Exact Simulation for Discrete Time Spin Systems and Unilateral Fields

In this paper we generalize the technique presented by Häggström and Steif (Comb. Probab. Comput. 9:425–439, 2000) for the exact simulation of finite sections of infinite-volume Gibbs random fields, to a more general class of discrete time nearest neighbour spin systems. The main role is played by an auxiliary binary field, which indicates the sampling region. Percolation bounds can be used to prove that the algorithm terminates a.s. In the simplest case this field is Bernoulli; however blocking techniques can be used that destroy the independence property but extend the validity of the algorithm. Finally, the connection with stationary unilateral fields in the plane considered by Pickard (Adv. Appl. Probab. 12:655–671, 1980) and Galbraith and Walley (J. Appl. Probab. 19:332–343, 1982) is discussed.     

混合幂次为2和3的整数变量非线性型的整数部分

证明了:假设λ_1,…,λ_6是正实数,λ_1/λ_2是无理数,Dirichlet L函数满足黎曼猜想,x_1,…,x_6是正整数,那么,λ_1x_1~2+λ_2x_2~2+λ_3x_3~3+λ_4x_4~3+λ_5x_2~3+λ_3x_3~3的整数部分可表示无穷多素数.     

Droplet Excitations for the Spin-1/2 XXZ Chain with Kink Boundary Conditions

We give a precise definition for excitations consisting of a droplet of size n in the XXZ chain with various choices of boundary conditions, including kink boundary conditions and prove that, for each n, the droplet energies converge to a boundary condition independent value in the thermodynamic limit. We rigorously compute an explicit formula for this limiting value using the Bethe Ansatz. Submitted: January 26, 2006; Accepted: April 10, 2006     

Exact Results for the Isotropic Spin-1/2 Heisenberg Chain With Dissipative Boundary Driving

Theoretical and Mathematical Physics - We consider the open isotropic spin-1/2 Heisenberg quantum spin chain with a finite number N of sites coupled at the ends to a dissipative environment that...     

Emptiness Formation Probability in the Spin-1/2 XXZ Heisenberg Chain

We present several explicit results for the emptiness formation probability in the spin-1/2 XXZ Heisenberg chain. We obtain these results using new integral representations for the correlation functions.     

K-Theory for the Integer Heisenberg Groups

We give a closed formula for topological K-theory of the homogeneous space N/, where is the standard integer lattice in the simply connected Heisenberg Lie group N of dimension 2n+1, n . The main tools in our calculations are obtained by computing diagonal forms for certain incidence matrices that arise naturally in combinatorics.     

On Z3-Actions on Spin 4-Manifolds

Let X be a closed, simply-connected, smooth, spin 4-manifold whose intersection form is isomorphic to 2 k(-E_8)⊕lH, where H is the hyperbolic form. In this paper, the authors prove that if there exists a locally linear pseudofree Z_3-action on X,then Sign(g, X) ≡-k mod 3. They also investigate the smoothability of locally linear Z_3-action satisfying above congruence. In particular, it is proved that there exist some nonsmoothable locally linear Z_3-actions on certain elliptic surfaces.     

Integer programming models for the q-mode problem

The q-mode problem is a combinatorial optimization problem that requires partitioning of objects into clusters. We discuss theoretical properties of an existing mixed integer programming (MIP) model for this problem and offer alternative models and enhancements. Through a comprehensive experiment we investigate computational properties of these MIP models. This experiment reveals that, in practice, the MIP approach is more effective for instances containing strong natural clusters and it is not as effective for instances containing weak natural clusters. The experiment also reveals that one of the MIP models that we propose is more effective than the other models for solving larger instances of the problem.     

Integer linear programming models for 2-staged two-dimensional Knapsack problems

 We are given a unique rectangular piece of stock material S, with height H and width W, and a list of m rectangular shapes to be cut from S. Each shape's type i (i = 1, ..., m) is characterized by a height , a width , a profit , and an upper bound ub _i indicating the maximum number of items of type i which can be cut. We refer to the Two-Dimensional Knapsack (TDK) as the problem of determining a cutting pattern of S maximizing the sum of the profits of the cut items. In particular, we consider the classical variant of TDK in which the maximum number of cuts allowed to obtain each item is fixed to 2, and we refer to this problem as 2-staged TDK (2TDK). For the 2TDK problem we present two new Integer Linear Programming models, we discuss their properties, and we compare them with other formulations in terms of the LP bound they provide. Finally, both models are computationally tested within a standard branch-and-bound framework on a large set of instances from the literature by reinforcing them with the addition of linear inequalities to eliminate symmetries. Received: October 17, 2000 / Accepted: December 19, 2001 Published online: September 27, 2002 Key words. packing – cutting – integer linear programming     

Spin Representations and Powers of 2

The elementary divisors of the reduced spin 2–decomposition matrix for the double covers of the finite symmetric groups are described. In addition, the maximal power of 2 dividing all spin character values on a fixed conjugacy class corresponding to a cycle type with odd parts only is determined.     

Mordell方程y~3=x~2+2p~4的正整数解

设p是奇素数.对于非负整数r,设U_(2r+1)=(α~(2r+1)+β~(2r+1))/2~(1/2),V_(2r+1)=(α~(2r+1)-β~(2r+1))/6~(1/2),其中α=(1+3~(1/2))/2~(1/2),β=(1-3~(1/2))/2~(1/2).运用初等数论方法证明了:方程y~3=x~2+2p~4有适合gcd(x,y)=1的正整数解(x,y)的充要条件是p=U_(2m+1),其中m是正整数.当上述条件成立时,方程仅有正整数解(x,y)=(V(2m+1)(V_(2m+1)~2-6),V_(2m+1)~2+2)适合gcd(x,y)=1.由此可知:当p10000时,方程仅有正整数解(p,x,y)=(5,9,11),(19,1265,123),(71,68675,1683)和(3691,9677201305,4541163)适合gcd(x,y)=1.     

关于丢番图方程 χ^3一У^6=3pz^2的整数解

设p=5（mod 6）为素数．证明了丢番图方程χ^3一У^6=3pz^2。在p=5（mod 12）为素数时均无正整数解;在P=11（mod 12）为素数时均有无穷多组正整数解,并且还获得了该方程全部正整数解的通解公式,同时还给出了该方程的部分整数解．     

Volume-preserving Fields and Reeb Fields on 3-manifolds

Volume-preserving field X on a 3-manifold is the one that satisfies LxΩ = 0 for some volume Ω. The Reeb vector field of a contact form is of volume-preserving, but not conversely. On the basis of Geiges-Gonzalo＇s parallelization results, we obtain a volume-preserving sphere, which is a triple of everywhere linearly independent vector fields such that all their linear combinations with constant coefficients are volume-preserving fields. From many aspects, we discuss the distinction between volume-preserving fields and Reeb-like fields. We establish a duality between volume-preserving fields and h-closed 2-forms to understand such distinction. We also give two kinds of non-Reeb-like but volume-preserving vector fields to display such distinction.     

Functional Equations and Quantum Separation of Variables for 3d Spin Models

The recently proposed invariant formulation of the auxiliary linear problem for 3d integrable models provides several new ideas for solving the spectral problem of 3d spin models, e.g., the Zamolodchikov–Bazhanov–Baxter model in its vertex formulation. This paper announces results following from the invariant formulation. We formulate the class of 3d spin models that are essentially appropriately parameterized inhomogeneous Zamolodchikov–Bazhanov–Baxter models, present an expression for the generating function of the complete set of matrices commuting with the transfer matrix of this model (integrals of motion), give the functional equations defining the eigenvalues of the integrals of motion and the transfer matrices, explicitly describe the groupoid of isospectral transformations of the initial system of integrals of motion, and finally give an explicit parameterization of a projection operator onto the separated states in the sense of the quantum separation of variables (functional Bethe ansatz).     

Integer Polynomials with Roots mod p for all Primes p

Let f(X) be an integer polynomial which is a product of two irreducible factors. Assume that f(X) has a root mod p for all primes p. If the splitting field of f(X) over the rationals is a cyclic extension of the stem fields, then the Galois group of f(X) over the rationals is soluble and of bounded Fitting length. Moreover, the fixed groups of the stem extensions are in, some sense, unique.     

Optimal Base Station Positioning and Channel Assignment for 3G Mobile Networks by Integer Programming

In this paper, discrete mathematical programming approaches are used to solve the frequency allocation and cell site selection problem in an integrated setup. Both CDMA (code division multiple access) and FD/TDMA (frequency/time division multiple access) technologies will be important for 3rd generation mobile systems. If all users share the same bandwidth, base transmitter stations should be placed such that a maximum of traffic can be carried at low interference rates. The expected traffic is represented by spatially scattered weighted nodes. The problem to select an optimal set of base station locations from a given pool of configurations is formulated as an integer linear program and solved by combinatorial optimization methods. For systems which employ FD/TDMA schemes, the cell site optimization process depends on the assignment of channels. We suggest an integrated linear programming approach to solve both objectives in a single planning step. Because of the problems' tremendous complexity, special branch-and-bound procedures are developed as exact and approximate solution methods. An examples is given for a typical urban scenario with base transmitters below roof tops.     

R-Matrices for SU(2)-Invariant Two-Leg Spin Ladders

A list of R-matrices for the set of eight integrable spin ladders is presented. Four of these matrices are new. Bibliography: 13 titles.