A hilly poor noncrossing partition is a noncrossing partition with the properties : (1) each block has at most two elements, (2) in its linear representation, any isolated vertex is covered by some arc. This paper defines basic pairs as a combinatorial object and gives the number of hilly poor noncrossing partitions with n blocks, which is closely related to Maximal Davenport-Schinzel sequences...

Building on work of Dubickas and Smyth regarding the metric Mahler measure and the authors regarding extremal norms associated to the Mahler measure, the authors introduce a new set of norms associated to the Mahler measure of algebraic numbers which allow for an equivalent reformulation of problems like the Lehmer problem and the Schinzel-Zassenhaus conjecture on a single spectrum. We present ...

The Josephson effect and superconducting proximity effect were observed in superconductor-graphenesuperconductor SGS Josephson junctions with coherence lengths comparable to the distance between the superconducting leads. By comparing the measured gate dependence of the proximity induced subgap features multiple Andreev reflections and of the supercurrent to theoretical predictions, we find tha...

In this paper, we enumerate the number of combinatorial changes of the the Euclidean minimum spanning tree (EMST) of a set of n moving points in 2dimensional space. We assume that the motion of the points in the plane, is defined by algebraic functions of maximum degree s of time. We prove an upper bound of O(nβ2s(n )) for the number of the combinatorial changes of the EMST, where βs(n)=λs(n)/n...

We introduce a containment relation of hypergraphs which respects linear orderings of vertices and investigate associated extremal functions. We extend, by means of a more generally applicable theorem, the n log n upper bound on the ordered graph extremal function of F = ({1, 3}, {1, 5}, {2, 3}, {2, 4}) due to Füredi to the n(log n)2(log log n)3 upper bound in the hypergraph case. We use Davenp...

We describe an iterative method of constructing some favorable auxiliary polynomials used to obtain lower bounds in some problems of algebraic number theory. With this method we improve a lower bound on Mahler’s measure of a polynomial with no cyclotomic factors whose coefficients are all congruent to 1 modulo m for some integer m ≥ 2, raise a lower bound in the problem of Schinzel and Zassenha...