Fractional differentiation in the self-affine case I – Random functions

AbstractThe invariance structure of self-affine functions and measures leads to the concept of fractional Cesáro derivatives and densities, respectively. In the present paper the case of random functions from Rp into Rq is considered. It is shown that the corresponding derivatives exist a.s. and equal a constant in the ergodic case. Part II will deal with the class of self-similar extremal processes and certain extensions. In Part III the fractional density of the Cantor measure will be evaluated, and arbitrary self-similar random measures will be treated in Part IV. There exists a deeper connection to fractional differentiation in the theory of function spaces which will be established elsewhere

Fractional differentiability of nowhere differentiable functions and dimensions

Weierstrass's everywhere continuous but nowhere differentiable function is shown to be locally continuously fractionally differentiable everywhere for all orders below the `critical order' 2-s and not so for orders between 2-s and 1, where s, 1<s<2 is the box dimension of the graph of the function. This observation is consolidated in the general result showing a direct connection between local fractional differentiability and the box dimension/ local Holder exponent. Levy index for one dimensional Levy flights is shown to be the critical order of its characteristic function. Local fractional derivatives of multifractal signals (non-random functions) are shown to provide the local Holder exponent. It is argued that Local fractional derivatives provide a powerful tool to analyze pointwise behavior of irregular signals.Comment: minor changes, 19 pages, Late

Holder exponents of irregular signals and local fractional derivatives

It has been recognized recently that fractional calculus is useful for handling scaling structures and processes. We begin this survey by pointing out the relevance of the subject to physical situations. Then the essential definitions and formulae from fractional calculus are summarized and their immediate use in the study of scaling in physical systems is given. This is followed by a brief summary of classical results. The main theme of the review rests on the notion of local fractional derivatives. There is a direct connection between local fractional differentiability properties and the dimensions/ local Holder exponents of nowhere differentiable functions. It is argued that local fractional derivatives provide a powerful tool to analyse the pointwise behaviour of irregular signals and functions.Comment: 20 pages, Late

Precursor engineering of hydrotalcite-derived redox sorbents for reversible and stable thermochemical oxygen storage

Chemical looping processes based on multiple-step reduction and oxidation of metal oxides hold great promise for a variety of energy applications, such as CO2 capture and conversion, gas separation, energy storage, and redox catalytic processes. Copper-based mixed oxides are one of the most promising candidate materials with a high oxygen storage capacity. However, the structural deterioration and sintering at high temperatures is one key scientific challenge. Herein, we report a precursor engineering approach to prepare durable copper-based redox sorbents for use in thermochemical looping processes for combustion and gas purification. Calcination of the CuMgAl hydrotalcite precursors formed mixed metal oxides consisting of CuO nanoparticles dispersed in the Mg-Al oxide support which inhibited the formation of copper aluminates during redox cycling. The copper-based redox sorbents demonstrated enhanced reaction rates, stable O2 storage capacity over 500 redox cycles at 900 °C, and efficient gas purification over a broad temperature range. We expect that our materials design strategy has broad implications on synthesis and engineering of mixed metal oxides for a range of thermochemical processes and redox catalytic applications

Fractional differentiation in the self-affine case II - Extremal processes

In Part I we introduced the concept of fractional Cesàro derivatives of random processes. We proved that they exist for self-affine functions at Lebesgue-a.a. points. In the present part we consider together with the random process a random measure and give conditions which ensure that the fractional Cesàro derivative exists at almost all points w.r.t. this random measure. Our conditions are satisfied by the measure associated with the maximal process of a self-affine process, so we deduce that the Cesàro derivative exists at almost all points of increase.

Fractional differentiation in the self-affine case I - Random functions

The invariance structure of self-affine functions and measures leads to the concept of fractional Cesáro derivatives and densities, respectively. In the present paper the case of random functions from p into q is considered. It is shown that the corresponding derivatives exist a.s. and equal a constant in the ergodic case. Part II will deal with the class of self-similar extremal processes and certain extensions. In Part III the fractional density of the Cantor measure will be evaluated, and arbitrary self-similar random measures will be treated in Part IV. There exists a deeper connection to fractional differentiation in the theory of function spaces which will be established elsewhere.