arXiv subjects
Publications and source records attributed to Ali Taghavi.
In this note we present a survey on some classical and modern approaches on Pythagorean triples. Some questions are also posed in direction of some materials under review. In particular some non commutative and operator theoretical approaches of Pythagorean triples are discussed
We present some questions and suggestion on the second part of the Hilbert 16th problem
We give a precise example of a polynomial vector field on $\mathbb{R}^2$ whose corresponding singular holomorphic foliation of $\mathbb{C}^2$ possesses a complex limit cycle which does not intersect the real plane $\mathbb{R}^2$.
In this paper we give an alternative proof for a vanishing result about flat functions proved in G.Stoica, "When must a flat function be identically zero", The American Mathematical Monthly 125(7)648-649,2018. With a dynamical approach we give a generalization of this result to multidimensional variables.
Let $\mathcal{A}$ be a prime $\ast$-algebra and $Φ$ preserves triple $\ast$-Jordan derivation on $\mathcal{A}$, that is, for every $A,B \in \mathcal{A}$, $$Φ(A\diamond B \diamond C)=Φ(A)\diamond B\diamond C+A\diamond Φ(B)\diamond C+A\diamond B\diamond Φ(C)$$ where $A\diamond B = AB + BA^{\ast}$ then $Φ$ is additive. Moreover, if $Φ(αI)$ is self-adjoint for $α\in\{1,i\}$ then $Φ$ is a $\ast$-derivation.
Let $\mathcal{A}$ be a prime $\ast$-algebra. In this paper, we suppose that $Φ:\mathcal{A}\to\mathcal{A}$ satisfies $$Φ(A\diamond B)=Φ(A)\diamond B+A\diamondΦ(B)$$ where $A\diamond B = A^{*}B - B^{*}A$ for all $A,B\in\mathcal{A}$ .We will show that if $Φ(α\frac{I}{2})$ is self-adjoint for $α\in\{1,i\}$ then $Φ$ is additive $\ast$-derivation.
In this paper, we prove some Hermite-Hadamard type inequalities for operator geometrically convex functions for non-commutative operators. Keywords: Operator geometrically convex function, Hermite-Hadamard inequality.
A special subcomplex of the singular chain complex for a topological space, historically called oriented singular chain complex is used here with the new name "alternative" singular chain complex. It was already known that this subcomplex and so its dual complex are chain homotopy equivalent to singular chains and cochains respectively and thus have the same homology and cohomology. Here, in addition to revisiting some aspects of this subcomplex, it is shown that alternative singular cochains (dual of alternative singular chains) with coefficients in rational or real numbers are indeed summands of singular cochains through a natural splitting. It is shown that this natural splitting also hold for cohomologies: At any order, the singular cohomology splits into the alternative cohomology and another summand which is zero if the considered topological space is compact. Also in this case similar to the wedge product for differential forms, a modified cup product can be defined with the same algebraic properties as in the wedge product in differential forms. This provides an idea to investigate some topological and structure-free aspects of nonlinear global differential equations on manifolds..
In this paper, we introduce the concept of operator arithmetic-geometrically convex functions for positive linear operators and prove some Hermite-Hadamard type inequalities for these functions. As applications, we obtain trace inequalities for operators which give some refinements of previous results. Moreover, some unitarily invariant norm inequalities are established.
In this paper, we extend the definition of hyperinner product defined on weak hypervector spaces with a hyperoperation scalar product to weak hypervector spaces with the hyperoperations sum and scalar products.
Let $x=a+ib$ be a complex number, so we have the following inequality $$(1/\sqrt{2})|a+b|\leq |x|\leq |a|+|b|$$ We give an operator version of above inequality. Also we obtain some results for normal operators.
Let $A$ be a $C^{*}$ algebra and $T: A\rightarrow A$ be a linear map which satisfies the functional equation $\begin{cases}T(x)T(y)=T^{2}(xy)\\T(x^{*})=T(x)^{*} \end{cases}$ We prove that under each of the following conditions, $T$ must be the trivial map $T(x)=λx$ for some $λ\in \mathbb{R}:$\\ \begin{enumerate} \item $A$ is a simple $C^{*}$-algebra. \item $A$ is unital with trivial center and has a faithful trace such that each zero-trace element lies in the closure of the span of commutator elements. \item $A=B(H)$ where H is a separable Hilbert space. \end{enumerate} For a given field $F$, we consider a similar functional equation $$\begin{cases}T(x)T(y)=T^{2}(xy)\\T(x^{tr})=T(x)^{tr} \end{cases}$$ where $T$ is a linear map on $M_{n}(F)$ and "tr" is the transpose operator. We prove that this functional equation has trivial solution for all $n\in \mathbb{N}$ if and only if $F$ is a formally real field.
In this paper, we introduce the concept of operator geometrically convex functions for positive linear operators and prove some Hermite-Hadamard type inequalities for these functions. As applications, we obtain trace inequalities for operators which give some refinements of previous results.
Let $\mathcal{A}$ be a factor von Neumann algebra and $ϕ$ be the $\ast$-Jordan derivation on $A$, that is, for every $A,B \in \mathcal{A}$, $ϕ(A\diamond_{1} B) = ϕ(A)\diamond_{1} B + A\diamond_{1}ϕ( B)$ where $A\diamond_{1} B = AB + BA^{\ast}$, then $ϕ$ is additive $\ast$-derivation.
Let $\mathcal{A}$ and $\mathcal{B}$ be two $C^{*}$-algebras such that $\mathcal{B}$ is prime. In this paper, we investigate the additivity of map $Φ$ from $\mathcal{A}$ onto $\mathcal{B}$ that are bijective unital and satisfies $$Φ(AP+ηPA^{*})=Φ(A)Φ(P)+ηΦ(P)Φ(A)^{*},$$ for all $A\in\mathcal{A}$ and $P\in\{P_{1},I_{\mathcal{A}}-P_{1}\}$ where $P_{1}$ is a nontrivial projection in $\mathcal{A}$. Let $η$ be a non-zero complex number such that $|η|\neq1$, then $Φ$ is additive. Moreover, if $η$ is rational then $Φ$ is $\ast$-additive.
Let $\mathcal{A}$ and $\mathcal{B}$ be two prime $C^{*}$-algebras. In this paper, we investigate the additivity of map $Φ$ from $\mathcal{A}$ onto $\mathcal{B}$ that are bijective unital and satisfies $$Φ(AP+λPA^{*})=Φ(A)Φ(P)+λΦ(P)Φ(A)^{*},$$ for all $A\in\mathcal{A}$ and $P\in\{P_{1},I_{\mathcal{A}}-P_{1}\}$ where $P_{1}$ is a nontrivial projection in $\mathcal{A}$ and $λ\in\{-1,+1\}$. Then, $Φ$ is $*$-additive.
Let X and Y be Banach spaces with dim X greater than 3. Let A and B be standard operator algebras on X and Y. We characterize the form of maps from A onto B such that completely preserve involution.
Using methods from the theory of commutative graded Banach algebras, we obtain a generalization of the two dimensional Borsuk-Ulam theorem as follows: Let $ϕ:S^{2} \rightarrow S^{2}$ be a homeomorphism of order n and $λ\neq 1$ be an nth root of the unity, then for every complex valued continuous function $f$ on $S^{2}$ the function $\sum_{i=0}^{n-1} λ^{i}f(ϕ^{i}(x))$ must be vanished at some point of $S^{2}$. We give a generalization in term of action of compact groups. We also discuss about some noncommutative versions of the Borsuk- Ulam theorem