Asymptotic multipartite version of the Alon-Yuster theorem

In this paper, we prove the asymptotic multipartite version of the Alon-Yuster theorem, which is a generalization of the Hajnal-Szemer\'edi theorem: If $k\geq 3$ is an integer, $H$ is a $k$-colorable graph and $\gamma>0$ is fixed, then, for every sufficiently large $n$, where $|V(H)|$ divides $n$, and for every balanced $k$-partite graph $G$ on $kn$ vertices with each of its corresponding $\binom{k}{2}$ bipartite subgraphs having minimum degree at least $(k-1)n/k+\gamma n$, $G$ has a subgraph consisting of $kn/|V(H)|$ vertex-disjoint copies of $H$. The proof uses the Regularity method together with linear programming.Comment: 22 pages, 1 figur

Algae Living in Salamanders, Friend or Foe?

Roughly speaking, our bodies use energy from the sun, but we can\u27t use sunlight directly. Instead, plants and algae collect sunlight and store it as chemical energy through the process of photosynthesis. We can access that fuel directly when we eat plants, or indirectly when we eat other animals that eat plants. However, in some invertebrate animals (those without a backbone) the relationships to algae are more intimate. Tiny single-celled algal symbionts can actually live inside the cells of living corals and small animals like hydra that live in water. The algae live in a safe environment inside animal cells and are provided with building block materials to function. They use sunlight to convert the building block materials into larger molecules to store energy and build cellular structures. At the same time some of that stored solar energy is directly transferred to the host animal, allowing it to live in otherwise nutrient poor environments. Thus the algae and their hosts depend on one another to live and thrive. These mutually beneficial relationships are called photosymbioses. [excerpt

Nevanlinna-Pick Interpolation and Factorization of Linear Functionals

If \fA is a unital weak-$*$ closed algebra of multiplication operators on a reproducing kernel Hilbert space which has the property \bA_1(1), then the cyclic invariant subspaces index a Nevanlinna-Pick family of kernels. This yields an NP interpolation theorem for a wide class of algebras. In particular, it applies to many function spaces over the unit disk including Bergman space. We also show that the multiplier algebra of a complete NP space has \bA_1(1), and thus this result applies to all of its subalgebras. A matrix version of this result is also established. It applies, in particular, to all unital weak-$*$ closed subalgebras of $H^\infty$ acting on Hardy space or on Bergman space.Comment: 26 pages; minor revisions; to appear in Integral Equations and Operator Theor