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https://mathoverflow.net/questions/271235 | 1 | Let $A, B, C..X, Y$ range over potentially proper classes in NBG set theory (or adding class quantification to ZFC in obvious way) and $x,y,z$ range over sets.
Since I don't know the proper symbols for formula classes that have $n$ proper class quantifiers I'll just continue the convention of add 1 to the upper subs... | https://mathoverflow.net/users/23648 | In NBG (ZFC + classes) if $P$ nonempty predicate of classes must $P$ have definable solution? | Your superscripts are all off by one with respect to the usual notation. That is, $\Sigma^0\_n$ in the language of set theory means we have first-order quantifiers only, that is, over sets, whereas $\Sigma^1\_n$ means we have $n$ alternations of second-order quantifiers (over classes).
If one is speaking of first-or... | 3 | https://mathoverflow.net/users/1946 | 271240 | 121,459 |
https://mathoverflow.net/questions/266179 | 8 | Let $\bar D$ denote the closed unit disc in the complex plane.
Consider the function $f:\bar D\longrightarrow \mathbb{C}$, defined as $f(z)=z$ for all $z\in \bar D$.
Let $n\in \mathbb{N}$. For $1\leq i\leq n$, let $p\_i:\bar D\longrightarrow \mathbb{C}$ be monic complex polynomials on $\bar{D}$, such that $p\_i$’s ha... | https://mathoverflow.net/users/93709 | Sum of the norm of polynomials | The question basically boils down to how large the coefficient at $z$ of a polynomial $p(z)$ that has no zeroes in the circle can be compared to the uniform norm of the polynomial. Indeed, one direction is clear. For the other direction, notice that if we have a polynomial $P$ of degree $n$ with the coefficient $1$ at ... | 6 | https://mathoverflow.net/users/1131 | 271247 | 121,461 |
https://mathoverflow.net/questions/271236 | 9 | $\newcommand{\v}{\operatorname{vol}}$The [isoperimetric dimension](https://en.wikipedia.org/wiki/Isoperimetric_dimension) is the maximum $d$ s.t.
$$\v(D)\leq C\cdot \v(\partial D)^{d/d-1}$$
for all open with smooth boundary $D\subset M$, differentiable manifold $M$, and universal constant $C$ for $M$. But it has be... | https://mathoverflow.net/users/99863 | Isoperimetric dimension for any (metric) measure space? | The main problem in defining isoperimetric inequalities on general measure metric spaces is that in the classical manifold case one actually has to deal not just with the original "volume" measure on state space, but also with a completely different "area" measure on appropriately defined "codimension 1 surfaces", whic... | 4 | https://mathoverflow.net/users/8588 | 271251 | 121,462 |
https://mathoverflow.net/questions/271254 | -1 | Let $A,B\neq \emptyset$ be disjoint and suppose $G = (A\cup B, E)$ is bipartite where for all $e\in E$ we have $e\cap A \neq \emptyset\neq e\cap B$. For $a\in A$ we set $N\_G(a) = \{b\in B: (\exists e\in E)\{a,b\}\in e\}$, and for $S\subseteq A$ let $N\_G(S) = \bigcup\{N\_G(a):a\in S\}$.
A *matching* is a set $M$ of... | https://mathoverflow.net/users/8628 | Marriages in infinite bipartite graphs with many neighbors | No. You may modify your example. Join $(k,0)$ with $(m,1)$ for all $m<k$ and also join $(k,0)$ with all $(m,1)$ for $k\leqslant n$. Then any matching must match $\{0,\dots,N\}\times 0$ with $\{0,\dots,N-1\}\times 1$ for large enough $N$. This is impossible.
| 2 | https://mathoverflow.net/users/4312 | 271257 | 121,464 |
https://mathoverflow.net/questions/271272 | 4 | Consider $n\times n$ matrices with entries in $\{0,1\}$. The determinants of these ranges from $0$ to the Hadamard bound $\frac{(n+1)^{\frac{n+1}2}}{2^n}$. Assume $n$ is large enough.
**What does the distribution of the determinants look like? Is it normal or skewed?**
**What proportion of such $n\times n$ determin... | https://mathoverflow.net/users/110332 | Distribution of $0$-$1$ matrices | Much is known about these Hadamard matrices, this [talk](http://mypage.iu.edu/~worrick/slides/SpectrumANU.pdf) by William Orrick gives a good overview:
The first few moments of the determinant $\det M$ are known exactly, for large $n$ the distribution of $|\det M|$ is conjectured to be log-normal.
For the number of... | 4 | https://mathoverflow.net/users/11260 | 271273 | 121,468 |
https://mathoverflow.net/questions/271246 | 4 | Suppose we are given positive semidefinite matrices $P\_1, P\_2, \dots, P\_n \in \mathbb{C}^{m \times m}$.
* How to characterize the set $S$ of their common lower bounds $$S = \{Q \mid 0 \leq Q\leq P\_i, \forall i\}$$ where $A \leq B$ means $B-A$ is positive semidefinite?
* The set is convex. How to describe all the... | https://mathoverflow.net/users/4987 | Common lower bounds for positive semidefinite matrices | We have $n+1$ [linear matrix inequalities](https://en.wikipedia.org/wiki/Linear_matrix_inequality) (LMIs) in $\mathrm X$, namely,
$$\mathrm X \succeq \mathrm O\_m, \qquad \mathrm P\_1 - \mathrm X \succeq \mathrm O\_m, \qquad \mathrm P\_2 - \mathrm X \succeq \mathrm O\_m, \qquad \cdots \qquad \mathrm P\_n - \mathrm X ... | 1 | https://mathoverflow.net/users/91764 | 271315 | 121,481 |
https://mathoverflow.net/questions/271253 | 3 | Consider $V(x)$ a one dimensional polynomial, confining, symmetric double well potential i.e.
1. $V(x)=V(-x)$ for all $x\in\mathbb{R}$
2. $\displaystyle \lim\_{x\to\pm\infty} V(x)=+\infty$
3. $V(x)\in \mathbb{R}\_{2n}[x]$
4. $V$ has two local minima $x\_1,x\_2$ such that $x\_1=-x\_2$.
Then the operator $H=-\frac{d... | https://mathoverflow.net/users/109757 | Ground state for a double well potential (Schrödinger) | A power bound on $M\_k$ is easy. Here's a crude argument, with no attempt made to obtain good bounds. If $\phi\_k$ is a normalized eigenfunction with eigenvalue $E=E\_k$, and $\phi\_k(x)=M\gg 1$ at some local maximum, then $\phi\_k$ must decrease to values $\le M/2$ within distance $\lesssim 1/M^2$, or otherwise we wou... | 5 | https://mathoverflow.net/users/48839 | 271317 | 121,483 |
https://mathoverflow.net/questions/271286 | 16 | I stumbled on the following identity, which has been checked numerically.
>
> **Question.** Is this true? If so, any proof?
> $$\sum\_{j=0}^{\lfloor\frac{k}2\rfloor}\binom{n-2k+j}{j,k-2j,n-3k+2j}
> =\sum\_{j=0}^{\lfloor\frac{k}2\rfloor}\binom{n-k-2j-1}{k-2j}.$$
>
>
>
Here, $\binom{m}{a,b,c}$ is understood as ... | https://mathoverflow.net/users/66131 | Sum of multinomals = sum of binomials: why? | For convenience set $m=n-2k$. Then
\begin{equation}
\begin{split}
\binom{n-2k+j}{j,k-2j,n-3k+2j} &= \binom{m+j}{j,k-2j,m-k+2j} \\
&= \binom{m+j}{m} \binom{m}{k-2j} \\
&= [t^j](1-t)^{-(m+1)} \cdot [t^{k-2j}](1+t)^m \\
&= [t^{2j}](1-t^2)^{-(m+1)} \cdot [t^{k-2j}](1+t)^m
\end{split}
\end{equation}
where $[t^a]p$ is th... | 13 | https://mathoverflow.net/users/88133 | 271322 | 121,485 |
https://mathoverflow.net/questions/271325 | 1 | I've seen many trigonometric identities but here is one that I encountered for which I did not find a reference.
In case you wonder where this came from, I was investigating certain $q$-series [in this paper](https://arxiv.org/pdf/1203.6736.pdf) on Eulerian polynomials.
>
> **Question.** Is this true? Can you pro... | https://mathoverflow.net/users/66131 | One trig "survives" a binomial summation: why? | The sum from k=-n to k=n-1 equals zero by pairing off the end terms and by using the sign difference together with the evenness of the function sec.
| 5 | https://mathoverflow.net/users/104791 | 271329 | 121,489 |
https://mathoverflow.net/questions/271274 | 2 | Assume that we have a vector bundle $E$ over $S^n$.
>
> Is there a continuous family of invertible linear maps $T\_x:E\_x \to E\_{-x}$?
>
>
>
Here continuity has the obvious meaning as soon as we have trivialization for the bundle around $x$ and $-x$
There is an obvious affirmative answer for the non orienta... | https://mathoverflow.net/users/36688 | Invertible (isometric) sections of certain hom bundles over sphere | There is no such family of *complex* linear maps for the tautological bundle on $\mathbb CP^1$. The antipodal map is orientation reversing, hence the first Chern class changes its sign. The same argument should work for the positive complex spinor bundle on any even-dimensional sphere.
The same argument also works fo... | 2 | https://mathoverflow.net/users/70808 | 271339 | 121,492 |
https://mathoverflow.net/questions/270597 | 11 | Suppose we have a finitely presented group $G$ with solvable word problem. (For instance, the command RWSGroup in Magma terminates giving us a finite [but possibly gigantic] rewrite system.) Is there then an algorithm to determine whether $G$ has an element of order 2?
If so, where is the best place to find such an i... | https://mathoverflow.net/users/3199 | Identifying a group without 2-torsion | Using a mixture of computation and thought I believe that I have established that this group is indeed torsion-free. I don't know of any general approach to solving that particular problem. Even if the group is hyperbolic (which this example is not, because it has free abelian subgroups of rank $2$), I am not aware of ... | 11 | https://mathoverflow.net/users/35840 | 271348 | 121,495 |
https://mathoverflow.net/questions/271347 | 3 | In Freitag and Kiehl's etale cohomology book p206,$H(X\_e,Rj\_\*(\Lambda|\_{X\_{\eta}}))\to H(X\_{es},i^\* Rj\_\*(\Lambda|\_{X\_{\eta}})$is a isomorphism,where X is a proper scheme and smooth over the base scheme S at all the point of Y, the complement of the scheme $X\_e$.
If $X\_e$ is proper,it is easily derived fr... | https://mathoverflow.net/users/85353 | How to prove this formula for nearby cycle? | In the case Freitag and Kiehl are describing ($X$ a family of quadrics degenerating to a node, $Y$ the divisor at $\infty$), $X$ and $Y$ are both smooth in a neighborhood of $Y$. This is sufficient for the base change property as it is a local condition.
| 5 | https://mathoverflow.net/users/18060 | 271351 | 121,496 |
https://mathoverflow.net/questions/271350 | 5 | Let $C$ be a stable $\infty$-category (presentable, if you like) and let $map(-,-)$ denote the simplicial mapping space. If $X \to Y \to Z$ is a fiber sequence, and $W$ is an object, when is $map(W,X) \to map(W,Y) \to map(W,Z)$ a fiber sequence?
I suspect that this does not come for free. I'm more willing to believe ... | https://mathoverflow.net/users/11540 | Simplicial mapping spaces, stable $\infty$-categories, and triangles | This is always true, even without the hypothesis of stability. In an ∞-category a fiber sequence $X\to Y\to Z$ is a pullback square
$$\require{AMScd}
\begin{CD}
X @>>> Y\\
@VVV @VVV \\
\* @>>> Z
\end{CD}\,.$$
So you are pretty much asking whether the functor $\mathrm{Map}(W,-)$ preserves pullback squares. In fact it ... | 8 | https://mathoverflow.net/users/43054 | 271355 | 121,498 |
https://mathoverflow.net/questions/271343 | 3 | Given two sets of variables $X=\{x\_1,\cdots,x\_n\}$, $Y=\{y\_1,\cdots,y\_m\}$, and two partitions $\lambda$ and $\mu$. Is there a formula for the product of the Schur functions $s\_{\lambda}(X) s\_{\mu}(Y)$?
If $X=Y$ then the answer is given by the Littlewood–Richardson rule, but what if they are not the same?
(I... | https://mathoverflow.net/users/110684 | Product of Schur functions | If $X = (u\_1, u\_2, \ldots, u\_a, w\_1, w\_2, \ldots, w\_c)$ and $Y = (v\_1, v\_2, \ldots, v\_b, w\_1, \ldots, w\_c)$ with $u$'s, $v$'s and $w$'s disjoint, then
$$s\_{\lambda}(u,w) s\_{\mu}(v,w) = \left( \sum\_{\alpha,\ \beta} c\_{\alpha \beta}^{\lambda} s\_{\alpha}(u) s\_{\beta}(w) \right) \left( \sum\_{\gamma,\ \de... | 4 | https://mathoverflow.net/users/297 | 271356 | 121,499 |
https://mathoverflow.net/questions/271318 | 6 | In something I've been thinking about recently, the following object appears:
$$\mathcal{F}\_{g} = \sum\_{n=0}^{\infty} Q^{n} T\_{n}\big( \phi\_{2g-2}(\tau, z) \big)$$
where $T\_{n}$ is the $n$-th Hecke operator and $\phi\_{2g-2}(\tau, z)$ is a weak Jacobi form of weight $2g-2$ and index one. For context, this aris... | https://mathoverflow.net/users/105661 | Geometry of Hecke Operators on Jacobi Forms? | The interpretation of Hecke operators on modular forms and Jacobi forms in terms of a sum over isogenies is classical - this is the worldsheet interpretation. Their "target space" interpretation in terms of genera of symmetric products and second quantization of strings was given by [Dijkgraaf-Moore-Verlinde-Verlinde](... | 4 | https://mathoverflow.net/users/121 | 271359 | 121,500 |
https://mathoverflow.net/questions/271373 | 1 | Assume that there are two distinct parallel lines on a Euclidean plane. Is there a name for the zone between these two lines?
| https://mathoverflow.net/users/36379 | Name of area between two parallel lines | *Strip*.
This is how it is used in complex analysis, where strips are common (as domains for various transcendental functions). I have seen the same term in other contexts...
| 5 | https://mathoverflow.net/users/13923 | 271374 | 121,505 |
https://mathoverflow.net/questions/271379 | 15 | Consider the matrix $2\times2$ symmetric matrix:
$$
A\_2=\begin{pmatrix} 1 & a\_1 \\ a\_1 & 1\end{pmatrix}.
$$
It's clear that the restriction $|a\_1|<1$ implies that $\det(A\_2)>0$. Moreover, this is the best restriction on the modulus of $a\_1$ with this property, for $\det\begin{pmatrix} 1 & 1\\ 1 & 1\end{pmatrix}=0... | https://mathoverflow.net/users/100740 | On the determinant of a class symmetric matrices | Your guess is correct. If the elements outside the diagonal have absolute values less than $1/(n-1)$, the matrix has 'diagonal dominance', thus it is nonsingular.
To make the answer self-contained, I give a proof. If $x=(x\_1,\dots,x\_n)^t$ satisfies $Ax=0$, take $k$ such that $|x\_k|$ is maximal and look at $\sum a... | 17 | https://mathoverflow.net/users/4312 | 271380 | 121,507 |
https://mathoverflow.net/questions/271395 | 1 | Suppose $\{Z\_{i}\}\_{i=1,2,\ldots}$ are normally distributed (identically and independent) random variables with mean $\mu>0$ and positive variance $\sigma^{2}$. Suppose we want to calculate the probability, that
$$U(n)=u+\sum\_1^nZ\_i\le0 \text{ for some }n\in\mathbb{N}\text{, where }u>0\text{.}$$
According to [ris... | https://mathoverflow.net/users/110735 | Random walk with gaussian increments - Probability that it falls below 0 | They simply take the exact formula for the ruin probability of a simple random walk and use it as an approximation. Ruin probability of a simple random walk that jumps to $+1$ with probability $p$ and to $-1$ with probability q is given by $(q/p)^z$ if $p>q$. Then, take the random walk that starts at $u=Cz$ and jumps t... | 0 | https://mathoverflow.net/users/85303 | 271398 | 121,514 |
https://mathoverflow.net/questions/271327 | 6 | It is well-known that the completion of a metric space which is homeomorphic to a ball can be very wild; in fact, I think, every compact manifold is the closure of an open ball!
But CAT(0) spaces are very different from general manifolds. If I have a CAT(0) with the path metric, that is totally bounded, and that is h... | https://mathoverflow.net/users/27933 | Is the completion of a CAT(0) open ball a closed ball? | The answer is "no".
Let $\Sigma$ be the suspension over Poincaré homology sphere.
It admits a polyhedral $\mathrm{CAT}[1]$-metric.
Let $B$ be the unit ball in the Euclidean cone $\mathrm{Cone}\,\Sigma$.
Note that $B$ is a compact $\mathrm{CAT}[0]$-space;
its interior is homeomorphic to the ball in $\mathbb{E}^5$ ... | 11 | https://mathoverflow.net/users/1441 | 271400 | 121,515 |
https://mathoverflow.net/questions/271405 | 7 | Is it (for some reason) true that
$\lim\_{c\to 0^+}\int\_c^{\pi/2}\frac{c}{t}\sqrt\frac{1+t^2}{t^2-c^2}dt=\frac{\pi}{2}$?
**Numerical evidence (from Mathematica):**
* when $c=1/5$, the integral is $\approx 1.578$.
* when $c=1/10$, the integral is $\approx 1.575$.
* when $c=1/100$, the integral is $\approx 1.571$.
... | https://mathoverflow.net/users/2819 | Identity involving an improper integral (with geometric application) | Since the main contribution to the integral comes from $t<<1$, analytically one has
\begin{align}
\lim\_{c\to 0^+}\int\_c^{\pi/2}\frac{c}{t}\sqrt\frac{1+t^2}{t^2-c^2}dt&=\lim\_{c\to 0^+}\int\_c^{\pi/2}\frac{c}{t}\sqrt\frac{1}{t^2-c^2}dt\\
&=\lim\_{c\to 0^+}\int\_{2/\pi}^{1/c}c\sqrt\frac{1}{1-c^2t^2}dt\\
&=\lim\_{c\to 0... | 7 | https://mathoverflow.net/users/82588 | 271408 | 121,518 |
https://mathoverflow.net/questions/271413 | 3 | As a non functional analyst, I stumbled over the following question:
Given a self-adjoint Operator $T:D(T) \subset H \rightarrow H.$ Assume we know that $T$ has some eigenvalue $\lambda$ which is isolated in the spectrum of $T$. The eigenvalue may be non-degenerate.
We do have an approximating sequence of eigenfunc... | https://mathoverflow.net/users/108483 | Spectrum of self-adjoint operator | Since $\lambda$ is an isolated point in the spectrum of $T$, the distance from $x\_n$ to the eigenspace corresponding to $\lambda$ tends to zero. Since we may assume that this eigenvalue is non-degenerate, we may force $(x\_n)$ to converge to some eigenvector $x$, multiplying by appropriate scalars if necessary.
Now ... | 4 | https://mathoverflow.net/users/24953 | 271446 | 121,528 |
https://mathoverflow.net/questions/271394 | 6 | Let $(\mathfrak{g},[-,-])$ be a pronilpotent Lie algebra (considered of degree zero). We can consider $(\mathfrak{g},[-,-])$ as a differential graded Lie algebra endowed with the $0$-differential. Let $A$ be a commutative non-negatively graded differential algebra. Then $A\widehat{\otimes }\mathfrak{g}$ is a differenti... | https://mathoverflow.net/users/41970 | Homotopy equivalence vs gauge equivalence | The two conditions are always equivalent, since $A\widehat{\otimes}\mathfrak{g}$ is complete. This is present in the literature, I think a possible reference is [A tale of three homotopies](https://arxiv.org/abs/1208.4695) by Dotsenko and Poncin, as well as my own article [Representing the Deligne-Hinich-Getzler ∞-grou... | 1 | https://mathoverflow.net/users/44134 | 271452 | 121,530 |
https://mathoverflow.net/questions/271439 | 2 | The elements of the closure of $\{ \sum\_{j=1}^n a\_j e^{i\nu\_j x}: a\_j\in \mathbb{C}, \nu\_j\in \mathbb{R} \}$ in the supremum-norm are called almost periodic functions. An almost periodic function $f$ is called quasi-periodic iff the frequency module
$$ \mathcal{M}(f)=\left\langle \left\{ \nu\in \mathbb{R}: \lim\... | https://mathoverflow.net/users/91098 | Approximation of quasi-periodic function by trigonometric polynomials | Yes. This is Satz XIV, p. 160 of Harald Bohr, [Zur Theorie der Fastperiodischen Funktionen: II, Acta Math. **46** (1925) 101-214](https://projecteuclid.org/euclid.acta/1485887596). (Page 162 attributes the quasi-periodic case to Bohl.)
| 3 | https://mathoverflow.net/users/19276 | 271453 | 121,531 |
https://mathoverflow.net/questions/104981 | 7 | Let $G$ be a discrete group which is amenable (i.e. it admits an left-invariant mean, i.e. a continuous positive normalised linear functional $m:\ell^\infty(G) \to \mathbb{R}$ such that $\forall g \in G, m \circ \lambda\_g = m$ where $\lambda\_g : \ell^\infty(G) \to \ell^\infty(G)$ is the left-regular action of $G$). L... | https://mathoverflow.net/users/18974 | The kernel of all invariant means | (Essentially from Narutaka Ozawa's comments)
Let $K = \bigcap\_{m \in \mathcal{M}} \ker m$. The trivial answer would be that $K$ is the closed linear span of
$$S =\{f-\lambda\_g(f):\;g\in G,\;f\in\ell^\infty(G) \}.$$
For a non-amenable group this is all of $\ell^\infty(G)$. For an amenable group with Følner net $(F... | 2 | https://mathoverflow.net/users/14094 | 271455 | 121,533 |
https://mathoverflow.net/questions/271485 | 4 | Let $\leq\_1$ and $\leq\_2$ be recursive well-orderings of $\omega$ that have the same order type. Is there necessarily a recursive bijection $f$ such that $f(x)\leq\_1f(y)\iff x\leq\_2y$? If this does not hold in general, are there natural assumptions under which it does, such as if the well-orderings are primitive re... | https://mathoverflow.net/users/83073 | Are two recursive well-orderings with the same order type recursively isomorphic? | The answer is no, not even for relations with order type $\omega$, and not even for primitive recursive relations.
To see this, let $\leq$ be the usual order relation on the natural numbers. And let $\unlhd$ be the order arising from the following computable procedure. We order the even numbers as usual. Next, we sta... | 8 | https://mathoverflow.net/users/1946 | 271488 | 121,547 |
https://mathoverflow.net/questions/271489 | 2 | Suppose a curve of degree $d$ in the plane passes through $\frac{d^2+3d}{2}$ lattice points. Must it pass through another lattice point?
| https://mathoverflow.net/users/110581 | Must a curve containing $(d^2+3d)/2$ lattice points contain another? | Yes if $d=1$, but no for $d\ge 2$.
[Schinzel's Theorem](http://mathworld.wolfram.com/CircleLatticePoints.html) says that for every $n\in \mathbb{N}$, there is a circle with exactly $n$ lattice points on the circumference. In particular, there is a circle passing through exactly $5$ lattice points, not $6$, which is ... | 8 | https://mathoverflow.net/users/2954 | 271493 | 121,548 |
https://mathoverflow.net/questions/271499 | 4 | I wonder if a smooth manifolds $M$ which admits an affine structure must necessarily satisfy $\pi\_2(M)=0$.
By affine structure I mean an atlas all of whose change of coordinates maps are affine maps. This kind of manifolds occur as bases of Lagrangian fibrations. If the answer to the above question is yes, this woul... | https://mathoverflow.net/users/58620 | Smooth manifold with affine structure: aspherical? | No they are not, you can consider $\mathbb{R}^3-\{0\}$ or the quotient of $\mathbb{R}^3-\{0\}$ by the map $h\_c(x)=cx, c>1$ which is $S^2\times S^1$.
| 7 | https://mathoverflow.net/users/80891 | 271500 | 121,550 |
https://mathoverflow.net/questions/271403 | 21 | Let $\Sigma\_g$ be the fundamental group of the closed orientable surface of genus $g\ge 2$; let $B\_n$ be the braid group on $n\ge 3$ braids; let $S\_n$ be the symmetric group on $n$ letters; let $p:B\_n\to S\_n$ be the canonical epimorphism.
Does every homomorphism $f:\Sigma\_g\to S\_n$ lift to $B\_n$? That is, is ... | https://mathoverflow.net/users/105095 | Morphism from a surface group to a symmetric group, lifted to the braid group | In certain very special cases, I think this can be answered.
In particular, if $f:\Sigma\_g \twoheadrightarrow S\_n$ is onto, $n\geq 4$ and $g \gg 0$, then Theorem 6.20 of [Dunfield-Thurston](http://rdcu.be/tf30) implies that the map $f$ is determined up to the action of the mapping class group by the image $f\_\ast:H... | 6 | https://mathoverflow.net/users/1345 | 271502 | 121,551 |
https://mathoverflow.net/questions/271213 | 6 | I'm trying to find the geodesic that connects the identity with some circulant, symmetric matrix $U\in\mathrm{GL}(N,\mathbb{R})$, meaning we have
\begin{align}
U=\left(\begin{array}{ccc}
u\_1 & u\_2 & \cdots\\
u\_2 & u\_1 & \cdots\\
\vdots & \vdots & \ddots
\end{array}\right)\quad\text{with}\quad b\_{N-i}=b\_{1+i}... | https://mathoverflow.net/users/80903 | Geodesic in space of circulant matrices | The answer is as follows: When $U$ is a positive-definite, symmetric, circulant matrix in $\mathrm{GL}(n,\mathbb{R})$, then there is a symmetric circulant matrix $u$ such that $U = e^u$ and the curve $\gamma(t) = e^{tu}$ (which is a positive-definite, symmetric, circulant matrix for all $t$) is a geodesic in the metric... | 10 | https://mathoverflow.net/users/13972 | 271505 | 121,552 |
https://mathoverflow.net/questions/271492 | 5 | How are the Goldman Lie algebra of a closed surface $\overline{S}$ and the bordered surface $S$ obtained by taking $\overline{S}$ and removing an open disc (or more generally, $n$ disjoint discs) related?
| https://mathoverflow.net/users/105615 | Goldman Lie algebra of a bordered surface vs. a closed surface? | I am not really sure what you are asking, but the main theorem in Section 5.17 in Goldman's paper [Invariant functions on Lie groups and Hamiltonian flows of surface group representations](http://www.math.umd.edu/~wmg/InvariantFunctions.pdf) seems to imply that adding boundary components to $\overline{S}$ introduces ce... | 3 | https://mathoverflow.net/users/12218 | 271514 | 121,555 |
https://mathoverflow.net/questions/271292 | 8 | I've read that the Tietze's extension theorem was still valid for continuous applications from a closed subspace of a normal topological space to a contractible topological manifold (understood as Hausdorf and 2nd countable).
But I can't find any clear reference for this result.
What I have found is that the theore... | https://mathoverflow.net/users/74372 | Tietze's extension theorem for contractible manifolds | Let us say that a topological space $Y$ is a *normal absolute extensor* if each continuous map $f:Z\to Y$ defined on a closed subspace $Z$ of a normal topological space $X$ extends to a continuous function $\bar f:X\to Y$. By the Tietze Theorem, the real line is a normal absolute extensor and so is the countable produc... | 6 | https://mathoverflow.net/users/61536 | 271519 | 121,558 |
https://mathoverflow.net/questions/271536 | 10 | [GAP](https://en.wikipedia.org/wiki/GAP_(computer_algebra_system)) is computer algebra system which allows to make calculations with finite groups. (See wikipedia link for an example).
Is there web interface for it ? (I cannot google it.)
Or may be some other computer algebra systems which allows to calculate with fi... | https://mathoverflow.net/users/10446 | Web interface for GAP (or other computer algebra system dealing with finite groups)? | There is the [Magma calculator](http://magma.maths.usyd.edu.au/calc/) which can be used to do calculations in finite groups.
One problem is that you have to type in all of your input before executing it, but with practice you can do quite complicated calculations.
For example, you can carry out the calculation from... | 11 | https://mathoverflow.net/users/35840 | 271538 | 121,560 |
https://mathoverflow.net/questions/271529 | 3 | This is a math problem whose solution would be great for physics, especially space flight planning.
Given two ellipses sharing a focus, could there be a pencil of ellipses sharing the same focus and tangent to the two given ellipses, or is there only one such ellipse? This problem has two versions: 2D (coplanar) and ... | https://mathoverflow.net/users/105172 | Pencil of ellipses | It is a pencil (one parametric family). Instead of ellipses with one common focus,
consider ellipses with the same center. The reduction is achieved by the following:
Lemma. The map of the complex plane $z\mapsto z^2$ maps ellipses centered at $0$ to ellipses with one focus at zero, bijectively.
Now consider ellips... | 11 | https://mathoverflow.net/users/25510 | 271540 | 121,561 |
https://mathoverflow.net/questions/271195 | 6 | For a compact topological space $X$, denote by $\mathcal{M}(X)$ the Banach space of finite signed Borel (Radon) measures on $X$ with the total variation norm. This is canonically isometric to the dual space of the space of continuous functions $C(X)$.
If $X$ and $Y$ are compact spaces, then we have the isometric iso... | https://mathoverflow.net/users/16702 | Tensor product of measure spaces | Let me work with general Banach spaces (good references are [Vector Measures](http://www.ams.org/mathscinet-getitem?mr=0453964) by Diestel and Uhl, or [Introduction to Tensor Products of Banach Spaces](http://www.ams.org/mathscinet-getitem?mr=1888309) by Ryan).
For Banach spaces $E,F$ the bounded bilinear maps on $E\... | 4 | https://mathoverflow.net/users/406 | 271544 | 121,563 |
https://mathoverflow.net/questions/271543 | -1 | **Edit:** After the answer of Prof. Eremenko to the previous version, I realized that a weaker assumption works for the main motivation of this post. so I revise the question.
The unit disk in the plane is denoted by $\mathbb{D}=\{(x,y) \in \mathbb{R}^2\mid x^2+y^2 <1\}$
Assume that $f:\mathbb{D} \to \mathbb{R}$ i... | https://mathoverflow.net/users/36688 | Does this function belong to $L^2(\mathbb{D})$? | No, it does not. Take a curve tangent to $S$ from inside, for example an arc of a circle. Then take a narrow neighborhood $V$ of this curve. Then there exists a function, $C^\infty$ in the open disk, zero outside $V$ and growing as much as you want in $V$, so it does not belong to any $L^p$.
| 2 | https://mathoverflow.net/users/25510 | 271547 | 121,566 |
https://mathoverflow.net/questions/271523 | 2 | Sometimes we meet with situation when we don't know asymptotics of a given but we know its average behaviour on $[1,n]$. For example, we can easily get that
$\sum\_{i=1}^n \tau(n) = n \log (n) (1 + o(1))$
but we have only upper bound on $\tau(n)$ itself.
So my question is the next: is there some research (maybe... | https://mathoverflow.net/users/61438 | Asymptotics of number-theory functions and its averages | It is known that $\sum\_{x<n\le x+h(x)}\tau(n)$ is asymptotic to $h(x)\log x$ for $h(x)=x^{\theta}$ with $\theta>131/416$, and it is conjectured that the asymptotic holds for $h(x)\gg x^{\epsilon}$. See, e.g., Danilo Bazzanella, On the divisor function in short intervals, Arch Math 97 (2011) 453-458.
| 7 | https://mathoverflow.net/users/3684 | 271563 | 121,573 |
https://mathoverflow.net/questions/271570 | 1 | I conjecture following statement in the Banach space $X$ with norm $\|.\|$.
Let $B \subseteq X$ be the unit closed ball in $X$ then,
$\|.\|$ is smooth, in sense $\|.\|$ is differentiable at any nonzero point in $X$, if and only if for any point, say $ x\_0 \in \text{bd}(B)$ there is a unique $f \in X^\*$ such that ... | https://mathoverflow.net/users/108824 | Can $X^*$ characterize a smooth norm on $X$? | This is a known result even without any reflexivity assumption, see for instance [Smoothness and renormings in Banach spaces](https://books.google.de/books/about/Smoothness_and_renormings_in_Banach_spac.html?id=ZR_vAAAAMAAJ) by Deville, Godefroy and Zizler, Chapter I, Corollary 1.5. The proof there is based on a charac... | 4 | https://mathoverflow.net/users/85906 | 271572 | 121,576 |
https://mathoverflow.net/questions/271491 | 3 | It's well known that the simply-typed lambda calculus corresponds to a cartesian closed category. How would substructural type systems be characterized in category theory?
For example, linear type systems correspond to closed symmetric monoidal categories. But what category-theoretic constructions correspond to affi... | https://mathoverflow.net/users/110807 | Substructural types, the lambda calculus, and CCCs | Dan Licata, Mitchell Riley, and Mike Shulman have recently proposed a rather general answer to this, in *A Fibrational Framework for Substructural and Modal Logics* ([extended preprint](http://dlicata.web.wesleyan.edu/pubs/lsr17multi/lsr17multi-ex.pdf)).
This probably isn’t the main answer you want, for several reaso... | 1 | https://mathoverflow.net/users/2273 | 271574 | 121,577 |
https://mathoverflow.net/questions/271551 | 11 | Let $f$ be a modular form with $q$-coefficients $a\_n$, and let $L=\mathbf Q(a\_n:n\ge 0)$ be the Fourier coefficient field.
Does anyone know of any necessary or sufficient conditions for $L/\mathbf Q$ to be a Galois extension?
I know that by Momose, if $f$ is a non-CM newform with weight $\ge 2$ then you can embe... | https://mathoverflow.net/users/nan | When is a Fourier coefficient field Galois? | As Will Sawin points out, it is not true in general that the Hecke field of a modular form is a Galois number field. The first example in weight $2$ and level $\Gamma\_0(N)$ appears at $N=41$: the space $S\_2(\Gamma\_0(41))$ has dimension $3$ and is generated by the conjugates of a newform $f$ with Hecke field $L=\math... | 5 | https://mathoverflow.net/users/6506 | 271580 | 121,579 |
https://mathoverflow.net/questions/271561 | 1 | Let $f(x): \mathbf{C} \to \mathbf{C}$ be a complex valued polynomial of degree $n$, and suppose that for a random point $z$ sampled uniformly from the unit disk $|z|\leq 1$ we have
$$\mathbf{P}\_{z}( |f(z)| \leq \epsilon ) \geq 1 - n^{-k},$$
for some large positive $k>0$.
Under what conditions can we conclude that $|f(... | https://mathoverflow.net/users/36272 | Bounding function by random sampling | As pointed out by fedja, one can not expect the inequality yo hold with the same epsilon. However, the following holds : if $\mathbf{P}\_{z}( |f(z)| \leq \epsilon ) \geq 1 - \delta$, with $\delta \ll n^{-2} (\log n)^{-2}$, then
$$
\forall z \in \mathbb D, \ |f(z)| \leq \epsilon \left(1 + O\left(n \sqrt{\delta} \log \fr... | 2 | https://mathoverflow.net/users/21724 | 271582 | 121,581 |
https://mathoverflow.net/questions/271593 | 6 | Let $\mathcal{C}$ be a category and $\mathcal{W} \subseteq \text{Arr}(\mathcal{C})$ a set (or class) of arrows.
There are (at least) two notions of localization of $\mathcal{C}$ with respect to $\mathcal{W}$, a strict and a weak one.
In the strict one, a functor $\lambda: \mathcal{C} \rightarrow \mathcal{D}$ is a loc... | https://mathoverflow.net/users/3824 | Are localization functors always essentially surjective? | For any weak version you have a stong one where $\lambda$ is the identity on objects. Moreover, the target of a (weak) localization functor is essentially unique (up to equivalence). Essential surjectivity is invariant by equivalences, hence the answer to your question is positive.
| 6 | https://mathoverflow.net/users/12166 | 271595 | 121,585 |
https://mathoverflow.net/questions/271586 | 5 | In a [finite projective plane](https://en.wikipedia.org/wiki/Finite_geometry#Finite_projective_planes), each two points appear together in exactly one line, and each two lines intersect in exactly one point. It is known that, if each line contains $n+1$ points, then the total number of points and of lines is $n^2+n+1$.... | https://mathoverflow.net/users/34461 | Generalization of finite-projective-plane with more than one intersection point | Let me adjust notation slightly -- the $k$ in the original post is more usually a $\lambda$ in the literature. Thus the concept you want is this:
>
> **Definition.** A *symmetric $2-(v,k,\lambda)$ design* is a pair $(\Omega, \mathcal{B})$ where $\Omega$ is a set of size $v$ and $\mathcal{B}$ is a set of $k$-subsets... | 4 | https://mathoverflow.net/users/801 | 271600 | 121,587 |
https://mathoverflow.net/questions/271602 | 7 | Consider a central extension of groups $\mathcal{E}$ :
$$1\rightarrow A\rightarrow \widetilde{G}\rightarrow G \rightarrow 1$$with $A$ finite cyclic. Does there always exist a finite index subgroup $H$ of $G$ such that the restriction of $\mathcal{E}$ to $H$ becomes trivial?
| https://mathoverflow.net/users/40297 | Does a central extension split over some finite index subgroup? | No. Deligne gave a famous example of a central extension
$$
1\to\mathbb{Z}/2\to \widetilde{G}\to G\to 1
$$
such that $G$ is a finite-index subgroup of $PSp(2n,\mathbb{Z})$ but $\widetilde{G}$ is not residually finite. (See, for instance, the references in [this](https://mathoverflow.net/questions/48656/is-delignes-cent... | 7 | https://mathoverflow.net/users/1463 | 271605 | 121,588 |
https://mathoverflow.net/questions/271599 | 3 | Given $\theta\in [0,2\pi)$ and let $a,b$ be two nonnegative integers. Consider the following integral:
$$I=\int\_0^{\pi}\sin^{2a+1}x\sin^{2b+1}(x+\theta)dx.$$
Since
$$\int\_0^{\pi}\sin^mx\cos^{2n+1}xdx=0$$
for all nonnegative integers $m,n$, we have
\begin{eqnarray\*}
&&\int\_0^{\pi}\sin^{2a+1}x\sin^{2b+1}(x+\theta)dx... | https://mathoverflow.net/users/58096 | The sign of the following integral | Replace $x$ to $\pi-x$, write $2I=2\int\_0^\pi f(x)dx=\int\_0^\pi f(x)+f(\pi-x)dx$ and note that
1) the sign of $\sin(x+\theta)+\sin(\pi-x+\theta)=2\sin x\cos \theta$ is the same as the sign of $\cos \theta$;
2) the sign of $p^{2b+1}+q^{2b+1}$ is the same as the sign of $p+q$.
| 7 | https://mathoverflow.net/users/4312 | 271606 | 121,589 |
https://mathoverflow.net/questions/271607 | 3 | Let $A$ be a separable unital quasidiagonal $C^\*$-algebra.
What can be said about the $K$-theory of $A$, for example some properties? Especially, are there some criterions to decide whether or not $K\_\*(A)$ has torsion?
I appreciate any reference request in this direction. Thank you.
| https://mathoverflow.net/users/110864 | K-group properties of quasi-diagonal $C^*$-algebras | This is not necessarily an answer, but it was too long for a comment:
Note that for any separable unital $C^\*$-algebra $A$ its suspension $SA := C\_0(\mathbb{R}) \otimes A$ is quasidiagonal. This can be found as Corollary 7.3.7 in the (excellent) book "$C^\*$-algebras and Finite-dimensional Approximations" by Brown ... | 6 | https://mathoverflow.net/users/3995 | 271613 | 121,592 |
https://mathoverflow.net/questions/271597 | 2 | Let $p\_n$ denote the $n$'th prime number. Can anything be said about the existence of a continuous differentiable function $f(x)$ and a positive constant $D > 0$ such that $$\frac{f(p\_{n+1})}{f(p\_n)} < D\cdot \frac{f^{\prime}(p\_n)}{f(p\_n) \log(p\_n)}$$ for all $p\_n$ with $n > N$ where $N$ is an arbitrary fixed po... | https://mathoverflow.net/users/110322 | The existence of a function with application in finding bounded gaps between primes | As you observe, if $f$ and $f'$ are positive and increasing,
$$\frac{f(p\_{n+1})}{f'(p\_n)}>p\_{n+1}-p\_n.$$
The right hand side is not bounded (there exist large gaps between primes), hence the left hand side is not bounded either. This contradicts your assumption, even when you omit $\log(p\_n)$.
| 2 | https://mathoverflow.net/users/11919 | 271618 | 121,596 |
https://mathoverflow.net/questions/271603 | 6 | Let $K$ be a function field over $\mathbb{C}$, i.e. a finitely generated extension of $\mathbb{C}$ of transcendence degree 1. Suppose that $x, y \in K^\ast$ are such that $x + y = 1$. Then the Stothers-Mason Theorem states that if $x \not \in \mathbb{C}$ we have
$$
H\_K(x) \leq |S| + 2g\_K - 2,
$$
where $g\_K$ is the g... | https://mathoverflow.net/users/96891 | A variant of the Stothers-Mason Theorem | No.
Take the curve with equation $z^n = x (x-1)$ for $n$ a large odd number. This is a degree $n$ covering of $\mathbb P^1$, totally ramified over $x=0,x=1$, and $x=\infty$. Those $3$ points are the only places where the valuation of $x$ or $y$ is nonzero. Hence $|S|=3$
Furthermore, the map $z$ is a degree $2$ map ... | 8 | https://mathoverflow.net/users/18060 | 271632 | 121,600 |
https://mathoverflow.net/questions/271526 | 29 | For $n\geqslant m>1$, the integral $$I\_{n,m}:=\int\limits\_0^\infty\dfrac{\tanh^n(x)}{x^m}dx$$ converges. If $m$ and $n$ are both even or both odd, [we can use the residue theorem](https://math.stackexchange.com/a/1583085) to easily evaluate it in terms of odd zeta values, since the integrand then is a nice even funct... | https://mathoverflow.net/users/29783 | Is there a closed form for $\int_0^\infty\frac{\tanh^3(x)}{x^2}dx$? | Following the suggestion I made in a [comment](https://mathoverflow.net/questions/271526/is-there-a-closed-form-for-int-0-infty-frac-tanh3xx2dx#comment672036_271526), the integral can be rewritten as the contour integral
$$
I\_{3,2} = \frac{1}{2\pi i} \oint \frac{\operatorname{tanh}^3 z}{z^2} \log(-z) \, dz ,
$$
where... | 43 | https://mathoverflow.net/users/2622 | 271633 | 121,601 |
https://mathoverflow.net/questions/271584 | 3 | I am studying the general theory of moduli spaces of stable maps, in particuar of the moduli spaces $\overline{M}\_{0,n}(\mathbb{P}^r,d)$ of degree $d$ stable maps from a rational curve with $n$ marked points to $\mathbb{P}^r$.
I see that there are morphisms $f\_I:\overline{M}\_{0,n}(\mathbb{P}^r,d)\rightarrow \overl... | https://mathoverflow.net/users/nan | Linear systems on moduli spaces of stable maps | I am posting this as an answer because the comment thread is too long. A proper morphism of integral, Noetherian schemes, $f:X\to Y$, is a *contraction* if the sheaf homomorphism $f^\#:\mathcal{O}\_Y\to f\_\*\mathcal{O}\_X$ is an isomorphism. Typically we only talk about this when $X$, or at least $Y$, is normal, becau... | 1 | https://mathoverflow.net/users/13265 | 271637 | 121,603 |
https://mathoverflow.net/questions/271644 | 0 | Is there a closed form solution to the expression below? Or, if there is no closed form solution but the series converges, is there some upper bound on this expression?
$$\mathbb E\_{i \sim Q}[i] = \sum\_{i=k}^\infty i Q(i) = \sum\_{i=k}^\infty i h {i \choose {k-1}} h^{k-1} (1-h)^{i - (k-1)}$$
The assumptions on co... | https://mathoverflow.net/users/110890 | Closed form of $\sum_{i=k}^\infty i h {i \choose {k-1}} h^{k-1} (1-h)^{i - (k-1)}$? | Consider an infinite sequence of independent coin-tosses with probability $h$ of "heads" in each one. $h {i \choose k-1} h^{k-1} (1-h)^{i-(k-1)}$ is the
probability that in the first $i$ coin-tosses there are exactly $k-1$ heads and the $i+1$'th toss is also heads, i.e. that the $k$'th heads occurs on the $i+1$'th tos... | 2 | https://mathoverflow.net/users/13650 | 271647 | 121,606 |
https://mathoverflow.net/questions/267975 | 1 | Let $k$ be a field and $A$ a finitely generated algebra over $k$. I know that if $A$ is finite dimensional as a vector space over $k$, then $A$ can be decomposed as a product of indecomposable $k$-algebras: $$A=A\_1\times\ldots\times A\_n$$ My question is if there is a similar decomposition of an arbitrary $k$-algebra.... | https://mathoverflow.net/users/103420 | Decomposition of finitely generated algebras | As I cannot mark a comment as an answer, I'll follow the ideas of YCor and Benjamin Steinberg to answer this question.
Let $A$ an arbitrary noetherian ring, then the boolean algebra of idempotents of $A$, say $Idem(A)$, (I choose this way to use the suggestion of Benjamin) is noetherian. Moreover, this boolean algebr... | 1 | https://mathoverflow.net/users/103420 | 271659 | 121,612 |
https://mathoverflow.net/questions/271652 | 7 | Let $\xi : M^3 \to F$ be an orientable circle bundle over a closed orientable surface $F$ of genus $g \geq 2$. I am mostly interested to the case where the bundle $\xi$ is non-trivial. My question is about the mapping class group $\mathrm{MCG} (M) = \pi\_0(\mathrm{Diff}\_+(M))$. In particular, is this group generated b... | https://mathoverflow.net/users/23193 | Mapping class group of certain 3-manifolds | Since you write ${\rm Diff}\_+(M)$ you are probably assuming $M$ is orientable and diffeomorphisms of $M$ are orientation-preserving. Every diffeomorphism of $M$ can be isotoped to take fibers to fibers. This is proved using the assumption that the base surface $F$ has genus at least 2, so $M$ contains vertical incompr... | 13 | https://mathoverflow.net/users/23571 | 271668 | 121,618 |
https://mathoverflow.net/questions/271306 | 6 | I have the following questions on monoidal model structure(s) for the motivic stable homotopy category $SH(k)$ (where $k$ is a field); certainly, I am also interested in general statements concerning these matters.
1. Which "models" for $SH(k)$ are monoidal model categories (with the extra monoid axiom fulfilled) suc... | https://mathoverflow.net/users/2191 | More on categories of modules over the algebraic cobordism spectrum | For (1), I recommend the paper Motivic Functors by Dundas-Rondigs-Ostvaer. They compare many different models for SH(k), with an eye towards monoidal properties (i.e. pushout product axiom, monoid axiom, axiom that cofibrant objects are flat).
For (2), you probably know that these categories of modules are Quillen e... | 3 | https://mathoverflow.net/users/11540 | 271695 | 121,627 |
https://mathoverflow.net/questions/271694 | 3 | **Question**
Let $f,g\in L^1(\mathbb{R}^n)$ be nonnegative and assume $\int f\,g \; dx <\infty$. Let $\Phi\_\varepsilon$ be the heat kernel at time $\varepsilon$. Denote with $f\_\varepsilon = f \ast \Phi\_\varepsilon$ and $g\_\varepsilon = g \ast \Phi\_\varepsilon$ the convolved functions.
Is the functional $F(f,... | https://mathoverflow.net/users/13400 | Continuity of convolution on L¹ | No. Take $C=\bigcup\_{n\in\mathbb N} [1/(2n+1),1/(2n))$ and let $D=[0,1]\setminus C$. Now take take $f(x)=x^{-1/2}1\_C$ and $g(x)=x^{-1/2}1\_D$. Now $f\_\epsilon$ and $g\_\epsilon$ are both approximately $\epsilon^{-1/2}$ on $[0,\epsilon]$, so that there is no convergence of the integrals.
| 3 | https://mathoverflow.net/users/11054 | 271697 | 121,629 |
https://mathoverflow.net/questions/271614 | 2 | I have two questions on the 2-category of prederivators $\bf PDer$:
1. Does $\bf PDer$ "admit the construction of algebras" (ATCOA) in the sense of Street's Formal theory of monads? I recall that by ATCOA for a 2-category $\cal K$ I mean that the inclusion functor ${\cal K} \to {\bf Mnd}({\cal K})$ has a right 2-adjo... | https://mathoverflow.net/users/7952 | The formal theory of monads on $\bf PDer$ | The 2-category of prederivators is just the functor 2-category $[\mathrm{Dia}^{op},\mathrm{Cat}]\_p$ with pseudonatural transformations as morphisms. As long as the domain 2-category $\mathrm{Dia}$ is small with respect to the codomain $\mathrm{Cat}$, this is of the form $T$-$\mathrm{Alg}\_p$, i.e. the 2-category of st... | 3 | https://mathoverflow.net/users/49 | 271717 | 121,637 |
https://mathoverflow.net/questions/271713 | 2 | Let G=Gr(k,n) the Grassmannian of $k$-dimensional subspaces of $\mathbb{C}^n$ and denote by $T$ the (rank $k$) tautological bundle over $G$, and by $Sym^p T$ its $p$-th symmetric power. Is there any known formula for computing the exterior powers $$\wedge^q(Sym^pT)$$ in terms of Schur powers $\Sigma^{\lambda}T$?
I am... | https://mathoverflow.net/users/52811 | Exterior powers of $Sym^p T$ over Gr(k,n) | For $p=2$, the plethysm is known. A reference is Macdonald's book "Symmetric Functions and Hall Polynomials" (p.138, example 6d in the second edition).
$\wedge^\bullet( Sym^2) = \bigoplus\_\lambda \Sigma^\lambda$ where the sum is over all $\lambda$ with the following property: for each box on the main diagonal of its... | 4 | https://mathoverflow.net/users/321 | 271719 | 121,638 |
https://mathoverflow.net/questions/271709 | 4 | Wang tiles are interesting in that they can simulate Turing machines. My question is whether anyone has studied their game theoretic properties?
In particular, we could imagine a game in which you have a plane with some Wang tiles on it, and players take turns placing tiles adjacent to tiles already on the board. The... | https://mathoverflow.net/users/65915 | Game theoretic aspects of Wang tiles? | Perhaps the key search phrase is "domino-tiling games." This 1986 paper
seems to be a source, subsequently cited ~100 times.
>
> Chlebus, Bogdan S. "Domino-tiling games." *Journal of Computer and System Sciences* 32.3 (1986): 374-392. [Journal link](http://www.sciencedirect.com/science/article/pii/002200008690036X)... | 6 | https://mathoverflow.net/users/6094 | 271722 | 121,639 |
https://mathoverflow.net/questions/271691 | 3 | Let $A$ be a C\*-algebra. According to operator algebraists, it is well known that $A$ embeds into the atomic part of its double dual in the following sense: if $z$ is the central projection in $A^{\*\*}$ onto its atomic part, then the composition of the embedding $A \hookrightarrow A^{\*\*}$ with multiplication by $z$... | https://mathoverflow.net/users/46472 | Reference request for c*-algebra embedding into the atomic part of its double dual | If you are only interested in seeing the statement written down in the literature, then see the first paragraph of the paper *[A Gelfand representation theory for C$^\*$-algebras](http://dx.doi.org/10.2140/pjm.1971.39.1)* by C. Akemann. This in turn refers to Dixmier's text *Les C$^\*$-algebres et leurs représentations... | 3 | https://mathoverflow.net/users/778 | 271723 | 121,640 |
https://mathoverflow.net/questions/271724 | 6 | I hope this question is not too trivial for mathoverfolw.
Let $R$ be a commutative ring (with $0\neq 1$) and $D\_{Perf}(R)$ the triangulated category of perfect complexes. Let $C$ be a thick subcategory of $D\_{Perf}(R)$
is it true that there exists a (commutative ?) ring $A$ such that $C$ is equivalent to $D\_{perf}... | https://mathoverflow.net/users/110929 | Thick subcategories | The answer is no, in principle. By Thomason's classification of thick subcategories, these correspond to certain stable for specialization subsets, i.e. arbitrary unions of closed subsets (with quasi-compact complement).
Even for a closed subset $Z \subset \mathrm{Spec}(R)$ with quasi-compact complement the correspo... | 6 | https://mathoverflow.net/users/6348 | 271726 | 121,641 |
https://mathoverflow.net/questions/271725 | 2 | Let $U$ be the infinite unitary group $\lim\_{n\to\infty}U(n)$. It is well known that, over the rationals, $BU$ is homotopy equivalent to $\prod\_{n=1}^\infty K(\mathbb{Z}, 2n)$.
**Question:** Is it true that, for any $k\geq 0$, the homotopy group $\pi\_{2k+1}(BU(m))$ is 0 for $m$ sufficiently large?
| https://mathoverflow.net/users/85722 | Odd homotopy group of $BU(m)$ | From the [long exact sequence of homotopy groups](https://en.wikipedia.org/wiki/Homotopy_group#Long_exact_sequence_of_a_fibration) associated to the fibration $U\to EU\to BU$, one has that $\pi\_{2k+1}(BU(m))=\pi\_{2k}(U(m))$ (since $EU$ is contractible, and for $k>0$). [From the proof of](https://ncatlab.org/nlab/show... | 7 | https://mathoverflow.net/users/1345 | 271727 | 121,642 |
https://mathoverflow.net/questions/271650 | 7 | **Preliminaries:**
Let $[\omega]^{\omega}$ be the set of all infinite subsets of $\omega$, the first countable ordinal (the set of the natural numbers).
* We say that $\mathcal A\subset [\omega]^{\omega}$ is open if for all $A, B \in [\omega]^{\omega}$ then if $A\subset^\*B\in \mathcal A$ then $A\in \mathcal A$.
*... | https://mathoverflow.net/users/nan | Existence of a specific mad family | You could define $\mathcal{A}$ recursively:
Enumerate all elements of $[2^{<\omega}]^\omega$ as $(X\_\alpha)\_{\alpha< \mathfrak{c}}$. Then whenever elements $(A\_{\alpha})\_{\alpha<\gamma}$ have already been constructed, check if $X\_\gamma$ is almost disjoint from them. If not, skip step $\gamma$. If yes then find... | 1 | https://mathoverflow.net/users/103802 | 271743 | 121,646 |
https://mathoverflow.net/questions/271720 | 4 | Let $X$, $Y$ be smooth, connected, compact manifolds (for instance, projective varieties) and $f \colon X \longrightarrow Y$ be a finite, branched cover of degree $n$, with branch locus $B \subset Y$. We can then associate to $f$ its monodromy representation $$\theta\_f \, \colon \pi\_1(Y-B) \longrightarrow S\_n,$$ so ... | https://mathoverflow.net/users/7460 | Monodromy representation of elementary simple covers | I am just writing my comments as an answer. For every subgroup $H$ of the symmetric group $\mathfrak{S}\_n$, define a relation on $\{1,\dots,n\}$ by $a\sim b$ if either $a$ equals $b$ or if the transposition $(a,b)$ is contained in $H$. By definition, this is symmetric and reflexive. Let $a,b,c$ be pairwise distinct el... | 6 | https://mathoverflow.net/users/13265 | 271744 | 121,647 |
https://mathoverflow.net/questions/271731 | 4 | Let $p$ be a prime and let $M\_p$ be the $\mathrm{GL}\_2(\mathbb{F}\_p)$-module of $2 \times 2$ matrices over $\mathbb{F}\_p$ with trace $0$ (the action is by conjugation).
Is it true that for $p$ large enough we have $H^2(\mathrm{GL}\_2(\mathbb{F}\_p), M\_p) \neq \{0\}$ ?
| https://mathoverflow.net/users/38889 | Second cohomology of the adjoint representation | Yes it has dimension $1$ for $p \ge 5$.
This module is actually the natural module for ${\rm SO}(3,p)$, which is isomorphic to ${\rm PGL}(2,p)$ (the scalars of ${\rm GL}(2,p)$ act trivially), so the result is probably in the literature somewhere.
But it's easy to do it by direct calculation. A Sylow $p$-subgroup $P... | 9 | https://mathoverflow.net/users/35840 | 271751 | 121,651 |
https://mathoverflow.net/questions/271649 | 4 | Let
$$\psi(x;q,a)=\sum\_{n\leq x\atop n\equiv a\pmod q}\Lambda(n)$$
where $\Lambda$ denotes the von Mangoldt function and $\phi$ to be Euler's totient function.
Then the Siegel-Wafisz theorem states that given any real number $N$ there exists a positive constant $C\_N$ depending only on $N$ such that
$$\psi(x;q,a)=... | https://mathoverflow.net/users/84272 | Siegel-Walfisz Theorem with smooth weights | Smoothing often removes technical issues in a problem, but fundamental obstructions cannot be removed merely by smoothing. It may be helpful to consider some examples.
1. The divisor problem. Here one asks for the error term with a sharp cutoff $\sum\_{n\le x} d(n)$, and this error is supposed to be $O(x^{\frac 14+\... | 10 | https://mathoverflow.net/users/38624 | 271777 | 121,660 |
https://mathoverflow.net/questions/271774 | 3 | Let $X$ be a set equipped with the discrete topology and consider the space $C\_c(X)$ of complex valued functions of finite support. It comes with a natural locally convex inductive limit topology. Is this space reflexive or not?
| https://mathoverflow.net/users/nan | Is $C_c(X)$ reflexive? | Yes, it is. Indeed,
1º) As inductive limit over finite sets $K\subset X$ of the finite-dimensional vector spaces $C\_c(K)$, on which *every* seminorm is continuous, $C\_c(X)$ has the topology defined by *all* seminorms on it, a.k.a. the *finest locally convex topology* (see e.g. [Robertson & Robertson 1964](http://ww... | 5 | https://mathoverflow.net/users/19276 | 271782 | 121,663 |
https://mathoverflow.net/questions/271711 | 12 | The [Sylvester–Gallai theorem](https://en.wikipedia.org/wiki/Sylvester%E2%80%93Gallai_theorem) in geometry states that, given a finite number of points in the Euclidean plane, either
1. All the points are collinear; or
2. There is a line which contains exactly two of the points.
I think the theorem is also true for... | https://mathoverflow.net/users/88319 | Sylvester–Gallai theorem with circle version, plane version and curve version? | The plane version is false as stated. Choose two skew lines $\ell\_1$ and $\ell\_2$ in $\mathbb{R}^3$ and place $\geq 3$ points on each of them. Any three of these points will either all lie on the same $\ell\_i$ (and thus be collinear) or two of them will lie on the same $\ell\_i$, so the plane they span contains that... | 7 | https://mathoverflow.net/users/297 | 271794 | 121,666 |
https://mathoverflow.net/questions/271790 | 4 | Let $G$ be a semisimple group over $\mathbb C$, and $X=G/H$ be a spherical homogeneous space of $G$.
Let $T\subset B\subset G$ be a maximal torus and a Borel subgroup.
Let $S=S(G,T,B)$ denote the corresponding set of simple roots.
Let ${\mathcal{P}}(S)$ denote the set of subsets of $S$.
Let $M$ denote the weight latt... | https://mathoverflow.net/users/4149 | Action of $N(H)/H$ on the colors of a spherical homogeneous space $G/H$ | The answer to both questions is "yes" as was communicated to me by Losev.
The main reason is that already $\varsigma$ is almost injective. More precisely, the sets $\varsigma(D)$ are always non-empty and they are pairwise disjoint except for a very well controlled situation.
Assume $I:=\varsigma(D\_1)\cap\varsigma(... | 3 | https://mathoverflow.net/users/89948 | 271795 | 121,667 |
https://mathoverflow.net/questions/271793 | 9 | I have seen many answers to the converse question (which seems to be difficult in general), but I would like to ask the following:
Let $T: L^2 \rightarrow L^2$ be a trace-class operator that is also an integral operator
$$Tf = \int K(\cdot,y)f(y)dy.$$
Since $T$ is trace-class $\operatorname{tr}(T)$ exists. Now, ... | https://mathoverflow.net/users/nan | Trace of integral trace-class operator | As js21 says in the comnent, this is always the case if we understand a restriction $K(x,x)$ of the kernel $K(x,y)$ onto diagonal in appropriate way. In general, a square-summable function of two variables does not have a well-defined restriction to sets of zero measure, like diagonal. But the kernels of nuclear operat... | 6 | https://mathoverflow.net/users/4312 | 271798 | 121,668 |
https://mathoverflow.net/questions/271801 | 21 | Now that [James Davis](http://www.popularmechanics.com/science/math/news/a26815/why-13532385396179-is-a-magic-number/) has found a counter example, [13532385396179](https://oeis.org/A195264), to John Conway's [climb-to-a-prime conjecture](http://gladhoboexpress.blogspot.co.at/2014/10/climb-to-prime.html), I would be in... | https://mathoverflow.net/users/11260 | Implications of the disproof of the "climb-to-a-prime" conjecture | Hans said that Conway's point in asking it was that there exist problems easy to state but impossible to prove. The point I took away was that there exist problems that look so hard, nobody has tried anything easy.
Read the letter to Conway on Numberphile if you want to see how it was easy. I'll ask Conway if it has... | 46 | https://mathoverflow.net/users/110974 | 271804 | 121,671 |
https://mathoverflow.net/questions/271820 | 3 | Let $K$ be a field, let $L$ be a field containing $K$, and let $G$ be a reductive group over $K$. Does there always exist a torus $T$ of $G$ so that $T\_{/L}$ is a maximal split torus of $G\_{/L}$? If such a torus does not exist in general, are there assumptions on $K$ and $L$ that guarantee that such a torus does exis... | https://mathoverflow.net/users/93798 | Does a reductive group over $K$ always have a torus that becomes maximal split over $L$? | Suppose that
* $K = \mathbb Q\_5$, say;
* $L$ is a cubic, totally ramified extension of $K$; and
* $G$ is the algebraic group underlying the multiplicative group of a cubic division algebra over $K$.
Then $G$ is $L$-split. On the other hand, because the splitting field of a torus in $G$ is a Galois extension of $K$... | 3 | https://mathoverflow.net/users/2383 | 271825 | 121,681 |
https://mathoverflow.net/questions/271814 | 3 | Given two algebraic varieties $X$ and $Y$, the Künneth theorem implies that there is a relation between $H^\*(X) \otimes H^\*(Y)$ and $H^\*(X \times Y)$, and in fact in many cases they are equal.
Given an open subvariety $U \subseteq X \times Y$ and a closed subvariety $Z \subseteq U$, is there a way to extend this a... | https://mathoverflow.net/users/37861 | "Künneth bigrading" for subsets of $X \times Y$? | In the topology case, it is the Kunneth theorem for excision pair.In the algebraic geometry(etale cohomology) case , it can be reduced to prove that for $Z\_1$,$Z\_2$(if $Z=Z\_1\times Z\_2)$and sheaf $Ri\_1^! F$and $Ri\_2^!G$ since $R\Gamma(Z,Ri^!F)=R\Gamma(X,R\Gamma\_ZF)$.Kunneth theorem needs the projection formula a... | 3 | https://mathoverflow.net/users/85353 | 271827 | 121,682 |
https://mathoverflow.net/questions/271800 | 5 | Suppose $f: \widetilde{X} \to X$ is a finite dominant morphism between connected, normal, Noetherian schemes, and that this morphism induces a dominant morphism $f\_W: \widetilde{W} \to W$ between connected normal subschemes of $\widetilde{X}$ and $X$, respectively.
**My question is:** can we bound $\deg(f\_W)$ in t... | https://mathoverflow.net/users/39777 | Does the degree of a finite dominant morphism bound the induced degree on subschemes? | In your case, it is bounded. Notice that, it suffices to show that for any point $p\in X$, the cardinality of $f^{-1}(p)$ is bounded. You can replace $f$ by a separable map, since purely inseparable maps are bijection on points. Then, you can replace $\widetilde{X}$ by the Galois closure, say $Z$. Thus you are reduced ... | 5 | https://mathoverflow.net/users/9502 | 271828 | 121,683 |
https://mathoverflow.net/questions/271789 | 10 | this is my first question here! Hopefully it is appropriate. Let $\mathbb{A}$ be the punctured plane, i.e. the 'standard' annulus. For compact, connected subsets of the plane (planar continua) $X \subset \mathbb{R}^2$, I know it's not necessarily true that $\mathbb{R}^2 \setminus X \simeq \mathbb{A}$ (homeomorphism), f... | https://mathoverflow.net/users/110965 | Complements of Simply Connected Subsets of the Plane | Moore's theorem says that if $\sim$ is an equivalence relation on $\mathbb{S}^2$ such that any equivalence class is closed connected and has connected complement then the quotient space $\mathbb{S}^2/\sim$ is homeomorphic to $\mathbb{S}^2$.
In particular the answer to your first question is "yes".
| 5 | https://mathoverflow.net/users/1441 | 271837 | 121,688 |
https://mathoverflow.net/questions/271770 | 7 | Let $H$ be a $n \times n$ real symmetric matrix that has eigenvalues with absolute value less than 1. Define the matrix $M = \prod\_{i=1}^n (I - e\_ie\_i^{\top}H)$ where $e\_i$ denotes the $i^\text{th}$ canonical basis vector of $\mathbb{R}^n$.
Assuming that $H$ has at least one negative eigenvalue, is it true that $... | https://mathoverflow.net/users/nan | Bound the eigenvalue of product of matrices? | OK, looks like I've got it. Since it is pretty late here now, it would be nice if someone could check that I haven't written some nonsense (I apologize in advance if I have).
Let $N(H)$ be the kernel of $H$, let $Q$ be the orthogonal projector to $N(H)$ and let $P$ be the orthogonal projector to $N(H)^\perp$. Note th... | 7 | https://mathoverflow.net/users/1131 | 271847 | 121,693 |
https://mathoverflow.net/questions/271823 | 4 | Consider a set of $n$ red lines and $m$ blue lines, suppose there are $nm$ distinct red-blue intersections.
What is the minimum number of lines $L\_1,L\_2,\dots, L\_n$ such that the union contains all $nm$ intersection points except for exactly one?
A trivial construction is to take $n-1$ red lines and $m-1$ blue l... | https://mathoverflow.net/users/24478 | Covering all except one of the purple intersection points of $n$ red and $m$ blue lines efficiently | The very nice paper "Cayley-Bacharach theorems and conjectures" by David Eisenbud, Mark Green, and Joe Harris (<http://www.ams.org/journals/bull/1996-33-03/S0273-0979-96-00666-0/home.html>) gives an introduction to some related theory, including in particular a theorem which they number Theorem CB4: If $X\_1$ and $X\_2... | 3 | https://mathoverflow.net/users/88133 | 271849 | 121,695 |
https://mathoverflow.net/questions/271490 | 3 | Can every càdlàg semi-martingale be written as a sequence of diffusions? That is, is the set of continuous semi-martingales dense in some Skorohod space?
| https://mathoverflow.net/users/36886 | Sequence of diffusions | $C=C[0,1]$, the space of continuous functions on $[0,1]$, is a closed subset in Skorohod space $D=D[0,1]$
(cf. P. Billingsley, *Convergence of Probability Measures*, 2nd ed.(1999) $\S\!\!12$, p.124), hence the set of continuous semi-martingales
**could not be dense** in Skorohod space $D=D[0,1]$.
| 1 | https://mathoverflow.net/users/103256 | 271858 | 121,698 |
https://mathoverflow.net/questions/271732 | 3 | A vector bundle $E$ on a smooth projective variety $X$ is called Ulrich bundle if it is Arithmetically Cohen-Macaulay , i.e., $H^i(E(t)) = 0 $ for all $t \in Z$ and $0 < i < k$ and with Hilbert polynomial $cr\binom{t+k}{k}$ for some linear projection $X\to \mathbb P^k$
The following statement appeard in the survey pa... | https://mathoverflow.net/users/110932 | Semi-stability of Ulrich bundle | Adding my comments as answer
From the definition of Ulrich bundle, for some linear projection $\pi : X → \mathbb P^k$ we have $π\_∗E \cong \mathcal O\_{\mathbb P^k}^{cr}$ so its direct image is Mumford semi-stable(since it is trivial) and if a direct image of vector bundle be Mumford semi-stable then that vector bund... | 4 | https://mathoverflow.net/users/nan | 271859 | 121,699 |
https://mathoverflow.net/questions/271841 | 41 | (This is a restatement of a [question asked on the Mathematics.SE](https://math.stackexchange.com/questions/1700734/find-a-succinct-problem-whose-solution-requires-methods-from-many-sub-branches-o), where the solutions were a bit disappointing. I'm hoping that professional mathematicians here might have a better soluti... | https://mathoverflow.net/users/89654 | What problem in pure mathematics required solution techniques from the widest range of math sub-disciplines? | The proof of the [Ramanujan conjecture](https://en.wikipedia.org/wiki/Ramanujan%E2%80%93Petersson_conjecture) by Deligne. It uses:
* number theory
* algebraic geometry
* topology
* representation theory
* commutative algebra
* complex analysis
| 36 | https://mathoverflow.net/users/37214 | 271863 | 121,701 |
https://mathoverflow.net/questions/271776 | 3 | An ordered pair $\ \mathbf X := (X\ d)\ $ is called a chain group $\ \Leftarrow:\Rightarrow\ X\ $ is an abelian group, $\ d:X\rightarrow X\ $ is an abelian group endomorphism, and $\ d\circ d= 0$.
A chain homomorphism $\ f:\mathbf X\rightarrow \mathbf X'\ $ of chain groups $\ \mathbf X := (X\ d)\ $ and $\ \mathbf X' ... | https://mathoverflow.net/users/110389 | Classification of finitely generated chain groups | The general classification problem seems to be a [wild](https://mathoverflow.net/questions/10481/when-is-a-classification-problem-wild/10484#10484) problem (i.e., contains the classification of pairs of square matrices over a field up to simultaneous conjugacy as a subproblem), and so is probably intractable.
Fix a p... | 3 | https://mathoverflow.net/users/22989 | 271866 | 121,702 |
https://mathoverflow.net/questions/271865 | 4 | You roll a dice with $s$ sides for $n$ times. What is the probability that $t$ sides appeared once?
Example with $s=2$, $n=3$: you roll a dice with 2 sides for 3 times.
possible outcomes: '1' and '2'.
* $p(t=0)$ is the probability that '1' appeared 0 times and '2' appeared 3 times + the probability that '2' appeare... | https://mathoverflow.net/users/110942 | What is the probability that you roll a dice with s sides for n times and t sides appeared once? | This is a variation of the combinatorial problem considered in section 5 of [Some new aspects of the coupon collector’s problem](http://www.brynmawr.edu/math/people/anmyers/PAPERS/Coupons.pdf) (2003).
The $t$ singleton sides (sides which appear once) can be chosen as an ordered sequence in $t! {s\choose t}$ ways; thi... | 3 | https://mathoverflow.net/users/11260 | 271870 | 121,703 |
https://mathoverflow.net/questions/271868 | 3 | I saw the following theorem in a very old paper of Bendixson. Does anybody know a shorter and beautiful proof of that?
**Theorem**. If $A$ is a real matrix, then for each of its eigenvalues $(\lambda)$, the following inequality holds:
$ m \leq Re(\lambda) \leq M $,
where $m$ and $M$ are the minimum and maximum ei... | https://mathoverflow.net/users/111007 | Relation between real part of eigenvalues of $A$ and $(A+A^{T})/2$ | Replacing $A$ to $A-m$ or to $M-A$ we may reduce to the following partial case: $(A+A^T)/2$ is non-negative definite, and we have to prove that $a\geqslant 0$ for any eigenvalue $\lambda=a+bi$ of $A$. We have $Av=(a+bi)v$ for a certain complex eigenvector $v$. Therefore $0\leqslant ((A+A^T)v,v)=(Av,v)+(v,Av)=(a+bi)(v,v... | 11 | https://mathoverflow.net/users/4312 | 271875 | 121,705 |
https://mathoverflow.net/questions/271864 | 3 | The elliptic curve $y^2=x^3+6x^2+x$ is associated with the Rational Distance problem, which asks whether there exists a point in the plane, that is at rational distances from the four vertices of the square (with rational side). This particular elliptic curve encodes the existence of rational points on the line bisecti... | https://mathoverflow.net/users/111001 | Does there exist a rational point on the elliptic curve: $y^2=x^3+6x^2+x$ ? If yes, how to find one? (relations to the 'rational distance problem') | Here is the $2$-descent, following Silverman, *The Arithmetic of Elliptic Curves*, Proposition X.4.9 and Example X.4.10.
Let $E: y^2 = x^3 + 6x^2 + x$ and $E': Y^2= X^3-12X^2+32X$ and $\phi: E \to E', (x,y) \mapsto(y^2/x^2,y(1-x^2)/x^2)$. This has the $\mathbf{Q}$-rational $2$-torsion point $(0,0)$. Then in Silverman... | 5 | https://mathoverflow.net/users/nan | 271876 | 121,706 |
https://mathoverflow.net/questions/271900 | 4 | Let $p$ be a prime and consider the field $\mathbb{F}\_p$. Fix $f\in\mathbb{F}\_p[X]$ a polynomial of degree $d\ge 2$. Define
$$
K(x,y)=\frac{1}{\sqrt{p}}\sum\_{z\in\mathbb{F}\_p}e\_p(xz+yf(z)),
$$
where $e\_p(t):=e^{2\pi i \frac{t}{p}}$.
It is well-known that if $p\nmid d$, then $|K|$ is bounded by $1$ (up to some c... | https://mathoverflow.net/users/4519 | estimate for a sum of products of Weil's sum | Here is a different approach than Denis Chaperon de Lauzières's.
Opening everything and using orthogonality of characters to remove the $x$ variable, we see that your sum is $1/p$ times $$\sum\_{z\_1,z\_2,z\_3,z\_4 \in \mathbb F\_p, z\_1-z\_2-z\_3+z\_4 =0 } e\_p \left( y f(z\_1) + t z\_2 - (y+t) f(z\_2) - y' f(z\_3) ... | 6 | https://mathoverflow.net/users/18060 | 271903 | 121,717 |
https://mathoverflow.net/questions/210730 | 3 | Let $C(x\_1,\ldots,x\_n)$ be a nonsigular cubic form with integral coefficients.
In his proof that $C$ fulfills the Hasse-Principle, if $n\geq 9$, Hooley used the following estimate that was provided by Katz in *Perversity and Exponential Sums*:
There exists a subset $\mathcal{P}$ of primes, having positive Dirichlet... | https://mathoverflow.net/users/75661 | Twisting by a multiplicative Character in Katz, Perversity and Exponential sums | First, an elementary manipulation of your sum. $\sum\_{t \mod w} \chi(t) e\_w (t C(\mathbf{x}))$ is some fixed Gauss sum times $\chi^{-1} ( C(\mathbf{x}))$.
You want to take in the notation of Katz's Section 4:
$R = \mathbb Z$, $S = \operatorname{Spec} \mathbb Z$, $r= n$, $E = \mathbb Z^{n}$, $X = \mathbb A^n$, $V=... | 3 | https://mathoverflow.net/users/18060 | 271910 | 121,721 |
https://mathoverflow.net/questions/271884 | 1 | Let $H=(V,E)$ be a $k$-uniform hypergraph with $n$ vertices, that is, $V:=V(H)$ is a $n$-element finite set of vertices and $E:=E(H)\subset\binom{V}{k}$ is a family of $k$-element subsets of $V$.
Given $d$ vertices $v\_1,v\_2,\ldots,v\_d\in V(H)$, $1\leq d\leq k-1$, we denote by $deg\_H(v\_1,v\_2,\ldots,v\_d)$ the d... | https://mathoverflow.net/users/81507 | Inequality about the minimum vertex degree in $k$-uniform hypergraphs | Double counting (as the form of the answer more or less says it has to be). Fix a $(d-1)$-set $X$; sum over vertices $v\not\in X$ the number of edges containing $X$ and $v$. You obviously count every edge containing $X$, and each edge is counted $k-d+1$ times. Since the sum is over $n-d+1$ vertices not in $X$, plugging... | 3 | https://mathoverflow.net/users/36212 | 271911 | 121,722 |
https://mathoverflow.net/questions/271938 | 3 | Let $G$ be a connected, reductive Lie group, and $W\mathfrak g = (S[\mathfrak g^\vee] \otimes \Lambda[\mathfrak g^\vee],\delta)$ the associated Weil algebra. This is a CDGA equipped with an action of $\mathfrak g$ by interior multiplication $\iota$ and a resulting action $\mathcal L$ of $\mathfrak g$ by Lie derivatives... | https://mathoverflow.net/users/5792 | Show the Cartan 3-form transgresses to the Killing form in the Weil algebra | This is the standard Chern-Simons calculation. Let me use gauge-theoretical notation and write $A$ and $F$ for the odd and even, respectively, generators of the Weyl algebra. If we choose canonical dual bases $\theta^a$ for $\mathfrak{g}^\*$ and $X\_a$ for $\mathfrak{g}$, then
$$ A = A^a X\_a \qquad\text{and}\qquad F =... | 1 | https://mathoverflow.net/users/394 | 271940 | 121,730 |
https://mathoverflow.net/questions/270668 | 2 | I am currently reading this [paper](https://arxiv.org/pdf/1704.01224.pdf) where the author classifies the pretzel links up to link homotopy using a quasi-trivial quandle $\mathbb{Z}\_{k}[t^{\pm 1}]\diagup\_{(t-1)^{2}}$, and I find it difficult to understand the proof of lemma 5.6. Can anyone help with this proof?
| https://mathoverflow.net/users/110173 | Classification of pretzel links up to link homotopy using alexander quandle |
>
> Beginning with coloring the $k$th box using
> ${\mathbb{Z}\_k[t^{\pm1}]}/\_{(t-1)^2}$, we conclude that if the link is
> homotopically trivial, then $p\_k$ divides all $p\_i$'s.
>
>
>
It seems that there is a typo here: "${\mathbb{Z}\_k[t^{\pm1}]}/\_{(t-1)^2}$" should be "${\mathbb{Z}\_{p\_k}[t^{\pm1}]}/\_... | 2 | https://mathoverflow.net/users/15770 | 271941 | 121,731 |
https://mathoverflow.net/questions/271943 | 8 | Let $B$ be an $n\times n$ matrix, and define $f$ to be the function that maps positive semidefinite (PSD) $n\times n$ matrices $A$ to real numbers by
$$
f(A) = \mathrm{trace}( (B^\*A^2B)^{1/3}).
$$
In other words, $f$ maps $A$ to the sum of $1/3$-powers of the eigenvalues of the PSD matrix $B^\*A^2B$.
>
> Is ... | https://mathoverflow.net/users/35733 | Concavity of the trace of a matrix power | Unfortunately, the conjectured function is not concave. Here is a ~~simple~~ *simpler* counterexample.
\begin{equation\*}
B = \begin{bmatrix} 1 & 2 \\ 3 & 4\end{bmatrix},\quad
A = \begin{bmatrix} 2 & 0 \\ 0 & 3\end{bmatrix},\quad
C = \begin{bmatrix} 5 & 0 \\ 0 & 2\end{bmatrix}.
\end{equation\*}
With this choice,... | 10 | https://mathoverflow.net/users/8430 | 271945 | 121,732 |
https://mathoverflow.net/questions/271931 | 1 | A doubly regular tournament is a tournament such that every two vertices have $j$ common out-neighbours. How can we prove such a tournament is $2j+1$-regular?
| https://mathoverflow.net/users/111007 | How to prove that doubly regular tournaments are regular? | Let $v$ be any vertex, $U$ be a set of its out-neighbours. The restriction of your tournament to $U$ is $j$-out-regular, thus $|U|=2j+1$.
| 1 | https://mathoverflow.net/users/4312 | 271947 | 121,733 |
https://mathoverflow.net/questions/271784 | 5 | It is another question about the category $\mathcal{D}\mathcal{K}$ of all small diagrams over all small categories, its definition is here : [About the category of all small diagrams](https://mathoverflow.net/q/266597). I suppose $\mathcal{K}$ locally $\lambda$-presentable.
>
> Question: Take a diagram $F:I\to \ma... | https://mathoverflow.net/users/24563 | Presentable small diagrams over a locally presentable category | Maybe I have this wrong, but I think that in $\mathcal{DK}$, filtered colimits at least are constructed in the following way. Let $I$ be a $\lambda$-directed poset for simplicity, and let $G: I \to \mathcal{DK}$, $i \mapsto G\_i: D\_i \to \mathcal{K}$ be a functor, with transition maps $(D\_{ii'}: D\_i \to D\_{i'}, \ga... | 2 | https://mathoverflow.net/users/2362 | 271951 | 121,735 |
https://mathoverflow.net/questions/271950 | 11 | The bound in Merel's solution to the Uniform Boundedness conjecture is not explicit, as it relies on Falting's work on the Mordell conjecture. I think this still is the case.
But there are known explicit bounds for the largest prime divisor. The best one seems to be $(1+3^{d/2})^2$, where $d$ is the degree of the num... | https://mathoverflow.net/users/43108 | Oesterlé's unpublished bound on Uniform Boundedness | Yes, this is published as appendix A to chapter 3 in Derickx' PhD thesis available here: <https://openaccess.leidenuniv.nl/handle/1887/43186> . The thesis contains, of course, many more interesting results.
| 11 | https://mathoverflow.net/users/5015 | 271952 | 121,736 |
https://mathoverflow.net/questions/217888 | 4 | Let $\mathcal{G}$ denote a (stable) tangential structure such as $O$, $SO$, $Spin$, or $Pin^\pm$. Which bordism classes $[M,f]\in\Omega\_\*^\mathcal{G}(X)$ are represented by an $f:M\rightarrow X$ where the $\mathcal{G}$-manifold $M$ fibers over $S^1$?
| https://mathoverflow.net/users/51107 | Which bordism classes fiber over the circle? | A manifold $M$ fibres over $S^1$ with fibre $F$ if and only if it is isomorphic to the mapping torus $$T(h)=F \times [0,1]/\{(x,0) \sim (h(x),1)\vert x \in F\}$$ of an automorphism $h:F \to F$. Mapping tori are particular examples of open books. Walter Neumann's result for $G=SO$ extends to arbitrary $X$ with the signa... | 10 | https://mathoverflow.net/users/732 | 271954 | 121,737 |
https://mathoverflow.net/questions/271968 | 7 | I am intrigued by the notion of dual connections: two affine connections $\nabla$ and $\nabla^\*$ are called dual if they satisfy $$X(g(Y,Z))=g(\nabla\_XY,Z)+g(Y,\nabla^\*\_XZ)$$
for a given (pseudo)-riemannian metric $g$.
>
> What is the motivation and the deep results behind this notion?
>
>
> What are the main... | https://mathoverflow.net/users/56980 | Motivations for the study of dual connections | There may be many motivations to study dual (or maybe rather: adjoint) connections. Let me give one. The metric $g$ identifies a vector bundle $V$ and its dual $V^\*$. Any connection $\nabla$ on $V$ induces one on $V^\*$, and the pullback of this by $g$ is the dual connection. They agree if and only if $\nabla$ and $g$... | 10 | https://mathoverflow.net/users/70808 | 271982 | 121,748 |
https://mathoverflow.net/questions/271986 | 0 | Following situation: If $\mathcal{O}\_\infty$ is the Cuntz algebra in infinitely many generators and $\mathcal{K}$ the compact operators on a separable Hilbert space, let $v\in \mathcal{O}\_\infty\otimes \mathcal{K}$ be a partial isometry.
Questions:
1. Just to make shure: $v(\mathcal{O}\_\infty\otimes \mathcal{K})v... | https://mathoverflow.net/users/110864 | Is there an embedding $v(\mathcal{O}_\infty\otimes \mathcal{K})v^*\to \mathcal{O}_\infty$? | The answers to both questions is Yes.
For the first one, recall that a partial isometry satisfies the equation $v=vv^\*v$ with $p=vv^\*$ being a projection (with same range as $v$ if viewed as an operator on a Hilbert space), which easily prompts
$$v(\mathcal O\_\infty\otimes\mathcal K)v^\* = p(\mathcal O\_\infty\ot... | 4 | https://mathoverflow.net/users/29404 | 271999 | 121,755 |
https://mathoverflow.net/questions/271992 | 6 | In [Lectures by Dan Bump on Modular representation theory](http://web.stanford.edu/~tonyfeng/mod_rep_theory.pdf),
Theorem 13.14 states that whenever $G$ is a non-abelian simple group of order $|G|=p^aq^br$ for distinct primes $p$,$q$, and $r$, every $r$-Sylow $R$ is equal to its own centralizer.
He states Corollary 1... | https://mathoverflow.net/users/111070 | Group of order $5p^aq^b$ | Note that Bump has his own notes in his webpage: <http://sporadic.stanford.edu/modrep>
While the contents are in different order with that of Feng's notes, he mentioned in Section~6.3 that this is a result of Brauer.
I found that it is in this paper of Brauer: <http://www.ams.org/journals/bull/1968-74-05/S0002-9904... | 9 | https://mathoverflow.net/users/56217 | 272006 | 121,759 |
https://mathoverflow.net/questions/272011 | 2 | Let $X$ be a Noetherian irreducible scheme of dimension $n$. Let $Y,Z$ be its closed irreducible subschemes of dimensions $k,l$ respectively.
>
> Under what technical conditions the dimension of each irreducible component of $Y\cap Z$ is at least $k+l-n$?
>
>
>
Say is that true for any regular (variety over a... | https://mathoverflow.net/users/16183 | Basic question on dimension of intersection of subschemes | This is true for $X$ a non-singular variety, see Fulton, *Intersection Theory*, section 8.2, p. 137.
| 2 | https://mathoverflow.net/users/nan | 272012 | 121,762 |
https://mathoverflow.net/questions/272013 | 3 | Suppose that $f:X\rightarrow S$ is a proper, separated morphism of complex spaces (with $S$ reduced) and $\mathcal{F}$ a is $f$-flat coherent sheaf on $X$.
From (well-)known results it is known that the set of points $U$ where $\mathcal{F}$ is cohomologically flat in dimension $q$ is a Zariski-open subset of $S$.
D... | https://mathoverflow.net/users/37150 | Example of a morphism of complex spaces or "nice schemes" that is not cohomologically flat in any point | For simplicity, assume that $S$ is affine (every algebraic space has a dense open subset that is an affine scheme). In the case of a proper morphism of Noetherian algebraic spaces,$$ f:X\to S,$$ for $\mathcal{F}$ an $f$-flat coherent sheaf, there exist (locally on $S$) a bounded complex $K^\bullet$ of locally free $\ma... | 2 | https://mathoverflow.net/users/13265 | 272020 | 121,765 |
https://mathoverflow.net/questions/272014 | 5 | Anyone knows if there is a chance of getting a copy of the following:
Proceedings of the Conference on p-adic Analysis.
Held in Nijmegen, January 16–20, 1978. Report, 7806. Katholieke Universiteit, Mathematisch Instituut, Nijmegen, 1978. ii+224 pp.
The link to MR is
<http://www.ams.org/mathscinet-getitem?mr=5221... | https://mathoverflow.net/users/109085 | Nijmegen 1978 $p$-adic analysis proceedings | In the Nijmegen University Repository I only found pages 193-204, [Non-archimedean differentiation](http://repository.ubn.ru.nl/handle/2066/57030). Since I presume a copy for private use is OK, I have scanned Amice's and Morita's contributions, you can find them here:
* [Duals,
Yvette Amice](https://ilorentz.org/been... | 8 | https://mathoverflow.net/users/11260 | 272021 | 121,766 |
https://mathoverflow.net/questions/272022 | 3 | Consider the $n\times n$ matrix $F$ defined by the following expression
$$
F=A-\varepsilon B
$$
where $A$ is a constant matrix such that $a\_{ij}=a>0$ for all $i,j$ and where $B$ is a symmetric matrix such that $b\_{ij}\geq 0$ for all $i,j$.
I would like to find sufficient conditions on $A$ and $B$ such that $F$ is p... | https://mathoverflow.net/users/91545 | Positive semi-definite in the limit | Let $P$ denote an orthogonal projection onto the hyperplane $H:\sum x\_i=0$. If $A-\varepsilon B$ is positive semi-definite, so is $P(A-\varepsilon B)P=-\varepsilon PBP$, thus we get a **necessary condition**: $PBP$ should be non-positive definite.
If $PBP$ is negative definite on $H$ (that is, the quadratic form $(... | 3 | https://mathoverflow.net/users/4312 | 272023 | 121,767 |
https://mathoverflow.net/questions/271998 | 6 | H$\vphantom{a}$i. Consider the Laplacian on $\mathbb R^n$,
$$
\Delta=\partial\_i^2
$$
It is easy to prove that the most general differential operator that commutes with rotations and translations is of the form
$$
\sum\_k a\_k\Delta^k
$$
for some constants $a\_k$.
**Is there a similar result for non-trivial manifol... | https://mathoverflow.net/users/106114 | Laplace-Beltrami and the isometry group | For the maximally symmetric case with the second order operator, the proof is very simple.
A second order differential operator acting on scalars can be written in the form
$$ b^{ij} \nabla^2\_{ij} + c^i \nabla\_i + e $$
Consider first the action of all isometries that fixes a point $p\in M$. You must have that $b^... | 7 | https://mathoverflow.net/users/3948 | 272035 | 121,770 |
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