Related papers: A diophantine problem concerning third order matri…
Reciprocal matrices are tridiagonal matrices $(a_{ij})_{i,j=1}^n$ with constant main diagonal and such that $a_{i,i+1}a_{i+1,i}=1$ for $i=1,\ldots,n-1$. For these matrices, criteria are established under which their Kippenhahn curves…
This monograph starts with an upper triangular matrix with integer entries and 1's on the diagonal. It develops from this a spectrum of structures, which appear in different contexts, in algebraic geometry, representation theory and the…
We construct a set of positive integers A in {1,.., n} with |A|>> n^{2/3} that does not contain Hilbert cubes of dimension 3.
We classify all integrable 3-dimensional scalar discrete quasilinear equations Q=0 on an elementary cubic cell of the 3-dimensional lattice. An equation Q=0 is called integrable if it may be consistently imposed on all 3-dimensional…
In this paper we deal with a non-linear Diophantine equation which arises from the determinant computation of an integer matrix. We show how to find a solution, when it exists. We define an equivalence relation and show how the set of all…
This paper presents a bijection between ascent sequences and upper triangular matrices whose non-negative entries are such that all rows and columns contain at least one non-zero entry. We show the equivalence of several natural statistics…
We determine those k-tuples of conjugacy classes of matrices, from which it is possible to choose matrices which have no common invariant subspace and have sum zero. This is an additive version of the Deligne-Simpson problem. We deduce the…
We show that a matrix is a Hermitian positive semidefinite matrix whose nonzero entries have modulus 1 if and only if it similar to a direct sum of all $1's$ matrices and a 0 matrix via a unitary monomial similarity. In particular, the only…
We show that the complement of the ring of integers in a number field K is Diophantine. This means the set of ring of integers in K can be written as {t in K | for all x_1, ..., x_N in K, f(t,x_1, ..., x_N) is not 0}. We will use global…
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…
A perfect cuboid is a rectangular parallelepiped with integer edges, integer face diagonals, and integer space diagonal. Such cuboids have not yet been found, but nor has their existence been disproved. Perfect cuboids are described by a…
In this work, we construct the algebra of differential forms with the cube of exterior differential equal to zero on one-dimensional space. We prove that this algebra is a graded q-differential algebra where q is a cubic root of unity.…
We study copositive matrices which admit a decomposition into a sum of a positive semidefinite matrix and a matrix with nonnegative entries. Our main result shows that if the off-diagonal entries of a copositive matrix are nondecreasing in…
Let A= (a_{ij}) be a non-negative integer k x k matrix. A is a homogeneous matrix if a_{ij} + a_{kl}=a_{il} + a_{kj} for any choice of the four indexes. We ask: If A is a homogeneous matrix and if F is a form in C[x_1, \dots x_n] with…
This paper is dedicated to the problem of verification of matrices for unitary similarity. For the case of nonderogatory matrices, we have been able to present the new solution for this problem based on geometric approach. The main…
For a nonzero integer n, a set of m distinct nonzero integers {a_1,a_2,...,a_m} such that a_i a_j + n is a perfect square for all 1 <= i < j <= m, is called a D(n)-m-tuple. In this paper, by using properties of so-called regular Diophantine…
A perfect Euler cuboid is a rectangular parallelepiped with integer edges and integer face diagonals whose space diagonal is also integer. Such cuboids are not yet discovered and their non-existence is also not proved. Perfect Euler cuboids…
In a simple integer chain, if $u_{i-1}$, $u_i$, and $u_{i+1}$ are three consecutive terms of the chain, and the pair $(u_{i-1}, u_i)$ has a certain property, then the next pair $(u_i, u_{i+1})$ also has the same property. We extend the idea…
We investigate a variation of Nicomachus's identity in which one term in the cubic sum is replaced by a different cube. Specifically, we study the Diophantine identity \[ \sum_{j=1}^{n} j^3 + x^3 - k^3 = \left( \sum_{j=1}^{n} j + x - k…
We construct a unital locally matrix algebra of uncountable dimension that (1) does not admit a primary decomposition, (2) has an infinite locally finite Steinitz number. It gives negative answers to questions from \cite{BezOl} and…