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.     

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 the value distribution of <Emphasis Type="Italic">ƒ</Emphasis> + <Emphasis Type="Italic">a</Emphasis>(<Emphasis Type="Italic">ƒ</Emphasis>′)<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let ƒ be a transcendental meromorphic function, a a nonzero finite complex number, and n ⩾ 2 a positive integer. Then ƒ + a(ƒ′) ⁿ assumes every complex value infinitely often. This answers a question of Ye for n = 2. A related normality criterion is also given. This work was supported by the National Natural Science Foundation of China (Grant No. 10771076), the Natural Science Foundation of Guangdong Province, China (Grant No. 07006700) and by the German-Israeli Foundation for Scientific Research and Development (Grant No. G-809-234.6/2003)     

A hybrid calculus for logic <Emphasis Type="Italic">N</Emphasis><Superscript><Emphasis Type="Italic">*</Emphasis></Superscript>: residual finiteness and decidability

It is proved that logic N ^* is residually finite and decidable. A hybrid calculus for the logic is constructed based on a tabular calculus for intuitionistic logic. It is shown that the hybrid calculus is sound and complete.     

Harnack estimate for the <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">k</Emphasis></Superscript>-flow

We obtain the Harnack estimate of the solution to Hk-flow in Euclidean space Rn 1,for k ＞ 0.By using this estimate,we get some corollaries about the translation soliton.     

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)     

SOME NEW PROPERTIES ON L^p（Ω, H^n）

Let L^p（Ω, H^n） indicate the L^P-space of the maps for Heisenberg group target. In this paper some new properties are obtained for the space L^p（Ω, H^n）     

A generalization of Hardy spaces <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> by using atoms

Let X = （X, d,μ） The purpose of this paper is to be a space of homogeneous type in the sense of Coifman and Weiss. generalize the definition of Hardy space H^P（X） and prove that the generalized Hardy spaces have the same property as H^P（X）. Our definition includes a kind of Hardy- Orlicz spaces and a kind of Hardy spaces with variable exponent. The results are new even for the R^n case. Let （X, δ, μ） be the normalized space of （X, d, μ） in the sense of Macias and Segovia. We also study the relations of our function spaces for （X, d, μ） and （X, δ,μ）.     

Two sided norm estimate of the Bergman projection on <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

We give some explicit values of the constants C ₁ and C ₂ in the inequality C ₁/sin(π/p) ⩽ |P| _p ⩽ C ₂/sin(π/p) where |P| _p denotes the norm of the Bergman projection on the L ^p space. Partially supported by MNZZS Grant No. ON144010.     

Summation of Fourier-Laplace Series in the Space <Emphasis Type="Italic">L</Emphasis>(<Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>)

We establish estimates for the rate of convergence of a group of deviations on a sphere in the space L(S ^m ), m ≥ 3. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 4, pp. 496–504, April, 2005.     

Piercing Families of Convex Sets with the <Emphasis Type="Italic">d</Emphasis>-Intersection Property in ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

In this paper, we consider finite families of convex sets in ℝ ^d such that every d or fewer sets of the family have a common point. For some families of this type, we give upper bounds on the size of a finite set intersecting all sets of the family. This research was supported by the Russian Foundation for the Basic Research Grants No. 03-01-00801 and 06-01-00648, and by the President of the Russian Federation Grant No. MK-5724.2006.1.     

The first <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-cohomology of some groups with one end

Let p be a real number greater than one. In this paper we study the vanishing and nonvanishing of the first L ^p -cohomology space of some groups that have one end. We also make a connection between the first L ^p -cohomolgy space and the Floyd boundary of the Cayley graph of a group. We apply the result about Floyd boundaries to show that there exists a real number p such that the first L ^p -cohomology space of a nonelementary hyperbolic group does not vanish. Received: 4 August 2006 Revised: 2 November 2006     

The Structure of the Closure of the Rational Functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>(μ)

Let K be a compact subset in the complex plane and let A(K) be the uniform closure of the functions continuous on K and analytic on K°. Let μ be a positive finite measure with its support contained in K. For 1 ≤ q < ∞, let A^q(K, μ) denote the closure of A(K) in L^q(μ). The aim of this work is to study the structure of the space A^q(K, μ). We seek a necessary and sufficient condition on K so that a Thomson-type structure theorem for A^q(K, μ) can be established. Our theorem deduces J. Thomson’s structure theorem for P^q(μ), the closure of polynomials in L^q(μ), as the special case when K is a closed disk containing the support of μ.     

Pointwise behaviour of <Emphasis Type="Italic">M</Emphasis><Superscript>1,1</Superscript> Sobolev functions

Our main objective is to study Haj?asz type Sobolev functions with the exponent one on metric measure spaces equipped with a doubling measure. We show that a discrete maximal function is bounded in the Haj?asz space with the exponent one. This implies that every such function has Lebesgue points outside a set of capacity zero. We also show that every Haj?asz function coincides with a Hölder continuous Haj?asz function outside a set of small Hausdorff content. Our proofs are based on Sobolev space estimates for maximal functions.     

On Optimality Conditions for Some Nonsmooth Optimization Problems over <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Spaces

The paper deals with the minimization of an integral functional over an L^p space subject to various types of constraints. For such optimization problems, new necessary optimality conditions are derived, based on several concepts of nonsmooth analysis. In particular, we employ the generalized differential calculus of Mordukhovich and the fuzzy calculus of proximal subgradients. The results are specialized to nonsmooth two-stage and multistage stochastic programs.The authors express their gratitude to Boris Mordukhovich (Detroit) for his extensive support during this research and to Marian Fabian (Prague) and Alexander Kruger (Ballarat) for valuable discussions. They are indebted also to two anonymous referees for helpful suggestions.The research of this author was partly supported by Grant 1075005 of the Czech Academy of SciencesThe research of this author was supported by the Deutsche Forschungsgemeinschaft     

On the completeness of algebraic polynomials in the spaces <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(ℝ, <Emphasis Type="Italic">d</Emphasis>μ)

We prove that the theorem on the incompleteness of polynomials in the space C ₀ ^w established by de Branges in 1959 is not true for the space L _p (ℝ, dμ) if the support of the measure μ is sufficiently dense.     

The Dichotomy Problem for Orbital Measures of <Emphasis Type="Italic">SU</Emphasis>(<Emphasis Type="Italic">n</Emphasis>)

For many orbital measures μ, on SU(n), we show that either μ^k ∈ L² or μ^k is singular to L¹. The least k for which μ^k ∈ L² is determined and is shown to be the minimum k for which the k-fold product of the conjugacy class supporting the measure has positive measure. It would be interesting to know if all orbital measures satisfy this dichotomy.     

On fuzzy semi δ-<Emphasis Type="Italic">V</Emphasis> continuity in fuzzy δ-<Emphasis Type="Italic">V</Emphasis> topological space

New concepts of fuzzy semi δ-V and fuzzy semi δ-Λ sets were introduced in our work “On fuzzy semi δ-Λ sets and fuzzy semi δ-V sets V-6,” J. Trip. Math. Soc., 6, 81–88 (2004). It was shown that the family of all fuzzy semi δ-V sets forms a fuzzy supra topological space on X denoted by (X, FS ^δV ). The aim of this paper is to introduce the concept of fuzzy semi δ-V continuity in a fuzzy δ-V topological space. Finally, some properties, preservation theorems, etc., are studied. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 5, pp. 712–717, May, 2008.     

On elements of order <Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">s</Emphasis></Superscript> in the plane Cremona group over a field of characteristic <Emphasis Type="Italic">p</Emphasis>

We show that the plane Cremona group over a field of characteristic p > 0 does not contain elements of order of power of p larger than 2. We also describe conjugacy classes of elements of order p ². To the memory of Vasily Iskovskikh