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Mathematik ist die Liebe zur Weisheit, die Philosophie des Unendlich-Vielfältigen. Daher ist es auch kein Wunder, dass der erste Philosoph, - wie Aristoteles sagte -, auch ein Mathematiker ist, nämlich Thales von Milet (625-547 v.Chr.), der die Sonnenfinsternis vom 28. Mai 585 v. Chr. richtig vorhersagte.
Seit über zweieinhalb Jahrtausenden beschäftigt sich also die Menschheit schon mit geometrischen Gebilden, wie Geraden, Dreiecke, Vierecke oder Pyramiden! Unter den alten Geometern finden sich der um 600 v. Chr. geborene, schulbekannte Pythagoras, der eine geheime Bruderschaft gründete; Zenon von Elea (490-430), der mit scharfsinnigen Paradoxien durch reine Überlegung schon der 'Quantennatur der Geometrie' auf die Schliche kam; Platon (427-347), ein Schüler Sokrates, der nur den Ideen eigentliche Realität zusprach, und unsere Sicht der Welt im Höhlengleichnis als nur schattenhaft erkannte; der um 300 v. Chr. in Alexandria lebende Euklid, der schliesslich das erste axiomatisch aufgebaute 13-bändige, mathematische Werk verfasste, nach dessen Geometrie noch heute alle Schüler unterrichtet werden, -nur das Beweisen scheint heute an den Schulen ausser Mode gekommen zu sein; Archimedes von Syrakus (285-212), der nicht nur die Kreiszahl ?, sonder beispielsweise auch äusserst elegant das Kugelvolumen berechnete; und die vielen, vielen anderen. Alle Gelehrten und Kosmologen beschäftigten sich mit dieser idealisierten Welt der Zahlen und des Raumes, angefangen von Aristarchos von Samos (320-250), der als erster das heliozentrische Weltbild lehrte, nachdem sich die Erde um die Sonne dreht, bis hin zu dem im 2. Jahrhundert nach Christus in Alexandria lebenden Claudius Ptolemäus, dessen geozentrisches Weltbild sich für Jahrhunderte durchsetzen sollte, (- würde sich nicht jede Fliege als Mittelpunkt der Welt betrachten? -), bis Kopernikus, Galilei und Kepler uns endgültig eines besseren belehren sollten.
Bald wird wohl die Anzahl der heute auf der Erde lebenden Mathematiker grösser sein, als alle einst in den vergangenen Jahrtausenden lebenden bzw. gestorbenen, zusammen genommen! Und sie haben sich alle schon mit Dreiecken beschäftigt! Könnte da noch etwas über das Dreieck unentdeckt geblieben sein?
Können Sie sich vorstellen, dass es für das Dreieck noch Formeln gibt, die in keinem Buch und keiner Formelsammlung zu finden sind?
Ja, dies ist der Fall, oder kennen Sie etwa die Formel, dass das Produkt der Dreiecksseiten dividiert durch seine Summe (auch Umfang genannt) gleich dem doppelten Produkt seiner beiden Radien des In- und Umkreises, - die sog. Wehrle-Zahl des Dreiecks-, ist? Oder wissen Sie, dass im rechtwinkligen Dreieck der Inkreis-Durchmesser gleich der um die grösste Seite (auch Hypotenuse genannt) verminderte Summe der kleineren Seiten (auch Katheten genannt) ist, dass die Summe der am rechten Winkel anliegenden Seiten gleich der Summe der Durchmesser ist?
Und dass das halbe Produkt dieser zwei Seiten, -die Dreiecksfläche also-, gleich der Summe der Wehrle-Zahl und dem vierten Teil der Wehrle-Zahl der Differenzen ist: A = w + Â¼w*. Dieser letztere 'Differenzen-Wehrle' ist das Quadrat des Durchmessers des Inkreises!
Kennen Sie das kleinste, diskrete gleichschenklige Dreieck, oder das kleinste, nicht-rechtwinklige, rationale Dreieck, das aus nur natürlichen Seitenlängen besteht?
Wissen Sie, welche Vierecke einen In- und Umkreis haben, oder kennen Sie deren doppelte Radienprodukte? Kennen sie diskrete Kreisvierecke, diskrete Sehnenvierecke ohne Inkreis gar?
Wie heisst der dreidimensionale Satz des Pythagoras, oder wissen Sie, welche rechtwinklige Pyramide mit ganzzahligen Katheten den Inkugelradius r=1 hat?
Wissen Sie, dass der Inkugelmittelpunkt rechtwinkliger Tetraeder
Mi = (r; r; r) mit
r = abc / [(ab+ac+bc)+?(aÂ²bÂ²+aÂ²cÂ²+bÂ²cÂ²)] ist;
und das Umkugelzentrum
Mu = (a/2; b/2; c/2) mit Radius R = Âö?(aÂ²+bÂ²+cÂ²) ist?
Und was gilt für ...
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IntroductionAlmost all human lives and all locations on earth are affected by the presence of chemical substances. However, besides its benefits, a substance usually also has a negative effect on the environmentand human health. If a substance escapes into the environment, it must not exceed the concentration that leads to adverse effects. A concentration at a target is called exposure (TRAPP AND MATTHIES 1998). Hence, the environmental hazard can be the product of (environmental) exposure and (eco-)toxicological response. If the probability for the occurrence of damage is included, the term risk is used (risk: damage multiplied by the probability of occurrence). It is a common procedure to use the quotient of the Predicted Environmental Concentration (PEC) and the Predicted No-Effect Concentration (PNEC) as a measure for the risk of a substance. Basic principles on the risk assessment of substances are given by LEEUWEN AND VAN HERMENS (1995). Obviously, there are two different aspects defining risk: the toxicological properties of a substance andthe predicted environmental concentrations. This work focuses on the latter aspect. Numerous works have already dealt with the methods to estimate environmental concentrations of substances. In 1979, D. Mackay presented his concept of an ideal sample world, the Unit World, and simultaneously derived models simulating the behaviour of organic substances released into the environment (some exemplary publications are MACKAY 1979, MACKAY AND PATERSON 1981,MACKAY ET AL. 1985, MACKAY ET AL. 1992). The European Centre for Ecotoxicology and Toxicology of Chemicals (ECETOC) discussed different modelling approaches for the estimation of environmentalconcentrations (ECETOC 1992) and concluded that multimedia models are of particular interest and that 'environmental fate and exposure models can be powerful tools for evaluating the potential hazard of chemicals released in the environment'. In the European Union, both new substances and previously notified ones (so-called 'existing' chemicales) are subject to a risk assessment.The basic principles for this procedure are laid down in the Commission Directive 93/67/EEC and the Commission Regulation (EC) No. 1488/94. But these do not contain the technical detailsconcerning hazard assessment, dose (concentration) - response (effect) assessment, exposure estimation and risk characterisation for human health and the environment. [...]...
Seminar paper from the year 2002 in the subject Statistics, grade: 5 pkt (1-6: 1), Helsinki School of Economics (Haaga Instituuti), - entries in the bibliography, language: English, abstract: This paper will tell about the customer satisfaction in a café downtown Helsinki. The reason why we chose this particular research, started out with personal experience, or better the wish for improvement. Since we are interested in marketing we started approaching our own wishes in a marketer's way, long before we were instructed to conduct this project. So we asked ourselves how we would do it better. But to get anywhere close to an answer, one needs to know about what other customers think, with what they might agree or disagree. So our second question was: how do people see it? Now, as we were assigned to this project, it seemed rather interesting to find out, if our own wishes first of all would be of significance compared to other customers. And secondly, to learn about how simple or difficult it would be for an owner or marketer of such an establishment, to find out about the needs for improvement from the customers point of view. The problem setting of this customer satisfaction survey is divided in four parts: The management needs to know whether the personnel are considered customer-service oriented. Further, how customers think about the place itself, the pricing and the products offered. The background of this work is the idea of how to obtain or improve a market position. In order to do so, it is vital to know about the market itself and the customers and their wishes. In this particular field, everything stands and falls with customer satisfaction. Therefore it is desirable to gain knowledge about the customers and their point of view. Although this is only a part of the marketing mix, which has to be considered for a healthy business, next to the knowledge about competition, in this field it might be the most important one.
Seminar paper from the year 2003 in the subject Statistics, grade: 2.0 (B), European University Viadrina Frankfurt (Oder), language: English, abstract: Today the financial market becomes more complex and includes more competition. Reasons are trends like globalization, liberalization and lower-cost trading mechanism. The market microstructure research has the aim of an efficient market. It is focused on the structure of the financial market. The investigation becomes possible through the availability of high- frequency data. Those data exist especially in the United States and like that most of the research focuses this market. To explain the phenomena, which have been found adequate, models that fit the characteristics of high- frequency data have to be developed. The research is important to understand actions on the market as well as develop new efficient mechanism. One part of the market microstructure field is the bid-ask spread. It will be focus of this paper. In the first two parts it will be discussed theoretically. In the last part one model will be empirically analyzed and tested on its usefulness and validity. The second part of this paper explains the basic elements surrounding the research of bid-ask spread. Those are the financial market, market microstructure as well as high-frequency data. In the following part the bid-ask spread itself, approaches, researches and models focussing the spread will be discussed. The model of Roll (1984) will be explained in detail. The last part will be the empirical analysis of the model of Roll. It is analyzed with data from the NASDAQ.
Diploma Thesis from the year 2003 in the subject Statistics, grade: 1,0, University of Cologne (Seminar für Wirtschafts- und Sozialstatistik), 86 entries in the bibliography, language: English, abstract: Finance experts and statisticians still have considerable difficulties to understand extremal movements in stock prices. Basically, there are two approaches to shed some light on this question: 1. Tail inference based on full parametric assumptions 2. 'Letting the tails speak for themselves' This paper discusses both approaches, the stable Paretian distribution serving as a conceptual framework for the analysis.
Diploma Thesis from the year 2004 in the subject Mathematics - Applied Mathematics, grade: 1.3, Brandenburg Technical University Cottbus (Institut für Mathematik), 60 entries in the bibliography, language: English, abstract: Since Karl Benz built the first motor driven vehicle in 1886 a new industry was born which is still one of the most important and influential sectors of economy. His 0.9 horse-power engine only reached 15 kilometers per hour but gave the starting shot for more and more inventions and a rapidly growing automobile industry. First cars where used for racing only but during the first period of the twentieth century it established its position as a new and individual means of transport. In 1936 the first Volkswagen called Beetle, a vehicle for the masses, was developed by Ferdinand Porsche and has been produced for 67 years. The last Beetle was built in the summer of 2003 in Mexico. Worldwide 100.000 new cars are being produced every day and in 2010 there will be over 1 billion private cars all over the world. This development requires perfectly organized and well optimized production processes and still calls for new inventions and improvements. The production process in automobile manufacturing splits into three major stages. The moulding and welding of the car body in the body shop, the painting of the welded car bodies in the paint shop and the customization of the painted car bodies in the assembly shop. The work presented here deals with the last two stages. The production plant contains a system of production lines which sometimes split into parallel lines and reunite to a non-parallel line. Each of these lines can require some set of restrictions on the cars sequenced on that line. In the paint shop the cars are being painted on a line by a robot which through jets sprays the color onto the surface of the car body. Whenever the next car on the line has the same color the jets can be re-used. Otherwise they have to be cleaned which requires time and causes pollution. Therefore, minimizing the color changes that occur in the paint shop can save non negligible costs. In the assembly shop separate teams install different options into the cars such as sun-roof, air-conditioning, central locking system etc. Therefore the order of the cars on the line has to fulfill some restrictions considering that each option requires a fixed time for installation and resources which have to be available at the time of installation....
Diploma Thesis from the year 2006 in the subject Statistics, grade: 1,4, University of Applied Sciences Kufstein Tirol, 45 entries in the bibliography, language: English, abstract: The discussion about the existence of theHot Hand,a widespread belief that success breeds success, has been unsettled since it was initially mentioned in 1985. The authors of the study originally concluded that any observed streakiness was simply a misperception and that the sequences did not significantly differ from the outcome of a randomly generated process. The reasons why the discussion became this extensive may only be assumed. One reason definitely is that the main application areas for theHot Handbelief are sports and the financial market, two areas that a vast number of people get in touch with. A large number of men and women at least occasionally watch sports on television or attend a live game. More and more people also get involved in actively managing their savings by a growing number of different financial instruments like shares and bonds. Another reason for the amount of existing analyses and studies about topics related to theHot Handeffect is that arguments for and against theHot Handrange from statistical problems, e.g. which statistical test to apply, to deep psychological issues like the perception of random processes. Knowing whether the believers in theHot Handor the sceptics are right is essential to everybody because in many everyday situations, people act according to which of the two groups they belong to. This is especially important in situations where money is at stake. Therefore, the aim of this paper is to answer the following research question: Is the belief in theHot Handeffect a fallacy and what is its impact on human behaviour in an economic environment as the betting market? While a lot of research has been done in the search for the existence of theHot Hand,it has rarely been attempted to take into account a wide range of suggestions concerning the way of examination and inference from data, and to conduct an empirical study that satisfies most of the criticism. Thus, the analysis of data will be conducted in a way that avoids errors and uncertainties that have been encountered in more than 20 years of discussion about the Hot Hand effect. The observations of gambling behaviour in the sports betting market shall indicate how a belief in theHot Handinfluences economic decisions. In the betting market, as well as in every other economic environment it is essential that decisions are made on a rational basis....
Diploma Thesis from the year 2006 in the subject Statistics, grade: 1.3, European University Viadrina Frankfurt (Oder) (Wirtschaftswissenschaftliche Fakultät), 43 entries in the bibliography, language: English, abstract: Regression Analysis is an important statistical tool for many applications. The most
frequently used approach to Regression Analysis is the method of Ordinary Least Squares.
But this method is vulnerable to outliers; even a single outlier can spoil the estimation
completely. How can this vulnerability be described by theoretical concepts and are there alternatives? This thesis gives an overview over concepts and alternative approaches.
The three fundamental approaches to Robustness (qualitative-, infinitesimal- and quantitative Robustness) are introduced in this thesis and are applied to different estimators. The estimators under study are measures of location, scale and regression. The Robustness approaches are important for the theoretical judgement of certain estimators but as well for the development of alternatives to classical estimators. This thesis focuses on the (Robustness-) performance of estimators if outliers occur within the data set. Measures of location and scale provide necessary steppingstones into the topic of Regression Analysis. In particular the median and trimming approaches are found to produce very robust results.
These results are used in Regression Analysis to find alternatives to the method of Ordinary Least Squares. Its vulnerability can be overcome by applying the methods of Least Median of Squares or Least Trimmed Squares. Different outlier diagnostic tools are introduced to improve the poor efficiency of these Regression Techniques. Furthermore, this thesis delivers a simulation of some Regression Techniques on different situations in Regression Analysis.
This simulation focuses in particular on changes in regression estimates if outliers occur in the data.
Theoretically derived results as well as the results of the simulation lead to the
recommendation of the method of Reweighted Least Squares. Applying this method
frequently on problems of Regression Analysis provides outlier resistant and efficient
Seminar paper from the year 2006 in the subject Statistics, grade: 1,7, University of Paderborn (Fakultät für Wirtschaftswissenschaften), course: Statistisch-okonometrisches Praktikum, 21 entries in the bibliography, language: English, abstract: Outdated education models, technological advances and increasing enrollment of students have led to involve Web-based education in some economics classes of various universities. The options range from Web-based applications in traditional classes to complete online courses without any face-to-face contact. Two facts are stressed with special regard to statistics and econometrics classes in this paper. These are, firstly, the problems tutors have to teach students the essential contents of the courses (this refers also to many aspects of economic undergraduate courses) and, secondly, the problems tutors face to find the right way to teach by using the possibilities the technological advance offers to education methods. Given that only a few written sources on teaching statistics and econometrics exist and having in mind that econometrics is part of the economics education, articles regarding research in overall economic education will also be analyzed in this paper. Due to the latest articles the aim of the paper is to compare and combine the findings of different studies carried out in order to find the best way of teaching econometrics and statistics. After a short introduction the main part of the paper gives an overview of the conventional way of teaching statistics and econometrics and indicates the problems involved. This is followed by a section on new initiatives in the education of econometrics and statistics. Thereinafter, three forms of teaching - traditional (off-line) instruction, hybrid Internet-based instruction and Internet-based instruction - will be dealt with to see in how far the proposed initiatives already have been applied on the subjects. In the last part a conclusion is drawn to summarize the main findings and to show the direction of future teaching in this field. As it already became clear in the headline subject matter of this paper is the aspect of teaching and not learning; the work deals exclusively with the perspectives of the teaching institutions and not of those on the receiving end of the instruction....
Seminar paper from the year 2007 in the subject Statistics, grade: sehr gut (6.0), University of Fribourg (Departement für Mathematik), course: Freies Seminar des Departements für Mathematik, 9 entries in the bibliography, language: English, abstract: There are many settings in which the outcome we seek to model is a discrete choice among a set of alternatives. Almost non of these models can be consistently estimated with linear regression methods. Other methods have been devised for these cases such as the logistic regression model. This paper presents some basic principles of the logistic regression model and explains its estimation using the maximum likelihood method. An econometric application at the end highlights the importance of the theoretical issues discussed.