فرض کنید ‎$mathcal{B(X)}$‎ جبر شامل تمام عملگرهای خطی کران‌دار روی فضای باناخ ‎$mathcal{X}$‎ و ‎$phi:mathcal{B(X)}longrightarrow mathcal{B(X)}$‎ یک نگاشت جمعی دوسویی باشد که جفت عملگری باناخ را از دو طرف حفظ می کند. در این مقاله، نشان می دهیم که به ازای هر ‎$A in mathcal{B(X)}$‎ و ‎$x in mathcal{X}$‎، اسکالرهای ‎$alpha‎ , ...

In this article, we develop a notion of Quillen bifibration which combines the two notions of Grothendieck bifibration and of Quillen model structure. In particular, given a bifibration $p:\mathcal E\to\mathcal B$, we describe when a family of model structures on the fibers $\mathcal E_A$ and on the basis category $\mathcal B$ combines into a model structure on the total category $\mathcal E$, ...

A compact metric space $(X, \rho)$ is given. Let $\mu$ be a Borel measure on $X$. By $r$-cluster we mean a measurable subset of $X$ with diameter at most $r$. A family of $k$ $2r$-clusters is called a $r$-cluster structure of order $k$ if any two clusters from the family are separated by a distance at least $r$. By measure of a cluster structure we mean a sum of clusters measures from the clust...

We further the classification of rational surface singularities. Suppose $(S, \mathfrak{n}, \mathcal{k})$ is a strictly Henselian regular local ring mixed characteristic $(0, p > 5)$. classify functions $f$ for which $S/(f)$ has an isolated singularity at maximal ideal $\mathfrak{n}$. The such are used to show that if $(R, \mathfrak{m}, excellent, Henselian, Gorenstein dimension $2$ and 5)$, th...

In this paper, we will study the theory of cyclic homology for regular multiplier Hopf algebras. We associate a cyclic module to a triple $(mathcal{R},mathcal{H},mathcal{X})$ consisting of a regular multiplier Hopf algebra $mathcal{H}$, a left $mathcal{H}$-comodule algebra $mathcal{R}$, and a unital left $mathcal{H}$-module $mathcal{X}$ which is also a unital algebra. First, we construct a para...

This note is devoted to the study of the links between the Hilbert function of a subscheme X of the projective space IPn and its geometric properties. We will assume X to be arithmetically Cohen-Macaulay (ACM). This allows us to characterize the Hilbert function φX of X by an increasing sequence of d integers (m0, . . . ,md−1), called the (absolute) characteristic sequence of X, d being the deg...

Let X be a smooth projective hypersurface of dimension at least three. We show that every automorphism the Hilbert square X[2] is induced by some X.