Newest Questions
159,042 questions
3
votes
0
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99
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Comparing computable structures via Kleene and Skolem
Below, by "structure" I mean "computable structure in a finite language with domain $\omega$," and by "sentence" I mean "finitary first-order sentence containing no ...
- 20.5k
21
votes
0
answers
770
views
+300
Snakes on a plane
A sleeping bag for a baby snake in $d$ dimensions (no, really) is a subset of $\mathbb{R}^d$ which can cover (via translation and rotation) every (piecewise-smooth for concreteness) curve of unit ...
- 20.5k
1
vote
1
answer
152
views
Compact locus in (ordered) configuration spaces
Let $\mathit{Conf}_n(\mathbb{R}^2)$ be the configuration space of $n$ ordered distinct points in the plane. I'd like to know if the topological subspace $C_n$ consisting of points $(p_1,...,p_n)$ with ...
- 1,125
5
votes
0
answers
104
views
Is it always possible to find a conjugate optical function?
Optical functions (functions with null gradients) and double null foliations (foliations of a spacetime by two related optical functions) play a large roll in modern mathematical relativity research. ...
- 419
3
votes
0
answers
185
views
Relations between some categories of étale sheaves
I asked this question on math.stackexchange but nobody answers, so I try here even if I'm not sure my question is a research level one..
Let $X$ be a scheme over a number field $k$. Feel free to add ...
- 1,270
0
votes
1
answer
317
views
Extensionality, regularity, NBG--
Suppose you in a more comprehensive framework has the result that U is the least set such that all axioms of NBG-- (NBG, minus extensionality and regularity), hold; replacement is here, as expected, ...
- 3,574
4
votes
1
answer
256
views
Irreducible integral polynomials having roots module primes in arithmetic progressions
Let $f(x)$ be an irreducible polynomial with integer coefficients. One can show (see Exercise 7.2 in this paper of Lenstra) that if $f(x)=0$ has a solution mod $p$ for all but finitely many primes $p$...
- 6,224
1
vote
1
answer
113
views
The Fourier projection mappings $\{ P_N \}$ form an equicontinuous family of linear maps on $E'(S^1)$ as well?
Let $S^1=\mathbb{R}/\mathbb{Z}$ and define the Fourier projection operator $P_N$ for each $N \in \mathbb{N}$ as
\begin{equation}
P_N(f)=\sum_{n=-N}^N \langle f, e_n \rangle_{L^2} e_n
\end{equation}
...
- 3,477
4
votes
1
answer
264
views
Is the left derivative of this theta function zero at $1$?
Is it true that
$$f(x):=\sum_{j=1}^\infty(-1)^j j^2 x^{j^2}\to0$$
as $x\uparrow1$?
(One may note that $f(x)=xh'(x)$, where $h(x):=\vartheta _4(0,x)/2$ and $\vartheta _4$ is a theta function, so that $...
- 128k
1
vote
1
answer
213
views
Qualitative values between two electrons in an atom or how to interpret these values?
This question is a little bit trying to understand physics through geometry of simplex:
Let $E_{i,j}$ be the ionization energy in times the number of hydrogen ionization energy for an element with ...
- 3,074
0
votes
0
answers
108
views
How to show that the map $ R $ here is measure-preserving
Assume that $ (X,\mathcal{B},m,T) $ is a measure-preserving dynamical system, where $ (X,\mathcal{B},m) $ is a probability space, $ \mathcal{B} $ denotes all the measurable sets in $ X $, $ m $ is the ...
- 1,219
2
votes
1
answer
251
views
A weird property of odd positive integers $n$ with $\sigma(n)\sim2n $
When one looks at positive odd integers $n$ for which $|\sigma(n)-2n|\le\log n$, (sequence A088012) it appears that for all seven known numbers of this type the abundance, $\sigma(n)-2n$ is $\equiv 6\...
- 433
6
votes
0
answers
201
views
Dependence of Neumann eigenvalues on the domain
I have the following problem in hands, in the context of a broader investigation:
Let $V\in L^{n/2}$ compactly supported, where $n\geq 3$ is the dimension. I want to prove the following:
For any $\...
- 101
1
vote
1
answer
127
views
Simultaneous rational approximations of multiples of the golden ratio
My question concerns potential simultaneous rational approximations of irrational numbers.
Let $\alpha = \frac{1 + \sqrt{5}}{2}$ be the golden ratio, and $k \in \mathbb{N}$ a positive integer. In what ...
- 2,696
6
votes
1
answer
445
views
Branched coverings of non-orientable 3-manifolds
A continuous map of 3-dimensional manifolds $f \colon M^3 \to N^3$ is called a branched covering if there is a link $L \subset N^3$, such that the restriction $f \colon M \setminus f^{-1}(L) \to N \...
1
vote
0
answers
128
views
Log resolution of ideal and associated dual graph
Let $(X,0)$ be a complex surface germ with an isolated singularity and $I$ be an $\mathfrak{m}\text{-primary}$ ideal (contains a power of the maximal ideal $\mathfrak{m}$) of the local ring $\...
- 25
1
vote
1
answer
173
views
Who introduced the concept of beyond planar graphs?
The concept of planar graphs seems to be standard (I'm also not sure who first used this term), and recently, beyond planar graphs attract a lot of interest in the field of graph drawing. I know that ...
- 1,851
4
votes
1
answer
427
views
"The index is independent of the Dirac operator"
Fix a Clifford module bundle $E$ on a compact Riemannian manifold $M$ and let $D_0$ and $D_1$ be two Dirac operators (compatible with the Clifford action). The proof of the Atiyah-Singer index theorem ...
- 339
0
votes
1
answer
108
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If $a_1=1$ and $a_n=\sec (a_{n-1})$ then what does the proportion of positive terms approach, as $n\to\infty$?
Consider the sequence $a_1=1$ and $a_n=\sec (a_{n-1})$ for $n>1$.
What does the proportion of positive terms approach, as $n\to\infty$?
At first I thought the limiting proportion might be $\frac{...
- 3,577
2
votes
0
answers
70
views
A lemma in the application of Lions's concentration compactness pricnciple in Hardy-Littlewood-Sobolev inequality
I'm encountering some problems when reading Lions' paper "the concentration-compactness principle in the calculus of variations. The limit case, Part 2".
The Hardy-Littlewood-Sobolev (HLS) ...
- 121
13
votes
1
answer
468
views
Four new series for $\pi$ and related identities involving harmonic numbers
Recently, I discovered the following four new (conjectural) series for $\pi$:
\begin{align}\sum_{k=1}^\infty\frac{(5k^2-4k+1)8^k\binom{3k}k}{k(3k-1)(3k-2)\binom{2k}k\binom{4k}{2k}}&=\frac{3\pi}2,\...
- 15.6k
5
votes
0
answers
328
views
Are there examples of different knots with identical Jones polynomials and different Seifert Genus?
I had asked this question on math.stackexchange 2 days back but came up empty handed so I wanted to ask it here.
Are there known examples of $2$ non equivalent knots that have identical jones ...
- 2,689
1
vote
1
answer
209
views
Borel functions in C*-algebras
Is there a way of defining representations of separable $C^*$-algebras, say $\Phi$, so that
$\Phi(A)$ is faithful representation of $A$ on a separable Hilbert space.
There is a closure operation $A\...
- 13
4
votes
1
answer
447
views
WZW primary correlations in terms of current algebra?
Given the
$\mathfrak{u}(N)$ algebra
with generators $L^a$ and commutation relations
$ [L^a,L^b] = \sum_c f^{a,b}_{c} L^c $ ,
the WZW currents of $U(N)_k$
$$ J(z) = \sum_{n \in \mathbb{Z}} J^a_n z^{-n-...
- 545
2
votes
1
answer
271
views
Apropos of two groups being globally isomorphic iff they are isomorphic
Denote by $\mathcal P(S)$ the semigroup obtained by endowing the non-empty subsets of a "ground semigroup" $S$ (written multiplicatively) with the operation of setwise multiplication induced ...
- 10.5k
1
vote
1
answer
179
views
Gentle(-er) Introduction to Erdős–Bollobás's solution to Ramsey–Turán Type Problem
I am currently trying to understand the construction of maximal graph which contains no $K_4$ and sub-linear number of independent points in the graph. The original paper On a Ramsey–Turán type ...
1
vote
2
answers
262
views
A linear representation of the group of jets at 0 under composition
Let $G$ be the set of sequences $(f'(0), f''(0), f'''(0), \ldots)$ of derivatives at zero of functions $f : \mathbb{R} \to \mathbb{R}$ with $f(0) = 0$ and $f'(0) \ne 0 $. The set is a group under ...
- 19
5
votes
1
answer
300
views
Weak mixing and entering time
Let $X$ be a compact metric space and $f$ a continuous map from $X$ to $X$. Is it true, that if $f$ is weakly mixing, then the entering time $$N(U,V) = \{n \in \mathbb{N}\mid f^n(U) \cap V \neq \...
4
votes
0
answers
276
views
Special case of Eichler–Shimura
I'm reading ‘Rational Points on Elliptic Curves’ by Silverman and Tate, and the exercise 4.6 is about the following special case of the Eichler–Shimura theorem. Let $C$ be the elliptic curve given by ...
- 41
4
votes
2
answers
515
views
Questions about the results about $\Delta u + e^u=0$, $3 \le n \le 9$: no finite Morse index solution, $n \ge 10$: stable radial symmetric solution
I just read the celebrated paper Farina and Dancer, which talks about the following PDE in $\mathbb{R}^n$
$$\Delta u + e^u=0.$$
They proved that when $3 \le n \le 9$, there is no finite Morse index ...
- 809
0
votes
0
answers
145
views
Stalk at isolated point of proper map with $f_*\mathcal{O}_X=\mathcal{O}_Y$
Let $f:X \to Y$ a proper surjective map between schemes $X,Y$ with additional assumption $f_*\mathcal{O}_X=\mathcal{O}_Y$. Let $y \in Y$ with a 'isolated point fiber', i.e., $f^{-1}(y)=\{x\}$ as set. ...
- 6,048
5
votes
1
answer
272
views
Is the local maximal function bounded from $W^{1, 1}$ to $L^1$?
Let $f \in W^{1, 1} (\mathbb R^d)$. For every $\varepsilon > 0$, we consider the local maximal function $M_\varepsilon f: \mathbb R^d \to \mathbb R$, defined by
$$M f_{\varepsilon} (x) = \sup_{r \...
- 6,323
1
vote
0
answers
243
views
Pull and push formula for degree for non-flat morphism
Let $\varphi\colon X_1\to X_2$ be dominant proper morphism of finite degree (in particular $\dim X_1=\dim X_2$) between varieties.
Let $D \subset X_2$ be a Cartier divisor.
Is it true that $$\varphi_*...
- 219
2
votes
0
answers
137
views
Is the GIT quotient of a finite map of varieties again a finite map?
Let $K$ be an algebraically closed field of characteristic $0$, let $X/K$ and $Y/K$ be quasi-projective varieties, and let $f:X\to Y$ be a morphism. Let $G/K$ be a reductive group that acts stably on $...
- 47.4k
1
vote
0
answers
178
views
Is there a characterization for graphs with independence number two?
An independent set is a set of vertices in a graph, no two of which are adjacent. A maximum independent set is an independent set of the largest possible size for a given graph. The size of a maximum ...
- 1,851
1
vote
1
answer
90
views
Chern character of a super-connection (Heat kernels and Dirac operators)
Let $A$ be a super-connection on a super-bundle $E\to M$, then the differential form
\begin{equation}
\mathrm{ch}(A)=\mathrm{Str}(e^{-A^2})
\end{equation}
is called the chern character of $A$ on page ...
- 339
2
votes
0
answers
216
views
When do these ODE have positive solutions?
Consider the ODE \begin{equation} x'' + q(t) x = 0 \end{equation} in the unit interval $(0,1)$, with a potential function $q(t) = 4\pi^2 - \frac{Ct}{(1 - t)^2}$ depending on a positive constant $C >...
- 5,048
9
votes
0
answers
1k
views
Some questions about Clausen's third IHES lecture on Efimov K-theory
I have some questions about the last theorem stated by Clausen at https://youtu.be/2xNG4rHUC6U?si=yw9eYiygLegH0nQK&t=4319. I'm not very familar with the definitions, so please correct me about any ...
- 2,356
0
votes
0
answers
136
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Understanding the relations without the knowledge of Plucker relations [duplicate]
Consider the grassmannian $\mathrm{Gr}(2,5)$. We know there is an embedding of $\mathrm{Gr}(2,5)$ into $\mathbb{P}^9$ by using the 10 Plucker coordinates, and they satisfy 5 Plucker relations. And, so ...
- 839
4
votes
0
answers
92
views
Do there exist any cuboids, five of whose seven distinct inter-vertex lengths are integers and the product of whose other two is also an integer?
There don’t seem to be any small ones. Perfect cuboids would be examples, so a proof that no examples exist that are not perfect cuboids would also be very interesting.
6
votes
0
answers
190
views
Is Vopěnka's principle inherited by Grothendieck topoi?
I call the Vopěnka's principle:
Every subfunctor of an accessible functor is accessible
but other formulations (which may lose equivalence in weak contexts?) are also interesting to me.
If this is ...
- 6,962
8
votes
1
answer
324
views
3-divisibility of Manin constant for elliptic curves with 3-torsion
Let $E/\mathbb{Q}$ be an elliptic curve with $E(\mathbb{Q}) \cong \mathbb{Z}/3\mathbb{Z}$ (not necessarily $\Gamma_0$-optimal). Does $3$ necessarily divide one of: the Manin constant (not necessarily $...
- 91
3
votes
0
answers
162
views
Why is this constrained quadratic-over-linear integer program separable?
Consider the following splitting problem. Given $Y$ balls of which $X\leqslant Y$ of them are blue balls. The goal is to split the balls by placing them in $K$ baskets based on the following quadratic-...
- 31
2
votes
0
answers
401
views
Extended Collatz conjecture
As you all know, the Collatz conjecture claims that any positive integer will eventrually be reduced to 1 by appllying the sequence $n_{i+1} = x*n_{i} + 1$, when $n_{i}$ is odd, and $n_{i+1} = n_{i} / ...
- 161
37
votes
1
answer
1k
views
Errata for Fulton's "Young tableaux"
Fulton's Young tableaux is one of the best texts on the subject, one which I
often recommend and cite for reference. Unlike Fulton/Lang and
Fulton/Harris,
it is neither an early-dawn draft nor a ...
6
votes
4
answers
674
views
Moduli of smooth curves
Why is the Moduli of smooth curves of a fixed genus not compact/proper?
I know that there is a compactification using stable curves. But is it easy to see that the Moduli of smooth curves is not ...
- 153
3
votes
2
answers
979
views
Are $L^p$ norms absolutely continuous?
Let $1 < K \leq \infty$, and suppose $f \in L^p (X)$ for all $1 \leq p \leq K$, for $X$ some $\sigma$-finite measure space with no atoms.
Question: Is the function $p \to \|f\|_{L^p}$ absolutely ...
- 6,323
10
votes
2
answers
1k
views
Why are the source-target rules of composition always strictly defined?
General categorical definitions always have two variants, a strict one, in which associativity and unity hold as equalities, and a weak one, in which they hold up to equivalence. However, every ...
3
votes
2
answers
254
views
Number of atoms of a probability measure
Let $P\mathbb{R}$ be the space of probability measures on $\mathbb{R}$. Is the function
\begin{align*}
P\mathbb{R} &\to \mathbb{N} \cup \{\infty\}\\
\mu &\mapsto \#\{ x \in \mathbb{R} \mid \mu\...
- 239
1
vote
0
answers
109
views
What is known about the average growth rate of the denominators of $n$ Egyptian fractions summing to one?
Motivation
In the following question posted here on MO and over at MSE, user Noah Schweber asks about a weighted count on Egyptian fraction representations (EFRs). To that end, he defines the ...
- 4,829