An extension of the proximal point algorithm with Bregman distances on Hadamard manifolds

In this paper we present an extension of the proximal point algorithm with Bregman distances to solve constrained minimization problems with quasiconvex and convex objective function on Hadamard manifolds. The proposed algorithm is a modified and extended version of the one presented in Papa Quiroz and Oliveira (J Convex Anal 16(1): 49–69, 2009). An advantage of the proposed algorithm, for the nonconvex case, is that in each iteration the algorithm only needs to find a stationary point of the proximal function and not a global minimum. For that reason, from the computational point of view, the proposed algorithm is more practical than the earlier proximal method. Another advantage, for the convex case, is that using minimal condition on the problem data as well as on the proximal parameters we get the same convergence results of the Euclidean proximal algorithm using Bregman distances.     

Hopf–Galois Extensions with Central Invariants and Their Geometric Properties

We set up a general framework to study representation theory of certain algebras whichusually appear in the study of restricted Lie algebras or various quantum objects at roots of unity.The object of the study is a Hopf–Galois extension with central invariants. It turns out that theseextensions possess some geometric properties which are close to those of principal bundles andFrobenius manifolds. We define Hopf–Galois extensions of not necessarily affine schemes andprove that the classification problem of such extensions leads to a stack.     

Optimal discrete Morse functions for 2-manifolds

Morse theory is a powerful tool in its applications to computational topology, computer graphics and geometric modeling. It was originally formulated for smooth manifolds. Recently, Robin Forman formulated a version of this theory for discrete structures such as cell complexes. It opens up several categories of interesting objects (particularly meshes) to applications of Morse theory. Once a Morse function has been defined on a manifold, then information about its topology can be deduced from its critical elements. The main objective of this paper is to introduce a linear algorithm to define optimal discrete Morse functions on discrete 2-manifolds, where optimality entails having the least number of critical elements. The algorithm presented is also extended to general finite cell complexes of dimension at most 2, with no guarantee of optimality.     

The analysis of the understanding of the three-dimensional (Euclidian) space and the two-variable function concept by university students

Conceptual understanding is being emphasized in mathematics education. Students often have difficulty understanding the multi-variable function, a key concept. Based on the APOS theory, which analyzes the cognitive structures formed by individuals in learning a mathematical concept and produces components related to that learning, this study analyzes the conceptual understanding of three-dimensional spaces and two-variable functions by university students. The genetic decomposition of these concepts proposed by Trigueros and Martinez-Planell is also considered. The analyzes results revealed that only one student constructed the concept of three-dimensional space as an object within the framework of genetic decomposition. Some students could not relate the concepts of two-variable function and three-dimensional space. Students who could perform algebraic operations had problems related to geometric representation. This study suggests the refinement of genetic decomposition to include, e.g., mental construction steps for writing algebraic equations of special surfaces whose graphs are given in R³.     

Random Čech complexes on Riemannian manifolds

In this paper we study the homology of a random ?ech complex generated by a homogeneous Poisson process in a compact Riemannian manifold M. In particular, we focus on the phase transition for “homological connectivity” where the homology of the complex becomes isomorphic to that of M. The results presented in this paper are an important generalization of 7 , from the flat torus to general compact Riemannian manifolds. In addition to proving the statements related to homological connectivity, the methods we develop in this paper can be used as a framework for translating results for random geometric graphs and complexes from the Euclidean setting into the more general Riemannian one.     

Fundamental groups of compact manifolds and symmetric geometry of noncompact type

We introduce the notion of geometrical engagement for actions of semisimple Lie groups and their lattices as a concept closely related to Zimmer's topological engagement condition. Our notion is a geometrical criterion in the sense that it makes use of Riemannian distances. However, it can be used together with the foliated harmonic map techniques introduced in [8] to establish foliated geometric superrigidity results for both actions and geometric objects. In particular, we improve the applications of the main theorem in [9] to consider nonpositively curved compact manifolds (not necessarily with strictly negative curvature). We also establish topological restrictions for Riemannian manifolds whose universal cover have a suitable symmetric de Rham factor (Theorem B), as well as geometric obstructions for nonpositively curved compact manifolds to have fundamental groups isomorphic to certain groups build out of cocompact lattices in higher rank simple Lie groups (Corollary 4.5). Received: October 22, 1997     

Calculating particle-to-wall distances in unstructured computational fluid dynamic models

Knowledge of particle deposition is relevant in biomedical engineering situations such as computational modeling of aerosols in the lungs and blood particles in diseased arteries. To determine particle deposition distributions, one must track particles through the flow field, and compute each particle's distance to the wall as it approaches the geometric surface. For complex geometries, unstructured tetrahedral grids are a powerful tool for discretizing the model, but they complicate the particle-to-wall distance calculation, especially when non-linear mesh elements are used. In this paper, a general algorithm for finding minimum particle-to-wall distances in complex geometries constructed from unstructured tetrahedral grids will be presented. The algorithm is validated with a three-dimensional 90° bend geometry, and a comparison in accuracy is made between the use of linear and quadratic tetrahedral elements to calculate the minimum particle-to-wall distance.     

Lower Dimensional Representation of Text Data Based on Centroids and Least Squares

Dimension reduction in today's vector space based information retrieval system is essential for improving computational efficiency in handling massive amounts of data. A mathematical framework for lower dimensional representation of text data in vector space based information retrieval is proposed using minimization and a matrix rank reduction formula. We illustrate how the commonly used Latent Semantic Indexing based on the Singular Value Decomposition (LSI/SVD) can be derived as a method for dimension reduction from our mathematical framework. Then two new methods for dimension reduction based on the centroids of data clusters are proposed and shown to be more efficient and effective than LSI/SVD when we have a priori information on the cluster structure of the data. Several advantages of the new methods in terms of computational efficiency and data representation in the reduced space, as well as their mathematical properties are discussed.Experimental results are presented to illustrate the effectiveness of our methods on certain classification problems in a reduced dimensional space. The results indicate that for a successful lower dimensional representation of the data, it is important to incorporate a priori knowledge in the dimension reduction algorithms.     

Some developments in Nielsen fixed point theory

We give a brief survey of some developments in Nielsen fixed point theory. After a look at early history and a digress to various generalizations, we confine ourselves to several topics on fixed points of self-maps on manifolds and polyhedra. Special attention is paid to connections with geometric group theory and dynamics, as well as some formal approaches.     

Positivity of direct images of positively curved volume forms

We discuss Kiselman–Berndtsson’s minimum principle for plurisubharmonic functions in terms of the positivity of direct images of certain positively curved volume forms, and generalize it to holomorphically convex manifolds with compact group actions. With this generalization and other techniques, we establish a minimum principle for positively curved volume forms from the point of view of geometric invariant theory on Stein manifolds. Minimum principle with some noncompact group actions is also considered.     

The Interval Geometric Machine Model

This paper introduces the interval version of the Geometric Machine (GM) model, to model the semantics of algorithms of interval mathematics. Based on coherence spaces, the set of values storable in the GM memory is represented by the bi-structured coherence space of rational intervals, a constructive computational representation of the set of real intervals. Over the inductive ordered structure called the coherence space of processes, the representation of parallel and nondeterministic processes operating on the array structures of the GM memory is obtained. The infinite GM memory, supporting a coherence space of states, is conceived as the set of points of a geometric space. Using this framework, a domain-theoretic semantics of interval algorithms is presented.     

Image Projection and Representation on S<Superscript>n</Superscript>

In signal processing, communications, and other branches of information technologies, it is often desirable to map the higher-dimensional signals on Sⁿ. In this article we introduce a novel method of representing signals on Sⁿ. This approach is based on geometric function theory, in particular on the theory of quasiregular mappings. The importance of sampling is underlined, and new geometric sampling theorems for general manifolds are presented.     

Logistic biplot for nominal data

Classical biplot methods allow for the simultaneous representation of individuals (rows) and variables (columns) of a data matrix. For binary data, logistic biplots have been recently developed. When data are nominal, both classical and binary logistic biplots are not adequate and techniques such as multiple correspondence analysis (MCA), latent trait analysis (LTA) or item response theory (IRT) for nominal items should be used instead. In this paper we extend the binary logistic biplot to nominal data. The resulting method is termed “nominal logistic biplot”(NLB), although the variables are represented as convex prediction regions rather than vectors. Using the methods from computational geometry, the set of prediction regions is converted to a set of points in such a way that the prediction for each individual is established by its closest “category point”. Then interpretation is based on distances rather than on projections. We study the geometry of such a representation and construct computational algorithms for the estimation of parameters and the calculation of prediction regions. Nominal logistic biplots extend both MCA and LTA in the sense that they give a graphical representation for LTA similar to the one obtained in MCA.     

Asymptotic expansions using blow-up

The method of matched asymptotic expansions and geometric singular perturbation theory are the most common and successful approaches to singular perturbation problems. In this work we establish a connection between the two approaches in the context of the simple fold problem. Using the blow-up technique [5], [12] and the tools of geometric singular perturbation theory we derive asymptotic expansions of slow manifolds continued beyond the fold point. Our analysis explains the structure of the expansion and gives an algorithm for computing its coefficients.     

A Report on Canonical Null Curves and Screen Distributions for Lightlike Geometry

The general theory of lightlike submanifolds makes use of a non-degenerate screen distribution which is not unique and, therefore, the induced objects (starting from null curves) depend on the choice of a screen, which creates a problem. The purpose of this paper is to report on the existence of a canonical representation of null curves of Lorentzian manifolds and the choice of a canonical or a good screen for large classes of lightlike hypersurfaces of semi-Riemannian manifolds. We also prove a new theorem on the existence of an integrable canonical screen, subject to a geometric condition, and supported by a physical application.      

A new theory of complex rays

A new approach to the theory of complex rays is presented. Itis shown that the three-dimensional Minkowski space, the variantof the well known four-dimensional space–time Minkowskispace of the special theory of relativity, is more appropriatefor describing both real and complex rays than the usual Euclideanspace. It turns out that in this space complex rays, as realones, may have quite definite directions and magnitudes. Thisallows us to understand the geometrical meaning of the complexmagnitudes such as complex distances and complex angles, intensivelydiscussed over the last several decades. From this point ofview a new interpretation of the Gaussian beams and reflectionlaws is presented.     

Methods of generalized Hermitian geometry in the theory of almost-contact manifolds

Almost-contact manifolds are considered as generalized almost-Hermitian manifolds of defect 1, which permits the use of the apparatus of the geometry of generalized almost-Hermitian manifolds to get a number of strong results, in particular to get a complete classification of important types of almost-contact manifolds. The theory of Q-algebras, which plays an important role in the apparatus mentioned, is presented in expanded form.Translated from Itogi Nauk i Tekhniki, Seriya Problemy Geometrii, Vol. 18, pp. 25–71, 1986.     

Monge-Ampère Equations on (Para-)K?hler Manifolds: from Characteristic Subspaces to Special Lagrangian Submanifolds

We present the basic notions and results of the geometric theory of second order PDEs in the framework of contact and symplectic manifolds including characteristics, formal integrability, existence and uniqueness of formal solutions of non-characteristic Cauchy problems. Then, we focus our attention to Monge-Ampère equations (MAEs) and discuss a natural class of MAEs arising in K?hler and para-K?hler geometry whose solutions are special Lagrangian submanifolds.     

Spontaneous Surgery on the Borromean Rings

We study the cone-manifolds whose singular sets are obtained by orbifold and spontaneous surgeries on components of the Borromean rings. We establish existence of geometric structures on these manifolds. For manifolds with hyperbolic structure we obtain an integral representation for volumes.