What is Mathematics?
Lee Peng Yee
21 November 2014
What is mathematics?
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•
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A book by Courant and Robbins (1941)
Last two chapters: calculus
Calculus was the palace of modern math
Now calculus is the gate to the palace
Calculus: Taylor series
An example:
• sin 𝑥 = 𝑥
• sin 𝑥 ≈ 𝑥
𝑥3
−
3!
𝑥3
−
3!
𝑥5
𝑥7
+ − +⋯
5!
7!
𝑥5
+ for𝑥near 0
5!
for all 𝑥
• If we can expand, we can approximate
• To expand, a function must be infinitely
differentiable
Power series
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Taylor series is one special case
We use it to solve differential equations
We used to express solutions in closed form
Also by special functions and huge volumes
of tables
Analysis: Fourier series
• Sawtooth function:
𝑥=2
𝑛−1 sin 𝑛𝑥
(−1)
∞
𝑛=1
𝑛
for −𝜋 < 𝑥 < 𝜋
and defined periodically over 𝑅
• Fourier series approximates a given function
globally, whereas Taylor series locally
Functional analysis
• Convergence of Fourier series is a problem
• It was solved by Hilbert space 𝐿2 [−𝜋, 𝜋]
• Taylor series converges to the given function
uniformly inside the interval of convergence
• Fourier series converges under the norm in
𝐿2 [−𝜋, 𝜋]
Mother functions
• Taylor series lives in an infinite dimensional space
with coordinates 1, 𝑥, 𝑥 2 , … , 𝑥 𝑛 , …
• Fourier series lives in a space with coordinates
1
, sin x, cos x, sin 2x, cos 2x, … , sin nx, cos nx, …
2
• x and sin 𝑥 are called mother functions
Further comparison
Taylor series / Fourier series
• Coefficients: by differentiation / by integration
• Coordinates: non-orthogonal / orthogonal
• Better approximation: locally / globally
Question: Find a series … … both locally and
globally
Haar wavelet
• Mother function
ψ 𝑡 = 1 for 0 ≤ 𝑡 <
1
,
2
1
2
–1 for ≤ 𝑡 < 1.
0 otherwise
• Coordinates
𝑛
2
ψ𝑛,𝑘 𝑡 = 2 ψ(2𝑛 𝑡 − 𝑘) for all t
Properties of Haar wavelet
• Haar wavelet lives in 𝐿2 𝑅
• The Haar coordinate system is orthogonal
• Haar wavelet converges to a given function
locally and globally
Wavelets
• Haar wavelet is one special case
• “Whatever your curves, we’ve you wrapped”
• Wavelets help send images fast and far away
among other applications
Stochastic equations
• Nobel prize was awarded for the works on the
Black-Scholes equation
• Stochastic equations contain nondeterministic
terms called noises
• Hence possible to model events in the
financial world
Wiener and Itô
• Norbert Wiener (1894 – 1964) provided the
concept known as Wiener process
• Kiyoshi Itô (1915 – 2008) developed the
operational tools
• French school led by Paul-André Meyer (1934
– 2003)
Big data analytics
• Latest challenges due to the size and the
speed of trade data
• Big data technology is to collect, analyse and
make use of the data
• Big data is not well-defined
Big data Singapore
Two examples:
• A bank with 25 million transactions a month
uses the technology to plan for ATM cash
reloading
• Singapore Civil Defence Force is looking into
using the technology to slash response times
for ambulance and fire engines
Modern history of mathematics
• Calculus helped make better locomotives
• Analysis was a product of the nineteenth
century
• Mathematics is being used in the nontraditional areas
• The latest: big data
What we miss
• Statistics, computational mathematics.
discrete mathematics … …
• Discrete mathematics came in only after
World War II
• Now we have moved from the world of
telephones to that of internet
• Mathematics outside is not the same as inside
the classroom
END
pengyee.lee@nie.edu.sg