On the integrable discrete versions of the Leznov lattice: Determinant solutions and pfaffianization

In this paper, we present the Casorati form of the N-soliton solution for an integrable fully-discrete version and two integrable semi-discrete versions of the Leznov lattice, which arise from the integrable discretization of the two-dimensional Leznov lattice. By using the pfaffianization procedure of Hirota and Ohta, a new integrable coupled system is generated from the semi-discrete version of the Leznov lattice in the y-direction.     

Pfaffianization of the two-dimensional Toda lattice

Pfaffianization procedure due to Hirota and Ohta is applied to the two-dimensional Toda lattice. As a result, a Pfaffianized version of the two-dimensional Toda lattice is found.     

Generalizing the KP Hierarchies: Pfaffian Hierarchies

We derive the Pfaffian analogues of the equations in the single-component KP hierarchies and the modified KP hierarchies and present an example of a system derived by reduction of some of the equations in these Pfaffianized hierarchies.     

Quantum group structure and local fields in the algebraic approach to 2D gravity

This review contains a summary of the work by J.-L. Gervais and the author on the operator approach to 2d gravity. Special emphasis is placed on the construction of local observables — the Liouville exponentials and the Liouville field itself — and the underlying algebra of chiral vertex operators. The double quantum group structure arising from the presence of two screening charges is discussed and the generalized algebra and field operators are derived. In the last part, we show that our construction gives rise to a natural definition of a quantum tau function, which is a noncommutative version of the classical group-theoretic representation of the Liouville fields by Leznov and Saveliev.Supported by DFG.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 1, pp. 158–191, July, 1995.     

Groups whose lattice of centralizers is a sublattice of the subgroup lattice

An earlier conjecture suggests that the lattice of centralizers is modular in the title groups. We prove it to hold for two classes of groups whose lattice of centralizers has finite length — groups with the normalizer condition and locally finite groups.Translated fromAlgebra i Logika, Vol. 33, No. 5, pp. 475–513, September–October, 1994.     

Non-covering in the interpretability lattice of equational theories

We prove a general theorem which implies that many members of the interpretability lattice — including the least element of the lattice, the element represented by the theory with one constant and no axioms, and the equational theory of semilattices — have no cover in the lattice.Presented by R. W. Quackenbush.Research supported by National Science Foundation Grant DMS 89 04014.     

The skew Schubert polynomials

We obtain a tableau definition of the skew Schubert polynomials named by Lascoux, which are defined as flagged double skew Schur functions. These polynomials are in fact Schubert polynomials in two sets of variables indexed by 321-avoiding permutations. From the divided difference definition of the skew Schubert polynomials, we construct a lattice path interpretation based on the Chen–Li–Louck pairing lemma. The lattice path explanation immediately leads to the determinantal definition and the tableau definition of the skew Schubert polynomials. For the case of a single variable set, the skew Schubert polynomials reduce to flagged skew Schur functions as studied by Wachs and by Billey, Jockusch, and Stanley. We also present a lattice path interpretation for the isobaric divided difference operators, and derive an expression of the flagged Schur function in terms of isobaric operators acting on a monomial. Moreover, we find lattice path interpretations for the Giambelli identity and the Lascoux–Pragacz identity for super-Schur functions. For the super-Lascoux–Pragacz identity, the lattice path construction is related to the code of the partition which determines the directions of the lines parallel to the y-axis in the lattice.     

Finite Intervals in the Lattice of Topologies

We discuss the question whether every finite interval in the lattice of all topologies on some set is isomorphic to an interval in the lattice of all topologies on a finite set – or, equivalently, whether the finite intervals in lattices of topologies are, up to isomorphism, exactly the duals of finite intervals in lattices of quasiorders. The answer to this question is in the affirmative at least for finite atomistic lattices. Applying recent results about intervals in lattices of quasiorders, we see that, for example, the five-element modular but non-distributive lattice cannot be an interval in the lattice of topologies. We show that a finite lattice whose greatest element is the join of two atoms is an interval of T ₀-topologies iff it is the four-element Boolean lattice or the five-element non-modular lattice. But only the first of these two selfdual lattices is an interval of orders because order intervals are known to be dually locally distributive.     

Correlation Function of the Two-Dimensional Ising Model on a Finite Lattice: I

The spin–spin correlation function of the two-dimensional Ising model with nearest-neighbor interaction is calculated on a finite lattice with periodic boundary conditions. The representations, which are analogous to the form-factor representation, are obtained for the ferromagnetic and paramagnetic domains of the interaction parameter. We discuss the effects of the finiteness of the system. We investigate the asymptotic dependence of the corresponding quantities on the lattice size.     

Evidence for a Phase Transition in Three-Dimensional Lattice Models

It was recently discovered that an eigenvector structure of commutative families of layer-to-layer matrices in three-dimensional lattice models is described by a two-dimensional spin lattice generalizing the notion of one-dimensional spin chains. We conjecture the relations between the two-dimensional spin lattice in the thermodynamic limit and the phase structure of three-dimensional lattice models. We consider two simplest cases: the homogeneous spin lattice related to the Zamolodchikov–Bazhanov–Baxter model and a chess spin lattice related to the Stroganov–Mangazeev elliptic solution of the modified tetrahedron equation. Evidence for the phase transition is obtained in the second case.     

Sublattices of complete lattices with continuity conditions

Various embedding problems of lattices into complete lattices are solved. We prove that for any join-semilattice S with the minimal join-cover refinement property, the ideal lattice Id S of S is both algebraic and dually algebraic. Furthermore, if there are no infinite D-sequences in J(S), then Id S can be embedded into a direct product of finite lower bounded lattices. We also find a system of infinitary identities that characterize sublattices of complete, lower continuous, and join-semidistributive lattices. These conditions are satisfied by any (not necessarily finitely generated) lower bounded lattice and by any locally finite, join-semidistributive lattice. Furthermore, they imply M. Erné’s dual staircase distributivity.On the other hand, we prove that the subspace lattice of any infinite-dimensional vector space cannot be embedded into any ℵ₀-complete, ℵ₀-upper continuous, and ℵ₀-lower continuous lattice. A similar result holds for the lattice of all order-convex subsets of any infinite chain.Dedicated to the memory of Ivan RivalReceived April 4, 2003; accepted in final form June 16, 2004.This revised version was published online in August 2005 with a corrected cover date.     

Incomparability and Intersection Properties of Boolean Interval Lattices and Chain Posets

In a canonical way, we establish an AZ-identity (see [2]) and its consequences, the LYM-inequality and the Sperner property, for the Boolean interval lattice. Furthermore, the Bollobas inequality for the Boolean interval lattice turns out to be just the LYM-inequality for the Boolean lattice. We also present an Intersection Theorem for this lattice.Perhaps more surprising is that by our approach the conjecture of P. L. Erdöset al.[7] and Z. Füredi concerning an Erdös–Ko–Rado-type intersection property for the poset of Boolean chains could also be established. In fact, we give two seemingly elegant proofs.     

The rapidly decreasing solution of the Cauchy problem for the Toda lattice

We prove the existence of a rapidly decreasing solution of the Cauchy problem for the Toda lattice. We find a class of initial data guaranteeing the existence of such a solution.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 1, pp. 5–12, January, 2005.     

On the Linear Complexity of the Power Generator

We obtain a lower bound on the linear complexity of the powergenerator of pseudo-random numbers, which in some special cases is alsoknown as the RSA generator and as the Blum–Blum–Shubgenerator. In some very important cases this bound is essentially thebest possible. In particular, this implies that lattice reductionattacks on such generators are not feasible.     

On the Coordinate of a Singular Point of the Time Correlation Function for a Spin System on a Simple Hypercubic Lattice at High Temperatures

Using the d ^–1 expansion method (d is the space dimension), we estimate the coordinate of the time-dependent autocorrelation function singular point on the imaginary time axis for the spin 1/2 Heisenberg model on a simple hypercubic lattice at high temperatures. We represent the coefficients of the time expansion (the spectral moments) for the autocorrelation function as the sums of the weighted lattice figures in which the trees constructed from the double bonds give the leading contributions with respect to d ^–1 and the same trees with the built-in squares from six bonds or diagrams with the fourfold bonds give the contribution of the next-to-leading order. We find the corrections to the coordinate of the autocorrelation function singular point that are due to the latter contributions.     

A higher-order moment method of the lattice Boltzmann model for the conservation law equation

In this paper, we proposed a higher-order moment method in the lattice Boltzmann model for the conservation law equation. In contrast to the lattice Bhatnagar–Gross–Krook (BGK) model, the higher-order moment method has a wide flexibility to select equilibrium distribution function. This method is based on so-called a series of partial differential equations obtained by using multi-scale technique and Chapman–Enskog expansion. According to Hirt’s heuristic stability theory, the stability of the scheme can be controlled by modulating some special moments to design the third-order dispersion term and the fourth-order dissipation term. As results, the conservation law equation is recovered with higher-order truncation error. The numerical examples show the higher-order moment method can be used to raise the accuracy of the truncation error of the lattice Boltzmann scheme for the conservation law equation.     

The boundary characteristic and the volume of lattice polyhedra

The boundary characteristic — introduced by Ding and Reay — is a functional defined for a given planar tiling which associates with a given lattice figure, some integer. It appeared to be a very useful parameter to determine the area of lattice figures in the planar tilings with congruent regular polygons. The purpose of this paper is to extend the notion of the boundary characteristic to lattice polyhedra inR³. Studying some of its properties we show, in particular, that it can be applied to determine the volume of lattice polyhedra.     

On the Lattice Approximation for Certain Generalized Vector Markov Fields in Four Spacetime Dimensions

We extend previous results by Albeverio, Iwata and Schmidt on the construction of a convergent lattice approximation for invariant scalar 3-vector generalized random fields F of an infinitely divisible type and apply them to the construction of convergent lattice approximation for the generalized random vector field A determined by the stochastic quaternionic Cauchy–Riemann equation A = F.     

Fixed point theorems for nonexpansive mappings and Suzuki-generalized nonexpansive mappings on a Banach lattice

We prove the existence of fixed points for multivalued nonexpansive nonself-mappings on a weakly orthogonal reflexive Banach lattice with uniformly monotone norm. Moreover, for single-valued mappings, we extend Betiuk-Pilarska and Prus’s result [A. Betiuk-Pilarska, S. Prus, Banach lattices which are order uniformly noncreasy, J. Math. Anal. Appl. 342 (2008) 1271–1279] on the weak fixed point property to continuous mappings satisfying condition (C) on a w-weakly orthogonal OUNC Banach lattice.     

Regular-Norm Balls can be Closed in the Strong Operator Topology

The main results obtained are:– A Dedekind complete Banach lattice Y has a Fatou norm if and only if, for any Banach lattice X, the regular-norm unit ball Ur = {T Lr(X,Y): ||T||r 1} is closed in the strong operator topology on the space of all regular operators, Lr(X,Y).– A Dedekind complete Banach lattice Y has a norm which is both Fatou and Levi if and only if, for any Banach lattice X, the regular-norm unit ball Ur is closed in the strong operator topology on the space of all bounded operators, L(X,Y).– A Banach lattice Y has a Fatou–Levi norm if and only if for every L-space X the space L(X,Y) is a Banach lattice under the operator norm.– A Banach lattice Y is isometrically order isomorphic to C(S) with the supremum norm, for some Stonean space S, if and only if, for every Banach lattice X, L(X,Y) is a Banach lattice under the operator norm.Several examples demonstrating that the hypotheses may not be removed, as well as some applications of the results obtained to the spaces of operators are also given. For instance:– If X = Lp() and Y = Lq(), where 1 < p,q < , then Lr(X,Y) is a first category subset of L(X,Y).