A comment on some new definitions of fractional derivative

After reviewing the definition of two differential operators which have been recently introduced by Caputo and Fabrizio and, separately, by Atangana and Baleanu, we present an argument for which these two integro-differential operators can be understood as simple realizations of a much broader class of fractional operators, i.e. the theory of Prabhakar fractional integrals. Furthermore, we also provide a series expansion of the Prabhakar integral in terms of Riemann-Liouville integrals of variable order. Then, by using this last result we finally argue that the operator introduced by Caputo and Fabrizio cannot be regarded as fractional. Besides, we also observe that the one suggested by Atangana and Baleanu is indeed fractional, but it is ultimately related to the ordinary Riemann-Liouville and Caputo fractional operators. All these statements are then further supported by a precise analysis of differential equations involving the aforementioned operators. To further strengthen our narrative, we also show that these new operators do not add any new insight to the linear theory of viscoelasticity when employed in the constitutive equation of the Scott-Blair model.Comment: 10 pages, 1 figure, to appear in Nonlinear Dynamics, comment adde

Matter and gravitons in the gravitational collapse

We consider the effects of gravitons in the collapse of baryonic matter that forms a black hole. We first note that the effective number of (soft off-shell) gravitons that account for the (negative) Newtonian potential energy generated by the baryons is conserved and always in agreement with Bekenstein's area law of black holes. Moreover, their (positive) interaction energy reproduces the expected post-Newtonian correction and becomes of the order of the total ADM mass of the system when the size of the collapsing object approaches its gravitational radius. This result supports a scenario in which the gravitational collapse of regular baryonic matter produces a corpuscular black hole without central singularity, in which both gravitons and baryons are marginally bound and form a Bose-Einstein condensate at the critical point. The Hawking emission of baryons and gravitons is then described by the quantum depletion of the condensate and we show the two energy fluxes are comparable, albeit negligibly small on astrophysical scales.Comment: 4 pages, no figures. Minor changes and typos fixe

Prabhakar-like fractional viscoelasticity

The aim of this paper is to present a linear viscoelastic model based on Prabhakar fractional operators. In particular, we propose a modification of the classical fractional Maxwell model, in which we replace the Caputo derivative with the Prabhakar one. Furthermore, we also discuss how to recover a formal equivalence between the new model and the known classical models of linear viscoelasticity by means of a suitable choice of the parameters in the Prabhakar derivative. Moreover, we also underline an interesting connection between the theory of Prabhakar fractional integrals and the recently introduced Caputo-Fabrizio differential operator.Comment: 9 page

The relativistic Green's function model in charged-current quasielastic neutrino and antineutrino scattering at MINERν\nuA kinematics

The analysis of charged-current quasielastic neutrino and antineutrino-nucleus scattering cross sections requires relativistic theoretical descriptions also accounting for the role of final-state interactions. We compare the results of the relativistic Green's function model with the data recently published by the MINER$\nu$A Collaboration. The model is able to describe both MINER$\nu$A and MiniBooNE data.Comment: 6 pages, 3 figure

The role of collapsed matter in the decay of black holes

We try to shed some light on the role of matter in the final stages of black hole evaporation from the fundamental frameworks of classicalization and the black-to-white hole bouncing scenario. Despite being based on very different grounds, these two approaches attempt at going beyond the background field method and treat black holes as fully quantum systems rather than considering quantum field theory on the corresponding classical manifolds. They also lead to the common prediction that the semiclassical description of black hole evaporation should break down and the system be disrupted by internal quantum pressure, but they both arrive at this conclusion neglecting the matter that formed the black hole. We instead estimate this pressure from the bootstrapped description of black holes, which allows us to express the total Arnowitt-Deser-Misner mass in terms of the baryonic mass still present inside the black hole. We conclude that, although these two scenarios provide qualitatively similar predictions for the final stages, the corpuscular model does not seem to suggest any sizeable deviation from the semiclassical time scale at which the disruption should occur, unlike the black-to-white hole bouncing scenario. This, in turn, makes the phenomenology of corpuscular black holes more subtle from an astrophysical perspective.Comment: 5 pages, no figur

Final-state interactions effects in neutral-current neutrino and antineutrino cross sections at MiniBooNE kinematics

The analysis of the recent neutral-current elastic neutrino and antineutrino-nucleus scattering cross sections measured by the MiniBooNE Collaboration requires relativistic theoretical descriptions also accounting for the role of final-state interactions. In this work we investigate the sensitivity to final-state interactions and compare the MiniBooNE data with the results obtained in the relativistic Green's function model with different parameterizations for the phenomenological relativistic optical potential.Comment: 4 pages, 3 figures. version published in Physical Review

Early recovery of microvascular perfusion induced by t-PA in combination with abciximab or eptifibatide during postischemic reperfusion

BACKGROUND: GPIIb/IIIa inhibitors abciximab and eptifibatide have been shown to inhibit platelet aggregation in ischemic heart disease. Our aim was to test the efficacy of abiciximab (Reo Pro) or eptifibatide (Integrilin) alone or in combination with plasminogen activator (t-PA) in an experimental model of ischemia reperfusion (I/R) in hamster cheek pouch microcirculation visualized by fluorescence microscopy. Hamsters were treated with saline, or abiciximab or eptifibatide or these drugs combined with t-PA infused intravenously 10 minutes before ischemia and through reperfusion. We measured the microvessel diameter changes, the arteriolar red blood cell (RBC) velocity, the increase in permeability, the perfused capillary length (PCL), and the platelet and leukocyte adhesion on microvessels. RESULTS: I/R elicited large increases in the platelet and leukocyte adhesion and a decrease in microvascular perfusion. These responses were significantly attenuated by abiciximab or eptifibatide (PCL:70 and 65% at 5–10 mins of reperfusion and 85 and 87% at 30 mins of reperfusion, respectively, p < 0.001) while t-PA combined with abiciximab or eptifibatide, was more effective and microvascular perfusion recovered immediately after postischemic reperfusion. CONCLUSIONS: Platelets are crucial in I/R injury, as shown by the treatment with abicixmab or eptifibatide, which decreased platelet aggregation in microvessels, and also decreased leukocyte adhesion in venules. Arterial vasoconstriction, decreased arterial RBC velocity and alterations in the endothelial barrier with increased permeability delayed the complete restoration of blood flow, while t-PA combined with inhibition of platelet aggregation speeded up the capillary perfusion after reperfusion