Relationships Between Various Outer Measures Induced by Lattice Measures

In this paper, we work with lattice spaces. (A lattice space is an ordered pair consisting of a set X and a lattice on X). As is known, each lattice measure µ induces six outer measures on P(X), one of which is the well-known "induced outer measure" µ^*(in case µ is countably additive); the others are µ, µ, µ,, and .In the first part of the paper, we consider an arbitrary lattice measure and discover various relationships involving two or more of the seven set functions under consideration, on P(X) and on various special subsets of P(X), under various conditions on the measure or on the lattice.In the second part of the paper, we consider two lattice measures related by "domination" and obtain similar results.In the third part of the paper, we consider two lattice measures related by "restriction" and obtain similar results.     

Sensitivity of risk measures with respect to the normal approximation of total claim distributions

A simple and commonly used method to approximate the total claim distribution of a (possibly weakly dependent) insurance collective is the normal approximation. In this article, we investigate the error made when the normal approximation is plugged in a fairly general distribution-invariant risk measure. We focus on the rate of convergence of the error relative to the number of clients, we specify the relative error’s asymptotic distribution, and we illustrate our results by means of a numerical example. Regarding the risk measure, we take into account distortion risk measures as well as distribution-invariant coherent risk measures.     

A new linear ordering of fuzzy numbers on subsets of {{\mathcal F}({\pmb{\mathbb{R}}}})

We introduce a novel linear order on every family of fuzzy numbers which satisfies the assumption that their modal values must be all different and must form a compact subset of . A distinct new feature is that our linear determined procedure employs the corresponding order of a class interval associated with a confidence measure which seems intuitively anticipated. It is worthy noting that although we start from an entirely different rationale, we introduce a fuzzy ordering which initially coincides with the one established earlier by Ramik and Rimanek. However, this fuzzy ordering does not apply when the supports of the fuzzy numbers overlap. In order to cover such cases we extent this initial fuzzy ordering to the “extended fuzzy order” (XFO). This new XFO method includes a possibility and a necessity measure which are compared with the widely accepted PD and NSD indices of D. Dubois and H. Prade. The comparison shows that our possibility and necessity measures comply better with our intuition.     

Dimension of hyperbolic measures of random diffeomorphisms

We consider dynamics of compositions of stationary random diffeomorphisms. We will prove that the sample measures of an ergodic hyperbolic invariant measure of the system are exact dimensional. This is an extension to random diffeomorphisms of the main result of Barreira, Pesin and Schmeling (1999), which proves the Eckmann-Ruelle dimension conjecture for a deterministic diffeomorphism.

     

The Wigner–Yanase information can increase under phase sensitive incoherent operations

We found that the Wigner-Yanase skew information, which has been recently proposed as a measure of coherence in Girolami (2014), can increase under a class of operations which may be interpreted as incoherent following the framework of Baumgratz et al., while being phase sensitive.     

T1 THEOREM FOR BESOV SPACES ON NONHOMOGENEOUS SPACES

Suppose μ is a Radon measure on Rd, which may be non doubling. The only condition assumed on μ is a growth condition, namely, there is a constant Co ＞ 0 such that for all x ∈ supp(μ) and r ＞ 0,μ(B(x,r)) ≤ Corn, where 0 ＜ n ≤ d. We prove T1 theorem for non doubling measures with weak kernel conditions. Our approach yields new results for kernels satisfying weakened regularity conditions, while recovering previously known Tolsa's results. We also prove T1 theorem for Besov spaces on nonhomogeneous spaces with weak kernel conditions given in [7].     

A predictive framework for quantum gravity and black hole to white hole transition

The apparent incompatibility between quantum theory and general relativity has long hampered efforts to find a quantum theory of gravity. The recently proposed positive formalism for quantum theory purports to remove this incompatibility. We showcase the power of the positive formalism by applying it to the black hole to white hole transition scenario that has been proposed as a possible effect of quantum gravity. We show how the characteristic observable of this scenario, the bounce time, can be predicted within the positive formalism, while a traditional S-matrix approach fails at this task. Our result also involves a conceptually novel use of positive operator valued measures.     

The initial value problems of first order impulsive differential equations in Banach spaces

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Large Deviation Principle for Additive Functionals of Brownian Motion Corresponding to Kato Measures

Let (B _t ,P ^W _x ) be the Brownian motion. Let be a Radon measure in the Kato class and A _t the additive functional associated with . We prove that A _t /t obeys the large deviation principle.     

Notes on equivalent stationary policies in Markov decision processes with total rewards

We construct examples of Markov Decision Processes for which, for a given initial state and for a given nonstationary transient policy, there is no equivalent (randomized) stationary policy, i.e. there is no stationary policy which occupation measure is equal to the occupation measure of a given policy. We also investigate the relation between the existence of equivalent stationary policies in special models and the existence of equivalent strategies in various classes of nonstationary policies in general models.