Variable-order fractional calculus: A change of perspective

Several approaches to the formulation of a fractional theory of calculus of “variable order” have appeared in the literature over the years. Unfortunately, most of these proposals lack a rigorous mathematical framework. We consider an alternative view on the problem, originally proposed by G. Scarpi in the early seventies, based on a naive modification of the representation in the Laplace domain of standard kernels functions involved in (constant-order) fractional calculus. We frame Scarpi's ideas within recent theory of General Fractional Derivatives and Integrals, that mostly rely on the Sonine condition, and investigate the main properties of the emerging variable-order operators. Then, taking advantage of powerful and easy-to-use numerical methods for the inversion of Laplace transforms of functions defined in the Laplace domain, we discuss some practical applications of the variable-order Scarpi integral and derivative

Stability of systems of fractional-order differential equations with caputo derivatives

Systems of fractional-order differential equations present stability properties which differ in a substantial way from those of systems of integer order. In this paper, a detailed analysis of the stability of linear systems of fractional differential equations with Caputo derivative is proposed. Starting from the well-known Matignon’s results on stability of single-order systems, for which a different proof is provided together with a clarification of a limit case, the investigation is moved towards multi-order systems as well. Due to the key role of the Mittag–Leffler function played in representing the solution of linear systems of FDEs, a detailed analysis of the asymptotic behavior of this function and of its derivatives is also proposed. Some numerical experiments are presented to illustrate the main results

Good (and Not So Good) practices in computational methods for fractional calculus

The solution of fractional-order differential problems requires in the majority of cases the use of some computational approach. In general, the numerical treatment of fractional differential equations is much more difficult than in the integer-order case, and very often non-specialist researchers are unaware of the specific difficulties. As a consequence, numerical methods are often applied in an incorrect way or unreliable methods are devised and proposed in the literature. In this paper we try to identify some common pitfalls in the use of numerical methods in fractional calculus, to explain their nature and to list some good practices that should be followed in order to obtain correct results

Why fractional derivatives with nonsingular kernels should not be used

In recent years, many papers discuss the theory and applications of new fractional-order derivatives that are constructed by replacing the singular kernel of the Caputo or Riemann-Liouville derivative by a non-singular (i.e., bounded) kernel. It will be shown here, through rigorous mathematical reasoning, that these non-singular kernel derivatives suffer from several drawbacks which should forbid their use. They fail to satisfy the fundamental theorem of fractional calculus since they do not admit the existence of a corresponding convolution integral of which the derivative is the left-inverse; and the value of the derivative at the initial time t = 0 is always zero, which imposes an unnatural restriction on the differential equations and models where these derivatives can be used. For the particular cases of the so-called Caputo-Fabrizio and Atangana-Baleanu derivatives, it is shown that when this restriction holds the derivative can be simply expressed in terms of integer derivatives and standard Caputo fractional derivatives, thus demonstrating that these derivatives contain nothing new

An analysis of solutions to fractional neutral differential equations with delay

This paper discusses some properties of solutions to fractional neutral delay differential equations. By combining a new weighted norm, the Banach fixed point theorem and an elegant technique for extending solutions, results on existence, uniqueness, and growth rate of global solutions under a mild Lipschitz continuous condition of the vector field are first established. Be means of the Laplace transform the solution of some delay fractional neutral differential equations are derived in terms of three-parameter Mittag–Leffler functions; their stability properties are hence studied by using use Rouché’s theorem to describe the position of poles of the characteristic polynomials and the final value theorem to detect the asymptotic behavior. By means of numerical simulations the theoretical findings on the asymptotic behavior are verified

Multi-Frequency Observations of the Candidate Neutrino Emitting Blazar BZB J0955+3551

The recent spatial and temporal coincidence of the blazar TXS 0506+056 with the IceCube detected neutrino event IC-170922A has opened up a realm of multi-messenger astronomy with blazar jets as a plausible site of cosmic-ray acceleration. After TXS 0506+056, a second blazar, BZB J0955+3551, has recently been found to be spatially coincident with the IceCube detected neutrino event IC-200107A and undergoing its brightest X-ray flare measured so far. Here, we present the results of our multi-frequency campaign to study this peculiar event that includes observations with the NuSTAR, Swift, NICER, and 10.4 m Gran Telescopio Canarias (GTC). The optical spectroscopic observation from GTC secured its redshift as $z=0.55703^{+0.00033}_{-0.00021}$ and the central black hole mass as 10$^{8.90\pm0.16}~M_{\odot}$. Both NuSTAR and NICER data reveal a rapid flux variability albeit at low-significance ($\lesssim3.5\sigma$). We explore the origin of the target photon field needed for the photo-pion production using analytical calculations and considering the observed optical-to-X-ray flux level. We conclude that seed photons may originate from outside the jet, similar to that reported for TXS 0506+056, although a scenario invoking a co-moving target photon field (e.g., electron-synchrotron) can not be ruled out. The electromagnetic output from the neutrino-producing photo-hadronic processes are likely to make only a sub-dominant contribution to the observed spectral energy distribution suggesting that the X-ray flaring event may not be directly connected with IC-200107A.Comment: Accepted for publication in the Astrophysical journa

Patterns in the multi-wavelength behavior of candidate neutrino blazars

Motivated by the identification of the blazar TXS 0506+056 as the first promising high-energy neutrino counterpart candidate, we search for additional neutrino blazars candidates among the Fermi-LAT detected blazars. We investigate the multi-wavelength behavior from radio to GeV gamma rays of blazars found to be in spatial coincidence with single high-energy neutrinos and lower-energy neutrino flare candidates. In addition, we compare the average gamma-ray emission of the potential neutrino-emitting sources to the entire sample of gamma-ray blazars. We find that neutrino-emitting blazar candidates are statistically compatible with both hypothesis of a linear correlation and of no correlation between neutrino and gamma-ray energy flux.Comment: accepted for publication by Ap

Measurements and tests on FBK silicon sensors with an optimized electronic design for a CTA camera

In October 2013, the Italian Ministry approved the funding of a Research & Development (R&D) study, within the "Progetto Premiale TElescopi CHErenkov made in Italy (TECHE)", devoted to the development of a demonstrator for a camera for the Cherenkov Telescope Array (CTA) consortium. The demonstrator consists of a sensor plane based on the Silicon Photomultiplier (SiPM) technology and on an electronics designed for signal sampling. Preliminary tests on a matrix of sensors produced by the Fondazione Bruno Kessler (FBK-Trento, Italy) and on electronic prototypes produced by SITAEL S.p.A. will be presented. In particular, we used different designs of the electronics in order to optimize the output signals in terms of tail cancellation. This is crucial for applications where a high background is expected, as for the CTA experiment.Comment: 5 pages, 6 figures; Proceedings of the 10th Workshop on Science with the New Generation of High-Energy Gamma-ray experiments (SciNeGHE) - PoS(Scineghe2014)00

Internal alignment and position resolution of the silicon tracker of DAMPE determined with orbit data

The DArk Matter Particle Explorer (DAMPE) is a space-borne particle detector designed to probe electrons and gamma-rays in the few GeV to 10 TeV energy range, as well as cosmic-ray proton and nuclei components between 10 GeV and 100 TeV. The silicon-tungsten tracker-converter is a crucial component of DAMPE. It allows the direction of incoming photons converting into electron-positron pairs to be estimated, and the trajectory and charge (Z) of cosmic-ray particles to be identified. It consists of 768 silicon micro-strip sensors assembled in 6 double layers with a total active area of 6.6 m$^2$. Silicon planes are interleaved with three layers of tungsten plates, resulting in about one radiation length of material in the tracker. Internal alignment parameters of the tracker have been determined on orbit, with non-showering protons and helium nuclei. We describe the alignment procedure and present the position resolution and alignment stability measurements

Establishing accretion flares from massive black holes as a major source of high-energy neutrinos

High-energy neutrinos have thus far been observed in coincidence with time-variable emission from three different accreting black holes: a gamma-ray flare from a blazar (TXS 0506+056), an optical transient following a stellar tidal disruption (AT2019dsg), and an optical outburst from an active galactic nucleus (AT2019fdr). Here we present a unified explanation for the latter two of these sources: accretion flares that reach the Eddington limit. A signature of these events is a luminous infrared reverberation signal from circumnuclear dust that is heated by the flare. Using this property we construct a sample of similar sources, revealing a third event coincident with a PeV-scale neutrino. This sample of three accretion flares is correlated with high-energy neutrinos at a significance of 3.7 sigma. Super-Eddington accretion could explain the high particle acceleration efficiency of this new population.Comment: 46 pages, 11 figures, 3 table