Most active questions
215 questions from the last 7 days
32
votes
1
answer
3k
views
Closed formula for the factorial over naturals
How to prove that there is no formula for $n!$ that does only use the binary operations $+,-,*,/$ on natural numbers, powers of natural numbers, and fixed natural numbers?
The same question over the ...
- 19.1k
13
votes
4
answers
2k
views
The ten most fundamental topics in geometric group theory
What are the ten most fundamental topics in geometric group theory?
This is a pedagogical question prompted by the fact that I am teaching geometric group theory to undergraduates. They are expected ...
16
votes
3
answers
1k
views
Is there a natural topology for sets of topological spaces?
The Gromov–Hausdorff metric makes a set of compact metric spaces into a metric space itself. I am wondering what some natural generalizations there are for arbitrary topological spaces. Namely, is ...
- 719
17
votes
1
answer
1k
views
Are integers conservatively embedded in the field of complex numbers?
I am looking for a reference to the fact that $\mathbb{Z}$ is conservatively embedded into the field $\mathbb{C}$ of complex numbers, that is anything in $\mathbb{Z}$ which is definable in $(\mathbb{C}...
- 301
6
votes
2
answers
739
views
Shifting an irrational binary sequence
Let $\newcommand{\tn}{\{0,1\}^\mathbb{N}}\tn$ be the collection of all infinite binary sequences. For $s\in\tn$ and $k\in\mathbb{N}$ let the left-shift of $s$ by $k$ positions, $\ell_k(s)\in \tn$, be ...
- 48.9k
6
votes
1
answer
2k
views
Difficulty with "A new elementary proof of the Prime Number Theorem" by Richter
I'm studying Richter's "A new elementary proof of the Prime Number Theorem" paper, and I'm finding some problems understanding some parts of it. For example, I don't see how to get, in Lemma ...
- 95
6
votes
1
answer
648
views
Is decomposability of polynomials over a field an undecidable problem?
By a decomposition of a polynomial $F(x)$ over a field $K$ we mean writing $F(x)$ as
$$
F(x)=G(H(x)) \quad(G(x), H(x) \in K[x]),
$$
which is nontrivial if $\operatorname{deg} G(x)>1$ and $\...
- 310
8
votes
1
answer
849
views
What is the smallest and "best" 27 lines configuration? And what is its symmetry group?
I was this past year working with a bright high-schooler on algebraic geometry following Reid's book Undergraduate Algebraic Geometry, and we got all the way to proving that there is at least one line ...
- 35.5k
5
votes
3
answers
343
views
An uncountable measurable subset of $\Bbb R$ containing no nonempty perfect set
$\newcommand\R{\Bbb R}$Assuming the axiom of choice, is there an uncountable Lebesgue-measurable subset $S$ of $\R$ that contains no nonempty perfect set?
Of course, such a set $S$, if it exists, ...
- 128k
7
votes
1
answer
498
views
Invertibility of a matrix defined using inner product
Let $n,m \geq 1$. We fix $n$ distinct vectors $x_1, ... , x_n \in \mathbb{R}^m$. We define $A \in \mathbb{R}^{n\times n}$ as
\begin{equation}
A_{ij} = x_i^T \left(n x_j - \sum_{1 \leq k \leq n} x_k \...
- 2,306
6
votes
2
answers
403
views
closed form for an alternating cosecant sum
Is there any closed form for the following finite sum
$$\sum_{j=1}^{n-1}\frac{(-1)^j}{\sin (\frac{j\pi}{n})}$$
where $n$ is an even number?
Any comment or reference is welcome.
- 701
3
votes
3
answers
403
views
Huygens' principle or finite speed of propagation?
I am reading the paper Large global solutions for energy supercritical nonlinear wave equations on $\mathbb{R}^{3+1}$ by Krieger and Schlag and am confused by one of their steps.
For context, $v(t,r)$ ...
- 890
5
votes
3
answers
296
views
Asymptotics for minimum of a sequence of random variables
This is a question that I'm sure has been investigated before, but I have found no good search terms for.
Let $X_i$ be a sequence of random variables, independent and uniformly distributed in $[0,1]$. ...
- 28.2k
5
votes
1
answer
238
views
Galois action on Borovoi's algebraic fundamental group
In Borovoi's paper Abelian Galois cohomology of reductive groups, the algebraic fundamental group of a connected reductive group $G$ over a field $K$ of characteristic zero is defined as
$$\pi_1(G, T):...
- 53
4
votes
1
answer
715
views
Can the Pythagorean theorem be proved using imaginary numbers?
Can the Pythagorean theorem be proved using imaginary numbers? The proof must avoid circular reasoning, of course.
I asked essentially the same question at MSE, but did not receive a definitive answer,...
- 3,567
9
votes
1
answer
330
views
An elementary proof of the equivalence of the Bol and Moufang identities
By a well-known result of Bol (1937) and Bruck (1946), for any loop the following two identities are equivalent:
B: $x(y(xz))=((xy)x)z$
M: $(xy)(zx)=(x(yz))x$.
A proof of the equivalence (B)$\...
- 42k
3
votes
1
answer
316
views
Which abelian varieties over a local field can be globalized?
As the title says, if $\mathcal{A}$ is an abelian variety over $\mathbb{Q}_p$, is there a criterion as to if I should expect there to exist $A$ over $\mathbb{Q}$ such that
$$\mathcal{A}\cong A\times_{\...
- 4,418
8
votes
1
answer
318
views
Fibers of generic smooth maps between manifolds of equal dimension
I have heard that the following is a "well-known"
Claim. Let $M$ and $N$ be smooth manifolds with equal dimensions and $M$ compact. Then a generic smooth map $f\colon M\to N$ has finite ...
- 501
12
votes
1
answer
590
views
+200
Fundamental group of the complement of a codimension two submanifold
Let $M$ denote an arbitrary smooth, closed, connected, n-dimensional manifold for $n\geq 4$. For every such $M$, does there exist a closed (not necessarily connected!) codimension two submanifold $S \...
- 1,174
8
votes
1
answer
230
views
Preserving non-conjugacy of loxodromic isometries in a Dehn filling
Suppose that $g$ and $h$ are non-conjugate loxodromic isometries in a cusped hyperbolic $3$-manifold $M$ of finite volume. Fix a cusp $T$ of $M$. Can I choose a hyperbolic Dehn filling of $M$ along $...
6
votes
1
answer
563
views
Destroying scales
Suppose $\lambda$ is a cardinal with uncountable cofinality and $C\subseteq\lambda$ is a club of ordertype cf$(\lambda)$. Let $f=\langle f_\xi\mid\xi<\lambda^+\rangle$ be a scale consisting of ...
- 533
4
votes
1
answer
275
views
Maximum density of sum-free sets with respect to Knuth's "addition"
A subset $S\subseteq\mathbb{N}$ is said to be sum-free if whenever $s,t\in S$, then $s+t\notin S$. For instance the set of odd numbers is sum-free and has (lower and upper) asymptotic density 1/2.
...
- 48.9k
3
votes
1
answer
167
views
Theory of $n$-truncated $A_\infty$ categories/functors?
One can define certain higher categorical truncations. For example, a discussion from the quasi-categories point of view can be found in HTT 5.5.6.
On the other hand, as a model of linear $\infty$-...
- 169
7
votes
1
answer
203
views
Is there a more natural way to define the Young symmetrizer and the Specht module?
It's well known that Young symmetrizer is a fundamental tool in the representation theory of symmetric groups.
For instance, for every Young diagram $\lambda\vdash n$, we construct a Young tableau $T_\...
3
votes
2
answers
187
views
Algorithms (or packages) to find recurrence relations for given sequence of q-polynomials?
Assume we have sequence of polynomials : $P_i(q)$ - each term is polynomial in $q$. (With integer coefficients, but hopefully it is not important).
We expect that there exists recurrence relation a ...
- 24.9k
5
votes
1
answer
153
views
Does there exist a section of $\mathcal{P}(\kappa)\to\mathcal{P}(\kappa)/(\text{fin})$ that is "nearly Boolean"?
The following might be a somewhat esoteric question:
Does there exist an infinite cardinal $\kappa$ and a section $f$ of the quotient map $\pi:\mathcal{P}(\kappa)\to\mathcal{P}(\kappa)/(\text{fin})$ (...
- 2,830
1
vote
1
answer
170
views
On the condition of preadditive categories being locally small
The theory of categories is more flexible when not adding the (quite common) condition of being locally small. So the general notion of a category is the following (assuming we have a suitable ...
- 63.1k
7
votes
1
answer
313
views
Kobayashi-Nomizu "Foundations of differential geometry" on page 117 wrong?
$\DeclareMathOperator\GL{GL}$Let $M$ be a smooth manifold, $G$ a Lie group and $P(M,G)$ a principal $G$-bundle and $\rho: G \to \GL(V)$ of $G$ a representation with $V$ finite-dimensional $\mathbb{F}$-...
- 331
4
votes
1
answer
181
views
Projective automorphisms of a plane cubic curves
Let $E$ be a smooth cubic curve in $\mathbb{P}^2$ over an algebraically closed field $k$.
What is the group of the projective transformations preserving $E$ ?
In characteristic $0$ the answer is known ...
- 486
7
votes
1
answer
226
views
Size doubling amoeba
Note: Here time is assumed to be discrete and indexed by the naturals $\mathbb N$.
A curious species of amoeba undergoes the following evolution rule - at each time step, with probability $0 < p &...
- 6,321
5
votes
1
answer
140
views
$\ell$-adic analogue of Kedlaya-Mochizuki
There is a well-known analogy between holonomic $\mathcal{D}$-modules on complex algebraic varieties and $\ell$-adic perverse sheaves on varieties over finite fields. Many theorems in one setting have ...
- 771
4
votes
1
answer
283
views
Eigenvalue of a convolution and a restriction?
Let $\epsilon>0$ be small. Let $\eta(t) = \frac{2\epsilon}{\epsilon^2+(2\pi t)^2}$ (the Fourier transform of $x\mapsto e^{-\epsilon |x|}$). Let $V$ be the space of integrable, bounded functions $f:\...
- 20.2k
2
votes
2
answers
127
views
Existence of k-complete uniform ultrafilter over a regular cardinal, k is strongly compact
This is a question about set theory. Let $\kappa\leq \lambda$ be infinite cardinals such that $\kappa$ is strongly compact and $\lambda$ is regular. My question is: how to construct a $\kappa$-...
- 61
2
votes
1
answer
145
views
Exotic Hopf algebra structures on the $p$-fold direct product in characteristic $p > 0$
Let $k$ be an algebraically closed field of characteristic $p > 0 $ and let $A$ be an algebra over $k$, which is a local ring.
There is an isomorphism of algebras $\prod_{i=1}^p A \cong A \otimes k[...
- 542
-4
votes
1
answer
163
views
What are all the complex structures on $\mathbb{R}^2$ which live inside $\mathrm{SL}_2(\mathbb{Z})$? [closed]
By "complex structure" I am referring to 2x2 matrices which square to $-\mathrm{Id}_2$. I need to know those with integer entries and determinant equal to 1.
Thank you
- 3
3
votes
1
answer
142
views
Whitney stratifications of hypersurfaces
Suppose that $X$ is a Whitney stratified algebraic variety with strata $\{S_i\}.$
Suppose that $Z$ is a hypersurface of $X$ which transversely intersects all strata of $X$, i.e. $S_i \cap Z$ is a ...
- 55
5
votes
2
answers
67
views
On the continuity a function given by evaluating compact subsets of smooth functions
Let $M$ be a compact connected smooth manifold. Write $C^{\infty}(M)$ for the Frechet space of the smooth real-valued functions on $M$ equipped with the usual $C^{\infty}$-topology.
Given a compact ...
- 557
3
votes
1
answer
141
views
Surjectivity of pushforward on image
Let $\mathcal X\subseteq\mathbb R^m$ be a Borel measurable set. $\Phi:\mathcal X\to\mathbb R^n$ be a continuous mapping and $\mathcal Y = \Phi(\mathcal X)\subseteq\mathbb R^n$ its image. Let $\mathcal ...
- 345
8
votes
0
answers
365
views
Closed formula for the factorial over reals
How to prove that there is no formula for $n!$ that does only use the binary operations $+,-,*,/$ on real numbers, powers of real numbers, and fixed real numbers?
Similar question have been asked ...
- 19.1k
5
votes
1
answer
128
views
Algebras over the trivial $\infty$-operad
I'm learning the concept of algebras over $\infty$-operads, following Higher Algebra. The simplest case is when the operad being the trivial operad $\mathrm{Triv}^\otimes$, defined as the 1-full ...
- 165
4
votes
1
answer
170
views
Diamonds at $\omega_2$ under PFA
Let $\lambda$ be a cardinal, and $S\subset \lambda^+$ be stationary. A $\diamondsuit^+(S)$-sequence is a sequence $\langle \mathcal{A}_\alpha\mid \alpha\in S\rangle$ such that each $\mathcal{A}_\alpha\...
- 61
-3
votes
1
answer
104
views
when does $h$ exist?
Let $\zeta(s)$ denote the Reimann zeta function in the critical strip. It is easy to see that $$ \zeta(s) = 0 \Longleftrightarrow \Re(\zeta(s))+\Im(\zeta(s)) = 0 ~~~~ \text{and} ~~~~~~ \Re(\zeta(s)) \...
1
vote
1
answer
119
views
Rational functions on elliptic curves over global fields with given support
Let $E$ be an elliptic curve over a global field $k$. Let $x_1, \dots, x_r$ be a set of generators of $E(k) / E(k)_{tor}$ (or more generally, a $\mathbb Q$-basis of $E(k)_{\mathbb Q}$), and let $x_0$ ...
- 77
2
votes
1
answer
150
views
Converse of Scherk–Segre theorem on the number of vertices of a convex space curve
It is well known that any smooth simple closed convex curve $\gamma$ in $\mathbb{R}^{3}$ that meets no plane in more than 4 points has exactly 4 vertices, i.e., points of vanishing torsion; here "...
- 985
13
votes
0
answers
304
views
What is the status of this conjecture on symplectic forms "standard-at-infinity" on $\mathbb{R}^{2n}$?
In McDuff and Salamon's Introduction to Symplectic Topology, the following open problem is mentioned.
Problem 50 (Standard-at-infinity)
Let $n \ge 3$ and let $\omega$ be a symplectic form on $\mathbb{...
5
votes
1
answer
80
views
Measure dependance of groupoid von Neumann algebra
Let $(G,\mu)$ be a measured groupoid and denote by $\nu,\nu^{-1}$ the measures on all of $G$ induced by $\mu$ and Haar system $\{\lambda^x\}$.
I have a question regarding the dependance of the ...
- 311
7
votes
1
answer
80
views
A syntactic characterisation of morphisms of algebraic theories whose induced algebraic functors admit right adjoints
Let $f : S \to T$ be a morphism of algebraic theories. Such a morphism induces a monadic functor $f^* : \mathrm{Mod}(T) \to \mathrm{Mod}(S)$ (hence $f^*$ has a left adjoint). We may view $f$ ...
- 10.7k
0
votes
1
answer
105
views
If $u \in H^2(\mathbb{R}^3)$, does $r^{-1} u \in H^{\alpha}(\mathbb{R}^3)$ for some $\alpha > 0$?
Let $u$ belong to the Sobolev space $H^1(\mathbb{R}^3)$. We have the classical Hardy inequality
\begin{equation*}
\int_{\mathbb{R}^3} \frac{|u|^2}{|x|^2} dx \le 4\int_{\mathbb{R}^3} |\nabla u(x)|^2 dx,...
- 481
2
votes
1
answer
142
views
Does this result above six points follow have a name?
Does this result above six points follow have a name?
Let $A$, $B$, $C$, $D$, $E$, $F$ be six points in the plane and $AB, CF, ED$ are concurrent and $BC, DA, FE$ are concurrent then $CD, EB, AF$ ...
- 1,776
5
votes
1
answer
109
views
Do the order statistics give a good approximation of uniform random variables?
Let $\{X_i\}_{i \geq 1}$ be a sequence of iid uniform random variables on $[0, 1]$. Define, for each $n$, the order statistic $O_n$ of $X_n$ by
$$O_n := \frac{1}{n}\#\{1 \leq k \leq n \, \, | \, X_k \...
- 6,321