Related papers: A note on ANR's

We study ideals $\mathcal{I}$ on $\mathbb{N}$ satisfying the following Baire-type property: if $X$ is a complete metric space and $\{X_{A} \colon A \in \mathcal{I} \}$ is a family of nowhere dense subsets of $X$ with $X_{A} \subset X_{B}$…

We introduce a new norm, called $N^{p}$-norm $(1\leq{p}<\infty)$ on a space $N^{p}(V,W)$ where $V$ and $W$ are abstract operator spaces. By proving some fundamental properties of the space $N^{p}(V,W)$, we also obtain that if $W$ is…

Let (E_0,E_1) and (H_0,H_1) be a pair of Banach spaces with dense and continuous embeddings E_1 into E_0, H_1 into H_0. For $\theta \in [0,1]$ denote by $B_\theta(0,R)$ the ball of radius R centered at zero in the interpolation spaces…

A topological space $X$ is Baire if the intersection of any sequence of open dense subsets of $X$ is dense in $X$. Let $C_p(X,[0,1])$ denote the space of all continuous $[0,1]$-valued functions on a Tychonoff space $X$ with the topology of…

We show that a plane continuum X is indecomposable iff X has a sequence (U_n) of not necessarily distinct complementary domains satisfying what we call the double-pass condition: If one draws an open arc A_n in each U_n whose ends limit…

We show that, as conjectured by Adrien Douady back in 1972, every complete metric space is homeomorphic (moreover, isometric) to the locus of zeros of an analytic map between two Banach spaces. As a corollary, a paracompact topological…

Let $\|\cdot\|$ be a norm on $\mathbb{R}^N$ and let $M$ be a closed $C^1$-submanifold of $\mathbb{R}^N$. Consider the pointed metric space $(M,d)$, where $d$ is the metric given by $d(x,y)=\|x-y\|$, $x,y\in M$. Then the Lipschitz-free space…

We isolate conditions on the relative size of sets of natural numbers $A,B$ that guarantee a nonempty intersection $\Delta(A)\cap\Delta(B)\ne\emptyset$ of the corresponding sets of distances. Such conditions apply to a large class of zero…

Let $(X_1,d_1),\dots, (X_N,d_N)$ be metric spaces, where $d_i: X_i \times X_i \rightarrow [0,1]$ is a distance function for $i=1,\dots,N$. Let $\mathcal{X}$ denote the set theoretic product $X_1\times \cdots \times X_N$. Let $\mathcal{G} =…

A topological space $Y$ is said to have (AEEP) if the following condition is fulfilled. Whenever $(X,\mathfrak{M})$ is a measurable space and $f, g: X \to Y$ are two measurable functions, then the set $\Delta(f,g) = \{x \in X:\ f(x) =…

We introduce the notion of strongly orthogonal set relative to an element in the sense of Birkhoff-James in a normed linear space to find a necessary and sufficient condition for an element $ x $ of the unit sphere $ S_{X}$ to be an exposed…

In this paper, we provide a proof that functions belonging to Besov spaces $B^{r}_{q,\infty}(\mathbb{R}^N,\mathbb{R}^d)$, $q\in [1,\infty)$, $r\in(0,1)$, satisfy the following formula under a certain condition: \begin{equation}…

We prove that for every integer $k$, there exists $\varepsilon > 0$ such that for every n-vertex graph $G$ with no pivot-minor isomorphic to $C_k$, there exist disjoint sets $A,B \subseteq V(G)$ such that $|A|,|B| \geq \varepsilon n$, and…

Using the notion of formal ball, we present a few new results in the theory of quasi-metric spaces. With no specific order: every continuous Yoneda-complete quasi-metric space is sober and convergence Choquet-complete hence Baire in its…

In this paper, we define the angle of a bounded linear operator $A$ along an unbounded path emanating from the origin and use it to characterize range-kernel complementarity. In particular we show that if $0$ faces the unbounded component…

Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \colon \mathcal{M}(X) \to \R$ by $I(mu) = \int_X \int_X d(x,y) d\mu(x) d\mu(y)$, and set $M(X) = \sup…

Let $(X,d)$ be an $n$-dimensional Alexandrov space whose Hausdorff measure $\mathcal{H}^n$ satisfies a condition giving the metric measure space $(X,d,\mathcal{H}^n)$ a notion of having nonnegative Ricci curvature. We examine the influence…

Given a point set $S=\{s_1,\ldots , s_n\}$ in the unit square $U=[0,1]^2$, an anchored square packing is a set of $n$ interior-disjoint empty squares in $U$ such that $s_i$ is a corner of the $i$th square. The reach $R(S)$ of $S$ is the set…

The configuration manifold $M$ of a mechanical system consisting of two unconstrained rigid bodies in $\mathbb{R}^n$, $n\geq 1$, is a manifold with boundary (typically with singularities.) A complete description of the system requires…

Let X be a metric space with metric d and T,S be two commutative measure-preserving maps of X. In this paper we obtain numerical results about multiple recurrence of almost every point of this dynamical system. On other words we study the…