Antiplectic Structure, Diffeomorphism and Generalized KdV Family

In this paper we show that the generalized KdV, generalized Camassa–Holm equations and the corresponding Möbius invariant generalized Schwarzian KdV, Schwarzian CH equations can be realized in terms of flows induced by on the space of differential operators and on the space of immersion curves, respectively. These are Euler–Poincaré type flows, and one of the flow takes place on an infinite-dimensional Poisson manifold and the other on a slightly degenerate infinite-dimensional Symplectic manifold. They form an Antiplectic pair. We also study Euler–Poincaré flow with respect to metric, and this induces generalized Camassa–Holm equation. In the final section we discuss the Antiplectic pair in dimensions.Dedicated to Professor George Wilson on his 65th birthday with great respect and admiration.     

Ladder systems on trees

We formulate the notion of uniformization of colorings of ladder systems on subsets of trees. We prove that Suslin trees have this property and also Aronszajn trees in the presence of Martin's Axiom. As an application we show that if a tree has this property, then every countable discrete family of subsets of the tree can be separated by a family of pairwise disjoint open sets. Such trees are then normal and hence countably paracompact. As a dual result for special Aronszajn trees we prove that the weak diamond, , implies that no special Aronszajn tree can be countably paracompact.

     

A bijection between Littlewood-Richardson tableaux and rigged configurations

We define a bijection from Littlewood-Richardson tableaux to rigged configurations and show that it preserves the appropriate statistics. This proves in particular a quasi-particle expression for the generalized Kostka polynomials labeled by a partition and a sequence of rectangles R. The generalized Kostka polynomials are q-analogues of multiplicities of the irreducible -module of highest weight in the tensor product .     

Valuative analysis of planar plurisubharmonic functions

We show that valuations on the ring R of holomorphic germs in dimension 2 may be naturally evaluated on plurisubharmonic functions, giving rise to generalized Lelong numbers in the sense of Demailly. Any plurisubharmonic function thus defines a real-valued function on the set of valuations on R and – by way of a natural Laplace operator defined in terms of the tree structure on – a positive measure on . This measure contains a great deal of information on the singularity at the origin. Under mild regularity assumptions, it yields an exact formula for the mixed Monge-Ampère mass of two plurisubharmonic functions. As a consequence, any generalized Lelong number can be interpreted as an average of valuations. Using our machinery we also show that the singularity of any positive closed (1,1) current T can be attenuated in the following sense: there exists a finite composition of blowups such that the pull-back of T decomposes into two parts, the first associated to a divisor with normal crossing support, the second having small Lelong numbers. Mathematics Subject Classification (1991) 32U25, 13A18, 13H05     

Multiresolution Analysis and Multivariate Approximation of Smooth Signals in CB(Rd

In this paper we derive rates of approximation for a class of linear operators on associated with a multiresolution analysis We show that for a uniformly bounded sequence of linear operators satisfying on the subspace a lower bound for the approximation order is determined by the number of vanishing moments of a prewavelet set. We consider applications to extensions of generalized projection operators as well as to sampling series.     

Integration by parts formulas involving generalized Fourier-Feynman transforms on function space

In an upcoming paper, Chang and Skoug used a generalized Brownian motion process to define a generalized analytic Feynman integral and a generalized analytic Fourier-Feynman transform. In this paper we establish several integration by parts formulas involving generalized Feynman integrals, generalized Fourier-Feynman transforms, and the first variation of functionals of the form where denotes the Paley-Wiener-Zygmund stochastic integral .

     

On the prize-collecting generalized minimum spanning tree problem

The prize-collecting generalized minimum spanning tree problem (PC-GMSTP), is a generalization of the generalized minimum spanning tree problem (GMSTP) and belongs to the hard core of -hard problems. We describe an exact exponential time algorithm for the problem, as well we present several mixed integer and integer programming formulations of the PC-GMSTP. Moreover, we establish relationships between the polytopes corresponding to their linear relaxations and present an efficient solution procedure that finds the optimal solution of the PC-GMSTP for graphs with up 240 nodes.     

Homotopy Method for a General Multiobjective Programming Problem

In this paper, we study a class of vector minimization problems on a complete metric space X which is identified with the corresponding complete metric space of objective functions . We do not impose any compactness assumption on X. We show that, for most (in the sense of Baire category) functions , the corresponding vector optimization problem has a solution.     

Topological Structures in Colombeau Algebras: Topological $\widetilde{\mathbb{C}}$ -modules and Duality Theory

We study modules over the ring of complex generalized numbers from a topological point of view, introducing the notions of -linear topology and locally convex -linear topology. In this context particular attention is given to completeness, continuity of -linear maps and elements of duality theory for topological -modules. As main examples we consider various Colombeau algebras of generalized functions.Mathematics Subject Classifications (2000) 46F30, 13J99, 46A20.Claudia Garetto: Current address: Institut für Technische Mathematik, Geometrie und Bauinformatik, Universität Innsbruck, 6020 Insbruck, Austria. e-mail: claudia@mat1.uibk.ac.at     

Minimal spanning trees

The main result is that a recursive weighted graph having a minimal spanning tree has a minimal spanning tree that is . This leads to a proof of the failure of a conjecture of Clote and Hirst (1998) concerning Reverse Mathematics and minimal spanning trees.

     

Regularity, Local and Microlocal Analysis in Theories of Generalized Functions

We introduce a general context involving a presheaf and a subpresheaf ℬ of  . We show that all previously considered cases of local analysis of generalized functions (defined from duality or algebraic techniques) can be interpretated as the ℬ-local analysis of sections of  . But the microlocal analysis of the sections of sheaves or presheaves under consideration is dissociated into a “frequential microlocal analysis” and into a “microlocal asymptotic analysis”. The frequential microlocal analysis based on the Fourier transform leads to the study of propagation of singularities under only linear (including pseudodifferential) operators in the theories described here, but has been extended to some non linear cases in classical theories involving Sobolev techniques. The microlocal asymptotic analysis is a new spectral study of singularities. It can inherit from the algebraic structure of ℬ some good properties with respect to nonlinear operations.      

Shift coordinates,stretch lines and polyhedral structures for Teichmüller space

This paper has two parts. In the first part, we study shift coordinates on a sphere S equipped with three distinguished points and a triangulation whose vertices are the distinguished points. These coordinates parametrize a space that we call an unfolded Teichmüller space. This space contains Teichmüller spaces of the sphere with boundary components and cusps (which we call generalized pairs of pants), for all possible values of and satisfying . The parametrization of by shift coordinates equips this space with a natural polyhedral structure, which we describe more precisely as a cone over an octahedron in . Each cone over a simplex of this octahedron is interpreted as a Teichmüller space of the sphere with boundary components and cusps, for fixed and , the sphere being furthermore equipped with an orientation on each boundary component. There is a natural linear action of a finite group on whose quotient is an augmented Teichmüller space in the usual sense. We describe several aspects of the geometry of the space . Stretch lines and earthquakes can be defined on this space. In the second part of the paper, we use the shift coordinates to obtain estimates on the behaviour of stretch lines in the Teichmüller space of a surface obtained by gluing hyperbolic pairs of pants. We also use the shift coordinates to give formulae that express stretch lines in terms of Fenchel-Nielsen coordinates. We deduce the disjointness of some stretch lines in Teichmüller space. We study in more detail the case of a closed surface of genus 2. Authors’ addresses: A. Papadopoulos, Institut de Recherche Mathématique Avancée, Université Louis Pasteur and CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France and Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany; G. Théret, Institut de Recherche Mathématique Avancée, Université Louis Pasteur and CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France and Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, Building 1530, DK-8000 Aarhus C, Denmark     

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.     

Bent and generalized bent Boolean functions

In this paper, we investigate the properties of generalized bent functions defined on ${\mathbb{Z}_2^n}$ with values in ${\mathbb{Z}_q}$ , where q ≥ 2 is any positive integer. We characterize the class of generalized bent functions symmetric with respect to two variables, provide analogues of Maiorana–McFarland type bent functions and Dillon’s functions in the generalized set up. A class of bent functions called generalized spreads is introduced and we show that it contains all Dillon type generalized bent functions and Maiorana–McFarland type generalized bent functions. Thus, unification of two different types of generalized bent functions is achieved. The crosscorrelation spectrum of generalized Dillon type bent functions is also characterized. We further characterize generalized bent Boolean functions defined on ${\mathbb{Z}_2^n}$ with values in ${\mathbb{Z}_4}$ and ${\mathbb{Z}_8}$ . Moreover, we propose several constructions of such generalized bent functions for both n even and n odd.     

Generalized Bessel Functions for p-Radial Functions

Suppose that and p > 0. In this paper we study the generalized Bessel functions for the surface , introduced by D.St.P. Richards. We derive a recurrence relation for these functions and utilize a series representation to relate them to the classical symmetric functions. These generalized Bessel functions are symmetric with respect to the action of the hyperoctahedral group W_d, which is the symmetry group of the unit sphere. By means of this symmetry under W_d, we further express these generalized Bessel functions in terms of Bessel functions for certain finite reflection groups. For the case in which p = 2, our representations lead to known relations for the classical Bessel functions of order (d - 2)/2. For the case in which p = 1, the generalized Bessel functions have been studied by Berens and Xu in the analysis of summability problems for 1-radial functions, and we show how their results may be framed within our more general context.     

A generalization of Euler's hypergeometric transformation

Euler's transformation formula for the Gauss hypergeometric function  is extended to hypergeometric functions of higher order. Unusually, the generalized transformation constrains the hypergeometric function parameters algebraically but not linearly. Its consequences for hypergeometric summation are explored. It has as a corollary a summation formula of Slater. From this formula new one-term evaluations of and are derived by applying transformations in the Thomae group. Their parameters are also constrained nonlinearly. Several new one-term evaluations of with linearly constrained parameters are derived as well.

     

Duality of chordal SLE

We derive some geometric properties of chordal SLE(κ;) processes. Using these results and the method of coupling two SLE processes, we prove that the outer boundary of the final hull of a chordal SLE(κ;) process has the same distribution as the image of a chordal SLE(κ’;’) trace, where κ>4, κ’=16/κ, and the forces and ’ are suitably chosen. We find that for κ≥8, the boundary of a standard chordal SLE(κ) hull stopped on swallowing a fixed is the image of some SLE(16/κ;) trace started from x. Then we obtain a new proof of the fact that chordal SLE(κ) trace is not reversible for κ>8. We also prove that the reversal of SLE(4;) trace has the same distribution as the time-change of some SLE(4;’) trace for certain values of and ’.     

Riesz and Poisson-Jensen Representation Formulas for a Class of Ultraparabolic Operators on Lie Groups

The aim of this paper is to give some representation formulas of Riesz and Poisson-Jensen type for super-solutions to a class of hypoelliptic ultraparabolic operators on a homogeneous Lie group . Our results complete the ones obtained in Cinti (Math Scand 100:1–21, 2007). We also provide a suitable theory for -Green functions and for -Green potentials of Radon measures. The proofs mostly rely on the use of appropriate techniques relevant to the Potential Theory for . Investigation supported by University of Bologna. Funds for selected research topics.