Nonzero Real Numbers Research Articles

Solution of a Solvable System of Difference Equation

In this study we give solutions for the following difference equation sytem 
 x_{n+1}= (a.x_{n}y_{n-3}/y_{n-2}-\alpha)+\beta y_{n+1}=(b.x_{n-3}y_{n}/x_{n-2}-\beta) +\alpha n ∈N0
 where the parameters a,b,, and initial values x_{-i}, y_{-i}, i=0,1,2,3 are non-zero real numbers. We show the asymptotic behavior of the system of equation.

The expressions and behavior of solutions for nonlinear systems of rational difference equations

In this paper, we investigate the form of the solutions of two systems of rational difference equations of second order, where the initial conditions are arbitrary nonzero real numbers.

On Inhomogeneous Extension of Thue–Roth’s Type Inequality with Moving Targets

AbstractLet $\Gamma \subset \overline{\mathbb Q}^{\times }$ be a finitely generated multiplicative group of algebraic numbers. Let $\delta ,\linebreak\beta \in \overline{\mathbb Q}^\times $ be algebraic numbers with $\beta $ irrational. In this paper, we prove that there exist only finitely many triples $(u, q, p)\in \Gamma \times \mathbb{Z}^2$ with $d = [\mathbb{Q}(u):\mathbb{Q}]$ such that $|\delta qu|>1$ and $$\begin{align*} & 0<|\delta qu+\beta-p|<\frac{1}{H(u)^\varepsilon q^{d+\varepsilon}}, \end{align*}$$where $H(u)$ denotes the absolute Weil height. This is an inhomogeneous analogue of the main theorem in [ 2]. As an application of our result, we also prove a transcendence result, which states as follows: let $\alpha>1$ be a real number. Let $\beta $ be an algebraic irrational number and $\lambda $ be a non-zero real algebraic number. For a given real number $\varepsilon>0$, if there are infinitely many natural numbers $n$ for which $||\lambda \alpha ^n+\beta || < 2^{- \varepsilon n}$ holds true, then $\alpha $ is transcendental, where $||x||$ denotes the distance from its nearest integer. When $\alpha $ and $\beta $ both are algebraic numbers satisfying same conditions, then a particular result of Kulkarni et al. [ 3] asserts that $\alpha ^d$ is a Pisot number. When $\beta $ is an algebraic irrational, our result implies that no algebraic number $\alpha $ satisfies the inequality for infinitely many natural numbers $n$. Also, our result strengthens a result of Wagner and Ziegler [ 7].

Diophantine approximation over primes with different powers

Assume non-zero real numbers κ1,…,κ6 are not all negative. Assume also κ1κ2 is algebraic and not rational, the sequence G is well-spaced and ϑ>0. In this work, we showed that the quantity of γ∈G satisfying 1≤γ≤X and making the Diophantine inequality|κ1p12+κ2p22+κ3p33+κ4p43+κ5p54+κ6p64−γ|<γ−ϑunsolvable in primes p1,…,p6 is not more than O(X67+2ϑ+ϵ) for any ϵ>0.

Some Upper Bounds on the First General Zagreb Index

The first general Zagreb index M γ G or zeroth-order general Randić index of a graph G is defined as M γ G = ∑ v ∈ V d v γ where γ is any nonzero real number, d v is the degree of the vertex v and γ = 2 gives the classical first Zagreb index. The researchers investigated some sharp upper and lower bounds on zeroth-order general Randić index (for γ &lt; 0 ) in terms of connectivity, minimum degree, and independent number. In this paper, we put sharp upper bounds on the first general Zagreb index in terms of independent number, minimum degree, and connectivity for γ . Furthermore, extremal graphs are also investigated which attained the upper bounds.

Solvable three-dimensional system of higher-order nonlinear difference equations

In this work, we indicate three-dimensional system of difference equations xn = ayn?k + dyn?kxn?k?l/?bxn?k?l+?czn?l, yn = ?zn?k + ?zn?kyn?k?l/??yn?k?l + ??xn?l, zn = exn?k + hxn?kzn?k?l/?fzn?k?l +?g?yn?l , n ? N0, where k and l are positive integers, the parameters a,?b, ?c, d, ?, ??, ??, ?, e, ?f ,?g, h and the initial values x?j, y?j, z?j j = ?1,k+l, are non-zero real numbers, can be solved in closed form. In addition, we obtain explicit formulas for the well-defined solutions of the aforementioned system for the case l = 1. Also, the set of undefinable solutions of the system is found. Finally, an application about a three-dimensional system of difference equations is given.

The form of the solutions of fourth order rational systems of difference equations

In this paper, we get the form of the solutions of the following difference equation systems of order four where the initial conditions z−3, z−2, z−1, z0, w−3, w−2, w−1, w0 are arbitrary nonzero real numbers.

On a system of difference equations of third order solved in closed form

In this work, we show that the system of difference equationsxn+1=(ayn-2xn-1yn+bxn-1yn-2+cyn-2+d)/(yn-2xn-1yn),yn+1=(axn-2yn-1xn+byn-1xn-2+cxn-2+d)/(xn-2yn-1xn),where n belongs to the set of positive integer numbers, x-2, x-1, x0, y-2, y-1 and y0 are arbitrary nonzero real numbers, and the parameters a, b, c and d are arbitrary real numbers with d nonzero can be solved in a closed form.We will see that when a = b = c = d = 1, the solutions are expressed using the famous Tetranacci numbers. In particular, the results obtained here extend those in our recent work.

Hyperreal Delta Functions as a New General Tool for Modeling Systems with Infinitely High Densities

In general, the state of a system in which a physical quantity such as mass is distributed over space can be modeled by a function that represents the density distribution. The purpose of this paper is to introduce special functions that can be applied when in the system to be modeled, where the quantity is distributed over isolated points. For that matter, the expanded real numbers are introduced as an ordered subring of the hyperreal number field that does not contain any infinitesimals, and hyperreal delta functions are defined as special functions from the real numbers to the expanded real numbers satisfying the condition that (i) the support is a singleton, and (ii) the integral over the reals is a nonzero real number. These newly defined hyperreal delta functions, and tensor products thereof, then provide a general tool, applicable for the mathematical modeling of physical systems in which infinitely high densities occur.

Correction to: Dynamical behavior of rational difference equation $$x_{n+1}=\frac{x_{n-17}}{\pm 1\pm x_{n-2}x_{n-5}x_{n-8}x_{n-11}x_{n-14}x_{n-17}}$$

In this paper, we study the qualitative behavior of the rational recursive sequences $$\begin{aligned} x_{n+1}=\frac{x_{n-17}}{\pm 1\pm x_{n-2}x_{n-5}x_{n-8}x_{n-11}x_{n-14}x_{n-17}}, \quad n \in \mathbb {N}_{0} \end{aligned}$$ where the initial conditions are arbitrary nonzero positive real numbers. Also, we give the numerical examples and solutions graphs of some cases of difference equations.

DIOPHANTINE INEQUALITIES OF FRACTIONAL DEGREE

This paper is concerned with the study of diagonal Diophantine inequalities of fractional degree θ, where θ > 2 is real and non-integral. For fixed non-zero real numbers λ i not all of the same sign, we write F ( x ) = λ 1 x 1 θ + ⋯ + λ s x s θ . For a fixed positive real number τ, we give an asymptotic formula for the number of positive integer solutions of the inequality | F ( x ) | < τ inside a box of side length P. Moreover, we investigate the problem of representing a large positive real number by a positive definite generalised polynomial of the above shape. A key result in our approach is an essentially optimal mean value estimate for exponential sums involving fractional powers of integers.

Periodicity of the solutions of general system of rational difference equations related to fibonacci numbers

In this paper, we deal with the periodicity of solutions of the following general system rational of difference equations: [Formula: see text] where [Formula: see text] [Formula: see text] and the initial conditions are arbitrary nonzero real numbers.

Hyers—Ulam Stability of Second-Order Linear Dynamic Equations on Time Scales

We investigate the Hyers—Ulam stability (HUS) of certain second-order linear constant coefficient dynamic equations on time scales, building on recent results for first-order constant coefficient time-scale equations. In particular, for the case where the roots of the characteristic equation are non-zero real numbers that are positively regressive on the time scale, we establish that the best HUS constant in this case is the reciprocal of the absolute product of these two roots. Conditions for instability are also given.

On the Solutions of Three-Dimensional Rational Difference Equation Systems

In this paper, we are interested in a technique for solving some nonlinear rational systems of difference equations of third order, in three-dimensional case as a special case of the following system: x n + 1 = y n z n − 1 / y n ± x n − 2 , y n + 1 = z n x n − 1 / z n ± y n − 2 , and z n + 1 = x n y n − 1 / x n ± z n − 2 with initial conditions x − 2 , x − 1 , x 0 , y − 2 , y − 1 , y 0 , z − 2 , z − 1 , and z 0 are nonzero real numbers. Moreover, we study some behavior of the systems such as the boundedness of solutions for such systems. Finally, we present some numerical examples by giving some numerical values for the initial values of each case. Some figures have been given to explain the behavior of the obtained solutions in the case of numerical examples by using the mathematical program MATLAB to confirm the obtained results.

Development of a Mathematical Model and Validation for Methane Production Using Cow Dung as Substrate in the Underground Biogas Digester

South Africa is highly dependent on conventional fuels from non-renewable energy sources such as coal and fossil fuels. The methane from biogas is a renewable energy source for generating electricity using a combined heat and power plant. Therefore, the aim of the study is to develop and validate a mathematical model for predicting methane production in an underground biogas digester. The developed model was able to predict the production of methane gas as separate entity differing from other models. A total of 286 datasets were used as a trained dataset for the model development, and 144 datasets served as test data for the validation of the model, making a total of 430 measured datasets of all the predictors. The determination coefficient (R2) and the p-value of the predicted and calculated methane yield were 0.962 and 0.920, respectively. The high R2 in the present study confirms a good correlation between the model and experimental value. Hence, the model is of significance because it is applicable in predicting the performance of methane production of systems of the same design used in different locations, thereby arriving at the same constant values. From the study, the ambient weather factors (ambient temperature, relative humidity, and global horizontal irradiance) affected the methane production. Additionally, the indoor parameters (pH, gas temperature, slurry bottom and slurry top temperature) impacted on the yield of the methane production because the scaling factors associated with these quantities are non-zero real numbers. Hence, the scope of the study did not consider the volume of the biogas digester as an input parameter to the response.

A Diophantine Problem with Unlike Powers of Primes

Let k be an integer with 4 ≤ k ≤ 6 and η be any real number. Suppose that λ 1 , λ 2 , … , λ 5 are nonzero real numbers, not all of them have the same sign, and λ 1 / λ 2 is irrational. It is proved that the inequality λ 1 p 1 + λ 2 p 2 2 + λ 3 p 3 3 + λ 4 p 4 4 + λ 5 p 5 k + η &lt; max 1 ≤ j ≤ 5 p j − σ k has infinitely many solutions in prime variables p 1 , p 2 , p 3 , p 4 , and p 5 , where 0 &lt; σ 4 &lt; 1 / 36 , 0 &lt; σ 5 &lt; 4 / 189 , and 0 &lt; σ 6 &lt; 1 / 54 . This gives an improvement of the recent results.

On the New Double Binomial Sequence Space

The aim of this paper is to present the new double Binomial sequence space $\mathcal{B}_{p}^{r,s}$ which consists of all sequences whose double Binomial transforms of orders $r,s$ ($r$ and $s$ are nonzero real numbers with $r+s \neq 0$) are in the space $\mathcal{L}_p$, where $0&amp;lt;p&amp;lt;\infty$. We examine its topological and algebraic properties and inclusion relations. Furthermore, the $\alpha-$, $\beta(bp)-$ and $\gamma-$duals of the space $\mathcal{B}_{p}^{r,s}$ are determined and finally, some 4-dimensional matrix mapping classes related to this space are characterized.

On the maximum general sum-connectivity index of trees with a fixed order and maximum degree

The general sum-connectivity index of a graph [Formula: see text], denoted by [Formula: see text], is defined as [Formula: see text], where [Formula: see text] is the edge connecting the vertices [Formula: see text], [Formula: see text] denotes the degree of the vertex [Formula: see text], and [Formula: see text] is a nonzero real number. This paper is devoted to characterizing the graphs having the maximal [Formula: see text] value among all the trees with order [Formula: see text] and fixed maximum degree.

Coulson-type integral formulas for the sum of powers of absolute values of roots of polynomials

Coulson integral formula was initiated in the research of energy of graphs in 1940 by Coulson, which provides an efficient way to calculate the energy of graphs without knowing any eigenvalues of graphs. Recently, Qiao et al. generalized the concept of energy from graphs to polynomials, which is referred to as the general energy of polynomials. For a polynomial ϕ(z) and a nonzero real number α, the general energy of polynomial ϕ(z), denoted by Eα(ϕ), is defined as the sum of αth powers of absolute values of all (nonzero) roots of ϕ(z), i.e., Eα(ϕ)=∑zk≠0|zk|α, where zk represents the nonzero roots of ϕ(z). In a series of three publications by Qiao et al., they gave some beautiful Coulson-type integral formulas for Eα(ϕ) when the roots of ϕ(z) are all nonnegative numbers for any rational number α and any irrational number α with 0<|α|<1, and when the roots of ϕ(z) are all real numbers for any rational number α and any irrational number α with 0<|α|<2. For the remaining α, their integral formulas are invalid. In this paper, we set up a research procedure covering all the nonzero real numbers α. That is to say, it not only combines all the Coulson-type integral formulas established by Qiao et al., but also involves in all the remaining unsolved α. As an extension and generalization of known integral formulas, the Coulson-type integral formulas for Eα(ϕ), when the roots of ϕ(z) are all pure imaginary numbers, are also presented, for any nonzero real number α. Moreover, we also give an alternative form of integral formulas expressed by the coefficients of polynomials.

Continuous Wong–Zakai Approximations of Random Attractors for Quasi-linear Equations with Nonlinear Noise

We consider a family of random quasi-linear equations driven by nonlinear Wong–Zakai noise and parameterized by the non-zero size $$\lambda $$ of noise. After proving the existence of a random attractor $$A_\lambda (\omega )$$ in the square Lebesgue space, we then show that there is a residual dense subset of the space of nonzero real numbers such that, under the Hausdorff metric, the map $$\lambda \rightarrow A_\lambda (\theta _s\omega )$$ is continuous at all points of the residual dense set, where $$\theta _s$$ is a group of self-transformations on the probability space. We also prove that as $$\lambda \rightarrow \pm \infty $$ the random attractor converges upper-semicontinuously to the global attractor of the deterministic quasi-linear equation. The upper semi-continuity result is new for nonlinear noise, while, the lower semi-continuity result is new even for linear noise. The theory of Baire category is the main tool used to prove the residual continuity.