All Questions
152,897
questions
1
vote
1
answer
98
views
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<...
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 ...
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.$ ...
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 ...
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 ...
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 ...
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 ...
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. ...
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{...
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 ...
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 $\...
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 \...
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,...
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 ...
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 ...
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 ...
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 ...
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}_{\...
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 ...
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
$$g=f_1f_2-f_3f_4$$
...
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 ...
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 ...
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 $\Delta^*$.
...
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 $\Omega$...
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-...
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 (...
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 ...
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 ...
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 ...
-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 ...
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 $M := H \setminus H^\times$,...
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 $k$-...
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 $\...
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 $\...
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 $\mathcal{H}$ ...
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 ...
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 ...
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 ...
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 ...
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. (...
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 $$XY := \{xy: x \in X,\, y \in Y\}.$$
...
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
$$\vert \nu(a)\vert \ge -\varepsilon \...
12
votes
2
answers
677
views
Is the fundamental group of any compact hyperbolic 3-manifold embeddable into a p-adic group?
Is it true that for every compact hyperbolic $3$-manifold $M$ there exists a prime $p$, a finite field extension $K/\mathbb{Q}_p$, and an injective group homomorphism $$\tau \colon \pi_1(M) \to \...
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 ...
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 $...
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: ...
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
$$
E:= \{f\in L^2(\textbf{R}): ...
15
votes
0
answers
385
views
References on Discrete field theory vs Discrete differential geometry vs Combinatorial topology
Let me ask several related questions on discretization of classical field theory:
In topological folklore, it is known that cochains are "discrete analogues" of differential forms, and coboundary ...
5
votes
0
answers
103
views
On the embedding of manifolds into infinite-dimensional spaces
Let $X$ be a (connected, finitely dimensional) topological/smooth/complex manifold and let $i$ be a weakly continuous/continuous/smooth/holomorphic map from $X$ into the dual $F^{*}$ of a real or ...
5
votes
1
answer
270
views
Question about Fourier coefficients of a newform at primes
For $q:=e^{2\pi i z},$ let $f(z)=\sum_{n\ge 1}\lambda(n)n^{(k-1)/2}q^n$ be a normalized newform of type $(k,\chi)$ and level $N$. For any prime $p,$ we have
$$\lambda(p)=2\cos(\theta_p)\;\;\;\text{...