A representation of bivariate extreme value distributions via norms on \mathbb{R}^{2}

It is known that a bivariate extreme value distribution (EVD) with reverse exponential margins can be represented as , , where is a suitable norm on . We prove in this paper the converse implication, i.e., given an arbitrary norm on , , , defines an EVD with reverse exponential margins, if and only if the norm satisfies for the condition . This result is extended to bivariate EVDs with arbitrary margins as well as to extreme value copulas. By identifying an EVD , , with the unit ball corresponding to the generating norm , we obtain a characterization of the class of EVDs in terms of compact and convex subsets of .     

Extremal stochastic integrals: a parallel between max-stable processes and α-stable processes

We construct extremal stochastic integrals of a deterministic function with respect to a random Fréchet () sup-measure. The measure is sup-additive rather than additive and is defined over a general measure space , where is a deterministic control measure. The extremal integral is constructed in a way similar to the usual stable integral, but with the maxima replacing the operation of summation. It is well-defined for arbitrary , and the metric metrizes the convergence in probability of the resulting integrals.This approach complements the well-known de Haan's spectral representation of max-stable processes with Fréchet marginals. De Haan's representation can be viewed as the max-stable analog of the LePage series representation of stable processes, whereas the extremal integrals correspond to the usual stable stochastic integrals. We prove that essentially any strictly stable process belongs to the domain of max-stable attraction of an Fréchet, max-stable process. Moreover, we express the corresponding Fréchet processes in terms of extremal stochastic integrals, involving the kernel function of the stable process. The close correspondence between the max-stable and stable frameworks yields new examples of max-stable processes with non-trivial dependence structures.This research was partially supported by a fellowship of the Horace H. Rackham School of Graduate Studies at the University of Michigan and the NSF Grant DMS-0505747 at Boston University.     

Accounting for threshold uncertainty in extreme value estimation

Tail data are often modelled by fitting a generalized Pareto distribution (GPD) to the exceedances over high thresholds. In practice, a threshold is fixed and a GPD is fitted to the data exceeding . A difficulty in this approach is the selection of the threshold above which the GPD assumption is appropriate. Moreover the estimates of the parameters of the GPD may depend significantly on the choice of the threshold selected. Sensitivity with respect to the threshold choice is normally studied but typically its effects on the properties of estimators are not accounted for. In this paper, to overcome the difficulties of the fixed-threshold approach, we propose to model extreme and non-extreme data with a distribution composed of a piecewise constant density from a low threshold up to an unknown end point and a GPD with threshold for the remaining tail part. Since we estimate the threshold together with the other parameters of the GPD we take naturally into account the threshold uncertainty. We will discuss this model from a Bayesian point of view and the method will be illustrated using simulated data and a real data set.     

Limit distributions of upper order statistics for families of multivariate distributions

We consider a portfolio of dependent exchangeable random variables , where the dependence structure is generated by a mixture model (Archimedean copulas belong to this class of models). Define the ordered sample . We prove results of the following type: fix and choose appropriately, then converges in distribution to a random vector as , for which we can explicitly give the distribution.     

Admissible Galois Structures and Coverings in Regular Mal’cev Categories

Given a regular Gumm category such that any regular epimorphism is effective for descent, we prove that any Birkhoff subcategory in gives rise to an admissible Galois structure. This result allows one to consider some new applications of the categorical Galois theory in the context of topological algebras. Given a regular Mal’cev category , we first characterize the coverings of the Galois structure induced by the subcategory of the abelian objects in . Then we consider as a subcategory of the category of the equivalence relations in , and we characterize the coverings of the corresponding Galois structure . By composing the Galois structures and we obtain the Galois structure induced by as a subcategory of . We give the characterization of the -coverings in terms of the coverings of and .     

Extreme values of a portfolio of Gaussian processes and a trend

We consider the extreme values of a portfolio of independent continuous Gaussian processes ( ) which are asymptotically locally stationary, with expectations and variances , and a trend for some constants with . We derive the probability for , which may be interpreted as ruin probability. AMS 2000 Subject Classifications Primary—60G15, 62G32, 91B28     

Regular quantizations and covering maps

Let $\tilde{M} \rightarrow MLet be a holomorphic (unbranched) covering map between two compact complex manifolds, with . We prove that if and M both admit regular K?hler forms and ω respectively then, up to homotheties, and (M, ω) are biholomorphically isometric. This work was supported by the M.I.U.R. Project “Geometric Properties of Real and Complex Manifolds”.     

Universality of Coproducts in Categories of Lax Algebras

Categories of lax -algebras are shown to have pullback-stable coproducts if preserves inverse images. The general result not only gives a common proof of this property in many topological categories but also shows that important topological categories, like the category of uniform spaces, are not presentable as a category of lax -algebras, with preserving inverse images. Moreover, we show that any such category of -algebras has a concrete, coproduct preserving functor into the category of topological spaces.     

Upward Three-Dimensional Grid Drawings of Graphs

A three-dimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line segments representing the edges do not cross. Our aim is to produce three-dimensional grid drawings with small bounding box volume. Our first main result is that every -vertex graph with bounded degeneracy has a three-dimensional grid drawing with volume. This is the largest known class of graphs that have such drawings. A three-dimensional grid drawing of a directed acyclic graph (dag) is upward if every arc points up in the -direction. We prove that every dag has an upward three-dimensional grid drawing with volume, which is tight for the complete dag. The previous best upper bound was . Our main result concerning upward drawings is that every -colourable dag ( constant) has an upward three-dimensional grid drawing with volume. This result matches the bound in the undirected case, and improves the best known bound from for many classes of dags, including planar, series parallel, and outerplanar. Improved bounds are also obtained for tree dags. We prove a strong relationship between upward three-dimensional grid drawings, upward track layouts, and upward queue layouts. Finally, we study upward three-dimensional grid drawings with bends in the edges.Research of Vida Dujmovi is supported by NSERC. Research of David Wood is supported by the Government of Spain grant MEC SB2003-0270 and by the projects MCYT-FEDER BFM2003-00368 and Gen. Cat 2001SGR00224.     

Skew Distributions Generated from Different Families

Following the recent paper by Gupta et al. [8], skew pdfs of the form are generated, where the pdf and the cdf are taken to be different and to come from normal, Student's , Cauchy, Laplace, logistic or the uniform distribution. The properties of the resulting distributions are studied. In particular, expressions for the th moment and the characteristic function are derived. Graphical illustrations are also provided.     

Asymptotic properties of type I elliptical random vectors

Let be an elliptical random vector with a non-singular square matrix and a spherical random vector in , and let be a sequence of vectors in such that . We assume in this paper that the associated random radius R _k =(S ₁ + S ₂ +...+S _k )^1/2 is almost surely positive, and it has distribution function in the Gumbel max-domain of attraction. Relying on extreme value theory we obtain an exact asymptotic expansion of the tail probability for converging as to a boundary point. Further we discuss density convergence under a suitable transformation. We apply our results to obtain an asymptotic approximation of the distribution of partial excess above a high threshold, and to derive a conditional limiting result. Further, we investigate the asymptotic behaviour of concomitants of order statistics, and the tail asymptotics of associated random radius for subvectors of .      

Using cross-product matrices to compute the SVD

This paper concerns accurate computation of the singular value decomposition (SVD) of an matrix . As is well known, cross-product matrix based SVD algorithms compute large singular values accurately but generally deliver poor small singular values. A new novel cross-product matrix based SVD method is proposed: (a) Use a backward stable algorithm to compute the eigenpairs of and take the square roots of the large eigenvalues of it as the large singular values of ; (b) form the Rayleigh quotient of with respect to the matrix consisting of the computed eigenvectors associated with the computed small eigenvalues of ; (c) compute the eigenvalues of the Rayleigh quotient and take the square roots of them as the small singular values of . A detailed quantitative error analysis is conducted on the method. It is proved that if small singular values are well separated from the large ones then the method can compute the small ones accurately up to the order of the unit roundoff . An algorithm is developed that is not only cheaper than the standard Golub–Reinsch and Chan SVD algorithms but also can update or downdate a new SVD by adding or deleting a row and compute certain refined Ritz vectors for large matrix eigenproblems at very low cost. Several variants of the algorithm are proposed that compute some or all parts of the SVD. Typical numerical examples confirm the high accuracy of our algorithm.Supported in part by the National Science Foundation of China (No. 10471074).     

The Birman–Craggs–Johnson homomorphism and abelian cycles in the Torelli group

In the 1970s, Birman–Craggs–Johnson (BCJ) (Trans AMS 237: 283–309, 1978; Trans AMS 261(1):423–422, 1980) used Rochlin’s invariant for homology 3-spheres to construct a remarkable surjective homomorphism , where is the Torelli group and B ₃ is a certain -vector space of Boolean (square-free) polynomials. By pulling back cohomology classes and evaluating them on abelian cycles, we construct dimensions worth of nontrivial elements of which cannot be detected rationally. These classes in fact restrict to nontrivial classes in the cohomology of the subgroup generated by Dehn twists about separating curves. We also use the “Casson–Morita algebra” and Morita’s integral lift of the BCJ map restricted to to give the same lower bound on . The first author is partially supported by NSF grant DMS-0606882 and was also supported in part by NSF grant DMS-0504208 and by a VIGRE postdoc under NSF grant 9983660 to Cornell University. The second author is supported in part by NSF grant DMS-0244542.     

From light tails to heavy tails through multiplier

Let X and Y be two independent nonnegative random variables, of which X has a distribution belonging to the class or for some γ ≥ 0 and Y is unbounded. We study how their product XY inherits the tail behavior of X. Under some mild technical assumptions we prove that the distribution of XY belongs to the class or accordingly. Hence, the multiplier Y builds a bridge between light tails and heavy tails.      

Epi-topology and Epi-convergence for Archimedean Lattice-ordered Groups with Unit

is the category of archimedean -groups with distinguished weak order unit, with -group homomorphisms which preserve unit. This category includes all rings of continuous functions and all rings of measurable functions modulo null functions, with ring homomorphisms. The authors, and others, have studied previously the epimorphisms (right-cancellable morphisms) in . There is a rich theory. In this paper, we describe a topological approach to the analysis of these epimorphisms. On each – object, we define a topology and a convergence . These have the same closure operator, and this closure “captures epics” in the sense: a divisible subobject of is dense iff is epically embedded. The topology is , but only sometimes Hausdorff or an -group topology. The convergence is a Hausdorff -group convergence, but only sometimes topological. The associations of to , and to , are functorial. Dedicated to Bernhard Banaschewski for his 80th birthday.     

Distances in random graphs with infinite mean degrees

We study random graphs with an i.i.d. degree sequence of which the tail of the distribution function is regularly varying with exponent . In particular, the degrees have infinite mean. Such random graphs can serve as models for complex networks where degree power laws are observed. The minimal number of edges between two arbitrary nodes, also called the graph distance or the hopcount, is investigated when the size of the graph tends to infinity. The paper is part of a sequel of three papers. The other two papers study the case where , and , respectively. The main result of this paper is that the graph distance for converges in distribution to a random variable with probability mass exclusively on the points and . We also consider the case where we condition the degrees to be at most for some , where is the size of the graph. For fixed and such that , the hopcount converges to in probability, while for , the hopcount converges to the same limit as for the unconditioned degrees. The proofs use extreme value theory. AMS 2000 Subject Classifications Primary—60G70; Secondary—05C80     

On Non-M-Cosingular Completely ⊕-Supplemented Modules

In this paper, it is shown that any non--cosingular -supplemented module is if and only if has the summand intersection property. Let be any module such that has a coclosure in . Then we prove that is (completely) -supplemented if and only if for some submodule of such that and both are (completely) -supplemented.     

Uniform Estimates of the Fundamental Solution for a Family of Hypoelliptic Operators

In this paper we are concerned with a family of elliptic operators represented as sum of square vector fields: , in where is the Laplace operator, , and the limit operator is hypoelliptic. It is well known that admits a fundamental solution . Here we establish some a priori estimates uniform in of it, using a modification of the lifting technique of Rothschild and Stein. As a consequence we deduce some a priori estimates uniform in , for solutions of the approximated equation . These estimates can be used in particular while studying regularity of viscosity solutions of nonlinear equations represented in terms of vector fields.     

Higher Level Affine Crystals and Young Walls

Using combinatorics of Young walls, we give a new realization of arbitrary level irreducible highest weight crystals for quantum affine algebras of type , , , , , and . The irreducible highest weight crystals are realized as the affine crystals consisting of reduced proper Young walls. The notion of slices and splitting of blocks plays a crucial role in the construction of crystals.Presented by Peter Littelman.