Newest Questions
159,019 questions
6
votes
1
answer
479
views
How to define a fractal from the lexicographic sorting on the prime factorization of natural numbers?
Consider on the natural number the lexicographic ordering on the prime factorization:
If $m = p_1^{a_1}\cdots p_r^{a_r},n = q_1^{b_1}\cdots q_s^{b_s}$ then we define:
$$m \vartriangleleft n :\iff [(...
- 3,074
1
vote
0
answers
101
views
Explicit central elements of $\mathcal{U}(\mathfrak{so}(4,1))$
I am interested in finding the central elements of the universal enveloping algebra of the Lie algebra $\mathfrak{so}(4,1)$.
Notation: the 10 generators are $D, J_i, P_i, K_i$ ($i=1,2,3$), satisfying ...
- 163
2
votes
1
answer
299
views
An example of a geometrically simply connected variety with infinite Brauer group (modulo constants)
$\DeclareMathOperator\Br{Br}$Let $X$ be a smooth, geometrically integral, geometrically simply connected variety over a numberfield $k$. Is it possible to have $\Br(X)/{\Br(k)}$ being an infinite ...
- 177
2
votes
1
answer
76
views
Finite pair-splitting family of $\mathbb{N}$
This is a kind of "dual" of an older question.
Is there a finite family ${\frak F}\subseteq {\cal P}(\mathbb{N})$ such that for all $a\neq b\in\mathbb{N}$ there is $S\in{\frak F}$ with $|S\...
- 48.9k
5
votes
0
answers
497
views
Is $\int \operatorname{sn}^2u\,\mathrm du$ really irreducible?
Let $\operatorname{sn}$, $\operatorname{cn}$, $\operatorname{dn}$ be Jacobian elliptic functions (https://dlmf.nist.gov/22). According to Greenhill,
The integrals
$$\operatorname{sn}^2u,\operatorname{...
- 317
16
votes
1
answer
770
views
Find a special integer coefficients polynomial which takes small absolute value on [0,4]
The question is easy to state: Is there a non-constant $f\in\mathbb{Z}[x]$ such that for all $x\in [0,4]$, we have $|f(x)|\leq 1$? I do not know where to find a useful reference for it.
I did a few ...
- 287
1
vote
0
answers
186
views
Converse of the relative canonical bundle $-K_{X/Z}$ of smooth morphism can not be ample?
Here is my question which is a classical result:
Let $f:X\to Z$ be a surjective smooth morphism between smooth projective varieties over an algebraically closed field $k$ of characteristic zero. Let $...
- 249
1
vote
0
answers
52
views
Complexity of the TSP for hypercube graphs
Question:
what is known about the complexity of finding the Hamilton cycle of minimum weight in graphs that resemble hypercubes with weighted edges?
- 13.2k
5
votes
1
answer
356
views
Diophantine equation $\cos(2\pi x)\cos(2\pi y) = \cos(2\pi z)$
While working on finite order elements of $\operatorname{SO}_n$, I meet this question:
Find all identities of the form $\cos(2\pi x)\cos(2\pi y) = \cos(2\pi z)$ with $x, y, z$ rational numbers.
As ...
- 3,432
1
vote
1
answer
108
views
If all mixed partials of a $C^1$ function exist and are continuous, is the function $C^2$? [closed]
For $n \geq 2$, let $f: \mathbb R^n \to \mathbb R$ be a $C^1$ function such that the mixed partial derivatives $\partial_i \partial_j f$ exist and are continuous for all $i \neq j$. Is it true that $f$...
- 6,275
2
votes
1
answer
192
views
Does every real number $r\in [0,1]$ have a rational sequence $q_n\to r$ s.t. $q_n$ has (simplified) denominator $n$? [closed]
This seems pretty trivial but I can't seem to figure it out. I think it's obviously true, given an unconstrained convergent sequence we just have to add some filler elements, but I'm having trouble ...
- 291
2
votes
0
answers
73
views
Derivative of a functional involving integral and level set
Let $\Omega$ be a bounded smooth domain. For $u\colon \Omega \to \mathbb{R}$, define the functional
$$F(u) = \int_{\{u=0\}}g(x) \; \mathrm{d}x$$
where eg. $u \in H^2(\Omega) \cap C^0(\bar\Omega)$ and ...
- 29
2
votes
1
answer
107
views
To find the convex planar region minimizing diameter when area and perimeter are given
The basic question is to find that planar convex region for which diameter is a minimum when area and perimeter are specified.
A partial answer is given here: http://nandacumar.blogspot.com/2012/11/...
- 5,979
4
votes
1
answer
299
views
Non-vanishing of archimedean integral representations
Let $\psi$ denote a non-trivial additive character of $\mathbb{R}$ and $n$ be a positive integer. Let $(\pi,V)$ and $(\pi',V')$ be two irreducible generic Casselman-Wallach representations of $G_n=\...
- 145
3
votes
0
answers
177
views
What is the expected size of the complement of the union of random cosets of the prime ideals of $\mathbb{Z}$?
For each rational prime $p$ let $\mathbf{X}_p$ denote the random variable uniformly distributed in $\{0, 1, ..., p-1\}$, with all the $\mathbf{X}_p$ independent of each other. Define the coset $\...
- 2,858
1
vote
0
answers
240
views
Examples of when $X$ is homotopy equivalent to $X\times X$
I was thinking about this question the other day: When is a topological space $X$ homotopy equivalent to $X\times X$ (with the product topology)? This is essentially a cross-post of this MSE question.....
3
votes
0
answers
82
views
Dirichlet-to-Neumann map is analytic
Let $M^n$, $n \geq 2$, be a compact smooth manifold with boundary and let $I \ni t \mapsto g_t$ be an analytic (with respect to t) $1$-parameter family of Riemannian metrics on $M$. For each $t \in I$,...
- 2,095
4
votes
1
answer
451
views
Does limitation of size imply axiom of powerset in Morse-Kelly if the generalized continuum hypothesis is included in Morse-Kelley set theory?
Suppose Morse-Kelley set theory consists of class comprehension, class foundation, class extension, axiom of infinity , limitation of size, and the general continuum hypothesis. Can the axiom of ...
- 481
0
votes
0
answers
178
views
Order of elements in amalgamated free products
Reading the book "A Course in the Theory of Groups" by D. J. S. Robinson, I was looking at the proof of 6.4.3 (iii), which states (suppose we are in the case of two groups): if $G_1$ and $...
- 119
8
votes
0
answers
178
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Do there exist Calabi-Yau 3-folds that contain a finite number of elliptic curves?
The moduli space $M_1(X, e)$ of degree $e$ elliptic curves on $X$ has virtual dimension zero if $X$ is a Calabi-Yau 3-fold. I am wondering if there is an example of such an $X$ so that each $M_1(X, e)$...
- 3,645
0
votes
0
answers
165
views
Are all infinite-dimensional Lie groups noncompact?
Basically what the title says — if a Lie group is infinite-dimensional, is it necessarily noncompact?
- 194
15
votes
0
answers
398
views
Will a unit disk be completely covered by randomly placed disks of area $\pi,\frac{\pi}{2},\frac{\pi}{3},\dots$ with probability $1$?
On a "bottom" disk of area $\pi$, we place "top" disks of area $\pi,\frac{\pi}{2},\frac{\pi}{3},\dots$ such that the centre of each top disk is an independent uniformly random ...
- 3,567
9
votes
1
answer
306
views
Two notions of generalized quotient/substructure
Given a language $\Sigma$ and a $\Sigma$-algebra (in the sense of universal algebra) $\mathcal{A}=(A;\dotsc)$ and a function $f:A\rightarrow A$, let $\mathcal{A}_f$ be the $\Sigma$-algebra whose ...
- 20.5k
2
votes
1
answer
94
views
Are the injections of a coproduct a cover in the canonical pretopology?
Assume we're in a category $C$ with all pullbacks and finite coproducts.
Recall that the canonical coverage of $C$ is the finest Grothendieck (pre) topology for which all representables are sheaves. A ...
- 237
3
votes
0
answers
83
views
Embedding theorems for Dini continuous functions
Are there embedding theorems for the space of Dini continuous functions on a Euclidean domain, or even just on an interval? Ideally, I am looking for something like the classical Morrey inequalities ...
- 3,388
1
vote
1
answer
142
views
Diagonalizability, orthogonal diagonalizability of higher order tensors and their being or not being dense in some suitable topology
For our discussion, we'll assume that we're working with $\mathbb{R}^m$ only, but much or all of the following discussion should be carried over immediately to any finite dimensional inner product ...
- 1,467
14
votes
2
answers
2k
views
Are Berkeley cardinals easier to refute in ZFC than Reinhardt cardinals?
Kunen showed that Reinhardt cardinals are inconsistent in ZFC. But his proof is a bit technical for a non-set-theorist to follow. Berkeley cardinals are stronger than Reinhardt cardinals. You can ...
- 64k
4
votes
1
answer
211
views
Ergodic actions and deviation from invariance
Let $M$ be a von Neumann algebra and let $(\phi_t)$ be an ergodic point-$\sigma$-weakly continuous one-parameter group of automorphisms $\phi_t\in \mathrm{Aut}(M)$, i.e., $\Vert\omega-\omega\circ\...
- 769
0
votes
0
answers
142
views
Calculating the expected hitting time of a specific birth and death chain
I am working with a specific birth and death chain, defined as follows.
Consider a set of states $X = \{0,1,2,...,n\}$, where $x^* \in (0,n)$ is a recurrent state. Transition probabilities are defined ...
2
votes
0
answers
109
views
Automorphism group of the first Weyl field
A related question is this one (Automorphism group of the quantum Weyl field).
Let $A_1$ denote the rank 1 Weyl algebra (over the complex numbers), and $D_1$ its skew field of fractions, called the ...
- 3,318
2
votes
0
answers
133
views
Rigorous QFT from integration over subspace
Many perturbative QFTs suffer from the lack of a rigorous
definition of a "good enough" measure over the space of paths (or
fields) $P$,
$$\mathcal{Z} = \int_{{x \in P}} e^{iS(x)} Dx$$
There ...
- 5,230
3
votes
0
answers
130
views
Series acceleration for $\sum_{k=0}^\infty\left(\frac{H^k}{k!}\right)^\beta$, $\beta\ll 1$
The probability mass of the Conway-Maxwell-Poisson variable $K$ is given by
$$
\mathsf P(K=k)=\frac{1}{Z(H,\beta)}\left(\frac{H^k}{k!}\right)^\beta
$$
where
$$
Z(H,\beta)=\sum_{k=0}^\infty\left(\frac{...
1
vote
0
answers
33
views
Eigendecomposition of hyper-complex multiplication
There is an isomorphism between quaternions and $4\times 4$ matrices:
$$
\phi: a+bi+cj+dk \longmapsto \begin{pmatrix}
a&b&c&d \\
-b&a&-d&c\\
-c&d&a&-b\\
-d&-c&...
- 1,171
2
votes
1
answer
202
views
Finite $k$-set-respecting splitting of $\mathbb{N}$
Motivation. My sons participated in a large football tournament recently; everyone wanted to be in a team with everyone else at least once. Tricky!
Formulation of the question. For any positive ...
- 48.9k
0
votes
1
answer
117
views
Decompositions of $\partial_i$ to the radial direction and rotations in higher dimensions
We know in dimension $3$,
\begin{align}
\partial_{i}= \frac{x_i}{r} \partial_{r} - \varepsilon_{ijk} \frac{x^j}{r} \frac{R^k}{r} ,
\end{align}
where $\varepsilon_{ijk}$ are Levi-Civita symbols ...
- 89
2
votes
0
answers
137
views
Tensor product of objectwise weak homotopy equivalences of $\mathcal{M}$-spaces
I consider the enriched category $[\mathcal{M}^{op},\mathrm{Top}]$ of enriched functors (I call them $\mathcal{M}$-spaces) from the enriched small category $\mathcal{M}^{op}$ to the enriched category $...
- 3,901
3
votes
0
answers
102
views
List of techniques that have been used to prove topological properties of locus in the deformation ring or the Hecke algebra
My question is maybe going to be a bit vague. My apologies if so.
The setting:
Let $\overline{\rho}$ be a residual representation and $R$ be a deformation ring of $\overline{\rho}$.
Let $\mathbb{T}$ ...
- 1,270
1
vote
0
answers
76
views
subsets of $\mathbb{N}$ whose shifts have finite intersection property in density
I am interested in proving the statement:
Let $S\subseteq\mathbb{N}$ such that for every $r\in\mathbb{N}$ and for every $k_{1}$, $k_{2}$, $\ldots$, $k_{r}\in\mathbb{N}$, the set $\big(S-k_{1}\big) \...
- 119
17
votes
3
answers
2k
views
Why is an internal proof of consistency satisfactory for some systems?
I've only a shallow understanding of the relevant theory, but I don't understand how any internal proof of consistency is in any way satisfactory (even for systems that are so weak Gödel's ...
- 171
3
votes
1
answer
349
views
Does Bernoulli imply exponential mixing?
This question comes from this paper where the authors proved that exponential mixing implies Bernoulli. They also mentioned in the introduction that Bernoulli is the strongest ergodic property and ...
3
votes
1
answer
235
views
On infinity-morphisms between algebras over algebraic operads
I posted this question in the "Mathematics" stack exchange, but it hasn't got much attention... I hope it will get more here.
Let $P$ be a Koszul operad.
In the book of Loday-Vallette "...
- 215
12
votes
1
answer
1k
views
A problem in commutative algebra whose solution requires algebraic geometry (resp., noncommutative algebra)?
One can argue that commutative algebra is affine algebraic geometry. However, a great deal of commutative algebra generalizes to non-commutative algebra, and in that setting there is little geometry, ...
- 4,985
1
vote
1
answer
107
views
When do faithfully semiinjective complexes exist?
Question:
For which (perhaps noncommutative but always unital and associative) rings $R$ do faithfully semiinjective complexes of right or left $R$-modules exist?
Hopefully the answer is: "for ...
- 1,020
2
votes
0
answers
73
views
Example of a ruled, CM, $ \mathbb{Q} $-factorial, normal, Mori dream space whose Cox ring is integral but not CM,
This question is related to one I asked here in Example of a $ \mathbb{Q} $-factorial, CM normal, projective, Mori dream space $ Z $ such that $ \operatorname{Cox}(Z) $ is integral and not CM. In ...
- 912
3
votes
1
answer
174
views
$n$-th root of meromorphic functions of several complex variables
Let $f:\Omega\rightarrow\mathbb{C}$ be a "nice" meromorphic function of several complex variables on some domain $\Omega$. I wonder if the following claim is true.
Claim. $f$ admits a global ...
- 1,024
1
vote
0
answers
103
views
Estimating the entropy of the solution to an SDE
Forgive me for the poorly researched question. I'm currently working on a computer science project involving training a neural stochastic differential equation, and I've run into a problem while ...
- 451
0
votes
0
answers
235
views
Reference book on the relation between modular forms and elliptic curves
What is a modern reference book to understand the relation between modular forms and elliptic curves after the proof of the Taniyama–Shimura theorem?
- 43
3
votes
2
answers
360
views
Proof of the Dunford-Pettis theorem in the context of probability spaces
I'd like to know if there's a proof of the Dunford-Pettis theorem without using relatively advanced theorems of functional analysis such as Eberlein–Smulian Theorem. Since I'm only interested in ...
- 649
0
votes
0
answers
128
views
Approximating all spanning trees with their sample
In a complete graph with $n$ vertices there are $n^{n-2}$ trees.
In my research I'm analyzing trees in the following way (each edge has a weight):
Get a tree.
Build a complete graph, by the following ...
- 49
2
votes
0
answers
164
views
$H^s$-mild solution for Navier–Stokes : why do we restrict attention to the function spaces "without Fourier zero mode"? (Related to Terence Tao blog)
This question has been triggered by the Definition 32 and Remark 33 in the blog of Terence Tao.
There, every function space is restricted to ones without the Fourier zeroth mode. And the Remark 33 ...
- 3,477