Related papers: Ahmed's Integral: the maiden solution
Two results are presented for reduced Yang-Mills integrals with different symmetry groups and dimensions: the first is a compact integral representation in terms of the relevant variables of the integral, the second is a method to…
An infinite dimensional algebra, which is useful for deriving exact solutions of the generalized pairing problem, is introduced. A formalism for diagonalizing the corresponding Hamiltonian is also proposed. The theory is illustrated with…
This is an introductory document surveying several results in polynomial approximation, known as the Bramble-Hilbert lemma.
For a set of curves, Ahn et al. introduced the notion of a middle curve and gave algorithms computing these with run time exponential in the number of curves. Here we study the computational complexity of this problem: we show that it is…
Here we present an algorithm to find elementary first integrals of rational second order ordinary differential equations (SOODEs). In \cite{PS2}, we have presented the first algorithmic way to deal with SOODEs, introducing the basis for the…
In this paper a double integral containing two Gaussian hypergeometric functions is discussed. The integral is not found in the literature and a direct computation is not (yet) possible. Therefore, a complete different integral is computed…
Recently, Chen, Hou and Jin used both Abel's lemma on summation by parts and Zeilberger's algorithm to generate recurrence relations for definite summations. Meanwhile, they proposed the Abel-Gosper method to evaluate some indefinite sums…
Algorithms for numerical computation of symmetric elliptic integrals of all three kinds are improved in several ways and extended to complex values of the variables (with some restrictions in the case of the integral of the third kind).…
The formal study of completely prime modules was initiated by N. J. Groenewald and the current author in the paper; Completely prime submodules, {\it Int. Elect. J. Algebra}, {\bf 13}, (2013), 1--14. In this paper, the study of completely…
It is a well-known result of Etingof, Nikshych and Ostrik that there are finitely many inequivalent integral modular categories of any fixed rank $n$. This follows from a double-exponential bound on the maximal denominator in an Egyptian…
A brief introduction is given to the topic of Smith normal forms of incidence matrices. A general discussion of techniques is illustrated by some classical examples. Some recent advances are described and the limits of our current…
We consider a multiple arithmetical sum involving the Moebius function which despite its elementary appearance is in fact of a highly intriguing nature. We establish an asymptotic formula for the quadruple case that raises the first…
This purpose of this paper is to note an interesting identity derived from an integral in Gradshteyn and Ryzhik using techniques from George Boros'(deceased) Ph.D thesis. The idenity equates a sum to a product by evaluating an integral in…
For a positive integer $m>1$, if the generalized Markoff equation $a^2+b^2+c^2=3abc+m$ has a solution triple, then it has infinitely many solutions. We show that all positive solution triples are generated by a finite set of triples that we…
They run our lives, if you believe the hype in the news, but there is no precise definition of "algorithms" which is generally accepted by the mathematicians, logicians and computer scientists who create and study them. My main aims here…
We determine the numbers of integral tetrahedra with diameter $d$ up to isomorphism for all $d\le 1000$ via computer enumeration. Therefore we give an algorithm that enumerates the integral tetrahedra with diameter at most $d$ in $O(d^5)$…
We introduce and study the arithmetic function E_m(n), defined as the sum of the remainders of n when divided by the first m positive integers. Although the definition is elementary, the function encodes rich arithmetic structure. In this…
In a recent article, Apagodu and Zeilberger (http://arxiv.org/abs/1606.03351)discuss some applications of an algorithm for finding and proving congruence identities (modulo primes) of indefinite sums of many combinatorial sequence. At the…
Multiplication and exponentiation can be defined by equations in which one of the operands is written as the sum of powers of two. When these powers are non-negative integers, the operand is integer; without this restriction it is a…
This paper introduces some efficient initials for a well-known algorithm (an inverse iteration) for computing the maximal eigenpair of a class of real matrices. The initials not only avoid the collapse of the algorithm but are also…