Most active questions
781 questions from the last 30 days
1
vote
1
answer
185
views
+50
On a probabilistic integer factorization algorithm given bounds for one prime factor
We got a probabilistic integer factorization algorithm and experimental evidence with large
integers given bounds for one factor.
Let $D \ge 2$ be real number and let $p,q$ be primes and $N=pq$.
...
- 25.4k
1
vote
0
answers
375
views
How to verify if sets satisfying cardinality condition exist? [migrated]
I am trying to find out if sets satisfying the following properties exist:
Call the sets $A_1, \ldots, A_{20}$ and $B_1, \ldots, B_{20}$.
For each $i \in \{1, \ldots, 20\}$, $|A_i| \in \{1,2\}, |B_i| ...
- 119
2
votes
1
answer
144
views
Inequality for hermitian matrices
Let $p_1$ and $p_2$ be a complete system of orthogonal projections on $\mathbf R^n, n \geq 2$ (i.e., $p_i^2=p_i=p_i^*$ and $p_1+p_2=\bf{1}$) and $S_1, S_2$ be two hermitian operators such that $S_i \...
- 73
3
votes
1
answer
161
views
How to check whether a triangulated subcategory is admissible?
Let $\mathcal{T}' \subseteq \mathcal{T}$ be a full triangulated subcategory. Recall, $\mathcal{T}'$ is called $\textit{right admissible}$ if the inclusion $\mathcal{T}' \hookrightarrow \mathcal{T}$ ...
- 127
3
votes
1
answer
226
views
Finite generativity of algebra with valuation
Let $C$ be a commutative finitely generated algebra with no zero divisors. If necessary, we can assume it to be graded and a unique factorization domain. Let $a\in C$ be a prime element.
Let's also ...
5
votes
0
answers
366
views
A Collatz-like map?
Consider the map $\psi$ acting on triples $(a\leq b\leq c)$ of three positive natural integers with $\mathrm{gcd}(a,b,c)=1$ as follows:
Set $$(a',b',c')=\left(\frac{a}{\mathrm{gcd}(a,bc)},\frac{b}{\...
- 17.9k
14
votes
0
answers
392
views
Can the axiom of choice be expressed in 4 quantifiers?
This 2007 paper presents a 5-quantifier $(\in, =)$-expression that is ZF-equivalent to the axiom of choice, but leaves open the 4-quantifier case:
Thus the gap is reduced to the undecided case of a 4 ...
- 2,213
6
votes
1
answer
131
views
Condition for a functor to induce a cartesian closed functor between categories of presheaves
We denote the category of presheaves on a small category ${\cal C}$ (set-valued functor-category) by $$\widehat{\cal C}:=[{\cal C}^{op},{\bf Set}].$$
Such a category is cartesian closed, i.e. it ...
- 567
6
votes
2
answers
238
views
Is the (inverse) Dold-Kan functor fully faithful on chain complexes of commutative monoids?
My question
The background/notation for all of the content of this post is in Lurie, Higher Algebra [HA], Ch. 1.2.3. Everything is purely 1-categorical.
Let $\mathcal{A}$ a semiadditive category (with ...
- 391
2
votes
1
answer
239
views
Self-adjointness of generator and semigroup of an SDE
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bT}{\mathbb{T}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\eps}{\...
- 825
3
votes
1
answer
141
views
Surjectivity of pushforward on image
Let $\mathcal X\subseteq\mathbb R^m$ be a Borel measurable set. $\Phi:\mathcal X\to\mathbb R^n$ be a continuous mapping and $\mathcal Y = \Phi(\mathcal X)\subseteq\mathbb R^n$ its image. Let $\mathcal ...
- 345
5
votes
1
answer
128
views
Algebras over the trivial $\infty$-operad
I'm learning the concept of algebras over $\infty$-operads, following Higher Algebra. The simplest case is when the operad being the trivial operad $\mathrm{Triv}^\otimes$, defined as the 1-full ...
- 165
3
votes
1
answer
133
views
Is the interval topology on ${\cal P}(\omega)/(\text{fin})$ connected?
If $(P,\leq)$ is a poset and $x\in X$, we let $\downarrow x = \{p\in P: p \leq x\}$, and $\uparrow x$ is defined dually. The collection $$\Big\{P\setminus (\downarrow x): x\in P\Big\} \cup \Big\{P\...
- 48.9k
0
votes
2
answers
100
views
Is there an uncountable extension of the Ramsey set $[\omega]^2$?
We say that a family ${\cal A}\subseteq {\cal P}(\omega)$ is Ramsey
if for every map $c:{\cal A}\to\{0,1\}$ there is an infinite set $X\subseteq \omega$
with the following properties:
${\cal A}\cap {\...
- 48.9k
4
votes
1
answer
172
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
2
votes
1
answer
157
views
Does the support of a smooth compactly supported function have a smooth boundary?
Let $U$ be a bounded domain in $\mathbb{R}^n$ and $f:U\to \mathbb{R}^n$ an infinitely differentiable function with compact support. My question is whether $\operatorname{supp}(f)$ has a smooth ...
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
1
vote
1
answer
158
views
Comparison of special metrics on Riemann surfaces with the hyperbolic one
Let $X$ be a complex algebraic curve, or equivalently a compact Riemann surface, of genus $a\ge2$. Denote $\omega_1, \dots , \omega_a$ a basis of holomorphic differentials, and consider the Riemaniann ...
- 31
3
votes
1
answer
103
views
Action of point stabilizers in finite doubly transitive groups
Suppose that $(H,X)$ is a finite faithful doubly transitive permutation group (where $H$ acts on the set $X$). Moreover, suppose that $H$ also acts doubly transitively (and faithfully) on a set $Y$, ...
- 4,575
-3
votes
1
answer
105
views
when does $h$ exist?
Let $\zeta(s)$ denote the Reimann zeta function in the critical strip. It is easy to see that $$ \zeta(s) = 0 \Longleftrightarrow \Re(\zeta(s))+\Im(\zeta(s)) = 0 ~~~~ \text{and} ~~~~~~ \Re(\zeta(s)) \...
2
votes
1
answer
201
views
Section 3 of Atiyah's "On analytic surfaces with double points" — some questions
I have some questions about section 3 of Atiyah's "On analytic surfaces with double points," a short 9 page paper. Section 3 is all dedicated to proving lemma 4.
Near the end of section 3, ...
- 129
0
votes
1
answer
143
views
Why are the homeomorphisms from the unit circle to the unit circle preserving measure affine? [closed]
Why are the homeomorphisms from the unit circle to the unit circle preserving measure affine? The affine is composition of rotation and continue automorphism.
- 83
2
votes
1
answer
156
views
Question about coverings of zero Hausdorff measure compact sets
Consider a compact set $E\subset \mathbb{R}^n$ ad assume that ${\cal H}^{n-1}(E)=0$.
If $\epsilon>0$, there is a finite covering of $E$ made of $N_\epsilon$ open balls $B_{r_{\epsilon,k}}(x_{\...
- 205
3
votes
1
answer
144
views
Jordan plane curve such that $\frac{d(g(x),g(y))}{d(x,y)}\to0$?
Write $g$ as the inverse of $f$.
Is there a continuous injective $f:S^1\to C\subset\mathbb{R}^2$ such that
$$
\displaystyle\sup_{d(x,y)<r}\dfrac{d(g(x),g(y))}{d(x,y)}\to0
$$ as $r\to0$?
If you like,...
- 211
0
votes
2
answers
126
views
Unique coupling
Let $X$ be a Polish metric space, and let $\mu,\nu$ be two Borel probability measures on $X$, when is the product measure the only coupling of $\mu$ and $\nu$. More formally, let $$\Gamma(\mu,\nu):=\{\...
- 245
4
votes
2
answers
165
views
Connectedness of degeneracy loci
Let $E, F$ be vector bundles of ranks $e, f$ on a smooth variety $X$ of dimension $n$.
Let $\varphi : E \to F$ be a morphism and let $k$ be such that $n > (e-k)(f-k)$.
Fulton-Lazarsfeld's theorem ...
- 331
2
votes
1
answer
131
views
Strong ultralimits?
I was going through the book Ultrafilters Throughout Mathematics and I came across the notion of ultralimits, defined below.
Ultralimit. Let $(X,\tau)$ be a topological space, $(x_i)_{i\in I}$ be a ...
- 23
2
votes
1
answer
162
views
Are Cohen Generics Minimal Covers?
Are Cohen generics (in $2^\omega$) minimal covers?
I'm ultimately interested in this question for some more effective notion of forcing but I realized I wasn't sure how to show this even assuming full ...
- 3,029
1
vote
1
answer
120
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
1
vote
1
answer
137
views
About dimensions of quotients of quasi projective varieties
This question is related to this one. If I have an locally closed, quasi projective scheme $X$ contained in an affine space, and a linearly reductive group $G$ acting freely on $X$, are there examples ...
- 558
4
votes
1
answer
281
views
A question in spin geometry in dimension 8
$\DeclareMathOperator\trace{trace}\DeclareMathOperator\End{End}\DeclareMathOperator\Trace{Trace}$This is to understand a very specific isomorphism in dimension $8$. In dimension $4$ for a spin$^c$ ...
- 954
2
votes
1
answer
155
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
2
votes
1
answer
169
views
Ratio of inscribed/circumscribed ellipsoids: geometrical proof?
Let $K$ be a convex subset of ${\mathbb R}^n$, with non-void interior. The Löwner-John theorem states that there are a minimal volume ellipsoid $\cal E$ containing $K$, a maximal one $\cal F$ ...
- 52.4k
13
votes
0
answers
305
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{...
0
votes
1
answer
188
views
Does the second Bourgain–Delbaen space belong to C_p?
The second Bourgain–Delbaen space $Y$ is a separable $\mathcal{L}_\infty$ space such that $Y$ contains no copy of $c_0$ or $\ell_1$. The dual space is isomorphic to $\ell_1$.
An operator $T:X\to Y$ ...
0
votes
1
answer
106
views
Curious about methods for finding Goldbach pairs for large even numbers
I am exploring the question of efficiently identifying two prime numbers that sum to a given large even number, particularly for even numbers exceeding 100 digits. While brute force and precomputed ...
- 1
11
votes
0
answers
333
views
+50
Sobolev's PDE Scottish Book Problem (Problem 188)
In 1940 Sobolev recorded the following problem in the Scottish Book, and offered a bottle of wine for a solution.
In 2015, when the second edition of the Scottish Book with updates and commentary on ...
- 13k
4
votes
0
answers
267
views
Is a pro-algebraic group over $\mathbb{Q}_p$ with Galois action the inverse limit of Galois-equivariant quotients?
Let $\mathcal{G}$ be a pro-algebraic group over $\mathbb{Q}_p$ with a continuous action of $G_K$ for a field $K$ (if $\mathcal{G}$ were an abelian unipotent group, this is precisely a $p$-adic Galois ...
- 15.4k
5
votes
1
answer
132
views
Relation between Tor amplitude and $p$-complete Tor amplitude for a ring of characteristic $p$
Fix a prime number $p$. Let $A$ be a commutative ring, and consider an $A$-algbera $B$ of characteristic $p$. So we have a sequence of ring homomorphisms
$$
A \to A/pA \to B.
$$
Assume that we want to ...
- 125
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
2
votes
1
answer
135
views
Properness of quotient map
I am new to algebraic spaces and stacks. My question is the following:
Let $X$ be a scheme and $G$ be a group scheme action on $X$. Let $[X/G]$ be the quotient stack. Then when the natural map $\pi: ...
- 613
1
vote
1
answer
214
views
Derived completeness of the inverse perfection
Fix a prime number $p$, and let $R$ be a ring of positive characteristic $p$. Consider the inverse perfection of $R$, which is defined as the inverse limit
$$
R^\flat = \varprojlim(\cdots \xrightarrow{...
- 125
12
votes
0
answers
344
views
Does every finite group have a small projective representation (over some ring)?
Question. Let $G$ be a finite group. Can we find some (commutative) ring $R$ and some positive integer $d=O(\log\lvert G\rvert)$ such that $G$ can be found as a subgroup of $\operatorname{PGL}_d(R)$?
...
- 602
1
vote
0
answers
265
views
Fourier transform of fat Cantor set
Let $C_n$ be the set obtained in the $n$-th iteration of the construction of the Smith-Volterra Cantor set, obtained by removing at the $n$-th step $2^{n-1}$ middle intervals of amplitude $1/4^n$. ...
- 213
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
143
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
8
votes
0
answers
282
views
Meaning of the Ehrhart polynomial at $-1/2$?
I am studying a large collection of lattice polytopes, all of them being simple and empty. The dimension can be any integer. The dilatation by $2$ gives non-empty polytopes.
For many of these ...
- 3,587
3
votes
1
answer
138
views
Can the coefficients of a Taylor series be expressed as rational functions for an affine variety?
Let $k$ be an algebraically closed field and $V \subseteq k^n$ an affine variety corresponding to a prime ideal $P \subseteq k[t_1, \dots, t_n]$. For $x\in V$ let $O_x = \{p/q \mid p,q\in k[t_1, \dots,...
- 696
1
vote
0
answers
267
views
Are there connections between Calabi-Yau manifolds and number theory?
I am interested in understanding whether there are any significant connections between Calabi-Yau manifolds and number theory. Calabi-Yau manifolds are central objects in algebraic geometry and string ...
4
votes
0
answers
277
views
Derek the Differentiable Dinosaur
I’ve come across several fond references to some semi-published lecture notes from Warwick in the 80s, by Bill Breckon (and, in some mentions, I. Harrison), Differentiating functions of lots of ...
- 21.3k