T-invariance,CPT-invariance,and local commutativity for the quantum (cosh ϕ)2-model

One gives a short proof of certain identities that are necessary for the quantization of the (cosh )₂-model.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 146, pp. 9–19, 1985.     

A reliable approach to the solution of Navier–Stokes equations

In this paper we will demonstrate an affective approach of solving Navier–Stokes equations by using a very reliable transformation method known as the Cole–Hopf transformation, which reduces the problem from nonlinear into a linear differential equation which, in turn, can be solved effectively.     

A filter-and-fan approach to the 2D HP model of?the?protein folding problem

We examine a prominent and widely-studied model of the protein folding problem, the two-dimensional (2D) HP model, by means of a filter-and-fan (F&F) solution approach. Our method is designed to generate compound moves that explore the solution space in a dynamic and adaptive fashion. Computational results for standard sets of benchmark problems show that the F&F algorithm is highly competitive with the current leading algorithms, requiring only a single solution trial to obtain best known solutions to all problems tested, in contrast to a hundred or more trials required in the typical case to evaluate the performance of the best of the alternative methods.     

A method for the numerical solution of the Painlevé equations

A numerical method for solving the Cauchy problem for all the six Painlevé equations is proposed. The difficulty of solving these equations is that the unknown functions can have movable (that is, dependent on the initial data) singular points of the pole type. Moreover, the Painlevé III–VI equations may have singularities at points where the solution takes certain finite values. The positions of all these singularities are not a priori known and are determined in the process of solving the equation. The proposed method is based on the transition to auxiliary systems of differential equations in neighborhoods of the indicated points. The equations in these systems and their solutions have no singularities at the corresponding point and its neighborhood. Such auxiliary equations are derived for all Painlevé equations and for all types of singularities. Efficient criteria for transition to auxiliary systems are formulated, and numerical results illustrating the potentials of the method are presented.     

A 8-neighbor model lattice Boltzmann method applied to mathematical–physical equations

A lattice Boltzmann method (LBM) 8-neighbor model (9-bit model) is presented to solve mathematical–physical equations, such as, Laplace equation, Poisson equation, Wave equation and Burgers equation. The 9-bit model has been verified by several test cases. Numerical simulations, including 1D and 2D cases, of each problem are shown, respectively. Comparisons are made between numerical predictions and analytic solutions or available numerical results from previous researchers. It turned out that the 9-bit model is computationally effective and accurate for all different mathematical–physical equations studied. The main benefits of the new model proposed is that it is faster than the previous existing models and has a better accuracy.     

A subjective Bayesian approach to the theory of queues I — Modeling

This is the first of an expository two-part paper which outlines a point of view different from that currently used in queueing theory. In both parts, the focus is on concepts. Here we adopt a personal probability point of view to all sources of uncertainty in the theory of queues and explore the consequences of our approach by comparing our results to those that are currently available. For ease of exposition, we confine attention to the M/M/1/ and the M/M/1/K queues. In Part I we outline the general strategy and focus on model development. In Part II we address the problem of inference in queues within the subjective Bayesian paradigm and introduce a use of Shannon's measure of information for assessing the amount of information conveyed by the various types of data from queues.     

A remark on the global existence of weak solutions to the compressible quantum Navier–Stokes equations

In this paper, we obtain the global existence of weak solutions to the compressible quantum Navier–Stokes equations. By virtue of a useful identity and an interesting estimate, we solve the critical case that the viscosity equals the dispersive coefficient. This result removes the restrictions on the coefficients and improves the recent work of Antonell and Spirito (2017) in some senses.     

A new approach to the stabilization of a fluid–structure interaction model

This article is a continuation of our work on a linear fluid–structure interaction model [Grobbelaar-Van Dalsen, On a fluid–structure model in which the dynamics of the structure involves the shear stress due to the fluid, J. Math. Fluid Mech. 10(3) (2008), pp. 388–401; Grobbelaar-Van Dalsen, Strong stability for a fluid––structure model, Math. Methods Appl. Sci., 32(2009) pp. 1452–1466]. The model describes the interaction between a 3-D incompressible fluid and a 2-D plate, the interface, which coincides with a flat flexible part of the surface of the vessel containing the fluid. The mathematical model comprises the Stokes equations and the equations for the longitudinal deflections of the plate with the inclusion of the shear stress that the fluid exerts on the plate. A dissipative damping mechanism of Kelvin–Voigt type is applied to the interior of the plate. While our earlier work shows that weak solutions in a space of finite energy are strongly asymptotically stable under no-slip transmission conditions at the interface with uniform exponential stability only attainable under an additional domination condition, the present research is directed at achieving uniform exponential stability of weak solutions without imposing the domination condition. Using energy methods we establish uniform exponential decay under a modified transmission condition at the interface. This condition entails that the fluid velocity at the interface is coupled to a linear combination of the plate velocity and displacement.     

A dynamical approach to the large-time behavior of solutions to weakly coupled systems of Hamilton–Jacobi equations

We investigate the large-time behavior of the value functions of the optimal control problems on the n-dimensional torus which appear in the dynamic programming for the system whose states are governed by random changes. From the point of view of the study on partial differential equations, it is equivalent to consider viscosity solutions of quasi-monotone weakly coupled systems of Hamilton–Jacobi equations. The large-time behavior of viscosity solutions of this problem has been recently studied by the authors and Camilli, Ley, Loreti, and Nguyen for some special cases, independently, but the general cases remain widely open. We establish a convergence result to asymptotic solutions as time goes to infinity under rather general assumptions by using dynamical properties of value functions.     

The (-1/2λϕ2) model of quantum field theory

The model of a scalar field with interaction potential exp (-1/2²) ind-dimensional spacetime (d2) is considered. It is shown that the Green's functions of the model are the same as the Green's functions of the free field.Physicotechnical Institute, Moscow. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 3, pp. 411–417, June, 1992.     

Applications of nonstandard analysis to partial differential equations—I. the diffusion equation

This paper begins by developing a basis for using ¹ finite difference equations to model physical phenomena. Under appropriate conditions the solution F of a ¹ finite difference equation has S-continuous “finite difference derivatives” up to order r. In these circumstances we can show that the standard function °F is a C^r-function and satisfies the differential equation corresponding to the original finite difference equation. The second part of the paper illustrates these techniques by applying them to the heat equation. In particular, we obtain a very nice model of the heat equation with initial conditions corresponding to all the heat concentrated at a single point.     

CTE method to the interaction solutions of Boussinesq–Burgers equations

Under investigation in this paper is the Boussinesq–Burgers equations, which describe the propagation of shallow water waves. Via the truncated Painlevé analysis and the consistent tanh expansion (CTE) method, some exact interaction solutions among different nonlinear excitations such as multiple resonant soliton solutions, soliton–error function waves, soliton–periodic waves, soliton–rational waves, and soliton–potential Burgers waves are explicitly given.     

New approach to calculating the spectrum of a quantum space–time

We study the dynamics of a massive pointlike particle coupled to gravity in four space–time dimensions. It has the same degrees of freedom as an ordinary particle: its coordinates with respect to a chosen origin (observer) and the canonically conjugate momenta. The effect of gravity is that such a particle is a black hole: its momentum becomes spacelike at a distances to the origin less than the Schwarzschild radius. This happens because the phase space of the particle has a nontrivial structure: the momentum space has curvature, and this curvature depends on the position in the coordinate space. The momentum space curvature in turn leads to the coordinate operator in quantum theory having a nontrivial spectrum. This spectrum is independent of the particle mass and determines the accessible points of space–time.     

The Fujita–Kato approach to the Navier–Stokes equations in the rotational framework

Consider the Navier-Stokes equations in the rotational framework. It is proved that these equations possess a unique global mild solution for arbitrary speed of rotation provided the initial data u ₀ is small enough in the H^\frac12_s(\mathbbR³){H^{\frac12}_{\sigma}(\mathbb{R}^3)} -norm.     

Convergence of the quantum Navier–Stokes–Poisson equations to the incompressible Euler equations for general initial data

In this paper, we are concerned with the rigorous proof of the convergence of the quantum Navier–Stokes-Poisson system to the incompressible Euler equations via the combined quasi-neutral, vanishing damping coefficient and inviscid limits in the three-dimensional torus for general initial data. Furthermore, the convergence rates are obtained.     

A constructive method for the stabilization of the wave equation with localized Kelvin–Voigt damping

We consider the wave equation with Kelvin–Voigt damping in a bounded domain. The damping is localized in a suitable open subset of the domain under consideration. The exponential stability result proposed by Liu and Rao for that system assumes that the damping is localized in a neighborhood of the whole boundary, and the damping coefficient is continuously differentiable with a bounded Laplacian. We propose a new solution to the exponential stability problem based on the introduction of a new variable, and a constructive frequency domain approach. The main features of our method are: (i) the damping region need not be a neighborhood of the whole boundary; (ii) the damping coefficient is assumed to be bounded measurable with bounded measurable gradient only; (iii) the introduction of a new variable. These features enable us to improve on the damping coefficient smoothness and more especially on the feedback control region. Further, when combined with a recent result of Borichev and Tomilov on the polynomial decay of bounded semigroups, the new method enables us to prove a polynomial decay estimate of the energy when the damping coefficient is bounded measurable only.     

A remark on weak solutions to the barotropic compressible quantum Navier–Stokes equations

In [A. Jüngel, Global weak solutions to compressible Navier–Stokes equations for quantum fluids, SIAM J. Math. Anal. 42 (2010) 1025–1045], Jüngel proved the global existence of the barotropic compressible quantum Navier–Stokes equations for when the viscosity constant is bigger than the scaled Planck constant. Recently, Dong [J. Dong, A note on barotropic compressible quantum Navier–Stokes equations, Nonlinear Anal. TMA 73 (2010) 854–856] extended Jüngel’s result to the case where the viscosity constant is equal to the scaled Planck constant by using a new estimate of the square root of the solutions. In this paper we show that Jüngel’s existence result still holds when the viscosity constant is bigger than the scaled Planck constant. Consequently, with our result, the existence for all physically interesting cases of the scaled Planck and viscosity constants is obtained.