We construct a Spanier-Whitehead type duality functor relating finite $\mathcal{C}$-spectra to $\mathcal{C}^{\mathrm{op}}$-spectra and prove that every $\mathcal{C}$-homology theory is given by taking the homotopy groups of balanced smash product with fixed $\mathcal{C}^{\mathrm{op}}$-spectrum. use this Chern characters for certain rational theories.

‎Let $mathcal{A}$ be a unital Banach algebra‎, ‎$mathcal{M}$ be a left $mathcal{A}$-module‎, ‎and $W$ in $mathcal{Z}(mathcal{A})$ be a left separating point of $mathcal{M}$‎. ‎We show that if $mathcal{M}$ is a unital left $mathcal{A}$-module and $delta$ is a linear mapping from $mathcal{A}$ into $mathcal{M}$‎, ‎then the following four conditions are equivalent‎: ‎(i) $delta$ is a Jordan left de...

Let $$M$$ be a closed manifold and $$L$$ an exact magnetic Lagrangian. In this paper we prove that there exists residual set $$\mathcal{G}$$ of $$H^{1}\left(M;\mathbb{R}\right)$$ such the property $${\widetilde{\mathcal{M}}}\left(c\right)={\widetilde{\mathcal{A}}}\left(c\right)={\widetilde{\mathcal{N}}}\left(c\right),\forall c\in\mathcal{G},$$ with $${\widetilde{\mathcal{M}}}\left(c\right)$$ su...

Given any modular category $\mathcal{C}$ over an algebraically closed field $k$, we extract a sequence $(M_g)_{g\geq 0}$ of $\mathcal{C}$-bimodules. We show that the Hochschild chain complex $CH(\mathcal{C};M_g)$ with coefficients in $M_g$ carries canonical homotopy coherent projective action mapping class group surface genus $g+1$. The ordinary corresponds to $CH(\mathcal{C};M_0)$. This result...

let $varphi(z)=z^m, z in mathbb{u}$, for some positive integer $m$, and $c_varphi$ be the composition operator on the bergman space $mathcal{a}^2$ induced by $varphi$. in this article, we completely determine the point spectrum, spectrum, essential spectrum, and essential norm of the operators $c^*_varphi c_varphi, c_varphi c^*_varphi$ as well as self-commutator and anti-self-commutators of $c_...

We investigate abelian quotients arising from extriangulated categories via morphism categories, which is a unified treatment for both exact and triangulated categories. Let $(\mathcal {C},\mathbb {E},\mathfrak {s})$ be an category with enough projectives $\mathcal {P}$ ${\mathscr{M}}$ full subcategory of {C}$ containing . show that certain quotient $\mathfrak {s}\textup {-def}({\mathscr{M}})$ ...

in this work‎, ‎an iterative method based on a matrix form of lsqr algorithm is constructed for solving the linear operator equation $mathcal{a}(x)=b$‎ ‎and the minimum frobenius norm residual problem $||mathcal{a}(x)-b||_f$‎ ‎where $xin mathcal{s}:={xin textsf{r}^{ntimes n}~|~x=mathcal{g}(x)}$‎, ‎$mathcal{f}$ is the linear operator from $textsf{r}^{ntimes n}$ onto $textsf{r}^{rtimes s}$‎, ‎$ma...

Let $mathcal{A}$ be a Banach algebra and $mathcal{M}$ be a Banach $mathcal{A}$-bimodule. We say that a linear mapping $delta:mathcal{A} rightarrow mathcal{M}$ is a generalized $sigma$-derivation whenever there exists a $sigma$-derivation $d:mathcal{A} rightarrow mathcal{M}$ such that $delta(ab) = delta(a)sigma(b) + sigma(a)d(b)$, for all $a,b in mathcal{A}$. Giving some facts concerning general...