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On the basis of the generalized argument principle, here we develop a numerical scheme for locating zeros and poles of a meromorphic function. A subdivision-transformation-calculation scheme is proposed to ensure the algorithm stability. A…
We present a derivation of the propagator for a particle in the presence of the step and delta function potentials. These propagators are known, but we present a direct path integral derivation, based on the path decomposition expansion and…
The approximation of the eigenvalues and eigenfunctions of an elliptic operator is a key computational task in many areas of applied mathematics and computational physics. An important case, especially in quantum physics, is the computation…
Quartic eigenvalue problem $(\lambda^4 A + \lambda^3 B + \lambda^2C + \lambda D + E)x = \mathbf{0}$ naturally arises e.g. when solving the Orr-Sommerfeld equation in the analysis of the stability of the {Poiseuille} flow, in theoretical…
The method of spectral decimation is applied to an infinite collection of self--similar fractals. The sets considered belong to the class of nested fractals, and are thus very symmetric. An explicit construction is given to obtain formulas…
We show that the factorization problem $\theta (z)=\theta_2(z)\theta_1(z)$ is solvable in the class of Hilbert space operator-valued functions holomorphic on some neighbourhood of $z=0$ in $\nspace{C}{N}$ and having a zero at $z=0$ (here…
We consider the electromagnetic field in the presence of polarizable point dipoles. In the corresponding effective Maxwell equation these dipoles are described by three dimensional delta function potentials. We review the approaches…
The exact exponential Foldy-Wouthuysen transformation operator applicable for a particle with an arbitrary spin is derived. It can be successfully utilized for verifying any Foldy-Wouthuysen transformation method based on the exponential…
We study 2-point and 3-point functions in CFT at finite temperature for large dimension operators using holography. The 2-point function leads to a universal formula for the holographic free energy in $d$ dimensions in terms of the…
A path-integral approach for delta-function potentials is presented. Particular attention is paid to the two-dimensional case, which illustrates the realization of a quantum anomaly for a scale invariant problem in quantum mechanics. Our…
Direct solvers are a common method for solving finite difference frequency domain (FDFD) systems that arise in numerical solutions of Maxwell's equations. In a direct solver, one factorizes the system matrix. Since the system matrix is…
Factor Analysis is an effective way of dimensionality reduction achieved by revealing the low-rank plus sparse structure of the data covariance matrix. The corresponding model identification task is often formulated as an optimization…
We propose a method for Monte Carlo simulations of systems with a complex action. The method has the advantages of being in principle applicable to any such system and provides a solution to the overlap problem. We apply it in random matrix…
Matrix factorization is an inference problem that has acquired importance due to its vast range of applications that go from dictionary learning to recommendation systems and machine learning with deep networks. The study of its fundamental…
Some calculations in supersymmetric theories, made with the higher derivative regularization, show that the beta-function is given by integrals of total derivatives. This is qualitatively explained for the N=1 supersymmetric electrodynamics…
Based on the fact that the Hamiltonians of the Coulomb many-particle systems are always factorized we develop the two different approaches for analytical solution of the Schr\"{o}dinger equation written for arbitrary few- and many-particle…
We propose a new method to accelerate the convergence of optimization algorithms. This method simply adds a power coefficient $\gamma\in[0,1)$ to the gradient during optimization. We call this the Powerball method and analyze the…
Randomized sampling has recently been demonstrated to be an efficient technique for computing approximate low-rank factorizations of matrices for which fast methods for computing matrix vector products are available. This paper describes an…
A factorization formula for wave functions, which is basic in the inverse spectral transform approach to initial-boundary value problems, is proved in greater generality than before. Applications follow. Related compatibility questions for…
We find all factorized duality functions for a class of interacting particle systems. The functions we recover are self-duality functions for interacting particle systems such as zero-range processes, symmetric inclusion and exclusion…