Fixed Points of Closed and Compact Composite Sequences of Operators and Projectors in a Class of Banach Spaces

Some results on fixed points related to the contractive compositions of bounded operators in a class of complete metric spaces which can be also considered as Banach's spaces are discussed through the paper. The class of composite operators under study can include, in particular, sequences of projection operators under, in general, oblique projective operators. In this paper we are concerned with composite operators which include sequences of pairs of contractive operators involving, in general, oblique projection operators. The results are generalized to sequences of, in general, nonconstant bounded closed operators which can have bounded, closed, and compact limit operators, such that the relevant composite sequences are also compact operators. It is proven that in both cases, Banach contraction principle guarantees the existence of unique fixed points under contractive conditions.

Convergence Properties and Fixed Points of Two General Iterative Schemes with Composed Maps in Banach Spaces with Applications to Guaranteed Global Stability

This paper investigates the boundedness and convergence properties of two general iterative processes which involve sequences of self-mappings on either complete metric or Banach spaces. The sequences of self-mappings considered in the first iterative scheme are constructed by linear combinations of a set of self-mappings, each of them being a weighted version of a certain primary self-mapping on the same space. The sequences of self-mappings of the second iterative scheme are powers of an iteration-dependent scaled version of the primary self-mapping. Some applications are also given to the important problem of global stability of a class of extended nonlinear polytopic-type parameterizations of certain dynamic systems.

On Best Proximity Point Theorems and Fixed Point Theorems for p-Cyclic Hybrid Self-Mappings in Banach Spaces

This paper relies on the study of fixed points and best proximity points of a class of so-called generalized point-dependent (K-Lambda)hybrid p-cyclic self-mappings relative to a Bregman distance Df, associated with a Gâteaux differentiable proper strictly convex function f in a smooth Banach space, where the real functions Lambda and K quantify the point-to-point hybrid and nonexpansive (or contractive) characteristics of the Bregman distance for points associated with the iterations through the cyclic self-mapping.Weak convergence results to weak cluster points are obtained for certain average sequences constructed with the iterates of the cyclic hybrid self-mappings.

Classification of neocortical interneurons using affinity propagation

In spite of over a century of research on cortical circuits, it is still unknown how many classes of cortical neurons exist. Neuronal classification has been a difficult problem because it is unclear what a neuronal cell class actually is and what are the best characteristics are to define them. Recently, unsupervised classifications using cluster analysis based on morphological, physiological or molecular characteristics, when applied to selected datasets, have provided quantitative and unbiased identification of distinct neuronal subtypes. However, better and more robust classification methods are needed for increasingly complex and larger datasets. We explored the use of affinity propagation, a recently developed unsupervised classification algorithm imported from machine learning, which gives a representative example or exemplar for each cluster. As a case study, we applied affinity propagation to a test dataset of 337 interneurons belonging to four subtypes, previously identified based on morphological and physiological characteristics. We found that affinity propagation correctly classified most of the neurons in a blind, non-supervised manner. In fact, using a combined anatomical/physiological dataset, our algorithm differentiated parvalbumin from somatostatin interneurons in 49 out of 50 cases. Affinity propagation could therefore be used in future studies to validly classify neurons, as a first step to help reverse engineer neural circuits.

Evidence for classification of c.1852_1853AA > GC in MLH1 as a neutral variant for Lynch syndrome

Background: Lynch syndrome (LS) is an autosomal dominant inherited cancer syndrome characterized by early onset cancers of the colorectum, endometrium and other tumours. A significant proportion of DNA variants in LS patients are unclassified. Reports on the pathogenicity of the c.1852_1853AA>GC (p.Lys618Ala) variant of the MLH1 gene are conflicting. In this study, we provide new evidence indicating that this variant has no significant implications for LS. Methods: The following approach was used to assess the clinical significance of the p.Lys618Ala variant: frequency in a control population, case-control comparison, co-occurrence of the p.Lys618Ala variant with a pathogenic mutation, co-segregation with the disease and microsatellite instability in tumours from carriers of the variant. We genotyped p.Lys618Ala in 1034 individuals (373 sporadic colorectal cancer [CRC] patients, 250 index subjects from families suspected of having LS [revised Bethesda guidelines] and 411 controls). Three well-characterized LS families that fulfilled the Amsterdam II Criteria and consisted of members with the p.Lys618Ala variant were included to assess co-occurrence and co-segregation. A subset of colorectal tumour DNA samples from 17 patients carrying the p.Lys618Ala variant was screened for microsatellite instability using five mononucleotide markers. Results: Twenty-seven individuals were heterozygous for the p.Lys618Ala variant; nine had sporadic CRC (2.41%), seven were suspected of having hereditary CRC (2.8%) and 11 were controls (2.68%). There were no significant associations in the case-control and case-case studies. The p.Lys618Ala variant was co-existent with pathogenic mutations in two unrelated LS families. In one family, the allele distribution of the pathogenic and unclassified variant was in trans, in the other family the pathogenic variant was detected in the MSH6 gene and only the deleterious variant co-segregated with the disease in both families. Only two positive cases of microsatellite instability (2/17, 11.8%) were detected in tumours from p.Lys618Ala carriers, indicating that this variant does not play a role in functional inactivation of MLH1 in CRC patients. Conclusions: The p.Lys618Ala variant should be considered a neutral variant for LS. These findings have implications for the clinical management of CRC probands and their relatives.

Image-based delineation and classification of built heritage masonry

[EN]Fundación Zain is developing new built heritage assessment protocols. The goal is to objectivize and standardize the analysis and decision process that leads to determining the degree of protection of built heritage in the Basque Country. The ultimate step in this objectivization and standardization effort will be the development of an information and communication technology (ICT) tool for the assessment of built heritage. This paper presents the ground work carried out to make this tool possible: the automatic, image-based delineation of stone masonry. This is a necessary first step in the development of the tool, as the built heritage that will be assessed consists of stone masonry construction, and many of the features analyzed can be characterized according to the geometry and arrangement of the stones. Much of the assessment is carried out through visual inspection. Thus, this process will be automated by applying image processing on digital images of the elements under inspection. The principal contribution of this paper is the automatic delineation the framework proposed. The other contribution is the performance evaluation of this delineation as the input to a classifier for a geometrically characterized feature of a built heritage object. The element chosen to perform this evaluation is the stone arrangement of masonry walls. The validity of the proposed framework is assessed on real images of masonry walls.

Image-Based Delineation and Classification of Built Heritage Masonry

Fundacion Zain is developing new built heritage assessment protocols. The goal is to objectivize and standardize the analysis and decision process that leads to determining the degree of protection of built heritage in the Basque Country. The ultimate step in this objectivization and standardization effort will be the development of an information and communication technology (ICT) tool for the assessment of built heritage. This paper presents the ground work carried out to make this tool possible: the automatic, image-based delineation of stone masonry. This is a necessary first step in the development of the tool, as the built heritage that will be assessed consists of stone masonry construction, and many of the features analyzed can be characterized according to the geometry and arrangement of the stones. Much of the assessment is carried out through visual inspection. Thus, this process will be automated by applying image processing on digital images of the elements under inspection. The principal contribution of this paper is the automatic delineation the framework proposed. The other contribution is the performance evaluation of this delineation as the input to a classifier for a geometrically characterized feature of a built heritage object. The element chosen to perform this evaluation is the stone arrangement of masonry walls. The validity of the proposed framework is assessed on real images of masonry walls.

Some Results on Fixed and Best Proximity Points of Multivalued

This paper is devoted to investigate the fixed points and best proximity points of multivalued cyclic self-mappings on a set of subsets of complete metric spaces endowed with a partial order under a generalized contractive condition involving a Hausdorff distance. The existence and uniqueness of fixed points of both the cyclic self-mapping and its associate composite self-mappings on each of the subsets are investigated, if the subsets in the cyclic disposal are nonempty, bounded and of nonempty convex intersection. The obtained results are extended to the existence of unique best proximity points in uniformly convex Banach spaces.

Some Results on Fixed and Best Proximity Points of Precyclic Self-Mappings

12 p.

Best Proximity Points of Generalized Semicyclic Impulsive Self-Mappings: Applications to Impulsive Differential and Difference Equations

This paper is devoted to the study of convergence properties of distances between points and the existence and uniqueness of best proximity and fixed points of the so-called semicyclic impulsive self-mappings on the union of a number of nonempty subsets in metric spaces. The convergences of distances between consecutive iterated points are studied in metric spaces, while those associated with convergence to best proximity points are set in uniformly convex Banach spaces which are simultaneously complete metric spaces. The concept of semicyclic self-mappings generalizes the well-known one of cyclic ones in the sense that the iterated sequences built through such mappings are allowed to have images located in the same subset as their pre-image. The self-mappings under study might be in the most general case impulsive in the sense that they are composite mappings consisting of two self-mappings, and one of them is eventually discontinuous. Thus, the developed formalism can be applied to the study of stability of a class of impulsive differential equations and that of their discrete counterparts. Some application examples to impulsive differential equations are also given.

Economic Valuation of Ecosystem Services: Conflicts in Classification

4 p.

Properties of convergence of a class of iterative processes generated by sequences of self-mappings with applications to switched dynamic systems

This article investigates the convergence properties of iterative processes involving sequences of self-mappings of metric or Banach spaces. Such sequences are built from a set of primary self-mappings which are either expansive or non-expansive self-mappings and some of the non-expansive ones can be contractive including the case of strict contractions. The sequences are built subject to switching laws which select each active self-mapping on a certain activation interval in such a way that essential properties of boundedness and convergence of distances and iterated sequences are guaranteed. Applications to the important problem of stability of dynamic switched systems are also given.