Research Publications of Ramanujan
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Analytic Vortex Solutions on Compact Hyperbolic Surfaces
Analytic vortex solutions on compact hyperbolic surfaces Rafael Maldonado∗ and Nicholas S. Mantony Department of Applied Mathematics and Theoretical Physics, Wilberforce Road, Cambridge CB3 0WA, U.K. August 27, 2018 Abstract We construct, for the first time, Abelian-Higgs vortices on certain compact surfaces of constant negative curvature. Such surfaces are represented by a tessellation of the hyperbolic plane by regular polygons. The Higgs field is given implicitly in terms of Schwarz triangle functions and analytic solutions are available for certain highly symmetric configurations. 1 Introduction Consider the Abelian-Higgs field theory on a background surface M with metric ds2 = Ω(x; y)(dx2+dy2). At critical coupling the static energy functional satisfies a Bogomolny bound 2 1 Z B Ω 2 E = + jD Φj2 + 1 − jΦj2 dxdy ≥ πN; (1) 2 2Ω i 2 where the topological invariant N (the `vortex number') is the number of zeros of Φ counted with multiplicity [1]. In the notation of [2] we have taken e2 = τ = 1. Equality in (1) is attained when the fields satisfy the Bogomolny vortex equations, which are obtained by completing the square in (1). In complex coordinates z = x + iy these are 2 Dz¯Φ = 0;B = Ω (1 − jΦj ): (2) This set of equations has smooth vortex solutions. As we explain in section 2, analytical results are most readily obtained when M is hyperbolic, having constant negative cur- vature K = −1, a case which is of interest in its own right due to the relation between arXiv:1502.01990v1 [hep-th] 6 Feb 2015 hyperbolic vortices and SO(3)-invariant instantons, [3]. -
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Science Horizon Volume 6 Issue 4 April, 2021 President, Odisha Bigyan Academy Editorial Board Sri Umakant Swain Prof. Niranjan Barik Prof. Ramesh Chandra Parida Editor Er. Mayadhar Swain Dr. Choudhury Satyabrata Nanda Dr. Rajballav Mohanty Managing Editor Dr. Nilambar Biswal Er. Bhagat Charan Mohanty Language Expert Secretary, Odisha Bigyan Academy Prof. Sachidananda Tripathy HJCONTENTS HJ Subject Author Page 1. Editorial : Uttarakhand Glacial Burst-A Warning Er. Mayadhar Swain 178 2. The Star World Prof. B.B. Swain 180 3. Role of Physics in Information and Communication Dr. Sadasiva Biswal 186 Technology 4. National Mathematics Day, National Mathematics Binapani Saikrupa 188 Park & Srinivasa Ramanujan 5. A Brief History of Indian Math-Magicians (Part IV) Purusottam Sahoo 192 6. Glacier Dr. Manas Ranjan Senapati 197 7. Do Plants Recognise their Kins? Dr. Taranisen Panda 198 Dr. Raj Ballav Mohanty 8. Paris Climate Agreement and India’s Response Dr. Sundara Narayana Patro 201 9. Fluoride Contamination in Groundwater with Jayant Kumar Sahoo 205 Special Reference to Odisha 10. Electricity from Trees Dr. Ramesh Chandra Parida 209 11. Hydrogen as a Renewable Energy for Sustainable Prof. (Dr.) L.M. Das 212 Future 12. Nomophobia Dr. Namita Kumari Das 217 13. Quiz: Mammals Dr. Kedareswar Pradhan 220 14. Recent News on Science & Technology 222 The Cover Page depicts : Green Hydrogen Cover Design : Kalakar Sahoo APRIL, 2021 // EDITORIAL // UTTARAKHAND GLACIAL BURST- A WARNING Er. Mayadhar Swain On 7th February 2021, a flash flood occurred in Chamoli district of Uttarakhand villagers, mostly women, were uprooted in causing devastating damage and killing 204 this flood. At Reni, Risiganga joins the river persons. -
Ramanujan and Labos Primes, Their Generalizations, and Classifications
1 2 Journal of Integer Sequences, Vol. 15 (2012), 3 Article 12.1.1 47 6 23 11 Ramanujan and Labos Primes, Their Generalizations, and Classifications of Primes Vladimir Shevelev Department of Mathematics Ben-Gurion University of the Negev Beer-Sheva 84105 Israel [email protected] Abstract We study the parallel properties of the Ramanujan primes and a symmetric counter- part, the Labos primes. Further, we study all primes with these properties (generalized Ramanujan and Labos primes) and construct two kinds of sieves for them. Finally, we give a further natural generalization of these constructions and pose some conjectures and open problems. 1 Introduction The very well-known Bertrand postulate (1845) states that, for every x > 1, there exists a prime in the interval (x, 2x). This postulate quickly became a theorem when, in 1851, it was unexpectedly proved by Chebyshev (for a very elegant version of his proof, see Theorem 9.2 in [8]). In 1919, Ramanujan ([6]; [7, pp. 208–209]) gave a quite new and very simple proof of Bertrand’s postulate. Moreover, in his proof of a generalization of Bertrand’s, the following sequence of primes appeared ([11], A104272) 2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 127, 149, 151, 167,... (1) Definition 1. For n ≥ 1, the n-th Ramanujan prime is the largest prime (Rn) for which π(Rn) − π(Rn/2) = n. Let us show that, equivalently, Rn is the smallest positive integer g(n) with the property that, if x ≥ g(n), then π(x) − π(x/2) ≥ n. -
Applications of Elliptic Functions in Classical and Algebraic Geometry
APPLICATIONS OF ELLIPTIC FUNCTIONS IN CLASSICAL AND ALGEBRAIC GEOMETRY Jamie Snape Collingwood College, University of Durham Dissertation submitted for the degree of Master of Mathematics at the University of Durham It strikes me that mathematical writing is similar to using a language. To be understood you have to follow some grammatical rules. However, in our case, nobody has taken the trouble of writing down the grammar; we get it as a baby does from parents, by imitation of others. Some mathe- maticians have a good ear; some not... That’s life. JEAN-PIERRE SERRE Jean-Pierre Serre (1926–). Quote taken from Serre (1991). i Contents I Background 1 1 Elliptic Functions 2 1.1 Motivation ............................... 2 1.2 Definition of an elliptic function ................... 2 1.3 Properties of an elliptic function ................... 3 2 Jacobi Elliptic Functions 6 2.1 Motivation ............................... 6 2.2 Definitions of the Jacobi elliptic functions .............. 7 2.3 Properties of the Jacobi elliptic functions ............... 8 2.4 The addition formulæ for the Jacobi elliptic functions . 10 2.5 The constants K and K 0 ........................ 11 2.6 Periodicity of the Jacobi elliptic functions . 11 2.7 Poles and zeroes of the Jacobi elliptic functions . 13 2.8 The theta functions .......................... 13 3 Weierstrass Elliptic Functions 16 3.1 Motivation ............................... 16 3.2 Definition of the Weierstrass elliptic function . 17 3.3 Periodicity and other properties of the Weierstrass elliptic function . 18 3.4 A differential equation satisfied by the Weierstrass elliptic function . 20 3.5 The addition formula for the Weierstrass elliptic function . 21 3.6 The constants e1, e2 and e3 ..................... -
Ramanujan Primes: Bounds, Runs, Twins, and Gaps
Ramanujan Primes: Bounds, Runs, Twins, and Gaps Jonathan Sondow 209 West 97th Street New York, NY 10025 USA [email protected] John W. Nicholson P. O. Box 2423 Arlington, TX 76004 USA [email protected] Tony D. Noe 14025 NW Harvest Lane Portland, OR 97229 USA [email protected] arXiv:1105.2249v2 [math.NT] 1 Aug 2011 Abstract The nth Ramanujan prime is the smallest positive integer Rn such that if x ≥ Rn, 1 then the interval 2 x,x contains at least n primes. We sharpen Laishram’s theorem that Rn < p3n by proving that the maximum of Rn/p3n is R5/p15 = 41/47. We give statistics on the length of the longest run of Ramanujan primes among all primes p < 10n, for n ≤ 9. We prove that if an upper twin prime is Ramanujan, then so is the lower; a table gives the number of twin primes below 10n of three types. Finally, we relate runs of Ramanujan primes to prime gaps. Along the way we state several conjectures and open problems. The Appendix explains Noe’s fast algorithm for computing R1,R2,...,Rn. 1 1 Introduction For n ≥ 1, the nth Ramanujan prime is defined as the smallest positive integer Rn with 1 the property that for any x ≥ Rn, there are at least n primes p with 2 x<p ≤ x. By its 1 minimality, Rn is indeed a prime, and the interval 2 Rn, Rn contains exactly n primes [10]. In 1919 Ramanujan proved a result which implies that Rn exists, and he gave the first five Ramanujan primes. -
V. Analytic Continuation and Riemann Sur- Faces
MTH6111 Complex Analysis 2009-10 Lecture Notes c Shaun Bullett 2009 V. Analytic Continuation and Riemann Sur- faces This section will introduce various ideas that are important for more advanced work in complex analysis. 1. Continuation along paths A holomorphic function f defined on some open disc containing z0, has a Taylor series ∞ (n) X f (z0) (z − z )n n! 0 n=0 We can use this series to define f at any points inside the disc D(z0, r0) of convergence of the series where f has not already been defined. Now choosing a new point z1 near the edge of D(z0, r0) we can repeat the same procedure for the Taylor series ∞ (n) X f (z1) (z − z )n n! 1 n=0 and this will enable us to define f in a new disc D(z1, r1), overlapping the first disc. This process is known as analytic continuation of f. An analytic function is one that can be written as the sum of a power series: note that we can only carry out the continuation process because of Taylor’s Theorem, which says that a holomorphic function is an analytic function. The most interesting examples of this process occur when f is one branch of a multi-valued function, and we follow the analytic continuation of this branch along some path, or even round a loop back to where we started. This is best illustrated with an example. √ Let f(z) = z. Around the point z = 1, one branch of this function can be defined by the power series: (1/2) (1/2)(−1/2) (1/2)(−1/2)(−3/2) (1 + (z − 1))1/2 = 1 + (z − 1) + (z − 1)2 + (z − 1)3 + .. -
13. Trigonometric Functions, Elliptic Functions 1. Reconsideration of Sinx
(February 24, 2021) 13. Trigonometric functions, elliptic functions Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ [This document is http:=/www.math.umn.edu/egarrett/m/complex/notes 2020-21/13 elliptic.pdf] 1. Reconsideration of sin x 2. Construction of singly-periodic functions 3. Arc length of ellipses: elliptic integrals, elliptic functions 4. Construction of doubly-periodic functions 5. Fields of elliptic functions 1. Reconsideration of sin x Although we already know quite a bit about trigonometric functions and their role in calculus, their treatment can be redone to emphasize parallels for elliptic integrals and elliptic functions. [1.1] Integral defining arcsin x The length of arc of a piece of a circle is x2 + y2 = 1 is Z x Z x r p d p 2 arc length of circle fragment from 0 to x = 1 + y02 dt = 1 + 1 − t2 dt 0 0 dt x s 1 x r x Z · (−2t)2 Z t2 Z dt = 1 + 2p dt = 1 + dt = p = arcsin x 2 2 2 0 1 − t 0 1 − t 0 1 − t [1.2] Periodicity The periodicity of sin x comes from the multi-valuedness of arcsin x. The multi-valuedness is ascertainable from this integral, since the path from 0 to x can meander through the complex plane, going around the two points ±1 special for the integral. The basic unit of this multi-valuedness is Z dζ = ±2π (counter-clockwise circular path γ enclosing both ±1) p 2 γ 1 − ζ That is, a path integral from 0 to x could go from 0 to x along the real axis, but then add to the path a vertical line segment from x out to a circle of radius 2, traverse the circle an arbitrary number of times, come back along the same segment (thus cancelling the contribution from this segment). -
1. Basic Properties of Elliptic Function Let Ω1,Ω2 Be Two Complex Numbers
1. Basic Properties of Elliptic Function Let !1;!2 be two complex numbers such that the ratio !2=!1 is not purely real. Without loss of generality, we may assume Im τ > 0; where τ = !2=!1: A function f on C is called a doubly periodic function with periods !1;!2 if f(z + !1) = f(z); f(z + !2) = f(z): A doubly periodic function is called an elliptic function if it is meromorphic. Let P be the parallelogram P = fx!1 + y!2 : 0 ≤ x < 1; 0 ≤ y < 1g: We call P a fundamental period-parallelogram for f: The set Λ = fm!1 + n!2 : m; n 2 Zg is called the period lattice for f: Remark. The set Λ is a free abelian group generate by f!1;!2g: Two complex numbers z1; z2 are said to be congruent modulo Λ if z1 − z2 2 Λ: The congruence of z1; z2 is expressed by the notation z1 ≡ z2 mod Λ: Notice that any two points in P are not congruent. In general, a fundamental parallelogram for Λ is a parallelogram P in C such that distinct points of P are not congruent and [ C = (! + P ): !2Λ A cell P for an elliptic function f is a fundamental parallelogram such that f has no zeros and no poles on @P: Theorem 1.1. An elliptic function without poles is a constant. Proof. An elliptic function f with poles is a bounded entire function. By Liouville's theorem, f must be a constant. Proposition 1.1. The number of zeros / poles of an elliptic function in any cell is finite. -
Numbers Quantification
Concept Map Integers Binary System Indices & Logarithms Primes & factorisation Encryption & Computer Cryptography Real Life Applications 1.1 Introduction: In this modern age of computers, primes play an important role to make our communications secure. So understanding primes are very important. In Section 1.1., we give an introduction to primes and its properties and give its application to encryption in 1.2. The concept of binary numbers and writing a number in binary and converting a number in binary to decimal is presented in 1.3. The Section 1.4 introduces the concept of complex numbers with some basic properties. In Sections 1.5, 1.6 and 1.7. logarithms, its properties and applications to real life problems are introduced. Finally in Section 1.8 we give a number of word problems which we come across in everyday life. [2] 1.2 Prime Numbers We say integer a divides b, and write a|b in symbols, if b = ac with c ∈ Z. For example 9|27. If a|b, we say a is a factor or a is divisor of b. And we also say b is a multiple of a. It is easy to see that 1 and –1 are divisors of any integer and every non-zero integer is a divisor of 0. We write a|b if a does not divide b. For example 5|13. Prime numbers are the building blocks of number system. A positive integer p > 1 is called a prime if 1 and p are the only positive divisors of p. For example, 2, 3, 5, 7, 11, 13, .... -
Vedic Mathematics Indian Mathematics from Vedic Period Until
Vedic Mathematics Indian Mathematics from Vedic Period until today is ‘Vedic Mathematics’ Ravi Kumar Iyer Mob. +91 8076 4843 56 Vedic Mathematics I am sorry I am not able to meet you Physically. But once the pandemic is over, let us meet and learn VM properly. Today is only a TRILER I need your cooperation If possible pl sit with your school going children above the age of 12. They pick up very quickly I have conducted VM workshops in many leading universities in USA, Canada, Holland, Norway, Australia, New Zealand etc. Also in Royal Society My 5 Sessions on VM in Radio Sydney won maximum attendance award [email protected], www.hindugenius.blogspot.com Quotes on Indian Mathematics We owe a lot to Indians, who taught us how to count, without which no worthwhile scientific discovery could have been made. Albert Einstein Let Noble Thoughts come from all 4 directions. Rig Veda Ancient Vedic Shloka over 5,000 years back "Om purna mada purna midam Purnaat purnam udachyate Purnasya purnam adaaya Purnam eva vasishyate Om shanti shanti shantih” (Isha Upanishad) Guillaume de l'Hôpital Which translates into: 1661- 1704 France, Paris INFINITY ÷÷ INFINITY = INFINITY "That is the whole, this is the Whole; from the Whole, the Whole arises; taking away the Whole from the Whole, the Whole remains“ (Replace Whole by Infinity) Great Mathematicians of Vedic Period Indian Mathematics from Vedic Period until today is ‘Vedic Mathematics’ How old is Vedic Civilization ?? How old is Vedic Civilization ?? Thomas Alva Edison (1847 – 1931) Gramaphone 1877 Max Müller (1823 – 1900) I worship Agni who is the priest, the one who leads us from the front, who is the deity subject matter of a ritual, a yajna who is the one who makes the formal invocations in the yajna who is the source, storehouse and the bestower of all wealth, gems, precious stones etc 1 -1-1 of Rigvedam. -
Elliptic Functions: Introduction Course
Elliptic functions: Introduction course Vladimir G. TKACHEV Department of Mathematics, Royal Institute of Technology Lindstedtsv¨agen 25, 10044 Stockholm, Sweden email: [email protected] URL: http://www.math.kth.se/˜tkatchev Contents Chapter 1. Elliptic integrals and Jacobi’s theta functions 5 1.1. Elliptic integrals and the AGM: real case 5 1.2. Lemniscates and elastic curves 11 1.3. Euler’s addition theorem 18 1.4. Theta functions: preliminaries 24 Chapter 2. General theory of doubly periodic functions 31 2.1. Preliminaries 31 2.2. Periods of analytic functions 33 2.3. Existence of doubly periodic functions 36 2.4. Liouville’s theorems 38 2.5. The Weierstrass function ℘(z) 43 2.6. Modular forms 51 Bibliography 61 3 CHAPTER 1 Elliptic integrals and Jacobi’s theta functions 1.1. Elliptic integrals and the AGM: real case 1.1.1. Arclength of ellipses. Consider an ellipse with major and minor arcs 2a and 2b and eccentricity e := (a2 b2)/a2 [0, 1), e.g., − ∈ x2 y2 + = 1. a2 b2 What is the arclength `(a; b) of the ellipse, as a function of a and b? There are two easy observations to be made: (1) `(ra; rb) = r`(a; b), because rescaling by a factor r increases the arclength by the same factor; (2) `(a; a) = 2πa, because we know the circumference of a circle. Of course, π is transcendental so it is debatable how well we understand it! 1 0.5 ±2 ±1 0 1 2 ±0.5 ±1 2 y2 Figure 1. Ellipse x + 4 = 1 The total arclength is four times the length of the piece in the first quadrant, where we have the relations 2 xb 1 y = b 1 (x/a) , y0(x)= . -
Existence of Ramanujan Primes for GL(3)
Existence of Ramanujan primes for GL(3) Dinakar Ramakrishnan 253-37 Caltech, Pasadena, CA 91125 To Joe Shalika with admiration Introduction Let π beacuspformonGL(n)/Q , i.e., a cuspidal automophic representation of GL(n, A), where A denotes the adele ring of Q . We will say that a prime p is a Ramanujan prime for π iff the corresponding πp is tempered.The local component πp will necessarily be unramified for almost all p, determined by an unordered n-tuple {α1,p,α2,p,...,αn,p} of non-zero complex numbers, often represented by the corresponding diagonal matrix Ap(π)inGL(n, C), unique up to permutation of the diagonal entries. The L-factor of π at p is given by n −s −1 −s −1 L(s, πp)=det(I − Ap(π)p ) = (1 − αj,pp ) . j=1 As π is unitary, πp is tempered (in the unramified case) iff each αj,p is of absolute value 1. It was shown in [Ra] that for n = 2,thesetR(π)of Ramanujan primes for π is of lower density at least 9/10. When one applies in addition the deep recent results of H. Kim and F. Shahidi ([KSh]) on the symmetric cube and the symmetric fourth power liftings for GL(2), the lower bound improves from 9/10 to 34/35 (loc. cit.), which is 0.971428 .... For n>2 there is a dearth of results for general π, though for cusp forms of regular infinity type, assumed for n>3 to have a discrete series component at a finite place, one knows by [Pic] for n = 3, which relies on the works of J.