ds.dynamical systems - Local linearization of ODE at singular point

I would like the simplest example of the failure of an ODE to be locally diffeomorphic to its linearization, despite being locally homeomorphic to it. More precisely, consider x' = f(x) with f(0) = 0 in R^n. Let A = f'(0) so that the local linearization is x' = Ax. Suppose the eigenvalues of A all have nonzero real part (i.e., 0 is a hyperbolic critical point).

The Hartman-Grobman theorem tells us that there is a homeomorphism of a neighborhood of 0 which conjugates the system x'=f(x) to its linearization x'=Ax. If one reads the elementary `differential equations from the dynamical systems point of view' literature, however, you will gain the false impression that there is a diffeomorphism h : U --> U of a nhood of 0 which does this, and that further, one can even do this with h'(0) = I, the identity matrix. The point of this is to ensure that the trajectories of the nonlinear system are tangent to the trajectories of the linearization: if $h'(0)neq I$ then this may be false.

Smale's stable manifold theorem gives partial information in this direction, saying that the stable manifolds of the system and its linearization are tangent, and similarly for the unstable manifolds. In 2D, at a saddle, this is sufficient to imply that the separatrices of the original system are tangent to those of the linearization. For a node in 2D, or in higher dimensions, I am under the impression this need not hold. I even think I had worked out an example many years ago, which I no longer recall.

Any enlightenment on this issue would be much appreciated. I am not at all expert in these matters, so welcome any corrections, if I have distorted the facts. I am hoping for a 2 dimensional example.

Added later:
Yuri: resonances and normal forms are definitely relevant. When I get time I will look into the references you suggest.

Here's an example of what I am trying to avoid. Consider a flow on the unit disk with trajectories the radial lines y = mx. Conjugate by $(r,theta) mapsto (r,f(r,theta))$ where $f(0,theta)$ is constant on, say, $[-pi/2,pi/2]$, e.g. $f(r,theta) = rtheta$ on $[-pi/2,pi/2]$ and $(2theta-pi) + r(pi-theta)$ in the left half plane. Now all the trajectories leaving the unit circle in the right half plane approach 0 along the positive x-axis. So, conjugating with such a homeomorphism has replaced a single trajectory with horizontal tangent by an entire interval of such.

I strongly suspect that this sort of pathology doesn't happen with polynomial flows. Perhaps the normal forms will show this. What I would hope is that for each slope, there is a 1-1 correspondence between the trajectories in the original flow and its linearization approaching or leaving the singular point at that slope. In particular, that you can't have a single trajectory in the linearization but a whole interval of them in the nonlinear flow. A counterexample to this hope would be disappointing, but would settle the matter.

linear algebra - subspace separation and M-matrices

The separation between two square matrices $A$ and $B$, often used as a measure of the sensitivity of invariant subspace problems, is defined as
$$
operatorname{sep}(A,B)=min_{Xneq 0}frac{leftVert AX-XBrightVert_F}{leftVert XrightVert_F}
$$
(see e.g. Golub and Van Loan, Section 7.2.4).

I would like to express the separation between two M-matrices in terms of their Perron vector and values $Au=lambda u$, $Bv=mu v$. All I can do is estimating $operatorname{sep}(A,B)geq lambda+mu$. Is there any better bound, maybe from above? The question looks like an "eigenvalues vs. singular values" bound, so I am not sure that the answer is positive.

Alternatively, do you know any "handier" way to deal with $operatorname{sep}(A,B)$, other than using its definition and its alternative formulation as the smallest singular value of a Kronecker sum $sigma_{min}(B otimes I + I otimes A^T)$? I always find it unwieldy to use this definition, it is not easy to squeeze something simple to compute/estimate out of it. For instance, if $A$, $A'$, $B$ are M-matrices and $A'geq A$, does $operatorname{sep}(A',B)geq operatorname{sep}(A,B)$ hold?

ag.algebraic geometry - Is there a general projection formula for morphisms of ringed topoi?

In the context of sheaves of $mathcal O_X$-modules,
there is the following reference: Prop. 3.9.4 in Lipman's
Notes on derived functors and Grothendieck duality.
A closely related result is in Neeman's paper The Grothendieck duality theorem ...; see Prop. 5.3.

I'm not sure that analogous results should be expected to hold in arbitrary generality;
for example, both references place a restriction on the base scheme, and require quasi-coherence assumptions. (In some sense, one has to reduce to the locally free case,
where the statement is obvious. Quasi-coherent sheaves then admit locally free resolutions.
The proofs of the cited results apply some form of this argument in rather subtle and sophisticated ways.)

random matrices - A formula for moments of the limit distribution of singular values in the proof of the circular law

One of the steps in the proof of the circular law in random matrix theory is obtaining the limiting spectral distribution for the matrix

$(frac{1}{sqrt{n}} X_n - zI)(frac{1}{sqrt{n}} X_n - zI)^ast$,

where $X_n$ is an $n times n$ matrix with i.i.d. square integrable coefficients.

This is done with the Stieltjes transform, however Bai and Silverstein remark in their book that (under some assumptions on the random variables), one can also prove that moments of the empirical distribution converge almost surely to $mu_k(|z|^2)$ (they leave the details as an exercise).

It is indeed not difficult to show that the moments converge to a number which depends only on $|z|^2$. My question is whether there is a nice formula for $mu_k(|z|^2)$. For z = 0, we have just the Marchenko-Pastur distribution and there is a relatively good-looking formula involving binomial coefficients. For other values of $z$ we have some additional terms corresponding to some graphs (depending on $k$). This is enough to provide a moment proof of the convergence of the empirical spectral distribution, but I wonder if there is a nicer formula for the moments, e.g. involving only binomial coefficients and algebraic operations (no summation over combinatorial objects).

Why do Humans not produce Vitamin C like other mammals?

Humans do not produce Vitamin C due to a mutation in the GULO (gulonolactone oxidase) gene, which results in the inability to synthesize the protein. Normal GULO is an enzyme that catalyses the reaction of D-glucuronolactone with oxygen to L-xylo-hex-3-gulonolactone. This then spontaneously forms Ascorbic Acid (Vitamin C). However without the GULO enzyme, no vitamin C is produced.

This has not been selected against in natural selection as we are able to consume more than enough vitamin C from our diet. It is also suggested that organisms without a functional GULO gene have a method of "recycling" the vitamin C that they obtain from their diets using red blood cells (see Montel-Hagen et al. 2008).

A 2008 published study (Li et al. 2008) claimed to have successfully re-instated the ability to produce vitamin C in mice.

Simply as trivia: other than humans; guinea pigs, bats and dry-nosed primates have lost their ability to produce vitamin C in the same way.

---

References

• Li, Y., Shi, C.-X., Mossman, K.L., Rosenfeld, J., Boo, Y.C. &
  Schellhorn, H.E. (2008) Restoration of vitamin C synthesis in
  transgenic Gulo-/- mice by helper-dependent adenovirus-based
  expression of gulonolactone oxidase. Human gene therapy. [Online] 19
  (12), 1349–1358. Available from: doi:10.1089/hgt.2008.106 [Accessed:
  31 December 2011].




• Montel-Hagen, A., Kinet, S., Manel, N., Mongellaz, C., Prohaska, R.,
  Battini, J.-L., Delaunay, J., Sitbon, M. & Taylor, N. (2008)
  Erythrocyte Glut1 Triggers Dehydroascorbic Acid Uptake in Mammals
  Unable to Synthesize Vitamin C. Cell. [Online] 132 (6), 1039–1048.
  Available from: doi:10.1016/j.cell.2008.01.042 [Accessed: 31 December
  2011].

oc.optimization control - Optimization over permutation?

It may be the case that simulated annealing and genetic algorithms are relatively complicated to understand, bound and implement in this instance.

Instead, a very easy starting point would be a simple hill-climbing algorithm.

Start with an arbitrary (or better, random) initial permutation $pi$.

The set of moves is the set $M$ of permutations that you can reach by transposing two elements of the permutation.

While there is a move that decreases $f$,

• Make the move to reach a new current permutation.




• Compute the new set of moves (or rather, their profits $f(pi) - f(pi')$ for a move reaching $pi'$).

This will get you to a local minimum at a cost of $O(n^2)cdot C(n)$, per move, where $C(n)$ is the cost of calculating $f(pi)$ for a permutation of $[n]$.

Extremely simple and probably not too costly as a first step. You may be able to prove some sort of worst case bound between a local optimum and a global optimum.

fa.functional analysis - Subspaces of $L^{2}$

[In what follows $0^{0}$= 1 by convention.]

Is there some closed infinite dimensional linear subspace $F$ of $L^{2}(0,1)$
such that $left|fright|^{left|fright|}$ belongs to $L^{2}(0,1)$
for all $f$ in $F$ ?

This problem is related to the Erdos - Shapiro - Shields paper [ESS].
From this paper it follows that the answer is negative if $left|fright|^{left|fright|}$ is replaced by $left|fright|^{left|fright|^{2}}$.

Some thoughts. Suppose that such an $F$ exists, and take some $p > 2$.
Let $f$ be in $F$.

Then clearly $g:=(p/2)cdot f$ is in $F$, too, hence $h :=left|gright|^{left|gright|}$
belongs to $L^{2}(0,1)$.

Next, it is easy to see that $leftVert frightVert _{p}^{p}leq1+leftVert hrightVert _{2}^{2}<+infty$.

Therefore, $F$ is contained in $L^{p}(0,1)$ as a linear subspace
(i.e., algebraically).

Now, applying the Closed Graph Theorem to the natural linear embedding
$j:(F, ||.||_{2})rightarrow L^{p}(0,1)$, it follows that $j$ is
continuous. Consequently, the Hilbertian 2-norm and the $p$-norm
are equivalent on $F$. Moreover, it follows that $F$ is complete
w.r.t. the $p$-norm, and, in turn, it is a closed subspace of $L^{p}(0,1)$.

And this is true for all $p > 2$.

[ESS] http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.mmj/1028999306