Supersymmetry and 't Hooft's twisted functional integral

't Hooft's twisted functional integral is exactly calculable in the supersymmetric version of the SU(N) gauge theories, provided that one uses periodic (up to gauge transformations) boundary conditions for fermions. The solution has in a sense light magnetic fluxes and heavy electric ones, and thus, it is consistent with confinement (but not a proof of it). This is an evidence of the fact that confinement is not ruled out by the absence of vacuum condensates in these supersymmetric gauge theories.     

On the finite-size scalling equation for the spherical model

The mean spherical model with an arbitrary interaction potential, the Fourier transform of which has a long-wavelength exponent , 0<2, is considered under periodic boundary conditions and fully finite geometry ind dimensions, when <d<2. A new form of the finite-size scaling equation for the spherical field in the critical region is derived, which relates the temperature shift to Madelung-type lattice constants. The method of derivation makes use of the Poisson summation formula and a Laplace transformation of the momentumspace correlation function.On leave of absence from Institute of Mechanics and Biomechanics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria.     

On the integral form of the equation of transfer for a homogeneous sphere

Basic considerations are used to derive the integral form of the equation of transfer relevant to a homogeneous sphere with internal sources as well as external illumination and a specularly and diffusely reflecting surface.     

A class of integral formulations of the Helmholtz equation for an inhomogenous medium

Summary It is shown how the Helmholtz equation (and, as a particular case, the Laplace equation) for an inhomogenous medium can be transformed into several integral formulations. Thus the already known formulations, used in numerical studies of wave diffraction by harbours and islands, are seen in the context of a more general theory.

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Riassunto Si mostra come l'equazione di Helmholtz (e come caso particolare l'equazione di Laplace) per un mezzo inomogeneo può essere trasformata in forma integrale in diversi modi. Così le formulazioni già note, usate nello studio numerico della diffrazione di onde da parte di isole o porti, sono viste nel contesto di una teoria più generale.

     

On the naturalness of Einstein’s equation

We compute all 2-covariant tensors naturally constructed from a semi-Riemannian metric which are divergence-free and have weight greater than −2.     

Singularity spectrum of the resolvent in a nonmarkovian master equation

The well-known standard nonmarkovian master equation is used to describe the damping of a harmonic macroscopic coordinate coupled to a thermally equilibrated Fermi gas. The time evolution of the phonon distribution is expressed in terms of a resolvent whose spectrum of singularities is computed and analyzed. It is seen that an apparent limitation of the model related to the appearance of undesired oscillations does not inhibit the expected behavior of the average phonon number and phonon number dispersion.     

On the solution of Polyakov's ansatz

I show that the functional differential equations for conserved currents in loop space are consistent with the integrability condition.     

麦克斯韦积分方程的一种数值解法

直接从麦克斯韦积分方程出发,采用FIT算法,模拟电磁场的分布情况。     

用近似轨道方程计算地球轨道的偏心率

对地球的椭圆轨道方程进行近似处理,然后应用开普勒面积定律和季节长度计算其轨道偏心率.     

On the path integral solutions of the master equation

The lagrangian in the path integral solution of the master equation of a stationary Markov process is derived by application of the Ehrenfest-type theorem of quantum mechanics and the Cauchy method of finding inverse functions. Applied to the non-linear Fokker-Planck equation we reproduce the result obtained by integrating over Fourier series coefficients and by other methods.     

Radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems

In graded index medium, the ray goes along a curved path determined by Fermat principle, and the curved ray-tracing is very difficult and complex. To avoid the complicated and time-consuming computation of curved ray trajectory, the methods not based on ray-tracing technique need to be developed for the solution of radiative transfer in graded index medium. For this purpose, in this paper the streaming operator along a curved ray trajectory in original radiative transfer equation for graded index medium is transformed and expressed in spatial and angular ordinates and the radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems are derived. The conservative and the non-conservative forms of radiative transfer equation for three-dimensional graded index medium are given, which can be used as base equations to develop the numerical simulation methods, such as finite volume method, discrete ordinates method, and finite element method, for radiative transfer in graded index medium in cylindrical and spherical coordinate systems.     

Fast integral equation methods for the modified Helmholtz equation

We present integral equation methods for the solution to the two-dimensional, modified Helmholtz equation, u(x) ? α²Δu(x) = 0, in bounded or unbounded multiply-connected domains. We consider both Dirichlet and Neumann problems. We derive well-conditioned Fredholm integral equations of the second kind, which are discretized using high-order, hybrid Gauss-trapezoid rules. Our fast multipole-based iterative solution procedure requires only O(N) operations, where N is the number of nodes in the discretization of the boundary. We demonstrate the performance of our methods on several numerical examples, and we show that they have both the ability to handle highly complex geometry and the potential to solve large-scale problems.     

On the Bardeen-Cooper-Schrieffer integral equation in the theory of superconductivity

The Bardeen-Cooper-Schrieffer integral equation with a positive kernel is studied in full generality. It is shown that, there exists a unique finite transition temperature, T _c so that, if T _c ,the equation possesses a positive solution, representing the onset of the superconducting phase, while if T>T _c ,the only solution of the equation is the trivial one, indicating the occurrence of the normal phase. Moreover, it is demonstrated that such a positive solution may be approximated by a sequence of solutions of the equation restricted on bounded domains. This latter result provides a useful computational scheme for the problem.     

Kinetic equation for electrons of a semiconductor in a strong electric field: Green's resolvent

Using the Laplace transform of Boltzmann's equation, a system of linking equations is obtained for the series expansion of the mapping of the distribution function in powers of the field. Since the form of the equations is identical, instead of direct integration it is possible to use the method of Green's functions for the solution of the system. Using the method of perturbances Green's resolvent is constructed in the form of an infinite series. Using the example of the evaluation of the distribution function in a high-frequency field, the convergence of the proposed method is demonstrated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, Vol. 16, No. 7, pp. 22–25, July, 1973.     

Random integral equation formulation of a generalized Langevin equation

In this paper, the generalized Langevin equation introduced by Kubo and Mori is formulated as a random integral equation. We consider (1) the existence and uniqueness of the solution, (2) moments of the solution process, (3) a comparison theorem for solution processes, and (4) the Cauchy polygonal approximation to the solution.     

Green’s function-stochastic methods framework for probing nonlinear evolution problems: Burger’s equation, the nonlinear Schrödinger’s equation, and hydrodynamic organization of near-molecular-scale vorticity

A framework which combines Green’s function (GF) methods and techniques from the theory of stochastic processes is proposed for tackling nonlinear evolution problems. The framework, established by a series of easy-to-derive equivalences between Green’s function and stochastic representative solutions of linear drift–diffusion problems, provides a flexible structure within which nonlinear evolution problems can be analyzed and physically probed. As a preliminary test bed, two canonical, nonlinear evolution problems – Burgers’ equation and the nonlinear Schrödinger’s equation – are first treated. In the first case, the framework provides a rigorous, probabilistic derivation of the well known Cole–Hopf ansatz. Likewise, in the second, the machinery allows systematic recovery of a known soliton solution. The framework is then applied to a fairly extensive exploration of physical features underlying evolution of randomly stretched and advected Burger’s vortex sheets. Here, the governing vorticity equation corresponds to the Fokker–Planck equation of an Ornstein–Uhlenbeck process, a correspondence that motivates an investigation of sub-sheet vorticity evolution and organization. Under the assumption that weak hydrodynamic fluctuations organize disordered, near-molecular-scale, sub-sheet vorticity, it is shown that these modes consist of two weakly damped counter-propagating cross-sheet acoustic modes, a diffusive cross-sheet shear mode, and a diffusive cross-sheet entropy mode. Once a consistent picture of in-sheet vorticity evolution is established, a number of analytical results, describing the motion and spread of single, multiple, and continuous sets of Burger’s vortex sheets, evolving within deterministic and random strain rate fields, under both viscous and inviscid conditions, are obtained. In order to promote application to other nonlinear problems, a tutorial development of the framework is presented. Likewise, time-incremental solution approaches and construction of approximate, though otherwise difficult-to-obtain backward-time GF’s (useful in solution of forward-time evolution problems) are discussed.     

On stationary solutions of the Schrödinger equation with a self-consistent potential satisfying boussinesq's equation

We derive an equation for a dense perturbation valid in a resonance region, M → 1, associated with the Schrödinger equation for a complex amplitude of the high-frequency Langmuir-oscillation electrical field. A solution of the above system in the linear approximation is obtained and discussed.     

圆柱体相对转动动力学方程的积分解

针对圆柱体任意两个横截面间的相对转动动力学方程，运用解耦方法获得方程的解析解，由于方程的特殊性，利用Jordan标准形求得可逆矩阵.从而得到了圆柱体相对转动动力学方程的积分形式的解.根据工程应用，给出了冲击性和周期性两类典型载荷作用下的解析解. 关键词： 相对转动 相似模拟 动力学方程 解析解