相关论文: Density of bounded maps in Sobolev spaces into com…

We formulate and discuss a necessary and sufficient condition for polynomials to be dense in a space of continuous functions on the real line, with respect to Bernstein's weighted uniform norm. Equivalently, for a positive finite measure…

The existence of a strict deformation quantization of $X_k=S(M_k({\mathbb{C}}))$, the state space of the $k\times k$ matrices $M_k({\mathbb{C}})$ which is canonically a compact Poisson manifold (with stratified boundary) has recently been…

In this paper, we will count the number of cusps of complete Riemannian manifolds $M$ with finite volume. When $M$ is a complete smooth metric measure spaces, we show that the number of cusps in bounded by the volume $V$ of $M$ if some…

We prove density results for neural-network classes on compact sets \(K\subset X\), where \(X\) is a topological vector space whose continuous dual \(X^*\) separates points. Let \(\Psi:\mathbb R\to\mathbb R\) be a continuous squashing…

Newton-Sobolev spaces, as presented by N. Shanmugalingam, describe a way to extend Sobolev spaces to the metric setting via upper gradients, for metric spaces with `sufficient' paths of finite length. Sometimes, as is the case of parabolic…

The quotient set, or ratio set, of a set of integers $A$ is defined as $R(A) := \left\{a/b : a,b \in A,\; b \neq 0\right\}$. We consider the case in which $A$ is the image of $\mathbb{Z}^+$ under a polynomial $f \in \mathbb{Z}[X]$, and we…

We continue our study of ends of non-compact manifolds, with a focus on the inward tameness condition. For manifolds with compact boundary, inward tameness, has significant implications. For example, such manifolds have stable homology at…

A well-known application of the dependent random choice asserts that any $n$-vertex graph $G$ with positive edge density contains a `rich' vertex subset $U$ of size $n^{1-o(1)}$ such that every pair of vertices in $U$ has at least…

The purpose of this paper is to present, for all $n\ge 3$, very simple examples of continuous maps $f:M^{n-1} \to M^{n}$ from closed $(n-1)$-manifolds $M^{n-1}$ into closed $n$-manifold $M^n$ such that even though the singular set $S(f)$ of…

Let $N\subset GL(n,R)$ be the group of upper triangular matrices with $1$s on the diagonal, equipped with the standard Carnot group structure. We show that quasiconformal homeomorphisms between open subsets of $N$, and more generally…

We show that every Sobolev function in $W^{1,p}_{\textrm{loc}}(U)$ on a $p$-quasiopen set $U \subset {\bf R}^n$ with a.e.-vanishing $p$-fine gradient is a.e.-constant if and only if $U$ is $p$-quasiconnected. To prove this we use the theory…

Given a pointed metric space $M$, we study when there exist $n$-dimensional linear subspaces of $\operatorname{Lip}_0(M)$ consisting of strongly norm-attaining Lipschitz functionals, for $n\in\mathbb{N}$. We show that this is always the…

We study graph-theoretic properties of random $0/1$ polytopes. Specifically, let $Q_p^n \subseteq \{0,1\}^n$ be a random subset where each point is included independently with probability $p$, and consider the graph $G_p$ of the polytope…

We show that smooth maps are $C^1$-dense among $C^1$ volume preserving maps.

We show that for each $n \geq 2$, the systoles of closed hyperbolic $n$-manifolds form a dense subset of $(0, +\infty)$. We also show that for any $n\geq 2$ and any Salem number $\lambda$, there is a closed arithmetic hyperbolic…

We study global regularity properties of Sobolev homeomorphisms on $n$-dimensional Riemannian manifolds under the assumption of $p$-integrability of its first weak derivatives in degree $p\geq n-1$. We prove that inverse homeomorphisms have…

Every open manifold L of dimension greater than one has complete Riemannian metrics g with bounded geometry such that (L,g) is not quasi-isometric to a leaf of a codimension one foliation of a closed manifold. Hence no conditions on the…

We investigate the validity and the failure of modular density of smooth maps on compact manifolds.

The present paper is devoted to the study of the well-posedness issue for the density-dependent Euler equations in the whole space. We establish local-in-time results for the Cauchy problem pertaining to data in the Besov spaces embedded in…

In this paper, we introduce a new energy density function $\mathscr Y$ on the projective bundle $\mathbb{P}(T_M)\>M$ for a smooth map $f:(M,h)\>(N,g)$ between Riemannian manifolds $$\mathscr Y=g_{ij}f^i_\alpha f^j_\beta \frac{W^\alpha…