ADirect Method for the Problem of the Solid Cylinder in Elasticity

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A Markoff-type chain for the view-obstruction problem for spheres in ℝ4

In the view-obstruction problem, congruent, closed, convex bodies centred at the points in ⁿ are expanded uniformly until they block all rays from the origin into the open positive cone. The central problem is to determine the minimal blocking size. In the case of spheres of diameter 1, this value is denoted byv(n) and is known for dimensionsn=2,3. Here we show that and obtain a Markoff type chain of isolated extreme values.     

Limit theorems for the best polynomial approximations in the L∞ metric

A limiting equality is established between the best approximations in L of functions of several variables by algebraic polynomials and entire functions of exponential type.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 336–342, March, 1991.     

Conditions on the Pressure for Vanishing Velocity in the Incompressible Fluid Flows in ℝ N

In this paper we derive various sufficient conditions on the pressure for vanishing velocity in the incompressible Navier-Stokes and the Euler equations in ? ^N .     

Limiting theorems for the nodes in binary search trees

We consider three random variables X_n, Y_n and Z_n, which represent the numbers of the nodes with 0, 1, and 2 children, in the binary search trees of size n. The expectation and variance of the three above random variables are got, and it is also shown that X_n, Y_n and Z_n are all asymptotically normal as n→∞by applying the contraction method.     

Explicit Formulae for the Wave Kernels for the Laplacians Δαβ in the Bergman Ball Bn,n≥1

For the perturbed Bergman Laplacians given by in the unit ball B ⁿ of C ⁿ we establish explicit formulae for the corresponding wave equations in B _n. The formulae obtained generalise, for arbitrary , the formulae given in [2] and [5] for the wave equation associated to the shifted Bergman Laplacian =₀₀ in B _n. Moreover, using an analytic continuation argument, we are able to give explicit formulae for the solutions of the wave equation associated to a two parameter family of Laplacians _, on C ⁿ which are natural deformations of the Fubini-Study Laplacian on the Projective space P ⁿ(C) , n 1, viewed as the dual space of the Bergman ball B _n.     

Limit theorems for the best polynomial approximations in the L∞ metric

A limiting equality is established between the best approximations in L of functions of several variables by algebraic polynomials and entire functions of exponential type.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 336–342, March, 1991.     

Testing for the maximum cell probabilities in multinomial distributions

This paper investigates one-sided hypotheses testing for p, the largest cell probability of multinomial distribution. A small sample test of Ethier (1982) is extended to the general cases. Based on an estimator of p, a kind of large sample tests is proposed. The asymptotic power of the above tests under local alternatives is derived. An example is presented at the end of this paper.     

On the Exact Constant in the Jackson–Stechkin Inequality for the Uniform Metric

The classical Jackson–Stechkin inequality estimates the value of the best uniform approximation of a 2π-periodic function f by trigonometric polynomials of degree ≤n−1 in terms of its r-th modulus of smoothness ω _r (f,δ). It reads

Global well-posedness in the energy space for the Benjamin–Ono equation on the circle

We prove that the Benjamin-Ono equation is locally well-posed in . This leads to a global well-posedeness result in thanks to the energy conservation.     

Refined distributional approximations for the uncovered set in the Johnson–Mehl model

Let Φ^z be the uncovered set (i.e., the complement of the union of intervals) at time z in the one-dimensional Johnson–Mehl model. We derive a bound for the total variation distance between the distribution of the number of components of Φ^z∩(0,t] and a compound Poisson-geometric distribution, which is sharper and simpler than an earlier bound obtained by Erhardsson. We also derive a previously unavailable bound for the total variation distance between the distribution of the Lebesgue measure of Φ^z∩(0,t] and a compound Poisson-exponential distribution. Both bounds are O(z_β(t)/t) as t→∞, where z_β(t) is defined so that the expected number of components of Φ^z_β(t)∩(0,t] converges to β>0 as t→∞, and the parameters of the approximating distributions are explicitly calculated.     

On the Feynman–Kac representation for solutions of the Cauchy problem for parabolic equations in divergence form

We consider the Cauchy problem for general second–order uniformly elliptic linear equation in divergence form. We give a stochastic representation of bounded weak solutions of the problem in terms of solutions of associated linear backward stochastic differential equations. Our representation may be considered as an extension of the classical Feynman–Kac formula.     

Perron–Frobenius theorem for hypermatrices in the max algebra

Recently, there have been many intriguing new developments in the study of hypermatrices and their associated eigenvalue problems. In particular, results coming from the matrix setting when studying the max algebra have shown especially attractive combinatorial features. We now extend this max algebra setting into the realm of hypermatrices. Considering that the max algebra has shown particular significance in optimization problems for the matrix setting, we look to examine and extend these results in the higher order conditions. Furthermore, we establish some algebraic properties for hypermatrices and then proceed to extend the Perron–Frobenius Theorem for this setting and prove the existence of a unique eigenvalue. We continue by stating a result from Nussbaum, that the Min–Max theorem holds, and provide a proof for completeness. For strongly increasing hypermatrices, an iterative algorithm which converges to our unique eigenvalue is given. Finally, we conclude with an analysis of our results in the hypergraph setting.     

An Alexandrov–Fenchel-type inequality for hypersurfaces in the sphere

We find a monotone quantity along the inverse mean curvature flow and use it to prove an Alexandrov–Fenchel-type inequality for strictly convex hypersurfaces in the n-dimensional sphere, \(n \ge 3\).