Let $X$ be a nonempty set. Denote by $\mathcal{F}^n_k$ the class of associative operations $F\colon X^n\to X$ satisfying condition $F(x_1,\ldots,x_n)\in\{x_1,\ldots,x_n\}$ whenever at least $k$ elements $x_1,\ldots,x_n$ are equal to each other. The $\mathcal{F}^n_1$ said quasitrivial and those $\mathcal{F}^n_n$ idempotent. We show that $\mathcal{F}^n_1=\cdots =\mathcal{F}^n_{n-2}\subseteq\mathc...

We propose a conjectural framework for computing Gorenstein measures and stringy Hodge numbers in terms of motivic integration over arcs smooth Artin stacks, we verify this the case fantastacks, which are certain toric stacks that provide (non-separated) resolutions singularities varieties. Specifically, let $\mathcal{X}$ be stack admitting good moduli space $\pi: \mathcal{X} \to X$, assume $X$...

In the present paper, we consider generalized equation $0\in f(x)+g(x)+\mathcal F(x)$, where $f:\mathcal X\to \mathcal Y$ is Fr\'{e}chet differentiable on a neighborhood $\Omega$ of point $\bar{x}$ in $\mathcal X$, $g:\mathcal at and linear as well F$ set-valued mapping with closed graph acting between two Banach spaces X$ Y$. We study above help extended Newton-type method, introduced [ M. Z. ...

In this article the notions of quasi mono (epi) as a generalization (epi), (quasi weakly hereditary) general closure operator $mathbf{C}$ on category $mathcal{X}$ with respect to class $mathcal{M}$ morphisms, and factorization structures in are introduced. It is shown that under certain conditions, if $(mathcal{E}, mathcal{M})$ structure $mathcal{X}$, then has right $mathcal{M}$-factorization l...

In this work we study the monodromy group of covers $\varphi \circ \psi$ curves \linebreak $\mathcal{Y}\xrightarrow {\quad {\psi}} \mathcal{X} \xrightarrow \varphi} \mathbb{P}^{1}$, where $\psi$ is a $q$-fold cyclic \'etale cover and $\varphi$ totally ramified $p$-fold cover, with $p$ $q$ different prime numbers odd. We show that Galois $\mathcal{G}$ closure $\mathcal{Z}$ form $ \mathcal{G} = \...

We prove regularity estimates in weighted Sobolev spaces for the $L^2$-eigenfunctions of Schr\"odinger type operators whose potentials have inverse square singularities and uniform radial limits at infinity. In particular, usual $N$-body Hamiltonians with Coulomb-type singular are covered by our result: that case, weight is $\delta_{\mathcal{F}}(x) := \min \{ d(x, \bigcup \mathcal{F}), 1\}$, wh...

For a random set $\mathcal{S} \subset U(d)$ of quantum gates we provide bounds on the probability that $\mathcal{S}$ forms $\delta$-approximate $t$-design. In particular have found for drawn from an exact $t$-design it satisfies inequality $\mathbb{P}\left(\delta \geq x \right)\leq 2D_t \, \frac{e^{-|\mathcal{S}| \mathrm{arctanh}(x)}}{(1-x^2)^{|\mathcal{S}|/2}} = O\left( \left( \frac{e^{-x^2}}{...

In this paper the continuity of set valued map $p\rightarrow B_{\Omega,\mathcal{X},p}(r),$ $p\in (1,+\infty),$ is proved where $B_{\Omega,\mathcal{X},p}(r)$ closed ball space $L_{p}\left(\Omega,\Sigma,\mu; \mathcal{X}\right)$ centered at origin with radius $r,$ $\left(\Omega,\Sigma,\mu\right)$ a finite and positive measure space, $\mathcal{X}$ separable Banach space. An application to input-out...

‎let $g$ be a graph without an isolated vertex‎, ‎the normalized laplacian matrix $tilde{mathcal{l}}(g)$‎‎is defined as $tilde{mathcal{l}}(g)=mathcal{d}^{-frac{1}{2}}mathcal{l}(g) mathcal{d}^{-frac{1}{2}}$‎, where ‎$‎mathcal{‎d}‎$ ‎is a‎ diagonal matrix whose entries are degree of ‎vertices ‎‎of ‎$‎g‎$‎‎. ‎the eigenvalues of‎‎$tilde{mathcal{l}}(g)$ are ‎called ‎ ‎ as ‎the ‎normalized laplacian ...

Generalizing Christensen's notion of a Haar-null set and Darji's Haar-meager set, we introduce study the Haar-$\mathcal I$ in Polish group. Here $\mathcal is an ideal subsets some compact metrizable space $K$. A Borel subset $B\subset X$ group $X$ called if there exists continuous map $f:K\to such that $f^{-1}(B+x)\in\mathcal for all $x\in X$. Moreover, $B$ generically witness functions $\{f\in...