Nonzero Real Numbers Research Articles

Dynamical analysis and solutions of nonlinear difference equations of thirty order

Discrete-time systems are sometimes used to explain natural phenomena that happen in nonlinear sciences. We study the periodicity, boundedness, oscillation, stability, and certain exact solutions of nonlinear difference equations in this paper. Using the standard iteration method, exact solutions are obtained. Some well-known theorems are used to test the stability of the equilibrium points. Some numerical examples are also provided to confirm the theoretical work’s validity. The numerical component is implemented with Wolfram Mathematica. The method presented may be simply applied to other rational recursive issues. \par In this paper, we explore the dynamics of adhering to rational difference formula \begin{equation*} x_{n+1}=\frac{x_{n-29}}{\pm1\pm x_{n-5}x_{n-11}x_{n-17}x_{n-23}x_{n-29}}, \end{equation*} where the initials are arbitrary nonzero real numbers.

Classifications of THA-surfaces in I^3

In classical differential geometry, the problem of obtaining Gaussian and mean curvatures of a surface is one of the most important problems. A surface M2 in I3 is a THA-surface of first type if it can be parameterized by r(s, t) = (s, t, Af(s + at)g(t) + B(f(s + at) + g(t))). A surface M2 in I3 is a THA- surface of second type if it can be parameterized by r(s, t) = (s, Af(s + at)g(t) + B(f(s + at) + g(t)), t), where A and B are non-zero real numbers [16, 17, 18]. In this paper, we classify two types THA-surfaces in the 3-dimensional isotropic space I3 and study THA-surfaces with zero curvature in I3.

Analysis and qualitative behaviour of a tenth-order rational difference equation

In this article, we examine the qualitative behavior of the solutionsof the following di¤erence equationzn+1 = aZn-4 +bZn-4/cZn-4-dzn4; n = 0,1,....where the initial conditions Z_9; Z_8; Z_7; Z_6; Z_5; Z_4; Z_3; Z_2; Z_1;Z0 are arbitrary non-zero real numbers and a, b, c, d are positive constants.

ROUGH LACUNARY STATISTICAL ANALYSIS OF ORDER OF η OF TRIPLE DIFFERENCE SEQUENCE SPACES

We generalized the concepts in the probability of rough lacunary statistical analysis of order η by introducing the difference operator Δ_γ^α of fractional order, where α is a proper fraction and γ=(γ_mnk ) is any fixed sequence of nonzero real or complex numbers. We study some properties of this operator involving lacunary sequence θ and arbitrary sequence p=(p_rst ) of strictly positive real numbers and investigate the topological structures of related with triple difference sequence spaces. The main focus of the present paper is to generalize rough lacunary statistical analysis of order η of triple difference sequence spaces and give the relations between statistical analysis and lacunary statistical analysis of order η.

Solvability of a third-order system of nonlinear difference equations via a generalized Fibonacci sequence

In this paper, we solve in closed-form the following third-order system of nonlinear difference equations xn+1 = ynyn−1xn−1p xn(anyn−2q + bnynyn−1),yn+1 = xnxn−1yn−1q yn(cnxn−2p + dnxnxn−1),p,q ∈ ℕ,n ∈ ℕ0 where the initial values x−i,y−i,i = 0,1,2 and the parameters (an)n∈ℕ0,(bn)n∈ℕ0,(cn)n∈ℕ0, (dn)n∈ℕ0 are non-zero real numbers. The form of the solutions of the one dimensional case of our system and a more general system defined by one to one functions are also presented.

On the Global Asymptotic Stability and 4-Period Oscillation of the Third-Order Difference Equation

The main objective of this paper is to study the global behavior and oscillation of the following third-order rational difference equation x n + 1 = α x n x n − 1 x n − 2 / β x n − 1 2 + γ x n − 2 2 , where the initial conditions x − 2 , x − 1 , x 0 are nonzero real numbers and α , β , γ are positive constants such that α ≤ β + γ . Visual examples supporting solutions are given at the end of the study. The figures are found with the help of MATLAB.

Analytical study of the pantograph equation using Jacobi theta functions

The aim of this paper is to use the analytic theory of linear q-difference equations for the study of the functional-differential equation y′(x)=ay(qx)+by(x), where a and b are two non-zero real or complex numbers. When 0<q<1 and y(0)=1, the associated Cauchy problem admits a unique power series solution, ∑n≥0(−a/b;q)nn!(bx)n, that converges in the whole complex x-plane. The principal result obtained in the paper explains how to express this entire function solution into a linear combination of solutions at infinity with the help of integral representations involving Jacobi theta functions. As a by-product, this connection formula between zero and infinity allows one to rediscover the classic theorem of Kato and McLeod on the asymptotic behavior of the solutions over the real axis.

Anisotropic dark energy universe in f(Q, T) gravity with observational constraints

Aim of this paper is to investigate an anisotropic locally rotationally symmetric (LRS) Bianchi type-I space–time in the context of the recently proposed f( Q, T) gravity, where Q is the non-metricity scalar and T is energy–momentum tensor. We have considered f( Q, T) = α Q + β T a linear form, where α and β are model parameters. We have analyzed the exact solution of LRS Bianchi type-I space–time by assuming relation between metric potential A = B n , where n is arbitrary non-zero real number. To study the anisotropic nature of the dynamical dark energy, we assume that the skewness parameters are time dependent and n ≠ 1. We have constrained to our model by using observational Hubble dataset. Onwards, discussed the physical behavior of cosmological parameters such as energy density, pressure, EoS parameter, deceleration parameter and, Energy conditions.

Strong Asymptotics of Planar Orthogonal Polynomials: Gaussian Weight Perturbed by Finite Number of Point Charges

AbstractWe consider the orthogonal polynomial pn(z) with respect to the planar measure supported on the whole complex plane where dA is the Lebesgue measure of the plane, N is a positive constant, {c1, …, cν} are nonzero real numbers greater than −1 and are distinct points inside the unit disk. In the scaling limit when n/N = 1 and n → ∞ we obtain the strong asymptotics of the polynomial pn(z). We show that the support of the roots converges to what we call the “multiple Szegő curve,” a certain connected curve having ν + 1 components in its complement. We apply the nonlinear steepest descent method [9,10] on the matrix Riemann‐Hilbert problem of size (ν + 1) × (ν + 1) posed in [22]. © 2023 Wiley Periodicals, LLC.

The Solution and Dynamic Behaviour of Difference Equations of Twenty-First Order

We explore the dynamics of adhering to rational difference formula $$ \psi_{m+1}=\frac{\psi_{m-20}}{\pm 1 \pm \psi_{m-2}\psi_{m-5}\psi_{m-8}\psi_{m-11}\psi_{m-14}\psi_{m-17}\psi_{m-20}}, \quad m \in \mathbb{N}_{0} $$ where the initials are arbitrary nonzero real numbers. Specifically, we examine global asymptotically stability. Additionally, we provide examples and solutions graphs of some special cases.

The periodic nature and expression on solutions of some rational systems of difference equations

Our aim in this paper is to obtain formulas expressions for solutions of the following fractional systems of difference equationsLn+1=Ln-3Mn-4Mn±1-Ln-3Mn-4Wn-1Xn-2,n=0,1,2,…,Mn+1=Mn-3Wn-4Wn±1-Mn-3Wn-4Xn-1Ln-2,Wn+1=Wn-3Xn-4Xn±1±Wn-3Xn-4Ln-1Mn-2,Xn+1=Xn-3Ln-4Ln±1±Xn-3Ln-4Mn-1Wn-2,where the initial values are non-zero real numbers. In addition we prove that some of these systems are periodic with different period. We also verify our theoretical results for every system with some numerical examples and draw by using some famous mathematical programs (Matlab program) to illustrate the results.

Dynamical behavior of one rational fifth-order difference equation

In this paper, we study the qualitative behavior of the rational recursive equation \begin{equation*} x_{n+1}=\frac{x_{n-4}}{\pm1\pm x_{n}x_{n-1}x_{n-2}x_{n-3}x_{n-4}}, \quad n \in \mathbb{N}_{0}:=\{0\}\cup\mathbb N, \end{equation*} where the initial conditions are arbitrary nonzero real numbers. The main goal of this paper, is to obtain the forms of the solutions of the nonlinear fifth-order difference equations, where the initial conditions are arbitrary positive real numbers. Moreover, we investigate stability, boundedness, oscillation and the periodic character of these solutions. The results presented in this paper improve and extend some corresponding results in the literature.

The Form of Solutions and Periodic Nature for Some System of Difference Equations

In this paper, we study the form of the solution of the following systems of difference equations of order two w_{n+1}=\frac{w_{n}s_{n-1}}{w_{n}+s_{n-1}},~~~ s_{n+1}=\frac{s_{n}w_{n-1}}{\pm s_{n}\pm w_{n-1}},with nonzero real numbers initial conditions.

On geometric circulant matrices with geometric sequence

In this paper, we consider a geometric circulant matrix with geometric sequence i.e. a geometric circulant matrix whose first row is ( g , gq , g q 2 , … , g q n − 1 ) , where g is a nonzero complex number and q is a nonzero real number. The determinant, the Euclidean norm and bounds for the spectral norm of such matrix are obtained. Also, in the case when such matrix is singular, we obtain the Moore-Penrose inverse of such matrix and show that the group inverse of such matrix does not exist (in most cases). The obtained results are illustrated by examples.

On Some Solvable Systems of Some Rational Difference Equations of Third Order

Our aim in this paper is to obtain formulas for solutions of rational difference equations such as xn+1=1±xn−1yn/1−yn,yn+1=1±yn−1xn/1−xn, and xn+1=1±xn−1yn−2/1−yn,yn+1=1±yn−1xn−2/1−xn, where the initial conditions x−2, x−1, x0, y−2, y−1, y0 are non-zero real numbers. In addition, we show that the some of these systems are periodic with different periods. We also verify our theoretical outcomes at the end with some numerical applications and draw it by using some mathematical programs to illustrate the results.

AN INVESTIGATION OF THE SOLUTIONS AND THE DYNAMIC BEHAVIOR OF SOME RATIONAL DIFFERENCE EQUATIONS

The main purpose of this work is to investigate the solutions, the periodic character and study the global asymptotic behavior of this so- lutions of the following rational di↵erence equation x_(n+1) = x_(n-2)*x_(n-6) / x_(n-3)*(±1 ± x_(n-2)*x_(n-6)) with a non-zero real numbers initial conditions.

New presentations of real numbers with k-Lucas numbers

In this paper, we first obtain a series of k-Lucas numbers using k-Lucas numbers. We give new presentations of any real number u ≠ 0 using this obtained k-Lucas series and show polynominal repsesentations that every nonzero real number can be uniquely represented as the sum of the squares of consecutive k-Lucas numbers. To do this, we give new presentation theorems for any real number using k-Lucas series. Finally, to support these theorems, we give examples where we obtain the roots of the polynomial representations of a selected real number u ≠ 0, as well as the values representing the first ten prime numbers corresponding to a chosen k-Lucas polynomial.

ON THE NUMBER OF ALGEBRAIC POINTS ON THE GRAPH OF THE WEIERSTRASS SIGMA FUNCTIONS

AbstractLet $\Omega =\mathbb {Z}\omega _1+\mathbb {Z}\omega _2$ be a lattice in $\mathbb {C}$ with invariants $g_2,g_3$ and $\sigma _{\Omega }(z)$ the associated Weierstrass $\sigma $ -function. Let $\eta _1$ and $\eta _2$ be the quasi-periods associated to $\omega _1$ and $\omega _2$ , respectively. Assuming $\eta _2/\eta _1$ is a nonzero real number, we give an upper bound for the number of algebraic points on the graph of $\sigma _{\Omega }(z)$ of bounded degrees and bounded absolute Weil heights in some unbounded region of $\mathbb {C}$ in the following three cases: (i) $\omega _1$ and $\omega _2$ algebraic; (ii) $g_2$ and $g_3$ algebraic; (iii) the algebraic points are far from the lattice points.

On two tensors associated to the structure Jacobi operator of a real hypersurface in complex projective space

A real hypersurface M in a complex projective space inherits an almost contact metric structure from the Kählerian structure of the ambient space. This almost contact metric structure allows us to define, for any nonzero real number k, the so-called k-th generalized Tanaka–Webster connection. With this connection and the Levi-Civita one we can associate two tensors of type (1,2) to the structure Jacobi operator R_{xi } of M. We classify real hypersurfaces in complex projective space for which such tensors satisfy a cyclic property.

Approximating an advanced multi-dimensional reciprocal-quadratic mapping via a fixed point approach

There are many results on stability of various forms of functional equations available in the theory of functional equations. The intention of this paper is to introduce an advanced and a new multi-dimensional reciprocal-quadratic functional equation involving $p&gt;1$ variables. It is interesting to note that it has two different solutions, namely, quadratic and multiplicative inverse quadratic functions. We solve its various stability problems in the setting of non-zero real numbers and non-Archimedean fields via fixed point approach.