# All Questions

**1**

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11 views

### Construction of differentials in the spectral sequence for double complexes

I was reading through Ravi Vakil's book/lecture notes on spectral sequences, but I came to an impasse. He leaves as an exercise the construction of the $d_2$ differentials of the spectral sequence (...

**0**

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4 views

### Probability Distribution on permutations with factor structure

Say I define a probability measure over the symmetric group $S_n$ as follows:
I specify $n$ positive `potential' functions $G_i : \{1, 2, \cdots, N\} \to (0,\infty)$
I then set
$$\mathbb{P}(\sigma) =...

**0**

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24 views

### Prove $\int_0^{\infty}{\frac{1}{e^{sx}\sqrt{1+s^2}}}ds < \arctan\left(\frac1x\right),\quad\forall x\ge1$

The question is to prove:
$$
\int_0^{\infty}{\frac{1}{e^{sx}\sqrt{1+s^2}}}ds < \arctan\left(\frac1x\right),\quad\forall x\ge1.
$$
Numerically it seems to hold true. So I have made some attempts to ...

**2**

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**0**answers

23 views

### $MK+CC$ as a foundation for category theory

Has any work been done on what $MK+CC$ looks like as a foundation for category theory? Is it 'the same' as restricting to inaccessibles in some precise manner?
According to wikipedia, any category ...

**0**

votes

**1**answer

50 views

### Is Quillen's bracket a “universal enveloping” something?

$\newcommand{\G}{\mathcal{G}}$
In K-theory, there is a construction due to Quillen as follows. Let $(\G, \oplus, 0)$ be a monoidal groupoid. Then the bracket $\langle \G, \G \rangle$, sometimes also ...

**0**

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23 views

### Inverse limit of $p^n$-torsion abelian groups

Let $p$ be a prime and let $\{A_n\}_{n > 0}$ be an inverse limit of abelian groups such that $A_n$ is $p^n$-torsion with $A_n/p^{n - 1} \cong A_{n - 1}$ (these isomorphisms are part of the data). ...

**0**

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9 views

### Halin Graphs with Highest Number of Hamilton Cycles

Halin graphs contain a Hamilton cycle and have the interesting property, that the shortest of all contained Hamilton cycles can be calculated in $O(n)$ time ("G. Cornuejols, D. Naddef, and W.R. ...

**-2**

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**0**answers

34 views

### Matrices M and N [on hold]

Matrices M and N Matricesrepresent reflections in the lines y = 2x and 3y = x, respectively. Verify that MN is not equal to NM, and explain why this should have been expected. What transformations do ...

**4**

votes

**1**answer

55 views

### Number guessing game with lying oracle

You are probably already familiar with the usual number guessing game. But for concreteness I restate it.
The usual game
The Oracle chooses a positive integer $n$ between 1 and 1024 (or any power of ...

**0**

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**0**answers

51 views

### Derivative of trace of matrix product

Let $G$ be an $m$-by-$n$ fat matrix and $S$ be an $n$-by-$n$ symmetric matrix. What will the following be?
$$\frac{d}{dG} \mbox{Tr} \big( A^TG^T(GSG^T)^{âˆ’1}GA \big)$$

**0**

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24 views

### small perturbation of BV function

consider an interesting real analysis question:
define average operator on $[0,1]$:
$A_{\epsilon} f (x) = \frac{1}{2\epsilon}\int_{x-\epsilon}^{x+\epsilon} f(y) dy , f \in BV[0,1] $
( may clarify ...

**4**

votes

**1**answer

96 views

### Can this self-adjoint operator have an infinite-dimensional compression with compact inverse?

The following might be quite straightforward, but I very rarely work in detail with unbounded operators, so I thought it would be worth seeing quickly if I have overlooked an example that is obvious ...

**1**

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**0**answers

64 views

### Schreier conjecture — without a simple proof? and sporadic simple groups

The Schreier conjecture asserts that $\mathrm{Out}(G)$ is always a solvable group when $G$ is a finite simple group. This result is known to be true as a corollary of the classification of finite ...

**1**

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**0**answers

14 views

### Deformation sublevel sets of functions which preserve boundary

I'm interested in proving the following fact, which seems to naturally arise from gradient flow deformations, but appears to be a bit tricky.
Consider a smooth family
$$f_s : M \to \mathbb{R}, \quad ...

**-3**

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**0**answers

45 views

### On Fourier transforms of even functions [on hold]

Suppose that $F$ is a real-valued Fourier transform of the function $G$, and that $F$ is even. That is $F(x)=F(-x)$ for all $x \in \mathbb{R}$. Does this necessarily mean that $G$ is also even/odd ? ...

**-5**

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**0**answers

56 views

### Are HOTT evaluated as alternatives to foundations of mathematics completely free from Gödel's incompleteness theorem?

I wonder how mathematicians who study Homotopy Type Theory think of this. Are they well aware of GÃ¶del's incompleteness theorem?

**2**

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**0**answers

83 views

### Hyperbolic subgroups of general linear groups

Is there a classification of hyperbolic subgroups of $GL_n(A)$ for $A$ some ring of characteristic $p$? $A$ for me is a finitely presented algebra over a finite field.
More precisely I'm looking for ...

**-2**

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**0**answers

74 views

### Maurer or Poincare

Is the Maurer Cartan form the same at the poincare cartan form with just different credit being given by different authors? Or is there a difference? The Poincare Cartan form is referred to by ...

**10**

votes

**1**answer

265 views

### What are the possible Stiefel-Whitney numbers of a five-manifold?

On a compact five-manifold, the Stiefel-Whitney number $w_2w_3$ can be nonzero.
An example is the manifold $SU(3)/SO(3)$, and also another example is a $\mathbb{CP}^2$ bundle over a circle where the ...

**2**

votes

**1**answer

82 views

### How do modular functions of level $N>1$ transform under the full modular group?

Let $f$ be a modular function of level $N>1$ and let $\gamma\in SL_2(\mathbf Z)$. Write $$f(\gamma\tau)=j(\gamma,\tau)f(\tau).$$
First question
What can we say in general about the factor $j(\...

**2**

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**0**answers

24 views

### Extreme points in $d$-dimensional quadrant hull of $n$ random points in a halfplane

Finding the asymptotic growth of the expected number of extreme points of the quadrant hull in $d$ dimensions seems to be a well studied problem. Interestingly, the solution appears to vary widely by ...

**1**

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**0**answers

75 views

### Atlas of gerbe over stack

Suppose $F:\mathcal{X}\rightarrow\mathcal{Y}$ is gerbe over stack and $p:X\rightarrow \mathcal{X}$ is an atlas $\mathcal{X}$.
Does this imply $F\circ p:X\rightarrow \mathcal{Y}$ is an atlas for $\...

**0**

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**0**answers

26 views

### Minimal steps of construction for constructible number

It is known that a real number $\alpha$ is constructible if and only if it lies in a number field $K=K_{n}$ s.t. there exists a tower of field extension $\mathbb{Q}\subset K_{1}\subset K_{2}\subset \...

**-4**

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**0**answers

63 views

### Question About 1/e Definition

I was thinking of a problem and have an answer through computer programming, but am unable to prove it mathematically. If you have the following:
(1/2)(1/2 + 1/3) and replace every summed number (in ...

**15**

votes

**2**answers

744 views

### Are there examples of conjectures supported by heuristic arguments that have been finally disproved?

The idea for this comes from the twin prime conjecture, where the heuristic evidence seems just so overwhelming, especially in the light of Zhang's famous result from 2014 about Bounded gaps between ...

**-1**

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**0**answers

51 views

### Galois action on Chern classes in $l$-adic cohomology of arithmetic varieties

What happens to Chern classes in the $l$-adic cohomology of varieties over an arithmetic field $K$ under the action of the absolute Galois group of $K$?

**1**

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58 views

### Categorical representations of absolute Galois groups

I am looking for interesting examples of categorical representations of absolute Galois groups of arithmetic fields. Pointers to the literature would be appreciated.

**2**

votes

**1**answer

60 views

### Flatness in a neighborhood of a point condition

Suppose that we have a Riemannian Manifold $(M,g)$ whose
curvature vanishes in an open neighborhood U of a point p.
When does this imply that the metric is Flat ?
In particular, does it happen ...

**0**

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35 views

### Multiple integral over $S^{n-1}$ exploiting spherical symmetry of the integrand

Note: I asked this on MSE as well, but I started thinking MO might be a better place to post the question.
$\textbf{Notation:}$
$\bullet$ For $N$ vectors $\textbf{u}_1, \ldots, \textbf{u}_N \in S^{...

**2**

votes

**1**answer

66 views

### Closure of tensor product /tensor product semigroup

In this reference the following claim is made in Remark 2
Let $A,B$ be closable operators on Banach spaces $X,Y$, then $A \otimes 1$ and $1 \otimes B$ are closable operators on the Banach space $X \...

**1**

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60 views

### Question on a proof of density of periodic orbits

In page 215 and 216 of the book "Introduction to the Modern Theory of Dynamical Systems" by Anatole Katok, Boris Hasselblatt, there is a theorem stated as following:
Theorem: Let $\Gamma$ be a ...

**3**

votes

**1**answer

53 views

### Convergence of a real sequence (stochastic approximation)

Let $(\gamma_n)_{n \geq 1}$ be a sequence of positive real numbers satisfying $$\sum_{n \in \mathbb{N}}\gamma_n = + \infty \text{ and }\sum_{n \in \mathbb{N}}\gamma_n^2 < + \infty$$
I would like ...

**4**

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**0**answers

107 views

### What does it mean to say the first Goodwillie derivative of $TC$ is $THH$?

A paradox:
Goodwillie calculus considers only finitary functors.
$TC$ isn't finitary.
Yet in some sense $\partial(TC) = \partial(K) = THH$ is the crux of the Dundas-Goodwillie-McCarthy theorem.
(...

**2**

votes

**1**answer

120 views

### Understanding a group of transformations of the plane $\mathbb{Z} \times \mathbb{Z}$

I'm doing some research in visualizing arithmetic sets (resp. properties, resp. sequences of integers). I try to create patterns (in which I hope to observe some symmetries) by injectively mapping $\...

**0**

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43 views

### Solvability of finite group from indices of commutator and abelian normal subgroup

Suppose finite directly indecomposable group $G$ has $\frac {|[G, G]|}{|G|} < \alpha$ and $\frac {|A|} {|G|} > \beta$, where $A \lhd G$ abelian. Are there some nontrivial bounds on $\alpha, \...

**0**

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57 views

### Generalized Sard's lemma

Let $f: X \to \mathbb{R}$ be a $C^{1,1}$ (that is $C^1$ with Lipschitz differential) function on a manifold $X$. Suppose that $f$ is smooth at all points of a subset $C \subset \text{Crit}f$ of ...

**2**

votes

**1**answer

89 views

### On the Calabi-Yau conjecture for minimal surfaces

Colding and Minicozzi proved that any embedded minimal surface in $\mathbb{R}^3$ with finite topology must be proper and thus it can not be bounded.
Is it possible to remove the assumption "finite ...

**-1**

votes

**1**answer

109 views

### The power of a prime in the prime factorization of a factorial [on hold]

How do we findâ€”for exampleâ€”how many $5$s are in the prime factorization of $n!$? I've read that it is $\lfloor n/5 \rfloor$, but why is that?

**-4**

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32 views

### Passage of the limit under the integral sign [on hold]

What articles or monographs on passage of the limit under the Riemann (Lebesgue) integral sign (with respect to a real parameter) do you know?
$$\lim \limits_{y \to y_0} \int \limits_a^b f(x,y)dx=\int ...

**-3**

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86 views

### No of relations which are symmetric but not reflexive [on hold]

How many relations are there in a set with n elements that are symmetric but not reflexive?

**0**

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27 views

### Initial value theorem for Fourier transform

Initial value theorem states that for a bounded function $f(t) = O(e^{ct})$ and an existing initial value, one-sided Laplace transform $F(s) = \int\limits^\infty_{0^-}f(\tau)e^{-s\tau}\mathrm{d\tau}$ ...

**10**

votes

**1**answer

280 views

### Homotopy orbits, spectra and infinite loop spaces

Let $X$ be an (naive) $O(n)$-spectrum (I'm choosing to work with orthogonal spectra). I've recently come across the following results,
$$(S^{n-1} \wedge X)_{hO(n)} \simeq X_{hO(n-1)}$$
and
$$\Omega^...

**12**

votes

**2**answers

204 views

### Can we realize a graph as the skeleton of a polytope that has the same symmetries?

Given a graph $G$, a realization of $G$ as a polytope is a convex polytope $P\subseteq \Bbb R^n$ with $G$ as its 1-skeleton.
A realization $P\subseteq \Bbb R^n$ is said to realize the symmetries of $...

**2**

votes

**0**answers

106 views

### The name for injective map $f:\mathbb{N}\rightarrow\mathbb{N}$ with $f(n)\geq n$ property

What is the name for map $f:\mathbb{N}\rightarrow\mathbb{N}$ (from natural numbers into natural numbers) with the following propeties:
1) $f$ is injective
2) $f(n)\geq n$ for every $n$?

**2**

votes

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65 views

### What's an example of a merely finitely additive probability that has Plachky's approximation property?

In this paper by Plachky, he characterizes the extreme points of the finitely additive probabilistic extensions from one field $\Sigma$ to a larger field $\Sigma'$ as those finitely additive ...

**3**

votes

**2**answers

135 views

### On convergent sequences in locally convex topological vector spaces

Assume that a sequence $(x_n)_{n\in\omega}$ of points of a locally convex topological vector space converges to zero. Is it always possible to find increasing number sequences $(n_k)_{k\in\omega}$ and ...

**4**

votes

**1**answer

78 views

### Two disjoint trees

Let $G$ be a graph and let $A_1, A_2 \subseteq V(G)$ be disjoint sets of vertices. Let us call $(A_1, A_2)$ independent if there exist vertex-disjoint trees $T_1, T_2 \subseteq G$ within $G$ which ...

**-1**

votes

**0**answers

40 views

### Collecting figurines [on hold]

You can buy n items from a shop containing a random figurine. There are k random figurines available and the goal is to collect them all. Assume that the probability is constant and equal for getting ...

**3**

votes

**0**answers

93 views

### Modular root of $-1$

Take two distinct coprime integers $a,b$ and take $q=a^2+b^2$. Consider the equation $(xy^{-1})^2\equiv-1\bmod q$. The solutions are two roots which correspond to $(x,y)=(a,b)$ and $(x,y)=(b,a)$. Look ...

**1**

vote

**0**answers

26 views

### Invariance of simple functions

Let $(A(s),D(A(s)))_{s\in\mathbb{R}}$ be a family of unbounded operators on a Banach space $X$ and $g:\mathbb{R}\rightarrow X$ be a simple function, i.e.,
\begin{align*}
g=\sum_{i=1}^n{x_i\textbf{1}_{...