Graph von Neumann Algebras

In this paper, we will define a graph von Neumann algebra over a fixed von Neumann algebra M, where G is a countable directed graph, by a crossed product algebra = M × _α , where is the graph groupoid of G and α is the graph-representation. After defining a certain conditional expectation from onto its M-diagonal subalgebra we can see that this crossed product algebra is *-isomorphic to an amalgamated free product where = vN(M × _α where is the subset of consisting of all reduced words in {e, e ^–1} and M × _α is a W ^*-subalgebra of as a new graph von Neumann algebra induced by a graph G _e . Also, we will show that, as a Banach space, a graph von Neumann algebra is isomorphic to a Banach space ⊕ where is a certain subset of the set E(G)* of all words in the edge set E(G) of G. The author really appreciates to Prof F. Radulescu and Prof P. Jorgensen for the valuable discussion and kind advice. Also, he appreciates all supports from St. Ambrose Univ.. In particular, he thanks to Prof T. Anderson and Prof V. Vega for the useful conversations and suggestions.     

On Some Properties of Randomly Indexed Sequences of Random Elements

Let be a probability space with a nonatomic measure P and let (S,ρ) be a separable complete metric space. Let {N _n ,n≥1} be an arbitrary sequence of positive-integer valued random variables. Let {F _k ,k≥1} be a family of probability laws and let X be some random element defined on and taking values in (S,ρ). In this paper we present necessary and sufficient conditions under which one can construct an array of random elements {X _n,k ,n,k≥1} defined on the same probability space and taking values in (S,ρ), and such that , and moreover as  n→∞. Furthermore, we consider the speed of convergence to X as n→∞.      

Constructions of p-ary Quadratic Bent Functions

Bent functions have many applications in the fields of coding theory, communications and cryptography. This paper studies the constructions of bent functions having the form for odd n and for even n, over the finite field of odd characteristic p, where . Based on the irreducibility of some polynomials on , we focus on characterizing the bent functions for n=p ^v q ^r and n=2p ^v q ^r , where is an odd prime and p a primitive root modulo q ². Moreover, the enumerations of those functions are also considered. Partially supported by the NSF of China under Grants No. 60603012 and No. 60573053.     

Canonical Representations on Lobachevsky Spaces: An Interaction with an Overalgebra

We define canonical representations R _λ , , for the Lobachevsky space ℒ=G/K of dimension n−1 where G=SO₀(n−1,1), K=SO(n−1), as the restriction to G of maximal degenerate series representations of the overgroup . We determine explicitly the interaction of Lie operators of with operators intertwining canonical representations and representations of G associated with a cone. Supported by the Russian Foundation for Basic Research: grants No. 05-01-00074a and No. 05-01-00001a, the Netherlands Organization for Scientific Research (NWO): grant 047-017-015, the Scientific Program “Devel. Sci. Potent. High. School”: project RNP.2.1.1.351 and Templan No. 1.2.02.     

Boundary Limits for Bounded Quasiregular Mappings

In this paper we establish results on the existence of nontangential limits for weighted -harmonic functions in the weighted Sobolev space , for some q>1 and w in the Muckenhoupt A _q class, where is the unit ball in . These results generalize the ones in Sect. 3 of Koskela et al., Trans. Am. Math. Soc. 348(2), 755–766, 1996, where the weight was identically equal to one. Weighted -harmonic functions are weak solutions of the partial differential equation

The Invariant and Quasi-invariant Transformations of the Stable Lévy Processes

Let ξ(t),t∈[0,1] be a strictly stable Lévy process with the index of stability α∈(0,2). By ℘ _ξ we denote the law of ξ in the Skorokhod space . For arbitrary ξ we construct ℘ _ξ -quasi-invariant semigroup of transformations of . Under some nondegeneracy condition on the spectral measure of the stable process we construct ℘ _ξ -quasi-invariant group of transformations of . In symmetric case this group is a group of the invariant transformations.      

Redefined fuzzy B-algebras

By two relations belonging to and quasi-coincidence (q) between fuzzy points and fuzzy sets, we define the concept of (α, β)-fuzzy subalgebras where α, β are any two of with . We state and prove some theorems in (α, β)-fuzzy B-algebras.     

Nonlinear Parabolic Equations with Regularized Derivatives in Colombeau Algebra

We consider several types of nonlinear parabolic equations with singular like potential and initial data. To prove the existence-uniqueness theorems we employ regularized derivatives. As a framework we use Colombeau space and Colombeau vector space      

Map Preserving Local Properties of a Digital Image

In order to study digital topological properties of a k-surface in Z ⁿ , we generalize the topological number in Bertrand (Pattern Recogn. Lett. 15:1003–1011, 1994). Furthermore, we show that a local (k ₀,k ₁)-isomorphism preserves some digital-topological properties, such as a generalized topological number and a simple k ₀-point, and prove that a local (k ₀,k ₁)-isomorphism takes a simple k ₀-surface in into a simple k ₁-surface in .      

Shellable Complexes and Topology of Diagonal Arrangements

We prove that if a simplicial complex Δ is shellable, then the intersection lattice L _Δ for the corresponding diagonal arrangement is homotopy equivalent to a wedge of spheres. Furthermore, we describe precisely the spheres in the wedge, based on the data of shelling. Also, we give some examples of diagonal arrangements  where the complement is K(π,1), coming from rank-3 matroids. This work forms part of the author’s doctoral dissertation at the University of Minnesota, supervised by Vic Reiner, and partially supported by NSF grant DMS-0245379.     

Inner Product Space and Concept Classes Induced by Bayesian Networks

Bayesian networks have become one of the major models used for statistical inference. In this paper we discuss the properties of the inner product spaces and concept class induced by some special Bayesian networks and the problem whether there exists a Bayesian network such that lower bound on dimensional inner product space just is some positive integer. We focus on two-label classification tasks over the Boolean domain. As main results we show that lower bound on the dimension of the inner product space induced by a class of Bayesian networks without v-structures is where m _i denotes the number of parents for ith variable. As the variable’s number of Bayesian network is n≤5, we also show that for each integer m∈[n+1,2 ⁿ −1] there is a Bayesian network such that VC dimension of concept class and lower bound on dimensional inner product space induced by all are m. This work was supported by National Natural Science Foundation of China (60574075).     

On the rate of convergence of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> norms in the CLT for poisson and gamma random variables

In the paper, we present upper bounds of L _p norms of order ( X)^-1/2 for all 1 ≤ p ≤ ∞ in the central limit theorem for a standardized random variable (X− X)/ √ X, where a random variable X is distributed by the Poisson distribution with parameter λ > 0 or by the standard gamma distribution Γ(α, 0, 1) with parameter α > 0. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-70/09.     

Characterization of the Newtonian Free Particle System in m\geqslant 2 Dependent Variables

We treat the problem of linearizability of a system of second order ordinary differential equations. The criterion we provide has applications to nonlinear Newtonian mechanics, especially in three-dimensional space. Let or , let , let , let and let

On automorphisms of A-groups

Let G be an A-group (i.e. a group in which xx ^α  = x ^α x for all and let denote the subgroup of Aut(G) consisting of all automorphisms that leave invariant the centralizer of each element of G. The quotient is an elementary abelian 2-group and natural analogies exist to suggest that it might always be trivial. It is shown that, in fact, for any odd prime p and any positive integer r, there exist infinitely many finite pA-groups G for which has rank r. Received: 23 March 2008, Revised: 20 May 2008     

Representations of Some Hopf Algebras Associated to the Symmetric Group S n

We prove that all irreducible representations of the bismash product have Frobenius–Schur indicator +1, where k is an algebraically closed field of characteristic 0. If n = p, a prime, we find all indicators for . We also study more general bismash products. Both authors were supported by NSF grants DMS-07-01291 and DMS-04-01399.     

Regularity, Local and Microlocal Analysis in Theories of Generalized Functions

We introduce a general context involving a presheaf and a subpresheaf ℬ of  . We show that all previously considered cases of local analysis of generalized functions (defined from duality or algebraic techniques) can be interpretated as the ℬ-local analysis of sections of  . But the microlocal analysis of the sections of sheaves or presheaves under consideration is dissociated into a “frequential microlocal analysis” and into a “microlocal asymptotic analysis”. The frequential microlocal analysis based on the Fourier transform leads to the study of propagation of singularities under only linear (including pseudodifferential) operators in the theories described here, but has been extended to some non linear cases in classical theories involving Sobolev techniques. The microlocal asymptotic analysis is a new spectral study of singularities. It can inherit from the algebraic structure of ℬ some good properties with respect to nonlinear operations.      

Multiple positive solutions for time scale boundary value problems on infinite intervals

The study of dynamic equations on time scales is an area of mathematics. It has been created in order to unify the study of differential and difference equations. In this paper, we consider the time-scale boundary value problems

Symmetric Tensors And Geometry of {\mathbb{P}} ^N Subvarieties

This paper using a geometric approach produces vanishing and nonvanishing results concerning the spaces of twisted symmetric differentials on subvarieties , with k ≤ m. Emphasis is given to the case of k = m which is special and whose nonvanishing results on the dimensional range dim X > 2/3(N − 1) are related to the space of quadrics containing X and the variety of all tangent trisecant lines of X. The paper ends with an application showing that the twisted symmetric plurigenera, along smooth families of projective varieties X_t are not invariant even for α arbitrarily large. Received: September 2006, Revision: May 2007, Accepted: June 2007     

Optimisation of Measures on a Hyperfinite Adapted Probability Space

The minimisation problem for a functional is considered, where is an ℝ ⁿ -valued stochastic process, defined on some filtered probability space , and P is an admissible probability measure in the sense that it obeys (1) some uniform equivalence condition with respect to the given measure ℙ on Γ, and (2) a finite number (possibly zero) of arbitrarily given other conditions that require the expectation (with respect to P) of some continuous bounded function φ of , for t ₁,…,t _k ∈[0,1], to lie within some closed set. We assume that u can be formulated through finite compositions of conditional expectations and bounded continuous functions. Under the assumption of |φ| being uniformly bounded from below and some condition on the dimension of , the existence of a solution on hyperfinite adapted probability spaces, as well as its minimality among admissible measures on any other adapted probability space, is proven. Also, a coarseness result for the Loeb operation is established. The main result of this paper, however, is a “standard result”: It does not include any reference to nonstandard analysis and can be perfectly understood without any familiarity with nonstandard analysis.      

Regularity of the Extremal Solution of Some Nonlinear Elliptic Problems Involving the <Emphasis Type="Italic">p</Emphasis>-Laplacian

We consider the equation on a smooth bounded domain of with zero Dirichlet boundary conditions where p ≥ 2, λ > 0 and f satisfies typical assumptions in the subject of extremal solutions. We prove that, for such general nonlinearities f, the extremal solution u ^* belongs to L ^∞(Ω) if N < p + p/(p − 1) and if N < p(1 + p/(p − 1)). This work was partially supported by MCyT BMF 2002-04613-CO3-02.