User ogerard - MathOverflow

Current status of Bloch Constant and Landau Constant bounds

2010-05-11T19:32:55Z

The Bloch constant B (based on a theorem introduced by André Bloch in 1925 on the maximum radius of a one-to-one disk in the image of a normalized analytic function of the unit disk, see for instance Remmert Funktionentheorie II or Steven Finch marvelous "Mathematical Constants") was conjectured by Ahlfors to be

$$ \frac{1}{\sqrt{1+\sqrt{3}}}\frac{\Gamma(\frac{1}{3})\Gamma(\frac{11}{12})}{\Gamma(\frac{1}{4})}$$

(This value, if I remember well Ahlfors' article corresponds to a particular function that he constructed for this purpose).

The Bloch Constant $B$ is currently known to be at least slightly greater than $\frac{\sqrt{3}}{4}$ (several articles improving upon each other by Mario Bonk, Chen and Gauthier, Xiong).

Has there been some progress since 1998 on the lower bound ?

Same question for the closely related (univalent) Landau constant (quite often called Bloch-Landau constant, sometimes seen as $B_\infty$) ?

The conjectured upper bound is

$$\frac{\Gamma(\frac{1}{3})\Gamma(\frac{5}{6})}{\Gamma(\frac{1}{6})}$$

What can be said of the various adaptations or specializations of this constant to various class of functions, and extensions of these constants to several complex variables or other functional spaces ?

I give as background the original article from Bloch, Ahlfors and Grunsky.

(1) A. Bloch, Les théorèmes de M. Valiron sur les fonctions entières et la théorie de l'uniformisation, Ann. Fac. Sci. Univ. Toulouse, vol. 17, (1925), pp1-22.

(2) L. V. Ahlfors and H. Grunsky, Über die Blochsche Konstante, Math. Zeitschrift 42 (1937), pp671–673.

(3) L. V. Ahlfors, An extension of Schwarz's lemma, Trans. Amer. Math. Soc. 43 (1938), pp359–364.

(these two are reprinted in Ahlfors Works vol 1)

Ahlfors life and works are evocated in an AMS Notices of 1998.

What is exactly the (singularity) confinement property ?

2010-05-04T20:19:55Z

This property seems to be used both in the context of differential equations and several kinds of discrete equation systems or automata.

It seems to be related in certain case to the Painlevé Property first discovered for Painlevé equations and their solutions.

I have seen several definitions, notations, criteria, which do not always seem to match each other or to be easily applicable to all context. I would appreciate some clarification about it.

I am mostly interested in the discrete applications but all explanations about the continuous cases are welcome especially if there are links to the discrete setting.

What are the oldest illustrations of "Venn" diagrams?

2010-05-05T13:03:37Z

Graphical representations of intersection of sets as logical combinations are much older than Venn. Euler and Leibniz are often quoted and the current Wikipedia article also quotes Ramon Llull but I do not really find the illustrations provided in the Wiki Commons for Llull very compelling.

I expect that these kind of ideas can be found in many other places and even older times, perhaps in disguise.

In this context I find the heraldic uses of theological diagrams such as shown here quite fascinating as a kind of medieval fashion statement.

Do you know of older examples of graphical representation of logical and/or set relations, for instance of Chinese, Arabic and Greek origin ?

(ps: at least one of the tags is a joke)

Answer by ogerard for Which mathematicians have influenced you the most?

2010-05-14T06:52:36Z

Louis Comtet, through his book "Analyse Combinatoire vol 1 and 2", now republished in english translation with additions and corrections as "Advanced Combinatorics".

When ? My first year in Paris University while I was attending boring courses in Analysis and Linear Algebra that were very inferior to what I have been exposed in high school the year before.

These two little pocket books were relatively easy and cheap to find and gave a wealth of packed information and links to the existing litterature on combinatorics. Combinatorial Mathematics were not in fashion in France in the 1970s, neither in the 1980s. Among many things I liked were the fancy notations, the diagrams, the density of results, the careful index, the intersection with so many other mathematical theories such as set theory, differential equations, topology, group theory. And it was also my first contact with a slightly formalized graph theory, Eulerian numbers, integer partitions, multiple summation, etc.

Is there any documented study of geometry in contemporary primates ?

2010-05-09T07:26:08Z

There are many studies of language learning abilities of primates (mostly chimpanzee, bonobo) and studies of tool use, knowledge transmission and number sense.

Are there studies or documented cases of drawing, any form of abstract graphical representation, use of concrete objects as representatives, symbols for other things not present, hints of ideal shapes such as circles or lines, uses of markings or pebbles for counting, etc ?

The less influenced by trainers and observers, the better.

Answer by ogerard for A place to find original papers

2010-05-17T15:32:14Z

I would also advise (especially if they were published in french journals, such as the articles by Elie Cartan, Frechet, Henri Cartan)

GALLICA

and

NUMDAM

You can often download whole articles, depending on date and copyright.

Here you have an obituary for Sophus Lie in the Weekly Accounts of the French Science Academy in 1899 (CRAS 1899, p525).

3 important original articles from Sophus Lie, probably on GDZ for the first 2.

Sophus Lie, Über Complexe, insbesondere Linien- und Kugel-Complexe, mit Anwendung auf die Theorie partieller Differentialgleichungen; Mathematische Annalen Vol 5, pp145- 256 (1872)

Sophus Lie, Untersuchungen über Transformationsgruppen. II; Archiv for Mathematik og Naturvidenskab vol 10, pp353-413 (Kristiania 1886)

Sophus Lie, unter Mitwirkung von Friedrich Engel, Theorie der Transformationsgruppen III, 1895. Printed as a book I think. I have it as chapters in the Chelsea reprint.

Answer by ogerard for Given a finite field $K$, what are the possible degrees of a polynomial $p\in K[x]$ such that $x\longmapsto p(x)$ is one-to-one?

2010-05-17T16:52:49Z

For permutation polynomials, you can also look into the relevant part of "Finite fields", by Rudolf Lidl and Harald Niederreiter, CUP.

Answer by ogerard for Non-real constants

2010-05-15T17:00:33Z

Since we do not want to restrict ourselves to values in traditional number systems...

Each remarquable/exceptional finite algebraic structure, graph, can be described/encoded as a specific value in a numbering system. For instance as a series of generating matrices, multiplication table, representation tables, etc.
We can consider each sporadic group as a remarquable constant. To me the Monster Group is a mathematical attraction point that can be compared to $\pi$ or $e$. And it is well hidden in the armies of soluble groups around it.

Answer by ogerard for Non-real constants

2010-05-15T16:52:07Z

Since all the known non-trivial zeroes of Riemann Zeta function are on the Re(z)=1/2 line we only give their imaginary parts, but in fact their are complex,

$$1/2+ i*14.1347251417346937904572519835624702707842571156992431756855674601499...$$

being the first above the real line. If Riemann Hypothesis is true we will never have to mention a different real part.

Answer by ogerard for Motives versus Motifs

2010-05-15T12:56:34Z

Motif in french has both the meaning of english "motive" and of "pattern". It is still actively used in decorative arts and art history "Cette tasse est ornée d'un très joli motif", "This cup is decorated with a very pretty pattern", and so on for tapestry, greek freeze, wallpaper, etc. And still used when describing a police case : "Il a un alibi et n'a aucun motif".

So I believe that Grothendieck was well aware of this ambiguity.

Answer by ogerard for "negative" vs "minus"

2010-05-15T12:50:11Z

As far as I can check, in french this issue does not arise. The only available description is "moins x" or "moins 3", corresponding to "minus x" or "minus 3". The closest equivalent of "negative x" would be "x négatif" and it would only be used in phrases such as "pour x négatif", meaning "for x < 0" and not as a description of part of such a formula. "Négatif" seems to me mostly used in french in a categorical way "températures négatives", "nombres négatifs".

It would be informative if native german, russian, chinese, etc. speakers could comment on this.

Answer by ogerard for What practical applications does set theory have?

2010-05-05T12:37:21Z

To complement some of the previous answers, notably the one by Zach Cont.

For question 1, mostly for finite sets, but not only: At the most elementary level as at the most involved or most internal to mathematics, set theory, classical logic and combinatorics are deeply related. Most applications of any two of them could be seen actually as an application of the third.

Many use of diagrams outside mathematics use a combination of naive set theory and classical boolean logic. The technical language of many disciplines use the terms union, intersection, complement of sets, and use a correspondence between (logical) combination of conditions on elements and combination of subset creation and operations. This can be traced at least to Leibniz and probably to medieval times (scholastic tradition for instance). Traditional names in this area are mostly from the 19th such as De Morgan, Boole, Pierce, Grassmann, Venn, Cayley.

In this context, it makes sense to study more precise treatment of set theory so that it reinforces intuition of what is reasonable and expressible in this context. It gives clean conceptual tools and refined language to analyze problems and reports opinions and facts. Usually with high school students, the discussion of the classical paradoxes such as Russel and the axiom of foundation leads to better appreciation for the art of defining and for the way to use (even in non mathematical contexts) informal quantifiers and adverbs such as all, always, never, none, nobody, everywhere, everytime, at least, etc.

This might not look very spectacular, but when one considers the usual sloppiness (sometimes voluntary) in newspapers, books and general conversation, this strikes me as very practical for non-mathematicians.

Sound notions of set theory, and the ability to think in terms of cartesian products, relations as quotients, etc. are the basis of a good grasp of probability (see measure theory in other answers) and statistics (and basically experimental data measurement, quantum physics, actuarial techniques, and from the 1960s data mining, database query languages, ...). I certainly do not rule out that we could have developped similar science and technologies by other roads, but it would have given them a very different aspect and to learn all these subjects (instead of recreating them with other foundations) without knowing set theory is especially difficult and limiting for the learner.

Answer by ogerard for Seeking reference for the enumerative "mass formula" concept

2010-05-14T05:16:06Z

Still another formulation: I recall hearing the whole idea being referred by a metaphor of skeleton and flesh. The "mass" of your example would be the "skeleton weight" or "bone mass" of the collection, the amount of "flesh" around each "bone" (the radius of the muscle ?) being the size of the automorphism group.

I find the groupoid cardinality sensible too and perfectly compatible with the basic results of Polya theory, and will certainly use it in the future.

I note that this is the converse operation of counting things with multiplicities such as roots.

Answer by ogerard for An integral that somehow equals pi^2/6 and involves dilogarithms?

2010-05-13T17:53:23Z

About learning about logarithmic integral and polylogarithm identities, there are large compendium of formulae on everything (Gradstein Ryzhik Jeffreys for instance), a few classic monographies such as Nielsen, Lewin, some new books (some of them with a physicist point of view since polylogarithm appear when dealing with Feynman path integrals among others) and also this online resource. You can print the pdf version of this list of dilogarithm identities for instance.

Answer by ogerard for Fundamental Examples

2010-05-09T10:08:34Z

In combinatorics, the (Pascal) Binomial Triangle more or less started the entwining of combinatorics, probability and algebra.

The other two most seminal and ubiquituous triangles of numbers are the Stirling (duals between 1st and 2nd sort, permutation world and set world, the most often generalized family of numbers) and the Eulerian numbers (permutations seen as words on ordered alphabet), the reference statistics on permutations.

For links between combinatorics and number theory, I think the first prize would be the Bernoulli numbers.

Answer by ogerard for Would Euler's proofs get published in a modern math Journal, especially considering his treatment of the Infinite?

2010-05-13T07:05:34Z

Just to stress a few points already addressed in comments and answers:

Euler in his time discovered many important facts and solutions to classical questions, advanced rigor and gave examples of the power of the recently created methods (infinitesimal calculus), popularized the science of his day (notably books dedicated to a German Princess), wrote some of the first textbooks in analysis (still pleasant reading today), gave strength to the prussian and russian academy of science, courtized by two of the most powerful powers of the day (the King of Prussia and the Czar of Russia), filled international academic journals, some of them he edited himself, with quality articles (in fact up to several decades after his death because of the sheer size of his output), fostered international cooperation, wrote in the most important languages of his day (latin, french, german, I think he also learned russian), published in applied science, was part of state scientific advisory commission, etc.

In fact Euler's work has been instrumental in progressively establishing the "rigor" some of us are so proud of.

So a better equivalent of his investigation of what we call now Zeta(2 n) and the Gamma function would be the solution of outstanding problems by one of the most recognized mathematician of his day building on recent work by one of his even more famous and established mathematician, Bernoulli, who was his PhD advisor and whose several family members have established positions in the scientific community.

I think he would have no difficulty publishing it. And his work would be quickly read and commented upon by many other mathematicians.

Even if we imagine a Leonard Euler finding himself straight-jacketed by the mathematical discourse and style of the XXIst century, he would pair up with another good mathematician to write scholarly articles, as Ramanujan and Hardy used to do at the beginning of the XXth in a mutually benefical couple.

Answer by ogerard for Are there situations when regarding isomorphic objects as identical leads to mistakes?

2010-05-12T06:20:40Z

In Group Theory there is a notion stronger than isomorphism, i.e. similarity. For two permutation groups to be "similar" there must exist both an isomorphism between the group and a compatible (commuting in the categorical sense) isomorphism between the sets being permuted. The couple of these two isomorphisms is called a similarity. Two permutation groups can be isomorphic but not similar.

Similarity is useful to compare group constructions such as the wreath product, group presentations and classifications of subgroups of the symmetric groups.

In these situations, using only isomorphism would be a mistake and would lose information.

Answer by ogerard for japanese/chinese for mathematicians?

2010-05-11T12:52:02Z

I am also interested in learning Chinese and Japanese enough to read mathematical articles.

While also learning basic language notions with traditional courses, I am practicing by reading small and elementary mathematical wikipedia articles in english, chinese and japanese, that I sometimes translate back with google translate to match with the english version. I making slowly my own quadri-lingual dictionary (with english and french) with an electronic card system (Anki), so that key hanzi/kanji for mathematics allow me to progressively guess the subject of an article for instance. I devote some time to writing correctly by hand each new characters many times to reinforce memory by gesture and concentration.

Answer by ogerard for What are dessins d'enfants?

2010-05-11T12:39:50Z

There is a french talk by Alexander Zvonkin which can be a good introduction to this subject as well.

If readers are interested I can translate parts of it in english.

Answer by ogerard for Is there any documented study of geometry in contemporary primates ?

2010-05-11T05:50:19Z

By following links provided by contributors to the meta-discussion, I have found these two resources of interest:

The Comparative Cognition Society

and the book Animal Spatial Cognition

Reading the rather strait-jacketed Wikipedia article on Animal Cognition, the idea of Cephalopodic Mathematics comes to mind as another pursuit.

Answer by ogerard for Mathematicians who were late learners?-list

2010-04-24T15:52:45Z

One could perhaps also cite George Green, miller and mainly autodidact mathematician as an unconventional and relatively late bloomer. He entered University only at 40, one year or so before his death. See for instance Green's Biography at MathTutor

Answer by ogerard for Reference Request: Perspective Painting

2010-05-09T08:04:19Z

Answer by ogerard for Books about history of recent mathematics

2010-05-06T03:08:49Z

Other examples (just to give an idea of the choice), thematic sample, with scholarly work, popular science, and other types :

Recent theorems

Four Colours Suffice: How the Map Problem Was Solved de Robin J. Wilson

Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture de David M. Bressoud, William Watkins, Gerald L. Alexanderson, and Dipa Choudhury

Kepler's Conjecture: How Some of the Greatest Minds in History Helped Solve One of the Oldest Math Problems in the World de George G. Szpiro

20th century

The Honors Class: Hilbert's Problems and Their Solvers de Benjamin H. Yandell

Mathematical Analysis During the 20th Century de Jean-Paul Pier

Development of Mathematics 1900-1950 de Jean-Paul Pier

The Mathematical Century: The 30 Greatest Problems of the Last 100 Years de Piergiorgio Odifreddi

Biography

Ludwig Wittgenstein: The Duty of Genius de Ray Monk

Von Neumann, Morgenstern, and the Creation of Game Theory: From Chess to Social Science, 1900-1960 de Robert J. Leonard

The Random Walks of George Polya de Gerald L. Alexanderson

Logic's Lost Genius: The Life of Gerhard Gentzen de Eckart Menzler-Trott

Auto biography

Indiscrete Thoughts de Gian-Carlo Rota et Fabrizio Palombi

Discrete Thoughts: Essays on Mathematics, Science, and Philosophy de M. Kac

A Mathematician Grappling With His Century de Laurent Schwartz = "Un mathematicien aux prises avec le siecle" original french title

Photographs

Mathematical People: Profiles and Interviews de Donald J. Albers

Source books

(because it is invaluable to read the original articles)

From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931 de Jean van Heijenoort

Yearly compilation

What's Happening in the Mathematical Sciences 200x-200x+1 de Barry Cipra

Math and society

The Rise of Statistical Thinking 1820-1900 de Theodore M. Porter

other

Abrégé d'histoire des mathématiques, 1700-1900

6000 Jahre Mathematik: Eine Kulturgeschichtliche Zeitreise - 2. Von Euler Bis Zur Gegenwart de Hans Wuaing

Answer by ogerard for Does pi contain 1000 consecutive zeroes (in base 10)?

2010-05-05T13:15:13Z

A similar question (1 million consecutive 7 in the decimal expansion of pi) has been discussed by Timothy Gowers in a text published in 2006 (see Reuben Hersh: 18 unconventional essays).

His (quite classical) heuristic arguments in favor of yes were even used for a study on the influence of autority on persuasiveness in mathematics (See Matthew Inglis and Juan Pablo Mejia-Ramos, Cognition and Instruction Journal, Routledge, 2009).

Answer by ogerard for Sum of digits of number A which equals sum of digits of x*A

2010-05-05T04:06:03Z

@Wadim Zudilin: Indeed. It looks like a problem from the Euler project.

Answer by ogerard for Are There Primes of Every Hamming Weight?

2010-04-27T14:28:54Z

This is not an answer but an observation.

Let $n$ be an integer and $H(n)$ its Hamming weight.

$H(n) <= 1+ \max( { H(d) |\ d\ {\rm divisor\ of\ } n-1 })$

in particular for $p$ a prime greater than 2

$H(p) <= 1+ \max( { H(d) |\ d\ {\rm proper\ divisor\ of }\ p-1 })$

It could suggest ways to attack this and related questions.

Answer by ogerard for Which mathematical ideas have done most to change history?

2010-04-24T16:18:44Z

Two simple ideas I attribute to a pre-mathematical thought in this respect

1= The closed line (more or less a circle), and the idea of a boundary, of an inside and outside with its many derivations in life strikes me as a very ancient concept with very deep implications on thought, culture and society.

At the same time this idea is still fruitful in contemporary mathematics with homology, limits, inequalities, etc. as well as in our society.

2= The line as a path, a track, linked with time, that one follows, step by step, joining a start and an end exactly for the same reasons as above, with multiple current incarnations in today mathematics.

Answer by ogerard for What are some examples of colorful language in serious mathematics papers?

2010-04-24T14:05:44Z

There is the famous (and with contradictory interpretations) cry from Jean Dieudonné "à bas Euclide !", "Down with Euclide !". His books and prefaces are good sources for strong (and dated) opinions on what was "good" or "productive" mathematics and what was not.

Doron Zeilberger papers may contain also some colorful language.

Comment by ogerard

2013-02-08T20:29:56Z

And there is the Syracusan Square which is a special case of Graeco-Latin square ...

Comment by ogerard

2010-05-29T06:03:58Z

@Omar: This definition by subset of the product of domain and codomain is not only a trick. This is a good basis for combinatorial thinking and fits nicely with the various ways of counting fundamental objects.

Comment by ogerard

2010-05-29T05:53:10Z

@Christos: As previously commented there is the 3. about partitions of a set. There are cases where you better use def 1 to prove there is one. In fact we bother discussing many kinds of relations. Before introducing equivalence relations, you can take some time to define strict and inclusive order relations, which are antisymmetric. It prepares you to the fact that all these concepts are related. And is a good preparation for lattice theory.

Comment by ogerard

2010-05-21T07:37:23Z

I am afraid these are (mathematically) of anecdotical interest and that Grothendieck has again recently vetoed translations and publications in any form of his magnus opus.

Comment by ogerard

2010-05-19T19:05:15Z

Except for some classes of presentable groups which are decidable.

Comment by ogerard

2010-05-18T22:33:22Z

Of course I agree. But it should also don't hide that their are also functional and difference equations.

Comment by ogerard

2010-05-17T18:36:17Z

By looking I found that most articles of Sophus Lie were reprinted in a collected works series, edited among others by Engels and published by Teubner in the 1930s.

Comment by ogerard

2010-05-17T18:27:18Z

I hope that at least your german is correct because most articles from Sophus Lie on algebras and differential transformations were written in german. I will list the most important of them in editing my answer. They have been reprinted in volumes by Chelsea, and this volume should be in most good mathematical libraries.

Comment by ogerard

2010-05-17T16:15:02Z

Perhaps one solution would require to embed the plane into a larger space where some components of the construction of $\sqrt(A)$ could live in analogy to the imaginary dimension of the complex plane and the idea of fractals as quotients of figures between spaces of larger dimension.

Comment by ogerard

2010-05-16T08:54:37Z

I used to believe that the O came as the geometrical approximation of a physical ring, from the local ring idea.

Comment by ogerard

2010-05-16T08:51:54Z

Certainly thanks to the chosen word (ring, anneau in french) I have always thought of a ring as a torus, like the product of one operation (+) by the other (x) and closed in these two dimensions. I guess that I would also like to see sub-rings and modules as small rings with the initial ring like a thread passing in their holes.

Comment by ogerard

2010-05-16T08:45:18Z

You mean touché . Touche without accent is not an adjective but the word for a piano key, the equivalent of the english "touch" as in "the painter's delicate touch" or a hit on a target.

Comment by ogerard

2010-05-15T16:46:18Z

How do you know that it is pitched at the right difficulty ? You will know that only when you solve it. =/= I will also second Karl's remark. Do you need the solution in order to go to the "further work and questions" ?

Comment by ogerard

2010-05-15T14:03:29Z

The common point of the two meanings is movement: motive is what moves someone to act, and a freeze motif is running around the edges of a plate, like footsteps of an animal and can also give an illusion of movement when following it visually.

Comment by ogerard

2010-05-15T13:56:07Z

@Xandi: I would like to see more non-native french speaker publish mathematical articles in french, but the arrogant style of your answer is certainly not going to help.