Numerical simulation for the solitary wave of Zakharov–Kuznetsov equation based on lattice Boltzmann method

The Zakharov–Kuznetsov equation is considered, which is an equation describing two dimensional weakly nonlinear ion-acoustic waves in plasma. We focus on using the lattice Boltzmann method to study the Zakharov–Kuznetsov equation. A lattice Boltzmann model is constructed. In numerical experiments, the propagation of the single solitary wave and the collision of double solitary waves are simulated. The results with different parameters are investigated and compared.     

Modeling and simulation of plasma jet by lattice Boltzmann method

In order to find a simple and efficient simulation for plasma spray process, an attempt of modeling was made to calculate velocity and temperature field of the plasma jet by hexagonal 7-bit lattice Boltzmann method (LBM) in this paper. Utilizing the methods of Chapman–Enskog expansion and multi-scale expansion, the authors derived the macro equations of the plasma jet from the lattice Boltzmann evolution equations on the basis of selecting two opportune equilibrium distribution functions. The present model proved to be valid when the predictions of the current model were compared with both experimental and previous model results. It is found that the LBM is simpler and more efficient than the finite difference method (FDM). There is no big variation of the flow characteristics, and the isotherm distribution of the turbulent plasma jet is compared with the changed quantity of the inlet velocity. Compared with the velocity at the inlet, the temperature at the inlet has a less influence on the characteristics of plasma jet.     

General propagation lattice Boltzmann model for variable-coefficient non-isospectral KdV equation

In this paper, a general propagation lattice Boltzmann model for variable-coefficient non-isospectral Korteweg–de Vries (vc-nKdV) equation, which can describe the interfacial waves in a two layer liquid and Alfvén waves in a collisionless plasma, is proposed by selecting appropriate equilibrium distribution function and adding the compensate function. The Chapman–Enskog analysis shows that the vc-nKdV equation can be recovered correctly from the present model. Numerical simulation for the non-propagating one soliton of this equation in different situations is conducted as validation. It is found that the numerical results match well with the analytical solutions, which demonstrates that the current general propagation lattice Boltzmann model is a satisfactory and efficient method, and could be more stable and accurate than the standard lattice Bhatnagar–Gross–Krook model.     

Enumeration of the self-avoiding polygons on a lattice by the Schwinger-Dyson equations

We show how to compute the generating function of the self-avoiding polygons on a lattice by using the statistical mechanics Schwinger-Dyson equations for the correlation functions of theN-vector spin model on that lattice.     

Complex-temperatures zeros of the partition function in spin-glass models

We propose a method for approximating the density of partition zeros for real-dimensional spin-glass models. The method uses the complex-temperature data of the ferromagnetic model on the same lattice. We obtain the universality line in the complex-temperature space. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 3, pp. 520–528, September, 2000.     

Lattice Boltzmann solver of Rossler equation

We proposed a lattice Boltzmann model for the Rossler equation. Using a method of multiscales in the lattice Boltzmann model, we get the diffusion reaction as a special case. If the diffusion effect disappeared, we can obtain the lattice Boltzmann solution of the Rossler equation on the mesescopic scale. The numerical results show the method can be used to simulate Rossler equation.     

Structural decompositions of multivariate distributions with applications in moment and cumulant

We provide lattice decompositions for multivariate distributions. The lattice decompositions reveal the structural relationship between the Lancaster/Bahadur model and the model of Streitberg (Ann. Statist. 18 (1990) 1878). For multivariate categorical data, the decompositions allows modeling strategy for marginal inference. The theory discussed in this paper illustrates the concept of reproducibility, which was discussed in Liang et al. (J. Roy. Statist. Soc. Ser. B 54 (1992) 3). For the purpose of delineating the relationship between the various types of decompositions of distributions, we develop a theory of polytypefication, the generality of which is exploited to prove results beyond interaction.     

Diffusion of Loops

We study a lattice model that is closely related to the Ising model and can be regarded as describing diffusion of loops in two dimensions. The time development is given by a transfer matrix for a random surface model on a three-dimensional lattice. The transfer matrix is indexed by loops and is invariant under a group of motions in the loop space. The eigenvalues of the transfer matrix are calculated in terms of the partition function and the correlation functions of the Ising model.     

The lattice of integer partitions and its infinite extension

In this paper, we use a simple discrete dynamical model to study integer partitions and their lattice. The set of reachable configurations of the model, with the order induced by the transition rule defined on it, is the lattice of all partitions of a positive integer, equipped with a dominance ordering. We first explain how this lattice can be constructed by an algorithm in linear time with respect to its size by showing that it has a self-similar structure. Then, we define a natural extension of the model to infinity, which we compare with the Young lattice. Using a self-similar tree, we obtain an encoding of the obtained lattice which makes it possible to enumerate easily and efficiently all the partitions of a given integer. This approach also gives a recursive formula for the number of partitions of an integer, and some informations on special sets of partitions, such as length bounded partitions.     

Quasi-Relativism,the Narrow-Gap Property,and Forced Electron Dynamics in Solids

Narrow-gap semiconductors, used in quantum network engineering, are characterized by small effective electron masses on the Fermi level and hence by high electron mobility in the lattice. We construct an explicitly solvable model that clarifies one possible mechanism for small effective masses to appear. Another mathematical model constructed here describes a possible mechanism for using a traveling wave to control an alternating quantum current in a one-dimensional lattice.     

An Improved Vortex Lattice Method for Nonstationary Problems

The vortex lattice method is improved for modeling nonlinear highly nonstationary processes appearing in an interaction of bodies that undergo an irregular motion in a proximity of solid boundaries with large-scale vortex structures. We show that keeping the condition of freezing the vortex buildups in the medium leads, in the vortex lattice method, to eliminating the arbitrariness in the calculated time step, singularity radius, and the buffer-zone radius.For the ensemble of discrete vortices that model the surface of tangential discontinuity of the velocity, we propose an economical method for solving the Cauchy problem. The method decreases the discretization error related to the replacement of this surface with a system of discrete vortices.A test for the improved vortex lattice method has been conducted for a nonstationary nonlinear problem on nonharmonic angular oscillations of a wing in a stationary medium near a solid surface in the case where there is no gap between the wing and the surface.By using the improved vortex lattice method, one succeeded, for the first time, in obtaining a solution of a problem of this type that converges from the numerical point of view. A comparison of the obtained results with known experimental data shows a good agreement.     

Convergence of a Force‐Based Hybrid Method in Three Dimensions

We study a force‐based hybrid method that couples an atomistic model with the Cauchy‐Born elasticity model. We show that the proposed scheme converges to the solution of the atomistic model with second‐order accuracy, since the ratio between lattice parameter and the characteristic length scale of the deformation tends to 0. Convergence is established for the three‐dimensional system without defects, with general finite‐range atomistic potential and simple lattice structure. The proof is based on consistency and stability analysis. General tools for stability analysis are developed in the framework opseudodifference operators in arbitrary dimensions. © 2012 Wiley Periodicals, Inc.     

Scattering from a spatially restricted atom,I

We prove the existence and completeness of the wave operators for a model describing the elastic scattering of a neutron from the nucleus of an atom which is harmonically bound to a certain site in an infinite lattice. We then solve a similar problem for proton scattering under the assumption that the nuclear charge distribution has a nontrivial symmetry group, and finally consider dissipative effects due to the other lattice atoms on the scattering.     

A mixed integer linear programming model for optimal sovereign debt issuance

Governments borrow funds to finance the excess of cash payments or interest payments over receipts, usually by issuing fixed income debt and index-linked debt. The goal of this work is to propose a stochastic optimization-based approach to determine the composition of the portfolio issued over a series of government auctions for the fixed income debt, to minimize the cost of servicing debt while controlling risk and maintaining market liquidity. We show that this debt issuance problem can be modeled as a mixed integer linear programming problem with a receding horizon. The stochastic model for the interest rates is calibrated using a Kalman filter and the future interest rates are represented using a recombining trinomial lattice for the purpose of scenario-based optimization. The use of a latent factor interest rate model and a recombining lattice provides us with a realistic, yet very tractable scenario generator and allows us to do a multi-stage stochastic optimization involving integer variables on an ordinary desktop in a matter of seconds. This, in turn, facilitates frequent re-calibration of the interest rate model and re-optimization of the issuance throughout the budgetary year allows us to respond to the changes in the interest rate environment. We successfully demonstrate the utility of our approach by out-of-sample back-testing on the UK debt issuance data.     

From the Bethe Ansatz to the Gessel-Viennot theorem

We state and prove several theorems that demonstrate how the coordinate Bethe Ansatz for the eigenvectors of suitable transfer matrices of a generalized inhomogeneous, five-vertex model on the square lattice, given certain conditions hold, is equivalent to the Gessel-Viennot determinant for the number of configurations ofN non-intersecting directed lattice paths, or vicious walkers, with various boundary conditions. Our theorems are sufficiently general to allow generalisation to any regular planar lattice.     

A Monte Carlo study of the dependence of the growth parameter for trees on the lattice dimension in the Eden model

We use the Monte Carlo method to compute the number of trees with n edges in the Eden model on d-dimensional simple cubic lattices for d=2,3,4,6,8,10. We compare these numbers with the exact data derived by the enumeration method up to n=12 on the square lattice and up to n=10 on the cubic lattice. We find that for d3, the computed values of the growth parameter for trees agree with the values that we derived earlier by the expansion in inverse powers of 2d-1.     

Four-vertex model and random tilings

We consider the exactly solvable four-vertex model on a square lattice with different boundary conditions. Using the algebraic Bethe ansatz method allows calculating the partition function of the model. For fixed boundary conditions, we establish the connection between the scalar product of the state vectors and the generating function of the column-and row-strict boxed plane partitions. We discuss the tiling model on a periodic lattice. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 1, pp. 25–38, April, 2008.     

A fixed point characterization of the dominance-solvability of lattice games with strategic substitutes

We demonstrate that a lattice game with strategic substitutes is dominance-solvable if and only if there exists a unique fixed point of the function that results from an iteration of the best response function. This finding complements a result of Milgrom and Roberts’ (Econometrica 58:1255–1277, 1990) by which a lattice game with strategic complements is dominance-solvable if and only if there exists a unique Nash equilibrium. We illustrate our main result by an application to a model of Cournot outcome-competition.      

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

We calculate the pair correlation function and the magnetic susceptibility in the anisotropic Ising model on the lattice with one infinite and one finite dimension with periodic boundary conditions imposed along the second dimension. Using the exact expressions for lattice form factors, we propose formulas for arbitrary spin matrix elements, thus providing a possibility to calculate all multipoint correlation functions in the anisotropic Ising model on cylindrical and toroidal lattices. We analyze passing to the scaling limit.