# Tagged Questions

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

**6**

votes

**2**answers

180 views

### Literature on “real” $C^*$-algebras

I am trying to get a better understanding of "real" $C^*$-algebras. I encountered them in the paper
D. Voiculescu, Dual algebraic structures, J. Operator Theory 17(1987), 85-98,
which cites
G.G. ...

**9**

votes

**1**answer

299 views

### Applications of SCH outside of set theory

Recall that the Singular Cardinals Hypothesis (SCH) says that if $\kappa$ is a singular cardinal and $2^{cf(\kappa)}<\kappa,$ then $\kappa^{cf(\kappa)}=\kappa^+.$ Clearly it has many applications ...

**0**

votes

**0**answers

69 views

### Exposition of the Calabi complex

I am interested in a complex derived by Eugenio Calabi in his article "On compact Riemannian manifolds with constant curvature".
The complex is referenced as "Calabi complex" in various citing ...

**4**

votes

**1**answer

99 views

### Generalized permutahedron and random polytopes

The Birkhoff polytope $B_n$ is defined as the convex hull of the set of permutation matrices, which gives us the set of doubly stochastic matrices. A concept which is intimately related is that of the ...

**4**

votes

**2**answers

143 views

### Embedding a linearly ordered free monoid into a linearly ordered group

A linearly ordered (shortly, l.o.) monoid is a triple $\mathbb M = (M, \cdot, \le)$ for which $(M, \cdot)$ is a (multiplicatively written) monoid and $\le$ is a total order on $M$ such that $xy < ...

**5**

votes

**2**answers

332 views

### Looking for paper: The Cauchy integral by M. Privalov

I am looking for a paper titled The Cauchy integral by M. Privalov (Saratov, 1919).
The paper is written in Russian. It is being cited (for instance) by A. Kolmogorov in the paper Sur les fonctions ...

**4**

votes

**4**answers

359 views

### Reference for Kronecker-Weyl theorem in full generality

The Kronecker-Weyl theorem asserts the following: fix real numbers $\theta_1,\dots,\theta_d$, and consider the infinite ray $t(\theta_1,\dots,\theta_d)$ $(t\in\Bbb R)$ inside the $d$-dimensional torus ...

**2**

votes

**1**answer

242 views

### Looking for paper: Weil's original 1952 “Sur les formules explicites de la théorie des nombres premiers”

I am looking for a source (preferably online) for Weil's original 1952 paper on the explicit formula. I am aware of an english translation available here, but would like to have access to the original ...

**7**

votes

**4**answers

478 views

### Conjugation Quandles and… “Quandle-Groups”? From quandles to Groups

This question is already asked MathSE
A quandle $(Q,*,/ )$ is a idempotent right-distributive and right invertible structure.
1) $a*a=a$
2) $(a*b)*c=(a*c)*(b*c)$
3) $(a*b) ...

**3**

votes

**0**answers

81 views

### Bicategorical limits with parameters

(This question was asked in http://math.stackexchange.com/questions/741334/bicategorical-limits-with-parameters with no answer.)
Let $F(-,-)\colon \mathcal{A}\times \mathcal{B}\to \mathcal{C}$ be a ...

**2**

votes

**1**answer

159 views

### Classical theory for the incompressible Euler equation (reference request)

I have recently been interested in the incompressible Euler equation, but since I am new to the topic, I would like to inquire what are the standard sources/references (for self-study) regarding the ...

**7**

votes

**0**answers

99 views

### Algebraic construction of the modular representation of $\mathrm{SL}_2(\hat{\mathbf Z})$

The answer to this question is probably to be found in the theory of automorphic forms, but (I don't know much about it and consequently) after some tries, I did not catch it. Thus I'd be grateful if ...

**4**

votes

**1**answer

112 views

### Graph presentation of Lexicographic shifts

Consider a finite alphabet $\{0,1, \ldots, n-1\}$. Let $\Sigma_n = \mathop{\prod}\limits_{j=1}^{\infty}\{0, \ldots n-1\}$ be the set of infinite one sided sequences and $\prec$ the lexicographic ...

**3**

votes

**1**answer

144 views

### Understanding a particular approximation for Stirling's number of the second kind

I noticed and employed (without a problem) an approximation for Stirling's number of the second kind found on Wikipedia (http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind), in ...

**3**

votes

**1**answer

53 views

### Reference request: results on the asymptotic distribution of singular values related to a random orthogonal matrix

Let $Q$ be a random variable taking as its values the set of $n \times k$ real matrices with orthogonal columns, and whose distribution is the Haar measure on the Stiefel manifold $O(n)/O(n-k)$. This ...

**4**

votes

**0**answers

114 views

### Is there a version of Serre's modularity conjecture for projective representations?

Serre's modularity conjecture asserts that a continuous odd irreducible representation $$\overline{\rho} : G_\mathbb{Q} \rightarrow \mathrm{GL}_2(\mathbb{F}_q)$$ must be modular, in the sense that ...

**0**

votes

**0**answers

46 views

### Discrete Fourier tranform on $L^2(\mathbb{R})$

I'm studying on the following construction:
For $\lambda\in\mathbb{R}$, denote
$$Y_{\lambda}=\left\{g\in L^{2,loc}(\mathbb{R})\,:\,g(t+2\pi)=e^{2\pi i\lambda}g(t)\right\}$$
We give to $Y_{\lambda}$ ...

**0**

votes

**0**answers

83 views

### Ref request: If an affine subspace $V$ of $\mathbb R^n$ meets an $n$-dimensional polytope $P$, then it meets $P$ in a face of dimension $\le n-\dim V$

Let $P$ and $V$ be, respectively, a bounded full-dimensional polytope and an $m$-dimensional affine subspace of $\mathbb R^n$, and assume $P \cap V$ is non-empty.
Q1. Could you provide me with a ...

**1**

vote

**5**answers

376 views

### Character table of $S_7$

Is there any reference where I can find the character table of the symmetric group $S_7$? A simple search in google gave me a GAP program that computes the character table, but I don't understand the ...

**4**

votes

**3**answers

324 views

### Systematic brute-force searches for counterexamples

This is getting nowhere on math.stackexchange.com, so I'm putting it here.
Gödel's completeness theorem says that for every statement in first-order predicate calculus with equality, there is either ...

**1**

vote

**2**answers

201 views

### Lattices in general totally disconnected locally compact groups

Besides automorphism groups of trees and buildings, I was wondering if the lattices in general totally disconnected locally compact groups have been studied in the literature? I appreciate if you ...

**3**

votes

**1**answer

268 views

### A.J. Galitzer's Ph.D. thesis: On the moduli space of closed polygonal linkages on the 2-sphere

Recently I became curious about moduli spaces of linkages and so I found and began reading some papers of Kapovich and Millson. In the paper Hodge theory and the art of paper folding, the Ph.D. ...

**8**

votes

**2**answers

331 views

### Covering the space by disjoint unit circles

Sierpinski has proved the following two interesting theorems.
Theorem 1. The Euclidean plane $\mathbb{R}^2$ is not a union of nondegenerate disjoint circles.
Theorem 2. The Euclidean space ...

**5**

votes

**1**answer

145 views

### Making additive envelopes of monoidal categories monoidal

I have a monoidal category $(\mathcal{C},\otimes)$ enriched over abelian groups, for which I want to take the additive envelope $\mathcal{M}at\,\mathcal{C}$.
(This is defined as the category with ...

**0**

votes

**2**answers

97 views

### reference needed for sobolev type estimates

I'm reading a paper and the authors applied the following sobolev type estimates
$$
||(Dv)^{2}||_{H^{3k-2}(\Omega)}\leq C||v||_{H^{3k-1+\alpha}(\Omega)}^{2}
$$
for $\alpha>\frac{1}{4}$,
where $v$ ...

**1**

vote

**1**answer

93 views

### Linearly ordered set arithmetic: reference request

A lot has been written about the arithmetic of ordinal numbers. However, we can also do arithmetic with linearly ordered sets.
Question. Is there an article or book where I can learn the basics of ...

**1**

vote

**0**answers

56 views

### the validity of a basic statement involving the Hausdorff distance

Let $\Omega_1 \supset \Omega_2 \supset \cdots$ be a sequence of nonempty, open, bounded and convex sets in $R^n.$
Define $\Omega = \operatorname{int} \Bigl( \overline{\bigcap_{k=1}^{\infty} \Omega_k ...

**4**

votes

**1**answer

121 views

### Estimating the probability that $\|Av\| \ge \|v\|$

Given a diagonalizable matrix $A \in \mathbb{R}^{n \times n}$ with real eigenvalues, satisfying $1+c_1 \le \rho(A) \le 1+c_2$ $(0<c_1 \le c_2)$, obviously there exists a $v \in \mathbb{R}^{n}$ such ...

**0**

votes

**2**answers

110 views

### Analyticity of Logarithmic Integrals

Assume $f\in L^2[0,1]$ and let $g(x)=\int_0^1f(y)\ln|x-y|dy$. Is it true that $g\in C^\infty(0,1)$? Is it true that $g$ is analytic in $(0,1)$? Can you refer me to a right reference to look up such ...

**2**

votes

**2**answers

160 views

### Looking for a reference to a classical formula for the sum of the base-$b$ digits of an integer

Pick an integer $b \ge 2$, and for $n \in \mathbf N$ let $s_b(n)$ denote the sum of the base-$b$ digits of $n$. It is a nice exercise to prove that $$s_b(n) = (b-1) \sum_{i=1}^\infty ...

**21**

votes

**2**answers

1k views

### Was Vinogradov's 1937 proof of the three-prime theorem effective?

Was Vinogradov's first proof of the three-prime theorem effective?
Reasons for my question: Vinogradov presented his proof in 1937 in a monograph; the English translation by K.F. Roth and A. ...

**2**

votes

**0**answers

93 views

### An equivariant Hahn Embedding Theorem?

The Hahn Embedding Theorem asserts that for any (linearly) ordered abelian group $\Lambda$, there exists a linearly ordered indexing set $\Omega$ such that $\Lambda$ admits an order-preserving group ...

**0**

votes

**0**answers

80 views

### Chain-complex and Co-chain-complex valued Topological Quantum field theories

My research has led to me considering chain-complex and co-chain-complex valued topological quantum field theories. However, I am unable to find any literature that extensively studies this. Is there ...

**0**

votes

**0**answers

104 views

### Third variation of area of a minimal surface

There is a formula for the third variation of area on page 96 of Nitsche's book,
Lectures on Minimal Surfaces, vol. 1 (English version). He says at the bottom of the
page it is good for normal ...

**4**

votes

**2**answers

353 views

### Infinite dimensional Riemannian geometry

My current research has brought me into an area the requires me to learn some infinite dimensional Riemannian and Kähler geometry. Can someone recommend some good books or survey articles to help me ...

**7**

votes

**1**answer

315 views

### Is this a new formula? $\Delta^d x^n/d! = \sum_k \left[ x \atop k\right]{ k+n \brace x + d}(-1)^{x+k}$

$$\frac{\Delta^d x^n}{d!} = \sum_k \left[ x \atop k\right]{ k+n \brace x + d}(-1)^{x+k}$$
Where $x$, $n$ and $d$ are non-negative integers, $\Delta^d$ is the $d$-th difference with respect to ...

**3**

votes

**0**answers

216 views

### Hirzebruch's ICM talk

Is there an English translation of Hirzebruch's 1958 ICM lecture note:
KOMPLEXE MANNIGFALTIGKEITEN
Thank you very much!

**2**

votes

**0**answers

150 views

### J-function of cotangent bundle of complete flag variety

Givental and Kim showed that the $J$-function of the complete flag variety $Fl_n=SL_{n}/B$ becomes an eigenfunction of the Toda Hamiltonian. How about the $J$-function of the cotangent bundle ...

**3**

votes

**0**answers

75 views

### Geometry of ends of a finite volume negatively curved manifold

Is there a survey of the geometry of manifolds with finite volume Riemannian metrics of negative sectional curvature? More specifically, I am interested in the geometry of cusp ends of such manifolds, ...

**11**

votes

**2**answers

211 views

### Subset of vectors whose sum has a large norm

In Rudin - Real & Complex Analysis we have the following
Lemma 6.3. If $z_1, \ldots, z_n \in \mathbb{C}$ then there is a subset $S \subseteq \{1,\ldots,n\}$ for which
$$\left|\sum_{k \in S} ...

**1**

vote

**1**answer

96 views

### Convergence of a sequence of dependent binomial trials

Consider a sequence of the the following stochastic process.
Let $b_0=1$, and let $n >1$. At each step $t$, let $b_t \sim Bin(n,\frac{b_{t-1}}{n})$.
The process stops when either $b_t=0$ $b_t=n$. ...

**5**

votes

**1**answer

287 views

### A question on generalized Einstein metrics on four-dimensional manifolds

I am thinking of a possible generalization of Einstein metrics (or a possible characterization of Einstein metrics) on four-dimensional manifolds,
\begin{equation*}
...

**6**

votes

**1**answer

215 views

### Is $G_{\operatorname{red}}$ normal in $G$?

Let $G$ be an affine group scheme of finite type over a field $k$. It is well known that the associated reduced subscheme $G_{\operatorname{red}}$ of $G$ is a subgroup if $k$ is perfect. So let us ...

**0**

votes

**1**answer

79 views

### Fractional Laplacian on compact hypersurface/manifold via harmonic extension?

Let $M$ be a sufficiently smooth compact hypersurface of dimension $n-1$ in $\mathbb{R}^n$.
In pages 10-11 of this paper, the authors define $\mathcal{M} = M \times (0,\infty)$ and consider the ...

**6**

votes

**6**answers

471 views

### Furstenberg $\times 2 \times 3$ conjecture, bibliography

Furstenberg $\times 2 \times 3$ original conjecture states that the unique continuous invariant probability measure for $2x$ mod $1$ and $3x$ mod $1$ is the Lebesgue measure.
I wanted to have a ...

**6**

votes

**1**answer

170 views

### Does the totality of Ackermann's function prove the consistency of $\Sigma_1$-induction?

It is well known that Ackermann's function is not primitive recursive. Therefore, the theories of primitive recursive arithmetic (PRA) and of $\Sigma_1$-induction ($I\Sigma_1$) cannot prove the ...

**2**

votes

**0**answers

45 views

### Local energy decay for variable-speed, divergence-form wave equation in non-trapping medium without obstacles

I'm looking for a reference in the literature describing local energy decay for solutions of a smooth-coefficient, variable-speed wave equation, in divergence form, with compactly-supported initial ...

**0**

votes

**0**answers

168 views

### Game Theory - need references on analysis of particular game

My hobby AI research have led me to a thorethical game of particular design. As design is pretty simple, I was sure that such game has well-known name. But my question on math.stackexchange, where I ...

**5**

votes

**1**answer

163 views

### Equivariant Formality

Let $G$ be a finite group and $\mathcal{A}$ be a $dg$-algebra. Assume $G$ acts on $\mathcal{A}$, i.e. there exists a homomorphism $G\to {\rm Aut}_{dg}(\mathcal{A})$.
Assume further there exists a ...

**1**

vote

**0**answers

56 views

### Periodic solution of first order ODE

There is a famous result shows that for every continuous function $f:{\mathbb R}\rightarrow {\mathbb R}$, the first order autonomous system
$$
\left\{
\begin{array}{l}
\dot{x}=f(x), \\
x(t_0)=x_0,
...