Two Forms of Inconsistency in Quantum Foundations

Recently, there has been some discussion of how Dutch Book arguments might be used to demonstrate the rational incoherence of certain hidden variable models of quantum theory (Feintzeig and Fletcher 2017). In this paper, we argue that the 'form of inconsistency' underlying this alleged irrationality is deeply and comprehensively related to the more familiar 'inconsistency' phenomenon of contextuality. Our main result is that the hierarchy of contextuality due to Abramsky and Brandenburger (2011) corresponds to a hierarchy of additivity/convexity-violations which yields formal Dutch Books of different strengths. We then use this result to provide a partial assessment of whether these formal Dutch Books can be interpreted normatively.Comment: 26 pages, 5 figure

Representation of maxitive measures: an overview

Idempotent integration is an analogue of Lebesgue integration where $\sigma$-maxitive measures replace $\sigma$-additive measures. In addition to reviewing and unifying several Radon--Nikodym like theorems proven in the literature for the idempotent integral, we also prove new results of the same kind.Comment: 40 page

Resource theories of knowledge

How far can we take the resource theoretic approach to explore physics? Resource theories like LOCC, reference frames and quantum thermodynamics have proven a powerful tool to study how agents who are subject to certain constraints can act on physical systems. This approach has advanced our understanding of fundamental physical principles, such as the second law of thermodynamics, and provided operational measures to quantify resources such as entanglement or information content. In this work, we significantly extend the approach and range of applicability of resource theories. Firstly we generalize the notion of resource theories to include any description or knowledge that agents may have of a physical state, beyond the density operator formalism. We show how to relate theories that differ in the language used to describe resources, like micro and macroscopic thermodynamics. Finally, we take a top-down approach to locality, in which a subsystem structure is derived from a global theory rather than assumed. The extended framework introduced here enables us to formalize new tasks in the language of resource theories, ranging from tomography, cryptography, thermodynamics and foundational questions, both within and beyond quantum theory.Comment: 28 pages featuring figures, examples, map and neatly boxed theorems, plus appendi

A decomposition theorem for fuzzy set-valued random variables and a characterization of fuzzy random translation

Let $X$ be a fuzzy set--valued random variable (\frv{}), and \huku{X} the family of all fuzzy sets $B$ for which the Hukuhara difference X\HukuDiff B exists $\mathbb{P}$--almost surely. In this paper, we prove that $X$ can be decomposed as X(\omega)=C\Mink Y(\omega) for $\mathbb{P}$--almost every $\omega\in\Omega$, $C$ is the unique deterministic fuzzy set that minimizes $\mathbb{E}[d_2(X,B)^2]$ as $B$ is varying in \huku{X}, and $Y$ is a centered \frv{} (i.e. its generalized Steiner point is the origin). This decomposition allows us to characterize all \frv{} translation (i.e. X(\omega) = M \Mink \indicator{\xi(\omega)} for some deterministic fuzzy convex set $M$ and some random element in \Banach). In particular, $X$ is an \frv{} translation if and only if the Aumann expectation $\mathbb{E}X$ is equal to $C$ up to a translation. Examples, such as the Gaussian case, are provided.Comment: 12 pages, 1 figure. v2: minor revision. v3: minor revision; references, affiliation and acknowledgments added. Submitted versio

Characterizations of discrete Sugeno integrals as polynomial functions over distributive lattices

We give several characterizations of discrete Sugeno integrals over bounded distributive lattices, as particular cases of lattice polynomial functions, that is, functions which can be represented in the language of bounded lattices using variables and constants. We also consider the subclass of term functions as well as the classes of symmetric polynomial functions and weighted minimum and maximum functions, and present their characterizations, accordingly. Moreover, we discuss normal form representations of these functions

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

Some Novel Estimates of Hermite–Hadamard and Jensen Type Inequalities for (h1,h2)-Convex Functions Pertaining to Total Order Relation

There are different types of order relations that are associated with interval analysis for determining integral inequalities. The purpose of this paper is to connect the inequalities terms to total order relations, often called (CR)-order. In contrast to classical interval-order relations, total order relations are quite different and novel in the literature and are calculated as ω=⟨ωc,ωr⟩=⟨ω¯¯+ω−−2,ω¯¯−ω−−2⟩. A major benefit of total order relations is that they produce more efficient results than other order relations. This study introduces the notion of CR-(h1,h2)-convex function using total order relations. Center and Radius order relations are a powerful tool for studying inequalities based on their properties and widespread application. Using this novel notion, we first developed some variants of Hermite–Hadamard inequality and then constructed Jensen inequality. Based on the results, this new concept is extremely useful in connection with a variety of inequalities. There are many new and well-known convex functions unified by this type of convexity. These results will stimulate further research on inequalities for fractional interval-valued functions and fuzzy interval-valued functions, as well as the optimization problems associated with them. For the purpose of verifying our main findings, we provide some nontrivial examples