Some Mathematicians

JL
1
Ulisse Dini, 1845-1918
Pisa, Italy
Dini’s theorem (not in book)
Let (fn : R → R)n∈N a sequence of continuous functions pointwisely converging
to 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 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:
E RIC C OOKE
2
Thomas Joannes Stieltjes, 18651894
The 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 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.
S YD F REDERICK
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.
J OHN G ORDOS
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.
K JERSTI J ACOBSON
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:
X INXIN J IANG
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
∞ (k )
X
f (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.
S IYUAN L IN
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
f 0 (x1 ) and f 0 (x2 ), then there exists (at least one) x in (x1 , x2 ) such that f 0 (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 upper
Darboux sum U(f , P) of f with P is the sum
!
n
X
sup f (x) (tk − tk −1 ) .
U(f , P) =
k =1
x∈[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 Ţiţeica and Stanisław Zaremba. So
he deserves five stars.
S AMUEL L OOS
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.
C ARLY M EYER
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 : R → R)k ∈N be a sequence ofP
functions and (Mk P
)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 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.
S AMUEL M ORTELLARO
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.
K ATHERINE PAINE
11
Augustin-Louis Cauchy, 1789-1857
France
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 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.
L AWRENCE P ELO
12
Emile Borel, 1871-1956
France
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 French
Resistance 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 at
random on a typewriter keyboard for an infinite amount of time will almost surely type a
given 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.
M IKAEL S PETH
13
Eduard Heine, 1821-1881
Germany
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 more
information about him.
J ENNA T SEDENSODNOM
14
Niels Henrik Abel, 1802-1829
Norway
From notebook of Niels Abel
Abel’s theorem
(26.6, p.212)
P
Let f (x) =
an x n 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 series
converges 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, extending
Euler’s result which had only held for rationals.
How many stars you give to your mathematicians:
C HANG WANG
15
Isaac Newton, 1643-1727
England
Newton’s Method (31.8, p.259)
Newton’s method for finding an approximate solution to f (x) = 0 is to begin with
a reasonable initial guess x0 and then compute
xn = xn−1 −
f (xn−1 )
,
f 0 (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, Newton
was 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 and
astronomer. He was an all-round talent who laid the foundation for physics, mathematics,
and engineering.
L IXIN WANG
16
Joseph-Louis Lagrange, 1736-1813
France
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 for
each x 6= c in (a, b) there is some y between c and y such that:
Rn (x) =
f (n) (y )
(x − c)n .
n!
One interesting fact about this mathematician:
He has said that “if I had been rich, I probably would not have devoted myself to
mathematics”.
How many stars you give to your mathematicians:
Lagrange made significant contributions to the fields of analysis, number theory, and both
classical and celestial mechanics. We will meet many theorem and methods of him in our
math courses.
S HUXIAN YANG
17
Jacques Hadamard, 1865-1963
France
Cauchy-Hadamard Theorem
(23.1,p.188)
P
For the power series
an x n , let
1
β = lim sup |an | n
and
R=
1
.
β
[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:
Y UNQUN Y I
18
Guillaume de l’Hôpital, 1661-1704
Paris, 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→s
f 0 (x)
g 0 (x)
= L. (Note that this hypothesis includes some implicit
assumptions: f and g must be defined and differentiable “near” s and g 0 (x) must be nonzero “near” s.)
f (x)
If limx→s f (x) = limx→s g(x) = 0 or if limx→s |g(x)| = +∞, then limx→s g(x) = L.
One interesting fact about this mathematician:
L’Hôpital abandoned a military career due to poor eyesight. In 1691 he met young Johann
Bernoulli, who was visiting France and agreed to supplement his Paris talks on
infinitesimal 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 is
very useful. On the other hand, he works very hard even though he has poor eyesight.