Related papers: Some observations on the properness of Identity pl…

Motivated by the fundamental lower bounds questions in proof complexity, we initiate the study of matrix identities as hard instances for strong proof systems. A matrix identity of $d \times d$ matrices over a field $\mathbb{F}$, is a…

We provide precise formulations and proofs of two theorems from Darboux's lectures on orthogonal systems. These results provide local existence and uniqueness of solutions to certain types of first order PDE systems where each equation…

The integral identity conjecture of Kontsevich and Soibelman plays an important role in proving the existence of motivic Donaldson-Thomas invariants for three-dimensional noncommutative Calabi-Yau manifolds. There are a number of different…

We are concerned with the behavior of the polynomial maps $F=(P,Q)$ of $\mathbb{C}^2$ with finite fibres and satisfying the condition that all of the curves $aP+bQ=0$, $(a:b)\in \mathbb{P}^1$, are irreducible rational curves. The obtained…

For self-similar sets $X,Y\subseteq \mathbb{R}$, we obtain new results towards the affine embeddings conjecture of Feng-Huang-Rao (2014), and the equivalent weak intersections conjecture. We show that the conjecture holds when the defining…

Let $Y:\R^n\to\R^n$ be a polynomial local diffeomorphism and let $S_Y$ denote the set of not proper points of $Y$. The Jelonek's real Jacobian Conjecture states that if $\codim(S_Y)\geq2$, then $Y$ is bijective. We prove a weak version of…

It is shown that the $n$-dimensional Jacobian conjecture over algebraic number fields may be considered as an existence problem of integral points on affine curves. More specially, if the Jacobian conjecture over $\mathbb{C}$ is false, then…

In this paper we use matrices, whose entries satisfy certain linear conditions, to obtain composition identities $f(x_i)f(y_i)=f(z_i)$, where $f(x_i)$ is an irreducible form, with integer coefficients, of degree $n$ in $n$ variables ($n$…

If L, respectively R are matrices with entries binom{i-1,j-1}, respectively binom{i-1,n-j}, it is known that L^2 = I (mod 2), respectively R^3 = I (mod 2), where I is the identity matrix of dimension n > 1 (see P10735-May 1999 issue of the…

Let X be a normal projective variety defined over an algebraically closed field of arbitrary characteristic. We study the sequence of intermediate degrees of the iterates of a dominant rational selfmap of X, recovering former results by…

Let ${\bf M}=(M_1,\ldots, M_k)$ be a tuple of real $d\times d$ matrices. Under certain irreducibility assumptions, we give checkable criteria for deciding whether ${\bf M}$ possesses the following property: there exist two constants…

In 1936, Stanislaw Ja\'skowski gave a construction of an interesting sequence of what he called "matrices", which we would today call "finite Heyting Algebras". He then gave a very brief sketch of a proof that if a propositional formula…

Given an integer mxn matrix A satisfying certain regularity assumptions, we consider the set F(A) of all integer vectors b such that the associated knapsack polytope P(A,b)={x: Ax=b, x>=0} contains an integer point. When m=1 the set F(A) is…

Let $f_1,...,f_s \in \mathbb{K}[x_1,...,x_m]$ be a system of polynomials generating a zero-dimensional ideal $\I$, where $\mathbb{K}$ is an arbitrary algebraically closed field. Assume that the factor algebra $\A=\mathbb{K}[x_1,...,x_m]/\I$…

For given real or complex $m \times n$ data matrices $X$, $Y$, we investigate when there is a matrix $A$ such that $AX = Y$, and $A$ is invertible, Hermitian, positive (semi)definite, unitary, an orthogonal projection, a reflection, complex…

In this paper, we derive some new combinatorial inequalities by applying well known real analytic results like H\"{o}lder's inequality, Young's inequality, and Minkowiski's inequality to the recursively defined sequence $f_n$ of functions…

The main result of this paper is the following version of the real Jacobian conjecture: "Let $F=(p,q):\R^2\to\R^2$ be a polynomial map with nowhere zero Jacobian determinant. If the degree of $p$ is less than or equal to $4$, then $F$ is…

The two-dimensional Jacobian Conjecture says that a $\mathbb{C}$-algebra endomorphism $F:\mathbb{C}[x,y] \to \mathbb{C}[x,y]$ that has an invertible Jacobian is an automorphism. We show that if a $\mathbb{C}$-algebra endomorphism…

Let $A$ be a separable, unital and exact $C^*$-algebra satisfying the universal coefficient theorem. We prove uniqueness theorems up to unitary conjugacy for unital, full and nuclear maps from $A$ into ultraproducts of finite von Neumann…

In this article, for generalized projective spaces with any weights, we prove four main theorems in three different contexts where the Unital Set Condition USC (Definition $2.8$) on ideals is further examined. In the first context we prove,…