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 |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/314475 | 1 | In Bushnell and Henniart's *The Local Langland's Conjecture for GL(2)* they define a right Haar integral on a locally profinite group $G$ as being a non-zero linear functional
$$
I: C^{\infty}\_{c}(G) \to \mathbb{C},
$$
where the space on the left is the space of all locally constant complex valued functions with compa... | https://mathoverflow.net/users/130895 | Definition of Haar integral in Bushnell and Henniart | Right, the second condition only really makes sense for real valued functions. In reality, it doesn't matter so much: since a Haar integral is linear it's determined by values on a basis of $C\_c^\infty(G)$, and such a basis is given by characteristic functions of open subsets of G. So the condition could just be rephr... | 1 | https://mathoverflow.net/users/128771 | 314481 | 136,620 |
https://mathoverflow.net/questions/314480 | 0 | Let $S$ = $\{v\_1, v\_2, ..., v\_n\}$ denote a random subset of a Hamming hypercube of dimension $d$, where $n = |S|$ and $n \leq 2^d$. If $v\_i$ = $\langle x^i\_1x^i\_2... x^i\_d\rangle$ for all $i \in [1,n]$, then what's the expected sum of all $x^i\_1$ (the first "coordinate" of a vertex $v\_i$) for all $i \in [1,n]... | https://mathoverflow.net/users/130894 | Expected sum of chosen coordinates in a random subset of a Hamming hypercube | Yes. There is an involution $\phi(x\_1,\cdots,x\_d)=(1-x\_1,\cdots,1-x\_d)$ which can extend to subsets by defining $\phi(S)=\{\phi(v\_i)\vert v\_i\in S\}$. If $X$ is the random variable which takes the value of the sum of all first coordinates then we have
$$\mathbb E[X]=\frac{1}{2}\mathbb E[X+\phi(X)]=\frac{n}{2}$$
a... | 3 | https://mathoverflow.net/users/2384 | 314486 | 136,621 |
https://mathoverflow.net/questions/314466 | 11 | In the first answer for this [question](https://mathoverflow.net/questions/5538/why-drinfeld-jimbo-type-quantum-groups) is writen, about the braided category of representation of the enveloping algebra $U(\frak{g})$, for $\frak{g}$ a semisimple Lie algebra:
>
> The space of deformations of the braided tensor struc... | https://mathoverflow.net/users/121660 | Quantum groups and deformations of the monoidal category of $U(\frak{g})$-modules | That the only monoidal deformations of the category of representations of $U(\mathfrak{g})$ is the category of representations of $U\_q(\mathfrak{g})$ is known in Type A from Kazhdan-Wenzl (Adv. Soviet Math, 1993), for the category of vector representations in Type BCD by combining results of [Wenzl-Tuba](https://arxiv... | 12 | https://mathoverflow.net/users/22 | 314487 | 136,622 |
https://mathoverflow.net/questions/314483 | 6 | Are there models of first order Zermelo set theory (axiomatized by: Extensionaity, Foundation, empty set, pairing, set union, power, Separation, infinity) in which $\in$-induction fail?
I asked this question on Mathematics Stack Exchange [here](https://math.stackexchange.com/questions/2977520/is-epsilon-induction-a-t... | https://mathoverflow.net/users/95347 | Is $\in$-induction provable in first order Zermelo set theory? | The answer is no. Take the standard model of Z and add in a $\mathbb{Z}$-sequence of objects, each of whose only element is the previous one. I.e., define
$M=\bigcup\_{n<\omega} \bigcup\_{m<\omega}\mathcal{P}^n(V\_{\omega} \cup (\{\omega + \omega\}\times[-m, \infty))),$
where we adjust the $\mathcal{P}$ operator to r... | 11 | https://mathoverflow.net/users/109573 | 314490 | 136,624 |
https://mathoverflow.net/questions/314511 | 0 | Let $K$ be compact, Hausdorff space but not necessarily metrizable. Let $\mathfrak{M}$ be the Borel $\sigma$ field over $K$ and $\mu$ be a positive Regular Borel measure on $K$. Let $S$ be a subset of $K$ not necessarily in $\mathfrak{M}$. Suppose for all Baire sets $E\subseteq K\setminus S$, $\mu(E)=0$. Can I conclude... | https://mathoverflow.net/users/76412 | Support of a regular measure Reg | Let $K=[0,1]$, $\mathfrak{R}$ its usual Borel $\sigma$-field. and $\mu$ Lebesgue measure, $S= K\setminus \mathbb{Q}$.
Then, for all Baire sets $E \subset \mathbb{Q}, \mu(E)=0$, while $$\text{supp}(\mu)= K \ne S$$
| 2 | https://mathoverflow.net/users/115814 | 314517 | 136,633 |
https://mathoverflow.net/questions/314534 | 4 | Let $G$ be a compact abelian group. A compact abelian group is said to have dimension $n$ if $\dim\_\mathbb{Q} \mathbb{Q}\otimes \hat G = n$. Equivalently one can show that this holds if $G$ is isomorphic to $(\mathbb{R}^n\times \Delta)/\Gamma$ where $\Delta$ is a zero dimensional compact abelian group and $\Gamma$ is ... | https://mathoverflow.net/users/130921 | Finite dimensional compact abelian group that is not a product of connected and a totally disconnected | Here's, for any positive integer $k$, a second countable, compact group, of dimension $k$, whose zero connected component is not a topological direct factor. (Edit: below I construct such groups with some additional requirement.)
Let $I$ be any infinite set of primes (all primes if you like). Consider $G=\prod\_{p\in... | 3 | https://mathoverflow.net/users/14094 | 314536 | 136,636 |
https://mathoverflow.net/questions/314532 | 28 | Let $G$ be a finite group and $K$ a field with $\mathbb{Q} \subseteq K \subseteq \mathbb{C}$.
For $K=\mathbb{C}$ the number of irreducible representations of $KG$ is equal to the number of conjugacy classes of $G$.
For $K=\mathbb{R}$ the number of irreducible representations of $KG$ is equal to $\frac{r+s}{2}$, whe... | https://mathoverflow.net/users/61949 | Number of irreducible representations of a finite group over a field of characteristic 0 | There is a characterization due to Berman for any field. In your case let $n$ be the least common multiple of the orders of elements of $G$. Let $\zeta$ be a primitive $n^{th}$ root of unity. Let $H=Gal(K(\zeta)/K)$. We can identify $H$ with a subgroup of $(\mathbb Z/n\mathbb Z)^\*$. Call $a,b\in G$ $K$-conjugate if $b... | 30 | https://mathoverflow.net/users/15934 | 314537 | 136,637 |
https://mathoverflow.net/questions/314509 | 2 | Let $\Lambda$ denote the Iwasawa algebra and $M$ a finitely generated torsion $\Lambda$ module. Does there exist a number field $K$ and a $\mathbb{Z}\_p$-extension $K\_{\infty}/K$ such that the $p$-Hilbert class field $\Lambda$-module $X\_{\infty}$ is pseudo-isomorphic to $M$? What about the same question for $\mathbb{... | https://mathoverflow.net/users/nan | Inverse Problem for Iwasawa Modules | I guess $X\_{\infty}$ stands for the Galois group of the maximal abelian unramified $p$-extension of $K\_{\infty}$, in other words the projective limit of the $p$-primary part of the class groups. Since the class group of $K$ is finite, it is impossible that $X\_{\infty}$ is pseudo-isomorphic to $\mathbb{Z}\_p$ with tr... | 1 | https://mathoverflow.net/users/5015 | 314540 | 136,639 |
https://mathoverflow.net/questions/314408 | 9 | In a given coxeter group $(W,S)$, a reflection is an element of $W$ that can be written with a symmetric word in the generators $S$.
In multiple sources, I found the following formula:
$$
\mathrm{dp}(\alpha) = \frac{1}{2}(l(t\_\alpha) + 1)
$$
where $\alpha$ is a positive root, $t\_\alpha$ the corresponding reflectio... | https://mathoverflow.net/users/130860 | Reference Request: Length of a reflection in a Coxeter group can be achieved by symmetric word | Let $\beta$ be a positive root. Let $u$ be an element of length $\mathrm{dp}(\beta)$ such that $u(\beta) <0$. Then $u(\beta)=-\alpha$ for some simple root $\alpha$ (because otherwise we could multiply $u$ by a left descent to get an element of shorter length inverting $\beta$), so $u^{-1}(\alpha)=-\beta$ and hence $\el... | 5 | https://mathoverflow.net/users/62135 | 314542 | 136,641 |
https://mathoverflow.net/questions/314143 | 7 | Let $P,Q,R$ be the Fourier series of the Eisenstein series $E\_2,E\_4,E\_6$, that is,
$$
P(q)=1-24\sum\_{n=1}^{\infty}\sigma\_1(n)q^n,
$$
$$
Q(q)=1+240\sum\_{n=1}^{\infty}\sigma\_3(n)q^n,
$$
$$
R(q)=1-504\sum\_{n=1}^{\infty}\sigma\_5(n)q^n.
$$
I heard that $P,Q,R$ are algebraically independent over $\mathbb C(q... | https://mathoverflow.net/users/123157 | Algebraic independence of $P,Q,R$ or $E_2,E_4,E_6$ over $\mathbb C(z)$ | I think the follwing is the answer of this question. It is not depend on the above comments because I didn't understand them. This is just my approach.
(But not fully my idea because it depends on a strong proposition which is already known, and the remaining part is just a corollary.)
>
> Proposition. Let $f$ be a... | 10 | https://mathoverflow.net/users/123157 | 314548 | 136,643 |
https://mathoverflow.net/questions/314560 | 9 | It is probably the most well-known result in quantum mechanics that the harmonic oscillator can be solved by supersymmetry.
More precisely, the operator
$$-\frac{d^2}{dx^2}+x^2$$
can be decomposed as
$$-\frac{d^2}{dx^2}+x^2 = \left(-\frac{d}{dx}-x\right)\left(\frac{d}{dx}-x\right)=:a^\*a.$$
When I tried t... | https://mathoverflow.net/users/130944 | Harmonic oscillator in spherical coordinates | Indeed, the supersymmetric operators do not factorize the Hamiltonian of the three-dimensional harmonic oscillator, there is an additional term. See [Creation and annihilation operators, symmetry and supersymmetry of the 3D isotropic harmonic oscillator](https://www.researchgate.net/publication/228813691_Creation_and_a... | 7 | https://mathoverflow.net/users/11260 | 314566 | 136,646 |
https://mathoverflow.net/questions/314598 | 1 | In [S. Shelah. Uncountable constructions for B.A. e.c. groups and Banach spaces. Israel J. Math. 51 (1985), 273-297], the existence of a special Banach space is proved, assuming the diamond principle for $\aleph\_1$.
Is this diamond principle for $\aleph\_1$ compatible with the continuum hypothesis?
| https://mathoverflow.net/users/39421 | The continuum hypothesis and the diamond principle for $\aleph_1$ | Jensen's diamond implies CH. See <https://math.stackexchange.com/a/2073421/17929>
<https://en.wikipedia.org/wiki/Diamond_principle>
| 8 | https://mathoverflow.net/users/15129 | 314599 | 136,658 |
https://mathoverflow.net/questions/314360 | 3 | Let $f\_i\in S\_{k\_i}(\Gamma\_0(N\_i))$ be normalized cuspidal eigenforms for $i=1,2$ and let $K$ be the composite of the fields of Fourier coefficients generated by $f\_1$ and $f\_2$ and let $\mathfrak{p}\_1$ and $\mathfrak{p}\_2$ are distinct prime ideals in $K$. Under what conditions can one find $f\_3\in S\_{k\_3}... | https://mathoverflow.net/users/nan | Congruence of normalized eigenforms at two primes | I assume you allow $f\_3$ to have arbitrary Fourier coefficients (i.e., $\mathfrak{p}\_1$ and $\mathfrak{p}\_2$ are prime ideals in a suitable number field that contains $K$ and the coefficient field of $f\_3$).
In the case of a single pair $(f\_1,\mathfrak{p}\_1)=(f\_2,\mathfrak{p}\_2)$, this is the famous level low... | 2 | https://mathoverflow.net/users/14967 | 314600 | 136,659 |
https://mathoverflow.net/questions/314597 | 9 | For the Lie algebra $\frak{sl}\_n$, its fundamental representations can be realised as the exterior powers of the first fundamental representation. From this we can see that their weight spaces are all $1$-dimensional. Is this true for the other series - are the weight spaces of the fundamental representations always $... | https://mathoverflow.net/users/126606 | Fundamental representations and weight space dimension | Let $\mathfrak{g}$ be a simple Lie algebra over an algebraically closed field of characteristic $0$ and denote the fundamental highest weights by $\varpi\_1, \ldots, \varpi\_l$, where $l$ is the rank of $\mathfrak{g}$ and the ordering of the $\varpi\_i$ is the usual one (Bourbaki).
The cases where an irreducible repr... | 11 | https://mathoverflow.net/users/10146 | 314604 | 136,661 |
https://mathoverflow.net/questions/314432 | 6 | Let $\mathbb{F}\_2=\{0,1\}$ be the field with two elements. I wonder if there is any known algorithm/construction that, given any $n\geq 1$, returns a boolean function $f:\mathbb{F}^n\_2\rightarrow \mathbb{F}\_2^m$ (for some $m\geq 1$) such that:
1. $f$ is *injective*;
2. for each $S\subseteq \mathbb{F}^n\_2$, with $... | https://mathoverflow.net/users/61364 | Nonlinear boolean functions | A relatively obvious, but possibly inefficient construction would be to identify the space $\Bbb{F}\_2^n$ with the extension field $K=\Bbb{F}\_{2^n}$.
With $m=n^2$ we can then similarly identify $\Bbb{F}\_2^m$ with $K^n$.
The mapping
$$
f:K\to K^n, x\mapsto (1,x,x^2,\ldots,x^{n-1})
$$
will then work. The reason is th... | 1 | https://mathoverflow.net/users/15503 | 314607 | 136,663 |
https://mathoverflow.net/questions/314553 | 16 | A convex surface is a connected open subset of the boundary of a convex body in $\mathbb{R}^3$.
An "infinitesimal bending" of a convex surface $S$ is a deformation of $S$ given by a velocity field $v:S\rightarrow \mathbb{R}^3$ such that the length of every curve on the surface is preserved. Vector field $v$ is called... | https://mathoverflow.net/users/126581 | What is known about sufficient conditions for the rigidity of a convex surface? | Answer to question 1: proper compact subsets of convex surfaces are infinitesimally flexible.
In his book Pogorelov proves infinitesimal flexibility of the graph of a convex function over a strictly convex compact subset of the plane. (Basically the idea is that in order to find an infinitesimal bending of a surface ... | 12 | https://mathoverflow.net/users/98590 | 314608 | 136,664 |
https://mathoverflow.net/questions/314611 | 3 | We say a space $(X,\tau)$ is *homogeneous* if for any $x,y\in X$ there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x) = y$.
What is an example of a connected, homogeneous $T\_2$-space $(X,\tau)$ with $|X| = 2^{\aleph\_0}$ such that $(X,\tau)$ is not homeomorphic to a subspace of $\mathbb{R}^\omega$?
| https://mathoverflow.net/users/8628 | Is every uncountable, homogeneous connected $T_2$-space isomorphic to a subspace of $\mathbb{R}^\omega$? | According to [this search](https://topology.pi-base.org/spaces?q=%7EFirst%20Countable%20%2B%20%24T_2%24%20%2B%20Connected%20%2B%20Homogeneous) in $\pi$-base, the *sigma product of incountably many copies of* $\mathbb{R}$ (no longer in pi-Base as of 2023 March) and the [*boolean product topology on* $\mathbb{R}^{\omega}... | 2 | https://mathoverflow.net/users/7460 | 314615 | 136,667 |
https://mathoverflow.net/questions/314396 | 9 | I already asked this on [math.stackexchange.com](https://math.stackexchange.com/questions/2979095/divided-power-algebra-is-artinian-as-a-module-over-the-polynomial-ring), but did not receive much responses. I hope this is also appropriate for mathoverflow.
In the paper *Homological algebra on a complete intersection,... | https://mathoverflow.net/users/111897 | Divided power algebra is artinian as a module over the polynomial ring | So we solved this today, in a way similar to how the Hilbert basis theorem might be proved. First show the following lemma, then the claim that $D^n(B)$ is artinian as a module over $B[t\_1,\dotsc,t\_n]$ is a direct consequence.
**Lemma** If $M$ is an artinian $R$-module, then $D^1 \otimes M$ is an artinian $R[t]$ mo... | 1 | https://mathoverflow.net/users/111897 | 314622 | 136,670 |
https://mathoverflow.net/questions/314612 | 6 | Given a Hermitian positive semi-definite $n \times n$ matrix $A$ and a rectangular $m \times n$ matrix $B$, is there anything that can be said about the eigenvalues of the matrix $B A B^T$?
It seems to me like one can regroup the product with a test vector $x$ to show that $(x^T B)A(B^T x)$ is at least the smallest ... | https://mathoverflow.net/users/130975 | Bounding the eigenvalues of $B A B^T$ with the eigenvalues of $A$ | The following paper studies relations between $\lambda(BAB^T)$ and $\lambda(A)$:
Li, Mathias (1999). [The Lidskii-Mirsky-Wielandt theorem – additive and multiplicative versions](https://link.springer.com/article/10.1007/s002110050397). Numerische Mathematik. January 1999, Volume 81, Issue 3, pp 377–413.
| 7 | https://mathoverflow.net/users/8430 | 314628 | 136,671 |
https://mathoverflow.net/questions/314627 | 6 | What follows is, as far as I can tell, totally standard folklore. I have one particular point of confusion, other than that, I wanted to confirm that I am uttering the incantations correctly.
The smooth 4-dimensional Poincare conjecture (SPC4) is the statement that any smooth 4-dimensional manifold $\Sigma$ that is ... | https://mathoverflow.net/users/99414 | smooth homotopy 4-balls with sphere boundary in dimension 4 | Yes, you can perform that ambient isotopy: any oriented embedding $i: B^n \to M^n$ is isotopic to any other. (This is a lemma proven independently by Cerf and Palais1, but the idea is quite clear: shrink the image of $i$ until it's contained in the chart, then take the limit that defines the derivative of a map.)
In ... | 6 | https://mathoverflow.net/users/40804 | 314637 | 136,673 |
https://mathoverflow.net/questions/314565 | 1 | According to Theorem 2.7 in the paper <https://arxiv.org/pdf/1704.07560.pdf>, we have the following classical results.
Let $s \in (0,1)$ and $1<p<\infty$. Then for any $F \in L^p(\mathbb{R}^n)$ there exists a unique weak solution $u$ of the equation $(-\Delta)^s u=F$ in $\mathbb{R}^n$, where $$(-\Delta)^s(x) = c\_{n,... | https://mathoverflow.net/users/130257 | Existence and regularity for fractional Poisson-type equation | [This is an extension of my comment, and it does not really answer the original question].
The question includes the following statement: there is a unique solution to $(-\Delta)^s u = F$ for any $f \in L^p(\mathbb{R}^n)$. This is not quite correct: the solution need not exist, and if it exists, then it is unique.
... | 0 | https://mathoverflow.net/users/108637 | 314639 | 136,675 |
https://mathoverflow.net/questions/314645 | 5 | Consider a set of $n$ points in the plane. Among all the connected graphs (trees) $T$ in the plane that have these $n$ points among their vertices, I am looking to find one such that the sum of its edge lengths is minimum. Note that this question is different from Minimum Spanning Tree as we allow $T$ to have vertices ... | https://mathoverflow.net/users/5259 | Length minimizing graphs between a finite set of points | This is the so-called [Steiner Tree Problem](https://en.wikipedia.org/wiki/Steiner_tree_problem).
| 7 | https://mathoverflow.net/users/11142 | 314649 | 136,677 |
https://mathoverflow.net/questions/314651 | 5 | I am recently studying on eigenvalues of a (random) correltion matrix. For a $N\times N$ correlation matrix (with a given meaning of randomness), its (1st, 2nd, etc.) eigenvalues have some distributions depending on $N$. For the case $N=2$, a correlation matrix is like
$$\begin{bmatrix}1&x\\x&1\end{bmatrix}$$
with $x\i... | https://mathoverflow.net/users/115114 | Largest eigenvalues of a (random) correlation matrix? | It sounds like you are asking for the distribution of the eigenvalues of a random symmetric matrix with i.i.d. centered entries. This is a huge subject, and you should read the oeuvre of Mehta, or Terry Tao's blog posts, or Anderson/Guionnet/Zeitouni, or..
*Mehta, Madan Lal*, Random matrices., Pure and Applied Mathem... | 4 | https://mathoverflow.net/users/11142 | 314654 | 136,680 |
https://mathoverflow.net/questions/314663 | 0 | Please change the title if needed.
Let $p$ and $q$ be distinct primes and $G\cong(\underbrace{\mathbb{Z}\_{q}\times\mathbb{Z}\_{q}\times\dots\times\mathbb{Z}\_{q}}\_{n\,\,times})\rtimes\mathbb{Z}\_{p}$, where a subgroup of order $p$ acts irreducibly on the kernel( means $G$ has no proper subgroup of order $pq^{i}$, f... | https://mathoverflow.net/users/97247 | A question on Frobenius groups | Assume that $p \neq q$, and let $r$ be the order of $q$ modulo $p$. The set of irreducible representations of $G = \mathbb{F}\_p$ over $\mathbb{F}\_q$ is in bijection with Frobenius-orbits of irreducible representations of $G$ over $\overline{\mathbb{F}\_q}$. Here, the non-trivial irreps over $\overline{\mathbb{F}\_q}$... | 2 | https://mathoverflow.net/users/21724 | 314669 | 136,688 |
https://mathoverflow.net/questions/314658 | 1 | Let $\mathfrak{g}$ be a basic classical simple Lie super algebra. I want to prove that every finite dimensional module over $\mathfrak{g}$ has a highest weight vector.
My feeling is, since $e\_i$'s are rising operators it will kill a non-zero vector and this will give us a highest weight vector and may be we need to... | https://mathoverflow.net/users/33047 | finite dimensional modules are highest weight modules | This is contained in section 1.5.3 of the book "Dualities and representations of Lie superalgebras" by Cheng and Wang. Chapter 1 of their book happens to be available for free on the AMS bookstore website: <https://bookstore.ams.org/gsm-144>.
| 3 | https://mathoverflow.net/users/12858 | 314670 | 136,689 |
https://mathoverflow.net/questions/314618 | 5 | I am looking for an expression that gives the determinant of a matrix of the form
\begin{bmatrix} A & B & 0 & \dots & 0 & C \\
B & A & B & & 0 & 0 \\
0 & B & A & \ddots & 0 & \vdots \\
0 & & & \ddots & A & B \\
C & 0 & \dots & \dots & B & A \\
\end{bmatrix}
Just to clarify. The above matrix is a block tridiagonal m... | https://mathoverflow.net/users/130979 | Is there a formula for the determinant of a block matrix of this kind? | Just a sketch of an idea that seems to work:
* You can get rid of the corrections $C$ using the [matrix determinant lemma](https://en.wikipedia.org/wiki/Matrix_determinant_lemma) (or, better, replace them with $B$, which makes the matrix block circulant).
* Once you have made those corrections, you can change basis u... | 4 | https://mathoverflow.net/users/1898 | 314674 | 136,692 |
https://mathoverflow.net/questions/314489 | 8 | I've been doing some work with saturated Bruhat paths in a Coxeter group between two elements $u\leq w$. It seems to me that if $\ell(u) =0$, then there are at most $\ell(w)! $. I haven't tried to prove this, it's more of an empirical observation. (Now that I think about it, it's pretty easy to prove, because if you fi... | https://mathoverflow.net/users/62135 | How many maximal length Bruhat paths from $u$ to $w$ can there be? | If you'll allow me to ignore the restriction to finite groups, there is a conjectured upper bound, phrased in terms of polytopes, depending on both $\ell(u)$ and $\ell(w)$: Conjecture 7.3 of the paper "The cd-index of Bruhat intervals" (Electronic journal of combinatorics 11 (2004), #R74) is that the cd-index is maximi... | 3 | https://mathoverflow.net/users/5519 | 314676 | 136,693 |
https://mathoverflow.net/questions/310641 | 0 | Original question (without additional information from Wendy):
Using 192 of the 240 roots of E8 (vertices of 4\_21), Wendy Krieger has defined 48 disjoint tetrahedra this way:
>
> Taking the E8 as {128,112}, of radius 2, we get
>
>
> 16 tetrahedra at (1,1,1,1)E, (1,1,1,1)E
>
>
> 16 tetrahedra at (1,1,1,1)O (1... | https://mathoverflow.net/users/118372 | The generalized Kronecker delta and three sets of 16 tetrahedra defined by 192 of the 240 roots of E8 (vertices of Gosset's 8-polytope 4_21) | Based on the comments from Wendy Krieger below, I accept this observation from მამუკა ჯიბლაძე as an acceptable answer to the question
"product of all coordinates is 1 for E, -1 for O and 0 for A"
Comments from Wendy Krieger 11/06/2018:
```
There is a geometric description here, which has to deal with
reflection... | -1 | https://mathoverflow.net/users/118372 | 314688 | 136,696 |
https://mathoverflow.net/questions/247426 | 4 | the reason for my question is the following:
the two-dimensional canonical singularities are the ADE-singularities, which all are quotients of either affine space or another ADE-singularity by finite abelian Groups and even more, quotients of affine 2-space by finite solvable Groups but one - namely $E\_8$, which is fa... | https://mathoverflow.net/users/61732 | Is the factorial cDV-singularity $T_1^2 + T_2^3 + T_3^4T_4$ any quotient of any affine space by any group? | Let $B=k[x,y,z,u]$ be a polynomial ring in four variables over a field $k$ of characteristic $0$, and let $X$ be 4-dimensional affine space over $k$. A locally nilpotent derivation $D$ of $B$ induces an algebraic action of the additive group $G=(k,+)$ on $X$ via the exponential mapping ${\rm exp} (tD)$, $t \in k$. The ... | 4 | https://mathoverflow.net/users/131028 | 314691 | 136,698 |
https://mathoverflow.net/questions/314678 | 5 | Let $X$ be a variety defined over a number field $K$. The *Hasse principle*, or the local-to-global principle, asserts that $X(K) \ne \emptyset$ if and only if for each completion $K\_v$ of $K$, we have $X(K\_v) \ne \emptyset$. This is known to hold for every quadric hypersurface, by the work of Hasse and Minkowski. In... | https://mathoverflow.net/users/10898 | Hasse principle and its failure for a special class of plane cubics | The case of plane cubics is actually *easier* than the family $ax^3 + by^3 + cz^3$. The reason being that a positive proportion of plane cubics are everywhere locally soluble, when ordered by the height of their coefficients.
However the problem, which one would naively expect to be simpler, of determining the number... | 10 | https://mathoverflow.net/users/5101 | 314694 | 136,699 |
https://mathoverflow.net/questions/314613 | 59 | Let $1 \leq k < n$ be natural numbers. Given orthonormal vectors $u\_1,\dots,u\_k$ in ${\bf R}^n$, one can always find an additional unit vector $v \in {\bf R}^n$ that is orthogonal to the preceding $k$. My question is: under what conditions on $k,n$ is it possible to make $v$ depend *continuously* on $u\_1,\dots,u\_k$... | https://mathoverflow.net/users/766 | When can one continuously prescribe a unit vector orthogonal to a given orthonormal system? | $\def\RR{\mathbb{R}}$ This problem was solved by
*Whitehead, G. W.*, [**Note on cross-sections in Stiefel manifolds**](https://eudml.org/doc/139260), Comment. Math. Helv. 37, 239-240 (1963). [ZBL0118.18702](https://zbmath.org/?q=an:0118.18702).
Such sections exist only in the cases $(k,n) = (1,2m)$, $(n-1, n)$, $(... | 68 | https://mathoverflow.net/users/297 | 314695 | 136,700 |
https://mathoverflow.net/questions/314690 | 15 | A quasicategory ($\infty$-category) $\mathcal{C}$ is finite if there is a finite simplicial set $K$ and a categorical equivalence $K\rightarrow\mathcal{C}$.
On the other hand, a Kan complex (space) $X$ is finite if there is a finite simplicial set $K$ and a weak homotopy equivalence $K\rightarrow X$. Finite Kan compl... | https://mathoverflow.net/users/47658 | Why are finite cell complexes also finite as infinity-categories? | Start from a finite simplicial set $K$ which is homotopicaly equivalent to a Kan complex $X$.
Then by applying a finite number of pushout of outer horn inclusion to $K$, you can build homotopy equivalences $K \hookrightarrow K' \rightarrow X$ such that all the $1$-cells of $K'$ are "invertible" (in the sense that "fo... | 13 | https://mathoverflow.net/users/22131 | 314697 | 136,701 |
https://mathoverflow.net/questions/314693 | 7 | I was wondering if it is possible to classify the finite groups which have no outer automorphisms?
I am currently only aware of the Symmetric Groups ($n \neq 6$) as an infinite class of examples. If there is no classification I would still appreciate any further example of groups which have no outer automorphisms.
... | https://mathoverflow.net/users/128120 | Finite groups which have trivial outer automorphism group | Wielandt's automorphism tower theorem states that if $G$ is any finite group with $Z(G) = 1$, then the sequence of groups $G\_{n}$ with $G\_{0}= G$ and $G\_{n+1} = {\rm Aut}(G\_{n})$ is eventually stable (and it follows that the sequence stabilizes with a complete group- ie a group without outer automorphisms).
Late... | 7 | https://mathoverflow.net/users/14450 | 314702 | 136,704 |
https://mathoverflow.net/questions/314714 | 1 | I am looking for a translation of Fakeev's "Optimal Stopping Rules for Stochastic Processes with Continuous Parameter" from 1970.
I can only find it in Russian. Does anyone know where to find this?
| https://mathoverflow.net/users/83682 | Translation of Fakeev's Optimal Stopping Rules for Stochastic Processes with Continuous Parameter | I have simply tried to put ["Optimal Stopping Rules for Stochastic Processes with Continuous Parameter"](https://www.google.com/search?q=%22Optimal+Stopping+Rules+for+Stochastic+Processes+with+Continuous+Parameter%22) into Google. One of the first results was: <http://mi.mathnet.ru/eng/tvp1835>
This website contains ... | 3 | https://mathoverflow.net/users/8250 | 314716 | 136,706 |
https://mathoverflow.net/questions/314711 | 8 | It turns out that a special case of something I'm working on gives, as a corollary, a rather 19th-century-looking elementary statement about the rank of a certain symmetric matrix. I thought I would post it here in the hope that someone might recognize it as something familiar.
Let $A$ be a $2\times n$ matrix over a ... | https://mathoverflow.net/users/5045 | A symmetric bilinear form and a Plücker identity | $\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\set}[1]{\left\{ #1 \right\}}
\newcommand{\abs}[1]{\left| #1 \right|}
\newcommand{\tup}[1]{\left( #1 \right)}
\newcommand{\ive}[1]{\left[ #1 \right]}
\newcommand{\rank}{\operatorname{rank}}$
I let $\NN$ be the set $\set{0, 1, 2, \ldots}$. For each ... | 8 | https://mathoverflow.net/users/2530 | 314720 | 136,707 |
https://mathoverflow.net/questions/313624 | 13 | $\require{AMScd}$
Weinstein's neighborhood theorem says that every Lagrangian has a standard neighborhood. The more precise statement goes like this.
>
> **Theorem 1:** (Lagrangian Neighborhood Theorem) Let $(X,\omega)$ be a symplectic manifold and $L \subset X$ be a closed Lagrangian. Then there exists a neighbo... | https://mathoverflow.net/users/123015 | Weinstein neighborhood theorem for Lagrangians with Legendrian boundary | Theorem 2 is true, verbatim. I will give an outline of the proof here, since the details make it kind of long. If you would like a detailed write-up and you don't want to do it yourself, DM or email me.
The actual statement that is true is more general: you do not need the boundary $\partial X$ to be contact or for t... | 9 | https://mathoverflow.net/users/123015 | 314732 | 136,712 |
https://mathoverflow.net/questions/314516 | 2 | Let $f : \mathbb{R} / \mathbb{Z} \to \mathbb{C}$ be a trigonometric polynomial of degree $n$ and $m-1 \geq n$ be an integer. The Marcinkiewicz-Zygmund inequality asserts $$\int |f|^p \leq \frac{C\_p}{m} \sum\_{j=1}^m |f(j/m)|^p , \ \ \ 1 < p < \infty.$$
My question is known about the behavior $C\_p$, as a function of... | https://mathoverflow.net/users/50426 | Constant in the Marcinkiewicz-Zygmund inequality | D. Lubinsky in this [paper](http://www.ams.org/journals/proc/2014-142-10/S0002-9939-2014-12270-2/S0002-9939-2014-12270-2.pdf) shows that the best constant in the Marcinkiewicz-Zygmund inequality is equal to the best constant in the Polya-Plancherel inequality. This observation and an upper bound for the constant in R. ... | 1 | https://mathoverflow.net/users/118731 | 314734 | 136,713 |
https://mathoverflow.net/questions/277299 | 8 | **Question:** What are the currently known best *provable and unconditional* lower and upper bounds for Brun's constant $B$, corresponding to the sum of the reciprocals of the twin primes?
**Remark.** According to Dominic Klyve's thesis "Explicit bounds on twin primes and Brun's Constant" (2007, p.23) the best *prova... | https://mathoverflow.net/users/66889 | Best provable and unconditional lower and upper bounds for Brun's constant | In March 7, 2018 (several months after you asked this question!), Dave Platt and Tim Trudgian published a paper called [Improved bounds on Brun’s constant](https://arxiv.org/pdf/1803.01925.pdf), where they show the following improved bounds:
$$1.840503 < B < 2.288513$$
| 7 | https://mathoverflow.net/users/91539 | 314737 | 136,715 |
https://mathoverflow.net/questions/314723 | 5 | I am reading a note at Page 63
<ftp://ftp.math.ethz.ch/users/pink/FGS/CompleteNotes.pdf>
It says whenever $G$ is finite and flat over $S$ the functor ${\rm Hom}(G,H)$ is representable. But it does not give the proof or any references.
Can anybody help me with this question?
[The original statement](https://i.st... | https://mathoverflow.net/users/98747 | Representability of Hom of two finite flat group schemes | The typical way statements like this are proven is by deducing them from the representability of Hilbert schemes. For example, we can use:
>
> **Theorem** (Grothendieck). *Let $S$ be a Noetherian scheme, let $X$ and $Y$ be $S$-schemes of finite type, and assume $X$ is flat and projective, and $Y$ is quasi-projectiv... | 8 | https://mathoverflow.net/users/82179 | 314738 | 136,716 |
https://mathoverflow.net/questions/314551 | 3 | I am trying to solve a problem on additive categories, that gives the following question on (non-commutative unital associative) rings: if for elements $a$ and $b$ of a ring $R$ we have $ab=0$ and $a+b=t$ is invertible then how can one verify the existence of certain $c$ and $c'$ in $R$ such that $ca+bc'=1$ (and so, $c... | https://mathoverflow.net/users/2191 | If the sum of a right (principal) ideal with a left one contains an invertible element and the product is zero then do they contain idempotents? | **Q: If, for elements $A$ and $B$ of a ring $R$,
we have
(i) $AB=0$, and
(ii) $T:=A+B$ is invertible,
then how can we verify the existence of $C$ and $D$ in $R$
such that $CA+BD=1$?**
Let me start by
describing an explicit example of a ring $R$
having elements $A$ and $B$ satisfying (i) and (ii), yet
the desi... | 1 | https://mathoverflow.net/users/75735 | 314741 | 136,717 |
https://mathoverflow.net/questions/223359 | 22 | Consider an irreducible $\mathrm{mod}$ $p$ representation:
$$\rho: \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}\_2(\bar{\mathbb{F}}\_p)$$
If $\rho$ is odd, it was conjectured by Serre in the 70s and proved by Khare-Wintenberger in the 00s that $\rho$ comes from a modular form.
But much like in the char... | https://mathoverflow.net/users/43108 | Even Galois representations "mod p" | To lift $\bar{\rho}$ to a geometric representation in the sense of Fontaine-Mazur, the standard technique requires that $\bar{\rho}$ is balanced, i.e. the dimension of a certain Selmer group must equal the dimension of its dual Selmer group (associated to a certain deformation problem). This is not the case for even re... | 2 | https://mathoverflow.net/users/nan | 314743 | 136,718 |
https://mathoverflow.net/questions/314745 | 3 | Let $\Delta$ be a simplicial complex on $n$ vertices, and $\phi$ a simplicial map that identifies two vertices $x$ and $y$ of $\Delta$. I want to show that the Betti numbers of $\phi(\Delta)$ cannot increase much from those of $\Delta$.
For instance, trivially $b\_0(\phi(\Delta))\leq b\_0(\Delta)$, and I could prove ... | https://mathoverflow.net/users/69171 | Change of Betti numbers under simplicial maps | Let $\Delta$ be the complex with vertices
$$
x, y, a, b\_1, b\_2, \dots, b\_k, c\_1, c\_2, \dots, c\_k
$$
generated by the following $2$-simplices:
$$
(a,b\_i,c\_i), (a,b\_i, y), (b\_i, c\_i, y), (a, c\_i, x)
$$
For any individual $i$, these 2-simplices generate a contractible subcomplex $\Delta\_i$ whose realization i... | 6 | https://mathoverflow.net/users/360 | 314753 | 136,719 |
https://mathoverflow.net/questions/314744 | 6 | I've studied some fundamentals of algebraic geometry and number theory, and now I want to read papers which seem to be the "main stream" of frontier research on arithmetic.
I've heard that Mazur's "Modular curves and the Eisenstein ideal" is one of such papers (and I've also heard that it is good for people who have ... | https://mathoverflow.net/users/128235 | Prerequisites for reading papers of arithmetic such as Ribet, Mazur, Faltings, Wiles | I don't know what you mean by Modular forms of moduli stack, I think maybe you mean modular forms *on* moduli stacks. Either way, you should probably have a look at the book by Katz--Mazur titled "Arithmetic Moduli of Elliptic Curves". It should explain how to think about modular curves in the correct setting you want.... | 11 | https://mathoverflow.net/users/131069 | 314756 | 136,720 |
https://mathoverflow.net/questions/314755 | -1 | For any set $X$ and cardinal $\kappa$ let $[X]^\kappa$ be the collection of all subsets of $X$ of cardinality $\kappa$.
I was looking for $T\_2$-spaces $(X,\tau)$ with the property that
>
> $(P)$ there is an injective function $f:[X]^\omega\to \tau$ such that for all $s\in [X]^\omega$ we have $s\subseteq f(s)$. ... | https://mathoverflow.net/users/8628 | Injective choice function for non-separable $T_2$-spaces | Here is a partial answer: if $|X|^{\aleph\_0}=|X|$ then the answer is yes.
First take an injective function $F:[X\times X]^\omega\to X$ and then take some
function $G:[X\times X]^\omega\to X$ such that for all $A\in[X\times X]^\omega$ the point $\langle F(A),G(A)\rangle$ is not in $A$.
Translate this via a bijection be... | 3 | https://mathoverflow.net/users/5903 | 314758 | 136,721 |
https://mathoverflow.net/questions/314757 | 17 | Let $\operatorname{Gr}(k, n)$ and $\operatorname{Gr}^+(k, n)$ denote the unoriented and oriented grassmannians respectively.
The $\mathbb{Z}\_2$ cohomology of the unoriented grassmannian is
$$H^\*(\operatorname{Gr}(k, n); \mathbb{Z}\_2) \cong \mathbb{Z}\_2[w\_1(\gamma), \dots, w\_k(\gamma)]/(\overline{w}\_{n-k+1},... | https://mathoverflow.net/users/21564 | What is the $\mathbb{Z}_2$ cohomology of an oriented grassmannian? | I was surprised to learn that the ring structure of $H^\*({\rm Gr}^+(k,n);\mathbb{Z}\_2)$ seems to be unknown, in general. The ring structure in the case $k=2$ is given in
*Korbaš, Július; Rusin, Tomáš*, [**A note on the $\mathbb Z\_2$-cohomology algebra of oriented Grassmann manifolds**](http://dx.doi.org/10.1007/s1... | 16 | https://mathoverflow.net/users/8103 | 314762 | 136,724 |
https://mathoverflow.net/questions/314748 | 8 | I know several papers that treat this, but it seems that most of these papers do things very differently with quite different conclusions, so I am confused.
Basically, when one tries to do classical field theory (as in, the branch of physics) in a mathematically precise manner, one considers a field $\psi$ to be a se... | https://mathoverflow.net/users/85500 | Lifting a diffeomorphism into a spinor bundle automorphism | There is an intrinsic ambiguity in lifting diffeomorphisms to spinors, since as you say spinor bundles are not natural bundles. But the ambiguity is not large and has more to do with the global properties of spin structures.
First, recall that the oriented orthogonal frame bundle $\mathcal{P}\_g M$ of a pseudo-Rieman... | 6 | https://mathoverflow.net/users/2622 | 314769 | 136,725 |
https://mathoverflow.net/questions/314780 | 1 | That $\in$-induction fails to be a theorem schema of first order Zermelo + Foundation (see [here](https://mathoverflow.net/questions/314483/is-in-induction-provable-in-first-order-zermelo-set-theory)), then it appears that it is more eligible to replace axiom of Regularity (Foundation) by a Regularity schema:
**Axiom... | https://mathoverflow.net/users/95347 | Does Regularity schema imply $\in$-induction when added to first order Zermelo set theory? | **Yes**; the axiom schemas of regularity and $\in$-induction are equivalent, in fact by first-order logic alone.
The argument is clearest presented in the language of classes, in the style of NBG set theory. To make this an argument in $Z$, just replace each mention of a class $C$ with a formula $\varphi(x,y\_1,\ldot... | 7 | https://mathoverflow.net/users/2273 | 314783 | 136,729 |
https://mathoverflow.net/questions/314768 | 8 | Let
$$\displaystyle f(x) = a\_d x^d + a\_{d-1} x^{d-1} + \cdots + a\_0.$$
Consider the *discriminant* of $f$, denoted by $\Delta(f)$, defined as
$$\displaystyle \Delta(f) = a\_d^{2d-2} \prod\_{i < j} (\theta\_i - \theta\_j)^2,$$
where $\theta\_1, \cdots, \theta\_d$ are the roots of $f(x) = 0$ (over some algebra... | https://mathoverflow.net/users/10898 | Can discriminant polynomials become perfect powers on hyperplanes? | For any $d \geq 2$, there are hyperplanes on which $\Delta\_d$ ramifies, but for $d \geq 3$, it never ramifies **completely**. I guess there are many proofs of this fact, let me give one based on projective duality.
First note that $\Delta\_d$ parametrizes polynomials of degree $d$ having a multiple root. Let me give... | 4 | https://mathoverflow.net/users/37214 | 314786 | 136,730 |
https://mathoverflow.net/questions/314436 | 3 | This is in continuation of the question asked in this earlier post [here](https://mathoverflow.net/questions/314083/reference-request-for-anti-palindromic-polynomials). Given an anti palindromic polynomial of degree $n$ with odd coefficients, does it have roots on the unit circle?
| https://mathoverflow.net/users/68232 | Roots of anti-palindromic polynomial if coefficients are odd. | $P$ has no roots in $\mathbb{U}$. Ad absurdum, assume that there exists $P \in \mathbb{Z}[X]$ with degree $n \ge 1$, such that :
* $P(-X) = X^nP\Big(\frac{1}{X}\Big)$,
* all the coefficients of $P$ from degree $0$ to $n$ are odd
* there exists $\lambda \in \mathbb{U}$ such that $P(\lambda) = 0$.
$ $
Note that $P(... | 2 | https://mathoverflow.net/users/105382 | 314790 | 136,731 |
https://mathoverflow.net/questions/250596 | 32 | It can be proven that if in a quasicompact scheme $X$ any finite subset is contained in an affine open subset then for any sheaf $\mathcal{F}$ on $X$ its Cech cohomology $\hat{H\_{et}^{\bullet}}(X,\mathcal{F})$ is naturally isomorphic to etale cohomology $H^{\bullet}\_{et}(X,\mathcal{F})$.
Is there a counterexaple fo... | https://mathoverflow.net/users/39304 | Etale cohomology can not be computed by Cech | Let $k$ be an algebraically closed field. Glue two copies of $\text{Spec}(k[[x]])$ along $\text{Spec}(k((x)))$. This gives a scheme $X = U \cup V$ such that any etale covering of $X$ can be refined by the Zariski covering $X = U \cup V$. (Hint: use that $k[[t]]$ is strictly henselian.) Thus the etale Cech cohomology is... | 14 | https://mathoverflow.net/users/131088 | 314796 | 136,732 |
https://mathoverflow.net/questions/314805 | 4 | Suppose $\kappa$ is an infinite cardinal and $\alpha$ is an ordinal of cardinality $\kappa$. Is it possible to find a bijection $f : \kappa \to \alpha$ such that for all $x \subseteq\kappa$, $\mathrm{ot}(x) \leq \mathrm{ot}(f[x])$? (Here, $\mathrm{ot}(y)$ is the order type of a set of ordinals $y$.)
| https://mathoverflow.net/users/11145 | bijections and order types | When $\kappa = \aleph\_0$, any bijection works.
When $\kappa$ is uncountable with uncountable cofinality, there is no order-type preserving bijection $f\colon \kappa\to \kappa+\omega$. Indeed, let $X = f^{-1}((\kappa+\omega)\setminus \kappa)$. Then $X$ is not cofinal in $\kappa$, so we can pick some $\alpha\in \kapp... | 9 | https://mathoverflow.net/users/2126 | 314809 | 136,736 |
https://mathoverflow.net/questions/314800 | 3 | I'm currently working on my undergraduate dissertation. I'm working on covering sapces of Riemann surfaces so my supervisor asked me to read the book I mention in the title: "Algébre et Théories Galoisiennes" by Régine and Adrien Douady. The final scope is to prove the following theorem in the book:
>
> Let $ B$ be... | https://mathoverflow.net/users/130117 | English literature close to "Algébre et Théories Galoisiennes" by Régine and Adrien Douady | O. Forster "Lectures on Riemann Surfaces" (Springer) is a good starting point before taking on T. Szamuely "Galois Groups and Fundamental Groups". After all, as Szamuely writes on page 65 at the beginning of Chapter 3, parts of his exposition in this chapter were inspired by Douady and Forster.
| 3 | https://mathoverflow.net/users/3903 | 314811 | 136,737 |
https://mathoverflow.net/questions/314631 | 4 | Let $\beta\in(1,2)$ and $\gamma\in(1,2)$ be Galois conjugates of height 1. That is, there exists a polynomial $p$ with coefficients $-1,0,1$ such that $p(\beta)=p(\gamma)=0$ (not necessarily minimal).
Numerically, there appears to be an absolute constant $C>0$ such that $|\gamma-\beta|\ge C$. Is this true/known? If ... | https://mathoverflow.net/users/8131 | Positive real root separation | The polynomial
$$1 + x^n + x^{2n} - x^{3n} - x^{5n} - x^{6n} + x^{7n}$$
is irreducible and has two Galois conjugate roots $\beta\_n$ and $\gamma\_n$ in $(1,2)$ with
$$| \beta\_n - \gamma\_n| \sim \frac{\log(\beta\_1/\gamma\_1)}{n} \rightarrow 0.$$
| 14 | https://mathoverflow.net/users/131093 | 314813 | 136,738 |
https://mathoverflow.net/questions/314788 | 1 | Let $P$ be a polytope. Is anything known about the set of linear relations that hold between the volumes of the (not-necessarily proper) faces of $P$ as $P$ “varies slightly”? By varies slightly I mean without changing the face lattice—so, it makes sense for a linear functional to vanish at each vector $(vol(F))\_{F \t... | https://mathoverflow.net/users/36720 | Linear relations between volume of a polytope and its faces | First, a linear relation can involve only volumes of faces of equal dimensions because scaling a polytope by $\lambda$ multiplies the volume of a $k$-face by $\lambda^k$.
For a polytope in $\mathbb{R}^d$ there are $d$ linear relations between the volumes of facets which follow from the Minkowski relation:
$$
\sum\_i ... | 2 | https://mathoverflow.net/users/98590 | 314840 | 136,745 |
https://mathoverflow.net/questions/314820 | 0 | Here is the generating function I'm studying.
$f=\prod^N\_{j=1}\left(1+e^{i\cdot j\varphi}z\right)$.
$\varphi$ is a phase related to a quantum optics problem.
And I want to know the analytical result of:
$\lim\_{z\to 0}\frac{1}{X!}\frac{d^Xf}{dz^X}$
with $1\le X\le N$.
| https://mathoverflow.net/users/131101 | Analytical result of a combination like generating function | The limit in question equals the coefficient of $z^X$ in $f(\varphi,z)$. For $1\leq X\leq N$, [series multisection](https://en.wikipedia.org/wiki/Series_multisection) allows to express this coefficient in the following closed form:
\begin{split}
& \frac{1}{N}\sum\_{k=0}^{N-1} e^{-X\frac{2\pi ik}{N}} f(\varphi,e^{\frac{... | 1 | https://mathoverflow.net/users/7076 | 314848 | 136,748 |
https://mathoverflow.net/questions/314844 | 12 | $\DeclareMathOperator\Imm{Imm}$I am looking for a proof in English or French of Schur's theorem that, for every $H$ in the space $\mathbb H\_n^+$ of positive semi-definite Hermitian matrices, and every irreducible character $\chi$ of $\mathfrak S\_n$, $\chi(e)\det H\le\Imm\_\chi(H)$, where the *immanant* $\Imm\_\chi$ i... | https://mathoverflow.net/users/8799 | Schur's Theorem about immanants | Many thanks to **Denis** for pointing out my erroneous initial "proof". This time around the proof is correct, and directly proves the assertion in line 3 of the OP, i.e., $\chi(e)\det(A)\le d\_\chi(A)$ (I will write $d\_\chi(I)$ instead of $\chi(e)$ for uniformity).
The explicit notation is cumbersome, so I am just... | 7 | https://mathoverflow.net/users/8430 | 314855 | 136,749 |
https://mathoverflow.net/questions/314846 | 2 | I have been thinking about multisets for a while. These are sets where elements can repeat, so $S =\{ a,a,b,c,b\}$ is a multiset on the set $A = \{a,b,c\}$.
I have also been looking into morphisms between multisets. Take two multisets $S\_A, S\_B$ with underlying sets $A, B$. I would like to define a morphism between... | https://mathoverflow.net/users/10007 | The category of Multisets and Spans: morphism composition and tensor product | One possibility is as follows. I'll think of a multiset as a finite set $X$ equipped with a multiplicity function $m\_X \colon X \to \{1,2,3,\dotsc\}$. We can then define a morphism from $X$ to $Y$ to be a function such that $m\_Y(y)=\sum\_{x\in f^{-1}\{y\}}m\_X(x)$ for all $y$. These can be thought of as "bijections u... | 6 | https://mathoverflow.net/users/10366 | 314856 | 136,750 |
https://mathoverflow.net/questions/314839 | 4 | Consider a function $F(x, y)$ of two complex variables. For $\Re(y)>0$, we know the analytic structure of the function. In that case, the function is meromorphic, with simple poles in $x$ at locations $x=w\_i$, for $i=0, 1, ...$ . The locations of the poles do not depend on $y$, however the residues do depend on $y$.
... | https://mathoverflow.net/users/131118 | Analytic continuation of 2 variable function | There is a related idea sometimes known as "Bochner's Lemma" (I first saw this as part of an appendix in R. Langlands' Springer Lecture Notes in Math 544). Let $f(z,w)$ be a *holomorphic* function of two complex variables $z,w$ (I feel more comfortable having complex variables $z,w$ rather than $x,y$) in the region whe... | 2 | https://mathoverflow.net/users/15629 | 314865 | 136,753 |
https://mathoverflow.net/questions/313967 | 4 | Inspired by the recent success of my "soft question" [here](https://mathoverflow.net/questions/313808/reference-request-oldest-calculus-real-analysis-books-with-exercises), I also have to ask, what are some of the oldest linear algebra books out there with exercises? I'm fine with or without solutions, either way.
Ag... | https://mathoverflow.net/users/126532 | Reference request: Oldest linear algebra books with exercises? | The title of the question (oldest textbook) is somehow at odds with its description (textbook closest to Grassmann's life time). Since I understand the motivation of the OP is "to read Grassmann in order to learn linear algebra properly", the obvious textbook would be Peano's "Geometrical Calculus, according to the Aus... | 7 | https://mathoverflow.net/users/11260 | 314873 | 136,756 |
https://mathoverflow.net/questions/314847 | 1 | I came across this very complex equation (calculating the Gaussian curvature of a surface):
\begin{align\*}
1 \not\equiv &-\frac{m}{2}\Bigl(\frac{3}{2}C+Su^{-1}-Tu^{-1}+2+Qu^{-1}\Bigr)\\
&\qquad\times(u^3C+Su^2-Tu^2)^{(-2u-2Q-5S+5T)/(6uC+4S-4T)} \\
&+\frac{m}{2}\Bigl[u^3(4C^2-3C)+u^2\Bigl(\frac{1}{3}TC-\frac{1}{3}SC-2S... | https://mathoverflow.net/users/111304 | Gaussian curvature of a surface does not take the constant value 1? | First of all, we introduce $R = T-S$, since $T$ and $S$ only appear in that combination. Secondly, we introduce the notation
$$
\alpha = \frac{-2 u - 2Q + 5 R}{6 C u - 4 R}
$$
which is the first exponent: the second exponent is $\alpha -1$.
If I have not made an error, I find that the RHS to the expression in the pos... | 5 | https://mathoverflow.net/users/394 | 314876 | 136,757 |
https://mathoverflow.net/questions/314861 | 5 | I have read the sections on the Bend & Break Lemmas in Koll\'ar-Mori and Debarre and have the following question. (See below for background and what I do know.)
**Question:** I would like to know if the following is true: if $X$ is a normal (projective) variety and $-K\_X$ is $\mathbb{Q}$-Cartier and ample, for the *... | https://mathoverflow.net/users/110746 | Degree Bound in Bend and Break Lemmas | The bound $-K\_X \cdot C \le \dim X + 1$ can be guaranteed if $X$ has local complete intersection singularities, and the curve to which you are applying bend and break intersects the smooth locus of $X$; see [[Kollár 1996](https://doi.org/10.1007/978-3-662-03276-3), Thm. II.5.14 and Rem. II.5.15]. The reason is that yo... | 4 | https://mathoverflow.net/users/33088 | 314880 | 136,758 |
https://mathoverflow.net/questions/314759 | 8 | Let $G$ be a locally compact group and $\Gamma$ a lattice in $G$.
Is it known whether the space
$$
\mathrm{Hom}\_G\left(\pi,L^2(\Gamma\backslash G)\right)
$$
is finite dimensional for $\pi\in\widehat G$?
This is true if $\Gamma$ is cocompact, but in general?
Here Hom$\_G$ refers to $G$-equivariant, continuous linear... | https://mathoverflow.net/users/nan | Finite multiplicities | For $\Gamma$ and $G$ to which Langlands SLN 544 applies, and to adele groups as in Moeglin-Waldspurger, I think we essentially know this finite multiplicity result, from the complete spectral decomposition of $L^2$.
First, we know it for the subspace of ($L^2$) cuspforms, by the compactness on that space of the integ... | 1 | https://mathoverflow.net/users/15629 | 314882 | 136,759 |
https://mathoverflow.net/questions/314892 | 0 | In a von Neumann algebra, if $A\_{\alpha}$ converges to $0$ in the $\sigma$-weak topology, do the positive parts $(A\_{\alpha})\_{+}$ necessarily converge to $0$ in the $\sigma$-weak topology?
| https://mathoverflow.net/users/127523 | Convergence of self-adjoint elements in $\sigma$-weak topology | Nope, not even in the abelian case. Work in $L^\infty[0,1]$. Let $f\_n$ be the function which is alternately plus and minus $1$ on the subintervals $[\frac{i}{2^n}, \frac{i+1}{2^n}]$. Then $f\_n \to 0$ weak\* but the positive parts converge to $1/2$.
| 3 | https://mathoverflow.net/users/23141 | 314894 | 136,763 |
https://mathoverflow.net/questions/314885 | -2 | Let $X,Y,Z$ be three discrete random variables which I can numerically sample. I need to numerically estimate the probability that $X=Y$ and the probability that $X=Z$. I would like to know whether the algorithm that I describe below is suited to this goal. In particular, I would like to know whether it is acceptable t... | https://mathoverflow.net/users/108252 | Using common samples to numerically estimate pairwise equality of three random variables | Your two estimators $c\_{XY}/n$ and $c\_{XZ}/n$ are unbiased estimators of $P(X=Y)$ and $P(X=Z)$ respectively. They are not independent, however. Whether that is "acceptable" might depend on what you're using them for.
| 0 | https://mathoverflow.net/users/13650 | 314901 | 136,764 |
https://mathoverflow.net/questions/314908 | 0 | Let $f:\mathbb{R^2}\mapsto\mathbb{R}$ be continuous and have partial derivatives in $D=\{(x,y):x^2+y^2\leq1\}$, and let $\mathscr{H}$ the set of such functions for which $\sup\_D |f|\leq1$.
Could someone calculate
$$
\sup\limits\_{f \in \mathscr{H}} \inf\limits\_{z\in D} [f\_x^2(z)+f\_y^2(z)]\;?
$$
Current work:
... | https://mathoverflow.net/users/83917 | $\sup_{f} \inf_{z\in D} [f_x^2(z)+f_y^2(z)]$ for $|f|\leq1$ on a unit disk | The supremum is indeed equal to $1$.
Suppose that $F$ is continuously differentiable and let $\Gamma$ be a maximal gradient line of $F$ in the unit disk (that is, $\Gamma$ is tangent to $\nabla F$ at each point of $\Gamma$, and the endpoints of $\Gamma$ either lie on the boundary of the disk or are critical points of... | 3 | https://mathoverflow.net/users/108637 | 314909 | 136,766 |
https://mathoverflow.net/questions/314903 | 4 | Let $K$ be a number field and $E$ be an elliptic curve defined over $\mathbb{Q}$. Consider the localization map
$$
E(K)\otimes \mathbb{Q}\_p/ \mathbb{Z}\_p \rightarrow \bigoplus\_{v|p} E(K\_v)\otimes \mathbb{Q}\_p/ \mathbb{Z}\_p.
$$
Can we say when is the above map injective. I am wondering if there is some properties ... | https://mathoverflow.net/users/116598 | When is this localization map injective, if at all? | Often it is, but not always. For instance if $K=\mathbb{Q}$ then the map is injective if and only if the rank of $E(\mathbb{Q})$ is at most $1$. This is because the $p$-adic elliptic logarithm of a non-torsion point $P\in E(\mathbb{Q})$ is non-zero.
The corank of the target is equal to $[K:\mathbb{Q}]$ while the rank... | 7 | https://mathoverflow.net/users/5015 | 314912 | 136,767 |
https://mathoverflow.net/questions/314913 | 18 | This is related to my [question](https://mathoverflow.net/q/314844), but it concerns a specific point of the proof of Schur's Theorem.
Let $G$ be a finite group and $\chi$ an irreducible character of $G$. Is it true that
$$\forall g\in G,\qquad\sum\_{h\in G}\overline{\chi(h)}\chi(gh)=\frac{|G|}{\chi(e)}\,\chi(g)\quad... | https://mathoverflow.net/users/8799 | A character identity | Yes. Let
$$ F(g) = \sum\_{h \in G} \overline{\chi(h)}\chi(gh).$$
I claim that $F$ is a class function. Indeed,
$$F(s^{-1} g s) = \sum\_{h \in G} \overline{\chi(h)}\chi(s^{-1}gsh)=\sum\_{h \in G} \overline{\chi(h)}\chi(g(shs^{-1})) = \sum\_{h' \in G} \overline{\chi(s^{-1}h's )}\chi(gh') = \sum\_{h' \in G} \overline{\ch... | 21 | https://mathoverflow.net/users/31469 | 314917 | 136,770 |
https://mathoverflow.net/questions/314918 | 2 | Fix a dimension $n \geq 1$ and a number $k \geq 1$ and suppose that $a\_1,...,a\_k \in \mathbb{R}^n$ are points in $n$-dimensional space such that among all the $nk$ coordinates there is a subset of size at least $nk - \binom{n+1}{2}$ that is algebraically independent (over $\mathbb{Q}$).
Is there an isometry $T$ of ... | https://mathoverflow.net/users/61813 | Algebraic independence of points under isometry | The answer is no.
A rational edge length is an algebraic dependence between the coordinates that you cannot get rid of by an isometry.
One can find $k$ points in $\mathbb{R}^n$ with all but one coordinate algebraically independent and $\sum\_{i=1}^n (a\_1^i - a\_2^i)^2 = 1$.
(For this, take $k$ generic points with $a... | 2 | https://mathoverflow.net/users/98590 | 314921 | 136,772 |
https://mathoverflow.net/questions/156735 | 12 | As well-known, a Hadamard matrix is a square matrix with all coefficients $\pm 1$
and pairwise orthogonal rows or columns. Such matrices exist conjecturally
in every dimension divisible by $4$. Call a matrix with an odd number $n$ of columns an "almost Hadamard matrix" if all its coefficients are $\pm 1$ and if all sc... | https://mathoverflow.net/users/4556 | Almost Hadamard matrices | You can find most useful information on [this](http://www.indiana.edu/~maxdet/) page, which however has not been updated since 2012. The initial question is different, but the partial solution they have found is relevant here.
Quote:
**The *Hadamard maximal determinant problem* asks when a matrix of a given order... | 7 | https://mathoverflow.net/users/29783 | 314923 | 136,774 |
https://mathoverflow.net/questions/309583 | 6 | **Definition 1.** A compactification $c\mathbb N$ of the discrete space $\mathbb N$ is called *soft* if for any disjoint sets $A,B\subset\mathbb N$ with $\bar A\cap\bar B\ne\emptyset$ there exists a homeomorphism $h:c\mathbb N\to c\mathbb N$ such that $h(x)=x$ for all $x\in c\mathbb N\setminus\mathbb N$ and the set $\{... | https://mathoverflow.net/users/61536 | Is each Parovichenko compact space homeomorphic to the remainder of a soft compactification of $\mathbb N$? | Here is a partial answer: the [Continuum Hypothesis implies that all Parovichenko spaces are soft-Parovichenko](http://fa.its.tudelft.nl/~hart/37/publications/the_papers/soft-compactifications.pdf); the proof is a bit long, so I put it in a PDF-file on my website.
Also, I retract my claim in the comments that all com... | 6 | https://mathoverflow.net/users/5903 | 314926 | 136,775 |
https://mathoverflow.net/questions/314911 | 0 | If $(X,\tau)$ is a connected $T\_2$-space with $|X|=\aleph\_0$, what values can $|\tau|$ take?
| https://mathoverflow.net/users/8628 | Cardinality of the topology in countable connected $T_2$-spaces | Continuum. Connected or not, every $T\_2$ space has an infinite pairwise disjoint family of (non-empty) open sets. All unions of all possible subfamilies will give you $\mathfrak{c}$ many open sets (and more is not possible in a countable space).
| 2 | https://mathoverflow.net/users/5903 | 314927 | 136,776 |
https://mathoverflow.net/questions/314312 | 2 | Given two independent random walks $S$ and $S'$ with different distributions for the random variables $X\_1$ and $X\_1'$, I am interested in studying the conditions that make their sum either a recurrent or transient random walk. Could anyone suggest relevant references?
| https://mathoverflow.net/users/68232 | Sum of independent random walks | A good criterion is Chung Fuchs Theorem ("On the distribution of values of sums of
independent random variables" 1951, it is explained in "Probability Theory, example and application" by Durrett). It tells you that S is transient iff $Re (1-E[e^{itX\_1}])^{-1}$ is integrable near 0.
As explained by M. Dus, in the cas... | 5 | https://mathoverflow.net/users/128582 | 314929 | 136,777 |
https://mathoverflow.net/questions/314938 | 4 | Let $S=\{1,2,\cdots,2n\}$, and $S\_i \subseteq S(i=1,2,\cdots,n+1)$ be $n+1$ subsets, each of which contains half of the $2n$ elements, namely $|S\_i|=n$. Consider the following expression:
$$M=\max\_{1\le i<j \le n+1} |S\_i \cap S\_j|$$
1. When $n$ is even, it seems that $M\ge n/2$ for any choice of $S\_i$. I think ... | https://mathoverflow.net/users/120302 | The maximal size of intersection of two sets | Here is an answer to the first point and shows how you can also do the third point.
Let $\displaystyle \bigcup\_{i=1}^{n+1} S\_i=\{b\_{1},b\_{2},\ldots, b\_{m}\}$.
We consider the following table with rows indexed by the sets $S\_1,S\_2,\ldots, S\_{n+1}$ and columns indexed by the elements $b\_1,b\_2,\ldots, b\_m$. W... | 4 | https://mathoverflow.net/users/41010 | 314941 | 136,779 |
https://mathoverflow.net/questions/314928 | 1 | Let $\Delta$ be an open triangle in $\mathbb{R}^2$ and $u\in C^0(\overline{\Delta})\cap C^\infty(\Delta)$ be the convex function satisfying
$$
\det D^2u=1,\quad u|\_{\partial\Delta}=0.
$$
Classical results on Monge-Ampère equations imply that there exists a unique such $u$. I need informations on the boundary regularit... | https://mathoverflow.net/users/17294 | Boundary regularity for the Monge-Ampère equation $\det D^2u=1$ | The answer to both questions is no.
For Question $2$: After subtracting a function of the form $cx\_1$ we may assume that $u \geq 0$ and $u(t,0) = o(t)$. (Take $c$ to be the slope of the tangent line to $u(t,0)$ at $t = 0$; then $u - cx\_1 \geq 0$ on the $x\_1$ axis and vanishes on the $x\_2$ axis, so by convexity is... | 3 | https://mathoverflow.net/users/16659 | 314943 | 136,780 |
https://mathoverflow.net/questions/314942 | 4 | I would like to understand how approximations by monomials and approximations by other kinds of functions are related which I illustrate with an example.
Consider the interval $[-\pi,\pi]$ let's say.
The Stone Weierstrass theorem tells us that the linear hull of functions
$$f\_n(x)=(\arctan(x))^n$$ for $n \in \m... | https://mathoverflow.net/users/119875 | Power series in functions other than monomials | Suppose you want to expand the function $f$ on an interval $I$ into powers of another function $h$ on $I$, which is invertible and maps $I$ onto an interval $J$. Let $H:=h^{-1}$ and $g:=f\circ H$, so that $f=g\circ h$. Thus, you want to expand $g(u)$ into powers of $u-u\_0$ for some $u\_0$ in the interior of $J$. The p... | 4 | https://mathoverflow.net/users/36721 | 314953 | 136,781 |
https://mathoverflow.net/questions/314860 | 6 | This question assumes familiarity with [combinatorial cardinal characteristics of the continuum](https://en.wikipedia.org/wiki/Cardinal_characteristic_of_the_continuum). It is abstracted out of a question in a joint research with Jialiang He. I hope we've got the abstraction right.
A family of subsets of $\mathbb{N}$... | https://mathoverflow.net/users/2415 | A ridiculous combinatorial cardinal characteristic of the continuum? | The cardinal $\mathfrak{ridiculous}$ is equal to $\mathfrak p$ (which is equal to the smallest character of a free filter without infinite pseudointesection on $\omega$). It suffices to prove that a free filter $\mathcal F$ on $\omega$ has infinite pseudointersection if $\mathcal F$ has a base $\mathcal B$ of cardinali... | 13 | https://mathoverflow.net/users/61536 | 314958 | 136,782 |
https://mathoverflow.net/questions/314292 | 5 | Let $N$ be a ${\rm II}\_1$ factor. A maximal abelian self-adjoint subalgebra (MASA) is a $\*$-subalgebra $A \subset N$ such that $A' \cap N = A$. It is called a *Cartan subalgebra* if moreover $\mathcal{N}(A)''=N$, with the normalizer $$\mathcal{N}(A) = \{u \in N \ | \ uAu^\* = u^\*Au = A, \ u \text{ unitary} \}.$$
L... | https://mathoverflow.net/users/34538 | Cartan subalgebra and group measure space construction | First of all, there is a projection $p \in M$ of finite trace such that $N \simeq pMp$, and its equivalent class depends at most on $tr(p)$. Because $G$ is countable and $M$ a ${\rm II}\_{\infty}$ factor, the measure $\mu$ must be infinite, then we can choose a subspace $Y \subset X$ with $\mu(Y)=tr(p)$, and assume tha... | 0 | https://mathoverflow.net/users/34538 | 314961 | 136,784 |
https://mathoverflow.net/questions/314893 | 1 | Let $F \in \mathbb{Z}[x\_0, \cdots, x\_n]$ be a homogeneous polynomial. Let $V \subset \mathbb{P}^n(\mathbb{C})$ be a hypersurface (defined over $\mathbb{Q}$ say), given by a homogeneous polynomial $G(x\_0, \cdots, x\_n)$ say.
We say that $F$ *ramifies completely* on $V$ if there exists a positive integer $r > 1$ an... | https://mathoverflow.net/users/10898 | Homogeneous polynomials which ramify completely on a hypersurface | The following result is completely standard in projective geometry.
>
> **Prop :** Let $S$ and $V$ are two smooth hypersurfaces in $\mathbb{P}^n$ with $n \geq 2$. Assume that $\deg S \neq \deg V$, then $V \cap S$ is singular at most in a finite number of points.
>
>
>
**Proof :** Assume that $v = \deg V < \deg... | 4 | https://mathoverflow.net/users/37214 | 314963 | 136,785 |
https://mathoverflow.net/questions/314971 | 3 | We say that a field $K$ is $C\_{m}$ if it satisfies the following property: for every positive integer $n$ and every sequence of positive integers $(d\_{1},\dotsc,d\_{r})$ satisfying $d\_{1}^{m} + \dotsb + d\_{r}^{m} \le n$, every sequence of $(F\_{1},\dotsc,F\_{r})$ of homogeneous polynomials in $K[x\_{0},\dotsc,x\_{n... | https://mathoverflow.net/users/15505 | Are local fields $C_{2}$? | No: see Guy Terjanian, "Un contre-example à une conjecture d'Artin", *C. R. Acad. Sci. Paris Sér. A–B* **262** (1966) A612 for an example of homogeneous form of degree $4$ in $18$ variables over the $2$-adics that has non non-trivial zero, showing that $\mathbb{Q}\_2$ is not $C\_2$ (Artin had conjectured that the $\mat... | 10 | https://mathoverflow.net/users/17064 | 314973 | 136,787 |
https://mathoverflow.net/questions/314972 | 7 | In general there seems no way to naturally define the tensor product of two $A\_\infty$ algebras $A$ and $B$. But, if $(A, m^A\_1,m^A\_2)$ is only a DGA(differential graded algebra) and $(B, m^B\_k, k\ge 1) $ is an $A\_\infty$ algebra, then is there a natural way to get an $A\_\infty$ algebra structure on the tensor pr... | https://mathoverflow.net/users/69190 | Tensor product of a DGA and an $A_\infty$ algebra | In fact the tensor product of two $A\_\infty$ algebras can be made into an $A\_\infty$ algebra in an explicit way: there are two constructions, one by Saneblidze-Umble and one by Loday. See the paper <https://arxiv.org/abs/0710.0572>
(For cofibrancy reasons one also knows abstractly that there is such a tensor produc... | 11 | https://mathoverflow.net/users/1310 | 314974 | 136,788 |
https://mathoverflow.net/questions/314977 | 5 | Let $X$ be a smooth projective variety over $\mathbf{F}\_q$ and $\overline{X}$ its base change to $\overline{\mathbf{F}\_q}$.
By Deligne’s Weil I, the eigenvalues of the geometric Frobenius acting on $H^{2j}(\overline{X},{\mathbf{Q}}\_{\ell})$ are all algebraic numbers.
Is it true that the eigenvalues of the geome... | https://mathoverflow.net/users/nan | Frobenius eigenvalues algebraic numbers | It seems you have mostly figured this out by yourself. I can happily confirm you are on the right track.
>
> Are they not just the eigenvalues of geometric Frobenius acting on $H^{2j}(\overline{X},{\mathbf{Q}}\_{\ell})$, renormalized by $q^{-j}$?
>
>
>
They are indeed. So they are in fact algebraic numbers.
... | 5 | https://mathoverflow.net/users/18060 | 314978 | 136,789 |
https://mathoverflow.net/questions/314936 | 0 | In posting "[Does Regularity schema imply $\in$-induction when added to first order Zermelo set theor?](https://mathoverflow.net/questions/314780/does-regularity-schema-imply-in-induction-when-added-to-first-order-zermelo-s)"
the answer was that they are equivalent in classical first order logic with membership "$\in... | https://mathoverflow.net/users/95347 | Are Regularity schema and $\in$-induction schema equivalent in intuitionistic logic? | As discussed in the [nLab link](https://ncatlab.org/nlab/show/well-founded+relation) I posted in my comment, the existence of a non-trivial relation satisfying regularity implies the law of excluded middle. We can use this to show that regularity and $\in$-induction are not intuitionistically equivalent. I'll transcrib... | 4 | https://mathoverflow.net/users/83901 | 314985 | 136,792 |
https://mathoverflow.net/questions/314986 | 2 | Let $X$ be a smooth projective variety over the algebraic closure of a finite field with Galois group $G$.
>
> Is it true that the vector space $H^{2k}(X,\mathbf{Q}\_{\ell}(k))^G$ has always positive dimension?
>
>
>
The reason why I ask, is that when $X$ is a complex algebraic variety then the vector space of... | https://mathoverflow.net/users/nan | Classes of hyperplane sections in cohomology | The deduction goes the other way around. We want to check that for $H$ a hyperplane class in the group of codimension $1$ cycles, the class induced by $H^k$ in $H^{2k} (X, \mathbb Q\_\ell(-k))$ is nonzero for $k$ from $0$ to $\dim X$.
Because the cycle class map is compatible with the intersection product (i.e. the c... | 3 | https://mathoverflow.net/users/18060 | 314989 | 136,793 |
https://mathoverflow.net/questions/314998 | 4 | Denote $g(n)=\sum\_{a,b=1}^n\gcd(a,b)$, can we prove that
$$g(n)=\frac6{\pi^2}n^2\ln n+Cn^2+O(n\ln n)$$, where $C=-\frac12+\frac{6}{\pi^2}(-\frac12+\gamma-\ln(2\pi)+12\ln A),$ where $\gamma$ denotes Euler's constant and $A$ denotes Glaisher's constant?
**Failed Attempt**
$$g(n)=\sum\_{k=1}^n\varphi(k)\left[\frac... | https://mathoverflow.net/users/130344 | On approximation of $\sum_{a,b=1}^n\gcd(a,b)$ | The asymptotic you want does not hold just because the "last-term fluctuation"
$$ g(n)-g(n-1) = 2\sum\_{a=1}^n \gcd(a,n)-n $$
is too large. Indeed, denoting the sum in the right-hand side by $\sigma(n)$, we have
$$ \sigma(n) =\sum\_{d\mid n} d\varphi(n/d) = n \sum\_{d\mid n} \prod\_{p\mid(n/d)} \Big(1-\frac1p\Big) ... | 12 | https://mathoverflow.net/users/9924 | 314999 | 136,796 |
https://mathoverflow.net/questions/314858 | 2 | I strive to prove that the following system of differential equations:
$$\begin{cases} x'=x-u(t)xy\\ y'= -y+u(t)xy \\ x(0)=x\_0>0\\ y(0)=y\_0>0 \end{cases}$$
has a unique Caratheodory solution on a given interval $[0,T]$, where $u:[0,T]\to [0,1]$ is a control, lets say measurable or continuous if necessary. I canno... | https://mathoverflow.net/users/61629 | Lotka Volterra existence of Caratheodory solution | Assuming that $u$ is Lebesgue integrable, $f$ **does** satisfy a Lipschitz-like condition, so we have (local) existence and uniqueness theorem.
Whatever the controls, the sets $\{(0,0)\}$, $\{\, (x, 0): x > 0\,\}$ and $\{\, (0, y): y > 0\,\}$ are invariant. So, by uniqueness, any nonextendible solution starting in $... | 3 | https://mathoverflow.net/users/121784 | 315002 | 136,798 |
https://mathoverflow.net/questions/315000 | 3 | Is it consistent in $\mathsf{ZF}$ that there is an infinite cardinal $\kappa$, cardinals $\alpha, \beta\in\kappa$ and a function $f:\kappa\to \alpha$ such that for each $x\in\alpha$ there is an injective map $i:f^{-1}(\{x\})\to \beta$?
| https://mathoverflow.net/users/8628 | Function $f:\kappa\to\alpha$ with small fibers where $\alpha\in\kappa$ | Yes, this is consistent.
Consider a (transitive) model of $\mathrm{ZF}$ in which $\omega\_1$ has countable cofinality. Fix a strictly increasing, cofinal sequence $(\xi\_n \mid n < \omega)$ in $\omega\_1$. Consider
$$
f \colon \omega\_1 \to \omega, x \mapsto \min \{n < \omega \mid x < \xi\_n \}.
$$
$f^{-1}(\{n\})$ i... | 5 | https://mathoverflow.net/users/57114 | 315006 | 136,800 |
https://mathoverflow.net/questions/314968 | 1 | Let $B$ be a finite dimensional selfinjective algebra over a field $k$ with a finite dimensional non-projective $B$-module $M$ and $$A=\pmatrix{k&M\\0&B}.$$
A module $N$ over an algebra $C$ is called Gorenstein projective in case $Ext\_C^i(N,C)=0=Ext\_C^i(D(C),\tau(N))$ for all $i >0$.
Questions:
>
> 1. Is ther... | https://mathoverflow.net/users/61949 | Gorenstein projective modules of a certain triangular matrix algebra | A (right) $A$-module consists of a pair $\pmatrix{V&X}$, where $V$ is a vector space and $X$ a $B$-module, together with a $B$-module map $V\otimes\_kM\to X$.
Another criterion for $\pmatrix{V&X}$ to be Gorenstein projective is that it has a complete projective resolution: i.e., an acyclic complex
$$P^\bullet:=\dots\... | 3 | https://mathoverflow.net/users/22989 | 315008 | 136,802 |
https://mathoverflow.net/questions/315004 | 8 | For a knot $K$, let $\Sigma\_K$ be the double cyclic branched cover of a knot.
By the classical work of **Casson** and **Gordon**, we know that if $K$ is smoothly slice, then $\Sigma\_K$ bounds a rational homology ball.
Is there any well-known counter-example for the reversed direction?
**EDIT** More general sta... | https://mathoverflow.net/users/131172 | A counter-example for the reversed direction of Casson-Gordon's theorem | Here's a particularly subtle counterexample, from the work of Kirk and Livingston (Topology Vol. 38, No. 3, pp. 663--671, 1999). They show that the pretzel knots $J = P(-3,5,7,2)$ and $K = P(5,-3,7,2)$ are not concordant (even locally flat). These two knots are related by mutation (switch the first two pairs of twists ... | 10 | https://mathoverflow.net/users/3460 | 315015 | 136,804 |
https://mathoverflow.net/questions/315027 | 2 | Making a standard bureocracy (using Word tables), I arrived to the following
>
> **Problem.** Assume that we have a table with $n+1$ rows. The first $n$ rows are filled with names of students (and say topics of their Master works) and the last row is empty. It is required to sort this table in alphabetic order (of... | https://mathoverflow.net/users/61536 | The complexity of sorting a list having one free cell | No, $\frac32n$ is optimal. Consider the case in which the initial order has the elements swapped in pairs, $BADCFEHG...$. Take any possible sequence of moves that sorts them; we shall show that this sequence contains at least $\frac32n$ moves.
Focus on the first two locations 1 and 2 (those that contain $BA$ initiall... | 4 | https://mathoverflow.net/users/1898 | 315032 | 136,808 |
https://mathoverflow.net/questions/315031 | 10 | Let $X$ be a spectral space ([en.wikipedia.org/wiki/Spectral\_space](https://en.wikipedia.org/wiki/Spectral_space)), i.e. a space of the form $\textrm{Spec}(A)$ for some commutative ring $A$. If $X$ is noetherian, does there also exist a noetherian ring $B$ such that $X=\textrm{Spec}(B)$?
| https://mathoverflow.net/users/36563 | Noetherian spectral space comes from noetherian ring? | Graph $N\_5$ with poset order topology (i.e. poset $M=\{p,q,r\}, P\_2=\{p,q\}, P\_1=\{p\}, Q=\{r\}, N=\phi$) is not Spec($A$) for Noetherian $A$ because if $a \in Q-P\_2$ then 1 = dim$(A/a)$ = dim$(A)-1$ = 2 by the principal ideal theorem.
| 9 | https://mathoverflow.net/users/59248 | 315040 | 136,812 |
https://mathoverflow.net/questions/315016 | 5 | Consider a random walk $S\_n=\sum\_{i=1}^n X\_i$ where $P(X\_i=+1)=P(X\_i=-1)=1/2$ with $n$ large. By Chernoff's bound we know that, for example, $\sum\_{i=1}^{n/2} X\_i=O(\sqrt{n})$ with high probability.
Now say I told you that $S\_n=0$, or more generally that $S\_n$ is ``close to'' its expectation of 0, e.g. that ... | https://mathoverflow.net/users/106377 | Concentration of closed random walks | I believe the following coupling argument shows that (in particular) if we specify that the random walk ends at 0 then halfway through the walk the probability that we're within distance $\lambda$ of the origin is at least as large as the probability that an unconditional walk is within distance $\lambda$ of the origin... | 2 | https://mathoverflow.net/users/106377 | 315047 | 136,814 |
https://mathoverflow.net/questions/315042 | 9 | Let $k$ be a local field (if necessary, assume characteristic zero). In general, if $X$ is a smooth variety of finite type over $k$ of dimension $n$, then the set of $k$-rational points $X(k)$ is an analytic manifold over $k$ of dimension $n$. I was thinking about the passage $X \mapsto X(k)$ from smooth varieties to m... | https://mathoverflow.net/users/38145 | $G(k)/H(k)$ as a submanifold of $G/H(k)$ | If $k$ has characteristic zero, then $H^1(k,H)$ is finite (Borel-Serre:
<https://mathscinet.ams.org/mathscinet-getitem?mr=181643>)
and the map $G(k)/H(k) \rightarrow (G/H)(k)$ has open image (can also be proved by the implicit function theorem). Moreover, a Baire category argument shows that this map is a homeomorphis... | 7 | https://mathoverflow.net/users/23291 | 315057 | 136,818 |
https://mathoverflow.net/questions/315046 | 5 | Let $F(x) = \prod\_{i=1}^{k} (a\_i x +b\_i)$ be a product of $k$ linear polynomials, where $a\_i,b\_i$ are integers. Under very reasonable conditions, it is known that a constant $C\_k$ exists with the following property: $F(n)$ is divisible by at most $C\_k$ primes, for infinitely many $n$-s. (This is proven in Chapte... | https://mathoverflow.net/users/31469 | Almost-prime values attained by a product of quadratic polynomials | A statement of this type follows from Selberg's sieve,
details are in Halberstam and Richert, Sieve methods, section 10.3 and 10.5.
(In the meantime there may be numerically somehwat stronger estimates,
but the flavour might still be the same.)
Let me quote Theorem 10.11. (hence $r$ is your $C\_k$ and $g$ is your $k... | 4 | https://mathoverflow.net/users/36707 | 315058 | 136,819 |
https://mathoverflow.net/questions/315005 | 11 |
>
> Let us define the *Parovichenko cardinal* $\mathfrak{P}$ as the largest cardinal $\kappa$ such that each compact Hausdorff space $K$ of weight $w(K)<\kappa$ is the continuous image of the remainder $\beta\mathbb N\setminus\mathbb N$ of the Stone-Cech compactification of the discrete space of positive integers $\m... | https://mathoverflow.net/users/61536 | The Parovichenko cardinal, is it equal to $\max\{\aleph_2,\mathfrak p\}$? | No -- it is consistent that $\mathsf{CH}$ fails, and that every compact Hausdorff space of weight $\leq\!\mathfrak{c}$ is a continuous image of $\beta \mathbb N \setminus \mathbb N$. (This is due to Baumgartner, who mentions it off-hand in his article in the Handbook of Set Theoretic Topology; the mutual consistency wi... | 9 | https://mathoverflow.net/users/70618 | 315067 | 136,822 |
https://mathoverflow.net/questions/315023 | 5 | We recall the notion of a $k$-space (or compactly generated space) to fix our notations. For every topological space $X$, we can define a category $\mathfrak{M}\_X$. The class of objects of $\mathfrak{M}\_X$ is the class of continuous mappings $u: K \to X$ from a compact Hausdorff space $K$ to $X$. Let $v : C \to X$ be... | https://mathoverflow.net/users/128540 | Does the notion of a compactly generated space (or $k$-space) depend on the choice of universe? | The following part of 5.9.1 of [Topology and Groupoids](http://groupoids.org.uk/topgpds.html) shows for a particular space $X$ how to reduce the role of a universe.
If $X$ is a **k**-space, there is a set $\mathcal C\_{X}$ of maps $t : C\_{t} \to X$ for compact Hausdorff
spaces $C\_t$ such that a set $A$ is closed i... | 4 | https://mathoverflow.net/users/19949 | 315068 | 136,823 |
https://mathoverflow.net/questions/314964 | 2 | I would like to know results on the structure of a finite group $G$ which possesses a maximal subgroup $H$, with $H$ solvable. More precisely, about
supplements of $H$, that is, decompositions $G=HK$ where the intersection of $H$ and $K$ is not necessarily the trivial subgroup. I know the question is a bit vague, but I... | https://mathoverflow.net/users/91103 | Groups with a maximal subgroup which is solvable | The following references answer the question in case of almost simple groups:
1. [The maximal factorizations of the finite simple groups and their automorphism groups](https://bookstore.ams.org/memo-86-432/) by M. W. Liebeck, C. E. Praeger, and J. Saxl.
The book describes all possible factorizations by maximal subg... | 4 | https://mathoverflow.net/users/40723 | 315071 | 136,824 |
https://mathoverflow.net/questions/315048 | 14 | This question is about two definitions of enriched monoidal categories I have:
Let $\mathcal{V}$ be a symmetric monoidal closed category.
The first definition: a $\mathcal{V}$-enriched category $\mathcal{C}$ is a pseudomonoid object in the Day-convolution monoidal category $(\mathcal{V}\text{-}\mathbf{Cat}, \otimes... | https://mathoverflow.net/users/109318 | Definitions of enriched monoidal category | The two definitions are equivalent, for monoidal structures on $\mathcal{V}$-categories that are tensored over $\mathcal{V}$. I'll describe how the tensor product corresponds to the distributivity map:
The tensor product for a monoidal $\mathcal{V}$-category $\mathcal{C}$ in the first sense is given by specifying a ... | 9 | https://mathoverflow.net/users/1100 | 315075 | 136,825 |
https://mathoverflow.net/questions/315036 | 3 | I've been trying to prove the following claim, but am now unsure about its truth. Is it true, and if so, where can I find a proof?
**Claim:** For any categories C, D, E such that C and D are equivalent,
(i) The set $Hom($C, E$)$ of functors from C to E is in bijective correspondence with the set $Hom($D, E$)$ of ... | https://mathoverflow.net/users/45570 | Bijection between hom sets of equivalent categories? | I will turn my comment into an answer.
If $X$ is any nonempty set, then let ${\mathcal C}\_X$ be the category
which has $X$ as its class of objects and, for each pair
$(x\_1,x\_2)\in X^2$, has exactly one morphism
$\varphi\_{x\_1,x\_2}:x\_1\to x\_2$. Every morphism
in ${\mathcal C}\_X$ is necessarily an isomorphism.
... | 5 | https://mathoverflow.net/users/75735 | 315080 | 136,827 |
https://mathoverflow.net/questions/315052 | 6 | Let $\binom{n}{k}:=\frac{\Gamma(n+1)}{\Gamma(k+1)\Gamma(1-k+n)}$ be the generalized binomial coefficient then I noticed by playing around with Mathematica that the function $f:[0,n/2] \rightarrow \mathbb R$
$$f(x) = \log\left(\binom{n}{n/2+x} \right)-n \alpha(1-(2x/n)^2)$$
has very interesting properties.
For $\a... | https://mathoverflow.net/users/131229 | Interesting behaviour of binomial coefficients | We have
\begin{equation}
f'(x)=\frac{8 \alpha x}{n}+\psi\left(\frac{n}{2}-x+1\right)-\psi\left(\frac{n}{2}+x+1\right),
\end{equation}
where $\psi:=(\ln\Gamma)'$.
By the [Gauss formula, Theorem 1.6.1, page 26](https://www.cambridge.org/core/books/special-functions/1F1C575CEA780EE774F5518C7963BF08)
\begin{equation}
\... | 6 | https://mathoverflow.net/users/36721 | 315081 | 136,828 |
https://mathoverflow.net/questions/314900 | 5 | Suppose we have a compact connected Lie group $G$ acting as isometries on a compact manifold $M^n.$ Then is it necessarily true that the Hausdorff dimension of the union of singular and exceptional orbits are no larger than the Hausdorff dimension of individual principal orbits? If not so, what about the union of singu... | https://mathoverflow.net/users/nan | The Hausdorff dimension of the union of singular orbits and exceptional orbits | According to the paper "ORBITS OF HIGHEST DIMENSION" by Montgomery and Yang, by "singular orbits" one means orbits that have dimension less than the generic (or principle) ones.
For this reason, the answer to your question is rather negative. Indeed, take the round $S^n$ and consider on it the action of $S^1$ that fi... | 2 | https://mathoverflow.net/users/943 | 315090 | 136,830 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.