Some Problems In Algebra And Topology A Study Of Fuzzy Convexity With Special Reference To Separation Properties

This thesis is a study of abstract fuzzy convexity spaces and fuzzy topology fuzzy convexity spaces No attempt seems to have been made to develop a fuzzy convexity theoryin abstract situations. The purpose of this thesis is to introduce fuzzy convexity theory in abstract situations

Second-order negative-curvature methods for box-constrained and general constrained optimization

A Nonlinear Programming algorithm that converges to second-order stationary points is introduced in this paper. The main tool is a second-order negative-curvature method for box-constrained minimization of a certain class of functions that do not possess continuous second derivatives. This method is used to define an Augmented Lagrangian algorithm of PHR (Powell-Hestenes-Rockafellar) type. Convergence proofs under weak constraint qualifications are given. Numerical examples showing that the new method converges to second-order stationary points in situations in which first-order methods fail are exhibited.

Structured minimal-memory inexact quasi-Newton method and secant preconditioners for augmented Lagrangian optimization

Augmented Lagrangian methods for large-scale optimization usually require efficient algorithms for minimization with box constraints. On the other hand, active-set box-constraint methods employ unconstrained optimization algorithms for minimization inside the faces of the box. Several approaches may be employed for computing internal search directions in the large-scale case. In this paper a minimal-memory quasi-Newton approach with secant preconditioners is proposed, taking into account the structure of Augmented Lagrangians that come from the popular Powell-Hestenes-Rockafellar scheme. A combined algorithm, that uses the quasi-Newton formula or a truncated-Newton procedure, depending on the presence of active constraints in the penalty-Lagrangian function, is also suggested. Numerical experiments using the Cute collection are presented.

Improving ultimate convergence of an augmented Lagrangian method

Optimization methods that employ the classical Powell-Hestenes-Rockafellar augmented Lagrangian are useful tools for solving nonlinear programming problems. Their reputation decreased in the last 10 years due to the comparative success of interior-point Newtonian algorithms, which are asymptotically faster. In this research, a combination of both approaches is evaluated. The idea is to produce a competitive method, being more robust and efficient than its `pure` counterparts for critical problems. Moreover, an additional hybrid algorithm is defined, in which the interior-point method is replaced by the Newtonian resolution of a Karush-Kuhn-Tucker (KKT) system identified by the augmented Lagrangian algorithm. The software used in this work is freely available through the Tango Project web page:http://www.ime.usp.br/similar to egbirgin/tango/.

Augmented Lagrangian methods under the constant positive linear dependence constraint qualification

Two Augmented Lagrangian algorithms for solving KKT systems are introduced. The algorithms differ in the way in which penalty parameters are updated. Possibly infeasible accumulation points are characterized. It is proved that feasible limit points that satisfy the Constant Positive Linear Dependence constraint qualification are KKT solutions. Boundedness of the penalty parameters is proved under suitable assumptions. Numerical experiments are presented.

Weak local Nash equilibrium

In this paper, we consider a concept of local Nash equilibrium for non-cooperative games - the so-called weak local Nash equilibrium. We prove its existence for a significantly more general class of sets of strategies than compact convex sets. The theorems on existence of the weak local equilibrium presented here are applications of Brouwer and Lefschetz fixed point theorems. © 2013 Juliusz Schauder Centre for Nonlinear Studies Nicolaus Copernicus University.

Distributional potential of the Triatoma brasiliensis species complex at present and under scenarios of future climate conditions

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Multiasset Derivatives and Joint Distributions of Asset Prices

Several of multiasset derivatives like basket options or options on the weighted maximum of assets exhibit the property that their prices determine uniquely the underlying asset distribution. Related to that the question how to retrieve this distributions from the corresponding derivatives quotes will be discussed. On the contrary, the prices of exchange options do not uniquely determine the underlying distributions of asset prices and the extent of this non-uniqueness can be characterised. The discussion is related to a geometric interpretation of multiasset derivatives as support functions of convex sets. Following this, various symmetry properties for basket, maximum and exchange options are discussed alongside with their geometric interpretations and some decomposition results for more general payoff functions.

Smallest disentangling state spaces for general entangled bipartite quantum states

Entangled quantum states can be given a separable decomposition if we relax the restriction that the local operators be quantum states. Motivated by the construction of classical simulations and local hidden variable models, we construct `smallest' local sets of operators that achieve this. In other words, given an arbitrary bipartite quantum state we construct convex sets of local operators that allow for a separable decomposition, but that cannot be made smaller while continuing to do so. We then consider two further variants of the problem where the local state spaces are required to contain the local quantum states, and obtain solutions for a variety of cases including a region of pure states around the maximally entangled state. The methods involve calculating certain forms of cross norm. Two of the variants of the problem have a strong relationship to theorems on ensemble decompositions of positive operators, and our results thereby give those theorems an added interpretation. The results generalise those obtained in our previous work on this topic [New J. Phys. 17, 093047 (2015)].

Matching image sets via adaptive multi convex hull

Traditional nearest points methods use all the samples in an image set to construct a single convex or affine hull model for classification. However, strong artificial features and noisy data may be generated from combinations of training samples when significant intra-class variations and/or noise occur in the image set. Existing multi-model approaches extract local models by clustering each image set individually only once, with fixed clusters used for matching with various image sets. This may not be optimal for discrimination, as undesirable environmental conditions (eg. illumination and pose variations) may result in the two closest clusters representing different characteristics of an object (eg. frontal face being compared to non-frontal face). To address the above problem, we propose a novel approach to enhance nearest points based methods by integrating affine/convex hull classification with an adapted multi-model approach. We first extract multiple local convex hulls from a query image set via maximum margin clustering to diminish the artificial variations and constrain the noise in local convex hulls. We then propose adaptive reference clustering (ARC) to constrain the clustering of each gallery image set by forcing the clusters to have resemblance to the clusters in the query image set. By applying ARC, noisy clusters in the query set can be discarded. Experiments on Honda, MoBo and ETH-80 datasets show that the proposed method outperforms single model approaches and other recent techniques, such as Sparse Approximated Nearest Points, Mutual Subspace Method and Manifold Discriminant Analysis.

Aspects of atomic decompositions and Bergman projections

The concept of an atomic decomposition was introduced by Coifman and Rochberg (1980) for weighted Bergman spaces on the unit disk. By the Riemann mapping theorem, functions in every simply connected domain in the complex plane have an atomic decomposition. However, a decomposition resulting from a conformal mapping of the unit disk tends to be very implicit and often lacks a clear connection to the geometry of the domain that it has been mapped into. The lattice of points, where the atoms of the decomposition are evaluated, usually follows the geometry of the original domain, but after mapping the domain into another this connection is easily lost and the layout of points becomes seemingly random. In the first article we construct an atomic decomposition directly on a weighted Bergman space on a class of regulated, simply connected domains. The construction uses the geometric properties of the regulated domain, but does not explicitly involve any conformal Riemann map from the unit disk. It is known that the Bergman projection is not bounded on the space L-infinity of bounded measurable functions. Taskinen (2004) introduced the locally convex spaces LV-infinity consisting of measurable and HV-infinity of analytic functions on the unit disk with the latter being a closed subspace of the former. They have the property that the Bergman projection is continuous from LV-infinity onto HV-infinity and, in some sense, the space HV-infinity is the smallest possible substitute to the space H-infinity of analytic functions. In the second article we extend the above result to a smoothly bounded strictly pseudoconvex domain. Here the related reproducing kernels are usually not known explicitly, and thus the proof of continuity of the Bergman projection is based on generalised Forelli-Rudin estimates instead of integral representations. The minimality of the space LV-infinity is shown by using peaking functions first constructed by Bell (1981). Taskinen (2003) showed that on the unit disk the space HV-infinity admits an atomic decomposition. This result is generalised in the third article by constructing an atomic decomposition for the space HV-infinity on a smoothly bounded strictly pseudoconvex domain. In this case every function can be presented as a linear combination of atoms such that the coefficient sequence belongs to a suitable Köthe co-echelon space.

Dimensions of projections of sets on Riemannian surfaces of constant curvature

We apply the theory of Peres and Schlag to obtain generic lower bounds for Hausdorff dimension of images of sets by orthogonal projections on simply connected two-dimensional Riemannian manifolds of constant curvature. As a conclusion we obtain appropriate versions of Marstrand's theorem, Kaufman's theorem, and Falconer's theorem in the above geometrical settings.

Planar point sets with large minimum convex decompositions

We show the existence of sets with n points (n ? 4) for which every convex decomposition contains more than (35/32)n?(3/2) polygons,which refutes the conjecture that for every set of n points there is a convex decomposition with at most n+C polygons. For sets having exactly three extreme pointswe show that more than n+sqr(2(n ? 3))?4 polygons may be necessary to form a convex decomposition.