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JL 1

Ulisse Dini, 1845-1918Pisa, Italy

Dinis theorem (not in book)Let (fn : R R)nN a sequence of continuous functions pointwisely convergingto a continuous function and such that n N, x [a, b], fn+1(x) fn(x). Then(fn : R R)nN converges uniformly.

One interesting fact about this mathematician:Beside being a mathematician, Dini reached the highest office in university administrationwhen he became rector of the University of Pisa, he was elected to the national Italianparliament in 1880 as a representative from Pisa. He was the chair of infinitesimalanalysis.

Another interesting fact about this mathematician:The implicit function theorem is known in Italy as the Dinis theorem.

How many stars you give to your mathematicians:

ERIC COOKE 2

Thomas Joannes Stieltjes, 1865-1894The Netherlands

Definition of the Riemann-Stieltjes sum (35.24, p.320)Let f be bounded on [a, b], and let P = {a = t0 < t1 < . . . < tn = b} , a partitionof [a, b]. A Riemann-Stieltjes sum of f associated with P and F is a sum of theform

nk=0

f (tk )(F (t+k ) F (t

k ))+

nk=1

f (xk )(F (tk ) F (t

+k1)

).

where xk is in (tk1, 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 inmathematics 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 inthe field for less than ten years.

SYD FREDERICK 3

Michel Rolle, 1652-1719France

Rolles Theorem (29.2, p.233)Let f be a continuous function on [a, b] that is differentiable on (a, b) andsatisfies f (a) = f (b). There exists [at least one] x in (a, b) such that f (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 ValueTheorem. He also was one of the first mathematicians to publish Gaussian elimination inEurope.

JOHN GORDOS 4

Julius Wilhelm Richard Dedekind,1831-1916Germany

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 Dedekindacknowledged he had weaknesses in advanced mathematics after receiving his doctorate.From here, he spent two years studying to compensate. I sympathize but my weaknessexists on a foundational level.

KJERSTI JACOBSON 5

Georg Cantor, 1845-1918Germany

Cantor set (Example 5, p.89)In 1883, he introduced the concept of the Cantor set. The Cantor set is simply asubset of the interval [0, 1], but the set has some very interesting properties: forinstance, the set is compact, uncountable, and contains no intervals. The mostcommon modern construction of a Cantor set is the Cantor ternary set, which isbuilt by removing the middle thirds of a line segment.

One interesting fact about this mathematician:Cantor believed that Francis Bacon wrote Shakespeares plays. He studied intenselyElizabethan literature to try to prove his theory. In 1896-97 he published pamphlets on thesubject.

How many stars you give to your mathematicians:

XINXIN JIANG 6

Brook Taylor, 1685-1731England

Taylor series (31.2,p.250)Let f be a function defined on some open interval containing c. If f possesses derivativesof all orders at c, then the Taylor series for f about c is

k=0

f (k)(c)k!

(x c)k .

One interesting fact about this mathematician:As a mathematician, he was the only Englishman after Sir Isaac Newton and Roger Cotescapable of holding his own with the Bernoullis; but a great part of the effect of hisdemonstrations 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 mathematicalresearch, a good grasp of communications skills is also vital. And unfortunately, he is notgood at expressing himself despite his brilliant thinking process.

SIYUAN LIN 7

Jean Gaston Darboux, 1842-1917France

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 betweenf (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 = {a = t0 < t1 < . . . < tn = b}, the upperDarboux sum U(f ,P) of f with P is the sum

U(f ,P) =n

k=1

(sup

x[tk1,tk ]f (x)

)(tk tk1) .

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 mustbe really brilliant. And he must be a great professor, because he taught many highly reputed Europeanmathematicians, for example, mile Borel, lie Cartan, Gheorghe Titeica and Stanisaw Zaremba. Sohe deserves five stars.

SAMUEL LOOS 8

Georg Friedrich Bernhard Riemann,1826-1866German

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 thiscase, we write b

af = L(f ) = U(f ).

One interesting fact about this mathematician:The base of Einsteins Theory of Relativity was set up in 1854 when Riemann gave his firstlectures 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 isimmense. He also had a lot of influence with the development of prime numbers.

CARLY MEYER 9

Karl Weierstrass, 1815-1897German

Bolzano-Weierstrass Theorem (11.5, p 72)Every bounded sequence has a convergent subsequence.Weierstrass M-test (25.7, p 205)Let (gk : R R)kN be a sequence of functions and (Mk )kN a sequence of real numbers such that(1) for all x R, |gk (x)| Mk and (2)

MK

SAMUEL MORTELLARO 10

Bernard Bolzano, 1781-1848Prague, 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 exiledto the county side and not allowed to publish in mainstream journals. For this reason, mostof 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. Igave Bolzano a four because he went beyond just the field of mathematics, and appliedmathematical thinking to philosophy. He developed a rigorous theory of science andbecame a formative influence on analytic philosophy; a philosophical movement which Ithink 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, 1789-1857France

Cauchy sequence (10.8, p.62)A sequence (sn)nN of real numbers is called a Cauchy sequence if and only if

> 0, N, m > N, n > N, |sn sm| < .

One interesting fact about this mathematician:There exist sixteen concepts and theorems named after him, more than any othermathematician.

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.

LAWRENCE PELO 12

Emile Borel, 1871-1956France

Heine-Borel Theorem (13.12, p.90)A subset E of Rk is compact if and only if it is closed and bounded.

One interesting fact about this mathematician:He served for 12 years in the French National Assembly, and was a member of the FrenchResistance during World War II.

A second interesting fact about this mathematician:Borel worked on the Infinite Monkey Theorem, which states that a monkey hitting keys atrandom on a typewriter keyboard for an infinite amount of time will almost surely type agiven text, such as the complete works of William Shakespeare.

How many stars you give to your mathematicians:5 stars, for his founding work in probability.

MIKAEL SPETH 13

Eduard Heine, 1821-1881Germany

Heine-Borel Theorem (13.12, p.90)A subset E of Rk is compact if and only if it is closed and bounded.Heine-Cantor Theorem (19.2, p.143)If f is a continuous function on [a, b], then f is uniformly continuous on [a, b].

One interesting fact about this mathematician:His advisor for his studies was another famous mathematician Peter Dirichlet

How many stars you give to your mathematicians:He came up with some interesting theorems/results but I did not find much moreinformation about him.

JENNA TSEDENSODNOM 14

Niels Henrik Abel, 1802-1829Norway