We consider the problem of broadcasting multiple messages from one processor to many processors in the k-port model for message passing systems. In such systems, processors communicate in rounds, where in every round, each processor can send k messages to k processors and receive k messages from k processors. In this paper, we first present a simple and practical algorithm based on variations o...

Hierarchical classification is supervised multi-class classification problem over the set of class labels organized according to a hierarchy. In this report, we study the work by Ramaswamy et. al. on hierarchical classification over symmetric tree distance loss. We extend the consistency of hierarchical classification algorithm over asymmetric tree distance loss. We design a $\mathcal{O}(nk\log...

Recently, Kitaev [Ki2] introduced partially ordered generalized patterns (POGPs) in the symmetric group, which further generalize the generalized permutation patterns introduced by Babson and Steingŕımsson [BS]. A POGP p is a GP some of whose letters are incomparable. In this paper, we study the generating functions (g.f.) for the number of k-ary words avoiding some POGPs. We give analogues, ex...

Molodtsov introduced the concept of soft set theory, which can be seen and used as a new mathematical tool for dealing with uncertainly. In this paper we introduce and initiate the study of soft m-ary semihypergroups by using the soft set theory. The notions of soft m-ary semihypergroups, soft m-ary subsemihypergroups, soft j-hyperideals, soft hyperideals, soft quasi-hyperideals and soft bi-hyp...

Recently, Han obtained two hook length formulas for binary trees and asked for combinatorial proofs. One of Han’s formulas has been generalized to k-ary trees by Yang. Sagan has found a probabilistic proof of Yang’s extension. We give combinatorial proofs of Yang’s formula for k-ary trees and the other formula of Han for binary trees. Our bijections are based on the structure of k-ary trees ass...

We show that, for any fixed ε > 0 and almost all primes p, the gary expansion of any fraction m/p with gcd(m, p) = 1 contains almost all g-ary strings of length k < (17/72 − ε) logg p. This complements a result of J. Bourgain, S. V. Konyagin, and I. E. Shparlinski that asserts that, for almost all primes, all g-ary strings of length k < (41/504− ε) logg p occur in the g-ary expansion of m/p.

Recently, Kitaev [9] introduced partially ordered generalized patterns (POGPs) in the symmetric group, which further generalize the generalized permutation patterns introduced by Babson and Steingrı́msson [1]. A POGP p is a GP some of whose letters are incomparable. In this paper, we study the generating functions (g.f.) for the number of k-ary words avoiding some POGPs. We give analogues, exten...

Many data models do not directly support -ary relationships. In most cases, they are either reduced to some of their binary projections or directly translated into an -ary “relationship relation” in the relational model. This paper addresses the reification of an -ary relationship into a new class with binary relationships and studies the preservation of semantics in the translation. It shows t...