Twofold 2-perfect bowtie systems with an extra property

A bowtie is a closed trail whose graph consists of two 3-cycles with exactly one vertex in common. A 2-fold bowtie system of order n is an edge-disjoint decomposition of 2K _n into bowties. A 2-fold bowtie system is said to be 2-perfect provided that every pair of distinct vertices is joined by two paths of length 2. It is said to be extra provided these two paths always have distinct midpoints. The extra property guarantees that the two paths x, a, y and x, b, y between every pair of vertices form a 4-cycle (x, a, y, b), and that the collection of all such 4-cycles is a four-fold 4-cycle system. We show that the spectrum for extra 2-perfect 2-fold bowtie systems is precisely the set of all n ?? 0 or 1 (mod 3), ${n\,\geqslant\,6}$ . Additionally, with an obvious definition, we show that the spectrum for extra 2-perfect 2-fold maximum packings of 2K _n with bowties is precisely the set of all n ?? 2 (mod 3), ${n\,\geqslant\,8}$ .     

Lambda-fold 2-perfect 6-cycle systems in equipartite graphs

A 6-cycle system of a graph G is an edge-disjoint decomposition of G into 6-cycles. Graphs G, for which necessary and sufficient conditions for existence of a 6-cycle system have been found, include complete graphs and complete equipartite graphs. A 6-cycle system of G is said to be 2-perfect if the graph formed by joining all vertices distance 2 apart in the 6-cycles is again an edge-disjoint decomposition of G, this time into 3-cycles, since the distance 2 graph in any 6-cycle is a pair of disjoint 3-cycles.Necessary and sufficient conditions for existence of 2-perfect 6-cycle systems of both complete graphs and complete equipartite graphs are known, and also of λ-fold complete graphs. In this paper, we complete the problem, giving necessary and sufficient conditions for existence of λ-fold 2-perfect 6-cycle systems of complete equipartite graphs.     

Varieties of algebras arising from K-perfect m-cycle systems

A class of algebras forms a variety if it is characterised by a collection of identities. There is a well-known method, often called the standard construction, which gives rise to algebras from m-cycle systems. It is known that the algebras arising from {1}-perfect m-cycle systems form a variety for m∈{3,5} only, and that the algebras arising from {1,2}-perfect m-cycle systems form a variety for m∈{3,5,7} only. Here we give, for any set K of positive integers, necessary and sufficient conditions under which the algebras arising from K-perfect m-cycle systems form a variety.     

The intersection problem for m-cycle systems

Let I_m(v) denote the set of integers k for which a pair of m-cycle systems of K_v, exist, on the same vertex set, having k common cycles. Let J_m(v) = {0, 1, 2,…, t_v ?2, t_v} where t_v = v(v ? 1)/2m. In this article, if 2mn + x is an admissible order of an m-cycle system, we investigate when I_m(2mn + x) = J_m(2mn + x), for both m even and m odd. Results include J^m(2mn + 1) = I^m(2mn + 1) for all n > 1 if m is even, and for all n > 2 if n is odd. Moreover, the intersection problem for even cycle systems is completely solved for an equivalence class x (mod 2m) once it is solved for the smallest in that equivalence class and for K_2m+1. For odd cycle systems, results are similar, although generally the two smallest values in each equivalence class need to be solved. We also completely solve the intersection problem for m = 4, 6, 7, 8, and 9. (The cased m = 5 was done by C-M. K. Fu in 1987.) © 1993 John Wiley & Sons, Inc.     

On the doyen-wilson theorem for m-cycle systems

In this article we consider the embedding of m-cycle systems of order u in m-cycle systems of order v when m is odd. When u and v are 1 or m (mod 2m) we completely settle this problem, except possibly for the smallest such embedding in some cases when u ≡ v ≡ m (mod 2m). In particular, there are no exceptions if m ∈ {7,9}, so the generalization of the Doyen-Wilson Theorem is now settled for all odd m with m ≤ 9. © 1994 John Wiley & Sons, Inc.     

The module theory of semisymmetric quasigroups,totally symmetric quasigroups,and triple systems

Journal of Algebraic Combinatorics - This paper initiates a unified module theory for four varieties of quasigroups: semisymmetric, semisymmetric idempotent, totally symmetric, and totally...     

Completing partial commutative quasigroups constructed from partial Steiner triple systems is NP-complete

Deciding whether an arbitrary partial commutative quasigroup can be completed is known to be NP-complete. Here, we prove that it remains NP-complete even if the partial quasigroup is constructed, in the standard way, from a partial Steiner triple system. This answers a question raised by Rosa in [A. Rosa, On a class of completable partial edge-colourings, Discrete Appl. Math. 35 (1992) 293-299]. To obtain this result, we prove necessary and sufficient conditions for the existence of a partial Steiner triple system of odd order having a leave L such that E(L)=E(G) where G is any given graph.     

Dynamical systems arising from random substitutions

Random substitutions are a natural generalisation of their classical ‘deterministic’ counterpart, whereby at every step of iterating the substitution, instead of replacing a letter with a predetermined word, every letter is independently replaced by a word from a finite set of possible words according to a probability distribution. We discuss the subshifts associated with such substitutions and explore the dynamical and ergodic properties of these systems in order to establish the groundwork for their systematic study. Among other results, we show under reasonable conditions that such systems are topologically transitive, have either empty or dense sets of periodic points, have dense sets of linearly repetitive elements, are rarely strictly ergodic, and have positive topological entropy.     

Solving Hamiltonian systems arising from ODE eigenproblems

This paper presents an algorithm for solving a linear Hamiltonian system arising in the study of certain ODE eigenproblems. The method follows the phase angles of an associated unitary matrix, which are essential for correct indexing of the eigenvalues of the ODE. Compared to the netlib code SL11F [11] the new method has the property that on many important problems – in particular, on matrix–vector Schrödinger equations – the cost of the integration is bounded independently of the eigenparameter λ. This allows large eigenvalues to be found much more efficiently. Numerical results show that our implementation of the new algorithm is substantially faster than the netlib code SL11F.     

Random dynamical systems arising from iterated function systems with place-dependent probabilities

We show that if an iterated function system with place-dependent probabilities admits an invariant and attractive measure, then it has the structure of a random dynamical system (in the sense of Ludwig Arnold).     

Graph problems arising from parameter identification of discrete dynamical systems

This paper focuses on combinatorial feasibility and optimization problems that arise in the context of parameter identification of discrete dynamical systems. Given a candidate parametric model for a physical system and a set of experimental observations, the objective of parameter identification is to provide estimates of the parameter values for which the model can reproduce the experiments. To this end, we define a finite graph corresponding to the model, to each arc of which a set of parameters is associated. Paths in this graph are regarded as feasible only if the sets of parameters corresponding to the arcs of the path have nonempty intersection. We study feasibility and optimization problems on such feasible paths, focusing on computational complexity. We show that, under certain restrictions on the sets of parameters, some of the problems become tractable, whereas others are NP-hard. In a similar vein, we define and study some graph problems for experimental design, whose goal is to support the scientist in optimally designing new experiments.     

Transitive resolvable idempotent quasigroups and large sets of resolvable Mendelsohn triple systems

We first define a transitive resolvable idempotent quasigroup (TRIQ), and show that a TRIQ of order v exists if and only if 3∣v and . Then we use TRIQ to present a tripling construction for large sets of resolvable Mendelsohn triple systems s, which improves an earlier version of tripling construction by Kang. As an application we obtain an for any integer n≥1, which provides an infinite family of even orders.     

The spectrum of dynamical systems arising from substitutions of constant length

Probability Theory and Related Fields - Minimal flows and dynamical systems arising from substitutions are considered. In the case of substitutions of constant length the trace relation of the flow...     

Similarity solutions for systems arising from an Aedes aegypti model

In a recent paper a new model for the Aedes aegypti mosquito dispersal dynamics was proposed and its Lie point symmetries were investigated. According to the carried group classification, the maximal symmetry Lie algebra of the nonlinear cases is reached whenever the advection term vanishes. In this work we analyze the family of systems obtained when the wind effects on the proposed model are neglected. Wide new classes of solutions to the systems under consideration are obtained.