Related papers: Uniform approximation problems of expanding Markov…

Let \((M^n,g)\) be a smooth closed Riemannian manifold of dimension \(n \ge 5\) with positive Yamabe invariant and semi-positive \(Q\)-curvature. We establish a precompactness result in the \(C^{\alpha}\)-H\"older topologie on the space of…

The universal approximation theorem is generalised to uniform convergence on the (noncompact) input space $\mathbb{R}^n$. All continuous functions that vanish at infinity can be uniformly approximated by neural networks with one hidden…

A simple \(P_\lambda\)-point on a regular cardinal \(\kappa\) is a uniform ultrafilter on \(\kappa\) with a mod-bounded decreasing generating sequence of length \(\lambda\). We prove that if there is a simple $P_\lambda$-point ultrafilter…

Let $(X, d)$ be a compact metric space, and let $Q \subset X$ be countable. Given functions $R: Q \to \mathbb{R}^+$ and $\phi: \mathbb{R}^+ \to \mathbb{R}^+$, we consider the set $E(Q, R, \phi)$ of points $x \in X$ that ``hit'' the…

The electromagnetic response of topological insulators and superconductors is governed by a modified set of Maxwell equations that derive from a topological Chern-Simons (CS) term in the effective Lagrangian with coupling constant $\kappa$.…

We make use of the fact that a two-sided whole-plane Schramm-Loewner evolution (SLE$_\kappa$) curve $\gamma$ for $\kappa\in(0,8)$ from $\infty$ to $\infty$ through $0$ may be parametrized by its $d$-dimensional Minkowski content, where…

We consider random iteration of exponential entire functions, i.e. of the form ${\mathbb C}\ni z\mapsto f_\lambda(z):=\lambda e^z\in\mathbb C$, $\lambda\in{\mathbb C}\setminus \{0\}$. Assuming that $\lambda$ is in a bounded closed interval…

In this paper we study a p harmonic measure, associated with a positive p harmonic function \hat{u} defined in an open set O, subset of R^n, and vanishing on a portion \Gamma of boundary of O. If p>n we show that this p harmonic measure is…

For a metrizable space $X$ of density $\kappa$, let $PM(X)$ be the space of continuous bounded pseudometrics on $X$ endowed with the uniform convergence topology. In this paper, its topology shall be classified as follows: (i) If $X$ is…

In this paper, we study the metrical theory of Cartesian products of exact approximation sets in $\beta$-expansions. More precisely, for an integer $d \ge 2$ and real numbers $\beta_i > 1$ $(1 \le i \le d)$, we consider the set of points…

We will show in this paper that if $\lambda$ is very close to 1, then $$I(M,\lambda,m)= \sup_{u\in H^{1,n}_0(M) ,\int_M|\nabla u|^ndV=1}\int_\Omega (e^{\alpha_n |u|^\frac{n}{n-1}}-\lambda\sum\limits_{k=1}^m\frac{|\alpha_nu^\frac{n}{n-1}|^k}…

Let $\{a_n\}_{n\in\mathbb{N}}$, $\{b_n\}_{n\in \mathbb{N}}$ be two infinite subsets of positive integers and $\psi:\mathbb{N}\to \mathbb{R}_{>0}$ be a positive function. We completely determine the Hausdorff dimensions of the set of all…

In this paper, we study inhomogeneous Diophantine approximation over the completion $K_v$ of a global function field $K$ (over a finite field) for a discrete valuation $v$, with affine algebra $R_v$. We obtain an effective upper bound for…

The measure of the full dimension for a general Sierpi\'{n}ski carpet is studied. In the first part of this study, we give a criterion for the measure of the full Hausdorff dimension of a Sierpi\'{n}ski carpet. Meanwhile, it is the…

Let $T:[0,1]^d \rightarrow[0,1]^d$ be a piecewise expanding map with an absolutely continuous (with respect to the $d$-dimensional Lebesgue measure $m_d$) $T$-invariant probability measure $\mu$. Let $\left\{\mathbf{r}_n\right\}$ be a…

We study the long-term behavior of the iteration of a random map consisting of Lipschitz transformations on a compact metric space, independently and randomly selected according to a fixed probability measure. Such a random map is said to…

In this paper, we use the thermodynamical formalism to show that there exists a unique equilibrium state $\mu_\phi$ for each expanding Thurston map $f: S^2\rightarrow S^2$ together with a real-valued H\"older continuous potential $\phi$.…

Given $0<s<\frac d2$ with $s\leq 1$, we are interested in the large $N$-behavior of the optimal constant $\kappa_N$ in the Hardy inequality $\sum_{n=1}^N (-\Delta_n)^s \geq \kappa_N \sum_{n<m} |X_n-X_m|^{-2s}$, when restricted to…

For $\kappa$ a cardinal, a space $X=(X,\sT)$ is $\kappa$-{\it resolvable} if $X$ admits $\kappa$-many pairwise disjoint $\sT$-dense subsets; $(X,\sT)$ is {\it exactly} $\kappa$-{\it resolvable} if it is $\kappa$-resolvable but not…

We study the existence of eigenvalues of the generalized Friedrichs model $H_\mu(p)$, with a rank-one perturbation, depending on parameters $\mu>0$ and $p\in\mathbb{T}^2$, and found an absolutely convergent expansions for eigenvalues at…