# Mathematics - General Mathematics Publications (50)

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## Mathematics - General Mathematics Publications

We study the geometry of Finsler submanifolds using the pulled-back approach. We define the Finsler normal pulled-back bundle and obtain the induced geometric objects, namely, induced pullback Finsler connection, normal pullback Finsler connection, second fundamental form and shape operator. Under a certain condition, we prove that induced and intrinsic Hashiguchi connections coincide on the pulled-back bundle of Finsler submanifold. Read More

The warped product $M_1 \times_F M_2$ of two Riemannian manifolds $(M_1,g_1)$ and $(M_2,g_2)$ is the product manifold $M_1 \times M_2$ equipped with the warped product metric $g=g_1 + F^2 g_2$, where $F$ is a positive function on $M_1$. The notion of warped product manifolds is one of the most fruitful generalizations of Riemannian products. Such a notion plays very important roles in differential geometry as well as in physics, especially in general relativity. Read More

Let $E$ be an elliptic curve of rank $\text{rk}(E) \geq 1$, and let $P \in E(\mathbb{Q})$ be a point of infinite order. The number of elliptic primes $p \leq x$ for which $\langle P\rangle=E(\mathbb{F}_p)$ is expected to be $\pi(x,E,P)=\delta(E,P)x/\log x+o(x/\log x)$, where $\delta(E,P)\geq 0$ is a constant. This note proves the lower bound $\pi(x,E,P) \gg x/\log x$. Read More

**Affiliations:**

^{1}NSC KIPT, Ukraine

**Category:**Mathematics - General Mathematics

Results of research of possibility of transformation of a difference equation into a system of the first-order difference equation are presented. In contrast to the method used previously, an unknown grid function is split into two new auxiliary functions, which have definite properties. Several examples show that proposed approach can be useful in solving different physical problems. Read More

In 1898, Riemann had announced the following conjecture : the nontrivial roots (zeros) $s=\alpha+i\beta$ of the zeta function, defined by: $$\zeta(s) = \sum_{n=1}^{+\infty}\frac{1}{n^s},\,\mbox{for}\quad \Re(s)>1$$ have real part $\alpha= \frac{1}{2}$. We give a proof that $0<\alpha \leq \frac{1}{2}$ using an equivalent statement of Riemann Hypothesis. Read More

We give a Goldbach's pair for even numbers that are in a specific form. Consider the following two functions where $x$ and $m$ are odd integers greater than $1$. $g(x,m) = x+2m-(x-m)mod(2m),$ and $f(x)$ returns the smallest prime factor of $x$. Read More

It is well known the concept of the condition number $\kappa(A) = \|A\|\|A^{-1}\|$, where $A$ is a $n \times n$ real or complex matrix and the norm used is the spectral norm. Although it is very common to think in $\kappa(A)$ as "the" condition number of $A$, the truth is that condition numbers are associated to problems, not just instance of problems. Our goal is to clarify this difference. Read More

We prove Riemann hypothesis. Method is to show the convexity of function which has zeros critical strip the same as zeta function. Read More

This paper sheds light on the essential characteristics of geodesics, which frequently occur in considerations from motion in Euclidean space. Focus is mainly on a method of obtaining them from the calculus of variations, and an explicit geodesic computation for a Riemannian hypersurface. Read More

We investigate whether there exists an arithmetic progression or geometric progression consisting only palindromic numbers. In this paper we show that the answer to this question is NO. Given the first and final term we will also give an estimate for how large that AP could be and so for the GP given its first term and common ratio. Read More

A lacunary sequence is an increasing integer sequence $\theta=(k_r)$ such that $k_r-k_{r-1}\rightarrow \infty$ as $r\rightarrow \infty.$ In this article we introduce arithmetic statistically convergent sequence space $ASC$ and lacunary arithmetic statistically convergent sequence space $ASC_{\theta}$ and study some inclusion properties between the two spaces. Finally we introduce lacunary arithmetic statistical continuity and establish some interesting results. Read More

In this paper, we give Smarandache curves according to the asymptotic orthonormal frame in null cone Q^2. By using cone frame formulas, we present some characterizations of Smarandache curves and calculate cone frenet invariants of these curves. Also, we illustrate these curves with an example. Read More

In this study, a computational method referred to as Perturbation Iteration Transform Method (PITM), which is a combination of the conventional Laplace Transform Method (LTM) and the Perturbation Iteration Algorithm (PIA) is applied for the solution of Newell-Whitehead- Segel Equations (NWSEs). Three unique examples are considered and the results obtained are compared with their exact solutions graphically. Also, the results agree with those obtained via other semi-analytical methods viz: New Iterative Method and Adomian Decomposition Method. Read More

In this paper, we introduce the notion of the rainbow neighbourhood and a related graph parameter namely, the rainbow neighbourhood number of a graph $G$. We report on preliminary results thereof. We also establish a necessary and sufficient condition for the existence of a rainbow neighbourhood in the line graph of a graph $G$. Read More

This article explores the geometric algebra of Minkowski spacetime, and its relationship to the geometric algebra of Euclidean 4-space. Both of these geometric algebras are algebraically isomorphic to the 2x2 matrix algebra over Hamilton's famous quaternions, and provide a rich geometric framework for various important topics in mathematics and physics, including stereographic projection and spinors, and both spherical and hyperbolic geometry. In addition, by identifying the time-like Minkowski unit vector with the extra dimension of Euclidean 4-space, David Hestenes' Space-Time Algebra of Minkowski spacetime is unified with William Baylis' Algebra of Physical Space. Read More

The non-elementary integrals $\text{Si}_{\beta,\alpha}=\int [\sin{(\lambda x^\beta)}/(\lambda x^\alpha)] dx$ and $\text{Ci}_{\beta,\alpha}=\int [\cos{(\lambda x^\beta)}/(\lambda x^\alpha)] dx$, where $\beta\ge1$ and $\alpha\ge1$, are evaluated in terms of the hypergeometric functions $_{1}F_2$ and $_{2}F_3$ respectively, and their asymptotic expressions for $|x|\gg1$ are derived. Integrals of the form $\int [\sin^n{(\lambda x^\beta)}/(\lambda x^\alpha)] dx$ and $\int [\cos^n{(\lambda x^\beta)}/(\lambda x^\alpha)] dx$, where n is a positive integer, are expressed in terms $\text{Si}_{\beta,\alpha}$ and $\text{Ci}_{\beta,\alpha}$, and then evaluated. On the other hand, $\text{Si}_{\beta,\alpha}$ and $\text{Ci}_{\beta,\alpha}$ are evaluated in terms of the hypergeometric function $_{2}F_2$. Read More

In this paper, we present a way to measure the intelligence (or the interest) of an approximation of a given real number in a given model of approximation. Basing on the idea of the complexity of a number, defined as the number of its digits, we introduce a function noted $\mu$ (called a measure of intelligence) associating to any approximation $\mathbf{app}$ of a given real number in a given model a positive number $\mu(\mathbf{app})$, which characterises the intelligence of that approximation. Precisely, the approximation $\mathbf{app}$ is intelligent if and only if $\mu(\mathbf{app}) \geq 1$. Read More

In this article, we give the explicit solutions to the Laplace equations associated to the Dirac operator, Euler operator and the harmonic oscillator in R. Read More

In this note we find the least order of a noncommutative even square ring and we note that it is a nil ring having characteristic four. In addition we note that if R be an even square ring such that 2ab+2ba=0, for all a, b in R then R is not necessarily a commutative ring. Read More

Certain analytical expressions which "feel" the divisors of natural numbers are investigated. We show that these expressions encode to some extent the well-known algorithm of the sieve of Eratosthenes. Most part of the text is written in pedagogical style, however some formulas are new. Read More

We report an inconsistency found in probability theory (also referred to as measure-theoretic probability). For probability measures induced by real-valued random variables, we deduce an "equality" such that one side of the "equality" is a probability, but the other side is not. For probability measures induced by extended random variables, we deduce an "equality" such that its two sides are unequal probabilities. Read More

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

We examine the prime gaps using a statistical approach. It is first shown that the Andrica's conjecture is true for half or more cases. Using the arguments of averages, it is further shown that Andrica's conjecture is true. 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