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.

City College Southampton: Striving for Gold with Moodle benchmarking scheme

.How do you answer the question: “How well is your VLE being used?” City College Southampton rose to the challenge with the creation, development and implementation of a highly motivational ‘Gold, Silver and Bronze’ benchmarking system to evaluate and recognise quality of content and activity on its College Moodle system. The results are impressive.

Olympic Coast National Marine Sanctuary Habitat Mapping: Survey report and classification of side scan sonar data from surveys HMPR-114-2004-02 and HMPR-116-2005-01.

The Olympic Coast National Marine Sanctuary (OCNMS) continues to invest significant resources into seafloor mapping activities along Washington’s outer coast (Intelmann and Cochrane 2006; Intelmann et al. 2006; Intelmann 2006). Results from these annual mapping efforts offer a snapshot of current ground conditions, help to guide research and management activities, and provide a baseline for assessing the impacts of various threats to important habitat. During the months of August 2004 and May and July 2005, we used side scan sonar to image several regions of the sea floor in the northern OCNMS, and the data were mosaicked at 1-meter pixel resolution. Video from a towed camera sled, bathymetry data, sedimentary samples and side scan sonar mapping were integrated to describe geological and biological aspects of habitat. Polygon features were created and attributed with a hierarchical deep-water marine benthic classification scheme (Greene et al. 1999). For three small areas that were mapped with both side scan sonar and multibeam echosounder, we made a comparison of output from the classified images indicating little difference in results between the two methods. With these considerations, backscatter derived from multibeam bathymetry is currently a costefficient and safe method for seabed imaging in the shallow (<30 meters) rocky waters of OCNMS. The image quality is sufficient for classification purposes, the associated depths provide further descriptive value and risks to gear are minimized. In shallow waters (<30 meters) which do not have a high incidence of dangerous rock pinnacles, a towed multi-beam side scan sonar could provide a better option for obtaining seafloor imagery due to the high rate of acquisition speed and high image quality, however the high probability of losing or damaging such a costly system when deployed as a towed configuration in the extremely rugose nearshore zones within OCNMS is a financially risky proposition. The development of newer technologies such as intereferometric multibeam systems and bathymetric side scan systems could also provide great potential for mapping these nearshore rocky areas as they allow for high speed data acquisition, produce precisely geo-referenced side scan imagery to bathymetry, and do not experience the angular depth dependency associated with multibeam echosounders allowing larger range scales to be used in shallower water. As such, further investigation of these systems is needed to assess their efficiency and utility in these environments compared to traditional side scan sonar and multibeam bathymetry. (PDF contains 43 pages.)

Generating cluster submodels from a multistage stochastic mixed integer optimization model using break stage

We present a scheme to generate clusters submodels with stage ordering from a (symmetric or a nonsymmetric one) multistage stochastic mixed integer optimization model using break stage. We consider a stochastic model in compact representation and MPS format with a known scenario tree. The cluster submodels are built by storing first the 0-1 the variables, stage by stage, and then the continuous ones, also stage by stage. A C++ experimental code has been implemented for reordering the stochastic model as well as the cluster decomposition after the relaxation of the non-anticipativiy constraints until the so-called breakstage. The computational experience shows better performance of the stage ordering in terms of elapsed time in a randomly generated testbed of multistage stochastic mixed integer problems.

On a Class of Self-Adjoint Compact Operators in Hilbert Spaces and Their Relations with Their Finite-Range Truncations

This paper investigates a class of self-adjoint compact operators in Hilbert spaces related to their truncated versions with finite-dimensional ranges. The comparisons are established in terms of worst-case norm errors of the composite operators generated from iterated computations. Some boundedness properties of the worst-case norms of the errors in their respective fixed points in which they exist are also given. The iterated sequences are expanded in separable Hilbert spaces through the use of numerable orthonormal bases.

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.

An observer-based vaccination control law for an SEIR epidemic model based on feedback linearization techniques for nonlinear systems

This paper presents a vaccination strategy for fighting against the propagation of epidemic diseases. The disease propagation is described by an SEIR (susceptible plus infected plus infectious plus removed populations) epidemic model. The model takes into account the total population amounts as a refrain for the illness transmission since its increase makes the contacts among susceptible and infected more difficult. The vaccination strategy is based on a continuous-time nonlinear control law synthesised via an exact feedback input-output linearization approach. An observer is incorporated into the control scheme to provide online estimates for the susceptible and infected populations in the case when their values are not available from online measurement but they are necessary to implement the control law. The vaccination control is generated based on the information provided by the observer. The control objective is to asymptotically eradicate the infection from the population so that the removed-by-immunity population asymptotically tracks the whole one without precise knowledge of the partial populations. The model positivity, the eradication of the infection under feedback vaccination laws and the stability properties as well as the asymptotic convergence of the estimation errors to zero as time tends to infinity are investigated.