The classification problem for simple unital finite rank dimension groups

The Borel complexity of the isomorphism problem for finite-rank unital simple dimension groups increases with rank. This implies that the isomorphism problems for the corresponding classes of Bratteli diagrams and LDA-groups also increase with rank.     

Carathéodory - Fejér Problem for Pairs of Meromorphic Functions

We continue our study of families of pairs of matrix-valued meromorphic functions P(ρ,P) depending on two parameters p and P introduced in [2]. These include as special cases the projective Schur, Nevanlinna and Carathéodory classes. A two sided Carathéodory Fejér interpolation problem is defined and solved in P(ρ,P), using the fundamental matrix inequality method. A corresponding Schur algorithm is studied. Finally we also consider the case of functions (as opposed to pairs).     

Dimension Groups for Polynomial Odometers

Here we study a class of dynamical systems we call polynomial odometers. These are adic maps on regularly structured Bratteli diagrams and include the Pascal and Stirling adic maps as examples. We describe the dimension groups associated with these systems and use this to study spaces of invariant measures. For many, but not all, the space of invariant measures is affinely homeomorphic to the space of Borel probability measures on a closed interval in $\mathbb{R}$ , we call such polynomial odometers reasonable. We describe the possible isomorphisms between dimension groups for reasonable polynomial odometers, and use this to prove a version of a result of Giordano, Putnam and Skau for this situation. Namely, we show that there is an isomorphism between unital ordered groups associated with two reasonable polynomial odometers if and only if there is a special kind of orbit equivalence between the two.     

Simple factor dressing and the López–Ros deformation of minimal surfaces in Euclidean 3-space

The aim of this paper is to investigate a new link between integrable systems and minimal surface theory. The dressing operation uses the associated family of flat connections of a harmonic map to construct new harmonic maps. Since a minimal surface in 3-space is a Willmore surface, its conformal Gauss map is harmonic and a dressing on the conformal Gauss map can be defined. We study the induced transformation on minimal surfaces in the simplest case, the simple factor dressing, and show that the well-known López–Ros deformation of minimal surfaces is a special case of this transformation. We express the simple factor dressing and the López–Ros deformation explicitly in terms of the minimal surface and its conjugate surface. In particular, we can control periods and end behaviour of the simple factor dressing. This allows to construct new examples of doubly-periodic minimal surfaces arising as simple factor dressings of Scherk’s first surface.

     

A binary decision diagram based algorithm for solving a class of binary two-stage stochastic programs

We consider a special class of two-stage stochastic integer programming problems with binary variables appearing in both stages. The class of problems we consider constrains the second-stage variables to belong to the intersection of sets corresponding to first-stage binary variables that equal one. Our approach seeks to uncover strong dual formulations to the second-stage problems by transforming them into dynamic programming (DP) problems parameterized by first-stage variables. We demonstrate how these DPs can be formed by use of binary decision diagrams, which then yield traditional Benders inequalities that can be strengthened based on observations regarding the structure of the underlying DPs. We demonstrate the efficacy of our approach on a set of stochastic traveling salesman problems.

     

GENERALIZED BRAUER ALGEBRAS

ABSTRACT

Using some ideas of Brauer, we introduce what we call generalized Brauer algebras and, as a special case, Brauer orders. We show that many well-known classes of so-called crossed product algebras, and in particular, the well-known crossed product orders, can be obtained as special instances of our construction. We prove several results showing when Brauer orders are Azumaya, maximal, hereditary or Gorenstein.     

Approximating Networks and Extended Ritz Method for the Solution of Functional Optimization Problems

Functional optimization problems can be solved analytically only if special assumptions are verified; otherwise, approximations are needed. The approximate method that we propose is based on two steps. First, the decision functions are constrained to take on the structure of linear combinations of basis functions containing free parameters to be optimized (hence, this step can be considered as an extension to the Ritz method, for which fixed basis functions are used). Then, the functional optimization problem can be approximated by nonlinear programming problems. Linear combinations of basis functions are called approximating networks when they benefit from suitable density properties. We term such networks nonlinear (linear) approximating networks if their basis functions contain (do not contain) free parameters. For certain classes of d-variable functions to be approximated, nonlinear approximating networks may require a number of parameters increasing moderately with d, whereas linear approximating networks may be ruled out by the curse of dimensionality. Since the cost functions of the resulting nonlinear programming problems include complex averaging operations, we minimize such functions by stochastic approximation algorithms. As important special cases, we consider stochastic optimal control and estimation problems. Numerical examples show the effectiveness of the method in solving optimization problems stated in high-dimensional settings, involving for instance several tens of state variables.     

Variationsprobleme in Orliczräumen und Splines

Two convex variational problems in Orlicz spaces are considered. We give sufficient conditions for existence and uniqueness of solutions and present several characterizations of these solutions. We show that the best interpolation property of certain nonlinear classes of spline functions is a special case of our results. As an application we consider the problem of Hermite-Birkhoff-interpolation with linear inequality constraints and illustrate the results by a simple example.

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Diese Arbeit ist eine gekürzte Fassung des zweiten Teils der Dissertation des Verfassers (Fakultät für Mathematik der Ludwig-Maximilians-Universität).     

The KOH terms and classes of unimodal N-modular diagrams

We show how certain suitably modified N-modular diagrams of integer partitions provide a nice combinatorial interpretation for the general term of Zeilberger?s KOH identity. This identity is the reformulation of O?Hara?s famous proof of the unimodality of the Gaussian polynomial as a combinatorial identity. In particular, we determine, using different bijections, two main natural classes of modular diagrams of partitions with bounded parts and length, having the KOH terms as their generating functions. One of our results greatly extends recent theorems of J. Quinn et al., which presented striking applications to quantum physics.     

Fractional square functions and potential spaces

Carlos Segovia and Richard Wheeden defined fractional square functions involving fractional derivatives. They obtained characterizations of potential spaces via square functions. Our aim in this paper is to reconsider the ideas of Segovia and Wheeden under the light of the semigroups of operators. We develop a quite general theory of fractional square functions associated to certain classes of operators. We present some examples of differential operators where our theory applies. We recover in a more compact way the results of Segovia and Wheeden and we obtain new characterizations of the potential spaces associated to the harmonic oscillator and Ornstein–Uhlenbeck operators.     

Flat surfaces,Bratteli diagrams and unique ergodicity à la Masur

Recalling the construction of a flat surface from a Bratteli diagram, this paper considers the dynamics of the shift map on the space of all bi-infinite Bratteli diagrams as the renormalizing dynamics on a moduli space of flat surfaces of finite area. A criterion of unique ergodicity similar to that of Masur’s for flat surface holds: if there is a subsequence of the renormalizing dynamical system which has a good accumulation point, the translation flow or Bratteli–Vershik transformation is uniquely ergodic. Related questions are explored.     

Trees as Brelot spaces

A Brelot space is a connected, locally compact, noncompact Hausdorff space together with the choice of a sheaf of functions on this space which are called harmonic. We prove that by considering functions on a tree to be functions on the edges as well as on the vertices (instead of just on the vertices), a tree becomes a Brelot space. This leads to many results on the potential theory of trees. By restricting the functions just to the vertices, we obtain several new results on the potential theory of trees considered in the usual sense. We study trees whose nearest-neighbor transition probabilities are defined by both transient and recurrent random walks. Besides the usual case of harmonic functions on trees (the kernel of the Laplace operator), we also consider as “harmonic” the eigenfunctions of the Laplacian relative to a positive eigenvalue showing that these also yield a Brelot structure and creating new classes of functions for the study of potential theory on trees.     

On the best mean square approximations by entire functions of exponential type in L 2(ℝ) and mean ν-widths of some functional classes

We consider some extremal problems of approximation theory of functions on the whole real axis ? by entire functions of the exponential type. In particular, we find the exact values of the mean ν-widths of classes of functions, defined by the modules of continuity of the mth order ω _m and majorants ψ satisfying the special type of restriction.     

Trajectories, first return limiting notions and rings of H-connected and iteratively H-connected functions

In the paper the existing results concerning a special kind of trajectories and the theory of first return continuous functions connected with them are used to examine some algebraic properties of classes of functions. To that end we define a new class of functions (denoted Conn*) contained between the families (widely described in literature) of Darboux Baire 1 functions (DB₁) and connectivity functions (Conn). The solutions to our problems are based, among other, on the suitable construction of the ring, which turned out to be in some senses an “optimal construction”. These considerations concern mainly real functions defined on [0, 1] but in the last chapter we also extend them to the case of real valued iteratively H-connected functions defined on topological spaces.     

Bloch constants for planar harmonic mappings

We give a lower estimate for the Bloch constant for planar harmonic mappings which are quasiregular and for those which are open. The latter includes the classical Bloch theorem for holomorphic functions as a special case. Also, for bounded planar harmonic mappings, we obtain results similar to a theorem of Landau on bounded holomorphic functions.

     

Ultradistributional boundary values of harmonic functions on the sphere

We present a theory of ultradistributional boundary values for harmonic functions defined on the Euclidean unit ball. We also give a characterization of ultradifferentiable functions and ultradistributions on the sphere in terms of their spherical harmonic expansions. To this end, we obtain explicit estimates for partial derivatives of spherical harmonics, which are of independent interest and refine earlier estimates by Calderón and Zygmund. We apply our results to characterize the support of ultradistributions on the sphere via Abel summability of their spherical harmonic expansions.     

Exact solutions of a novel integrable partial differential equation

We consider the derivation of exact solutions of a novel integrable partial differential equation (PDE). This equation was introduced with the aim that it mirror properties of the second Painlevé equation (P_II), and it has the remarkable property that, in addition to the usual kind of auto-Bäcklund transformation that one would expect of an integrable PDE, it also admits an auto-Bäcklund transformation of ordinary differential equation (ODE) type, i.e., a mapping between solutions involving shifts in coefficient functions, and which is an exact analogue of that of P_II with its shift in parameter.We apply three methods of obtaining exact solutions. First of all we consider the Lie symmetries of our PDE, this leading to a variety of solutions including in terms of the second Painlevé transcendent, elliptic functions and hyperbolic functions. Our second approach involves the use of our ODE-type auto-Bäcklund transformation applied to solutions arising as solutions of an equation analogous to the special integral of P_II. It turns out that our PDE has a second remarkable property, namely, that special functions defined by any linear second order ODE can be used to obtain a solution of our PDE. In particular, in the case of solutions defined by Bessel’s equation, iteration using our ODE-type auto-Bäcklund transformation is possible and yields a chain of solutions defined in terms of Bessel functions. We also consider the use of this transformation in order to derive solutions rational in x. Finally, we consider the standard auto-Bäcklund transformation, obtaining a nonlinear superposition formula along with one- and two-soliton solutions. Velocities are found to depend on coefficients appearing in the equation and this leads to a wide range of interesting behaviours.