DOCSLIB.ORG

Fundamental Theorems of Integral Calculus on R Introduction

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Fundamental Theorems of Integral Calculus on R Introduction Part VI Fundamental theorems of integral calculus on R Introduction The Riemann integral can be considered an evolution of Cauchy’s integral, in that certain functions that were not integrable according to Cauchy became integrable in Riemann’s theory. At the same time, alas, in the new framework integration is no longer the inverse operation to differentiation. As a matter of fact, there exist Riemann ( ) = x ( ) integrable functions f whose integral map F x a f t dt is not differentiable (not even on one side only) at certain points. Thus the fundamental theorem of calculus, in the version for continuous maps proved by Cauchy, loses its status of calculus’ highest pinnacle and becomes a mere special case of a much bigger picture. Realising this fact led mathematicians to view differentiation and integration with a more critical eye, and prompted them to explore the depths of the relationship between continuity, differentiability and integrability. Hermann Hankel was one among many academics trying to unravel the connection differentiability–continuity. In 1871 Hankel used Riemann’s integral function1 R(t) to define the map x G(x) = k + R(t)dt, a which is continuous everywhere, but not differentiable at infinitely many points. The example raised much incredulity among the scientific community, for it went against the conventional wisdom that continuity implied differentiability but at a finite number of points. Many textbooks of the time, to prove that a continuous map f (x) on some interval had to be differentiable, used geometrically-flavoured arguments involving tangents to the curve representing f (x), without fathoming that there do exist continuous maps not allowing for a pictorial description of that kind. It was Karl Weierstraß who cleared away all doubts when he constructed a function of one real variable that is everywhere continuous but never differentiable. 1Here R(t) will denote the function Riemann employed to show his notion of integral was sophis- ticated enough to allow the integration of maps with infinitely many discontinuities. 348 Fundamental theorems of integral calculus on R 349 Liberating the concept of derivative from continuity allowed to study differenti- ation independently, and thus extend its scope to discontinuous functions as well. For this reason Ulisse Dini, to make derivatives and integrals both rigorous and more general, and encompass maps with dense discontinuity sets (which he called “discon- tinuous dot-functions”2) introduced in the early 1880s the so-called Dini derivatives. These numbers were created to extend the notion of derivative to pathological functions. Explicitly, the (upper and lower) right Dini derivatives of F at a point x ∈[a, b) are + F(x + h) − F(x) F(x + h) − F(x) D F(x) = lim sup and D+ F(x) = lim inf ; → + h→0+ h h 0 h the (upper and lower) left Dini derivatives at x ∈ (a, b] are − F(x) − F(x − h) F(x) − F(x − h) D F(x) = lim sup and D− F(x) = lim inf . → + h→0+ h h 0 h + − + Note that D F(x) ≥ D+ F(x) and D F(x) ≥ D− F(x). When D F(x) = − D+ F(x) = D F(x) = D− F(x) ∈ R the map F is differentiable at x and its derivative is f (x), precisely the common value of the four Dini derivatives. Among other things, these new concepts allowed Dini to state the fundamental theorem of calculus for the Riemann integral in the following form: if F is continuous and one of Dini’s derivatives DFis Riemann integrable, then the integral map F(x) = x DF(t)d(t) has all four Dini derivatives bounded and integrable, and moreover a b ( ) = ( ) − ( ) a DF x dx F b F a . Dini, moreover, explicitly emphasised the necessity of assuming the integrability of DF, and conjectured the existence non-constant, continuous primitives F that become constant over a suitable subinterval of any given interval. This would imply DF is not integrable, because we would not be able to enclose its discontinuities using a finite number of arbitrarily small intervals. By the way, the surmise was proved in 1881 by Vito Volterra, a student of Dini at Scuola Normale Superiore in Pisa, who gave an explicit example (see p.xxx ff.). The existence of continuous maps with no derivative, and the ensuing demise of the fundamental theorem of Cauchy’s integral calculus, persuaded many mathemati- cians, most notably Lebesgue, to investigate the relationship between integrals and primitives. The first result of Lebesgue’s theory is that “If f ∈ L1[a, b] the integral map F(x) = f (t)(dt) [a,x] is continuous and has derivative F (x) = f (x) a.e. on [a, b].” 2‘funzioni punteggiate discontinue’ in Italian. 350 Fundamental theorems of integral calculus on R Lebesgue also proved, conversely, “If F :[a, b] → R has bounded derivative f , then f is Lebesgue integrable and the equality F(x) − F(a) = f (t)(dt) [a,x] holds everywhere on [a, b].” These results entitled him to declare in 1902 that the problem of integral calcu- lus, conjectured by Dini and proved by Volterra within Riemann’s set-up, had been completely solved in the case of bounded derivatives. However, things get more complicated if the derivative f is not bounded, for then f might not be integrable. Lebesgue tried to pin down the features a primitive F should have in order for f to be well defined almost everywhere on [a, b] and integrable on the entire interval. Restricting to the case where one Dini derivative of F is almost everywhere finite, Lebesgue showed that F must have bounded variation. It is at this point that the notion of bounded variation, for which we will provide a brief account, entered the stage. The idea was elaborated by Jordan for other reasons, namely in relationship to conditions granting that a function F(x) has a Fourier series converging to F(x). Jordan proved a sufficient condition for convergence in terms of a function with bounded variation. To define bounded variation he took a map F :[a, b] → R and fixed a finite partition P ={a = x0, x1,...,xn−1, xn = b} of [a, b]. He then called variation of F(x) over [a, b] with respect to P ∈ P the quantity n b( , ) = | ( ) − ( )|. Va P F F xi F xi−1 i=1 | ( ) − ( )| = ,..., The difference F xi F xi−1 , i 1 n, represents the largest variation of [ , ] n | ( ) − ( )|= { } F over xi−1 xi .If i=1 F xi F xi−1 k, the set of values k P∈P can, as P varies, have either finite or infinite least upper bound. Let us call total variation of F on [a, b] the supremum. When that is finite we say F has bounded variation on [a, b]: b( ) = b( , ) : [ , ] ≤∞. Va F sup Va P F P partition of a b P∈P A function with bounded variation is bounded, whereas there exist bounded maps whose variation is not bounded. To see this just take the Dirichlet function f : [0, 1] → R: albeit bounded, since f (x) ∈{0, 1}, it has oscillations of width 1 for every x ∈ Q ∩[0, 1]. Therefore functions with bounded variation form a proper Fundamental theorems of integral calculus on R 351 subfamily of bounded maps. Moreover, there exist discontinuous maps with bounded variation, such as discontinuous monotone functions, and there also are continuous maps with unbounded variation. But going back to the issue, Jordan proved that a function with bounded variation on [a, b] can be written as difference of two increasing functions (Cours d’analyse de l’École Polytechnique, first edition in three volumes published in 1882-1887). On the matter Lebesgue observed that if f ∈ L1 is positive, the integral map F(x) = f (t)(dt), x ∈[a, b] [a,x] is increasing. Moreover, since a summable function f is the difference of positive summable functions f +, f −, any integral map is the difference of increasing func- tions F(x) = F +(x) − F −(x). Based on this, in 1904 Lebesgue established the following: if F :[a, b] → R has bounded variation, then it is differentiable a.e. on [a, b], and f is summable on [a, b]. He stopped short of saying |F(x) − F(a)|= | f (t)|(dt) [a,x] for every x ∈[a, b], because in that case the difference of the two sides would be a monotone map G with bounded variation and zero derivative -almost everywhere on [a, x]. ( ) = ≥ The point is that Lebesgue viewed F x [a,x] fd (clearly for f 0) not as afunctionofx, but rather as being defined on an interval, or more generally as a set function μF : S(X) →[0, ∞], where μF (E) = fd, E ∈ S(X) E designates a measure on the measurable space (X, S(X)). In practice μF is a weighted measure (weighted by the non-negative measurable map f ) determined by the for- mula dμF = fd. Now the question is: is there a way to characterise, among all set functions μF ([a, b]), indefinite integrals that satisfy 352 Fundamental theorems of integral calculus on R F(x) − F(a) = f (t)(dt) [a,x] for every x ∈[a, b] ? The answer is yes, there is indeed, and it relies on two measure- theoretical notions that are central to the whole of real analysis: complete additivity and absolute continuity with respect to a given measure. The punchline is this: among functions with bounded variation, the only ones satisfying F(x) − F(a) = f (t)(dt) [a,x] must be both • completely additive – in the words of De la Vallée Poussin: “Given E ⊆ X measurable, a set function μF (E) is completely additive if, irre- spective of how one partitions E in a finite/countable number of measurable subsets Ei , ∞ ∞ μF (E) = μF Ei = μF (Ei ) ; i=1 i=1 • absolutely continuous, as defined by the mathematician Giuseppe Vitali thus: “Given E ⊆ X measurable, a set function μF (E) is absolutely continuous with respect to measure if μF (E) tends to 0 as (E) tends to 0.” Lebesgue went on to prove that “If a set function with bounded variation μF ([a, b]) is absolutely continuous with respect and completely additive, then it is differentiable a.e.
Load more

Recommended publications
  • Fabio Tonini –
    Fabio Tonini Personal Data Date of Birth May 16, 1984 Citzenship Italy Current Position Since Researcher (RTDA) at University of Florence November 2018 Positions Held October 2018 Scholarship at Scuola Normale Superiore of Pisa October 2015 Post Doc at the Freie University of Berlin - September 2018 April 2013 - Post Doc at the Humboldt University of Berlin September 2015 January 2012 Scholarship at Scuola Normale Superiore of Pisa - January 2013 January 2009 Ph.D. student at Scuola Normale Superiore of Pisa under the supervision of - January Prof. Angelo Vistoli 2012 Italian National Scientific Habilitation (ASN) May 2021 - Professore II fascia May 2030 Education May 2013 Ph.D. at Scuola Normale Superiore of Pisa December Diploma at Scuola Normale Superiore of Pisa 2008 September Master’s Degree in Pure Mathematics, University of Pisa, with honors 2008 July 2006 Bachelor’s Degree in Pure Mathematics, University of Pisa, with honors Universitá degli Studi di Firenze, Dipartimento di Matematica e Informatica ’Ulisse Dini’, Viale Morgagni, 67/a, Firenze, 50134 Italy B fabio.tonini@unifi.it • Í people.dimai.unifi.it/tonini/ PhD’s Thesis Title Stacks of ramified Galois covers defended the 2 May 2013 pdf online link Advisor Angelo Vistoli Master’s Thesis Title Rivestimenti di Gorenstein (Gorenstein covers) pdf online link Advisor Angelo Vistoli Research Interests { Algebraic Geometry { Algebraic stacks, Moduli theory { Action of algebraic groups and Galois covers { Representation theory { Algebraic fundamental groups and gerbes Memberships { GNSAGA, INdAM, Gruppo Nazionale per le Strutture Algebriche, Geomet- riche e le loro Applicazioni Teaching Experience 2020/21 Course Title: Matematica e statistica, First year at “Scienze Farmaceutiche”, University of Florence 2020/21 Course Title: Matematica con elementi di Statistica, First year at “Scienze Naturali”, University of Florence.
    [Show full text]
  • A Historical Outline of the Theorem of Implicit Nctions
    Divulgaciones Matem acute-a t icas Vol period 10 No period 2 open parenthesis 2002 closing parenthesis commannoindent pp periodDivulgaciones 171 endash 180 Matem $ nacutefag $ t icas Vol . 10No . 2 ( 2002 ) , pp . 171 −− 180 A .... Historical .... Outline .... of the .... Theorem .... of nnoindentImplicit ..A F-un h nctions f i l l H i s t o r i c a l n h f i l l Outline n h f i l l o f the n h f i l l Theorem n h f i l l o f Un Bosquejo Hist acute-o rico del Teorema de las Funciones Impl dotlessi-acute citas n centerlineGiovanni Mingarif I m pScarpello l i c i t nquad open parenthesis$ F−u $ giovannimingari n c t i o n s g at l ibero period it closing parenthesis Daniele Ritelli dr itelli at e c onomia period unibo period it n centerlineDipartimentofUn di Matematica BosquejoDivulgaciones per Hist Matem le Scienzea´ $ tn icasacute Economiche Vol .f 10og No$ .e 2 Sociali( rico 2002 )delcomma , pp . Teorema 171 { 180 de las Funciones Impl $ nBolognaacutefn ItalyimathgA$ c Historical i t a s g Outline of the Theorem Abstract n centerline f Giovanniof Mingari Scarpello ( giovannimingari $ @ $ l ibero . it ) g In this article a historical outline ofImplicit the implicit functionsF theory− u nctions is presented startingUn from Bosquejo the wiewpoint Hist ofo´ Descartesrico del Teorema algebraic geometry de las Funciones Impl ´{ citas n centerline f Daniele Ritelli dr itelli $@$ e c onomia . unibo . it g open parenthesisGiovanni 1 637 closing Mingari parenthesis Scarpello and Leibniz ( giovannimingari open parenthesis 1 676@ l or ibero 1 677 closing .
    [Show full text]
  • 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.
    [Show full text]
  • Continuous Nowhere Differentiable Functions
    2003:320 CIV MASTER’S THESIS Continuous Nowhere Differentiable Functions JOHAN THIM MASTER OF SCIENCE PROGRAMME Department of Mathematics 2003:320 CIV • ISSN: 1402 - 1617 • ISRN: LTU - EX - - 03/320 - - SE Continuous Nowhere Differentiable Functions Johan Thim December 2003 Master Thesis Supervisor: Lech Maligranda Department of Mathematics Abstract In the early nineteenth century, most mathematicians believed that a contin- uous function has derivative at a significant set of points. A. M. Amp`ereeven tried to give a theoretical justification for this (within the limitations of the definitions of his time) in his paper from 1806. In a presentation before the Berlin Academy on July 18, 1872 Karl Weierstrass shocked the mathematical community by proving this conjecture to be false. He presented a function which was continuous everywhere but differentiable nowhere. The function in question was defined by ∞ X W (x) = ak cos(bkπx), k=0 where a is a real number with 0 < a < 1, b is an odd integer and ab > 1+3π/2. This example was first published by du Bois-Reymond in 1875. Weierstrass also mentioned Riemann, who apparently had used a similar construction (which was unpublished) in his own lectures as early as 1861. However, neither Weierstrass’ nor Riemann’s function was the first such construction. The earliest known example is due to Czech mathematician Bernard Bolzano, who in the years around 1830 (published in 1922 after being discovered a few years earlier) exhibited a continuous function which was nowhere differen- tiable. Around 1860, the Swiss mathematician Charles Cell´erieralso discov- ered (independently) an example which unfortunately wasn’t published until 1890 (posthumously).
    [Show full text]
  • On History of Epsilontics
    ANTIQUITATES MATHEMATICAE Vol. 10(1) 2016, p. xx–zz doi: 10.14708/am.v10i0.805 Galina Ivanovna Sinkevich∗ (St. Petersburg) On History of Epsilontics Abstract. This is a review of genesis of − δ language in works of mathematicians of the 19th century. It shows that although the sym- bols and δ were initially introduced in 1823 by Cauchy, no functional relationship for δ as a function of was ever ever specied by Cauchy. It was only in 1861 that the epsilon-delta method manifested itself to the full in Weierstrass denition of a limit. The article gives various interpretations of these issues later provided by mathematicians. This article presents the text [Sinkevich, 2012d] of the same author which is slightly redone and translated into English. 2010 Mathematics Subject Classication: 01A50; 01A55; 01A60. Key words and phrases: History of mathematics, analysis, continu- ity, Lagrange, Ampére, Cauchy, Bolzano, Heine, Cantor, Weierstrass, Lebesgue, Dini.. It is mere feedback-style ahistory to read Cauchy (and contemporaries such as Bernard Bolzano) as if they had read Weierstrass already. On the contrary, their own pre-Weierstrassian muddles need historical reconstruction. [Grattan-Guinness 2004, p. 176]. Since the early antiquity the concept of continuity was described throgh the notions of time, motion, divisibility, contact1 . The ideas about functional accuracy came with the extension of mathematical interpretation to natural-science observations. Physical and geometrical notions of continuity became insucient, ultimately ∗ Galina Ivanovna Sinkeviq 1The 'continuous' is a subdivision of the contiguous: things are called continuous when the touching limits of each become one and the same and are, as the word implies, contained in each other: continuity is impossible if these extremities are two.
    [Show full text]
  • 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
    [Show full text]
  • Ph. D. Programme in Mathematics
    Ph.D School in Mathematics Department of Mathematics, Pisa University, Pisa, Italy Coordinator: Sergio Spagnolo History In Italy, before the year 1980 there was only one university degree awarded, namely, the "laurea". After a study of four years, the students had to write a dessertation consisting of an original work, the "tesi di laurea": which required about an year of work, and there were no postgraduate studies as such, except for the “Perfezionamento” of the Scuola Normale Superiore of Pisa and the courses of the "Alta Matematica" at Rome. At the beginning of the 80's the “Dottorato di Ricerca” was officially established by law. About ten Ph.D. programmes in Mathematics were set up, mainly at some of the major Italian Universities (Roma I, Roma II, Milano, Torino, Bologna, Padova, Pisa, Firenze, Trento, Messina, Napoli). Some of the “peripherical” universities joined with these major centers to co-sponsor the Ph. D. degree as associate Universities. In addition, one should also consider the somewhat special Ph. D. programmes of the Scuola Normale, the Institute of Alta Matematica, and the SISSA at Trieste. The Ph. D. programme in Mathematics of Pisa, which included also the Universities of Bari, Parma, Lecce and Ferrara as associated Universities, was established in the year 1983. The institutional structure of all the programmes was partially centralized, for instance all the theses had to be defended before a national panel of examiners, in Rome, while organization of courses and periodical evaluations were left to the single universities. At the end of nineties, a new national legislation under the title “Autonomy of the Universities” was introduced, which allowed, in particular, each University to institute and manage financially its own Ph.
    [Show full text]
  • On the Role Played by the Work of Ulisse Dini on Implicit Function
    On the role played by the work of Ulisse Dini on implicit function theory in the modern differential geometry foundations: the case of the structure of a differentiable manifold, 1 Giuseppe Iurato Department of Physics, University of Palermo, IT Department of Mathematics and Computer Science, University of Palermo, IT 1. Introduction The structure of a differentiable manifold defines one of the most important mathematical object or entity both in pure and applied mathematics1. From a traditional historiographical viewpoint, it is well-known2 as a possible source of the modern concept of an affine differentiable manifold should be searched in the Weyl’s work3 on Riemannian surfaces, where he gave a new axiomatic description, in terms of neighborhoods4, of a Riemann surface, that is to say, in a modern terminology, of a real two-dimensional analytic differentiable manifold. Moreover, the well-known geometrical works of Gauss and Riemann5 are considered as prolegomena, respectively, to the topological and metric aspects of the structure of a differentiable manifold. All of these common claims are well-established in the History of Mathematics, as witnessed by the work of Erhard Scholz6. As it has been pointed out by the Author in [Sc, Section 2.1], there is an initial historical- epistemological problem of how to characterize a manifold, talking about a ‘’dissemination of manifold idea’’, and starting, amongst other, from the consideration of the most meaningful examples that could be taken as models of a manifold, precisely as submanifolds of a some environment space n, like some projective spaces ( m) or the zero sets of equations or inequalities under suitable non-singularity conditions, in this last case mentioning above all of the work of Enrico Betti on Combinatorial Topology [Be] (see also Section 5) but also that of R.
    [Show full text]
  • Levi-Civita,Tullio Francesco Dell’Isola, Emilio Barchiesi, Luca Placidi
    Levi-Civita,Tullio Francesco Dell’Isola, Emilio Barchiesi, Luca Placidi To cite this version: Francesco Dell’Isola, Emilio Barchiesi, Luca Placidi. Levi-Civita,Tullio. Encyclopedia of Continuum Mechanics, 2019, 11 p. hal-02099661 HAL Id: hal-02099661 https://hal.archives-ouvertes.fr/hal-02099661 Submitted on 15 Apr 2019 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. 2 Levi-Civita, Tullio dating back to the fourteenth century. Giacomo the publication of one of his best known results Levi-Civita had also been a counselor of the in the field of analytical mechanics. We refer to municipality of Padua from 1877, the mayor of the Memoir “On the transformations of dynamic Padua between 1904 and 1910, and a senator equations” which, due to the importance of the of the Kingdom of Italy since 1908. A bust of results and the originality of the proceedings, as him by the Paduan sculptor Augusto Sanavio well as to its possible further developments, has has been placed in the council chamber of the remained a classical paper. In 1897, being only municipality of Padua after his death. According 24, Levi-Civita became in Padua full professor to Ugo Amaldi, Tullio Levi-Civita drew from in rational mechanics, a discipline to which he his father firmness of character, tenacity, and his made important scientific original contributions.
    [Show full text]
  • Science and Fascism
    Science and Fascism Scientific Research Under a Totalitarian Regime Michele Benzi Department of Mathematics and Computer Science Emory University Outline 1. Timeline 2. The ascent of Italian mathematics (1860-1920) 3. The Italian Jewish community 4. The other sciences (mostly Physics) 5. Enter Mussolini 6. The Oath 7. The Godfathers of Italian science in the Thirties 8. Day of infamy 9. Fascist rethoric in science: some samples 10. The effect of Nazism on German science 11. The aftermath: amnesty or amnesia? 12. Concluding remarks Timeline • 1861 Italy achieves independence and is unified under the Savoy monarchy. Venice joins the new Kingdom in 1866, Rome in 1870. • 1863 The Politecnico di Milano is founded by a mathe- matician, Francesco Brioschi. • 1871 The capital is moved from Florence to Rome. • 1880s Colonial period begins (Somalia, Eritrea, Lybia and Dodecanese). • 1908 IV International Congress of Mathematicians held in Rome, presided by Vito Volterra. Timeline (cont.) • 1913 Emigration reaches highest point (more than 872,000 leave Italy). About 75% of the Italian popu- lation is illiterate and employed in agriculture. • 1914 Benito Mussolini is expelled from Socialist Party. • 1915 May: Italy enters WWI on the side of the Entente against the Central Powers. More than 650,000 Italian soldiers are killed (1915-1918). Economy is devastated, peace treaty disappointing. • 1921 January: Italian Communist Party founded in Livorno by Antonio Gramsci and other former Socialists. November: National Fascist Party founded in Rome by Mussolini. Strikes and social unrest lead to political in- stability. Timeline (cont.) • 1922 October: March on Rome. Mussolini named Prime Minister by the King.
    [Show full text]
  • A Brief Account Op the Life and Work of the Late Professor Ulisse Dini
    1920.3 LIFE AND WOEK OF ULISSE DINI. 173 A BRIEF ACCOUNT OP THE LIFE AND WORK OF THE LATE PROFESSOR ULISSE DINI. BY PROFESSOR WALTER B. FORD. (Read before the American Mathematical Society September 3,1919.) THE death of Professor Ulisse Dini in his native city of Pisa, Italy, on the 28th of last October marked the passing of one whose name has long been familiar to the mathematical fraternity of the entire world and it therefore seems fitting, despite the wide separation geographically between the scene of his labors and our own, that a brief account of his life and work be given at this time in America and a small measure at least of tribute be rendered to his genius. Dini was born on the 14th of November, 1845, of parents highly respected but of very moderate circumstances. He early manifested an unusual activity of both mind and body, thus commanding the attention and admiration of his in­ structors and masters who foresaw for him a future of extraordinary promise. Upon entering the neighboring uni­ versity of Pisa his marked capabilities in mathematics Were soon recognized and he shortly became the favorite pupil of Professor Betti, well known as one of the leaders in mathe­ matical instruction and research at this period in Italy. Under such auspices and at the uncommonly early age of nine­ teen years, Dini attained the laureate and received directly afterward a government scholarship enabling him to continue his studies for a year at Paris. During this brief foreign sojourn he came chiefly under the influences of Bertrand and Hermite and formed an intimate friendship with them which extended long into later years.
    [Show full text]
  • 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.
    [Show full text]