Dual Heterotic Black-Holes in Four and Two Dimensions

We consider a class of extremal and non-extremal four-dimensional black-hole solutions occuring in toroidally compactified heterotic string theory, whose ten-dimensional interpretation involves a Kaluza-Klein monopole and a five-brane. We show that these four-dimensional solutions can be connected to extremal and non-extremal two-dimensional heterotic black-hole solutions through a change in the asymptotic behaviour of the harmonic functions associated with the Kaluza-Klein monopole and with the five-brane. This change in the asymptotic behaviour can be achieved by a sequence of S and T-S-T duality transformations in four dimensions. These transformations are implemented by performing a reduction on a two-torus with Lorentzian signature. We argue that the same mechanism can be applied to extremal and non-extremal black-hole solutions in the FHSV model.Comment: 10 pages, latex, 1 reference adde

Extremal dichotomy for uniformly hyperbolic systems

We consider the extreme value theory of a hyperbolic toral automorphism $T: \mathbb{T}^2 \to \mathbb{T}^2$ showing that if a H\"older observation $\phi$ which is a function of a Euclidean-type distance to a non-periodic point $\zeta$ is strictly maximized at $\zeta$ then the corresponding time series $\{\phi\circ T^i\}$ exhibits extreme value statistics corresponding to an iid sequence of random variables with the same distribution function as $\phi$ and with extremal index one. If however $\phi$ is strictly maximized at a periodic point $q$ then the corresponding time-series exhibits extreme value statistics corresponding to an iid sequence of random variables with the same distribution function as $\phi$ but with extremal index not equal to one. We give a formula for the extremal index (which depends upon the metric used and the period of $q$). These results imply that return times are Poisson to small balls centered at non-periodic points and compound Poisson for small balls centered at periodic points.Comment: 21 pages, 4 figure

The integrated periodogram of a dependent extremal event sequence

We investigate the asymptotic properties of the integrated periodogram calculated from a sequence of indicator functions of dependent extremal events. An event in Euclidean space is extreme if it occurs far away from the origin. We use a regular variation condition on the underlying stationary sequence to make these notions precise. Our main result is a functional central limit theorem for the integrated periodogram of the indicator functions of dependent extremal events. The limiting process is a continuous Gaussian process whose covari- ance structure is in general unfamiliar, but in the iid case a Brownian bridge appears. In the general case, we propose a stationary bootstrap procedure for approximating the distribution of the limiting process. The developed theory can be used to construct classical goodness-of-fit tests such as the Grenander- Rosenblatt and Cram\'{e}r-von Mises tests which are based only on the extremes in the sample. We apply the test statistics to simulated and real-life data

Extremes of periodic moving averages of random variables with regularly varying tail probabilities

We define a family of local mixing conditions that enable the computation of the extremal index of periodic sequences from the joint distributions of kconsecutive variables of the sequence. By applying results, under local and global mixing conditions, to the ( 2m – 1)–dependent periodic sequence X(m) n = Pm – 1 j = –m cj Zn – j, n ≥ 1, we compute the extremal index of the periodic moving average sequence Xn= P∞ j=–∞ cj Zn – j, n ≥ 1, of random variables with regularly varying tail probabilities. This paper generalizes the theory for extremes of stationary moving averages with regularly varying tail probabilities.Peer Reviewe

Convex Hull Realizations of the Multiplihedra

We present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the nth polytope in the sequence known as the multiplihedra. This answers the open question of whether the multiplihedra could be realized as convex polytopes. We use this realization to unite the approach to A_n-maps of Iwase and Mimura to that of Boardman and Vogt. We include a review of the appearance of the nth multiplihedron for various n in the studies of higher homotopy commutativity, (weak) n-categories, A_infinity-categories, deformation theory, and moduli spaces. We also include suggestions for the use of our realizations in some of these areas as well as in related studies, including enriched category theory and the graph associahedra.Comment: typos fixed, introduction revise