Decomposition rank and absorbing extensions of type I algebras

We prove the following: Let A and B be separable C^*-algebras. Suppose that B is a type I C^*-algebra such that

(i)

B has only infinite dimensional irreducible *-representations, and

(ii)

B has finite decomposition rank.

If

0→B→C→A→0     

Binary relations and reduced hypergroups

Different partial hypergroupoids are associated with binary relations defined on a set H. In this paper we find sufficient and necessary conditions for these hypergroupoids in order to be reduced hypergroups. Given two binary relations ρ and σ on H we investigate when the hypergroups associated with the relations ρ∩σ, ρ∪σ and ρσ are reduced. We also determine when the cartesian product of two hypergroupoids associated with a binary relation is a reduced hypergroup.     

Remark on uniqueness of mild solutions to the Navier-Stokes equations

We investigate a limiting uniqueness criterion to the Navier-Stokes equations. We prove that the mild solution is unique under the class , where bmo^-1 is the “critical” space including Lⁿ. As an application of uniqueness theorem, we also consider the local well-posedness of Navier-Stokes equations in bmo^-1.     

Minimum monopoly in regular and tree graphs

In this paper we consider a graph optimization problem called minimum monopoly problem, in which it is required to find a minimum cardinality set S⊆V, such that, for each u∈V, |N[u]∩S|?|N[u]|/2 in a given graph G=(V,E). We show that this optimization problem does not have a polynomial-time approximation scheme for k-regular graphs (k?5), unless P=NP. We show this by establishing two L-reductions (an approximation preserving reduction) from minimum dominating set problem for k-regular graphs to minimum monopoly problem for 2k-regular graphs and to minimum monopoly problem for (2k-1)-regular graphs, where k?3. We also show that, for tree graphs, a minimum monopoly set can be computed in linear time.     

A class of essential representations of product systems

We construct a family of essential representations of an arbitrary product system by generalizing some techniques introduced by M. Skeide and W. Arveson. We then classify the resulting E₀-semigroups up to conjugacy, by identifying their tail flows as periodic W^∗-dynamical systems acting on factors of type I_∞. The conjugacy classes of these E₀-semigroups correspond to the orbits of the action of the automorphism group of the product system on unital vectors. In the sequel, this classification shows explicitly that any E₀-semigroup admits uncountably many non-conjugate cocycle perturbations.     

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).     

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.     

On upper and lower bounds of higher order derivatives for solutions to the 2D micropolar fluid equations

The present paper is concerned with asymptotic behaviours of the solutions to the micropolar fluid motion equations in R². Upper and lower bounds are derived for the L² decay rates of higher order derivatives of solutions to the micropolar fluid flows. The findings are mainly based on the basic estimates of the linearized micropolar fluid motion equations and generalized Gronwall type argument.     

On symmetric graphs of valency five

A graph X, with a subgroup G of the automorphism group of X, is said to be (G,s)-transitive, for some s≥1, if G is transitive on s-arcs but not on (s+1)-arcs, and s-transitive if it is -transitive. Let X be a connected (G,s)-transitive graph, and G_v the stabilizer of a vertex v∈V(X) in G. If X has valency 5 and G_v is solvable, Weiss [R.M. Weiss, An application of p-factorization methods to symmetric graphs, Math. Proc. Camb. Phil. Soc. 85 (1979) 43-48] proved that s≤3, and in this paper we prove that G_v is isomorphic to the cyclic group Z₅, the dihedral group D₁₀ or the dihedral group D₂₀ for s=1, the Frobenius group F₂₀ or F₂₀×Z₂ for s=2, or F₂₀×Z₄ for s=3. Furthermore, it is shown that for a connected 1-transitive Cayley graph of valency 5 on a non-abelian simple group G, the automorphism group of is the semidirect product , where R(G) is the right regular representation of G and .     

Global existence for the one-dimensional second order semilinear hyperbolic equations

The aim of the paper is to study necessary and sufficient conditions for the existence of the global solution of the one-dimensional semilinear equation appearing in the boundary value problems of gas dynamics. We investigate the Cauchy problem for such equation in the domain where the operator is weakly hyperbolic. We obtain the necessary condition for the existence of the self-similar solutions for the semilinear Gellerstedt-type equation. The approach used in the paper is based on the fundamental solution of the linear Gellerstedt operator and the Lp-Lq estimates.     

Further improved recursions for a class of compound Poisson distributions

In the present paper we develop more efficient recursive formulae for the evaluation of the t-order cumulative function Γ^th(x) and the t-order tail probability Λ^th(x) of the class of compound Poisson distributions in the case where the derivative of the probability generating function of the claim amounts can be written as a ratio of two polynomials. These efficient recursions can be applied for the exact evaluation of the probability function (given by De Pril [De Pril, N., 1986a. Improved recursions for some compound Poisson distributions. Insurance Math. Econom. 5, 129-132]), distribution function, tail probability, stop-loss premiums and t-order moments of stop-loss transforms of compound Poisson distributions. Also, efficient recursive algorithms are given for the evaluation of higher-order moments and r-order factorial moments about any point for this class of compound Poisson distributions. Finally, several examples of discrete claim size distributions belonging to this class are also given.     

Property (w) and perturbations II

This note is a continuation of a previous article [P. Aiena, M.T. Biondi, Property (w) and perturbations, J. Math. Anal. Appl. 336 (2007) 683-692] concerning the stability of property (w), a variant of Weyl's theorem, for a bounded operator T acting on a Banach space, under finite-dimensional perturbations K commuting with T. A counterexample shows that property (w) in general is not preserved under finite-dimensional perturbations commuting with T, also under the assumption that T is a-isoloid.     

Construction of a high order fluid-structure interaction solver

Accuracy is critical if we are to trust simulation predictions. In settings such as fluid-structure interaction, it is all the more important to obtain reliable results to understand, for example, the impact of pathologies on blood flows in the cardiovascular system. In this paper, we propose a computational strategy for simulating fluid structure interaction using high order methods in space and time.First, we present the mathematical and computational core framework, Life, underlying our multi-physics solvers. Life is a versatile library allowing for 1D, 2D and 3D partial differential solves using h/p type Galerkin methods. Then, we briefly describe the handling of high order geometry and the structure solver. Next we outline the high-order space-time approximation of the incompressible Navier-Stokes equations and comment on the algebraic system and the preconditioning strategy. Finally, we present the high-order Arbitrary Lagrangian Eulerian (ALE) framework in which we solve the fluid-structure interaction problem as well as some initial results.     

Unitary part of a contraction

For a contraction A on a Hilbert space H, we define the index j(A) (resp., k(A)) as the smallest nonnegative integer j (resp., k) such that ker(I−Aj∗Aj) (resp., ker(I−Ak*Ak)∩ker(I−AkAk∗)) equals the subspace of H on which the unitary part of A acts. We show that if , then j(A)?n (resp., k(A)?⌈n/2⌉), and the equality holds if and only if A is of class Sn (resp., one of the three conditions is true: (1) A is of class Sn, (2) n is even and A is completely nonunitary with ‖An−2‖=1 and ‖An−1‖<1, and (3) n is even and A=U⊕A^′, where U is unitary on a one-dimensional space and A^′ is of class Sn−1).     

A degree sum condition for graphs to be prism hamiltonian

Win, in 1975, and Jackson and Wormald, in 1990, found the best sufficient conditions on the degree sum of a graph to guarantee the properties of “having a k-tree” and “having a k-walk”, respectively. The property of “being prism hamiltonian” is an intermediate property between “having a 2-tree” and “having a 2-walk”. Thus, it is natural to ask what is the best degree sum condition for graphs to be prism hamiltonian. As an answer to this problem, in this paper, we show that a connected graph G of order n with σ₃(G)≥n is prism hamiltonian. The degree sum condition “σ₃(G)≥n” is best possible.     

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 .     

Mapping properties for oscillatory integrals in d-dimensions

For aj,bj?1, j=1,2,…,d, we prove that the operator maps into itself for , where , and k(x,y)=φ(x,y)eig(x,y), φ(x,y) satisfies (1.2) (e.g. φ(x,y)=|x−y|^iτ,τ real) and the phase g(x,y)=xa⋅yb. We study operators with more general phases and for these operators we require that aj,bj>1, j=1,2,…,d, or al=bl?1 for some l∈{1,2,…,d}.