Searcharxiv⌕ Search

arXiv subjects

Kazem Haghnejad Azar

Publications and source records attributed to Kazem Haghnejad Azar.

At least 19 recordsLinked to original sources
arXiv2025

The Relationship Between Euler Numbers and Bernoulli Numbers with Ordered Partitions

In this paper, for every $n \in \mathbb{N}$, the following relationships between the functions $K_{b}(n)$ and $K_{e}(n)$ and the Bernoulli and Euler numbers are proved: \[ B_{2n} = -\,\frac{(2n)!}{2^{2n}-2}\, K_{b}(n), \qquad E_{2n} = (2n)!\, K_{e}(n). \] The functions $K_{b}$ and $K_{e}$ are defined recursively by \[ K_{b}(0) = K_{e}(0) = 1, \] \[ K_{b}(n) = - \sum_{n'=0}^{\,n-1} \frac{K_{b}(n')}{\bigl( 2(n-n') + 1 \bigr)!}, \qquad n \ge 1, \] \[ K_{e}(n) = - \sum_{n'=0}^{\,n-1} \frac{K_{e}(n')}{\bigl( 2(n-n') \bigr)!}, \qquad n \ge 1. \] Furthermore, we present combinatorial interpretations of these functions in terms of ordered partitions of $n$: \[ K_{b}(n) = \sum_{λ\vDash n} \frac{(-1)^{\ell(λ)}} {\displaystyle\prod_{i=1}^{\ell(λ)} (2b_i + 1)!}, \qquad n \ge 1, \] \[ K_{e}(n) = \sum_{λ\vDash n} \frac{(-1)^{\ell(λ)}} {\displaystyle\prod_{i=1}^{\ell(λ)} (2b_i)!}, \qquad n \ge 1, \] where $λ= (b_1,b_2,\ldots,b_k) \vDash n$ and $\ell(λ)=k$.

math.GM↗
arXiv2024

The Interplay Between Logical Phenomena and the Cognitive System of the Mind

In this article, we employ mathematical concepts as a tool to examine the phenomenon of consciousness experience and logical phenomena. Through our investigation, we aim to demonstrate that our experiences, while not confined to limitations, cannot be neatly encapsulated within a singular collection. Our conscious experience emerges as a result of the developmental and augmentative trajectory of our cognitive system. As our cognitive abilities undergo refinement and advancement, our capacity for logical thinking likewise evolves, thereby manifesting a heightened level of conscious experience. The primary objective of this article is to embark upon a profound exploration of the concept of logical experience, delving into the intricate process by which these experiences are derived from our mind.

q-bio.NC↗
arXiv2024

The quest for the definition of life

The intricacy and diversity inherent in living organisms present a formidable obstacle to the establishment of a universally accepted definition. Life manifests in a multitude of forms, exhibiting various attributes such as growth, reproduction, responsiveness to stimuli, adaptation, and homeostasis. However, each of these characteristics can also be observed to some degree within certain non-living systems, leading to a blurring of boundaries and generating conceptual complexities. In this manuscript, I demonstrate that the transformation of a non-living entity into a living organism does not adhere to a specific temporal boundary that unequivocally designates the onset of life. Through mathematical analysis, I have demonstrated that a comprehensive definition of living beings does not exist, which means that there are no clear boundaries in the chemical processes that turn non-living entities into living ones. In other words, living organisms do not possess unique characteristics that can completely set them apart from non-living entities. Therefore, no definitive definition exists that unequivocally distinguishes living things from non-living things.

q-bio.NC↗
arXiv2022

$\tilde{o}$rder-norm continuous operators and $\tilde{o}$rder weakly compact operators

Let $E$ be a sublattice of a vector lattice $F$. A continuous operator $T$ from the vector lattice $E$ into a normed vector space $X$ is said to be $\tilde{o}$rder-norm continuous whenever $x_α\xrightarrow{Fo}0$ implies $Tx_α\xrightarrow{\Vert.\Vert}0$ for each $(x_α)_α\subseteq E$. Our mean from the convergence $ x_α\stackrel{Fo} {\longrightarrow} x $ is that there exists another net $ \left(y_α\right) $ in $F $ with the same index set satisfying $ y_α\downarrow 0 $ in $F$ and $ \vert x_α- x \vert \leq y_α$ for all indexes $ α$. In this paper, we will study some properties of this new class of operators and its relationships with some known classifications of operators. We also define the new class of operators that named $\tilde{o}$rder weakly compact operators. A continuous operator $T: E \rightarrow X $ is said to be $\tilde{o}$rder weakly compact, if $ T(A) $ in $X$ is a relatively weakly compact set for each $Fo$-bounded $A\subseteq E$. In this manuscript, we study some properties of this class of operators and its relationships with $\tilde{o}$rder-norm continuous operators.

math.FA↗
arXiv2022

On the properties of the Aron-Berner regularity of bounded tri-linear maps

Let $f:X\times Y\times Z\longrightarrow W $ be a bounded tri-linear map on normed spaces. We say that $f$ is close-to-regular when $f^{t****s}=f^{s****t}$ and $f$ is Aron-Berener regular when all natural extensions are equal. In this manuscript, we have some results on the Aron-Berner regular maps. We investigate the relation between Arens regularity of bounded bilinear maps and Aron-Berner regularity of bounded tri-linear maps. We also give a simple criterion for the Aron-Berner regularity of tri-linear maps.

math.FA↗
arXiv2022

Almost order-weakly compact operators on Banach lattices

A continuous operator $T$ between two Banach lattices $E$ and $F$ is called almost order-weakly compact, whenever for each almost order bounded subset $A$ of $E$, $T(A)$ is a relatively weakly compact subset of $F$. In Theorem 4, we show that the positive operator $T$ from $E$ into Dedekind complete $F$ is almost order-weakly compact if and only if $T(x_n) \xrightarrow{\|.\|}0$ in $F$ for each disjoint almost order bounded sequence $\{x_n\}$ in $E$. In this manuscript, we study some properties of this class of operators and its relationships with others known operators.

math.FA↗
arXiv2021

Semi unbounded order convergent in ordered vector spaces

Let $X$ be an ordered vector space. The net $\{x_α\}\subseteq X$ is semi unbounded order convergent to $x$ (in symbol $x_α\xrightarrow{suo}x$), if there is a net $\{y_β\}$, possibly over a different index set, such that $y_β\downarrow 0$ and for every $β$ there exists $α_0$ such that $\{\{\pm(x_α- x)\}^u,y\}^l\subseteq \{y_β\}^l$, whenever $α\geq α_0$ and for all $0\leq y \in X$. In vector lattice $E$, semi unbounded order convergence is equivalent with unbounded order convergence. We study some properties of this convergence and some of its relationships with others known order convergence.

math.FA↗
arXiv2021

Unbounded $M$-weakly and unbounded $L$-weakly compact operators

We introduce the class of unbounded $M$-weakly operators and the class of unbounded $L$-weakly compact operators. We investigate some properties for these new classification of operators and we study relation between them and $M$-weakly compact and $L$-weakly compact operators. We also present an operator characterization of Banach lattices with order continuous norm. \keywords{unbounded $M$-weakly compact \and unbounded $L$-weakly compact \and unbounded norm convergence \and $M$-weakly compact \and $L$-weakly compact

math.FA↗
arXiv2020

Weak tolpological centers and cohomological properties

Let $B$ be a Banach $A-bimodule$. We introduce the weak topological centers of left module action and we show it by $\tilde{Z}^\ell_{B^{**}}(A^{**})$. For a compact group, we show that $L^1(G)=\tilde{Z}_{M(G)^{**}}^\ell(L^1(G)^{**})$ and on the other hand we have $\tilde{Z}_1^\ell{(c_0^{**})}\neq c_0^{**}$. Thus the weak topological centers are different with topological centers of left or right module actions. In this manuscript, we investigate the relationships between two concepts with some conclusions in Banach algebras. We also have some application of this new concept and topological centers of module actions in the cohomological properties of Banach algebras, spacial, in the weak amenability and $n$-weak amenability of Banach algebras.

math.FA↗
arXiv2020

Cohomological properties and Arens regularity of Banach algebras

In this paper, we study some cohomlogical properties of Banach algebras. For a Banach algebra $A$ and a Banach $A$-bimodule $B$, we investigate the vanishing of the first Hochschild cohomology groups $H^1(A^n,B^m)$ and $H_{w^*}^1(A^n,B^m)$, where $0\leq m,n\leq 3$. For amenable Banach algebra $A$, we show that there are Banach $A$-bimodules $C$, $D$ and elements $\mathfrak{a}, \mathfrak{b}\in A^{**}$ such that $$Z^1(A,C^*)=\{R_{D^{\prime\prime}(\mathfrak{a})}:~D\in Z^1(A,C^*)\}=\{L_{D^{\prime\prime}(\mathfrak{b})}:~D\in Z^1(A,D^*)\}.$$ where, for every $b\in B$, $L_{b}(a)=ba$ and $R_{b}(a)=a b,$ for every $a\in A$. Moreover, under a condition, we show that if the second transpose of a continuous derivation from the Banach algebra $A$ into $A^*$ i.e., a continuous linear map from $A^{**}$ into $A^{***}$, is a derivation, then $A$ is Arens regular. Finally, we show that if $A$ is a dual left strongly irregular Banach algebra such that its second dual is amenable, then $A$ is reflexive.

math.FA↗
arXiv2020

Weak Unbounded Norm Topology and Dounford-Pettis Operators

In this paper, we study $un$-dual (in symbol, $\ud{E}$) of Banach lattice $E$ and compare it with topological dual $E^*$. If $E^*$ has order continuous norm, then $E^* = \ud{E}$. We introduce and study weakly unbounded norm topology ($wun$-topology) on Banach lattices and compare it with weak topology and $uaw$-topology. In the final, we introduce and study $wun$-Dunford-Pettis opertors from a Banach lattice $E$ into Banach space $X$ and we investigate some of its properties and its relationships with others known operators.

math.FA↗
arXiv2020

Semi-order continuous operators on vector spaces

In this manuscript, we will study both $\tilde{o}$-convergence in (partially) ordered vector spaces and a kind of convergence in a vector space $V$. A vector space $V$ is called semi-order vector space (in short semi-order space), if there exist an ordered vector space $W$ and an operator $T$ from $V$ into $W$. In this way, we say that $V$ is semi-order space with respect to $\{W, T\}$. A net $\{x_α\}\subseteq V$ is said to be ${\{W,T\}}$-order convergent to a vector $x\in V$ (in short we write $x_α\xrightarrow {\{W, T\}}x$), whenever there exists a net $\{y_β\}$ in $W$ satisfying $y_β\downarrow 0$ in $W$ and for each $β$, there exists $α_0$ such that $\pm (Tx_α-Tx) \leq y_β$ whenever $α\geq α_0$. In this manuscript, we study and investigate some properties of $\{W,T\}$-convergent nets and its relationships with other order convergence in partially ordered vector spaces. Assume that $V_1$ and $V_2$ are semi-order spaces with respect to $\{{W_1}, T_1\}$ and $\{W_2, T_2\}$, respectively. An operator $S$ from $V_1$ into $V_2$ is called semi-order continuous, if $x_α\xrightarrow {\{{W_1}, T_1\}}x$ implies $Sx_α\xrightarrow {\{W_2, T_2\}}Sx$ whenever $\{x_α\}\subseteq V_1$. We study some properties of this new classification of operators.

math.FA↗
arXiv2019

Some properties of bounded tri-linear maps

Let $X,Y,Z$ and $W$ be normed spaces and $f:X\times Y\times Z\longrightarrow W $ be a bounded tri-linear mapping. In this Article, we define the topological centers for bounded tri-linear mapping and we invistagate thier properties. We study the relationships between weakly compactenss of bounded linear mappings and regularity of bounded tri-linear mappings. For both bounded tri-linear mappings $f$ and $g$, let $f$ factors through $g$, we present necessary and suficient condition such that the extensions of $f$ factors through extensions of $g$. Also we establish relations between regularity and factorization property of bounded tri-linear mappings.

math.FA↗
arXiv2019

A generalization of order continuous operators

Let $E$ be a sublattice of a vector lattice $F$. A net $\{ x_α\}_{α\in \mathcal{A}}\subseteq E$ is said to be $ F $-order convergent to a vector $ x \in E$ (in symbols $ x_α\xrightarrow{Fo} x $ in $E$), whenever there exists a net $ \{y_β\}_{β\in \mathcal{B}} $ in $F $ satisfying $ y_β\downarrow 0 $ in $F$ and for each $β$, there exists $α_0$ such that $ \vert x_α- x \vert \leq y_β$ whenever $ α\geq α_0 $. In this manuscript, first we study some properties of $F$-order convergence nets and we extend some results to the general cases. Let $E$ and $G$ be sublattices of vector lattices $F$ and $H$ respectively. We introduce $FH$-order continuous operators, that is, an operator $T$ between two vector lattices $E$ and $G$ is said to be $FH$-order continuous, if $x_α\xrightarrow{Fo} 0$ in $E$ implies $Tx_α\xrightarrow{Ho} 0$ in $G$. We will study some properties of this new classification of operators and its relationships with order continuous operators.

math.FA↗
arXiv2019

Regularity of bounded tri-linear maps and the fourth adjiont of a tri-derivation

In this Article, we give a simple criterion for the regularity of a tri-linear mapping. We provide if $f:X\times Y\times Z\longrightarrow W $ is a bounded tri-linear mapping and $h:W\longrightarrow S$ is a bounded linear mapping, then $f$ is regular if and only if $hof$ is regular. We also shall give some necessary and sufficient conditions such that the fourth adjoint $D^{****}$ of a tri-derivation $D$ is again tri-derivation.

math.FA↗
arXiv2019

Close-to-regularity and completely regularity of bounded tri-linear maps

Let $f:X\times Y\times Z\longrightarrow W $ be a bounded tri-linear map on normed spaces. We say that $f$ is close-to-regular when $f^{t****s}=f^{s****t}$ and we say that $f$ is completely regular when all natural extensions are equal. In this manuscript, we have some results on the close-to-regular maps and investigate the close-to-regularity of tri-linear maps. We investigate the relation between Arens regularity of bounded bilinear maps and close-to-regularity bounded tri-linear maps. We give a simple criterion for the completely regularity of tri-linear maps. We provide a necessary and sufficient condition such that the fourth adjoint $D^{****}$ of a tri-derivation is again a tri-derivation.

math.FA↗
arXiv2019

Unbounded $σ$-order-to-norm continuous and $un$-continuous operators

An operator $T $ from a vector lattice $E$ into a normed lattice $F$ is called unbounded $σ$-order-to-norm continuous whenever $x_{n}\xrightarrow{uo}0$ implies $\| Tx_{n}\|\rightarrow 0$, for each sequence $(x_{n})_n\subseteq E$. For a net $(x_α)_α\subseteq E$, if $x_α\xrightarrow{un}0$ implies $Tx_α\xrightarrow{un}0$, then $T$ is called an unbounded norm continuous operator. In this manuscript, we study some properties of these classes of operators and their relationships with the other classes of operators.

math.FA↗