Fire containment in grids of dimension three and higher

We consider a deterministic discrete-time model of fire spread introduced by Hartnell [Firefighter! an application of domination, Presentation, in: 20th Conference on Numerical Mathematics and Computing, University of Manitoba in Winnipeg, Canada, September 1995] and the problem of minimizing the number of burnt vertices when a fixed number of vertices can be defended by firefighters per time step. While only two firefighters per time step are needed in the two-dimensional lattice to contain any outbreak, we prove a conjecture of Wang and Moeller [Fire control on graphs, J. Combin. Math. Combin. Comput. 41 (2002) 19-34] that 2d-1 firefighters per time step are needed to contain a fire outbreak starting at a single vertex in the d-dimensional square lattice for d?3; we also prove that in the d-dimensional lattice, d?3, for each positive integer f there is some outbreak of fire such that f firefighters per time step are insufficient to contain the outbreak. We prove another conjecture of Wang and Moeller that the proportion of elements in the three-dimensional grid P_n×P_n×P_n which can be saved with one firefighter per time step when an outbreak starts at one vertex goes to 0 as n gets large. Finally, we use integer programming to prove results about the minimum number of time steps needed and minimum number of burnt vertices when containing a fire outbreak in the two-dimensional square lattice with two firefighters per time step.     

Computation of the Highest Coefficients of Weighted Ehrhart Quasi-polynomials of Rational Polyhedra

This article concerns the computational problem of counting the lattice points inside convex polytopes, when each point must be counted with a weight associated to it. We describe an efficient algorithm for computing the highest degree coefficients of the weighted Ehrhart quasi-polynomial for a rational simple polytope in varying dimension, when the weights of the lattice points are given by a polynomial function h. Our technique is based on a refinement of an algorithm of A.?Barvinok in the unweighted case (i.e., h≡1). In contrast to Barvinok’s method, our method is local, obtains an approximation on the level of generating functions, handles the general weighted case, and provides the coefficients in closed form as step polynomials of the dilation. To demonstrate the practicality of our approach, we report on computational experiments which show that even our simple implementation can compete with state-of-the-art software.     

Uniform Anderson localization,unimodal eigenstates and simple spectra in a class of “haarsh” deterministic potentials

We consider a particular class of families of multi-dimensional lattice Schrödinger operators with deterministic (e.g., quasi-periodic) potentials generated by the “hull” given by an orthogonal series over the Haar wavelet basis on the torus of arbitrary dimension, with expansion coefficients considered as independent parameters. In the strong disorder regime, we prove Anderson localization for generic operator families and show that all localized eigenfunctions are unimodal and feature uniform exponential decay away from their respective localization centers. We also prove a variant of the Minami estimate for deterministic potentials and simplicity of the spectrum.     

Chaotic transport of a matter-wave soliton in a biperiodically driven optical superlattice

Under the effective particle approximation, we study the temporal ratchet effect for chaotic transport of a matter-wave soliton consisting of an attractive Bose–Einstein condensate held in a quasi-one-dimensional symmetric optical superlattice with biperiodic driving. It is known that chaos can substitute for disorder in Anderson’s scenario [Wimberger S, Krug A, Buchleitner A. Phys Rev Lett 2002;89:263601] and only a higher level of disorder can induce Anderson localization for some special systems [Schwartz T, Bartal G, Fishman S, Segev M. Nature 2007;46:52], and a matter-wave soliton can transit to chaos with high or low probability in a high- or low-chaoticity region [Zhu Q, Hai W, Rong S. Phys Rev E 2009;80:016203]. Here we demonstrate that varying the driving phase to break the time reversal symmetry of the system can increase the size of the high-chaoticity region for low- and moderate-frequency regions. Consequently, the parameter region of the exponential spatial localization increases to the same size, and the low-chaoticity and delocalization region, which includes subregions of the ratchet effect and its inverse effect, correspondingly decreases. The positive dependence of the localization on the driving frequency is also revealed. The results indicate that a high-chaoticity region could replace higher disorder and assists in Anderson localization. From the results we suggest a method for controlling directed motion of a matter-wave soliton by adjusting the driving frequency and amplitude to strengthen or suppress, or even reverse, the temporal ratchet effect.     

Simulation of a Hybrid-Forming Process using Temperature- and Rate-Dependent Viscoplastic Material Models

Hybrid-forming processes for graded structures are quite innovative methods for the production of components with tailored properties, particularly tailored material properties and geometrical shape. In this contribution a hybrid-forming process based on the utilization of locally varying thermo-mechanical effects is investigated [1]. For process optimization and improvement of the resulting work piece the simulation of the entire forming process is necessary in modern engineering. The main topics of this contribution are the simulation of the cyclic thermal loaded forming tool and the simulation of the work piece treated at large deformations with phase transformations. For both materials temperature- and rate-dependent viscoplastic material models are applied and parameter identification using cyclic tensile-compression tests for the forming tool material and phase transformation tests for a low-alloy steel similar to the work piece material is presented. For validation of finite-element-calculations for the forming tool thermal shock experiments are performed with optical deformation measurements. For validation of finite-element-calculations for the work piece numerical results of geometry and structure after heating, forming and cooling are compared to experimental micro sections. Results concerning the forming tool will be used for future lifetime prediction and results concerning the work piece will be used for future specific setting of graded material properties. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two-particle scattering in a periodic medium

We consider a two-particle Schrödinger difference operator with a periodic potential perturbed by an exponentially decaying interaction potential for particles on a one-dimensional lattice. We obtain rigorous results for the two-particle scattering problem in the case of a small interaction and low velocities. Here, as in other quasi-one-dimensional models, small interactions can significantly affect the scattering pattern. In particular, we find the probability that the velocities of two particles in a periodic medium (e.g., they can be ultracold atoms in a one-dimensional optical lattice) change their signs during a collision. This probability increases as the relative velocity decreases and also as the absolute value of the matrix element between single-particle unperturbed Bloch states increases.     

Congruences via abelian groups

Given a group G acting on a set S, Möbius inversion over the lattice of subgroups can be used to obtain congruences relating the number of elements of S stabilized by each subgroup. By taking S to be a set of subsets, partitions, or permutations congruences for binomial and multinomial coefficients, Stirling numbers of both kinds, and various other combinatorial sequences are derived. Congruences for different moduli are obtained by varying the order of G.     

Generalized coherent states and nonlinear dynamics of a lattice quasispin model

The nonlinear lattice equation of the ϕ⁶ theory is studied by using the technique of generalized coherent states associated to a SU(2) Lie group. We analyze the discrete nonlinear equation with weak interaction between sites. The existence of saddles and centers is shown. The qualitative parametric domains which contain kinks, bubbles and plane waves were obtained. The specific implications of saddles and centers to the parametric first- and second-order phase transitions are identified and analyzed.     

Effect of self-interaction on the phase diagram of a Gibbs-like measure derived by a reversible Probabilistic Cellular Automata

Cellular Automata are discrete-time dynamical systems on a spatially extended discrete space which provide paradigmatic examples of nonlinear phenomena. Their stochastic generalizations, i.e., Probabilistic Cellular Automata (PCA), are discrete time Markov chains on lattice with finite single-cell states whose distinguishing feature is the parallel character of the updating rule. We study the ground states of the Hamiltonian and the low-temperature phase diagram of the related Gibbs measure naturally associated with a class of reversible PCA, called the cross PCA. In such a model the updating rule of a cell depends indeed only on the status of the five cells forming a cross centered at the original cell itself. In particular, it depends on the value of the center spin (self-interaction). The goal of the paper is that of investigating the role played by the self-interaction parameter in connection with the ground states of the Hamiltonian and the low-temperature phase diagram of the Gibbs measure associated with this particular PCA.     

Lattice geometry and pythagorean triangles

By drawing a Pythagorean triangle in a quadratic lattice and attaching a congruent lattice at the hypotenuse there will occur a Moiré effect with a new quadratic lattice of enlarged scale in the superposition. This new lattice is related to the parameterization of the Pythagorean triangle. A similar effect occurs with triangles with integer side lengths and an angle of 120° in a regular triangular lattice. We work with dot lattices on transparencies to visualize the optical effects.     

Minimal condition for shortest vectors in lattices of low dimension

For a lattice, finding a nonzero shortest vector is computationally difficult in general. The problem becomes quite complicated even when the dimension of the lattice is five. There are two related notions of reduced bases, say, Minkowski-reduced basis and greedy-reduced basis. When the dimension becomes d = 5, there are greedy-reduced bases without achieving the first minimum while any Minkowski-reduced basis contains the shortest four linearly independent vectors. This suggests that the notion of Minkowski-reduced basis is somewhat strong and the notion of greedy-reduced basis is too weak for a basis to achieve the first minimum of the lattice. In this work, we investigate a more appropriate condition for a basis to achieve the first minimum for d = 5. We present a minimal sufficient condition, APG⁺, for a five dimensional lattice basis to achieve the first minimum in the sense that any proper subset of the required inequalities is not sufficient to achieve the first minimum.     

On a Theorem of Bandt and Wang and Its Extension to <Emphasis Type="Italic">p</Emphasis>2-tiles

We study tilings of the plane by a single prototile with respect to the lattice and to the crystallographic group p2. We are interested in the connection between the neighbors of a tile in the tiling and its topology. We show that lattice and p2-tiles always have at least six neighbors. We characterize self-affine tiles that are homeomorphic to a disk in a rather easy way by the set and number of neighbors of the central tile in the tiling. This extends the work of Bandt and Wang devoted to lattice self-affine disk-like tiles of the plane.     

Spatiotemporal quasiperiodicity transition to chaos in the one-way coupled bistability lattice system

One-way coupled optical bistability lattice (OCBL) system for open flow is investigated. By using numerical simulations spatiotemporal quasiperiodicity is found. A rough phase diagram indicating the main spatiotemporal dynamic behavior in weakly coupled lattice is given. The transitions between spatiotemporal quasiperiodicity and chaos are observed. This result is important for the understanding of turbulence. Project supported by the Natural Science Foundation of Hebei Province.     

Mesures pluriharmoniques et comparaison de deux théories du potentiel au bord

To explore the possibility of developing a phase transition theory in quantum lattice systems within the C¹-framework, the notion of phase transition free states on UHF algebras is introduced and the relationship between phase transition free states and unique KMS states are studied.     

Analysis of a Polynomial System Arising in the Design of an Optical Lattice Filter Useful in Channelization

In this work, we consider design questions for an active optical lattice filter, which is being manufactured at the University of Texas at Dallas, and which has proven to be useful in the signal processing task of RF channelization. The filter can be described by a linear, discrete time state space model. The controlling agents, the gains, are embedded in the matrices intervening in this state space model. Consequently, techniques from linear feedback control theory do not apply. We concentrate on the question of finding real valued gains so that the A matrix of the state space model has a prescribed characteristic polynomial. We find that three of the coefficients can be arbitrarily picked, but that the remaining are constrained by these and the other system parameters. Our techniques use methods from constructive algebraic geometry.     

An existence result for a two‐phase Stefan problem arising in metal casting

We present a model arising from the thermal modelling of two metal casting processes. We consider an enthalpy formulation for this two‐phase Stefan problem in a time varying three‐dimensional domain and consider convective heat transfer in the liquid phase. Then, we introduce a weak formulation in a fixed domain, by means of a suitable transformation. Existence of solution is obtained by applying an abstract theorem. The proof of this theorem is done by taking an implicit discretization in time together with a regularization. By passing to the limit in the regularization parameter and in the time step, we obtain the existence of solution of the continuous problem. Copyright © 2005 John Wiley & Sons, Ltd.     

A Second Look at the Toric h-Polynomial of a Cubical Complex

We provide an explicit formula for the toric h-contribution of each cubical shelling component, and a new combinatorial model to prove Chan??s result on the non-negativity of these contributions. Our model allows for a variant of the Gessel-Shapiro result on the g-polynomial of the cubical lattice, this variant may be shown by simple inclusion-exclusion. We establish an isomorphism between our model and Chan??s model and provide a reinterpretation in terms of noncrossing partitions. By discovering another variant of the Gessel-Shapiro result in the work of Denise and Simion, we find evidence that the toric h-polynomials of cubes are related to the Morgan-Voyce polynomials via Viennot??s combinatorial theory of orthogonal polynomials.     

Higher cohomology for Anosov actions on certain homogeneous spaces

We study the smooth untwisted cohomology with real coefficients for the action on [SL(2,?)×…×SL(2,?)]/Γ by the subgroup of diagonal matrices, where Γ is an irreducible lattice. We show that in the top degree, the obstructions to solving the coboundary equation come from distributions that are invariant under the action. We also show that in intermediate degrees, the cohomology trivializes. It has been conjectured by A. Katok and S. Katok that, analogously to Liv?ic’s theorem for Anosov flows for a standard partially hyperbolic ? ^d - or ? ^d - action, the obstructions to solving the top-degree coboundary equation are given by periodic orbits, and also that the intermediate cohomology trivializes, as it is known to do in the first degree by work of Katok and Spatzier. Katok and Katok proved their conjecture for abelian groups of toral automorphisms. Our results verify the “intermediate cohomology” part of the conjecture for diagonal subgroup actions on SL(2,?) ^d /Γ and are a step in the direction of the “top-degree cohomology” part.     

Lattice path moments by cut and paste

In the coordinate plane consider those lattice paths whose step types consist of (1,1), (1,−1), and perhaps one or more horizontal steps. For the set of such paths running from (0,0) to (n+2,0) and remaining strictly elevated above the horizontal axis elsewhere, we define a zeroth moment (cardinality), a first moment (essentially, the total area), and a second moment, each in terms of the ordinates of the lattice points traced by the paths. We then establish a bijection relating these moments to the cardinalities of sets of selected marked unrestricted paths running from (0,0) to (n,0). Roughly, this bijection acts by cutting each elevated path into well-defined subpaths and then pasting the subpaths together in a specified order to form an unrestricted path.     

On the involutions fixing the class of a lattice

With any integral lattice Λ in n-dimensional Euclidean space we associate an elementary abelian 2-group I(Λ) whose elements represent parts of the dual lattice that are similar to Λ. There are corresponding involutions on modular forms for which the theta series of Λ is an eigenform; previous work has focused on this connection. In the present paper I(Λ) is considered as a quotient of some finite 2-subgroup of . We establish upper bounds, depending only on n, for the order of I(Λ), and we study the occurrence of similarities of specific types.