Interpolation and convergence of Bernstein-Bézier coefficients

In this paper, two ways of the proof are given for the fact that the Bernstein-Bézier coefficients (BB-coefficients) of a multivariate polynomial converge uniformly to the polynomial under repeated degree elevation over the simplex. We show that the partial derivatives of the inverse Bernstein polynomial A _n (g) converge uniformly to the corresponding partial derivatives of g at the rate 1/n. We also consider multivariate interpolation for the BB-coefficients, and provide effective interpolation formulas by using Bernstein polynomials with ridge form which essentially possess the nature of univariate polynomials in computation, and show that Bernstein polynomials with ridge form with least degree can be constructed for interpolation purpose, and thus a computational algorithm is provided correspondingly.     

Some Properties of The Automorphisms of a Bernstein Algebra

In this short note we show that if A is a nuclear Bernstein algebra then the group of automorphisms of M(A), its multiplication algebra, has a proper subgroup isomorphic to Aut A.     

Asymptotic convergence of degree‐raising

It is well known that the degree‐raised Bernstein–Bézier coefficients of degree n of a polynomial g converge to g at the rate 1/n. In this paper we consider the polynomial A _n(g) of degree ⩼ n interpolating the coefficients. We show how A _n can be viewed as an inverse to the Bernstein polynomial operator and that the derivatives A _n(g)(^r) converge uniformly to g(^r) at the rate 1/n for all r. We also give an asymptotic expansion of Voronovskaya type for A _n(g) and discuss some shape preserving properties of this polynomial. This revised version was published online in June 2006 with corrections to the Cover Date.     

Polynomial least squares fitting in the Bernstein basis

The problem of polynomial least squares fitting in which the usual monomial basis is replaced by the Bernstein basis is considered. The coefficient matrix of the overdetermined system to be solved in the least squares sense is then a rectangular Bernstein-Vandermonde matrix. In order to use the method based on the QR decomposition of A, the first stage consists of computing the bidiagonal decomposition of the coefficient matrix A. Starting from that bidiagonal decomposition, an algorithm for obtaining the QR decomposition of A is the applied. Finally, a triangular system is solved by using the bidiagonal decomposition of the R-factor of A. Some numerical experiments showing the behavior of this approach are included.     

Inverses of Boolean matrices

For a Boolean matrix A, a g-inverse of A is a Boolean matrix G satisfying AGA=A, and a Vagner inverse is a g-inverse which in addition satisfies GAG=G. We give algorithms for finding all g-inverses, all Vagner inverses, and all of several other types of inverses including Moore-Penrose inverses. We give a criterion for a Boolean matrix to be regular, and criteria for the various types of inverse to exist. We count the numbers of Boolean matrices having Moore-Penrose and related types of inverses.     

Sets of periods for automorphisms of compact Riemann surfaces

Let G=〈f〉 be a finite cyclic group of order N that acts by conformal automorphisms on a compact Riemann surface S of genus g≥2. Associated to this is a set A of periods defined to be the subset of proper divisors d of N such that, for some x∈S, x is fixed by f^d but not by any smaller power of f. For an arbitrary subset A of proper divisors of N, there is always an associated action and, if g_A denotes the minimal genus for such an action, an algorithm is obtained here to determine g_A. Furthermore, a set A_max is determined for which g_A is maximal.     

The generalized entropic property for a pair of operations

In this paper a generalized entropic property is defined for a pair of operations. We show that for an idempotent algebra A = (A, f, g) with two ternary operations, if one of f or g is commutative and the pair of operations (f, g) satisfies the generalized entropic property, then (f, g) is entropic. Also, it is proved that every idempotent, commutative algebra A = (A, f, g) with a ternary and a binary operation, satisfying the generalized entropic property, is entropic.     

A note on generalized derivations in Banach algebras

In this note we characterize a complex Banach algebra A admitting a generalized derivation g such that the cardinality of the spectrum σ(g(x)) is exactly one for all x∈A.     

On the structure of the class A(ϕ)

In this paper, we study whether the set A(?) is closed under multiplication f · g, where f and g belong to the class A(?). We also study the problem of the existence of a solution of the equation Bx = C (where B,C ∈ A(?) and B ≠ 0) on the set A(?).     

On pre-periods of discrete influence systems

We investigate mappings of the form $\text{g = ƒA}$ where ƒ is a cyclically monotonous mapping of finite range and A is a linear mapping given by a symmetric matrix. We give some upper bounds on the pre-period of g, i.e. the maximum q for which all g(x),g²(x),…,g^q(x) are distinct.     

An oblique projection iterative method to compute Drazin inverse and group inverse

In this paper, an oblique projection iterative method is presented to compute matrix equation AXA=A of a square matrix A with ind(A)=1. By this iterative method, when taken the initial matrix X₀=A, the group inverse A_g can be obtained in absence of the roundoff errors. If we use this iterative method to the matrix equation A^kXA^k=A^k, a group inverse (A^k)_g of matrix A^k is got, then we use the formulae A_d=A^k-1(A^k)_g, the Drazin inverse A_d can be obtained.     

Integer valued polynomials over function fields

Let A be an integrally closed subring of a function field K defined over a finite field. In this paper we investigate whether the subring of K[X], consisting of those polynomials ƒ with ƒ[A]⊂A, has an A-basis {g_i: i ∈ ℤZ_≥0}, with deg (g_i) = i.     

Invariance of dimension of p-subspaces in bernstein algebras *

The purpose of this paper is to prove that some vector subspaces, called p-subspaces, obtained from the Peirce decomposition of a Bernstein algebra A relative to an idempotent have dimensions which are independent of the idempotent used to decompose A. In particular, for Bernstein-Jordan algebras, this fact is true for every such subspace and this implies that all p-subspaces of a Bernstein algebra, contained in V, for A = Ke + U + V, have invariant dimension. Finally we classify all p-subspaces of degree ≥ 3, contained in U, in a Bernstein algebra A, relative to the invariance (or not) of dimension.     

Topologically inseparable functions II: Infinitary case

Given a set A and a function A: A → A, we study the set of all functions g: A → A that are continuous for all topologies for which f continuous. We prove that in a sense to be made precise in the text, for any essentially infinitary function f, any non-constant such g equals f ⁿ , for some n∈ ?. We also prove a similar result for the clone of n-ary functions from A ⁿ → A.     

Abelian group matrices over the p-adic and rational integers

Let G be a finite abelian group of order n. Let Z and Q denote the rational integers and rationals, respectively. A group matrix for G over Z (or Q) is an n-square matrix of the form Σ_{g ∈ G}a_gP(g), where a_g ∈ Z (or Q) and P is the regular representation of G so that P(g) is an n-square permutation matrix and P(gh) = P(g)P(h) for all g, h ∈ G. It is known that if M is an arbitrary positive definite unimodular matrix over Z then there exists a matrix A over Q such that M = A^τA, where τ denotes transposition. This paper proves that the exact analogue of this theorem holds if one demands that M and A be group matrices for G over Z and Q, respectively. Furthermore, if M is a group matrix for G over the p-adic integers then necessary and sufficient conditions are given for the existence of a group matrix A for G over the p-adic numbers such that M = A^τA.     

Bounded, attractive and repulsive Markov specifications on trees and on the one-dimensional lattice

Let Π be a homogenous Markov specification associated with a countable state space S and countably infinite parameter space A possessing a neighbor relation ~ such that (A,~) is the regular tree with d +1 edges meeting at each vertex. Let $\text{g}$(π)be the simplex of corresponding Markov random fields. We show that if Π satisfies a ‘boundedness’ condition then $\text{g}$(π).We further study the structure of $\text{g}$(π) when Π is either attractive or repulsive with respect to a linear ordering on S. When d = 1, so that (A, ~) is the one-dimensional lattice, we relax the requirement of homogeneity to that of stationarity; here we give sufficient conditions for $\text{g}$(π) and for $\text{g}$(π)to have precisely one member.     

Lie theory and the Chern-Weil homomorphism

Let P→B be a principal G-bundle. For any connection θ on P, the Chern-Weil construction of characteristic classes defines an algebra homomorphism from the Weil algebra Wg=Sg^∗⊗∧g^∗ into the algebra of differential forms A=Ω(P). Invariant polynomials inv(Sg^∗)⊂Wg map to cocycles, and the induced map in cohomology inv(Sg^∗)→H(Abasic) is independent of the choice of θ. The algebra Ω(P) is an example of a commutativeg-differential algebra with connection, as introduced by H. Cartan in 1950. As observed by Cartan, the Chern-Weil construction generalizes to all such algebras.In this paper, we introduce a canonical Chern-Weil map Wg→A for possibly non-commutativeg-differential algebras with connection. Our main observation is that the generalized Chern-Weil map is an algebra homomorphism “up to g-homotopy”. Hence, the induced map inv(Sg^∗)→Hbasic(A) is an algebra homomorphism. As in the standard Chern-Weil theory, this map is independent of the choice of connection.Applications of our results include: a conceptually easy proof of the Duflo theorem for quadratic Lie algebras, a short proof of a conjecture of Vogan on Dirac cohomology, generalized Harish-Chandra projections for quadratic Lie algebras, an extension of Rouvière's theorem for symmetric pairs, and a new construction of universal characteristic forms in the Bott-Shulman complex.     

On the Gröbner complexity of matrices

In this paper we show that if for an integer matrix A the universal Gröbner basis of the associated toric ideal I_A coincides with the Graver basis of A, then the Gröbner complexity u(A) and the Graver complexity g(A) of its higher Lawrence liftings agree, too. In fact, if the universal Gröbner basis of I_A coincides with the Graver basis of A, then also the more general complexities u(A,B) and g(A,B) agree for arbitrary B. We conclude that for the matrices A_3×3 and A_3×4, defining the 3×3 and 3×4 transportation problems, we have u(A_3×3)=g(A_3×3)=9 and u(A_3×4)=g(A_3×4)≥27. Moreover, we prove that u(A_a,b)=g(A_a,b)=2(a+b)/gcd(a,b) for positive integers a,b and .     

On rational invariants of a set of symmetric matrices

Some identities resulting from the Cayley-Hamilton theorem are derived. Some applications include: (a) for k = 1,2,…,n ? 1 a condition is found for a pair (A,B) of symmetric operators acting in Euclidean n-space to have common invariant k-subspace (provided that A does not have multiple eigenvalues); (b) it is shown that the field of rational invariants of (A,B) is isomorphic to a subfield of a rational function field with n(n+3)/2 generators consisting of elements symmetric with respect to the permutaion group P_n; (c) it is shown that any rational invariant of (g+2) symmetric operators A,B,C₁,C₂,…, C_g can be expressed as a rational function of invariants of one or two operators that are taken for pairs (A,B), (A,C₂),…, (A,C_g, (A,B+C₁), (A,B+C₂),…,(A,B+C_g).