On semi-R-boundedness and its applications

R-Boundedness is a randomized boundedness condition for sets of operators which in recent years has found many applications in the maximal regularity theory of evolution equations, stochastic evolution equations, spectral theory and vector-valued harmonic analysis. However, in some situations additional geometric properties such as Pisier's property (α) are required to guaranty the R-boundedness of a relevant set of operators. In this paper we show that a weaker property called semi-R-boundedness can be used to avoid these geometric assumptions in the context of Schauder decompositions and the H^∞-calculus. Furthermore, we give weaker conditions for stochastic integrability of certain convolutions.     

Convolutions of singular measures and applications to the Zakharov system

Uniform L²-estimates for the convolution of singular measures with respect to transversal submanifolds are proved in arbitrary space dimension. The results of Bennett-Bez are used to extend previous work of Bejenaru-Herr-Tataru. As an application, it is shown that the 3D Zakharov system is locally well-posed in the full subcritical regime.     

On cocycle twisting of compact quantum groups

We provide a class of examples of compact quantum groups and unitary 2-cocycles on them, such that the twisted quantum groups are non-compact, but still locally compact quantum groups (in the sense of Kustermans and Vaes). This also gives examples of cocycle twists where the underlying C^∗-algebra of the quantum group changes.     

On the complexity of the multicut problem in bounded tree-width graphs and digraphs

Given an edge- or vertex-weighted graph or digraph and a list of source-sink pairs, the minimum multicut problem consists in selecting a minimum weight set of edges or vertices whose removal leaves no path from each source to the corresponding sink. This is a classical NP-hard problem, and we show that the edge version becomes tractable in bounded tree-width graphs if the number of source-sink pairs is fixed, but remains NP-hard in directed acyclic graphs and APX-hard in bounded tree-width and bounded degree unweighted digraphs. The vertex version, although tractable in trees, is proved to be NP-hard in unweighted cacti of bounded degree and bounded path-width.     

A cohomological description of property (T) for quantum groups

We prove a Delorme-Guichardet type theorem for discrete quantum groups expressing property (T) of the quantum group in question in terms of its first cohomology groups. As an application, we show that the first L²-Betti number of a discrete property (T) quantum group vanishes.     

Twisted first homology groups of the automorphism group of a free group

The automorphism group and outer automorphism group of a free group F_n of rank n act on the abelianized group H of F_n and the dual group H^* of H. The twisted first homology groups of and with coefficients in H and H^* are calculated.     

On constants in the Füredi-Hajnal and the Stanley-Wilf conjecture

For a given permutation matrix P, let fP(n) be the maximum number of 1-entries in an n×n(0,1)-matrix avoiding P and let SP(n) be the set of all n×n permutation matrices avoiding P. The Füredi-Hajnal conjecture asserts that cP:=limn→∞fP(n)/n is finite, while the Stanley-Wilf conjecture asserts that is finite.In 2004, Marcus and Tardos proved the Füredi-Hajnal conjecture, which together with the reduction introduced by Klazar in 2000 proves the Stanley-Wilf conjecture.We focus on the values of the Stanley-Wilf limit (sP) and the Füredi-Hajnal limit (cP). We improve the reduction and obtain which decreases the general upper bound on sP from sP?constconstO(klog(k)) to sP?constO(klog(k)) for any k×k permutation matrix P. In the opposite direction, we show .For a lower bound, we present for each k a k×k permutation matrix satisfying cP=Ω(k²).     

Endomorphisms of the shift dynamical system, discrete derivatives, and applications

All continuous endomorphisms f_∞ of the shift dynamical system S on the 2-adic integers Z₂ are induced by some , where n is a positive integer, B_n is the set of n-blocks over {0, 1}, and f_∞(x)=y₀y₁y₂… where for all i∈N, y_i=f(x_ix_i+1…x_i+n−1). Define D:Z₂→Z₂ to be the endomorphism of S induced by the map {(00,0),(01,1),(10,1),(11,0)} and V:Z₂→Z₂ by V(x)=−1−x. We prove that D, V°D, S, and V°S are conjugate to S and are the only continuous endomorphisms of S whose parity vector function is solenoidal. We investigate the properties of D as a dynamical system, and use D to construct a conjugacy from the 3x+1 function T:Z₂→Z₂ to a parity-neutral dynamical system. We also construct a conjugacy R from D to T. We apply these results to establish that, in order to prove the 3x+1 conjecture, it suffices to show that for any m∈Z⁺, there exists some n∈N such that R⁻¹(m) has binary representation of the form or .     

Some applications of free Fisher information on frame theory

We introduce the notion of operator-valued free Fisher information with respect to a positive map of a random variable in an operator-valued noncommutative probability space and point out its close relations to the modular frames arising from conditional expectations. Then we can apply this notion on the study of frame theory, especially on the disjointness problem of modular frames arising from conditional expectations.     

Strong characterizing sequences of countable groups

András Biró and Vera Sós prove that for any subgroup G of T generated freely by finitely many generators there is a sequence A⊆N such that for all β∈T we have (‖.‖ denotes the distance to the nearest integer)

     

On the base dimension I and the property of universality

In Iliadis (2005) [13] for an ordinal α the notion of the so-called (bn-Ind?α)-dimensional normal base C for the closed subsets of a space X was introduced. This notion is defined similarly to the classical large inductive dimension Ind. In this case we shall write here I(X,C)?α and say that the base dimension I of the space X by the normal base C is less than or equal to α. The classical large inductive dimension Ind of a normal space X, the large inductive dimension Ind₀ of a Tychonoff space X defined independently by Charalambous and Filippov, as well as, the relative inductive dimension defined by Chigogidze for a subspace X of a Tychonoff space Y may be considered as the base dimension I of X by normal bases Z(X) (all closed subsets of X), Z(X) (all functionally closed subsets of X), and , respectively.In the present paper, we shall consider normal bases of spaces consisting of functionally closed subsets. In particular, we introduce new dimension invariant : for a space X, is the minimal element α of the class O∪{−1,∞}, where O is the class of all ordinals, for which there exists a normal base C on X consisting of functionally closed subsets such that I(X,C)?α. We prove that in the class of all completely regular spaces X of weight less than or equal to a given infinite cardinal τ such that there exist universal spaces. However, the following questions are open.(1) Are there universal elements in the class of all normal (respectively, of all compact) spaces X of weight ?τ with ?(2) Are there universal elements in the class of all Tychonoff (respectively, of all normal) spaces X of weight ?τ with Ind₀(X)?n∈ω? (Note that for a compact space X.)     

Parseval frames for ICC groups

We analyze Parseval frames generated by the action of an ICC group on a Hilbert space. We parametrize the set of all such Parseval frames by operators in the commutant of the corresponding representation. We characterize when two such frames are strongly disjoint. We prove an undersampling result showing that if the representation has a Parseval frame of equal norm vectors of norm , the Hilbert space is spanned by an orthonormal basis generated by a subgroup. As applications we obtain some sufficient conditions under which a unitary representation admits a Parseval frame which is spanned by a Riesz sequences generated by a subgroup. In particular, every subrepresentation of the left-regular representation of a free group has this property.     

Realizations of AF-algebras as graph algebras, Exel-Laca algebras, and ultragraph algebras

We give various necessary and sufficient conditions for an AF-algebra to be isomorphic to a graph C^∗-algebra, an Exel-Laca algebra, and an ultragraph C^∗-algebra. We also explore consequences of these results. In particular, we show that all stable AF-algebras are both graph C^∗-algebras and Exel-Laca algebras, and that all simple AF-algebras are either graph C^∗-algebras or Exel-Laca algebras. In addition, we obtain a characterization of AF-algebras that are isomorphic to the C^∗-algebra of a row-finite graph with no sinks.     

Fourier analysis on domains in compact groups

Let Ω be a measurable subset of a compact group G of positive Haar measure. Let be a non-negative function defined on the dual space and let L²(μ) be the corresponding Hilbert space which consists of elements (ξπ)_π∈suppμ satisfying , where ξπ is a linear operator on the representation space of π, and is equipped with the inner product: . We show that the Fourier transform gives an isometric isomorphism from L²(Ω) onto L²(μ) if and only if the restrictions to Ω of all matrix coordinate functions , π∈suppμ, constitute an orthonormal basis for L²(Ω). Finally compact connected Lie groups case is studied.     

On holomorphic functions of slow growth on coverings of strongly pseudoconvex manifolds

In this paper we extend some results of the paper [M. Gromov, G. Henkin, M. Shubin, Holomorphic L²-functions on coverings of pseudoconvex manifolds, Geom. Funct. Anal. 8 (1998) 552-585].     

On a generalization of the Evans Conjecture

The Evans Conjecture states that a partial Latin square of order n with at most n-1 entries can be completed. In this paper we generalize the Evans Conjecture by showing that a partial r-multi Latin square of order n with at most n-1 entries can be completed. Using this generalization, we confirm a case of a conjecture of Häggkvist.     

Computations of cohomology groups and nontrivial periodic solutions of Hamiltonian systems

By computing the E-critical groups at θ and infinity of the corresponding functional of Hamiltonian systems, we proved the existence of nontrivial periodic solutions for the systems which may be resonant at θ and infinity under some new conditions. Some results in the literature are extended and some new type of theorems are proved. The main tool is the E-Morse theory developed by Kryszewski and Szulkin.     

On Tutte polynomial uniqueness of twisted wheels

A graph G is called T-unique if any other graph having the same Tutte polynomial as G is isomorphic to G. Recently, there has been much interest in determining T-unique graphs and matroids. For example, de Mier and Noy [A. de Mier, M. Noy, On graphs determined by their Tutte polynomials, Graphs Combin. 20 (2004) 105-119; A. de Mier, M. Noy, Tutte uniqueness of line graphs, Discrete Math. 301 (2005) 57-65] showed that wheels, ladders, Möbius ladders, square of cycles, hypercubes, and certain class of line graphs are all T-unique. In this paper, we prove that the twisted wheels are also T-unique.