WorldWideScience

Sample records for bivariate polynomial systems

  1. Solving Bivariate Polynomial Systems on a GPU

    International Nuclear Information System (INIS)

    Moreno Maza, Marc; Pan Wei

    2012-01-01

    We present a CUDA implementation of dense multivariate polynomial arithmetic based on Fast Fourier Transforms over finite fields. Our core routine computes on the device (GPU) the subresultant chain of two polynomials with respect to a given variable. This subresultant chain is encoded by values on a FFT grid and is manipulated from the host (CPU) in higher-level procedures. We have realized a bivariate polynomial system solver supported by our GPU code. Our experimental results (including detailed profiling information and benchmarks against a serial polynomial system solver implementing the same algorithm) demonstrate that our strategy is well suited for GPU implementation and provides large speedup factors with respect to pure CPU code.

  2. Darboux polynomials and first integrals of natural polynomial Hamiltonian systems

    International Nuclear Information System (INIS)

    Maciejewski, Andrzej J.; Przybylska, Maria

    2004-01-01

    We show that for a natural polynomial Hamiltonian system the existence of a single Darboux polynomial (a partial polynomial first integral) is equivalent to the existence of an additional first integral functionally independent with the Hamiltonian function. Moreover, we show that, in a case when the degree of potential is odd, the system does not admit any proper Darboux polynomial, i.e., the only Darboux polynomials are first integrals

  3. Polynomials

    Science.gov (United States)

    Apostol, Tom M. (Editor)

    1991-01-01

    In this 'Project Mathematics! series, sponsored by California Institute for Technology (CalTech), the mathematical concept of polynomials in rectangular coordinate (x, y) systems are explored. sing film footage of real life applications and computer animation sequences, the history of, the application of, and the different linear coordinate systems for quadratic, cubic, intersecting, and higher degree of polynomials are discussed.

  4. Control to Facet for Polynomial Systems

    DEFF Research Database (Denmark)

    Sloth, Christoffer; Wisniewski, Rafael

    2014-01-01

    This paper presents a solution to the control to facet problem for arbitrary polynomial vector fields defined on simplices. The novelty of the work is to use Bernstein coefficients of polynomials for determining certificates of positivity. Specifically, the constraints that are set up...... for the controller design are solved by searching for polynomials in Bernstein form. This allows the controller design problem to be formulated as a linear programming problem. Examples are provided that demonstrate the efficiency of the method for designing controls for polynomial systems....

  5. 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)

  6. Beta-integrals and finite orthogonal systems of Wilson polynomials

    International Nuclear Information System (INIS)

    Neretin, Yu A

    2002-01-01

    The integral is calculated and the system of orthogonal polynomials with weight equal to the corresponding integrand is constructed. This weight decreases polynomially, therefore only finitely many of its moments converge. As a result the system of orthogonal polynomials is finite. Systems of orthogonal polynomials related to 5 H 5 -Dougall's formula and the Askey integral is also constructed. All the three systems consist of Wilson polynomials outside the domain of positiveness of the usual weight

  7. A bivariate optimal replacement policy for a multistate repairable system

    International Nuclear Information System (INIS)

    Zhang Yuanlin; Yam, Richard C.M.; Zuo, Ming J.

    2007-01-01

    In this paper, a deteriorating simple repairable system with k+1 states, including k failure states and one working state, is studied. It is assumed that the system after repair is not 'as good as new' and the deterioration of the system is stochastic. We consider a bivariate replacement policy, denoted by (T,N), in which the system is replaced when its working age has reached T or the number of failures it has experienced has reached N, whichever occurs first. The objective is to determine the optimal replacement policy (T,N)* such that the long-run expected profit per unit time is maximized. The explicit expression of the long-run expected profit per unit time is derived and the corresponding optimal replacement policy can be determined analytically or numerically. We prove that the optimal policy (T,N)* is better than the optimal policy N* for a multistate simple repairable system. We also show that a general monotone process model for a multistate simple repairable system is equivalent to a geometric process model for a two-state simple repairable system in the sense that they have the same structure for the long-run expected profit (or cost) per unit time and the same optimal policy. Finally, a numerical example is given to illustrate the theoretical results

  8. Constructing the Lyapunov Function through Solving Positive Dimensional Polynomial System

    Directory of Open Access Journals (Sweden)

    Zhenyi Ji

    2013-01-01

    Full Text Available We propose an approach for constructing Lyapunov function in quadratic form of a differential system. First, positive polynomial system is obtained via the local property of the Lyapunov function as well as its derivative. Then, the positive polynomial system is converted into an equation system by adding some variables. Finally, numerical technique is applied to solve the equation system. Some experiments show the efficiency of our new algorithm.

  9. 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

  10. Orthogonal polynomials describing polarization aberration for rotationally symmetric optical systems.

    Science.gov (United States)

    Xu, Xiangru; Huang, Wei; Xu, Mingfei

    2015-10-19

    Optical lithography has approached a regime of high numerical aperture and wide field, where the impact of polarization aberration on imaging quality turns to be serious. Most of the existing studies focused on the distribution rule of polarization aberration on the pupil, and little attention had been paid to the field. In this paper, a new orthonormal set of polynomials is established to describe the polarization aberration of rotationally symmetric optical systems. The polynomials can simultaneously reveal the distribution rules of polarization aberration on the exit pupil and the field. Two examples are given to verify the polynomials.

  11. Polynomial Poisson algebras for two-dimensional classical superintegrable systems and polynomial associative algebras for quantum superintegrable systems

    International Nuclear Information System (INIS)

    Daskaloyannis, C.

    2000-01-01

    The integrals of motion of the classical two-dimensional superintegrable systems close in a restrained polynomial Poisson algebra, whose general form is discussed. Each classical superintegrable problem has a quantum counterpart, a quantum superintegrable system. The polynomial Poisson algebra is deformed to a polynomial associative algebra, the finite-dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique. It is conjectured that the finite-dimensional representations of the polynomial algebra are determined by the energy eigenvalues of the superintegrable system. The calculation of energy eigenvalues is reduced to the solution of algebraic equations, which are universal for a large number of two-dimensional superintegrable systems. (author)

  12. Orthonormal aberration polynomials for optical systems with circular and annular sector pupils.

    Science.gov (United States)

    Díaz, José Antonio; Mahajan, Virendra N

    2013-02-20

    Using the Zernike circle polynomials as the basis functions, we obtain the orthonormal polynomials for optical systems with circular and annular sector pupils by the Gram-Schmidt orthogonalization process. These polynomials represent balanced aberrations yielding minimum variance of the classical aberrations of rotationally symmetric systems. Use of the polynomials obtained is illustrated with numerical examples.

  13. Polynomials-Based Terminal Control of Affine Systems

    Directory of Open Access Journals (Sweden)

    A. E. Golubev

    2015-01-01

    Full Text Available One of the approaches to solving terminal control problems for affine dynamical systems is based on the use of polynomials of degree 2n − 1, where n is the order of the system in question. In this paper, we investigate the terminal control problem for which the final state of the system coincides with the origin in the phase space. We seek a set of initial states such that the solution of the terminal control problem can be constructed by using a polynomial of degree 2n − 2.Note that solution of the terminal control problem in question can be used to solve the problem of stabilizing the zero equilibrium in a finite time.For the second-order systems we prove the necessary and sufficient conditions for existence of the polynomial of the second degree which determines the solution of the terminal problem. The solutions of the terminal control problem based on the polynomials of second and third degree are given. As an example, the terminal control problem is considered for the simple pendulum.We also discuss solution of the terminal problem for affine systems of the third order, based on the use of the fourth and fifth degree polynomials. The necessary and sufficient conditions for existence of the fourth-degree polynomial such that its phase graph connects an arbitrary initial state of the system and the origin are obtained.For systems of arbitrary order n we obtain the necessary and sufficient conditions for existence of a solution of the terminal problem using the polynomial of degree 2n − 2. We also give the solution of the problem by means of the polynomial of degree 2n − 1.Further research can be focused on extending the results obtained in this note to terminal control problems where the desired final state of the system is not necessarily the origin.One of the potential application areas for the obtained theoretical results is automatic control of technical plants like unmanned aerial vehicles and mobile robots.

  14. Tricomplex Dynamical Systems Generated by Polynomials of Odd Degree

    Science.gov (United States)

    Parisé, Pierre-Olivier; Rochon, Dominic

    In this paper, we give the exact interval of the cross section of the Multibrot sets generated by the polynomial zp + c where z and c are complex numbers and p > 2 is an odd integer. Furthermore, we show that the same Multibrots defined on the hyperbolic numbers are always squares. Moreover, we give a generalized 3D version of the hyperbolic Multibrot set and prove that our generalization is an octahedron for a specific 3D slice of the dynamical system generated by the tricomplex polynomial ηp + c where p > 2 is an odd integer.

  15. Computer Algebra Systems and Theorems on Real Roots of Polynomials

    Science.gov (United States)

    Aidoo, Anthony Y.; Manthey, Joseph L.; Ward, Kim Y.

    2010-01-01

    A computer algebra system is used to derive a theorem on the existence of roots of a quadratic equation on any bounded real interval. This is extended to a cubic polynomial. We discuss how students could be led to derive and prove these theorems. (Contains 1 figure.)

  16. 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.

  17. Application of ANNs approach for solving fully fuzzy polynomials system

    Directory of Open Access Journals (Sweden)

    R. Novin

    2017-11-01

    Full Text Available In processing indecisive or unclear information, the advantages of fuzzy logic and neurocomputing disciplines should be taken into account and combined by fuzzy neural networks. The current research intends to present a fuzzy modeling method using multi-layer fuzzy neural networks for solving a fully fuzzy polynomials system. To clarify the point, it is necessary to inform that a supervised gradient descent-based learning law is employed. The feasibility of the method is examined using computer simulations on a numerical example. The experimental results obtained from the investigation of the proposed method are valid and delivers very good approximation results.

  18. Solving polynomial systems using no-root elimination blending schemes

    KAUST Repository

    Barton, Michael

    2011-12-01

    Searching for the roots of (piecewise) polynomial systems of equations is a crucial problem in computer-aided design (CAD), and an efficient solution is in strong demand. Subdivision solvers are frequently used to achieve this goal; however, the subdivision process is expensive, and a vast number of subdivisions is to be expected, especially for higher-dimensional systems. Two blending schemes that efficiently reveal domains that cannot contribute by any root, and therefore significantly reduce the number of subdivisions, are proposed. Using a simple linear blend of functions of the given polynomial system, a function is sought after to be no-root contributing, with all control points of its BernsteinBézier representation of the same sign. If such a function exists, the domain is purged away from the subdivision process. The applicability is demonstrated on several CAD benchmark problems, namely surfacesurfacesurface intersection (SSSI) and surfacecurve intersection (SCI) problems, computation of the Hausdorff distance of two planar curves, or some kinematic-inspired tasks. © 2011 Elsevier Ltd. All rights reserved.

  19. Neural Systems with Numerically Matched Input-Output Statistic: Isotonic Bivariate Statistical Modeling

    Directory of Open Access Journals (Sweden)

    Simone Fiori

    2007-07-01

    Full Text Available Bivariate statistical modeling from incomplete data is a useful statistical tool that allows to discover the model underlying two data sets when the data in the two sets do not correspond in size nor in ordering. Such situation may occur when the sizes of the two data sets do not match (i.e., there are “holes” in the data or when the data sets have been acquired independently. Also, statistical modeling is useful when the amount of available data is enough to show relevant statistical features of the phenomenon underlying the data. We propose to tackle the problem of statistical modeling via a neural (nonlinear system that is able to match its input-output statistic to the statistic of the available data sets. A key point of the new implementation proposed here is that it is based on look-up-table (LUT neural systems, which guarantee a computationally advantageous way of implementing neural systems. A number of numerical experiments, performed on both synthetic and real-world data sets, illustrate the features of the proposed modeling procedure.

  20. On Continued Fraction Expansion of Real Roots of Polynomial Systems

    DEFF Research Database (Denmark)

    Mantzaflaris, Angelos; Mourrain, Bernard; Tsigaridas, Elias

    2011-01-01

    We elaborate on a correspondence between the coefficients of a multivariate polynomial represented in the Bernstein basis and in a tensor-monomial basis, which leads to homography representations of polynomial functions that use only integer arithmetic (in contrast to the Bernstein basis) and are...

  1. Orthonormal aberration polynomials for anamorphic optical imaging systems with circular pupils.

    Science.gov (United States)

    Mahajan, Virendra N

    2012-06-20

    In a recent paper, we considered the classical aberrations of an anamorphic optical imaging system with a rectangular pupil, representing the terms of a power series expansion of its aberration function. These aberrations are inherently separable in the Cartesian coordinates (x,y) of a point on the pupil. Accordingly, there is x-defocus and x-coma, y-defocus and y-coma, and so on. We showed that the aberration polynomials orthonormal over the pupil and representing balanced aberrations for such a system are represented by the products of two Legendre polynomials, one for each of the two Cartesian coordinates of the pupil point; for example, L(l)(x)L(m)(y), where l and m are positive integers (including zero) and L(l)(x), for example, represents an orthonormal Legendre polynomial of degree l in x. The compound two-dimensional (2D) Legendre polynomials, like the classical aberrations, are thus also inherently separable in the Cartesian coordinates of the pupil point. Moreover, for every orthonormal polynomial L(l)(x)L(m)(y), there is a corresponding orthonormal polynomial L(l)(y)L(m)(x) obtained by interchanging x and y. These polynomials are different from the corresponding orthogonal polynomials for a system with rotational symmetry but a rectangular pupil. In this paper, we show that the orthonormal aberration polynomials for an anamorphic system with a circular pupil, obtained by the Gram-Schmidt orthogonalization of the 2D Legendre polynomials, are not separable in the two coordinates. Moreover, for a given polynomial in x and y, there is no corresponding polynomial obtained by interchanging x and y. For example, there are polynomials representing x-defocus, balanced x-coma, and balanced x-spherical aberration, but no corresponding y-aberration polynomials. The missing y-aberration terms are contained in other polynomials. We emphasize that the Zernike circle polynomials, although orthogonal over a circular pupil, are not suitable for an anamorphic system as

  2. 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

  3. 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.)

  4. 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

  5. Nodal-line dynamics via exact polynomial solutions for coherent waves traversing aberrated imaging systems

    Science.gov (United States)

    Paganin, David M.; Beltran, Mario A.; Petersen, Timothy C.

    2018-03-01

    We obtain exact polynomial solutions for two-dimensional coherent complex scalar fields propagating through arbitrary aberrated shift-invariant linear imaging systems. These are used to model nodal-line dynamics of coherent fields output by such systems.

  6. Mean-square filter design for stochastic polynomial systems with Gaussian and Poisson noises

    Science.gov (United States)

    Basin, Michael; Rodriguez-Ramirez, Pablo

    2014-07-01

    This paper addresses the mean-square finite-dimensional filtering problem for polynomial system states with both, Gaussian and Poisson, white noises over linear observations. A constructive procedure is established to design the mean-square filtering equations for system states described by polynomial equations of an arbitrary finite degree. An explicit closed form of the designed filter is obtained in case of a third-order polynomial system. The theoretical result is complemented with an illustrative example verifying performance of the designed filter.

  7. Orthonormal polynomials describing polarization aberration for M-fold optical systems.

    Science.gov (United States)

    Xu, Xiangru; Huang, Wei; Xu, Mingfei

    2016-03-07

    Polarization aberration (PA) is a serious issue that affects imaging quality for optical systems with high numerical aperture. Numerous studies have focused on the distribution rule of PA on the pupil, but the field remains poorly studied. We previously developed an orthonormal set of polynomials to reveal the pupil and field dependences of PA in rotationally symmetric optical systems. However, factors, such as intrinsic birefringence of cubic crystalline material in deep ultraviolet optics and tolerance, break the rotational symmetry of PA. In this paper, we extend the polynomials from rotationally symmetric to M-fold to describe the PA of M-fold optical systems. Two examples are presented to verify the polynomials.

  8. Numerical solutions of the nonlinear fractional-order brusselator system by Bernstein polynomials.

    Science.gov (United States)

    Khan, Hasib; Jafari, Hossein; Khan, Rahmat Ali; Tajadodi, Haleh; Johnston, Sarah Jane

    2014-01-01

    In this paper we propose the Bernstein polynomials to achieve the numerical solutions of nonlinear fractional-order chaotic system known by fractional-order Brusselator system. We use operational matrices of fractional integration and multiplication of Bernstein polynomials, which turns the nonlinear fractional-order Brusselator system to a system of algebraic equations. Two illustrative examples are given in order to demonstrate the accuracy and simplicity of the proposed techniques.

  9. New families of superintegrable systems from Hermite and Laguerre exceptional orthogonal polynomials

    Energy Technology Data Exchange (ETDEWEB)

    Marquette, Ian [School of Mathematics and Physics, The University of Queensland, Brisbane QLD 4072 (Australia); Quesne, Christiane [Physique Nucleaire Theorique et Physique Mathematique, Universite Libre de Bruxelles, Campus de la Plaine CP229, Boulevard du Triomphe, B-1050 Brussels (Belgium)

    2013-04-15

    In recent years, many exceptional orthogonal polynomials (EOP) were introduced and used to construct new families of 1D exactly solvable quantum potentials, some of which are shape invariant. In this paper, we construct from Hermite and Laguerre EOP and their related quantum systems new 2D superintegrable Hamiltonians with higher-order integrals of motion and the polynomial algebras generated by their integrals of motion. We obtain the finite-dimensional unitary representations of the polynomial algebras and the corresponding energy spectrum. We also point out a new type of degeneracies of the energy levels of these systems that is associated with holes in sequences of EOP.

  10. 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)

  11. Chaos, Fractals, and Polynomials.

    Science.gov (United States)

    Tylee, J. Louis; Tylee, Thomas B.

    1996-01-01

    Discusses chaos theory; linear algebraic equations and the numerical solution of polynomials, including the use of the Newton-Raphson technique to find polynomial roots; fractals; search region and coordinate systems; convergence; and generating color fractals on a computer. (LRW)

  12. Information Decomposition in Bivariate Systems: Theory and Application to Cardiorespiratory Dynamics

    Directory of Open Access Journals (Sweden)

    Luca Faes

    2015-01-01

    Full Text Available In the framework of information dynamics, the temporal evolution of coupled systems can be studied by decomposing the predictive information about an assigned target system into amounts quantifying the information stored inside the system and the information transferred to it. While information storage and transfer are computed through the known self-entropy (SE and transfer entropy (TE, an alternative decomposition evidences the so-called cross entropy (CE and conditional SE (cSE, quantifying the cross information and internal information of the target system, respectively. This study presents a thorough evaluation of SE, TE, CE and cSE as quantities related to the causal statistical structure of coupled dynamic processes. First, we investigate the theoretical properties of these measures, providing the conditions for their existence and assessing the meaning of the information theoretic quantity that each of them reflects. Then, we present an approach for the exact computation of information dynamics based on the linear Gaussian approximation, and exploit this approach to characterize the behavior of SE, TE, CE and cSE in benchmark systems with known dynamics. Finally, we exploit these measures to study cardiorespiratory dynamics measured from healthy subjects during head-up tilt and paced breathing protocols. Our main result is that the combined evaluation of the measures of information dynamics allows to infer the causal effects associated with the observed dynamics and to interpret the alteration of these effects with changing experimental conditions.

  13. 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.

  14. General Reducibility and Solvability of Polynomial Equations ...

    African Journals Online (AJOL)

    General Reducibility and Solvability of Polynomial Equations. ... Unlike quadratic, cubic, and quartic polynomials, the general quintic and higher degree polynomials cannot be solved algebraically in terms of finite number of additions, ... Galois Theory, Solving Polynomial Systems, Polynomial factorization, Polynomial Ring ...

  15. Number of solutions of systems of homogeneous polynomial equations over finite fields

    DEFF Research Database (Denmark)

    Datta, Mrinmoy; Ghorpade, Sudhir Ramakant

    2017-01-01

    We consider the problem of determining the maximum number of common zeros in a projective space over a finite field for a system of linearly independent multivariate homogeneous polynomials defined over that field. There is an elaborate conjecture of Tsfasman and Boguslavsky that predicts the max...

  16. Polynomial integrability of Hamiltonian systems with homogeneous potentials of degree −k

    Energy Technology Data Exchange (ETDEWEB)

    Oliveira, Regilene, E-mail: regilene@icmc.usp.br [Departamento de Matemática, Instituto de Ciências Matemáticas e Computação, Universidade de São Paulo, Avenida Trabalhador São-carlense, 400, 13566-590, São Carlos, SP (Brazil); Valls, Claudia, E-mail: cvalls@math.ist.utl.pt [Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa (Portugal)

    2016-12-01

    In this paper we shall answer positively two open problems proposed by Llibre–Mahdi–Valls (2011) in [9]. More precisely, we characterize the polynomial integrability of Hamiltonian system with potentials given by the inverse of a homogeneous potential of degree k.

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

    Science.gov (United States)

    Colón, Diego; Balthazar, José M.; dos Reis, Célia A.; Bueno, Átila M.; Diniz, Ivando S.; de S. R. F. Rosa, Suelia

    2014-12-01

    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.

  18. 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.

  19. Control of linear systems subject to time-domain constraints with polynomial pole placement and LMIs

    Czech Academy of Sciences Publication Activity Database

    Henrion, D.; Tarbouriech, S.; Kučera, Vladimír

    2005-01-01

    Roč. 50, č. 9 (2005), s. 1360-1364 ISSN 0018-9286 R&D Projects: GA MŠk 1M0567; GA ČR GA102/05/0011 Institutional research plan: CEZ:AV0Z10750506 Keywords : linear matrix inequality (LMI) * linear systems * pole placement * polynomials * time-domain constraints Subject RIV: BC - Control Systems Theory Impact factor: 2.159, year: 2005

  20. Homogeneous piecewise polynomial Lyapunov function for robust stability of uncertain piecewise linear system

    International Nuclear Information System (INIS)

    BenAbdallah, Abdallah; Hammami, Mohamed Ali; Kallel, Jalel

    2009-01-01

    In this paper we present some sufficient conditions for the robust stability and stabilization of time invariant uncertain piecewise linear system using homogenous piecewise polynomial Lyapunov function. The proposed conditions are given in terms of linear matrix inequalities which can be numerically solved. An application of the obtained result is given. It consists in resolving the stabilization of piecewise uncertain linear control systems by using a state piecewise linear feedback.

  1. Distributed stabilisation of spatially invariant systems: positive polynomial approach

    Czech Academy of Sciences Publication Activity Database

    Augusta, Petr; Hurák, Z.

    2013-01-01

    Roč. 24, Č. 1 (2013), s. 3-21 ISSN 1573-0824 R&D Projects: GA MŠk(CZ) 1M0567 Institutional research plan: CEZ:AV0Z10750506 Institutional support: RVO:67985556 Keywords : Multidimensional systems * Algebraic approach * Control design * Positiveness Subject RIV: BC - Control Systems Theory http://library.utia.cas.cz/separaty/2013/TR/augusta-0382623.pdf

  2. 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.

  3. 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.

  4. 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.

  5. Novel application of continuously variable transmission system using composite recurrent Laguerre orthogonal polynomials modified PSO NN control system.

    Science.gov (United States)

    Lin, Chih-Hong

    2016-09-01

    Because the V-belt continuously variable transmission system spurred by permanent magnet (PM) synchronous motor 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 composite recurrent Laguerre orthogonal polynomials modified particle swarm optimization (PSO) neural network (NN) control system which has online learning capability to come back to the nonlinear and time-varying of system, is developed for controlling PM synchronous motor servo-driven V-belt continuously variable transmission system with the lumped nonlinear load disturbances. The composite recurrent Laguerre orthogonal polynomials NN control system consists of an inspector control, a recurrent Laguerre orthogonal polynomials NN control with adaptation law and a recouped control with estimation law. Moreover, the adaptation law of online parameters in the recurrent Laguerre orthogonal polynomials NN is originated from Lyapunov stability theorem. Additionally, two optimal learning rates of the parameters by means of modified PSO are posed in order to achieve better convergence. At last, comparative studies shown by experimental results are illustrated to demonstrate the control performance of the proposed control scheme. Copyright © 2016 ISA. Published by Elsevier Ltd. All rights reserved.

  6. Chebyshev polynomials

    CERN Document Server

    Mason, JC

    2002-01-01

    Chebyshev polynomials crop up in virtually every area of numerical analysis, and they hold particular importance in recent advances in subjects such as orthogonal polynomials, polynomial approximation, numerical integration, and spectral methods. Yet no book dedicated to Chebyshev polynomials has been published since 1990, and even that work focused primarily on the theoretical aspects. A broad, up-to-date treatment is long overdue.Providing highly readable exposition on the subject''s state of the art, Chebyshev Polynomials is just such a treatment. It includes rigorous yet down-to-earth coverage of the theory along with an in-depth look at the properties of all four kinds of Chebyshev polynomials-properties that lead to a range of results in areas such as approximation, series expansions, interpolation, quadrature, and integral equations. Problems in each chapter, ranging in difficulty from elementary to quite advanced, reinforce the concepts and methods presented.Far from being an esoteric subject, Chebysh...

  7. Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials

    International Nuclear Information System (INIS)

    Odake, Satoru; Sasaki, Ryu

    2009-01-01

    Two sets of infinitely many exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials are presented. They are derived as the eigenfunctions of shape invariant and thus exactly solvable quantum mechanical Hamiltonians, which are deformations of those for the Wilson and Askey-Wilson polynomials in terms of a degree l (l=1,2,...) eigenpolynomial. These polynomials are exceptional in the sense that they start from degree l≥1 and thus not constrained by any generalisation of Bochner's theorem.

  8. 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.

  9. Polynomial Asymptotes

    Science.gov (United States)

    Dobbs, David E.

    2010-01-01

    This note develops and implements the theory of polynomial asymptotes to (graphs of) rational functions, as a generalization of the classical topics of horizontal asymptotes and oblique/slant asymptotes. Applications are given to hyperbolic asymptotes. Prerequisites include the division algorithm for polynomials with coefficients in the field of…

  10. ROBUST REGULATION FOR SYSTEMS WITH POLYNOMIAL NONLINEARITY APPLIED TO RAPID THERMAL PROCESSES

    Directory of Open Access Journals (Sweden)

    S. V. Aranovskiy

    2014-07-01

    Full Text Available Abstract. A problem of output robust control for a system with power nonlinearity is considered. The considered problem can be rewritten as a stabilization problem for a system with polynomial nonlinearity by introducing the error term. The problem of temperature regulation is considered as application; the rapid thermal processes in vapor deposition processing are studied. Modern industrial equipment uses complex sensors and control systems; these devices are not available for laboratory setups. The limited amount of available sensors and other technical restrictions for laboratory setups make it an actual problem to design simple low-order output control laws. The problem is solved by the consecutive compensator approach. The paper deals with a new type of restriction which is a combination of linear and power restrictions. It is shown that the polynomial nonlinearity satisfies this restriction. Asymptotical stability of the closed-loop system is proved by the Lyapunov functions approach for the considered nonlinear function; this contribution extends previously known results. Numerical simulation of the vapor deposition processing illustrates that the proposed approach results in zero-mean tracking error with standard deviation less than 1K.

  11. 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.

  12. Ordinal Bivariate Inequality

    DEFF Research Database (Denmark)

    Sonne-Schmidt, Christoffer Scavenius; Tarp, Finn; Østerdal, Lars Peter Raahave

    2016-01-01

    This paper introduces a concept of inequality comparisons with ordinal bivariate categorical data. In our model, one population is more unequal than another when they have common arithmetic median outcomes and the first can be obtained from the second by correlation-increasing switches and....../or median-preserving spreads. For the canonical 2 × 2 case (with two binary indicators), we derive a simple operational procedure for checking ordinal inequality relations in practice. As an illustration, we apply the model to childhood deprivation in Mozambique....

  13. Ordinal bivariate inequality

    DEFF Research Database (Denmark)

    Sonne-Schmidt, Christoffer Scavenius; Tarp, Finn; Østerdal, Lars Peter Raahave

    This paper introduces a concept of inequality comparisons with ordinal bivariate categorical data. In our model, one population is more unequal than another when they have common arithmetic median outcomes and the first can be obtained from the second by correlationincreasing switches and/or median......-preserving spreads. For the canonical 2x2 case (with two binary indicators), we derive a simple operational procedure for checking ordinal inequality relations in practice. As an illustration, we apply the model to childhood deprivation in Mozambique....

  14. Efficient Temperature-Dependent Green's Functions Methods for Realistic Systems: Compact Grids for Orthogonal Polynomial Transforms.

    Science.gov (United States)

    Kananenka, Alexei A; Phillips, Jordan J; Zgid, Dominika

    2016-02-09

    The Matsubara Green's function that is used to describe temperature-dependent behavior is expressed on a numerical grid. While such a grid usually has a couple of hundred points for low-energy model systems, for realistic systems with large basis sets the size of an accurate grid can be tens of thousands of points, constituting a severe computational and memory bottleneck. In this paper, we determine efficient imaginary time grids for the temperature-dependent Matsubara Green's function formalism that can be used for calculations on realistic systems. We show that, because of the use of an orthogonal polynomial transform, we can restrict the imaginary time grid to a few hundred points and reach micro-Hartree accuracy in the electronic energy evaluation. Moreover, we show that only a limited number of orthogonal polynomial expansion coefficients are necessary to preserve accuracy when working with a dual representation of the Green's function or self-energy and transforming between the imaginary time and frequency domain.

  15. A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus

    Science.gov (United States)

    Dunkl, Charles F.

    2017-06-01

    For each irreducible module of the symmetric group S_{N} there is a set of parametrized nonsymmetric Jack polynomials in N variables taking values in the module. These polynomials are simultaneous eigenfunctions of a commutative set of operators, self-adjoint with respect to two Hermitian forms, one called the contravariant form and the other is with respect to a matrix-valued measure on the N-torus. The latter is valid for the parameter lying in an interval about zero which depends on the module. The author in a previous paper [SIGMA 12 (2016), 033, 27 pages] proved the existence of the measure and that its absolutely continuous part satisfies a system of linear differential equations. In this paper the system is analyzed in detail. The N-torus is divided into (N-1)! connected components by the hyperplanes x_{i}=x_{j}, isystem. The main result is that the orthogonality ! measure has no singular part with respect to Haar measure, and thus is given by a matrix function times Haar measure. This function is analytic on each of the connected components.

  16. 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.

  17. M-Polynomials And Topological Indices Of Zigzag And Rhombic Benzenoid Systems

    Directory of Open Access Journals (Sweden)

    Ali Ashaq

    2018-02-01

    Full Text Available M-polynomial of different molecular structures helps to calculate many topological indices. This polynomial is a new idea and its beauty is the wealth of information it contains about the closed forms of degree-based topological indices of molecular graph G of the structure. It is a well-known fact that topological indices play significant role in determining properties of the chemical compound [1, 2, 3, 4]. In this article, we computed the closed form of M-polynomial of zigzag and rhombic benzenoid systemsbecause of their extensive usages in industry. Moreover we give graphs of M-polynomials and their relations with the parameters of structures.

  18. The Trigonometric Polynomial Like Bernstein Polynomial

    Science.gov (United States)

    2014-01-01

    A symmetric basis of trigonometric polynomial space is presented. Based on the basis, symmetric trigonometric polynomial approximants like Bernstein polynomials are constructed. Two kinds of nodes are given to show that the trigonometric polynomial sequence is uniformly convergent. The convergence of the derivative of the trigonometric polynomials is shown. Trigonometric quasi-interpolants of reproducing one degree of trigonometric polynomials are constructed. Some interesting properties of the trigonometric polynomials are given. PMID:25276845

  19. The Trigonometric Polynomial Like Bernstein Polynomial

    Directory of Open Access Journals (Sweden)

    Xuli Han

    2014-01-01

    Full Text Available A symmetric basis of trigonometric polynomial space is presented. Based on the basis, symmetric trigonometric polynomial approximants like Bernstein polynomials are constructed. Two kinds of nodes are given to show that the trigonometric polynomial sequence is uniformly convergent. The convergence of the derivative of the trigonometric polynomials is shown. Trigonometric quasi-interpolants of reproducing one degree of trigonometric polynomials are constructed. Some interesting properties of the trigonometric polynomials are given.

  20. 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

  1. Monitoring bivariate process

    Directory of Open Access Journals (Sweden)

    Marcela A. G. Machado

    2009-12-01

    Full Text Available The T² chart and the generalized variance |S| chart are the usual tools for monitoring the mean vector and the covariance matrix of multivariate processes. The main drawback of these charts is the difficulty to obtain and to interpret the values of their monitoring statistics. In this paper, we study control charts for monitoring bivariate processes that only requires the computation of sample means (the ZMAX chart for monitoring the mean vector, sample variances (the VMAX chart for monitoring the covariance matrix, or both sample means and sample variances (the MCMAX chart in the case of the joint control of the mean vector and the covariance matrix.Os gráficos de T² e da variância amostral generalizada |S| são as ferramentas usualmente utilizadas no monitoramento do vetor de médias e da matriz de covariâncias de processos multivariados. A principal desvantagem desses gráficos é a dificuldade em obter e interpretar os valores de suas estatísticas de monitoramento. Neste artigo, estudam-se gráficos de controle para o monitoramento de processos bivariados que necessitam somente do cálculo de médias amostrais (gráfico ZMAX para o monitoramento do vetor de médias, ou das variâncias amostrais (gráfico VMAX para o monitoramento da matriz de covariâncias, ou então das médias e variâncias amostrais (gráfico MCMAX para o caso do monitoramento conjunto do vetor de médias e da matriz de covariâncias.

  2. Orthogonal polynomials

    CERN Document Server

    Szegő, G

    1939-01-01

    The general theory of orthogonal polynomials was developed in the late 19th century from a study of continued fractions by P. L. Chebyshev, even though special cases were introduced earlier by Legendre, Hermite, Jacobi, Laguerre, and Chebyshev himself. It was further developed by A. A. Markov, T. J. Stieltjes, and many other mathematicians. The book by Szegő, originally published in 1939, is the first monograph devoted to the theory of orthogonal polynomials and its applications in many areas, including analysis, differential equations, probability and mathematical physics. Even after all the

  3. Modeling State-Space Aeroelastic Systems Using a Simple Matrix Polynomial Approach for the Unsteady Aerodynamics

    Science.gov (United States)

    Pototzky, Anthony S.

    2008-01-01

    A simple matrix polynomial approach is introduced for approximating unsteady aerodynamics in the s-plane and ultimately, after combining matrix polynomial coefficients with matrices defining the structure, a matrix polynomial of the flutter equations of motion (EOM) is formed. A technique of recasting the matrix-polynomial form of the flutter EOM into a first order form is also presented that can be used to determine the eigenvalues near the origin and everywhere on the complex plane. An aeroservoelastic (ASE) EOM have been generalized to include the gust terms on the right-hand side. The reasons for developing the new matrix polynomial approach are also presented, which are the following: first, the "workhorse" methods such as the NASTRAN flutter analysis lack the capability to consistently find roots near the origin, along the real axis or accurately find roots farther away from the imaginary axis of the complex plane; and, second, the existing s-plane methods, such as the Roger s s-plane approximation method as implemented in ISAC, do not always give suitable fits of some tabular data of the unsteady aerodynamics. A method available in MATLAB is introduced that will accurately fit generalized aerodynamic force (GAF) coefficients in a tabular data form into the coefficients of a matrix polynomial form. The root-locus results from the NASTRAN pknl flutter analysis, the ISAC-Roger's s-plane method and the present matrix polynomial method are presented and compared for accuracy and for the number and locations of roots.

  4. 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)

  5. Modelling Trends in Ordered Correspondence Analysis Using Orthogonal Polynomials.

    Science.gov (United States)

    Lombardo, Rosaria; Beh, Eric J; Kroonenberg, Pieter M

    2016-06-01

    The core of the paper consists of the treatment of two special decompositions for correspondence analysis of two-way ordered contingency tables: the bivariate moment decomposition and the hybrid decomposition, both using orthogonal polynomials rather than the commonly used singular vectors. To this end, we will detail and explain the basic characteristics of a particular set of orthogonal polynomials, called Emerson polynomials. It is shown that such polynomials, when used as bases for the row and/or column spaces, can enhance the interpretations via linear, quadratic and higher-order moments of the ordered categories. To aid such interpretations, we propose a new type of graphical display-the polynomial biplot.

  6. 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.

  7. On the Atkin Polynomials

    OpenAIRE

    El-Guindy, Ahmad; Ismail, Mourad E. H.

    2013-01-01

    We identify the Atkin polynomials in terms of associated Jacobi polynomials. Our identificationthen takes advantage of the theory of orthogonal polynomials and their asymptotics to establish many new properties of the Atkin polynomials. This shows that co-recursive polynomials may lead to interesting sets of orthogonal polynomials.

  8. Polynomial expansion of the precoder for power minimization in large-scale MIMO systems

    KAUST Repository

    Sifaou, Houssem

    2016-07-26

    This work focuses on the downlink of a single-cell large-scale MIMO system in which the base station equipped with M antennas serves K single-antenna users. In particular, we are interested in reducing the implementation complexity of the optimal linear precoder (OLP) that minimizes the total power consumption while ensuring target user rates. As most precoding schemes, a major difficulty towards the implementation of OLP is that it requires fast inversions of large matrices at every new channel realizations. To overcome this issue, we aim at designing a linear precoding scheme providing the same performance of OLP but with lower complexity. This is achieved by applying the truncated polynomial expansion (TPE) concept on a per-user basis. To get a further leap in complexity reduction and allow for closed-form expressions of the per-user weighting coefficients, we resort to the asymptotic regime in which M and K grow large with a bounded ratio. Numerical results are used to show that the proposed TPE precoding scheme achieves the same performance of OLP with a significantly lower implementation complexity. © 2016 IEEE.

  9. Irreducible multivariate polynomials obtained from polynomials in ...

    Indian Academy of Sciences (India)

    over K(X)? We provided some methods to construct irreducible multivariate polynomials over an arbi- trary field, starting from arbitrary irreducible polynomials in fewer variables, of which we mention the following two results: Theorem A. If we write an irreducible polynomial f ∈ K[X] as a sum of polynomials a0,..., an ∈ K[X] ...

  10. Irreducible multivariate polynomials obtained from polynomials in ...

    Indian Academy of Sciences (India)

    We provided some methods to construct irreducible multivariate polynomials over an arbi- trary field, starting from arbitrary irreducible polynomials in fewer variables, of which we mention the following two results: Theorem A. If we write an irreducible polynomial f ∈ K[X] as a sum of polynomials a0,..., an ∈ K[X] with deg a0 ...

  11. Interactive Graphics System for the Study of Variance/Covariance Structures of Bivariate and Multivariate Normal Populations

    Science.gov (United States)

    1992-09-01

    students (who later become workers) who are complacent and uncreative . What is needed Is an educational process that promotes active learning. An educational...concepts Figure 1.1. Typical Mathematics Learning Process the student and encourages passive and uncreative behavior. Such a learning system promotes...text(7); If length(number) = 0 then Cell_Rite(strg(miatrx.data[i~j])) E8 else Cell Rite(numiber); end; { Re Writ ) procedure initialize; var lij: integer

  12. 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.

  13. 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.

  14. 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.

  15. Classification of the centers, their cyclicity and isochronicity for a class of polynomial differential systems generalizing the linear systems with cubic homogeneous nonlinearities

    Science.gov (United States)

    Llibre, Jaume; Valls, Clàudia

    In this paper we classify the centers, the cyclicity of its Hopf bifurcation and their isochronicity for the polynomial differential systems in R of arbitrary degree d⩾3 odd that in complex notation z=x+iy can be written as z˙=(λ+i)z+((Az+Bzz¯+Czz+Dz), where λ∈R and A,B,C,D∈C. If d=3 we obtain the well-known class of all polynomial differential systems of the form a linear system with cubic homogeneous nonlinearities.

  16. Complex Polynomial Vector Fields

    DEFF Research Database (Denmark)

    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...

  17. 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

  18. 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)

  19. Trade off between variable and fixed size normalization in orthogonal polynomials based iris recognition system.

    Science.gov (United States)

    Krishnamoorthi, R; Anna Poorani, G

    2016-01-01

    Iris normalization is an important stage in any iris biometric, as it has a propensity to trim down the consequences of iris distortion. To indemnify the variation in size of the iris owing to the action of stretching or enlarging the pupil in iris acquisition process and camera to eyeball distance, two normalization schemes has been proposed in this work. In the first method, the iris region of interest is normalized by converting the iris into the variable size rectangular model in order to avoid the under samples near the limbus border. In the second method, the iris region of interest is normalized by converting the iris region into a fixed size rectangular model in order to avoid the dimensional discrepancies between the eye images. The performance of the proposed normalization methods is evaluated with orthogonal polynomials based iris recognition in terms of FAR, FRR, GAR, CRR and EER.

  20. Bivariate value-at-risk

    Directory of Open Access Journals (Sweden)

    Giuseppe Arbia

    2007-10-01

    Full Text Available In this paper we extend the concept of Value-at-risk (VaR to bivariate return distributions in order to obtain measures of the market risk of an asset taking into account additional features linked to downside risk exposure. We first present a general definition of risk as the probability of an adverse event over a random distribution and we then introduce a measure of market risk (b-VaR that admits the traditional b of an asset in portfolio management as a special case when asset returns are normally distributed. Empirical evidences are provided by using Italian stock market data.

  1. 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.

  2. Improved polynomial remainder sequences for Ore polynomials.

    Science.gov (United States)

    Jaroschek, Maximilian

    2013-11-01

    Polynomial remainder sequences contain the intermediate results of the Euclidean algorithm when applied to (non-)commutative polynomials. The running time of the algorithm is dependent on the size of the coefficients of the remainders. Different ways have been studied to make these as small as possible. The subresultant sequence of two polynomials is a polynomial remainder sequence in which the size of the coefficients is optimal in the generic case, but when taking the input from applications, the coefficients are often larger than necessary. We generalize two improvements of the subresultant sequence to Ore polynomials and derive a new bound for the minimal coefficient size. Our approach also yields a new proof for the results in the commutative case, providing a new point of view on the origin of the extraneous factors of the coefficients.

  3. Prediction of thermophysical and transport properties of ternary organic non-electrolyte systems including water by polynomials

    Directory of Open Access Journals (Sweden)

    Đorđević Bojan D.

    2013-01-01

    Full Text Available The description and prediction of the thermophysical and transport properties of ternary organic non-electrolyte systems including water by the polynomial equations are reviewed. Empirical equations of Radojković et al. (also known as Redlich-Kister, Kohler, Jacob-Fitzner, Colinet, Tsao-Smith, Toop, Scatchard et al. and Rastogi et al. are compared with experimental data of available papers appeared in well know international journals (Fluid Phase Equilibria, Journal of Chemical and Engineering Data, Journal of Chemical Thermodynamics, Journal of Solution Chemistry, Journal of the Serbian Chemical Society, The Canadian Journal of Chemical Engineering, Journal of Molecular Liquids, Thermochimica Acta, etc.. The applicability of empirical models to estimate excess molar volumes, VE, excess viscosities, ηE, excess free energies of activation of a viscous flow,

  4. 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....

  5. 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....

  6. 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 ...

  7. Poly-Frobenius-Euler polynomials

    Science.gov (United States)

    Kurt, Burak

    2017-07-01

    Hamahata [3] defined poly-Euler polynomials and the generalized poly-Euler polynomials. He proved some relations and closed formulas for the poly-Euler polynomials. By this motivation, we define poly-Frobenius-Euler polynomials. We give some relations for this polynomials. Also, we prove the relationships between poly-Frobenius-Euler polynomials and Stirling numbers of the second kind.

  8. 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.

  9. Coherent orthogonal polynomials

    International Nuclear Information System (INIS)

    Celeghini, E.; Olmo, M.A. del

    2013-01-01

    We discuss a fundamental characteristic of orthogonal polynomials, like the existence of a Lie algebra behind them, which can be added to their other relevant aspects. At the basis of the complete framework for orthogonal polynomials we include thus–in addition to differential equations, recurrence relations, Hilbert spaces and square integrable functions–Lie algebra theory. We start here from the square integrable functions on the open connected subset of the real line whose bases are related to orthogonal polynomials. All these one-dimensional continuous spaces allow, besides the standard uncountable basis (|x〉), for an alternative countable basis (|n〉). The matrix elements that relate these two bases are essentially the orthogonal polynomials: Hermite polynomials for the line and Laguerre and Legendre polynomials for the half-line and the line interval, respectively. Differential recurrence relations of orthogonal polynomials allow us to realize that they determine an infinite-dimensional irreducible representation of a non-compact Lie algebra, whose second order Casimir C gives rise to the second order differential equation that defines the corresponding family of orthogonal polynomials. Thus, the Weyl–Heisenberg algebra h(1) with C=0 for Hermite polynomials and su(1,1) with C=−1/4 for Laguerre and Legendre polynomials are obtained. Starting from the orthogonal polynomials the Lie algebra is extended both to the whole space of the L 2 functions and to the corresponding Universal Enveloping Algebra and transformation group. Generalized coherent states from each vector in the space L 2 and, in particular, generalized coherent polynomials are thus obtained. -- Highlights: •Fundamental characteristic of orthogonal polynomials (OP): existence of a Lie algebra. •Differential recurrence relations of OP determine a unitary representation of a non-compact Lie group. •2nd order Casimir originates a 2nd order differential equation that defines the

  10. Complex Polynomial Vector Fields

    DEFF Research Database (Denmark)

    Dias, Kealey

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

  11. Incomplete q-Chebyshev Polynomials

    OpenAIRE

    Ercan, Elif; Firengiz, Mirac Cetin; Tuglu, Naim

    2016-01-01

    In this paper, we get the generating functions of q-Chebyshev polynomials using operator. Also considering explicit formulas of q-Chebyshev polynomials, we give new generalizations of q-Chebyshev polynomials called incomplete q-Chebyshev polynomials of the first and second kind. We obtain recurrence relations and several properties of these polynomials. We show that there are connections between incomplete q-Chebyshev polynomials and the some well-known polynomials. Keywords: q-Chebyshev poly...

  12. 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

  13. Equivalences of the multi-indexed orthogonal polynomials

    International Nuclear Information System (INIS)

    Odake, Satoru

    2014-01-01

    Multi-indexed orthogonal polynomials describe eigenfunctions of exactly solvable shape-invariant quantum mechanical systems in one dimension obtained by the method of virtual states deletion. Multi-indexed orthogonal polynomials are labeled by a set of degrees of polynomial parts of virtual state wavefunctions. For multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson, and Askey-Wilson types, two different index sets may give equivalent multi-indexed orthogonal polynomials. We clarify these equivalences. Multi-indexed orthogonal polynomials with both type I and II indices are proportional to those of type I indices only (or type II indices only) with shifted parameters

  14. The approximation of bivariate Chlodowsky-Sz?sz-Kantorovich-Charlier-type operators

    OpenAIRE

    Agrawal, Purshottam Narain; Baxhaku, Behar; Chauhan, Ruchi

    2017-01-01

    Abstract In this paper, we introduce a bivariate Kantorovich variant of combination of Szász and Chlodowsky operators based on Charlier polynomials. Then, we study local approximation properties for these operators. Also, we estimate the approximation order in terms of Peetre’s K-functional and partial moduli of continuity. Furthermore, we introduce the associated GBS-case (Generalized Boolean Sum) of these operators and study the degree of approximation by means of the Lipschitz class of Bög...

  15. Kriging and local polynomial methods for blending satellite-derived and gauge precipitation estimates to support hydrologic early warning systems

    Science.gov (United States)

    Verdin, Andrew; Funk, Christopher C.; Rajagopalan, Balaji; Kleiber, William

    2016-01-01

    Robust estimates of precipitation in space and time are important for efficient natural resource management and for mitigating natural hazards. This is particularly true in regions with developing infrastructure and regions that are frequently exposed to extreme events. Gauge observations of rainfall are sparse but capture the precipitation process with high fidelity. Due to its high resolution and complete spatial coverage, satellite-derived rainfall data are an attractive alternative in data-sparse regions and are often used to support hydrometeorological early warning systems. Satellite-derived precipitation data, however, tend to underrepresent extreme precipitation events. Thus, it is often desirable to blend spatially extensive satellite-derived rainfall estimates with high-fidelity rain gauge observations to obtain more accurate precipitation estimates. In this research, we use two different methods, namely, ordinary kriging and κ-nearest neighbor local polynomials, to blend rain gauge observations with the Climate Hazards Group Infrared Precipitation satellite-derived precipitation estimates in data-sparse Central America and Colombia. The utility of these methods in producing blended precipitation estimates at pentadal (five-day) and monthly time scales is demonstrated. We find that these blending methods significantly improve the satellite-derived estimates and are competitive in their ability to capture extreme precipitation.

  16. Irreversible data compression concepts with polynomial fitting in time-order of particle trajectory for visualization of huge particle system

    International Nuclear Information System (INIS)

    Ohtani, H; Ito, A M; Hagita, K; Kato, T; Saitoh, T; Takeda, T

    2013-01-01

    We propose in this paper a data compression scheme for large-scale particle simulations, which has favorable prospects for scientific visualization of particle systems. Our data compression concepts deal with the data of particle orbits obtained by simulation directly and have the following features: (i) Through control over the compression scheme, the difference between the simulation variables and the reconstructed values for the visualization from the compressed data becomes smaller than a given constant. (ii) The particles in the simulation are regarded as independent particles and the time-series data for each particle is compressed with an independent time-step for the particle. (iii) A particle trajectory is approximated by a polynomial function based on the characteristic motion of the particle. It is reconstructed as a continuous curve through interpolation from the values of the function for intermediate values of the sample data. We name this concept ''TOKI (Time-Order Kinetic Irreversible compression)''. In this paper, we present an example of an implementation of a data-compression scheme with the above features. Several application results are shown for plasma and galaxy formation simulation data

  17. Polynomial Graphs and Symmetry

    Science.gov (United States)

    Goehle, Geoff; Kobayashi, Mitsuo

    2013-01-01

    Most quadratic functions are not even, but every parabola has symmetry with respect to some vertical line. Similarly, every cubic has rotational symmetry with respect to some point, though most cubics are not odd. We show that every polynomial has at most one point of symmetry and give conditions under which the polynomial has rotational or…

  18. 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...

  19. 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.

  20. Lowering and raising operators for some special orthogonal polynomials

    NARCIS (Netherlands)

    Koornwinder, T.H.

    2006-01-01

    Abstract: This paper discusses operators lowering or raising the degree but preserving the parameters of special orthogonal polynomials. Results for one-variable classical (q-)orthogonal polynomials are surveyed. For Jacobi polynomials associated with root system BC2 a new pair of lowering and

  1. Bivariate hard thresholding in wavelet function estimation

    OpenAIRE

    Piotr Fryzlewicz

    2007-01-01

    We propose a generic bivariate hard thresholding estimator of the discrete wavelet coefficients of a function contaminated with i.i.d. Gaussian noise. We demonstrate its good risk properties in a motivating example, and derive upper bounds for its mean-square error. Motivated by the clustering of large wavelet coefficients in real-life signals, we propose two wavelet denoising algorithms, both of which use specific instances of our bivariate estimator. The BABTE algorithm uses basis averaging...

  2. 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.

  3. 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.

  4. Superiority of legendre polynomials to Chebyshev polynomial in ...

    African Journals Online (AJOL)

    In this paper, we proved the superiority of Legendre polynomial to Chebyshev polynomial in solving first order ordinary differential equation with rational coefficient. We generated shifted polynomial of Chebyshev, Legendre and Canonical polynomials which deal with solving differential equation by first choosing Chebyshev ...

  5. Extended biorthogonal matrix polynomials

    Directory of Open Access Journals (Sweden)

    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.

  6. 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...

  7. More on rotations as spin matrix polynomials

    International Nuclear Information System (INIS)

    Curtright, Thomas L.

    2015-01-01

    Any nonsingular function of spin j matrices always reduces to a matrix polynomial of order 2j. The challenge is to find a convenient form for the coefficients of the matrix polynomial. The theory of biorthogonal systems is a useful framework to meet this challenge. Central factorial numbers play a key role in the theoretical development. Explicit polynomial coefficients for rotations expressed either as exponentials or as rational Cayley transforms are considered here. Structural features of the results are discussed and compared, and large j limits of the coefficients are examined

  8. Small oscillations, Sturm sequences, and orthogonal polynomials

    International Nuclear Information System (INIS)

    Baake, M.

    1986-07-01

    The relation between small oscillations of one-dimensional mechanical systems and the theory of orthogonal polynomials is investigated. It is shown how the polynomials provide a natural tool to determine the eigenfrequencies and eigencoordinates completely, where the existence of a certain two-termed recurrence formula is essential. Physical and mathematical statements are formulated in terms of the recursion coefficients which can directly be obtained from the corresponding secular equation. Several known as well as new results on Sturm sequences and orthogonal polynomials are presented with respect to the treatment of small oscillations. (orig.)

  9. Context-free grammars for several polynomials associated with Eulerian polynomials

    OpenAIRE

    Ma, Shi-Mei; Ma, Jun; Yeh, Yeong-Nan; Zhu, Bao-Xuan

    2016-01-01

    In this paper, we present grammatical descriptions of several polynomials associated with Eulerian polynomials, including q-Eulerian polynomials, alternating run polynomials and derangement polynomials. As applications, we get several convolution formulas involving these polynomials.

  10. Classifying polynomials and identity testing

    Indian Academy of Sciences (India)

    m = nO(log n). [2]. This provides a excellent classification of easy polynomials. Hence, any polynomial that cannot be written as a determinant of a small sized matrix is not easy. A simple counting argument shows that there exist many such polynomials. However, prov- ing an explicitly given polynomial to be hard has.

  11. 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...

  12. 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.

  13. Fast polynomial multiplication on a GPU

    International Nuclear Information System (INIS)

    Maza, Marc Moreno; Pan Wei

    2010-01-01

    We present CUDA implementations of Fast Fourier Transforms over finite fields. This allows us to develop GPU support for dense univariate polynomial multiplication leading to speedup factors in the range 21 - 37 with respect to the best serial C-code available to us, for our largest input data sets. Since dense univariate polynomial multiplication is a core routine in symbolic computation, this is promising result for the integration of GPU support into computer algebra systems.

  14. Orthonormal polynomials for annular pupil including a cross-shaped obstruction.

    Science.gov (United States)

    Dai, Fengzhao; Wang, Xiangzhao; Sasaki, Osami

    2015-04-01

    By nonrecursive matrix method using the Zernike circle polynomials as the basis functions, we derived a set of polynomials up to fourth order which is approximately orthonormal for optical systems with an annular pupil having a cross-shaped obstruction. The performance of the polynomials is compared with the strictly orthonormal polynomials with some numerical examples.

  15. 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

  16. The Bivariate (Complex) Fibonacci and Lucas Polynomials: An Historical Investigation with the Maple's Help

    Science.gov (United States)

    Alves, Francisco Regis Vieira; Catarino, Paula Maria Machado Cruz

    2016-01-01

    The current research around the Fibonacci's and Lucas' sequence evidences the scientific vigor of both mathematical models that continue to inspire and provide numerous specializations and generalizations, especially from the sixties. One of the current of research and investigations around the Generalized Sequence of Lucas, involves it's…

  17. 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

  18. Fast methods for resumming matrix polynomials and Chebyshev matrix polynomials

    International Nuclear Information System (INIS)

    Liang Wanzen; Baer, Roi; Saravanan, Chandra; Shao Yihan; Bell, Alexis T.; Head-Gordon, Martin

    2004-01-01

    Fast and effective algorithms are discussed for resumming matrix polynomials and Chebyshev matrix polynomials. These algorithms lead to a significant speed-up in computer time by reducing the number of matrix multiplications required to roughly twice the square root of the degree of the polynomial. A few numerical tests are presented, showing that evaluation of matrix functions via polynomial expansions can be preferable when the matrix is sparse and these fast resummation algorithms are employed

  19. The approximation of bivariate Chlodowsky-Szász-Kantorovich-Charlier-type operators

    Directory of Open Access Journals (Sweden)

    Purshottam Narain Agrawal

    2017-08-01

    Full Text Available Abstract In this paper, we introduce a bivariate Kantorovich variant of combination of Szász and Chlodowsky operators based on Charlier polynomials. Then, we study local approximation properties for these operators. Also, we estimate the approximation order in terms of Peetre’s K-functional and partial moduli of continuity. Furthermore, we introduce the associated GBS-case (Generalized Boolean Sum of these operators and study the degree of approximation by means of the Lipschitz class of Bögel continuous functions. Finally, we present some graphical examples to illustrate the rate of convergence of the operators under consideration.

  20. The approximation of bivariate Chlodowsky-Szász-Kantorovich-Charlier-type operators.

    Science.gov (United States)

    Agrawal, Purshottam Narain; Baxhaku, Behar; Chauhan, Ruchi

    2017-01-01

    In this paper, we introduce a bivariate Kantorovich variant of combination of Szász and Chlodowsky operators based on Charlier polynomials. Then, we study local approximation properties for these operators. Also, we estimate the approximation order in terms of Peetre's K-functional and partial moduli of continuity. Furthermore, we introduce the associated GBS-case (Generalized Boolean Sum) of these operators and study the degree of approximation by means of the Lipschitz class of Bögel continuous functions. Finally, we present some graphical examples to illustrate the rate of convergence of the operators under consideration.

  1. Approximations of orthogonal polynomials in terms of Hermite polynomials

    NARCIS (Netherlands)

    J.L. López; N.M. Temme (Nico)

    1999-01-01

    textabstractSeveral orthogonal polynomials have limit forms in which Hermite polynomials show up. Examples are limits with respect to certain parameters of the Jacobi and Laguerre polynomials. In this paper we are interested in more details of these limits and we give asymptotic representations of

  2. Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials

    NARCIS (Netherlands)

    Koornwinder, T.H.; Bouzeffour, F.

    2011-01-01

    Nonsymmetric Askey-Wilson polynomials are usually written as Laurent polynomials. We write them equivalently as 2-vector-valued symmetric Laurent polynomials. Then the Dunkl-Cherednik operator of which they are eigenfunctions, is represented as a 2 × 2 matrix-valued operator. As a new result made

  3. Cyclotomy and Cyclotomic Polynomials

    Indian Academy of Sciences (India)

    Home; Journals; Resonance – Journal of Science Education; Volume 4; Issue 12. Cyclotomy and Cyclotomic Polynomials - The Story of how Gauss Narrowly Missed Becoming a Philologist. B Sury. General Article Volume 4 Issue 12 December 1999 pp 41-53 ...

  4. Reliability for some bivariate beta distributions

    Directory of Open Access Journals (Sweden)

    Nadarajah Saralees

    2005-01-01

    Full Text Available In the area of stress-strength models there has been a large amount of work as regards estimation of the reliability R=Pr( Xbivariate distribution with dependence between X and Y . In particular, we derive explicit expressions for R when the joint distribution is bivariate beta. The calculations involve the use of special functions.

  5. Reliability for some bivariate gamma distributions

    Directory of Open Access Journals (Sweden)

    Nadarajah Saralees

    2005-01-01

    Full Text Available In the area of stress-strength models, there has been a large amount of work as regards estimation of the reliability R=Pr( Xbivariate distribution with dependence between X and Y . In particular, we derive explicit expressions for R when the joint distribution is bivariate gamma. The calculations involve the use of special functions.

  6. 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....

  7. Covariate analysis of bivariate survival data

    Energy Technology Data Exchange (ETDEWEB)

    Bennett, L.E.

    1992-01-01

    The methods developed are used to analyze the effects of covariates on bivariate survival data when censoring and ties are present. The proposed method provides models for bivariate survival data that include differential covariate effects and censored observations. The proposed models are based on an extension of the univariate Buckley-James estimators which replace censored data points by their expected values, conditional on the censoring time and the covariates. For the bivariate situation, it is necessary to determine the expectation of the failure times for one component conditional on the failure or censoring time of the other component. Two different methods have been developed to estimate these expectations. In the semiparametric approach these expectations are determined from a modification of Burke's estimate of the bivariate empirical survival function. In the parametric approach censored data points are also replaced by their conditional expected values where the expected values are determined from a specified parametric distribution. The model estimation will be based on the revised data set, comprised of uncensored components and expected values for the censored components. The variance-covariance matrix for the estimated covariate parameters has also been derived for both the semiparametric and parametric methods. Data from the Demographic and Health Survey was analyzed by these methods. The two outcome variables are post-partum amenorrhea and breastfeeding; education and parity were used as the covariates. Both the covariate parameter estimates and the variance-covariance estimates for the semiparametric and parametric models will be compared. In addition, a multivariate test statistic was used in the semiparametric model to examine contrasts. The significance of the statistic was determined from a bootstrap distribution of the test statistic.

  8. Financial Applications of Bivariate Markov Processes

    OpenAIRE

    Ortobelli Lozza, Sergio; Angelelli, Enrico; Bianchi, Annamaria

    2011-01-01

    This paper describes a methodology to approximate a bivariate Markov process by means of a proper Markov chain and presents possible financial applications in portfolio theory, option pricing and risk management. In particular, we first show how to model the joint distribution between market stochastic bounds and future wealth and propose an application to large-scale portfolio problems. Secondly, we examine an application to VaR estimation. Finally, we propose a methodology...

  9. Adaptive-Mesh-Refinement for hyperbolic systems of conservation laws based on a posteriori stabilized high order polynomial reconstructions

    Science.gov (United States)

    Semplice, Matteo; Loubère, Raphaël

    2018-02-01

    In this paper we propose a third order accurate finite volume scheme based on a posteriori limiting of polynomial reconstructions within an Adaptive-Mesh-Refinement (AMR) simulation code for hydrodynamics equations in 2D. The a posteriori limiting is based on the detection of problematic cells on a so-called candidate solution computed at each stage of a third order Runge-Kutta scheme. Such detection may include different properties, derived from physics, such as positivity, from numerics, such as a non-oscillatory behavior, or from computer requirements such as the absence of NaN's. Troubled cell values are discarded and re-computed starting again from the previous time-step using a more dissipative scheme but only locally, close to these cells. By locally decrementing the degree of the polynomial reconstructions from 2 to 0 we switch from a third-order to a first-order accurate but more stable scheme. The entropy indicator sensor is used to refine/coarsen the mesh. This sensor is also employed in an a posteriori manner because if some refinement is needed at the end of a time step, then the current time-step is recomputed with the refined mesh, but only locally, close to the new cells. We show on a large set of numerical tests that this a posteriori limiting procedure coupled with the entropy-based AMR technology can maintain not only optimal accuracy on smooth flows but also stability on discontinuous profiles such as shock waves, contacts, interfaces, etc. Moreover numerical evidences show that this approach is at least comparable in terms of accuracy and cost to a more classical CWENO approach within the same AMR context.

  10. Interpolation and Polynomial Curve Fitting

    Science.gov (United States)

    Yang, Yajun; Gordon, Sheldon P.

    2014-01-01

    Two points determine a line. Three noncollinear points determine a quadratic function. Four points that do not lie on a lower-degree polynomial curve determine a cubic function. In general, n + 1 points uniquely determine a polynomial of degree n, presuming that they do not fall onto a polynomial of lower degree. The process of finding such a…

  11. Connection between quantum systems involving the fourth Painlevé transcendent and k-step rational extensions of the harmonic oscillator related to Hermite exceptional orthogonal polynomial

    Energy Technology Data Exchange (ETDEWEB)

    Marquette, Ian, E-mail: i.marquette@uq.edu.au [School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072 (Australia); Quesne, Christiane, E-mail: cquesne@ulb.ac.be [Physique Nucléaire Théorique et Physique Mathématique, Université Libre de Bruxelles, Campus de la Plaine CP229, Boulevard du Triomphe, B-1050 Brussels (Belgium)

    2016-05-15

    The purpose of this communication is to point out the connection between a 1D quantum Hamiltonian involving the fourth Painlevé transcendent P{sub IV}, obtained in the context of second-order supersymmetric quantum mechanics and third-order ladder operators, with a hierarchy of families of quantum systems called k-step rational extensions of the harmonic oscillator and related with multi-indexed X{sub m{sub 1,m{sub 2,…,m{sub k}}}} Hermite exceptional orthogonal polynomials of type III. The connection between these exactly solvable models is established at the level of the equivalence of the Hamiltonians using rational solutions of the fourth Painlevé equation in terms of generalized Hermite and Okamoto polynomials. We also relate the different ladder operators obtained by various combinations of supersymmetric constructions involving Darboux-Crum and Krein-Adler supercharges, their zero modes and the corresponding energies. These results will demonstrate and clarify the relation observed for a particular case in previous papers.

  12. Open Problems Related to the Hurwitz Stability of Polynomials Segments

    Directory of Open Access Journals (Sweden)

    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.

  13. Fourier series in orthogonal polynomials

    CERN Document Server

    Osilenker, Boris

    1999-01-01

    This book presents a systematic course on general orthogonal polynomials and Fourier series in orthogonal polynomials. It consists of six chapters. Chapter 1 deals in essence with standard results from the university course on the function theory of a real variable and on functional analysis. Chapter 2 contains the classical results about the orthogonal polynomials (some properties, classical Jacobi polynomials and the criteria of boundedness).The main subject of the book is Fourier series in general orthogonal polynomials. Chapters 3 and 4 are devoted to some results in this topic (classical

  14. Generalized q-Hermite polynomials

    International Nuclear Information System (INIS)

    Berg, C.; Ruffing, A.

    2001-01-01

    We consider two operators A and A + in a Hilbert space of functions on the exponential lattice {q n ,-q n vertical stroke n element of Z}, where 0 0 and that A and A + are ladder operators for the sequence ψ n =(A + ) n ψ 0 . The sequence (ψ n /ψ 0 ) is shown to be a family of orthogonal polynomials, which we identify as symmetrized q-Laguerre polynomials. We obtain in this way a new proof of the orthogonality for these polynomials. When γ=0 the polynomials are the discrete q-Hermite polynomials of type II, studied in several papers on q-quantum mechanics. (Orig.) (orig.)

  15. Comparative assessment of freeform polynomials as optical surface descriptions.

    Science.gov (United States)

    Kaya, Ilhan; Thompson, Kevin P; Rolland, Jannick P

    2012-09-24

    Slow-servo single-point diamond turning as well as advances in computer controlled small lap polishing enables the fabrication of freeform optics, or more specifically, optical surfaces for imaging applications that are not rotationally symmetric. Various forms of polynomials for describing freeform optical surfaces exist in optical design and to support fabrication. A popular method is to add orthogonal polynomials onto a conic section. In this paper, recently introduced gradient-orthogonal polynomials are investigated in a comparative manner with the widely known Zernike polynomials. In order to achieve numerical robustness when higher-order polynomials are required to describe freeform surfaces, recurrence relations are a key enabler. Results in this paper establish the equivalence of both polynomial sets in accurately describing freeform surfaces under stringent conditions. Quantifying the accuracy of these two freeform surface descriptions is a critical step in the future application of these tools in both advanced optical system design and optical fabrication.

  16. A Characterization of Polynomials

    DEFF Research Database (Denmark)

    Andersen, Kurt Munk

    1996-01-01

    Given the problem:which functions f(x) are characterized by a relation of the form:f[x1,x2,...,xn]=h(x1+x2+...+xn), where n>1 and h(x) is a given function? Here f[x1,x2,...,xn] denotes the divided difference on n points x1,x2,...,xn of the function f(x).The answer is: f(x) is a polynomial of degr...

  17. Chemical Reaction Networks for Computing Polynomials.

    Science.gov (United States)

    Salehi, Sayed Ahmad; Parhi, Keshab K; Riedel, Marc D

    2017-01-20

    Chemical reaction networks (CRNs) provide a fundamental model in the study of molecular systems. Widely used as formalism for the analysis of chemical and biochemical systems, CRNs have received renewed attention as a model for molecular computation. This paper demonstrates that, with a new encoding, CRNs can compute any set of polynomial functions subject only to the limitation that these functions must map the unit interval to itself. These polynomials can be expressed as linear combinations of Bernstein basis polynomials with positive coefficients less than or equal to 1. In the proposed encoding approach, each variable is represented using two molecular types: a type-0 and a type-1. The value is the ratio of the concentration of type-1 molecules to the sum of the concentrations of type-0 and type-1 molecules. The proposed encoding naturally exploits the expansion of a power-form polynomial into a Bernstein polynomial. Molecular encoders for converting any input in a standard representation to the fractional representation as well as decoders for converting the computed output from the fractional to a standard representation are presented. The method is illustrated first for generic CRNs; then chemical reactions designed for an example are mapped to DNA strand-displacement reactions.

  18. Spectral density regression for bivariate extremes

    KAUST Repository

    Castro Camilo, Daniela

    2016-05-11

    We introduce a density regression model for the spectral density of a bivariate extreme value distribution, that allows us to assess how extremal dependence can change over a covariate. Inference is performed through a double kernel estimator, which can be seen as an extension of the Nadaraya–Watson estimator where the usual scalar responses are replaced by mean constrained densities on the unit interval. Numerical experiments with the methods illustrate their resilience in a variety of contexts of practical interest. An extreme temperature dataset is used to illustrate our methods. © 2016 Springer-Verlag Berlin Heidelberg

  19. Bivariate Kumaraswamy Models via Modified FGM Copulas: Properties and Applications

    Directory of Open Access Journals (Sweden)

    Indranil Ghosh

    2017-11-01

    Full Text Available A copula is a useful tool for constructing bivariate and/or multivariate distributions. In this article, we consider a new modified class of FGM (Farlie–Gumbel–Morgenstern bivariate copula for constructing several different bivariate Kumaraswamy type copulas and discuss their structural properties, including dependence structures. It is established that construction of bivariate distributions by this method allows for greater flexibility in the values of Spearman’s correlation coefficient, ρ and Kendall’s τ .

  20. 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.

  1. Polynomial intelligent states

    Energy Technology Data Exchange (ETDEWEB)

    Milks, Matthew M; Guise, Hubert de [Department of Physics, Lakehead University, Thunder Bay, ON, P7B 5E1 (Canada)

    2005-12-01

    The construction of su(2) intelligent states is simplified using a polynomial representation of su(2). The cornerstone of the new construction is the diagonalization of a 2 x 2 matrix. The method is sufficiently simple to be easily extended to su(3), where one is required to diagonalize a single 3 x 3 matrix. For two perfectly general su(3) operators, this diagonalization is technically possible but the procedure loses much of its simplicity owing to the algebraic form of the roots of a cubic equation. Simplified expressions can be obtained by specializing the choice of su(3) operators. This simpler construction will be discussed in detail.

  2. 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....... In this thesis we study the real roots of the chromatic polynomial, termed chromatic roots, and focus on how certain properties of a graph affect the location of its chromatic roots. Firstly, we investigate how the presence of a certain spanning tree in a graph affects its chromatic roots. In particular we prove...... of certain minor-closed families of graphs. Later, we study the Tutte polynomial of a graph, which contains the chromatic polynomial as a specialisation. We discuss a technique of Thomassen using which it is possible to deduce that the roots of the chromatic polynomial are dense in certain intervals. We...

  3. Non polynomial B-splines

    Science.gov (United States)

    Laksâ, Arne

    2015-11-01

    B-splines are the de facto industrial standard for surface modelling in Computer Aided design. It is comparable to bend flexible rods of wood or metal. A flexible rod minimize the energy when bending, a third degree polynomial spline curve minimize the second derivatives. B-spline is a nice way of representing polynomial splines, it connect polynomial splines to corner cutting techniques, which induces many nice and useful properties. However, the B-spline representation can be expanded to something we can call general B-splines, i.e. both polynomial and non-polynomial splines. We will show how this expansion can be done, and the properties it induces, and examples of non-polynomial B-spline.

  4. Orthonormal polynomials for elliptical wavefronts with an arbitrary orientation.

    Science.gov (United States)

    Díaz, José A; Navarro, Rafael

    2014-04-01

    We generalize the analytical form of the orthonormal elliptical polynomials for any arbitrary aspect ratio to arbitrary orientation and give expression for them up to the 4th order. The utility of the polynomials is demonstrated by obtaining the expansion up to the 8th order in two examples of an off-axis wavefront exiting from an optical system with a vignetted pupil.

  5. Bivariate Rayleigh Distribution and its Properties

    Directory of Open Access Journals (Sweden)

    Ahmad Saeed Akhter

    2007-01-01

    Full Text Available Rayleigh (1880 observed that the sea waves follow no law because of the complexities of the sea, but it has been seen that the probability distributions of wave heights, wave length, wave induce pitch, wave and heave motions of the ships follow the Rayleigh distribution. At present, several different quantities are in use for describing the state of the sea; for example, the mean height of the waves, the root mean square height, the height of the “significant waves” (the mean height of the highest one-third of all the waves the maximum height over a given interval of the time, and so on. At present, the ship building industry knows less than any other construction industry about the service conditions under which it must operate. Only small efforts have been made to establish the stresses and motions and to incorporate the result of such studies in to design. This is due to the complexity of the problem caused by the extensive variability of the sea and the corresponding response of the ships. Although the problem appears feasible, yet it is possible to predict service conditions for ships in an orderly and relatively simple manner Rayleigh (1980 derived it from the amplitude of sound resulting from many independent sources. This distribution is also connected with one or two dimensions and is sometimes referred to as “random walk” frequency distribution. The Rayleigh distribution can be derived from the bivariate normal distribution when the variate are independent and random with equal variances. We try to construct bivariate Rayleigh distribution with marginal Rayleigh distribution function and discuss its fundamental properties.

  6. D-stability of polynomial matrices

    Czech Academy of Sciences Publication Activity Database

    Henrion, Didier; Bachelier, O.; Šebek, M.

    2001-01-01

    Roč. 74, č. 8 (2001), s. 845-856 ISSN 0020-7179 R&D Projects: GA AV ČR KSK1019101 Institutional research plan: AV0Z1075907 Keywords : polynomial methods * robust control * linear matrix inequalities Subject RIV: BC - Control Systems Theory Impact factor: 0.736, year: 2001

  7. Polynomial computation of Hankel singular values

    NARCIS (Netherlands)

    Kwakernaak, H.

    1992-01-01

    A revised and improved version of a polynomial algorithm is presented. It was published by N.J. Young (1990) for the computation of the singular values and vectors of the Hankel operator defined by a linear time-invariant system with a rotational transfer matrix. Tentative numerical experiments

  8. Exceptional Askey-Wilson-type polynomials through Darboux-Crum transformations

    International Nuclear Information System (INIS)

    Odake, S; Sasaki, R

    2010-01-01

    An alternative derivation is presented of the infinitely many exceptional Wilson and Askey-Wilson polynomials, which were introduced by the present authors in 2009. Darboux-Crum transformations intertwining the discrete quantum mechanical systems of the original and the exceptional polynomials play an important role. Infinitely many continuous Hahn polynomials are derived in the same manner. The present method provides a simple proof of the shape invariance of these systems as in the corresponding cases of the exceptional Laguerre and Jacobi polynomials.

  9. 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.

  10. Polynomials formalism of quantum numbers

    International Nuclear Information System (INIS)

    Kazakov, K.V.

    2005-01-01

    Theoretical aspects of the recently suggested perturbation formalism based on the method of quantum number polynomials are considered in the context of the general anharmonicity problem. Using a biatomic molecule by way of example, it is demonstrated how the theory can be extrapolated to the case of vibrational-rotational interactions. As a result, an exact expression for the first coefficient of the Herman-Wallis factor is derived. In addition, the basic notions of the formalism are phenomenologically generalized and expanded to the problem of spin interaction. The concept of magneto-optical anharmonicity is introduced. As a consequence, an exact analogy is drawn with the well-known electro-optical theory of molecules, and a nonlinear dependence of the magnetic dipole moment of the system on the spin and wave variables is established [ru

  11. Classifying polynomials and identity testing

    Indian Academy of Sciences (India)

    hard to compute [3,4]! Therefore, the solution to. PIT problem has a key role in our attempt to com- putationally classify polynomials. In this article, we will focus on this connection between PIT and polynomial classification. We now formally define arithmetic circuits and the identity testing problem. 1.1 Problem definition.

  12. Polynomial methods in combinatorics

    CERN Document Server

    Guth, Larry

    2016-01-01

    This book explains some recent applications of the theory of polynomials and algebraic geometry to combinatorics and other areas of mathematics. One of the first results in this story is a short elegant solution of the Kakeya problem for finite fields, which was considered a deep and difficult problem in combinatorial geometry. The author also discusses in detail various problems in incidence geometry associated to Paul Erdős's famous distinct distances problem in the plane from the 1940s. The proof techniques are also connected to error-correcting codes, Fourier analysis, number theory, and differential geometry. Although the mathematics discussed in the book is deep and far-reaching, it should be accessible to first- and second-year graduate students and advanced undergraduates. The book contains approximately 100 exercises that further the reader's understanding of the main themes of the book. Some of the greatest advances in geometric combinatorics and harmonic analysis in recent years have been accompl...

  13. On the limit from q-Racah polynomials to big q-Jacobi polynomials

    NARCIS (Netherlands)

    Koornwinder, T.H.

    2011-01-01

    A limit formula from q-Racah polynomials to big q-Jacobi polynomials is given which can be considered as a limit formula for orthogonal polynomials. This is extended to a multi-parameter limit with 3 parameters, also involving (q-)Hahn polynomials, little q-Jacobi polynomials and Jacobi polynomials.

  14. Application of a composition of generating functions for obtaining explicit formulas of polynomials

    OpenAIRE

    Kruchinin, Vladimir; Kruchinin, Dmitry

    2012-01-01

    Using notions of composita and composition of generating functions we obtain explicit formulas for Chebyshev polynomials, Legendre polynomials, Gegenbauer polynomials, Associated Laguerre polynomials, Stirling polynomials, Abel polynomials, Bernoulli Polynomials of the Second Kind, Generalized Bernoulli polynomials, Euler Polynomials, Peters polynomials, Narumi polynomials, Humbert polynomials, Lerch polynomials and Mahler polynomials.

  15. 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.

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

    Directory of Open Access Journals (Sweden)

    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.

  17. Steam gasification of wood biomass in a fluidized biocatalytic system bed gasifier: A model development and validation using experiment and Boubaker Polynomials Expansion Scheme BPES

    Directory of Open Access Journals (Sweden)

    Luigi Vecchione

    2015-07-01

    Full Text Available One of the most important issues in biomass biocatalytic gasification is the correct prediction of gasification products, with particular attention to the Topping Atmosphere Residues (TARs. In this work, performedwithin the European 7FP UNIfHY project, we develops and validate experimentally a model which is able of predicting the outputs,including TARs, of a steam-fluidized bed biomass gasifier. Pine wood was chosen as biomass feedstock: the products obtained in pyrolysis tests are the relevant model input. Hydrodynamics and chemical properties of the reacting system are considered: the hydrodynamic approach is based on the two phase theory of fluidization, meanwhile the chemical model is based on the kinetic equations for the heterogeneous and homogenous reactions. The derived differentials equations for the gasifier at steady state were implemented MATLAB. Solution was consecutively carried out using the Boubaker Polynomials Expansion Scheme by varying steam/biomass ratio (0.5-1 and operating temperature (750-850°C.The comparison between models and experimental results showed that the model is able of predicting gas mole fractions and production rate including most of the representative TARs compounds

  18. 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

  19. 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....

  20. Sum rules for zeros of polynomials and generalized Lucas polynomials

    Science.gov (United States)

    Ricci, Paolo Emilio

    1993-10-01

    A representation formula in terms of generalized Lucas polynomials of the second kind [see formula (4.3)], for the sum rules Js(i) introduced by Case [J. Math. Phys. 21, 702 (1980)] and studied by Dehesa et al. [J. Math. Phys. 26, 1547 (1985); 26, 2729 (1985)] in order to obtain information about the zeros' distribution of eigenfunctions of a class of ordinary polynomial differential operators, is derived.

  1. Bell-Type Inequalities for Bivariate Maps on Orthomodular Lattices

    Science.gov (United States)

    Pykacz, Jarosław; Valášková, L'ubica; Nánásiová, Ol'ga

    2015-08-01

    Bell-type inequalities on orthomodular lattices, in which conjunctions of propositions are not modeled by meets but by maps for simultaneous measurements (-maps), are studied. It is shown, that the most simple of these inequalities, that involves only two propositions, is always satisfied, contrary to what happens in the case of traditional version of this inequality in which conjunctions of propositions are modeled by meets. Equivalence of various Bell-type inequalities formulated with the aid of bivariate maps on orthomodular lattices is studied. Our investigations shed new light on the interpretation of various multivariate maps defined on orthomodular lattices already studied in the literature. The paper is concluded by showing the possibility of using -maps and -maps to represent counterfactual conjunctions and disjunctions of non-compatible propositions about quantum systems.

  2. Polynomial Regressions and Nonsense Inference

    Directory of Open Access Journals (Sweden)

    Daniel Ventosa-Santaulària

    2013-11-01

    Full Text Available Polynomial specifications are widely used, not only in applied economics, but also in epidemiology, physics, political analysis and psychology, just to mention a few examples. In many cases, the data employed to estimate such specifications are time series that may exhibit stochastic nonstationary behavior. We extend Phillips’ results (Phillips, P. Understanding spurious regressions in econometrics. J. Econom. 1986, 33, 311–340. by proving that an inference drawn from polynomial specifications, under stochastic nonstationarity, is misleading unless the variables cointegrate. We use a generalized polynomial specification as a vehicle to study its asymptotic and finite-sample properties. Our results, therefore, lead to a call to be cautious whenever practitioners estimate polynomial regressions.

  3. Stress-strength reliability for general bivariate distributions

    Directory of Open Access Journals (Sweden)

    Alaa H. Abdel-Hamid

    2016-10-01

    Full Text Available An expression for the stress-strength reliability R=P(X1bivariate distribution. Such distribution includes bivariate compound Weibull, bivariate compound Gompertz, bivariate compound Pareto, among others. In the parametric case, the maximum likelihood estimates of the parameters and reliability function R are obtained. In the non-parametric case, point and interval estimates of R are developed using Govindarajulu's asymptotic distribution-free method when X1 and X2 are dependent. An example is given when the population distribution is bivariate compound Weibull. Simulation is performed, based on different sample sizes to study the performance of estimates.

  4. Non-existence criteria for Laurent polynomial first integrals

    Directory of Open Access Journals (Sweden)

    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.

  5. 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

  6. 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....

  7. Stochastic Estimation via Polynomial Chaos

    Science.gov (United States)

    2015-10-01

    processes. As originally developed by Norbert Weiner, a polynomial chaos represents key properties of a stochastic process through the application of...polynomial chaos method for representing the properties of second order stochastic processes. As originally developed by Norbert Weiner, a...any real or complex functional ][xF in 2L converges to ][xF in the 2L sense with Wiener measure.[3] In the same reference, the orthogonality of

  8. On Modular Counting with Polynomials

    DEFF Research Database (Denmark)

    Hansen, Kristoffer Arnsfelt

    2006-01-01

    For any integers m and l, where m has r sufficiently large (depending on l) factors, that are powers of r distinct primes, we give a construction of a (symmetric) polynomial over Z_m of degree O(\\sqrt n) that is a generalized representation (commonly also called weak representation) of the MODl...... function. We give a detailed study of the case when m has exactly two distinct prime factors, and classify the minimum possible degree for a symmetric representing polynomial....

  9. 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

  10. 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.

  11. A 'missing' family of classical orthogonal polynomials

    International Nuclear Information System (INIS)

    Vinet, Luc; Zhedanov, Alexei

    2011-01-01

    We study a family of 'classical' orthogonal polynomials which satisfy (apart from a three-term recurrence relation) an eigenvalue problem with a differential operator of Dunkl type. These polynomials can be obtained from the little q-Jacobi polynomials in the limit q = -1. We also show that these polynomials provide a nontrivial realization of the Askey-Wilson algebra for q = -1.

  12. 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

  13. 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 prove...

  14. On Multiple Polynomials of Capelli Type

    Directory of Open Access Journals (Sweden)

    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.

  15. STUDI PERBANDINGAN ANTARA ALGORITMA BIVARIATE MARGINAL DISTRIBUTION DENGAN ALGORITMA GENETIKA

    Directory of Open Access Journals (Sweden)

    Chastine Fatichah

    2006-01-01

    Full Text Available Bivariate Marginal Distribution Algorithm is extended from Estimation of Distribution Algorithm. This heuristic algorithm proposes the new approach for recombination of generate new individual that without crossover and mutation process such as genetic algorithm. Bivariate Marginal Distribution Algorithm uses connectivity variable the pair gene for recombination of generate new individual. Connectivity between variable is doing along optimization process. In this research, genetic algorithm performance with one point crossover is compared with Bivariate Marginal Distribution Algorithm performance in case Onemax, De Jong F2 function, and Traveling Salesman Problem. In this research, experimental results have shown performance the both algorithm is dependence of parameter respectively and also population size that used. For Onemax case with size small problem, Genetic Algorithm perform better with small number of iteration and more fast for get optimum result. However, Bivariate Marginal Distribution Algorithm perform better of result optimization for case Onemax with huge size problem. For De Jong F2 function, Genetic Algorithm perform better from Bivariate Marginal Distribution Algorithm of a number of iteration and time. For case Traveling Salesman Problem, Bivariate Marginal Distribution Algorithm have shown perform better from Genetic Algorithm of optimization result. Abstract in Bahasa Indonesia : Bivariate Marginal Distribution Algorithm merupakan perkembangan lebih lanjut dari Estimation of Distribution Algorithm. Algoritma heuristik ini mengenalkan pendekatan baru dalam melakukan rekombinasi untuk membentuk individu baru, yaitu tidak menggunakan proses crossover dan mutasi seperti pada Genetic Algorithm. Bivariate Marginal Distribution Algorithm menggunakan keterkaitan pasangan variabel dalam melakukan rekombinasi untuk membentuk individu baru. Keterkaitan antar variabel tersebut ditemukan selama proses optimasi berlangsung. Aplikasi yang

  16. Approximation of bivariate copulas by patched bivariate Fréchet copulas

    KAUST Repository

    Zheng, Yanting

    2011-03-01

    Bivariate Fréchet (BF) copulas characterize dependence as a mixture of three simple structures: comonotonicity, independence and countermonotonicity. They are easily interpretable but have limitations when used as approximations to general dependence structures. To improve the approximation property of the BF copulas and keep the advantage of easy interpretation, we develop a new copula approximation scheme by using BF copulas locally and patching the local pieces together. Error bounds and a probabilistic interpretation of this approximation scheme are developed. The new approximation scheme is compared with several existing copula approximations, including shuffle of min, checkmin, checkerboard and Bernstein approximations and exhibits better performance, especially in characterizing the local dependence. The utility of the new approximation scheme in insurance and finance is illustrated in the computation of the rainbow option prices and stop-loss premiums. © 2010 Elsevier B.V.

  17. A note on finding peakedness in bivariate normal distribution using Mathematica

    Directory of Open Access Journals (Sweden)

    Anwer Khurshid

    2007-07-01

    Full Text Available Peakedness measures the concentration around the central value. A classical standard measure of peakedness is kurtosis which is the degree of peakedness of a probability distribution. In view of inconsistency of kurtosis in measuring of the peakedness of a distribution, Horn (1983 proposed a measure of peakedness for symmetrically unimodal distributions. The objective of this paper is two-fold. First, Horn’s method has been extended for bivariate normal distribution. Secondly, to show that computer algebra system Mathematica can be extremely useful tool for all sorts of computation related to bivariate normal distribution. Mathematica programs are also provided.

  18. Design of robust differential microphone arrays with orthogonal polynomials.

    Science.gov (United States)

    Pan, Chao; Benesty, Jacob; Chen, Jingdong

    2015-08-01

    Differential microphone arrays have the potential to be widely deployed in hands-free communication systems thanks to their frequency-invariant beampatterns, high directivity factors, and small apertures. Traditionally, they are designed and implemented in a multistage way with uniform linear geometries. This paper presents an approach to the design of differential microphone arrays with orthogonal polynomials, more specifically with Jacobi polynomials. It first shows how to express the beampatterns as a function of orthogonal polynomials. Then several differential beamformers are derived and their performance depends on the parameters of the Jacobi polynomials. Simulations show the great flexibility of the proposed method in terms of designing any order differential microphone arrays with different beampatterns and controlling white noise gain.

  19. Classification of Knee Joint Vibration Signals Using Bivariate Feature Distribution Estimation and Maximal Posterior Probability Decision Criterion

    Directory of Open Access Journals (Sweden)

    Fang Zheng

    2013-04-01

    Full Text Available Analysis of knee joint vibration or vibroarthrographic (VAG signals using signal processing and machine learning algorithms possesses high potential for the noninvasive detection of articular cartilage degeneration, which may reduce unnecessary exploratory surgery. Feature representation of knee joint VAG signals helps characterize the pathological condition of degenerative articular cartilages in the knee. This paper used the kernel-based probability density estimation method to model the distributions of the VAG signals recorded from healthy subjects and patients with knee joint disorders. The estimated densities of the VAG signals showed explicit distributions of the normal and abnormal signal groups, along with the corresponding contours in the bivariate feature space. The signal classifications were performed by using the Fisher’s linear discriminant analysis, support vector machine with polynomial kernels, and the maximal posterior probability decision criterion. The maximal posterior probability decision criterion was able to provide the total classification accuracy of 86.67% and the area (Az of 0.9096 under the receiver operating characteristics curve, which were superior to the results obtained by either the Fisher’s linear discriminant analysis (accuracy: 81.33%, Az: 0.8564 or the support vector machine with polynomial kernels (accuracy: 81.33%, Az: 0.8533. Such results demonstrated the merits of the bivariate feature distribution estimation and the superiority of the maximal posterior probability decision criterion for analysis of knee joint VAG signals.

  20. 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.

  1. Plain Polynomial Arithmetic on GPU

    International Nuclear Information System (INIS)

    Haque, Sardar Anisul; Maza, Marc Moreno

    2012-01-01

    As for serial code on CPUs, parallel code on GPUs for dense polynomial arithmetic relies on a combination of asymptotically fast and plain algorithms. Those are employed for data of large and small size, respectively. Parallelizing both types of algorithms is required in order to achieve peak performances. In this paper, we show that the plain dense polynomial multiplication can be efficiently parallelized on GPUs. Remarkably, it outperforms (highly optimized) FFT-based multiplication up to degree 2 12 while on CPU the same threshold is usually at 2 6 . We also report on a GPU implementation of the Euclidean Algorithm which is both work-efficient and runs in linear time for input polynomials up to degree 2 18 thus showing the performance of the GCD algorithm based on systolic arrays.

  2. Orthogonal Polynomials and their Applications

    CERN Document Server

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

    1988-01-01

    The Segovia meeting set out to stimulate an intensive exchange of ideas between experts in the area of orthogonal polynomials and its applications, to present recent research results and to reinforce the scientific and human relations among the increasingly international community working in orthogonal polynomials. This volume contains original research papers as well as survey papers about fundamental questions in the field (Nevai, Rakhmanov & López) and its relationship with other fields such as group theory (Koornwinder), Padé approximation (Brezinski), differential equations (Krall, Littlejohn) and numerical methods (Rivlin).

  3. Recursive formula to compute Zernike radial polynomials.

    Science.gov (United States)

    Honarvar Shakibaei, Barmak; Paramesran, Raveendran

    2013-07-15

    In optics, Zernike polynomials are widely used in testing, wavefront sensing, and aberration theory. This unique set of radial polynomials is orthogonal over the unit circle and finite on its boundary. This Letter presents a recursive formula to compute Zernike radial polynomials using a relationship between radial polynomials and Chebyshev polynomials of the second kind. Unlike the previous algorithms, the derived recurrence relation depends neither on the degree nor on the azimuthal order of the radial polynomials. This leads to a reduction in the computational complexity.

  4. Super-A-polynomial for knots and BPS states

    International Nuclear Information System (INIS)

    Fuji, Hiroyuki; Gukov, Sergei; Sułkowski, Piotr

    2013-01-01

    We introduce and compute a 2-parameter family deformation of the A-polynomial that encodes the color dependence of the superpolynomial and that, in suitable limits, reduces to various deformations of the A-polynomial studied in the literature. These special limits include the t-deformation which leads to the “refined A-polynomial” introduced in the previous work of the authors and the Q-deformation which leads, by the conjecture of Aganagic and Vafa, to the augmentation polynomial of knot contact homology. We also introduce and compute the quantum version of the super-A-polynomial, an operator that encodes recursion relations for S r -colored HOMFLY homology. Much like its predecessor, the super-A-polynomial admits a simple physical interpretation as the defining equation for the space of SUSY vacua (= critical points of the twisted superpotential) in a circle compactification of the effective 3d N=2 theory associated to a knot or, more generally, to a 3-manifold M. Equivalently, the algebraic curve defined by the zero locus of the super-A-polynomial can be thought of as the space of open string moduli in a brane system associated with M. As an inherent outcome of this work, we provide new interesting formulas for colored superpolynomials for the trefoil and the figure-eight knot.

  5. Global Polynomial Kernel Hazard Estimation

    DEFF Research Database (Denmark)

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

    2015-01-01

    This paper introduces a new bias reducing method for kernel hazard estimation. The method is called global polynomial adjustment (GPA). It is a global correction which is applicable to any kernel hazard estimator. The estimator works well from a theoretical point of view as it asymptotically redu...

  6. Polynomial Regressions and Nonsense Inference

    DEFF Research Database (Denmark)

    Ventosa-Santaulària, Daniel; Rodríguez-Caballero, Carlos Vladimir

    Polynomial specifications are widely used, not only in applied economics, but also in epidemiology, physics, political analysis, and psychology, just to mention a few examples. In many cases, the data employed to estimate such estimations are time series that may exhibit stochastic nonstationary ...

  7. Dirichlet polynomials, majorization, and trumping

    International Nuclear Information System (INIS)

    Pereira, Rajesh; Plosker, Sarah

    2013-01-01

    Majorization and trumping are two partial orders which have proved useful in quantum information theory. We show some relations between these two partial orders and generalized Dirichlet polynomials, Mellin transforms, and completely monotone functions. These relations are used to prove a succinct generalization of Turgut’s characterization of trumping. (paper)

  8. Two polynomial division inequalities in

    Directory of Open Access Journals (Sweden)

    Goetgheluck P

    1998-01-01

    Full Text Available This paper is a first attempt to give numerical values for constants and , in classical estimates and where is an algebraic polynomial of degree at most and denotes the -metric on . The basic tools are Markov and Bernstein inequalities.

  9. Aspects of the Tutte polynomial

    DEFF Research Database (Denmark)

    Ok, Seongmin

    This thesis studies various aspects of the Tutte polynomial, especially focusing on the Merino-Welsh conjecture. We write T(G;x,y) for the Tutte polynomial of a graph G with variables x and y. In 1999, Merino and Welsh conjectured that if G is a loopless 2-connected graph, then T(G;1,1) ≤ max{T(G;2......,0), T(G;0,2)}. The three numbers, T(G;1,1), T(G;2,0) and T(G;0,2) are respectively the numbers of spanning trees, acyclic orientations and totally cyclic orientations of G. First, I extend Negami's splitting formula to the multivariate Tutte polynomial. Using the splitting formula, Thomassen and I found......-Welsh conjecture has a stronger version claiming that the Tutte polynomial is convex on the line segment between (2,0) and (0,2) for loopless 2-connected graphs. Chavez-Lomeli et al. proved that this holds for coloopless paving matroids, and I provide a shorter proof of their theorem. I also prove it for minimally...

  10. 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.

  11. Tables of properties of airfoil polynomials

    Science.gov (United States)

    Desmarais, Robert N.; Bland, Samuel R.

    1995-01-01

    This monograph provides an extensive list of formulas for airfoil polynomials. These polynomials provide convenient expansion functions for the description of the downwash and pressure distributions of linear theory for airfoils in both steady and unsteady subsonic flow.

  12. Random walk polynomials and random walk measures

    NARCIS (Netherlands)

    van Doorn, Erik A.; Schrijner, Pauline

    1993-01-01

    Random walk polynomials and random walk measures play a prominent role in the analysis of a class of Markov chains called random walks. Without any reference to random walks, however, a random walk polynomial sequence can be defined (and will be defined in this paper) as a polynomial sequence{Pn(x)}

  13. Positive trigonometric polynomials and signal processing applications

    CERN Document Server

    Dumitrescu, Bogdan

    2007-01-01

    Presents the results on positive trigonometric polynomials within a unitary framework; the theoretical results obtained partly from the general theory of real polynomials, partly from self-sustained developments. This book provides information on the theory of sum-of-squares trigonometric polynomials in two parts: theory and applications.

  14. 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

  15. Sibling curves of polynomials | Wiggins | Quaestiones Mathematicae

    African Journals Online (AJOL)

    Sibling curves were demonstrated in papers [2, 3] as a novel way to visualize the zeros of complex valued functions. In this paper, we continue the work done in those papers by focusing solely on polynomials. We proceed to prove that the number of sibling curves of a polynomial is the degree of the polynomial. Keywords: ...

  16. Restricted Schur polynomials and finite N counting

    International Nuclear Information System (INIS)

    Collins, Storm

    2009-01-01

    Restricted Schur polynomials have been posited as orthonormal operators for the change of basis from N=4 SYM to type IIB string theory. In this paper we briefly expound the relationship between the restricted Schur polynomials and the operators forwarded by Brown, Heslop, and Ramgoolam. We then briefly examine the finite N counting of the restricted Schur polynomials.

  17. On polynomial and multiplicative radicals | Tumurbat | Quaestiones ...

    African Journals Online (AJOL)

    We show that polynomial and multiplicative radicals in [1] are special cases of radicals defined by means of elements. We scrutinize the way of defining a radical γG by a subset G of polynomials in noncommuting indeterminates. Defining polynomial radicals, Drazin and Roberts [1] required that the set G be closed

  18. Hermite polynomials in asymptotic representations of generalized Bernoulli,Euler, Bessel and Buchholz polynomials

    NARCIS (Netherlands)

    J.L. López; N.M. Temme (Nico)

    1999-01-01

    textabstractThis is a second paper on finite exact representations of certain polynomials in terms of Hermite polynomials. The representations have asymptotic properties and include new limits of the polynomials, again in terms of Hermite polynomials. This time we consider the generalized Bernoulli,

  19. Special polynomials associated with rational solutions of some hierarchies

    NARCIS (Netherlands)

    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

  20. Dirac(-Pauli), Fokker-Planck equations and exceptional Laguerre polynomials

    International Nuclear Information System (INIS)

    Ho, Choon-Lin

    2011-01-01

    Research highlights: → Physical examples involving exceptional orthogonal polynomials. → Exceptional polynomials as deformations of classical orthogonal polynomials. → Exceptional polynomials from Darboux-Crum transformation. - Abstract: An interesting discovery in the last two years in the field of mathematical physics has been the exceptional X l Laguerre and Jacobi polynomials. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have lowest degree l = 1, 2, and ..., and yet they form complete set with respect to some positive-definite measure. While the mathematical properties of these new X l polynomials deserve further analysis, it is also of interest to see if they play any role in physical systems. In this paper we indicate some physical models in which these new polynomials appear as the main part of the eigenfunctions. The systems we consider include the Dirac equations coupled minimally and non-minimally with some external fields, and the Fokker-Planck equations. The systems presented here have enlarged the number of exactly solvable physical systems known so far.

  1. 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

  2. Polynomial Mappings with Small Degree

    Directory of Open Access Journals (Sweden)

    Jelonek Zbigniew

    2015-06-01

    Full Text Available Let Xn be an affine variety of dimension n and Yn be a quasi-projective variety of the same dimension. We prove that for a quasi-finite polynomial mapping ƒ : Xn → Yn, every non-empty component of the set Yn\\ ƒ (Xn is closed and it has dimension greater or equal to n μ(ƒ, where μ(ƒ is a geometric degree of ƒ. Moreover, we prove that generally, if ƒ : Xn → Yn is any polynomial mapping, then either every non-empty component of the set is of dimension ≥ n μ(ƒ or ƒ contracts a subvariety of dimension ≥ n μ(ƒ + 1.

  3. 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.

  4. A bivariate model for analyzing recurrent multi-type automobile failures

    Science.gov (United States)

    Sunethra, A. A.; Sooriyarachchi, M. R.

    2017-09-01

    The failure mechanism in an automobile can be defined as a system of multi-type recurrent failures where failures can occur due to various multi-type failure modes and these failures are repetitive such that more than one failure can occur from each failure mode. In analysing such automobile failures, both the time and type of the failure serve as response variables. However, these two response variables are highly correlated with each other since the timing of failures has an association with the mode of the failure. When there are more than one correlated response variables, the fitting of a multivariate model is more preferable than separate univariate models. Therefore, a bivariate model of time and type of failure becomes appealing for such automobile failure data. When there are multiple failure observations pertaining to a single automobile, such data cannot be treated as independent data because failure instances of a single automobile are correlated with each other while failures among different automobiles can be treated as independent. Therefore, this study proposes a bivariate model consisting time and type of failure as responses adjusted for correlated data. The proposed model was formulated following the approaches of shared parameter models and random effects models for joining the responses and for representing the correlated data respectively. The proposed model is applied to a sample of automobile failures with three types of failure modes and up to five failure recurrences. The parametric distributions that were suitable for the two responses of time to failure and type of failure were Weibull distribution and multinomial distribution respectively. The proposed bivariate model was programmed in SAS Procedure Proc NLMIXED by user programming appropriate likelihood functions. The performance of the bivariate model was compared with separate univariate models fitted for the two responses and it was identified that better performance is secured by

  5. Poly-Cauchy and Peters mixed-type polynomials

    OpenAIRE

    Kim, Dae San; Kim, Taekyun

    2013-01-01

    The Peters polynomials are a generalization of Boole polynomials. In this paper, we consider Peters and poly-Cauchy mixed type polynomials and investigate the properties of those polynomials which are derived from umbral calculus. Finally, we give various identities of those polynomials associated with special polynomials.

  6. Evolutionary Design of Polynomial Controller

    OpenAIRE

    R. Matousek; S. Lang; P. Minar; P. Pivonka

    2011-01-01

    In the control theory one attempts to find a controller that provides the best possible performance with respect to some given measures of performance. There are many sorts of controllers e.g. a typical PID controller, LQR controller, Fuzzy controller etc. In the paper will be introduced polynomial controller with novel tuning method which is based on the special pole placement encoding scheme and optimization by Genetic Algorithms (GA). The examples will show the perform...

  7. Space complexity in polynomial calculus

    Czech Academy of Sciences Publication Activity Database

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

    2015-01-01

    Roč. 44, č. 4 (2015), s. 1119-1153 ISSN 0097-5397 R&D Projects: GA AV ČR IAA100190902; GA ČR GBP202/12/G061 Institutional support: RVO:67985840 Keywords : proof complexity * polynomial calculus * lower bounds Subject RIV: BA - General Mathematics Impact factor: 0.841, year: 2015 http://epubs.siam.org/doi/10.1137/120895950

  8. Codimensions of generalized polynomial identities

    International Nuclear Information System (INIS)

    Gordienko, Aleksei S

    2010-01-01

    It is proved that for every finite-dimensional associative algebra A over a field of characteristic zero there are numbers C element of Q + and t element of Z + such that gc n (A)∼Cn t d n as n→∞, where d=PI exp(A) element of Z + . Thus, Amitsur's and Regev's conjectures hold for the codimensions gc n (A) of the generalized polynomial identities. Bibliography: 6 titles.

  9. 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.

  10. A C++ package for assembling quantum circuits and generating associated polynomial sets

    International Nuclear Information System (INIS)

    Gerdt, V.P.; Sever'yanov, V.M.

    2007-01-01

    Recently it has been shown that elements of the unitary matrix determined by a quantum circuit can be computed by counting the number of common roots in the finite field Z 2 for a certain set of multivariate polynomials over Z 2 . In this paper we present a C++ package that allows a user to assemble a quantum circuit and to generate the multivariate polynomial set associated with the circuit. The generated polynomial system can further be converted to the canonical triangular involutive basis form which is appropriate for counting the number of common roots of the polynomials

  11. mitants of Order Statistics from Bivariate Inverse Rayleigh Distribution

    Directory of Open Access Journals (Sweden)

    Muhammad Aleem

    2006-01-01

    Full Text Available The probability density function (pdf of the rth, 1 r n and joint pdf of the rth and sth, 1 rBivariate Inverse Rayleigh Distribution and their moments, product moments are obtained. Its percentiles are also obtained.

  12. GIS-based bivariate statistical techniques for groundwater potential ...

    Indian Academy of Sciences (India)

    Groundwater potential analysis prepares better comprehension of hydrological settings of different regions. This study shows the potency of two GIS-based data driven bivariate techniques namely statistical index (SI) and Dempster–Shafer theory (DST) to analyze groundwater potential in Broujerd region of Iran.

  13. Dissecting the correlation structure of a bivariate phenotype ...

    Indian Academy of Sciences (India)

    Home; Journals; Journal of Genetics; Volume 84; Issue 2. Dissecting the correlation structure of a bivariate phenotype: common genes or shared environment? ... High correlations between two quantitative traits may be either due to common genetic factors or common environmental factors or a combination of both.

  14. Modelling of Uncertainty and Bi-Variable Maps

    Science.gov (United States)

    Nánásiová, Ol'ga; Pykacz, Jarosław

    2016-05-01

    The paper gives an overview and compares various bi-varilable maps from orthomodular lattices into unit interval. It focuses mainly on such bi-variable maps that may be used for constructing joint probability distributions for random variables which are not defined on the same Boolean algebra.

  15. An assessment on the use of bivariate, multivariate and soft ...

    Indian Academy of Sciences (India)

    Conditional probability (CP), logistic regression (LR) and artificial neural networks (ANN) models representing the bivariate, multivariate and soft computing techniques were used in GIS based collapse susceptibility mapping in an area from Sivas basin (Turkey). Collapse-related factors, directly or indirectly related to the ...

  16. An assessment on the use of bivariate, multivariate and soft ...

    Indian Academy of Sciences (India)

    The paper presented herein compares and discusses the use of bivariate, multivariate and soft computing techniques for ... map is a useful tool in urban planning. ..... 381. Table 1. Frequency ratio of geological factors to collapse occurrences and results of the P(A/Bi) obtained from the. Conditional Probability model. Class.

  17. Comparison between two bivariate Poisson distributions through the ...

    African Journals Online (AJOL)

    To remedy this problem, Berkhout and Plug proposed a bivariate Poisson distribution accepting the correlation as well negative, equal to zero, that positive. In this paper, we show that these models are nearly everywhere asymptotically equal. From this survey that the ø-divergence converges toward zero, both models are ...

  18. About some properties of bivariate splines with shape parameters

    Science.gov (United States)

    Caliò, F.; Marchetti, E.

    2017-07-01

    The paper presents and proves geometrical properties of a particular bivariate function spline, built and algorithmically implemented in previous papers. The properties typical of this family of splines impact the field of computer graphics in particular that of the reverse engineering.

  19. The matrix and polynomial approaches to Lanczos-type algorithms

    Science.gov (United States)

    Brezinski, C.; Redivo-Zaglia, M.; Sadok, H.

    2000-11-01

    Lanczos method for solving a system of linear equations can be derived by using formal orthogonal polynomials. It can be implemented by several recurrence relationships, thus leading to several algorithms. In this paper, the Lanczos/Orthodir algorithm will be derived in two different ways. The first one is based on a matrix approach and on the recursive computation of two successive regular matrices. We will show that it can be directly obtained from the orthogonality conditions and the fact that Lanczos method is a Krylov subspace method. The second approach is based on formal orthogonal polynomials. The case of breakdowns will be treated similarly.

  20. On Chebyshev Polynomials of Matrices

    Czech Academy of Sciences Publication Activity Database

    Faber, V.; Liesen, J.; Tichý, Petr

    2010-01-01

    Roč. 31, č. 4 (2010), s. 2205-2221 ISSN 0895-4798 R&D Projects: GA AV ČR IAA100300802 Grant - others:GA AV ČR(CZ) M100300901 Institutional research plan: CEZ:AV0Z10300504 Source of funding: I - inštitucionálna podpora na rozvoj VO Keywords : matrix approximation problems * Chebyshev polynomials * complex approximation theory * Krylov subspace methods * Arnoldi's method Subject RIV: BA - General Mathematics Impact factor: 1.725, year: 2010

  1. 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.

  2. 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.

  3. On classical orthogonal polynomials and differential operators

    International Nuclear Information System (INIS)

    Miranian, L

    2005-01-01

    It is well known that the four families of classical orthogonal polynomials (Jacobi, Bessel, Hermite and Laguerre) each satisfy an equation FP n (x) = λ n P n (x), n ≥ 0, for an appropriate second-order differential operator F. In this paper it is shown that any linear differential operator U which has the Jacobi, Bessel, Hermite or Laguerre polynomials as eigenfunctions has to be a polynomial with constant coefficients in the classical second-order operator F

  4. Model selection for univariable fractional polynomials.

    Science.gov (United States)

    Royston, Patrick

    2017-07-01

    Since Royston and Altman's 1994 publication ( Journal of the Royal Statistical Society, Series C 43: 429-467), fractional polynomials have steadily gained popularity as a tool for flexible parametric modeling of regression relationships. In this article, I present fp_select, a postestimation tool for fp that allows the user to select a parsimonious fractional polynomial model according to a closed test procedure called the fractional polynomial selection procedure or function selection procedure. I also give a brief introduction to fractional polynomial models and provide examples of using fp and fp_select to select such models with real data.

  5. Acceleration of computation of φ-polynomials.

    Science.gov (United States)

    Kaya, Ilhan; Rolland, Jannick

    2013-11-18

    The benefits of making an effective use of impressive computational power offered by multi-core platforms are investigated for the computation of φ-polynomials used in the description of freeform surfaces. Specifically, we devise parallel algorithms based upon the recurrence relations of both Zernike polynomials and gradient orthogonal Q-polynomials and implement these parallel algorithms on Graphical Processing Units (GPUs) respectively. The results show that more than an order of magnitude improvement is achieved in computational time over a sequential implementation if these recurrence-based parallel algorithms are adopted in the computation of the φ-polynomials.

  6. The q-Laguerre matrix polynomials.

    Science.gov (United States)

    Salem, Ahmed

    2016-01-01

    The Laguerre polynomials have been extended to Laguerre matrix polynomials by means of studying certain second-order matrix differential equation. In this paper, certain second-order matrix q-difference equation is investigated and solved. Its solution gives a generalized of the q-Laguerre polynomials in matrix variable. Four generating functions of this matrix polynomials are investigated. Two slightly different explicit forms are introduced. Three-term recurrence relation, Rodrigues-type formula and the q-orthogonality property are given.

  7. Orthonormal polynomials in wavefront analysis: analytical solution.

    Science.gov (United States)

    Mahajan, Virendra N; Dai, Guang-ming

    2007-09-01

    Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. In recent papers, we derived closed-form polynomials that are orthonormal over a hexagonal pupil, such as the hexagonal segments of a large mirror. We extend our work to elliptical, rectangular, and square pupils. Using the circle polynomials as the basis functions for their orthogonalization over such pupils, we derive closed-form polynomials that are orthonormal over them. These polynomials are unique in that they are not only orthogonal across such pupils, but also represent balanced classical aberrations, just as the Zernike circle polynomials are unique in these respects for circular pupils. The polynomials are given in terms of the circle polynomials as well as in polar and Cartesian coordinates. Relationships between the orthonormal coefficients and the corresponding Zernike coefficients for a given pupil are also obtained. The orthonormal polynomials for a one-dimensional slit pupil are obtained as a limiting case of a rectangular pupil.

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

    Directory of Open Access Journals (Sweden)

    Ryoo CS

    2010-01-01

    Full Text Available The purpose of this paper is to give some properties of several Bernstein type polynomials to represent the fermionic -adic integral on . From these properties, we derive some interesting identities on the Euler numbers and polynomials.

  9. Bivariate extreme value with application to PM10 concentration analysis

    Science.gov (United States)

    Amin, Nor Azrita Mohd; Adam, Mohd Bakri; Ibrahim, Noor Akma; Aris, Ahmad Zaharin

    2015-05-01

    This study is focus on a bivariate extreme of renormalized componentwise maxima with generalized extreme value distribution as a marginal function. The limiting joint distribution of several parametric models are presented. Maximum likelihood estimation is employed for parameter estimations and the best model is selected based on the Akaike Information Criterion. The weekly and monthly componentwise maxima series are extracted from the original observations of daily maxima PM10 data for two air quality monitoring stations located in Pasir Gudang and Johor Bahru. The 10 years data are considered for both stations from year 2001 to 2010. The asymmetric negative logistic model is found as the best fit bivariate extreme model for both weekly and monthly maxima componentwise series. However the dependence parameters show that the variables for weekly maxima series is more dependence to each other compared to the monthly maxima.

  10. Univariate and Bivariate Empirical Mode Decomposition for Postural Stability Analysis

    Directory of Open Access Journals (Sweden)

    Jacques Duchêne

    2008-05-01

    Full Text Available The aim of this paper was to compare empirical mode decomposition (EMD and two new extended methods of  EMD named complex empirical mode decomposition (complex-EMD and bivariate empirical mode decomposition (bivariate-EMD. All methods were used to analyze stabilogram center of pressure (COP time series. The two new methods are suitable to be applied to complex time series to extract complex intrinsic mode functions (IMFs before the Hilbert transform is subsequently applied on the IMFs. The trace of the analytic IMF in the complex plane has a circular form, with each IMF having its own rotation frequency. The area of the circle and the average rotation frequency of IMFs represent efficient indicators of the postural stability status of subjects. Experimental results show the effectiveness of these indicators to identify differences in standing posture between groups.

  11. Probability distributions with truncated, log and bivariate extensions

    CERN Document Server

    Thomopoulos, Nick T

    2018-01-01

    This volume presents a concise and practical overview of statistical methods and tables not readily available in other publications. It begins with a review of the commonly used continuous and discrete probability distributions. Several useful distributions that are not so common and less understood are described with examples and applications in full detail: discrete normal, left-partial, right-partial, left-truncated normal, right-truncated normal, lognormal, bivariate normal, and bivariate lognormal. Table values are provided with examples that enable researchers to easily apply the distributions to real applications and sample data. The left- and right-truncated normal distributions offer a wide variety of shapes in contrast to the symmetrically shaped normal distribution, and a newly developed spread ratio enables analysts to determine which of the three distributions best fits a particular set of sample data. The book will be highly useful to anyone who does statistical and probability analysis. This in...

  12. Spectrum-based estimators of the bivariate Hurst exponent

    Czech Academy of Sciences Publication Activity Database

    Krištoufek, Ladislav

    2014-01-01

    Roč. 90, č. 6 (2014), art. 062802 ISSN 1539-3755 R&D Projects: GA ČR(CZ) GP14-11402P Institutional support: RVO:67985556 Keywords : bivariate Hurst exponent * power- law cross-correlations * estimation Subject RIV: AH - Economics Impact factor: 2.288, year: 2014 http://library.utia.cas.cz/separaty/2014/E/kristoufek-0436818.pdf

  13. A reduced polynomial chaos expansion method for the stochastic ...

    Indian Academy of Sciences (India)

    The stochastic finite element analysis of elliptic type partial differential equations is considered. A reduced method of the spectral stochastic finite element method using polynomial chaos is proposed. The method is based on the spectral decomposition of the deterministic system matrix. The reduction is achieved by ...

  14. 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…

  15. A general polynomial solution to convection–dispersion equation ...

    Indian Academy of Sciences (India)

    Jiao Wang

    A number of models have been established to simulate the behaviour of solute transport due to chemical pollution, both in croplands and groundwater systems. An approximate polynomial solution to convection–dispersion equation (CDE) based on boundary layer theory has been verified for the use to describe solute ...

  16. Entropic functionals of Laguerre and Gegenbauer polynomials with large parameters

    NARCIS (Netherlands)

    N.M. Temme (Nico); I.V. Toranzo; J.S. Dehesa

    2017-01-01

    textabstractThe determination of the physical entropies (Rényi, Shannon, Tsallis) of high-dimensional quantum systems subject to a central potential requires the knowledge of the asymptotics of some power and logarithmic integral functionals of the hypergeometric orthogonal polynomials which control

  17. A new type of hermite matrix polynomial series | Defez | Quaestiones ...

    African Journals Online (AJOL)

    Conventional Hermite polynomials emerge in a great diversity of applications in mathematical physics, engineering, and related fields. However, in physical systems with higher degrees of freedom it will be of practical interest to extend the scalar Hermite functions to their matrix analogue. This work introduces various new ...

  18. Orthogonal polynomials from Hermitian matrices. II

    Science.gov (United States)

    Odake, Satoru; Sasaki, Ryu

    2018-01-01

    This is the second part of the project "unified theory of classical orthogonal polynomials of a discrete variable derived from the eigenvalue problems of Hermitian matrices." In a previous paper, orthogonal polynomials having Jackson integral measures were not included, since such measures cannot be obtained from single infinite dimensional Hermitian matrices. Here we show that Jackson integral measures for the polynomials of the big q-Jacobi family are the consequence of the recovery of self-adjointness of the unbounded Jacobi matrices governing the difference equations of these polynomials. The recovery of self-adjointness is achieved in an extended ℓ2 Hilbert space on which a direct sum of two unbounded Jacobi matrices acts as a Hamiltonian or a difference Schrödinger operator for an infinite dimensional eigenvalue problem. The polynomial appearing in the upper/lower end of the Jackson integral constitutes the eigenvector of each of the two-unbounded Jacobi matrix of the direct sum. We also point out that the orthogonal vectors involving the q-Meixner (q-Charlier) polynomials do not form a complete basis of the ℓ2 Hilbert space, based on the fact that the dual q-Meixner polynomials introduced in a previous paper fail to satisfy the orthogonality relation. The complete set of eigenvectors involving the q-Meixner polynomials is obtained by constructing the duals of the dual q-Meixner polynomials which require the two-component Hamiltonian formulation. An alternative solution method based on the closure relation, the Heisenberg operator solution, is applied to the polynomials of the big q-Jacobi family and their duals and q-Meixner (q-Charlier) polynomials.

  19. Integral of exponent of a polynomial is a generalized hypergeometric function of the coefficients of the polynomial

    OpenAIRE

    Stoyanovsky, Alexander

    2010-01-01

    We show that the integral \\int e^{S(x_1,...,x_n)}dx_1...dx_n, for an arbitrary polynomial S, satisfies a generalized hypergeometric system of differential equations in the sense of I. M. Gelfand et al.

  20. 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.

  1. 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.

  2. 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)

  3. Integral inequalities for self-reciprocal polynomials

    Indian Academy of Sciences (India)

    that is, the coefficients satisfy ak = an−k for k = 0, 1,...,n. We denote the class of self-reciprocal polynomials of order n by Pn. The properties of self-reciprocal polynomials have been studied by several mathemati- cians. We refer to [1–5] and the references given therein. Here, we are concerned with the following interesting ...

  4. 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.

  5. 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 ...

  6. 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

  7. Exceptional polynomials and SUSY quantum mechanics

    Indian Academy of Sciences (India)

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

  8. 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)

  9. 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

  10. 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

  11. Block orthogonal polynomials: I. Definitions and properties

    International Nuclear Information System (INIS)

    Normand, Jean-Marie

    2007-01-01

    Constrained orthogonal polynomials have been recently introduced in the study of the Hohenberg-Kohn functional to provide basis functions satisfying particle number conservation for an expansion of the particle density. More generally, we define block orthogonal (BO) polynomials which are orthogonal, with respect to a first Euclidean inner product, to a given i-dimensional subspace E i of polynomials associated with the constraints. In addition, they are mutually orthogonal with respect to a second Euclidean inner product. We recast the determination of these polynomials into a general problem of finding particular orthogonal bases in an Euclidean vector space endowed with distinct inner products. An explicit two step Gram-Schmidt orthogonalization (G-SO) process to determine these bases is given. By definition, the standard block orthogonal (SBO) polynomials are associated with a choice of E i equal to the subspace of polynomials of degree less than i. We investigate their properties, emphasizing similarities to and differences from the standard orthogonal polynomials. Applications to classical orthogonal polynomials will be given in forthcoming papers

  12. The Medusa Algorithm for Polynomial Matings

    DEFF Research Database (Denmark)

    Boyd, Suzanne Hruska; Henriksen, Christian

    2012-01-01

    The Medusa algorithm takes as input two postcritically finite quadratic polynomials and outputs the quadratic rational map which is the mating of the two polynomials (if it exists). Specifically, the output is a sequence of approximations for the parameters of the rational map, as well as an image...

  13. On the zeros of a polynomial

    Indian Academy of Sciences (India)

    For a polynomial of degree , we have obtained an upper bound involving coefficients of the polynomial, for moduli of its zeros of smallest moduli, and then a refinement of the well-known Eneström–Kakeya theorem (under certain conditions).

  14. Polynomial approximation approach to transient heat conduction ...

    African Journals Online (AJOL)

    This work reports polynomial approximation approach to transient heat conduction in a long slab, long cylinder and sphere with linear internal heat generation. It has been shown that the polynomial approximation method is able to calculate average temperature as a function of time for higher value of Biot numbers.

  15. Orthonormal polynomials in wavefront analysis: error analysis.

    Science.gov (United States)

    Dai, Guang-Ming; Mahajan, Virendra N

    2008-07-01

    Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. However, they are not appropriate for noncircular pupils, such as annular, hexagonal, elliptical, rectangular, and square pupils, due to their lack of orthogonality over such pupils. We emphasize the use of orthonormal polynomials for such pupils, but we show how to obtain the Zernike coefficients correctly. We illustrate that the wavefront fitting with a set of orthonormal polynomials is identical to the fitting with a corresponding set of Zernike polynomials. This is a consequence of the fact that each orthonormal polynomial is a linear combination of the Zernike polynomials. However, since the Zernike polynomials do not represent balanced aberrations for a noncircular pupil, the Zernike coefficients lack the physical significance that the orthonormal coefficients provide. We also analyze the error that arises if Zernike polynomials are used for noncircular pupils by treating them as circular pupils and illustrate it with numerical examples.

  16. 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.

  17. Generating polynomials using function composition | Aichinger ...

    African Journals Online (AJOL)

    We characterize the integral polynomials that can be obtained from x and x2 (or from 1, x, and x3) by using +;-, and function composition. Keywords: Near-rings, integer polynomials, subnear-ring membership problem. Quaestiones Mathematicae 36(2013), 39-46 ...

  18. 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 ...

  19. Fuzzy Morphological Polynomial Image Representation

    Directory of Open Access Journals (Sweden)

    Chin-Pan Huang

    2010-01-01

    Full Text Available A novel signal representation using fuzzy mathematical morphology is developed. We take advantage of the optimum fuzzy fitting and the efficient implementation of morphological operators to extract geometric information from signals. The new representation provides results analogous to those given by the polynomial transform. Geometrical decomposition of a signal is achieved by windowing and applying sequentially fuzzy morphological opening with structuring functions. The resulting representation is made to resemble an orthogonal expansion by constraining the results of opening to equate adapted structuring functions. Properties of the geometric decomposition are considered and used to calculate the adaptation parameters. Our procedure provides an efficient and flexible representation which can be efficiently implemented in parallel. The application of the representation is illustrated in data compression and fractal dimension estimation temporal signals and images.

  20. 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...... that are subcodes of the binary Reed-Muller codes and can be very simply instrumented, 3) a new class of constacyclic codes that are subcodes of thep-ary "Reed-Muller codes," 4) two new classes of binary convolutional codes with large "free distance" derived from known binary cyclic codes, 5) two new classes...... 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....

  1. Differential expansion for link polynomials

    Science.gov (United States)

    Bai, C.; Jiang, J.; Liang, J.; Mironov, A.; Morozov, A.; Morozov, An.; Sleptsov, A.

    2018-03-01

    The differential expansion is one of the key structures reflecting group theory properties of colored knot polynomials, which also becomes an important tool for evaluation of non-trivial Racah matrices. This makes highly desirable its extension from knots to links, which, however, requires knowledge of the 6j-symbols, at least, for the simplest triples of non-coincident representations. Based on the recent achievements in this direction, we conjecture a shape of the differential expansion for symmetrically-colored links and provide a set of examples. Within this study, we use a special framing that is an unusual extension of the topological framing from knots to links. In the particular cases of Whitehead and Borromean rings links, the differential expansions are different from the previously discovered.

  2. 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.

  3. Tutte polynomial in functional magnetic resonance imaging

    Science.gov (United States)

    García-Castillón, Marlly V.

    2015-09-01

    Methods of graph theory are applied to the processing of functional magnetic resonance images. Specifically the Tutte polynomial is used to analyze such kind of images. Functional Magnetic Resonance Imaging provide us connectivity networks in the brain which are represented by graphs and the Tutte polynomial will be applied. The problem of computing the Tutte polynomial for a given graph is #P-hard even for planar graphs. For a practical application the maple packages "GraphTheory" and "SpecialGraphs" will be used. We will consider certain diagram which is depicting functional connectivity, specifically between frontal and posterior areas, in autism during an inferential text comprehension task. The Tutte polynomial for the resulting neural networks will be computed and some numerical invariants for such network will be obtained. Our results show that the Tutte polynomial is a powerful tool to analyze and characterize the networks obtained from functional magnetic resonance imaging.

  4. 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)

  5. Improved polynomial remainder sequences for Ore polynomials☆

    Science.gov (United States)

    Jaroschek, Maximilian

    2013-01-01

    Polynomial remainder sequences contain the intermediate results of the Euclidean algorithm when applied to (non-)commutative polynomials. The running time of the algorithm is dependent on the size of the coefficients of the remainders. Different ways have been studied to make these as small as possible. The subresultant sequence of two polynomials is a polynomial remainder sequence in which the size of the coefficients is optimal in the generic case, but when taking the input from applications, the coefficients are often larger than necessary. We generalize two improvements of the subresultant sequence to Ore polynomials and derive a new bound for the minimal coefficient size. Our approach also yields a new proof for the results in the commutative case, providing a new point of view on the origin of the extraneous factors of the coefficients. PMID:26523087

  6. Resolving capacity of the circular Zernike polynomials.

    Science.gov (United States)

    Svechnikov, M V; Chkhalo, N I; Toropov, M N; Salashchenko, N N

    2015-06-01

    Circular Zernike polynomials are often used for approximation and analysis of optical surfaces. In this paper, we analyse their lateral resolving capacity, illustrating the effects of a lack of approximation by a finite set of polynomials and answering the following questions: What is the minimum number of polynomials that is necessary to describe a local deformation of a certain size? What is the relationship between the number of approximating polynomials and the spatial spectrum of the approximation? What is the connection between the mean-square error of approximation and the number of polynomials? The main results of this work are the formulas for calculating the error of fitting the relief and the connection between the width of the spatial spectrum and the order of approximation.

  7. Block orthogonal polynomials: II. Hermite and Laguerre standard block orthogonal polynomials

    International Nuclear Information System (INIS)

    Normand, Jean-Marie

    2007-01-01

    The standard block orthogonal (SBO) polynomials P i;n (x), 0 ≤ i ≤ n are real polynomials of degree n which are orthogonal with respect to a first Euclidean scalar product to polynomials of degree less than i. In addition, they are mutually orthogonal with respect to a second Euclidean scalar product. Applying the general results obtained in a previous paper, we determine and investigate these polynomials when the first scalar product corresponds to Hermite (resp. Laguerre) polynomials. These new sets of polynomials, which we call Hermite (resp. Laguerre) SBO polynomials, provide a basis of functional spaces well suited for some applications requiring to take into account special linear constraints which can be recast into an Euclidean orthogonality relation

  8. Recurrence relations on the generelizad poly-Genocchi polynomials

    Science.gov (United States)

    Kurt, Burak; Bilgic, Secil

    2017-07-01

    In this work, we introduce and investigate poly-Genocchi numbers and polynomials. We give recurrence relations for the poly-Genocchi numbers. Also, we prove a relation between multi-poly-Genocchi polynomial and multi-poly-Bernoulli polynomial.

  9. Computational approach to Thornley's problem by bivariate operational calculus

    Science.gov (United States)

    Bazhlekova, E.; Dimovski, I.

    2012-10-01

    Thornley's problem is an initial-boundary value problem with a nonlocal boundary condition for linear onedimensional reaction-diffusion equation, used as a mathematical model of spiral phyllotaxis in botany. Applying a bivariate operational calculus we find explicit representation of the solution, containing two convolution products of special solutions and the arbitrary initial and boundary functions. We use a non-classical convolution with respect to the space variable, extending in this way the classical Duhamel principle. The special solutions involved are represented in the form of fast convergent series. Numerical examples are considered to show the application of the present technique and to analyze the character of the solution.

  10. 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.)

  11. Polynomial fitting of tight-binding method in carbon

    Science.gov (United States)

    Haa, Wai Kang; Yeak, Su Hoe

    2017-04-01

    Carbon is very unique in among the elements and its ability to form strong chemical bonds with a variety number such as two carbons (graphene) and four carbons (diamond). This combination of strong bonds with tight mass and high melting point makes them technologically and scientifically important in nanoscience development. Tight-binding model (TB) is one of the semi-empirical approximations used in quantum mechanical world which is restricted to the Linear Combinations of Localized Atomic Orbitals (LCAO). Currently, there are many approaches in tight-binding calculation. In this paper, we have reproduced a polynomial scaling function by fitting to the TB model. The model is then applied into carbon molecules and obtained the energy bands of the system. The elements of the overlap Hamiltonian matrix in the model will be depending on the parameter of the polynomials. Our purpose is to find out a set of parameters in the polynomial which were commonly fit to an independently calculated band structure. We used minimization approach to calculate the polynomial coefficients which involves differentiation of eigenvalues in the eigensystem. The algorithm of fitting the parameters is carried out in FORTRAN.

  12. 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.)

  13. Exact multi-restricted Schur polynomial correlators

    International Nuclear Information System (INIS)

    Bhattacharyya, Rajsekhar; Koch, Robert de Mello; Stephanou, Michael

    2008-01-01

    We derive a product rule satisfied by restricted Schur polynomials. We focus mostly on the case that the restricted Schur polynomial is built using two matrices, although our analysis easily extends to more than two matrices. This product rule allows us to compute exact multi-point correlation functions of restricted Schur polynomials, in the free field theory limit. As an example of the use of our formulas, we compute two point functions of certain single trace operators built using two matrices and three point functions of certain restricted Schur polynomials, exactly, in the free field theory limit. Our results suggest that gravitons become strongly coupled at sufficiently high energy, while the restricted Schur polynomials for totally antisymmetric representations remain weakly interacting at these energies. This is in perfect accord with the half-BPS (single matrix) results of hep-th/0512312. Finally, by studying the interaction of two restricted Schur polynomials we suggest a physical interpretation for the labels of the restricted Schur polynomial: the composite operator χ R,(r n ,r m ) (Z,X) is constructed from the half BPS 'partons' χ r n (Z) and χ r m (X).

  14. Multiple imputation methods for bivariate outcomes in cluster randomised trials.

    Science.gov (United States)

    DiazOrdaz, K; Kenward, M G; Gomes, M; Grieve, R

    2016-09-10

    Missing observations are common in cluster randomised trials. The problem is exacerbated when modelling bivariate outcomes jointly, as the proportion of complete cases is often considerably smaller than the proportion having either of the outcomes fully observed. Approaches taken to handling such missing data include the following: complete case analysis, single-level multiple imputation that ignores the clustering, multiple imputation with a fixed effect for each cluster and multilevel multiple imputation. We contrasted the alternative approaches to handling missing data in a cost-effectiveness analysis that uses data from a cluster randomised trial to evaluate an exercise intervention for care home residents. We then conducted a simulation study to assess the performance of these approaches on bivariate continuous outcomes, in terms of confidence interval coverage and empirical bias in the estimated treatment effects. Missing-at-random clustered data scenarios were simulated following a full-factorial design. Across all the missing data mechanisms considered, the multiple imputation methods provided estimators with negligible bias, while complete case analysis resulted in biased treatment effect estimates in scenarios where the randomised treatment arm was associated with missingness. Confidence interval coverage was generally in excess of nominal levels (up to 99.8%) following fixed-effects multiple imputation and too low following single-level multiple imputation. Multilevel multiple imputation led to coverage levels of approximately 95% throughout. © 2016 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd. © 2016 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.

  15. Bivariate generalized Pareto distribution for extreme atmospheric particulate matter

    Science.gov (United States)

    Amin, Nor Azrita Mohd; Adam, Mohd Bakri; Ibrahim, Noor Akma; Aris, Ahmad Zaharin

    2015-02-01

    The high particulate matter (PM10) level is the prominent issue causing various impacts to human health and seriously affecting the economics. The asymptotic theory of extreme value is apply for analyzing the relation of extreme PM10 data from two nearby air quality monitoring stations. The series of daily maxima PM10 for Johor Bahru and Pasir Gudang stations are consider for year 2001 to 2010 databases. The 85% and 95% marginal quantile apply to determine the threshold values and hence construct the series of exceedances over the chosen threshold. The logistic, asymmetric logistic, negative logistic and asymmetric negative logistic models areconsidered as the dependence function to the joint distribution of a bivariate observation. Maximum likelihood estimation is employed for parameter estimations. The best fitted model is chosen based on the Akaike Information Criterion and the quantile plots. It is found that the asymmetric logistic model gives the best fitted model for bivariate extreme PM10 data and shows the weak dependence between two stations.

  16. Fixed-point bifurcation analysis in biological models using interval polynomials theory.

    Science.gov (United States)

    Rigatos, Gerasimos G

    2014-06-01

    The paper proposes a systematic method for fixed-point bifurcation analysis in circadian cells and similar biological models using interval polynomials theory. The stages for performing fixed-point bifurcation analysis in such biological systems comprise (i) the computation of fixed points as functions of the bifurcation parameter and (ii) the evaluation of the type of stability for each fixed point through the computation of the eigenvalues of the Jacobian matrix that is associated with the system's nonlinear dynamics model. Stage (ii) requires the computation of the roots of the characteristic polynomial of the Jacobian matrix. This problem is nontrivial since the coefficients of the characteristic polynomial are functions of the bifurcation parameter and the latter varies within intervals. To obtain a clear view about the values of the roots of the characteristic polynomial and about the stability features they provide to the system, the use of interval polynomials theory and particularly of Kharitonov's stability theorem is proposed. In this approach, the study of the stability of a characteristic polynomial with coefficients that vary in intervals is equivalent to the study of the stability of four polynomials with crisp coefficients computed from the boundaries of the aforementioned intervals. The efficiency of the proposed approach for the analysis of fixed-point bifurcations in nonlinear models of biological neurons is tested through numerical and simulation experiments.

  17. 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.

  18. The Translated Dowling Polynomials and Numbers.

    Science.gov (United States)

    Mangontarum, Mahid M; Macodi-Ringia, Amila P; Abdulcarim, Normalah S

    2014-01-01

    More properties for the translated Whitney numbers of the second kind such as horizontal generating function, explicit formula, and exponential generating function are proposed. Using the translated Whitney numbers of the second kind, we will define the translated Dowling polynomials and numbers. Basic properties such as exponential generating functions and explicit formula for the translated Dowling polynomials and numbers are obtained. Convexity, integral representation, and other interesting identities are also investigated and presented. We show that the properties obtained are generalizations of some of the known results involving the classical Bell polynomials and numbers. Lastly, we established the Hankel transform of the translated Dowling numbers.

  19. Orthonormal curvature polynomials over a unit circle: basis set derived from curvatures of Zernike polynomials.

    Science.gov (United States)

    Zhao, Chunyu; Burge, James H

    2013-12-16

    Zernike polynomials are an orthonormal set of scalar functions over a circular domain, and are commonly used to represent wavefront phase or surface irregularity. In optical testing, slope or curvature of a surface or wavefront is sometimes measured instead, from which the surface or wavefront map is obtained. Previously we derived an orthonormal set of vector polynomials that fit to slope measurement data and yield the surface or wavefront map represented by Zernike polynomials. Here we define a 3-element curvature vector used to represent the second derivatives of a continuous surface, and derive a set of orthonormal curvature basis functions that are written in terms of Zernike polynomials. We call the new curvature functions the C polynomials. Closed form relations for the complete basis set are provided, and we show how to determine Zernike surface coefficients from the curvature data as represented by the C polynomials.

  20. Adapted polynomial chaos expansion for failure detection

    International Nuclear Information System (INIS)

    Paffrath, M.; Wever, U.

    2007-01-01

    In this paper, we consider two methods of computation of failure probabilities by adapted polynomial chaos expansions. The performance of the two methods is demonstrated by a predator-prey model and a chemical reaction problem

  1. 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...

  2. Twisted Polynomials and Forgery Attacks on GCM

    DEFF Research Database (Denmark)

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

    2015-01-01

    nonce misuse resistance, such as POET. The algebraic structure of polynomial hashing has given rise to security concerns: At CRYPTO 2008, Handschuh and Preneel describe key recovery attacks, and at FSE 2013, Procter and Cid provide a comprehensive framework for forgery attacks. Both approaches rely...... heavily on the ability to construct forgery polynomials having disjoint sets of roots, with many roots (“weak keys”) each. Constructing such polynomials beyond naïve approaches is crucial for these attacks, but still an open problem. In this paper, we comprehensively address this issue. We propose to use...... in an improved key recovery algorithm. As cryptanalytic applications of our twisted polynomials, we develop the first universal forgery attacks on GCM in the weak-key model that do not require nonce reuse. Moreover, we present universal weak-key forgeries for the nonce-misuse resistant AE scheme POET, which...

  3. Some results for extended Jacobi polynomials

    International Nuclear Information System (INIS)

    Khan, I.A.

    1991-06-01

    In this paper, some contiguous function relations and generating functions have been established for extended Jacobi polynomials. The results obtained here, are of very general nature. (author). 5 refs

  4. Schur Stability Regions for Complex Quadratic Polynomials

    Science.gov (United States)

    Cheng, Sui Sun; Huang, Shao Yuan

    2010-01-01

    Given a quadratic polynomial with complex coefficients, necessary and sufficient conditions are found in terms of the coefficients such that all its roots have absolute values less than 1. (Contains 3 figures.)

  5. Some generating functions of Laguerre polynomials

    Directory of Open Access Journals (Sweden)

    Bidyut Kumar Guha Thakurta

    1987-01-01

    Full Text Available In this note a class of interesting generating relation, which is stated in the form of theorem, involving Laguerre polynomials is derived. Some applications of the theorem are also given here.

  6. On linear equations with general polynomial solutions

    Science.gov (United States)

    Laradji, A.

    2018-04-01

    We provide necessary and sufficient conditions for which an nth-order linear differential equation has a general polynomial solution. We also give necessary conditions that can directly be ascertained from the coefficient functions of the equation.

  7. Handbook on semidefinite, conic and polynomial optimization

    CERN Document Server

    Anjos, Miguel F

    2012-01-01

    This book offers the reader a snapshot of the state-of-the-art in the growing and mutually enriching areas of semidefinite optimization, conic optimization and polynomial optimization. It covers theory, algorithms, software and applications.

  8. Polynomial analysis of ambulatory blood pressure measurements

    NARCIS (Netherlands)

    Zwinderman, A. H.; Cleophas, T. A.; Cleophas, T. J.; van der Wall, E. E.

    2001-01-01

    In normotensive subjects blood pressures follow a circadian rhythm. A circadian rhythm in hypertensive patients is less well established, and may be clinically important, particularly with rigorous treatments of daytime blood pressures. Polynomial analysis of ambulatory blood pressure monitoring

  9. 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

  10. A Polynomial Estimate of Railway Line Delay

    DEFF Research Database (Denmark)

    Cerreto, Fabrizio; Harrod, Steven; Nielsen, Otto Anker

    2017-01-01

    by numerical analysis with simulation. This paper proposes an analytical estimate of aggregate delay with a polynomial form. The function returns the aggregate delay of a railway line resulting from an initial, primary, delay. Analysis of the function demonstrates that there should be a balance between the two...... remedial measures, supplement and buffer. Numerical analysis of a Copenhagen Sbane line shows that the polynomial function is valid even when theoretical assumptions are violated....

  11. 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.

  12. Integral inequalities for self-reciprocal polynomials

    Indian Academy of Sciences (India)

    Integral inequalities for self-reciprocal polynomials. HORST ALZER. Morsbacher Str. 10, D-51545 Waldbröl, Germany. E-mail: H.Alzer@gmx.de. MS received 14 November 2007. Abstract. Let n ≥ 1 be an integer and let Pn be the class of polynomials P of degree at most n satisfying znP(1/z) = P (z) for all z ∈ C. Moreover, ...

  13. 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.

  14. Local fibred right adjoints are polynomial

    DEFF Research Database (Denmark)

    Kock, Anders; Kock, Joachim

    2013-01-01

    For any locally cartesian closed category E, we prove that a local fibred right adjoint between slices of E is given by a polynomial. The slices in question are taken in a well known fibred sense......For any locally cartesian closed category E, we prove that a local fibred right adjoint between slices of E is given by a polynomial. The slices in question are taken in a well known fibred sense...

  15. Efficient estimation of semiparametric copula models for bivariate survival data

    KAUST Repository

    Cheng, Guang

    2014-01-01

    A semiparametric copula model for bivariate survival data is characterized by a parametric copula model of dependence and nonparametric models of two marginal survival functions. Efficient estimation for the semiparametric copula model has been recently studied for the complete data case. When the survival data are censored, semiparametric efficient estimation has only been considered for some specific copula models such as the Gaussian copulas. In this paper, we obtain the semiparametric efficiency bound and efficient estimation for general semiparametric copula models for possibly censored data. We construct an approximate maximum likelihood estimator by approximating the log baseline hazard functions with spline functions. We show that our estimates of the copula dependence parameter and the survival functions are asymptotically normal and efficient. Simple consistent covariance estimators are also provided. Numerical results are used to illustrate the finite sample performance of the proposed estimators. © 2013 Elsevier Inc.

  16. Selection effects in the bivariate brightness distribution for spiral galaxies

    International Nuclear Information System (INIS)

    Phillipps, S.; Disney, M.

    1986-01-01

    The joint distribution of total luminosity and characteristic surface brightness (the bivariate brightness distribution) is investigated for a complete sample of spiral galaxies in the Virgo cluster. The influence of selection and physical limits of various kinds on the apparent distribution are detailed. While the distribution of surface brightness for bright galaxies may be genuinely fairly narrow, faint galaxies exist right across the (quite small) range of accessible surface brightnesses so no statement can be made about the true extent of the distribution. The lack of high surface brightness bright galaxies in the Virgo sample relative to an overall RC2 sample (mostly field galaxies) supports the contention that the star-formation rate is reduced in the inner region of the cluster for environmental reasons. (author)

  17. Vector-Valued Jack Polynomials from Scratch

    Directory of Open Access Journals (Sweden)

    Jean-Gabriel Luque

    2011-03-01

    Full Text Available Vector-valued Jack polynomials associated to the symmetric group S_N are polynomials with multiplicities in an irreducible module of S_N and which are simultaneous eigenfunctions of the Cherednik-Dunkl operators with some additional properties concerning the leading monomial. These polynomials were introduced by Griffeth in the general setting of the complex reflections groups G(r,p,N and studied by one of the authors (C. Dunkl in the specialization r=p=1 (i.e. for the symmetric group. By adapting a construction due to Lascoux, we describe an algorithm allowing us to compute explicitly the Jack polynomials following a Yang-Baxter graph. We recover some properties already studied by C. Dunkl and restate them in terms of graphs together with additional new results. In particular, we investigate normalization, symmetrization and antisymmetrization, polynomials with minimal degree, restriction etc. We give also a shifted version of the construction and we discuss vanishing properties of the associated polynomials.

  18. Polynomial-Time Classical Simulation of Quantum Ferromagnets

    Science.gov (United States)

    Bravyi, Sergey; Gosset, David

    2017-09-01

    We consider a family of quantum spin systems which includes, as special cases, the ferromagnetic X Y model and ferromagnetic Ising model on any graph, with or without a transverse magnetic field. We prove that the partition function of any model in this family can be efficiently approximated to a given relative error ɛ using a classical randomized algorithm with runtime polynomial in ɛ-1, system size, and inverse temperature. As a consequence, we obtain a polynomial time algorithm which approximates the free energy or ground energy to a given additive error. We first show how to approximate the partition function by the perfect matching sum of a finite graph with positive edge weights. Although the perfect matching sum is not known to be efficiently approximable in general, the graphs obtained by our method have a special structure which facilitates efficient approximation via a randomized algorithm due to Jerrum and Sinclair.

  19. Strong asymptotics for extremal polynomials associated with weights on ℝ

    CERN Document Server

    Lubinsky, Doron S

    1988-01-01

    0. The results are consequences of a strengthened form of the following assertion: Given 0 1. Auxiliary results include inequalities for weighted polynomials, and zeros of extremal polynomials. The monograph is fairly self-contained, with proofs involving elementary complex analysis, and the theory of orthogonal and extremal polynomials. It should be of interest to research workers in approximation theory and orthogonal polynomials.

  20. 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

  1. 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.

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

    Directory of Open Access Journals (Sweden)

    Paul Barry

    2013-01-01

    Full Text Available The Chebyshev-Boubaker polynomials are the orthogonal polynomials whose coefficient arrays are defined by ordinary Riordan arrays. Examples include the Chebyshev polynomials of the second kind and the Boubaker polynomials. We study the connection coefficients of this class of orthogonal polynomials, indicating how Riordan array techniques can lead to closed-form expressions for these connection coefficients as well as recurrence relations that define them.

  3. On the connection coefficients of the Chebyshev-Boubaker polynomials.

    Science.gov (United States)

    Barry, Paul

    2013-01-01

    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.

  4. 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.

  5. 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.

  6. Complete commutative subalgebras in polynomial poisson algebras: A proof of the Mischenko-Fomenko conjecture

    Directory of Open Access Journals (Sweden)

    Bolsinov Alexey V.

    2016-01-01

    Full Text Available The Mishchenko-Fomenko conjecture says that for each real or complex finite-dimensional Lie algebra g there exists a complete set of commuting polynomials on its dual space g*. In terms of the theory of integrable Hamiltonian systems this means that the dual space g* endowed with the standard Lie-Poisson bracket admits polynomial integrable Hamiltonian systems. This conjecture was proved by S. T. Sadetov in 2003. Following his idea, we give an explicit geometric construction for commuting polynomials on g* and consider some examples. (This text is a revised version of my paper published in Russian: A. V. Bolsinov, Complete commutative families of polynomials in Poisson–Lie algebras: A proof of the Mischenko–Fomenko conjecture in book: Tensor and Vector Analysis, Vol. 26, Moscow State University, 2005, 87–109.

  7. 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.

  8. 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

  9. 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.].

  10. Extending a Property of Cubic Polynomials to Higher-Degree Polynomials

    Science.gov (United States)

    Miller, David A.; Moseley, James

    2012-01-01

    In this paper, the authors examine a property that holds for all cubic polynomials given two zeros. This property is discovered after reviewing a variety of ways to determine the equation of a cubic polynomial given specific conditions through algebra and calculus. At the end of the article, they will connect the property to a very famous method…

  11. 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.

  12. An integrated user-friendly ArcMAP tool for bivariate statistical modeling in geoscience applications

    Science.gov (United States)

    Jebur, M. N.; Pradhan, B.; Shafri, H. Z. M.; Yusof, Z.; Tehrany, M. S.

    2014-10-01

    Modeling and classification difficulties are fundamental issues in natural hazard assessment. A geographic information system (GIS) is a domain that requires users to use various tools to perform different types of spatial modeling. Bivariate statistical analysis (BSA) assists in hazard modeling. To perform this analysis, several calculations are required and the user has to transfer data from one format to another. Most researchers perform these calculations manually by using Microsoft Excel or other programs. This process is time consuming and carries a degree of uncertainty. The lack of proper tools to implement BSA in a GIS environment prompted this study. In this paper, a user-friendly tool, BSM (bivariate statistical modeler), for BSA technique is proposed. Three popular BSA techniques such as frequency ratio, weights-of-evidence, and evidential belief function models are applied in the newly proposed ArcMAP tool. This tool is programmed in Python and is created by a simple graphical user interface, which facilitates the improvement of model performance. The proposed tool implements BSA automatically, thus allowing numerous variables to be examined. To validate the capability and accuracy of this program, a pilot test area in Malaysia is selected and all three models are tested by using the proposed program. Area under curve is used to measure the success rate and prediction rate. Results demonstrate that the proposed program executes BSA with reasonable accuracy. The proposed BSA tool can be used in numerous applications, such as natural hazard, mineral potential, hydrological, and other engineering and environmental applications.

  13. Time-dependent generalized polynomial chaos

    International Nuclear Information System (INIS)

    Gerritsma, Marc; Steen, Jan-Bart van der; Vos, Peter; Karniadakis, George

    2010-01-01

    Generalized polynomial chaos (gPC) has non-uniform convergence and tends to break down for long-time integration. The reason is that the probability density distribution (PDF) of the solution evolves as a function of time. The set of orthogonal polynomials associated with the initial distribution will therefore not be optimal at later times, thus causing the reduced efficiency of the method for long-time integration. Adaptation of the set of orthogonal polynomials with respect to the changing PDF removes the error with respect to long-time integration. In this method new stochastic variables and orthogonal polynomials are constructed as time progresses. In the new stochastic variable the solution can be represented exactly by linear functions. This allows the method to use only low order polynomial approximations with high accuracy. The method is illustrated with a simple decay model for which an analytic solution is available and subsequently applied to the three mode Kraichnan-Orszag problem with favorable results.

  14. 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.

  15. Real-time task recognition in cataract surgery videos using adaptive spatiotemporal polynomials.

    Science.gov (United States)

    Quellec, Gwénolé; Lamard, Mathieu; Cochener, Béatrice; Cazuguel, Guy

    2015-04-01

    This paper introduces a new algorithm for recognizing surgical tasks in real-time in a video stream. The goal is to communicate information to the surgeon in due time during a video-monitored surgery. The proposed algorithm is applied to cataract surgery, which is the most common eye surgery. To compensate for eye motion and zoom level variations, cataract surgery videos are first normalized. Then, the motion content of short video subsequences is characterized with spatiotemporal polynomials: a multiscale motion characterization based on adaptive spatiotemporal polynomials is presented. The proposed solution is particularly suited to characterize deformable moving objects with fuzzy borders, which are typically found in surgical videos. Given a target surgical task, the system is trained to identify which spatiotemporal polynomials are usually extracted from videos when and only when this task is being performed. These key spatiotemporal polynomials are then searched in new videos to recognize the target surgical task. For improved performances, the system jointly adapts the spatiotemporal polynomial basis and identifies the key spatiotemporal polynomials using the multiple-instance learning paradigm. The proposed system runs in real-time and outperforms the previous solution from our group, both for surgical task recognition ( Az = 0.851 on average, as opposed to Az = 0.794 previously) and for the joint segmentation and recognition of surgical tasks ( Az = 0.856 on average, as opposed to Az = 0.832 previously).

  16. Epileptic seizure prediction based on a bivariate spectral power methodology.

    Science.gov (United States)

    Bandarabadi, Mojtaba; Teixeira, Cesar A; Direito, Bruno; Dourado, Antonio

    2012-01-01

    The spectral power of 5 frequently considered frequency bands (Alpha, Beta, Gamma, Theta and Delta) for 6 EEG channels is computed and then all the possible pairwise combinations among the 30 features set, are used to create a 435 dimensional feature space. Two new feature selection methods are introduced to choose the best candidate features among those and to reduce the dimensionality of this feature space. The selected features are then fed to Support Vector Machines (SVMs) that classify the cerebral state in preictal and non-preictal classes. The outputs of the SVM are regularized using a method that accounts for the classification dynamics of the preictal class, also known as "Firing Power" method. The results obtained using our feature selection approaches are compared with the ones obtained using minimum Redundancy Maximum Relevance (mRMR) feature selection method. The results in a group of 12 patients of the EPILEPSIAE database, containing 46 seizures and 787 hours multichannel recording for out-of-sample data, indicate the efficiency of the bivariate approach as well as the two new feature selection methods. The best results presented sensitivity of 76.09% (35 of 46 seizures predicted) and a false prediction rate of 0.15(-1).

  17. Joint association analysis of bivariate quantitative and qualitative traits.

    Science.gov (United States)

    Yuan, Mengdie; Diao, Guoqing

    2011-11-29

    Univariate genome-wide association analysis of quantitative and qualitative traits has been investigated extensively in the literature. In the presence of correlated phenotypes, it is more intuitive to analyze all phenotypes simultaneously. We describe an efficient likelihood-based approach for the joint association analysis of quantitative and qualitative traits in unrelated individuals. We assume a probit model for the qualitative trait, under which an unobserved latent variable and a prespecified threshold determine the value of the qualitative trait. To jointly model the quantitative and qualitative traits, we assume that the quantitative trait and the latent variable follow a bivariate normal distribution. The latent variable is allowed to be correlated with the quantitative phenotype. Simultaneous modeling of the quantitative and qualitative traits allows us to make more precise inference on the pleiotropic genetic effects. We derive likelihood ratio tests for the testing of genetic effects. An application to the Genetic Analysis Workshop 17 data is provided. The new method yields reasonable power and meaningful results for the joint association analysis of the quantitative trait Q1 and the qualitative trait disease status at SNPs with not too small MAF.

  18. Preparation and bivariate analysis of suspensions of human chromosomes

    Energy Technology Data Exchange (ETDEWEB)

    van den Engh, G.J.; Trask, B.J.; Gray, J.W.; Langlois, R.G.; Yu, L.C.

    1985-01-01

    Chromosomes were isolated from a variety of human cell types using a HEPES-buffered hypotonic solution (pH 8.0) containing KCl, MgSO/sub 4/ dithioerythritol, and RNase. The chromosomes isolated by this procedure could be stained with a variety of fluorescent stains including propidium iodide, chromomycin A3, and Hoeschst 33258. Addition of sodium citrate to the stained chromosomes was found to improve the total fluorescence resolution. High-quality bivariate Hoeschst vs. chromomycin fluorescence distributions were obtained for chromosomes isolated from a human fibroblast cell strain, a human colon carcinoma cell line, and human peripheral blood lymphocyte cultures. Good flow karyotypes were also obtained from primary amniotic cell cultures. The Hoeschst vs. chromomycin flow karyotypes of a given cell line, made at different times and at dye concentrations varying over fourfold ranges, show little variation in the relative peak positions of the chromosomes. The size of the DNA in chromosomes isolated using this procedure ranges from 20 to 50 kilobases. The described isolation procedure is simple, it yields high-quality flow karyotypes, and it can be used to prepare chromosomes from clinical samples. 22 references, 7 figures, 1 table.

  19. Nonlinear 1D DPCM image prediction using polynomial neural networks

    Science.gov (United States)

    Liatsis, Panos; Hussain, Abir J.

    1999-03-01

    This work presents a novel polynomial neural network approach to 1D differential pulse code modulation (DPCM) design for image compression. This provides an alternative to current tradition and neural networks techniques, by allowing the incremental construction of higher-order polynomials of different orders. The proposed predictor utilizes Ridge Polynomial Neural Networks (RPNs), which allow the use of linear and non-linear terms, and avoid the problem of the combinatorial explosion of the higher-order terms. In RPNs, there is no requirement to select the number of hidden units or the order of the network. Extensive computer simulations have demonstrated that the resulting encoders work very well. At a transmission rate of 1 bit/pixel, the 1D RPN system provides on average a 13 dB improvement in SNR over the standard linear DPCM and a 9 dB improvement when compared to HONNs. A further result of the research was that third-order RPNs can provide very good predictions in a variety of images.

  20. From Jack to Double Jack Polynomials via the Supersymmetric Bridge

    Science.gov (United States)

    Lapointe, Luc; Mathieu, Pierre

    2015-07-01

    The Calogero-Sutherland model occurs in a large number of physical contexts, either directly or via its eigenfunctions, the Jack polynomials. The supersymmetric counterpart of this model, although much less ubiquitous, has an equally rich structure. In particular, its eigenfunctions, the Jack superpolynomials, appear to share the very same remarkable combinatorial and structural properties as their non-supersymmetric version. These super-functions are parametrized by superpartitions with fixed bosonic and fermionic degrees. Now, a truly amazing feature pops out when the fermionic degree is sufficiently large: the Jack superpolynomials stabilize and factorize. Their stability is with respect to their expansion in terms of an elementary basis where, in the stable sector, the expansion coefficients become independent of the fermionic degree. Their factorization is seen when the fermionic variables are stripped off in a suitable way which results in a product of two ordinary Jack polynomials (somewhat modified by plethystic transformations), dubbed the double Jack polynomials. Here, in addition to spelling out these results, which were first obtained in the context of Macdonal superpolynomials, we provide a heuristic derivation of the Jack superpolynomial case by performing simple manipulations on the supersymmetric eigen-operators, rendering them independent of the number of particles and of the fermionic degree. In addition, we work out the expression of the Hamiltonian which characterizes the double Jacks. This Hamiltonian, which defines a new integrable system, involves not only the expected Calogero-Sutherland pieces but also combinations of the generators of an underlying affine {widehat{sl}_2} algebra.

  1. Zernike olivary polynomials for applications with olivary pupils.

    Science.gov (United States)

    Zheng, Yi; Sun, Shanshan; Li, Ying

    2016-04-20

    Orthonormal polynomials have been extensively applied in optical image systems. One important optical pupil, which is widely processed in lateral shearing interferometers (LSI) and subaperture stitch tests (SST), is the overlap region of two circular wavefronts that are displaced from each other. We call it an olivary pupil. In this paper, the normalized process of an olivary pupil in a unit circle is first presented. Then, using a nonrecursive matrix method, Zernike olivary polynomials (ZOPs) are obtained. Previously, Zernike elliptical polynomials (ZEPs) have been considered as an approximation over an olivary pupil. We compare ZOPs with their ZEPs counterparts. Results show that they share the same components but are in different proportions. For some low-order aberrations such as defocus, coma, and spherical, the differences are considerable and may lead to deviations. Using a least-squares method to fit coefficient curves, we present a power-series expansion form for the first 15 ZOPs, which can be used conveniently with less than 0.1% error. The applications of ZOP are demonstrated in wavefront decomposition, LSI interferogram reconstruction, and SST overlap domain evaluation.

  2. Quantum chaotic dynamics and random polynomials

    International Nuclear Information System (INIS)

    Bogomolny, E.; Bohigas, O.; Leboeuf, P.

    1995-11-01

    The distribution of roots of polynomials of high degree with random coefficients is investigated which, among others, appear naturally in the context of 'quantum chaotic dynamics'. It is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane. In order to further increase this tendency, the particular case of self-inverse random polynomials is studied, and it is shown that for them a finite portion of all roots lies exactly on the unit circle. Correlation functions of these roots are also computed analytically, and compared to the correlations of eigenvalues of random matrices. The problem of ergodicity of chaotic wavefunctions is also considered. Special attention is devoted to the role of symmetries in the distribution of roots of random polynomials. (author)

  3. Markov and Bernstein type inequalities for polynomials

    Directory of Open Access Journals (Sweden)

    Mohapatra RN

    1999-01-01

    Full Text Available In an answer to a question raised by chemist Mendeleev, A. Markov proved that if is a real polynomial of degree , then The above inequality which is known as Markov's Inequality is best possible and becomes equality for the Chebyshev polynomial . Few years later, Serge Bernstein needed the analogue of this result for the unit disk in the complex plane instead of the interval and the following is known as Bernstein's Inequality. If is a polynomial of degree then This inequality is also best possible and is attained for , being a complex number. The above two inequalities have been the starting point of a considerable literature in Mathematics and in this article we discuss some of the research centered around these inequalities.

  4. Dominating Sets and Domination Polynomials of Paths

    Directory of Open Access Journals (Sweden)

    Saeid Alikhani

    2009-01-01

    Full Text Available Let G=(V,E be a simple graph. A set S⊆V is a dominating set of G, if every vertex in V\\S is adjacent to at least one vertex in S. Let 𝒫ni be the family of all dominating sets of a path Pn with cardinality i, and let d(Pn,j=|𝒫nj|. In this paper, we construct 𝒫ni, and obtain a recursive formula for d(Pn,i. Using this recursive formula, we consider the polynomial D(Pn,x=∑i=⌈n/3⌉nd(Pn,ixi, which we call domination polynomial of paths and obtain some properties of this polynomial.

  5. Some Inequalities for the Derivative of Polynomials

    Directory of Open Access Journals (Sweden)

    Sunil Hans

    2014-01-01

    Full Text Available If pz=∑υ=0ncυzυ is a polynomial of degree n, having no zeros in z<1, then Aziz (1989 proved maxz=1p′z≤n/2Mα2+Mα+π21/2, where Mα=max1≤k≤npeiα+2kπ/n. In this paper, we consider a class of polynomial Pnμ of degree n, defined as pz=a0+∑υ=μnaυzυ and present certain generalizations of above inequality and some other well-known results.

  6. 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....

  7. Some Relationships between the Analogs of Euler Numbers and Polynomials

    Directory of Open Access Journals (Sweden)

    Kim T

    2007-01-01

    Full Text Available We construct new twisted Euler polynomials and numbers. We also study the generating functions of the twisted Euler numbers and polynomials associated with their interpolation functions. Next we construct twisted Euler zeta function, twisted Hurwitz zeta function, twisted Dirichlet -Euler numbers and twisted Euler polynomials at non-positive integers, respectively. Furthermore, we find distribution relations of generalized twisted Euler numbers and polynomials. By numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the twisted -Euler polynomials. Finally, we give a table for the solutions of the twisted -Euler polynomials.

  8. Some Relationships between the Analogs of Euler Numbers and Polynomials

    Directory of Open Access Journals (Sweden)

    C. S. Ryoo

    2007-10-01

    Full Text Available We construct new twisted Euler polynomials and numbers. We also study the generating functions of the twisted Euler numbers and polynomials associated with their interpolation functions. Next we construct twisted Euler zeta function, twisted Hurwitz zeta function, twisted Dirichlet l-Euler numbers and twisted Euler polynomials at non-positive integers, respectively. Furthermore, we find distribution relations of generalized twisted Euler numbers and polynomials. By numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the twisted q-Euler polynomials. Finally, we give a table for the solutions of the twisted q-Euler polynomials.

  9. Location of zeros of Wiener and distance polynomials.

    Science.gov (United States)

    Dehmer, Matthias; Ilić, Aleksandar

    2012-01-01

    The geometry of polynomials explores geometrical relationships between the zeros and the coefficients of a polynomial. A classical problem in this theory is to locate the zeros of a given polynomial by determining disks in the complex plane in which all its zeros are situated. In this paper, we infer bounds for general polynomials and apply classical and new results to graph polynomials namely Wiener and distance polynomials whose zeros have not been yet investigated. Also, we examine the quality of such bounds by considering four graph classes and interpret the results.

  10. 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)

  11. Orthogonal polynomials of compact simple Lie groups: branching rules for polynomials

    International Nuclear Information System (INIS)

    Nesterenko, M; Patera, J; Szajewska, M; Tereszkiewicz, A

    2010-01-01

    Polynomials in this paper are defined starting from a compact semisimple Lie group. A known classification of maximal, semisimple subgroups of simple Lie groups is used to select the cases to be considered here. A general method is presented and all the cases of rank ≤3 are explicitly studied. We derive the polynomials of simple Lie groups B 3 and C 3 as they are not available elsewhere. The results point to far reaching Lie theoretical connections to the theory of multivariable orthogonal polynomials.

  12. Airy polynomials, three-variable Hermite polynomials and the paraxial wave equation

    International Nuclear Information System (INIS)

    Torre, A

    2012-01-01

    Within the context of an investigation of the potential of functions, which directly or indirectly relate to the Airy functions, to yield solutions of the 2D paraxial wave equation, we consider here the Airy polynomials. We see that appropriate Gaussian-modulated versions of such polynomials, which could in a sense parallel the Hermite–Gaussian wavefunctions of both elegant and standard type, yield, when assumed as initial functions, solutions of the 2D PWE shaping in the form of a complex Gaussian modulation of three-variable Hermite polynomials. A basic characterization of these wavefunctions is provided, having as a touchstone the Hermite–Gaussian wavefunctions. (paper)

  13. Staircase tableaux, the asymmetric exclusion process, and Askey-Wilson polynomials.

    Science.gov (United States)

    Corteel, Sylvie; Williams, Lauren K

    2010-04-13

    We introduce some combinatorial objects called staircase tableaux, which have cardinality 4(n)n!, and connect them to both the asymmetric exclusion process (ASEP) and Askey-Wilson polynomials. The ASEP is a model from statistical mechanics introduced in the late 1960s, which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites with open boundaries. It has been cited as a model for traffic flow and translation in protein synthesis. In its most general form, particles may enter and exit at the left with probabilities alpha and gamma, and they may exit and enter at the right with probabilities beta and delta. In the bulk, the probability of hopping left is q times the probability of hopping right. Our first result is a formula for the stationary distribution of the ASEP with all parameters general, in terms of staircase tableaux. Our second result is a formula for the moments of (the weight function of) Askey-Wilson polynomials, also in terms of staircase tableaux. Since the 1980s there has been a great deal of work giving combinatorial formulas for moments of classical orthogonal polynomials (e.g. Hermite, Charlier, Laguerre); among these polynomials, the Askey-Wilson polynomials are the most important, because they are at the top of the hierarchy of classical orthogonal polynomials.

  14. Inequalities for a Polynomial and its Derivative

    Indian Academy of Sciences (India)

    Annual Meetings · Mid Year Meetings · Discussion Meetings · Public Lectures · Lecture Workshops · Refresher Courses · Symposia · Live Streaming. Home; Journals; Proceedings – Mathematical Sciences; Volume 110; Issue 2. Inequalities for a Polynomial and its Derivative. V K Jain. Volume 110 Issue 2 May 2000 pp 137- ...

  15. Integral Inequalities for Self-Reciprocal Polynomials

    Indian Academy of Sciences (India)

    Annual Meetings · Mid Year Meetings · Discussion Meetings · Public Lectures · Lecture Workshops · Refresher Courses · Symposia · Live Streaming. Home; Journals; Proceedings – Mathematical Sciences; Volume 120; Issue 2. Integral Inequalities for Self-Reciprocal Polynomials. Horst Alzer. Volume 120 Issue 2 April 2010 ...

  16. Integral inequalities for self-reciprocal polynomials

    Indian Academy of Sciences (India)

    Annual Meetings · Mid Year Meetings · Discussion Meetings · Public Lectures · Lecture Workshops · Refresher Courses · Symposia · Live Streaming. Home; Journals; Proceedings – Mathematical Sciences; Volume 120; Issue 2. Integral Inequalities for Self-Reciprocal Polynomials. Horst Alzer. Volume 120 Issue 2 April 2010 ...

  17. Bernoulli Polynomials, Fourier Series and Zeta Numbers

    DEFF Research Database (Denmark)

    Scheufens, Ernst E

    2013-01-01

    Fourier series for Bernoulli polynomials are used to obtain information about values of the Riemann zeta function for integer arguments greater than one. If the argument is even we recover the well-known exact values, if the argument is odd we find integral representations and rapidly convergent...

  18. Euler Polynomials, Fourier Series and Zeta Numbers

    DEFF Research Database (Denmark)

    Scheufens, Ernst E

    2012-01-01

    Fourier series for Euler polynomials is used to obtain information about values of the Riemann zeta function for integer arguments greater than one. If the argument is even we recover the well-known exact values, if the argument is odd we find integral representations and rapidly convergent series....

  19. Odd Degree Polynomials on Real Banach Spaces

    Czech Academy of Sciences Publication Activity Database

    Aron, R. M.; Hájek, Petr Pavel

    2007-01-01

    Roč. 11, č. 1 (2007), s. 143-153 ISSN 1385-1292 R&D Projects: GA AV ČR IAA100190502; GA ČR GA201/04/0090 Institutional research plan: CEZ:AV0Z10190503 Keywords : odd degree polynomials * zero sets Subject RIV: BA - General Mathematics Impact factor: 0.356, year: 2007

  20. Quantum Hilbert matrices and orthogonal polynomials

    DEFF Research Database (Denmark)

    Andersen, Jørgen Ellegaard; Berg, Christian

    2009-01-01

    Using the notion of quantum integers associated with a complex number q≠0 , we define the quantum Hilbert matrix and various extensions. They are Hankel matrices corresponding to certain little q -Jacobi polynomials when |q|... of reciprocal Fibonacci numbers called Filbert matrices. We find a formula for the entries of the inverse quantum Hilbert matrix....

  1. Characteristic polynomials of linear polyacenes and their ...

    Indian Academy of Sciences (India)

    Administrator

    Abstract. Coefficients of characteristic polynomials (CP) of linear polyacenes (LP) have been shown to be obtainable from Pascal's triangle by using a graph factorisation and squaring technique. Strong subspectrality existing among the members of the linear polyacene series has been shown from the derivation of the CP's.

  2. Characteristic polynomials of linear polyacenes and their ...

    Indian Academy of Sciences (India)

    Coefficients of characteristic polynomials (CP) of linear polyacenes (LP) have been shown to be obtainable from Pascal's triangle by using a graph factorisation and squaring technique. Strong subspectrality existing among the members of the linear polyacene series has been shown from the derivation of the CP's. Thus it ...

  3. Coherent states for polynomial su(2) algebra

    International Nuclear Information System (INIS)

    Sadiq, Muhammad; Inomata, Akira

    2007-01-01

    A class of generalized coherent states is constructed for a polynomial su(2) algebra in a group-free manner. As a special case, the coherent states for the cubic su(2) algebra are discussed. The states so constructed reduce to the usual SU(2) coherent states in the linear limit

  4. On zeros of certain Dirichlet polynomials

    OpenAIRE

    BLLACA, KAJTAZ H.

    2015-01-01

    In this article we establish the zero-free region of certain Dirichlet polynomials $L_{F, X}$ arising in approximate functional equation for functions in the Selberg class and we prove an asymptotic formula for the number of zeros of $L_{F, X}$.

  5. Exceptional polynomials and SUSY quantum mechanics

    Indian Academy of Sciences (India)

    Exceptional polynomials and SUSY quantum mechanics. K V S SHIV CHAITANYA1,∗, S SREE RANJANI2,. PRASANTA K PANIGRAHI3, R RADHAKRISHNAN4 and V SRINIVASAN4. 1BITS Pilani, Hyderabad Campus, Jawahar Nagar, Shameerpet Mandal, Hyderabad 500 078, India. 2Faculty of Science and Technology, ...

  6. Deformation Prediction Using Linear Polynomial Functions ...

    African Journals Online (AJOL)

    The predictions are compared with measured data reported in literature and the results are discussed. The computational aspects of implementation of the model are also discussed briefly. Keywords: Linear Polynomial, Structural Deformation, Prediction Journal of the Nigerian Association of Mathematical Physics, Volume ...

  7. Subintegrality, invertible modules and Laurent polynomial extensions

    Indian Academy of Sciences (India)

    Indian Acad. Sci. (Math. Sci.) Vol. 125, No. 2, May 2015, pp. 149–160. c Indian Academy of Sciences. Subintegrality, invertible modules and Laurent polynomial extensions. VIVEK SADHU. Department of Mathematics ...... comments which have improved the exposition. Further, he would like to thank CSIR,. India for financial ...

  8. 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.

  9. Error Minimization of Polynomial Approximation of Delta

    Indian Academy of Sciences (India)

    2016-01-27

    Jan 27, 2016 ... The difference between Universal time (UT) and Dynamical time (TD), known as Delta ( ) is tabulated for the first day of each year in the Astronomical Almanac. During the last four centuries it is found that there are large differences between its values for two consecutive years. Polynomial ...

  10. On Arithmetic-Geometric-Mean Polynomials

    Science.gov (United States)

    Griffiths, Martin; MacHale, Des

    2017-01-01

    We study here an aspect of an infinite set "P" of multivariate polynomials, the elements of which are associated with the arithmetic-geometric-mean inequality. In particular, we show in this article that there exist infinite subsets of probability "P" for which every element may be expressed as a finite sum of squares of real…

  11. On the zeros of the classical polynomials

    International Nuclear Information System (INIS)

    Calogero, F.

    1977-01-01

    Some theorems and conjectures concerning the zeros of the classical polynomials (of Hermite, Laguerre and Jacobi) are announced. The theorems, whose proof is not reported, have been obtained from the relationship between the motion of the zeros of special solutions of simple linear partial differential equations (in x and t), and one-dimensional many-body problems. (author)

  12. Hermite polynomials and quasi-classical asymptotics

    Czech Academy of Sciences Publication Activity Database

    Ali, S. T.; Engliš, Miroslav

    2014-01-01

    Roč. 55, č. 4 (2014), 042102 ISSN 0022-2488 R&D Projects: GA ČR GA201/09/0473 Institutional support: RVO:67985840 Keywords : quantization * Hermite polynomials * reproducing kernel Subject RIV: BA - General Mathematics Impact factor: 1.243, year: 2014 http://scitation.aip.org/content/aip/journal/jmp/55/4/10.1063/1.4869325

  13. Polynomial Asymptotes of the Second Kind

    Science.gov (United States)

    Dobbs, David E.

    2011-01-01

    This note uses the analytic notion of asymptotic functions to study when a function is asymptotic to a polynomial function. Along with associated existence and uniqueness results, this kind of asymptotic behaviour is related to the type of asymptote that was recently defined in a more geometric way. Applications are given to rational functions and…

  14. The 6 Vertex Model and Schubert Polynomials

    Directory of Open Access Journals (Sweden)

    Alain Lascoux

    2007-02-01

    Full Text Available We enumerate staircases with fixed left and right columns. These objects correspond to ice-configurations, or alternating sign matrices, with fixed top and bottom parts. The resulting partition functions are equal, up to a normalization factor, to some Schubert polynomials.

  15. Polynomial Vector Fields in One Complex Variable

    DEFF Research Database (Denmark)

    Branner, Bodil

    In recent years Adrien Douady was interested in polynomial vector fields, both in relation to iteration theory and as a topic on their own. This talk is based on his work with Pierrette Sentenac, work of Xavier Buff and Tan Lei, and my own joint work with Kealey Dias....

  16. Nonclassical Orthogonal Polynomials and Corresponding Quadratures

    CERN Document Server

    Fukuda, H; Alt, E O; Matveenko, A V

    2004-01-01

    We construct nonclassical orthogonal polynomials and calculate abscissas and weights of Gaussian quadrature for arbitrary weight and interval. The program is written by Mathematica and it works if moment integrals are given analytically. The result is a FORTRAN subroutine ready to utilize the quadrature.

  17. Indecomposability of polynomials via Jacobian matrix

    International Nuclear Information System (INIS)

    Cheze, G.; Najib, S.

    2007-12-01

    Uni-multivariate decomposition of polynomials is a special case of absolute factorization. Recently, thanks to the Ruppert's matrix some effective results about absolute factorization have been improved. Here we show that with a jacobian matrix we can get sharper bounds for the special case of uni-multivariate decomposition. (author)

  18. transformation of independent variables in polynomial regression ...

    African Journals Online (AJOL)

    Ada

    preferable when possible to work with a simple functional form in transformed variables rather than with a more complicated form in the original variables. In this paper, it is shown that linear transformations applied to independent variables in polynomial regression models affect the t ratio and hence the statistical ...

  19. 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.

  20. On the Symmetric Properties for the Generalized Twisted Bernoulli Polynomials

    Directory of Open Access Journals (Sweden)

    Kim Taekyun

    2009-01-01

    Full Text Available We study the symmetry for the generalized twisted Bernoulli polynomials and numbers. We give some interesting identities of the power sums and the generalized twisted Bernoulli polynomials using the symmetric properties for the -adic invariant integral.

  1. 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.

  2. On a Family of Multivariate Modified Humbert Polynomials

    Science.gov (United States)

    Aktaş, Rabia; Erkuş-Duman, Esra

    2013-01-01

    This paper attempts to present a multivariable extension of generalized Humbert polynomials. The results obtained here include various families of multilinear and multilateral generating functions, miscellaneous properties, and also some special cases for these multivariable polynomials. PMID:23935411

  3. Irreducibility results for compositions of polynomials in several ...

    Indian Academy of Sciences (India)

    We obtain explicit upper bounds for the number of irreducible factors for a class of compositions of polynomials in several variables over a given field. In particular, some irreducibility criteria are given for this class of compositions of polynomials.

  4. Recurrence relations for the Cartesian derivatives of the Zernike polynomials.

    Science.gov (United States)

    Stephenson, Philip C L

    2014-04-01

    A recurrence relation for the first-order Cartesian derivatives of the Zernike polynomials is derived. This relation is used with the Clenshaw method to determine an efficient method for calculating the derivatives of any linear series of Zernike polynomials.

  5. Comparison of Model Reliabilities from Single-Step and Bivariate Blending Methods

    DEFF Research Database (Denmark)

    Taskinen, Matti; Mäntysaari, Esa; Lidauer, Martin

    2013-01-01

    the production trait evaluation of Nordic Red dairy cattle. Genotyped bulls with daughters are used as training animals, and genotyped bulls and producing cows as candidate animals. For simplicity, size of the data is chosen so that the full inverses of the mixed model equation coefficient matrices can......Model based reliabilities in genetic evaluation are compared between three methods: animal model BLUP, single-step BLUP, and bivariate blending after genomic BLUP. The original bivariate blending is revised in this work to better account animal models. The study data is extracted from...... be calculated. Model reliabilities by the single-step and the bivariate blending methods were higher than by animal model due to genomic information. Compared to the single-step method, the bivariate blending method reliability estimates were, in general, lower. Computationally bivariate blending method was...

  6. Some symmetric identities for the generalized Bernoulli, Euler and Genocchi polynomials associated with Hermite polynomials.

    Science.gov (United States)

    Khan, Waseem A; Haroon, Hiba

    2016-01-01

    In 2008, Liu and Wang established various symmetric identities for Bernoulli, Euler and Genocchi polynomials. In this paper, we extend these identities in a unified and generalized form to families of Hermite-Bernoulli, Euler and Genocchi polynomials. The procedure followed is that of generating functions. Some relevant connections of the general theory developed here with the results obtained earlier by Pathan and Khan are also pointed out.

  7. 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

  8. Weighted inequalities for generalized polynomials with doubling weights.

    Science.gov (United States)

    Joung, Haewon

    2017-01-01

    Many weighted polynomial inequalities, such as the Bernstein, Marcinkiewicz, Schur, Remez, Nikolskii inequalities, with doubling weights were proved by Mastroianni and Totik for the case [Formula: see text], and by Tamás Erdélyi for [Formula: see text]. In this paper we extend such polynomial inequalities to those for generalized trigonometric polynomials. We also prove the large sieve for generalized trigonometric polynomials with doubling weights.

  9. Structural identifiability of polynomial and rational systems

    NARCIS (Netherlands)

    J. Nemcová (Jana)

    2010-01-01

    htmlabstractSince analysis and simulation of biological phenomena require the availability of their fully specified models, one needs to be able to estimate unknown parameter values of the models. In this paper we deal with identifiability of parametrizations which is the property of one-to-one

  10. 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…

  11. 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

  12. 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

  13. 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.

  14. 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.

  15. 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.

  16. The quality of zero bounds for complex polynomials.

    Science.gov (United States)

    Dehmer, Matthias; Tsoy, Yury Robertovich

    2012-01-01

    In this paper, we evaluate the quality of zero bounds on the moduli of univariate complex polynomials. We select classical and recently developed bounds and evaluate their quality by using several sets of complex polynomials. As the quality of priori bounds has not been investigated thoroughly, our results can be useful to find optimal bounds to locate the zeros of complex polynomials.

  17. Multiple Representations and the Understanding of Taylor Polynomials

    Science.gov (United States)

    Habre, Samer

    2009-01-01

    The study of Maclaurin and Taylor polynomials entails the comprehension of various new mathematical ideas. Those polynomials are initially discussed at the college level in a calculus class and then again in a course on numerical methods. This article investigates the understanding of these polynomials by students taking a numerical methods class…

  18. The Gibbs Phenomenon for Series of Orthogonal Polynomials

    Science.gov (United States)

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

    2006-01-01

    This note considers the four classes of orthogonal polynomials--Chebyshev, Hermite, Laguerre, Legendre--and investigates the Gibbs phenomenon at a jump discontinuity for the corresponding orthogonal polynomial series expansions. The perhaps unexpected thing is that the Gibbs constant that arises for each class of polynomials appears to be the same…

  19. Root and critical point behaviors of certain sums of polynomials

    Indian Academy of Sciences (India)

    Seon-Hong Kim

    2018-04-24

    Apr 24, 2018 ... Introduction. There is an extensive literature concerning roots of sums of polynomials. Many authors. [5–7] 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?' ...

  20. 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 ...

  1. 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.

  2. An integrated user-friendly ArcMAP tool for bivariate statistical modelling in geoscience applications

    Science.gov (United States)

    Jebur, M. N.; Pradhan, B.; Shafri, H. Z. M.; Yusoff, Z. M.; Tehrany, M. S.

    2015-03-01

    Modelling and classification difficulties are fundamental issues in natural hazard assessment. A geographic information system (GIS) is a domain that requires users to use various tools to perform different types of spatial modelling. Bivariate statistical analysis (BSA) assists in hazard modelling. To perform this analysis, several calculations are required and the user has to transfer data from one format to another. Most researchers perform these calculations manually by using Microsoft Excel or other programs. This process is time-consuming and carries a degree of uncertainty. The lack of proper tools to implement BSA in a GIS environment prompted this study. In this paper, a user-friendly tool, bivariate statistical modeler (BSM), for BSA technique is proposed. Three popular BSA techniques, such as frequency ratio, weight-of-evidence (WoE), and evidential belief function (EBF) models, are applied in the newly proposed ArcMAP tool. This tool is programmed in Python and created by a simple graphical user interface (GUI), which facilitates the improvement of model performance. The proposed tool implements BSA automatically, thus allowing numerous variables to be examined. To validate the capability and accuracy of this program, a pilot test area in Malaysia is selected and all three models are tested by using the proposed program. Area under curve (AUC) is used to measure the success rate and prediction rate. Results demonstrate that the proposed program executes BSA with reasonable accuracy. The proposed BSA tool can be used in numerous applications, such as natural hazard, mineral potential, hydrological, and other engineering and environmental applications.

  3. Reducing the number of variables of a polynomial

    OpenAIRE

    Carlini, Enrico

    2005-01-01

    In this paper, we consider two basic questions about presenting a homogeneous polynomial f: how many variables are needed for presenting f? How can one find a presentation of f involving as few variables as possible? We give a complete answer to both questions, determining the minimal number of variables needed, NEssVar(f), and describing these variables through their linear span, EssVar(f). Our results give rise to effective algorithms which we implemented in the computer algebra system CoCoA.

  4. Polynomial mechanics and optimal control

    OpenAIRE

    Srinivasan, Akshay; Venkadesan, Madhusudhan

    2014-01-01

    We describe a new algorithm for trajectory optimization of mechanical systems. Our method combines pseudo-spectral methods for function approximation with variational discretization schemes that exactly preserve conserved mechanical quantities such as momentum. We thus obtain a global discretization of the Lagrange-d'Alembert variational principle using pseudo-spectral methods. Our proposed scheme inherits the numerical convergence characteristics of spectral methods, yet preserves momentum-c...

  5. 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.

  6. 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

  7. 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...... 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...

  8. Planar Harmonic and Monogenic Polynomials of Type A

    Directory of Open Access Journals (Sweden)

    Charles F. Dunkl

    2016-10-01

    Full Text Available Harmonic polynomials of type A are polynomials annihilated by the Dunkl Laplacian associated to the symmetric group acting as a reflection group on R N . The Dunkl operators are denoted by T j for 1 ≤ j ≤ N , and the Laplacian Δ κ = ∑ j = 1 N T j 2 . This paper finds the homogeneous harmonic polynomials annihilated by all T j for j > 2 . The structure constants with respect to the Gaussian and sphere inner products are computed. These harmonic polynomials are used to produce monogenic polynomials, those annihilated by a Dirac-type operator.

  9. Zernike polynomials for photometric characterization of LEDs

    International Nuclear Information System (INIS)

    Velázquez, J L; Ferrero, A; Pons, A; Campos, J; Hernanz, M L

    2016-01-01

    We propose a method based on Zernike polynomials to characterize photometric quantities and descriptors of light emitting diodes (LEDs) from measurements of the angular distribution of the luminous intensity, such as total luminous flux, BA, inhomogeneity, anisotropy, direction of the optical axis and Lambertianity of the source. The performance of this method was experimentally tested for 18 high-power LEDs from different manufacturers and with different photometric characteristics. A small set of Zernike coefficients can be used to calculate all the mentioned photometric quantities and descriptors. For applications not requiring a great accuracy such as those of lighting design, the angular distribution of the luminous intensity of most of the studied LEDs can be interpolated with only two Zernike polynomials. (paper)

  10. Orthogonal polynomials in neutron transport theory

    Energy Technology Data Exchange (ETDEWEB)

    Dehesa, J.S. (Granada Univ. (Spain). Facultad de Ciencias)

    1982-01-01

    The asymptotic average properties of zeros of the polynomials gsub(k)sup(m) (x), which play a fundamental role in neutron transport and radiative transfer theories, are investigated analytically in terms of the angular expansion coefficients wsub(k) of the scattering kernel for three wide classes of scattering models. In particular it is found that the scattering models of Eccleston-McCormick (J. Nucl. Energy.; 24:23 (1970)), Shultis et al (Nucl. Sci. Eng.; 59:53 (1976)) and Henyey-Greenstein (Astrophys. J.; 93:70 (1941)) belong in one of the above-mentioned classes, and their associated polynomials gsub(k)sup(m) (x) have the same asymptotic density of zeros.

  11. Polynomial chaos representation of databases on manifolds

    Energy Technology Data Exchange (ETDEWEB)

    Soize, C., E-mail: christian.soize@univ-paris-est.fr [Université Paris-Est, Laboratoire Modélisation et Simulation Multi-Echelle, MSME UMR 8208 CNRS, 5 bd Descartes, 77454 Marne-La-Vallée, Cedex 2 (France); Ghanem, R., E-mail: ghanem@usc.edu [University of Southern California, 210 KAP Hall, Los Angeles, CA 90089 (United States)

    2017-04-15

    Characterizing the polynomial chaos expansion (PCE) of a vector-valued random variable with probability distribution concentrated on a manifold is a relevant problem in data-driven settings. The probability distribution of such random vectors is multimodal in general, leading to potentially very slow convergence of the PCE. In this paper, we build on a recent development for estimating and sampling from probabilities concentrated on a diffusion manifold. The proposed methodology constructs a PCE of the random vector together with an associated generator that samples from the target probability distribution which is estimated from data concentrated in the neighborhood of the manifold. The method is robust and remains efficient for high dimension and large datasets. The resulting polynomial chaos construction on manifolds permits the adaptation of many uncertainty quantification and statistical tools to emerging questions motivated by data-driven queries.

  12. Polynomial structures in one-loop amplitudes

    International Nuclear Information System (INIS)

    Britto, Ruth; Feng Bo; Yang Gang

    2008-01-01

    A general one-loop scattering amplitude may be expanded in terms of master integrals. The coefficients of the master integrals can be obtained from tree-level input in a two-step process. First, use known formulas to write the coefficients of (4-2ε)-dimensional master integrals; these formulas depend on an additional variable, u, which encodes the dimensional shift. Second, convert the u-dependent coefficients of (4-2ε)-dimensional master integrals to explicit coefficients of dimensionally shifted master integrals. This procedure requires the initial formulas for coefficients to have polynomial dependence on u. Here, we give a proof of this property in the case of massless propagators. The proof is constructive. Thus, as a byproduct, we produce different algebraic expressions for the scalar integral coefficients, in which the polynomial property is apparent. In these formulas, the box and pentagon contributions are separated explicitly.

  13. A Deterministic and Polynomial Modified Perceptron Algorithm

    Directory of Open Access Journals (Sweden)

    Olof Barr

    2006-01-01

    Full Text Available We construct a modified perceptron algorithm that is deterministic, polynomial and also as fast as previous known algorithms. The algorithm runs in time O(mn3lognlog(1/ρ, where m is the number of examples, n the number of dimensions and ρ is approximately the size of the margin. We also construct a non-deterministic modified perceptron algorithm running in timeO(mn2lognlog(1/ρ.

  14. Polynomials and identities on real Banach spaces

    Czech Academy of Sciences Publication Activity Database

    Hájek, Petr Pavel; Kraus, M.

    2012-01-01

    Roč. 385, č. 2 (2012), s. 1015-1026 ISSN 0022-247X R&D Projects: GA ČR(CZ) GAP201/11/0345 Institutional research plan: CEZ:AV0Z10190503 Keywords : Polynomials on Banach spaces Subject RIV: BA - General Mathematics Impact factor: 1.050, year: 2012 http://www.sciencedirect.com/science/article/pii/S0022247X11006743

  15. Piecewise polynomial solutions to linear inverse problems

    DEFF Research Database (Denmark)

    Hansen, Per Christian; Mosegaard, K.

    1996-01-01

    We have presented a new algorithm PP-TSVD that computes piecewise polynomial solutions to ill-posed problems, without a priori knowledge about the positions of the break points. In particular, we can compute piecewise constant functions that describe layered models. Such solutions are useful, e.g.......g., in seismological problems, and the algorithm can also be used as a preprocessor for other methods where break points/discontinuities must be incorporated explicitly....

  16. Polynomial Variables and the Jacobian Problem

    Indian Academy of Sciences (India)

    Most readers would be familiar with the technique of shifting the origin to simplify the equation -of a locus. This process is the same as choosing new coordinate axes, ..... 1982 by H Bass, E H Connell and D Wright. Points at Infinity. Let f be a nonzero polynomial in X, Y, and let d = deg (f). Then f can be written in the form f =.

  17. 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,

  18. Zero sets of polynomials in several variables

    Czech Academy of Sciences Publication Activity Database

    Aron, R. M.; Hájek, Petr Pavel

    2006-01-01

    Roč. 86, č. 6 (2006), s. 561-568 ISSN 0003-889X R&D Projects: GA ČR(CZ) GA201/01/1198; GA ČR(CZ) GA201/04/0090; GA AV ČR(CZ) IAA1019205 Institutional research plan: CEZ:AV0Z10190503 Keywords : polynomial * zero set Subject RIV: BA - General Mathematics Impact factor: 0.341, year: 2006

  19. Trigonometric Polynomials For Estimation Of Spectra

    Science.gov (United States)

    Greenhall, Charles A.

    1990-01-01

    Orthogonal sets of trigonometric polynomials used as suboptimal substitutes for discrete prolate-spheroidal "windows" of Thomson method of estimation of spectra. As used here, "windows" denotes weighting functions used in sampling time series to obtain their power spectra within specified frequency bands. Simplified windows designed to require less computation than do discrete prolate-spheroidal windows, albeit at price of some loss of accuracy.

  20. Detecting Prime Numbers via Roots of Polynomials

    Science.gov (United States)

    Dobbs, David E.

    2012-01-01

    It is proved that an integer n [greater than or equal] 2 is a prime (resp., composite) number if and only if there exists exactly one (resp., more than one) nth-degree monic polynomial f with coefficients in Z[subscript n], the ring of integers modulo n, such that each element of Z[subscript n] is a root of f. This classroom note could find use in…

  1. Multivariate Krawtchouk Polynomials and Composition Birth and Death Processes

    Directory of Open Access Journals (Sweden)

    Robert Griffiths

    2016-05-01

    Full Text Available This paper defines the multivariate Krawtchouk polynomials, orthogonal on the multinomial distribution, and summarizes their properties as a review. The multivariate Krawtchouk polynomials are symmetric functions of orthogonal sets of functions defined on each of N multinomial trials. The dual multivariate Krawtchouk polynomials, which also have a polynomial structure, are seen to occur naturally as spectral orthogonal polynomials in a Karlin and McGregor spectral representation of transition functions in a composition birth and death process. In this Markov composition process in continuous time, there are N independent and identically distributed birth and death processes each with support 0 , 1 , … . The state space in the composition process is the number of processes in the different states 0 , 1 , … . Dealing with the spectral representation requires new extensions of the multivariate Krawtchouk polynomials to orthogonal polynomials on a multinomial distribution with a countable infinity of states.

  2. Doubling (Dual) Hahn Polynomials: Classification and Applications

    Science.gov (United States)

    Oste, Roy; Van der Jeugt, Joris

    2016-01-01

    We classify all pairs of recurrence relations in which two Hahn or dual Hahn polynomials with different parameters appear. Such couples are referred to as (dual) Hahn doubles. The idea and interest comes from an example appearing in a finite oscillator model [Jafarov E.I., Stoilova N.I., Van der Jeugt J., J. Phys. A: Math. Theor. 44 (2011), 265203, 15 pages, arXiv:1101.5310]. Our classification shows there exist three dual Hahn doubles and four Hahn doubles. The same technique is then applied to Racah polynomials, yielding also four doubles. Each dual Hahn (Hahn, Racah) double gives rise to an explicit new set of symmetric orthogonal polynomials related to the Christoffel and Geronimus transformations. For each case, we also have an interesting class of two-diagonal matrices with closed form expressions for the eigenvalues. This extends the class of Sylvester-Kac matrices by remarkable new test matrices. We examine also the algebraic relations underlying the dual Hahn doubles, and discuss their usefulness for the construction of new finite oscillator models.

  3. 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

  4. Polynomial expansions and transition strengths

    International Nuclear Information System (INIS)

    Draayer, J.P.

    1980-01-01

    The subject is statistical spectroscopy applied to determining strengths and strength sums of excitation processes in nuclei. The focus will be on a ds-shell isoscalar E2 study with detailed shell-model results providing the standard for comparison; similar results are available for isovector E2 and M1 and E4 transitions as well as for single-particle transfer and ν +- decay. The present study is intended to serve as a tutorial for applications where shell-model calculations are not feasible. The problem is posed and a schematic theory for strengths and sums is presented. The theory is extended to include the effect of correlations between H, the system Hamiltonian, and theta, the excitation operator. Associated with correlation measures is a geometry that can be used to anticipate the goodness of a symmetry. This is illustrated for pseudo SU(3) in the fp-shell. Some conclusions about fluctuations and collectivity that one can deduce from the statistical results for strengths are presented

  5. Non-axisymmetric Aberration Patterns from Wide-field Telescopes Using Spin-weighted Zernike Polynomials

    Science.gov (United States)

    Kent, Stephen M.

    2018-04-01

    If the optical system of a telescope is perturbed from rotational symmetry, the Zernike wavefront aberration coefficients describing that system can be expressed as a function of position in the focal plane using spin-weighted Zernike polynomials. Methodologies are presented to derive these polynomials to arbitrary order. This methodology is applied to aberration patterns produced by a misaligned Ritchey–Chrétien telescope and to distortion patterns at the focal plane of the DESI optical corrector, where it is shown to provide a more efficient description of distortion than conventional expansions.

  6. A multiple-scale Pascal polynomial for 2D Stokes and inverse Cauchy-Stokes problems

    Science.gov (United States)

    Liu, Chein-Shan; Young, D. L.

    2016-05-01

    The polynomial expansion method is a useful tool for solving both the direct and inverse Stokes problems, which together with the pointwise collocation technique is easy to derive the algebraic equations for satisfying the Stokes differential equations and the specified boundary conditions. In this paper we propose two novel numerical algorithms, based on a third-first order system and a third-third order system, to solve the direct and the inverse Cauchy problems in Stokes flows by developing a multiple-scale Pascal polynomial method, of which the scales are determined a priori by the collocation points. To assess the performance through numerical experiments, we find that the multiple-scale Pascal polynomial expansion method (MSPEM) is accurate and stable against large noise.

  7. A bivariate optimal replacement policy with cumulative repair cost ...

    Indian Academy of Sciences (India)

    Min-Tsai Lai

    Shock model; cumulative damage model; cumulative repair cost limit; preventive maintenance model. 1. Introduction. Most production systems are repaired or replaced when they have already failed. However, they may require much time and high expenses to repair a failed system, so maintaining a system to prevent ...

  8. On the matched pairs sign test using bivariate ranked set sampling ...

    African Journals Online (AJOL)

    BVRSS) is introduced and investigated. We show that this test is asymptotically more efficient than its counterpart sign test based on a bivariate simple random sample (BVSRS). The asymptotic null distribution and the efficiency of the test are derived.

  9. From $r$-Linearized Polynomial Equations to $r^m$-Linearized Polynomial Equations

    OpenAIRE

    Fernando, Neranga; Hou, Xiang-dong

    2015-01-01

    Let $r$ be a prime power and $q=r^m$. For $0\\le i\\le m-1$, let $f_i\\in \\mathbb{F}_r[x]$ be $q$-linearized and $a_i\\in \\mathbb{F}_q$. Assume that $z\\in \\mathbb{\\bar{F}}_r$ satisfies the equation $\\sum_{i=0}^{m-1}a_if_i(z)^{r^i}=0$, where $\\sum_{i=0}^{m-1}a_if_i^{r_i}\\in \\mathbb{F}_q[x]$ is an $r$-linearized polynomial. It is shown that $z$ satisfies a $q$-linearized polynomial equation with coefficients in $\\mathbb{F}_r$. This result provides an explanation for numerous permutation polynomials...

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

    KAUST Repository

    Butler, 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.

  11. Phase wavefront aberration modeling using Zernike and pseudo-Zernike polynomials.

    Science.gov (United States)

    Rahbar, Kambiz; Faez, Karim; Attaran Kakhki, Ebrahim

    2013-10-01

    Orthogonal polynomials can be used for representing complex surfaces on a specific domain. In optics, Zernike polynomials have widespread applications in testing optical instruments, measuring wavefront distributions, and aberration theory. This orthogonal set on the unit circle has an appropriate matching with the shape of optical system components, such as entrance and exit pupils. The existence of noise in the process of representation estimation of optical surfaces causes a reduction of precision in the process of estimation. Different strategies are developed to manage unwanted noise effects and to preserve the quality of the estimation. This article studies the modeling of phase wavefront aberrations in third-order optics by using a combination of Zernike and pseudo-Zernike polynomials and shows how this combination may increase the robustness of the estimation process of phase wavefront aberration distribution.

  12. Polynomial invariants for discrimination and classification of four-qubit entanglement

    International Nuclear Information System (INIS)

    Viehmann, Oliver; Eltschka, Christopher; Siewert, Jens

    2011-01-01

    The number of entanglement classes in stochastic local operations and classical communication (SLOCC) classifications increases with the number of qubits and is already infinite for four qubits. Criteria for explicitly discriminating and classifying pure states of four and more qubits are highly desirable and therefore at the focus of intense theoretical research. We develop a general criterion for the discrimination of pure N-partite entangled states in terms of polynomial SL(d,C) xN invariants. By means of this criterion, existing SLOCC classifications of four-qubit entanglement are reproduced. Based on this we propose a polynomial classification scheme in which entanglement types are identified through 'tangle patterns'. This scheme provides a practicable way to classify states of arbitrary multipartite systems. Moreover, the use of polynomials induces a corresponding quantification of the different types of entanglement.

  13. The Use of Generalized Laguerre Polynomials in Spectral Methods for Solving Fractional Delay Differential Equations.

    Science.gov (United States)

    Khader, M M

    2013-10-01

    In this paper, an efficient numerical method for solving the fractional delay differential equations (FDDEs) is considered. The fractional derivative is described in the Caputo sense. The proposed method is based on the derived approximate formula of the Laguerre polynomials. The properties of Laguerre polynomials are utilized to reduce FDDEs to a linear or nonlinear system of algebraic equations. Special attention is given to study the error and the convergence analysis of the proposed method. Several numerical examples are provided to confirm that the proposed method is in excellent agreement with the exact solution.

  14. Exponential operators, generalized polynomials and evolution problems

    International Nuclear Information System (INIS)

    Dattoli, G.; Mancho, A.M.; Quattromini, M.; Torre, A.

    2001-01-01

    The operator (d/dx) χ d/dx plays a central role in the theory of operational calculus. Its exponential form is crucial in problems relevant to solutions of Fokker-Planck and Schroedinger equations. We explore the formal properties of the evolution operators associated to (d/dx) χ d/dx, discuss its link to special forms of Laguerre polynomials and Laguerre-based functions. The obtained results are finally applied to specific problems concerning the solution of Fokker-Planck equations relevant to the beam lifetime in storage rings

  15. 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.

  16. 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

  17. Explicitly integrable polynomial Hamiltonians and evaluation of Lie transformations

    International Nuclear Information System (INIS)

    Shi, J.; Yan, Y.T.

    1993-01-01

    We have found that any homogeneous polynomial can be written as a sum of integrable polynomials of the same degree, with which each associated polynomial Hamiltonian is integrable, and the associated Lie transformation can be evaluated exactly. An integrable polynomial factorization has thus been developed to convert a sympletic map in the form of a Dragt-Finn factorization into a product of exactly evaluable Lie transformations associated with integrable polynomials. Having a small number of factorization bases of integrable polynomials enables one to consider a factorization with the use of high-order symplectic integrators so that a symplectic map can always be evaluated with the desired accuracy. The results are significant for studying the long-term stability of beams in accelerators

  18. Polynomials for torus links from Chern-Simons gauge theories

    International Nuclear Information System (INIS)

    Isidro, J.M.; Labastida, J.M.F.; Ramallo, A.V.

    1993-01-01

    Invariant polynomials for torus links are obtained in the framework of the Chern-Simons topological gauge theory. The polynomials are computed as vacuum expectation values on the three-sphere of Wilson line operators representing the Verlinde algebra of the corresponding rational conformal field theory. In the case of the SU(2) gauge theory our results provide explicit expressions for the Jones polynomial as well as for the polynomials associated to the N-state (N>2) vertex models (Akutsu-Wadati polynomials). By means of the Chern-Simons coset construction, the minimal unitary models are analyzed, showing that the corresponding link invariants factorize into two SU(2) polynomials. A method to obtain skein rules from the Chern-Simons knot operators is developed. This procedure yields the eigenvalues of the braiding matrix of the corresponding conformal field theory. (orig.)

  19. 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

  20. Recurrence relations and weight functions for approximation by orthonormal polynomials

    International Nuclear Information System (INIS)

    Bisoi, A.K.

    1989-12-01

    The popular three-term recurrence relation is found to be a numerically unstable algorithm for generation of higher order orthogonal polynomials. For stability, a k-term recurrence relation for generating the k th order polynomial is suggested. Instead of assigning equal weight to data points with unspecified errors, the use of the Jacobian weight function is suggested for generating the orthogonal polynomials to effect a closer representation. (author). 6 refs, 1 tab

  1. Vector polynomials for direct analysis of circular wavefront slope data.

    Science.gov (United States)

    Mahajan, Virendra N; Acosta, Eva

    2017-10-01

    In the aberration analysis of a circular wavefront, Zernike circle polynomials are used to obtain its wave aberration coefficients. To obtain these coefficients from the wavefront slope data, we need vector functions that are orthogonal to the gradients of the Zernike polynomials, and are irrotational so as to propagate minimum uncorrelated random noise from the data to the coefficients. In this paper, we derive such vector functions, which happen to be polynomials.

  2. 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 ...

  3. 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.

  4. GIS-based bivariate statistical techniques for groundwater potential ...

    Indian Academy of Sciences (India)

    ... complicated relation between groundwater occurrence and groundwater conditioning factors, which permits investigation of both systemic and stochastic uncertainty. Finally, it can be realized that these techniques are very beneficial for groundwater potential analyzing and can be practical for water-resource management ...

  5. GIS-based bivariate statistical techniques for groundwater potential ...

    Indian Academy of Sciences (India)

    Ali Haghizadeh

    2017-11-23

    Nov 23, 2017 ... of both systemic and stochastic uncertainty. Finally, it can be realized that these techniques are very beneficial for groundwater potential analyzing and can be practical for water-resource management experts. Keywords. Groundwater; statistical index; Dempster–Shafer theory; water resource management; ...

  6. An assessment on the use of bivariate, multivariate and soft ...

    Indian Academy of Sciences (India)

    map is a useful tool in urban planning. There have been few studies carried out on collapse hazard ... application of a scoring system to several control- ling factors. The probabilistic method for assess- ment of .... of conditions and processes controlling collapse event. In order to predict collapse, it is necessary to assume ...

  7. An Analytic Formula for the A_2 Jack Polynomials

    Directory of Open Access Journals (Sweden)

    Vladimir V. Mangazeev

    2007-01-01

    Full Text Available In this letter I shall review my joint results with Vadim Kuznetsov and Evgeny Sklyanin [Indag. Math. 14 (2003, 451-482] on separation of variables (SoV for the $A_n$ Jack polynomials. This approach originated from the work [RIMS Kokyuroku 919 (1995, 27-34] where the integral representations for the $A_2$ Jack polynomials was derived. Using special polynomial bases I shall obtain a more explicit expression for the $A_2$ Jack polynomials in terms of generalised hypergeometric functions.

  8. Tabulating knot polynomials for arborescent knots

    International Nuclear Information System (INIS)

    Mironov, A; Morozov, A; Morozov, A; Sleptsov, A; Ramadevi, P; Singh, Vivek Kumar

    2017-01-01

    Arborescent knots are those which can be represented in terms of double fat graphs or equivalently as tree Feynman diagrams. This is the class of knots for which the present knowledge is sufficient for lifting topological description to the level of effective analytical formulas. The paper describes the origin and structure of the new tables of colored knot polynomials, which will be posted at the dedicated site (http://knotebook.org). Even if formal expressions are known in terms of modular transformation matrices, the computation in finite time requires additional ideas. We use the ‘family’ approach, suggested in Mironov and Morozov (2015 Nucl. Phys . B 899 395–413), and apply it to arborescent knots in the Rolfsen table by developing a Feynman diagram technique, associated with an auxiliary matrix model field theory. Gauge invariance in this theory helps to provide meaning to Racah matrices in the case of non-trivial multiplicities and explains the need for peculiar sign prescriptions in the calculation of [21]-colored HOMFLY-PT polynomials. (paper)

  9. 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...

  10. Tabulating knot polynomials for arborescent knots

    Science.gov (United States)

    Mironov, A.; Morozov, A.; Morozov, A.; Ramadevi, P.; Singh, Vivek Kumar; Sleptsov, A.

    2017-02-01

    Arborescent knots are those which can be represented in terms of double fat graphs or equivalently as tree Feynman diagrams. This is the class of knots for which the present knowledge is sufficient for lifting topological description to the level of effective analytical formulas. The paper describes the origin and structure of the new tables of colored knot polynomials, which will be posted at the dedicated site (http://knotebook.org). Even if formal expressions are known in terms of modular transformation matrices, the computation in finite time requires additional ideas. We use the ‘family’ approach, suggested in Mironov and Morozov (2015 Nucl. Phys. B 899 395-413), and apply it to arborescent knots in the Rolfsen table by developing a Feynman diagram technique, associated with an auxiliary matrix model field theory. Gauge invariance in this theory helps to provide meaning to Racah matrices in the case of non-trivial multiplicities and explains the need for peculiar sign prescriptions in the calculation of [21]-colored HOMFLY-PT polynomials.

  11. Sphere-cone-polynomial special window with good aberration characteristic

    International Nuclear Information System (INIS)

    Wang Chao; Zhang Xin; Qu He-Meng; Wang Ling-Jie; Wang Yu

    2013-01-01

    Optical windows with external surfaces shaped to satisfy operational environment needs are known as special windows. A novel special window, a sphere-cone-polynomial (SCP) window, is proposed. The formulas of this window shape are given. An SCP MgF 2 window with a fineness ratio of 1.33 is designed as an example. The field-of-regard (FOR) angle is ±75°. From the window system simulation results obtained with the calculated fluid dynamics (CFD) and optical design software, we find that compared to the conventional window forms, the SCP shape can not only introduce relatively less drag in the airflow, but also have the minimal effect on imaging. So the SCP window optical system can achieve a high image quality across a super wide FOR without adding extra aberration correctors. The tolerance analysis results show that the optical performance can be maintained with a reasonable fabricating tolerance to manufacturing errors

  12. Polynomial Modeling of Child and Adult Intonation in German Spontaneous Speech

    Science.gov (United States)

    de Ruiter, Laura E.

    2011-01-01

    In a data set of 291 spontaneous utterances from German 5-year-olds, 7-year-olds and adults, nuclear pitch contours were labeled manually using the GToBI annotation system. Ten different contour types were identified.The fundamental frequency (F0) of these contours was modeled using third-order orthogonal polynomials, following an approach similar…

  13. GloptiPoly: Global optimization over polynomials with Matlab and SeDuMi

    Czech Academy of Sciences Publication Activity Database

    Henrion, Didier; Lasserre, J.-B.

    č. 2 (2003), s. 165-194 ISSN 0098-3500 R&D Projects: GA MŠk ME 496 Institutional research plan: CEZ:AV0Z1075907 Keywords : polynomial programming * semidefinite programming * linear matrix inequality Subject RIV: BC - Control Systems Theory Impact factor: 0.979, year: 2003

  14. GIS-based bivariate statistical techniques for groundwater potential analysis (an example of Iran)

    Science.gov (United States)

    Haghizadeh, Ali; Moghaddam, Davoud Davoudi; Pourghasemi, Hamid Reza

    2017-12-01

    Groundwater potential analysis prepares better comprehension of hydrological settings of different regions. This study shows the potency of two GIS-based data driven bivariate techniques namely statistical index (SI) and Dempster-Shafer theory (DST) to analyze groundwater potential in Broujerd region of Iran. The research was done using 11 groundwater conditioning factors and 496 spring positions. Based on the ground water potential maps (GPMs) of SI and DST methods, 24.22% and 23.74% of the study area is covered by poor zone of groundwater potential, and 43.93% and 36.3% of Broujerd region is covered by good and very good potential zones, respectively. The validation of outcomes displayed that area under the curve (AUC) of SI and DST techniques are 81.23% and 79.41%, respectively, which shows SI method has slightly a better performance than the DST technique. Therefore, SI and DST methods are advantageous to analyze groundwater capacity and scrutinize the complicated relation between groundwater occurrence and groundwater conditioning factors, which permits investigation of both systemic and stochastic uncertainty. Finally, it can be realized that these techniques are very beneficial for groundwater potential analyzing and can be practical for water-resource management experts.

  15. Comparative assessment of orthogonal polynomials for wavefront reconstruction over the square aperture.

    Science.gov (United States)

    Ye, Jingfei; Gao, Zhishan; Wang, Shuai; Cheng, Jinlong; Wang, Wei; Sun, Wenqing

    2014-10-01

    Four orthogonal polynomials for reconstructing a wavefront over a square aperture based on the modal method are currently available, namely, the 2D Chebyshev polynomials, 2D Legendre polynomials, Zernike square polynomials and Numerical polynomials. They are all orthogonal over the full unit square domain. 2D Chebyshev polynomials are defined by the product of Chebyshev polynomials in x and y variables, as are 2D Legendre polynomials. Zernike square polynomials are derived by the Gram-Schmidt orthogonalization process, where the integration region across the full unit square is circumscribed outside the unit circle. Numerical polynomials are obtained by numerical calculation. The presented study is to compare these four orthogonal polynomials by theoretical analysis and numerical experiments from the aspects of reconstruction accuracy, remaining errors, and robustness. Results show that the Numerical orthogonal polynomial is superior to the other three polynomials because of its high accuracy and robustness even in the case of a wavefront with incomplete data.

  16. Bivariable analysis of ventricular late potentials in high resolution ECG records

    International Nuclear Information System (INIS)

    Orosco, L; Laciar, E

    2007-01-01

    In this study the bivariable analysis for ventricular late potentials detection in high-resolution electrocardiographic records is proposed. The standard time-domain analysis and the application of the time-frequency technique to high-resolution ECG records are briefly described as well as their corresponding results. In the proposed technique the time-domain parameter, QRSD and the most significant time-frequency index, EN QRS are used like variables. A bivariable index is defined, that combines the previous parameters. The propose technique allows evaluating the risk of ventricular tachycardia in post-myocardial infarct patients. The results show that the used bivariable index allows discriminating between the patient's population with ventricular tachycardia and the subjects of the control group. Also, it was found that the bivariable technique obtains a good valuation as diagnostic test. It is concluded that comparatively, the valuation of the bivariable technique as diagnostic test is superior to that of the time-domain method and the time-frequency technique evaluated individually

  17. 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

  18. 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.

  19. Optimal Approximation of Biquartic Polynomials by Bicubic Splines

    Directory of Open Access Journals (Sweden)

    Kačala Viliam

    2018-01-01

    The goal of this paper is to resolve this problem. Unlike the spline curves, in the case of spline surfaces it is insufficient to suppose that the grid should be uniform and the spline derivatives computed from a biquartic polynomial. We show that the biquartic polynomial coefficients have to satisfy some additional constraints to achieve optimal approximation by bicubic splines.

  20. 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 top...

  1. A Combinatorial Proof of a Result on Generalized Lucas Polynomials

    Directory of Open Access Journals (Sweden)

    Laugier Alexandre

    2016-09-01

    Full Text Available We give a combinatorial proof of an elementary property of generalized Lucas polynomials, inspired by [1]. These polynomials in s and t are defined by the recurrence relation 〈n〉 = s〈n-1〉+t〈n-2〉 for n ≥ 2. The initial values are 〈0〉 = 2; 〈1〉= s, respectively.

  2. 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…

  3. On peculiar properties of generating functions of some orthogonal polynomials

    International Nuclear Information System (INIS)

    Szabłowski, Paweł J

    2012-01-01

    We prove that for |x| ⩽ |t| ≥( t i )/(q) i h n+i ( x|q) =h n (x|t,q) Σ i≥0 (t i )/(q) i h i (x|q), where h n (x|q) and h n (x|t, q) are respectively the so-called q-Hermite and the big q-Hermite polynomials, and (q) n denotes the so-called q-Pochhammer symbol. We prove similar equalities involving big q-Hermite and Al-Salam–Chihara polynomials, and Al-Salam–Chihara and the so-called continuous dual q-Hahn polynomials. Moreover, we are able to relate in this way some other ‘ordinary’ orthogonal polynomials such as, e.g., Hermite, Chebyshev or Laguerre. These equalities give a new interpretation of the polynomials involved and moreover can give rise to a simple method of generating more and more general (i.e. involving more and more parameters) families of orthogonal polynomials. We pose some conjectures concerning Askey–Wilson polynomials and their possible generalizations. We prove that these conjectures are true for the cases q = 1 (classical case) and q = 0 (free case), thus paving the way to generalization of Askey–Wilson polynomials at least in these two cases. (paper)

  4. On an inequality concerning the polar derivative of a polynomial

    Indian Academy of Sciences (India)

    We also extend Zygmund's inequality to the polar derivative of a polynomial. Keywords. Zygmund's inequality; polar derivative; Lp-norm inequalities. 1. Introduction and statement of results. Let P (z) be a polynomial of degreen and let P (z) be its derivative. Then according to the famous result known as Bernstein's inequality ...

  5. Numerical Performances of Two Orthogonal Polynomials in the Tau ...

    African Journals Online (AJOL)

    In this work, efforts are made at comparing the numerical effectiveness of the two most accurate orthogonal polynomials; Chebyshev and Legendre polynomials. Although the two have different weight functions, but they most time give close results especially when they are considered in the same interval. This work has ...

  6. 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 ...

  7. 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.

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

    Indian Academy of Sciences (India)

    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 [ f n ( z ) f ( q z + c ) ] ( k ) and [ f n ( z ) ( f ( q z + c ) − f ( z ) ) ] ( k ) , where () is a transcendental function of zero order.

  9. Conditional Density Approximations with Mixtures of Polynomials

    DEFF Research Database (Denmark)

    Varando, Gherardo; López-Cruz, Pedro L.; Nielsen, Thomas Dyhre

    2015-01-01

    Mixtures of polynomials (MoPs) are a non-parametric density estimation technique especially designed for hybrid Bayesian networks with continuous and discrete variables. Algorithms to learn one- and multi-dimensional (marginal) MoPs from data have recently been proposed. In this paper we introduce...... is found. We illustrate and study the methods using data sampled from known parametric distributions, and we demonstrate their applicability by learning models based on real neuroscience data. Finally, we compare the performance of the proposed methods with an approach for learning mixtures of truncated...... basis functions (MoTBFs). The empirical results show that the proposed methods generally yield models that are comparable to or significantly better than those found using the MoTBF-based method....

  10. Kinetic term anarchy for polynomial chaotic inflation

    Science.gov (United States)

    Nakayama, Kazunori; Takahashi, Fuminobu; Yanagida, Tsutomu T.

    2014-09-01

    We argue that there may arise a relatively flat inflaton potential over super-Planckian field values with an approximate shift symmetry, if the coefficients of the kinetic terms for many singlet scalars are subject to a certain random distribution. The inflation takes place along the flat direction with a super-Planckian length, whereas the other light directions can be stabilized by the Hubble-induced mass. The inflaton potential generically contains various shift-symmetry breaking terms, leading to a possibly large deviation of the predicted values of the spectral index and tensor-to-scalar ratio from those of the simple quadratic chaotic inflation. We revisit a polynomial chaotic inflation in supergravity as such.

  11. Dynamic normal forms and dynamic characteristic polynomial

    DEFF Research Database (Denmark)

    Frandsen, Gudmund Skovbjerg; Sankowski, Piotr

    2011-01-01

    We present the first fully dynamic algorithm for computing the characteristic polynomial of a matrix. In the generic symmetric case, our algorithm supports rank-one updates in O(n2logn) randomized time and queries in constant time, whereas in the general case the algorithm works in O(n2klogn......) randomized time, where k is the number of invariant factors of the matrix. The algorithm is based on the first dynamic algorithm for computing normal forms of a matrix such as the Frobenius normal form or the tridiagonal symmetric form. The algorithm can be extended to solve the matrix eigenproblem...... with relative error 2−b in additional O(nlog2nlogb) time. Furthermore, it can be used to dynamically maintain the singular value decomposition (SVD) of a generic matrix. Together with the algorithm, the hardness of the problem is studied. For the symmetric case, we present an Ω(n2) lower bound for rank...

  12. Causal networks clarify productivity-richness interrelations, bivariate plots do not

    Science.gov (United States)

    Grace, James B.; Adler, Peter B.; Harpole, W. Stanley; Borer, Elizabeth T.; Seabloom, Eric W.

    2014-01-01

    Perhaps no other pair of variables in ecology has generated as much discussion as species richness and ecosystem productivity, as illustrated by the reactions by Pierce (2013) and others to Adler et al.'s (2011) report that empirical patterns are weak and inconsistent. Adler et al. (2011) argued we need to move beyond a focus on simplistic bivariate relationships and test mechanistic, multivariate causal hypotheses. We feel the continuing debate over productivity–richness relationships (PRRs) provides a focused context for illustrating the fundamental difficulties of using bivariate relationships to gain scientific understanding.

  13. 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.

  14. Structured matrix based methods for approximate polynomial GCD

    CERN Document Server

    Boito, Paola

    2011-01-01

    Defining and computing a greatest common divisor of two polynomials with inexact coefficients is a classical problem in symbolic-numeric computation. The first part of this book reviews the main results that have been proposed so far in the literature. As usual with polynomial computations, the polynomial GCD problem can be expressed in matrix form: the second part of the book focuses on this point of view and analyses the structure of the relevant matrices, such as Toeplitz, Toepliz-block and displacement structures. New algorithms for the computation of approximate polynomial GCD are presented, along with extensive numerical tests. The use of matrix structure allows, in particular, to lower the asymptotic computational cost from cubic to quadratic order with respect to polynomial degree. .

  15. 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)

  16. 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.

  17. Jack polynomial fractional quantum Hall states and their generalizations

    International Nuclear Information System (INIS)

    Baratta, Wendy; Forrester, Peter J.

    2011-01-01

    In the study of fractional quantum Hall states, a certain clustering condition involving up to four integers has been identified. We give a simple proof that particular Jack polynomials with α=-(r-1)/(k+1), (r-1) and (k+1) relatively prime, and with partition given in terms of its frequencies by [n 0 0 (r-1)s k0 r-1 k0 r-1 k...0 r-1 m] satisfy this clustering condition. Our proof makes essential use of the fact that these Jack polynomials are translationally invariant. We also consider nonsymmetric Jack polynomials, symmetric and nonsymmetric generalized Hermite and Laguerre polynomials, and Macdonald polynomials from the viewpoint of the clustering.

  18. 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.

  19. Exact spectrum and wave functions of the hyperbolic Scarf potential in terms of finite Romanovsky polynomials

    International Nuclear Information System (INIS)

    Alvarez C, D.E.; Kirchbach, M.

    2007-01-01

    The Schroedinger equation with the hyperbolic Scarf potential reported so far in the literature is somewhat artificially manipulated into the form of the Jacobi equation with an imaginary argument and parameters that are complex conjugate to each other. Instead we here solve the former equation anew and make the case that it reduces straightforward to a particular form of the generalized real hypergeometric equation whose solutions are referred to in the mathematics literature as the finite Romanovsky polynomials, in reference to the observation that for any parameter set only a finite number of such polynomials appear to be orthogonal. This is a qualitatively new integral property that does not copy any of the features of the Jacobi polynomials. In this manner, the finite number of bound states within the hyperbolic Scarf potential is brought into correspondence with a finite system of a new class of orthogonal polynomials. This work adds a new example to the circle of the problems on the Schroedinger equation. The techniques used by us extend the teachings on the Sturm-Liouville theory of ordinary differential equations beyond their standard presentation in the textbooks on mathematical methods in physics. (Author)

  20. Strehl ratio and amplitude-weighted generalized orthonormal Zernike-based polynomials.

    Science.gov (United States)

    Mafusire, Cosmas; Krüger, Tjaart P J

    2017-03-10

    The concept of orthonormal polynomials is revisited by developing a Zernike-based orthonormal set for a non-circular pupil that is transmitting an aberrated, non-uniform field. We refer to this pupil as a general pupil. The process is achieved by using the matrix form of the Gram-Schmidt procedure on Zernike circle polynomials and is interpreted as a process of balancing each Zernike circle polynomial by adding those of lower order in the general pupil, a procedure which was previously performed using classical aberrations. We numerically demonstrate this concept by comparing the representation of phase in a square-Gaussian pupil using the Zernike-Gauss square and Zernike circle polynomials. As expected, using the Strehl ratio, we show that only specific lower-order aberrations can be used to balance specific aberrations, for example, tilt cannot be used to balance spherical aberration. In the process, we present a possible definition of the Maréchal criterion for the analysis of the tolerance of systems with apodized pupils.

  1. Mobile device camera design with Q-type polynomials to achieve higher production yield.

    Science.gov (United States)

    Ma, Bin; Sharma, Katelynn; Thompson, Kevin P; Rolland, Jannick P

    2013-07-29

    The camera lenses that are built into the current generation of mobile devices are extremely stressed by the excessively tight packaging requirements, particularly the length. As a result, the aspheric departures and slopes on the lens surfaces, when designed with conventional power series based aspheres, are well beyond those encountered in most optical systems. When the as-manufactured performance is considered, the excessive aspheric slopes result in unusually high sensitivity to tilt and decenter and even despace resulting in unusually low manufacturing yield. Q(bfs) polynomials, a new formulation for nonspherical optical surfaces introduced by Forbes, not only build on orthogonal polynomials, but their unique normalization provides direct access to the RMS slope of the aspheric departure during optimization. Using surface shapes with this description in optimization results in equivalent performance with reduced alignment sensitivity and higher yield. As an additional approach to increasing yield, mechanically imposed external pivot points, introduced by Bottema, can be used as a design technique to further reduce alignment sensitivity and increase yield. In this paper, the Q-type polynomials and external pivot points were applied to a mobile device camera lens designed using an active RMS slope constraint that was then compared to a design developed using conventional power series surface descriptions. Results show that slope constrained Q-type polynomial description together with external pivot points lead directly to solutions with significantly higher manufacturing yield.

  2. Accuracy of body mass index in predicting pre-eclampsia: bivariate meta-analysis

    NARCIS (Netherlands)

    Cnossen, J. S.; Leeflang, M. M. G.; de Haan, E. E. M.; Mol, B. W. J.; van der Post, J. A. M.; Khan, K. S.; ter Riet, G.

    2007-01-01

    OBJECTIVE: The objective of this study was to determine the accuracy of body mass index (BMI) (pre-pregnancy or at booking) in predicting pre-eclampsia and to explore its potential for clinical application. DESIGN: Systematic review and bivariate meta-analysis. SETTING: Medline, Embase, Cochrane

  3. Parameter estimation and statistical test of geographically weighted bivariate Poisson inverse Gaussian regression models

    Science.gov (United States)

    Amalia, Junita; Purhadi, Otok, Bambang Widjanarko

    2017-11-01

    Poisson distribution is a discrete distribution with count data as the random variables and it has one parameter defines both mean and variance. Poisson regression assumes mean and variance should be same (equidispersion). Nonetheless, some case of the count data unsatisfied this assumption because variance exceeds mean (over-dispersion). The ignorance of over-dispersion causes underestimates in standard error. Furthermore, it causes incorrect decision in the statistical test. Previously, paired count data has a correlation and it has bivariate Poisson distribution. If there is over-dispersion, modeling paired count data is not sufficient with simple bivariate Poisson regression. Bivariate Poisson Inverse Gaussian Regression (BPIGR) model is mix Poisson regression for modeling paired count data within over-dispersion. BPIGR model produces a global model for all locations. In another hand, each location has different geographic conditions, social, cultural and economic so that Geographically Weighted Regression (GWR) is needed. The weighting function of each location in GWR generates a different local model. Geographically Weighted Bivariate Poisson Inverse Gaussian Regression (GWBPIGR) model is used to solve over-dispersion and to generate local models. Parameter estimation of GWBPIGR model obtained by Maximum Likelihood Estimation (MLE) method. Meanwhile, hypothesis testing of GWBPIGR model acquired by Maximum Likelihood Ratio Test (MLRT) method.

  4. Analysis of Blood Transfusion Data Using Bivariate Zero-Inflated Poisson Model: A Bayesian Approach.

    Science.gov (United States)

    Mohammadi, Tayeb; Kheiri, Soleiman; Sedehi, Morteza

    2016-01-01

    Recognizing the factors affecting the number of blood donation and blood deferral has a major impact on blood transfusion. There is a positive correlation between the variables "number of blood donation" and "number of blood deferral": as the number of return for donation increases, so does the number of blood deferral. On the other hand, due to the fact that many donors never return to donate, there is an extra zero frequency for both of the above-mentioned variables. In this study, in order to apply the correlation and to explain the frequency of the excessive zero, the bivariate zero-inflated Poisson regression model was used for joint modeling of the number of blood donation and number of blood deferral. The data was analyzed using the Bayesian approach applying noninformative priors at the presence and absence of covariates. Estimating the parameters of the model, that is, correlation, zero-inflation parameter, and regression coefficients, was done through MCMC simulation. Eventually double-Poisson model, bivariate Poisson model, and bivariate zero-inflated Poisson model were fitted on the data and were compared using the deviance information criteria (DIC). The results showed that the bivariate zero-inflated Poisson regression model fitted the data better than the other models.

  5. The Effects of Selection Strategies for Bivariate Loglinear Smoothing Models on NEAT Equating Functions

    Science.gov (United States)

    Moses, Tim; Holland, Paul W.

    2010-01-01

    In this study, eight statistical strategies were evaluated for selecting the parameterizations of loglinear models for smoothing the bivariate test score distributions used in nonequivalent groups with anchor test (NEAT) equating. Four of the strategies were based on significance tests of chi-square statistics (Likelihood Ratio, Pearson,…

  6. First-order dominance: stronger characterization and a bivariate checking algorithm

    DEFF Research Database (Denmark)

    Range, Troels Martin; Østerdal, Lars Peter Raahave

    2018-01-01

    distributions. Utilizing that this problem can be formulated as a transportation problem with a special structure, we provide a stronger characterization of multivariate first-order dominance and develop a linear time complexity checking algorithm for the bivariate case. We illustrate the use of the checking...

  7. On the Construction of Bivariate Exponential Distributions with an Arbitrary Correlation Coefficient

    DEFF Research Database (Denmark)

    Bladt, Mogens; Nielsen, Bo Friis

    2010-01-01

    In this article we use the concept of multivariate phase-type distributions to define a class of bivariate exponential distributions. This class has the following three appealing properties. Firstly, we may construct a pair of exponentially distributed random variables with any feasible correlation...

  8. Semi-automated detection of aberrant chromosomes in bivariate flow karyotypes

    NARCIS (Netherlands)

    Boschman, G. A.; Manders, E. M.; Rens, W.; Slater, R.; Aten, J. A.

    1992-01-01

    A method is described that is designed to compare, in a standardized procedure, bivariate flow karyotypes of Hoechst 33258 (HO)/Chromomycin A3 (CA) stained human chromosomes from cells with aberrations with a reference flow karyotype of normal chromosomes. In addition to uniform normalization of

  9. Bivariate analysis of sensitivity and specificity produces informative summary measures in diagnostic reviews

    NARCIS (Netherlands)

    Reitsma, Johannes B.; Glas, Afina S.; Rutjes, Anne W. S.; Scholten, Rob J. P. M.; Bossuyt, Patrick M.; Zwinderman, Aeilko H.

    2005-01-01

    Background and Objectives: Studies of diagnostic accuracy most often report pairs of sensitivity and specificity. We demonstrate the advantage of using bivariate meta-regression models to analyze such data. Methods: We discuss the methodology of both the summary Receiver Operating Characteristic

  10. A simple approximation to the bivariate normal distribution with large correlation coefficient

    NARCIS (Netherlands)

    Albers, Willem/Wim; Kallenberg, W.C.M.

    1994-01-01

    The bivariate normal distribution function is approximated with emphasis on situations where the correlation coefficient is large. The high accuracy of the approximation is illustrated by numerical examples. Moreover, exact upper and lower bounds are presented as well as asymptotic results on the

  11. A comparison of bivariate and univariate QTL mapping in livestock populations

    Directory of Open Access Journals (Sweden)

    Sorensen Daniel

    2003-11-01

    Full Text Available Abstract This study presents a multivariate, variance component-based QTL mapping model implemented via restricted maximum likelihood (REML. The method was applied to investigate bivariate and univariate QTL mapping analyses, using simulated data. Specifically, we report results on the statistical power to detect a QTL and on the precision of parameter estimates using univariate and bivariate approaches. The model and methodology were also applied to study the effectiveness of partitioning the overall genetic correlation between two traits into a component due to many genes of small effect, and one due to the QTL. It is shown that when the QTL has a pleiotropic effect on two traits, a bivariate analysis leads to a higher statistical power of detecting the QTL and to a more precise estimate of the QTL's map position, in particular in the case when the QTL has a small effect on the trait. The increase in power is most marked in cases where the contributions of the QTL and of the polygenic components to the genetic correlation have opposite signs. The bivariate REML analysis can successfully partition the two components contributing to the genetic correlation between traits.

  12. New Colors for Histology: Optimized Bivariate Color Maps Increase Perceptual Contrast in Histological Images.

    Directory of Open Access Journals (Sweden)

    Jakob Nikolas Kather

    Full Text Available Accurate evaluation of immunostained histological images is required for reproducible research in many different areas and forms the basis of many clinical decisions. The quality and efficiency of histopathological evaluation is limited by the information content of a histological image, which is primarily encoded as perceivable contrast differences between objects in the image. However, the colors of chromogen and counterstain used for histological samples are not always optimally distinguishable, even under optimal conditions.In this study, we present a method to extract the bivariate color map inherent in a given histological image and to retrospectively optimize this color map. We use a novel, unsupervised approach based on color deconvolution and principal component analysis to show that the commonly used blue and brown color hues in Hematoxylin-3,3'-Diaminobenzidine (DAB images are poorly suited for human observers. We then demonstrate that it is possible to construct improved color maps according to objective criteria and that these color maps can be used to digitally re-stain histological images.To validate whether this procedure improves distinguishability of objects and background in histological images, we re-stain phantom images and N = 596 large histological images of immunostained samples of human solid tumors. We show that perceptual contrast is improved by a factor of 2.56 in phantom images and up to a factor of 2.17 in sets of histological tumor images.Thus, we provide an objective and reliable approach to measure object distinguishability in a given histological image and to maximize visual information available to a human observer. This method could easily be incorporated in digital pathology image viewing systems to improve accuracy and efficiency in research and diagnostics.

  13. New Colors for Histology: Optimized Bivariate Color Maps Increase Perceptual Contrast in Histological Images.

    Science.gov (United States)

    Kather, Jakob Nikolas; Weis, Cleo-Aron; Marx, Alexander; Schuster, Alexander K; Schad, Lothar R; Zöllner, Frank Gerrit

    2015-01-01

    Accurate evaluation of immunostained histological images is required for reproducible research in many different areas and forms the basis of many clinical decisions. The quality and efficiency of histopathological evaluation is limited by the information content of a histological image, which is primarily encoded as perceivable contrast differences between objects in the image. However, the colors of chromogen and counterstain used for histological samples are not always optimally distinguishable, even under optimal conditions. In this study, we present a method to extract the bivariate color map inherent in a given histological image and to retrospectively optimize this color map. We use a novel, unsupervised approach based on color deconvolution and principal component analysis to show that the commonly used blue and brown color hues in Hematoxylin-3,3'-Diaminobenzidine (DAB) images are poorly suited for human observers. We then demonstrate that it is possible to construct improved color maps according to objective criteria and that these color maps can be used to digitally re-stain histological images. To validate whether this procedure improves distinguishability of objects and background in histological images, we re-stain phantom images and N = 596 large histological images of immunostained samples of human solid tumors. We show that perceptual contrast is improved by a factor of 2.56 in phantom images and up to a factor of 2.17 in sets of histological tumor images. Thus, we provide an objective and reliable approach to measure object distinguishability in a given histological image and to maximize visual information available to a human observer. This method could easily be incorporated in digital pathology image viewing systems to improve accuracy and efficiency in research and diagnostics.

  14. Analysis of input variables of an artificial neural network using bivariate correlation and canonical correlation

    International Nuclear Information System (INIS)

    Costa, Valter Magalhaes; Pereira, Iraci Martinez

    2011-01-01

    The monitoring of variables and diagnosis of sensor fault in nuclear power plants or processes industries is very important because a previous diagnosis allows the correction of the fault and, like this, to prevent the production stopped, improving operator's security and it's not provoking economics losses. The objective of this work is to build a set, using bivariate correlation and canonical correlation, which will be the set of input variables of an artificial neural network to monitor the greater number of variables. This methodology was applied to the IEA-R1 Research Reactor at IPEN. Initially, for the input set of neural network we selected the variables: nuclear power, primary circuit flow rate, control/safety rod position and difference in pressure in the core of the reactor, because almost whole of monitoring variables have relation with the variables early described or its effect can be result of the interaction of two or more. The nuclear power is related to the increasing and decreasing of temperatures as well as the amount radiation due fission of the uranium; the rods are controls of power and influence in the amount of radiation and increasing and decreasing of temperatures; the primary circuit flow rate has the function of energy transport by removing the nucleus heat. An artificial neural network was trained and the results were satisfactory since the IEA-R1 Data Acquisition System reactor monitors 64 variables and, with a set of 9 input variables resulting from the correlation analysis, it was possible to monitor 51 variables. (author)

  15. Analysis of input variables of an artificial neural network using bivariate correlation and canonical correlation

    Energy Technology Data Exchange (ETDEWEB)

    Costa, Valter Magalhaes; Pereira, Iraci Martinez, E-mail: valter.costa@usp.b [Instituto de Pesquisas Energeticas e Nucleares (IPEN/CNEN-SP), Sao Paulo, SP (Brazil)

    2011-07-01

    The monitoring of variables and diagnosis of sensor fault in nuclear power plants or processes industries is very important because a previous diagnosis allows the correction of the fault and, like this, to prevent the production stopped, improving operator's security and it's not provoking economics losses. The objective of this work is to build a set, using bivariate correlation and canonical correlation, which will be the set of input variables of an artificial neural network to monitor the greater number of variables. This methodology was applied to the IEA-R1 Research Reactor at IPEN. Initially, for the input set of neural network we selected the variables: nuclear power, primary circuit flow rate, control/safety rod position and difference in pressure in the core of the reactor, because almost whole of monitoring variables have relation with the variables early described or its effect can be result of the interaction of two or more. The nuclear power is related to the increasing and decreasing of temperatures as well as the amount radiation due fission of the uranium; the rods are controls of power and influence in the amount of radiation and increasing and decreasing of temperatures; the primary circuit flow rate has the function of energy transport by removing the nucleus heat. An artificial neural network was trained and the results were satisfactory since the IEA-R1 Data Acquisition System reactor monitors 64 variables and, with a set of 9 input variables resulting from the correlation analysis, it was possible to monitor 51 variables. (author)

  16. Meta-analysis of studies with bivariate binary outcomes: a marginal beta-binomial model approach.

    Science.gov (United States)

    Chen, Yong; Hong, Chuan; Ning, Yang; Su, Xiao

    2016-01-15

    When conducting a meta-analysis of studies with bivariate binary outcomes, challenges arise when the within-study correlation and between-study heterogeneity should be taken into account. In this paper, we propose a marginal beta-binomial model for the meta-analysis of studies with binary outcomes. This model is based on the composite likelihood approach and has several attractive features compared with the existing models such as bivariate generalized linear mixed model (Chu and Cole, 2006) and Sarmanov beta-binomial model (Chen et al., 2012). The advantages of the proposed marginal model include modeling the probabilities in the original scale, not requiring any transformation of probabilities or any link function, having closed-form expression of likelihood function, and no constraints on the correlation parameter. More importantly, because the marginal beta-binomial model is only based on the marginal distributions, it does not suffer from potential misspecification of the joint distribution of bivariate study-specific probabilities. Such misspecification is difficult to detect and can lead to biased inference using currents methods. We compare the performance of the marginal beta-binomial model with the bivariate generalized linear mixed model and the Sarmanov beta-binomial model by simulation studies. Interestingly, the results show that the marginal beta-binomial model performs better than the Sarmanov beta-binomial model, whether or not the true model is Sarmanov beta-binomial, and the marginal beta-binomial model is more robust than the bivariate generalized linear mixed model under model misspecifications. Two meta-analyses of diagnostic accuracy studies and a meta-analysis of case-control studies are conducted for illustration. Copyright © 2015 John Wiley & Sons, Ltd.

  17. SU-C-BRC-04: Efficient Dose Calculation Algorithm for FFF IMRT with a Simplified Bivariate Gaussian Source Model

    Energy Technology Data Exchange (ETDEWEB)

    Li, F; Park, J; Barraclough, B; Lu, B; Li, J; Liu, C; Yan, G [University Florida, Gainesville, FL (United States)

    2016-06-15

    Purpose: To develop an efficient and accurate independent dose calculation algorithm with a simplified analytical source model for the quality assurance and safe delivery of Flattening Filter Free (FFF)-IMRT on an Elekta Versa HD. Methods: The source model consisted of a point source and a 2D bivariate Gaussian source, respectively modeling the primary photons and the combined effect of head scatter, monitor chamber backscatter and collimator exchange effect. The in-air fluence was firstly calculated by back-projecting the edges of beam defining devices onto the source plane and integrating the visible source distribution. The effect of the rounded MLC leaf end, tongue-and-groove and interleaf transmission was taken into account in the back-projection. The in-air fluence was then modified with a fourth degree polynomial modeling the cone-shaped dose distribution of FFF beams. Planar dose distribution was obtained by convolving the in-air fluence with a dose deposition kernel (DDK) consisting of the sum of three 2D Gaussian functions. The parameters of the source model and the DDK were commissioned using measured in-air output factors (Sc) and cross beam profiles, respectively. A novel method was used to eliminate the volume averaging effect of ion chambers in determining the DDK. Planar dose distributions of five head-and-neck FFF-IMRT plans were calculated and compared against measurements performed with a 2D diode array (MapCHECK™) to validate the accuracy of the algorithm. Results: The proposed source model predicted Sc for both 6MV and 10MV with an accuracy better than 0.1%. With a stringent gamma criterion (2%/2mm/local difference), the passing rate of the FFF-IMRT dose calculation was 97.2±2.6%. Conclusion: The removal of the flattening filter represents a simplification of the head structure which allows the use of a simpler source model for very accurate dose calculation. The proposed algorithm offers an effective way to ensure the safe delivery of FFF-IMRT.

  18. 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.

  19. 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

  20. Coherent states and Hermite polynomials on quaternionic Hilbert spaces

    Energy Technology Data Exchange (ETDEWEB)

    Thirulogasanthar, K; Krzyzak, A [Department of Computer Science and Software Engineering, Concordia University, 1455 de Maisonneuve Blvd. W, Montreal, Quebec H3G 1M8 (Canada); Honnouvo, G, E-mail: santhar@cs.concordia.c, E-mail: krzyzak@cs.concordia.c, E-mail: g-honnouvo@yahoo.f [Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, Quebec H3A 2K6 (Canada)

    2010-09-24

    Coherent states, similar to the canonical coherent states of the harmonic oscillator, labeled by quaternions are established on the right and left quaternionic Hilbert spaces. On the left quaternionic Hilbert space reproducing kernels are established. As was claimed by Adler and Millard (J. Math. Phys. 1997 38 2117-26) it is proved that these coherent states cannot be realized as a group action on a quaternionic Hilbert space. As an extension of the complex Hermite polynomials, quaternionic Hermite polynomials are defined and these polynomials are associated with an operator as eigenfunctions.

  1. Local polynomial Whittle estimation covering non-stationary fractional processes

    DEFF Research Database (Denmark)

    Nielsen, Frank

    This paper extends the local polynomial Whittle estimator of Andrews & Sun (2004) to fractionally integrated processes covering stationary and non-stationary regions. We utilize the notion of the extended discrete Fourier transform and periodogram to extend the local polynomial Whittle estimator...... to the non-stationary region. By approximating the short-run component of the spectrum by a polynomial, instead of a constant, in a shrinking neighborhood of zero we alleviate some of the bias that the classical local Whittle estimators is prone to. This bias reduction comes at a cost as the variance is in...

  2. Chromatic Polynomials Of Some (m,l-Hyperwheels

    Directory of Open Access Journals (Sweden)

    Julian A. Allagan

    2014-03-01

    Full Text Available In this paper, using a standard method of computing the chromatic polynomial of hypergraphs, we introduce a new reduction theorem which allows us to find explicit formulae for the chromatic polynomials of some (complete non-uniform $(m,l-$hyperwheels and non-uniform $(m,l-$hyperfans. These hypergraphs, constructed through a ``join" graph operation, are some generalizations of the well-known wheel and fan graphs, respectively. Further, we revisit some results concerning these graphs and present their chromatic polynomials in a standard form that involves the Stirling numbers of the second kind.

  3. 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...

  4. Asymptotic distribution of zeros of polynomials satisfying difference equations

    Science.gov (United States)

    Krasovsky, I. V.

    2003-01-01

    We propose a way to find the asymptotic distribution of zeros of orthogonal polynomials pn(x) satisfying a difference equation of the formB(x)pn(x+[delta])-C(x,n)pn(x)+D(x)pn(x-[delta])=0.We calculate the asymptotic distribution of zeros and asymptotics of extreme zeros of the Meixner and Meixner-Pollaczek polynomials. The distribution of zeros of Meixner polynomials shows some delicate features. We indicate the relation of our technique to the approach based on the Nevai-Dehesa-Ullman distribution.

  5. Robust stability of fractional order polynomials with complicated uncertainty structure.

    Science.gov (United States)

    Matušů, Radek; Şenol, Bilal; Pekař, Libor

    2017-01-01

    The main aim of this article is to present a graphical approach to robust stability analysis for families of fractional order (quasi-)polynomials with complicated uncertainty structure. More specifically, the work emphasizes the multilinear, polynomial and general structures of uncertainty and, moreover, the retarded quasi-polynomials with parametric uncertainty are studied. Since the families with these complex uncertainty structures suffer from the lack of analytical tools, their robust stability is investigated by numerical calculation and depiction of the value sets and subsequent application of the zero exclusion condition.

  6. On an Ordering-Dependent Generalization of the Tutte Polynomial

    Science.gov (United States)

    Geloun, Joseph Ben; Caravelli, Francesco

    2017-09-01

    A generalization of the Tutte polynomial involved in the evaluation of the moments of the integrated geometric Brownian in the Itô formalism is discussed. The new combinatorial invariant depends on the order in which the sequence of contraction-deletions have been performed on the graph. Thus, this work provides a motivation for studying an order-dependent Tutte polynomial in the context of stochastic differential equations. We show that in the limit of the control parameters encoding the ordering going to zero, the multivariate Tutte-Fortuin-Kasteleyn polynomial is recovered.

  7. Polynomial Chaos Surrogates for Bayesian Inference

    KAUST Repository

    Le Maitre, Olivier

    2016-01-06

    The Bayesian inference is a popular probabilistic method to solve inverse problems, such as the identification of field parameter in a PDE model. The inference rely on the Bayes rule to update the prior density of the sought field, from observations, and derive its posterior distribution. In most cases the posterior distribution has no explicit form and has to be sampled, for instance using a Markov-Chain Monte Carlo method. In practice the prior field parameter is decomposed and truncated (e.g. by means of Karhunen- Lo´eve decomposition) to recast the inference problem into the inference of a finite number of coordinates. Although proved effective in many situations, the Bayesian inference as sketched above faces several difficulties requiring improvements. First, sampling the posterior can be a extremely costly task as it requires multiple resolutions of the PDE model for different values of the field parameter. Second, when the observations are not very much informative, the inferred parameter field can highly depends on its prior which can be somehow arbitrary. These issues have motivated the introduction of reduced modeling or surrogates for the (approximate) determination of the parametrized PDE solution and hyperparameters in the description of the prior field. Our contribution focuses on recent developments in these two directions: the acceleration of the posterior sampling by means of Polynomial Chaos expansions and the efficient treatment of parametrized covariance functions for the prior field. We also discuss the possibility of making such approach adaptive to further improve its efficiency.

  8. Dynamics of polynomial Chaplygin gas warm inflation

    Energy Technology Data Exchange (ETDEWEB)

    Jawad, Abdul [COMSATS Institute of Information Technology, Department of Mathematics, Lahore (Pakistan); Chaudhary, Shahid [Sharif College of Engineering and Technology, Department of Mathematics, Lahore (Pakistan); Videla, Nelson [Pontificia Universidad Catolica de Valparaiso, Instituto de Fisica, Valparaiso (Chile)

    2017-11-15

    In the present work, we study the consequences of a recently proposed polynomial inflationary potential in the context of the generalized, modified, and generalized cosmic Chaplygin gas models. In addition, we consider dissipative effects by coupling the inflation field to radiation, i.e., the inflationary dynamics is studied in the warm inflation scenario. We take into account a general parametrization of the dissipative coefficient Γ for describing the decay of the inflaton field into radiation. By studying the background and perturbative dynamics in the weak and strong dissipative regimes of warm inflation separately for the positive and negative quadratic and quartic potentials, we obtain expressions for the most relevant inflationary observables as the scalar power spectrum, the scalar spectral, and the tensor-to-scalar ratio. We construct the trajectories in the n{sub s}-r plane for several expressions of the dissipative coefficient and compare with the two-dimensional marginalized contours for (n{sub s}, r) from the latest Planck data. We find that our results are in agreement with WMAP9 and Planck 2015 data. (orig.)

  9. 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

  10. A summation procedure for expansions in orthogonal polynomials

    International Nuclear Information System (INIS)

    Garibotti, C.R.; Grinstein, F.F.

    1977-01-01

    Approximants to functions defined by formal series expansions in orthogonal polynomials are introduced. They are shown to be convergent even out of the elliptical domain where the original expansion converges

  11. Force prediction in cold rolling mills by polynomial methods

    Directory of Open Access Journals (Sweden)

    Nicu ROMAN

    2007-12-01

    Full Text Available A method for steel and aluminium strip thickness control is provided including a new technique for predictive rolling force estimation method by statistic model based on polynomial techniques.

  12. on the performance of Autoregressive Moving Average Polynomial

    African Journals Online (AJOL)

    Timothy Ademakinwa

    Using numerical example, DL, PDL, ARPDL and. ARMAPDL models were fitted. Autoregressive Moving Average Polynomial Distributed Lag Model. (ARMAPDL) model performed better than the other models. Keywords: Distributed Lag Model, Selection Criterion, Parameter Estimation, Residual Variance. ABSTRACT. 247.

  13. Almost normed spaces of functions with polynomial asymptotic behaviour

    International Nuclear Information System (INIS)

    Kudryavtsev, L D

    2003-01-01

    We consider a linear space of functions asymptotically approaching polynomials of degree not higher than a fixed one as the independent variable approaches infinity. Such a space cannot be normed if the functions in it possess a certain smoothness. For this reason the concept of almost normed space is introduced and the spaces in question, namely, spaces of functions asymptotically or strongly asymptotically approaching polynomials, are shown to be almost normed. The completeness of these spaces in the metric generated by their almost norm is also proved, the connection between the asymptotic approach and the strong asymptotic approach of functions to polynomials is studied, and a new (and shorter) proof of the criterion for the asymptotic approach of functions to polynomials is presented

  14. 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...

  15. A general polynomial solution to convection–dispersion equation ...

    Indian Academy of Sciences (India)

    Jiao Wang

    s12040-017-0820-4. A general polynomial solution to convection–dispersion equation using ... water pollution of groundwater, numerical models are increasingly used in .... to convective transport by water flow is negligi- ble. Equation (4) is ...

  16. An Elementary Proof of the Polynomial Matrix Spectral Factorization Theorem

    OpenAIRE

    Ephremidze, Lasha

    2010-01-01

    A very simple and short proof of the polynomial matrix spectral factorization theorem (on the unit circle as well as on the real line) is presented, which relies on elementary complex analysis and linear algebra.

  17. 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.

  18. 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 ...

  19. Orthogonal sets of data windows constructed from trigonometric polynomials

    Science.gov (United States)

    Greenhall, Charles A.

    1990-01-01

    Suboptimal, easily computable substitutes for the discrete prolate-spheroidal windows used by Thomson for spectral estimation are given. Trigonometric coefficients and energy leakages of the window polynomials are tabulated.

  20. Cubic Polynomials with Rational Roots and Critical Points

    Science.gov (United States)

    Gupta, Shiv K.; Szymanski, Waclaw

    2010-01-01

    If you want your students to graph a cubic polynomial, it is best to give them one with rational roots and critical points. In this paper, we describe completely all such cubics and explain how to generate them.