Lattices of Order-Convex Sets of Forests

We give a characterization of the class Co(F)\mathbf{Co}(\mathcal{F}) [Co(F_n)\mathrm{\mathbf{Co}}(\mathcal{F}_n), n < ω, respectively] of lattices isomorphic to convexity lattices of posets which are forests [forests of length at most n, respectively], as well as of the class Co(L)\mathbf{Co}(\mathcal{L}) of lattices isomorphic to convexity lattices of linearly ordered posets. This characterization yields that the class of finite members from Co(F)\mathbf{Co}(\mathcal{F}) [from Co(F_n)\mathbf{Co}(\mathcal{F}_n), n < ω, or from Co(L)\mathbf{Co}(\mathcal{L})] is finitely axiomatizable within the class of finite lattices.     

On finite retract lattices of monounary algebras

For a monounary algebra (A, f) we denote R ^∅(A, f) the system of all retracts (together with the empty set) of (A, f) ordered by inclusion. This system forms a lattice. We prove that if (A, f) is a connected monounary algebra and R ^∅(A, f) is finite, then this lattice contains no diamond. Next distributivity of R ^∅(A, f) is studied. We find a representation of a certain class of finite distributive lattices as retract lattices of monounary algebras.     

Commensurability invariants for nonuniform tree lattices

We study nonuniform lattices in the automorphism groupG of a locally finite simplicial treeX. In particular, we are interested in classifying lattices up to commensurability inG. We introduce two new commensurability invariants:quotient growth, which measures the growth of the noncompact quotient of the lattice; andstabilizer growth, which measures the growth of the orders of finite stabilizers in a fundamental domain as a function of distance from a fixed basepoint. WhenX is the biregular treeX _m,n, we construct lattices realizing all triples of covolume, quotient growth, and stabilizer growth satisfying some mild conditions. In particular, for each positive real numberν we construct uncountably many noncommensurable lattices with covolumeν. Supported in part by NSF grants DMS-9704640 and DMS-0244542. Supported in part by an NSF postdoctoral research fellowship.     

Cancellation in Skew Lattices

Distributive lattices are well known to be precisely those lattices that possess cancellation: x úy = x úzx \lor y = x \lor z and x ùy = x ùzx \land y = x \land z imply y = z. Cancellation, in turn, occurs whenever a lattice has neither of the five-element lattices M ₃ or N ₅ as sublattices. In this paper we examine cancellation in skew lattices, where the involved objects are in many ways lattice-like, but the operations ù\land and ú\lor no longer need be commutative. In particular, we find necessary and sufficient conditions involving the nonoccurrence of potential sub-objects similar to M ₃ or N ₅ that ensure that a skew lattice is left cancellative (satisfying the above implication) right cancellative (x úz = y úzx \lor z = y \lor z and x ùz = y ùzx \land z = y \land z imply x = y) or just cancellative (satisfying both implications). We also present systems of identities showing that left [right or fully] cancellative skew lattices form varieties. Finally, we give some positive characterizations of cancellation.     

Gabor pairs, and a discrete approach to wave-front sets

We introduce admissible lattices and Gabor pairs to define discrete versions of wave-front sets with respect to Fourier–Lebesgue and modulation spaces. We prove that these wave-front sets agree with each other and with corresponding wave-front sets of “continuous type”. This implies that the coefficients of a Gabor frame expansion of f are parameter dependent, and describe the wave-front set of f.     

Modal operators on commutative residuated lattices

We prove some fundamental properties of monotone modal operators on bounded commutative integral residuated lattices (CRL). Moreover we give a positive answer to the problem left open in [RACHŮNEK, J.—ŠALOUNOV á, D.: Modal operators on bounded commutative residuated Rℓ-monoids, Math. Slovaca 57 (2007), 321–332].     

Discrete isoperimetric and Poincaré-type inequalities

We study some discrete isoperimetric and Poincaré-type inequalities for product probability measures μ ⁿ on the discrete cube {0, 1} ⁿ and on the lattice Z ⁿ . In particular we prove sharp lower estimates for the product measures of boundaries of arbitrary sets in the discrete cube. More generally, we characterize those probability distributions μ on Z which satisfy these inequalities on Z ⁿ . The class of these distributions can be described by a certain class of monotone transforms of the two-sided exponential measure. A similar characterization of distributions on R which satisfy Poincaré inequalities on the class of convex functions is proved in terms of variances of suprema of linear processes. Received: 30 April 1997 / Revised version: 5 June 1998     

On convergences of measurable functions in Archimedean vector lattices

For Archimedean vector lattices X, Y and the positive cone \mathbbL{\mathbb{L}} of all regular linear operators L : X → Y, a theory of sequential convergences of functions connected with an \mathbbL{\mathbb{L}} -valued measure is introduced and investigated.     

Perron Conditions for Uniform Exponential Expansiveness of Linear Skew-Product Flows

 We present necessary and sufficient conditions for uniform exponential expansiveness of discrete skew-product flows, in terms of uniform complete admissibility of the pair (c ₀(N, X), c ₀(N, X)). We give discrete and continuous characterizations for uniform exponential expansiveness of linear skew-product flows, using the uniform complete admissibility of the pairs (c ₀(N, X), c ₀(N, X)) and (C ₀(R ₊, X), C ₀(R ₊, X)), respectively. We generalize an expansiveness theorem due to Van Minh, R?biger and Schnaubelt, for the case of linear skew-product flows. Received August 10, 2001; in revised form June 25, 2002     

Perron Conditions for Uniform Exponential Expansiveness of Linear Skew-Product Flows

 We present necessary and sufficient conditions for uniform exponential expansiveness of discrete skew-product flows, in terms of uniform complete admissibility of the pair (c ₀(N, X), c ₀(N, X)). We give discrete and continuous characterizations for uniform exponential expansiveness of linear skew-product flows, using the uniform complete admissibility of the pairs (c ₀(N, X), c ₀(N, X)) and (C ₀(R ₊, X), C ₀(R ₊, X)), respectively. We generalize an expansiveness theorem due to Van Minh, R?biger and Schnaubelt, for the case of linear skew-product flows.     

Large hyperbolic lattices

For a fundamental group of a compact orientable manifold, a condition is specified that is sufficient to guarantee the presence of a “virtual” epimorphism onto a free non-Abelian group. A consequence is deriving a strong Tits alternative. An arbitrary noncompact finitely generated discrete subgroup in PO(3, 1) either is large or is virtually Abelian. An application is provided to the problem of uniform exponential growth for lattices in a 3-dimensional hyperbolic space and of growth of Betti numbers for lattices in a hyperbolic n-dimensional space, where n is an odd number. Supported by RFBR (project No. 08-01-00067), by DFG grant Gr 627-11, and by Forschergruppe “Spektrale Analysis, asymptotical Verteilungen und stochastische Dynamiken,” Billfold University. (G. A. Noskov) Translated from Algebra i Logika, Vol. 48, No. 2, pp. 174–189, March–April, 2009.     

Linear Stochastic Systems: A White Noise Approach

Using the white noise setting, in particular the Wick product, the Hermite transform, and the Kondratiev space, we present a new approach to study linear stochastic systems, where randomness is also included in the transfer function. We prove BIBO type stability theorems for these systems, both in the discrete and continuous time cases. We also consider the case of dissipative systems for both discrete and continuous time systems.We further study ℓ ₁-ℓ ₂ stability in the discrete time case, and L ₂-L _∞ stability in the continuous time case.     

Discrete spectrum of the perturbed Dirac operator

In this paper we study the asymptotics of the discrete spectrum in the gap (−1, 1) of the perturbed Dirac operatorD(α)=D ₀−αV1 acting inL ₂(R ³;C ⁴) with large coupling constant α. In particular some “non-standard” asymptotic formulae are obtained.     

Imbedded Circle Patterns with the Combinatorics of the Square Grid and Discrete Painlevé Equations

   Abstract. A discrete analogue of the holomorphic maps z ^γ and log(z) is studied. These maps are given by Schramm's circle pattern with the combinatorics of the square grid. It is shown that the corresponding circle patterns are imbedded and described by special separatrix solutions of discrete Painlevé equations. Global properties of these solutions, as well as of the discrete z ^γ and log(z) , are established.     

On discrete α-unimodality

A continuous composition semigroup of probability generating functionsF≔(F _t ,t≥0) and the corresponding multiplication ⊙ _F of van Harnet al. (1982,Z. Wahrsch. Verw. Gebiete,61, 97–118) are used to introduce the concept of [F; α]-unimodality which generalizes the discrete α-unimodality due to Abouammoh (1987,Statist. Neerlandica,41, 239–244) and Alamatsaz (1993,Statist. Neerlandica,47, 245–252). We offer various characterizations and other properties of [F;α]-unimodality. Notably, several convolution results are presented. Moreover, we explore the relationship between [F;α]-unimodality and the concepts of discrete self-decomposability and stability. Finally, lower bounds for variances of [F;α]-monotone and [F;α]-unimodal random variables are derived and some examples are also mentioned. Research supported by Grant SS024 of the Research Center of Kuwait University.     

Ergodicity of generalised Patterson-Sullivan measures in higher rank symmetric spaces

Let X = G/K be a higher rank symmetric space of noncompact type and a discrete Zariski dense group. In a previous article, we constructed for each G-invariant subset of the regular limit set of Γ a family of measures, the so-called (b, Γ · ξ)-densities. Our main result here states that these densities are Γ-ergodic with respect to an important subset of the limit set which we choose to call the ``ray limit set'. In the particular case of uniform lattices and products of convex cocompact groups acting on the product of rank one symmetric spaces every limit point belongs to the ray limit set, hence our result is most powerful for these examples. For nonuniform lattices, however, it is a priori not clear whether the ray limit set has positive measure with respect to a (b, Γ · ξ)-density. Using a counting theorem of Eskin and McMullen, we are able to prove that the ray limit set has full measure in each G-invariant subset of the limit set.     

On a problem involving bounded epimorphisms of lattices

A latticeL satisfies thebounded epimorphism condition if wheneverM is a lattce and ϕ:M →L is a bounded epimorphism, there exists a homomorphismι:L →M such that ιϕ=id _L . we show that the class of finite lattices satisfying the bounded epimorphism condition is properly contained in the class of finite lattices satisfying Whitman's condition (W). We also introduce a property defined for finite lattices that is sufficient to imply the bounded epimorphism condition. Presented by B. Jónsson.     

Anosov representations: domains of discontinuity and applications

The notion of Anosov representations has been introduced by Labourie in his study of the Hitchin component for SL(n,R). Subsequently, Anosov representations have been studied mainly for surface groups, in particular in the context of higher Teichmüller spaces, and for lattices in SO(1,n). In this article we extend the notion of Anosov representations to representations of arbitrary word hyperbolic groups and start the systematic study of their geometric properties. In particular, given an Anosov representation Γ→G we explicitly construct open subsets of compact G-spaces, on which Γ acts properly discontinuously and with compact quotient.