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The present article is focused on the study of a special class of systems of nonlinear transcendental equations for which classical algebraic and symbolic methods are inapplicable. For the purpose of the study of such systems, we develop a…
Working in the p-adic analog of the complex numbers, we'll define a line integral on a small arc of a circle. This allows new versions of the Residue Theorem, the Cauchy-Goursat Theorem on discs with and without holes, Cauchy's Integral…
Let $f$ be a function on the real line. The Fourier transform inversion theorem is proved under the assumption that $f$ is absolutely continuous such that $f$ and $f'$ are Lebesgue integrable. A function $g$ is defined by…
Stieltjes integral theorem is more commonly known by the phrase 'integration by parts' and enables rearrangement of an otherwise intractable integral to a more amenable form; often permitting completion of an integral in closed form.…
The paper presents an interesting mathematical feedback between the formalism of coherent states and the field of integrals and integral representations involving special functions. This materializes through an easy and fast method to…
We show that the centered discrete Hilbert transform on integers applied to a function can be written as the conditional expectation of a transform of stochastic integrals, where the stochastic processes considered have jump components. The…
A previously established correspondence between definite-parity real functions and inner analytic functions is generalized to real functions without definite parity properties. The set of inner analytic functions that corresponds to the set…
We describe algorithms for computing central values of twists of $L$-functions associated to Hilbert modular forms, carry out such computations for a number of examples, and compare the results of these computations to some heuristics and…
This is a discussion of miscellaneous summation, integration and transformation formulas obtained using Fourier analysis. The topics covered are: Series of the form $\sum_{n\in\mathbb{Z}} c_ne^{\pi i \gamma n^2}$; Fusion of integrals, and…
In this paper, we describe a framework to compute expected convergence rates for residuals based on the Calder\'on identities for general second order differential operators for which fundamental solutions are known. The idea is that these…
We extend the Eruguin result exposed in the paper "Construction of the whole set of ordinary differential equations with a given integral curve" published in 1952 and construct a differential system in $\Bbb{R}^N$ which admits a given set…
We show how to transform any set of prioritized propositional defaults into an equivalent set of parallel (i.e., unprioritized) defaults, in circumscription. We give an algorithm to implement the transform. We show how to use the transform…
Eigenvalue transformations appear ubiquitously in scientific computation, ranging from matrix polynomials to differential equations, and are beyond the reach of the quantum singular value transformation framework. In this work, we study the…
We study fractional differential equations of Riemann-Liouville and Caputo type in Hilbert spaces. Using exponentially weighted spaces of functions defined on $\mathbb{R}$, we define fractional operators by means of a functional calculus…
The aim of this article is to design a moment transformation for Student- t distributed random variables, which is able to account for the error in the numerically computed mean. We employ Student-t process quadrature, an instance of…
In this work we focus on substantial fractional integral and differential operators which play an important role in modeling anomalous diffusion. We introduce a new generalized substantial fractional integral. Generalizations of fractional…
The construction of synthetic complex-valued signals from real-valued observations is an important step in many time series analysis techniques. The most widely used approach is based on the Hilbert transform, which maps the real-valued…
We survey recent contributions to finite element exterior calculus on manifolds and surfaces within a comprehensive formalism for the error analysis of vector-valued partial differential equations on manifolds. Our primary focus is on…
The problem of inverting the total divergence operator is central to finding components of a given conservation law. This might not be taxing for a low-order conservation law of a scalar partial differential equation, but integrable systems…
This paper focuses on a wide class of Collatz-type arithmetic dynamics, and presents a systematic derivation of recursive formulas and functional equations satisfied by the associated generating functions. The main tools belong to complex…