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...

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.

1. While the present paper is concerned chiefly with continuous groups, its results may be readily utilized for discontinuous groups.* Indeed, the finite form of the general transformation of the group is known ah initio. Further, the method is applicable to the construction of a linear Cm> q -ary group isomorphic to an arbitrary m-ary linear group. 2. The formula of composition of m-ary linear...

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...

In this paper, given a composite integer n, we propose a method of constructing p-ary low correlation zone(LCZ) sequences of period p − 1 from p-ary m-sequences of the same length. The new construction method is a generalized form of the quaternary LCZ sequence by Kim, Jang, No, and Chung in the view of the alphabet size. The correlation distribution of these new p-ary LCZ sequences is derived.

In this paper, given a q-ary LCZ sequence set with parameters (N,M,L, ǫ) and even q, we construct another q-ary LCZ sequence set with parameters (2N, 2M,L, 2ǫ) when q is even. Especially for odd L such that L|N , if a q-ary optimal LCZ sequence set with parameters (N,M,L, 1) is used, the new set with parameters (2N, 2M,L, 2) can be optimal in terms of the set size.

N -ary queries in trees select sets of n-tuples of nodes. We propose and investigate representation formalisms for n-ary queries by tree automata, both for ranked and unranked trees. We show that existential run-based queries capture MSO in the n-ary case, as well as universal run-based queries. We then characterize queries by runs of unambiguous tree automata, and show how to decide whether an...