Let $f\_i\colon X \rightarrow \mathbf{C}$ , for $i$ integer between $1$ and $p$ be analytic functions defined on a complex variety $X$. us consider $\mathcal{D}\_X$ the ring of linear differential operators $\mathcal{D}\_X \[s\_1, \ldots, s\_p] = \mathbf{C}X \otimes{\mathbf{C}} \mathcal{D}\_X$. $m$ section holonomic -module. We denote $\mathcal{B}(m, x\_0, f\_1, f\_p)$ ideal $\mathbf{C} s\_p]$ ...

We establish the following Hilbert-space analogue of Gleason-Kahane-\.Zelazko theorem. If $\mathcal{H}$ is a reproducing kernel Hilbert space with normalized complete Pick kernel, and if $\Lambda$ linear functional on such that $\Lambda(1)=1$ $\Lambda(f)\ne0$ for all cyclic functions $f\in\mathcal{H}$, then multiplicative, in sense $\Lambda(fg)=\Lambda(f)\Lambda(g)$ $f,g\in\mathcal{H}$ $fg\in\m...

We deal with the question of Masayoshi Hata: is every Peano continuum a topological fractal? A compact space $X$ fractal if there exists $\mathcal{F}$ finite family self-maps on such that $X=\bigcup_{f\in\mathcal{F}}f(X)$ and for open cover $\mathcal{U}$ $n\in\mathbb{N}$ all maps $f_1,\dots,f_n\in\mathcal{F}$ set $f_1\circ\dots\circ f_n(X)$ contained in some $U\in\mathcal{U}$. In paper we prese...

Let $(X, \tau)$ and $(Y, \sigma)$ be topological spaces in which no separation axioms are assumed, unless explicitly stated if $\mathcal{I}$ is an ideal on $X$.Given a multifunction $F\colon (X, \tau)\rightarrow (Y, \sigma)$, $\alpha,\beta$ operators \tau)$, $\theta,\delta$ proper $X$. We introduce study upper lower $(\alpha, \beta,\theta,\delta,\mathcal{I})$-continuous multifunctions.A said to...

This paper presents a formula expressing Macaulay constants of numerical polynomial through its minimizing coefficients. From this, we obtain that Kolchin dimension polynomials do not decrease. For the minimal differential $$ {\omega}_{\mathcal{G}/\mathcal{F}} (this concept was introduced by W. Sitt) will prove criterion for to be equal. In this case, as our example shows, there are no bounds f...

Given a countable group $G$ splitting as free product $G=G_1\ast\dots\ast G_k\ast F_N$, we establish classification results for subgroups of the $Out(G,\mathcal{F})$ all outer automorphisms that preserve conjugacy classes each $G_i$. We show every finitely generated subgroup $H\subseteq Out(G,\mathcal{F})$ either contains relatively fully irreducible automorphism, or else it virtually preserves...

Let $X$ be a set and $\mathcal{T}_X$ the full transformation semigroup on $X$. For partition $\mathcal{P}$ of $X$, we consider semigroups $T(X, \mathcal{P}) = \{f\in \mathcal{T}_X| (\forall X_i\in (\exists X_j \in \mathcal{P})\;X_i f \subseteq X_j\}$, $\Sigma(X, T(X, \mathcal{P})|(\forall X_i \mathcal{P})\; Xf \cap \neq \emptyset\}$, $\Gamma(X, \mathcal{T}_X|(\forall \mathcal{P})(\exists X_j\in...