The multicovering radius problem for some types of discrete structures

The covering radius problem is a question in coding theory concerned with finding the minimum radius $r$ such that, given a code that is a subset of an underlying metric space, balls of radius $r$ over its code words cover the entire metric space. Klapper introduced a code parameter, called the multicovering radius, which is a generalization of the covering radius. In this paper, we introduce an analogue of the multicovering radius for permutation codes (cf. Keevash and Ku, 2006) and for codes of perfect matchings (cf. Aw and Ku, 2012). We apply probabilistic tools to give some lower bounds on the multicovering radii of these codes. In the process of obtaining these results, we also correct an error in the proof of the lower bound of the covering radius that appeared in Keevash and Ku (2006). We conclude with a discussion of the multicovering radius problem in an even more general context, which offers room for further research.Comment: To appear in Designs, Codes and Cryptography (2012

On the String Consensus Problem and the Manhattan Sequence Consensus Problem

In the Manhattan Sequence Consensus problem (MSC problem) we are given $k$ integer sequences, each of length $l$, and we are to find an integer sequence $x$ of length $l$ (called a consensus sequence), such that the maximum Manhattan distance of $x$ from each of the input sequences is minimized. For binary sequences Manhattan distance coincides with Hamming distance, hence in this case the string consensus problem (also called string center problem or closest string problem) is a special case of MSC. Our main result is a practically efficient $O(l)$-time algorithm solving MSC for $k\le 5$ sequences. Practicality of our algorithms has been verified experimentally. It improves upon the quadratic algorithm by Amir et al.\ (SPIRE 2012) for string consensus problem for $k=5$ binary strings. Similarly as in Amir's algorithm we use a column-based framework. We replace the implied general integer linear programming by its easy special cases, due to combinatorial properties of the MSC for $k\le 5$. We also show that for a general parameter $k$ any instance can be reduced in linear time to a kernel of size $k!$, so the problem is fixed-parameter tractable. Nevertheless, for $k\ge 4$ this is still too large for any naive solution to be feasible in practice.Comment: accepted to SPIRE 201

Vector-meson contributions do not explain the rate and spectrum in K_L -> pi0 gamma gamma

We analyze the recent NA48 data for the reaction K_L -> pi0 gamma gamma with and without the assumption of vector meson dominance (VMD). We find that the data is well described by a three-parameter expression inspired by O(p^6) chiral perturbation theory. We also find that it is impossible to fit the shape of the decay distribution and the overall rate simultaneously if one imposes the VMD constraints on the three parameters. We comment on the different fits and their implications for the CP-conserving component of the decay K_L -> pi0 e+ e-.Comment: Version accepted for publication on Phys. Rev. D. 19 pages, LaTeX, 8 figures, uses epsf.st

Quantum mechanics on non commutative spaces and squeezed states: a functional approach

We review here the quantum mechanics of some noncommutative theories in which no state saturates simultaneously all the non trivial Heisenberg uncertainty relations. We show how the difference of structure between the Poisson brackets and the commutators in these theories generically leads to a harmonic oscillator whose positions and momenta mean values are not strictly equal to the ones predicted by classical mechanics. This raises the question of the nature of quasi classical states in these models. We propose an extension based on a variational principle. The action considered is the sum of the absolute values of the expressions associated to the non trivial Heisenberg uncertainty relations. We first verify that our proposal works in the usual theory i.e we recover the known Gaussian functions. Besides them, we find other states which can be expressed as products of Gaussians with specific hyper geometrics. We illustrate our construction in two models defined on a four dimensional phase space: a model endowed with a minimal length uncertainty and the non commutative plane. Our proposal leads to second order partial differential equations. We find analytical solutions in specific cases. We briefly discuss how our proposal may be applied to the fuzzy sphere and analyze its shortcomings.Comment: 15 pages revtex. The title has been modified,the paper shortened and misprints have been corrected. Version to appear in JHE

Magnetic Field Stimulated Transitions of Excited States in Fast Muonic Helium Ions

It is shown that one can stimulate, by using the present-day laboratory magnetic fields, transitions between the $lm$ sub-levels of fast $\mu He^+$ ions formating in muon catalyzed fusion. Strong fields also cause the self-ionization from highly excited states of such muonic ions. Both effects are the consequence of the interaction of the bound muon with the oscillating field of the Stark term coupling the center-of-mass and muon motions of the $\mu He^+$ ion due to the non-separability of the collective and internal variables in this system. The performed calculations show a possibility to drive the population of the $lm$ sub-levels by applying a field of a few $Tesla$, which affects the reactivation rate and is especially important to the $K\alpha$ $x$-ray production in muon catalyzed fusion. It is also shown that the $2s-2p$ splitting in $\mu He^+$ due to the vacuum polarization slightly decreases the stimulated transition rates.Comment: 5 figure

Semiclassical force for electroweak baryogenesis: three-dimensional derivation

We derive a semiclassical transport equation for fermions propagating in the presence of a CP-violating planar bubble wall at a first order electroweak phase transition. Starting from the Kadanoff-Baym (KB) equation for the two-point (Wightman) function we perform an expansion in gradients, or equivalently in the Planck constant h-bar. We show that to first order in h-bar the KB equations have a spectral solution, which allows for an on-shell description of the plasma excitations. The CP-violating force acting on these excitations is found to be enhanced by a boost factor in comparison with the 1+1-dimensional case studied in a former paper. We find that an identical semiclassical force can be obtained by the WKB method. Applications to the MSSM are also mentioned.Comment: 19 page

Three Dimensional N=2 Supersymmetry on the Lattice

We show how 3-dimensional, N=2 supersymmetric theories, including super QCD with matter fields, can be put on the lattice with existing techniques, in a way which will recover supersymmetry in the small lattice spacing limit. Residual supersymmetry breaking effects are suppressed in the small lattice spacing limit by at least one power of the lattice spacing a.Comment: 21 pages, 2 figures, typo corrected, reference adde

Evolutionarily stable defence and signalling of that defence

We examine the evolution and maintenance of defence and conspicuousness in prey species using a game theoretic model. In contrast to previous works, predators can raise as well as lower their attack probabilities as a consequence of encountering moderately defended prey. Our model predicts four distinct possibilities for evolutionarily stable strategies (ESSs) featuring maximum crypsis. Namely that such a solution can exist with (1) zero toxicity, (2) a non-zero but non-aversive level of toxicity, (3) a high, aversive level of toxicity or (4) that no such maximally cryptic solution exists. Maximally cryptic prey may still invest in toxins, because of the increased chance of surviving an attack (should they be discovered) that comes from having toxins. The toxin load of maximally cryptic prey may be sufficiently strong that the predators will find them aversive, and seek to avoid similar looking prey in future. However, this aversiveness does not always necessarily trigger aposematic signalling, and highly toxic prey can still be maximally cryptic, because the increased initial rate of attack from becoming more conspicuous is not necessarily always compensated for by increased avoidance of aversive prey by predators. In other circumstances, the optimal toxin load may be insufficient to generate aversion but still be non-zero (because it increases survival), and in yet other circumstances, it is optimal to make no investment in toxins at all. The model also predicts ESSs where the prey are highly defended and aversive and where this defence is advertised at a cost of increased conspicuousness to predators. In many circumstances there is an infinite array of these aposematic ESSs, where the precise appearance is unimportant as long as it is highly visible and shared by all members of the population. Yet another class of solutions is possible where there is strong between-individual variation in appearance between conspicuous, poorly defended prey

K_S\rightarrow \gamma\gamma , K_L\rightarrow\pi^0\gamma\gamma$ and Unitarity

Agreement between the experimental value $\Gamma (K_S\rightarrow \gamma\gamma)$ and the number predicted via a one-loop chiral perturbation theory calculation has been cited as a success for the latter. On the other hand the one-loop prediction for the closely related process $K_L\rightarrow \pi^0\gamma\gamma$ has been found to be a factor three below the experimental value. Using the inputs of unitarity and dispersion relations, we demonstrate the importance of higher order loop effects to both of these processes.Comment: 20 pages (4 figures available on request), UMHEP-39

Role of Scalar Meson Resonances in $K_{L}^{0} \rightarrow \pi^{0} \gamma \gamma Decay

Corrections to $K_{L}^{0}\rightarrow \pi^{0} \gamma \gamma$ decay induced by scalar meson exchange are studied within chiral perturbation theory. In spite of bad knowledge of scalar-mesons parameters, the calculated branching ratio can be changed by a few percent.Comment: 18 pages of text, 2 figures (available upon request); preprint IJS-TP-16-94 , TUM-T31-63-94