Trending questions
159,035 questions
2
votes
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answers
28
views
Equation of a projective conic over a division ring
The equation of a conic section in homogeneous coordinates in the projective plane $\mathbb{RP}^2$ can be written as
$$
\left[\matrix{x&y&z\cr}\right]
\left[\matrix{a&b/2&d/2\cr b/2&...
- 51
2
votes
0
answers
134
views
Universal semistable curve
For the moduli of stable curve, we have the result by Deligne-Mumford-Knudsen that there's an isomorphism of moduli spaces
$$\Phi : \overline{\mathcal{X}}_{g,n} \xrightarrow{\sim} \overline{\mathcal{M}...
- 834
3
votes
0
answers
243
views
Multiplicative functions and Dirichlet characters
I am studying Dirichlet characters and modular functions as part of my research, specifically working through concepts found in the paper "An explicit hybrid estimate for $L(1/2 + it)$" by ...
- 111
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
2
votes
0
answers
59
views
Tensor product of two transcendental flat algebras is not a field?
I'm considering the correctness of the following assertion, which is related to linear disjointness (I'm trying to generalize it to subalgebras), What does "linearly disjoint" mean for ...
- 173
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
3
votes
0
answers
87
views
Linearization coefficients for Jacobi polynomials
In general, for families of polynomials $\{ Q_n\}, \{ R_n\},\{S_n\}$, there exist linearization coefficients such that one may write the product $Q_m R_n = \sum_k c_{m,n}^k S_k$.
Let $P^{\alpha,\beta}...
- 177
1
vote
0
answers
22
views
maximal length of saturated chains with a given terminal point in the lattice of partitions of an integer ordered by dominance
Let $Pr(n)$ be the set of partitions of the positive integer $n$. This is a lattice with respect to the dominance order: if $\lambda=(\lambda_1\geq\lambda_2\geq\cdots)$, $\mu=(\mu_1\geq\mu_2\geq\cdots)...
- 593
6
votes
0
answers
278
views
The history of Riemann-Roch theorem [closed]
I became interested in the history of the Riemann-Roch theorem, so I searched various materials.
So, I read Über die Wechselwirkungen zwischen der französischen Schule, Riemann und Weierstraß. Eine ...
- 245
2
votes
1
answer
153
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On generation of $A_n$ by elements of prime order
There is a question regarding generation of finite simple groups with elements of prime order. Recently, Guralnick, Shareshian, Woodroofe and Teräväinen made advances in this direction. We have, for ...
- 329
6
votes
1
answer
173
views
Mass minimizing current in real homology class
It is a well-known results by Federer and Fleming that there exists at least one mass-minimizing normal current in every real homology class of a closed $n$-dimensional Riemannian manifold $M$. Their ...
- 61
0
votes
0
answers
67
views
Are there aperiodic tilesets with 7-fold rotational symmetry?
In my explorations, I have seen non-periodic tilings with 7-fold rotational symmetry, and I've seen substitutions for such tilings, however I haven't seen anywhere a tileset which enforces such a ...
- 283
4
votes
1
answer
252
views
Connected Frobenius algebras non-semisimple as an object
A Frobenius algebra object $A$ in a tensor category $\mathcal C$ is said to be connected if $\text{Hom}_{\mathcal C}(\mathbb{1}, A)$ is a one dimensional vector space, where $\mathbb {1} $ denotes the ...
- 43
3
votes
1
answer
110
views
Lie subalgebra annihilated by all derivations
Let $k$ be a field and $\mathfrak{g}$ a Lie algebra over $k$. Put $K(\mathfrak{g}) = \bigcap_{f\in\mathrm{Der}(\mathfrak{g})} \mathrm{Ker}(f)$, which is a Lie subalgebra of $\mathfrak{g}$.
Question. ...
- 985
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
0
votes
0
answers
87
views
Curl-Div equation with singular matrix
I want to solve the equation:
$$
\begin{cases}
\nabla \times (A \mathbf v)=f, \quad x\in \Omega \\
\operatorname{div}(\mathbf v)=0,
\end{cases}
$$
where $\Omega \subset\mathbb{R}^n$, is an open set, $...
- 617
3
votes
1
answer
224
views
Interpretation of an asymptotic result in probability
A result in asymptotic theory says the following: Let $Y$ be a real random variable with full support and $E(|Y|)<\infty$. Assume that:
$$
(A)\quad \lim_{y\rightarrow \infty} \frac{\Pr(Y\geq y)}{\...
- 108
7
votes
1
answer
555
views
Example of continuous function which is not differentiable everywhere in a strong sense
Is there a continuous function
$$u\colon (0,1)\to \mathbb{R}$$
such that at every point $x\in (0,1)$ one has
$$\lim\sup_{y\to x+0}\frac{u(y)-u(x)}{y-x}=+\infty?$$
In particular $u$ is not ...
- 21.8k
2
votes
1
answer
185
views
Number of rational points of a quotient of connected linear algebraic groups
Let $H$ be a closed connected subgroup of a connected linear algebraic group $G$ over an algebraically closed field of characteristic $p>0$, and let $\sigma=\sigma_q$ be the standard Frobenius ...
- 23
8
votes
1
answer
1k
views
GAP cannot solve Rubik's cube 4x4x4 and higher ? (Practical limits of Schreier–Sims algorithm)
According to our practical experiments and literature search - computer algebra system GAP cannot "solve" Rubik's cube 4x4x4 and higher. That means cannot decompose given random element of ...
- 24.9k
2
votes
0
answers
84
views
Is it a property when a cohesive type is a manifold?
Let $X : Type$ in a type theory $T$ interpreting synthetic differential geometry - I don't believe it should matter too much if we have smooth stuff on hand except maybe at the end of this line, but ...
- 433
0
votes
0
answers
114
views
Minesweeper constructions in combinatorics
In a related question I asked if constructions based on Sudoku puzzles could be used to obtain any deep results in combinatorics and noted that there were papers of Greenfeld and Tao where Sudoku ...
- 5,146
1
vote
0
answers
29
views
Factoring semiprimes via sum of two squares? [migrated]
The following thoughts came into my head after watching Grant Sanderson's JBPM award lecture here, in which he discusses the fact that we can quickly factor 3599 by noticing it can be written as (60-1)...
- 61
16
votes
1
answer
977
views
Pedagogically intuitive reformulation of Zorn's Lemma for functional analysis
While teaching an applied functional analysis class, I’ve noticed that students often struggle to develop an intuitive understanding of Zorn’s lemma. It’s relatively straightforward to explain why ...
- 5,824
2
votes
1
answer
93
views
Does approximately null gradient imply approximately global minimum for convex functions?
Let $f: \mathbb{R}^{n} \rightarrow \mathbb{R}_{+}$ be a non-negative and differentiable convex function which vanishes in a non-empty convex set $\Omega$ - possibly unbounded. Usually, when one ...
- 225
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
4
votes
0
answers
109
views
Differential duality: Triangular codes vs. VT codes / Single-substitution vs. Single-deletion
Here is the introduction to my problem:
Codes correcting single-deletion. Let $q$ and $n$ be non-negative integers, and let $\mathbf{x}= \left ( x_{1}, x_{2}, \ldots, x_{n} \right )\in\mathbb{Z}_{q}^{...
- 41
0
votes
0
answers
41
views
Descend local system to the canonical model of Shimura varieties
Suppose $(G,X)$ is a Shimura data, and $E$ be its reflex field. In page 33 of this paper, it constructs an etale local system on the canonical model $\mathrm{Sh}(G,X)_{K,E}$ (variety over $E$) for any ...
- 785
-1
votes
0
answers
95
views
Relation between properties of functions/sets and Grzegorczyk's hierarchy
I know for example that the first level of the Grzegorczyk hierarchy contains the functions which enumerate the c.e sets and that it has an interesting relation to the provably total functions in ...
- 893
3
votes
3
answers
255
views
Continuum-distanced complete, ultrametric space
Our professor asked us to find a complete metric space where the intersection of nested closed balls can be empty.
The following space is such an example, and I would like to learn more on it (since ...
- 637
0
votes
0
answers
18
views
Third order estimate for linear elliptic equations
Let $\lambda < A < \Lambda$ be a constant symmetric matrix and $u$ be a $C^{\infty}$(elliptic regularity gives smooth solutions) solution of $\text{div} A \nabla u = 0$. Let $S_1$ be a sphere ...
- 455
19
votes
2
answers
2k
views
Can there exist a definable "ultrafilter" on the ordinals?
Can there exist a model $\textit M$ of $\textit{ZF}$ (or $\textit{ZFC}$) that contains a definable nontrivial "ultrafilter" on $\sf Ord^{\textit M}$? By this I mean that there is some ...
- 576
12
votes
1
answer
227
views
Is there a $\Pi_2$ sentence $A$ such that $\text{ZFC}^- + A$ proves powerset?
This is a follow-up to this question.
Let $\text{ZFC}^-$ be ZFC without powerset and with collection rather than replacement, as described here.
Is there a $\Pi_2$ (or perhaps $\Sigma_2$) sentence $A$ ...
- 2,213
0
votes
1
answer
83
views
Image and pre-image integer choice function
Let $\newcommand{\Nplus}{\mathbb{N}^+}\Nplus$ denote the set of positive integers. Is there a function $f:\Nplus\to\Nplus$ with the following property?
For all $(a,b)\in \Nplus\times\Nplus$ there is ...
- 48.9k
4
votes
1
answer
183
views
About the reduction type of the Kodaira symbol of elliptic curves defined over p-adic local fields
Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $E$ be an elliptic curve defined over $K$. Tate's algorithm can be used to compute the Kodaira symbol of the reduction type of $E$.
However, I ...
- 77
1
vote
0
answers
70
views
Quadratic forms with the same roots over GF(2) for low rank problems
Let $Q_1(x)=x^TA_1x$ and $Q_2(x)=x^TA_2x$ with $x\in GF(2)^n$, $A_i\in GF(2)^{n\times n}, i \in \{1, 2\}$. If $rank(A_1)=rank(A_2)=2$, is it possible that $Q_1(x)$ and $Q_2(x)$ can have the same roots ...
- 63
0
votes
1
answer
75
views
Singularities at the circle of convergence: generalization of Cauchy-Hadamard theorem
Consider a series $\sum a_n z^n$ with finite radius of convergence $R$. Cauchy-Hadamard theorem gives $1/R = lim\ sup |a_n|^{1/n}$.
Q: Suppose for some reason (e.g. numerical) we know that there is ...
- 593
0
votes
1
answer
115
views
Fourier transform of exponential over torus
I found the following formula for the Fourier transform on a flat 2-torus, but I don't quite know how to derive it. We have a variable $q=(q_x,q_y) \in [0,2\pi)^2$ and by considering it in polar ...
2
votes
0
answers
176
views
Eigenvalues of Laplace operator and Schrödinger operator
When reading the paper Control for Schrödinger operators on tori by N.Burq and M.Zworski (Math. Res. Lett., 19(2):309–324, 2012), an inequality confused me:
Define the flat torus $\mathbb{T}^2=\mathbb{...
- 21
3
votes
0
answers
107
views
The existence of a harmonic diffeomorphism on a punctured surface
Let $\bar{X}$ be a compact Riemann surface of genus $g$, and let $D:=\{p_{1},p_{2},\cdots,p_{n}\}$ be $n$ distinct points on $\bar{X}$. Define $X:=\bar{X}-D$ to be the punctured Riemann surface given ...
- 401
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
3
votes
0
answers
124
views
Helly number of closed unit balls in R^d
Does anyone know the Helly number of closed unit balls in $\mathbf{R}^d$ and a reference? This number is the smallest number $k(d)$ so that for any collection of of closed unit balls in $\mathbf{R}^d$,...
- 199
3
votes
2
answers
159
views
Is there a sparse almost disjoint family over $\omega$ of cardinality $2^{\aleph_0}$?
Is there an almost disjoint family $\mathcal{F}$ of subsets of $\omega$ of cardinality $2^{\aleph_0}$ satisfying the following property?
For all $A,B\in\mathcal{F}$ with $A\neq B$ and every $k\in\...
- 1,792
4
votes
1
answer
245
views
Group action on affine variety induces faithful action on tangent space
I have a queestion about the proof of Lemma 2.2 from the paper arxiv 1105.3739:
Let $G$ be a group acting faithfully on an irreducible affine variety $X=\operatorname{Spec}(A)$ over $k= \Bbb C$. ...
- 6,038
3
votes
1
answer
214
views
Geodesic flows and Killing fields
How well-known is the following result:
Let $(M,g)$ be a closed Riemannian manifold and define $D:C^\infty(M,Sym^mT^*M)\to C^\infty(M,Sym^{m+1}T^*M)$, $D\alpha:=\mathbb S(\nabla\alpha)$ where $Sym^m$ ...
3
votes
0
answers
95
views
What is the smallest known number of states that a one-way cellular automaton needs to be universal?
We know there is an elementary cellular automaton (ECA) with 2 states (Rule 110) that is universal, i.e. Turing-complete. One-way cellular automata (OCA's) are a subcategory of ECA's where the next ...
- 121
2
votes
0
answers
127
views
A property of matrices formed by pairing of roots and coroots
Let $A$ be an $n\times n$ integral matrix, define its level $l(A)$ as
$$l(A) := \begin{cases}0 &\det A = 0 \\ \text{smallest integer } N \text{ such that } NA^{-1} \text{ is integral} &\det A \...
- 528
5
votes
1
answer
267
views
UFD property for power series in characteristic 0
Samuel famously produced an example of a UFD, namely $S = R_{(x,y,z)}$, where $R = K[x,y,z]/(x^2+y^3+z^7)$ and $K$ has characteristic 2, such that the power series extension $S[[ x ]]$ is not a UFD. ...
2
votes
1
answer
215
views
Representing positive integers $n$ by binary forms $n=ax^2+by^2$, $a\geq 0$, $b\geq 0$
In there a finite set of $(a_k,b_k)\in\mathbb{Q}_{\geq 0}\times \mathbb{Q}_{\geq 0}$ so that any $n\in \mathbb{Z}_{>0}$
can be written as $n=a_k x^2+b_k y^2$ for some $k$ ?
This is related to ...
- 14k
14
votes
1
answer
565
views
What is the "schematic" point of view for regular polyhedra?
Last week, I read Wikipedia's article on Alexander Grothendieck. It lists his twelve greatest contributions to mathematics as accounted for in Grothendieck's own Récoltes et Semailles. The final item ...
