All Questions
152,891
questions
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140
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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 $p\geq 5$?
8
votes
0
answers
179
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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 ...
9
votes
0
answers
171
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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, ...
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 $G$ ...
7
votes
2
answers
402
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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 (...
7
votes
1
answer
396
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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 ...
3
votes
0
answers
94
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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) ...
2
votes
1
answer
182
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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,...
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
$p^m\equiv m\pmod{n}$?
2
votes
0
answers
131
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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 ...
7
votes
1
answer
270
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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 $\...
7
votes
0
answers
935
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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_{...
2
votes
0
answers
83
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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 ...
3
votes
0
answers
136
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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 ...
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 ...
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 ...
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 ...
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 ...
2
votes
0
answers
121
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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 ...
1
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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{...
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 ...
4
votes
2
answers
1k
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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 ...
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 $R^i_{et}~f_*F=0~~~\forall i>0$? ...
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 ...
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+...
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 ...
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 $a < b$ ...
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 ...
-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]
$\int_{0}^{1}u^k\cot{\frac{\pi(1-u)}{m}}\sin{\frac{2\pi n(1-u)}{m}}\,du$
-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 $...
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?
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{...
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:
$$M(n)= \max \{f(u,v): u,v\in \mathbb{R}^n, |u|=...
5
votes
1
answer
205
views
Curve-counting with fixed source
Suppose I fix a smooth projective curve $C$ of positive genus, and I have a smooth projective variety $X$. Do standard tools from GW theory (or any curve-counting theory for that matter) allow me to ...
1
vote
0
answers
143
views
Example of open manifold with no free integer homology non-homeomorphic to a ball
I would like to state that if an open oriented even-dimensional (complex) manifold $M$ is such that $dim(H_k(M,\mathbb{Z}))=0$ for $k>0$, and 1 for $k=0$, then $M$ is homeomorphic to an open ball.
...
2
votes
1
answer
214
views
Projective G-group
Let $G$ be a fixed group. By definition, a $G$-group is a group $X$ with a $G$-action that respects the group operation of $X$. A free $G$-group means a group freely generated by a free $G$-set. A &...
-2
votes
1
answer
66
views
I need help with snake's position bounds based on center point(rounded) and the length of the snake problem [closed]
First of all, if it's an existing problem just tell me the name, please. To solve the problem a formula/algorythm which receivs a center point of a snake (snake game type (points on a grid connected ...
9
votes
2
answers
448
views
Why might the Lawson minimal surface $\xi_{1,2}$ have a Morse index 9?
In page 15 of the article New Applications of Min-max Theory, Andre Neves said that: a wishful thinking would suggest the Lawson minimal surface $\xi_{1,2}$ in the 3-sphere to have a Morse index 9, ...
3
votes
0
answers
193
views
On some inequality involving the Riemann zeta function integral at $\Re(s)=1/2$ [closed]
I recently saw on p.$458$ of Montgomery-Vaugahn's ''Multiplicative number theory'' that the inequality
$$\int_{1}^{T} \zeta(1/2 + it) \mathrm{d}t = T + O(T^{1/2})$$
holds uniformly for $T\geq 2$, ...
35
votes
3
answers
1k
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Second Betti number of lattices in $\mathrm{SL}_3(\mathbf{R})$
We fix $G=\mathrm{SL}_3(\mathbf{R})$.
Let $\Gamma$ be a torsion-free cocompact lattice in $G$. Is $b_2(\Gamma)=0$?
Here the second Betti number $b_2(\Gamma)$ is both the dimension of the ...
1
vote
1
answer
107
views
Closed subsets in Coxeter groups
Let $W$ be a finite or infinite Coxeter group and $\Phi^+$ the set of its positive roots.
In the paper, a subset $A$ of $\Phi^+$ is closed if for all $a, b \in A$, $r_1 a + r_2 b \in \Phi^+$ for ...
2
votes
0
answers
130
views
Open problems in Monte Carlo Simulation [closed]
I want to know some open problems in Monte Carlo Simulation, which is being studied or in a stalemate.
Could you please give me some advice? Thanks a alot
1
vote
0
answers
75
views
how the borel subalgebras of p(n) look like except the standard one?
I am reading the book Musson: Lie superalgebras. In Chapter 3, on page 62, Lemma 3.6.8 tells about support of Borel subalgebra. I am confused about this.
I am trying to do the proof of the Proposition ...
6
votes
0
answers
235
views
A characterisation of certain $C^*$-algebras
I was wondering if there is a characterisation for $C^*$-algebras (unital) for which the bidual does not have any central atoms. It is not sufficient for example to demand that the $C^*$-algebra does ...
5
votes
1
answer
397
views
Random pairs of commuting permutations
Let $\Omega_n \subseteq \mathrm{Sym}(n)^4$ be the set of all $4$-tuples $(\sigma_1,\sigma_2,\tau_1,\tau_2)$ of permutations of $\{1,\ldots,n\}$ such that $\sigma_j \tau_k = \tau_k \sigma_j$ for each ...
2
votes
0
answers
214
views
Integral with product of two infinite sums
I am looking for references and results on integrals with product of two infinite sums:
$$S_1=\int_0^{\infty} \sum_{n=0}^{\infty} f(nx) \sum_{k=0}^{\infty} \overline{ g(kx)} dx$$
Above integral is ...
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 $...
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 ...