1
vote
0answers
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
votes
0answers
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
votes
0answers
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
votes
0answers
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
1answer
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
votes
0answers
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
votes
0answers
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
votes
0answers
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
1answer
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
votes
0answers
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? $$\frac{d}{dG} \mbox{Tr} \big( A^TG^T(GSG^T)^{−1}GA \big)$$
0
votes
0answers
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
1answer
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
vote
0answers
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
vote
0answers
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
votes
0answers
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
votes
0answers
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 Gödel's incompleteness theorem?
2
votes
0answers
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
votes
0answers
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
1answer
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
1answer
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
votes
0answers
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
vote
0answers
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
votes
0answers
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
votes
0answers
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
2answers
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
votes
0answers
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
vote
0answers
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
1answer
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
votes
0answers
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
1answer
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
vote
0answers
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
1answer
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
votes
0answers
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
1answer
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
votes
0answers
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
votes
0answers
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
1answer
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
1answer
109 views

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

How do we find—for example—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
votes
0answers
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
votes
0answers
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
votes
0answers
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
1answer
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
2answers
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
0answers
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
0answers
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
2answers
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
1answer
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
0answers
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
0answers
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
0answers
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}_{...

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