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bio website rybu.org
location Victoria, Canada
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I am a professor of mathematics at the University of Victoria, in Canada.


Jul
20
awarded  Enlightened
Jul
14
awarded  Enlightened
Jul
14
awarded  Nice Answer
Jul
7
comment Can every parameterised smooth curve be reparameterised by arc-length?
The answer depends on what you mean by "parametrised smooth curve" and "reparametrized by arc length" respectively. Could you be a little more precise on what you take those two terms to mean? Is the initial curve allowed to have zero speed at some number of times, and is a reparametrized curve supposed to be smooth or only piecewise smooth?
Jun
20
comment Can a proper Morse function $\mathbb{R}\to\mathbb{R}$ have infinitely many critical points?
The answer is yes, of course. And there's both even and odd proper morse functions with infinitely many critical points. And you have an example of one such. I suspect Nicaolescu just forgot to write down the a finiteness hypothesis.
Jun
16
comment Terminologies for induced connections
Thanks Ted, for confirming my suspicions!
Jun
16
revised Terminologies for induced connections
added 257 characters in body
Jun
16
asked Terminologies for induced connections
May
29
comment Are Exponential and Trigonometric Functions the Only Non-Trivial Solutions to $F'(x)=F(x+a)$?
For example, if $a<0$ is real, look at the solutions with $F(x)=0$ on $[0,-a)$. That's not trig or exponential, unless its the zero function.
May
29
comment Are Exponential and Trigonometric Functions the Only Non-Trivial Solutions to $F'(x)=F(x+a)$?
That seems like an unrealistic expectation. Delay differential equations have an entire interval of definition as input. So you could easily ensure that's not an exponential or trig function.
May
24
comment Are there other models for 2 dimensional hyperbolic geometry?
A generalization of Hubbard's "belt model" would be to take any k-dimensional totally geodesic hyperbolic subspace of hyperbolic n-dimensional space and on the normal bundle pull-back the metric via the exponential map.
May
24
comment Surface with constant Gaussian curvature $K > 0$
Yes. There's lots.
May
19
comment notation used in algebraic topology
There is no uniform standard in the subject. Ask your prof, or check the notation in your textbook.
May
19
comment Explicit expression for 1-forms which produce, by exterior derivative, a given 2-form in DR-cohomology for twice punctured plane
Look up the Poincare Lemma in whatever textbook you're reading.
May
19
comment Reference request-What is the prerequisite of S.S.Chern's proof of the generalised Gauss-Bonnet theorem?
Chern's proof of the original Gauss-Bonnet theorem is very pleasant and conceptual. It only requires manifold theory to the point of Stoke's theorem. There's a pretty good version of it in Berger's Panoramic View of Riemannian Geometry. I have a version written up in my differential geometry lecture notes, available on my webpage. The proof gives you a very good idea of what to expect in higher dimensions, and it's pleasantly conceptual.
Apr
28
reviewed Close Solve the equation 13 modulo 35 times x = 2
Apr
25
comment Are $C^{k}$ manifolds the same as $C^{\infty}$ manifolds?
Instead of using $|x|$ in the above definition, you could use a PL function which is linear on $[0,\infty)$ and $(-\infty,0]$ but with different (but positive) slopes.
Apr
25
comment Are $C^{k}$ manifolds the same as $C^{\infty}$ manifolds?
I think you need to re-think your questions. There is no notion of differentiability of a single chart in an atlas. You need to perhaps talk about the atlas with a chart removed, but then you're dealing with a different differentiable structure.
Apr
25
comment Are $C^{k}$ manifolds the same as $C^{\infty}$ manifolds?
The function $f : \mathbb R \to \mathbb R$ given by $f(x) = x^n |x|$ is $C^{n}$ but not $C^{n+1}$, and a diffeomorphism provided $n$ is odd. You can construct such diffeomorphisms quite easily like this.
Apr
25
answered Are $C^{k}$ manifolds the same as $C^{\infty}$ manifolds?