Spacebook: a facebook for space

I follow the work of Mark Changizi on vision. Previously I mentioned one of his early papers on this subject, “Harnessing vision for computation” .

One of the applications of computing with space  could be to SHARE THE SPATIAL EXPERIENCE ON THE WEB.

Background. When I was writing the paper on the problem of computing with space, I stumbled upon this article by Mark in Psychology Today

The Problem With the Web and E-Books Is That There’s No Space for Them

The title says a lot. I was intrigued by the following passage

“My personal library serves as extension of my brain. I may have read all my books, but I don’t remember most of the information. What I remember is where in my library my knowledge sits, and I can look it up when I need it. But I can only look it up because my books are geographically arranged in a fixed spatial organization, with visual landmarks. I need to take the integral of an arctangent? Then I need my Table of Integrals book, and that’s in the left bookshelf, upper middle, adjacent to the large, colorful Intro Calculus book.”

So I posted the following comment:  Is your library my library?

“Good point, but you have converted a lot of time into understanding, exploring and using the space of your library. To me the brain-spatial interface of your library is largely incomprehensible. I have to spend time in order to reconstruct it in my head.

Then, your excellent suggestion may give somebody the idea to do a “facebook” for our personal libraries. How to share spatial competences, that is a question!”

In the section 2.7 (“Spacebook”) of the paper on computing with space I mention this as an intriguing application of this type of computing (the name itself was suggested by Mark Changizi after I sent him a first version of the paper).

What more?Again from browsing Mark Changizi site, I learned that in fact this problem of non-spatiality (say) of e-books has measurable effects. Indeed, see this article by Maia Szalavitz

Do E-Books Make It Harder to Remember What You Just Read?

Nice! But in order to do a spacebook we need first to understand the primitives of space (as represented in the human brain) and then how to “port” them by using the web.

Curvdimension and curvature of a metric profile III

I continue from the previous post “Curvdimension and curvature of a metric profile II“.

Let’s see what is happening for (X,g), a sufficiently smooth (C^{4} for example),  complete, connected  riemannian manifold.  The letter “g” denotes the metric (scalar product on the tangent space) and the letter “d” will denote the riemannian distance, that is for any two points x,y \in X the distance  d(x,y) between them is the infimum of the length of absolutely continuous curves which start from x and end in y. The length of curves is computed with the help of the metric g.

Notations.   In this example X is a differential manifold, therefore it has tangent spaces at every point, in the differential geometric sense. Further on, when I write “tangent space” it will mean tangent space in this sense. Otherwise I shall write “metric tangent space” for the metric notion of tangent space.

Let u,v be vectors in the tangent space at $x \in X$. When the basepoint x is fixed by the context then I may renounce to mention it in the various notations. For example \|u\| means the norm of the vector u with respect to the scalar product  g_{x} on the tangent space T_{x} X  at the point x. Likewise,\langle u,v \rangle may be used instead of g_{x}(u,v);  the riemannian curvature tensor at x  may be denoted by R and not by R_{x}, and so on …

Remark 2. The smoothness of the riemannian manifold (X,g) should be just enough such that the curvature tensor is C^{1} and such that for any compact subset C \subset X of X, possibly by rescaling g, the geodesic exponential exp_{x} u makes sense (exists and it is uniquely defined) for any x \in C and for any  u \in T_{x} X with \|u\| \leq 2.

Let us fix such a compact set C and let’s take a  point x \in C.

Definition 5. For any \varepsilon \in (0,1) we define on the closed ball of radius 1 centered at x (with respect to the distance d) the following distance: for any u,v \in T_{x} X with \|u\| \leq 1, \| v\| \leq 1

d^{x}_{\varepsilon} (exp_{x} \, u, exp_{x} v) \, = \, \frac{1}{\varepsilon} d((exp_{x} \, \varepsilon u, exp_{x} \varepsilon v).

(The notation used here is in line with the one used in  dilation structures.)

Recall that the sectional curvature K_{x}(u,v) is defined for any pair of vectors   u,v \in T_{x} X which are linearly independent (i.e. non collinear).

Proposition 1. Let M > 0 be greater or equal than \mid K_{x}(u,v)\mid , for any x \in C and any non-collinear pair of vectors u,v \in T_{x} X with \|u\| \leq 1, \| v\| \leq 1.  Then for any  \varepsilon \in (0,1) and any x \in Cu,v \in T_{x} X with \|u\| \leq 1, \| v\| \leq 1 we have

\mid d^{x}_{\varepsilon} (exp_{x} \, u, exp_{x} v) - \|u-v\|_{x} \mid \leq \frac{1}{3} M \varepsilon^{2} \|u-v\|_{x} \|u\|_{x} \|v\|_{x} + \varepsilon^{2} \|u-v\|_{x} O(\varepsilon).

Corollary 1. For any x \in X the metric space (X,d) has a metric tangent space at x, which is the isometry class of the unit ball in T_{x}X with the distance d^{x}_{0}(u,v) = \|u - v\|_{x}.

Corollary 2. If the sectional curvature at x \in X is non trivial then the metric profile at x has curvdimension 2 and moreover

d_{GH}(P^{m}(\varepsilon, [X,d,x]), P^{m}(0, [X,d,x]) \leq \frac{2}{3} M \varepsilon^{2} + \varepsilon^{2} O(\varepsilon).

Proofs next time, but you may figure them out by looking at the section 1.3 of these notes on comparison geometry , available from the page of  Vitali Kapovitch.

Boycott Elsevier poster jpeg

Via this post by John Baez “Research Work act Dead – What Next?”.

Please read Baez post!

Here is the jpeg:

 

UPDATE:  Tim Gowers, who initiated this movement, has posted now about the fact that the mathematics department at TU Munich cancels its subscriptions to Elsevier journals. Here is quote from the post:

“A natural way that one might hope to bring about a genuine change to the current subscription model where libraries pay through the nose for journals is that (i) we all put our papers on the arXiv and (ii) the libraries conclude, correctly, that the benefits from their very expensive subscriptions do not justify the costs.”

In the comments, Andreas Caranti  points to “the following Memorandum on Journal Pricing by the Harvard Faculty Advisory Council to the Library: ”

http://isites.harvard.edu/icb/icb.do?keyword=k77982&tabgroupid=icb.tabgroup143448

The memorandum proposes to faculty members to, roughly: make sure that all  papers are accessible, try to submit to open-access journals, “if on the editorial board of a journal involved, determine if it can be published as open access material, or independently from publishers that practice pricing described above. If not, consider resigning“, and so on. They are NOT targeting Elsevier, instead they write:  “many large journal publishers have made the scholarly communication environment fiscally unsustainable and academically restrictive”.

Well, even officially it begins to be obvious that the emperor is naked.

 

 

 

 

 

Curvdimension and curvature of a metric profile, II

This continues the previous post Curvdimension and curvature of a metric profile, I.

Definition 3. (flat space) A locally compact metric space (X,d) is locally flat around x \in X if there exists a > 0 such that for any \varepsilon, \mu \in (0,a] we have P^{m}(\varepsilon , [X,d,x]) = P^{m}(\mu , [X,d.x]). A locally compact metric space is flat if the metric profile at any point is eventually constant.

Carnot groups  and, more generally, normed conical groups are flat.

Question 1. Metric tangent spaces  are locally flat but are they locally flat everywhere? I don’t think so, but I don’t have an example.

Definition 4. Let (X,d) be a  locally compact metric space and x \in X a point where the metric space admits a metric tangent space. The curvdimension of (X,d) at x is curvdim \, (X,d,x) = \sup M, where  M \subset [0,+\infty) is the set of all \alpha \geq 0 such that

\lim_{\varepsilon \rightarrow 0} \frac{1}{\varepsilon^{\alpha}} d_{GH}(P^{m}(\varepsilon , [X,d,x]) , P^{m}( 0 , [X,d,x])) = 0

Remark that the set M always contains 0. Also, according to this definition, if the space is locally flat around x then the curvdimension at x is + \infty.

Question 2. Is there any  metric space with infinite curvdimension at a point where the space  is not locally flat? (Most likely the answer is “yes”, a possible example would be the revolution surface obtained from a  graph of a infinitely differentiable function f such that f(0) = 0 and all derivatives of f at 0 are equal to 0. This surface is taken with the distance from the 3-dimensional space, but maybe I am wrong. )

We are going to see next that the curvdimension of a sufficiently smooth riemannian manifold  at any of its points where the sectional curvature is not trivial is equal to 2.