Trending questions
159,063 questions
1
vote
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How many non-overlapping, mutually non-antipodal unit spheres can be placed on the surface of four dimensional unit sphere?
If by linear or semi-definite programming this number could be shown to be less than 24, then this could be a route to showing that the 24-cell (consisting of 12 antipodal pairs of spheres) is the ...
- 11
2
votes
1
answer
104
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Minimum time required by a curve to reenter a closed ball with radius equal to the reciprocal of its maximum curvature
I have a continuously differentiable curve $r : \mathbb{R} \to \mathbb{R}^{n}$, $n \ge 2$, with the properties
\begin{align}
\lvert r'(t) \rvert = 1 \text{ for all } t \in \mathbb{R}, \\
\lvert r'(t) -...
- 351
0
votes
1
answer
49
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How to efficiently replace the Paterson-Stockmeyer algorithm and reduce non-scalar multiplications
Paterson-Stockmeyer algorithm
If we need to compute a high-degree polynomial expression, such as:
$$
P(y) = \sum_{k=0}^{B} a_k y^k
$$
the Paterson-Stockmeyer algorithm can process the powers in ...
- 3
6
votes
0
answers
140
views
Can an algebra be isomorphic to its own algebra of $n^2 \times n^2$ matrices but not its own algebra of $n \times n$ matrices?
Is there an associative unital algebra $A$ which is isomorphic to its own algebra of $n^2\times n^2$ matrices $\operatorname{Mat}_{n^2}(A)$, but not isomorphic to its algebra of $n \times n$ matrices $...
- 54.6k
7
votes
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193
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Constructing non-split $\mathbf{G}_m$-extensions of elliptic curves
I would like to find examples of abelian surfaces $A$ over a DVR $R$ with residue field $k$, so that the special fibre of $A$ is a non-split $\mathbf{G}_{m/k}$-extension of an elliptic curve over $k$.
...
- 265
1
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74
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Does Hermite's approach to the Bring quintic yield pairs of methods?
In a previous post, we mentioned $4+2=6$ methods to solve the Bring quintic. The first four uses the same quartic to find the elliptic modulus $k$ and the last two uses the same octic. For balance, we ...
- 12.6k
5
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290
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Why does a line bundle on an abelian variety give a group extension only if it is algebraically trivial?
If $X$ is an abelian variety and $L$ is a line bundle, deleting the zero-section one obtains a diagram
$$
0 \to \mathbb G_m \to Y \stackrel{\phi}{\to} X \to 0
$$
where $\mathbb G_m = \phi^{-1}(0)$,
...
5
votes
0
answers
252
views
Does a simple formal group give rise to a simple Lie algebra?
A formal group $F$ is an intermediate object between Lie group $G$ and Lie algebra $\mathfrak{g}$.
A formal group is called simple if it has no nontrivial sub-formal group. On the other hand, a simple ...
- 195
1
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70
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Degree axiom for P1 or P2
I am getting stuck on equation (7.33) on p. 192 of Cox-Katz's Mirror Symmetry and Algebraic Geometry. This concerns the degree of a cohomology class used as input for a Gromov-Witten invariant.
Let $X$...
- 511
6
votes
1
answer
661
views
On the martingale betting scheme
For a fixed probability $0 < p < 1$, let $X^p$ be the martingale that goes up by $1$ with probability $p$, and goes down by $\frac{p}{q}$ with probability $q := 1-p$.
Write $X$ for the ...
- 6,313
3
votes
2
answers
118
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Does the derivative of the antiderivative of a BV function $f$ agree with $f$ at all but countably many points of differentiability?
Let $f: (a, b) \to \mathbb R$ be a function of bounded variation, and write
$$F(x) := \int_a^x f(t) \, dt$$
for the antiderivative. Is it true that at all but countably points of differentiability of $...
- 6,313
3
votes
1
answer
239
views
A palindromic formula for simple convex polytopes
Let $P$ be a simple convex $d$-polytope (a $d$-dimensional convex polytope in which the number of edges incident to a vertex is $d$) and let $n_i$ be the number of $i$-faces of $P$. Is it true that ...
- 161
5
votes
2
answers
449
views
Centre of group with deficiency at least two (Progress on Murasugi's conjecture)
In 1965, Murasugi [1] conjectured that any finitely presented group with deficiency at least two has trivial centre. The year before, he had proved it true for one-relator groups, and in [1] he proved ...
17
votes
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359
views
The convergence domain of the function $\sum \{n!x\}$
This is a problem from a mathematics competition: Does there exist an irrational number $x$ such that the series
$$\sum_{n=1}^{\infty}\{n!x\}<+\infty$$
where $\{ \}$ means the fractional part of a ...
- 515
4
votes
0
answers
212
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When does a short exact sequence of abelian groups with $B\cong A\oplus C$ split?
$\hspace{20pt}$Duplicate on stackexchange.
This question, in a way, extends this one. The question is what are some sufficient conditions on the abelian group $B$ so that if $B\cong A\oplus C$ and a ...
- 171
1
vote
0
answers
33
views
M-char reduction inverse Theorem [closed]
I need a bit of help finishing this theorem I believe it to be true. First, set $\mathcal{B} = \{A, B, \dots, Z\}$ to be the set of capital letters in base 64, and define $\mathcal{A}_m$ as the set of ...
- 317
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votes
1
answer
51
views
Cycle-Sculpturing with Minimal Vertex-Deletion
given a simple, finite and symmetric graph $G(V,E)$ with $n$ vertices and at least $n$ edges
Question:
how can the smallest set of vertices $W\subset V$ be calculated for which the graph induced by $...
- 13.2k
0
votes
1
answer
124
views
Horospherical type of a spherical variety
In the following, I will fix $k$ a characteristic zero algebraically closed field, and $G$ a connected reductive group over $k$, $B$ a Borel subgroup of $G$, $T\subseteq B$ a maximal torus, $X$ a $G$-...
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votes
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answers
71
views
Laplace spectrum on $U(n)$
Consider $\psi:SU(n)\times U(1)\to U(n)$, $(w,z)\mapsto \bar{z}\cdot w$. One can show that $\psi$ serves as a projection and $SU(n)\times U(1)$ is a principal $\mathbb Z_n$-bundle over $U(n)$.
Suppose ...
63
votes
11
answers
7k
views
What are some ways to stay engaged with the mathematical community from outside academia?
I will be graduating with a Masters degree soon in mathematics. For various reasons I have decided not to pursue a career in academia for now and will instead be working a job in industry that will ...
1
vote
1
answer
65
views
Dual of blocking sets in finite geometry
Let $V$ be an $n$-dimensional vector space over the finite field of cardinality $q$ and let $W_1,\ldots,W_m$ be hyperplanes of $V$ such that
$$V=\bigcup_{i=1}^mW_i \,\,\hbox{ and }\,\,0=\bigcap_{i=...
- 934
4
votes
0
answers
155
views
Quasi-invariance of $\Phi_3^4$ under translation by nonsmooth shifts
In https://hairer.org/Phi4.pdf Hairer shows that the $\Phi_3^4$ measure is mutually singular with respect to any nonzero smooth shift. Is it also mutually singular with respect to any nonzero ...
- 1,924
4
votes
1
answer
256
views
Third page differential in the Lyndon–Hochschild–Serre Spectral Sequence
I am trying to understand the description of the group cohomology of $Q_8$ from Adem–Milgram’s “Cohomology of finite groups”. The main result is the following:
Theorem 2.9. In the Lyndon–Hochschild–...
- 141
8
votes
1
answer
322
views
Does every cancellative duo semigroup embed into a group?
Prompted by the comments to a recent answer by YCor to a related question (here), I'd like to ask the following:
Q. Does every cancellative duo semigroup embed into a group?
A (multiplicatively ...
- 10.5k
2
votes
1
answer
431
views
Shadows of partitions of lcm
$\DeclareMathOperator\lcm{lcm}$Fix an integer $n\geq1$. Denote the least common multiple $L_n=\lcm(1,2,\dots,n)$.
QUESTION. Is the following true? For each integer partition $\lambda=(\lambda_1,\...
- 43.2k
5
votes
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answers
50
views
Surjectivity of irreducible representation of Chevalley algebras
Let $A$ be an associative algebra over an algebraically closed field $k$. The following theorem holds (see, for instance, Theorem $2.5$ on page $24$ in Introduction to Representation Theory by Pavel ...
3
votes
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answers
267
views
Cohomology for quantum groups
I'm interested in quantum groups for two perspectives:
Compact quantum groups in the sense of Woronowicz.
Deformation of the universal enveloping algebra of a Lie algebra in the sense of Drinfeld &...
- 357
42
votes
2
answers
2k
views
How decreasing can a bijection $f:\mathbb{N}\to\mathbb{N}$ be?
This is a follow-up to this question by
Dominic van der Zypen. For each bijection $f:\mathbb{N}\to\mathbb{N}$, let
$$\operatorname{rc}(f) := \liminf_{N\to\infty} \frac{\left|\left\{(m,n)\in\{1,\dots,N\...
- 10.6k
0
votes
0
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112
views
An interesting unramified extension of imaginary quadratic fields
Let $K/\mathbb{Q}$ be an imaginary quadratic extension of discriminant $-D_K < -3$ and fix a prime $p > 3$ that is split in $K$ as $p\mathcal{O}_K = \mathfrak{p}\overline{\mathfrak{p}}$. ...
- 21
1
vote
0
answers
58
views
Turn a coherent subsheaf into a vector subbundle by blowing up
I have some confusions on reading Narasimhan's paper Hermitian-einstein metrics on parabolic stable bundles. The following statements appears between page 103-104.
Let $E$ be a rank 2 holomorphic ...
- 206
3
votes
1
answer
163
views
Is a pseudo-effective divisor on a rational surface numerically effective?
Let $D$ be a pseudo-effective $\mathbb{R}$-divisor on a rational surface. Can we find an example that the numerical class of $D$ contains no effective divisor?
- 393
3
votes
1
answer
80
views
Solution of $d Y_t/dt = A(t) Y_t, Y_0 = I_d$ is positive definite?
Let $\{A(t)\}_{t \in [0,1]}$ be time-varying symmetric matrices in $\mathbb{R}^{d\times d}$. We consider the following ODE for $Y_t \in \mathbb{R}^{d \times d}$
$$
\tag{1}
\frac{d Y_t}{dt} = A(t) Y_t, ...
- 399
2
votes
1
answer
276
views
Estimating a sum over set partitions
Let $[n]:=\{1,\dots,n\}$. Fix a set partition $\rho$ of $[n]$, with an abuse of notation we shall use $\rho\vdash [n]$.
I would like to estimate the following alternating sum.
QUESTION. Is this true?
...
- 43.2k
1
vote
0
answers
96
views
Determine equivalences in the generated collection of subgroups and quotients
Let $A$ be an abelian group, and $B_1, B_2, \dots, B_m$ be subgroups of $A$. Define the family of subgroups $\mathcal{D}_0 = \{ \{0\}, A, B_1, B_2, \dots, B_m \}$.
Let $\mathcal{C}_1$ be the ...
- 807
5
votes
2
answers
344
views
Non-commuting elements of finite orders in a reductive group over a p-adic field
Let $k$ be a $p$-adic field and $G$ be a connected non-abelian reductive algebraic group over $k$. I am asking for a proof of the following lemma:
Lemma. Assuming that $p$ is "good" for $G$,...
- 14.2k
1
vote
0
answers
207
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Jaw-breaking sum (related to quasi-analytic decompositions of unity — and Hörmander’s Lemma 1.3.6 from LinPDE vol.1)
$$F\left(\frac 1{\text e}\right) ≈ 1-\frac 1{\text e};\qquad\text{here }F(z)≔\sum_{k=1}^\infty \frac{(k-1)^{k-1}}{k!}z^k.$$
The match is at least with 800 decimal places (checked with ...
- 455
0
votes
0
answers
124
views
Counit map surjective
Let $X \to Y$ a (set theoretically) surjective morphism of schemes, $L$ a line bundle/invertible sheaf on $X$ (maybe more generally a locally free coh sheaf, but let's stick firstly on invertible ...
- 5,966
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votes
0
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35
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Describing the boundary of the feasible direction cone to a convex open subset of $\mathbb{R}^n$ at a boundary point: connection via subdifferential?
Let $U\subset \mathbb{R}^n$ be a convex, open set with nonempty boundary. Let $x_0\in \partial{U}.$ We can describe $U$ locally near $x_0$ as a super level set of a suitable continuous concave ...
- 1,467
1
vote
1
answer
130
views
Is every operator range a Baire space in the relative topology?
Let $X$ be a Banach space and let $U\subseteq X$ be a (not necessarily closed) linear subspace. One says that $U$ is an operator range if there is another Banach space $E$, and a bounded linear map $...
- 483
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votes
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258
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Forcing axiom for Mathias forcing
Given a poset $\mathbb{P}$, let $\mathsf{FA}(\kappa,\mathbb{P})$ denote the assertion that for every family of dense sets $\mathcal{D}$ with $|\mathcal{D}| = \kappa$, there is a filter $G \subseteq \...
- 1,442
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0
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22
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Alignment of unit vectors under graph-neighbor constraints with a global vector
Statement
Let $G = (V, E)$ be a connected, unweighted, and undirected graph with $n$ nodes, represented by its adjacency matrix $A$. Suppose each node $i$ is associated with a unit vector $ \mathbf{v}...
13
votes
2
answers
813
views
A dichotomy for everywhere differentiable eikonal functions
Let $f: \mathbb R^n \to \mathbb R$ be everywhere differentiable, with $|\nabla f| = 1$ almost everywhere. Is it true that $|\nabla f| = 0$ or $1$ everywhere?
- 6,313
3
votes
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169
views
equivalence of two categories
I am new to algebraic geometry and category theory. I am wondering about the following functor is equivalence of categories or not.
Let $X$ be irreducible scheme and $x$ be its unique generic point. ...
- 613
6
votes
1
answer
194
views
The most even partition of $\mathbb R$ into measure dense sets
Notation: $\mu$ denotes the Lebesgue measure. Let $\mathcal D$ be the set of Lebesgue measurable subsets of $\mathbb R$ such that itself and its complement have nonzero Lebesgue measure in every ...
- 6,313
3
votes
0
answers
144
views
Minimizing vertical integral of a Mellin transform
Let $\eta:[0,\infty)\to [0,\infty)$ satisfy $\eta(0)=1$ and $\int_0^\infty \eta(x) dx = 1$ (say).
Write $M\eta$ for the Mellin transform of $\eta$. Let $\epsilon>0$ be small.
What is the choice of $...
- 20.2k
0
votes
1
answer
60
views
Whether the fractional Sobolev seminorm of any smooth function with compact support is finite
Let $n\geqslant 1$, $p\in [1,\infty)$, and $s\in (0,1)$. Define the fractional Sobolev seminorm
\begin{equation*}
[f]_{\dot{W}^{s,p}(\mathbb{R}^n)}
:=\Bigl[\frac{f(x)-f(y)}{|x-y|^{\frac{n}{p}+s}}\...
- 3
1
vote
1
answer
197
views
Probability distribution on Python-dictionary-like objects?
I would like to examine information-theoretical properties of random variables that take as values objects which are akin to dictionaries in the Python programing language.
That is, each sample of the ...
- 11
1
vote
1
answer
73
views
In a ring with a $p$-derivation every $p$-power-torsion element is nilpotent
Let $p$ be a prime. The definition of $p$-derivation on a ring (aka $\delta$-ring structure) can be read in [K, Definition 2.1.1]. In short, a $\delta$-ring is a commutative ring with unity $A$ plus a ...
0
votes
0
answers
76
views
Existence solutions of the system of equations on Riemannian manifold
Is there a way to show that the following system of two equations has a solution? I don't want to find an explicit solution, but just verify its existence.
$$f''(r) + \beta \coth(r) f'(r) = \rho_0 e^{-...
- 272
2
votes
4
answers
213
views
Efficient algorithm for graph problem
Let $D=(V,E)$ be a directed graph, $S,T\subset V$ and $f:V\rightarrow \{1,\ldots, k\}$ a positive, bounded weight-function and $l\in \mathbb{N}$, find a path $v_1,\ldots, v_l\in V$ with $v_1\in S$ and ...
- 157