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Arnold Sommerfeld in Einigen Zitaten Von Ihm Und Über Ihn1
K.-P. Dostal, Arnold Sommerfeld in einigen Zitaten von ihm und über ihn Seite 1 Karl-Peter Dostal, Arnold Sommerfeld in einigen Zitaten von ihm und über ihn1 Kurze biographische Bemerkungen Arnold Sommerfeld [* 5. Dezember 1868 in Königsberg, † 26. April 1951 in München] zählt neben Max Planck, Albert Einstein und Niels Bohr zu den Begründern der modernen theoretischen Physik. Durch die Ausarbeitung der Bohrschen Atomtheorie, als Lehrbuchautor (Atombau und Spektrallinien, Vorlesungen über theoretische Physik) und durch seine „Schule“ (zu der etwa die Nobelpreisträger Peter Debye, Wolfgang Pauli, Werner Heisenberg und Hans Bethe gehören) sorgte Sommerfeld wie kein anderer für die Verbreitung der modernen Physik.2 Je nach Auswahl könnte Sommerfeld [aber] nicht nur als theoretischer Physiker, sondern auch als Mathematiker, Techniker oder Wissenschaftsjournalist porträtiert werden.3 Als Schüler der Mathematiker Ferdinand von Lindemann, Adolf Hurwitz, David Hilbert und Felix Klein hatte sich Sommerfeld zunächst vor allem der Mathematik zugewandt (seine erste Professur: 1897 - 1900 für Mathematik an der Bergakademie Clausthal). Als Professor an der TH Aachen von 1900 - 1906 gewann er zunehmendes Interesse an der Technik. 1906 erhielt er den seit Jahren verwaisten Lehrstuhl für theoretische Physik in München, an dem er mit wenigen Unterbrechungen noch bis 1940 (und dann wieder ab 19464) unterrichtete. Im Gegensatz zur etablierten Experimen- talphysik war die theoretische Physik anfangs des 20. Jh. noch eine junge Disziplin. Sie wurde nun zu -
Keynote Address : the International Society for Photogrammetry And
Keynote Address The International Society for Photogrammetry and Remote Sensing-75 Years Old, or 75 Years Young GOTTFRIEDKONECNY, Uniuersity c$ Hnnnooer, F.R. Ger~nany ADIES AND GENTLEMEN.I am grateful for the dis- sphere, and he determined its average radius within L tinction of having been asked to prepare the 1% of accuracy. He thus became the first geodesist. keynote speech for the 1985 Annual Meeting of the Mathematical thought and survey practices American Society of Photogrammetry and the spread from Mesopotamia and the Nile to the areas American Congress on Surveying and Mapping. of Greek and Roman culture. Roman land surveys As a photogrammetrist, I am naturally concerned were a highly developed technique. Its practice was about matters dear to me and therefore I have interrupted by the fall of the Roman Empire be- chosen as the title of my talk "The International cause of invasion from the North and the introduc- Society for Photogrammetry and Remote Sensing- tion of Christianity with its upcoming mysticism. 75 years old, or 75 years young." I have done this Therefore the oriental cultures then became re- for two reasons. Last year I became successor to sponsible for the further development of our disci- Fred Doyle, who ably represented the American plines. In the Arab world, astronomy and navigation Society of Photogrammetry as President of ISPRS became highly developed. Cartography, as a de- for four years from 1980 to 1984. Secondly, our or- scription of the earth's features on its surface, flour- ganization celebrates this year its 75th Anniversary. ished in China, long before this was the case in We photogrammetrists are officially in our third Europe. -
The Evolution of Rock Glacier Monitoring Using Terres
ZOBODAT - www.zobodat.at Zoologisch-Botanische Datenbank/Zoological-Botanical Database Digitale Literatur/Digital Literature Zeitschrift/Journal: Austrian Journal of Earth Sciences Jahr/Year: 2012 Band/Volume: 105_2 Autor(en)/Author(s): Kaufmann Viktor Artikel/Article: The evolution of rock glacier monitoring using terrestrial photogrammetry: the example of Äußeres Hochebenkar rock glacier (Austria) 63-77 © Österreichische Geologische Gesellschaft/Austria; download unter www.geol-ges.at/ und www.biologiezentrum.at Austrian Journal of Earth Sciences Volume 105/2 Vienna 2012 The evolution of rock glacier monitoring using terres- trial photogrammetry: the example of Äußeres Hoch- ebenkar rock glacier (Austria)____________________________ Viktor KAUFMANN KEYWORDS Terrestrial photogrammetry Deformation measurement Graz University of Technology, Institute of Remote Sensing and Photogrammetry, Äußeres Hochebenkar Steyrergasse 30, A-8010 Graz, Austria; Rock glacier Ötztal Alps [email protected] Monitoring Abstract Rock glaciers are creep phenomena of mountain permafrost and have been the subject of research for over 100 years. Rock gla- ciers are lobate or tongue-shaped bodies of perennially frozen ice-rich unconsolidated material. Active rock glaciers creep down- slope by force of gravity. Mean annual flow velocities are in the order of a few centimeters to several meters per year. Rock glacier surfaces typically show furrows and ridges which are visible expressions of active flow and cumulative deformation. The kinematics of rock glaciers can be determined by different measurement techniques. Terrestrial (ground-based or close-range) photogrammetry was one of the first successful methods for detecting and quantifying surface changes in rock glaciers. Flow velocity was a typical parameter derived from this. The 2D or even 3D kinematics of the rock glacier surface is needed for rheological models. -
50 Mathematical Ideas You Really Need to Know
50 mathematical ideas you really need to know Tony Crilly 2 Contents Introduction 01 Zero 02 Number systems 03 Fractions 04 Squares and square roots 05 π 06 e 07 Infinity 08 Imaginary numbers 09 Primes 10 Perfect numbers 11 Fibonacci numbers 12 Golden rectangles 13 Pascal’s triangle 14 Algebra 15 Euclid’s algorithm 16 Logic 17 Proof 3 18 Sets 19 Calculus 20 Constructions 21 Triangles 22 Curves 23 Topology 24 Dimension 25 Fractals 26 Chaos 27 The parallel postulate 28 Discrete geometry 29 Graphs 30 The four-colour problem 31 Probability 32 Bayes’s theory 33 The birthday problem 34 Distributions 35 The normal curve 36 Connecting data 37 Genetics 38 Groups 4 39 Matrices 40 Codes 41 Advanced counting 42 Magic squares 43 Latin squares 44 Money mathematics 45 The diet problem 46 The travelling salesperson 47 Game theory 48 Relativity 49 Fermat’s last theorem 50 The Riemann hypothesis Glossary Index 5 Introduction Mathematics is a vast subject and no one can possibly know it all. What one can do is explore and find an individual pathway. The possibilities open to us here will lead to other times and different cultures and to ideas that have intrigued mathematicians for centuries. Mathematics is both ancient and modern and is built up from widespread cultural and political influences. From India and Arabia we derive our modern numbering system but it is one tempered with historical barnacles. The ‘base 60’ of the Babylonians of two or three millennia BC shows up in our own culture – we have 60 seconds in a minute and 60 minutes in an hour; a right angle is still 90 degrees and not 100 grads as revolutionary France adopted in a first move towards decimalization. -
The Ubiquity of Phi in Human Culture & the Natural World
John Carroll University Carroll Collected Masters Essays Master's Theses and Essays 2020 THE UBIQUITY OF PHI IN HUMAN CULTURE & THE NATURAL WORLD Jennifer Bressler Follow this and additional works at: https://collected.jcu.edu/mastersessays Part of the Mathematics Commons THE UBIQUITY OF PHI IN HUMAN CULTURE & THE NATURAL WORLD An Essay Submitted to the Office of Graduate Studies College of Arts & Sciences of John Carroll University In Partial Fulfillment of the Requirements For the Degree of Master of Arts By Jennifer L. Bressler 2020 Table of Contents I. Introduction…………………………………………………………………………. 2 II. The Early Greeks…………………………………………………………………… 4 III. Algebraic Properties of the Golden Ratio………………………………………….. 11 IV. The Golden Rectangle…………………………………………………….……….. 20 V. Architecture & Design……………………………………………………………… 22 VI. Art………………………………………………………………………………….. 30 VII. Music……………………………………………………………………………….. 38 VIII. The Natural World………………………………………………………………….. 43 IX. Human Anatomy…………………………………………………………………… 52 X. Geometry…………………………………………………………………………… 56 XI. Conclusion……………………………………………………………………………65 1 I. INTRODUCTION What do rabbit breeding, tornadoes, the Chambered Nautilus, a pentagram, the rhythm of a heartbeat, apple seeds, the shape of a credit card, a pinecone, the human ear, DaVinci’s Last Supper, the structure of DNA, a light switch cover, and the structure of galaxies all have in common? Each relates to an extraordinary ratio that is highly efficient in nature, profoundly attractive to the human eye, and some claim, even divinely inspired. This special ratio is referred to as the “Golden Ratio” and is also known as the divine proportion, golden section, and golden mean. The Golden Ratio has a constant numeric value called “phi” (pronounced “FEE,” or “FI”) which is thought to be the most beautiful and astounding of all numbers. -
The Story of Π
And of its friend e Wheels Colin Adams in the video “The Great π/e Debate” mentions the wheel “arguably the greatest invention of all times,” as an example of how π appeared already in prehistoric times. But square wheels are possible. If the surface of the earth were not flat but rilled in a very special way. For this to work each little rill (arc) has to have the shape of a hyperbolic cosine, a so called cosh curve. By definition, the hyperbolic cosine is ex e x coshx 2 Another appearance by e. As we shall see, π and e are closely related, even though the relationship is far from being well understood. NOTATION In these notes r = radius of a circle d= its diameter, d = 2r C = its circumference (perimeter) A = its area When did people discover that C A ? d r 2 The Story Begins… There are records (skeletons, skulls and other such cheerful remains) that indicate that the human species existed as long as 300,000 years ago. Most of this was prehistory. The following graph compares history to prehistory: The green part is prehistory, the red history. The vertical black line indicates the time to which some of the oldest records of human activity were dated. Ahmes, the scribe Scribes were a very important class in ancient Egypt. The picture shows a statue of a scribe. Ahmes, the scribe of the Rhind papyrus (c. 1650 BC) may have looked much like this guy. In the Rhind papyrus, Ahmes writes: Cut off 1/9 of a diameter and construct a square upon the remainder; this has the same area of the circle. -
Carathéodory, Constantin
Constantin Carathéodory (September 13, 1873–February 2, 1950) by HEINZ KLAUS STRICK, Germany In 1994, the Greek postal service issued a double stamp devoted to two Greek mathematicians: CONSTANTIN CARATHÉODORY und THALES OF MILETUS. The name of the Greek scientist from antiquity (THALES was active in the sixth century B.C.E.) may sound familiar, for his name is forever linked with a famous theorem in geometry. But who was CARATHÉODORY? CONSTANTIN CARATHÉODORY was born in 1873 in Berlin, the son of a diplomat of Greek origin serving in the Ottoman Empire’s embassy in that city. At that time, a large portion of what is today the country of Greece was under the dominion of the Ottomans. After a temporary sojourn in Constantinople, the family moved to Brussels, where the father assumed the position of ambassador. After the death of CONSTANTIN’s mother, his grandmother took over the care of the boy and his sister, LOULIA. A German servant girl was engaged so that in addition to French and their native Greek, the children would become fluent in German. CONSTANTIN attended primary school in Brussels, but he also spent part of the year in Berlin. The family spent winters on the Italian Riviera. It was during his secondary-school years that his interest in mathematics developed. He twice won the first prize of the Concours généreaux, the Belgian national mathematics competition for advanced secondary-school pupils. CONSTANTIN CARATHÉODORY completed engineering studies at the École Militaire de Belgique and then worked as a civil engineer on Lesbos until the outbreak of the Greco-Turkish War in 1896. -
Indiana Pols Forced to Eat Humble Pi: the Curious History of an Irrational Number
The History of the Number \π" On Lambert's Proof of the Irrationality of π House Bill #246: the \Indiana Pi Bill" Indiana Pols Forced to Eat Humble Pi: The Curious History of an Irrational Number Edray Herber Goins D´ıade Pi Instituto de Matem´aticas Pontificia Universidad Cat´olica de Valpara´ıso(PUCV) Department of Mathematics Purdue University March 14, 2018 PUCV Pi Day Celebration Indiana Pols Forced to Eat Humble Pi The History of the Number \π" On Lambert's Proof of the Irrationality of π House Bill #246: the \Indiana Pi Bill" Outline of Talk 1 The History of the Number \π" Introduction What is \π"? Approximations to π 2 On Lambert's Proof of the Irrationality of π Bessel Functions Laczkovich's Theorem Lambert's Theorem 3 House Bill #246: the \Indiana Pi Bill" Lindemann's Theorem Squaring the Circle A Bill for an Act Introducing a New Mathematical Truth PUCV Pi Day Celebration Indiana Pols Forced to Eat Humble Pi The History of the Number \π" On Lambert's Proof of the Irrationality of π House Bill #246: the \Indiana Pi Bill" Abstract In 1897, Indiana physician Edwin J. Goodwin believed he had discovered a way to square the circle, and proposed a bill to Indiana Representative Taylor I. Record which would secure Indiana's the claim to fame for his discovery. About the time the debate about the bill concluded, Purdue University professor Clarence A. Waldo serendipitously came across the claimed discovery, and pointed out its mathematical impossibility to the lawmakers. It had only be shown just 15 years before, by the German mathematician Ferdinand von Lindemann, that it was impossible to square the circle because π is an irrational number. -
The Transcendence of Pi Has Been Known for About a Century
Results in Mathematics Contents 117 J. Bair/F. Jongmans Volume 7/No. 2/1984 Some remarks about recent result Pages 117-250 on the asymptotic cone 119 W. Beekmann/S. C Chang On the structure of summability fields 130 P. Bundschuh/1. Shiokawa A measure for the linear inde- pendence of certain numbers 145 P. L. Butzer/R. J. Nessel/E. L. Stark Eduard Helly (1884-1943) in memoriam 154 A. S. Cavaretta jr./H. P. Dikshit/ A. Sharma An extension of a theorem of Walsh 164 R. Fritsch The transcendence of ir has been known for about a century-but who was the man who discovered it? 184 Y. Hirano/H. Tominaga On simple ring extensions gener- ated by two idempotents 190 J. Joussen Eine Bemerkung zu einem Satz von Sylvester 192 H. Karzel/C. J. Maxson Fibered groups with non-trivial centers 209 H. Meiert G. Rosenberger Hecke-Integrale mit rationalen periodischen Funktionen und Dirichlet-Reihen mit Funk• tionalgleichung 234 G. Schiffels/M. Stemel Einbettung von topologischen Ringen in Quotientenringe Short Communications on Mathematical Dissertations 249 G. BaszenskijW. Schempp TAXT Konvergenzbeschleunigung von Orthogonal-Doppelreihen The Journal Copyright RESULTS IN MATHEMATICS It is a fundamental condition of publication that submitted RESULTATE DER MATHEMATIK manuscripts have not been published, nor will be simulta- publishes mainly research papers in all Heids of pure and applied mathematics. In addition, it publishes summaries of any mathematical neously submitted or published elsewhere. By submitting a field and surveys of any mathematical subject provided they are de- manuscript, the authors agree that the Copyright for their signed to advance some recent mathematical development. -
Science Discussion Started: 21 August 2017 C Author(S) 2017
Discussions Earth Syst. Sci. Data Discuss., https://doi.org/10.5194/essd-2017-85 Earth System Manuscript under review for journal Earth Syst. Sci. Data Science Discussion started: 21 August 2017 c Author(s) 2017. CC BY 4.0 License. Open Access Open Data The Rofental: a high Alpine research basin (1890 m – 3770 m a.s.l.) in the Ötztal Alps (Austria) with over 150 years of hydro- meteorological and glaciological observations Ulrich Strasser1, Thomas Marke1, Ludwig Braun3, Heidi Escher-Vetter3, Irmgard Juen2, Michael Kuhn2, 2 3 2 4 1 5 Fabien Maussion , Christoph Mayer , Lindsey Nicholson , Klaus Niedertscheider , Rudolf Sailer , Johann Stötter1, Markus Weber5, Georg Kaser2 1Institute of Geography, University of Innsbruck, Innsbruck, 6020, Austria 2Institute of Atmospheric and Cryospheric Sciences, University of Innsbruck, Innsbruck, 6020, Austria 3Geodesy and Glaciology, Bavarian Academy of Sciences and Humanities, Munich, 80539, Germany 10 4Hydrographic Service of Tyrol, Innsbruck, 6020, Austria 5Photogrammetry and Remote Sensing, Technical University of Munich, Munich, 80333, Germany Correspondence to: Ulrich Strasser ([email protected]) Abstract. A comprehensive hydrometeorological and glaciological data set is presented, originating from a multitude of 15 recordings at several intensively operated research sites in the Rofental (1891 – 3772 m a.s.l., Ötztal Alps, Austria). The data sets are spanning a period of 150 years and hence represent a unique, worldwide unprecedented pool of high mountain observations. Their collection has originally been initiated to support the scientific investigation of the glaciers Hintereis-, Kesselwand- and Vernagtferner. Later, additional measurements of meteorological and hydrological variables have been undertaken; data now comprise records of temperature, relative humidity, short- and longwave radiation, wind speed and 20 direction, air pressure, precipitation and water levels. -
Abstracts of Talks Presented to the Indiana Section of the Maa
ABSTRACTS OF TALKS PRESENTED TO THE INDIANA SECTION OF THE MAA 1. Introduction The Spring 2016 meeting of the Indiana Section of the Mathematical Association of America is at Franklin College, March 18–19. The abstracts appearing here are based on text electronically submitted by the presenters. Contributed talks are listed in alphabetical order by presenter. 2. Invited Talks Presenter: Edray Goins, Purdue University West Lafayette Indiana Pols Forced to Eat Humble Pi — The Curious History of an Irrational Number In 1897, Indiana physician Edwin J. Goodwin believed he had discovered a way to square the circle, and proposed a bill to Indiana Representative Taylor I. Record which would secure Indiana’s claim to fame for his discovery. About the time the debate about the bill concluded, Purdue University professor C. A. Waldo serendipitously came across the claimed discovery, and pointed out its mathematical impossibility to the lawmakers. It had only been shown just 15 years before, by the German mathematician Ferdinand von Lindemann, that it was impossible to square the circle because π is a transcendental number. This fodder became ignominiously known as the “Indiana Pi Bill” as Goodwin’s result would force π =3.2. In this talk, we review this humorous history of the irrationality of π.Weintro- duce a method to compute its digits, present Lindemann’s proof of its irrationality (following a simplification by Mikl´os Laczkovich), discuss the relationship with the Hermite-Lindemann-Weierstrass theorem, and explain how Edwin J. Goodwin came to his erroneous conclusion in the first place. Presenter: Colin McKinney, Wabash College Hoosiers, The MAA, and the Indiana Section This talk will chronicle the role some Hoosiers played in the founding of the MAA. -
The Evolution of Rock Glacier Monitoring Using Terres- Trial Photogrammetry: the Example of Äußeres Hoch- Ebenkar Rock Glacier (Austria)______
Austrian Journal of Earth Sciences Volume 105/2 Vienna 2012 The evolution of rock glacier monitoring using terres- trial photogrammetry: the example of Äußeres Hoch- ebenkar rock glacier (Austria)____________________________ Viktor KAUFMANN KEYWORDS Terrestrial photogrammetry Deformation measurement Graz University of Technology, Institute of Remote Sensing and Photogrammetry, Äußeres Hochebenkar Steyrergasse 30, A-8010 Graz, Austria; Rock glacier Ötztal Alps [email protected] Monitoring Abstract Rock glaciers are creep phenomena of mountain permafrost and have been the subject of research for over 100 years. Rock gla- ciers are lobate or tongue-shaped bodies of perennially frozen ice-rich unconsolidated material. Active rock glaciers creep down- slope by force of gravity. Mean annual flow velocities are in the order of a few centimeters to several meters per year. Rock glacier surfaces typically show furrows and ridges which are visible expressions of active flow and cumulative deformation. The kinematics of rock glaciers can be determined by different measurement techniques. Terrestrial (ground-based or close-range) photogrammetry was one of the first successful methods for detecting and quantifying surface changes in rock glaciers. Flow velocity was a typical parameter derived from this. The 2D or even 3D kinematics of the rock glacier surface is needed for rheological models. In recent years, active rock glaciers have also become the focus of climate change research. Atmospheric warming is supposed to influence flow/creep velocity of rock glaciers, which can thus be seen as indicators of environmental change in mountainous regions. Melting of the subsurface ice causes surface lowering, which in the worst case may lead to active landsliding and even a total collapse of the rock glacier surface.