The Paranormality of Products and Their Subsets

A topological space is called paranormal if any countable discrete system of closed sets {D_n:n = 1, 2, 3,...} can be expanded to a locally finite system of open sets {U_n:n = 1, 2, 3,...}, i.e., D_n is contained in U_n for all n, and D_m ∩ U_n≠ Ø if and only if D_m = D_n. It is proved that if X is a countably compact space whose cube is hereditarily paranormal, then X is metrizable.     

On the extent of star countable spaces

For a topological property P, we say that a space X is star Pif for every open cover Uof the space X there exists Y ? X such that St(Y,U) = X and Y has P. We consider star countable and star Lindelöf spaces establishing, among other things, that there exists first countable pseudocompact spaces which are not star Lindelöf. We also describe some classes of spaces in which star countability is equivalent to countable extent and show that a star countable space with a dense σ-compact subspace can have arbitrary extent. It is proved that for any ω ₁-monolithic compact space X, if C _p (X)is star countable then it is Lindelöf.     

Hankel and Berezin Type Operators on Weighted Besov Spaces of Holomorphic Functions on Polydisks

Let S be the space of functions of regular variation and let ω = (ω₁,..., ω_n), ω_j ∈ S. The weighted Besov space of holomorphic functions on polydisks, denoted by B _p (ω) (0 < p < +∞), is defined to be the class of all holomorphic functions f defined on the polydisk U ⁿ such that \(||f||_{{B_{P(\omega )}}}^P = \int_{{U^n}} {|Df(z){|^p}\prod\limits_{j = 1}^n {{\omega _j}{{(1 - |{z_j}{|^2})}^{P - 2}}dm{a_{2n}}(z) < \infty } } \), where dm_2n(z) is the 2ndimensional Lebesgue measure on U ⁿ and D stands for a special fractional derivative of f.We prove some theorems concerning boundedness of the generalized little Hankel and Berezin type operators on the spaces B _p (ω) and L _p (ω) (the weighted L _p -space).     

An improvement of the five halves theorem of J. Boardman

Let (M ^m , T) be a smooth involution on a closed smooth m-dimensional manifold and F = ∪ _j=0 ⁿ F ^j (n ≤ m) its fixed point set, where F ^j denotes the union of those components of F having dimension j. The famous Five Halves Theorem of J. Boardman, announced in 1967, establishes that, if F is nonbounding, then m ≤ 5/2n. In this paper we obtain an improvement of the Five Halves Theorem when the top dimensional component of F, F ⁿ , is nonbounding. Specifically, let ω = (i ₁, i ₂, …, i _r ) be a non-dyadic partition of n and s _ω (x ₁, x ₂, …, x _n ) the smallest symmetric polynomial over Z ₂ on degree one variables x ₁, x ₂, …, x _n containing the monomial \(x_1^{i_1 } x_2^{i_2 } \cdots x_r^{i_r }\). Write s _ω (F ⁿ ) ∈ H ⁿ (F ⁿ , Z ₂) for the usual cohomology class corresponding to s _ω (x ₁, x ₂, …, x _n ), and denote by ?(F ⁿ ) the minimum length of a nondyadic partition ω with s _ω (F ⁿ ) ≠ 0 (here, the length of ω = (i ₁, i ₂, …, i _r ) is r). We will prove that, if (M ^m , T) is an involution for which the top dimensional component of the fixed point set, F ⁿ , is nonbounding, then m ≤ 2n + ?(F ⁿ ); roughly speaking, the bound for m depends on the degree of decomposability of the top dimensional component of the fixed point set. Further, we will give examples to show that this bound is best possible.     

Balanced metrics on Hartogs domains

An n-dimensional strictly pseudoconvex Hartogs domain D _F can be equipped with a natural Kähler metric g _F . In this paper we prove that if m ₀ g _F is balanced for a given positive integer m ₀ then m ₀>n and (D _F ,g _F ) is holomorphically isometric to an open subset of the n-dimensional complex hyperbolic space.     

A note on transitively <Emphasis Type="Italic">D</Emphasis>-spaces

In this note, we show that if for any transitive neighborhood assignment φ for X there is a point-countable refinement ? such that for any non-closed subset A of X there is some V ∈ ? such that |V ∩ A| ? ω, then X is transitively D. As a corollary, if X is a sequential space and has a point-countable wcs*-network then X is transitively D, and hence if X is a Hausdorff k-space and has a point-countable k-network, then X is transitively D. We prove that if X is a countably compact sequential space and has a pointcountable wcs*-network, then X is compact. We point out that every discretely Lindelöf space is transitively D. Let (X, τ) be a space and let (X, ?) be a butterfly space over (X, τ). If (X, τ) is Fréchet and has a point-countable wcs*-network (or is a hereditarily meta-Lindelöf space), then (X, ?) is a transitively D-space.     

Topological entropy of sets of generic points for actions of amenable groups

Let G be a countable discrete infinite amenable group which acts continuously on a compact metric space X and let μ be an ergodic G-invariant Borel probability measure on X. For a fixed tempered F?lner sequence {Fn} in G with limn→+∞|Fn|/log n= ∞, we prove the following result:h_top~B(G_μ, {F_n}) = h_μ(X, G),where G_μ is the set of generic points for μ with respect to {F_n} and h_top~B(G_μ, {F_n}) is the Bowen topological entropy(along {F_n}) on G_μ. This generalizes the classical result of Bowen(1973).     

Thom-Sebastiani properties of Kohn-Rossi cohomology of compact connected strongly pseudoconvex CR manifolds

Let X_1 and X_2 be two compact connected strongly pseudoconvex embeddable Cauchy-Riemann(CR) manifolds of dimensions 2m-1 and 2n-1 in C~(m+1)and C~(n+1), respectively. We introduce the ThomSebastiani sum X = X_1 ⊕X_2which is a new compact connected strongly pseudoconvex embeddable CR manifold of dimension 2m+2n+1 in C~(m+n+2). Thus the set of all codimension 3 strongly pseudoconvex compact connected CR manifolds in Cn+1for all n 2 forms a semigroup. X is said to be an irreducible element in this semigroup if X cannot be written in the form X_1 ⊕ X_2. It is a natural question to determine when X is an irreducible CR manifold. We use Kohn-Rossi cohomology groups to give a necessary condition of the above question. Explicitly,we show that if X = X_1 ⊕ X_2, then the Kohn-Rossi cohomology of the X is the product of those Kohn-Rossi cohomology coming from X_1 and X_2 provided that X_2 admits a transversal holomorphic S~1-action.     

A renewal approach to Markovian <Emphasis Type="Italic">U</Emphasis>-statistics

In this paper we describe a novel approach to the study of U-statistics in the Markovian setup based on the (pseudo-) regenerative properties of Harris Markov chains. Exploiting the fact that any sample path X ₁, ..., X _n of a general Harris chain X may be divided into asymptotically i.i.d. data blocks \(\mathcal{B}_1 , \ldots \mathcal{B}_N\) of random length corresponding to successive (pseudo-) regeneration times, we introduce the notion of regenerative U-statistic Ω _N = Σ _k≠l ω _h \(\left( {\mathcal{B}_k ,\mathcal{B}_l } \right)\)/(N(N ? 1)) related to a U-statistic U _n = Σ _i≠j h(X _i , X _j )/(n(n ? 1)). We show that, under mild conditions, these two statistics are asymptotically equivalent up to the order O _?(n?1). This result serves as a basis for establishing limit theorems related to statistics of the same form as U _n . Beyond its use as a technical tool for proving results of a theoretical nature, the regenerative method is also employed here in a constructive fashion for estimating the limiting variance or the sampling distribution of certain U-statistics through resampling. The proof of the asymptotic validity of this statistical methodology is provided, together with an illustrative simulation result.     

Almost everywhere divergence of lacunary subsequences of partial sums of fourier series

If an increasing sequence {n _m } of positive integers and a modulus of continuity ω satisfy the condition Σ _m=1 ^∞ ω(1/n _m )/m < ∞, then it is known that the subsequence of partial sums \(S_{n_m } \left( {f,x} \right)\) converges almost everywhere to f(x) for any function f ∈ H ₁ ^ω . We show that this sufficient convergence condition is close to a necessary condition for a lacunary sequence {n _m }.     

The Tukey Order and Subsets of <Emphasis Type="Italic">ω</Emphasis><Subscript>1</Subscript>

One partially ordered set, Q, is a Tukey quotient of another, P, if there is a map ? : P → Q carrying cofinal sets of P to cofinal sets of Q. Two partial orders which are mutual Tukey quotients are said to be Tukey equivalent. Let X be a space and denote by \(\mathcal {K}(X)\) the set of compact subsets of X, ordered by inclusion. The principal object of this paper is to analyze the Tukey equivalence classes of \(\mathcal {K}(S)\) corresponding to various subspaces S of ω ₁, their Tukey invariants, and hence the Tukey relations between them. It is shown that ω ^ω is a strict Tukey quotient of \({\Sigma }(\omega ^{\omega _{1}})\) and thus we distinguish between two Tukey classes out of Isbell’s ten partially ordered sets from (Isbell, J. R.: J. London Math Society 4(2), 394–416, 1972). The relationships between Tukey equivalence classes of \(\mathcal {K}(S)\), where S is a subspace of ω ₁, and \(\mathcal {K}(M)\), where M is a separable metrizable space, are revealed. Applications are given to function spaces.     

Degeneracy of holomorphic curves in surfaces

LetX be a complex projective algebraic manifold of dimension 2 and let D₁, ..., D_u be distinct irreducible divisors onX such that no three of them share a common point. Let\(f:{\mathbb{C}} \to X\backslash ( \cup _{1 \leqslant i \leqslant u} D_i )\) be a holomorphic map. Assume thatu ? 4 and that there exist positive integers n₁, ... ,n_u,c such that n_in_J D _i.D_j) =c for all pairsi,j. Thenf is algebraically degenerate, i.e. its image is contained in an algebraic curve onX.     

Khinchin’s inequality,Dunford-Pettis and compact operators on the space <Emphasis Type="Italic">C</Emphasis>([0, 1], <Emphasis Type="Italic">X</Emphasis>)

We prove that if X, Y are Banach spaces, Ω a compact Hausdorff space and U:C(Ω, X) → Y is a bounded linear operator, and if U is a Dunford-Pettis operator the range of the representing measure G(Σ) ? DP(X, Y) is an uniformly Dunford-Pettis family of operators and ∥G∥ is continuous at Ø. As applications of this result we give necessary and/or sufficient conditions that some bounded linear operators on the space C([0, 1], X) with values in c ₀ or l _p, (1 ≤ p < ∞) be Dunford-Pettis and/or compact operators, in which, Khinchin’s inequality plays an important role.     

Semistability of Frobenius Direct Image of Representations of Cotangent Bundles

Let k be an algebraically closed field of characteristic p > 0, X a smooth projective variety over k with a fixed ample divisor H, F_X : X → X the absolute Frobenius morphism on X. Let E be a rational GL_n(k)-bundle on X, and ρ : GL_n(k) → GL_m(k) a rational GL_n(k)-representation of degree at most d such that ρ maps the radical RGL_n(k)) of GL_n(k) into the radical R(GL_m(k)) of GL_m(k). We show that if \(F_X^{N*}(E)\) is semistable for some integer \(N \ge {\max {_{0 < r < m}}}(_r^m) \cdot {\log _p}(dr)\), then the induced rational GL_m(k)-bundle E(GL_m(k)) is semistable. As an application, if dimX = n, we get a sufficient condition for the semistability of Frobenius direct image \(F_{X*}(\rho*(\Omega_X^1))\), where \(\rho*(\Omega_X^1)\) is the vector bundle obtained from \(\Omega_X^1\) via the rational representation ρ.     

Rectangular <Emphasis Type="Italic">R</Emphasis>-Transform as the Limit of Rectangular Spherical Integrals

In this paper, we connect rectangular free probability theory and spherical integrals. We prove the analogue, for rectangular or square non-Hermitian matrices, of a result that Guionnet and Maïda proved for Hermitian matrices in (J. Funct. Anal. 222(2):435–490, 2005). More specifically, we study the limit, as n and m tend to infinity, of \(\frac{1}{n}\log\mathbb{E}\{\exp[\sqrt{nm}\theta X_{n}]\}\), where θ∈?, X _n is the real part of an entry of U _n M _n V _m and M _n   is a certain n×m deterministic matrix and U _n and V _m are independent Haar-distributed orthogonal or unitary matrices with respective sizes n×n and m×m. We prove that when the singular law of M _n converges to a probability measure μ, for θ small enough, this limit actually exists and can be expressed with the rectangular R-transform of μ. This gives an interpretation of this transform, which linearizes the rectangular free convolution, as the limit of a sequence of log-Laplace transforms.     

Low distortion embeddings of some metric graphs into Banach spaces

We give a simple example of a countable metric graph M such that M Lipschitz embeds with distortion strictly less than 2 into a Banach space X only if X contains an isomorphic copy of l ₁. Further we show that, for each ordinal α < ω ₁, the space C([0, ω ^α ]) does not Lipschitz embed into C(K) with distortion strictly less than 2 unless K ^(α) ≠ 0. Also \(C\left( {\left[ {0,{\omega ^{{\omega ^\alpha }}}} \right]} \right)\) does not Lipschitz embed into a Banach space X with distortion strictly less than 2 unless Sz(X) ≥ ω ^α+1.     

Searching for a Best Least Absolute Deviations Solution of an Overdetermined System of Linear Equations Motivated by Searching for a Best Least Absolute Deviations Hyperplane on the Basis of Given Data

We consider the problem of searching for a best LAD-solution of an overdetermined system of linear equations Xa=z, X∈?^m×n, m≥n, \(\mathbf{a}\in \mathbb{R}^{n}, \mathbf {z}\in\mathbb{R}^{m}\). This problem is equivalent to the problem of determining a best LAD-hyperplane x?a ^T x, x∈? ⁿ on the basis of given data \((\mathbf{x}_{i},z_{i}), \mathbf{x}_{i}= (x_{1}^{(i)},\ldots,x_{n}^{(i)})^{T}\in \mathbb{R}^{n}, z_{i}\in\mathbb{R}, i=1,\ldots,m\), whereby the minimizing functional is of the form

$F(\mathbf{a})=\|\mathbf{z}-\mathbf{Xa}\|_1=\sum_{i=1}^m|z_i-\mathbf {a}^T\mathbf{x}_i|.$

An iterative procedure is constructed as a sequence of weighted median problems, which gives the solution in finitely many steps. A criterion of optimality follows from the fact that the minimizing functional F is convex, and therefore the point a ^?∈? ⁿ is the point of a global minimum of the functional F if and only if 0∈?F(a ^?).

Motivation for the construction of the algorithm was found in a geometrically visible algorithm for determining a best LAD-plane (x,y)?αx+βy, passing through the origin of the coordinate system, on the basis of the data (x _i ,y _i ,z _i ),i=1,…,m.     

The Law of the Iterated Logarithm for Finitely Inhomogeneous Random Walks

The Hartman–Wintner–Strassen law of the iterated logarithm states that if X ₁, X ₂,… are independent identically distributed random variables and S _n =X ₁+???+X _n , then

$\limsup_{n}S_{n}/\sqrt{2n\log \log n}=1\quad \text{a.s.},\qquad \liminf_{n}S_{n}/\sqrt{2n\log \log n}=-1\quad \text{a.s.}$

if and only if EX ₁ ² =1 and EX ₁=0. We extend this to the case where the X _n are no longer identically distributed, but rather their distributions come from a finite set of distributions.

     

Functional envelope of Cantor spaces

Given a topological dynamical system(X, T), where X is a compact metric space and T a continuous selfmap of X. Denote by S(X) the space of all continuous selfmaps of X with the compactopen topology. The functional envelope of(X, T) is the system(S(X), FT), where FT is defined by FT(?) = T ? ? for any ? ∈ S(X). We show that(1) If(Σ, T) is respectively weakly mixing, strongly mixing, diagonally transitive, then so is its functional envelope, where Σ is any closed subset of a Cantor set and T a selfmap of Σ;(2) If(S(Σ), F_σ) is transitive then it is Devaney chaos, where(Σ, σ) is a subshift of finite type;(3) If(Σ, T) has shadowing property, then(SU(Σ), FT) has shadowing property,where Σ is any closed subset of a Cantor set and T a selfmap of Σ;(4) If(X, T) is sensitive, where X is an interval or any closed subset of a Cantor set and T : X → X is continuous, then(SU(X), FT) is sensitive;(5) If Σ is a closed subset of a Cantor set with infinite points and T : Σ→Σ is positively expansive then the entropy ent U(FT) of the functional envelope of(Σ, T) is infinity.