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We investigate and compare different representations of the Riesz derivative, which plays an important role in anomalous diffusion and space fractional quantum mechanics. In particular, we show that a certain representation of the Riesz derivative that is generally given as also valid for order alpha equals 1, behaves no differently than the other definition given in terms of its Fourier transform. In the light of this, we discuss the alpha goes to 1 limit of the space fractional quantum mechanics and its consistency.

Recently, consistency of the infinite square well solution of the space fractional Schr\"odinger equation has been the subject of some controversy. In [J. Math. Phys. 54, 014101 (2013)], Hawkins and Schwarz objected to the way certain integrals are evaluated to show the consistency of the infinite square well solutions of the space fractional Schr\"odinger equation [J. Math. Phys. 53, 042105 (2012); J. Math. Phys. 53, 084101 (2012)]. Here, we show for general n that as far as the integral representation of the solution in the momentum space is concerned, there is no inconsistency. To pinpoint the source of a possible inconsistency, we also scrutinize the different representations of the Riesz derivative that plays a central role in this controversy and show that they all have the same Fourier transform, when evaluated with consistent assumptions. PACS numbers: 03.65.Ca, 02.50.Ey, 02.30.Gp, 03.65.Db, Comment: Misprints in Eqs.(68-70)corrected

Fractional order derivatives and integrals (differintegrals) are viewed from a frequency-domain perspective using the formalism of Riesz, providing a computational tool as well as a way to interpret the operations in the frequency domain. Differintegrals provide a logical extension of current techniques, generalizing the notion of integral and differential operators and acting as kind of frequency-domain filtering that has many of the advantages of a nonlocal linear operator. Several important properties of differintegrals are presented, and sample applications are given to one- and two-dimensional signals. Computer code to carry out the computations is made available on the author's website., Comment: 22 pages, 10 figures

Geometry of the universe has always intrigued mathematicians and cosmologists. Recent results from the Wilkinson Microwave Anisotropy Project (WMAP) indicate that the visible universe is incredibly flat. This apparent flatness could be due to the fact that only a small part of the universe is visible, thus indicating that the geometry of the universe is still an open and an interesting problem in cosmology. Assuming a profound connection between Friedmann, Robertson, Walker (FRW) geometries and thermodynamics, we construct a parameter free exploratory model that allows us to predict the geometry of the universe by thermodynamic arguments. The key parameters in this model are the concept of global equation of state and the concept of gravitational temperature. By comparing the equal time expansion of the Green function for the massless conformal scalar field in background FRW geometry with the thermal Green function in Minkowski space-time, we define the gravitational temperature. We also give the protocol for determining the global equation of state for a given local equation of state. Using a local equation of state that covers a wide range of physically acceptable cases, P={\alpha}{\rho},{\alpha}>0, and within the context of FRW thermodynamics, we predict that the geometry of the visible universe is Lobachevskian (open), albeit being very close to flat. This is consistent with the WMAP data, which indicates that the universe may deviate from flatness by as much as 1%. We also discuss the self-consistency of this suggestive model along with its possible connections with nonextensive thermodynamics and black hole thermodynamics.

Recently we have reanalyzed the consistency of the solutions of the space fractional Schr\"odinger equation found in a piecewise manner, and showed that an exact and a proper treatment of the relevant integrals prove that they are consistent. In this comment, for clarity, we present additional information about the critical integrals and describe how their analytic continuation is accomplished. [doi: 10.1063/1.4705268]

Recently it was pointed out that the solutions found in literature for the space fractional Schr\"odinger equation in a piecewise manner are wrong, except the case with the delta potential. We reanalyze this problem and show that an exact and a proper treatment of the relevant integral proves otherwise. We also discuss effective potential approach and present a free particle solution for the space and time fractional Schr\"odinger equation in general coordinates in terms of Fox's H-functions.

After introducing the formalism of the general space and time fractional Schr\"odinger equation, we concentrate on the time fractional Schr\"odinger equation and present new results via the elegant language of Fox's H-functions. We show that the general time dependent part of the wave function for the separable solutions of the time fractional Schr\"odinger equation is the Mittag-Leffler function with an imaginary argument by two different methods. After separating the Mittag-Leffler function into its real and imaginary parts, in contrast to existing works, we show that the total probability is less than one and decays with time. Introducing the effective potential approach, we also write the Mittag-Leffler function with an imaginary argument as the product of its purely decaying and purely oscillating parts. In the light of these, we reconsider the simple box problem. PACS numbers: 03.65.Ca, 02.50.Ey, 02.30.Gp, 03.65.Db

In a recent article we have introduced Friedmann thermodynamics, where certain geometric parameters in Friedmann models are treated like their thermodynamic counterparts (temperature, entropy, Gibbs potential etc.). This model has the advantage of allowing us to determine the geometry of the universe by thermodynamic stability arguments. In this article we review connections between thermodynamics, geometry and cosmology., Comment: Key note speech delivered in the International Conference on Efficiency, Costs, Optimization, Simulation and Environmental Impact of Energy Systems, ECOS'95 and ASME

We present evidence of direct electron tunneling across intrinsic insulating regions in sub-micrometer wide bridges of the phase separated ferromagnet (La,Pr,Ca)MnO$_3$. Upon cooling below the Curie temperature, a predominantly ferromagnetic supercooled state persists where tunneling across the intrinsic tunnel barriers (ITBs) results in metastable, temperature-independent, high-resistance plateaus over a large range of temperatures. Upon application of a magnetic field, our data reveal that the ITBs are extinguished resulting in sharp, colossal, low-field resistance drops. Our results compare well to theoretical predictions of magnetic domain walls coinciding with the intrinsic insulating phase., Comment: changed content; accepted for publication in Physical Review Letters, 5 pages, 4 figures