Comments on: “Fundamental” Theorems

By: Ars Mathematica » Blog Archive » Lax Attack

Ars Mathematica » Blog Archive » Lax Attack — Fri, 15 Jul 2005 18:44:59 +0000

[…] Last week, when Michael asked for a list of fundamental theorems in different branches of mathematics, “>Juan de Mairena suggested the Lax Equivalence Theorem as a candidate. Today on ArXiv I spotted a paper that makes the rather dramatic claim that the theorem is “wrong” — not that it is wrong in the strict mathematical sense, but that its conditions are not realistic for real-world problems. I’m not in a position to evaluate the claim (I never even heard of the result until Juan’s comment), but I thought it was interesting to see a paper on the subject so soon after we discussed it. […]

By: demairena

demairena — Sun, 03 Jul 2005 15:43:04 +0000

I just found a different proof of Lax E.Thm., by using Fourier methods:
http://www.acm.caltech.edu/~acm210/2005/WINTER/week3.pdf

By: sigfpe

sigfpe — Sun, 03 Jul 2005 15:35:11 +0000

Theoretical Physics: Noether’s Theorem (symmetries give rise to conservation laws)

By: demairena

demairena — Sun, 03 Jul 2005 15:25:00 +0000

Lax Equiv. Thm: a consistent finite difference scheme for a well posed initial value problem is convergent if and only if it is stable.
Is a very deep theorem, which follows from the Uniform boundedness principle.

By the way, Lax won the Abel Prize this year: http://www.abelprisen.no/en/prisvinnere/2005/documents/abelprize_2005_EN.pdf , (previously, Jean-Pierre Serre in 2003, and MF Atiyah and IM Singer in 2004)

By: Megan

Megan — Sun, 03 Jul 2005 06:26:43 +0000

Graph Theory: Menger's Theorem. Let G be a graph and A, B be vertices in G. Then the minimum number of vertices separating A from B in G is equal to the maximum number of A-B disjoint paths in G. There are three proofs available in Diestel. Also Number Theory (and my favorite): There are infinitely many primes...

By: Walt

Walt — Sun, 03 Jul 2005 06:17:03 +0000

What is the Lax equivalence theorem?

By: demairena

demairena — Sat, 02 Jul 2005 13:18:51 +0000

Functional Analysis: Hanh- Banach’s Theorem.

Numerical Analysis (PDE): Lax equivalence Theorem.

By: PeterMcB

PeterMcB — Sat, 02 Jul 2005 13:08:21 +0000

Algebraic Topology: Van Kampen’s Theorem.

By: PeterMcB

PeterMcB — Sat, 02 Jul 2005 11:47:30 +0000

Apologies — an anti-Cantorian (Weierstraussian?) gremlin crept into my post: I should have said:

Set Theory: Cantorâ€™s theorem that the real numbers are uncountable.

By: PeterMcB

PeterMcB — Sat, 02 Jul 2005 11:45:37 +0000

Mathematical Statistics: The Central Limit Theorem
Logic: Godel’s Incompleteness Theorems
Set Theory: Cantor’s theorem that the rationals are uncountable
Number Theory: The non-rationality of the square-root of 2 (due to Pythagoras?)