We investigate the black string in the context of the string theories. It is shown that the graviton is the only propagating mode in the (2+1)– dimensional extremal black string background. Both the dilation and axion turn out to be non-propagating modes. 1 Recently there has been much progress in understanding the microscopic origin for the black hole entropy. This was possible by using a new ...

This is a write-up of introductory lectures on black holes in string theory given at TASI-99. Topics discussed include: black holes, thermodynamics and the Bekenstein-Hawking entropy, the information problem; supergravity actions, conserved quantum numbers, supersymmetry and BPS states, units, duality and dimensional reduction, solution-generating; extremal M-branes and D-branes, smearing, prob...

in this article, we develop the distributed order fractional hybrid differential equations (dofhdes) with linear perturbations involving the fractional riemann-liouville derivative of order $0 < q < 1$ with respect to a nonnegative density function. furthermore, an existence theorem for the fractional hybrid differential equations of distributed order is proved under the mixed $varphi$-lipschit...

We give several pieces of evidence to show that extremal black holes cannot be obtained as limits of non-extremal black holes. We review arguments in the literature showing that the entropy of extremal black holes is zero, while that of near-extremal ones obey the Bekenstein-Hawking formula. However, from the counting of degeneracy of quantum (BPS) states of string theory the entropy of extrema...

The attractor mechanism for the four-dimensional N = 2 supergravity black hole solution is analyzed in the case of the D0-D4 system. Our analyses are based on newly derived exact solutions, which exhibit explicitly the attractor mechanism for extremal non-BPS black holes. Our solutions account for the moduli as general complex fields, while in almost all non-BPS solutions obtained previously, t...

In the traditional areas of graph theory (Ramsey theory, extremal graph theory, random graphs, etc.), graphs are regarded as abstract binary relations. The relevant methods are often incapable of providing satisfactory answers to questions arising in geometric applications. Geometric graph theory focuses on combinatorial and geometric properties of graphs drawn in the plane by straight-line edg...

We define the extremal length of vector measures. We show the reciprocal relation between extremal distance and extremal width of vector measures.

We consider the following extremal problem: Given three natural numbers n, m and l, what is the monotone DNF formula that has a minimal or maximal number of satisfying assignments over all monotone DNF formulas on n variables with m terms each of length l? We first show that the solution to the minimization problem can be obtained by the Kruskal-Katona theorem developed in extremal set theory. ...

Supersymmetric black holes in five-dimensional gauged supergravity must necessarily be rotating, and so in order to study the passage to black holes away from supersymmetry, it is of great interest to obtain non-extremal black holes that again have non-zero rotation. In this paper we find a simple framework for describing non-extremal rotating black holes in five-dimensional gauged supergraviti...

Let X be a compact Riemann surface of genus ≥ 2 of constant negative curvature −1. An extremal disk is an embedded (resp. covering) disk of maximal (resp. minimal) radius. A surface containing an extremal disk is an extremal surface. This paper gives formulas enumerating extremal surfaces of genus ≥ 4 up to isometry. We show also that the isometry group of an extremal surface is always cyclic o...