A note on the onset of synchrony in avian ovulation cycles

Spontaneous oscillator synchrony occurs when populations of interacting oscillators begin cycling together in the absence of environmental forcing. Synchrony has been documented in many physical and biological systems, including oestrus/menstrual cycles in rats and humans. In previous work we showed that Glaucous-winged Gulls (Larus glaucescens) can lay eggs synchronously on an every-other-day schedule, and that synchrony increases with colony density. Here we pose a discrete-time model of avian ovulation to study the dynamics of synchronization. We prove the existence and uniqueness of an equilibrium solution which bifurcates to increasingly synchronous cycles as colony density increases.     

Colony and evolutionary dynamics of a two‐stage model with brood cannibalism and division of labor in social insects

Division of labor (DOL) is a major factor for the great success of social insects because it increases the efficiency of a social group where different individuals perform different tasks repeatedly and presumably with increased performance. Cannibalism plays an important role in regulating colony growth and development by regulating the number of individuals in a colony and increasing survival by providing access to essential nutrients and minimizing competition among colony mates. To understand the synergy effects of DOL and cannibalistic behavior on colony dynamic outcomes, we propose and study a compartmental two‐stage model using ecological and evolutionary game theory settings. Our analytical results of the ecological and evolutionary models suggest that: (1) A noncannibalistic colony can survive if the efficiency of energy investment reflecting the DOL is greater than the relative death rate of the older population. (2) A cannibalistic colony can die out if both the efficiency of energy investment and the relative cannibalism rate (where each is also reflecting the DOL) are too large; or if the relative cannibalism rate alone is too small. (3) From our numerical analysis, cannibalism can increase or reduce the colony's total population size, which greatly depends on the benefit of egg cannibalism increasing or decreasing of adult's lifespan. (4) A cannibalistic and noncannibalistic colony can experience bistability due to cooperative behavior. (5) In the evolutionary settings, DOL can prevent colony death and natural selection can preserve strong Allee effects by selecting the traits with the largest investment on brood care and the lowest cannibalism rate. (6) Evolutionary dynamics may increase the fitness of the colony, i.e., the successful production of workforce which results in the increase of total worker population size, colony survival, and reproduction. Our results suggest both cannibalism and DOLs are adaptive strategies that increase the size of the worker population, and therefore, persistence of the colony.     

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.     

A characterization of diameter-2-critical graphs with no antihole of length four

A graph G is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. In this paper we characterize the diameter-2-critical graphs with no antihole of length four, that is, the diameter-2-critical graphs whose complements have no induced 4-cycle. Murty and Simon conjectured that the number of edges in a diameter-2-critical graph of order n is at most n ²/4 and that the extremal graphs are complete bipartite graphs with equal size partite sets. As a consequence of our characterization, we prove the Murty-Simon Conjecture for graphs with no antihole of length four.     

Multigrid algorithms for symmetric discontinuous Galerkin methods on graded meshes

We study a class of symmetric discontinuous Galerkin methods on graded meshes. Optimal order error estimates are derived in both the energy norm and the L ₂ norm, and we establish the uniform convergence of V-cycle, F-cycle and W-cycle multigrid algorithms for the resulting discrete problems. Numerical results that confirm the theoretical results are also presented.     

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.     

Equipartite Gregarious 5-Cycle Systems and Other Results

A k-cycle decomposition of a complete multipartite graph is said to be gregarious if each k-cycle in the decomposition has its vertices in k different partite sets. Equipartite 3-cycle systems are 3-GDDs (and so are automatically gregarious), and necessary and sufficient conditions for their existence are known. The cases of equipartite gregarious 4-, 6- and 8-cycle systems have also been dealt with (using techniques that could be applied in the case of any even length cycle). Here we give necessary and sufficient conditions for the existence of a gregarious 5-cycle decomposition of the complete equipartite graph K_m(n) (in effect the first odd length cycle case for which the gregarious constraint has real meaning). In doing so, we also define some general cyclic constructions for the decomposition of certain complete equipartite graphs into gregarious p-cycles (where p is an odd prime).     

Some sufficient conditions for a planar graph of maximum degree six to be Class 1

Let G be a planar graph of maximum degree 6. In this paper we prove that if G does not contain either a 6-cycle, or a 4-cycle with a chord, or a 5- and 6-cycle with a chord, then χ^′(G)=6, where χ^′(G) denotes the chromatic index of G.     

Equipartite gregarious 6- and 8-cycle systems

A k-cycle decomposition of a complete multipartite graph is said to be gregarious if each k-cycle in the decomposition has its vertices in k different partite sets. Equipartite gregarious 3-cycle systems are 3-GDDs, and necessary and sufficient conditions for their existence are known (see for instance the CRC Handbook of Combinatorial Designs, 1996, C.J. Colbourn, J.H. Dinitz (Eds.), Section III 1.3). The cases of equipartite and of almost equipartite 4-cycle systems were recently dealt with by Billington and Hoffman. Here, for both 6-cycles and for 8-cycles, we give necessary and sufficient conditions for existence of a gregarious cycle decomposition of the complete equipartite graph K_n(a) (with n parts, n?6 or n?8, of size a).     

The crossing number of C3 × Cn

The crossing number of the Cartesian product C₃ × C_n of a 3-cycle and an n-cycle is shown to be n.     

The taxicab locus of Apollonius: promoting exploratory routes to the punchline using rich undergraduate tasks

ABSTRACT

A key motivational tactic in undergraduate mathematics teaching is to launch topics with fundamental questions that originate from surprising or remarkable phenomena. Nonetheless, constructing a sequence of tasks that promotes students' own routes to resolving such questions is challenging. This note aims to address this challenge in two ways. First, to illustrate the motivational tactic, the taxicab manifestation of a locus attributed to Apollonius is introduced and a natural question arising from comparison with the analogous Euclidean locus is considered, namely, does the taxicab locus of Apollonius ever coincide with a taxicab circle? Second, a companion sequence of rich undergraduate tasks is elaborated using theoretical design principles, with the tasks culminating in this fundamental geometric question. This note therefore provides a design approach that can be replicated in undergraduate teaching contexts based around similarly motivating mathematical phenomena.     

A characterization of 1-cycle resonant graphs among bipartite 2-connected plane graphs

It is proved that a bipartite 2-connected plane graph in which the common boundary of adjacent faces is a simple curve is 1-cycle resonant if and only if the outer face of $G$ is alternating and each inner vertex has degree two. This extends a result from [X. Guo, F. Zhang, $k$-cycle resonant graphs, Discrete Math. 135 (1994) 113–20] that a hexagonal system is 1-cycle resonant if and only if it is catacondensed.     

A Note on Varieties of Groupoids Arising from m-Cycle Systems

Decompositions of the complete graph with n vertices K _n into edge disjoint cycles of length m whose union is K _n are commonly called m-cycle systems. Any m-cycle system gives rise to a groupoid defined on the vertex set of K _n via a well known construction. Here, it is shown that the groupoids arising from all m-cycle systems are precisely the finite members of a variety (of groupoids) for m = 3 and 5 only.     

Varieties of quasigroups arising from 2-perfect m-cycle systems

For m = 6 and for all odd composite integers m, as well as for all even integers m 10 that satisfy certain conditions, 2-perfect m-cycle systems are constructed whose quasigroups have a homomorphism onto quasigroups which do not correspond to a 2-perfect m-cycle systems. Thus it is shown that for these values of m the class of quasigroups arising from all 2-perfect m-cycle systems does not form a variety.     

Planar graphs without adjacent cycles of length at most seven are 3-colorable

We prove that every planar graph in which no i-cycle is adjacent to a j-cycle whenever 3≤i≤j≤7 is 3-colorable and pose some related problems on the 3-colorability of planar graphs.     

Hexagon-free subgraphs of hypercubes

It is shown (for all n ≥ 3) that the edges of the n-cube can be 3-colored in such a way that there is no monochromatic 4-cycle or 6-cycle. © 1993 John Wiley & Sons, Inc.     

On equationally defining m-perfect 2m + 1 cycle systems

In this article we give a complete solution to the problem of equationally defining m-perfect (2m + 1)-cycle systems and of equationally defining m-perfect directed (2m + 1)-cycle systems. © 1994 John Wiley & Sons, Inc.     

ALMOST OPTIMAL LOCAL WELL-POSEDNESS OF THE MAXWELL-KLEIN-GORDON EQUATIONS IN 1 + 4 DIMENSIONS

Abstract

We study strong solutions of the Navier–Stokes equations for nonhomogeneous incompressible fluids in Ω ? R ³. Deriving higher a priori estimates independent of the lower bounds of the density, we prove the existence and uniqueness of local strong solutions to the initial value problem (for Ω =R ³) or the initial boundary value problem (for Ω ? ? R ³) even though the initial density vanishes in an open subset of Ω, i.e., an initial vacuum exists. As an immediate consequence of the a priori estimates, we obtain a continuation theorem for the local strong solutions.     

Enclosings of λ-fold 4-cycle systems

In this paper we solve the problem of enclosing a λ-fold 4-cycle system of order v into a (λ + m)-fold 4-cycle system of order v + u for all m > 0 and u ≥ 1. An ingredient is constructed that is of interest on its own right, namely the problem of finding equitable partial 4-cycle systems of λ K _v . This supplementary solution builds on a result of Raines and Staniszlo.