Black Holes with Multiple Charges and the Correspondence Principle

We consider the entropy of near extremal black holes with multiple charges in the context of the recently proposed correspondence principle of Horowitz and Polchinski, including black holes with two, three and four Ramond-Ramond charges. We find that at the matching point the black hole entropy can be accounted for by massless open strings ending on the D-branes for all cases except a black hole with four Ramond-Ramond charges, in which case a possible resolution in terms of brane-antibrane excitations is considered.Comment: 26 pages, harvmac, minor correction

On the relationship between the Hurst exponent, the ratio of the mean square successive difference to the variance, and the number of turning points

The long range dependence of the fractional Brownian motion (fBm), fractional Gaussian noise (fGn), and differentiated fGn (DfGn) is described by the Hurst exponent $H$. Considering the realisations of these three processes as time series, they might be described by their statistical features, such as half of the ratio of the mean square successive difference to the variance, $\mathcal{A}$, and the number of turning points, $T$. This paper investigates the relationships between $\mathcal{A}$ and $H$, and between $T$ and $H$. It is found numerically that the formulae $\mathcal{A}(H)=a{\rm e}^{bH}$ in case of fBm, and $\mathcal{A}(H)=a+bH^c$ for fGn and DfGn, describe well the $\mathcal{A}(H)$ relationship. When $T(H)$ is considered, no simple formula is found, and it is empirically found that among polynomials, the fourth and second order description applies best. The most relevant finding is that when plotted in the space of $(\mathcal{A},T)$, the three process types form separate branches. Hence, it is examined whether $\mathcal{A}$ and $T$ may serve as Hurst exponent indicators. Some real world data (stock market indices, sunspot numbers, chaotic time series) are analyzed for this purpose, and it is found that the $H$'s estimated using the $H(\mathcal{A})$ relations (expressed as inverted $\mathcal{A}(H)$ functions) are consistent with the $H$'s extracted with the well known wavelet approach. This allows to efficiently estimate the Hurst exponent based on fast and easy to compute $\mathcal{A}$ and $T$, given that the process type: fBm, fGn or DfGn, is correctly classified beforehand. Finally, it is suggested that the $\mathcal{A}(H)$ relation for fGn and DfGn might be an exact (shifted) $3/2$ power-law.Comment: 20 pages in one-column format, 7 figures; matches the version accepted for publicatio

Regions of linearity, Lusztig cones and canonical basis elements for the quantized enveloping algebra of type A_4

Let U_q be the quantum group associated to a Lie algebra g of rank n. The negative part U^- of U has a canonical basis B with favourable properties, introduced by Kashiwara and Lusztig. The approaches of Kashiwara and Lusztig lead to a set of alternative parametrizations of the canonical basis, one for each reduced expression for the longest word in the Weyl group of g. We show that if g is of type A_4 there are close relationships between the Lusztig cones, canonical basis elements and the regions of linearity of reparametrization functions arising from the above parametrizations. A graph can be defined on the set of simplicial regions of linearity with respect to adjacency, and we further show that this graph is isomorphic to the graph with vertices given by the reduced expressions of the longest word of the Weyl group modulo commutation and edges given by long braid relations. Keywords: Quantum group, Lie algebra, Canonical basis, Tight monomials, Weyl group, Piecewise-linear functions.Comment: 61 pages, 17 figures, uses picte

Limits of bifractional Brownian noises

Let $B^{H,K}=(B^{H,K}_{t}, t\geq 0)$ be a bifractional Brownian motion with two parameters $H\in (0,1)$ and $K\in(0,1]$. The main result of this paper is that the increment process generated by the bifractional Brownian motion $(B^{H,K}_{h+t} -B^{H,K}_{h}, t\geq 0)$ converges when $h\to \infty$ to $(2^{(1-K)/{2}}B^{HK}_{t}, t\geq 0)$, where $(B^{HK}_{t}, t\geq 0)$ is the fractional Brownian motion with Hurst index $HK$. We also study the behavior of the noise associated to the bifractional Brownian motion and limit theorems to $B^{H,K}$