Journal of Linear and Topological Algebra
Volume:11 Issue: 1, Winter 2022

‎Let $left( H;leftlangle cdot‎ ,‎cdot rightrangle right)$ be a complex‎ ‎Hilbert space‎. ‎Denote by $mathcal{B}left( Hright)$ the Banach $C^{ast }$-‎algebra of bounded linear operators on $H$‎. ‎For $Ain mathcal{B}left(‎Hright)$ we define the modulus of $A$ by $leftvert Arightvert‎ :‎=left(‎A^{ast }Aright) ^{1/2}$ and $func{Re}A:=frac{1}{2}left( A^{ast‎‎}+Aright)‎.‎$ In this paper we show among other that‎, ‎if $A,$ $Bin mathcal{‎‎B}left( Hright)$ with $0leq mleq leftvert left( 1-tright)‎‎A+tBrightvert ^{2}leq M$ for all $tin left[ 0,1right]‎,‎$ then begin{align*}‎ ‎0& leq int_{0}^{1}fleft( leftvert left( 1-tright) A+tBrightvert‎‎^{2}right) dt-fleft( frac{leftvert Arightvert ^{2}+func{Re}left(‎‎B^{ast }Aright)‎ +‎leftvert Brightvert ^{2}}{3}right) \‎ ‎& leq 2left[ frac{fleft( mright)‎ +‎fleft( Mright) }{2}-fleft( frac{‎m+M}{2}right) right] 1_{H}‎ ‎end{align*} ‎for operator convex functions $f:[0,infty )rightarrow mathbb{R}$‎. ‎Applications for power and logarithmic functions are also provided‎.

‎Best proximity point‎ ‎theorems for self-mappings were investigated with different‎ ‎conditions on spaces for contraction mappings‎. ‎In this‎ ‎paper‎, ‎we prove best proximity point theorems for proximal $mathcal{F}^{*}$-weak contraction mappings‎.

‎The purpose of this work is to define and investigate a new class of sets termed fuzzy nano $ Z $-open sets and fuzzy nano $ Z $-closed sets in fuzzy nano topological spaces‎, ‎as well as their basic properties‎. ‎We also talk about fuzzy nano $ Z $-closure and $ Z $-interior‎, ‎as well as their connections to fuzzy nano topological spaces‎.

‎The aim of this work is to introduce the concepts of $(v‎, ‎u‎, ‎phi)$-contraction and $(q‎, ‎p‎, ‎phi)$-contraction‎, ‎and to obtain new results in fixed point theory for four mappings in $b$-metric spaces‎. ‎Finally‎, ‎we have developed an example and an application for a system of integral equations that protects the main theorems‎.

‎In this paper‎, ‎the main purpose is to calculate the conservation laws of Kuramoto-Sivashinsky equation using‎ the scaling method‎. ‎Linear algebra and calculus of variations are used in this algorithmic method‎. ‎Also the density of the conservation law is obtained by scaling symmetries of the equation and the flux corresponding to the density is calculated using the homotopy operator‎.

‎Digital topological methods are often used in computing the topological complexity of digital images‎. ‎We give new results on the relation between reducibility and digital contractibility in order to determine the topological complexity of a digitally connected finite digital image‎. ‎We present all possible cases of the topological complexity TC of a finite digital image in $mathbb{Z}$ and $mathbb{Z}^{2}$‎. ‎Finally‎, ‎we determine the higher topological complexity TC$_{n}$ of finite irreducible digital images independently of the number of points for $n > 1$‎.

In this article, we present a non-interactive key exchange protocol with a faster run time, which is based on a Module-LWE. The Structure of protocol is designed just by relating the error vectors of both sides, without any use of a reconciliation mechanism. The idea is that as error vectors get closer to each other the success probability of the protocol increases. The innovation in this scheme is the use of high-order bits in the keys computed by both sides. Compared to the existing lattice-based key-exchange protocols, this scheme leads to lower computational complexity and longer parameters.