The existence of thick triangulations -- an "elementary" proof

We provide an alternative, simpler proof of the existence of thick triangulations for noncompact $\mathcal{C}^1$ manifolds. Moreover, this proof is simpler than the original one given in \cite{pe}, since it mainly uses tools of elementary differential topology. The role played by curvatures in this construction is also emphasized.Comment: 7 pages Short not

Further results on snakes in powers of complete graphs

AbstractBy a snake in a finite graph G is meant a cycle without chords. Denoted by S(G) the length of a longest snake in G. In this paper we obtain a new lower bound for S(G) in the case where G is the product of d copies of the complete graph on n vertices

On the snake in the box problem

AbstractA snake in a graph is a simple cycle without chords. Denote by s(d) the length of a longest snake in the d-dimensional unit cube. We give a new proof of the theorem of Evdokimov that s(d) > λ2d, where λ is a positive constant

Non-Monochromatic and Conflict-Free Coloring on Tree Spaces and Planar Network Spaces

It is well known that any set of n intervals in $\mathbb{R}^1$ admits a non-monochromatic coloring with two colors and a conflict-free coloring with three colors. We investigate generalizations of this result to colorings of objects in more complex 1-dimensional spaces, namely so-called tree spaces and planar network spaces

The non-equilibrium work relation. Thermodynamic analysis and microscopic foundations

We discuss the conditions for which the non-equilibrium work relation is valid by means of thermodynamic and microscopic arguments.Comment: 16 pages 1 figur