This tag is used if a reference is needed in a paper or textbook on a specific result.

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29 views

### Approximation with Predefined Topology of Niveau Sets

Problem
given are
a finite, connected, undirected and, cycle-free graph (i.e. a "tree") $T(V,E)$, of which one of the vertices (w.l.o.g. $v_0$) is defined to be the root.
a planar imbedding ...

**11**

votes

**3**answers

1k views

### Source of quotation about the waste-baskets of physicists

In an article I'm writing I want to quote (with attribution) the original version of an aphorism that says that one can often find mathematical gold in the waste-baskets of physicists. Would someone ...

**9**

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**0**answers

278 views

### Short proof of $\frak p=t$

It is known for a while now that $\frak p=t$, as a result of Malliaris-Shelah. The original paper draws from model theoretic methods.
I've heard rumors that there was a proof which was purely set ...

**2**

votes

**2**answers

163 views

### How to calculate $P(\sum_{i=1}^{m}(A_i+S_i)\le L)$ with $A_i,L\sim\text{exp}(\lambda),S_i\sim\text{exp}(\mu)$ and positive integers $\lambda\neq\mu$?

Recently I was stumped by the calculation of the probability
$$\mathbb{P} \big(\sum_{i=1}^{m} (A_i + S_i) \le L < \sum_{i=1}^{m+1} (A_i + S_i) \big)$$
where $A_i \sim \text{exp}(\lambda), S_i \sim ...

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157 views

### “Naïve”cobordism?

The naive view of (say, complex) cobordism of a space $X$ is that it should be the group generated by continuous families of closed manifolds parametrized by $X$. Acting even more naively, one may try ...

**10**

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95 views

### Is there literature on approaching semisimplicity of l-adic cohomology using vanishing cycles

Let $K$ be a finitely generated field, and $X/K$ a smooth proper variety. Let $G$ denote the absolute Galois group $\mathrm{Gal}(\bar{K}/K)$. Let $\ell$ be a prime different from $\mathrm{char}(K)$. A ...

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votes

**1**answer

339 views

### Incomplete Failures of the Inverse Galois Problem

I thought of this question the other day and have not been able to get any traction on references or results along its lines, so I finally caved and decided to ask it here. I am no expert on Galois ...

**3**

votes

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128 views

### Reference request: construction of Chern classes

I am looking for a reference on splitting principle for etale cohomology of simplicial schemes (over arbitrary field k). I found a paper by Schechtman, "On the delooping of chern character and Adams ...

**16**

votes

**3**answers

358 views

### Real Lie groups versus real linear algebraic groups: differences in connexity and fundamental group

There are many introductory texts on real Lie groups, and many on linear algebraic groups in general, but fewer on the specific case of linear algebraic groups over the reals, and even fewer that try ...

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votes

**2**answers

181 views

### Hardy-Littlewood-Sobolev inequality on hyperbolic space

Let $I_\alpha = (-\Delta)^{-\alpha/2}$ be the Riesz potential on $\mathbb{R}^n$. The Hardy-Littlewood-Sobolev inequality on $\mathbb{R}^n$ says
$$||I_\alpha f||_{L^q} \leq C||f||_{L^p}$$
where $q = ...

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votes

**1**answer

70 views

### Meromorphic extension of local defining equations of a complex submanifold

let $M$ be a smooth compact complex manifold of dimension $m$ and $N\subset M$ a smooth complex submanifold of dimension $1\leq n \leq m-2$. Covering $N$ with well chosen open sets of $M$ we can ...

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votes

**0**answers

236 views

### The dogma of the natural numbers in physics

As is well known "God made the natural numbers; all else is the work of man" (Leopold Kronecker). However, "what would correspond more to the spirit of physics would be a mathematical theory of the ...

**0**

votes

**1**answer

176 views

### Trace map for sepeared morphism of non-singular varieties

I read about for any separable morphism of non-singular varieties $f:X'\to X$, one can define a homomorphism $\text{Tr}:f_*(\Omega_{X'}^q) \to \Omega_{X}^q$,so that the map $\Omega_{X}^q \to ...

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52 views

### Is this duality operation on simplicial complexes/Stanley-Reisner rings previously known?

Let $K$ be an abstract simplicial complex on vertices $x_1,\ldots,x_n$, then there is the familiar construction of the face ideal $I_K=\langle x_{i_1}\cdots x_{i_r} | ...

**3**

votes

**1**answer

383 views

### Questions on the proof of the Serrin condition for the regularity of Navier-Stokes equations and related issues for the incompressible Euler equation

Edit: The question has been substantially modified from the original one. The original question (see below) concerned with rigorously justifying the proof of the Serrin condition. These questions have ...

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votes

**4**answers

784 views

### Expressing the Lebesgue integral using categories + the difficulty of describing estimates in category theory

In this question of mine in a comment to the accepted answer, someone remarked:
There are ways to express even basic things in analysis, such as the
spectral theorem or the Lebesgue integral, ...

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votes

**0**answers

100 views

### access to Ramanujam's paper on vanishing theorem

I couldn't found the article
"C.P Ramanujam, Remarks on the kodaira vanishing theorem, J. Indian Math. Soc.36(1972) 41-50"
Can anyone help me to find that? Thanks!

**10**

votes

**1**answer

303 views

### Did Milnor and Thurston write anything else about characteristic numbers for 3-manifolds?

In Characteristic numbers for 3-manifolds Milnor and Thurston define a characteristic number and this is cited in ch. 6 of Thurston's notes when discussing the Gromov approach to Mostow rigidity.
The ...

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votes

**1**answer

118 views

### What do epimorphisms in noncommutative rings look like?

The question I want to ask is inspired by this mathoverflow post about epimorphisms in the category of commutative rings. I found the seminar (by P. Samuel) referenced by David Rydh particularly ...

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vote

**0**answers

153 views

### Reference for Local class field theory via witt vectors

I would like to find some books or lecture notes on geometric local class field theory via Witt vectors. I can't find any good paper on this subject.All approaches in the books to local class field ...

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votes

**0**answers

119 views

### n-ary quadratic forms with $S$-integer values

Let $Q(x_1,\ldots,x_n):=x_1^2+\cdots+x_n^2$ be an $n$-ary quadratic form.
Given a finite set of (rational) primes $S$ is there an algorithm or theorem that describes all solutions to ...

**9**

votes

**1**answer

238 views

### Control ramification in Noether Normalization

Let $X$ be an irreducible affine variety (integral, reduced scheme of finite type over an algebraically closed field $\Bbbk$ of characteristic zero) of dimension $n$. The well-known Noether ...

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**1**answer

95 views

### Complement of bifurcation variety

I am reading a seminal paper of Arnold "Normal forms of functions near degenerate critical points, the Weyl group of $A_k$, $D_k$, $E_k$ and lagrangian singularities".
Let $f\colon \mathbb{C}^n\to ...

**3**

votes

**0**answers

66 views

### Grunsky-Motzkin-Schoenberg formula

I found this formula in Brian McCartin's interesting book "Mysteries of the equilateral triangle" http://www.kettering.edu/news/mysteries-equilateral-triangle and it looks as follows:
Suppose that ...

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votes

**1**answer

101 views

### Why are all involutions conjugate in the special linear group of degree 2?

It appears to be standard that the set of non-identity involutions in $SL(2, 2^n) = PSL(2, 2^n)$ forms a single conjugacy class. What is the best reference for this?
I note that
...

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votes

**2**answers

213 views

### Only admissibles start gaps in clockable ordinals

This is a question about ITTM model introduced by Hamkins et al. In this paper it is proven that no admissible ordinal is clockable, so it either starts or lies within a gap in clockable ordinals. I ...

**2**

votes

**2**answers

174 views

### Negative real order Sobolev spaces: density and representation

First, I give my motivation to ask this question. The generalised Neumann trace can be defined as
$$
{}_{H^{-1/2}(\partial\Omega)}\langle\frac{\partial ...

**1**

vote

**1**answer

57 views

### Approaches to implicitly defining generating function

First,every language in Chomsky hierarchy(or c.e.language) corresponds to a generating function,the set of the functions is GF,now,a question : is every generating function with integral coefficient ...

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vote

**0**answers

47 views

### Some resources about minimum-length generator sequences

In the group theory I want to know what are the best results known for problem of finding minimum-length generator sequences. This problam have different titles in articles that cause difficality in ...

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**0**answers

193 views

### Irreducibility of Galois representations attached to unitary groups

If $G$ is a unitary group in $n$ variables over $\mathbb Q$, attached to an hermitian form for an imaginary quadratic extension $E/\mathbb Q$ and if we suppose that the hermitian form is definite over ...

**1**

vote

**1**answer

106 views

### Formula for the Ordinal Number of k-Sets of Positive Integers

Background of my question is, that I would like to store flags indicating the relation between a pairs of non-adjacent edges of a graph (that relation could for example be, whether the edges cross, ...

**3**

votes

**1**answer

363 views

### Levi's book on Leibnizian calculus

Raphael Levi learned from Leibniz at a late stage in Leibniz's career. This might be a definite advantage for understanding Leibniz. Leibniz did not elaborate some of the philosophical principles ...

**3**

votes

**2**answers

103 views

### Conformal invariants of planar pairs of pants

Consider a planar pair of pants
$$P = \left\lbrace z \in \mathbb{C}: |z| \le 1, |z-x| \ge r_1, |z+x| \ge r_2 \right\rbrace$$
where $-1 < -x-r_2 < -x+r_2 < 0 < x-r_1 < x+r_1 < 1$.
...

**3**

votes

**1**answer

69 views

### reference for existence and blow up results in transport-like PDEs

This question was originally posted by me on math.stackexchange but I didn't get any answers and I thought that perhaps it would be better off here. I hope it's appropriate, I've encountered the ...

**2**

votes

**0**answers

60 views

### Strictly Convex Smoothing of a function defined on an affine manifold

A function defined on an interval is strongly convex if it has positive definite second derivative. A function on an affine manifold is strongly convex
if its restriction to each line is. An affine ...

**23**

votes

**1**answer

689 views

### How strong is this conjecture? $(Z/nZ)^*$ is generated by “small” elements

Conjecture: There are constants $c,k$ such that every $(Z/nZ)^*$ is generated by its elements smaller than $k (\log n)^c$.
Where $(Z/nZ)^*$ is the multiplicative group of integers mod $n$. My ...

**6**

votes

**1**answer

302 views

### Class groups of orders

In Cox's book "Primes of the form $x^2 + ny^2$", he proves that in a quadratic imaginary field $K$, if $\mathcal O$ is an order of conductor $f \in \mathbb Z$, we have that the class group ...

**5**

votes

**3**answers

296 views

### Decomposition of $L^2(\Gamma \backslash G)$

Let $G$ be a semisimple Lie group, and $\Gamma$ be an lattice (arithmetic) - typical examples I am thinking about would be $(SL_2(\mathbb{R}), SL_2(\mathbb{Z})$, or $(SL_2(\mathbb{C}), PGL_2(O_F))$ ...

**1**

vote

**1**answer

122 views

### Tables of data associated to reductive algebraic groups?

I am looking for a reference that contains lots of calculations for specific examples of various objects one can associate to a reductive algebraic group. For example, given a (specific) linear ...

**2**

votes

**1**answer

210 views

### Absolutness of $\Pi_1^1$ statements

Shoenfield absoluteness is well known for $\Pi_2^1$-statements, but it does not hold between a countable transitive model of ZFC and the universe.
But it is also known that $\Pi_1^1$ statements are ...

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votes

**1**answer

187 views

### Examples of groups such that order isomorphism of the subgroups of $G\times G$ and $H\times H$ does not imply isomorphism of $G$ and $H$

Let $G$ and $H$ be groups, $\operatorname{Sub}(G\times G)$ be the set of all subgroups of $G\times G$ and $\operatorname{Sub}(H\times H)$ be the set of all subgroups of $H\times H$. Assume there ...

**12**

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**3**answers

432 views

### Formal/rigorous treatment of (im)predicativity/predicativism

There are several places on the web where one may find quite intuitively understandable accounts of (im)predicativity; here on MO I found two questions with very good detailed answers (Predicative ...

**12**

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**0**answers

251 views

### Connes' idea to use the hyperfinite $III_1$ factor for the archimedian place of Spec(Z)?

I recently gave a talk, where I talked about the tensor category
of (all, not just finite index) bimodules over the hyperfinite $III_1$ factor.
Vaughan Jones, who was in the audience, later told me ...

**1**

vote

**1**answer

228 views

### Is the Lie quadric $Q^3$ isomorphic to the Lagrangian Grassmannian $LG(2,4)$?

On the paper
http://arxiv.org/abs/1009.1364
(published on Proc. London Math. Soc.) I've found an interesting statement:
The Lie quadric $Q^3$, i.e., the space of all points, lines and
circles ...

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**0**answers

337 views

### Last Status of Feferman's Conjecture on Indefinite Value of Continuum

The "true" value of $2^{\aleph_0}$ is one of the most fundamental open questions of mathematics and its philosophy. Hundreds of set theoretic results during the last century don't say anything more ...

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votes

**1**answer

138 views

### A generalization of covering spaces to fiber bundles with totally path-disconnected fibers

There is a classical theorem about covering spaces and the actions of the fundamental group.
Theorem 1: Let $B$ be a non-empty locally path-connected and path-connected space. The category of ...

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69 views

### Unions of orbits of dimension $\leq n$

Let $G$ be a complex linear algebraic group acting on a smooth complex projective variety $X$ with finitely many orbits. Note that each $G$-orbit is a smooth locally closed subvariety of $X$.
For a ...

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vote

**2**answers

166 views

### Defining degeneracies for semi-simplicial sets with inner Kan conditions

Suppose we are given a category enriched over semi-simplicial sets, i.e. for the simplices in this category we have well-defined boundary maps, but no degeneracy maps. Suppose also that in this ...

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vote

**1**answer

102 views

### Small resolutions are automatically crepant?

Page 17 of the following survey:
http://arxiv.org/abs/1103.5380
makes the claim that small resolutions, meaning resolutions such that the exceptional set is in codimension at least two, are ...

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67 views

### Orthogonal basis for the multilinear polynomials with zero “trace”

We say that a multilinear polynomial $P(x_1,\ldots,x_n)$ in $n$ commuting variables over $\mathbb{R}$ has zero trace if
$$ \frac{d}{dt} P(t,\ldots,t) = 0. $$
Equivalently,
$$ \left(\sum_{i=1}^n ...