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7
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4answers
736 views

Errors, oversights, and misunderstandings in mathematical research [duplicate]

Possible Duplicate: Examples of common false beliefs in mathematics. Hopefully this is not overly controversial, but I thought it would be instructive to compile a list of errors which are ...
19
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9answers
2k views

real symmetric matrix has real eigenvalues - elementary proof

Every real symmetric matrix has at least one real eigenvalue. Does anyone know how to prove this elementary, that is without the notion of complex numbers?
3
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1answer
352 views

What was Lambert's solution to $x^m+x=q$?

I've been thinking about Lambert's Trinomial Equation quite a bit, and I want to see his solution. The only solution I could find was in Euler's form, and I still don't quite understand how he got ...
5
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0answers
249 views

Sophus Lie on the symplectic foliation theorem

Given a Poisson manifold $(P,\{\cdot,\cdot\})$, its characteristic distribution $\mathcal C$ is the singular tangent distribution on $M$ generated by the Hamiltonian vector fields,i.e. $$\mathcal ...
15
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0answers
403 views

Authorship of Grothendieck universes

Universes seem to first appear in Grothendieck's work in SGA 1, which is credited to Grothendieck, and a lengthy discussion is in the chapter on Prefaisceaux (presheaves) in SGA 4. That chapter is ...
51
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61answers
7k views

Old books still used

It's a commonplace to state that while other sciences (like biology) may always need the newest books, we mathematicians also use to use older books. While this is a qualitative remark, I would like ...
11
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3answers
2k views

Why is a ring called a “ring”?

Why is a ring called "ring" (or Zahlring in German)? There seems to (naive) me nothing more ring-like to a ring than there is to a group or a field. I am particularly interested to learn why the ...
33
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4answers
2k views

The origin of sets?

The history of set theory from Cantor to modern times is well documented. However, the origin of the idea of sets is not so clear. A few years ago, I taught a set theory course and I did some digging ...
12
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4answers
1k views

Statements which were given as axioms, which later turned out to be false.

I know that early axiomatizations of real arithmetic (in the first half of the nineteenth century) were often inadequate. For example, the earliest axiomatizations did not include a completeness ...
6
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1answer
326 views

Where do the Kähler Identities first appear?

The Kähler identities (sometimes known as the Hodge identities) are an important collection of relationships between operators on the exterior algebra of a Kähler manifold. These relationships ...
6
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1answer
763 views

Deligne's letter to Looijenga from 1974

Hello, I wonder if anyone has a copy of Deligne's letter to Looijenga from 1974 mentioned as reference [26] in Bessis' paper Finite complex reflection arrangements are $K(\pi,1)$ from 2006, see ...
11
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1answer
924 views

At what point does number theory stop playing with finite rings?

Basic results in number theory, like the Chinese remainder theorem, the Euclidean algorithm and Euler's theorem, are really about finite structures, namely the rings $\mathbb{Z}/n\mathbb{Z}$ for ...
43
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3answers
2k views

Groups that do not exist

In the long process that resulted in the classification of finite simple groups, some of the exceptional groups were only shown to exist after people had computed (most of) their character tables and ...
1
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0answers
320 views

Area Under Generalized Parabolas and Hyperbolas without Calculus.

This is shorter and more specific version of certain questions about a rather simple quadrature method. The answers I got were great but not what I asked. The terms in the title for $y=x^p$ look ...
14
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1answer
421 views

The Riemann zeros and the heat equation

The Riemann xi function $\Xi(x)$ is defined, with $s=1/2+ix$, as $$ \Xi(x)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)=2\int_0^\infty \Phi(u)\cos(ux) \, du, $$ where $\Phi(u)$ is defined as $$ ...
8
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1answer
1k views

What is “Teichmüller Theory” and its history ?

What is "Teichmüller Theory" ? What part has been worked out / forseen by O. Teichmüller himself and what is further development ? Is there some current work which might be considered as ...
9
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4answers
804 views

Integrating Powers without much Calculus

I'll jump into the question and then back off into qualifications and context Using the definition of a definite integral as the limit of Riemann sums, what is the best way (or the very good ways) ...
11
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7answers
1k views

What are some Applications of Teichmüller Theory?

I'm trying to collect some specific examples of applications of Teichmüller Theory. Here are some things I have collected thus far: No-wandering-domain Theorem (Sullivan) Theorems of Thurston ...
7
votes
9answers
981 views

Examples where adding complexity made a problem simpler

I can think of a few situations in math where a problem becomes easier or an object becomes simpler when some complexity is added. Examples: $S^n$ is never contractible, but $S^{\infty}$ is. The ...
5
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0answers
339 views

Who was Hermann Künneth?

Question as in the title: Who was Hermann Künneth? Where can I find some biographical information beyond what is available on Wikipedia? The well-known Künneth formula, for example in the form of ...
3
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0answers
229 views

A Question Regarding Boolean-valued Models

What were the intuitions motivating the creation (or discovery, if you will) of Boolean-valued models? I have searched for the Scott-Solovay paper on the subject, but to no avail. There also seems to ...
2
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2answers
535 views

Has the notion of “space” been reconsidered in 20th century?

The original title, "has the bases of geometry been reconsidered in 20th century" of this question refers to Riemann's paper "On the Hypotheses which lie at the Bases of Geometry", an English version ...
6
votes
2answers
584 views

Why is Gauss credited with his connection?

Let $\pi : X \to B$ be a family of compact Kähler manifolds over a smooth base $B$. We then have a local system $\mathcal R^k \pi_* \mathbb Z$ (for your favorite $k$) of abelian groups over $B$, whose ...
8
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2answers
447 views

The history of the geometrization of closed surfaces

Who first recognized that the torus supports a flat structure? Who first characterized the moduli space of flat structures on the torus? Who first recognized that the closed, orientable genus 2 ...
22
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6answers
3k views

Who wrote up Banach's Thesis?

Some time ago I read somewhere (and I don't remember where it was) that Stefan Banach -- a highly creative and great mathematician -- did not always write down his ideas. Allegedly, he did not write ...
33
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5answers
4k views

Why might André Weil have named Carl Ludwig Siegel the greatest mathematician of the 20th century?

According to Steven Krantz's Mathematical Apocrypha (pg. 186): As was custom, Weil often attended tea at [Princeton] University . Graduate student Steven Weintrab one day went about the room ...
5
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1answer
671 views

Historical questions on the term “general abstract nonsense”

Saunders Mac Lane reports that the contents of his 1942 paper (joint with Samuel Eilenberg), that first introduced categories, were then referred to (in the words of prominent representatives of the ...
3
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3answers
412 views

Origin of the theorem on the existence of the smallest field of definition of an affine variety

Weil proved the following theorem in his book Foundations of Algebraic Geometry, p.19. The proof is somewhat involved. I wonder if the theorem is his original. Theorem Let $K[X_1,\dots, X_n]$ be the ...
15
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1answer
949 views

On a theorem of Galois

I am currently teaching Galois theory and this week, I mentioned the following theorem of Galois : Let $P(x) \in \mathbf{Q}[x]$ be an irreducible polynomial of prime degree. Then $P$ is solvable by ...
30
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10answers
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Believing the Conjectures

In Believing the axioms (I and II), Penelope Maddy proposes five "rules of thumb" that she then uses to justify large cardinal axioms in set theory. These extrinsic rules are modeled after the ...
6
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3answers
615 views

Classification of Platonic solids

My question is very basic: where can I find a complete (and hopefully self-contained) proof of the classification of Platonic solids? In all the references that I found, they use Euler's formula ...
16
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2answers
2k views

Location of Archimedes' grave in Syracuse (math/archaelogy trivia)

This is really a question for our archaelogist friends, but I could not find an "archaelogy overflow" and some mathematicians might find it interesting. In a few weeks I am giving a talk in which I ...
10
votes
1answer
922 views

Did Gauss know Dirichlet's class number formula in 1801?

Let $h_d$ be the number of $SL_{2}(\mathbb{Z})$ classes of primitive binary quadratic forms of discriminant $d$. It's natural to impose the hypothesis that $d$ is not at square, as we do below. In ...
5
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1answer
211 views

constant averages along orbits

What should one say to describe the situation in which a function $T$ from some set $X$ to itself, and a function $f$ from $X$ to some characteristic-zero field $K$, have the property that the average ...
10
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1answer
455 views

Smallest dimension of nontrivial representation of a simple Lie algebra over `$\mathbb{C}$`

The question involved here is natural and very classical, but I'm unsure what has been formally stated and proved in the literature. The only approach I know involves assembling facts that apparently ...
5
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1answer
366 views

Numerical Methods for ODEs - History

Wikipedia presents a timeline of important developments in Numerical Methods for ODEs, namely: ...
7
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0answers
212 views

What is the historical connection between Zeeman's twist spinning and Fox's Examples?

Both Ralph Fox and (at that time, yet to be knighted) Sir Christopher Zeeman attended the 1961 Georgia topology conference. Fox's paper from that conference was his seminal work, "A Quick Trip through ...
6
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1answer
1k views

Origin of square-and-multiply algorithm

I'm teaching an introductory course in cryptography and explained the square-and-multiply algorithm to the class. http://en.wikipedia.org/wiki/Square-and-multiply_algorithm Someone asked who ...
5
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3answers
1k views

First known proof of $\sqrt 2$ is irrational with prime factorization?

Do any of you happen to know the history of the standard prime factorization proof of $\sqrt 2$ is irrational? I know this theorem was known to Aristotle, and that the Fundamental Theorem of ...
15
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2answers
1k views

Who named it the Snake Lemma?

What is the history behind the colorful name of this result? Cartan-Eilenberg states it without any particular fanfare.
26
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7answers
2k views

At what times were people interested in prime numbers

While prime numbers are central objects in mathematics it looks that they were ignored and forgotten for long periods of time. I am interested to get some facts and insights about this matter, in ...
11
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1answer
544 views

Equivariant homotopy theory: some history questions

I have sometimes wondered about the following: (1) Who was the first articulate that in dealing with $G$-equivariant cohomology theories ($G$ a finite group or a compact Lie group), it is best to ...
29
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2answers
1k views

Definitive source about Dirichlet finally proving the Unit Theorem in the Sistine Chapel

(This question was posted on math.stackexchange a week ago at http://math.stackexchange.com/questions/187315/definitive-source-about-dirichlet-finally-proving-the-unit-theorem-in-the-sistinbut and ...
7
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0answers
166 views

what's the role of $H^{p}(\mathbb{R}^{n})$ in modern (harmonic) analysis?

The classical theory of $H^p$,due to it's heavy dependence on the complex function theory(such as Blaschke products), seemed to have an insurmountable obstacle barrying its extension to several ...
3
votes
4answers
788 views

Another Chicken or Egg: Sequence or Series

This is a side question which is more motivated by teaching than research. First, I am trying to convince myself that sequences appear before series (as numerical approximations to "interesting" ...
41
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1answer
2k views

Thurston's senior thesis at New College

I've been collecting some of the many unpublished manuscripts of Bill Thurston over the years. His recent passing inspired me to ask the following. I've seen a number of references (for instance, in ...
11
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0answers
3k views

Wikipedia story about Bill Thurston's death [closed]

I did not know whom to ask about this rather unsettling piece of news. Apparently Wikipedia has announced Bill Thurston's death on August 21, 2012. I could not independently verify it. Is this true?
3
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1answer
132 views

Fourier-Möller transform

In the book by Antosik, Mikusiński and Sikorski named Theory of Distributions, The Sequential Approach (russian translation, page 217) one can read: Таким образом, мы видим, что для произвольного ...
7
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2answers
821 views

Mathematical computer desk [closed]

D. Gibb, from the Mathematical Laboratory, University of Edinburgh, describes a Computer Desk in his book A course in interpolation and numerical integration for the mathematical laboratory, G. Bell ...
2
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2answers
572 views

Why unknowns are usually denoted by “X” ? [closed]

Why unknowns are usually denoted by "X" ? More precisely: is this answer really a serious answer or might be a 1 April joke ? Let me sketch it. But please watch it, it is really fun and cool and ...