parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/381828 | 4 | It is well known how to present solutions on the heat equation using the path integral (including the case of Riemannian manifold).
**Is there a way to present solutions of the Dirac equation using path integral?**
Here I assume that there is a distinguished time coordinate orthogonal to spatial coordinates. If nec... | https://mathoverflow.net/users/16183 | Path integral presentation of solutions of Dirac equation | There are several relevant papers:
* [Path Integral Approach to Relativistic Quantum Mechanics: Two-Dimensional Dirac Equation](https://academic.oup.com/ptps/article/doi/10.1143/PTPS.92.144/1934972) (1987)
* [Path Integral for
Relativistic Equations of Motion](https://arxiv.org/abs/hep-th/9708121) (1997)
* [Path Inte... | 7 | https://mathoverflow.net/users/11260 | 381829 | 158,922 |
https://mathoverflow.net/questions/381843 | 9 | Define $\sigma(N)=\sum\_{d|n} d$. A superabundant number is a positive integer $u$ for which $\frac{\sigma(u)}{u} > \frac{\sigma(v)}{v}$ for every positive integer $v<u$.
Similarly, do there exist infinitely many positive integers $n$ for which $\frac{\sigma(n^2)}{n^2}>\frac{\sigma(m^2)}{m^2}$ for every positive inte... | https://mathoverflow.net/users/480516 | On superabundant-like numbers | The answer is yes because $\sigma(n^2)/n^2$ is unbounded. To see this, take the product of the first $k$ primes $n=p\_1p\_2\cdots p\_k$. We have
$$\frac{\sigma(n^2)}{n^2}=\prod\_{i=1}^k \frac{p\_i^{2}+p\_i+1}{p\_i^2}>\prod\_{i=1}^k\left(1+\frac{1}{p\_1}\right)>\sum\_{i=1}^k\frac{1}{p\_i}$$
and the final sum diverges as... | 14 | https://mathoverflow.net/users/2384 | 381845 | 158,931 |
https://mathoverflow.net/questions/381834 | 6 | My adviser recently shared a problem with me that seeks to establish non-elementary\* hyperbolic quotients for mapping class groups. They told me that this could be useful for establishing results on separability or omnipotence, and that these could be relevant for examining profinite rigidity of hyperbolic 3-manifolds... | https://mathoverflow.net/users/151664 | What does it matter if a group has a non-elementary hyperbolic quotient? | You don't get anything *just* from knowing that a non-elementary hyperbolic quotient exists. However, the problem of constructing such quotients of mapping class groups appears very difficult, and can be viewed as a step on the way to constructing other interesting classes of quotients that you might hope to study, lik... | 7 | https://mathoverflow.net/users/1463 | 381852 | 158,934 |
https://mathoverflow.net/questions/381848 | 23 | My question is whether the construction of higher Witt groups of a scheme in Schlichting's [Hermitian K-theory of Exact Categories](https://homepages.warwick.ac.uk/%7Emasiap/research/HermitianKth.pdf) agrees with the definition in Balmer's [chapter in the Handbook of K-theory](https://faculty.math.illinois.edu/%7Edan/K... | https://mathoverflow.net/users/172742 | Do Schlichting's and Balmer's definitions of higher Witt groups of a scheme agree when 2 is inverted? | No, the definition in Schlichting's first paper are not the "correct" definition of higher Witt groups (in any case they are not the analogue of Balmer's Witt groups), rather they are some shifted higher Grothendieck-Witt groups. He provided later a different definition that does coincide with Balmer's (but which is de... | 16 | https://mathoverflow.net/users/43054 | 381856 | 158,935 |
https://mathoverflow.net/questions/381855 | 4 | Let $\mathcal{M}$ be a countable transitive standard-model of ZFC.
Let $B \in \mathcal{M}$ be a boolean algebra that is complete in $\mathcal{M}$.
Further, let $\mathcal{M}^{(B)}$ be the corresponding boolean model of ZFC.
Now we consider an $\mathcal{M}$-generic ultrafilter $U$ on $B$.
According to Jech (Se... | https://mathoverflow.net/users/171884 | Obtaining elements of a generic extension from a Boolean-valued model of ZFC | Simply use induction on $\mathbb{B}$-names (or the rank of $\mathbb{B}$-names, if you are not familiar with applying induction directly to $\mathbb{B}$-names.) Suppose that $i(z)=z^U$ holds for all $z\in\operatorname{dom} x$.
Since $x(z)\le \|z\in x\|$, we have $x^U\subseteq i(x)$. Conversely, assume that $\|y\in x\|... | 4 | https://mathoverflow.net/users/48041 | 381859 | 158,936 |
https://mathoverflow.net/questions/381858 | -2 | Let $f$ be a function on some real interval $[a,b]$. Suppose that $\forall x\in [a,b]$, there exists a positive constant $C$ such that
$$ |f(x)-f(y)| \leq C|x-y| $$
for all $y \in [a,b]$.
Does each $x \in [a,b]$ have a neighborhood $U$ such that
$$ |f(t)-f(s)| \leq C'|t-s| $$
for some $C' > 0$ and $\forall s,t \i... | https://mathoverflow.net/users/152618 | Question about Lipschitz conditions | $f(t) = t\sin(1/t)$ on $[0,1]$.
| 4 | https://mathoverflow.net/users/23141 | 381860 | 158,937 |
https://mathoverflow.net/questions/381877 | 2 | Given an ample line bundle $L$ on a smooth projective variety of dimension $\geq 2$, let $C$ be the category of vector bundles that are direct sums of powers of $L$. Two related questions:
1. Given a surjection in $C$ does the kernel have a filtration by line bundles? (a filtration that the successive quotients are l... | https://mathoverflow.net/users/127776 | Are splitting vector bundles closed under kernel or cokernels? | Definitely not. For instance, consider the projective space $\mathbb{P}^n$ and the Euler sequence
$$
0 \to \Omega \to \mathcal{O}(-1)^{\oplus (n + 1)} \to \mathcal{O} \to 0.
$$
Its second and third terms are in $C$ (if $L = \mathcal{O}(1)$), but the first term has no filtration by line bundles. Indeed, the first cohomo... | 7 | https://mathoverflow.net/users/4428 | 381878 | 158,942 |
https://mathoverflow.net/questions/381880 | 1 | I'm trying to understand Theorem 1.1 in [Limit laws for random matrix products](https://arxiv.org/pdf/1712.03698.pdf).
It states that a specific product of random matrices converges to a set matrix but I don't know which kind of convergence of random variables is meant here. Is it almost sure or even sure convergence?
... | https://mathoverflow.net/users/171867 | Limit laws for random matrix products covergence | Theorem 1.1 does not refer to a probabilistic limit, it holds elementwise for any given series of matrices that satisfies the conditions stated in the theorem.
| 1 | https://mathoverflow.net/users/11260 | 381891 | 158,943 |
https://mathoverflow.net/questions/381710 | 3 | I'm looking to study the existence solutions of the following coupled equation:
\begin{equation}
\left\{\begin{matrix}
x(t)&=&\int\_{0}^{t} K\big(t, s\big) f\big(s, x(s),y(s)\big) d s, \quad t \in[0,1) \\
y(t)&=&\int\_{0}^{t} K\big(t, s\big) f\big(s, y(s),x(s)\big) d s, \quad t \in[0,1)
\end{matrix}\right.
\end{equ... | https://mathoverflow.net/users/102228 | Applications of coupled Volterra-Hammerstein in Banach space | [Solvability of Coupled Systems of Generalized Hammerstein-Type Integral Equations in the Real Line](https://www.mdpi.com/2227-7390/8/1/111/htm) (2020): section 4 gives an application in mechanics, a study of the coupling between bending and torsion of two coupled beams on an elastic foundation, motivated by the dynami... | 1 | https://mathoverflow.net/users/11260 | 381896 | 158,945 |
https://mathoverflow.net/questions/381903 | 1 | Let $\Omega \subset \mathbb R^3$ denote an open, bounded and simply connected set with smooth boundary. The Helmholtz decomposition
$$ L^2(\Omega) = \nabla H^1\_0(\Omega) \oplus L^2(\operatorname{div}=0; \Omega)$$ entails the existence of two linear and bounded natural projections $\pi\_1 \in \mathcal{L}(L^2(\Omega); \... | https://mathoverflow.net/users/123407 | Is there any quantitative relationship between the two terms of a Helmholtz decomposition? | I don't know much about the curl spaces but I think I can answer in the negative for pure Hilbert space reasons.
If indeed $\hat{\pi}\_1$ and $\hat{\pi}\_2$ are projections, denote their ranges by $V\_1$ and $V\_2$, which we'll assume are also Hilbert subspaces of $V = V\_1 + V\_2$ ($= H\_0(curl;\Omega)$ for your spe... | 1 | https://mathoverflow.net/users/73890 | 381906 | 158,950 |
https://mathoverflow.net/questions/381302 | 9 | The most well-known construction of a non-measurable set is the Vitali set. The idea behind Vitali sets is to split up the space (such as $[0,1]$) into equal-sized copies (guaranteed by translation invariance), by looking at something like $\mathbb{R}/\mathbb{Q}$. This same idea is used in ["Visualizing a Nonmeasurable... | https://mathoverflow.net/users/111894 | Non-measurable sets on groups from translation invariance | The proof for the reals can be generalized to any non-discrete locally compact group $G$. We let $K \subset G$ be any compact set with positive Haar measure $\lambda(K) > 0$ (e.g., $K = [0, 1]$ when $G = \mathbb R$), and we let $\Lambda < G$ be any subgroup such that $\Lambda \cap KK^{-1}$ is countably infinite (e.g., ... | 4 | https://mathoverflow.net/users/6460 | 381916 | 158,956 |
https://mathoverflow.net/questions/381648 | 5 | Sorry if this question is naive, I am not very well versed in recursion theory.
Does it exist a formula $\phi$ such that:
* $\phi$ is provable in Peano arithmetic
* $\phi \in \Sigma^0\_n$ or $\phi \in \Pi^0\_n$, but
* provably, every proof of $\phi$ from the axioms must involve a step that does not belong to $\Sigm... | https://mathoverflow.net/users/828 | Formula that requires a higher complexity to be proved | First, let me mention that the only way a formula may require proofs using complex formulas is that it requires complex *axioms* to prove: the [cut elimination theorem](https://en.wikipedia.org/wiki/Cut-elimination_theorem) implies that if a $\Sigma\_n$ formula is provable from $\Sigma\_n$ axioms, it has a proof from t... | 6 | https://mathoverflow.net/users/12705 | 381917 | 158,957 |
https://mathoverflow.net/questions/381870 | 14 | Let $\{a\_n\}\_{n\ge1}$ be a real sequence that decays faster than any algebraic speed, that is, $\lim\_{n\to \infty} n^pa\_n = 0$ for every positive integer $p$. Assume that $$\sum\_{n\ge 1}(n+1)^kn^ka\_n = 0$$ for every integer $k \ge 0$.
**Question:** Can we conclude that $a\_n \equiv 0$?
| https://mathoverflow.net/users/114951 | A question on a real sequence | Counterexample. Consider the analytic function in the unit disk
$$f(z)=\exp\left(-\sqrt{\frac{1}{1-z}}\right)=a\_0+a\_1z+\ldots,\quad |z|<1,$$
where the principal branch of the $\sqrt{\;}$ is used.
This is the definition of our sequence $a\_n$. Function $f$ extends to a $C^\infty$ function on the unit circle, which evi... | 14 | https://mathoverflow.net/users/25510 | 381919 | 158,959 |
https://mathoverflow.net/questions/381908 | 46 | Many mathematical subfields often use the axiom of choice and proofs by contradiction. I heard from people supporting constructive mathematics that often one can rewrite the definitions and theorems so that both the axiom of choice and proofs by contradiction aren't needed anymore.
An example is the theory of [locale... | https://mathoverflow.net/users/172789 | How to rewrite mathematics constructively? | If you want a "general method" that "always works" to turn a classical theorem into a constructive one, there are [double-negation translations](https://en.wikipedia.org/wiki/Double-negation_translation): if you add enough $\neg\neg$s to a classical theorem, you can make a constructively provable statement. However, th... | 61 | https://mathoverflow.net/users/49 | 381920 | 158,960 |
https://mathoverflow.net/questions/378471 | 1 | I was just reading Mike Shulman's [blog post](https://homotopytypetheory.org/2013/07/24/cohomology/) on how to define cohomology in homotopy type theory (HoTT), and I was curious if we can similarly define cohomology with local coefficients in HoTT as well?
I know that a local system can be viewed as a locally consta... | https://mathoverflow.net/users/56938 | Cohomology with local coefficients in homotopy type theory | *(I suppose this is actually an answer, so I should post it as one.)*
Yes, this generalization is described in the [next blog post](https://homotopytypetheory.org/2013/08/08/spectral-sequences/), since it's needed for the Serre spectral sequence. Make sure you read the version [here](https://ncatlab.org/homotopytypet... | 4 | https://mathoverflow.net/users/49 | 381928 | 158,964 |
https://mathoverflow.net/questions/381927 | 5 | Suppose that $\Omega \subset \mathbb R^3$ is a domain with smooth boundary $\partial \Omega$ and suppose that $0\in \Omega$. Given any $f \in C^{\infty}(\partial \Omega)$ let $u^f$ denote the unique harmonic function on $\Omega$ with Dirichlet data $f$. Finally, suppose that given any smooth $f$ there holds:
$$ u^f(0)=... | https://mathoverflow.net/users/50438 | Mean value principle reversed | *Edit: I misread the question. New answer:*
The question asks whether the Poisson kernel $P\_\Omega(0, \cdot)$ is constant only when the domain $\Omega$ is a ball centred at $0$.
This is indeed true: let $r$ be the radius of the largest ball $B(0, r)$ contained in $\Omega$, and $R$ the radius of the smallest ball $... | 6 | https://mathoverflow.net/users/108637 | 381931 | 158,966 |
https://mathoverflow.net/questions/381530 | 3 | Let $\rightarrowtail$ denote a monomorphism.
Given a morphism $A \stackrel{j}{\to} B$, I am interested in the (not necessarily unique) existence of a factorization $A \stackrel{j'}{\rightarrowtail} X \stackrel{s}{\to} B$ of $j$ (so $j = s \circ j'$) such that for any other factorization $A \stackrel{i}{\rightarrowtai... | https://mathoverflow.net/users/152371 | Subobject- and factorization-preserving typings | One general way to construct such an $X$ is as a [partial map classifier](https://ncatlab.org/nlab/show/partial+map+classifier) for $j:A\to B$, regarded as an object of the slice category $\mathcal{E}/B$.
To see this, note first that if $s:X\to B$ is such a partial map classifier, then it comes with a canonical parti... | 2 | https://mathoverflow.net/users/49 | 381933 | 158,967 |
https://mathoverflow.net/questions/381840 | 11 | For a number field $K$, we write $\Delta\_K$ for its absolute discriminant. I was hoping for a Siegel--Walfisz type theorem of the following type:
Let $A > 0$. Then for every $X > 0$, every number field $K$ and every Galois extension $L/K$ with $\Delta\_L \leq (\log X)^A$ and every conjugacy class $C$ of $\text{Gal}(... | https://mathoverflow.net/users/96891 | Siegel--Walfisz for number fields | There is enough in the literature to extract a result of this form, but it might not appear explicitly. I will reference recent work of Thorner and Zaman instead of Lagarias-Odlyzko, since it gives a substantial improvement:
Let $L/F$ be a Galois extension with group G, and let $C\subseteq G$ be a conjugacy class. Le... | 3 | https://mathoverflow.net/users/111215 | 381941 | 158,970 |
https://mathoverflow.net/questions/381935 | 1 | When [counting the number of integers $n(x)$](https://math.stackexchange.com/questions/468334/an-integer-counting-function-nx) below a certain non-integer number $x$, the following series could be used:
$$n(x) = x-\frac12 + \sum\_{n=1}^{\infty} \left(\frac{e^{x \mu\_n}} {\mu\_n}+\frac{e^{x \overline{\mu\_n}}} {\overl... | https://mathoverflow.net/users/12489 | Deriving the functional equation for $\zeta(s)$ from summing the powers of the zeros required to count the integers | The intermediary seems to be the Bernoulli number sequence which was originally birthed in summing up powers of the integers and in turn eventually gave birth, via the midwife the Mellin transform, to the Riemann and Hurwitz zeta functions. The MO-Q to which you link on motivating derivations of the functional equation... | 4 | https://mathoverflow.net/users/12178 | 381942 | 158,971 |
https://mathoverflow.net/questions/381944 | 0 | By some analogy, the integral and differential can be extend to factorial differintegral, see <https://en.wikipedia.org/wiki/Differintegral>
My question is, what is the he geometry interpretation of fractional differintegral?
| https://mathoverflow.net/users/14024 | what is the geometry interpretation of fractional differintegral? | I recommend you [I.Podlubny Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation](https://arxiv.org/abs/math/0110241)
| 1 | https://mathoverflow.net/users/nan | 381951 | 158,974 |
https://mathoverflow.net/questions/381956 | 1 | I am trying to develop a theory explaining analytic continuation of a holomorphic function $f(z)$ defined on an open set $D \subset \mathbb{C}$. Recently, I was looking at the last chapter of Lars Ahlfors Complex Analysis book and I discovered striking similarities between my approach and that of Weierstrass.
First, ... | https://mathoverflow.net/users/8435 | Global theory of holomorphic functions | In general, the holomorphic extension to $U\_{z\_0}$ that you envision does not exist. Take, for example, the principal branch of the logarithm on the disk $B(1,1)$. It has a holomorphic extension to $\mathbb{C}\setminus i[0,\infty)$, and also a holomorphic extension to $\mathbb{C}\setminus -i[0,\infty)$. Yet, it has n... | 4 | https://mathoverflow.net/users/11919 | 381957 | 158,975 |
https://mathoverflow.net/questions/381993 | 2 | For a bounded function $F: \mathbb R\_{\ge 0} \to \mathbb R$ (*not necessarily non-negative*), is it true that
$$\int\_0^\infty \frac{x^ks}{(s^2+x^2)^{(k+3)/2}} F(x) dx = 0 \text{ for all $s >0$} \iff F \equiv 0$$
where $k \in \mathbb{N}$ is a positive constant? Of course, one implication ($\leftarrow$) is true. What a... | https://mathoverflow.net/users/110835 | Injectivity of an integral transform | Think that $|x| F(|x|)$ ($x \in \mathbb R^{k+2}$) is the boundary value of a harmonic function $u$ in the half-space $\mathbb R^{k+2} \times (0, \infty)$, given by an appropriate [Poisson integral](https://en.wikipedia.org/wiki/Poisson_kernel#On_the_upper_half-space):
$$ u(x,s) = c\_k \int\_{\mathbb R^{k+2}} |y| F(|y|)... | 2 | https://mathoverflow.net/users/108637 | 382001 | 158,986 |
https://mathoverflow.net/questions/381984 | 7 | Let $R$ be an associative ring with identity and $\mathrm{mod}R$ be the category of finitely presented $R$-modules. I would like to know when the category $\mathrm{mod}R$ is abelian.
I know that if $R$ is noetherian, then $\mathrm{mod}R$ is an abelian category. Maybe, it could be the case that if $\mathrm{mod}R$ is a... | https://mathoverflow.net/users/156726 | When is the category of finitely presented modules abelian? | Wojowu's idea is right:
**Lemma.** *Let $R$ be a ring, let $\mathbf{Mod}\_R$ be the category of (left) $R$-modules, and let $\mathbf{Mod}\_R^{\text{fp}}$ be the subcategory of finitely presented modules. Then the following are equivalent:*
1. *$R$ is left coherent, i.e. every finitely generated left ideal is finite... | 17 | https://mathoverflow.net/users/82179 | 382005 | 158,987 |
https://mathoverflow.net/questions/382004 | 16 | For a paper I am writing related to the history of combinatorics, I am
looking for the year of birth of Craige Eugene
Schensted, the eponym for the
Schensted correspondence. According to [this
site](https://ancestors.familysearch.org/en/L23J-X2F/roy-eugene-schensted-1904-1%5C%0A988), a Craige Eugene Schensted was born ... | https://mathoverflow.net/users/2807 | Year of birth of Craige Schensted | Craige Eugene Schensted was born on April 12, 1927, in Mayfield, North Dakota, according to the 1940-1947 US Draft Card on [ancestry.com.](https://www.ancestry.com/search/?name=Craige+E_Schensted)
(I have a scan of the draft card, you can email me for a copy, not sure if posting it here is legit.)
| 16 | https://mathoverflow.net/users/11260 | 382006 | 158,988 |
https://mathoverflow.net/questions/382011 | 8 | [This question came up while idly thinking about [this other one](https://mathoverflow.net/q/381068/17064), but it is not directly related.]
**Definitions:** If $X$ is a topological space, let $C(X)$ stand for the $\mathbb{R}$-algebra of continuous real-valued functions $X\to\mathbb{R}$ where $\mathbb{R}$ has its usu... | https://mathoverflow.net/users/17064 | Can locally constant real functions on a space be made into continuous functions (on a different space)? | $D(\mathbb{Q})$ is not isomorphic to $C(X)$ (as an $\mathbb{R}$-algebra) for any topological space $X$. For both $D(X)$ and $C(X)$, you can recover the (extended) uniform norm from the $\mathbb{R}$-algebraic structure with
$$ \left\| x \right\| = \sup\{|\lambda| : x - \lambda 1\text{ is not invertible}\},$$
where $1$ i... | 6 | https://mathoverflow.net/users/83901 | 382013 | 158,991 |
https://mathoverflow.net/questions/381991 | 20 | It seems that, almost all computer programs assume GRH to calculate $\mathbb{Q}(\zeta\_p)$ for $p > 23$. I'm very curious how assuming the GRH, helps us to calculate class groups in practice. Can anyone give an explicit example of a number field (not necessarily $\mathbb{Q}(\zeta\_p)$'s), and explicit calculation of th... | https://mathoverflow.net/users/68462 | How does assuming GRH help us calculate class group? | In general, to compute the class group of a number field $K$ of degree $n$ and discriminant $\Delta$, we need to find some bound $N$ such that the class group of $K$ is generated by primes of norm at most $N$. Unconditionally, we have the Minkowski bound
$$M\_K = \sqrt{\lvert \Delta \rvert} \left( \frac{4}{\pi} \right)... | 37 | https://mathoverflow.net/users/31308 | 382016 | 158,992 |
https://mathoverflow.net/questions/382028 | 10 | In 2009, Jochen Koenigsmann [showed](https://annals.math.princeton.edu/wp-content/uploads/Koenigsmann.pdf) that $\mathbb{Z}$ is universally definable in the field $\mathbb{Q}$. My question is, are there any other number fields in which $\mathbb{Z}$ is universally definable?
Or failing that, what is the lowest complex... | https://mathoverflow.net/users/5017 | Is $\mathbb{Z}$ universally definable in any number fields other than $\mathbb{Q}$? | Koenigsmann's result was generlized by [Jennifer Park](https://arxiv.org/abs/1202.6371) to number fields, giving a universal definition of the ring of integers $\mathcal{O}\_K$ in $K$. Then there is a series of results proving that $\mathbb{Z}$ is existentially definable in $\mathcal{O}\_K$ for certain $K$, starting I ... | 12 | https://mathoverflow.net/users/50351 | 382036 | 159,001 |
https://mathoverflow.net/questions/381975 | -1 | Let $(X,d)$ be a separable complete [geodesic metric space](https://topospaces.subwiki.org/wiki/Geodesic_metric_space) and let $K$ be a compact (non-empty) subset of $X$. Without assuming things like linearity, the convexity of $K$, and locally convexity, under what conditions can we guarantee that there is a continuou... | https://mathoverflow.net/users/170917 | Existence of continuous selection for metric projection | In general, $argmin$ is not continuous. Even on the real line, if I take $K$ to be two distinct points, say $K=\{-1,1\}$, then $argmin\_{k\in K} d^2(x,k)$ is not continuous. This is why convexity is so important. Without assuming convexity, little is guaranteed in this case.
| 1 | https://mathoverflow.net/users/160011 | 382040 | 159,002 |
https://mathoverflow.net/questions/382015 | 2 | The following question is related to the families of high rank elliptic curves with torsion subgroup $\mathbb{Z}/6\mathbb{Z}$.
The SageMath/Python code below produces a list of small fractions $a$ for which $e=\sqrt{a(a+1)}$ is a multiple of $\sqrt{r}=\sqrt{2}$.
```
from time import time
t0 = time()
r = 2
listA = ... | https://mathoverflow.net/users/95511 | A new simple formula is needed | I found
$$b = 16 \frac{a^2+a}{8a+9}$$
| 1 | https://mathoverflow.net/users/158462 | 382045 | 159,003 |
https://mathoverflow.net/questions/381996 | 8 | Consider the representation of $\textrm{SO}(4)$ on $\textrm{Sym}^2(\wedge^2\mathbb{R}^4)$ induced by the standard representation of $\textrm{SO}(4)$ on $\mathbb{R}^4$. I am interested in the ring of invariants of this representation, i.e. the ring of all polynomial functions on $\textrm{Sym}^2(\wedge^2\mathbb{R}^4)$ th... | https://mathoverflow.net/users/36563 | Invariant ring of $\textrm{Sym}^2(\wedge^2\mathbb{R}^4)$ under $\textrm{SO}(4)$ | The answer is 'no', though I don't know an easy way to see this without doing an explicit calculation. Here is where to look though, if you want to do the calculation yourself:
Things work out a bit better if one uses indeterminates $z = x+y$ and $w = x-y$. Then one has an expansion
$$
\det\bigl(x{\cdot}I + y{\cdot}\... | 7 | https://mathoverflow.net/users/13972 | 382052 | 159,005 |
https://mathoverflow.net/questions/381963 | 3 | Let $k$ be a field complete with respect to a non-trivial non-archimedean
absolute value (so that rigid $k$-space makes sense). Denote $K$ a finite field extension of $k$.
Denote $X\rightsquigarrow X^{\mathrm{an}/k}$ the analytification functor from the category of locally of finite type $k$-schemes to the category o... | https://mathoverflow.net/users/110471 | How to show analytification functor commutes with forgetful functor? | I believe I got an answer. Denote $X^{\mathrm{an}/K},X^{\mathrm{an}/k}$ the analytifications of $X$ over $K$ and $k$ respectively. Denote $K$-maps (resp. $k$-maps) the maps of locally $G$-ringed $K$-spaces (resp. $k$-spaces).
**Lemma:** Let $X$ be a locally of finite type $K$-schemes. Then $X^{\mathrm{an}/K}\cong X^{... | 0 | https://mathoverflow.net/users/110471 | 382059 | 159,009 |
https://mathoverflow.net/questions/382060 | 10 | Let $G\_1 \to G\_2 \to \cdots$ be a sequence of epimorphisms of finitely generated residually finite groups. Does it eventually stabilize? That is, are all but finitely many epimorphisms actually isomorphisms?
Note that finitely generated residually finite groups are Hopfian, so this excludes the simple counterexampl... | https://mathoverflow.net/users/145915 | Sequence of epimorphisms of residually finite groups stabilizes | The answer is "no". The lamplighter group (which is infinitely presented) is a limit of a sequence of virtually free groups and surjective homomorphisms (see, for example, this [question and answers there](https://mathoverflow.net/questions/75784/is-there-a-non-hopfian-lacunary-hyperbolic-group)). All virtually free gr... | 12 | https://mathoverflow.net/users/157261 | 382063 | 159,010 |
https://mathoverflow.net/questions/382076 | 1 | Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Let $\{E\_i\}\_{i = 1}^N,$ with $E\_i \in\mathcal{F}$ be a set of events and let $i(X)$ be a R.V. assuming values in $\{1,...,N\}$
Is there a way to bound the following quantity?
$$\mathbb{P}\left[\bigcup\_{i\in[N]: i \neq i(X)} E\_i\right].$$
I am loo... | https://mathoverflow.net/users/156139 | Union bound probability of random union | Let $n:=N$ and $J:=i(X)$. Then the probability to bound is
\begin{aligned}
P\Big(\bigcup\_{i\in[n]\setminus\{J\}}E\_i\Big)
&=\sum\_{j\in[n]}P\Big(\{J=j\}\cap \bigcup\_{i\in[n]\setminus\{j\}}E\_i\Big) \\
&=\sum\_{j\in[n]}P\Big(\bigcup\_{i\in[n]\setminus\{j\}}\big(\{J=j\}\cap E\_i\big)\Big) \\
&\le\sum\_{j\in[n]}\sum\_... | 3 | https://mathoverflow.net/users/36721 | 382080 | 159,018 |
https://mathoverflow.net/questions/382090 | 0 | Let $f:\mathbb R \to \mathbb R$ be a $BV$ function and $g:\mathbb R \to \mathbb R$ be a diffeomorphism. What is the total variation of $f \circ g$?
My guess is
$$
TV(f\circ g) \le TV(f) \Vert (g^{-1})'\Vert\_{L^\infty}
$$
but I don't have a proof.
| https://mathoverflow.net/users/110835 | Total variation of composition of BV function and diffeomorphism | $\newcommand\R{\mathbb R}$Since $g$ is a diffeomorphism of $\R$, it is a homeomorphism of $\R$. So, $g$ is either increasing or decreasing.
Suppose that $g$ is increasing. Take any real $x\_1,\dots,x\_n$ such that $x\_1<\dots<x\_n$. Let $y:=g(x\_i)$ for all $i$. Then $y\_1<\dots<y\_n$. So,
$$\sum\_{i=1}^{n-1}|(f\circ... | 3 | https://mathoverflow.net/users/36721 | 382095 | 159,022 |
https://mathoverflow.net/questions/382094 | 6 | Let $F$ be a field, let $E$ be a field extension of $F$, and let $U$ be an ultrafilter. Then my question is, what is the relationship between the Galois groups $Gal(\Pi\_U E/\Pi\_U F)$ and $Gal(E/F)$?
Or if that's too general, is it at least possible to say something in the case when $E$ is a number field and $F=\mat... | https://mathoverflow.net/users/5017 | What is the Galois group of one ultrapower over another ultrapower? | $\newcommand{\Gal}{\operatorname{Gal}}$If $E/F$ is a finite Galois extension, then $\Gal(\prod\_UE/\prod\_UF)$ is canonically isomorphic to $\Gal(E/F)$. Indeed, by the primitive element theorem, $E=F(\alpha)$ for some $\alpha\in E$. This means every element of $E$ can be written as a polynomial in $\alpha$ with coeffic... | 9 | https://mathoverflow.net/users/30186 | 382102 | 159,024 |
https://mathoverflow.net/questions/382033 | 2 | Let $M$ be a compact and connected manifold without boundary.
My question is how to prove the following fact which I believe is true:
If $f : M \to \mathbb{R}$ is a continuous function that attains the values $a < b$, then for any $c\in [a,b]$ and any $1\leq p<\infty$, there is diffeomorphism $\varphi : M\to M$ such ... | https://mathoverflow.net/users/94097 | Approximating continuous functions via diffeomorphisms on compact manifolds | The answer to the last question follows from the following result:
>
> **Theorem.** *If $f:\mathcal{M}\to\mathbb{R}$ is a continuous function on a smooth compact connected manifold without boundary and if
> $$
> \inf\_{\mathcal{M}} f\leq c\leq \sup\_{\mathcal{M}} f,
> $$
> then for any $1\leq p<\infty$ and any $\va... | 1 | https://mathoverflow.net/users/121665 | 382103 | 159,025 |
https://mathoverflow.net/questions/382101 | 0 | Let $f \in C^k(0,1)$ and assume that the $k$-th derivative is $\alpha$-Hölder continuous. Assume that $f(x) = 0$ in a fixed interval $(a,b) \subset (0,1)$. Can we characterize (or at least find some examples of) non-constant functions $f$ as above such that
$$|f^{(k)}|\_{C^{0,\alpha}(0,1)} \le \Vert f \Vert\_{L^1(0,1)}... | https://mathoverflow.net/users/139843 | Functions for which $|f^{(k)}|_{C^{0,\alpha}(0,1)} \le \Vert f \Vert_{L^1(0,1)}$ | The answer is no.
Indeed, let $t:=\alpha\in(0,1)$ and $c:=\|f\|\_1:=\|f\|\_{L^1(0,1)}\in(0,\infty)$. Suppose that $f\in C^k(0,1)$ and $f=0$ on $(a,b)$, with $0\le a<b\le1$. Suppose that the inequality in question holds.
Then for $x\in[b,1]$ we have $|f^{(k)}(x)|=|f^{(k)}(x)-f^{(k)}(b)|\le c(x-b)^t\le c$, $|f^{(k-1)... | 2 | https://mathoverflow.net/users/36721 | 382110 | 159,028 |
https://mathoverflow.net/questions/382084 | 4 | I’m studying symplectic manifolds and almost complex structures. This lead to two propositions:
**Proposition 1** (from da Silva’s *Lectures on Symplectic Geometry*): If $J\_0$ and $J\_1$ are almost complex structures compatible with a symplectic manifold $(M,\omega)$, then there is a family of almost complex structu... | https://mathoverflow.net/users/153883 | (Contradiction) All symplectic manifolds are holomorphic | The comments of @JHM and @Ivan Solonenko contain an answer to OP's question.
(1) The OP is correct that $j\_0:=\phi\_\* \circ J \circ \phi^{-1}\_\*$ is an almost-complex structure on the image $V:=\phi(U) \subset \mathbb{R}^{2n}$ which is compatible with the standard symplectic form $\omega\_{std}$ satisfying $\omega... | 5 | https://mathoverflow.net/users/20516 | 382145 | 159,038 |
https://mathoverflow.net/questions/382131 | 1 | Is there a (non-constant) function $f \in C^4((0,1))$ that is zero in an interval $(a,b) \subset (0,1)$ and such that the inequality
$$\Vert\tfrac{d^4}{dx^4}f\Vert\_{L^2(0,1)} < \sqrt{2}\Vert f \Vert\_{L^1(0,1)}$$
holds?
| https://mathoverflow.net/users/139843 | Functions such that $ \Vert\tfrac{d^4}{dx^4}f\Vert_{L^2(0,1)} < \sqrt{2} \Vert f \Vert_{L^1(0,1)}$ | The answer is no.
Indeed, let $c:=\sqrt2\,\|f\|\_1\in(0,\infty)$, where $\|f\|\_p:=\|f\|\_{L^p(0,1)}$. Suppose that $f\in C^4(0,1)$ and $f=0$ on $(a,b)$, where $0\le a<b\le1$. Suppose that the inequality in question holds:
$$\|f''''\|\_2<c.$$
Then, using the Cauchy--Schwarz inequality, for $x\in[b,1]$ we have
$$|f'... | 2 | https://mathoverflow.net/users/36721 | 382146 | 159,039 |
https://mathoverflow.net/questions/371948 | 0 | For commutative rings $R \subseteq S$,
recall that $S$ is [separable](https://en.wikipedia.org/wiki/Separable_algebra) over $R$, if $S$ is a projective $S \otimes\_R S$-module, via $f: S \otimes\_R S \to S$ given by: $f(s\_1 \otimes\_R s\_2)=s\_1s\_2$.
>
> **Question 1:** Is $\mathbb{C}[x]$ separable over $\mathbb{... | https://mathoverflow.net/users/72288 | Separability of $\mathbb{C}[x]$ over its $\mathbb{C}$-subalgebras | As I wrote in my comment, these extensions are known in algebraic geometry as *unramified*, which can be tested computationally by the vanishing of $\Omega\_{S/R}$. (In differential geometry, this means $Y \to X$ is an immersion, i.e. injective on tangent spaces.)
**Lemma.** *Let $h \in \mathbf C[x]$ be a polynomial ... | 2 | https://mathoverflow.net/users/82179 | 382150 | 159,041 |
https://mathoverflow.net/questions/382151 | 5 | In remark 1.2.6.2 (HTT), Lurie states that
>
> Another possible approach to the problem of homotopy
> coherence is to restrict our attention to simplicial (or topological) categories
> C in which every homotopy coherent diagram is equivalent to a strictly commutative
> diagram. For example, this is always true when... | https://mathoverflow.net/users/140013 | Homotopy coherent colimits in chain complexes | The result is not only true for simplicial model categories, but for plain combinatorial model categories too - this is *Higher Algebra* 1.3.4.25..
In fact, for this you can *reduce* to the case of simplicial model categories, by noting that a combinatorial model category is always Quillen equivalent to a simplicial ... | 13 | https://mathoverflow.net/users/102343 | 382154 | 159,043 |
https://mathoverflow.net/questions/382164 | 7 | Suppose that for a finite collection of planar convex sets $\mathcal F$ the following holds.
For any six members of $\mathcal F$ there are two points such that every set contains (at least) one of the points.
Does it follow that all members of $\mathcal F$ can be stabbed by two points?
I am sure that this is kn... | https://mathoverflow.net/users/955 | Two-point Helly | This is true for special families of convex sets, for example axis parallel rectangles, but it is false for general convex sets, even if $6$ is replaced by any other finite number. This was shown by M. Katchalski and D. Nashtir in the paper *On a conjecture of Danzer and Grünbaum* (Proc. Amer. Math. Soc. 124 (1996), 32... | 9 | https://mathoverflow.net/users/2384 | 382168 | 159,049 |
https://mathoverflow.net/questions/382172 | 0 | Recall that, given an extended real-valued function $f: \mathbb{R}^n \to (-\infty, \infty]$
Its effective domain is,
$$\text{dom}(f) = \{x \in \mathbb{R}^n : f(x) < +\infty\}$$
The subdifferential is, $$\partial f(x) = \{v \in \mathbb{R}^n: f(x^\prime) \geq f(x) + v^\top (x^\prime - x), \forall x^\prime \in \mathbb... | https://mathoverflow.net/users/74540 | When does strict inclusion holds for the domain of subdifferential? | Define $f\colon\mathbb R\to\mathbb R$ by letting $f(x):=-\sqrt x$ if $x\ge0$ and $f(x):=\infty$ if $x<0$. Then $f$ is a [closed](https://en.wikipedia.org/wiki/Closed_convex_function) [proper](https://en.wikipedia.org/wiki/Proper_convex_function) convex function.
However, $f(0)=0<\infty$, so that $0\in\text{dom}(f)$. ... | 3 | https://mathoverflow.net/users/36721 | 382182 | 159,053 |
https://mathoverflow.net/questions/382203 | 1 | A function $f(x)$ is called low-dimensional if there exists non-zero vector $v\in R^n$ such that $f(x)=f(x+cv)$ for all $c\in R$. I'm wondering whether any finite sum of continuous low-dimensional functions $f\_i(x)$ is either non-integrable or being the zero function. It's trivially true for $n=1$ and if we restrict $... | https://mathoverflow.net/users/173021 | Finite sum of low-dimensional functions in R^n | I assume that by integrable you mean $\int |f|<\infty$.
Assume that $f=f\_1+\ldots+f\_n$ where each $f\_i$ is a continuous periodic function: $f\_i(x+v\_i)=f\_i(x)$ for certain $v\_i\in \mathbb{R}^n\setminus \{0\}$. This is weaker condition then being low-dimensional. I claim that if $\int |f|<\infty$ then $f\equiv 0... | 1 | https://mathoverflow.net/users/4312 | 382208 | 159,059 |
https://mathoverflow.net/questions/382207 | 3 | Let $u, \eta$ be smooth functions and $\eta$ compactly supported in $(0,1)$. Integrating by parts, we can easily prove $$-\int\_0^1 u\_{xxx}u\_x \eta = \int\_0^1 (u\_{xx})^2\eta - \int\_0^1 \frac{1}{2} (u\_x)^2 \eta\_{xx}$$
Can we obtain the same result using only the Fourier series
$$u(x) = \sum\_{n=1}^\infty \sq... | https://mathoverflow.net/users/nan | Using Fourier series to prove $-\int_0^1 u_{xxx}u_x \eta = \int_0^1 (u_{xx})^2\eta - \int_0^1 \frac{1}{2} (u_x)^2 \eta_{xx}$ | term on left hand side:
$$L=-\int\_0^1 dx\, u\_{xxx}u\_x \eta =2^{3/2}\pi^4 \sum\_{n,m,k=1}^\infty n^3m a\_na\_mb\_k \int\_0^1 dx\,\sin(n\pi x)\sin(m\pi x)\cos(k\pi x)=$$
$$=2^{3/2}\pi^4 \frac{1}{4}\sum\_{n,m,k=1}^\infty n^3m a\_na\_mb\_k \left(\delta\_{m,n+k}+\delta\_{n,m+k}-\delta\_{k,n+m}\right).$$
two terms on righ... | 2 | https://mathoverflow.net/users/11260 | 382212 | 159,060 |
https://mathoverflow.net/questions/381165 | 2 | **Background:** I'm facing the computation of the zeta regularization of the infinite product given by
$$\prod\_{m=-\infty}^\infty (km+u)$$
for a real positive $k$ and $\Im(u)\neq 0$. From [J. R. Quine, S. H. Heydari and R. Y. Song](https://www.ams.org/journals/tran/1993-338-01/S0002-9947-1993-1100699-1/home.html#S... | https://mathoverflow.net/users/125244 | The zeta regularization of $\prod_{m=-\infty}^\infty (km+u)$ | I think there are no problems but one should be careful with analytic continuation.
Up until this line nothing happens:
$$\sum\_{m=0}^\infty(m+uk^{-1})^{-s} - (uk^{-1})^{-s} + e^{\pi i s}\sum\_{m=0}^\infty(m-uk^{-1})^{-s}$$
Then you claim that to equal:
$$\zeta(s,uk^{-1}) -(uk^{-1})^{-s} + e^{\pi is}\zeta(s,-uk... | 2 | https://mathoverflow.net/users/148223 | 382214 | 159,061 |
https://mathoverflow.net/questions/382194 | 12 | $\newcommand\FinSet{\mathit{FinSet}}\newcommand\FinBool{\mathit{FinBool}}\newcommand\FreeFinBool{\mathit{FreeFinBool}}\newcommand\Set{\mathit{Set}}\newcommand\Psh{\mathit{Psh}}$It's [well-known](https://ncatlab.org/nlab/show/generic+interval#remark) that the topos of presheaves on the category $\FinSet$ of finite sets ... | https://mathoverflow.net/users/2362 | Stone duality for the algebra of Boolean functions such that $f(\top,\dots,\top) = \top$, or: What does the presheaf topos on $FinSet_\ast$ classify? | I might be missing something, but I think you are overcomplicating things.
I clain that your topos classifies the theory $T$ of pairs $(B,\phi)$ where $B$ is a boolean algebra and $\phi : B \to \{0,1\}$ is a boolean algebra morphism.
Indeed, this theory is given by a finite limit sketch, so it is classified by the ... | 8 | https://mathoverflow.net/users/22131 | 382232 | 159,066 |
https://mathoverflow.net/questions/382240 | 5 | Assume that $K/\Bbb Q$ is a cyclic Galois extension, and $\mathfrak{p}$ a prime ideal of $K$ and $\sigma$ an element of the Galois group. What can we say about the classes $[\mathfrak{p}]$ and $[\sigma(\mathfrak{p})]$ in the ideal class group? Let $\left \langle [\mathfrak{p}] \right \rangle$ be the subgroup of the ide... | https://mathoverflow.net/users/166540 | Action of the Galois group on the ideal class group | This is true if $K$ is quadratic, since then $[\sigma(\mathfrak p)] = [\mathfrak p^{-1}]$.
It should be false for every other degree.
The only relation that the Galois action should satisfy is (for cyclic fields of degree $n$) that $ \prod\_{i=0}^{n-1} [\sigma^{i}(\mathfrak p)] $ is trivial in the class group, sinc... | 8 | https://mathoverflow.net/users/18060 | 382268 | 159,075 |
https://mathoverflow.net/questions/382229 | 4 | $\newcommand\Ind{\mathsf{Ind}}\newcommand\Ord{\mathsf{Ord}}\newcommand\Psh{\mathsf{Psh}}$For $\kappa \leq \lambda \leq \Ord$ regular cardinals and $\mathcal C$ an essentially small category, let $\Ind\_\kappa^\lambda(\mathcal C)$ be the free completion of $\mathcal C$ under $\lambda$-small, $\kappa$-filtered colimits. ... | https://mathoverflow.net/users/2362 | Does $\mathsf{Ind}_\lambda^\mu(\mathsf{Ind}_\kappa^\lambda(\mathcal C)) = \mathsf{Ind}_\kappa^\mu(\mathcal C)$? | I asked myself the same questions while working on [this paper](https://arxiv.org/abs/1402.6659) [[TAC](http://www.tac.mta.ca/tac/volumes/31/2/31-02abs.html)].
It seemed to me that the way to answer questions like this is to embed $\textbf{Ind}\_\kappa^\lambda (\mathcal{C})$ into the usual $\textbf{Ind}\_\kappa (\mathc... | 4 | https://mathoverflow.net/users/11640 | 382273 | 159,077 |
https://mathoverflow.net/questions/382274 | 0 | Let's denote
* $F\_{t\_u}^{-1}(x)$ the quantile function of the Student's t-distribution $t\_u$ with $u$ degrees of freedom and
* $F\_{t\_v}(x)$ the cumulative distribution function of the t-distribution $t\_v$ with $v$ degrees of freedom
where $u \ne v$ and $u,v >2$.
What is the asymptotic behavior of the functi... | https://mathoverflow.net/users/62193 | Asymptotic behavior of the Student's t-quantile function of Student's t-cumulative distribution function | Let $F\_u:=F\_{t\_u}$ and $G\_u:=1-F\_u$. Let $f\_u:=F'\_u$, the pdf of $t\_u$, so that
$$f\_u(x)=c\_u(1+x^2/u)^{-(u+1)/2},\quad c\_u:=\frac{\Gamma((u+1)/2)}{\Gamma(u/2)\sqrt{\pi u}}.$$
So, for $x\to\infty$
$$f\_u(x)\sim c\_u\,u^{(u+1)/2}x^{-u-1},$$
whence
$$G\_u(x)=\int\_x^\infty f\_u(y)\,dy
\sim c\_u\,u^{(u+1)/2}\int... | 1 | https://mathoverflow.net/users/36721 | 382276 | 159,078 |
https://mathoverflow.net/questions/382242 | 8 | Let $C$ be a (hyperelliptic) genus $2$ curve over a number field $K$ with a $K$-rational Weierstrass point $\infty$. We embed $C$ in its Jacobian $J$ via $\infty$.
**Question:** Is there a quadratic extension $L/K$ and a point $x\in C(L)$ which is non-degenerate in $J$, i.e. such that $\mathbb{Z}x$ is dense in $J$?
... | https://mathoverflow.net/users/173042 | Non-degenerate points on a Jacobian surface | The product $(E \times C) / \sigma$, where $\sigma$ acts by inversion on $E$ and the hypereliptic involution on $C$, is an elliptic surface over $C/\sigma = \mathbb P^1$.
This surface has two sections, which are given by the two maps $C \to E$ we get from the Abel-Jacobi map composed with the two projections $J \to E... | 2 | https://mathoverflow.net/users/18060 | 382283 | 159,082 |
https://mathoverflow.net/questions/382284 | 0 | During his investigation of zeta Riemann defined the $\xi$ function as $\xi(s):= \Gamma(\frac{s}{2})(s-1)\pi^{-s/2}\zeta(s)$ which is an entire function that is invariant under the substitution $s \to 1-s$. Moreover $\xi$ shares its zeros with Riemann zeta function $\zeta$.
Riemann wanted to write $\xi(s)$ in the for... | https://mathoverflow.net/users/8435 | Convergence of Riemann's Product representation of Xi | You need to group the complex conjugates pairs of non-trivial zeros together
$$2\sum\_{\rho} \log(1-\frac{s}{\rho})=
2\sum\_{\Im(\rho)\le 2|s|} \log(1-\frac{s}{\rho})+
2\sum\_{\Im(\rho)> 2|s|}\log(1-\frac{s}{\rho})+\log(1-\frac{s}{\overline{\rho}})$$
$$=2\sum\_{\Im(\rho)\le 2|s|} \log(1-\frac{s}{\rho})+
2\sum\_{\Im(\... | 2 | https://mathoverflow.net/users/84768 | 382290 | 159,085 |
https://mathoverflow.net/questions/382299 | 1 | (Cross posted at [MSE](https://math.stackexchange.com/questions/3999840/is-there-a-maximal-translation-invariant-extension-of-lebesgue-measure).)
The answer to [this question](https://math.stackexchange.com/questions/209532/extension-of-the-lebesgue-measurable-sets) shows that there are translation-invariant extensio... | https://mathoverflow.net/users/96899 | Is there a maximal translation-invariant extension of Lebesgue measure? | The answer is no. This is the main result of "Extensions of invariant measures on Euclidean spaces" by Ciesielski and Pelc.
| 4 | https://mathoverflow.net/users/109573 | 382300 | 159,088 |
https://mathoverflow.net/questions/382279 | 4 | Let $X$ be a Seifert fiber space, that is, a 3-manifold which is a circle bundle over a 2-orbifold. Suppose all generic fiber of $X$ is homotopically trivial, can we prove that the universal cover of $X$ is homeomorphic to $S^3$?
| https://mathoverflow.net/users/105900 | Seifert fiber space with homotopically trivial generic fiber | Yes, this follows from classification of Seifert fibered spaces. In fact, you can change the hypothesis to "the fiber is torsion in $\pi\_1(X)$" and still get the result. $\newcommand{\RR}{\mathbb{R}}$
Suppose that $X$ is reducible. Thus $X$ has $S^2 \times \RR$ geometry and the fiber is not torsion.
Suppose that $... | 6 | https://mathoverflow.net/users/1650 | 382304 | 159,090 |
https://mathoverflow.net/questions/382310 | 12 | I started to read the HoTT book. I'm now on chapter 1 and I have several questions concerning not even homotopical, but "regular" type theory.
1. On page 24, where the universes are introduced, there is a sequence:
$$\mathcal U\_0:\mathcal U\_1:\mathcal U\_2:\cdots$$
Everything here makes sense, but I don't under... | https://mathoverflow.net/users/143549 | 3 questions about basics of Martin-Löf type theory | Universe levels usually trip up newcomers to type theory since there is no straightforward intuition for them. What I found helpful is to think of them as a *merely technical device to prevent impredicativity*, and only dive deeper into the technicalities when necessary.
The first recognition is that we need a univer... | 16 | https://mathoverflow.net/users/167839 | 382316 | 159,093 |
https://mathoverflow.net/questions/382225 | 3 | Recall that the Hawaiian earring group, $\mathbb G$, is the fundamental group of the Hawaiian Earing using the point at the origin. It can be understood more combinatorially as a subgroup of the inverse limit $\varprojlim F\_n$ where $F\_n$ is the free group, say with $n$ generators, call them $a^n\_0,...,a^n\_{n-1}$ a... | https://mathoverflow.net/users/114946 | Does the Hawaiian Earring Group embed into the permutation group of $\mathbb N$? | $\DeclareMathOperator\S{\mathfrak{S}}\DeclareMathOperator\N{\mathbf{N}}$Yes, because:
1. as a subgroup of a projective limit of a sequence of finitely generated free groups $F\_n$, it embeds into the product $\prod\_n F\_n$.
2. each countable group embeds into $\S(\N)$ (just consider the left action)
3. If $(G\_n)$ i... | 4 | https://mathoverflow.net/users/14094 | 382318 | 159,095 |
https://mathoverflow.net/questions/382320 | 0 | I have a bunch of iid $\{X\_i\}$ with $X\_i \sim \exp(\lambda)$ - let's say $\lambda = 1$. Now, classic version of CLT tells me:
\begin{equation}
\sqrt{n}\left(1-\bar{X}\_n\right) \rightarrow \mathcal{N}\left(0,\frac{1}{\lambda^2}\right)
\end{equation}
in distribution. But doesn't the convergence to a standard normal i... | https://mathoverflow.net/users/166974 | CLT for random variables with positive support (e.g. exponential) | You need to be careful with the **order of quantifiers** in understanding what (this version of) CLT is claiming. In particular, the order of $x$ (the point where you evaluate your CDF) and $n$ (sample size).
Convergence in distribution means that *if* you pick a value $x$ (yes, it can be negative), and consider the ... | 0 | https://mathoverflow.net/users/171662 | 382326 | 159,097 |
https://mathoverflow.net/questions/382206 | 3 | In the 1960's, Dana Scott constructed the domain $D\_{\infty}$ which has the property
$D\_{\infty} \cong D\_{\infty}{}^{D\_{\infty}}$.
Its construction is based on a cumulative hierarchy of infinite sequences.
For an exposition of its construction one can read the Stenlund (1972) book, “Combinators, $\lambda$-terms... | https://mathoverflow.net/users/11555 | What is the cardinality of Dana Scott's $D_{\infty}$? | If you take $D\_0$ to be the two-element chain, $D\_1\cong(D\_0\to D\_0)$ is the **three**-element chain consisting of **order-preserving** endofunctions of $D\_0$ (not a $4=2^2$-element set). Then $D\_2$ is a lattice with **ten** elements (not $27=3^3$).
It is then a combinatorial question how big the subsequent lat... | 11 | https://mathoverflow.net/users/2733 | 382328 | 159,098 |
https://mathoverflow.net/questions/382288 | 2 | Let $M$ be an orientable surface **without** boundary$($**I am not assuming $M$ is compact, it can be non-compact**$)$. Let $\Phi: M\to M$ be a *proper* homotopy-equivalnce$($A proper homotopy-equivalence can be defined analog way as homotopy-equivalence, but here we need to assume all maps, *including homotopies, are ... | https://mathoverflow.net/users/172285 | Lifting of a proper map in the cover is a proper map | The two squares are pullbacks. This follows from the following more general result:
**Lemma.** *Let $f \colon X \to Y$ be a continuous map of topological spaces that induces an isomorphism $f\_\* \colon \pi\_1(X) \stackrel\sim\to \pi\_1(Y)$. Let $G \subseteq \pi\_1(X)$ be a subgroup with image $H = f\_\*(G)$, and let... | 2 | https://mathoverflow.net/users/82179 | 382352 | 159,107 |
https://mathoverflow.net/questions/382353 | 3 | Suppose we have a (commutative, unital) ring $R$ and a (commutative, unital) $R$-algebra $A$ such that $A$ is projective of constant rank $n$ as an $R$-module. This condition is equivalent to there existing $r\_1,\dots, r\_k\in R$ that together generate the unit ideal and for which each localization $A\_{r\_i}$ is isom... | https://mathoverflow.net/users/1474 | Is a tower of locally-free modules locally a tower of free modules? | The answer to your question is "yes." See EGA II, Prop. 6.1.12.
That proposition tells you something a bit more general:
Let $A$ be a finite $R$-algebra (finite as $R$-module), and let $M$ be an $A$-module. Then $M$ is locally free of rank $m$ over $A$ if and only if there is a list $r\_1, \dots, r\_k$ of elements ... | 3 | https://mathoverflow.net/users/173111 | 382363 | 159,110 |
https://mathoverflow.net/questions/382360 | 3 | Let $K$ be a finitely generated field over $\mathbb{Q}$ of transcendence degree 1, and take a curve $C$ over a number field $k$ such that $k(C)=K$. In "Arithmetic height functions over finitely generated fields" (*Invent. Math.* 140 (2000), no. 1, 101–142), Moriwaki states that one can use points on $C$ to define non-a... | https://mathoverflow.net/users/146401 | What is the "geometric height" mentioned by Moriwaki? | The geometric height is easiest to define for points on $\mathbb P^n(K)$. These define maps $C \to \mathbb P^n$ and we take the line bundle $\mathcal O(1)$ on $\mathbb P^n$, pull back to $C$, and take the degree.
We can express this with valuations by fixing coordinates $(a\_0,\dots, a\_n)$ and taking $-\sum\_v \min ... | 5 | https://mathoverflow.net/users/18060 | 382367 | 159,113 |
https://mathoverflow.net/questions/382355 | 7 | Suppose $M$ is a countable transitive model of some fragment of $\mathbf{ZFC}$, $\mathbb{P}\in M$ is a forcing notion and $G, H$ are $\mathbb{P}$-generic such that $M[G]=M[H]$. Does it then follow that there is some automorphism $\pi:\mathbb{P}\longrightarrow\mathbb{P}$ such that $\pi\in M$ and $\pi[G]=H$?
If the ans... | https://mathoverflow.net/users/138274 | Are generic filters that produce the same forcing extension related by a ground-model automorphism? | This works much better in terms of complete Boolean algebras. If $\mathbb B$ and $\mathbb B'$ are complete Boolean algebras and a $\mathbb B$-generic filter $G$ and a $\mathbb B'$-generic filter $G'$ generate the same forcing extension, then there are $b\in G$ and $b'\in G'$ such that the part of $\mathbb B$ below $b$ ... | 12 | https://mathoverflow.net/users/6794 | 382369 | 159,114 |
https://mathoverflow.net/questions/382081 | 7 | Let $(R/A)\_\Delta$ be the prismatic site over $R$ relative to a prism $(A, I)$, then it is known that $(R/A)\_\Delta$ admits finite non-empty coproduct, for instance, by Cor. 5.2 in [Bhatt's lecture notes V on prismatic cohomology](http://www-personal.umich.edu/%7Ebhattb/teaching/prismatic-columbia/lecture5-prismatic-... | https://mathoverflow.net/users/90253 | Finite non-empty coproduct in the absolute prismatic site | Yes, it does admit nonempty finite coproducts. If you have two prisms $(A\_1,I\_1)$ and $(A\_2,I\_2)$ with maps $R\to A\_i/I\_i$, you need to find the initial prism $(A,I)$ with maps from both $(A\_i,I\_i)$ such that the two induced maps $R\to A\_i/I\_i\to A/I$ agree. For this, start with $A\_0=A\_1\hat{\otimes}\_{\mat... | 7 | https://mathoverflow.net/users/6074 | 382374 | 159,116 |
https://mathoverflow.net/questions/382372 | 1 | Lehmer's totient problem asks if there are any composite integers $n$ with $\phi(n) \ | \ n-1$.
It is known that any such $n$ must be odd. It must also be a charmichael number.
Assume $n=4m+3$ then $\phi(n) \ | \ n-1=2(2m+1)$ . Because $n$ is a carmichael number, we have $2^3 \ | \ \phi(n)$. It follows that $2^3 \ ... | https://mathoverflow.net/users/166404 | A new perspective on Lehmer's totient problem | This is false. Take $n$ equal to the product of first $k$ odd primes, or 3 times larger (so that $n-1$ is divisible by 4). For large $k$, $\varphi(n)/n=(1-1/3)(1-1/5) \ldots (1-1/p\_{k+1})$ tends to 0. The same with conjecture 2.
| 9 | https://mathoverflow.net/users/4312 | 382375 | 159,117 |
https://mathoverflow.net/questions/382241 | 3 | Let $P = \{1,\dots,p\}$ be a set of people. Consider partitioning $P$ into two disjoint sets, $A$ (of cardinality $a$) and $A^c = P-A$. Let us index $A$ as $A = \{A\_1,\dots,A\_a\}$. Each person in $A$ can choose at most $a$ people from $A^c$ to be friends with. Formally, $A\_i$ can be friends with at most $a$ people d... | https://mathoverflow.net/users/83070 | Maximum number of subsets in which people co-exist with their friends | The proof of the maximum is rather straight forward.
The cardinality
\begin{split}
&\left|\left\{S\in\binom{P}{r}: |A\cap S| \geq r' \text{ and } (A^c\cap S)\subseteq \bigcap\limits\_{i\in A\cap S}F\_i\right\}\right| \\
=& \sum\_{i=r'}^a \left|\left\{S\in\binom{P}{r}: |A\cap S| = i \text{ and } (A^c\cap S)\subseteq \... | 3 | https://mathoverflow.net/users/7076 | 382382 | 159,121 |
https://mathoverflow.net/questions/381936 | 2 | I am trying to prove continuity of the maximal tensor product functor. I have a problem in the proof that I cannot see how to handle; If anyone could give me a clue on how to go on from here, I would really appreciate it. So here it goes:
Let $B$ be a $C^\*$-algebra. I am trying to show that if $A\_1\xrightarrow{\var... | https://mathoverflow.net/users/164203 | The maximal tensor product is a continuous functor | $\otimes\_{\max}$ is continuous by universality: $\varinjlim (A\_n\otimes\_{\max}B)$ is a C\*-completion of the algebraic tensor product $A\otimes\_{\mathrm{alg}} B$ and one is left to check maximality (which is easy).
On the other hand, $A\_n\otimes\_{\max}B \to A\otimes\_{\max}B$ may not be faithful. For example, c... | 5 | https://mathoverflow.net/users/7591 | 382395 | 159,126 |
https://mathoverflow.net/questions/382389 | 5 | I have a collection $\{v\_1,...,v\_k\}$ of vectors in $\{\pm 1\}^n$ with the property that for all $i\neq j$ we have $\langle v\_i, v\_j \rangle \le c\log\_2(n)$. I am looking for an upper bound on $k$ in terms of $n$.
I am aware that given instead unit vectors $v\_i$ in $\mathbb{R}^n$, and the bound $\langle v\_i, v... | https://mathoverflow.net/users/173121 | Maximum number of vectors with upper bound on pairwise inner products | This is an interesting twist on the usual question.
Relevant results are due to Welch, Kabatianski, Levenshtein, Sidelnikov. Welch's applies to arbitrary vectors, real or complex. The others apply to vectors constructed from complex roots of unity of some finite order.
Welch's bound states (I will apply it to $\pm 1$... | 4 | https://mathoverflow.net/users/17773 | 382399 | 159,128 |
https://mathoverflow.net/questions/354981 | 2 | For a given rational $c\ne-1$, I need to find a rational $x\ne20$ with a small denominator such that $(5cx+100)(5cx-64c+36)$ is a perfect square.
I start with
$y^2=(5cx+100)(5cx-64c+36)$
and complete the square to transform it to
$y^2+(32c+32)^2=(5cx-32c+68)^2$
A standard approach for non-primitive Pythagorea... | https://mathoverflow.net/users/95511 | Perfect square quadratic expression | 1. The following Maple code is the fastest I could do.
```
restart:
with(numtheory):
mins1:=10^10:
mins2:=10^10:
c:=-37178488/89505763;
N0:=64*abs(numer(c+1)):
time0:=time():
for h from 1 to 10^10 do
N:=h*N0:
listN:=divisors(N):
for i from 1 to nops(listN) do
K:=listN[i]:
P:=N/K:
... | 1 | https://mathoverflow.net/users/95511 | 382402 | 159,129 |
https://mathoverflow.net/questions/382422 | 2 | The answer to [this](https://mathoverflow.net/questions/371948/separability-of-mathbbcx-over-its-mathbbc-subalgebras/382150?noredirect=1#comment972149_382150) MO question says the following:
**Lemma 1.** *Let $h \in \mathbf C[x]$ be a polynomial of degree $n \geq 2$. Then $\mathbf C+(h) \subseteq \mathbf C[x]$ is unr... | https://mathoverflow.net/users/72288 | Separability of $\mathbb{C}[x,y_1,\ldots,y_r]$ over $\mathbb{C} + (h,y_1,\ldots,y_r)$ | In Lemma 2, I'm not sure what "squarefree" means. However, the meaning is clear if $h\in\mathbb{C}[x]$, and then Lemma 2 is true. Indeed, the inclusion $\mathbb{C}+(h)\subset \mathbb{C}[x]$ induces an injection $A\otimes\_\mathbb{C}(\mathbb{C}+(h))\subset A[x]$. The image is easily checked to be $A+(h)$. The lemma foll... | 2 | https://mathoverflow.net/users/7666 | 382424 | 159,133 |
https://mathoverflow.net/questions/382416 | 0 | Let $\mathbf{A}$ be a matrix of size $N\times N$ whose elements $A\_{ij}$ (with $1\leq i,j\leq N$) are I.I.D following some distribution.
If we set set $\langle A\_{ij}\rangle=0$ and $\langle {A\_{ij}}^2\rangle=\frac{1}{N}$ (where $\langle \cdot\rangle$ denotes the average over the distribution) then we know that for... | https://mathoverflow.net/users/142153 | Why is the determinant of a large random matrix equal to zero? (Heuristics) | The determinant ${\rm det}\,A$ is a polynomial $P(\{a\_{nm}\})$ in the $N^2$ elements of the matrix, which we can consider as a point in $\mathbb{R}^{N^2}$. The probability distribution of the random matrix gives you some measure in this space. Your question for the probability that ${\rm det}\,A=0$ amounts to the ques... | 6 | https://mathoverflow.net/users/11260 | 382427 | 159,135 |
https://mathoverflow.net/questions/382378 | 12 | In Langlands' [review](https://publications.ias.edu/sites/default/files/hida-ps.pdf) of Hida's book "$p$-adic automorphic forms on Shimura varieties", he discusses a nexus of 4 areas of modern number theory: automorphic representations, motives, spaces of $p$-adic Galois representations, and a fourth less well-defined ... | https://mathoverflow.net/users/136098 | Eigenvarieties and functoriality | You have asked a lot of questions at once, and it is impossible to give more than a hint at a small subset of these questions.
I think the general theme here is: the existence of eigenvarieties doesn't "create information from nowhere" about the core questions of global Langlands (functoriality and reciprocity); but ... | 5 | https://mathoverflow.net/users/2481 | 382428 | 159,136 |
https://mathoverflow.net/questions/382407 | 1 | Given a smooth pseudo-Riemannian manifold $(M,g)$ one can define the conformal group as the set of smooth diffeomorphisms $\varphi:M\to M$ such that there is a positive smooth function $u$ with $\varphi^\ast g=ug$. One could also define it as the set of all $C^1$-diffeomorphisms $\varphi:M\to M$ such that there is a po... | https://mathoverflow.net/users/156492 | Smoothness of conformal transformations | Answers to your question are somewhat dimension and signature dependent.
For example, in dimension $2$, if $g$ is definite, then every $C^1$ conformal diffeomorphism is, in fact, real-analytic, because, locally, $g$ can be written in the form $g = F\,\mathrm{d}z{\circ}\mathrm{d}\bar z$ for some complex-valued coordin... | 4 | https://mathoverflow.net/users/13972 | 382429 | 159,137 |
https://mathoverflow.net/questions/382384 | 4 | I need to show that the property of being a domain of holomorphy is the same as being a holomorphically convex domain (this result is known as Cartan-Thullen theorem). However, the proofs I found in textbooks (e.g. Shabat) look ugly and are hard to digest.
Is there a reference with a better proof? Can you share your ... | https://mathoverflow.net/users/98037 | Characterization of a domain of holomorphy | I can't really anticipate what you will find ugly and hard to digest, but at least I found the proof in Jiří Lebls book "Tasty Bits of Several Complex Variables", Theorem 2.6.3, to be nicely presented (but I don't think it is very different from what is found in many other sources).
In general, I think this book is a... | 2 | https://mathoverflow.net/users/49151 | 382432 | 159,139 |
https://mathoverflow.net/questions/382423 | 3 | Let $A$ be a finite dimensional Frobenius algebra and $e$ and idempotent of $A$.
It is well known that the algebra $eAe$ does not have to be a Frobenius algebra. But if $A$ is additionally symmetric, then $eAe$ is also a symmetric Frobenius algebra for any idempotent $e$.
The Frobenius algebra $A$ is called weakly sy... | https://mathoverflow.net/users/61949 | Weakly symmetric Frobenius algebras | Yes.
Using left modules, the indecomposable projective $eAe$-modules are $eP$ for $P$ an indecomposable projective $A$-module such that $e\operatorname{top}(P)=\operatorname{top}(P)$, and in this case $\operatorname{top}(eP)=\operatorname{top}(P)$.
For any finite-dimensional $A$-module $M$ such that $\operatorname{... | 3 | https://mathoverflow.net/users/21483 | 382434 | 159,141 |
https://mathoverflow.net/questions/382442 | 22 | Let $\mathbb CP^n$ denotes the complex projective space of dimension $n$, we have a standard complex structure of $\mathbb CP^n$, and my question is: is this complex structure unique?
Or equivalently, let $X$ be a complex manifold diffeomorphic to $\mathbb CP^n$, is $X$ biholomorphic to $\mathbb CP^n$?
What I know ... | https://mathoverflow.net/users/99826 | Is the complex structure of $\mathbb CP^n$ unique? | Let me write this too long comment as an answer.
As abx says, what we do know is
**Theorem 1.** *If a Kähler manifold $X$ is homeomorphic to $\mathbb{CP}^n$, then $X$ is biholomorphic to it.*
This is due to Hirzebruch and Kodaira for $n$ odd (but with the strongest assumption for $X$ to be diffeomorphic to $\math... | 26 | https://mathoverflow.net/users/9871 | 382455 | 159,148 |
https://mathoverflow.net/questions/382469 | 6 | A 3-manifold $M$ is *irreducible* if every embedded 2-sphere bounds a 3-ball. Thanks to Papakyriakopoulos's sphere theorem, irreducibility is the same as having $\pi\_2(M)=0$. Does irreduciblity imply that the manifold is in fact aspherical, i.e. that $\pi\_k(M)=0$ for all $k \geq 2$?
(Or maybe I should say that the ... | https://mathoverflow.net/users/151664 | Higher homotopy groups of irreducible 3-manifolds | An irreducible 3-manifold $M$ is aspherical if and only if it's not a finite quotient of $S^3$, which in turn is equivalent to having infinite fundamental group. Essentially you've already outlined the proof: the universal cover $\tilde M$ is a simply-connected 3-manifold with trivial $\pi\_2$, and so also $H\_2(\tilde... | 12 | https://mathoverflow.net/users/13119 | 382471 | 159,154 |
https://mathoverflow.net/questions/382449 | 4 | Let $(X,T)$ be a topological dynamical system ($X$ is compact metric space and $T\colon X\to X$ a homeomorphism). Recall that its *automorphism group* is
$$ \mathrm{Aut}(X,T) = \{g\colon X\to X : \text{$g$ is hoemomorphism and $g\circ T = T\circ g$}\}.$$
Observe that when $(X,T^k)$ is not minimal, we can decompose $X... | https://mathoverflow.net/users/166847 | Is it true that $(X,T^k)$ minimal for all $k\geq1$ implies $\mathrm{Aut}(X,T) = \mathrm{Aut}(X,T^k)$ for all $k\geq1$? | Here's a subshift counterexample. Let $E : \{0,1\}^\* \to \{0,1\}^\*$ be the map on finite words that flips every second bit, preserving word length, e.g. $E(01000) = 11101$, and let $O$ flip the even positions. This gives an action of the four-group $V = \langle E, O \rangle \cong (\mathbb{Z}/2\mathbb{Z})^2$ on binary... | 2 | https://mathoverflow.net/users/123634 | 382475 | 159,155 |
https://mathoverflow.net/questions/382463 | 3 | I'm looking for **rigorous** discussions on the derivation of the Euler-Lagrange equation for **field** as it is usually discussed in classical field theory books. More precisely, if the action is given by:
$$S(\phi) = \int \mathcal{L}(\phi, \partial\_{x\_{i}}\phi) d^{4}\vec{x}$$
where $\vec{x} = (x\_{1},x\_{2},x\_{3... | https://mathoverflow.net/users/152094 | Rigorous Euler-Lagrange equations for fields | See page 16 and following of [Coordinate-free derivation of the Euler–Lagrange equations and identification of global solutions via local behavior](https://mast.queensu.ca/~andrew/papers/pdf/2005a.pdf) by Elsa Hansen (2005).
>
> Results concerning $C^2$-minimizing curves on manifolds are presented.
> A coordinate- ... | 1 | https://mathoverflow.net/users/11260 | 382476 | 159,156 |
https://mathoverflow.net/questions/382482 | 5 | $\newcommand{\unsim}{\mathord{\sim}}$Let $G$ be a group. What is
$$
G/\left(ab\sim ba\ \middle|\ a,b\in G\right)?
$$
Answer: not $G^{\mathrm{ab}}$, [but](https://math.stackexchange.com/a/765583) the set of conjugacy classes of $G$.
When passing to monoids, the situation gets more complicated: the equivalence relatio... | https://mathoverflow.net/users/130058 | Conjugacy classes of monoids II: Abelianising a monoid, wrongly | Defining conjugacy for monoids is a dicey subject because many different notions that are equivalent for groups are different for monoids and it is not clear which of these is interesting. The one you call 3 is probably the most commonly studied one, although it varies depending on the context how useful its.
I am no... | 7 | https://mathoverflow.net/users/15934 | 382486 | 159,159 |
https://mathoverflow.net/questions/381881 | 2 | Pg.248 of "Textbook in Tensor Calculus and Differential Geometry" by Prasun Nayak.
---
Let us suppose that $\lambda\_{h|}^i$
is not a unit vector and therefore, the mean curvature $M\_h$ in
this case is given by
$$M\_h=-\frac{R\_{ij}\lambda\_{h|}^i\lambda\_{h|}^j}{g\_{ij}\lambda^i\_{h|}\lambda^j\_{h|}} \tag{1}$... | https://mathoverflow.net/users/172764 | Mean Gaussian curvature from non-unit vector | It's really quite simple to obtain your equation (1) for a non-unit vector from the unit-vector result (your second, unnumbered equation).
Start from your second equation which gives the mean curvature (Ricci curvature) in terms of the Ricci tensor $\mathbf{R}$ and a unit vector ${\lambda}\_{h|}$:
$$M\_h=-\sum\_{i,j}... | 2 | https://mathoverflow.net/users/11260 | 382487 | 159,160 |
https://mathoverflow.net/questions/382458 | 3 | $\DeclareMathOperator\SL{SL}\DeclareMathOperator\Aut{Aut}$Fix $d < 0$, a fundamental quadratic discriminant and $n$ a positive integer. Suppose $Q$ is a primitive binary quadratic form of discriminant $d$. Let us define the following,
1. Representations of $n$ by $Q$: $R(Q, n) = \{ (x, y) \in \mathbb{Z}^2 | Q(x, y) =... | https://mathoverflow.net/users/167999 | Correspondence between binary quadratic representations and proper ideals of quadratic number fields | As explained below, there is a correspondence between non-zero locally principal ideals $I$ in the order of disc. $D$ and quadratic forms $Q$ of disc. $D$. The form $Q$ is naturally defined on $I$ via $x \in I \mapsto N(x)/N(I).$ So $Q(x) = n$ iff $N(x) = n N(I)$ if $N(x I^{-1}) = n,$ so representations of $n$ by $Q$ c... | 3 | https://mathoverflow.net/users/169863 | 382493 | 159,162 |
https://mathoverflow.net/questions/382497 | 5 | Is it known whether for a generalized complex flag variety $X$ (that is, $G/P$ for a complex semisimple Lie group $G$ and a parabolic $P$), the homology of the free loop space $H\_\*(\Lambda X, \mathbb{Q})$ is degree-wise finite-dimensional? Is it known at least for type A (i.e. classical) flag varieties?
This is kno... | https://mathoverflow.net/users/114985 | Homology of the free loop space of generalized flag varieties | Serre proved that for any simply-connected $X$, if $X$ has finitely generated homology groups in each degree, then the loop space of $X$ has finitely generated homology groups in each degree. (Proposition 9 of chapter IV of Homologie singulière des espaces fibrés. Applications. Ann. of Math., 54,
1951, p. 425-505.)
S... | 9 | https://mathoverflow.net/users/18060 | 382499 | 159,166 |
https://mathoverflow.net/questions/382488 | 4 | Let $X^4$ be a simply-connected closed smooth 4-manifold. Then every element $x \in H\_2(X; \mathbb{Z})$ can be represented by an embedded orientable surface and the minimal genus of such a surface is called the genus of $x$, denoted $g(x)$. An element $x \in H\_2(X; \mathbb{Z})$ is called characteristic if the reducti... | https://mathoverflow.net/users/99414 | Minimal genus of characteristic surfaces? | If you want something specific to characteristic classes, the only thing I know you can leverage on is the fact that the complement of any surface representing a characteristic class is spin. Let's fix a 4-manifold $X$ and a characteristic class $x$.
The first result in this direction is probably the Kervaire-Milnor ... | 3 | https://mathoverflow.net/users/13119 | 382506 | 159,168 |
https://mathoverflow.net/questions/382473 | 10 | Here's a question I was wondering about this week. Not sure how interesting it is, but I thought it was kind of curious.
**Question:** Given $k$, is there a number $N=N(k)$ such that if a closed orientable hyperbolic surface X is the union of at most $k$ embedded metric balls, then the genus of $X$ is at most $N$?
... | https://mathoverflow.net/users/74169 | Can you cover a genus a billion hyperbolic surface with 15 balls? | Your conjecture is false. Every nonorientable closed connected surface of negative Euler characteristic, admits a hyperbolic metric such that the surface is covered by 3 embedded disks. Hence, for each $p\ge 2$, there is a closed connected orientable hyperbolic genus $p$ surface covered by 6 embedded disks. (One can li... | 11 | https://mathoverflow.net/users/39654 | 382526 | 159,176 |
https://mathoverflow.net/questions/382406 | 2 | Consider the standard second order cone programming problem:
\begin{equation}
\begin{array}{ll}
\operatorname{maximize} & \bar{p}^{T} x \\
\text { subject to } & \bar{p}^{T} x+\Phi^{-1}(\beta)\left\|\Sigma^{1 / 2} x\right\|\_2 \geq \alpha.
\end{array}
\end{equation}
This is a problem appearing in portfolio optimiza... | https://mathoverflow.net/users/156139 | Convex optimization closed-form solution | For this special case an explicit solution is possible. To simplify notation let $A := \Sigma^{1/2}$ and let $A^+$ be the pseudoinverse of $A$ (see f.i. Golub/Van Loan: Matrix Computations (1996), 3rd ed., p. 257). Let $y := Ax$, $z := (E - A^+A)x$, thus $x = A^+y +z$ and the problem is
$$p^T(A^+y + z) = max!\\p^T(A^+y... | 1 | https://mathoverflow.net/users/100904 | 382528 | 159,177 |
https://mathoverflow.net/questions/382530 | 4 | Let $\mathcal{A}\_\Gamma$ be the space of convex (non-degenerate) Euclidean polyhedra with $1$-skeleton a certain polyhedral graph $\Gamma$. This space can be seen as a subset of $\mathcal{Gr}\_2(\mathbb{R}^3)^F$ where $F$ is the number of faces of $\Gamma$.
It is a well known fact (for example in Proposition 17 of [... | https://mathoverflow.net/users/42912 | Is the space of Euclidean polyhedra with a fixed $1$-skeleton connected? | I believe your question asks about *isotopy*:
>
> **Isotopy property**: A combinatorial structure (such as a combinatorial type of
> polytope) has the isotopy property if any two realizations with the same orientation can be deformed into each other by a continuous deformation that maintains
> the combinatorial typ... | 5 | https://mathoverflow.net/users/6094 | 382536 | 159,179 |
https://mathoverflow.net/questions/382547 | -1 | Suppose that $X\_1,X\_2...$ is a sequence of **non-negative** real random variables. I have that $\mathbb{E}(X\_i^2) \to 0$ as $i \to +\infty$, therefore my sequence converges at least in distribution to the random vairbale that is identically zero.
Does it converge in proba ? almost surely ?
| https://mathoverflow.net/users/143783 | Which type of convergence for this sequence of random variables? | Yes in probability: The definitions of convergence in distribution to a constant random variable and convergence in probability to a constant random variable are the same.
No almost surely: If we let $\alpha$ be randomly distributed between $[0,1]$ and we let $X\_n$ be $1$ if $$\alpha + \sum\_{i=1}^{n-1} (1/i) \mod 1... | 5 | https://mathoverflow.net/users/18060 | 382550 | 159,187 |
https://mathoverflow.net/questions/382504 | 2 | Let $n \geq 2$ be an integer, and let $f(x) = \prod\limits\_{k = 1}^n(x - \alpha\_k)$ be a monic irreducible polynomial in $\mathbb Z[x]$, with the property that $f(-\alpha\_k) \neq 0$ for any $k = 1, 2, \ldots, n$.
Is there anything meaningful that we can say about $\operatorname{Res}(f(x), f(-x))$, the resultant of... | https://mathoverflow.net/users/22733 | Resultant of $f(x)$ and $f(-x)$ | We have $\mathrm{Res}(f(x),f(-x))=2^n a\_n P(\alpha)^2$, where
$P(\alpha)=\prod\_{1\leq i<j\leq n}(\alpha\_i+\alpha\_j)$. By
e.g. the case $d=2$ of Exercise 7.30 in *Enumerative Combinatorics*,
vol. 2, we have $P(\alpha)=s\_{n-1,n-2,\dots,1}(\alpha)$, where
$s\_{n-1,n-2,\dots,1}$ is a Schur function. By the dual Jacobi... | 10 | https://mathoverflow.net/users/2807 | 382552 | 159,188 |
https://mathoverflow.net/questions/382489 | 1 | Let $X$ be a threefold with Kodaira dimension 2 such that the Iitaka map $\Phi :X \to Y$ is not isotrivial. The generic fiber of $\Phi$ is an elliptic curve.
**Q1.** How many such threefolds exist, and how many explicit examples can be given?
| https://mathoverflow.net/users/172177 | Threefolds with Kodaira dimension 2 and non-isotrivial Iitaka map | Here is a way to construct such threefolds.
Take $E \to \mathbb P^1$ a non-isotrivial elliptic surface. Take $S$ another surface, say a general type surface. Any rational function on $S$ gives a rational map $f\colon S \to \mathbb P^1$, which may not be well-defined everywhere. However, the graph of $f$ (in generic s... | 2 | https://mathoverflow.net/users/18060 | 382556 | 159,189 |
https://mathoverflow.net/questions/382529 | 0 | Suppose that $X\_1,..,X\_n$ are i.i.d real random variables with density $f \in L\_2(\mathbb R)$, and that $g\_i$ are function forming an orthonormal basis of $L\_2(\mathbb R)$, i.e :
$$f(x) = \sum\limits\_{i} a\_i g\_i(x) \text{ for } a\_i = \int g\_i(x) f(x) dx$$
Set the Monte-Carlo coefficients to be $\widehat{a... | https://mathoverflow.net/users/143783 | Convergence of an orthormal expansion of the density | 1. The (more) correct definition of the $\widehat{a\_i}$'s should be
$$\widehat{a\_{n,i}}:=\frac1n\,\sum\_{j=1}^n g\_i(X\_j).$$
2. So,
$$\widehat{a\_{n,i}}=\int\_{\mathbb R}\mu\_n(t) g\_i(t)\,dt,$$
where
$$\mu\_n(t):=\frac1n\,\sum\_{j=1}^n \delta\_{X\_j}(t)$$
and $\delta\_x$ is the Dirac probability measure at $x$, vie... | 1 | https://mathoverflow.net/users/36721 | 382557 | 159,190 |
https://mathoverflow.net/questions/382548 | 0 | Most (stochastic) "gradient descent" type algorithms (such as Nesterov-accelerated gradient-descent or ADAM) seem to be well-defined only for functions which are either:
* lower semi-continuous + convex (these sub-gradient methods),
* non-smooth but locally-lipschitz (Clarke's generalized methods),
The first of the... | https://mathoverflow.net/users/36886 | Gradient-descent "type" Methods for non-convex and non-smooth functions | Several splitting methods fit the bill: Often the non-convexity and the non-smoothness come from different parts of the objective and one can split the objective like $ f(x)=g(x) +h(x)$ with a convex but non-smooth $g$ and a non-convex but smooth $h$. In this case one can, for example, try a proximal gradient method wh... | 1 | https://mathoverflow.net/users/9652 | 382560 | 159,191 |
https://mathoverflow.net/questions/382170 | 9 | In the literature, I've mostly seen two quasicategories coming from $\text{Ch}\_R$:
1. By considering $\text{Ch}\_R$ with weak equivalences $\mathcal W = \text{quasi-isomorphisms}$, we can consider its Dwyer-Kan localization $L^H(\text{Ch}\_R)$, a simplicial category. Then, from the Quillen equivalence $$|-|:\text{sS... | https://mathoverflow.net/users/123439 | Two $\infty$-categories of chain complexes | The two categories you describe are not equivalent in the fashion that you hope.
No matter what kind of simplicial category $C$ is, the quasicategory $N\_\Delta(C)$ has an explicit description of its homotopy category: namely, it has the same objects as $C$, and
$$
Hom\_{hN\_\Delta C}(X,Y) = \pi\_0 Hom\_C(X,Y).
$$
(T... | 13 | https://mathoverflow.net/users/360 | 382564 | 159,193 |
https://mathoverflow.net/questions/182278 | 5 | I've run across something that surprises me, so I'm wondering (1) Is it true? and (2) Is it well known? (And if the answers are affirmative, why didn't I know this already?)
Let $G$ be a compact Lie group and let $H$ and $K$ be closed subgroups. So that the question isn't trivial, you can assume that $H$ is subconjug... | https://mathoverflow.net/users/58888 | Fixed sets of orbit spaces | To answer my own question: I happened to run across the answer just now. Yes, it's true and known. tom Dieck quotes it as II.5.7 in Bredon's *Introduction to compact transformation groups* (which I don't happen to have handy).
| 2 | https://mathoverflow.net/users/58888 | 382574 | 159,196 |
https://mathoverflow.net/questions/382485 | 4 | The so-called *$\ell$-sequences* are defined by $a\_0=0, a\_1=1$ and $a\_n=\ell\,a\_{n-1}-a\_{n-2}$. The *Generalized Lecture Hall Theorem* (due to Mireille BousquetMelou and Kimmo Eriksson) depends on a polynomial analogue of $\ell$-sequences.
I've scaled down the question from its earlier version to read as follows... | https://mathoverflow.net/users/66131 | Integrality of ratios of $\ell$-sequences | Yes, this is true for any odd exponent $2k-1$ on place of 3.
First of all, $a\_m$ is a monic polynomial in $\ell$ of degree $m-1$, and these polynomials are known as Chebyshev polynomials of second kind: if $\ell=2\cos x$, then $a\_m=\frac{\sin mx}{\sin x}$. So it suffices to prove that $A:=\prod\_{j=1}^n (a\_j^{2k-1... | 4 | https://mathoverflow.net/users/4312 | 382576 | 159,198 |
https://mathoverflow.net/questions/382569 | 14 | Long time listener, first time caller!
Suppose that I have a locally free sheaf $\mathcal{E}$ on an smooth algebraic variety $X/k$. Let $\Delta^{(1)}\subset X\times X$ denote the first-order neighbourhood of the diagonal, with projection maps $p\_1,p\_2:\Delta^{(1)}\to X$. Then there are a few different ways one can ... | https://mathoverflow.net/users/143797 | Relation between flatness and integrability of an algebraic connection | (In characteristic zero) Flatness implies the other two definitions; integrability and formal lifting are very weak conditions (in fact if I haven't made a mistake I think this notion of integrability is automatic. As such it seems like a weird definition to me; I usually use the words "integrable" and "flat" as synony... | 6 | https://mathoverflow.net/users/51424 | 382584 | 159,200 |
https://mathoverflow.net/questions/382585 | 1 | Suppose I have a closed convex cone $C\subseteq \mathbb R^n$ and suppose that for every $x$ in the non-negative orthant $\mathbb R\_{0+}^n$ there is a $y\in C$ such that $x\cdot y>0$ (with the standard scalar product). Does it follow that intersection of $C$ with the non-negative orthant contains more than just the ori... | https://mathoverflow.net/users/26809 | Intersection of a closed convex cone with the non-negative orthant | Assume $C$ and $\mathbb{R}\_{\ge 0}^n$ can be (non-strictly) separated by a subspace of dimension $n-1$.
Then a normal vector $x$ to that subspace lies in $\mathbb{R}\_{\ge 0}^n$; see e.g. [here](https://math.stackexchange.com/questions/106312/intersection-between-orthogonal-complement-of-a-subspace-and-a-set) for ... | 2 | https://mathoverflow.net/users/122628 | 382587 | 159,201 |
https://mathoverflow.net/questions/382578 | 0 | Let $f \in BV(\mathbb R)$ and $g: \mathbb R \to \mathbb R$ be Lipschitz. How can I estimate the total variation of $f\circ g$, that is
$$
\int\_{\mathbb R} \left|\frac{d}{dx}f(g(x))\right| dx \ ?
$$
For example is it true that
$$
\int\_{\mathbb R} \left|\frac{d}{dx}f(g(x))\right| dx \le TV(f) \Vert g' \Vert\_{L^\infty}... | https://mathoverflow.net/users/110835 | Estimate on total variation of composition of functions | If $f$ has a jump at 0, and $g:[0,1]\to\mathbb R$ crosses zero infinitely often, then var$(f\circ g)=\infty$.
| 1 | https://mathoverflow.net/users/11054 | 382594 | 159,204 |
https://mathoverflow.net/questions/382568 | 2 | I am describing the question details, though the main question is short as below.
Let $O$ be the ring of integers of the finite extension $K$ of the $p$-adic field $\mathbb{Q}\_p$. Let $R$ be a finite $\mathbb{Z}\_p$-algebra. Let $\bar K$ be the algebraic closure of $K$ and $G\_K:=\text{Gal}(\bar K/K)$ Then consider ... | https://mathoverflow.net/users/122445 | How to use $5$-lemma to prove that $F(M) \otimes_RM' \overset{\simeq}{\longrightarrow} F(M \otimes_R M') $ is a (natural) isomorphism? | Okay, I might as well answer. Whoever gave the hint "5-lemma" might have been using that term as a shorthand for "apply some standard homological algebra result", knowing that some application or other of the 5-lemma would get the job done. But that hint doesn't seem optimized.
Let $R$ be a commutative ring. For any ... | 2 | https://mathoverflow.net/users/2926 | 382600 | 159,206 |
https://mathoverflow.net/questions/382601 | 6 |
>
> Is there a standard method for showing that a functor $F:\mathcal{C}\to\mathcal{D}$ is a fibration, aside from constructing a cleavage?
>
>
>
In the proof of the Grothendieck construction, the fibration we obtain from an indexed category $\Psi:\mathcal{B}^{op}\to\mathfrak{Cat}$ is automatically cloven since ... | https://mathoverflow.net/users/92164 | Can we show that a functor is a fibration without choosing a cleavage? | Just as an example, given a category $\mathcal{C}$ with finite limits, showing $\mathrm{cod}\colon \mathcal{C}^\mathbf{2}\to \mathcal{C}$ is a fibration does not involve choosing a cleaving. All that you need is that a pullback square *exists* for each piece of relevant data. A cleaving would be a specified choice of p... | 9 | https://mathoverflow.net/users/4177 | 382606 | 159,209 |
https://mathoverflow.net/questions/382597 | 7 | This question is similar to (but more specific than) this one: [When are two proofs of the same theorem really different proofs](https://mathoverflow.net/questions/3776/when-are-two-proofs-of-the-same-theorem-really-different-proofs)
I do not know very much about homotopy type theory, but I am trying to understand ho... | https://mathoverflow.net/users/173267 | Explicit different proofs of the same identity type in MLTT | Martin-Löf type theory contains no such type because it is consistent with [uniqueness of identity proofs](https://ncatlab.org/nlab/show/axiom+UIP) which states precisely that what you are looking for is not there.
Martin-Löf type theory is also consistent with the [univalence axiom](https://ncatlab.org/nlab/show/uni... | 11 | https://mathoverflow.net/users/1176 | 382608 | 159,210 |
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