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Results tagged with reference-request
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user 5371
This tag is used if a reference is needed in a paper or textbook on a specific result.
13
votes
Accepted
Who discovered the winding number?
Grünbaum and Shephard suggest that the winding numbers (for closed polygons) have been discussed in the literature at least since 1769.
See
A.L.F. Meister, Generalia de genesi figurarum planarum …
- 22.3k
3
votes
Accepted
Knaster-Kuratowski-Mazurkiewicz (KKM ) Thoerem
There is a book devoted entirely to KKM theory and its applications. It contains
an outline of the classical KKM theorem as well as a number of generalizations. Another standard reference is "Fixed p …
- 22.3k
3
votes
Accepted
Invertibility of the Laplacian
A careful exposition can be found in "Analytic Semigroups and Semilinear Initial Boundary Value Problems" by Kazuaki Taira.
He considers the mixed boundary value problem
$$\begin{cases} A u=f & \mb …
- 22.3k
9
votes
Accepted
The classical Krein-Rutman theorem
"Topological Vector Spaces" by Helmut Schaefer contains a thorough treatment of the classical Krein-Rutman theorem for compact positive operators in an ordered Banach space along with several general …
- 22.3k
5
votes
Accepted
Equivalent definitions of Gaussian curvature
I like the presentation of the Theorema Egregium in A Comprehensive Introduction to Differential Geometry (Volume 2) by Michael Spivak. A translation of the original paper by Gauss and the historical …
- 22.3k
7
votes
Area of cross-section (at midpoint perpendicular to longest diagonal) in the unit cube of di...
This is a very old problem and there is a classical analytic approach to it. You can express the volume of sections of a convex body in terms of the Fourier transform of powers of the Minkowski functi …
- 22.3k
2
votes
Accepted
Approximation by analytic functions
The answer to the modified question is given by Jackson-type theorems.
The classic book by N.I. Akhiezer which is quoted in the Wikipedia article contains a number of specialised results on optimal a …
- 22.3k
10
votes
What are Central Limit Theorems and why are they called so?
From the introduction to History of the Central Limit Theorem: From Laplace to Donsker by Hans Fischer:
The term “central limit theorem” most likely traces back to Georg Pólya. As he
recapitulat …
- 22.3k
9
votes
Accepted
Quantitative bounds for multivariate central limit theorem
There is a bunch of such statements which can be obtained by Stein's method.
You might be interested in the paper "On the Rate of Convergence in the Multivariate CLT" by Gotze, which is specifically …
- 22.3k
8
votes
Accepted
Dirichlet series expansion of an analytic function
A.F. Leont'ev continued to work on general Dirichlet series well into 1980s (until his death in 1987). Actually, he published three monographs on the subject from 1976 to 1983! He made a short summary …
- 22.3k
9
votes
Accepted
English or French translation of Gauss' "Summatio Quarumdam Serierum Singularium"
"The determination of Gauss sums" by Berndt and Evans (Bull. Amer. Math. Soc., Vol. 5, Number 2 (1981), 107-129.) contains an exposition of the original proof due to Gauss. It also includes a short hi …
- 22.3k
9
votes
How to "globalize" the inverse function theorem?
There exist several known global implicit function theorems. Those results tend to be tailored for specific applications. It seems to be rather difficult to state a universally useful one-size-fits-al …
- 22.3k
15
votes
What are "classical groups"?
Such a definition (but not the definition, I suppose) can be found in Clifford Algebras and the Classical Groups by Ian Porteous (see Chapt. 13).
It is based on the classification of real algebra an …
- 22.3k
32
votes
Accepted
Are there smooth bodies of constant width?
Fillmore showed that there are sets of constant width in $\mathbb R^d$ with analytic boundaries which have a trivial symmetry group (so these are very different from spheres;
see "Symmetries of surfac …
- 22.3k
7
votes
Accepted
Approximation by exponential polynomials
This follows from the fact that the set of $n\times n$ matrices with simple spectrum is dense in the space of all $n\times n$ matrices ${\bf M}_n(\mathbb C)$ (or that the set of polynomials of degree …
- 22.3k