The following chapters summarize lectures given in March 2001 during the summerschool on Hyperbolic Partial Differential Equations which took place at the Technical University of Hamburg-Harburg in Germany. This type of meeting is originally funded by the Volkswa­ genstiftung in Hannover (Germany) with the aim to bring together well-known leading experts from special mathematical, physical and engineering fields of interest with PhD­ students, members of Scientific Research Institutes as well as people from Industry, in order to learn and discuss modern theoretical and numerical developments. Hyperbolic partial differential equations play an important role in various applications from natural sciences and engineering. Starting from the classical Euler equations in fluid dynamics, several other hyperbolic equations arise in traffic flow problems, acoustics, radiation transfer, crystal growth etc. The main interest is concerned with nonlinear hyperbolic problems and the special structures, which are characteristic for solutions of these equations, like shock and rarefaction waves as well as entropy solutions. As a consequence, even numerical schemes for hyperbolic equations differ significantly from methods for elliptic and parabolic equations: the transport of information runs along the characteristic curves of a hyperbolic equation and consequently the direction of transport is of constitutive importance. This property leads to the construction of upwind schemes and the theory of Riemann solvers. Both concepts are combined with explicit or implicit time stepping techniques whereby the chosen order of accuracy usually depends on the expected dynamic of the underlying solution.

Th e variational spline theory which originates f rom th e well-known paper by J.e. Holliday (1957) is today a well-developed field in approxi- mat ion theory. The gene ra l definition of splines in the Hilbert space, ex- istence, uniqueness, and characterization theorems were obt ained about 35 years ago by M. Atteia, P.J. L aur ent, and P.M. Anselone, but in rec ent years impor tant new results have been obtained in th e abst ract variational spline theory. Th ese concern the convergence in th e Hilb ert spaces, general techniques for error es tim ation of abs trac t splines includ- ing multi-dimensional splines on scatt ered meshes, new kinds of charac- teriz ation th eorems based on reproducing kernels and m appin gs, and the theory and algorithms for the splines on subspaces (finite element me th od for complic at ed non-polynomial splines). New variatio na l for- mulations arose for vec tor splines, rational splines, te nsor and blending splines, trac es of splines on the smooth manifold, discontinuous splines, etc. Optim al approxi mat ions of linear functionals and ope rat ors are also developed from th ese concepts.

The Ninth EPSRC Numerical Analysis Summer School was held at the Uni­ versity of Durharn, UK, from the 10th to the 21st of July 2000. This was the first of these schools to be held in Durharn, having previously been hosted, initially by the University of Lancaster and latterly by the University of Leicester. The purpose of the summer school was to present high quality in­ structional courses on topics at the forefront of numerical analysis research to postgraduate students. Eminent figures in numerical analysis presented lectures and provided high quality lecture notes. At the time of writing it is now more than two years since we first con­ tacted the guest speakers and during that period they have given significant portions of their time to making the summer school, and this volume, a suc­ cess. We would like to thank all six of them for the care which they took in the preparation and delivery of their lectures.

In questo testo si introducono i concetti di base per la modellistica numerica di problemi differenziali alle derivate parziali. Si considerano le classiche equazioni lineari ellittiche, paraboliche ed iperboliche, ma anche altre equazioni, quali quelle di diffusione e trasporto, di Navier-Stokes e le leggi di conservazione; si forniscono inoltre numerosi esempi fisici che stanno alla base di tali equazioni. Quindi si analizzano metodi di risoluzione numerica basati su elementi finiti (continui e discontinui), differenze finite, volumi finiti, metodi spettrali (continui e discontinui), nonché strategie di approssimazione più avanzate basate sui metodi di decomposizione di domini o quelli di risoluzione di problemi di controllo ottimale. In particolare vengono discussi gli aspetti algoritmici e di implementazione al calcolatore e si forniscono diversi programmi di semplice utilizzo. Il testo non presuppone una approfondita conoscenza matematica delle equazioni alle derivate parziali: iconcetti rigorosamente indispensabili al riguardo sono riportati nell'Appendice. Esso è pertanto adatto agli studenti dei corsi di laurea di indirizzo scientifico (Ingegneria, Matematica, Fisica, Scienze dell'Informazione) e consigliabile a ricercatori del mondo accademico ed extra-accademico che vogliano avvicinarsi a questo interessante ramo della matematica applicata e delle scienze computazionali.

Excel Financial Modeling from Scratch: Financial Statements, Discounted Cash Flow (DCF), and Accounting

Beginning with realistic mathematical or verbal models of physical or biological phenomena, the author derives tractable mathematical models that are amenable to further mathematical analysis or to elucidating computer simulations. For the most part, derivations are based on perturbation methods. Because of this, the majority of the text is devoted to careful derivations of implicit function theorems, the method of averaging, and quasi-static state approximation methods. The duality between stability and perturbation is developed and used, relying heavily on the concept of stability under persistent disturbances. This explains why stability results developed for quite simple problems are often useful for more complicated, even chaotic, ones. Relevant topics about linear systems, nonlinear oscillations, and stability methods for difference, differential-delay, integro- differential and ordinary and partial differential equations are developed throughout the book. For the second edition, the author has restructured the chapters, placing special emphasis on introductory materials in Chapters 1 and 2 as distinct from presentation materials in Chapters 3 through 8. In addition, more material on bifurcations from the point of view of canonical models, sections on randomly perturbed systems, and several new computer simulations have been added.

This book develops spatial interaction models for the analysis of human interaction within space, in terms of both accessibility and land use. Presenting case studies on the Azores and Morocco, it covers applications in various regions of Europe and Africa. The respective models simulate land use, employment, households, commuting and shopping movements and land values, employment distribution for basic activities, changes in accessibility, and changes in land suitability due to climate change.

The book presents the winners of the Abel Prize in mathematics for the period 2018–2022: