Abstract:
We provide simple proofs describing the behavior of the largest component of the Erdos-Renyi random graph G(n,p) outside of the scaling window, p={1+\eps(n) \over n} where \eps(n) tends to 0, but \eps(n)n^{1/3} tends to \infty.

Abstract:
We have studied two-body charmless hadronic decays of $B$ mesons into the final states phi K and phi K^*. Using 9.7 million $B\bar{B}$ pairs collected with the CLEO II detector, we observe the decays B- -> phi K- and B0 -> phi K*0 with the following branching fractions: BR(B- -> phi K-)=(5.5 +2.1-1.8 +- 0.6) x 10^{-6} and BR(B0 -> phi K*0)=(11.5 +4.5-3.7 +1.8-1.7) x 10^{-6}. We also see evidence for the decays B0 -> phi K0 and B- -> phi K*-. However, since the statistical significance is not overwhelming for these modes we determine upper limits of <12.3 x 10^{-6} and <22.5 x 10^{-6} (90% C.L.) respectively.

Abstract:
We prove sharper versions of theorems of Linial-Meshulam and Meshulam-Wallach which describe the behavior for (Z/2)-cohomology of a random k-dimensional simplicial complex within a narrow transition window. In particular, we show that within this window the (k-1)st Betti number is in the limit Poisson distributed. For k=2 we also prove that in an accompanying growth process, with high probability, first cohomology vanishes exactly at the moment when the last isolated (k-1)-simplex gets covered by a k-simplex.

Abstract:
We introduce four-point functions in the hadronic ladder resummation approach to large $N_c$ QCD Green functions. We determine the relevant one to calculate the $B_K$ kaon parameter in the chiral limit. This four-point function contains both the large momenta QCD OPE and the small momenta ChPT at NLO limits, analytically. We get $\hat B_K^\chi = 0.38 \pm 0.15$. We also give the ChPT result at NLO for the relevant four-point function to calculate $B_K$ outside the chiral limit, while the leading QCD OPE is the same as the chiral limit one.

Abstract:
This is the second in a series of papers in which we analyze spectra of over 400 K and M giants in Baade's Window, including most of the stars with proper motions measured by Spaenhauer et al. [AJ, 103, 297 (1992)]. In our first paper, we measured line--strength indices of Fe, Mg, CN and H$\beta$ and calibrated them on the system of Faber et al. [ApJS, 57, 711 (1985)]. Here, we use the $\langle{\rm Fe}\rangle$ index to derive an abundance distribution in [Fe/H] for 322 stars with effective temperatures between 3900 K and 5160 K. Our derived values of [Fe/H] agree well with those measured from high--resolution echelle spectra (e.g., McWilliam \& Rich [ApJS, 91, 749 (1994)]) for the small number of stars in common. We find a mean abundance $\langle{\rm [Fe/H]}\rangle = -0.11 \pm 0.04$ for our sample of Baade's Window K giants. More than half the sample lie in the range $-0.4 <$ \feh\ $<+0.3$. We estimate line--of--sight distances for individual stars in our sample and confirm that, in Baade's Window, most K giants with $V < 15.5$ are foreground disk stars, but the great majority (more than 80\%) with $V > 16$ belong to the bulge. We also compare the metallicities derived from the CN and Mg$_2$ indices to those from iron. Most of the metal--rich stars in our sample appear to be CN--weak, in contrast to the situation in metal--rich globular clusters and elliptical galaxies. The metal--poor half of our sample ([Fe/H] $< 0$) shows evidence for a mild Mg overenhancement ([Mg/Fe] $\sim +0.2$); but this is not seen in the more metal--rich stars ([Fe/H] $\geq$ 0). The K giants in Baade's Window therefore share some, but not all, of the characteristics of stars in elliptical galaxies as inferred from their integrated light.

Abstract:
The near equality between the measured value for $|V_{ub}|/|V_{cb}|$ and its unitarity triangle lower bound from $\sin 2\beta$ measurements suggests that $\alpha \approx 90^{\circ}$ and $\sin\gamma \approx \cos\beta = 0.91$ and a large direct CP violation in B decays would be possible. In this talk, I would like to discuss a recent analysis of the non resonant and resonant CP asymmetry for the decays $B^- \to \pi^+ \pi^- K^-$ and $B^- \to K^+ K^- K^-$ where the CP asymmetry at the $\chi_{c0}$ resonance comes from the interference of the non resonant amplitude with the resonant amplitude $B^\pm \to \chi_{c0} K^\pm $ $ \to\pi^+ \pi^- K^\pm $ and $B^\pm \to \chi_{c0} K^\pm$ $ \to K^+ K^- K^\pm$. Using the the Babar and Belle measurements of the $B^-\to \chi_{c0} K^- $ branching ratios, we predict that partial width asymmetry at the $\chi_{c0}$ resonance could reach 10% for the $B^- \to \pi^+ \pi^- K^-$ decay and 16% for the $B^- \to K^+ K^- K^-$decay.

Abstract:
We report results of the Dalitz analysis of the three-body charmless B->K+pi+pi- and B->K+K+K- decays based on a 140fb^{-1} data sample collected with the Belle detector. Measurements of branching fractions for quasi-two-body decays to scalar-pseudoscalar states: B-> f_0(980)K^+, B-> K^*_0(1430)^0 pi+, and to vector-pseudoscalar states: B-> K^*(892)^0 pi+, B-> rho^0 K+, B-> phi K+ are presented. Upper limits on decays to some pseudoscalar-tensor final states are reported. We also report the new measurement of the B-> chi_c K+ branching fraction in two chi_c decay channels: chi_c-> pi+ pi- and chi_c-> K+K-.

Abstract:
We study $B\to \phi K$ and $B\to \phi X_s$ decays in the heavy quark limit using perturbative QCD. The next leading order corrections introduce substantial modifications to the naive factorization results (more than 50%). The branching ratio $Br(B\to \phi K)$ is predicted to be in the range $(F^{B\to K}_1(m^2_\phi)/0.33)^2(3.5\sim 4.2) \times 10^{-6}$ which is within the one $\sigma$ allowed region from the central value of $6.2\times 10^{-6}$ measured by CLEO, but outside the one $\sigma$ allowed region from the central value of $17.2\times 10^{-6}$ measured by BELLE for reasonable $F_1^{B\to K}$. For the semi-inclusive decay $B\to \phi X_s$ we also include initial bound state effect in the heavy quark limit which decreases the branching ratio by about 10%. $Br(B\to \phi X_s)$ is predicted to be in the range $(5.1\sim 6.3)\times 10^{-5}$.

Abstract:
The most recent calculations of pi+ pi+ and K+ K+ scattering by the NPLQCD collaboration using domain-wall valence quarks on staggered MILC configurations are presented. In addition, a quenched calculation of the potentials between two B-mesons is discussed.