Compact hypothesis and extremal set estimators

Extremal problems on set systems

Extremal Pseudocompact Abelian Groups Are Compact Metrizable

Every pseudocompact Abelian group of uncountable weight has both a proper dense pseudocompact subgroup and a strictly finer pseudocompact group topology.

Extremal Isosystolic Metrics for Compact Surfaces

Given a closed, orientable surface M of genus ≥ 2, one seeks an extremal isosystolic metric on M : this is a Riemannian metric that induces on M the smallest possible area, subject to the constraint that the corresponding systole, or shortest length of any non-contractible closed curve, is a fixed, positive number. The geometric problem is rendered into an analytic one by reducing it to solving...

Compact Forbidden-Set Routing

We study labelling schemes for X-constrained path problems. Given a graph (V,E) and X ⊆ V , a path is X-constrained if all intermediate vertices avoidX. We study the problem of assigning labels J(x) to vertices so that given {J(x) : x ∈ X} for any X ⊆ V , we can route on the shortest X-constrained path between x, y ∈ X. This problem is motivated by Internet routing, where the presence of routin...

Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes

We consider a family of meta-Fibonacci sequences which arise in studying the number of leaves at the largest level in certain infinite sequences of binary trees, restricted compositions of an integer, and binary compact codes. For this family of meta-Fibonacci sequences and two families of related sequences we derive ordinary generating functions and recurrence relations. Included in these fami...