Intersections of Leray complexes and regularity of monomial ideals

For a simplicial complex X and a field K, let .It is shown that if X,Y are complexes on the same vertex set, then for k?0

     

Hyperbolic dynamics in graph-directed IFS

A cone space is a complete metric space (X,d) with a pair of functions cs,cu:X×X→R, such that there exists K>0 satisfying

     

Normality and modulability indices. Part II: Convex cones in Hilbert spaces

Let K be a closed convex cone in a Hilbert space X. Let BX be the closed unit ball of X and K_•=(BX+K)∩(BX−K). The normality index

     

An asymptotic formula for the ranks of homotopy groups

Let X be a finite simply connected CW complex of dimension n. The loop space homology H_∗(ΩX;Q) is the universal enveloping algebra of a graded Lie algebra LX isomorphic with π∗−1(X)⊗Q. Let QX⊂LX be a minimal generating subspace, and set .Theorem: If dimLX=∞ and , then

     

The local variational principle of topological pressure for sub-additive potentials

Let (X,T) be a topological dynamical system and be a sub-additive potential on C(X,R). Let U be an open cover of X. Then for any T-invariant measure μ, let . The topological pressure for open covers U is defined for sub-additive potentials. Then we have a variational principle:

     

Approximation of the Hausdorff distance by the distance of continuous surjections

The aim of this paper is to answer the following question: let (X,?) and (Y,d) be metric spaces, let A,B⊂Y be continuous images of the space X and let be a fixed continuous surjection. When is the inequality

     

Distances from selectors to spaces of Baire one functions

Given a metric space X and a Banach space (E,‖⋅‖) we study distances from the set of selectors Sel(F) of a set-valued map to the space B₁(X,E) of Baire one functions from X into E. For this we introduce the d-τ-semioscillation of a set-valued map with values in a topological space (Y,τ) also endowed with a metric d. Being more precise we obtain that

     

An extension of the Vu-Sine theorem and compact-supercyclicity

If (Tt)_t?0 is a bounded C₀-semigroup in a Banach space X and there exists a compact subset K⊆X such that

     

Random martingales and localization of maximal inequalities

Let (X,d,μ) be a metric measure space. For ∅≠R⊆(0,∞) consider the Hardy-Littlewood maximal operator

     

Disjointness preserving operators between little Lipschitz algebras

Given a real number α∈(0,1) and a metric space (X,d), let Lipα(X) be the algebra of all scalar-valued bounded functions f on X such that

     

Functional calculi for convolution operators on a discrete, periodic, solvable group

Suppose T is a bounded self-adjoint operator on the Hilbert space L²(X,μ) and let

     

Precise rates in complete moment convergence for ρ-mixing sequences

Let X₁,X₂,… be a strictly stationary sequence of ρ-mixing random variables with mean zeros and positive, finite variances, set Sn=X₁+?+Xn. Suppose that , , where q>2δ+2. We prove that, if for any 0<δ?1, then

     

Canonical conservative state/signal shift realizations of passive discrete time behaviors

A passive linear discrete time invariant s/s (state/signal) system Σ=(V;X,W) consists of a Hilbert (state) space X, a Kre?n (signal) space W, a maximal nonnegative (generating) subspace V of the Kre?n space K:=−X[?]X[?]W. The sets of trajectories (x(⋅);w(⋅)) generated by V on the discrete time intervals I⊂Z are defined by

     

Continuity modulus of stochastic homeomorphism flows for SDEs with non-Lipschitz coefficients

Let Xt(x) solve the following Itô-type SDE (denoted by EQ.(σ,b,x)) in Rd

     

Inducing sensitivity on hyperspaces

Let (X,d) be a compact metric space and (K(X),dH) be the space of all non-empty compact subsets of X equipped with the Hausdorff metric dH. The dynamical system (X,f) induces another dynamical system . We study the relations between the various forms of sensitivity of the systems (X,f) and . We prove that all forms of sensitivity of partly imply the same for (X,f), and the converse holds in some cases.     

On the spectral radius of positive operators on Banach sequence spaces

Let K₁,…,K_n be (infinite) non-negative matrices that define operators on a Banach sequence space. Given a function f:[0,∞)×…×[0,∞)→[0,∞) of n variables, we define a non-negative matrix and consider the inequality

     

Orthogonality of matrices

Let X be a real finite-dimensional normed space with unit sphere S_X and let L(X) be the space of linear operators from X into itself. It is proved that X is an inner product space if and only if for A,C∈L(X)

     

Precise rates in the law of the logarithm in the Hilbert space

Let be a sequence of i.i.d. random variables taking values in a real separable Hilbert space (H,‖⋅‖) with covariance operator Σ, and set Sn=X₁+?+Xn, n?1. Let . We prove that, for any 1<r<3/2 and a>−d/2,