Reducibility of hyperplane arrangements

Certain problems on reducibility of central hyperplane arrangements are settled. Firstly, a necessary and sufficient condition on reducibility is obtained. More precisely, it is proved that the number of irreducible components of a central hyperplane arrangement equals the dimension of the space consisting of the logarithmic derivations of the arrangement with degree zero or one. Secondly, it is proved that the decomposition of an arrangement into a direct sum of its irreducible components is unique up to an isomorphism of the ambient space. Thirdly, an effective algorithm for determining the number of irreducible components and decomposing an arrangement into a direct sum of its irreducible components is offered. This algorithm can decide whether an arrangement is reducible, and if it is the case, what the defining equations of irreducible components are.     

Addition–deletion results for the minimal degree of logarithmic derivations of hyperplane arrangements and maximal Tjurina line arrangements

Journal of Algebraic Combinatorics - We study the change of the minimal degree of a logarithmic derivation of a hyperplane arrangement under the addition or the deletion of a hyperplane and give a...     

Approximation with Bernstein–Szegö polynomials

We present approximation kernels for orthogonal expansions with respect to Bernstein–Szegö polynomials. Theconstruction is derived from known results for Chebyshev polynomials of the first kind and does not pose any restrictions on the Bernstein–Szegö polynomials.     

On approximation by some Bernstein–Kantorovich exponential-type polynomials

Periodica Mathematica Hungarica - Since the introduction of Bernstein operators, many authors defined and/or studied Bernstein type operators and their generalizations, among them are Morigi and...     

Bézier form of dual bivariate Bernstein polynomials

Dual Bernstein polynomials of one or two variables have proved to be very useful in obtaining Bézier form of the L ₂-solution of the problem of best polynomial approximation of Bézier curve or surface. In this connection, the Bézier coefficients of dual Bernstein polynomials are to be evaluated at a reasonable cost. In this paper, a set of recurrence relations satisfied by the Bézier coefficients of dual bivariate Bernstein polynomials is derived and an efficient algorithm for evaluation of these coefficients is proposed. Applications of this result to some approximation problems of Computer Aided Geometric Design (CAGD) are discussed.     

Constrained degree reduction of polynomials in Bernstein–Bézier form over simplex domain

In this paper we show that the orthogonal complement of a subspace in the polynomial space of degree n over d  -dimensional simplex domain with respect to the _L2$L_2$-inner product and the weighted Euclidean inner product of BB (Bézier–Bernstein) coefficients are equal. Using it we also prove that the best constrained degree reduction of polynomials over the simplex domain in BB form equals the best approximation of weighted Euclidean norm of coefficients of given polynomial in BB form from the coefficients of polynomials of lower degree in BB form.     

On multivariate polynomials in Bernstein–Bézier form and tensor algebra

The Bernstein–Bézier representation of polynomials is a very useful tool in computer aided geometric design. In this paper we make use of (multilinear) tensors to describe and manipulate multivariate polynomials in their Bernstein–Bézier form. As an application we consider Hermite interpolation with polynomials and splines.     

Kazhdan–Lusztig polynomials of boolean elements

We give closed combinatorial product formulas for Kazhdan–Lusztig polynomials and their parabolic analogue of type q in the case of boolean elements, introduced in (Marietti in J. Algebra 295:1–26, 2006), in Coxeter groups whose Coxeter graph is a tree. Such formulas involve Catalan numbers and use a combinatorial interpretation of the Coxeter graph of the group. In the case of classical Weyl groups, this combinatorial interpretation can be restated in terms of statistics of (signed) permutations. As an application of the formulas, we compute the intersection homology Poincaré polynomials of the Schubert varieties of boolean elements.     

Specializations of nonsymmetric Macdonald–Koornwinder polynomials

The purpose of this article is to work out the details of the Ram–Yip formula for nonsymmetric Macdonald–Koornwinder polynomials for the double affine Hecke algebras of not-necessarily reduced affine root systems. It is shown that the \(t\rightarrow 0\) equal-parameter specialization of nonsymmetric Macdonald polynomials admits an explicit combinatorial formula in terms of quantum alcove paths, generalizing the formula of Lenart in the untwisted case. In particular, our formula yields a definition of quantum Bruhat graph for all affine root systems. For mixed type, the proof requires the Ram–Yip formula for the nonsymmetric Koornwinder polynomials. A quantum alcove path formula is also given at \(t\rightarrow \infty \). As a consequence, we establish the positivity of the coefficients of nonsymmetric Macdonald polynomials under this limit, as conjectured by Cherednik and the first author. Finally, an explicit formula is given at \(q\rightarrow \infty \), which yields the p-adic Iwahori–Whittaker functions of Brubaker, Bump, and Licata.     

A note on constrained degree reduction of polynomials in Bernstein–Bézier form over simplex domain

In the paper [H.S. Kim, Y.J. Ahn, Constrained degree reduction of polynomials in Bernstein–Bézier form over simplex domain, J. Comput. Appl. Math. 216 (2008) 14–19], Kim and Ahn proved that the best constrained degree reduction of a polynomial over d$d$-dimensional simplex domain in _L2$L_2$-norm equals the best approximation of weighted Euclidean norm of the Bernstein–Bézier coefficients of the given polynomial. In this paper, we presented a counterexample to show that the approximating polynomial of lower degree to a polynomial is virtually non-existent when d≥2$d\ge 2$. Furthermore, we provide an assumption to guarantee the existence of solution for the constrained degree reduction.     

Asymptotic Properties of Bernstein–Durrmeyer Operators

It is known that Szász–Durrmeyer operator is the limit, in an appropriate sense, of Bernstein–Durrmeyer operators. In this paper, we adopt a new technique that comes from the representation of operator semigroups to study the approximation issue as mentioned above. We provide some new results on approximating Szász–Durrmeyer operator by Bernstein–Durrmeyer operators. Our results improve the corresponding results of Adell and De La Cal (Comput Math Appl 30:1–14, 1995).     

Numerical solution of stochastic It?-Volterra integral equations based on Bernstein multi-scaling polynomials

In this paper, first, Bernstein multi-scaling polynomials(BMSPs) and their properties are introduced. These polynomials are obtained by compressing Bernstein polynomials(BPs) under sub-intervals. Then, by using these polynomials, stochastic operational matrices of integration are generated. Moreover, by transforming the stochastic integral equation to a system of algebraic equations and solving this system using Newton's method, the approximate solution of the stochastic It?-Volterra integral equation is obtained. To illustrate the efficiency and accuracy of the proposed method, some examples are presented and the results are compared with other methods.     

A generalization of the category O of Bernstein–Gelfand–Gelfand

In the study of simple modules over a simple complex Lie algebra, Bernstein, Gelfand and Gelfand introduced a category of modules which provides a natural setting for highest weight modules. In this note, we define a family of categories which generalizes the BGG category. We classify the simple modules for some of these categories. As a consequence we show that these categories are semisimple.