Related papers: Proof of Cramer's rule with Dirac Delta Function
We demonstrate how to make the coordinate transformation or change of variables from Cartesian coordinates to curvilinear coordinates by making use of a convolution of a function with Dirac delta functions whose arguments are determined by…
In this paper, we present and analyze methods for solving a system of linear equations over idempotent semifields. The first method is based on the pseudo-inverse of the system matrix. We then present a specific version of Cramer's rule…
The Relativistic Dynamical Inversion technique, a novel tool for finding analytical solutions to the Dirac equation, is written in explicitly covariant form. It is then shown how the technique can be used to make a change from Cartesian to…
Electromagnetic phenomena are mathematically described by solutions of boundary value problems. For exploiting symmetries of these boundary value problems in a way that is offered by techniques of dimensional reduction, it needs to be…
We study the theory of systems with constraints from the point of view of the formal theory of partial differential equations. For finite-dimensional systems we show that the Dirac algorithm completes the equations of motion to an…
We prove multidimensional integration by parts formulas for generalized fractional derivatives and integrals. The new results allow us to obtain optimality conditions for multidimensional fractional variational problems with Lagrangians…
While the definition of a fractional integral may be codified by Riemann and Liouville, an agreed-upon fractional derivative has eluded discovery for many years. This is likely a result of integral definitions including numerous constants…
An integro-differential Dirac system with an integral term in the form of convolution is considered. We suppose that the convolution kernel is known a priori on a part of the interval, and recover it on the remaining part, using a part of…
New definitions of determinant functionals over the quaternion skew field are given in this paper. The inverse matrix over the quaternion skew field is represented by analogues of the classical adjoint matrix. Cramer rule for right and left…
In analogy with the Poisson summation formula, we identify when the fractional Fourier transform, applied to a Dirac comb in dimension one, gives a discretely supported measure. We describe the resulting series of complex multiples of delta…
Previously (A. Akhmeteli, J. Math. Phys., v. 52, p. 082303 (2011)), the Dirac equation in an arbitrary electromagnetic field was shown to be generally equivalent to a fourth-order equation for just one component of the four-component Dirac…
We provide a simple approach for the evaluation of inverse integral transforms that does not require any knowledge of complex analysis. The central idea behind the method is to reduce the inverse transform to the solution of an ordinary…
It is shown that the contraction mapping principle with the involvement of a Carleman Weight Function works for a Coefficient Inverse Problem for a 1D hyperbolic equation. Using a Carleman estimate, the global convergence of the…
Anderson's theorem asserting, that symmetry of dynamic equations written in the relativisitically covariant form is determined by symmetry of its absolute objects, is applied to the free Dirac equation. Dirac matrices are the only absolute…
Dirac delta function of matrix argument is employed frequently in the development of diverse fields such as Random Matrix Theory, Quantum Information Theory, etc. The purpose of the article is pedagogical, it begins by recalling detailed…
The Drazin inverse solutions of the matrix equations ${\rm {\bf A}}{\rm {\bf X}} = {\rm {\bf B}}$, ${\rm {\bf X}}{\rm {\bf A}} = {\rm {\bf B}}$ and ${\rm {\bf A}}{\rm {\bf X}}{\rm {\bf B}} ={\rm {\bf D}} $ are considered in this paper. We…
We give a half-page proof of the Lagrange-Good formula, using the Fourier representation of Dirac delta function.
Carleman linearization is a technique that embeds systems of ordinary differential equations with polynomial nonlinearities into infinite dimensional linear systems in a procedural way. In this paper we generalize the method for systems of…
This note contains a new combinatorial proof of Cramer's rule based on the Gessel-Viennot-Lindstrom Lemma.
Integration of the Dirac equation with an external electromagnetic field is explored in the framework of the method of separation of variables and of the method of noncommutative integration. We have found a new type of solutions that are…