Sequential closure in product spaces

Let X denote the product of m-many second countable Hausdorff spaces. Main theorems: (1) If S?X is invariant under compositions, m is weakly accessible (resp., nonmeasurable), and F?S is sequentially closed and a sequential G_σ-set which is invariant under projections for finite sets (resp., F?S is sequentially open and sequentially closed), then F is closed. (2) If S?X is invariant under projections and m is nonmeasurable, then every sequentially continuous {0, 1} valued function on S is continuous. (3) A sequentially continuous {0, 1}-valued function on an m-adic space of nonmeasurable weight is continuous. Now let X denote the product of arbitrarily many W-spaces and S?X be invariant under compositions. (4) Then in S, the closure of any Q-open subset coincides with its sequential closure.     

Series of tail distributions of finitely inhomogeneous random walks

Let Sn=X₁+?+Xn be a random walk, where the steps Xn are independent random variables having a finite number of possible distributions, and consider general series of the form

(∗)     

Hankel operators on planar domains

Let Ω be a bounded domain in the plane whose boundary consists of a finite number of disjoint analytic simple closed curves LetA denote the space of analytic functions on Ω which are square integrable over Ω with respect to area measure and letP denote the orthogonal projection ofL ²(Ω,dA) ontoA. A functionb inA induces a Hankel operator (densely defined) onA by the ruleH _b (g)=(I?P)bg. This paper continues earlier investigations of the authors and others by determining conditions under whichH _b is bounded, compact, or lies in the Schatten-von Neumann idealS _p , 1<p<∞     

A characterization of certain weak1-closed subalgebras of L∞

Let $\text{R}$ denote the real line and L^∞(R), the class of all Borel measurable L^∞-functions of $\text{R}$. Let S ≠ {0} or φ, be a linear subspace of L^∞(R) which is (i) translation invariant, (ii) weak¹-closed, (iii) self-adjoint, i.e., f?S implies f?S, and (iv) an algebra. Then either (a) S = all constant functions in L^∞; or (b) S = L^∞; or (c) there is a unique c > 0 such that S consists of all L^∞-functions which are periodic of period c.Extension of the above characterization of periodic subalgebras of L^∞ to LCA groups are presented. Also it is shown that the above characterization is in various ways best possible.     

Density of sets of natural numbers and the Lévy group

Let N denote the set of positive integers. The asymptotic density of the set A⊆N is d(A)=limn→∞|A∩[1,n]|/n, if this limit exists. Let AD denote the set of all sets of positive integers that have asymptotic density, and let SN denote the set of all permutations of the positive integers N. The group L^? consists of all permutations f∈SN such that A∈AD if and only if f(A)∈AD, and the group L^* consists of all permutations f∈L^? such that d(f(A))=d(A) for all A∈AD. Let be a one-to-one function such that d(f(N))=1 and, if A∈AD, then f(A)∈AD. It is proved that f must also preserve density, that is, d(f(A))=d(A) for all A∈AD. Thus, the groups L^? and L^* coincide.     

The Arcsine Law

Let N _n denote the number of positive sums in the first n trials in a random walk (S _i) and let L _n denote the first time we obtain the maximum in S ₀,..., S _n. Then the classical equivalence principle states that N _n and L _n have the same distribution and the classical arcsine law gives necessary and sufficient condition for (1/n) L _n or (1/n) N _n to converge in law to the arcsine distribution. The objective of this note is to provide a simple and elementary proof of the arcsine law for a general class of integer valued random variables (T _n) and to provide a simple an elementary proof of the equivalence principle for a general class of integer valued random vectors (N _n, L _n).     

Estimates for interval probabilities of the sums of random variables with locally subexponential distributions

Let {_i} _i=1 be a sequence of independent identically distributed nonnegative random variables, S _n = ξ₁ + ? +ξ_n. Let Δ = (0, T] and x + Δ = (x, x + T]. We study the ratios of the probabilities P(S _n ε x + Δ)/P(ξ ₁ ε x + Δ) for all n and x. The estimates uniform in x for these ratios are known for the so-called Δ-subexponential distributions. Here we improve these estimates for two subclasses of Δ-subexponential distributions; one of them is a generalization of the well-known class LC to the case of the interval (0, T] with an arbitrary T ≤ ∞. Also, a characterization of the class LC is given.     

Cartesian Closed Subcategories of $CONT_{\ll }^{{\kern 8pt}\ast }$

Let CONT _? be the category of continuous domains and Scott continuous mappings that preserve the way-below relation on domains. Let ω-ALG _? be the full subcategory of CONT _? consisting of all countably based algebraic domains, and F I N be the category of finite posets and monotone mappings. The main result proved in this paper is that F I N is the largest Cartesian closed full subcategory of ω-ALG _?. On the other hand, it is shown that the algebraic L-domains form a Cartesian closed full subcategory of ALG _?.     

On Heyde’s Theorem for Probability Distributions on a Discrete Abelian Group

Let X be a countable discrete Abelian group containing no elements of order 2. Let α be an automorphism of X. Let ξ₁ and ξ₂ be independent random variables with values in the group X and distributions μ₁ and μ₂. The main result of the article is the following statement. The symmetry of the conditional distribution of the linear form L₂ = ξ₁ + αξ₂ given L₁ = ξ₁ + ξ₂ implies that μ _j are shifts of the Haar distribution of a finite subgroup of X if and only if α satisfies the condition Ker(I + α)= {0}. Some generalisations of this theorem are also proved.     

S-units and periodicity of continued fractions in hyperelliptic fields

Let L be the function field of a hyperelliptic curve defined over any field of characteristic different from 2, and let S be a set consisting of an infinite and a finite valuation of L. A relationship between the problem of the existence of nontrivial S-units in the field L and the periodicity of the continued fraction expansion of certain key elements of L is discovered for the first time for finite valuations.     

Periodic Groups Saturated with the Linear Groups of Degree 2 and the Unitary Groups of Degree 3 over Finite Fields of Odd Characteristic

Let M denote the set of the simple 3-dimensional unitary groups U₃ and the simple linear groups L₂ over finite fields of odd characteristic.We prove that each periodic group saturated with groups in M is locally finite and isomorphic to either U₃(Q) or L₂(Q) for a suitable locally finite field Q of odd characteristic.     

Asymptotic behavior of product of two heavy-tailed dependent random variables

Let X and Y be positive weakly negatively dependent(WND)random variables with finite expectations and continuous distribution functions F and G with heavy tails, respectively. The asymptotic behavior of the tail of distribution of XY is studied and some closure properties under some suitable conditions on ˉ F(x)= 1 F(x)and ˉ G(x)= 1 G(x)are provided. Moreover, subexponentiality of XY when X and Y are WND random variables is derived.     

Uniform Uniform Exponential Growth of Subgroups of the Mapping Class Group

Let Mod(S) denote the mapping class group of a compact, orientable surface S. We prove that finitely generated subgroups of Mod(S) which are not virtually abelian have uniform exponential growth with minimal growth rate bounded below by a constant depending only, and necessarily, on S. For the proof, we find in any such subgroup explicit free group generators which are “short” in any word metric. Besides bounding growth, this allows a bound on the return probability of simple random walks.     

The spectrum of a finite pseudocomplemented lattice

Let L be a finite pseudocomplemented lattice. Every interval [0, a] in L is pseudocomplemented, so by Glivenko’s theorem, the set S(a) of all pseudocomplements in [0, a] forms a boolean lattice. Let B _i denote the finite boolean lattice with i atoms. We describe all sequences (s ₀, s ₁, . . . , s _n ) of integers for which there exists a finite pseudocomplemented lattice L with s _i = |{ a ∈ L | S(a) ? B _i }|, for all i, and there is no a ∈ L with S(a) ? B _n+1. This result settles a problem raised by the first author in 1971.     

A general iterative method for hierarchical variational inequality problems in Hilbert spaces and applications

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let ?? >?0 and let A be an ??-inverse-strongly monotone mapping of C into H and let B be a maximal monotone operator on H. Let F be a maximal monotone operator on H such that the domain of F is included in C. Let 0?< k?<?1 and let g be a k-contraction of H into itself. Let V be a ${\overline{\gamma}}$ -strongly monotone and L-Lipschitzian continuous operator with ${\overline{\gamma} >0 }$ and L >?0. Take ${\mu, \gamma \in \mathbb R}$ as follows: $${0 < \mu < \frac{2\overline{\gamma}}{L^2}, \quad 0 < \gamma < \frac{\overline{\gamma}-\frac{L^2 \mu}{2}}{k}.}$$ In this paper, under the assumption ${(A+B)^{-1}0 \cap F^{-1}0 \neq \emptyset}$ , we prove a strong convergence theorem for finding a point ${z_0\in (A+B)^{-1}0\cap F^{-1}0}$ which is a unique solution of the hierarchical variational inequality $${\langle (V-\gamma g)z_0, q-z_0 \rangle \geq 0, \quad \forall q\in (A+B)^{-1}0 \cap F^{-1}0.}$$ Using this result, we obtain new and well-known strong convergence theorems in a Hilbert space which are useful in nonlinear analysis and optimization.     

A Generalization of Strassen’s Functional LIL

Let X ₁,X ₂,… be a sequence of i.i.d. mean zero random variables and let S _n denote the sum of the first n random variables. We show that whenever we have with probability one, lim?sup? _n→∞|S _n |/c _n =α ₀<∞ for a regular normalizing sequence {c _n }, the corresponding normalized partial sum process sequence is relatively compact in C[0,1] with canonical cluster set. Combining this result with some LIL type results in the infinite variance case, we obtain Strassen type results in this setting.     

Asymptotic Results for Tail Probabilities of Sums of Dependent and Heavy-Tailed Random Variables

Abstract Let X1,X2,...be a sequence of dependent and heavy-tailed random variables with distributions F1,F2,…. on (-∞,∞),and let т be a nonnegative integer-valued random variable independent of the seq...     

Closure in a Hilbert space of a prehilbert space Chebyshev set

Let E denote the real inner product space that is the union of all finite dimensional Euclidean spaces. There is a bounded nonconvex set S, that is a subset of E, such that each point of E has a unique nearest point in S. Let H denote the separable Hilbert space that is the completion of space E. A condition is given in order that a point in H have a unique nearest point in the closure of S. We shall also provide an example where the condition fails.     

Tridiagonal pairs and the quantum affine algebra {\boldmath U_q({\widehat {sl}}_2)}

Let ${\mathbb K}$ denote an algebraically closed field and let q denote a nonzero scalar in ${\mathbb K}$ that is not a root of unity. Let V denote a vector space over ${\mathbb K}$ with finite positive dimension and let A,A* denote a tridiagonal pair on V. Let θ₀, θ₁,…, θ _d (resp. θ*₀, θ*₁,…, θ* _d ) denote a standard ordering of the eigenvalues of A (resp. A*). We assume there exist nonzero scalars a, a* in ${\mathbb K}$ such that θ _i = aq ^2i?d and θ* _i = a*q ^d?2i for 0 ≤ i ≤ d. We display two irreducible ${\boldmath U_q({\widehat {sl}}_2)}$ -module structures on V and discuss how these are related to the actions of A and A*.