Related papers: A refined Lusin type theorem for gradients

If one thinks of a Riemannian metric, $g_1$, analogously as the gradient of the corresponding distance function, $d_1$, with respect to a background Riemannian metric, $g_0$, then a natural question arises as to whether a corresponding…

We show that for any infinite set $A$ in ${\mathbb R}$, there exists a compact set $E \subseteq \mathbb{R}$ of positive Lebesgue measure that does not contain any non-trivial affine copy of $A$. This proves the Erd\"os similarity…

The Bernstein approximation problem is to determine whether or not the space of all polynomials is dense in a given weighted $C_0$-space on the real line. A theorem of L. de Branges characterizes non--density by existence of an entire…

The "Universality Theorem" for gravity shows that f(R) theories (in their metric-affine formulation) in vacuum are dynamically equivalent to vacuum Einstein equations with suitable cosmological constants. This holds true for a generic (i.e.…

Lapid and M\'{i}nguez gave a criterion of the irreducibility of the parabolic induction $\sigma \times \pi$, where $\sigma$ is a ladder representation and $\pi$ is an arbitrary irreducible representation of the general linear group over a…

Let $(X, d, \mu)$ be a metric measure space, with $\mu$ a Borel regular measure. In this paper, we prove that, if $u\in L^1_{{\mathop\mathrm{\,loc\,}}}(X)$ and $g$ is a Haj{\l}asz gradient of $u$, then there exists $\widetilde u$ such that…

We introduce a one-parameter family of Borel regular measures on $\mathbb{R}^n$ that enhances Lebesgue measure by incorporating a scale-invariant penalty for codimension-1 boundary structures. Utilizing Carath\'eodory's outer measure…

We state the Brumer-Stark conjecture and motivate it from two perspectives. Stark's perspective arose in his attempts to generalize the classical Dirichlet class number formula for the leading term of the Dedekind zeta function at $s=1$…

Let g be an integer greater than 1. A uniform version of the Parshin-Arakelov theorem on the finiteness of the set of non-isotrivial curves of genus g over a function field, with fixed degeneracy locus, is proved. This is applied to obtain…

Let $(\lambda_k)$ be a strictly increasing sequence of positive numbers such that $\sum_{k=1}^{\infty} \frac{1}{\lambda_k} < \infty.$ Let $f $ be a bounded smooth function and denote by $u= u^f$ the bounded classical solution to $u(x) -…

We consider a novel class of $f(\R)$ gravity theories where the connection is related to the conformally scaled metric $\hat g_{\mu\nu}=C(\R)g_{\mu\nu}$ with a scaling that depends on the scalar curvature $\R$ only. We call them C-theories…

The Falconer conjecture asserts that if E is a planar set with Hausdorff dimension strictly greater than 1, then its Euclidean distance set has positive one-dimensional Lebesgue measure. We discuss the analogous question with the Euclidean…

Let $X$ be a smooth projective variety of dimension $n$ over the algebraic closure of a finite field $\mathbb{F}_p$. Assuming the standard conjecture $D$, we prove a weaker form of the Dynamical Degree Comparison conjecture; equivalence of…

The error on a real quantity Y due to the graduation of the measuring instrument may be represented, when the graduation is regular and fines down, by a Dirichlet form on R whose square field operator do not depend on the probability law of…

We address the study of nonlocal gradients defined through general radial kernels $\rho$. Our investigation focuses on the properties of the associated function spaces, which depend on the characteristics of the kernel function.…

Kings, Lei, Loeffler and Zerbes constructed a three-variable Euler system $\kappa({\bf g},{\bf h})$ of Beilinson-Flach elements associated to a pair of Hida families $({\bf g},{\bf h})$ and exploited it to obtain applications to the…

We prove an abstract Fubini-type theorem in the context of monoidal and enriched category theory, and as a corollary we establish a Fubini theorem for integrals on arbitrary convergence spaces that generalizes (and entails) the classical…

Let $E$ and $F$ be finite graphs with no sinks, and $k$ any field. We show that shift equivalence of the adjacency matrices $A_E$ and $A_F$, together with an additional compatibility condition, implies that the Leavitt path algebras…

We construct a $\mathbb{Z}_2 \times \mathbb{Z}_2$ gauge theory coupled to matter on a one-dimensional chain, aiming to study the ground-state physics in the Gauss law subspace. We show that the theory in the Gauss law subspace has a U$(1)$…

This article provides a new approach to address Mosco convergence of gradient-type Dirichlet forms, $\mathcal E^N$ on $L^2(E,\mu_N)$ for $N\in\mathbb N$, in the framework of converging Hilbert spaces by K.~Kuwae and T.~Shioya. The basic…