Given $$(X,\omega )$$ compact Kähler manifold and $$\psi \in \mathcal {M}^{+}\subset PSH(X,\omega a model type envelope with non-zero mass, i.e. fixed potential determining singularity such that $$\int _{X}(\omega +dd^{c}\psi )^{n}>0$$ , we prove the -$$ relative finite energy class $$\mathcal {E}^{1}(X,\omega ,\psi becomes complete metric space if endowed distance d which generalizes well-know...

We consider Galton-Watson branching processes with countable typeset $\mathcal{X}$. study the vectors ${\bf q}(A)=(q_x(A))_{x\in\mathcal{X}}$ recording conditional probabilities of extinction in subsets types $A\subseteq \mathcal{X}$, given that type initial individual is $x$. first investigate location q}(A)$ set fixed points progeny generating vector and prove $q_x(\{x\})$ larger than or equa...

In this note, we construct a new set \boldsymbol{S} of primitive sets such that for any real number x\geq 60 get: \begin{equation*} \sum\limits_{a\in \mathcal{A}}\frac{1}{a(\log a+x)}>\sum\limits_{p\in \mathcal{P}}\frac{1}{p(\log p+x)},\text{ }\mathcal{A\in }{\boldsymbol{S}}, \end{equation*} where \mathcal{P} denotes the prime numbers.

Consider the optimal subspace expansion problem for matrix eigenvalue $Ax=\lambda x$: Which vector $w$ in current $\mathcal{V}$, after multiplied by $A$, provides an approximating a desired eigenvector $x$ sense that has smallest angle with expanded $\mathcal{V}_w=\mathcal{V}+{span}\{Aw\}$, i.e., $w_{opt}=\arg\max_{w\in\mathcal{V}}\cos\angle(\mathcal{V}_w,x)$? This is important as many iterativ...

Let $\mathcal A$ be a unital $C^*$-algebra. We consider Jordan $*$-homomorphisms on $C(X, \mathcal A)$ and Lip$(X,\mathcal A)$. More precisely, for any $C^*$-algebra A$, we prove that every $*$-homomorph

A nonsingular rational curve $C$ in a complex manifold $X$ whose normal bundle is isomorphic to $\mathcal{O}_{\mathbb{P}^1}(1)^{\oplus p} \oplus \mathcal{O}_{\mathbb{P}^1}^{\oplus q}$ for some nonnegative integers $p$ and $q$ called an unbendable on $X$. Associated with it the variety of minimal tangents (VMRT) at point $x \in C$, which germ submanifolds $\mathcal{C}^C_{x} \subset \mathbb{P} T_...

Abstract Let G be a countably infinite discrete amenable group. It should noted that -system $(X,G)$ naturally induces $(\mathcal {M}(X),G)$ , where $\mathcal {M}(X)$ denotes the space of Borel probability measures on compact metric X endowed with weak*-topology. A factor map $\pi : (X,G)\to (Y,G)$ between two -systems $\widetilde {\pi }:(\mathcal {M}(X),G)\to (\mathcal {M}(Y),G)$ . turns out }...