General Mathematics
The Poincar\'{e} lemma (or Volterra theorem) is of utmost importance both in theory and in practice. It tells us every differential form which is closed, is locally exact. In other words, on a contractible manifold all closed forms are…
Cographs--defined most simply as complete graphs with colored lines--both dualize and generalize ordinary graphs, and promise a comparably wide range of applications. This article introduces them by examples, catalogues, and elementary…
In this note, we extend the connection between the hydrogen atom and $\pi$ to the number $e$ via the Lerch's transcendent.
The present paper deals with a study of curves on a smooth surface whose position vector always lies in the tangent plane of the surface and it is proved that such curves remain invariant under isometry of surfaces. It is also shown that…
Some trapezoid and mid-point type inequalities related to the Hermite-Hadamard inequality for the mappings defined on a ball in the space are obtained.
In these notes we introduce the Clifford algebra of a quadratic space using techniques from universal algebra and algebraic theory of quadratic forms. We also define the Clifford, Pin and Spin groups associated to the algebra, and study how…
A novel kind of self-referential square matrix is introduced. A certain subset of the matrix entries record the frequencies of occurrence of each distinct number appearing within the entire matrix. Such squares are necessarily elusive. Our…
Dual numbers and their higher order version are important tools for numerical computations, and in particular for finite difference calculus. Based upon the relevant algebraic rules and matrix realizations of dual numbers, we will present a…
There is given a characterization of hyperbolic secant distribution by the independence of linear forms with random coefficients. We provide a characterization by the identic distribution property. Keywords: hyperbolic secant distribution;…
Let $G$ be a molecular graph. The total-eccentricity index of graph $G$ is defined as the sum of eccentricities of all vertices of $G$. %In [R. Farooq, M.A. Malik, J. Rada, Extremal graphs with respect to total-eccentricity index, 2017,…
Why is it that after so many years matrices continue to play such an important roll in Physics and mathematics? Is there a geometric way of looking at matrices, and linear transformations in general, that lies at the roots of their success?…
Let $A=[a_{ij}]\in O_3(\mathbb{R})$. We give several different proofs of the fact that the vector $$ V:=\left[\begin{array}{ccc} \displaystyle \frac{1}{a_{23}+a_{32}} & \displaystyle \frac{1}{a_{13}+a_{31}} & \displaystyle…
The fundamental purpose of this paper is to propose a new Laplace-type integral transform (NL-TIT) for solving steady heat-transfer problems. The proposed integral transform is a generalization of the Sumudu, and the Laplace transforms and…
Blurring of a photographic image by a wrong focus can be modeled by convolution. Is inversion a possible answer? This paper adds complements to a foregoing paper discussing convolution-inversion of some measures.
It is well known that the Leibniz rule for the integer derivative of order one does not hold for the fractional derivative case when the fractional order lies between 0 and 1. Thus it poses a great difficulty in the calculation of…
To allow for Division By Zero, we develop a new algebraic structure containing addition and multiplication called an S-Extension of a Field. This unique structure extends a Field so that the equation $0\cdot s=x$ has exactly one solution…
In this paper we will relate hyperstructures and the general $\mathscr{H}$-principle to known mathematical structures, and also discuss how they may give rise to new mathematical structures. The main purpose is to point out new ideas and…
In this article, we introduce equivariant formal deformation theory of associative algebra morphisms. We introduce an equivariant deformation cohomology of associative algebra morphisms and using this we study the equivariant formal…
Elementary transformations of equations $A\psi=\lambda\psi$ are considered. The invertibility condition (Theorem 1) is established and similar transformations of Riccati equations in the case of second order differential operator $A$ are…
We consider the Euler--Darboux equation with parameters modulo 1/2 and generalization to the space 3D analogue. Due to the fact that the Cauchy problem in its classical formulation is incorrect for such parameter values, the authors propose…