Oscillation phenomena and experimental determination of exact mathematical stability zones for magneto-conductivity in metals having complicated Fermi surfaces

We consider the problem of exact experimental determination of the boundaries of Stability Zones for magneto-conductivity in normal metals in the space of directions of magnetic field B. As can be shown, this problem turns out to be nontrivial since the exact boundaries of Stability Zones are in fact unobservable in direct measurements of conductivity. However, this problem can be effectively solved with the aid of the study of oscillation phenomena (cyclotron resonance, quantum oscillations) in normal metals, which reveal a singular behavior on the mathematical boundary of a Stability Zone.     

Single point velocity distributions in homogeneous isotropic turbulence

The starting point for this paper lies in the results obtained by Tatsumi (2004) for isotropic turbulence with the self-preserving hypothesis. A careful consideration of the mathematical structure of the one-point velocity distribution function equation obtained by Tatsumi (2004) leads to an exact analysis of all possible cases and to all admissible solutions of the problem. This paper revisits this interesting problem from a new point of view, and obtains a new complete set of solutions. Based on these exact solutions, some physically significant consequences of recent advances in the theory of homogenous statistical solution of the Navier--Stokes equations are presented. The comparison with former theory was also made. The origin of non--Gaussian character could be deduced from the above exact solutions.     

New Sedov-Type Solution of Isotropic Turbulence

The starting point lies in the results obtained by Sedov （1944） for isotropic turbulence with a self-preserving hypothesis. A careful consideration of the mathematical structure of the Karman-Howarth equation leads to an exact analysis of all cases possible and to all admissible solutions of the problem. I study this interesting problem from a new point of view. New solutions are obtained. Based on these exact solutions, some physical significant consequences of recent advances in the theory of self-preserved homogeneous statistical solution of the Navier-Stokes equations are presented.     

The Landau-Lifschitz equation and the Riemann boundary problem on a torus

The Cauchy problem for the Landau-Lifschitz equation is studied by the inverse transform method. We reduce this problem to the matrix Riemann problem on a torus and then to a certain Fredholm integral equation. An exact N-soliton solution is constructed. The reduction group approach is a main tool of our study.     

Toward Analytic Solution of Nonlinear Differential Difference Equations via Extended Sensitivity Approach

In this paper an efficient computational method based on extending the sensitivity approach(SA) is proposed to find an analytic exact solution of nonlinear differential difference equations.In this manner we avoid solving the nonlinear problem directly.By extension of sensitivity approach for differential difference equations(DDEs),the nonlinear original problem is transformed into infinite linear differential difference equations,which should be solved in a recursive manner.Then the exact solution is determined in the form of infinite terms series and by intercepting series an approximate solution is obtained.Numerical examples are employed to show the effectiveness of the proposed approach.     

Causality in the Relativistic Bound-State Problem

Although the exact Bethe–Salpeter equation is certainly the appropriate field-theoretic framework to describe the non-perturbative problem of scattering and bound states, the inevitable truncations introduce inconsistencies such as loss of symmetries or incorrect one-body limit. I conjecture that these problem can be overcome if the truncation preserves the field-redefinition invariance of the exact equation. A sum rule for light-by-light scattering can provide a testing ground of this conjecture.     

Two interacting spins in external fields. Four‐level systems

In the present article, we consider the so‐called two‐spin equation that describes four‐level quantum systems. Recently, these systems attract attention due to their relation to the problem of quantum computation. We study general properties of the two‐spin equation and show that the problem for certain external backgrounds can be identified with the problem of one spin in an appropriate background. This allows one to generate a number of exact solutions for two‐spin equations on the basis of already known exact solutions of the one‐spin equation. Besides, we present some exact solutions for the two‐spin equation with an external background different for each spin but having the same direction. We study the eigenvalue problem for a time‐independent spin interaction and a time‐independent external background. A possible analogue of the Rabi problem for the two‐spin equation is defined. We present its exact solution and demonstrate the existence of magnetic resonances in two specific frequencies, one of them coinciding with the Rabi frequency, and the other depending on the rotating field magnitude. The resonance that corresponds to the second frequency is suppressed with respect to the first one.     

A Modified Kernel Method for Solving Cauchy Problem of Two-Dimensional Heat Conduction Equation

In this paper, a Cauchy problem of two-dimensional heat conduction equation is investigated. This is a severely ill-posed problem. Based on the solution of Cauchy problem of two-dimensional heat conduction equation, we propose to solve this problem by modifying the kernel, which generates a well-posed problem. Error estimates between the exact solution and the regularized solution are given. We provide a numerical experiment to illustrate the main results.     

Numerical solution of 3D Green's function for the dynamic system of anisotropic elasticity

In this Letter, we used homotopy perturbation method to obtain numerical solution of the 3D Green's function for the dynamic system of anisotropic elasticity. Application of homotopy perturbation method to this problem shows the rapid convergence of the sequence constructed by this method to the exact solution. The numerical results obtained from convolution of Green's function and data of the Cauchy problem are compared with the exact solution for cubic media. The results reveal that the proposed method is very effective and simple.     

大平板瞬态热传导问题的一种新的近似解法

１引言大平板瞬态热传导问题有着广泛的工程应用背景。对于复杂的初边值条件或含内热源问题,以及工程上常见的多层复合平壁对象,分析求解难度很大甚至无法求解。在此类情况下往往采用数值方法。但是单纯的数值解不便于理解影响该问题的各种参数的物理意义。因此,各种近似分析方法得到了发展［１,２］。但在近似精度上,往往难以对整个时间坐标范围都达到较高的精度,这就使得近似解更多地局限于定性分析。此外,对于不同的初边值条件或含内热源问题,近似解的形式相异,降低了解的通用性。增加了求解的工作量。本文提出一种基于矩阵理论…     

Exact diffraction calculation from fields specified over arbitrary curved surfaces

Calculation of the scalar diffraction field over the entire space from a given field over a surface is an important problem in computer generated holography. A straightforward approach to compute the diffraction field from field samples given on a surface is to superpose the emanated fields from each such sample. In this approach, possible mutual interactions between the fields at these samples are omitted and the calculated field may be significantly in error. In the proposed diffraction calculation algorithm, mutual interactions are taken into consideration, and thus the exact diffraction field can be calculated. The algorithm is based on posing the problem as the inverse of a problem whose formulation is straightforward. The problem is then solved by a signal decomposition approach. The computational cost of the proposed method is high, but it yields the exact scalar diffraction field over the entire space from the data on a surface.     

Monomer-dimer model on a scale-free small-world network

The explicit determination of the number of monomer-dimer arrangements on a network is a theoretical challenge, and exact solutions to monomer-dimer problem are available only for few limiting graphs with a single monomer on the boundary, e.g., rectangular lattice and quartic lattice; however, analytical research (even numerical result) for monomer-dimer problem on scale-free small-world networks is still missing despite the fact that a vast variety of real systems display simultaneously scale-free and small-world structures. In this paper, we address the monomer-dimer problem defined on a scale-free small-world network and obtain the exact formula for the number of all possible monomer-dimer arrangements on the network, based on which we also determine the asymptotic growth constant of the number of monomer-dimer arrangements in the network. We show that the obtained asymptotic growth constant is much less than its counterparts corresponding to two-dimensional lattice and Sierpinski fractal having the same average degree as the studied network, which indicates from another aspect that scale-free networks have a fundamentally distinct architecture as opposed to regular lattices and fractals without power-law behavior.     

El-Nabulsi动力学模型下非Chetaev型非完整系统的精确不变量与绝热不变量

研究El-Nabulsi动力学模型下非Chetaev型非完整系统精确不变量与绝热不变量问题. 首先, 导出El-Nabulsi-d'Alembert-Lagrange原理并建立系统的运动微分方程. 其次, 建立El-Nabulsi模型下未受扰动的非Chetaev 型非完整系统的Noether对称性与Noether对称性导致的精确不变量之间的关系; 再次, 引入力学系统的绝热不变量概念, 研究受小扰动作用下非Chetaev型非完整系统Noether对称性的摄动导致绝热不变量问题, 给出了绝热不变量存在的条件及其形式. 作为特例, 本文讨论了El-Nabulsi模型下Chetaev型非完整系统的精确不变量与绝热不变量问题. 最后分别给出非Chetaev型和Chetaev型两种约束下的算例以说明结果的应用.     

Large bending strains in an orthotropic beam with a preliminarily stretched or compressed layer: Exact solution

The model of an orthotropic nonlinear elastic compressible material generalizing the known model of an isotropic semilinear (harmonic) material and admitting a number of exact solutions at large strains is proposed. For this model of the material, an exact solution of the plane problem on large bending strains of a compound rectangular beam consisting of two layers, one of which is preliminarily deformed, is obtained. When formulating and solving the problem, the theory of superimposed large strains is used. On the basis of the calculations carried out, the effect of anisotropy on a strong bending of the compound beam with a preliminarily deformed layer is analyzed.     

Sense and nonsense in the renormalization-scheme-dependence problem

I discuss in detail the issues involved in the renormalization-scheme-dependence problem in perturbative QCD. The problem is not to find a good universal scheme; nor is it a question of how to make coefficients small. The problem is that finite-order results depend on the choice of scheme, even though it is “arbitrary” (in that the exact result is scheme independent). Only the “principle of minimal sensitivity” approach gets to the heart of the problem, and attempts to reconcile this contradiction. I explain the motivation for this approach, defend it against recent criticisms, and explain to what extent it provides a solution to the problem.     

Model of a degenerate four-wave mixing phase-conjugate mirror without slowly varying envelope approximation

The exact solution to the boundary value problem for a simple conjugator model is presented. The analysis shows that, for this model, it is necessary to introduce two reflected and two transmitted waves. No SVEA approximation is used to obtain the exact formulae for the amplitudes of the reflected waves. The role of the boundaries in the phase-conjugation process is stressed.     

Exact solution of the Riemann problem for the shallow water equations with discontinuous bottom geometry

In this paper we present the exact solution of the Riemann problem for the non-linear shallow water equations with a step-like bottom. The solution has been obtained by solving an enlarged system that includes an additional equation for the bottom geometry and then using the principles of conservation of mass and momentum across the step. The resulting solution is unique and satisfies the principle of dissipation of energy across the shock wave. We provide examples of possible wave patterns. Numerical solution of a first-order dissipative scheme as well as an implementation of our Riemann solver in the second-order upwind method are compared with the proposed exact Riemann problem solution. A practical implementation of the proposed exact Riemann solver in the framework of a second-order upwind TVD method is also illustrated.     

Nonlinear dispersion of a pollutant ejected into a channel flow

In this paper, we study the nonlinear coupled boundary value problem arising from the nonlinear dispersion of a pollutant ejected by an external source into a channel flow. We obtain exact solutions for the steady flow for some special cases and an implicit exact solution for the unsteady flow. Additionally, we obtain analytical solutions for the transient flow. From the obtained solutions, we are able to deduce the qualitative influence of the model parameters on the solutions. Furthermore, we are able to give both exact and analytical expressions for the skin friction and wall mass transfer rate as functions of the model parameters. The model considered can be useful for understanding the polluting situations of an improper discharge incident and evaluating the effects of decontaminating measures for the water bodies.     

On Pollard's wave solution at the Equator

In this paper we present a dynamical study of the exact nonlinear Pollard wave solution to the geophysical water-wave problem in the f-plane approximation. We deduce an exact dispersion relation and we discuss some properties of this solution.     

Theory of frequency modulation spectroscopy of coherent dark resonances

The interaction of a three-level atom in the A configuration with a frequency-modulated (FM) field is analyzed theoretically for the first time. The two-frequency model for solving the problem of frequency modulation spectroscopy of coherent dark resonances is described and analyzed for a three-level A system. The effectiveness of the two-frequency model is demonstrated by comparing the results obtained using this model with the results of solving the exact problem of interaction of a A system with an FM field, which can be solved by the density matrix method. It is shown that the simple two-frequency model corresponds to the exact solution and is in qualitative agreement with experimental data.