Ramanujan, His Lost Notebook, Its Importance
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Notices of the American Mathematical Society June/July 2006
of the American Mathematical Society ... (I) , Ate._~ f.!.o~~Gffi·u. .4-e.e..~ ~~~- •i :/?I:(; $~/9/3, Honoring J ~ rt)d ~cLra-4/,:e~ o-n. /'~7 ~ ~<A at a Gift from fL ~ /i: $~ "'7/<J/3. .} -<.<>-a.-<> ~e.Lz?-1~ CL n.y.L;; ro'T>< 0 -<>-<~:4z_ I Kumbakonam li .d. ~ ~~d a. v#a.d--??">ovt<.·c.-6 ~~/f. t:JU- Lo,.,do-,......) ~a page 640 ~!! ?7?.-L ..(; ~7 Ca.-uM /3~~-d~ .Y~~:Li: ~·e.-l a:.--nd '?1.-d- p ~ .di.,r--·c/~ C(c£~r~~u . J~~~aq_ f< -e-.-.ol ~ ~ ~/IX~ ~ /~~ 4)r!'a.. /:~~c~ •.7~ The Millennium Grand Challenge .(/.) a..Lu.O<"'? ...0..0~ e--ne_.o.AA/T..C<.r~- /;;; '7?'E.G .£.rA-CLL~ ~ ·d ~ in Mathematics C>n.A..U-a.A-d ~~. J /"-L .h. ?n.~ ~?(!.,£ ~ ~ &..ct~ /U~ page 652 -~~r a-u..~~r/a.......<>l/.k> 0?-t- ~at o ~~ &~ -~·e.JL d ~~ o(!'/UJD/ J;I'J~~Lcr~~ 0 ??u£~ ifJ>JC.Qol J ~ ~ ~ -0-H·d~-<.() d Ld.orn.J,k, -F-'1-. ~- a-o a.rd· J-c~.<-r:~ rn-u-{-r·~ ~'rrx ~~/ ~-?naae ~~ a...-'XS.otA----o-n.<l C</.J.d:i. ~~~ ~cL.va- 7 ??.L<A) ~ - Ja/d ~~ ./1---J- d-.. ~if~ ~0:- ~oj'~ t1fd~u: - l + ~ _,. :~ _,. .~., -~- .. =- ~ ~ d.u. 7 ~'d . H J&."dIJ';;;::. cL. r ~·.d a..L- 0.-n(U. jz-o-cn-...l- o~- 4; ~ .«:... ~....£.~.:: a/.l~!T cLc.·£o.-4- ~ d.v. /-)-c~ a;- ~'>'T/JH'..,...~ ~ d~~ ~u ~ ~ a..t-4. l& foLk~ '{j ~~- e4 -7'~ -£T JZ~~c~ d.,_ .&~ o-n ~ -d YjtA:o ·C.LU~ ~or /)-<..,.,r &-. -
Indian Mathematics
Indian Mathemtics V. S. Varadarajan University of California, Los Angeles, CA, USA UCLA, March 3-5, 2008 Abstract In these two lectures I shall talk about some Indian mathe- maticians and their work. I have chosen two examples: one from the 7th century, Brahmagupta, and the other, Ra- manujan, from the 20th century. Both of these are very fascinating figures, and their histories illustrate various as- pects of mathematics in ancient and modern times. In a very real sense their works are still relevant to the mathe- matics of today. Some great ancient Indian figures of Science Varahamihira (505–587) Brahmagupta (598-670) Bhaskara II (1114–1185) The modern era Ramanujan, S (1887–1920) Raman, C. V (1888–1970) Mahalanobis, P. C (1893–1972) Harish-Chandra (1923–1983) Bhaskara represents the peak of mathematical and astro- nomical knowledge in the 12th century. He reached an un- derstanding of calculus, astronomy, the number systems, and solving equations, which were not to be achieved any- where else in the world for several centuries...(Wikipedia). Indian science languished after that, the British colonial occupation did not help, but in the 19th century there was a renaissance of arts and sciences, and Indian Science even- tually reached a level comparable to western science. BRAHMAGUPTA (598–670c) Some quotations of Brahmagupta As the sun eclipses the stars by its brilliancy, so the man of knowledge will eclipse the fame of others in assemblies of the people if he proposes algebraic problems, and still more, if he solves them. Quoted in F Cajori, A History of Mathematics A person who can, within a year, solve x2 92y2 =1, is a mathematician. -
Zero-One Laws
1 THE LOGIC IN COMPUTER SCIENCE COLUMN by 2 Yuri GUREVICH Electrical Engineering and Computer Science UniversityofMichigan, Ann Arb or, MI 48109-2122, USA [email protected] Zero-One Laws Quisani: I heard you talking on nite mo del theory the other day.Itisinteresting indeed that all those famous theorems ab out rst-order logic fail in the case when only nite structures are allowed. I can understand that for those, likeyou, educated in the tradition of mathematical logic it seems very imp ortantto ndoutwhichof the classical theorems can b e rescued. But nite structures are to o imp ortant all by themselves. There's got to b e deep nite mo del theory that has nothing to do with in nite structures. Author: \Nothing to do" sounds a little extremist to me. Sometimes in nite ob jects are go o d approximations of nite ob jects. Q: I do not understand this. Usually, nite ob jects approximate in nite ones. A: It may happ en that the in nite case is cleaner and easier to deal with. For example, a long nite sum may b e replaced with a simpler integral. Returning to your question, there is indeed meaningful, inherently nite, mo del theory. One exciting issue is zero- one laws. Consider, say, undirected graphs and let be a propertyofsuch graphs. For example, may b e connectivity. What fraction of n-vertex graphs have the prop erty ? It turns out that, for many natural prop erties , this fraction converges to 0 or 1asn grows to in nity. If the fraction converges to 1, the prop erty is called almost sure. -
After Ramanujan Left Us– a Stock-Taking Exercise S
Ref: after-ramanujanls.tex Ver. Ref.: : 20200426a After Ramanujan left us– a stock-taking exercise S. Parthasarathy [email protected] 1 Remembering a giant This article is a sequel to my article on Ramanujan [14]. April 2020 will mark the death centenary of the legendary Indian mathe- matician – Srinivasa Ramanujan (22 December 1887 – 26 April 1920). There will be celebrations of course, but one way to honour Ramanujan would be to do some introspection and stock-taking. This is a short survey of notable achievements and contributions to mathematics by Indian institutions and by Indian mathematicians (born in India) and in the last hundred years since Ramanujan left us. It would be highly unfair to compare the achievements of an individual, Ramanujan, during his short life span (32 years), with the achievements of an entire nation over a century. We should also consider the context in which Ramanujan lived, and the most unfavourable and discouraging situation in which he grew up. We will still attempt a stock-taking, to record how far we have moved after Ramanujan left us. Note : The table below should not be used to compare the relative impor- tance or significance of the contributions listed there. It is impossible to list out the entire galaxy of mathematicians for a whole century. The table below may seem incomplete and may contain some inad- vertant errors. If you notice any major lacunae or omissions, or if you have any suggestions, please let me know at [email protected]. 1 April 1920 – April 2020 Year Name/instit. Topic Recognition 1 1949 Dattatreya Kaprekar constant, Ramchandra Kaprekar number Kaprekar [1] [2] 2 1968 P.C. -
Arxiv:2108.08639V1 [Math.CO]
Generalizations of Dyson’s Rank on Overpartitions Alice X.H. Zhao College of Science Tianjin University of Technology, Tianjin 300384, P.R. China [email protected] Abstract. We introduce a statistic on overpartitions called the k-rank. When there are no overlined parts, this coincides with the k-rank of a partition introduced by Garvan. Moreover, it reduces to the D-rank of an overpartition when k = 2. The generating function for the k-rank of overpartitions is given. We also establish a relation between the generating function of self-3-conjugate overpartitions and the tenth order mock theta functions X(q) and χ(q). Keywords: Dyson’s rank, partitions, overpartitions, mock theta functions. AMS Classifications: 11P81, 05A17, 33D15 1 Introduction Dyson’s rank of a partition is defined to be the largest part minus the number of parts [8]. In 1944, Dyson conjectured that this partition statistic provided combinatorial interpre- tations of Ramanujan’s congruences p(5n + 4) ≡ 0 (mod 5) and p(7n + 5) ≡ 0 (mod 7), where p(n) is the number of partitions of n. Let N(m, n) denote the number of partitions of n with rank m. He also found the following generating function of N(m, n) [8, eq. arXiv:2108.08639v1 [math.CO] 19 Aug 2021 (22)]: ∞ 1 ∞ N(m, n)qn = (−1)n−1qn(3n−1)/2+|m|n(1 − qn). (1.1) (q; q) n=0 ∞ n=1 X X Here and in the sequel, we use the standard notation of q-series: ∞ (a; q) (a; q) := (1 − aqn), (a; q) := ∞ , ∞ n (aqn; q) n=0 ∞ Y j(z; q) := (z; q)∞(q/z; q)∞(q; q)∞. -
On the Normality of Numbers
ON THE NORMALITY OF NUMBERS Adrian Belshaw B. Sc., University of British Columbia, 1973 M. A., Princeton University, 1976 A THESIS SUBMITTED 'IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of Mathematics @ Adrian Belshaw 2005 SIMON FRASER UNIVERSITY Fall 2005 All rights reserved. This work may not be reproduced in whole or in part, by photocopy or other means, without the permission of the author. APPROVAL Name: Adrian Belshaw Degree: Master of Science Title of Thesis: On the Normality of Numbers Examining Committee: Dr. Ladislav Stacho Chair Dr. Peter Borwein Senior Supervisor Professor of Mathematics Simon Fraser University Dr. Stephen Choi Supervisor Assistant Professor of Mathematics Simon Fraser University Dr. Jason Bell Internal Examiner Assistant Professor of Mathematics Simon Fraser University Date Approved: December 5. 2005 SIMON FRASER ' u~~~snrllbrary DECLARATION OF PARTIAL COPYRIGHT LICENCE The author, whose copyright is declared on the title page of this work, has granted to Simon Fraser University the right to lend this thesis, project or extended essay to users of the Simon Fraser University Library, and to make partial or single copies only for such users or in response to a request from the library of any other university, or other educational institution, on its own behalf or for one of its users. The author has further granted permission to Simon Fraser University to keep or make a digital copy for use in its circulating collection, and, without changing the content, to translate the thesislproject or extended essays, if technically possible, to any medium or format for the purpose of preservation of the digital work. -
Diophantine Approximation and Transcendental Numbers
Diophantine Approximation and Transcendental Numbers Connor Goldstick June 6, 2017 Contents 1 Introduction 1 2 Continued Fractions 2 3 Rational Approximations 3 4 Transcendental Numbers 6 5 Irrationality Measure 7 6 The continued fraction for e 8 6.1 e's Irrationality measure . 10 7 Conclusion 11 A The Lambert Continued Fraction 11 1 Introduction Suppose we have an irrational number, α, that we want to approximate with a rational number, p=q. This question of approximating an irrational number is the primary concern of Diophantine approximation. In other words, we want jα−p=qj < . However, this method of trying to approximate α is boring, as it is possible to get an arbitrary amount of precision by making q large. To remedy this problem, it makes sense to vary with q. The problem we are really trying to solve is finding p and q such that p 1 1 jα − j < which is equivalent to jqα − pj < (1) q q2 q Solutions to this problem have applications in both number theory and in more applied fields. For example, in signal processing many of the quickest approximation algorithms are given by solutions to Diophantine approximation problems. Diophantine approximations also give many important results like the continued fraction expansion of e. One of the most interesting aspects of Diophantine approximations are its relationship with transcendental 1 numbers (a number that cannot be expressed of the root of a polynomial with rational coefficients). One of the key characteristics of a transcendental number is that it is easy to approximate with rational numbers. This paper is separated into two categories. -
Srinivasa Ramanujan Iyengar
Srinivasa Ramanujan Iyengar Srinivasa Ramanujan Iyengar (Srinivāsa Rāmānujan Iyengār)() (December 22, 1887 – April 26, 1920) was an Indian mathematician widely regarded as one of the greatest mathematical minds in recent history. With almost no formal training in mathematics, he made profound contributions in the areas of analysis and number theory. A child prodigy, Ramanujan was largely self-taught and compiled nearly 3,884 theorems during his short lifetime. Although a small number of these theorems were actually false, most of his statements have now been proven to be correct. His deep intuition and uncanny algebraic manipulative ability enabled him to state highly original and unconventional results that have inspired a vast amount of research; however, some of his discoveries have been slow to enter the mathematical mainstream. Recently his formulae have begun to be applied in the field of crystallography and physics. The Ramanujan Journal was launched specifically to publish work "in areas of mathematics influenced by Ramanujan". With the encouragement of friends, he wrote to mathematicians in Cambridge seeking validation of his work. Twice he wrote with no response; on the third try, he found the English Mathematician G. H. Hardy. Ramanujan's arrival at Cambridge (March 1914) was the beginning of a very successful five-year collaboration with Hardy. One remarkable result of the Hardy-Ramanujan collaboration was a formula for the number p(n) exp(π 2n /3) of partitions of a number n: pn()∼ asn→∞.A partition of a positive integer n is just an expression for n as a sum of 43n positive integers, regardless of order. -
Secondary Indian Culture and Heritage
Culture: An Introduction MODULE - I Understanding Culture Notes 1 CULTURE: AN INTRODUCTION he English word ‘Culture’ is derived from the Latin term ‘cult or cultus’ meaning tilling, or cultivating or refining and worship. In sum it means cultivating and refining Ta thing to such an extent that its end product evokes our admiration and respect. This is practically the same as ‘Sanskriti’ of the Sanskrit language. The term ‘Sanskriti’ has been derived from the root ‘Kri (to do) of Sanskrit language. Three words came from this root ‘Kri; prakriti’ (basic matter or condition), ‘Sanskriti’ (refined matter or condition) and ‘vikriti’ (modified or decayed matter or condition) when ‘prakriti’ or a raw material is refined it becomes ‘Sanskriti’ and when broken or damaged it becomes ‘vikriti’. OBJECTIVES After studying this lesson you will be able to: understand the concept and meaning of culture; establish the relationship between culture and civilization; Establish the link between culture and heritage; discuss the role and impact of culture in human life. 1.1 CONCEPT OF CULTURE Culture is a way of life. The food you eat, the clothes you wear, the language you speak in and the God you worship all are aspects of culture. In very simple terms, we can say that culture is the embodiment of the way in which we think and do things. It is also the things Indian Culture and Heritage Secondary Course 1 MODULE - I Culture: An Introduction Understanding Culture that we have inherited as members of society. All the achievements of human beings as members of social groups can be called culture. -
Network Intrusion Detection with Xgboost and Deep Learning Algorithms: an Evaluation Study
2020 International Conference on Computational Science and Computational Intelligence (CSCI) Network Intrusion Detection with XGBoost and Deep Learning Algorithms: An Evaluation Study Amr Attia Miad Faezipour Abdelshakour Abuzneid Computer Science & Engineering Computer Science & Engineering Computer Science & Engineering University of Bridgeport, CT 06604, USA University of Bridgeport, CT 06604, USA University of Bridgeport, CT 06604, USA [email protected] [email protected] [email protected] Abstract— This paper introduces an effective Network Intrusion In the KitNET model introduced in [2], an unsupervised Detection Systems (NIDS) framework that deploys incremental technique is introduced for anomaly-based intrusion statistical damping features of the packets along with state-of- detection. Incremental statistical feature extraction of the the-art machine/deep learning algorithms to detect malicious packets is passed through ensembles of autoencoders with a patterns. A comprehensive evaluation study is conducted predefined threshold. The model calculates the Root Mean between eXtreme Gradient Boosting (XGBoost) and Artificial Neural Networks (ANN) where feature selection and/or feature Square (RMS) error to detect anomaly behavior. The higher dimensionality reduction techniques such as Principal the calculated RMS at the output, the higher probability of Component Analysis (PCA) and Linear Discriminant Analysis suspicious activity. (LDA) are also integrated into the models to decrease the system Supervised learning has achieved very decent results with complexity for achieving fast responses. Several experimental algorithms such as Random Forest, ZeroR, J48, AdaBoost, runs confirm how powerful machine/deep learning algorithms Logit Boost, and Multilayer Perceptron [3]. Machine/deep are for intrusion detection on known attacks when combined learning-based algorithms for NIDS have been extensively with the appropriate features extracted. -
The Prime Number Theorem a PRIMES Exposition
The Prime Number Theorem A PRIMES Exposition Ishita Goluguri, Toyesh Jayaswal, Andrew Lee Mentor: Chengyang Shao TABLE OF CONTENTS 1 Introduction 2 Tools from Complex Analysis 3 Entire Functions 4 Hadamard Factorization Theorem 5 Riemann Zeta Function 6 Chebyshev Functions 7 Perron Formula 8 Prime Number Theorem © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 2 Introduction • Euclid (300 BC): There are infinitely many primes • Legendre (1808): for primes less than 1,000,000: x π(x) ' log x © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 3 Progress on the Distribution of Prime Numbers • Euler: The product formula 1 X 1 Y 1 ζ(s) := = ns 1 − p−s n=1 p so (heuristically) Y 1 = log 1 1 − p−1 p • Chebyshev (1848-1850): if the ratio of π(x) and x= log x has a limit, it must be 1 • Riemann (1859): On the Number of Primes Less Than a Given Magnitude, related π(x) to the zeros of ζ(s) using complex analysis • Hadamard, de la Vallée Poussin (1896): Proved independently the prime number theorem by showing ζ(s) has no zeros of the form 1 + it, hence the celebrated prime number theorem © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 4 Tools from Complex Analysis Theorem (Maximum Principle) Let Ω be a domain, and let f be holomorphic on Ω. (A) jf(z)j cannot attain its maximum inside Ω unless f is constant. (B) The real part of f cannot attain its maximum inside Ω unless f is a constant. Theorem (Jensen’s Inequality) Suppose f is holomorphic on the whole complex plane and f(0) = 1. -
On Dyson's Crank Conjecture and the Uniform Asymptotic
ON DYSON’S CRANK CONJECTURE AND THE UNIFORM ASYMPTOTIC BEHAVIOR OF CERTAIN INVERSE THETA FUNCTIONS KATHRIN BRINGMANN AND JEHANNE DOUSSE Abstract. In this paper we prove a longstanding conjecture by Freeman Dyson concerning the limiting shape of the crank generating function. We fit this function in a more general family of inverse theta functions which play a key role in physics. 1. Introduction and statement of results Dyson’s crank was introduced to explain Ramanujan’s famous partition congru- ences with modulus 5, 7, and 11. Denoting for n 2 N by p(n) the number of integer partitions of n, Ramanujan [22] proved that for n ≥ 0 p (5n + 4) ≡ 0 (mod 5); p (7n + 5) ≡ 0 (mod 7); p (11n + 6) ≡ 0 (mod 11): A key ingredient of his proof is the modularity of the partition generating function 1 1 X 1 q 24 P (q) := p(n)qn = = ; (q; q) η(τ) n=0 1 Qj−1 ` 2πiτ where for j 2 N0 [ f1g we set (a)j = (a; q)j := `=0(1 − aq ), q := e , and 1 1 1 24 Q n η(τ) := q n=1(1 − q ) is Dedekind’s η-function, a modular form of weight 2 . Ramanujan’s proof however gives little combinatorial insight into why the above congruences hold. In order to provide such an explanation, Dyson [8] famously intro- duced the rank of a partition, which is defined as its largest part minus the number of its parts. He conjectured that the partitions of 5n + 4 (resp. 7n + 5) form 5 (resp.