Pros and cons of higher order Pansu derivatives

This interesting question from mathoverflow

Higher order Pansu derivative

is asked by nil (no website, no location). I shall try to explain the pros and cons of higher order derivatives in Carnot groups. As for a real answer to nil’s question, I could tell him but then …

For “Pansu derivative” see the paper: (mentioned in this previous post)

Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, The Annals of Mathematics Second Series, Vol. 129, No. 1 (Jan., 1989), pp. 1-60

Such derivatives can be done in any metric space with dilations, or in any normed group with dilations in particular (see definition in this previous post).

Pros/cons: It would be interesting to have a higher order differential calculus with Pansu derivatives, for all the reasons which make higher derivatives interesting in more familiar situations. Three examples come to my mind: convexity, higher order differential operators and curvature.

1. Convexity pro: the positivity of the hessian of a function implies convexity. In the world of Carnot groups the most natural definition of convexity (at least that is what I think) is the following: a function f: N \rightarrow \mathbb{R}, defined on a Carnot group N with (homogeneous) dilations \displaystyle \delta_{\varepsilon}, is convex if for any x,y \in N and for any \varepsilon \in [0,1] we have

f( x \delta_{\varepsilon}(x^{-1} y)) \leq f(x) + \varepsilon (-f(x) + f(y)) .

There are conditions in terms of higher order horizontal derivatives (if the function is derivable in the classical sense) which are sufficient for the function to be convex (in the mentioned sense). Note that the positivity of the horizontal hessian is not enough! It would be nice to have a more intrinsic differential condition, which does not use classical horizontal derivatives. Con: as in classical analysis, we can do well without second order derivatives when we study convexity. In fact convex analysis is so funny because we can do it without the need of differentiability.

2. Differential operators Pro: Speaking about higher order horizontal derivatives, notice that the horizontal laplacian is not expressed in an intrinsic manner (i.e. as a combinaion of higher order Pansu derivatives). It would be interesting to have such a representation for the horizontal laplacian, at least for not having to use “coordinates” (well, these are families of horizontal vector fields which span the distribution) in order to be able to define the operator. Con: nevertheless the horizontal hessian can be defined intrinsically in a weak sense, using only the sub-riemannian distance (and the energy functional associated to it, as in the classical case). Sobolev spaces and others are a flourishing field of research, without the need to appeal to higher order Pansu derivatives. (pro: this regards the existence of solutions in a weak sense, but to be honest, what about the regularity business?)

3. Curvature Pro: What is the curvature of a level set of a function defined on a Carnot group? Clearly higher order derivatives are needed here. Con: level set are not even rectifiable in the Carnot world!

Besides all this, there is a general:

Con: There are not many functions, from a Carnot group to itself, which are Pansu derivable everywhere, with continuous derivative. Indeed, for most Carnot groups (excepting the Heisenberg type and the jet type) only left translations are “smooth” in this sense. So even if we could define higher order derivatives, there is not much room to apply them.

However, I think that it is possible to define derivatives of Pansu type such that always there are lots of functions derivable in this sense and moreover it is possible to introduce higher order derivatives of Pansu type (i.e. which can be expressed with dilations).

UPDATE:  This should be read in conjunction with this post. Please look at Lemma 11   from the   last post of Tao    and also at the notations made previously in that post.  Now, relation (4) contains an estimate of a kind of discretization of a second order derivative. Based on Lemma 11 and on what I explained in the linked post, the relation (4) cannot hold in the sub-riemannian world, that is there is surely no bump  function \phi such that d_{\phi} is equivalent with a sub-riemannian distance (unless the metric is riemannian). In conclusion, there are no “interesting” nontrivial C^{1,1} bump functions (say quadratic-like, see in the post of Tao how he constructs his bump function by using the distance).

There must be something going wrong with the “Taylor expansion” from the end of the proof of Lemma 11,  if instead of a norm with respect to a bump function we put a sub-riemannian distance. Presumably instead of “n”  and  “n^{2}” we have to put something else, like   “n^{a}”    and  “n^{b}” respectively, with coefficients  a, b/2 <1 and also functions of (a kind of  degree,  say) of g. Well, the coefficient b will be very interesting, because related to some notion of curvature to be discovered.

Topographica, the neural map simulator

The following speaks for itself:

 Topographica neural map simulator 

“Topographica is a software package for computational modeling of neural maps, developed by the Institute for Adaptive and Neural Computation at the University of Edinburgh and the Neural Networks Research Group at the University of Texas at Austin. The project is funded by the NIMH Human Brain Project under grant 1R01-MH66991. The goal is to help researchers understand brain function at the level of the topographic maps that make up sensory and motor systems.”

From the Introduction to the user manual:

“The cerebral cortex of mammals primarily consists of a set of brain areas organized as topographic maps (Kaas et al. 1997Vanessen et al. 2001). These maps contain systematic two-dimensional representations of features relevant to sensory, motor, and/or associative processing, such as retinal position, sound frequency, line orientation, or sensory or motor motion direction (Blasdel 1992Merzenich et al. 1975Weliky et al. 1996). Understanding the development and function of topographic maps is crucial for understanding brain function, and will require integrating large-scale experimental imaging results with single-unit studies of individual neurons and their connections.”

One of the Tutorials is about the Kohonen model of self-organizing maps, mentioned in the post  Maps in the brain: fact and explanations.

Numbers for biology, are them enough?

Very impressed by this post:

Numb or numbered?

from the blog of Stephen Curry.

Two reactions, opposite somehow, could be triggered by the parallel between physics (now a field respected by any  layman) and biology (the new challenger).

The glory of physics, as well as the industrial revolution, are a consequence of the discovery of infinitesimal calculus by  the Lucasian Professor of Mathematics Isaac Newton  and by the   philosopher, lawyer and mathematician Gottfried Leibniz. All of this started from the extraordinary creation of a gifted generation of thinkers. We may like this or not, but this is TRUE.

The reactions:

1. Positive: yes, definitely some mathematical literacy would do a lot of good to students from the biological sciences. In fact I am shocked that apparently there is resistance to this in the field. (Yes, mathematicians can be and are arrogant when interacting with other scientists, but in most of the cases that means that (a) they are bad mathematicians anyway, except when they are not, or  (b) that they react to the misconceptions of the other scientists (which, by manifesting such narrowness of view, are bad scientists, except when they are not))

2. Negative: Numeracy and preadolescent recipes (at least this is (or was)  the level of mathematics knowledge in the school curriculum in the part of the world where I grown up) are not enough. Mathematics was highly developed before infinitesimal calculus, but this was not sufficient for the newtonian revolution.

To finish,  Robert Hooke was in the same generation with Newton and Leibniz. So maybe biology could hurry up a bit in this respect.

Noncommutative Baker-Campbell-Hausdorff formula: the problem

I come back to a problem alluded in a previous post, where the proof of the Baker-Campbell-Hausdorff formula from this post by Tao is characterized as “commutative”, because of the “radial homogeneity” condition in his Theorem 1 , which forces commutativity.

Now I am going to try to explain this, as well as what the problem of a “noncommutative” BCH formula would be.

Take a Lie group G and identify a neighbourhood of its neutral element with a neighbourhood of the 0 element of its Lie algebra. This is standard for Carnot groups (connected, simply connected nilpotent groups which admit a one parameter family of contracting automorphisms), where the exponential is bijective, so the identification is global. The advantage of this identification is that we get rid of log’s and exp’s in formulae.

For every s > 0 define a deformation of the group operation (which is denoted multiplicatively), by the formula

(1)                s(x *_{s} y) = (sx) (sy)

Then we have x *_{s} y \rightarrow x+y as s \rightarrow 0.

Denote by [x,y] the Lie bracket of the (Lie algebra of the) group G with initial operation and likewise denote by [x,y]_{s} the Lie bracket of the operation *_{s}.

The relation between these brackets is: [x,y]_{s} = s [x,y].

From the Baker-Campbell-Hausdorff formula we get:

-x + (x *_{s} y) - y = \frac{s}{2} [x,y] + o(s),

(for reasons which will be clear later, I am not using the commutativity of addition), therefore

(2)         \frac{1}{s} ( -x + (x *_{s} y) - y ) \rightarrow \frac{1}{2} [x,y]       as        s \rightarrow 0.

Remark that (2) looks like a valid definition of the Lie bracket which is not related to the group commutator. Moreover, it is a formula where we differentiate only once, so to say. In the usual derivation of the Lie bracket from the group commutator we have to differentiate twice!

Let us now pass to a slightly different context: suppose G is a normed group with dilations (the norm is for simplicity, we can do without; in the case of “usual” Lie groups, taking a norm corresponds to taking a left invariant Riemannian distance on the group).

G is a normed group with dilations if

  • it is a normed group, that is there is a norm function defined on G with values in [0,+\infty), denoted by \|x\|, such that

\|x\| = 0 iff x = e (the neutral element)

\| x y \| \leq \|x\| + \|y\|

\| x^{-1} \| = \| x \|

– “balls” \left\{ x \mid \|x\| \leq r \right\} are compact in the topology induced by the distance $d(x,y) = \|x^{-1} y\|$,

  • and a “multiplication by positive scalars” (s,x) \in (0,\infty) \times G \mapsto sx \in G with the properties:

s(px) = (sp)x , 1x = x and sx \rightarrow e as $s \rightarrow 0$; also s(x^{-1}) = (sx)^{-1},

– define x *_{s} y as previously, by the formula (1) (only this time use the multiplication by positive scalars). Then

x *_{s} y \rightarrow x \cdot y      as      s \rightarrow 0

uniformly with respect to x, y in an arbitrarry closed ball.

\frac{1}{s} \| sx \| \rightarrow \|x \|_{0}, uniformly with respect to x in a closed ball, and moreover \|x\|_{0} = 0 implies x = e.

Comments:

1. In truth, everything is defined in a neighbourhood of the neutral element, also G has only to be a local group.

2. the operation x \cdot y is a (local) group operation and the function \|x\|_{0} is a norm for this operation, which is also “homogeneous”, in the sense

\|sx\|_{0} = s \|x\|_{0}.

Also we have the distributivity property s(x \cdot y) = (sx) \cdot (sy), but generally the dot operation is not commutative.

3. A Lie group with a left invariant Riemannian distance d and with the usual multiplication by scalars (after making the identification of a neighbourhood of the neutral element with a neighbourhood in the Lie algebra) is an example of a normed group with dilations, with the norm \|x\| = d(e,x).

4. Any Carnot group can be endowed with a structure of a group with dilations, by defining the multiplication by positive scalars with the help of its intrinsic dilations. Indeed, take for example a Heisenberg group G = \mathbb{R}^{3} with the operation

(x_{1}, x_{2}, x_{3}) (y_{1}, y_{2}, y_{3}) = (x_{1} + y_{1}, x_{2} + y_{2}, x_{3} + y_{3} + \frac{1}{2} (x_{1}y_{2} - x_{2} y_{1}))

multiplication by positive scalars

s (x_{1},x_{2},x_{3}) = (sx_{1}, sx_{2}, s^{2}x_{3})

and norm given by

\| (x_{1}, x_{2}, x_{3}) \|^{2} = (x_{1})^{2} + (x_{2})^{2} + \mid x_{3} \mid

Then we have X \cdot Y = XY, for any X,Y \in G and \| X\|_{0} = \|X\| for any X \in G.

Carnot groups are therefore just a noncommutative generalization of vector spaces, with the addition operation $+$ replaced by a noncommutative operation!

5. There are many groups with dilations which are not Carnot groups. For example endow any Lie group with a left invariant sub-riemannian structure and hop, this gives a norm group with dilations structure.

In such a group with dilations the “radial homogeneity” condition of Tao implies that the operation x \cdot y is commutative! (see the references given in this previous post). Indeed, this radial homogeneity is equivalent with the following assertion: for any s \in (0,1) and any x, y \in G

x s( x^{-1} ) = (1-s)x

which is called elsewhere “barycentric condition”. This condition is false in any noncommutative Carnot group! What it is true is the following: let, in a Carnot group, x be any solution of the equation

x s( x^{-1} ) = y

for given y \in G and $s \in (0,1)$. Then

x = \sum_{k=0}^{\infty} (s^{k}) y ,

(so the solution is unique) where the sum is taken with respect to the group operation (noncommutative series).

Problem of the noncommutative BCH formula: In a normed group with dilations, express the group operation xy as a noncommutative series, by using instead of “+” the operation “\cdot” and by using a definition of the “noncommutative Lie bracket” in the same spirit as (2), that is something related to the asymptotic behaviour of the “approximate bracket”

(3)         [x,y]_{s} = (s^{-1}) ( x^{-1} \cdot (x *_{s} y) \cdot y^{-1} ).

Notice that there is NO CHANCE to have a limit like the one in (2), so the problem seems hard also from this point of view.

Also notice that if G is a Carnot group then

[x,y]_{s} = e (that is like it is equal to o, remember)

which is normal, if we think about G as being a kind of noncommutative vector space, even of G may be not commutative.

So this noncommutative Lie bracket is not about commutators!

Topological substratum of the derivative

Until recently, on my home page was a link to the scan of the paper

 The topological substratum of the derivative (I), Math. Reports (Stud. Cerc. Mat.) 45, 6,       (1993), 453-465

which is no longer visible now. But maybe it deserves a post here, because is my oldest attempt to understand differential calculus as an abstract matter and to look to new forms of it.

To me it became clear that differential calculus admits variants, in the same spirit as euclidean geometry admitting non-euclidean variants. At that moment I had no really intersting examples of such a “non-euclidean” differential calculus, so I switched to other research subjects. Nobody pointed to me the huge paper

Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang unThe Annals of Mathematics Second Series, Vol. 129, No. 1 (Jan., 1989), pp. 1-60

by Pierre Pansu. It was only luck that in 2000, at Lausanne, I met Sergey Vodop’yanov (from Sobolev Institute of Mathematics). He started to explain to me what Carnot groups are and I was thrilled to   learn that examples I needed previously are numerous in sub-riemannian geometry.

With the right frame of mind (at least I think so), that of intrinsic dilations, I  started then to study sub-riemannian geometry.

Planar rooted trees and Baker-Campbell-Hausdorff formula

Today on arXiv was posted the paper

Posetted trees and Baker-Campbell-Hausdorff product, by Donatella Iacono, Marco Manetti

with the abstract

We introduce the combinatorial notion of posetted trees and we use it in order to write an explicit expression of the Baker-Campbell-Hausdorff formula.

The paper may be relevant (check also the bibliography!) for the subject of writing “finitary“, “noncommutative” BCH formulae, from self-similarity arguments using dilations.

computing with space | open notebook

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