Many key phenomena in physics and engineering are described as singularities in the solutions to the differential equations describing them. Examples covered thoroughly in this book include the formation of drops and bubbles, the propagation of a crack and the formation of a shock in a gas. Aimed at a broad audience, this book provides the mathematical tools for understanding singularities and explains the many common features in their mathematical structure. Part I introduces the main concepts and techniques, using the most elementary mathematics possible so that it can be followed by readers with only a general background in differential equations. Parts II and III require more specialised methods of partial differential equations, complex analysis and asymptotic techniques. The book may be used for advanced fluid mechanics courses and as a complement to a general course on applied partial differential equations.

Many key phenomena in physics and engineering are described as singularities in the solutions to the differential equations describing them. Examples covered thoroughly in this book include the formation of drops and bubbles, the propagation of a crack and the formation of a shock in a gas. Aimed at a broad audience, this book provides the mathematical tools for understanding singularities and explains the many common features in their mathematical structure. Part I introduces the main concepts and techniques, using the most elementary mathematics possible so that it can be followed by readers with only a general background in differential equations. Parts II and III require more specialised methods of partial differential equations, complex analysis and asymptotic techniques. The book may be used for advanced fluid mechanics courses and as a complement to a general course on applied partial differential equations.

Early-stage venture capital is underdeveloped in many European countries like Germany and Italy. After 20 years of analysis one culprit, supply of funds, is identified and policy action has been instituted for change. Analysis however is limited by lack and quality of data and a missing industry perspective including supply and demand. This study therefore summarizes current research on macro, micro and policy level and discusses its main arguments. It compiles (new) long-term data on "true" venture capital for Germany and Italy, analyzes data in an industry life-cycle perspective, and compares these results to those currently referred to in the literature. It introduces a new measure for available funds. As a result, it becomes evident that the hitherto all but forgotten demand side - namely good entrepreneurs and their ideas - of venture capital requires much more attention. This book is for all those, researchers, politicians, industry representatives and venture capital managers as well as entrepreneurs who want to seriously discuss all aspects in order to improve early-stage venture capital and education in Europe.