Global Demand for Environmental Goods and Services on the Rise: Good Growth Opportunities for German Suppliers

According to conservative calculations, over $580 billion was spent worldwide on environmental goods and services and renewable energy technologies1 in 2004. So-called "green spending" is set for strong growth in the future on account of the long-term expansion of the global economy and mounting environmental challenges. Significant opportunities for growth and employment in Germany are also offered by forecasted trends in the market for green technologies. DIW Berlin has developed a method to quantify future global demand for environmental goods and services based on alternative economic scenarios. The method places a key focus on the international trade of environmental goods and services. Our calculations predict that the effective annual demand for environmental goods and services in Germany will increase from$75 billion in 2004 to between $106 and 171 billion by 2020 (at 2004 prices and exchange rates).The high growth in German exports is responsible in particular for this trend. Nevertheless, sensitivity calculations indicate that demand could also grow at a much slower rate through 2020 under unfavorable economic conditions.Environmental Goods and Services Sector, World Trade, Scenarios

An Application of Kolmogorov's Superposition Theorem to Function Reconstruction in Higher Dimensions

In this thesis we present a Regularization Network approach to reconstruct a continuous function ƒ:[0,1]n→R from its function values ƒ(xj) on discrete data points xj, j=1,…,P. The ansatz is based on a new constructive version of Kolmogorov's superposition theorem. Typically, the numerical solution of mathematical problems underlies the so--called curse of dimensionality. This term describes the exponential dependency of the involved numerical costs on the dimensionality n. To circumvent the curse at least to some extend, typically higher regularity assumptions on the function ƒ are made which however are unrealistic in most cases. Therefore, we employ a representation of the function as superposition of one--dimensional functions which does not require higher smoothness assumptions on ƒ than continuity. To this end, a constructive version of Kolmogorov's superposition theorem which is based on D. Sprecher is adapted in such a manner that one single outer function Φ and a universal inner function ψ suffice to represent the function ƒ. Here, ψ is the extension of a function which was defined by M. Köppen on a dense subset of the real line. The proofs of existence, continuity, and monotonicity are presented in this thesis. To compute the outer function Φ, we adapt a constructive algorithm by Sprecher such that in each iteration step, depending on ƒ, an element of a sequence of univariate functions { Φr}r is computed. It will be shown that this sequence converges to a continuous limit Φ:R→R. This constructively proves Kolmogorov's superposition theorem with a single outer and inner function. Due to the fact that the numerical complexity to compute the outer function Φ by the algorithm grows exponentially with the dimensionality, we alternatively present a Regularization Network approach which is based on this representation. Here, the outer function is computed from discrete function samples (xj,ƒ(xj)), j=1,…,P. The model to reconstruct ƒ will be introduced in two steps. First, the outer function Φ is represented in a finite basis with unknown coefficients which are then determined by a variational formulation, i.e. by the minimization of a regularized empirical error functional. A detailed numerical analysis of this model shows that the dimensionality of ƒ is transformed by Kolmogorov's representation into oscillations of Φ. Thus, the use of locally supported basis functions leads to an exponential growth of the complexity since the spatial mesh resolution has to resolve the strong oscillations. Furthermore, a numerical analysis of the Fourier transform of Φ shows that the locations of the relevant frequencies in Fourier space can be determined a priori and are independent of ƒ. It also reveals a product structure of the outer function and directly motivates the definition of the final model. Therefore, Φ is replaced in the second step by a product of functions for which each factor is expanded in a Fourier basis with appropriate frequency numbers. Again, the coefficients in the expansions are determined by the minimization of a regularized empirical error functional. For both models, the underlying approximation spaces are developed by means of reproducing kernel Hilbert spaces and the corresponding norms are the respective regularization terms in the empirical error functionals. Thus, both approaches can be interpreted as Regularization Networks. However, it is important to note that the error functional for the second model is not convex and that nonlinear minimizers have to be used for the computation of the model parameters. A detailed numerical analysis of the product model shows that it is capable of reconstructing functions which depend on up to ten variables.Eine Anwendung von Kolmogorovs Superpositionen Theorem zur Funktionsrekonstruktion in höheren Dimensionen In der vorliegenden Arbeit wird ein Regularisierungsnetzwerk zur Rekonstruktion von stetigen Funktionen ƒ:[0,1]n→R vorgestellt, welches direkt auf einer neuen konstruktiven Version von Kolmogorovs Superpositionen Theorem basiert. Dabei sind lediglich die Funktionswerte ƒ(xj) an diskreten Datenpunktenxj, j=1,…,P bekannt. Typischerweise leidet die numerische Lösung mathematischer Probleme unter dem sogenannten Fluch der Dimension. Dieser Begriff beschreibt das exponentielle Wachstum der Komplexität des verwendeten Verfahrens mit der Dimension n. Um dies zumindest teilweise zu vermeiden, werden üblicherweise höhere Regularitätsannahmen an die Lösung des Problems gemacht, was allerdings häufig unrealistisch ist. Daher wird in dieser Arbeit eine Darstellung der Funktion ƒ als Superposition eindimensionaler Funktionen verwendet, welche keiner höheren Regularitätsannahmen als Stetigkeit bedarf. Zu diesem Zweck wird eine konstruktive Variante des Kolmogorov Superpositionen Theorems, welche auf D. Sprecher zurückgeht, so angepasst, dass nur eine äußere Funktion Φ sowie eine universelle innere Funktion ψ zur Darstellung von ƒ notwendig ist. Die Funktion ψ ist nach einer Definition von M. Köppen explizit und unabhängig von ƒ als Fortsetzung einer Funktion, welche auf einer Dichten Teilmenge der reellen Achse definiert ist, gegeben. Der fehlende Beweis von Existenz, Stetigkeit und Monotonie von ψ wird in dieser Arbeit geführt. Zur Berechnung der äußeren Funktion Φ wird ein iterativer Algorithmus von Sprecher so modifiziert, dass jeder Iterationsschritt, abhängig von ƒ, ein Element einer Folge univariater Funktionen{ Φr}r liefert. Es wird gezeigt werden, dass die Folge gegen einen stetigen Grenzwert Φ:R→R konvergiert. Dies liefert einen konstruktiven Beweis einer neuen Version des Kolmogorov Superpositionen Theorems mit einer äußeren und einer inneren Funktion. Da die numerische Komplexität des Algorithmus zur Berechnung von Φ exponentiell mit der Dimension wächst, stellen wir alternativ ein Regularisierungsnetzwerk, basierend auf dieser Darstellung, vor. Dabei wird die äußere Funktion aus gegebenen Daten (xj,ƒ(xj)), j=1,…,P berechnet. Das Modell zur Rekonstruktion von ƒ wird in zwei Schritten eingeführt. Zunächst wird zur Definition eines vorläufigen Modells die äußere Funktion, bzw. eine Approximation an Φ, in einer endlichen Basis mit unbekannten Koeffizienten dargestellt. Diese werden dann durch eine Variationsformulierung bestimmt, d.h. durch die Minimierung eines regularisierten empirischen Fehlerfunktionals. Eine detaillierte numerische Analyse zeigt dann, dass Kolmogorovs Darstellung die Dimensionalität von ƒ in Oszillationen von F transformiert. Somit ist die Verwendung von Basisfunktionen mit lokalem Träger nicht geeignet, da die räumliche Auflösung der Approximation die starken Oszillationen erfassen muss. Des Weiteren zeigt eine Analyse der Fouriertransformation von Φ, dass die relevanten Frequenzen, unabhängig von ƒ, a priori bestimmbar sind, und dass die äußere Funktion Produktstruktur aufweist. Dies motiviert die Definition des endgültigen Modells. Dazu wird Φ nun durch ein Produkt von Funktionen ersetzt und jeder Faktor in einer Fourierbasis entwickelt. Die Koeffizienten werden ebenfalls durch Minimierung eines regularisierten empirischen Fehlerfunktionals bestimmt. Für beide Modelle wird ein theoretischer Rahmen in Form von Hilberträumen mit reproduzierendem Kern entwickelt. Die zugehörigen Normen bilden dabei jeweils den Regularisierungsterm der entsprechenden Fehlerfunktionale. Somit können beide Ansätze als Regularisierungsnetzwerke interpretiert werden. Allerdings ist zu beachten, dass das Fehlerfunktional für den Produktansatz nicht konvex ist und nichtlineare Minimierungsverfahren zur Berechnung der Koeffizienten notwendig sind. Weitere ausführliche numerische Tests zeigen, dass dieses Modell in der Lage ist Funktionen zu rekonstruieren welche von bis zu zehn Variablen abhängen

Effects of the Running of the QCD Coupling on the Energy Loss in the Quark-Gluon Plasma

Finite temperature modifies the running of the QCD coupling alpha_s(k,T) with resolution k. After calculating the thermal quark and gluon masses selfconsistently, we determine the quark-quark and quark-gluon cross sections in the plasma based on the running coupling. We find that the running coupling enhances these cross sections by factors of two to four depending on the temperature. We also compute the energy loss dE/dx of a high-energy quark in the plasma as a function of temperature. Our study suggests that, beside t-channel processes, inverse Compton scattering is a relevant process for a quantitative understanding of the energy loss of an incident quark in a hot plasma.Comment: 14 pages, 6 figure

Faraday-rotation fluctuation spectroscopy with static and oscillating magnetic fields

By Faraday-rotation fluctuation spectroscopy one measures the spin noise via Faraday-induced fluctuations of the polarization plane of a laser transmitting the sample. In the fist part of this paper, we present a theoretical model of recent experiments on alkali gas vapors and semiconductors, done in the presence of a {\em static} magnetic field. In a static field, the spin noise shows a resonance line, revealing the Larmor frequency and the spin coherence time $T_2$ of the electrons. Second, we discuss the possibility to use an {\em oscillating} magnetic field in the Faraday setup. With an oscillating field applied, one can observe multi-photon absorption processes in the spin noise. Furthermore an oscillating field could also help to avoid line broadening due to structural or chemical inhomogeneities in the sample, and thereby increase the precision of the spin-coherence time measurement.Comment: 5 pages, 7 figure

The Impact of Spin-Orbit Interaction on the Image States of High-Z Materials

Due to many important technical developments over the past two decades angle-resolved (inverse) photoemission has become the method of choice to study experimentally the bulk and surface-related electronic states of solids in the most detailed way. Due to new powerful photon sources as well as efficient analyzers and detectors extremely high energy and angle resolution are achieved nowadays for spin-integrated and also for spin-resolved measurements. These developments allow in particular to explore the influence of spin-orbit coupling on image potential states of simple metals like Ir, Pt, or Au with a high atomic number as well as new types of materials as for example topological insulators. Herein, fully relativistic angle- and spin-resolved inverse photoemission calculations are presented that make use of the spin-density matrix formulation of the one-step model. This way a quantitative analysis of all occupied and unoccupied electronic features in the vicinity of the Fermi level is achieved for a wide range of excitation energies. Using this approach, in addition, it is possible to deal with arbitrarily ordered but also disordered systems. Because of these features, the one-step or spectral function approach to photoemission permits detailed theoretical studies on a large variety of interesting solid-state systems.y

Economic Effects of Renewable Energy Expansion: A Model-Based Analysis for Germany

Increasing utilization of renewable energy sources (RES) is a priority worldwide. Germany has been a forerunner in the deployment of RES and has ambitious goals for the future. The support and use of renewables affects the economy: It creates business opportunities in sectors producing renewable energy facilities, but also comes along with costs for supporting the deployment of renewables. This paper analyses and quantifies the net balance of economic effects associated with renewable energy deployment in Germany until 2030. To this end, we use a novel model, the 'Sectoral Energy-Economic Econometric Model' (SEEEM). SEEEM is an econometric multi-country model which, for Germany, contains a detailed representation of industries, including 14 renewable energy technology sectors. Our results show that renewable energy expansion can be achieved without compromising growth or employment. The analysis reveals a positive net effect on economic growth in Germany. Net employment effects are positive. Their size depends strongly on labour market conditions and policies. Results at the industry level indicate the size and direction of the need for restructuring across the sectors of the Germany economy.