Khayyam Journal of Mathematics
https://www.kjm-math.org/
Khayyam Journal of Mathematicsendaily1Sun, 01 Jan 2023 00:00:00 +0330Sun, 01 Jan 2023 00:00:00 +0330Reviewed techniques in automatic continuity of linear functionals
https://www.kjm-math.org/article_164399.html
Some techniques which were already used to derive automatic continuity results are chosen, they are modified, and extended results as well as generalized results are obtained. A technique of using the open mapping theorem and a technique of using the Hahn Banach extension theorem are explained. Results in connection with measurable cardinals are also obtained. Results for multiplicative linear functionals, positive linear functionals and uniqueness of topology are obtained. For example, sequential continuity of real multiplicative linear functionals on sequentially complete LMC algebras is obtained, when Michael's open problem is concerned only with boundedness of multiplicative linear functionals. The continuity of positive linear functionals on F-algebras with identity elements and involution is derived, when these functionals are continuous on the set of all involution-symmetric elements. Possibilities of extending the concept of positive linear functionals are considered to derive results for the continuity of such functionals on topological groups and topological vector spaces with additional structures. The technique for the Carpenter's uniqueness theorem is modified to derive boundedness of some homomorphisms. The entire article is oriented towards Michael's problem.Anti-derivations on triangular rings
https://www.kjm-math.org/article_164403.html
&lrm;The aim of this paper is to give necessary and sufficient conditions for anti-derivations to be zero on 2-torsion free triangular rings. As application of our main result&lrm;, &lrm;we present sufficient conditions for anti-derivations to be zero on block upper triangular matrix rings&lrm;. The aim of this paper is to give necessary and sufficient conditions for anti-derivations to be zero on 2-torsion free triangular rings. As application of our main result&lrm;, &lrm;we present sufficient conditions for anti-derivations to be zero on block upper triangular matrix rings&lrm;.Sharp bounds of third Hankel determinant for a class of starlike functions and a subclass of q-starlike functions
https://www.kjm-math.org/article_164407.html
Following the trend of coefficient bound problems in Geometric Function Theory, in the present paper, the authors obtain the sharp bound of third Hankel determinant for the classes $S^*$, of starlike functions and $SL^*_q$, of $q$-starlike functions related with lemniscate of Bernoulli. Bound on the functions in the initial class, apart from being sharp is also an improvement over the known existing bound and the bound on the latter class generalizes the prior known outcome. Further, the extremal functions of classes $S^*$ and $SL^*_q$ are deduced to prove the sharpness of these results.On strongly star semi-compactness of topological spaces
https://www.kjm-math.org/article_164474.html
Sabah et al. developed strongly star semi-compactness of a topological space in 2016, which is a variant of star-compactness in which semi-open covers are employed instead of open covers. The goal of this study is to compare the structure of strongly star semi-compactness to that of other topological characteristics with similar structures. Furthermore, the nature of a strongly star semi-compact space's subspace and the features of a strongly star semi-compact subset relative to a space are examined.On φ-δ-S-primary ideals of commutative rings
https://www.kjm-math.org/article_164475.html
Let $R$ be a commutative ring with unity $(1\not=0)$ and let $\mathfrak{J}(R)$ be the set of all ideals of $R$. Let $\phi:\mathfrak{J}(R)\rightarrow\mathfrak{J}(R)\cup\{\emptyset\}$ be a reduction function of ideals of $R$ and let $\delta:\mathfrak{J}(R)\rightarrow\mathfrak{J}(R)$ be an expansion function of ideals of $R$. We recall that a proper ideal $I$ of $R$ is called a $\phi$-$\delta$-primary ideal of $R$ if whenever $a,b\in R$ and $ab\in I-\phi(I)$, then $a\in I$ or $b\in\delta(I)$. In this paper, we introduce a new class of ideals that is a generalization to the class of $\phi$-$\delta$-primary ideals. Let $S$ be a multiplicative subset of $R$ such that $1\in S$ and let $I$ be a proper ideal of $R$ with $S\cap I=\emptyset$, then $I$ is called a $\phi$-$\delta$-$S$-primary ideal of $R$ associated to $s\in S$ if whenever $a,b\in R$ and $ab\in I-\phi(I)$, then $sa\in I$ or $sb\in\delta(I)$. In this paper, we have presented a range of different examples, properties, characterizations of this new class of ideals.Generalized derivations on Lie ideals with annihilating Engel conditions
https://www.kjm-math.org/article_164476.html
Let $\mathcal{R}$ be a non-commutative prime ring with characteristic different from $2$, $\mathcal{U}$ be the Utumi quotient ring of $\mathcal{R}$ and $\mathcal{C}$ be the extended centroid of $\mathcal{R}$. Let $\mathcal{G}$ be a generalized derivation on $\mathcal{R}$, $\mathcal{L}$ be a non-central Lie ideal of $\mathcal{R}$, $0 \neq c \in\mathcal{R}$ and $ n, r, s,t$ are fixed positive integers. If $c u^s[\mathcal{G}(u^n),u^r]_ku^t =0$, for all $u \in\mathcal{L}$, then one of the following holds:
(1) $\mathcal{R}$ satisfies $s_4$.
(2) There exists $\lambda \in \mathcal{C}$ s.t. $\mathcal{G}(\zeta)= \lambda \zeta$ for all $\zeta \in \mathcal{R}$.
(3) If $\mathcal{C}$ is finite field, $\mathcal{R} \cong M_{l}(\mathcal{C})$, a $l \times l$ matrix ring over $\mathcal{C}$ for $l&gt;2$.On matrix-valued Gabor Bessel sequences and dual frames over locally compact abelian groups
https://www.kjm-math.org/article_164478.html
We study matrix-valued Gabor Bessel sequences and frames in the matrix-valued space $L^2(G, \mathbb{C}^{n\times n})$, where $G$ is a locally compact abelian group and $n$ is a positive integer. Firstly, we show that the Bessel condition (or upper frame condition) can be extended from $L^2(G)$ to its associated matrix-valued signal space $L^2(G,\mathbb{C}^{n\times n})$, and conversely. However, this is not true for the lower frame condition. Secondly, we give sufficient conditions for the extension of a pair of matrix-valued Bessel sequences to matrix-valued dual frames over LCA groups. A special class of matrix-valued dual generators is given. It is shown that the symmetric windows associated with a given matrix-valued Gabor frames constitutes a Gabor frame in matrix-valued spaces over LCA groups.A study of Bessel sequences and frames via perturbations of constant multiples of the identity
https://www.kjm-math.org/article_164487.html
&lrm;We study those Bessel sequences $\{f_k\}_{k=1}^{\infty}$ in an infinite-dimensional, separable Hilbert space $H$ for which the operator $S$ defined by $Sf:=\sum_{k=1}^{\infty} \langle f,f_k\rangle f_k$ is of the form $cI+T$&lrm;, &lrm;for some real number $c$ and a bounded linear operator $T$&lrm;, &lrm;where $I$ denotes the identity operator&lrm;. &lrm;We use a reverse Schwarz inequality to provide conditions on $T$ and $c$ that allow $\{f_k\}_{k=1}^{\infty}$ to be a frame&lrm;. &lrm;Moreover&lrm;, &lrm;we introduce and study frames whose frame operators are compact (respectively&lrm;, &lrm;finite-rank) perturbations of constant multiples of the identity&lrm;, &lrm;frames to which we refer as compact-tight (respectively&lrm;, &lrm;finite-rank-tight) frames&lrm;. &lrm;As our final result&lrm;, &lrm;we prove a theorem on the weaving of certain compact-tight frames&lrm;.On $S$-finite conductor rings
https://www.kjm-math.org/article_164488.html
Let $R$ be a commutative ring with nonzero identity and $S \subseteq R$ be a multiplicatively closed subset of $R$. In this paper, we introduce and study $S$-finite conductor rings. $R$ is said to be an $S$-finite conductor ring if $(0:a)$ and $Ra\cap Rb$ are $S$-finite ideals of $R$ for each $a,b\in R.$ Some basic properties of $S$-finite conductor rings are studied. For instance, we give necessary and sufficient conditions for a ring to be $S$-finite conductor. Also, we prove that every pre-Schreier $S$-finite conductor domain is an $S$-$GCD$ domain and the converse is true for some particular cases of $S$. Further, we examine the stability of these rings in localization and study the possible transfer to direct product, trivial ring extension and amalgamated algebra along an ideal.Iterated function systems over arbitrary shift spaces
https://www.kjm-math.org/article_164489.html
The orbit of a point $x\in X$ in a classical iterated function system (IFS) can be defined as $\{f_u(x)=f_{u_n}\circ\cdots \circ f_{u_1}(x):$ $u=u_1\cdots u_n$ is a word of a full shift $\Sigma$ on finite symbols and $f_{u_i}$ is a continuous self map on $X \}$. One also can associate to $\sigma=\sigma_1\sigma_2\cdots\in\Sigma$ a non-autonomous system $(X,\,f_\sigma)$ where the trajectory of $x\in X$ is defined as $x,\,f_{\sigma_1}(x),\,f_{\sigma_1\sigma_2}(x),\ldots$. Here instead of the full shift, we consider an arbitrary shift space $\Sigma$. Then we investigate basic properties related to this IFS and the associated non-autonomous systems. In particular, we look for sufficient conditions that guarantees that in a transitive IFS one may have a transitive $(X,\,f_\sigma)$ for some $\sigma\in\Sigma$ and how abundance are such $\sigma$'s.Hilbert-Schmidtness of Fourier integral operators in $\mathbf{SG}$ classes
https://www.kjm-math.org/article_164490.html
In this paper, we define a particular class of Fourier integral operators with $\mathbf{SG}$-symbol. These class of operators turn out to be bounded on the spaces $\mathcal{S}\left(\mathbb{R}^{n}\right)$ of rapidly decreasing functions and turn out to be Hilbert-Schmidt on $L^2\left(\mathbb{R}^{n}\right)$. Mainly, we prove that the Fourier integral operators with $\mathbf{SG}$-symbol are a Hilbert-Schmidt operators.Fully $S$-idempotent modules
https://www.kjm-math.org/article_164491.html
&lrm;Let $R$ be a commutative ring with identity&lrm;, &lrm;$S$ be a multiplicatively closed subset of $R$&lrm;, &lrm;and $M$ be an $R$-module&lrm;. A submodule $N$ of $M$ is said to be \emph{idempotent} if $N=(N:_RM)^2M$&lrm;. &lrm;Also&lrm;, &lrm;$M$ is said to be \emph{fully idempotent} if every submodule of $M$ is idempotent&lrm;. The aim of this paper is to introduce the notion of fully $S$-idempotent modules as a generalization of fully idempotent modules and investigate some properties of this class of modules&lrm;.