Invariant Lattice Research Articles

Ultraproducts of Z with an Application to Many-Valued Logics

Up to categorical equivalence, abelian lattice-ordered groups with strong unit coincide with Chang's MV-algebras—the Lindenbaum algebras of the infinite-valued Łukasiewicz calculus. While the property of being a strong unit is not definable even in first-order logic, MV-algebras form an equational class. On the other hand, the addition operation and the translation invariant lattice order of a lattice-ordered group are more amenable than the truncated addition operation of an MV-algebra. In this paper MV-algebraic and group-theoretical techniques are combined to classify and axiomatize all universal classes generated by an infinite totally ordered MV-algebra A such that the quotient of A by its unique maximal ideal is finite. The number of elements of this quotient, and that of the largest finite subalgebra of A turns out to be a complete classifier. The main tool for our results is given by order preserving embeddings of totally ordered groups G into ultrapowers of the additive group of integers, that also preserve the nondivisibility properties of prescribed elements of G.

Möbius invariant integrable lattice equations associated with KP and 2DTL hierarchies

The integrable lattice equations arising in the context of singular manifold equations for scalar, multicomponent KP hierarchies and 2D Toda lattice hierarchy are considered. They generate the corresponding continuous hierarchy of singular manifold equations, its Bäcklund transformations and different forms of superposition principles; their distinctive feature is invariance under the action of Möbius transformation. Geometric interpretation of these discrete equations is given.

The image of the Galois group for some crystalline representations

Let be the field of fractions of the ring of Witt vectors, where is an algebraically closed field of characteristic , and let . If is a -invariant lattice in a continuous -module of finite dimension over and if the set of characters of the semisimple envelope of satisfies some additional assumptions, then one can associate to a function containing a considerable amount of information about the image of in . In this paper we describe the set of functions arising from crystalline modules with Hodge-Tate weights in the interval . Moreover, we explicitly express these functions in terms of the corresponding filtered modules. This is applied to the description of the image , where is the Tate module of a 1-dimensional formal group over of finite height.

Symplectic group lattices

Let p p be an odd prime. It is known that the symplectic group S p 2 n ( p ) Sp_{2n}(p) has two (algebraically conjugate) irreducible representations of degree ( p n + 1 ) / 2 (p^{n}+1)/2 realized over Q ( ϵ p ) \mathbb {Q}(\sqrt {\epsilon p}) , where ϵ = ( − 1 ) ( p − 1 ) / 2 \epsilon = (-1)^{(p-1)/2} . We study the integral lattices related to these representations for the case p n ≡ 1 mod 4 p^{n} \equiv 1 \bmod 4 . (The case p n ≡ 3 mod 4 p^{n} \equiv 3 \bmod 4 has been considered in a previous paper.) We show that the class of invariant lattices contains either unimodular or p p -modular lattices. These lattices are explicitly constructed and classified. Gram matrices of the lattices are given, using a discrete analogue of Maslov index.

Thermodynamic Automaton Simulations of Fluid Flow and Diffusion in Porous Media

A thermodynamic automaton model of fluid flow in porous media is presented. The model is a nonrelativistic version of a Lorentz invariant lattice gas model constructed by Udey et al. (1998). In the previous model it was shown that the energy momentum tensor and the relativistic Boltzman equation can be rigorously derived from the collision and propagation rules. In the present paper we demonstrate that this nonrelativistic model can be used to accurately simulate well known results involving single phase flow and diffusion in porous media. The simulation results show that (1) one-phase flow simulations in porous media are consistent with Darcy's law; (2) the apparent diffusion coefficient decreases with a decrease in permeability; (3) small scale heterogeneity does not affect diffusion significantly in the cases considered.

Automorphism groups of invariant lattices in the Steinberg module of groups of Lie type of odd characteristic

The purpose of this article is to compute the automorphism groups of G-invariant lattices in the Steinberg module V, where G is a finite simple group of Lie type over a field of characteristic p>2.

Rank One Subspaces of Bimodules over Maximal Abelian Selfadjoint Algebras

Spaces of operators that are left and right modules over maximal abelian selfadjoint algebras (masa bimodules for short) are natural generalizations of algebras with commutative subspace lattices. This paper is concerned with density properties of finite rank operators and of various classes of compact operators in such modules. It is shown that every finite rank operator of a norm closed masa bimodule M is in the trace norm closure of the rank one subspace of M. An important consequence is that the rank one subspace of a strongly reflexive masa bimodule (that is, one which is the reflexive hull of its rank one operators) is dense in the module in the weak operator topology. However, in contrast to the situation for algebras, it is shown that such density need not hold in the ultraweak topology. A new method of representing masa bimodules is introduced. This uses a novel concept of anω-topology. With the appropriate notion ofω-support, a correspondence is established between reflexive masa bimodules and theirω-supports. It is shown that, if a C2-closed masa bimodule contains a trace class operator then it must contain rank one operators; indeed, every such operator is in the C2-norm closure of the rank one subspace of the module. Consequently the weak closure of any masa bimodule of trace class operators is strongly reflexive. However, the trace norm closure of the rank one subspace need not contain all trace class operators of the module. Also, it is shown that there exists a CSL algebra which contains no trace class operators yet contains an operator belonging to Cpfor allp>1. From this it follows that a transitive bimodule spanned by the rank one operators it contains need not be dense in Cpfor 1⩽p<∞. As an application, it is shown that there exists a commutative subspace lattice L such that L is non-synthetic but every weakly closed algebra which contains a masa and has invariant lattice L coincides with Alg L.

Damage spreading in the ( Nα, Nβ) model: exact results

The ( N α , N β ) model is investigated within the damage-spreading framework. Exact relations involving thermodynamic quantities (order parameters and correlation functions) and conveniently defined combinations of damages are obtained, which are valid for any regular (translationally invariant) lattice. The corresponding existing results for particular cases of the model, e.g., Ising, Potts, Ashkin–Teller and discrete N-vector models, are all recovered by the present relations.

Finite quaternionic matrix groups

Let D \mathcal {D} be a definite quaternion algebra such that its center has degree d d over Q \mathbb {Q} . A subgroup G G of G L n ( D ) GL_n(\mathcal {D}) is absolutely irreducible if the Q \mathbb {Q} -algebra spanned by the matrices in G G is D n × n \mathcal {D}^{n\times n} . The finite absolutely irreducible subgroups of G L n ( D ) GL_n(\mathcal {D}) are classified for n d ≤ 10 nd \leq 10 by constructing representatives of the conjugacy classes of the maximal finite ones. Methods to construct the groups and to deal with the quaternion algebras are developed. The investigation of the invariant rational lattices yields quaternionic structures for many interesting lattices.

SU(2) chiral Yang–Mills model on a lattice. Witten anomaly eaten by the lattice

Using a regularisation of the chiral SU(2) Yang–Mills model by an infinite number of Pauli–Villars fields the doubler free gauge invariant lattice model is constructed. Its continuum limit provides the recently proposed mechanism for cancelation of Witten anomaly.

μ +SR in the random bond spin glass URh 2Ge 2

As-cast single crystals of URh 2Ge 2 are heavy fermions which also behave as random-bond spin glasses. This is unusual among compounds with a translationally invariant lattice of magnetic ions. We have carried out μ + SR experiments on a single crystal in both the as-grown and annealed states. Strong spin depolarization is observed when the temperature is lowered below 15 K, but no spontaneous precession has been found. Experiments in a longitudinal field showed this relaxation to be of a static origin. This indicates the presence of large and random dipolar fields at the muon site, thus of large and randomly ordered magnetic moments. The depolarization function can be described by a stretched exponential, characteristic for spin glasses. Experiments with different angles between the crystalline axes and the initial μ + spin failed to show any anisotropy of the depolarization rate. Also, the effect of annealing (one week at 900°C) is very small.

The Fourier binest algebra

The Fourier binest algebra is defined as the intersection of the Volterra nest algebra on L2(ℝ) with its conjugate by the Fourier transform. Despite the absence of nonzero finite rank operators this algebra is equal to the closure in the weak operator topology of the Hilbert–Schmidt bianalytic pseudo-differential operators. The (non-distributive) invariant subspace lattice is determined as an augmentation of the Volterra and analytic nests (the Fourier binest) by a continuum of nests associated with the unimodular functions exp(−isx2/2) for s&gt;0. This multinest is the reflexive closure of the Fourier binest and, as a topological space with the weak operator topology, it is shown to be homeomorphic to the unit disc. Using this identification the unitary automorphism group of the algebra is determined as the semi-direct product ℝ2×κℝ for the action κt(λ, μ) =(etλ, e−t μ).

Theory of kink and domain-wall excitations in translationally invariant lattices with quartic anharmonicity

A theoretical investigation has been made of localized excitations having the form of kinks or domain walls in translationally invariant monatomic lattices with harmonic and quartic anharmonic interactions between nearest-neighbor atoms. Antisymmetric kinks are found for negative values of the anharmonic coefficient. Symmetric kinks are found to be poorly defined for translationally invariant anharmonic potentials. In the case of domain walls, stable solutions were obtained only for negative values of the anharmonic coefficient. These results are compared with those for on-site quartic anharmonicity.

Modular permutation representations of

Invariant lattices in the irreducible module associated with a complex permutation representation of the group , are studied. The submodule structure of a permutation module of this group over a finite field is described.

A cross section from invariant subspaces to inner functions

Beurling's well known theorem connects the study of invariant subspaces to that of inner functions over the unit disc. In this paper, we will further explore this connection and, as a corollary of the result, show a one to one correspondence between the components of the invariant subspace lattice and the components of the space of inner functions.

Real-space renormalization-group study of the phase transition in a Gaussian model of fractals

In this paper the phase transition of the Gaussian model on m-sheet fractals (mSG${)}_{\mathrm{l}}$ and (mDH${)}_{\mathrm{l}}$ is investigated by the real-space renormalization-group method, i.e., decimation following a spin rescaling. The latter is introduced to keep the parameter b constant. Fixed points of the renormalization-group transformation are found and discussed. Our results show the existence of different properties of phase transition between the Gaussian model and the Ising model on fractals. In addition, we find that the critical point k=b/4 in a regular Sierpinski gasket is identified, with result of k=b/d (d is the coordination number) in Euclidean space. This indicates that the critical point of the Gaussian model may be uniquely determined by the coordination number whether on homogeneous fractals or translationally invariant lattices.

Renormalization group approach to the bond percolation on Sierpinski carpets

The critical behaviors of bond percolation on a family of Sierpinski carpets (SCs) are studied. We distinguish two sorts of bonds and assign them to two kinds of occupation probabilities. We develop the usual choice of cell on translationally invariant lattices and choose suitable cells to cover the fractal lattice. On this basis we construct a new real-space renormalization group (RG) transformation scheme and use it to solve the percolation problems. Phase transitions of percolation on such fractals with infinite order of ramification are found at non-trivial bond occupation probabilities. The percolation threshold values, correlation length exponents ν, and the RG flow diagrams are obtained. The flow diagrams are remarkably similar to those of Ising model and Potts model. This agrees with the correspondence between the pure bond percolation and Potts model.

Time dependent Ginzburg - Landau model in the absence of translational invariance. Non-conserved order parameter domain growth

We have determined the static and dynamical properties of the Ginzburg - Landau model, with global coupling of the spherical type, on some non-translationally invariant lattices. Our solutions show that, in agreement with general theorems, fractal lattices with finite ramification do not display a finite temperature phase transition for any embedding dimension, d. On the other hand, the dynamical behaviour associated with the phase ordering dynamics of a non-conserved order parameter is non-trivial. Our analysis reveals that the domain size R grows in time as and relates this exponent to the three exponents which characterize the static and dynamical properties of fractal structures, namely the fractal dimension of the lattice , the random walk dimension and the spectral dimension . We also present a brief renormalization group treatment of the model. Finally, we have considered lattices with infinite ramification numbers which have spectral dimensions larger that 2 and show a finite temperature phase transition.

Invariant hermitian lattices in the steinberg module and their isometry groups

The invariant Hermitian lattices in the Steinberg module of SL2(q) are described. These lattices are connected with generalized quadratic residue codes over a field of four elements. The isometry groups of in-variant lattices are calculated. In particular, lbdimensional unimodular lattices over Eisenstein numbers with minimum norm 3 and automor- phism group Z6x PSp6(3) are obtained.

On the class of norm limits of nilpotents

It is known that every operator on a Hilbert space H whose invariant subspace lattice is possibly {(0),H} is a norm-limit of a sequence of nilpotent operators. In this note we study properties of such approximating sequences.