All Questions
152,887
questions
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.,
$$ f(x) + \langle\nabla f(x), y- x\rangle + \frac{\alpha}{2}\|y-x\|^2
\leq f(y) \leq f(x) ...
8
votes
3
answers
360
views
monochromatic subset
Suppose we have $n^2$ red points and $n(n-1)$ blue points in the plane in general position. Is it possible to find a subset $S$ of red points such that the convex hull of $S$ does not contain any blue ...
6
votes
1
answer
197
views
How big can the index inside the root lattice of the lattice generated by a subset of roots be?
Let $\Phi$ be an irreducible crystallographic root system in a Euclidean vector space $V$. Let $S\subseteq \Phi$ be some subset of roots for which $\mathrm{Span}_{\mathbb{R}}(S)=V$.
Question: How big ...
1
vote
0
answers
215
views
Tensor product decomposition of commuting representations
If $\mathscr{X}$ is a Hilbert space, we denote by $\mathrm{GL}(\mathscr{X})$ the group of all bounded operators on $\mathscr{X}$ with bounded inverses. Let $\mathbb{F}_2$ be the free group on two ...
8
votes
0
answers
294
views
"Complementarity" between homotopy and cohomology [duplicate]
I was browsing MO and I have stumbled upon this answer which discusses why we should expect homotopy groups of spheres to be complicated. One heuristic argument given is that "the theory needs to ...
6
votes
1
answer
982
views
Unbounded version of continuous functional calculus
For a normal operator $T$ on a Hilbert space ${\cal H}$, it is well known that for any continuous complex valued function $f$ on the spectrum of $T$, we have a well-defined operator $f(T) \in B({\cal ...
3
votes
1
answer
113
views
Combinatorial problem about binary arrays with certain mutual distinctions
If there are m binary arrays (with 0 and 1) of length n, and between any two of these m arrays, there are k and only k same numbers (with the same site index in two different arrays). For example, if ...
4
votes
2
answers
171
views
Monotonicity of infimum of the Willmore energy with prescribed genus
Let
$$
\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 ...
1
vote
1
answer
65
views
Existence of bisolutions for wave operators on globally hyperbolic Lorentzian manifolds
Let $(M,g)$ be a globally hyperbolic Lorentzian manifold with Lorentzian volume density $dV$ and $\Sigma$ a Cauchy hypersurface, i.e. each inextendible causal curve hits $\Sigma$ exactly once. ...
12
votes
0
answers
263
views
If two group actions lead to the same orbifold, are they conjugate?
In The Geometry and Topology of Three-Manifolds, Thurston says: "In these examples, it was not hard to construct the quotient space from the group action. In order to go in the opposite direction, we ...
6
votes
0
answers
151
views
$\mathbb{F}_q$-rational elements in unipotent classes of a finite group of Lie-type
I've tried posting this question on MSE, but didn't manage to get an answer there, so I'm trying again here. Sorry in advance if this question is trivial or trivially false. I haven't managed to find ...
4
votes
1
answer
138
views
Asymptotic colouring of edges and vertices, and untwisting cocycles
This question regards colourings on edges and vertices on countable directed multigraphs.
We start with an example. Let $G=\mathbb Z^2$. We define two functions $a_h$ and $a_v$ from $\mathbb Z^2$ to $\...
2
votes
0
answers
514
views
Convergence Based on Recurrence Relation
I am studying a sequence based on the following recurrence:
$$X[t] = \sqrt{\alpha X[t-1]^2+(X[t-1]^2-\alpha X[t-2]^2)\frac{(2-X[t-1])^2}{X[t-1]^2}}$$
$$X[0]=0$$
$$X[1]>0$$
$$\alpha \in (0,1)$$
I ...
7
votes
2
answers
411
views
On the Fourier-Laplace transform of compactly supported distributions
Let $\mathcal{E}'(\mathbb{R})$ be the space of all compactly supported distributions on $\mathbb{R}$.
For $f\in \mathcal{E}'(\mathbb{R})$, let $\widehat{f}$ denote the entire extension of the ...
10
votes
0
answers
218
views
Colimits of algebras for $\infty$-Monad
I would like to know in anyone has developed method for constructing colimits in the category of algebra for a monad in the $(\infty,1)$-categorical framework, using transfinite constructions.
I have ...
1
vote
1
answer
142
views
Optimal estimate in trace norm
Let $x,y$ be vectors of some Hilbert space of unit length.
Then we can consider the projection $P_x:=\langle \bullet, x \rangle x$ and similarly $P_y.$
Assume then that we know that $\left\lVert x-...
2
votes
0
answers
198
views
An example of non trivial projections in a group von Neumann algebra
Let $G$ and $\text{vN}(G)$ be a torsion free group and its group von Neumann algebra. Is there a characterization of non trivial projections in $\text{vN}(G)$? If not, is a certain class of them ...
1
vote
0
answers
49
views
Representability of smooth invertible Lipschitz functions by a finite composition of near-identity functions
Theorem 1 of this paper shows that
For every positve integer $K$ and for every nonempty bounded domain $\mathcal X \subseteq \mathbb R^d$, the restriction on $\mathcal X$ of any function $h: \...
1
vote
0
answers
119
views
Completed Tensorproduct
I am trying to understand the completed tensorproduct. This can be defined as follows:
Given a topological ring $R$ and two linearly topologized rings $A$ and $B$ with fundamental systems of open ...
1
vote
1
answer
85
views
monochromatic induced subgraph in a complete 3-partite graph
$H=(V_1\cup V_2 \cup V_3, E)$ is a complete $3$ partite graph such that $|V_1|=|V_2|=|V_3|=n$ . Color the edges with three colors.
My question is: Is it possible to find sets $V_1' \subset V_1, V_2' ...
4
votes
1
answer
838
views
Evans-Krylov theorem
Do there exist estimates for nonconcave functionals similar to Evans-Krylov theorem in chapter 6 of Fully nonlinear elliptic equations by Luis A.C affarelli and Cabre? Perhaps there is a ...
5
votes
2
answers
193
views
$k$-invariants as extension classes
Let $X$ be a connected cell complex with fundamental group $G$
and $(n-1)$-connected universal covering space. Let $\Pi=\pi_n(X)$.
We may construct a $K(G,1)$ complex $K$ by adjoining cells of ...
5
votes
2
answers
861
views
Searching for a proof for a series identity
The below identity I have found experimentally.
Question. Is this true? If so, may you provide a "slick" (or any) proof.
$$6\sum_{k=1}^{\infty}\frac{k^2q^k}{(1-q^k)^2}+12\left(\sum_{k=1}^{\infty}...