Asian-European Journal of Mathematics

A linear tree language of type $\tau$ is a set of linear terms, terms containing no multiple occurrences of the same variable, of that type. Instead of the usual generalized superposition of tree languages, we define the generalized linear superposition to deal with linear tree languages and study its properties. Using this superposition, we define the product of linear tree languages. This product is not associative on the collection of all linear tree languages, but it is associative on some subsets of this collection whose products of any element in the subsets are nonempty. We classify such subsets and study properties of the obtained semigroup especially idempotent elements, regular elements, and Green’s relations ${ℒ}$ and ${ℛ}$.

In this paper, some generalization of weighted Ostrowski type integral inequalities for mappings of bounded variation are obtained and some interesting inequalities as special cases are given.

Let $L$ be a Filippov algebra. A derivation of $L$ is called an $\mathrm{\text{ID}}$-derivation if its image is contained in the derived algebra of $L$. Let ${\mathrm{\text{ID}}}^{\ast }(L)$ be the set of all $\mathrm{\text{ID}}$-derivations which map central elements to $0$. We prove that ${\mathrm{\text{ID}}}^{\ast }(L)$ and ${\mathrm{\text{ID}}}^{\ast }(M)$ are isomorphic for any two isoclinic Filippov algebras $L$ and $M$.

Every multiplicative $\delta$-derivation of an alternative algebra $A$ is additive if there exists an idempotent $e^{\prime }(e^{\prime }\ne 0,1)$ in $A$ satisfying the following conditions: (i) $e^{\prime }Au=0$ implies $A=0$; (ii) $e^{\prime }ueA(1-e)=0$ implies $e^{\prime }ue=0$; (iii) $uA=0$ implies $u=0$ for $e^{\prime }=\delta (e)$. In particular, every $\delta$-derivation of a prime alternative algebra with a nontrivial idempotent is additive. This generalizes the known result obtained by Rodrigues, Guzzo and Ferreira for $\delta$-derivations. As an application, we apply multiplicative $\delta$-derivation to an alternative complex algebra $M_n(ℂ)$ of all $n\times n$ complex matrices to see how it decomposes into a sum of $\delta$-inner derivation and a $\delta$-derivation on $M_n(ℂ)$ given by an additive derivation $\gamma$ on $ℂ$.

Let $I$ be a regular ideal in noetherian ring $A$. Mc Adam and Ratliff showed the existence of the unique minimal reduction number of $I$, noted $e=e(I)\in \mathbb{N}$, such that for every minimal reduction $Y$ of $I$, $Y{I}^e=I^{e+1}$ and $Y{I}^{e-1}\ne I^e$. They showed that the set of integers $\{e(I^n),n\in \mathbb{N}\}$ is bounded in terms of the analytic spread of $I$. Here, we extend these results to good filtrations. Let $f={(I_n)}_{n\in \mathbb{N}}$ be a good filtration on $A$, we show that the set of integers $\{e(I_n),n\in \mathbb{N}\}$ is bounded.

In this paper, we show some Second Main Theorems for zero-order meromorphic mappings intersecting slowly moving targets in ${\mathbb{P}}^m(ℂ)$ by considering their $p$-Casorati determinant. Our results are $p$-difference analogues of Cartan’s Second Main Theorem for moving targets. As an application, we give an unicity theorem for meromorphic mappings of ${ℂ}^m$ into ${\mathbb{P}}^m(ℂ)$ under the growth condition “order $=0$”.

This paper is devoted to study some chaotic properties of iterated function systems (IFSs). Specially, a new notion named thick chaotic IFSs is introduced. The relationship between thick chaos and another properties of some notions in dynamical systems are studied.

Let $R$ be a noncommutative prime ring with its Utumi ring of quotients $U$, $C=Z(U)$ the extended centroid of $R$, $G$ a generalized derivation of $R$ and $I$ a nonzero ideal of $R$. If $I$ satisfies any one of the following conditions:

• ⁽ⁱ⁾$[[G({[x,y]}_n)$, ${[x,y]}_n]$, $G({[x,y]}_n)]\in C$,
• ⁽ⁱⁱ⁾$(G(x{\circ }_{n}y)\circ (x{\circ }_{n}y))\circ G(x{\circ }_{n}y)\in C$,

where $n\ge 1$ is a fixed integer, then one of the following holds:

• ⁽¹⁾there exists $\lambda \in C$ such that $G(x)=\lambda x$ for all $x\in R$;
• ⁽²⁾$R$ satisfies $s_4$ and there exist $a\in U$ and $\lambda \in C$ such that $G(x)=ax+xa+\lambda x$ for all $x\in R$;
• ⁽³⁾char $(R)=2$, $R$ satisfies $s_4$ and there exist $\lambda \in C$ and an outer derivation $d$ of $R$ such that $G(x)=\lambda x+d(x)$ for all $x\in R$.

Let $a,b,c$ be any positive integers such that $c|ab$ and $d_i^{\pm }$ is a square free positive integer of the form $d_i^{\pm }=a^{2k}{b}^{2l}\pm i{c}^m$ where $k,l\ge m$ and $i=1,2.$ The main focus of this paper is to find the fundamental solution of the equation $x^2-d_i^{\pm }{y}^2=1,$ with the help of the continued fraction of $\sqrt{d_i^{\pm }}.$ We also obtain all the positive solutions of the equations $x^2-d_i^{\pm }{y}^2=\pm 1$ and $x^2-d_i^{\pm }{y}^2=\pm 4$ by means of the Fibonacci and Lucas sequences.

Furthermore, in this work, we derive some algebraic relations on the Pell form $F_{d_i^{\pm }}(x,y)=x^2-d_i^{\pm }{y}^2$ including cycle, proper cycle, reduction and proper automorphism of it. We also determine the integer solutions of the Pell equation $F_{{\mathrm{\Delta }}_{d_i^{\pm }}}(x,y)=1$ in terms of $d_i^{\pm }.$ We extend all the results of the papers [3, 10, 27, 37].

We investigate the first differential cohomology space associated with the embedding of the affine Lie superalgebra $𝔞𝔣𝔣(m|1)$ on the $(1,m)$-dimensional supercircle $S^{1|m}$ in the Lie superalgebra ${𝒮}\mathrm{\Psi }{𝒟}{𝒪}(S^{1|m})$ of superpseudodifferential operators with smooth coefficients, where $m=1,2$. Following Ovsienko and Roger, we give explicit expressions of the basis cocycles. We study the deformations of the structure of the $𝔞𝔣𝔣(2|1)$-module ${𝒮}\mathrm{\Psi }{𝒟}{𝒪}(S^{1|2})$. We prove that any formal deformation is equivalent to its infinitesimal part.

Let $R$ be a commutative ring with identity. The essential ideal graph of $R$, denoted by ${{ℰ}}_R$, is a graph whose vertex set is the set of all nonzero proper ideals of $R$ and two vertices $I$ and $J$ are adjacent whenever $I+J$ is an essential ideal. In this paper, we initiate the study of the essential ideal graph of a commutative ring and we investigate its properties.

In this paper, we study properties of the endomorphism monoids of strong semilattices of groups. In Sec. 2, several properties for endomorphism monoids of finite semilattices are investigated. In Sec. 3, we collect some results on endomorphism monoids of strong semilattices of groups, i.e. Clifford semigroups.

We consider discrete and continuous cyclic refinements of Jensen’s inequality and generalize them from convex function to higher order convex function by means of Lagrange Green’s function and Montgomery identity. We give application of our results by formulating the monotonicity of the linear functionals obtained from generalized identities utilizing the theory of inequalities for $n$-convex functions at a point. We compute Grüss and Ostrowski type bounds for generalized identities associated with the obtained inequalities. Finally, we investigate the properties of linear functionals regarding exponential convexity log convexity and mean value theorems.

This study aimed to give definitions and relations between strip theory and harmonic curvatures of the strip in Minkowski space. Previously, the same was done in Euclidean Space (see [F. Ertem Kaya, Y. Yayli and H. H. Hacısalihoglu, A characterization of cylindrical helix strip, Commun. Fac. Sci. Univ. Ank. Ser. A1 59(2) (2010) 37–51]). The present paper gives for the first time a generic characterization of the harmonic curvatures of the strip, helix strip and inclined strip in Minkowski space.

In this paper, Pachpatte’s inequality is employed to discuss the Ulam–Hyers stabilities for Volterra integrodifferential equations and Volterra delay integrodifferential equations in Banach spaces on both finite and infinite intervals. Examples are given to show the applicability of our obtained results.

It is known that an ideal of a direct product of commutative unitary rings is directly decomposable into ideals of the corresponding factors. We show that this does not hold in general for commutative rings and we find necessary and sufficient conditions for direct decomposability of ideals. For varieties of commutative rings, we derive a Mal’cev type condition characterizing direct decomposability of ideals and we determine explicitly all varieties satisfying this condition.