Multifunctions determined by integrable functions

Integral properties of multifunctions determined by vector valued functions are presented. Such multifunctions quite often serve as examples and counterexamples. In particular it can be observed that the properties of being integrable in the sense of Bochner, McShane or Birkhoff can be transferred to the generated multifunction while Henstock integrability does not guarantee i

A Girsanov Result through Birkhoff Integral

A vector-valued version of the Girsanov theorem is presented, for a scalar process with respect to a Banach-valued measure. Previously, a short discussion about the Birkhoff-type integration is outlined, as for example integration by substitution, in order to fix the measure-theoretic tools needed for the main result, Theorem 6, where a martingale equivalent to the underlying vector probability has been obtained in order to represent the modified process as a martingale with the same marginals as the original one

Stochastic processes and applications to countably additive restrictions of group-valued finitely additive measures

As an application of a theorem concerning a general stochastic process in a finitely additive probability space, the existence of non-atomic countably additive restrictions with large range is obtained for group-valued finitely additive measures

Quantum probes for universal gravity corrections

We address estimation of the minimum length arising from gravitational theories. In particular, we provide bounds on precision and assess the use of quantum probes to enhance the estimation performances. At first, we review the concept of minimum length and show how it induces a perturbative term appearing in the Hamiltonian of any quantum system, which is proportional to a parameter depending on the minimum length. We then systematically study the effects of this perturbation on different state preparations for several 1-dimensional systems, and we evaluate the Quantum Fisher Information in order to find the ultimate bounds to the precision of any estimation procedure. Eventually, we investigate the role of dimensionality by analysing the use of two-dimensional square well and harmonic oscillator systems to probe the minimal length. Our results show that quantum probes are convenient resources, providing potential enhancement in precision. Additionally, our results provide a set of guidelines to design possible future experiments to detect minimal length.Comment: 11 pages, 4 figure

Role of topology in determining the precision of a finite thermometer

Temperature fluctuations of a finite system follow the Landau bound \u3b4T 2 = T 2/C(T ) where C(T ) is the heat capacity of the system. In turn, the same bound sets a limit to the precision of temperature estimation when the system itself is used as a thermometer. In this paper, we employ graph theory and the concept of Fisher information to assess the role of topology on the thermometric performance of a given system. We find that low connectivity is a resource to build precise thermometers working at low temperatures, whereas highly connected systems are suitable for higher temperatures. Upon modeling the thermometer as a set of vertices for the quantum walk of an excitation, we compare the precision achievable by position measurement to the optimal one, which itself corresponds to energy measurement

Stieltjes-Type Integrals for Metric Semigroup-Valued Functions Defined on Unbounded Intervals

Abstract We introduce the GH k integral for functions defined on (possibly) unbounded subintervals of the extended real line and with values in metric semigroups. Basic properties and convergence theorems for this integral are deduced