A polynomial time algorithm for solving the maximum flow problem in directed networks

International Nuclear Information System (INIS)

Tlas, M.

2015-01-01

An efficient polynomial time algorithm for solving maximum flow problems has been proposed in this paper. The algorithm is basically based on the binary representation of capacities; it solves the maximum flow problem as a sequence of O(m) shortest path problems on residual networks with nodes and m arcs. It runs in O(m"2r) time, where is the smallest integer greater than or equal to log B , and B is the largest arc capacity of the network. A numerical example has been illustrated using this proposed algorithm.(author)

Algebraic calculations for spectrum of superintegrable system from exceptional orthogonal polynomials

Science.gov (United States)

Hoque, Md. Fazlul; Marquette, Ian; Post, Sarah; Zhang, Yao-Zhong

2018-04-01

We introduce an extended Kepler-Coulomb quantum model in spherical coordinates. The Schrödinger equation of this Hamiltonian is solved in these coordinates and it is shown that the wave functions of the system can be expressed in terms of Laguerre, Legendre and exceptional Jacobi polynomials (of hypergeometric type). We construct ladder and shift operators based on the corresponding wave functions and obtain their recurrence formulas. These recurrence relations are used to construct higher-order, algebraically independent integrals of motion to prove superintegrability of the Hamiltonian. The integrals form a higher rank polynomial algebra. By constructing the structure functions of the associated deformed oscillator algebras we derive the degeneracy of energy spectrum of the superintegrable system.

Many-body orthogonal polynomial systems

International Nuclear Information System (INIS)

Witte, N.S.

1997-03-01

The fundamental methods employed in the moment problem, involving orthogonal polynomial systems, the Lanczos algorithm, continued fraction analysis and Pade approximants has been combined with a cumulant approach and applied to the extensive many-body problem in physics. This has yielded many new exact results for many-body systems in the thermodynamic limit - for the ground state energy, for excited state gaps, for arbitrary ground state avenges - and are of a nonperturbative nature. These results flow from a confluence property of the three-term recurrence coefficients arising and define a general class of many-body orthogonal polynomials. These theorems constitute an analytical solution to the Lanczos algorithm in that they are expressed in terms of the three-term recurrence coefficients α and β. These results can also be applied approximately for non-solvable models in the form of an expansion, in a descending series of the system size. The zeroth order order this expansion is just the manifestation of the central limit theorem in which a Gaussian measure and hermite polynomials arise. The first order represents the first non-trivial order, in which classical distribution functions like the binomial distributions arise and the associated class of orthogonal polynomials are Meixner polynomials. Amongst examples of systems which have infinite order in the expansion are q-orthogonal polynomials where q depends on the system size in a particular way. (author)

Finite-Time Stability and Controller Design of Continuous-Time Polynomial Fuzzy Systems

Directory of Open Access Journals (Sweden)

Xiaoxing Chen

2017-01-01

Full Text Available Finite-time stability and stabilization problem is first investigated for continuous-time polynomial fuzzy systems. The concept of finite-time stability and stabilization is given for polynomial fuzzy systems based on the idea of classical references. A sum-of-squares- (SOS- based approach is used to obtain the finite-time stability and stabilization conditions, which include some classical results as special cases. The proposed conditions can be solved with the help of powerful Matlab toolbox SOSTOOLS and a semidefinite-program (SDP solver. Finally, two numerical examples and one practical example are employed to illustrate the validity and effectiveness of the provided conditions.

Operational matrices with respect to Hermite polynomials and their applications in solving linear dierential equations with variable coecients

Directory of Open Access Journals (Sweden)

A. Aminataei

2014-05-01

Full Text Available In this paper, a new and ecient approach is applied for numerical approximation of the linear dierential equations with variable coecients based on operational matrices with respect to Hermite polynomials. Explicit formulae which express the Hermite expansioncoecients for the moments of derivatives of any dierentiable function in terms of the original expansion coecients of the function itself are given in the matrix form. The mainimportance of this scheme is that using this approach reduces solving the linear dierentialequations to solve a system of linear algebraic equations, thus greatly simplifying the problem. In addition, two experiments are given to demonstrate the validity and applicability of the method

Computer programme for the derivation of transfer functions for multivariable systems (solutions of determinants with polynomial elements)

International Nuclear Information System (INIS)

Guppy, C.B.

1962-03-01

In the methods adopted in this report transfer functions in the form of the ratio of two polynomials of the complex variable s are derived from sets of laplace transformed simultaneous differential equations. The set of algebraic simultaneous equations are solved using Cramer's Rule and this gives rise to determinants having polynomial elements. It is shown how the determinants are formed when transfer functions are specified. The procedure for finding the polynomial coefficients from a given determinant is fully described. The first method adopted is a direct one and reduces a determinant with first degree polynomial elements to secular form and follows this by an application of the similarity transformation to reduce the determinant to a form from which the polynomial coefficients can be read out directly. The programme is able to solve a single determinant with polynomial elements and this can be used to reduce an eigenvalue problem in the form of a secular determinant to polynomial form if the need arises. A description is given of the way in which the data is to be set out for solution by the programme. A description is also given of a method used in an earlier programme for solving polynomial determinants by curve fitting techniques using Chebyshev Polynomials. In this method determinants with polynomial elements of any degree can be solved. (author)

Vortices and polynomials: non-uniqueness of the Adler–Moser polynomials for the Tkachenko equation

International Nuclear Information System (INIS)

Demina, Maria V; Kudryashov, Nikolai A

2012-01-01

Stationary and translating relative equilibria of point vortices in the plane are studied. It is shown that stationary equilibria of any system containing point vortices with arbitrary choice of circulations can be described with the help of the Tkachenko equation. It is also obtained that translating relative equilibria of point vortices with arbitrary circulations can be constructed using a generalization of the Tkachenko equation. Roots of any pair of polynomials solving the Tkachenko equation and the generalized Tkachenko equation are proved to give positions of point vortices in stationary and translating relative equilibria accordingly. These results are valid even if the polynomials in a pair have multiple or common roots. It is obtained that the Adler–Moser polynomial provides non-unique polynomial solutions of the Tkachenko equation. It is shown that the generalized Tkachenko equation possesses polynomial solutions with degrees that are not triangular numbers. (paper)

Design of polynomial fuzzy observer-controller for nonlinear systems with state delay: sum of squares approach

Science.gov (United States)

Gassara, H.; El Hajjaji, A.; Chaabane, M.

2017-07-01

This paper investigates the problem of observer-based control for two classes of polynomial fuzzy systems with time-varying delay. The first class concerns a special case where the polynomial matrices do not depend on the estimated state variables. The second one is the general case where the polynomial matrices could depend on unmeasurable system states that will be estimated. For the last case, two design procedures are proposed. The first one gives the polynomial fuzzy controller and observer gains in two steps. In the second procedure, the designed gains are obtained using a single-step approach to overcome the drawback of a two-step procedure. The obtained conditions are presented in terms of sum of squares (SOS) which can be solved via the SOSTOOLS and a semi-definite program solver. Illustrative examples show the validity and applicability of the proposed results.

A Symbolic Computation Approach to Parameterizing Controller for Polynomial Hamiltonian Systems

Directory of Open Access Journals (Sweden)

Zhong Cao

2014-01-01

Full Text Available This paper considers controller parameterization method of H∞ control for polynomial Hamiltonian systems (PHSs, which involves internal stability and external disturbance attenuation. The aims of this paper are to design a controller with parameters to insure that the systems are H∞ stable and propose an algorithm for solving parameters of the controller with symbolic computation. The proposed parameterization method avoids solving Hamilton-Jacobi-Isaacs equations, and thus the obtained controllers with parameters are relatively simple in form and easy in operation. Simulation with a numerical example shows that the controller is effective as it can optimize H∞ control by adjusting parameters. All these results are expected to be of use in the study of H∞ control for nonlinear systems with perturbations.

Guaranteed cost control of polynomial fuzzy systems via a sum of squares approach.

Science.gov (United States)

Tanaka, Kazuo; Ohtake, Hiroshi; Wang, Hua O

2009-04-01

This paper presents the guaranteed cost control of polynomial fuzzy systems via a sum of squares (SOS) approach. First, we present a polynomial fuzzy model and controller that are more general representations of the well-known Takagi-Sugeno (T-S) fuzzy model and controller, respectively. Second, we derive a guaranteed cost control design condition based on polynomial Lyapunov functions. Hence, the design approach discussed in this paper is more general than the existing LMI approaches (to T-S fuzzy control system designs) based on quadratic Lyapunov functions. The design condition realizes a guaranteed cost control by minimizing the upper bound of a given performance function. In addition, the design condition in the proposed approach can be represented in terms of SOS and is numerically (partially symbolically) solved via the recent developed SOSTOOLS. To illustrate the validity of the design approach, two design examples are provided. The first example deals with a complicated nonlinear system. The second example presents micro helicopter control. Both the examples show that our approach provides more extensive design results for the existing LMI approach.

Orthogonal polynomials derived from the tridiagonal representation approach

Science.gov (United States)

Alhaidari, A. D.

2018-01-01

The tridiagonal representation approach is an algebraic method for solving second order differential wave equations. Using this approach in the solution of quantum mechanical problems, we encounter two new classes of orthogonal polynomials whose properties give the structure and dynamics of the corresponding physical system. For a certain range of parameters, one of these polynomials has a mix of continuous and discrete spectra making it suitable for describing physical systems with both scattering and bound states. In this work, we define these polynomials by their recursion relations and highlight some of their properties using numerical means. Due to the prime significance of these polynomials in physics, we hope that our short expose will encourage experts in the field of orthogonal polynomials to study them and derive their properties (weight functions, generating functions, asymptotics, orthogonality relations, zeros, etc.) analytically.

Global stability and quadratic Hamiltonian structure in Lotka-Volterra and quasi-polynomial systems

Energy Technology Data Exchange (ETDEWEB)

Szederkenyi, Gabor; Hangos, Katalin M

2004-04-26

We show that the global stability of quasi-polynomial (QP) and Lotka-Volterra (LV) systems with the well-known logarithmic Lyapunov function is equivalent to the existence of a local generalized dissipative Hamiltonian description of the LV system with a diagonal quadratic form as a Hamiltonian function. The Hamiltonian function can be calculated and the quadratic dissipativity neighborhood of the origin can be estimated by solving linear matrix inequalities.

Global stability and quadratic Hamiltonian structure in Lotka-Volterra and quasi-polynomial systems

Science.gov (United States)

Szederkényi, Gábor; Hangos, Katalin M.

2004-04-01

We show that the global stability of quasi-polynomial (QP) and Lotka-Volterra (LV) systems with the well-known logarithmic Lyapunov function is equivalent to the existence of a local generalized dissipative Hamiltonian description of the LV system with a diagonal quadratic form as a Hamiltonian function. The Hamiltonian function can be calculated and the quadratic dissipativity neighborhood of the origin can be estimated by solving linear matrix inequalities.

Global stability and quadratic Hamiltonian structure in Lotka-Volterra and quasi-polynomial systems

International Nuclear Information System (INIS)

Szederkenyi, Gabor; Hangos, Katalin M.

2004-01-01

We show that the global stability of quasi-polynomial (QP) and Lotka-Volterra (LV) systems with the well-known logarithmic Lyapunov function is equivalent to the existence of a local generalized dissipative Hamiltonian description of the LV system with a diagonal quadratic form as a Hamiltonian function. The Hamiltonian function can be calculated and the quadratic dissipativity neighborhood of the origin can be estimated by solving linear matrix inequalities

Algebraic limit cycles in polynomial systems of differential equations

International Nuclear Information System (INIS)

Llibre, Jaume; Zhao Yulin

2007-01-01

Using elementary tools we construct cubic polynomial systems of differential equations with algebraic limit cycles of degrees 4, 5 and 6. We also construct a cubic polynomial system of differential equations having an algebraic homoclinic loop of degree 3. Moreover, we show that there are polynomial systems of differential equations of arbitrary degree that have algebraic limit cycles of degree 3, as well as give an example of a cubic polynomial system of differential equations with two algebraic limit cycles of degree 4

Expansion methods for solving integral equations with multiple time lags using Bernstein polynomial of the second kind

Directory of Open Access Journals (Sweden)

Mahmoud Paripour

2014-08-01

Full Text Available In this paper, the Bernstein polynomials are used to approximatethe solutions of linear integral equations with multiple time lags (IEMTL through expansion methods (collocation method, partition method, Galerkin method. The method is discussed in detail and illustrated by solving some numerical examples. Comparison between the exact and approximated results obtained from these methods is carried out

Random polynomials and expected complexity of bisection methods for real solving

DEFF Research Database (Denmark)

Emiris, Ioannis Z.; Galligo, André; Tsigaridas, Elias

2010-01-01

, and by Edelman and Kostlan in order to estimate the real root separation of degree d polynomials with i.i.d. coefficients that follow two zero-mean normal distributions: for SO(2) polynomials, the i-th coefficient has variance (d/i), whereas for Weyl polynomials its variance is 1/i!. By applying results from....... The second part of the paper shows that the expected number of real roots of a degree d polynomial in the Bernstein basis is √2d ± O(1), when the coefficients are i.i.d. variables with moderate standard deviation. Our paper concludes with experimental results which corroborate our analysis....

Parallel Construction of Irreducible Polynomials

DEFF Research Database (Denmark)

Frandsen, Gudmund Skovbjerg

Let arithmetic pseudo-NC^k denote the problems that can be solved by log space uniform arithmetic circuits over the finite prime field GF(p) of depth O(log^k (n + p)) and size polynomial in (n + p). We show that the problem of constructing an irreducible polynomial of specified degree over GF(p) ...... of polynomials is in arithmetic NC^3. Our algorithm works over any field and compared to other known algorithms it does not assume the ability to take p'th roots when the field has characteristic p....

On polynomial solutions of the Heun equation

International Nuclear Information System (INIS)

Gurappa, N; Panigrahi, Prasanta K

2004-01-01

By making use of a recently developed method to solve linear differential equations of arbitrary order, we find a wide class of polynomial solutions to the Heun equation. We construct the series solution to the Heun equation before identifying the polynomial solutions. The Heun equation extended by the addition of a term, -σ/x, is also amenable for polynomial solutions. (letter to the editor)

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.

Polynomial fuzzy observer designs: a sum-of-squares approach.

Science.gov (United States)

Tanaka, Kazuo; Ohtake, Hiroshi; Seo, Toshiaki; Tanaka, Motoyasu; Wang, Hua O

2012-10-01

This paper presents a sum-of-squares (SOS) approach to polynomial fuzzy observer designs for three classes of polynomial fuzzy systems. The proposed SOS-based framework provides a number of innovations and improvements over the existing linear matrix inequality (LMI)-based approaches to Takagi-Sugeno (T-S) fuzzy controller and observer designs. First, we briefly summarize previous results with respect to a polynomial fuzzy system that is a more general representation of the well-known T-S fuzzy system. Next, we propose polynomial fuzzy observers to estimate states in three classes of polynomial fuzzy systems and derive SOS conditions to design polynomial fuzzy controllers and observers. A remarkable feature of the SOS design conditions for the first two classes (Classes I and II) is that they realize the so-called separation principle, i.e., the polynomial fuzzy controller and observer for each class can be separately designed without lack of guaranteeing the stability of the overall control system in addition to converging state-estimation error (via the observer) to zero. Although, for the last class (Class III), the separation principle does not hold, we propose an algorithm to design polynomial fuzzy controller and observer satisfying the stability of the overall control system in addition to converging state-estimation error (via the observer) to zero. All the design conditions in the proposed approach can be represented in terms of SOS and are symbolically and numerically solved via the recently developed SOSTOOLS and a semidefinite-program solver, respectively. To illustrate the validity and applicability of the proposed approach, three design examples are provided. The examples demonstrate the advantages of the SOS-based approaches for the existing LMI approaches to T-S fuzzy observer designs.

Minimal residual method stronger than polynomial preconditioning

Energy Technology Data Exchange (ETDEWEB)

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

1994-12-31

Two popular methods for solving symmetric and nonsymmetric systems of equations are the minimal residual method, implemented by algorithms such as GMRES, and polynomial preconditioning methods. In this study results are given on the convergence rates of these methods for various classes of matrices. It is shown that for some matrices, such as normal matrices, the convergence rates for GMRES and for the optimal polynomial preconditioning are the same, and for other matrices such as the upper triangular Toeplitz matrices, it is at least assured that if one method converges then the other must converge. On the other hand, it is shown that matrices exist for which restarted GMRES always converges but any polynomial preconditioning of corresponding degree makes no progress toward the solution for some initial error. The implications of these results for these and other iterative methods are discussed.

AZTEC: A parallel iterative package for the solving linear systems

Energy Technology Data Exchange (ETDEWEB)

Hutchinson, S.A.; Shadid, J.N.; Tuminaro, R.S. [Sandia National Labs., Albuquerque, NM (United States)

1996-12-31

We describe a parallel linear system package, AZTEC. The package incorporates a number of parallel iterative methods (e.g. GMRES, biCGSTAB, CGS, TFQMR) and preconditioners (e.g. Jacobi, Gauss-Seidel, polynomial, domain decomposition with LU or ILU within subdomains). Additionally, AZTEC allows for the reuse of previous preconditioning factorizations within Newton schemes for nonlinear methods. Currently, a number of different users are using this package to solve a variety of PDE applications.

General quantum polynomials: irreducible modules and Morita equivalence

International Nuclear Information System (INIS)

Artamonov, V A

1999-01-01

In this paper we continue the investigation of the structure of finitely generated modules over rings of general quantum (Laurent) polynomials. We obtain a description of the lattice of submodules of periodic finitely generated modules and describe the irreducible modules. We investigate the problem of Morita equivalence of rings of general quantum polynomials, consider properties of division rings of fractions, and solve Zariski's problem for quantum polynomials

Polynomial algebra of discrete models in systems biology.

Science.gov (United States)

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.

Orthogonal polynomials

CERN Document Server

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

Note on Generating Orthogonal Polynomials and Their Application in Solving Complicated Polynomial Regression Tasks

Czech Academy of Sciences Publication Activity Database

Knížek, J.; Tichý, Petr; Beránek, L.; Šindelář, Jan; Vojtěšek, B.; Bouchal, P.; Nenutil, R.; Dedík, O.

2010-01-01

Roč. 7, č. 10 (2010), s. 48-60 ISSN 0974-5718 Grant - others:GA MZd(CZ) NS9812; GA ČR(CZ) GAP304/10/0868 Institutional research plan: CEZ:AV0Z10300504; CEZ:AV0Z10750506 Keywords : polynomial regression * orthogonalization * numerical methods * markers * biomarkers Subject RIV: BA - General Mathematics

Applications of polynomial optimization in financial risk investment

Science.gov (United States)

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.

Ratio asymptotics of Hermite-Pade polynomials for Nikishin systems

International Nuclear Information System (INIS)

Aptekarev, A I; Lopez, Guillermo L; Rocha, I A

2005-01-01

The existence of ratio asymptotics is proved for a sequence of multiple orthogonal polynomials with orthogonality relations distributed among a system of m finite Borel measures with support on a bounded interval of the real line which form a so-called Nikishin system. For m=1 this result reduces to Rakhmanov's celebrated theorem on the ratio asymptotics for orthogonal polynomials on the real line.

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.

About the solvability of matrix polynomial equations

OpenAIRE

Netzer, Tim; Thom, Andreas

2016-01-01

We study self-adjoint matrix polynomial equations in a single variable and prove existence of self-adjoint solutions under some assumptions on the leading form. Our main result is that any self-adjoint matrix polynomial equation of odd degree with non-degenerate leading form can be solved in self-adjoint matrices. We also study equations of even degree and equations in many variables.

Polynomial chaos expansion with random and fuzzy variables

Science.gov (United States)

Jacquelin, E.; Friswell, M. I.; Adhikari, S.; Dessombz, O.; Sinou, J.-J.

2016-06-01

A dynamical uncertain system is studied in this paper. Two kinds of uncertainties are addressed, where the uncertain parameters are described through random variables and/or fuzzy variables. A general framework is proposed to deal with both kinds of uncertainty using a polynomial chaos expansion (PCE). It is shown that fuzzy variables may be expanded in terms of polynomial chaos when Legendre polynomials are used. The components of the PCE are a solution of an equation that does not depend on the nature of uncertainty. Once this equation is solved, the post-processing of the data gives the moments of the random response when the uncertainties are random or gives the response interval when the variables are fuzzy. With the PCE approach, it is also possible to deal with mixed uncertainty, when some parameters are random and others are fuzzy. The results provide a fuzzy description of the response statistical moments.

STABILITY SYSTEMS VIA HURWITZ POLYNOMIALS

Directory of Open Access Journals (Sweden)

BALTAZAR AGUIRRE HERNÁNDEZ

2017-01-01

Full Text Available To analyze the stability of a linear system of differential equations  ẋ = Ax we can study the location of the roots of the characteristic polynomial pA(t associated with the matrix A. We present various criteria - algebraic and geometric - that help us to determine where the roots are located without calculating them directly.

Tensor calculus in polar coordinates using Jacobi polynomials

Science.gov (United States)

Vasil, Geoffrey M.; Burns, Keaton J.; Lecoanet, Daniel; Olver, Sheehan; Brown, Benjamin P.; Oishi, Jeffrey S.

2016-11-01

Spectral methods are an efficient way to solve partial differential equations on domains possessing certain symmetries. The utility of a method depends strongly on the choice of spectral basis. In this paper we describe a set of bases built out of Jacobi polynomials, and associated operators for solving scalar, vector, and tensor partial differential equations in polar coordinates on a unit disk. By construction, the bases satisfy regularity conditions at r = 0 for any tensorial field. The coordinate singularity in a disk is a prototypical case for many coordinate singularities. The work presented here extends to other geometries. The operators represent covariant derivatives, multiplication by azimuthally symmetric functions, and the tensorial relationship between fields. These arise naturally from relations between classical orthogonal polynomials, and form a Heisenberg algebra. Other past work uses more specific polynomial bases for solving equations in polar coordinates. The main innovation in this paper is to use a larger set of possible bases to achieve maximum bandedness of linear operations. We provide a series of applications of the methods, illustrating their ease-of-use and accuracy.

Scattering theory and orthogonal polynomials

International Nuclear Information System (INIS)

Geronimo, J.S.

1977-01-01

The application of the techniques of scattering theory to the study of polynomials orthogonal on the unit circle and a finite segment of the real line is considered. The starting point is the recurrence relations satisfied by the polynomials instead of the orthogonality condition. A set of two two terms recurrence relations for polynomials orthogonal on the real line is presented and used. These recurrence relations play roles analogous to those satisfied by polynomials orthogonal on unit circle. With these recurrence formulas a Wronskian theorem is proved and the Christoffel-Darboux formula is derived. In scattering theory a fundamental role is played by the Jost function. An analogy is deferred of this function and its analytic properties and the locations of its zeros investigated. The role of the analog Jost function in various properties of these orthogonal polynomials is investigated. The techniques of inverse scattering theory are also used. The discrete analogues of the Gelfand-Levitan and Marchenko equations are derived and solved. These techniques are used to calculate asymptotic formulas for the orthogonal polynomials. Finally Szego's theorem on toeplitz and Hankel determinants is proved using the recurrence formulas and some properties of the Jost function. The techniques of inverse scattering theory are used to calculate the correction terms

Solving the Rational Polynomial Coefficients Based on L Curve

Science.gov (United States)

Zhou, G.; Li, X.; Yue, T.; Huang, W.; He, C.; Huang, Y.

2018-05-01

The rational polynomial coefficients (RPC) model is a generalized sensor model, which can achieve high approximation accuracy. And it is widely used in the field of photogrammetry and remote sensing. Least square method is usually used to determine the optimal parameter solution of the rational function model. However the distribution of control points is not uniform or the model is over-parameterized, which leads to the singularity of the coefficient matrix of the normal equation. So the normal equation becomes ill conditioned equation. The obtained solutions are extremely unstable and even wrong. The Tikhonov regularization can effectively improve and solve the ill conditioned equation. In this paper, we calculate pathological equations by regularization method, and determine the regularization parameters by L curve. The results of the experiments on aerial format photos show that the accuracy of the first-order RPC with the equal denominators has the highest accuracy. The high order RPC model is not necessary in the processing of dealing with frame images, as the RPC model and the projective model are almost the same. The result shows that the first-order RPC model is basically consistent with the strict sensor model of photogrammetry. Orthorectification results both the firstorder RPC model and Camera Model (ERDAS9.2 platform) are similar to each other, and the maximum residuals of X and Y are 0.8174 feet and 0.9272 feet respectively. This result shows that RPC model can be used in the aerial photographic compensation replacement sensor model.

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 highresol......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...

SOLVING THE HAMILTONIAN CYCLE PROBLEM USING SYMBOLIC DETERMINANTS

OpenAIRE

Ejov, V.; Filar, J. A.; Lucas, S. K.; Nelson, J. L.

2006-01-01

In this note we show how the Hamiltonian Cycle problem can be reduced to solving a system of polynomial equations related to the adjacency matrix of a graph. This system of equations can be solved using the method of Gröbner bases, but we also show how a symbolic determinant related to the adjacency matrix can be used to directly decide whether a graph has a Hamiltonian cycle.

Classification of polynomial integrable systems of mixed scalar and vector evolution equations: I

International Nuclear Information System (INIS)

Tsuchida, Takayuki; Wolf, Thomas

2005-01-01

We perform a classification of integrable systems of mixed scalar and vector evolution equations with respect to higher symmetries. We consider polynomial systems that are homogeneous under a suitable weighting of variables. This paper deals with the KdV weighting, the Burgers (or potential KdV or modified KdV) weighting, the Ibragimov-Shabat weighting and two unfamiliar weightings. The case of other weightings will be studied in a subsequent paper. Making an ansatz for undetermined coefficients and using a computer package for solving bilinear algebraic systems, we give the complete lists of second-order systems with a third-order or a fourth-order symmetry and third-order systems with a fifth-order symmetry. For all but a few systems in the lists, we show that the system (or, at least a subsystem of it) admits either a Lax representation or a linearizing transformation. A thorough comparison with recent work of Foursov and Olver is made

Classification of polynomial integrable systems of mixed scalar and vector evolution equations: I

Energy Technology Data Exchange (ETDEWEB)

Tsuchida, Takayuki [Department of Physics, Kwansei Gakuin University, 2-1 Gakuen, Sanda 669-1337 (Japan); Wolf, Thomas [Department of Mathematics, Brock University, St Catharines, ON L2S 3A1 (Canada)

2005-09-02

We perform a classification of integrable systems of mixed scalar and vector evolution equations with respect to higher symmetries. We consider polynomial systems that are homogeneous under a suitable weighting of variables. This paper deals with the KdV weighting, the Burgers (or potential KdV or modified KdV) weighting, the Ibragimov-Shabat weighting and two unfamiliar weightings. The case of other weightings will be studied in a subsequent paper. Making an ansatz for undetermined coefficients and using a computer package for solving bilinear algebraic systems, we give the complete lists of second-order systems with a third-order or a fourth-order symmetry and third-order systems with a fifth-order symmetry. For all but a few systems in the lists, we show that the system (or, at least a subsystem of it) admits either a Lax representation or a linearizing transformation. A thorough comparison with recent work of Foursov and Olver is made.

Polynomial approach method to solve the neutron point kinetics equations with use of the analytic continuation

Energy Technology Data Exchange (ETDEWEB)

Tumelero, Fernanda; Petersen, Claudio Zen; Goncalves, Glenio Aguiar [Universidade Federal de Pelotas, Capao do Leao, RS (Brazil). Programa de Pos Graduacao em Modelagem Matematica; Schramm, Marcelo [Universidade Federal do Rio Grande do Sul, Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica

2016-12-15

In this work, we report a solution to solve the Neutron Point Kinetics Equations applying the Polynomial Approach Method. The main idea is to expand the neutron density and delayed neutron precursors as a power series considering the reactivity as an arbitrary function of the time in a relatively short time interval around an ordinary point. In the first interval one applies the initial conditions and the analytical continuation is used to determine the solutions of the next intervals. A genuine error control is developed based on an analogy with the Rest Theorem. For illustration, we also report simulations for different approaches types (linear, quadratic and cubic). The results obtained by numerical simulations for linear approximation are compared with results in the literature.

A polynomial expansion method and its application in the coupled Zakharov-Kuznetsov equations

International Nuclear Information System (INIS)

Huang Wenhua

2006-01-01

A polynomial expansion method is presented to solve nonlinear evolution equations. Applying this method, the coupled Zakharov-Kuznetsov equations in fluid system are studied and many exact travelling wave solutions are obtained. These solutions include solitary wave solutions, periodic solutions and rational type solutions

Skew-orthogonal polynomials, differential systems and random matrix theory

International Nuclear Information System (INIS)

Ghosh, S.

2007-01-01

We study skew-orthogonal polynomials with respect to the weight function exp[-2V (x)], with V (x) = Σ K=1 2d (u K /K)x K , u 2d > 0, d > 0. A finite subsequence of such skew-orthogonal polynomials arising in the study of Orthogonal and Symplectic ensembles of random matrices, satisfy a system of differential-difference-deformation equation. The vectors formed by such subsequence has the rank equal to the degree of the potential in the quaternion sense. These solutions satisfy certain compatibility condition and hence admit a simultaneous fundamental system of solutions. (author)

A general U-block model-based design procedure for nonlinear polynomial control systems

Science.gov (United States)

Zhu, Q. M.; Zhao, D. Y.; Zhang, Jianhua

2016-10-01

The proposition of U-model concept (in terms of 'providing concise and applicable solutions for complex problems') and a corresponding basic U-control design algorithm was originated in the first author's PhD thesis. The term of U-model appeared (not rigorously defined) for the first time in the first author's other journal paper, which established a framework for using linear polynomial control system design approaches to design nonlinear polynomial control systems (in brief, linear polynomial approaches → nonlinear polynomial plants). This paper represents the next milestone work - using linear state-space approaches to design nonlinear polynomial control systems (in brief, linear state-space approaches → nonlinear polynomial plants). The overall aim of the study is to establish a framework, defined as the U-block model, which provides a generic prototype for using linear state-space-based approaches to design the control systems with smooth nonlinear plants/processes described by polynomial models. For analysing the feasibility and effectiveness, sliding mode control design approach is selected as an exemplary case study. Numerical simulation studies provide a user-friendly step-by-step procedure for the readers/users with interest in their ad hoc applications. In formality, this is the first paper to present the U-model-oriented control system design in a formal way and to study the associated properties and theorems. The previous publications, in the main, have been algorithm-based studies and simulation demonstrations. In some sense, this paper can be treated as a landmark for the U-model-based research from intuitive/heuristic stage to rigour/formal/comprehensive studies.

Improved Polynomial Fuzzy Modeling and Controller with Stability Analysis for Nonlinear Dynamical Systems

Directory of Open Access Journals (Sweden)

Hamed Kharrati

2012-01-01

Full Text Available This study presents an improved model and controller for nonlinear plants using polynomial fuzzy model-based (FMB systems. To minimize mismatch between the polynomial fuzzy model and nonlinear plant, the suitable parameters of membership functions are determined in a systematic way. Defining an appropriate fitness function and utilizing Taylor series expansion, a genetic algorithm (GA is used to form the shape of membership functions in polynomial forms, which are afterwards used in fuzzy modeling. To validate the model, a controller based on proposed polynomial fuzzy systems is designed and then applied to both original nonlinear plant and fuzzy model for comparison. Additionally, stability analysis for the proposed polynomial FMB control system is investigated employing Lyapunov theory and a sum of squares (SOS approach. Moreover, the form of the membership functions is considered in stability analysis. The SOS-based stability conditions are attained using SOSTOOLS. Simulation results are also given to demonstrate the effectiveness of the proposed method.

Inelastic scattering with Chebyshev polynomials and preconditioned conjugate gradient minimization.

Science.gov (United States)

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.

A novel stabilization condition for T-S polynomial fuzzy system with time-delay:A sum-of-squares approach

OpenAIRE

Tsai, Shun Hung; Chen, Yu-An; Chen, Yu-Wen; Lo, Ji-Chang; Lam, Hak-Keung

2017-01-01

A novel stabilization problem for T-S polynomial fuzzy system with time-delay is investigated in this paper. Firstly, a polynomial fuzzy controller for T-S polynomial fuzzy system with time-delay is proposed. In addition, based on polynomial Lyapunov-Krasovskii function and the developed polynomial slack variable matrices, a novel stabilization condition for T-S polynomial fuzzy system with time-delay is presented in terms of sum-of-square (SOS) form. Lastly, nonlinear system with time-delay ...

A Posteriori Error Analysis of Stochastic Differential Equations Using Polynomial Chaos Expansions

KAUST Repository

Butler, T.; Dawson, C.; Wildey, T.

2011-01-01

We develop computable a posteriori error estimates for linear functionals of a solution to a general nonlinear stochastic differential equation with random model/source parameters. These error estimates are based on a variational analysis applied to stochastic Galerkin methods for forward and adjoint problems. The result is a representation for the error estimate as a polynomial in the random model/source parameter. The advantage of this method is that we use polynomial chaos representations for the forward and adjoint systems to cheaply produce error estimates by simple evaluation of a polynomial. By comparison, the typical method of producing such estimates requires repeated forward/adjoint solves for each new choice of random parameter. We present numerical examples showing that there is excellent agreement between these methods. © 2011 Society for Industrial and Applied Mathematics.

Solving Multi-variate Polynomial Equations in a Finite Field

Science.gov (United States)

2013-06-01

hardware to encrypt and decrypt messages. Many of the AES predecessors use this Feistel structure (i.e. DES, Lucifer , Blowfish). However, AES does not...However, then it is very effective . The interesting aspect about the agreeing algorithm is that it can gain momentum to solve the system once RHSs are...columns from Lh can now be removed. This can create a ‘cascade effect ’ on the system and the system quickly reduces its size and complexity. Agreeing

Polynomial-time solution of prime factorization and NP-complete problems with digital memcomputing machines

Science.gov (United States)

Traversa, Fabio L.; Di Ventra, Massimiliano

2017-02-01

We introduce a class of digital machines, we name Digital Memcomputing Machines, (DMMs) able to solve a wide range of problems including Non-deterministic Polynomial (NP) ones with polynomial resources (in time, space, and energy). An abstract DMM with this power must satisfy a set of compatible mathematical constraints underlying its practical realization. We prove this by making a connection with the dynamical systems theory. This leads us to a set of physical constraints for poly-resource resolvability. Once the mathematical requirements have been assessed, we propose a practical scheme to solve the above class of problems based on the novel concept of self-organizing logic gates and circuits (SOLCs). These are logic gates and circuits able to accept input signals from any terminal, without distinction between conventional input and output terminals. They can solve boolean problems by self-organizing into their solution. They can be fabricated either with circuit elements with memory (such as memristors) and/or standard MOS technology. Using tools of functional analysis, we prove mathematically the following constraints for the poly-resource resolvability: (i) SOLCs possess a global attractor; (ii) their only equilibrium points are the solutions of the problems to solve; (iii) the system converges exponentially fast to the solutions; (iv) the equilibrium convergence rate scales at most polynomially with input size. We finally provide arguments that periodic orbits and strange attractors cannot coexist with equilibria. As examples, we show how to solve the prime factorization and the search version of the NP-complete subset-sum problem. Since DMMs map integers into integers, they are robust against noise and hence scalable. We finally discuss the implications of the DMM realization through SOLCs to the NP = P question related to constraints of poly-resources resolvability.

Design of a Polynomial Fuzzy Observer Controller With Sampled-Output Measurements for Nonlinear Systems Considering Unmeasurable Premise Variables

OpenAIRE

Liu, Chuang; Lam, H. K.

2015-01-01

In this paper, we propose a polynomial fuzzy observer controller for nonlinear systems, where the design is achieved through the stability analysis of polynomial-fuzzy-model-based (PFMB) observer-control system. The polynomial fuzzy observer estimates the system states using estimated premise variables. The estimated states are then employed by the polynomial fuzzy controller for the feedback control of nonlinear systems represented by the polynomial fuzzy model. The system stability of the P...

Complex centers of polynomial differential equations

Directory of Open Access Journals (Sweden)

Mohamad Ali M. Alwash

2007-07-01

Full Text Available We present some results on the existence and nonexistence of centers for polynomial first order ordinary differential equations with complex coefficients. In particular, we show that binomial differential equations without linear terms do not have complex centers. Classes of polynomial differential equations, with more than two terms, are presented that do not have complex centers. We also study the relation between complex centers and the Pugh problem. An algorithm is described to solve the Pugh problem for equations without complex centers. The method of proof involves phase plane analysis of the polar equations and a local study of periodic solutions.

Efficient Method to Approximately Solve Retrial Systems with Impatience

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Jose Manuel Gimenez-Guzman

2012-01-01

Full Text Available We present a novel technique to solve multiserver retrial systems with impatience. Unfortunately these systems do not present an exact analytic solution, so it is mandatory to resort to approximate techniques. This novel technique does not rely on the numerical solution of the steady-state Kolmogorov equations of the Continuous Time Markov Chain as it is common for this kind of systems but it considers the system in its Markov Decision Process setting. This technique, known as value extrapolation, truncates the infinite state space using a polynomial extrapolation method to approach the states outside the truncated state space. A numerical evaluation is carried out to evaluate this technique and to compare its performance with previous techniques. The obtained results show that value extrapolation greatly outperforms the previous approaches appeared in the literature not only in terms of accuracy but also in terms of computational cost.

Numerical Solutions for Convection-Diffusion Equation through Non-Polynomial Spline

Directory of Open Access Journals (Sweden)

Ravi Kanth A.S.V.

2016-01-01

Full Text Available In this paper, numerical solutions for convection-diffusion equation via non-polynomial splines are studied. We purpose an implicit method based on non-polynomial spline functions for solving the convection-diffusion equation. The method is proven to be unconditionally stable by using Von Neumann technique. Numerical results are illustrated to demonstrate the efficiency and stability of the purposed method.

Automatic Control Systems Modeling by Volterra Polynomials

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S. V. Solodusha

2012-01-01

Full Text Available The problem of the existence of the solutions of polynomial Volterra integral equations of the first kind of the second degree is considered. An algorithm of the numerical solution of one class of Volterra nonlinear systems of the first kind is developed. Numerical results for test examples are presented.

Smoothing optimization of supporting quadratic surfaces with Zernike polynomials

Science.gov (United States)

Zhang, Hang; Lu, Jiandong; Liu, Rui; Ma, Peifu

2018-03-01

A new optimization method to get a smooth freeform optical surface from an initial surface generated by the supporting quadratic method (SQM) is proposed. To smooth the initial surface, a 9-vertex system from the neighbor quadratic surface and the Zernike polynomials are employed to establish a linear equation system. A local optimized surface to the 9-vertex system can be build by solving the equations. Finally, a continuous smooth optimization surface is constructed by stitching the above algorithm on the whole initial surface. The spot corresponding to the optimized surface is no longer discrete pixels but a continuous distribution.

Stability analysis of nonlinear Roesser-type two-dimensional systems via a homogenous polynomial technique

Science.gov (United States)

Zhang, Tie-Yan; Zhao, Yan; Xie, Xiang-Peng

2012-12-01

This paper is concerned with the problem of stability analysis of nonlinear Roesser-type two-dimensional (2D) systems. Firstly, the fuzzy modeling method for the usual one-dimensional (1D) systems is extended to the 2D case so that the underlying nonlinear 2D system can be represented by the 2D Takagi—Sugeno (TS) fuzzy model, which is convenient for implementing the stability analysis. Secondly, a new kind of fuzzy Lyapunov function, which is a homogeneous polynomially parameter dependent on fuzzy membership functions, is developed to conceive less conservative stability conditions for the TS Roesser-type 2D system. In the process of stability analysis, the obtained stability conditions approach exactness in the sense of convergence by applying some novel relaxed techniques. Moreover, the obtained result is formulated in the form of linear matrix inequalities, which can be easily solved via standard numerical software. Finally, a numerical example is also given to demonstrate the effectiveness of the proposed approach.

Stability analysis of nonlinear Roesser-type two-dimensional systems via a homogenous polynomial technique

International Nuclear Information System (INIS)

Zhang Tie-Yan; Zhao Yan; Xie Xiang-Peng

2012-01-01

This paper is concerned with the problem of stability analysis of nonlinear Roesser-type two-dimensional (2D) systems. Firstly, the fuzzy modeling method for the usual one-dimensional (1D) systems is extended to the 2D case so that the underlying nonlinear 2D system can be represented by the 2D Takagi—Sugeno (TS) fuzzy model, which is convenient for implementing the stability analysis. Secondly, a new kind of fuzzy Lyapunov function, which is a homogeneous polynomially parameter dependent on fuzzy membership functions, is developed to conceive less conservative stability conditions for the TS Roesser-type 2D system. In the process of stability analysis, the obtained stability conditions approach exactness in the sense of convergence by applying some novel relaxed techniques. Moreover, the obtained result is formulated in the form of linear matrix inequalities, which can be easily solved via standard numerical software. Finally, a numerical example is also given to demonstrate the effectiveness of the proposed approach. (general)

Linear precoding based on polynomial expansion: reducing complexity in massive MIMO

KAUST Repository

Mueller, Axel

2016-02-29

Massive multiple-input multiple-output (MIMO) techniques have the potential to bring tremendous improvements in spectral efficiency to future communication systems. Counterintuitively, the practical issues of having uncertain channel knowledge, high propagation losses, and implementing optimal non-linear precoding are solved more or less automatically by enlarging system dimensions. However, the computational precoding complexity grows with the system dimensions. For example, the close-to-optimal and relatively “antenna-efficient” regularized zero-forcing (RZF) precoding is very complicated to implement in practice, since it requires fast inversions of large matrices in every coherence period. Motivated by the high performance of RZF, we propose to replace the matrix inversion and multiplication by a truncated polynomial expansion (TPE), thereby obtaining the new TPE precoding scheme which is more suitable for real-time hardware implementation and significantly reduces the delay to the first transmitted symbol. The degree of the matrix polynomial can be adapted to the available hardware resources and enables smooth transition between simple maximum ratio transmission and more advanced RZF. By deriving new random matrix results, we obtain a deterministic expression for the asymptotic signal-to-interference-and-noise ratio (SINR) achieved by TPE precoding in massive MIMO systems. Furthermore, we provide a closed-form expression for the polynomial coefficients that maximizes this SINR. To maintain a fixed per-user rate loss as compared to RZF, the polynomial degree does not need to scale with the system, but it should be increased with the quality of the channel knowledge and the signal-to-noise ratio.

Improved Polynomial Fuzzy Modeling and Controller with Stability Analysis for Nonlinear Dynamical Systems

OpenAIRE

Hamed Kharrati; Sohrab Khanmohammadi; Witold Pedrycz; Ghasem Alizadeh

2012-01-01

This study presents an improved model and controller for nonlinear plants using polynomial fuzzy model-based (FMB) systems. To minimize mismatch between the polynomial fuzzy model and nonlinear plant, the suitable parameters of membership functions are determined in a systematic way. Defining an appropriate fitness function and utilizing Taylor series expansion, a genetic algorithm (GA) is used to form the shape of membership functions in polynomial forms, which are afterwards used in fuzzy m...

Polynomial chaos functions and stochastic differential equations

International Nuclear Information System (INIS)

Williams, M.M.R.

2006-01-01

The Karhunen-Loeve procedure and the associated polynomial chaos expansion have been employed to solve a simple first order stochastic differential equation which is typical of transport problems. Because the equation has an analytical solution, it provides a useful test of the efficacy of polynomial chaos. We find that the convergence is very rapid in some cases but that the increased complexity associated with many random variables can lead to very long computational times. The work is illustrated by exact and approximate solutions for the mean, variance and the probability distribution itself. The usefulness of a white noise approximation is also assessed. Extensive numerical results are given which highlight the weaknesses and strengths of polynomial chaos. The general conclusion is that the method is promising but requires further detailed study by application to a practical problem in transport theory

Nonnegativity of uncertain polynomials

Directory of Open Access Journals (Sweden)

Šiljak 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.

Algorithm for solving the linear Cauchy problem for large systems of ordinary differential equations with the use of parallel computations

Energy Technology Data Exchange (ETDEWEB)

Moryakov, A. V., E-mail: sailor@orc.ru [National Research Centre Kurchatov Institute (Russian Federation)

2016-12-15

An algorithm for solving the linear Cauchy problem for large systems of ordinary differential equations is presented. The algorithm for systems of first-order differential equations is implemented in the EDELWEISS code with the possibility of parallel computations on supercomputers employing the MPI (Message Passing Interface) standard for the data exchange between parallel processes. The solution is represented by a series of orthogonal polynomials on the interval [0, 1]. The algorithm is characterized by simplicity and the possibility to solve nonlinear problems with a correction of the operator in accordance with the solution obtained in the previous iterative process.

A polynomial approach for generating a monoparametric family of chaotic attractors via switched linear systems

International Nuclear Information System (INIS)

Aguirre-Hernández, B.; Campos-Cantón, E.; López-Renteria, J.A.; Díaz González, E.C.

2015-01-01

In this paper, we consider characteristic polynomials of n-dimensional systems that determine a segment of polynomials. One parameter is used to characterize this segment of polynomials in order to determine the maximal interval of dissipativity and unstability. Then we apply this result to the generation of a family of attractors based on a class of unstable dissipative systems (UDS) of type affine linear systems. This class of systems is comprised of switched linear systems yielding strange attractors. A family of these chaotic switched systems is determined by the maximal interval of perturbation of the matrix that governs the dynamics for still having scroll attractors

Confluent hypergeometric orthogonal polynomials related to the rational quantum Calogero system with harmonic confinement

International Nuclear Information System (INIS)

van Diejen, J.F.

1997-01-01

Two families (type A and type B) of confluent hypergeometric polynomials in several variables are studied. We describe the orthogonality properties, differential equations, and Pieri-type recurrence formulas for these families. In the one-variable case, the polynomials in question reduce to the Hermite polynomials (type A) and the Laguerre polynomials (type B), respectively. The multivariable confluent hypergeometric families considered here may be used to diagonalize the rational quantum Calogero models with harmonic confinement (for the classical root systems) and are closely connected to the (symmetric) generalized spherical harmonics investigated by Dunkl. (orig.)

Efficient algorithms for construction of recurrence relations for the expansion and connection coefficients in series of Al-Salam-Carlitz I polynomials

International Nuclear Information System (INIS)

Doha, E H; Ahmed, H M

2005-01-01

Two formulae expressing explicitly the derivatives and moments of Al-Salam-Carlitz I polynomials of any degree and for any order in terms of Al-Salam-Carlitz I themselves are proved. Two other formulae for the expansion coefficients of general-order derivatives D p q f(x), and for the moments x l D p q f(x), of an arbitrary function f(x) in terms of its original expansion coefficients are also obtained. Application of these formulae for solving q-difference equations with varying coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained. An algebraic symbolic approach (using Mathematica) in order to build and solve recursively for the connection coefficients between Al-Salam-Carlitz I polynomials and any system of basic hypergeometric orthogonal polynomials, belonging to the q-Hahn class, is described

On Sequences of Numbers and Polynomials Defined by Linear Recurrence Relations of Order 2

Directory of Open Access Journals (Sweden)

Tian-Xiao He

2009-01-01

Full Text Available Here we present a new method to construct the explicit formula of a sequence of numbers and polynomials generated by a linear recurrence relation of order 2. The applications of the method to the Fibonacci and Lucas numbers, Chebyshev polynomials, the generalized Gegenbauer-Humbert polynomials are also discussed. The derived idea provides a general method to construct identities of number or polynomial sequences defined by linear recurrence relations. The applications using the method to solve some algebraic and ordinary differential equations are presented.

Polynomial Heisenberg algebras

International Nuclear Information System (INIS)

Carballo, Juan M; C, David J Fernandez; Negro, Javier; Nieto, Luis M

2004-01-01

Polynomial deformations of the Heisenberg algebra are studied in detail. Some of their natural realizations are given by the higher order susy partners (and not only by those of first order, as is already known) of the harmonic oscillator for even-order polynomials. Here, it is shown that the susy partners of the radial oscillator play a similar role when the order of the polynomial is odd. Moreover, it will be proved that the general systems ruled by such kinds of algebras, in the quadratic and cubic cases, involve Painleve transcendents of types IV and V, respectively

Recurrences and explicit formulae for the expansion and connection coefficients in series of Bessel polynomials

International Nuclear Information System (INIS)

Doha, E H; Ahmed, H M

2004-01-01

A formula expressing explicitly the derivatives of Bessel polynomials of any degree and for any order in terms of the Bessel polynomials themselves is proved. Another explicit formula, which expresses the Bessel expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of its original Bessel coefficients, is also given. A formula for the Bessel coefficients of the moments of one single Bessel polynomial of certain degree is proved. A formula for the Bessel coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its Bessel coefficients is also obtained. Application of these formulae for solving ordinary differential equations with varying coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained. An algebraic symbolic approach (using Mathematica) in order to build and solve recursively for the connection coefficients between Bessel-Bessel polynomials is described. An explicit formula for these coefficients between Jacobi and Bessel polynomials is given, of which the ultraspherical polynomial and its consequences are important special cases. Two analytical formulae for the connection coefficients between Laguerre-Bessel and Hermite-Bessel are also developed

Polynomials associated with equilibria of affine Toda-Sutherland systems

International Nuclear Information System (INIS)

Odake, S; Sasaki, R

2004-01-01

An affine Toda-Sutherland system is a quasi-exactly solvable multi-particle dynamics based on an affine simple root system. It is a 'cross' between two well-known integrable multi-particle dynamics, an affine Toda molecule (exponential potential, periodic nearest-neighbour interaction) and a Sutherland system (inverse sine-square interaction). Polynomials describing the equilibrium positions of affine Toda-Sutherland systems are determined for all affine simple root systems

Efficient linear precoding for massive MIMO systems using truncated polynomial expansion

KAUST Repository

Müller, Axel

2014-06-01

Massive multiple-input multiple-output (MIMO) techniques have been proposed as a solution to satisfy many requirements of next generation cellular systems. One downside of massive MIMO is the increased complexity of computing the precoding, especially since the relatively \\'antenna-efficient\\' regularized zero-forcing (RZF) is preferred to simple maximum ratio transmission. We develop in this paper a new class of precoders for single-cell massive MIMO systems. It is based on truncated polynomial expansion (TPE) and mimics the advantages of RZF, while offering reduced and scalable computational complexity that can be implemented in a convenient parallel fashion. Using random matrix theory we provide a closed-form expression of the signal-to-interference-and-noise ratio under TPE precoding and compare it to previous works on RZF. Furthermore, the sum rate maximizing polynomial coefficients in TPE precoding are calculated. By simulation, we find that to maintain a fixed peruser rate loss as compared to RZF, the polynomial degree does not need to scale with the system, but it should be increased with the quality of the channel knowledge and signal-to-noise ratio. © 2014 IEEE.

The structure of solutions of the matrix linear unilateral polynomial equation with two variables

Directory of Open Access Journals (Sweden)

N. S. Dzhaliuk

2017-07-01

Full Text Available We investigate the structure of solutions of the matrix linear polynomial equation $A(\\lambdaX(\\lambda+B(\\lambdaY(\\lambda=C(\\lambda,$ in particular, possible degrees of the solutions. The solving of this equation is reduced to the solving of the equivalent matrix polynomial equation with matrix coefficients in triangular forms with invariant factors on the main diagonals, to which the matrices $A (\\lambda, B(\\lambda$ \\ and \\ $C(\\lambda$ are reduced by means of semiscalar equivalent transformations. On the basis of it, we have pointed out the bounds of the degrees of the matrix polynomial equation solutions. Necessary and sufficient conditions for the uniqueness of a solution with a minimal degree are established. An effective method for constructing minimal degree solutions of the equations is suggested. In this article, unlike well-known results about the estimations of the degrees of the solutions of the matrix polynomial equations in which both matrix coefficients are regular or at least one of them is regular, we have considered the case when the matrix polynomial equation has arbitrary matrix coefficients $A(\\lambda$ and $B(\\lambda.$ 

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.)

Mirror symmetry, toric branes and topological string amplitudes as polynomials

International Nuclear Information System (INIS)

Alim, Murad

2009-01-01

The central theme of this thesis is the extension and application of mirror symmetry of topological string theory. The contribution of this work on the mathematical side is given by interpreting the calculated partition functions as generating functions for mathematical invariants which are extracted in various examples. Furthermore the extension of the variation of the vacuum bundle to include D-branes on compact geometries is studied. Based on previous work for non-compact geometries a system of differential equations is derived which allows to extend the mirror map to the deformation spaces of the D-Branes. Furthermore, these equations allow the computation of the full quantum corrected superpotentials which are induced by the D-branes. Based on the holomorphic anomaly equation, which describes the background dependence of topological string theory relating recursively loop amplitudes, this work generalizes a polynomial construction of the loop amplitudes, which was found for manifolds with a one dimensional space of deformations, to arbitrary target manifolds with arbitrary dimension of the deformation space. The polynomial generators are determined and it is proven that the higher loop amplitudes are polynomials of a certain degree in the generators. Furthermore, the polynomial construction is generalized to solve the extension of the holomorphic anomaly equation to D-branes without deformation space. This method is applied to calculate higher loop amplitudes in numerous examples and the mathematical invariants are extracted. (orig.)

How many invariant polynomials are needed to decide local unitary equivalence of qubit states?

International Nuclear Information System (INIS)

Maciążek, Tomasz; Oszmaniec, Michał; Sawicki, Adam

2013-01-01

Given L-qubit states with the fixed spectra of reduced one-qubit density matrices, we find a formula for the minimal number of invariant polynomials needed for solving local unitary (LU) equivalence problem, that is, problem of deciding if two states can be connected by local unitary operations. Interestingly, this number is not the same for every collection of the spectra. Some spectra require less polynomials to solve LU equivalence problem than others. The result is obtained using geometric methods, i.e., by calculating the dimensions of reduced spaces, stemming from the symplectic reduction procedure

Optimization of Cubic Polynomial Functions without Calculus

Science.gov (United States)

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…

Constructing general partial differential equations using polynomial and neural networks.

Science.gov (United States)

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.

Stability Analysis of Positive Polynomial Fuzzy-Model-Based Control Systems with Time Delay under Imperfect Premise Matching

OpenAIRE

Li, Xiaomiao; Lam, Hak Keung; Song, Ge; Liu, Fucai

2017-01-01

This paper deals with the stability and positivity analysis of polynomial-fuzzy-model-based ({PFMB}) control systems with time delay, which is formed by a polynomial fuzzy model and a polynomial fuzzy controller connected in a closed loop, under imperfect premise matching. To improve the design and realization flexibility, the polynomial fuzzy model and the polynomial fuzzy controller are allowed to have their own set of premise membership functions. A sum-of-squares (SOS)-based stability ana...

Wilson polynomials/functions and intertwining operators for the generic quantum superintegrable system on the 2-sphere

Science.gov (United States)

Miller, W., Jr.; Li, Q.

2015-04-01

The Wilson and Racah polynomials can be characterized as basis functions for irreducible representations of the quadratic symmetry algebra of the quantum superintegrable system on the 2-sphere, HΨ = EΨ, with generic 3-parameter potential. Clearly, the polynomials are expansion coefficients for one eigenbasis of a symmetry operator L2 of H in terms of an eigenbasis of another symmetry operator L1, but the exact relationship appears not to have been made explicit. We work out the details of the expansion to show, explicitly, how the polynomials arise and how the principal properties of these functions: the measure, 3-term recurrence relation, 2nd order difference equation, duality of these relations, permutation symmetry, intertwining operators and an alternate derivation of Wilson functions - follow from the symmetry of this quantum system. This paper is an exercise to show that quantum mechancal concepts and recurrence relations for Gausian hypergeometrc functions alone suffice to explain these properties; we make no assumptions about the structure of Wilson polynomial/functions, but derive them from quantum principles. There is active interest in the relation between multivariable Wilson polynomials and the quantum superintegrable system on the n-sphere with generic potential, and these results should aid in the generalization. Contracting function space realizations of irreducible representations of this quadratic algebra to the other superintegrable systems one can obtain the full Askey scheme of orthogonal hypergeometric polynomials. All of these contractions of superintegrable systems with potential are uniquely induced by Wigner Lie algebra contractions of so(3, C) and e(2,C). All of the polynomials produced are interpretable as quantum expansion coefficients. It is important to extend this process to higher dimensions.

Wilson polynomials/functions and intertwining operators for the generic quantum superintegrable system on the 2-sphere

International Nuclear Information System (INIS)

Miller, W Jr; Li, Q

2015-01-01

The Wilson and Racah polynomials can be characterized as basis functions for irreducible representations of the quadratic symmetry algebra of the quantum superintegrable system on the 2-sphere, HΨ = EΨ, with generic 3-parameter potential. Clearly, the polynomials are expansion coefficients for one eigenbasis of a symmetry operator L 2 of H in terms of an eigenbasis of another symmetry operator L 1 , but the exact relationship appears not to have been made explicit. We work out the details of the expansion to show, explicitly, how the polynomials arise and how the principal properties of these functions: the measure, 3-term recurrence relation, 2nd order difference equation, duality of these relations, permutation symmetry, intertwining operators and an alternate derivation of Wilson functions - follow from the symmetry of this quantum system. This paper is an exercise to show that quantum mechancal concepts and recurrence relations for Gausian hypergeometrc functions alone suffice to explain these properties; we make no assumptions about the structure of Wilson polynomial/functions, but derive them from quantum principles. There is active interest in the relation between multivariable Wilson polynomials and the quantum superintegrable system on the n-sphere with generic potential, and these results should aid in the generalization. Contracting function space realizations of irreducible representations of this quadratic algebra to the other superintegrable systems one can obtain the full Askey scheme of orthogonal hypergeometric polynomials. All of these contractions of superintegrable systems with potential are uniquely induced by Wigner Lie algebra contractions of so(3, C) and e(2,C). All of the polynomials produced are interpretable as quantum expansion coefficients. It is important to extend this process to higher dimensions. (paper)

Sum-of-squares based observer design for polynomial systems with a known fixed time delay

Czech Academy of Sciences Publication Activity Database

Rehák, Branislav

2015-01-01

Roč. 51, č. 5 (2015), s. 858-873 ISSN 0023-5954 R&D Projects: GA ČR GA13-02149S Institutional support: RVO:67985556 Keywords : sum-of-squares polynomial * observer * polynomial system Subject RIV: BC - Control Systems Theory Impact factor: 0.628, year: 2015 http://www.kybernetika.cz/content/2015/5/856

On the Evaluation of Computational Results Obtained from Solving System of linear Equations With matlab The Dual affine Scalling interior Point

International Nuclear Information System (INIS)

Murfi, Hendri; Basaruddin, T.

2001-01-01

The interior point method for linear programming has gained extraordinary interest as an alternative to simplex method since Karmarkar presented a polynomial-time algorithm for linear programming based on interior point method. In implementation of the algorithm of this method, there are two important things that have impact heavily to performance of the algorithm; they are data structure and used method to solve linear equation system in the algorithm. This paper describes about solving linear equation system in variants of the algorithm called dual-affine scaling algorithm. Next, we evaluate experimentally results of some used methods, either direct method or iterative method. The experimental evaluation used Matlab

Polynomial approximation on polytopes

CERN Document Server

Totik, Vilmos

2014-01-01

Polynomial approximation on convex polytopes in \\mathbf{R}^d is considered in uniform and L^p-norms. For an appropriate modulus of smoothness matching direct and converse estimates are proven. In the L^p-case so called strong direct and converse results are also verified. The equivalence of the moduli of smoothness with an appropriate K-functional follows as a consequence. The results solve a problem that was left open since the mid 1980s when some of the present findings were established for special, so-called simple polytopes.

Variational Iteration Method for Fifth-Order Boundary Value Problems Using He's Polynomials

Directory of Open Access Journals (Sweden)

Muhammad Aslam Noor

2008-01-01

Full Text Available We apply the variational iteration method using He's polynomials (VIMHP for solving the fifth-order boundary value problems. The proposed method is an elegant combination of variational iteration and the homotopy perturbation methods and is mainly due to Ghorbani (2007. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The proposed iterative scheme finds the solution without any discritization, linearization, or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that the proposed technique solves nonlinear problems without using Adomian's polynomials can be considered as a clear advantage of this algorithm over the decomposition method.

Solution of linear transport equation using Chebyshev polynomials and Laplace transform

International Nuclear Information System (INIS)

Cardona, A.V.; Vilhena, M.T.M.B. de

1994-01-01

The Chebyshev polynomials and the Laplace transform are combined to solve, analytically, the linear transport equation in planar geometry, considering isotropic scattering and the one-group model. Numerical simulation is presented. (author)

A Polynomial Optimization Approach to Constant Rebalanced Portfolio Selection

NARCIS (Netherlands)

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

A polynomial optimization approach to constant rebalanced portfolio selection

NARCIS (Netherlands)

Takano, Y.; Sotirov, R.

2012-01-01

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

Fitting method of pseudo-polynomial for solving nonlinear parametric adjustment

Institute of Scientific and Technical Information of China (English)

陶华学; 宫秀军; 郭金运

2001-01-01

The optimal condition and its geometrical characters of the least-square adjustment were proposed. Then the relation between the transformed surface and least-squares was discussed. Based on the above, a non-iterative method, called the fitting method of pseudo-polynomial, was derived in detail. The final least-squares solution can be determined with sufficient accuracy in a single step and is not attained by moving the initial point in the view of iteration. The accuracy of the solution relys wholly on the frequency of Taylor's series. The example verifies the correctness and validness of the method.

Bifurcation in Z2-symmetry quadratic polynomial systems with delay

International Nuclear Information System (INIS)

Zhang Chunrui; Zheng Baodong

2009-01-01

Z 2 -symmetry systems are considered. Firstly the general forms of Z 2 -symmetry quadratic polynomial system are given, and then a three-dimensional Z 2 equivariant system is considered, which describes the relations of two predator species for a single prey species. Finally, the explicit formulas for determining the Fold and Hopf bifurcations are obtained by using the normal form theory and center manifold argument.

Arabic text classification using Polynomial Networks

Directory of Open Access Journals (Sweden)

Mayy M. Al-Tahrawi

2015-10-01

Full Text Available In this paper, an Arabic statistical learning-based text classification system has been developed using Polynomial Neural Networks. Polynomial Networks have been recently applied to English text classification, but they were never used for Arabic text classification. In this research, we investigate the performance of Polynomial Networks in classifying Arabic texts. Experiments are conducted on a widely used Arabic dataset in text classification: Al-Jazeera News dataset. We chose this dataset to enable direct comparisons of the performance of Polynomial Networks classifier versus other well-known classifiers on this dataset in the literature of Arabic text classification. Results of experiments show that Polynomial Networks classifier is a competitive algorithm to the state-of-the-art ones in the field of Arabic text classification.

A companion matrix for 2-D polynomials

International Nuclear Information System (INIS)

Boudellioua, M.S.

1995-08-01

In this paper, a matrix form analogous to the companion matrix which is often encountered in the theory of one dimensional (1-D) linear systems is suggested for a class of polynomials in two indeterminates and real coefficients, here referred to as two dimensional (2-D) polynomials. These polynomials arise in the context of 2-D linear systems theory. Necessary and sufficient conditions are also presented under which a matrix is equivalent to this companion form. (author). 6 refs

Chromatic polynomials of random graphs

International Nuclear Information System (INIS)

Van Bussel, Frank; Fliegner, Denny; Timme, Marc; Ehrlich, Christoph; Stolzenberg, Sebastian

2010-01-01

Chromatic polynomials and related graph invariants are central objects in both graph theory and statistical physics. Computational difficulties, however, have so far restricted studies of such polynomials to graphs that were either very small, very sparse or highly structured. Recent algorithmic advances (Timme et al 2009 New J. Phys. 11 023001) now make it possible to compute chromatic polynomials for moderately sized graphs of arbitrary structure and number of edges. Here we present chromatic polynomials of ensembles of random graphs with up to 30 vertices, over the entire range of edge density. We specifically focus on the locations of the zeros of the polynomial in the complex plane. The results indicate that the chromatic zeros of random graphs have a very consistent layout. In particular, the crossing point, the point at which the chromatic zeros with non-zero imaginary part approach the real axis, scales linearly with the average degree over most of the density range. While the scaling laws obtained are purely empirical, if they continue to hold in general there are significant implications: the crossing points of chromatic zeros in the thermodynamic limit separate systems with zero ground state entropy from systems with positive ground state entropy, the latter an exception to the third law of thermodynamics.

Homogenous polynomially parameter-dependent H∞ filter designs of discrete-time fuzzy systems.

Science.gov (United States)

Zhang, Huaguang; Xie, Xiangpeng; Tong, Shaocheng

2011-10-01

This paper proposes a novel H(∞) filtering technique for a class of discrete-time fuzzy systems. First, a novel kind of fuzzy H(∞) filter, which is homogenous polynomially parameter dependent on membership functions with an arbitrary degree, is developed to guarantee the asymptotic stability and a prescribed H(∞) performance of the filtering error system. Second, relaxed conditions for H(∞) performance analysis are proposed by using a new fuzzy Lyapunov function and the Finsler lemma with homogenous polynomial matrix Lagrange multipliers. Then, based on a new kind of slack variable technique, relaxed linear matrix inequality-based H(∞) filtering conditions are proposed. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed approach.

Some Identities Involving the Derivative of the First Kind Chebyshev Polynomials

Directory of Open Access Journals (Sweden)

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.

Application of Least-Squares Spectral Element Methods to Polynomial Chaos

NARCIS (Netherlands)

Vos, P.E.J.; Gerritsma, M.I.

2006-01-01

This papers describes the use of the Least-Squares Spectral Element Method to polynomial Chaos to solve stochastic partial differential equations. The method will be described in detail and a comparison will be presented between the least-squares projection and the conventional Galerkin projection.

Hybrid Recurrent Laguerre-Orthogonal-Polynomial NN Control System Applied in V-Belt Continuously Variable Transmission System Using Particle Swarm Optimization

Directory of Open Access Journals (Sweden)

Chih-Hong Lin

2015-01-01

Full Text Available Because the V-belt continuously variable transmission (CVT system driven by permanent magnet synchronous motor (PMSM has much unknown nonlinear and time-varying characteristics, the better control performance design for the linear control design is a time consuming procedure. In order to overcome difficulties for design of the linear controllers, the hybrid recurrent Laguerre-orthogonal-polynomial neural network (NN control system which has online learning ability to respond to the system’s nonlinear and time-varying behaviors is proposed to control PMSM servo-driven V-belt CVT system under the occurrence of the lumped nonlinear load disturbances. The hybrid recurrent Laguerre-orthogonal-polynomial NN control system consists of an inspector control, a recurrent Laguerre-orthogonal-polynomial NN control with adaptive law, and a recouped control with estimated law. Moreover, the adaptive law of online parameters in the recurrent Laguerre-orthogonal-polynomial NN is derived using the Lyapunov stability theorem. Furthermore, the optimal learning rate of the parameters by means of modified particle swarm optimization (PSO is proposed to achieve fast convergence. Finally, to show the effectiveness of the proposed control scheme, comparative studies are demonstrated by experimental results.

Exact solution of Chern-Simons-matter matrix models with characteristic/orthogonal polynomials

International Nuclear Information System (INIS)

Tierz, Miguel

2016-01-01

We solve for finite N the matrix model of supersymmetric U(N) Chern-Simons theory coupled to N f fundamental and N f anti-fundamental chiral multiplets of R-charge 1/2 and of mass m, by identifying it with an average of inverse characteristic polynomials in a Stieltjes-Wigert ensemble. This requires the computation of the Cauchy transform of the Stieltjes-Wigert polynomials, which we carry out, finding a relationship with Mordell integrals, and hence with previous analytical results on the matrix model. The semiclassical limit of the model is expressed, for arbitrary N f , in terms of a single Hermite polynomial. This result also holds for more general matter content, involving matrix models with double-sine functions.

Contributions to fuzzy polynomial techniques for stability analysis and control

OpenAIRE

Pitarch Pérez, José Luis

2014-01-01

The present thesis employs fuzzy-polynomial control techniques in order to improve the stability analysis and control of nonlinear systems. Initially, it reviews the more extended techniques in the field of Takagi-Sugeno fuzzy systems, such as the more relevant results about polynomial and fuzzy polynomial systems. The basic framework uses fuzzy polynomial models by Taylor series and sum-of-squares techniques (semidefinite programming) in order to obtain stability guarantees...

Parallel Optimization of Polynomials for Large-scale Problems in Stability and Control

Science.gov (United States)

Kamyar, Reza

In this thesis, we focus on some of the NP-hard problems in control theory. Thanks to the converse Lyapunov theory, these problems can often be modeled as optimization over polynomials. To avoid the problem of intractability, we establish a trade off between accuracy and complexity. In particular, we develop a sequence of tractable optimization problems --- in the form of Linear Programs (LPs) and/or Semi-Definite Programs (SDPs) --- whose solutions converge to the exact solution of the NP-hard problem. However, the computational and memory complexity of these LPs and SDPs grow exponentially with the progress of the sequence - meaning that improving the accuracy of the solutions requires solving SDPs with tens of thousands of decision variables and constraints. Setting up and solving such problems is a significant challenge. The existing optimization algorithms and software are only designed to use desktop computers or small cluster computers --- machines which do not have sufficient memory for solving such large SDPs. Moreover, the speed-up of these algorithms does not scale beyond dozens of processors. This in fact is the reason we seek parallel algorithms for setting-up and solving large SDPs on large cluster- and/or super-computers. We propose parallel algorithms for stability analysis of two classes of systems: 1) Linear systems with a large number of uncertain parameters; 2) Nonlinear systems defined by polynomial vector fields. First, we develop a distributed parallel algorithm which applies Polya's and/or Handelman's theorems to some variants of parameter-dependent Lyapunov inequalities with parameters defined over the standard simplex. The result is a sequence of SDPs which possess a block-diagonal structure. We then develop a parallel SDP solver which exploits this structure in order to map the computation, memory and communication to a distributed parallel environment. Numerical tests on a supercomputer demonstrate the ability of the algorithm to

A new numerical treatment based on Lucas polynomials for 1D and 2D sinh-Gordon equation

Science.gov (United States)

Oruç, Ömer

2018-04-01

In this paper, a new mixed method based on Lucas and Fibonacci polynomials is developed for numerical solutions of 1D and 2D sinh-Gordon equations. Firstly time variable discretized by central finite difference and then unknown function and its derivatives are expanded to Lucas series. With the help of these series expansion and Fibonacci polynomials, matrices for differentiation are derived. With this approach, finding the solution of sinh-Gordon equation transformed to finding the solution of an algebraic system of equations. Lucas series coefficients are acquired by solving this system of algebraic equations. Then by plugginging these coefficients into Lucas series expansion numerical solutions can be obtained consecutively. The main objective of this paper is to demonstrate that Lucas polynomial based method is convenient for 1D and 2D nonlinear problems. By calculating L2 and L∞ error norms of some 1D and 2D test problems efficiency and performance of the proposed method is monitored. Acquired accurate results confirm the applicability of the method.

Moments, positive polynomials and their applications

CERN Document Server

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,

Imaging characteristics of Zernike and annular polynomial aberrations.

Science.gov (United States)

Mahajan, Virendra N; Díaz, José Antonio

2013-04-01

The general equations for the point-spread function (PSF) and optical transfer function (OTF) are given for any pupil shape, and they are applied to optical imaging systems with circular and annular pupils. The symmetry properties of the PSF, the real and imaginary parts of the OTF, and the modulation transfer function (MTF) of a system with a circular pupil aberrated by a Zernike circle polynomial aberration are derived. The interferograms and PSFs are illustrated for some typical polynomial aberrations with a sigma value of one wave, and 3D PSFs and MTFs are shown for 0.1 wave. The Strehl ratio is also calculated for polynomial aberrations with a sigma value of 0.1 wave, and shown to be well estimated from the sigma value. The numerical results are compared with the corresponding results in the literature. Because of the same angular dependence of the corresponding annular and circle polynomial aberrations, the symmetry properties of systems with annular pupils aberrated by an annular polynomial aberration are the same as those for a circular pupil aberrated by a corresponding circle polynomial aberration. They are also illustrated with numerical examples.

A quasi-static polynomial nodal method for nuclear reactor analysis

International Nuclear Information System (INIS)

Gehin, J.C.

1992-09-01

Modern nodal methods are currently available which can accurately and efficiently solve the static and transient neutron diffusion equations. Most of the methods, however, are limited to two energy groups for practical application. The objective of this research is the development of a static and transient, multidimensional nodal method which allows more than two energy groups and uses a non-linear iterative method for efficient solution of the nodal equations. For both the static and transient methods, finite-difference equations which are corrected by the use of discontinuity factors are derived. The discontinuity factors are computed from a polynomial nodal method using a non-linear iteration technique. The polynomial nodal method is based upon a quartic approximation and utilizes a quadratic transverse-leakage approximation. The solution of the time-dependent equations is performed by the use of a quasi-static method in which the node-averaged fluxes are factored into shape and amplitude functions. The application of the quasi-static polynomial method to several benchmark problems demonstrates that the accuracy is consistent with that of other nodal methods. The use of the quasi-static method is shown to substantially reduce the computation time over the traditional fully-implicit time-integration method. Problems involving thermal-hydraulic feedback are accurately, and efficiently, solved by performing several reactivity/thermal-hydraulic updates per shape calculation

A quasi-static polynomial nodal method for nuclear reactor analysis

Energy Technology Data Exchange (ETDEWEB)

Gehin, Jess C. [Massachusetts Inst. of Tech., Cambridge, MA (United States)

1992-09-01

Modern nodal methods are currently available which can accurately and efficiently solve the static and transient neutron diffusion equations. Most of the methods, however, are limited to two energy groups for practical application. The objective of this research is the development of a static and transient, multidimensional nodal method which allows more than two energy groups and uses a non-linear iterative method for efficient solution of the nodal equations. For both the static and transient methods, finite-difference equations which are corrected by the use of discontinuity factors are derived. The discontinuity factors are computed from a polynomial nodal method using a non-linear iteration technique. The polynomial nodal method is based upon a quartic approximation and utilizes a quadratic transverse-leakage approximation. The solution of the time-dependent equations is performed by the use of a quasi-static method in which the node-averaged fluxes are factored into shape and amplitude functions. The application of the quasi-static polynomial method to several benchmark problems demonstrates that the accuracy is consistent with that of other nodal methods. The use of the quasi-static method is shown to substantially reduce the computation time over the traditional fully-implicit time-integration method. Problems involving thermal-hydraulic feedback are accurately, and efficiently, solved by performing several reactivity/thermal-hydraulic updates per shape calculation.

Using Chebyshev polynomials and approximate inverse triangular factorizations for preconditioning the conjugate gradient method

Science.gov (United States)

Kaporin, I. E.

2012-02-01

In order to precondition a sparse symmetric positive definite matrix, its approximate inverse is examined, which is represented as the product of two sparse mutually adjoint triangular matrices. In this way, the solution of the corresponding system of linear algebraic equations (SLAE) by applying the preconditioned conjugate gradient method (CGM) is reduced to performing only elementary vector operations and calculating sparse matrix-vector products. A method for constructing the above preconditioner is described and analyzed. The triangular factor has a fixed sparsity pattern and is optimal in the sense that the preconditioned matrix has a minimum K-condition number. The use of polynomial preconditioning based on Chebyshev polynomials makes it possible to considerably reduce the amount of scalar product operations (at the cost of an insignificant increase in the total number of arithmetic operations). The possibility of an efficient massively parallel implementation of the resulting method for solving SLAEs is discussed. For a sequential version of this method, the results obtained by solving 56 test problems from the Florida sparse matrix collection (which are large-scale and ill-conditioned) are presented. These results show that the method is highly reliable and has low computational costs.

Planar real polynomial differential systems of degree n > 3 having a weak focus of high order

International Nuclear Information System (INIS)

Llibre, J.; Rabanal, R.

2008-06-01

We construct planar polynomial differential systems of even (respectively odd) degree n > 3, of the form linear plus a nonlinear homogeneous part of degree n having a weak focus of order n 2 -1 (respectively (n 2 -1)/2 ) at the origin. As far as we know this provides the highest order known until now for a weak focus of a polynomial differential system of arbitrary degree n. (author)

Polynomial fuzzy model-based control systems stability analysis and control synthesis using membership function dependent techniques

CERN Document Server

Lam, Hak-Keung

2016-01-01

This book presents recent research on the stability analysis of polynomial-fuzzy-model-based control systems where the concept of partially/imperfectly matched premises and membership-function dependent analysis are considered. The membership-function-dependent analysis offers a new research direction for fuzzy-model-based control systems by taking into account the characteristic and information of the membership functions in the stability analysis. The book presents on a research level the most recent and advanced research results, promotes the research of polynomial-fuzzy-model-based control systems, and provides theoretical support and point a research direction to postgraduate students and fellow researchers. Each chapter provides numerical examples to verify the analysis results, demonstrate the effectiveness of the proposed polynomial fuzzy control schemes, and explain the design procedure. The book is comprehensively written enclosing detailed derivation steps and mathematical derivations also for read...

Parallel multigrid smoothing: polynomial versus Gauss-Seidel

International Nuclear Information System (INIS)

Adams, Mark; Brezina, Marian; Hu, Jonathan; Tuminaro, Ray

2003-01-01

Gauss-Seidel is often the smoother of choice within multigrid applications. In the context of unstructured meshes, however, maintaining good parallel efficiency is difficult with multiplicative iterative methods such as Gauss-Seidel. This leads us to consider alternative smoothers. We discuss the computational advantages of polynomial smoothers within parallel multigrid algorithms for positive definite symmetric systems. Two particular polynomials are considered: Chebyshev and a multilevel specific polynomial. The advantages of polynomial smoothing over traditional smoothers such as Gauss-Seidel are illustrated on several applications: Poisson's equation, thin-body elasticity, and eddy current approximations to Maxwell's equations. While parallelizing the Gauss-Seidel method typically involves a compromise between a scalable convergence rate and maintaining high flop rates, polynomial smoothers achieve parallel scalable multigrid convergence rates without sacrificing flop rates. We show that, although parallel computers are the main motivation, polynomial smoothers are often surprisingly competitive with Gauss-Seidel smoothers on serial machines

Parallel multigrid smoothing: polynomial versus Gauss-Seidel

Science.gov (United States)

Adams, Mark; Brezina, Marian; Hu, Jonathan; Tuminaro, Ray

2003-07-01

Gauss-Seidel is often the smoother of choice within multigrid applications. In the context of unstructured meshes, however, maintaining good parallel efficiency is difficult with multiplicative iterative methods such as Gauss-Seidel. This leads us to consider alternative smoothers. We discuss the computational advantages of polynomial smoothers within parallel multigrid algorithms for positive definite symmetric systems. Two particular polynomials are considered: Chebyshev and a multilevel specific polynomial. The advantages of polynomial smoothing over traditional smoothers such as Gauss-Seidel are illustrated on several applications: Poisson's equation, thin-body elasticity, and eddy current approximations to Maxwell's equations. While parallelizing the Gauss-Seidel method typically involves a compromise between a scalable convergence rate and maintaining high flop rates, polynomial smoothers achieve parallel scalable multigrid convergence rates without sacrificing flop rates. We show that, although parallel computers are the main motivation, polynomial smoothers are often surprisingly competitive with Gauss-Seidel smoothers on serial machines.

Global sensitivity analysis by polynomial dimensional decomposition

Energy Technology Data Exchange (ETDEWEB)

Rahman, Sharif, E-mail: rahman@engineering.uiowa.ed [College of Engineering, The University of Iowa, Iowa City, IA 52242 (United States)

2011-07-15

This paper presents a polynomial dimensional decomposition (PDD) method for global sensitivity analysis of stochastic systems subject to independent random input following arbitrary probability distributions. The method involves Fourier-polynomial expansions of lower-variate component functions of a stochastic response by measure-consistent orthonormal polynomial bases, analytical formulae for calculating the global sensitivity indices in terms of the expansion coefficients, and dimension-reduction integration for estimating the expansion coefficients. Due to identical dimensional structures of PDD and analysis-of-variance decomposition, the proposed method facilitates simple and direct calculation of the global sensitivity indices. Numerical results of the global sensitivity indices computed for smooth systems reveal significantly higher convergence rates of the PDD approximation than those from existing methods, including polynomial chaos expansion, random balance design, state-dependent parameter, improved Sobol's method, and sampling-based methods. However, for non-smooth functions, the convergence properties of the PDD solution deteriorate to a great extent, warranting further improvements. The computational complexity of the PDD method is polynomial, as opposed to exponential, thereby alleviating the curse of dimensionality to some extent.

A Fast lattice-based polynomial digital signature system for m-commerce

Science.gov (United States)

Wei, Xinzhou; Leung, Lin; Anshel, Michael

2003-01-01

The privacy and data integrity are not guaranteed in current wireless communications due to the security hole inside the Wireless Application Protocol (WAP) version 1.2 gateway. One of the remedies is to provide an end-to-end security in m-commerce by applying application level security on top of current WAP1.2. The traditional security technologies like RSA and ECC applied on enterprise's server are not practical for wireless devices because wireless devices have relatively weak computation power and limited memory compared with server. In this paper, we developed a lattice based polynomial digital signature system based on NTRU's Polynomial Authentication and Signature Scheme (PASS), which enabled the feasibility of applying high-level security on both server and wireless device sides.

Irreducible multivariate polynomials obtained from polynomials in ...

Indian Academy of Sciences (India)

Hall, 1409 W. Green Street, Urbana, IL 61801, USA. E-mail: Nicolae. ... Theorem A. If we write an irreducible polynomial f ∈ K[X] as a sum of polynomials a0,..., an ..... This shows us that deg ai = (n − i) deg f2 for each i = 0,..., n, so min k>0.

A family of orthogonal polynomials to solve the inverse equation of punctual kinetics

International Nuclear Information System (INIS)

Suescun D, D.; Bonilla L, H. F.; Figueroa J, J. H.

2015-09-01

This paper proposes a new method to reduce fluctuations in the reactivity calculation, a discrete family of orthogonal polynomials that can soften the nuclear power is used, regarded as the set of data needed to calculate the reactivity in a nuclear reactor. The final results show an improvement in the calculation precision of reactivity, compared to some papers presented in the literature. (Author)

Global structure of a polynomial autonomous system on the plane

International Nuclear Information System (INIS)

Nguyen Van Chau.

1991-10-01

This note is to study the global behaviour of a polynomial autonomous system on the plane with divergence non-positive outside a bounded set. It is shown that in some certain conditions the global structure of such system can be simple. The main result here can be seen as an improvement of the result of Olech and Meister concerning with the global asymptotical stable conjecture of Markur and Yamable and the Jacobian Conjecture. (author). 13 refs

Complex Polynomial Vector Fields

DEFF Research Database (Denmark)

Dias, Kealey

vector fields. Since the class of complex polynomial vector fields in the plane is natural to consider, it is remarkable that its study has only begun very recently. There are numerous fundamental questions that are still open, both in the general classification of these vector fields, the decomposition...... of parameter spaces into structurally stable domains, and a description of the bifurcations. For this reason, the talk will focus on these questions for complex polynomial vector fields.......The two branches of dynamical systems, continuous and discrete, correspond to the study of differential equations (vector fields) and iteration of mappings respectively. In holomorphic dynamics, the systems studied are restricted to those described by holomorphic (complex analytic) functions...

Transport equation solving methods

International Nuclear Information System (INIS)

Granjean, P.M.

1984-06-01

This work is mainly devoted to Csub(N) and Fsub(N) methods. CN method: starting from a lemma stated by Placzek, an equivalence is established between two problems: the first one is defined in a finite medium bounded by a surface S, the second one is defined in the whole space. In the first problem the angular flux on the surface S is shown to be the solution of an integral equation. This equation is solved by Galerkin's method. The Csub(N) method is applied here to one-velocity problems: in plane geometry, slab albedo and transmission with Rayleigh scattering, calculation of the extrapolation length; in cylindrical geometry, albedo and extrapolation length calculation with linear scattering. Fsub(N) method: the basic integral transport equation of the Csub(N) method is integrated on Case's elementary distributions; another integral transport equation is obtained: this equation is solved by a collocation method. The plane problems solved by the Csub(N) method are also solved by the Fsub(N) method. The Fsub(N) method is extended to any polynomial scattering law. Some simple spherical problems are also studied. Chandrasekhar's method, collision probability method, Case's method are presented for comparison with Csub(N) and Fsub(N) methods. This comparison shows the respective advantages of the two methods: a) fast convergence and possible extension to various geometries for Csub(N) method; b) easy calculations and easy extension to polynomial scattering for Fsub(N) method [fr

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.......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...

On the use of a spatial Chebyshev polynomials together with the collocation method in solving radiative transfer problem in a slab

International Nuclear Information System (INIS)

Haggag, M.H.; Al-Gorashi, A.K.; Machali, H.M.

2013-01-01

In this study, the integral form of the radiative transfer equation in planar slab with isotropic scattering has been studied by using the Chebyshev polynomial approximation which is called TN method. The scalar flux is expanded in terms of Chebyshev polynomials in the space variable. The expansion coefficients are solutions to a system of linear algebraic equations. Analytical expressions are given for the scalar and angular flux everywhere in the slab. Numerical calculations are done for the transmissivity and reflectivity of slabs with various values of the single scattering albedo. Calculations are also carried out for the transmitted and reflected angular intensity at the slab boundaries. Our numerical results are in a very good agreement with other results, as shown in the tables

Stability analysis of polynomial fuzzy models via polynomial fuzzy Lyapunov functions

OpenAIRE

Bernal Reza, Miguel Ángel; Sala, Antonio; JAADARI, ABDELHAFIDH; Guerra, Thierry-Marie

2011-01-01

In this paper, the stability of continuous-time polynomial fuzzy models by means of a polynomial generalization of fuzzy Lyapunov functions is studied. Fuzzy Lyapunov functions have been fruitfully used in the literature for local analysis of Takagi-Sugeno models, a particular class of the polynomial fuzzy ones. Based on a recent Taylor-series approach which allows a polynomial fuzzy model to exactly represent a nonlinear model in a compact set of the state space, it is shown that a refinemen...

Numerical study of nonlinear singular fractional differential equations arising in biology by operational matrix of shifted Legendre polynomials

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.

Firefly Algorithm for Polynomial Bézier Surface Parameterization

Directory of Open Access Journals (Sweden)

Akemi Gálvez

2013-01-01

reality, medical imaging, computer graphics, computer animation, and many others. Very often, the preferred approximating surface is polynomial, usually described in parametric form. This leads to the problem of determining suitable parametric values for the data points, the so-called surface parameterization. In real-world settings, data points are generally irregularly sampled and subjected to measurement noise, leading to a very difficult nonlinear continuous optimization problem, unsolvable with standard optimization techniques. This paper solves the parameterization problem for polynomial Bézier surfaces by applying the firefly algorithm, a powerful nature-inspired metaheuristic algorithm introduced recently to address difficult optimization problems. The method has been successfully applied to some illustrative examples of open and closed surfaces, including shapes with singularities. Our results show that the method performs very well, being able to yield the best approximating surface with a high degree of accuracy.

Global Monte Carlo Simulation with High Order Polynomial Expansions

International Nuclear Information System (INIS)

William R. Martin; James Paul Holloway; Kaushik Banerjee; Jesse Cheatham; Jeremy Conlin

2007-01-01

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 convergence

Application of polynomial preconditioners to conservation laws

NARCIS (Netherlands)

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

Poincare map for some polynomial systems of differential equations

International Nuclear Information System (INIS)

Varin, V P

2004-01-01

One approach to the classical problem of distinguishing between a centre and a focus for a system of differential equations with polynomial right-hand sides in the plane is discussed. For a broad class of such systems necessary and sufficient conditions for a centre are expressed in terms of equations in variations of higher order. By contrast with the existing methods of investigation, attention is concentrated on the explicit calculation of the asymptotic behaviour of the Poincare map rather than on finding sufficient centre conditions as such; this also enables one to study bifurcations of birth of arbitrarily strongly degenerate cycles.

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

A comparison of companion matrix methods to find roots of a trigonometric polynomial

Science.gov (United States)

Boyd, John P.

2013-08-01

A trigonometric polynomial is a truncated Fourier series of the form fN(t)≡∑j=0Naj cos(jt)+∑j=1N bj sin(jt). It has been previously shown by the author that zeros of such a polynomial can be computed as the eigenvalues of a companion matrix with elements which are complex valued combinations of the Fourier coefficients, the "CCM" method. However, previous work provided no examples, so one goal of this new work is to experimentally test the CCM method. A second goal is introduce a new alternative, the elimination/Chebyshev algorithm, and experimentally compare it with the CCM scheme. The elimination/Chebyshev matrix (ECM) algorithm yields a companion matrix with real-valued elements, albeit at the price of usefulness only for real roots. The new elimination scheme first converts the trigonometric rootfinding problem to a pair of polynomial equations in the variables (c,s) where c≡cos(t) and s≡sin(t). The elimination method next reduces the system to a single univariate polynomial P(c). We show that this same polynomial is the resultant of the system and is also a generator of the Groebner basis with lexicographic ordering for the system. Both methods give very high numerical accuracy for real-valued roots, typically at least 11 decimal places in Matlab/IEEE 754 16 digit floating point arithmetic. The CCM algorithm is typically one or two decimal places more accurate, though these differences disappear if the roots are "Newton-polished" by a single Newton's iteration. The complex-valued matrix is accurate for complex-valued roots, too, though accuracy decreases with the magnitude of the imaginary part of the root. The cost of both methods scales as O(N3) floating point operations. In spite of intimate connections of the elimination/Chebyshev scheme to two well-established technologies for solving systems of equations, resultants and Groebner bases, and the advantages of using only real-valued arithmetic to obtain a companion matrix with real-valued elements

Non-linear triangle-based polynomial expansion nodal method for hexagonal core analysis

International Nuclear Information System (INIS)

Cho, Jin Young; Cho, Byung Oh; Joo, Han Gyu; Zee, Sung Qunn; Park, Sang Yong

2000-09-01

This report is for the implementation of triangle-based polynomial expansion nodal (TPEN) method to MASTER code in conjunction with the coarse mesh finite difference(CMFD) framework for hexagonal core design and analysis. The TPEN method is a variation of the higher order polynomial expansion nodal (HOPEN) method that solves the multi-group neutron diffusion equation in the hexagonal-z geometry. In contrast with the HOPEN method, only two-dimensional intranodal expansion is considered in the TPEN method for a triangular domain. The axial dependence of the intranodal flux is incorporated separately here and it is determined by the nodal expansion method (NEM) for a hexagonal node. For the consistency of node geometry of the MASTER code which is based on hexagon, TPEN solver is coded to solve one hexagonal node which is composed of 6 triangular nodes directly with Gauss elimination scheme. To solve the CMFD linear system efficiently, stabilized bi-conjugate gradient(BiCG) algorithm and Wielandt eigenvalue shift method are adopted. And for the construction of the efficient preconditioner of BiCG algorithm, the incomplete LU(ILU) factorization scheme which has been widely used in two-dimensional problems is used. To apply the ILU factorization scheme to three-dimensional problem, a symmetric Gauss-Seidel Factorization scheme is used. In order to examine the accuracy of the TPEN solution, several eigenvalue benchmark problems and two transient problems, i.e., a realistic VVER1000 and VVER440 rod ejection benchmark problems, were solved and compared with respective references. The results of eigenvalue benchmark problems indicate that non-linear TPEN method is very accurate showing less than 15 pcm of eigenvalue errors and 1% of maximum power errors, and fast enough to solve the three-dimensional VVER-440 problem within 5 seconds on 733MHz PENTIUM-III. In the case of the transient problems, the non-linear TPEN method also shows good results within a few minute of

Analytical-Algebraic Approach to Solving Chaotic System

Czech Academy of Sciences Publication Activity Database

Beran, Zdeněk; Čelikovský, Sergej

2016-01-01

Roč. 26, č. 3 (2016), č. článku 1650051. ISSN 0218-1274 R&D Projects: GA ČR GA13-20433S Institutional support: RVO:67985556 Keywords : Laplace transform * Laplace-Adomian decomposition * Adomian polynomials * nonlinear systems * chaos Subject RIV: BC - Control Systems Theory Impact factor: 1.329, year: 2016 http://library.utia.cas.cz/separaty/2016/TR/beran-0458430.pdf

Okounkov's BC-Type Interpolation Macdonald Polynomials and Their q=1 Limit

NARCIS (Netherlands)

Koornwinder, T.H.

2015-01-01

This paper surveys eight classes of polynomials associated with A-type and BC-type root systems: Jack, Jacobi, Macdonald and Koornwinder polynomials and interpolation (or shifted) Jack and Macdonald polynomials and their BC-type extensions. Among these the BC-type interpolation Jack polynomials were

Positive trigonometric polynomials and signal processing applications

CERN Document Server

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...

Synchronization of generalized Henon map using polynomial controller

International Nuclear Information System (INIS)

Lam, H.K.

2010-01-01

This Letter presents the chaos synchronization of two discrete-time generalized Henon map, namely the drive and response systems. A polynomial controller is proposed to drive the system states of the response system to follow those of the drive system. The system stability of the error system formed by the drive and response systems and the synthesis of the polynomial controller are investigated using the sum-of-squares (SOS) technique. Based on the Lyapunov stability theory, stability conditions in terms of SOS are derived to guarantee the system stability and facilitate the controller synthesis. By satisfying the SOS-based stability conditions, chaotic synchronization is achieved. The solution of the SOS-based stability conditions can be found numerically using the third-party Matlab toolbox SOSTOOLS. A simulation example is given to illustrate the merits of the proposed polynomial control approach.

Control design and robustness analysis of a ball and plate system by using polynomial chaos

Energy Technology Data Exchange (ETDEWEB)

Colón, Diego [University of São Paulo, Polytechnic School, LAC -PTC, São Paulo (Brazil); Balthazar, José M. [São Paulo State University - Rio Claro Campus, Rio Claro (Brazil); Reis, Célia A. dos [São Paulo State University - Bauru Campus, Bauru (Brazil); Bueno, Átila M.; Diniz, Ivando S. [São Paulo State University - Sorocaba Campus, Sorocaba (Brazil); Rosa, Suelia de S. R. F. [University of Brasilia, Brasilia (Brazil)

2014-12-10

In this paper, we present a mathematical model of a ball and plate system, a control law and analyze its robustness properties by using the polynomial chaos method. The ball rolls without slipping. There is an auxiliary robot vision system that determines the bodies' positions and velocities, and is used for control purposes. The actuators are to orthogonal DC motors, that changes the plate's angles with the ground. The model is a extension of the ball and beam system and is highly nonlinear. The system is decoupled in two independent equations for coordinates x and y. Finally, the resulting nonlinear closed loop systems are analyzed by the polynomial chaos methodology, which considers that some system parameters are random variables, and generates statistical data that can be used in the robustness analysis.

The Nodal Polynomial Expansion method to solve the multigroup diffusion equations

International Nuclear Information System (INIS)

Ribeiro, R.D.M.

1983-03-01

The methodology of the solutions of the multigroup diffusion equations and uses the Nodal Polynomial Expansion Method is covered. The EPON code was developed based upon the above mentioned method for stationary state, rectangular geometry, one-dimensional or two-dimensional and for one or two energy groups. Then, one can study some effects such as the influence of the baffle on the thermal flux by calculating the flux and power distribution in nuclear reactors. Furthermore, a comparative study with other programs which use Finite Difference (CITATION and PDQ5) and Finite Element (CHD and FEMB) Methods was undertaken. As a result, the coherence, feasibility, speed and accuracy of the methodology used were demonstrated. (Author) [pt

Rodrigues formulas for the non-symmetric multivariable polynomials associated with the BCN-type root system

International Nuclear Information System (INIS)

Nishino, Akinori; Ujino, Hideaki; Komori, Yasushi; Wadati, Miki

2000-01-01

The non-symmetric Macdonald-Koornwinder polynomials are joint eigenfunctions of the commuting Cherednik operators which are constructed from the representation theory for the affine Hecke algebra corresponding to the BC N -type root system. We present the Rodrigues formula for the non-symmetric Macdonald-Koornwinder polynomials. The raising operators are derived from the realizations of the corresponding double affine Hecke algebra. In the quasi-classical limit, the above theory reduces to that of the BC N -type Sutherland model which describes many particles with inverse-square long-range interactions on a circle with one impurity. We also present the Rodrigues formula for the non-symmetric Jacobi polynomials of type BC N which are eigenstates of the BC N -type Sutherland model

Computation of the Likelihood in Biallelic Diffusion Models Using Orthogonal Polynomials

Directory of Open Access Journals (Sweden)

Claus Vogl

2014-11-01

Full Text Available In population genetics, parameters describing forces such as mutation, migration and drift are generally inferred from molecular data. Lately, approximate methods based on simulations and summary statistics have been widely applied for such inference, even though these methods waste information. In contrast, probabilistic methods of inference can be shown to be optimal, if their assumptions are met. In genomic regions where recombination rates are high relative to mutation rates, polymorphic nucleotide sites can be assumed to evolve independently from each other. The distribution of allele frequencies at a large number of such sites has been called “allele-frequency spectrum” or “site-frequency spectrum” (SFS. Conditional on the allelic proportions, the likelihoods of such data can be modeled as binomial. A simple model representing the evolution of allelic proportions is the biallelic mutation-drift or mutation-directional selection-drift diffusion model. With series of orthogonal polynomials, specifically Jacobi and Gegenbauer polynomials, or the related spheroidal wave function, the diffusion equations can be solved efficiently. In the neutral case, the product of the binomial likelihoods with the sum of such polynomials leads to finite series of polynomials, i.e., relatively simple equations, from which the exact likelihoods can be calculated. In this article, the use of orthogonal polynomials for inferring population genetic parameters is investigated.

Methods of solving nonstandard problems

CERN Document Server

Grigorieva, Ellina

2015-01-01

This book, written by an accomplished female mathematician, is the second to explore nonstandard mathematical problems – those that are not directly solved by standard mathematical methods but instead rely on insight and the synthesis of a variety of mathematical ideas.   It promotes mental activity as well as greater mathematical skills, and is an ideal resource for successful preparation for the mathematics Olympiad. Numerous strategies and techniques are presented that can be used to solve intriguing and challenging problems of the type often found in competitions.  The author uses a friendly, non-intimidating approach to emphasize connections between different fields of mathematics and often proposes several different ways to attack the same problem.  Topics covered include functions and their properties, polynomials, trigonometric and transcendental equations and inequalities, optimization, differential equations, nonlinear systems, and word problems.   Over 360 problems are included with hints, ...

Strategy Iteration Is Strongly Polynomial for 2-Player Turn-Based Stochastic Games with a Constant Discount Factor

DEFF Research Database (Denmark)

Hansen, Thomas Dueholm; Miltersen, Peter Bro; Zwick, Uri

2013-01-01

Ye [2011] showed recently that the simplex method with Dantzig’s pivoting rule, as well as Howard’s policy iteration algorithm, solve discounted Markov decision processes (MDPs), with a constant discount factor, in strongly polynomial time. More precisely, Ye showed that both algorithms terminate...... terminates after at most O(m1−γ log n1−γ) iterations. Second, and more importantly, we show that the same bound applies to the number of iterations performed by the strategy iteration (or strategy improvement) algorithm, a generalization of Howard’s policy iteration algorithm used for solving 2-player turn-based...... for 2-player turn-based stochastic games; it is strongly polynomial for a fixed discount factor, and exponential otherwise....

A parallel additive Schwarz preconditioned Jacobi-Davidson algorithm for polynomial eigenvalue problems in quantum dot simulation

International Nuclear Information System (INIS)

Hwang, F-N; Wei, Z-H; Huang, T-M; Wang Weichung

2010-01-01

We develop a parallel Jacobi-Davidson approach for finding a partial set of eigenpairs of large sparse polynomial eigenvalue problems with application in quantum dot simulation. A Jacobi-Davidson eigenvalue solver is implemented based on the Portable, Extensible Toolkit for Scientific Computation (PETSc). The eigensolver thus inherits PETSc's efficient and various parallel operations, linear solvers, preconditioning schemes, and easy usages. The parallel eigenvalue solver is then used to solve higher degree polynomial eigenvalue problems arising in numerical simulations of three dimensional quantum dots governed by Schroedinger's equations. We find that the parallel restricted additive Schwarz preconditioner in conjunction with a parallel Krylov subspace method (e.g. GMRES) can solve the correction equations, the most costly step in the Jacobi-Davidson algorithm, very efficiently in parallel. Besides, the overall performance is quite satisfactory. We have observed near perfect superlinear speedup by using up to 320 processors. The parallel eigensolver can find all target interior eigenpairs of a quintic polynomial eigenvalue problem with more than 32 million variables within 12 minutes by using 272 Intel 3.0 GHz processors.

New formulae between Jacobi polynomials and some fractional Jacobi functions generalizing some connection formulae

Science.gov (United States)

Abd-Elhameed, W. M.

2017-07-01

In this paper, a new formula relating Jacobi polynomials of arbitrary parameters with the squares of certain fractional Jacobi functions is derived. The derived formula is expressed in terms of a certain terminating hypergeometric function of the type _4F3(1) . With the aid of some standard reduction formulae such as Pfaff-Saalschütz's and Watson's identities, the derived formula can be reduced in simple forms which are free of any hypergeometric functions for certain choices of the involved parameters of the Jacobi polynomials and the Jacobi functions. Some other simplified formulae are obtained via employing some computer algebra algorithms such as the algorithms of Zeilberger, Petkovsek and van Hoeij. Some connection formulae between some Jacobi polynomials are deduced. From these connection formulae, some other linearization formulae of Chebyshev polynomials are obtained. As an application to some of the introduced formulae, a numerical algorithm for solving nonlinear Riccati differential equation is presented and implemented by applying a suitable spectral method.

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....

Automorphisms of Algebras and Bochner's Property for Vector Orthogonal Polynomials

Science.gov (United States)

Horozov, Emil

2016-05-01

We construct new families of vector orthogonal polynomials that have the property to be eigenfunctions of some differential operator. They are extensions of the Hermite and Laguerre polynomial systems. A third family, whose first member has been found by Y. Ben Cheikh and K. Douak is also constructed. The ideas behind our approach lie in the studies of bispectral operators. We exploit automorphisms of associative algebras which transform elementary vector orthogonal polynomial systems which are eigenfunctions of a differential operator into other systems of this type.

On progress of the solution of the stationary 2-dimensional neutron diffusion equation: a polynomial approximation method with error analysis

International Nuclear Information System (INIS)

Ceolin, C.; Schramm, M.; Bodmann, B.E.J.; Vilhena, M.T.

2015-01-01

Recently the stationary neutron diffusion equation in heterogeneous rectangular geometry was solved by the expansion of the scalar fluxes in polynomials in terms of the spatial variables (x; y), considering the two-group energy model. The focus of the present discussion consists in the study of an error analysis of the aforementioned solution. More specifically we show how the spatial subdomain segmentation is related to the degree of the polynomial and the Lipschitz constant. This relation allows to solve the 2-D neutron diffusion problem for second degree polynomials in each subdomain. This solution is exact at the knots where the Lipschitz cone is centered. Moreover, the solution has an analytical representation in each subdomain with supremum and infimum functions that shows the convergence of the solution. We illustrate the analysis with a selection of numerical case studies. (author)

On progress of the solution of the stationary 2-dimensional neutron diffusion equation: a polynomial approximation method with error analysis

Energy Technology Data Exchange (ETDEWEB)

Ceolin, C., E-mail: celina.ceolin@gmail.com [Universidade Federal de Santa Maria (UFSM), Frederico Westphalen, RS (Brazil). Centro de Educacao Superior Norte; Schramm, M.; Bodmann, B.E.J.; Vilhena, M.T., E-mail: celina.ceolin@gmail.com [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica

2015-07-01

Recently the stationary neutron diffusion equation in heterogeneous rectangular geometry was solved by the expansion of the scalar fluxes in polynomials in terms of the spatial variables (x; y), considering the two-group energy model. The focus of the present discussion consists in the study of an error analysis of the aforementioned solution. More specifically we show how the spatial subdomain segmentation is related to the degree of the polynomial and the Lipschitz constant. This relation allows to solve the 2-D neutron diffusion problem for second degree polynomials in each subdomain. This solution is exact at the knots where the Lipschitz cone is centered. Moreover, the solution has an analytical representation in each subdomain with supremum and infimum functions that shows the convergence of the solution. We illustrate the analysis with a selection of numerical case studies. (author)

A Gradient Weighted Moving Finite-Element Method with Polynomial Approximation of Any Degree

Directory of Open Access Journals (Sweden)

Ali R. Soheili

2009-01-01

Full Text Available A gradient weighted moving finite element method (GWMFE based on piecewise polynomial of any degree is developed to solve time-dependent problems in two space dimensions. Numerical experiments are employed to test the accuracy and effciency of the proposed method with nonlinear Burger equation.

A harmonic polynomial cell (HPC) method for 3D Laplace equation with application in marine hydrodynamics

Energy Technology Data Exchange (ETDEWEB)

Shao, Yan-Lin, E-mail: yanlin.shao@dnvgl.com; Faltinsen, Odd M.

2014-10-01

We propose a new efficient and accurate numerical method based on harmonic polynomials to solve boundary value problems governed by 3D Laplace equation. The computational domain is discretized by overlapping cells. Within each cell, the velocity potential is represented by the linear superposition of a complete set of harmonic polynomials, which are the elementary solutions of Laplace equation. By its definition, the method is named as Harmonic Polynomial Cell (HPC) method. The characteristics of the accuracy and efficiency of the HPC method are demonstrated by studying analytical cases. Comparisons will be made with some other existing boundary element based methods, e.g. Quadratic Boundary Element Method (QBEM) and the Fast Multipole Accelerated QBEM (FMA-QBEM) and a fourth order Finite Difference Method (FDM). To demonstrate the applications of the method, it is applied to some studies relevant for marine hydrodynamics. Sloshing in 3D rectangular tanks, a fully-nonlinear numerical wave tank, fully-nonlinear wave focusing on a semi-circular shoal, and the nonlinear wave diffraction of a bottom-mounted cylinder in regular waves are studied. The comparisons with the experimental results and other numerical results are all in satisfactory agreement, indicating that the present HPC method is a promising method in solving potential-flow problems. The underlying procedure of the HPC method could also be useful in other fields than marine hydrodynamics involved with solving Laplace equation.

On the number of polynomial solutions of Bernoulli and Abel polynomial differential equations

Science.gov (United States)

Cima, A.; Gasull, A.; Mañosas, F.

2017-12-01

In this paper we determine the maximum number of polynomial solutions of Bernoulli differential equations and of some integrable polynomial Abel differential equations. As far as we know, the tools used to prove our results have not been utilized before for studying this type of questions. We show that the addressed problems can be reduced to know the number of polynomial solutions of a related polynomial equation of arbitrary degree. Then we approach to these equations either applying several tools developed to study extended Fermat problems for polynomial equations, or reducing the question to the computation of the genus of some associated planar algebraic curves.

On generalized Fibonacci and Lucas polynomials

Energy Technology Data Exchange (ETDEWEB)

Nalli, Ayse [Department of Mathematics, Faculty of Sciences, Selcuk University, 42075 Campus-Konya (Turkey)], E-mail: aysenalli@yahoo.com; Haukkanen, Pentti [Department of Mathematics, Statistics and Philosophy, 33014 University of Tampere (Finland)], E-mail: mapehau@uta.fi

2009-12-15

Let h(x) be a polynomial with real coefficients. We introduce h(x)-Fibonacci polynomials that generalize both Catalan's Fibonacci polynomials and Byrd's Fibonacci polynomials and also the k-Fibonacci numbers, and we provide properties for these h(x)-Fibonacci polynomials. We also introduce h(x)-Lucas polynomials that generalize the Lucas polynomials and present properties of these polynomials. In the last section we introduce the matrix Q{sub h}(x) that generalizes the Q-matrix whose powers generate the Fibonacci numbers.

Dynamic Harmony Search with Polynomial Mutation Algorithm for Valve-Point Economic Load Dispatch

Directory of Open Access Journals (Sweden)

M. Karthikeyan

2015-01-01

mutation (DHSPM algorithm to solve ORPD problem. In DHSPM algorithm the key parameters of HS algorithm like harmony memory considering rate (HMCR and pitch adjusting rate (PAR are changed dynamically and there is no need to predefine these parameters. Additionally polynomial mutation is inserted in the updating step of HS algorithm to favor exploration and exploitation of the search space. The DHSPM algorithm is tested with three power system cases consisting of 3, 13, and 40 thermal units. The computational results show that the DHSPM algorithm is more effective in finding better solutions than other computational intelligence based methods.

Solutions of interval type-2 fuzzy polynomials using a new ranking method

Science.gov (United States)

Rahman, Nurhakimah Ab.; Abdullah, Lazim; Ghani, Ahmad Termimi Ab.; Ahmad, Noor'Ani

2015-10-01

A few years ago, a ranking method have been introduced in the fuzzy polynomial equations. Concept of the ranking method is proposed to find actual roots of fuzzy polynomials (if exists). Fuzzy polynomials are transformed to system of crisp polynomials, performed by using ranking method based on three parameters namely, Value, Ambiguity and Fuzziness. However, it was found that solutions based on these three parameters are quite inefficient to produce answers. Therefore in this study a new ranking method have been developed with the aim to overcome the inherent weakness. The new ranking method which have four parameters are then applied in the interval type-2 fuzzy polynomials, covering the interval type-2 of fuzzy polynomial equation, dual fuzzy polynomial equations and system of fuzzy polynomials. The efficiency of the new ranking method then numerically considered in the triangular fuzzy numbers and the trapezoidal fuzzy numbers. Finally, the approximate solutions produced from the numerical examples indicate that the new ranking method successfully produced actual roots for the interval type-2 fuzzy polynomials.

Polynomial pseudosupersymmetry underlying a two-level atom in an external electromagnetic field

International Nuclear Information System (INIS)

Samsonov, B.F.; Shamshutdinova, V.V.; Gitman, D.M.

2005-01-01

Chains of transformations introduced previously were studied in order to obtain electric fields with a time-dependent frequency for which the equation of motion of a two-level atom in the presence of these fields can be solved exactly. It is shown that a polynomial pseudosupersymmetry may be associated to such chains

Testing reachability and stabilizability of systems over polynomial rings using Gröbner bases

NARCIS (Netherlands)

Habets, L.C.G.J.M.

1993-01-01

Conditions for the reachability and stabilizability of systems over polynomial rings are well-known in the literature. For a system $ \\Sigma = (A,B)$ they can be expressed as right-invertibility cconditions on the matrix $(zI - A \\mid B)$. Therefore there is quite a strong algebraic relationship

Modified Chebyshev Collocation Method for Solving Differential Equations

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M Ziaul Arif

2015-05-01

Full Text Available This paper presents derivation of alternative numerical scheme for solving differential equations, which is modified Chebyshev (Vieta-Lucas Polynomial collocation differentiation matrices. The Scheme of modified Chebyshev (Vieta-Lucas Polynomial collocation method is applied to both Ordinary Differential Equations (ODEs and Partial Differential Equations (PDEs cases. Finally, the performance of the proposed method is compared with finite difference method and the exact solution of the example. It is shown that modified Chebyshev collocation method more effective and accurate than FDM for some example given.

Coexistence of algebraic and non-algebraic limit cycles for quintic polynomial differential systems

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Ahmed Bendjeddou

2017-03-01

Full Text Available In the work by Gine and Grau [11], a planar differential system of degree nine admitting a nested configuration formed by an algebraic and a non-algebraic limit cycles explicitly given was presented. As an improvement, we obtain by a new method a similar result for a family of quintic polynomial differential systems.

LMI-based stability analysis of fuzzy-model-based control systems using approximated polynomial membership functions.

Science.gov (United States)

Narimani, Mohammand; Lam, H K; Dilmaghani, R; Wolfe, Charles

2011-06-01

Relaxed linear-matrix-inequality-based stability conditions for fuzzy-model-based control systems with imperfect premise matching are proposed. First, the derivative of the Lyapunov function, containing the product terms of the fuzzy model and fuzzy controller membership functions, is derived. Then, in the partitioned operating domain of the membership functions, the relations between the state variables and the mentioned product terms are represented by approximated polynomials in each subregion. Next, the stability conditions containing the information of all subsystems and the approximated polynomials are derived. In addition, the concept of the S-procedure is utilized to release the conservativeness caused by considering the whole operating region for approximated polynomials. It is shown that the well-known stability conditions can be special cases of the proposed stability conditions. Simulation examples are given to illustrate the validity of the proposed approach.

Better polynomials for GNFS

OpenAIRE

Bai , Shi; Bouvier , Cyril; Kruppa , Alexander; Zimmermann , Paul

2016-01-01

International audience; The general number field sieve (GNFS) is the most efficient algo-rithm known for factoring large integers. It consists of several stages, the first one being polynomial selection. The quality of the selected polynomials can be modelled in terms of size and root properties. We propose a new kind of polynomials for GNFS: with a new degree of freedom, we further improve the size property. We demonstrate the efficiency of our algorithm by exhibiting a better polynomial tha...

Computation of rectangular source integral by rational parameter polynomial method

International Nuclear Information System (INIS)

Prabha, Hem

2001-01-01

Hubbell et al. (J. Res. Nat Bureau Standards 64C, (1960) 121) have obtained a series expansion for the calculation of the radiation field generated by a plane isotropic rectangular source (plaque), in which leading term is the integral H(a,b). In this paper another integral I(a,b), which is related with the integral H(a,b) has been solved by the rational parameter polynomial method. From I(a,b), we compute H(a,b). Using this method the integral I(a,b) is expressed in the form of a polynomial of a rational parameter. Generally, a function f (x) is expressed in terms of x. In this method this is expressed in terms of x/(1+x). In this way, the accuracy of the expression is good over a wide range of x as compared to the earlier approach. The results for I(a,b) and H(a,b) are given for a sixth degree polynomial and are found to be in good agreement with the results obtained by numerically integrating the integral. Accuracy could be increased either by increasing the degree of the polynomial or by dividing the range of integration. The results of H(a,b) and I(a,b) are given for values of b and a up to 2.0 and 20.0, respectively

Sparse grid-based polynomial chaos expansion for aerodynamics of an airfoil with uncertainties

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Xiaojing WU

2018-05-01

Full Text Available The uncertainties can generate fluctuations with aerodynamic characteristics. Uncertainty Quantification (UQ is applied to compute its impact on the aerodynamic characteristics. In addition, the contribution of each uncertainty to aerodynamic characteristics should be computed by uncertainty sensitivity analysis. Non-Intrusive Polynomial Chaos (NIPC has been successfully applied to uncertainty quantification and uncertainty sensitivity analysis. However, the non-intrusive polynomial chaos method becomes inefficient as the number of random variables adopted to describe uncertainties increases. This deficiency becomes significant in stochastic aerodynamic analysis considering the geometric uncertainty because the description of geometric uncertainty generally needs many parameters. To solve the deficiency, a Sparse Grid-based Polynomial Chaos (SGPC expansion is used to do uncertainty quantification and sensitivity analysis for stochastic aerodynamic analysis considering geometric and operational uncertainties. It is proved that the method is more efficient than non-intrusive polynomial chaos and Monte Carlo Simulation (MSC method for the stochastic aerodynamic analysis. By uncertainty quantification, it can be learnt that the flow characteristics of shock wave and boundary layer separation are sensitive to the geometric uncertainty in transonic region. The uncertainty sensitivity analysis reveals the individual and coupled effects among the uncertainty parameters. Keywords: Non-intrusive polynomial chaos, Sparse grid, Stochastic aerodynamic analysis, Uncertainty sensitivity analysis, Uncertainty quantification

Novel algebraic aspects of Liouvillian integrability for two-dimensional polynomial dynamical systems

Science.gov (United States)

Demina, Maria V.

2018-05-01

The general structure of irreducible invariant algebraic curves for a polynomial dynamical system in C2 is found. Necessary conditions for existence of exponential factors related to an invariant algebraic curve are derived. As a consequence, all the cases when the classical force-free Duffing and Duffing-van der Pol oscillators possess Liouvillian first integrals are obtained. New exact solutions for the force-free Duffing-van der Pol system are constructed.

Families of superintegrable Hamiltonians constructed from exceptional polynomials

International Nuclear Information System (INIS)

Post, Sarah; Tsujimoto, Satoshi; Vinet, Luc

2012-01-01

We introduce a family of exactly-solvable two-dimensional Hamiltonians whose wave functions are given in terms of Laguerre and exceptional Jacobi polynomials. The Hamiltonians contain purely quantum terms which vanish in the classical limit leaving only a previously known family of superintegrable systems. Additional, higher-order integrals of motion are constructed from ladder operators for the considered orthogonal polynomials proving the quantum system to be superintegrable. (paper)

On Multiple Polynomials of Capelli Type

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S.Y. Antonov

2016-03-01

Full Text Available This paper deals with the class of Capelli polynomials in free associative algebra F{Z} (where F is an arbitrary field, Z is a countable set generalizing the construction of multiple Capelli polynomials. The fundamental properties of the introduced Capelli polynomials are provided. In particular, decomposition of the Capelli polynomials by means of the same type of polynomials is shown. Furthermore, some relations between their T -ideals are revealed. A connection between double Capelli polynomials and Capelli quasi-polynomials is established.

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...

A reachability test for systems over polynomial rings using Gröbner bases

NARCIS (Netherlands)

Habets, L.C.G.J.M.

1992-01-01

Conditions for the reachability of a system over a polynomial ring are well known in the literature. However, the verification of these conditions remained a difficult problem in general. Application of the Gröbner Basis method from constructive commutative algebra makes it possible to carry out

Polynomial fuzzy model-based approach for underactuated surface vessels

DEFF Research Database (Denmark)

Khooban, Mohammad Hassan; Vafamand, Navid; Dragicevic, Tomislav

2018-01-01

The main goal of this study is to introduce a new polynomial fuzzy model-based structure for a class of marine systems with non-linear and polynomial dynamics. The suggested technique relies on a polynomial Takagi–Sugeno (T–S) fuzzy modelling, a polynomial dynamic parallel distributed compensation...... surface vessel (USV). Additionally, in order to overcome the USV control challenges, including the USV un-modelled dynamics, complex nonlinear dynamics, external disturbances and parameter uncertainties, the polynomial fuzzy model representation is adopted. Moreover, the USV-based control structure...... and a sum-of-squares (SOS) decomposition. The new proposed approach is a generalisation of the standard T–S fuzzy models and linear matrix inequality which indicated its effectiveness in decreasing the tracking time and increasing the efficiency of the robust tracking control problem for an underactuated...

Roots of the Chromatic Polynomial

DEFF Research Database (Denmark)

Perrett, Thomas

The chromatic polynomial of a graph G is a univariate polynomial whose evaluation at any positive integer q enumerates the proper q-colourings of G. It was introduced in connection with the famous four colour theorem but has recently found other applications in the field of statistical physics...... 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...

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.

An asymptotic formula for polynomials orthonormal with respect to a varying weight. II

International Nuclear Information System (INIS)

Komlov, A V; Suetin, S P

2014-01-01

This paper gives a proof of the theorem announced by the authors in the preceding paper with the same title. The theorem states that asymptotically the behaviour of the polynomials which are orthonormal with respect to the varying weight e −2nQ(x) p g (x)/√(∏ j=1 2p (x−e j )) coincides with the asymptotic behaviour of the Nuttall psi-function, which solves a special boundary-value problem on the relevant hyperelliptic Riemann surface of genus g=p−1. Here e 1 polynomial of even degree 2m and p g is a certain auxiliary polynomial of degree p−1. Bibliography: 23 titles

Contractions of 2D 2nd Order Quantum Superintegrable Systems and the Askey Scheme for Hypergeometric Orthogonal Polynomials

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Ernest G. Kalnins

2013-10-01

Full Text Available We show explicitly that all 2nd order superintegrable systems in 2 dimensions are limiting cases of a single system: the generic 3-parameter potential on the 2-sphere, S9 in our listing. We extend the Wigner-Inönü method of Lie algebra contractions to contractions of quadratic algebras and show that all of the quadratic symmetry algebras of these systems are contractions of that of S9. Amazingly, all of the relevant contractions of these superintegrable systems on flat space and the sphere are uniquely induced by the well known Lie algebra contractions of e(2 and so(3. By contracting function space realizations of irreducible representations of the S9 algebra (which give the structure equations for Racah/Wilson polynomials to the other superintegrable systems, and using Wigner's idea of ''saving'' a representation, we obtain the full Askey scheme of hypergeometric orthogonal polynomials. This relationship directly ties the polynomials and their structure equations to physical phenomena. It is more general because it applies to all special functions that arise from these systems via separation of variables, not just those of hypergeometric type, and it extends to higher dimensions.

Closed-form estimates of the domain of attraction for nonlinear systems via fuzzy-polynomial models.

Science.gov (United States)

Pitarch, José Luis; Sala, Antonio; Ariño, Carlos Vicente

2014-04-01

In this paper, the domain of attraction of the origin of a nonlinear system is estimated in closed form via level sets with polynomial boundaries, iteratively computed. In particular, the domain of attraction is expanded from a previous estimate, such as a classical Lyapunov level set. With the use of fuzzy-polynomial models, the domain of attraction analysis can be carried out via sum of squares optimization and an iterative algorithm. The result is a function that bounds the domain of attraction, free from the usual restriction of being positive and decrescent in all the interior of its level sets.

Using CAS to Solve Classical Mathematics Problems

Science.gov (United States)

Burke, Maurice J.; Burroughs, Elizabeth A.

2009-01-01

Historically, calculus has displaced many algebraic methods for solving classical problems. This article illustrates an algebraic method for finding the zeros of polynomial functions that is closely related to Newton's method (devised in 1669, published in 1711), which is encountered in calculus. By exploring this problem, precalculus students…

Polynomial stabilization of some dissipative hyperbolic systems

Czech Academy of Sciences Publication Activity Database

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

2014-01-01

Roč. 34, č. 11 (2014), s. 4371-4388 ISSN 1078-0947 R&D Projects: GA ČR GA201/09/0917 Institutional support: RVO:67985840 Keywords : exponential stability * polynomial stability * observability inequality Subject RIV: BA - General Mathematics Impact factor: 0.826, year: 2014 http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=9924

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.].

Application of the Generalized Differential Quadrature Method in Solving Burgers' Equations

International Nuclear Information System (INIS)

Mokhtari, R.; Toodar, A. Samadi; Chegini, N.G.

2011-01-01

The aim of this paper is to obtain numerical solutions of the one-dimensional, two-dimensional and coupled Burgers' equations through the generalized differential quadrature method (GDQM). The polynomial-based differential quadrature (PDQ) method is employed and the obtained system of ordinary differential equations is solved via the total variation diminishing Runge-Kutta (TVD-RK) method. The numerical solutions are satisfactorily coincident with the exact solutions. The method can compete against the methods applied in the literature. (general)

Space-time spectral collocation algorithm for solving time-fractional Tricomi-type equations

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Abdelkawy M.A.

2016-01-01

Full Text Available We introduce a new numerical algorithm for solving one-dimensional time-fractional Tricomi-type equations (T-FTTEs. We used the shifted Jacobi polynomials as basis functions and the derivatives of fractional is evaluated by the Caputo definition. The shifted Jacobi Gauss-Lobatt algorithm is used for the spatial discretization, while the shifted Jacobi Gauss-Radau algorithmis applied for temporal approximation. Substituting these approximations in the problem leads to a system of algebraic equations that greatly simplifies the problem. The proposed algorithm is successfully extended to solve the two-dimensional T-FTTEs. Extensive numerical tests illustrate the capability and high accuracy of the proposed methodologies.

Polynomial optimization : Error analysis and applications

NARCIS (Netherlands)

Sun, Zhao

2015-01-01

Polynomial optimization is the problem of minimizing a polynomial function subject to polynomial inequality constraints. In this thesis we investigate several hierarchies of relaxations for polynomial optimization problems. Our main interest lies in understanding their performance, in particular how

The Use of Transformations in Solving Equations

Science.gov (United States)

Libeskind, Shlomo

2010-01-01

Many workshops and meetings with the US high school mathematics teachers revealed a lack of familiarity with the use of transformations in solving equations and problems related to the roots of polynomials. This note describes two transformational approaches to the derivation of the quadratic formula as well as transformational approaches to…

Permutation invariant polynomial neural network approach to fitting potential energy surfaces. II. Four-atom systems

Energy Technology Data Exchange (ETDEWEB)

Li, Jun; Jiang, Bin; Guo, Hua, E-mail: hguo@unm.edu [Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, New Mexico 87131 (United States)

2013-11-28

A rigorous, general, and simple method to fit global and permutation invariant potential energy surfaces (PESs) using neural networks (NNs) is discussed. This so-called permutation invariant polynomial neural network (PIP-NN) method imposes permutation symmetry by using in its input a set of symmetry functions based on PIPs. For systems with more than three atoms, it is shown that the number of symmetry functions in the input vector needs to be larger than the number of internal coordinates in order to include both the primary and secondary invariant polynomials. This PIP-NN method is successfully demonstrated in three atom-triatomic reactive systems, resulting in full-dimensional global PESs with average errors on the order of meV. These PESs are used in full-dimensional quantum dynamical calculations.

Limit cycles bifurcating from the periodic annulus of cubic homogeneous polynomial centers

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Jaume Llibre

2015-10-01

Full Text Available We obtain an explicit polynomial whose simple positive real roots provide the limit cycles which bifurcate from the periodic orbits of any cubic homogeneous polynomial center when it is perturbed inside the class of all polynomial differential systems of degree n.

Birth-death processes and associated polynomials

NARCIS (Netherlands)

van Doorn, Erik A.

2003-01-01

We consider birth-death processes on the nonnegative integers and the corresponding sequences of orthogonal polynomials called birth-death polynomials. The sequence of associated polynomials linked with a sequence of birth-death polynomials and its orthogonalizing measure can be used in the analysis

Extended biorthogonal matrix polynomials

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Ayman Shehata

2017-01-01

Full Text Available The pair of biorthogonal matrix polynomials for commutative matrices were first introduced by Varma and Tasdelen in [22]. The main aim of this paper is to extend the properties of the pair of biorthogonal matrix polynomials of Varma and Tasdelen and certain generating matrix functions, finite series, some matrix recurrence relations, several important properties of matrix differential recurrence relations, biorthogonality relations and matrix differential equation for the pair of biorthogonal matrix polynomials J(A,B n (x, k and K(A,B n (x, k are discussed. For the matrix polynomials J(A,B n (x, k, various families of bilinear and bilateral generating matrix functions are constructed in the sequel.

Numerical Solution of a Fractional Order Model of HIV Infection of CD4+T Cells Using Müntz-Legendre Polynomials

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Mojtaba Rasouli Gandomani

2016-06-01

Full Text Available In this paper, the model of HIV infection of CD4+ T cells is considered as a system of fractional differential equations. Then, a numerical method by using collocation method based on the Müntz-Legendre polynomials to approximate solution of the model is presented. The application of the proposed numerical method causes fractional differential equations system to convert into the algebraic equations system. The new system can be solved by one of the existing methods. Finally, we compare the result of this numerical method with the result of the methods have already been presented in the literature.

vs. a polynomial chaos-based MCMC

KAUST Repository

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

Bannai-Ito polynomials and dressing chains

OpenAIRE

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

2012-01-01

Schur-Delsarte-Genin (SDG) maps and Bannai-Ito polynomials are studied. SDG maps are related to dressing chains determined by quadratic algebras. The Bannai-Ito polynomials and their kernel polynomials -- the complementary Bannai-Ito polynomials -- are shown to arise in the framework of the SDG maps.

Some Applications of Algebraic System Solving

Science.gov (United States)

Roanes-Lozano, Eugenio

2011-01-01

Technology and, in particular, computer algebra systems, allows us to change both the way we teach mathematics and the mathematical curriculum. Curiously enough, unlike what happens with linear system solving, algebraic system solving is not widely known. The aim of this paper is to show that, although the theory lying behind the "exact…

Open Problems Related to the Hurwitz Stability of Polynomials Segments

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Baltazar Aguirre-Hernández

2018-01-01

Full Text Available In the framework of robust stability analysis of linear systems, the development of techniques and methods that help to obtain necessary and sufficient conditions to determine stability of convex combinations of polynomials is paramount. In this paper, knowing that Hurwitz polynomials set is not a convex set, a brief overview of some results and open problems concerning the stability of the convex combinations of Hurwitz polynomials is then provided.

Solution of Dirac equation for modified Poschl Teller plus trigonometric Scarf potential using Romanovsky polynomials method

International Nuclear Information System (INIS)

Prastyaningrum, I.; Cari, C.; Suparmi, A.

2016-01-01

The approximation analytical solution of Dirac equation for Modified Poschl Teller plus Trigonometric Scarf Potential are investigated numerically in terms of finite Romanovsky Polynomial. The combination of two potentials are substituted into Dirac Equation then the variables are separated into radial and angular parts. The Dirac equation is solved by using Romanovsky Polynomial Method. The equation that can reduce from the second order of differential equation into the differential equation of hypergeometry type by substituted variable method. The energy spectrum is numerically solved using Matlab 2011. Where the increase in the radial quantum number nr and variable of modified Poschl Teller Potential causes the energy to decrease. The radial and the angular part of the wave function also visualized with Matlab 2011. The results show, by the disturbance of a combination between this potential can change the wave function of the radial and angular part. (paper)

Investigation of the Practical Possibility of Solving Problems on Generalized Cellular Automata Associated with Cryptanalysis by Mean Algebraic Methods

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P. G. Klyucharev

2017-01-01

Full Text Available A number of previous author’s papers proposed methods for constructing various cryptographic algorithms, including block ciphers and cryptographic hash functions, based on generalized cellular automata. This one is aimed at studying a possibility to use the algebraic cryptanalysis methods related to the construction of Gröbner bases for the generalized cellular automata to be applied in cryptography, i.e. this paper studies the possibility for using algebraic cryptanalysis methods to solve the problems of inversion of a generalized cellular automaton and recovering the key of such an automaton.If the cryptographic algorithm is represented as a system of polynomial equations over a certain finite field, then its breach is reduced to solving this system with respect to the key. Although the problem of solving a system of polynomial equations in a finite field is NP-difficult in the general case, the solution of a particular system can have low computational cost.Cryptanalysis based on the construction of a system of polynomial equations that links plain text, cipher-text and key, and its solution by algebraic methods, is usually called algebraic cryptanalysis. Among the main methods to solve systems of polynomial equations are those to construct Gröbner bases.Cryptanalysis of ciphers and hash functions based on generalized cellular automata can be reduced to various problems. We will consider two such problems: the problem of inversion of a generalized cellular automaton, which, in case we know the values of the cells after k iterations, enables us to find the initial values. And the task of recovering the key, which is to find the initial values of the remaining cells, using the cell values after k steps and the initial values of a part of the cells.A computational experiment was carried out to solve the two problems above stated in order to determine the maximum size of a generalized cellular automaton for which the solution of these

About several classes of bi-orthogonal polynomials and discrete integrable systems

International Nuclear Information System (INIS)

Chang, Xiang-Ke; Chen, Xiao-Min; Hu, Xing-Biao; Tam, Hon-Wah

2015-01-01

By introducing some special bi-orthogonal polynomials, we derive the so-called discrete hungry quotient-difference (dhQD) algorithm and a system related to the QD-type discrete hungry Lotka–Volterra (QD-type dhLV) system, together with their Lax pairs. These two known equations can be regarded as extensions of the QD algorithm. When this idea is applied to a higher analogue of the discrete-time Toda (HADT) equation and the quotient–quotient-difference (QQD) scheme proposed by Spicer, Nijhoff and van der Kamp, two extended systems are constructed. We call these systems the hungry forms of the higher analogue discrete-time Toda (hHADT) equation and the quotient-quotient-difference (hQQD) scheme, respectively. In addition, the corresponding Lax pairs are provided. (paper)

Supersymmetric quantum mechanics: Engineered hierarchies of integrable potentials and related orthogonal polynomials

International Nuclear Information System (INIS)

Balondo Iyela, Daddy; Govaerts, Jan; Hounkonnou, M. Norbert

2013-01-01

Within the context of supersymmetric quantum mechanics and its related hierarchies of integrable quantum Hamiltonians and potentials, a general programme is outlined and applied to its first two simplest illustrations. Going beyond the usual restriction of shape invariance for intertwined potentials, it is suggested to require a similar relation for Hamiltonians in the hierarchy separated by an arbitrary number of levels, N. By requiring further that these two Hamiltonians be in fact identical up to an overall shift in energy, a periodic structure is installed in the hierarchy which should allow for its resolution. Specific classes of orthogonal polynomials characteristic of such periodic hierarchies are thereby generated, while the methods of supersymmetric quantum mechanics then lead to generalised Rodrigues formulae and recursion relations for such polynomials. The approach also offers the practical prospect of quantum modelling through the engineering of quantum potentials from experimental energy spectra. In this paper, these ideas are presented and solved explicitly for the cases N= 1 and N= 2. The latter case is related to the generalised Laguerre polynomials, for which indeed new results are thereby obtained. In the context of dressing chains and deformed polynomial Heisenberg algebras, some partial results for N⩾ 3 also exist in the literature, which should be relevant to a complete study of the N⩾ 3 general periodic hierarchies

Supersymmetric quantum mechanics: Engineered hierarchies of integrable potentials and related orthogonal polynomials

Energy Technology Data Exchange (ETDEWEB)

Balondo Iyela, Daddy [International Chair in Mathematical Physics and Applications (ICMPA–UNESCO Chair), University of Abomey–Calavi, 072 B. P. 50 Cotonou, Republic of Benin (Benin); Centre for Cosmology, Particle Physics and Phenomenology (CP3), Institut de Recherche en Mathématique et Physique (IRMP), Université catholique de Louvain U.C.L., 2, Chemin du Cyclotron, B-1348 Louvain-la-Neuve (Belgium); Département de Physique, Université de Kinshasa (UNIKIN), B.P. 190 Kinshasa XI, Democratic Republic of Congo (Congo, The Democratic Republic of the); Govaerts, Jan [International Chair in Mathematical Physics and Applications (ICMPA–UNESCO Chair), University of Abomey–Calavi, 072 B. P. 50 Cotonou, Republic of Benin (Benin); Centre for Cosmology, Particle Physics and Phenomenology (CP3), Institut de Recherche en Mathématique et Physique (IRMP), Université catholique de Louvain U.C.L., 2, Chemin du Cyclotron, B-1348 Louvain-la-Neuve (Belgium); Hounkonnou, M. Norbert [International Chair in Mathematical Physics and Applications (ICMPA–UNESCO Chair), University of Abomey–Calavi, 072 B. P. 50 Cotonou, Republic of Benin (Benin)

2013-09-15

Within the context of supersymmetric quantum mechanics and its related hierarchies of integrable quantum Hamiltonians and potentials, a general programme is outlined and applied to its first two simplest illustrations. Going beyond the usual restriction of shape invariance for intertwined potentials, it is suggested to require a similar relation for Hamiltonians in the hierarchy separated by an arbitrary number of levels, N. By requiring further that these two Hamiltonians be in fact identical up to an overall shift in energy, a periodic structure is installed in the hierarchy which should allow for its resolution. Specific classes of orthogonal polynomials characteristic of such periodic hierarchies are thereby generated, while the methods of supersymmetric quantum mechanics then lead to generalised Rodrigues formulae and recursion relations for such polynomials. The approach also offers the practical prospect of quantum modelling through the engineering of quantum potentials from experimental energy spectra. In this paper, these ideas are presented and solved explicitly for the cases N= 1 and N= 2. The latter case is related to the generalised Laguerre polynomials, for which indeed new results are thereby obtained. In the context of dressing chains and deformed polynomial Heisenberg algebras, some partial results for N⩾ 3 also exist in the literature, which should be relevant to a complete study of the N⩾ 3 general periodic hierarchies.

An iterative method for the solution of nonlinear systems using the Faber polynomials for annular sectors

Energy Technology Data Exchange (ETDEWEB)

Myers, N.J. [Univ. of Durham (United Kingdom)

1994-12-31

The author gives a hybrid method for the iterative solution of linear systems of equations Ax = b, where the matrix (A) is nonsingular, sparse and nonsymmetric. As in a method developed by Starke and Varga the method begins with a number of steps of the Arnoldi method to produce some information on the location of the spectrum of A. This method then switches to an iterative method based on the Faber polynomials for an annular sector placed around these eigenvalue estimates. The Faber polynomials for an annular sector are used because, firstly an annular sector can easily be placed around any eigenvalue estimates bounded away from zero, and secondly the Faber polynomials are known analytically for an annular sector. Finally the author gives three numerical examples, two of which allow comparison with Starke and Varga`s results. The third is an example of a matrix for which many iterative methods would fall, but this method converges.

APPROX, 1-D and 2-D Function Approximation by Polynomials, Splines, Finite Elements Method

International Nuclear Information System (INIS)

Tollander, Bengt

1975-01-01

1 - Nature of physical problem solved: Approximates one- and two- dimensional functions using different forms of the approximating function, as polynomials, rational functions, Splines and (or) the finite element method. Different kinds of transformations of the dependent and (or) the independent variables can easily be made by data cards using a FORTRAN-like language. 2 - Method of solution: Approximations by polynomials, Splines and (or) the finite element method are made in L2 norm using the least square method by which the answer is directly given. For rational functions in one dimension the result given in L(infinite) norm is achieved by iterations moving the zero points of the error curve. For rational functions in two dimensions, the norm is L2 and the result is achieved by iteratively changing the coefficients of the denominator and then solving the coefficients of the numerator by the least square method. The transformation of the dependent and (or) independent variables is made by compiling the given transform data card(s) to an array of integers from which the transformation can be made

Spherical harmonics and energy polynomial solution of the Boltzmann equation for neutrons, 1

International Nuclear Information System (INIS)

Toledo, P.S. de

1974-01-01

The approximate solution of the source-free energy-dependent Boltzmann transport equation for neutrons in plane geometry and isotropic scattering case was given by Leonard and Ferziger using a truncated development in a series of energy-polynomials for the energy dependent neutron flux and solving exactly for the angular dependence. The presence in the general solution of eigenfunctions belonging to a continuous spectrum gives rise to difficult analytical problems in the application of their method even to simple problems. To avoid such difficulties, the angular dependence is treated by a spherical harmonics method and a general solution of the energy-dependent transport equation in plane geometry and isotropic scattering is obtained, in spite of the appearance of matrices as argument of the angular polynomials [pt

Generalizations of orthogonal polynomials

Science.gov (United States)

Bultheel, A.; Cuyt, A.; van Assche, W.; van Barel, M.; Verdonk, B.

2005-07-01

We give a survey of recent generalizations of orthogonal polynomials. That includes multidimensional (matrix and vector orthogonal polynomials) and multivariate versions, multipole (orthogonal rational functions) variants, and extensions of the orthogonality conditions (multiple orthogonality). Most of these generalizations are inspired by the applications in which they are applied. We also give a glimpse of these applications, which are usually generalizations of applications where classical orthogonal polynomials also play a fundamental role: moment problems, numerical quadrature, rational approximation, linear algebra, recurrence relations, and random matrices.

Analytical and numerical construction of vertical periodic orbits about triangular libration points based on polynomial expansion relations among directions

Science.gov (United States)

Qian, Ying-Jing; Yang, Xiao-Dong; Zhai, Guan-Qiao; Zhang, Wei

2017-08-01

Innovated by the nonlinear modes concept in the vibrational dynamics, the vertical periodic orbits around the triangular libration points are revisited for the Circular Restricted Three-body Problem. The ζ -component motion is treated as the dominant motion and the ξ and η -component motions are treated as the slave motions. The slave motions are in nature related to the dominant motion through the approximate nonlinear polynomial expansions with respect to the ζ -position and ζ -velocity during the one of the periodic orbital motions. By employing the relations among the three directions, the three-dimensional system can be transferred into one-dimensional problem. Then the approximate three-dimensional vertical periodic solution can be analytically obtained by solving the dominant motion only on ζ -direction. To demonstrate the effectiveness of the proposed method, an accuracy study was carried out to validate the polynomial expansion (PE) method. As one of the applications, the invariant nonlinear relations in polynomial expansion form are used as constraints to obtain numerical solutions by differential correction. The nonlinear relations among the directions provide an alternative point of view to explore the overall dynamics of periodic orbits around libration points with general rules.

Fast beampattern evaluation by polynomial rooting

Science.gov (United States)

Häcker, P.; Uhlich, S.; Yang, B.

2011-07-01

Current automotive radar systems measure the distance, the relative velocity and the direction of objects in their environment. This information enables the car to support the driver. The direction estimation capabilities of a sensor array depend on its beampattern. To find the array configuration leading to the best angle estimation by a global optimization algorithm, a huge amount of beampatterns have to be calculated to detect their maxima. In this paper, a novel algorithm is proposed to find all maxima of an array's beampattern fast and reliably, leading to accelerated array optimizations. The algorithm works for arrays having the sensors on a uniformly spaced grid. We use a general version of the gcd (greatest common divisor) function in order to write the problem as a polynomial. We differentiate and root the polynomial to get the extrema of the beampattern. In addition, we show a method to reduce the computational burden even more by decreasing the order of the polynomial.

Stochastic Estimation via Polynomial Chaos

Science.gov (United States)

2015-10-01

AFRL-RW-EG-TR-2015-108 Stochastic Estimation via Polynomial Chaos Douglas V. Nance Air Force Research...COVERED (From - To) 20-04-2015 – 07-08-2015 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER Stochastic Estimation via Polynomial Chaos ...This expository report discusses fundamental aspects of the polynomial chaos method for representing the properties of second order stochastic

A New Generalisation of Macdonald Polynomials

Science.gov (United States)

Garbali, Alexandr; de Gier, Jan; Wheeler, Michael

2017-06-01

We introduce a new family of symmetric multivariate polynomials, whose coefficients are meromorphic functions of two parameters ( q, t) and polynomial in a further two parameters ( u, v). We evaluate these polynomials explicitly as a matrix product. At u = v = 0 they reduce to Macdonald polynomials, while at q = 0, u = v = s they recover a family of inhomogeneous symmetric functions originally introduced by Borodin.

Special polynomials associated with some hierarchies

International Nuclear Information System (INIS)

Kudryashov, Nikolai A.

2008-01-01

Special polynomials associated with rational solutions of a hierarchy of equations of Painleve type are introduced. The hierarchy arises by similarity reduction from the Fordy-Gibbons hierarchy of partial differential equations. Some relations for these special polynomials are given. Differential-difference hierarchies for finding special polynomials are presented. These formulae allow us to obtain special polynomials associated with the hierarchy studied. It is shown that rational solutions of members of the Schwarz-Sawada-Kotera, the Schwarz-Kaup-Kupershmidt, the Fordy-Gibbons, the Sawada-Kotera and the Kaup-Kupershmidt hierarchies can be expressed through special polynomials of the hierarchy studied

A Summation Formula for Macdonald Polynomials

Science.gov (United States)

de Gier, Jan; Wheeler, Michael

2016-03-01

We derive an explicit sum formula for symmetric Macdonald polynomials. Our expression contains multiple sums over the symmetric group and uses the action of Hecke generators on the ring of polynomials. In the special cases {t = 1} and {q = 0}, we recover known expressions for the monomial symmetric and Hall-Littlewood polynomials, respectively. Other specializations of our formula give new expressions for the Jack and q-Whittaker polynomials.

Polynomial Chaos Expansion of Random Coefficients and the Solution of Stochastic Partial Differential Equations in the Tensor Train Format

KAUST Repository

Dolgov, Sergey; Khoromskij, Boris N.; Litvinenko, Alexander; Matthies, Hermann G.

2015-01-01

We apply the tensor train (TT) decomposition to construct the tensor product polynomial chaos expansion (PCE) of a random field, to solve the stochastic elliptic diffusion PDE with the stochastic Galerkin discretization, and to compute some

Solving Sparse Polynomial Optimization Problems with Chordal Structure Using the Sparse, Bounded-Degree Sum-of-Squares Hierarchy

NARCIS (Netherlands)

Marandi, Ahmadreza; de Klerk, Etienne; Dahl, Joachim

The sparse bounded degree sum-of-squares (sparse-BSOS) hierarchy of Weisser, Lasserre and Toh [arXiv:1607.01151,2016] constructs a sequence of lower bounds for a sparse polynomial optimization problem. Under some assumptions, it is proven by the authors that the sequence converges to the optimal

Universal localizing bounds for compact invariant sets of natural polynomial Hamiltonian systems

International Nuclear Information System (INIS)

Starkov, Konstantin E.

2008-01-01

In this Letter we study the localization problem of compact invariant sets of natural Hamiltonian systems with a polynomial Hamiltonian. Our results are based on applying the first order extremum conditions. We compute universal localizing bounds for some domain containing all compact invariant sets of a Hamiltonian system by using one quadratic function of a simple form. These bounds depend on the value of the total energy of the system, degree and some coefficients of a potential and, in addition, some positive number got as a result of a solution of one maximization problem. Besides, under some quasihomogeneity condition(s) we generalize our construction of the localization set

Universal localizing bounds for compact invariant sets of natural polynomial Hamiltonian systems

Energy Technology Data Exchange (ETDEWEB)

Starkov, Konstantin E. [CITEDI-IPN, Av. del Parque 1310, Mesa de Otay, Tijuana, BC (Mexico)], E-mail: konst@citedi.mx

2008-10-06

In this Letter we study the localization problem of compact invariant sets of natural Hamiltonian systems with a polynomial Hamiltonian. Our results are based on applying the first order extremum conditions. We compute universal localizing bounds for some domain containing all compact invariant sets of a Hamiltonian system by using one quadratic function of a simple form. These bounds depend on the value of the total energy of the system, degree and some coefficients of a potential and, in addition, some positive number got as a result of a solution of one maximization problem. Besides, under some quasihomogeneity condition(s) we generalize our construction of the localization set.

Quantum entanglement via nilpotent polynomials

International Nuclear Information System (INIS)

Mandilara, Aikaterini; Akulin, Vladimir M.; Smilga, Andrei V.; Viola, Lorenza

2006-01-01

We propose a general method for introducing extensive characteristics of quantum entanglement. The method relies on polynomials of nilpotent raising operators that create entangled states acting on a reference vacuum state. By introducing the notion of tanglemeter, the logarithm of the state vector represented in a special canonical form and expressed via polynomials of nilpotent variables, we show how this description provides a simple criterion for entanglement as well as a universal method for constructing the invariants characterizing entanglement. We compare the existing measures and classes of entanglement with those emerging from our approach. We derive the equation of motion for the tanglemeter and, in representative examples of up to four-qubit systems, show how the known classes appear in a natural way within our framework. We extend our approach to qutrits and higher-dimensional systems, and make contact with the recently introduced idea of generalized entanglement. Possible future developments and applications of the method are discussed

A Novel Operational Matrix of Caputo Fractional Derivatives of Fibonacci Polynomials: Spectral Solutions of Fractional Differential Equations

Directory of Open Access Journals (Sweden)

Waleed M. Abd-Elhameed

2016-09-01

Full Text Available Herein, two numerical algorithms for solving some linear and nonlinear fractional-order differential equations are presented and analyzed. For this purpose, a novel operational matrix of fractional-order derivatives of Fibonacci polynomials was constructed and employed along with the application of the tau and collocation spectral methods. The convergence and error analysis of the suggested Fibonacci expansion were carefully investigated. Some numerical examples with comparisons are presented to ensure the efficiency, applicability and high accuracy of the proposed algorithms. Two accurate semi-analytic polynomial solutions for linear and nonlinear fractional differential equations are the result.

Weierstrass polynomials for links

DEFF Research Database (Denmark)

Hansen, Vagn Lundsgaard

1997-01-01

There is a natural way of identifying links in3-space with polynomial covering spaces over thecircle. Thereby any link in 3-space can be definedby a Weierstrass polynomial over the circle. Theequivalence relation for covering spaces over thecircle is, however, completely different from...

On Symmetric Polynomials

OpenAIRE

Golden, Ryan; Cho, Ilwoo

2015-01-01

In this paper, we study structure theorems of algebras of symmetric functions. Based on a certain relation on elementary symmetric polynomials generating such algebras, we consider perturbation in the algebras. In particular, we understand generators of the algebras as perturbations. From such perturbations, define injective maps on generators, which induce algebra-monomorphisms (or embeddings) on the algebras. They provide inductive structure theorems on algebras of symmetric polynomials. As...

Associated polynomials and birth-death processes

NARCIS (Netherlands)

van Doorn, Erik A.

2001-01-01

We consider sequences of orthogonal polynomials with positive zeros, and pursue the question of how (partial) knowledge of the orthogonalizing measure for the {\\it associated polynomials} can lead to information about the orthogonalizing measure for the original polynomials, with a view to

P A M Dirac meets M G Krein: matrix orthogonal polynomials and Dirac's equation

International Nuclear Information System (INIS)

Duran, Antonio J; Gruenbaum, F Alberto

2006-01-01

The solution of several instances of the Schroedinger equation (1926) is made possible by using the well-known orthogonal polynomials associated with the names of Hermite, Legendre and Laguerre. A relativistic alternative to this equation was proposed by Dirac (1928) involving differential operators with matrix coefficients. In 1949 Krein developed a theory of matrix-valued orthogonal polynomials without any reference to differential equations. In Duran A J (1997 Matrix inner product having a matrix symmetric second order differential operator Rocky Mt. J. Math. 27 585-600), one of us raised the question of determining instances of these matrix-valued polynomials going along with second order differential operators with matrix coefficients. In Duran A J and Gruenbaum F A (2004 Orthogonal matrix polynomials satisfying second order differential equations Int. Math. Res. Not. 10 461-84), we developed a method to produce such examples and observed that in certain cases there is a connection with the instance of Dirac's equation with a central potential. We observe that the case of the central Coulomb potential discussed in the physics literature in Darwin C G (1928 Proc. R. Soc. A 118 654), Nikiforov A F and Uvarov V B (1988 Special Functions of Mathematical Physics (Basle: Birkhauser) and Rose M E 1961 Relativistic Electron Theory (New York: Wiley)), and its solution, gives rise to a matrix weight function whose orthogonal polynomials solve a second order differential equation. To the best of our knowledge this is the first instance of a connection between the solution of the first order matrix equation of Dirac and the theory of matrix-valued orthogonal polynomials initiated by M G Krein

Maximum Power Point Tracking Control of Photovoltaic Systems: A Polynomial Fuzzy Model-Based Approach

DEFF Research Database (Denmark)

Rakhshan, Mohsen; Vafamand, Navid; Khooban, Mohammad Hassan

2018-01-01

This paper introduces a polynomial fuzzy model (PFM)-based maximum power point tracking (MPPT) control approach to increase the performance and efficiency of the solar photovoltaic (PV) electricity generation. The proposed method relies on a polynomial fuzzy modeling, a polynomial parallel......, a direct maximum power (DMP)-based control structure is considered for MPPT. Using the PFM representation, the DMP-based control structure is formulated in terms of SOS conditions. Unlike the conventional approaches, the proposed approach does not require exploring the maximum power operational point...

Weakly and strongly polynomial algorithms for computing the maximum decrease in uniform arc capacities

Directory of Open Access Journals (Sweden)

Ghiyasvand Mehdi

2016-01-01

Full Text Available In this paper, a new problem on a directed network is presented. Let D be a feasible network such that all arc capacities are equal to U. Given a t > 0, the network D with arc capacities U - t is called the t-network. The goal of the problem is to compute the largest t such that the t-network is feasible. First, we present a weakly polynomial time algorithm to solve this problem, which runs in O(log(nU maximum flow computations, where n is the number of nodes. Then, an O(m2n time approach is presented, where m is the number of arcs. Both the weakly and strongly polynomial algorithms are inspired by McCormick and Ervolina (1994.

Quality Parameters Defined by Chebyshev Polynomials in Cold Rolling Process Chain

International Nuclear Information System (INIS)

Judin, Mika; Nylander, Jari; Larkiola, Jari; Verho, Martti

2011-01-01

The thickness profile of hot strip is of importance to profile, flatness and shape of the final cold rolled product. In this work, strip thickness and flatness profiles are decomposed into independent components by solving Chebyshev polynomials coefficients using matrix calculation. Four terms are used to characterize most common shapes of thickness and flatness profile. The calculated Chebyshev coefficients from different line measurements are combined together and analysed using neural network tools. The most common types of shapes are classified.

Fermionic formula for double Kostka polynomials

OpenAIRE

Liu, Shiyuan

2016-01-01

The $X=M$ conjecture asserts that the $1D$ sum and the fermionic formula coincide up to some constant power. In the case of type $A,$ both the $1D$ sum and the fermionic formula are closely related to Kostka polynomials. Double Kostka polynomials $K_{\\Bla,\\Bmu}(t),$ indexed by two double partitions $\\Bla,\\Bmu,$ are polynomials in $t$ introduced as a generalization of Kostka polynomials. In the present paper, we consider $K_{\\Bla,\\Bmu}(t)$ in the special case where $\\Bmu=(-,\\mu'').$ We formula...

Relations between Möbius and coboundary polynomials

NARCIS (Netherlands)

Jurrius, R.P.M.J.

2012-01-01

It is known that, in general, the coboundary polynomial and the Möbius polynomial of a matroid do not determine each other. Less is known about more specific cases. In this paper, we will investigate if it is possible that the Möbius polynomial of a matroid, together with the Möbius polynomial of

Matrix product formula for Macdonald polynomials

Science.gov (United States)

Cantini, Luigi; de Gier, Jan; Wheeler, Michael

2015-09-01

We derive a matrix product formula for symmetric Macdonald polynomials. Our results are obtained by constructing polynomial solutions of deformed Knizhnik-Zamolodchikov equations, which arise by considering representations of the Zamolodchikov-Faddeev and Yang-Baxter algebras in terms of t-deformed bosonic operators. These solutions are generalized probabilities for particle configurations of the multi-species asymmetric exclusion process, and form a basis of the ring of polynomials in n variables whose elements are indexed by compositions. For weakly increasing compositions (anti-dominant weights), these basis elements coincide with non-symmetric Macdonald polynomials. Our formulas imply a natural combinatorial interpretation in terms of solvable lattice models. They also imply that normalizations of stationary states of multi-species exclusion processes are obtained as Macdonald polynomials at q = 1.

Matrix product formula for Macdonald polynomials

International Nuclear Information System (INIS)

Cantini, Luigi; Gier, Jan de; Michael Wheeler

2015-01-01

We derive a matrix product formula for symmetric Macdonald polynomials. Our results are obtained by constructing polynomial solutions of deformed Knizhnik–Zamolodchikov equations, which arise by considering representations of the Zamolodchikov–Faddeev and Yang–Baxter algebras in terms of t-deformed bosonic operators. These solutions are generalized probabilities for particle configurations of the multi-species asymmetric exclusion process, and form a basis of the ring of polynomials in n variables whose elements are indexed by compositions. For weakly increasing compositions (anti-dominant weights), these basis elements coincide with non-symmetric Macdonald polynomials. Our formulas imply a natural combinatorial interpretation in terms of solvable lattice models. They also imply that normalizations of stationary states of multi-species exclusion processes are obtained as Macdonald polynomials at q = 1. (paper)

Using Systemic Problem Solving (SPS) to Assess Student ...

African Journals Online (AJOL)

This paper focuses on the uses of systemic problem solving in chemistry at the tertiary level. Traditional problem solving (TPS) is a useful tool to help teachers examine recall of information, comprehension, and application. However, systemic problem solving (SPS) can challenge students and probe higher cognitive skills ...

Euler polynomials and identities for non-commutative operators

Science.gov (United States)

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.

A New Approach and Solution Technique to Solve Time Fractional Nonlinear Reaction-Diffusion Equations

Directory of Open Access Journals (Sweden)

Inci Cilingir Sungu

2015-01-01

Full Text Available A new application of the hybrid generalized differential transform and finite difference method is proposed by solving time fractional nonlinear reaction-diffusion equations. This method is a combination of the multi-time-stepping temporal generalized differential transform and the spatial finite difference methods. The procedure first converts the time-evolutionary equations into Poisson equations which are then solved using the central difference method. The temporal differential transform method as used in the paper takes care of stability and the finite difference method on the resulting equation results in a system of diagonally dominant linear algebraic equations. The Gauss-Seidel iterative procedure then used to solve the linear system thus has assured convergence. To have optimized convergence rate, numerical experiments were done by using a combination of factors involving multi-time-stepping, spatial step size, and degree of the polynomial fit in time. It is shown that the hybrid technique is reliable, accurate, and easy to apply.

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.

Vertex models, TASEP and Grothendieck polynomials

International Nuclear Information System (INIS)

Motegi, Kohei; Sakai, Kazumitsu

2013-01-01

We examine the wavefunctions and their scalar products of a one-parameter family of integrable five-vertex models. At a special point of the parameter, the model investigated is related to an irreversible interacting stochastic particle system—the so-called totally asymmetric simple exclusion process (TASEP). By combining the quantum inverse scattering method with a matrix product representation of the wavefunctions, the on-/off-shell wavefunctions of the five-vertex models are represented as a certain determinant form. Up to some normalization factors, we find that the wavefunctions are given by Grothendieck polynomials, which are a one-parameter deformation of Schur polynomials. Introducing a dual version of the Grothendieck polynomials, and utilizing the determinant representation for the scalar products of the wavefunctions, we derive a generalized Cauchy identity satisfied by the Grothendieck polynomials and their duals. Several representation theoretical formulae for the Grothendieck polynomials are also presented. As a byproduct, the relaxation dynamics such as Green functions for the periodic TASEP are found to be described in terms of the Grothendieck polynomials. (paper)

An Efficient Numerical Approach for Solving Nonlinear Coupled Hyperbolic Partial Differential Equations with Nonlocal Conditions

Directory of Open Access Journals (Sweden)

A. H. Bhrawy

2014-01-01

Full Text Available One of the most important advantages of collocation method is the possibility of dealing with nonlinear partial differential equations (PDEs as well as PDEs with variable coefficients. A numerical solution based on a Jacobi collocation method is extended to solve nonlinear coupled hyperbolic PDEs with variable coefficients subject to initial-boundary nonlocal conservation conditions. This approach, based on Jacobi polynomials and Gauss-Lobatto quadrature integration, reduces solving the nonlinear coupled hyperbolic PDEs with variable coefficients to a system of nonlinear ordinary differential equation which is far easier to solve. In fact, we deal with initial-boundary coupled hyperbolic PDEs with variable coefficients as well as initial-nonlocal conditions. Using triangular, soliton, and exponential-triangular solutions as exact solutions, the obtained results show that the proposed numerical algorithm is efficient and very accurate.

On the Laurent polynomial rings

International Nuclear Information System (INIS)

Stefanescu, D.

1985-02-01

We describe some properties of the Laurent polynomial rings in a finite number of indeterminates over a commutative unitary ring. We study some subrings of the Laurent polynomial rings. We finally obtain two cancellation properties. (author)

Finding higher order Darboux polynomials for a family of rational first order ordinary differential equations

Science.gov (United States)

Avellar, J.; Claudino, A. L. G. C.; Duarte, L. G. S.; da Mota, L. A. C. P.

2015-10-01

For the Darbouxian methods we are studying here, in order to solve first order rational ordinary differential equations (1ODEs), the most costly (computationally) step is the finding of the needed Darboux polynomials. This can be so grave that it can render the whole approach unpractical. Hereby we introduce a simple heuristics to speed up this process for a class of 1ODEs.

A Polynomial Subset-Based Efficient Multi-Party Key Management System for Lightweight Device Networks.

Science.gov (United States)

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.

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

KAUST Repository

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.

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

KAUST Repository

Ait-Haddou, Rachid; Goldman, Ron

2015-01-01

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.

Computing the Alexander Polynomial Numerically

DEFF Research Database (Denmark)

Hansen, Mikael Sonne

2006-01-01

Explains how to construct the Alexander Matrix and how this can be used to compute the Alexander polynomial numerically.......Explains how to construct the Alexander Matrix and how this can be used to compute the Alexander polynomial numerically....

Strategy iteration is strongly polynomial for 2-player turn-based stochastic games with a constant discount factor

DEFF Research Database (Denmark)

Hansen, Thomas Dueholm; Miltersen, Peter Bro; Zwick, Uri

2011-01-01

Ye showed recently that the simplex method with Dantzig pivoting rule, as well as Howard's policy iteration algorithm, solve discounted Markov decision processes (MDPs), with a constant discount factor, in strongly polynomial time. More precisely, Ye showed that both algorithms terminate after...... iterations. Second, and more importantly, we show that the same bound applies to the number of iterations performed by the strategy iteration (or strategy improvement) algorithm, a generalization of Howard's policy iteration algorithm used for solving 2-player turn-based stochastic games with discounted zero...

Density of Real Zeros of the Tutte Polynomial

DEFF Research Database (Denmark)

Ok, Seongmin; Perrett, Thomas

2018-01-01

The Tutte polynomial of a graph is a two-variable polynomial whose zeros and evaluations encode many interesting properties of the graph. In this article we investigate the real zeros of the Tutte polynomials of graphs, and show that they form a dense subset of certain regions of the plane. This ....... This is the first density result for the real zeros of the Tutte polynomial in a region of positive volume. Our result almost confirms a conjecture of Jackson and Sokal except for one region which is related to an open problem on flow polynomials.......The Tutte polynomial of a graph is a two-variable polynomial whose zeros and evaluations encode many interesting properties of the graph. In this article we investigate the real zeros of the Tutte polynomials of graphs, and show that they form a dense subset of certain regions of the plane...

Density of Real Zeros of the Tutte Polynomial

DEFF Research Database (Denmark)

Ok, Seongmin; Perrett, Thomas

2017-01-01

The Tutte polynomial of a graph is a two-variable polynomial whose zeros and evaluations encode many interesting properties of the graph. In this article we investigate the real zeros of the Tutte polynomials of graphs, and show that they form a dense subset of certain regions of the plane. This ....... This is the first density result for the real zeros of the Tutte polynomial in a region of positive volume. Our result almost confirms a conjecture of Jackson and Sokal except for one region which is related to an open problem on flow polynomials.......The Tutte polynomial of a graph is a two-variable polynomial whose zeros and evaluations encode many interesting properties of the graph. In this article we investigate the real zeros of the Tutte polynomials of graphs, and show that they form a dense subset of certain regions of the plane...

Chebyshev polynomial functions based locally recurrent neuro-fuzzy information system for prediction of financial and energy market data

Directory of Open Access Journals (Sweden)

A.K. Parida

2016-09-01

Full Text Available In this paper Chebyshev polynomial functions based locally recurrent neuro-fuzzy information system is presented for the prediction and analysis of financial and electrical energy market data. The normally used TSK-type feedforward fuzzy neural network is unable to take the full advantage of the use of the linear fuzzy rule base in accurate input–output mapping and hence the consequent part of the rule base is made nonlinear using polynomial or arithmetic basis functions. Further the Chebyshev polynomial functions provide an expanded nonlinear transformation to the input space thereby increasing its dimension for capturing the nonlinearities and chaotic variations in financial or energy market data streams. Also the locally recurrent neuro-fuzzy information system (LRNFIS includes feedback loops both at the firing strength layer and the output layer to allow signal flow both in forward and backward directions, thereby making the LRNFIS mimic a dynamic system that provides fast convergence and accuracy in predicting time series fluctuations. Instead of using forward and backward least mean square (FBLMS learning algorithm, an improved Firefly-Harmony search (IFFHS learning algorithm is used to estimate the parameters of the consequent part and feedback loop parameters for better stability and convergence. Several real world financial and energy market time series databases are used for performance validation of the proposed LRNFIS model.

Optimization over polynomials : Selected topics

NARCIS (Netherlands)

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

Non-intrusive uncertainty quantification in structural-acoustic systems using polynomial chaos expansion method

Directory of Open Access Journals (Sweden)

Wang Mingjie

2017-01-01

Full Text Available A framework of non-intrusive polynomial chaos expansion method (PC was proposed to investigate the statistic characteristics of the response of structural-acoustic system containing random uncertainty. The PC method does not need to reformulate model equations, and the statistics of the response can be evaluated directly. The results show that compared to the direct Monte Carlo method (MCM based on the original numerical model, the PC method is effective and more efficient.

Confluent-Functional solving systems

Directory of Open Access Journals (Sweden)

V.N. Koval

2001-08-01

Full Text Available The paper proposes a statistical knowledge-acquision approach. The solving systems are considered, which are able to find unknown structural dependences between situational and transforming variables on the basis of statistically analyzed input information. Situational variables describe features, states and relations between environment objects. Transforming variables describe transforming influences, exerted by a goal-oriented system onto an environment. Unknown environment rules are simulated by a structural equations system, associating situational and transforming variables.

Numerical solution of the Schroedinger equation with a polynomial potential

International Nuclear Information System (INIS)

Campoy, G.; Palma, A.

1986-01-01

A numerical method for solving the Schroedinger equation for a potential expressed as a polynomial is proposed. The basic assumption relies on the asymptotic properties of the solution of this equation. It is possible to obtain the energies and the stationary state functions simultaneously. They analyze, in particular, the cases of the quartic anharmonic oscillator and a hydrogen atom perturbed by a quadratic term, obtaining its energy eigenvalues for some values of the perturbation parameter. Together with the Hellmann-Feynman theorem, they use their algorithm to calculate expectation values of x'' for arbitrary positive values of n. 4 tables

Efficient computation of Laguerre polynomials

NARCIS (Netherlands)

A. Gil (Amparo); J. Segura (Javier); N.M. Temme (Nico)

2017-01-01

textabstractAn efficient algorithm and a Fortran 90 module (LaguerrePol) for computing Laguerre polynomials . Ln(α)(z) are presented. The standard three-term recurrence relation satisfied by the polynomials and different types of asymptotic expansions valid for . n large and . α small, are used

Orthogonal polynomials in transport theories

International Nuclear Information System (INIS)

Dehesa, J.S.

1981-01-01

The asymptotical (k→infinity) behaviour of zeros of the polynomials gsub(k)sup((m)(ν)) encountered in the treatment of direct and inverse problems of scattering in neutron transport as well as radiative transfer theories is investigated in terms of the amplitude antiwsub(k) of the kth Legendre polynomial needed in the expansion of the scattering function. The parameters antiwsub(k) describe the anisotropy of scattering of the medium considered. In particular, it is shown that the asymptotical density of zeros of the polynomials gsub(k)sup(m)(ν) is an inverted semicircle for the anisotropic non-multiplying scattering medium

New polynomial-based molecular descriptors with low degeneracy.

Directory of Open Access Journals (Sweden)

Matthias Dehmer

Full Text Available In this paper, we introduce a novel graph polynomial called the 'information polynomial' of a graph. This graph polynomial can be derived by using a probability distribution of the vertex set. By using the zeros of the obtained polynomial, we additionally define some novel spectral descriptors. Compared with those based on computing the ordinary characteristic polynomial of a graph, we perform a numerical study using real chemical databases. We obtain that the novel descriptors do have a high discrimination power.

P A M Dirac meets M G Krein: matrix orthogonal polynomials and Dirac's equation

Energy Technology Data Exchange (ETDEWEB)

Duran, Antonio J [Departamento de Analisis Matematico, Universidad de Sevilla, Apdo (PO BOX) 1160, 41080 Sevilla (Spain); Gruenbaum, F Alberto [Department of Mathematics, University of California, Berkeley, CA 94720 (United States)

2006-04-07

The solution of several instances of the Schroedinger equation (1926) is made possible by using the well-known orthogonal polynomials associated with the names of Hermite, Legendre and Laguerre. A relativistic alternative to this equation was proposed by Dirac (1928) involving differential operators with matrix coefficients. In 1949 Krein developed a theory of matrix-valued orthogonal polynomials without any reference to differential equations. In Duran A J (1997 Matrix inner product having a matrix symmetric second order differential operator Rocky Mt. J. Math. 27 585-600), one of us raised the question of determining instances of these matrix-valued polynomials going along with second order differential operators with matrix coefficients. In Duran A J and Gruenbaum F A (2004 Orthogonal matrix polynomials satisfying second order differential equations Int. Math. Res. Not. 10 461-84), we developed a method to produce such examples and observed that in certain cases there is a connection with the instance of Dirac's equation with a central potential. We observe that the case of the central Coulomb potential discussed in the physics literature in Darwin C G (1928 Proc. R. Soc. A 118 654), Nikiforov A F and Uvarov V B (1988 Special Functions of Mathematical Physics (Basle: Birkhauser) and Rose M E 1961 Relativistic Electron Theory (New York: Wiley)), and its solution, gives rise to a matrix weight function whose orthogonal polynomials solve a second order differential equation. To the best of our knowledge this is the first instance of a connection between the solution of the first order matrix equation of Dirac and the theory of matrix-valued orthogonal polynomials initiated by M G Krein.

Need for higher order polynomial basis for polynomial nodal methods employed in LWR calculations

International Nuclear Information System (INIS)

Taiwo, T.A.; Palmiotti, G.

1997-01-01

The paper evaluates the accuracy and efficiency of sixth order polynomial solutions and the use of one radial node per core assembly for pressurized water reactor (PWR) core power distributions and reactivities. The computer code VARIANT was modified to calculate sixth order polynomial solutions for a hot zero power benchmark problem in which a control assembly along a core axis is assumed to be out of the core. Results are presented for the VARIANT, DIF3D-NODAL, and DIF3D-finite difference codes. The VARIANT results indicate that second order expansion of the within-node source and linear representation of the node surface currents are adequate for this problem. The results also demonstrate the improvement in the VARIANT solution when the order of the polynomial expansion of the within-node flux is increased from fourth to sixth order. There is a substantial saving in computational time for using one radial node per assembly with the sixth order expansion compared to using four or more nodes per assembly and fourth order polynomial solutions. 11 refs., 1 tab

Event-Triggered Fault Detection of Nonlinear Networked Systems.

Science.gov (United States)

Li, Hongyi; Chen, Ziran; Wu, Ligang; Lam, Hak-Keung; Du, Haiping

2017-04-01

This paper investigates the problem of fault detection for nonlinear discrete-time networked systems under an event-triggered scheme. A polynomial fuzzy fault detection filter is designed to generate a residual signal and detect faults in the system. A novel polynomial event-triggered scheme is proposed to determine the transmission of the signal. A fault detection filter is designed to guarantee that the residual system is asymptotically stable and satisfies the desired performance. Polynomial approximated membership functions obtained by Taylor series are employed for filtering analysis. Furthermore, sufficient conditions are represented in terms of sum of squares (SOSs) and can be solved by SOS tools in MATLAB environment. A numerical example is provided to demonstrate the effectiveness of the proposed results.

Algebraic invariant curves of plane polynomial differential systems

Science.gov (United States)

Tsygvintsev, Alexei

2001-01-01

We consider a plane polynomial vector field P(x,y) dx + Q(x,y) dy of degree m>1. With each algebraic invariant curve of such a field we associate a compact Riemann surface with the meromorphic differential ω = dx/P = dy/Q. The asymptotic estimate of the degree of an arbitrary algebraic invariant curve is found. In the smooth case this estimate has already been found by Cerveau and Lins Neto in a different way.

Use of Zernike polynomials and interferometry in the optical design and assembly of large carbon-dioxide laser systems

International Nuclear Information System (INIS)

Viswanathan, V.K.

1981-01-01

This paper describes the need for non-raytracing schemes in the optical design and analysis of large carbon-dioxide lasers like the Gigawatt, Gemini, and Helios lasers currently operational at Los Alamos, and the Antares laser fusion system under construction. The scheme currently used at Los Alamos involves characterizing the various optical components with a Zernike polynomial set obtained by the digitization of experimentally produced interferograms of the components. A Fast Fourier Transform code then propagates the complex amplitude and phase of the beam through the whole system and computes the optical parameters of interest. The analysis scheme is illustrated through examples of the Gigawatt, Gemini, and Helios systems. A possible way of using the Zernike polynomials in optical design problems of this type is discussed. Comparisons between the computed values and experimentally obtained results are made and it is concluded that this appears to be a valid approach. As this is a review article, some previously published results are also used where relevant

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.

Design of reinforced areas of concrete column using quadratic polynomials

Science.gov (United States)

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.

Stabilization of nonlinear systems using sampled-data output-feedback fuzzy controller based on polynomial-fuzzy-model-based control approach.

Science.gov (United States)

Lam, H K

2012-02-01

This paper investigates the stability of sampled-data output-feedback (SDOF) polynomial-fuzzy-model-based control systems. Representing the nonlinear plant using a polynomial fuzzy model, an SDOF fuzzy controller is proposed to perform the control process using the system output information. As only the system output is available for feedback compensation, it is more challenging for the controller design and system analysis compared to the full-state-feedback case. Furthermore, because of the sampling activity, the control signal is kept constant by the zero-order hold during the sampling period, which complicates the system dynamics and makes the stability analysis more difficult. In this paper, two cases of SDOF fuzzy controllers, which either share the same number of fuzzy rules or not, are considered. The system stability is investigated based on the Lyapunov stability theory using the sum-of-squares (SOS) approach. SOS-based stability conditions are obtained to guarantee the system stability and synthesize the SDOF fuzzy controller. Simulation examples are given to demonstrate the merits of the proposed SDOF fuzzy control approach.

Generalized Pseudospectral Method and Zeros of Orthogonal Polynomials

Directory of Open Access Journals (Sweden)

Oksana Bihun

2018-01-01

Full Text Available Via a generalization of the pseudospectral method for numerical solution of differential equations, a family of nonlinear algebraic identities satisfied by the zeros of a wide class of orthogonal polynomials is derived. The generalization is based on a modification of pseudospectral matrix representations of linear differential operators proposed in the paper, which allows these representations to depend on two, rather than one, sets of interpolation nodes. The identities hold for every polynomial family pνxν=0∞ orthogonal with respect to a measure supported on the real line that satisfies some standard assumptions, as long as the polynomials in the family satisfy differential equations Apν(x=qν(xpν(x, where A is a linear differential operator and each qν(x is a polynomial of degree at most n0∈N; n0 does not depend on ν. The proposed identities generalize known identities for classical and Krall orthogonal polynomials, to the case of the nonclassical orthogonal polynomials that belong to the class described above. The generalized pseudospectral representations of the differential operator A for the case of the Sonin-Markov orthogonal polynomials, also known as generalized Hermite polynomials, are presented. The general result is illustrated by new algebraic relations satisfied by the zeros of the Sonin-Markov polynomials.

On the Connection Coefficients of the Chebyshev-Boubaker Polynomials

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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.

Linear precoding based on polynomial expansion: reducing complexity in massive MIMO

KAUST Repository

Mueller, Axel; Kammoun, Abla; Bjö rnson, Emil; Debbah, Mé rouane

2016-01-01

By deriving new random matrix results, we obtain a deterministic expression for the asymptotic signal-to-interference-and-noise ratio (SINR) achieved by TPE precoding in massive MIMO systems. Furthermore, we provide a closed-form expression for the polynomial coefficients that maximizes this SINR. To maintain a fixed per-user rate loss as compared to RZF, the polynomial degree does not need to scale with the system, but it should be increased with the quality of the channel knowledge and the signal-to-noise ratio.

Resource-constrained project scheduling: computing lower bounds by solving minimum cut problems

NARCIS (Netherlands)

Möhring, R.H.; Nesetril, J.; Schulz, A.S.; Stork, F.; Uetz, Marc Jochen

1999-01-01

We present a novel approach to compute Lagrangian lower bounds on the objective function value of a wide class of resource-constrained project scheduling problems. The basis is a polynomial-time algorithm to solve the following scheduling problem: Given a set of activities with start-time dependent

Control Synthesis of Discrete-Time T-S Fuzzy Systems via a Multi-Instant Homogenous Polynomial Approach.

Science.gov (United States)

Xie, Xiangpeng; Yue, Dong; Zhang, Huaguang; Xue, Yusheng

2016-03-01

This paper deals with the problem of control synthesis of discrete-time Takagi-Sugeno fuzzy systems by employing a novel multiinstant homogenous polynomial approach. A new multiinstant fuzzy control scheme and a new class of fuzzy Lyapunov functions, which are homogenous polynomially parameter-dependent on both the current-time normalized fuzzy weighting functions and the past-time normalized fuzzy weighting functions, are proposed for implementing the object of relaxed control synthesis. Then, relaxed stabilization conditions are derived with less conservatism than existing ones. Furthermore, the relaxation quality of obtained stabilization conditions is further ameliorated by developing an efficient slack variable approach, which presents a multipolynomial dependence on the normalized fuzzy weighting functions at the current and past instants of time. Two simulation examples are given to demonstrate the effectiveness and benefits of the results developed in this paper.

Policy Iteration for $H_\\infty $ Optimal Control of Polynomial Nonlinear Systems via Sum of Squares Programming.

Science.gov (United States)

Zhu, Yuanheng; Zhao, Dongbin; Yang, Xiong; Zhang, Qichao

2018-02-01

Sum of squares (SOS) polynomials have provided a computationally tractable way to deal with inequality constraints appearing in many control problems. It can also act as an approximator in the framework of adaptive dynamic programming. In this paper, an approximate solution to the optimal control of polynomial nonlinear systems is proposed. Under a given attenuation coefficient, the Hamilton-Jacobi-Isaacs equation is relaxed to an optimization problem with a set of inequalities. After applying the policy iteration technique and constraining inequalities to SOS, the optimization problem is divided into a sequence of feasible semidefinite programming problems. With the converged solution, the attenuation coefficient is further minimized to a lower value. After iterations, approximate solutions to the smallest -gain and the associated optimal controller are obtained. Four examples are employed to verify the effectiveness of the proposed algorithm.

Polynomial sequences generated by infinite Hessenberg matrices

Directory of Open Access Journals (Sweden)

Verde-Star Luis

2017-01-01

Full Text Available We show that an infinite lower Hessenberg matrix generates polynomial sequences that correspond to the rows of infinite lower triangular invertible matrices. Orthogonal polynomial sequences are obtained when the Hessenberg matrix is tridiagonal. We study properties of the polynomial sequences and their corresponding matrices which are related to recurrence relations, companion matrices, matrix similarity, construction algorithms, and generating functions. When the Hessenberg matrix is also Toeplitz the polynomial sequences turn out to be of interpolatory type and we obtain additional results. For example, we show that every nonderogative finite square matrix is similar to a unique Toeplitz-Hessenberg matrix.

Special polynomials associated with rational solutions of some hierarchies

International Nuclear Information System (INIS)

Kudryashov, Nikolai A.

2009-01-01

New special polynomials associated with rational solutions of the Painleve hierarchies are introduced. The Hirota relations for these special polynomials are found. Differential-difference hierarchies to find special polynomials are presented. These formulae allow us to search special polynomials associated with the hierarchies. It is shown that rational solutions of the Caudrey-Dodd-Gibbon, the Kaup-Kupershmidt and the modified hierarchy for these ones can be obtained using new special polynomials.

The structure of the polynomials in preconditioned BiCG algorithms and the switching direction of preconditioned systems

OpenAIRE

Itoh, Shoji; Sugihara, Masaaki

2016-01-01

We present a theorem that defines the direction of a preconditioned system for the bi-conjugate gradient (BiCG) method, and we extend it to preconditioned bi-Lanczos-type algorithms. We show that the direction of a preconditioned system is switched by construction and by the settings of the initial shadow residual vector. We analyze and compare the polynomial structures of four preconditioned BiCG algorithms.

Inverse kinematics algorithm for a six-link manipulator using a polynomial expression

International Nuclear Information System (INIS)

Sasaki, Shinobu

1987-01-01

This report is concerned with the forward and inverse kinematics problem relevant to a six-link robot manipulator. In order to derive the kinematic relationships between links, the vector rotation operator was applied instead of the conventional homogeneous transformation. The exact algorithm for solving the inverse problem was obtained by transforming kinematics equations into a polynomial. As shown in test calculations, the accuracies of numerical solutions obtained by means of the present approach are found to be quite high. The algorithm proposed permits to find out all feasible solutions for the given inverse problem. (author)

Cosmographic analysis with Chebyshev polynomials

Science.gov (United States)

Capozziello, Salvatore; D'Agostino, Rocco; Luongo, Orlando

2018-05-01

The limits of standard cosmography are here revised addressing the problem of error propagation during statistical analyses. To do so, we propose the use of Chebyshev polynomials to parametrize cosmic distances. In particular, we demonstrate that building up rational Chebyshev polynomials significantly reduces error propagations with respect to standard Taylor series. This technique provides unbiased estimations of the cosmographic parameters and performs significatively better than previous numerical approximations. To figure this out, we compare rational Chebyshev polynomials with Padé series. In addition, we theoretically evaluate the convergence radius of (1,1) Chebyshev rational polynomial and we compare it with the convergence radii of Taylor and Padé approximations. We thus focus on regions in which convergence of Chebyshev rational functions is better than standard approaches. With this recipe, as high-redshift data are employed, rational Chebyshev polynomials remain highly stable and enable one to derive highly accurate analytical approximations of Hubble's rate in terms of the cosmographic series. Finally, we check our theoretical predictions by setting bounds on cosmographic parameters through Monte Carlo integration techniques, based on the Metropolis-Hastings algorithm. We apply our technique to high-redshift cosmic data, using the Joint Light-curve Analysis supernovae sample and the most recent versions of Hubble parameter and baryon acoustic oscillation measurements. We find that cosmography with Taylor series fails to be predictive with the aforementioned data sets, while turns out to be much more stable using the Chebyshev approach.

Multilevel weighted least squares polynomial approximation

KAUST Repository

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.

Optimization of polynomials in non-commuting variables

CERN Document Server

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.

Relations between zeros of special polynomials associated with the Painleve equations

International Nuclear Information System (INIS)

Kudryashov, Nikolai A.; Demina, Maria V.

2007-01-01

A method for finding relations of roots of polynomials is presented. Our approach allows us to get a number of relations between the zeros of the classical polynomials as well as the roots of special polynomials associated with rational solutions of the Painleve equations. We apply the method to obtain the relations for the zeros of several polynomials. These are: the Hermite polynomials, the Laguerre polynomials, the Yablonskii-Vorob'ev polynomials, the generalized Okamoto polynomials, and the generalized Hermite polynomials. All the relations found can be considered as analogues of generalized Stieltjes relations

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.

Orthogonal Polynomials and Special Functions

CERN Document Server

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.

Multiple Meixner polynomials and non-Hermitian oscillator Hamiltonians

International Nuclear Information System (INIS)

Ndayiragije, F; Van Assche, W

2013-01-01

Multiple Meixner polynomials are polynomials in one variable which satisfy orthogonality relations with respect to r > 1 different negative binomial distributions (Pascal distributions). There are two kinds of multiple Meixner polynomials, depending on the selection of the parameters in the negative binomial distribution. We recall their definition and some formulas and give generating functions and explicit expressions for the coefficients in the nearest neighbor recurrence relation. Following a recent construction of Miki, Tsujimoto, Vinet and Zhedanov (for multiple Meixner polynomials of the first kind), we construct r > 1 non-Hermitian oscillator Hamiltonians in r dimensions which are simultaneously diagonalizable and for which the common eigenstates are expressed in terms of multiple Meixner polynomials of the second kind. (paper)

Colouring and knot polynomials

International Nuclear Information System (INIS)

Welsh, D.J.A.

1991-01-01

These lectures will attempt to explain a connection between the recent advances in knot theory using the Jones and related knot polynomials with classical problems in combinatorics and statistical mechanics. The difficulty of some of these problems will be analysed in the context of their computational complexity. In particular we shall discuss colourings and groups valued flows in graphs, knots and the Jones and Kauffman polynomials, the Ising, Potts and percolation problems of statistical physics, computational complexity of the above problems. (author). 20 refs, 9 figs

Investigation of magneto-hemodynamic flow in a semi-porous channel using orthonormal Bernstein polynomials

Science.gov (United States)

Hosseini, E.; Loghmani, G. B.; Heydari, M.; Rashidi, M. M.

2017-07-01

In this paper, the problem of the magneto-hemodynamic laminar viscous flow of a conducting physiological fluid in a semi-porous channel under a transverse magnetic field is investigated numerically. Using a Berman's similarity transformation, the two-dimensional momentum conservation partial differential equations can be written as a system of nonlinear ordinary differential equations incorporating Lorentizian magneto-hydrodynamic body force terms. A new computational method based on the operational matrix of derivative of orthonormal Bernstein polynomials for solving the resulting differential systems is introduced. Moreover, by using the residual correction process, two types of error estimates are provided and reported to show the strength of the proposed method. Graphical and tabular results are presented to investigate the influence of the Hartmann number ( Ha) and the transpiration Reynolds number ( Re on velocity profiles in the channel. The results are compared with those obtained by previous works to confirm the accuracy and efficiency of the proposed scheme.

Robust ∞ Filtering of 2D Roesser Discrete Systems: A Polynomial Approach

Directory of Open Access Journals (Sweden)

Chakir El-Kasri

2012-01-01

procedure for generating conditions for the existence of a 2D discrete filter such that, for all admissible uncertainties, the error system is asymptotically stable, and the ∞ norm of the transfer function from the noise signal to the estimation error is below a prespecified level. These conditions are expressed as parameter-dependent linear matrix inequalities. Using homogeneous polynomially parameter-dependent filters of arbitrary degree on the uncertain parameters, the proposed method extends previous results in the quadratic framework and the linearly parameter-dependent framework, thus reducing its conservatism. Performance of the proposed method, in comparison with that of existing methods, is illustrated by two examples.

Uniqueness and zeros of q-shift difference polynomials

Indian Academy of Sciences (India)

In this paper, we consider the zero distributions of -shift difference polynomials of meromorphic functions with zero order, and obtain two theorems that extend the classical Hayman results on the zeros of differential polynomials to -shift difference polynomials. We also investigate the uniqueness problem of -shift ...

Applying Groebner bases to solve reduction problems for Feynman integrals

International Nuclear Information System (INIS)

Smirnov, Alexander V.; Smirnov, Vladimir A.

2006-01-01

We describe how Groebner bases can be used to solve the reduction problem for Feynman integrals, i.e. to construct an algorithm that provides the possibility to express a Feynman integral of a given family as a linear combination of some master integrals. Our approach is based on a generalized Buchberger algorithm for constructing Groebner-type bases associated with polynomials of shift operators. We illustrate it through various examples of reduction problems for families of one- and two-loop Feynman integrals. We also solve the reduction problem for a family of integrals contributing to the three-loop static quark potential

Applying Groebner bases to solve reduction problems for Feynman integrals

Energy Technology Data Exchange (ETDEWEB)

Smirnov, Alexander V. [Mechanical and Mathematical Department and Scientific Research Computer Center of Moscow State University, Moscow 119992 (Russian Federation); Smirnov, Vladimir A. [Nuclear Physics Institute of Moscow State University, Moscow 119992 (Russian Federation)

2006-01-15

We describe how Groebner bases can be used to solve the reduction problem for Feynman integrals, i.e. to construct an algorithm that provides the possibility to express a Feynman integral of a given family as a linear combination of some master integrals. Our approach is based on a generalized Buchberger algorithm for constructing Groebner-type bases associated with polynomials of shift operators. We illustrate it through various examples of reduction problems for families of one- and two-loop Feynman integrals. We also solve the reduction problem for a family of integrals contributing to the three-loop static quark potential.

Completeness of the ring of polynomials

DEFF Research Database (Denmark)

Thorup, Anders

2015-01-01

Consider the polynomial ring R:=k[X1,…,Xn]R:=k[X1,…,Xn] in n≥2n≥2 variables over an uncountable field k. We prove that R   is complete in its adic topology, that is, the translation invariant topology in which the non-zero ideals form a fundamental system of neighborhoods of 0. In addition we pro...

Factoring polynomials over arbitrary finite fields

NARCIS (Netherlands)

Lange, T.; Winterhof, A.

2000-01-01

We analyse an extension of Shoup's (Inform. Process. Lett. 33 (1990) 261–267) deterministic algorithm for factoring polynomials over finite prime fields to arbitrary finite fields. In particular, we prove the existence of a deterministic algorithm which completely factors all monic polynomials of

Additive and polynomial representations

CERN Document Server

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

Free vibration of Euler and Timoshenko nanobeams using boundary characteristic orthogonal polynomials

Science.gov (United States)

Behera, Laxmi; Chakraverty, S.

2014-03-01

Vibration analysis of nonlocal nanobeams based on Euler-Bernoulli and Timoshenko beam theories is considered. Nonlocal nanobeams are important in the bending, buckling and vibration analyses of beam-like elements in microelectromechanical or nanoelectromechanical devices. Expressions for free vibration of Euler-Bernoulli and Timoshenko nanobeams are established within the framework of Eringen's nonlocal elasticity theory. The problem has been solved previously using finite element method, Chebyshev polynomials in Rayleigh-Ritz method and using other numerical methods. In this study, numerical results for free vibration of nanobeams have been presented using simple polynomials and orthonormal polynomials in the Rayleigh-Ritz method. The advantage of the method is that one can easily handle the specified boundary conditions at the edges. To validate the present analysis, a comparison study is carried out with the results of the existing literature. The proposed method is also validated by convergence studies. Frequency parameters are found for different scaling effect parameters and boundary conditions. The study highlights that small scale effects considerably influence the free vibration of nanobeams. Nonlocal frequency parameters of nanobeams are smaller when compared to the corresponding local ones. Deflection shapes of nonlocal clamped Euler-Bernoulli nanobeams are also incorporated for different scaling effect parameters, which are affected by the small scale effect. Obtained numerical solutions provide a better representation of the vibration behavior of short and stubby micro/nanobeams where the effects of small scale, transverse shear deformation and rotary inertia are significant.

A Determinant Expression for the Generalized Bessel Polynomials

Directory of Open Access Journals (Sweden)

Sheng-liang Yang

2013-01-01

Full Text Available Using the exponential Riordan arrays, we show that a variation of the generalized Bessel polynomial sequence is of Sheffer type, and we obtain a determinant formula for the generalized Bessel polynomials. As a result, the Bessel polynomial is represented as determinant the entries of which involve Catalan numbers.

Solving the Traveling Salesman Problem Based on The Genetic Reactive Bone Route Algorithm whit Ant Colony System

Directory of Open Access Journals (Sweden)

Majid Yousefikhoshbakht

2016-07-01

Full Text Available The TSP is considered one of the most well-known combinatorial optimization tasks and researchers have paid so much attention to the TSP for many years. In this problem, a salesman starts to move from an arbitrary place called depot and after visits all of the nodes, finally comes back to the depot. The objective is to minimize the total distance traveled by the salesman.  Because this problem is a non-deterministic polynomial (NP-hard problem in nature, a hybrid meta-heuristic algorithm called REACSGA is used for solving the TSP. In REACSGA, a reactive bone route algorithm that uses the ant colony system (ACS for generating initial diversified solutions and the genetic algorithm (GA as an improved procedure are applied. Since the performance of the Metaheuristic algorithms is significantly influenced by their parameters, Taguchi Method is used to set the parameters of the proposed algorithm. The proposed algorithm is tested on several standard instances involving 24 to 318 nodes from the literature. The computational result shows that the results of the proposed algorithm are competitive with other metaheuristic algorithms for solving the TSP in terms of better quality of solution and computational time respectively. In addition, the proposed REACSGA is significantly efficient and finds closely the best known solutions for most of the instances in which thirteen best known solutions are also found.

A generalization of the Bernoulli polynomials

Directory of Open Access Journals (Sweden)

Pierpaolo Natalini

2003-01-01

Full Text Available A generalization of the Bernoulli polynomials and, consequently, of the Bernoulli numbers, is defined starting from suitable generating functions. Furthermore, the differential equations of these new classes of polynomials are derived by means of the factorization method introduced by Infeld and Hull (1951.

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.

Transversals of Complex Polynomial Vector Fields

DEFF Research Database (Denmark)

Dias, Kealey

Vector fields in the complex plane are defined by assigning the vector determined by the value P(z) to each point z in the complex plane, where P is a polynomial of one complex variable. We consider special families of so-called rotated vector fields that are determined by a polynomial multiplied...... by rotational constants. Transversals are a certain class of curves for such a family of vector fields that represent the bifurcation states for this family of vector fields. More specifically, transversals are curves that coincide with a homoclinic separatrix for some rotation of the vector field. Given...... a concrete polynomial, it seems to take quite a bit of work to prove that it is generic, i.e. structurally stable. This has been done for a special class of degree d polynomial vector fields having simple equilibrium points at the d roots of unity, d odd. In proving that such vector fields are generic...

On Multiple Interpolation Functions of the -Genocchi Polynomials

Directory of Open Access Journals (Sweden)

Jin Jeong-Hee

2010-01-01

Full Text Available Abstract Recently, many mathematicians have studied various kinds of the -analogue of Genocchi numbers and polynomials. In the work (New approach to q-Euler, Genocchi numbers and their interpolation functions, "Advanced Studies in Contemporary Mathematics, vol. 18, no. 2, pp. 105–112, 2009.", Kim defined new generating functions of -Genocchi, -Euler polynomials, and their interpolation functions. In this paper, we give another definition of the multiple Hurwitz type -zeta function. This function interpolates -Genocchi polynomials at negative integers. Finally, we also give some identities related to these polynomials.

Polynomial Chaos Expansion of Random Coefficients and the Solution of Stochastic Partial Differential Equations in the Tensor Train Format

KAUST Repository

Dolgov, Sergey

2015-11-03

We apply the tensor train (TT) decomposition to construct the tensor product polynomial chaos expansion (PCE) of a random field, to solve the stochastic elliptic diffusion PDE with the stochastic Galerkin discretization, and to compute some quantities of interest (mean, variance, and exceedance probabilities). We assume that the random diffusion coefficient is given as a smooth transformation of a Gaussian random field. In this case, the PCE is delivered by a complicated formula, which lacks an analytic TT representation. To construct its TT approximation numerically, we develop the new block TT cross algorithm, a method that computes the whole TT decomposition from a few evaluations of the PCE formula. The new method is conceptually similar to the adaptive cross approximation in the TT format but is more efficient when several tensors must be stored in the same TT representation, which is the case for the PCE. In addition, we demonstrate how to assemble the stochastic Galerkin matrix and to compute the solution of the elliptic equation and its postprocessing, staying in the TT format. We compare our technique with the traditional sparse polynomial chaos and the Monte Carlo approaches. In the tensor product polynomial chaos, the polynomial degree is bounded for each random variable independently. This provides higher accuracy than the sparse polynomial set or the Monte Carlo method, but the cardinality of the tensor product set grows exponentially with the number of random variables. However, when the PCE coefficients are implicitly approximated in the TT format, the computations with the full tensor product polynomial set become possible. In the numerical experiments, we confirm that the new methodology is competitive in a wide range of parameters, especially where high accuracy and high polynomial degrees are required.

Polynomial regression analysis and significance test of the regression function

International Nuclear Information System (INIS)

Gao Zhengming; Zhao Juan; He Shengping

2012-01-01

In order to analyze the decay heating power of a certain radioactive isotope per kilogram with polynomial regression method, the paper firstly demonstrated the broad usage of polynomial function and deduced its parameters with ordinary least squares estimate. Then significance test method of polynomial regression function is derived considering the similarity between the polynomial regression model and the multivariable linear regression model. Finally, polynomial regression analysis and significance test of the polynomial function are done to the decay heating power of the iso tope per kilogram in accord with the authors' real work. (authors)

The modified Gauss diagonalization of polynomial matrices

International Nuclear Information System (INIS)

Saeed, K.

1982-10-01

The Gauss algorithm for diagonalization of constant matrices is modified for application to polynomial matrices. Due to this modification the diagonal elements become pure polynomials rather than rational functions. (author)

Approximating Exponential and Logarithmic Functions Using Polynomial Interpolation

Science.gov (United States)

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…

Numerical Simulation of Polynomial-Speed Convergence Phenomenon

Science.gov (United States)

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.

Exceptional polynomials and SUSY quantum mechanics

Indian Academy of Sciences (India)

Abstract. We show that for the quantum mechanical problem which admit classical Laguerre/. Jacobi polynomials as solutions for the Schrödinger equations (SE), will also admit exceptional. Laguerre/Jacobi polynomials as solutions having the same eigenvalues but with the ground state missing after a modification of the ...

Analytic Expression of Arbitrary Matrix Elements for Boson Exponential Quadratic Polynomial Operators

Institute of Scientific and Technical Information of China (English)

XU Xiu-Wei; REN Ting-Qi; LIU Shu-Yan; MA Qiu-Ming; LIU Sheng-Dian

2007-01-01

Making use of the transformation relation among usual, normal, and antinormal ordering for the multimode boson exponential quadratic polynomial operators (BEQPO's), we present the analytic expression of arbitrary matrix elements for BEQPO's. As a preliminary application, we obtain the exact expressions of partition function about the boson quadratic polynomial system, matrix elements in particle-number, coordinate, and momentum representation, and P representation for the BEQPO's.

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.

Fast exhaustive search for polynomial systems in F2

NARCIS (Netherlands)

Bouillaguet, C.; Chen, H.-C.; Cheng, C.M.; Chou, T.; Niederhagen, R.F.; Shamir, A.; Yang, B.Y.

2010-01-01

Abstract. We analyze how fast we can solve general systems of multivariate equations of various low degrees over F2; this is a well known hard problem which is important both in itself and as part of many types of algebraic cryptanalysis. Compared to the standard exhaustive-search technique, our

Fast exhaustive search for polynomial systems in F2

NARCIS (Netherlands)

Bouillaguet, C.; Chen, H.-C.; Cheng, C.M.; Chou, T.; Niederhagen, R.F.; Shamir, A.; Yang, B.Y.; Mangard, S.; Standaert, F.X.

2010-01-01

Abstract: We analyze how fast we can solve general systems of multivariate equations of various low degrees over $F_2$; this is a well known hard problem which is important both in itself and as part of many types of algebraic cryptanalysis. Compared to the standard exhaustive search technique, our

Polynomial meta-models with canonical low-rank approximations: Numerical insights and comparison to sparse polynomial chaos expansions

International Nuclear Information System (INIS)

Konakli, Katerina; Sudret, Bruno

2016-01-01

The growing need for uncertainty analysis of complex computational models has led to an expanding use of meta-models across engineering and sciences. The efficiency of meta-modeling techniques relies on their ability to provide statistically-equivalent analytical representations based on relatively few evaluations of the original model. Polynomial chaos expansions (PCE) have proven a powerful tool for developing meta-models in a wide range of applications; the key idea thereof is to expand the model response onto a basis made of multivariate polynomials obtained as tensor products of appropriate univariate polynomials. The classical PCE approach nevertheless faces the “curse of dimensionality”, namely the exponential increase of the basis size with increasing input dimension. To address this limitation, the sparse PCE technique has been proposed, in which the expansion is carried out on only a few relevant basis terms that are automatically selected by a suitable algorithm. An alternative for developing meta-models with polynomial functions in high-dimensional problems is offered by the newly emerged low-rank approximations (LRA) approach. By exploiting the tensor–product structure of the multivariate basis, LRA can provide polynomial representations in highly compressed formats. Through extensive numerical investigations, we herein first shed light on issues relating to the construction of canonical LRA with a particular greedy algorithm involving a sequential updating of the polynomial coefficients along separate dimensions. Specifically, we examine the selection of optimal rank, stopping criteria in the updating of the polynomial coefficients and error estimation. In the sequel, we confront canonical LRA to sparse PCE in structural-mechanics and heat-conduction applications based on finite-element solutions. Canonical LRA exhibit smaller errors than sparse PCE in cases when the number of available model evaluations is small with respect to the input

Polynomial meta-models with canonical low-rank approximations: Numerical insights and comparison to sparse polynomial chaos expansions

Energy Technology Data Exchange (ETDEWEB)

Konakli, Katerina, E-mail: konakli@ibk.baug.ethz.ch; Sudret, Bruno

2016-09-15

The growing need for uncertainty analysis of complex computational models has led to an expanding use of meta-models across engineering and sciences. The efficiency of meta-modeling techniques relies on their ability to provide statistically-equivalent analytical representations based on relatively few evaluations of the original model. Polynomial chaos expansions (PCE) have proven a powerful tool for developing meta-models in a wide range of applications; the key idea thereof is to expand the model response onto a basis made of multivariate polynomials obtained as tensor products of appropriate univariate polynomials. The classical PCE approach nevertheless faces the “curse of dimensionality”, namely the exponential increase of the basis size with increasing input dimension. To address this limitation, the sparse PCE technique has been proposed, in which the expansion is carried out on only a few relevant basis terms that are automatically selected by a suitable algorithm. An alternative for developing meta-models with polynomial functions in high-dimensional problems is offered by the newly emerged low-rank approximations (LRA) approach. By exploiting the tensor–product structure of the multivariate basis, LRA can provide polynomial representations in highly compressed formats. Through extensive numerical investigations, we herein first shed light on issues relating to the construction of canonical LRA with a particular greedy algorithm involving a sequential updating of the polynomial coefficients along separate dimensions. Specifically, we examine the selection of optimal rank, stopping criteria in the updating of the polynomial coefficients and error estimation. In the sequel, we confront canonical LRA to sparse PCE in structural-mechanics and heat-conduction applications based on finite-element solutions. Canonical LRA exhibit smaller errors than sparse PCE in cases when the number of available model evaluations is small with respect to the input

Monomiality principle, Sheffer-type polynomials and the normal ordering problem

International Nuclear Information System (INIS)

Penson, K A; Blasiak, P; Dattoli, G; Duchamp, G H E; Horzela, A; Solomon, A I

2006-01-01

We solve the boson normal ordering problem for (q(a † )a + v(a † )) n with arbitrary functions q(x) and v(x) and integer n, where a and a † are boson annihilation and creation operators, satisfying [a, a † ] = 1. This consequently provides the solution for the exponential e λ(q(a † )a+v(a † )) generalizing the shift operator. In the course of these considerations we define and explore the monomiality principle and find its representations. We exploit the properties of Sheffer-type polynomials which constitute the inherent structure of this problem. In the end we give some examples illustrating the utility of the method and point out the relation to combinatorial structures

Clock Math — a System for Solving SLEs Exactly

Directory of Open Access Journals (Sweden)

Jakub Hladík

2013-01-01

Full Text Available In this paper, we present a GPU-accelerated hybrid system that solves ill-conditioned systems of linear equations exactly. Exactly means without rounding errors due to using integer arithmetics. First, we scale floating-point numbers up to integers, then we solve dozens of SLEs within different modular arithmetics and then we assemble sub-solutions back using the Chinese remainder theorem. This approach effectively bypasses current CPU floating-point limitations. The system is capable of solving Hilbert’s matrix without losing a single bit of precision, and with a significant speedup compared to existing CPU solvers.

Complex Polynomial Vector Fields

DEFF Research Database (Denmark)

The two branches of dynamical systems, continuous and discrete, correspond to the study of differential equations (vector fields) and iteration of mappings respectively. In holomorphic dynamics, the systems studied are restricted to those described by holomorphic (complex analytic) functions...... or meromorphic (allowing poles as singularities) functions. There already exists a well-developed theory for iterative holomorphic dynamical systems, and successful relations found between iteration theory and flows of vector fields have been one of the main motivations for the recent interest in holomorphic...... vector fields. Since the class of complex polynomial vector fields in the plane is natural to consider, it is remarkable that its study has only begun very recently. There are numerous fundamental questions that are still open, both in the general classification of these vector fields, the decomposition...

Degenerate r-Stirling Numbers and r-Bell Polynomials

Science.gov (United States)

Kim, T.; Yao, Y.; Kim, D. S.; Jang, G.-W.

2018-01-01

The purpose of this paper is to exploit umbral calculus in order to derive some properties, recurrence relations, and identities related to the degenerate r-Stirling numbers of the second kind and the degenerate r-Bell polynomials. Especially, we will express the degenerate r-Bell polynomials as linear combinations of many well-known families of special polynomials.

Commutators with idempotent values on multilinear polynomials in ...

Indian Academy of Sciences (India)

Multilinear polynomial; derivations; generalized polynomial identity; prime ring; right ideal. Abstract. Let R be a prime ring of characteristic different from 2, C its extended centroid, d a nonzero derivation of R , f ( x 1 , … , x n ) a multilinear polynomial over C , ϱ a nonzero right ideal of R and m > 1 a fixed integer such that.

Polynomial weights and code constructions

DEFF Research Database (Denmark)

Massey, J; Costello, D; Justesen, Jørn

1973-01-01

polynomial included. This fundamental property is then used as the key to a variety of code constructions including 1) a simplified derivation of the binary Reed-Muller codes and, for any primepgreater than 2, a new extensive class ofp-ary "Reed-Muller codes," 2) a new class of "repeated-root" cyclic codes...... of long constraint length binary convolutional codes derived from2^r-ary Reed-Solomon codes, and 6) a new class ofq-ary "repeated-root" constacyclic codes with an algebraic decoding algorithm.......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...

The generalized Yablonskii-Vorob'ev polynomials and their properties

International Nuclear Information System (INIS)

Kudryashov, Nikolai A.; Demina, Maria V.

2008-01-01

Rational solutions of the generalized second Painleve hierarchy are classified. Representation of the rational solutions in terms of special polynomials, the generalized Yablonskii-Vorob'ev polynomials, is introduced. Differential-difference relations satisfied by the polynomials are found. Hierarchies of differential equations related to the generalized second Painleve hierarchy are derived. One of these hierarchies is a sequence of differential equations satisfied by the generalized Yablonskii-Vorob'ev polynomials

2-variable Laguerre matrix polynomials and Lie-algebraic techniques

International Nuclear Information System (INIS)

Khan, Subuhi; Hassan, Nader Ali Makboul

2010-01-01

The authors introduce 2-variable forms of Laguerre and modified Laguerre matrix polynomials and derive their special properties. Further, the representations of the special linear Lie algebra sl(2) and the harmonic oscillator Lie algebra G(0,1) are used to derive certain results involving these polynomials. Furthermore, the generating relations for the ordinary as well as matrix polynomials related to these matrix polynomials are derived as applications.

The semantic system is involved in mathematical problem solving.

Science.gov (United States)

Zhou, Xinlin; Li, Mengyi; Li, Leinian; Zhang, Yiyun; Cui, Jiaxin; Liu, Jie; Chen, Chuansheng

2018-02-01

Numerous studies have shown that the brain regions around bilateral intraparietal cortex are critical for number processing and arithmetical computation. However, the neural circuits for more advanced mathematics such as mathematical problem solving (with little routine arithmetical computation) remain unclear. Using functional magnetic resonance imaging (fMRI), this study (N = 24 undergraduate students) compared neural bases of mathematical problem solving (i.e., number series completion, mathematical word problem solving, and geometric problem solving) and arithmetical computation. Direct subject- and item-wise comparisons revealed that mathematical problem solving typically had greater activation than arithmetical computation in all 7 regions of the semantic system (which was based on a meta-analysis of 120 functional neuroimaging studies on semantic processing). Arithmetical computation typically had greater activation in the supplementary motor area and left precentral gyrus. The results suggest that the semantic system in the brain supports mathematical problem solving. Copyright © 2017 Elsevier Inc. All rights reserved.

Optimal non-coherent data detection for massive SIMO wireless systems: A polynomial complexity solution

KAUST Repository

Alshamary, Haider Ali Jasim

2016-01-04

© 2015 IEEE. This paper considers the joint maximum likelihood (ML) channel estimation and data detection problem for massive SIMO (single input multiple output) wireless systems. We propose efficient algorithms achieving the exact ML non-coherent data detection, for both constant-modulus constellations and nonconstant-modulus constellations. Despite a large number of unknown channel coefficients in massive SIMO systems, we show that the expected computational complexity is linear in the number of receive antennas and polynomial in channel coherence time. To the best of our knowledge, our algorithms are the first efficient algorithms to achieve the exact joint ML channel estimation and data detection performance for massive SIMO systems with general constellations. Simulation results show our algorithms achieve considerable performance gains at a low computational complexity.

Optimal non-coherent data detection for massive SIMO wireless systems: A polynomial complexity solution

KAUST Repository

Alshamary, Haider Ali Jasim; Al-Naffouri, Tareq Y.; Zaib, Alam; Xu, Weiyu

2016-01-01

© 2015 IEEE. This paper considers the joint maximum likelihood (ML) channel estimation and data detection problem for massive SIMO (single input multiple output) wireless systems. We propose efficient algorithms achieving the exact ML non-coherent data detection, for both constant-modulus constellations and nonconstant-modulus constellations. Despite a large number of unknown channel coefficients in massive SIMO systems, we show that the expected computational complexity is linear in the number of receive antennas and polynomial in channel coherence time. To the best of our knowledge, our algorithms are the first efficient algorithms to achieve the exact joint ML channel estimation and data detection performance for massive SIMO systems with general constellations. Simulation results show our algorithms achieve considerable performance gains at a low computational complexity.

Describing Quadratic Cremer Point Polynomials by Parabolic Perturbations

DEFF Research Database (Denmark)

Sørensen, Dan Erik Krarup

1996-01-01

We describe two infinite order parabolic perturbation proceduresyielding quadratic polynomials having a Cremer fixed point. The main ideais to obtain the polynomial as the limit of repeated parabolic perturbations.The basic tool at each step is to control the behaviour of certain externalrays.......Polynomials of the Cremer type correspond to parameters at the boundary of ahyperbolic component of the Mandelbrot set. In this paper we concentrate onthe main cardioid component. We investigate the differences between two-sided(i.e. alternating) and one-sided parabolic perturbations.In the two-sided case, we prove...... the existence of polynomials having an explicitlygiven external ray accumulating both at the Cremer point and at its non-periodicpreimage. We think of the Julia set as containing a "topologists double comb".In the one-sided case we prove a weaker result: the existence of polynomials havingan explicitly given...

A note on some identities of derangement polynomials.

Science.gov (United States)

Kim, Taekyun; Kim, Dae San; Jang, Gwan-Woo; Kwon, Jongkyum

2018-01-01

The problem of counting derangements was initiated by Pierre Rémond de Montmort in 1708 (see Carlitz in Fibonacci Q. 16(3):255-258, 1978, Clarke and Sved in Math. Mag. 66(5):299-303, 1993, Kim, Kim and Kwon in Adv. Stud. Contemp. Math. (Kyungshang) 28(1):1-11 2018. A derangement is a permutation that has no fixed points, and the derangement number [Formula: see text] is the number of fixed-point-free permutations on an n element set. In this paper, we study the derangement polynomials and investigate some interesting properties which are related to derangement numbers. Also, we study two generalizations of derangement polynomials, namely higher-order and r -derangement polynomials, and show some relations between them. In addition, we express several special polynomials in terms of the higher-order derangement polynomials by using umbral calculus.

Topological quantum information, virtual Jones polynomials and Khovanov homology

International Nuclear Information System (INIS)

Kauffman, Louis H

2011-01-01

In this paper, we give a quantum statistical interpretation of the bracket polynomial state sum 〈K〉, the Jones polynomial V K (t) and virtual knot theory versions of the Jones polynomial, including the arrow polynomial. We use these quantum mechanical interpretations to give new quantum algorithms for these Jones polynomials. In those cases where the Khovanov homology is defined, the Hilbert space C(K) of our model is isomorphic with the chain complex for Khovanov homology with coefficients in the complex numbers. There is a natural unitary transformation U:C(K) → C(K) such that 〈K〉 = Trace(U), where 〈K〉 denotes the evaluation of the state sum model for the corresponding polynomial. We show that for the Khovanov boundary operator ∂:C(K) → C(K), we have the relationship ∂U + U∂ = 0. Consequently, the operator U acts on the Khovanov homology, and we obtain a direct relationship between the Khovanov homology and this quantum algorithm for the Jones polynomial. (paper)

Polynomial solutions of the Monge-Ampère equation

Energy Technology Data Exchange (ETDEWEB)

Aminov, Yu A [B.Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, Khar' kov (Ukraine)

2014-11-30

The question of the existence of polynomial solutions to the Monge-Ampère equation z{sub xx}z{sub yy}−z{sub xy}{sup 2}=f(x,y) is considered in the case when f(x,y) is a polynomial. It is proved that if f is a polynomial of the second degree, which is positive for all values of its arguments and has a positive squared part, then no polynomial solution exists. On the other hand, a solution which is not polynomial but is analytic in the whole of the x, y-plane is produced. Necessary and sufficient conditions for the existence of polynomial solutions of degree up to 4 are found and methods for the construction of such solutions are indicated. An approximation theorem is proved. Bibliography: 10 titles.

Zeros and uniqueness of Q-difference polynomials of meromorphic ...

Indian Academy of Sciences (India)

Meromorphic functions; Nevanlinna theory; logarithmic order; uniqueness problem; difference-differential polynomial. Abstract. In this paper, we investigate the value distribution of -difference polynomials of meromorphic function of finite logarithmic order, and study the zero distribution of difference-differential polynomials ...

Laguerre polynomials by a harmonic oscillator

Science.gov (United States)

Baykal, Melek; Baykal, Ahmet

2014-09-01

The study of an isotropic harmonic oscillator, using the factorization method given in Ohanian's textbook on quantum mechanics, is refined and some collateral extensions of the method related to the ladder operators and the associated Laguerre polynomials are presented. In particular, some analytical properties of the associated Laguerre polynomials are derived using the ladder operators.

Galerkin projection methods for solving multiple related linear systems

Energy Technology Data Exchange (ETDEWEB)

Chan, T.F.; Ng, M.; Wan, W.L.

1996-12-31

We consider using Galerkin projection methods for solving multiple related linear systems A{sup (i)}x{sup (i)} = b{sup (i)} for 1 {le} i {le} s, where A{sup (i)} and b{sup (i)} are different in general. We start with the special case where A{sup (i)} = A and A is symmetric positive definite. The method generates a Krylov subspace from a set of direction vectors obtained by solving one of the systems, called the seed system, by the CG method and then projects the residuals of other systems orthogonally onto the generated Krylov subspace to get the approximate solutions. The whole process is repeated with another unsolved system as a seed until all the systems are solved. We observe in practice a super-convergence behaviour of the CG process of the seed system when compared with the usual CG process. We also observe that only a small number of restarts is required to solve all the systems if the right-hand sides are close to each other. These two features together make the method particularly effective. In this talk, we give theoretical proof to justify these observations. Furthermore, we combine the advantages of this method and the block CG method and propose a block extension of this single seed method. The above procedure can actually be modified for solving multiple linear systems A{sup (i)}x{sup (i)} = b{sup (i)}, where A{sup (i)} are now different. We can also extend the previous analytical results to this more general case. Applications of this method to multiple related linear systems arising from image restoration and recursive least squares computations are considered as examples.

Image Compression Based On Wavelet, Polynomial and Quadtree

Directory of Open Access Journals (Sweden)

Bushra A. SULTAN

2011-01-01

Full Text Available In this paper a simple and fast image compression scheme is proposed, it is based on using wavelet transform to decompose the image signal and then using polynomial approximation to prune the smoothing component of the image band. The architect of proposed coding scheme is high synthetic where the error produced due to polynomial approximation in addition to the detail sub-band data are coded using both quantization and Quadtree spatial coding. As a last stage of the encoding process shift encoding is used as a simple and efficient entropy encoder to compress the outcomes of the previous stage.The test results indicate that the proposed system can produce a promising compression performance while preserving the image quality level.

A convex optimization approach for solving large scale linear systems

Directory of Open Access Journals (Sweden)

Debora Cores

2017-01-01

Full Text Available The well-known Conjugate Gradient (CG method minimizes a strictly convex quadratic function for solving large-scale linear system of equations when the coefficient matrix is symmetric and positive definite. In this work we present and analyze a non-quadratic convex function for solving any large-scale linear system of equations regardless of the characteristics of the coefficient matrix. For finding the global minimizers, of this new convex function, any low-cost iterative optimization technique could be applied. In particular, we propose to use the low-cost globally convergent Spectral Projected Gradient (SPG method, which allow us to extend this optimization approach for solving consistent square and rectangular linear system, as well as linear feasibility problem, with and without convex constraints and with and without preconditioning strategies. Our numerical results indicate that the new scheme outperforms state-of-the-art iterative techniques for solving linear systems when the symmetric part of the coefficient matrix is indefinite, and also for solving linear feasibility problems.

Julia Sets of Orthogonal Polynomials

DEFF Research Database (Denmark)

Christiansen, Jacob Stordal; Henriksen, Christian; Petersen, Henrik Laurberg

2018-01-01

For a probability measure with compact and non-polar support in the complex plane we relate dynamical properties of the associated sequence of orthogonal polynomials fPng to properties of the support. More precisely we relate the Julia set of Pn to the outer boundary of the support, the lled Julia...... set to the polynomial convex hull K of the support, and the Green's function associated with Pn to the Green's function for the complement of K....

An introduction to orthogonal polynomials

CERN Document Server

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 selection in number field sieve for integer factorization

Directory of Open Access Journals (Sweden)

Gireesh Pandey

2016-09-01

Full Text Available The general number field sieve (GNFS is the fastest algorithm for factoring large composite integers which is made up by two prime numbers. Polynomial selection is an important step of GNFS. The asymptotic runtime depends on choice of good polynomial pairs. In this paper, we present polynomial selection algorithm that will be modelled with size and root properties. The correlations between polynomial coefficient and number of relations have been explored with experimental findings.

A polynomial chaos ensemble hydrologic prediction system for efficient parameter inference and robust uncertainty assessment

Science.gov (United States)

Wang, S.; Huang, G. H.; Baetz, B. W.; Huang, W.

2015-11-01

This paper presents a polynomial chaos ensemble hydrologic prediction system (PCEHPS) for an efficient and robust uncertainty assessment of model parameters and predictions, in which possibilistic reasoning is infused into probabilistic parameter inference with simultaneous consideration of randomness and fuzziness. The PCEHPS is developed through a two-stage factorial polynomial chaos expansion (PCE) framework, which consists of an ensemble of PCEs to approximate the behavior of the hydrologic model, significantly speeding up the exhaustive sampling of the parameter space. Multiple hypothesis testing is then conducted to construct an ensemble of reduced-dimensionality PCEs with only the most influential terms, which is meaningful for achieving uncertainty reduction and further acceleration of parameter inference. The PCEHPS is applied to the Xiangxi River watershed in China to demonstrate its validity and applicability. A detailed comparison between the HYMOD hydrologic model, the ensemble of PCEs, and the ensemble of reduced PCEs is performed in terms of accuracy and efficiency. Results reveal temporal and spatial variations in parameter sensitivities due to the dynamic behavior of hydrologic systems, and the effects (magnitude and direction) of parametric interactions depending on different hydrological metrics. The case study demonstrates that the PCEHPS is capable not only of capturing both expert knowledge and probabilistic information in the calibration process, but also of implementing an acceleration of more than 10 times faster than the hydrologic model without compromising the predictive accuracy.

Polynomial solutions of nonlinear integral equations

International Nuclear Information System (INIS)

Dominici, Diego

2009-01-01

We analyze the polynomial solutions of a nonlinear integral equation, generalizing the work of Bender and Ben-Naim (2007 J. Phys. A: Math. Theor. 40 F9, 2008 J. Nonlinear Math. Phys. 15 (Suppl. 3) 73). We show that, in some cases, an orthogonal solution exists and we give its general form in terms of kernel polynomials

Polynomial solutions of nonlinear integral equations

Energy Technology Data Exchange (ETDEWEB)

Dominici, Diego [Department of Mathematics, State University of New York at New Paltz, 1 Hawk Dr. Suite 9, New Paltz, NY 12561-2443 (United States)], E-mail: dominicd@newpaltz.edu

2009-05-22

We analyze the polynomial solutions of a nonlinear integral equation, generalizing the work of Bender and Ben-Naim (2007 J. Phys. A: Math. Theor. 40 F9, 2008 J. Nonlinear Math. Phys. 15 (Suppl. 3) 73). We show that, in some cases, an orthogonal solution exists and we give its general form in terms of kernel polynomials.

Laguerre polynomials by a harmonic oscillator

International Nuclear Information System (INIS)

Baykal, Melek; Baykal, Ahmet

2014-01-01

The study of an isotropic harmonic oscillator, using the factorization method given in Ohanian's textbook on quantum mechanics, is refined and some collateral extensions of the method related to the ladder operators and the associated Laguerre polynomials are presented. In particular, some analytical properties of the associated Laguerre polynomials are derived using the ladder operators. (paper)

Remarks on determinants and the classical polynomials

International Nuclear Information System (INIS)

Henning, J.J.; Kranold, H.U.; Louw, D.F.B.

1986-01-01

As motivation for this formal analysis the problem of Landau damping of Bernstein modes is discussed. It is shown that in the case of a weak but finite constant external magnetic field, the analytical structure of the dispersion relations is of such a nature that longitudinal waves propagating orthogonal to the external magnetic field are also damped, contrary to normal belief. In the treatment of the linearized Vlasov equation it is found convenient to generate certain polynomials by the problem at hand and to explicitly write down expressions for these polynomials. In the course of this study methods are used that relate to elementary but fairly unknown functional relationships between power sums and coefficients of polynomials. These relationships, also called Waring functions, are derived. They are then used in other applications to give explicit expressions for the generalized Laguerre polynomials in terms of determinant functions. The properties of polynomials generated by a wide class of generating functions are investigated. These relationships are also used to obtain explicit forms for the cumulants of a distribution in terms of its moments. It is pointed out that cumulants (or moments, for that matter) do not determine a distribution function

Problem solving using soft systems methodology.

Science.gov (United States)

Land, L

This article outlines a method of problem solving which considers holistic solutions to complex problems. Soft systems methodology allows people involved in the problem situation to have control over the decision-making process.

Multiple Meixner polynomials and non-Hermitian oscillator Hamiltonians

OpenAIRE

Ndayiragije, François; Van Assche, Walter

2013-01-01

Multiple Meixner polynomials are polynomials in one variable which satisfy orthogonality relations with respect to $r>1$ different negative binomial distributions (Pascal distributions). There are two kinds of multiple Meixner polynomials, depending on the selection of the parameters in the negative binomial distribution. We recall their definition and some formulas and give generating functions and explicit expressions for the coefficients in the nearest neighbor recurrence relation. Followi...

Multivariable biorthogonal continuous--discrete Wilson and Racah polynomials

International Nuclear Information System (INIS)

Tratnik, M.V.

1990-01-01

Several families of multivariable, biorthogonal, partly continuous and partly discrete, Wilson polynomials are presented. These yield limit cases that are purely continuous in some of the variables and purely discrete in the others, or purely discrete in all the variables. The latter are referred to as the multivariable biorthogonal Racah polynomials. Interesting further limit cases include the multivariable biorthogonal Hahn and dual Hahn polynomials

Primitive polynomials selection method for pseudo-random number generator

Science.gov (United States)

Anikin, I. V.; Alnajjar, Kh

2018-01-01

In this paper we suggested the method for primitive polynomials selection of special type. This kind of polynomials can be efficiently used as a characteristic polynomials for linear feedback shift registers in pseudo-random number generators. The proposed method consists of two basic steps: finding minimum-cost irreducible polynomials of the desired degree and applying primitivity tests to get the primitive ones. Finally two primitive polynomials, which was found by the proposed method, used in pseudorandom number generator based on fuzzy logic (FRNG) which had been suggested before by the authors. The sequences generated by new version of FRNG have low correlation magnitude, high linear complexity, less power consumption, is more balanced and have better statistical properties.

Neck curve polynomials in neck rupture model

International Nuclear Information System (INIS)

Kurniadi, Rizal; Perkasa, Yudha S.; Waris, Abdul

2012-01-01

The Neck Rupture Model is a model that explains the scission process which has smallest radius in liquid drop at certain position. Old fashion of rupture position is determined randomly so that has been called as Random Neck Rupture Model (RNRM). The neck curve polynomials have been employed in the Neck Rupture Model for calculation the fission yield of neutron induced fission reaction of 280 X 90 with changing of order of polynomials as well as temperature. The neck curve polynomials approximation shows the important effects in shaping of fission yield curve.

The finite Fourier transform of classical polynomials

OpenAIRE

Dixit, Atul; Jiu, Lin; Moll, Victor H.; Vignat, Christophe

2014-01-01

The finite Fourier transform of a family of orthogonal polynomials $A_{n}(x)$, is the usual transform of the polynomial extended by $0$ outside their natural domain. Explicit expressions are given for the Legendre, Jacobi, Gegenbauer and Chebyshev families.

Superintegrability in two-dimensional Euclidean space and associated polynomial solutions

International Nuclear Information System (INIS)

Kalnins, E.G.; Miller, W. Jr; Pogosyan, G.S.

1996-01-01

In this work we examine the basis functions for those classical and quantum mechanical systems in two dimensions which admit separation of variables in at least two coordinate systems. We do this for the corresponding systems defined in Euclidean space and on the two dimensional sphere. We present all of these cases from a unified point of view. In particular, all of the spectral functions that arise via variable separation have their essential features expressed in terms of their zeros. The principal new results are the details of the polynomial base for each of the nonsubgroup base, not just the subgroup cartesian and polar coordinate case, and the details of the structure of the quadratic algebras. We also study the polynomial eigenfunctions in elliptic coordinates of the N-dimensional isotropic quantum oscillator. 28 refs., 1 tab

From sequences to polynomials and back, via operator orderings

Energy Technology Data Exchange (ETDEWEB)

Amdeberhan, Tewodros, E-mail: tamdeber@tulane.edu; Dixit, Atul, E-mail: adixit@tulane.edu; Moll, Victor H., E-mail: vhm@tulane.edu [Department of Mathematics, Tulane University, New Orleans, Louisiana 70118 (United States); De Angelis, Valerio, E-mail: vdeangel@xula.edu [Department of Mathematics, Xavier University of Louisiana, New Orleans, Louisiana 70125 (United States); Vignat, Christophe, E-mail: vignat@tulane.edu [Department of Mathematics, Tulane University, New Orleans, Louisiana 70118, USA and L.S.S. Supelec, Universite d' Orsay (France)

2013-12-15

Bender and Dunne [“Polynomials and operator orderings,” J. Math. Phys. 29, 1727–1731 (1988)] showed that linear combinations of words q{sup k}p{sup n}q{sup n−k}, where p and q are subject to the relation qp − pq = ı, may be expressed as a polynomial in the symbol z=1/2 (qp+pq). Relations between such polynomials and linear combinations of the transformed coefficients are explored. In particular, examples yielding orthogonal polynomials are provided.

Connection coefficients between Boas-Buck polynomial sets

Science.gov (United States)

Cheikh, Y. Ben; Chaggara, H.

2006-07-01

In this paper, a general method to express explicitly connection coefficients between two Boas-Buck polynomial sets is presented. As application, we consider some generalized hypergeometric polynomials, from which we derive some well-known results including duplication and inversion formulas.

New realisation of Preisach model using adaptive polynomial approximation

Science.gov (United States)

Liu, Van-Tsai; Lin, Chun-Liang; Wing, Home-Young

2012-09-01

Modelling system with hysteresis has received considerable attention recently due to the increasing accurate requirement in engineering applications. The classical Preisach model (CPM) is the most popular model to demonstrate hysteresis which can be represented by infinite but countable first-order reversal curves (FORCs). The usage of look-up tables is one way to approach the CPM in actual practice. The data in those tables correspond with the samples of a finite number of FORCs. This approach, however, faces two major problems: firstly, it requires a large amount of memory space to obtain an accurate prediction of hysteresis; secondly, it is difficult to derive efficient ways to modify the data table to reflect the timing effect of elements with hysteresis. To overcome, this article proposes the idea of using a set of polynomials to emulate the CPM instead of table look-up. The polynomial approximation requires less memory space for data storage. Furthermore, the polynomial coefficients can be obtained accurately by using the least-square approximation or adaptive identification algorithm, such as the possibility of accurate tracking of hysteresis model parameters.

Least squares orthogonal polynomial approximation in several independent variables

International Nuclear Information System (INIS)

Caprari, R.S.

1992-06-01

This paper begins with an exposition of a systematic technique for generating orthonormal polynomials in two independent variables by application of the Gram-Schmidt orthogonalization procedure of linear algebra. It is then demonstrated how a linear least squares approximation for experimental data or an arbitrary function can be generated from these polynomials. The least squares coefficients are computed without recourse to matrix arithmetic, which ensures both numerical stability and simplicity of implementation as a self contained numerical algorithm. The Gram-Schmidt procedure is then utilised to generate a complete set of orthogonal polynomials of fourth degree. A theory for the transformation of the polynomial representation from an arbitrary basis into the familiar sum of products form is presented, together with a specific implementation for fourth degree polynomials. Finally, the computational integrity of this algorithm is verified by reconstructing arbitrary fourth degree polynomials from their values at randomly chosen points in their domain. 13 refs., 1 tab

New families of superintegrable systems from k-step rational extensions, polynomial algebras and degeneracies

International Nuclear Information System (INIS)

Marquette, Ian

2015-01-01

Four new families of two-dimensional quantum superintegrable systems are constructed from k-step extension of the harmonic oscillator and the radial oscillator. Their wavefunctions are related with Hermite and Laguerre exceptional orthogonal polynomials (EOP) of type III. We show that ladder operators obtained from alternative construction based on combinations of supercharges in the Krein-Adler and Darboux Crum (or state deleting and creating) approaches can be used to generate a set of integrals of motion and a corresponding polynomial algebra that provides an algebraic derivation of the full spectrum and total number of degeneracies. Such derivation is based on finite dimensional unitary representations (unirreps) and doesn't work for integrals build from standard ladder operators in supersymmetric quantum mechanics (SUSYQM) as they contain singlets isolated from excited states. In this paper, we also rely on a novel approach to obtain the finite dimensional unirreps based on the action of the integrals of motion on the wavefunctions given in terms of these EOP. We compare the results with those obtained from the Daskaloyannis approach and the realizations in terms of deformed oscillator algebras for one of the new families in the case of 1-step extension. This communication is a review of recent works. (paper)

Diffusion Coefficient Calculations With Low Order Legendre Polynomial and Chebyshev Polynomial Approximation for the Transport Equation in Spherical Geometry

International Nuclear Information System (INIS)

Yasa, F.; Anli, F.; Guengoer, S.

2007-01-01

We present analytical calculations of spherically symmetric radioactive transfer and neutron transport using a hypothesis of P1 and T1 low order polynomial approximation for diffusion coefficient D. Transport equation in spherical geometry is considered as the pseudo slab equation. The validity of polynomial expansionion in transport theory is investigated through a comparison with classic diffusion theory. It is found that for causes when the fluctuation of the scattering cross section dominates, the quantitative difference between the polynomial approximation and diffusion results was physically acceptable in general

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].

Multistage Spectral Relaxation Method for Solving the Hyperchaotic Complex Systems

Directory of Open Access Journals (Sweden)

Hassan Saberi Nik

2014-01-01

Full Text Available We present a pseudospectral method application for solving the hyperchaotic complex systems. The proposed method, called the multistage spectral relaxation method (MSRM is based on a technique of extending Gauss-Seidel type relaxation ideas to systems of nonlinear differential equations and using the Chebyshev pseudospectral methods to solve the resulting system on a sequence of multiple intervals. In this new application, the MSRM is used to solve famous hyperchaotic complex systems such as hyperchaotic complex Lorenz system and the complex permanent magnet synchronous motor. We compare this approach to the Runge-Kutta based ode45 solver to show that the MSRM gives accurate results.

Topological string partition functions as polynomials

International Nuclear Information System (INIS)

Yamaguchi, Satoshi; Yau Shingtung

2004-01-01

We investigate the structure of the higher genus topological string amplitudes on the quintic hypersurface. It is shown that the partition functions of the higher genus than one can be expressed as polynomials of five generators. We also compute the explicit polynomial forms of the partition functions for genus 2, 3, and 4. Moreover, some coefficients are written down for all genus. (author)

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

q-analogue of the Krawtchouk and Meixner orthogonal polynomials

International Nuclear Information System (INIS)

Campigotto, C.; Smirnov, Yu.F.; Enikeev, S.G.

1993-06-01

The comparative analysis of Krawtchouk polynomials on a uniform grid with Wigner D-functions for the SU(2) group is presented. As a result the partnership between corresponding properties of the polynomials and D-functions is established giving the group-theoretical interpretation of the Krawtchouk polynomials properties. In order to extend such an analysis on the quantum groups SU q (2) and SU q (1,1), q-analogues of Krawtchouk and Meixner polynomials of a discrete variable are studied. The total set of characteristics of these polynomials is calculated, including the orthogonality condition, normalization factor, recurrent relation, the explicit analytic expression, the Rodrigues formula, the difference derivative formula and various particular cases and values. (R.P.) 22 refs.; 2 tabs

Skew-orthogonal polynomials and random matrix theory

CERN Document Server

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 ...

Some properties of generalized self-reciprocal polynomials over finite fields

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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.

Polynomial Digital Control of a Series Equal Liquid Tanks

Directory of Open Access Journals (Sweden)

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.

Coherent states for a polynomial su(1, 1) algebra and a conditionally solvable system

International Nuclear Information System (INIS)

Sadiq, Muhammad; Inomata, Akira; Junker, Georg

2009-01-01

In a previous paper (2007 J. Phys. A: Math. Theor. 40 11105), we constructed a class of coherent states for a polynomially deformed su(2) algebra. In this paper, we first prepare the discrete representations of the nonlinearly deformed su(1, 1) algebra. Then we extend the previous procedure to construct a discrete class of coherent states for a polynomial su(1, 1) algebra which contains the Barut-Girardello set and the Perelomov set of the SU(1, 1) coherent states as special cases. We also construct coherent states for the cubic algebra related to the conditionally solvable radial oscillator problem.

Modified homotopy perturbation method for solving hypersingular integral equations of the first kind.

Science.gov (United States)

Eshkuvatov, Z K; Zulkarnain, F S; Nik Long, N M A; Muminov, Z

2016-01-01

Modified homotopy perturbation method (HPM) was used to solve the hypersingular integral equations (HSIEs) of the first kind on the interval [-1,1] with the assumption that the kernel of the hypersingular integral is constant on the diagonal of the domain. Existence of inverse of hypersingular integral operator leads to the convergence of HPM in certain cases. Modified HPM and its norm convergence are obtained in Hilbert space. Comparisons between modified HPM, standard HPM, Bernstein polynomials approach Mandal and Bhattacharya (Appl Math Comput 190:1707-1716, 2007), Chebyshev expansion method Mahiub et al. (Int J Pure Appl Math 69(3):265-274, 2011) and reproducing kernel Chen and Zhou (Appl Math Lett 24:636-641, 2011) are made by solving five examples. Theoretical and practical examples revealed that the modified HPM dominates the standard HPM and others. Finally, it is found that the modified HPM is exact, if the solution of the problem is a product of weights and polynomial functions. For rational solution the absolute error decreases very fast by increasing the number of collocation points.

on the performance of Autoregressive Moving Average Polynomial

African Journals Online (AJOL)

Timothy Ademakinwa

Distributed Lag (PDL) model, Autoregressive Polynomial Distributed Lag ... Moving Average Polynomial Distributed Lag (ARMAPDL) model. ..... Global Journal of Mathematics and Statistics. Vol. 1. ... Business and Economic Research Center.

Simulation of aspheric tolerance with polynomial fitting

Science.gov (United States)

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.

Symmetric functions and orthogonal polynomials

CERN Document Server

Macdonald, I G

1997-01-01

One of the most classical areas of algebra, the theory of symmetric functions and orthogonal polynomials has long been known to be connected to combinatorics, representation theory, and other branches of mathematics. Written by perhaps the most famous author on the topic, this volume explains some of the current developments regarding these connections. It is based on lectures presented by the author at Rutgers University. Specifically, he gives recent results on orthogonal polynomials associated with affine Hecke algebras, surveying the proofs of certain famous combinatorial conjectures.

Hermite Polynomials and the Inverse Problem for Collisionless Equilibria

Science.gov (United States)

Allanson, O.; Neukirch, T.; Troscheit, S.; Wilson, F.

2017-12-01

It is long established that Hermite polynomial expansions in either velocity or momentum space can elegantly encode the non-Maxwellian velocity-space structure of a collisionless plasma distribution function (DF). In particular, Hermite polynomials in the canonical momenta naturally arise in the consideration of the 'inverse problem in collisionless equilibria' (IPCE): "for a given macroscopic/fluid equilibrium, what are the self-consistent Vlasov-Maxwell equilibrium DFs?". This question is of particular interest for the equilibrium and stability properties of a given macroscopic configuration, e.g. a current sheet. It can be relatively straightforward to construct a formal solution to IPCE by a Hermite expansion method, but several important questions remain regarding the use of this method. We present recent work that considers the necessary conditions of non-negativity, convergence, and the existence of all moments of an equilibrium DF solution found for IPCE. We also establish meaningful analogies between the equations that link the microscopic and macrosopic descriptions of the Vlasov-Maxwell equilibrium, and those that solve the initial value problem for the heat equation. In the language of the heat equation, IPCE poses the pressure tensor as the 'present' heat distribution over an infinite domain, and the non-Maxwellian features of the DF as the 'past' distribution. We find sufficient conditions for the convergence of the Hermite series representation of the DF, and prove that the non-negativity of the DF can be dependent on the magnetisation of the plasma. For DFs that decay at least as quickly as exp(-v^2/4), we show non-negativity is guaranteed for at least a finite range of magnetisation values, as parameterised by the ratio of the Larmor radius to the gradient length scale. 1. O. Allanson, T. Neukirch, S. Troscheit & F. Wilson: From one-dimensional fields to Vlasov equilibria: theory and application of Hermite polynomials, Journal of Plasma Physics, 82

Polynomially Riesz elements | Živković-Zlatanović | Quaestiones ...

African Journals Online (AJOL)

A Banach algebra element ɑ ∈ A is said to be "polynomially Riesz", relative to the homomorphism T : A → B, if there exists a nonzero complex polynomial p(z) such that the image Tp ∈ B is quasinilpotent. Keywords: Homomorphism of Banach algebras, polynomially Riesz element, Fredholm spectrum, Browder element, ...

Higher order polynomial expansion nodal method for hexagonal core neutronics analysis

International Nuclear Information System (INIS)

Jin, Young Cho; Chang, Hyo Kim

1998-01-01

A higher-order polynomial expansion nodal(PEN) method is newly formulated as a means to improve the accuracy of the conventional PEN method solutions to multi-group diffusion equations in hexagonal core geometry. The new method is applied to solving various hexagonal core neutronics benchmark problems. The computational accuracy of the higher order PEN method is then compared with that of the conventional PEN method, the analytic function expansion nodal (AFEN) method, and the ANC-H method. It is demonstrated that the higher order PEN method improves the accuracy of the conventional PEN method and that it compares very well with the other nodal methods like the AFEN and ANC-H methods in accuracy

Symmetric integrable-polynomial factorization for symplectic one-turn-map tracking

International Nuclear Information System (INIS)

Shi, Jicong

1993-01-01

It was found that any homogeneous polynomial can be written as a sum of integrable polynomials of the same degree which Lie transformations can be evaluated exactly. By utilizing symplectic integrators, an integrable-polynomial factorization is developed to convert a symplectic map in the form of Dragt-Finn factorization into a product of Lie transformations associated with integrable polynomials. A small number of factorization bases of integrable polynomials enable one to use high order symplectic integrators so that the high-order spurious terms can be greatly suppressed. A symplectic map can thus be evaluated with desired accuracy

A fast numerical test of multivariate polynomial positiveness with applications

Czech Academy of Sciences Publication Activity Database

Augusta, Petr; Augustová, Petra

2018-01-01

Roč. 54, č. 2 (2018), s. 289-303 ISSN 0023-5954 Institutional support: RVO:67985556 Keywords : stability * multidimensional systems * positive polynomials * fast Fourier transforms * numerical algorithm Subject RIV: BC - Control Systems Theory OBOR OECD: Automation and control systems Impact factor: 0.379, year: 2016 https://www.kybernetika.cz/content/2018/2/289/paper.pdf

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.

On Generalisation of Polynomials in Complex Plane

Directory of Open Access Journals (Sweden)

Maslina Darus

2010-01-01

Full Text Available The generalised Bell and Laguerre polynomials of fractional-order in complex z-plane are defined. Some properties are studied. Moreover, we proved that these polynomials are univalent solutions for second order differential equations. Also, the Laguerre-type of some special functions are introduced.

Technique for image interpolation using polynomial transforms

NARCIS (Netherlands)

Escalante Ramírez, B.; Martens, J.B.; Haskell, G.G.; Hang, H.M.

1993-01-01

We present a new technique for image interpolation based on polynomial transforms. This is an image representation model that analyzes an image by locally expanding it into a weighted sum of orthogonal polynomials. In the discrete case, the image segment within every window of analysis is

Interlacing of zeros of quasi-orthogonal meixner polynomials | Driver ...

African Journals Online (AJOL)

... interlacing of zeros of quasi-orthogonal Meixner polynomials Mn(x;β; c) with the zeros of their nearest orthogonal counterparts Mt(x;β + k; c), l; n ∈ ℕ, k ∈ {1; 2}; is also discussed. Mathematics Subject Classication (2010): 33C45, 42C05. Key words: Discrete orthogonal polynomials, quasi-orthogonal polynomials, Meixner

Localization of periodic orbits of polynomial vector fields of even degree by linear functions

Energy Technology Data Exchange (ETDEWEB)

Starkov, Konstantin E. [CITEDI-IPN, Av. del Parque 1310, Mesa de Otay, Tijuana, BC (Mexico)] e-mail: konst@citedi.mx

2005-08-01

This paper is concerned with the localization problem of periodic orbits of polynomial vector fields of even degree by using linear functions. Conditions of the localization of all periodic orbits in sets of a simple structure are obtained. Our results are based on the solution of the conditional extremum problem and the application of homogeneous polynomial forms of even degrees. As examples, the Lanford system, the jerky system with one quadratic monomial and a quartically perturbed harmonic oscillator are considered.

Localization of periodic orbits of polynomial vector fields of even degree by linear functions

International Nuclear Information System (INIS)

Starkov, Konstantin E.

2005-01-01

This paper is concerned with the localization problem of periodic orbits of polynomial vector fields of even degree by using linear functions. Conditions of the localization of all periodic orbits in sets of a simple structure are obtained. Our results are based on the solution of the conditional extremum problem and the application of homogeneous polynomial forms of even degrees. As examples, the Lanford system, the jerky system with one quadratic monomial and a quartically perturbed harmonic oscillator are considered

Discriminants and functional equations for polynomials orthogonal on the unit circle

International Nuclear Information System (INIS)

Ismail, M.E.H.; Witte, N.S.

2000-01-01

We derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and q-difference equations for these polynomials. A general functional equation is found which allows one to relate the zeros of the orthogonal polynomials to the stationary values of an explicit quasi-energy and implies recurrences on the orthogonal polynomial coefficients. We also evaluate the discriminants and quantized discriminants of polynomials orthogonal on the unit circle

On the Lorentz degree of a product of polynomials

KAUST Repository

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.

Strong result for real zeros of random algebraic polynomials

Directory of Open Access Journals (Sweden)

T. Uno

2001-01-01

Full Text Available An estimate is given for the lower bound of real zeros of random algebraic polynomials whose coefficients are non-identically distributed dependent Gaussian random variables. Moreover, our estimated measure of the exceptional set, which is independent of the degree of the polynomials, tends to zero as the degree of the polynomial tends to infinity.

Large degree asymptotics of generalized Bessel polynomials

NARCIS (Netherlands)

J.L. López; N.M. Temme (Nico)

2011-01-01

textabstractAsymptotic expansions are given for large values of $n$ of the generalized Bessel polynomials $Y_n^\\mu(z)$. The analysis is based on integrals that follow from the generating functions of the polynomials. A new simple expansion is given that is valid outside a compact neighborhood of the

Study of solving a Toda dynamic system with loop algebra

International Nuclear Information System (INIS)

Zhu Qiao; Yang Zhanying; Shi Kangjie; Wen Junqing

2006-01-01

The authors construct a Toda system with Loop algebra, and prove that the Lax equation L=[L,M] can be solved by means of solving a regular Riemann-Hilbert problem. In our system, M in Lax pair is an antisymmetrical matrix, while L=L + + M, and L + is a quasi-upper triangular matrix of loop algebra. In order to check our result, the authors exactly solve an R-H problem under a given initial condition as an example. (authors)

A new Identity Based Encryption (IBE) scheme using extended Chebyshev polynomial over finite fields Zp

International Nuclear Information System (INIS)

Benasser Algehawi, Mohammed; Samsudin, Azman

2010-01-01

We present a method to extract key pairs needed for the Identity Based Encryption (IBE) scheme from extended Chebyshev polynomial over finite fields Z p . Our proposed scheme relies on the hard problem and the bilinear property of the extended Chebyshev polynomial over Z p . The proposed system is applicable, secure, and reliable.

Linear operator pencils on Lie algebras and Laurent biorthogonal polynomials

International Nuclear Information System (INIS)

Gruenbaum, F A; Vinet, Luc; Zhedanov, Alexei

2004-01-01

We study operator pencils on generators of the Lie algebras sl 2 and the oscillator algebra. These pencils are linear in a spectral parameter λ. The corresponding generalized eigenvalue problem gives rise to some sets of orthogonal polynomials and Laurent biorthogonal polynomials (LBP) expressed in terms of the Gauss 2 F 1 and degenerate 1 F 1 hypergeometric functions. For special choices of the parameters of the pencils, we identify the resulting polynomials with the Hendriksen-van Rossum LBP which are widely believed to be the biorthogonal analogues of the classical orthogonal polynomials. This places these examples under the umbrella of the generalized bispectral problem which is considered here. Other (non-bispectral) cases give rise to some 'nonclassical' orthogonal polynomials including Tricomi-Carlitz and random-walk polynomials. An application to solutions of relativistic Toda chain is considered

Higher order branching of periodic orbits from polynomial isochrones

Directory of Open Access Journals (Sweden)

B. Toni

1999-09-01

Full Text Available We discuss the higher order local bifurcations of limit cycles from polynomial isochrones (linearizable centers when the linearizing transformation is explicitly known and yields a polynomial perturbation one-form. Using a method based on the relative cohomology decomposition of polynomial one-forms complemented with a step reduction process, we give an explicit formula for the overall upper bound of branch points of limit cycles in an arbitrary $n$ degree polynomial perturbation of the linear isochrone, and provide an algorithmic procedure to compute the upper bound at successive orders. We derive a complete analysis of the nonlinear cubic Hamiltonian isochrone and show that at most nine branch points of limit cycles can bifurcate in a cubic polynomial perturbation. Moreover, perturbations with exactly two, three, four, six, and nine local families of limit cycles may be constructed.

Insight and analysis problem solving in microbes to machines.

Science.gov (United States)

Clark, Kevin B

2015-11-01

A key feature for obtaining solutions to difficult problems, insight is oftentimes vaguely regarded as a special discontinuous intellectual process and/or a cognitive restructuring of problem representation or goal approach. However, this nearly century-old state of art devised by the Gestalt tradition to explain the non-analytical or non-trial-and-error, goal-seeking aptitude of primate mentality tends to neglect problem-solving capabilities of lower animal phyla, Kingdoms other than Animalia, and advancing smart computational technologies built from biological, artificial, and composite media. Attempting to provide an inclusive, precise definition of insight, two major criteria of insight, discontinuous processing and problem restructuring, are here reframed using terminology and statistical mechanical properties of computational complexity classes. Discontinuous processing becomes abrupt state transitions in algorithmic/heuristic outcomes or in types of algorithms/heuristics executed by agents using classical and/or quantum computational models. And problem restructuring becomes combinatorial reorganization of resources, problem-type substitution, and/or exchange of computational models. With insight bounded by computational complexity, humans, ciliated protozoa, and complex technological networks, for example, show insight when restructuring time requirements, combinatorial complexity, and problem type to solve polynomial and nondeterministic polynomial decision problems. Similar effects are expected from other problem types, supporting the idea that insight might be an epiphenomenon of analytical problem solving and consequently a larger information processing framework. Thus, this computational complexity definition of insight improves the power, external and internal validity, and reliability of operational parameters with which to classify, investigate, and produce the phenomenon for computational agents ranging from microbes to man-made devices. Copyright

On the estimation of the degree of regression polynomial

International Nuclear Information System (INIS)

Toeroek, Cs.

1997-01-01

The mathematical functions most commonly used to model curvature in plots are polynomials. Generally, the higher the degree of the polynomial, the more complex is the trend that its graph can represent. We propose a new statistical-graphical approach based on the discrete projective transformation (DPT) to estimating the degree of polynomial that adequately describes the trend in the plot

On Some Extensions of Szasz Operators Including Boas-Buck-Type Polynomials

Directory of Open Access Journals (Sweden)

Sezgin Sucu

2012-01-01

Full Text Available This paper is concerned with a new sequence of linear positive operators which generalize Szasz operators including Boas-Buck-type polynomials. We establish a convergence theorem for these operators and give the quantitative estimation of the approximation process by using a classical approach and the second modulus of continuity. Some explicit examples of our operators involving Laguerre polynomials, Charlier polynomials, and Gould-Hopper polynomials are given. Moreover, a Voronovskaya-type result is obtained for the operators containing Gould-Hopper polynomials.

Generalized neurofuzzy network modeling algorithms using Bézier-Bernstein polynomial functions and additive decomposition.

Science.gov (United States)

Hong, X; Harris, C J

2000-01-01

This paper introduces a new neurofuzzy model construction algorithm for nonlinear dynamic systems based upon basis functions that are Bézier-Bernstein polynomial functions. This paper is generalized in that it copes with n-dimensional inputs by utilising an additive decomposition construction to overcome the curse of dimensionality associated with high n. This new construction algorithm also introduces univariate Bézier-Bernstein polynomial functions for the completeness of the generalized procedure. Like the B-spline expansion based neurofuzzy systems, Bézier-Bernstein polynomial function based neurofuzzy networks hold desirable properties such as nonnegativity of the basis functions, unity of support, and interpretability of basis function as fuzzy membership functions, moreover with the additional advantages of structural parsimony and Delaunay input space partition, essentially overcoming the curse of dimensionality associated with conventional fuzzy and RBF networks. This new modeling network is based on additive decomposition approach together with two separate basis function formation approaches for both univariate and bivariate Bézier-Bernstein polynomial functions used in model construction. The overall network weights are then learnt using conventional least squares methods. Numerical examples are included to demonstrate the effectiveness of this new data based modeling approach.

Quantum groups, orthogonal polynomials and applications to some dynamical systems; Groupes quantiques, polynomes orthogonaux et applications a quelques systemes dynamiques

Energy Technology Data Exchange (ETDEWEB)

Campigotto, C

1993-12-01

The first part is concerned with the introduction of quantum groups as an extension of Lie groups. In particular, we study the case of unitary enveloping algebras in dimension 2. We then connect the quantum group formalism to the construction of g CGC recurrent relations. In addition, we construct g-deformed Krawtchouck and Meixner orthogonal polynomials and list their respective main characteristics. The second part deals with some dynamical systems from a classical, a quantum and a gp-analogue point of view. We investigate the Coulomb Kepler system by using the canonical namical systems which contain as special cases some interesting systems for nuclear of atomic physics and for quantum chemistry, such as the Hartmann system, the ring-shaped oscillator, the Smarodinsky-Winternitz system, the Aharonov-Bohen system and the dyania of Dirac and Schroedinger. (author). 291 refs.

On associated polynomials and decay rates for birth-death processes

NARCIS (Netherlands)

van Doorn, Erik A.

2001-01-01

We consider sequences of orthogonal polynomials and pursue the question of how (partial) knowledge of the orthogonalizing measure for the {\\it associated polynomials} can lead to information about the orthogonalizing measure for the original polynomials. In particular, we relate the supports of the

On associated polynomials and decay rates for birth-death processes

NARCIS (Netherlands)

van Doorn, Erik A.

2003-01-01

We consider sequences of orthogonal polynomials and pursue the question of how (partial) knowledge of the orthogonalizing measure for the associated polynomials can lead to information about the orthogonalizing measure for the original polynomials. In particular, we relate the supports of the two

Quantized vortices in the ideal bose gas: a physical realization of random polynomials.

Science.gov (United States)

Castin, Yvan; Hadzibabic, Zoran; Stock, Sabine; Dalibard, Jean; Stringari, Sandro

2006-02-03

We propose a physical system allowing one to experimentally observe the distribution of the complex zeros of a random polynomial. We consider a degenerate, rotating, quasi-ideal atomic Bose gas prepared in the lowest Landau level. Thermal fluctuations provide the randomness of the bosonic field and of the locations of the vortex cores. These vortices can be mapped to zeros of random polynomials, and observed in the density profile of the gas.

Statistics of stationary points of random finite polynomial potentials

International Nuclear Information System (INIS)

Mehta, Dhagash; Niemerg, Matthew; Sun, Chuang

2015-01-01

The stationary points (SPs) of the potential energy landscapes (PELs) of multivariate random potentials (RPs) have found many applications in many areas of Physics, Chemistry and Mathematical Biology. However, there are few reliable methods available which can find all the SPs accurately. Hence, one has to rely on indirect methods such as Random Matrix theory. With a combination of the numerical polynomial homotopy continuation method and a certification method, we obtain all the certified SPs of the most general polynomial RP for each sample chosen from the Gaussian distribution with mean 0 and variance 1. While obtaining many novel results for the finite size case of the RP, we also discuss the implications of our results on mathematics of random systems and string theory landscapes. (paper)

Some Polynomials Associated with the r-Whitney Numbers

Indian Academy of Sciences (India)

26

Abstract. In the present article we study three families of polynomials associated with ... [29, 39] for their relations with the Bernoulli and generalized Bernoulli polynomials and ... generating functions in a similar way as in the classical cases.

The Bessel polynomials and their differential operators

International Nuclear Information System (INIS)

Onyango Otieno, V.P.

1987-10-01

Differential operators associated with the ordinary and the generalized Bessel polynomials are defined. In each case the commutator bracket is constructed and shows that the differential operators associated with the Bessel polynomials and their generalized form are not commutative. Some applications of these operators to linear differential equations are also discussed. (author). 4 refs

A new VLSI complex integer multiplier which uses a quadratic-polynomial residue system with Fermat numbers

Science.gov (United States)

Shyu, H. C.; Reed, I. S.; Truong, T. K.; Hsu, I. S.; Chang, J. J.

1987-01-01

A quadratic-polynomial Fermat residue number system (QFNS) has been used to compute complex integer multiplications. The advantage of such a QFNS is that a complex integer multiplication requires only two integer multiplications. In this article, a new type Fermat number multiplier is developed which eliminates the initialization condition of the previous method. It is shown that the new complex multiplier can be implemented on a single VLSI chip. Such a chip is designed and fabricated in CMOS-Pw technology.

X-ray multiaxial stress analysis by means of polynomial approximation and an application to plane stress problem

International Nuclear Information System (INIS)

Yoshioka, Yasuo; Sasaki, Toshihiko; Kuramoto, Makoto.

1984-01-01

A new polynomial approximation method was proposed for the X-ray multiaxial stress analysis, in which the effect of stress gradient along the penetration depth of X-rays was taken into account. Three basic assumptions were made; (1) the stress gradient is linear in respect to the depth from the specimen surface, (2) the ponetration depth of X-rays is a function of Sin 2 phi and (3) the strain measured by X-rays corresponds to the weighted average strain on the intensity of the diffracted X-rays. Consequently, the stress state within the thin layer near the surface was expressed by making use of three surface stresses and six stress gradients in the present method. The average strains by X-rays were approximated by the third order polynomial equations of sin 2 phi using a least square method at several phi angles on the coordinate system of specimen. Since the coefficients of these polynomials include these nine stress components mentioned above, it is possible to solve them as simultaneous equations. The calculating process of this method is simpler than that of the integral method. An X-ray plane stress problem was analyzed as an application of the present method, and the residual stress distribution on a shot-peened steel plate was actually measured by use of Cr-Kα X-rays to verify the analysis. The result showed that the compressive residual stress near the surface determined by the present method was smaller than the weighted average stress by the Sin 2 phi method because of the steep stress gradient. The present method is useful to obtain a reasonable value of stress for such a specimen with steep stress gradients near the surface. (author)

Conference on Commutative rings, integer-valued polynomials and polynomial functions

CERN Document Server

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...

A software package to construct polynomial sets over Z2 for determining the output of quantum computations

International Nuclear Information System (INIS)

Gerdt, Vladimir P.; Severyanov, Vasily M.

2006-01-01

A C package is presented that allows a user for an input quantum circuit to generate a set of multivariate polynomials over the finite field Z 2 whose total number of solutions in Z 2 determines the output of the quantum computation defined by the circuit. The generated polynomial system can further be converted to the canonical Grobner basis form which provides a universal algorithmic tool for counting the number of common roots of the polynomials

Integrable couplings of relativistic Toda lattice systems in polynomial form and rational form, their hierarchies and bi-Hamiltonian structures

Energy Technology Data Exchange (ETDEWEB)

Xu Xixiang [College of Science, Shandong University of Science and Technology, Qingdao 266510 (China)], E-mail: xixiang_xu@yahoo.com.cn

2009-10-02

Integrable couplings of relativistic Toda lattice systems in polynomial form and rational form, and their hierarchies, are derived from a four-by-four discrete matrix eigenvalue problem. The bi-Hamiltonian structure for every integrable coupling in the two hierarchies obtained is established by means of the discrete variational identity. Ultimately, Liouvolle integrability of the obtained integrable couplings is demonstrated.

Experimental quantum computing to solve systems of linear equations.

Science.gov (United States)

Cai, X-D; Weedbrook, C; Su, Z-E; Chen, M-C; Gu, Mile; Zhu, M-J; Li, Li; Liu, Nai-Le; Lu, Chao-Yang; Pan, Jian-Wei

2013-06-07

Solving linear systems of equations is ubiquitous in all areas of science and engineering. With rapidly growing data sets, such a task can be intractable for classical computers, as the best known classical algorithms require a time proportional to the number of variables N. A recently proposed quantum algorithm shows that quantum computers could solve linear systems in a time scale of order log(N), giving an exponential speedup over classical computers. Here we realize the simplest instance of this algorithm, solving 2×2 linear equations for various input vectors on a quantum computer. We use four quantum bits and four controlled logic gates to implement every subroutine required, demonstrating the working principle of this algorithm.

Solving the 0/1 Knapsack Problem by a Biomolecular DNA Computer

Directory of Open Access Journals (Sweden)

Hassan Taghipour

2013-01-01

Full Text Available Solving some mathematical problems such as NP-complete problems by conventional silicon-based computers is problematic and takes so long time. DNA computing is an alternative method of computing which uses DNA molecules for computing purposes. DNA computers have massive degrees of parallel processing capability. The massive parallel processing characteristic of DNA computers is of particular interest in solving NP-complete and hard combinatorial problems. NP-complete problems such as knapsack problem and other hard combinatorial problems can be easily solved by DNA computers in a very short period of time comparing to conventional silicon-based computers. Sticker-based DNA computing is one of the methods of DNA computing. In this paper, the sticker based DNA computing was used for solving the 0/1 knapsack problem. At first, a biomolecular solution space was constructed by using appropriate DNA memory complexes. Then, by the application of a sticker-based parallel algorithm using biological operations, knapsack problem was resolved in polynomial time.

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 ...

Dual exponential polynomials and linear differential equations

Science.gov (United States)

Wen, Zhi-Tao; Gundersen, Gary G.; Heittokangas, Janne

2018-01-01

We study linear differential equations with exponential polynomial coefficients, where exactly one coefficient is of order greater than all the others. The main result shows that a nontrivial exponential polynomial solution of such an equation has a certain dual relationship with the maximum order coefficient. Several examples illustrate our results and exhibit possibilities that can occur.

A semi-analytical iterative technique for solving chemistry problems

Directory of Open Access Journals (Sweden)

Majeed Ahmed AL-Jawary

2017-07-01

Full Text Available The main aim and contribution of the current paper is to implement a semi-analytical iterative method suggested by Temimi and Ansari in 2011 namely (TAM to solve two chemical problems. An approximate solution obtained by the TAM provides fast convergence. The current chemical problems are the absorption of carbon dioxide into phenyl glycidyl ether and the other system is a chemical kinetics problem. These problems are represented by systems of nonlinear ordinary differential equations that contain boundary conditions and initial conditions. Error analysis of the approximate solutions is studied using the error remainder and the maximal error remainder. Exponential rate for the convergence is observed. For both problems the results of the TAM are compared with other results obtained by previous methods available in the literature. The results demonstrate that the method has many merits such as being derivative-free, and overcoming the difficulty arising in calculating Adomian polynomials to handle the non-linear terms in Adomian Decomposition Method (ADM. It does not require to calculate Lagrange multiplier in Variational Iteration Method (VIM in which the terms of the sequence become complex after several iterations, thus, analytical evaluation of terms becomes very difficult or impossible in VIM. No need to construct a homotopy in Homotopy Perturbation Method (HPM and solve the corresponding algebraic equations. The MATHEMATICA® 9 software was used to evaluate terms in the iterative process.

Generalized Freud's equation and level densities with polynomial

Indian Academy of Sciences (India)

Home; Journals; Pramana – Journal of Physics; Volume 81; Issue 2. Generalized Freud's equation and level densities with polynomial potential. Akshat Boobna Saugata Ghosh. Research Articles Volume 81 ... Keywords. Orthogonal polynomial; Freud's equation; Dyson–Mehta method; methods of resolvents; level density.

Method for calculating anisotropic neutron transport using scattering kernel without polynomial expansion

International Nuclear Information System (INIS)

Takahashi, Akito; Yamamoto, Junji; Ebisuya, Mituo; Sumita, Kenji

1979-01-01

A new method for calculating the anisotropic neutron transport is proposed for the angular spectral analysis of D-T fusion reactor neutronics. The method is based on the transport equation with new type of anisotropic scattering kernels formulated by a single function I sub(i) (μ', μ) instead of polynomial expansion, for instance, Legendre polynomials. In the calculation of angular flux spectra by using scattering kernels with the Legendre polynomial expansion, we often observe the oscillation with negative flux. But in principle this oscillation disappears by this new method. In this work, we discussed anisotropic scattering kernels of the elastic scattering and the inelastic scatterings which excite discrete energy levels. The other scatterings were included in isotropic scattering kernels. An approximation method, with use of the first collision source written by the I sub(i) (μ', μ) function, was introduced to attenuate the ''oscillations'' when we are obliged to use the scattering kernels with the Legendre polynomial expansion. Calculated results with this approximation showed remarkable improvement for the analysis of the angular flux spectra in a slab system of lithium metal with the D-T neutron source. (author)

A note on the zeros of Freud-Sobolev orthogonal polynomials

Science.gov (United States)

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.

Two polynomial representations of experimental design

OpenAIRE

Notari, Roberto; Riccomagno, Eva; Rogantin, Maria-Piera

2007-01-01

In the context of algebraic statistics an experimental design is described by a set of polynomials called the design ideal. This, in turn, is generated by finite sets of polynomials. Two types of generating sets are mostly used in the literature: Groebner bases and indicator functions. We briefly describe them both, how they are used in the analysis and planning of a design and how to switch between them. Examples include fractions of full factorial designs and designs for mixture experiments.

Stable piecewise polynomial vector fields

Directory of Open Access Journals (Sweden)

Claudio Pessoa

2012-09-01

Full Text Available Let $N={y>0}$ and $S={y<0}$ be the semi-planes of $mathbb{R}^2$ having as common boundary the line $D={y=0}$. Let $X$ and $Y$ be polynomial vector fields defined in $N$ and $S$, respectively, leading to a discontinuous piecewise polynomial vector field $Z=(X,Y$. This work pursues the stability and the transition analysis of solutions of $Z$ between $N$ and $S$, started by Filippov (1988 and Kozlova (1984 and reformulated by Sotomayor-Teixeira (1995 in terms of the regularization method. This method consists in analyzing a one parameter family of continuous vector fields $Z_{epsilon}$, defined by averaging $X$ and $Y$. This family approaches $Z$ when the parameter goes to zero. The results of Sotomayor-Teixeira and Sotomayor-Machado (2002 providing conditions on $(X,Y$ for the regularized vector fields to be structurally stable on planar compact connected regions are extended to discontinuous piecewise polynomial vector fields on $mathbb{R}^2$. Pertinent genericity results for vector fields satisfying the above stability conditions are also extended to the present case. A procedure for the study of discontinuous piecewise vector fields at infinity through a compactification is proposed here.

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

Hermite Functional Link Neural Network for Solving the Van der Pol-Duffing Oscillator Equation.

Science.gov (United States)

Mall, Susmita; Chakraverty, S

2016-08-01

Hermite polynomial-based functional link artificial neural network (FLANN) is proposed here to solve the Van der Pol-Duffing oscillator equation. A single-layer hermite neural network (HeNN) model is used, where a hidden layer is replaced by expansion block of input pattern using Hermite orthogonal polynomials. A feedforward neural network model with the unsupervised error backpropagation principle is used for modifying the network parameters and minimizing the computed error function. The Van der Pol-Duffing and Duffing oscillator equations may not be solved exactly. Here, approximate solutions of these types of equations have been obtained by applying the HeNN model for the first time. Three mathematical example problems and two real-life application problems of Van der Pol-Duffing oscillator equation, extracting the features of early mechanical failure signal and weak signal detection problems, are solved using the proposed HeNN method. HeNN approximate solutions have been compared with results obtained by the well known Runge-Kutta method. Computed results are depicted in term of graphs. After training the HeNN model, we may use it as a black box to get numerical results at any arbitrary point in the domain. Thus, the proposed HeNN method is efficient. The results reveal that this method is reliable and can be applied to other nonlinear problems too.

Computing Galois Groups of Eisenstein Polynomials Over P-adic Fields

Science.gov (United States)

Milstead, Jonathan

The most efficient algorithms for computing Galois groups of polynomials over global fields are based on Stauduhar's relative resolvent method. These methods are not directly generalizable to the local field case, since they require a field that contains the global field in which all roots of the polynomial can be approximated. We present splitting field-independent methods for computing the Galois group of an Eisenstein polynomial over a p-adic field. Our approach is to combine information from different disciplines. We primarily, make use of the ramification polygon of the polynomial, which is the Newton polygon of a related polynomial. This allows us to quickly calculate several invariants that serve to reduce the number of possible Galois groups. Algorithms by Greve and Pauli very efficiently return the Galois group of polynomials where the ramification polygon consists of one segment as well as information about the subfields of the stem field. Second, we look at the factorization of linear absolute resolvents to further narrow the pool of possible groups.

The theory of contractions of 2D 2nd order quantum superintegrable systems and its relation to the Askey scheme for hypergeometric orthogonal polynomials

International Nuclear Information System (INIS)

Miller, Willard Jr

2014-01-01

We describe a contraction theory for 2nd order superintegrable systems, showing that all such systems in 2 dimensions are limiting cases of a single system: the generic 3-parameter potential on the 2-sphere, S9 in our listing. Analogously, all of the quadratic symmetry algebras of these systems can be obtained by a sequence of contractions starting from S9. By contracting function space realizations of irreducible representations of the S9 algebra (which give the structure equations for Racah/Wilson polynomials) to the other superintegrable systems one obtains the full Askey scheme of orthogonal hypergeometric polynomials.This relates the scheme directly to explicitly solvable quantum mechanical systems. Amazingly, all of these contractions of superintegrable systems with potential are uniquely induced by Wigner Lie algebra contractions of so(3, C) and e(2, C). The present paper concentrates on describing this intimate link between Lie algebra and superintegrable system contractions, with the detailed calculations presented elsewhere. Joint work with E. Kalnins, S. Post, E. Subag and R. Heinonen.

Guts of surfaces and the colored Jones polynomial

CERN Document Server

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...

Computing Tutte polynomials of contact networks in classrooms

Science.gov (United States)

Hincapié, Doracelly; Ospina, Juan

2013-05-01

Objective: The topological complexity of contact networks in classrooms and the potential transmission of an infectious disease were analyzed by sex and age. Methods: The Tutte polynomials, some topological properties and the number of spanning trees were used to algebraically compute the topological complexity. Computations were made with the Maple package GraphTheory. Published data of mutually reported social contacts within a classroom taken from primary school, consisting of children in the age ranges of 4-5, 7-8 and 10-11, were used. Results: The algebraic complexity of the Tutte polynomial and the probability of disease transmission increases with age. The contact networks are not bipartite graphs, gender segregation was observed especially in younger children. Conclusion: Tutte polynomials are tools to understand the topology of the contact networks and to derive numerical indexes of such topologies. It is possible to establish relationships between the Tutte polynomial of a given contact network and the potential transmission of an infectious disease within such network

Polynomial model inversion control: numerical tests and applications

OpenAIRE

Novara, Carlo

2015-01-01

A novel control design approach for general nonlinear systems is described in this paper. The approach is based on the identification of a polynomial model of the system to control and on the on-line inversion of this model. Extensive simulations are carried out to test the numerical efficiency of the approach. Numerical examples of applicative interest are presented, concerned with control of the Duffing oscillator, control of a robot manipulator and insulin regulation in a type 1 diabetic p...

Evaluating the Performance of Polynomial Regression Method with Different Parameters during Color Characterization

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.

Exponential time paradigms through the polynomial time lens

NARCIS (Netherlands)

Drucker, A.; Nederlof, J.; Santhanam, R.; Sankowski, P.; Zaroliagis, C.

2016-01-01

We propose a general approach to modelling algorithmic paradigms for the exact solution of NP-hard problems. Our approach is based on polynomial time reductions to succinct versions of problems solvable in polynomial time. We use this viewpoint to explore and compare the power of paradigms such as

Root and Critical Point Behaviors of Certain Sums of Polynomials

Indian Academy of Sciences (India)

13

There is an extensive literature concerning roots of sums of polynomials. Many papers and books([5], [6],. [7]) have written about these polynomials. Perhaps the most immediate question of sums of polynomials,. A + B = C, is “given bounds for the roots of A and B, what bounds can be given for the roots of C?” By. Fell [3], if ...

Predicting physical time series using dynamic ridge polynomial neural networks.

Directory of Open Access Journals (Sweden)

Dhiya Al-Jumeily

Full Text Available 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.

Generalized Bilinear Differential Operators, Binary Bell Polynomials, and Exact Periodic Wave Solution of Boiti-Leon-Manna-Pempinelli Equation

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.

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....

Design of a polynomial ring based symmetric homomorphic encryption scheme

Directory of Open Access Journals (Sweden)

Smaranika Dasgupta

2016-09-01

Full Text Available Security of data, especially in clouds, has become immensely essential for present-day applications. Fully homomorphic encryption (FHE is a great way to secure data which is used and manipulated by untrusted applications or systems. In this paper, we propose a symmetric FHE scheme based on polynomial over ring of integers. This scheme is somewhat homomorphic due to accumulation of noise after few operations, which is made fully homomorphic using a refresh procedure. After certain amount of homomorphic computations, large ciphertexts are refreshed for proper decryption. The hardness of the scheme is based on the difficulty of factorizing large integers. Also, it requires polynomial addition which is computationally cost effective. Experimental results are shown to support our claim.

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.

Large N Penner matrix model and a novel asymptotic formula for the generalized Laguerre polynomials

International Nuclear Information System (INIS)

Deo, N

2003-01-01

The Gaussian Penner matrix model is re-examined in the light of the results which have been found in double-well matrix models. The orthogonal polynomials for the Gaussian Penner model are shown to be the generalized Laguerre polynomials L (α) n (x) with α and x depending on N, the size of the matrix. An asymptotic formula for the orthogonal polynomials is derived following closely the orthogonal polynomial method of Deo (1997 Nucl. Phys. B 504 609). The universality found in the double-well matrix model is extended to include non-polynomial potentials. An asymptotic formula is also found for the Laguerre polynomial using the saddle-point method by rescaling α and x with N. Combining these results a novel asymptotic formula is found for the generalized Laguerre polynomials (different from that given in Szego's book) in a different asymptotic regime. This may have applications in mathematical and physical problems in the future. The density-density correlators are derived and are the same as those found for the double-well matrix models. These correlators in the smoothed large N limit are sensitive to odd and even N where N is the size of the matrix. These results for the two-point density-density correlation function may be useful in finding eigenvalue effects in experiments in mesoscopic systems or small metallic grains. There may be applications to string theory as well as the tunnelling of an eigenvalue from one valley to the other being an important quantity there

SOLUTION OF SINGULAR INTEGRAL EQUATION FOR ELASTICITY THEORY WITH THE HELP OF ASYMPTOTIC POLYNOMIAL FUNCTION

Directory of Open Access Journals (Sweden)

V. P. Gribkova

2014-01-01

Full Text Available The paper offers a new method for approximate solution of one type of singular integral equations for elasticity theory which have been studied by other authors. The approximate solution is found in the form of asymptotic polynomial function of a low degree (first approximation based on the Chebyshev second order polynomial. Other authors have obtained a solution (only in separate points using a method of mechanical quadrature  and though they used also the Chebyshev polynomial of the second order they applied another system of junctures which were used for the creation of the required formulas.The suggested method allows not only to find an approximate solution for the whole interval in the form of polynomial, but it also makes it possible to obtain a remainder term in the form of infinite expansion where coefficients are linear functional of the given integral equation and basis functions are the Chebyshev polynomial of the second order. Such presentation of the remainder term of the first approximation permits to find a summand of the infinite series, which will serve as a start for fulfilling the given solution accuracy. This number is a degree of the asymptotic polynomial (second approximation, which will give the approximation to the exact solution with the given accuracy. The examined polynomial functions tend asymptotically to the polynomial of the best uniform approximation in the space C, created for the given operator.The paper demonstrates a convergence of the approximate solution to the exact one and provides an error estimation. The proposed algorithm for obtaining of the approximate solution and error estimation is easily realized with the help of computing technique and does not require considerable preliminary preparation during programming.

Positive polynomials and robust stabilization with fixed-order controllers

Czech Academy of Sciences Publication Activity Database

Henrion, Didier; Šebek, M.; Kučera, V.

2003-01-01

Roč. 48, č. 7 (2003), s. 1178-1186 ISSN 0018-9286 R&D Projects: GA ČR GA102/02/0709; GA MŠk ME 496 Institutional research plan: CEZ:AV0Z1075907 Keywords : fixed-order control lers * linear matrix inequality * polynomials, robust control Subject RIV: BC - Control Systems Theory Impact factor: 1.896, year: 2003

Multiparameter extrapolation and deflation methods for solving equation systems

Directory of Open Access Journals (Sweden)

A. J. Hughes Hallett

1984-01-01

Full Text Available Most models in economics and the applied sciences are solved by first order iterative techniques, usually those based on the Gauss-Seidel algorithm. This paper examines the convergence of multiparameter extrapolations (accelerations of first order iterations, as an improved approximation to the Newton method for solving arbitrary nonlinear equation systems. It generalises my earlier results on single parameter extrapolations. Richardson's generalised method and the deflation method for detecting successive solutions in nonlinear equation systems are also presented as multiparameter extrapolations of first order iterations. New convergence results are obtained for those methods.

Bernoulli numbers and polynomials from a more general point of view

International Nuclear Information System (INIS)

Dattoli, G.; Cesarano, C.; Lorenzutta, S.

2000-01-01

In this work it is applied the method of generating function, to introduce new forms of Bernoulli numbers and polynomials, which are exploited to derive further classes of partial sums involving generalized many index many variable polynomials. Analogous considerations are developed for the Euler numbers and polynomials [it

Generalizations of an integral for Legendre polynomials by Persson and Strang

NARCIS (Netherlands)

Diekema, E.; Koornwinder, T.H.

2012-01-01

Persson and Strang (2003) evaluated the integral over [−1,1] of a squared odd degree Legendre polynomial divided by x2 as being equal to 2. We consider a similar integral for orthogonal polynomials with respect to a general even orthogonality measure, with Gegenbauer and Hermite polynomials as

Eye aberration analysis with Zernike polynomials

Science.gov (United States)

Molebny, Vasyl V.; Chyzh, Igor H.; Sokurenko, Vyacheslav M.; Pallikaris, Ioannis G.; Naoumidis, Leonidas P.

1998-06-01

New horizons for accurate photorefractive sight correction, afforded by novel flying spot technologies, require adequate measurements of photorefractive properties of an eye. Proposed techniques of eye refraction mapping present results of measurements for finite number of points of eye aperture, requiring to approximate these data by 3D surface. A technique of wave front approximation with Zernike polynomials is described, using optimization of the number of polynomial coefficients. Criterion of optimization is the nearest proximity of the resulted continuous surface to the values calculated for given discrete points. Methodology includes statistical evaluation of minimal root mean square deviation (RMSD) of transverse aberrations, in particular, varying consecutively the values of maximal coefficient indices of Zernike polynomials, recalculating the coefficients, and computing the value of RMSD. Optimization is finished at minimal value of RMSD. Formulas are given for computing ametropia, size of the spot of light on retina, caused by spherical aberration, coma, and astigmatism. Results are illustrated by experimental data, that could be of interest for other applications, where detailed evaluation of eye parameters is needed.

Animating Nested Taylor Polynomials to Approximate a Function

Science.gov (United States)

Mazzone, Eric F.; Piper, Bruce R.

2010-01-01

The way that Taylor polynomials approximate functions can be demonstrated by moving the center point while keeping the degree fixed. These animations are particularly nice when the Taylor polynomials do not intersect and form a nested family. We prove a result that shows when this nesting occurs. The animations can be shown in class or…

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...

Discovering Steiner Triple Systems through Problem Solving

Science.gov (United States)

Sriraman, Bharath

2004-01-01

An attempt to implement problem solving as a teacher of ninth grade algebra is described. The problems selected were not general ones, they involved combinations and represented various situations and were more complex which lead to the discovery of Steiner triple systems.

Differential recurrence formulae for orthogonal polynomials

Directory of Open Access Journals (Sweden)

Anton L. W. von Bachhaus

1995-11-01

Full Text Available Part I - By combining a general 2nd-order linear homogeneous ordinary differential equation with the three-term recurrence relation possessed by all orthogonal polynomials, it is shown that sequences of orthogonal polynomials which satisfy a differential equation of the above mentioned type necessarily have a differentiation formula of the type: gn(xY'n(x=fn(xYn(x+Yn-1(x. Part II - A recurrence formula of the form: rn(xY'n(x+sn(xY'n+1(x+tn(xY'n-1(x=0, is derived using the result of Part I.

New approach to solve symmetric fully fuzzy linear systems

Indian Academy of Sciences (India)

concepts of fuzzy set theory and then define a fully fuzzy linear system of equations. .... To represent the above problem as fully fuzzy linear system, we represent x .... Fully fuzzy linear systems can be solved by Linear programming approach, ...

Workflow Agents vs. Expert Systems: Problem Solving Methods in Work Systems Design

Science.gov (United States)

Clancey, William J.; Sierhuis, Maarten; Seah, Chin

2009-01-01

During the 1980s, a community of artificial intelligence researchers became interested in formalizing problem solving methods as part of an effort called "second generation expert systems" (2nd GES). How do the motivations and results of this research relate to building tools for the workplace today? We provide an historical review of how the theory of expertise has developed, a progress report on a tool for designing and implementing model-based automation (Brahms), and a concrete example how we apply 2nd GES concepts today in an agent-based system for space flight operations (OCAMS). Brahms incorporates an ontology for modeling work practices, what people are doing in the course of a day, characterized as "activities." OCAMS was developed using a simulation-to-implementation methodology, in which a prototype tool was embedded in a simulation of future work practices. OCAMS uses model-based methods to interactively plan its actions and keep track of the work to be done. The problem solving methods of practice are interactive, employing reasoning for and through action in the real world. Analogously, it is as if a medical expert system were charged not just with interpreting culture results, but actually interacting with a patient. Our perspective shifts from building a "problem solving" (expert) system to building an actor in the world. The reusable components in work system designs include entire "problem solvers" (e.g., a planning subsystem), interoperability frameworks, and workflow agents that use and revise models dynamically in a network of people and tools. Consequently, the research focus shifts so "problem solving methods" include ways of knowing that models do not fit the world, and ways of interacting with other agents and people to gain or verify information and (ultimately) adapt rules and procedures to resolve problematic situations.

Considering a non-polynomial basis for local kernel regression problem

Science.gov (United States)

Silalahi, Divo Dharma; Midi, Habshah

2017-01-01

A common used as solution for local kernel nonparametric regression problem is given using polynomial regression. In this study, we demonstrated the estimator and properties using maximum likelihood estimator for a non-polynomial basis such B-spline to replacing the polynomial basis. This estimator allows for flexibility in the selection of a bandwidth and a knot. The best estimator was selected by finding an optimal bandwidth and knot through minimizing the famous generalized validation function.

Application of Homotopy Analysis Method to Solve Relativistic Toda Lattice System

International Nuclear Information System (INIS)

Wang Qi

2010-01-01

In this letter, the homotopy analysis method is successfully applied to solve the Relativistic Toda lattice system. Comparisons are made between the results of the proposed method and exact solutions. Analysis results show that homotopy analysis method is a powerful and easy-to-use analytic tool to solve systems of differential-difference equations. (general)

Quantitative Boltzmann-Gibbs Principles via Orthogonal Polynomial Duality

Science.gov (United States)

Ayala, Mario; Carinci, Gioia; Redig, Frank

2018-06-01

We study fluctuation fields of orthogonal polynomials in the context of particle systems with duality. We thereby obtain a systematic orthogonal decomposition of the fluctuation fields of local functions, where the order of every term can be quantified. This implies a quantitative generalization of the Boltzmann-Gibbs principle. In the context of independent random walkers, we complete this program, including also fluctuation fields in non-stationary context (local equilibrium). For other interacting particle systems with duality such as the symmetric exclusion process, similar results can be obtained, under precise conditions on the n particle dynamics.

The computation of bond percolation critical polynomials by the deletion–contraction algorithm

International Nuclear Information System (INIS)

Scullard, Christian R

2012-01-01

Although every exactly known bond percolation critical threshold is the root in [0,1] of a lattice-dependent polynomial, it has recently been shown that the notion of a critical polynomial can be extended to any periodic lattice. The polynomial is computed on a finite subgraph, called the base, of an infinite lattice. For any problem with exactly known solution, the prediction of the bond threshold is always correct for any base containing an arbitrary number of unit cells. For unsolved problems, the polynomial is referred to as the generalized critical polynomial and provides an approximation that becomes more accurate with increasing number of bonds in the base, appearing to approach the exact answer. The polynomials are computed using the deletion–contraction algorithm, which quickly becomes intractable by hand for more than about 18 bonds. Here, I present generalized critical polynomials calculated with a computer program for bases of up to 36 bonds for all the unsolved Archimedean lattices, except the kagome lattice, which was considered in an earlier work. The polynomial estimates are generally within 10 −5 –10 −7 of the numerical values, but the prediction for the (4,8 2 ) lattice, though not exact, is not ruled out by simulations. (paper)

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....

On the Lorentz degree of a product of polynomials

KAUST Repository

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

Generalized Freud's equation and level densities with polynomial potential

Science.gov (United States)

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.

Zeros and logarithmic asymptotics of Sobolev orthogonal polynomials for exponential weights

Science.gov (United States)

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.

Current advances on polynomial resultant formulations

Science.gov (United States)

Sulaiman, Surajo; Aris, Nor'aini; Ahmad, Shamsatun Nahar

2017-08-01

Availability of computer algebra systems (CAS) lead to the resurrection of the resultant method for eliminating one or more variables from the polynomials system. The resultant matrix method has advantages over the Groebner basis and Ritt-Wu method due to their high complexity and storage requirement. This paper focuses on the current resultant matrix formulations and investigates their ability or otherwise towards producing optimal resultant matrices. A determinantal formula that gives exact resultant or a formulation that can minimize the presence of extraneous factors in the resultant formulation is often sought for when certain conditions that it exists can be determined. We present some applications of elimination theory via resultant formulations and examples are given to explain each of the presented settings.

Some Results on the Independence Polynomial of Unicyclic Graphs

Directory of Open Access Journals (Sweden)

Oboudi Mohammad Reza

2018-05-01

Full Text Available Let G be a simple graph on n vertices. An independent set in a graph is a set of pairwise non-adjacent vertices. The independence polynomial of G is the polynomial I(G,x=∑k=0ns(G,kxk$I(G,x = \\sum\

Polynomial Poisson algebras: Gel'fand-Kirillov problem and Poisson spectra

OpenAIRE

Lecoutre, César

2014-01-01

We study the fields of fractions and the Poisson spectra of polynomial Poisson algebras.\\ud \\ud First we investigate a Poisson birational equivalence problem for polynomial Poisson algebras over a field of arbitrary characteristic. Namely, the quadratic Poisson Gel'fand-Kirillov problem asks whether the field of fractions of a Poisson algebra is isomorphic to the field of fractions of a Poisson affine space, i.e. a polynomial algebra such that the Poisson bracket of two generators is equal to...

Robustness analysis of a parallel two-box digital polynomial predistorter for an SOA-based CO-OFDM system

Science.gov (United States)

Diouf, C.; Younes, M.; Noaja, A.; Azou, S.; Telescu, M.; Morel, P.; Tanguy, N.

2017-11-01

The linearization performance of various digital baseband pre-distortion schemes is evaluated in this paper for a coherent optical OFDM (CO-OFDM) transmitter employing a semiconductor optical amplifier (SOA). In particular, the benefits of using a parallel two-box (PTB) behavioral model, combining a static nonlinear function with a memory polynomial (MP) model, is investigated for mitigating the system nonlinearities and compared to the memoryless and MP models. Moreover, the robustness of the predistorters under different operating conditions and system uncertainties is assessed based on a precise SOA physical model. The PTB scheme proves to be the most effective linearization technique for the considered setup, with an excellent performance-complexity tradeoff over a wide range of conditions.

Finite element method for solving Kohn-Sham equations based on self-adaptive tetrahedral mesh

International Nuclear Information System (INIS)

Zhang Dier; Shen Lihua; Zhou Aihui; Gong Xingao

2008-01-01

A finite element (FE) method with self-adaptive mesh-refinement technique is developed for solving the density functional Kohn-Sham equations. The FE method adopts local piecewise polynomials basis functions, which produces sparsely structured matrices of Hamiltonian. The method is well suitable for parallel implementation without using Fourier transform. In addition, the self-adaptive mesh-refinement technique can control the computational accuracy and efficiency with optimal mesh density in different regions

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.

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...

Can a polynomial interpolation improve on the Kaplan-Yorke dimension?

International Nuclear Information System (INIS)

Richter, Hendrik

2008-01-01

The Kaplan-Yorke dimension can be derived using a linear interpolation between an h-dimensional Lyapunov exponent λ (h) >0 and an h+1-dimensional Lyapunov exponent λ (h+1) <0. In this Letter, we use a polynomial interpolation to obtain generalized Lyapunov dimensions and study the relationships among them for higher-dimensional systems

Macdonald polynomials from Sklyanin algebras: A conceptual basis for the p-adics-quantum group connection

International Nuclear Information System (INIS)

Freund, P.G.O.

1992-01-01

We establish a previously conjectured connection between p-adics and quantum groups. We find in Sklyanin's two parameter elliptic quantum algebra and its generalizations, the conceptual basis for the Macdonald polynomials, which 'interpolate' between the zonal spherical functions of related real and p-adic symmetric spaces. The elliptic quantum algebras underlie the Z n -Baxter models. We show that in the n→∞ limit, the Jost function for the scattering of first level excitations in the Z n -Baxter model coincides with the Harish-Chandra-like c-function constructed from the Macdonald polynomials associated to the root system A 1 . The partition function of the Z 2 -Baxter model itself is also expressed in terms of this Macdonald-Harish-Chandra c-function albeit in a less simple way. We relate the two parameters q and t of the Macdonald polynomials to the anisotropy and modular parameters of the Baxter model. In particular the p-acid 'regimes' in the Macdonald polynomials correspond to a discrete sequence of XXZ models. We also discuss the possibility of 'q-deforming' Euler products. (orig.)

New approach to solve fully fuzzy system of linear equations using ...

Indian Academy of Sciences (India)

Known example problems are solved to illustrate the efficacy and ... The concept of fuzzy set and fuzzy number were first introduced by Zadeh .... (iii) Fully fuzzy linear systems can be solved by linear programming approach, Gauss elim-.

O(N) symmetries, sum rules for generalized Hermite polynomials and squeezed states

International Nuclear Information System (INIS)

Daboul, Jamil; Mizrahi, Salomon S

2005-01-01

Quantum optics has been dealing with coherent states, squeezed states and many other non-classical states. The associated mathematical framework makes use of special functions as Hermite polynomials, Laguerre polynomials and others. In this connection we here present some formal results that follow directly from the group O(N) of complex transformations. Motivated by the squeezed states structure, we introduce the generalized Hermite polynomials (GHP), which include as particular cases, the Hermite polynomials as well as the heat polynomials. Using generalized raising operators, we derive new sum rules for the GHP, which are covariant under O(N) transformations. The GHP and the associated sum rules become useful for evaluating Wigner functions in a straightforward manner. As a byproduct, we use one of these sum rules, on the operator level, to obtain raising and lowering operators for the Laguerre polynomials and show that they generate an sl(2, R) ≅ su(1, 1) algebra

On integral and finite Fourier transforms of continuous q-Hermite polynomials

International Nuclear Information System (INIS)

Atakishiyeva, M. K.; Atakishiyev, N. M.

2009-01-01

We give an overview of the remarkably simple transformation properties of the continuous q-Hermite polynomials H n (x vertical bar q) of Rogers with respect to the classical Fourier integral transform. The behavior of the q-Hermite polynomials under the finite Fourier transform and an explicit form of the q-extended eigenfunctions of the finite Fourier transform, defined in terms of these polynomials, are also discussed.

On polynomial selection for the general number field sieve

Science.gov (United States)

Kleinjung, Thorsten

2006-12-01

The general number field sieve (GNFS) is the asymptotically fastest algorithm for factoring large integers. Its runtime depends on a good choice of a polynomial pair. In this article we present an improvement of the polynomial selection method of Montgomery and Murphy which has been used in recent GNFS records.

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.

Introduction to the spectral theory of polynomial operator pencils

CERN Document Server

Markus, A S

1988-01-01

This monograph contains an exposition of the foundations of the spectral theory of polynomial operator pencils acting in a Hilbert space. Spectral problems for polynomial pencils have attracted a steady interest in the last 35 years, mainly because they arise naturally in such diverse areas of mathematical physics as differential equations and boundary value problems, controllable systems, the theory of oscillations and waves, elasticity theory, and hydromechanics. In this book, the author devotes most of his attention to the fundamental results of Keldysh on multiple completeness of the eigenvectors and associate vectors of a pencil, and on the asymptotic behavior of its eigenvalues and generalizations of these results. The author also presents various theorems on spectral factorization of pencils which grew out of known results of M. G. Kreibreven and Heinz Langer. A large portion of the book involves the theory of selfadjoint pencils, an area having numerous applications. Intended for mathematicians, resea...

A generalized polynomial chaos based ensemble Kalman filter with high accuracy

International Nuclear Information System (INIS)

Li Jia; Xiu Dongbin

2009-01-01

As one of the most adopted sequential data assimilation methods in many areas, especially those involving complex nonlinear dynamics, the ensemble Kalman filter (EnKF) has been under extensive investigation regarding its properties and efficiency. Compared to other variants of the Kalman filter (KF), EnKF is straightforward to implement, as it employs random ensembles to represent solution states. This, however, introduces sampling errors that affect the accuracy of EnKF in a negative manner. Though sampling errors can be easily reduced by using a large number of samples, in practice this is undesirable as each ensemble member is a solution of the system of state equations and can be time consuming to compute for large-scale problems. In this paper we present an efficient EnKF implementation via generalized polynomial chaos (gPC) expansion. The key ingredients of the proposed approach involve (1) solving the system of stochastic state equations via the gPC methodology to gain efficiency; and (2) sampling the gPC approximation of the stochastic solution with an arbitrarily large number of samples, at virtually no additional computational cost, to drastically reduce the sampling errors. The resulting algorithm thus achieves a high accuracy at reduced computational cost, compared to the classical implementations of EnKF. Numerical examples are provided to verify the convergence property and accuracy improvement of the new algorithm. We also prove that for linear systems with Gaussian noise, the first-order gPC Kalman filter method is equivalent to the exact Kalman filter.

Raising and Lowering Operators for Askey-Wilson Polynomials

Directory of Open Access Journals (Sweden)

Siddhartha Sahi

2007-01-01

Full Text Available In this paper we describe two pairs of raising/lowering operators for Askey-Wilson polynomials, which result from constructions involving very different techniques. The first technique is quite elementary, and depends only on the ''classical'' properties of these polynomials, viz. the q-difference equation and the three term recurrence. The second technique is less elementary, and involves the one-variable version of the double affine Hecke algebra.

The application of polynomial chaos methods to a point kinetics model of MIPR: An Aqueous Homogeneous Reactor

International Nuclear Information System (INIS)

Cooling, C.M.; Williams, M.M.R.; Nygaard, E.T.; Eaton, M.D.

2013-01-01

Highlights: • A point kinetics model for the Medical Isotope Production Reactor is formulated. • Reactivity insertions are simulated using this model. • Polynomial chaos is used to simulate uncertainty in reactor parameters. • The computational efficiency of polynomial chaos is compared to that of Monte Carlo. -- Abstract: This paper models a conceptual Medical Isotope Production Reactor (MIPR) using a point kinetics model which is used to explore power excursions in the event of a reactivity insertion. The effect of uncertainty of key parameters is modelled using intrusive polynomial chaos. It is found that the system is stable against reactivity insertions and power excursions are all bounded and tend towards a new equilibrium state due to the negative feedbacks inherent in Aqueous Homogeneous Reactors (AHRs). The Polynomial Chaos Expansion (PCE) method is found to be much more computationally efficient than that of Monte Carlo simulation in this application

Asymptotically extremal polynomials with respect to varying weights and application to Sobolev orthogonality

Science.gov (United States)

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.

Mathematical Use Of Polynomials Of Different End Periods Of ...

African Journals Online (AJOL)

This paper focused on how polynomials of different end period of random numbers can be used in the application of encryption and decryption of a message. Eight steps were used in generating information on how polynomials of different end periods of random numbers in the application of encryption and decryption of a ...

Computational Technique for Teaching Mathematics (CTTM): Visualizing the Polynomial's Resultant

Science.gov (United States)

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…

Computing derivative-based global sensitivity measures using polynomial chaos expansions

International Nuclear Information System (INIS)

Sudret, B.; Mai, C.V.

2015-01-01

In the field of computer experiments sensitivity analysis aims at quantifying the relative importance of each input parameter (or combinations thereof) of a computational model with respect to the model output uncertainty. Variance decomposition methods leading to the well-known Sobol' indices are recognized as accurate techniques, at a rather high computational cost though. The use of polynomial chaos expansions (PCE) to compute Sobol' indices has allowed to alleviate the computational burden though. However, when dealing with large dimensional input vectors, it is good practice to first use screening methods in order to discard unimportant variables. The derivative-based global sensitivity measures (DGSMs) have been developed recently in this respect. In this paper we show how polynomial chaos expansions may be used to compute analytically DGSMs as a mere post-processing. This requires the analytical derivation of derivatives of the orthonormal polynomials which enter PC expansions. Closed-form expressions for Hermite, Legendre and Laguerre polynomial expansions are given. The efficiency of the approach is illustrated on two well-known benchmark problems in sensitivity analysis. - Highlights: • Derivative-based global sensitivity measures (DGSM) have been developed for screening purpose. • Polynomial chaos expansions (PC) are used as a surrogate model of the original computational model. • From a PC expansion the DGSM can be computed analytically. • The paper provides the derivatives of Hermite, Legendre and Laguerre polynomials for this purpose

Adomian decomposition method for solving the telegraph equation in charged particle transport

International Nuclear Information System (INIS)

Abdou, M.A.

2005-01-01

In this paper, the analysis for the telegraph equation in case of isotropic small angle scattering from the Boltzmann transport equation for charged particle is presented. The Adomian decomposition is used to solve the telegraph equation. By means of MAPLE the Adomian polynomials of obtained series (ADM) solution have been calculated. The behaviour of the distribution function are shown graphically. The results reported in this article provide further evidence of the usefulness of Adomain decomposition for obtaining solution of linear and nonlinear problems

Exact polynomial solutions of second order differential equations and their applications

International Nuclear Information System (INIS)

Zhang Yaozhong

2012-01-01

We find all polynomials Z(z) such that the differential equation where X(z), Y(z), Z(z) are polynomials of degree at most 4, 3, 2, respectively, has polynomial solutions S(z) = ∏ n i=1 (z − z i ) of degree n with distinct roots z i . We derive a set of n algebraic equations which determine these roots. We also find all polynomials Z(z) which give polynomial solutions to the differential equation when the coefficients of X(z) and Y(z) are algebraically dependent. As applications to our general results, we obtain the exact (closed-form) solutions of the Schrödinger-type differential equations describing: (1) two Coulombically repelling electrons on a sphere; (2) Schrödinger equation from the kink stability analysis of φ 6 -type field theory; (3) static perturbations for the non-extremal Reissner–Nordström solution; (4) planar Dirac electron in Coulomb and magnetic fields; and (5) O(N) invariant decatic anharmonic oscillator. (paper)

The Jones polynomial as a new invariant of topological fluid dynamics

International Nuclear Information System (INIS)

Ricca, Renzo L; Liu, Xin

2014-01-01

A new method based on the use of the Jones polynomial, a well-known topological invariant of knot theory, is introduced to tackle and quantify topological aspects of structural complexity of vortex tangles in ideal fluids. By re-writing the Jones polynomial in terms of helicity, the resulting polynomial becomes then function of knot topology and vortex circulation, providing thus a new invariant of topological fluid dynamics. Explicit computations of the Jones polynomial for some standard configurations, including the Whitehead link and the Borromean rings (whose linking numbers are zero), are presented for illustration. In the case of a homogeneous, isotropic tangle of vortex filaments with same circulation, the new Jones polynomial reduces to some simple algebraic expression, that can be easily computed by numerical methods. This shows that this technique may offer a new setting and a powerful tool to detect and compute topological complexity and to investigate relations with energy, by tackling fundamental aspects of turbulence research. (paper)

PLOTNFIT.4TH, Data Plotting and Curve Fitting by Polynomials

International Nuclear Information System (INIS)

Schiffgens, J.O.

1990-01-01

1 - Description of program or function: PLOTnFIT is used for plotting and analyzing data by fitting nth degree polynomials of basis functions to the data interactively and printing graphs of the data and the polynomial functions. It can be used to generate linear, semi-log, and log-log graphs and can automatically scale the coordinate axes to suit the data. Multiple data sets may be plotted on a single graph. An auxiliary program, READ1ST, is included which produces an on-line summary of the information contained in the PLOTnFIT reference report. 2 - Method of solution: PLOTnFIT uses the least squares method to calculate the coefficients of nth-degree (up to 10. degree) polynomials of 11 selected basis functions such that each polynomial fits the data in a least squares sense. The procedure incorporated in the code uses a linear combination of orthogonal polynomials to avoid 'i11-conditioning' and to perform the curve fitting task with single-precision arithmetic. 3 - Restrictions on the complexity of the problem - Maxima of: 225 data points per job (or graph) including all data sets 8 data sets (or tasks) per job (or graph)

Multivariate Local Polynomial Regression with Application to Shenzhen Component Index

Directory of Open Access Journals (Sweden)

Liyun Su

2011-01-01

Full Text Available This study attempts to characterize and predict stock index series in Shenzhen stock market using the concepts of multivariate local polynomial regression. Based on nonlinearity and chaos of the stock index time series, multivariate local polynomial prediction methods and univariate local polynomial prediction method, all of which use the concept of phase space reconstruction according to Takens' Theorem, are considered. To fit the stock index series, the single series changes into bivariate series. To evaluate the results, the multivariate predictor for bivariate time series based on multivariate local polynomial model is compared with univariate predictor with the same Shenzhen stock index data. The numerical results obtained by Shenzhen component index show that the prediction mean squared error of the multivariate predictor is much smaller than the univariate one and is much better than the existed three methods. Even if the last half of the training data are used in the multivariate predictor, the prediction mean squared error is smaller than the univariate predictor. Multivariate local polynomial prediction model for nonsingle time series is a useful tool for stock market price prediction.

First-Order Polynomial Heisenberg Algebras and Coherent States

International Nuclear Information System (INIS)

Castillo-Celeita, M; Fernández C, D J

2016-01-01

The polynomial Heisenberg algebras (PHA) are deformations of the Heisenberg- Weyl algebra characterizing the underlying symmetry of the supersymmetric partners of the Harmonic oscillator. When looking for the simplest system ruled by PHA, however, we end up with the harmonic oscillator. In this paper we are going to realize the first-order PHA through the harmonic oscillator. The associated coherent states will be also constructed, which turn out to be the well known even and odd coherent states. (paper)

Twisted Polynomials and Forgery Attacks on GCM

DEFF Research Database (Denmark)

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

2015-01-01

Polynomial hashing as an instantiation of universal hashing is a widely employed method for the construction of MACs and authenticated encryption (AE) schemes, the ubiquitous GCM being a prominent example. It is also used in recent AE proposals within the CAESAR competition which aim at providing...... in an improved key recovery algorithm. As cryptanalytic applications of our twisted polynomials, we develop the first universal forgery attacks on GCM in the weak-key model that do not require nonce reuse. Moreover, we present universal weak-key forgeries for the nonce-misuse resistant AE scheme POET, which...

A polynomial based model for cell fate prediction in human diseases.

Science.gov (United States)

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.

Effective quadrature formula in solving linear integro-differential equations of order two

Science.gov (United States)

Eshkuvatov, Z. K.; Kammuji, M.; Long, N. M. A. Nik; Yunus, Arif A. M.

2017-08-01

In this note, we solve general form of Fredholm-Volterra integro-differential equations (IDEs) of order 2 with boundary condition approximately and show that proposed method is effective and reliable. Initially, IDEs is reduced into integral equation of the third kind by using standard integration techniques and identity between multiple and single integrals then truncated Legendre series are used to estimate the unknown function. For the kernel integrals, we have applied Gauss-Legendre quadrature formula and collocation points are chosen as the roots of the Legendre polynomials. Finally, reduce the integral equations of the third kind into the system of algebraic equations and Gaussian elimination method is applied to get approximate solutions. Numerical examples and comparisons with other methods reveal that the proposed method is very effective and dominated others in many cases. General theory of existence of the solution is also discussed.

Fractional order differentiation by integration with Jacobi polynomials

KAUST Repository

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.

Fractional order differentiation by integration with Jacobi polynomials

KAUST Repository

Liu, Dayan; Gibaru, O.; Perruquetti, Wilfrid; Laleg-Kirati, Taous-Meriem

2012-01-01

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.