Trending questions
159,032 questions
3
votes
1
answer
271
views
$\mathrm{SL}(2,\mathbb{Z})$ finitely generated by using the Schwarz-Milnor lemma
Recently, I have been studying the modular group $G=\mathrm{SL}(2,\mathbb{Z})$, and I am trying to prove $G$ is finitely generated by using the Schwarz-Milnor lemma in geometric group theory.I am ...
- 111
4
votes
1
answer
155
views
Kirby diagram of the complement of a subhandlebody of a smooth closed 4-manifold
Let $X$ be a smooth closed connected 4-manifold. It admits a handlebody structure, having a unique 0- and a unique 4-handle. We can express the handlebody structure as a Kirby diagram (https://en....
- 213
2
votes
0
answers
159
views
About the 7.3.5. Corollary of the book "Measure Theory" by V.I. Bogachev
According to the 7.3.5. Corollary of the book "Measure Theory" by V.I. Bogachev we have the following result:
Let $(X,\tau)$ be a completely regular space and let $\Gamma$ be a
family of ...
- 649
0
votes
0
answers
103
views
On the form of algebraic numbers belonging to a specific field extension
Let $m>1$ be an integer and set $\theta=10^{-1/m}$. For a $\gamma\in \mathbb{Q}(\theta)$, there exists $a_0,\ldots,a_{m-1}\in \mathbb{Q}$ such that
$$
\gamma=a_0+a_1\theta+\cdots+a_{m-1}\theta^{m-1}...
- 515
0
votes
1
answer
90
views
How to calculate the maximum dimensions of a rectangle inside two concentric circles? [closed]
If I have a rectangle ABCD such that A and B touch two points of the outer circle and CD's touches one point of the inner circle, how could the maximum dimensions of the rectangle be calculated?
...
- 17
3
votes
1
answer
153
views
Lower bound in the singularity of random Bernoulli matrices
Let $A_n$ be a random $n \times n$ matrix with entries in $\{-1, +1\}$. As usual, "random" here means with respect to the uniform measure over such matrices.
The strong version of the ...
- 380
12
votes
1
answer
701
views
Does synonymy seep down to the fragments of theories?
IF we have a synonymous interpretation between two theories $T$ and $H$ that uses translation $\tau$ from the language of $T$ to the language of $H$. Then I'd expect that for a sentence $\mu$ in the ...
- 11.3k
-3
votes
2
answers
196
views
Which self homeomorphisms preserve measure on a torus, apart from affine? [closed]
Which self homeomorphisms preserve measure on a torus, apart from affines? Affine is the composition of rotation and automorphism. Measure is the Lebesgue measure.
- 83
2
votes
0
answers
76
views
When is a first-order delay differential equation equivalent to a higher-order ordinary differential equation?
The proportional delay differential equation
$$
xf'(x)+2xf'(x/2)+C+4f(x/2)-5f(x)=0
$$
with initial condition $f(0)=C$ expresses that Simpson's rule exactly integrates $f$ over any interval $[0,x]$ and ...
- 3,065
5
votes
0
answers
95
views
$\text{Rep}(D_4)$ and its three fiber functors
It is well-known that the fusion category $\text{Rep}(D_4)$ of representations of the dihedral group $D_4$ of order 8 admits three distinct fiber functors. Therefore, there are three different Hopf ...
5
votes
1
answer
369
views
Are PA and Counting Theory synonymous\bi-interpretable?
The following question is whether $\sf PA$ is synonymous or even bi-interpretable with a theory about counting objects in finite sets.
Counting Theory:
$\textbf{Logic:}$ Bi-sorted first order logic ...
- 11.3k
1
vote
0
answers
36
views
induced module of hyperoctahedral group
Let $H$ be the subgroup of the symmetric group $\mathfrak{S}_n$. Let $W_n$ be the group algebra of the hyperoctahedral group $\mathbb{Z}/2\mathbb{Z} \wr \mathfrak{S}_n$.The induced module $M:=\mathrm{...
- 179
0
votes
0
answers
61
views
Is the new method used by the GIMPS project applicable to non-Mersenne primes?
For years, there was a simple reason why the largest known prime is of the form $2^{p}-1$: We had the Lucas-Lehmer test which was specific to Mersenne numbers, and faster than all other known methods.
...
- 233
5
votes
1
answer
261
views
Diagonal analogue of symmetric functions
Let $n$ be a positive integer and consider the ring $R$ of power series over $\mathbb{Q}$ in commuting variables $x_1,y_1,x_2,y_2,...$. Let the symmetric group $\mathfrak{S}$ of permutations of the ...
- 1,500
1
vote
0
answers
82
views
Galois group of shimura varieties with different level structure
Let $(G,X)$ be a shimura data, and $K$ an open compact neat subgroup of $G(\mathbb A_f)$. Suppose $K'\subset K$ is open and normal, then, I see in many references that the finite etale cover $Sh(G,X)_{...
- 785
2
votes
0
answers
101
views
An inequality related to Problem 10210 AMM 1992 No. 3
Problem. Let $A$ be a $N \times N$ real matrix whose $(i,j)$ entry is $a_{ij} \ge 0, \forall i, j$. Let $1$ denote $N\times 1$ all-ones vector. Prove that
$$N^2 1^\top A^\top A A^\top 1 \ge (1^\top A ...
- 1,053
-5
votes
1
answer
88
views
Why is the second order correction to energy zero for a fully degenerate eigensystem? [closed]
Consider the system given by,
$$ H|n\rangle = E|n\rangle$$
where:
$H$ is the hamiltonian.
$|n\rangle$ is the eigenstate.
$E$ is the energy of the eigenstate.
Using degenerate perturbation theory and ...
- 23
7
votes
4
answers
500
views
Distinguishing finite families of sets by algebras of bounded size
Say that an algebra of sets $K$ distinguishes set $B$ from set $C$ provided that for some $A\in K$, we have exactly one of $A\cap B$ and $A\cap C$ non-empty. Given families $F$ and $G$ of sets, say ...
- 2,555
2
votes
1
answer
124
views
Questions about elliptic curves with level-$n$ structure
Let $n$ be a positive integer, which is $4$ or a prime number $l$.
Let $E$ be an elliptic curve defined over a number field $K$.
Assume that all the $n$-torsion points of $E$ are defined over $K$, i.e....
- 77
6
votes
0
answers
59
views
Connectedness of the space of negatively curved metrics of a compact 3-manifold
Is the space of metrics of negative sectional curvature over a closed 3-manifold connected? If so, in what paper is this result stated?
Note: as the Ricci flow hyperbolizes negatively curved metrics, ...
- 430
-1
votes
2
answers
373
views
Are these polynomials the same? [closed]
Let $p=2^{127}-1,P,Q\in \mathbb F_p[x]$ with $P(x)=x^2+1$ and $Q(x)=x^2+2$.
Are there some polynomial $H \in \mathbb F_p[x]$ bijectif on $\mathbb F_p$ with $\forall x \in \mathbb F_p, H(P(x))=Q(H(x))...
- 4,074
7
votes
1
answer
289
views
Group cohomology valued in a bimodule
The usual setup for group cohomology of a group $G$ is as follows. One takes a $G$-module $M$, and considers the space of all maps
$$\ell : G \times \cdots \times G \longrightarrow M $$
together with ...
- 9,853
5
votes
1
answer
368
views
Groups with no proper non-trivial fully invariant subgroup
Let $G$ be a finite group. A subgroup $H$ of $G$ is said to be characteristic if $\phi(H)\subseteq H$, $\forall \phi \in \operatorname{Aut}(G)$ and fully invariant if $\phi(H)\subseteq H$, $\forall \...
- 361
14
votes
1
answer
501
views
Is there an 'unnatural' topological construction of an algebraically closed field of positive characteristic?
It's well known that while there is a natural topological construction of a nearly algebraically closed field of characteristic $0$, algebraically closed fields of positive characteristic seemingly ...
- 13.1k
13
votes
2
answers
664
views
Categories in which isomorphism of stalks does not imply isomorphism of sheaves
Let $\mathcal{A}$ be a locally small category with colimits of small filtered diagrams.
For the purposes of this question, an $\mathcal{A}$-presheaf on a topological space $X$ is a functor $\Omega (X)^...
- 15.9k
7
votes
1
answer
161
views
When is a non-linear first-order ODE equivalent to a linear second-order ODE?
The Riccati equation $y'(x)+y(x)^2=f(x)$ is non-linear, but can be transformed into the linear equation $-u''(x)+f(x)u(x)=0$ via $y(x)=\frac{u'(x)}{u(x)}$.
Is there a general statement known about ...
- 3,065
2
votes
0
answers
96
views
Galois representations attached to discrete automorphic representations
Let $F$ be a totally real field. Let $G$ be a (split) connected reductive group over $F$. Let $\pi$ be an irreducible automorphic representation of $G$.
Recall in the work of Buzzard and Gee "The ...
- 6,622
3
votes
0
answers
62
views
On the relative growth rates of occupancy times in ergodic theory
Let $(X, \mathcal{F}, \mu)$ be a general measure space, and let $T: X \to X$ be a measure-preserving transformation on $X$. Assume that $T$ is ergodic and satisfies the property that, for any set $A \...
- 105
5
votes
0
answers
93
views
Rational maps from the circle to the unitary group (energy-preserving convolutive mixtures)
Consider a rational map $A : S^1 \to U(n)$, i.e. a matrix of rational functions such that evaluation at any $z \in S^1 \subset \mathbf C$ is unitary. These objects show up in digital signal processing ...
- 151
3
votes
2
answers
366
views
Rational divisors on Calabi–Yau threefolds
Following the construction of [2], consider the full subcategory $\mathcal{D}_0 \subset D^\flat(\operatorname{Coh} \omega_{\mathbb{P}^1})$ consisting of complexes whose cohomology objects are ...
- 317
-2
votes
0
answers
24
views
Conditions for a cubic function to be quasiconcave or quasiconvex [closed]
I would like to understand under what conditions a cubic function $f(x)=ax^3+bx^2+cx+d$ can be considered quasiconcave or quasiconvex. Specifically, I am interested in finding conditions on the ...
- 13
0
votes
1
answer
82
views
A representable functor that does not admit an adjoint? [closed]
What is a simple example of a representable functor that does not admit an (left) adjoint?
6
votes
0
answers
77
views
About path-connected components of the Bohr compactification of $\mathbb{R}^d$
Let ${\rm b}(\mathbb{R}^d)$ denote the Bohr compactification of $\mathbb{R}^d$, with $d\in\mathbb{N}$. This is the Pontryagin dual of the group $\mathbb{R}^d_d$, corresponding to $\mathbb{R}^d$ with ...
- 193
3
votes
0
answers
105
views
Jacobian of a reducible curve with arbitrary singularities
Let X be a reduced, reducible curve over $\mathbb{C}$ with locally planar singularities, and let $\widetilde{X}$ be its normalization. I am interested in the Jacobian varieties $\mathrm{Jac}(X)$ and $\...
- 31
2
votes
0
answers
80
views
Question about lattice with dense projection
Let $H\subset \operatorname{GL}(n,\mathbb{C})$ be a connected, semisimple algebraic group defined over $\mathbb{Q}$. Fix a number field $K$ with $[K:\mathbb{Q}]=3$ that is not totally real. Denote its ...
- 305
1
vote
1
answer
187
views
Existence of Finite Amicable Groups
I'm interested in exploring the concept of "amicable groups" as follows:
Definition. Two finite groups $G$ and $H$ are called amicable groups if:
$G$ is the direct sum of proper subgroups ...
3
votes
2
answers
158
views
Restriction of a locally finitely supported function on an ordinal is finitely supported?
This is a question about set theory. Let $f:\delta\rightarrow H$ be a function from an ordinal $\delta< \omega_1$ to an arbitrary abelian group $H$. Endow $\delta$ with the order topology. Let $f$ ...
- 61
3
votes
0
answers
110
views
Bertini's theorem at a fixed point
Recently, I am learning Bertini's theorem because I encounter "generic smooth" problem during my research. I'm not an algebraic geometer and I read the Hartshorne Chapter 3 Theorem 10.8 to ...
- 51
5
votes
0
answers
142
views
Automorphism-invariant probability measure on the space of minimal colorings of a graph
Let $\Gamma$ be a connected graph of chromatic number $d$, and let $C_n(\Gamma)$ denote the set of $n$-colorings, $n\in \mathbb{N}$. Thinking of this as a subset of $[n]^{V(\Gamma)}$ which may be ...
- 68.9k
13
votes
2
answers
393
views
What does $\mathrm{Conf}_n(M)^{h S_n}$ look like?
$\DeclareMathOperator\Conf{Conf}$Let $M$ be a manifold, and $\Conf_n M$ the ordered configuration space of n points on $M$. The symmetric group $S_n$ acts by permuting the points.
Is there a simple ...
- 275
4
votes
0
answers
92
views
Hölder stability of the PDE $\partial_t u (t, x) = \operatorname{div} \{ a (t, x) \nabla u(t, x) \}$
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bT}{\mathbb{T}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\eps}{\...
- 825
7
votes
0
answers
78
views
Colimits of underlying $n$-categories vs. $n$-uple categories
A double category is a category object in categories. There is a functor (of $1$-categories) $u : \mathrm{Cat}(1\mathrm{Cat}) \to 2\mathrm{Cat}$ from double categories to $2$-categories which is right ...
- 495
2
votes
1
answer
126
views
"Bad" valid edge contractions
In this paper, an edge contraction of a simplicial complex $\Gamma$ is defined as the operation of removing the neighborhood $N_e\Gamma$ of the edge $e=\{0,1\}$ and identifying $N_0\partial N_e\Gamma$ ...
- 21
7
votes
0
answers
220
views
Is there a Cayley graph with end space infinite and discrete?
A Cayley graph of a finitely generated group must be locally finite, and we know end spaces of locally finite graphs must be compact - so we can't have an infinite and discrete end space in this ...
- 407
2
votes
0
answers
49
views
Are maps between cohomology of homogeneous vector bundles morphisms of representations?
Let $X = G/P$ a rational homogeneous variety, e.g. a grassmannian. Consider a short exact sequence $$ 0 \longrightarrow E_1 \longrightarrow E_2 \longrightarrow E_3 \longrightarrow 0$$
where $E_i$ are ...
- 41
0
votes
0
answers
46
views
Amenability of locally convex algebras
Let $A$ be an amenable Banach algebra, and let $A_w$ denote $A$ with the weak topology. Clearly, $A_w$ is a Hausdorff locally convex algebra (l.c.a.).
Q0: Is $A_w$ amenable as a l.c.a. in the sense ...
- 2,605
8
votes
1
answer
437
views
Function $\phi$ such that $f(\phi(x,y)) = f(x) + f(y)$
I have a continuous function $f:\mathbb{R}^n\to\mathbb{R}$, and I am looking for a continuous (or at least measurable) function $\phi:\mathbb{R}^{2n}\to\mathbb{R}^n$ such that $f(\phi(x,y))=f(x)+f(y)$....
- 3,065
0
votes
0
answers
136
views
What are the automorphisms of finite commutative groups？ [migrated]
What are the automorphisms of finite commutative groups？Is there a relatively complete conclusion? Although it can be decomposed into the direct product of cyclic groups, this question still seems ...
- 83
3
votes
0
answers
110
views
Wedge of curvature and subsequent trace
I am currently reading https://arxiv.org/abs/1901.10322. More specifically, I am interested in understanding the equation
$$i\partial\overline{\partial}\omega = \frac{\alpha'}{4}Tr(R\wedge R-F\wedge F)...
2
votes
0
answers
85
views
On the trajectory followed by a point P on a planar convex region C when P is mapped repeatedly to the farthest point to it on C
Consider a planar convex region $C$. Let us define a mapping of a point $P$ on $C$ to that point on C that is farthest from $P$. Obviously, if from an initial position of $P$, we do this mapping ...
- 5,979