DOCSLIB.ORG

Chen Prime Liczby Pierwsze Chena

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chen Prime Liczby Pierwsze Chena Chen Prime Liczby pierwsze Chena Chen Jingrun • Data urodzenia: 22 maj 1933 • Data śmierci: 19 marzec 1996 Pochodzi z wielodzietnej rodziny z Fuzhou, Fujian, Chiny. W 1953 roku skończył wydział matematyki na Uniwersytecie w Xiamen. Jego prace nad przypuszczeniem o bliźniaczych liczbach pierwszych oraz hipotezą Goldbacha doprowadziły do postępu analitycznej teorii liczb. Największym jego osiągnięciem było tzw. twierdzenie Chena stanowiące słabszą wersję słynnej hipotezy Goldbacha. Nazwiskiem Chen Jingruna została nazwana planetoida 7681 Chenjingrun odkryta w 1996 roku Hipoteza Goldbacha • jeden z najstarszych nierozwiązanych problemów w teorii liczb, liczy sobie ponad 250 lat • W 1742 roku, w liście do Leonharda Eulera, Christian Goldbach postawił hipotezę: każda liczba naturalna większa niż 2 może być przedstawiona w postaci sumy trzech liczb pierwszych (ta sama liczba pierwsza może być użyta dwukrotnie) Euler po otrzymaniu listu stwierdził iż hipotezę Goldbacha można uprościć i przedstawić ją w następujący sposób: każda liczba naturalna parzysta większa od 2 jest sumą dwóch liczb pierwszych Powyższą hipotezę do dzisiaj nazywaną "hipotezą Goldbacha" sformułował w rezultacie Euler, jednak nazwa nie została zmieniona. Oto kilka prostych przykładów: 4=2+2 6=3+3 8=3+5 10=3+7=5+5 … 100=53+47… Dzięki użyciu komputerów udało się pokazać, że hipoteza Goldbacha jest prawdziwa dla liczb naturalnych mniejszych niż 4 × 1017 (przez przedstawienie każdej z tych liczb w postaci sumy dwóch liczb pierwszych). Co więcej, większość współczesnych matematyków uważa, iż jest ona prawdziwa, ponieważ ze względu na stosunkowo gęsty rozkład liczb pierwszych wydaje się, że większe liczby parzyste coraz łatwiej jest przedstawić w postaci sumy dwóch liczb pierwszych. Hipoteza Goldbacha pozostaje do dnia dzisiejszego nierozstrzygnięta Liczby półpierwsze • liczby posiadające dokładnie dwa czynniki pierwsze, odgrywają one znaczącą rolę w kryptografii • liczby półpierwsze występują maksymalnie po trzy obok siebie Wynika to z podzielności przez 4 (nie może być czterech kolejnych liczb pierwszych, bo jedna z nich byłaby podzielna przez 4, a więc podzielna także przez 2) Przykładowe trójki liczb półpierwszych: (33,34,35) (85,86,87) (93,94,95) (121,122,123) (141,142,143) (201,202,203) (213,214,215) Twierdzenie Chena Zostało udowodnione w roku 1966 i nie jest hipotezą. Twierdzenie Chena różni się jedynie tym od hipotezy Goldbacha, że drugi składnik sumy może być liczbą półpierwszą. Liczby pierwsze Chena Liczba pierwsza Chena jest liczba pierwszą postaci: p+2 gdzie: p jest dowolną liczbą pierwszą, p+2 natomiast może być liczbą pierwszą bądź półpierwszą. początkowe elementy ciągu liczb pierwszych Chena: 2,3,5,7,11,13,17,19,23,29,31,37,41,47,53,59,67,71,83,89,101 Największą znaną liczbą Chena (odkryta w październiku 2005) posiadającą 70301 cyfr jest: (1 284 991 359x298305 +1)x(96 060 285x2135170 +1)-2 Rudolf Ondrejka odkrył następujący magiczny kwadrat (3x3) złożony z 9 liczb pierwszych chena: 17 89 71 113 59 5 47 29 101 Home Prime Liczbę home pierwszą HP(n), osiągamy w następujący sposób: - zaczynamy od liczby n - liczbę n zapisujemy jako iloczyn jej czynników pierwszych - następnie łączymy czynniki pierwsze w jedną liczbę - czynność powtarzamy, az do osiągnięcia liczby pierwszej np. dla n=9 9=3*3→33=3*11→311 tak więc liczba 311 jest home liczbą pierwszą dla 9 Dla n=2,3,4,…, kilkoma pierwszymi home liczbami są liczby: 2,3,211,5,23,7,3331113965338635107,311,773,….. Tak więc home liczba pierwsza powinna istnieć dla każdej dodatniej liczby całkowitej. Odkąd liczby pierwsze posiadają proste home liczby pierwsze, uwagę skupia się na liczbach złożonych. Liczba kroków do osiągnięcia home liczby pierwszej dla złożonych liczb, Odkąd liczby pierwsze posiadają proste home liczby pierwsze, uwagę skupia się liczbach złożonych. Liczba kroków do osiągnięcia home liczby pierwszej dla złożonych liczb, takich jak: 4, 6, 8, 9, 10, … wynosi: 2, 1, 13, 2, 4, … a osiągnięte home liczby pierwsze wynoszą odpowiednio: 211, 23, 3331113965338635107, 311, 773, … takich jak: 4, 6, 8, 9, 10, ... wynosi: 2, 1, 13, 2, 4, … a osiągane home liczby pierwsze to odpowiednio: 211, 23, 3331113965338635107, 311, 773, … Największą home liczbą pierwszą dla n<100 jest liczba: HP(49) = HP(77) Kilka pierwszych liczb w sekwencji home liczby dla n=49 to: 49, 77, 711, 3379, 31109, 132393, 344131, … Obliczanie sekwencji dla tej cyfry zakończyło się aktualnie na setnym kroku. Bibliografia: http://en.wikipedia.org/wiki/Chen_Jingrun http://en.wikipedia.org/wiki/Chen_prime http://pl.wikipedia.org/wiki/Hipoteza_Goldbacha http://mathworld.wolfram.com/ChenPrime.html http://mathworld.wolfram.com/HomePrime.html Dziękuję .
Load more

Recommended publications
  • Goldbach, Christian
    CHRISTIAN GOLDBACH (March 18, 1690 – November 20, 1764) by HEINZ KLAUS STRICK, Germany One of the most famous, still unproven conjectures of number theory is: • Every even number greater than 2 can be represented as the sum of two prime numbers. The scholar CHRISTIAN GOLDBACH made this simple mathematical statement to his pen pal LEONHARD EULER in 1742 as an assumption. (In the original version it said: Every natural number greater than 2 can be represented as the sum of three prime numbers, since at that time the number 1 was still considered a prime number.) All attempts to prove this theorem have so far failed. Even the offer of a prize of one million dollars hardly led to any progress. CHEN JINGRUN (1933-1996, Chinese stamp on the left), student of HUA LUOGENG (1910-1985, stamp on the right), the most important Chinese mathematician of the 20th century, succeeded in 1966 in making the "best approximation" to GOLDBACH's conjecture. CHEN JINGRUN was able to prove that any sufficiently large even number can be represented as the sum of a prime number and another number that has at most two prime factors. Among the first even numbers are those that have only one GOLDBACH decomposition: 4 = 2 + 2; 6 = 3 + 3; 8 = 3 + 5; 12 = 5 + 7. For larger even numbers one finds a "tendency" to increase the number of possibilities, but even then there is always a number that has only a few decompositions, such as 98 = 19 + 79 = 31 + 67 = 37 + 61. See the graph below and The On-Line Encyclopedia of Integer Sequences A045917.
    [Show full text]
  • Conjecture of Twin Primes (Still Unsolved Problem in Number Theory) an Expository Essay
    Surveys in Mathematics and its Applications ISSN 1842-6298 (electronic), 1843-7265 (print) Volume 12 (2017), 229 { 252 CONJECTURE OF TWIN PRIMES (STILL UNSOLVED PROBLEM IN NUMBER THEORY) AN EXPOSITORY ESSAY Hayat Rezgui Abstract. The purpose of this paper is to gather as much results of advances, recent and previous works as possible concerning the oldest outstanding still unsolved problem in Number Theory (and the most elusive open problem in prime numbers) called "Twin primes conjecture" (8th problem of David Hilbert, stated in 1900) which has eluded many gifted mathematicians. This conjecture has been circulating for decades, even with the progress of contemporary technology that puts the whole world within our reach. So, simple to state, yet so hard to prove. Basic Concepts, many and varied topics regarding the Twin prime conjecture will be cover. Petronas towers (Twin towers) Kuala Lumpur, Malaysia 2010 Mathematics Subject Classification: 11A41; 97Fxx; 11Yxx. Keywords: Twin primes; Brun's constant; Zhang's discovery; Polymath project. ****************************************************************************** http://www.utgjiu.ro/math/sma 230 H. Rezgui Contents 1 Introduction 230 2 History and some interesting deep results 231 2.1 Yitang Zhang's discovery (April 17, 2013)............... 236 2.2 "Polymath project"........................... 236 2.2.1 Computational successes (June 4, July 27, 2013)....... 237 2.2.2 Spectacular progress (November 19, 2013)........... 237 3 Some of largest (titanic & gigantic) known twin primes 238 4 Properties 240 5 First twin primes less than 3002 241 6 Rarefaction of twin prime numbers 244 7 Conclusion 246 1 Introduction The prime numbers's study is the foundation and basic part of the oldest branches of mathematics so called "Arithmetic" which supposes the establishment of theorems.
    [Show full text]
  • N N Ln N N Ln N N Ln N N Ln
    Practice and Understanding of the Goldbach Problem (A) Tong Xinping (童 信 平) [email protected] Abstract Practice shows: Without enough parameters, 1+2 cannot estimate 1+1 and 1+1×1. At the same time, it is pointed out that among the prime numbers not greater than N, the prime number 6t-1 is more than the prime number 6t+1, (this is the "detail" that Hardy and the others did not notice.) "detail" (= "remainder") affects Represents the negligible oscillation in the accuracy curve of the number r2(N) of Hardy-Litttlewood conjecture (A) in 1921. (In 1989, Hua Luogeng (华罗庚) confirmed that conjecture (A) is the "main term" of even Goldbach's problem.) [Tong Xinping(童信平)Practice and Understanding of theGoldbachProblem(A)AcademArena2021;13(4):71-75]. ISSN1553-992X(print);ISSN2158-771X(online).http://www.sciencepub.net/academia. 7. doi:10.7537/marsaaj130421. 07. Keywords: Practice; Understanding; Goldbach; Problem In 1977, from a logical and philosophical point of view, I thought that "1+2" could not reach "1+1" (short The c(N) in the formula is represented c in the other for even Goldbach conjecture). Starting from interest, I book. am determined to find mathematical evidence that "1+2" Wang Yuan said that to prove "1+1" is to prove cannot reach "1+1" under the guidance of philosophical (1+O(1))→1, which is the "remainder" O(1)→0. The viewpoints. Seeing Xu Chi's article, I dare not act. -0.665 here is a definite value, not →0, and it cannot be However, the passion in my heart cannot be extinguished.
    [Show full text]
  • Download Article (PDF)
    Advances in Social Science, Education and Humanities Research, volume 120 World Conference on Management Science and Human Social Development (MSHSD 2017) On the Infiltration of Ideological and Moral Education in Primary Mathematics Teaching Qin Wang1, a and JiangJing Xia 2,b 1 Wuhan University of Technology, Wuhan 430070, China 2 Wuhan University of Technology, Wuhan 430070, China [email protected], [email protected] Keywords: Ideological and moral education, primary school mathematics, infiltration Abstract. Primary school is the initial stage of the formation of a person's outlook on life, values and morality. During this period, the ideological and moral education of students is directly related to the students' ideological and moral standards and personality construction. Primary mathematics teaching not only has a certain role of knowledge inheritance, but also has a certain role in moral education. The infiltration of moral education in mathematics teaching not only improves the ability of mathematical knowledge, but also improves the comprehensive quality of students effectively. According to the students' existing knowledge base, actively looking for the best combination of moral education and mathematics knowledge, well-designed teaching content, grasp the opportunity of teaching, utilizing the characteristics of mathematics to excavate the ideological and moral education contents contained in mathematics, and realizing mathematics teaching and the combination of ideological and moral education is an important issue in primary school mathematics education. 1 Introduction Syllabus for primary school: "when students master basic knowledge, they develop intelligence, improve their ability and receive ideological and moral education.". Primary school mathematics is a very important discipline in elementary education.
    [Show full text]
  • Visiting Mathematicians Jon Barwise, in Setting the Tone for His New Column, Has Incorporated Three Articles Into This Month's Offering
    OTICES OF THE AMERICAN MATHEMATICAL SOCIETY The Growth of the American Mathematical Society page 781 Everett Pitcher ~~ Centennial Celebration (August 8-12) page 831 JULY/AUGUST 1988, VOLUME 35, NUMBER 6 Providence, Rhode Island, USA ISSN 0002-9920 Calendar of AMS Meetings and Conferences This calendar lists all meetings which have been approved prior to Mathematical Society in the issue corresponding to that of the Notices the date this issue of Notices was sent to the press. The summer which contains the program of the meeting. Abstracts should be sub­ and annual meetings are joint meetings of the Mathematical Associ­ mitted on special forms which are available in many departments of ation of America and the American Mathematical Society. The meet­ mathematics and from the headquarters office of the Society. Ab­ ing dates which fall rather far in the future are subject to change; this stracts of papers to be presented at the meeting must be received is particularly true of meetings to which no numbers have been as­ at the headquarters of the Society in Providence, Rhode Island, on signed. Programs of the meetings will appear in the issues indicated or before the deadline given below for the meeting. Note that the below. First and supplementary announcements of the meetings will deadline for abstracts for consideration for presentation at special have appeared in earlier issues. sessions is usually three weeks earlier than that specified below. For Abstracts of papers presented at a meeting of the Society are pub­ additional information, consult the meeting announcements and the lished in the journal Abstracts of papers presented to the American list of organizers of special sessions.
    [Show full text]
  • Chen Jingrun, China's Famous Mathematician
    NEUROSURGICAL FOCUS Neurosurg Focus 41 (1):E11, 2016 Chen Jingrun, China’s famous mathematician: devastated by brain injuries on the doorstep to solving a fundamental mathematical puzzle Ting Lei, MD,1 Evgenii Belykh, MD,1 Alexander B. Dru, BA,2 Kaan Yagmurlu, MD,1 Ali M. Elhadi, MD,1 Peter Nakaji, MD,1 and Mark C. Preul, MD1 1Department of Neurosurgery, Barrow Neurological Institute, St. Joseph’s Hospital and Medical Center, Phoenix; and 2University of Arizona College of Medicine, Tucson, Arizona Chen Jingrun (1933–1996), perhaps the most prodigious mathematician of his time, focused on the field of analytical number theory. His work on Waring’s problem, Legendre’s conjecture, and Goldbach’s conjecture led to progress in ana- lytical number theory in the form of “Chen’s Theorem,” which he published in 1966 and 1973. His early life was ravaged by the Second Sino-Japanese War and the Chinese Cultural Revolution. On the verge of solving Goldbach’s conjecture in 1984, Chen was struck by a bicyclist while also bicycling and suffered severe brain trauma. During his hospitalization, he was also found to have Parkinson’s disease. Chen suffered another serious brain concussion after a fall only a few months after recovering from the bicycle crash. With significant deficits, he remained hospitalized for several years with- out making progress while receiving modern Western medical therapies. In 1988 traditional Chinese medicine experts were called in to assist with his treatment. After a year of acupuncture and oxygen therapy, Chen could control his basic bowel and bladder functions, he could walk slowly, and his swallowing and speech improved.
    [Show full text]
  • Chapter 1: Matrix Online Characteristic Number and Goldbach’S Conjecture
    Chapter 1: Matrix Online Characteristic Number and Goldbach’s Conjecture Baoshan Zhang 1 (School of Mathematics and Statistics, Nanjing Audit University, Nanjing, China , Zip code: 211815) Email: [email protected], [email protected] Abstract：In this paper, it is determined by the consecutive odd numbers between 3 to 2n +1, and study to the intrinsic properties of a class of matrix sequence {}An . Through the establishment of matrix online number concept, characteristics and the online number column use mathematical induction to prove the some properties of this kind of matrix on the number of online features (Theorem 1). Finally, it is given a trial to prove the Goldbach’s conjecture (Theorem 6). This is the author in the years to explore prime properties in the process of research and discovery, and believe that this finding is of great significance. Keywords Prime numbers；Online character; Online characteristic columns; Goldbach's conjecture. Classification：O15.156 MSC：11A41 §1 Introduction Goldbach 's Conjecture is one of the three Difficult mathematical problems in modern world, It was first found in teaching by C.Goldbach(1690-1764）, who was the German high school teachers [1~3]. On June 7, 1742 , Goldbach write to the great mathematician Euler (1707～1783), and asked that his experiments have found. Euler said to Goldbach in a reply letter, that he believes that the conclusion is correct, but he can't prove. And then Euler confirmed the following two guess [1~8]: (1) Strong Goldbach's Conjecture: any more than 4 even can be always expressed as the sum of the two odd primes.
    [Show full text]
  • Chen Jingrun, China's Famous Mathematician: Devastated by Brain
    NEUROSURGICAL FOCUS Neurosurg Focus 41 (1):E11, 2016 Chen Jingrun, China’s famous mathematician: devastated by brain injuries on the doorstep to solving a fundamental mathematical puzzle Ting Lei, MD,1 Evgenii Belykh, MD,1 Alexander B. Dru, BA,2 Kaan Yagmurlu, MD,1 Ali M. Elhadi, MD,1 Peter Nakaji, MD,1 and Mark C. Preul, MD1 1Department of Neurosurgery, Barrow Neurological Institute, St. Joseph’s Hospital and Medical Center, Phoenix; and 2University of Arizona College of Medicine, Tucson, Arizona Chen Jingrun (1933–1996), perhaps the most prodigious mathematician of his time, focused on the field of analytical number theory. His work on Waring’s problem, Legendre’s conjecture, and Goldbach’s conjecture led to progress in ana- lytical number theory in the form of “Chen’s Theorem,” which he published in 1966 and 1973. His early life was ravaged by the Second Sino-Japanese War and the Chinese Cultural Revolution. On the verge of solving Goldbach’s conjecture in 1984, Chen was struck by a bicyclist while also bicycling and suffered severe brain trauma. During his hospitalization, he was also found to have Parkinson’s disease. Chen suffered another serious brain concussion after a fall only a few months after recovering from the bicycle crash. With significant deficits, he remained hospitalized for several years with- out making progress while receiving modern Western medical therapies. In 1988 traditional Chinese medicine experts were called in to assist with his treatment. After a year of acupuncture and oxygen therapy, Chen could control his basic bowel and bladder functions, he could walk slowly, and his swallowing and speech improved.
    [Show full text]
  • Download Article
    Advances in Intelligent Systems Research (AISR), volume 160 3rd International Conference on Modelling, Simulation and Applied Mathematics (MSAM 2018) The Mathematical Modeling and Proof of the Goldbach Conjecture Yu Wang College of Information Science & Technology, Chengdu University of Technology, China Abstract—The Goldbach conjecture declares that any even number 2m=2n+2>4 can be expressed as the sum of two prime I. INTRODUCTION numbers. The mathematical modeling of the conjecture is: any Three major mathematical conjectures include Fermat’s even number 2m=2n+2 greater than 4 can be expressed as Conjecture, Four Color Conjecture and Goldbach Conjecture. 2n+2=a+b, 2≤a≤n+1, n+1≤b≤2n. With the modeling, let c be a The former two conjectures have been proved as Fermat’s Last composite number in 2~2n, a mapping number is 2m-c or 2n+2-c. Theorem and Four color theorem. The Goldbach conjecture A complete composite pair is a pair (c, 2m-c) that both c and 2m-c are composite numbers. The composite numbers one-to-one (referred as the conjecture) is a bright pearl in the field of correspond to the mapping numbers. Using an induction to mathe matics and is regarded as unproved till now. It was raised absurdity, suppose the Goldbach conjecture is wrong, so that in 1742 in a letter fro m Go ldbach to Euler. In 1900, D. Hilbert 2n+2 cannot be expressed as the sum of two primes. With the proposed the Goldbach conjecture as one of the twenty-three mathematical modeling for the even number 2n+2, numbers 2~2n unsolved problems.
    [Show full text]
  • Los Saberes Asiaticos.Indd
    LOS SABERES ASIÁTICOS CIENCIA Y TÉCNICA EN CHINA, INDIA Y JAPÓN DESDE LA ANTIGÜEDAD HASTA NUESTROS DÍAS SERIE PRETEXTOS N.º 42 LOS SABERES ASIÁTICOS CIENCIA Y TÉCNICA EN CHINA, INDIA Y JAPÓN DESDE LA ANTIGÜEDAD HASTA NUESTROS DÍAS Pío García Roberto Restrepo Sandra Salamanca Ricardo Tokunaga Hiroshi Wago Bogotá, diciembre de 2012 Los saberes asiáticos. Ciencia y técnica en China, India y Japón desde la antigüedad hasta nuestros días / Pío García... [et al.]. – Bogotá: Universidad Externado de Colombia, 2012. 519 p.: il. Incluye bibliografía. ISBN: 978958710809 5 1. Ciencia y tecnología – Japón 2. Cambio social – Japón 3. Tecnología y civilización – Japón 4. Ciencia y tecnología – India 5. Cambio social – India 6. Tecnología y civilización – India 7. Ciencia y tecnología – China 8. Cambio social – China 9. Tecnología y civilización – China 10. Historia social – Japón 11. Historia social – India 12. Historia social – China I. García, Pío II. Universidad Externado de Colombia 303.483 SCDD 21 Catalogación en la fuente – Universidad Externado de Colombia. Biblioteca Agosto de 2012 ISBN 978-958-710-809-5 © 2012, PÍO GARCÍA, roberto restrepo, SANDRA SALAMANCA, ricardo toKUNAGA, HIROSHI WAGO © 2012, UNIVERSIDAD EXTERNADO DE COLOMBIA Calle 12 n.º 1-17 este, Bogotá Teléfono (57 1) 342 0288 www.uexternado.edu.co [email protected] Primera edición: diciembre de 2012 Diseño de cubierta: Departamento de Publicaciones Composición: David Alba Impresión y encuadernación: Cadena S. A. Tiraje de 1 a 1.000 ejemplares Impreso en Colombia Printed in Colombia Prohibida la reproducción o cita impresa o electrónica total o parcial de esta obra, sin autorización expresa y por escrito del Departamento de Publicaciones de la Universidad Externado de Colombia.
    [Show full text]
  • Pedidos De Patente Sobre Energia Solar Publicados No 2º Semestre De 2012
    Pedidos de Patente sobre Energia Solar - Nº 8 Pedidos publicados no 2 º semestre de 201 2 Diretoria de Cooperação para o Desenvolvimento – DICOD Centro de Disseminação da Informação Tecnológica – CEDIN Coordenação de Estudos e Programas – CEPRO abril de 2013 INSTITUTO NACIONAL DA PROPRIEDADE INDUSTRIAL - INPI Presidente: Jorge de Paula Costa Ávila Vice-Presidente: Ademir Tardelli DIRETORIA DE COOPERAÇÃO PARA O DESENVOLVIMENTO - DICOD Diretor: Denise Nogueira Gregory CENTRO DE DISSEMINAÇÃO DA INFORMAÇÃO TECNOLÓGICA - CEDIN Chefe: Raul Suster COORDENAÇÃO DE ESTUDOS E PROGRAMAS - CEPRO Chefe: Sérgio Barcelos Theotonio Autor: Sérgio Barcelos Theotonio – Pesquisador em Propriedade Industrial SUMÁRIO 1. INTRODUÇÃO ....................................................................................................4 1.1 - Alerta Tecnológico ..........................................................................................4 1.2 - Pedidos de patentes sobre Energia Solar.......................................................6 2. RESULTADOS....................................................................................................9 2.1 - Mundo.............................................................................................................9 2.2 - Brasil.............................................................................................................14 Anexo 1: Códigos dos Principais Países ...............................................................331 LISTA DE GRÁFICOS Grafico 1: Países de prioridade dos
    [Show full text]
  • View This Volume's Front and Back Matter
    Number Theory and Its Applications in China AMERICAn MATHEMATICAL SOCIETY u0 LU mE 77 http://dx.doi.org/10.1090/conm/077 Titles in This Series Volume 1 Markov random fields and their 18 Fixed points and nonexpansive applications, Ross Kindermann and mappings. Robert C. Sine, Editor J. Laurie Snell 19 Proceedings of the Northwestern 2 Proceedings of the conference homotopy theory conference, on integration, topology, and Haynes R. Miller and Stewart B. geometry in linear spaces, Priddy, Editors William H. Graves. Editor 20 Low dimensional topology, 3 The closed graph and P-closed Samuel J. Lomonaco. Jr., Editor graph properties in general 21 Topological methods in nonlinear topology, T. R. Hamlett and functional analysis, S. P. Singh, L. L. Herrington S. Thomeier, and B. Watson. Editors 4 Problems of elastic stability and 22 Factorizations of b" ± 1, b = vibrations, Vadim Komkov, Editor 2. 3. 5. 6. 7, 10, 5 Rational constructions of 11. 12 up to high powers, modules for simple Lie algebras, John Brillhart. D. H. Lehmer. George B. Seligman J. L. Selfridge, Bryant Tuckerman, and 6 Umbral calculus and Hopf algebras, S. S. Wagstaff, Jr. Robert Morris, Editor 23 Chapter 9 of Ramanujan's second 7 Complex contour integral notebook-Infinite series identities, representation of cardinal spline transformations, and evaluations, functions, Walter Schempp Bruce C. Berndt and Padmini T. Joshi 8 Ordered fields and real algebraic 24 Central extensions, Galois groups, geometry, D. W. Dubois and and ideal class groups of numiJer T. Recio. Editors fields, A. Frohlich 9 Papers in algebra, analysis and 25 Value distribution theory and its statistics.
    [Show full text]