JL 1
Ulisse Dini, 1845-1918Pisa, Italy
Dini’s theorem (not in book)Let (fn : R→ R)n∈N 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)n∈N 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 Dini’s 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
n∑k=0
f (tk )(F (t+k )− F (t−k )
)+
n∑k=1
f (xk )(F (t−k )− F (t+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 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
Rolle’s 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 Shakespeare’s 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∈[tk−1,tk ]f (x)
)(tk − tk−1) .
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, Émile Borel, Élie Cartan, Gheorghe Titeica and Stanisław 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 Einstein’s 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)k∈N be a sequence of functions and (Mk )k∈N a sequence of real numbers such that(1) for all x ∈ R, |gk (x)| ≤ Mk and (2)
∑MK <∞, 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 wehave learned this semester. Without the Bolzano-Weierstrass theorem, a lot of our proofs would fallapart. The Weierstrass M-test is also quite a strong theorem–without knowing the pointwiseconvergence of a sequence of functions, we still have the ability to conclude if a sequence of functionsconverges uniformly.
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)n∈N of real numbers is called a Cauchy sequence if and only if
∀ε > 0, ∃N, ∀m > N, ∀n > N, |sn − sm| < ε.
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
From notebook of Niels Abel
Abel’s theorem (26.6, p.212)Let f (x) =
∑anxn be a power series with finite positive radius of convergence
R. If the series converges at x = R, then f is continuous at x = R. If the seriesconverges at x = −R, then f is continuous at x = −R.
One interesting fact about this mathematician:At the age of 16, Abel gave a proof of the binomial theorem valid for all numbers, extendingEuler’s result which had only held for rationals.
How many stars you give to your mathematicians:
CHANG WANG 15
Isaac Newton, 1643-1727England
Newton’s Method (31.8, p.259)Newton’s method for finding an approximate solution to f (x) = 0 is to begin witha reasonable initial guess x0 and then compute
xn = xn−1 −f (xn−1)
f ′(xn−1), for n ≥ 1.
Often the sequence (xn)n∈N converges rapidly to a solution of f (x) = 0.
One interesting fact about this mathematician:Newton believed in magic. In addition to his more respectable scientific pursuits, Newtonwas a student of alchemy and the occult.How many stars you give to your mathematicians:5 stars. He was not only a great mathematician but also a great physical scientist andastronomer. He was an all-round talent who laid the foundation for physics, mathematics,and engineering.
LIXIN WANG 16
Joseph-Louis Lagrange, 1736-1813France
Taylor’s theorem with Lagrange remainder (31.3,p.250)Let f : R→ R be defined on (a, b). Suppose the n-th derivative f (n) exists on(a, b), we denote Rn(x) the remainder of the Taylor series of f about c. Then foreach x 6= c in (a, b) there is some y between c and y such that:
Rn(x) =f (n)(y)
n!(x − c)n .
One interesting fact about this mathematician:He has said that “if I had been rich, I probably would not have devoted myself tomathematics”.How many stars you give to your mathematicians:Lagrange made significant contributions to the fields of analysis, number theory, and bothclassical and celestial mechanics. We will meet many theorem and methods of him in ourmath courses.
SHUXIAN YANG 17
Jacques Hadamard, 1865-1963France
Cauchy-Hadamard Theorem (23.1,p.188)For the power series
∑anxn, let
β = lim sup |an|1n and R =
1β.
[If β = 0 we set R = +∞, and if β = +∞ we set R = 0.] Then(i) The power series converges for |x | < R;(ii) The power series diverges for |x | > R.
One interesting fact about this mathematician:He married his childhood sweetheart.
How many stars you give to your mathematicians:
YUNQUN YI 18
Guillaume de l’Hôpital, 1661-1704Paris, France
l’Hôpital’s Rule (30.2,p.241)Let s signify a, a+, a−,∞ or −∞ where a ∈ R, and suppose f and g are differentiable functions for
which the following limit exists: limx→sf ′(x)g′(x) = L. (Note that this hypothesis includes some implicit
assumptions: f and g must be defined and differentiable “near” s and g′(x) must be nonzero “near” s.)If limx→s f (x) = limx→s g(x) = 0 or if limx→s |g(x)| = +∞, then limx→s
f (x)g(x) = L.
One interesting fact about this mathematician:L’Hôpital abandoned a military career due to poor eyesight. In 1691 he met young JohannBernoulli, who was visiting France and agreed to supplement his Paris talks oninfinitesimal calculus with private lectures to l’Hôpital at his estate at Oucques.How many stars you give to your mathematicians:I will give 5 stars to him. Because he is very smart and his method about the calculus isvery useful. On the other hand, he works very hard even though he has poor eyesight.
