Related papers: Rado Numbers of Regular Nonhomogeneous Equations

A solution of the Abel equation $\dot{x}=A(t)x^3+B(t)x^2$ such that $x(0)=x(1)$ is called a periodic orbit of the equation. Our main result proves that if there exist two real numbers $a$ and $b$ such that the function $aA(t)+bB(t)$ is not…

For a fixed positive integer $t$, we consider the graph colouring problem in which edges at distance at most $t$ are given distinct colours. We obtain sharp lower bounds for the distance-$t$ chromatic index, the least number of colours…

There is the classical Radon theorem. Given integer $d \geq 1$ and $d+2$ points in d-dimensional space $R^d$. Then these points can be divided into two disjoint subsets whose convex hulls have a non-empty intersection. The original proof of…

Motivated by an Erd\H{o}s--Ko--Rado type problem on sets of strongly orthogonal roots in the $A_{\ell}$ root system, we estimate bounds for the size of a family of pairs $(A_{i}, B_{i})$ of $k$-subsets in $\{ 1, 2, \ldots, n\}$ such that…

An edge-coloured graph is said to be rainbow if no colour appears more than once. Extremal problems involving rainbow objects have been a focus of much research over the last decade as they capture the essence of a number of interesting…

\noindent Using the techniques connected with the measure of noncompactness we investigate the neutral difference equation of the following form \begin{equation*} \Delta \left(r_{n}\left(\Delta \left(x_{n}+p_{n}x_{n-k}\right) \right)…

We show that, for $n$ large, there must exist at least \[\frac{n^t}{C^{(1+o(1))t^2}}\] monochromatic $K_t$s in any two-colouring of the edges of $K_n$, where $C \approx 2.18$ is an explicitly defined constant. The old lower bound, due to…

In this article, we discuss when one can extend an r-regular graph to an r + 1 regular by adding edges. Different conditions on the num- ber of vertices n and regularity r are developed. We derive an upper bound of r, depending on n, for…

Let $\mathcal{H}$ be the class of normalized complex valued harmonic functions $ f = h + \overline{g}$ defined on the unit disk $\mathbb{D}$, where $h$ and $g$ are analytic functions with the normalization conditions $h(0) = h'(0) - 1 = 0$…

We are concerned with positive solutions of equation (E) $(-\Delta)^s u=f(u)$ in a domain $\Omega \subset \mathbb{R}^N$ ($N>2s$), where $s \in (\frac{1}{2},1)$ and $f\in C^{\alpha}_{loc}(\mathbb{R})$ for some $\alpha \in(0,1)$. We establish…

We prove regularity results up to the boundary for time independent generalized Maxwell equations on Riemannian manifolds with boundary using the calculus of alternating differential forms. We discuss homogeneous and inhomogeneous boundary…

We give two lower bound formulas for multicolored Ramsey numbers. These formulas improve the bounds for several small multicolored Ramsey numbers.

The \textit{set-coloring Ramsey number} $\mathrm{R}_{r, s}(G_1,G_2,...,G_r)$ is the least $n \in \mathbb{N}$ such that every coloring $\chi: E\left(K_n\right) \rightarrow\binom{[r]}{s}$ contains a monochromatic copy of $G_i$, that is, a…

For an arbitrary representation $\rho$ of a complex finite-dimensional Lie algebra, we construct a collection of numbers that we call the Jordan-Kronecker invariants of $\rho$. Among other interesting properties, these numbers provide lower…

We study the Abel differential equation x0 = A(t)x3 + B(t)x2 +C(t)x. Specifically, we find bounds on the number of its rational solutions when A(t), B(t) and C(t) are polynomials with real or complex coefficients; and on the number of…

Given any integer d >= 3, let k be the smallest integer such that d < 2k log k. We prove that with high probability the chromatic number of a random d-regular graph is k, k+1, or k+2, and that if (2k-1) \log k < d < 2k \log k then the…

This paper studies quantitative homogenization of elliptic equations with random, uniformly elliptic coefficients that vanish in a union of random holes. Assuming an upper bound on the size of the holes and a separation condition between…

An existence of a nontrivial solution in some `weaker' sense of the following system of equations \begin{align*} (-\Delta)^{s}u+l(x)\phi u+w(x)|u|^{k-1}u&=\mu~\text{in}~\Omega\nonumber\\ (-\Delta)^{s}\phi&=…

The aim of this paper is to prove the existence of solution for a partial differential equation involving a singularity with a general nonnegative, Radon measure $\mu$ as its nonhomogenous term which is given as \begin{eqnarray} -\Delta…

Let $F(x,y)$ be an irreducible form of degree $r\geq 3$ and having $s+1$ non-zero coefficients. Let $h\geq 1$ be an integer and consider the Thue inequality $$|F(x,y)|\leq h.$$ Following the seminal work of Thue in 1909, several papers were…