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Some Mathematicians

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Some Mathematicians JL 1 Ulisse Dini, 1845-1918 Pisa, Italy Dini’s theorem (not in book) Let (fn : R ! R)n2N a sequence of continuous functions pointwisely converging to a continuous function and such that 8n 2 N; 8x 2 [a; b]; fn+1(x) ≥ fn(x). Then (fn : R ! R)n2N converges uniformly. One interesting fact about this mathematician: Beside being a mathematician, Dini reached the highest office in university administration when he became rector of the University of Pisa, he was elected to the national Italian parliament in 1880 as a representative from Pisa. He was the chair of “infinitesimal analysis”. Another interesting fact about this mathematician: The implicit function theorem is known in Italy as the Dini’s theorem. How many stars you give to your mathematicians: ERIC COOKE 2 Thomas Joannes Stieltjes, 1865- 1894 The Netherlands Definition of the Riemann-Stieltjes sum (35.24, p.320) Let f be bounded on [a; b], and let P = fa = t0 < t1 < : : : < tn = bg ; a partition of [a; b].A Riemann-Stieltjes sum of f associated with P and F is a sum of the form n n X + − X − + f (tk ) F(tk ) − F(tk ) + f (xk ) F(tk ) − F(tk−1) : k=0 k=1 where xk is in (tk−1; tk ) for k = 1; 2;:::; n. One interesting fact about this mathematician: Stieltjes never graduated college and in fact failed out twice. It was his achievements in mathematics that earned him an honorary degree. How many stars you give to your mathematicians: I gave this mathematician four stars, mainly because he died so young and only worked in the field for less than ten years. SYD FREDERICK 3 Michel Rolle, 1652-1719 France Rolle’s Theorem (29.2, p.233) Let f be a continuous function on [a; b] that is differentiable on (a; b) and satisfies f (a) = f (b). There exists [at least one] x in (a; b) such that f 0(x) = 0. One interesting fact about this mathematician: Educated himself in Mathematics, no formal training. How many stars you give to your mathematicians: 5 out of 5, because his theorem is very fundamental and helps to prove the Mean Value Theorem. He also was one of the first mathematicians to publish Gaussian elimination in Europe. JOHN GORDOS 4 Julius Wilhelm Richard Dedekind, 1831-1916 Germany Dedekind Cuts (§6, p.30) Dedekind Cuts are a way to define the real numbers from the rational numbers. A Dedekind cut A is a subset of Q satisfying these properties: 1. A is neither ; nor Q; 2. If r is in A, s is in Q and s < r, then s is in A; 3. A contains no largest rational. The set of all possible Dedekind cuts can be used as the definition of R. One interesting fact about this mathematician: Dedekind was the last student of Gauss. How many stars you give to your mathematicians: Building the reals like this is mindblowing to think about, more so because Dedekind acknowledged he had weaknesses in advanced mathematics after receiving his doctorate. From here, he spent two years studying to compensate. I sympathize but my weakness exists on a foundational level. KJERSTI JACOBSON 5 Georg Cantor, 1845-1918 Germany Cantor set (Example 5, p.89) In 1883, he introduced the concept of the Cantor set. The Cantor set is simply a subset of the interval [0; 1], but the set has some very interesting properties: for instance, the set is compact, uncountable, and contains no intervals. The most common modern construction of a Cantor set is the Cantor ternary set, which is built by removing the middle thirds of a line segment. One interesting fact about this mathematician: Cantor believed that Francis Bacon wrote Shakespeare’s plays. He studied intensely Elizabethan literature to try to prove his theory. In 1896-97 he published pamphlets on the subject. How many stars you give to your mathematicians: XINXIN JIANG 6 Brook Taylor, 1685-1731 England Taylor series (31.2,p.250) Let f be a function defined on some open interval containing c. If f possesses derivatives of all orders at c, then the Taylor series for f about c is 1 X f (k)(c) (x − c)k : k! k=0 One interesting fact about this mathematician: As a mathematician, he was the only Englishman after Sir Isaac Newton and Roger Cotes capable of holding his own with the Bernoullis; but a great part of the effect of his demonstrations was lost through his failure to express his ideas fully and clearly. How many stars you give to your mathematicians: I give him 4. Though it is very important for a mathematician to focus on mathematical research, a good grasp of communications skills is also vital. And unfortunately, he is not good at expressing himself despite his brilliant thinking process. SIYUAN LIN 7 Jean Gaston Darboux, 1842-1917 France Intermediate Value Theorem for Derivatives (29.8,p.236) Let f :(a; b) ! R be a differentiable function. If a < x1 < x2 < b, and if c lies between 0 0 0 f (x1) and f (x2), then there exists (at least one) x in (x1; x2) such that f (x) = c. Upper Darboux sums (p.270) Given a f : R ! R, given a partition of [a; b], P = fa = t0 < t1 < : : : < tn = bg, the upper Darboux sum U(f ; P) of f with P is the sum n ! X U(f ; P) = sup f (x) (tk − tk−1) : k=1 x2[tk−1;tk ] One interesting fact about this mathematician: In 1902, he was elected to the Royal Society; in 1916, he received the Sylvester Medal from the Society. How many stars you give to your mathematicians: His theorems seem really complicated since we learn it at the very end of this book, so I guess he must be really brilliant. And he must be a great professor, because he taught many highly reputed European mathematicians, for example, Émile Borel, Élie Cartan, Gheorghe ¸Ti¸teicaand Stanisław Zaremba. So he deserves five stars. SAMUEL LOOS 8 Georg Friedrich Bernhard Riemann, 1826-1866 German Riemann integral (p.270) Given L(f ) (resp. U(f )) the lower (resp. upper) Darboux integral of f over [a; b], we say that f is (Riemann) integrable on [a; b] provided that L(f ) = U(f ). In this case, we write Z b f = L(f ) = U(f ): a One interesting fact about this mathematician: The base of Einstein’s Theory of Relativity was set up in 1854 when Riemann gave his first lectures on the geometry of space. How many stars you give to your mathematicians: I would give Riemann 4 stars. His contribution to numerous areas in mathematics is immense. He also had a lot of influence with the development of prime numbers. CARLY MEYER 9 Karl Weierstrass, 1815-1897 German Bolzano-Weierstrass Theorem (11.5, p 72) Every bounded sequence has a convergent subsequence. Weierstrass M-test (25.7, p 205) Let (gk : ! )k2 be a sequence of functions and (Mk )k2 a sequence of real numbers such that R R N P P N (1) for all x 2 R, jgk (x)j ≤ Mk and (2) MK < 1, then gK converges uniformly. Weierstrass’s Approximation Theorem (27.5, p 220) Every continuous function on a closed interval [a,b] can be uniformly approximated by polynomials on [a.b]. One interesting fact about this mathematician: Along with teaching mathematics, he taught physics, gymnastics, geography, history, German, calligraphy and botanics at the Lyceum Hosianum in Braunsberg, Poland. A second interesting fact about this mathematician: Weierstrass has a lunar crater named after him. How many stars you give to your mathematicians: I give Weierstrass 5 out of 5 stars because he played a significant role in a lot of the content that we have learned this semester. Without the Bolzano-Weierstrass theorem, a lot of our proofs would fall apart. The Weierstrass M-test is also quite a strong theorem–without knowing the pointwise convergence of a sequence of functions, we still have the ability to conclude if a sequence of functions converges uniformly. SAMUEL MORTELLARO 10 Bernard Bolzano, 1781-1848 Prague, Kingdom of Bohemia Bolzano-Weierstrass theorem (11.5, p.72) Every bounded sequence has a convergent subsequence. One interesting fact about this mathematician: Because he argued adamantly that war was a human and economic waste, he was exiled to the county side and not allowed to publish in mainstream journals. For this reason, most of his works only became well known posthumously. How many stars you give to your mathematicians: Four, I would give a random moderately famous and important mathematician a three. I gave Bolzano a four because he went beyond just the field of mathematics, and applied mathematical thinking to philosophy. He developed a rigorous theory of science and became a formative influence on analytic philosophy; a philosophical movement which I think deserves credit for removing the nonsense and ambiguity from continental philosophy (please note: that is a lot of nonsense) and has survived to this day. KATHERINE PAINE 11 Augustin-Louis Cauchy, 1789-1857 France Cauchy sequence (10.8, p.62) A sequence (sn)n2N of real numbers is called a Cauchy sequence if and only if 8" > 0; 9N; 8m > N; 8n > N; jsn − smj < ": One interesting fact about this mathematician: There exist sixteen concepts and theorems named after him, more than any other mathematician. How many stars you give to your mathematicians: 5 stars because we consantly see his definition/theorems show up throughout the class. The Cauchy sequence concept has showed up for sequences, series, uniform continuity, uniform convergence.
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