Fractional Vector Calculus and Fractional Maxwell's Equations

The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The history of fractional vector calculus (FVC) has only 10 years. The main approaches to formulate a FVC, which are used in the physics during the past few years, will be briefly described in this paper. We solve some problems of consistent formulations of FVC by using a fractional generalization of the Fundamental Theorem of Calculus. We define the differential and integral vector operations. The fractional Green's, Stokes' and Gauss's theorems are formulated. The proofs of these theorems are realized for simplest regions. A fractional generalization of exterior differential calculus of differential forms is discussed. Fractional nonlocal Maxwell's equations and the corresponding fractional wave equations are considered.Comment: 42 pages, LaTe

Remarks on Bell and higher order Bell polynomials and numbers

We recover a recurrence relation for representing in an easy form the coefficients $A_{n,k}$ of the Bell polynomials, which are known in literature as the partial Bell polynomials. Several applications in the framework of classical calculus are derived, avoiding the use of operational techniques. Furthermore, we generalize this result to the coefficients $A^{[2]}_{n,k}$ of the second-order Bell polynomials, i.e. of the Bell polynomials relevant to nth derivative of a composite function of the type f(g(h(t))). The second-order Bell polynomials $B_n^{[2]}$ and the relevant Bell numbers $b_n^{[2]}$ are introduced. Further extension of the nth derivative of M-nested functions is also touched on

Nonassociative differential geometry and gravity with non-geometric fluxes

We systematically develop the metric aspects of nonassociative differential geometry tailored to the parabolic phase space model of constant locally non-geometric closed string vacua, and use it to construct preliminary steps towards a nonassociative theory of gravity on spacetime. We obtain explicit expressions for the torsion, curvature, Ricci tensor and Levi-Civita connection in nonassociative Riemannian geometry on phase space, and write down Einstein field equations. We apply this formalism to construct R-flux corrections to the Ricci tensor on spacetime, and comment on the potential implications of these structures in non-geometric string theory and double field theory.Comment: 50 pages; v2: corrected comparison of curvature to ref. [14]; v3: clarifying comments added; Final version to be published in JHE

N=1,2 Super-NLS Hierarchies as Super-KP Coset Reductions

We define consistent finite-superfields reductions of the $N=1,2$ super-KP hierarchies via the coset approach we already developped for reducing the bosonic KP-hierarchy (generating e.g. the NLS hierarchy from the $sl(2)/U(1)-{\cal KM}$ coset). We work in a manifestly supersymmetric framework and illustrate our method by treating explicitly the $N=1,2$ super-NLS hierarchies. W.r.t. the bosonic case the ordinary covariant derivative is now replaced by a spinorial one containing a spin ${\textstyle {1\over 2}}$ superfield. Each coset reduction is associated to a rational super-\cw algebra encoding a non-linear super-\cw_\infty algebra structure. In the $N=2$ case two conjugate sets of superLax operators, equations of motion and infinite hamiltonians in involution are derived. Modified hierarchies are obtained from the original ones via free-fields mappings (just as a m-NLS equation arises by representing the $sl(2)-{\cal KM}$ algebra through the classical Wakimoto free-fields).Comment: 27 pages, LaTex, Preprint ENSLAPP-L-467/9