Algorithmic and Hardness Results for the Colorful Components Problems

In this paper we investigate the colorful components framework, motivated by applications emerging from comparative genomics. The general goal is to remove a collection of edges from an undirected vertex-colored graph $G$ such that in the resulting graph $G'$ all the connected components are colorful (i.e., any two vertices of the same color belong to different connected components). We want $G'$ to optimize an objective function, the selection of this function being specific to each problem in the framework. We analyze three objective functions, and thus, three different problems, which are believed to be relevant for the biological applications: minimizing the number of singleton vertices, maximizing the number of edges in the transitive closure, and minimizing the number of connected components. Our main result is a polynomial time algorithm for the first problem. This result disproves the conjecture of Zheng et al. that the problem is $NP$-hard (assuming $P \neq NP$). Then, we show that the second problem is $APX$-hard, thus proving and strengthening the conjecture of Zheng et al. that the problem is $NP$-hard. Finally, we show that the third problem does not admit polynomial time approximation within a factor of $|V|^{1/14 - \epsilon}$ for any $\epsilon > 0$, assuming $P \neq NP$ (or within a factor of $|V|^{1/2 - \epsilon}$, assuming $ZPP \neq NP$).Comment: 18 pages, 3 figure

Unfolding symmetric Bogdanov-Takens bifurcations for front dynamics in a reaction-diffusion system

This manuscript extends the analysis of a much studied singularly perturbed three-component reaction-diffusion system for front dynamics in the regime where the essential spectrum is close to the origin. We confirm a conjecture from a preceding paper by proving that the triple multiplicity of the zero eigenvalue gives a Jordan chain of length three. Moreover, we simplify the center manifold reduction and computation of the normal form coefficients by using the Evans function for the eigenvalues. Finally, we prove the unfolding of a Bogdanov-Takens bifurcation with symmetry in the model. This leads to stable periodic front motion, including stable traveling breathers, and these results are illustrated by numerical computations.Comment: 39 pages, 7 figure

Exchange Bias Induced by the Fe3O4 Verwey transition

We present a study of the exchange bias in different configurations of V2O3 thin films with ferromagnetic layers. The exchange bias is accompanied by a large vertical shift in the magnetization. These effects are only observed when V2O3 is grown on top of Ni80Fe20 permalloy. The magnitude of the vertical shift is as large as 60% of the total magnetization which has never been reported in any system. X-Ray diffraction studies show that the growth conditions promote the formation of a ferrimagnetic Fe3O4 interlayer. The change in the easy magnetization axis of Fe3O4 across the Verwey transition at 120 K is correlated with the appearance of exchange bias and vertical shift in magnetization. Both phenomena disappear above 120 K, indicating for the first time a direct relationship between the magnetic signature of the Verwey transition and exchange bias.Comment: Accepted for publication Physical Review

Review of the Laguerre-Gauss mode technology research program at Birmingham

Gravitational wave detectors from the advanced generation onwards are expected to be limited in sensitivity by thermal noise of the optics, making the reduction of this noise a key factor in the success of such detectors. A proposed method for reducing the impact of this noise is to use higher-order Laguerre-Gauss (LG) modes for the readout beam, as opposed to the currently used fundamental mode. We present here a synopsis of the research program undertaken by the University of Birmingham into the suitability of LG mode technology for future gravitational wave detectors. This will cover our previous and current work on this topic, from initial simulations and table-top LG mode experiments up to implementation in a prototype scale suspended cavity and high-power laser bench

Fungal Peptaibiotics: Assessing Potential Meteoritic Amino Acid Contamination

The presence of non-protein alpha-dialkyl-amino acids such as alpha-aminoisobutyric acid (alpha-A1B) and isovaline (Iva), which are relatively rare in the terrestrial biosphere, has long been used as an indication of the indigeneity of meteoritic amino acids, however, the discovery of alpha-AIB in peptides producers by a widespread group of filamentous fungi indicates the possibility of a terrestrial biotic source for the alpha-AIB observed in some meteorites. The alpha-AIB-containing peptides produced by these fungi are dubbed peptaibiotics. We measured the molecular distribution and stable carbon and nitrogen isotopic ratios for amino acids found in the total hydrolysates of four biologically synthesized peptaibiotics. We compared these aneasurenetts with those from the CM2 carbonaceous chondrite Murchison and from three Antarctic CR2 carbonaceous chondrites in order to understand the peptaibiotics as a potential source of meteoritic contamination

Diffusive stability for periodic metric graphs

Analysis and Stochastic