Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups

We study equivalence relations and II₁ factors associated with (quotients of) generalized Bernoulli actions of Kazhdan groups. Specific families of these actions are entirely classified up to isomorphism of II₁ factors. This yields explicit computations of outer automorphism and fundamental groups. In particular, every finitely presented group is concretely realized as the outer automorphism group of a continuous family of non stably isomorphic II₁ factors.     

Variations on Weyl's theorem

In this note we study the property (w), a variant of Weyl's theorem introduced by Rako?evi?, by means of the localized single-valued extension property (SVEP). We establish for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which property (w) holds. We also relate this property with Weyl's theorem and with another variant of it, a-Weyl's theorem. We show that Weyl's theorem, a-Weyl's theorem and property (w) for T (respectively T^*) coincide whenever T^* (respectively T) satisfies SVEP. As a consequence of these results, we obtain that several classes of commonly considered operators have property (w).     

Turán's theorem for pseudo-random graphs

The generalized Turán number ex(G,H) of two graphs G and H is the maximum number of edges in a subgraph of G not containing H. When G is the complete graph Km on m vertices, the value of ex(Km,H) is , where o(1)→0 as m→∞, by the Erd?s-Stone-Simonovits theorem.In this paper we give an analogous result for triangle-free graphs H and pseudo-random graphs G. Our concept of pseudo-randomness is inspired by the jumbled graphs introduced by Thomason [A. Thomason, Pseudorandom graphs, in: Random Graphs '85, Poznań, 1985, North-Holland, Amsterdam, 1987, pp. 307-331. MR 89d:05158]. A graph G is (q,β)-bi-jumbled if

     

Completing an extension of Tunnell's theorem

We look at a special case of a familiar problem: Given a locally compact group G, a subgroup H and a complex representation π₊ of G how does π₊ decompose on restriction to H. Here G is GL⁺(2,F), where F is a nonarchimedian local field of characteristic not two, K a separable quadratic extension of F, GL⁺(2,F) the subgroup of index 2 in GL(2,F) consisting of those matrices whose determinant is in NK/F(K^∗), π₊ is an irreducible, admissible supercuspidal representation of GL⁺(2,F) and H=K^∗ under an embedding of K^∗ into GL(2,F).     

Maximal ergodic theorems for some group actions

We prove maximal ergodic inequalities for a sequence of operators and for their averages in the noncommutative Lp-space. We also obtain the corresponding individual ergodic theorems. Applying these results to actions of a free group on a von Neumann algebra, we get noncommutative analogues of maximal ergodic inequalities and pointwise ergodic theorems of Nevo-Stein.     

The substitution theorem for semilinear stochastic partial differential equations

In this article we establish a substitution theorem for semilinear stochastic evolution equations (see's) depending on the initial condition as an infinite-dimensional parameter. Due to the infinite-dimensionality of the initial conditions and of the stochastic dynamics, existing finite-dimensional results do not apply. The substitution theorem is proved using Malliavin calculus techniques together with new estimates on the underlying stochastic semiflow. Applications of the theorem include dynamic characterizations of solutions of stochastic partial differential equations (spde's) with anticipating initial conditions and non-ergodic stationary solutions. In particular, our result gives a new existence theorem for solutions of semilinear Stratonovich spde's with anticipating initial conditions.     

Turán’s theorem inverted

In this note we complete an investigation started by Erd?s in 1963 that aims to find the strongest possible conclusion from the hypothesis of Turán’s theorem in extremal graph theory.Let be the complete r-partite graph with parts of sizes s₁≥2,s₂,…,s_r with an edge added to the first part. Letting t_r(n) be the number of edges of the r-partite Turán graph of order n, we prove that:For all r≥2 and all sufficiently small c>0, every graph of sufficiently large order n with t_r(n)+1 edges contains a .We also give a corresponding stability theorem and two supporting results of wider scope.     

A new proof of convergence of MCMC via the ergodic theorem

A key result underlying the theory of MCMC is that any η-irreducible Markov chain having a transition density with respect to η and possessing a stationary distribution π is automatically positive Harris recurrent. This paper provides a short self-contained proof of this fact using the ergodic theorem in its standard form as the most advanced tool.     

Applications of operator identities to the multiple q-binomial theorem and q-Gauss summation theorem

In this paper, we first give an interesting operator identity. Furthermore, using the q-exponential operator technique to the multiple q-binomial theorem and q-Gauss summation theorem, we obtain some transformation formulae and summation theorems of multiple basic hypergeometric series.     

The rate of quasi sure convergence in the functional limit theorem for increments of a Brownian motion

We investigate the quasi sure convergence of the functional limit for increments of a Brownian motion. The rate of quasi sure convergence in the functional limit for increments of a d-dimensional Brownian motion is derived. The main tool in the proof is large deviation and small deviation for Brownian motion in terms of (r,p)-capacity.     

Colorings of k-balanced matrices and integer decomposition property of related polyhedra

We show that a class of polyhedra, arising from certain 0,1 matrices introduced by Truemper and Chandrasekaran, has the integer decomposition property. This is accomplished by proving certain coloring properties of these matrices.     

Common fixed point theorem for hybrid generalized multi-valued contraction mappings

In this paper, we introduce the notion of a hybrid generalized multi-valued contraction mapping and establish the common fixed point theorem for this mapping. Our results generalize, unify, extend and complement several common fixed point theorems of many authors in the literature.     

A new construction of vertex algebras and quasi-modules for vertex algebras

In this paper, a new construction of vertex algebras from more general vertex operators is given and a notion of quasimodule for vertex algebras is introduced and studied. More specifically, a notion of quasilocal subset(space) of for any vector space W is introduced and studied, generalizing the notion of usual locality in the most possible way, and it is proved that on any maximal quasilocal subspace there exists a natural vertex algebra structure and that any quasilocal subset of generates a vertex algebra. Furthermore, it is proved that W is a quasimodule for each of the vertex algebras generated by quasilocal subsets of . A notion of Γ-vertex algebra is also introduced and studied, where Γ is a subgroup of the multiplicative group C^× of nonzero complex numbers. It is proved that any maximal quasilocal subspace of is naturally a Γ-vertex algebra and that any quasilocal subset of generates a Γ-vertex algebra. It is also proved that a Γ-vertex algebra exactly amounts to a vertex algebra equipped with a Γ-module structure which satisfies a certain compatibility condition. Finally, two families of examples are given, involving twisted affine Lie algebras and certain quantum torus Lie algebras.     

A Morita theorem for dual operator algebras

We prove that two dual operator algebras are weak^∗ Morita equivalent in the sense of [D.P. Blecher, U. Kashyap, Morita equivalence of dual operator algebras, J. Pure Appl. Algebra 212 (2008) 2401-2412] if and only if they have equivalent categories of dual operator modules via completely contractive functors which are also weak^∗-continuous on appropriate morphism spaces. Moreover, in a fashion similar to the operator algebra case, we characterize such functors as the module normal Haagerup tensor product with an appropriate weak^∗ Morita equivalence bimodule. We also develop the theory of the W^∗-dilation, which connects the non-selfadjoint dual operator algebra with the W^∗-algebraic framework. In the case of weak^∗ Morita equivalence, this W^∗-dilation is a W^∗-module over a von Neumann algebra generated by the non-selfadjoint dual operator algebra. The theory of the W^∗-dilation is a key part of the proof of our main theorem.     

Faulhaber’s theorem on power sums

We observe that the classical Faulhaber’s theorem on sums of odd powers also holds for an arbitrary arithmetic progression, namely, the odd power sums of any arithmetic progression a+b,a+2b,…,a+nb is a polynomial in na+n(n+1)b/2. While this assertion can be deduced from the original Fauhalber’s theorem, we give an alternative formula in terms of the Bernoulli polynomials. Moreover, by utilizing the central factorial numbers as in the approach of Knuth, we derive formulas for r-fold sums of powers without resorting to the notion of r-reflective functions. We also provide formulas for the r-fold alternating sums of powers in terms of Euler polynomials.     

Non-simultaneous blow-up and blow-up rates for reaction-diffusion equations

This paper considers blow-up solutions for reaction-diffusion equations, complemented by homogeneous Dirichlet boundary conditions. It is proved that there exist initial data such that one block or two (separated or contiguous) blocks of n components blow up simultaneously while the others remain bounded. As a corollary, a necessary and sufficient condition is obtained such that any blow-up must be the case for at least two components blowing up simultaneously. We also show some other exponent regions, where any blow-up of k(∈{1,2,…,n}) components must be simultaneous. Moreover, the corresponding blow-up rates and sets are discussed. The results extend those in Liu and Li [B.C. Liu, F.J. Li, Non-simultaneous blow-up of n components for nonlinear parabolic systems, J. Math. Anal. Appl. 356 (2009) 215-231].     

Conservation laws for some compacton equations using the multiplier approach

This paper is an application of the variational derivative method to the derivation of the conservation laws for partial differential equations. The conservation laws for (1+1) dimensional compacton k(2,2) and compacton k(3,3) equations are studied via multiplier approach. Also the conservation laws for (2+1) dimensional compacton Zk(2,2) equation are established by first computing the multipliers.     

Halphen pencils on quartic threefolds

For any smooth quartic threefold in P⁴ we classify pencils on it whose general element is an irreducible surface birational to a surface of Kodaira dimension zero.