Kronecker Constants for Finite Subsets of Integers

A set of integers S is called ε-Kronecker if every function on S of modulus one can be approximated uniformly to within ε by a character. The least such ε is called the ε-Kronecker constant.     

Methods to compare expensive stochastic optimization algorithms with random restarts

We consider the challenge of numerically comparing optimization algorithms that employ random-restarts under the assumption that only limited test data is available. We develop a bootstrapping technique to estimate the incumbent solution of the optimization problem over time as a stochastic process. The asymptotic properties of the estimator are examined and the approach is validated by an out-of-sample test. Finally, three methods for comparing the performance of different algorithms based on the estimator are proposed and demonstrated with data from a real-world optimization problem.     

Smoothness of convolution powers of orbital measures on the symmetric space SU(n)/SO(n)

We prove that if ${\mu _{a}\,{=}\,m_{K}*\delta _{a}*m_{K}}$ is the K-bi-invariant measure supported on the double coset ${KaK\subseteq SU(n)}$ , for K = SO(n), then ${\mu _{a}^{k}}$ is absolutely continuous with respect to the Haar measure on SU(n) for all a not in the normalizer of K if and only if k ≥ n. The measure, μ _a , supported on the minimal dimension double coset has the property that ${\mu _{a}^{n-1}}$ is singular to the Haar measure.     

On the spectra of Pisot-cyclotomic numbers

We investigate the complex spectra

$$\begin{aligned} X^\mathcal A(\beta )=\left\{ \sum _{j=0}^na_j\beta ^j : n\in \mathbb N,\ a_j\in \mathcal A\right\} \end{aligned}$$

where \(\beta \) is a quadratic or cubic Pisot-cyclotomic number and the alphabet \(\mathcal A\) is given by 0 along with a finite collection of roots of unity. Such spectra are discrete aperiodic structures with crystallographically forbidden symmetries. We discuss in general terms under which conditions they possess the Delone property required for point sets modeling quasicrystals. We study the corresponding Voronoi tilings and we relate these structures to quasilattices arising from the cut-and-project method.

     

Sparse color-critical graphs and hypergraphs with no short cycles

We give constructions of color-critical graphs and hypergraphs with no short cycles and with relatively few edges. In particular, we show that, for each n ≧ 3, the smallest number of edges in a 3-critical triangle-free n-graph (hypergraph) with m vertices is m + o(m) as m → ∞. Also, for each r ≧ 4, there exists an r-critical triangle-free 2-graph (graph) with m vertices and at most (r ? (7/3))m + o(m) edges. Weaker results are obtained for the existence of r-critical graphs containing no cycle of length at most / > 3.     

Feasibility study of pulse compression by two-photon self-induced transparency

The possibility of pulse compression by the process of two-photon self-induced transparency (TPSIT) is considered. Numerical solutions to the two-photon Bloch-Maxwell equations are presented, and limits established for the maximum amount of detuning from exact resonance for which the compression process will still occur.     

Direct in vivo imaging of ferrous iron dyshomeostasis in ageing Caenorhabditis elegans

Iron is essential for eukaryotic biochemistry. Systematic trafficking and storage is required to maintain supply of iron while preventing it from catalysing unwanted reactions, particularly the generation of oxidising reactive species. Iron dyshomeostasis has been implicated in major age-associated diseases including cancers, neurodegeneration and heart disease. Here, we employ population-level X-ray fluorescence imaging and native-metalloproteomic analysis to determine that altered iron coordination and distribution is a pathological imperative of ageing in the nematode, Caenorhabditis elegans. Our approach provides a method to simultaneously study iron metabolism across different scales of biological organisation, from populations to cells. Here we report how and where iron homeostasis is lost during C. elegans ageing, and its relationship to the age-related elevation of damaging reactive oxygen species. We find that wild types utilise ferritin to sustain longevity, buffering against exogenous iron and showing rapid ageing if ferritin is ablated. After reproduction, escape of iron from safe-storage in ferritin raised cellular Fe²⁺ load in the ageing C. elegans, and increased generation of reactive species. These findings support the hypothesis that iron-mediated processes drive senescence. We propose that loss of iron homeostasis may be a fundamental and inescapable consequence of ageing that could represent a critical target for therapeutic strategies to improve health outcomes in ageing.     

Vertex transversals that dominate

For any graph, there is a largest integer k such that given any partition of the vertex set with at most k elements in each class of the partition, there is transversal of the partition that is a dominating set in the graph. Some basic results about this parameter, the partition domination number, are obtained. In particular, it is shown that its value is 2 for the two-dimensional infinite grid, and that determining whether a given vertex partition admits a dominating transversal is NP-complete, even for a graph which is a simple path. The existence of various dominating transversals in certain partitions in regular graphs is studied as well. © 1996 John Wiley & Sons, Inc.     

A new type of Littlewood-Paley partition

We define a partition of Z into intervals {I _j} and prove the Littlewood-Paley inequality ‖f‖ _p ≦C _p‖Sf‖ _p , 2≦p<∞. Heref is a function on [o, 2π) and \(Sf = (\sum |\Delta _j |^2 )^{1/2} , \hat \Delta j = \hat f\chi _{Ij} \) . This is a new example of a partition having the Littlewood-Paley property since the {I _j} are not of the type obtained by iterating lacunary partitions finitely many times.     

Models and algorithms to improve earthwork operations in road design using mixed integer linear programming

In road construction, earthwork operations account for about 25% of the construction costs. Existing linear programming models for earthwork optimization are designed to minimize the hauling costs and to balance the earth across the construction site. However, these models do not consider the removal of physical blocks that may influence the earthwork process. As such, current models may result in inaccurate estimates of optimal earthwork costs, leading to poor choices in road design. In this research, we extend the classical linear program model of earthwork operations to a mixed integer linear program model that accounts for blocks. We examine the economic impact of incorporating blocks via mixed integer linear programming, and find significant savings for most road designs in our test-set. However, the resulting model is considerably harder to solve than the original linear program. Based on structural observations, we introduce a set of algorithms that theoretically reduce the solving time of the model. We confirm this reduction in solve time with numerical experiments.