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Most of the problems in mathematical and engineering sciences can be studied in the context of nonlinear equations. In this paper, we develop a new family of iterative methods for the approximation of the zeros of mathematical models whose governing equations are nonlinear in nature. The proposed methods are based on decomposition technique due to Daftardar-Gejji and Jaffri [1]. The new family gives several iterative schemes as special cases. The convergence analysis of proposed methods is also presented. In order to determine the performance of newly developed methods, numerical as well as graphical analysis of four complex mathematical models from diverse fields of science and engineering are considered. We also consider some complex polynomials to visualize the roots through polynomiography in the context of proposed methods.

In this paper, we obtain several results on generalized derivations and partial generalized automorphisms in prime rings. Also, some examples are given to show that the restrictions imposed on the hypothesis of the various results cannot be omitted.

A topological index is a numerical value associated with the chemical constitution for the correlation of chemical structures with various physical properties, chemical reactivity or biological act...

Let R be a prime ring of characteristic different from 2, U its Utumi quotient ring, C the center of U, F a non-zero generalized derivation of R and L a non-commutative Lie ideal of R. Suppose that there exists $$0\ne a\in R$$ such that $$a(u^s[F(u),u]u^t)^n=0$$ for all $$u \in L$$ , where $$s\ge 0, t\ge 0, n\ge 1$$ are fixed integers. Then either $$F(x)=\alpha x$$ for all $$x\in R$$ with $$\alpha \in C$$ or R satisfies $$s_4(x_1,\ldots ,x_4)$$ , the standard identity in four variables, and $$F(x)=bx+xb+\alpha x$$ for all $$x\in R$$ , for some $$b\in U$$ and $$\alpha \in C$$ .

Let $R$ be a prime ring with center $Z$ and $I$ be a nonzero ideal of $R$. In this manuscript, we investigate the action of skew derivation $(\delta,\varphi)$ of $R$ which acts as a homomorphism or an anti-homomorphism on $I$. Moreover, we provide an example for semiprime case.

Let R be a prime ring with center Z(R) and I a nonzero right ideal of R. Suppose that R admits a generalized reverse derivation (F, d) such that d(Z(R)) ≠ 0. In the present paper, we shall prove that if one of the following conditions holds: (i) F (xy) ± xy ∈ Z(R) (ii) F ([x, y]) ± [F (x), y] ∈ Z(R) (iii) F ([x, y]) ± [F (x), F (y)] ∈ Z(R) (iv) F (x ο y) ± F (x) ο F (y) ∈ Z(R) (v) [F (x), y] ± [x, F (y)] ∈ Z(R) (vi) F (x) ο y ± x ο F (y) ∈ Z(R) for all x, y ∈ I, then R is commutative.

Let R be a semiprime ring with characteristic different from two and L a noncentral square-closed Lie ideal of R. Suppose that R admits a multiplicative generalized derivation (F,d) satisfying d(L) ⊆ L. In the present paper, we shall prove that d is commuting on L if one of the following conditions holds: (i) F([x,y]) = ∓ [x,d(y)]; (ii) F(x ∘ y) = ∓ (x ∘ d(y)); (iii) F([x,y]) = ∓ (F(y)x); (iv) F([x,y]) = ∓ (F(x)y); (v) F(x ∘ y) = ∓ (F(y)x); (vi) F(x ∘ y) = ∓ (F(x)y) for all x,y ∈ L.

Let $R$ be a prime ring, $I$ a nonzero ideal of $R$ and $m, n$ fixed positive integers. If $R$ admits a generalized derivation $F$ associated with a nonzero derivation $d$ such that $(F([x,y])^{m}=[x,y]_{n}$ for all $x,y\in I$, then $R$ is commutative. Moreover we also examine the case when $R$ is a semiprime ring.

Let Γ be a non-abelian group and Ω ⊆ Γ. We define the commuting graph G = 𝒞(Γ, Ω) with vertex set Ω and two distinct elements of Ω are joined by an edge when they commute in Γ. In this article, among some properties of commuting graphs, we investigate distant properties as well as detour distant properties of commuting graph on D2n. We also study the metric dimension of commuting graph on D2n and compute its resolving polynomial.

Let R be a prime ring, I a nonzero ideal of R, and a ∈ R. Suppose that σ is a nontrivial automorphism of R such that a{(σ(x ∘ y))n − (x ∘ y)m} = 0 or a{(σ([x,y]))n − ([x,y])m} = 0 for all x,y ∈ I, where n and m are fixed positive integers. We prove that if char(R) > n + 1 or char(R) = 0, then either a = 0 or R is commutative.

Let \(R\) be a prime ring of characteristic different from \(2\) and \(k\ge 1\) fixed positive integer. If \(R\) admits a generalized derivation \(G\) associated with a deviation \(d\), here we study the following cases: (1) \(G([r_1,r_2]_k)\circ ([r_1,r_2]_k)=0\) (2) \(G([r_1,r_2]_k)\circ G([r_3,r_4]_k)=[r_1,r_2]_k\circ [r_3,r_4]_k\) (3) \(G([r_1,r_2])\circ _k G([r_3,r_4])=0\) (4) \(G([r_1,r_2])\circ _k G([r_3,r_4])=[r_1,r_2]\circ _k [r_3,r_4]\). We obtain a description of the structure of \(R\) and information on the form of \(G\) in terms of the commutativity of \(R\) and the multiplication by a specific element from the extended centroid of \(R\).

Bu makalenin amacı yarıasal bir gamma halkasının kesirli gamma halkalarını tanımlamak ve kesirli gamma halkasının bazı özelliklerini araştırmaktır., The aim of this paper is to define the gamma rings of quotients of a semiprime gamma ring and investigate some properties of gamma ring of quotients.

In this paper, using the system of coupled equations involving an auxiliary function, we introduce some new efficient higher order iterative methods based on modified homotopy perturbation method. We study the convergence analysis and also present various numerical examples to demonstrate the validity and efficiency of our methods.

Let R be a 6-torsion free semiprime *-ring, \tau an endomorphism of R, \sigam an epimorphism of R and f : R ! R an additive mapping. In this paper we proved the following result: f is a generalized Jordan ( \sigma,\tau) *-iderivation if and only if f is a generalized Jordan triple ( \sigma,\tau) *-derivation.

Let [Formula: see text] be a prime ring with characteristic different from two, [Formula: see text] a derivation of [Formula: see text], [Formula: see text] a noncentral Lie ideal of [Formula: see text], and [Formula: see text]. In the present paper, it is shown that if one of the following conditions holds: (i) [Formula: see text], (ii) [Formula: see text], (iii) [Formula: see text] and (iv) [Formula: see text] for all [Formula: see text], where [Formula: see text] are fixed positive integers, then [Formula: see text] unless [Formula: see text] satisfies [Formula: see text], the standard polynomial identity in four variables.

Let R be a prime ring with Utumi quotient ring U and m, n fixed positive integers. If R admits a generalized derivation F associated with a derivation d such that (F([x, y]))m = [x, y]n for all x, y ∈ R, then R is commutative or d = 0. Moreover, we also examine the cases R is a semiprime ring. Finally, we apply the above result to Banach algebras, and we obtain a noncommutative version of the Singer–Wermer theorem.

During the steam condensation, the presence of non-condensable gases is an important issue affecting the efficiency of the whole thermodynamic process. For this reason, many researchers investigated it by theoretical or experimental methods. A heat and mass transfer analogy model on steam condensation in presence of air over the vertical external surface based on the diffusion layer model is modified in the present paper. Based on previous authors’ experience, the suction effect at the gas-liquid interface and other analogy drawbacks are identified and overcome by supplementing it with more detailed analysis as well as targeted experiments. The experimental data obtained for condensation, outside vertical tube with an external diameter of 38 mm, of air/steam and helium/air/steam mixture, have been used to verify the present heat and mass transfer analogy formulation. By comparing against different available experimental data and previous formulations, the heat and mass transfer analogy formulation is demonstrated to be a accurate enough theoretical approximation. The deviation between predicted values of the new model and experiment results of this paper is less than 15% which has relative higher precision.

Let M be a prime Γ-ring with center Z(M), I a nonzero ideal of M and F be a generalized derivation with associated nonzero derivation d. In the present paper, our purpose is to produce commutativity results for prime Γ-rings M admitting a generalized derivation F satisfying any one of the properties: (i) F(xαy)∓xαy ∈ Z(M), (ii) F(xαy)∓yαx ∈ Z(M), (iii) F(x)αF(y)∓xαy ∈ Z(M), (iv) F(x)αF(y)∓yαx ∈ Z(M), (v) F([x, y]α)=[F(x), y]α, (vi) [F(x), y]α∓[x, F(y)]α=0, (vii) F([x, y]α)∓[d(x), d(y)]α=0, for all x,y ∈ I and α ∈ Γ. Also, some examples are given to show that the primeness of the various results is not superfluous.

Let R be a prime ring with center Z(R), J a nonzero left ideal, \(\alpha \) an automorphism of R and R admits a generalized \((\alpha ,\alpha )\)-derivation F associated with a nonzero \({(}\alpha ,\alpha {)}\)-derivation d such that \(d(Z(R))\ne (0)\). In the present paper, we prove that if any one of the following holds: \(\textit{(i)}\) \(F([x,y])-\alpha ([x,y])\in Z(R)\) (ii) \(F([x,y])+\alpha ([x,y])\in Z(R)\) (iii) \(F(x \circ y)-\alpha (x \circ y)\in Z(R)\) (iv) \(F(x \circ y)-\alpha (x \circ y)\in Z(R)\) for all \(x,y\in J\), then R is commutative. Also some related results have been obtained.

Let (R, *) be a 2-torsion free *-prime ring with involution *, L ≠ 0 be a square closed *-Lie ideal of R and α, β automorphisms of R commuting with *. An additive mapping F: R → R is called a generalized (α, β)-derivation on R if there exists an (α, β)-derivation d such that F(xy) = F(x)α(y) + β(x)d(y) holds for all \({x, y \in R}\). In the present paper, we shall show that \({L\subseteq Z(R)}\) such that R is a *-prime ring admits a generalized (α, β)-derivation satisfying several conditions, but associated with an (α, β)-derivation commuting with *.

The distance, $d(u,v)$, between two vertices $u$ and $v$ of a connected graph $G$ is the length of a $u-v$ geodesic in $G$. A large number of graph-distance-based topological indices in various families of graphs and networks have been computed. In this paper, we consider circulant networks and compute three distance-based topological indices, namely the Wiener index, hyper-Wiener index and Schultz molecular topological index on these networks.

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d[x, y])m = [x, y]n for all x, y ∈ I, then R is commutative. (ii) If Char R ≠ 2 and [d(x), d(y)]m = [x, y]n for all x, y ∈ I, then R is commutative. Moreover, we also examine the case when R is a semiprime ring.

Let R be a prime ring, I a nonzero ideal of R and n a fixed positive integer. If R admits a generalized derivation F associated with a derivation d such that c for all x, . Then either R is commutative or n = 1, d = 0 and F is the identity map on R. Moreover in case R is a semiprime ring and for all x, , then either R is commutative or n = 1, , R contains a non-zero central ideal and for all .

In this paper, we present some results concerning orthogonal generalized ()-derivations in semiprime near-rings. These results are a generalization of result of Bresar and Vukman, which are related to a theorem of Posner for the product of two derivations in prime rings.

Let \(R\) be a prime ring of characteristic different from \(2\) and \(3\), \(L\) a non-central Lie ideal of \(R\), \((d,\sigma )\) a nonzero skew derivation of \(R\), \(n\) a fixed positive integer. If \([d(x),x]^{n}=0\) for all \(x\in L\), then \(R\) satisfies \(s_{4}\).

The concept of $\Gamma$-ring is a generalization of ring. Twoimportant classes of $\Gamma$-rings are prime and semi-prime$\Gamma$-rings. In this paper, we consider the concept of derivations on prime and semi-prime $\Gamma$-rings and we study some of their properties.

Let R be a prime ring, let L a noncentral Lie ideal, and let f be a nonzero semiderivation associated with an automorphism σ such that f(u)ⁿ = 0 for all u ∈ L; where n is a fixed positive integer. If either Char R > n + 1 or Char R = 0; then R satisfies s₄; the standard identity in four variables. Нехай R — просте кільцє, L — нецентральний ідеал Лі та f — ненульова напівпохідна, асоційована з автоморфiзмом a таким, що f(u)ⁿ=0 для всіх uЄL, де n — фіксоване натуральне число. Якщо Char R>n+1 або Char R=0, то R задовольняє стандартну тотожність s₄ у чотирьох змінних.

Let R be a semiprime ring with center Z(R) and µ be a surjective homomorphism. In this paper, we prove that T is a µ-centralizer if one of the following holds: (i) T(x)µ(y) = µ(x)T(y) for all x;y 2 R, where T is a mapping. (ii) [T(x);µ(x)] = 0 for all x 2 R, where T is a left µ-centralizer. (iii) 2T(xyx) = T(x)µ(y)µ(x) + µ(x)µ(y)T(x) for all x;y 2 R, where T is an additive mapping, µ(Z(R)) = Z(R) and R is 2-torsion free. 1

WOS: 000338514500017, Let (r, *) be a 2-torsion free *-prime ring with involution * and center Z (R), U a nonzero square closed *-Lie ideal of R. An additive mapping F W R! R is called a generalized derivation if there exits a derivation d WR R such that F. xy/DF. x/yCxd. y/. In the present paper, we prove that U * Z. R/if any one of following conditions holds: 1) O F. u/; u _ D 0; 2) O d. u/; F. v/_ D 0; 3) d. u/oF. v/D 0; 4) O d. u/; F. v/ D * O u; v; 5) d. u/oF (v) D * uov; 6) d. u/F. v/* uv 2 Z. R/; for all u; v 2 U: Furthermore, an example is given to demonstrate that the *-primeness hypothesis is not superfluous.