepines, Tutte polynomials are computed and some numeric characteristics are obtained, such as the number of spanning trees and the number of spanning forests. Computations are done using the computer algebra Maple's GraphTheory package. The obtained analytical results are of great importance in pharmaceutical engineering. As a future research line, the usage of the chemistry computational program named Spartan, will be used to extent and compare it with the obtained results from the Tutte polynomials of benzodiazepines.
Numerical Simulation of Polynomial-Speed Convergence Phenomenon
Li, Yao; Xu, Hui
2017-11-01
We provide a hybrid method that captures the polynomial speed of convergence and polynomial speed of mixing for Markov processes. The hybrid method that we introduce is based on the coupling technique and renewal theory. We propose to replace some estimates in classical results about the ergodicity of Markov processes by numerical simulations when the corresponding analytical proof is difficult. After that, all remaining conclusions can be derived from rigorous analysis. Then we apply our results to seek numerical justification for the ergodicity of two 1D microscopic heat conduction models. The mixing rate of these two models are expected to be polynomial but very difficult to prove. In both examples, our numerical results match the expected polynomial mixing rate well.
Properties of the Exceptional (Xl) Laguerre and Jacobi Polynomials
National Research Council Canada - National Science Library
Choon-Lin Ho; Satoru Odake; Ryu Sasaki
2011-01-01
We present various results on the properties of the four infinite sets of the exceptional Xl polynomials discovered recently by Odake and Sasaki [Phys. Lett. B 679 (2009), 414-417; Phys. Lett. B 684 (2010), 173-176...
Quasiregular mappings of polynomial type in R^{2}
Fletcher, Alastair; Goodman, Dan
Complex dynamics deals with the iteration of holomorphic functions. As is well known, the first functions to be studied which gave non-trivial dynamics were quadratic polynomials, which produced beautiful computer generated pictures of Julia sets and the Mandelbrot set. In the same spirit, this article aims to study the dynamics of the simplest non-trivial quasiregular mappings. These are mappings in R^{2} which are a composition of a quadratic polynomial and an affine stretch.
Quasiregular mappings of polynomial type in R^2
Fletcher, Alastair; Goodman, Dan
2010-01-01
Complex dynamics deals with the iteration of holomorphic functions. As is well- known, the first functions to be studied which gave non-trivial dynamics were quadratic polynomials, which produced beautiful computer generated pictures of Julia sets and the Mandelbrot set. In the same spirit, this article aims to study the dynamics of the simplest non-trivial quasiregular mappings. These are mappings in R^2 which are a composition of a quadratic polynomial and an affine stretch.
Multi-variable Gould-Hopper and Laguerre polynomials
Directory of Open Access Journals (Sweden)
Caterina Cassisa
2007-12-01
Full Text Available The monomiality principle was introduced by G. Dattoli, in order to derive the properties of special or generalized polynomials starting from the corresponding ones of monomials. In this article we show a general technique to extend themonomiality approach tomulti-index polynomials in several variables. Application to the case of Hermite, Laguerre-type and mixed-type (i.e. between Laguerre and Hermite are derived.
Limit cycles for a class of polynomial differential systems
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Jianyuan Qiao
2016-02-01
Full Text Available In this paper, we consider the limit cycles of a class of polynomial differential systems of the form $\\dot{x}=-y^{2p-1},\\ \\dot{y}=x^{2mp-1}+\\varepsilon(px^{2mp}+qy^{2p}(g(x,y-A$, where $g(x,y$ is a polynomial. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of a center using the averaging theory of first order.
Non-existence criteria for Laurent polynomial first integrals
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Shaoyun Shi
2003-01-01
Full Text Available In this paper we derived some simple criteria for non-existence and partial non-existence Laurent polynomial first integrals for a general nonlinear systems of ordinary differential equations $\\dot x = f(x$, $x \\in \\mathbb{R}^n$ with $f(0 = 0$. We show that if the eigenvalues of the Jacobi matrix of the vector field $f(x$ are $\\mathbb{Z}$-independent, then the system has no nontrivial Laurent polynomial integrals.
Bounds and asymptotics for orthogonal polynomials for varying weights
Levin, Eli
2018-01-01
This book establishes bounds and asymptotics under almost minimal conditions on the varying weights, and applies them to universality limits and entropy integrals. Orthogonal polynomials associated with varying weights play a key role in analyzing random matrices and other topics. This book will be of use to a wide community of mathematicians, physicists, and statisticians dealing with techniques of potential theory, orthogonal polynomials, approximation theory, as well as random matrices. .
Nobile, Fabio
2015-01-07
We consider a general problem F(u, y) = 0 where u is the unknown solution, possibly Hilbert space valued, and y a set of uncertain parameters. We specifically address the situation in which the parameterto-solution map u(y) is smooth, however y could be very high (or even infinite) dimensional. In particular, we are interested in cases in which F is a differential operator, u a Hilbert space valued function and y a distributed, space and/or time varying, random field. We aim at reconstructing the parameter-to-solution map u(y) from random noise-free or noisy observations in random points by discrete least squares on polynomial spaces. The noise-free case is relevant whenever the technique is used to construct metamodels, based on polynomial expansions, for the output of computer experiments. In the case of PDEs with random parameters, the metamodel is then used to approximate statistics of the output quantity. We discuss the stability of discrete least squares on random points show convergence estimates both in expectation and probability. We also present possible strategies to select, either a-priori or by adaptive algorithms, sequences of approximating polynomial spaces that allow to reduce, and in some cases break, the curse of dimensionality
Grandati, Y.; Quesne, C.
2013-07-01
The power of the disconjugacy properties of second-order differential equations of Schrödinger type to check the regularity of rationally extended quantum potentials connected with exceptional orthogonal polynomials is illustrated by re-examining the extensions of the isotonic oscillator (or radial oscillator) potential derived in kth-order supersymmetric quantum mechanics or multistep Darboux-Bäcklund transformation method. The function arising in the potential denominator is proved to be a polynomial with a nonvanishing constant term, whose value is calculated by induction over k. The sign of this term being the same as that of the already known highest degree term, the potential denominator has the same sign at both extremities of the definition interval, a property that is shared by the seed eigenfunction used in the potential construction. By virtue of disconjugacy, such a property implies the nodeless character of both the eigenfunction and the resulting potential.
An overview on polynomial approximation of NP-hard problems
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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.
On Zeros of Self-Reciprocal Random Algebraic Polynomials
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K. Farahmand
2008-01-01
Full Text Available This paper provides an asymptotic estimate for the expected number of level crossings of a trigonometric polynomial TN(ÃŽÂ¸=Ã¢ÂˆÂ‘j=0NÃ¢ÂˆÂ’1{ÃŽÂ±NÃ¢ÂˆÂ’jcos(j+1/2ÃŽÂ¸+ÃŽÂ²NÃ¢ÂˆÂ’jsin(j+1/2ÃŽÂ¸}, where ÃŽÂ±j and ÃŽÂ²j, j=0,1,2,Ã¢Â€Â¦, NÃ¢ÂˆÂ’1, are sequences of independent identically distributed normal standard random variables. This type of random polynomial is produced in the study of random algebraic polynomials with complex variables and complex random coefficients, with a self-reciprocal property. We establish the relation between this type of random algebraic polynomials and the above random trigonometric polynomials, and we show that the required level crossings have the functionality form of cos(N+ÃŽÂ¸/2. We also discuss the relationship which exists and can be explored further between our random polynomials and random matrix theory.
Polynomial asymptotic stability of damped stochastic differential equations
Directory of Open Access Journals (Sweden)
John Appleby
2004-08-01
Full Text Available The paper studies the polynomial convergence of solutions of a scalar nonlinear It\\^{o} stochastic differential equation\\[dX(t = -f(X(t\\,dt + \\sigma(t\\,dB(t\\] where it is known, {\\it a priori}, that $\\lim_{t\\rightarrow\\infty} X(t=0$, a.s. The intensity of the stochastic perturbation $\\sigma$ is a deterministic, continuous and square integrable function, which tends to zero more quickly than a polynomially decaying function. The function $f$ obeys $\\lim_{x\\rightarrow 0}\\mbox{sgn}(xf(x/|x|^\\beta = a$, for some $\\beta>1$, and $a>0$.We study two asymptotic regimes: when $\\sigma$ tends to zero sufficiently quickly the polynomial decay rate of solutions is the same as for the deterministic equation (when $\\sigma\\equiv0$. When $\\sigma$ decays more slowly, a weaker almost sure polynomial upper bound on the decay rate of solutions is established. Results which establish the necessity for $\\sigma$ to decay polynomially in order to guarantee the almost sure polynomial decay of solutions are also proven.
Polynomials for crystal frameworks and the rigid unit mode spectrum
Power, S. C.
2014-01-01
To each discrete translationally periodic bar-joint framework in , we associate a matrix-valued function defined on the d-torus. The rigid unit mode (RUM) spectrum of is defined in terms of the multi-phases of phase-periodic infinitesimal flexes and is shown to correspond to the singular points of the function and also to the set of wavevectors of harmonic excitations which have vanishing energy in the long wavelength limit. To a crystal framework in Maxwell counting equilibrium, which corresponds to being square, the determinant of gives rise to a unique multi-variable polynomial . For ideal zeolites, the algebraic variety of zeros of on the d-torus coincides with the RUM spectrum. The matrix function is related to other aspects of idealized framework rigidity and flexibility, and in particular leads to an explicit formula for the number of supercell-periodic floppy modes. In the case of certain zeolite frameworks in dimensions two and three, direct proofs are given to show the maximal floppy mode property (order N). In particular, this is the case for the cubic symmetry sodalite framework and some other idealized zeolites. PMID:24379422
Real analysis proof of Fundamental Theorem of Algebra using polynomial interlacing
Basu, Soham
2017-01-01
The existence of a real quadratic polynomial factor, given any polynomial with real coefficients, is proven using only elementary real analysis. The aim is to provide an approachable proof to anyone familiar with the least upper property for real numbers, continuity and growth property of polynomials. Complex numbers naturally arise as solutions to the general real quadratic divisors of any polynomial.
Some identities of the q-Laguerre polynomials on q-Umbral calculus
Dere, Rahime
2017-07-01
Some interesting identities of Sheffer polynomials was given by Roman [11], [12]. In this paper, we study a q-analogue of Laguerre polynomials which are also q-Sheffer polynomials. Furthermore, we give some new properties and formulas of these q-Laguerre polynomials of higher order by using the theory of the umbral algebra and umbral calculus.
Bayes Node Energy Polynomial Distribution to Improve Routing in Wireless Sensor Network.
Palanisamy, Thirumoorthy; Krishnasamy, Karthikeyan N
2015-01-01
Wireless Sensor Network monitor and control the physical world via large number of small, low-priced sensor nodes. Existing method on Wireless Sensor Network (WSN) presented sensed data communication through continuous data collection resulting in higher delay and energy consumption. To conquer the routing issue and reduce energy drain rate, Bayes Node Energy and Polynomial Distribution (BNEPD) technique is introduced with energy aware routing in the wireless sensor network. The Bayes Node Energy Distribution initially distributes the sensor nodes that detect an object of similar event (i.e., temperature, pressure, flow) into specific regions with the application of Bayes rule. The object detection of similar events is accomplished based on the bayes probabilities and is sent to the sink node resulting in minimizing the energy consumption. Next, the Polynomial Regression Function is applied to the target object of similar events considered for different sensors are combined. They are based on the minimum and maximum value of object events and are transferred to the sink node. Finally, the Poly Distribute algorithm effectively distributes the sensor nodes. The energy efficient routing path for each sensor nodes are created by data aggregation at the sink based on polynomial regression function which reduces the energy drain rate with minimum communication overhead. Experimental performance is evaluated using Dodgers Loop Sensor Data Set from UCI repository. Simulation results show that the proposed distribution algorithm significantly reduce the node energy drain rate and ensure fairness among different users reducing the communication overhead.
Covariance of the number of real zeros of a random trigonometric polynomial
Directory of Open Access Journals (Sweden)
K. Farahmand
2006-01-01
Full Text Available For random coefficients aj and bj we consider a random trigonometric polynomial defined as Tn(θ=∑j=0n{ajcosjθ+bjsinjθ}. The expected number of real zeros of Tn(θ in the interval (0,2π can be easily obtained. In this note we show that this number is in fact n/3. However the variance of the above number is not known. This note presents a method which leads to the asymptotic value for the covariance of the number of real zeros of the above polynomial in intervals (0,π and (π,2π. It can be seen that our method in fact remains valid to obtain the result for any two disjoint intervals. The applicability of our method to the classical random trigonometric polynomial, defined as Pn(θ=∑j=0naj(ωcosjθ, is also discussed. Tn(θ has the advantage on Pn(θ of being stationary, with respect to θ, for which, therefore, a more advanced method developed could be used to yield the results.
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.
The value of value congruence.
Edwards, Jeffrey R; Cable, Daniel M
2009-05-01
Research on value congruence has attempted to explain why value congruence leads to positive outcomes, but few of these explanations have been tested empirically. In this article, the authors develop and test a theoretical model that integrates 4 key explanations of value congruence effects, which are framed in terms of communication, predictability, interpersonal attraction, and trust. These constructs are used to explain the process by which value congruence relates to job satisfaction, organizational identification, and intent to stay in the organization, after taking psychological need fulfillment into account. Data from a heterogeneous sample of employees from 4 organizations indicate that the relationships that link individual and organizational values to outcomes are explained primarily by the trust that employees place in the organization and its members, followed by communication, and, to a lesser extent, interpersonal attraction. Polynomial regression analyses reveal that the relationships emanating from individual and organizational values often deviated from the idealized value congruence relationship that underlies previous theory and research. The authors' results also show that individual and organizational values exhibited small but significant relationships with job satisfaction and organizational identification that bypassed the mediators in their model, indicating that additional explanations of value congruence effects should be pursued in future research. (c) 2009 APA, all rights reserved.
Euler polynomials and identities for non-commutative operators
De Angelis, Valerio; Vignat, Christophe
2015-12-01
Three kinds of identities involving non-commutating operators and Euler and Bernoulli polynomials are studied. The first identity, as given by Bender and Bettencourt [Phys. Rev. D 54(12), 7710-7723 (1996)], expresses the nested commutator of the Hamiltonian and momentum operators as the commutator of the momentum and the shifted Euler polynomial of the Hamiltonian. The second one, by Pain [J. Phys. A: Math. Theor. 46, 035304 (2013)], links the commutators and anti-commutators of the monomials of the position and momentum operators. The third appears in a work by Figuieira de Morisson and Fring [J. Phys. A: Math. Gen. 39, 9269 (2006)] in the context of non-Hermitian Hamiltonian systems. In each case, we provide several proofs and extensions of these identities that highlight the role of Euler and Bernoulli polynomials.
Limit cycles for discontinuous generalized Lienard polynomial differential equations
Directory of Open Access Journals (Sweden)
Jaume Llibre
2013-09-01
Full Text Available We divide $\\mathbb{R}^2$ into sectors $S_1,\\dots ,S_l$, with $l>1$ even, and define a discontinuous differential system such that in each sector, we have a smooth generalized Lienard polynomial differential equation $\\ddot{x}+f_i(x\\dot{x} +g_i(x=0$, $i=1, 2$ alternatively, where $f_i$ and $g_i$ are polynomials of degree n-1 and m respectively. Then we apply the averaging theory for first-order discontinuous differential systems to show that for any $n$ and $m$ there are non-smooth Lienard polynomial equations having at least max{n,m} limit cycles. Note that this number is independent of the number of sectors.
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.
Local polynomial Whittle estimation of perturbed fractional processes
DEFF Research Database (Denmark)
Frederiksen, Per; Nielsen, Frank; Nielsen, Morten Ørregaard
We propose a semiparametric local polynomial Whittle with noise (LPWN) estimator of the memory parameter in long memory time series perturbed by a noise term which may be serially correlated. The estimator approximates the spectrum of the perturbation as well as that of the short-memory component...... of the signal by two separate polynomials. Including these polynomials we obtain a reduction in the order of magnitude of the bias, but also in‡ate the asymptotic variance of the long memory estimate by a multiplicative constant. We show that the estimator is consistent for d 2 (0; 1), asymptotically normal...... for d ε (0, 3/4), and if the spectral density is infinitely smooth near frequency zero, the rate of convergence can become arbitrarily close to the parametric rate, pn. A Monte Carlo study reveals that the LPWN estimator performs well in the presence of a serially correlated perturbation term...
Polynomial algebra of discrete models in systems biology.
Veliz-Cuba, Alan; Jarrah, Abdul Salam; Laubenbacher, Reinhard
2010-07-01
An increasing number of discrete mathematical models are being published in Systems Biology, ranging from Boolean network models to logical models and Petri nets. They are used to model a variety of biochemical networks, such as metabolic networks, gene regulatory networks and signal transduction networks. There is increasing evidence that such models can capture key dynamic features of biological networks and can be used successfully for hypothesis generation. This article provides a unified framework that can aid the mathematical analysis of Boolean network models, logical models and Petri nets. They can be represented as polynomial dynamical systems, which allows the use of a variety of mathematical tools from computer algebra for their analysis. Algorithms are presented for the translation into polynomial dynamical systems. Examples are given of how polynomial algebra can be used for the model analysis. alanavc@vt.edu Supplementary data are available at Bioinformatics online.
Recurrence Relations for Orthogonal Polynomials on Triangular Domains
Directory of Open Access Journals (Sweden)
Abedallah Rababah
2016-04-01
Full Text Available In Farouki et al, 2003, Legendre-weighted orthogonal polynomials P n , r ( u , v , w , r = 0 , 1 , … , n , n ≥ 0 on the triangular domain T = { ( u , v , w : u , v , w ≥ 0 , u + v + w = 1 } are constructed, where u , v , w are the barycentric coordinates. Unfortunately, evaluating the explicit formulas requires many operations and is not very practical from an algorithmic point of view. Hence, there is a need for a more efficient alternative. A very convenient method for computing orthogonal polynomials is based on recurrence relations. Such recurrence relations are described in this paper for the triangular orthogonal polynomials, providing a simple and fast algorithm for their evaluation.
A probabilistic approach of sum rules for heat polynomials
Energy Technology Data Exchange (ETDEWEB)
Vignat, C; Leveque, O, E-mail: christophe.vignat@epfl.ch, E-mail: olivier.leveque@epfl.ch [L.T.H.I., E.P.F.L. (Switzerland)
2012-01-13
In this paper, we show that the sum rules for generalized Hermite polynomials derived by Daboul and Mizrahi (2005 J. Phys. A: Math. Gen. http://dx.doi.org/10.1088/0305-4470/38/2/010) and by Graczyk and Nowak (2004 C. R. Acad. Sci., Ser. 1 338 849) can be interpreted and easily recovered using a probabilistic moment representation of these polynomials. The covariance property of the raising operator of the harmonic oscillator, which is at the origin of the identities proved in Daboul and Mizrahi and the dimension reduction effect expressed in the main result of Graczyk and Nowak are both interpreted in terms of the rotational invariance of the Gaussian distributions. As an application of these results, we uncover a probabilistic moment interpretation of two classical integrals of the Wigner function that involve the associated Laguerre polynomials. (paper)
A probabilistic approach of sum rules for heat polynomials
Vignat, C.; Lévêque, O.
2012-01-01
In this paper, we show that the sum rules for generalized Hermite polynomials derived by Daboul and Mizrahi (2005 J. Phys. A: Math. Gen. http://dx.doi.org/10.1088/0305-4470/38/2/010) and by Graczyk and Nowak (2004 C. R. Acad. Sci., Ser. 1 338 849) can be interpreted and easily recovered using a probabilistic moment representation of these polynomials. The covariance property of the raising operator of the harmonic oscillator, which is at the origin of the identities proved in Daboul and Mizrahi and the dimension reduction effect expressed in the main result of Graczyk and Nowak are both interpreted in terms of the rotational invariance of the Gaussian distributions. As an application of these results, we uncover a probabilistic moment interpretation of two classical integrals of the Wigner function that involve the associated Laguerre polynomials.
Examining the BMI-mortality relationship using fractional polynomials
Directory of Open Access Journals (Sweden)
Wong Edwin S
2011-12-01
Full Text Available Abstract Background Many previous studies estimating the relationship between body mass index (BMI and mortality impose assumptions regarding the functional form for BMI and result in conflicting findings. This study investigated a flexible data driven modelling approach to determine the nonlinear and asymmetric functional form for BMI used to examine the relationship between mortality and obesity. This approach was then compared against other commonly used regression models. Methods This study used data from the National Health Interview Survey, between 1997 and 2000. Respondents were linked to the National Death Index with mortality follow-up through 2005. We estimated 5-year all-cause mortality for adults over age 18 using the logistic regression model adjusting for BMI, age and smoking status. All analyses were stratified by sex. The multivariable fractional polynomials (MFP procedure was employed to determine the best fitting functional form for BMI and evaluated against the model that includes linear and quadratic terms for BMI and the model that groups BMI into standard weight status categories using a deviance difference test. Estimated BMI-mortality curves across models were then compared graphically. Results The best fitting adjustment model contained the powers -1 and -2 for BMI. The relationship between 5-year mortality and BMI when estimated using the MFP approach exhibited a J-shaped pattern for women and a U-shaped pattern for men. A deviance difference test showed a statistically significant improvement in model fit compared to other BMI functions. We found important differences between the MFP model and other commonly used models with regard to the shape and nadir of the BMI-mortality curve and mortality estimates. Conclusions The MFP approach provides a robust alternative to categorization or conventional linear-quadratic models for BMI, which limit the number of curve shapes. The approach is potentially useful in estimating
Polynomials-Based Terminal Control of Affine Systems
Directory of Open Access Journals (Sweden)
A. E. Golubev
2015-01-01
Full Text Available One of the approaches to solving terminal control problems for affine dynamical systems is based on the use of polynomials of degree 2n − 1, where n is the order of the system in question. In this paper, we investigate the terminal control problem for which the final state of the system coincides with the origin in the phase space. We seek a set of initial states such that the solution of the terminal control problem can be constructed by using a polynomial of degree 2n − 2.Note that solution of the terminal control problem in question can be used to solve the problem of stabilizing the zero equilibrium in a finite time.For the second-order systems we prove the necessary and sufficient conditions for existence of the polynomial of the second degree which determines the solution of the terminal problem. The solutions of the terminal control problem based on the polynomials of second and third degree are given. As an example, the terminal control problem is considered for the simple pendulum.We also discuss solution of the terminal problem for affine systems of the third order, based on the use of the fourth and fifth degree polynomials. The necessary and sufficient conditions for existence of the fourth-degree polynomial such that its phase graph connects an arbitrary initial state of the system and the origin are obtained.For systems of arbitrary order n we obtain the necessary and sufficient conditions for existence of a solution of the terminal problem using the polynomial of degree 2n − 2. We also give the solution of the problem by means of the polynomial of degree 2n − 1.Further research can be focused on extending the results obtained in this note to terminal control problems where the desired final state of the system is not necessarily the origin.One of the potential application areas for the obtained theoretical results is automatic control of technical plants like unmanned aerial vehicles and mobile robots.
Polynomial Approximation on Unbounded Subsets and the Markov Moment Problem
Directory of Open Access Journals (Sweden)
Octav Olteanu
2013-09-01
Full Text Available We start this review paper by recalling some known and relatively recent results in polynomial approximation on unbounded subsets. These results allow approximation of nonnegative continuous functions with compact support contained in the first quadrant by sums of tensor products of positive polynomials in each separate variable, on the positive semiaxes. Consequently, we characterize the existence of solution of a two dimensional Markov moment problem in terms of products of quadratic forms. Secondly, one proves some applications of abstract results on the extension of linear operators with two constraints to the Markov moment problem. Two applications related to this last part are considered.
Skew-orthogonal polynomials, differential systems and random matrix theory
Energy Technology Data Exchange (ETDEWEB)
Ghosh, Saugata [Abdus Salam ICTP, Strada Costiera 11, 34100, Trieste (Italy)
2007-01-26
We study skew-orthogonal polynomials with respect to the weight function exp [ - 2V(x)], with V(x) = {sigma}{sup 2d}{sub K=1}(u{sub K}/K)x{sup K}, u{sub 2d} > 0, d > 0. A finite subsequence of such skew-orthogonal polynomials arising in the study of orthogonal and symplectic ensembles of random matrices satisfies a system of differential-difference-deformation equation. The vectors formed by such subsequence have the rank equal to the degree of the potential in the quaternion sense. These solutions satisfy certain compatibility condition and hence admit a simultaneous fundamental system of solutions.
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.
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.
Orthogonal polynomials on the unit circle part 2 spectral theory
Simon, Barry
2013-01-01
This two-part book is a comprehensive overview of the theory of probability measures on the unit circle, viewed especially in terms of the orthogonal polynomials defined by those measures. A major theme involves the connections between the Verblunsky coefficients (the coefficients of the recurrence equation for the orthogonal polynomials) and the measures, an analog of the spectral theory of one-dimensional Schrödinger operators. Among the topics discussed along the way are the asymptotics of Toeplitz determinants (Szegő's theorems), limit theorems for the density of the zeros of orthogonal po
The Laplace transform and polynomial approximation in L2
DEFF Research Database (Denmark)
Labouriau, Rodrigo
2016-01-01
This short note gives a sufficient condition for having the class of polynomials dense in the space of square integrable functions with respect to a finite measure dominated by the Lebesgue measure in the real line, here denoted by L2. It is shown that if the Laplace transform of the measure...... concerning the polynomial approximation is original, even thought the proof is relatively simple. Additionally, an alternative stronger condition (easier to be verified) not involving the calculation of the Laplace transform is given. The condition essentially says that the density of the measure should have...
Polynomial Approximation Techniques for Differential Equations in Electrochemical Problems
1981-01-15
LEYE L I oFFIPE-W NAVAL RESEARCH ConracEi NP014-8--0107 Task No. NR 359-718 1>. -/ TECHNICAL REP&T O. 4 Polynomial Approximation Techniques for...If the number of points xn is increased, a new polynomial may thus be found which de - scribes O(x) at these new points. Therefore, in the limit, O...has been used to simulate O(x) in the interval [xlXn1. Certain of these quadrature formulas lead to the well known Newton -Cotes, trapezoidal, and
Orthogonal polynomials on the unit circle part 1 classical theory
2009-01-01
This two-part book is a comprehensive overview of the theory of probability measures on the unit circle, viewed especially in terms of the orthogonal polynomials defined by those measures. A major theme involves the connections between the Verblunsky coefficients (the coefficients of the recurrence equation for the orthogonal polynomials) and the measures, an analog of the spectral theory of one-dimensional Schrodinger operators. Among the topics discussed along the way are the asymptotics of Toeplitz determinants (SzegÅ‘'s theorems), limit theorems for the density of the zeros of orthogonal po
Paraxial and nonparaxial polynomial beams and the analytic approach to propagation.
Dennis, Mark R; Götte, Jörg B; King, Robert P; Morgan, Michael A; Alonso, Miguel A
2011-11-15
We construct solutions of the paraxial and Helmholtz equations that are polynomials in their spatial variables. These are derived explicitly by using the angular spectrum method and generating functions. Paraxial polynomials have the form of homogeneous Hermite and Laguerre polynomials in Cartesian and cylindrical coordinates, respectively, analogous to heat polynomials for the diffusion equation. Nonparaxial polynomials are found by substituting monomials in the propagation variable z with reverse Bessel polynomials. These explicit analytic forms give insight into the mathematical structure of paraxially and nonparaxially propagating beams, especially in regard to the divergence of nonparaxial analogs to familiar paraxial beams.
Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials
Odake, Satoru; Sasaki, Ryu
2017-03-01
The multi-indexed Laguerre and Jacobi polynomials form a complete set of orthogonal polynomials. They satisfy second-order differential equations but not three term recurrence relations, because of the 'holes' in their degrees. The multi-indexed Laguerre and Jacobi polynomials have Wronskian expressions originating from multiple Darboux transformations. For the ease of applications, two different forms of simplified expressions of the multi-indexed Laguerre and Jacobi polynomials are derived based on various identities. The parity transformation property of the multi-indexed Jacobi polynomials is derived based on that of the Jacobi polynomial.
Some results about a special nonlinear difference equation and uniqueness of difference polynomial
Directory of Open Access Journals (Sweden)
Ding Jie
2011-01-01
Full Text Available Abstract In this paper, we continue to study a special nonlinear difference equation solutions of finite order entire function. We also continue to investigate the value distribution and uniqueness of difference polynomials of meromorphic functions. Our results which improve the results of Yang and Laine [Proc. Jpn. Acad. Ser. A Math. Sci. 83:50-55 (2007]; Qi et al. [Comput. Math. Appl. 60:1739-1746 (2010 ]. Mathematics Subject Classification (2000: 30D35, 39B32, 34M05.
An estimation of the p-adic sizes of common zeros of partial derivative polynomials of degree six
Aminudin, S. S.; Sapar, S. H.; Atan, K. A. Mohd
2014-07-01
Let x¯ = (x1,x2,...,xn) be a vector in Zn with Z ring of integers and q be a positive integer, f a polynomial in x with coefficient in Z. The exponential sum associated with f is defined as S(f;q) = ∑ xmodq e2πif(x)/q, where the sum is taken over a complete set of residues modulo q. The value of S (f; q) depends on the estimate of cardinality |V|, the number of elements contained in the set V = {x¯ modq| f¯ x¯ ≡0¯modq} where f¯ x¯ is the partial derivatives of f with respect to x. To determine the cardinality of V, the p-adic sizes of common zeros of the partial derivative polynomials need to be obtained. In this paper, we estimate the p-adic sizes of common zeros of partial derivative polynomials of f(x,y) in Zp[x,y] with a sixth degree form by using Newton polyhedron technique. The polynomial is of the form f(x,y) = ax6+bx5y+cx4y2+sx+ty+k.
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....
Some Polynomials Associated with the r-Whitney Numbers
Indian Academy of Sciences (India)
26
Introduction. The r-Whitney numbers of the first kind wm,r(n, k) and the second kind Wm,r(n, k) were defined by Mez˝o [29] as the connecting coefficients between some particular polynomials. Note that the ...... the Scientific Research Foundation of Nanjing University of Information Science & Technol- ogy, the Startup ...
Shell polynomials and dual birth-death processes
van Doorn, Erik A.
2016-01-01
This paper aims to clarify certain aspects of the relations between birth-death processes, measures solving a Stieltjes moment problem, and sets of parameters defining polynomial sequences that are orthogonal with respect to such a measure. Besides giving an overview of the basic features of these
Shell polynomials and dual birth-death processes
van Doorn, Erik A.
2015-01-01
This paper aims to clarify certain aspects of the relations between birth-death processes, measures solving a Stieltjes moment problem, and sets of parameters defining polynomial sequences that are orthogonal with respect to such a measure. Besides giving an overview of the basic features of these
The Critical Orbit Structure of Quadratic Polynomials in Zp
Mullen, Cara
In this thesis, we develop a non-Archimedean analog to the Hubbard tree, a well-understood object from classical dynamics studied over the complex numbers. To that end, we explore the critical orbit structure of quadratic polynomials ƒc(z) = z 2 + c with parameters c in the ring of p-adic rational integers, Zp. All such polynomials are post-critically bounded (PCB), and some are post-critically finite (PCF), which means that the forward orbit of the critical point, 0, is finite. If the orbit of 0 is finite, there exist minimal integers m and n such that ƒcm+n(0)=ƒc m(0), and we call (m,n) the critical orbit type of ƒc. All PCF polynomials f defined over the complex numbers have an associated Hubbard tree which illustrates the orbit type of the critical points, and the geometry of those orbits within the filled Julia set of f. In order to define a p-adic analog, we give a description of the exact critical orbit type for PCF polynomials ƒc defined over Z p, and then use the proximity of PCB parameters to PCF points in order to prove statements about the structure of the infinite critical orbit as visualized in the Zp tree.
Polynomials constant on a hyperplane and CR maps of spheres
Lebl, J.; Peters, H.
2012-01-01
We prove a sharp degree bound for polynomials constant on a hyperplane with a fixed number of nonnegative distinct monomials. This bound was conjectured by John P. D’Angelo, proved in two dimensions by D’Angelo, Kos and Riehl and in three dimensions by the authors. The current work builds upon these
Generalized Freud's equation and level densities with polynomial ...
Indian Academy of Sciences (India)
The generalized Freud's equations for = 3, 4 and 5 are derived and using this R = h / h − 1 is obtained, where h is the normalization constant for the corresponding orthogonal polynomials. Moments of the density functions, expressed in terms of R , are obtained using Freud's equation and using this, explicit ...
Generalized Freud's equation and level densities with polynomial ...
Indian Academy of Sciences (India)
5 are derived and using this Rμ = hμ/hμ−1 is obtained, where hμ is the normalization constant for the corresponding orthogonal polynomials. Moments of the density functions, expressed in terms of Rμ, are obtained using Freud's equation and using this, explicit results of level densities as N → ∞ are derived using the ...
Quantum homomorphic encryption for polynomial-sized circuits
Y. Dulek (Yfke); C. Schaffner (Christian); F. Speelman (Florian)
2016-01-01
textabstractWe present a new scheme for quantum homomorphic encryption which is compact and allows for efficient evaluation of arbitrary polynomial-sized quantum circuits. Building on the framework of Broadbent and Jeffery [BJ15] and recent results in the area of instantaneous non-local quantum
On the complex oscillation of differential polynomials generated by ...
Indian Academy of Sciences (India)
Home; Journals; Proceedings – Mathematical Sciences; Volume 120; Issue 4. On the Complex Oscillation of Differential Polynomials Generated by Meromorphic Solutions of Differential Equations in the Unit Disc. Ting-Bin Cao Hong-Yan Xu Chuan-Xi Zhu. Volume 120 Issue 4 September 2010 pp 481-493 ...
A reduced polynomial chaos expansion method for the stochastic ...
Indian Academy of Sciences (India)
The stochastic ﬁnite element analysis of elliptic type partial differential equations is considered. A reduced method of the spectral stochastic ﬁnite element method using polynomial chaos is proposed. The method is based on the spectral decomposition of the deterministic system matrix. The reduction is achieved by ...
Polynomial skew-products with wandering Fatou-disks
Peters, H.; Vivas, L.R.
2016-01-01
Until recently, little was known about the existence of wandering Fatou components for rational maps in more than one complex variables. In 2014, examples of wandering Fatou components were constructed in Astorg et al. [1] for polynomial skew-products with an invariant parabolic fiber. In 2004 Lilov
A Genetic algorithm for evaluating the zeros (roots) of polynomial ...
African Journals Online (AJOL)
This paper presents a Genetic Algorithm software (which is a computational, search technique) for finding the zeros (roots) of any given polynomial function, and optimizing and solving N-dimensional systems of equations. The software is particularly useful since most of the classic schemes are not all embracing.
Model-robust experimental designs for the fractional polynomial ...
African Journals Online (AJOL)
can give a good a fit to the data and much more plausible behavior between de- sign points than the polynomial models. In this ... often resorted to the theory of optimal designs to get practical designs for those models. However the complexity of these nonlinear ..... on Statistics in Industry and Tech- nology, Dayton, Ohio. 18 ...
New r-Matrices for Lie Bialgebra Structures over Polynomials
Pop, Iulia; Yermolova-Magnusson, Julia
2010-08-01
For a finite dimensional simple complex Lie algebra {mathfrak{g}} , Lie bialgebra structures on {mathfrak{g}left[left[u right]right]} and {mathfrak{g}left[uright]} were classified by Montaner, Stolin and Zelmanov. In our paper, we provide an explicit algorithm to produce r-matrices which correspond to Lie bialgebra structures over polynomials.
Mirror symmetry, toric branes and topological string amplitudes as polynomials
Energy Technology Data Exchange (ETDEWEB)
Alim, Murad
2009-07-13
The central theme of this thesis is the extension and application of mirror symmetry of topological string theory. The contribution of this work on the mathematical side is given by interpreting the calculated partition functions as generating functions for mathematical invariants which are extracted in various examples. Furthermore the extension of the variation of the vacuum bundle to include D-branes on compact geometries is studied. Based on previous work for non-compact geometries a system of differential equations is derived which allows to extend the mirror map to the deformation spaces of the D-Branes. Furthermore, these equations allow the computation of the full quantum corrected superpotentials which are induced by the D-branes. Based on the holomorphic anomaly equation, which describes the background dependence of topological string theory relating recursively loop amplitudes, this work generalizes a polynomial construction of the loop amplitudes, which was found for manifolds with a one dimensional space of deformations, to arbitrary target manifolds with arbitrary dimension of the deformation space. The polynomial generators are determined and it is proven that the higher loop amplitudes are polynomials of a certain degree in the generators. Furthermore, the polynomial construction is generalized to solve the extension of the holomorphic anomaly equation to D-branes without deformation space. This method is applied to calculate higher loop amplitudes in numerous examples and the mathematical invariants are extracted. (orig.)
Polynomial probability distribution estimation using the method of moments
Mattsson, Lars; Rydén, Jesper
2017-01-01
We suggest a procedure for estimating Nth degree polynomial approximations to unknown (or known) probability density functions (PDFs) based on N statistical moments from each distribution. The procedure is based on the method of moments and is setup algorithmically to aid applicability and to ensure rigor in use. In order to show applicability, polynomial PDF approximations are obtained for the distribution families Normal, Log-Normal, Weibull as well as for a bimodal Weibull distribution and a data set of anonymized household electricity use. The results are compared with results for traditional PDF series expansion methods of Gram–Charlier type. It is concluded that this procedure is a comparatively simple procedure that could be used when traditional distribution families are not applicable or when polynomial expansions of probability distributions might be considered useful approximations. In particular this approach is practical for calculating convolutions of distributions, since such operations become integrals of polynomial expressions. Finally, in order to show an advanced applicability of the method, it is shown to be useful for approximating solutions to the Smoluchowski equation. PMID:28394949
Entire functions sharing one polynomial with their derivatives
Indian Academy of Sciences (India)
Home; Journals; Proceedings – Mathematical Sciences; Volume 118; Issue 1. Entire Functions Sharing One Polynomial with their Derivatives ... restriction of the hyper order less than 1/2, and obtain some uniqueness theorems of a nonconstant entire function and its derivative sharing a finite nonzero complex number CM.
Zeros and uniqueness of Q-difference polynomials of meromorphic ...
Indian Academy of Sciences (India)
(Math. Sci.) Vol. 124, No. 4, November 2014, pp. 533–549. c Indian Academy of Sciences. Zeros and uniqueness of Q-difference polynomials of meromorphic functions with zero order. TING-BIN CAO. ∗. , KAI LIU and NA XU. Department of Mathematics, Nanchang University, Nanchang,. Jiangxi 330031, People's Republic ...
Numerical Performances of Two Orthogonal Polynomials in the Tau ...
African Journals Online (AJOL)
Although the two have different weight functions, but they most time give close results especially when they are considered in the same interval. This work has therefore used the two polynomials, within the same interval, as bases functions in the Ortiz's Recursive Formulation of Lanczos' Tau method. Numerical experiments ...
Application of grafted polynomial function in forecasting cotton ...
African Journals Online (AJOL)
A study was conducted to forecast cotton production trend with the application of a grafted polynomial function in Nigeria from 1985 through 2013. Grafted models are used in econometrics to embark on economic analysis involving time series. In economic time series, the paucity of data and their availability has always ...
Entropic functionals of Laguerre and Gegenbauer polynomials with large parameters
N.M. Temme (Nico); I.V. Toranzo; J.S. Dehesa
2017-01-01
textabstractThe determination of the physical entropies (Rényi, Shannon, Tsallis) of high-dimensional quantum systems subject to a central potential requires the knowledge of the asymptotics of some power and logarithmic integral functionals of the hypergeometric orthogonal polynomials which control
Polynomial probability distribution estimation using the method of moments.
Munkhammar, Joakim; Mattsson, Lars; Rydén, Jesper
2017-01-01
We suggest a procedure for estimating Nth degree polynomial approximations to unknown (or known) probability density functions (PDFs) based on N statistical moments from each distribution. The procedure is based on the method of moments and is setup algorithmically to aid applicability and to ensure rigor in use. In order to show applicability, polynomial PDF approximations are obtained for the distribution families Normal, Log-Normal, Weibull as well as for a bimodal Weibull distribution and a data set of anonymized household electricity use. The results are compared with results for traditional PDF series expansion methods of Gram-Charlier type. It is concluded that this procedure is a comparatively simple procedure that could be used when traditional distribution families are not applicable or when polynomial expansions of probability distributions might be considered useful approximations. In particular this approach is practical for calculating convolutions of distributions, since such operations become integrals of polynomial expressions. Finally, in order to show an advanced applicability of the method, it is shown to be useful for approximating solutions to the Smoluchowski equation.
Polynomial embedding algorithms for controllers in a behavioral framework
Trentelman, H.L.; Zavala Yoe, R.; Praagman, C.; Zavala Yoé, 27772
2007-01-01
In this correspondence, we will establish polynomial algorithms for computation of controllers in the behavioral approach to control, in particular for the computation of controllers that regularly implement a given desired behavior and for controllers that achieve pole placement and stabilization
Constructing irreducible polynomials with prescribed level curves over finite fields
Mihai Caragiu
2001-01-01
We use Eisenstein's irreducibility criterion to prove that there exists an absolutely irreducible polynomial P(X,Y)∈GF(q)[X,Y] with coefficients in the finite field GF(q) with q elements, with prescribed level curves Xc:={(x,y)∈GF(q)2|P(x,y)=c}.
A Polynomial Optimization Approach to Constant Rebalanced Portfolio Selection
Takano, Y.; Sotirov, R.
2010-01-01
We address the multi-period portfolio optimization problem with the constant rebalancing strategy. This problem is formulated as a polynomial optimization problem (POP) by using a mean-variance criterion. In order to solve the POPs of high degree, we develop a cutting-plane algorithm based on
Polynomials constant on a hyperplane and CR maps of hyperquadrics
Lebl, J.; Peters, H.
2011-01-01
We prove a sharp degree bound for polynomials constant on a hyperplane with a fixed number of distinct monomials for dimensions 2 and 3. We study the connection with monomial CR maps of hyperquadrics and prove similar bounds in this setup with emphasis on the case of spheres. The results support
A general polynomial solution to convection–dispersion equation ...
Indian Academy of Sciences (India)
A number of models have been established to simulate the behaviour of solute transport due to chemical pollution, both in croplands and groundwater systems. An approximate polynomial solution to convection–dispersion equation (CDE) based on boundary layer theory has been verified for the use to describe solute ...
PICK'S FORMULA AND GENERALIZED EHRHART QUASI-POLYNOMIALS
Directory of Open Access Journals (Sweden)
Takayuki Hibi
2015-11-01
Full Text Available By virtue of Pick's formula, the generalized Ehrhart quasi-polynomial of the triangulation $\\mathcal{P} \\subset \\mathbb{R}^2$ with the vertices $(0,0, (u(n,0, (0,v(n$, where $u(x$ and $v(x$ belong to $\\mathbb{Z}[x]$ and where $n=1,2, \\ldots$, will be computed.
Solving Multi-variate Polynomial Equations in a Finite Field
2013-06-01
both industry and government agencies showed a dislike for Rijndael compared to its competitors due to its simplicity [16]. Rijndael Serpent Twofish... Serpent ) [24]. 3.3.2 Gröbner Bases The second method to solve large systems of multivariate polynomial equations is to use the ideal derived from the
Computational Technique for Teaching Mathematics (CTTM): Visualizing the Polynomial's Resultant
Alves, Francisco Regis Vieira
2015-01-01
We find several applications of the Dynamic System Geogebra--DSG related predominantly to the basic mathematical concepts at the context of the learning and teaching in Brasil. However, all these works were developed in the basic level of Mathematics. On the other hand, we discuss and explore, with DSG's help, some applications of the polynomial's…
Superiority of Bessel function over Zernicke polynomial as base ...
Indian Academy of Sciences (India)
... for radial expansion over Zernicke polynomial in the tomographic reconstruction technique. The causes for the superiority have been described in detail. The superiority has been shown both with simulated data for Kadomtsev's model for saw-tooth oscillation and real experimental x-ray data from W7-AS Stellarator.
Bounding the number of remarkable values via Jouanolou's theorem
Chèze, Guillaume
2015-01-01
In this article we bound the number of remarkable values of a polynomial vector field. The proof is short and based on Jouanolou's theorem about rational first integrals of planar polynomial derivations. Our bound is given in term of the size of a Newton polygon associated to the vector field. We prove that this bound is almost reached.
Bounding the number of remarkable values via Jouanolou's theorem
Chèze, Guillaume
2015-05-01
In this article we bound the number of remarkable values of a polynomial vector field. The proof is short and based on Jouanolou's theorem about rational first integrals of planar polynomial derivations. Our bound is given in term of the size of a Newton polygon associated to the vector field. We prove that this bound is almost reached.
Directory of Open Access Journals (Sweden)
Graciela Foglia
2011-10-01
Full Text Available Resumo: O objetivo deste trabalho é comentar os epílogos de cada uma das edições de Operación masacre (1957, 1964, 1969, 1972 de Rodolfo Walsh e mais um capítulo, o 37, “Aramburu y el juicio histórico” (1972, tendo como base teórica as reflexões sobre as relações entre literatura e testemunho. A opção por pensar cada um desses capítulos como parte da mesma seqüência se deve a que é aí onde melhor pode se acompanhar o percurso que indica uma convergência entre as instancias do narrador, do autor e do cidadão.Palavras-chave: Rodolfo Walsh; Operación masacre; literatura; testemunho.Resumen: El objetivo de este trabajo es comentar los epílogos de cada una de las ediciones de Operación masacre (1957, 1964, 1969, 1972 de Rodolfo Walsh y un capítulo más, el 37, “Aramburu y el juicio histórico” (1972, teniendo como base teórica la reflexión sobre las relaciones entre literatura y testimonio. La opción por pensar cada uno de esos capítulos como parte de la misma secuencia se debe a que es allí donde mejor se acompaña el recorrido que indica una convergencia entre las instancias de narrador, autor y ciudadano.Palabras-clave: Rodolfo Walsh; Operación masacre; literatura; testimonio.Keywords: Rodolfo Walsh; Operación masacre; literature; testimony.
Application of Chybeshev Polynomials in Factorizations of Balancing and Lucas-Balancing Numbers
Directory of Open Access Journals (Sweden)
Prasanta Kumar Ray
2012-01-01
Full Text Available In this paper, with the help of orthogonal polynomial especially Chybeshev polynomials of first and second kind, number theory and linear algebra intertwined to yield factorization of the balancing and Lucas-balancing numbers.
On some permutation polynomials over finite fields
Directory of Open Access Journals (Sweden)
Amir Akbary
2005-10-01
Full Text Available Let p be prime, q=pm, and qÃ¢ÂˆÂ’1=7s. We completely describe the permutation behavior of the binomial P(x=xr(1+xes (1Ã¢Â‰Â¤eÃ¢Â‰Â¤6 over a finite field Fq in terms of the sequence {an} defined by the recurrence relation an=anÃ¢ÂˆÂ’1+2anÃ¢ÂˆÂ’2Ã¢ÂˆÂ’anÃ¢ÂˆÂ’3 (nÃ¢Â‰��¥3 with initial values a0=3, a1=1, and a2=5.
A high-order q-difference equation for q-Hahn multiple orthogonal polynomials
DEFF Research Database (Denmark)
Arvesú, J.; Esposito, Chiara
2012-01-01
A high-order linear q-difference equation with polynomial coefficients having q-Hahn multiple orthogonal polynomials as eigenfunctions is given. The order of the equation coincides with the number of orthogonality conditions that these polynomials satisfy. Some limiting situations when are studie....... Indeed, the difference equation for Hahn multiple orthogonal polynomials given in Lee [J. Approx. Theory (2007), ), doi: 10.1016/j.jat.2007.06.002] is obtained as a limiting case....
Some properties of generalized self-reciprocal polynomials over finite fields
Directory of Open Access Journals (Sweden)
Ryul Kim
2014-07-01
Full Text Available Numerous results on self-reciprocal polynomials over finite fields have been studied. In this paper we generalize some of these to a-self reciprocal polynomials defined in [4]. We consider some properties of the divisibility of a-reciprocal polynomials and characterize the parity of the number of irreducible factors for a-self reciprocal polynomials over finite fields of odd characteristic.
Interval Continuous Plant Identification from Value Sets
Directory of Open Access Journals (Sweden)
R. Hernández
2012-01-01
Full Text Available This paper shows how to obtain the values of the numerator and denominator Kharitonov polynomials of an interval plant from its value set at a given frequency. Moreover, it is proven that given a value set, all the assigned polynomials of the vertices can be determined if and only if there is a complete edge or a complete arc lying on a quadrant. This algorithm is nonconservative in the sense that if the value-set boundary of an interval plant is exactly known, and particularly its vertices, then the Kharitonov rectangles are exactly those used to obtain these value sets.
Directory of Open Access Journals (Sweden)
D. Jabari Sabeg
2016-10-01
Full Text Available In this paper, we present a new computational method for solving nonlinear singular boundary value problems of fractional order arising in biology. To this end, we apply the operational matrices of derivatives of shifted Legendre polynomials to reduce such problems to a system of nonlinear algebraic equations. To demonstrate the validity and applicability of the presented method, we present some numerical examples.
On the unimodality of independence polynomial of certain classes of graphs
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Fatemeh Jafari
2013-09-01
Full Text Available The independence polynomial of a graph G is the polynomial $sum i_kx^k$, where $i_k$ denote the number of independent sets of cardinality k in G. In this paper we study unimodality problem for the independence polynomial of certain classes of graphs.
On the Computation of the Maximum Uncertainty Volume of Stable Polynomials
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F.M. Al-Sunni
2003-06-01
Full Text Available In this paper we propose a non-linear optimization based approach for the computation of the stability region for uncertain polynomials. Both box of polynomials and diamond of polynomials are addressed. Examples are presented as an illustration.
A note on the zeros of Freud-Sobolev orthogonal polynomials
Moreno-Balcazar, Juan J.
2007-10-01
We prove that the zeros of a certain family of Sobolev orthogonal polynomials involving the Freud weight function e-x4 on are real, simple, and interlace with the zeros of the Freud polynomials, i.e., those polynomials orthogonal with respect to the weight function e-x4. Some numerical examples are shown.
A matrix Rodrigues formula for classical orthogonal polynomials in two variables
Álvarez de Morales, María; Fernández, Lidia; Pérez, Teresa E.; Piñar, Miguel A.
2006-01-01
Classical orthogonal polynomials in one variable can be characterized as the only orthogonal polynomials satisfying a Rodrigues formula. In this paper, using the second kind Kronecker power of a matrix, a Rodrigues formula is introduced for classical orthogonal polynomials in two variables.
Picard Iteration, Chebyshev Polynomials and Chebyshev-Picard Methods: Application in Astrodynamics
Junkins, John L.; Bani Younes, Ahmad; Woollands, Robyn M.; Bai, Xiaoli
2013-12-01
This paper extends previous work on parallel-structured Modified Chebyshev Picard Iteration (MCPI) Methods. The MCPI approach iteratively refines path approximation of the state trajectory for smooth nonlinear dynamical systems and this paper shows that the approach is especially suitable for initial value problems of astrodynamics. Using Chebyshev polynomials, as the orthogonal approximation basis, it is straightforward to distribute the computation of force functions needed in MCPI to generate the polynomial coefficients (approximating the path iterations) to different processors. Combining Chebyshev polynomials with Picard iteration, MCPI methods iteratively refines path estimates over large time intervals chosen to be within the domain of convergence of Picard iteration. The developed vector-matrix form makes MCPI methods computationally efficient and a more systematic approach is given, leading to a modest correction to results in the published dissertation by Bai. The power of MCPI methods for solving IVPs is clearly illustrated using a simple nonlinear differential equation with a known analytical solution. Compared with the most common integration scheme, the standard Runge-Kutta 4-5 method as implemented in MATLAB, MCPI methods generate solutions with better accuracy as well as orders of magnitude speedups, on a serial machine. MCPI performance is also compared to state of the art integrators such as the Runge-Kutta Nystrom 12(10) methods applied to the relevant orbit mechanics problems. The MCPI method is shown to be well-suited to solving these problems in serial processors with over an order of magnitude speedup relative to known methods. Furthermore, the approach is parallel-structured so that it is suited for parallel implementation and further speedups. When used in conjunction with the recently developed local gravity approximations in conjunction with parallel computation, we anticipate MCPI will enable revolutionary speedups while ensuring
Uncertainty Quantification for Combined Polynomial Chaos Kriging Surrogate Models
Weinmeister, Justin; Gao, Xinfeng; Krishna Prasad, Aditi; Roy, Sourajeet
2017-11-01
Surrogate modeling techniques are currently used to perform uncertainty quantification on computational fluid dynamics (CFD) models for their ability to identify the most impactful parameters on CFD simulations and help reduce computational cost in engineering design process. The accuracy of these surrogate models depends on a number of factors, such as the training data created from the CFD simulations, the target functions, the surrogate model framework, and so on. Recently, we have combined polynomial chaos expansions (PCE) and Kriging to produce a more accurate surrogate model, polynomial chaos Kriging (PCK). In this talk, we analyze the error convergence rate for the Kriging, PCE, and PCK model on a convection-diffusion-reaction problem, and validate the statistical measures and performance of the PCK method for its application to practical CFD simulations.
A robust regularization algorithm for polynomial networks for machine learning
Jaenisch, Holger M.; Handley, James W.
2011-06-01
We present an improvement to the fundamental Group Method of Data Handling (GMDH) Data Modeling algorithm that overcomes the parameter sensitivity to novel cases presented to derived networks. We achieve this result by regularization of the output and using a genetic weighting that selects intermediate models that do not exhibit divergence. The result is the derivation of multi-nested polynomial networks following the Kolmogorov-Gabor polynomial that are robust to mean estimators as well as novel exemplars for input. The full details of the algorithm are presented. We also introduce a new method for approximating GMDH in a single regression model using F, H, and G terms that automatically exports the answers as ordinary differential equations. The MathCAD 15 source code for all algorithms and results are provided.
Polynomial mixture method of solving ordinary differential equations
Shahrir, Mohammad Shazri; Nallasamy, Kumaresan; Ratnavelu, Kuru; Kamali, M. Z. M.
2017-11-01
In this paper, a numerical solution of fuzzy quadratic Riccati differential equation is estimated using a proposed new approach that provides mixture of polynomials where iteratively the right mixture will be generated. This mixture provide a generalized formalism of traditional Neural Networks (NN). Previous works have shown reliable results using Runge-Kutta 4th order (RK4). This can be achieved by solving the 1st Order Non-linear Differential Equation (ODE) that is found commonly in Riccati differential equation. Research has shown improved results relatively to the RK4 method. It can be said that Polynomial Mixture Method (PMM) shows promising results with the advantage of continuous estimation and improved accuracy that can be produced over Mabood et al, RK-4, Multi-Agent NN and Neuro Method (NM).
Uncertainty Quantification in Simulations of Epidemics Using Polynomial Chaos
Santonja, F.; Chen-Charpentier, B.
2012-01-01
Mathematical models based on ordinary differential equations are a useful tool to study the processes involved in epidemiology. Many models consider that the parameters are deterministic variables. But in practice, the transmission parameters present large variability and it is not possible to determine them exactly, and it is necessary to introduce randomness. In this paper, we present an application of the polynomial chaos approach to epidemiological mathematical models based on ordinary differential equations with random coefficients. Taking into account the variability of the transmission parameters of the model, this approach allows us to obtain an auxiliary system of differential equations, which is then integrated numerically to obtain the first-and the second-order moments of the output stochastic processes. A sensitivity analysis based on the polynomial chaos approach is also performed to determine which parameters have the greatest influence on the results. As an example, we will apply the approach to an obesity epidemic model. PMID:22927889
Level crossings and turning points of random hyperbolic polynomials
Directory of Open Access Journals (Sweden)
K. Farahmand
1999-01-01
Full Text Available In this paper, we show that the asymptotic estimate for the expected number of K-level crossings of a random hyperbolic polynomial a1sinhx+a2sinh2x+⋯+ansinhnx, where aj(j=1,2,…,n are independent normally distributed random variables with mean zero and variance one, is (1/πlogn. This result is true for all K independent of x, provided K≡Kn=O(n. It is also shown that the asymptotic estimate of the expected number of turning points for the random polynomial a1coshx+a2cosh2x+⋯+ancoshnx, with aj(j=1,2,…,n as before, is also (1/πlogn.
On high-order polynomial heat-balance integral implementations
Directory of Open Access Journals (Sweden)
Wood Alastair S.
2009-01-01
Full Text Available This article reconsiders aspects of the analysis conventionally used to establish accuracy, performance and limitations of the heat balance integral method: theoretical and practical rates of convergence are confirmed for a familiar piecewise heat-balance integral based upon mesh refinement, and the use of boundary conditions is discussed with respect to fixed and moving boundaries. Alternates to mesh refinement are increased order of approximation or non-polynomial approximants. Here a physically intuitive high-order polynomial heat balance integral formulation is described that exhibits high accuracy, rapid convergence, and desirable qualitative solution properties. The simple approach combines a global approximant of prescribed degree with spatial sub-division of the solution domain. As a variational-type method, it can be argued that heat-balance integral is simply 'one amongst many'. The approach is compared with several established variational formulations and performance is additionally assessed in terms of 'smoothness'.
M-Polynomials and Topological Indices of Titania Nanotubes
Directory of Open Access Journals (Sweden)
Mobeen Munir
2016-10-01
Full Text Available Titania is one of the most comprehensively studied nanostructures due to their widespread applications in the production of catalytic, gas sensing, and corrosion-resistant materials. M-polynomial of nanotubes has been vastly investigated, as it produces many degree-based topological indices, which are numerical parameters capturing structural and chemical properties. These indices are used in the development of quantitative structure-activity relationships (QSARs in which the biological activity and other properties of molecules, such as boiling point, stability, strain energy, etc., are correlated with their structure. In this report, we provide M-polynomials of single-walled titania (SW TiO2 nanotubes and recover important topological degree-based indices to theoretically judge these nanotubes. We also plot surfaces associated to single-walled titania (SW TiO2 nanotubes.
Novel application of the kernel polynomial method to inhomogeneous superconductivity
Covaci, Lucian; Berciu, Mona
2009-05-01
Inhomogeneities (surface, interfaces, impurities, etc.) in superconductors give rise to interesting phenomena, like broken time-reversal states, bound states near surfaces, etc. Numerical solutions of the self-consistent Bogoliubov-de Gennes mean field equations become computationally intensive for systems whose translational symmetry is broken. We propose a new method of solving the mean-field equations based on the Kernel Polynomial Method by expanding the Green's functions in terms of Chebyshev polynomials and calculating the order parameters self-consistently. The benefits of this method are multiple: usage of large systems, easy implementation of symmetries, multiple bands. Although we apply this method to a specific example (formation of Andreev states in 2D superconductors), it is applicable to any mean-field calculation.
New graph polynomials in parametric QED Feynman integrals
Golz, Marcel
2017-10-01
In recent years enormous progress has been made in perturbative quantum field theory by applying methods of algebraic geometry to parametric Feynman integrals for scalar theories. The transition to gauge theories is complicated not only by the fact that their parametric integrand is much larger and more involved. It is, moreover, only implicitly given as the result of certain differential operators applied to the scalar integrand exp(-ΦΓ /ΨΓ) , where ΨΓ and ΦΓ are the Kirchhoff and Symanzik polynomials of the Feynman graph Γ. In the case of quantum electrodynamics we find that the full parametric integrand inherits a rich combinatorial structure from ΨΓ and ΦΓ. In the end, it can be expressed explicitly as a sum over products of new types of graph polynomials which have a combinatoric interpretation via simple cycle subgraphs of Γ.
Optimization of polynomials in non-commuting variables
Burgdorf, Sabine; Povh, Janez
2016-01-01
This book presents recent results on positivity and optimization of polynomials in non-commuting variables. Researchers in non-commutative algebraic geometry, control theory, system engineering, optimization, quantum physics and information science will find the unified notation and mixture of algebraic geometry and mathematical programming useful. Theoretical results are matched with algorithmic considerations; several examples and information on how to use NCSOStools open source package to obtain the results provided. Results are presented on detecting the eigenvalue and trace positivity of polynomials in non-commuting variables using Newton chip method and Newton cyclic chip method, relaxations for constrained and unconstrained optimization problems, semidefinite programming formulations of the relaxations and finite convergence of the hierarchies of these relaxations, and the practical efficiency of algorithms.
Uncertainty Quantification in Simulations of Epidemics Using Polynomial Chaos
Directory of Open Access Journals (Sweden)
F. Santonja
2012-01-01
Full Text Available Mathematical models based on ordinary differential equations are a useful tool to study the processes involved in epidemiology. Many models consider that the parameters are deterministic variables. But in practice, the transmission parameters present large variability and it is not possible to determine them exactly, and it is necessary to introduce randomness. In this paper, we present an application of the polynomial chaos approach to epidemiological mathematical models based on ordinary differential equations with random coefficients. Taking into account the variability of the transmission parameters of the model, this approach allows us to obtain an auxiliary system of differential equations, which is then integrated numerically to obtain the first-and the second-order moments of the output stochastic processes. A sensitivity analysis based on the polynomial chaos approach is also performed to determine which parameters have the greatest influence on the results. As an example, we will apply the approach to an obesity epidemic model.
Chebyshev recursion methods: Kernel polynomials and maximum entropy
Energy Technology Data Exchange (ETDEWEB)
Silver, R.N.; Roeder, H.; Voter, A.F.; Kress, J.D. [Los Alamos National Lab., NM (United States). Theoretical Div.
1995-10-01
The authors describe two Chebyshev recursion methods for calculations with very large sparse Hamiltonians, the kernel polynomial method (KPM) and the maximum entropy method (MEM). They are especially applicable to physical properties involving large numbers of eigenstates, which include densities of states, spectral functions, thermodynamics, total energies, as well as forces for molecular dynamics and Monte Carlo simulations. The authors apply Chebyshev methods to the electronic structure of Si, the thermodynamics of Heisenberg antiferromagnets, and a polaron problem.
Quaternionic polynomials with multiple zeros: A numerical point of view
Falcão, M. I.; Miranda, F.; Severino, R.; Soares, M. J.
2017-01-01
In the ring of quaternionic polynomials there is no easy solution to the problem of finding a suitable definition of multiplicity of a zero. In this paper we discuss different notions of multiple zeros available in the literature and add a computational point of view to this problem, by taking into account the behavior of the well known Newton's method in the presence of such roots.
On an inequality concerning the polar derivative of a polynomial
Indian Academy of Sciences (India)
For polynomials P (z) which does not vanish in the unit disk, the right-hand side of (2) can be improved. In fact , in ... authors on page 624 in lines 12 to 16 by using Lemma 2.3 is incorrect. The reason being ... iθ )]θ=0 and use the same argument as used by Govil et al (page 624, line 10 of [5]), then in view of the inequality.
A new approach to the theory of classical hypergeometric polynomials
Marco, José Manuel; Parcet, Javier
2003-01-01
In this paper we present a unified approach to the spectral analysis of an hypergeometric type operator whose eigenfunctions include the classical orthogonal polynomials. We write the eigenfunctions of this operator by means of a new Taylor formula for operators of Askey-Wilson type. This gives rise to some expressions for the eigenfunctions, which are unknown in such a general setting. Our methods also give a general Rodrigues formula from which several well known formulas of Rodrigues type ...
Null Steering Applications of Polynomials with Unimodular Coefficients.
1987-03-23
for any P<G, the Parseval , Theorem implies that the La norm of P on the unit circle C is (n +) = Furthermore, since IPlz)l a is expressible as z...Given the magnitude of the coefficients of a polynomial P, a finite subset S of the unit circle C , and a point p<C distinct from those in S, choose the
Recognition of Arabic Sign Language Alphabet Using Polynomial Classifiers
Directory of Open Access Journals (Sweden)
M. Al-Rousan
2005-08-01
Full Text Available Building an accurate automatic sign language recognition system is of great importance in facilitating efficient communication with deaf people. In this paper, we propose the use of polynomial classifiers as a classification engine for the recognition of Arabic sign language (ArSL alphabet. Polynomial classifiers have several advantages over other classifiers in that they do not require iterative training, and that they are highly computationally scalable with the number of classes. Based on polynomial classifiers, we have built an ArSL system and measured its performance using real ArSL data collected from deaf people. We show that the proposed system provides superior recognition results when compared with previously published results using ANFIS-based classification on the same dataset and feature extraction methodology. The comparison is shown in terms of the number of misclassified test patterns. The reduction in the rate of misclassified patterns was very significant. In particular, we have achieved a 36% reduction of misclassifications on the training data and 57% on the test data.
Hierarchical polynomial network approach to automated target recognition
Kim, Richard Y.; Drake, Keith C.; Kim, Tony Y.
1994-02-01
A hierarchical recognition methodology using abductive networks at several levels of object recognition is presented. Abductive networks--an innovative numeric modeling technology using networks of polynomial nodes--results from nearly three decades of application research and development in areas including statistical modeling, uncertainty management, genetic algorithms, and traditional neural networks. The systems uses pixel-registered multisensor target imagery provided by the Tri-Service Laser Radar sensor. Several levels of recognition are performed using detection, classification, and identification, each providing more detailed object information. Advanced feature extraction algorithms are applied at each recognition level for target characterization. Abductive polynomial networks process feature information and situational data at each recognition level, providing input for the next level of processing. An expert system coordinates the activities of individual recognition modules and enables employment of heuristic knowledge to overcome the limitations provided by a purely numeric processing approach. The approach can potentially overcome limitations of current systems such as catastrophic degradation during unanticipated operating conditions while meeting strict processing requirements. These benefits result from implementation of robust feature extraction algorithms that do not take explicit advantage of peculiar characteristics of the sensor imagery, and the compact, real-time processing capability provided by abductive polynomial networks.
Inelastic scattering with Chebyshev polynomials and preconditioned conjugate gradient minimization.
Temel, Burcin; Mills, Greg; Metiu, Horia
2008-03-27
We describe and test an implementation, using a basis set of Chebyshev polynomials, of a variational method for solving scattering problems in quantum mechanics. This minimum error method (MEM) determines the wave function Psi by minimizing the least-squares error in the function (H Psi - E Psi), where E is the desired scattering energy. We compare the MEM to an alternative, the Kohn variational principle (KVP), by solving the Secrest-Johnson model of two-dimensional inelastic scattering, which has been studied previously using the KVP and for which other numerical solutions are available. We use a conjugate gradient (CG) method to minimize the error, and by preconditioning the CG search, we are able to greatly reduce the number of iterations necessary; the method is thus faster and more stable than a matrix inversion, as is required in the KVP. Also, we avoid errors due to scattering off of the boundaries, which presents substantial problems for other methods, by matching the wave function in the interaction region to the correct asymptotic states at the specified energy; the use of Chebyshev polynomials allows this boundary condition to be implemented accurately. The use of Chebyshev polynomials allows for a rapid and accurate evaluation of the kinetic energy. This basis set is as efficient as plane waves but does not impose an artificial periodicity on the system. There are problems in surface science and molecular electronics which cannot be solved if periodicity is imposed, and the Chebyshev basis set is a good alternative in such situations.
Design of reinforced areas of concrete column using quadratic polynomials
Arif Gunadi, Tjiang; Parung, Herman; Rachman Djamaluddin, Abd; Arwin Amiruddin, A.
2017-11-01
Designing of reinforced concrete columns mostly carried out by a simple planning method which uses column interaction diagram. However, the application of this method is limited because it valids only for certain compressive strenght of the concrete and yield strength of the reinforcement. Thus, a more applicable method is still in need. Another method is the use of quadratic polynomials as a basis for the approach in designing reinforced concrete columns, where the ratio of neutral lines to the effective height of a cross section (ξ) if associated with ξ in the same cross-section with different reinforcement ratios is assumed to form a quadratic polynomial. This is identical to the basic principle used in the Simpson rule for numerical integral using quadratic polynomials and had a sufficiently accurate level of accuracy. The basis of this approach to be used both the normal force equilibrium and the moment equilibrium. The abscissa of the intersection of the two curves is the ratio that had been mentioned, since it fulfill both of the equilibrium. The application of this method is relatively more complicated than the existing method but provided with tables and graphs (N vs ξN ) and (M vs ξM ) so that its used could be simplified. The uniqueness of these tables are only distinguished based on the compresssive strength of the concrete, so in application it could be combined with various yield strenght of the reinforcement available in the market. This method could be solved by using programming languages such as Fortran.
A Knot Polynomial Invariant for Analysis of Topology of RNA Stems and Protein Disulfide Bonds
Directory of Open Access Journals (Sweden)
Tian Wei
2017-01-01
Full Text Available Knot polynomials have been used to detect and classify knots in biomolecules. Computation of knot polynomials in DNA and protein molecules have revealed the existence of knotted structures, and provided important insight into their topological structures. However, conventional knot polynomials are not well suited to study RNA molecules, as RNA structures are determined by stem regions which are not taken into account in conventional knot polynomials. In this study, we develop a new class of knot polynomials specifically designed to study RNA molecules, which considers stem regions. We demonstrate that our knot polynomials have direct structural relation with RNA molecules, and can be used to classify the topology of RNA secondary structures. Furthermore, we point out that these knot polynomials can be used to model the topological effects of disulfide bonds in protein molecules.
Directory of Open Access Journals (Sweden)
Qihua Tan
2011-01-01
Full Text Available Identifying the various gene expression response patterns is a challenging issue in expression microarray time-course experiments. Due to heterogeneity in the regulatory reaction among thousands of genes tested, it is impossible to manually characterize a parametric form for each of the time-course pattern in a gene by gene manner. We introduce a growth curve model with fractional polynomials to automatically capture the various time-dependent expression patterns and meanwhile efficiently handle missing values due to incomplete observations. For each gene, our procedure compares the performances among fractional polynomial models with power terms from a set of fixed values that offer a wide range of curve shapes and suggests a best fitting model. After a limited simulation study, the model has been applied to our human in vivo irritated epidermis data with missing observations to investigate time-dependent transcriptional responses to a chemical irritant. Our method was able to identify the various nonlinear time-course expression trajectories. The integration of growth curves with fractional polynomials provides a flexible way to model different time-course patterns together with model selection and significant gene identification strategies that can be applied in microarray-based time-course gene expression experiments with missing observations.
Directory of Open Access Journals (Sweden)
R. Lal
2012-01-01
Full Text Available The free transverse vibrations of thin nonhomogeneous rectangular plates of variable thickness have been studied using boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method. Gram-Schmidt process has been used to generate these orthogonal polynomials in two variables. The thickness variation is bidirectional and is the cartesian product of linear variations along two concurrent edges of the plate. The nonhomogeneity of the plate is assumed to arise due to linear variations in Young's modulus and density of the plate material with the in-plane coordinates. Numerical results have been computed for four different combinations of clamped, simply supported and free edges. Effect of the nonhomogeneity and thickness variation with varying values of aspect ratio on the natural frequencies of vibration is illustrated for the first three modes of vibration. Three dimensional mode shapes for all the four boundary conditions have been presented. A comparison of results with those available in the literature has been made.
Global Monte Carlo Simulation with High Order Polynomial Expansions
Energy Technology Data Exchange (ETDEWEB)
William R. Martin; James Paul Holloway; Kaushik Banerjee; Jesse Cheatham; Jeremy Conlin
2007-12-13
The functional expansion technique (FET) was recently developed for Monte Carlo simulation. The basic idea of the FET is to expand a Monte Carlo tally in terms of a high order expansion, the coefficients of which can be estimated via the usual random walk process in a conventional Monte Carlo code. If the expansion basis is chosen carefully, the lowest order coefficient is simply the conventional histogram tally, corresponding to a flat mode. This research project studied the applicability of using the FET to estimate the fission source, from which fission sites can be sampled for the next generation. The idea is that individual fission sites contribute to expansion modes that may span the geometry being considered, possibly increasing the communication across a loosely coupled system and thereby improving convergence over the conventional fission bank approach used in most production Monte Carlo codes. The project examined a number of basis functions, including global Legendre polynomials as well as “local” piecewise polynomials such as finite element hat functions and higher order versions. The global FET showed an improvement in convergence over the conventional fission bank approach. The local FET methods showed some advantages versus global polynomials in handling geometries with discontinuous material properties. The conventional finite element hat functions had the disadvantage that the expansion coefficients could not be estimated directly but had to be obtained by solving a linear system whose matrix elements were estimated. An alternative fission matrix-based response matrix algorithm was formulated. Studies were made of two alternative applications of the FET, one based on the kernel density estimator and one based on Arnoldi’s method of minimized iterations. Preliminary results for both methods indicate improvements in fission source convergence. These developments indicate that the FET has promise for speeding up Monte Carlo fission source
AND and/or OR: Uniform Polynomial-Size Circuits
Directory of Open Access Journals (Sweden)
Niall Murphy
2013-09-01
Full Text Available We investigate the complexity of uniform OR circuits and AND circuits of polynomial-size and depth. As their name suggests, OR circuits have OR gates as their computation gates, as well as the usual input, output and constant (0/1 gates. As is the norm for Boolean circuits, our circuits have multiple sink gates, which implies that an OR circuit computes an OR function on some subset of its input variables. Determining that subset amounts to solving a number of reachability questions on a polynomial-size directed graph (which input gates are connected to the output gate?, taken from a very sparse set of graphs. However, it is not obvious whether or not this (restricted reachability problem can be solved, by say, uniform AC^0 circuits (constant depth, polynomial-size, AND, OR, NOT gates. This is one reason why characterizing the power of these simple-looking circuits in terms of uniform classes turns out to be intriguing. Another is that the model itself seems particularly natural and worthy of study. Our goal is the systematic characterization of uniform polynomial-size OR circuits, and AND circuits, in terms of known uniform machine-based complexity classes. In particular, we consider the languages reducible to such uniform families of OR circuits, and AND circuits, under a variety of reduction types. We give upper and lower bounds on the computational power of these language classes. We find that these complexity classes are closely related to tallyNL, the set of unary languages within NL, and to sets reducible to tallyNL. Specifically, for a variety of types of reductions (many-one, conjunctive truth table, disjunctive truth table, truth table, Turing we give characterizations of languages reducible to OR circuit classes in terms of languages reducible to tallyNL classes. Then, some of these OR classes are shown to coincide, and some are proven to be distinct. We give analogous results for AND circuits. Finally, for many of our OR circuit
Multicomplex algebras on polynomials and generalized Hamilton dynamics
Yamaleev, Robert M.
2006-10-01
Generator of the complex algebra within the framework of general formulation obeys the quadratic equation. In this paper we explore multicomplex algebra with the generator obeying n-order polynomial equation with real coefficients. This algebra induces generalized trigonometry ((n+1)-gonometry), underlies of the nth order oscillator model and nth order Hamilton equations. The solution of an evolution equation generated by (nxn) matrix is represented via the set of (n+1)-gonometric functions. The general form of the first constant of motion of the evolution equation is established.
Statistical properties of quadratic polynomials with a neutral fixed point
Avila, Artur; Cheraghi, Davoud
2012-01-01
We describe the statistical properties of the dynamics of the quadratic polynomials P_a(z):=e^{2\\pi a i} z+z^2 on the complex plane, with a of high return times. In particular, we show that these maps are uniquely ergodic on their measure theoretic attractors, and the unique invariant probability is a physical measure describing the statistical behavior of typical orbits in the Julia set. This confirms a conjecture of Perez-Marco on the unique ergodicity of hedgehog dynamics, in this class of...
The A-polynomial in Chern-Simons theory
DEFF Research Database (Denmark)
Malusà, Alessandro
One of the most amusing aspects of mathematical physics is the great variety of areas of mathematics it relates to, and builds bridges between. The world of TQFT’s, and in particular Chern-Simons, relates to algebraic geometry via the theory of moduli spaces: one example of this is given by the A......-polynomial. This knot invariant is obtained from the algebraic geometry of character varieties, and takes the meaning of the equation of a constraint central in Chern-Simons theory. In my poster I wish to expose the construction of this invariant, and highlight its strong ties with mathematical physics....
A neural feedforward network with a polynomial nonlinearity
DEFF Research Database (Denmark)
Hoffmann, Nils
1992-01-01
A novel neural network based on the Wiener model is proposed. The network is composed of a hidden layer of preprocessing neurons followed by a polynomial nonlinearity and a linear output neuron. The author tries to solve the problem of finding an appropriate preprocessing method by using a modified...... backpropagation algorithm. It is shown by the use of calculation trees that the proposed approach is simple to implement, and that the computational complexity is not much larger than for the alternative method of using PCA to determine the weights in the preprocessing network. A simulation is given which...... indicates superior performance of the proposed network compared to the PCA network...
Simplified polynomial digital predistortion for multimode software defined radios
DEFF Research Database (Denmark)
Kardaras, Georgios; Soler, José; Dittmann, Lars
2010-01-01
Increasing the efficiency of mobile communication networks while keeping operational and maintenance cost at reasonable level is a major task for operators. Multimode radios capable of operating according to various mobile broadband standards, such as WCDMA, WiMAX and LTE, represent the new trend...... a simplified approach using polynomial digital predistortion in the intermediated frequency (IF) domain. It is fully implementable in software and no hardware changes are required on the digital or analog platform. The adaptation algorithm selected was Least Mean Squares because of its relevant simplicity...
Robust and fast computation for the polynomials of optics.
Forbes, G W
2010-06-21
Mathematical methods that are poorly known in the field of optics are adapted and shown to have striking significance. Orthogonal polynomials are common tools in physics and optics, but problems are encountered when they are used to higher orders. Applications to arbitrarily high orders are shown to be enabled by remarkably simple and robust algorithms that are derived from well known recurrence relations. Such methods are demonstrated for a couple of familiar optical applications where, just as in other areas, there is a clear trend to higher orders.
Application of ANNs approach for solving fully fuzzy polynomials system
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R. Novin
2017-11-01
Full Text Available In processing indecisive or unclear information, the advantages of fuzzy logic and neurocomputing disciplines should be taken into account and combined by fuzzy neural networks. The current research intends to present a fuzzy modeling method using multi-layer fuzzy neural networks for solving a fully fuzzy polynomials system. To clarify the point, it is necessary to inform that a supervised gradient descent-based learning law is employed. The feasibility of the method is examined using computer simulations on a numerical example. The experimental results obtained from the investigation of the proposed method are valid and delivers very good approximation results.
A Polynomial Term Structure Model with Macroeconomic Variables
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José Valentim Vicente
2007-06-01
Full Text Available Recently, a myriad of factor models including macroeconomic variables have been proposed to analyze the yield curve. We present an alternative factor model where term structure movements are captured by Legendre polynomials mimicking the statistical factor movements identified by Litterman e Scheinkmam (1991. We estimate the model with Brazilian Foreign Exchange Coupon data, adopting a Kalman filter, under two versions: the first uses only latent factors and the second includes macroeconomic variables. We study its ability to predict out-of-sample term structure movements, when compared to a random walk. We also discuss results on the impulse response function of macroeconomic variables.
On the Distance to a Root of Polynomials
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Somjate Chaiya
2011-01-01
Full Text Available In 2002, Dierk Schleicher gave an explicit estimate of an upper bound for the number of iterations of Newton's method it takes to find all roots of polynomials with prescribed precision. In this paper, we provide a method to improve the upper bound given by D. Schleicher. We give here an iterative method for finding an upper bound for the distance between a fixed point z in an immediate basin of a root α to α, which leads to a better upper bound for the number of iterations of Newton's method.
Universality of Mesoscopic Fluctuations for Orthogonal Polynomial Ensembles
Breuer, Jonathan; Duits, Maurice
2016-03-01
We prove that the fluctuations of mesoscopic linear statistics for orthogonal polynomial ensembles are universal in the sense that two measures with asymptotic recurrence coefficients have the same asymptotic mesoscopic fluctuations (under an additional assumption on the local regularity of one of the measures). The convergence rate of the recurrence coefficients determines the range of scales on which the limiting fluctuations are identical. Our main tool is an analysis of the Green's function for the associated Jacobi matrices.As a particular consequencewe obtain a central limit theorem for the modified Jacobi Unitary Ensembles on all mesoscopic scales.
Higher Cell Probe Lower Bounds for Evaluating Polynomials
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Larsen, Kasper Green
2012-01-01
In this paper, we study the cell probe complexity of evaluating an $n$-degree polynomial $P$ over a finite field $F$ of size at least $n^{1+Omega(1)}$. More specifically, we show that any static data structure for evaluating $P(x)$, where $x in F$, must use $Omega(lg |F|/lg(Sw/nlg|F|))$ cell probes...... to answer a query, where $S$ denotes the space of the data structure in number of cells and $w$ the cell size in bits. This bound holds in expectation for randomized data structures with any constant error probability $delta...
Refining Multivariate Value Set Bounds
Smith, Luke Alexander
Over finite fields, if the image of a polynomial map is not the entire field, then its cardinality can be bounded above by a significantly smaller value. Earlier results bound the cardinality of the value set using the degree of the polynomial, but more recent results make use of the powers of all monomials. In this paper, we explore the geometric properties of the Newton polytope and show how they allow for tighter upper bounds on the cardinality of the multivariate value set. We then explore a method which allows for even stronger upper bounds, regardless of whether one uses the multivariate degree or the Newton polytope to bound the value set. Effectively, this provides an alternate proof of Kosters' degree bound, an improved Newton polytope-based bound, and an improvement of a degree matrix-based result given by Zan and Cao.
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Chi Yaodan
2017-08-01
Full Text Available Crosstalk in wiring harness has been studied extensively for its importance in the naval ships electromagnetic compatibility field. An effective and high-efficiency method is proposed in this paper for analyzing Statistical Characteristics of crosstalk in wiring harness with random variation of position based on Polynomial Chaos Expansion (PCE. A typical 14-cable wiring harness was simulated as the object of research. Distance among interfering cable, affected cable and GND is synthesized and analyzed in both frequency domain and time domain. The model of naval ships wiring harness distribution parameter was established by utilizing Legendre orthogonal polynomials as basis functions along with prediction model of statistical characters. Detailed mean value, mean square error, probability density function and reasonable varying range of crosstalk in naval ships wiring harness are described in both time domain and frequency domain. Numerical experiment proves that the method proposed in this paper, not only has good consistency with the MC method can be applied in the naval ships EMC research field to provide theoretical support for guaranteeing safety, but also has better time-efficiency than the MC method. Therefore, the Polynomial Chaos Expansion method.
Karlita, Tita; Yuniarno, Eko Mulyanto; Purnama, I. Ketut Eddy; Purnomo, Mauridhi Hery
2017-06-01
Analyzing ultrasound (US) images to get the shapes and structures of particular anatomical regions is an interesting field of study since US imaging is a non-invasive method to capture internal structures of a human body. However, bone segmentation of US images is still challenging because it is strongly influenced by speckle noises and it has poor image quality. This paper proposes a combination of local phase symmetry and quadratic polynomial fitting methods to extract bone outer contour (BOC) from two dimensional (2D) B-modes US image as initial steps of three-dimensional (3D) bone surface reconstruction. By using local phase symmetry, the bone is initially extracted from US images. BOC is then extracted by scanning one pixel on the bone boundary in each column of the US images using first phase features searching method. Quadratic polynomial fitting is utilized to refine and estimate the pixel location that fails to be detected during the extraction process. Hole filling method is then applied by utilize the polynomial coefficients to fill the gaps with new pixel. The proposed method is able to estimate the new pixel position and ensures smoothness and continuity of the contour path. Evaluations are done using cow and goat bones by comparing the resulted BOCs with the contours produced by manual segmentation and contours produced by canny edge detection. The evaluation shows that our proposed methods produces an excellent result with average MSE before and after hole filling at the value of 0.65.
Salleh, Nur Hanim Mohd; Ali, Zalila; Noor, Norlida Mohd.; Baharum, Adam; Saad, Ahmad Ramli; Sulaiman, Husna Mahirah; Ahmad, Wan Muhamad Amir W.
2014-07-01
Polynomial regression is used to model a curvilinear relationship between a response variable and one or more predictor variables. It is a form of a least squares linear regression model that predicts a single response variable by decomposing the predictor variables into an nth order polynomial. In a curvilinear relationship, each curve has a number of extreme points equal to the highest order term in the polynomial. A quadratic model will have either a single maximum or minimum, whereas a cubic model has both a relative maximum and a minimum. This study used quadratic modeling techniques to analyze the effects of environmental factors: temperature, relative humidity, and rainfall distribution on the breeding of Aedes albopictus, a type of Aedes mosquito. Data were collected at an urban area in south-west Penang from September 2010 until January 2011. The results indicated that the breeding of Aedes albopictus in the urban area is influenced by all three environmental characteristics. The number of mosquito eggs is estimated to reach a maximum value at a medium temperature, a medium relative humidity and a high rainfall distribution.
Odake, Satoru
2017-12-01
In our previous papers [S. Odake and R. Sasaki, J. Phys. A 46, 245201 (2013) and S. Odake and R. Sasaki, J. Approx. Theory 193, 184 (2015)], the Wronskian identities for the Hermite, Laguerre, and Jacobi polynomials and the Casoratian identities for the Askey-Wilson polynomial and its reduced-form polynomials were presented. These identities are naturally derived through quantum-mechanical formulation of the classical orthogonal polynomials: ordinary quantum mechanics for the former and discrete quantum mechanics with pure imaginary shifts for the latter. In this paper we present the corresponding identities for the discrete quantum mechanics with real shifts. Infinitely many Casoratian identities for the q-Racah polynomial and its reduced-form polynomials are obtained.
Díaz Mendoza, C.; Orive, R.; Pijeira Cabrera, H.
2008-10-01
We study the asymptotic behavior of the zeros of a sequence of polynomials whose weighted norms, with respect to a sequence of weight functions, have the same nth root asymptotic behavior as the weighted norms of certain extremal polynomials. This result is applied to obtain the (contracted) weak zero distribution for orthogonal polynomials with respect to a Sobolev inner product with exponential weights of the form e-[phi](x), giving a unified treatment for the so-called Freud (i.e., when [phi] has polynomial growth at infinity) and Erdös (when [phi] grows faster than any polynomial at infinity) cases. In addition, we provide a new proof for the bound of the distance of the zeros to the convex hull of the support for these Sobolev orthogonal polynomials.
Frisch, Sophie; Glaz, Sarah; Tartarone, Francesca; Zanardo, Paolo
2017-01-01
This volume presents a collection of articles highlighting recent developments in commutative algebra and related non-commutative generalizations. 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, module theory and integer-valued polynomials along with connections to algebraic number theory, algebraic geometry, topology and homological algebra. Most of the eighteen contributions are authored by attendees of the two conferences in commutative algebra that were held in the summer of 2016: “Recent Advances in Commutative Ring and Module Theory,” Bressanone, Italy; “Conference on Rings and Polynomials” Graz, Austria. There is also a small collection of invite...
Real-time flight altitude estimation using phase correlation with Gram polynomial decimation
Choudhry, Aadil Jaleel; Badshah, Amir; Amin, Saadullah
2017-03-01
The paper presents a passive technique for real-time altitude above ground level estimation for aerial vehicles using a monocular camera, a GPS receiver and an inertial measurement unit. The paper discusses a robust method for featureless registration of successive images through phase correlation using Gram polynomial decimation. Altitude is estimated by formulating the shift in pixels between the images in terms of distance travelled, calculated using corresponding GPS latitudes and longitudes. Resultant value is compensated for changes in pitch before being passed through Savitzky-Golay filter. The system can generate results every 300ms on a lowcost commercial digital signal processor with mean error of 2m and standard deviation of 13m. The proposed system is suitable for speeds up to 300m/s and altitudes up to 3000m.
Typical Orbits of Quadratic Polynomials with a Neutral Fixed Point: Brjuno Type
Cheraghi, Davoud
2013-09-01
We describe the topological behavior of typical orbits of complex quadratic polynomials {P_{α}(z) = e^{2 π α {i}} z + z2}, with α of high return type. Here we prove that for such Brjuno values of α the closure of the critical orbit, which is the measure theoretic attractor of the map, has zero area. Then we show that the limit set of the orbit of a typical point in the Julia set of P α is equal to the closure of the critical orbit. Our method is based on the near parabolic renormalization of Inou-Shishikura, and a uniform optimal estimate on the derivative of the Fatou coordinate that we prove here.
A Geometric Method for Model Reduction of Biochemical Networks with Polynomial Rate Functions.
Samal, Satya Swarup; Grigoriev, Dima; Fröhlich, Holger; Weber, Andreas; Radulescu, Ovidiu
2015-12-01
Model reduction of biochemical networks relies on the knowledge of slow and fast variables. We provide a geometric method, based on the Newton polytope, to identify slow variables of a biochemical network with polynomial rate functions. The gist of the method is the notion of tropical equilibration that provides approximate descriptions of slow invariant manifolds. Compared to extant numerical algorithms such as the intrinsic low-dimensional manifold method, our approach is symbolic and utilizes orders of magnitude instead of precise values of the model parameters. Application of this method to a large collection of biochemical network models supports the idea that the number of dynamical variables in minimal models of cell physiology can be small, in spite of the large number of molecular regulatory actors.
Valuing Equality in Irish Social Care
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Niall Hanlon
2009-01-01
Full Text Available In this article the author critiques Irish social care by presenting an equality perspective on practice. An equality perspective involves developing emancipatory practices, that is, ways of helping that provide egalitarian solutions and outcomes. Although emancipatory values are often contrasted with traditional social care values, the author seeks a pragmatic and integrated approach to emancipatory practices rather than a restatement of traditional dichotomies. Emancipatory practice begins with an appreciation of the nature and relevance of inequalities on the lives of diverse social care users. Building a commitment to equality within social care education and practice is an important step in altering many individual and institutional social care practices by focussing on equality processes and outcomes as central social care objectives. Using a well credited framework that outlines five dimensions of inequality (Baker, Lynch, Cantillon and Walsh, 2004, the author argues that social care educators and practitioners need to debate the issues raised and develop emancipatory practices.
Polynomial Digital Control of a Series Equal Liquid Tanks
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Bobála Vladimír
2016-01-01
Full Text Available Time-delays are mainly caused by the time required to transport mass, energy or information, but they can also be caused by processing time or accumulation. Typical examples of such processes are e.g. pumps, liquid storing tanks, distillation columns or some types of chemical reactors. In many cases time-delay is caused by the effect produced by the accumulation of a large number of low-order systems. Several industrial processes have the time-delay effect produced by the accumulation of a great number of low-order systems with the identical dynamic. The dynamic behavior of series these low-order systems is expressed by high-order system. One of possibilities of control of such processes is their approximation by low-order model with time-delay. The paper is focused on the design of the digital polynomial control of a set of equal liquid cylinder atmospheric tanks. The designed control algorithms are realized using the digital Smith Predictor (SP based on polynomial approach – by minimization of the Linear Quadratic (LQ criterion. The LQ criterion was combined with pole assignment.
High Resolution of the ECG Signal by Polynomial Approximation
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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.
Solving polynomial systems using no-root elimination blending schemes
Barton, Michael
2011-12-01
Searching for the roots of (piecewise) polynomial systems of equations is a crucial problem in computer-aided design (CAD), and an efficient solution is in strong demand. Subdivision solvers are frequently used to achieve this goal; however, the subdivision process is expensive, and a vast number of subdivisions is to be expected, especially for higher-dimensional systems. Two blending schemes that efficiently reveal domains that cannot contribute by any root, and therefore significantly reduce the number of subdivisions, are proposed. Using a simple linear blend of functions of the given polynomial system, a function is sought after to be no-root contributing, with all control points of its BernsteinBézier representation of the same sign. If such a function exists, the domain is purged away from the subdivision process. The applicability is demonstrated on several CAD benchmark problems, namely surfacesurfacesurface intersection (SSSI) and surfacecurve intersection (SCI) problems, computation of the Hausdorff distance of two planar curves, or some kinematic-inspired tasks. © 2011 Elsevier Ltd. All rights reserved.
Developing the Polynomial Expressions for Fields in the ITER Tokamak
Sharma, Stephen
2017-10-01
The two most important problems to be solved in the development of working nuclear fusion power plants are: sustained partial ignition and turbulence. These two phenomena are the subject of research and investigation through the development of analytic functions and computational models. Ansatz development through Gaussian wave-function approximations, dielectric quark models, field solutions using new elliptic functions, and better descriptions of the polynomials of the superconducting current loops are the critical theoretical developments that need to be improved. Euler-Lagrange equations of motion in addition to geodesic formulations generate the particle model which should correspond to the Dirac dispersive scattering coefficient calculations and the fluid plasma model. Feynman-Hellman formalism and Heaviside step functional forms are introduced to the fusion equations to produce simple expressions for the kinetic energy and loop currents. Conclusively, a polynomial description of the current loops, the Biot-Savart field, and the Lagrangian must be uncovered before there can be an adequate computational and iterative model of the thermonuclear plasma.
Filtrations on Springer fiber cohomology and Kostka polynomials
Bellamy, Gwyn; Schedler, Travis
2017-09-01
We prove a conjecture which expresses the bigraded Poisson-de Rham homology of the nilpotent cone of a semisimple Lie algebra in terms of the generalized (one-variable) Kostka polynomials, via a formula suggested by Lusztig. This allows us to construct a canonical family of filtrations on the flag variety cohomology, and hence on irreducible representations of the Weyl group, whose Hilbert series are given by the generalized Kostka polynomials. We deduce consequences for the cohomology of all Springer fibers. In particular, this computes the grading on the zeroth Poisson homology of all classical finite W-algebras, as well as the filtration on the zeroth Hochschild homology of all quantum finite W-algebras, and we generalize to all homology degrees. As a consequence, we deduce a conjecture of Proudfoot on symplectic duality, relating in type A the Poisson homology of Slodowy slices to the intersection cohomology of nilpotent orbit closures. In the last section, we give an analogue of our main theorem in the setting of mirabolic D-modules.
Classification by using Prony's method with a polynomial model
Mueller, R.; Lee, W.; Okamitsu, J.
2012-06-01
Prony's Method with a Polynomial Model (PMPM) is a novel way of doing classification. Given a number of training samples with features and labels, it assumes a Gaussian mixture model for each feature, and uses Prony's method to determine a method of moments solution for the means and priors of the Gaussian distributions in the Gaussian mixture model. The features are then sorted in descending order by their relative performance. Based on the Gaussian mixture model of the first feature, training samples are partitioned into clusters by determining which Gaussian distribution each training sample is most likely from. Then with the training samples in each cluster, a new Gaussian mixture model is built for the next most powerful feature. This process repeats until a Gaussian mixture model is built for each feature, and a tree is thus grown with the training data partitioned into several final clusters. A "leaf" model for each final cluster is the weighted least squares solution (regression) for approximating a polynomial function of the features to the truth labels. Testing consists of determining for each testing sample a likelihood that the testing sample belongs to each cluster, and then regressions are weighted by their likelihoods and averaged to produce the test confidence. Evaluation of PMPM is done by extracting features from data collected by both Ground Penetrating Radar and Metal Detector of a robot-mounted land-mine detection system, training PMPM models, and testing in a cross-validation fashion.
Anomalies in non-polynomial closed string field theory
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Kaku, Michio (Institute for Advanced Study, Princeton, NJ (USA))
1990-11-01
The complete classical action for the non-polynomial closed string field theory was written down last year by the author and the Kyoto group. It successfully reproduces all closed string tree diagrams, but fails to reproduce modular invariant loop amplitudes. In this paper we show that the classical action is also riddled with gauge anomalies. Thus, the classical action is not really gauge invariant and fails as a quantum theory. The presence of gauge anomalies and the violation of modular invariance appear to be a disaster for the theory. Actually, this is a blessing in disguise. We show that by adding new non-polynomial terms to the action, we can simultaneously eliminate both the gauge anomalies and the modular-violating loop diagrams. We show this explicitly at the one loop level and also for an infinite class of p-puncture, genus-g amplitudes, making use of a series of non-trivial identities. The theory is thus an acceptable quantum theory. We comment on the origin of this strange link between local gauge anomalies and global modular invariance. (orig.).
Positive polynomials, convex integral polytopes, and a random walk problem
Handelman, David E
1987-01-01
Emanating from the theory of C*-algebras and actions of tori theoren, the problems discussed here are outgrowths of random walk problems on lattices. An AGL (d,Z)-invariant (which is a partially ordered commutative algebra) is obtained for lattice polytopes (compact convex polytopes in Euclidean space whose vertices lie in Zd), and certain algebraic properties of the algebra are related to geometric properties of the polytope. There are also strong connections with convex analysis, Choquet theory, and reflection groups. This book serves as both an introduction to and a research monograph on the many interconnections between these topics, that arise out of questions of the following type: Let f be a (Laurent) polynomial in several real variables, and let P be a (Laurent) polynomial with only positive coefficients; decide under what circumstances there exists an integer n such that Pnf itself also has only positive coefficients. It is intended to reach and be of interest to a general mathematical audience as we...
Polynomial Chaos Expansion Approach to Interest Rate Models
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Luca Di Persio
2015-01-01
Full Text Available The Polynomial Chaos Expansion (PCE technique allows us to recover a finite second-order random variable exploiting suitable linear combinations of orthogonal polynomials which are functions of a given stochastic quantity ξ, hence acting as a kind of random basis. The PCE methodology has been developed as a mathematically rigorous Uncertainty Quantification (UQ method which aims at providing reliable numerical estimates for some uncertain physical quantities defining the dynamic of certain engineering models and their related simulations. In the present paper, we use the PCE approach in order to analyze some equity and interest rate models. In particular, we take into consideration those models which are based on, for example, the Geometric Brownian Motion, the Vasicek model, and the CIR model. We present theoretical as well as related concrete numerical approximation results considering, without loss of generality, the one-dimensional case. We also provide both an efficiency study and an accuracy study of our approach by comparing its outputs with the ones obtained adopting the Monte Carlo approach, both in its standard and its enhanced version.
A Burge tree of Virasoro-type polynomial identities
Foda, O E; Welsh, Trevor A; Foda, Omar; Lee, Keith S. M.; Welsh, Trevor A.
1997-01-01
Using a summation formula due to Burge, and a combinatorial identity between partition pairs, we obtain an infinite tree of q-polynomial identities for the Virasoro characters \\chi^{p, p'}_{r, s}, dependent on two finite size parameters M and N, in the cases where: (i) p and p' are coprime integers that satisfy 0 = 1, then the pair (s, r) has a continued fraction (c_1, c_2, ... , c_{u-1}, d), where 1 = infinity, for fixed N, and the limit N -> infinity, for fixed M, lead to two independent boson-fermion-type q-polynomial identities: in one case, the bosonic side has a conventional dependence on the parameters that characterise the corresponding character. In the other, that dependence is not conventional. In each case, the fermionic side can also be cast in either of two different forms. Taking the remaining finite size parameter to infinity in either of the above identities, so that M -> infinity and N -> infinity, leads to the same q-series identity for the corresponding character.
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Hjalmar Rosengren
2006-12-01
Full Text Available We study multivariable Christoffel-Darboux kernels, which may be viewed as reproducing kernels for antisymmetric orthogonal polynomials, and also as correlation functions for products of characteristic polynomials of random Hermitian matrices. Using their interpretation as reproducing kernels, we obtain simple proofs of Pfaffian and determinant formulas, as well as Schur polynomial expansions, for such kernels. In subsequent work, these results are applied in combinatorics (enumeration of marked shifted tableaux and number theory (representation of integers as sums of squares.
On Linear Combinations of Two Orthogonal Polynomial Sequences on the Unit Circle
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Suárez C
2010-01-01
Full Text Available Let be a monic orthogonal polynomial sequence on the unit circle. We define recursively a new sequence of polynomials by the following linear combination: , , . In this paper, we give necessary and sufficient conditions in order to make be an orthogonal polynomial sequence too. Moreover, we obtain an explicit representation for the Verblunsky coefficients and in terms of and . Finally, we show the relation between their corresponding Carathéodory functions and their associated linear functionals.
Real Scalar Field Scattering with Polynomial Approximation around Schwarzschild-de Sitter Black-hole
Liu, Molin; Liu, Hongya; Zhang, Jingfei; Yu, Fei
2008-01-01
As one of the fitting methods, the polynomial approximation is effective to process sophisticated problem. In this paper, we employ this approach to handle the scattering of scalar field around the Schwarzschild-de Sitter black-hole. The complex relationship between tortoise coordinate and radial coordinate is replaced by the approximate polynomial. The Schr$\\ddot{o}$dinger-like equation, the real boundary conditions and the polynomial approximation construct a full Sturm-Liouville type probl...
Exact Output Response Computation of RC Interconnects Under General Polynomial Input Waveforms
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L. M. Patnaik
2000-01-01
Full Text Available Accurate output response computation of RC interconnects under various input excitations is a key issue in deep submicron delay analysis. In this paper, we present an exact analysis of output response computation of a distributed RC interconnect under input signals that are polynomial in time (tn. A simple, recursive equation that helps us to calculate the interconnect response under higher order polynomial inputs in terms of the lower order polynomial responses is derived. To the best of our knowledge, this is the first exact output response analysis of RC interconnects under generalized polynomial inputs.
Szász-Durrmeyer operators involving Boas-Buck polynomials of blending type
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Manjari Sidharth
2017-05-01
Full Text Available Abstract The present paper introduces the Szász-Durrmeyer type operators based on Boas-Buck type polynomials which include Brenke type polynomials, Sheffer polynomials and Appell polynomials considered by Sucu et al. (Abstr. Appl. Anal. 2012:680340, 2012. We establish the moments of the operator and a Voronvskaja type asymptotic theorem and then proceed to studying the convergence of the operators with the help of Lipschitz type space and weighted modulus of continuity. Next, we obtain a direct approximation theorem with the aid of unified Ditzian-Totik modulus of smoothness. Furthermore, we study the approximation of functions whose derivatives are locally of bounded variation.
Zeros and logarithmic asymptotics of Sobolev orthogonal polynomials for exponential weights
Díaz Mendoza, C.; Orive, R.; Pijeira Cabrera, H.
2009-12-01
We obtain the (contracted) weak zero asymptotics for orthogonal polynomials with respect to Sobolev inner products with exponential weights in the real semiaxis, of the form , with [gamma]>0, which include as particular cases the counterparts of the so-called Freud (i.e., when [phi] has a polynomial growth at infinity) and Erdös (when [phi] grows faster than any polynomial at infinity) weights. In addition, the boundness of the distance of the zeros of these Sobolev orthogonal polynomials to the convex hull of the support and, as a consequence, a result on logarithmic asymptotics are derived.
Orthonormal aberration polynomials for anamorphic optical imaging systems with circular pupils.
Mahajan, Virendra N
2012-06-20
In a recent paper, we considered the classical aberrations of an anamorphic optical imaging system with a rectangular pupil, representing the terms of a power series expansion of its aberration function. These aberrations are inherently separable in the Cartesian coordinates (x,y) of a point on the pupil. Accordingly, there is x-defocus and x-coma, y-defocus and y-coma, and so on. We showed that the aberration polynomials orthonormal over the pupil and representing balanced aberrations for such a system are represented by the products of two Legendre polynomials, one for each of the two Cartesian coordinates of the pupil point; for example, L(l)(x)L(m)(y), where l and m are positive integers (including zero) and L(l)(x), for example, represents an orthonormal Legendre polynomial of degree l in x. The compound two-dimensional (2D) Legendre polynomials, like the classical aberrations, are thus also inherently separable in the Cartesian coordinates of the pupil point. Moreover, for every orthonormal polynomial L(l)(x)L(m)(y), there is a corresponding orthonormal polynomial L(l)(y)L(m)(x) obtained by interchanging x and y. These polynomials are different from the corresponding orthogonal polynomials for a system with rotational symmetry but a rectangular pupil. In this paper, we show that the orthonormal aberration polynomials for an anamorphic system with a circular pupil, obtained by the Gram-Schmidt orthogonalization of the 2D Legendre polynomials, are not separable in the two coordinates. Moreover, for a given polynomial in x and y, there is no corresponding polynomial obtained by interchanging x and y. For example, there are polynomials representing x-defocus, balanced x-coma, and balanced x-spherical aberration, but no corresponding y-aberration polynomials. The missing y-aberration terms are contained in other polynomials. We emphasize that the Zernike circle polynomials, although orthogonal over a circular pupil, are not suitable for an anamorphic system as
An Alternative Definition of the Hermite Polynomials Related to the Dunkl Laplacian
Directory of Open Access Journals (Sweden)
Hendrik De Bie
2008-12-01
Full Text Available We introduce the so-called Clifford-Hermite polynomials in the framework of Dunkl operators, based on the theory of Clifford analysis. Several properties of these polynomials are obtained, such as a Rodrigues formula, a differential equation and an explicit relation connecting them with the generalized Laguerre polynomials. A link is established with the generalized Hermite polynomials related to the Dunkl operators (see [Rösler M., Comm. Math. Phys. 192 (1998, 519-542, q-alg/9703006.] as well as with the basis of the weighted $L^2$ space introduced by Dunkl.
Hydrodynamic-type systems describing 2-dimensional polynomially integrable geodesic flows
Manno, Gianni; Pavlov, Maxim V.
2017-03-01
Starting from a homogeneous polynomial in momenta of arbitrary order we extract multi-component hydrodynamic-type systems which describe 2-dimensional geodesic flows admitting the initial polynomial as integral. All these hydrodynamic-type systems are semi-Hamiltonian, thus implying that they are integrable according to the generalized hodograph method. Moreover, they are integrable in a constructive sense as polynomial first integrals allow to construct generating equations of conservation laws. According to the multiplicity of the roots of the polynomial integral, we separate integrable particular cases.
A new derivation of the highest-weight polynomial of a unitary lie algebra
Energy Technology Data Exchange (ETDEWEB)
P Chau, Huu-Tai; P Van, Isacker [Grand Accelerateur National d' Ions Lourds (GANIL), 14 - Caen (France)
2000-07-01
A new method is presented to derive the expression of the highest-weight polynomial used to build the basis of an irreducible representation (IR) of the unitary algebra U(2J+1). After a brief reminder of Moshinsky's method to arrive at the set of equations defining the highest-weight polynomial of U(2J+1), an alternative derivation of the polynomial from these equations is presented. The method is less general than the one proposed by Moshinsky but has the advantage that the determinantal expression of the highest-weight polynomial is arrived at in a direct way using matrix inversions. (authors)
An extended class of orthogonal polynomials defined by a Sturm-Liouville problem
Gómez-Ullate, David; Kamran, Niky; Milson, Robert
2008-01-01
We present two infinite sequences of polynomial eigenfunctions of a Sturm-Liouville problem. As opposed to the classical orthogonal polynomial systems, these sequences start with a polynomial of degree one. We denote these polynomials as $X_1$-Jacobi and $X_1$-Laguerre and we prove that they are orthogonal with respect to a positive definite inner product defined over the the compact interval $[-1,1]$ or the half-line $[0,\\infty)$, respectively, and they are a basis of the corresponding $L^2$...
On trees with the same restricted U-polynomial and the Prouhet-Tarry-Escott problem
Aliste-Prieto, José; de Mier, Anna; Zamora, José
2015-01-01
This paper focuses on the well-known problem due to Stanley of whether two non-isomorphic trees can have the same U-polynomial (or, equivalently, the same chromatic symmetric function). We consider the Uk-polynomial, which is a restricted version of U-polynomial, and construct, for any given kk, non-isomorphic trees with the same Uk-polynomial. These trees are constructed by encoding solutions of the Prouhet–Tarry–Escott problem. As a consequence, we find a new class of trees that are disting...
Chen, Zhixiang; Fu, Bin
This paper is our third step towards developing a theory of testing monomials in multivariate polynomials and concentrates on two problems: (1) How to compute the coefficients of multilinear monomials; and (2) how to find a maximum multilinear monomial when the input is a ΠΣΠ polynomial. We first prove that the first problem is #P-hard and then devise a O *(3 n s(n)) upper bound for this problem for any polynomial represented by an arithmetic circuit of size s(n). Later, this upper bound is improved to O *(2 n ) for ΠΣΠ polynomials. We then design fully polynomial-time randomized approximation schemes for this problem for ΠΣ polynomials. On the negative side, we prove that, even for ΠΣΠ polynomials with terms of degree ≤ 2, the first problem cannot be approximated at all for any approximation factor ≥ 1, nor "weakly approximated" in a much relaxed setting, unless P=NP. For the second problem, we first give a polynomial time λ-approximation algorithm for ΠΣΠ polynomials with terms of degrees no more a constant λ ≥ 2. On the inapproximability side, we give a n (1 - ɛ)/2 lower bound, for any ɛ> 0, on the approximation factor for ΠΣΠ polynomials. When the degrees of the terms in these polynomials are constrained as ≤ 2, we prove a 1.0476 lower bound, assuming Pnot=NP; and a higher 1.0604 lower bound, assuming the Unique Games Conjecture.
Laguerre polynomial excited coherent state: generation and nonclassical properties
Ye, Wei; Zhou, Weidong; Zhang, Haoliang; Liu, Cunjin; Huang, Jiehui; Hu, Liyun
2017-11-01
We propose a theoretical protocol to generate a kind of non-Gaussian state—a Laguerre polynomial excited coherent state (LPECS) by exploiting a two-mode squeezing transformation and a conditional measurement with a coherent state input. Then we investigate the nonclassical features of the LPECS according to the Glauber-Sudarshan P(α ) function, photon number distribution, Mandel’s Q parameter, second-order correlation function, and squeezing properties as well as negative Wigner distribution. Our results show that the generated output state presents obvious nonclassical properties which can be modulated by a coherent amplitude, a squeezing parameter and a conditional measurement. In particular, the squeezing and negative Wigner function are clear.
Coulomb branch Hilbert series and Hall-Littlewood polynomials
Cremonesi, Stefano; Mekareeya, Noppadol; Zaffaroni, Alberto
2014-01-01
There has been a recent progress in understanding the chiral ring of 3d $\\mathcal{N}=4$ superconformal gauge theories by explicitly constructing an exact generating function (Hilbert series) counting BPS operators on the Coulomb branch. In this paper we introduce Coulomb branch Hilbert series in the presence of background magnetic charges for flavor symmetries, which are useful for computing the Hilbert series of more general theories through gluing techniques. We find a simple formula of the Hilbert series with background magnetic charges for $T_\\rho(G)$ theories in terms of Hall-Littlewood polynomials. Here $G$ is a classical group and $\\rho$ is a certain partition related to the dual group of $G$. The Hilbert series for vanishing background magnetic charges show that Coulomb branches of $T_\\rho(G)$ theories are complete intersections. We also demonstrate that mirror symmetry maps background magnetic charges to baryonic charges.
Learning Mixtures of Polynomials of Conditional Densities from Data
DEFF Research Database (Denmark)
L. López-Cruz, Pedro; Nielsen, Thomas Dyhre; Bielza, Concha
2013-01-01
Mixtures of polynomials (MoPs) are a non-parametric density estimation technique for hybrid Bayesian networks with continuous and discrete variables. We propose two methods for learning MoP ap- proximations of conditional densities from data. Both approaches are based on learning MoP approximations......- ods with the approach for learning mixtures of truncated basis functions from data....... of the joint density and the marginal density of the conditioning variables, but they differ as to how the MoP approximation of the quotient of the two densities is found. We illustrate the methods using data sampled from a simple Gaussian Bayesian network. We study and compare the performance of these meth...
On Continued Fraction Expansion of Real Roots of Polynomial Systems
DEFF Research Database (Denmark)
Mantzaflaris, Angelos; Mourrain, Bernard; Tsigaridas, Elias
2011-01-01
) and are feasible over unbounded regions. Then, we study an algorithm to split this representation and obtain a subdivision scheme for the domain of multivariate polynomial functions. This implies a new algorithm for real root isolation, MCF, that generalizes the Continued Fraction (CF) algorithm of univariate......, 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...... number of the system. Examples computed with our C++ implementation illustrate the practical aspects of our method....
TreeCmp: Comparison of Trees in Polynomial Time
Bogdanowicz, Damian; Giaro, Krzysztof; Wróbel, Borys
2012-01-01
When a phylogenetic reconstruction does not result in one tree but in several, tree metrics permit finding out how far the reconstructed trees are from one another. They also permit to assess the accuracy of a reconstruction if a true tree is known. TreeCmp implements eight metrics that can be calculated in polynomial time for arbitrary (not only bifurcating) trees: four for unrooted (Matching Split metric, which we have recently proposed, Robinson-Foulds, Path Difference, Quartet) and four for rooted trees (Matching Cluster, Robinson-Foulds cluster, Nodal Splitted and Triple). TreeCmp is the first implementation of Matching Split/Cluster metrics and the first efficient and convenient implementation of Nodal Splitted. It allows to compare relatively large trees. We provide an example of the application of TreeCmp to compare the accuracy of ten approaches to phylogenetic reconstruction with trees up to 5000 external nodes, using a measure of accuracy based on normalized similarity between trees.
Optimized polynomial expansions: potentials and phase shift analyses
Energy Technology Data Exchange (ETDEWEB)
Marker, D.; Rijken, T.; Bohannon, G.; Signell, P.
1980-03-18
The Cutkosky-Deo-Ciulli-Chao Optimized Polynomial Expansion as applied to the proton-proton scattering amplitudes was examined. Chao's positive numerical results for intermediate energy scattering data could not be reproduced. Instead, a negative conclusion as to the method's usefulness for that application was reached. Source(s) of the discrepancies with Chao's results could not be located definitely, but there were indications of a problem with a relative phase. The ability of the method to predict the higher angular momentum parts of the amplitudes for three different potentials having realistic parts was also examined. Again, the method was found to have very meager successes. 7 tables.
Boson normal ordering via substitutions and Sheffer-type polynomials
Energy Technology Data Exchange (ETDEWEB)
Blasiak, P. [H.Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, ul. Eliasza-Radzikowskiego 152, PL-31342 Cracow (Poland) and Laboratoire de Physique Theorique des Liquides, Universite Pierre et Marie Curie, CNRS UMR 7600, Tour 24 - 2e et., 4 pl. Jussieu, F-75252 Paris Cedex 05 (France)]. E-mail: blasiak@lptl.jussieu.fr; Horzela, A. [H.Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, ul. Eliasza-Radzikowskiego 152, PL-31342 Cracow (Poland)]. E-mail: andrzej.horzela@ifj.edu.pl; Penson, K.A. [Laboratoire de Physique Theorique des Liquides, Universite Pierre et Marie Curie, CNRS UMR 7600, Tour 24 - 2e et., 4 pl. Jussieu, F-75252 Paris Cedex 05 (France)]. E-mail: penson@lptl.jussieu.fr; Duchamp, G.H.E. [Institut Galilee, LIPN, 99 Av. J.-B. Clement, F-93430 Villetaneuse (France)]. E-mail: ghed@lipn-univ.paris13.fr; Solomon, A.I. [Laboratoire de Physique Theorique des Liquides, Universite Pierre et Marie Curie, CNRS UMR 7600, Tour 24 - 2e et., 4 pl. Jussieu, F-75252 Paris Cedex 05 (France) and Open University, Physics and Astronomy Department, Milton Keynes MK7 6AA (United Kingdom)]. E-mail: a.i.solomon@open.ac.uk
2005-04-25
We solve the boson normal ordering problem for (q(a{sup -}bar )a+v(a{sup -}bar )){sup n} with arbitrary functions q and v and integer n, where a and a{sup -}bar are boson annihilation and creation operators, satisfying [a,a{sup -}bar ]=1. This leads to exponential operators generalizing the shift operator and we show that their action can be expressed in terms of substitutions. Our solution is naturally related through the coherent state representation to the exponential generating functions of Sheffer-type polynomials. This in turn opens a vast arena of combinatorial methodology which is applied to boson normal ordering and illustrated by a few examples.
Uncertainty Analysis via Failure Domain Characterization: Polynomial Requirement Functions
Crespo, Luis G.; Munoz, Cesar A.; Narkawicz, Anthony J.; Kenny, Sean P.; Giesy, Daniel P.
2011-01-01
This paper proposes an uncertainty analysis framework based on the characterization of the uncertain parameter space. This characterization enables the identification of worst-case uncertainty combinations and the approximation of the failure and safe domains with a high level of accuracy. Because these approximations are comprised of subsets of readily computable probability, they enable the calculation of arbitrarily tight upper and lower bounds to the failure probability. A Bernstein expansion approach is used to size hyper-rectangular subsets while a sum of squares programming approach is used to size quasi-ellipsoidal subsets. These methods are applicable to requirement functions whose functional dependency on the uncertainty is a known polynomial. Some of the most prominent features of the methodology are the substantial desensitization of the calculations from the uncertainty model assumed (i.e., the probability distribution describing the uncertainty) as well as the accommodation for changes in such a model with a practically insignificant amount of computational effort.
Virasoro constraints and polynomial recursion for the linear Hodge integrals
Guo, Shuai; Wang, Gehao
2017-04-01
The Hodge tau-function is a generating function for the linear Hodge integrals. It is also a tau-function of the KP hierarchy. In this paper, we first present the Virasoro constraints for the Hodge tau-function in the explicit form of the Virasoro equations. The expression of our Virasoro constraints is simply a linear combination of the Virasoro operators, where the coefficients are restored from a power series for the Lambert W function. Then, using this result, we deduce a simple version of the Virasoro constraints for the linear Hodge partition function, where the coefficients are restored from the Gamma function. Finally, we establish the equivalence relation between the Virasoro constraints and polynomial recursion formula for the linear Hodge integrals.
Delta T: Polynomial Approximation of Time Period 1620–2013
Directory of Open Access Journals (Sweden)
M. Khalid
2014-01-01
Full Text Available The difference between the Uniform Dynamical Time and Universal Time is referred to as ΔT (delta T. Delta T is used in numerous astronomical calculations, that is, eclipses,and length of day. It is additionally required to reduce quantified positions of minor planets to a uniform timescale for the purpose of orbital determination. Since Universal Time is established on the basis of the variable rotation of planet Earth, the quantity ΔT mirrors the unevenness of that rotation, and so it changes slowly, but rather irregularly, as time passes. We have worked on empirical formulae for estimating ΔT and have discovered a set of polynomials of the 4th order with nine intervals which is accurate within the range of ±0.6 seconds for the duration of years 1620–2013.
Predicting Cutting Forces in Aluminum Using Polynomial Classifiers
Kadi, H. El; Deiab, I. M.; Khattab, A. A.
Due to increased calls for environmentally benign machining processes, there has been focus and interest in making processes more lean and agile to enhance efficiency, reduce emissions and increase profitability. One approach to achieving lean machining is to develop a virtual simulation environment that enables fast and reasonably accurate predictions of various machining scenarios. Polynomial Classifiers (PCs) are employed to develop a smart data base that can provide fast prediction of cutting forces resulting from various combinations of cutting parameters. With time, the force model can expand to include different materials, tools, fixtures and machines and would be consulted prior to starting any job. In this work, first, second and third order classifiers are used to predict the cutting coefficients that can be used to determine the cutting forces. Predictions obtained using PCs are compared to experimental results and are shown to be in good agreement.
Some Relations between Admissible Monomials for the Polynomial Algebra
Directory of Open Access Journals (Sweden)
Mbakiso Fix Mothebe
2015-01-01
Full Text Available Let P(n=F2[x1,…,xn] be the polynomial algebra in n variables xi, of degree one, over the field F2 of two elements. The mod-2 Steenrod algebra A acts on P(n according to well known rules. A major problem in algebraic topology is of determining A+P(n, the image of the action of the positively graded part of A. We are interested in the related problem of determining a basis for the quotient vector space Q(n=P(n/A+P(n. Q(n has been explicitly calculated for n=1,2,3,4 but problems remain for n≥5. Both P(n=⨁d≥0Pd(n and Q(n are graded, where Pd(n denotes the set of homogeneous polynomials of degree d. In this paper, we show that if u=x1m1⋯xn-1mn-1∈Pd′(n-1 is an admissible monomial (i.e., u meets a criterion to be in a certain basis for Q(n-1, then, for any pair of integers (j,λ, 1≤j≤n, and λ≥0, the monomial hjλu=x1m1⋯xj-1mj-1xj2λ-1xj+1mj⋯xnmn-1∈Pd′+(2λ-1(n is admissible. As an application we consider a few cases when n=5.
Ren, Gang; Du, Jian-ming; Zhang, Wen-hai; Yu, Hai-jun
2017-05-01
We examine nonclassical properties of the quantum state generated by applying Hermite polynomials photon-added operator on the even/odd coherent state (HPECS/HPOCS). Explicit expressions for its nonclassical properties, such as quantum statistical properties and squeezing phenomenon, are obtained. It is interesting to find that the HPECS/HPOCS exhibits sub-Poissonian distribution, anti-bunching effects and negative values of the Wigner function. Thus, we confirm the HPPECS/HPPOCS is a new nonclassical state. Finally, we reveal that the HPPECS/HPPOCS is a novel intelligent state by its squeezing effects in position distribution and quadrature squeezing.
Directory of Open Access Journals (Sweden)
Huanhe Dong
2014-01-01
Full Text Available We introduce how to obtain the bilinear form and the exact periodic wave solutions of a class of (2+1-dimensional nonlinear integrable differential equations directly and quickly with the help of the generalized Dp-operators, binary Bell polynomials, and a general Riemann theta function in terms of the Hirota method. As applications, we solve the periodic wave solution of BLMP equation and it can be reduced to soliton solution via asymptotic analysis when the value of p is 5.
Nonnegative Polynomial with no Certificate of Nonnegativity in the Simplicial Bernstein Basis
DEFF Research Database (Denmark)
Sloth, Christoffer
2017-01-01
This paper presents a nonnegative polynomial that cannot be represented with nonnegative coefficients in the simplicial Bernstein basis by subdividing the standard simplex. The example shows that Bernstein Theorem cannot be extended to certificates of nonnegativity for polynomials with zeros at i...
Numerical solution of nonlinear Volterra-Fredholm integral equation by using Chebyshev polynomials
Directory of Open Access Journals (Sweden)
R. Ezzati
2011-03-01
Full Text Available In this paper, we have used Chebyshev polynomials to solve linearand nonlinear Volterra-Fredholm integral equations, numerically.First we introduce these polynomials, then we use them to changethe Volterra-Fredholm integral equation to a linear or nonlinearsystem. Finally, the numerical examples illustrate the efficiencyof this method.
E. de Klerk (Etienne); M. Laurent (Monique)
2010-01-01
htmlabstractThe Lasserre hierarchy of semidefinite programming approximations to convex polynomial optimization problems is known to converge finitely under some assumptions. [J.B. Lasserre. Convexity in semialgebraic geometry and polynomial optimization. SIAM J. Optim. 19, 1995-2014, 2009.] We give
M. Laurent (Monique); E. de Klerk (Etienne)
2011-01-01
htmlabstractThe Lasserre hierarchy of semidenite programming approximations to convex polynomial optimization problems is known to converge nitely under some assumptions. [J.B. Lasserre. Convexity in semialgebraic geometry and polynomial optimization. SIAM J. Optim. 19, 1995{2014,2009.] We give a
de Klerk, E.; Laurent, M.
2011-01-01
The Lasserre hierarchy of semidefinite programming approximations to convex polynomial optimization problems is known to converge finitely under some assumptions. [J. B. Lasserre, Convexity in semialgebraic geometry and polynomial optimization, SIAM J. Optim., 19 (2009), pp. 1995–2014]. We give a
The Approximation Szász-Chlodowsky Type Operators Involving Gould-Hopper Type Polynomials
Baxhaku, Behar; Berisha, Artan
2017-01-01
We introduce the Szász and Chlodowsky operators based on Gould-Hopper polynomials and study the statistical convergence of these operators in a weighted space of functions on a positive semiaxis. Further, a Voronovskaja type result is obtained for the operators containing Gould-Hopper polynomials. Finally, some graphical examples for the convergence of this type of operator are given.
Continuous and discrete best polynomial degree reduction with Jacobi and Hahn weights
Ait-Haddou, Rachid
2016-03-02
We show that the weighted least squares approximation of Bézier coefficients with Hahn weights provides the best polynomial degree reduction in the Jacobi L2L2-norm. A discrete analogue of this result is also provided. Applications to Jacobi and Hahn orthogonal polynomials are presented.
Directory of Open Access Journals (Sweden)
Martin Hallnäs
2007-03-01
Full Text Available We review a recent construction of an explicit analytic series representation for symmetric polynomials which up to a groundstate factor are eigenfunctions of Calogero-Sutherland type models. We also indicate a generalisation of this result to polynomials which give the eigenfunctions of so-called 'deformed' Calogero-Sutherland type models.
2D-Zernike Polynomials and Coherent State Quantization of the Unit Disc
Energy Technology Data Exchange (ETDEWEB)
Thirulogasanthar, K., E-mail: santhar@gmail.com [Concordia University, Department of Comuter Science and Software Engineering (Canada); Saad, Nasser, E-mail: nsaad@upei.ca [University of Prince Edward Island, Department of mathematics and Statistics (Canada); Honnouvo, G., E-mail: g-honnouvo@yahoo.fr [McGill University, Department of Mathematics and Statistics (Canada)
2015-12-15
Using the orthonormality of the 2D-Zernike polynomials, reproducing kernels, reproducing kernel Hilbert spaces, and ensuring coherent states attained. With the aid of the so-obtained coherent states, the complex unit disc is quantized. Associated upper symbols, lower symbols and related generalized Berezin transforms also obtained. A number of necessary summation formulas for the 2D-Zernike polynomials proved.
Fast computation of the roots of polynomials over the ring of power series
DEFF Research Database (Denmark)
Neiger, Vincent; Rosenkilde, Johan; Schost, Éric
2017-01-01
We give an algorithm for computing all roots of polynomials over a univariate power series ring over an exact field K. More precisely, given a precision d, and a polynomial Q whose coefficients are power series in x, the algorithm computes a representation of all power series f(x) such that Q(f(x...
Polynomial Approximation Algorithms for the TSP and the QAP with a Factorial Domination Number
DEFF Research Database (Denmark)
Gutin, Gregory; Yeo, Anders
2002-01-01
Glover and Punnen (J. Oper. Res. Soc. 48 (1997) 502) asked whether there exists a polynomial time algorithm that always produces a tour which is not worse than at least n!/p(n) tours for some polynomial p(n) for every TSP instance on n cities. They conjectured that, unless P = NP, the answer to t...
Discrete Least-norm Approximation by Nonnegative (Trigonomtric) Polynomials and Rational Functions
Siem, A.Y.D.; de Klerk, E.; den Hertog, D.
2005-01-01
Polynomials, trigonometric polynomials, and rational functions are widely used for the discrete approximation of functions or simulation models.Often, it is known beforehand, that the underlying unknown function has certain properties, e.g. nonnegative or increasing on a certain region.However, the
Modal Analysis Using the Singular Value Decomposition and Rational Fraction Polynomials
2017-04-06
number from 0 to 9, which will appear in the rectangular box labeled RFP fit order. Sometimes it is necessary to “click” the mouse on the Matlab GUI ...graphical interface for the MATLAB program siftSVD with a unit fit order...23 Figure 10. “Snapshot” of the graphical interface for the MATLAB program siftSVD with a fit order of 1
Directory of Open Access Journals (Sweden)
Amirhossein Amiri
2012-03-01
Full Text Available Profile monitoring is a new research subject in Statistical Process Control which has been recently considered by many researchers. Profile describes the relationship between a response variable and one or more independent variables. This relationship is often modeled using regression which can be simple linear, multiple linear, polynomial or sometimes nonlinear. To monitor polynomial profiles, some methods have been developed, the best of which is the orthogonal polynomial approach. One of the disadvantages of this method is the large number of control charts which are simultaneously used. In this paper, a new approach has been proposed based on orthogonal polynomial approach, in which only two control charts are used to monitor a kth-order polynomial profile. Simulation findings and average run length (ARL curve analysis imply better performance of the proposed approach compared to the existing approach. Also, using the proposed approach is much easier in practice.
Bounds for the Hilbert function of polynomial ideals and for the degrees in the Nullstellensatz
Sombra, M
1996-01-01
We present a new effective Nullstellensatz with bounds for the degrees which depend not only on the number of variables and on the degrees of the input polynomials but also on an additional parameter called the {\\it geometric degree of the system of equations}. The obtained bound is polynomial in these parameters. It is essentially optimal in the general case, and it substantially improves the existent bounds in some special cases. The proof of this result is combinatorial, and it relies on global estimations for the Hilbert function of homogeneous polynomial ideals. In this direction, we obtain a lower bound for the Hilbert function of an arbitrary homogeneous polynomial ideal, and an upper bound for the Hilbert function of a generic hypersurface section of an unmixed radical polynomial ideal.
Jiménez, L; Angulo, V; Caparrós, S; Ariza, J
2007-12-01
The influence of operational variables in the pulping of vine shoots by use of ethanolamine [viz. temperature (155-185 degrees C), cooking time (30-90min) and ethanolamine concentration (50-70% v/v)] on the properties of the resulting pulp (viz. yield, kappa index, viscosity and drainability) was studied. A central composite factorial design was used in conjunction with the software BMDP and ANFIS Edit Matlab 6.5 to develop polynomial and fuzzy neural models that reproduced the experimental results of the dependent variables with errors less than 10%. Both types of models are therefore effective with a view to simulating the ethanolamine pulping process. Based on the proposed equations, the best choice is to use values of the operational valuables resulting in near-optimal pulp properties while saving energy and immobilized capital on industrial facilities by using lower temperatures and shorter processing times. One combination leading to near-optimal properties with reduced costs is using a temperature of 180 degrees C and an ethanolamine concentration of 60% for 60min, to obtain pulp with a viscosity of 6.13% lower than the maximum value (932.8ml/g) and a drainability of 5.49% lower than the maximum value (71 (o)SR).
Mota, L F M; Martins, P G M A; Littiere, T O; Abreu, L R A; Silva, M A; Bonafé, C M
2017-08-14
The objective was to estimate (co)variance functions using random regression models (RRM) with Legendre polynomials, B-spline function and multi-trait models aimed at evaluating genetic parameters of growth traits in meat-type quail. A database containing the complete pedigree information of 7000 meat-type quail was utilized. The models included the fixed effects of contemporary group and generation. Direct additive genetic and permanent environmental effects, considered as random, were modeled using B-spline functions considering quadratic and cubic polynomials for each individual segment, and Legendre polynomials for age. Residual variances were grouped in four age classes. Direct additive genetic and permanent environmental effects were modeled using 2 to 4 segments and were modeled by Legendre polynomial with orders of fit ranging from 2 to 4. The model with quadratic B-spline adjustment, using four segments for direct additive genetic and permanent environmental effects, was the most appropriate and parsimonious to describe the covariance structure of the data. The RRM using Legendre polynomials presented an underestimation of the residual variance. Lesser heritability estimates were observed for multi-trait models in comparison with RRM for the evaluated ages. In general, the genetic correlations between measures of BW from hatching to 35 days of age decreased as the range between the evaluated ages increased. Genetic trend for BW was positive and significant along the selection generations. The genetic response to selection for BW in the evaluated ages presented greater values for RRM compared with multi-trait models. In summary, RRM using B-spline functions with four residual variance classes and segments were the best fit for genetic evaluation of growth traits in meat-type quail. In conclusion, RRM should be considered in genetic evaluation of breeding programs.
Hounga, C.; Hounkonnou, M. N.; Ronveaux, A.
2006-10-01
In this paper, we give Laguerre-Freud equations for the recurrence coefficients of discrete semi-classical orthogonal polynomials of class two, when the polynomials in the Pearson equation are of the same degree. The case of generalized Charlier polynomials is also presented.
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Jang Leechae
2010-01-01
Full Text Available Abstract We define multiple Nörlund-type twisted -Euler polynomials and numbers and give interpolation functions of multiple Nörlund-type twisted -Euler polynomials at negative integers. Furthermore, we investigate some identities related to these polynomials and interpolation functions.
Energy Technology Data Exchange (ETDEWEB)
Gonzalez, H.E. [ININ, 52045 Ocoyoacac, Estado de Mexico (Mexico)
2006-07-01
In linear algebra, one can associate an equation to each square matrix: its characteristic equation or secular equation. Starting from this equation, the one characteristic polynomial that codes several important properties of the matrix is obtained: its own values, it determinant and it appearance. The first method to calculate those coefficients of this polynomial were proposed by the french astronomer Urbain Jean Joseph Le Verrier (1811-1877), from then on, many methods have intended to calculate these coefficients. In this work the author proposes a new one method and a bibliographical citation is given where the calculations with others methods that know each other for it, taking like reference the matrix used by Le Verrier are explained. It was concluded that it here proposed, besides being the only mexican method that is knew, has the advantage of being very easy of understanding and of calculating well, in the operations that it carries out, it doesn't use the division and it avoids fractions in matrices whose entrances are whole. This has a great importance for their use in the classroom for their great didactic value and in nuclear reactors and Genetic Engineering. (Author)
An Accurate Projector Calibration Method Based on Polynomial Distortion Representation
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Miao Liu
2015-10-01
Full Text Available In structure light measurement systems or 3D printing systems, the errors caused by optical distortion of a digital projector always affect the precision performance and cannot be ignored. Existing methods to calibrate the projection distortion rely on calibration plate and photogrammetry, so the calibration performance is largely affected by the quality of the plate and the imaging system. This paper proposes a new projector calibration approach that makes use of photodiodes to directly detect the light emitted from a digital projector. By analyzing the output sequence of the photoelectric module, the pixel coordinates can be accurately obtained by the curve fitting method. A polynomial distortion representation is employed to reduce the residuals of the traditional distortion representation model. Experimental results and performance evaluation show that the proposed calibration method is able to avoid most of the disadvantages in traditional methods and achieves a higher accuracy. This proposed method is also practically applicable to evaluate the geometric optical performance of other optical projection system.
An Accurate Projector Calibration Method Based on Polynomial Distortion Representation
Liu, Miao; Sun, Changku; Huang, Shujun; Zhang, Zonghua
2015-01-01
In structure light measurement systems or 3D printing systems, the errors caused by optical distortion of a digital projector always affect the precision performance and cannot be ignored. Existing methods to calibrate the projection distortion rely on calibration plate and photogrammetry, so the calibration performance is largely affected by the quality of the plate and the imaging system. This paper proposes a new projector calibration approach that makes use of photodiodes to directly detect the light emitted from a digital projector. By analyzing the output sequence of the photoelectric module, the pixel coordinates can be accurately obtained by the curve fitting method. A polynomial distortion representation is employed to reduce the residuals of the traditional distortion representation model. Experimental results and performance evaluation show that the proposed calibration method is able to avoid most of the disadvantages in traditional methods and achieves a higher accuracy. This proposed method is also practically applicable to evaluate the geometric optical performance of other optical projection system. PMID:26492247
Fast Minimum Variance Beamforming Based on Legendre Polynomials.
Bae, MooHo; Park, Sung Bae; Kwon, Sung Jae
2016-09-01
Currently, minimum variance beamforming (MV) is actively investigated as a method that can improve the performance of an ultrasound beamformer, in terms of the lateral and contrast resolution. However, this method has the disadvantage of excessive computational complexity since the inverse spatial covariance matrix must be calculated. Some noteworthy methods among various attempts to solve this problem include beam space adaptive beamforming methods and the fast MV method based on principal component analysis, which are similar in that the original signal in the element space is transformed to another domain using an orthonormal basis matrix and the dimension of the covariance matrix is reduced by approximating the matrix only with important components of the matrix, hence making the inversion of the matrix very simple. Recently, we proposed a new method with further reduced computational demand that uses Legendre polynomials as the basis matrix for such a transformation. In this paper, we verify the efficacy of the proposed method through Field II simulations as well as in vitro and in vivo experiments. The results show that the approximation error of this method is less than or similar to those of the above-mentioned methods and that the lateral response of point targets and the contrast-to-speckle noise in anechoic cysts are also better than or similar to those methods when the dimensionality of the covariance matrices is reduced to the same dimension.
Stochastic Polynomial Dynamic Models of the Yeast Cell Cycle
Mitra, Indranil; Dimitrova, Elena; Jarrah, Abdul S.
2010-03-01
In the last decade a new holistic approach for tackling biological problems, systems biology, which takes into account the study of the interactions between the components of a biological system to predict function and behavior has emerged. The reverse-engineering of biochemical networks from experimental data have increasingly become important in systems biology. Based on Boolean networks, we propose a time-discrete stochastic framework for the reverse engineering of the yeast cell cycle regulatory network from experimental data. With a suitable choice of state set, we have used powerful tools from computational algebra, that underlie the reverse-engineering algorithm, avoiding costly enumeration strategies. Stochasticity is introduced by choosing at each update step a random coordinate function for each variable, chosen from a probability space of update functions. The algorithm is based on a combinatorial structure known as the Gr"obner fans of a polynomial ideal which identifies the underlying network structure and dynamics. The model depicts a correct dynamics of the yeast cell cycle network and reproduces the time sequence of expression patterns along the biological cell cycle. Our findings indicate that the methodolgy has high chance of success when applied to large and complex systems to determine the dynamical properties of corresponding networks.
Optimization of LED-based non-imaging optics with orthogonal polynomial shapes
Brick, Peter; Wiesmann, Christopher
2012-10-01
Starting with a seminal paper by Forbes [1], orthogonal polynomials have received considerable interest as descriptors of lens shapes for imaging optics. However, there is little information on the application of orthogonal polynomials in the field of non-imaging optics. Here, we consider fundamental cases related to LED primary and secondary optics. To make it most realistic, we avoid many of the simplifications of non-imaging theory and consider the full complexity of LED optics. In this framework, the benefits of orthogonal polynomial surface description for LED optics are evaluated in comparison to a surface description by widely used monomials.
A zero-free interval for chromatic polynomials of graphs with 3-leaf spanning trees
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Perrett, Thomas
2016-01-01
It is proved that if G is a graph containing a spanning tree with at most three leaves, then the chromatic polynomial of G has no roots in the interval (1,t1], where t1≈1.2904 is the smallest real root of the polynomial (t-2)6+4(t-1)2 (t-2)3-(t-1)4. We also construct a family of graphs containing...... such spanning trees with chromatic roots converging to t1 from above. We employ the Whitney 2-switch operation to manage the analysis of an infinite class of chromatic polynomials....
Multiple Twisted q-Euler Numbers and Polynomials Associated with p-Adic q-Integrals
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Lee-Chae Jang
2008-04-01
Full Text Available By using p-adic q-integrals on Ã¢Â„Â¤p, we define multiple twisted q-Euler numbers and polynomials. We also find Witt's type formula for multiple twisted q-Euler numbers and discuss some characterizations of multiple twisted q-Euler Zeta functions. In particular, we construct multiple twisted Barnes' type q-Euler polynomials and multiple twisted Barnes' type q-Euler Zeta functions. Finally, we define multiple twisted Dirichlet's type q-Euler numbers and polynomials, and give Witt's type formula for them.
Polynomial reduction and evaluation of tree- and loop-level CHY amplitudes
Zlotnikov, Michael
2016-08-01
We develop a polynomial reduction procedure that transforms any gauge fixed CHY amplitude integrand for n scattering particles into a σ-moduli multivariate polynomial of what we call the standard form. We show that a standard form polynomial must have a specific ladder type monomial structure, which has finite size at any n, with highest multivariate degree given by ( n - 3)( n - 4) /2. This set of monomials spans a complete basis for polynomials with rational coefficients in kinematic data on the support of scattering equations. Subsequently, at tree and one-loop level, we employ the global residue theorem to derive a prescription that evaluates any CHY amplitude by means of collecting simple residues at infinity only. The prescription is then applied explicitly to some tree and one-loop amplitude examples.
Wavefront reconstruction for lateral shearing interferometry based on difference polynomial fitting
Li, Jie; Tang, Feng; Wang, Xiangzhao; Dai, Fengzhao; Feng, Peng; Li, Sikun
2015-06-01
The wavefront reconstruction method for shearing interferometry using difference Zernike polynomial fitting has been the easiest algorithm to implement up to now. The method is extended to using general basis functions in this paper. Simulations and experiments verify that highly accurate reconstructions can be achieved based on difference polynomial fitting regardless, of the pupil shape and the orthogonality of the basis functions. The reconstruction accuracy mainly depends on whether the used terms of the polynomials are enough to represent the wavefront. When the used terms cannot perfectly represent the wavefront, the reconstruction accuracy of Taylor monomials is a little higher than that of Zernike polynomials. It is also presented and proved that the reconstruction accuracy can be estimated using the deviation between the reconstructed difference fronts and the measured difference fronts.
Polynomial reduction and evaluation of tree- and loop-level CHY amplitudes
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Zlotnikov, Michael [Department of Physics, Brown University,182 Hope St, Providence, RI, 02912 (United States)
2016-08-24
We develop a polynomial reduction procedure that transforms any gauge fixed CHY amplitude integrand for n scattering particles into a σ-moduli multivariate polynomial of what we call the standard form. We show that a standard form polynomial must have a specific ladder type monomial structure, which has finite size at any n, with highest multivariate degree given by (n−3)(n−4)/2. This set of monomials spans a complete basis for polynomials with rational coefficients in kinematic data on the support of scattering equations. Subsequently, at tree and one-loop level, we employ the global residue theorem to derive a prescription that evaluates any CHY amplitude by means of collecting simple residues at infinity only. The prescription is then applied explicitly to some tree and one-loop amplitude examples.
A polynomial bound on solutions of quadratic equations in free groups
Lysenok, Igor; Myasnikov, Alexei
2011-01-01
We provide polynomial upper bounds on the size of a shortest solution for quadratic equations in a free group. A similar bound is given for parametric solutions in the description of solutions sets of quadratic equations in a free group.
Su, Liyun; Zhao, Yanyong; Yan, Tianshun; Li, Fenglan
2012-01-01
Multivariate local polynomial fitting is applied to the multivariate linear heteroscedastic regression model. Firstly, the local polynomial fitting is applied to estimate heteroscedastic function, then the coefficients of regression model are obtained by using generalized least squares method. One noteworthy feature of our approach is that we avoid the testing for heteroscedasticity by improving the traditional two-stage method. Due to non-parametric technique of local polynomial estimation, it is unnecessary to know the form of heteroscedastic function. Therefore, we can improve the estimation precision, when the heteroscedastic function is unknown. Furthermore, we verify that the regression coefficients is asymptotic normal based on numerical simulations and normal Q-Q plots of residuals. Finally, the simulation results and the local polynomial estimation of real data indicate that our approach is surely effective in finite-sample situations.
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Bolsinov Alexey V.
2016-01-01
Full Text Available The Mishchenko-Fomenko conjecture says that for each real or complex finite-dimensional Lie algebra g there exists a complete set of commuting polynomials on its dual space g*. In terms of the theory of integrable Hamiltonian systems this means that the dual space g* endowed with the standard Lie-Poisson bracket admits polynomial integrable Hamiltonian systems. This conjecture was proved by S. T. Sadetov in 2003. Following his idea, we give an explicit geometric construction for commuting polynomials on g* and consider some examples. (This text is a revised version of my paper published in Russian: A. V. Bolsinov, Complete commutative families of polynomials in Poisson–Lie algebras: A proof of the Mischenko–Fomenko conjecture in book: Tensor and Vector Analysis, Vol. 26, Moscow State University, 2005, 87–109.
Some Identities Involving the Derivative of the First Kind Chebyshev Polynomials
Tingting Wang; Han Zhang
2015-01-01
We use the combinatorial method and algebraic manipulations to obtain several interesting identities involving the power sums of the derivative of the first kind Chebyshev polynomials. This solved an open problem proposed by Li (2015).
Some Identities Involving the Derivative of the First Kind Chebyshev Polynomials
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Tingting Wang
2015-01-01
Full Text Available We use the combinatorial method and algebraic manipulations to obtain several interesting identities involving the power sums of the derivative of the first kind Chebyshev polynomials. This solved an open problem proposed by Li (2015.
Periodic perturbations of quadratic planar polynomial vector fields
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MARCELO MESSIAS
2002-06-01
Full Text Available In this work are studied periodic perturbations, depending on two parameters, of quadratic planar polynomial vector fields having an infinite heteroclinic cycle, which is an unbounded solution joining two saddle points at infinity. The global study envolving infinity is performed via the Poincaré compactification. The main result obtained states that for certain types of periodic perturbations, the perturbed system has quadratic heteroclinic tangencies and transverse intersections between the local stable and unstable manifolds of the hyperbolic periodic orbits at infinity. It implies, via the Birkhoff-Smale Theorem, in a complex dynamical behavior of the solutions of the perturbed system, in a finite part of the phase plane.Neste trabalho são estudadas perturbações periódicas, dependendo de dois parâmetros, de campos vetoriais polinomiais planares que possuem um ciclo heteroclínico infinito, que consiste de uma solução ilimitada, que conecta dois pontos de sela no infinito. O estudo global do campo vetorial, envolvendo o infinito, foi elaborado utilizando-se a compactificação de Poincaré. O resultado principal estabelece que, para certo tipo de perturbação periódica, o sistema apresenta tangências heteroclínicas e intersecção transversal das variedades invariantes de órbitas periódicas no infinito, o que implica, via o Teorema de Birkhoff-Smale, em um comportamento dinâmico bastante complexo das soluções do sistema perturbado, em uma região finita do plano de fase.
On the non-hyperbolicity of a class of exponential polynomials
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Gaspar Mora
2017-10-01
Full Text Available In this paper we have constructed a class of non-hyperbolic exponential polynomials that contains all the partial sums of the Riemann zeta function. An exponential polynomial been also defined to illustrate the complexity of the structure of the set defined by the closure of the real projections of its zeros. The sensitivity of this set, when the vector of delays is perturbed, has been analysed. These results have immediate implications in the theory of the neutral differential equations.