This paper provides a review of the Fractal Market Hypothesis (FMH) focusing on financial times series analysis. In order to put the FMH into a broader perspective, the Random Walk and Efficient Market Hypotheses are considered together with the basic principles of fractal geometry. After exploring the historical developments associated with different financial hypotheses, an overview of the basic mathematical modelling is provided. The principal goal of this paper is to consider the intrinsic scaling properties that are characteristic for each hypothesis. In regard to the FMH, it is explained why a financial time series can be taken to be characterised by a $1/t^{1-1/\gamma }$ scaling law, where $\gamma >0$ is the Lévy index, which is able to quantify the likelihood of extreme changes in price differences occurring (or otherwise). In this context, the paper explores how the Lévy index, coupled with other metrics, such as the Lyapunov Exponent and the Volatility, can be combined to provide long-term forecasts. Using these forecasts as a quantification for risk assessment, short-term price predictions are considered using a machine learning approach to evolve a nonlinear formula that simulates price values. A short case study is presented which reports on the use of this approach to forecast Bitcoin exchange rate values.

Before the effect of the COVID-19 Pandemic, there had been continued debate about the future of Higher Education (HE) in the UK. It is now accepted that the effect of the pandemic will have a long-lasting effect on HE in the UK and elsewhere. This paper addresses the changes that are currently taking place, based on a strategy that aims to develop a future knowledge-based economy, following the UK governments 2019 landmark review of HE. It explores the underlying parallels between the current situation and certain historical events that catalysed the development of a new approach to HE in the past, which is very relevant today. In this context, the paper discusses why major changes in UK HE provision is now required as a response to the fact that although the cost of education is rising, employers are reporting that graduates are increasingly unprepared for the workplace. In this respect, the paper addresses a model for HE that focuses on `earn-as-you-learn’ apprenticeships and work-place-based learning. The key to this is the emphasis that the UK government is now placing on funding new `Technological Colleges’, in which students are trained by experts from the industry on a contractual basis, rather than by university academics with tenured positions.

This paper presents trend prediction results based on backtesting of the European Union Emissions Trading Scheme futures market. This is based on the Intercontinental Exchange from 2005 to 2019. An alternative trend prediction strategy is taken that is predicated on an application of the Fractal Market Hypothesis (FMH) in order to develop an indicator that is predictive of short term future behaviour. To achieve this, we consider that a change in the polarity of the Lyapunov-to-Volatility Ratio precedes an associated change in the trend of the European Union Allowances (EUAs) price signal. The application of the FMH in this case is demonstrated to provide a useful tool in order to assess the likelihood of the market becoming bear or bull dominant, thereby helping to inform carbon trading investment decisions. Under specific conditions, Evolutionary Computing methods are utilised in order to optimise specific trading execution points within a trend and improve the potential profitability of trading returns. Although the approach may well be of value for general energy commodity futures trading (and indeed the wider financial and commodity derivative markets), this paper presents the application of an investment indicator for EUA carbon futures risk modelling and investment trend analysis only.

This paper examines a range of results that can be derived from Einstein’s evolution equation focusing on the effect of introducing a Lévy distribution into the evolution equation. In this context, we examine the derivation (derived exclusively from the evolution equation) of the classical and fractional diffusion equations, the classical and generalised Kolmogorov–Feller equations, the evolution of self-affine stochastic fields through the fractional diffusion equation, the fractional Poisson equation (for the time independent case), and, a derivation of the Lyapunov exponent and volatility. In this way, we provide a collection of results (which includes the derivation of certain fractional partial differential equations) that are fundamental to the stochastic modelling associated with elastic scattering problems obtained under a unifying theme, i.e., Einstein’s evolution equation. This includes an analysis of stochastic fields governed by a symmetric (zero-mean) Gaussian distribution, a Lévy distribution characterised by the Lévy index $\gamma \in [0,2]$ and the derivation of two impulse response functions for each case. The relationship between non-Gaussian distributions and fractional calculus is examined and applications to financial forecasting under the fractal market hypothesis considered, the reader being provided with example software functions (written in MATLAB) so that the results presented may be reproduced and/or further investigated.

The ‘diffraction in space’ and the ‘diffraction in time’ phenomena are considered in regard to a continuously open, and a closed shutter that is opened at an instant in time, respectively. The purpose of this is to provide a background to the principal theme of this article, which is to extend the ‘quantum shutter problem’ for the case when the wave function is determined by the fundamental solution to a partial differential equation with a fractional derivative of space or of time. This involves the development of Green’s function solutions for the space- and time-fractional Schrödinger equation and the time-fractional Klein–Gordon equation (for the semi-relativistic case). In each case, the focus is on the development of primarily one-dimensional solutions, subject to an initial condition which controls the dynamical behaviour of the wave function. Coupled with variations in the fractional order of the fractional derivatives, illustrative example results are provided that are based on presenting space-time maps of the wave function; specifically, the probability density of the wave function. In this context, the paper provides a case study of fractional quantum mechanics and control using fractional calculus.

Fractal antennas have and are continuing to receive attention in regard to the future of wireless communications. This is because of their wide- and multi-band capabilities, the opportunity of fractal geometries to drive multiple resonances, and, the ability to make smaller and lighter antennas with fewer components and radiative elements with higher gains. Small scale (i.e. on the micro- and nano-scale) and ultra high frequency (in the Terahertz or THz range) fractal antennas composed of Graphene have the potential to enhance wireless communications at a data rate that is unprecedented, i.e. ∼ 10¹² bits per second. A Fractal Graphene antenna is a high-frequency tuneable antenna for radio communications in the THz spectrum, enabling unique applications such as wireless nano-networks. This is because (mono-layer) Graphene is a one-atom-thick two-dimensional allotrope of Carbon with the highest known electrical conductivity that is currently unavailable in any other material, including metals such as Gold and Silver. Thus, combining the properties of Graphene with the self-affine characteristics of a fractal at the micro- and nano-scale, provides the potential to revolutionise communications, at least in the near field (the order of a few metres) for low power systems. In this paper, we consider the basic physics and some of the principle mathematical models associated with the development of this new disruptive technology in order to provide a guide to those engaged in current and future research, a fractal Graphene antenna being an example of an advanced material for demanding applications. This includes some example simulations on the THz field patterns generated by a fractal patch antenna composed of Graphene whose conductivity is taken to scale with the inverse of the frequency according to a ‘Drude’ model. The approach to generating THz sources using Graphene is also explored based on Infrared laser pumping to induce a THz photo-current.