All Questions
152,902
questions
3
votes
1
answer
108
views
Cut on hypersurfaces and angular defects
I like very much the elementary property that if one cuts a geodesic triangle onto a sphere (one can use 3 plans that contain $0$). The cut surface of the sphere is given by the sum of the angles of ...
2
votes
2
answers
589
views
About block of category $\mathcal{O}$
In the book "Representation of Semisimple Lie Algebra in the BGG Category $\mathcal{O}$".
Exercise 1.13. Suppose $\lambda\not\in\Lambda$, so the linkage class $W\cdot\lambda$ is the disjoint union of ...
1
vote
1
answer
171
views
Linear matrix inequality
I have the following linear matrix inequality:
$$F^T P + PF < 0,$$
where $P$ is a positive definite matrix and $F$ is a matrix with appropriate dimension.
Let $Q$ be a positive definite matrix ...
6
votes
2
answers
1k
views
Manifolds with negative dimension – Definition, References
Does the concept of differential manifold with negative dimension make sense, in differential geometry?
If yes, how is it defined? Do you have any reference to recommend?
My problem was born in ...
2
votes
1
answer
1k
views
JSJ decomposition and classification of 3-manifolds
I need some philosophical explanation for JSJ decomposition theorem. It says that closed orientable irreducible 3-manifold can be cut along set of incompressible tori onto pieces which are:
atoroidal ...
1
vote
0
answers
118
views
Dense versus sequentially dense in $\mathcal{E}’$
Endow the dual space $\mathcal{E}’$ of smooth functions $C^\infty$ (with its metrizable topology described by uniform convergence on compacts for convergent sequences) with the weak* topology. Let $D$ ...
3
votes
1
answer
265
views
If $A$ is a (shifted) Poisson algebra, what does $A[\varepsilon]$ represent?
I have a question which is not really precise, unfortunately.
Let $A$ be a Poisson $n$-algebra, i.e. a graded commutative algebra with a Lie bracket of degree $n-1$ s.t. the bracket is a biderivation ...
5
votes
2
answers
404
views
Frobenius coordinate expansion of character
Let $\lambda$ be the partition of integer $d$. The Frobenius coordinate of $\lambda$ is given
$$ (a_1 ,\ldots,a_{d(\lambda)}|b_1,\ldots,b_{d(\lambda)}),$$
where $d(\lambda)$ denote the diagonal of $\...
3
votes
4
answers
1k
views
A conjecture regarding odd perfect numbers
(Note: I asked this question in MSE this June 2018 but did not receive any responses there. I have therefore cross-posted it here, hoping that it gets answered.)
Let $\sigma(z)$ denote the sum of ...
5
votes
0
answers
251
views
Bad forcing permutations
Let $P$ be the finite-support product of the Cohen forcing. It adjoins a sequence of Cohen-generic reals say $a_n$, $n<\omega$, which one naturally calls a $P$-generic sequence. Suppose that $\pi$ ...
2
votes
1
answer
2k
views
Closed-form formula for Wasserstein distance between uniform discrete distribution and discrete distribution with same support
Let $x_1,\ldots,x_n$ be $n \ge 1$ distinct points in $\mathbb R^d$ and consider two discrete distributions on these points $\mu = (1/n)\sum_{i=1}^n\delta_{x_i}$, and $\nu = \sum_{i=1}^n\nu_i\delta_{...
7
votes
1
answer
510
views
The probability that two elements of a finite nonabelian simple group commute
It is mentioned in here (last paragraph of the first page) that Dixon proved the following result: the probability that two elements of a finite nonabelian simple group commute is at most $\frac{1}{12}...
3
votes
0
answers
185
views
Lattice points in a rotated product-of-balls
Fix $U$ unitary over $\mathbb{R}^{K},$ take $U_n=I_{n\times n}\otimes U$ and denote the unit ball at 0 in $\mathbb{R}^n$ as $B^n$. For $d_1,\dots,d_K>0$, fix $S_n:=U_n\left(\prod_{k=1}^K d_k B^n\...
5
votes
2
answers
326
views
Codimension-1 subgroups of 3-manifold groups
Let $G$ be a finitely generated group and let $H$ be a subgroup of $G$. $H$ is a codimension-1 subgroup of $G$ if $C_{G}/H$ has more than one end, where $C_{G}$ is the Cayley graph of $G$.
Do all ...
16
votes
1
answer
561
views
When is the category of models of a limit theory a topos?
If $\mathcal{E}$ is a Grothendieck topos on a small base, then it is locally presentable, and hence is equivalent to the category of models of some limit theory.
Is there a characterization of ...
5
votes
0
answers
140
views
Self-additive posets
We say that a partially ordered set $(P,<)$ is self-additive if the two natural embeddings of $P$ in $P\oplus P$ (the linear sum of $P$ and itself) are elementary.
We have the following.
...
3
votes
0
answers
182
views
Level sets of strongly convex and smooth functions
Let $f: \mathbb{R}^N \to \mathbb{R}$ be a $\alpha$-strongly convex and $\beta$-strongly smooth function, i.e.,
$$ f(x) + \langle\nabla f(x), y- x\rangle + \frac{\alpha}{2}\|y-x\|^2
\leq f(y) \leq f(x) ...
8
votes
3
answers
360
views
monochromatic subset
Suppose we have $n^2$ red points and $n(n-1)$ blue points in the plane in general position. Is it possible to find a subset $S$ of red points such that the convex hull of $S$ does not contain any blue ...
6
votes
1
answer
197
views
How big can the index inside the root lattice of the lattice generated by a subset of roots be?
Let $\Phi$ be an irreducible crystallographic root system in a Euclidean vector space $V$. Let $S\subseteq \Phi$ be some subset of roots for which $\mathrm{Span}_{\mathbb{R}}(S)=V$.
Question: How big ...
1
vote
0
answers
215
views
Tensor product decomposition of commuting representations
If $\mathscr{X}$ is a Hilbert space, we denote by $\mathrm{GL}(\mathscr{X})$ the group of all bounded operators on $\mathscr{X}$ with bounded inverses. Let $\mathbb{F}_2$ be the free group on two ...
8
votes
0
answers
294
views
"Complementarity" between homotopy and cohomology [duplicate]
I was browsing MO and I have stumbled upon this answer which discusses why we should expect homotopy groups of spheres to be complicated. One heuristic argument given is that "the theory needs to ...
6
votes
1
answer
983
views
Unbounded version of continuous functional calculus
For a normal operator $T$ on a Hilbert space ${\cal H}$, it is well known that for any continuous complex valued function $f$ on the spectrum of $T$, we have a well-defined operator $f(T) \in B({\cal ...
3
votes
1
answer
113
views
Combinatorial problem about binary arrays with certain mutual distinctions
If there are m binary arrays (with 0 and 1) of length n, and between any two of these m arrays, there are k and only k same numbers (with the same site index in two different arrays). For example, if ...
4
votes
2
answers
171
views
Monotonicity of infimum of the Willmore energy with prescribed genus
Let
$$
\beta_g:=\inf\{\frac14\int_\Sigma H^2 d\mu \hspace{0.2cm} | \hspace{0.2cm} \Sigma\subset \mathbb
R^{3}, \operatorname{genus}(\Sigma)=g \}
$$
be the infimum of the Willmore energy of embedded ...
1
vote
0
answers
49
views
Proving a property of tame spectra
Let $U$ be a universe. Let $D$ be tame spectra, and let $f:E_1\rightarrow E_2$ be a map of spectra that is a spacewise homotopy equivalence. It is supposed to hold that $f^*: h\mathcal{S}U(D,E_1) \...
7
votes
1
answer
726
views
higher Casimirs for $\mathfrak{sl}$
The Wikipedia universal enveloping algebra suggest a way to obtain higher Casimir operators (e.g. generators of the center of $\mathfrak{U(g)}$ for $\mathfrak{g}$ semisimple) by evaluating certain ...
1
vote
0
answers
68
views
Nodal domains on a surface
What is it known about the topology of nodal domains of eigenfunctions of self-adjoint operators?
In particular I'm interested in self-adjoint operators on a complete, non-compact, surface $\Sigma \...
8
votes
1
answer
438
views
Is an open subscheme of a rationally connected variety, rationally connected?
Let $X$ be a projective, irreducible variety over an algebraically closed field (of characteristic zero) which is rationally connected. Is it true that any open dense subvariety of $X$ is rationally ...
2
votes
1
answer
157
views
Tail condition (Varadhan's lemma)
I would like your help with the following tail condition, which arises in the theory of large deviations.
Let $P(\mathbb{R}^{d})$ the space of probability measures on $\mathbb{R}^{d}$, $ G:P(\mathbb{...
11
votes
0
answers
154
views
Known obstruction for efficient computation of Stable homotopy groups?
Computation of stable homotopy groups (for example of sphere) is hard, but still, not as hard as unstable ones.
For unstable homotopy groups there are some results showing that there cannot be ...
3
votes
1
answer
910
views
Local ring of infinite dimension
Short version: Let $R$ be a commutative ring such that all chains of primes of $R$ with the same extremities have the same finite cardinality. Is $R$ locally finite-dimensional?
Longer version: Let $R$...
5
votes
1
answer
202
views
Two graphs with the same number of walks but without a common equitable partition
Consider two undirected graphs $G$ and $H$ of the same order (same number of vertices).
If $G$ and $H$ have a common equitable partition, then it is known (see e.g., Chapter 6 in 1) that these ...
2
votes
0
answers
95
views
Diagonal operator and infinite wedge space formalism
Let $\bigwedge^{\infty /2}V$ denote semiinfinte wedge space. The followin article section 2 gives a good description about the space and the operator on it.
https://arxiv.org/pdf/math/0207233.pdf
...
18
votes
1
answer
692
views
What non-standard model of arithmetic does Hofstadter reference in GEB?
Following some of the coolest bits of Hofstadter's Gödel, Escher, Bach, extensions of the standard model of arithmetic are described. A ways in, the paragraph "Supernatural Addition and Multiplication"...
4
votes
1
answer
269
views
Resultants for compactly represented product form polynomials?
Typically computing resultnt of $n+1$ different $n+1$-variate homogeneous polynomials takes $O(poly(\prod_{i=1}^{d_{i}}))$ time where $d_i$ is degree of $i$th polynomial. In certain cases the ...
23
votes
4
answers
2k
views
Are infinite groups in which most elements have order $\leq 2$ commutative?
The starting point of this question is the following:
If $G$ is a group such that all elements have order at most $2$, then $G$ is commutative.
If $G$ is any group, let $G_{>2}$ denote the set ...
16
votes
2
answers
476
views
What is the largest known Dehn function of f.p. subgroup of a f.p. group with quadratic Dehn function?
Is it true that the Baumslag-Solitar groups, say, $BS(1,n)$, $|n|\ge 2$, are finitely presented groups with largest Dehn functions (namely, exponential growth) known to be inside finitely presented ...
2
votes
0
answers
124
views
About newforms of half-integral weight
(Sorry for my poor english..)
I have some questions about newforms of half-integral weight. In Mao's paper ("A generalized Shimura correspondence for newforms"), he said: "Ueda defined the set of ...
6
votes
2
answers
711
views
Gauge integral versus path integral
According to this paper, "The gauge integral [a.k.a. Henstock-Kurzweil integral] provides the only formal framework that is close to the original development of the Feynman path integral", and also "...
3
votes
1
answer
320
views
improvement of flatness in the regularity of minimal surfaces
Recently,I am reading Savin's celebrated theorem about improvement of flatness in proving the regularity of minimal surfaces. I have some questions.
1.How to show the boundary of a minimal set ...
1
vote
4
answers
903
views
PDE with Laplacian and squared of the gradient
Let $u$ be a real function in $\mathbb{R}^2$. Does anybody know that the following PDE
$$\Delta u+|\nabla u|^2=0$$
has any non-constant general solution or not? It would be appreciated if any one ...
16
votes
2
answers
1k
views
Are mapping class groups of orientable surfaces good in the sense of Serre?
A group G is called ‘good’ if the canonical map $G\to\hat{G}$ to the profinite completion induces isomorphisms $H^i(\hat{G},M)\to H^i(G,M)$ for any finite $G$-module $M$. I’ve had multiple academics ...
1
vote
0
answers
47
views
Does a weighted sequence derived from a sequence matching the Siegel lemma BV bound behave as an uniformly random sequence?
Pick integers $r_1',\dots,r_t'$ that achieve the Bombieri Vaaler bound for the Siegel lemma and find an $m$ (it exists by Dirichlet Pigeonhole) such that given prime $T$ gets $m(r_1',\dots,r_t')\equiv(...
11
votes
1
answer
770
views
Cyclic cubic extensions and Kummer theory
The Galois cohomology group $H^1(\mathbb{Q}, \mathbb{Z}/3\mathbb{Z})$ classifies cyclic cubic extensions $K/\mathbb{Q}$ (specifically: the non-trivial elements correspond to Galois cubic field ...
3
votes
0
answers
197
views
derived symmetric powers of an ideal
Let $R$ be the polynomial algebra in $n$ variables over a field $F$ of characteristic $0$. Let $m$ be the ideal of the origin: $m=(x_1,...,x_n)$.
We have a canonical map $Lsym^k(m)\to m^k$ from the ...
0
votes
0
answers
86
views
On the dimension of the cohomology of toric manifolds
Let $M$ be a toric manifold. I'm not sure what conditions on $M$ are required, but one can assume, if needed, that it is compact, smooth, etc. We consider $M$ as a quotient given by the momentum map $...
1
vote
1
answer
251
views
Under What assumptions on $p$, $\mathcal{O}_K^* \simeq \mathbb{Z}_p^{*} \oplus \mathbb{Z}_p^{*}$
Let $p$ be a fixed prime number and $\mathbb{Q}_p$ be the field of $p$-adic numbers and $K$ be an extension of degree $2$ of $\mathbb{Q}_p$. Let $\mathcal{O}_K$ be the ring of integers of $K$ and $\...
1
vote
2
answers
324
views
Use Importance sampling for multimodal and multivariate distribution draws, how to choose proposal distribution?
I'm in trouble trying to generate samples following a particular distribution which is not numerically known perfectly. Let us consider a $R^n$ space provided with an orthonormal base $( e_{1},...,e_{...
4
votes
1
answer
836
views
Which inner products preserve positive correlation?
Suppose we have a symmetric PD or PSD matrix M which induces an inner product $\langle \cdot, \cdot \rangle_M$. If we have that $\langle x, y \rangle > 0$ for two unit vectors $x$, $y$, are there ...
4
votes
1
answer
584
views
Computing relative cohomology class of differential form
When dealing with a top degree differential form $\mu$ in a manifold $M$, a way of "computing" its cohomology class is integrating it through the whole manifold. For instance, if the integral $ \int_M ...