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Simple formulae for Lebesgue constants arising in the classical Fourier series approximation are obtained. Both even and odd cases are addressed, extending Fejér's results. Asymptotic formulae are also obtained. [ABSTRACT FROM AUTHOR]

Duality is one of the most interesting properties of the Laplace and Fourier transforms associated with the integer-order derivative. Here, we will generalize it for fractional derivatives and extend the results to the Mellin, Z and discrete-time Fourier transforms. The scale and nabla derivatives are used. Some consequences are described. [ABSTRACT FROM AUTHOR]

The fractional scale-invariant systems are introduced and studied, using an operational formalism. It is shown that the impulse and step responses of such systems belong to the vector space generated by some special functions here introduced. For these functions, the fractional scale derivative is a decremental index operator, allowing the construction of an algebraic framework that enables to compute the impulse and step responses of such systems. The effectiveness and accuracy of the method are demonstrated through various numerical simulations. [ABSTRACT FROM AUTHOR]

Fractionalisation and solution of the Ambartsumian equation is considered. The general approach to fractional calculus suitable for applications in physics and engineering is described. It is shown that Liouville-type derivatives are the necessary ones, because they fully preserve backward compatibility with classical results. Such derivatives are used to define and solve the fractional Ambartsumian equation. First, a solution in terms of a slowly convergent fractional Taylor series is obtained. Then, a simple solution expressed in terms of an infinite linear combination of Mittag–Leffler functions is deduced. A fast algorithm, based on a bilinear transformation and using the fast Fourier transform, is described and demonstrated for its approximate numerical realisation. [ABSTRACT FROM AUTHOR]

We calculate the primordial power spectrum of tensor perturbations, within the emergent universe scenario, incorporating a version of the Continuous Spontaneous Localization (CSL) model as a mechanism capable of: breaking the initial symmetries of the system, generating the perturbations, and also achieving the quantum-to-classical transition of such perturbations. We analyze how the CSL model modifies the characteristics of the B-mode CMB polarization power spectrum, and we explore their differences with current predictions from the standard concordance cosmological model. We have found that, regardless of the CSL mechanism, a confirmed detection of primordial B-modes that fits to a high degree of precision the shape of the spectrum predicted from the concordance Λ CDM model, would rule out one of the distinguishing features of the emergent universe. Namely, achieving a best fit to the data consistent with the suppression observed in the low multipoles of the angular power spectrum of the temperature anisotropy of the CMB. On the contrary, a confirmed detection that accurately exhibits a suppression of the low multipoles in the B-modes, would be a new feature that could be considered as a favorable evidence for the emergent scenario. In addition, we have been able to establish an upper bound on the collapse parameter of the specific CSL model used. [ABSTRACT FROM AUTHOR]

We propose a novel realization for the natural extrapolation of the continuous spontaneous localization (CSL) model, in order to account for the origin of primordial inhomogeneities during inflation. This particular model is based on three main elements: (i) the semiclassical gravity framework, (ii) a collapse-generating operator associated to a relativistic invariant scalar of the energy-momentum tensor, and (iii) an extension of the CSL parameter(s) as a function of the spacetime curvature. Furthermore, employing standard cosmological perturbation theory at linear order, and for a reasonable range within the parameter space of the model, we obtain a nearly scale invariant power spectrum consistent with recent observational CMB data. This opens a vast landscape of different options for the application of the CSL model to the cosmological context, and possibly sheds light on searches for a full covariant version of the CSL theory. [ABSTRACT FROM AUTHOR]

In this work we analyze how the spectrum of primordial scalar perturbations is modified, within the emergent universe scenario, when a particular version of the Continuous Spontaneous Localization (CSL) model is incorporated as the generating mechanism of initial perturbations, providing also an explanation to the quantum-to-classical transition of such perturbations. On the other hand, a phase of super-inflation, prior to slow-roll inflation, is a characteristic feature of the emergent universe hypothesis. In recent works, it was shown that the super-inflation phase could generically induce a suppression of the temperature anisotropies of the CMB at large angular scales. We study here under what conditions the CSL maintains or modifies these characteristics of the emergent universe and their compatibility with the CMB observations. [ABSTRACT FROM AUTHOR]

The substantial, tempered, and shifted fractional derivatives, useful in medium range systems, are reviewed and highlighted in a unified framework. Their historical evolution is described and their properties studied. Moreover, they are characterized and assessed under the light of the strict sense criterion for the definition of derivatives. The relationship between these derivatives and several formulations developed in fractional calculus are explored. Additionally, the interpretations in the time and frequency domains are included. In the scope of the framework, new tempered linear systems and transfer functions are introduced. As particular applications, several systems for compensation and control purposes are revisited. [ABSTRACT FROM AUTHOR]

We present an operational method suitable for solving fractional differential equations with two or more noncommensurate orders. The method can be applied in a recursive way in the sense that one can add or remove pseudo‐poles or pseudo‐zeros and recompute the output from pre‐existing solutions. In order to illustrate how the method is applied and its efficiency and simplicity, we present several examples. [ABSTRACT FROM AUTHOR]

In this work we consider a wide variety of alternatives opened when applying the continuous spontaneous localization (CSL) dynamical collapse theory to the inflationary era. The definitive resolution of many of the issues discussed here will have to await, not only for a general relativistic CSL theory, but for a fully workable theory of quantum gravity. Our concern here is to explore these issues, and to warn against premature conclusions. This exploration includes: two different approaches to deal with quantum field theory and gravitation, the identification of the collapse-generating operator and the general nature and values of the parameters of the CSL theory. All the choices connected with these issues have the potential to dramatically alter the conclusions one can draw. We also argue that the incompatibilities found in a recent paper, between the CSL parameter values and the cosmic microwave background observational data, are associated with specific choices made for the extrapolation to the cosmological context of the CSL theory (as it is known to work in non-relativistic laboratory situations) which do not represent the most natural ones. [ABSTRACT FROM AUTHOR]

Recently it has been argued that a correct reading of the quantum fluctuations of the vacuum could lead to a solution to the cosmological constant problem. In this work we critically examine such a proposal, finding it questionable due to conceptual and self-consistency problems, as well as issues with the actual calculations. We conclude that the proposal is inadequate as a solution to the cosmological constant problem. [ABSTRACT FROM AUTHOR]

We apply a general algebraic operational method to obtain solutions of ordinary differential equations. The solutions are expressed as series of scaled Hermite polynomials. We present some examples that show that the solutions obtained as truncated Hermite series give acceptable approximations to the exact solutions on intervals larger than the corresponding intervals for the solutions obtained as truncated Taylor series. Our method is algebraic and does not use any integral transforms. [ABSTRACT FROM AUTHOR]

In this paper, an operational method for solving linear and nonlinear systems described by ordinary differential equations is presented. The construction is based on the generalized derivative in the sense of distribution theory. The approach allows the response computation without needing the use of any integral transform as in the Mikusiński operational calculus. A general description of the algorithmis done and some illustrating examples are presented. The algorithm is recursive allowing to add and remove any pole or zero contribution. The extension to nonlinear systems is done by means of the Adomian polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we present a method for solving the Emden-Fowler equation using the Adomian polynomials and the operational calculus introduced by the authors. [ABSTRACT FROM AUTHOR]

We prove that the cofinite dual of the Hopf algebra of polynomials in several variables can be represented as a Hopf algebra ℰ of exponential polynomials that contains the polynomials as a Hopf subalgebra. We also present some algebras isomorphic to ℰ whose elements are rational functions or multi-sequences. [ABSTRACT FROM AUTHOR]

We construct an operational calculus supported on the algebraic operational calculus introduced by Bengochea and Verde [Linear algebraic foundations of the operational calculi. Adv Appl Math. 2011;47:330–351]. With this operational calculus we study the solution of certain Bessel-type equations. [ABSTRACT FROM PUBLISHER]

This paper proposed a definition of the fractional line integral, generalising the concept of the fractional definite integral. The proposal replicated the properties of the classic definite integral, namely the fundamental theorem of integral calculus. It was based on the concept of the fractional anti-derivative used to generalise the Barrow formula. To define the fractional line integral, the Grünwald–Letnikov and Liouville directional derivatives were introduced and their properties described. The integral was defined for a piecewise linear path first and, from it, for any regular curve. [ABSTRACT FROM AUTHOR]

The bilateral tempered fractional derivatives are introduced generalising previous works on the one-sided tempered fractional derivatives and the two-sided fractional derivatives. An analysis of the tempered Riesz potential is done and shows that it cannot be considered as a derivative. [ABSTRACT FROM AUTHOR]