All Questions
152,891
questions
2
votes
0
answers
208
views
Cyclotomic ring of integers proof via matrix theory
Not sure of a better title (and I'm open to suggestions). But when following Laffey's notes on Integer Matrices, found here (and in particular, starting on page 13), I came across an alternative proof ...
2
votes
0
answers
141
views
Two definitions of super-Virasoro algebra
Let $A=\mathbb C[x,\epsilon]$ where $x$ is an even variable and $\epsilon$ is an odd variable (thus $A$ is a commutative super-algebra). Let $\mathfrak g$ denote the Lie super-algebra of vector fields ...
1
vote
0
answers
64
views
A question on separable algebras
First let us recall some terminologies. Let $A$ be a finite dimensional algebra over a field $k$. A linear map $t: A\to k$ is called symmetric (resp. non-degenerate) if the bilinear map $Q_t: A\times ...
8
votes
1
answer
150
views
Are there Type III codes with small but nonzero "index"?
Recall that a Type III code of rank $r$ is a linear subspace $C \subset \mathbb F_3^r$ which is self-dual for the standard inner product. (These occur only when $r$ is divisible by $4$.) Elements of $...
6
votes
1
answer
288
views
Generating prime knots (in order)
In this really cool paper https://arxiv.org/abs/1612.03368, A. Malyutin shows that the probability that a random prime knot of up to $N$ crossings (as $N$ goes to infinity) is not generically ...
12
votes
2
answers
2k
views
A necessary and sufficient condition for a space curve to lie on a ellipsoid
Any (arc-length parametrized) space curve is uniquely determined (up to rigid motion) by its curvature and its torsion.
For instance we know that a necessary and sufficient condition for a space ...
3
votes
0
answers
123
views
Characterization of a meromorphic function as arithmetic zeta function
I'd like to know if there is a (conjectural) criterion for a meromorphic function on $\mathbb{C}$ to be the zeta function of an arithmetic scheme, i.e., a statement of the form
"If a meromorphic ...
5
votes
3
answers
459
views
Is there a dense subset on closed Jordan curve $C$ which its points make intersections under certain rotations?
Is it true that for any given closed Jordan curve of $C \subset \mathbb{R}^2$ there is a dense subset $A$ such that for every point $p\in A$ we have the following property:
If we rotate $C$ around $p$...
8
votes
0
answers
197
views
Slicing satellite knots
Call a knot L a "braid-pattern satellite" of a knot K if it is a satellite of K and the pattern on which it is based is a closed braid in the solid torus. Is there a knot K so that no braid-pattern ...
5
votes
1
answer
168
views
Reference for Weyl's law for higher order operators on closed Riemannian manifolds
I am looking at page 32 (beginning of Chapter 5) here. We are given a formally self-adjoint, metrically defined differential operator $A$ on $(M^n,g)$ of order $2l$ with positive definite leading ...
20
votes
8
answers
4k
views
Mathematical theory of aesthetics
The notion of beauty has historically led many mathematicians to fruitful work. Yet, I have yet to find a mathematical text which has attempted to elucidate what exactly makes certain geometric ...
2
votes
3
answers
534
views
Is product/coproduct in category with only one object possible? [closed]
Let's say we have category with one object $N$ and infinite number of arrows, which are named as natural numbers, with the same law of composition, where $id$ arrow 0.
I try to understand 1) if ...
1
vote
1
answer
383
views
Is fractional Laplacian invariant under rotation?
If $\Delta u=0$, then $\Delta u(Ox)=0$, where $O$ is an orthogonal matrix. From here, do we know whether fractional Laplacian is invariant under rotation? We use the usual definition of fractional ...
3
votes
2
answers
944
views
Composition of Riesz potentials
For $0<\alpha<n$ and $n\geq 2$ we define the Riesz potential by
$$
(I_\alpha f)(x) = \frac{1}{\gamma(\alpha)}
\int_{\mathbb{R}^n} \frac{f(y)}{|x-y|^{n-\alpha}}\, dy\, ,
\quad
\text{where}
\quad
\...
6
votes
0
answers
445
views
Local isometry of complete length spaces that is not a covering map
Let $\pi:\widetilde{M}\to M$ be a surjective local isometry between complete length spaces (local isometry means that every point $x\in \widetilde{M}$ has a neighborhood which is isometrically mapped ...
3
votes
1
answer
289
views
Contragredient of a cuspidal representation
Let $G$ be a reductive group over a nonarchimedean local field $F$. Let $\pi$ be an irreducible, cuspidal representation of $G$, with contragredient $\tilde{\pi}$. Then $\tilde{\pi}$ is cuspidal.
A ...
4
votes
1
answer
326
views
Do these sets all contain an infinite number of prime numbers? Or, at least, provably (unconditionally), one of them?
Let us define $A_1$ to be a set of all numbers of the form $p_1+q(p_1)$ where $p_1$ goes through a set of primes $\mathbb P$ and $q(p_1)$ is $1$ if $p_1$ is even prime and $2$ if $p_1$ is odd prime.
...
3
votes
1
answer
273
views
The nilpotency class and the derived length of a $p$-group
Let $G$ be a finite $p$-group of nilpotency class $c$ and of derived length $d$.
As is well known, we have $d\leq \lfloor\log_2 c\rfloor+1$,
(https://groupprops.subwiki.org/wiki/...
4
votes
2
answers
144
views
Factorizations in terms of characters
I have asked this in math.SE (https://math.stackexchange.com/questions/2698772/factorizations-in-terms-of-characters) but it was barely viewed.
I have seen mention in different places that the number ...
6
votes
1
answer
326
views
Bruhat-Tits building of $SL_n(\mathbb{Q})$, hyperbolic isometries and its axis
Consider $G=SL_n(\mathbb{Q})$ and $p$ a prime integer. Associated to $G$ and $p$ we have its Bruhat-Tits building $\Delta$.
It is well known that $\Delta$ can be provided with a canonical $CAT(0)$ ...
3
votes
0
answers
306
views
Roots of polynomials over $\mathbb{Z}/p^k\mathbb{Z}$
Over a finite field, such as $\mathbb{Z}/p\mathbb{Z}$, the number of roots of a polynomial is no larger than the degree. I'm interested in how does this generalize to $\mathbb{Z}/p^k\mathbb{Z}$.
I'm ...
1
vote
0
answers
212
views
Torsion line bundle on hyperelliptic curves and Weierstrass points
Let $C$ be an hyperelliptic curve of genus $g$ and let $f:C\rightarrow
\mathbb{P}^1$ be the corresponding 2 to 1 covering
ramified in $2g+2$ points.
Let $L$ be a line bundle on $C$ such that either $...
2
votes
2
answers
514
views
Regarding minimal elementary generators for $GL(n, \mathbb{Z})$
I have a result concerning the minimal number of elementary generators (and by this I mean generators which are elementary matrices) for $GL(3, \mathbb{Z})$. I'm currently working on extending the ...
0
votes
0
answers
58
views
$N-$Green function in $\mathbb R^N$
Let $N \geq 3$. Does there exist solution of the following equation
$$-\Delta_N G + G^{N-1} = \delta_0,$$
where $-\Delta_N = - \text{ div}(|\nabla \cdot |^{N-2} \nabla \cdot )$ denotes $N-$Laplace ...
10
votes
0
answers
277
views
Hochschild homology of a Hopf algebra
Let $A$ be a Hopf algebra over the complex numbers.
Denote by $\mathcal{M}$ the dg-category of dg-$A$-modules.
The Hochschild homology of $\mathcal{M}$ is not going to be the Hochschild homology of $A$...
42
votes
2
answers
3k
views
Abel and Galois (and Arnold)
Question Is there a connection between Abel and Galois theories of polynomial equations?
Recall that for every polynomial $p(x)\in \mathbb{Q}[x]$ (say, without the free coefficient), Abel considered ...
5
votes
2
answers
326
views
The largest disk contained by a 'product' of two simply connected plane regions with unit conformal radii
Consider a pair of holomorphic functions $f,g \in \mathcal{O}(\Delta)$ on the complex unit disk $\Delta = \{|z| < 1\}$ that both satisfy $f(0) = g(0) = 0$ and $f'(0) = g'(0) = 1$. Does the domain
$$...
2
votes
1
answer
152
views
Finding a similarities and differences of sent of matrices
Suppose we have a set of rank deficient covariance matrices. How can I know the similarities and differences between those set of matrices?
Regards,
16
votes
2
answers
1k
views
Is there a universal way to force the Axiom of Choice to be true?
Given a model of set theory $V$ there are various ways to construct a model in which the Axiom of Choice holds, such as Gödel's constructible universe $L^V$ or by using forcing*. I'm wondering if any ...
2
votes
1
answer
184
views
Sobolev inequalities for Banach-valued functions
To what extend do the various different Sobolev inequalites hold if I replace the usual target space $\mathbb R$ by an arbitrary Banach space, the notion of derivative by Frechét derivative and the ...
7
votes
2
answers
260
views
Examples of schemes $S$ with $H^{3}(S,\mathbb G_{m})=0$
The examples I have are: $S$ is equal to the spectrum of a global field; or a proper non-empty open subscheme of the spectrum of the ring of integers $\mathcal O_{K}$ of a number field $K$ (proper ...
4
votes
0
answers
193
views
Geometric fundamental group and algebraically closed residue field
my questions relates to the following talk of Tsuji:
https://www.youtube.com/watch?v=2brDj26phP0
At around 10:30 of the video, Tsuji is interrupted by a man stating that his construction does not ...
1
vote
1
answer
116
views
Can every Banach space with the Schur property embed into $L_{1}(\mu)$ for some $\mu$?
In 1974, W. B. Johnson and E. Odell observed that there are subspaces $X$ of $L_{1}$ with the Schur property. In 1980, J. Bourgain and H. P. Rosenthal constructed a subspace $X$ of $L_{1}$ such that $...
6
votes
1
answer
167
views
Is this morphism of regular local rings regular?
Let $R$ be a regular, local $\mathbb{Q}$-algebra with a regular system of parameters $x_1, \dotsc, x_n$, and let
$$f \colon \mathbb{Q}[X_1, \dotsc, X_n]_{(X_1, \dotsc, X_n)} \rightarrow R$$
be the ...
1
vote
1
answer
313
views
A Liouville theorem for a uniformly elliptic equation in divergence form
I would like to know if there exists a Liouville theorem for solutions $u : \mathbb{R}^n \to \mathbb{R}$ of uniformly elliptic equations of the kind
$$
D_i \left( a_{ij} D_j u \right) + b_i D_i u = 0.
...
2
votes
1
answer
378
views
Alfred van der Poorten--rational functions paper
Does anybody has a copy of the following paper:
Alfred van der Poorten, Some facts that should be better known, especially about rational functions; Number Theory and Applications”, Richard A. Mollin ...
1
vote
1
answer
243
views
Classical solution of fractional laplacian
What is meant by a classical solution of a fractional laplacian in $ (-\Delta)^su= f(u)$ in $\mathbb{R}^N$ with no condition at infinity? If one can show that $u$ is a weak solution of the above ...
2
votes
0
answers
327
views
Structure of Derived Category of Sheaves on a curve
Let $C$ be a smooth projective curve over $\mathbb{C}$. We will be working in the topological category. Let $A \in D^b(C):=$ bounded derived category of sheaves on $C$. Is it true that $$A \cong \...
12
votes
1
answer
275
views
$1$ as difference of composites with same number of prime factors
I noticed and found only first three cases:
We can write $1$ as difference of two composites that have one prime factor $$3^2-2^3=1$$
and as difference of two composites that have two prime factors $...
3
votes
1
answer
316
views
Positive and Null recurrence of Markov Chains on a General State Space
Suppose $X_n$ is an irreducible, aperiodic and Harris recurrent Markov chain. It is well known that in this case, $X_n$ has a stationary distribution $\pi$.
Are there any conditions that are ...
8
votes
2
answers
162
views
Neighborhood fingerprint of a graph
Let $G=(V,E)$ be a finite, simple, undirected graph. For $v\in V$ we set $N_0(v) = N(v) = $ $\{w\in V: \{v,w\} \in E\}$ and for $k\in \omega$ let $$N_{k+1}(v) = N_k(v) \cup \bigcup\big\{N(z): z\in N_k(...
0
votes
1
answer
196
views
How to find all orthogonal coordinates in a space of dimension $n$
I have been thinking for a while how to determine all the orthogonal coordinate systems in linear spaces of an arbitrary dimension $n$.
The motivation for such a task comes from physics: I am ...
4
votes
1
answer
191
views
A version of the Martin–Solovay tree for $H_\kappa$
Consider a fixed $\Pi^1_2$ property of reals, $A(x)$.
Is it true that relative to a regular cardinal $\kappa$, one can define a version of the Martin–Solovay tree $T_2$ for $A$ with the ...
2
votes
0
answers
41
views
Solving a bilinear form equation over a cubic field
(This question was asked earlier on math stack exchange, see: https://math.stackexchange.com/questions/2687760/solving-a-bilinear-form-equation-over-a-cubic-field)
Let $K/\mathbb{Q}$ be a cyclic ...
12
votes
2
answers
1k
views
Graph automorphism group
Let $A_w$ denote such set of positive integer $n$ that: for any two permutations $\pi_0,\pi_1\in S_n$, if $\pi_1$ is not a power of $\pi_0$, then there exists a (labeled non oriented) graph $G$ of ...
1
vote
1
answer
137
views
Upper bound of the dimension of automorphism group of compact Kähler manifolds
It is well-known that the dimension of the isometry group of an $n$-dimensional compact Riemannian manifold is no larger than $\frac{1}{2}n(n+1)$, which is attained precisely by $S^n$ and $\mathbb{R}P^...
5
votes
1
answer
155
views
Example of a prime action on a compact Hausdorff Space
Suppose that a discrete group $\Gamma$ acts on a compact Hausdorff space $X$ via homeomorphisms. This action induces an action on $C(X)$, the space of all continuous functions from $X$ to $\mathbb{C}$,...
0
votes
0
answers
66
views
Suggestions to solve an optimization problem that involves quadratic forms
I am in a crucial part of my research, I have arrived at an optimization problem that I can not solve, I need to solve it to be able to perform simulations and thus complete my research, due to this ...
3
votes
1
answer
139
views
Is there a geometric interpretation of a Zariski dense surface subgroup?
Is there a geometric interpretation of a surface subgroup being Zariski dense? Or, conversely, given a $\Pi_1$ injective surface in a 3-manifold, is there a geometric or topological requirement on the ...
7
votes
1
answer
291
views
Reflexive subspaces of bidual Banach spaces
The answer to the question is almost surely negative (as almost always in Banach space theory) but I cannot find a relevant example.
Is there an example of an infinite-dimensional Banach space $X$ ...