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Logical thinking, the analysis of complex relationships, the recognition of und- lying simple structures which are common to a multitude of problems &#8212; these are the skills which are needed to do mathematics, and their development is the main goal of mathematics education. Of course, these skills cannot be learned &#8216;in a vacuum&#8217;. Only a continuous struggle with concrete problems and a striving for deep understanding leads to success. A good measure of abstraction is needed to allow one to concentrate on the essential, without being distracted by appearances and irrelevancies. The present book strives for clarity and transparency. Right from the beg- ning, it requires from the reader a willingness to deal with abstract concepts, as well as a considerable measure of self-initiative. For these e?orts, the reader will be richly rewarded in his or her mathematical thinking abilities, and will possess the foundation needed for a deeper penetration into mathematics and its applications. Thisbookisthe?rstvolumeofathreevolumeintroductiontoanalysis.It- veloped from courses that the authors have taught over the last twenty six years at theUniversitiesofBochum,Kiel,Zurich,BaselandKassel.Sincewehopethatthis book will be used also for self-study and supplementary reading, we have included far more material than can be covered in a three semester sequence. This allows us to provide a wide overview of the subject and to present the many beautiful and important applications of the theory. We also demonstrate that mathematics possesses, not only elegance and inner beauty, but also provides e?cient methods for the solution of concrete problems.

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Seminar paper from the year 2011 in the subject Economics - Finance, Christian-Albrechts-University of Kiel (Department of Economics), course: Seminar in Public Economics and Social Policy: Federalism and (De)Centralization, language: English, abstract: An implemented tax system causes distortions which leads to a minor overall welfare level compared to a system without taxes. This deviation in social welfare is often denoted by excess burden or dead weight loss (DWL) of taxation. So the traditional optimal taxation approach comprises the implementation of a tax system which minimizes the excess burden and hence the distortions caused by the levied taxes. Therefore, the policy maker has to anticipate possible behavioral adjustments of the market participants when choosing its optimal tax policy. Assuming the policy maker will do so all effects (i.e. distortions) caused by the tax system will be internalized which means that no fiscal externalities would arise from implementing the (optimal) tax system. However, the traditional optimal taxation approach abstracts from any intergovernmental relations as the existence of only one government and accordingly only one level with fiscal jurisdiction is assumed. The question here is whether and to what extent federal structures (i.e. multileveled government structures) affect the optimal tax policy decision.The first attempt to take into account the characteristics of a federal systemrelated to optimal tax policy goes back to Gordon (1983) who applied the methodologyof the traditional optimal taxation approach to fiscal federalism. Therein eachunit of government (i.e. the federal and usually several state governments) decidesindependently how much of public goods to provide and in particular which taxpolicy to use in funding the provided public goods. Hence, we now consider a decentralizedform of decision-making in which each unit of government chooses theoptimal tax policy in the best interest of its own residents. As a consequence of thissolely intrajurisdictional externalities are internalized analogous to the traditionaloptimization approach. Though, it isn't obvious whether this solution is also optimalin the sense of an inter jurisdictional point of view. Sobel (1997), Wrede (1999)and also Keen/Kotsogiannis (2002) stated that a common pool problem emergesgiven that subordinated governments (i.e. state governments) are allowed to levytaxes as well as the federal government. This means that taxation at multiple levelslead to a shared tax base which is the fiscal analogue to the common propertyresource. Due to this overlap in tax bases any separately considered optimal taxpolicy at a certain level may affect the optimality character of the ...

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This book has developed from lectures that the author gave for mathematics students at the Ruhr-Universitat Bochum and the Christian-Albrechts-Uni versitat Kiel. This edition is the result of the translation and correction of the German edition entitled Theone und Numenk elliptischer Differential gleichungen. The present work is restricted to the theory of partial differential equa tions of elliptic type, which otherwise tends to be given a treatment which is either too superficial or too extensive. The following sketch shows what the problems are for elliptic differential equations. A: Theory of B: Discretisation: c: Numerical analysis elliptic Difference Methods, convergence, equations finite elements, etc. stability Elliptic Discrete boundary value equations f-------- ----- problems E:Theory of D: Equation solution: iteration Direct or with methods iteration methods The theory of elliptic differential equations (A) is concerned with ques tions of existence, uniqueness, and properties of solutions. The first problem of VI Foreword numerical treatment is the description of the discretisation procedures (B), which give finite-dimensional equations for approximations to the solu tions. The subsequent second part of the numerical treatment is numerical analysis (0) of the procedure in question. In particular it is necessary to find out if, and how fast, the approximation converges to the exact solution.

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