We review what has been learnt and what remains unknown about the physics of hot enhancons following studies in supergravity. We recall a rather general family of static, spherically symmetric, non-extremal enhancon solutions describing D4 branes wrapped on K3 and discuss physical aspects of the solutions. We embed these solutions in the six dimensional supergravity describing type IIA strings on K3 and generalize them to have arbitrary charge vector. This allows us to demonstrate the equivalence with a known family of hot fractional D0 brane solutions, to widen the class of solutions of this second type and to carry much of the discussion across from the D4 brane analysis. In particular we argue for the existence of a horizon branch for these branes.
It is straightforward to find an expression for the radius of the @math -shell solutions: We could also rewrite this in terms of the parameters: @math , @math , @math , in order to put the solution exactly in terms of the language of previous studies @cite_0 .
@cite_0
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We speed up previous (1 + e)-factor approximation algorithms for a number of geometric optimization problems in fixed dimensions: diameter, width, minimum-radius enclosing cylinder, minimum-width enclosing annulus, minimum-width enclosing cylindrical shell, etc. Linear time bounds were known before; we further improve the dependence of the "constants" in terms of e.We next consider the data-stream model and present new (1 + e)-factor approximation algorithms that need only constant space for all of the above problems in any fixed dimension. Previously, such a result was known only for diameter.Both sets of results are obtained using the core-set framework recently proposed by Agarwal, Har-Peled, and Varadarajan. Published by Elsevier B.V. Given a polygon @math , for two points @math and @math contained in the polygon, their is the length of the shortest @math -path within @math . A of radius @math centered at a point @math is the set of points in @math whose geodesic distance to @math is at most @math . We present a polynomial time @math -approximation algorithm for finding a densest geodesic unit disk packing in @math . Allowing arbitrary radii but constraining the number of disks to be @math , we present a @math -approximation algorithm for finding a packing in @math with @math geodesic disks whose minimum radius is maximized. We then turn our focus on of @math and present a @math -approximation algorithm for covering @math with @math geodesic disks whose maximal radius is minimized. Furthermore, we show that all these problems are @math -hard in polygons with holes. Lastly, we present a polynomial time exact algorithm which covers a polygon with two geodesic disks of minimum maximal radius. Given a set of @math n points, each is painted by one of the @math k given colors, we want to choose @math k points with distinct colors to form a color spanning set. For each color spanning set, we can construct the convex hull and the smallest axis-aligned enclosing rectangle, etc. Assuming that each point is chosen independently and identically from the subset of points of the same color, we propose an @math O ( n 2 ) time algorithm to compute the expected area of convex hulls of the color spanning sets and an @math O ( n 2 ) time algorithm to compute the expected perimeter of convex hulls of the color spanning sets. For the expected perimeter (resp. area) of the smallest perimeter (resp. area) axis-aligned enclosing rectangles of the color spanning sets, we present an @math O ( n log n ) (resp. @math O ( n 2 ) ) time algorithm. We also propose a simple approximation algorithm to compute the expected diameter of the color spanning sets. For the expected distance of the closest pair, we show that it is @math # P-complete to compute and there exists no polynomial time @math 2 n 1 - ? approximation algorithm to compute the probability that the closest pair distance of all color spanning sets equals to a given value @math d unless @math P = N P , even in one dimension and when each color paints two points. Abstract The problem of finding the maximum diameter of n equal mutually disjoint circles inside a unit square is addressed in this paper. Exact solutions exist for only n = 1, …, 9,10,16,25,36 while for other n only conjectural solutions have been reported. In this work a max-min optimization approach is introduced which matches the best reported solutions in the literature for all n ⩽ 30, yields a better configuration for n = 15, and provides new results for n = 28 and 29.