General order multivariate Padé approximants for pseudo-multivariate functions

Although general order multivariate Padé approximants were introduced some decades ago, very few explicit formulas for special functions have been given. We explicitly construct some general order multivariate Padé approximants to the class of so-called pseudo-multivariate functions, using the Padé approximants to their univariate versions. We also prove that the constructed approximants inherit the normality and consistency properties of their univariate relatives, which do not hold in general for multivariate Padé approximants. Examples include the multivariate forms of the exponential and the -exponential functions

and

as well as the Appell function

and the multivariate form of the partial theta function

     

Counting integral Lamé equations by means of dessins d'enfants

We obtain an explicit formula for the number of Lamé equations (modulo linear changes of variable) with index and projective monodromy group of order , for given and . This is done by performing the combinatorics of the `dessins d'enfants' associated to the Belyi covers which transform hypergeometric equations into Lamé equations by pull-back.

     

The Poincaré metric and isoperimetric inequalities for hyperbolic polygons

We prove several isoperimetric inequalities for the conformal radius (or equivalently for the Poincaré density) of polygons on the hyperbolic plane. Our results include, as limit cases, the isoperimetric inequality for the conformal radius of Euclidean -gons conjectured by G. Pólya and G. Szegö in 1951 and a similar inequality for the hyperbolic -gons of the maximal hyperbolic area conjectured by J. Hersch. Both conjectures have been proved in previous papers by the third author.

Our approach uses the method based on a special triangulation of polygons and weighted inequalities for the reduced modules of trilaterals developed by A. Yu. Solynin. We also employ the dissymmetrization transformation of V. N. Dubinin. As an important part of our proofs, we obtain monotonicity and convexity results for special combinations of the Euler gamma functions, which appear to have a significant interest in their own right.

     

Domaine numérique du produit et de la bimultiplication

In this paper, we present an extension of Bouldin's result (1970) concerning the numerical range of the product of two operators and that are commuting and for which one of the set or consists of positive numbers. We also prove that if or is a subnormal operator on a separable Hilbert space, then

where is the operator bimultiplication and is the convex hull.

RÉSUMÉ. Dans ce travail, nous améliorons un résultat de Bouldin (1970) concernant la localisation de le domaine numérique du produit de deux opérateurs et sur un espace de Hilbert lorsque et commutent et est constitué de réels strictement positifs. Dans le cas où ou est un opérateur sous normal sur un espace de Hilbert séparable, nous montrons que

où est l'opérateur produit ou bimultiplication et est l'enveloppe convexe.

     

Harnack inequality for the linearized parabolic Monge-Ampère equation

In this paper we prove the Harnack inequality for nonnegative solutions of the linearized parabolic Monge-Ampère equation

on parabolic sections associated with , under the assumption that the Monge-Ampère measure generated by satisfies the doubling condition on sections and the uniform continuity condition with respect to Lebesgue measure. The theory established is invariant under the group , where denotes the group of all invertible affine transformations on .

     

Minimizing multi-homogeneous Bézout numbers by a local search method

Consider the multi-homogeneous homotopy continuation method for solving a system of polynomial equations. For any partition of variables, the multi-homogeneous Bézout number bounds the number of isolated solution curves one has to follow in the method. This paper presents a local search method for finding a partition of variables with minimal multi-homogeneous Bézout number. As with any other local search method, it may give a local minimum rather than the minimum over all possible homogenizations. Numerical examples show the efficiency of this local search method.

     

A Fejér type theorem to determine jumps in terms of the Abel-Poisson mean of double Fourier series

We extend from single to double Fourier series a theorem of Zygmund to determine the generalized jumps of a periodic integrable function at a simple discontinuity point. As a by-product of the proof, we obtain an estimate of the fourth mixed partial derivative of the Abel-Poisson mean of any integrable function at such a point where is smooth. We also consider the extension of the Zygmund classes and to the two-dimensional torus .

     

Higher Order Turán Inequalities

The celebrated Turán inequalities , where denotes the Legendre polynomial of degree , are extended to inequalities for sums of products of four classical orthogonal polynomials. The proof is based on an extension of the inequalities , which hold for the Maclaurin coefficients of the real entire function in the Laguerre-Pólya class, .

     

Generalizations of Gonçalves' inequality

Let be a polynomial with complex coefficients and roots , ..., , let denote its norm over the unit circle, and let denote Mahler's measure of . Gonçalves' inequality asserts that

We prove that

for , where is an explicit constant, and that

for . We also establish additional lower bounds on the norms of a polynomial in terms of its coefficients.

     

Critical points of real entire functions and a conjecture of Pólya

Let be a nonconstant real entire function of genus and assume that all the zeros of are distributed in some infinite strip , . It is shown that (1) if has nonreal zeros in the region , and has nonreal zeros in the same region, and if the points and are located outside the Jensen disks of , then has exactly critical zeros in the closed interval , (2) if is at most of order , , and minimal type, then for each positive constant there is a positive integer such that for all has only real zeros in the region , and (3) if is of order less than , then has just as many critical points as couples of nonreal zeros.

     

A generalization of Carleman's uniqueness theorem and a discrete Phragmén-Lindelöf theorem

Let be a Borel measure on and be its moments. T. Carleman found sharp conditions on the magnitude of for to be uniquely determined by its moments. We show that the same conditions ensure a stronger property: if are the moments of another measure, with then the measure is supported on the interval This result generalizes both the Carleman theorem and a theorem of J. Mikusi\'{n}ski. We also present an application of this result by establishing a discrete version of a Phragmén-Lindelöf theorem.

     

PA���е�H\'{a}jek-R\'{e}nyi�Ͳ���ʽ��ǿ��������

In this paper, a new H\'{a}jek-R\'{e}nyi-type inequality for mean zero associated random variables is obtained, which generalizes and improves the result of Theorem 2.2 of \ncite{9}. In addition, a Brunk-Prokhorov-type strong law of large numbers is also given.     

�Ǿֲ��������������˥���ٶ�

In this paper, the asymptotic behavior of the weak solution (u_t)_{t\ge0} to the non-local Cauchy problems as stated in (1) is considered. Only using lower bounds of jumping kernel J(x,y) for large |x-y|, it is obtained that \|u_t\|_p\le c(t)\|u_0\|_q with any 1\le q相似文献   

The transversal homoclinic points are dense in the codimension-1 Hénon-like strange attractors

We consider the codimension-1 Hénon-like strange attractors . We prove that the transversal homoclinic points are dense in , and that hyperbolic periodic points are dense in . Moreover the hyperbolic periodic points that are heteroclinically related to the primary fixed point ( transversal intersection of stable and unstable manifolds) are dense in .

     

三角B\''{e}zier曲面和四边B\''{e}zier曲面之间的相互转化

B\'{e}zier曲面有两种不同的形式:三角B\'{e}zier曲面和四边B\'{e}zier曲面,它们有着不同的基底和不同的几何拓扑结构, 但是它们也有很多共同的性质,因此三角B\'{e}zier曲面和四边B\'{e}zier曲面之间的相互转化就成为CAGD 里一个重要研究课题.在本文中, 我们用函数复合的方法实现两者之间的相互转化.被复合的两个函数, 一个用Polar形式表示,另一个用常见的Bernstein基形式表示.     

A minimum energy problem and Dirichlet spaces

We analyze a minimum energy problem for a discrete electrostatic model in the complex plane and discuss some applications. A natural characteristic distinguishing the state of minimum energy from other equilibrium states is established. It enables us to gain insight into the structure of positive trigonometric polynomials and Dirichlet spaces associated with finitely atomic measures. We also derive a related family of linear second order differential equations with polynomial solutions.

     

A probabilistic proof of the fundamental theorem of algebra

We use Lévy's theorem on invariance of planar Brownian motion under conformal maps and the support theorem for Brownian motion to show that the range of a non-constant polynomial of a complex variable consists of the whole complex plane. In particular, we obtain a probabilistic proof of the fundamental theorem of algebra.

     

Sz\''asz-Kantorovich-B\''{e}zier算子在$L_p[0, \infty)$上的逼近定理

本文利用Ditzian-Totik模得到了Sz\'{a}sz-Kantorovich-B\'{e}zier算子在$L_{p}[0,\infty)$空间逼近的正逆定理及等价定理.     

A new proof of D. Popescu's theorem on smoothing of ring homomorphisms

We give a new proof of D. Popescu's theorem which says that if is a regular homomorphism of noetherian rings, then is a filtered inductive limit of smooth finite type -algebras. We strengthen Popescu's theorem in two ways. First, we show that a finite type -algebra , mapping to , has a desingularization which is smooth wherever possible (roughly speaking, above the smooth locus of ). Secondly, we give sufficient conditions for to be a filtered inductive limit of its smooth finite type -subalgebras. We also give counterexamples to the latter statement in cases when our sufficient conditions do not hold.