Continuity for s-convex fuzzy processes

In a previous paper we introduced the concept of s-convex fuzzy mapping and established some properties. In this work we study the continuity for s-convex fuzzy processes

Normal Cones and Thompson Metric

The aim of this paper is to study the basic properties of the Thompson metric $d_T$ in the general case of a real linear space $X$ ordered by a cone $K$. We show that $d_T$ has monotonicity properties which make it compatible with the linear structure. We also prove several convexity properties of $d_T$ and some results concerning the topology of $d_T$, including a brief study of the $d_T$-convergence of monotone sequences. It is shown most of the results are true without any assumption of an Archimedean-type property for $K$. One considers various completeness properties and one studies the relations between them. Since $d_T$ is defined in the context of a generic ordered linear space, with no need of an underlying topological structure, one expects to express its completeness in terms of properties of the ordering, with respect to the linear structure. This is done in this paper and, to the best of our knowledge, this has not been done yet. The Thompson metric $d_T$ and order-unit (semi)norms $|\cdot|_u$ are strongly related and share important properties, as both are defined in terms of the ordered linear structure. Although $d_T$ and $|\cdot|_u$ are only topological (and not metrical) equivalent on $K_u$, we prove that the completeness is a common feature. One proves the completeness of the Thompson metric on a sequentially complete normal cone in a locally convex space. At the end of the paper, it is shown that, in the case of a Banach space, the normality of the cone is also necessary for the completeness of the Thompson metric.Comment: 36 page

Recent progress in random metric theory and its applications to conditional risk measures

The purpose of this paper is to give a selective survey on recent progress in random metric theory and its applications to conditional risk measures. This paper includes eight sections. Section 1 is a longer introduction, which gives a brief introduction to random metric theory, risk measures and conditional risk measures. Section 2 gives the central framework in random metric theory, topological structures, important examples, the notions of a random conjugate space and the Hahn-Banach theorems for random linear functionals. Section 3 gives several important representation theorems for random conjugate spaces. Section 4 gives characterizations for a complete random normed module to be random reflexive. Section 5 gives hyperplane separation theorems currently available in random locally convex modules. Section 6 gives the theory of random duality with respect to the locally $L^{0}-$convex topology and in particular a characterization for a locally $L^{0}-$convex module to be $L^{0}-$pre$-$barreled. Section 7 gives some basic results on $L^{0}-$convex analysis together with some applications to conditional risk measures. Finally, Section 8 is devoted to extensions of conditional convex risk measures, which shows that every representable $L^{\infty}-$type of conditional convex risk measure and every continuous $L^{p}-$type of convex conditional risk measure ($1\leq p<+\infty$) can be extended to an $L^{\infty}_{\cal F}({\cal E})-$type of $\sigma_{\epsilon,\lambda}(L^{\infty}_{\cal F}({\cal E}), L^{1}_{\cal F}({\cal E}))-$lower semicontinuous conditional convex risk measure and an $L^{p}_{\cal F}({\cal E})-$type of ${\cal T}_{\epsilon,\lambda}-$continuous conditional convex risk measure ($1\leq p<+\infty$), respectively.Comment: 37 page

Exact controllability of non-Lipschitz semilinear systems

We present sufficient conditions for exact controllability of a semilinear infinite-dimensional dynamical system. The system mild solution is formed by a noncompact semigroup and a nonlinear disturbance that does not need to be Lipschitz continuous. Our main result is based on a fixed point-type application of the Schmidt existence theorem and illustrated by a nonlinear transport partial differential equation

The troubling concept of class: reflecting on our ‘failure’ to encourage sociology students to re-cognise their classed locations using autobiographical methods

The troubling concept of class: reflecting on our ‘failure’ to encourage sociology students to re-cognise their classed locations using autobiographical methods Abstract This paper provides a narrative of the four authors‟ commitment to auto/biographical methods as teachers and researchers in „new‟ universities. As they went about their work, they observed that, whereas students engage with the gendered, sexualised and racialised processes when negotiating their identities, they are reluctant or unable to conceptualise „class-ifying‟ processes as key determinants of their life chances. This general inability puzzled the authors, given the students‟ predominantly working-class backgrounds. Through application of their own stories, the authors explore the sociological significance of this pedagogical „failure‟ to account for the troubling concept of class not only in the classroom but also in contemporary society

Set optimization - a rather short introduction

Recent developments in set optimization are surveyed and extended including various set relations as well as fundamental constructions of a convex analysis for set- and vector-valued functions, and duality for set optimization problems. Extensive sections with bibliographical comments summarize the state of the art. Applications to vector optimization and financial risk measures are discussed along with algorithmic approaches to set optimization problems

Learning to Think Iconically in the Human and Social Sciences: Iconic Standards of Understanding as a Pivotal Challenge for Method Development

Theoretically as well as alongside an empirical research idea, this paper outlines conditions for the development of social scientific empirical methods able to further exploit the iconic potential of the image. Reconstructing the role of formal pictorial elements for the standards of understanding within the medium “image” is considered pivotal in this endeavor. Within the context of language, standards of communication have already been extensively researched. The linguistic format of the narrative, for instance, is well studied. Up to now, though, comparable formal vehicles of iconic semantics have only been examined in aesthetics and art history. Nevertheless, standards of iconic understanding are part of our implicit knowledge, are incessantly in use in everyday practice and, thus, the basis of everyday identity formation. With the help of empirical methods based on an iconic logos we can deepen our understanding of orientations, longings, and anxieties of our time that are often silently conveyed by images. Fashion will be outlined as a prototypical field, in which an empirically based development of such methods might start off

Der virtuelle Raum als Double - oder: zur Persistenz hierarchischer Gesellschaftsstruktur im Netz

Der raumsoziologische Beitrag diskutiert, was die mit der Technologie des Internets einhergehende Virtualisierung für eine sich ändernde Gesellschaftsordnung bedeuten kann. Anhand ihres methodologischen RaumZeit-Modells zeigt die Autorin, dass die virtuelle Realität des Internets eine doppelte materiale Gestalt hervorbringt. Als Double ergänzt der virtuelle Raum den realen. Die Ausgangsannahme lautet, dass die bürgerlich-moderne Gesellschaft Raum als Zweidimensionalität und als Behälter sowie Zeit als messbar und linear hervorgebracht hat. Entgegen vielfach geäußerter Einschätzungen revolutioniert das Internet nicht Vorstellungen und Praxis von Raum und Zeit, sondern es perfektioniert die bürgerliche Konstruktion des ideal beherrschbaren Lebens. Der Hypothese folgend, dass die neuen virtuellen Realitäten veränderte Materialitäten etablieren, werden verschiedene Zukunftsszenarien am Beispiel des Geschlechterverhältnisses durchgespielt