10,319 reputation
14673
bio website vmm.math.uci.edu
location Univ. of California at Irvine
age 83
visits member for 4 years, 3 months
seen Sep 25 at 23:12
I'm a Professor at UC Irvine, but spent most of my career at Brandeis. My research areas: Differential Topology, Transformation Groups, and Global Analysis. Recently I developed a math visualization program, 3D-XplorMath, freely available at http://3D-XplorMath.org) and a companion website, the Virtual Math Museum at http://VirtualMathMuseum.org. Last year I co-authored a differential equations text with my son Bob, most of which is downloadable from http://ode-math.com .

Sep
30
comment About Palais' remark that an isometry of Riemannian manifolds does not induce an isometry of the Hilbert manifolds of curves
The problem lies in the line above defining $g_\sigma(λ,μ)$. This is the metric for $H_0$, NOT the metric for $H_1$. The correct definition of the metric for $H_1$ uses $⟨λ'(t),μ'(t)⟩$ rather than $⟨λ(t),μ(t)⟩$ (see page 222 of my article where both metrics are defined), and with this change it is fairly obvious why my remark in the cited paper is in fact correct.
Aug
23
comment Algorithm for finding inverse images of a local diffeomorphism
Yes! Newton's Method is clearly the way to go. For $k=1$, the algorithm I wrote in fact did use Newton's Method, (rather than bisection) since it is much faster. But I guess I forgot that Newton's Method works in higher dimensions too. Thanks Ryan.
Aug
22
comment Algorithm for finding inverse images of a local diffeomorphism
@Nanda For the application I have in mind, F will be given by a formula.
Jul
11
comment What is a good method to find random points on the n-sphere when n is large?
@Mark Meckes Many thanks, Mark. In retrospect, if I had been more adept with my Googling I might have discovered that the Gaussian RV approach was a "well-known" answer to my question, and would not have needed to ask it on MO. But as usual, I have come away very impressed with the utility of this great website and learned a lot from the answers I received.
May
30
comment A Canonical Form Theorem for $n$-forms?
@Francesco YES ! I don't think you are missing anything---that is essentially the same proof as mine. I'll have to get Kobayashi's book from the library and have a look. Many thanks for this clue.
May
30
comment A Canonical Form Theorem for $n$-forms?
@Mariano. Right, that is the way I think of it, and it was Darboux Theorem that made me think it might be true.
May
22
comment Why don't more mathematicians improve Wikipedia articles?
@Mark M YES! That is a very good approximation of what I was asking for. Thanks for telling me about it. It is now one of my "pinned" tabs and I will try to help with the project.
May
17
comment Are there proofs that you feel you did not “understand” for a long time?
How about the proof that Einstein is said to have (re)discovered as a teen-ager? It is clear that the areas of similar right triangles is proportional to the squares of corresponding sides---in particular to the squares of their hypotenuses. Dropping a perpendicular from the vertex of the right triangle onto the hypotenuse c divides it into two similar triangles with hypotenuses a and b, hence: kc^2 = ka^2 + k*b^2 so c^2 = a^2 + b^2
Apr
20
comment Which hard mathematical problems do you have to solve to earn bitcoins ?
Is it known whether there is an algorithm using a quantum computer (i.e., qubits instead of bits) for simplifying the Bitcoin mining?
Apr
11
comment Fixed point theorems
...has a unique solution, provided ε is sufficiently small.
Jan
13
comment Lie group action with no slice
@Yves Cormulier I don't think that's so Yves. If it were then even a compact group action wouldn't have a slice at an isolated fixed point p, whereas in fact an invariant neighborhood of p is a slice at p in that case.
Jan
13
comment Lie group action with no slice
After thinking about my answer a bit I realized that an even simpler example is the action of the group Z of integers on the circle generated by rotation through an irrational angle.
Dec
7
comment Notes for Bott's 1963 lectures on Morse theory
I did some more searching with Google and found that Amazon has a reference to: "Lectures on Morse theory: [revised and expanded version of notes of lectures delivered at Professor R. Bott's topology seminar at Harvard in February and March of 1963 [Unknown Binding] Richard S Palais (Author)" with Brandeis Univ, listed as the publisher. I suspect this must have been an early preprint version of "Morse Theory on Hilbert Manifolds"
Oct
16
comment Proof synopsis collection
@mathahada Where do you see a geometric series in my proof? (One of the points I had in publishing the above proof was to show that the geometric series argument (used since Banach's original proof) is really not necessary.
Oct
14
comment Can the level set of a critical value be a regular submanifold?
You probably should say amend the statement of the theorem to say that a non-empty level set of a regular value of a smooth function f:M→ℝ on a smooth manifold is a regular submanifold of codimension one. (That takes care of the problem with f identically zero.)
Sep
19
comment Collapsing of Riemannian manifolds with a group action
"...Consider the fixed point set F, it is of course a submanifold of M by the slice theorem". Note that it is really simpler than that; in geodesic coordinates at a point p of F, the fixed point set is locally the linear subspace left fixed by the linearized action at p.
Sep
15
comment First known proof of $\sqrt 2$ is irrational with prime factorization?
Your right Franz, it doesn't. It's just that there seems to be a belief that you NEED unique prime factorization to prove the irrationality of non-square integers, and when I first saw this (much more elementary) proof I found it an eye-opening experience.
Sep
3
comment Area of union of random circles in a plane
You will probably get a more "natural" answer if you choose a "torus", i.e., identify opposite edges of a square, to eliminate edge effects.
Aug
25
comment Square root of a positive $C^\infty$ function.
Yes, and functions of this type are discussed in section 2 of the reference I gave in my answer.
Aug
15
comment About a letter by Richard Palais of 1965.
@Mariano: Yes, I guess that was a pretty verbose "no". :-) Dick