User jérôme jean-charles - MathOverflow

Enumerating 0-1 finite boxes without null rays.

2013-03-29T14:48:42Z

Here rays are called lines. Call $M(a_1,a_2)$ the number for matrices of length $a_1$ and height $a_2$, made of $0$ and $1$, having neither null vector nor null co-vector. In other words any line (row or column) contains at least one $1$.
QUESTION 1 : How to count them?
QUESTION 2 : Same as above in higher dimensions.

It looks like a standard problem:

For example $M(1,p) =1$ , $M(2,p) = 3^p -2$ , $M(3,p) = 7^p -3.3^p +3$ , $M(4,p) = 15^p -4.7^p + 6.3^p -4$ The closed formula being clear but I have no clean proof for it.

Question for higher dimensions; for a cube $M(a_1,a_2,a_3)$ in $0-1$ ($2^{a_1a_2a_3}$ of them) with no null lines (in any of the three directions ).
For example $M(2,2,2) = 35$ (by hand).

Is there a closed formula for M(a_1,a_2,...a_k) in particular k = 3,4?

It seems that things change a lot when passing from k=2 to k=3, Even for k=3 and constraining last dimension to 2 : $M(a_1,a_2,2)=? $

Answer by Jérôme JEAN-CHARLES for Difficult examples for Frankl's union-closed conjecture

2013-03-01T16:23:08Z

i) Definitions:
Let $F$ be a union closed family (U.C. family for short)
Let $S$ := $\cup_{ \Omega \in F} \Omega $ (the support of the family).
Let $F_{x} $ denote the members of $F$ containing $x$.

A) Union Closed Average variation : ( with $S$ being $T_0$-separated by $F$)

$ \sum_{x \in S}{ |F_{x}| } is \ge |F|.|S|/2 $.
( That is average $F_x$ size (on $S$) is greater than $|F|/2$)

Note: I cannot find any counter-example.
The $T_0$ separation of $ S $ by $ F $ means that for any two different points ${x,y} $ there exist a member of $F$ containing one point and not the other(you can choose which!) .

Another avenue of generalization that I cannot dismiss is a the multiset variation.

B) Mutltiset variation :
F is a family of $(\Omega,\eta_{\Omega})$ where:
- the $\Omega$ forms an U.C family say $F_0$
- $\eta : F_0 \rightarrow \mathbb {N}^{\gt 0}$ ( or $\mathbb {R^{\gt 0}}$ or $ [1,2,..,p]$ ordered naturally )
- for any $ \Omega_1 , \Omega_2$ of $F_0$ : $ \eta_{\Omega_1 \cup \Omega_2} \ge Sup ( \eta_{\Omega_1} ,\eta_{\Omega_2} )$

Now the conjecture is: The best measured $F_x$ is at least half that of $F$, where measure means the sum of members measure : the standard U.C. conjecture appears when $\eta$ is the constant one function.

C) The general point of view: (not strictly a generalization as asked by M.O.),
The U.C. problem is I believe of a fundamental nature, it pertains to very weak structures in the following sense: in algebra the weak (or basic or atomic) structures are the ideals, in the case of an order (or lattice) it is an upset (a family stable by overclusion , otherwise said an over-set of any member is a member), whereas we have in this problem something even weaker or more basic: stable by max only. All this comes of observing first that U.C. conjecture is easy for upsets family and still unsolved otherwise and second that algebraic techniques are not very helpful.

Which finite group is not the automorphism group of some rooted finite trees

2013-02-08T11:00:02Z

The question is in the title, a rephrasing could be is any finite group representable as the automorphism group of a finite tree, if not what is typically unrepresentable?

In case of ambiguity :
An homomorphism of finite rooted trees must preserved the root, and so does an isomorphism which is called an automorphism.

Contexte :
The cause/spirit of the question is : I make the isomorphism class of finite graphs smaller by specifying a group acting on the graph's vertices, that is an isomorphism must respect the group action (instead of the bigger $S(n)$ action).
Do I loose something by restricting myself to trees automorphism instead of group?

Is "Napkin conjecture" open ? (ORIGAMI)

2010-11-21T21:40:44Z

If false the following conjecture would be a nice counter intuitive fact.

Given a square sheet of perimeter $P$ when folding it along Origami moves you end up with some polygonal flat figure with perimeter $P^'$ :
Napkin conjecture : You always have $P^' \leq P$.

In other words you cannot increase the perimeter using any finite sequence of origami folds.

Q1: Intuition tells us it is true ( how on hell can it increase?). Yet I think I read somewhere that there was some weird folding (called "mountain urchin"??) that strictly increases the perimeter. Is this true?
Note1 : I am not even sure that the initial sheet's squareness is required.

I cannot find any reference on the net, maybe the name has changed, I heard about this 20 years ago.

The second question is about generalizing the conjecture.

Q2: With the idea of generalizing the conjecture to continuous folds or bends ( using some average shadow as a perimeter) I stumble on how you can mathematically define bending a sheet, alternatively : how do you say "a sheet is untearable" in maths?
Note2: It might also be a matter of physics about how much we idealize bending mathematically.

Possible semantics for categorical co-constness

2011-03-05T13:48:05Z

In category theory a morphism is constant IIF it is absorbing (for left composition).
That is a morphism $k$ from $k:A\rightarrow B$ is constant if an only if for any two parrallel (same domain and same codomain) morphisms $f$ and $g$ of codomain $A$ we have $k(f) = k(g)$.

Now $c$ is co-constant IIF it is constant in the opposite category (equationally $f(c) = g(c)$ for any $f$ ...) .
Semantically (in Set at least) constness is rather clear , that is a non mathematician understands the idea of "unchanging", the question(s) is:

Q1: For constness are there other semantic appearing in something else than $Set$.

Q2 :What does it mean (semantically) to be coconstant in $Set$ and elsewhere ?

Note: I guess there are several (categorically prototypical) answers in both questions.

Remark : I thought of something like an unchanging measure of noise (constant) versus an intrinsically flat noise (coconstant). But this is a rather foggy intuition, moreover I cannot categorify it properly.

Answer by Jérôme JEAN-CHARLES for Elementary+Short+Useful

2011-04-11T00:51:21Z

I would tell them "What is real maths". To achieve this use Lakatos way about Euler's formula ( $ V - E + F = 2 $ ).
It is a set of successive reformulations (more and more precise) each followed by a counter example justifying the next reformulation.

Reference is : I. Lakatos, "Proofs and Refutations: The Logic of Mathematical Discovery

Theorems true but wrong.

2010-10-05T00:12:47Z

Many theorems have the form : Premise(es) implies Conclusion(s)

Example A of wrongness:
There are many examples in which a theorem is stated without mentioning that part of the premise is not necessary to reach the conclusion.
Usually it is simple (and much better) to add a remark stating that the result is not sharp (ideally providing an example of weaker premise holding with the solution).

But there is another type of bias :

Added Note: Below composition means the AND of two relations ( for classical composition the transitivity does not compose! ( thanks to HenrikRüping remark).

Example B of wrongness:
Theorem 1 : The composition of 2 equivalence relations is an equivalence relation.
Or in fewer words : Equivalence relations are stable under composition.

Actually there is a much finer version of B :

Theorem A: For relations each of the following properties are stable under composition : Reflexive , Transitive , Symmetric.
By conjunction of the above we obtain:
Corollary B: Equivalence relations are stable under composition

Note: The second form is not only more precise but it also makes the mention "left as an easy exercise" more acceptable.

The "WRONG" notion:
I called theorem 1 (or its statement) wrong as it induced the reader to think that the conjunction of the 3 properties plays a role in proving the conclusion.

Of course only true theorems may be qualified as wrong.

Taking an absolute stance you may call wrong any theorem that is not a tautology.
A less absolute stance would call wrong any theorem that is not a tautology and in which you forget to mention non-sharpness.

Question 1: is there a better / more adequate term than wrong ( the subtext is: do you think it is a good notion?) .

Question 2: Do you know examples that follow a pattern like B or some variation in lack of tautology?

ADDED TO BE MORE SPECIFIC:

Question 3: More specifically : Are there other types of patterns showing a distance between premise and conclusion. The types need to be common in the mathematical literature, not purely logical types ( of course those are more countable).

Is there a conjunction bias?

2011-02-16T02:10:02Z

This is slightly related to question http://mathoverflow.net/questions/33366/the-unprecedented-success-of-the-intersection-operator .

Apart from a set of maths books of null measure, most have the following property:

Objects definitions are presented as a conjunction of properties.

Most axiomatic are also clearly conjunctive in their presentation.
It is uncommon to have say "By definition a Zorglub is a red zorg or a white zorg".

Q1 : Do you agree with the bias (if not, give enough examples)?.

Q2: Is this bias mainly a discourse convention or does it lie deeper (where?) ?

Who is this guy : Z.A. Melzak (wrote Companion to Concrete Mathematics) ?

2011-02-05T00:12:27Z

Author : Z.A. Melzak
Book Title : Companion to Concrete Mathematics.
Publication : Dover renewed 2004 2 volumes in one. Copyright 1972/1976.

I found this book extremely nice.
To whet your appetite he talks about reification as well as some plain and less plain way to accelerate series convergence. In a few words it is a rare blend of concreteness and conceptualization. It smells a bit like Concrete Mathematics (by Knuth and co..) but the only information I found was an an depth review by Klamkin (not entirely positive though).

QUESTION : I would like to find connections : information about the author, and for those who appreciate this book or heard loadably about it: what others books/works are in the same vein ?

Answer by Jérôme JEAN-CHARLES for A Book You Would Like to Write

2011-02-04T22:24:45Z

"Thinking with categories" a small introduction for the layman.
May be a more commercial title would be "Functorial Thinking".
A small book (circa 120 p.) with the goal of explaining basic category theory using plenty of examples but mostly non mathematical ones.
Intended for an audience of linguists, philosophers, computer designers and any curious intellectual.

The book presuppose a reader not adverse to a minimum of algebra, yet it should mostly contains basic defining algebraic equations for categories, functors , natural transformations and adjunctions.

The goal of this book: It should enable a philosopher (not necessarily specialized in logic) to grasp properly what an adjunction is in 2 to 4 hours.

The basic motivation: Find proper real-life examples (as in elementary set theory) for category theory.

To illustrate : A 5-subset of a football team can be made by picking some players randomly, but a sub-object is a set of 5 players that can play together! In fact common language would call it sub-team. So far when trying to design examples in real life you end up too often with groupoids and thin category(posets).

Any suggestions of places from which to draw material/inspiration would be most welcome.

Answer by Jérôme JEAN-CHARLES for Computer Science for Mathematicians

2011-01-09T02:32:52Z

This question is meaningless if you don't specify the goal(s) you have in mind. Goals are teaching , using , understanding , linking .....
Some people like me think that Knuth is bad from a certain point of view: his writings are interesting, of high quality and contains precise and accurate facts yet it is "not functional" and it is misleading in a way.

As a mathematician try to read some of Wadler below. It is specific but may show you what computing is about. (see http://homepages.inf.ed.ac.uk/wadler/) .

Computing is about neatly describing parts of the real world, it is not about encoding in zeroes and ones. This confusion is akin to someone saying he 'understood maths' when he has mastered calculus.

Answer by Jérôme JEAN-CHARLES for What would you want to see at the Museum of Mathematics?

2011-01-09T00:13:46Z

Working mathematicians live!!

Movies showing sessions of working mathematicians, with some comments and explanations along it.

So at last the general public (and sadly the not so general one as well) will be aware that mathematics has more to do with art and understanding than with formulas and logic.

Five to ten "movies" would do, it is not easy but neither hard nor expensive to produce, Some people are very good at producing documentaries. Those professionals should be asked/used of course.

Answer by Jérôme JEAN-CHARLES for Ingenuity in mathematics

2010-11-30T03:18:52Z

A recent reference is "Street fighting mathematics" by Sanjoy Mahajan.

AT http://www.amazon.com/Street-Fighting-Mathematics-Educated-Guessing-Opportunistic/dp/026251429X

I have browse and read some part. It look as if it almost manages to render NavierStokes equation edible for an hard die finitist. It uses mainly dimensionality but is full of ingenuity.

Answer by Jérôme JEAN-CHARLES for How To Present Mathematics To Non-Mathematicians?

2010-11-25T03:12:26Z

Though I am not teaching on a regular basis, I often explain what mathematics is to laymen. My explanations tend to converge toward the following lines:

1) Mathematics is poetry: 2 quotes:
Quote 1 : Mathematics is the art of giving two names to the same thing AND the same name to two different things ( HENRI POINCARE).

Quote 2 : In mathematics you have an absolute liberty, the price to pay for this is that you have to be very precise ( YURI MANIN).

2) Maths is made of observations and rendering them with an eventual need to make up a new language, just as anybody would need one in a complex and professional field (say dancing).

3) THE WORLD OF A SURFACE IN ITSELF

Then show them a band a paper,make it a cylinder (2 faces, 2 circle boundaries) . Then link it with a twist and ask them to count the boundaries and then faces. ( They will be astonished and see the difference for themselves ...)

3.a) On a Moebius band a river drawn in the middle ( a blue pencil will do) has only one bank( Let them check it), it is a different world if you live on it ( you are little bugs with no sense of the third dimension)

3.b) Tell them about the game of cylindrical chess (played on a torus, abstracting the game if necessary ( just moving pieces) those who know the rules of chess feel more at ease. Show them the game while remaining flat, then tell them that for someone really dumb you could imagine to produce a real torus by bending the board and using magnetic pieces. As a world (a fighting world for example) make the observation that proximity is changed.... (chess serves as a surrogate topology in this, but you do not have to pronounce the frightening word topology).

3.b) Back on the Moebius band : the game of chess on it is not the same as the cylindrical one: a piece does not move the same way and does not aspects the sames cells...

4) DIFFERENT WORLD AND VIEWS :
Now take a band make it a cylinder with a knot first ( need a band long and thin enough) put it side by side with the normal cylinder and ask them is it the same. After their answer your is of course yes AND no (the ambient or the embedded surface) . A matter of point of view.

5) USEFULNESS OF MATHEMATICS:
Of course there lots of applications but the killing example is : In 400 BC Greeks were doing land regrouping (consolidation),each land was measured by willing geometers. A year later there was plenty of lawyers at work because the pieces of land had been measured by perimeter!! Tell them that it might seem obviously stupid to do so, yet basic school told them about surface concept. Moreover using the perimeter might be a good way to do things if the goal was not farming but showing off with high flags and poles. Again many points of views blablabla...

NOTE : The interactivity is essential at least when checking the boundary of MB with the finger or sight for some. This is a close call for ten minutes, part 5 can be removed.
Try it on some none mathematical friends first after three times you will be probably quite sleek. It is also important to have the right length and width for the band paper 12 inches by less than one roughly usually the side of a sheet of paper...

Answer by Jérôme JEAN-CHARLES for Suggestions for good notation

2010-11-16T01:54:44Z

A) Two notations I love are the rising factorial $x^\overline n$ and its falling factorial twin $x^\underline n$. They are used and advocated in the great book see http://en.wikipedia.org/wiki/Concrete_Mathematics . In passing this book uses great notations.

B) A general trick with binomials to reuse them with sets instead of numbers, here are some typical examples.

1) $\binom S k $ to denote the set of all $k$-sets of the base set $S$ .

2) $S^\underline 2$ to denote the pairs $(x,y)$ of $S$ where $x$ and $y$ are different.

3) $S^\underline k $ to denote the $k$- uplets of $S$ (each uplet has $k$ different elements).

C) Another notation I find useful when listing some (big) families of examples in a combinatorial setting. Use as variables the very numerals $1$ $2$ .. themselves instead of $x_1$ , $x_2$ ... . For example ( very untelling because too small an example) : the intersection of $123$ and $34$ is $3$.

D) I also often use {{ a,a,b,c}} for multiset. Any other standard or suggestion (or a way to avoid speaking about multiset) is welcome.

Answer by Jérôme JEAN-CHARLES for Most memorable titles

2010-11-01T02:12:50Z

The missing axiom of matroid theory is lost forever

A emotional variation on absolute negative results.
Refs : Vámos, Peter (1978), "The missing axiom of matroid theory is lost forever", Journal of the London Mathematical Society, II. Ser. 18: AT : http://jlms.oxfordjournals.org/content/s2-18/3/403.extract

Answer by Jérôme JEAN-CHARLES for Combinatorial results without known combinatorial proofs

2010-10-07T22:50:42Z

PROBLEM Of splitting a necklace between two thieves:

Two thieves want to share equally the stones of a necklace ( an open circle).
The necklace has $s$ types of stones ( each type of stone appears an even number of time.).

They want to minimize the number of cuts ( the link are costly and they do not want to make a mess of it).
Show that it is always possible to achieve the split using $s$ cuts.

SOLUTIONS:

For $s=2$ a combinatorial solution is not too difficult.

For any $s$, a topological/linear algebra proof exists ( a nice exposition by Jiri Matousek in reference below.)

http://www.amazon.com/Using-Borsuk-Ulam-Theorem-Combinatorics-Universitext/dp/3540003622

Though by now there seem to be a combinatorial proof.
AT : http://www.combinatorics.org/Volume_16/PDF/v16i1r79.pdf

Yet I believe it might be of interest as a problem that had no combinatorial proof for a while.

Answer by Jérôme JEAN-CHARLES for Pacing for learning new material

2010-10-05T00:48:08Z

Of course things may vary considerably from one person to another. Yet I can give you my experience of the last two years with category theory. For applied category theory anything goes as standard maths : rest from time to time etc... But for categories theory itself (or its first applications) I believe the pace is a bit special because it "recables" your brain in a way different from that other of fields (in which re-cabling is due to focusing on one type of object).

Typically I tend to describe categorical definitions as rather short or almost trivial yet THICK! You have to use them several time before being cabled.

I believe that the reason lies in the 'abstract nonsense': a categorical definition make sense only when applied to specific examples which are yet abstract. Hope it helps.

Answer by Jérôme JEAN-CHARLES for Collection of equivalent forms of Riemann Hypothesis

2010-09-25T15:36:16Z

This one is not too bad though not big .

http://aimath.org/pl/rhequivalences

Yet there are many (above a hundred at least) and it depends on the type you are looking for. Analytic elementary number theory ....

ADDED LATER : My favorite is very elementary:
Among the square free integers below $N$:
Let $D(N)$ denote the absolute value of the difference between the number of those divisible by an even number of primes and the number of those divisible by an odd number of primes .

R.H. says that $D(N)$ comes close to the square root of $N$.

More precisely: for any $\epsilon > 0 $ there is $N_0$ such that any $N > N_0$ verifies $ {D(N)} <= N^{1/2+\epsilon}$.

Answer by Jérôme JEAN-CHARLES for Most harmful heuristic?

2010-10-02T23:54:17Z

The excluded middle ( A Law or an Heuristic) .

On a more general level given any closed question: Is it A or B ? , the heuristic says it is one or the other disregarding the option : the question is wrong or stupid or irrelevant or incomplete.

The principle of excluded middle disregards intuitionist logic. And has been harmful in not providing direct (constructive) proofs which are often more clear - yet can be harder to find.

Intuitionism is is also rather natural : being against anti-communists does not means you are a communist.

Answer by Jérôme JEAN-CHARLES for Math puzzles for dinner

2010-10-01T01:21:35Z

I understand your feeling , I myself know lots of them . Among original ones

http://www.amazon.com/Mathematical-Puzzles-Connoisseurs-Peter-Winkler/dp/1568812019

This Peter Winkler does something that is rarely done and is a must not only for a mathematician but for a connoisseur: He produces declination of a problem.

In fact there are two books of his.

Another source of problems that you may like is "IBM ponder this" AT

http://domino.research.ibm.com/Comm/wwwr_ponder.nsf/pages/index.html

Answer by Jérôme JEAN-CHARLES for Examples of common false beliefs in mathematics.

2010-09-26T23:45:18Z

"The image of a category under a functor is a category."

This is a small one, but it lasted for six months when I was starting in category theory.

A finite counterexample exists, with just 3 objects. Even under a connectivity requirement, a small finite counterexample still exists. In fact, it is dead wrong to think anything like this holds.

Answer by Jérôme JEAN-CHARLES for Is monomorphism going in both directions sufficient for isomorphism?

2010-09-27T00:34:24Z

One of the simplest example , lightest in structure and very easy in checking is :

In the category of monoids take the canonical injection $i$ from $(N,+,0)$ to $(Z,+,0)$.

This is a monomorphism that is also an epimorphism yet not an iso ($i$ is not a surjection).

($N$ and $Z$ are the positive integers and integers respectively)

Answer by Jérôme JEAN-CHARLES for Is there a general setting for self-reference?

2010-09-25T16:28:23Z

For pleasure only I can at least give you the shortest definition of self reference.

You need only to look in a good dictionary ( from Borges world of course) it says:

Self-reference : see self-reference.

Answer by Jérôme JEAN-CHARLES for Which mathematical ideas have done most to change history?

2010-08-13T02:28:37Z

There is a very nice category of mathematical results (that are also relevant to culture) : the negative results.

For example there is no solution problem X (Say Fermat last Theorem) is negative and usually the result is not very interesting or motivated for laymen. But think of the following : There no program checking that a program has no bugs ( by classical diagonal argument: how would this program test itself). In this case we prove something is impossible and so we save a hell of a lot of time: no need to search any more. Negative results are extremely useful : in a negative way you avoid loosing money and in fact it is a good justification for pure research.

Yet beware that tough simple negative statements are not always understood some people say : "Oh Fermat equation has no solution , that is because they did not try hard enough , I will do it". This is akin to the trisectors and other poor souls looking for perpetual motion machines.

Answer by Jérôme JEAN-CHARLES for Thinking and Explaining

2010-09-22T11:50:11Z

The well known situation of language translation is I believe akin to the tension between thinking and explaining.

I am French, I can understand, write and explained myself in English yet I am a bit at a loss when required to translate some piece of English into French so I call this translating ability a third language. My guess or feeling or personal view is that the thinking is more semantic and - for weird (and sad) cultural reasons- mathematical explaining is too often required to be on a syntactic level.

3 directions of infinity ?

2010-09-22T00:52:49Z

$N$ the positive natural numbers has one infinity. $Z$ the integers has 2 infinities.

What object would as "naturally" as possible have 3 infinities?

This probably can be answered in many ways. Yet for me the algebraic side would be more important than the topological one, though this does not exclude both.

What troubles me is that $Z$ is natural as being final in the category of rings) and moreover it is the completion ( in fractional sense) of $N$.

A mathematical idea "abstract enough to be useless for physics"

2010-09-07T01:09:33Z

Grothendieck (if it was him) said somewhere :

This XXX, at least, is an idea that will not be used in physics.

Q1 : Is XXX an n-groupoid? a stack? Can someone supply the precise quote, either in French or in English?

Q2: Predecessors of this quote in the same vein would also be of interest. Thank you.

Can knowing ahead the length of 3-SAT instance really help?

2010-08-29T17:04:01Z

If I say I can solve 3-SAT ( known to be NP-complete) in polynomial time, yet with the following 'little' proviso: Give me first $n$ the length of your 3-SAT formula, then give me some time on my own , then as soon as you give me your formula, I will answer in less that $n^k$.

The $k$ will be constant independent of $n$ (this is not parametrized complexity)

Implicitly: after you give me $n$, I may pre-calculate as much as I want (say $n^n$ or even much more) and I may also store some results as much as I want.

Question : is this equivalent to 3-SAT?

Comment : I cannot find a polynomial solution like : calculate all solutions store them on a tree and then retrieve on question . So it seems to be as 'difficult' as 3-SAT.

Note : I took 3 SAT but any NP-complete problem Q will do : define generically the variation Q' with the length of the instance of the problem Q given ahead of the instance.

Convergence of a general Bertrand series

2010-01-05T02:17:03Z

Let $ S= \sum 1/n log^1n log^2n log^3n ..log^{TL(n)}n $.

Is it convergent when $n$ runs on integers say above 2 ?

$log^i n$ denotes the i'th iterate of $log$ (in base 2 ) of $n$, $log^2n$ means $loglogn$ .

$T(n)$ is the tower of $n$ (stack of $n$ 2's) that is $T(1)=2$ , $T(n+1)=2^{T(n)}$.

$TL(n)$ is the towerian log:
$ TL(n) = Sup ( k : T(k) <= n < T(k+1) ) $.

MOTIVATION : Generalizing the following that are called Bertrand series (I think):

$\sum 1/n$ is the harmonic serie , $\sum 1/nlogn$ , $\sum 1/nlognlog^2n $ and $\sum 1/nlognlog^2nlog^3n $ are all known to be divergent.

Here the product of iterated logs is pushed as far as possible and its size depends on the parameter $n$.

Comment by Jérôme JEAN-CHARLES

2013-03-30T18:21:49Z

@Casteels : OK you are right there was a typo the 4 and the 6 where exchanged in front of the powers ^p.

Comment by Jérôme JEAN-CHARLES

2013-02-08T14:21:21Z

Thank you very much , there is still an aspect of "density". Do we loose kind of "half" when restricting to (FTAG) FiniteTree-Automorphism -Groups ?

Comment by Jérôme JEAN-CHARLES

2013-02-08T12:51:09Z

How funny that my context is in fact graph isomorphism and Brendan is the first to answer this. Mysterious or funny ?

Comment by Jérôme JEAN-CHARLES

2013-02-08T11:15:09Z

Too short and too partial for an answer: I think that functions are on the technical side whereas relation are on the conceptual side. A function (partial) "is" a partition whereas a relation thought as a bipartite graph is much more complicated. A related (no pun) question is that partial functions should be used instead of function.

Comment by Jérôme JEAN-CHARLES

2013-01-24T01:20:29Z

@jim It is a detail : in galois.pdf the property is symetric (not reflexive as given).

Comment by Jérôme JEAN-CHARLES

2012-12-25T01:57:18Z

@Todd : I have seen a very nice documentary about professional dancers at the Opéra de Paris ( from Reichenbach ? ) . You see them talk and act (dance) for training and interpreting, yet the "truth" they reach through successive corrections is felt but not understood (unless you are semi professional). So the idea is that people may understand the general aim and the kind of logic used by mathematicians without a proper grasp.

Comment by Jérôme JEAN-CHARLES

2012-02-24T00:08:11Z

If I remember well it was almost a bit of a cut and paste of sets product. I often view things as an idealized world ( computer point of view) and I think you know it to.

Comment by Jérôme JEAN-CHARLES

2011-11-29T01:28:08Z

If you are looking for connections between infinite (not even small) categories with graphs you must look into infinite graphs whose smell is definitely different from that beloved smell of finite graphs. For example : the Rado graph (en.wikipedia.org/wiki/Rado_graph) contains any finite graph as a subgraph.

Comment by Jérôme JEAN-CHARLES

2011-11-21T14:36:41Z

@darij grinberg : Yet you must show that that this is not significant : Is the empty graph an initial object in the category of graphs ? In which respect does and empty set have an even number of elements ?

Comment by Jérôme JEAN-CHARLES

2011-09-28T11:54:28Z

for unknowngoogle : this is even better without the parenthesis around a , yet it does not work with variables ${{a1, a2}}$ where you don't know if $a1$ equals $a2$.

Comment by Jérôme JEAN-CHARLES

2011-04-20T11:19:01Z

@Finn :I found a counter example (C.E.): Let C has two objects A and B, an arrow f from A to B and the two identities arrows then f is nLab constant (vacuously) but NOT Lawvere. Another C.E. is A=x,y ,B={u,v} ,C={u',v'} , T={t} as morphisms add all applications to T (the terminal element). Add morphisms f,g,h with f(u)=u',f(v)=v',g(x)=u ,g(y)=v,h(x)=v ,h(y)=u. Now clearly f(g)=f(h):f is nLab-constant but not Lawvere. More generally if g and h have the same B-fibers it gives a C.E. The idea is nLab means "absorbing" , Lawvere is "one valued".

Comment by Jérôme JEAN-CHARLES

2011-04-20T11:15:02Z

@Finn :I found a counter example (C.E.): Let $C$ has two objects $A$ and $B$, an arrow $f$ from $A$ to $B$ and the two identities arrows then $f$ is nLab constant (vacuously) but NOT Lawvere. Another C.E. is $A={x,y}$ ,B={u,v} ,C={u',v'} , T={t}$ as morphisms add all applications to $T$ (the terminal element). Add morphisms $f,g,h$ with $f(u)=u',f(v)=v'$ , $g(x)=u ,g(y)=v$ , $h(x)=v ,h(y)=u$. Now clearly $f(g)=f(h)$ : $f$ is nLab-constant but not Lawvere. More generally if $g$ and $h$ have the same $B$ fibers it gives a C.E.. The idea is nLab means "absorbing" , Lawvere is "one valued".

Comment by Jérôme JEAN-CHARLES

2011-04-20T10:34:02Z

@Todd Trimble : Very nice!

Comment by Jérôme JEAN-CHARLES

2011-04-11T00:36:42Z

Sorry I don't read Russian but the drawings are still worth looking at thank you.

Comment by Jérôme JEAN-CHARLES

2011-04-11T00:33:03Z

I was a bit surprised by fact 2, after some reflection OK. Moreover I suddenly realize that something is dreadfully missing from the question: the figure at start and at the end should be flat ( a closed polygonal line made of finitely many segments). In you first lines of 3 drawings can you confirm that the first two are in volumes and the last one is flat?