Related papers: On the explicit representation of orthonormal Bern…
We investigate the representation of arbitrary polynomials using probabilistic Bernoulli and degenerate Bernoulli polynomials associated with a random variable $Y$, whose moment generating function exists in a neighborhood of the origin. In…
We generalize Block-G\"ottsche polynomials, originally defined for toric del Pezzo surfaces, to arbitrary surfaces. To do this, we show that these polynomials arise as special cases of BPS polynomials, defined for any surface $S$ as Laurent…
We show that each member of a doubly infinite sequence of highly nonlinear expressions of Bernoulli polynomials, which can be seen as linear combinations of certain higher-order convolutions, is a multiple of a specific product of linear…
Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. It consists of a four-parameter polynomial with continuous spectrum on the whole real line…
We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in…
Let $(p_n)_n$ be either the $q$-Meixner or the $q$-Laguerre polynomials. We form a new sequence of polynomials $(q_n)_n$ by considering a linear combination of two consecutive $p_n$: $q_n=p_n+\beta_np_{n-1}$, $\beta_n\in \RR$. Using the…
In this paper we address the issue of designing developable surfaces with Bezier patches. We show that developable surfaces with a polynomial edge of regression are the set of developable surfaces which can be constructed with Aumann's…
There are some distinguished ensembles of non-Hermitian random matrices for which the joint PDF can be written down explicitly, is unchanged by rotations, and furthermore which have the property that the eigenvalues form a Pfaffian point…
Skew orthogonal polynomials arise in the calculation of the $n$-point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry. In particular, the distribution functions are completely…
For bivariate polynomials of degree $n\le 5$ we give fast numerical constructions of determinantal representations with $n\times n$ matrices. Unlike some other available constructions, our approach returns matrices of the smallest possible…
We show that the discrete Painlev\'e-type equations arising from quantum minimal surfaces are equations for recurrence coefficients of orthogonal polynomials for indefinite hermitian products. As a consequence, we obtain an explicit formula…
Employing the random matrix formulation of Chern-Simons theory on Seifert manifolds, we show how the Stieltjes-Wigert orthogonal polynomials are useful in exact computations in Chern-Simons matrix models. We construct a biorthogonal…
The Bernstein polynomial basis sees significant use owing to its unique properties, particularly in the field of optimal control. However, the basis is known to have a slow rate of convergence to the function it approximates. With this in…
We obtain approximation results for general positive linear operators satisfying mild conditions, when acting on discontinuous functions and absolutely continuous functions having discontinuous derivatives. The upper bounds, given in terms…
We prove that several forms of the Bernstein polynomials with integer coefficients possess the property of simultaneous approximation, that is, they approximate not only the function but also its derivatives. We establish direct estimates…
The representations of the kernels of the transmutation operator and of its inverse relating the one-dimensional Schr\"odinger operator with the second derivative are obtained in terms of the eigenfunctions of a corresponding…
We establish simultaneous approximation properties of weighted first-order Sobolev orthogonal projectors onto spaces of polynomials of bounded total degree in the Euclidean unit ball. The simultaneity is in the sense that we provide bounds…
Main purpose of this paper is to reconstruct generating function of the Bernstein type polynomials. Some properties this generating functions are given. By applying this generating function, not only derivative of these polynomials but also…
The Bernstein-B\'ezier form of a polynomial is widely used in the fields of computer aided geometric design, spline approximation theory and, more recently, for high order finite element methods for the solution of partial differential…
In this paper, we examine linear programming (LP) relaxations based on Bernstein polynomials for polynomial optimization problems (POPs). We present a progression of increasingly more precise LP relaxations based on expressing the given…