Trending questions
159,065 questions
6
votes
3
answers
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Boolean rings with many automorphisms
Does there exist an infinite Boolean ring $R$ (not assume unital, only associative) with the property that for any nonzero $x,y\in R$, there is a ring automorphism $\varphi\colon R\to R$ such that $\...
- 131
16
votes
2
answers
1k
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Identifying short introductory book on non-commutative geometry I read c.2008
I’m trying to identify a book I remember reading and enjoying in grad school around 2008–9; I’ve forgotten its author and title, and haven’t been able to find a book matching my memories in half an ...
- 21.3k
2
votes
0
answers
97
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Free, easy-to-use program for noncommutative algebra over finite fields
I am looking for a computer program that can handle computations in noncommutative algebra over a finite field of prime order $p$.
My requirements are:
The program should be free, as I do not have ...
- 1,013
13
votes
2
answers
802
views
For which rationals is this exponential sum bounded?
Given $x \in [0, 1]$, we denote by $e(x)$ the complex number $e^{2 \pi i x}$.
Can we characterise the set of rationals $x$ for which the sum
$$A_N(x)\, :=\, \sum_{n = 0}^N e(2^n x)$$
remains bounded ...
- 6,313
175
votes
39
answers
31k
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Short exact sequences every mathematician should know
I'd like to have a big-list of "great" short exact sequences that capture some vital phenomena. I'm learning module theory, so I'd like to get a good stock of examples to think about. An ...
4
votes
1
answer
327
views
Is the pushforward of a closed immersion ever fully-faithful at the level of Derived Categories?
Let $i: Z \rightarrow X$ be a closed immersion of schemes. Then, for any $\mathcal{O}_{Z}$-module $\mathcal{G}$, the counit of adjunction $i^{*}i_{*}\mathcal{G} \rightarrow \mathcal{G}$ is an ...
- 137
1
vote
0
answers
146
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Can we find curves with many rational points using linear algebra?
Probably this is impossible, but let us try.
Working over $\mathbb{Q}[x_1,...,x_n]$.
Let $T_i$ be $n$ sets of rationals with cardinality $B$.
Assume we are given $n-2$ linear equations $f_i$ which are ...
- 25.4k
12
votes
2
answers
868
views
Sets that project to zero measure on all lines except one
It is a (difficult) exercise to show that there exists a measurable set $E \subset [0,1]^2$ (necessarily with zero 2-dimensional Lebesgue measure) such that the projection on every line passing ...
- 355
7
votes
3
answers
708
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Properties of $P_{n}(x)={e}^{-x}\sum_{k=0}^{\infty}\frac{a_{k,n}{x}^{k}}{k!}$
I know this will sound like a general question, but given the structure $$P_{n}(x)={e}^{-x}\sum_{k=0}^{\infty}\frac{a_{k,n}{x}^{k}}{k!}$$ where $$a_{k,n} = \frac{1}{\prod_{i=1}^{n} (k+2i) }, $$ what ...
5
votes
1
answer
367
views
Check that a Sheaf is Invertible Etale Locally
A question about following statement from Martin Olsson's book on Stacks. In the proof of Proposition 13.2.9. (p 269) is claimed that certain sheaf $K$ on a nodal curve $C$ is invertible it suffice to ...
- 5,966
7
votes
0
answers
140
views
Average number of $\mathbb{F}_p$-points over twists of a variety
Let $p \gg 1$ be a sufficiently large prime. I recently stumbled across a fascinating fact about the number of $\mathbb{F}_p$-points on elliptic curves over finite fields. Specifically, we have:
Fact ...
0
votes
1
answer
141
views
Existence of infinite rank compact operator
Given any separable Banach space $X$, we know that always there exists a Banach space $Y$ such that there is an injective compact operator from $X$ to $Y$. Can we show that given any infinite ...
- 585
5
votes
1
answer
205
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Computing the Second Exterior Power of Certain Ideals in $\mathbb{Z}[\sqrt{-5}]$ and $\mathbb{Z}[\sqrt{5}]$ as Modules
I'm working on a problem involving the computation of the second exterior power of certain ideals within the rings $R_1 = \mathbb{Z}[\sqrt{-5}]$ and $R_2 = \mathbb{Z}[\sqrt{5}]$. The problem is as ...
- 93
2
votes
2
answers
172
views
Can the Category of that kind of small sets in $\sf NFU$ be Cartesian closed?
Working in Quine's $\sf NFU$, with urelements being at least as many as sets. Formally the latter is: $|Ur| \geq |Set|$.
Where $Ur$ is the set of all urelements and $Set$ is the set of all sets. We ...
- 11.3k
27
votes
8
answers
3k
views
Object of proven finiteness, yet with no algorithm discovered?
I explain my title by two examples in number theory:
The rational points on elliptic curve over number fields forms a finitely generated abelian group, so its rank is an integer, but so far we do not ...
- 1,053
2
votes
1
answer
277
views
Does the Apéry-like sequence $A_n=(n!)^2\sum_{k=0}^n { \rho \choose k}^2 { \rho+n-k \choose n-k}, \rho=e^{2 \pi i/3}$ change signs infinitely often?
This is an integer sequence OEIS sequence A217703.
It satisfies an order 3 recurrence which is the constant term $A_n=u_n(0)$ of a three term recurrent sequence of polynomial defined by
$$u_0(x)=1,u_1(...
- 1,362
8
votes
1
answer
229
views
Examples of anti-classical theories in iFOL
An anti-classical axiom $\phi$ is one which is inconsistent with LEM
Are there any sources for good examples of anti-classical theories in intuitionstic first-order logic? There are many examples of ...
- 183
8
votes
1
answer
653
views
Status of a conjecture in Grothendieck's "Crystals and the de Rham Cohomology of Schemes"
Let $X/\mathbb{C}$ be a scheme over the complex numbers. In "Crystals and the de Rham cohomology of schemes," Grothendieck constructs the infinitesimal ringed site $(X_{\operatorname{inf}}, \...
- 333
1
vote
0
answers
60
views
Tiling with one of each 3D shape
Encouraged by the positive solutions to my question,
Tiling with one of each shape,
I'd like to pose the $\mathbb{R}^3$ equivalent:
Q. Is there a tiling of $\mathbb{R}^3$ by (bounded) polyhedra, one ...
- 151k
3
votes
0
answers
96
views
Deeper reason for why classical orthogonal polynomials have simple generating functions?
Is there a known reason why all classical families of orthogonal polynomials have simple generating functions? I was wondering whether one could get an explanation using the connection with Sturm-...
- 312
10
votes
1
answer
230
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Is there a concise description of the $\infty$-category $\mathrm{Mod}_A^\mathcal{O}(\mathcal{C})$ of modules over an algebra over an $\infty$-operad?
[Cross-posted from this Math SE question.]
In Higher Algebra, Section 3.3 Lurie constructs the $\infty$-operads $\newcommand{\Mod}{\mathrm{Mod}}\newcommand{\cO}{\mathcal{O}}\newcommand{\cC}{\mathcal{C}...
- 201
0
votes
1
answer
60
views
Optimizing sum of discrete minimum
Please consider the following optimization problem: Given a fixed positive natural $n < N$, and a set of functions $f_i$ over a finite domain of nonnegative outputs, s.t. $1 \le i \le N$, then we ...
- 103
0
votes
0
answers
37
views
separator and vertex-connectivity
A definition of "separator" is the following: Let $G$ is an $n$-vertex graph, then $S\subseteq V(G)$ is a separator if there is a partition $V=A\cup B\cup S$ such that $|A|,|B|\le 2n/3$ and ...
- 281
4
votes
2
answers
590
views
Is this function injective?
For all given ordered lists $$\mathcal A=\big\{\{a_\mu\mid\mu=1,\cdots,N\}\mid\forall\mu,\nu> \mu,\ a_\mu > a_\nu\big\},$$ the function on the quotient space $$ G_\mu(a+\mathbb R: \mathcal A / \...
- 87
9
votes
1
answer
625
views
The reals: a topological lattice in more than the obvious way?
Define a topological lattice as a (not necessarily bounded) lattice in $\textbf{Top}$, i.e. meet and join are continuous maps $X^2 \rightarrow X$. There are two obvious topological lattice structures ...
- 631
0
votes
1
answer
128
views
Chern Classes of $\mathcal{O}_E(1)$ on $\mathbb{P}(E)$ for $E = \mathcal{O} \oplus \mathcal{O}(n) \to \mathbb{P}^2$
Let $E =\mathcal{O} \oplus \mathcal{O}(n) \to \mathbb{P}^2$ and denote by $\mathcal{O}_E(1)$ the dual of the tautological bundle.
How can I compute $c_1^2(\mathcal{O}_E(1)), c_1^3(\mathcal{O}_E(1))$, $...
- 233
1
vote
1
answer
100
views
Is there any known upper bound for the local crossing number of a graph drawing in the plane?
The local crossing number ${\rm LCR(G)}$ of a graph $G$ is defined as the least nonnegative integer $k$ such that the graph has a $k$-planar drawing. In other words, it is the smallest possible number ...
- 1,190
3
votes
2
answers
156
views
On nowhere differentiability of functions that just barely fail to be Lipschitz
By Rademacher’s theorem, Lipschitz functions are differentiable almost everywhere. I am wondering how badly this pointwise differentiability fails for functions that “just barely” fail to be Lipschitz....
- 6,313
7
votes
1
answer
131
views
Classification of modules all whose weight spaces are $1$-dimensional
In type $A$, the simple modules all of whose weight spaces are $1$-dimensional are the $L(n\varpi_1)$ and $L(\varpi_k)$. This can be seen from the fact that dimensions of weight spaces are given by ...
- 820
2
votes
0
answers
127
views
Nonabelian Hodge correspondence for $\mathbb{G}_m$
Please excuse me if this question is too naive. I know very little about the nonabelian Hodge correspondence but I am trying to understand how the correspondence works in the simplest case of the ...
- 3,446
1
vote
1
answer
150
views
Resource request (probability theory, computability theory, algebra)
I'm a first year graduate student trying to explore specific topics I might be interested in researching. Currently, I enjoy algebra, probability theory, and the computability theory side of logic, ...
- 121
0
votes
0
answers
71
views
Second order PDE with Hessian
I am wondering if there is a existence/uniqueness result for the solution to PDE
$$
D^2 u = F (x, u, Du)
$$
with appropriate initial value conditions.
(Just to clarify, $u : \mathbb R^d \to \mathbb R$ ...
- 51
11
votes
3
answers
765
views
Uniform distribution of sequence mod 1
Is it known whether "for most $r$" the sequence $$r \cdot 2^k \bmod 1, \qquad k \in \mathbb N $$ is uniformly disributed in $[0,1]$?
- 355
4
votes
0
answers
130
views
mod $p$ local Galois representation attached to elliptic curves
In the paper, lemma 4.4. The author gives the form of the representation of $G_p$ on $E[p]$ of the form
$$\begin{pmatrix} \varepsilon\chi & *\\0 & \chi^{-1} \end{pmatrix}.$$
Do they assumed ...
- 275
4
votes
0
answers
180
views
Subgroups that conjugate-cover the ambient group
Let $G$ be a finite group, and suppose that a set of proper subgroups $H_1,\dotsc,H_n$ satisfy $G=\bigcup_{g\in G}\bigcup_{i=1}^nH_i^g$, where $H_i^g$ is the conjugate of $H_i$ by $g$. In this case, ...
3
votes
1
answer
213
views
$\nabla \times (F\times \mathbf v)=g$, $\operatorname{div}(\mathbf v)=0$
I want to solve the equation:
$$
\begin{cases}
\nabla \times (F\times\mathbf v)=g, \\
\operatorname{div}(\mathbf v)=0,
\end{cases}\label{1}\tag{1}
$$ where $F$ and $g$ are given vector fields. The ...
- 617
5
votes
1
answer
174
views
Commutativity of pairs of reflective localizations
Suppose there are two classes of morphisms $w_1, w_2$ in $C$ and two
two reflective localizations $L_1: C \overset{\rightarrow}{\hookleftarrow} C^\text{$w_1$-local}: i_1$ and $L_2: C \overset{\...
- 719
6
votes
2
answers
390
views
Continuity of perimeter with respect to metric
Let $\Omega$ be an open set in a closed manifold, $(M^n, g)$. We can define the perimeter as
$$\text{Per}_g(\Omega) = \sup\bigg\{\int_{\Omega} \text{div}_g(T) dVol_g, \; : \; T \in C^1(M, T M), \quad \...
- 337
1
vote
0
answers
38
views
Metric entropy of an ellipsoid
Let $B^d_2$ denote the unit ball of $\ell_2^d$ and let $T$ be an invertible linear map.
Consider the function
$$
H(T) := \log M(TB_2^d, B_2^d),
$$
which is the packing entropy for $TB_2^d$ by $B_2^d$....
- 380
1
vote
1
answer
292
views
General algebraic definition of mirror symmetry
I'm trying to understand the following statement of Hori-Vafa from the algebraic perspective:
The mirror of the Hirzebruch surface $\mathbb{F}_{n}$ is the Landau-Ginzburg model $x+y+\frac{a}{x}+\frac{...
- 305
1
vote
0
answers
55
views
Real-holomorphic Hamiltonian vector fields
Consider a Kähler manifold with complex structure $J$. Is there a characterization of real-valued functions $H$ for which the corresponding Hamiltonian vector field $X_H$ is real-holomorphic, that is, ...
- 121
1
vote
1
answer
76
views
Determinant formula for a certain parametrized M-matrix
Let $P_{ij}$ be variables, and let $A \in \mathbb{R}^{n\times n}$ be the matrix defined by
$$
A_{ij} = \begin{cases}
-P_{ij} & i \neq j,\\
P_{i1} + P_{i2} + \dots + P_{in} & i=j.
\end{cases}
$$...
- 20.2k
296
votes
125
answers
93k
views
What are some examples of colorful language in serious mathematics papers?
The popular MO question "Famous mathematical quotes" has turned
up many examples of witty, insightful, and humorous writing by
mathematicians. Yet, with a few exceptions such as Weyl's "angel of
...
1
vote
0
answers
32
views
$\alpha$ stable processes without jumps
Levy processes with jumps can be formulated following the Levy-kinchkine representation, which provide a decomposition of the characteristic function into three factors corresponding to the diffusion (...
- 305
0
votes
1
answer
89
views
Singular continuous ergodic measures for the map $z \to z^2$
Where can I find the details of constructing singular continuous ergodic measures for the map $z \to z^2$ on the unit circle? I know that it was done by Furstenberg, but I could not find it explicitly ...
21
votes
2
answers
2k
views
Boundedness of sum of sin(sin(n))
Playing with desmos I have accidentally noticed that the sequence of partial sums
$$\left\{ \sum_{n=1}^{N}\sin(\sin(n)) : N\geq 1 \right\}$$
is bounded.
However, I did not succeed in proving this ...
-1
votes
0
answers
53
views
convergence of convolution in Bochner space
I want to prove a well-known fact in $L^p(R^n)$ namely that, the convolution of an element in $L^p$ with an element of $L^1$ is in $L^p$
let: if $u∈L^p (R;X) , f∈L^1 (R)$ and $X$ is Separable and ...
5
votes
1
answer
198
views
Base change for module categories? ($E_\infty$-modules in $\mathrm{Cat}$)
I'm working on a project where I would like to consider the category of symmetric monoidal categories. Though I suspect it will be easier easier to consider the $\infty$-category of symmetric monoidal ...
- 291
1
vote
0
answers
219
views
Quotient of K3 surfaces by non-symplectic automorphism of finite order
Let $X$ be a $K3$ surface and $f: X \to X$ a non-symplectic morphism (ie non symplectic in sense of that that the induced action on $H(X,K_X=H^0(X, \Omega_X^2)$ is not trivial) of finite order.
...
- 5,966
3
votes
1
answer
147
views
Descriptive set theoretic complexity of computable maps with respect to the Turing jump of the input
For natural numbers $e$, $n$ and elements of Cantor space $X$ let $\{e\}^X(n)$ be the result of running the $e$th Turing machine with oracle $X$ on input $n$. Let $X'$ be the Turing jump of X.
Suppose ...
- 2,221