An uniqueness theorem for characteristic functions

Suppose that f is the characteristic function of a probability measure on the real line \(\mathbb R\). In this paper, we deal with the following problem posed by N.G. Ushakov: Is it true that f is never determined by its imaginary part \(\mathfrak {I}f\)? In other words, is it true that for any characteristic function f there exists a characteristic function g such that \(\mathfrak {I}f\equiv \mathfrak {I}g\) but \( f\not \equiv g\)? We study this question in the more general case of the characteristic function defined on an arbitrary locally compact abelian group. A characterization of what characteristic functions are uniquely determined by their imaginary parts are given. As a consequence of this characterization, we obtain that several frequently used characteristic functions on the classical locally compact abelian groups are uniquely determined by their imaginary parts.     

On a theorem of Hermite and Hurwitz

The Hermite-Hurwitz theorem computes the degree, over $\text{R}$, of a real rational function ? in terms of the signature of an associated quadratic form—known today as the Hankel matrix of ?. This formula, which Hermite was led to by his work on the problem of representing integers as sums of squares, gave rise to striking applications in the theory of equations and in the stability theory of ordinary differential equations. In this paper, this theorem and various generalizations to the matrix-valued case are discussed and described in terms of signature formulae. These include its relation to stability theory and the matrix Hermite-Hurwitz theorem of Bitmead-Anderson as applied to questions of circuit synthesis. This also includes a global form of Hörmander's signature formula for the Maslov index of a rational loop in a Lagrangian Grassmannian, due to Byrnes and Duncan, and applications to the topology of spaces of rational matrix-valued functions, following the work of Brockett, Byrnes, and Duncan. This includes, in particular, a topological proof of the matrix Hermite-Hurwitz theorem.     

An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation

We prove an infinite dimensional KAM theorem. As an application, we use the theorem to study the two dimensional nonlinear Schrödinger equation with periodic boundary conditions. We obtain for the equation a Whitney smooth family of small-amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamical system.     

An application of the sampling theorem

Let B_σ,p,1 <-p<-∞, be the set of all functions from L₈(R) which can be continued to entire functions of exponential type <-σ. The well known Shannon sampling theorem and its generalization [1] state that every f∈B_σ,p, 1<p<∞, can be represented as

An application of the sampling theorem

Let B_σ,p,1 <-p<-∞, be the set of all functions from L₈(R) which can be continued to entire functions of exponential type <-σ. The well known Shannon sampling theorem and its generalization [1] state that every f∈B_σ,p, 1 $$f(x) = \mathop \Sigma \limits_{j \in z} f(j\pi /\sigma )\tfrac{{sin\sigma (x - j\pi /\sigma )}}{{\sigma (x - j\pi /\sigma )}}, \sigma > 0$$ It is well known that the Shannon sampling theorem plays an important role in many applied fields [1,2,7]. In this paper we give an application of sampling theorem to approximation theory.     

An application of the Riemann-Roch theorem

Applying the Riemann-Roch theorem,we calculate the dimension of a kind of mero- morphicλ-differentials' space on compact Riemann surfaces.And we also construct a basis of theλ-differentials' space.As the main result,the Cauchy type of integral formula on compact Riemann surfaces is established.     

Harmonic morphisms of graphs and the Riemann–Hurwitz theorem

Several versions of the Riemann–Hurwitz theorem for branched coverings of graphs are presented. A larger class of graphs which may have not only multiple edges but also loops and darts is considered. This makes it possible to render the class of graphs closed with respect to morphisms arising as quotient maps for actions of finite groups.     

An application of Kronecker's theorem to rational functions

Dedicated to the memory of S.K. Pichorides     

Generalizations of a discrete universality theorem for Hurwitz zeta-functions

It is well known that the Hurwitz zeta-function ζ(s, α) with transcendental or rational parameter α is universal in the sense that the shifts ζ(s + iτ ), t ? \mathbbR \tau \in \mathbb{R} (continuous case), and ζ(s + imh), m ? \mathbbN è{ 0 } m \in \mathbb{N} \cup \left\{ 0 \right\} , with fixed h > 0 (discrete case) approximate any analytic function. In the paper, the discrete universality is extended for some classes of the functions F(ζ(s, α)).     

An application of the Krasnoselskii theorem to systems of algebraic equations

Based on the Krasnoselskii theorem, we study the existence, multiplicity and nonexistence of positive solutions of general systems of nonlinear algebraic equations under superlinearity and sublinearity conditions. Systems of nonlinear algebraic equations often arise from studies of differential and difference equations. Our results significantly extend and improve those in the literature. A?number of examples and open questions are given to illustrate these results.     

An optimal compactness theorem and application to elliptic-parabolic systems

We give a new optimal compactness criterion which insures that time dependent bounded sequences in suitable Hilbert spaces contain convergent subsequences. Our proof is related to PDE techniques. We then give an abstract application of the result.     

An application of the Putnam-Fuglede theorem to normal products of self-adjoint operators

We prove that if we have two self-adjoint operators (bounded or not) and if their product is normal, then it is self-adjoint provided a certain condition is satisfied.

     

An application of the projective spectral theorem to commuting families of operators

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 40, No. 4, pp. 469–481, July–August, 1988.     

A refined Hurwitz theorem for imbeddings of irredundant Cayley graphs

Hurwitz's theorem states that the order of any finite group acting on a surface of genus γ > 1 is bounded by 168(γ ? 1). It can be refined to give useful information about groups whose order is near this bound. In this paper, similar results are obtained for Cayley graphs imbedded in a surface of genus γ. These results have important implications for the classification of Cayley graphs of low genus and the number of Cayley graphs of a given genus.     

An application of Shoenfield's absoluteness theorem to the theory of uniform distribution

IfC is a Polish probability space, a Borel set whose sectionsW _x ( have measure one and are decreasing , then we show that the set _x W _x has measure one. We give two proofs of this theorem—one in the language of set theory, the other in the language of probability theory, and we apply the theorem to a question on completely uniformly distributed sequences.Supported by DFG grant Ko 490/7-1.