0
votes
0answers
8 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 ...
0
votes
0answers
14 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
11 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 ...
2
votes
1answer
103 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
14 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
30 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
24 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 ...
-1
votes
0answers
57 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
18 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 ...
0
votes
0answers
37 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
35 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? ...
2
votes
0answers
78 views

Is it possible to assume that an ´etale neighborhood is connected?

I am new to ´etale 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 $X=\textrm{Spec}R$ (over a ...
2
votes
0answers
39 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?
7
votes
0answers
172 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
153 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 ...
6
votes
0answers
62 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
99 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 $L$ ...
1
vote
0answers
26 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
234 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 ...
3
votes
2answers
156 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
25 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
176 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
111 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 ...
2
votes
2answers
50 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 ...
2
votes
0answers
47 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. A real finite-dimensional vector space $V$ defines the following two spaces, also real. $\mathcal{P}(V)$ generated by ...
2
votes
1answer
61 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
40 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
46 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
0answers
39 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 ...
5
votes
0answers
77 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 Let's ...
6
votes
0answers
61 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?
19
votes
2answers
439 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
140 views

Constructing a particular Cayley Graph

Pick a positive number $P$. Pick $M$ positive numbers $g_1<\dots<g_M$ each less than $P-1$. Denote $\mathcal{S}=g_1,\dots,g_M$. Denote $G=G_{P}[\mathcal{S}]$ to be Cayley graph (considered ...
0
votes
0answers
19 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 ...
-5
votes
0answers
42 views

Multi-dimensional space [on hold]

Let $x(t):{\rm I\!R} \to {{\rm I\!R}^n}$ and $f(t,x(t)):{\rm I\!R} \times {{\rm I\!R}^n} \to {{\rm I\!R}^n}$ What norm is Appropriate to show the following inequality? $$\left\| f(t,x) \right\| \le ...
3
votes
1answer
56 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
28 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
86 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
16 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
27 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
80 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
43 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
47 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
32 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
64 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
18 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 ...
-3
votes
0answers
86 views

What is the interpretation of f'(x)/f(x)? [on hold]

I'm studying Fisher information and the function d/dx ln(f(x)) thus f'(x)/f(x) appears. I'm trying to interpretate what this is quantier could represent in a function. Thank you.
1
vote
2answers
288 views

Popular books written by great mathematicians [on hold]

I read: H. Poincare. Value of science F. Klein. Development of Mathematics in the 19th Century J.E. Littlewood. A Mathematicians Miscellany G.H. Hardy. A Mathematician’s Apology R. Courant, ...
3
votes
0answers
53 views

reference for higher spin - not gravitational nor stringy

Other than the papers of Berends,Burgers and van Dam, are there any papers that study the general case of deforming a free field theory with higher spin fields to be interactive?
2
votes
0answers
31 views

Does there exist duality/symmetry between Fuchsian differential equations, like in the case of confluent hypergeometric?

My question is about whether certain dualities or symmetries hold between pairs of Fuchsian differential equations, but I need to give a bit of background to explain what I want to ask. Thank you for ...

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