Related papers: Yet Another Table of Integrals
A novel method of summation for power series is developed. The method is based on the self-similar approximation theory. The trick employed is in transforming, first, a series expansion into a product expansion and in applying the…
We consider a generalized Gauss sum supported on matrices over a number field. We evaluate this Gauss sum and relate it to the number of totally isotropic subspaces of related quadratic spaces. Then we consider a further generalization of…
Recent advances in large language models (LLMs) have led to new summarization strategies, offering an extensive toolkit for extracting important information. However, these approaches are frequently limited by their reliance on isolated…
An integral equation is a way to encapsulate the relationships between a function and its integrals. We develop a systematic way of describing Volterra integral equations -- specifically an algorithm that reduces any separable Volterra…
Graph compositions generalize both integer compositions and partitions of a finite set. We develop formulas, generating functions and recurrence relations for composition counting functions for several families of graphs.
Frank Harary introduced the concepts of sum and integral sum graphs. A graph $G$ is a \textit{sum graph} if the vertices of $G$ can be labeled with distinct positive integers so that $e = uv$ is an edge of $G$ if and only if the sum of the…
This note provides formula for determinant and inverse of r-circulant matrices with general sequences of third order. In other words, the study combines many papers in the literature.
A bibliographic database containing studies on recurrence plots and related methods is analyzed from various perspectives. This allows a detailed view of the field's development, showcasing the continuous growth in the method's popularity,…
The classical table of integrals by I. S. Gradshteyn and I. M. Ryzhik contains some elementary integrals. We discuss their evaluations.
Recent advances in data science, machine learning, and artificial intelligence, such as the emergence of large language models, are leading to an increasing demand for data that can be processed by such models. While data sources are…
A novel type of approximants is introduced, being based on the ideas of self-similar approximation theory. The method is illustrated by the examples possessing the structure typical of many problems in applied mathematics. Good numerical…
Computer programs are often factored into pure components -- simple, total functions from inputs to outputs -- and components that may have side effects -- errors, changes to memory, parallel threads, abortion of the current loop, etc. We…
Matrices are very popular and widely used in mathematics and other fields of science. Every mathematician has known the properties of finite-sized matrices since the time of study. In this paper, we consider the basic theory of infnite…
A matrix approach to continuous iteration is proposed for general formal series. It leads, in particular, to an order{to{order iteration of the exponential function, and consequently to an algorithmic approach to tetration. Lower{order…
A family of general integral identities is derived and several applications of physical interest are presented
We continue the investigation of tabular algebras with trace (a certain class of associative ${\Bbb Z}[v, v^{-1}]$-algebras equipped with distinguished bases) by determining the extent to which the tabular structure may be recovered from a…
We show how to underapproximate the procedure summaries of recursive programs over the integers using off-the-shelf analyzers for non-recursive programs. The novelty of our approach is that the non-recursive program we compute may capture…
Recently people started to understand that applications of the mathematical formalism of quantum theory are not reduced to physics. Nowadays, this formalism is widely used outside of quantum physics, in particular, in cognition, psychology,…
Why and why-not provenance have been studied extensively in recent years. However, why-not provenance, and to a lesser degree why provenance, can be very large resulting in severe scalability and usability challenges. In this paper, we…
In this paper, we give explicit evaluation for some infinite series involving generalized (alternating) harmonic numbers. In addition, some formulas for generalized (alternating) harmonic numbers will also be derived.