Choosability of the square of planar subcubic graphs with large girth

We show that the choice number of the square of a subcubic graph with maximum average degree less than 18/7 is at most 6. As a corollary, we get that the choice number of the square of a subcubic planar graph with girth at least 9 is at most 6. We then show that the choice number of the square of a subcubic planar graph with girth at least 13 is at most 5.     

Hitchin-Thorpe inequality for noncompact Einstein 4-manifolds

We prove a Hitchin-Thorpe inequality for noncompact Einstein 4-manifolds with specified asymptotic geometry at infinity. The asymptotic geometry at infinity is either a cusp bundle over a compact space (the fibered cusps) or a fiber bundle over a cone with a compact fiber (the fibered boundary). Many noncompact Einstein manifolds come with such a geometry at infinity.     

Graphs with Branchwidth at Most Three

In this paper we investigate both the structure of graphs with branchwidth at most three, as well as algorithms to recognise such graphs. We show that a graph has branchwidth at most three if and only if it has treewidth at most three and does not contain the three-dimensional binary cube graph as a minor. A set of four graphs is shown to be the obstruction set for the class of graphs with branchwidth at most three. Moreover, we give a safe and complete set of reduction rules for the graphs with branchwidth at most three. Using this set, a linear time algorithm is given that verifies if a given graph has branchwidth at most three, and, if so, outputs a minimum width branch decomposition.     

On Wiener’s Theorem for functions periodic at infinity

We consider the functions periodic at infinity with values in a complex Banach space. The notions are introduced of the canonical and generalized Fourier series of a function periodic at infinity. We prove an analog of Wiener’s Theorem on absolutely convergent Fourier series for functions periodic at infinity whose Fourier series are summable with weight. The two criteria are given: for the function periodic at infinity to be the sum of a purely periodic function and a function vanishing at infinity and for a function to be periodic at infinity. The results of the article base on substantially use on spectral theory of isometric representations.     

A linear system of differential equations with small parameter at a part of derivatives,a deviation of the argument,and a turning point

For a system of differential equations with small parameter at a part of derivatives, a linear deviation of the argument, and a turning point, we obtained conditions, under which its solutions are solutions of a system of differential equations with small parameter at a part of derivatives such that its matrices possess the asymptotic expansions at |ε| ≤ ε₀ with the coefficients holomorphic at |x| ≤ x ₀ . The existence and the infinite differentiability of a solution of the system of differential equations with small parameter at a part of derivatives and with a linear deviation of the argument in the presence of a turning point are proved.     

Relaxations for two-level multi-item lot-sizing problems

We consider several variants of the two-level lot-sizing problem with one item at the upper level facing dependent demand, and multiple items or clients at the lower level, facing independent demands. We first show that under a natural cost assumption, it is sufficient to optimize over a stock-dominant relaxation. We further study the polyhedral structure of a strong relaxation of this problem involving only initial inventory variables and setup variables. We consider several variants: uncapacitated at both levels with or without start-up costs, uncapacitated at the upper level and constant capacity at the lower level, constant capacity at both levels. We finally demonstrate how the strong formulations described improve our ability to solve instances with up to several dozens of periods and a few hundred products.     

Twice Q-polynomial distance-regular graphs of diameter 4

It is known that a distance-regular graph with valency k at least three admits at most two Qpolynomial structures. We show that all distance-regular graphs with diameter four and valency at least three admitting two Q-polynomial structures are either dual bipartite or almost dual bipartite. By the work of Dickie(1995) this implies that any distance-regular graph with diameter d at least four and valency at least three admitting two Q-polynomial structures is, provided it is not a Hadamard graph, either the cube H(d, 2)with d even, the half cube 1/2H(2d + 1, 2), the folded cube?H(2d + 1, 2), or the dual polar graph on [2A2d-1(q)]with q 2 a prime power.     

Even subgraphs of bridgeless graphs and 2-factors of line graphs

By Petersen's theorem, a bridgeless cubic multigraph has a 2-factor. Fleischner generalised this result to bridgeless multigraphs of minimum degree at least three by showing that every such multigraph has a spanning even subgraph. Our main result is that every bridgeless simple graph with minimum degree at least three has a spanning even subgraph in which every component has at least four vertices. We deduce that if G is a simple bridgeless graph with n vertices and minimum degree at least three, then its line graph has a 2-factor with at most max{1,(3n-4)/10} components. This upper bound is best possible.     

Vertex covers and eternal dominating sets

The eternal domination problem requires a graph to be protected against an infinitely long sequence of attacks on vertices by guards located at vertices, the configuration of guards inducing a dominating set at all times. An attack at a vertex with no guard is defended by sending a guard from a neighboring vertex to the attacked vertex. We allow any number of guards to move to neighboring vertices at the same time in response to an attack. We compare the eternal domination number with the vertex cover number of a graph. One of our main results is that the eternal domination number is less than the vertex cover number of any graph of minimum degree at least two having girth at least nine.     

Spanning Trees with a Bounded Number of Branch Vertices in a Claw-Free Graph

Let k be a non-negative integer. A branch vertex of a tree is a vertex of degree at least three. We show two sufficient conditions for a connected claw-free graph to have a spanning tree with a bounded number of branch vertices: (i) A connected claw-free graph has a spanning tree with at most k branch vertices if its independence number is at most 2k + 2. (ii) A connected claw-free graph of order n has a spanning tree with at most one branch vertex if the degree sum of any five independent vertices is at least n ? 2. These conditions are best possible. A related conjecture also is proposed.     

QUADRATIC SYSTEMS WITH A 3RD-ORDER (OR 2ND-ORDER) WEAK FOCUS

1 IntroductionSince a quadratic system has no limit cycle around a 3rd-order weak focu,[1]and has at most one limit cycle surrounding a 2nd-order weak fOcus['], study-ing the number of limit cycles of a p1anar quadratic system with a 3rd-order(or 2nd-order) weak focus we only need to study the number of limit cyclessurrounding the strong focus for the system. Without loss of generality thequadratic system with a 3rd--order (or 2nd--order) weak foclls and a strong focuscan be written in the fo.…     

Homogenization and propagation of chaos to a nonlinear diffusion with sticky reflection

Summary We study a process reflecting in a domain. The process follows Wentzell non-sticky boundary conditions while being adsorbed at the boundary at a certain rate with respect to local time and desorbed at a rate with respect to natural time. We show that when the rates go to infinity with a converging ratio, the process converges to a process with sticky reflection having the limit ratio as the sojourn coefficient. We then study a mean-field interacting system of such particles. We show propagation of chaos to a nonlinear diffusion with sticky reflection when we perform this homogenization simultaneously as the number of particles goes to infinity.     

Periodic waves with constant vorticity in water of infinite depth

Periodic waves propagating at a constant velocity at the surfaceof a fluid with constant vorticity in water of infinite depthare considered. The problem is solved numerically by a boundary-integral-equationmethod. Simmen & Saffman (Stud. Appl. Maths 75, 35, 1985)showed that there are families of solutions which have limitingconfigurations with a 120 angle at their crests or a trappedbubble at their troughs. It is shown that there are additionalfamilies of solutions. These families have limiting configurationswith trapped bubbles at their crests. Each bubble is circularand contains fluid in rigid-body rotation. The results are consistentwith previous calculations for solitary waves in water of finitedepth.     

Tandem Behaviour of a Telecommunication System with Repeated Calls: A Markovian Case with Buffers

The performance of a telecommunication system consisting of a set of transmitters with finite capacity buffers is modelled with a Markovian queueing network, and its tandem behaviour is approximated, in steady state. In this system, a fraction of the units which, at the instants of their arrival at each transmitter, find it busy may retry to be processed by merging with the incoming arrival units at the same transmitter after a fixed delay time. The performance of this system is approximated by a recursive algorithm, in steady state. Furthermore, the approximation outcomes are compared against those from a simulation study. In summary, our numerical results indicate that approximating the non-renewal superposition arrival, the non-renewal overflow and the non-renewal departure processes at each node of the network can be approximated with compatible Poisson processes.     

Optimal vaccination patterns in age-structured populations: Endemic case

We consider a simple age-structured SIR model with vaccination and separable mixing. We define costs for a vaccination strategy (number of vaccinations per time) and effect (reduction of prevalence of disease or reduction of the number of risk cases). We consider two optimization problems: to find the strategy with minimal costs at a given level for the effect, and also to find the strategy with maximal effect at given costs. It turns out that there is an optimal strategy that is nonzero only in at most three age classes, i.e., consists of a sum of at most three delta-peaks.     

On the vibrations described by the telegraph equation in the case of a system consisting of several parts of different densities and elasticities

We consider mixed initial-boundary value problems for longitudinal vibrations described by the telegraph equation in the case of a system consisting of several parts with different densities and elasticities but with equal impedances. We consider the cases of control by displacements at both endpoints of the rod, by elastic forces at both endpoints, and by an elastic force at one endpoint and a displacement at the other endpoint. We find closed-form expressions for the solutions of these mixed problems.     

Finite element calculations of flow past a spherical bubble rising on the axis of a cylindrical tube

The velocity and pressure fields of a Newtonian fluid with homogeneous and constant physical properties flowing around a sphere on the axis of a cylindrical tube with no slip, free slip and partial slip at the sphere surface and no slip at the cylinder wall have been calculated by solving the Navier-Stokes equations and the continuity equation using the finite element technique with the penalty function method. Terminal rise velocities of spherical air bubbles in water have been calculated as function of the bubble radius and some conclusions have been drawn about the nature of the interface. Finally, the influence of the presence of a cylindrical wall on the drag force has been determined and a new empirical equation is derived for the wall correction factor for a sphere rising with free slip at its surface at low Reynolds number.     

Chromatic numbers of quadrangulations on closed surfaces

It has been shown that every quadrangulation on any nonspherical orientable closed surface with a sufficiently large representativity has chromatic number at most 3. In this paper, we show that a quadrangulation G on a nonorientable closed surface N_k has chromatic number at least 4 if G has a cycle of odd length which cuts open N_k into an orientable surface. Moreover, we characterize the quadrangulations on the torus and the Klein bottle with chromatic number exactly 3. By our characterization, we prove that every quadrangulation on the torus with representativity at least 9 has chromatic number at most 3, and that a quadrangulation on the Klein bottle with representativity at least 7 has chromatic number at most 3 if a cycle cutting open the Klein bottle into an annulus has even length. As an application of our theory, we prove that every nonorientable closed surface N_k admits an eulerian triangulation with chromatic number at least 5 which has arbitrarily large representativity. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 100–114, 2001     

On critical graphs with chromatic index 4

It is shown that the number of vertices of valency 2 in a critical graph with chromatic index 4 is at most 1/3 of the total number of vertices, and that there exist critical graphs with just one vertex of valency 2, but none with exactly two vertices of valency 2. From this bounds for the number of edges are deduced. The paper ends with a presentation of a catalogue of all critical graphs with chromatic index 4 and at most 8 vertices, and all simple critical graphs with chromatic index 4 and at most 10 vertices.     

A Mathematical Programming Model of Decentralized Multi-Level Systems

In decentralized systems, the objectives at the different levels often differ. Each level controls only a subset of the decision variables but is affected by the decisions made at the other levels. Recent methods in bi-level linear programming solve such systems better than decomposition methods. Here, the bi-level LP is extended to a bi-level system with many decision-makers at the lower level, and a tri-level system with one decision-maker at each level. In both of these cases, the higher level acts as the leader and the lower level as the follower in the Stackelberg game. The resulting techniques are illustrated with simple numerical examples.