Integrable models for vicious and friendly walkers

We consider random walks of two essentially different classes of random walkers, namely, of vicious and friendly ones, on one-dimensional lattices with periodic boundary conditions. The walkers are called vicious since, arriving at a lattice site, they annihilate not only one another but all the remaining walkers as well. On the contrary, an arbitrary number of friendly walkers can share the same lattice sites. It is shown that a natural model describing the behavior of friendly walkers is an integrable model of the boson type. A representation of the generating function for the number of the lattice paths performed by a fixed number of friendly walkers for a certain number of steps is obtained. Bibliography: 22 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 335, 2006, pp. 59–74.     

Order topology in vector lattices with closure by one step

An order topology in vector lattices and Boolean algebras is studied under the additional condition of “closure by one step” that generalizes the well-known “regularity” property of Boolean algebras and K-spaces. It is proved that in a vector lattice or a Boolean algebra possessing such a property there exists a basis of solid neighborhoods of zero with respect to an order topology. An example of a Boolean algebra without basis of solid neighborhoods of zero (an algebra of regular open subsets of the interval (0, 1)) is given. Bibliography: 3 titles. Translated fromProblemy Matematicheskogo Analiza, No. 15 1995, pp. 213–220.     

A Shadow identity and an application to isoduality

The identity derived here from the theta transformation law replaces the “Atkin-Lehner identity” when the level decomposes into two factors which are not coprime. An application is given to the study of modular lattices of level 4, connected with modular forms for the classical theta group. CONWAY and Sloane have determined the maximal Hermite number of a self-dual lattice in ℝⁿ for alln ≤ 33, and their result generalizes to the isodual case considered here in most of these dimensions.     

Algebraic axiomatization of tense intuitionistic logic

We introduce two unary operators G and H on a relatively pseudocomplemented lattice which form an algebraic axiomatization of the tense quantifiers “it is always going to be the case that” and “it has always been the case that”. Their axiomatization is an extended version for the classical logic and it is in accordance with these operators on many-valued Łukasiewicz logic. Finally, we get a general construction of these tense operators on complete relatively pseudocomplemented lattice which is a power lattice via the so-called frame.     

What are all the best sphere packings in low dimensions?

We describe what may beall the best packings of nonoverlapping equal spheres in dimensionsn ≤10, where “best” means both having the highest density and not permitting any local improvement. For example, the best five-dimensional sphere packings are parametrized by the 4-colorings of the one-dimensional integer lattice. We also find what we believe to be the exact numbers of “uniform” packings among these, that is, those in which the automorphism group acts transitively. These assertions depend on certain plausible but as yet unproved postulates. Our work may be regarded as a continuation of László Fejes Tóth's work on solid packings.     

An anyon model

We construct an infinite-dimensional dynamical Hamiltonian system that can be interpreted as a localized structure (“quasiparticle”) on the plane E ₂. The model is based on the theory of an infinite string in the Minkowski space E _1,3 formulated in terms of the second fundamental forms of the worldsheet. The model phase space H is parameterized by the coordinates, which are interpreted as “internal” (E(2)-invariant) and “external” (elements of T*E ₂) degrees of freedom. The construction is nontrivial because H contains a finite number of constraints entangling these two groups of coordinates. We obtain the expressions for the energy and for the effective mass of the constructed system and the formula relating the proper angular momentum and the energy. We consider a possible interpretation of the proposed construction as an anyon model.     

Poles of L-functions,periods and unitary theta. Part I: Odd to even

In this paper we prove the equivalence between the non-vanishing of the Θ correspondence on an irreducible, generic, cuspidal representation of U_2n+1, the non-vanishing of a certain generalized period and the existence of a pole of a twisted partial L-function. Supported by contract number HPRN-CT-2000-00120 of the programme “Improving the Human Potential and the Socio-economic Knowledge Base” from the European Community, network “Arithmetic Algebraic Geometry”.     

Synaptic algebras

A synaptic algebra is both a special Jordan algebra and a spectral order-unit normed space satisfying certain natural conditions suggested by the partially ordered Jordan algebra of bounded Hermitian operators on a Hilbert space. The adjective “synaptic”, borrowed from biology, is meant to suggest that such an algebra coherently “ties together” the notions of a Jordan algebra, a spectral order-unit normed space, a convex effect algebra, and an orthomodular lattice.     

A complexity measure for families of binary sequences

In earlier papers finite pseudorandom binary sequences were studied, quantitative measures of pseudorandomness of them were introduced and studied, and large families of “good” pseudorandom sequences were constructed. In certain applications (cryptography) it is not enough to know that a family of “good” pseudorandom binary sequences is large, it is a more important property if it has a “rich”, “complex” structure. Correspondingly, the notion of “f-complexity” of a family of binary sequences is introduced. It is shown that the family of “good” pseudorandom binary sequences constructed earlier is also of high f-complexity. Finally, the cardinality of the smallest family achieving a prescibed f-complexity and multiplicity is estimated. This revised version was published online in August 2006 with corrections to the Cover Date.     

r-entropy,equipartition, and Ornstein’s isomorphism theorem inR n

A new approach is given to the entropy of a probability-preserving group action (in the context ofZ and ofR _n ), by defining an approximate “r-entropy”, 0<r<1, and lettingr → 0. If the usual entropy may be described as the growth rate of the number of essential names, then ther-entropy is the growth rate of the number of essential “groups of names” of width≦r, in an appropriate sense. The approach is especially useful for actions of continuous groups. We apply these techniques to state and prove a “second order” equipartition theorem forZ _m ×R _n and to give a “natural” proof of Ornstein’s isomorphism theorem for Bernoulli actions ofZ _m ×R _n , as well as a characterization of such actions which seems to be the appropriate generalization of “finitely determined”.     

Multiple levels of symmetry breaking

In the previous paper in this volume we have studied the p-spin interaction model just below the critical temperature, and we have rigorously proved several aspects of the physicists prediction that this model exhibits “one level of symmetry breaking”. In the present paper we show how to construct systems that exhibit an arbitrarily large, but finite number of “levels of symmetry-breaking”. As the temperature decreases, such systems exhibit many phase transitions, as the structure of the overlaps gains complexity. This phenomenon does not seem to have been described previously, even in the physics literature. Received: 15 January 1998 / Revised version: 10 November 1999 / Published online: 21 June 2000     

The <Emphasis Type="Italic">R</Emphasis>-observability and <Emphasis Type="Italic">R</Emphasis>-controllability of linear differential-algebraic systems

We study the R-controllability (the controllability within the attainability set) and the R-observability of time-varying linear differential-algebraic equations (DAE). We analyze DAE under assumptions guaranteeing the existence of a structural form (which is called “equivalent”) with separated “differential” and “algebraic” subsystems. We prove that the existence of this form guarantees the solvability of the corresponding conjugate system, and construct the corresponding “equivalent form” for the conjugate DAE. We obtain conditions for the R-controllability and R-observability, in particular, in terms of controllability and observability matrices. We prove theorems that establish certain connections between these properties.     

Counting Overlattices in Automorphism Groups of Trees

We give an upper bound for the number u _Γ(n) of “overlattices” in the automorphism group of a tree, containing a fixed lattice Γ with index n. For an example of Γ in the automorphism group of a 2p-regular tree whose quotient is a loop, we obtain a lower bound of the asymptotic behavior as well.     

The Number of Hierarchical Orderings

An ordered set-partition (or preferential arrangement) of n labeled elements represents a single “hierarchy” these are enumerated by the ordered Bell numbers. In this note we determine the number of “hierarchical orderings” or “societies”, where the n elements are first partitioned into m ≤ n subsets and a hierarchy is specified for each subset. We also consider the unlabeled case, where the ordered Bell numbers are replaced by the composition numbers. If there is only a single hierarchy, we show that the average rank of an element is asymptotic to n/(4 log 2) in the labeled case and to n/4 in the unlabeled case. This revised version was published online in September 2006 with corrections to the Cover Date.     

Topological flatness of orthogonal local models in the split,even case. I

Local models are schemes, defined in terms of linear algebra, that were introduced by Rapoport and Zink to study the étale-local structure of integral models of certain PEL Shimura varieties over p-adic fields. A basic requirement for the integral models, or equivalently for the local models, is that they be flat. In the case of local models for even orthogonal groups, Genestier observed that the original definition of the local model does not yield a flat scheme. In a recent article, Pappas and Rapoport introduced a new condition to the moduli problem defining the local model, the so-called spin condition, and conjectured that the resulting “spin” local model is flat. We prove a preliminary form of their conjecture in the split, Iwahori case, namely that the spin local model is topologically flat. An essential combinatorial ingredient is the equivalence of μ-admissibility and μ-permissibility for two minuscule cocharacters μ in root systems of type D.     

An Inequality from Moment Theory

We show how certain simple ℓ^p–inequalities may be proved by “ignoring the p.” An application to moment sequences is considered.     

An outranking-based fuzzy logic model for collaborative group preferences

Some ideas coming from Multi Criteria Decision Aid are here extended to group decision. We present a logical model in order to reflect the degree of truth of the following predicate “group G considers that option x is at least as good as y.” The proposal considers the strength of the majority using its number and the intensity of its preference, and it also acknowledges the importance of the minorities of a certain numerical significance that manifest an intense disagreement with the predicate of the outranking. The effects of the “majority dictatorship” are restricted. Since it considers simultaneously the strength of the majority, the importance of unhappy minorities, and the intensity of the preference/opposition, this model exhibits desirable qualities of the classic methods by Condorcet and Borda. This model can be used to solve problems regarding selection, ranking, classification, and sorting. Various examples are given, which show the quality of the solutions that were obtained.     

From the Bethe Ansatz to the Gessel-Viennot theorem

We state and prove several theorems that demonstrate how the coordinate Bethe Ansatz for the eigenvectors of suitable transfer matrices of a generalized inhomogeneous, five-vertex model on the square lattice, given certain conditions hold, is equivalent to the Gessel-Viennot determinant for the number of configurations ofN non-intersecting directed lattice paths, or vicious walkers, with various boundary conditions. Our theorems are sufficiently general to allow generalisation to any regular planar lattice.     

Weakly Gibbsian representations for joint measures of quenched lattice spin models

Can the joint measures of quenched disordered lattice spin models (with finite range) on the product of spin-space and disorder-space be represented as (suitably generalized) Gibbs measures of an “annealed system”? - We prove that there is always a potential (depending on both spin and disorder variables) that converges absolutely on a set of full measure w.r.t. the joint measure (“weak Gibbsianness”). This “positive” result is surprising when contrasted with the results of a previous paper [K6], where we investigated the measure of the set of discontinuity points of the conditional expectations (investigation of “a.s. Gibbsianness”). In particular we gave natural “negative” examples where this set is even of measure one (including the random field Ising model). Further we discuss conditions giving the convergence of vacuum potentials and conditions for the decay of the joint potential in terms of the decay of the disorder average over certain quenched correlations. We apply them to various examples. From this one typically expects the existence of a potential that decays superpolynomially outside a set of measure zero. Our proof uses a martingale argument that allows to cut (an infinite-volume analogue of) the quenched free energy into local pieces, along with generalizations of Kozlov's constructions. Received: 11 November 1999 / Revised version: 18 April 2000 / Published online: 22 November 2000 RID="*" ID="*" Work supported by the DFG Schwerpunkt `Wechselwirkende stochastische Systeme hoher Komplexit?t'     

Codescente en K-théorie étale et corps de nombres

In this paper, we give a codescent criterion for the higher tame kernelK _2i ^−2/ét O _E for a Galoisp-extensionE/F of algebraic number fields (p odd). As an application, we give fori∈ℤ a “going-up” theorem for certain property called (p, i)-regularity, which allows us in particular to construct examples of number fields verifying “twisted” Leopoldt conjectures.