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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 ...
HumbabaOReilly's user avatar
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 ...
Alexander Braverman's user avatar
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 ...
G.-S. Zhou's user avatar
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 $...
Theo Johnson-Freyd's user avatar
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 ...
Igor Rivin's user avatar
  • 95.6k
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 ...
Niven Zhao's user avatar
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 ...
user avatar
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 $p$...
MasM's user avatar
  • 289
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 ...
Michael Freedman's user avatar
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 ...
kt77's user avatar
  • 153
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 ...
Sych's user avatar
  • 23
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 ...
PG_One's user avatar
  • 11
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 \...
Piotr Hajlasz's user avatar
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 ...
Dmitrii Korshunov's user avatar
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 ...
rj7k8's user avatar
  • 716
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. ...
Shalom's user avatar
  • 513
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/...
sebastian's user avatar
  • 457
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 ...
thedude's user avatar
  • 1,417
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 $CAT(0)$ ...
Luis Jorge's user avatar
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 ...
user122270's user avatar
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 $...
user43198's user avatar
  • 1,949
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 ...
Bogdan Luchian's user avatar
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 ...
nguyen0610's user avatar
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 $A$...
Lukas Woike's user avatar
  • 1,372
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 ...
user avatar
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 $$...
Vesselin Dimitrov's user avatar
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,
User11441's user avatar
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 ...
Oscar Cunningham's user avatar
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 ...
H1ghfiv3's user avatar
  • 1,225
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 ...
Cristian D. Gonzalez-Aviles's user avatar
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 ...
Konstantin's user avatar
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 $...
Dongyang Chen's user avatar
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 ...
O-Ren Ishii's user avatar
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. ...
Onil90's user avatar
  • 823
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 ...
Sam Taylor's user avatar
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 ...
sadiaz's user avatar
  • 402
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 \...
random123's user avatar
  • 411
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 $...
Shalom's user avatar
  • 513
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 ...
joeyg's user avatar
  • 339
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(...
Dominic van der Zypen's user avatar
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 ...
Filip Nemec's user avatar
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 ...
Trevor Wilson's user avatar
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 ...
Stanley Yao Xiao's user avatar
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 ...
Jiayi Liu's user avatar
  • 909
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^...
Kevin's user avatar
  • 483
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 $\mathbb{C}$,...
tattwamasi amrutam's user avatar
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 ...
Diego Fonseca's user avatar
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 ...
user1831's user avatar
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 $X$ ...
user32141's user avatar

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