Journal of Frame and Matrix Theory
Volume:1 Issue: 1, Winter 2023

We establish a maximal probability inequality for a class of random variables in the framework of measure-free, Riesz spaces. In the ``other" Kolmogorov's inequality, we consider an upper bound for independent random variables and estimate the lower bound for the sums of random variables in Riesz spaces setting. Furthermore, we get an upper bound for the variance of the sum of random variables.

Error estimate and rate of convergence are two important topics in the field of numerical analysis. A convenient normed space corresponding to the problem under regard can have better upper bounds. This paper introduces a weighted normed space $L_{p,\omega}$ which from the measure theory point of view, is a special case of $L^{p}$ space. This space is a modification of $L_{p,\alpha}$ space, which is introduced before in \cite{Baghani}. Next, by using $L_{p,\alpha}$-norm, we compute a two-variable upper bound of the triangular function.

The purpose of the paper is to analyze frames $\{f_k\}_{k \in \mathbb{Z}}$ having the form $\{T^{k}f_{0}\}_{k \in \mathbb{Z}}$; so called iteration operator frames for some bounded linear operator $T$ and a fixed vector $f_0$. We state the duality of such frames with respect to their generators. Moreover, we characterize all duals of iteration operator frames with the same structure. We also show that the duals of two iteration operator frames are close to each other provided that the original frames are sufficiently close to each other and vise versa.

Motivated by definitions of countable completely regular spaces and completely below relations of frames, we define what we call a $c$-completely below relation, denoted by $\prec\!\!\prec_c$, in between two elements of a frame. We show that $a\prec\!\!\prec_c b$ for two elements $a, b$ of a frame $L$ if and only if there is $\alpha\in\mathcal{R}L$ such that $\coz\alpha\wedge a=0$ and $\coz(\alpha-{\bf1})\leq b$ where the set $\{r\in\mathbb{R} : \coz(\alpha-{\bf r})\ne 1\}$ is countable. We say a frame $L$ is a $c$-completely regular frame if $a=\bigvee \limits_{x\prec\!\!\prec_ca}x$ for any $a\in L$. It is shown that a frame $L$ is a $c$-completely regular frame if and only if it is a zero-dimensional frame. An ideal $I$ of a frame $L$ is said to be $c$-completely regular if $a\in I$ implies $a\prec\!\!\prec_c b$ for some $b\in I$. The set of all $c$-completely regular ideals of a frame $L$, denoted by ${\mathrm{c-CRegId}}(L)$, is a compact regular frame and it is a compactification for $L$ whenever $it$ is a $c$-completely regular frame. We denote this compactification by $\beta_cL$ and it is isomorphic to the frame $\beta_0L$, that is, Stone-Banaschewski compactification of $L$. Finally, we show that open and closed quotients of a $c$-completely regular frame are $c$-completely regular.

In this article, we introduce (strongly) 2-absorbing ideals of monoids and generalize them to (strongly) 2-absorbing subacts of an act over monoids. Among some useful lemmas, we show that the radical ideal of a strongly 2-absorbing ideal is either a prime ideal or the intersection of two ideals which are only distinct prime ideals minimal over it. Also, we prove that for each strongly 2-absorbing ideal I of a monoid, there exists a strongly 2-absorbing S-act A such that Ann(A) = I and vice versa. Moreover, some of their basic properties are developed.

The paper is devoted to 2-local higher derivations on some algebras. It is shown that continuous 2-local higher derivations on B(H), for an infinite dimensional separable Hilbert space H are higher derivations and each 2-local inner higher derivation on F(H) (the ideal of all finite-dimensional operators from B(H)) is a higher derivation. Also, we prove that every 2-local higher derivation from a commutative ∗-subalgebra of the matrix algebra Mn over C is a higher derivation.