Edge-disjoint Hamilton cycles in graphs

In this paper we give an approximate answer to a question of Nash-Williams from 1970: we show that for every \alpha > 0, every sufficiently large graph on n vertices with minimum degree at least (1/2 + \alpha)n contains at least n/8 edge-disjoint Hamilton cycles. More generally, we give an asymptotically best possible answer for the number of edge-disjoint Hamilton cycles that a graph G with minimum degree \delta must have. We also prove an approximate version of another long-standing conjecture of Nash-Williams: we show that for every \alpha > 0, every (almost) regular and sufficiently large graph on n vertices with minimum degree at least $(1/2 + \alpha)n$ can be almost decomposed into edge-disjoint Hamilton cycles.Comment: Minor Revisio

Boundary Treatment and Multigrid Preconditioning for Semi-Lagrangian Schemes Applied to Hamilton-Jacobi-Bellman Equations

We analyse two practical aspects that arise in the numerical solution of Hamilton-Jacobi-Bellman (HJB) equations by a particular class of monotone approximation schemes known as semi-Lagrangian schemes. These schemes make use of a wide stencil to achieve convergence and result in discretization matrices that are less sparse and less local than those coming from standard finite difference schemes. This leads to computational difficulties not encountered there. In particular, we consider the overstepping of the domain boundary and analyse the accuracy and stability of stencil truncation. This truncation imposes a stricter CFL condition for explicit schemes in the vicinity of boundaries than in the interior, such that implicit schemes become attractive. We then study the use of geometric, algebraic and aggregation-based multigrid preconditioners to solve the resulting discretised systems from implicit time stepping schemes efficiently. Finally, we illustrate the performance of these techniques numerically for benchmark test cases from the literature

Prolactin delays hair regrowth in mice

Mammalian hair growth is cyclic, with hair-producing follicles alternating between active (anagen) and quiescent (telogen) phases. The timing of hair cycles is advanced in prolactin receptor (PRLR) knockout mice, suggesting that prolactin has a role in regulating follicle cycling. In this study, the relationship between profiles of circulating prolactin and the first post-natal hair growth cycle was examined in female Balb/c mice. Prolactin was found to increase at 3 weeks of age, prior to the onset of anagen 1 week later. Expression of PRLR mRNA in skin increased fourfold during early anagen. This was followed by upregulation of prolactin mRNA, also expressed in the skin. Pharmacological suppression of pituitary prolactin advanced dorsal hair growth by 3.5 days. Normal hair cycling was restored by replacement with exogenous prolactin for 3 days. Increasing the duration of prolactin treatment further retarded entry into anagen. However, prolactin treatments, which began after follicles had entered anagen at 26 days of age, did not alter the subsequent progression of the hair cycle. Skin from PRLR-deficient mice grafted onto endocrine-normal hosts underwent more rapid hair cycling than comparable wild-type grafts, with reduced duration of the telogen phase. These experiments demonstrate that prolactin regulates the timing of hair growth cycles in mice via a direct effect on the skin, rather than solely via the modulation of other endocrine factors

Finding a Maximum 2-Matching Excluding Prescribed Cycles in Bipartite Graphs

We introduce a new framework of restricted 2-matchings close to Hamilton cycles. For an undirected graph (V,E) and a family U of vertex subsets, a 2-matching F is called U-feasible if, for each setU in U, F contains at most |setU|-1 edges in the subgraph induced by U. Our framework includes C_{= 5. For instance, in bipartite graphs in which every cycle of length six has at least two chords, our algorithm solves the maximum C_{<=6}-free 2-matching problem in O(n^2 m) time, where n and m are the numbers of vertices and edges, respectively

Resolution of the Oberwolfach problem

The Oberwolfach problem, posed by Ringel in 1967, asks for a decomposition of $K_{2n+1}$ into edge-disjoint copies of a given $2$-factor. We show that this can be achieved for all large $n$. We actually prove a significantly more general result, which allows for decompositions into more general types of factors. In particular, this also resolves the Hamilton-Waterloo problem for large $n$.Comment: 28 page

Impacts of financial regulation on the cyclicality of banks’ capital requirements and on financial stability

One of the main functions of the central bank is to strengthen the stability of the financial system, an important aspect of which is to take an active part in the legislation process to improve the regulatory environment and to assess the potential impacts of new regulatory measures. In the summer of 2007 substantial changes took place in the governance of financial institutions with the introduction of regulations based on the new Basel capital standards (Basel II). The objective of this study is to investigate the likely consequences of such new bank regulations and their potential impact on financial stability. To this end, the study analyses the foreseeable developments in the cyclicality of capital requirements of banks based on the corporate credit portfolio of internationally active large banks, and points out that bank regulations are not always capable of fulfilling their intended function of enhancing financial stability in times of economic distress. Notably, the prospective increase in the cyclicality of capital requirements could well lead to a deepening of economic problems and to instability in the banking system, if the banking system appears undercapitalised relative to the risks assumed. All of this highlights the need for the development of a forward-looking risk assessment system and a supportive regulatory regime providing proper incentives.Basel II, credit risk, capital requirement, regulation, cyclicality, financial stability.

Cycle factors and renewal theory

For which values of $k$ does a uniformly chosen $3$-regular graph $G$ on $n$ vertices typically contain $n/k$ vertex-disjoint $k$-cycles (a $k$-cycle factor)? To date, this has been answered for $k=n$ and for $k \ll \log n$; the former, the Hamiltonicity problem, was finally answered in the affirmative by Robinson and Wormald in 1992, while the answer in the latter case is negative since with high probability most vertices do not lie on $k$-cycles. Here we settle the problem completely: the threshold for a $k$-cycle factor in $G$ as above is $\kappa_0 \log_2 n$ with $\kappa_0=[1-\frac12\log_2 3]^{-1}\approx 4.82$. Precisely, we prove a 2-point concentration result: if $k \geq \kappa_0 \log_2(2n/e)$ divides $n$ then $G$ contains a $k$-cycle factor w.h.p., whereas if $k<\kappa_0\log_2(2n/e)-\frac{\log^2 n}n$ then w.h.p. it does not. As a byproduct, we confirm the "Comb Conjecture," an old problem concerning the embedding of certain spanning trees in the random graph $G(n,p)$. The proof follows the small subgraph conditioning framework, but the associated second moment analysis here is far more delicate than in any earlier use of this method and involves several novel features, among them a sharp estimate for tail probabilities in renewal processes without replacement which may be of independent interest.Comment: 45 page