Coincidence and common fixed point theorems for Suzuki type hybrid contractions and applications

Coincidence and common fixed point theorems for a class of Suzuki hybrid contractions involving two pairs of single-valued and multivalued maps in a metric space are obtained. In addition, the existence of a common solution for a certain class of functional equations arising in a dynamic programming is also discussed.

On turbulent, erratic and other dynamical properties of Zadeh's extensions

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Chaos induced by turbulent and erratic functions

Let (X, d) be a compact metric space and f: X → X a continuous function and consider the hyperspace (K(X), H) of all nonempty compact subsets of X endowed with the Hausdorff metric induced by d. Let f̄: K(X) → K (X) be defined by f̄(A) = {f(a)/a ∈ A} the natural extension of f to K(X), then the aim of this work is to study the dynamics of f when f is turbulent (erratic, respectively) and its relationships.

Curriculum vita of Prof. Vasil Atanasov Popov

Our primary goal in this preamble is to highlight the best of Vasil Popov’s mathematical achievements and ideas. V. Popov showed his extraordinary talent for mathematics in his early papers in the (typically Bulgarian) area of approximation in the Hausdorff metric. His results in this area are very well presented in the monograph of his advisor Bl. Sendov, “Hausdorff Approximation”.

Spline Subdivision Schemes for Compact Sets. A Survey

Dedicated to the memory of our colleague Vasil Popov January 14, 1942 – May 31, 1990 * Partially supported by ISF-Center of Excellence, and by The Hermann Minkowski Center for Geometry at Tel Aviv University, Israel

On the Approximation of the Generalized Cut Function of Degree p+1 By Smooth Sigmoid Functions

We introduce a modification of the familiar cut function by replacing the linear part in its definition by a polynomial of degree p + 1 obtaining thus a sigmoid function called generalized cut function of degree p + 1 (GCFP). We then study the uniform approximation of the (GCFP) by smooth sigmoid functions such as the logistic and the shifted logistic functions. The limiting case of the interval-valued Heaviside step function is also discussed which imposes the use of Hausdorff metric. Numerical examples are presented using CAS MATHEMATICA.

Dimension Distortion by Sobolev Mappings in Foliated Metric Spaces

We quantify the extent to which a supercritical Sobolev mapping can increase the dimension of subsets of its domain, in the setting of metric measure spaces supporting a Poincaré inequality. We show that the set of mappings that distort the dimensions of sets by the maximum possible amount is a prevalent subset of the relevant function space. For foliations of a metric space X defined by a David–Semmes regular mapping Π : X → W, we quantitatively estimate, in terms of Hausdorff dimension in W, the size of the set of leaves of the foliation that are mapped onto sets of higher dimension. We discuss key examples of such foliations, including foliations of the Heisenberg group by left and right cosets of horizontal subgroups.

Hausdorff Distances for Searching in Binary Text Images

This work has been partially supported by Grant No. DO 02-275, 16.12.2008, Bulgarian NSF, Ministry of Education and Science.

Dynamic load sharing for heavy vehicles : a new metric

Dynamic load sharing can be defined as a measure of the ability of a heavy vehicle multi-axle group to equalise load across its wheels under typical travel conditions; i.e. in the dynamic sense at typical travel speeds and operating conditions of that vehicle. Various attempts have been made to quantify the ability of heavy vehicles to equalise the load across their wheels during travel. One of these was the concept of the load sharing coefficient (LSC). Other metrics such as the dynamic load coefficient (DLC) have been used to compare one heavy vehicle suspension with another for potential road damage. This paper compares these metrics and determines a relationship between DLC and LSC with sensitivity analysis of this relationship. The shortcomings of these presently-available metrics are discussed with a new metric proposed - the dynamic load equalisation (DLE) measure.

Using incomplete online metric maps for topological exploration with the Gap Navigation Tree

This paper presents a general, global approach to the problem of robot exploration, utilizing a topological data structure to guide an underlying Simultaneous Localization and Mapping (SLAM) process. A Gap Navigation Tree (GNT) is used to motivate global target selection and occluded regions of the environment (called “gaps”) are tracked probabilistically. The process of map construction and the motion of the vehicle alters both the shape and location of these regions. The use of online mapping is shown to reduce the difficulties in implementing the GNT.

Enterprise information security policy assessment : an extended framework for metrics development utilising the goal-question-metric approach

Effective enterprise information security policy management requires review and assessment activities to ensure information security policies are aligned with business goals and objectives. As security policy management involves the elements of policy development process and the security policy as output, the context for security policy assessment requires goal-based metrics for these two elements. However, the current security management assessment methods only provide checklist types of assessment that are predefined by industry best practices and do not allow for developing specific goal-based metrics. Utilizing theories drawn from literature, this paper proposes the Enterprise Information Security Policy Assessment approach that expands on the Goal-Question-Metric (GQM) approach. The proposed assessment approach is then applied in a case scenario example to illustrate a practical application. It is shown that the proposed framework addresses the requirement for developing assessment metrics and allows for the concurrent undertaking of process-based and product-based assessment. Recommendations for further research activities include the conduct of empirical research to validate the propositions and the practical application of the proposed assessment approach in case studies to provide opportunities to introduce further enhancements to the approach.

A quantitative metric to identify critical elements within seafood supply networks

A theoretical basis is required for comparing key features and critical elements in wild fisheries and aquaculture supply chains under a changing climate. Here we develop a new quantitative metric that is analogous to indices used to analyse food-webs and identify key species. The Supply Chain Index (SCI) identifies critical elements as those elements with large throughput rates, as well as greater connectivity. The sum of the scores for a supply chain provides a single metric that roughly captures both the resilience and connectedness of a supply chain. Standardised scores can facilitate cross-comparisons both under current conditions as well as under a changing climate. Identification of key elements along the supply chain may assist in informing adaptation strategies to reduce anticipated future risks posed by climate change. The SCI also provides information on the relative stability of different supply chains based on whether there is a fairly even spread in the individual scores of the top few key elements, compared with a more critical dependence on a few key individual supply chain elements. We use as a case study the Australian southern rock lobster Jasus edwardsii fishery, which is challenged by a number of climate change drivers such as impacts on recruitment and growth due to changes in large-scale and local oceanographic features. The SCI identifies airports, processors and Chinese consumers as the key elements in the lobster supply chain that merit attention to enhance stability and potentially enable growth. We also apply the index to an additional four real-world Australian commercial fishery and two aquaculture industry supply chains to highlight the utility of a systematic method for describing supply chains. Overall, our simple methodological approach to empirically-based supply chain research provides an objective method for comparing the resilience of supply chains and highlighting components that may be critical.

Homogeneous model theory of metric structures

This thesis studies homogeneous classes of complete metric spaces. Over the past few decades model theory has been extended to cover a variety of nonelementary frameworks. Shelah introduced the abstact elementary classes (AEC) in the 1980s as a common framework for the study of nonelementary classes. Another direction of extension has been the development of model theory for metric structures. This thesis takes a step in the direction of combining these two by introducing an AEC-like setting for studying metric structures. To find balance between generality and the possibility to develop stability theoretic tools, we work in a homogeneous context, thus extending the usual compact approach. The homogeneous context enables the application of stability theoretic tools developed in discrete homogeneous model theory. Using these we prove categoricity transfer theorems for homogeneous metric structures with respect to isometric isomorphisms. We also show how generalized isomorphisms can be added to the class, giving a model theoretic approach to, e.g., Banach space isomorphisms or operator approximations. The novelty is the built-in treatment of these generalized isomorphisms making, e.g., stability up to perturbation the natural stability notion. With respect to these generalized isomorphisms we develop a notion of independence. It behaves well already for structures which are omega-stable up to perturbation and coincides with the one from classical homogeneous model theory over saturated enough models. We also introduce a notion of isolation and prove dominance for it.