Related papers: Hilbert's tenth problem for systems of diagonal qu…
We establish quantum and classical exact solvability for two large classes of maximally superintegrable Benenti systems in $n$ dimensions with arbitrarily large $n$. Namely, we solve the Hamilton--Jacobi and Schr\"odinger equations for the…
Hilbert's Tenth Problem over the field $\mathbb Q$ of rational numbers is one of the biggest open problems in the area of undecidability in number theory. In this paper we construct new, computably presentable subrings $R$ of $\mathbb Q$…
We prove the algebraicity of the Hilbert functor, the Hilbert stack, the Quot functor and the stack of coherent sheaves on an algebraic stack X with (quasi-)finite diagonal without any finiteness assumptions on X. We also give similar…
We develop numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus. These homotopies are optimal in that generically no paths diverge. For problems defined by hypersurface…
We consider a variety of Euler's conjecture, i.e., whether the Diophantine system \[\begin{cases} n=a_{1}+a_{2}+\cdots+a_{s-1}, a_{1}a_{2}\cdots a_{s-1}(a_{1}+a_{2}+\cdots+a_{s-1})=b^{s} \end{cases}\] has solutions…
Under general conditions we show that the solution of a stochastic parabolic partial differential equation of the form \[ \partial_t u = \mathrm{div} (A \nabla u) + f(t,x, u) + g_i (t,x,u) \dot{w}^i_t \] is almost surely H\"older continuous…
A canonical factorization is given for a quadratic pencil of accretive operators in a Hilbert space. Also, we establish some relationships between an m-accretive operator and its Moore-Penorse inverse. As an application, we study a result…
The square-free word problem relative to a system of two defining relations is decidable.
All non-equivalent integrable evolution equations of the fifth order of the form $u_t=D_x\frac{\delta H}{\delta u}$ are found.
Let $f(x_1, x_2, x_3)$ be an indefinite anisotropic integral quadratic form with determinant $d(f)$, and $t$ a non-zero integer such that $d(f)t$ is square-free. It is proved in this paper that, as long as there is one integral solution to…
We examine the solubility of a diagonal, translation invariant, quadratic equation system in arbitrary (dense) subsets A \subset Z and show quantitative bounds on the size of A if there are no non-trivial solutions. We use the circle method…
For a smooth projective variety $X$ with exceptional structure sheaf, and $\operatorname{Hilb}^2X$ the Hilbert scheme of two points on $X$, we show that the Fourier-Mukai functor $\mathbf{D}^{\mathrm{b}}(X)…
We study the problem of solvability of linear differential systems with small coefficients in the Liouvillian sense (or, by generalized quadratures). For a general system, this problem is equivalent to that of solvability of the Lie algebra…
About ten years ago, a paper proposed the first integer linear programming formulation for the constrained two-dimensional guillotine cutting problem (with unlimited cutting stages). Since, six other formulations followed, five of them in…
We investigate here the representability of integers as sums of triangular numbers, where the $n$-th triangular number is given by $T_n = n(n + 1)/2$. In particular, we show that $f(x_1,x_2,..., x_k) = b_1 T_{x_1} +...+ b_k T_{x_k}$, for…
It is shown that all PDM Schroedinger equations admitting more than five dimensional Lie symmetry algebras (whose completed list can be found in paper~[{\it J.~Math. Phys.} {\bf 58}, , 083508 (2017)] are exactly solvable. The corresponding…
The problem of finding perfect Euler cuboids or proving their non-existence is an old unsolved problem in mathematics. The third cuboid conjecture is the last of the three propositions suggested as intermediate stages in proving the…
In this paper we prove that the Diophantine problem in iterated restricted wreath products $G$ of arbitrary non-trivial free abelian groups $A_1,\ldots, A_k$, $k>1$ of finite ranks is undecidable, i.e., there is no algorithm that given a…
Consider the equation $q_1\alpha^{x_1}+\dots+q_k\alpha^{x_k} = q$, with constants $\alpha \in \overline{\mathbb{Q}} \setminus \{0,1\}$, $q_1,\ldots,q_k,q\in\overline{\mathbb{Q}}$ and unknowns $x_1,\ldots,x_k$, referred to in this paper as…
Given a prime $p\ge5$ and an integer $s\ge1$, we show that there exists an integer $M$ such that for any quadratic polynomial $f$ with coefficients in the ring of integers modulo $p^s$, such that $f$ is not a square, if a sequence…