# Mathematics - General Mathematics Publications (50)

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## Mathematics - General Mathematics Publications

We present a prime-generating polynomial $(1+2n)(p -2n) + 2$ where $p>2$ is a lower member of a pair of twin primes less than $41$ and the integer $n$ is such that $\: \frac {1-p}{2} < n < p-1$. Read More

Let $E$ be an elliptic curve of rank $\text{rk}(E) \geq 1$, and let $E(\mathbb{F}_p)$ be the elliptic group of order $\#E(\mathbb{F}_p)=n$. The number of primes $p\leq x$ such that $n$ is prime is expected to be $\pi(x,E)=\delta(E)x/\log^2 x+o(x/\log^2 x)$, where $\delta(E)\geq 0$ is the density constant. This note proves a lower bound $\pi(x,E) \gg x/\log^3 x$. Read More

In this article, utilizing the concept of w-distance, we prove the celebrated Banach's fixed point theorem in metric spaces equipped with an arbitrary binary relation. Necessarily our findings unveil another direction of relation-theoretic metrical fixed point theory. Also, our paper consists of several non-trivial examples which signify the motivation for such investigations. Read More

The Collatz variations pattern seems not to have any recurrence relation between numbers. But knowing that there is at least a natural number that converges after several iterations we construct a function $f_{X,Y}$ that is equal to the value of convergence for all convergent sequences. A canonical decomposition can be expressed for such numbers. Read More

Closed form expressions are given for computing the parameters and vectors that identify and define the $n-1$ dimensional conic section that results from the intersection of a hyperplane with an $n$-dimensional conic section: cone, hyperboloid of two sheets, ellipsoid or paraboloid. The conic sections are assumed to be symmetric about their major axis, but may have any orientation and center. A class of hyperboloids are identified with the property that the parameters and vectors of the intersection of all hyperboloids in a subset of the class can be computed efficiently. Read More

Given a nonempty set $X$ and a function $f:X \rightarrow X$, three fuzzy topological spaces are introduced. Some properties of these spaces and relation among them are studied and discussed. Read More

Using a simple Vi\`ete-like formula for $\pi$ based on the nested radicals $a_k = \sqrt{2 + a_{k-1}}$ and $a_1 = \sqrt{2}$, we derive a set of the recurrence relations for the constant $1$. Computational test shows that application of this set of the Vi\`ete-like recurrence relations results in a rapid convergence to unity. Read More

For a colour cluster $\C =(\mathcal{C}_1,\mathcal{C}_2, \mathcal{C}_3,\dots,\mathcal{C}_\ell)$, $\mathcal{C}_i$ is a colour class, and $|\mathcal{C}_i|=r_i \geq 1$, we investigate a simple connected graph structure $G^{\C}$, which represents a graphical embodiment of the colour cluster such that the chromatic number $\chi(G^{\C})= \ell,$ and the number of edges is a maximum, denoted $\varepsilon^+(G^{\C})$. We also extend the study by inducing new colour clusters recursively by blending the colours of all pairs of adjacent vertices. Recursion repeats until a maximal homogeneous blend between all $\ell$ colours is obtained. Read More

A random variable X that is base b Benford will not in general be base c Benford when c is not equal to b. (There are known exceptions to this statement.) Let X be a base b Benford random variable and let U be the fractional part of the base c logarithm of X. Read More

In this short note we prove that a matrix $A\in\mathbb{R}^{n,n}$ is self-adjoint if and only if it is equivariant with respect to the action of a group $\Gamma\subset {\bf O}(n)$ which is isomorphic to $\otimes_{k=1}^n\mathbf{Z}_2$. Moreover we discuss potential applications of this result, and we use it in particular for the approximation of higher order derivatives for smooth real valued functions of several variables. Read More

In this paper we consider the set of all bounded subsets of totally ordered Dedekind complete Riesz spaces, equipped with the order topology. We show the existence of bounded linear functions on this set, that are invariant under group actions of the symmetric group of it. Read More

We present a function that tests for primality, factorizes composites and builds a closed form expression of $\pi(n^2)$ in terms of $\sum_{3 \leq p \leq n} \frac{1}{p}$ and a weaker version of $\omega(n)$. Read More

The purpose of this note is to give an accessible proof of Moliens Theorem in Invariant Theory, in the language of today's Linear Algebra and Group Theory, in order to prevent this beautiful theorem from being forgotten. Read More

In our recent publication we obtained a series expansion of the arctangent function involving complex numbers. In this work we show that this formula can also be expressed as a real rational function. Read More

Let \(x\geq 1\) be a large number, and let $1 \leq a 0$ constant. This note proves that
the counting function for the number of primes $p \in \{p=qn+a: n \geq1 \}$
with a fixed primitive root $u\ne \pm 1, v^2$ has the asymptotic formula
$\pi_u(x,q,a)=\delta(u,q,a)x/ \log x +O(x/\log^b x),$ where $\delta(u,q,a)>0$
is the density, and $b>c+1$ is a constant.
Read More

In 1976 Appel and Haken achieved a major break through by thoroughly establishing the Four Color Theorem (4CT). Their proof is based on studying a large number of cases for which a computer-assisted search for hours is required. In 1997 the 4CT was reproved with less need for computer verification by Robertson, Sanders, Seymour and Thomas. Read More

The aim of this paper is to establish some metrical coincidence and common fixed point theorems with an arbitrary relation under an implicit contractive condition which is general enough to cover a multitude of well known contraction conditions in one go besides yielding several new ones. We also provide an example to demonstrate the generality of our results over several well known corresponding results of the existing literature. Finally, we utilize our results to prove an existence theorem for ensuring the solution of an integral equation. Read More

In an article, E. Grafarend and B. Schaffrin studied the geometry of non-linear adjustment of the planar trisection problem using the Gauss Markov model and the method of the least squares. Read More

Let $X$ be a non-empty ground set and $\mathscr{P}(X)$ be its power set. A set-labeling (or a set-valuation) of a graph $G$ is an injective set-valued function $f:V(G)\to \mathscr{P}(X)$ such that the induced function $f^*:E(G) \to \mathscr{P}(X)$ is defined by $f^*(uv)=f(u)\ast f(v)$, where $f(u)\ast f(v)$ is a binary operation of the sets $f(u)$ and $f(v)$. A graph which admits a set-labeling is known to be a set-labeled graph. Read More

Two open problems in the fixed point theory of quasi metric spaces posed in [Berinde, V. and Choban, M. M. Read More

We introduce and study a general concept of multiple fixed point for mappings defined on partially ordered distance spaces in the presence of a contraction type condition and appropriate monotonicity properties. This notion and the obtained results complement the corresponding ones from [Choban, M., Berinde, V. Read More

Our main aim in this paper is to introduce a general concept of multidimensional fixed point of a mapping in spaces with distance and establish various multidimensional fixed point results. This new concept simplifies the similar notion from [A. Roldan, J. Read More

Hintikka and Sandu's independence-friendly (IF) logic is a conservative extension of first-order logic that allows one to consider semantic games with imperfect information. In the present article, we first show how several variants of the Monty Hall problem can be modeled as semantic games for IF sentences. In the process, we extend IF logic to include semantic games with chance moves and dub this extension stochastic IF logic. Read More

We transformed the generalized exponential power series to another functional form suitable for further analysis. By applying the Cauchy-Euler differential operator in the form of an exponential operator, the series became a sum of exponential differential operators acting on a simple exponential (exp(-x). In the process we found new relations for the operator and a new polynomial with some interesting properties. Read More

**Category:**Mathematics - General Mathematics

In this paper we give a new definition of soft topology using elementary union and elementary intersection although these operations are not distributive. Also we have shown that this soft topology is different from Naz's soft topology and studied some basic properties of this new type of soft topology. Here we use elementary complement of soft sets, though law of excluded middle is not valid in general for this type of complementation. Read More

We determine the probability $P$ of two independent events $A$ and $B$, which occur randomly $n_A$ and $n_B$ times during a total time $T$ and last for $t_A$ and $t_B$, to occur simultaneously at some point during $T$. Therefore we first prove the precise equation \begin{equation*} P^* = \dfrac{t_A+t_B}{T} - \dfrac{t_A^2+t_B^2}{2T^2} \end{equation*} for the case $n_A = n_B = 1$ and continue to establish a simple approximation equation \begin{equation*} P \approx 1 - \left( 1 - n_A \dfrac{t_A + t_B}{T} \right)^{n_B} \end{equation*} for any given value of $n_A$ and $n_B$. Finally we prove the more complex universal equation \begin{equation*} P = 1 - \dfrac{ \left( T^+ - t_A n_A - t_B n_B \right)^{n_A + n_B} }{ \left( T^+ - t_A n_A \right)^{n_A} \left( T^+ - t_B n_B \right)^{n_B} } \pm E^\pm, \end{equation*} which yields the probability for $A$ and $B$ to overlap at some point for any given parameter, with $T^+ := T + \frac{t_A + t_B}{2}$ and a small error term $E^\pm$. Read More

The congruence lattices of all algebras defined on a fixed finite set $A$ ordered by inclusion form a finite atomistic lattice $\mathcal E$. We describe the atoms and coatoms. Each meet-irreducible element of $\mathcal E$ being determined by a single unary mapping on A, we characterize completely those which are determined by a permutation or by an acyclic mapping on the set $A$. Read More

The concept of Type-2 soft sets had been proposed as a generalization of Molodstov's soft sets. In this paper some shortcomings of some existing distance measures for Type-1 soft sets have been shown and accordingly some new distance measures have been proposed. The axiomatic definitions for distance, entropy and similarity measures for Type-2 soft sets have been introduced and a couple of such measures have been defined. Read More

In this paper, we extend the Banach contraction principle to metric-like as well as partial metric spaces (not essentially complete) equipped with an arbitrary binary relation. Thereafter, we derive some fixed point results which are sharper versions of the corresponding known results of the existing literature. Finally, we use an example to demonstrate our main result. Read More

This paper establishes grounds for deeper exploration into the question of dual nature of mathematics as an abstract discipline and as a concrete science. It is argued, as one of the consequences of the discussion, that the division into "pure" and "applied" mathematics is artificial. The criterion of creativity and applicability outside of the original context is used as a litmus test. Read More

In this paper, a new type of colouring called Johan colouring is introduced. This colouring concept is motivated by the newly introduced invariant called the rainbow neighbourhood number of a graph. The study ponders on maximal colouring opposed to minimum colouring. Read More

**Affiliations:**

^{1}Department of Mathematics and Statistics, Ohio Northern University, Ada, OH

**Category:**Mathematics - General Mathematics

In this note we compare two formulas for the higher order derivatives of the function 1/(exp(x) -1). We also provide an integral representation for these derivatives and obtain a classical formula relating zeta values and Bernoulli numbers. Read More

In the present study, we derive the problem of constructing a hypersurface family from a given isogeodesic curve in the 4D Galilean space $\mathbf{G}_{4}.$ We obtain the hypersurface as a linear combination of the Frenet frame in $\mathbf{G}_{4}$ and examine the necessary and sufficient conditions for the curve as a geodesic curve$.$ Finally, some examples related to our method are given for the sake of clarity. Read More

In this study, we investigate a new type of a surface curve called a new D-type special curve. Also, we show that this special curve is more generally than a geodesic curve or an asymptotic curve. Then, we give the necessary and sufficient conditions for a curve to be the new D-type special curve using Frenet frame in Galilean space. Read More

**Category:**Mathematics - General Mathematics

In this paper, we introduce notion of fuzzy soft number. Here defined fuzzy soft number and four arithmetric operations $ \tilde{+}, \tilde{-}, \tilde{\times}, \tilde{\div} $ and related properties. Also introduce Hausdorff distance, Fuzzy soft metric space, convergence sequence, Cauchy sequence, Continuity and uniform continuity of fuzzy soft numbers. Read More

This paper revisits formulas for $\pi$ involving nested radicals in iterative forms by discussing a method of deriving an infinite number of them.This method involves deriving a limit for $\pi$ from the formula expression, circumference, $C=2r k$. In consequence the limit derived is transformed into formula expressions consisting of infinitely nested radicals. Read More

The purpose of this paper is to emulate the process used in defining and learning about the algebraic structure known as a Field in order to create a new algebraic structure which contains numbers that can be used to define Division By Zero, just as $i$ can be used to define $\sqrt{-1}$. This method of Division By Zero is different from other previous attempts in that each ${\alpha\over 0}$ has a different unique, numerical solution for every possible $\alpha$, albeit these numerical solutions are not any numbers we have ever seen. To do this, the reader will be introduced to an algebraic structure called an S-Structure and will become familiar with the operations of addition, subtraction, multiplication and division in particular S-Structures. Read More

The paper deals with the process of mathematical modeling representations of exponential and logarithmic functions hypercomplex number system of generalized quaternions via determining a linear differential equation with hypercomplex coefficients. Simulation is performed using symbolic computation system Maple. Some properties of these concepts and their relation to the exponential representations of specific non-commutative hypercomplex number systems of dimension four. Read More

By using Modified simple equation method, we study the Cahn Allen equation which arises in many scientific applications such as mathematical biology, quantum mechanics and plasma physics. As a result, the existence of solitary wave solutions of the Cahn Allen equation is obtained. Exact explicit solutions interms of hyperbolic solutions of the associated Cahn Allen equation are characterized with some free parameters. Read More

In this article we introduce a simple straightforward and powerful method involving symbolic manipulation, Picard iteration, and auxiliary variables for approximating solutions of partial differential boundary value problems. The method is easy to implement, computationally efficient, and it is highly accurate. The output of the method is a function that approximates the exact solution. Read More

This analysis which uses new mathematical methods aims at proving the Riemann hypothesis and figuring out an approximate base for imaginary non-trivial zeros of zeta function at very large numbers, in order to determine the path that those numbers would take. This analysis will prove that there is a relation links the non-trivial zeros of zeta with the prime numbers, as well as approximately pointing out the shape of this relationship, which is going to be a totally valid one at numbers approaching infinity. Read More

In 1918 S. Ramanujan defined a family of trigonometric sum now known as Ramanujan sums. In the last few years, Ramanujan sums have inspired the signal processing community. Read More

In this note the authors have raised the question regarding the validity of the main result in [1] by setting an example. Read More

In this paper, we characterize the positive integers $n$ for which intersection graph of ideals of $\mathbb{Z}_n$ is perfect. Read More

For a colour cluster $\mathbb{C} =(\mathcal{C}_1,\mathcal{C}_2, \mathcal{C}_3,\ldots,\mathcal{C}_\ell)$, where $\mathcal{C}_i$ is a colour class such that $|\mathcal{C}_i|=r_i$, a positive integer, we investigate two types of simple connected graph structures $G^{\mathbb{C}}_1$, $G^{\mathbb{C}}_2$ which represent graphical embodiments of the colour cluster such that the chromatic numbers $\chi(G^{\mathbb{C}}_1)=\chi(G^{\mathbb{C}}_2)=\ell$ and $\min\{\varepsilon(G^{\mathbb{C}}_1)\}=\min\{\varepsilon(G^{\mathbb{C}}_2)\} =\sum\limits_{i=1}^{\ell}r_i-1$. Therefore, the problem is the edge-minimality inverse to finding the chromatic number of a given simple connected graph. In this paper, we also discuss the chromatic Zagreb indices corresponding to $G^{\mathbb{C}}_1$, $G^{\mathbb{C}}_2$. Read More

In this paper we present a new identity and some of its variants which can be used for finding solutions while solving fractional infinite and finite series. We introduce another simple identity which is capable of generating solutions for some finite series. We demonstrate a method for generation of variants of the identities based on the findings. Read More

The aim of this paper is to construct a Riemann-Lagrange geometry (in the sense of d-linear connection, d-torsions and d-curvatures) for a quadratic multi-time Lagrangian. Read More

We present a generalized $\text{Vi\'ete's}$-like formula for pi with rapid convergence. This formula is based on the arctangent function identity with argument $x=\sqrt{2-{{a}_{K-1}}}/{{a}_{K}}$, where \[ {{a}_{K}}=\underbrace{\sqrt{2+\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}_{K\,\,\text{square}\,\,\text{roots}} \] is a radical consisting of $K$ nested square roots of twos. The computational test we performed reveals that the generalized $\text{Vi\'ete's}$-like formula provides a significant improvement in accuracy as the integer $K$ increases. Read More

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. Read More

An algebraic structure underlying the quantity calculus is proposed consisting in an algebraic fiber bundle, that is, a base structure which is a free Abelian group together with fibers which are one dimensional vector spaces, all of them bound by algebraic restrictions. Subspaces, tensor product and quotient spaces are considered as well as homomorphisms to end with a classification theorem of these structures. The new structure provides an axiomatic foundation for quantity calculus and gives complete justification within its framework of the way that quantity calculus is actually performed. Read More