Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie A-matematicas 

Abstract: According to the great mathematician Henri Lebesgue, making direct comparisons of objects with regard to a property is a fundamental mathematical process for deriving measurements. Measuring objects by using a known scale first then comparing the measurements works well for properties for which scales of measurement exist. The theme of this paper is that direct comparisons are necessary to establish measurements for intangible properties that have no scales of measurement. In that case the value derived for each element depends on what other elements it is compared with. We show how relative scales can be derived by making pairwise comparisons using numerical judgments from an absolute scale of numbers. Such measurements, when used to represent comparisons can be related and combined to define a cardinal scale of absolute numbers that is stronger than a ratio scale. They are necessary to use when intangible factors need to be added and multiplied among themselves and with tangible factors. To derive and synthesize relative scales systematically, the factors are arranged in a hierarchic or a network structure and measured according to the criteria represented within these structures. The process of making comparisons to derive scales of measurement is illustrated in two types of practical real life decisions, the Iran nuclear show-down with the West in this decade and building a Disney park in Hong Kong in 2005. It is then generalized to the case of making a continuum of comparisons by using Fredholm’s equation of the second kind whose solution gives rise to a functional equation. The Fourier transform of the solution of this equation in the complex domain is a sum of Dirac distributions demonstrating that proportionate response to stimuli is a process of firing and synthesis of firings as neurons in the brain do. The Fourier transform of the solution of the equation in the real domain leads to nearly inverse square responses to natural influences. Various generalizations and critiques of the approach are included.

Abstract: This paper is a survey of some classical contributions and recent progress in identifying optimal dividend payment strategies in the framework of collective risk theory. In particular, available mathematical tools are discussed and some challenges are described that occur under various objective functions and model assumptions. Finally, some open research problems in this field are stated.

Abstract: There have been many attempts to generalize the definition of a metric space in order to obtain possibilities for more general fixed point results. In this paper, we give a survey of recent results on reducing fixed point theorems on generalized metric spaces to fixed point theorems on metric spaces and then investigate this fact in other generalized metric spaces. We show that many generalized metric spaces are topologically equivalent to certain metric spaces or to previously generalized metric spaces. Also, the fixed point theory in these generalized metric spaces may be a consequence of the fixed point theory in certain metric spaces or in previously generalized metric spaces.

Abstract: In a paper written some 25 years ago, I distinguished three contexts in which one might wish to combine expert judgements of uncertainty: the expert problem, the group decision problem and the textbook problem. Over the intervening years much has been written on the first two, which have the focus of a single decision context, but little on the third, though the closely related field of meta-analysis has developed considerably. With many developments in internet technology, particularly in relation to interactivity and communication, the textbook problem is gaining in importance since data and expert judgements can be made available over the web to be used by many different individuals to shape their own beliefs in many different contexts. Moreover, applications such as web-based decision support, e-participation and e-democracy are making algorithmic 'solutions' to the group decision problem attractive, despite many results showing we know that such solutions are, at best, rare and, at worst, illusory. In this paper I survey developments since my earlier paper and note some unresolved issues. Then I turn to how expert judgement might be used within web-based group decision support, as well as in e-participation and e-democracy contexts. The latter points to a growing importance of the textbook problem and suggests that Cooke's principles for scientific reporting of expert judgement studies may need enhancing for such studies to be used by a wider audience. © 2011 Springer-Verlag.

Abstract: Rus (Approx. Convexity 3:171–178, 2005) introduced the concept of cyclic contraction mapping. Pacurar and Rus (Nonlinear Anal. 72:1181–1187, 2010) proved some fixed point results for cyclic \({\phi }\)-contraction mappings on a metric space. Karapinar (Appl. Math. Lett. 24:822–825, 2011) obtained a unique fixed point of cyclic weak \({\phi }\)- contraction mappings and studied well-posedness problem for such mappings. On the other hand, Matthews (Ann. New York Acad. Sci. 728:183–197, 1994) introduced the concept of a partial metric as a part of the study of denotational semantics of dataflow networks. He gave a modified version of the Banach contraction principle, more suitable in this context. In this paper, we initiate the study of fixed points of generalized cyclic contraction in the framework of partial metric spaces. We also present some examples to validate our results.