Novel Threshold Changeable Secret Sharing Schemes Based on Polynomial Interpolation.
Yuan, Lifeng; Li, Mingchu; Guo, Cheng; Choo, Kim-Kwang Raymond; Ren, Yizhi
2016-01-01
After any distribution of secret sharing shadows in a threshold changeable secret sharing scheme, the threshold may need to be adjusted to deal with changes in the security policy and adversary structure. For example, when employees leave the organization, it is not realistic to expect departing employees to ensure the security of their secret shadows. Therefore, in 2012, Zhang et al. proposed (t → t', n) and ({t1, t2,⋯, tN}, n) threshold changeable secret sharing schemes. However, their schemes suffer from a number of limitations such as strict limit on the threshold values, large storage space requirement for secret shadows, and significant computation for constructing and recovering polynomials. To address these limitations, we propose two improved dealer-free threshold changeable secret sharing schemes. In our schemes, we construct polynomials to update secret shadows, and use two-variable one-way function to resist collusion attacks and secure the information stored by the combiner. We then demonstrate our schemes can adjust the threshold safely.
Schubert calculus and threshold polynomials of affine fusion
International Nuclear Information System (INIS)
Irvine, S.E.; Walton, M.A.
2000-01-01
We show how the threshold level of affine fusion, the fusion of Wess-Zumino-Witten (WZW) conformal field theories, fits into the Schubert calculus introduced by Gepner. The Pieri rule can be modified in a simple way to include the threshold level, so that calculations may be done for all (non-negative integer) levels at once. With the usual Giambelli formula, the modified Pieri formula deforms the tensor product coefficients (and the fusion coefficients) into what we call threshold polynomials. We compare them with the q-deformed tensor product coefficients and fusion coefficients that are related to q-deformed weight multiplicities. We also discuss the meaning of the threshold level in the context of paths on graphs
Stability analysis of polynomial fuzzy models via polynomial fuzzy Lyapunov functions
Bernal Reza, Miguel Ángel; Sala, Antonio; JAADARI, ABDELHAFIDH; Guerra, Thierry-Marie
2011-01-01
In this paper, the stability of continuous-time polynomial fuzzy models by means of a polynomial generalization of fuzzy Lyapunov functions is studied. Fuzzy Lyapunov functions have been fruitfully used in the literature for local analysis of Takagi-Sugeno models, a particular class of the polynomial fuzzy ones. Based on a recent Taylor-series approach which allows a polynomial fuzzy model to exactly represent a nonlinear model in a compact set of the state space, it is shown that a refinemen...
Orthogonal Polynomials and Special Functions
Assche, Walter
2003-01-01
The set of lectures from the Summer School held in Leuven in 2002 provide an up-to-date account of recent developments in orthogonal polynomials and special functions, in particular for algorithms for computer algebra packages, 3nj-symbols in representation theory of Lie groups, enumeration, multivariable special functions and Dunkl operators, asymptotics via the Riemann-Hilbert method, exponential asymptotics and the Stokes phenomenon. The volume aims at graduate students and post-docs working in the field of orthogonal polynomials and special functions, and in related fields interacting with orthogonal polynomials, such as combinatorics, computer algebra, asymptotics, representation theory, harmonic analysis, differential equations, physics. The lectures are self-contained requiring only a basic knowledge of analysis and algebra, and each includes many exercises.
On Multiple Interpolation Functions of the -Genocchi Polynomials
Directory of Open Access Journals (Sweden)
Jin Jeong-Hee
2010-01-01
Full Text Available Abstract Recently, many mathematicians have studied various kinds of the -analogue of Genocchi numbers and polynomials. In the work (New approach to q-Euler, Genocchi numbers and their interpolation functions, "Advanced Studies in Contemporary Mathematics, vol. 18, no. 2, pp. 105–112, 2009.", Kim defined new generating functions of -Genocchi, -Euler polynomials, and their interpolation functions. In this paper, we give another definition of the multiple Hurwitz type -zeta function. This function interpolates -Genocchi polynomials at negative integers. Finally, we also give some identities related to these polynomials.
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)
Approximating Exponential and Logarithmic Functions Using Polynomial Interpolation
Gordon, Sheldon P.; Yang, Yajun
2017-01-01
This article takes a closer look at the problem of approximating the exponential and logarithmic functions using polynomials. Either as an alternative to or a precursor to Taylor polynomial approximations at the precalculus level, interpolating polynomials are considered. A measure of error is given and the behaviour of the error function is…
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)
Symmetric functions and orthogonal polynomials
Macdonald, I G
1997-01-01
One of the most classical areas of algebra, the theory of symmetric functions and orthogonal polynomials has long been known to be connected to combinatorics, representation theory, and other branches of mathematics. Written by perhaps the most famous author on the topic, this volume explains some of the current developments regarding these connections. It is based on lectures presented by the author at Rutgers University. Specifically, he gives recent results on orthogonal polynomials associated with affine Hecke algebras, surveying the proofs of certain famous combinatorial conjectures.
Animating Nested Taylor Polynomials to Approximate a Function
Mazzone, Eric F.; Piper, Bruce R.
2010-01-01
The way that Taylor polynomials approximate functions can be demonstrated by moving the center point while keeping the degree fixed. These animations are particularly nice when the Taylor polynomials do not intersect and form a nested family. We prove a result that shows when this nesting occurs. The animations can be shown in class or…
Discriminants and functional equations for polynomials orthogonal on the unit circle
International Nuclear Information System (INIS)
Ismail, M.E.H.; Witte, N.S.
2000-01-01
We derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and q-difference equations for these polynomials. A general functional equation is found which allows one to relate the zeros of the orthogonal polynomials to the stationary values of an explicit quasi-energy and implies recurrences on the orthogonal polynomial coefficients. We also evaluate the discriminants and quantized discriminants of polynomials orthogonal on the unit circle
Papadopoulos, Anthony
2009-01-01
The first-degree power-law polynomial function is frequently used to describe activity metabolism for steady swimming animals. This function has been used in hydrodynamics-based metabolic studies to evaluate important parameters of energetic costs, such as the standard metabolic rate and the drag power indices. In theory, however, the power-law polynomial function of any degree greater than one can be used to describe activity metabolism for steady swimming animals. In fact, activity metabolism has been described by the conventional exponential function and the cubic polynomial function, although only the power-law polynomial function models drag power since it conforms to hydrodynamic laws. Consequently, the first-degree power-law polynomial function yields incorrect parameter values of energetic costs if activity metabolism is governed by the power-law polynomial function of any degree greater than one. This issue is important in bioenergetics because correct comparisons of energetic costs among different steady swimming animals cannot be made unless the degree of the power-law polynomial function derives from activity metabolism. In other words, a hydrodynamics-based functional form of activity metabolism is a power-law polynomial function of any degree greater than or equal to one. Therefore, the degree of the power-law polynomial function should be treated as a parameter, not as a constant. This new treatment not only conforms to hydrodynamic laws, but also ensures correct comparisons of energetic costs among different steady swimming animals. Furthermore, the exponential power-law function, which is a new hydrodynamics-based functional form of activity metabolism, is a special case of the power-law polynomial function. Hence, the link between the hydrodynamics of steady swimming and the exponential-based metabolic model is defined.
Directory of Open Access Journals (Sweden)
Anthony Papadopoulos
Full Text Available The first-degree power-law polynomial function is frequently used to describe activity metabolism for steady swimming animals. This function has been used in hydrodynamics-based metabolic studies to evaluate important parameters of energetic costs, such as the standard metabolic rate and the drag power indices. In theory, however, the power-law polynomial function of any degree greater than one can be used to describe activity metabolism for steady swimming animals. In fact, activity metabolism has been described by the conventional exponential function and the cubic polynomial function, although only the power-law polynomial function models drag power since it conforms to hydrodynamic laws. Consequently, the first-degree power-law polynomial function yields incorrect parameter values of energetic costs if activity metabolism is governed by the power-law polynomial function of any degree greater than one. This issue is important in bioenergetics because correct comparisons of energetic costs among different steady swimming animals cannot be made unless the degree of the power-law polynomial function derives from activity metabolism. In other words, a hydrodynamics-based functional form of activity metabolism is a power-law polynomial function of any degree greater than or equal to one. Therefore, the degree of the power-law polynomial function should be treated as a parameter, not as a constant. This new treatment not only conforms to hydrodynamic laws, but also ensures correct comparisons of energetic costs among different steady swimming animals. Furthermore, the exponential power-law function, which is a new hydrodynamics-based functional form of activity metabolism, is a special case of the power-law polynomial function. Hence, the link between the hydrodynamics of steady swimming and the exponential-based metabolic model is defined.
On the existence of polynomial Lyapunov functions for rationally stable vector fields
DEFF Research Database (Denmark)
Leth, Tobias; Wisniewski, Rafal; Sloth, Christoffer
2018-01-01
This paper proves the existence of polynomial Lyapunov functions for rationally stable vector fields. For practical purposes the existence of polynomial Lyapunov functions plays a significant role since polynomial Lyapunov functions can be found algorithmically. The paper extents an existing result...... on exponentially stable vector fields to the case of rational stability. For asymptotically stable vector fields a known counter example is investigated to exhibit the mechanisms responsible for the inability to extend the result further....
Optimization of Cubic Polynomial Functions without Calculus
Taylor, Ronald D., Jr.; Hansen, Ryan
2008-01-01
In algebra and precalculus courses, students are often asked to find extreme values of polynomial functions in the context of solving an applied problem; but without the notion of derivative, something is lost. Either the functions are reduced to quadratics, since students know the formula for the vertex of a parabola, or solutions are…
Conference on Commutative rings, integer-valued polynomials and polynomial functions
Frisch, Sophie; Glaz, Sarah; Commutative Algebra : Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions
2014-01-01
This volume presents a multi-dimensional collection of articles highlighting recent developments in commutative algebra. It also includes an extensive bibliography and lists a substantial number of open problems that point to future directions of research in the represented subfields. The contributions cover areas in commutative algebra that have flourished in the last few decades and are not yet well represented in book form. Highlighted topics and research methods include Noetherian and non- Noetherian ring theory as well as integer-valued polynomials and functions. Specific topics include: · Homological dimensions of Prüfer-like rings · Quasi complete rings · Total graphs of rings · Properties of prime ideals over various rings · Bases for integer-valued polynomials · Boolean subrings · The portable property of domains · Probabilistic topics in Intn(D) · Closure operations in Zariski-Riemann spaces of valuation domains · Stability of do...
A note on generating functions for Rogers-Szego polynomials | Cao ...
African Journals Online (AJOL)
Szego polynomials and their generalizations are given by Carlitz's q-operators. In addition, an additional proof of Srivastava-Jain's generating function is shown. At last, generalizations of Srivastava-Agarwal type generating functions are obtained.
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.].
Superiority of Bessel function over Zernicke polynomial as base ...
Indian Academy of Sciences (India)
Abstract. Here we describe the superiority of Bessel function as base function for radial expan- sion over Zernicke polynomial in the tomographic reconstruction technique. The causes for the superiority have been described in detail. The superiority has been shown both with simulated data for Kadomtsev's model for ...
Hong, X; Harris, C J
2000-01-01
This paper introduces a new neurofuzzy model construction algorithm for nonlinear dynamic systems based upon basis functions that are Bézier-Bernstein polynomial functions. This paper is generalized in that it copes with n-dimensional inputs by utilising an additive decomposition construction to overcome the curse of dimensionality associated with high n. This new construction algorithm also introduces univariate Bézier-Bernstein polynomial functions for the completeness of the generalized procedure. Like the B-spline expansion based neurofuzzy systems, Bézier-Bernstein polynomial function based neurofuzzy networks hold desirable properties such as nonnegativity of the basis functions, unity of support, and interpretability of basis function as fuzzy membership functions, moreover with the additional advantages of structural parsimony and Delaunay input space partition, essentially overcoming the curse of dimensionality associated with conventional fuzzy and RBF networks. This new modeling network is based on additive decomposition approach together with two separate basis function formation approaches for both univariate and bivariate Bézier-Bernstein polynomial functions used in model construction. The overall network weights are then learnt using conventional least squares methods. Numerical examples are included to demonstrate the effectiveness of this new data based modeling approach.
Stabilisation of discrete-time polynomial fuzzy systems via a polynomial lyapunov approach
Nasiri, Alireza; Nguang, Sing Kiong; Swain, Akshya; Almakhles, Dhafer
2018-02-01
This paper deals with the problem of designing a controller for a class of discrete-time nonlinear systems which is represented by discrete-time polynomial fuzzy model. Most of the existing control design methods for discrete-time fuzzy polynomial systems cannot guarantee their Lyapunov function to be a radially unbounded polynomial function, hence the global stability cannot be assured. The proposed control design in this paper guarantees a radially unbounded polynomial Lyapunov functions which ensures global stability. In the proposed design, state feedback structure is considered and non-convexity problem is solved by incorporating an integrator into the controller. Sufficient conditions of stability are derived in terms of polynomial matrix inequalities which are solved via SOSTOOLS in MATLAB. A numerical example is presented to illustrate the effectiveness of the proposed controller.
Abd-Elhameed, W. M.
2017-07-01
In this paper, a new formula relating Jacobi polynomials of arbitrary parameters with the squares of certain fractional Jacobi functions is derived. The derived formula is expressed in terms of a certain terminating hypergeometric function of the type _4F3(1) . With the aid of some standard reduction formulae such as Pfaff-Saalschütz's and Watson's identities, the derived formula can be reduced in simple forms which are free of any hypergeometric functions for certain choices of the involved parameters of the Jacobi polynomials and the Jacobi functions. Some other simplified formulae are obtained via employing some computer algebra algorithms such as the algorithms of Zeilberger, Petkovsek and van Hoeij. Some connection formulae between some Jacobi polynomials are deduced. From these connection formulae, some other linearization formulae of Chebyshev polynomials are obtained. As an application to some of the introduced formulae, a numerical algorithm for solving nonlinear Riccati differential equation is presented and implemented by applying a suitable spectral method.
Application of grafted polynomial function in forecasting cotton ...
African Journals Online (AJOL)
A study was conducted to forecast cotton production trend with the application of a grafted polynomial function in Nigeria from 1985 through 2013. Grafted models are used in econometrics to embark on economic analysis involving time series. In economic time series, the paucity of data and their availability has always ...
Zhao, Ke; Ji, Yaoyao; Pan, Boan; Li, Ting
2018-02-01
The continuous-wave Near-infrared spectroscopy (NIRS) devices have been highlighted for its clinical and health care applications in noninvasive hemodynamic measurements. The baseline shift of the deviation measurement attracts lots of attentions for its clinical importance. Nonetheless current published methods have low reliability or high variability. In this study, we found a perfect polynomial fitting function for baseline removal, using NIRS. Unlike previous studies on baseline correction for near-infrared spectroscopy evaluation of non-hemodynamic particles, we focused on baseline fitting and corresponding correction method for NIRS and found that the polynomial fitting function at 4th order is greater than the function at 2nd order reported in previous research. Through experimental tests of hemodynamic parameters of the solid phantom, we compared the fitting effect between the 4th order polynomial and the 2nd order polynomial, by recording and analyzing the R values and the SSE (the sum of squares due to error) values. The R values of the 4th order polynomial function fitting are all higher than 0.99, which are significantly higher than the corresponding ones of 2nd order, while the SSE values of the 4th order are significantly smaller than the corresponding ones of the 2nd order. By using the high-reliable and low-variable 4th order polynomial fitting function, we are able to remove the baseline online to obtain more accurate NIRS measurements.
Directory of Open Access Journals (Sweden)
V. P. Gribkova
2014-01-01
Full Text Available The paper offers a new method for approximate solution of one type of singular integral equations for elasticity theory which have been studied by other authors. The approximate solution is found in the form of asymptotic polynomial function of a low degree (first approximation based on the Chebyshev second order polynomial. Other authors have obtained a solution (only in separate points using a method of mechanical quadrature and though they used also the Chebyshev polynomial of the second order they applied another system of junctures which were used for the creation of the required formulas.The suggested method allows not only to find an approximate solution for the whole interval in the form of polynomial, but it also makes it possible to obtain a remainder term in the form of infinite expansion where coefficients are linear functional of the given integral equation and basis functions are the Chebyshev polynomial of the second order. Such presentation of the remainder term of the first approximation permits to find a summand of the infinite series, which will serve as a start for fulfilling the given solution accuracy. This number is a degree of the asymptotic polynomial (second approximation, which will give the approximation to the exact solution with the given accuracy. The examined polynomial functions tend asymptotically to the polynomial of the best uniform approximation in the space C, created for the given operator.The paper demonstrates a convergence of the approximate solution to the exact one and provides an error estimation. The proposed algorithm for obtaining of the approximate solution and error estimation is easily realized with the help of computing technique and does not require considerable preliminary preparation during programming.
Miller, W., Jr.; Li, Q.
2015-04-01
The Wilson and Racah polynomials can be characterized as basis functions for irreducible representations of the quadratic symmetry algebra of the quantum superintegrable system on the 2-sphere, HΨ = EΨ, with generic 3-parameter potential. Clearly, the polynomials are expansion coefficients for one eigenbasis of a symmetry operator L2 of H in terms of an eigenbasis of another symmetry operator L1, but the exact relationship appears not to have been made explicit. We work out the details of the expansion to show, explicitly, how the polynomials arise and how the principal properties of these functions: the measure, 3-term recurrence relation, 2nd order difference equation, duality of these relations, permutation symmetry, intertwining operators and an alternate derivation of Wilson functions - follow from the symmetry of this quantum system. This paper is an exercise to show that quantum mechancal concepts and recurrence relations for Gausian hypergeometrc functions alone suffice to explain these properties; we make no assumptions about the structure of Wilson polynomial/functions, but derive them from quantum principles. There is active interest in the relation between multivariable Wilson polynomials and the quantum superintegrable system on the n-sphere with generic potential, and these results should aid in the generalization. Contracting function space realizations of irreducible representations of this quadratic algebra to the other superintegrable systems one can obtain the full Askey scheme of orthogonal hypergeometric polynomials. All of these contractions of superintegrable systems with potential are uniquely induced by Wigner Lie algebra contractions of so(3, C) and e(2,C). All of the polynomials produced are interpretable as quantum expansion coefficients. It is important to extend this process to higher dimensions.
International Nuclear Information System (INIS)
Miller, W Jr; Li, Q
2015-01-01
The Wilson and Racah polynomials can be characterized as basis functions for irreducible representations of the quadratic symmetry algebra of the quantum superintegrable system on the 2-sphere, HΨ = EΨ, with generic 3-parameter potential. Clearly, the polynomials are expansion coefficients for one eigenbasis of a symmetry operator L 2 of H in terms of an eigenbasis of another symmetry operator L 1 , but the exact relationship appears not to have been made explicit. We work out the details of the expansion to show, explicitly, how the polynomials arise and how the principal properties of these functions: the measure, 3-term recurrence relation, 2nd order difference equation, duality of these relations, permutation symmetry, intertwining operators and an alternate derivation of Wilson functions - follow from the symmetry of this quantum system. This paper is an exercise to show that quantum mechancal concepts and recurrence relations for Gausian hypergeometrc functions alone suffice to explain these properties; we make no assumptions about the structure of Wilson polynomial/functions, but derive them from quantum principles. There is active interest in the relation between multivariable Wilson polynomials and the quantum superintegrable system on the n-sphere with generic potential, and these results should aid in the generalization. Contracting function space realizations of irreducible representations of this quadratic algebra to the other superintegrable systems one can obtain the full Askey scheme of orthogonal hypergeometric polynomials. All of these contractions of superintegrable systems with potential are uniquely induced by Wigner Lie algebra contractions of so(3, C) and e(2,C). All of the polynomials produced are interpretable as quantum expansion coefficients. It is important to extend this process to higher dimensions. (paper)
The computation of bond percolation critical polynomials by the deletion–contraction algorithm
International Nuclear Information System (INIS)
Scullard, Christian R
2012-01-01
Although every exactly known bond percolation critical threshold is the root in [0,1] of a lattice-dependent polynomial, it has recently been shown that the notion of a critical polynomial can be extended to any periodic lattice. The polynomial is computed on a finite subgraph, called the base, of an infinite lattice. For any problem with exactly known solution, the prediction of the bond threshold is always correct for any base containing an arbitrary number of unit cells. For unsolved problems, the polynomial is referred to as the generalized critical polynomial and provides an approximation that becomes more accurate with increasing number of bonds in the base, appearing to approach the exact answer. The polynomials are computed using the deletion–contraction algorithm, which quickly becomes intractable by hand for more than about 18 bonds. Here, I present generalized critical polynomials calculated with a computer program for bases of up to 36 bonds for all the unsolved Archimedean lattices, except the kagome lattice, which was considered in an earlier work. The polynomial estimates are generally within 10 −5 –10 −7 of the numerical values, but the prediction for the (4,8 2 ) lattice, though not exact, is not ruled out by simulations. (paper)
A New Wavelet Threshold Function and Denoising Application
Directory of Open Access Journals (Sweden)
Lu Jing-yi
2016-01-01
Full Text Available In order to improve the effects of denoising, this paper introduces the basic principles of wavelet threshold denoising and traditional structures threshold functions. Meanwhile, it proposes wavelet threshold function and fixed threshold formula which are both improved here. First, this paper studies the problems existing in the traditional wavelet threshold functions and introduces the adjustment factors to construct the new threshold function basis on soft threshold function. Then, it studies the fixed threshold and introduces the logarithmic function of layer number of wavelet decomposition to design the new fixed threshold formula. Finally, this paper uses hard threshold, soft threshold, Garrote threshold, and improved threshold function to denoise different signals. And the paper also calculates signal-to-noise (SNR and mean square errors (MSE of the hard threshold functions, soft thresholding functions, Garrote threshold functions, and the improved threshold function after denoising. Theoretical analysis and experimental results showed that the proposed approach could improve soft threshold functions with constant deviation and hard threshold with discontinuous function problems. The proposed approach could improve the different decomposition scales that adopt the same threshold value to deal with the noise problems, also effectively filter the noise in the signals, and improve the SNR and reduce the MSE of output signals.
QCD analysis of structure functions in terms of Jacobi polynomials
International Nuclear Information System (INIS)
Krivokhizhin, V.G.; Kurlovich, S.P.; Savin, I.A.; Sidorov, A.V.; Skachkov, N.B.; Sanadze, V.V.
1987-01-01
A new method of QCD-analysis of singlet and nonsinglet structure functions based on their expansion in orthogonal Jacobi polynomials is proposed. An accuracy of the method is studied and its application is demonstrated using the structure function F 2 (x,Q 2 ) obtained by the EMC Collaboration from measurements with an iron target. (orig.)
Expansion of Sobolev functions in series in Laguerre polynomials
International Nuclear Information System (INIS)
Selyakov, K.I.
1985-01-01
The solution of the integral equation for the Sobolev functions is represented in the form of series in Laguerre polynomials. The coefficients of these series are simultaneously the coefficients of the power series for the Ambartsumyan-Chandrasekhar H functions. Infinite systems of linear algebraic equations with Toeplitz matrices are given for the coefficients of the series. Numerical results and approximate expressions are given for the case of isotropic scattering
Discrete-time state estimation for stochastic polynomial systems over polynomial observations
Hernandez-Gonzalez, M.; Basin, M.; Stepanov, O.
2018-07-01
This paper presents a solution to the mean-square state estimation problem for stochastic nonlinear polynomial systems over polynomial observations confused with additive white Gaussian noises. The solution is given in two steps: (a) computing the time-update equations and (b) computing the measurement-update equations for the state estimate and error covariance matrix. A closed form of this filter is obtained by expressing conditional expectations of polynomial terms as functions of the state estimate and error covariance. As a particular case, the mean-square filtering equations are derived for a third-degree polynomial system with second-degree polynomial measurements. Numerical simulations show effectiveness of the proposed filter compared to the extended Kalman filter.
A New Generalisation of Macdonald Polynomials
Garbali, Alexandr; de Gier, Jan; Wheeler, Michael
2017-06-01
We introduce a new family of symmetric multivariate polynomials, whose coefficients are meromorphic functions of two parameters ( q, t) and polynomial in a further two parameters ( u, v). We evaluate these polynomials explicitly as a matrix product. At u = v = 0 they reduce to Macdonald polynomials, while at q = 0, u = v = s they recover a family of inhomogeneous symmetric functions originally introduced by Borodin.
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.
Nuclear threshold effects and neutron strength function
International Nuclear Information System (INIS)
Hategan, Cornel; Comisel, Horia
2003-01-01
One proves that a Nuclear Threshold Effect is dependent, via Neutron Strength Function, on Spectroscopy of Ancestral Neutron Threshold State. The magnitude of the Nuclear Threshold Effect is proportional to the Neutron Strength Function. Evidence for relation of Nuclear Threshold Effects to Neutron Strength Functions is obtained from Isotopic Threshold Effect and Deuteron Stripping Threshold Anomaly. The empirical and computational analysis of the Isotopic Threshold Effect and of the Deuteron Stripping Threshold Anomaly demonstrate their close relationship to Neutron Strength Functions. It was established that the Nuclear Threshold Effects depend, in addition to genuine Nuclear Reaction Mechanisms, on Spectroscopy of (Ancestral) Neutron Threshold State. The magnitude of the effect is proportional to the Neutron Strength Function, in their dependence on mass number. This result constitutes also a proof that the origins of these threshold effects are Neutron Single Particle States at zero energy. (author)
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)
Polynomial optimization : Error analysis and applications
Sun, Zhao
2015-01-01
Polynomial optimization is the problem of minimizing a polynomial function subject to polynomial inequality constraints. In this thesis we investigate several hierarchies of relaxations for polynomial optimization problems. Our main interest lies in understanding their performance, in particular how
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
Narimani, Mohammand; Lam, H K; Dilmaghani, R; Wolfe, Charles
2011-06-01
Relaxed linear-matrix-inequality-based stability conditions for fuzzy-model-based control systems with imperfect premise matching are proposed. First, the derivative of the Lyapunov function, containing the product terms of the fuzzy model and fuzzy controller membership functions, is derived. Then, in the partitioned operating domain of the membership functions, the relations between the state variables and the mentioned product terms are represented by approximated polynomials in each subregion. Next, the stability conditions containing the information of all subsystems and the approximated polynomials are derived. In addition, the concept of the S-procedure is utilized to release the conservativeness caused by considering the whole operating region for approximated polynomials. It is shown that the well-known stability conditions can be special cases of the proposed stability conditions. Simulation examples are given to illustrate the validity of the proposed approach.
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
Nonnegativity of uncertain polynomials
Directory of Open Access Journals (Sweden)
iljak Dragoslav D.
1998-01-01
Full Text Available The purpose of this paper is to derive tests for robust nonnegativity of scalar and matrix polynomials, which are algebraic, recursive, and can be completed in finite number of steps. Polytopic families of polynomials are considered with various characterizations of parameter uncertainty including affine, multilinear, and polynomic structures. The zero exclusion condition for polynomial positivity is also proposed for general parameter dependencies. By reformulating the robust stability problem of complex polynomials as positivity of real polynomials, we obtain new sufficient conditions for robust stability involving multilinear structures, which can be tested using only real arithmetic. The obtained results are applied to robust matrix factorization, strict positive realness, and absolute stability of multivariable systems involving parameter dependent transfer function matrices.
Polynomial chaos functions and stochastic differential equations
International Nuclear Information System (INIS)
Williams, M.M.R.
2006-01-01
The Karhunen-Loeve procedure and the associated polynomial chaos expansion have been employed to solve a simple first order stochastic differential equation which is typical of transport problems. Because the equation has an analytical solution, it provides a useful test of the efficacy of polynomial chaos. We find that the convergence is very rapid in some cases but that the increased complexity associated with many random variables can lead to very long computational times. The work is illustrated by exact and approximate solutions for the mean, variance and the probability distribution itself. The usefulness of a white noise approximation is also assessed. Extensive numerical results are given which highlight the weaknesses and strengths of polynomial chaos. The general conclusion is that the method is promising but requires further detailed study by application to a practical problem in transport theory
Uniform approximations of Bernoulli and Euler polynomials in terms of hyperbolic functions
J.L. López; N.M. Temme (Nico)
1998-01-01
textabstractBernoulli and Euler polynomials are considered for large values of the order. Convergent expansions are obtained for $B_n(nz+1/2)$ and $E_n(nz+1/2)$ in powers of $n^{-1$, with coefficients being rational functions of $z$ and hyperbolic functions of argument $1/2z$. These expansions are
Lam, Hak-Keung
2016-01-01
This book presents recent research on the stability analysis of polynomial-fuzzy-model-based control systems where the concept of partially/imperfectly matched premises and membership-function dependent analysis are considered. The membership-function-dependent analysis offers a new research direction for fuzzy-model-based control systems by taking into account the characteristic and information of the membership functions in the stability analysis. The book presents on a research level the most recent and advanced research results, promotes the research of polynomial-fuzzy-model-based control systems, and provides theoretical support and point a research direction to postgraduate students and fellow researchers. Each chapter provides numerical examples to verify the analysis results, demonstrate the effectiveness of the proposed polynomial fuzzy control schemes, and explain the design procedure. The book is comprehensively written enclosing detailed derivation steps and mathematical derivations also for read...
Asymptotics and Numerics of Polynomials Used in Tricomi and Buchholz Expansions of Kummer functions
J.L. López; N.M. Temme (Nico)
2010-01-01
textabstractExpansions in terms of Bessel functions are considered of the Kummer function ${}_1F_1(a;c,z)$ (or confluent hypergeometric function) as given by Tricomi and Buchholz. The coefficients of these expansions are polynomials in the parameters of the Kummer function and the asymptotic
Generalizations of orthogonal polynomials
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.
Threshold Signature Schemes Application
Directory of Open Access Journals (Sweden)
Anastasiya Victorovna Beresneva
2015-10-01
Full Text Available This work is devoted to an investigation of threshold signature schemes. The systematization of the threshold signature schemes was done, cryptographic constructions based on interpolation Lagrange polynomial, elliptic curves and bilinear pairings were examined. Different methods of generation and verification of threshold signatures were explored, the availability of practical usage of threshold schemes in mobile agents, Internet banking and e-currency was shown. The topics of further investigation were given and it could reduce a level of counterfeit electronic documents signed by a group of users.
Methods of geometric function theory in classical and modern problems for polynomials
International Nuclear Information System (INIS)
Dubinin, Vladimir N
2012-01-01
This paper gives a survey of classical and modern theorems on polynomials, proved using methods of geometric function theory. Most of the paper is devoted to results of the author and his students, established by applying majorization principles for holomorphic functions, the theory of univalent functions, the theory of capacities, and symmetrization. Auxiliary results and the proofs of some of the theorems are presented. Bibliography: 124 titles.
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...
AMDLIBAE, IBM 360 Subroutine Library, Special Function, Polynomials, Differential Equation
International Nuclear Information System (INIS)
Wang, Jesse Y.
1980-01-01
Description of problem or function: AMDLIBAE is a subset of the IBM 360 Subroutine Library at the Applied Mathematics Division at Argonne National Laboratory. This subset includes library categories A-E: Identification/Description: A152S A MPA: Mult. prec. floating point arith. package; B156S A ARSIN: Arcsine, arccosine; B158S A DSIN/DCOS: DP sine, cosine; B159S A DTAN/DCOT: DP tangent, cotangent; B252S A SINH/COSH: Hyperbolic sine, cosine; B353S A ALOG: SP logarithm; B354S A DEXP: DP exponential; B355S A DLOG: DP logarithm; B456S A DCUBRT: DP cube root; B457S A ARGPOWER: X Y ; B458S A ARGFDXPD: DP X Y ; C150S F DQD: Q. D. algorithm applied to a power series; C151S F DCONF1: Eval. cont. fract. Q. D. of power series; C250S F CUBIC: Roots of cubic polynomial equation; C251S F QUARTIC: Roots of quartic polynomial equation; C252S F RSSR: All roots of poly eqs. with real coef.; C253S F POLDRV: Driver for C254S; C254S F CPOLY: Roots arb. poly. Jenkins-Traub algorithm; C353S F1 CLEBSH: Ang. mom. coef. - Clebsch, Racah, Wigner; C365S A ALGAMA: Logarithm of the gamma function; C366S A DGAMMA/DLGAMA: DP gamma and log(gamma) functions; C368S F EONE: Exponential integral E1; C370S F BESJY: Bessel functions J and Y; C371S F BESIK: Bessel functions I and K; C372S F CHIPRB: Chi-square integral; C380S F DRZETA: Long precision zeta, zeta-1 functions; C382S F DCGAM: Long precision complex gamma; C383S A DERF/DERFC: DP error function; C384S F BFJ1: Bessel function J1; C385S F COULMB: Regular Coulomb wave functions; C386S F1 DSGMAL: Coulomb phase shift; C387S F BFJYR: Bessel functions J0,J1,Y0,Y1; C388S F IRCOUL: LP irregular Coulomb wave functions; C389S F GAMIN: Incomplete gamma function; C390S F LQ: Assoc. Legendre functions of 2. kind; C392S A DAERF: Inverse error function; C393S F CDEONE: Modified complex exponential integral; D153S F DROMB: Two-dimensional Romberg quadrature; D153S P DROMBP: Two-dimensional Romberg quadrature; D158S F ANC4: Adap. quad. using 4. order Newton
International Nuclear Information System (INIS)
Olesov, A V
2014-01-01
New inequalities are established for analytic functions satisfying Meiman's majorization conditions. Estimates for values of and differential inequalities involving rational trigonometric functions with an integer majorant on an interval of length less than the period and with prescribed poles which are symmetrically positioned relative to the real axis, as well as differential inequalities for trigonometric polynomials in some classes, are given as applications. These results improve several theorems due to Meiman, Genchev, Smirnov and Rusak. Bibliography: 27 titles
Optimization over polynomials : Selected topics
Laurent, M.; Jang, Sun Young; Kim, Young Rock; Lee, Dae-Woong; Yie, Ikkwon
2014-01-01
Minimizing a polynomial function over a region defined by polynomial inequalities models broad classes of hard problems from combinatorics, geometry and optimization. New algorithmic approaches have emerged recently for computing the global minimum, by combining tools from real algebra (sums of
Zeros and uniqueness of Q-difference polynomials of meromorphic ...
Indian Academy of Sciences (India)
Meromorphic functions; Nevanlinna theory; logarithmic order; uniqueness problem; difference-differential polynomial. Abstract. In this paper, we investigate the value distribution of -difference polynomials of meromorphic function of finite logarithmic order, and study the zero distribution of difference-differential polynomials ...
Localization of periodic orbits of polynomial vector fields of even degree by linear functions
Energy Technology Data Exchange (ETDEWEB)
Starkov, Konstantin E. [CITEDI-IPN, Av. del Parque 1310, Mesa de Otay, Tijuana, BC (Mexico)] e-mail: konst@citedi.mx
2005-08-01
This paper is concerned with the localization problem of periodic orbits of polynomial vector fields of even degree by using linear functions. Conditions of the localization of all periodic orbits in sets of a simple structure are obtained. Our results are based on the solution of the conditional extremum problem and the application of homogeneous polynomial forms of even degrees. As examples, the Lanford system, the jerky system with one quadratic monomial and a quartically perturbed harmonic oscillator are considered.
Localization of periodic orbits of polynomial vector fields of even degree by linear functions
International Nuclear Information System (INIS)
Starkov, Konstantin E.
2005-01-01
This paper is concerned with the localization problem of periodic orbits of polynomial vector fields of even degree by using linear functions. Conditions of the localization of all periodic orbits in sets of a simple structure are obtained. Our results are based on the solution of the conditional extremum problem and the application of homogeneous polynomial forms of even degrees. As examples, the Lanford system, the jerky system with one quadratic monomial and a quartically perturbed harmonic oscillator are considered
Remarks on determinants and the classical polynomials
International Nuclear Information System (INIS)
Henning, J.J.; Kranold, H.U.; Louw, D.F.B.
1986-01-01
As motivation for this formal analysis the problem of Landau damping of Bernstein modes is discussed. It is shown that in the case of a weak but finite constant external magnetic field, the analytical structure of the dispersion relations is of such a nature that longitudinal waves propagating orthogonal to the external magnetic field are also damped, contrary to normal belief. In the treatment of the linearized Vlasov equation it is found convenient to generate certain polynomials by the problem at hand and to explicitly write down expressions for these polynomials. In the course of this study methods are used that relate to elementary but fairly unknown functional relationships between power sums and coefficients of polynomials. These relationships, also called Waring functions, are derived. They are then used in other applications to give explicit expressions for the generalized Laguerre polynomials in terms of determinant functions. The properties of polynomials generated by a wide class of generating functions are investigated. These relationships are also used to obtain explicit forms for the cumulants of a distribution in terms of its moments. It is pointed out that cumulants (or moments, for that matter) do not determine a distribution function
International Nuclear Information System (INIS)
Lee, M.W.; Bigeleisen, J.
1978-01-01
The MINIMAX finite polynomial approximation to an arbitrary function has been generalized to include a weighting function (WINIMAX). It is suggested that an exponential is a reasonable weighting function for the logarithm of the reduced partition function of a harmonic oscillator. Comparison of the error function for finite orthogonal polynomial (FOP), MINIMAX, and WINIMAX expansions of the logarithm of the reduced vibrational partition function show WINIMAX to be the best of the three approximations. A condensed table of WINIMAX coefficients is presented. The FOP, MINIMAX, and WINIMAX approximations are compared with exact calculations of the logarithm of the reduced partition function ratios for isotopic substitution in H 2 O, CH 4 , CH 2 O, C 2 H 4 , and C 2 H 6 at 300 0 K. Both deuterium and heavy atom isotope substitution are studied. Except for a third order expansion involving deuterium substitution, the WINIMAX method is superior to FOP and MINIMAX. At the level of a second order expansion WINIMAX approximations to ln(s/s')f are good to 2.5% and 6.5% for deuterium and heavy atom substitution, respectively
Comparison Between Polynomial, Euler Beta-Function and Expo-Rational B-Spline Bases
Kristoffersen, Arnt R.; Dechevsky, Lubomir T.; Laksa˚, Arne; Bang, Børre
2011-12-01
Euler Beta-function B-splines (BFBS) are the practically most important instance of generalized expo-rational B-splines (GERBS) which are not true expo-rational B-splines (ERBS). BFBS do not enjoy the full range of the superproperties of ERBS but, while ERBS are special functions computable by a very rapidly converging yet approximate numerical quadrature algorithms, BFBS are explicitly computable piecewise polynomial (for integer multiplicities), similar to classical Schoenberg B-splines. In the present communication we define, compute and visualize for the first time all possible BFBS of degree up to 3 which provide Hermite interpolation in three consecutive knots of multiplicity up to 3, i.e., the function is being interpolated together with its derivatives of order up to 2. We compare the BFBS obtained for different degrees and multiplicities among themselves and versus the classical Schoenberg polynomial B-splines and the true ERBS for the considered knots. The results of the graphical comparison are discussed from analytical point of view. For the numerical computation and visualization of the new B-splines we have used Maple 12.
Method of applying single higher order polynomial basis function over multiple domains
CSIR Research Space (South Africa)
Lysko, AA
2010-03-01
Full Text Available A novel method has been devised where one set of higher order polynomial-based basis functions can be applied over several wire segments, thus permitting to decouple the number of unknowns from the number of segments, and so from the geometrical...
Additive and polynomial representations
Krantz, David H; Suppes, Patrick
1971-01-01
Additive and Polynomial Representations deals with major representation theorems in which the qualitative structure is reflected as some polynomial function of one or more numerical functions defined on the basic entities. Examples are additive expressions of a single measure (such as the probability of disjoint events being the sum of their probabilities), and additive expressions of two measures (such as the logarithm of momentum being the sum of log mass and log velocity terms). The book describes the three basic procedures of fundamental measurement as the mathematical pivot, as the utiliz
Skew-orthogonal polynomials and random matrix theory
Ghosh, Saugata
2009-01-01
Orthogonal polynomials satisfy a three-term recursion relation irrespective of the weight function with respect to which they are defined. This gives a simple formula for the kernel function, known in the literature as the Christoffel-Darboux sum. The availability of asymptotic results of orthogonal polynomials and the simple structure of the Christoffel-Darboux sum make the study of unitary ensembles of random matrices relatively straightforward. In this book, the author develops the theory of skew-orthogonal polynomials and obtains recursion relations which, unlike orthogonal polynomials, depend on weight functions. After deriving reduced expressions, called the generalized Christoffel-Darboux formulas (GCD), he obtains universal correlation functions and non-universal level densities for a wide class of random matrix ensembles using the GCD. The author also shows that once questions about higher order effects are considered (questions that are relevant in different branches of physics and mathematics) the ...
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 ...
Polynomial asymptotic stability of damped stochastic differential equations
Directory of Open Access Journals (Sweden)
John Appleby
2004-08-01
Full Text Available The paper studies the polynomial convergence of solutions of a scalar nonlinear It\\^{o} stochastic differential equation\\[dX(t = -f(X(t\\,dt + \\sigma(t\\,dB(t\\] where it is known, {\\it a priori}, that $\\lim_{t\\rightarrow\\infty} X(t=0$, a.s. The intensity of the stochastic perturbation $\\sigma$ is a deterministic, continuous and square integrable function, which tends to zero more quickly than a polynomially decaying function. The function $f$ obeys $\\lim_{x\\rightarrow 0}\\mbox{sgn}(xf(x/|x|^\\beta = a$, for some $\\beta>1$, and $a>0$.We study two asymptotic regimes: when $\\sigma$ tends to zero sufficiently quickly the polynomial decay rate of solutions is the same as for the deterministic equation (when $\\sigma\\equiv0$. When $\\sigma$ decays more slowly, a weaker almost sure polynomial upper bound on the decay rate of solutions is established. Results which establish the necessity for $\\sigma$ to decay polynomially in order to guarantee the almost sure polynomial decay of solutions are also proven.
Julia Sets of Orthogonal Polynomials
DEFF Research Database (Denmark)
Christiansen, Jacob Stordal; Henriksen, Christian; Petersen, Henrik Laurberg
2018-01-01
For a probability measure with compact and non-polar support in the complex plane we relate dynamical properties of the associated sequence of orthogonal polynomials fPng to properties of the support. More precisely we relate the Julia set of Pn to the outer boundary of the support, the lled Julia...... set to the polynomial convex hull K of the support, and the Green's function associated with Pn to the Green's function for the complement of K....
An introduction to orthogonal polynomials
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
Function approximation with polynomial regression slines
International Nuclear Information System (INIS)
Urbanski, P.
1996-01-01
Principles of the polynomial regression splines as well as algorithms and programs for their computation are presented. The programs prepared using software package MATLAB are generally intended for approximation of the X-ray spectra and can be applied in the multivariate calibration of radiometric gauges. (author)
APPROX, 1-D and 2-D Function Approximation by Polynomials, Splines, Finite Elements Method
International Nuclear Information System (INIS)
Tollander, Bengt
1975-01-01
1 - Nature of physical problem solved: Approximates one- and two- dimensional functions using different forms of the approximating function, as polynomials, rational functions, Splines and (or) the finite element method. Different kinds of transformations of the dependent and (or) the independent variables can easily be made by data cards using a FORTRAN-like language. 2 - Method of solution: Approximations by polynomials, Splines and (or) the finite element method are made in L2 norm using the least square method by which the answer is directly given. For rational functions in one dimension the result given in L(infinite) norm is achieved by iterations moving the zero points of the error curve. For rational functions in two dimensions, the norm is L2 and the result is achieved by iteratively changing the coefficients of the denominator and then solving the coefficients of the numerator by the least square method. The transformation of the dependent and (or) independent variables is made by compiling the given transform data card(s) to an array of integers from which the transformation can be made
The modified Gauss diagonalization of polynomial matrices
International Nuclear Information System (INIS)
Saeed, K.
1982-10-01
The Gauss algorithm for diagonalization of constant matrices is modified for application to polynomial matrices. Due to this modification the diagonal elements become pure polynomials rather than rational functions. (author)
Imaging characteristics of Zernike and annular polynomial aberrations.
Mahajan, Virendra N; Díaz, José Antonio
2013-04-01
The general equations for the point-spread function (PSF) and optical transfer function (OTF) are given for any pupil shape, and they are applied to optical imaging systems with circular and annular pupils. The symmetry properties of the PSF, the real and imaginary parts of the OTF, and the modulation transfer function (MTF) of a system with a circular pupil aberrated by a Zernike circle polynomial aberration are derived. The interferograms and PSFs are illustrated for some typical polynomial aberrations with a sigma value of one wave, and 3D PSFs and MTFs are shown for 0.1 wave. The Strehl ratio is also calculated for polynomial aberrations with a sigma value of 0.1 wave, and shown to be well estimated from the sigma value. The numerical results are compared with the corresponding results in the literature. Because of the same angular dependence of the corresponding annular and circle polynomial aberrations, the symmetry properties of systems with annular pupils aberrated by an annular polynomial aberration are the same as those for a circular pupil aberrated by a corresponding circle polynomial aberration. They are also illustrated with numerical examples.
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 ...
Arbuckle, P. D.; Sliwa, S. M.; Roy, M. L.; Tiffany, S. H.
1985-01-01
A computer program for interactively developing least-squares polynomial equations to fit user-supplied data is described. The program is characterized by the ability to compute the polynomial equations of a surface fit through data that are a function of two independent variables. The program utilizes the Langley Research Center graphics packages to display polynomial equation curves and data points, facilitating a qualitative evaluation of the effectiveness of the fit. An explanation of the fundamental principles and features of the program, as well as sample input and corresponding output, are included.
Nahay, John Michael
2008-01-01
We present a new approach to solving polynomial ordinary differential equations by transforming them to linear functional equations and then solving the linear functional equations. We will focus most of our attention upon the first-order Abel differential equation with two nonlinear terms in order to demonstrate in as much detail as possible the computations necessary for a complete solution. We mention in our section on further developments that the basic transformation idea can be generali...
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
A generalization of the Bernoulli polynomials
Directory of Open Access Journals (Sweden)
Pierpaolo Natalini
2003-01-01
Full Text Available A generalization of the Bernoulli polynomials and, consequently, of the Bernoulli numbers, is defined starting from suitable generating functions. Furthermore, the differential equations of these new classes of polynomials are derived by means of the factorization method introduced by Infeld and Hull (1951.
Orthogonal polynomials derived from the tridiagonal representation approach
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.
A note on the zeros of Freud-Sobolev orthogonal polynomials
Moreno-Balcazar, Juan J.
2007-10-01
We prove that the zeros of a certain family of Sobolev orthogonal polynomials involving the Freud weight function e-x4 on are real, simple, and interlace with the zeros of the Freud polynomials, i.e., those polynomials orthogonal with respect to the weight function e-x4. Some numerical examples are shown.
Freud, Géza
1971-01-01
Orthogonal Polynomials contains an up-to-date survey of the general theory of orthogonal polynomials. It deals with the problem of polynomials and reveals that the sequence of these polynomials forms an orthogonal system with respect to a non-negative m-distribution defined on the real numerical axis. Comprised of five chapters, the book begins with the fundamental properties of orthogonal polynomials. After discussing the momentum problem, it then explains the quadrature procedure, the convergence theory, and G. Szegő's theory. This book is useful for those who intend to use it as referenc
q-analogue of the Krawtchouk and Meixner orthogonal polynomials
International Nuclear Information System (INIS)
Campigotto, C.; Smirnov, Yu.F.; Enikeev, S.G.
1993-06-01
The comparative analysis of Krawtchouk polynomials on a uniform grid with Wigner D-functions for the SU(2) group is presented. As a result the partnership between corresponding properties of the polynomials and D-functions is established giving the group-theoretical interpretation of the Krawtchouk polynomials properties. In order to extend such an analysis on the quantum groups SU q (2) and SU q (1,1), q-analogues of Krawtchouk and Meixner polynomials of a discrete variable are studied. The total set of characteristics of these polynomials is calculated, including the orthogonality condition, normalization factor, recurrent relation, the explicit analytic expression, the Rodrigues formula, the difference derivative formula and various particular cases and values. (R.P.) 22 refs.; 2 tabs
Global sensitivity analysis by polynomial dimensional decomposition
Energy Technology Data Exchange (ETDEWEB)
Rahman, Sharif, E-mail: rahman@engineering.uiowa.ed [College of Engineering, The University of Iowa, Iowa City, IA 52242 (United States)
2011-07-15
This paper presents a polynomial dimensional decomposition (PDD) method for global sensitivity analysis of stochastic systems subject to independent random input following arbitrary probability distributions. The method involves Fourier-polynomial expansions of lower-variate component functions of a stochastic response by measure-consistent orthonormal polynomial bases, analytical formulae for calculating the global sensitivity indices in terms of the expansion coefficients, and dimension-reduction integration for estimating the expansion coefficients. Due to identical dimensional structures of PDD and analysis-of-variance decomposition, the proposed method facilitates simple and direct calculation of the global sensitivity indices. Numerical results of the global sensitivity indices computed for smooth systems reveal significantly higher convergence rates of the PDD approximation than those from existing methods, including polynomial chaos expansion, random balance design, state-dependent parameter, improved Sobol's method, and sampling-based methods. However, for non-smooth functions, the convergence properties of the PDD solution deteriorate to a great extent, warranting further improvements. The computational complexity of the PDD method is polynomial, as opposed to exponential, thereby alleviating the curse of dimensionality to some extent.
Polynomial sequences generated by infinite Hessenberg matrices
Directory of Open Access Journals (Sweden)
Verde-Star Luis
2017-01-01
Full Text Available We show that an infinite lower Hessenberg matrix generates polynomial sequences that correspond to the rows of infinite lower triangular invertible matrices. Orthogonal polynomial sequences are obtained when the Hessenberg matrix is tridiagonal. We study properties of the polynomial sequences and their corresponding matrices which are related to recurrence relations, companion matrices, matrix similarity, construction algorithms, and generating functions. When the Hessenberg matrix is also Toeplitz the polynomial sequences turn out to be of interpolatory type and we obtain additional results. For example, we show that every nonderogative finite square matrix is similar to a unique Toeplitz-Hessenberg matrix.
Constructing general partial differential equations using polynomial and neural networks.
Zjavka, Ladislav; Pedrycz, Witold
2016-01-01
Sum fraction terms can approximate multi-variable functions on the basis of discrete observations, replacing a partial differential equation definition with polynomial elementary data relation descriptions. Artificial neural networks commonly transform the weighted sum of inputs to describe overall similarity relationships of trained and new testing input patterns. Differential polynomial neural networks form a new class of neural networks, which construct and solve an unknown general partial differential equation of a function of interest with selected substitution relative terms using non-linear multi-variable composite polynomials. The layers of the network generate simple and composite relative substitution terms whose convergent series combinations can describe partial dependent derivative changes of the input variables. This regression is based on trained generalized partial derivative data relations, decomposed into a multi-layer polynomial network structure. The sigmoidal function, commonly used as a nonlinear activation of artificial neurons, may transform some polynomial items together with the parameters with the aim to improve the polynomial derivative term series ability to approximate complicated periodic functions, as simple low order polynomials are not able to fully make up for the complete cycles. The similarity analysis facilitates substitutions for differential equations or can form dimensional units from data samples to describe real-world problems. Copyright © 2015 Elsevier Ltd. All rights reserved.
Vertex models, TASEP and Grothendieck polynomials
International Nuclear Information System (INIS)
Motegi, Kohei; Sakai, Kazumitsu
2013-01-01
We examine the wavefunctions and their scalar products of a one-parameter family of integrable five-vertex models. At a special point of the parameter, the model investigated is related to an irreversible interacting stochastic particle system—the so-called totally asymmetric simple exclusion process (TASEP). By combining the quantum inverse scattering method with a matrix product representation of the wavefunctions, the on-/off-shell wavefunctions of the five-vertex models are represented as a certain determinant form. Up to some normalization factors, we find that the wavefunctions are given by Grothendieck polynomials, which are a one-parameter deformation of Schur polynomials. Introducing a dual version of the Grothendieck polynomials, and utilizing the determinant representation for the scalar products of the wavefunctions, we derive a generalized Cauchy identity satisfied by the Grothendieck polynomials and their duals. Several representation theoretical formulae for the Grothendieck polynomials are also presented. As a byproduct, the relaxation dynamics such as Green functions for the periodic TASEP are found to be described in terms of the Grothendieck polynomials. (paper)
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...
Does the Polynomial Hierarchy Collapse if Onto Functions are Invertible?
Czech Academy of Sciences Publication Activity Database
Buhrman, H.; Fortnow, L.; Koucký, Michal; Rogers, J.D.; Vereshchagin, N.K.
2010-01-01
Roč. 46, č. 1 (2010), s. 143-156 ISSN 1432-4350. [2nd International Computer Science Symposium in Russia ( CSR 2007). Ekaterinburg, 03.09.2007-07.09.2007] R&D Projects: GA ČR GP201/07/P276; GA MŠk(CZ) 1M0545 Institutional research plan: CEZ:AV0Z10190503 Keywords : one-way functions * polynomial hierarchy * Kolmogorov generic oracles Subject RIV: BA - General Mathematics Impact factor: 0.600, year: 2010 http://link.springer.com/article/10.1007%2Fs00224-008-9160-8
Optimal threshold functions for fault detection and isolation
DEFF Research Database (Denmark)
Stoustrup, J.; Niemann, Hans Henrik; Cour-Harbo, A. la
2003-01-01
Fault diagnosis systems usually comprises two parts: a filtering part and a decision part, the latter typically based on threshold functions. In this paper, systematic ways to choose the threshold values are proposed. Two different test functions for the filtered signals are discussed and a method...
Application of Improved Wavelet Thresholding Function in Image Denoising Processing
Directory of Open Access Journals (Sweden)
Hong Qi Zhang
2014-07-01
Full Text Available Wavelet analysis is a time – frequency analysis method, time-frequency localization problems are well solved, this paper analyzes the basic principles of the wavelet transform and the relationship between the signal singularity Lipschitz exponent and the local maxima of the wavelet transform coefficients mold, the principles of wavelet transform in image denoising are analyzed, the disadvantages of traditional wavelet thresholding function are studied, wavelet threshold function, the discontinuity of hard threshold and constant deviation of soft threshold are improved, image is denoised through using the improved threshold function.
International Nuclear Information System (INIS)
Dattoli, G.; Torre, A.; Mancho, A.M.
2000-01-01
The theory of generalized Bessel functions has found significant applications in the analysis of radiation phenomena, associated with charges moving in magnetic devices. In this paper we exploit the monomiality principle to discuss the theory of two-variable Laguerre polynomials and introduce the associated Laguerre-Bessel functions. We study their properties (addition and multiplication theorems, generating function, recurrence relations and so on) and discuss the link with the ordinary case. The usefulness of the obtained results to treat problems relevant to the paraxial propagation of electromagnetic waves is also discussed.
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.
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
Multiple Meixner polynomials and non-Hermitian oscillator Hamiltonians
Ndayiragije, François; Van Assche, Walter
2013-01-01
Multiple Meixner polynomials are polynomials in one variable which satisfy orthogonality relations with respect to $r>1$ different negative binomial distributions (Pascal distributions). There are two kinds of multiple Meixner polynomials, depending on the selection of the parameters in the negative binomial distribution. We recall their definition and some formulas and give generating functions and explicit expressions for the coefficients in the nearest neighbor recurrence relation. Followi...
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)
Thresholding projection estimators in functional linear models
Cardot, Hervé; Johannes, Jan
2010-01-01
We consider the problem of estimating the regression function in functional linear regression models by proposing a new type of projection estimators which combine dimension reduction and thresholding. The introduction of a threshold rule allows to get consistency under broad assumptions as well as minimax rates of convergence under additional regularity hypotheses. We also consider the particular case of Sobolev spaces generated by the trigonometric basis which permits to get easily mean squ...
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)
Large degree asymptotics of generalized Bessel polynomials
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
Some Polynomials Associated with the r-Whitney Numbers
Indian Academy of Sciences (India)
26
Abstract. In the present article we study three families of polynomials associated with ... [29, 39] for their relations with the Bernoulli and generalized Bernoulli polynomials and ... generating functions in a similar way as in the classical cases.
Modeling DPOAE input/output function compression: comparisons with hearing thresholds.
Bhagat, Shaum P
2014-09-01
Basilar membrane input/output (I/O) functions in mammalian animal models are characterized by linear and compressed segments when measured near the location corresponding to the characteristic frequency. A method of studying basilar membrane compression indirectly in humans involves measuring distortion-product otoacoustic emission (DPOAE) I/O functions. Previous research has linked compression estimates from behavioral growth-of-masking functions to hearing thresholds. The aim of this study was to compare compression estimates from DPOAE I/O functions and hearing thresholds at 1 and 2 kHz. A prospective correlational research design was performed. The relationship between DPOAE I/O function compression estimates and hearing thresholds was evaluated with Pearson product-moment correlations. Normal-hearing adults (n = 16) aged 22-42 yr were recruited. DPOAE I/O functions (L₂ = 45-70 dB SPL) and two-interval forced-choice hearing thresholds were measured in normal-hearing adults. A three-segment linear regression model applied to DPOAE I/O functions supplied estimates of compression thresholds, defined as breakpoints between linear and compressed segments and the slopes of the compressed segments. Pearson product-moment correlations between DPOAE compression estimates and hearing thresholds were evaluated. A high correlation between DPOAE compression thresholds and hearing thresholds was observed at 2 kHz, but not at 1 kHz. Compression slopes also correlated highly with hearing thresholds only at 2 kHz. The derivation of cochlear compression estimates from DPOAE I/O functions provides a means to characterize basilar membrane mechanics in humans and elucidates the role of compression in tone detection in the 1-2 kHz frequency range. American Academy of Audiology.
Recurrence coefficients for discrete orthonormal polynomials and the Painlevé equations
International Nuclear Information System (INIS)
Clarkson, Peter A
2013-01-01
We investigate semi-classical generalizations of the Charlier and Meixner polynomials, which are discrete orthogonal polynomials that satisfy three-term recurrence relations. It is shown that the coefficients in these recurrence relations can be expressed in terms of Wronskians of modified Bessel functions and confluent hypergeometric functions, respectively for the generalized Charlier and generalized Meixner polynomials. These Wronskians arise in the description of special function solutions of the third and fifth Painlevé equations. (paper)
Energy Technology Data Exchange (ETDEWEB)
Deogracias, E.C.; Wood, J.L.; Wagner, E.C.; Kearfott, K.J
1999-02-11
The CEPXS/ONEDANT code package was used to produce a library of depth-dose profiles for monoenergetic electrons in various materials for energies ranging from 500 keV to 5 MeV in 10 keV increments. The various materials for which depth-dose functions were derived include: lithium fluoride (LiF), aluminium oxide (Al{sub 2}O{sub 3}), beryllium oxide (BeO), calcium sulfate (CaSO{sub 4}), calcium fluoride (CaF{sub 2}), lithium boron oxide (LiBO), soft tissue, lens of the eye, adiopose, muscle, skin, glass and water. All materials data sets were fit to five polynomials, each covering a different range of electron energies, using a least squares method. The resultant three dimensional, fifth-order polynomials give the dose as a function of depth and energy for the monoenergetic electrons in each material. The polynomials can be used to describe an energy spectrum by summing the doses at a given depth for each energy, weighted by the spectral intensity for that energy. An application of the polynomial is demonstrated by explaining the energy dependence of thermoluminescent detectors (TLDs) and illustrating the relationship between TLD signal and actual shallow dose due to beta particles.
International Nuclear Information System (INIS)
Yuste, Santos Bravo; Abad, Enrique
2011-01-01
We present an iterative method to obtain approximations to Bessel functions of the first kind J p (x) (p > -1) via the repeated application of an integral operator to an initial seed function f 0 (x). The class of seed functions f 0 (x) leading to sets of increasingly accurate approximations f n (x) is considerably large and includes any polynomial. When the operator is applied once to a polynomial of degree s, it yields a polynomial of degree s + 2, and so the iteration of this operator generates sets of increasingly better polynomial approximations of increasing degree. We focus on the set of polynomial approximations generated from the seed function f 0 (x) = 1. This set of polynomials is useful not only for the computation of J p (x) but also from a physical point of view, as it describes the long-time decay modes of certain fractional diffusion and diffusion-wave problems.
Lagrange polynomial interpolation method applied in the calculation of the J({xi},{beta}) function
Energy Technology Data Exchange (ETDEWEB)
Fraga, Vinicius Munhoz; Palma, Daniel Artur Pinheiro [Centro Federal de Educacao Tecnologica de Quimica de Nilopolis, RJ (Brazil)]. E-mails: munhoz.vf@gmail.com; dpalma@cefeteq.br; Martinez, Aquilino Senra [Universidade Federal do Rio de Janeiro (UFRJ), RJ (Brazil). Coordenacao dos Programas de Pos-graduacao de Engenharia (COPPE) (COPPE). Programa de Engenharia Nuclear]. E-mail: aquilino@lmp.ufrj.br
2008-07-01
The explicit dependence of the Doppler broadening function creates difficulties in the obtaining an analytical expression for J function . The objective of this paper is to present a method for the quick and accurate calculation of J function based on the recent advances in the calculation of the Doppler broadening function and on a systematic analysis of its integrand. The methodology proposed, of a semi-analytical nature, uses the Lagrange polynomial interpolation method and the Frobenius formulation in the calculation of Doppler broadening function . The results have proven satisfactory from the standpoint of accuracy and processing time. (author)
Lagrange polynomial interpolation method applied in the calculation of the J(ξ,β) function
International Nuclear Information System (INIS)
Fraga, Vinicius Munhoz; Palma, Daniel Artur Pinheiro; Martinez, Aquilino Senra
2008-01-01
The explicit dependence of the Doppler broadening function creates difficulties in the obtaining an analytical expression for J function . The objective of this paper is to present a method for the quick and accurate calculation of J function based on the recent advances in the calculation of the Doppler broadening function and on a systematic analysis of its integrand. The methodology proposed, of a semi-analytical nature, uses the Lagrange polynomial interpolation method and the Frobenius formulation in the calculation of Doppler broadening function . The results have proven satisfactory from the standpoint of accuracy and processing time. (author)
Uncertainty Analysis via Failure Domain Characterization: Polynomial Requirement Functions
Crespo, Luis G.; Munoz, Cesar A.; Narkawicz, Anthony J.; Kenny, Sean P.; Giesy, Daniel P.
2011-01-01
This paper proposes an uncertainty analysis framework based on the characterization of the uncertain parameter space. This characterization enables the identification of worst-case uncertainty combinations and the approximation of the failure and safe domains with a high level of accuracy. Because these approximations are comprised of subsets of readily computable probability, they enable the calculation of arbitrarily tight upper and lower bounds to the failure probability. A Bernstein expansion approach is used to size hyper-rectangular subsets while a sum of squares programming approach is used to size quasi-ellipsoidal subsets. These methods are applicable to requirement functions whose functional dependency on the uncertainty is a known polynomial. Some of the most prominent features of the methodology are the substantial desensitization of the calculations from the uncertainty model assumed (i.e., the probability distribution describing the uncertainty) as well as the accommodation for changes in such a model with a practically insignificant amount of computational effort.
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
PLOTNFIT.4TH, Data Plotting and Curve Fitting by Polynomials
International Nuclear Information System (INIS)
Schiffgens, J.O.
1990-01-01
1 - Description of program or function: PLOTnFIT is used for plotting and analyzing data by fitting nth degree polynomials of basis functions to the data interactively and printing graphs of the data and the polynomial functions. It can be used to generate linear, semi-log, and log-log graphs and can automatically scale the coordinate axes to suit the data. Multiple data sets may be plotted on a single graph. An auxiliary program, READ1ST, is included which produces an on-line summary of the information contained in the PLOTnFIT reference report. 2 - Method of solution: PLOTnFIT uses the least squares method to calculate the coefficients of nth-degree (up to 10. degree) polynomials of 11 selected basis functions such that each polynomial fits the data in a least squares sense. The procedure incorporated in the code uses a linear combination of orthogonal polynomials to avoid 'i11-conditioning' and to perform the curve fitting task with single-precision arithmetic. 3 - Restrictions on the complexity of the problem - Maxima of: 225 data points per job (or graph) including all data sets 8 data sets (or tasks) per job (or graph)
Discrimination Power of Polynomial-Based Descriptors for Graphs by Using Functional Matrices.
Dehmer, Matthias; Emmert-Streib, Frank; Shi, Yongtang; Stefu, Monica; Tripathi, Shailesh
2015-01-01
In this paper, we study the discrimination power of graph measures that are based on graph-theoretical matrices. The paper generalizes the work of [M. Dehmer, M. Moosbrugger. Y. Shi, Encoding structural information uniquely with polynomial-based descriptors by employing the Randić matrix, Applied Mathematics and Computation, 268(2015), 164-168]. We demonstrate that by using the new functional matrix approach, exhaustively generated graphs can be discriminated more uniquely than shown in the mentioned previous work.
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 ...
Optimization Problems on Threshold Graphs
Directory of Open Access Journals (Sweden)
Elena Nechita
2010-06-01
Full Text Available During the last three decades, different types of decompositions have been processed in the field of graph theory. Among these we mention: decompositions based on the additivity of some characteristics of the graph, decompositions where the adjacency law between the subsets of the partition is known, decompositions where the subgraph induced by every subset of the partition must have predeterminate properties, as well as combinations of such decompositions. In this paper we characterize threshold graphs using the weakly decomposition, determine: density and stability number, Wiener index and Wiener polynomial for threshold graphs.
Välimäki, Vesa; Pekonen, Jussi; Nam, Juhan
2012-01-01
Digital subtractive synthesis is a popular music synthesis method, which requires oscillators that are aliasing-free in a perceptual sense. It is a research challenge to find computationally efficient waveform generation algorithms that produce similar-sounding signals to analog music synthesizers but which are free from audible aliasing. A technique for approximately bandlimited waveform generation is considered that is based on a polynomial correction function, which is defined as the difference of a non-bandlimited step function and a polynomial approximation of the ideal bandlimited step function. It is shown that the ideal bandlimited step function is equivalent to the sine integral, and that integrated polynomial interpolation methods can successfully approximate it. Integrated Lagrange interpolation and B-spline basis functions are considered for polynomial approximation. The polynomial correction function can be added onto samples around each discontinuity in a non-bandlimited waveform to suppress aliasing. Comparison against previously known methods shows that the proposed technique yields the best tradeoff between computational cost and sound quality. The superior method amongst those considered in this study is the integrated third-order B-spline correction function, which offers perceptually aliasing-free sawtooth emulation up to the fundamental frequency of 7.8 kHz at the sample rate of 44.1 kHz. © 2012 Acoustical Society of America.
Quantum Hurwitz numbers and Macdonald polynomials
Harnad, J.
2016-11-01
Parametric families in the center Z(C[Sn]) of the group algebra of the symmetric group are obtained by identifying the indeterminates in the generating function for Macdonald polynomials as commuting Jucys-Murphy elements. Their eigenvalues provide coefficients in the double Schur function expansion of 2D Toda τ-functions of hypergeometric type. Expressing these in the basis of products of power sum symmetric functions, the coefficients may be interpreted geometrically as parametric families of quantum Hurwitz numbers, enumerating weighted branched coverings of the Riemann sphere. Combinatorially, they give quantum weighted sums over paths in the Cayley graph of Sn generated by transpositions. Dual pairs of bases for the algebra of symmetric functions with respect to the scalar product in which the Macdonald polynomials are orthogonal provide both the geometrical and combinatorial significance of these quantum weighted enumerative invariants.
Multilevel weighted least squares polynomial approximation
Haji-Ali, Abdul-Lateef
2017-06-30
Weighted least squares polynomial approximation uses random samples to determine projections of functions onto spaces of polynomials. It has been shown that, using an optimal distribution of sample locations, the number of samples required to achieve quasi-optimal approximation in a given polynomial subspace scales, up to a logarithmic factor, linearly in the dimension of this space. However, in many applications, the computation of samples includes a numerical discretization error. Thus, obtaining polynomial approximations with a single level method can become prohibitively expensive, as it requires a sufficiently large number of samples, each computed with a sufficiently small discretization error. As a solution to this problem, we propose a multilevel method that utilizes samples computed with different accuracies and is able to match the accuracy of single-level approximations with reduced computational cost. We derive complexity bounds under certain assumptions about polynomial approximability and sample work. Furthermore, we propose an adaptive algorithm for situations where such assumptions cannot be verified a priori. Finally, we provide an efficient algorithm for the sampling from optimal distributions and an analysis of computationally favorable alternative distributions. Numerical experiments underscore the practical applicability of our method.
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.
Irreducible multivariate polynomials obtained from polynomials in ...
Indian Academy of Sciences (India)
Hall, 1409 W. Green Street, Urbana, IL 61801, USA. E-mail: Nicolae. ... Theorem A. If we write an irreducible polynomial f ∈ K[X] as a sum of polynomials a0,..., an ..... This shows us that deg ai = (n − i) deg f2 for each i = 0,..., n, so min k>0.
Bernoulli numbers and polynomials from a more general point of view
International Nuclear Information System (INIS)
Dattoli, G.; Cesarano, C.; Lorenzutta, S.
2000-01-01
In this work it is applied the method of generating function, to introduce new forms of Bernoulli numbers and polynomials, which are exploited to derive further classes of partial sums involving generalized many index many variable polynomials. Analogous considerations are developed for the Euler numbers and polynomials [it
Branched polynomial covering maps
DEFF Research Database (Denmark)
Hansen, Vagn Lundsgaard
2002-01-01
A Weierstrass polynomial with multiple roots in certain points leads to a branched covering map. With this as the guiding example, we formally define and study the notion of a branched polynomial covering map. We shall prove that many finite covering maps are polynomial outside a discrete branch ...... set. Particular studies are made of branched polynomial covering maps arising from Riemann surfaces and from knots in the 3-sphere. (C) 2001 Elsevier Science B.V. All rights reserved.......A Weierstrass polynomial with multiple roots in certain points leads to a branched covering map. With this as the guiding example, we formally define and study the notion of a branched polynomial covering map. We shall prove that many finite covering maps are polynomial outside a discrete branch...
Complex Polynomial Vector Fields
DEFF Research Database (Denmark)
Dias, Kealey
vector fields. Since the class of complex polynomial vector fields in the plane is natural to consider, it is remarkable that its study has only begun very recently. There are numerous fundamental questions that are still open, both in the general classification of these vector fields, the decomposition...... of parameter spaces into structurally stable domains, and a description of the bifurcations. For this reason, the talk will focus on these questions for complex polynomial vector fields.......The two branches of dynamical systems, continuous and discrete, correspond to the study of differential equations (vector fields) and iteration of mappings respectively. In holomorphic dynamics, the systems studied are restricted to those described by holomorphic (complex analytic) functions...
Interpretation of stream programs: characterizing type 2 polynomial time complexity
Férée , Hugo; Hainry , Emmanuel; Hoyrup , Mathieu; Péchoux , Romain
2010-01-01
International audience; We study polynomial time complexity of type 2 functionals. For that purpose, we introduce a first order functional stream language. We give criteria, named well-founded, on such programs relying on second order interpretation that characterize two variants of type 2 polynomial complexity including the Basic Feasible Functions (BFF). These charac- terizations provide a new insight on the complexity of stream programs. Finally, we adapt these results to functions over th...
Non-Abelian integrable hierarchies: matrix biorthogonal polynomials and perturbations
Ariznabarreta, Gerardo; García-Ardila, Juan C.; Mañas, Manuel; Marcellán, Francisco
2018-05-01
In this paper, Geronimus–Uvarov perturbations for matrix orthogonal polynomials on the real line are studied and then applied to the analysis of non-Abelian integrable hierarchies. The orthogonality is understood in full generality, i.e. in terms of a nondegenerate continuous sesquilinear form, determined by a quasidefinite matrix of bivariate generalized functions with a well-defined support. We derive Christoffel-type formulas that give the perturbed matrix biorthogonal polynomials and their norms in terms of the original ones. The keystone for this finding is the Gauss–Borel factorization of the Gram matrix. Geronimus–Uvarov transformations are considered in the context of the 2D non-Abelian Toda lattice and noncommutative KP hierarchies. The interplay between transformations and integrable flows is discussed. Miwa shifts, τ-ratio matrix functions and Sato formulas are given. Bilinear identities, involving Geronimus–Uvarov transformations, first for the Baker functions, then secondly for the biorthogonal polynomials and its second kind functions, and finally for the τ-ratio matrix functions, are found.
Two polynomial representations of experimental design
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.
Cosmographic analysis with Chebyshev polynomials
Capozziello, Salvatore; D'Agostino, Rocco; Luongo, Orlando
2018-05-01
The limits of standard cosmography are here revised addressing the problem of error propagation during statistical analyses. To do so, we propose the use of Chebyshev polynomials to parametrize cosmic distances. In particular, we demonstrate that building up rational Chebyshev polynomials significantly reduces error propagations with respect to standard Taylor series. This technique provides unbiased estimations of the cosmographic parameters and performs significatively better than previous numerical approximations. To figure this out, we compare rational Chebyshev polynomials with Padé series. In addition, we theoretically evaluate the convergence radius of (1,1) Chebyshev rational polynomial and we compare it with the convergence radii of Taylor and Padé approximations. We thus focus on regions in which convergence of Chebyshev rational functions is better than standard approaches. With this recipe, as high-redshift data are employed, rational Chebyshev polynomials remain highly stable and enable one to derive highly accurate analytical approximations of Hubble's rate in terms of the cosmographic series. Finally, we check our theoretical predictions by setting bounds on cosmographic parameters through Monte Carlo integration techniques, based on the Metropolis-Hastings algorithm. We apply our technique to high-redshift cosmic data, using the Joint Light-curve Analysis supernovae sample and the most recent versions of Hubble parameter and baryon acoustic oscillation measurements. We find that cosmography with Taylor series fails to be predictive with the aforementioned data sets, while turns out to be much more stable using the Chebyshev approach.
Hamed Kharrati; Sohrab Khanmohammadi; Witold Pedrycz; Ghasem Alizadeh
2012-01-01
This study presents an improved model and controller for nonlinear plants using polynomial fuzzy model-based (FMB) systems. To minimize mismatch between the polynomial fuzzy model and nonlinear plant, the suitable parameters of membership functions are determined in a systematic way. Defining an appropriate fitness function and utilizing Taylor series expansion, a genetic algorithm (GA) is used to form the shape of membership functions in polynomial forms, which are afterwards used in fuzzy m...
Polynomial Asymptotes of the Second Kind
Dobbs, David E.
2011-01-01
This note uses the analytic notion of asymptotic functions to study when a function is asymptotic to a polynomial function. Along with associated existence and uniqueness results, this kind of asymptotic behaviour is related to the type of asymptote that was recently defined in a more geometric way. Applications are given to rational functions and…
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.
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
Directory of Open Access Journals (Sweden)
Hamed Kharrati
2012-01-01
Full Text Available This study presents an improved model and controller for nonlinear plants using polynomial fuzzy model-based (FMB systems. To minimize mismatch between the polynomial fuzzy model and nonlinear plant, the suitable parameters of membership functions are determined in a systematic way. Defining an appropriate fitness function and utilizing Taylor series expansion, a genetic algorithm (GA is used to form the shape of membership functions in polynomial forms, which are afterwards used in fuzzy modeling. To validate the model, a controller based on proposed polynomial fuzzy systems is designed and then applied to both original nonlinear plant and fuzzy model for comparison. Additionally, stability analysis for the proposed polynomial FMB control system is investigated employing Lyapunov theory and a sum of squares (SOS approach. Moreover, the form of the membership functions is considered in stability analysis. The SOS-based stability conditions are attained using SOSTOOLS. Simulation results are also given to demonstrate the effectiveness of the proposed method.
Review and Analysis of Cryptographic Schemes Implementing Threshold Signature
Directory of Open Access Journals (Sweden)
Anastasiya Victorovna Beresneva
2015-03-01
Full Text Available This work is devoted to the study of threshold signature schemes. The systematization of the threshold signature schemes was done, cryptographic constructions based on interpolation Lagrange polynomial, ellipt ic curves and bilinear pairings were investigated. Different methods of generation and verification of threshold signatures were explored, e.g. used in a mobile agents, Internet banking and e-currency. The significance of the work is determined by the reduction of the level of counterfeit electronic documents, signed by certain group of users.
O(N) symmetries, sum rules for generalized Hermite polynomials and squeezed states
International Nuclear Information System (INIS)
Daboul, Jamil; Mizrahi, Salomon S
2005-01-01
Quantum optics has been dealing with coherent states, squeezed states and many other non-classical states. The associated mathematical framework makes use of special functions as Hermite polynomials, Laguerre polynomials and others. In this connection we here present some formal results that follow directly from the group O(N) of complex transformations. Motivated by the squeezed states structure, we introduce the generalized Hermite polynomials (GHP), which include as particular cases, the Hermite polynomials as well as the heat polynomials. Using generalized raising operators, we derive new sum rules for the GHP, which are covariant under O(N) transformations. The GHP and the associated sum rules become useful for evaluating Wigner functions in a straightforward manner. As a byproduct, we use one of these sum rules, on the operator level, to obtain raising and lowering operators for the Laguerre polynomials and show that they generate an sl(2, R) ≅ su(1, 1) algebra
Linear operator pencils on Lie algebras and Laurent biorthogonal polynomials
International Nuclear Information System (INIS)
Gruenbaum, F A; Vinet, Luc; Zhedanov, Alexei
2004-01-01
We study operator pencils on generators of the Lie algebras sl 2 and the oscillator algebra. These pencils are linear in a spectral parameter λ. The corresponding generalized eigenvalue problem gives rise to some sets of orthogonal polynomials and Laurent biorthogonal polynomials (LBP) expressed in terms of the Gauss 2 F 1 and degenerate 1 F 1 hypergeometric functions. For special choices of the parameters of the pencils, we identify the resulting polynomials with the Hendriksen-van Rossum LBP which are widely believed to be the biorthogonal analogues of the classical orthogonal polynomials. This places these examples under the umbrella of the generalized bispectral problem which is considered here. Other (non-bispectral) cases give rise to some 'nonclassical' orthogonal polynomials including Tricomi-Carlitz and random-walk polynomials. An application to solutions of relativistic Toda chain is considered
Recurrence approach and higher order polynomial algebras for superintegrable monopole systems
Hoque, Md Fazlul; Marquette, Ian; Zhang, Yao-Zhong
2018-05-01
We revisit the MIC-harmonic oscillator in flat space with monopole interaction and derive the polynomial algebra satisfied by the integrals of motion and its energy spectrum using the ad hoc recurrence approach. We introduce a superintegrable monopole system in a generalized Taub-Newman-Unti-Tamburino (NUT) space. The Schrödinger equation of this model is solved in spherical coordinates in the framework of Stäckel transformation. It is shown that wave functions of the quantum system can be expressed in terms of the product of Laguerre and Jacobi polynomials. We construct ladder and shift operators based on the corresponding wave functions and obtain the recurrence formulas. By applying these recurrence relations, we construct higher order algebraically independent integrals of motion. We show that the integrals form a polynomial algebra. We construct the structure functions of the polynomial algebra and obtain the degenerate energy spectra of the model.
On Linear Combinations of Two Orthogonal Polynomial Sequences on the Unit Circle
Directory of Open Access Journals (Sweden)
Suárez C
2010-01-01
Full Text Available Let be a monic orthogonal polynomial sequence on the unit circle. We define recursively a new sequence of polynomials by the following linear combination: , , . In this paper, we give necessary and sufficient conditions in order to make be an orthogonal polynomial sequence too. Moreover, we obtain an explicit representation for the Verblunsky coefficients and in terms of and . Finally, we show the relation between their corresponding Carathéodory functions and their associated linear functionals.
Considering a non-polynomial basis for local kernel regression problem
Silalahi, Divo Dharma; Midi, Habshah
2017-01-01
A common used as solution for local kernel nonparametric regression problem is given using polynomial regression. In this study, we demonstrated the estimator and properties using maximum likelihood estimator for a non-polynomial basis such B-spline to replacing the polynomial basis. This estimator allows for flexibility in the selection of a bandwidth and a knot. The best estimator was selected by finding an optimal bandwidth and knot through minimizing the famous generalized validation function.
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....
On the number of polynomial solutions of Bernoulli and Abel polynomial differential equations
Cima, A.; Gasull, A.; Mañosas, F.
2017-12-01
In this paper we determine the maximum number of polynomial solutions of Bernoulli differential equations and of some integrable polynomial Abel differential equations. As far as we know, the tools used to prove our results have not been utilized before for studying this type of questions. We show that the addressed problems can be reduced to know the number of polynomial solutions of a related polynomial equation of arbitrary degree. Then we approach to these equations either applying several tools developed to study extended Fermat problems for polynomial equations, or reducing the question to the computation of the genus of some associated planar algebraic curves.
On generalized Fibonacci and Lucas polynomials
Energy Technology Data Exchange (ETDEWEB)
Nalli, Ayse [Department of Mathematics, Faculty of Sciences, Selcuk University, 42075 Campus-Konya (Turkey)], E-mail: aysenalli@yahoo.com; Haukkanen, Pentti [Department of Mathematics, Statistics and Philosophy, 33014 University of Tampere (Finland)], E-mail: mapehau@uta.fi
2009-12-15
Let h(x) be a polynomial with real coefficients. We introduce h(x)-Fibonacci polynomials that generalize both Catalan's Fibonacci polynomials and Byrd's Fibonacci polynomials and also the k-Fibonacci numbers, and we provide properties for these h(x)-Fibonacci polynomials. We also introduce h(x)-Lucas polynomials that generalize the Lucas polynomials and present properties of these polynomials. In the last section we introduce the matrix Q{sub h}(x) that generalizes the Q-matrix whose powers generate the Fibonacci numbers.
Bai , Shi; Bouvier , Cyril; Kruppa , Alexander; Zimmermann , Paul
2016-01-01
International audience; The general number field sieve (GNFS) is the most efficient algo-rithm known for factoring large integers. It consists of several stages, the first one being polynomial selection. The quality of the selected polynomials can be modelled in terms of size and root properties. We propose a new kind of polynomials for GNFS: with a new degree of freedom, we further improve the size property. We demonstrate the efficiency of our algorithm by exhibiting a better polynomial tha...
Families of superintegrable Hamiltonians constructed from exceptional polynomials
International Nuclear Information System (INIS)
Post, Sarah; Tsujimoto, Satoshi; Vinet, Luc
2012-01-01
We introduce a family of exactly-solvable two-dimensional Hamiltonians whose wave functions are given in terms of Laguerre and exceptional Jacobi polynomials. The Hamiltonians contain purely quantum terms which vanish in the classical limit leaving only a previously known family of superintegrable systems. Additional, higher-order integrals of motion are constructed from ladder operators for the considered orthogonal polynomials proving the quantum system to be superintegrable. (paper)
Zea Vera, Alonso; Aungaroon, Gewalin; Horn, Paul S; Byars, Anna W; Greiner, Hansel M; Tenney, Jeffrey R; Arthur, Todd M; Crone, Nathan E; Holland, Katherine D; Mangano, Francesco T; Arya, Ravindra
2017-10-01
To examine current thresholds and their determinants for language and motor mapping with extra-operative electrical cortical stimulation (ECS). ECS electrocorticograph recordings were reviewed to determine functional thresholds. Predictors of functional thresholds were found with multivariable analyses. In 122 patients (age 11.9±5.4years), average minimum, frontal, and temporal language thresholds were 7.4 (± 3.0), 7.8 (± 3.0), and 7.4 (± 3.1) mA respectively. Average minimum, face, upper and lower extremity motor thresholds were 5.4 (± 2.8), 6.1 (± 2.8), 4.9 (± 2.3), and 5.3 (± 3.3) mA respectively. Functional and after-discharge (AD)/seizure thresholds were significantly related. Minimum, frontal, and temporal language thresholds were higher than AD thresholds at all ages. Minimum motor threshold was higher than minimum AD threshold up to 8.0years of age, face motor threshold was higher than frontal AD threshold up to 11.8years age, and lower subsequently. UE motor thresholds remained below frontal AD thresholds throughout the age range. Functional thresholds are frequently above AD thresholds in younger children. These findings raise concerns about safety and neurophysiologic validity of ECS mapping. Functional and AD/seizure thresholds relationships suggest individual differences in cortical excitability which cannot be explained by clinical variables. Copyright © 2017 International Federation of Clinical Neurophysiology. Published by Elsevier B.V. All rights reserved.
Global sensitivity analysis using sparse grid interpolation and polynomial chaos
International Nuclear Information System (INIS)
Buzzard, Gregery T.
2012-01-01
Sparse grid interpolation is widely used to provide good approximations to smooth functions in high dimensions based on relatively few function evaluations. By using an efficient conversion from the interpolating polynomial provided by evaluations on a sparse grid to a representation in terms of orthogonal polynomials (gPC representation), we show how to use these relatively few function evaluations to estimate several types of sensitivity coefficients and to provide estimates on local minima and maxima. First, we provide a good estimate of the variance-based sensitivity coefficients of Sobol' (1990) [1] and then use the gradient of the gPC representation to give good approximations to the derivative-based sensitivity coefficients described by Kucherenko and Sobol' (2009) [2]. Finally, we use the package HOM4PS-2.0 given in Lee et al. (2008) [3] to determine the critical points of the interpolating polynomial and use these to determine the local minima and maxima of this polynomial. - Highlights: ► Efficient estimation of variance-based sensitivity coefficients. ► Efficient estimation of derivative-based sensitivity coefficients. ► Use of homotopy methods for approximation of local maxima and minima.
Generalized Freud's equation and level densities with polynomial potential
Boobna, Akshat; Ghosh, Saugata
2013-08-01
We study orthogonal polynomials with weight $\\exp[-NV(x)]$, where $V(x)=\\sum_{k=1}^{d}a_{2k}x^{2k}/2k$ is a polynomial of order 2d. We derive the generalised Freud's equations for $d=3$, 4 and 5 and using this obtain $R_{\\mu}=h_{\\mu}/h_{\\mu -1}$, where $h_{\\mu}$ is the normalization constant for the corresponding orthogonal polynomials. Moments of the density functions, expressed in terms of $R_{\\mu}$, are obtained using Freud's equation and using this, explicit results of level densities as $N\\rightarrow\\infty$ are derived.
H∞ Control of Polynomial Fuzzy Systems: A Sum of Squares Approach
Directory of Open Access Journals (Sweden)
Bomo W. Sanjaya
2014-07-01
Full Text Available This paper proposes the control design ofa nonlinear polynomial fuzzy system with H∞ performance objective using a sum of squares (SOS approach. Fuzzy model and controller are represented by a polynomial fuzzy model and controller. The design condition is obtained by using polynomial Lyapunov functions that not only guarantee stability but also satisfy the H∞ performance objective. The design condition is represented in terms of an SOS that can be numerically solved via the SOSTOOLS. A simulation study is presented to show the effectiveness of the SOS-based H∞ control designfor nonlinear polynomial fuzzy systems.
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.
International Nuclear Information System (INIS)
Ramazanov, A.-R K
2005-01-01
Necessary and sufficient conditions for the best polynomial approximation with an arbitrary and, generally speaking, unbounded sign-sensitive weight to a continuous function are obtained; the components of the weight can also take infinite values, therefore the conditions obtained cover, in particular, approximation with interpolation at fixed points and one-sided approximation; in the case of the weight with components equal to 1 one arrives at Chebyshev's classical alternation theorem.
Mandal, Sudhansu S.; Mukherjee, Sutirtha; Ray, Koushik
2018-03-01
A method for determining the ground state of a planar interacting many-electron system in a magnetic field perpendicular to the plane is described. The ground state wave-function is expressed as a linear combination of a set of basis functions. Given only the flux and the number of electrons describing an incompressible state, we use the combinatorics of partitioning the flux among the electrons to derive the basis wave-functions as linear combinations of Schur polynomials. The procedure ensures that the basis wave-functions form representations of the angular momentum algebra. We exemplify the method by deriving the basis functions for the 5/2 quantum Hall state with a few particles. We find that one of the basis functions is precisely the Moore-Read Pfaffian wave function.
International Nuclear Information System (INIS)
Konakli, Katerina; Sudret, Bruno
2016-01-01
The growing need for uncertainty analysis of complex computational models has led to an expanding use of meta-models across engineering and sciences. The efficiency of meta-modeling techniques relies on their ability to provide statistically-equivalent analytical representations based on relatively few evaluations of the original model. Polynomial chaos expansions (PCE) have proven a powerful tool for developing meta-models in a wide range of applications; the key idea thereof is to expand the model response onto a basis made of multivariate polynomials obtained as tensor products of appropriate univariate polynomials. The classical PCE approach nevertheless faces the “curse of dimensionality”, namely the exponential increase of the basis size with increasing input dimension. To address this limitation, the sparse PCE technique has been proposed, in which the expansion is carried out on only a few relevant basis terms that are automatically selected by a suitable algorithm. An alternative for developing meta-models with polynomial functions in high-dimensional problems is offered by the newly emerged low-rank approximations (LRA) approach. By exploiting the tensor–product structure of the multivariate basis, LRA can provide polynomial representations in highly compressed formats. Through extensive numerical investigations, we herein first shed light on issues relating to the construction of canonical LRA with a particular greedy algorithm involving a sequential updating of the polynomial coefficients along separate dimensions. Specifically, we examine the selection of optimal rank, stopping criteria in the updating of the polynomial coefficients and error estimation. In the sequel, we confront canonical LRA to sparse PCE in structural-mechanics and heat-conduction applications based on finite-element solutions. Canonical LRA exhibit smaller errors than sparse PCE in cases when the number of available model evaluations is small with respect to the input
Energy Technology Data Exchange (ETDEWEB)
Konakli, Katerina, E-mail: konakli@ibk.baug.ethz.ch; Sudret, Bruno
2016-09-15
The growing need for uncertainty analysis of complex computational models has led to an expanding use of meta-models across engineering and sciences. The efficiency of meta-modeling techniques relies on their ability to provide statistically-equivalent analytical representations based on relatively few evaluations of the original model. Polynomial chaos expansions (PCE) have proven a powerful tool for developing meta-models in a wide range of applications; the key idea thereof is to expand the model response onto a basis made of multivariate polynomials obtained as tensor products of appropriate univariate polynomials. The classical PCE approach nevertheless faces the “curse of dimensionality”, namely the exponential increase of the basis size with increasing input dimension. To address this limitation, the sparse PCE technique has been proposed, in which the expansion is carried out on only a few relevant basis terms that are automatically selected by a suitable algorithm. An alternative for developing meta-models with polynomial functions in high-dimensional problems is offered by the newly emerged low-rank approximations (LRA) approach. By exploiting the tensor–product structure of the multivariate basis, LRA can provide polynomial representations in highly compressed formats. Through extensive numerical investigations, we herein first shed light on issues relating to the construction of canonical LRA with a particular greedy algorithm involving a sequential updating of the polynomial coefficients along separate dimensions. Specifically, we examine the selection of optimal rank, stopping criteria in the updating of the polynomial coefficients and error estimation. In the sequel, we confront canonical LRA to sparse PCE in structural-mechanics and heat-conduction applications based on finite-element solutions. Canonical LRA exhibit smaller errors than sparse PCE in cases when the number of available model evaluations is small with respect to the input
Fast beampattern evaluation by polynomial rooting
Häcker, P.; Uhlich, S.; Yang, B.
2011-07-01
Current automotive radar systems measure the distance, the relative velocity and the direction of objects in their environment. This information enables the car to support the driver. The direction estimation capabilities of a sensor array depend on its beampattern. To find the array configuration leading to the best angle estimation by a global optimization algorithm, a huge amount of beampatterns have to be calculated to detect their maxima. In this paper, a novel algorithm is proposed to find all maxima of an array's beampattern fast and reliably, leading to accelerated array optimizations. The algorithm works for arrays having the sensors on a uniformly spaced grid. We use a general version of the gcd (greatest common divisor) function in order to write the problem as a polynomial. We differentiate and root the polynomial to get the extrema of the beampattern. In addition, we show a method to reduce the computational burden even more by decreasing the order of the polynomial.
Astroidal geometry of hypocycloids and the Hessian topology of hyperbolic polynomials
International Nuclear Information System (INIS)
Arnol'd, Vladimir I
2001-01-01
The Hessian topology has just begun to be developed (in connection with the study of parabolic curves on smooth surfaces in Euclidean or projective space), in contrast to the symplectic and contact topologies related to it. For instance, it is not known how many (compact) parabolic curves can belong to the graph of a polynomial of a given (even of the fourth) degree in two variables or to a smooth algebraic surface of a given degree. The astroid is a hypocycloid with four cusp points. A hyperbolic polynomial is a homogeneous polynomial whose second differential has the signature (+,-) at any non-zero point. Hyperbolic polynomials and functions are connected with Morse theory and Sturm theory and with hypocycloids via caustics (and wave fronts) of periodic functions. The astroid is the caustic of the cosine of a double angle. The caustic of any periodic function has at least four cusp points, and if there are four of them, as is the case for the astroid, then these points form a parallelogram. The theory developed in this paper, based on the study of envelopes and inequalities between derivatives of smooth functions, proves that hyperbolic polynomials of degree four form a connected set and those of degree six form a disconnected set. These topological generalizations of the Sturm and Hurwitz theorems about the zeros of Fourier series give algebraic-geometric results on caustics and wave fronts as well and also establish relationships between these results and the Morse theory of anti-Rolle functions (whose zeros alternate with those of their derivatives)
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...
Roots of the Chromatic Polynomial
DEFF Research Database (Denmark)
Perrett, Thomas
The chromatic polynomial of a graph G is a univariate polynomial whose evaluation at any positive integer q enumerates the proper q-colourings of G. It was introduced in connection with the famous four colour theorem but has recently found other applications in the field of statistical physics...... extend Thomassen’s technique to the Tutte polynomial and as a consequence, deduce a density result for roots of the Tutte polynomial. This partially answers a conjecture of Jackson and Sokal. Finally, we refocus our attention on the chromatic polynomial and investigate the density of chromatic roots...
Caglayan, Günhan
2014-01-01
This study investigates prospective secondary mathematics teachers' visual representations of polynomial and rational inequalities, and graphs of exponential and logarithmic functions with GeoGebra Dynamic Software. Five prospective teachers in a university in the United States participated in this research study, which was situated within a…
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)
International Nuclear Information System (INIS)
Lobanov, Yu.Yu.; Shidkov, E.P.
1987-01-01
The method for numerical evaluation of path integrals in Eucledean quantum mechanics without lattice discretization is elaborated. The method is based on the representation of these integrals in the form of functional integrals with respect to the conditional Wiener measure and on the use of the derived approximate exact on a class of polynomial functionals of a given degree. By the computations of non-perturbative characteristics, concerned the topological structure of vacuum, the advantages of this method versus lattice Monte-Carlo calculations are demonstrated
Ilić, Aleksandar D.; Pavlović, Vlastimir D.
2011-01-01
A new original formulation of all pole low-pass filter functions is proposed in this article. The starting point in solving the approximation problem is a direct application of the Christoffel-Darboux formula for the set of orthogonal polynomials, including Gegenbauer orthogonal polynomials in the finite interval [-1, +1] with the application of a weighting function with a single free parameter. A general solution for the filter functions is obtained in a compact explicit form, which is shown to enable generation of the Gegenbauer filter functions in a simple way by choosing the value of the free parameter. Moreover, the proposed solution with the same criterion of approximation could be used to generate Legendre and Chebyshev filter functions of the first and second kind as well. The examples of proposed filter functions of even (10th) and odd (11th) order are illustrated. The approximation is shown to yield a good compromise solution with respect to the filter frequency characteristics (magnitude as well as phase characteristics). The influence of tolerance of the filter critical component (inductor) on the proposed magnitude and group delay characteristics of a resistively terminated LC lossless ladder filter is analysed as well. The proposed filter functions are superior in terms of the excellent magnitude characteristic, which approximates an ideal filter almost perfectly over the entire pass-band range and exhibits the summed sensitivity function better than that of a Butterworth filter. In the article, we present the filter function solution that exhibits optimum amplitude as well as optimum group delay characteristics that are of crucial importance for implementation of digital processing as well as RF analogue parts of communication networks. Derivation of the other band range filter functions, which could be realised either by continuous or digital filters, is also generally possible with the procedure proposed in this article.
H∞ Control of Polynomial Fuzzy Systems: A Sum of Squares Approach
Bomo W. Sanjaya; Bambang Riyanto Trilaksono; Arief Syaichu-Rohman
2014-01-01
This paper proposes the control design ofa nonlinear polynomial fuzzy system with H∞ performance objective using a sum of squares (SOS) approach. Fuzzy model and controller are represented by a polynomial fuzzy model and controller. The design condition is obtained by using polynomial Lyapunov functions that not only guarantee stability but also satisfy the H∞ performance objective. The design condition is represented in terms of an SOS that can be numerically solved via the SOSTOOLS. A simul...
A Formally Verified Conflict Detection Algorithm for Polynomial Trajectories
Narkawicz, Anthony; Munoz, Cesar
2015-01-01
In air traffic management, conflict detection algorithms are used to determine whether or not aircraft are predicted to lose horizontal and vertical separation minima within a time interval assuming a trajectory model. In the case of linear trajectories, conflict detection algorithms have been proposed that are both sound, i.e., they detect all conflicts, and complete, i.e., they do not present false alarms. In general, for arbitrary nonlinear trajectory models, it is possible to define detection algorithms that are either sound or complete, but not both. This paper considers the case of nonlinear aircraft trajectory models based on polynomial functions. In particular, it proposes a conflict detection algorithm that precisely determines whether, given a lookahead time, two aircraft flying polynomial trajectories are in conflict. That is, it has been formally verified that, assuming that the aircraft trajectories are modeled as polynomial functions, the proposed algorithm is both sound and complete.
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 ...
A Kantorovich Type of Szasz Operators Including Brenke-Type Polynomials
Directory of Open Access Journals (Sweden)
Fatma Taşdelen
2012-01-01
convergence properties of these operators by using Korovkin's theorem. We also present the order of convergence with the help of a classical approach, the second modulus of continuity, and Peetre's -functional. Furthermore, an example of Kantorovich type of the operators including Gould-Hopper polynomials is presented and Voronovskaya-type result is given for these operators including Gould-Hopper polynomials.
Polynomial Heisenberg algebras
International Nuclear Information System (INIS)
Carballo, Juan M; C, David J Fernandez; Negro, Javier; Nieto, Luis M
2004-01-01
Polynomial deformations of the Heisenberg algebra are studied in detail. Some of their natural realizations are given by the higher order susy partners (and not only by those of first order, as is already known) of the harmonic oscillator for even-order polynomials. Here, it is shown that the susy partners of the radial oscillator play a similar role when the order of the polynomial is odd. Moreover, it will be proved that the general systems ruled by such kinds of algebras, in the quadratic and cubic cases, involve Painleve transcendents of types IV and V, respectively
Directory of Open Access Journals (Sweden)
Hjalmar Rosengren
2006-12-01
Full Text Available We study multivariable Christoffel-Darboux kernels, which may be viewed as reproducing kernels for antisymmetric orthogonal polynomials, and also as correlation functions for products of characteristic polynomials of random Hermitian matrices. Using their interpretation as reproducing kernels, we obtain simple proofs of Pfaffian and determinant formulas, as well as Schur polynomial expansions, for such kernels. In subsequent work, these results are applied in combinatorics (enumeration of marked shifted tableaux and number theory (representation of integers as sums of squares.
Fourier series and orthogonal polynomials
Jackson, Dunham
2004-01-01
This text for undergraduate and graduate students illustrates the fundamental simplicity of the properties of orthogonal functions and their developments in related series. Starting with a definition and explanation of the elements of Fourier series, the text follows with examinations of Legendre polynomials and Bessel functions. Boundary value problems consider Fourier series in conjunction with Laplace's equation in an infinite strip and in a rectangle, with a vibrating string, in three dimensions, in a sphere, and in other circumstances. An overview of Pearson frequency functions is followe
Birth-death processes and associated polynomials
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
SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos
Energy Technology Data Exchange (ETDEWEB)
Ahlfeld, R., E-mail: r.ahlfeld14@imperial.ac.uk; Belkouchi, B.; Montomoli, F.
2016-09-01
A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrix is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the Hankel matrix of moments. It can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions. The extension to more input variables suggested in this work, is an anisotropic and adaptive version of Smolyak's algorithm that is solely based on the moments of the input probability distributions. It is referred to as SAMBA (PC), which is short for Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos. It is illustrated that for moderately high-dimensional problems (up to 20 different input variables or histograms) SAMBA can significantly simplify the calculation of sparse Gaussian quadrature rules. SAMBA's efficiency for multivariate functions with regard to data availability is further demonstrated by analysing higher order convergence and accuracy for a set of nonlinear test functions with 2, 5
SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos
International Nuclear Information System (INIS)
Ahlfeld, R.; Belkouchi, B.; Montomoli, F.
2016-01-01
A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrix is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the Hankel matrix of moments. It can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions. The extension to more input variables suggested in this work, is an anisotropic and adaptive version of Smolyak's algorithm that is solely based on the moments of the input probability distributions. It is referred to as SAMBA (PC), which is short for Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos. It is illustrated that for moderately high-dimensional problems (up to 20 different input variables or histograms) SAMBA can significantly simplify the calculation of sparse Gaussian quadrature rules. SAMBA's efficiency for multivariate functions with regard to data availability is further demonstrated by analysing higher order convergence and accuracy for a set of nonlinear test functions with 2, 5 and 10
Analysis of Discrete L2 Projection on Polynomial Spaces with Random Evaluations
Migliorati, Giovanni; Nobile, Fabio; von Schwerin, Erik; Tempone, Raul
2014-01-01
We analyze the problem of approximating a multivariate function by discrete least-squares projection on a polynomial space starting from random, noise-free observations. An area of possible application of such technique is uncertainty quantification for computational models. We prove an optimal convergence estimate, up to a logarithmic factor, in the univariate case, when the observation points are sampled in a bounded domain from a probability density function bounded away from zero and bounded from above, provided the number of samples scales quadratically with the dimension of the polynomial space. Optimality is meant in the sense that the weighted L2 norm of the error committed by the random discrete projection is bounded with high probability from above by the best L∞ error achievable in the given polynomial space, up to logarithmic factors. Several numerical tests are presented in both the univariate and multivariate cases, confirming our theoretical estimates. The numerical tests also clarify how the convergence rate depends on the number of sampling points, on the polynomial degree, and on the smoothness of the target function. © 2014 SFoCM.
Analysis of Discrete L2 Projection on Polynomial Spaces with Random Evaluations
Migliorati, Giovanni
2014-03-05
We analyze the problem of approximating a multivariate function by discrete least-squares projection on a polynomial space starting from random, noise-free observations. An area of possible application of such technique is uncertainty quantification for computational models. We prove an optimal convergence estimate, up to a logarithmic factor, in the univariate case, when the observation points are sampled in a bounded domain from a probability density function bounded away from zero and bounded from above, provided the number of samples scales quadratically with the dimension of the polynomial space. Optimality is meant in the sense that the weighted L2 norm of the error committed by the random discrete projection is bounded with high probability from above by the best L∞ error achievable in the given polynomial space, up to logarithmic factors. Several numerical tests are presented in both the univariate and multivariate cases, confirming our theoretical estimates. The numerical tests also clarify how the convergence rate depends on the number of sampling points, on the polynomial degree, and on the smoothness of the target function. © 2014 SFoCM.
Numerical Solutions for Convection-Diffusion Equation through Non-Polynomial Spline
Directory of Open Access Journals (Sweden)
Ravi Kanth A.S.V.
2016-01-01
Full Text Available In this paper, numerical solutions for convection-diffusion equation via non-polynomial splines are studied. We purpose an implicit method based on non-polynomial spline functions for solving the convection-diffusion equation. The method is proven to be unconditionally stable by using Von Neumann technique. Numerical results are illustrated to demonstrate the efficiency and stability of the purposed method.
Polynomial realization of the Uq (sl(3)) Gel'fand-(Weyl)-Zetlin basis
International Nuclear Information System (INIS)
Dobrev, V.K.; Truini, P.
1996-01-01
We give an explicit realization of the U ≡ U q (sl(3)) Gel'fand-(Weyl)-Zetlin (GWZ) basis as polynomial functions in three variables. This realization is obtained in two complementary ways. First we establish a 1-to-1 correspondence between the abstract GWZ basis and explicit polynomials in the quantum subgroup U + of the raising generators. We then use an explicit construction of arbitrary lowest weight (holomorphic) representations of U in terms of three variables on which the generators of U are realized as q-difference operators. Applying the GWZ corresponding polynomials in this realization to the lowest weight vector (the function 1) produces one realization of our GWZ basis. Another realization of the GWZ polynomial basis is found by the explicit diagonalization of the operators of isospin I-circumflex 2 , third component of isospin I-circumflex z , and hypercharge Y-circumflex, in the same realization as q-difference operators. The result is that the eigenvectors can be written in terms of q-hypergeometric polynomials in our three variables. Finally we construct an explicit scalar product (adapting the Shapovalov form to our setting). Using it we prove the orthogonality of our GWZ polynomials for which we use both realizations. This provides a polynomial construction for the orthonormal GWZ basis. We work for generic q, leaving the root of unity case for a following paper. It seems that our results are new also in the classical situation (q=1). (author). 20 refs
Bannai-Ito polynomials and dressing chains
Derevyagin, Maxim; Tsujimoto, Satoshi; Vinet, Luc; Zhedanov, Alexei
2012-01-01
Schur-Delsarte-Genin (SDG) maps and Bannai-Ito polynomials are studied. SDG maps are related to dressing chains determined by quadratic algebras. The Bannai-Ito polynomials and their kernel polynomials -- the complementary Bannai-Ito polynomials -- are shown to arise in the framework of the SDG maps.
Díaz Mendoza, C.; Orive, R.; Pijeira Cabrera, H.
2008-10-01
We study the asymptotic behavior of the zeros of a sequence of polynomials whose weighted norms, with respect to a sequence of weight functions, have the same nth root asymptotic behavior as the weighted norms of certain extremal polynomials. This result is applied to obtain the (contracted) weak zero distribution for orthogonal polynomials with respect to a Sobolev inner product with exponential weights of the form e-[phi](x), giving a unified treatment for the so-called Freud (i.e., when [phi] has polynomial growth at infinity) and Erdös (when [phi] grows faster than any polynomial at infinity) cases. In addition, we provide a new proof for the bound of the distance of the zeros to the convex hull of the support for these Sobolev orthogonal polynomials.
Szász-Durrmeyer operators involving Boas-Buck polynomials of blending type.
Sidharth, Manjari; Agrawal, P N; Araci, Serkan
2017-01-01
The present paper introduces the Szász-Durrmeyer type operators based on Boas-Buck type polynomials which include Brenke type polynomials, Sheffer polynomials and Appell polynomials considered by Sucu et al. (Abstr. Appl. Anal. 2012:680340, 2012). We establish the moments of the operator and a Voronvskaja type asymptotic theorem and then proceed to studying the convergence of the operators with the help of Lipschitz type space and weighted modulus of continuity. Next, we obtain a direct approximation theorem with the aid of unified Ditzian-Totik modulus of smoothness. Furthermore, we study the approximation of functions whose derivatives are locally of bounded variation.
Szász-Durrmeyer operators involving Boas-Buck polynomials of blending type
Directory of Open Access Journals (Sweden)
Manjari Sidharth
2017-05-01
Full Text Available Abstract The present paper introduces the Szász-Durrmeyer type operators based on Boas-Buck type polynomials which include Brenke type polynomials, Sheffer polynomials and Appell polynomials considered by Sucu et al. (Abstr. Appl. Anal. 2012:680340, 2012. We establish the moments of the operator and a Voronvskaja type asymptotic theorem and then proceed to studying the convergence of the operators with the help of Lipschitz type space and weighted modulus of continuity. Next, we obtain a direct approximation theorem with the aid of unified Ditzian-Totik modulus of smoothness. Furthermore, we study the approximation of functions whose derivatives are locally of bounded variation.
International Nuclear Information System (INIS)
Rashid, M.A.
1984-08-01
Integrals involving powers of (1-x 2 ) and two associated Legendre functions or two Gegenbauer polynomials are evaluated as finite sums which can be expressed in terms of terminating hypergeometric function 4 F 3 . The integrals which are evaluated are ∫sub(-1)sup(1)[Psub(l)sup(m)(x)Psub(k)sup(n)(x)]/[(1-x 2 )sup(p+1)]dx and ∫sub(-1)sup(1)Csub(l)sup(α)(x)Csub(k)sup(β)(x)[(1-x 2 )sup[(α+β-3)/2-p
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
The neighbourhood polynomial of some families of dendrimers
Nazri Husin, Mohamad; Hasni, Roslan
2018-04-01
The neighbourhood polynomial N(G,x) is generating function for the number of faces of each cardinality in the neighbourhood complex of a graph and it is defined as (G,x)={\\sum }U\\in N(G){x}|U|, where N(G) is neighbourhood complex of a graph, whose vertices of the graph and faces are subsets of vertices that have a common neighbour. A dendrimers is an artificially manufactured or synthesized molecule built up from branched units called monomers. In this paper, we compute this polynomial for some families of dendrimer.
Polynomials in algebraic analysis
Multarzyński, Piotr
2012-01-01
The concept of polynomials in the sense of algebraic analysis, for a single right invertible linear operator, was introduced and studied originally by D. Przeworska-Rolewicz \\cite{DPR}. One of the elegant results corresponding with that notion is a purely algebraic version of the Taylor formula, being a generalization of its usual counterpart, well known for functions of one variable. In quantum calculus there are some specific discrete derivations analyzed, which are right invertible linear ...
Vortices and polynomials: non-uniqueness of the Adler–Moser polynomials for the Tkachenko equation
International Nuclear Information System (INIS)
Demina, Maria V; Kudryashov, Nikolai A
2012-01-01
Stationary and translating relative equilibria of point vortices in the plane are studied. It is shown that stationary equilibria of any system containing point vortices with arbitrary choice of circulations can be described with the help of the Tkachenko equation. It is also obtained that translating relative equilibria of point vortices with arbitrary circulations can be constructed using a generalization of the Tkachenko equation. Roots of any pair of polynomials solving the Tkachenko equation and the generalized Tkachenko equation are proved to give positions of point vortices in stationary and translating relative equilibria accordingly. These results are valid even if the polynomials in a pair have multiple or common roots. It is obtained that the Adler–Moser polynomial provides non-unique polynomial solutions of the Tkachenko equation. It is shown that the generalized Tkachenko equation possesses polynomial solutions with degrees that are not triangular numbers. (paper)
The Jones polynomial as a new invariant of topological fluid dynamics
International Nuclear Information System (INIS)
Ricca, Renzo L; Liu, Xin
2014-01-01
A new method based on the use of the Jones polynomial, a well-known topological invariant of knot theory, is introduced to tackle and quantify topological aspects of structural complexity of vortex tangles in ideal fluids. By re-writing the Jones polynomial in terms of helicity, the resulting polynomial becomes then function of knot topology and vortex circulation, providing thus a new invariant of topological fluid dynamics. Explicit computations of the Jones polynomial for some standard configurations, including the Whitehead link and the Borromean rings (whose linking numbers are zero), are presented for illustration. In the case of a homogeneous, isotropic tangle of vortex filaments with same circulation, the new Jones polynomial reduces to some simple algebraic expression, that can be easily computed by numerical methods. This shows that this technique may offer a new setting and a powerful tool to detect and compute topological complexity and to investigate relations with energy, by tackling fundamental aspects of turbulence research. (paper)
Stochastic Estimation via Polynomial Chaos
2015-10-01
AFRL-RW-EG-TR-2015-108 Stochastic Estimation via Polynomial Chaos Douglas V. Nance Air Force Research...COVERED (From - To) 20-04-2015 – 07-08-2015 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER Stochastic Estimation via Polynomial Chaos ...This expository report discusses fundamental aspects of the polynomial chaos method for representing the properties of second order stochastic
Special polynomials associated with some hierarchies
International Nuclear Information System (INIS)
Kudryashov, Nikolai A.
2008-01-01
Special polynomials associated with rational solutions of a hierarchy of equations of Painleve type are introduced. The hierarchy arises by similarity reduction from the Fordy-Gibbons hierarchy of partial differential equations. Some relations for these special polynomials are given. Differential-difference hierarchies for finding special polynomials are presented. These formulae allow us to obtain special polynomials associated with the hierarchy studied. It is shown that rational solutions of members of the Schwarz-Sawada-Kotera, the Schwarz-Kaup-Kupershmidt, the Fordy-Gibbons, the Sawada-Kotera and the Kaup-Kupershmidt hierarchies can be expressed through special polynomials of the hierarchy studied
Bernoulli numbers and polynomials from a more general point of view
Energy Technology Data Exchange (ETDEWEB)
Dattoli, G. [ENEA, Centro Ricerche Frascati, Frascati, RM(Italy). Div. Fisica Applicata; Cesarano, C. [Ulm Univ., Ulm (Germany). Dept. of Mathematics; Lonzellutta, S. [ENEA, Centro Ricerche E. Clementel, Bologna (Italy). Div. Fisica Applicata
2000-07-01
In this work it is applied the method of generating function, to introduce new forms of Bernoulli numbers and polynomials, which are exploited to derive further classes of partial sums involving generalized many index many variable polynomials. Analogous considerations are developed for the Euler numbers and polynomials. [Italian] Si applica il metodo della funzione generatrice per introdurre nuove forme di numeri e polinomi di Bernoulli che vengono utilizzati per sviluppare e per calcolare somme parziali che coinvolgono polinomi a piu' indici ed a piu' variabili. Si sviluppano considerazioni analoghe per i polinomi ed i numeri di Eulero.
International Nuclear Information System (INIS)
Freund, P.G.O.
1992-01-01
We establish a previously conjectured connection between p-adics and quantum groups. We find in Sklyanin's two parameter elliptic quantum algebra and its generalizations, the conceptual basis for the Macdonald polynomials, which 'interpolate' between the zonal spherical functions of related real and p-adic symmetric spaces. The elliptic quantum algebras underlie the Z n -Baxter models. We show that in the n→∞ limit, the Jost function for the scattering of first level excitations in the Z n -Baxter model coincides with the Harish-Chandra-like c-function constructed from the Macdonald polynomials associated to the root system A 1 . The partition function of the Z 2 -Baxter model itself is also expressed in terms of this Macdonald-Harish-Chandra c-function albeit in a less simple way. We relate the two parameters q and t of the Macdonald polynomials to the anisotropy and modular parameters of the Baxter model. In particular the p-acid 'regimes' in the Macdonald polynomials correspond to a discrete sequence of XXZ models. We also discuss the possibility of 'q-deforming' Euler products. (orig.)
Tsai, Shun Hung; Chen, Yu-An; Chen, Yu-Wen; Lo, Ji-Chang; Lam, Hak-Keung
2017-01-01
A novel stabilization problem for T-S polynomial fuzzy system with time-delay is investigated in this paper. Firstly, a polynomial fuzzy controller for T-S polynomial fuzzy system with time-delay is proposed. In addition, based on polynomial Lyapunov-Krasovskii function and the developed polynomial slack variable matrices, a novel stabilization condition for T-S polynomial fuzzy system with time-delay is presented in terms of sum-of-square (SOS) form. Lastly, nonlinear system with time-delay ...
A Summation Formula for Macdonald Polynomials
de Gier, Jan; Wheeler, Michael
2016-03-01
We derive an explicit sum formula for symmetric Macdonald polynomials. Our expression contains multiple sums over the symmetric group and uses the action of Hecke generators on the ring of polynomials. In the special cases {t = 1} and {q = 0}, we recover known expressions for the monomial symmetric and Hall-Littlewood polynomials, respectively. Other specializations of our formula give new expressions for the Jack and q-Whittaker polynomials.
Weighted Polynomial Approximation for Automated Detection of Inspiratory Flow Limitation
Directory of Open Access Journals (Sweden)
Sheng-Cheng Huang
2017-01-01
Full Text Available Inspiratory flow limitation (IFL is a critical symptom of sleep breathing disorders. A characteristic flattened flow-time curve indicates the presence of highest resistance flow limitation. This study involved investigating a real-time algorithm for detecting IFL during sleep. Three categories of inspiratory flow shape were collected from previous studies for use as a development set. Of these, 16 cases were labeled as non-IFL and 78 as IFL which were further categorized into minor level (20 cases and severe level (58 cases of obstruction. In this study, algorithms using polynomial functions were proposed for extracting the features of IFL. Methods using first- to third-order polynomial approximations were applied to calculate the fitting curve to obtain the mean absolute error. The proposed algorithm is described by the weighted third-order (w.3rd-order polynomial function. For validation, a total of 1,093 inspiratory breaths were acquired as a test set. The accuracy levels of the classifications produced by the presented feature detection methods were analyzed, and the performance levels were compared using a misclassification cobweb. According to the results, the algorithm using the w.3rd-order polynomial approximation achieved an accuracy of 94.14% for IFL classification. We concluded that this algorithm achieved effective automatic IFL detection during sleep.
The Schur algorithm for generalized Schur functions III : J-unitary matrix polynomials on the circle
Alpay, Daniel; Azizov, Tomas; Dijksma, Aad; Langer, Heinz
2003-01-01
The main result is that for J = ((1)(0) (0)(-1)) every J-unitary 2 x 2-matrix polynomial on the unit circle is an essentially unique product of elementary J-unitary 2 x 2-matrix polynomials which are either of degree 1 or 2k. This is shown by means of the generalized Schur transformation introduced
Li, Xiaomiao; Lam, Hak Keung; Song, Ge; Liu, Fucai
2017-01-01
This paper deals with the stability and positivity analysis of polynomial-fuzzy-model-based ({PFMB}) control systems with time delay, which is formed by a polynomial fuzzy model and a polynomial fuzzy controller connected in a closed loop, under imperfect premise matching. To improve the design and realization flexibility, the polynomial fuzzy model and the polynomial fuzzy controller are allowed to have their own set of premise membership functions. A sum-of-squares (SOS)-based stability ana...
Exponential-Polynomial Families and the Term Structure of Interest Rates
Filipovic, Damir
2000-01-01
Exponential-polynomial families like the Nelson-Siegel or Svensson family are widely used to estimate the current forward rate curve. We investigate whether these methods go well with inter-temporal modelling. We characterize the consistent Ito processes which have the property to provide an arbitrage free interest rate model when representing the parameters of some bounded exponential-polynomial type function. This includes in particular diffusion processes. We show that there is a strong li...
Introduction to Real Orthogonal Polynomials
1992-06-01
uses Green’s functions. As motivation , consider the Dirichlet problem for the unit circle in the plane, which involves finding a harmonic function u(r...xv ; a, b ; q) - TO [q-N ab+’q ; q, xq b. Orthogoy RMotion O0 (bq :q)x p.(q* ; a, b ; q) pg(q’ ; a, b ; q) (q "q), (aq)x (q ; q), (I -abq) (bq ; q... motivation and justi- fication for continued study of the intrinsic structure of orthogonal polynomials. 99 LIST OF REFERENCES 1. Deyer, W. M., ed., CRC
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...
International Nuclear Information System (INIS)
Dattoli, G.; Lorenzutta, S.; Cesarano, C.; Sacchetti, D.
2000-01-01
Operational methods are used to provide a different point of view on the theory of Kampe' de Feriet-Bell polynomials. The method here proposed can be exploited to simplify the formalism associated with this family of polynomials and may provide advantages in computation. It suggests their extension to classes of multi-index families, whole role in application is discussed [it
Associated polynomials and birth-death processes
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
Characterizing the Lyα forest flux probability distribution function using Legendre polynomials
Energy Technology Data Exchange (ETDEWEB)
Cieplak, Agnieszka M.; Slosar, Anže, E-mail: acieplak@bnl.gov, E-mail: anze@bnl.gov [Brookhaven National Laboratory, Bldg 510, Upton, NY, 11973 (United States)
2017-10-01
The Lyman-α forest is a highly non-linear field with considerable information available in the data beyond the power spectrum. The flux probability distribution function (PDF) has been used as a successful probe of small-scale physics. In this paper we argue that measuring coefficients of the Legendre polynomial expansion of the PDF offers several advantages over measuring the binned values as is commonly done. In particular, the n -th Legendre coefficient can be expressed as a linear combination of the first n moments, allowing these coefficients to be measured in the presence of noise and allowing a clear route for marginalisation over mean flux. Moreover, in the presence of noise, our numerical work shows that a finite number of coefficients are well measured with a very sharp transition into noise dominance. This compresses the available information into a small number of well-measured quantities. We find that the amount of recoverable information is a very non-linear function of spectral noise that strongly favors fewer quasars measured at better signal to noise.
Characterizing the Lyα forest flux probability distribution function using Legendre polynomials
Cieplak, Agnieszka M.; Slosar, Anže
2017-10-01
The Lyman-α forest is a highly non-linear field with considerable information available in the data beyond the power spectrum. The flux probability distribution function (PDF) has been used as a successful probe of small-scale physics. In this paper we argue that measuring coefficients of the Legendre polynomial expansion of the PDF offers several advantages over measuring the binned values as is commonly done. In particular, the n-th Legendre coefficient can be expressed as a linear combination of the first n moments, allowing these coefficients to be measured in the presence of noise and allowing a clear route for marginalisation over mean flux. Moreover, in the presence of noise, our numerical work shows that a finite number of coefficients are well measured with a very sharp transition into noise dominance. This compresses the available information into a small number of well-measured quantities. We find that the amount of recoverable information is a very non-linear function of spectral noise that strongly favors fewer quasars measured at better signal to noise.
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...
a Unified Matrix Polynomial Approach to Modal Identification
Allemang, R. J.; Brown, D. L.
1998-04-01
One important current focus of modal identification is a reformulation of modal parameter estimation algorithms into a single, consistent mathematical formulation with a corresponding set of definitions and unifying concepts. Particularly, a matrix polynomial approach is used to unify the presentation with respect to current algorithms such as the least-squares complex exponential (LSCE), the polyreference time domain (PTD), Ibrahim time domain (ITD), eigensystem realization algorithm (ERA), rational fraction polynomial (RFP), polyreference frequency domain (PFD) and the complex mode indication function (CMIF) methods. Using this unified matrix polynomial approach (UMPA) allows a discussion of the similarities and differences of the commonly used methods. the use of least squares (LS), total least squares (TLS), double least squares (DLS) and singular value decomposition (SVD) methods is discussed in order to take advantage of redundant measurement data. Eigenvalue and SVD transformation methods are utilized to reduce the effective size of the resulting eigenvalue-eigenvector problem as well.
A probabilistic approach of sum rules for heat polynomials
International Nuclear Information System (INIS)
Vignat, C; Lévêque, O
2012-01-01
In this paper, we show that the sum rules for generalized Hermite polynomials derived by Daboul and Mizrahi (2005 J. Phys. A: Math. Gen. http://dx.doi.org/10.1088/0305-4470/38/2/010) and by Graczyk and Nowak (2004 C. R. Acad. Sci., Ser. 1 338 849) can be interpreted and easily recovered using a probabilistic moment representation of these polynomials. The covariance property of the raising operator of the harmonic oscillator, which is at the origin of the identities proved in Daboul and Mizrahi and the dimension reduction effect expressed in the main result of Graczyk and Nowak are both interpreted in terms of the rotational invariance of the Gaussian distributions. As an application of these results, we uncover a probabilistic moment interpretation of two classical integrals of the Wigner function that involve the associated Laguerre polynomials. (paper)
International Nuclear Information System (INIS)
Doha, E H; Ahmed, H M
2004-01-01
A formula expressing explicitly the derivatives of Bessel polynomials of any degree and for any order in terms of the Bessel polynomials themselves is proved. Another explicit formula, which expresses the Bessel expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of its original Bessel coefficients, is also given. A formula for the Bessel coefficients of the moments of one single Bessel polynomial of certain degree is proved. A formula for the Bessel coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its Bessel coefficients is also obtained. Application of these formulae for solving ordinary differential equations with varying coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained. An algebraic symbolic approach (using Mathematica) in order to build and solve recursively for the connection coefficients between Bessel-Bessel polynomials is described. An explicit formula for these coefficients between Jacobi and Bessel polynomials is given, of which the ultraspherical polynomial and its consequences are important special cases. Two analytical formulae for the connection coefficients between Laguerre-Bessel and Hermite-Bessel are also developed
Fermionic formula for double Kostka polynomials
Liu, Shiyuan
2016-01-01
The $X=M$ conjecture asserts that the $1D$ sum and the fermionic formula coincide up to some constant power. In the case of type $A,$ both the $1D$ sum and the fermionic formula are closely related to Kostka polynomials. Double Kostka polynomials $K_{\\Bla,\\Bmu}(t),$ indexed by two double partitions $\\Bla,\\Bmu,$ are polynomials in $t$ introduced as a generalization of Kostka polynomials. In the present paper, we consider $K_{\\Bla,\\Bmu}(t)$ in the special case where $\\Bmu=(-,\\mu'').$ We formula...
Relations between Möbius and coboundary polynomials
Jurrius, R.P.M.J.
2012-01-01
It is known that, in general, the coboundary polynomial and the Möbius polynomial of a matroid do not determine each other. Less is known about more specific cases. In this paper, we will investigate if it is possible that the Möbius polynomial of a matroid, together with the Möbius polynomial of
Matrix product formula for Macdonald polynomials
Cantini, Luigi; de Gier, Jan; Wheeler, Michael
2015-09-01
We derive a matrix product formula for symmetric Macdonald polynomials. Our results are obtained by constructing polynomial solutions of deformed Knizhnik-Zamolodchikov equations, which arise by considering representations of the Zamolodchikov-Faddeev and Yang-Baxter algebras in terms of t-deformed bosonic operators. These solutions are generalized probabilities for particle configurations of the multi-species asymmetric exclusion process, and form a basis of the ring of polynomials in n variables whose elements are indexed by compositions. For weakly increasing compositions (anti-dominant weights), these basis elements coincide with non-symmetric Macdonald polynomials. Our formulas imply a natural combinatorial interpretation in terms of solvable lattice models. They also imply that normalizations of stationary states of multi-species exclusion processes are obtained as Macdonald polynomials at q = 1.
Matrix product formula for Macdonald polynomials
International Nuclear Information System (INIS)
Cantini, Luigi; Gier, Jan de; Michael Wheeler
2015-01-01
We derive a matrix product formula for symmetric Macdonald polynomials. Our results are obtained by constructing polynomial solutions of deformed Knizhnik–Zamolodchikov equations, which arise by considering representations of the Zamolodchikov–Faddeev and Yang–Baxter algebras in terms of t-deformed bosonic operators. These solutions are generalized probabilities for particle configurations of the multi-species asymmetric exclusion process, and form a basis of the ring of polynomials in n variables whose elements are indexed by compositions. For weakly increasing compositions (anti-dominant weights), these basis elements coincide with non-symmetric Macdonald polynomials. Our formulas imply a natural combinatorial interpretation in terms of solvable lattice models. They also imply that normalizations of stationary states of multi-species exclusion processes are obtained as Macdonald polynomials at q = 1. (paper)
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.
The threshold of cortical electrical stimulation for mapping sensory and motor functional areas.
Guojun, Zhang; Duanyu, Ni; Fu, Paul; Lixin, Cai; Tao, Yu; Wei, Du; Liang, Qiao; Zhiwei, Ren
2014-02-01
This study aimed to investigate the threshold of cortical electrical stimulation (CES) for functional brain mapping during surgery for the treatment of rolandic epilepsy. A total of 21 patients with rolandic epilepsy who underwent surgical treatment at the Beijing Institute of Functional Neurosurgery between October 2006 and March 2008 were included in this study. Their clinical data were retrospectively collected and analyzed. The thresholds of CES for motor response, sensory response, and after discharge production along with other threshold-related factors were investigated. The thresholds (mean ± standard deviation) for motor response, sensory response, and after discharge production were 3.48 ± 0.87, 3.86 ± 1.31, and 4.84 ± 1.38 mA, respectively. The threshold for after discharge production was significantly higher than those of both the motor and sensory response (both pstimulation frequency of 50 Hz and a pulse width of 0.2 ms, the threshold of sensory and motor responses were similar, and the threshold of after discharge production was higher than that of sensory and motor response. Copyright © 2013 Elsevier Ltd. All rights reserved.
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....
Mirror symmetry, toric branes and topological string amplitudes as polynomials
Energy Technology Data Exchange (ETDEWEB)
Alim, Murad
2009-07-13
The central theme of this thesis is the extension and application of mirror symmetry of topological string theory. The contribution of this work on the mathematical side is given by interpreting the calculated partition functions as generating functions for mathematical invariants which are extracted in various examples. Furthermore the extension of the variation of the vacuum bundle to include D-branes on compact geometries is studied. Based on previous work for non-compact geometries a system of differential equations is derived which allows to extend the mirror map to the deformation spaces of the D-Branes. Furthermore, these equations allow the computation of the full quantum corrected superpotentials which are induced by the D-branes. Based on the holomorphic anomaly equation, which describes the background dependence of topological string theory relating recursively loop amplitudes, this work generalizes a polynomial construction of the loop amplitudes, which was found for manifolds with a one dimensional space of deformations, to arbitrary target manifolds with arbitrary dimension of the deformation space. The polynomial generators are determined and it is proven that the higher loop amplitudes are polynomials of a certain degree in the generators. Furthermore, the polynomial construction is generalized to solve the extension of the holomorphic anomaly equation to D-branes without deformation space. This method is applied to calculate higher loop amplitudes in numerous examples and the mathematical invariants are extracted. (orig.)
Mirror symmetry, toric branes and topological string amplitudes as polynomials
International Nuclear Information System (INIS)
Alim, Murad
2009-01-01
The central theme of this thesis is the extension and application of mirror symmetry of topological string theory. The contribution of this work on the mathematical side is given by interpreting the calculated partition functions as generating functions for mathematical invariants which are extracted in various examples. Furthermore the extension of the variation of the vacuum bundle to include D-branes on compact geometries is studied. Based on previous work for non-compact geometries a system of differential equations is derived which allows to extend the mirror map to the deformation spaces of the D-Branes. Furthermore, these equations allow the computation of the full quantum corrected superpotentials which are induced by the D-branes. Based on the holomorphic anomaly equation, which describes the background dependence of topological string theory relating recursively loop amplitudes, this work generalizes a polynomial construction of the loop amplitudes, which was found for manifolds with a one dimensional space of deformations, to arbitrary target manifolds with arbitrary dimension of the deformation space. The polynomial generators are determined and it is proven that the higher loop amplitudes are polynomials of a certain degree in the generators. Furthermore, the polynomial construction is generalized to solve the extension of the holomorphic anomaly equation to D-branes without deformation space. This method is applied to calculate higher loop amplitudes in numerous examples and the mathematical invariants are extracted. (orig.)
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.
A pair of biorthogonal polynomials for the Szegö-Hermite weight function
Directory of Open Access Journals (Sweden)
N. K. Thakare
1988-01-01
Full Text Available A pair of polynomial sequences {Snμ(x;k} and {Tmμ(x;k} where Snμ(x;k is of degree n in xk and Tmμ(x;k is of degree m in x, is constructed. It is shown that this pair is biorthogonal with respect to the Szegö-Hermite weight function |x|2μexp(−x2, (μ>−1/2 over the interval (−∞,∞ in the sense that∫−∞∞|x|2μexp(−x2Snμ(x;kTmμ(x;kdx=0, ifm≠n ≠0, ifm=nwhere m,n=0,1,2,… and k is an odd positive integer.
On Dual Gabor Frame Pairs Generated by Polynomials
DEFF Research Database (Denmark)
Christensen, Ole; Rae Young, Kim
2010-01-01
We provide explicit constructions of particularly convenient dual pairs of Gabor frames. We prove that arbitrary polynomials restricted to sufficiently large intervals will generate Gabor frames, at least for small modulation parameters. Unfortunately, no similar function can generate a dual Gabo...
Machado, Fabiana Andrade; Nakamura, Fábio Yuzo; Moraes, Solange Marta Franzói De
2012-01-01
This study examined the influence of the regression model and initial intensity of an incremental test on the relationship between the lactate threshold estimated by the maximal-deviation method and the endurance performance. Sixteen non-competitive, recreational female runners performed a discontinuous incremental treadmill test. The initial speed was set at 7 km · h⁻¹, and increased every 3 min by 1 km · h⁻¹ with a 30-s rest between the stages used for earlobe capillary blood sample collection. Lactate-speed data were fitted by an exponential-plus-constant and a third-order polynomial equation. The lactate threshold was determined for both regression equations, using all the coordinates, excluding the first and excluding the first and second initial points. Mean speed of a 10-km road race was the performance index (3.04 ± 0.22 m · s⁻¹). The exponentially-derived lactate threshold had a higher correlation (0.98 ≤ r ≤ 0.99) and smaller standard error of estimate (SEE) (0.04 ≤ SEE ≤ 0.05 m · s⁻¹) with performance than the polynomially-derived equivalent (0.83 ≤ r ≤ 0.89; 0.10 ≤ SEE ≤ 0.13 m · s⁻¹). The exponential lactate threshold was greater than the polynomial equivalent (P performance index that is independent of the initial intensity of the incremental test and better than the polynomial equivalent.
On the non-hyperbolicity of a class of exponential polynomials
Directory of Open Access Journals (Sweden)
Gaspar Mora
2017-10-01
Full Text Available In this paper we have constructed a class of non-hyperbolic exponential polynomials that contains all the partial sums of the Riemann zeta function. An exponential polynomial been also defined to illustrate the complexity of the structure of the set defined by the closure of the real projections of its zeros. The sensitivity of this set, when the vector of delays is perturbed, has been analysed. These results have immediate implications in the theory of the neutral differential equations.
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 ...
Bounds on the degree of APN polynomials: the case of x −1 + g(x)
DEFF Research Database (Denmark)
Leander, Gregor; Rodier, François
2011-01-01
In this paper we consider APN functions $${f:\\mathcal{F}_{2^m}\\to \\mathcal{F}_{2^m}}$$ of the form f(x) = x −1 + g(x) where g is any non $${\\mathcal{F}_{2}}$$-affine polynomial. We prove a lower bound on the degree of the polynomial g. This bound in particular implies that such a function f is AP...
International Nuclear Information System (INIS)
Takahashi, Akito; Yamamoto, Junji; Ebisuya, Mituo; Sumita, Kenji
1979-01-01
A new method for calculating the anisotropic neutron transport is proposed for the angular spectral analysis of D-T fusion reactor neutronics. The method is based on the transport equation with new type of anisotropic scattering kernels formulated by a single function I sub(i) (μ', μ) instead of polynomial expansion, for instance, Legendre polynomials. In the calculation of angular flux spectra by using scattering kernels with the Legendre polynomial expansion, we often observe the oscillation with negative flux. But in principle this oscillation disappears by this new method. In this work, we discussed anisotropic scattering kernels of the elastic scattering and the inelastic scatterings which excite discrete energy levels. The other scatterings were included in isotropic scattering kernels. An approximation method, with use of the first collision source written by the I sub(i) (μ', μ) function, was introduced to attenuate the ''oscillations'' when we are obliged to use the scattering kernels with the Legendre polynomial expansion. Calculated results with this approximation showed remarkable improvement for the analysis of the angular flux spectra in a slab system of lithium metal with the D-T neutron source. (author)
On the Laurent polynomial rings
International Nuclear Information System (INIS)
Stefanescu, D.
1985-02-01
We describe some properties of the Laurent polynomial rings in a finite number of indeterminates over a commutative unitary ring. We study some subrings of the Laurent polynomial rings. We finally obtain two cancellation properties. (author)
Cosine and sine operators related to orthogonal polynomial sets on the interval [-1, 1
International Nuclear Information System (INIS)
Appl, Thomas; Schiller, Diethard H
2005-01-01
The quantization of phase is still an open problem. In the approach of Susskind and Glogower, the so-called cosine and sine operators play a fundamental role. Their eigenstates in the Fock representation are related to the Chebyshev polynomials of the second kind. Here we introduce more general cosine and sine operators whose eigenfunctions in the Fock basis are related in a similar way to arbitrary orthogonal polynomial sets on the interval [-1, 1]. To each polynomial set defined in terms of a weight function there corresponds a pair of cosine and sine operators. Depending on the symmetry of the weight function, we distinguish generalized or extended operators. Their eigenstates are used to define cosine and sine representations and probability distributions. We also consider the arccosine and arcsine operators and use their eigenstates to define cosine-phase and sine-phase distributions, respectively. Specific, numerical and graphical results are given for the classical orthogonal polynomials and for particular Fock and coherent states
Polynomial approximation on polytopes
Totik, Vilmos
2014-01-01
Polynomial approximation on convex polytopes in \\mathbf{R}^d is considered in uniform and L^p-norms. For an appropriate modulus of smoothness matching direct and converse estimates are proven. In the L^p-case so called strong direct and converse results are also verified. The equivalence of the moduli of smoothness with an appropriate K-functional follows as a consequence. The results solve a problem that was left open since the mid 1980s when some of the present findings were established for special, so-called simple polytopes.
Topological classification of trigonometric polynomials related to affine Coxeter group A-tilde2
International Nuclear Information System (INIS)
Arnold, V.I.
2006-06-01
The family of trigonometric polynomials, is defined by the six?Cparametrical expression f(x, y) = a cos x + b sin x + c cos y + d sin y + p cos(x + y) + q sin(x + y). The trigonometric polynomials of this family, having the most complicated topological structure, have 6 critical points. These functions are classified up to the actions of the following groups: two functions are called to be topologically equivalent, if one is transformed to the other by two smooth diffeomorphisms of the manifolds T 2 and R (of the preimages and of the images of mapping f : T 2 → R). We suppose, that the images diffeomorphisms (dependent variable changes) preserve the orientation of the real line, and that the preimages spaces diffeomorphism is homotopic to the identity mapping of the torus. We shall see, that the trigonometric polynomials which have 6 nondegenerated critical points and six different critical values (that might be fixed at points {1, 2, 3, 4, 5, 6 }) form 6 equivalence classes of topologically different functions, while the general Morse functions on the torus, having six critical points and six fixed critical values, form an infinite set of the equivalence classes of functions. With respect to the Diff-equivalence all these functions form only 16 classes. All these unexpected results suggest, that in the 16 Cth Hilbert's problem (on the topological classification of real algebraic manifolds) one would ask to classify topologically rather the defining polynomials, than the real hypersurfaces of their zeros
Skew-orthogonal polynomials, differential systems and random matrix theory
International Nuclear Information System (INIS)
Ghosh, S.
2007-01-01
We study skew-orthogonal polynomials with respect to the weight function exp[-2V (x)], with V (x) = Σ K=1 2d (u K /K)x K , u 2d > 0, d > 0. A finite subsequence of such skew-orthogonal polynomials arising in the study of Orthogonal and Symplectic ensembles of random matrices, satisfy a system of differential-difference-deformation equation. The vectors formed by such subsequence has the rank equal to the degree of the potential in the quaternion sense. These solutions satisfy certain compatibility condition and hence admit a simultaneous fundamental system of solutions. (author)
On the Lojasiewicz exponent at infinity of real polynomials
International Nuclear Information System (INIS)
Ha Huy Vui; Pham Tien Son
2007-07-01
Let f : R n → R be a nonconstant polynomial function. In this paper, using the information from 'the curve of tangency' of f, we provide a method to determine the Lojasiewicz exponent at infinity of f. As a corollary, we give a computational criterion to decide if the Lojasiewicz exponent at infinity is finite or not. Then, we obtain a formula to calculate the set of points at which the polynomial f is not proper. Moreover, a relation between the Lojasiewicz exponent at infinity of f with the problem of computing the global optimum of f is also established. (author)
Best polynomial degree reduction on q-lattices with applications to q-orthogonal polynomials
Ait-Haddou, Rachid
2015-06-07
We show that a weighted least squares approximation of q-Bézier coefficients provides the best polynomial degree reduction in the q-L2-norm. We also provide a finite analogue of this result with respect to finite q-lattices and we present applications of these results to q-orthogonal polynomials. © 2015 Elsevier Inc. All rights reserved.
Best polynomial degree reduction on q-lattices with applications to q-orthogonal polynomials
Ait-Haddou, Rachid; Goldman, Ron
2015-01-01
We show that a weighted least squares approximation of q-Bézier coefficients provides the best polynomial degree reduction in the q-L2-norm. We also provide a finite analogue of this result with respect to finite q-lattices and we present applications of these results to q-orthogonal polynomials. © 2015 Elsevier Inc. All rights reserved.
Computing the Alexander Polynomial Numerically
DEFF Research Database (Denmark)
Hansen, Mikael Sonne
2006-01-01
Explains how to construct the Alexander Matrix and how this can be used to compute the Alexander polynomial numerically.......Explains how to construct the Alexander Matrix and how this can be used to compute the Alexander polynomial numerically....
Institute of Scientific and Technical Information of China (English)
XU Xiu-Wei; REN Ting-Qi; LIU Shu-Yan; MA Qiu-Ming; LIU Sheng-Dian
2007-01-01
Making use of the transformation relation among usual, normal, and antinormal ordering for the multimode boson exponential quadratic polynomial operators (BEQPO's), we present the analytic expression of arbitrary matrix elements for BEQPO's. As a preliminary application, we obtain the exact expressions of partition function about the boson quadratic polynomial system, matrix elements in particle-number, coordinate, and momentum representation, and P representation for the BEQPO's.
Polynomial estimation of the smoothing splines for the new Finnish reference values for spirometry.
Kainu, Annette; Timonen, Kirsi
2016-07-01
Background Discontinuity of spirometry reference values from childhood into adulthood has been a problem with traditional reference values, thus modern modelling approaches using smoothing spline functions to better depict the transition during growth and ageing have been recently introduced. Following the publication of the new international Global Lung Initiative (GLI2012) reference values also new national Finnish reference values have been calculated using similar GAMLSS-modelling, with spline estimates for mean (Mspline) and standard deviation (Sspline) provided in tables. The aim of this study was to produce polynomial estimates for these spline functions to use in lieu of lookup tables and to assess their validity in the reference population of healthy non-smokers. Methods Linear regression modelling was used to approximate the estimated values for Mspline and Sspline using similar polynomial functions as in the international GLI2012 reference values. Estimated values were compared to original calculations in absolute values, the derived predicted mean and individually calculated z-scores using both values. Results Polynomial functions were estimated for all 10 spirometry variables. The agreement between original lookup table-produced values and polynomial estimates was very good, with no significant differences found. The variation slightly increased in larger predicted volumes, but a range of -0.018 to +0.022 litres of FEV1 representing ± 0.4% of maximum difference in predicted mean. Conclusions Polynomial approximations were very close to the original lookup tables and are recommended for use in clinical practice to facilitate the use of new reference values.
Expressing the remainder of the Taylor polynomial when the function lacks smoothness
Czech Academy of Sciences Publication Activity Database
Hošek, Radim
2017-01-01
Roč. 72, č. 3 (2017), s. 126-130 ISSN 0013-6018 Institutional support: RVO:67985840 Keywords : Taylor polynomial * Taylor theorem Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics http://www.ems-ph.org/doi/10.4171/EM/335
Density of Real Zeros of the Tutte Polynomial
DEFF Research Database (Denmark)
Ok, Seongmin; Perrett, Thomas
2018-01-01
The Tutte polynomial of a graph is a two-variable polynomial whose zeros and evaluations encode many interesting properties of the graph. In this article we investigate the real zeros of the Tutte polynomials of graphs, and show that they form a dense subset of certain regions of the plane. This ....... This is the first density result for the real zeros of the Tutte polynomial in a region of positive volume. Our result almost confirms a conjecture of Jackson and Sokal except for one region which is related to an open problem on flow polynomials.......The Tutte polynomial of a graph is a two-variable polynomial whose zeros and evaluations encode many interesting properties of the graph. In this article we investigate the real zeros of the Tutte polynomials of graphs, and show that they form a dense subset of certain regions of the plane...
Density of Real Zeros of the Tutte Polynomial
DEFF Research Database (Denmark)
Ok, Seongmin; Perrett, Thomas
2017-01-01
The Tutte polynomial of a graph is a two-variable polynomial whose zeros and evaluations encode many interesting properties of the graph. In this article we investigate the real zeros of the Tutte polynomials of graphs, and show that they form a dense subset of certain regions of the plane. This ....... This is the first density result for the real zeros of the Tutte polynomial in a region of positive volume. Our result almost confirms a conjecture of Jackson and Sokal except for one region which is related to an open problem on flow polynomials.......The Tutte polynomial of a graph is a two-variable polynomial whose zeros and evaluations encode many interesting properties of the graph. In this article we investigate the real zeros of the Tutte polynomials of graphs, and show that they form a dense subset of certain regions of the plane...
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....
Tensor calculus in polar coordinates using Jacobi polynomials
Vasil, Geoffrey M.; Burns, Keaton J.; Lecoanet, Daniel; Olver, Sheehan; Brown, Benjamin P.; Oishi, Jeffrey S.
2016-11-01
Spectral methods are an efficient way to solve partial differential equations on domains possessing certain symmetries. The utility of a method depends strongly on the choice of spectral basis. In this paper we describe a set of bases built out of Jacobi polynomials, and associated operators for solving scalar, vector, and tensor partial differential equations in polar coordinates on a unit disk. By construction, the bases satisfy regularity conditions at r = 0 for any tensorial field. The coordinate singularity in a disk is a prototypical case for many coordinate singularities. The work presented here extends to other geometries. The operators represent covariant derivatives, multiplication by azimuthally symmetric functions, and the tensorial relationship between fields. These arise naturally from relations between classical orthogonal polynomials, and form a Heisenberg algebra. Other past work uses more specific polynomial bases for solving equations in polar coordinates. The main innovation in this paper is to use a larger set of possible bases to achieve maximum bandedness of linear operations. We provide a series of applications of the methods, illustrating their ease-of-use and accuracy.
A comparison of high-order polynomial and wave-based methods for Helmholtz problems
Lieu, Alice; Gabard, Gwénaël; Bériot, Hadrien
2016-09-01
The application of computational modelling to wave propagation problems is hindered by the dispersion error introduced by the discretisation. Two common strategies to address this issue are to use high-order polynomial shape functions (e.g. hp-FEM), or to use physics-based, or Trefftz, methods where the shape functions are local solutions of the problem (typically plane waves). Both strategies have been actively developed over the past decades and both have demonstrated their benefits compared to conventional finite-element methods, but they have yet to be compared. In this paper a high-order polynomial method (p-FEM with Lobatto polynomials) and the wave-based discontinuous Galerkin method are compared for two-dimensional Helmholtz problems. A number of different benchmark problems are used to perform a detailed and systematic assessment of the relative merits of these two methods in terms of interpolation properties, performance and conditioning. It is generally assumed that a wave-based method naturally provides better accuracy compared to polynomial methods since the plane waves or Bessel functions used in these methods are exact solutions of the Helmholtz equation. Results indicate that this expectation does not necessarily translate into a clear benefit, and that the differences in performance, accuracy and conditioning are more nuanced than generally assumed. The high-order polynomial method can in fact deliver comparable, and in some cases superior, performance compared to the wave-based DGM. In addition to benchmarking the intrinsic computational performance of these methods, a number of practical issues associated with realistic applications are also discussed.
A polynomial based model for cell fate prediction in human diseases.
Ma, Lichun; Zheng, Jie
2017-12-21
Cell fate regulation directly affects tissue homeostasis and human health. Research on cell fate decision sheds light on key regulators, facilitates understanding the mechanisms, and suggests novel strategies to treat human diseases that are related to abnormal cell development. In this study, we proposed a polynomial based model to predict cell fate. This model was derived from Taylor series. As a case study, gene expression data of pancreatic cells were adopted to test and verify the model. As numerous features (genes) are available, we employed two kinds of feature selection methods, i.e. correlation based and apoptosis pathway based. Then polynomials of different degrees were used to refine the cell fate prediction function. 10-fold cross-validation was carried out to evaluate the performance of our model. In addition, we analyzed the stability of the resultant cell fate prediction model by evaluating the ranges of the parameters, as well as assessing the variances of the predicted values at randomly selected points. Results show that, within both the two considered gene selection methods, the prediction accuracies of polynomials of different degrees show little differences. Interestingly, the linear polynomial (degree 1 polynomial) is more stable than others. When comparing the linear polynomials based on the two gene selection methods, it shows that although the accuracy of the linear polynomial that uses correlation analysis outcomes is a little higher (achieves 86.62%), the one within genes of the apoptosis pathway is much more stable. Considering both the prediction accuracy and the stability of polynomial models of different degrees, the linear model is a preferred choice for cell fate prediction with gene expression data of pancreatic cells. The presented cell fate prediction model can be extended to other cells, which may be important for basic research as well as clinical study of cell development related diseases.
Parallel multigrid smoothing: polynomial versus Gauss-Seidel
International Nuclear Information System (INIS)
Adams, Mark; Brezina, Marian; Hu, Jonathan; Tuminaro, Ray
2003-01-01
Gauss-Seidel is often the smoother of choice within multigrid applications. In the context of unstructured meshes, however, maintaining good parallel efficiency is difficult with multiplicative iterative methods such as Gauss-Seidel. This leads us to consider alternative smoothers. We discuss the computational advantages of polynomial smoothers within parallel multigrid algorithms for positive definite symmetric systems. Two particular polynomials are considered: Chebyshev and a multilevel specific polynomial. The advantages of polynomial smoothing over traditional smoothers such as Gauss-Seidel are illustrated on several applications: Poisson's equation, thin-body elasticity, and eddy current approximations to Maxwell's equations. While parallelizing the Gauss-Seidel method typically involves a compromise between a scalable convergence rate and maintaining high flop rates, polynomial smoothers achieve parallel scalable multigrid convergence rates without sacrificing flop rates. We show that, although parallel computers are the main motivation, polynomial smoothers are often surprisingly competitive with Gauss-Seidel smoothers on serial machines
Parallel multigrid smoothing: polynomial versus Gauss-Seidel
Adams, Mark; Brezina, Marian; Hu, Jonathan; Tuminaro, Ray
2003-07-01
Gauss-Seidel is often the smoother of choice within multigrid applications. In the context of unstructured meshes, however, maintaining good parallel efficiency is difficult with multiplicative iterative methods such as Gauss-Seidel. This leads us to consider alternative smoothers. We discuss the computational advantages of polynomial smoothers within parallel multigrid algorithms for positive definite symmetric systems. Two particular polynomials are considered: Chebyshev and a multilevel specific polynomial. The advantages of polynomial smoothing over traditional smoothers such as Gauss-Seidel are illustrated on several applications: Poisson's equation, thin-body elasticity, and eddy current approximations to Maxwell's equations. While parallelizing the Gauss-Seidel method typically involves a compromise between a scalable convergence rate and maintaining high flop rates, polynomial smoothers achieve parallel scalable multigrid convergence rates without sacrificing flop rates. We show that, although parallel computers are the main motivation, polynomial smoothers are often surprisingly competitive with Gauss-Seidel smoothers on serial machines.
On characteristic polynomials for a generalized chiral random matrix ensemble with a source
Fyodorov, Yan V.; Grela, Jacek; Strahov, Eugene
2018-04-01
We evaluate averages involving characteristic polynomials, inverse characteristic polynomials and ratios of characteristic polynomials for a N× N random matrix taken from a L-deformed chiral Gaussian Unitary Ensemble with an external source Ω. Relation to a recently studied statistics of bi-orthogonal eigenvectors in the complex Ginibre ensemble, see Fyodorov (2017 arXiv:1710.04699), is briefly discussed as a motivation to study asymptotics of these objects in the case of external source proportional to the identity matrix. In particular, for an associated complex bulk/chiral edge scaling regime we retrieve the kernel related to Bessel/Macdonald functions.
A Threshold Pseudorandom Function Construction and Its Applications
DEFF Research Database (Denmark)
Nielsen, Jesper Buus
2002-01-01
We give the first construction of a practical threshold pseudo- random function.The protocol for evaluating the function is efficient enough that it can be used to replace random oracles in some protocols relying on such oracles. In particular, we show how to transform the efficient...... cryptographically secure Byzantine agreement protocol by Cachin, Kursawe and Shoup for the random oracle model into a cryptographically secure protocol for the complexity theoretic model without loosing efficiency or resilience,thereby constructing an efficient and optimally resilient Byzantine agreement protocol...
Efficient computation of Laguerre polynomials
A. Gil (Amparo); J. Segura (Javier); N.M. Temme (Nico)
2017-01-01
textabstractAn efficient algorithm and a Fortran 90 module (LaguerrePol) for computing Laguerre polynomials . Ln(α)(z) are presented. The standard three-term recurrence relation satisfied by the polynomials and different types of asymptotic expansions valid for . n large and . α small, are used
Optimal Conformal Polynomial Projections for Croatia According to the Airy/Jordan Criterion
Directory of Open Access Journals (Sweden)
Dražen Tutić
2009-05-01
Full Text Available The paper describes optimal conformal polynomial projections for Croatia according to the Airy/Jordan criterion. A brief introduction of history and theory of conformal mapping is followed by descriptions of conformal polynomial projections and their current application. The paper considers polynomials of degrees 1 to 10. Since there are conditions in which the 1st degree polynomial becomes the famous Mercator projection, it was not considered specifically for Croatian territory. The area of Croatia was defined as a union of national territory and the continental shelf. Area definition data were taken from the Euro Global Map 1:1 000 000 for Croatia, as well as from two maritime delimitation treaties. Such an irregular area was approximated with a regular grid consisting of 11 934 ellipsoidal trapezoids 2' large. The Airy/Jordan criterion for the optimal projection is defined as minimum of weighted mean of Airy/Jordan measure of distortion in points. The value of the Airy/Jordan criterion is calculated from all 11 934 centres of ellipsoidal trapezoids, while the weights are equal to areas of corresponding ellipsoidal trapezoids. The minimum is obtained by Nelder and Mead’s method, as implemented in the fminsearch function of the MATLAB package. Maps of Croatia representing the distribution of distortions are given for polynomial degrees 2 to 6 and 10. Increasing the polynomial degree results in better projections considering the criterion, and the 6th degree polynomial provides a good ratio of formula complexity and criterion value.
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.
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.
International Nuclear Information System (INIS)
Cari, C; Suparmi, A
2013-01-01
The energy eigenvalues and eigenfunctions of Schrodinger equation for three dimensional harmonic oscillator potential plus Rosen-Morse non-central potential are investigated using NU method and Romanovski polynomial. The bound state energy eigenvalues are given in a closed form and corresponding radial wave functions are expressed in associated Laguerre polynomials while angular eigen functions are given in terms of Romanovski polynomials. The Rosen-Morse potential is considered to be a perturbation factor to the three dimensional harmonic oscillator potential that causes the increase of radial wave function amplitude and decrease of angular momentum length. Keywords: Schrodinger Equation, Three dimensional Harmonic Oscillator potential, Rosen-morse non-central potential, NU method, Romanovski Polynomials
Development of a polynomial nodal model to the multigroup transport equation in one dimension
International Nuclear Information System (INIS)
Feiz, M.
1986-01-01
A polynomial nodal model that uses Legendre polynomial expansions was developed for the multigroup transport equation in one dimension. The development depends upon the least-squares minimization of the residuals using the approximate functions over the node. Analytical expressions were developed for the polynomial coefficients. The odd moments of the angular neutron flux over the half ranges were used at the internal interfaces, and the Marshak boundary condition was used at the external boundaries. Sample problems with fine-mesh finite-difference solutions of the diffusion and transport equations were used for comparison with the model
On factorization of generalized Macdonald polynomials
International Nuclear Information System (INIS)
Kononov, Ya.; Morozov, A.
2016-01-01
A remarkable feature of Schur functions - the common eigenfunctions of cut-and-join operators from W ∞ - is that they factorize at the peculiar two-parametric topological locus in the space of time variables, which is known as the hook formula for quantum dimensions of representations of U q (SL N ) and which plays a big role in various applications. This factorization survives at the level of Macdonald polynomials. We look for its further generalization to generalized Macdonald polynomials (GMPs), associated in the same way with the toroidal Ding-Iohara-Miki algebras, which play the central role in modern studies in Seiberg-Witten-Nekrasov theory. In the simplest case of the first-coproduct eigenfunctions, where GMP depend on just two sets of time variables, we discover a weak factorization - on a one- (rather than four-) parametric slice of the topological locus, which is already a very non-trivial property, calling for proof and better understanding. (orig.)
On factorization of generalized Macdonald polynomials
Energy Technology Data Exchange (ETDEWEB)
Kononov, Ya. [Landau Institute for Theoretical Physics, Chernogolovka (Russian Federation); HSE, Math Department, Moscow (Russian Federation); Morozov, A. [ITEP, Moscow (Russian Federation); Institute for Information Transmission Problems, Moscow (Russian Federation); National Research Nuclear University MEPhI, Moscow (Russian Federation)
2016-08-15
A remarkable feature of Schur functions - the common eigenfunctions of cut-and-join operators from W{sub ∞} - is that they factorize at the peculiar two-parametric topological locus in the space of time variables, which is known as the hook formula for quantum dimensions of representations of U{sub q}(SL{sub N}) and which plays a big role in various applications. This factorization survives at the level of Macdonald polynomials. We look for its further generalization to generalized Macdonald polynomials (GMPs), associated in the same way with the toroidal Ding-Iohara-Miki algebras, which play the central role in modern studies in Seiberg-Witten-Nekrasov theory. In the simplest case of the first-coproduct eigenfunctions, where GMP depend on just two sets of time variables, we discover a weak factorization - on a one- (rather than four-) parametric slice of the topological locus, which is already a very non-trivial property, calling for proof and better understanding. (orig.)
Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies
International Nuclear Information System (INIS)
Hampton, Jerrad; Doostan, Alireza
2015-01-01
Sampling orthogonal polynomial bases via Monte Carlo is of interest for uncertainty quantification of models with random inputs, using Polynomial Chaos (PC) expansions. It is known that bounding a probabilistic parameter, referred to as coherence, yields a bound on the number of samples necessary to identify coefficients in a sparse PC expansion via solution to an ℓ 1 -minimization problem. Utilizing results for orthogonal polynomials, we bound the coherence parameter for polynomials of Hermite and Legendre type under their respective natural sampling distribution. In both polynomial bases we identify an importance sampling distribution which yields a bound with weaker dependence on the order of the approximation. For more general orthonormal bases, we propose the coherence-optimal sampling: a Markov Chain Monte Carlo sampling, which directly uses the basis functions under consideration to achieve a statistical optimality among all sampling schemes with identical support. We demonstrate these different sampling strategies numerically in both high-order and high-dimensional, manufactured PC expansions. In addition, the quality of each sampling method is compared in the identification of solutions to two differential equations, one with a high-dimensional random input and the other with a high-order PC expansion. In both cases, the coherence-optimal sampling scheme leads to similar or considerably improved accuracy
Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies
Hampton, Jerrad; Doostan, Alireza
2015-01-01
Sampling orthogonal polynomial bases via Monte Carlo is of interest for uncertainty quantification of models with random inputs, using Polynomial Chaos (PC) expansions. It is known that bounding a probabilistic parameter, referred to as coherence, yields a bound on the number of samples necessary to identify coefficients in a sparse PC expansion via solution to an ℓ1-minimization problem. Utilizing results for orthogonal polynomials, we bound the coherence parameter for polynomials of Hermite and Legendre type under their respective natural sampling distribution. In both polynomial bases we identify an importance sampling distribution which yields a bound with weaker dependence on the order of the approximation. For more general orthonormal bases, we propose the coherence-optimal sampling: a Markov Chain Monte Carlo sampling, which directly uses the basis functions under consideration to achieve a statistical optimality among all sampling schemes with identical support. We demonstrate these different sampling strategies numerically in both high-order and high-dimensional, manufactured PC expansions. In addition, the quality of each sampling method is compared in the identification of solutions to two differential equations, one with a high-dimensional random input and the other with a high-order PC expansion. In both cases, the coherence-optimal sampling scheme leads to similar or considerably improved accuracy.
Need for higher order polynomial basis for polynomial nodal methods employed in LWR calculations
International Nuclear Information System (INIS)
Taiwo, T.A.; Palmiotti, G.
1997-01-01
The paper evaluates the accuracy and efficiency of sixth order polynomial solutions and the use of one radial node per core assembly for pressurized water reactor (PWR) core power distributions and reactivities. The computer code VARIANT was modified to calculate sixth order polynomial solutions for a hot zero power benchmark problem in which a control assembly along a core axis is assumed to be out of the core. Results are presented for the VARIANT, DIF3D-NODAL, and DIF3D-finite difference codes. The VARIANT results indicate that second order expansion of the within-node source and linear representation of the node surface currents are adequate for this problem. The results also demonstrate the improvement in the VARIANT solution when the order of the polynomial expansion of the within-node flux is increased from fourth to sixth order. There is a substantial saving in computational time for using one radial node per assembly with the sixth order expansion compared to using four or more nodes per assembly and fourth order polynomial solutions. 11 refs., 1 tab
Exact solution of Chern-Simons-matter matrix models with characteristic/orthogonal polynomials
International Nuclear Information System (INIS)
Tierz, Miguel
2016-01-01
We solve for finite N the matrix model of supersymmetric U(N) Chern-Simons theory coupled to N f fundamental and N f anti-fundamental chiral multiplets of R-charge 1/2 and of mass m, by identifying it with an average of inverse characteristic polynomials in a Stieltjes-Wigert ensemble. This requires the computation of the Cauchy transform of the Stieltjes-Wigert polynomials, which we carry out, finding a relationship with Mordell integrals, and hence with previous analytical results on the matrix model. The semiclassical limit of the model is expressed, for arbitrary N f , in terms of a single Hermite polynomial. This result also holds for more general matter content, involving matrix models with double-sine functions.
Large level crossings of a random polynomial
Directory of Open Access Journals (Sweden)
Kambiz Farahmand
1987-01-01
Full Text Available We know the expected number of times that a polynomial of degree n with independent random real coefficients asymptotically crosses the level K, when K is any real value such that (K2/nÃ¢Â†Â’0 as nÃ¢Â†Â’Ã¢ÂˆÂž. The present paper shows that, when K is allowed to be large, this expected number of crossings reduces to only one. The coefficients of the polynomial are assumed to be normally distributed. It is shown that it is sufficient to let KÃ¢Â‰Â¥exp(nf where f is any function of n such that fÃ¢Â†Â’Ã¢ÂˆÂž as nÃ¢Â†Â’Ã¢ÂˆÂž.
Sheffer and Non-Sheffer Polynomial Families
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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.
Sugisaki, Kenji; Yamamoto, Satoru; Nakazawa, Shigeaki; Toyota, Kazuo; Sato, Kazunobu; Shiomi, Daisuke; Takui, Takeji
2016-08-18
Quantum computers are capable to efficiently perform full configuration interaction (FCI) calculations of atoms and molecules by using the quantum phase estimation (QPE) algorithm. Because the success probability of the QPE depends on the overlap between approximate and exact wave functions, efficient methods to prepare accurate initial guess wave functions enough to have sufficiently large overlap with the exact ones are highly desired. Here, we propose a quantum algorithm to construct the wave function consisting of one configuration state function, which is suitable for the initial guess wave function in QPE-based FCI calculations of open-shell molecules, based on the addition theorem of angular momentum. The proposed quantum algorithm enables us to prepare the wave function consisting of an exponential number of Slater determinants only by a polynomial number of quantum operations.
Complex Polynomial Vector Fields
DEFF Research Database (Denmark)
The two branches of dynamical systems, continuous and discrete, correspond to the study of differential equations (vector fields) and iteration of mappings respectively. In holomorphic dynamics, the systems studied are restricted to those described by holomorphic (complex analytic) functions...... or meromorphic (allowing poles as singularities) functions. There already exists a well-developed theory for iterative holomorphic dynamical systems, and successful relations found between iteration theory and flows of vector fields have been one of the main motivations for the recent interest in holomorphic...... vector fields. Since the class of complex polynomial vector fields in the plane is natural to consider, it is remarkable that its study has only begun very recently. There are numerous fundamental questions that are still open, both in the general classification of these vector fields, the decomposition...
On continuous lifetime distributions with polynomial failure rate with an application in reliability
International Nuclear Information System (INIS)
Csenki, Attila
2011-01-01
It is shown that the Laplace transform of a continuous lifetime random variable with a polynomial failure rate function satisfies a certain differential equation. This generates a set of differential equations which can be used to express the polynomial coefficients in terms of the derivatives of the Laplace transform at the origin. The technique described here establishes a procedure for estimating the polynomial coefficients from the sample moments of the distribution. Some special cases are worked through symbolically using computer algebra. Real data from the literature recording bus motor failures is used to compare the proposed approach with results based on the least squares procedure.
Guaranteed cost control of polynomial fuzzy systems via a sum of squares approach.
Tanaka, Kazuo; Ohtake, Hiroshi; Wang, Hua O
2009-04-01
This paper presents the guaranteed cost control of polynomial fuzzy systems via a sum of squares (SOS) approach. First, we present a polynomial fuzzy model and controller that are more general representations of the well-known Takagi-Sugeno (T-S) fuzzy model and controller, respectively. Second, we derive a guaranteed cost control design condition based on polynomial Lyapunov functions. Hence, the design approach discussed in this paper is more general than the existing LMI approaches (to T-S fuzzy control system designs) based on quadratic Lyapunov functions. The design condition realizes a guaranteed cost control by minimizing the upper bound of a given performance function. In addition, the design condition in the proposed approach can be represented in terms of SOS and is numerically (partially symbolically) solved via the recent developed SOSTOOLS. To illustrate the validity of the design approach, two design examples are provided. The first example deals with a complicated nonlinear system. The second example presents micro helicopter control. Both the examples show that our approach provides more extensive design results for the existing LMI approach.
Generalized Pseudospectral Method and Zeros of Orthogonal Polynomials
Directory of Open Access Journals (Sweden)
Oksana Bihun
2018-01-01
Full Text Available Via a generalization of the pseudospectral method for numerical solution of differential equations, a family of nonlinear algebraic identities satisfied by the zeros of a wide class of orthogonal polynomials is derived. The generalization is based on a modification of pseudospectral matrix representations of linear differential operators proposed in the paper, which allows these representations to depend on two, rather than one, sets of interpolation nodes. The identities hold for every polynomial family pνxν=0∞ orthogonal with respect to a measure supported on the real line that satisfies some standard assumptions, as long as the polynomials in the family satisfy differential equations Apν(x=qν(xpν(x, where A is a linear differential operator and each qν(x is a polynomial of degree at most n0∈N; n0 does not depend on ν. The proposed identities generalize known identities for classical and Krall orthogonal polynomials, to the case of the nonclassical orthogonal polynomials that belong to the class described above. The generalized pseudospectral representations of the differential operator A for the case of the Sonin-Markov orthogonal polynomials, also known as generalized Hermite polynomials, are presented. The general result is illustrated by new algebraic relations satisfied by the zeros of the Sonin-Markov polynomials.
On the Connection Coefficients of the Chebyshev-Boubaker Polynomials
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Paul Barry
2013-01-01
Full Text Available The Chebyshev-Boubaker polynomials are the orthogonal polynomials whose coefficient arrays are defined by ordinary Riordan arrays. Examples include the Chebyshev polynomials of the second kind and the Boubaker polynomials. We study the connection coefficients of this class of orthogonal polynomials, indicating how Riordan array techniques can lead to closed-form expressions for these connection coefficients as well as recurrence relations that define them.
Adaptive method for multi-dimensional integration and selection of a base of chaos polynomials
International Nuclear Information System (INIS)
Crestaux, T.
2011-01-01
This research thesis addresses the propagation of uncertainty in numerical simulations and its processing within a probabilistic framework by a functional approach based on random variable functions. The author reports the use of the spectral method to represent random variables by development in polynomial chaos. More precisely, the author uses the method of non-intrusive projection which uses the orthogonality of Chaos Polynomials to compute the development coefficients by approximation of scalar products. The approach is applied to a cavity and to waste storage [fr
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.
Solving polynomial systems using no-root elimination blending schemes
Barton, Michael
2011-12-01
Searching for the roots of (piecewise) polynomial systems of equations is a crucial problem in computer-aided design (CAD), and an efficient solution is in strong demand. Subdivision solvers are frequently used to achieve this goal; however, the subdivision process is expensive, and a vast number of subdivisions is to be expected, especially for higher-dimensional systems. Two blending schemes that efficiently reveal domains that cannot contribute by any root, and therefore significantly reduce the number of subdivisions, are proposed. Using a simple linear blend of functions of the given polynomial system, a function is sought after to be no-root contributing, with all control points of its BernsteinBézier representation of the same sign. If such a function exists, the domain is purged away from the subdivision process. The applicability is demonstrated on several CAD benchmark problems, namely surfacesurfacesurface intersection (SSSI) and surfacecurve intersection (SCI) problems, computation of the Hausdorff distance of two planar curves, or some kinematic-inspired tasks. © 2011 Elsevier Ltd. All rights reserved.
International Nuclear Information System (INIS)
Holmstroem, E.; Kuronen, A.; Nordlund, K.
2008-01-01
We studied threshold displacement energies for creating stable Frenkel pairs in silicon using density functional theory molecular dynamics simulations. The average threshold energy over all lattice directions was found to be 36±2 STAT ±2 SYST eV, and thresholds in the directions and were found to be 20±2 SYST eV and 12.5±1.5 SYST eV, respectively. Moreover, we found that in most studied lattice directions, a bond defect complex is formed with a lower threshold than a Frenkel pair. The average threshold energy for producing either a bond defect or a Frenkel pair was found to be 24±1 STAT ±2 SYST eV
Computation of the Likelihood in Biallelic Diffusion Models Using Orthogonal Polynomials
Directory of Open Access Journals (Sweden)
Claus Vogl
2014-11-01
Full Text Available In population genetics, parameters describing forces such as mutation, migration and drift are generally inferred from molecular data. Lately, approximate methods based on simulations and summary statistics have been widely applied for such inference, even though these methods waste information. In contrast, probabilistic methods of inference can be shown to be optimal, if their assumptions are met. In genomic regions where recombination rates are high relative to mutation rates, polymorphic nucleotide sites can be assumed to evolve independently from each other. The distribution of allele frequencies at a large number of such sites has been called “allele-frequency spectrum” or “site-frequency spectrum” (SFS. Conditional on the allelic proportions, the likelihoods of such data can be modeled as binomial. A simple model representing the evolution of allelic proportions is the biallelic mutation-drift or mutation-directional selection-drift diffusion model. With series of orthogonal polynomials, specifically Jacobi and Gegenbauer polynomials, or the related spheroidal wave function, the diffusion equations can be solved efficiently. In the neutral case, the product of the binomial likelihoods with the sum of such polynomials leads to finite series of polynomials, i.e., relatively simple equations, from which the exact likelihoods can be calculated. In this article, the use of orthogonal polynomials for inferring population genetic parameters is investigated.
International Nuclear Information System (INIS)
Feng Yi-Fu; Zhang Qing-Ling; Feng De-Zhi
2012-01-01
The global stability problem of Takagi—Sugeno (T—S) fuzzy Hopfield neural networks (FHNNs) with time delays is investigated. Novel LMI-based stability criteria are obtained by using Lyapunov functional theory to guarantee the asymptotic stability of the FHNNs with less conservatism. Firstly, using both Finsler's lemma and an improved homogeneous matrix polynomial technique, and applying an affine parameter-dependent Lyapunov—Krasovskii functional, we obtain the convergent LMI-based stability criteria. Algebraic properties of the fuzzy membership functions in the unit simplex are considered in the process of stability analysis via the homogeneous matrix polynomials technique. Secondly, to further reduce the conservatism, a new right-hand-side slack variables introducing technique is also proposed in terms of LMIs, which is suitable to the homogeneous matrix polynomials setting. Finally, two illustrative examples are given to show the efficiency of the proposed approaches
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
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
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
On polynomial solutions of the Heun equation
International Nuclear Information System (INIS)
Gurappa, N; Panigrahi, Prasanta K
2004-01-01
By making use of a recently developed method to solve linear differential equations of arbitrary order, we find a wide class of polynomial solutions to the Heun equation. We construct the series solution to the Heun equation before identifying the polynomial solutions. The Heun equation extended by the addition of a term, -σ/x, is also amenable for polynomial solutions. (letter to the editor)
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.
Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion
International Nuclear Information System (INIS)
Oladyshkin, S.; Nowak, W.
2012-01-01
We discuss the arbitrary polynomial chaos (aPC), which has been subject of research in a few recent theoretical papers. Like all polynomial chaos expansion techniques, aPC approximates the dependence of simulation model output on model parameters by expansion in an orthogonal polynomial basis. The aPC generalizes chaos expansion techniques towards arbitrary distributions with arbitrary probability measures, which can be either discrete, continuous, or discretized continuous and can be specified either analytically (as probability density/cumulative distribution functions), numerically as histogram or as raw data sets. We show that the aPC at finite expansion order only demands the existence of a finite number of moments and does not require the complete knowledge or even existence of a probability density function. This avoids the necessity to assign parametric probability distributions that are not sufficiently supported by limited available data. Alternatively, it allows modellers to choose freely of technical constraints the shapes of their statistical assumptions. Our key idea is to align the complexity level and order of analysis with the reliability and detail level of statistical information on the input parameters. We provide conditions for existence and clarify the relation of the aPC to statistical moments of model parameters. We test the performance of the aPC with diverse statistical distributions and with raw data. In these exemplary test cases, we illustrate the convergence with increasing expansion order and, for the first time, with increasing reliability level of statistical input information. Our results indicate that the aPC shows an exponential convergence rate and converges faster than classical polynomial chaos expansion techniques.
A Polynomial Estimate of Railway Line Delay
DEFF Research Database (Denmark)
Cerreto, Fabrizio; Harrod, Steven; Nielsen, Otto Anker
2017-01-01
Railway service may be measured by the aggregate delay over a time horizon or due to an event. Timetables for railway service may dampen aggregate delay by addition of additional process time, either supplement time or buffer time. The evaluation of these variables has previously been performed...... 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...
Colouring and knot polynomials
International Nuclear Information System (INIS)
Welsh, D.J.A.
1991-01-01
These lectures will attempt to explain a connection between the recent advances in knot theory using the Jones and related knot polynomials with classical problems in combinatorics and statistical mechanics. The difficulty of some of these problems will be analysed in the context of their computational complexity. In particular we shall discuss colourings and groups valued flows in graphs, knots and the Jones and Kauffman polynomials, the Ising, Potts and percolation problems of statistical physics, computational complexity of the above problems. (author). 20 refs, 9 figs
Phase unwrapping algorithm using polynomial phase approximation and linear Kalman filter.
Kulkarni, Rishikesh; Rastogi, Pramod
2018-02-01
A noise-robust phase unwrapping algorithm is proposed based on state space analysis and polynomial phase approximation using wrapped phase measurement. The true phase is approximated as a two-dimensional first order polynomial function within a small sized window around each pixel. The estimates of polynomial coefficients provide the measurement of phase and local fringe frequencies. A state space representation of spatial phase evolution and the wrapped phase measurement is considered with the state vector consisting of polynomial coefficients as its elements. Instead of using the traditional nonlinear Kalman filter for the purpose of state estimation, we propose to use the linear Kalman filter operating directly with the wrapped phase measurement. The adaptive window width is selected at each pixel based on the local fringe density to strike a balance between the computation time and the noise robustness. In order to retrieve the unwrapped phase, either a line-scanning approach or a quality guided strategy of pixel selection is used depending on the underlying continuous or discontinuous phase distribution, respectively. Simulation and experimental results are provided to demonstrate the applicability of the proposed method.
P A M Dirac meets M G Krein: matrix orthogonal polynomials and Dirac's equation
International Nuclear Information System (INIS)
Duran, Antonio J; Gruenbaum, F Alberto
2006-01-01
The solution of several instances of the Schroedinger equation (1926) is made possible by using the well-known orthogonal polynomials associated with the names of Hermite, Legendre and Laguerre. A relativistic alternative to this equation was proposed by Dirac (1928) involving differential operators with matrix coefficients. In 1949 Krein developed a theory of matrix-valued orthogonal polynomials without any reference to differential equations. In Duran A J (1997 Matrix inner product having a matrix symmetric second order differential operator Rocky Mt. J. Math. 27 585-600), one of us raised the question of determining instances of these matrix-valued polynomials going along with second order differential operators with matrix coefficients. In Duran A J and Gruenbaum F A (2004 Orthogonal matrix polynomials satisfying second order differential equations Int. Math. Res. Not. 10 461-84), we developed a method to produce such examples and observed that in certain cases there is a connection with the instance of Dirac's equation with a central potential. We observe that the case of the central Coulomb potential discussed in the physics literature in Darwin C G (1928 Proc. R. Soc. A 118 654), Nikiforov A F and Uvarov V B (1988 Special Functions of Mathematical Physics (Basle: Birkhauser) and Rose M E 1961 Relativistic Electron Theory (New York: Wiley)), and its solution, gives rise to a matrix weight function whose orthogonal polynomials solve a second order differential equation. To the best of our knowledge this is the first instance of a connection between the solution of the first order matrix equation of Dirac and the theory of matrix-valued orthogonal polynomials initiated by M G Krein
Homogenous polynomially parameter-dependent H∞ filter designs of discrete-time fuzzy systems.
Zhang, Huaguang; Xie, Xiangpeng; Tong, Shaocheng
2011-10-01
This paper proposes a novel H(∞) filtering technique for a class of discrete-time fuzzy systems. First, a novel kind of fuzzy H(∞) filter, which is homogenous polynomially parameter dependent on membership functions with an arbitrary degree, is developed to guarantee the asymptotic stability and a prescribed H(∞) performance of the filtering error system. Second, relaxed conditions for H(∞) performance analysis are proposed by using a new fuzzy Lyapunov function and the Finsler lemma with homogenous polynomial matrix Lagrange multipliers. Then, based on a new kind of slack variable technique, relaxed linear matrix inequality-based H(∞) filtering conditions are proposed. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed approach.
Quadratic polynomial interpolation on triangular domain
Li, Ying; Zhang, Congcong; Yu, Qian
2018-04-01
In the simulation of natural terrain, the continuity of sample points are not in consonance with each other always, traditional interpolation methods often can't faithfully reflect the shape information which lie in data points. So, a new method for constructing the polynomial interpolation surface on triangular domain is proposed. Firstly, projected the spatial scattered data points onto a plane and then triangulated them; Secondly, A C1 continuous piecewise quadric polynomial patch was constructed on each vertex, all patches were required to be closed to the line-interpolation one as far as possible. Lastly, the unknown quantities were gotten by minimizing the object functions, and the boundary points were treated specially. The result surfaces preserve as many properties of data points as possible under conditions of satisfying certain accuracy and continuity requirements, not too convex meantime. New method is simple to compute and has a good local property, applicable to shape fitting of mines and exploratory wells and so on. The result of new surface is given in experiments.
Lu, Zhaoyang; Xu, Wei; Sun, Decai; Han, Weiguo
2009-10-01
In this paper, the discounted penalty (Gerber-Shiu) functions for a risk model involving two independent classes of insurance risks under a threshold dividend strategy are developed. We also assume that the two claim number processes are independent Poisson and generalized Erlang (2) processes, respectively. When the surplus is above this threshold level, dividends are paid at a constant rate that does not exceed the premium rate. Two systems of integro-differential equations for discounted penalty functions are derived, based on whether the surplus is above this threshold level. Laplace transformations of the discounted penalty functions when the surplus is below the threshold level are obtained. And we also derive a system of renewal equations satisfied by the discounted penalty function with initial surplus above the threshold strategy via the Dickson-Hipp operator. Finally, analytical solutions of the two systems of integro-differential equations are presented.
A Design-Adaptive Local Polynomial Estimator for the Errors-in-Variables Problem
Delaigle, Aurore
2009-03-01
Local polynomial estimators are popular techniques for nonparametric regression estimation and have received great attention in the literature. Their simplest version, the local constant estimator, can be easily extended to the errors-in-variables context by exploiting its similarity with the deconvolution kernel density estimator. The generalization of the higher order versions of the estimator, however, is not straightforward and has remained an open problem for the last 15 years. We propose an innovative local polynomial estimator of any order in the errors-in-variables context, derive its design-adaptive asymptotic properties and study its finite sample performance on simulated examples. We provide not only a solution to a long-standing open problem, but also provide methodological contributions to error-invariable regression, including local polynomial estimation of derivative functions.
A Posteriori Error Analysis of Stochastic Differential Equations Using Polynomial Chaos Expansions
Butler, T.; Dawson, C.; Wildey, T.
2011-01-01
We develop computable a posteriori error estimates for linear functionals of a solution to a general nonlinear stochastic differential equation with random model/source parameters. These error estimates are based on a variational analysis applied to stochastic Galerkin methods for forward and adjoint problems. The result is a representation for the error estimate as a polynomial in the random model/source parameter. The advantage of this method is that we use polynomial chaos representations for the forward and adjoint systems to cheaply produce error estimates by simple evaluation of a polynomial. By comparison, the typical method of producing such estimates requires repeated forward/adjoint solves for each new choice of random parameter. We present numerical examples showing that there is excellent agreement between these methods. © 2011 Society for Industrial and Applied Mathematics.
An efficient coupled polynomial interpolation scheme for shear mode sandwich beam finite element
Directory of Open Access Journals (Sweden)
Litesh N. Sulbhewar
Full Text Available An efficient piezoelectric sandwich beam finite element is presented here. It employs the coupled polynomial field interpolation scheme for field variables which incorporates electromechanical coupling at interpolation level itself; unlike conventional sandwich beam theory (SBT based formulations available in the literature. A variational formulation is used to derive the governing equations, which are used to establish the relationships between field variables. These relations lead to the coupled polynomial field descriptions of variables, unlike conventional SBT formulations which use assumed independent polynomials. The relative axial displacement is expressed only by coupled terms containing contributions from other mechanical and electrical variables, thus eliminating use of the transverse displacement derivative as a degree of freedom. A set of coupled shape function based on these polynomials has shown the improvement in the convergence characteristics of the SBT based formulation. This improvement in the performance is achieved with one nodal degree of freedom lesser than the conventional SBT formulations.
Factoring polynomials over arbitrary finite fields
Lange, T.; Winterhof, A.
2000-01-01
We analyse an extension of Shoup's (Inform. Process. Lett. 33 (1990) 261–267) deterministic algorithm for factoring polynomials over finite prime fields to arbitrary finite fields. In particular, we prove the existence of a deterministic algorithm which completely factors all monic polynomials of
A Determinant Expression for the Generalized Bessel Polynomials
Directory of Open Access Journals (Sweden)
Sheng-liang Yang
2013-01-01
Full Text Available Using the exponential Riordan arrays, we show that a variation of the generalized Bessel polynomial sequence is of Sheffer type, and we obtain a determinant formula for the generalized Bessel polynomials. As a result, the Bessel polynomial is represented as determinant the entries of which involve Catalan numbers.
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...
Numerical Simulation of Polynomial-Speed Convergence Phenomenon
Li, Yao; Xu, Hui
2017-11-01
We provide a hybrid method that captures the polynomial speed of convergence and polynomial speed of mixing for Markov processes. The hybrid method that we introduce is based on the coupling technique and renewal theory. We propose to replace some estimates in classical results about the ergodicity of Markov processes by numerical simulations when the corresponding analytical proof is difficult. After that, all remaining conclusions can be derived from rigorous analysis. Then we apply our results to seek numerical justification for the ergodicity of two 1D microscopic heat conduction models. The mixing rate of these two models are expected to be polynomial but very difficult to prove. In both examples, our numerical results match the expected polynomial mixing rate well.
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 ...
A companion matrix for 2-D polynomials
International Nuclear Information System (INIS)
Boudellioua, M.S.
1995-08-01
In this paper, a matrix form analogous to the companion matrix which is often encountered in the theory of one dimensional (1-D) linear systems is suggested for a class of polynomials in two indeterminates and real coefficients, here referred to as two dimensional (2-D) polynomials. These polynomials arise in the context of 2-D linear systems theory. Necessary and sufficient conditions are also presented under which a matrix is equivalent to this companion form. (author). 6 refs
Inelastic scattering with Chebyshev polynomials and preconditioned conjugate gradient minimization.
Temel, Burcin; Mills, Greg; Metiu, Horia
2008-03-27
We describe and test an implementation, using a basis set of Chebyshev polynomials, of a variational method for solving scattering problems in quantum mechanics. This minimum error method (MEM) determines the wave function Psi by minimizing the least-squares error in the function (H Psi - E Psi), where E is the desired scattering energy. We compare the MEM to an alternative, the Kohn variational principle (KVP), by solving the Secrest-Johnson model of two-dimensional inelastic scattering, which has been studied previously using the KVP and for which other numerical solutions are available. We use a conjugate gradient (CG) method to minimize the error, and by preconditioning the CG search, we are able to greatly reduce the number of iterations necessary; the method is thus faster and more stable than a matrix inversion, as is required in the KVP. Also, we avoid errors due to scattering off of the boundaries, which presents substantial problems for other methods, by matching the wave function in the interaction region to the correct asymptotic states at the specified energy; the use of Chebyshev polynomials allows this boundary condition to be implemented accurately. The use of Chebyshev polynomials allows for a rapid and accurate evaluation of the kinetic energy. This basis set is as efficient as plane waves but does not impose an artificial periodicity on the system. There are problems in surface science and molecular electronics which cannot be solved if periodicity is imposed, and the Chebyshev basis set is a good alternative in such situations.
Simulation of aspheric tolerance with polynomial fitting
Li, Jing; Cen, Zhaofeng; Li, Xiaotong
2018-01-01
The shape of the aspheric lens changes caused by machining errors, resulting in a change in the optical transfer function, which affects the image quality. At present, there is no universally recognized tolerance criterion standard for aspheric surface. To study the influence of aspheric tolerances on the optical transfer function, the tolerances of polynomial fitting are allocated on the aspheric surface, and the imaging simulation is carried out by optical imaging software. Analysis is based on a set of aspheric imaging system. The error is generated in the range of a certain PV value, and expressed as a form of Zernike polynomial, which is added to the aspheric surface as a tolerance term. Through optical software analysis, the MTF of optical system can be obtained and used as the main evaluation index. Evaluate whether the effect of the added error on the MTF of the system meets the requirements of the current PV value. Change the PV value and repeat the operation until the acceptable maximum allowable PV value is obtained. According to the actual processing technology, consider the error of various shapes, such as M type, W type, random type error. The new method will provide a certain development for the actual free surface processing technology the reference value.
Computation of rectangular source integral by rational parameter polynomial method
International Nuclear Information System (INIS)
Prabha, Hem
2001-01-01
Hubbell et al. (J. Res. Nat Bureau Standards 64C, (1960) 121) have obtained a series expansion for the calculation of the radiation field generated by a plane isotropic rectangular source (plaque), in which leading term is the integral H(a,b). In this paper another integral I(a,b), which is related with the integral H(a,b) has been solved by the rational parameter polynomial method. From I(a,b), we compute H(a,b). Using this method the integral I(a,b) is expressed in the form of a polynomial of a rational parameter. Generally, a function f (x) is expressed in terms of x. In this method this is expressed in terms of x/(1+x). In this way, the accuracy of the expression is good over a wide range of x as compared to the earlier approach. The results for I(a,b) and H(a,b) are given for a sixth degree polynomial and are found to be in good agreement with the results obtained by numerically integrating the integral. Accuracy could be increased either by increasing the degree of the polynomial or by dividing the range of integration. The results of H(a,b) and I(a,b) are given for values of b and a up to 2.0 and 20.0, respectively
A Synoptic of Software Implementation for Shift Registers Based on 16th Degree Primitive Polynomials
Directory of Open Access Journals (Sweden)
Mirella Amelia Mioc
2016-08-01
Full Text Available Almost all of the major applications in the specific Fields of Communication used a well-known device called Linear Feedback Shift Register. Usually LFSR functions in a Galois Field GF(2n, meaning that all the operations are done with arithmetic modulo n degree Irreducible and especially Primitive Polynomials. Storing data in Galois Fields allows effective and manageable manipulation, mainly in computer cryptographic applications. The analysis of functioning for Primitive Polynomials of 16th degree shows that almost all the obtained results are in the same time distribution.
Degenerate r-Stirling Numbers and r-Bell Polynomials
Kim, T.; Yao, Y.; Kim, D. S.; Jang, G.-W.
2018-01-01
The purpose of this paper is to exploit umbral calculus in order to derive some properties, recurrence relations, and identities related to the degenerate r-Stirling numbers of the second kind and the degenerate r-Bell polynomials. Especially, we will express the degenerate r-Bell polynomials as linear combinations of many well-known families of special polynomials.
Commutators with idempotent values on multilinear polynomials in ...
Indian Academy of Sciences (India)
Multilinear polynomial; derivations; generalized polynomial identity; prime ring; right ideal. Abstract. Let R be a prime ring of characteristic different from 2, C its extended centroid, d a nonzero derivation of R , f ( x 1 , … , x n ) a multilinear polynomial over C , ϱ a nonzero right ideal of R and m > 1 a fixed integer such that.
Polynomial weights and code constructions
DEFF Research Database (Denmark)
Massey, J; Costello, D; Justesen, Jørn
1973-01-01
polynomial included. This fundamental property is then used as the key to a variety of code constructions including 1) a simplified derivation of the binary Reed-Muller codes and, for any primepgreater than 2, a new extensive class ofp-ary "Reed-Muller codes," 2) a new class of "repeated-root" cyclic codes...... of long constraint length binary convolutional codes derived from2^r-ary Reed-Solomon codes, and 6) a new class ofq-ary "repeated-root" constacyclic codes with an algebraic decoding algorithm.......For any nonzero elementcof a general finite fieldGF(q), it is shown that the polynomials(x - c)^i, i = 0,1,2,cdots, have the "weight-retaining" property that any linear combination of these polynomials with coefficients inGF(q)has Hamming weight at least as great as that of the minimum degree...
The generalized Yablonskii-Vorob'ev polynomials and their properties
International Nuclear Information System (INIS)
Kudryashov, Nikolai A.; Demina, Maria V.
2008-01-01
Rational solutions of the generalized second Painleve hierarchy are classified. Representation of the rational solutions in terms of special polynomials, the generalized Yablonskii-Vorob'ev polynomials, is introduced. Differential-difference relations satisfied by the polynomials are found. Hierarchies of differential equations related to the generalized second Painleve hierarchy are derived. One of these hierarchies is a sequence of differential equations satisfied by the generalized Yablonskii-Vorob'ev polynomials
2-variable Laguerre matrix polynomials and Lie-algebraic techniques
International Nuclear Information System (INIS)
Khan, Subuhi; Hassan, Nader Ali Makboul
2010-01-01
The authors introduce 2-variable forms of Laguerre and modified Laguerre matrix polynomials and derive their special properties. Further, the representations of the special linear Lie algebra sl(2) and the harmonic oscillator Lie algebra G(0,1) are used to derive certain results involving these polynomials. Furthermore, the generating relations for the ordinary as well as matrix polynomials related to these matrix polynomials are derived as applications.
Describing Quadratic Cremer Point Polynomials by Parabolic Perturbations
DEFF Research Database (Denmark)
Sørensen, Dan Erik Krarup
1996-01-01
We describe two infinite order parabolic perturbation proceduresyielding quadratic polynomials having a Cremer fixed point. The main ideais to obtain the polynomial as the limit of repeated parabolic perturbations.The basic tool at each step is to control the behaviour of certain externalrays.......Polynomials of the Cremer type correspond to parameters at the boundary of ahyperbolic component of the Mandelbrot set. In this paper we concentrate onthe main cardioid component. We investigate the differences between two-sided(i.e. alternating) and one-sided parabolic perturbations.In the two-sided case, we prove...... the existence of polynomials having an explicitlygiven external ray accumulating both at the Cremer point and at its non-periodicpreimage. We think of the Julia set as containing a "topologists double comb".In the one-sided case we prove a weaker result: the existence of polynomials havingan explicitly given...
A note on some identities of derangement polynomials.
Kim, Taekyun; Kim, Dae San; Jang, Gwan-Woo; Kwon, Jongkyum
2018-01-01
The problem of counting derangements was initiated by Pierre Rémond de Montmort in 1708 (see Carlitz in Fibonacci Q. 16(3):255-258, 1978, Clarke and Sved in Math. Mag. 66(5):299-303, 1993, Kim, Kim and Kwon in Adv. Stud. Contemp. Math. (Kyungshang) 28(1):1-11 2018. A derangement is a permutation that has no fixed points, and the derangement number [Formula: see text] is the number of fixed-point-free permutations on an n element set. In this paper, we study the derangement polynomials and investigate some interesting properties which are related to derangement numbers. Also, we study two generalizations of derangement polynomials, namely higher-order and r -derangement polynomials, and show some relations between them. In addition, we express several special polynomials in terms of the higher-order derangement polynomials by using umbral calculus.
Topological quantum information, virtual Jones polynomials and Khovanov homology
International Nuclear Information System (INIS)
Kauffman, Louis H
2011-01-01
In this paper, we give a quantum statistical interpretation of the bracket polynomial state sum 〈K〉, the Jones polynomial V K (t) and virtual knot theory versions of the Jones polynomial, including the arrow polynomial. We use these quantum mechanical interpretations to give new quantum algorithms for these Jones polynomials. In those cases where the Khovanov homology is defined, the Hilbert space C(K) of our model is isomorphic with the chain complex for Khovanov homology with coefficients in the complex numbers. There is a natural unitary transformation U:C(K) → C(K) such that 〈K〉 = Trace(U), where 〈K〉 denotes the evaluation of the state sum model for the corresponding polynomial. We show that for the Khovanov boundary operator ∂:C(K) → C(K), we have the relationship ∂U + U∂ = 0. Consequently, the operator U acts on the Khovanov homology, and we obtain a direct relationship between the Khovanov homology and this quantum algorithm for the Jones polynomial. (paper)
Polynomial solutions of the Monge-Ampère equation
Energy Technology Data Exchange (ETDEWEB)
Aminov, Yu A [B.Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, Khar' kov (Ukraine)
2014-11-30
The question of the existence of polynomial solutions to the Monge-Ampère equation z{sub xx}z{sub yy}−z{sub xy}{sup 2}=f(x,y) is considered in the case when f(x,y) is a polynomial. It is proved that if f is a polynomial of the second degree, which is positive for all values of its arguments and has a positive squared part, then no polynomial solution exists. On the other hand, a solution which is not polynomial but is analytic in the whole of the x, y-plane is produced. Necessary and sufficient conditions for the existence of polynomial solutions of degree up to 4 are found and methods for the construction of such solutions are indicated. An approximation theorem is proved. Bibliography: 10 titles.
Tsallis p, q-deformed Touchard polynomials and Stirling numbers
Herscovici, O.; Mansour, T.
2017-01-01
In this paper, we develop and investigate a new two-parametrized deformation of the Touchard polynomials, based on the definition of the NEXT q-exponential function of Tsallis. We obtain new generalizations of the Stirling numbers of the second kind and of the binomial coefficients and represent two new statistics for the set partitions.
Laguerre polynomials by a harmonic oscillator
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.
A Genetic algorithm for evaluating the zeros (roots) of polynomial ...
African Journals Online (AJOL)
This paper presents a Genetic Algorithm software (which is a computational, search technique) for finding the zeros (roots) of any given polynomial function, and optimizing and solving N-dimensional systems of equations. The software is particularly useful since most of the classic schemes are not all embracing.
On the optimal polynomial approximation of stochastic PDEs by galerkin and collocation methods
Beck, Joakim; Tempone, Raul; Nobile, Fabio; Tamellini, Lorenzo
2012-01-01
In this work we focus on the numerical approximation of the solution u of a linear elliptic PDE with stochastic coefficients. The problem is rewritten as a parametric PDE and the functional dependence of the solution on the parameters is approximated by multivariate polynomials. We first consider the stochastic Galerkin method, and rely on sharp estimates for the decay of the Fourier coefficients of the spectral expansion of u on an orthogonal polynomial basis to build a sequence of polynomial subspaces that features better convergence properties, in terms of error versus number of degrees of freedom, than standard choices such as Total Degree or Tensor Product subspaces. We consider then the Stochastic Collocation method, and use the previous estimates to introduce a new class of Sparse Grids, based on the idea of selecting a priori the most profitable hierarchical surpluses, that, again, features better convergence properties compared to standard Smolyak or tensor product grids. Numerical results show the effectiveness of the newly introduced polynomial spaces and sparse grids. © 2012 World Scientific Publishing Company.
On the optimal polynomial approximation of stochastic PDEs by galerkin and collocation methods
Beck, Joakim
2012-09-01
In this work we focus on the numerical approximation of the solution u of a linear elliptic PDE with stochastic coefficients. The problem is rewritten as a parametric PDE and the functional dependence of the solution on the parameters is approximated by multivariate polynomials. We first consider the stochastic Galerkin method, and rely on sharp estimates for the decay of the Fourier coefficients of the spectral expansion of u on an orthogonal polynomial basis to build a sequence of polynomial subspaces that features better convergence properties, in terms of error versus number of degrees of freedom, than standard choices such as Total Degree or Tensor Product subspaces. We consider then the Stochastic Collocation method, and use the previous estimates to introduce a new class of Sparse Grids, based on the idea of selecting a priori the most profitable hierarchical surpluses, that, again, features better convergence properties compared to standard Smolyak or tensor product grids. Numerical results show the effectiveness of the newly introduced polynomial spaces and sparse grids. © 2012 World Scientific Publishing Company.
Hybrid threshold adaptable quantum secret sharing scheme with reverse Huffman-Fibonacci-tree coding.
Lai, Hong; Zhang, Jun; Luo, Ming-Xing; Pan, Lei; Pieprzyk, Josef; Xiao, Fuyuan; Orgun, Mehmet A
2016-08-12
With prevalent attacks in communication, sharing a secret between communicating parties is an ongoing challenge. Moreover, it is important to integrate quantum solutions with classical secret sharing schemes with low computational cost for the real world use. This paper proposes a novel hybrid threshold adaptable quantum secret sharing scheme, using an m-bonacci orbital angular momentum (OAM) pump, Lagrange interpolation polynomials, and reverse Huffman-Fibonacci-tree coding. To be exact, we employ entangled states prepared by m-bonacci sequences to detect eavesdropping. Meanwhile, we encode m-bonacci sequences in Lagrange interpolation polynomials to generate the shares of a secret with reverse Huffman-Fibonacci-tree coding. The advantages of the proposed scheme is that it can detect eavesdropping without joint quantum operations, and permits secret sharing for an arbitrary but no less than threshold-value number of classical participants with much lower bandwidth. Also, in comparison with existing quantum secret sharing schemes, it still works when there are dynamic changes, such as the unavailability of some quantum channel, the arrival of new participants and the departure of participants. Finally, we provide security analysis of the new hybrid quantum secret sharing scheme and discuss its useful features for modern applications.
Polynomial selection in number field sieve for integer factorization
Directory of Open Access Journals (Sweden)
Gireesh Pandey
2016-09-01
Full Text Available The general number field sieve (GNFS is the fastest algorithm for factoring large composite integers which is made up by two prime numbers. Polynomial selection is an important step of GNFS. The asymptotic runtime depends on choice of good polynomial pairs. In this paper, we present polynomial selection algorithm that will be modelled with size and root properties. The correlations between polynomial coefficient and number of relations have been explored with experimental findings.
Polynomial solutions of nonlinear integral equations
International Nuclear Information System (INIS)
Dominici, Diego
2009-01-01
We analyze the polynomial solutions of a nonlinear integral equation, generalizing the work of Bender and Ben-Naim (2007 J. Phys. A: Math. Theor. 40 F9, 2008 J. Nonlinear Math. Phys. 15 (Suppl. 3) 73). We show that, in some cases, an orthogonal solution exists and we give its general form in terms of kernel polynomials
Polynomial solutions of nonlinear integral equations
Energy Technology Data Exchange (ETDEWEB)
Dominici, Diego [Department of Mathematics, State University of New York at New Paltz, 1 Hawk Dr. Suite 9, New Paltz, NY 12561-2443 (United States)], E-mail: dominicd@newpaltz.edu
2009-05-22
We analyze the polynomial solutions of a nonlinear integral equation, generalizing the work of Bender and Ben-Naim (2007 J. Phys. A: Math. Theor. 40 F9, 2008 J. Nonlinear Math. Phys. 15 (Suppl. 3) 73). We show that, in some cases, an orthogonal solution exists and we give its general form in terms of kernel polynomials.
Laguerre polynomials by a harmonic oscillator
International Nuclear Information System (INIS)
Baykal, Melek; Baykal, Ahmet
2014-01-01
The study of an isotropic harmonic oscillator, using the factorization method given in Ohanian's textbook on quantum mechanics, is refined and some collateral extensions of the method related to the ladder operators and the associated Laguerre polynomials are presented. In particular, some analytical properties of the associated Laguerre polynomials are derived using the ladder operators. (paper)
General quantum polynomials: irreducible modules and Morita equivalence
International Nuclear Information System (INIS)
Artamonov, V A
1999-01-01
In this paper we continue the investigation of the structure of finitely generated modules over rings of general quantum (Laurent) polynomials. We obtain a description of the lattice of submodules of periodic finitely generated modules and describe the irreducible modules. We investigate the problem of Morita equivalence of rings of general quantum polynomials, consider properties of division rings of fractions, and solve Zariski's problem for quantum polynomials
Multivariable biorthogonal continuous--discrete Wilson and Racah polynomials
International Nuclear Information System (INIS)
Tratnik, M.V.
1990-01-01
Several families of multivariable, biorthogonal, partly continuous and partly discrete, Wilson polynomials are presented. These yield limit cases that are purely continuous in some of the variables and purely discrete in the others, or purely discrete in all the variables. The latter are referred to as the multivariable biorthogonal Racah polynomials. Interesting further limit cases include the multivariable biorthogonal Hahn and dual Hahn polynomials
Primitive polynomials selection method for pseudo-random number generator
Anikin, I. V.; Alnajjar, Kh
2018-01-01
In this paper we suggested the method for primitive polynomials selection of special type. This kind of polynomials can be efficiently used as a characteristic polynomials for linear feedback shift registers in pseudo-random number generators. The proposed method consists of two basic steps: finding minimum-cost irreducible polynomials of the desired degree and applying primitivity tests to get the primitive ones. Finally two primitive polynomials, which was found by the proposed method, used in pseudorandom number generator based on fuzzy logic (FRNG) which had been suggested before by the authors. The sequences generated by new version of FRNG have low correlation magnitude, high linear complexity, less power consumption, is more balanced and have better statistical properties.
Neck curve polynomials in neck rupture model
International Nuclear Information System (INIS)
Kurniadi, Rizal; Perkasa, Yudha S.; Waris, Abdul
2012-01-01
The Neck Rupture Model is a model that explains the scission process which has smallest radius in liquid drop at certain position. Old fashion of rupture position is determined randomly so that has been called as Random Neck Rupture Model (RNRM). The neck curve polynomials have been employed in the Neck Rupture Model for calculation the fission yield of neutron induced fission reaction of 280 X 90 with changing of order of polynomials as well as temperature. The neck curve polynomials approximation shows the important effects in shaping of fission yield curve.
The finite Fourier transform of classical polynomials
Dixit, Atul; Jiu, Lin; Moll, Victor H.; Vignat, Christophe
2014-01-01
The finite Fourier transform of a family of orthogonal polynomials $A_{n}(x)$, is the usual transform of the polynomial extended by $0$ outside their natural domain. Explicit expressions are given for the Legendre, Jacobi, Gegenbauer and Chebyshev families.
Santos-Concejero, Jordan; Tucker, Ross; Granados, Cristina; Irazusta, Jon; Bidaurrazaga-Letona, Iraia; Zabala-Lili, Jon; Gil, Susana María
2014-01-01
This study investigated the influence of the regression model and initial intensity during an incremental test on the relationship between the lactate threshold estimated by the maximal-deviation method and performance in elite-standard runners. Twenty-three well-trained runners completed a discontinuous incremental running test on a treadmill. Speed started at 9 km · h(-1) and increased by 1.5 km · h(-1) every 4 min until exhaustion, with a minute of recovery for blood collection. Lactate-speed data were fitted by exponential and polynomial models. The lactate threshold was determined for both models, using all the co-ordinates, excluding the first and excluding the first and second points. The exponential lactate threshold was greater than the polynomial equivalent in any co-ordinate condition (P performance and is independent of the initial intensity of the test.
Algebraic limit cycles in polynomial systems of differential equations
International Nuclear Information System (INIS)
Llibre, Jaume; Zhao Yulin
2007-01-01
Using elementary tools we construct cubic polynomial systems of differential equations with algebraic limit cycles of degrees 4, 5 and 6. We also construct a cubic polynomial system of differential equations having an algebraic homoclinic loop of degree 3. Moreover, we show that there are polynomial systems of differential equations of arbitrary degree that have algebraic limit cycles of degree 3, as well as give an example of a cubic polynomial system of differential equations with two algebraic limit cycles of degree 4
From sequences to polynomials and back, via operator orderings
Energy Technology Data Exchange (ETDEWEB)
Amdeberhan, Tewodros, E-mail: tamdeber@tulane.edu; Dixit, Atul, E-mail: adixit@tulane.edu; Moll, Victor H., E-mail: vhm@tulane.edu [Department of Mathematics, Tulane University, New Orleans, Louisiana 70118 (United States); De Angelis, Valerio, E-mail: vdeangel@xula.edu [Department of Mathematics, Xavier University of Louisiana, New Orleans, Louisiana 70125 (United States); Vignat, Christophe, E-mail: vignat@tulane.edu [Department of Mathematics, Tulane University, New Orleans, Louisiana 70118, USA and L.S.S. Supelec, Universite d' Orsay (France)
2013-12-15
Bender and Dunne [“Polynomials and operator orderings,” J. Math. Phys. 29, 1727–1731 (1988)] showed that linear combinations of words q{sup k}p{sup n}q{sup n−k}, where p and q are subject to the relation qp − pq = ı, may be expressed as a polynomial in the symbol z=1/2 (qp+pq). Relations between such polynomials and linear combinations of the transformed coefficients are explored. In particular, examples yielding orthogonal polynomials are provided.
Connection coefficients between Boas-Buck polynomial sets
Cheikh, Y. Ben; Chaggara, H.
2006-07-01
In this paper, a general method to express explicitly connection coefficients between two Boas-Buck polynomial sets is presented. As application, we consider some generalized hypergeometric polynomials, from which we derive some well-known results including duplication and inversion formulas.
EEG-based functional networks evoked by acupuncture at ST 36: A data-driven thresholding study
Li, Huiyan; Wang, Jiang; Yi, Guosheng; Deng, Bin; Zhou, Hexi
2017-10-01
This paper investigates how acupuncture at ST 36 modulates the brain functional network. 20 channel EEG signals from 15 healthy subjects are respectively recorded before, during and after acupuncture. The correlation between two EEG channels is calculated by using Pearson’s coefficient. A data-driven approach is applied to determine the threshold, which is performed by considering the connected set, connected edge and network connectivity. Based on such thresholding approach, the functional network in each acupuncture period is built with graph theory, and the associated functional connectivity is determined. We show that acupuncturing at ST 36 increases the connectivity of the EEG-based functional network, especially for the long distance ones between two hemispheres. The properties of the functional network in five EEG sub-bands are also characterized. It is found that the delta and gamma bands are affected more obviously by acupuncture than the other sub-bands. These findings highlight the modulatory effects of acupuncture on the EEG-based functional connectivity, which is helpful for us to understand how it participates in the cortical or subcortical activities. Further, the data-driven threshold provides an alternative approach to infer the functional connectivity under other physiological conditions.
Intermediate structure and threshold phenomena
International Nuclear Information System (INIS)
Hategan, Cornel
2004-01-01
The Intermediate Structure, evidenced through microstructures of the neutron strength function, is reflected in open reaction channels as fluctuations in excitation function of nuclear threshold effects. The intermediate state supporting both neutron strength function and nuclear threshold effect is a micro-giant neutron threshold state. (author)
Multi-Valued Logic Gates, Continuous Sensitivity, Reversibility, and Threshold Functions
İlhan, Aslı Güçlükan; Ünlü, Özgün
2016-01-01
We define an invariant of a multi-valued logic gate by considering the number of certain threshold functions associated with the gate. We call this invariant the continuous sensitivity of the gate. We discuss a method for analysing continuous sensitivity of a multi-valued logic gate by using experimental data about the gate. In particular, we will show that this invariant provides a lower bound for the sensitivity of a boolean function considered as a multi-valued logic gate. We also discuss ...
International Nuclear Information System (INIS)
Yasa, F.; Anli, F.; Guengoer, S.
2007-01-01
We present analytical calculations of spherically symmetric radioactive transfer and neutron transport using a hypothesis of P1 and T1 low order polynomial approximation for diffusion coefficient D. Transport equation in spherical geometry is considered as the pseudo slab equation. The validity of polynomial expansionion in transport theory is investigated through a comparison with classic diffusion theory. It is found that for causes when the fluctuation of the scattering cross section dominates, the quantitative difference between the polynomial approximation and diffusion results was physically acceptable in general
On Roots of Polynomials and Algebraically Closed Fields
Directory of Open Access Journals (Sweden)
Schwarzweller Christoph
2017-10-01
Full Text Available In this article we further extend the algebraic theory of polynomial rings in Mizar [1, 2, 3]. We deal with roots and multiple roots of polynomials and show that both the real numbers and finite domains are not algebraically closed [5, 7]. We also prove the identity theorem for polynomials and that the number of multiple roots is bounded by the polynomial’s degree [4, 6].
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
Some properties of generalized self-reciprocal polynomials over finite fields
Directory of Open Access Journals (Sweden)
Ryul Kim
2014-07-01
Full Text Available Numerous results on self-reciprocal polynomials over finite fields have been studied. In this paper we generalize some of these to a-self reciprocal polynomials defined in [4]. We consider some properties of the divisibility of a-reciprocal polynomials and characterize the parity of the number of irreducible factors for a-self reciprocal polynomials over finite fields of odd characteristic.
Energy Technology Data Exchange (ETDEWEB)
Degroote, M. [Rice Univ., Houston, TX (United States); Henderson, T. M. [Rice Univ., Houston, TX (United States); Zhao, J. [Rice Univ., Houston, TX (United States); Dukelsky, J. [Consejo Superior de Investigaciones Cientificas (CSIC), Madrid (Spain). Inst. de Estructura de la Materia; Scuseria, G. E. [Rice Univ., Houston, TX (United States)
2018-01-03
We present a similarity transformation theory based on a polynomial form of a particle-hole pair excitation operator. In the weakly correlated limit, this polynomial becomes an exponential, leading to coupled cluster doubles. In the opposite strongly correlated limit, the polynomial becomes an extended Bessel expansion and yields the projected BCS wavefunction. In between, we interpolate using a single parameter. The e ective Hamiltonian is non-hermitian and this Polynomial Similarity Transformation Theory follows the philosophy of traditional coupled cluster, left projecting the transformed Hamiltonian onto subspaces of the Hilbert space in which the wave function variance is forced to be zero. Similarly, the interpolation parameter is obtained through minimizing the next residual in the projective hierarchy. We rationalize and demonstrate how and why coupled cluster doubles is ill suited to the strongly correlated limit whereas the Bessel expansion remains well behaved. The model provides accurate wave functions with energy errors that in its best variant are smaller than 1% across all interaction stengths. The numerical cost is polynomial in system size and the theory can be straightforwardly applied to any realistic Hamiltonian.
on the performance of Autoregressive Moving Average Polynomial
African Journals Online (AJOL)
Timothy Ademakinwa
Distributed Lag (PDL) model, Autoregressive Polynomial Distributed Lag ... Moving Average Polynomial Distributed Lag (ARMAPDL) model. ..... Global Journal of Mathematics and Statistics. Vol. 1. ... Business and Economic Research Center.
Application of polynomial preconditioners to conservation laws
Geurts, Bernardus J.; van Buuren, R.; Lu, H.
2000-01-01
Polynomial preconditioners which are suitable in implicit time-stepping methods for conservation laws are reviewed and analyzed. The preconditioners considered are either based on a truncation of a Neumann series or on Chebyshev polynomials for the inverse of the system-matrix. The latter class of
International Nuclear Information System (INIS)
Sulbhewar, Litesh N; Raveendranath, P
2014-01-01
An efficient piezoelectric smart beam finite element based on Reddy’s third-order displacement field and layerwise linear potential is presented here. The present formulation is based on the coupled polynomial field interpolation of variables, unlike conventional piezoelectric beam formulations that use independent polynomials. Governing equations derived using a variational formulation are used to establish the relationship between field variables. The resulting expressions are used to formulate coupled shape functions. Starting with an assumed cubic polynomial for transverse displacement (w) and a linear polynomial for electric potential (φ), coupled polynomials for axial displacement (u) and section rotation (θ) are found. This leads to a coupled quadratic polynomial representation for axial displacement (u) and section rotation (θ). The formulation allows accommodation of extension–bending, shear–bending and electromechanical couplings at the interpolation level itself, in a variationally consistent manner. The proposed interpolation scheme is shown to eliminate the locking effects exhibited by conventional independent polynomial field interpolations and improve the convergence characteristics of HSDT based piezoelectric beam elements. Also, the present coupled formulation uses only three mechanical degrees of freedom per node, one less than the conventional formulations. Results from numerical test problems prove the accuracy and efficiency of the present formulation. (paper)
Applications of polynomial optimization in financial risk investment
Zeng, Meilan; Fu, Hongwei
2017-09-01
Recently, polynomial optimization has many important applications in optimization, financial economics and eigenvalues of tensor, etc. This paper studies the applications of polynomial optimization in financial risk investment. We consider the standard mean-variance risk measurement model and the mean-variance risk measurement model with transaction costs. We use Lasserre's hierarchy of semidefinite programming (SDP) relaxations to solve the specific cases. The results show that polynomial optimization is effective for some financial optimization problems.
P A M Dirac meets M G Krein: matrix orthogonal polynomials and Dirac's equation
Energy Technology Data Exchange (ETDEWEB)
Duran, Antonio J [Departamento de Analisis Matematico, Universidad de Sevilla, Apdo (PO BOX) 1160, 41080 Sevilla (Spain); Gruenbaum, F Alberto [Department of Mathematics, University of California, Berkeley, CA 94720 (United States)
2006-04-07
The solution of several instances of the Schroedinger equation (1926) is made possible by using the well-known orthogonal polynomials associated with the names of Hermite, Legendre and Laguerre. A relativistic alternative to this equation was proposed by Dirac (1928) involving differential operators with matrix coefficients. In 1949 Krein developed a theory of matrix-valued orthogonal polynomials without any reference to differential equations. In Duran A J (1997 Matrix inner product having a matrix symmetric second order differential operator Rocky Mt. J. Math. 27 585-600), one of us raised the question of determining instances of these matrix-valued polynomials going along with second order differential operators with matrix coefficients. In Duran A J and Gruenbaum F A (2004 Orthogonal matrix polynomials satisfying second order differential equations Int. Math. Res. Not. 10 461-84), we developed a method to produce such examples and observed that in certain cases there is a connection with the instance of Dirac's equation with a central potential. We observe that the case of the central Coulomb potential discussed in the physics literature in Darwin C G (1928 Proc. R. Soc. A 118 654), Nikiforov A F and Uvarov V B (1988 Special Functions of Mathematical Physics (Basle: Birkhauser) and Rose M E 1961 Relativistic Electron Theory (New York: Wiley)), and its solution, gives rise to a matrix weight function whose orthogonal polynomials solve a second order differential equation. To the best of our knowledge this is the first instance of a connection between the solution of the first order matrix equation of Dirac and the theory of matrix-valued orthogonal polynomials initiated by M G Krein.
Polynomially Riesz elements | Živković-Zlatanović | Quaestiones ...
African Journals Online (AJOL)
A Banach algebra element ɑ ∈ A is said to be "polynomially Riesz", relative to the homomorphism T : A → B, if there exists a nonzero complex polynomial p(z) such that the image Tp ∈ B is quasinilpotent. Keywords: Homomorphism of Banach algebras, polynomially Riesz element, Fredholm spectrum, Browder element, ...
Chay, Junegone; Kim, Chul
2018-05-01
We reanalyze the factorization theorems for the Drell-Yan process and for deep inelastic scattering near threshold, as constructed in the framework of the soft-collinear effective theory (SCET), from a new, consistent perspective. In order to formulate the factorization near threshold in SCET, we should include an additional degree of freedom with small energy, collinear to the beam direction. The corresponding collinear-soft mode is included to describe the parton distribution function (PDF) near threshold. The soft function is modified by subtracting the contribution of the collinear-soft modes in order to avoid double counting on the overlap region. As a result, the proper soft function becomes infrared finite, and all the factorized parts are free of rapidity divergence. Furthermore, the separation of the relevant scales in each factorized part becomes manifest. We apply the same idea to the dihadron production in e+e- annihilation near threshold, and show that the resultant soft function is also free of infrared and rapidity divergences.
Symmetric integrable-polynomial factorization for symplectic one-turn-map tracking
International Nuclear Information System (INIS)
Shi, Jicong
1993-01-01
It was found that any homogeneous polynomial can be written as a sum of integrable polynomials of the same degree which Lie transformations can be evaluated exactly. By utilizing symplectic integrators, an integrable-polynomial factorization is developed to convert a symplectic map in the form of Dragt-Finn factorization into a product of Lie transformations associated with integrable polynomials. A small number of factorization bases of integrable polynomials enable one to use high order symplectic integrators so that the high-order spurious terms can be greatly suppressed. A symplectic map can thus be evaluated with desired accuracy
Energy Technology Data Exchange (ETDEWEB)
Suparmi, A., E-mail: suparmiuns@gmail.com; Cari, C., E-mail: suparmiuns@gmail.com [Physics Department, Post Graduate Study, Sebelas Maret University (Indonesia); Angraini, L. M. [Physics Department, Mataram University (Indonesia)
2014-09-30
The bound state solutions of Dirac equation for Hulthen and trigonometric Rosen Morse non-central potential are obtained using finite Romanovski polynomials. The approximate relativistic energy spectrum and the radial wave functions which are given in terms of Romanovski polynomials are obtained from solution of radial Dirac equation. The angular wave functions and the orbital quantum number are found from angular Dirac equation solution. In non-relativistic limit, the relativistic energy spectrum reduces into non-relativistic energy.
Polynomial fuzzy model-based approach for underactuated surface vessels
DEFF Research Database (Denmark)
Khooban, Mohammad Hassan; Vafamand, Navid; Dragicevic, Tomislav
2018-01-01
The main goal of this study is to introduce a new polynomial fuzzy model-based structure for a class of marine systems with non-linear and polynomial dynamics. The suggested technique relies on a polynomial Takagi–Sugeno (T–S) fuzzy modelling, a polynomial dynamic parallel distributed compensation...... surface vessel (USV). Additionally, in order to overcome the USV control challenges, including the USV un-modelled dynamics, complex nonlinear dynamics, external disturbances and parameter uncertainties, the polynomial fuzzy model representation is adopted. Moreover, the USV-based control structure...... and a sum-of-squares (SOS) decomposition. The new proposed approach is a generalisation of the standard T–S fuzzy models and linear matrix inequality which indicated its effectiveness in decreasing the tracking time and increasing the efficiency of the robust tracking control problem for an underactuated...
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.
Technique for image interpolation using polynomial transforms
Escalante Ramírez, B.; Martens, J.B.; Haskell, G.G.; Hang, H.M.
1993-01-01
We present a new technique for image interpolation based on polynomial transforms. This is an image representation model that analyzes an image by locally expanding it into a weighted sum of orthogonal polynomials. In the discrete case, the image segment within every window of analysis is
Okounkov's BC-Type Interpolation Macdonald Polynomials and Their q=1 Limit
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
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
Contributions to fuzzy polynomial techniques for stability analysis and control
Pitarch Pérez, José Luis
2014-01-01
The present thesis employs fuzzy-polynomial control techniques in order to improve the stability analysis and control of nonlinear systems. Initially, it reviews the more extended techniques in the field of Takagi-Sugeno fuzzy systems, such as the more relevant results about polynomial and fuzzy polynomial systems. The basic framework uses fuzzy polynomial models by Taylor series and sum-of-squares techniques (semidefinite programming) in order to obtain stability guarantees...
On the Lorentz degree of a product of polynomials
Ait-Haddou, Rachid
2015-01-01
In this note, we negatively answer two questions of T. Erdélyi (1991, 2010) on possible lower bounds on the Lorentz degree of product of two polynomials. We show that the correctness of one question for degree two polynomials is a direct consequence of a result of Barnard et al. (1991) on polynomials with nonnegative coefficients.
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.
Directory of Open Access Journals (Sweden)
Liyun Su
2012-01-01
Full Text Available We introduce the extension of local polynomial fitting to the linear heteroscedastic regression model. Firstly, the local polynomial fitting is applied to estimate heteroscedastic function, then the coefficients of regression model are obtained by using generalized least squares method. One noteworthy feature of our approach is that we avoid the testing for heteroscedasticity by improving the traditional two-stage method. Due to nonparametric technique of local polynomial estimation, we do not need to know the heteroscedastic function. Therefore, we can improve the estimation precision, when the heteroscedastic function is unknown. Furthermore, we focus on comparison of parameters and reach an optimal fitting. Besides, we verify the asymptotic normality of parameters based on numerical simulations. Finally, this approach is applied to a case of economics, and it indicates that our method is surely effective in finite-sample situations.
Threshold quantum cryptography
International Nuclear Information System (INIS)
Tokunaga, Yuuki; Okamoto, Tatsuaki; Imoto, Nobuyuki
2005-01-01
We present the concept of threshold collaborative unitary transformation or threshold quantum cryptography, which is a kind of quantum version of threshold cryptography. Threshold quantum cryptography states that classical shared secrets are distributed to several parties and a subset of them, whose number is greater than a threshold, collaborates to compute a quantum cryptographic function, while keeping each share secretly inside each party. The shared secrets are reusable if no cheating is detected. As a concrete example of this concept, we show a distributed protocol (with threshold) of conjugate coding
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
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.
On Some Extensions of Szasz Operators Including Boas-Buck-Type Polynomials
Directory of Open Access Journals (Sweden)
Sezgin Sucu
2012-01-01
Full Text Available This paper is concerned with a new sequence of linear positive operators which generalize Szasz operators including Boas-Buck-type polynomials. We establish a convergence theorem for these operators and give the quantitative estimation of the approximation process by using a classical approach and the second modulus of continuity. Some explicit examples of our operators involving Laguerre polynomials, Charlier polynomials, and Gould-Hopper polynomials are given. Moreover, a Voronovskaya-type result is obtained for the operators containing Gould-Hopper polynomials.
On associated polynomials and decay rates for birth-death processes
van Doorn, Erik A.
2001-01-01
We consider sequences of orthogonal polynomials and pursue the question of how (partial) knowledge of the orthogonalizing measure for the {\\it associated polynomials} can lead to information about the orthogonalizing measure for the original polynomials. In particular, we relate the supports of the
On associated polynomials and decay rates for birth-death processes
van Doorn, Erik A.
2003-01-01
We consider sequences of orthogonal polynomials and pursue the question of how (partial) knowledge of the orthogonalizing measure for the associated polynomials can lead to information about the orthogonalizing measure for the original polynomials. In particular, we relate the supports of the two
Chkifa, Abdellah
2015-04-08
Motivated by the numerical treatment of parametric and stochastic PDEs, we analyze the least-squares method for polynomial approximation of multivariate functions based on random sampling according to a given probability measure. Recent work has shown that in the univariate case, the least-squares method is quasi-optimal in expectation in [A. Cohen, M A. Davenport and D. Leviatan. Found. Comput. Math. 13 (2013) 819–834] and in probability in [G. Migliorati, F. Nobile, E. von Schwerin, R. Tempone, Found. Comput. Math. 14 (2014) 419–456], under suitable conditions that relate the number of samples with respect to the dimension of the polynomial space. Here “quasi-optimal” means that the accuracy of the least-squares approximation is comparable with that of the best approximation in the given polynomial space. In this paper, we discuss the quasi-optimality of the polynomial least-squares method in arbitrary dimension. Our analysis applies to any arbitrary multivariate polynomial space (including tensor product, total degree or hyperbolic crosses), under the minimal requirement that its associated index set is downward closed. The optimality criterion only involves the relation between the number of samples and the dimension of the polynomial space, independently of the anisotropic shape and of the number of variables. We extend our results to the approximation of Hilbert space-valued functions in order to apply them to the approximation of parametric and stochastic elliptic PDEs. As a particular case, we discuss “inclusion type” elliptic PDE models, and derive an exponential convergence estimate for the least-squares method. Numerical results confirm our estimate, yet pointing out a gap between the condition necessary to achieve optimality in the theory, and the condition that in practice yields the optimal convergence rate.
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
Lattice Boltzmann method for bosons and fermions and the fourth-order Hermite polynomial expansion.
Coelho, Rodrigo C V; Ilha, Anderson; Doria, Mauro M; Pereira, R M; Aibe, Valter Yoshihiko
2014-04-01
The Boltzmann equation with the Bhatnagar-Gross-Krook collision operator is considered for the Bose-Einstein and Fermi-Dirac equilibrium distribution functions. We show that the expansion of the microscopic velocity in terms of Hermite polynomials must be carried to the fourth order to correctly describe the energy equation. The viscosity and thermal coefficients, previously obtained by Yang et al. [Shi and Yang, J. Comput. Phys. 227, 9389 (2008); Yang and Hung, Phys. Rev. E 79, 056708 (2009)] through the Uehling-Uhlenbeck approach, are also derived here. Thus the construction of a lattice Boltzmann method for the quantum fluid is possible provided that the Bose-Einstein and Fermi-Dirac equilibrium distribution functions are expanded to fourth order in the Hermite polynomials.
Dual exponential polynomials and linear differential equations
Wen, Zhi-Tao; Gundersen, Gary G.; Heittokangas, Janne
2018-01-01
We study linear differential equations with exponential polynomial coefficients, where exactly one coefficient is of order greater than all the others. The main result shows that a nontrivial exponential polynomial solution of such an equation has a certain dual relationship with the maximum order coefficient. Several examples illustrate our results and exhibit possibilities that can occur.
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.
Polynomial fuzzy observer designs: a sum-of-squares approach.
Tanaka, Kazuo; Ohtake, Hiroshi; Seo, Toshiaki; Tanaka, Motoyasu; Wang, Hua O
2012-10-01
This paper presents a sum-of-squares (SOS) approach to polynomial fuzzy observer designs for three classes of polynomial fuzzy systems. The proposed SOS-based framework provides a number of innovations and improvements over the existing linear matrix inequality (LMI)-based approaches to Takagi-Sugeno (T-S) fuzzy controller and observer designs. First, we briefly summarize previous results with respect to a polynomial fuzzy system that is a more general representation of the well-known T-S fuzzy system. Next, we propose polynomial fuzzy observers to estimate states in three classes of polynomial fuzzy systems and derive SOS conditions to design polynomial fuzzy controllers and observers. A remarkable feature of the SOS design conditions for the first two classes (Classes I and II) is that they realize the so-called separation principle, i.e., the polynomial fuzzy controller and observer for each class can be separately designed without lack of guaranteeing the stability of the overall control system in addition to converging state-estimation error (via the observer) to zero. Although, for the last class (Class III), the separation principle does not hold, we propose an algorithm to design polynomial fuzzy controller and observer satisfying the stability of the overall control system in addition to converging state-estimation error (via the observer) to zero. All the design conditions in the proposed approach can be represented in terms of SOS and are symbolically and numerically solved via the recently developed SOSTOOLS and a semidefinite-program solver, respectively. To illustrate the validity and applicability of the proposed approach, three design examples are provided. The examples demonstrate the advantages of the SOS-based approaches for the existing LMI approaches to T-S fuzzy observer designs.
Solutions of interval type-2 fuzzy polynomials using a new ranking method
Rahman, Nurhakimah Ab.; Abdullah, Lazim; Ghani, Ahmad Termimi Ab.; Ahmad, Noor'Ani
2015-10-01
A few years ago, a ranking method have been introduced in the fuzzy polynomial equations. Concept of the ranking method is proposed to find actual roots of fuzzy polynomials (if exists). Fuzzy polynomials are transformed to system of crisp polynomials, performed by using ranking method based on three parameters namely, Value, Ambiguity and Fuzziness. However, it was found that solutions based on these three parameters are quite inefficient to produce answers. Therefore in this study a new ranking method have been developed with the aim to overcome the inherent weakness. The new ranking method which have four parameters are then applied in the interval type-2 fuzzy polynomials, covering the interval type-2 of fuzzy polynomial equation, dual fuzzy polynomial equations and system of fuzzy polynomials. The efficiency of the new ranking method then numerically considered in the triangular fuzzy numbers and the trapezoidal fuzzy numbers. Finally, the approximate solutions produced from the numerical examples indicate that the new ranking method successfully produced actual roots for the interval type-2 fuzzy polynomials.
de Klerk, Etienne; Laurent, Monique; Sun, Zhao
We consider the problem of minimizing a continuous function f over a compact set K. We analyze a hierarchy of upper bounds proposed by Lasserre (SIAM J Optim 21(3):864–885, 2011), obtained by searching for an optimal probability density function h on K which is a sum of squares of polynomials, so
Polynomial Chaos Expansion Approach to Interest Rate Models
Directory of Open Access Journals (Sweden)
Luca Di Persio
2015-01-01
Full Text Available The Polynomial Chaos Expansion (PCE technique allows us to recover a finite second-order random variable exploiting suitable linear combinations of orthogonal polynomials which are functions of a given stochastic quantity ξ, hence acting as a kind of random basis. The PCE methodology has been developed as a mathematically rigorous Uncertainty Quantification (UQ method which aims at providing reliable numerical estimates for some uncertain physical quantities defining the dynamic of certain engineering models and their related simulations. In the present paper, we use the PCE approach in order to analyze some equity and interest rate models. In particular, we take into consideration those models which are based on, for example, the Geometric Brownian Motion, the Vasicek model, and the CIR model. We present theoretical as well as related concrete numerical approximation results considering, without loss of generality, the one-dimensional case. We also provide both an efficiency study and an accuracy study of our approach by comparing its outputs with the ones obtained adopting the Monte Carlo approach, both in its standard and its enhanced version.
An overview on polynomial approximation of NP-hard problems
Directory of Open Access Journals (Sweden)
Paschos Vangelis Th.
2009-01-01
Full Text Available The fact that polynomial time algorithm is very unlikely to be devised for an optimal solving of the NP-hard problems strongly motivates both the researchers and the practitioners to try to solve such problems heuristically, by making a trade-off between computational time and solution's quality. In other words, heuristic computation consists of trying to find not the best solution but one solution which is 'close to' the optimal one in reasonable time. Among the classes of heuristic methods for NP-hard problems, the polynomial approximation algorithms aim at solving a given NP-hard problem in poly-nomial time by computing feasible solutions that are, under some predefined criterion, as near to the optimal ones as possible. The polynomial approximation theory deals with the study of such algorithms. This survey first presents and analyzes time approximation algorithms for some classical examples of NP-hard problems. Secondly, it shows how classical notions and tools of complexity theory, such as polynomial reductions, can be matched with polynomial approximation in order to devise structural results for NP-hard optimization problems. Finally, it presents a quick description of what is commonly called inapproximability results. Such results provide limits on the approximability of the problems tackled.
An asymptotic formula for polynomials orthonormal with respect to a varying weight. II
International Nuclear Information System (INIS)
Komlov, A V; Suetin, S P
2014-01-01
This paper gives a proof of the theorem announced by the authors in the preceding paper with the same title. The theorem states that asymptotically the behaviour of the polynomials which are orthonormal with respect to the varying weight e −2nQ(x) p g (x)/√(∏ j=1 2p (x−e j )) coincides with the asymptotic behaviour of the Nuttall psi-function, which solves a special boundary-value problem on the relevant hyperelliptic Riemann surface of genus g=p−1. Here e 1
Peng, Mei; Jaeger, Sara R; Hautus, Michael J
2014-03-01
Psychometric functions are predominately used for estimating detection thresholds in vision and audition. However, the requirement of large data quantities for fitting psychometric functions (>30 replications) reduces their suitability in olfactory studies because olfactory response data are often limited (ASTM) E679. The slope parameter of the individual-judge psychometric function is fixed to be the same as that of the group function; the same-shaped symmetrical sigmoid function is fitted only using the intercept. This study evaluated the proposed method by comparing it with 2 available methods. Comparison to conventional psychometric functions (fitted slope and intercept) indicated that the assumption of a fixed slope did not compromise precision of the threshold estimates. No systematic difference was obtained between the proposed method and the ASTM method in terms of group threshold estimates or threshold distributions, but there were changes in the rank, by threshold, of judges in the group. Overall, the fixed-slope psychometric function is recommended for obtaining relatively reliable individual threshold estimates when the quantity of data is limited.
About the solvability of matrix polynomial equations
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.
Stable piecewise polynomial vector fields
Directory of Open Access Journals (Sweden)
Claudio Pessoa
2012-09-01
Full Text Available Let $N={y>0}$ and $S={y<0}$ be the semi-planes of $mathbb{R}^2$ having as common boundary the line $D={y=0}$. Let $X$ and $Y$ be polynomial vector fields defined in $N$ and $S$, respectively, leading to a discontinuous piecewise polynomial vector field $Z=(X,Y$. This work pursues the stability and the transition analysis of solutions of $Z$ between $N$ and $S$, started by Filippov (1988 and Kozlova (1984 and reformulated by Sotomayor-Teixeira (1995 in terms of the regularization method. This method consists in analyzing a one parameter family of continuous vector fields $Z_{epsilon}$, defined by averaging $X$ and $Y$. This family approaches $Z$ when the parameter goes to zero. The results of Sotomayor-Teixeira and Sotomayor-Machado (2002 providing conditions on $(X,Y$ for the regularized vector fields to be structurally stable on planar compact connected regions are extended to discontinuous piecewise polynomial vector fields on $mathbb{R}^2$. Pertinent genericity results for vector fields satisfying the above stability conditions are also extended to the present case. A procedure for the study of discontinuous piecewise vector fields at infinity through a compactification is proposed here.
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.
Computing Galois Groups of Eisenstein Polynomials Over P-adic Fields
Milstead, Jonathan
The most efficient algorithms for computing Galois groups of polynomials over global fields are based on Stauduhar's relative resolvent method. These methods are not directly generalizable to the local field case, since they require a field that contains the global field in which all roots of the polynomial can be approximated. We present splitting field-independent methods for computing the Galois group of an Eisenstein polynomial over a p-adic field. Our approach is to combine information from different disciplines. We primarily, make use of the ramification polygon of the polynomial, which is the Newton polygon of a related polynomial. This allows us to quickly calculate several invariants that serve to reduce the number of possible Galois groups. Algorithms by Greve and Pauli very efficiently return the Galois group of polynomials where the ramification polygon consists of one segment as well as information about the subfields of the stem field. Second, we look at the factorization of linear absolute resolvents to further narrow the pool of possible groups.
International Nuclear Information System (INIS)
Cerveri, P.; Forlani, C.; Borghese, N.A.; Ferrigno, G.
2002-01-01
In this paper we present two novel techniques, namely a local unwarping polynomial (LUP) and a hierarchical radial basis function (HRBF) network, to correct geometric distortions in XRII images. The two techniques have been implemented and compared, in terms of residual error measured at control and intermediate points, with local and global methods reported in the previous literature. In particular, LUP rests on a locally optimized 3rd degree polynomial applied within each quadrilateral cell on the rectilinear calibration grid of points. HRBF, based on a feed-forward neural network paradigm, is constituted by a set of hierarchical layers at increasing cut-off frequency, each characterized by a set of Gaussian functions. Extensive experiments have been performed both on simulated and real data. In simulation, we tested the effect of pincushion, sigmoidal and local distortions, along with the number of calibration points. Provided that a sufficient number of cells of the calibration grid is available, the obtained accuracy for both LUP and HRBF is comparable to or better than that of global polynomial technique. Tests on real data, carried out by using two different (12 in. and 16 in.) XRIIs, showed that the global polynomial accuracy (0.16±0.08 pixels) is slightly worse than that of LUP (0.07±0.05 pixels) and HRBF (0.08±0.04 pixels). The effects of the discontinuity at the border of the local areas and the decreased accuracy at intermediate points, typical of local techniques, have been proved to be smoothed for both LUP and HRBF
Xie, Xiangpeng; Yue, Dong; Zhang, Huaguang; Xue, Yusheng
2016-03-01
This paper deals with the problem of control synthesis of discrete-time Takagi-Sugeno fuzzy systems by employing a novel multiinstant homogenous polynomial approach. A new multiinstant fuzzy control scheme and a new class of fuzzy Lyapunov functions, which are homogenous polynomially parameter-dependent on both the current-time normalized fuzzy weighting functions and the past-time normalized fuzzy weighting functions, are proposed for implementing the object of relaxed control synthesis. Then, relaxed stabilization conditions are derived with less conservatism than existing ones. Furthermore, the relaxation quality of obtained stabilization conditions is further ameliorated by developing an efficient slack variable approach, which presents a multipolynomial dependence on the normalized fuzzy weighting functions at the current and past instants of time. Two simulation examples are given to demonstrate the effectiveness and benefits of the results developed in this paper.
Huang, Yi-Jen
2016-04-07
The combination of nonvolatile memory switching and volatile threshold switching functions of transition metal oxides in crossbar memory arrays is of great potential for replacing charge-based flash memory in very-large-scale integration. Here, we show that the resistive switching material structure, (amorphous TiOx)/(Ag nanoparticles)/(polycrystalline TiOx), fabricated on the textured-FTO substrate with ITO as the top electrode exhibits both the memory switching and threshold switching functions. When the device is used for resistive switching, it is forming-free for resistive memory applications with low operation voltage (<±1 V) and self-compliance to current up to 50 μA. When it is used for threshold switching, the low threshold current is beneficial for improving the device selectivity. The variation of oxygen distribution measured by energy dispersive X-ray spectroscopy and scanning transmission electron microscopy indicates the formation or rupture of conducting filaments in the device at different resistance states. It is therefore suggested that the push and pull actions of oxygen ions in the amorphous TiOx and polycrystalline TiOx films during the voltage sweep account for the memory switching and threshold switching properties in the device.
Guts of surfaces and the colored Jones polynomial
Futer, David; Purcell, Jessica
2013-01-01
This monograph derives direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses, we prove that the growth of the degree of the colored Jones polynomials is a boundary slope of an essential surface in the knot complement. We show that certain coefficients of the polynomial measure how far this surface is from being a fiber for the knot; in particular, the surface is a fiber if and only if a particular coefficient vanishes. We also relate hyperbolic volume to colored Jones polynomials. Our method is to generalize the checkerboard decompositions of alternating knots. Under mild diagrammatic hypotheses, we show that these surfaces are essential, and obtain an ideal polyhedral decomposition of their complement. We use normal surface theory to relate the pieces of the JSJ decomposition of the complement to the combinatorics of certain surface spines (state graphs). Since state graphs have p...
Computing Tutte polynomials of contact networks in classrooms
Hincapié, Doracelly; Ospina, Juan
2013-05-01
Objective: The topological complexity of contact networks in classrooms and the potential transmission of an infectious disease were analyzed by sex and age. Methods: The Tutte polynomials, some topological properties and the number of spanning trees were used to algebraically compute the topological complexity. Computations were made with the Maple package GraphTheory. Published data of mutually reported social contacts within a classroom taken from primary school, consisting of children in the age ranges of 4-5, 7-8 and 10-11, were used. Results: The algebraic complexity of the Tutte polynomial and the probability of disease transmission increases with age. The contact networks are not bipartite graphs, gender segregation was observed especially in younger children. Conclusion: Tutte polynomials are tools to understand the topology of the contact networks and to derive numerical indexes of such topologies. It is possible to establish relationships between the Tutte polynomial of a given contact network and the potential transmission of an infectious disease within such network
Canavos, G. C.
1974-01-01
A study is made of the extent to which the size of the sample affects the accuracy of a quadratic or a cubic polynomial approximation of an experimentally observed quantity, and the trend with regard to improvement in the accuracy of the approximation as a function of sample size is established. The task is made possible through a simulated analysis carried out by the Monte Carlo method in which data are simulated by using several transcendental or algebraic functions as models. Contaminated data of varying amounts are fitted to either quadratic or cubic polynomials, and the behavior of the mean-squared error of the residual variance is determined as a function of sample size. Results indicate that the effect of the size of the sample is significant only for relatively small sizes and diminishes drastically for moderate and large amounts of experimental data.
A note on envy-free cake cutting with polynomial valuations
DEFF Research Database (Denmark)
Branzei, Simina
2015-01-01
number of players. The only existing finite envy-free cake cutting protocol for any number of players, designed by Brams and Taylor [4], has the property that its runtime can be made arbitrarily large by setting up the valuation functions of the players appropriately. Moreover, there is no closed formula...... that relates the valuation functions to the number of queries required by the protocol. In this note we show that when the valuations can be represented as polynomial functions, there exists a protocol in the standard query model that is much simpler conceptually and has a runtime bound depending...
Directory of Open Access Journals (Sweden)
Bangyong Sun
2014-01-01
Full Text Available The polynomial regression method is employed to calculate the relationship of device color space and CIE color space for color characterization, and the performance of different expressions with specific parameters is evaluated. Firstly, the polynomial equation for color conversion is established and the computation of polynomial coefficients is analysed. And then different forms of polynomial equations are used to calculate the RGB and CMYK’s CIE color values, while the corresponding color errors are compared. At last, an optimal polynomial expression is obtained by analysing several related parameters during color conversion, including polynomial numbers, the degree of polynomial terms, the selection of CIE visual spaces, and the linearization.
Exponential time paradigms through the polynomial time lens
Drucker, A.; Nederlof, J.; Santhanam, R.; Sankowski, P.; Zaroliagis, C.
2016-01-01
We propose a general approach to modelling algorithmic paradigms for the exact solution of NP-hard problems. Our approach is based on polynomial time reductions to succinct versions of problems solvable in polynomial time. We use this viewpoint to explore and compare the power of paradigms such as
Methods in Symbolic Computation and p-Adic Valuations of Polynomials
Guan, Xiao
Symbolic computation has widely appear in many mathematical fields such as combinatorics, number theory and stochastic processes. The techniques created in the area of experimental mathematics provide us efficient ways of symbolic computing and verification of complicated relations. Part I consists of three problems. The first one focuses on a unimodal sequence derived from a quartic integral. Many of its properties are explored with the help of hypergeometric representations and automatic proofs. The second problem tackles the generating function of the reciprocal of Catalan number. It springs from the closed form given by Mathematica. Furthermore, three methods in special functions are used to justify this result. The third issue addresses the closed form solutions for the moments of products of generalized elliptic integrals , which combines the experimental mathematics and classical analysis. Part II concentrates on the p-adic valuations of polynomials from the perspective of trees. For a given polynomial f( n) indexed in positive integers, the package developed in Mathematica will create certain tree structure following a couple of rules. The evolution of such trees are studied both rigorously and experimentally from the view of field extension, nonparametric statistics and random matrix.
Global stability and quadratic Hamiltonian structure in Lotka-Volterra and quasi-polynomial systems
Energy Technology Data Exchange (ETDEWEB)
Szederkenyi, Gabor; Hangos, Katalin M
2004-04-26
We show that the global stability of quasi-polynomial (QP) and Lotka-Volterra (LV) systems with the well-known logarithmic Lyapunov function is equivalent to the existence of a local generalized dissipative Hamiltonian description of the LV system with a diagonal quadratic form as a Hamiltonian function. The Hamiltonian function can be calculated and the quadratic dissipativity neighborhood of the origin can be estimated by solving linear matrix inequalities.
Global stability and quadratic Hamiltonian structure in Lotka-Volterra and quasi-polynomial systems
Szederkényi, Gábor; Hangos, Katalin M.
2004-04-01
We show that the global stability of quasi-polynomial (QP) and Lotka-Volterra (LV) systems with the well-known logarithmic Lyapunov function is equivalent to the existence of a local generalized dissipative Hamiltonian description of the LV system with a diagonal quadratic form as a Hamiltonian function. The Hamiltonian function can be calculated and the quadratic dissipativity neighborhood of the origin can be estimated by solving linear matrix inequalities.
Global stability and quadratic Hamiltonian structure in Lotka-Volterra and quasi-polynomial systems
International Nuclear Information System (INIS)
Szederkenyi, Gabor; Hangos, Katalin M.
2004-01-01
We show that the global stability of quasi-polynomial (QP) and Lotka-Volterra (LV) systems with the well-known logarithmic Lyapunov function is equivalent to the existence of a local generalized dissipative Hamiltonian description of the LV system with a diagonal quadratic form as a Hamiltonian function. The Hamiltonian function can be calculated and the quadratic dissipativity neighborhood of the origin can be estimated by solving linear matrix inequalities
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 ...
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....
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.
Determination of the paraxial focal length using Zernike polynomials over different apertures
Binkele, Tobias; Hilbig, David; Henning, Thomas; Fleischmann, Friedrich
2017-02-01
The paraxial focal length is still the most important parameter in the design of a lens. As presented at the SPIE Optics + Photonics 2016, the measured focal length is a function of the aperture. The paraxial focal length can be found when the aperture approaches zero. In this work, we investigate the dependency of the Zernike polynomials on the aperture size with respect to 3D space. By this, conventional wavefront measurement systems that apply Zernike polynomial fitting (e.g. Shack-Hartmann-Sensor) can be used to determine the paraxial focal length, too. Since the Zernike polynomials are orthogonal over a unit circle, the aperture used in the measurement has to be normalized. By shrinking the aperture and keeping up with the normalization, the Zernike coefficients change. The relation between these changes and the paraxial focal length are investigated. The dependency of the focal length on the aperture size is derived analytically and evaluated by simulation and measurement of a strong focusing lens. The measurements are performed using experimental ray tracing and a Shack-Hartmann-Sensor. Using experimental ray tracing for the measurements, the aperture can be chosen easily. Regarding the measurements with the Shack-Hartmann- Sensor, the aperture size is fixed. Thus, the Zernike polynomials have to be adapted to use different aperture sizes by the proposed method. By doing this, the paraxial focal length can be determined from the measurements in both cases.
Bayes Node Energy Polynomial Distribution to Improve Routing in Wireless Sensor Network
Palanisamy, Thirumoorthy; Krishnasamy, Karthikeyan N.
2015-01-01
Wireless Sensor Network monitor and control the physical world via large number of small, low-priced sensor nodes. Existing method on Wireless Sensor Network (WSN) presented sensed data communication through continuous data collection resulting in higher delay and energy consumption. To conquer the routing issue and reduce energy drain rate, Bayes Node Energy and Polynomial Distribution (BNEPD) technique is introduced with energy aware routing in the wireless sensor network. The Bayes Node Energy Distribution initially distributes the sensor nodes that detect an object of similar event (i.e., temperature, pressure, flow) into specific regions with the application of Bayes rule. The object detection of similar events is accomplished based on the bayes probabilities and is sent to the sink node resulting in minimizing the energy consumption. Next, the Polynomial Regression Function is applied to the target object of similar events considered for different sensors are combined. They are based on the minimum and maximum value of object events and are transferred to the sink node. Finally, the Poly Distribute algorithm effectively distributes the sensor nodes. The energy efficient routing path for each sensor nodes are created by data aggregation at the sink based on polynomial regression function which reduces the energy drain rate with minimum communication overhead. Experimental performance is evaluated using Dodgers Loop Sensor Data Set from UCI repository. Simulation results show that the proposed distribution algorithm significantly reduce the node energy drain rate and ensure fairness among different users reducing the communication overhead. PMID:26426701
Bayes Node Energy Polynomial Distribution to Improve Routing in Wireless Sensor Network.
Palanisamy, Thirumoorthy; Krishnasamy, Karthikeyan N
2015-01-01
Wireless Sensor Network monitor and control the physical world via large number of small, low-priced sensor nodes. Existing method on Wireless Sensor Network (WSN) presented sensed data communication through continuous data collection resulting in higher delay and energy consumption. To conquer the routing issue and reduce energy drain rate, Bayes Node Energy and Polynomial Distribution (BNEPD) technique is introduced with energy aware routing in the wireless sensor network. The Bayes Node Energy Distribution initially distributes the sensor nodes that detect an object of similar event (i.e., temperature, pressure, flow) into specific regions with the application of Bayes rule. The object detection of similar events is accomplished based on the bayes probabilities and is sent to the sink node resulting in minimizing the energy consumption. Next, the Polynomial Regression Function is applied to the target object of similar events considered for different sensors are combined. They are based on the minimum and maximum value of object events and are transferred to the sink node. Finally, the Poly Distribute algorithm effectively distributes the sensor nodes. The energy efficient routing path for each sensor nodes are created by data aggregation at the sink based on polynomial regression function which reduces the energy drain rate with minimum communication overhead. Experimental performance is evaluated using Dodgers Loop Sensor Data Set from UCI repository. Simulation results show that the proposed distribution algorithm significantly reduce the node energy drain rate and ensure fairness among different users reducing the communication overhead.
A quasi-static polynomial nodal method for nuclear reactor analysis
International Nuclear Information System (INIS)
Gehin, J.C.
1992-09-01
Modern nodal methods are currently available which can accurately and efficiently solve the static and transient neutron diffusion equations. Most of the methods, however, are limited to two energy groups for practical application. The objective of this research is the development of a static and transient, multidimensional nodal method which allows more than two energy groups and uses a non-linear iterative method for efficient solution of the nodal equations. For both the static and transient methods, finite-difference equations which are corrected by the use of discontinuity factors are derived. The discontinuity factors are computed from a polynomial nodal method using a non-linear iteration technique. The polynomial nodal method is based upon a quartic approximation and utilizes a quadratic transverse-leakage approximation. The solution of the time-dependent equations is performed by the use of a quasi-static method in which the node-averaged fluxes are factored into shape and amplitude functions. The application of the quasi-static polynomial method to several benchmark problems demonstrates that the accuracy is consistent with that of other nodal methods. The use of the quasi-static method is shown to substantially reduce the computation time over the traditional fully-implicit time-integration method. Problems involving thermal-hydraulic feedback are accurately, and efficiently, solved by performing several reactivity/thermal-hydraulic updates per shape calculation
A quasi-static polynomial nodal method for nuclear reactor analysis
Energy Technology Data Exchange (ETDEWEB)
Gehin, Jess C. [Massachusetts Inst. of Tech., Cambridge, MA (United States)
1992-09-01
Modern nodal methods are currently available which can accurately and efficiently solve the static and transient neutron diffusion equations. Most of the methods, however, are limited to two energy groups for practical application. The objective of this research is the development of a static and transient, multidimensional nodal method which allows more than two energy groups and uses a non-linear iterative method for efficient solution of the nodal equations. For both the static and transient methods, finite-difference equations which are corrected by the use of discontinuity factors are derived. The discontinuity factors are computed from a polynomial nodal method using a non-linear iteration technique. The polynomial nodal method is based upon a quartic approximation and utilizes a quadratic transverse-leakage approximation. The solution of the time-dependent equations is performed by the use of a quasi-static method in which the node-averaged fluxes are factored into shape and amplitude functions. The application of the quasi-static polynomial method to several benchmark problems demonstrates that the accuracy is consistent with that of other nodal methods. The use of the quasi-static method is shown to substantially reduce the computation time over the traditional fully-implicit time-integration method. Problems involving thermal-hydraulic feedback are accurately, and efficiently, solved by performing several reactivity/thermal-hydraulic updates per shape calculation.
International Nuclear Information System (INIS)
Uosif, M.A.M.
2006-01-01
A new 9 th degree polynomial fit function has been constructed to calculate the absolute γ-ray detection efficiencies (ηth) of Ge(Li) and HPGe Detectors, for calculating the absolute efficiency at any interesting γ-energy in the energy range between 25 and 2000 keV and distance between 6 and 148 cm. The total absolute γ -ray detection efficiencies have been calculated for six detectors, three of them are Ge(Li) and three HPGe at different distances. The absolute efficiency of the different detectors was calculated at the specific energy of the standard sources for each measuring distances. In this calculation, experimental (η e xp) and fitting (η f it) efficiency have been calculated. Seven calibrated point sources Am-241, Ba-133, Co-57, Co-60, Cs-137, Eu-152 and Ra-226 were used. The uncertainties of efficiency calibration have been calculated also for quality control. The measured (η e xp) and (η f it) calculated efficiency values were compared with efficiency, which calculated, by Gray fit function (time)- The results obtained on the basis of (η e xp)and (η f it) seem to be in very good agreement
Threshold values of ankle dorsiflexion and gross motor function in 60 children with cerebral palsy
DEFF Research Database (Denmark)
Rasmussen, Helle M; Svensson, Joachim; Thorning, Maria
2018-01-01
Background and purpose - Threshold values defining 3 categories of passive range of motion are used in the Cerebral Palsy follow-Up Program to guide clinical decisions. The aim of this study was to investigate the threshold values by testing the hypothesis that passive range of motion in ankle...... dorsiflexion is associated with gross motor function and that function differs between the groups of participants in each category. Patients and methods - We analyzed data from 60 ambulatory children (aged 5-9 years) with spastic cerebral palsy. Outcomes were passive range of motion in ankle dorsiflexion...... with flexed and extended knee and gross motor function (Gait Deviation Index, Gait Variable Score of the ankle, peak dorsiflexion during gait, 1-minute walk, Gross Motor Function Measure, the Pediatric Quality of Life Inventory Cerebral Palsy Module, and Pediatric Outcomes Data Collection Instrument). Results...
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.
Generalizations of an integral for Legendre polynomials by Persson and Strang
Diekema, E.; Koornwinder, T.H.
2012-01-01
Persson and Strang (2003) evaluated the integral over [−1,1] of a squared odd degree Legendre polynomial divided by x2 as being equal to 2. We consider a similar integral for orthogonal polynomials with respect to a general even orthogonality measure, with Gegenbauer and Hermite polynomials as
Eye aberration analysis with Zernike polynomials
Molebny, Vasyl V.; Chyzh, Igor H.; Sokurenko, Vyacheslav M.; Pallikaris, Ioannis G.; Naoumidis, Leonidas P.
1998-06-01
New horizons for accurate photorefractive sight correction, afforded by novel flying spot technologies, require adequate measurements of photorefractive properties of an eye. Proposed techniques of eye refraction mapping present results of measurements for finite number of points of eye aperture, requiring to approximate these data by 3D surface. A technique of wave front approximation with Zernike polynomials is described, using optimization of the number of polynomial coefficients. Criterion of optimization is the nearest proximity of the resulted continuous surface to the values calculated for given discrete points. Methodology includes statistical evaluation of minimal root mean square deviation (RMSD) of transverse aberrations, in particular, varying consecutively the values of maximal coefficient indices of Zernike polynomials, recalculating the coefficients, and computing the value of RMSD. Optimization is finished at minimal value of RMSD. Formulas are given for computing ametropia, size of the spot of light on retina, caused by spherical aberration, coma, and astigmatism. Results are illustrated by experimental data, that could be of interest for other applications, where detailed evaluation of eye parameters is needed.
International Nuclear Information System (INIS)
Almeida Ferreira, A.C. de.
1984-01-01
For problems with azimuthal symmetry in velocity space, the distribution function depends only on the speed and on the pitch angle. The angular dependence of the distribution function is expanded in Legendre polynomials, and the expansions of the collision integrals describing two-body Coulomb interactions in a plasma are determined through the use of the Rosenbluth potentials. The electron distribution function is written as a Maxwellian plus a deviation, and the representation in Legendre polynomials of the electron-electron collision term is given for both its linear and nonlinear part. To determine the representation of the electron-ion collision term it is assumed that the ion distribution is much narrower in velocity space than the electron distribution, and shifted from the origin by a flow velocity. The equations are presented in a form that is suitable for their use in a computer. (Author) [pt
International Nuclear Information System (INIS)
Saber, Ahmed Yousuf; Chakraborty, Shantanu; Abdur Razzak, S.M.; Senjyu, Tomonobu
2009-01-01
This paper presents a modified particle swarm optimization (MPSO) for constrained economic load dispatch (ELD) problem. Real cost functions are more complex than conventional second order cost functions when multi-fuel operations, valve-point effects, accurate curve fitting, etc., are considering in deregulated changing market. The proposed modified particle swarm optimization (PSO) consists of problem dependent variable number of promising values (in velocity vector), unit vector and error-iteration dependent step length. It reliably and accurately tracks a continuously changing solution of the complex cost function and no extra concentration/effort is needed for the complex higher order cost polynomials in ELD. Constraint management is incorporated in the modified PSO. The modified PSO has balance between local and global searching abilities, and an appropriate fitness function helps to converge it quickly. To avoid the method to be frozen, stagnated/idle particles are reset. Sensitivity of the higher order cost polynomials is also analyzed visually to realize the importance of the higher order cost polynomials for the optimization of ELD. Finally, benchmark data sets and methods are used to show the effectiveness of the proposed method. (author)
Complex centers of polynomial differential equations
Directory of Open Access Journals (Sweden)
Mohamad Ali M. Alwash
2007-07-01
Full Text Available We present some results on the existence and nonexistence of centers for polynomial first order ordinary differential equations with complex coefficients. In particular, we show that binomial differential equations without linear terms do not have complex centers. Classes of polynomial differential equations, with more than two terms, are presented that do not have complex centers. We also study the relation between complex centers and the Pugh problem. An algorithm is described to solve the Pugh problem for equations without complex centers. The method of proof involves phase plane analysis of the polar equations and a local study of periodic solutions.
Differential recurrence formulae for orthogonal polynomials
Directory of Open Access Journals (Sweden)
Anton L. W. von Bachhaus
1995-11-01
Full Text Available Part I - By combining a general 2nd-order linear homogeneous ordinary differential equation with the three-term recurrence relation possessed by all orthogonal polynomials, it is shown that sequences of orthogonal polynomials which satisfy a differential equation of the above mentioned type necessarily have a differentiation formula of the type: gn(xY'n(x=fn(xYn(x+Yn-1(x. Part II - A recurrence formula of the form: rn(xY'n(x+sn(xY'n+1(x+tn(xY'n-1(x=0, is derived using the result of Part I.
Quantitative Boltzmann-Gibbs Principles via Orthogonal Polynomial Duality
Ayala, Mario; Carinci, Gioia; Redig, Frank
2018-06-01
We study fluctuation fields of orthogonal polynomials in the context of particle systems with duality. We thereby obtain a systematic orthogonal decomposition of the fluctuation fields of local functions, where the order of every term can be quantified. This implies a quantitative generalization of the Boltzmann-Gibbs principle. In the context of independent random walkers, we complete this program, including also fluctuation fields in non-stationary context (local equilibrium). For other interacting particle systems with duality such as the symmetric exclusion process, similar results can be obtained, under precise conditions on the n particle dynamics.
Energy Technology Data Exchange (ETDEWEB)
Li, Jun; Jiang, Bin; Guo, Hua, E-mail: hguo@unm.edu [Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, New Mexico 87131 (United States)
2013-11-28
A rigorous, general, and simple method to fit global and permutation invariant potential energy surfaces (PESs) using neural networks (NNs) is discussed. This so-called permutation invariant polynomial neural network (PIP-NN) method imposes permutation symmetry by using in its input a set of symmetry functions based on PIPs. For systems with more than three atoms, it is shown that the number of symmetry functions in the input vector needs to be larger than the number of internal coordinates in order to include both the primary and secondary invariant polynomials. This PIP-NN method is successfully demonstrated in three atom-triatomic reactive systems, resulting in full-dimensional global PESs with average errors on the order of meV. These PESs are used in full-dimensional quantum dynamical calculations.
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.
Deformation of the three-term recursion relation and generation of new orthogonal polynomials
International Nuclear Information System (INIS)
Alhaidari, A D
2002-01-01
We find solutions for a linear deformation of the three-term recursion relation. The orthogonal polynomials of the first and second kind associated with the deformed relation are obtained. The new density (weight) function is written in terms of the original one and the deformation parameters
Special functions for scientists and engineers
Bell, William Wallace
1968-01-01
Clear and comprehensive, this text provides undergraduates with a straightforward guide to special functions. It is equally suitable as a reference volume for professionals, and readers need no higher level of mathematical knowledge beyond elementary calculus. Topics include the solution of second-order differential equations in terms of power series; gamma and beta functions; Legendre polynomials and functions; Bessel functions; Hermite, Laguerre, and Chebyshev polynomials; Gegenbauer and Jacobi polynomials; and hypergeometric and other special functions. Three appendices offer convenient t
Closed-form estimates of the domain of attraction for nonlinear systems via fuzzy-polynomial models.
Pitarch, José Luis; Sala, Antonio; Ariño, Carlos Vicente
2014-04-01
In this paper, the domain of attraction of the origin of a nonlinear system is estimated in closed form via level sets with polynomial boundaries, iteratively computed. In particular, the domain of attraction is expanded from a previous estimate, such as a classical Lyapunov level set. With the use of fuzzy-polynomial models, the domain of attraction analysis can be carried out via sum of squares optimization and an iterative algorithm. The result is a function that bounds the domain of attraction, free from the usual restriction of being positive and decrescent in all the interior of its level sets.
Solving the interval type-2 fuzzy polynomial equation using the ranking method
Rahman, Nurhakimah Ab.; Abdullah, Lazim
2014-07-01
Polynomial equations with trapezoidal and triangular fuzzy numbers have attracted some interest among researchers in mathematics, engineering and social sciences. There are some methods that have been developed in order to solve these equations. In this study we are interested in introducing the interval type-2 fuzzy polynomial equation and solving it using the ranking method of fuzzy numbers. The ranking method concept was firstly proposed to find real roots of fuzzy polynomial equation. Therefore, the ranking method is applied to find real roots of the interval type-2 fuzzy polynomial equation. We transform the interval type-2 fuzzy polynomial equation to a system of crisp interval type-2 fuzzy polynomial equation. This transformation is performed using the ranking method of fuzzy numbers based on three parameters, namely value, ambiguity and fuzziness. Finally, we illustrate our approach by numerical example.
Directory of Open Access Journals (Sweden)
Maria Gabriela Campolina Diniz Peixoto
2014-05-01
Full Text Available The objective of this work was to compare random regression models for the estimation of genetic parameters for Guzerat milk production, using orthogonal Legendre polynomials. Records (20,524 of test-day milk yield (TDMY from 2,816 first-lactation Guzerat cows were used. TDMY grouped into 10-monthly classes were analyzed for additive genetic effect and for environmental and residual permanent effects (random effects, whereas the contemporary group, calving age (linear and quadratic effects and mean lactation curve were analized as fixed effects. Trajectories for the additive genetic and permanent environmental effects were modeled by means of a covariance function employing orthogonal Legendre polynomials ranging from the second to the fifth order. Residual variances were considered in one, four, six, or ten variance classes. The best model had six residual variance classes. The heritability estimates for the TDMY records varied from 0.19 to 0.32. The random regression model that used a second-order Legendre polynomial for the additive genetic effect, and a fifth-order polynomial for the permanent environmental effect is adequate for comparison by the main employed criteria. The model with a second-order Legendre polynomial for the additive genetic effect, and that with a fourth-order for the permanent environmental effect could also be employed in these analyses.
A high-order q-difference equation for q-Hahn multiple orthogonal polynomials
DEFF Research Database (Denmark)
Arvesú, J.; Esposito, Chiara
2012-01-01
A high-order linear q-difference equation with polynomial coefficients having q-Hahn multiple orthogonal polynomials as eigenfunctions is given. The order of the equation coincides with the number of orthogonality conditions that these polynomials satisfy. Some limiting situations when are studie....... Indeed, the difference equation for Hahn multiple orthogonal polynomials given in Lee [J. Approx. Theory (2007), ), doi: 10.1016/j.jat.2007.06.002] is obtained as a limiting case....
On the Lorentz degree of a product of polynomials
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
On Best Approximations of Polynomials in Matrices in the Matrix 2-Norm
Czech Academy of Sciences Publication Activity Database
Liesen, J.; Tichý, Petr
2009-01-01
Roč. 31, č. 2 (2009), s. 853-863 ISSN 0895-4798 R&D Projects: GA AV ČR IAA100300802 Institutional research plan: CEZ:AV0Z10300504 Keywords : matrix approximation problems * polynomials in matrices * matrix functions * matrix 2-norm * GMRES * Arnoldi's method Subject RIV: BA - General Mathematics Impact factor: 2.411, year: 2009
Zeros and logarithmic asymptotics of Sobolev orthogonal polynomials for exponential weights
Díaz Mendoza, C.; Orive, R.; Pijeira Cabrera, H.
2009-12-01
We obtain the (contracted) weak zero asymptotics for orthogonal polynomials with respect to Sobolev inner products with exponential weights in the real semiaxis, of the form , with [gamma]>0, which include as particular cases the counterparts of the so-called Freud (i.e., when [phi] has a polynomial growth at infinity) and Erdös (when [phi] grows faster than any polynomial at infinity) weights. In addition, the boundness of the distance of the zeros of these Sobolev orthogonal polynomials to the convex hull of the support and, as a consequence, a result on logarithmic asymptotics are derived.
Some Results on the Independence Polynomial of Unicyclic Graphs
Directory of Open Access Journals (Sweden)
Oboudi Mohammad Reza
2018-05-01
Full Text Available Let G be a simple graph on n vertices. An independent set in a graph is a set of pairwise non-adjacent vertices. The independence polynomial of G is the polynomial I(G,x=∑k=0ns(G,kxk$I(G,x = \\sum\
Bayes Node Energy Polynomial Distribution to Improve Routing in Wireless Sensor Network.
Directory of Open Access Journals (Sweden)
Thirumoorthy Palanisamy
Full Text Available Wireless Sensor Network monitor and control the physical world via large number of small, low-priced sensor nodes. Existing method on Wireless Sensor Network (WSN presented sensed data communication through continuous data collection resulting in higher delay and energy consumption. To conquer the routing issue and reduce energy drain rate, Bayes Node Energy and Polynomial Distribution (BNEPD technique is introduced with energy aware routing in the wireless sensor network. The Bayes Node Energy Distribution initially distributes the sensor nodes that detect an object of similar event (i.e., temperature, pressure, flow into specific regions with the application of Bayes rule. The object detection of similar events is accomplished based on the bayes probabilities and is sent to the sink node resulting in minimizing the energy consumption. Next, the Polynomial Regression Function is applied to the target object of similar events considered for different sensors are combined. They are based on the minimum and maximum value of object events and are transferred to the sink node. Finally, the Poly Distribute algorithm effectively distributes the sensor nodes. The energy efficient routing path for each sensor nodes are created by data aggregation at the sink based on polynomial regression function which reduces the energy drain rate with minimum communication overhead. Experimental performance is evaluated using Dodgers Loop Sensor Data Set from UCI repository. Simulation results show that the proposed distribution algorithm significantly reduce the node energy drain rate and ensure fairness among different users reducing the communication overhead.
Statistics of Data Fitting: Flaws and Fixes of Polynomial Analysis of Channeled Spectra
Karstens, William; Smith, David
2013-03-01
Starting from general statistical principles, we have critically examined Baumeister's procedure* for determining the refractive index of thin films from channeled spectra. Briefly, the method assumes that the index and interference fringe order may be approximated by polynomials quadratic and cubic in photon energy, respectively. The coefficients of the polynomials are related by differentiation, which is equivalent to comparing energy differences between fringes. However, we find that when the fringe order is calculated from the published IR index for silicon* and then analyzed with Baumeister's procedure, the results do not reproduce the original index. This problem has been traced to 1. Use of unphysical powers in the polynomials (e.g., time-reversal invariance requires that the index is an even function of photon energy), and 2. Use of insufficient terms of the correct parity. Exclusion of unphysical terms and addition of quartic and quintic terms to the index and order polynomials yields significantly better fits with fewer parameters. This represents a specific example of using statistics to determine if the assumed fitting model adequately captures the physics contained in experimental data. The use of analysis of variance (ANOVA) and the Durbin-Watson statistic to test criteria for the validity of least-squares fitting will be discussed. *D.F. Edwards and E. Ochoa, Appl. Opt. 19, 4130 (1980). Supported in part by the US Department of Energy, Office of Nuclear Physics under contract DE-AC02-06CH11357.
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.
Polynomial Poisson algebras: Gel'fand-Kirillov problem and Poisson spectra
Lecoutre, César
2014-01-01
We study the fields of fractions and the Poisson spectra of polynomial Poisson algebras.\\ud \\ud First we investigate a Poisson birational equivalence problem for polynomial Poisson algebras over a field of arbitrary characteristic. Namely, the quadratic Poisson Gel'fand-Kirillov problem asks whether the field of fractions of a Poisson algebra is isomorphic to the field of fractions of a Poisson affine space, i.e. a polynomial algebra such that the Poisson bracket of two generators is equal to...
Implementing fuzzy polynomial interpolation (FPI and fuzzy linear regression (LFR
Directory of Open Access Journals (Sweden)
Maria Cristina Floreno
1996-05-01
Full Text Available This paper presents some preliminary results arising within a general framework concerning the development of software tools for fuzzy arithmetic. The program is in a preliminary stage. What has been already implemented consists of a set of routines for elementary operations, optimized functions evaluation, interpolation and regression. Some of these have been applied to real problems.This paper describes a prototype of a library in C++ for polynomial interpolation of fuzzifying functions, a set of routines in FORTRAN for fuzzy linear regression and a program with graphical user interface allowing the use of such routines.
On an Inequality Concerning the Polar Derivative of a Polynomial
Indian Academy of Sciences (India)
Abstract. In this paper, we present a correct proof of an -inequality concerning the polar derivative of a polynomial with restricted zeros. We also extend Zygmund's inequality to the polar derivative of a polynomial.
Classification of complex polynomial vector fields in one complex variable
DEFF Research Database (Denmark)
Branner, Bodil; Dias, Kealey
2010-01-01
This paper classifies the global structure of monic and centred one-variable complex polynomial vector fields. The classification is achieved by means of combinatorial and analytic data. More specifically, given a polynomial vector field, we construct a combinatorial invariant, describing...... the topology, and a set of analytic invariants, describing the geometry. Conversely, given admissible combinatorial and analytic data sets, we show using surgery the existence of a unique monic and centred polynomial vector field realizing the given invariants. This is the content of the Structure Theorem......, the main result of the paper. This result is an extension and refinement of Douady et al. (Champs de vecteurs polynomiaux sur C. Unpublished manuscript) classification of the structurally stable polynomial vector fields. We further review some general concepts for completeness and show that vector fields...
On a polynomial inequality of P. Erdős and T. Grünwald
Directory of Open Access Journals (Sweden)
Rahman QI
1999-01-01
Full Text Available Let be a polynomial with only real zeros having , as consecutive zeros. It was proved by P. Erdős and T. Grünwald that if on , then the ratio of the area under the curve to the area of the tangential rectangle does not exceed . The main result of our paper is a multidimensional version of this result. First, we replace the class of polynomials considered by Erdős and Grünwald by the wider class consisting of functions of the form , where is logarithmically concave on , and show that their result holds for all functions in . More generally, we show that if and , then for all , the integral does not exceed . It is this result that is extended to higher dimensions. Our consideration of the class is crucial, since, unlike the narrower one of Erdős and Grünwald, its definition does not involve the distribution of zeros of its elements; besides, the notion of logarithmic concavity makes perfect sense for functions of several variables.
Random polynomials and expected complexity of bisection methods for real solving
DEFF Research Database (Denmark)
Emiris, Ioannis Z.; Galligo, André; Tsigaridas, Elias
2010-01-01
, and by Edelman and Kostlan in order to estimate the real root separation of degree d polynomials with i.i.d. coefficients that follow two zero-mean normal distributions: for SO(2) polynomials, the i-th coefficient has variance (d/i), whereas for Weyl polynomials its variance is 1/i!. By applying results from....... The second part of the paper shows that the expected number of real roots of a degree d polynomial in the Bernstein basis is √2d ± O(1), when the coefficients are i.i.d. variables with moderate standard deviation. Our paper concludes with experimental results which corroborate our analysis....
Euler polynomials and identities for non-commutative operators
De Angelis, Valerio; Vignat, Christophe
2015-12-01
Three kinds of identities involving non-commutating operators and Euler and Bernoulli polynomials are studied. The first identity, as given by Bender and Bettencourt [Phys. Rev. D 54(12), 7710-7723 (1996)], expresses the nested commutator of the Hamiltonian and momentum operators as the commutator of the momentum and the shifted Euler polynomial of the Hamiltonian. The second one, by Pain [J. Phys. A: Math. Theor. 46, 035304 (2013)], links the commutators and anti-commutators of the monomials of the position and momentum operators. The third appears in a work by Figuieira de Morisson and Fring [J. Phys. A: Math. Gen. 39, 9269 (2006)] in the context of non-Hermitian Hamiltonian systems. In each case, we provide several proofs and extensions of these identities that highlight the role of Euler and Bernoulli polynomials.
On integral and finite Fourier transforms of continuous q-Hermite polynomials
International Nuclear Information System (INIS)
Atakishiyeva, M. K.; Atakishiyev, N. M.
2009-01-01
We give an overview of the remarkably simple transformation properties of the continuous q-Hermite polynomials H n (x vertical bar q) of Rogers with respect to the classical Fourier integral transform. The behavior of the q-Hermite polynomials under the finite Fourier transform and an explicit form of the q-extended eigenfunctions of the finite Fourier transform, defined in terms of these polynomials, are also discussed.
On polynomial selection for the general number field sieve
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.
Automorphisms of Algebras and Bochner's Property for Vector Orthogonal Polynomials
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.
Luigi Gatteschi's work on asymptotics of special functions and their zeros
Gautschi, Walter; Giordano, Carla
2008-12-01
A good portion of Gatteschi's research publications-about 65%-is devoted to asymptotics of special functions and their zeros. Most prominently among the special functions studied figure classical orthogonal polynomials, notably Jacobi polynomials and their special cases, Laguerre polynomials, and Hermite polynomials by implication. Other important classes of special functions dealt with are Bessel functions of the first and second kind, Airy functions, and confluent hypergeometric functions, both in Tricomi's and Whittaker's form. This work is reviewed here, and organized along methodological lines.
Raising and Lowering Operators for Askey-Wilson Polynomials
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Siddhartha Sahi
2007-01-01
Full Text Available In this paper we describe two pairs of raising/lowering operators for Askey-Wilson polynomials, which result from constructions involving very different techniques. The first technique is quite elementary, and depends only on the ''classical'' properties of these polynomials, viz. the q-difference equation and the three term recurrence. The second technique is less elementary, and involves the one-variable version of the double affine Hecke algebra.
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 ...
Computing derivative-based global sensitivity measures using polynomial chaos expansions
International Nuclear Information System (INIS)
Sudret, B.; Mai, C.V.
2015-01-01
In the field of computer experiments sensitivity analysis aims at quantifying the relative importance of each input parameter (or combinations thereof) of a computational model with respect to the model output uncertainty. Variance decomposition methods leading to the well-known Sobol' indices are recognized as accurate techniques, at a rather high computational cost though. The use of polynomial chaos expansions (PCE) to compute Sobol' indices has allowed to alleviate the computational burden though. However, when dealing with large dimensional input vectors, it is good practice to first use screening methods in order to discard unimportant variables. The derivative-based global sensitivity measures (DGSMs) have been developed recently in this respect. In this paper we show how polynomial chaos expansions may be used to compute analytically DGSMs as a mere post-processing. This requires the analytical derivation of derivatives of the orthonormal polynomials which enter PC expansions. Closed-form expressions for Hermite, Legendre and Laguerre polynomial expansions are given. The efficiency of the approach is illustrated on two well-known benchmark problems in sensitivity analysis. - Highlights: • Derivative-based global sensitivity measures (DGSM) have been developed for screening purpose. • Polynomial chaos expansions (PC) are used as a surrogate model of the original computational model. • From a PC expansion the DGSM can be computed analytically. • The paper provides the derivatives of Hermite, Legendre and Laguerre polynomials for this purpose
Exact polynomial solutions of second order differential equations and their applications
International Nuclear Information System (INIS)
Zhang Yaozhong
2012-01-01
We find all polynomials Z(z) such that the differential equation where X(z), Y(z), Z(z) are polynomials of degree at most 4, 3, 2, respectively, has polynomial solutions S(z) = ∏ n i=1 (z − z i ) of degree n with distinct roots z i . We derive a set of n algebraic equations which determine these roots. We also find all polynomials Z(z) which give polynomial solutions to the differential equation when the coefficients of X(z) and Y(z) are algebraically dependent. As applications to our general results, we obtain the exact (closed-form) solutions of the Schrödinger-type differential equations describing: (1) two Coulombically repelling electrons on a sphere; (2) Schrödinger equation from the kink stability analysis of φ 6 -type field theory; (3) static perturbations for the non-extremal Reissner–Nordström solution; (4) planar Dirac electron in Coulomb and magnetic fields; and (5) O(N) invariant decatic anharmonic oscillator. (paper)
Multivariate Local Polynomial Regression with Application to Shenzhen Component Index
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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.
Twisted Polynomials and Forgery Attacks on GCM
DEFF Research Database (Denmark)
Abdelraheem, Mohamed Ahmed A. M. A.; Beelen, Peter; Bogdanov, Andrey
2015-01-01
Polynomial hashing as an instantiation of universal hashing is a widely employed method for the construction of MACs and authenticated encryption (AE) schemes, the ubiquitous GCM being a prominent example. It is also used in recent AE proposals within the CAESAR competition which aim at providing...... in an improved key recovery algorithm. As cryptanalytic applications of our twisted polynomials, we develop the first universal forgery attacks on GCM in the weak-key model that do not require nonce reuse. Moreover, we present universal weak-key forgeries for the nonce-misuse resistant AE scheme POET, which...
Sraj, Ihab; Le Maî tre, Olivier P.; Knio, Omar; Hoteit, Ibrahim
2015-01-01
using a coordinate transformation to account for the dependence with respect to the covariance hyper-parameters. Polynomial Chaos expansions are employed for the acceleration of the Bayesian inference using similar coordinate transformations, enabling us
All-Pole Recursive Digital Filters Design Based on Ultraspherical Polynomials
N. Stojanovic; N. Stamenkovic; V. Stojanovic
2014-01-01
A simple method for approximation of all-pole recursive digital filters, directly in digital domain, is described. Transfer function of these filters, referred to as Ultraspherical filters, is controlled by order of the Ultraspherical polynomial, nu. Parameter nu, restricted to be a nonnegative real number (nu ≥ 0), controls ripple peaks in the passband of the magnitude response and enables a trade-off between the passband loss and the group delay response of the resulting filter. Chebyshev f...
Flowchart Programs, Regular Expressions, and Decidability of Polynomial Growth-Rate
Ben-Amram, Amir M.; Pineles, Aviad
2014-01-01
We present a new method for inferring complexity properties for a class of programs in the form of flowcharts annotated with loop information. Specifically, our method can (soundly and completely) decide if computed values are polynomially bounded as a function of the input; and similarly for the running time. Such complexity properties are undecidable for a Turing-complete programming language, and a common work-around in program analysis is to settle for sound but incomplete solutions. In ...
Fractional order differentiation by integration with Jacobi polynomials
Liu, Dayan
2012-12-01
The differentiation by integration method with Jacobi polynomials was originally introduced by Mboup, Join and Fliess [22], [23]. This paper generalizes this method from the integer order to the fractional order for estimating the fractional order derivatives of noisy signals. The proposed fractional order differentiator is deduced from the Jacobi orthogonal polynomial filter and the Riemann-Liouville fractional order derivative definition. Exact and simple formula for this differentiator is given where an integral formula involving Jacobi polynomials and the noisy signal is used without complex mathematical deduction. Hence, it can be used both for continuous-time and discrete-time models. The comparison between our differentiator and the recently introduced digital fractional order Savitzky-Golay differentiator is given in numerical simulations so as to show its accuracy and robustness with respect to corrupting noises. © 2012 IEEE.
Synchronization of generalized Henon map using polynomial controller
International Nuclear Information System (INIS)
Lam, H.K.
2010-01-01
This Letter presents the chaos synchronization of two discrete-time generalized Henon map, namely the drive and response systems. A polynomial controller is proposed to drive the system states of the response system to follow those of the drive system. The system stability of the error system formed by the drive and response systems and the synthesis of the polynomial controller are investigated using the sum-of-squares (SOS) technique. Based on the Lyapunov stability theory, stability conditions in terms of SOS are derived to guarantee the system stability and facilitate the controller synthesis. By satisfying the SOS-based stability conditions, chaotic synchronization is achieved. The solution of the SOS-based stability conditions can be found numerically using the third-party Matlab toolbox SOSTOOLS. A simulation example is given to illustrate the merits of the proposed polynomial control approach.
Fractional order differentiation by integration with Jacobi polynomials
Liu, Dayan; Gibaru, O.; Perruquetti, Wilfrid; Laleg-Kirati, Taous-Meriem
2012-01-01
The differentiation by integration method with Jacobi polynomials was originally introduced by Mboup, Join and Fliess [22], [23]. This paper generalizes this method from the integer order to the fractional order for estimating the fractional order derivatives of noisy signals. The proposed fractional order differentiator is deduced from the Jacobi orthogonal polynomial filter and the Riemann-Liouville fractional order derivative definition. Exact and simple formula for this differentiator is given where an integral formula involving Jacobi polynomials and the noisy signal is used without complex mathematical deduction. Hence, it can be used both for continuous-time and discrete-time models. The comparison between our differentiator and the recently introduced digital fractional order Savitzky-Golay differentiator is given in numerical simulations so as to show its accuracy and robustness with respect to corrupting noises. © 2012 IEEE.
Crossover ensembles of random matrices and skew-orthogonal polynomials
International Nuclear Information System (INIS)
Kumar, Santosh; Pandey, Akhilesh
2011-01-01
Highlights: → We study crossover ensembles of Jacobi family of random matrices. → We consider correlations for orthogonal-unitary and symplectic-unitary crossovers. → We use the method of skew-orthogonal polynomials and quaternion determinants. → We prove universality of spectral correlations in crossover ensembles. → We discuss applications to quantum conductance and communication theory problems. - Abstract: In a recent paper (S. Kumar, A. Pandey, Phys. Rev. E, 79, 2009, p. 026211) we considered Jacobi family (including Laguerre and Gaussian cases) of random matrix ensembles and reported exact solutions of crossover problems involving time-reversal symmetry breaking. In the present paper we give details of the work. We start with Dyson's Brownian motion description of random matrix ensembles and obtain universal hierarchic relations among the unfolded correlation functions. For arbitrary dimensions we derive the joint probability density (jpd) of eigenvalues for all transitions leading to unitary ensembles as equilibrium ensembles. We focus on the orthogonal-unitary and symplectic-unitary crossovers and give generic expressions for jpd of eigenvalues, two-point kernels and n-level correlation functions. This involves generalization of the theory of skew-orthogonal polynomials to crossover ensembles. We also consider crossovers in the circular ensembles to show the generality of our method. In the large dimensionality limit, correlations in spectra with arbitrary initial density are shown to be universal when expressed in terms of a rescaled symmetry breaking parameter. Applications of our crossover results to communication theory and quantum conductance problems are also briefly discussed.
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.
Vanishing of Littlewood-Richardson polynomials is in P
Adve, Anshul; Robichaux, Colleen; Yong, Alexander
2017-01-01
J. DeLoera-T. McAllister and K. D. Mulmuley-H. Narayanan-M. Sohoni independently proved that determining the vanishing of Littlewood-Richardson coefficients has strongly polynomial time computational complexity. Viewing these as Schubert calculus numbers, we prove the generalization to the Littlewood-Richardson polynomials that control equivariant cohomology of Grassmannians. We construct a polytope using the edge-labeled tableau rule of H. Thomas-A. Yong. Our proof then combines a saturation...
Design and Use of a Learning Object for Finding Complex Polynomial Roots
Benitez, Julio; Gimenez, Marcos H.; Hueso, Jose L.; Martinez, Eulalia; Riera, Jaime
2013-01-01
Complex numbers are essential in many fields of engineering, but students often fail to have a natural insight of them. We present a learning object for the study of complex polynomials that graphically shows that any complex polynomials has a root and, furthermore, is useful to find the approximate roots of a complex polynomial. Moreover, we…
International Nuclear Information System (INIS)
Pask, J.E.; Klein, B.M.; Fong, C.Y.; Sterne, P.A.
1999-01-01
We present an approach to solid-state electronic-structure calculations based on the finite-element method. In this method, the basis functions are strictly local, piecewise polynomials. Because the basis is composed of polynomials, the method is completely general and its convergence can be controlled systematically. Because the basis functions are strictly local in real space, the method allows for variable resolution in real space; produces sparse, structured matrices, enabling the effective use of iterative solution methods; and is well suited to parallel implementation. The method thus combines the significant advantages of both real-space-grid and basis-oriented approaches and so promises to be particularly well suited for large, accurate ab initio calculations. We develop the theory of our approach in detail, discuss advantages and disadvantages, and report initial results, including electronic band structures and details of the convergence of the method. copyright 1999 The American Physical Society
From Jack to Double Jack Polynomials via the Supersymmetric Bridge
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.
Directory of Open Access Journals (Sweden)
Y. Yuliana
2011-07-01
Full Text Available The aim of an orthodontic treatment is to achieve aesthetic, dental health and the surrounding tissues, occlusal functional relationship, and stability. The success of an orthodontic treatment is influenced by many factors, such as diagnosis and treatment plan. In order to do a diagnosis and a treatment plan, medical record, clinical examination, radiographic examination, extra oral and intra oral photos, as well as study model analysis are needed. The purpose of this study was to evaluate the differences in dental arch form between level four polynomial and pentamorphic arch form and to determine which one is best suitable for normal occlusion sample. This analytic comparative study was conducted at Faculty of Dentistry Universitas Padjadjaran on 13 models by comparing the dental arch form using the level four polynomial method based on mathematical calculations, the pattern of the pentamorphic arch and mandibular normal occlusion as a control. The results obtained were tested using statistical analysis T student test. The results indicate a significant difference both in the form of level four polynomial method and pentamorphic arch form when compared with mandibular normal occlusion dental arch form. Level four polynomial fits better, compare to pentamorphic arch form.
Korff, Christian
2010-10-01
Starting from the Verma module of U_{q}\\mathfrak {sl}(2) we consider the evaluation module for affine U_{q}\\widehat{\\mathfrak {sl}}(2) and discuss its crystal limit (q → 0). There exists an associated integrable statistical mechanics model on a square lattice defined in terms of vertex configurations. Its transfer matrix is the generating function for noncommutative complete symmetric polynomials in the generators of the affine plactic algebra, an extension of the finite plactic algebra first discussed by Lascoux and Schützenberger. The corresponding noncommutative elementary symmetric polynomials were recently shown to be generated by the transfer matrix of the so-called phase model discussed by Bogoliubov, Izergin and Kitanine. Here we establish that both generating functions satisfy Baxter's TQ-equation in the crystal limit by tying them to special U_{q}\\widehat{ \\mathfrak {sl}}(2) solutions of the Yang-Baxter equation. The TQ-equation amounts to the well-known Jacobi-Trudi formula leading naturally to the definition of noncommutative Schur polynomials. The latter can be employed to define a ring which has applications in conformal field theory and enumerative geometry: it is isomorphic to the fusion ring of the \\widehat{\\mathfrak {sl}}(n)_{k} Wess-Zumino-Novikov-Witten model whose structure constants are the dimensions of spaces of generalized θ-functions over the Riemann sphere with three punctures.
Some Bounds for the Logarithmic Function
DEFF Research Database (Denmark)
Topsøe, Flemming
2007-01-01
Development in continued fraction, rational approximations and orthogonal polynomials in relation to the logarithmic function are discussed.......Development in continued fraction, rational approximations and orthogonal polynomials in relation to the logarithmic function are discussed....
Rational approximations of f(R) cosmography through Pad'e polynomials
Capozziello, Salvatore; D'Agostino, Rocco; Luongo, Orlando
2018-05-01
We consider high-redshift f(R) cosmography adopting the technique of polynomial reconstruction. In lieu of considering Taylor treatments, which turn out to be non-predictive as soon as z>1, we take into account the Pad&apose rational approximations which consist in performing expansions converging at high redshift domains. Particularly, our strategy is to reconstruct f(z) functions first, assuming the Ricci scalar to be invertible with respect to the redshift z. Having the so-obtained f(z) functions, we invert them and we easily obtain the corresponding f(R) terms. We minimize error propagation, assuming no errors upon redshift data. The treatment we follow naturally leads to evaluating curvature pressure, density and equation of state, characterizing the universe evolution at redshift much higher than standard cosmographic approaches. We therefore match these outcomes with small redshift constraints got by framing the f(R) cosmology through Taylor series around 0zsimeq . This gives rise to a calibration procedure with small redshift that enables the definitions of polynomial approximations up to zsimeq 10. Last but not least, we show discrepancies with the standard cosmological model which go towards an extension of the ΛCDM paradigm, indicating an effective dark energy term evolving in time. We finally describe the evolution of our effective dark energy term by means of basic techniques of data mining.
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.
Asymptotics for the ratio and the zeros of multiple Charlier polynomials
Ndayiragije, François; Van Assche, Walter
2011-01-01
We investigate multiple Charlier polynomials and in particular we will use the (nearest neighbor) recurrence relation to find the asymptotic behavior of the ratio of two multiple Charlier polynomials. This result is then used to obtain the asymptotic distribution of the zeros, which is uniform on an interval. We also deal with the case where one of the parameters of the various Poisson distributions depend on the degree of the polynomial, in which case we obtain another asymptotic distributio...
Euler Polynomials and Identities for Non-Commutative Operators
De Angelis, V.; Vignat, C.
2015-01-01
Three kinds of identities involving non-commutating operators and Euler and Bernoulli polynomials are studied. The first identity, as given by Bender and Bettencourt, expresses the nested commutator of the Hamiltonian and momentum operators as the commutator of the momentum and the shifted Euler polynomial of the Hamiltonian. The second one, due to J.-C. Pain, links the commutators and anti-commutators of the monomials of the position and momentum operators. The third appears in a work by Fig...
Local polynomial Whittle estimation of perturbed fractional processes
DEFF Research Database (Denmark)
Frederiksen, Per; Nielsen, Frank; Nielsen, Morten Ørregaard
We propose a semiparametric local polynomial Whittle with noise (LPWN) estimator of the memory parameter in long memory time series perturbed by a noise term which may be serially correlated. The estimator approximates the spectrum of the perturbation as well as that of the short-memory component...... of the signal by two separate polynomials. Including these polynomials we obtain a reduction in the order of magnitude of the bias, but also in‡ate the asymptotic variance of the long memory estimate by a multiplicative constant. We show that the estimator is consistent for d 2 (0; 1), asymptotically normal...... for d ε (0, 3/4), and if the spectral density is infinitely smooth near frequency zero, the rate of convergence can become arbitrarily close to the parametric rate, pn. A Monte Carlo study reveals that the LPWN estimator performs well in the presence of a serially correlated perturbation term...
Polynomial algebra of discrete models in systems biology.
Veliz-Cuba, Alan; Jarrah, Abdul Salam; Laubenbacher, Reinhard
2010-07-01
An increasing number of discrete mathematical models are being published in Systems Biology, ranging from Boolean network models to logical models and Petri nets. They are used to model a variety of biochemical networks, such as metabolic networks, gene regulatory networks and signal transduction networks. There is increasing evidence that such models can capture key dynamic features of biological networks and can be used successfully for hypothesis generation. This article provides a unified framework that can aid the mathematical analysis of Boolean network models, logical models and Petri nets. They can be represented as polynomial dynamical systems, which allows the use of a variety of mathematical tools from computer algebra for their analysis. Algorithms are presented for the translation into polynomial dynamical systems. Examples are given of how polynomial algebra can be used for the model analysis. alanavc@vt.edu Supplementary data are available at Bioinformatics online.
Polynomial chaos expansion with random and fuzzy variables
Jacquelin, E.; Friswell, M. I.; Adhikari, S.; Dessombz, O.; Sinou, J.-J.
2016-06-01
A dynamical uncertain system is studied in this paper. Two kinds of uncertainties are addressed, where the uncertain parameters are described through random variables and/or fuzzy variables. A general framework is proposed to deal with both kinds of uncertainty using a polynomial chaos expansion (PCE). It is shown that fuzzy variables may be expanded in terms of polynomial chaos when Legendre polynomials are used. The components of the PCE are a solution of an equation that does not depend on the nature of uncertainty. Once this equation is solved, the post-processing of the data gives the moments of the random response when the uncertainties are random or gives the response interval when the variables are fuzzy. With the PCE approach, it is also possible to deal with mixed uncertainty, when some parameters are random and others are fuzzy. The results provide a fuzzy description of the response statistical moments.
Polynomial Vector Fields in One Complex Variable
DEFF Research Database (Denmark)
Branner, Bodil
In recent years Adrien Douady was interested in polynomial vector fields, both in relation to iteration theory and as a topic on their own. This talk is based on his work with Pierrette Sentenac, work of Xavier Buff and Tan Lei, and my own joint work with Kealey Dias.......In recent years Adrien Douady was interested in polynomial vector fields, both in relation to iteration theory and as a topic on their own. This talk is based on his work with Pierrette Sentenac, work of Xavier Buff and Tan Lei, and my own joint work with Kealey Dias....
Smith, K; Abasolo, Daniel Emilio; Escudero, J
2016-01-01
The Cluster-Span Threshold (CST) is a recently introduced unbiased threshold for functional connectivity networks. This binarisation technique offers a natural trade-off of sparsity and density of information by balancing the ratio of closed to open triples in the network topology. Here we present findings comparing it with the Union of Shortest Paths (USP), another recently proposed objective method. We analyse standard network metrics of binarised networks for sensitivity to clinical Alzhei...
Directory of Open Access Journals (Sweden)
Litesh N. Sulbhewar
Full Text Available The convergence characteristic of the conventional two-noded Euler-Bernoulli piezoelectric beam finite element depends on the configuration of the beam cross-section. The element shows slower convergence for the asymmetric material distribution in the beam cross-section due to 'material-locking' caused by extension-bending coupling. Hence, the use of conventional Euler-Bernoulli beam finite element to analyze piezoelectric beams which are generally made of the host layer with asymmetrically surface bonded piezoelectric layers/patches, leads to increased computational effort to yield converged results. Here, an efficient coupled polynomial interpolation scheme is proposed to improve the convergence of the Euler-Bernoulli piezoelectric beam finite elements, by eliminating ill-effects of material-locking. The equilibrium equations, derived using a variational formulation, are used to establish relationships between field variables. These relations are used to find a coupled quadratic polynomial for axial displacement, having contributions from an assumed cubic polynomial for transverse displacement and assumed linear polynomials for layerwise electric potentials. A set of coupled shape functions derived using these polynomials efficiently handles extension-bending and electromechanical couplings at the field interpolation level itself in a variationally consistent manner, without increasing the number of nodal degrees of freedom. The comparison of results obtained from numerical simulation of test problems shows that the convergence characteristic of the proposed element is insensitive to the material configuration of the beam cross-section.
An extension of Krawtchouk\\'s polynomials to the contstruction of ...
African Journals Online (AJOL)
A simple method is described for the construction of a set of orthogonal polynomials for any case where the proportions of observations follow a binomial distribution. The least squares equation which fits the data is determined using the properties of orthogonal polynomials and the analysis of variance technique.
Prediction of zeolite-cement-sand unconfined compressive strength using polynomial neural network
MolaAbasi, H.; Shooshpasha, I.
2016-04-01
The improvement of local soils with cement and zeolite can provide great benefits, including strengthening slopes in slope stability problems, stabilizing problematic soils and preventing soil liquefaction. Recently, dosage methodologies are being developed for improved soils based on a rational criterion as it exists in concrete technology. There are numerous earlier studies showing the possibility of relating Unconfined Compressive Strength (UCS) and Cemented sand (CS) parameters (voids/cement ratio) as a power function fits. Taking into account the fact that the existing equations are incapable of estimating UCS for zeolite cemented sand mixture (ZCS) well, artificial intelligence methods are used for forecasting them. Polynomial-type neural network is applied to estimate the UCS from more simply determined index properties such as zeolite and cement content, porosity as well as curing time. In order to assess the merits of the proposed approach, a total number of 216 unconfined compressive tests have been done. A comparison is carried out between the experimentally measured UCS with the predictions in order to evaluate the performance of the current method. The results demonstrate that generalized polynomial-type neural network has a great ability for prediction of the UCS. At the end sensitivity analysis of the polynomial model is applied to study the influence of input parameters on model output. The sensitivity analysis reveals that cement and zeolite content have significant influence on predicting UCS.
Geometry of polynomials and root-finding via path-lifting
Kim, Myong-Hi; Martens, Marco; Sutherland, Scott
2018-02-01
Using the interplay between topological, combinatorial, and geometric properties of polynomials and analytic results (primarily the covering structure and distortion estimates), we analyze a path-lifting method for finding approximate zeros, similar to those studied by Smale, Shub, Kim, and others. Given any polynomial, this simple algorithm always converges to a root, except on a finite set of initial points lying on a circle of a given radius. Specifically, the algorithm we analyze consists of iterating where the t k form a decreasing sequence of real numbers and z 0 is chosen on a circle containing all the roots. We show that the number of iterates required to locate an approximate zero of a polynomial f depends only on log\\vert f(z_0)/ρ_\\zeta\\vert (where ρ_\\zeta is the radius of convergence of the branch of f-1 taking 0 to a root ζ) and the logarithm of the angle between f(z_0) and certain critical values. Previous complexity results for related algorithms depend linearly on the reciprocals of these angles. Note that the complexity of the algorithm does not depend directly on the degree of f, but only on the geometry of the critical values. Furthermore, for any polynomial f with distinct roots, the average number of steps required over all starting points taken on a circle containing all the roots is bounded by a constant times the average of log(1/ρ_\\zeta) . The average of log(1/ρ_\\zeta) over all polynomials f with d roots in the unit disk is \
The Combinatorial Rigidity Conjecture is False for Cubic Polynomials
DEFF Research Database (Denmark)
Henriksen, Christian
2003-01-01
We show that there exist two cubic polynomials with connected Julia sets which are combinatorially equivalent but not topologically conjugate on their Julia sets. This disproves a conjecture by McMullen from 1995.......We show that there exist two cubic polynomials with connected Julia sets which are combinatorially equivalent but not topologically conjugate on their Julia sets. This disproves a conjecture by McMullen from 1995....
Ratio asymptotics of Hermite-Pade polynomials for Nikishin systems
International Nuclear Information System (INIS)
Aptekarev, A I; Lopez, Guillermo L; Rocha, I A
2005-01-01
The existence of ratio asymptotics is proved for a sequence of multiple orthogonal polynomials with orthogonality relations distributed among a system of m finite Borel measures with support on a bounded interval of the real line which form a so-called Nikishin system. For m=1 this result reduces to Rakhmanov's celebrated theorem on the ratio asymptotics for orthogonal polynomials on the real line.
International Nuclear Information System (INIS)
Calogero, F.
1978-01-01
Let zsub(j)(α, β) be the jth zero of the Jacobi polynomial J sub(n)sup(α,β)(z), and xsub(j) the jth zero of the Hermite polynomial Hsub(n)(x). Then, as t→infinity, zsub(j)(at,bt)=(b-a)/(b+a)+t sup(-1/2)c x sub(j)+t -1 4/3(n+1/2+xsub(j) 2 )(a-b)/(a+b) 2 +0(t sup(-3/2)), with c=(ab)sup(1/2) [(a+b)/2]sup(-3/2) a>0, b>0. This formula implies the limit relation n exclamation mark lim sub(t→infinity) [t sup(-n/2)J sub(n)sup(at,bt) ((b-a)/(b+a)+t sup(-1/2)x)] = [(a+b)c/4]sup(n) Hsub(n)(chi/c). (author)
Congruences concerning Legendre polynomials III
Sun, Zhi-Hong
2010-01-01
Let $p>3$ be a prime, and let $R_p$ be the set of rational numbers whose denominator is coprime to $p$. Let $\\{P_n(x)\\}$ be the Legendre polynomials. In this paper we mainly show that for $m,n,t\\in R_p$ with $m\
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...
Real zeros of classes of random algebraic polynomials
Directory of Open Access Journals (Sweden)
K. Farahmand
2003-01-01
Full Text Available There are many known asymptotic estimates for the expected number of real zeros of an algebraic polynomial a0+a1x+a2x2+⋯+an−1xn−1 with identically distributed random coefficients. Under different assumptions for the distribution of the coefficients {aj}j=0n−1 it is shown that the above expected number is asymptotic to O(logn. This order for the expected number of zeros remains valid for the case when the coefficients are grouped into two, each group with a different variance. However, it was recently shown that if the coefficients are non-identically distributed such that the variance of the jth term is (nj the expected number of zeros of the polynomial increases to O(n. The present paper provides the value for this asymptotic formula for the polynomials with the latter variances when they are grouped into three with different patterns for their variances.
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
Quantum entanglement via nilpotent polynomials
International Nuclear Information System (INIS)
Mandilara, Aikaterini; Akulin, Vladimir M.; Smilga, Andrei V.; Viola, Lorenza
2006-01-01
We propose a general method for introducing extensive characteristics of quantum entanglement. The method relies on polynomials of nilpotent raising operators that create entangled states acting on a reference vacuum state. By introducing the notion of tanglemeter, the logarithm of the state vector represented in a special canonical form and expressed via polynomials of nilpotent variables, we show how this description provides a simple criterion for entanglement as well as a universal method for constructing the invariants characterizing entanglement. We compare the existing measures and classes of entanglement with those emerging from our approach. We derive the equation of motion for the tanglemeter and, in representative examples of up to four-qubit systems, show how the known classes appear in a natural way within our framework. We extend our approach to qutrits and higher-dimensional systems, and make contact with the recently introduced idea of generalized entanglement. Possible future developments and applications of the method are discussed
Directory of Open Access Journals (Sweden)
Ernest G. Kalnins
2013-10-01
Full Text Available We show explicitly that all 2nd order superintegrable systems in 2 dimensions are limiting cases of a single system: the generic 3-parameter potential on the 2-sphere, S9 in our listing. We extend the Wigner-Inönü method of Lie algebra contractions to contractions of quadratic algebras and show that all of the quadratic symmetry algebras of these systems are contractions of that of S9. Amazingly, all of the relevant contractions of these superintegrable systems on flat space and the sphere are uniquely induced by the well known Lie algebra contractions of e(2 and so(3. By contracting function space realizations of irreducible representations of the S9 algebra (which give the structure equations for Racah/Wilson polynomials to the other superintegrable systems, and using Wigner's idea of ''saving'' a representation, we obtain the full Askey scheme of hypergeometric orthogonal polynomials. This relationship directly ties the polynomials and their structure equations to physical phenomena. It is more general because it applies to all special functions that arise from these systems via separation of variables, not just those of hypergeometric type, and it extends to higher dimensions.
Directory of Open Access Journals (Sweden)
Chi Yaodan
2017-08-01
Full Text Available Crosstalk in wiring harness has been studied extensively for its importance in the naval ships electromagnetic compatibility field. An effective and high-efficiency method is proposed in this paper for analyzing Statistical Characteristics of crosstalk in wiring harness with random variation of position based on Polynomial Chaos Expansion (PCE. A typical 14-cable wiring harness was simulated as the object of research. Distance among interfering cable, affected cable and GND is synthesized and analyzed in both frequency domain and time domain. The model of naval ships wiring harness distribution parameter was established by utilizing Legendre orthogonal polynomials as basis functions along with prediction model of statistical characters. Detailed mean value, mean square error, probability density function and reasonable varying range of crosstalk in naval ships wiring harness are described in both time domain and frequency domain. Numerical experiment proves that the method proposed in this paper, not only has good consistency with the MC method can be applied in the naval ships EMC research field to provide theoretical support for guaranteeing safety, but also has better time-efficiency than the MC method. Therefore, the Polynomial Chaos Expansion method.
On generalized Rédei functions
Directory of Open Access Journals (Sweden)
R. Matthews
1988-01-01
Full Text Available A generalization of Rédei functions to polynomial vectors in n indeterminates over finite fields or residue class rings of integers is given by considering special types of polynomial vectors. Properties such as polynomial composition, change of basis, group structure and fixed points are studied together with applications in cryptography.
Nuclear-magnetic-resonance quantum calculations of the Jones polynomial
International Nuclear Information System (INIS)
Marx, Raimund; Spoerl, Andreas; Pomplun, Nikolas; Schulte-Herbrueggen, Thomas; Glaser, Steffen J.; Fahmy, Amr; Kauffman, Louis; Lomonaco, Samuel; Myers, John M.
2010-01-01
The repertoire of problems theoretically solvable by a quantum computer recently expanded to include the approximate evaluation of knot invariants, specifically the Jones polynomial. The experimental implementation of this evaluation, however, involves many known experimental challenges. Here we present experimental results for a small-scale approximate evaluation of the Jones polynomial by nuclear magnetic resonance (NMR); in addition, we show how to escape from the limitations of NMR approaches that employ pseudopure states. Specifically, we use two spin-1/2 nuclei of natural abundance chloroform and apply a sequence of unitary transforms representing the trefoil knot, the figure-eight knot, and the Borromean rings. After measuring the nuclear spin state of the molecule in each case, we are able to estimate the value of the Jones polynomial for each of the knots.
Alpay, D.; Dijksma, A.; Langer, H.
2004-01-01
We prove that a 2 × 2 matrix polynomial which is J-unitary on the real line can be written as a product of normalized elementary J-unitary factors and a J-unitary constant. In the second part we give an algorithm for this factorization using an analog of the Schur transformation.
Weierstrass method for quaternionic polynomial root-finding
Falcão, M. Irene; Miranda, Fernando; Severino, Ricardo; Soares, M. Joana
2018-01-01
Quaternions, introduced by Hamilton in 1843 as a generalization of complex numbers, have found, in more recent years, a wealth of applications in a number of different areas which motivated the design of efficient methods for numerically approximating the zeros of quaternionic polynomials. In fact, one can find in the literature recent contributions to this subject based on the use of complex techniques, but numerical methods relying on quaternion arithmetic remain scarce. In this paper we propose a Weierstrass-like method for finding simultaneously {\\sl all} the zeros of unilateral quaternionic polynomials. The convergence analysis and several numerical examples illustrating the performance of the method are also presented.
Characteristic Polynomials of Sample Covariance Matrices: The Non-Square Case
Kösters, Holger
2009-01-01
We consider the sample covariance matrices of large data matrices which have i.i.d. complex matrix entries and which are non-square in the sense that the difference between the number of rows and the number of columns tends to infinity. We show that the second-order correlation function of the characteristic polynomial of the sample covariance matrix is asymptotically given by the sine kernel in the bulk of the spectrum and by the Airy kernel at the edge of the spectrum. Similar results are g...
A set of sums for continuous dual q-2-Hahn polynomials
International Nuclear Information System (INIS)
Gade, R. M.
2009-01-01
An infinite set {τ (l) (y;r,z)} r,lisanelementofN 0 of linear sums of continuous dual q -2 -Hahn polynomials with prefactors depending on a complex parameter z is studied. The sums τ (l) (y;r,z) have an interpretation in context with tensor product representations of the quantum affine algebra U q ' (sl(2)) involving both a positive and a negative discrete series representation. For each l>0, the sum τ (l) (y;r,z) can be expressed in terms of the sum τ (0) (y;r,z), continuous dual q 2 -Hahn polynomials, and their associated polynomials. The sum τ (0) (y;r,z) is obtained as a combination of eight basic hypergeometric series. Moreover, an integral representation is provided for the sums τ (l) (y;r,z) with the complex parameter restricted by |zq| -2 -Hahn polynomials.
Fibonacci-like Differential Equations with a Polynomial Non-Homogeneous Part
Asveld, P.R.J.
1989-01-01
We investigate non-homogeneous linear differential equations of the form $x''(t) + x'(t) - x(t) = p(t)$ where $p(t)$ is either a polynomial or a factorial polynomial in $t$. We express the solution of these differential equations in terms of the coefficients of $p(t)$, in the initial conditions, and
Sparse DOA estimation with polynomial rooting
DEFF Research Database (Denmark)
Xenaki, Angeliki; Gerstoft, Peter; Fernandez Grande, Efren
2015-01-01
Direction-of-arrival (DOA) estimation involves the localization of a few sources from a limited number of observations on an array of sensors. Thus, DOA estimation can be formulated as a sparse signal reconstruction problem and solved efficiently with compressive sensing (CS) to achieve highresol......Direction-of-arrival (DOA) estimation involves the localization of a few sources from a limited number of observations on an array of sensors. Thus, DOA estimation can be formulated as a sparse signal reconstruction problem and solved efficiently with compressive sensing (CS) to achieve...... highresolution imaging. Utilizing the dual optimal variables of the CS optimization problem, it is shown with Monte Carlo simulations that the DOAs are accurately reconstructed through polynomial rooting (Root-CS). Polynomial rooting is known to improve the resolution in several other DOA estimation methods...
Hypergeometric series recurrence relations and some new orthogonal functions
International Nuclear Information System (INIS)
Wilson, J.A.
1978-01-01
A set of hypergeometric orthogonal polynomials, a set of biorthogonal rational functions generalizing them, and some new three-term relations for hypergeometric series containing properties of these functions are exhibited. The orthogonal polynomials depend on four free parameters, and their orthogonality relations include as special or limiting cases the orthogonalities for the classical polynomials, the Hahn and dual Hahn polynomials, Pollaczek's polynomials orthogonal on an infinite interval, and the 6-j symbols of angular momentum in quantum mechanics. Their properties include a second-order difference equation and a Rodrigues-type formula involving a divided difference operator
International Nuclear Information System (INIS)
Prastyaningrum, I.; Cari, C.; Suparmi, A.
2016-01-01
The approximation analytical solution of Dirac equation for Modified Poschl Teller plus Trigonometric Scarf Potential are investigated numerically in terms of finite Romanovsky Polynomial. The combination of two potentials are substituted into Dirac Equation then the variables are separated into radial and angular parts. The Dirac equation is solved by using Romanovsky Polynomial Method. The equation that can reduce from the second order of differential equation into the differential equation of hypergeometry type by substituted variable method. The energy spectrum is numerically solved using Matlab 2011. Where the increase in the radial quantum number nr and variable of modified Poschl Teller Potential causes the energy to decrease. The radial and the angular part of the wave function also visualized with Matlab 2011. The results show, by the disturbance of a combination between this potential can change the wave function of the radial and angular part. (paper)
Fabrication and correction of freeform surface based on Zernike polynomials by slow tool servo
Cheng, Yuan-Chieh; Hsu, Ming-Ying; Peng, Wei-Jei; Hsu, Wei-Yao
2017-10-01
Recently, freeform surface widely using to the optical system; because it is have advance of optical image and freedom available to improve the optical performance. For freeform optical fabrication by integrating freeform optical design, precision freeform manufacture, metrology freeform optics and freeform compensate method, to modify the form deviation of surface, due to production process of freeform lens ,compared and provides more flexibilities and better performance. This paper focuses on the fabrication and correction of the free-form surface. In this study, optical freeform surface using multi-axis ultra-precision manufacturing could be upgrading the quality of freeform. It is a machine equipped with a positioning C-axis and has the CXZ machining function which is also called slow tool servo (STS) function. The freeform compensate method of Zernike polynomials results successfully verified; it is correction the form deviation of freeform surface. Finally, the freeform surface are measured experimentally by Ultrahigh Accurate 3D Profilometer (UA3P), compensate the freeform form error with Zernike polynomial fitting to improve the form accuracy of freeform.
Polynomials in finite geometries and combinatorics
Blokhuis, A.; Walker, K.
1993-01-01
It is illustrated how elementary properties of polynomials can be used to attack extremal problems in finite and euclidean geometry, and in combinatorics. Also a new result, related to the problem of neighbourly cylinders is presented.
Lyapunov Function Synthesis - Algorithm and Software
DEFF Research Database (Denmark)
Leth, Tobias; Sloth, Christoffer; Wisniewski, Rafal
2016-01-01
In this paper we introduce an algorithm for the synthesis of polynomial Lyapunov functions for polynomial vector fields. The Lyapunov function is a continuous piecewisepolynomial defined on simplices, which compose a collection of simplices. The algorithm is elaborated and crucial features are ex...
A new derivation of the highest-weight polynomial of a unitary lie algebra
International Nuclear Information System (INIS)
P Chau, Huu-Tai; P Van, Isacker
2000-01-01
A new method is presented to derive the expression of the highest-weight polynomial used to build the basis of an irreducible representation (IR) of the unitary algebra U(2J+1). After a brief reminder of Moshinsky's method to arrive at the set of equations defining the highest-weight polynomial of U(2J+1), an alternative derivation of the polynomial from these equations is presented. The method is less general than the one proposed by Moshinsky but has the advantage that the determinantal expression of the highest-weight polynomial is arrived at in a direct way using matrix inversions. (authors)
Approximate solutions of dual fuzzy polynomials by feed-back neural networks
Directory of Open Access Journals (Sweden)
Ahmad Jafarian
2012-11-01
Full Text Available Recently, artificial neural networks (ANNs have been extensively studied and used in different areas such as pattern recognition, associative memory, combinatorial optimization, etc. In this paper, we investigate the ability of fuzzy neural networks to approximate solution of a dual fuzzy polynomial of the form $a_{1}x+ ...+a_{n}x^n =b_{1}x+ ...+b_{n}x^n+d,$ where $a_{j},b_{j},d epsilon E^1 (for j=1,...,n.$ Since the operation of fuzzy neural networks is based on Zadeh's extension principle. For this scope we train a fuzzified neural network by back-propagation-type learning algorithm which has five layer where connection weights are crisp numbers. This neural network can get a crisp input signal and then calculates its corresponding fuzzy output. Presented method can give a real approximate solution for given polynomial by using a cost function which is defined for the level sets of fuzzy output and target output. The simulation results are presented to demonstrate the efficiency and effectiveness of the proposed approach.
Analysis of the Level-Release Polynomial from a Hydroelectric Plant
Directory of Open Access Journals (Sweden)
Ieda Hidalgo
2012-02-01
Full Text Available The mathematic representation of the tailrace elevation as function of the water release can be modified, for example, by the geomorphologic impact of large floods. The level-release polynomial from a hydroelectric plant is important information to computational models used for optimization and simulation of the power generation systems operation. They depend on data quality to provide reliable results. Therefore, this paper presents a method for adjusting of the tailrace polynomial based on operation data recorded by the plant’s owner or company. The proposed method uses a non-linear regression tool, such as Trendline in Excel. A case study has been applied to the data from a large Brazilian hydroelectric plant whose operation is under the coordination of the Electric System ational Operator. The benefits of the data correction are analyzed using a simulation model for the hydroelectric plants operation. This simulator is used to reproduce the past operation of the plant, first with official data and second with adjusted data. The results show significant improvements in terms of quality of the data, contributing to bring the real and simulated operation closer.
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.
Lower bounds for the circuit size of partially homogeneous polynomials
Czech Academy of Sciences Publication Activity Database
Le, Hong-Van
2017-01-01
Roč. 225, č. 4 (2017), s. 639-657 ISSN 1072-3374 Institutional support: RVO:67985840 Keywords : partially homogeneous polynomials * polynomials Subject RIV: BA - General Mathematics OBOR OECD: Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8) https://link.springer.com/article/10.1007/s10958-017-3483-4
Generalized catalan numbers, sequences and polynomials
KOÇ, Cemal; GÜLOĞLU, İsmail; ESİN, Songül
2010-01-01
In this paper we present an algebraic interpretation for generalized Catalan numbers. We describe them as dimensions of certain subspaces of multilinear polynomials. This description is of utmost importance in the investigation of annihilators in exterior algebras.
Multilevel weighted least squares polynomial approximation
Haji-Ali, Abdul-Lateef; Nobile, Fabio; Tempone, Raul; Wolfers, Sö ren
2017-01-01
, 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
An algorithmic approach to solving polynomial equations associated with quantum circuits
International Nuclear Information System (INIS)
Gerdt, V.P.; Zinin, M.V.
2009-01-01
In this paper we present two algorithms for reducing systems of multivariate polynomial equations over the finite field F 2 to the canonical triangular form called lexicographical Groebner basis. This triangular form is the most appropriate for finding solutions of the system. On the other hand, the system of polynomials over F 2 whose variables also take values in F 2 (Boolean polynomials) completely describes the unitary matrix generated by a quantum circuit. In particular, the matrix itself can be computed by counting the number of solutions (roots) of the associated polynomial system. Thereby, efficient construction of the lexicographical Groebner bases over F 2 associated with quantum circuits gives a method for computing their circuit matrices that is alternative to the direct numerical method based on linear algebra. We compare our implementation of both algorithms with some other software packages available for computing Groebner bases over F 2
Canonical basis for type A4 (II) - Polynomial elements in one variable
International Nuclear Information System (INIS)
Hu Yuwang; Ye Jiachen
2003-12-01
All the 62 monomial elements in the canonical basis B of the quantized enveloping algebra for type A 4 have been determined. According to Lusztig's idea, the elements in the canonical basis B consist of monomials and linear combinations of monomials (for convenience, we call them polynomials). In this note, we compute all the 144 polynomial elements in one variable in the canonical basis B of the quantized enveloping algebra for type A 4 based on our joint note. We conjecture that there are other polynomial elements in two or three variables in the canonical basis B, which include independent variables and dependent variables. Moreover, it is conjectured that there are no polynomial elements in the canonical basis B with four or more variables. (author)
Discrete-Time Filter Synthesis using Product of Gegenbauer Polynomials
N. Stojanovic; N. Stamenkovic; I. Krstic
2016-01-01
A new approximation to design continuoustime and discrete-time low-pass filters, presented in this paper, based on the product of Gegenbauer polynomials, provides the ability of more flexible adjustment of passband and stopband responses. The design is achieved taking into account a prescribed specification, leading to a better trade-off among the magnitude and group delay responses. Many well-known continuous-time and discrete-time transitional filter based on the classical polynomial approx...
Bounds and asymptotics for orthogonal polynomials for varying weights
Levin, Eli
2018-01-01
This book establishes bounds and asymptotics under almost minimal conditions on the varying weights, and applies them to universality limits and entropy integrals. Orthogonal polynomials associated with varying weights play a key role in analyzing random matrices and other topics. This book will be of use to a wide community of mathematicians, physicists, and statisticians dealing with techniques of potential theory, orthogonal polynomials, approximation theory, as well as random matrices. .
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 f...
Fitting polynomial surfaces to triangular meshes with Voronoi squared distance minimization
Nivoliers, Vincent
2012-11-06
This paper introduces Voronoi squared distance minimization (VSDM), an algorithm that fits a surface to an input mesh. VSDM minimizes an objective function that corresponds to a Voronoi-based approximation of the overall squared distance function between the surface and the input mesh (SDM). This objective function is a generalization of the one minimized by centroidal Voronoi tessellation, and can be minimized by a quasi-Newton solver. VSDM naturally adapts the orientation of the mesh elements to best approximate the input, without estimating any differential quantities. Therefore, it can be applied to triangle soups or surfaces with degenerate triangles, topological noise and sharp features. Applications of fitting quad meshes and polynomial surfaces to input triangular meshes are demonstrated. © 2012 Springer-Verlag London.
Real-root property of the spectral polynomial of the Treibich-Verdier potential and related problems
Chen, Zhijie; Kuo, Ting-Jung; Lin, Chang-Shou; Takemura, Kouichi
2018-04-01
We study the spectral polynomial of the Treibich-Verdier potential. Such spectral polynomial, which is a generalization of the classical Lamé polynomial, plays fundamental roles in both the finite-gap theory and the ODE theory of Heun's equation. In this paper, we prove that all the roots of such spectral polynomial are real and distinct under some assumptions. The proof uses the classical concept of Sturm sequence and isomonodromic theories. We also prove an analogous result for a polynomial associated with a generalized Lamé equation, where we apply a new approach based on the viewpoint of the monodromy data.
On Sequences of Numbers and Polynomials Defined by Linear Recurrence Relations of Order 2
Directory of Open Access Journals (Sweden)
Tian-Xiao He
2009-01-01
Full Text Available Here we present a new method to construct the explicit formula of a sequence of numbers and polynomials generated by a linear recurrence relation of order 2. The applications of the method to the Fibonacci and Lucas numbers, Chebyshev polynomials, the generalized Gegenbauer-Humbert polynomials are also discussed. The derived idea provides a general method to construct identities of number or polynomial sequences defined by linear recurrence relations. The applications using the method to solve some algebraic and ordinary differential equations are presented.
Directory of Open Access Journals (Sweden)
S. Vukotic
2016-08-01
Full Text Available Digital polynomial-based interpolation filters implemented using the Farrow structure are used in Digital Signal Processing (DSP to calculate the signal between its discrete samples. The two basic design parameters for these filters are number of polynomial-segments defining the finite length of impulse response, and order of polynomials in each polynomial segment. The complexity of the implementation structure and the frequency domain performance depend on these two parameters. This contribution presents estimation formulae for length and polynomial order of polynomial-based filters for various types of requirements including attenuation in stopband, width of transitions band, deviation in passband, weighting in passband/stopband.
Invariant hyperplanes and Darboux integrability of polynomial vector fields
International Nuclear Information System (INIS)
Zhang Xiang
2002-01-01
This paper is composed of two parts. In the first part, we provide an upper bound for the number of invariant hyperplanes of the polynomial vector fields in n variables. This result generalizes those given in Artes et al (1998 Pac. J. Math. 184 207-30) and Llibre and Rodriguez (2000 Bull. Sci. Math. 124 599-619). The second part gives an extension of the Darboux theory of integrability to polynomial vector fields on algebraic varieties
Nobile, Fabio
2015-01-07
We consider a general problem F(u, y) = 0 where u is the unknown solution, possibly Hilbert space valued, and y a set of uncertain parameters. We specifically address the situation in which the parameterto-solution map u(y) is smooth, however y could be very high (or even infinite) dimensional. In particular, we are interested in cases in which F is a differential operator, u a Hilbert space valued function and y a distributed, space and/or time varying, random field. We aim at reconstructing the parameter-to-solution map u(y) from random noise-free or noisy observations in random points by discrete least squares on polynomial spaces. The noise-free case is relevant whenever the technique is used to construct metamodels, based on polynomial expansions, for the output of computer experiments. In the case of PDEs with random parameters, the metamodel is then used to approximate statistics of the output quantity. We discuss the stability of discrete least squares on random points show convergence estimates both in expectation and probability. We also present possible strategies to select, either a-priori or by adaptive algorithms, sequences of approximating polynomial spaces that allow to reduce, and in some cases break, the curse of dimensionality
On selfadjoint functors satisfying polynomial relations
DEFF Research Database (Denmark)
Agerholm, Troels; Mazorchuk, Volodomyr
2011-01-01
We study selfadjoint functors acting on categories of finite dimen- sional modules over finite dimensional algebras with an emphasis on functors satisfying some polynomial relations. Selfadjoint func- tors satisfying several easy relations, in particular, idempotents and square roots of a sum...
Bayer Demosaicking with Polynomial Interpolation.
Wu, Jiaji; Anisetti, Marco; Wu, Wei; Damiani, Ernesto; Jeon, Gwanggil
2016-08-30
Demosaicking is a digital image process to reconstruct full color digital images from incomplete color samples from an image sensor. It is an unavoidable process for many devices incorporating camera sensor (e.g. mobile phones, tablet, etc.). In this paper, we introduce a new demosaicking algorithm based on polynomial interpolation-based demosaicking (PID). Our method makes three contributions: calculation of error predictors, edge classification based on color differences, and a refinement stage using a weighted sum strategy. Our new predictors are generated on the basis of on the polynomial interpolation, and can be used as a sound alternative to other predictors obtained by bilinear or Laplacian interpolation. In this paper we show how our predictors can be combined according to the proposed edge classifier. After populating three color channels, a refinement stage is applied to enhance the image quality and reduce demosaicking artifacts. Our experimental results show that the proposed method substantially improves over existing demosaicking methods in terms of objective performance (CPSNR, S-CIELAB E, and FSIM), and visual performance.
Huang, Yi-Jen; Chao, Shih-Chun; Lien, Der-Hsien; Wen, Cheng-Yen; He, Jr-Hau; Lee, Si-Chen
2016-01-01
The combination of nonvolatile memory switching and volatile threshold switching functions of transition metal oxides in crossbar memory arrays is of great potential for replacing charge-based flash memory in very-large-scale integration. Here, we
Viewing the Roots of Polynomial Functions in Complex Variable: The Use of Geogebra and the CAS Maple
Alves, Francisco Regis Vieira
2013-01-01
Admittedly, the Fundamental Theorem of Calculus-TFA holds an important role in the Complex Analysis-CA, as well as in other mathematical branches. In this article, we bring a discussion about the TFA, the Rouché's theorem and the winding number with the intention to analyze the roots of a polynomial equation. We propose also a description for a…
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.)