Related papers: Element-Distinct Solution For Rado's Theorem

In this paper, we prove the existence of a unique positive entropy solution to a fractional Laplacian problem involving nonlinear singular terms and also a non-negative bounded Radon measure as a source term.

A well-known result by Graham in Euclidean Ramsey Theory states that, for every positive real number $A$, every coloring of the plane with finite number of colors contains a monochromatic triangle of area $A$. We consider canonical versions…

We consider the following problem: Given a matrix A, find minimal subsets of columns of A with cardinality no larger than a given bound that are linear dependent or nearly so. This problem arises in various forms in optimization, electrical…

We show the existence of several infinite monochromatic patterns in the integers obtained as values of suitable symmetric polynomials. The simplest example is the following. For every finite coloring of the natural numbers…

Hindman proved in 1979 that no matter how natural numbers are colored in r colors, for a fixed positive integer r, there is an infinite subset X of numbers and a color t such that for any finite non-empty subset X' of X, the color of the…

Given an $n\times n$ array $M$ ($n\ge 7$), where each cell is colored in one of two colors, we give a necessary and sufficient condition for the existence of a partition of $M$ into $n$ diagonals, each containing at least one cell of each…

For any primitive matrix $M\in\mathbb{R}^{n\times n}$ with positive diagonal entries, we prove the existence and uniqueness of a positive vector $\mathbf{x}=(x_1,\dots,x_n)^t$ such that $M\mathbf{x}=(\frac{1}{x_1},\dots,\frac{1}{x_n})^t$.…

We give a short, explicit proof of Hindman's Theorem that in every finite coloring of the integers, there is an infinite set all of whose finite sums have the same color. We give several exampls of colorings of the integers which do not…

We show that for every coloring of the rationals into finitely many colors, one of the colors contains a set of the form $\{x,y,xy,x+y\}$ for some nonzero $x$ and $y$.

A celebrated but non-effective theorem of Tibor Gallai states that for any finite set $A$ of $\Z^n$ and for any finite number of colors $c$ there is a minimal $m$ such that no coloring of the finite $m^n$-grid can avoid that a homothetic…

We prove an arithmetic analog of the induced graph removal lemma for complexity 1 patterns over finite fields. Informally speaking, we show that given a fixed collection of $r$-colored complexity 1 arithmetic patterns over $\mathbb F_q$,…

In this work we investigate the existence of solutions, their uniqueness and finally dependence on parameters for solutions of second order neutral nonlinear difference equations. The main tool which we apply is Darbo fixed point theorem.

Let $\mathcal{M}$ and $\mathcal{N}$ be two matroids on the same ground set $V$. Let $A_1,\dots,A_{2n-1}$ be sets which are independent in both $\mathcal{M}$ and $\mathcal{N}$, satisfying $|A_i|\geq \textrm{min}(i,n)$ for all $i$. We show…

The paper proves the existence of monochrome standard simplexes of a given volume on a multidimensional rational lattice painted in a finite number of colors.

Formalization of mathematics is a major topic, that includes in particular numerical analysis, towards proofs of scientific computing programs. The present study is about the finite element method, a popular method to numerically solve…

Let $x_0\in\Omega\subset\Bbb{R}^n$, $n\ge 2$, be a domain and let $m\ge 2$. We will prove that a solution $u$ of the polyharmonic equation $\Delta^mu=0$ in $\Omega\setminus\{x_0\}$ has a removable singularity at $x_0$ if and only if…

P. Erd\H{o}s proved that every 2-edge coloured complete graph on the natural numbers can be vertex decomposed into two monochromatic paths of different colour. This result was extended by R. Rado to an arbitrary finite number of colours. We…

The main objective of this talk is to develop a matrix pencil approach for the study of an initial value problem of a class of singular linear matrix differential equations whose coefficients are constant matrices. By using matrix pencil…

In this paper we show a variant of colorful Tverberg's theorem which is valid in any matroid: Let $S$ be a sequence of non-loops in a matroid $M$ of finite rank $m$ with closure operator cl. Suppose that $S$ is colored in such a way that…

We consider the algebraic Riccati equation for which the four coefficient matrices form an M-matrix K. When K is a nonsingular M-matrix or an irreducible singular M-matrix, the Riccati equation is known to have a minimal nonnegative…