Blockchain Papers

Karine Chemla

No abstract is available for this record.

Juliusz Brzeziński

No abstract is available for this record.

Weiqiong Zheng

The Riemann-Roch theorem is a useful tool to calculate the dimension of the space of meromorphic functions with prescribed zeros and poles. There are severals versions of the theorem such as the Riemann-Roch theorem for line bundles, for (algebraic) curves, for surfaces and for higher dimensions. In this thesis, we will focus on the Riemann-Roch theorem for algebraic curves over an algebraically closed eld, which is a very important result in complex analysis and algebraic geometry. The study of the elds of rational functions on curves can be very useful in the proof. So we will recall some pre-knowledges in commutative algebra and some facts about a ne varieties. Then talk about function elds, discrete valuation rings and Weil di erentials to prove the theorem, using the methods of Andre Weil.

Philippe R. Richard, Antonio M. Oller‐Marcén, Vicente Meavilla Seguí

No abstract is available for this record.

Katrine Frovin Gravesen, Niels Grønbæk, Carl Winsløw

No abstract is available for this record.

Alec D. Walen

The standard of proof in criminal trials in many liberal democracies is proof beyond a reasonable doubt, the BARD standard. It is customary to describe it, when putting a number on it, as requiring that the fact finder be at least 90% certain, after considering the evidence, that the defendant is guilty. Strikingly, no good reason has yet been offered in defense of using that standard. A number of non-consequentialist justifications that aim to support an even higher standard have been offered; all are morally unsound. Meanwhile, consequentialist arguments plausibly support a substantially lower standard — in some cases so low as to undermine the idea that punishment is what is at stake. In this paper, I offer a new retributive justification that supports excluding the instrumental benefits of punishment from the balance that sets the standard. The resulting balance supports a standard arguably in the ballpark of the customary understanding of BARD: a standard requiring that the fact finder have a high, though not maximally high, degree of confidence that the defendant is guilty.

Scott Aaronson

We're going to start by beating a retreat from QuantumLand, back onto the safe territory of computational complexity. In particular, we're going to see how, in the 1980s and 1990s, computational complexity theory reinvented the millennia-old concept of mathematical proof – making it probabilistic, interactive, and cryptographic. But then, having fashioned our new pruning-hooks (proving-hooks?), we're going to return to QuantumLand and reap the harvest. In particular, I’ll show you why, if you could see the entire trajectory of a hidden variable, then you could efficiently solve any problem that admits a “statistical zero-knowledge proof protocol,” including problems like Graph Isomorphism for which no efficient quantum algorithm is yet known. What is a proof? Historically, mathematicians have had two very different notions of “proof.” The first is that a proof is something that induces in the audience (or at least the prover!) an intuitive sense of certainty that the result is correct. In this view, a proof is an inner transformative experience – a way for your soul to make contact with the eternal verities of Platonic heaven.

Ronald Cramer

No abstract is available for this record.

Frank Smithies

In 1798, there appeared in the Philosophical Transactions of the Royal Society a paper by James Wood, purporting to prove the fundamental theorem of algebra, to the effect that every non-constant polynomial with real coefficients has at least one real or complex zero. Since the first generally accepted proof of this result was given by Gauss in 1799, Wood's paper deserves careful examination. After giving a brief outline of Wood's career, I describe the argument of his paper. His proof turns out to be incomplete as it stands, but it contains an original idea, which was to be used later, in the same context, by von Staudt, Gordan and others, without knowledge of Wood's work. After putting Wood's work in context, I conclude by showing how his idea can be used to prove the complex form of the fundamental theorem of algebra, stating that every non-constant polynomial with complex coefficients has at least one zero in the complex field.

Isabelle Bloch

The study of maths curriculum in the last grades of secondary schools in France along 30 years brings to light important variations that took place since 1962 in the contents of calculus at this level. These evolutions concern the objects of calculus that are taught as well as the procedures used by students and teachers. The suggested methods affect the knowledge that students are likely to use when doing the given tasks; and we observe that since the 90ths', the tasks given to students do not valorise validation. We study the possibilities of establishing an real relationship to the knowledge in calculus, at this level of teaching, and to allow the students to build appropriate methods.<br />We study the question of validation in teaching analysis through the following directions:<br />- the mathematical theory; its organisation; the methods of proof and the formalization; how these methods can be introduced in the teaching, in a way that students can understand;<br />- the existence of fundamental situations concerning the concepts of function and limit, and the possibility of implement such situations in the class.<br /><br />Besides, the study of the different settings of representation that are at stake to build a suitable environment for the teaching of function and limit makes new potentialities come to light, particularly in the graphic and formal settings.<br />The experimentation is carried through the building of situations with an a-didactical component for the teaching of function and limit, and through the observation of their implementation in a scientific class of 17 years-old students. This makes us first question the knowledge and professional knowing a teacher uses to manage a teaching situation in analysis, with an a-didactical component, and then draw a pattern to the teacher's milieu.<br />We also submit a test to the students and analyse the results with statistic tools so as to test the main features of the learning.<br />In the last chapter we study lectures at undergraduate level, and student's papers with lots of errors about calculus definitions. This leads us to question the knowledge that is compulsory at University level; we wonder how it is possible to link it with Secondary school's knowledge and habits.<br />As a conclusion, we shall suggest some remarks about the balance between definitive knowledge and what students must get as an experience in the teaching of a new mathematical theory; this balance affects the possibilities of validation and finally, the future prospects of teaching analysis from Secondary Schools to University.

Moritz Epple

No abstract is available for this record.

Paul H. Edelman

In these pages,' Steven Lubet recently reviewed A Tour of Calculus, by David Berlinski.2 Inspired by both beauty of calculus and Berlinski's description of it, Lubet waxes poetic on many parallels between law and calculus. It is completely understandable--even admirablethat one might be led to ruminations on relationship between calculus and one's own discipline. There is little doubt that subject of calculus stands as one of great intellectual feats of Western thought. It has had profound implications for physics, engineering, economics and many other disciplines-so why not law? Alas, these philosophical musings would be more persuasive had Professor Lubet better understood what it was that he was writing about. Lubet's errors come in two types. first is just a misunderstanding of history, but it is a misunderstanding that unfortunately forms basis for an entire section of his review. second type of error is more fundamentally mathematical: he does not distinguish between a definition and a theorem. Just as Lubet draws legal lessons from calculus, we can draw legal parallels from his mistakes. While some of these might be comforting, others will be more unsettling. As noted by Lubet, development of calculus was done more or less simultaneously in mid-17th century by Sir Isaac Newton and Gottfried Wilhelm Leibniz. Leibniz based much of his development of subject on idea of an infinitesimal, a class of numbers that are smaller than any other number. According to Lubet, the `infinitesimals' turn out to be a futile fiction, notwithstanding Liebnitz's [sic] own endorsement of them. In 1734, Bishop Berkeley that they do not and cannot exist.3 Lubet goes on in Part III to draw a number of legal parallels to this discrediting of idea of infinitesimals. While legal conclusions he draws from these events may well be true, Lubet cannot base them on invalidity of infinitesimals: fact of matter is that Leibniz was right. To be fair to Lubet, ultimate vindication of Leibniz's belief in infinitesimals is hidden in a footnote by Berlinski: The development of [non-Archimedean] fields by logician Abraham Robinson in twentieth century has made possible development of calculus entirely along lines anticipated by Leibnitz [sic].4 Nevertheless, anyone with serious mathematical training would not have needed Berlinski's footnote; Lubet's error highlights danger of relying on secondhand knowledge of a field quite different from one's own. Moreover, culpability aside, Lubet has lost foundation for legal insights he draws from purported invalidity of infinitesimals. And what of supposed proof' of Bishop Berkeley? Berlinski writes that [w]riting in 1734, Bishop Berkeley wasted no time in attacking very idea of infinitesimals, and later says that [1]ooking backward, we can see that Berkeley was entirely correct,5 but never claims that Bishop Berkeley proved conclusively anything about existence of infinitesimals. Indeed, he couldn't have, since by appropriately generalizing idea of a number, Abraham Robinson was able to define them. Lubet should be more careful in using term proof' in context of mathematics. Lubet sees more parallels between computation of area under a curve and way that legal trials proceed by means of accretion of detail.6 Surprisingly, Lubet doesn't draw obvious parallel, that just as sum of more and more rectangles gives better and better approximations for area under a curve, as a trial proceeds evidence presented gives a better and better approximation of truth. He instead focuses on error in mathematical approximation: An integral combines rectangles until limit of error approaches zero, but error-zone never actually becomes zero. …

Karine Chemla

The Argument To highlight speculative trends specific to the mathematical tradition that developed in China, the paper analyzes an excerpt of a third-century commentary on a mathematical classic, which arguably contains a proof. The paper shows that the following three tasks cannot be dissociated one from the other: (1) to discuss how the ancient text should be read; (2) to describe the practice of mathematical proof to which this text bears witness; (3) to bring to light connections between philosophy and mathematics that it demonstrates were established in China. To this end the paper defines its use of the word “proof” and outlines a program for an international history of mathematical proof. It describes the sense in which the text conveys a proof and shows how it simultaneously fulfills algorithmic ends, bringing to light a formal pattern that appears to be fundamental both for mathematics and for other domains of reality. The interest in transformations that mathematical writings demonstrate in China at that time seems to have been influenced by philosophical developments based on The Book of Changes (Yi-jing) , which the excerpt quotes. This quotation within a mathematical context makes it possible to suggest an interpretation for a rather difficult philosophical statement.

Stephen Silverman

1. The proof consists of the observation that Tr(AB)= Tr(BA), where A = [aij] and B = [bji]. Note that if the field had nonzero characteristic then the argument would merely establish that the sizes of the two bases differed by a multiple of the characteristic. 2. The same argument can be used in an obvious way to define the degree of a finite extension of a field of characteristic zero. This leads directly to a proof of the double extension theorem: if E is a field of characteristic zero, F is a finite extension of E and G is a finite extension of F, then G is a finite extension of E and [G: E] = [G: F][F: E]. This approach can be used to demonstrate the impossibility of the duplication of the cube and the trisection of the angle using straightedge and compass to students who have no knowledge of linear algebra.

Liliane Beaulieu

No abstract is available for this record.

Alkiviadis G. Akritas

Isolation of the real roots of a polynomial equation is the process of finding real, disjoint intervals such that each contains exactly one real root and every real root is contained in some interval. This process is quite important because, as J. B. J. Fourier pointed out, it constitutes the first step toward the solution of general equations of degree greater than four, the second step being the approximation of roots to any desired degree of accuracy. In the beginning of the 19th century F. D. Budan and J. B. J. Fourier presented two different (but equivalent) theorems which enable us to determine the maximum possible number of real roots that an equation has within a given interval. Budan's theorem appeared in 1807 in the memoir Nouvelle methode pour la resolution des equations numeriques [10, p. 219], whereas Fourier's theorem was first published in 1820 in Le Bulletin des sciences par la Societe Philomatique de Paris, pp. 156, 181 [10, p. 223]. Due to the importance of these two theorems, there was a great controversy regarding priority rights. In his book (1859) Biographies of distinguished scientific men, p. 383, F. Arago informs us that Fourier deemed it necessary to have recourse to the certificates of early students of the Polytechnic School or Professors of the University in order to prove that he had taught his theorem in 1796, 1797 and 1803 [10]. Based on Fourier's proposition, C. Sturm presented in 1829 an improved theorem whose application yields the exact number of real roots which a polynomial equation without multiple zeros has within a real interval; thus he solved the real root isolation problem. Since 1830 Sturm's method has been the only one widely known and used, and consequently Budan's theorem was pushed into oblivion. To our knowledge, Budan's theorem can be found only in [16] and [61 whereas Fourier's proposition appears in almost all texts on the theory of equations. We feel that Budan's theorem merits special attention because it constitutes the basis of Vincent's forgotten theorem of 1836 which, in turn, is the foundation of our method for the isolation of the real roots of an equation [1], a method which far surpasses Sturm's in efficiency [2], [3]. In the discussion which follows we present separately, and without proofs, the classical theorems by Fourier and Budan and we indicate how they lead to the corresponding real root isolation methods. Some empirical results are also presented for comparison.

Olive C. Hazlett

Part I. Introduction 1. Prologue.Thus far, very little has been published on the general theory of formal modular invariants or covariants.Workers have, on the whole, obtained results for special, more or less isolated, cases; and although some beautiful and important general theorems have been proved, they are more or less unrelated.This is, of course, only natural in any division of knowledge in its formative state.Nevertheless, no worker in the field could fail to be conscious of a certain uniformity common to the special cases that have been studied in detail; though (alas !) this uniformity usually appeared to be broken ruthlessly in the next case studied.This breaking of an apparent law signified, however, merely that we did not know these special cases with a sufficient thoroughness of illuminating detail, or were trying unwittingly to make the laws conform to certain standards, unconsciously preconceived.This latter handicap was laid on us naturally enough by our thorough knowledge of algebraic invariants and the fact that this newer kind of covariants is, in many ways, strikingly like the older, classic covariants, though so tantalisingly different.Their similarity and their difference show themselves in the very beginning of the study: in the definitions, in the simplest examples.Perhaps the differences that first come to mind are those which are inherent in the fields of definition, which, in the case of classic covariants, is the field of reals or ordinary complex numbers and, in the case of modular covariants, is a Galois field, GF[pn], of order pn.These differences are too obvious to mention in detail, but one who has studied the beautiful proofs given by the old masters of invariant theory has been forced to the conclusion that most of the proofs seemed to use the properties of a field of characteristic zero, not in some accidental manner, but rather in veriest necessity.Growing from the surface differences between the two fields are two very important distinguishing characteristics of the two kinds of covariants.It * Part II was presented to the Society, September 7, 1920; Part III, December 28, 1921; Parts IV and V, December 27, 1922.