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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 ...
RaphaelB4's user avatar
  • 4,296
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 ...
James Cheung's user avatar
  • 1,855
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 ...
khallouq abdelmounaim's user avatar
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 ...
MathDG's user avatar
  • 242
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 ...
user avatar
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 $D$ ...
Vikas's user avatar
  • 11
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 ...
Najib Idrissi's user avatar
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 $\...
GGT's user avatar
  • 685
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 ...
Jose Arnaldo Bebita Dris's user avatar
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 $\pi$ ...
Vladimir Kanovei's user avatar
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_{...
dohmatob's user avatar
  • 6,716
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}...
user129021's user avatar
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\...
Christian Chapman's user avatar
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 ...
Gus's user avatar
  • 85
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 ...
Jonathan Gallagher's user avatar
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. ...
tomasz's user avatar
  • 1,214
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) ...
corollary's user avatar
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 ...
Ken's user avatar
  • 397
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 ...
Sam Hopkins's user avatar
  • 22.7k
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 ...
burtonpeterj's user avatar
  • 1,689
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 ...
Wojowu's user avatar
  • 27.4k
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 ...
Dave Shulman's user avatar
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 ...
Unstandard Candle's user avatar
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 ...
user128470's user avatar
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) \...
user09127's user avatar
  • 765
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 ...
Vít Tuček's user avatar
  • 8,159
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 \...
Onil90's user avatar
  • 823
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 ...
Chen's user avatar
  • 1,583
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{...
john_b's user avatar
  • 165
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 ...
Simon Henry's user avatar
  • 39.9k
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 $R$...
Fred Rohrer's user avatar
  • 6,660
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 ...
Sirolf's user avatar
  • 493
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 ...
GGT's user avatar
  • 685
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"...
Dave Pritchard's user avatar
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 ...
Turbo's user avatar
  • 13.7k
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 ...
Dominic van der Zypen's user avatar
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 ...
user avatar
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 ...
ililiil's user avatar
  • 661
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 "...
Nik Weaver's user avatar
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 ...
user128943's user avatar
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 ...
Masoud's user avatar
  • 99
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 ...
Tsein32's user avatar
  • 343
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(...
Turbo's user avatar
  • 13.7k
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 ...
Daniel Loughran's user avatar
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 ...
S. carmeli's user avatar
  • 4,064
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 $...
BrianT's user avatar
  • 1,197
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 $\...
user89236's user avatar
  • 101
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_{...
Gavroche's user avatar
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 ...
B Merlot's user avatar
  • 269
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 ...
Reb's user avatar
  • 243

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