arXiv subjects
Publications and source records attributed to S. Albeverio.
Given a type I von Neumann algebra $M$ with a faithful normal semi-finite trace $τ,$ let $L(M, τ)$ be the algebra of all $τ$-measurable operators affiliated with $M.$ We give a complete description of all derivations on the algebra $L(M, τ).$ In particular, we prove that if $M$ is of type I$_\infty$ then every derivation on $L(M, τ)$ is inner.
We develop an analysis of wavelets and pseudodifferential operators on multidimensional ultrametric spaces which are defined as products of locally compact ultrametric spaces. We introduce bases of wavelets, spaces of generalized functions and Lizorkin generalized functions on multidimensional ultrametric spaces. We also consider some family of pseudodifferential operators on multidimensional ultrametric spaces. The notions of Cauchy problem for ultrametric pseudodifferential equations and of ultrametric characteristics are introduced. A theorem about existence and uniqueness of the solution for the Cauchy problem (the analogue of the Kovalevskaya theorem) is proven.
Polyadic arithmetics is a branch of mathematics related to $p$--adic theory. The aim of the present paper is to show that there are very close relations between polyadic arithmetics and the classic theory of commutative Banach algebras. Namely, let $\ms A$ be the algebra consisting of all complex periodic functions on $\Z$ with the uniform norm. Then the polyadic topological ring can be defined as the ring of all characters $\ms A\to\C$ with convolution operations and the Gelfand topology.
A $p$-adic Schrödinger-type operator $D^α+V_Y$ is studied. $D^α$ ($α>0$) is the operator of fractional differentiation and $V_Y=\sum_{i,j=1}^nb_{ij}<δ_{x_j}, \cdot>δ_{x_i}$ $(b_{ij}\in\mathbb{C})$ is a singular potential containing the Dirac delta functions $δ_{x}$ concentrated on points $\{x_1,...,x_n\}$ of the field of $p$-adic numbers $\mathbb{Q}_p$. It is shown that such a problem is well-posed for $α>1/2$ and the singular perturbation $V_Y$ is form-bounded for $α>1$. In the latter case, the spectral analysis of $η$-self-adjoint operator realizations of $D^α+V_Y$ in $L_2(\mathbb{Q}_p)$ is carried out.
In this work $n$-dimensional filiform Leibniz algebras admitting a gradation of length $(n-1)$ are classified. Derivations of such algebras are also described.
Let $M$ be a type I von Neumann algebra with the center $Z,$ a faithful normal semi-finite trace $τ.$ Let $L(M, τ)$ be the algebra of all $τ$-measurable operators affiliated with $M$ and let $S_0(M, τ)$ be the subalgebra in $L(M, τ)$ consisting of all operators $x$ such that given any $ε>0$ there is a projection $p\in\mathcal{P}(M)$ with $τ(p^{\perp})<\infty, xp\in M$ and $\|xp\|<ε.$ We prove that any $Z$-linear derivation of $S_0(M, τ)$ is spatial and generated by an element from $L(M, τ).$
Given a von Neumann algebra $M$ with a faithful normal semi-finite trace $τ,$ we consider the non commutative Arens algebra $L^ω(M, τ)=\bigcap\limits_{p\geq1}L^{p}(M, τ)$ and the related algebras $L^ω_2(M, τ)=\bigcap\limits_{p\geq2}L^{p}(M, τ)$ and $M+L^ω_2(M, τ)$ which are proved to be complete metrizable locally convex *-algebras. The main purpose of the present paper is to prove that any derivation of the algebra $M+L^ω_2(M, τ)$ is inner and all derivations of the algebras $L^ω(M,τ)$ and $L^ω_2(M, τ)$ are spatial and implemented by elements of $M+L^ω_2(M, τ).$
Finite rank perturbations of a semi-bounded self-adjoint operator A are studied in the scale of Hilbert spaces associated with A. A concept of quasi-boundary value space is used to describe self-adjoint operator realizations of regular and singular perturbations of A by the same formula. As an application the one-dimensional Schrödinger operator with generalized zero-range potential is considered in the Sobolev space W^p_2(\mathbb{R}), p\in\mathbb{N}.
We show that translations and dilations of a p-adic wavelet coincides (up to the multiplication by some root of one) with a vector from the known basis of discrete p-adic wavelets. In this sense the continuous p-adic wavelet transform coincides with the discrete p-adic wavelet transform. The p-adic multiresolution approximation is introduced and relation with the real multiresolution approximation is described. Relation of application of p-adic wavelets to spectral theory of p-adic pseudodifferential operators and the known results about sparsity of matrices of some real integral operators in the bases of multiresolution wavelets is discussed.
The spectral properties of Dirac operators on $(0,1)$ with potentials that belong entrywise to $L_p(0,1)$, for some $p\in[1,\infty)$, are studied. The algorithm of reconstruction of the potential from two spectra or from one spectrum and the corresponding norming constants is established, and a complete solution of the inverse spectral problem is provided.
The Leibniz algebras appeared as a generalization of the Lie algebras. In this work we deal with the classification of nilpotent complex Leibniz algebras of low dimensions. Namely, the classification of nilpotent complex Leibniz algebras dimensions less than 3 is extended to the dimension four. {\it AMS Subject Classifications}: 16D70, 17A30, 17A60, 17B30 {\it Key words:} Leibniz algebra, associative algebra, nilpotence, nulfiliform Leibniz algebra, filiform Leibniz algebra.
Spaces of differential forms over configuration spaces with Poisson measures are constructed. The corresponding Laplacians (of Bochner and de Rham type) on 1-forms and associated semigroups are considered. Their probabilistic interpretation is given.
The space $Γ_X$ of all locally finite configurations in a Riemannian manifold $X$ of infinite volume is considered. The deRham complex of square-integrable differential forms over $Γ_X$, equipped with the Poisson measure, and the corresponding deRham cohomology are studied. The latter is shown to be unitarily isomorphic to a certain Hilbert tensor algebra generated by the $L^2$-cohomology of the underlying manifold $X$.
We carry out analysis and geometry on a marked configuration space $Ω^M_X$ over a Riemannian manifold $X$ with marks from a space $M$. We suppose that $M$ is a homogeneous space $M$ of a Lie group $G$. As a transformation group $\frak A$ on $Ω_X^M$ we take the ``lifting'' to $Ω_X^M$ of the action on $X\times M$ of the semidirect product of the group $\operatorname{Diff}_0(X)$ of diffeomorphisms on $X$ with compact support and the group $G^X$ of smooth currents, i.e., all $C^\infty$ mappings of $X$ into $G$ which are equal to the identity element outside of a compact set. The marked Poisson measure $π_σ$ on $Ω_X^M$ with Lévy measure $σ$ on $X\times M$ is proven to be quasiinvariant under the action of $\frak A$. Then, we derive a geometry on $Ω_X^M$ by a natural ``lifting'' of the corresponding geometry on $X\times M$. In particular, we construct a gradient $\nabla^Ω$ and a divergence $\operatorname{div}^Ω$. The associated volume elements, i.e., all probability measures $μ$ on $Ω_X^M$ with respect to which $\nabla^Ω$ and $\operatorname{div}^Ω$ become dual operators on $L^2(Ω_X^M;μ)$, are identified as the mixed marked Poisson measures with mean measure equal to a multiple of $σ$. As a direct consequence of our results, we obtain marked Poisson space representations of the group $\frak A$ and its Lie algebra $\frak a$. We investigate also Dirichlet forms and Dirichlet operators connected with (mixed) marked Poisson measures.
Spaces of differential forms over configuration spaces with Poisson measures are constructed. The corresponding Laplacians (of Bochner and de Rham type) on forms and associated semigroups are considered. Their probabilistic interpretation is given.
Properties of the set $T_s$ of "particularly non-normal numbers" of the unit interval are studied in details ($T_s$ consists of real numbers $x$, some of whose s-adic digits have the asymptotic frequencies in the nonterminating $s-$ adic expansion of $x$, and some do not). It is proven that the set $T_s$ is residual in the topological sense (i.e., it is of the first Baire category) and it is generic in the sense of fractal geometry ($T_s$ is a superfractal set, i.e., its Hausdorff-Besicovitch dimension is equal to~1). A topological and fractal classification of sets of real numbers via analysis of asymptotic frequencies of digits in their s-adic expansions is presented.
We develop a metric and probabilistic theory for the Ostrogradsky representation of real numbers, i.e., the expansion of a real number $x$ in the following form: \begin{align*} x&= \sum_n\frac{(-1)^{n-1}}{q_1q_2... q_n}= &=\sum_n\frac{(-1)^{n-1}}{g_1(g_1+g_2)...(g_1+g_2+...+g_n)}\equiv \bO1(g_1,g_2,...,g_n,...), \end{align*} where $q_{n+1}>q_n\in\N$, $g_1=q_1$, $g_{k+1}=q_{k+1}-q_k$. We compare this representation with the corresponding one in terms of continued fractions. We establish basic metric relations (equalities and inequalities for ratios of the length of cylindrical sets). We also compute the Lebesgue measure of subsets belonging to some classes of closed nowhere dense sets defined by characteristic properties of the $\bO1$-representation. In particular, the conditions for the set $\Cset{V}$, consisting of real numbers whose $\bO1$-symbols take values from the set $V \subset N$, to be of zero resp. positive Lebesgue measure are found. For a random variable $ξ$ with independent $\bO1$-symbols $g_n(ξ)$ we prove the theorem establishing the purity of the distribution. In the case of singularity the conditions for such distributions to be of Cantor type are also found.
Euclidean quantum fields obtained as solutions of stochastic partial pseudo differential equations driven by a Poisson white noise have paths given by locally integrable functions. This makes it possible to define a class of ultra-violet finite local interactions for these models (in any space-time dimension). The corresponding interacting Euclidean quantum fields can be identified with systems of classical "charged" particles in the grand canonical ensemble with an interaction given by a nonlinear energy density of the "static field" generated by the particles' charges via a "generalized Poisson equation". The infinite volume limit of such systems is discussed for models with trigonometric interactions using a representation of such models as Widom-Rowlinson models associated with a (formal) Potts models at imaginary temperature. The continuum limit of the particle systems under consideration is also investigated and the formal analogy with the scaling limit of renormalization group theory is pointed out. In some simple cases the question of (non-) triviality of the continuum limits is clarified.