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Toeplitz matrices are characterized by their constant diagonals, have been extensively studied in various settings, including over real and complex numbers. However, their study over quaternions is quite sparse. In this paper, we…
Evaluation of the angular distribution function of particles scattered in an amorphous medium is improved by deforming the integration path in the Fourier integral representation into the complex plane. That allows us to present the…
The role of differential equations in the process of calculating Feynman integrals is reviewed. An example of a diagram is given for which the method of differential equations was introduced, the properties of the inverse-mass-expansion…
The Fast Fourier Transform is extended to functions on finite graphs whose edges are identified with intervals of finite length. Spectral and pseudospectral methods are developed to solve a wide variety of time dependent partial…
Random matrix theory is a well-developed area of probability theory that has numerous connections with other areas of mathematics and its applications. Much of the literature in this area is concerned with matrices that possess many exact…
A procedure to obtain differentiation matrices is extended straightforwardly to yield new differentiation matrices useful to obtain derivatives of complex rational functions. Such matrices can be used to obtain numerical solutions of some…
In this short note we collect together known results on the use of Random Matrix Theory in lattice statistical mechanics. The purpose here is two fold. Firstly the RMT analysis provides an intrinsic characterization of integrability, and…
This review presents an account of the major works done on spectra of adjacency matrices drawn on networks and the basic understanding attained so far. We have divided the review under three sections: (a) extremal eigenvalues, (b) bulk part…
Feynman's laws of quantum dynamics are concisely stated, discussed in comparison with other formulations of quantum mechanics and applied to selected problems in the physical optics of photons and massive particles as well as flavour…
We develop simple rules for performing integrals over products of distributions in coordinate space. Such products occur in perturbation expansions of path integrals in curvilinear coordinates, where the interactions contain terms of the…
We develop a unified construction of matrix-valued orthogonal polynomials associated with discrete weights, yielding bispectral sequences as eigenfunctions of second-order difference operators. This general framework extends the discrete…
The essence of the path integral method in quantum physics can be expressed in terms of two relations between unitary propagators, describing perturbations of the underlying system. They inherit the causal structure of the theory and its…
The Feynman path integral plays a crucial role in quantum mechanics, offering significant insights into the interaction between classical action and propagators, and linking quantum electrodynamics (QED) with Feynman diagrams. However, the…
Feynman's path integral is generalized to quantum mechanics on p-adic space and time. Such p-adic path integral is analytically evaluated for quadratic Lagrangians. Obtained result has the same form as that one in ordinary quantum…
Polyakov's spin factor enters as a weight in the path-integral description of particle-like modes propagating in Euclidean space-times, accounting for particle spin. The Fock-Feynman-Schwinger path integral applied to QCD accomodates…
A fractal approach to numerical analysis of electromagnetic space-time crystals, created by three standing plane harmonic waves with mutually orthogonal phase planes and the same frequency, is presented. Finite models of electromagnetic…
We give a unified interpretation of confluences, contiguity relations and Katz's middle convolutions for linear ordinary differential equations with polynomial coefficients and their generalization to partial differential equations. The…
We construct a path distribution representing the kinetic part of the Feynman path integral at discrete times similar to that defined by Thomas [1], but on a Hilbert space of paths rather than a nuclear sequence space. We also consider…
An extension of the method and results of A. Schwarz for evaluating the partition function of a quadratic functional is presented. This enables the partition functions to be evaluated for a wide class of quadratic functionals of interest in…
Score-based diffusion models have proven effective in image generation and have gained widespread usage; however, the underlying factors contributing to the performance disparity between stochastic and deterministic (i.e., the probability…