Ergodic properties of Poisson processes with almost periodic intensity

Summary We discuss in this paper a non-homogeneous Poisson process A driven by an almost periodic intensity function. We give the stationary version A ^* and the Palm version A ⁰ corresponding to A ^*. Let (T _i ,i) be the inter-point distance sequence in A and (T _i ⁰ ,i) in A ⁰. We prove that forj, the sequence (T _i+j,i) converges in distribution to (T _i ⁰ ,i). If the intensity function is periodic then the convergence is in variation.     

A Nonlinear Extension of the Coale–Lopez Theorem

For a sequence (T _n)_nof nonlinear positive operators on a real Banach space, certain monotonicity conditions are specified under which asymptotic equality holds for the inhomogeneous iterates T _n T _n-1 T ₁ T ₀. As a consequence, weak ergodicity or even path stability holds in the norm-topology. Finally, a nonlinear extension of the Coale–Lopez theorem in demography is shown.     

The associativity equation revisited

Summary Consideration of the Associativity Equation,x (y z) = (x y) z, in the case where:I × I I (I a real interval) is continuous and satisfies a cancellation property on both sides, provides a complete characterization of real continuous cancellation semigroups, namely that they are topologically order-isomorphic to addition on some real interval: ( – ,b), ( – ,b], –, +), (a, + ), or [a, + ) — whereb = 0 or –1 anda = 0 or 1. The original proof, however, involves some awkward handling of cases and has defied streamlining for some time. A new proof is given following a simpler approach, devised by Páles and fine-tuned by Craigen.     

Representations of distributive lattice-ordered semigroups with binary relations

LetR(, , ¦) denote the class of all algebras isomorphic to ones whose elements are binary relations and whose operations are union, intersection, and relation composition (or relative product) of relations. We prove thatR(, , ¦) is not a variety and is not finitely axiomatizable. LetDLOS denote the class of all structures (A, , , ) where (A, , ) is a distributive lattice, (A, ) is a semigroup and is additive w.r.t. . We prove thatDLOS is the variety generated byR(, , ¦), and moreover, if (A, , , ) DLOS then it is representable whenever we disregard one of its operations.Presented by Boris M. Schein.Research supported by Hungarian National Foundation for Scientific Research grant No. 1810.     

Ein Existenzsatz für Elementarinhalte

An elementary content I is a real valued, non-negative, invariant and monotonous homomorphism on a decomposition structure of elementary figures. The semigroup (H,+,) of abstract classes is introduced (§3) by using the relation of equidecomposability and it's natural generalizations. Each elementary content divides into I= where and are canonical homomorphisms with respect to the relations studied before and : H + is a monotonous homomorphism called content (cf. Satz 3, §3). In §4 (Satz 4) the Existence-Theorem on contents is stated and it is proved in §5. The last section §6 gives the application on Archimedean decomposition structures including the case of volume measurement on polyhedrons.     

Volumes des représentations sur un corps local

Let be a locally compact second countable group, F a local field of characteristic zero and G an F-almost-simple F-algebraic group. In this paper we study the space X(,G) of Zariski-dense representations : G = G(F) using the natural morphism of cohomological functors * : H*(G, ·) H*(, ·) (where H denotes the continuous cohomology).First let F be a p-adic field. We completely describe the relations between the geometry and the cohomology of G : using geometric properties of the Bruhat-Tits building of G we construct natural cocycles for any irreducible cohomological representation of G. We then adapt these results to the case where the field F is archimedean.Using these cocycles we obtain a simple cohomological characterization of representations with bounded image.Our main result is then the construction, using the previous cocycles and dynamical properties at infinity of , of cohomological invariants (called volumes) on the space X(,G). These volumes describe how the image () goes to infinity in G. They have coefficients in the natural universal infinite-dimensional representation L(, )$\mathbb{C}$ of .In the case where is a cocompact lattice of SO(n, 1) or SU(n, 1), we use these volumes to produce new non-trivial numerical invariants on X(,G), which refine previously known invariants.

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Volumes des représentations sur un corps local

     

Limites dégénérées de séries discrètes, formes automorphes et variétés de Griffiths–Schmid: le cas du groupe U (2, 1)

Let G be an inner anisotropic form of an unitary group of 3 variables over Q, such that G_RU(2,1), and be an automorphic representation of G(A) whose archimedean component is a degenerate limit of discrete series; such a never occurs in the cohomology (coherent or étale) of a Shimura variety. We show that however it does appear in the coherent cohomology of some line bundle over an associated Griffiths-Schmid variety. Moreover we study cup products between such cohomology classes and some other automorphic cohomology classes and we prove some non-vanishing results.     

Irreducible representations of Lie algebras of reductive groups and the Kac-Weisfeiler conjecture

Letg be the Lie algebra of a connected reductive groupG over an algebraically closed field of characteristicp>0. Suppose thatG ⁽¹⁾ is simply connected andp is good for the root system ofG. Ifp=2, suppose in addition thatg admits a nondegenerateG-invariant trace form. LetV be an irreducible and faithfulg-module withp-character g ^*. It is proved in the paper that dimV is divisible byp ^1/2dim() where () stands for the orbit of under the coadjoint action ofG.Oblatum 14-III-1994 & 17-XI-1994     

Compatible Almost Complex Structures on Quaternion Kähler Manifolds

Let (M⁴ⁿ,g,Q) be a quaternion Kähler manifold with reduced scalar curvature = K/4n(n + 2). Suppose J is an almost complex structure which is compatible with the quaternionic structure Q and let = – F J be the Lee form of J. We prove the following local results: (1) if J is conformally symplectic, then it is parallel and = 0; (2) if J is cosymplectic, then 0 with equality if and only if J is parallel; (3) if J is integrable, then d is Q-Hermitian and harmonic; and (4) any closed self-dual 2-form = f(g J) ² ₊ = g Q ² is parallel. In Section 5, extending previous results of Salamon [24], we describe a correspondence among conformally balanced J, Killing vector fields X and self-dual 2-forms satisfying the twistor equation.When M⁴ⁿ is compact our main global results are the following: (1) if > 0, then there exists no compatible almost complex structure J; (2) if the first Chern class c₁(T^(1,0) _J M) = 0, then = 0; (3) if = 0 a compatible complex structure J is parallel; and (4) if 0, then no compatible complex structure J exists. The last two results have been proved in [23] by twistor methods.     

Weakly isomorphic transformations that are not isomorphic

Summary A new method for construction of transformations T _i: (X _i, B _i, _i) , i=1,2, that are factors of each other but that are not measuretheoretically isomorphic is provided. This method uses ergodic product cocycles of the form S ⁱ ₁xS ⁱ ₂x...,, where : XZ ₂ is a cocycle, S belongs to the centralizer of T and T is an ergodic translation on a compact, monothetic group X.     

Iterative Pexider equation modulo a subset

Summary LetX, Y, Z be arbitrary nonempty sets,E be a nonempty subset ofZ ^z andK be a groupoid. Assume that {F _t} _{t K} Z ^X, {G _t} _{t K} Y ^X, {H _t} _{t K} Z ^Y are families of functions satisfying the functional equationF _st = k_(s,t) H_s G_t for (s, t) D(K), whereD(K) stands for the domain of the binary operation on the groupoidK andk _(s,t) E for (s, t) D(K). Conditions are established under which the equation can be reduced to the corresponding Cauchy equation. This paper generalizes some results from [4] and [1].     

Studies on the Painlevé equations

Summary In this series of papers, we study birational canonical transformations of the Painlevé system , that is, the Hamiltonian system associated with the Painlevé differential equations. We consider also -function related to and particular solutions of . The present article concerns the sixth Painlevé equation. By giving the explicit forms of the canonical transformations of associated with the affine transformations of the space of parameters of , we obtain the non-linear representation: GG_*, of the affine Weyl group of the exceptional root system of the type F₄ A canonical transformation of G_* can extend to the correspondence of the -functions related to . We show the certain sequence of -functions satisfies the equation of the Toda lattice. Solutions of , which can be written by the use of the hypergeometric functions, are studied in details.     

Commutativity of operators on the lattice of existence varieties

In a previous paper the authors introduced seven complete congruences on the lattice _ev(I of e-varieties of regular semigroups of the form _P :U _P VPU=PV, whereP is drawn from a small set of e-varieties: left zero, right zero, rectangular bands, groups, left groups, right groups and completely simple semigroups. Four new complete congruences are introduced here of the form _P :U _P VPU=PV, whereP is one of the following classes of regular semigroups: left monoids, right monoids, monoids, idempotent generated semigroups. For each complete congruence on _ev(I) and eachU_ev(I), the -class ofU is an interval [U ,U ] so that there is associated with each such congruence an idempotent operatorUU on _ev(I). This paper establishes numerous results concerning the commutativity of operators of this form.This work was supported in part by NSERC Grant 4044.     

A new characterization of <Emphasis Type="Italic">PSU</Emphasis>(<Emphasis Type="Italic">p,q</Emphasis>)

Summary Let G be a finite group. Order components of G were introduced in Chen [5]. Let OC(G) be the set of order components of G. Some finite groups are characterizable by their order components. This assertion was proved for the simple groups PSU(p,q), where p=3, 5, 7 and 11. In this paper, we prove that the simple groups PSU(p,q) can be uniquely determined by their order components, where p&ge;13 is a prime number. Main consequences of our results are the validity of a conjecture of J. G. Thompson and another conjecture of W. Shi and J. Bi for the groups under consideration.     

On the bestL p multipoint local approximation

P (f) — , f L ^p - , k . f 02k–2 P (f) 0.     

A generalization of second variation

f p- , l p . p=1 . . p - , f -.     

On pathwise behavior of multiserver queues

In this paper, by exploiting recent results on the pathwise behavior of the workload process in single server, work conserving queues of theG/G/1/ type, we show that the workload of multiserver, work conserving queues ofG/G/m/ (m<) (andG/G/) queues satisfies an o(t) growth condition, provided that the time average of the work brought into the system is less thanm form < (and finite form=).     

The Spectral Theory of Amenable Actions and Invariants of Discrete Groups

Let G denote a semisimple group, a discrete subgroup, B=G/P the Poisson boundary. Regarding invariants of discrete subgroups we prove, in particular, the following:(1) For any -quasi-invariant measure on B, and any probablity measure on , the norm of the operator () on L ²(B,) is equal to (), where is the unitary representation in L ²(X,), and is the regular representation of .(2) In particular this estimate holds when is Lebesgue measure on B, a Patterson–Sullivan measure, or a -stationary measure, and implies explicit lower bounds for the displacement and Margulis number of (w.r.t. a finite generating set), the dimension of the conformal density, the -entropy of the measure, and Lyapunov exponents of .(3) In particular, when G=PSL₂() and is free, the new lower bound of the displacement is somewhat smaller than the Culler–Shalen bound (which requires an additional assumption) and is greater than the standard ball-packing bound.We also prove that ()=_G() for any amenable action of G and L ¹(G), and conversely, give a spectral criterion for amenability of an action of G under certain natural dynamical conditions. In addition, we establish a uniform lower bound for the -entropy of any measure quasi-invariant under the action of a group with property T, and use this fact to construct an interesting class of actions of such groups, related to 'virtual' maximal parabolic subgroups. Most of the results hold in fact in greater generality, and apply for instance when G is any semi-simple algebraic group, or when is any word-hyperbolic group, acting on their Poisson boundary, for example.     

The Topology of the Coloring Complex

In a recent paper, E. Steingrímsson associated to each simple graph G a simplicial complex _G, referred to as the coloring complex of G. Certain nonfaces of _G correspond in a natural manner to proper colorings of G. Indeed, the h-vector is an affine transformation of the chromatic polynomial _G of G, and the reduced Euler characteristic is, up to sign, equal to |_G(–1)|–1. We show that _G is constructible and hence Cohen-Macaulay. Moreover, we introduce two subcomplexes of the coloring complex, referred to as polar coloring complexes. The h-vectors of these complexes are again affine transformations of _G, and their Euler characteristics coincide with _G(0) and –_G(1), respectively. We show for a large class of graphs—including all connected graphs—that polar coloring complexes are constructible. Finally, the coloring complex and its polar subcomplexes being Cohen-Macaulay allows for topological interpretations of certain positivity results about the chromatic polynomial due to N. Linial and I. M. Gessel.Research financed by ECs IHRP Programme, within the Research Training Network Algebraic Combinatorics in Europe, grant HPRN-CT-2001-00272.