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## Estimação de distribuições discretas via cópulas de Bernstein; Discrete Distributions Estimation via Bernstein Copulas

Fossaluza, Victor
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Tese de Doutorado Formato: application/pdf
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As relações de dependência entre variáveis aleatórias é um dos assuntos mais discutidos em probabilidade e estatística e a forma mais abrangente de estudar essas relações é por meio da distribuição conjunta. Nos últimos anos vem crescendo a utilização de cópulas para representar a estrutura de dependência entre variáveis aleatórias em uma distribuição multivariada. Contudo, ainda existe pouca literatura sobre cópulas quando as distribuições marginais são discretas. No presente trabalho será apresentada uma proposta não-paramétrica de estimação da distribuição conjunta bivariada de variáveis aleatórias discretas utilizando cópulas e polinômios de Bernstein.; The relations of dependence between random variables is one of the most discussed topics in probability and statistics and the best way to study these relationships is through the joint distribution. In the last years has increased the use of copulas to represent the dependence structure among random variables in a multivariate distribution. However, there is still little literature on copulas when the marginal distributions are discrete. In this work we present a non-parametric approach for the estimation of the bivariate joint distribution of discrete random variables using copulas and Bernstein polynomials.

## Numerical Solutions of the Nonlinear Fractional-Order Brusselator System by Bernstein Polynomials

Khan, Hasib; Jafari, Hossein; Khan, Rahmat Ali; Tajadodi, Haleh; Johnston, Sarah Jane
Fonte: Hindawi Publishing Corporation Publicador: Hindawi Publishing Corporation
Tipo: Artigo de Revista Científica
Português
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In this paper we propose the Bernstein polynomials to achieve the numerical solutions of nonlinear fractional-order chaotic system known by fractional-order Brusselator system. We use operational matrices of fractional integration and multiplication of Bernstein polynomials, which turns the nonlinear fractional-order Brusselator system to a system of algebraic equations. Two illustrative examples are given in order to demonstrate the accuracy and simplicity of the proposed techniques.

## A new approach to modified q-Bernstein polynomials for functions of two variables with their generating and interpolation functions

Acikgoz, Mehmet; Araci, Serkan
Tipo: Artigo de Revista Científica
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The aim of this paper is to give a new approach to modified q-Bernstein polynomials for functions of two variables. By using these type polynomials, we derive recurrence formulas and some new interesting identities related to the second kind Stirling numbers and generalized Bernoulli polynomials. Moreover, we give the generating function and interpolation function of these modified q-Bernstein polynomials of two variables and also give the derivatives of these polynomials and their generating function.; Comment: 11 pages

## Generating functions for the Bernstein polynomials: A unified approach to deriving identities for the Bernstein basis functions

Simsek, Yilmaz
Tipo: Artigo de Revista Científica
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The main aim of this paper is to provide a unified approach to deriving identities for the Bernstein polynomials using a novel generating function. We derive various functional equations and differential equations using this generating function. Using these equations, we give new proofs both for a recursive definition of the Bernstein basis functions and for derivatives of the nth degree Bernstein polynomials. We also find some new identities and properties for the Bernstein basis functions. Furthermore, we discuss analytic representations for the generalized Bernstein polynomials through the binomial or Newton distribution and Poisson distribution with mean and variance. Using this novel generating function, we also derive an identity which represents a pointwise orthogonality relation for the Bernstein basis functions. Finally, by using the mean and the variance, we generalize Szasz-Mirakjan type basis functions.

## Note on the Modified q-Bernstein Polynomials

Kim, Taekyun; Jang, Lee-Chae; Yi, Heungsu
Tipo: Artigo de Revista Científica
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In the present paper, we propose the modified q-Bernstein polynomials of degree n, which are different q-Bernstein polynomials of Phillips(see [4]). From these the modified q-Bernstein polynomials of degree n, we derive some interesting recurrence formulae for the modified q-Bernstein polynomials.; Comment: 11 pages

## Efficient discontinuous Galerkin finite element methods via Bernstein polynomials

Kirby, Robert C.
Tipo: Artigo de Revista Científica
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We consider the discontinuous Galerkin method for hyperbolic conservation laws, with some particular attention to the linear acoustic equation, using Bernstein polynomials as local bases. Adapting existing techniques leads to optimal-complexity computation of the element and boundary flux terms. The element mass matrix, however, requires special care. In particular, we give an explicit formula for its eigenvalues and exact characterization of the eigenspaces in terms of the Bernstein representation of orthogonal polynomials. We also show a fast algorithm for solving linear systems involving the element mass matrix to preserve the overall complexity of the DG method. Finally, we present numerical results investigating the accuracy of the mass inversion algorithms and the scaling of total run-time for the function evaluation needed in DG time-stepping.; Comment: 20 pages

## A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials

Araci, Serkan; Acikgoz, Mehmet
Tipo: Artigo de Revista Científica
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The present paper deals with Bernstein polynomials and Frobenius-Euler numbers and polynomials. We apply the method of generating function and fermionic p-adic integral representation on Zp, which are exploited to derive further classes of Bernstein polynomials and Frobenius-Euler numbers and polynomials. To be more precise we summarize our results as follows, we obtain some combinatorial relations between Frobenius-Euler numbers and polynomials. Furthermore, we derive an integral representation of Bernstein polynomials of degree n on Zp . Also we deduce a fermionic p-adic integral representation of product Bernstein polynomials of different degrees n1, n2,...on Zp and show that it can be written with Frobenius-Euler numbers which yields a deeper insight into the effectiveness of this type of generalizations. Our applications possess a number of interesting properties which we state in this paper.; Comment: 8 pages, submitted

## A note on the values of the weighted q-Bernstein polynomials and modified q-Genocchi numbers with weight alpha and beta via the p-adic q-integral on Zp

Araci, Serkan; Acikgoz, Mehmet
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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The rapid development of q-calculus has led to the discovery of new generalizations of Bernstein polynomials and Genocchi polynomials involving q-integers. The present paper deals with weighted q-Bernstein polynomials and q-Genocchi numbers with weight alpha and beta. We apply the method of generating function and p-adic q-integral representation on Zp, which are exploited to derive further classes of Bernstein polynomials and q-Genocchi numbers and polynomials. To be more precise we summarize our results as follows, we obtain some combinatorial relations between q-Genocchi numbers and polynomials with weight alpha and beta. Furthermore, we derive an integral representation of weighted q-Bernstein polynomials of degree n on Zp. Also we deduce a fermionic p-adic q-integral representation of product weighted q-Bernstein polynomials of different degrees n1,n2,...on Zp and show that it can be written with q-Genocchi numbers with weight alpha and beta which yields a deeper insight into the effectiveness of this type of generalizations. Our new generating function possess a number of interesting properties which we state in this paper.; Comment: 10 pages

## Bernstein polynomials on Simplex

Bayad, A.; Kim, T.; Rim, S. -H.
Tipo: Artigo de Revista Científica
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We prove two identities for multivariate Bernstein polynomials on simplex, which are considered on a pointwise. In this paper, we study good approximations of Bernstein polynomials for every continuous functions on simplex and the higher dimensional q-analogues of Bernstein polynomials on simplex; Comment: 8 pages

## A note on q-Bernstein polynomials

Kim, Taekyun
Tipo: Artigo de Revista Científica
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In this paper we constructed new q-extension of Bernstein polynomials. Fron those q-Berstein polynomials, we give some interesting properties and we investigate some applications related this q-Bernstein polynomials.; Comment: 13 pages

## Bernstein polynomials, Bergman kernels and toric K\"ahler varieties

Zelditch, Steve
Tipo: Artigo de Revista Científica
Português
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It does not seem to have been observed previously that the classical Bernstein polynomials $B_N(f)(x)$ are closely related to the Bergman-Szego kernels $\Pi_N$ for the Fubini-Study metric on $\CP^1$: $B_N(f)(x)$ is the Berezin symbol of the Toeplitz operator $\Pi_N f(N^{-1} D_{\theta})$. The relation suggests a generalization of Bernstein polynomials to any toric Kahler variety and Delzant polytope $P$. When $f$ is smooth, $B_N(f)(x)$ admits a complete asymptotic expansion. Integrating it over $P$ gives a complete asymptotic expansion for Dedekind-Riemann sums of smooth functions over lattice points in $N P$ related to Euler-MacLaurin sum formulae of Guillemin-Sternberg and others.; Comment: Fixed some more typos and added some detail

## Bezier curves based on Lupas (p,q)-analogue of Bernstein polynomials in CAGD

Khan, Khalid; Lobiyal, D. K.
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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In this paper, we use the blending functions of Lupas type (rational) (p,q)-Bernstein operators based on (p,q)-integers for construction of Lupas (p,q)-Beezier curves (rational curves) and surfaces (rational surfaces) with two shape parameters. We study the nature of degree elevation and degree reduction for Lupas (p,q)-Bezier Bernstein functions. Parametric curves are represented using Lupas (p,q)-Bernstein basis and the concept of total positivity is applied to investigate the shape properties of the curve. We get q-Bezier curve when we set the parameter p to the value 1: We also introduce a de Casteljau algorithm for Lupas type (p,q)-Bernstein Bezier curves. The new curves have some properties similar to q-Bezier curves. Moreover, we construct the corresponding tensor product surfaces over the rectangular domain (u,v) \in [0,1] \times [0,1] depending on four parameters. We also study the de Casteljau algorithm and degree evaluation properties of the surfaces for these generalization over the rectangular domain. Furthermore, some fundamental properties for Lupas type (p,q)-Bernstein Bezier curves are discussed. We get q-Bezier curves and surfaces for (u,v) \in [0,1] \times [0,1] when we set the parameter p1 = p2 = 1. In Comparison to q-Bezier curves and surfaces based on Phillips q-Bernstein polynomials...

## Modified Bernstein Polynomials and Jacobi Polynomials in q-Calculus

Tipo: Artigo de Revista Científica
Português
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We introduce here a generalization of the modified Bernstein polynomials for Jacobi weights using the $q$-Bernstein basis proposed by G.M. Phillips to generalize classical Bernstein Polynomials. The function is evaluated at points which are in geometric progression in $]0,1[$. Numerous properties of the modified Bernstein Polynomials are extended to their $q$-analogues: simultaneous approximation, pointwise convergence even for unbounded functions, shape-preserving property, Voronovskaya theorem, self-adjointness. Some properties of the eigenvectors, which are $q$-extensions of Jacobi polynomials, are given.

## Banded operational matrices for Bernstein polynomials and application to the fractional advection-dispersion equation

Jani, M.; Babolian, E.; Javadi, S.; Bhatta, D.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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In the papers dealing with derivation and applications of operational matrices of Bernstein polynomials, a basis transformation, commonly a transformation to power basis, is used. The main disadvantage of this method is that the transformation may be ill-conditioned. Moreover, when applied to the numerical simulation of a functional differential equation, it leads to dense operational matrices and so a dense coefficient matrix is obtained. In this paper, we present a new property for Bernstein polynomials. Using this property, we build exact banded operational matrices for derivatives of Bernstein polynomials. Next, as an application, we propose a new numerical method based on a Petrov-Galerkin variational formulation and the new operational matrices utilizing the dual Bernstein basis for the time-fractional advection-dispersion equation. Finally, we show that the proposed method leads to a narrow-banded linear system and so less computational effort is required to obtain the desired accuracy for the approximate solution. Some numerical examples are provided to demonstrate the efficiency of the method.; Comment: 15 pages

## On the explicit representation of orthonormal Bernstein polynomials

Bellucci, Michael A.
Tipo: Artigo de Revista Científica
Português
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In this work we present an explicit representation of the orthonormal Bernstein polynomials and demonstrate that they can be generated from a linear combination of non-orthonormal Bernstein polynomials. In addition, we report a set of $n$ Sturm-Liouville eigenvalue equations, where each of the $n$ eigenvalue equations have the orthonormal Bernstein polynomials of degree $n$ as their solution set. We also show that each of the $n$ Sturm-Liouville operators are naturally self-adjoint. While the orthonormal Bernstein polynomials can be used in a variety of different applications, we demonstrate the utility of these polynomials here by using them in a generalized Fourier series to approximate curves and surfaces. Using the orthonormal Bernstein polynomial basis, we show that highly accurate approximations to curves and surfaces can be obtained by using small sized basis sets. Finally, we demonstrate how the orthonormal Bernstein polynomials can be used to find the set of control points of Bezier curves or Bezier surfaces that best approximate a function.

## The Legendre polynomials associated with Bernoulli, Euler, Hermite and Bernstein polynomials

Araci, Serkan; Acikgoz, Mehmet; Bagdasaryan, Armen; Sen, Erdogan
Tipo: Artigo de Revista Científica
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In the present paper, we deal mainly with arithmetic properties of Legendre polynomials by using their orthogonality property. We show that Legendre polynomials are proportional with Bernoulli, Euler, Hermite and Bernstein polynomials.

## Bezier curves and surfaces based on modified Bernstein polynomials

Khan, Khalid; Lobiyal, D. K.; Kilicman, Adem
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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In this paper, we use the blending functions of Bernstein polynomials with shifted knots for construction of Bezier curves and surfaces. We study the nature of degree elevation and degree reduction for Bezier Bernstein functions with shifted knots. Parametric curves are represented using these modified Bernstein basis and the concept of total positivity is applied to investigate the shape properties of the curve. We get Bezier curve defined on [0, 1] when we set the parameter \alpha=\beta to the value 0. We also present a de Casteljau algorithm to compute Bernstein Bezier curves and surfaces with shifted knots. The new curves have some properties similar to Bezier curves. Furthermore, some fundamental properties for Bernstein Bezier curves and surfaces are discussed.; Comment: 11 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1507.04110

## Stabilization of polynomial dynamical systems using linear programming based on Bernstein polynomials

Sassi, Mohamed Amin Ben; Sankaranarayanan, Sriram
Tipo: Artigo de Revista Científica
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In this paper, we deal with the problem of synthesizing static output feedback controllers for stabilizing polynomial systems. Our approach jointly synthesizes a Lyapunov function and a static output feedback controller that stabilizes the system over a given subset of the state-space. Specifically, our approach is simultaneously targeted towards two goals: (a) asymptotic Lyapunov stability of the system, and (b) invariance of a box containing the equilibrium. Our approach uses Bernstein polynomials to build a linear relaxation of polynomial optimization problems, and the use of a so-called "policy iteration" approach to deal with bilinear optimization problems. Our approach can be naturally extended to synthesizing hybrid feedback control laws through a combination of state-space decomposition and Bernstein polynomials. We demonstrate the effectiveness of our approach on a series of numerical benchmark examples.

## q-Bernstein polynomials associated with q-stirling numbers and Carlitz's q-Bernoulli numbers

Kim, Taekyun; Kim, Younghee; Choi, Jongsoung
Tipo: Artigo de Revista Científica