A REPRESENTATION FORMULA FOR THE RESOLVENT OF CONFORMABLE FRACTIONAL STURM-LIOUVILLE OPERATOR

In this study, the resolvent of the conformable fractional Sturm–Liouville operator is considered. An integral representation for the resolvent of this operator is obtained

Spectral expansion for singular conformable fractional Sturm-Liouville problem

With this study, the spectral function for singular conformable fractional Sturm-Lioville problem is demonstrated. Further, we establish a Parseval equality and spectral expansion formula by terms of the spectral function

An Unexpectedly Swift Rise in the Gamma-ray Burst Rate

The association of long gamma-ray bursts with supernovae naturally suggests that the cosmic GRB rate should trace the star formation history. Finding otherwise would provide important clues concerning these rare, curious phenomena. Using a new estimate of Swift GRB energetics to construct a sample of 36 luminous GRBs with redshifts in the range z=0-4, we find evidence of enhanced evolution in the GRB rate, with ~4 times as many GRBs observed at z~4 than expected from star formation measurements. This direct and empirical demonstration of needed additional evolution is a new result. It is consistent with theoretical expectations from metallicity effects, but other causes remain possible, and we consider them systematically.Comment: 4 pages, 4 figures; minor changes to agree with published versio

Stringent Constraint on Galactic Positron Production

The intense 0.511 MeV gamma-ray line emission from the Galactic Center observed by INTEGRAL requires a large annihilation rate of nonrelativistic positrons. If these positrons are injected at even mildly relativistic energies, higher-energy gamma rays will also be produced. We calculate the gamma-ray spectrum due to inflight annihilation and compare to the observed diffuse Galactic gamma-ray data. Even in a simplified but conservative treatment, we find that the positron injection energies must be $\lesssim 3$ MeV, which strongly constrains models for Galactic positron production.Comment: 4 pages, 2 figures; minor revisions, accepted for publication in PR

Dynamic phase transition properties and hysteretic behavior of a ferrimagnetic core-shell nanoparticle in the presence of a time dependent magnetic field

We have presented dynamic phase transition features and stationary-state behavior of a ferrimagnetic small nanoparticle system with a core-shell structure. By means of detailed Monte Carlo simulations, a complete picture of the phase diagrams and magnetization profiles have been presented and the conditions for the occurrence of a compensation point $T_{comp}$ in the system have been investigated. According to N\'{e}el nomenclature, the magnetization curves of the particle have been found to obey P-type, N-type and Q-type classification schemes under certain conditions. Much effort has been devoted to investigation of hysteretic response of the particle and we observed the existence of triple hysteresis loop behavior which originates from the existence of a weak ferromagnetic core coupling $J_{c}/J_{sh}$, as well as a strong antiferromagnetic interface exchange interaction $J_{int}/J_{sh}$. Most of the calculations have been performed for a particle in the presence of oscillating fields of very high frequencies and high amplitudes in comparison with exchange interactions which resembles a magnetic system under the influence of ultrafast switching fields. Particular attention has also been paid on the influence of the particle size on the thermal and magnetic properties, as well as magnetic features such as coercivity, remanence and compensation temperature of the particle. We have found that in the presence of ultrafast switching fields, the particle may exhibit a dynamic phase transition from paramagnetic to a dynamically ordered phase with increasing ferromagnetic shell thickness.Comment: 12 pages, 12 figure

On the solution of conformable fractional heat conduction equation

In this article, we study a conformable fractional heat conduction equation. Applying the method of separation variables to this problem, we get a conformable fractional Sturm–Liouville eigenvalue problem. Later, we prove the existence of a countably infinite set of eigenvalues and eigenfunctions. Finally, we establish uniformly convergent expansions in the eigenfunctions

On the dissipative conformable fractional singular Sturm-Liouville operator

In this study, a dissipative conformable fractional singular Sturm–Liouville operator is studied. For this operator, a completeness theorem is proved by Krein’s theorem

A representation formula for the resolvent of conformable fractional Sturm-Liouville operator

In this study, the resolvent of the conformable fractional Sturm–Liouville operator is considered. An integral representation for the resolvent of this operator is obtained