ADAPTIVE MAJORITY PROBLEMS FOR RESTRICTED QUERY GRAPHS AND FOR WEIGHTED SETS

Suppose that the vertices of a graph G are colored with two colors in an unknown way. The color that occurs on more than half of the vertices is called the majority color (if it exists), and any vertex of this color is called a majority vertex. We study the problem of finding a majority vertex (or show that none exists), if we can query edges to learn whether their endpoints have the same or different colors. Denote the least number of queries needed in the worst case by m(G). It was shown by Saks and Werman that m(K-n) = n - b(n) where b(n) is the number of 1's in the binary representation of n. In this paper we initiate the study of the problem for general graphs. The obvious bounds for a connected graph G on n vertices are n - b(n) <= m(G) <= n - 1. We show that for any tree T on an even number of vertices we have m(T) = n - 1, and that for any tree T on an odd number of vertices, we have n - 65 <= m (T) <= n - 2. Our proof uses results about the weighted version of the problem for K-n, which may be of independent interest. We also exhibit a sequence G(n) of graphs with m(G(n)) = n - b(n) such that the number of edges in G(n) is O(nb(n))

Spectral functions in the sigma-channel near the critical end point

Spectral functions in the $\sigma$-channel are investigated near the chiral critical end point (CEP), that is, the point where the chiral phase transition ceases to be first-ordered in the $(\mu,T)$-plane of the QCD phase diagram. At that point the $\sigma$ meson becomes massless in spite of explicit breaking of the chiral symmetry. It is expected that experimental signatures peculiar to CEP can be observed through spectral changes in the presence of abnormally light $\sigma$ mesons. As a candidate, the invariant-mass spectrum for diphoton emission is estimated with the chiral quark model incorporated. The results show the characteristic shape with a peak in the low energy region, which may serve as a signal for CEP. However, we find that the diphoton multiplicity is highly suppressed by infrared behaviors of the $\sigma$ meson. Experimentally, in such a low energy region below the threshold of two pions, photons from $\pi^0\to2\gamma$ are major sources of the background for the signal.Comment: 12 pages, 8 figures, 1 figure replaced, minor modification

Scalar density fluctuation at critical end point in NJL model

Soft mode near the critical end point in the phase diagram of two-flavor Nambu--Jona-Lasinio (NJL) model is investigated within the leading 1/N_c approximation with N_c being the number of the colors. It is explicitly shown by studying the spectral function of the scalar channel that the relevant soft mode is the scalar density fluctuation, which is coupled with the quark number density, while the sigma meson mode stays massive.Comment: 9 pages, 4 figure

The O(N) linear sigma model at finite temperature beyond the Hartree approximation

We study the O(N) linear sigma model with spontaneous symmetry breaking, using a Hartree-like ansatz with a classical field and variational masses. We go beyond the Hartree approximation by including the two-loop contribution, the sunset diagram, using the 2PPI expansion. We have computed numerically the effective potential at finite temperature. We find a phase transition of second order, while it is first order in the Hartree approximation. We also discuss some implications of the fact that in this order, the decay of the sigma into two pions affects the thermal diagrams.Comment: 22 pages, 14 figures. v2: minor corrections, some more references. v3: added new set of data, new appendix. Submitted to Phys.Rev.

Flux tube dynamics in the dual superconductor

We study plasma oscillations in a flux tube of the dual superconductor model of 't Hooft and Mandelstam. A magnetic condensate is coupled to an electromagnetic field by its dual vector potential, and fixed electric charges set up a flux tube. An electrically charged fluid (a quark plasma) flows in the tube and screens the fixed charges via plasma oscillations. We investigate both Type I and Type II superconductors, with plasma frequencies both above and below the threshold for radiation into the Higgs vacuum. We find strong radiation of electric flux into the superconductor in all regimes, and argue that this invalidates the use of the simplest dual superconductor model for dynamical problems.Comment: 25 pages Revtex with 11 EPS figure

Rule-Based Context Assessment in Smart Cities

International audienceno abstrac

A Hilton-Milner theorem for vector spaces

We show for k � 2 that if q � 3 and n � 2k + 1, or q = 2 and n � 2k + 2, then any intersecting family F of k-subspaces of an n-dimensional vector space over GF(q) with ⋂ F ∈F F = 0 has size at most [] n−1 k−1 − qk(k−1) [] n−k−1 k−1 + qk. This bound is sharp as is shown by Hilton-Milner type families. As an application of this result, we determine the chromatic number of the corresponding q-Kneser graphs