On <Emphasis Type="Italic">k</Emphasis>-Spaces and <Emphasis Type="Italic">k</Emphasis><Subscript><Emphasis Type="Italic">R</Emphasis></Subscript>-Spaces

In this note we study the relation between k _R -spaces and k-spaces and prove that a k _R -space with a σ-hereditarily closure-preserving k-network consisting of compact subsets is a k-space, and that a k _R -space with a point-countable k-network consisting of compact subsets need not be a k-space. This work was supported by the NSF of China (10271056).     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-mixed intersection bodies

In this paper the author first introduce a new concept of L _p -dual mixed volumes of star bodies which extends the classical dual mixed volumes. Moreover, we extend the notions of L _p intersection body to L _p -mixed intersection body. Inequalities for L _p -dual mixed volumes of L _p -mixed intersection bodies are established and the results established here provide new estimates for these type of inequalities. This work was supported by the Natural Science Foundation of Zhejiang Province of China (Grant No. Y605065) and the Foundation of the Education Department of Zhejiang Province of China (Grant No. 20050392)     

Joint continuity of <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">h</Emphasis></Subscript><Emphasis Type="Italic">C</Emphasis>-functions with values in moore spaces

We introduce the notion of categorical cliquish mapping and show that, for each K _h C-mapping f: X × Y → Z, where X is a topological space, Y is a space with the first axiom of countability, and Z is a Moore space, with categorical-cliquish horizontal y-sections f _y , the sets C _y (f) are residual G _δ-type sets in X for every y ∈ Y. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 11, pp. 1539–1547, November, 2008.     

Localization of <Emphasis Type="Italic">LM</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-algebras

The aim of this paper is to define the localization LM _n -algebra of an LM _n —algebra L with respect to a topology F on L; in Section 5 we prove that the maximal LM _n -algebra of fractions (defined in [3]) and the LM _n -algebra of fractions relative to an Λ—closed system (defined in Section 2) are LM _n -algebras of localization.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Subscript>-Cohomology and Normal Solvability

We consider some problems concerning the L _p,q -cohomology of Riemannian manifolds. In the first part, we study the question of the normal solvability of the operator of exterior derivation on a surface of revolution M considered as an unbounded linear operator acting from L_p^k (M) into L^k+1_q (M). In the second part, we prove that the first L _p,q-cohomology of the general Heisenberg group is nontrivial, provided that p < q. Received: 17 January 2006 Supported by INTAS (Grant 03–51–3251) and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grants NSh 311.2003.1, NSh 8526.2006.1).     

A note on <Emphasis Type="Italic">p</Emphasis>-groups of order ≤ <Emphasis Type="Italic">p</Emphasis><Superscript>4</Superscript>

In [1], we defined c(G), q(G) and p(G). In this paper we will show that if G is a p-group, where p is an odd prime and |G| ≤ p ⁴, then c(G) = q(G) = p(G). However, the question of whether or not there is a p-group G with strict inequality c(G) = q(G) < p(G) is still open.     

The Characteristic Intersection Property of Line-Free Choquet Simplices in <Emphasis Type="Italic">E</Emphasis><Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

   Abstract. Generalizing the characteristic intersection property of Choquet simplices, it is proved that for line-free convex bodies B ₁ and B ₂ in E ^d , the following conditions are equivalent: (i) there is a line-free convex body B ⊂ E ^d such that every nonempty intersection B ₁ ∩ (v + B ₂ ) , v ∈ E ^d , is a homothetic copy of B , (ii) both B ₁ and B ₂ are Choquet simplices and the nonempty intersections B ₁ ∩ (v + B ₂ ) , v ∈ E ^d , are homothetic copies of a Choquet simplex B . All such triplets B ₁ ,B ₂ ,B are described.     

On noncommutative weighted local ergodic theorems on <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-spaces

In the present paper we consider a von Neumann algebra M with a faithful normal semi-finite trace τ, and {α _t }, a strongly continuous extension to L ^p (M, τ) of a semigroup of absolute contractions on L ¹(M, τ). By means of a non-commutative Banach Principle we prove for a Besicovitch function b and x ∊ L ^p (M, τ), that the averages 1/T ∫₀ ^T b(t)α _t (x)dt converge bilateral almost uniformly in L ^p (M, τ) as T → 0. Communicated by Dénes Petz     

An Algorithm to Construct <Emphasis Type="BoldItalic">k</Emphasis>-Regular <Emphasis Type="BoldItalic">k</Emphasis>-Connected Graphs with the Maximum <Emphasis Type="BoldItalic">k</Emphasis>-Diameter

 Let f(2m,k) be the Maximum k-diameter of k-regular k-connected graphs on 2m vertices. In this paper we give an algorithm and prove that we can construct k-regular k-connected graphs on 2m vertices with the maximum k-diameter using it. We also prove some known results about f(2m,k) and verify that we can get some unknown values of f(2m,k) by our algorithm. Received: December 1, 2000 Final version received: March 12, 2002 Acknowledgments. We thank the referee for many useful suggestions.     

Littlewood–Paley theorem on spaces <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">t</Emphasis>)</Superscript>(ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

We point out that if the Hardy–Littlewood maximal operator is bounded on the space L ^p(t)(ℝ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ, then the well-known characterization of the spaces L ^p (ℝ), 1 < p < ∞, by the Littlewood–Paley theory extends to the space L ^p(t)(ℝ). We show that, for n > 1 , the Littlewood–Paley operator is bounded on L ^p(t) (ℝ ⁿ ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ ⁿ , if and only if p(t) = const. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1709–1715, December, 2008.     

On <Emphasis Type="Italic">l</Emphasis><Subscript>2,<Emphasis Type="Italic">p</Emphasis></Subscript>-circle numbers

Circle numbers are defined to reflect the Euclidean area-content and, for p ≠ 2, suitably defined non-Euclidean circumference properties of the l _2,p -circles, p ∈ [1, ∞]. The resulting function is continuous, increasing, and takes all values from [2, 4]. The actually chosen dual l _2,p -geometry for measuring the arc-length is closely connected with a generalization of the method of indivisibles of Cavalieri and Torricelli in the sense that integrating such arc-lengths means measuring area content. Moreover, this approach enables one to look in a new way into the co-area formula of measure theory which says that integrating Euclidean arc-lengths does not yield area content except for p = 2. The new circle numbers play a natural role, e.g., as norming constants in geometric measure representation formulae for p-generalized uniform probability distributions on l _2,p -circles.     

Division and <Emphasis Type="Italic">k</Emphasis>-th root theorems for <Emphasis Type="Italic">Q</Emphasis>-manifolds

We prove that a locally compact ANR-space X is a Q-manifold if and only if it has the Disjoint Disk Property (DDP), all points of X are homological Z _∞-points and X has the countable-dimensional approximation property (cd-AP), which means that each map f: K → X of a compact polyhedron can be approximated by a map with the countable-dimensional image. As an application we prove that a space X with DDP and cd-AP is a Q-manifold if some finite power of X is a Q-manifold. If some finite power of a space X with cd-AP is a Q-manifold, then X ² and X × [0, 1] are Q-manifolds as well. We construct a countable family χ of spaces with DDP and cd-AP such that no space X ∈ χ is homeomorphic to the Hilbert cube Q whereas the product X × Y of any different spaces X, Y ∈ χ is homeomorphic to Q. We also show that no uncountable family χ with such properties exists. This work was supported by the Slovenian-Ukrainian (Grant No. SLO-UKR 04-06/07)     

<Emphasis Type="Italic">∂</Emphasis>-operations and <Emphasis Type="Italic">H</Emphasis><Subscript><Emphasis Type="Italic">V</Emphasis></Subscript>-fields

The hyperoperations, called theta-operations （δ）, are motivated from the usual property, which the derivative has on the derivation of a product of functions. Using any map on a set, one can define δ-operations. In this paper, we continue our study on the δ-operations on groupoids, rings, fields and vector spaces or on the corresponding hyperstructures. Using δ-operations one obtains, mainly, Hwstructures, which form the largest class of the hyperstructures. For representation theory of hyperstructures, by hypermatrices, one needs special Hv-rings or Hy-fields, so these hyperstructures can be used. Moreover, we study the relation of these δ-structures with other classes of hyperstructures, especially with the Hv-structures.     

<Emphasis Type="Italic">K</Emphasis>th Roots of <Emphasis Type="Italic">p</Emphasis>-Hyponormal Operators are Subscalar Operators of Order 4<Emphasis Type="Italic">k</Emphasis>

In this paper, we consider the special case of the question raised by Halmos (see below). In particular, we show that if T^k is p-hyponormal, then T is a subscalar operator of order 4k. As a corollary, we obtain that if T^k is p-hyponormal and σ(T) has nonempty interior in the plane, then T has a nontrivial invariant subspace.     

Cc （X） Spaces with X Locally Compact

In this paper, we show, among other results, that if X is a [separable] locally compact space X [satisfying the first countability axiom] then the space Cc （X） has countable tightness [if and only if it has bounding tightness] if and only if it is Frechet-Urysohn, if and only if Cc （X） contains a dense （LM） subspace and if and only if X is a-compact.     

Liftable <Emphasis Type="Italic">D</Emphasis><Subscript>4</Subscript>-covers

Let k be an algebraically closed field of characteristic p and let G \hookrightarrow Aut_k(k[[ t ]] ){G \hookrightarrow {\rm Aut}_k(k\left[\kern-0.15em\left[ t \right]\kern-0.15em\right] )} be a faithful action on a local power series ring over k. Let R be a discrete valuation ring of characteristic 0 with residue field k. One asks, whether it is possible to find a faithful action G \hookrightarrow Aut_R(R[[ t ]] ){G \hookrightarrow {\rm Aut}_R(R\left[\kern-0.15em\left[ t \right]\kern-0.15em\right] )} which reduces to the given action, i.e., a lift to characteristic 0. We show that liftable actions exists in the case that G  =  D ₄ and p  =  2. In fact we introduce a family, the supersimple D ₄-actions, which can always be lifted to characteristic 0.     

On the Lower Tail Probabilities of Some Random Sequences in <Emphasis Type="Italic">l</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We investigate the behaviour of the logarithmic small deviation probability of a sequence (σ _n θ _n ) in l _p , 0<p≤∞, where (θ _n ) are i.i.d. random variables and (σ _n ) is a decreasing sequence of positive numbers. In particular, the example σ _n ∼n ^−μ (1+log n)^−ν is studied thoroughly. Contrary to the existing results in the literature, the rate function and the small deviation constant are expressed expli- citly in the present treatment. The restrictions on the distribution of θ ₁ are kept to an absolute minimum. In particular, the usual variance assumption is removed. As an example, the results are applied to stable and Gamma-distributed random variables.     

Consecutive customer losses in regular and oscillating <Emphasis Type="Italic">M</Emphasis><Superscript><Emphasis Type="Italic">X</Emphasis></Superscript>/<Emphasis Type="Italic">G</Emphasis>/1/<Emphasis Type="Italic">n</Emphasis> systems

We derive fast recursions to compute the probability that k or more consecutive customer losses take place during a busy period of a queue, the so called k-CCL probability, for regular and oscillating M ^X /G/1/n systems.     

Reliability formula &; limit law of the failure time of “<Emphasis Type="Italic">m</Emphasis>-consecutive-<Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis>:<Emphasis Type="Italic">F</Emphasis> system”

An “m-consecutive-k-out-of-n:F system” consists of n components ordered on a line; the system fails if and only if there are at least m nonoverlapping runs of k consecutive failed components. In this paper, we give a recursive formula to compute the reliability of such a system. Thereafter, we state two asymptotic results concerning the failure time Z _n of the system. The first result concerns a limit theorem for Z _n when the failure times of components are not necessarily with identical failure distributions. In the second one, we prove that, for an arbitrary common failure distribution of components, the limit system failure distribution is always of the Poisson class.