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Polar coordinates of a set with different radius and angle

Let $M$ be a $2$-dimensional Riemannian manifold and let $U\subset M$ be an open set. Suppose there exist polar coordinates $(r,\theta)$ with center $q\in M$ such that $$U=\lbrace{ (r,\theta): 0<...
Sammyy Delbrin's user avatar
12 votes
3 answers
949 views

Fixed point set of smooth circle action

Suppose $M$ is a connected closed smooth $d$-dimensional manifold, and suppose $S^1 = SO(2)$ acts smoothly on $M$. Then the fixed point set $Y = M^{S^1}$ will be a submanifold of $M$ of even ...
Jens Reinhold's user avatar
2 votes
0 answers
115 views

Some operations on categories - nomenclature question

Suppose that we have a "diagram of categories", i.e. a map from some small category $J$, viewed as a 2-category (with trivial two-morphisms) to the category of categories, $j\mapsto \mathcal{C}_j.$ ...
Dmitry Vaintrob's user avatar
7 votes
1 answer
271 views

A Converse of the Skorokhod Embedding Theorem

I am wondering whether the following "sort of converse" of Skorokhod's embedding theorem holds: Suppose that $\{D_t\}_{t \geq 0}$ is a stochastic process with continuous paths, $D_0 = 0$, and suppose ...
Probabilist's user avatar
0 votes
2 answers
612 views

References on KKT Conditions applied to Calculus of Variations

I am looking for research references for KKT conditions and Slack Variables applied to variational calculus problems in things like Control Theory and such. For example, naively I might try solving a ...
user avatar
6 votes
1 answer
710 views

Relation - Anabelian geometry and Tate conjecture

A lot has been said about the relation between Anabelian algebraic geometry and Mordell conjecture. I would like to know what is the relation between Anabelian algebraic geometry and Tate ...
tttbase's user avatar
  • 1,700
5 votes
1 answer
179 views

The uniqueness of a $K$-fixed vector in a spinor representation

Consider $G=SO(2n)$ and $K=U(n)$. $(G,K)$ is a symmetric pair. I'm interested in (zonal) spherical functions on $G/K$ which are matrix elements with respect to $K$-fixed vectors in irreducible ...
Ninnat Dangniam's user avatar
3 votes
1 answer
738 views

Gauss' Circle Problem at $\left ( \frac{1}{2}, \frac{1}{2} \right ) $

GCP (Gauss' Circle Problem) asks for a closed form for the number of square-lattice points inside a circle, centered at the origin, of radius $r$. Let's denote by $N(r)$ the number of these points. ...
user3141592's user avatar
6 votes
1 answer
772 views

Yoneda extensions in exact categories and their derived categories

If $\mathcal{A}$ is any abelian category, then for all objects $X,Y$ in $\mathcal{A}$ and for all integers $i \geq 0$, there is an natural isomorphism $$\mathrm{Ext}_\mathcal{A}^i(X,Y) \simeq \mathrm{...
Arkandias's user avatar
  • 951
3 votes
1 answer
285 views

Counting Bipartitions

Numerical evidence suggests that $p_2(n) \geq n p(n)$ for large $n$. Here $p(n)$ is the number of partitions of $n$, and $p_2(n)$ is the number of bipartitions of $n$, i.e., ordered pairs of ...
Steven Spallone's user avatar
14 votes
1 answer
676 views

$\mathbb{Z}$-module structure of the subring generated by an algebraic number

Let $a$ and $b$ be algebraic numbers which are not necessarily algebraic integers. Is there some invariant that allows us to determine whether $\mathbb Z[a]$ and $\mathbb Z[b]$ are isomorphic as $\...
user108921's user avatar
22 votes
4 answers
2k views

Anti-large cardinal principles

I'm interested in axioms that prevent the existence of large cardinals. More precisely: (Informal definition) $\Phi$ is an anti-large-cardinal axiom iff $V \models \Phi \Rightarrow V \not \models \...
Neil Barton's user avatar
1 vote
2 answers
552 views

Cross-ratio and projective transformations

Let $P=\{p_1,\ldots,p_6\}\subset\mathbb{P}^1$ be a set of six general points of the projective line. In particular there are no two different subsets $\{p_{i_1},\ldots,p_{i_4}\}$ and $\{p_{j_1},\ldots,...
Puzzled's user avatar
  • 8,842
2 votes
1 answer
85 views

does finite dimensional representations of bialgebras separate elements?

let $B$ be a bialgebra over a field (i.e. associative, coassociative, unitary and counitary, maybe it has an antipode or maybe not). If $b\in B$ acts by zero on every finite dimensional ...
Marco Farinati's user avatar
3 votes
1 answer
198 views

Cartan Formula for Steenrod square on cocycles

Let $x_n,y_n,\cdots$ be cocycles in $Z^n(X,\mathbb{Z}_2)$ (not cohomology classes in $H^n(X,\mathbb{Z}_2)$). Let $Sq^k(x)\equiv x_n \cup_{n-k} x_n$ be the Steenrod square (This definition is valid for ...
Xiao-Gang Wen's user avatar
7 votes
5 answers
1k views

The Hopf invariant is an isomorphism for $\pi_3 (S^2)$

Does any one have a reference to a proof that the Hopf invariant classifies the homotopy classes of maps from $\mathbb{S}^3$ to $\mathbb{S}^2$. It is quite standard to find a proof that the Hopf ...
Jean Van Schaftingen's user avatar
10 votes
1 answer
271 views

Compact Lie group inclusions that are trivial on all homotopy groups

Is there a classification of closed subgroups $H\le SO(n)$ such that the inclusion $H\to SO(n)$ is trivial on all homotopy groups? This happen e.g. when the group $H$ is finite. Are there other ...
Igor Belegradek's user avatar
3 votes
0 answers
36 views

Is there a name for Lie superalgebras which are generated by the odd subspace?

Every Lie superalgebra $\mathfrak{g} = \mathfrak{g}_{\bar 0} \oplus \mathfrak{g}_{\bar 1}$ has a canonical ideal $\mathfrak{k} = [\mathfrak{g}_{\bar 1}, \mathfrak{g}_{\bar 1}] \oplus \mathfrak{g}_{\...
José Figueroa-O'Farrill's user avatar
7 votes
1 answer
446 views

Galois invariant line bundles on a product of varieties

Let $k$ be a field with separable algebraic closure $k^{\rm s}$ and corresponding absolute Galois group $\varGamma={\rm Gal}(k^{\rm s}/\,k)$ and let $X$ and $Y$ be geometrically connected and ...
Cristian D. Gonzalez-Aviles's user avatar
5 votes
0 answers
84 views

Smaller root of a difference of products of polynomials with integer bounded coefficients

Is there a positive constant $K>0$ such that for every polynomials $f_1,\dots,f_4 \in\mathbb{Z}[X]$ with coefficients in {-1,0,1}, every positive root $x$ of the polynomial $$g=f_1f_2-f_3f_4$$ ...
A. Lampadophore's user avatar
14 votes
3 answers
805 views

Undefinability of $\mathbb{Z}$ in the reals

It is a well-known fact that $\mathbb{Z}$ is not definable in the structure $\mathcal{R}=(\mathbb{R}, +, ., < , 0, 1)$. This follows from Tarski's quantifier elimination, and in fact, we can ...
Mohammad Golshani's user avatar
2 votes
1 answer
161 views

Bound of higher rank spherical Whittaker function

I am not much familiar with the literature about upper bound of the spherical Whittaker function on higher rank real groups. Any reference or answer to the following question would be appreciated. Let ...
Subhajit Jana's user avatar
6 votes
3 answers
558 views

Strong (Inverse of) Residue Theorem

Let $C$ be a compact Riemann surfaces of genus $g$. Let $p$ be a point, $\Delta$ a disc around $p$, and $\Delta^*$ the disc minus $p$. Let $\omega$ be a holomorphic one form defined on $\Delta^*$. ...
Giulio's user avatar
  • 2,324
7 votes
0 answers
213 views

Notation: Why Ω for the based loop functor?

This is just a question about notation - probably useless, but it's always baffled me: Why was $\Omega$ chosen to denote the based loop functor? I once heard someone speculate: "It's because $\Omega$...
user316092's user avatar
0 votes
0 answers
183 views

Dirac's theorem and the 1-factorization conjecture

Let $G=(V,E)$ be a simple, undirected graph. A matching is a subset $M\subseteq E$ such that all members of $M$ are pairwise disjoint; moreover we call $M$ perfect if $\bigcup M = V$. The 1-...
Dominic van der Zypen's user avatar
6 votes
2 answers
358 views

Sources for Alexandrov surfaces

There are two distinct notions in differential geometry associated with A. D. Alexandrov: (1) Alexandrov spaces of courvature bounded from below; (2) Alexandrov surfaces of bounded total curvature (...
Mikhail Katz's user avatar
  • 15.1k
3 votes
1 answer
184 views

Finite congruence-free semigroup without zero [closed]

I am reading the book " Fundamentals of semigroup theory" by John M. Howie $\textbf{(Section 3.7)}$. I want to prove $\textbf{Theorem 3.7.2}$ If $S$ is a finite congruence free semigroup ...
Struggler's user avatar
  • 153
1 vote
1 answer
76 views

Variant of Kneser hypergraph with elements appearing more than once

A Kneser hypergraph is a hypergraph with the vertices being the subsets of $M=\{1,2,\dots,m\}$ of size $l$ and the edges being the collections of size $r$ of these subsets such that any two subsets ...
Karo's user avatar
  • 247
0 votes
1 answer
577 views

Fixed point iteration using derivatives

Let $g(x)= \mathrm{\lim_{n \rightarrow \infty}}\,f^{\circ n} (x) $, where $f^{\circ n} (x)$ is the repeated application of $f$ to $x$, $n$ times. Why doesn't the following procedure work for ...
user3433489's user avatar
-1 votes
1 answer
62 views

Minimax solution but game has no value

Fix convex sets $\Delta,\Pi$ and let $r: \Pi \times \Delta \in [0,\infty]$ be linear (i.e., concave and convex) in its first parameter for every fixed second parameter. I'm looking for a situation ...
D.R.'s user avatar
  • 321
4 votes
1 answer
350 views

Values attained by the coheight of $(H \setminus H^\times)^k$ as a function of $H$ and $k$

Edit (Apr 24, 2017). I'm updating this post in the light of the latest developments of a related thread. Let $H$ be a multiplicatively written, commutative monoid, and set $M := H \setminus H^\times$,...
Salvo Tringali's user avatar
1 vote
1 answer
212 views

Decomposition of finitely generated algebras

Let $k$ be a field and $A$ a finitely generated algebra over $k$. I know that if $A$ is finite dimensional as a vector space over $k$, then $A$ can be decomposed as a product of indecomposable $k$-...
Luis Turcio's user avatar
2 votes
2 answers
242 views

$n$-distant permutations more than not

Let $\mathfrak{S}_{2n}$ be the permutation group of the letters $[2n]=\{1,2,\dots,2n\}$. Call a permutation $\pi\in\mathfrak{S}_{2n}$ has an $n$-distant pair if there is some $j\in [2n-1]$ such that $\...
T. Amdeberhan's user avatar
9 votes
2 answers
482 views

Comparing the growth of $f\circ g$ and $g\circ f$

I asked this Question on Math.StackExchange without success. Then I learned, that this might be the better place to ask. So, sorry for crossposting. I would agree on deleting my old question. Let $\...
M. Winter's user avatar
  • 12.5k
9 votes
1 answer
516 views

Canonical models of Shimura varieties for GL2

Let $N \ge 4$ and let $Y_1(N)$ be the complex manifold $\Gamma_1(N) \backslash \mathcal{H}$, where $\Gamma_1(N) \subset \mathrm{SL}_2(\mathbf{Z})$ is the usual congruence subgroup and $\mathcal{H}$ ...
David Loeffler's user avatar
4 votes
0 answers
159 views

sieving in function fields

When we want to find a high rank curve over $\mathbb{Q}$ among an elliptic curve over $\mathbb{Q}(t)$, we give some scores for the curves (the curves with bigger points on them over various primes in ...
user371596's user avatar
6 votes
1 answer
582 views

The spectral radius of a binary matrix

Let $\mathcal B_n$ denote the set of $n\times n$ matrices with entries in $\{0,1\}$. It follows from the spectral radius formula that if $M\in\mathcal B_n$ is not nilpotent, then $\rho(M)$, the ...
Nikita Sidorov's user avatar
5 votes
1 answer
368 views

explicit description of the product map in K-theory

Let $A$ and $B$ be unital $C^*$-algebras. Let $u\in M_n(A)$ be a unitary representing an element $[u]\in K_1(A)$ and $p\in M_m(B)$ be a projection representing an element $[p]\in K_0(B)$. Then the ...
user avatar
2 votes
1 answer
171 views

Generating an arbitrarily long sequence with decreasing Kolmogorov complexity of terms

Is there an algorithm which, given a string $s$, generates a sequence of $|s|$ strings, such that it can be proven in some axiomatic system $S$, that the Kolmogorov complexity of each successive ...
ARi's user avatar
  • 841
6 votes
1 answer
250 views

How to braid a ribbon knot

Is there any algorithm known for braiding ribbon knots? More specifically I need to braid a generic ribbon knot presented as boundary of a ribbon surface= union of two 0-handles and one 1-handle. (...
braid rep's user avatar
  • 143
3 votes
2 answers
161 views

Weak ideal systems $r$ for which the $r$-coheight satisfies a kind of triangle inequality

Let $H$ be a multiplicatively written, commutative monoid with identity $1_H$, and let $\mathcal P(H)$ be the power set of $H$. If $X, Y \subseteq H$, we will set $$XY := \{xy: x \in X,\, y \in Y\}.$$ ...
Salvo Tringali's user avatar
1 vote
0 answers
110 views

Are almost positive functionals close to positive functionals?

This is a bit of an open-ended question... Let $S$ be an operator algebra (or an operator system) and consider a functional $\nu:M\to \mathbb{C}$ that satisfies $$\vert \nu(a)\vert \ge -\varepsilon \...
Lambda's user avatar
  • 19
12 votes
2 answers
677 views

Is the fundamental group of any compact hyperbolic 3-manifold embeddable into a p-adic group?

Is it true that for every compact hyperbolic $3$-manifold $M$ there exists a prime $p$, a finite field extension $K/\mathbb{Q}_p$, and an injective group homomorphism $$\tau \colon \pi_1(M) \to \...
Pablo's user avatar
  • 11.2k
3 votes
3 answers
833 views

Good exposition of "Calabi ansatz"

As far as I understand, Calabi ansatz is (in particular) a way to produce Kähler metrics on total spaces of line bundles (or their disk subbudles) over Kähler manifolds of the following form: Calabi ...
aglearner's user avatar
  • 14k
2 votes
1 answer
150 views

A problem about normal distribution, independent random variables

Suppose $\alpha_1, ..., \alpha_n $ are independent identically distributed random variables, $ a_1, ..., a_n,b_1,...,b_n $ are non-zero constants. Is it true that if $ \sum_{i=1}^{n}a_i\alpha_i $ and $...
Martin Chow's user avatar
6 votes
2 answers
1k views

Proof of the holomorphic Frobenius theorem in Voisin's book on Hodge theory (Theorem 2.26)

I'm trying to understand the proof of the holomorphic version of the Frobenius integrability theorem given in p. 51-52 of Voisin's text "Hodge Theory and Complex Algebraic Geometry I". Statement: ...
Saal Hardali's user avatar
  • 7,549
1 vote
1 answer
94 views

Quantitative unique continuation for entire functions

Let $\widehat{f}$ be the Fourier transform of $f$ defined by $$ \widehat{f}(\xi) = \int_{\textbf{R}}e^{2\pi i x \xi}f(x)dx, \quad f\in L^2(\textbf{R}). $$ Define a set $$ E:= \{f\in L^2(\textbf{R}): ...
Wang Ming's user avatar
  • 415
15 votes
0 answers
385 views

References on Discrete field theory vs Discrete differential geometry vs Combinatorial topology

Let me ask several related questions on discretization of classical field theory: In topological folklore, it is known that cochains are "discrete analogues" of differential forms, and coboundary ...
Mikhail Skopenkov's user avatar
5 votes
0 answers
103 views

On the embedding of manifolds into infinite-dimensional spaces

Let $X$ be a (connected, finitely dimensional) topological/smooth/complex manifold and let $i$ be a weakly continuous/continuous/smooth/holomorphic map from $X$ into the dual $F^{*}$ of a real or ...
erz's user avatar
  • 5,385
5 votes
1 answer
270 views

Question about Fourier coefficients of a newform at primes

For $q:=e^{2\pi i z},$ let $f(z)=\sum_{n\ge 1}\lambda(n)n^{(k-1)/2}q^n$ be a normalized newform of type $(k,\chi)$ and level $N$. For any prime $p,$ we have $$\lambda(p)=2\cos(\theta_p)\;\;\;\text{...
square-free's user avatar

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