Special issue: Fractal functions and applications

This volume gathers some important advances in the fields of fractional calculus and fractal curves and functions. Fractional derivatives and integrals play an increasingly important role in applied science, and these types of models are ubiquitous in the current scientific literature. The references [1, 2] are devoted to fractional calculus and an application of it to a coronavirus spreading model. The first one studies three procedures of inverse Laplace Transforms: A Sinc–Thiele approximation, a Sinc and a Sinc–Gaussian (SG) method. Both Sinc versions are exact methods of inverse Laplace Transforms. The author proves that SG-based transformations present some advantages over the pure Sinc version regarding stability and convergence properties. The convergence is of exponential type. All the methods presented are applied to Mittag-Leffler functions depending on one, two and three parameters, and the author proves that the representation of this kind of functions is very effective. The author concludes that even for variable-order fractional differential or integral equations, the Sinc–Gaussian method is a powerful procedure..

A physical approach to Tsirelson's problem

Tsirelson's problem deals with how to model separate measurements in quantum mechanics. In addition to its theoretical importance, the resolution of Tsirelson's problem could have great consequences for device independent quantum key distribution and certified randomness. Unfortunately, understanding present literature on the subject requires a heavy mathematical background. In this paper, we introduce quansality, a new theoretical concept that allows to reinterpret Tsirelson's problem from a foundational point of view. Using quansality as a guide, we recover all known results on Tsirelson's problem in a clear and intuitive way.Comment: 11 pages, 2 figure

Unitary evolution and uniqueness of the Fock representation of Dirac fields in cosmological spacetimes

We present a privileged Fock quantization of a massive Dirac field in a closed Friedmann-Robertson-Walker cosmology, partially selected by the criteria of invariance of the vacuum under the symmetries of the field equations, and unitary implementation of the dynamics. When quantizing free scalar fields in homogeneous and isotropic spacetimes with compact spatial sections, these criteria have been shown to pick out a unique Fock representation (up to unitary equivalence). Here, we employ the same criteria for fermion fields and explore whether that uniqueness result can be extended to the case of the Fock quantization of fermions. For the massive Dirac field, we start by introducing a specific choice of the complex structure that determines the Fock representation. Such structure is invariant under the symmetries of the equations of motion. We then prove that the corresponding representation of the canonical anticommutation relations admits a unitary implementation of the dynamics. Moreover, we construct a rather general class of representations that satisfy the above criteria, and we demonstrate that they are all unitarily equivalent to our previous choice. The complex structures in this class are restricted only by certain conditions on their asymptotic behavior for modes in the ultraviolet sector of the Dirac operator. We finally show that, if one assumes that these asymptotic conditions are in fact trivial once our criteria are fulfilled, then the time-dependent scaling in the definition of the fermionic annihilation and creation-like variables is essentially unique.Comment: 24 page

Fractal functions on the real projective plane

Formerly the geometry was based on shapes, but since the last centuries this founding mathematical science deals with transformations, projections and mappings. Projective geometry identifies a line with a single point, like the perspective on the horizon line and, due to this fact, it requires a restructuring of the real mathematical and numerical analysis. In particular, the problem of interpolating data must be refocused. In this paper we define a linear structure along with a metric on a projective space, and prove that the space thus constructed is complete. Then we consider an iterated function system giving rise to a fractal interpolation function of a set of data.Comment: 25 pages, 18 figure

Uniqueness of the Fock quantization of Dirac fields in 2+1 dimensions

We study the Fock quantization of a free Dirac field in 2+1-dimensional backgrounds which are conformally ultrastatic, with a time-dependent conformal factor. As it is typical for field theories, there is an infinite ambiguity in the Fock representation of the canonical anticommutation relations. Different choices may lead to unitarily inequivalent theories that describe different physics. To remove this ambiguity one usually requires that the vacuum be invariant under the unitary transformations that implement the symmetries of the equations of motion. However, in non-stationary backgrounds, where time translation is not a symmetry transformation, the requirement of vacuum invariance is in general not enough to fix completely the Fock representation. We show that this problem is overcome in the considered scenario by demanding, in addition, a unitarily implementable quantum dynamics. The combined imposition of these conditions selects a unique family of equivalent Fock representations. Moreover, one also obtains an essentially unique splitting of the time variation of the Dirac field into an explicit dependence on the background scale factor and a quantum evolution of the corresponding creation and annihilation operators.Comment: 24 pages. Document replaced to match published versio

Almost quantum correlations

Quantum theory is not only successfully tested in laboratories every day but also constitutes a robust theoretical framework: small variations usually lead to implausible consequences, such as faster-than-light communication. It has even been argued that quantum theory may be special among possible theories. Here we report that, at the level of correlations among different systems, quantum theory is not so special. We define a set of correlations, dubbed 'almost quantum', and prove that it strictly contains the set of quantum correlations but satisfies all-but-one of the proposed principles to capture quantum correlations. We present numerical evidence that the remaining principle is satisfied too. © 2015 Macmillan Publishers Limited

Spectroscopy of Very Low Mass Stars and Brown Dwarfs in the Lambda Orionis Star Forming Region

Context. Most observational studies so far point towards brown dwarfs sharing a similar formation mechanism as the one accepted for low mass stars. However, larger databases and more systematic studies are needed before strong conclusions can be reached. Aims. In this second paper of a series devoted to the study of the spectroscopic properties of the members of the Lambda Orionis Star Forming Region, we study accretion, activity and rotation for a wide set of spectroscopically confirmed members of the central star cluster Collinder 69 to draw analogies and/or differences between the brown dwarf and stellar populations of this cluster. Moreover, we present comparisons with other star forming regions of similar and different ages to address environmental effects on our conclusions. Methods. We study prominent photospheric lines to derive rotational velocities and emission lines to distinguish between accretion processes and chromospheric activity. In addition, we include information about disk presence and X-ray emission. Results. We report very large differences in the disk fractions of low mass stars and brown dwarfs (~58%) when compared to higher mass stars (26+4-3%) with 0.6 Msun being the critical mass we find for this dichotomy. As a byproduct, we address the implications of the spatial distribution of disk and diskless members in the formation scenario of the cluster itself. We have used the Halpha emission to discriminate among accreting and non-accreting sources finding that 38+8-7% of sources harboring disks undergo active accretion and that his percentage stays similar in the substellar regime. For those sources we have estimated accretion rates. Finally, regarding rotational velocities, we find a high dispersion in vsin(i) which is even larger among the diskless population.Comment: Accepted for publication in A&A. 18 figs including the Appendix and an online tabl

Cyclic Meir-Keeler contraction and its fractals

In present times, there has been a substantial endeavor to generalize the classical notion of iterated function system (IFS). We introduce a new type of non-linear contraction namely cyclic Meir-Keeler contraction, which is a generalization of the famous Banach contraction. We show the existence and uniqueness of the fixed point for the cyclic Meir-Keeler contraction. Using this result, we propose the cyclic Meir-Keeler IFS in the literature for construction of fractals. Furthermore, we extend the theory of countable IFS and generalized IFS by using these cyclic Meir-Keeler contraction maps

Uniqueness of the Fock quantization of scalar fields in a Bianchi I cosmology with unitary dynamics

The Fock quantization of free scalar fields is subject to an infinite ambiguity when it comes to choosing a set of annihilation and creation operators, choice that is equivalent to the determination of a vacuum state. In highly symmetric situations, this ambiguity can be removed by asking vacuum invariance under the symmetries of the system. Similarly, in stationary backgrounds, one can demand time-translation invariance plus positivity of the energy. However, in more general situations, additional criteria are needed. For the case of free (test) fields minimally coupled to a homogeneous and isotropic cosmology, it has been proven that the ambiguity is resolved by introducing the criterion of unitary implementability of the quantum dynamics, as an endomorphism in Fock space. This condition determines a specific separation of the time dependence of the field, so that this splits into a very precise background dependence and a genuine quantum evolution. Furthermore, together with the condition of vacuum invariance under the spatial Killing symmetries, unitarity of the dynamics selects a unique Fock representation for the canonical commutation relations, up to unitary equivalence. In this work, we generalize these results to anisotropic spacetimes with shear, which are therefore not conformally symmetric, by considering the case of a free scalar field in a Bianchi I cosmology.Comment: 23 pages. Updated to match published versio