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Does there exist an unramified $PSL_2(\mathbb{F}_p)$ extension of a quadratic field $K$?

Does there exist an unramified $PSL_2(\mathbb{F}_p)$ extension of a quadratic field $K$ for $p\geq 5$?
user avatar
8 votes
0 answers
179 views

When is G->G/H a trivial bundle

Suppose that $H\subseteq G$ are connected lie groups. Then $G\mapsto G/H$ is a principal $H$-bundle. I would like to know if there is some non-tautological characterization of when this is a trivial ...
user130661's user avatar
9 votes
0 answers
171 views

Can every non-inner automorphism of a group with residually finite outer automorphisms be realised as an non-inner automorphism of a finite quotient?

First some motivation: most proofs that show that the group of outer automorphisms is residually finite do not only show that the subgroup of inner automorphisms is closed in the profinite topology, ...
Michal Ferov's user avatar
5 votes
1 answer
355 views

Is the following variant of Shafarevich's theorem known?

Let $Q$ be a finite simple group which may be realized as the Galois group of some extension of $\mathbb{Q}$ (like for instance $PSL_2(\mathbb{F}_p)$ for $p\geq 5$, or the monster group) and let $G$ ...
user avatar
7 votes
2 answers
402 views

Is every element of $Mod(S_{g,1})$ a composition of right handed Dehn twists?

Let $S_{g,1}$ be the surface of genus $g \geq 1$ and $1$ boundary component. Let $Mod(S_{g,1})$ be the mapping class group in which we allow isotopies to rotate the action on the boundary (...
Paul's user avatar
  • 1,379
7 votes
1 answer
396 views

Subtori of groups of type E6

Let $G$ be a semisimple algebraic group of type $E_6$, defined over a perfect field $k$ (so $G$ is a group scheme over $k$ and $G_{\bar{k}}$ is a semisimple algebraic group in the usual sense), and ...
Cehiju's user avatar
  • 81
3 votes
0 answers
94 views

Legendre transform on signed measure space

Let $X$ be an open set in $\mathbb{R}^n$ and $M(X)$ be the space of finite signed measures defined on $X$. $L(p)$ is a lower-semicontinuous convex functional defined on $M(X)$. My question is: (1) ...
Elliott's user avatar
  • 325
2 votes
1 answer
182 views

effectively distinguishing knots

It was proven, I think by Mijatović EDIT: by Waldhausen, that there is an effective algorithm for distinguishing knot complements (the effective constants were found by Coward and Lackenby). The bound,...
Dmitry Vaintrob's user avatar
4 votes
1 answer
201 views

On the solvability of the congruence $p^m\equiv m\pmod{n}$

Let $n,p\geq 1$ be integers, and assume that $p$ is a prime. Question. Does there always exist an integer $m\geq 1$ such that $p^m\equiv m\pmod{n}$?
Samvel Safaryan's user avatar
2 votes
0 answers
131 views

Tuples with same coordinate sum

Some $4$-tuples of positive real numbers $(a_1,b_1,c_1,d_1),\dots,(a_n,b_n,c_n,d_n)$ are given, with $$\sum_{i=1}^na_i=\sum_{i=1}^nb_i=\sum_{i=1}^nc_i=\sum_{i=1}^nd_i=3.$$ It is known that there ...
pi66's user avatar
  • 1,199
7 votes
1 answer
270 views

On the number $n_0$ in Shelah's construction of a Jonsson group

In the paper "On a Problem of Kurosh, Jonsson Groups, and Applications", Shelah proved the following Theorem 2.1. There exists a number $n_0$ such that for every infinite cardinal $\lambda$ with $\...
Taras Banakh's user avatar
  • 40.8k
7 votes
0 answers
935 views

A problem of Erdős on convergence of $\sum (-1)^nn/p_n$ and equidistribution of $\pi(n)$ modulo 2

Erdős asked1 whether the series $$\sum_{n=1}^\infty \frac{(-1)^n n}{p_n}$$ converges. Here, $p_n$ denotes the n-th prime. I can show that this series converges simultaneously with the series $\sum_{...
Mustafa Said's user avatar
  • 3,679
2 votes
0 answers
83 views

Is there a knot invariant robust to hiding one part of the diagram behind another?

Is there a well-known knot invariant that can be computed solely by inspecting an arbitrary projection of the knot into the plane (with a marking of each crossing as "over" or "under")? The reason ...
GMB's user avatar
  • 1,379
3 votes
0 answers
136 views

eta invariant and spectral flow

We know that for a family of first-order self adjoint elliptic (Fredholm) operator $A_t$, for $t\in [0,1]$ we have the formula $$\eta(A_1)-\eta(A_0)=spfl(A_t)_{t\in[0,1]}+\int^1_0 \omega(s)ds,$$ where ...
DLIN's user avatar
  • 1,905
8 votes
3 answers
252 views

Random reflections unexpectedly produce banded distributions

Start with $p_1$ a random point on the origin-centered unit circle $C$. At step $i$, select a random point $q_i$ on $C$, and a random mirror line $M_i$ through $q_i$, and reflect $p_i$ in $M_i$ to ...
Joseph O'Rourke's user avatar
1 vote
0 answers
74 views

Elliptic pde L^p theory via adjoint theory

Let $ T:X \rightarrow Y$ denote some linear operator and suppose we know its one to one (here $X$ and $Y$ are Banach spaces). I believe their is results that say $Ker(T^*)= (R(T))^\perp$ (where ...
Math604's user avatar
  • 1,363
3 votes
1 answer
145 views

Bounding the derived length of a solvable group given the degrees of the irreducible monomial characters

Much is known about the derived length of a solvable group given the degrees and cardinality of the set of degrees of the irreducible characters. Martin Isaacs and Donald Passman pretty much started ...
Joakim Færgeman's user avatar
4 votes
2 answers
848 views

Heuristics behind the Circle problem?

Is there a heuristic argument behind the exponent in the circle problem? The problem that I am referring to is the following: Consider a circle of radius $R$ centered at the origin in the plane and ...
Mustafa Said's user avatar
  • 3,679
2 votes
0 answers
121 views

A community effort: equilibrium in quitting games [closed]

This thread is in the spirit of the polymath project: a combined effort of the community to solve a difficult open problem. It is an activity of the European Network for Game Theory whose goal is to ...
Eilon's user avatar
  • 745
1 vote
0 answers
90 views

Some hypersurface has a positive second fundamental form potentially

Notation : $r^2=x^2+y^2$. Exercise : Define $$F_\sigma (x,y)= (f_\sigma (x,y),\frac{x^2-y^2}{2},xy)$$ where $$f_\sigma (x,y)=(1- \sigma^2 r^2)(x-\sigma x^3,y-\sigma y^3)$$ Define $ G_\sigma: \mathbb{...
Hee Kwon Lee's user avatar
  • 1,070
2 votes
0 answers
137 views

Hodge-Tate weights of etale cohomology groups

Given a smooth algebraic variety $X$ over a number field $F$, its $p$-adic cohomology groups $H^i(X \times_F \bar F, \mathbb Q_p)$ carries an action of $\mathrm{Gal}(\bar F/F)$, which gives a ...
Shawn's user avatar
  • 453
4 votes
2 answers
1k views

Reference request: Oldest linear algebra books with exercises?

Inspired by the recent success of my "soft question" here, I also have to ask, what are some of the oldest linear algebra books out there with exercises? I'm fine with or without solutions, either way....
8 votes
2 answers
939 views

How do fractional tensor products work?

[I asked and bountied this question on Math SE, where it got several upvotes and a comment suggesting it was research-level, but no answers. So I'm reposting here with slight edits, but please feel ...
tparker's user avatar
  • 1,243
4 votes
1 answer
781 views

etale higher direct image sheaf

Let $f:X\rightarrow Y$ be a smooth morphism of schemes such that all the fibres (for geometric points) are affine spaces. Let $F$ be a coherent sheaf on $X$. Is $R^i_{et}~f_*F=0~~~\forall i>0$? ...
user avatar
44 votes
3 answers
9k views

Publishing a Simple Paper as an Undergraduate

First off I apologize if this question does not belong here, I would be happy to hear about any better locations to post this on. I am a (first year) undergraduate mathematics student, and I recently ...
5 votes
1 answer
170 views

Uniformity in Wirsing's Mean Value Theorems

In two important papers of Wirsing, namely "Das asymptotische Verhalten von Summen über multiplikative Funktionen" (1961) and its follow up (1967), several results on mean values of multiplicative ...
Ofir Gorodetsky's user avatar
0 votes
1 answer
242 views

A peculiar operation on $M_2(\mathbb Z)$ which along with the usual matrix addition, makes $M_2(\mathbb Z)$ into a commutative ring with unity [closed]

For $A=\begin{pmatrix} a_1 & b_1 \\ c_1&d_1 \end{pmatrix}, B=\begin{pmatrix} a_2 & b_2 \\ c_2&d_2 \end{pmatrix}\in M_2(\mathbb Z)$, define $A*B:=a_1L_1BR_1+b_1L_1BR_2+c_1L_2BR_1+...
user521337's user avatar
  • 1,189
4 votes
0 answers
233 views

Class fields without class field theory

Is there an English reference for the analytic construction of the Hilbert class field of an imaginary quadratic field without using class field theory? I am in particular interested in a proof of the ...
Shimrod's user avatar
  • 2,335
8 votes
0 answers
199 views

Does anyone know of this manifestation of the Littlewood-Richardson coefficients for the complete flag variety?

This is the culmination of about 11 years of research but after I discovered it I found a proof that was extremely trivial, so I'm wondering if it's already known. Let $(a,b)$ with $a < b$ ...
Matt Samuel's user avatar
  • 2,008
2 votes
1 answer
177 views

For every table of interpolating nodes, there is a positive continuous function whose interpolating polynomials are not positive infinitely often

Fix an interval $[a,b]$. Is it true that for every table of interpolating nodes $\{x_{0,n},x_{1,n}...,x_{n,n}\}_{n=1}^{\infty}$, there exists a continuous function $f:[a,b]\to (0,\infty)$ such that ...
user521337's user avatar
  • 1,189
-2 votes
2 answers
123 views

What is the limit of this integral as $n$ approaches infinity for integer $k\geq 0$ and real $m\geq 1$? [closed]

$\int_{0}^{1}u^k\cot{\frac{\pi(1-u)}{m}}\sin{\frac{2\pi n(1-u)}{m}}\,du$
user avatar
-5 votes
1 answer
190 views

How do you prove the validity of this formula for $H(n)$? [closed]

I'm looking for a proof of the identity $$\sum_{k=1}^{n}\frac{1}{k}=\frac{1}{2n}+\pi\int_{0}^{1} (1-u)\cot{\pi u}\left(1-\cos{2\pi n u}\right)\,du.$$ There is a generalization of this formula for $...
user avatar
4 votes
1 answer
247 views

A variation on four-vertex theorem

Is it true that, if a closed, strictly convex curve has exactly four vertices (extrema of curvature), then any circle has at most four points of intersection with it?
Minghui Ouyang's user avatar
2 votes
1 answer
328 views

$L_2$ bounds for $\zeta(1/2 + it)$ and a related integral

Denote by $\zeta$ the Riemann zeta function. I just learnt from this question $L_2$ bounds for tails of $\zeta(s)$ on a vertical line that $\int_{T}^{\infty} \frac{\zeta(1/2 + it)}{1/4 + t^2}\mathrm{...
acc10's user avatar
  • 21
9 votes
2 answers
296 views

Maximum of a quantity for two normal orthogonal vectors in $\mathbb{R}^n$

Let's define for every pair of vectors $u,v\in\mathbb{R}^n$, a quantity as follows: $$f(u,v) = \sum_{1\leq i,j\leq n}|u_iu_j-v_iv_j|.$$ I want to find: $$M(n)= \max \{f(u,v): u,v\in \mathbb{R}^n, |u|=...
Mostafa's user avatar
  • 4,454
5 votes
1 answer
205 views

Curve-counting with fixed source

Suppose I fix a smooth projective curve $C$ of positive genus, and I have a smooth projective variety $X$. Do standard tools from GW theory (or any curve-counting theory for that matter) allow me to ...
Hans Sachs's user avatar
1 vote
0 answers
143 views

Example of open manifold with no free integer homology non-homeomorphic to a ball

I would like to state that if an open oriented even-dimensional (complex) manifold $M$ is such that $dim(H_k(M,\mathbb{Z}))=0$ for $k>0$, and 1 for $k=0$, then $M$ is homeomorphic to an open ball. ...
MathBug's user avatar
  • 258
2 votes
1 answer
214 views

Projective G-group

Let $G$ be a fixed group. By definition, a $G$-group is a group $X$ with a $G$-action that respects the group operation of $X$. A free $G$-group means a group freely generated by a free $G$-set. A &...
user49822's user avatar
  • 2,033
-2 votes
1 answer
66 views

I need help with snake's position bounds based on center point(rounded) and the length of the snake problem [closed]

First of all, if it's an existing problem just tell me the name, please. To solve the problem a formula/algorythm which receivs a center point of a snake (snake game type (points on a grid connected ...
Todam's user avatar
  • 9
9 votes
2 answers
448 views

Why might the Lawson minimal surface $\xi_{1,2}$ have a Morse index 9?

In page 15 of the article New Applications of Min-max Theory, Andre Neves said that: a wishful thinking would suggest the Lawson minimal surface $\xi_{1,2}$ in the 3-sphere to have a Morse index 9, ...
JSCB's user avatar
  • 1,610
3 votes
0 answers
193 views

On some inequality involving the Riemann zeta function integral at $\Re(s)=1/2$ [closed]

I recently saw on p.$458$ of Montgomery-Vaugahn's ''Multiplicative number theory'' that the inequality $$\int_{1}^{T} \zeta(1/2 + it) \mathrm{d}t = T + O(T^{1/2})$$ holds uniformly for $T\geq 2$, ...
user avatar
35 votes
3 answers
1k views

Second Betti number of lattices in $\mathrm{SL}_3(\mathbf{R})$

We fix $G=\mathrm{SL}_3(\mathbf{R})$. Let $\Gamma$ be a torsion-free cocompact lattice in $G$. Is $b_2(\Gamma)=0$? Here the second Betti number $b_2(\Gamma)$ is both the dimension of the ...
YCor's user avatar
  • 60.1k
1 vote
1 answer
107 views

Closed subsets in Coxeter groups

Let $W$ be a finite or infinite Coxeter group and $\Phi^+$ the set of its positive roots. In the paper, a subset $A$ of $\Phi^+$ is closed if for all $a, b \in A$, $r_1 a + r_2 b \in \Phi^+$ for ...
Jianrong Li's user avatar
  • 6,101
2 votes
0 answers
130 views

Open problems in Monte Carlo Simulation [closed]

I want to know some open problems in Monte Carlo Simulation, which is being studied or in a stalemate. Could you please give me some advice? Thanks a alot
Phát Đạt Nguyễn's user avatar
1 vote
0 answers
75 views

how the borel subalgebras of p(n) look like except the standard one?

I am reading the book Musson: Lie superalgebras. In Chapter 3, on page 62, Lemma 3.6.8 tells about support of Borel subalgebra. I am confused about this. I am trying to do the proof of the Proposition ...
Shushma Rani's user avatar
6 votes
0 answers
235 views

A characterisation of certain $C^*$-algebras

I was wondering if there is a characterisation for $C^*$-algebras (unital) for which the bidual does not have any central atoms. It is not sufficient for example to demand that the $C^*$-algebra does ...
Mark Roelands's user avatar
5 votes
1 answer
397 views

Random pairs of commuting permutations

Let $\Omega_n \subseteq \mathrm{Sym}(n)^4$ be the set of all $4$-tuples $(\sigma_1,\sigma_2,\tau_1,\tau_2)$ of permutations of $\{1,\ldots,n\}$ such that $\sigma_j \tau_k = \tau_k \sigma_j$ for each ...
burtonpeterj's user avatar
  • 1,689
2 votes
0 answers
214 views

Integral with product of two infinite sums

I am looking for references and results on integrals with product of two infinite sums: $$S_1=\int_0^{\infty} \sum_{n=0}^{\infty} f(nx) \sum_{k=0}^{\infty} \overline{ g(kx)} dx$$ Above integral is ...
Bertrand's user avatar
  • 1,121
2 votes
1 answer
41 views

Terminology for tree subgraphs where non-neighbouring vertices are not connected by single ambient edges

Suppose $G=(V,E)$ is a connected graph and $T=(V_T, E_T)$ is a subgraph of $G$ that is a tree. If we further suppose that any pair of vertices $v,w \in V_T$ that are not joined by a single edge in $...
Greg Egan's user avatar
  • 2,852
1 vote
0 answers
216 views

Weak elliptic maximum principle on manifolds without strict ellipticity

This question is not to be confused with the similarly titled question here. In the above lined question, I gave a complete answer, but noticed that things are apparently not so simple in the ...
Ryan Unger's user avatar

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