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For two not necessarily commutative topological groups G and T, let H(G,T) denote the space of all continuous homomorphisms from G to T with the compact-open topology. We prove that if G is metrizable and T is compact then H(G,T) is a…

Let $X$ be a compact real algebraic set of dimension $n$. We prove that every Euclidean continuous map from $X$ into the unit $n$-sphere can be approximated by regulous map. This strengthens and generalizes previously known results.

Let $(M^{n},g)$ be a compact Riemannian manifold with $Ric\geq-(n-1) $. It is well known that the bottom of spectrum $\lambda_{0}$ of its unverversal covering satisfies $\lambda_{0}\leq(n-1) ^{2}/4 $. We prove that equality holds iff $M$ is…

Suppose that the inverse image of the zero vector by a continuous map $f:{\mathbb R}^n\to{\mathbb R}^q$ has an isolated point $P$. There is a local obstruction to removing this isolated zero by a small perturbation, generalizing the notion…

Let $M$ be a complete Riemannian manifold. Suppose $M$ contains a bounded, concave, connected open set $U$ with $C^0$ boundary and $M\setminus U$ is connected. We assume that either the relative homotopy set $\pi_1(M,M\setminus U)=0$ or the…

We consider locally symmetric manifolds with a fixed universal covering, and construct for each such manifold M a simplicial complex R whose size is proportional to the volume of M. When M is non-compact, R is homotopically equivalent to M,…

We construct a new class of positive indecomposable maps in the algebra of `d x d' complex matrices. These maps are characterized by the `weakest' positivity property and for this reason they are called atomic. This class provides a new…

Machine Learning (ML) potentials such as Gaussian Approximation Potential (GAP) have demonstrated impressive capabilities in mapping structure to properties across diverse systems. Here, we introduce a GAP model for low-dimensional Ni…

Nearest neighbor methods are a popular class of nonparametric estimators with several desirable properties, such as adaptivity to different distance scales in different regions of space. Prior work on convergence rates for nearest neighbor…

In the paper we investigate the degree and the homotopy theory of Orlicz-Sobolev mappings $W^{1,P}(M,N)$ between manifolds, where the Young function $P$ satisfies a divergence condition and forms a slightly larger space than $W^{1,n}$,…

It is shown that each continuous transformation $h$ from Euclidean $m$-space ($m>1$) into Euclidean $n$-space that preserves the equality of distances (that is, fulfils the implication $|x-y|=|z-w|\Rightarrow|h(x)-h(y)|=|h(z)-h(w)|$) is a…

The main results of this paper show that various coarse (`large scale') geometric properties are closely related. In particular, we show that property A implies the operator norm localisation property, and thus that norms of operators…

In this paper, we establish the equivalence between the Haj{\l}asz-Sobolev spaces or classical Triebel-Lizorkin spaces and a class of grand Triebel-Lizorkin spaces on Euclidean spaces and also on metric spaces that are both doubling and…

Given a K\"ahler manifold $(Z,J,\omega)$ and a compact real submanifold $M\subset Z$, we study the properties of the gradient map associated with the action of a noncompact real reductive Lie group ${\rm G}$ on the space of probability…

Let Map(K,X) denote the space of pointed continuous maps from a finite cell complex K to a space X. Let E_* be a generalized homology theory. We use Goodwillie calculus methods to prove that under suitable conditions on K and X, Map(K, X)…

We study measurable spaces equipped with a $\sigma$-ideal of negligible sets. We find conditions under which they admit a localizable locally determined version -- a kind of fiber space that describes locally their directions -- defined by…

Let $f:G_{n,k}\longrightarrow G_{m,l}$ be any continuous map between any two distinct complex Grassmann manifolds of the same dimension where the target is not the complex projective space. We show that, for any given $k,l$, the degree of…

Thom-Pontrjagin constructions are used to give a computable necessary and sufficient condition when a homomorphism $\phi : H^n(L;Z)\to H^n(M;Z)$ can be realized by a map $f:M\to L$ of degree $k$ for closed $(n-1)$-connected $2n$-manifolds…

A homogeneous Riemannian space $(M= G/H,g)$ is called a geodesic orbit space (shortly, GO-space) if any geodesic is an orbit of one-parameter subgroup of the isometry group $G$. We study the structure of compact GO-spaces and give some…

Let $f\colon X\to Y$ be a $\sigma$-perfect $k$-dimensional surjective map of metrizable spaces such that $\dim Y\leq m$. It is shown that, for every positive integer $p\geq 1$ there exists a dense $G_{\delta}$-subset ${\mathcal H}(k,m,p)$…