Related papers: The Change of Variable Formula Integrals, do they …

In this note, we derive a finite summation formula and an infinite summation formula involving Harmonic numbers of order up to some order by means of several definite integrals

The article concerns the problem if a~given system of differential equations is identical with the Euler--Lagrange system of an~appropriate variational integral. Elementary approach is applied. The main results involve the determination of…

A counter-example to lower bounds for the singular values of the sum of two matrices in [1] and [2] is given. Correct forms of the bounds are pointed out.

This paper studies the integral of the Riemann xi-function. More generally, it studies a one-parameter family of functions given by Fourier integrals and satisfying a functional equation. Members of this family are shown to have only…

We give a direct proof of a cancellation formula raised in [7] on the level of differential forms. We also obtain more cancellation formulas for even dimensional Riemannian manifolds with a complex line bundle involved. Relations among…

Using a concept of filter we propose one generalization of Riemann integral, that is integration with respect to filter. We study this problem, demonstrate different properties and phenomena of filter integration.

In this paper, we establish new an inequality of weighted Ostrowski-type for double integrals involving functions of two independent variables by using fairly elementary analysis.

Two permutations in a class are Wilf-equivalent if, for every size, $n$, the number of permutations in the class of size $n$ containing each of them is the same. Those infinite classes that have only one equivalence class in each size for…

The sum formula for finite and symmetric multiple zeta values, established by Wakabayashi and the authors, implies that if the weight and depth are fixed and the specified component is required to be more than one, then the values sum up to…

In the present paper we consider a general family of two dimensional wave equations which represents a great variety of linear and nonlinear equations within the framework of the transformations of equivalence groups. We have investigated…

We give a survey of the Lagrange inversion formula, including different versions and proofs, with applications to combinatorial and formal power series identities.

It is shown that every Feynman integral can be interpreted as Green function of some linear differential operator with constant coefficients. This definition is equivalent to usual one but needs no regularization and application of…

A new derivative, called deformable derivative, is introduced here which is equivalent to ordinary derivative in the sense that one implies other. The deformable derivative is defined using limit approach like that of ordinary one but with…

We give an exposition of the Newton-Leibniz calculus. We begin by defining the integral as a limit of Riemann sums, verify the integrals of the standard catalog of functions by direct manipulation, prove the substitution lemmas as theorems…

We utilize a combination of integral transforms, including the Laplace transform, with some classical results in analytic number theory concerning the Riemann $\xi$-function, to obtain a new integral equation. We also provide a new proof of…

In this paper we introduce two new generalized variational inequalities, and we give some existence results of the solutions for these variational inequalities involving operators belonging to a recently introduced class of operators. We…

Let $n \in \mathbb{Z}_{\geq 3}$ be given. We prove Lebesgue-almost everywhere pointwise inversion formulae for the Siegel transforms in the geometry of numbers. These inversion formulae are quite general; for instance, they are valid for…

Einstein equations for several matter sources in Robertson-Walker and Bianchi I type metrics, are shown to reduce to a kind of second order nonlinear ordinary differential equation $\ddot{y}+\alpha f(y)\dot{y}+\beta f(y)\int{f(y) dy}+\gamma…

Motivated by Euler-Goldbach and Shallit-Zikan theorems, we introduce zeta-one functions with infinite sums of $n^{s}\pm1$ as an analogy of the Riemann zeta function. Then we compute values of these functions at positive even integers by the…

We give a number new examples analytically and numerically to confirm the Kohler conjecture. It turned out that for a rather large class of nonnegative functions the equality (A) hold.