Fractional Calculus and the Future of Science

Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding

Resolving the QCD phase structure

This thesis discusses the quantitative description of the phase structure of Quantum Chromo- dynamics (QCD). We find that, in strongly correlated theories such as QCD, even a qualitative investigation of the phase structure can require highly quantitative methods. Hence, the de- velopment of a method with systematic error control is essential. In the present work, we use functional renormalisation group (fRG) method to this aim. This work focusses on three ideas: Firstly, we identify quantitatively dominating and sub-leading scattering-processes in our approximations. This allows a formulation of low energy effective theories of the four-quark interaction, as well as the description of gluon condensation. For the former, we present results for meson and quark masses. The latter provides an estimate of the Yang-Mills mass gap. Secondly, we further develop the use of highly precise numerical methods from fluid-dynamics in the fRG. In particular we use Discontinuous Galerkin methods, which are able to capture shock-development. Shock-waves are found to play a big role in a possible creation-mechanism of first-order phase transitions. Lastly, we focus on general RG-transformations (gRGt). For example, they allow a real time formulation of fRG flows and hence give access to spectral functions. Furthermore, we use them to formulate complex RG-flows, which enables us to locate Lee-Yang singularities in the complex plane and extrapolate the position of (real) phase transitions. Finally, we also use gRGts to formulate significant qualitative improvements of current fRG approximation schemes by means of dynamical field transformations

Stepsize Restrictions for Nonlinear Stability Properties of Neutral Delay Differential Equations

The present paper is concerned with the relationship between stepsize restriction and nonlinear stability of Runge-Kutta methods for delay differential equations. We obtain a special stepsize condition guaranteeing global and asymptotical stability properties of numerical methods. Some confirmations of the conditions on Runge-Kutta methods are illustrated at last

Computational Engineering

This Workshop treated a variety of finite element methods and applications in computational engineering and expanded their mathematical foundation in engineering analysis. Among the 53 participants were mathematicians and engineers with focus on mixed and nonstandard finite element schemes and their applications

A Compact Difference Scheme for a Class of Variable Coefficient Quasilinear Parabolic Equations with Delay

A linearized compact difference scheme is provided for a class of variable coefficient parabolic systems with delay. The unique solvability, unconditional stability, and convergence of the difference scheme are proved, where the convergence order is four in space and two in time. A numerical test is presented to illustrate the theoretical results

Single and two-photon fluorescence studies of linear and non-linear optical chromophores.

The subject matter presented in this thesis concerns the structural and dynamic studies of new fluorescent probe molecules and the application of polarised fluorescence techniques and analysis to molecular motion, order and solvation in a highly ordered environment. Chapter 1 reviews recent group research providing a context to the work in this thesis, whilst Chapters 2 and 3 concern the study of fluorescent probe dynamics. Time resolved photoselection techniques were used to probe the order and full angular motion of Coumarin 6 and Coumarin 153 in the nematic and isotropic phases of the liquid crystal 5CB. The uptake of coumarin molecules into this host differs from previously studied (Xanthene) probes - in particular, Coumarin 6 is seen to adopt a disruptive position within the alkyl tails due to its size and hydrophobic nature this is discussed in Chapter 2. Furthermore, Coumarin 153 undergoes a substantial increase in dipole moment upon electronic excitation this led to a unique study of time dependent solvation dynamics in both a globally and locally structured environment. The presence of strong solvent- solute interactions necessitated the development of a new approach to the analysis of time resolved polarised fluorescence in ordered systems. This approach and the study of time dependent solvation dynamics in the isotropic and nematic phases of 5CB is presented in Chapter 3. Structural studies of new two-photon fluorescent probes in collaboration with CNRS Rennes and Los Alamos are described in the final two chapters. Large two-photon resonances in the green-visible were observed together with a fuller characterisation of those in the near IR. Polarised two-photon absorption and anisotropy measurements were used to examine the structure of the two-photon resonances. Finally, the stimulated emission depletion dynamics of a branched two-photon fluorophore were investigated and found to differ markedly from conventional (non-degenerate) fluorophores