Newest Questions
159,052 questions
6
votes
0
answers
239
views
Generalization of $H^*(\Gamma; \mathbb{Z\Gamma}) \cong H^*_c(X; \mathbb{Z})$?
Let $\Gamma$ be a group acting freely and cocompactly on an acyclic space $X$. I know that there is a isomorphism $H^*(\Gamma, \mathbb{Z\Gamma}) \cong H^*_c(X; \mathbb{Z})$ of $\Gamma$-modules. The ...
- 169
1
vote
1
answer
142
views
Foliation of spaces
It turns out that a very important idea to derive properties for a bigger space is to try to foliate the space, derive the same property for each leaf and patch everything up to get the desired ...
- 203
2
votes
0
answers
89
views
The space of immersions of a loop in a surface
Let $\Sigma$ be a compact oriented surface with boundary and $L = \mathrm{Imm}(\bigsqcup_{i=1}^n S^1,\Sigma)$ the space of all generic (i.e. transversally and at most doubly intersecting) immersions ...
- 985
6
votes
1
answer
362
views
A simple way to bound the density of sums of two odd squares
Define
$$S(x) ~=~ \# \left\{ n^2+m^2\leq x : n,m\in\mathbb{N}\right\}$$
Landau (1908) proved that with
$$ B(x) ~=~ K\,\frac{x}{ \sqrt{\log x}} ~~\text{ one has}~~~ \lim \limits_{x\to \infty} \frac{S(...
- 1,676
8
votes
3
answers
284
views
Generation of $\mathrm{SO}(n,\mathbb{Q})$ by coordinate subgroups
$\DeclareMathOperator\SO{SO}\SO(n,\mathbb{Q})$ is the group of $n\times n$ matrices $A$ with rational entries such that $AA^t=I$ and $\hbox{det}(A)=1$.
The $n$ coordinate subgroups of $\SO(n,\mathbb{Q}...
- 3,451
4
votes
0
answers
175
views
Intrinsic maps between complex integers modulo $p$ and integers modulo $p+2$
$\DeclareMathOperator\GF{GF}$Let $p$ and $p+2$ be twin primes. Let's assume that $-1$ is not a quadratic residue modulo $p$ (and therefore is a Q.R. modulo $p+2$).
Consider the complex numbers $a+bi$ ...
- 591
0
votes
1
answer
337
views
Schemes with open generic point
Let $X$ be any irreducible scheme with the property that the generic point $\eta$ of $X$ is an set open wrt underlying Zariski topology.
What can we say about the structure of such schemes? ...
- 6,048
1
vote
0
answers
89
views
Help understanding this proof of asymptotic bounds on solutions
In Boukanjime et al. " https://www.sciencedirect.com/science/article/pii/S0005109821004039 " I'm having difficulties understanding the proof of Theorem 5.1 after equation (13). It's a ...
- 121
5
votes
1
answer
328
views
Is the inclusion of the maximal torus in a simply connected compact Lie group null-homotopic?
Let $G$ be a simply connected compact Lie group and $T$ its maximal torus with inclusion $i:T \hookrightarrow G$.
By simply connectedness of the group $G$ and asphericity of the torus $T$, the induced ...
- 53
2
votes
2
answers
285
views
Dimension of Zariski closure of a locally closed subscheme
Let $S$ be a Dedekind scheme with function field $K=K(S)$ and $C$ a projective regular curve over $K$, so we can fix certain closed embedding $e:C \subset \mathbb{P}^n_K$.
Let compose this embedding ...
- 6,048
1
vote
1
answer
69
views
Local completion of bornological space
I recently stumbled across this old publication which defines the local completion of a Hausdorff locally convex space.
The construction is as follows:
A Hausdorff locally convex space $E$ is locally ...
- 661
10
votes
2
answers
1k
views
Size of maximal intersecting families
Let $X$ be a non-empty set, and let ${\cal S}\subseteq {\cal P}(X)$ be family of non-empty subsets of $X$. We say that ${\cal S}$ is intersecting if any two members of ${\cal S}$ have non-empty ...
- 48.9k
3
votes
2
answers
285
views
Cut a homotopy in two via a "frontier"
Consider a space $G$ obtained by glueing two disjoint cobordisms (the fact that they are might be irrelevant, assume they are topological spaces at first) $L$ and $R$ on a common boundary $C$.
(...
1
vote
1
answer
151
views
$(2^a-1)+b^2=2^c$ [closed]
$31+15^2=256$.
Are there infinitely many solutions to:
$(2^a-1)+b^2=2^c$ with a,b,c positive integer and a,b,c different each other.
- 11
7
votes
1
answer
1k
views
Can a field have an irreducible polynomial of any degree?
We all know that all the irreducible polynomials in $\mathbb{C}[x]$ are linear and in $\mathbb{R}[x]$ they aren't more than 2 degree. However,in $\mathbb{Q}[x]$ we can have an irreducible polynomial ...
- 515
2
votes
1
answer
246
views
Examples of non-discrete, cocompact subgroups
I am looking for non-trivial examples of the following:
$G$ is a locally compact group
$H\subset G$ a closed subgroup
Both are unimodular and non-discrete
The quotient space $G/H$ is compact, but $G$ ...
user473423
1
vote
0
answers
32
views
Any papers on the Lambek graph-$\lambda$ calculus-adjunction and the semantics of the Hindley Milner type system?
Joachim Lambek has described an adjunction between the category of graphs and the category of positive intuitionistic calculi with iteration, see e. g. Introduction to Higher Order Categorical Logic ...
5
votes
2
answers
350
views
How expressive is $e^A$ in the sense of universal approximation?
For any real matrix $B\in\mathbb{R}^{n\times n},n\ge 2$ and precision $\varepsilon$, is there a real matrix $A\in\mathbb{R}^{n\times n}$ such that $\|e^A-B\|_F<\varepsilon$? ($F$ refers to ...
- 109
1
vote
1
answer
218
views
Perturbation of matrices
Let $A(t)$ be a symmetric $n\times n$ matrix that continuously depend on $t\in [0,1]$. Let $\lambda_1(t)$ stand for the smallest eigenvalue for $A(t)$.
Question. Does there exist a Lebesgue measurable ...
- 4,115
9
votes
1
answer
498
views
Is there a shape-independent definition of (∞,1)-categories?
For all definitions of $\infty$-categories I am aware of, an $(\infty,1)$-category is defined via reference to some shape, be it simplices in a form of a quasi-category or a cubical analogue of a ...
- 91
3
votes
0
answers
100
views
Conjugate actions and isomorphic Zappa–Szép products
Let $A$ and $G$ be two cyclic groups. Let $\alpha:G\rightarrow\operatorname{Bij}(A)$ and $\beta: A\rightarrow\operatorname{Bij}(G)$ be two group homomorphisms satisfying some conditions given in ...
- 393
1
vote
1
answer
58
views
Possibility of bounding one functional by another functional (under certain constraints)
Suppose that we consider a class of $L^2(\mathbb{R}_+)$ functions $h$ such that $h$ can be expressed as a difference of two cumulative distribution functions $F$ and $G$ (whose corresponding densities ...
- 730
1
vote
0
answers
263
views
An algebraic stack is an algebraic space if and only if it has the trivial stabilizer group
Let $G\to S$ be a smooth affine group scheme over a scheme. Let $U$ be a scheme over $S$ with an action of $G$. Let $[U/G]$ be the quotient stack.
In Alper's note: Stacks and Moduli, there is a result ...
- 11
5
votes
0
answers
76
views
Is the choosability/list chromatic number of a circular arc graph equal to its chromatic number?
In 2003, Prowse and Woodall proved that for graphs $C_n^k$ which are powers of cycles,
$$\chi_\ell(C_n^k) = \chi(C_n^k).$$
They conjectured that this equality holds for the broader class of graphs ...
- 151
2
votes
1
answer
133
views
Can convergence in distribution necessarily be realised by almost-sure convergence?
Let $X$ be a Polish space. Let $(\mu_n)_{n \in \mathbb{N} \cup \{\infty\}}$ be a family of Borel probability measures $\mu_n$ on $X$ such that $\mu_n \to \mu_\infty$ weakly as $n \to \infty$. For each ...
- 708
1
vote
0
answers
59
views
A certain subalgebra of $\mathfrak{e}_8$ over a p-adic field
Can $\mathfrak{e}_8(\mathrm{k})$ have a maximal subalgebra isomorphic to $\mathfrak{sl}_1(\mathrm{D})\oplus\mathfrak{g}_2(\mathrm{K})$, where $\mathrm{k}$ is a finite extension of some $\mathbb{Q}_p$, ...
- 2,773
3
votes
1
answer
148
views
Homotopy groups of $K(n)$-local $E_n$-modules are $L$-complete
Let $E_n$ be the $n$-th Morava $E$-theory and let $K(n)$ denote the $n$-th Morava $K$-theory.
Question: If $M$ is a $K(n)$-local $E_n$-module, then are the homotopy groups $\pi_*(M)$ $L$-complete? (...
- 177
0
votes
0
answers
140
views
Bounding the difference of weighted-mean estimations
I recently asked a question related to a proof I'm working on as well as a follow-up to that. The detailed counter-examples I received elucidated a lot, but it quickly transpired (i) that I was ...
- 15
5
votes
1
answer
512
views
Norm inequality for the inclusion $L^2(\partial \Omega)\hookrightarrow H^{-1/2}(\partial \Omega)$
Let $\Omega \subset \mathbb{R}^3$ be a lipschitz domain. We then have the trace operator $\tau : H^1(\Omega) \to L^2(\partial \Omega)$ and can define the space $H^{1/2}(\partial \Omega) := \tau(H^1(\...
- 75
5
votes
1
answer
485
views
General algebraic result obtained from consideration on $\mathbb{Q}_p$
There are results in field theory which are obtained from, let's say, the complex numbers and then generalized to all algebraically closed fields.
For instance, the fact that a polynomial $P$ admits a ...
- 241
4
votes
0
answers
325
views
In which sense affine Grassmannian is "affine"
A pretty naïve question: Which meaning has the term "affine" in the notion of affine Grassmanian. Especially, I do not see any immediate connection to the concept of an "affine scheme&...
- 6,048
3
votes
0
answers
330
views
Can you prove and/or generalize this formula involving the Möbius function at n = square free numbers for elliptic curve related sequence in the OEIS?
Let $g(n)$ be the Dirichlet inverse of the Euler totient function:
$$g(n) = \sum\limits_{d|n} d \cdot \mu(d)$$
and let $f(x,y)$ be the elliptic equation:
$$f(x,y)=x^3 - x^2 - y^2 - y$$
Show that the ...
- 1,183
1
vote
0
answers
76
views
Help with understanding a proof of existence of solutions
In El Fatini and Boukanjime "Stochastic analysis of a two delayed epidemic model incorporating Lévy processes with a general non-linear transmission" paper, can someone give a detailed ...
- 121
1
vote
1
answer
218
views
Dimension of Zariski closure of a closed point of generic fiber
Let $S= \operatorname{Spec} A$ be a local Dedekind scheme of dimension $1$, (eg spectrum of localization at a prime of the ring of integers of a number field). Let $s \in S$ it's unique closed point ...
- 6,048
1
vote
0
answers
52
views
Does a substitution tiling being FLC depend on starting seed?
I've been trying to understand more on "geometric" substitutions rather than just symbolic ones. As symbolic substitutions always yield FLC tilings, I wanted to know whether a tiling coming ...
- 633
0
votes
0
answers
185
views
Sufficient condition for existence of a closest-point projection from a neighborhood onto a subset of a Riemannian manifold
Let $(M,g)$ be a Riemannian manifold and let $N$ be a subset of $M$.
On one hand, it is well known that if $N$ is an embedded submanifold of $M$, then it admits a tubular neighborhood, and, ...
- 144
3
votes
0
answers
120
views
Sequence which is related to the binary expansion of $n$ and partition numbers
Let $p(n)$ be A000041 i.e. the number of partitions of $n$ (the partition numbers).
Let
$$
\ell(n)=\left\lfloor\log_2 n\right\rfloor
$$
Let $\operatorname{wt}(n)$ be A000120 i.e. number of $1$'s in ...
- 4,938
0
votes
0
answers
119
views
About definition of stable solution. $Q_u(\phi) \ge 0$ for all $\phi \in C_c^1(\Omega)$ replaced by "for all $\phi \in W_0^{1,2}(\Omega)$"
I want to ask about a remark about the stable solution of elliptic PDE Remark 1.1.1.
We say $u$ is stable solution of $-\Delta u=f(u) \ \text { in } \Omega$ and $u=0$ on $\partial \Omega$ if it ...
- 809
2
votes
1
answer
320
views
Can you confirm the positivity of a quantity involving the Stirling numbers of the first kind
Let $s(m,n)$ denote the Stirling numbers of the first kind. For $m,n\in\mathbb{N}$, define
\begin{equation}
\mathcal{Q}(m,n)=(-1)^n\sum_{\ell=0}^{2n} \binom{m+\ell-1}{m-1} s(m+2n-1,m+\ell-1)\biggl(\...
- 1,101
0
votes
0
answers
151
views
Does the consistency of a large cardinal axiom imply the $\omega$-consistency of that axiom?
Let $P$ be some large cardinal property (or indeed any first-order formula in the language of set theory, but lets focus on large cardinals for now). Does the $\omega$-consistency of $\mathsf{ZFC}+P$ ...
- 2,206
2
votes
0
answers
156
views
Heat kernels and Dirac operators - Why are half densities invoked in the definition of heat kernels?
The authors of Heat kernels and Dirac operators chose to consider a generalized Laplacian $H$ on a bundle $E\otimes|\Lambda|^{1/2}$ and heat kernels of $H$ are defined to be sections of $(E\otimes|\...
- 339
3
votes
1
answer
469
views
Adic generic fiber of a small formal scheme in the sense of Faltings
$\DeclareMathOperator{\Spf}{Spf}\DeclareMathOperator{\Spa}{Spa}$In the Definition 8.5 of the paper "integral $p$ adic Hodge theory" by Bhatt-Morrow-Scholze, they define the adic generic ...
- 31
1
vote
0
answers
67
views
Limits of branched minimal immersions into the sphere
Can a sequence of branched minimal immersions $M_j^n$ in the round sphere $S^{n+1}$ converge to a smoothly embedded $\Sigma$, in the sense that $ M_j \to 2 \Sigma$ as currents or varifolds?
The case ...
- 5,048
1
vote
0
answers
59
views
Study of the properties of a non-local ODE
I am studying the following non-local ODE
$$\dot p(x) \nu_{\varepsilon, \alpha}(x) + \int_{x}^{2x_0}\frac{\dot p(s)}{s + \varepsilon} ds = c \quad \text{for } x \in [0,2x_0].$$
The number $x_0$ can ...
- 452
5
votes
1
answer
322
views
Do longest paths in 4-connected graphs intersect?
Is there for every $k$ a $k$-connected graph whose longest paths do not have a vertex in common?
This is known to be true for $k\le 3$, see
Ayesha Shabbir, Carol T. Zamfirescu., Tudor I. Zamfirescu: ...
- 19.1k
3
votes
0
answers
123
views
The group $G=\langle t,a \mid t^4=1, (t^{-1}at)^{-1} a (t^{-1} at)=a^2 \rangle$ is non-residually finite
A group $G= \langle t,a,| t^4=1, (t^{-1}at)^{-1} a (t^{-1}a t)= a^2 \rangle$ ( I do not remember the reference) was shown as a non -residually finite group. I do not know how to prove it. I will be ...
- 81
0
votes
0
answers
144
views
Dualising sheaf of a nodal curve
Let $X$ be a projective nodal curve. Why is the dualizing sheaf of $X$ isomorphic to the log-cotangent bundle of $X$?
- 1
2
votes
1
answer
119
views
Any theorem shows that flowmap $\phi_{\sum_{i=1}^n a_i f_i(x)}^\tau$ can be approximated by $\phi_{f_{\theta(t)}(x)}^{\tau'}$?
Given a control family $F:=\{f_1,\dotsc,f_n\}$, and $\phi_f^\tau(x)$ is the flowmap of the dynamical system
$$
\begin{cases}
z'(t)=f(z),\\
z(0)=x,
\end{cases}
$$ at end time point $\tau$.
Suppose $a_i&...
- 109
1
vote
0
answers
76
views
Is this extension of n-th derivatives to ordinal-indexed derivatives trivial? [duplicate]
Let $f$ be a function defined everywhere on the real line, which is infinitely differentiable everywhere, in other words, $f$ is everywhere smooth. I define the $\omega$-th derivative, where $\omega$ ...
- 2,023
1
vote
1
answer
187
views
Bound the distance between two vectors on the probability simplex
Let $a,b$ be two vectors with strictly positive elements and $\delta = 1 - \frac{\langle a,b \rangle}{\|a\|\|b\|}$. Bound the following optimization problem as a function of $\delta$
$$\sup_{x>0} \...
- 135