On a problem posed by Pauli

A problem posed in a remark by Pauli is discussed: Is it possible to recover the state vector of a quantum system from the distribution functions of the physical observables of this system?     

On a problem posed by Murray and Tent

In this short note, we give a positive answer to a problem posed by Murray and Tent (J. Algebra 398:318–328, 2014).     

On a problem posed by S. Li and J. Liu

A subgroup H of a finite group G is said to be Hall normally embedded in G if there is a normal subgroup N of G such that H is a Hall subgroup of N. The aim of this note is to prove that a group G has a Hall normally embedded subgroup of order |B| for each subgroup B of G if and only if G is soluble with nilpotent residual cyclic of square-free order. This is the answer to a problem posed by Li and Liu (J. Algebra 388:1–9, 2013).     

On a problem related to the Vandermonde determinant

This short note describes new properties of the elementary symmetric polynomials, and reveals that the properties give an answer to the conjecture raised by El-Mikkawy in [M.E.A. El-Mikkawy, On a connection between the Pascal, Vandermonde and Stirling matrices—II, Appl. Math. Comput. 146 (2003) 759-769].     

On certain exact relations for sojourn probabilities of a wiener process

New exact relations are proved for the sojourn probability of a Wiener process between two time-de-pendent boundaries. The proof is based on the investigation of the heat-conduction equation in the domain determined by these functions-boundaries. The relations are given in the form of series. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 6, pp. 842–846, June, 1999     

On a combinatorial problem related to permanents

Some numerical results will be presented concerning the largest and smallest values of square permanents containing the integers 1, 2, 3, ...,n ² n=1(1)8.Dedicated to Peter Naur on the occasion of his 60th birthday     

On the lower limits of maxima and minima of wiener process and partial sums

Let M ⁺ (t) and ?M ^? (t) be the maximum and minimum of a Wiener process on the interval (O,t). This paper gives an integral test for P(M ⁺(t)?(t)t i.o.)=0 or 1. The case of i.i.d. random variables is also treated here. If a(t)=b(t), then our result gives Chung's law of the iterated logarithm [5], while b(t)=∞ corresponds to Hirsch's theorem [9]. Finally, a converse to Chung's LIL is given.     

A GENERAL FORM OF THE INCREMENTS OF A TWO-PARAMETER WIENER PROCESS

In this paper,we consider a general form of the increments for a two-parameter Wiener process.Both the Csorgo-Revesz‘s increments and a class of the lag increments are the special cases of this general form of increments.Our results imply the theorem that have been given by Csorgo and Revesz(1978),and some of their conditions are removed.     

On a divisor problem related to the Epstein zeta-function, II

Recently by using the theory of modular forms and the Riemann zeta-function, Lü improved the estimates for the error term in a divisor problem related to the Epstein zeta-function established by Sankaranarayanan. In this short note, we are able to further sharpen some results of Sankaranarayanan and of Lü, and to establish corresponding Ω-estimates.     

On the behavior of solutions to certain parabolic SPDE's driven by wiener processes

In this article we prove new results concerning the structure and the stability properties of the global attractor associated with a class of nonlinear stochastic partial differential equations driven by finite-dimensional Wiener processes. This class encompasses important equations that occur in the mathematical analysis of certain migration phenomena in population dynamics and population genetics. The solutions to such equations are generalized random fields whose long-time behavior we investigate in detail. In particular, we unveil the mechanism whereby these random fields approach the global attractor by proving that their asymptotic behavior is entirely controlled by that of their spatial average. We also show how to determine explicitly the corresponding Lyapunov exponents when the nonlinearities of the noise-term of the equations are subordinated to the nonlinearity of the drift-term in some sense. The ultimate picture that emerges from our analysis is one that displays a phenomenon of exchange of stability between the components of the global attractor. We provide a very simple interpretation of this phenomenon in the case of Fisher's equation of population genetics. Our method of investigation rests upon the use of martingale arguments, of a comparison principle and of some simple ergodic properties for certain Lebesgue- and Itô integrals.     

On a problem of R. Devore

Acta Mathematica Hungarica -     

Oscillations of the random argument of a wiener process in the Skorokhod representation

A sequence of independent random variables with zero mean generates, by applying to it the A. V. Skorokhod representation, a sequence of random arguments of a Wiener process. At the proof of limit theorems for sums of independent random variables with the use of this representation, there arises the question of the stability of the sequence of the random arguments. In the paper the law of the iterated logarithms yields exact bounds for the oscillation of this sequence.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova Akademii Nauk SSSR, Vol. 177, pp. 83–91, 1989.     

On a problem of integral geometry related to the Dirichlet problem for an ultrahyperbolic equation

We study the nontrivial solvability of the homogeneous problem of integral geometry on the family of spheres formed by sections of the (n − 1)-dimensional unit sphere by hyperplanes perpendicular to the generators of a given cone. By using a relationship between this problem and the Dirichlet problem for an ultrahyperbolic equation in a ball and a criterion for the failure of uniqueness of the solution of the latter problem in terms of zeros of classical Jacobi polynomials, we obtain a criterion for the uniqueness of the solution of the former problem.