Classical solution for a nonlinear hybrid system modeling combustion in a multilayer porous medium

Combustion processes in porous media have been used by the petroleum engineering industry to extract heavy oil from reservoirs. This study focuses on a one-dimensional nonlinear hybrid system consisting of $n$ reaction–diffusion–convection equations coupled with $n$ ordinary differential equations, which models a combustion front moving through a porous medium with $n$ parallel layers. The state variables are the temperature and fuel concentration in each layer. Coupling occurs in both the reaction function and differential operator coefficients. We prove the existence of a classical solution, first locally and then globally over time, to an initial and boundary value problem for the corresponding system. The proof uses a new approach for combustion problems in porous media. The local solution is obtained by defining an operator in a set of Hölder continuous functions and using Schauder’s fixed-point theorem to find a fixed point as the desired solution. Using Zorn’s lemma, we extend the local solution to a global solution, proving that the first-order spatial derivative of the temperature in each layer is a bounded function.     

Bipullbacks of fractions and the snail lemma

We establish conditions giving the existence of bipullbacks in bicategories of fractions. We apply our results to construct a ${\pi }_0$-${\pi }_1$ exact sequence associated with a fractor between groupoids internal to a pointed exact category.     

Chase’s lemma and its context

Chase’s lemma provides a powerful tool for translating properties of (co)products in abelian categories into chain conditions. This note discusses the context in which the lemma is used, making explicit what is often neglected in the literature because of its technical nature.     

Homogeneous expansions of normalized biholomorphic convex mappings overB P

The power series expansions of normalized biholomorphic convex mappings on the Reinhardt domain are studied. It is proved that the first (k+1) terms of the expansions of the jth componentf j of such a mapf depend only onz _j , for 1 ⩽j⩽n, wherek is the natural number that satisfiesk < ρ ⩽k +I. Whenp→ ∞, this gives the result on the unit polydisc obtained by Suffridge in 1970. Project supported in part by the National Natural Science Foundation of China.     

Quadripartite version of the Hajnal-Szemerédi theorem

Let G be a quadripartite graph with N vertices in each vertex class and each vertex is adjacent to at least vertices in each of the other classes. There exists an N₀ such that, if N?N₀, then G contains a subgraph that consists of N vertex-disjoint copies of K₄.     

Spanning 3-colourable subgraphs of small bandwidth in dense graphs

A conjecture by Bollobás and Komlós states the following: For every γ>0 and integers r2 and Δ, there exists β>0 with the following property. If G is a sufficiently large graph with n vertices and minimum degree at least and H is an r-chromatic graph with n vertices, bandwidth at most βn and maximum degree at most Δ, then G contains a copy of H.This conjecture generalises several results concerning sufficient degree conditions for the containment of spanning subgraphs. We prove the conjecture for the case r=3.     

Schwarz—Pick Inequalities for Derivatives of Arbitrary Order

   Abstract. Let Ω and Π be two simply connected domains in the complex plane C which are not equal to the whole plane C and let λ _Ω and λ _Π denote the densities of the Poincare metric in Ω and Π , respectively. For f: Ω → Π analytic in Ω , inequalities of the type