0
votes
1answer
405 views

Cartan decomposition of a unitary group?

For local fields $F$, we consider two case 1) $E$=quadratic extension of $F$ , 2) $E = F \times F$. Let V be a 2-dim hermition space over E. In 1) case, by Cartan decompostion $U(2)$ can be ...
2
votes
1answer
155 views

Maximization of specific Likelihood function

N coins have probability $p_n = e^{-t_n/s}$ of heads, $t_n$ being specific for each coin. Coins 1 to m came up heads and m+1 to N came up tails. Now I'm trying to estimate $s$ using the Maximum ...
5
votes
1answer
256 views

Convex PBW bases

Given a reduced expression for the longest word $w_0$ in the Weyl group of $\mathfrak{g}=\mathfrak{n}^+\oplus\mathfrak{h}\oplus{n}^-$, one obtains a convex ordering on the set of positive roots, ...
2
votes
2answers
345 views

non-Identity operator on a separable Hilbert space

Suppose $\mathcal{H}$ is a separable Hilbert space over $\mathbb{C}$ (countable dimensions) with inner product $\langle,\rangle$. Let $A$ be a bounded linear operator on $\mathcal{H}$, i.e, in ...
4
votes
4answers
444 views

Coproduct on coordinate ring of finite algebraic group

I'm reading Mukai's book "An introduction to invariants and moduli", and I am having trouble understanding one of his examples. It is example 3.49 on page 101. The setup is as follows. Let $G$ be a ...
5
votes
1answer
1k views

binary code with constant hamming distance

I want as many 80-bits words as possible with the constraint that the hamming distance between any couple of words is exactly 40. How many can I generate? Is there a generic formula telling me how ...
8
votes
4answers
1k views

Is there much difference between Kronecker's and Dedekind's methods in algebraic number theory and commutative algebra?

Edwards, in his book "Divisor theory" says that Kronecker's methods are quite different to Dedekind's and those of today. Is there really much of a difference apart from Kronecker's methods being more ...
2
votes
3answers
502 views

Notion of internality in model theory

Good evening, Can someone explain to me the notion of internality in model theory (what it is, where it comes from...) ? Thank you
6
votes
2answers
605 views

Which semigroups can be linearly ordered?

As usual I consider a semigroup to be a structure $(A, +)$ such that $+$ is an associative binary function over the set $A$. The notion of linearly-ordered semigroup corresponds to structures of the ...
6
votes
3answers
2k views

Maximal ideal in polynomial ring

Is it true that the intersection of a maximal ideal in $A[x]$ with $A$ is a maximal ideal in $A$? Let's say A is Noetherian. I would be surprised if it isn't true but somehow I can't seem to show it. ...
0
votes
1answer
193 views

Embedding a semigroup into a divisible semigroup

The following is motivated by the fact that I'd like to have a way, much better if canonical, to isometrically embed a normed group into a normed divisible group. But semigroups are a much more ...
6
votes
3answers
266 views

Minimum separating subdivision in Plane

Hi I was thinking about the following problem: Given a planar Graph embedded in the plane and a set of points $P$ contained in the faces (no face contains more than one point) I want to determine ...
1
vote
1answer
103 views

examples of space of direction at a point in an infinite dim Alexandrov space compact

The space of direction at a point in an infinite dim Alexandrov space can be compact?Please give examples or prove it's wrong.
3
votes
1answer
161 views

Optimizing a stochastic “flip and prune” procedure for selecting a subset of coins

I place some number of coins, $(c_1, ..., c_N) \in C$ on a table, where each coin is originally tails up. Let's call the "tails" state $0$ and the "heads" state $1$. I then perform the following ...
2
votes
0answers
143 views

Mappings between Banach spaces

What is the definition of an analytic mapping between two Banach spaces? This is a problem I ran into when solving an integral equation. One of the related coefficients is represented as a functional ...
4
votes
3answers
317 views

closed meagre sets

A closed meagre subset of $[0,1]$ is either countable or homeomorphic to the Cantor set: either way it is $0$-dimensional. Q.1. Is every closed meagre subset of an $n$-dimensional locally compact ...
5
votes
2answers
804 views

Higher vanishing cycles

The generalisation of the vanishing cycle formalism in SGA 7 is apparently since the 1970's an issue, Morava mentioned a connection with Bousfield localization. I find the Morava's remarks ...
0
votes
1answer
228 views

Number theory question

Given $a$ and $b$ irrational numbers with $a/b$ also irrational, how do you prove that $( \{ na\} , \{ nb \})$ is dense in $[0,1] * [0,1]$ , where $n$ ranges over the integers? $\{x\}$ is the ...
4
votes
1answer
390 views

A Hölder like inequality

If $0< a_1\le a_2\le \cdots \le a_n\le a_{n+1}$ and $p>1$, is it true that ...
2
votes
1answer
531 views

Problem:Gromov-Witten;Moduli space

Let us consider a map from a $\Sigma_g \longrightarrow N$, where $N$ is a symplectic manifold. Then we define the moduli space as $M= \{ f | f \mbox{ is a pseudoholomorphic map } \Sigma_g \to N, ...
4
votes
1answer
1k views

Ask some matrix eigenvalue inequalities.

Let $ \begin{bmatrix} A& B \\\\ B^* &C \end{bmatrix}$ be positive semidefinite, $A,C$ are of size $n\times n$. Are the following plausible inequalities true? I have seen a lot of ...
7
votes
3answers
1k views

amenable equivalence relation generated by an action of a non-amenable group

Question. Give a (possibly elementary) example of a probability measure preserving action $\rho\colon G \curvearrowright X$ of a finitely-generated discrete group $G$ on a standard borel space $X$ ...
38
votes
1answer
3k views

Why are there 1024 Hamiltonian cycles on an icosahedron?

Fix one edge $e$ of the graph (1-skeleton) of an icosahedron. By a computer search, I found that there are 1024 Hamiltonian cycles that include $e$. [But see edit below re directed vs. undirected!] ...
2
votes
2answers
539 views

Infinite domain with finite number of prime ideals(elements)

While trying to prove one property of commutative rings with units I can't prove one fact without assuming existence of infinitely many different prime ideals or elements. I tried to test if it was ...
5
votes
1answer
412 views

Elementary end extension of a countable model for ZF

Theorem 2.2.18 in Chang and Kiesler uses omitting types to show that any countable model of ZF has an elementary end extension. Can we control the countable order type of such a model? for example, if ...
13
votes
2answers
761 views

Model category structure on categories enriched over quasi-coherent sheaves

Gonçalo Tabuada has shown that there is a Quillen model category structure on the category of small dg-categories, i.e. the category of small categories enriched over chain complexes (for a fixed ...
1
vote
1answer
174 views

Is a variety of algebras a set?

Let $K$ be a field and $K\{X\}$ be the free non-associative algebra, freely generated by the countably infinite set $X$. We consider elements of $K\{X\}$ as (non-associative) polynomials in the ...
1
vote
1answer
224 views

Question on localization technique

In the book "Local cohomology : An algebraic introduction with geometric application", page 289 there is a proof of the following theorem : Assume that $R=\bigoplus_{n}R_{n}$ is positive graded ...
1
vote
0answers
223 views

lifts of maps to $\mathcal{M}_{1,1}$

Hi, here's there's a construction about elliptic curves that I do not completely understand. Suppose I consider the two following families of elliptic curves over $\mathbb{C}^*$. The first, which I ...
14
votes
3answers
1k views

Striking applications of Baker's theorem

I saw that there are many "applications" questions in Mathoverflow; so hopefully this is an appropriate question. I was rather surprised that there were only five questions at Mathoverflow so far with ...
4
votes
1answer
262 views

Is there a good way to estimate the Fourier transform of $\frac{1}{\lambda-iP(\xi)}$

Assume that P is a real valued strong elliptic polynomial, then what do we know about the following $$ K(\lambda,x)=\int{\frac{e^{ix\xi}}{\lambda-iP(\xi)}}d\xi,\quad \lambda\in \mathbb{R}\0 $$ The ...
2
votes
0answers
2k views

Who will write the algebraic geometry texts that are needed? [closed]

Readers of MO are probably aware of the pedagogic need that would provoke such a query. It's around 60 years since Serre's FAC, and I imagine some people would say "you still have to read the original ...
4
votes
4answers
836 views

Quotient Surface of A Hyperelliptic Involution

Let $X$ be a hyperelliptic Riemann surface, and let $J$ be the hyperelliptic involution. Then consider the quotient surface $X/ < J > ,$ my question is whether $X/ < J > $ is a Riemann ...
2
votes
2answers
285 views

Does a generic curve inside the space of curves with a node at a specific point have only finitely many nodes?

Let $\mathcal{D} \approx \mathbb{P}^{\delta_d}$ be the space of homogeneous degree $d$ polynomials in three variables (up to scaling), where $\delta_d = \frac{d(d+3)}{2}$. Define $\mathcal{A}$ to be ...
1
vote
1answer
109 views

On the simply connectedness of Symmetric products and Hilbert schemes of points

My first question is whether $m$-th symmetric product of $\mathbb{C}^{n}$ is simply connected, where $n\geq 3$. The second question is whether $Hilb^{m}(\mathbb{C}^{n})$ is simply connected, where ...
0
votes
1answer
889 views

Finding linearly independent columns of a large sparse rectangular matrix

I have a problem that necessitates solving a large non-negative least-squares problem. My matrix A is large, sparse, highly rectangular (num rows >> num cols) and nearly binary. However, A is not ...
12
votes
3answers
2k views

Analysis from a categorical perspective

I have not studied category theory in extreme depth, so perhaps this question is a little naive, but I have always wondered if analysis could be taught naturally using categories. I ask this because ...
2
votes
1answer
2k views

sum of maxima vs the maximum of the sum

Consider the following integer program $$ \begin{align} \max &\sum\nolimits_{i}\sum\nolimits_{j} U_i(j)\cdot x_{i,j}\\ \text{subject to}& \sum_{i}x_{i,j}\cdot f\left(i,j\right)\leqslant ...
0
votes
1answer
97 views

Collages along composition of distributors

The construction of a collage of two categories $\bf A,B$ along a profunctor $\phi\colon \bf A\mid\hspace{-2mm}\to B$ gives a new category $\bf A \uplus_\phi B$ having as objects those of $\bf A\amalg ...
2
votes
2answers
784 views

Countable Fields with No Countable Extension

Let $\mathscr{S}$ be the set of all countable subfields of $\mathbb{C}$. Clearly, $\mathscr{S}$ is a partially ordered set under inclusion, and if $K_1\subseteq K_2 \subseteq \cdots$ is an ascending ...
8
votes
2answers
478 views

A characterisation of well-ordering ?

It is easy to prove that if $E$ is well-ordered, and if $f$ is a strictly increasing map from $E$ to $E$, then, for all $x$ in $E$, $f(x) \ge x$ (just consider the sequence $x$, $f(x)$, ...
3
votes
0answers
172 views

identity for number of monomials

Fix a positive integer $d \geq 2$, and let $n,k$ be natural numbers with $k \leq n$. Let $b(n,k)$ denote the number of monomials of degree $kd-(n+1)$ in $n+1$ variables $x_0,\ldots,x_n$ with all ...
5
votes
1answer
536 views

Decay of Relative Growth in Conway's Game of Life

Intro The question is about Game of Life. Let us denote the set of points obtained from initial configuration $A$ after $m$ steps as $A(m)$ (we are only interested in finite initial configuration, ...
18
votes
2answers
2k views

Are non-PL manifolds CW-complexes?

Can every topological (not necessarily smooth or PL) manifold be given the structure of a CW complex? I'm pretty sure that the answer is yes. However, I have not managed to find a reference for ...
3
votes
2answers
258 views

Is a representation sphere dualizable inside naive G-spectra?

Let $G$ be a finite group and let $G$ act on a vector space $V$. Let $S(V)$ be the representation sphere. In the monoidal category of $RO(G)$-graded spectra $S(V)$ is invertible, so it's dualizable ...
1
vote
1answer
219 views

crossed product

on Williams Crossed product book,on page 198, it is mentioned that there is only one regular representation for C_c(G), and that is the left regular representation. I know that this representation is ...
1
vote
2answers
203 views

How to prove H^2(g,J(g)) is nonzero for a semisimple Lie algebra g, where J(g) is the augmentation ideal of g?

Suppose g is a fiinte dimensional semisimple lie algebra over a field with characteristic 0. This question is related to Whitehead's second lemma, which says for finite dimensional g-module M, ...
6
votes
1answer
1k views

Non-analytic function with convergent Taylor series everywhere

Is there a smooth function on an interval in $\mathbb R$, not analytic on any subinterval, whose Taylor series at every point has positive radius of convergence? The Fabius function might be an ...
7
votes
1answer
272 views

What is the homotopy type of a free simplicial ring?

Is there a good description of the homotopy type of a free simplicial ring (or simplicial $R$-algebra) on a given simplicial set, in terms of the homotopy type of that simplicial set? (This is mostly ...
9
votes
1answer
683 views

Is there a mathematical explanation for the Aharonov-Casher effect?

Recall that the Aharonov-Bohm effect can be interpreted mathematically as follows. Consider an electromagnetic field A on some smooth manifold M, i.e., A is an element in the first differential ...

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