In Bevan et al. (2016), the study of patterns in forests of binary shrubs was introduced. A k-ary heap H is a k-ary tree labeled with {1, . . . , n} such that every child has a larger label than its parent. Given a k-ary heap H , we associate a permutation σH with H by recording the vertex labels as they are encountered in the breadth-first search of the tree. For example, in Figure 1, we pictu...

In this paper we investigate the structure and representation of n-ary algebras arising from DNA recombination, where n is a number of DNA segments participating in recombination. Our methods involve a generalization of the Jordan formalization of observables in quantum mechanics in n-ary splicing algebras. We show that the splicing algebras are an n-ary envelope for algebras of DNA recombinati...

The n-ary first and second recursion theorems formalize two distinct, yet similar, notions of self-reference. Roughly, the n-ary first recursion theorem says that, for any n algorithmic tasks (of an appropriate type), there exist n partial computable functions that use their own graphs in the manner prescribed by those tasks; the n-ary second recursion theorem says that, for any n algorithmic t...

The edge-bandwidth problem is an analog of the classical bandwidth problem, in which one has to label the edges of a graph by distinct integers such that the maximum difference of labels of any two incident edges is minimized. We prove tight bounds on the edge-bandwidth of hypercube and butterfly graphs and complete k-ary trees which extend and improve on previous known results. We also provide...

In this paper we introduce a notion of divisibility and primality on k-ary trees and we nd a relation between indecomposable preex code and prime trees. This relation allows to work on trees instead than directly on preex codes. In this way we can prove a density theorem for indecomposable preex codes and we can give an algorithm to test the indecomposability of a maximal preex code.

In this paper, we present an involution on some kind of colored k-ary trees which provides a combinatorial proof of a combinatorial sum involving the generalized Catalan numbers Ck,γ(n) = γ kn+γ ( kn+γ n ) . From the combinatorial sum, we refine the formula for k-ary trees and obtain an implicit formula for the generating function of the generalized Catalan numbers which obviously implies a Van...

We prove that any IN-rational sequence s = (s n) n1 of nonnega-tive integers satisfying the Kraft strict inequality P n1 s n k ?n < 1 is the enumerative sequence of leaves by height of a rational k-ary tree. We give an eecient algorithm to get a k-ary rational tree. Particular cases of this result had been previously proven. We give some partial results in the case of equality. Especially we so...

n-ary transit functions are introduced as a generalization of binary (2-ary) transit functions. We show that they can be associated with convexities in natural way and discuss the Steiner convexity as a natural n-ary generalization of geodesicaly convexity. Furthermore, we generalize the betweenness axioms to n-ary transit functions and discuss the connectivity conditions for underlying hypergr...

A residue number system based M -ary modem is proposed and its performance is evaluated over Gaussian channels. When one or two redundant moduli are employed, a signal-to-noise ratio gain of 1.2–2 dB was achieved for a 16-ary, 32-ary and 37-ary modem, respectively, at a bit error rate of 10 .