Counting Humps in Motzkin paths

In this paper we study the number of humps (peaks) in Dyck, Motzkin and Schr\"{o}der paths. Recently A. Regev noticed that the number of peaks in all Dyck paths of order $n$ is one half of the number of super Dyck paths of order $n$. He also computed the number of humps in Motzkin paths and found a similar relation, and asked for bijective proofs. We give a bijection and prove these results. Using this bijection we also give a new proof that the number of Dyck paths of order $n$ with $k$ peaks is the Narayana number. By double counting super Schr\"{o}der paths, we also get an identity involving products of binomial coefficients.Comment: 8 pages, 2 Figure

Hot Spots on the Fermi Surface of Bi2212: Stripes versus Superstructure

In a recent paper Saini et al. have reported evidence for a pseudogap around (pi,0) at room temperature in the optimally doped superconductor Bi2212. This result is in contradiction with previous ARPES measurements. Furthermore they observed at certain points on the Fermi surface hot spots of high spectral intensity which they relate to the existence of stripes in the CuO planes. They also claim to have identified a new electronic band along Gamma-M1 whose one dimensional character provides further evidence for stripes. We demonstrate in this Comment that all the measured features can be simply understood by correctly considering the superstructure (umklapp) and shadow bands which occur in Bi2212.Comment: 1 page, revtex, 1 encapsulated postscript figure (color

Manin-Olshansky triples for Lie superalgebras

Following V. Drinfeld and G. Olshansky, we construct Manin triples (\fg, \fa, \fa^*) such that \fg is different from Drinfeld's doubles of \fa for several series of Lie superalgebras \fa which have no even invariant bilinear form (periplectic, Poisson and contact) and for a remarkable exception. Straightforward superization of suitable Etingof--Kazhdan's results guarantee then the uniqueness of $q$-quantization of our Lie bialgebras. Our examples give solutions to the quantum Yang-Baxter equation in the cases when the classical YB equation has no solutions. To find explicit solutions is a separate (open) problem. It is also an open problem to list (\`a la Belavin-Drinfeld) all solutions of the {\it classical} YB equation for the Poisson superalgebras \fpo(0|2n) and the exceptional Lie superalgebra \fk(1|6) which has a Killing-like supersymmetric bilinear form but no Cartan matrix

Canonical Charmonium Interpretation for Y(4360) and Y(4660)

In this work, we consider the canonical charmonium assignments for Y(4360) and Y(4660). Y(4660) is good candidate of $\rm 5 ^3S_1$ $c\bar{c}$ state, the possibility of Y(4360) as a $\rm 3 ^3D_1$ $c\bar{c}$ state is studied, and the charmonium hybrid interpretation of Y(4360) can not be excluded completely. We evaluate the $e^{+}e^{-}$ leptonic widths, E1 transitions, M1 transitions and the open flavor strong decays of Y(4360) and Y(4660). Experimental tests for the charmonium assignments are suggested.Comment: 32 pages, 4 figure

Entanglement Entropy and Mutual Information in Bose-Einstein Condensates

In this paper we study the entanglement properties of free {\em non-relativistic} Bose gases. At zero temperature, we calculate the bipartite block entanglement entropy of the system, and find it diverges logarithmically with the particle number in the subsystem. For finite temperatures, we study the mutual information between the two blocks. We first analytically study an infinite-range hopping model, then numerically study a set of long-range hopping models in one-deimension that exhibit Bose-Einstein condensation. In both cases we find that a Bose-Einstein condensate, if present, makes a divergent contribution to the mutual information which is proportional to the logarithm of the number of particles in the condensate in the subsystem. The prefactor of the logarithmic divergent term is model dependent.Comment: 12 pages, 6 figure