A matrix integral solution to [P,Q]=P and matrix laplace transforms

In this paper we solve the following problems: (i) find two differential operatorsP andQ satisfying [P, Q]=P, whereP flows according to the KP hierarchy P/t _n =[(P ^n/p )₊,P], withp:=ordP2; (ii) find a matrix a integral representation for the associated -function. First we construct an infinite dimensional spaceW= span{ ₀(z, ₁(z,...)} of functions ofz invariant under the action of two operators, multiplication byz ^p andA _c :=z/z–z+c. This requirement is satisfied, for arbitraryp, if ₀ is a certain function generalizing the classical Hänkel function (forp=2); our representation of the generalized Hänkel function as adouble Laplace transform of a simple function, which was unknown even for thep=2 case, enables us to represent the -function associated with the KP time evolution of the spaceW as a double matrix Laplace transform in two different ways. One representation involves an integration over the space of matrices whose spectrum belongs to a wedge-shaped contour ^-+ ^- defined by ^± = ₊e^±i/p. The new integrals above relate to matrix Laplace transforms, in contrast with matrix Fourier transforms, which generalize the Kontsevich integrals and solve the operator equation [P, Q]=1.The support of a National Science Foundation grant #DMS-95-4-51179 is gratefully acknowledged.The hospitality of the Volterra Center at Brandeis University is gratefully acknowledged.The hospitality of the University of Louvain and Brandeis University is gratefully acknowledged.The support of a National Science Foundation grant #DMS-95-4-51179, a Nato, an FNRS and a Francqui Foundation grant is gratefully acknowledged.     

A riemann integral approach to Feynman's path integral

It is a well known result that the Feynman's path integral (FPI) approach to quantum mechanics is equivalent to Schrödinger's equation when we use as integration measure the Wiener-Lebesgue measure. This results in little practical applicability due to the great algebraic complexibity involved, and the fact is that almost all applications of (FPI) practical calculations — are done using a Riemann measure. In this paper we present an expansion to all orders in time of FPI in a quest for a representation of the latter solely in terms of differentiable trajetories and Riemann measure. We show that this expansion agrees with a similar expansion obtained from Schrödinger's equation only up to first order in a Riemann integral context, although by chance both expansions referred to above agree for the free particle and harmonic oscillator cases. Our results permit, from the mathematical point of view, to estimate the many errors done in practical calculations of the FPI appearing in the literature and, from the physical point of view, our results supports the stochastic approach to the problem.     

The adsorption integral equation: A new method of solution

A classification of the adsorption integral equation in to finite and infinite limit problems is made. A method of solution based on the theory of finite Hilbert transforms is proposed.     

A stable numerical inversion of Abel's integral equation using almost Bernstein operational matrix

Many problems in physics like reconstruction of the radially distributed emissivity from the line-of-sight projected intensity, the 3-D image reconstruction from cone-beam projections in computerized tomography, etc. lead naturally, in the case of radial symmetry, to the study of Abel's type integral equation. The aim of this communication is to modify the stable algorithm proposed in [Singh VK, Pandey RK, Singh OP. New stable numerical solution of singular integral equations of Abel type by using normalized Bernstein polynomials. Applied Mathematical Sciences 2009;3(5):241–255] which is based on normalized Bernstein polynomial approximation of the projected intensity profile. So, first we construct an orthonormal family of polynomials of degree 5 from the 5th degree Bernstein polynomials B_i5 and use them as a basis to approximate the projected intensity profile. Then, a 6×6 matrix P, named as almost Bernstein operational matrix of integration is constructed and used to reduce the integral equation to a system of algebraic equation which can be solved easily. The method is quite accurate and stable even though the approximations are performed by polynomials of degree 5, as illustrated by applying the method to intensity data with and without random noise to invert and compare it with those obtained by the other methods or with the known analytical inverse. Thus it is good method for applying to experimental intensities distorted by noise.     

A path integral to quantize spin

We present a model for a classical spinning particle, characterized by spin magnitude, arbitrary but fixed, and continuously varying direction. A gauge freedom of the model reflects the choice of canonical coordinates in the phase space, which is spherical. We formulate the path integral for the model and find, unexpectedly, that the phase space must be punctured at the poles. It then follows that both the total spin and spin projection along any axis are quantized. The model has rotational invariance and yields the usual quantum mechanics of spin, including commutation relations, in a simple way.     

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。     

The integral equations for the average stochastic scattering matrix

The integral equations describing the dependence of the average scattering matrix on the distribution of resonance parameters are derived for the large numbers of the levels and channels. In particular, the case of statistically nonequivalent channels is considered.The authors are grateful to Prof. V.M. Strutinsky for useful discussions.     

A functional integral approach to plasma turbulence

The study of weakly turbulent plasmas uses perturbative methods, in contrast to the strong turbulence case which is investigated numerically. A functional integral method is proposed as a unifying approach based on the examination of the structure of the space of exact phase-space particle orbits. The method is nonperturbative and easily obtains basic results of the weak turbulence theory.     

A path integral approach to inclusive processes

The single-particle inclusive differential cross-section for a reaction is written as the imaginary part of a correlation function in a forward scattering amplitude for in a modified effective theory. In this modified theory the interaction Hamiltonian equals in the original theory up to a certain time. Then there is a sign change and becomes nonlocal. This is worked out in detail for scalar field models and for QED plus the abelian gluon model. A suitable path integral for direct calculations of inclusive cross sections is presented. Received: 8 March 2000 / Published online: 6 July 2000     

A path integral way to option pricing

An efficient computational algorithm to price financial derivatives is presented. It is based on a path integral formulation of the pricing problem. It is shown how the path integral approach can be worked out in order to obtain fast and accurate predictions for the value of a large class of options, including those with path-dependent and early exercise features. As examples, the application of the method to European and American options in the Black–Scholes model is illustrated. A particularly simple and fast semi-analytical approximation for the price of American options is derived. The results of the algorithm are compared with those obtained with the standard procedures known in the literature and found to be in good agreement.     

A path integral approach to asset-liability management

Functional integrals constitute a powerful tool in the investigation of financial models. In the recent econophysics literature, this technique was successfully used for the pricing of a number of derivative securities. In the present contribution, we introduce this approach to the field of asset-liability management. We work with a representation of cash flows by means of a two-dimensional delta-function perturbation, in the case of a Brownian model and a geometric Brownian model. We derive closed-form solutions for a finite horizon ALM policy. The results are numerically and graphically illustrated.     

A mixed integral equation of mechanics and a generalized projection method of its solution

A mixed multidimensional integral equation containing integral operators of various types is studied. The case in which the equation has one compact, self-adjoint, and strongly positive operator (with constant limits of integration) and two non-self-adjoint integral Volterra operators (with a variable upper limit of integration) is considered. To solve the equation, an effective projection method allowing one to obtain the result in a form with explicitly distinguished principal singularities is proposed.     

Analytical solution to the material balance equation by integral transform for different cathode geometries

The 1-dimensional non-homogeneous material balance equation has been examined in the rectangular, spherical and cylindrical coordinate system. The solutions to this equation in the respective coordinate system has been determined analytically using the method of integral transform, together with the norms and eigen values suitable for galvanostatic discharge boundary conditions.     

Brownian-motion ensembles of random matrix theory: A classification scheme and an integral transform method

We study a class of Brownian-motion ensembles obtained from the general theory of Markovian stochastic processes in random-matrix theory. The ensembles admit a complete classification scheme based on a recent multivariable generalization of classical orthogonal polynomials and are closely related to Hamiltonians of Calogero–Sutherland-type quantum systems. An integral transform is proposed to evaluate the n-point correlation function for a large class of initial distribution functions. Applications of the classification scheme and of the integral transform to concrete physical systems are presented in detail.     

Asymptotics of collision integral matrix elements in the isotropic case

The method of nonlinear moments, when used to solve the Boltzmann equation, necessitates the calculation of collision integral matrix elements. The matrix elements are hard to calculate numerically, especially at large indices. The asymptotics of the matrix elements are constructed. In terms of the model of pseudopower particle interaction, a formula free of summation is derived. This makes it possible to find the asymptotic behavior of linear and nonlinear elements when two indices are large. For an arbitrary interaction cross section, asymptotic expansions of linear and nonlinear matrix elements in one index are obtained. For Maxwellian molecules, asymptotic formulas are derived for three large indices.     

A solution of density matrix equation for one mode of laser light

Time evolution of the density matrix equation for laser light was found under the assumption that some characteristics of the atomic system do not change in time. Two-time normal ordered correlation function was evaluated.     

Exact solution of the neutralino mass matrix

The eigenvalue problem for the neutralino mass matrix has been solved exactly and the eigenvalues are expressed in terms of thev ₁/v ₂, chargino and gluino masses which are directly measurable. An analytical formula for the lightest neutralino mass as a function of the above parameters is obtained. Formulae for the photino, zino and neutral higgsino contents of each of the physical neutralino mass eigenstates states have been found. Taking into account these formulae it has been possible to predict the upper (lower) bound on the mass of zino (photino) dominant neutralino states, including the lightest one. The neutralino-gluino and neutralino-chargino mass planes have also been constrained by using the latest LEP data.     

A path integral approach to effective non-linear medium

In this article, we propose a new method to compute the effective properties of non-linear disordered media. We use the fact that the effective constants can be defined through the minimum of an energy functional. We express this minimum in terms of a path integral allowing us to use many-body techniques. We obtain the perturbation expansion of the effective constants to second order in disorder, for any kind of non-linearity. We apply our method to the case of strong non-linearities (i.e. , where is fluctuating from point to point), and to the case of weak non-linearity (i.e. where and fluctuate from point to point). Our results are in agreement with previous ones, and could be easily extended to other types of non-linear problems in disordered systems. Received: 13 May 1998 / Accepted: 27 July 1998