Computing the Girth of a Planar Graph in Linear Time

The girth of a graph is the minimum weight of all simple cycles of the graph. We study the problem of determining the girth of an n-node unweighted undirected planar graph. The first non-trivial algorithm for the problem, given by Djidjev, runs in O(n^{5/4} log n) time. Chalermsook, Fakcharoenphol, and Nanongkai reduced the running time to O(n log^2 n). Weimann and Yuster further reduced the running time to O(n log n). In this paper, we solve the problem in O(n) time.Comment: 20 pages, 7 figures, accepted to SIAM Journal on Computin

Gauge Unification and Quark Masses in a Pati-Salam Model from Branes

We investigate the phase space of parameters in the Pati-Salam model derived in the context of D-branes scenarios, requiring low energy string scale. We find that a non-supersymmetric version complies with a string scale as low as 10 TeV, while in the supersymmetric version the string scale raises up to ~2 x 10^7 TeV. The limited energy region for RGE running demands a large tan(beta) in order to have experimentally acceptable masses for the top and bottom quarks.Comment: 11 pages, LaTeX, 7 figures include

Geometric CP Violation with Extra Dimensions

We discuss how CP symmetry can be broken geometrically through orbifold projections in hidden extra dimensions in the context of D-brane models for particle unifications. We present a few toy models to illustrate the idea and suggest ways to incorporate this technique in the context of realistic models.Comment: 6 pages, one figure; references updated and a new model adde

On theories of random variables

We study theories of spaces of random variables: first, we consider random variables with values in the interval $[0,1]$, then with values in an arbitrary metric structure, generalising Keisler's randomisation of classical structures. We prove preservation and non-preservation results for model theoretic properties under this construction: i) The randomisation of a stable structure is stable. ii) The randomisation of a simple unstable structure is not simple. We also prove that in the randomised structure, every type is a Lascar type

Theory of magnetic excitons in the heavy-fermion superconductor UPd2Al3UPd_{2}Al_{3}

We analyze the influence of unconventional superconductivity on the magnetic excitations in the heavy fermion compound UPd$_2$Al$_3$. We show that it leads to the formation of a bound state at energies well below 2$\Delta_0$ at the antiferromagnetic wave vector {\textbf Q}=$(0,0,\pi/c)$. Its signature is a resonance peak in the spectrum of magnetic excitations in good agreement with results from inelastic neutron scattering. Furthermore we investigate the influence of antiferromagnetic order on the formation of the resonance peak. We find that its intensity is enhanced due to intraband transitions induced by the reconstruction of Fermi surface sheets. We determine the dispersion of the resonance peak near {\textbf Q} and show that it is dominated by the magnetic exciton dispersion associated with local moments. We demonstrate by a microscopic calculation that UPd$_2$Al$_3$ is another example in which the unconventional nature of the superconducting order parameter can be probed by means of inelastic neutron scattering and determined unambiguously.Comment: 6 pages, 4 figure

Larkin-Ovchinnikov-Fulde-Ferrell phase in the superconductor (TMTSF)2ClO4: Theory versus experiment

We consider a formation of the Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) phase in a quasi-one-dimensional (Q1D) conductor in a magnetic field, parallel to its conducting chains, where we take into account both the paramagnetic spin-splitting and orbital destructive effects against superconductivity. We show that, due to a relative weakness of the orbital effects in a Q1D case, the LOFF phase appears in (TMTSF)$_2$ClO$_4$ superconductor for real values of its Q1D band parameters. We compare our theoretical calculations with the recent experimental data by Y. Maeno's group [S. Yonezawa et al., Phys. Rev. Lett. \textbf{100}, 117002 (2008)] and show that there is a good qualitative and quantitative agreement between the theory and experimental data.Comment: 4 pages, 1 figur

Vacuum Energy Density and Cosmological Constant in dS Brane World

We discuss the vacuum energy density and the cosmological constant of dS$_5$ brane world with a dilaton field. It is shown that a stable AdS$_4$ brane can be constructed and gravity localization can be realized. An explicit relation between the dS bulk cosmological constant and the brane cosmological constant is obtained. The discrete mass spectrum of the massive scalar field in the AdS$_4$ brane is used to acquire the relationship between the brane cosmological constant and the vacuum energy density. The vacuum energy density in the brane gotten by this method is in agreement with astronomical observations.Comment: 16 pages,4 figure

P and T Violation From Certain Dimension Eight Weinberg Operators

Dimension eight operators of the Weinberg type have been shown to give important contributions to CP violating phenomena, such as the electric dipole moment of the neutron. In this note we show how operators related to these (and expected to occur on equal footing) can give rise to time-reversal violating phenomena such as atomic electric dipole moments. We also estimate the induced parity violating phenomena such as small ``wrong'' parity admixtures in atomic states and find that they are negligible. Uses harvmac.tex and epsf.tex; one figure submitted as a uuencoded, compressed EPS file.Comment: 6 pages, EFI-92-5