On the Representation of Band Limited Functions Using Finitely Many Bits

In this paper, we consider the question of representing an entire function of finite order and type in terms of finitely many bits, and reconstructing the function from these. Instead of making any further assumptions about the function, we measure the error in reconstruction in a suitably weighted L^p norm. The optimal number of bits in order to obtain a given accuracy is given by the Kolmogorov entropy. We determine this entropy in the case of certain compact subsets of these weighted L^p spaces and obtain constructive algorithms to determine the asymptotically optimal bit representation from finitely many samples of the function. Our theory includes both equidistant and non-uniform sampling. The reconstructions are polynomials, having several other optimality properties.     

On Groups with Finitely Many Conradian Orderings

We study the space of left-orderings on groups with (only) finitely many Conradian orderings. We show that, within this class of groups, having an isolated left-ordering is equivalent to having finitely many left-orderings.     

有限个有理函数的随机复动力系统的Julia集

本文讨论有限个有理函数生成的随机复动力系统，得到Julia集有内点的充分条件和必要条件．证明了对任意的正数，可以构造有限个多项式，彼此的Julia集之间的距离大干L，但J(f1，…，fm)含有内点但不是全平面。     

On Smooth Maps with Finitely Many Critical Points

The minimum number of critical points of a small codimensionsmooth map between two manifolds is computed. Some partial resultsfor the case of higher codimension when the manifolds are spheresare also given.     

On Rings for Which Finitely Generated Ideals Have Only Finitely Many Minimal Components

For a commutative ring R we investigate the property that the sets of minimal primes of finitely generated ideals of R are always finite. We prove this property passes to polynomial ring extensions (in an arbitrary number of variables) over R as well as to R-algebras which are finitely presented as R-modules.     

迭代修复含有限个同类型间断函数的连续性

我们已证明具有一个间断点的函数有连续的二次迭代.它实际上表明在迭代之下它的间断点能被自己函数对修复为连续点.如果一个函数含至少两个间断点,那么,在迭代之下,它的间断点或者被它自己函数对修复为连续点或者被其它间断点的函数对修复为连续点.本文研究具有多于一个但是只含有限个同类型间断点的不连续函数,给出了这些函数二次迭代连续...     

On a Property of Functions on the Sphere

According to the Knaster conjecture, for any continuous function $f:S^{n - 1} \to \mathbb{R}$ and any $n$ -point subset of the sphere $S^{n - 1}$ , there exists a rotation mapping all the points of this subset to a level surface of the function $f$ . In the present paper, this conjecture is proved for the case in which ${n = p^1 }$ for an odd prime $p$ and the points lie on a circle and divide it into equal parts.     

On the Finiteness of Noetherian Rings with Finitely Many Regular Elements

Let R be a left Noetherian ring and ZD(R) be the set of all zero-divisors of R. In this paper, it is shown that if R \ ZD(R) is finite, then R is finite.     

On Smooth Maps with Finitely Many Critical Points: Addendum

The paper is an addendum to D. Andrica and L. Funar, ‘Onsmooth maps with finitely many critical points’, J. LondonMath. Soc. (2) 69 (2004) 783–800.     

具有有限组不相交压缩圆片的Heegaard分解

设$V\cup_SW$是一个闭的三维流形亏格为$g$的, 弱可约的Heegaard分解, 并且在合痕意义下只有有限组位于曲面不同侧的不相交的压缩圆片, 则它存在一个广义的Heegaard分解: $V\cup_SW=(V_1\cup_{S_1}W_1)\cup_F(W_2\cup_{S_2}V_2)$, 并且满足对于每个$i=1,2$, 压缩体$W_i$都只有一个分离的压缩圆片且$d(S_i)\geq 2$. 进一步的, 如果有有限且多于1组不相交的压缩圆片, 则至少一个$d(S_i)$等于2, 并且Heegaard曲面满足临界性质.     

Random Walks on Trees with Finitely Many Cone Types

This paper is devoted to the study of random walks on infinite trees with finitely many cone types (also called periodic trees). We consider nearest neighbour random walks with probabilities adapted to the cone structure of the tree, which include in particular the well studied classes of simple and homesick random walks. We give a simple criterion for transience or recurrence of the random walk and prove that the spectral radius is equal to 1 if and only if the random walk is recurrent. Furthermore, we study the asymptotic behaviour of return probabilitites and prove a local limit theorem. In the transient case, we also prove a law of large numbers and compute the rate of escape of the random walk to infinity, as well as prove a central limit theorem. Finally, we describe the structure of the boundary process and explain its connection with the random walk.     

On Jackson-Type Inequalities for Functions Defined on a Sphere

We obtain exact estimates for the approximation of functions defined on a sphere in the metrics of C and L ₂ by linear methods of summation of Fourier series in spherical harmonics in the case where differential and difference properties of these functions are defined in the space L ₂. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 3, pp. 291–304, March, 2005.     

On the Representation by Bivariate Ridge Functions

Ukrainian Mathematical Journal - We consider the problem of representation of bivariate functions by sums of ridge functions. It is shown that if a function from a certain class of smoothness is...     

A Functional Integral Representation for Many Boson Systems II: Correlation Functions

We derive functional integral representations for the partition function and correlation functions of many Boson systems for which the configuration space consists of finitely many points. Submitted: August 20, 2007. Accepted: July 1, 2008.     

On Integral Representation Formulae for Biharmonic Functions on the Ball

Biharmonic functions are solutions of the fourth order partial differential equation ΔΔu = 0. A simple method is proposed for deriving integral representation formulae for these functions u on the n‐dimensional ball. Poisson‐type representations in the setting of Hardy Spaces are obtained for biharmonic functions subject to Dirichlet, Riquier and other boundary conditions. The approach exploits algebraic properties of a first order partial differential operator and its resolvent.     

On the Representation of Functions with Gaussian Wave Packets

We introduce Gaussian wave packets in pursuit of representations of functions, in which the representation is invariant under translation, modulation, scale, rotation and anisotropic dilation. Properties of both continuous and discrete representations are discussed. For the discrete (two-dimensional) case, we develop fast algorithms for the application of the analysis and synthesis operators. A main objective for using Gaussian wave packets is to obtain sparse approximations of functions. However, due to the many invariance properties, the representations will have a high degree of redundancy. Therefore, we also introduce sparse methods for highly redundant representations, that employ some of the analytic properties of Gaussian wave packet for gaining computational efficiency.     

On Quadratic Integral Polynomials with only Finitely Many Roots in any Commutative Finite-Dimensional Algebra

The monic quadratic polynomials f with integer coefficients such that each commutative finite-dimensional algebra over a field contains only finitely many roots of f are determined as the polynomials of the form f = X ² + (2m + 1)X + m ² + m, where ${m \in \mathbb{Z}}$ .     

Weighted Approximation of Functions on the Unit Sphere

The direct and inverse theorems are established for the best approximation in the weighted L^p space on the unit sphere of R^d+1, in which the weight functions are invariant under finite reflection groups. The theorems are stated using a modulus of smoothness of higher order, which is proved to be equivalent to a K-functional defined using the power of the spherical h-Laplacian. Furthermore, similar results are also established for weighted approximation on the unit ball and on the simplex of R^d.