Trending questions
159,035 questions
14
votes
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565
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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 ...
5
votes
1
answer
274
views
Why "no wandering domain" fails in parabolic basin?
Theorem (Sullivan). Every Fatou component $U$ of $f$ rational map is eventually periodic, that is, there exist $n > m > 0$ such that $f^n(U) = f^m(U)$
I am familiar with the proof: spread around ...
18
votes
3
answers
1k
views
Is the contravariant power set functor more "natural" than the covariant power set functor?
There are two natural ways to make the power set operation into a functor $\mathbf{Set}\to\mathbf{Set}$: given a function $f:X\to Y$, we can send it to:
The function $\mathcal P(X)\to\mathcal P(Y)$ ...
- 181
3
votes
1
answer
180
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Left exact functor $F$ preserves quasi-isomorphism between $F$-acyclics
In this math overflow page, the poster gives a proof of the statement "an additive left exact functor $F$ preserves quasi-isomorphisms between $F$-acyclic objects." I'm having trouble ...
3
votes
1
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133
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Is a simply connected locally 2-connected complex a union of spheres and planes?
Let $X$ be a (potentially infinite) 2-dimensional simplicial complex. Then each link at a vertex $x\in X$ is a graph.
Question. If $X$ is simply connected and each link is 2-connected (in the sense ...
- 13.6k
5
votes
1
answer
325
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An inequality that may be of isoperimetric nature
I am trying to prove the following inequality: let $f,g:S^1\to R$ (here $S^1$ is the unit circle parametrized by arc-length) be differentiable and have zero mean. Then
$$
4\pi \int f(t) g(t)\, dt \le \...
- 797
0
votes
1
answer
141
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A lower bound for the largest prime divisor of an integer
I have often heard it stated that Erdős conjectured the following:
For any integer $n > 1$, there exists a prime divisor $p$ of $n$ such that
$$p > c \cdot \log \log n,$$
where $c > 0$ is a ...
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5
votes
1
answer
263
views
Central isogeny, Shimura varieties and exceptional cases
For a simple complex Lie algebra $\mathfrak g$, its weight lattice is not equal to the root lattice (i.e. the center of its simply connected form is a non-trivial finite group) iff $\mathfrak g$ is of ...
- 6,622
5
votes
0
answers
366
views
A Collatz-like map?
Consider the map $\psi$ acting on triples $(a\leq b\leq c)$ of three positive natural integers with $\mathrm{gcd}(a,b,c)=1$ as follows:
Set $$(a',b',c')=\left(\frac{a}{\mathrm{gcd}(a,bc)},\frac{b}{\...
- 17.9k
0
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0
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34
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Existence and uniqueness of heteroclinic solution of Allen–Cahn on $\mathbb R$ with driving-damping term
The Allen–Cahn equations on $\mathbb R$ are $u'' = u^3 - u$. It is well-known that all the solutions of this equation which satisfy the asymptotic boundary conditions $\lim_{x \to \pm \infty} u\left(x\...
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1
vote
0
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44
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Lower bound for restricted sumset in ordered groups
Recently in The restricted sumsets in finite abelian groups it is proved that
Suppose that $k \geq 2$ and $A$ is a non-empty subset of a finite abelian
group $G$ with $|G| > 1$. Then the ...
1
vote
0
answers
65
views
Relation between $-2/d$-norm and polynomial discriminant
Consider a homogeneous bivariate polynomial $f(x, y) = a_d x^d + a_{d-1} x^{d-1} y + \cdots + a_0 y^d$ of degree $d > 2$, and consider the “$-2/d$-norm”
$$\int_0^{2\pi} |f(\cos{\theta}, \sin{\theta}...
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9
votes
1
answer
402
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Conceptual understanding of the Néron–Severi group
I'm trying to understand the importance of the Néron–Severi group $\operatorname{NS}(X)$ when $X$ is, say a complex manifold. My background is in the analytic side so I'm much more familiar with line ...
- 137
4
votes
0
answers
240
views
What do we do when $G$ doesn't have a Shimura variety?
Let $G$ be a reductive group. If one can associate to $G$ a Shimura datum $(G,X)$, then the étale cohomology of the associated Shimura variety $\operatorname{Sh}(G,X)$ is a strong tool for the ...
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3
votes
0
answers
89
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Examples of (co)commutativity of Frobenius algebras via ambijunctions
This question is related to the paper "Frobenius algebras and ambidextrous adjunctions" by Aaron Lauda (https://arxiv.org/abs/math/0502550). Below $\Sigma\mathrm{Vect}$ is the one-object ...
- 131
3
votes
0
answers
77
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primes that ramify in division fields for hyperelliptic jacobians
Let $C$ be a hyperelliptic curve $y^2=f(x)$ of genus $g\geq 2$ and $\Delta$ the discriminant of $f(x)$. Let $\ell>2$ be a prime that divides $\Delta$ to the order $e:=\operatorname{ord}_\ell(\Delta)...
- 685
2
votes
0
answers
40
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Lie algebra of Hamiltonian (1,0) vector fields on 4-manifold
I have encountered a certain Lie subalgebra of the Lie algebra of vector fields on a 4-manifold that is also a complex manifold, distinct from the well-known Lie algebra of holomorphic vector fields. ...
- 223
0
votes
0
answers
62
views
Characterization of duals of Sobolev space
Proposition 8.14. in Brezis states that:$(W_0^{1,p} (Ω))^*=W^{-1,p^*} (Ω)$ and we have the representation:
$∀ F∈(W_0^{1,p} (Ω))^* ∃ f_0...f_n ∈L^{p^*} (Ω)$ such that $∀ u∈W_0^{1,p}(Ω)$
$F(u)=∫_Ω ...
1
vote
0
answers
116
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Can all congruences for a third-order recurrence relation hold for some composite $n$?
Let $p$ be a prime with $p \gt 3$. Consider the polynomial $f = x^3 - 3x -1$. Suppose $f$ is irreducible over $\mathbb{F}_{p}$. Let $E$ be the splitting field of $f$ over $\mathbb{F}_{p}$, and let $\...
0
votes
1
answer
66
views
Groups with $2$-transitive permutation representations of different degrees
Suppose $G$ is a finite group, and suppose that it acts $2$-transitively in each of the permutation representations $(G,X_i)$ ($i$ ranges over some index set $I$), where the $X_i$s all have different ...
- 4,555
4
votes
1
answer
262
views
Are renormalizability and the criticality of a PDE synonymous?
In the physics literature a quantum field theory is qualitatively classified as renormalizable, super-renormalizable, or non-renormalizable. This heuristic is based on how many Feynman diagrams ...
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3
votes
1
answer
151
views
Locally nilpotent derivations and triangularizability
If $ k $ is a field of characteristic zero and $ \delta \in T_{\mathbb{A}^{n}_{k}/k} $, then $ \delta $ is triangular if $ \delta = \sum_{i=2}^{n} f_{i}(x_{1},\dots,x_{i-1}) \frac{\partial}{\partial ...
- 912
3
votes
3
answers
492
views
In hyperreal field, can ln(ε) and ln(ω) be expressed as infinite sums?
In the hyperreal field, we can use Taylor series to express e^(ε) and e^(ω) as:
e^(ε) = 1 + ε + (ε^2)/2! + ...
e^(ω) = 1 + ω + (ω^2)/2! + ...
Is it similarly possible to express ln(ε) and ln(ω) as ...
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0
votes
1
answer
127
views
Holomorphic functions of certain blow up at origin
Suppose that $D=\{z\in \mathbb C\,:\, |z|\leq 1\}$ and let $f$ be holomorphic on $D\setminus\{0\}$ such that $|f(z)|\leq e^{\frac{1}{|z|}}$ for all $0<|z|\leq 1$ and assume additionally that $\lim\...
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8
votes
1
answer
246
views
Is $\operatorname{non}(\mathcal{M}) < \mathfrak{a}$ consistent?
Let $\operatorname{non}(\mathcal{M})$ be the least cardinality of a non-meagre subset of the reals. Let $\mathfrak{a}$ be the least cardinality of an infinite maximal almost disjoint family (i.e. $\...
- 1,442
4
votes
1
answer
256
views
First occurrence of formula for $\sum_{n\leq x} \mu(n) \log n$ in terms of $\psi(y)-\lfloor y\rfloor$?
The identity contained in the last two displayed equations in the following passage (from page 110 in Ayoub's An Introduction to the Analytic Theory of Numbers, 1963) gives us right away a simple ...
- 20.2k
7
votes
3
answers
773
views
Implicit uses of Countable or Dependent Choice
What are instances of implicit reliance on countable or dependent choice in classic books? Two examples are
Introduction to Commutative Algebra by M.F. Atiyah and I.G. MacDonald
where it is claimed,...
2
votes
0
answers
165
views
Nonabelian groups where every element has small order
Let $G$ be a finite nonabelian group with the property that if $g \in G$, then
$$\DeclareMathOperator{\ord}{ord} \ord(g) \leqslant 10 \log_2 |G|, $$
where $\ord(g)$ is the order of the element $g$, ...
- 1,354
5
votes
1
answer
209
views
Compactness in trace class operators space
Let $H$ be a separable Hilbert space. Let $L_1$ denote the space of trace class operators on $H$ with the trace-class norm $\|\cdot\|_1$, i.e. $\|K\|_1=Tr|K|$ for all $K\in L_1$.
Are there easy ...
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0
votes
0
answers
45
views
Artinian simple algebras with involution
Let $A$ be an Artinian simple $K$-algebra with involution $*$. The algebra $A$ has a primitive idempotent $e$, and $D_e:=eAe$ is a division $K$-algebra due to Schur's lemma and the isomorphism $(eAe)^{...
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0
votes
1
answer
99
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A question about G-Hewitt spaces
In the paper linked below, S. A. Antonyan gives the following proposition without proof (in fact all results are given without proof). I need a proof of this theorem. If anyone has information on this ...
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2
votes
0
answers
120
views
Analogs of Plücker relations in Clifford algebras, and Bott periodicity (?)
Classical Plücker relations can be viewed as conditions on coefficients of an element $x=\sum_Sc_Se_S$, $S=(i_1,...,i_k)$, $i_1<\cdots<i_k$, $\{i_1,...,i_k\}\subset\{1,...,n\}$ of an exterior ...
- 18.4k
4
votes
1
answer
93
views
Tight upper bound on a ratio involving symmetric PSD matrices and Kronecker products
Let $\mathbf{A}_i \in \mathbb{R}^{d \times d}$ ($i = 1, \dots, T$) be symmetric positive semidefinite (PSD) matrices. Define the quantity
$$
m = \frac{\lambda_{\max}\left(\sum_{i=1}^T \mathbf{A}_i^2\...
- 73
0
votes
1
answer
77
views
When is Laplace transform of a function power-law and relation to the behavior of the function near zero?
I want to see when the Laplace transform of a non-negative function $f$ defined on $[0, +\infty)$ is a power function in the loose sense, i.e.,
$$g(s) = \mathcal L\{f\}(x) = \int_0^\infty f(x) e^{-sx} ...
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14
votes
0
answers
392
views
Can the axiom of choice be expressed in 4 quantifiers?
This 2007 paper presents a 5-quantifier $(\in, =)$-expression that is ZF-equivalent to the axiom of choice, but leaves open the 4-quantifier case:
Thus the gap is reduced to the undecided case of a 4 ...
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1
vote
0
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264
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Fourier transform of fat Cantor set
Let $C_n$ be the set obtained in the $n$-th iteration of the construction of the Smith-Volterra Cantor set, obtained by removing at the $n$-th step $2^{n-1}$ middle intervals of amplitude $1/4^n$. ...
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1
vote
0
answers
57
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Discrepancy of general element of linear system
Let $X$ be a normal scheme and $|D|$ a linear system on $X$.
In "Singularity of Minimal Model Program" by Janos kollar p249, it says,
If $X$ is a variety over $\mathbb{C}$, and $E_j$ ...
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3
votes
0
answers
86
views
Asymptotics of number of running maxima of iid random variables
Let $\{X_i\}_{i \geq 1}$ be a sequence of iid non atomic random variables, that is, their CDF has no jump discontinuities.
Given a realisation $\omega$ of the random variables, we say that $X_i (\...
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votes
0
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90
views
How to show a point is a weak* -weak continuous for the identity map on $X_1^*$ or on $X_1^{**}$?
I am trying to understand the Remark 3.2 mentioned in the paper titled as "On Weak*
-Extreme Points in Banach Spaces" written by S. Dutta and T. S. S. R. K. Rao (http://library.isical.ac.in:...
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4
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0
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165
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Problem understanding the cup-products for the modified cohomology in David Harari's book "Galois Cohomology and Class Field Theory"
I'm recently reading the very beginning of David Harari's book Galois Cohomology and Class Field Theory, and I met a problem with the definition of cup-products for the modified cohomology.
Before ...
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7
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1
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833
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Why are some heuristics successful?
Mathematicians sometimes use heuristics to form expectations about what might be true or false. For examples, see Matthew Emerton's answer to Why should I believe the Mordell Conjecture?, this blog ...
3
votes
0
answers
171
views
Cellular structure of $F_4$
Is there the cellular structure of the Exceptional Lie group $F_4$?
Is there a reference to it?
Thanks
1
vote
0
answers
82
views
Markov Chain that maximises the entropy creation rate
I am working on MERW (Maximal entropy random walk) for a project.
I want to show that given a graph G, there is $\textbf{only one}$ aperiodic markov chain on G that maximises the entropy creation rate ...
- 11
1
vote
0
answers
148
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integral over the unit sphere of $\Bbb C^n$
Please, is there a way to calculate this integral
$$\int_{S_{2n-1}} \frac{e^{a \langle z, \zeta \rangle}}{|z - \zeta|^{\beta}} \, d\sigma(\zeta)$$
where $ z $ is a fixed point in the complex unit ball ...
- 531
41
votes
1
answer
2k
views
Implications and consequences of the recent proof of the geometric Langlands conjecture
I am a beginner in mathematical physics and geometric Langlands, having very limited knowledge in both fields so far.
The proof of geometric Langlands conjecture is published a few months ago. What ...
- 213
0
votes
1
answer
98
views
Intersection of sigma algebras generated by shifts
EDIT: Iosif's answer showed that my motivation for this question was mislead.
To keep this question interesting for a broader readership, let us forget about sequence spaces and tail algebras and ...
- 257
5
votes
2
answers
243
views
Expansion of key polynomials in terms of non-symmetric Hall-Littlewood polynomials and charge-like statistics
Edit: The problem I pose here is impossible to solve with the basis $H$, in the answer I made to this post I explain why. The only way I can think it to amend the situation would be to try with ...
- 161
0
votes
0
answers
71
views
Fourier decay implies what kind of regularity
We consider a function $f:\mathbb R^2 \to \mathbb C$ that is compactly supported and bounded. In addition, we know that
$$\lim_{\vert x\vert \to \infty} \vert x \vert^2 \vert \hat{f}(x)\vert =0,$$
...
4
votes
0
answers
101
views
There is only one reasonable $\sigma$-algebra on the space $\mathcal D'$ of distributions
Consider the space $\mathcal D'(M)$ of distributions on a manifold $M$.
Is there a ready reference for the fact that the Borel $\sigma$-algebra (for the strong dual topology) coincides with the weak ...
- 3,669
4
votes
0
answers
117
views
Convergence in probability results with still open point-wise versions
In ergodic theory and more generally in stochastic processes, often convergence in probability results precede convergence almost-surely results in quite a few years. Classical examples include the ...
- 41