Tag Archives: Pansu differentiability

Emergent algebras as combinatory logic (part IV)

In this post, following “Emergent algebras and combinatory logic (part III)“, I define the class of “computable” graphs in GRAPH and then I show you how differentiability (a generalization of Pansu notion of differentiability on Carnot groups) can be seen as related to this “computability”.

I am using the notations from the last post. See also the post Combinators and zippers for the other notions which I shall use here. The tutorial on graphic lambda calculus may be consulted, if necessary.

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1. The combinatory logic sector. This is the collection of graphs in GRAPH  denoted by COMBINL which is generated by:

  • untyped lambda calculus zippers,
  • arrows and loops,
  • the S diamond,

s_gate_1

  • the termination gate,

with the operation of linking outputs to inputs,under the constraint that we get graphs with a single output, which has to be either:

  • the upper side output of the S diamond,
  • or the upper side output of a zipper

and with  the moves:

 Remarks:

  1. this sector contains the untyped lambda calculus sector,
  2. conjecture: any graph is COMBINL can be transformed into a finite collection of graphs in the untyped lambda calculus sector.

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2. Emergent computable graphs.   I define the sector of EMERCOMPUT(A) as the collection of graphs in GRAPH which are generated by the reunion of EMER(A) and COMBINL, with the collection of moves which belong to one of the sectors, that is all moves excepting the dual of the graphic beta move and the ext1 move.

This sector can be described as the collection of graphs in GRAPH generated by:

  • graphs in COMBINL,
  • \Upsilon gate and the gates \bar{\varepsilon} for any \varepsilon \in \Gamma,
  • the approximate sum gate \Sigma_{a} and the approximate difference gate \Delta_{a}, for any a \in A,

with the operations of linking output to input arrows  and with the following list of moves:

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3. Computably differentiable graphs. A graph A \in GRAPH which has at least one input and one output is computably differentiable if the following graph is in EMERCOMPUT(A)

comp_differ

I shall explain in a future post the meaning of this definition, after writing a bit about differentiability in emergent algebras and about the Pansu differentiability.

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.

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.