Related papers: Subconvexity and the Hilbert-Kamke Problem

A common optimization problem is the minimization of a symmetric positive definite quadratic form $< x,Tx >$ under linear constrains. The solution to this problem may be given using the Moore-Penrose inverse matrix. In this work we extend…

In this note, we establish a multiplicity theorem for a nonlocal discrete problem of the type $$\cases{-\left(a\sum_{m=1}^{n+1}|x_m-x_{m-1}|^2+b\right)(x_{k+1}-2x_k+x_{k-1})=h_k(x_k)\hskip 10pt k=1,...,n, \cr & \cr x_0=x_{n+1}=0\cr}$$…

The following system of equations {x_1 \cdot x_1=x_2, x_2 \cdot x_2=x_3, 2^{2^{x_1}}=x_3, x_4 \cdot x_5=x_2, x_6 \cdot x_7=x_2} has exactly one solution in ({\mathbb N}\{0,1})^7, namely (2,4,16,2,2,2,2). Hypothesis 1 states that if a system…

Given two positive integers $n$ and $k$ and a parameter $t\in (0,1)$, we choose at random a vector subspace $V_{n}\subset \mathbb{C}^{k}\otimes\mathbb{C}^{n}$ of dimension $N\sim tnk$. We show that the set of $k$-tuples of singular values…

In this paper we consider a linear homogeneous system of $m$ equations in $n$ unknowns with integer coefficients over the reals. Assume that the sum of the absolute values of the coefficients of each equation does not exceed $k+1$ for some…

In this paper we theoretically show that interior-point methods based on self-concordant barriers possess favorable global complexity beyond their standard application area of convex optimization. To do that we propose first- and…

We consider an optimal control problem governed by a rate-inde\-pendent system with non-convex energy. The state equation is approximated by means of viscous regularization w.r.t.\ to hierarchy of two different Hilbert spaces. The…

For the truncated multidimensional moment problem we introduce a notion of a canonical solution. Namely, canonical solutions are those solutions which are generated by commuting self-adjoint extensions inside the associated Hilbert space.…

In this paper, we make significant progress on a conjecture proposed by Dan Archdeacon on the existence of integer Heffter arrays $H(m,n;s,k)$ whenever the necessary conditions hold, that is, $3\leqslant s \leqslant n$, $3\leqslant…

In this paper we consider the computation of approximate solutions for inverse problems in Hilbert spaces. In order to capture the special feature of solutions, non-smooth convex functions are introduced as penalty terms. By exploiting the…

In bounded domains, without any geometric conditions, we study the existence and uniqueness of globally Lipschitz and interior strong C^{1,1}, (and classical C^2), solutions of general semilinear oblique boundary value problems for…

We consider a continuous analogue of Babai et al.'s and Cai et al.'s problem of solving multiplicative matrix equations. Given $k+1$ square matrices $A_{1}, \ldots, A_{k}, C$, all of the same dimension, whose entries are real algebraic, we…

Kamke \cite{Kamke1921} solved an analog of Waring's problem with $n$th powers replaced by integer-valued polynomials. Larsen and Nguyen \cite{LN2019} explored the view of algebraic groups as a natural setting for Waring's problem. This…

In this paper we consider the defocusing Hartree nonlinear Schr\"odinger equations on $\mathbb T^3$ with real valued and even potential $V$ and Fourier multiplier decaying like $|k|^{-\beta}$. By relying on the method of random averaging…

Complex geometric optics solutions to a system of d-bar equations appearing in the context of electrical impedance tomography and the scattering theory of the integrable Davey-Stewartson II equations are studied for large values of the…

We prove existence of solutions and study the nonlocal-to-local asymptotics for nonlocal, convective, Cahn-Hilliard equations in the case of a W^{1,1} convolution kernel and under homogeneous Neumann conditions. Any type of potential,…

We consider the generalized two-dimensional Zakharov-Kuznetsov equation $u_t+\partial_x \Delta u+\partial_x(u^{k+1})=0$, where $k\geq3$ is an integer number. For $k\geq8$ we prove local well-posedness in the $L^2$-based Sobolev spaces…

For fixed integers $b\geq k$, a problem of relevant interest in computer science and combinatorics is that of determining the asymptotic growth, with $n$, of the largest set for which a $(b, k)$-hash family of $n$ functions exists.…

We provide new conditions under which the alternating projection sequence converges in norm for the convex feasibility problem where a linear subspace with finite codimension $N\geq 2$ and a lattice cone in a Hilbert space are considered.…

Let $n$ be a positive integer, and let $A$ be a set of $k\ge 2n-1$ integers. For the restricted sumset $$ S_n(A)=\{a_1+\cdots +a_n:\ a_1,\ldots,a_n\in A,\ \text{and}\ a_i^2\neq a_j^2\ \text{for} \ 1\le i<j\le n\}, $$ by a 2002 result of Liu…