Exactly solvable and unsolvable shortest network problems in 3D-space.
For any given ε > 0 we construct an ε-exhaustive normalized pathological submeasure. To this end we use potentially exhaustive submeasures and barriers of finite subsets of ℕ.
One can find in the mathematical literature many recent papers studying the generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, defined by means of generating functions. In this article we clarify the range of parameters in which these definitions are valid and when they provide essentially different families of polynomials. In particular, we show that, up to multiplicative constants, it is enough to take as the “main family” those given by and as an “exceptional family”...
In this paper, we recall the existence of graphs with bounded valency such that the simple random walk has a return probability at time at the origin of order for fixed and with Følner function . This result was proved by Erschler (see [4], [3]); we give a more detailed proof of this construction in the appendix. In the second part, we give an application of the existence of such graphs. We obtain bounds of the correct order for some functional of the local time of a simple random walk on...
We give a necessary and sufficient condition for the existence of perfect matchings in a plane bipartite graph in terms of elementary edge-cut, which extends the result for the existence of perfect matchings in a hexagonal system given in the paper of F. Zhang, R. Chen and X. Guo (1985).
We prove that the Cayley graphs of are expanders with respect to the projection of any fixed elements in generating a Zariski dense subgroup.