Reversibility of 1D Cellular Automata with Periodic Boundary over Finite Fields $\pmb{ {\mathbb{Z}}}_{p}$

The reversibility problem for linear cellular automata with null boundary defined by a rule matrix in the form of a pentadiagonal matrix was studied recently over the binary field ℤ₂ (del Rey and Rodriguez Sánchez in Appl. Math. Comput., 2011, doi:). In this paper, we study one-dimensional linear cellular automata with periodic boundary conditions over any finite field ℤ _p . For any given p≥2, we show that the reversibility problem can be reduced to solving a recurrence relation depending on the number of cells and the coefficients of the local rules defining the one-dimensional linear cellular automata. More specifically, for any given values (from any fixed field ℤ _p ) of the coefficients of the local rules, we outline a computer algorithm determining the recurrence relation which can be solved by testing reversibility of the cellular automaton for some finite number of cells. As an example, we give the full criteria for the reversibility of the one-dimensional linear cellular automata over the fields ℤ₂ and ℤ₃.     

A Condensed Matter Interpretation of SM Fermions and Gauge Fields

We present the bundle (Aff(3)⊗ℂ⊗Λ)(ℝ³), with a geometric Dirac equation on it, as a three-dimensional geometric interpretation of the SM fermions. Each (ℂ⊗Λ)(ℝ³) describes an electroweak doublet. The Dirac equation has a doubler-free staggered spatial discretization on the lattice space (Aff(3)⊗ℂ)(ℤ³). This space allows a simple physical interpretation as a phase space of a lattice of cells. We find the SM SU(3) _c ×SU(2) _L ×U(1) _Y action on (Aff(3)⊗ℂ⊗Λ)(ℝ³) to be a maximal anomaly-free gauge action preserving E(3) symmetry and symplectic structure, which can be constructed using two simple types of gauge-like lattice fields: Wilson gauge fields and correction terms for lattice deformations. The lattice fermion fields we propose to quantize as low energy states of a canonical quantum theory with ℤ₂-degenerated vacuum state. We construct anticommuting fermion operators for the resulting ℤ₂-valued (spin) field theory. A metric theory of gravity compatible with this model is presented too.     

Surface States and Spectra

Let ℤ₊ ^{d +1}= ℤ⁺×ℤ₊, let H ₀ be the discrete Laplacian on the Hilbert space l ²(ℤ₊ ^{d +1}) with a Dirichlet boundary condition, and let V be a potential supported on the boundary ∂ℤ₊ ^{d +1}. We introduce the notions of surface states and surface spectrum of the operator H=H ₀+V and explore their properties. Our main result is that if the potential V is random and if the disorder is either large or small enough, then in dimension two H has no surface spectrum on σ(H ₀) with probability one. To prove this result we combine Aizenman–Molchanov theory with techniques of scattering theory. Received: 18 September 2000 / Accepted: 21 November 2000     

Transitions in three-dimensional XY magnets with two chiral order parameters

Two models of classic XY antiferromagnets in three dimensions are studied by Monte Carlo simulation: the model on a simple cubic lattice with two extra intralayer exchanges and the model on a stackedtriangular lattice with an extra interlayer exchange. In suggested models, the order parameters are magnetization and two chiral parameters. A transition corresponds to breaking ℤ₂ ⊗ ℤ₂ ⊗ SO(2) symmetry. A distinct first order transition is found in both models.     

Inverse resonance scattering for Jacobi operators

The Jacobi operator (Jf) _n = a _n−1 f _n−1 +a _n f _n+1 + b _n f _n on ℤ with real finitely supported sequences (a _n − 1) _n∈ℤ and (b _n ) _n∈ℤ is considered. The inverse problem for two mappings (including their characterization): (a _n , b _n , n ∈ ℤ) → {the zeros of the reflection coefficient} and (a _n , b _n , n ∈ ℤ) → {the eigenvalues and the resonances} is solved. All Jacobi operators with the same eigenvalues and resonances are also described.     

Quantum Geometry of Algebra Factorisations¶and Coalgebra Bundles

We develop the noncommutative geometry (bundles, connections etc.) associated to algebras that factorise into two subalgebras. An example is the factorisation of matrices M ₂(ℂ)=ℂℤ₂·ℂℤ₂. We also further extend the coalgebra version of theory introduced previously, to include frame resolutions and corresponding covariant derivatives and torsions. As an example, we construct q-monopoles on all the Podleś quantum spheres S ² _q,s . Received: 25 September 1998 / Accepted: 23 February 2000     

Construction of the Incipient Infinite Cluster for Spread-out Oriented Percolation Above 4 + 1 Dimensions

 We construct the incipient infinite cluster measure (IIC) for sufficiently spread-out oriented percolation on ℤ ^d × ℤ₊, for d +1 > 4+1. We consider two different constructions. For the first construction, we define ℙ ⁿ (E) by taking the probability of the intersection of an event E with the event that the origin is connected to (x,n)  ℤ ^d × ℤ₊, summing this probability over x  ℤ ^d , and normalising the sum to get a probability measure. We let n → ∞ and prove existence of a limiting measure ℙ_∞, the IIC. For the second construction, we condition the connected cluster of the origin in critical oriented percolation to survive to time n, and let n → ∞. Under the assumption that the critical survival probability is asymptotic to a multiple of n ⁻¹, we prove existence of a limiting measure ℚ_∞, with ℚ_∞ = ℙ_∞. In addition, we study the asymptotic behaviour of the size of the level set of the cluster of the origin, and the dimension of the cluster of the origin, under ℙ_∞. Our methods involve minor extensions of the lace expansion methods used in a previous paper to relate critical oriented percolation to super-Brownian motion, for d+1 > 4+1. Received: 13 December 2001 / Accepted: 11 July 2002 Published online: 29 October 2002 RID="*" ID="*" Present address: Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. E-mail: rhofstad@win.tue.nl     

Infinite Volume Limit for the Stationary Distribution of Abelian Sandpile Models

We study the stationary distribution of the standard Abelian sandpile model in the box Λ_n = [-n, n] ^d ∩ ℤ ^d for d≥ 2. We show that as n→ ∞, the finite volume stationary distributions weakly converge to a translation invariant measure on allowed sandpile configurations in ℤ ^d . This allows us to define infinite volume versions of the avalanche-size distribution and related quantities. The proof is based on a mapping of the sandpile model to the uniform spanning tree due to Majumdar and Dhar, and the existence of the wired uniform spanning forest measure on ℤ ^d . In the case d > 4, we also make use of Wilson’s method. An erratum to this article is available at .     

Entropic Repulsion for the High Dimensional Gaussian Lattice Field Between Two Walls

The main aim of this paper is to discuss the entropic repulsion of random interfaces between two hard walls. We consider the d (≥ 3)-dimensional Gaussian lattice field on ℝ^λ _N , λ _N = [−N, N] ^d ∩ ℤ ^d and identify the repulsion of the field as N → ∞ under the condition that the field lies between two hard walls at the height level 0 and L in Λ _N where L is large enough but finite. We also study the same problem for two layered interfaces case.     

Extended q -euler numbers and polynomials associated with fermionic p -adic q -integral on Z p

The purpose of this paper is to construct extended q-Euler numbers and polynomials related to fermionic p-adic q-integral on ℤ _p . By evaluating a multivariate p-adic q-integral on ℤ _p , we give new explicit formulas related to these numbers and polynomials.     

A C# package for assembling quantum circuits and generating associated polynomial sets

Recently it has been shown that elements of the unitary matrix determined by a quantum circuit can be computed by counting the number of common roots in the finite field ℤ₂ for a certain set of multivariate polynomials over ℤ₂. In this paper we present a C# package that allows a user to assemble a quantum circuit and to generate the multivariate polynomial set associated with the circuit. The generated polynomial system can further be converted to the canonical triangular involutive basis form, which is appropriate for counting the number of common roots of the polynomials. The text was submitted by the authors in English.     

Adiabatic Times for Markov Chains and Applications

We state and prove a generalized adiabatic theorem for Markov chains and provide examples and applications related to Glauber dynamics of the Ising model over ℤ ^d /nℤ ^d . The theorems derived in this paper describe a type of adiabatic dynamics for l¹(\mathbbRⁿ₊)\ell^{1}(\mathbb{R}^{n}_{+}) norm preserving, time inhomogeneous Markov transformations, while quantum adiabatic theorems deal with ℓ ²(ℂ ⁿ ) norm preserving ones, i.e. gradually changing unitary dynamics in ℂ ⁿ .     

Continuous-Time Quantum Walk on Integer Lattices and Homogeneous Trees

This paper is concerned with the continuous-time quantum walk on ℤ, ℤ ^d , and infinite homogeneous trees. By using the generating function method, we compute the limit of the average probability distribution for the general isotropic walk on ℤ, and for nearest-neighbor walks on ℤ ^d and infinite homogeneous trees. In addition, we compute the asymptotic approximation for the probability of the return to zero at time t in all these cases.     

Symplectic transformations and entanglement in multipartite finite systems

Symplectic Sp(2l, ℤ_p) transformations in l-partite finite systems are explicitly constructed. The general method is applied to bi-partite and tri-partite systems. The effect of these transformations on the correlations and entanglement between the subsystems is discussed.     

The Behaviour of Aging Functions in One-Dimensional Bouchaud's Trap Model

Let τ_i be a collection of i.i.d. positive random variables with distribution in the domain of attraction of an α-stable law with α<1. The symmetric Bouchaud's trap model on ℤ is a Markov chain X(t) whose transition rates are given by w_xy=(2τ_x)⁻¹ if x, y are neighbours in ℤ. We study the behaviour of two correlation functions: ℙ[X(t_w+t)=X(t_w)] and It is well known that for any of these correlation functions a time-scale t=f(t_w) such that aging occurs can be found. We study these correlation functions on time-scales different from f(t_w), and we describe more precisely the behaviour of a singular diffusion obtained as the scaling limit of Bouchaud's trap model. Work supported by DFG Research Center Matheon ``Mathematics for key technologies'     

Realizability of Point Processes

There are various situations in which it is natural to ask whether a given collection of k functions, ρ _j (r ₁,…,r _j ), j=1,…,k, defined on a set X, are the first k correlation functions of a point process on X. Here we describe some necessary and sufficient conditions on the ρ _j ’s for this to be true. Our primary examples are X=ℝ ^d , X=ℤ ^d , and X an arbitrary finite set. In particular, we extend a result by Ambartzumian and Sukiasian showing realizability at sufficiently small densities ρ ₁(r). Typically if any realizing process exists there will be many (even an uncountable number); in this case we prove, when X is a finite set, the existence of a realizing Gibbs measure with k body potentials which maximizes the entropy among all realizing measures. We also investigate in detail a simple example in which a uniform density ρ and translation invariant ρ ₂ are specified on ℤ; there is a gap between our best upper bound on possible values of ρ and the largest ρ for which realizability can be established.     

An Asymptotic Expansion and Recursive Inequalities for the Monomer-Dimer Problem

Let λ _d (p) be the p monomer-dimer entropy on the d-dimensional integer lattice ℤ ^d , where p∈[0,1] is the dimer density. We give upper and lower bounds for λ _d (p) in terms of expressions involving λ _d−1(q). The upper bound is based on a conjecture claiming that the p monomer-dimer entropy of an infinite subset of ℤ ^d is bounded above by λ _d (p). We compute the first three terms in the formal asymptotic expansion of λ _d (p) in powers of  \frac1d\frac{1}{d}. We prove that the lower asymptotic matching conjecture is satisfied for λ _d (p). Converted to a power series in p, our “formal” expansion shows remarkable validity in low dimensions, d=1,2,3, in which dimensions we give some numerical studies.     

On Noncommutative Multi-Solitons

 We find the moduli space of multi-solitons in noncommutative scalar field theories at large θ, in arbitrary dimension. The existence of a non-trivial moduli space at leading order in 1/θ is a consequence of a Bogomolnyi bound obeyed by the kinetic energy of the θ=∞ solitons. In two spatial dimensions, the parameter space for k solitons is a K?hler de-singularization of the symmetric product (ℝ²) ^k /S _k . We exploit the existence of this moduli space to construct solitons on quotient spaces of the plane: ℝ²/ℤ _k , cylinder, and T ² . However, we show that tori of area less than or equal to 2πθ do not admit stable solitons. In four dimensions the moduli space provides an explicit K?hler resolution of (ℝ⁴) ^k /S _k . In general spatial dimension 2d, we show it is isomorphic to the Hilbert scheme of k points in ℂ ^d , which for d>2 (and k>3) is not smooth and can have multiple branches. Received: 29 May 2001 / Accepted: 16 August 2002 Published online: 7 November 2002 Communicated by R.H. Dijkgraaf     

The Global Renormalization Group Trajectory in a Critical Supersymmetric Field Theory on the Lattice ℤ3

We consider an Euclidean supersymmetric field theory in ℤ³ given by a supersymmetric Φ⁴ perturbation of an underlying massless Gaussian measure on scalar bosonic and Grassmann fields with covariance the Green’s function of a (stable) Lévy random walk in ℤ³. The Green’s function depends on the Lévy-Khintchine parameter with 0<α<2. For the Φ⁴ interaction is marginal. We prove for sufficiently small and initial parameters held in an appropriate domain the existence of a global renormalization group trajectory uniformly bounded on all renormalization group scales and therefore on lattices which become arbitrarily fine. At the same time we establish the existence of the critical (stable) manifold. The interactions are uniformly bounded away from zero on all scales and therefore we are constructing a non-Gaussian supersymmetric field theory on all scales. The interest of this theory comes from the easily established fact that the Green’s function of a (weakly) self-avoiding Lévy walk in ℤ³ is a second moment (two point correlation function) of the supersymmetric measure governing this model. The rigorous control of the critical renormalization group trajectory is a preparation for the study of the critical exponents of the (weakly) self-avoiding Lévy walk in ℤ³.