Functional analytic aspects of non-commutative harmonic analysis

Given a homogeneous space X = G/H with an invariant measure it is shown, using Grothendieck's inequality, that a G-invariant Hilbert subspace of the space of distributions of order zero on X is actually contained in L_loc²(X). Moreover, if θ is an automorphism on G appropriately related to H, it is shown that, under condition that H-orbits are smooth, an H-bi-invariant distribution of positive type on G satisfies the identity Ť^θ = T if the corresponding Hilbert space is contained in L_loc²(X). This shows that, under the smooth orbit condition, G-invariant Hilbert subspaces of L_loc² (X) have a unique decomposition into irreducible Hilbert spaces as in the case of generalized Gelfand pairs.     

On a nonstationary problem of group pursuit

Pontryagin’s classical nonstationary example with many participants and phase constraints on the evader’s states with equal dynamic and inertia capabilities of the players is considered. The case of a simple motion is considered separately. Solvability conditions for problems of pursuit and evasion are obtained.     

Simultaneous multiple capture in a simple pursuit problem

The necessary and sufficient conditions for simultaneous multiple capture in a simple group pursuit problem with different opportunities for the participants are obtained.     

A nonstationary group pursuit problem

We consider a linear nonstationary problem of conflict interaction of controlled objects, where the number of pursuers equals ν and the number of evaders equals µ. All participants are assumed to have equal dynamic abilities. The purpose of the pursuers is to catch all evaders, while the purpose of the latter is to avoid being caught for at least one of them. We establish sufficient solvability conditions for the local evasion problem.     

A group pursuit problem with phase constraints

We shall consider the problem of the pursuit of a group of controlled objects by a group of controlled objects, with phase constraints on the states of the evaders.     

The hilbert boundary value problem for generalized analytic functions in clifford analysis

Let ?_0,n be the real Clifford algebra generated by e₁, e₂,…, e_n satisfying e_ie_j+e_je_i=−2δ_ij, i, j=1,2,…, n. e₀ is the unit element. Let Ω be an open set. A function f is called left generalized analytic in Ω if f satisfies the equation

equation(0.1)

Lf=0,$Lf=0,$

where

L=q₀e₀∂x₀+ q₁e₁∂x₁+…+qnen_∂xn,$L=q_0{e}_0\partial x_0+\hspace{0.17em}{q}_1{e}_1\partial x_1+\dots +q_n{e}_n{\partial }_{x_n},$

qi <0, i=0,1,…, n. In this article, we first give the kernel function for the generalized analytic function. Further, the Hilbert boundary value problem for generalized analytic functions in ?ⁿ⁺¹₊ will be investigated.     

Measures induced by analytic functions and a problem of Walter Rudin

The measure induced by a bounded analytic function on the unit disk may be defined by , where is normalized Lebesgue measure on . We discuss the problem of characterizing such measures, and produce some necessary conditions which we conjecture are sufficient. Our main results are a construction showing that our conjectured sufficient conditions are sufficient for a measure to be weakly approximable by induced measures, and a construction of a function , not a constant multiple of an inner function, whose induced measure is rotationally symmetric. This function is not inner, but satisfies if , thus answering a question posed by Walter Rudin.

     

Riemann-Hilbert problem for analytic families of representations

Translated from Matematicheskie Zametki, Vol. 50, No. 2, pp. 89–97, August, 1991.     

A pursuit and evasion problem with measurement uncertainty

A pursuit-evasion process with uncertain state-dependent measurements, in finite, discrete time and in a finite, discrete state space, is considered. Three types of strategies which might be employed in such a process are compared, and attention is concentrated on the behavior strategy, the least well known of the three types, but often the simplest optimal strategy to employ.A variation of the Brown-Robinson fictitious play algorithm is presented which can be used to compute behavior strategies in the case of perfect recall processes.Two examples are given in which optimal behavior strategies are computed using the algorithm and compared with a type of plausible but nonoptimalseparation strategy.This work was partially supported by NASA Grant No. NGR-15-005-021, NSF Grant No. GK-1970, and AFOSR Grant No. 69-1776.     

Approximate analytic solution for critical parameters in a thermal explosion problem

In this paper, a new approximate analytical solution based on the homotopy perturbation method (HPM) is applied to predict critical conditions for a thermal explosion problem. Apart from the conceptual simplicity and flexibility, the method gives rise to an easy converging solution leading to the predictions of critical parameters for thermal explosion.     

The Schwarz problem for Douglis analytic functions

This paper considers the Schwarz problem that consists in finding a J-analytic function by its real part on the boundary. The Fredholm solvability of this problem is proved. The integral representation of J-analytic functions by Cauchy-type integrals with real density is obtained.