Trigonometry 3A


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  • Respected mathematician
    we all give great tribute and feel proud on all mathematician but still i wonder that still we are unable to value of tan 90
    brahmgupta even mahavira also confirm that N / 0 is N
    we still use N/0 is not define or infinite
    In maths every where we put condition that in fraction , denominator must be greater than zero why ?
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  • bt ganun? d ko maDL. kagabi pa!
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Trigonometry 3A

  1. 1. Parsci Films Incorporated
  2. 2. Makes Trigonometry Learning Easier
  3. 3. Through This Special Slide Presentation
  4. 4. <ul><li>Planned, </li></ul><ul><li>Researched, </li></ul><ul><li>And Prepared by: </li></ul><ul><li>“ Trigonometers ” </li></ul>
  5. 5. <ul><li>Dannicor Beatisula </li></ul><ul><li>Katryna Mae Ann Torres </li></ul><ul><li>Micaela Angela Mangubat </li></ul><ul><li>Floranil Pacina </li></ul><ul><li>Angelica Marie Pelaez </li></ul><ul><li>Aika Gail Vera Cruz </li></ul><ul><li>Justmine Joy Valiente </li></ul><ul><li>Frances Mae Dela Cruz </li></ul><ul><li>Christian Dominic Manga </li></ul><ul><li>Juan Jesus Dela Cruz </li></ul>
  6. 6. <ul><li>In Cooperation with </li></ul><ul><li>Mrs. Jaclyn Pobre </li></ul><ul><li>and </li></ul><ul><li>Mr. Gilbert Cruzado </li></ul>
  7. 7. Mathematicians
  8. 8. Luitzen Egbertus Jan Brouwer Born : 27 Feb 1881 in Overschie (now a suburb of Rotterdam), Netherlands Died : 2 Dec 1966 in Blaricum , Netherlands
  9. 10. Biography
  10. 11. <ul><li>usually known by L.E.J. Brouwer form of his name with full initials, but he was known to his friends as Bertus, an abbreviation of the second of his three forenames </li></ul>
  11. 12. <ul><li>Brouwer studied at the (municipal) University of Amsterdam where his most important teachers were Diederik Korteweg (of the Korteweg-de Vries equation) and, especially philosophically, Gerrit Mannoury. </li></ul>
  12. 13. Korteweg
  13. 14. Gerrit Mannoury
  14. 15. <ul><li>In philosophy, his brainchild is intuitionism , a revisionist foundation of mathematics. </li></ul>
  15. 16. <ul><li>The implications are twofold. </li></ul>
  16. 17. <ul><li>First, it leads to a form of constructive mathematics, in which large parts of classical mathematics are rejected. </li></ul>
  17. 18. <ul><li>Second, the reliance on a philosophy of mind introduces features that are absent from classical mathematics as well as from other forms of constructive mathematics: unlike those, intuitionistic mathematics is not a proper part of classical mathematics. </li></ul>
  18. 19. <ul><li>Brouwer's principal students were Maurits Belinfante and Arend Heyting; the latter, in turn, was the teacher of Anne Troelstra and Dirk van Dalen. </li></ul>
  19. 20. <ul><li>Brouwer's classes were also attended by Max Euwe, the later world chess champion, who published a game-theoretical paper on chess from the intuitionistic point of view (Euwe, 1929), and who would much later deliver Brouwer's funeral speech. </li></ul>
  20. 21. <ul><li>Among Brouwer's assistants were Heyting, Hans Freudenthal, Karl Menger, and Witold Hurewicz, the latter two of whom were not intuitionistically inclined. </li></ul>
  21. 22. <ul><li>The most influential supporter of Brouwer's intuitionism outside the Netherlands at the time was, for a number of years, Hermann Weyl . </li></ul>
  22. 23. Hermann Weyl
  23. 24. <ul><li>Brouwer seems to have been an independent and brilliant man of high moral standards, but with an exaggerated sense of justice, making him at times pugnacious. As a consequence, in his life he energetically fought many battles. </li></ul>
  24. 25. <ul><li>From 1914 to 1928, Brouwer was member of the editiorial board of the Mathematische Annalen , and he was the founding editor of Compositio Mathematica , which first appeared in 1934. </li></ul>
  25. 26. <ul><li>He was a member of, among others, the Royal Dutch Academy of Sciences, the Royal Society in London, the Preußische Akademie der Wissenschaften in Berlin, and the Akademie der Wissenschaften in Göttingen. </li></ul>
  26. 27. <ul><li>Brouwer received honorary doctorates from the universities of Oslo (1929) and Cambridge (1954), and was made Knight in the Order of the Dutch Lion in 1932. </li></ul>
  27. 28. <ul><li>Brouwer's archive is kept at the Department of Philosophy, Utrecht University, the Netherlands. An edition of correspondence and manuscripts is in preparation. </li></ul>
  28. 29. Contribution
  29. 30. Luitzen Egbertus Jan Brouwer <ul><li>founded modern topology by establishing, for example, the topological invariance of dimension and the fixpoint theorem . </li></ul>
  30. 31. <ul><li>gave the first correct definition of dimension </li></ul>
  31. 32. Brouwer's Fix Point Theorem
  32. 33. <ul><li>Theorem 1   Every continuous mapping f of a closed n -ball to itself has a fixed point. Alternatively, Let be a non empty compact convex set and a continuous function. Then f has a fix point, i.e. f ( x )= x for some </li></ul>
  33. 34. <ul><li>founded the doctrine of mathematical intuitionism, which views mathematics as the formulation of mental constructions that are governed by self-evident laws. He became an honorary member of the EMS in 1954. </li></ul>
  34. 35. Questions & Answer
  35. 36. What is topology ?
  36. 37. Topology <ul><li>the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects </li></ul>
  37. 38. What is intuitionism ?
  38. 39. <ul><li>views mathematics as a free activity of the mind, independent of any language or Platonic realm of objects, and therefore bases mathematics on a philosophy of mind. </li></ul>Intuitionism
  39. 40. Andrey Nikolaevich Kolmogorov Born : 25 April 1903 in Tambov, Tambov province, Russia Died : 20 Oct 1987 in Moscow, Russia
  40. 42. Biography
  41. 43. <ul><li>Andrei Nikolaevich Kolmogorov 's parents were not married and his father took no part in his upbringing. </li></ul>
  42. 44. <ul><li>His father Nikolai Kataev , the son of a priest, was an agriculturist who was exiled. </li></ul>
  43. 45. <ul><li>He returned after the Revolution to head a Department in the Agricultural Ministry but died in fighting in 1919. </li></ul>
  44. 46. <ul><li>Kolmogorov's mother also, tragically, took no part in his upbringing since she died in childbirth at Kolmogorov's birth. </li></ul>
  45. 47. <ul><li>His mother's sister, Vera Yakovlena, brought Kolmogorov up and he always had the deepest affection for her. </li></ul>
  46. 48. <ul><li>In fact it was chance that had Kolmogorov born in Tambov since the family had no connections with that place. </li></ul>
  47. 49. <ul><li>Kolmogorov's mother had been on a journey from the Crimea back to her home in Tunoshna near Yaroslavl and it was in the home of his maternal grandfather in Tunoshna that Kolmogorov spent his youth. </li></ul>
  48. 50. <ul><li>Kolmogorov's name came from his grandfather, Yakov Stepanovich Kolmogorov, and not from his own father. Yakov Stepanovich was from the nobility, a difficult status to have in Russia at this time, and there is certainly stories told that an illegal printing press was operated from his house. </li></ul>
  49. 51. <ul><li>Kolmogorov graduated from Moscow State University in 1925 and then became a professor there in 1931. </li></ul>
  50. 52. <ul><li>In 1939 he was elected to the Soviet Academy of Sciences, receiving the Lenin Prize in 1965 and the Order of Lenin on seven separate occasions. </li></ul>
  51. 53. Contribution
  52. 54. <ul><li>He laid the mathematical foundations of probability theory and the algorithmic theory of randomness and made crucial contributions to the foundations of statistical mechanics , stochastic processes , information theory , fluid mechanics , and nonlinear dynamics . </li></ul>
  53. 55. <ul><li>All of these areas, and their interrelationships, underlie complex systems, as they are studied today. </li></ul>
  54. 56. <ul><li>His work on reformulating probability started with a 1933 paper in which he built up probability theory in a rigorous way from fundamental axioms, similar to Euclid's treatment of geometry. </li></ul>
  55. 57. <ul><li>Kolmogorov went on to study the motion of the planets and turbulent fluid flows, later publishing two papers in 1941 on turbulence that even today are of fundamental importance. </li></ul>
  56. 58. <ul><li>In 1954 he developed his work on dynamical systems in relation to planetary motion, thus demonstrating the vital role of probability theory in physics and re-opening the study of apparent randomness in deterministic systems, much along the lines originally conceived by Henri Poincare . </li></ul>
  57. 59. <ul><li>In 1965 he introduced the algorithmic theory of randomness via a measure of complexity, now referred to Kolmogorov Complexity .  </li></ul>
  58. 60. <ul><li>According to Kolmogorov, the complexity of an object is the length of the shortest computer program that can reproduce the object. </li></ul>
  59. 61. <ul><li>Random objects, in his view, were their own shortest description. </li></ul>
  60. 62. <ul><li>Whereas, periodic sequences have low Kolmogorov complexity, given by the length of the smallest repeating &quot;template&quot; sequence they contain. </li></ul>
  61. 63. <ul><li>Kolmogorov's notion of complexity is a measure of randomness, one that is closely related to Claude Shannon 's entropy rate of an information source. </li></ul>
  62. 64. <ul><li>Kolmogorov had many interests outside mathematics research, notable examples being the quantitative analysis of structure in the poetry of the Russian author Pushkin, studies of agrarian development in 16th and 17th century Novgorod, and mathematics education. </li></ul>
  63. 65. Questions & Answer
  64. 66. What place did he occupy among all the Soviet mathematicians in the number of foreign academies and scientific societies that have elected him as member?
  65. 67. First Place
  66. 68. Where did he graduate?
  67. 69. Moscow University
  68. 70. Albert Einstein Born : 14 March 1879 in Ulm, Württemberg, Germany Died : 18 April 1955 in Princeton, New Jersey, USA
  69. 72. Biography
  70. 73. <ul><li>Six weeks later the family moved to Munich, where he later on began his schooling at the Luitpold Gymnasium. </li></ul>
  71. 74. <ul><li>Later, they moved to Italy and Albert continued his education at Aarau, Switzerland and in 1896 he entered the Swiss Federal Polytechnic School in Zurich to be trained as a teacher in physics and mathematics. </li></ul>
  72. 75. <ul><li>In 1901, the year he gained his diploma, he acquired Swiss citizenship and, as he was unable to find a teaching post, he accepted a position as technical assistant in the Swiss Patent Office. </li></ul>
  73. 76. <ul><li>In 1905 he obtained his doctor's degree. </li></ul>
  74. 77. <ul><li>Unhappy with life in Berlin, his wife Mileva returned to Switzerland with their sons near the beginning of World World I; their separation led to a divorce in 1919. Einstein married his second cousin, Elsa Lowenthal, later that year. </li></ul>
  75. 78. Contribution
  76. 79. <ul><li>Einstein worked as a professor of physics at universities in Prague and Zurich before moving to Berlin in 1914 with his wife and two sons, Hans Albert and Eduard. </li></ul>
  77. 80. <ul><li>He took a post at the Prussian Academy of Sciences, where he could continue his research and lecture at the University of Berlin. </li></ul>
  78. 81. <ul><li>In 1915, Einstein perfected his general theory of relativity, summing up his theory with the mathematical equation E=mc² (energy equals mass times the speed of light squared). </li></ul>
  79. 82. <ul><li>His findings on relativity were published in The Principle of Relativity , Sidelights on Relativity , and The Meaning of Relativity . </li></ul>
  80. 83. <ul><li>In November 1919, the Royal Society of London announced that their experiment conducted during the solar eclipse of that year had confirmed the predictions Einstein made in his general theory of relativity. </li></ul>
  81. 84. <ul><li>The implications of this announcement shook the world of science and earned Einstein the international acclaim he had long deserved. </li></ul>
  82. 85. What is Einstein's best-known equation?
  83. 86. E=mc²
  84. 87. According to this equation, any given amount of WHAT is equivalent to a certain amount of energy, and vice versa?
  85. 88. mass
  86. 89. Andre Weil Born : 6 May 1906 in Paris, France Died : 6 Aug 1998 in Princeton, New Jersey, USA
  87. 91. Biography
  88. 92. <ul><li>He studied Sanskrit as a child, loved to travel, taught at a Muslim university in India for two years (intending to teach French civilization), wrote as a young man under the famous pseudonym Nicolas Bourbaki , </li></ul>
  89. 93. <ul><li>spent time in prison during World War II as a Jewish objector, was almost executed as a spy, escaped to America, and eventually joined Princeton's Institute for Advanced Studies. </li></ul>
  90. 94. <ul><li>He once wrote: &quot;Every mathematician worthy of the name has experienced [a] lucid exaltation in which one thought succeeds another as if miraculously.&quot; </li></ul>
  91. 95. Contribution
  92. 96. Algebraic Geometry <ul><li>branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. </li></ul>
  93. 97. <ul><li>occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory </li></ul>
  94. 98. Weil’s Conjectures <ul><li>some highly-influential proposals from the late 1940s by André Weil on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields. </li></ul>
  95. 99. <ul><li>Weil proved a special case of the Riemann hypothesis </li></ul>
  96. 100. <ul><li>he contributed, at least indirectly, to the recent proof of &quot;Fermat's last Theorem;&quot; </li></ul>
  97. 101. <ul><li>He also worked in group theory, general and algebraic topology, differential geometry, sheaf theory, representation theory, and theta functions. </li></ul>
  98. 102. <ul><li>He invented several new concepts including vector bundles, and uniform space. </li></ul>
  99. 103. <ul><li>His work has found applications in particle physics and string theory. </li></ul>
  100. 104. <ul><li>He is considered to be one of the most influential of modern mathematicians. </li></ul>
  101. 105. Questions & Answer
  102. 106. What branches of mathematics are combined in algebraic geometry?
  103. 107. Algebra & Geometry
  104. 108. What is the symbol for theta ?
  105. 109. θ
  106. 110. Richard Dedekind Born : 6 Oct 1831 in Braunschweig, duchy of Braunschweig (now Germany) Died : 12 Feb 1916 in Braunschweig, duchy of Braunschweig (now Germany)
  107. 112. Biography
  108. 113. <ul><li>Much of his education took place in Brunswick as well, where he first attended school and then, for two years, the local technical university. </li></ul>
  109. 114. <ul><li>In 1850, he transferred to the University of Göttingen, a center for scientific research in Europe at the time. </li></ul>
  110. 115. <ul><li>Carl Friedrich Gauss, one of the greatest mathematicians of all time, taught in Göttingen, and Dedekind became his last doctoral student. </li></ul>
  111. 116. <ul><li>He wrote a dissertation in mathematics under Gauss, finished in 1852. </li></ul>
  112. 117. <ul><li>As was customary, he also wrote a second dissertation ( Habilitation ), completed in 1854, shortly after that of his colleague and friend Bernhard Riemann. </li></ul>
  113. 118. <ul><li>Dedekind stayed in Göttingen for four more years, as an unsalaried lecturer ( Privatdozent ). </li></ul>
  114. 119. <ul><li>During that time he was strongly influenced by P.G. Lejeune-Dirichlet, another renowned mathematician in Göttingen, and by Riemann, then a rising star. </li></ul>
  115. 120. <ul><li>(Later, Dedekind did important editorial work for Gauss, Dirichlet, and Riemann.) In 1858, he moved to the Polytechnic in Zurich (ETH Zürich), Switzerland, to take up his first salaried position. </li></ul>
  116. 121. <ul><li>He returned to Brunswick in 1862, where he became professor at the local university and taught until his retirement in 1896. </li></ul>
  117. 122. <ul><li>In this later period, he published most of his major works. </li></ul>
  118. 123. <ul><li>He also had further interactions with important mathematicians; thus, he was in correspondence with Georg Cantor, collaborated with Heinrich Weber, and developed an intellectual rivalry with Leopold Kronecker. </li></ul>
  119. 124. <ul><li>He stayed in his hometown until the end of his life, in 1916. </li></ul>
  120. 125. Contribution
  121. 126. <ul><li>developed a major redefinition of irrational numbers in terms of arithmetic concepts. </li></ul>
  122. 127. <ul><li>Although not fully recognized in his lifetime, his treatment of the ideas of the infinite and of what constitutes a real number continues to influence modern mathematics. </li></ul>
  123. 128. <ul><li>While teaching, Dedekind developed the idea that both rational and irrational numbers could form a continuum (with no gaps) of real numbers, provided that the real numbers have a one-to-one relationship with points on a line. </li></ul>
  124. 129. <ul><li>He said that an irrational number would then be that boundary value that separates two especially constructed collections of rational numbers. </li></ul>
  125. 130. Dedekind Cut <ul><li>concept advanced in 1872 by the German mathematician Richard Dedekind that combines an arithmetic formulation of the idea of continuity with a rigorous distinction between rational and irrational numbers </li></ul>
  126. 131. <ul><li>He reasoned that the real numbers form an ordered continuum , so that any two numbers x and y must satisfy one and only one of the conditions x  <  y , x  =  y , or x  >  y . </li></ul>
  127. 132. <ul><li>He postulated a cut that separates the continuum into two subsets, say X and Y , such that if x is any member of X and y is any member of Y , then x  <  y . </li></ul>
  128. 133. <ul><li>If the cut is made so that X has a largest rational member or Y a least member, then the cut corresponds to a rational number. </li></ul>
  129. 134. <ul><li>If, however, the cut is made so that X has no largest rational member and Y no least rational member, then the cut corresponds to an irrational number. </li></ul>
  130. 135. <ul><li>Dedekind developed his arithmetical rendering of irrational numbers in 1872 in his Stetigkeit und Irrationale Zahlen (Eng. trans., “Continuity and Irrational Numbers,” published in Essays on the Theory of Numbers ). </li></ul>
  131. 136. <ul><li>He also proposed, as did the German mathematician Georg Cantor, two years later, that a set—a collection of objects or components—is infinite if its components may be arranged in a one-to-one relationship with the components of one of its subsets. </li></ul>
  132. 137. <ul><li>By supplementing the geometric method in analysis, Dedekind contributed substantially to the modern treatment of the infinitely large and the infinitely small. </li></ul>
  133. 138. Questions & Answer
  134. 139. What is an irrational number ?
  135. 140. Irrational Number <ul><li>any real number that cannot be expressed as the quotient of two integers </li></ul>
  136. 141. Who was his teacher in the University of Göttingen?
  137. 142. Carl Friedrich Gauss
  138. 143. Stefan Banach Born : 30 March 1892 in Kraków, Austria-Hungary (now Poland) Died: 31 Aug 1945 in Lvov, (now Ukraine)
  139. 145. Biography
  140. 146. <ul><li>son of Stefan Greczek, a tax official and, possibly, of Katarzyna Banach who was in fact his mother remains uncertain </li></ul>
  141. 147. <ul><li>Banach was brought up in Krakow by Franciszka Plowa and received his early education from a French intellectual, Juliusz Mien, who was the guardian of Plowa’s daughter.  </li></ul>
  142. 148. <ul><li>In 1902, Banach finished primary school in Krakow and began his secondary education at the Henryk Sienkiewicz Gymnasium No. 4 in the same city.  </li></ul>
  143. 149. <ul><li>According to one of his colleagues, Banach was very good in mathematics and natural sciences, but was not interested in anything else.  </li></ul>
  144. 150. <ul><li>He finished the Gymnasium in 1910 without distinction.  </li></ul>
  145. 151. <ul><li>As he felt that nothing new can be discovered in mathematics, he chose to study engineering at the Lwow Polytechnic (1910-1916).  </li></ul>
  146. 152. <ul><li>His father did not want to support him financially, so he supported himself probably by tutoring.  </li></ul>
  147. 153. <ul><li>During this period, he frequently left Lwow to build roads, but also attended mathematics lectures at the Jagiellonian University in Krakow and taught in local schools.  </li></ul>
  148. 154. <ul><li>In the spring of 1916, he met in Krakow, by chance, Steinhaus, a mathematician who just got a position at Lwow University.  </li></ul>
  149. 155. <ul><li>Steinhaus, impressed by young Banach's talent for mathematics, told him about a problem he couldn't solve.  </li></ul>
  150. 156. <ul><li>Banach helped Steinhaus and the ensuing paper they wrote together was published in Krakow in 1918.  Since then Banach produced many papers. </li></ul>
  151. 157. <ul><li>Also thanks to Steinhaus he met Lucja Braus, whom he married in Zakopane in 1920.  </li></ul>
  152. 158. <ul><li>In the same year, Banach became an assistant to Lomnicki, a professor of mathematics at Lwow Polytechnic.  </li></ul>
  153. 159. <ul><li>Lomnicki served as Banach's major advisor for his doctoral thesis.  </li></ul>
  154. 160. <ul><li>In 1922, the Jan Kazimierz University in Lwow awarded Banach his habilitation (a degree allowing to teach at the university) for a Docent in Mathematics for his thesis on measure theory.  </li></ul>
  155. 161. <ul><li>In July 1922, he was appointed Extraordinary Professor.  </li></ul>
  156. 162. <ul><li>In 1924, he was promoted to Ordinary Professor (Full Professor).  </li></ul>
  157. 163. <ul><li>He spent the academic year 1924-25 in Paris.  </li></ul>
  158. 164. <ul><li>In 1929, he started, with Steinhaus, the journal Studia Mathematica. </li></ul>
  159. 165. <ul><li>In 1931, he started co-editing, together with Steinhaus, Knaster, Kuratowski, Mazurkiewicz and Sierpinski, a series titled Mathematical Monographs.  </li></ul>
  160. 166. <ul><li>In 1936, Banach gave a plenary address at the International Congress of Mathematicians in Oslo, Norway.  </li></ul>
  161. 167. <ul><li>From 1927 until 1934, he wrote some joint papers with Kuratowski.  </li></ul>
  162. 168. <ul><li>He also worked with Ulam and Turowicz. </li></ul>
  163. 169. <ul><li>In 1939, Banach was elected President of the Polish Mathematical Society.  </li></ul>
  164. 170. <ul><li>After the Soviets invaded Lwow in 1939, Banach was allowed to continue to hold his chair at the university and he became the Dean of the Faculty of Science, the university being renamed the Ivan Franko University.  </li></ul>
  165. 171. <ul><li>Famous Soviet mathematicians Sobolev and Aleksandrov visited Banach in Lwow in 1940.  </li></ul>
  166. 172. <ul><li>In the same period, Banach attended conferences in the Soviet Union. </li></ul>
  167. 173. <ul><li>He was in Kiev when Germany invaded the Soviet Union, but he returned immediately to his family in Lwow.  </li></ul>
  168. 174. <ul><li>He was arrested, but after few weeks he was released.  </li></ul>
  169. 175. <ul><li>He also survived the Nazi slaughter of Polish university professors.  </li></ul>
  170. 176. <ul><li>His advisor Lomnicki was among those who perished.  </li></ul>
  171. 177. <ul><li>From the end of 1941 through the remainder of the Nazi occupation (July 1944), Banach worked feeding lice in Prof. Weigel’s Institute in Lwow.  </li></ul>
  172. 178. <ul><li>After the Soviets reentered Lwow, Banach contacted his Soviet friend Sobolev, who wrote about this encounter: &quot;...and despite the grave illness that was undercutting his strength, Banach's eyes were still lively.  </li></ul>
  173. 179. <ul><li>He remained the same sociable, cheerful and extraordinarily well-meaning and charming Stefan Banach whom I had seen in Lvov before the war.  </li></ul>
  174. 180. <ul><li>That is how he remains in my memory: with a great sense of humor, an energetic human being, a beautiful soul and a great talent...&quot;  Banach died of lung cancer in Lwow in 1945. </li></ul>
  175. 181. Contribution
  176. 182. <ul><li>Founded the important modern mathematical field of functional analysis and made major contributions to the theory of topological vector spaces .  </li></ul>
  177. 183. <ul><li>In addition, he contributed to measure theory , integration , the theory of sets and orthogonal series . </li></ul>
  178. 184. Questions & Answer
  179. 185. What is the measure theory about?
  180. 186. Measure Theory <ul><li>the study of measures. </li></ul><ul><li>generalizes the intuitive notions of length, area, and volume. </li></ul>
  181. 187. What is a vector space ?
  182. 188. Vector Space <ul><li>a mathematical structure formed by a collection of vectors </li></ul>
  183. 189. Bernhard Riemann Born : 17 Sept 1826 in Breselenz, Hanover (now Germany) Died : 20 July 1866 in Selasca, Italy
  184. 191. Biography
  185. 192. <ul><li>His father, Friedrich Bernhard Riemann, was a poor Lutheran pastor in Breselenz who fought in the Napoleonic Wars. </li></ul>
  186. 193. <ul><li>His mother died before her children had reached adulthood. </li></ul>
  187. 194. <ul><li>Riemann was the second of six children, shy, and suffered from numerous nervous breakdowns. </li></ul>
  188. 195. <ul><li>Riemann exhibited exceptional mathematical skills, such as fantastic calculation abilities, from an early age, but suffered from timidity and a fear of speaking in public. </li></ul>
  189. 196. <ul><li>During 1840, Riemann went to Hanover to live with his grandmother and attend lyceum (middle school). </li></ul>
  190. 197. <ul><li>After the death of his grandmother in 1842, he attended high school at the Johanneum Lüneburg. </li></ul>
  191. 198. <ul><li>In high school, Riemann studied the Bible intensively, but he was often distracted by mathematics. </li></ul>
  192. 199. <ul><li>To this end, he even tried to prove mathematically the correctness of the Book of Genesis. </li></ul>
  193. 200. <ul><li>His teachers were amazed by his adept ability to solve complicated mathematical operations, in which he often outstripped his instructor's knowledge. </li></ul>
  194. 201. <ul><li>In 1846, at the age of 19, he started studying philology and theology in order to become a priest and help with his family's finances. </li></ul>
  195. 202. <ul><li>During the spring of 1846, his father (Friedrich Riemann), after gathering enough money to send Riemann to university, allowed him to stop studying theology and start studying mathematics. </li></ul>
  196. 203. <ul><li>He was sent to the renowned University of Göttingen, where he first met Carl Friedrich Gauss, and attended his lectures on the method of least squares. </li></ul>
  197. 204. <ul><li>In 1847, Riemann moved to Berlin, where Jacobi, Dirichlet, Steiner, and Eisenstein were teaching. </li></ul>
  198. 205. <ul><li>He stayed in Berlin for two years and returned to Göttingen in 1849. </li></ul>
  199. 206. <ul><li>Bernhard Riemann held his first lectures in 1854, which founded the field of Riemannian geometry and thereby set the stage for Einstein's general theory of relativity. </li></ul>
  200. 207. <ul><li>In 1857, there was an attempt to promote Riemann to extraordinary professor status at the University of Göttingen. </li></ul>
  201. 208. <ul><li>Although this attempt failed, it did result in Riemann finally being granted a regular salary. </li></ul>
  202. 209. <ul><li>In 1859, following Dirichlet's death, he was promoted to head the mathematics department at Göttingen. </li></ul>
  203. 210. <ul><li>He was also the first to suggest using dimensions higher than merely three or four in order to describe physical—an idea that was ultimately vindicated with Einstein's contribution in the early 20th century. </li></ul>
  204. 211. <ul><li>In 1862 he married Elise Koch and had a daughter. </li></ul>
  205. 212. <ul><li>Riemann fled Göttingen when the armies of Hanover and Prussia clashed there in 1866. </li></ul>
  206. 213. <ul><li>He died of tuberculosis during his third journey to Italy in Selasca (now a hamlet of Verbania on Lake Maggiore) where he was buried in the cemetery in Biganzolo (Verbania). </li></ul>
  207. 214. <ul><li>Meanwhile, in Göttingen his housekeeper tidied up some of the mess in his office, including much unpublished work. </li></ul>
  208. 215. <ul><li>Riemann refused to publish incomplete work and some deep insights may have been lost forever. </li></ul>
  209. 216. Contribution
  210. 217. <ul><li>Riemannian geometry </li></ul><ul><li>Riemann surface </li></ul><ul><li>Riemann integral </li></ul><ul><li>Riemann sum </li></ul><ul><li>Riemann–Liouville integral </li></ul><ul><li>Riemann zeta function </li></ul><ul><li>Riemann hypothesis </li></ul><ul><li>Riemannian metrics </li></ul>
  211. 218. Riemannian Geometry <ul><li>the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric , i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. </li></ul>
  212. 219. Riemannian Geometry <ul><li>This gives in particular local notions of angle, length of curves, surface area, and volume. </li></ul>
  213. 220. Riemannian Geometry <ul><li>From those some other global quantities can be derived by integrating local contributions. </li></ul>
  214. 221. Riemann Surface <ul><li>a one-dimensional complex manifold </li></ul>
  215. 222. <ul><li>can be thought of as &quot;deformed versions&quot; of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. </li></ul>Riemann Surface
  216. 223. <ul><li>For example, they can look like a sphere or a torus or a couple of sheets glued together. </li></ul>
  217. 224. Riemann Integral <ul><li>the first rigorous definition of the integral of a function on an interval. </li></ul>
  218. 225. <ul><li>While the Riemann integral is unsuitable for many theoretical purposes, it is one of the easiest integrals to define. </li></ul>
  219. 226. <ul><li>Some of these technical deficiencies can be remedied by the Riemann–Stieltjes integral, and most of them disappear in the Lebesgue integral. </li></ul>
  220. 227. <ul><li>a method for approximating the total area underneath a curve on a graph, otherwise known as an integral </li></ul>Riemann Sum
  221. 228. <ul><li>may also be used to define the integration operation </li></ul>
  222. 229. Riemann–Liouville Integral <ul><li>associates with a real function ƒ  :  R  ->  R another function I α ƒ of the same kind for each value of the parameter α > 0. </li></ul>
  223. 230. <ul><li>The integral is a manner of generalization of the repeated antiderivative of ƒ in the sense that for positive integer values of α, I α ƒ is an iterated antiderivative of ƒ of order α. </li></ul>Riemann–Liouville Integral
  224. 231. Riemann Zeta Function <ul><li>a prominent function of great significance in number theory because of its relation to the distribution of prime numbers. </li></ul>
  225. 232. Riemann Zeta Function <ul><li>also has applications in other areas such as physics, probability theory, and applied statistics. </li></ul>
  226. 233. Riemann Hypothesis <ul><li>a conjecture about the distribution of the zeros of the Riemann zeta-function stating that all non-trivial zeros of the Riemann zeta function have real part 1/2. </li></ul>
  227. 234. <ul><li>The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. </li></ul>Riemann Hypothesis
  228. 235. Riemannian Metric <ul><li>a generalization of the first fundamental form of a surface in three-dimensional Euclidean space — of the internal metric of the surface. </li></ul>
  229. 236. Question & Answer
  230. 237. Who asked his student Riemann to prepare a Habilitationsschrift on the foundations of geometry?
  231. 238. Carl Friedrich Gauss
  232. 239. Jules Henri Poincare Born : 29 April 1854 in Nancy, Lorraine, France Died : 17 July 1912 in Paris, France
  233. 241. Signature
  234. 242. Biography
  235. 243. <ul><li>a French mathematician, living at the turn of the century, who made many fundamental contributions to mathematics and was an influential philosopher of science </li></ul>
  236. 244. <ul><li>In the natural sciences he is best appreciated for his highly original work on celestial mechanics. </li></ul>
  237. 245. <ul><li>Residence : France </li></ul><ul><li>Nationality : French </li></ul><ul><li>Fields : Mathematician and physicist </li></ul>
  238. 246. <ul><li>Institutions : </li></ul><ul><ul><li>Corps des Mines </li></ul></ul><ul><ul><li>Caen University </li></ul></ul><ul><ul><li>La Sorbonne </li></ul></ul><ul><ul><li>Bureau des Longitudes </li></ul></ul>
  239. 247. <ul><li>Alma mater: </li></ul><ul><ul><li>Lycée Nancy </li></ul></ul><ul><ul><li>École Polytechnique </li></ul></ul><ul><ul><li>École des Mines </li></ul></ul>
  240. 248. Contribution
  241. 249. <ul><li>Poincaré conjecture </li></ul><ul><li>Topology </li></ul><ul><li>Special relativity </li></ul>
  242. 250. <ul><li>Poincaré–Hopf Theorem </li></ul><ul><li>Poincaré duality </li></ul><ul><li>Poincaré–Birkhoff–Witt theorem </li></ul><ul><li>Poincaré inequality </li></ul>
  243. 251. <ul><li>Hilbert–Poincaré series </li></ul><ul><li>Poincaré metric </li></ul><ul><li>Rotation number </li></ul><ul><li>Coining term 'Betti number‘ </li></ul>
  244. 252. <ul><li>Chaos theory </li></ul><ul><li>Sphere-world </li></ul><ul><li>Poincaré–Bendixson theorem </li></ul>
  245. 253. <ul><li>Poincaré–Lindstedt method </li></ul><ul><li>Poincaré Recurrence </li></ul>
  246. 254. <ul><li>RAS Gold Medal (1900) </li></ul><ul><li>Sylvester Medal (1901) </li></ul><ul><li>Matteucci Medal (1905) </li></ul><ul><li>Bruce Medal (1911) </li></ul>Notable Awards:
  247. 255. Poincaré Conjecture <ul><li>states that every simply connected closed three-manifold is homeomorphic to the three-sphere (in a topologist's sense) , where a three-sphere is simply a generalization of the usual sphere to one dimension higher. </li></ul>
  248. 256. <ul><li>More colloquially, the conjecture says that the three-sphere is the only type of bounded three-dimensional space possible that contains no holes. </li></ul>Poincaré Conjecture
  249. 257. <ul><li>the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects. </li></ul>Topology
  250. 258. <ul><li>generalizes Galileo's principle of relativity–that all uniform motion is relative, and that there is no absolute and well-defined state of rest (no privileged reference frames) </li></ul>Special Relativity
  251. 259. <ul><li>– from mechanics to all the laws of physics, including both the laws of mechanics and of electrodynamics, whatever they may be </li></ul>
  252. 260. <ul><li>incorporates the principle that the speed of light is the same for all inertial observers regardless of the state of motion of the source </li></ul>
  253. 261. <ul><li>illustrated by the special case of the Hairy ball theorem, which simply states that there is no smooth vector field on a sphere having no sources or sinks. </li></ul>Poincaré–Hopf theorem
  254. 262. <ul><li>states that if M is an n -dimensional oriented closed manifold (compact and without boundary), then the k th cohomology group of M is isomorphic to the ( n  −  k )th homology group of M , for all integers k </li></ul>Poincaré Duality
  255. 263. <ul><li>the product of canonical monomials in Y can be reduced uniquely to a linear combination of canonical monomials by repeatedly using the structure equations </li></ul>Poincaré–Birkhoff–Witt theorem
  256. 264. <ul><li>allows one to obtain bounds on a function using bounds on its derivatives and the geometry of its domain of definition. </li></ul>Poincaré Inequality
  257. 265. <ul><li>Such bounds are of great importance in the modern, direct methods of the calculus of variations. </li></ul>
  258. 266. <ul><li>a formal power series in one indeterminate, say t , where the coefficient of tn gives the dimension (or rank) of the sub-structure of elements homogeneous of degree  n . </li></ul>Hilbert–Poincaré Series
  259. 267. <ul><li>the metric tensor describing a two-dimensional surface of constant negative curvature. </li></ul>Poincaré Metric
  260. 268. <ul><li>It is the natural metric commonly used in a variety of calculations in hyperbolic geometry or Riemann surfaces. </li></ul>
  261. 269. <ul><li>an invariant of homeomorphisms of the circle. </li></ul>Rotation Number
  262. 270. <ul><li>It was first defined by Henri Poincaré in 1885, in relation to the precession of the perihelion of a planetary orbit. </li></ul>
  263. 271. <ul><li>Poincaré later proved a theorem characterizing the existence of periodic orbits in terms of rationality of the rotation number. </li></ul>
  264. 272. <ul><li>can be used to distinguish topological spaces. </li></ul>Betti Number
  265. 273. <ul><li>Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces. </li></ul>
  266. 274. Chaos Theory <ul><li>an area of inquiry in mathematics, physics, and philosophy studying the behavior of dynamical systems that are highly sensitive to initial conditions. </li></ul>
  267. 275. Sphere-world <ul><li>The idea of a sphere-world was constructed by Henri Poincaré while pursuing his argument for conventionalism , </li></ul>
  268. 276. <ul><li>offered a thought experiment about a sphere with strange properties. </li></ul>
  269. 277. Poincaré–Bendixson Theorem <ul><li>tells that the fate of any bounded solution of a differential equation in the is to convergence either to an attractive fixed point or to a limit cycle. </li></ul>
  270. 278. Poincaré–Lindstedt Method <ul><li>Provides us with a consistent perturbation scheme that identifies the frequency correction that is required to suppress secular generating terms </li></ul>
  271. 279. <ul><li>gives us an approximation for the displacement x(t) </li></ul>
  272. 280. Poincaré Recurrence <ul><li>states that certain systems will, after a sufficiently long time, return to a state very close to the initial state. </li></ul>
  273. 281. Question & Answer
  274. 282. What was founded by Jules Henri Poincare through his innovations and is the mathematical theory of dynamical systems?
  275. 283. Qualitative Dynamics
  276. 284. What new mathematical tool was used by Jules Henri Poincare to attempt to answer a very longstanding question “ Is the solar system stable ”?
  277. 285. Topology
  278. 286. <ul><li>At the end of the 19th century this question was re-posed by King Oscar II of Sweden with a cash prize promised to whomever answered it definitively. </li></ul>
  279. 287. King Oscar II of Sweden
  280. 288. <ul><li>In attacking the problem Poincare limited his sights to the restricted problem of just three bodies moving under their mutual gravitational attraction. </li></ul>
  281. 289. <ul><li>He won the prize with his publication of &quot;On The Problem of Three Bodies and the Equations of Equilibrium&quot;. </li></ul>
  282. 290. <ul><li>But through this investigation Poincare came to understand that infinitely complicated behaviors could arise in simple nonlinear systems. </li></ul>
  283. 291. <ul><li>Without the benefit of computers, only through his mathematical insight and his calculation abilities, he was able to describe many of the basic properties of deterministic chaos. </li></ul>
  284. 292. David Hilbert Born : January 23, 1862) Königsberg or Wehlau (today Znamensk, Kaliningrad Oblast ), Province of Prussia Died : February 14, 1943 (aged 81) Göttingen , Germany
  285. 294. Biography
  286. 295. <ul><li>Residence : Germany </li></ul><ul><li>Nationality : German </li></ul><ul><li>Fields : Mathematician and Philosopher </li></ul>
  287. 296. David Hilbert <ul><li>the first of two children and only son of Otto and Maria Therese (Erdtmann) Hilbert </li></ul>
  288. 297. <ul><li>born in either Königsberg (according to Hilbert's own statement) or in Wehlau (today Znamensk, Kaliningrad Oblast)) near Königsberg where his father was occupied at the time of his birth in the Province of Prussia. </li></ul>
  289. 298. <ul><li>In the fall of 1872, he entered the Friedrichskolleg Gymnasium (the same school that Immanuel Kant had attended 140 years before), </li></ul>
  290. 299. <ul><li>but after an unhappy duration he transferred (fall 1879) to and graduated from (spring 1880) the more science-oriented Wilhelm Gymnasium. </li></ul>
  291. 300. <ul><li>Upon graduation he enrolled (autumn 1880) at the University of Königsberg, the &quot;Albertina&quot;. </li></ul>
  292. 301. <ul><li>In the spring of 1882, Hermann Minkowski (two years younger than Hilbert and also a native of Königsberg but so talented he had graduated early from his gymnasium and gone to Berlin for three semesters), returned to Königsberg and entered the university. </li></ul>
  293. 302. <ul><li>&quot;Hilbert knew his luck when he saw it. In spite of his father's disapproval, he soon became friends with the shy, gifted Minkowski.&quot; </li></ul>
  294. 303. <ul><li>In 1884, Adolf Hurwitz arrived from Göttingen as an Extraordinarius, i.e., an associate professor. </li></ul>
  295. 304. <ul><li>An intense and fruitful scientific exchange between the three began and especially Minkowski and Hilbert would exercise a reciprocal influence over each other at various times in their scientific careers. </li></ul>
  296. 305. <ul><li>Hilbert obtained his doctorate in 1885, with a dissertation, written under Ferdinand von Lindemann, titled </li></ul>
  297. 306. <ul><li>Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen </li></ul><ul><li>(&quot; On the invariant properties of special binary forms, in particular the spherical harmonic functions &quot;). </li></ul>
  298. 307. <ul><li>Hilbert remained at the University of Königsberg as a professor from 1886 to 1895. </li></ul>
  299. 308. <ul><li>In 1892, Hilbert married Käthe Jerosch (1864–1945), &quot;the daughter of a Konigsberg merchant, an outspoken young lady with an independence of mind that matched his own&quot;. </li></ul>
  300. 309. <ul><li>While at Königsberg they had their one child Franz Hilbert (1893–1969). </li></ul>
  301. 310. <ul><li>In 1895, as a result of intervention on his behalf by Felix Klein he obtained the position of Chairman of Mathematics at the University of Göttingen, at that time the best research center for mathematics in the world and where he remained for the rest of his life. </li></ul>
  302. 311. <ul><li>His son Franz would suffer his entire life from an (undiagnosed) mental illness, his inferior intellect a terrible disappointment to his father and this tragedy a matter of distress to the mathematicians and students at Göttingen. </li></ul>
  303. 312. <ul><li>Sadly, Minkowski — Hilbert's &quot;best and truest friend” — would die prematurely of a ruptured appendix in 1909. </li></ul>
  304. 313. Contribution
  305. 314. <ul><li>Hilbert's basis theorem </li></ul><ul><li>Hilbert's axioms </li></ul><ul><li>Hilbert's problems </li></ul><ul><li>Hilbert's program </li></ul><ul><li>Einstein–Hilbert action </li></ul><ul><li>Hilbert space </li></ul>
  306. 315. Hilbert's basis Theorem <ul><li>states that every ideal in the ring of multivariate polynomials over a field is finitely generated. </li></ul>
  307. 316. <ul><li>This can be translated into algebraic geometry as follows: every algebraic set over a field can be described as the set of common roots of finitely many polynomial equations. </li></ul>
  308. 317. <ul><li>a set of 20 (originally 21) assumptions proposed by David Hilbert in 1899, as the foundation for a modern treatment of Euclidean geometry. </li></ul>Hilbert's Axioms
  309. 318. Hilbert's Problems <ul><li>a set of (originally) unsolved problems in mathematics proposed by Hilbert. </li></ul>
  310. 319. <ul><li>Of the 23 total appearing in the printed address, ten were actually presented at the Second International Congress in Paris on August 8, 1900. </li></ul>
  311. 320. Hilbert's Program <ul><li>a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies </li></ul>
  312. 321. Einstein–Hilbert Action <ul><li>he action that yields the Einstein's field equations through the principle of least action. </li></ul>
  313. 322. Hilbert Space <ul><li>an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. </li></ul>
  314. 323. <ul><li>David Hilbert was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. </li></ul>
  315. 324. <ul><li>He discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. </li></ul>
  316. 325. <ul><li>He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis. </li></ul>
  317. 326. <ul><li>Hilbert adopted and warmly defended Georg Cantor's set theory and transfinite numbers. </li></ul>
  318. 327. <ul><li>A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems that set the course for much of the mathematical research of the 20th century. </li></ul>
  319. 328. <ul><li>Hilbert and his students contributed significantly to establishing rigor and some tools to the mathematics used in modern physics. </li></ul>
  320. 329. <ul><li>He is also known as one of the founders of proof theory, mathematical logic and the distinction between mathematics and metamathematics. </li></ul>
  321. 330. Question & Answer
  322. 331. <ul><li>In what areas did David Hilbert discovered and developed a broad range of fundamental ideas in many areas? </li></ul>
  323. 332. <ul><li>Invariant theory and the axiomatization of geometry </li></ul>
  324. 333. <ul><li>David Hilbert is also known as one of the founders of what? </li></ul>
  325. 334. <ul><li>proof theory, mathematical logic and the distinction between mathematics and metamathematics </li></ul>
  326. 335. Alexander Grothendieck Born: March 28, 1928 in Berlin, Germany
  327. 337. Biography
  328. 338. <ul><li>Alexander Grothendieck is considered one of the greatest mathematicians of the 20th century. </li></ul>
  329. 339. <ul><li>Residence : France </li></ul><ul><li>Nationality : Stateless </li></ul><ul><li>Field : Mathematician </li></ul>
  330. 340. Alexander Grothendieck <ul><li>born in Berlin to anarchist parents: a Russian father from an ultimately Hassidic family, Alexander &quot;Sascha&quot; Shapiro a.k.a. Tanaroff, </li></ul>
  331. 341. <ul><li>and a mother from a German Protestant family, Johanna &quot;Hanka&quot; Grothendieck ; both of his parents had broken away from their early backgrounds in their teens </li></ul>