We say that X×Y satisfies the Uniquely Universal property (UU) iff there exists an open set U ⊆ X × Y such that for every open set W ⊆ Y there is a unique cross section of U with Ux = W . Michael Hrus̆ák raised the question of when does X × Y satisfy UU and noted that if Y is compact, then X must have an isolated point. We consider the problem when the parameter space X is either the Cantor spac...

Recent data identify distinct components of social cognition associated with five brain regions. In posterior temporal cortex, the extrastriate body area is associated with perceiving the form of other human bodies. A nearby region in the posterior superior temporal sulcus is involved in interpreting the motions of a human body in terms of goals. A distinct region at the temporo-parietal juncti...

In the Graph Realization Problem (GRP), one is given a graph G, a set of non-negative edge-weights, and an integer d. The goal is to find, if possible, a realization of G in the Euclidian space R, such that the distance between any two vertices is the assigned edge weight. The problem has many applications in mathematics and computer science, but is NP-hard when the dimension d is fixed. Charac...

The concept of a k-pairable graph was introduced by Chen (On k-pairable graphs, Discrete Mathematics 287 (2004), 11–15) as an extension of hypercubes and graphs with an antipodal isomorphism. In this talk, we introduce a special class of k-pairable graphs which are called uniquely pairable graphs. Then we give a characterization of uniquely pairable graphs in terms of the prime factor decomposi...

There exists a model in a countable language having a unique element which is not definable in Lω1,ω. We answer problem 10.12 from the updated version of Arnie Miller’s problem list, [3]. Definition For a structure M and a ∈ M, we say that a is definable over Lω1,ω if there is a formul ψ ∈ Lω1,ω such that for all b ∈ M a = b ⇔ M |= ψ(b). Theorem 0.1 There is a countable language L and a model M...