From Elasticity to Electromagnetism: Beyond the Mirror
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HOMOTECIA Nº 6-15 Junio 2017
HOMOTECIA Nº 6 – Año 15 Martes, 1º de Junio de 2017 1 Entre las expectativas futuras que se tienen sobre un docente en formación, está el considerar como indicativo de que logrará realizarse como tal, cuando evidencia confianza en lo que hace, cuando cree en sí mismo y no deja que su tiempo transcurra sin pro pósitos y sin significado. Estos son los principios que deberán pautar el ejercicio de su magisterio si aspira tener éxito en su labor, lo cual mostrará mediante su afán por dar lo bueno dentro de sí, por hacer lo mejor posible, por comprometerse con el porvenir de quienes confiadamente pondrán en sus manos la misión de enseñarles. Pero la responsabilidad implícita en este proceso lo debería llevar a considerar seriamente algunos GIACINTO MORERA (1856 – 1907 ) aspectos. Obtener una acreditación para enseñar no es un pergamino para exhib ir con petulancia ante familiares y Nació el 18 de julio de 1856 en Novara, y murió el 8 de febrero de 1907, en Turín; amistades. En otras palabras, viviendo en el mundo educativo, es ambas localidades en Italia. asumir que se produjo un cambio significativo en la manera de Matemático que hizo contribuciones a la dinámica. participar en este: pasó de ser guiado para ahora guiar. No es que no necesite que se le orie nte como profesional de la docencia, esto es algo que sucederá obligatoriamente a nivel organizacional, Giacinto Morera , hijo de un acaudalado hombre de pero el hecho es que adquirirá una responsabilidad mucho mayor negocios, se graduó en ingeniería y matemáticas en la porque así como sus preceptores universitarios tuvieron el compromiso de formarlo y const ruirlo cultural y Universidad de Turín, Italia, habiendo asistido a los académicamente, él tendrá el mismo compromiso de hacerlo con cursos por Enrico D'Ovidio, Angelo Genocchi y sus discípulos, sea cual sea el nivel docente donde se desempeñe. -
Real Proofs of Complex Theorems (And Vice Versa)
REAL PROOFS OF COMPLEX THEOREMS (AND VICE VERSA) LAWRENCE ZALCMAN Introduction. It has become fashionable recently to argue that real and complex variables should be taught together as a unified curriculum in analysis. Now this is hardly a novel idea, as a quick perusal of Whittaker and Watson's Course of Modern Analysis or either Littlewood's or Titchmarsh's Theory of Functions (not to mention any number of cours d'analyse of the nineteenth or twentieth century) will indicate. And, while some persuasive arguments can be advanced in favor of this approach, it is by no means obvious that the advantages outweigh the disadvantages or, for that matter, that a unified treatment offers any substantial benefit to the student. What is obvious is that the two subjects do interact, and interact substantially, often in a surprising fashion. These points of tangency present an instructor the opportunity to pose (and answer) natural and important questions on basic material by applying real analysis to complex function theory, and vice versa. This article is devoted to several such applications. My own experience in teaching suggests that the subject matter discussed below is particularly well-suited for presentation in a year-long first graduate course in complex analysis. While most of this material is (perhaps by definition) well known to the experts, it is not, unfortunately, a part of the common culture of professional mathematicians. In fact, several of the examples arose in response to questions from friends and colleagues. The mathematics involved is too pretty to be the private preserve of specialists. -
Complex Analysis
8 Complex Representations of Functions “He is not a true man of science who does not bring some sympathy to his studies, and expect to learn something by behavior as well as by application. It is childish to rest in the discovery of mere coincidences, or of partial and extraneous laws. The study of geometry is a petty and idle exercise of the mind, if it is applied to no larger system than the starry one. Mathematics should be mixed not only with physics but with ethics; that is mixed mathematics. The fact which interests us most is the life of the naturalist. The purest science is still biographical.” Henry David Thoreau (1817-1862) 8.1 Complex Representations of Waves We have seen that we can determine the frequency content of a function f (t) defined on an interval [0, T] by looking for the Fourier coefficients in the Fourier series expansion ¥ a0 2pnt 2pnt f (t) = + ∑ an cos + bn sin . 2 n=1 T T The coefficients take forms like 2 Z T 2pnt an = f (t) cos dt. T 0 T However, trigonometric functions can be written in a complex exponen- tial form. Using Euler’s formula, which was obtained using the Maclaurin expansion of ex in Example A.36, eiq = cos q + i sin q, the complex conjugate is found by replacing i with −i to obtain e−iq = cos q − i sin q. Adding these expressions, we have 2 cos q = eiq + e−iq. Subtracting the exponentials leads to an expression for the sine function. Thus, we have the important result that sines and cosines can be written as complex exponentials: 286 partial differential equations eiq + e−iq cos q = , 2 eiq − e−iq sin q = .( 8.1) 2i So, we can write 2pnt 1 2pint − 2pint cos = (e T + e T ). -
4 Complex Analysis
4 Complex Analysis “He is not a true man of science who does not bring some sympathy to his studies, and expect to learn something by behavior as well as by application. It is childish to rest in the discovery of mere coincidences, or of partial and extraneous laws. The study of geometry is a petty and idle exercise of the mind, if it is applied to no larger system than the starry one. Mathematics should be mixed not only with physics but with ethics; that is mixed mathematics. The fact which interests us most is the life of the naturalist. The purest science is still biographical.” Henry David Thoreau (1817 - 1862) We have seen that we can seek the frequency content of a signal f (t) defined on an interval [0, T] by looking for the the Fourier coefficients in the Fourier series expansion In this chapter we introduce complex numbers and complex functions. We a ¥ 2pnt 2pnt will later see that the rich structure of f (t) = 0 + a cos + b sin . 2 ∑ n T n T complex functions will lead to a deeper n=1 understanding of analysis, interesting techniques for computing integrals, and The coefficients can be written as integrals such as a natural way to express analog and dis- crete signals. 2 Z T 2pnt an = f (t) cos dt. T 0 T However, we have also seen that, using Euler’s Formula, trigonometric func- tions can be written in a complex exponential form, 2pnt e2pint/T + e−2pint/T cos = . T 2 We can use these ideas to rewrite the trigonometric Fourier series as a sum over complex exponentials in the form ¥ 2pint/T f (t) = ∑ cne , n=−¥ where the Fourier coefficients now take the form Z T −2pint/T cn = f (t)e dt. -
Sunti Delle Conferenze
Sunti delle Conferenze Analisi complessa a Pisa, 1860-1900 UMBERTO BOTTAZZINI (Università di Milano) Nel 1859 Enrico Betti inaugura gli studi di analisi complessa a Pisa (e di fatto in Italia) pubblicando la traduzione italiana della Inauguraldissertation (1851) di Riemann. L’incontro con il grande matematico conosciuto l’anno prima a Göttingen segna una svolta nella carriera scientifica di Betti, che fa dell’analisi complessa l’oggetto delle sue lezioni e delle sue pubblicazioni (1860/61 e 1862) che incontrano l’approvazione di Riemann, durante il suo soggiorno in Italia. Nella conferenza saranno discussi i contributi all’analisi complessa di Betti, Dini e Bianchi. Ulisse Dini raccolse l’eredità del maestro dapprima in articoli (1870/71, 1871/73, 1881) che suscitano l’interesse della comunità internazionale, e poi in lezioni litografate (1890) che hanno offerto a Luigi Bianchi il modello e il riferimento iniziale per le sue celebri lezioni sulla teoria delle funzioni di variabile complessa in due volumi, apparse prima in versione litografata (1898/99) e poi a stampa in diverse edizioni. Il periodo romano di Luigi Cremona: tra Statica Grafica e Geometria Algebrica, la Biblioteca Nazionale, i Lincei, il Senato ALDO BRIGAGLIA (Università di Palermo) Il periodo romano (1873 – 1903) è considerato il meno produttivo, dal punto di vista scientifico, della vita di Luigi Cremona. Un periodo quasi unicamente dedicato agli aspetti politico – istituzionali della sua attività. Senza voler capovolgere questo giudizio consolidato, anzi sottolineando -
Considerations Based on His Correspondence Rossana Tazzioli
New Perspectives on Beltrami’s Life and Work - Considerations Based on his Correspondence Rossana Tazzioli To cite this version: Rossana Tazzioli. New Perspectives on Beltrami’s Life and Work - Considerations Based on his Correspondence. Mathematicians in Bologna, 1861-1960, 2012, 10.1007/978-3-0348-0227-7_21. hal- 01436965 HAL Id: hal-01436965 https://hal.archives-ouvertes.fr/hal-01436965 Submitted on 22 Jan 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. New Perspectives on Beltrami’s Life and Work – Considerations based on his Correspondence Rossana Tazzioli U.F.R. de Math´ematiques, Laboratoire Paul Painlev´eU.M.R. CNRS 8524 Universit´ede Sciences et Technologie Lille 1 e-mail:rossana.tazzioliatuniv–lille1.fr Abstract Eugenio Beltrami was a prominent figure of 19th century Italian mathematics. He was also involved in the social, cultural and political events of his country. This paper aims at throwing fresh light on some aspects of Beltrami’s life and work by using his personal correspondence. Unfortunately, Beltrami’s Archive has never been found, and only letters by Beltrami - or in a few cases some drafts addressed to him - are available. -
Mathematik Für Physiker
Mathematik f¨urPhysiker III Michael Dreher Fachbereich f¨urMathematik und Statistik Universit¨atKonstanz Studienjahr 2012/13 2 Acknowledgements: These are the lecture notes to a third semester course on Mathematics for Physicists, and the author is indebted to Nicola Wurz, Maria Lindauer, Florian Franz, Philip Lindner, Simon Sch¨uz,Samuel Greiner, Christian Schoder, Pascal Gumbsheimer for remarks which helped to improve the presentation, and to the audience for appropriating this huge amount of knowledge. Some Legalese: This work is licensed under the Creative Commons Attribution { Noncommercial { No Derivative Works 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-nd/3.0/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA. I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea{shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me. Sir Isaac Newton (1642 { 1727) 1 1As quoted in [3]. 4 Contents I Ordinary Differential Equations7 1 Existence and Uniqueness Results9 1.1 An Introductory Example....................................9 1.2 General Considerations...................................... 12 1.3 The Theorem of Picard and Lindelof¨ ............................ 19 1.4 Comparison Principles...................................... 23 2 Special Solution Methods 25 2.1 Equations with Separable Variables............................... 25 2.2 Substitution and Homogeneous Differential Equations.................... 27 2.3 Power Series Expansions (Or How to Determine the Sound of a Drum).......... -
Gli Archivi Di Corrado Segre Presso L'università Di Torino
Gli Archivi di Corrado Segre presso l’Università di Torino LIVIA GIACARDI, ERIKA LUCIANO, CHIARA PIZZARELLI, CLARA SILVIA ROERO Alla scomparsa di Corrado Segre nel Maggio 1924, sua moglie Olga Michelli donò alla Biblioteca Speciale di Matematica della Facoltà di Scienze Matematiche Fisiche e Naturali dell’Università di Torino la sua raccolta di opuscoli, estratti e ritratti1, completando il lascito nel Marzo 1926, con la donazione dei celebri quaranta Quaderni delle sue lezioni e con una collezione di manoscritti e documenti scientifici del marito 2. In occasione delle Celebrazioni per il 150° anniversario della nascita di Segre3, nel 2013 Paola Gario (Università Statale di Milano) consegnò alla medesima Biblioteca Speciale di Matematica trentaquattro tavole di Geometria Proiettiva e Descrittiva, che Segre aveva realizzato nel 1878-79, all’epoca dei suoi studi superiori presso l’Istituto tecnico ‘G. Sommeiller’ di Torino. Gario aveva ricevuto tali tavole dagli eredi di Segre nel 1989. Nel Gennaio 2014, e successivamente nell’ottobre 2015, l’Università di Torino ricevette la donazione di una ricca raccolta di tavole, corrispondenze, necrologi e documenti di differente tipologia, già custoditi ad Ancona dai discendenti di Segre: i pronipoti Silvano e Daniele Fuà4. Si fornisce qui una preliminare ricognizione del complesso di questi fondi d’archivio, posticipandone l’analisi storiografica alla conclusione dell’opera di catalogazione e di schedatura. Gli Archivi Segre possono essere articolati nelle seguenti dieci Serie. UTo-ACS Università di Torino, Archivi di Corrado Segre I. Documenti di Carriera - Corrispondenza istituzionale - Attestati e lettere di Accademie e Società scientifiche - Premi di studio, diplomi e onorificenze II. Documenti di Famiglia - Indirizzario di Segre - Lettere di Segre a O. -
Real Proofs of Complex Theorems (And Vice Versa) Author(S): Lawrence Zalcman Source: the American Mathematical Monthly, Vol
Real Proofs of Complex Theorems (and Vice Versa) Author(s): Lawrence Zalcman Source: The American Mathematical Monthly, Vol. 81, No. 2 (Feb., 1974), pp. 115-137 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2976953 Accessed: 05-04-2017 21:47 UTC REFERENCES Linked references are available on JSTOR for this article: http://www.jstor.org/stable/2976953?seq=1&cid=pdf-reference#references_tab_contents You may need to log in to JSTOR to access the linked references. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://about.jstor.org/terms Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly This content downloaded from 108.179.181.50 on Wed, 05 Apr 2017 21:47:20 UTC All use subject to http://about.jstor.org/terms REAL PROOFS OF COMPLEX THEOREMS (AND VICE VERSA) LAWRENCE ZALCMAN Introduction. It has become fashionable recently to argue that real and complex variables should be taught together as a unified curriculum in analysis. Now this is hardly a novel idea, as a quick perusal of Whittaker and Watson's Course of Modern Analysis or either Littlewood's or Titchmarsh's Theory of Functions (not to mention any number of cours d'analyse of the nineteenth or twentieth century) will indicate. -
Eine Auswahl Weiterer Mathematiker, Die in Der Vorlesung Funktionentheorie Erw¨Ahnt Wurden
Eine Auswahl weiterer Mathematiker, die in der Vorlesung Funktionentheorie erw¨ahnt wurden John Wallis (* 3. Dezember 1616 in Ashford, Kent; y 8. November 1703 in Oxford) war ein englischer Mathematiker, der Beitr¨age zur Infinitesimalrechnung und zur Berechnung der Kreiszahl π leistete. Abraham de Moivre (* 26. Mai 1667 in Vitry-le-Fran¸cois; y 27. November 1754 in London) war ein franz¨osischer Mathematiker, der vor allem fur¨ den Satz von Moivre bekannt ist. James Stirling (* Mai 1692 in Garden bei Stirling; y 5. Dezember 1770 in Edinburgh) war ein schottischer Mathematiker. Joseph-Louis Lagrange (* 25. Januar 1736 in Turin als Giuseppe Lodovico Lag- rangia; y 10. April 1813 in Paris) war ein italienischer Mathematiker und Astronom. Er begrundete¨ die analytische Mechanik (Lagrangeformalismus mit der Lagrangefunktion), die er 1788 in seinem beruhmten¨ Lehrbuch M´ecaniqueanalytique\ darstellte. Weitere " Arbeitsgebiete waren das Dreik¨orperproblem der Himmelsmechanik (Lagrangepunkte), die Variationsrechnung und die Theorie der komplexen Funktionen. Lorenzo Mascheroni (* 13. Mai 1750 bei Bergamo; y 14. Juli 1800 in Paris) war ein italienischer Geistlicher und Mathematiker. Simeon´ Denis Poisson (* 21. Juni 1781 in Pithiviers (D´epartement Loiret); y 25. April 1840 in Paris) war ein franz¨osischer Physiker und Mathematiker. Augustin Jean Fresnel (* 10. Mai 1788 in Broglie (Eure); y 14. Juli 1827 in Ville- d'Avray bei Paris) war ein franz¨osischer Physiker und Ingenieur, der wesentlich zur Be- grundung¨ der Wellentheorie des Lichts und zur Optik beitrug. Augustin-Louis Cauchy (* 21. August 1789 in Paris; y 23. Mai 1857 in Sceaux) war ein franz¨osischer Mathematiker. Als ein Pionier der Analysis entwickelte er die von Gott- fried Wilhelm Leibniz und Sir Isaac Newton aufgestellten Grundlagen weiter, wo- bei er die fundamentalen Aussagen auch formal bewies. -
Notes for the Course in Analytic Number Theory G. Molteni
Notes for the course in Analytic Number Theory G. Molteni Fall 2019 revision 8.0 Disclaimer These are the notes I have written for the course in Analytical Number Theory in A.Y. 2011{'20. I wish to thank my former students (alphabetical order): Gu- glielmo Beretta, Alexey Beshenov, Alessandro Ghirardi, Davide Redaelli and Fe- derico Zerbini, for careful reading and suggestions improving these notes. I am the unique responsible for any remaining error in these notes. The image appearing on the cover shows a picture of the 1859 Riemann's scratch note where in 1932 C. Siegel recognized the celebrated Riemann{Siegel formula (an identity allowing to computed with extraordinary precision the values of the Riemann zeta function inside the critical strip). This image is a resized version of the image in H. M. Edwards Riemann's Zeta Function, Dover Publications, New York, 2001, page 156. The author has not been able to discover whether this image is covered by any Copyright and believes that it can appear here according to some fair use rule. He will remove it in case a Copyright infringement would be brought to his attention. Giuseppe Molteni This work is licensed under a Creative Commons Attribution-Non- Commercial-NoDerivatives 4.0 International License. This means that: (Attribu- tion) You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. (NonCommercial) You may not use the material for commercial purposes. -
Catalogo Dei Testi Di Scienza Delle Costruzioni
Catalogo dei testi di Scienza delle Costruzioni Claudio Franciosi1 September 2, 2019 1E-mail: [email protected] 2 Libri 1894 Cesàro Ernesto Cesàro: Introduzione alla teoria matematica dell’elasticità, Fratelli Bocca Editori, Torino 1927 Colonnetti Gustavo Colonnetti: I fondamenti della statica, UTET, Torino 1928 Colonnetti Gustavo Colonnetti: La Statica delle Costruzioni Vol.I, UTET, Torino 1929 Colonnetti Gustavo Colonnetti: Principii di Dinamica, UTET, Torino 1932 Ricci Carlo Luigi Ricci: Lezioni di Scienza delle Costruzioni, raccolte dall’Ing. Adriano Galli, Scuola Superiore di Architettura, Sezione Editoriale del G.U.F. Mussolini, Napoli Colonnetti Gustavo Colonnetti: La Statica delle Costruzioni Vol.II Parte Prima, UTET, Torino 1934 Colonnetti Gustavo Colonnetti: La Statica delle Costruzioni Vol.II, Parte Seconda UTET, Torino 1940 Guidi Camillo Guidi: Lezioni di Scienza delle Costruzioni Vol.I (Statica Grafica, Bona, Torino 1942 Ricci Carlo Luigi Ricci: Meccanica Applicata alle Costruzioni, E.P.S.A., Editrice Politecnica S.A., Napoli 1948 Magini Otello Magini: Esercizi di Scienza delle Costruzioni Vol.I, Colombo Cursi Editore, Pisa Magini Otello Magini: Esercizi di Scienza delle Costruzioni Vol.II, Colombo Cursi Editore, Pisa 1955 Donato Letterio F. Donato: Lezioni di Scienza delle Costruzioni Vol.I, Colombo Cursi Editore, Pisa 1956 Donato Letterio F. Donato: Lezioni di Scienza delle Costruzioni Vol.II, Colombo Cursi Editore, Pisa 1960 Signorini Antonio Signorini: Questioni di Elasticità non linearizzata, Conferenze