On a Definitizable Analog of the Trigonometric Moment Problem Generating an Indefinite Toeplitz Form

We prove the existence of an integro-polynomial representation for a sequence of numbers such that there exists a difference operator mapping this sequence to a sequence that generates the solvable trigonometric moment problem. A similar result related to the power moment problem was given in [12].     

Congruence-preserving functions on Stone and Kleene algebras

We investigate local polynomial functions on Stone algebras and on Kleene algebras. We find a generating set for the clone of all local polynomial functions. We also represent local polynomial functions on a given algebra by polynomial functions of some canonical extension of this algebra.     

Asymptotic Normality of Cross-correlogram Estimates of the Response Function

The problem of estimation of an unknown response function of a time-invariant continuous linear system is considered. Discrete-time sample input–output cross-correlograms are taken as estimates of the response function. The inputs are supposed to be zero-mean stationary Gaussian processes close, in some sense, to a white noise. Both asymptotic normality of finite-dimensional distributions of the estimates and their asymptotic normality in spaces of continuous functions are studied. Our basic tool is a new integral representation for cumulants of the estimate as a finite sum of integrals involving cyclic products of kernels. Some inequalities for these integrals are obtained and their asymptotic behaviour is studied.     

An asymptotic theory of higher-order operator differential equations with nonsmooth nonlinearities

We develop an asymptotic theory of nonlinear operator differential equations of an arbitrary order in Banach spaces. The nonlinear part of the equation is written in a divergent form. It is shown that the main term in an asymptotic representation of solutions at infinity satisfies a finite-dimensional dynamical system perturbed by a small nonlocal operator.     

Isomorphisms of Certain Locally Nilpotent Finitary Groups and Associated Rings

For any chain Γ the ring NT(Γ,K) of all finitary Γ-matrices ‖a _ij ‖ _i,jεΓ over an associative ring K with zeros on and above the main diagonal is locally nilpotent and hence radical. If R′=NT(Γ′,K′),R=NT(Γ,K) and either |Γ|<∞ or K is a ring with no zero-divisors, then isomorphisms between rings R and R′, their adjoint groups and associated Lie rings are described.     

Spatial No-Waiting Stations with Moving Customers

We analyze spatial MAP/G/∞-, spatial MAP/G/c/0¹ and spatial Cox/G/∞-stations with group arrivals over some Polish space X (with Borel σ-algebra X), including the aspect of customer motion in space. For models with MAP-input, characteristic differential equations are set up that describe the dynamics of phase dependent random functions Q _r;ij (u,t;S′), where Q _r;ij (u,t;S′) is the probability to observe, at time u≤t, the number r of those customers in some source set S′∈X, who will be in a destination set S∈X at time t. For Cox/G/∞-stations, i.e., infinite server stations with doubly stochastic input, the arrival intensities as well as service times may depend on some general stochastic process (J′ _t ) _t≥0 with countable state space. For that case we obtain explicit expressions for space–time distributions as well as stationary and non-stationary characteristics.     

Split Orthogonal Arrays and Maximum Independent Resilient Systems of Functions

A system of (Boolean) functions in variables is called randomized if the functions preserve the property of their variables to be independent and uniformly distributed random variables. Such a system is referred to as -resilient if for any substitution of constants for any variables, where 0 i t, the derived system of functions in variables will be also randomized. We investigate the problem of finding the maximum number of functions in variables of which any form a -resilient system. This problem is reduced to the minimization of the size of certain combinatorial designs, which we call split orthogonal arrays. We extend some results of design and coding theory, in particular, a duality in bounding the optimal sizes of codes and designs, in order to obtain upper and lower bounds on . In some cases, these bounds turn out to be very tight. In particular, for some infinite subsequences of integers they allow us to prove that , , , , . We also find a connection of the problem considered with the construction of unequal-error-protection codes and superimposed codes for multiple access in the Hamming channel.     

Numerical radius and zero pattern of matrices

Let A be an n×n complex matrix and r be the maximum size of its principal submatrices with no off-diagonal zero entries. Suppose A has zero main diagonal and x is a unit n-vector. Then, letting ‖A‖ be the Frobenius norm of A, we show that

|〈Ax,x〉^|2?(1−1/2r−1/2n)‖A^‖2.     

A spectral condition for odd cycles in graphs

Let G be a graph of sufficiently large order n, and let the largest eigenvalue μ(G) of its adjacency matrix satisfies . Then G contains a cycle of length t for every t?n/320This condition is sharp: the complete bipartite graph T₂(n) with parts of size ⌊n/2⌋ and ⌈n/2⌉ contains no odd cycles and its largest eigenvalue is equal to .This condition is stable: if μ(G) is close to and G fails to contain a cycle of length t for some t?n/321, then G resembles T₂(n).