Two-dimensional Meixner Random Vectors of Class ${mathcal{M}}_{L}$

The paper is divided into two parts. In the first part we lay down the foundation for defining the joint annihilation–preservation–creation decomposition of a finite family of not necessarily commutative random variables, and show that this decomposition is essentially unique. In the second part we show that any two, not necessarily commutative, random variables X and Y for which the vector space spanned by the identity and their annihilation, preservation, and creation operators equipped with the bracket given by the commutator forms a Lie algebra are equivalent up to an invertible linear transformation to two independent Meixner random variables with mixed preservation operators. In particular, if X and Y commute, then they are equivalent up to an invertible linear transformation to two independent classic Meixner random variables. To show this we start with a small technical condition called “non-degeneracy”.     

Discreteness criteria for Möbius groups acting on $$\mathcal{O}\mathcal{L}_p $$

In this paper, three new discreteness criteria for Möbius groups acting on\(\bar R^{n * } \) are obtained; they are generalizations of known results using the information of two-generator subgroups.     

Menger’s theorem in {{\Pi^1_1\tt{-CA}_0}}

We prove Menger’s theorem for countable graphs in ${{\Pi^1_1\tt{-CA}_0}}$ . Our proof in fact proves a stronger statement, which we call extended Menger’s theorem, that is equivalent to ${{\Pi^1_1\tt{-CA}_0}}$ over ${{\tt{RCA}_0}}$ .     

Die metrisierbaren linearen Teilräume des Raumes\mathfrak{O}_M von L. Schwartz

In Schwartz' terminology, a real or complex valued functionf, defined and infinitely differentiable on ? _n , belongs to \(\mathfrak{O}_M \) iff, as well as any of its derivatives, is at most of polynomial growth. The topology of \(\mathfrak{O}_M \) is defined by the seminorms sup{∣?(x)D ^p f(x)∣;x∈? ⁿ }, where ? belongs to \(\mathfrak{S}\) andD ^p is any derivative. It is well-known that \(\mathfrak{O}_M \) is non-metrisable. For any μ: ? ⁿ →?, let \(\mathfrak{B}_\mu \) be the space of all infinitely differentiable functionsf satisfying, for eachp, sup{∣(1+∣x∣²)^?μ(p) D ^p f(x)∣;x∈? ⁿ }<∞, with the obvious topology. These spaces, which are of very little use elsewhere in the theory of distributions, can be conveniently applied to characterise the metrisable linear subspaces of \(\mathfrak{O}_M \) : A linear subspace of \(\mathfrak{O}_M \) is metrisable if and only if it is, algebraically and topologically, a subspace of some \(\mathfrak{B}_\mu \) .     

$${\mathcal{L}_1}$$-spaces with the Radon-Nikodým property containing reflexive subspaces

We show that for every sequence \({(p_n)_{n\in\mathbb{N}}}\) with 1 ≤ p _n  ≤ 2 there exists an \({\mathcal{L}_1}\) -space with the Radon-Nikodým containing an isomorphic copy of \({\ell_1(\ell_{p_n})}\) .     

A control theorem for p-adic automorphic forms and Teitelbaum’s $$\mathcal {L}$$ L -invariant

The Ramanujan Journal - We extend the results of Kaneko–Zagier and Baba–Granath on relations of supersingular polynomials and solutions of certain second-order modular differential...     

Classes of semigroups modulo Green’s relation \mathcal{H}

Inverse semigroups and orthodox semigroups are either defined in terms of inverses, or in terms of the set of idempotents E(S). In this article, we study analogs of these semigroups defined in terms of inverses modulo Green’s relation \(\mathcal{H}\) , or in terms of the set of completely regular elements H(S). Results are obtained both for the regular and the non-regular cases. We then study the interplays between these new classes of semigroups, as well as with various known classes notably of inverse, orthodox, E-solid and cryptic semigroups.     

Basis Properties in $${\varvec{L}}_{{\varvec{p}}}$$ of Root Functions of Sturm–Liouville Problem with Spectral Parameter-Dependent Boundary Conditions

In this paper, we consider the Sturm–Liouville problem with spectral parameter in the boundary conditions. We associate this problem with a self-adjoint operator in the Pontryagin space \(\Pi _{2}\). Using this operator-theoretic formulation and analytic methods, we study the basis properties in the space \(L_{p} (0,1),\,1<p < \infty \), of systems of root functions of this problem.     

On $${\mathcal {I}}_{

Tóth  János T.  Filip  Ferdinánd  Bukor  József  Zsilinszky  László 《Periodica Mathematica Hungarica》2021,82(2):125-135

Approximation of the Classes C β ψ $$ {C}_{beta}^{psi } $$ H�� By Biharmonic Poisson Integrals

Ukrainian Mathematical Journal - We study the problem of approximation of functions (ψ, β)-differentiable (in the Stepanets sense) whose (ψ, β)-derivative belongs to the class...     

Nuclear dimension and $\mathcal{Z}$-stability of pure C∗-algebras

In this article I study a number of topological and algebraic dimension type properties of simple C^∗-algebras and their interplay. In particular, a simple C^∗-algebra is defined to be (tracially) (m,[`(m)])(m,\bar{m})-pure, if it has (strong tracial) m-comparison and is (tracially) [`(m)]\bar{m}-almost divisible. These notions are related to each other, and to nuclear dimension.     

Irreducible Modules Over Witt Algebras $\mathcal {W}_{n}$ and Over 𝖘𝖑n+1(ℂ)$\mathfrak {sl}_{n+1}(\mathbb {C})$

In this paper, by using the “twisting technique” we obtain a class of new modules A _b over the Witt algebras \(\mathcal {W}_{n}\) from modules A over the Weyl algebras \(\mathcal {K}_{n}\) (of Laurent polynomials) for any \(b\in \mathbb {C}\). We give necessary and sufficient conditions for A _b to be irreducible, and determine necessary and sufficient conditions for two such irreducible \(\mathcal {W}_{n}\)-modules to be isomorphic. Since \(\mathfrak {sl}_{n+1}(\mathbb {C})\) is a subalgebra of \(\mathcal {W}_{n}\), all the above irreducible \(\mathcal {W}_{n}\)-modules A _b can be considered as \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-modules. For a class of such \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-modules, denoted by Ω_1?a (λ ₁, λ ₂, ? ,λ _n ) where \(a\in \mathbb {C}, \lambda _{1},\lambda _{2},\cdots ,\lambda _{n} \in \mathbb {C}^{*}\), we determine necessary and sufficient conditions for these \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-modules to be irreducible. If the \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-module Ω_1?a (λ ₁, λ ₂,? ,λ _n ) is reducible, we prove that it has a unique nontrivial submodule W _1?a (λ ₁, λ ₂,...λ _n ) and the quotient module is the finite dimensional \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-module with highest weight mΛ _n for some non-negative integer \(m\in \mathbb {Z}_{+}\). We also determine necessary and sufficient conditions for two \(\mathfrak {sl}_{n+1}(\mathbb {C})\)-modules of the form Ω_1?a (λ ₁, λ ₂,? ,λ _n ) or of the form W _1?a (λ ₁, λ ₂,...λ _n ) to be isomorphic.     

Strict comparison and $$ \mathcal{Z} $$-absorption of nuclear C∗-algebras

For any unital separable simple infinite-dimensional nuclear C ^∗-algebra with finitely many extremal traces, we prove that -absorption, strict comparison and property (SI) are equivalent. We also show that any unital separable simple nuclear C ^∗-algebra with tracial rank zero is approximately divisible, and hence is -absorbing.     

A short proof of Cartan’s Nullstellensatz for entire functions in $${mathbb{C}^n}$$

Using the fact that the maximal ideals in the polydisk algebra are given by the kernels of point evaluations, we derive a simple formula that gives a solution to the Bézout equation in the space of all entire functions of several complex variables. Thus a short and easy analytic proof of Cartan’s Nullstellensatz is obtained.     

Itô’s formula for the L p -norm of stochastic {W^{1}_{p}} -valued processes

We prove Itô’s formula for the L _p -norm of a stochastic ${W^{1}_{p}}$ -valued processes appearing in the theory of SPDEs in divergence form.     

Classical Limit of the Scaled Elliptic Algebra \mathcal{A} ħ,η (\widetilde{\mathfrak{s}\mathfrak{l}}_2)

The classical limit of the scaled elliptic algebra $\mathcal{A}$ ?,η ( $\widetilde{\mathfrak{s}\mathfrak{l}}_2$ ) is investigated. The limiting Lie algebra is described in two equivalent ways: as a central extension of the algebra of generalized automorphic sl2 valued functions on a strip and as an extended algebra of decreasing automorphic sl2 valued functions on the real line. A bialgebra structure and an infinite-dimensional representation in the Fock space are studied. The classical limit of elliptic algebra $\mathcal{A}$ q,p ( $\widetilde{\mathfrak{s}\mathfrak{l}}_2$ ) is also briefly presented.     

The Aronszajn–Donoghue Theory for Rank One Perturbations of the $$\mathcal{H}_{-2} {\text{-Class}}$$

A singular rank one perturbation of a self-adjoint operator A in a Hilbert space is considered, where and but with the usual A–scale of Hilbert spaces. A modified version of the Aronszajn-Krein formula is given. It has the form where F denotes the regularized Borel transform of the scalar spectral measure of A associated with . Using this formula we develop a variant of the well known Aronszajn–Donoghue spectral theory for a general rank one perturbation of the class.Submitted: March 14, 2002 Revised: December 15, 2002     

Infinite time extensions of Kleene’s $${\mathcal{O}}$$

Using infinite time Turing machines we define two successive extensions of Kleene’s O{\mathcal{O}} and characterize both their height and their complexity. Specifically, we first prove that the one extension—which we will call O⁺{\mathcal{O}^{+}}—has height equal to the supremum of the writable ordinals, and that the other extension—which we will call O⁺⁺{\mathcal{O}}^{++}—has height equal to the supremum of the eventually writable ordinals. Next we prove that O⁺{\mathcal{O}^+} is Turing computably isomorphic to the halting problem of infinite time Turing computability, and that O⁺⁺{\mathcal{O}^{++}} is Turing computably isomorphic to the halting problem of eventual computability.     

A new optimal quaternary sequence family of length 2(2 n − 1) obtained from the orthogonal transformation of Families {\mathcal{B}} and {\mathcal{C}}

In this paper, a general orthogonal transformation on the optimal quaternary sequence Families ${\mathcal{B}}$ and ${\mathcal{C}}$ is presented. Consequently, the known optimal Family ${\mathcal{D}}$ and a new optimal Family ${\mathcal{E}}$ are produced in a uniform method. In contrast to the known optimal Family ${\mathcal{D}}$ , the new Family ${\mathcal{E}}$ has the same parameters such as the sequence length 2(2 ⁿ ? 1), the family size 2 ⁿ , and the maximal nontrivial correlation value ${2^{\frac{n+1}{2}}+2}$ , where n is a positive integer, but with a different correlation function.