Related papers: Ahmed's Integral: the maiden solution
These notes explore three amazing formulas proved by Abel in his 1826 Paris memoir on what we now call Abelian integrals. We discuss the first two formulas from the point of view of symbolic computation and explain their connection to…
In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are…
This habilitation thesis is intended to be a good introduction to enumeration, the problem of listing solutions. It focuses on the different ways of measuring complexity in enumeration, with a particular emphasis on my contributions to the…
The paper suggests a slightly more rigorous justification to Wang et al.'s work from 2007, and introduces the Slanted Line Integral.
This book has seven chapters. In chapter one we give the basics needed to make this book a self contained one. Chapter two introduces the notion of interval semigroups and interval semifields and are algebraically analysed. Chapter three…
The results of the computer hunt for the primes of the form $q = m^2+1$ up to $10^{20}$ are reported. The number of sign changes of the difference $\pi_q(x) - \frac{C_q}{2}\int_2^x{du \over \sqrt{u}\log(u)}$ and the error term for this…
Prime numbers or primes are man's eternal treasures that have been cherished for several millennia, until today. As their academic ancestors in ancient Mesopotamia, many mathematicians are still trying hard to see primes better. I shall…
{\bf In the fourth extended version of this article, we provide a comprehensive historical survey of 200 different proofs of famous Euclid's theorem on the infinitude of prime numbers (300 {\small B.C.}--2022)}. The author is trying to…
Given two combinatorial identities proved earlier, a new set of variations of these combinatorial identities is listed and proved with the integral representation method. Some identities from literature are shown to be special cases of…
The Inner Model Hypothesis (IMH) and the Strong Inner Model Hypothesis (SIMH) were introduced by the first author in ``Internal consistency and the inner model hypothesis'', Bulletin of Symbolic Logic, December 2006. In this article we…
This collection of sums and integrals has been harvested from the mathematical and physical literature in unstructured ways. Its main use is backtracking the original sources whenever an integral of the reader's application resembles one of…
In 2007, the first author gave an alternative proof of the refined alternating sign matrix theorem by introducing a linear equation system that determines the refined ASM numbers uniquely. Computer experiments suggest that the numbers…
This paper proposes a modification to the traditional binary search algorithm in which it checks the presence of the input element with the middle element of the given set of elements at each iteration. Modified binary search algorithm…
In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are…
We propose in this paper a unifying scheme for several algorithms from the literature dedicated to the solving of monotone inclusion problems involving compositions with linear continuous operators in infinite dimensional Hilbert spaces. We…
In recent years, although the Alternating Direction Method of Multipliers (ADMM) has been empirically applied widely to many multi-convex applications, delivering an impressive performance in areas such as nonnegative matrix factorization…
An alternative form of Fermats equation[1] is proposed. It represents a portion of the identity that includes three terms of Fermats original equation. This alternative form permits an elementary and compact proof of the first case of…
Extending a classical estimate of Mertens for the sum of the reciprocals of the first primes, we provide an explicit remainder formula for products of an arbitrary, but fixed, number of primes.
We give an identity which is conjectured and proved by using an implementation in Multi-WZ.
The polynomial Ramanujan sum was first introduced by Carlitz [7], and a generalized version by Cohen [10]. In this paper, we study the arithmetical and analytic properties of these sums, derive various fundamental identities, such as H…