A generalized moment problem

Let {λ _n} (n≧0) satisfy (1.1) we are considering the following problems: What are the necessary and sufficient conditions on a sequence {μ_n} (n≧0) in order that it should possess the representation (1.2) wherea(t) is of bounded variation or the representation (1.3) wheref(t) ∈L _M[0, 1] orf(t) is essentially bounded.     

A generalized moment problem

Mathematische Annalen -     

The moment mapping for unitary representations

For any unitary representation of an arbitrary Lie group I construct a moment mapping from the space of smooth vectors of the representation into the dual of the Lie algebra. This moment mapping is equivariant and smooth. For the space of analytic vectors the same construction is possible and leads to a real analytic moment mapping.     

Generalized moment representations of basis hypergeometric series

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 6, pp. 803–808, June, 1989.     

Convex hulls of generalized moment curves

We consider the family of curves inR ⁴: {fx115-1}, wherep andq are positive integers, and determine the facial structure of the convex hull of these curves.     

Coxian representations of generalized Erlang distributions

This paper studies Coxian representations of generalized Erlang distributions. A nonlinear program is derived for computing the parameters of minimal Coxian representations of generalized Erlang distributions. The nonlinear program is also used to characterize the triangular order and the admissible region of generalized Erlang distributions. It is shown that the admissible region associated with a triangular order may not be convex. For generalized Erlang distributions of ME-order 3, a minimal Coxian representation is found explicitly. In addition, an algorithm is developed for computing a special type of ordered Coxian representations - the bivariate Coxian representation - for generalized Erlang distributions.     

Indecomposable representations of generalized weyl algebras

For a class of generalized Weyl algebras which includes the Weyl algebras A_n a criterion is given as to when the category of indecomposable weight and generalized weight modules with supports from a fixed orbit is tame. In tame cases indecomposable modules are described.     

Wavelet transforms via generalized quasi-regular representations

The construction of the well-known continuous wavelet transform has been extended before to higher dimensions. Then it was generalized to a group which is topologically isomorphic to a homogeneous space of the semidirect product of an abelian locally compact group and a locally compact group. In this paper, we consider a more general case. We introduce a class of continuous wavelet transforms obtained from the generalized quasi-regular representations. To define such a representation of a group G, we need a homogeneous space with a relatively invariant Radon measure and a character of G.     

Realizing representations on generalized flag manifolds

Let G be a complex reductive linear algebraic group and a real form. Suppose P is a parabolic subgroup of G and assume that P has a Levi factor L such that is a real form of L.Using the minimal globalization V _min of a finite length admissible representation for L ₀, one can define a homogeneous analytic vector bundle on the G ₀ orbit S of P in the generalized flag manifold . Let denote the corresponding sheaf of polarized sections. In this article we analyze the G ₀ representations obtained on the compactly supported sheaf cohomology groups .     

BK-type inequalities and generalized random-cluster representations

Recently, van den Berg and Jonasson gave the first substantial extension of the BK inequality for non-product measures: they proved that, for $k$ -out-of- $n$ measures, the probability that two increasing events occur disjointly is at most the product of the two individual probabilities. We show several other extensions and modifications of the BK inequality. In particular, we prove that the antiferromagnetic Ising Curie–Weiss model satisfies the BK inequality for all increasing events. We prove that this also holds for the Curie–Weiss model with three-body interactions under the so-called negative lattice condition. For the ferromagnetic Ising model we show that the probability that two events occur ‘cluster-disjointly’ is at most the product of the two individual probabilities, and we give a more abstract form of this result for arbitrary Gibbs measures. The above cases are derived from a general abstract theorem whose proof is based on an extension of the Fortuin–Kasteleyn random-cluster representation for all probability distributions and on a ‘folding procedure’ which generalizes an argument of Reimer.     

A method of constructing generalized difference sets

On the elements of the ring of residues modulo v (zτ v, 3τ v) we construct cyclic PBIB-designs with τ(v)-1 classes of connectedness, where τ(v) is the number of divisors of v. We prove the existence of cyclic BIB-designs with parameters b, v, r, k, and λ such that: 1) λ=k (and also λ=k/2 if k is even), k≥4, and (k-1) ¦ (p-1) for each prime divisor p of the number v; 2) λ=(k?l)/2, k odd, k≥3, k ¦ (p?1) for each prime divisor p of the number v.     

Generalized moment representations and Padé-Chebyshev approximations

An approach to the application of Dzyadyk's generalized moment representations in problems of construction and investigation of the Padé-Chebyshev approximants is developed. With its help, certain properties of the Padé-Chebyshev approximants of a class of functions that is a natural analog of the class of Markov functions are studied. In particular, it is proved that the poles of the Padé-Chebyshev approximants of these functions lie outside their domain of analyticity.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 762–766, June, 1990.     

Methods of constructing optimal stabilizers

The problem of transforming a linear dynamical system in the neighbourhood of a state of equilibrium [1,2] is solved using the special problem of the damping of the system by controls of minimum intensity after a finite time interval. The possibility of using other problems of optimal control is discussed. The main attention is devoted to constructing algorithms of the operation of a device (a stabilizer) which is able, in real time, to generate a stabilizing control circulating in the closed optimal system when unknown perturbations operate constantly [3, 4]. The proposed method is based on the constructive theory of optimal control [5, 6]. Another form of this theory for solving the problem of stabilization is presented in [7](see also [8]).     

Gluing representations via idempotent modules and constructing endotrivial modules

Let G be a finite group and k be a field of characteristic p. We show how to glue Rickard idempotent modules for a pair of open subsets of the cohomology variety along an automorphism for their intersection. The result is an endotrivial module. An interesting aspect of the construction is that we end up constructing finite dimensional endotrivial modules using infinite dimensional Rickard idempotent modules. We prove that this construction produces a subgroup of finite index in the group of endotrivial modules. More generally, we also show how to glue any pair of kG-modules.     

A generalized moment problem for self-adjoint operators

Let the sequence {λ _i } (i≧0) satisfy condition (1.1) and let {A _n} (n≧0) be a sequence of bounded self-adjoint operators over a complex Hilbert spaceH. We give a necessary and sufficient condition in order that {A _n} (n≧0) should possess the representation (1.2).     

Generalized oscillator representations for generalized Calogero Hamiltonians

We construct generalized oscillator representations for all generalized Calogero Hamiltonians with the potential V (x) = g ₁ /x ² + g ₂ x ² , g ₁ ≥ ?1/4, g ₂ > 0. These representations are generically nonunique, but for each Hamiltonian, there exists an optimum representation explicitly determining the ground state and its energy. For generalized Calogero Hamiltonians with coupling constants g ₁ < ?1/4 or g ₂ < 0, generalized oscillator representations do not exist, which agrees with the fact that the corresponding Hamiltonians are not bounded from below.