3
votes
2answers
460 views

Co-Hausdorffification

Given a topological space $(X,\tau)$ we can define the "$T_2$-ification" of $X$ by setting $T_2(X,\tau) = X/\simeq$ where $x\simeq y$ in $X$ if and only if for every open neighborhood of $x$ has ...
3
votes
0answers
54 views

Connected sum of chiral manifolds

Let $M,N$ be two closed, smooth, orientable manifolds of the same dimension and assume that these manifolds are chiral, i.e. they do not admit an orientation reversing automorphism. Then there are two ...
-1
votes
1answer
153 views

Parabolic subgroup

I have a question about root set corresponding to $P_θ ∩ M_Ω$ where $θ$ is a subset of simple roots, $Ω=θ∪{α}$ where $α$ is a simple root and not in $θ$, $P_θ$ is a parabolic subgroup corresponding ...
1
vote
1answer
104 views

Does a line bundle on a normal Noetherian algebraic space come from a Weil divisor?

Let $X$ be a normal Noetherian algebraic space and $\mathscr{L}$ a line bundle on $X$. If $X$ is a scheme, then there is locally principal Weil divisor on $X$ that gives rise to $\mathscr{L}$. Is the ...
5
votes
0answers
163 views

Twisted equivariant modular forms?

I'd like to know where I can find information about a class of objects which I think deserve to be called twisted equivariant modular forms. Let me guess a definition, indicate how it can be made more ...
2
votes
1answer
163 views

When is a homogeneous space connected? [on hold]

Let $G$ be a Lie group (not necessarily connected) and let $H$ be a closed subgroup of $G$. I am after an algebraic (group theoretic) characterization of when the homogeneous space $G/H$ is connected. ...
2
votes
0answers
66 views

Chern character of Schubert structure sheaf

Let $X_\lambda \subset Gr = Gr(k,n)$ be a Schubert variety in the Grassmannian and $\mathrm{ch} : K_0(Gr) \otimes \mathbb{Q} \to A^\bullet(Gr) \otimes \mathbb{Q}$ the Chern character isomorphism. Is ...
4
votes
3answers
263 views

What is the difference between p-adic Lie groups and linear algebraic groups over p-adic fields?

I thought they were the same, just different names. Let me make question more precise: Let $G$ be any linear algebraic group over a p-adic field $\mathbb{Q}_p$, is $G$ a p-adic Lie group w.r.t. the ...
0
votes
0answers
39 views

Shuffle multiplication and generalized Leibniz rule in tensor calculus

The headline already says it: Is anybody (except me) aware of this formula for higher total covariant derivatives of tensor products? It is the simplest application of the commutative shuffle product ...
5
votes
1answer
303 views

How many geometric structures on manifolds are there?

Let $M$ be a smooth manifold, of dimension $n$. I know that there are many types of geometric structures on $M$. A large number of them are captured by the notion of a $G$-structure, which is a ...
0
votes
0answers
24 views

Projected Alternating Minimization

Assume that $f(x,y)$ is a non-convex function for $x,y\in \mathbb{R}$. Assume that we want to minimize this function (even locally) with respect to $x$ and $y$ such that $x \in \mathcal{X}$ and $y \in ...
4
votes
0answers
48 views

Global theta series on GL

Let $E/F$ be a quadratic extension of number field and let $V$ be a Hermitian space over $E$. Then we have Weil representations for the dual pair $U(n,n)\times U(V)$, and we can consider the theta ...
0
votes
0answers
45 views

Algorithm to determine if a set of sets can “cover” a range [on hold]

let's say you have a finite and arbitrary set of sets, and each inner set has can contain integers from 1 to 4 not repeating. So a set could be {{1}, {1,4}, {1,4}, {1,2,3,4,4}, {2,3,4}}. And suppose ...
0
votes
0answers
80 views

Prime number sequence in nature? [on hold]

Nature follows mathematical rules. We've been able to determine many of these underlying mathematical concepts. We've seen the Fibonacci sequence, etc...I'm curious about Prime numbers, though. Are ...
0
votes
0answers
30 views

Extension of a smooth function to a small neighborhood of a cone

Let $C\subset\mathbb{R}^n$ be an open polysimplicial cone. Let $f$ be a smooth function on $C$ such that all its derivatives extend by continuity to $\overline{C}$ (the closure of $C$). Does this ...
2
votes
1answer
140 views

An expectation of the product of random unitaries

I want to find the answer of $$\int dU \ U^m X \ U^{\dagger m}$$ Where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $U(n)$ and $dU$ is a normalized Haar measure. $X$ is a given self-adjoint ...
1
vote
1answer
56 views

About Central-by-finite subgroups

Let $G$ be a torsion group and $H \unlhd G$. Suppose that $H$ is a locally finite group and suppose that $H$ let be a FC-group. Let $x \in G$. Then is true that $[H,x]$ is a central-by-finite group? ...
3
votes
0answers
157 views

Is it possible to assume that an étale neighborhood is connected?

I am new to étale topology (though I've seen Grothendieck's sites before). Let $S:=\mathcal{O}^\textrm{sh}_{X,x}$ be the strict local ring of a point $x$ of a scheme $X=\operatorname{Spec}R$ (over a ...
5
votes
0answers
74 views

Centralizers of elements in free group algebras

Let $A$ be a group algebra of a free group, and $x \in A$. What is the centralizer of $x$? Is there something like Bergman's theorem for free associative algebras?
16
votes
2answers
596 views

A possibly surprising appearance of Lucas numbers

Let $S$ be the set of polynomials defined as follows: $0$ is in $S$, and if $p$ is in $S$, then $p + 1$ is in $S$ and $x \cdot p$ is in $S$, so that $S$ "grows" in generations: $g(0)=\{0\}$, ...
2
votes
1answer
198 views

Automorphisms of ideals of $\mathbb{C}[t]$

Let $f(t)\in\mathbb{C}[t]$, and let $I_f$ be the ideal in $\mathbb{C}[t]$ generated by $f(t)$. The ideal $I_f$ has a natural $\mathbb{C}$-algebra structure. My question is the following: For ...
7
votes
0answers
91 views

Asymmetric random walk on the line with barriers

The most commonly considered random walk on the line takes one step left or right with equal probability until a barrier is reached (if there are any barriers). More generally, suppose we fix any ...
0
votes
0answers
114 views

Linearization of line bundle [on hold]

In the definition of the linearization of a line bundle in Dolgashev book [Lectures on Invariant Theory, page 104], there are two conditions : 1) the action of the group $G$ on the line bundle $L$ ...
2
votes
0answers
36 views

“Prime” fusion rings

Surely this concept is known! (But I don't recall seeing it - maybe under another name? But "prime" is the obvious name choice.) Example. Open the Gepner/Kapustin paper at ...
7
votes
2answers
313 views

de Rham cohomology of smooth affine varieties

Let $U$ be a smooth variety over $\mathbb{C}$. We know that there exists a smooth compactification $X$ such that $X-U$ is a normal crossings divisor $D$ and that the de Rham cohomology of $U$ can be ...
5
votes
2answers
278 views

Expectation of trace of nth power of unitary matrices

I am trying to find the answer of $$\int dU \ |Tr(U^m)|^2$$ where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $\textit{U}(n)$ and $dU$ is a normalized Haar measure. In the case $m=1$, the ...
2
votes
0answers
35 views

Space of p-harmonic functions

Let $\Omega \subset\mathbb{R}^d$, $d \geq 2$, be a sufficiently nice set to make the following question meaningful. I am interested in the space of p-harmonic functions on $\Omega$; that is, the ...
3
votes
2answers
222 views

Fano manifold admit an smooth anti-canonical divisor?

Let $M$ ba a compact Kaehler Fano manifold. Under which conditions $M$ admit a smooth anti-canonical divisor $D$
4
votes
1answer
172 views

Can a Brownian motion be fast at its extrema?

After pondering this MO question > Location of maximum of Brownian motion with rough drift <, I wonder whether a Brownian motion can be fast (i.e. beats the law of the iterated logarithm) at its ...
5
votes
2answers
69 views

An example of two cofibrant dg categories whose tensor product is not cofibrant

I have been reading the paper by Toën "The homotopy theory of dg categories and derived Morita theory" where in chapter 4 it is stated that the tensor product of two cofibrant dg categories $C$ and ...
3
votes
0answers
90 views

Looking for the correct version of a wrong statement from Barvinok's book on convex polyhedra

The book I'm concerned with is "Integer Points in Polyhedra" by A. Barvinok, which, I must say, is turning out to be highly fascinating. A real finite-dimensional vector space $V$ defines the ...
2
votes
1answer
74 views

When the vertex covering number is smaller than the chromatic number

For any graph $G=(V,E)$ let $\tau(G)$ be the minimum cardinality of a vertex cover of $G$. As noted here, we have $\tau(G) \geq \chi(G) - 1$ for all finite graphs $G$. I'm interested in graphs $G$ ...
1
vote
0answers
68 views

A question related to metric Diophantine approximation

In metric Diophantine approximation you are often interested in finding conditions on $(\phi(q))_{q \geq 1}$ which guarantee that $$ \left| \alpha - \frac{p}{q} \right| < \frac{\phi(q)}{q} $$ has ...
1
vote
1answer
65 views

Proof of closed walk generating function identity

In 'Spectral Conditions for the Reconstructibility of a Graph' Godsil and McKay give a short proof of an identity (Lemma 2.1) that relates the generating function for the number of closed walks ...
0
votes
1answer
79 views

Uniform $L^p-L^{p'}$ bound of a Fourier multiplier

Let $(\tau,\xi)\in\mathbb{R}\times \mathbb{R}^n$, and consider the function $$ m_{\epsilon}(\tau,\xi)=\frac{1}{\tau+|\xi|^4+\epsilon|\xi|^2+i} $$ in $\mathbb{R}^{n+1}$. My first question is that does ...
15
votes
1answer
255 views

How to make the Capelli's identity less mysterious?

The formulation of the Capelli's identity is very elementary; it has important applications in invariant theory and representation theory, see http://en.wikipedia.org/wiki/Capelli%27s_identity To ...
6
votes
1answer
180 views

Do free profinite groups satisfy Howson's theorem?

Let $F$ be a free profinite group, and let $A,B \leq F$ be finitely generated closed subgroups. Must $A \cap B$ be finitely generated?
23
votes
2answers
827 views

What is the purpose of the flat/fppf/fpqc topologies?

There have been other similar questions before (e.g. What is your picture of the flat topology?), but none of them seem to have been answered fully. As someone who originally started in ...
0
votes
0answers
25 views

Way to parameterise sparse multi diagonal matrix

I have an NxN matrix S that looks like this: $$ S^{-1} = K^{-1} + \Lambda $$ where N is a multiple of 3, both K and S are positive definite matrices, and Lambda is $$ \Lambda = \begin{bmatrix} x ...
4
votes
1answer
65 views

Antoine's Necklace and positive Hausdorff/Lebesgue measure

I have the following question: The usual construction of the Antoine's Necklace produces a Cantor of $1$-dimensional Hausdorff measure in $\mathbb R^3$. I would like to know whether one could adapt ...
1
vote
2answers
34 views

Sensitivity of inverse normal cdf

Let $Q^{-1}$ be the inverse function of a standard normal CDF. For $0 < \epsilon < p,p' < 1 - \epsilon$, how much does the function $Q^{-1}$ change as a function of $|p - p'|$? Any useful ...
2
votes
1answer
104 views

recognising weak equivalences of simplicial sets

$\require{AMScd}$ I am interested in detecting using a lifting property when a map $f:X\to Y$ in $sSet$ (with the standard Kan model structure) is a weak equivalence. In the paper Weak Equivalences ...
-2
votes
0answers
19 views

Can we predict next sample using the existing samples? [on hold]

Suppose that I have 18 data points and I'm sampling 3 data points each time. Suppose that I have 60 samples (each has 3 data points). Can we predict the next sample (of 3 points) from the existing ...
0
votes
0answers
46 views

Can we conclude that $\varphi(L_xf_0)\neq0$ for every $x\in G$?

Let $H$ be a compact subgroup of locally compact topological group $G$ and $A=\{f\in L^1(G); R_hf=f(a,e)\forall h\in H\}$ as a subalgebra of $L^1(G)$ by convolution of $L^1(G)$. If $\varphi \in ...
2
votes
0answers
92 views

Does this symmetrization operator have a name? Any theory?

Consider a function $f(x_1,\ldots,x_n)$ of $n$ complex variables. Define $$f_{\mathrm{symm}}(x_1,\ldots,x_n) = 2^{-n}\sum_{\varepsilon_1,\ldots,\varepsilon_n=\pm 1} ...
1
vote
0answers
52 views

Are A and A^T unitarily equivalent over a p-adic field?

Let $E/F$ be a quadratic extension of p-adic field. Let $U=\{u\in GL_2(E): uu^*=1\}$ be the unitary group of rank 2. My question is: given a matrix $A\in GL_2(E)$ can we find $u_1,u_2\in U$ such ...
-4
votes
0answers
51 views

Convert 1-5 Grading Scale to 1-100 Grading System [on hold]

I am creating a formula in Excel to convert 1-5 Grading Scale to 1-100 Grading System Suppose that I have the following table: 97-100 = 1.00 94 - 96 = 1.25 91-93 = 1.50 88-90 = 1.75 85-87 = 2.00 ...
0
votes
0answers
35 views

Linear elliptic estimates

i am interested in solutions of the following $$-\Delta \phi =f \; \; \; A_\lambda, \qquad \phi=0 \; \; \partial A_\lambda,$$ where $ A_\lambda=\{ x \in R^N: \lambda <|x|<1 \}$ with $ ...
0
votes
0answers
81 views

Computations of derivations, $d^2$, of a (Grothendieck) spectral sequence

The maps $d^1:E_1\rightarrow E_1$ have a nice description. Is there any text providing us with a description of the higher derivations $d^2:E_2\rightarrow E_2$ arising from a Grothendieck spectral ...
0
votes
0answers
22 views

Examples of Sigma-Adequate Links that are not simply Adequate in A and B type sense?

I am looking to see if anyone has constructed explicit examples of $\sigma$-adequate links a la Makoto Ozawa's Essential State Surfaces for Knots and Links? This technique is centered around taking ...

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