Related papers: On some explicit integrals related to "fractal mou…
Fractal-like structures of varying complexity are common in nature, and measure-based dimensions (Minkowski, Hausdorff) supply their basic geometric characterization. However, at the level of fundamental dynamics, which is quantum,…
In this work we discuss techniques for the numerical computation of Fox functions that represent Feynman integrals. Illustrative examples based on Sinc numerical methods and Quasi-Monte Carlo methods are given
We examine the aggregate behavior of one-dimensional random walks in a model known as (one-dimensional) Internal Diffusion Limited Aggregation. In this model, a sequence of $n$ particles perform random walks on the integers, beginning at…
Quantum walks are widely and successfully used to model diverse physical processes. This leads to computation of the models, to explore their properties. Quantum walks have also been shown to be universal for quantum computing. This is a…
Intrinsic computation refers to how dynamical systems store, structure, and transform historical and spatial information. By graphing a measure of structural complexity against a measure of randomness, complexity-entropy diagrams display…
We define a random walk problem which admits analytic results, on a class of infinite periodic lattices which are directed and colored. Our approach is motivated from the fact that such lattices arise in string theoretic constructs of…
UD integrals published by N. Usyukina and A. Davydychev in 1992-1993 are integrals corresponding to ladder-type Feynman diagrams. The results are UD functions $\Phi^{(L)},$ where $L$ is the number of loops. They play an important role in…
In this note, we try to analyze and clarify the intriguing interplay between some counting problems related to specific thermalized weighted graphs and random walks consistent with such graphs.
In this paper, we explore spectral measures whose square integrable spaces admit a family of exponential functions as an orthonormal basis.Our approach involves utilizing the integral periodic zeros set of Fourier transform to characterize…
In this article the integration of the $\alpha$-fractal interpolation function $f^{\alpha}$ corresponding to any continuous function $f$ on a compact interval $I$ of $\mathbb{R}$ is estimated although there is no explicit form of…
In this article, we considered a fractal image as a fractal curve, that is, as a walk on a grid in Euclidean space $\R^d$. We placed integers on the generating vectors of a grid, such that opposite directions have opposite numbers. This…
We introduce fractional flat space, described by a continuous geometry with constant non-integer Hausdorff and spectral dimensions. This is the analogue of Euclidean space, but with anomalous scaling and diffusion properties. The basic tool…
Discrete time quantum walks are known to be universal for quantum computation. This has been proven by showing that they can simulate a universal quantum gate set. In this paper, we examine computation by quantum walks in terms of language…
A fractal bears a complex structure that is reflected in a scaling hierarchy, indicating that there are far more small things than large ones. This scaling hierarchy can be effectively derived using head/tail breaks - a clustering and…
We review the concept of $\tau$-function for simple analytic curves. The $\tau$-function gives a formal solution to the 2D inverse potential problem and appears as the $\tau$-function of the integrable hierarchy which describes conformal…
Intrinsic complexity of a relation on a given computable structure is captured by the notion of its degree spectrum - the set of Turing degrees of images of the relation in all computable isomorphic copies of that structure. We investigate…
We define a tau function for a generic Riemann-Hilbert problem posed on a union of non-intersecting smooth closed curves with jump matrices analytic in their neighborhood. The tau function depends on parameters of the jumps and is expressed…
We analyze the boundaries of multiply connected Fatou components of transcendental maps by means of universal covering maps and associated inner functions. A unified approach is presented, which includes invariant Fatou components (of any…
An approach to (normalized) infinite dimensional integrals, including normalized oscillatory integrals, through a sequence of evaluations in the spirit of the Monte Carlo method for probability measures is proposed. in this approach the…
We present a random walk approximation to fractional Brownian motion where the increments of the fractional random walk are defined as a weighted sum of the past increments of a Bernoulli random walk.