Fractional Derivative as Fractional Power of Derivative

Definitions of fractional derivatives as fractional powers of derivative operators are suggested. The Taylor series and Fourier series are used to define fractional power of self-adjoint derivative operator. The Fourier integrals and Weyl quantization procedure are applied to derive the definition of fractional derivative operator. Fractional generalization of concept of stability is considered.Comment: 20 pages, LaTe

Derivative Chameleons

We consider generalized chameleon models where the conformal coupling between matter and gravitational geometries is not only a function of the chameleon field \phi, but also of its derivatives via higher order co-ordinate invariants. Specifically we consider the first such non-trivial conformal factor A(\phi,X), where X is the canonical kinetic term for \phi. The associated phenomenology is investigated and we show that such theories have a new generic mass-altering mechanism, potentially assisting the generation of a sufficiently large chameleon mass in dense environments. The most general effective potential is derived for such derivative chameleon setups and explicit examples are given. Interestingly this points us to the existence of a purely derivative chameleon protected by a shift symmetry for \phi. We also discuss potential ghost-like instabilities associated with mass-lifting mechanisms and find another, mass-lowering and instability-free, branch of solutions. This suggests that, barring fine-tuning, stable derivative models are in fact typically anti-chameleons that suppress the field's mass in dense environments. Furthermore we investigate modifications to the thin-shell regime and prove a no-go theorem for chameleon effects in non-conformal geometries of the disformal type.Comment: 28 pages, 4 figure

Derivative Polynomials and Closed-Form Higher Derivative Formulae

In a recent paper, Adamchik [V.S. Adamchik, On the Hurwitz function for rational arguments, Appl. Math. Comp. 187 (2007) 3--12] expressed in a closed form symbolic derivatives of four functions belonging to the class of functions whose derivatives are polynomials in terms of the same functions. In this sequel, simple closed-form higher derivative formulae which involve the Carlitz-Scoville higher order tangent and secant numbers are derived for eight trigonometric and hyperbolic functions.Comment: 7 page

Reverse Derivative Categories

The reverse derivative is a fundamental operation in machine learning and automatic differentiation. This paper gives a direct axiomatization of a category with a reverse derivative operation, in a similar style to that given by Cartesian differential categories for a forward derivative. Intriguingly, a category with a reverse derivative also has a forward derivative, but the converse is not true. In fact, we show explicitly what a forward derivative is missing: a reverse derivative is equivalent to a forward derivative with a dagger structure on its subcategory of linear maps. Furthermore, we show that these linear maps form an additively enriched category with dagger biproducts.Comment: Extended version of paper to appear at CSL 202

Notion of a virtual derivative

Diagrams as a graphic expresion of derivatives is proposed for calculation of derivatives for composed function. The concret diagram is understood as a virtual derivative in contrast of concret derivative. In polynomial expression of functions derivative the concret derivative will be every monomic member, and the virtual derivative represent the sum of similar monomic members. The word virtual denotes that we dont need to know every virtual derivative, we don't write all the sequence of these virtual derivatives, and simply pick the needed one. This is in contrast of tradition to write the whole algebraic expresion as a denotion of whole function's derivative. Such graphic expresion can be helpful in the problems of differential geometry, in the various asymptotic expantions, also in the solution of some differential equations.Comment: The diagrams are drown with the help of xy-pic and can be automaticly generated for the derivative of large degry of more general composed functio