Other uncertainty theories based on capacities

International audienceThe two main uncertainty representations in the literature that tolerate imprecision are possibility distributions and random disjunctive sets. This chapter devotes special attention to the theories that have emerged from them. The first part of the chapter discusses epistemic logic and derives the need for capturing imprecision in information representations. It bridges the gap between uncertainty theories and epistemic logic showing that imprecise probabilities subsume modalities of possibility and necessity as much as probability. The second part presents possibility and evidence theories, their origins, assumptions and semantics, discusses the connections between them and the general framework of imprecise probability. Finally, chapter points out the remaining discrepancies between the different theories regarding various basic notions, such as conditioning, independence or information fusion and the existing bridges between them

Hardy spaces and quasiconformal maps in the Heisenberg group

We define Hardy spaces $H^p$, $0<p<\infty$, for quasiconformal mappings on the Kor\'{a}nyi unit ball $B$ in the first Heisenberg group $\mathbb{H}^1$. Our definition is stated in terms of the Heisenberg polar coordinates introduced by Kor\'{a}nyi and Reimann, and Balogh and Tyson. First, we prove the existence of $p_0(K)>0$ such that every $K$-quasiconformal map $f:B \to f(B) \subset \mathbb{H}^1$ belongs to $H^p$ for all $0<p<p_0(K)$. Second, we give two equivalent conditions for the $H^p$ membership of a quasiconformal map $f$, one in terms of the radial limits of $f$, and one using a nontangential maximal function of $f$. As an application, we characterize Carleson measures on $B$ via integral inequalities for quasiconformal mappings on $B$ and their radial limits. Our paper thus extends results by Astala and Koskela, Jerison and Weitsman, Nolder, and Zinsmeister, from $\mathbb{R}^n$ to $\mathbb{H}^1$. A crucial difference between the proofs in $\mathbb{R}^n$ and $\mathbb{H}^1$ is caused by the nonisotropic nature of the Kor\'{a}nyi unit sphere with its two characteristic points.Comment: 51 p

Représentation et combinaison d'informations incertaines : nouveaux résultats avec applications aux études de sûreté nucléaires

It often happens that the value of some parameters or variables of a system are imperfectly known, either because of the variability of the modelled phenomena, or because the availableinformation is imprecise or incomplete. Classical probability theory is usually used to treat these uncertainties. However, recent years have witnessed the appearance of arguments pointing to the conclusion that classical probabilities are inadequate to handle imprecise or incomplete information. Other frameworks have thus been proposed to address this problem: the three main are probability sets, random sets and possibility theory. There are many open questions concerning uncertainty treatment within these frameworks. More precisely, it is necessary to build bridges between these three frameworks to advance toward a unified handlingof uncertainty. Also, there is a need of practical methods to treat information, as using these framerowks can be computationally costly. In this work, we propose some answers to these two needs for a set of commonly encountered problems. In particular, we focus on the problems of:- Uncertainty representation- Fusion and evluation of multiple source information- Independence modellingThe aim being to give tools (both of theoretical and practical nature) to treat uncertainty. Some tools are then applied to some problems related to nuclear safety issues.Souvent, les valeurs de certains paramètres ou variables d'un système ne sont connues que de façon imparfaite, soit du fait de la variabilité des phénomènes physiques que l'on cherche à représenter,soit parce que l'information dont on dispose est imprécise, incomplète ou pas complètement fiable.Usuellement, cette incertitude est traitée par la théorie classique des probabilités. Cependant, ces dernières années ont vu apparaître des arguments indiquant que les probabilités classiques sont inadéquates lorsqu'il faut représenter l'imprécision présente dans l'information. Des cadres complémentaires aux probabilités classiques ont donc été proposés pour remédier à ce problème : il s'agit, principalement, des ensembles de probabilités, des ensembles aléatoires et des possibilités. Beaucoup de questions concernant le traitement des incertitudes dans ces trois cadres restent ouvertes. En particulier, il est nécessaire d'unifier ces approches et de comprendre les liens existants entre elles, et de proposer des méthodes de traitement permettant d'utiliser ces approches parfois cher en temps de calcul. Dans ce travail, nous nous proposons d'apporter des réponses à ces deux besoins pour une série de problème de traitement de l'incertain rencontré en analyse de sûreté. En particulier, nous nous concentrons sur les problèmes suivants :- Représentation des incertitudes- Fusion/évaluation de données venant de sources multiples- Modélisation de l'indépendanceL'objectif étant de fournir des outils, à la fois théoriques et pratiques, de traitement d'incertitude. Certains de ces outils sont ensuite appliqués à des problèmes rencontrés en sûreté nucléaire

On the Hofer–Zehnder Capacity of Twisted Tangent Bundles

In this thesis, we deal with the Hofer--Zehnder capacity of disc subbundles of twisted tangent bundles. While in the literature for most cases only the finiteness of this capacity is shown, we use symmetries to determine exact values of the capacity. We therefore restrict ourselves to a class of homogeneous Kähler manifolds, called Hermitian symmetric spaces. For these, we construct a symplectomorphism that identifies the twisted tangent bundle, or at least a neighborhood of the zero section that we can specify explicitly, and the Hermitian tangent bundle. The advantage of the Hermitian tangent bundle is that the fibers are symplectic. This makes it easier to study holomorphic curves, which we use to obtain an upper bound on the Hofer--Zehnder capacity. We get the lower bound by specifying a Hamiltonian that generates a circle action. The oscillation of such a Hamiltonian always yields a lower bound. We also clarify the relationship between the twisted, respectively Hermitian, symplectic structure to the hyperkähler structure in a neighborhood of the zero section of the tangent bundle of a Hermitian symmetric space. For various reasons, it is much harder to determine the Hofer--Zehnder capacity for standard tangential bundles than it is for the twisted case. Nevertheless, we were able to compute the Hofer--Zehnder capacity for the disc subbundle of the standard tangent bundle of the complex projective space and the real projective space. To obtain the lower bound it is for the former sufficient to consider the kinetic Hamiltonian, i.e. geodesic flow, while in the second case, geodesic billiards must be used. For the upper bound one uses the symmetries of the spaces to show that the disc subbundle of the tangent bundles compactify to the product of two complex projective spaces in the first case and the complex projective space in the second case. In these compact symplectic manifolds one can again study holomorphic spheres in order to construct upper bounds. In fact, we also show in the twisted case that the disc subbundle of the tangent bundle of the complex projective space compactifies to the product, but now with differently weighted factors. Furthermore, this thesis includes the computation of the Hofer--Zehnder capacity of Hermitian symmetric spaces of compact type. This exploits the fact that Hermitian symmetric spaces can be represented as coadjoint orbits. In this representation it is relatively easy to specify a Hamiltonian which generates a semi-free circle action and which attains its minimum at an isolated point. The oscillation of such a Hamiltonian provides both lower and upper bounds and thus determines the capacity

A Dempster-Shafer theory inspired logic.

Issues of formalising and interpreting epistemic uncertainty have always played a prominent role in Artificial Intelligence. The Dempster-Shafer (DS) theory of partial beliefs is one of the most-well known formalisms to address the partial knowledge. Similarly to the DS theory, which is a generalisation of the classical probability theory, fuzzy logic provides an alternative reasoning apparatus as compared to Boolean logic. Both theories are featured prominently within the Artificial Intelligence domain, but the unified framework accounting for all the aspects of imprecise knowledge is yet to be developed. Fuzzy logic apparatus is often used for reasoning based on vague information, and the beliefs are often processed with the aid of Boolean logic. The situation clearly calls for the development of a logic formalism targeted specifically for the needs of the theory of beliefs. Several frameworks exist based on interpreting epistemic uncertainty through an appropriately defined modal operator. There is an epistemic problem with this kind of frameworks: while addressing uncertain information, they also allow for non-constructive proofs, and in this sense the number of true statements within these frameworks is too large. In this work, it is argued that an inferential apparatus for the theory of beliefs should follow premises of Brouwer's intuitionism. A logic refuting tertium non daturìs constructed by defining a correspondence between the support functions representing beliefs in the DS theory and semantic models based on intuitionistic Kripke models with weighted nodes. Without addional constraints on the semantic models and without modal operators, the constructed logic is equivalent to the minimal intuitionistic logic. A number of possible constraints is considered resulting in additional axioms and making the proposed logic intermediate. Further analysis of the properties of the created framework shows that the approach preserves the Dempster-Shafer belief assignments and thus expresses modality through the belief assignments of the formulae within the developed logic

The Level Game: Architectures of Play in American Fiction and Theory, 1968–2018

This thesis investigates the theme of ‘levels’ in postmodern and contemporary American fiction, as manifested through levels of reality, levels of architecture and levels in games. Postmodern fiction engages levels in the ontological sense, employing literary devices such as reflexivity and narrative embedding in order to interrogate the nature of fictional worlds. In the later stages of postmodernism, approaching the millennium, technological developments contribute to a terminology of levels in video games. Here, levels come to be associated with goal-oriented hierarchies, and are adopted by the corporate world as motivating tools. Throughout these examples, the navigation of levels is associated with play, and I conceptualise the spatiality of levels through the phrase ‘architectures of play’. This applies both abstractly (architectures of narrative) and concretely (architectures in narrative). My introduction defines the concept of levels, detailing their role in my period of study. Chapter one discusses the work of Jean Baudrillard, interrogating the relationship between play and ontology through his remark that ‘reality has passed completely into the game of reality’. Chapter two analyses John Barth’s ‘Lost in the Funhouse’, where I suggest that the spatial navigation of architectural levels in physical funhouses corresponds with the conceptual navigation of narrative levels in this text. Comparing Barth’s story with David Foster Wallace’s ‘Westward the Course of Empire Takes its Way’, I illustrate how Wallace uses the same literary materials as Barth but experiments with their arrangement. This is exemplified by Wallace’s Infinite Jest, which my next chapter examines in relation to the mise en abyme and the play within the play. I conclude by suggesting that the physical traversal demanded by the novel is a means of restoring the boundaries of play to the infinite jest. Chapter four further probes the physicality of texts, studying the material levels of two formally experimental works: Mark Danielewski’s House of Leaves and Jonathan Safran Foer’s Tree of Codes. Chapter five contrastingly explores the thematisation of digitality in fiction, where levels are used in a teleological sense to denote progress in video games and commercial gamification strategies. Chapter six elaborates on the theme of technology by discussing levels in relation to networks, comparing Don DeLillo’s Underworld (1997) with Richard Powers’s The Overstory (2018). Both novels depict worlds structured as networks, but I draw attention to the prepositions of their titles, arguing that one must travel through levels in order to realise the network’s connections. Exploring the ludic capacity of levels, my study asks: what do levels do? How do we play with levels – architecturally, digitally, and narratively? How do these different media interact in postmodern and contemporary fiction? Through the above six case studies, I delineate the effects – and affects – associated with the figure of the level, identifying a pervasive ‘level game’ in postmodern and contemporary literature and culture.English Faculty Centenary Awar

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

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Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition