Ranks of partitions play an important role in the theory of partitions. They provide combinatorial interpretations for Ramanujan’s famous congruences for partition functions. We establish a family of congruences modulo powers of $5$ for ranks of partitions.

In this paper we examine solutions in the Gaussian integers to the Diophantine equation $ax^{4}+by^{4}=cz^{2}$ for different choices of $a,b$ and $c$. Elliptic curve methods are used to show that these equations have a finite number of solutions or have no solution.

The concept of dominance is used in the behavioral and biological sciences to describe outcomes in a variety of competitive interactions. In some taxa, a history of agonistic encounters among individuals modifies the course of future agonistic encounters such that the existence of a certain type of relationship can be inferred. If one is to characterize such relationships as dominance, however, then they must be distinguished from other kinds of interaction patterns for which the term tends to be used, as well as from factors such as territoriality and "trained" winners and losers, which may also influence the expression of agonistic behavior. Operational definitions based on causal, functional, evolutionary, and ontogenetic considerations have been proposed. Reliability and validity problems have been discussed, but the dominance concept has proved useful despite methodological difficulties. The confusion of dominance relationships (which involve two or more individuals) with dominance ranks (which are assigned to a single individual) has obscured the possible evolutionary basis of dominance relationships. If benefits accrue to dominant members of pairs, then those attributes which allow an animal to establish dominance can be selected. Dominance per se and dominance ranks, on the other hand, cannot be genetically transmitted since they constitute relationships with other individuals rather than absolute attributes. Dominance rankings in particular may be useful for describing behavioral patterns within a group, but they may reflect our own ability to count rather than any important variable in social organization.

We show that 17.9% of all elliptic curves over Q, ordered by their exponential height, are semistable, and that there is a positive density subset of elliptic curves for which the root numbers are uniformly distributed. Moreover, for any α > 1/6 (resp. α > 1/12) the set of Frey curves (resp. all elliptic curves) for which the generalized Szpiro Conjecture |Δ(E)| [Lt]α NE12α is false has density zero. This implies that the ABC Conjecture holds for almost all Frey triples. These results remain true if we use the logarithmic or the Faltings height. The proofs make use of the fibering argument in the square-free sieve of Gouvêa and Mazur. We also obtain conditional as well as unconditional lower bounds for the number of curves with Mordell–Weil rank 0 and [ges ]2, respectively.

We give a simple proof for the independence of record indices in Nevzorov's record model which is based on ranks. An application of these results to a probabilistic analysis of a particular searching algorithm with non-equiprobable orderings is also discussed.