Related papers: Relatively weakly open convex combinations of slic…

Given a finite point set $P\subset\mathbb{R}^d$, we call a multiset $A$ a one-sided weak $\varepsilon$-approximant for $P$ (with respect to convex sets), if $|P\cap C|/|P|-|A\cap C|/|A|\leq\varepsilon$ for every convex set $C$. We show…

We show that if $A$ is a closed subset of the Heisenberg group whose vertical projections are nowhere dense, then the complement of $A$ is quasiconvex. In particular, closed sets which are null sets for the cc-Hausdorff $3$-measure have…

We show that the twisted traces of CM values of weak Maass forms of weight 0 are Fourier coefficients of vector valued weak Maass forms of weight 3/2. These results generalize work by Zagier on traces of singular moduli. We utilize a…

Let $X$ and $Y$ be separable Banach spaces. Suppose $Y$ either has a shrinking basis or $Y$ is isomorphic to $C(2^\mathbb{N})$ and $A$ is a subset of weakly compact operators from $X$ to $Y$ which is analytic in the strong operator…

Given a topological space $X$, we study the structure of $\infty$-convex subsets in the space $SC_p(X)$ of scatteredly continuous functions on $X$. Our main result says that for a topological space $X$ with countable strong fan tightness,…

We initiate in this article the study of weakly exact structures, a generalization of Quillen exact structures. We introduce weak counterparts of one-sided exact structures and show that a left and a right weakly exact structure generate a…

In this paper, we prove some divisibility results for the Fourier coefficients of reduced modular forms of sign vectors. More precisely, we generalize a divisibility result of Siegel on constant terms when the weight is non-positive, which…

A countable, bounded degree graph is almost finite if it has a tiling with isomorphic copies of finitely many F\o lner sets, and we call it strongly almost finite, if the tiling can be randomized so that the probability that a vertex is on…

We introduce a new technique for the study of the local extension property (LEP) for boolean algebras and we use it to show that the clopen algebra of every compact Hausdorff space $K$ of finite height has LEP. This implies, under…

We prove that a Hausdorff space $X$ is very $\mathrm I$-favorable if and only if $X$ is the almost limit space of a $\sigma$-complete inverse system consisting of (not necessarily Hausdorff) second countable spaces and surjective d-open…

In these (not-completed) notes, we study the Hartogs extension phenomenon for holomorphic sections of holomorphic vector bundles over complex analytic varieties. Namely, we study properties of the Hartogs extension phenomenon with respect…

Several new characterizations of the Gelfand-Phillips property are given. We define a strong version of the Gelfand-Phillips property and prove that a Banach space has this stronger property iff it embeds into $c_0$. For an infinite compact…

The slice condition and the more general weak slice conditions are geometric conditions on Euclidean space domains which have evolved over the last several years as a tool in various areas of analysis. This paper examines some of their…

A combinatorial principle CECA is formulated and its equivalence with GCH+ certain weakenings of Box_lambda for singular lambda is proved. CECA is used to show that certain ``almost point- < tau'' families can be refined to point- < tau…

A Banach space $X$ is said to have Efremov's property ($\mathcal{E}$) if every element of the weak$^*$-closure of a convex bounded set $C \subseteq X^*$ is the weak$^*$-limit of a sequence in $C$. By assuming the Continuum Hypothesis, we…

We investigate the existence of compact Hausdorff spaces $X$ that are minimum with respect to $cX=K$ for some fixed covering operator $c$ and compact Hausdorff space $K$ with $cK=K$. Then, using the Yosida representation theorem, we show…

In this article, we consider minimal $L^2$ integrals on the sublevel sets of plurisubharmonic functions on weakly pseudoconvex K\"ahler manifolds with Lebesgue measurable gain related to modules at boundary points of the sublevel sets, and…

The Brouwer fixed point theorem states that the disk $D^n$ has the fixed point property. More generally, by the Lefschetz fixed point theorem any compact ANR with trivial rational homology has the fixed point property. In this note we prove…

For $0\leqslant \xi\leqslant \omega_1$, we define the notion of $\xi$-weakly precompact and $\xi$-weakly compact sets in Banach spaces and prove that a set is $\xi$-weakly precompact if and only if its weak closure is $\xi$-weakly compact.…

A Banach space $E$ has the Grothendieck property if every (linear bounded) operator from $E$ into $c_0$ is weakly compact. It is proved that, for an integer $k>1$, every $k$-homogeneous polynomial from $E$ into $c_0$ is weakly compact if…