Related papers: Subconvexity and the Hilbert-Kamke Problem

When $k$ and $s$ are natural numbers and $\mathbf h\in \mathbb Z^k$, denote by $J_{s,k}(X;\mathbf h)$ the number of integral solutions of the system \[ \sum_{i=1}^s(x_i^j-y_i^j)=h_j\quad (1\le j\le k), \] with $1\le x_i,y_i\le X$. When…

Fix $k,s,n\in \mathbb N$, and consider non-zero integers $c_1,\ldots ,c_s$, not all of the same sign. Provided that $s\ge k(k+1)$, we establish a Hasse principle for the existence of lines having integral coordinates lying on the affine…

When $\mathbf h\in \mathbb Z^3$, denote by $B(X;\mathbf h)$ the number of integral solutions to the system \[ \sum_{i=1}^6(x_i^j-y_i^j)=h_j\quad (1\le j\le 3), \] with $1\le x_i,y_i\le X$ $(1\le i\le 6)$. When $h_1\ne 0$ and appropriate…

We provide conditions ensuring that the KKT-type conditions characterizes the global optimality for quadratically constrained (possibly nonconvex) quadratic programming QCQP problems in Hilbert spaces. The key property is the convexity of a…

In this paper, we investigate the combinatorial structure and asymptotic distribution of the solution set of the equation $\sigma(n+1) = k\sigma(n)$ for a given integer $k>1$. From a combinatorial perspective, the solutions to this equation…

For fully nonlinear $k$-Hessian operators on bounded strictly $(k-1)$-convex domains $\Omega$ in ${\mathbb R}^N$, a characterization of the principal eigenvalue associated to a $k$-convex and negative principal eigenfunction will be given…

In this paper we consider systems of vectors in a Hilbert space $\mathcal{H}$ of the form $\{g_{jk}: j \in J, \, k\in K\}\subset \mathcal{H}$ where $J$ and $K$ are countable sets of indices. We find conditions under which the local…

Let $k({\bf x})=k(x_1,\ldots ,x_n)$ be the rational function field, and $k\subsetneqq L\subsetneqq k({\bf x})$ an intermediate field. Then, Hilbert's fourteenth problem asks whether the $k$-algebra $A:=L\cap k[x_1,\ldots ,x_n]$ is finitely…

Let $k\geq2$ and $s$ be positive integers. Let $\theta\in(0,1)$ be a real number. In this paper, we establish that if $s>k(k+1)$ and $\theta>0.55$, then every sufficiently large natural number $n$, subjects to certain congruence conditions,…

In this paper convex optimization techniques are employed for convex optimization problems in infinite dimensional Hilbert spaces. A first order optimality condition is given. Let $f : \mathbb{R}^{n}\rightarrow \mathbb{R}$ and let $x\in…

We prove bounds for the number of solutions to $$a_1 + \dots + a_k = a_1' + \dots + a_k'$$ over $N$-element sets of reals, which are sufficiently convex or near-convex. A near-convex set will be the image of a set with small additive…

Chv\'{a}tal and Klincsek (1980) gave an $O(n^3)$-time algorithm for the problem of finding a maximum-cardinality convex subset of an arbitrary given set $P$ of $n$ points in the plane. This paper examines a generalization of the problem,…

We obtain a new upper bound for $\sum_{h\le H}\Delta_k(N,h)$ for $1\le H\le N$, $k\in \N$, $k\ge3$, where $\Delta_k(N,h)$ is the (expected) error term in the asymptotic formula for $\sum_{N < n\le2N}d_k(n)d_k(n+h)$, and $d_k(n)$ is the…

We obtain asymptotic upper bounds for the number of natural solutions of the following diagonal Diophantine equations in a hypercube with side - $N$ in the paper: $x_1 = x_2^k+...+x_s^k$, $x_1^k = x_2^k+...+x_s^k$, $x_1 = \sum_{j=2}^s…

A successful computational approach for solving large-scale positive semidefinite (PSD) programs is to enforce PSD-ness on only a collection of submatrices. For our study, we let $\mathcal{S}^{n,k}$ be the convex cone of $n\times n$…

We study the exterior Dirichlet problem for the homogeneous $k$-Hessian equation. The prescribed asymptotic behavior at infinity of the solution is zero if $k<\frac{n}{2}$, it is $\log|x|+O(1)$ if $k=\frac{n}{2}$ and it is…

This paper presents a new extension of the classical Heron problem, termed the generalized $(k,m)$-Heron problem, which seeks an optimal configuration among $k$ feasible and $m$ target non-empty closed convex sets in $\mathbb{R}^n$. The…

Depth-3 circuit lower bounds and $k$-SAT algorithms are intimately related; the state-of-the-art $\Sigma^k_3$-circuit lower bound and the $k$-SAT algorithm are based on the same combinatorial theorem. In this paper we define a problem which…

Using the established $d$-concavity of the $k$-Hessian type functions $F_k(R)=\log(S_k(R)),$ whose variables are nonsymmetric matrices, we prove $ C^{2, \alpha}(\overline{\Omega}) $ estimates for strictly $(\delta, \widetilde{\gamma}_k)…

We show that for integers $k\geq 4$ and $s\geq k^2+(3k-1)/4$, we have an asymptotic formula for the number of solutions, in positive integers $x_i$, to the inequality $\left|(x_1-\theta_1)^k+\dotsc+(x_s-\theta_s)^k-\tau\right|<\eta$, where…