Related papers: A sharp Bernstein-type theorem for entire minimal …

Let $G$ be a graph. A good function is a function $f:V(G)\rightarrow \{-1,1\}$, satisfying $f(N(v))\geq 1$, for each $v\in V(G)$, where $ N(v)=\{u\in V(G)\, |\, uv\in E(G) \} $ and $f(S) = \sum_{u\in S} f(u)$ for every $S \subseteq V(G) $.…

In this article, we give a proof for a geometric presentation theorem for any irreducible scheme $X$ smooth projective over a discrete valuation ring $R$. As a consequence, for any reductive $R$-group scheme $\mathbf{G}$, we prove that any…

In Theorem 1, we generalize the results of Szabo for Berwald metrics that are not necessary strictly convex: we show that for every Berwald metric F there always exists a Riemannian metric affine equivalent to F. As an application we show…

Here we simplify the proof of the de Rham theorem for Schwartz functions on affine Nash manifolds and generalize the result to the case of non affine Nash manifolds.

We prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index (and volume if the ambient dimension is greater than three) in a Riemannian manifold…

This paper presents a complete classification of minimal graph surfaces that admit graphical transformations into other minimal surfaces. These transformations are functions that map the height function of a minimal graph surface to another…

Let $U$ be a bounded open subset of the complex plane and let $A_{\alpha}(U)$ denote the set of functions analytic on $U$ that also belong to the little Lipschitz class with Lipschitz exponent $\alpha$. It is shown that if $A_{\alpha}(U)$…

It is be shown that the sequence of Bernstein polynomials for a function of several variables converges to this function uniformly along with every partial derivative of any order, provided that the latter derivative is well defined and…

We prove that several invariants of a possibly singular complex affine or projective variety of degree $d$ in the affine space $\mathbb{A}^{n}$, or $\mathbb{P}^n$, are bounded by a function of $d$ alone, provided $b_{1}=0$ for a resolution…

In this paper, we study a second order variational problem for locally convex hypersurfaces, which is the affine invariant analogue of the classical Plateau problem for minimal surfaces. We prove existence, regularity and uniqueness results…

It is proved that if a smooth function $u(x)$, $x\in \mathbb{R}^3$, such that $\inf_{s\in S}|u_N(s)|>0$, where $u_N$ is the normal derivative of $u$ on $S$, has a closed smooth surface $S$ of zeros, then the function $u(x)+\epsilon v(x)$…

We prove that any affine ring $R$ over a field $k$ has full cosupport, i.e., the cosupport of $R$ is equal to ${\rm Spec}\, R$. Using this fact, we give a complete description of all terms in a minimal pure-injective resolution of $R$,…

We prove that in a closed Riemannian manifold with dimension between $3$ and $7$, either there are minimal hypersurfaces with arbitrarily large area, or there exist uncountably many stable minimal hypersurfaces. Moreover, the latter case…

Sharp bounds are given for solutions to the minimal surface equation with vanishing boundary values over domains containing sectors of opening bigger than pi.

Let $K_7^{\vee}$ denote the graph obtained from the complete graph on seven vertices by deleting two edges with a common end. Motivated by Hadwiger's conjecture, we prove that every graph with no $K_7^{\vee}$-minor is $6$-colorable.

This short note describes the connection between strong minimality of the differential equation satisfied by an complex analytic function and the real and imaginary parts of the function being Pfaffian. This connection combined with a…

In 1934, Reinhardt asked for the centrally symmetric convex domain in the plane whose best lattice packing has the lowest density. He conjectured that the unique solution up to an affine transformation is the smoothed octagon (an octagon…

We prove a conjecture of Kleinbock which gives a clear-cut classification of all extremal affine subspaces of $\mathbb{R}^n$. We also give an essentially complete classification of all Khintchine type affine subspaces, except for some…

We consider functions $f: \mathbb{Z} \to \mathbb{R}$ and kernels $u: \{-n, \cdots, n\} \to \mathbb{R}$ normalized by $\sum_{\ell = -n}^{n} u(\ell) = 1$, making the convolution $u \ast f$ a "smoother" local average of $f$. We identify which…

Assume $n=2m\geq 2$ and let $F(x)=x^{d_1}+x^{d_2}$ be a binomial vectorial function over $\F_{2^n}$ possessing the maximal number (i.e. $2^n-2^m$) of bent components. Suppose the $2$-adic Hamming weights $\wt_2(d_1)$ and $\wt_2(d_2)$ are…