General Mathematics
String art is an arrangement of pegs on a board with thread strung between these pegs to form beautiful geometric patterns. In this article, we consider a simple form of string art where pegs are placed on two diverging axes, and segments…
A structured approach for the Collatz conjecture is presented using just the odd integers that are, in turn, divided into categories based on the roles they play such as Starter, Intermediary and Terminal. The expression 4x+1 is used as a…
A tutorial introduction to projective geometric algebra (PGA), a modern, coordinate-free framework for doing euclidean geometry. PGA features: uniform representation of points, lines, and planes; robust, parallel-safe join and meet…
In this short note, we present the sharp estimate for the existence of a unique solution for a Hadamard-type fractional differential equations with two-point boundary value conditions. The method of analysis is obtained by using the…
The Riemann Hypothesis is the main open problem of Number Theory and several scientists are trying to solve this problem. In this regard, in a recent work [8], a difference equation has been proposed that calculates the nth non-trivial zero…
For $n$ positive numbers ($a_k$, $1\leq k \leq n$), enhanced inequalities about the arithmetic mean ($A_n \equiv \frac{\sum_ka_k}{n}$) and the geometric mean ($G_n\equiv \sqrt[n]{\Pi_ka_k}$) are found if some numbers are known, namely,…
Let $p_{-r}(n)$ denote the $r$-coloured partition function, and $\sigma(n)=\sum_{d|n}d$ denote the sum of positive divisors of $n$. The aim of this note is to prove the following $$…
In this paper we develop the Riemann-Lagrange geometry, in the sense of nonlinear connection, d-torsions, d-curvatures and jet Yang-Mills entity, associated with the dynamical system concerning social interaction in colonial organisms.
We consider several characterizations of $\mathbb R$-linear mappings. In particular, we give a characterization of linear mappings whose range is $\geq$ 2 dimensional, in terms of preservation of lines (and contraction of lines to a point)…
Focus of this study is to explore some aspects of mathematical foundations for using complex manifolds as a model for space-time. More specifically, certain equations of motions have been derived as a Projective geodesic on a real manifold…
Since the theory of rough sets was introduced by Zdzislaw Pawlak, several approaches have been proposed to combine rough set theory with fuzzy set theory. In this paper, we examine one of these approaches, namely fuzzy rough sets with crisp…
This paper presents a novel direct elementary proof for Fermat's Last Theorem. We use algebra, modular math, and binomial series to develop inherent mathematical relationships hidden within Fermat's Last Theorem. With these derived…
We showed that the prime gap for a prime number p is less than or equal to the prime count of the prime number.
Nullnorms with a zero element being at any point of a bounded lattice are an important generalization of triangular norms and triangular conorms. This paper obtains an equivalent characterization for the existence of idempotent nullnorms…
This paper attempts to prove the Sylvester's conjecture using Egyptian Fractions with two key ingredients. First, creating a set of operators that completely generates all possible Egyptian fraction of 1. And second, to detect patterns in…
Some properties of the optimal representation of numbers are investigated. This representation, which is to the base-e, is examined for coding of integers. An approximate representation without fractions that we call WF is introduced and…
We prove a~general form of Chebyshev type inequality for generalized upper Sugeno integral in the form of necessary and sufficient condition. A key role in our considerations is played by the~class of $m$-positively dependent functions…
The paper presents a counterexample to the Hodge conjecture.
Projection factors describe the contraction of Lebesgue measures in orthogonal projections between subspaces of a real or complex inner product space. They are connected to Grassmann's exterior algebra and the Grassmann angle between…
The main objective of present investigation to obtain some Minkowski-type fractional integral inequalities using generalised proportional Hadamard fractional integral operators which is introduced by Rahman et al in the paper (certain…