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https://mathoverflow.net/questions/306975 | 5 | What English translations are there of work done by the Italian school of algebraic geometry?
Perhaps I'm being too spoiled here, given that mathematical French, German, Italian are much easier to pick up on the fly than say, mathematical Russian or Japanese, for a native English speaker.
| https://mathoverflow.net/users/126532 | What English translations are there of work done by the Italian school of algebraic geometry? | Here is a [complete bibliography](http://math.unipa.it/~brig/sds/prima%20pagina/tirocinio/bibliografie.htm). It includes translations, but not in English (many in French, German, and Spanish).
From this bibliography I notice that only [Guido Fubini](http://math.unipa.it/~brig/sds/prima%20pagina/tirocinio/Fubini%20Gui... | 5 | https://mathoverflow.net/users/11260 | 306977 | 133,782 |
https://mathoverflow.net/questions/306937 | 14 | John Derbyshire in his book [PRIME OBSESSION](http://tomlr.free.fr/Math%E9matiques/Fichiers%20Claude/Nombres/Derbyshire%20-%20Prime%20Obsession%20-%20Bernhard%20Riemann%20and%20the%20Greatest%20Unsolved%20Problem%20in%20MathematicsAAA.pdf) says on page 343:
>
> "I’ll round off with a complete calculation of $\pi(1... | https://mathoverflow.net/users/116870 | How many zeta zeros are needed to accurately calculate five digits for π(1000000), where π(x) is the prime counting function? | [Monte-Carlo Approximation of the Prime Counting Function](https://wstein.org/edu/2010/414/projects/stueve.pdf)
This is a numerical study of the dependence of the error $\delta\pi(x)$ in the prime counting function when $N$ zeros $\rho$ of the zeta function are used in the sum $\sum\_\rho {\rm Li}\,(x^\rho)$.
Figu... | 7 | https://mathoverflow.net/users/11260 | 306983 | 133,783 |
https://mathoverflow.net/questions/306984 | 4 | Let $f: X \to S$ be an arithmetic surface, where $S=\operatorname{Spec } O\_K$ for a number field $K$. It is well known that if we want to introduce a reasonable intersection theory on $X$ we have to deal with Arakelov geometry. In this question I want to stick to the usual theory of schemes, so no fibres at infinity a... | https://mathoverflow.net/users/65980 | Pairing on arithmetic surfaces | As mentioned in the second comment, the "usual" construction requires that the target is the divisor class group, rather than the group of divisors. In the geometric case, this is clear: how do you define the pairing of $D$ and $E$ in the nontransverse case, i.e., when $E$ equals $D$? As a divisor class, there are seve... | 2 | https://mathoverflow.net/users/13265 | 306986 | 133,785 |
https://mathoverflow.net/questions/306995 | 4 | **Definitions and assumptions**
On a 2n-dimensional phase-space with coordinates $x$ and $p$, the Moyal product can be written explicitly as
$$g(x,p) \star h(x,p) = g(x,p) e^{\frac{i}{2}\left( \overleftarrow{\partial\_x} \cdot \overrightarrow{\partial\_p} - \overrightarrow{\partial\_x} \cdot \overleftarrow{\partial... | https://mathoverflow.net/users/37773 | Moyal $\star$-product of $\star$-exponentials | The question to resolve $h$ from
$e^f e^g=e^h$
amounts to the [Baker–Campbell–Hausdorff formula](https://en.wikipedia.org/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula) for the Moyal $\star$ product. One can use the same [formula](https://en.wikipedia.org/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula#A... | 2 | https://mathoverflow.net/users/11260 | 306999 | 133,791 |
https://mathoverflow.net/questions/306872 | 2 | In equation 6 of [Computing Persistent Homology](https://geometry.stanford.edu/papers/zc-cph-05/zc-cph-05.pdf) (page 8), the authors put forward the following identity:
$$\deg \hat{e\_i}+\deg M\_k (i,j)=\deg e\_j$$
Where $\hat{e\_i}$ and $e\_j$ are elements of homogeneous bases for the space of $(k-1)$-chains and $k$-c... | https://mathoverflow.net/users/126581 | On the entries of a matrix representation for a boundary operator of a persistence module | One can certainly verify Equation 6 on several instances, but this holds true in general as **a consequence of Theorem 3.1 (Correspondence Theorem)** on page 7.
Indeed, $\mathcal{M}\_k \Doteq \{C\_k^{l},\, f^l\}$ and $\mathcal{M}\_{k - 1} \Doteq \{C\_{k - 1}^{l},\, f^l\}$ are persistent modules in the sense of Defini... | 3 | https://mathoverflow.net/users/84349 | 307005 | 133,793 |
https://mathoverflow.net/questions/306987 | 0 | Assume that $g(re^{it}),$ and $h(re^{it})$ are smooth positive functions defined on the annulus $A=A(R,1)=\{z: R<|z|<1\}$. Assume also that $\int\_0^{2\pi}h(re^{it})dt\ge 2\pi c$ for every $r\in(R,1)$.
Can we state that for every $r$ there is $t\_r$ so that
$$I=\int\_A g\cdot h \, dxdy\ge 2\pi c\int\_R^1 rg(re^{it\_r... | https://mathoverflow.net/users/124426 | An inequality that involves integrals | This doesn't seem true. For instance, let $f : S^1 \rightarrow \mathbb{R}$ be a bump function with total mass $2\pi$ localized near $e^{i0}=1$. Define $h(re^{it}):= f(e^{2\pi i\frac{(r-R)}{1-R}}e^{it})$. Then, for any $r$, $\int\_0^{2\pi}h(re^{it})=2\pi$, so we can set $c=1$. Now we can take $g$ to be any smooth positi... | 2 | https://mathoverflow.net/users/45553 | 307007 | 133,795 |
https://mathoverflow.net/questions/307013 | 0 | As the following product is a bit unfamiliar to me:
>
> How do we compute the rational cohomology ring $H^\*(M;\mathbb{Q})$ of the product of Lie groups:
> $M=SO(n\_1)\times U(n\_2)\times SU(n\_3)\times (Sp(n\_4)\cdot Sp(1))\times Sp(n\_5)\times G\_2\times Spin(7)\times \left(\frac{SU(n\_6+1)}{S(U(1)\times U(n\_... | https://mathoverflow.net/users/98896 | Compute the rational cohomology ring $H^*(M;\mathbb{Q})$ of a product of Lie groups (and their quotients) | See Section 3 in Fok's [nice survey paper.](http://pi.math.cornell.edu/~ckfok/Cohomology_Lie_groups.pdf) (Cohomology and K-Theory of compact Lie groups), and all will be revealed.
NOTE: I just noticed that (as pointed out by Arun Debray) that yours is not a product of Lie groups, so maybe not all will be revealed.
| 4 | https://mathoverflow.net/users/11142 | 307014 | 133,797 |
https://mathoverflow.net/questions/307016 | 2 | Let $K\subset E\subset\bar{K}$ be field extensions, $\bar{K}$ an algebraic closure of $K$. Denote $E\_s$ the field of separable elements of $E$ over $K$, denote $\tilde{E}\subset\bar{K}$ the normal closure of $E/K$ (the minimal subfield of $\bar{K}$ which contains $E$ and is normal over $K$). Is it true that
$$
(E\_s)^... | https://mathoverflow.net/users/42571 | Normal closure and separable elements | Yes, it is true.
We will denote by $p(a,K)\in K[x]$ the minimal polynomial of $a\in \overline{K}$ over $K$: it is monic, irreducible and has $a$ as a root. An irreducible polynomial is called separable if it has no multiple roots on $\overline{K}$
Then $$E\_s:=\{a\in E\ | \ p(a,K) \mbox{ separable }\}$$ and $$\wid... | 3 | https://mathoverflow.net/users/24442 | 307028 | 133,800 |
https://mathoverflow.net/questions/306623 | 4 | Chambert-Loir and Ducros have introduced real differential forms and currents on Berkovich spaces.(See Gubler's [survey](https://arxiv.org/abs/1303.7364) for example). In that survey, a tropical chart $V$ is defined on an analytification $X^{\mathrm{an}}$, something like $ \mathrm{val}^{-1}(\Delta)$ for a polyhedron $\... | https://mathoverflow.net/users/69190 | Tropical charts (coordinates) and differential forms in non-archimedean geometry | I will try to explain what it is a tropical chart on an algebraic variety over a non-archimedean field $K$ (complete with respect to a non-archimedean absolute value, algebraically closed by simplicity).
First of all, for the algebraic tori (of the form $T:=\mathbb G\_m^n$, for $n\ge 1$), we have a continuous map $T... | 4 | https://mathoverflow.net/users/24442 | 307029 | 133,801 |
https://mathoverflow.net/questions/306392 | 2 | Given a natural number $n$ and an element $k \in \mathbb{Z}\_n$, how many solutions are there in $\mathbb{Z}\_n$ to the equation $x^2+y^2 =k$? That is, I'm wondering whether there is a mod-$n$ version of the sum of squares function discussed in [this post](https://mathoverflow.net/questions/29644/enumerating-ways-to-de... | https://mathoverflow.net/users/126785 | Number of ways to write an integer as a sum of squares modulo $k$ | A formula for the quantity you are considering is given is computed by elementary means in <https://arxiv.org/abs/1404.4214> .
In fact, they consider the same counting problem for general quadratic congruences of the form
$$ a\_1 x\_1^2 + \dots + a\_k x\_k^2 \equiv b \mod n .$$
| 2 | https://mathoverflow.net/users/127070 | 307032 | 133,803 |
https://mathoverflow.net/questions/306081 | 3 | The exponential map associated with the (-) - connection on a Lie group is generally not surjective. This is because, for this connection, the one-parameter subgroups and geodesics coincide. If we drop the requirement that one-parameter subgroups are geodesics, is it possible to put a left invariant connection on a Lie... | https://mathoverflow.net/users/14454 | Left invariant connections on a Lie group | Yes. Just take the Levi-Civita connection of any left-invariant Riemannian metric on the Lie group. The metric is complete, so any two points can be joined by a geodesic (Hopf-Rinow). Thus, the geodesic exponential map of that connection starting from the identity is surjective.
| 6 | https://mathoverflow.net/users/13972 | 307038 | 133,805 |
https://mathoverflow.net/questions/306978 | 4 | (**Note:** I've migrated this question from math.stackexchange, as the lack of answers there made me believe it was perhaps too advanced for that forum.)
Consider the one-dimensional heat equation
$$\partial\_t u(t,x)=\frac12\Delta u(t,x),\qquad t\geq0,~x\in I$$
on some interval $I=(-a,a)$, with some initial conditio... | https://mathoverflow.net/users/50406 | *Full proof* references for Markov generators with various boundary conditions | As I understand, we take for granted that the solution of the heat equation is given in terms of the appropriate heat kernel. This heat kernel is given in terms of the Gauss–Weierstrass kernel $k\_t(x) = (2 \pi t)^{-1/2} \exp(-x^2 / (2 t))$; namely, we have
$$ u(t, x) = \int\_{-a}^a p(t, x, y) f(y) dy, $$
where
$$ p(t,... | 4 | https://mathoverflow.net/users/108637 | 307054 | 133,809 |
https://mathoverflow.net/questions/307051 | 3 | The [Calkin algebra](https://en.wikipedia.org/wiki/Calkin_algebra) $C(H)$ is the quotient of $B(H)$, the ring of bounded linear operators on a separable infinite-dimensional Hilbert space $H$, by the ideal $K(H)$ of compact operators.
In 1977, Brown, Douglas, and Fillmore [1] asked whether it is possible for a Calki... | https://mathoverflow.net/users/82843 | What is the consistency strength of non-existence of outer automorphisms of Calkin algebra? | Perhaps surprisingly, the Open Coloring Axiom (even OCA + MA$\_{\aleph\_1}$) has no additional consistency strength. The situation is described in Velickovic's paper "Applications of the open coloring axiom," and you can see the relevant bit [here](https://books.google.com/books?id=mPLpBwAAQBAJ&pg=PA142&lpg=PA142&dq=co... | 8 | https://mathoverflow.net/users/8133 | 307056 | 133,810 |
https://mathoverflow.net/questions/307041 | 0 | Given two space curves defined by the equations
$P\_1(x\_1,x\_2)=0,\dots,P\_{n-1}(x\_{n-1},x\_n)=0$ and $P\_1(x\_1,x\_2)=0,\dots,P\_{n}(x\_{n},x\_{n+1})=0$, respectively, is the genus of the former, say $g\_1$, smaller than or equal to the genus of the latter,
say $g\_2$? $P\_k$ are polynomials. For example, if $g\_2=... | https://mathoverflow.net/users/4274 | Genus of space curves given by the intersection of bivariate polynomial equations | Forgetting the variable $x\_{n+1}$ defines a non-constant, so dominant,
morphism from one curve to the other, so yes.
| 2 | https://mathoverflow.net/users/8726 | 307064 | 133,811 |
https://mathoverflow.net/questions/307053 | 1 | Let us denote by $i:U\rightarrow X$ an open inclusion and $\mathcal{F}$ a coherent sheaf on $U$.
Is the map $i\_!\mathcal{F} \rightarrow i\_\*\mathcal{F}$ always injective ?
I ask this question because it seemed to me that we define the minimal extension $i\_{!\*}\mathcal{F}$ as the image of $i\_!\mathcal{F} \right... | https://mathoverflow.net/users/126483 | Proper direct image and minimal extension | Here’s what your misunderstanding: $i\_{!\*}\mathcal F$ is the image of $i\_!\mathcal F \to i\_\* \mathcal F$ *in the category of perverse sheaves*. It’s instructive to look at the case $U=\Bbb C^\*$ and $X=\Bbb C$. Try showing that the cone of $$i\_! \Bbb C\_U[1] \to i\_\* \Bbb C\_U[1]$$ is not perverse (in the middle... | 1 | https://mathoverflow.net/users/36720 | 307065 | 133,812 |
https://mathoverflow.net/questions/307047 | 17 | A monoidal category is a well-defined categorical object abstracting products to the categorical setting. The term *tensor category* is also used, and seems to mean a monoidal category with more structure, usually the structure of an abelian cateogry, but I can't find a precise definition. So I ask question: What is a ... | https://mathoverflow.net/users/125941 | What is a tensor category? | There seem to be many different definitions in the literature, based on individual papers. But, I think that might change, now that the textbook [Tensor Categories](http://www-math.mit.edu/~etingof/egnobookfinal.pdf), by Etingof, Gelaki, Nikshych, and Ostrik, has appeared. They define a tensor category as follows:
Le... | 13 | https://mathoverflow.net/users/11540 | 307068 | 133,814 |
https://mathoverflow.net/questions/307074 | 2 | Let $T$ be a smooth projective variety and $f\_T : \mathbb {P}^N\_T \rightarrow \mathbb {P}^N\_T $ be a family of dominant rational maps. The dynamical degree of a dominant rational map $f $ is defined by $\lim\_{n \to \infty}(\mathrm {deg}(f^n))^{1/n}$.
Let $E $ be a set of $t \in T $ such that the dynamical degree ... | https://mathoverflow.net/users/123157 | Is the exceptional set Zariski open? | It is not true that your exceptional set is always Zariski open. Here's an example. Consider the map
$$ f\_{a,b,c}(X,Y,Z) = [XY, XY+aZ^2, bYZ+cZ^2] :
\mathbb P^2 \longrightarrow\mathbb P^2
$$
parametrized by $(a,b,c)\in\mathbb A^3$. Then generically $\delta(f\_{a,b,c})=2$, but it's a fun calculation to show that $\del... | 8 | https://mathoverflow.net/users/11926 | 307077 | 133,817 |
https://mathoverflow.net/questions/306979 | 8 | I remember coming across this result some time ago but I am having trouble finding a reference for it. It goes something like this:
>
> Let $p$ be a(n odd?) prime, then the $p$-primary component of $\pi^S\_k$ is $\Bbb Z\_p$ when $k=2l(p-1)-1$ for $l=1,\dots,p-1$ and is trivial for all other $k<2p(p-1)-2$.
>
>
>
... | https://mathoverflow.net/users/54401 | Reference for: $p$-primary component of $\pi^S_k$ is $\Bbb Z_p$ when $k=2l(p-1)-1$ | This follows easily from Theorem 4.4.20 of Ravenel's book, Complex cobordism and stable homotopy groups of spheres". The elements in $Ext^i$ with $i>1$ have total degrees greater than or equal to $2p(p−1)−2$, so we only have to look at $Ext^1$, and we have $$\pi \_{ql-1}(S^0)\cong Ext ^{1,ql}\cong Z/p
\mbox{ where $q=... | 6 | https://mathoverflow.net/users/43326 | 307082 | 133,818 |
https://mathoverflow.net/questions/306889 | 9 | If we assume MA+¬CH, then every boolean algebra with cardinality smaller than the continuum embeds in ℘(ω)/Fin. A proof of this result can be found in Theorem 1.1, Chapter 8 of the book "Hausdorff gaps and limits". In the paper "R. Frankiewicz, Some remarks on embeddings of boolean algebras and topological spaces II, F... | https://mathoverflow.net/users/122189 | Embeddings of Boolean algebras in $\wp(\omega)/Fin$ | In the paper [Embedding of Boolean algebras in $Ρ(ω)/$fin](https://eudml.org/doc/211828) the following partial result is proved:
**Theorem** There is a model of $ZFC$ with arbitrarily large continuum in which each Boolean algebra $B$ of cardinality $\leq 2^{\aleph\_0}$ can be embedded into $P(ω)/$fin. In addition, Ma... | 4 | https://mathoverflow.net/users/11115 | 307085 | 133,820 |
https://mathoverflow.net/questions/307033 | 6 | I know by [Van Kampen](http://pi.math.cornell.edu/~hatcher/AT/ATchapters.html)'s Theorem that we can obtain $\pi\_1(S\_1 \vee S\_1) = \mathbb{Z} \* \mathbb{Z}$, so I am wondering if we can construct a surface or 3-manifold whose fundamental group is $\mathbb{Z}\_n \* \mathbb{Z}\_2$ or even $\mathbb{Z}\_m \rtimes \mathb... | https://mathoverflow.net/users/1358 | Find a surface or 3-manifold whose fundamental group is $(\mathbb{Z}/n\mathbb{Z}) \rtimes (\mathbb{Z}/2\mathbb{Z})$ | See pp. 449--457 of Peter Scott's article *The geometries of 3-manifolds* for a complete description of all 3-manifolds with finite fundamental group. The article is available on his [website](http://www.math.lsa.umich.edu/~pscott/). There don't seem to be any with dihedral fundamental groups (see Allen Hatcher's comme... | 6 | https://mathoverflow.net/users/1463 | 307097 | 133,823 |
https://mathoverflow.net/questions/221339 | 18 | For this question I will let the overly ambiguous word curve mean: smooth projective and connected curve over $\mathbb C$ (or equivalently a smooth compact Riemann-Surface).
Let $C$ be a curve over $\mathbb C$. And suppose that $f: D \to C$ and $g:D \to E$ are dominant maps of curves. Then the pair $(f,g)$ forms a c... | https://mathoverflow.net/users/23501 | Do all simple factors of jacobians of curves come from correspondences? | This argument is mostly contained in t3suji's comments, but with some of the proofs somewhat expanded.
As a convention (consistent with that of the theory of Chow motives), all actions of correspondences $\alpha \in \operatorname{CH}^\*(X \times Y)$ on cohomology (or Jacobians) are *covariant*: $\alpha\_\* \colon H^\... | 10 | https://mathoverflow.net/users/82179 | 307105 | 133,828 |
https://mathoverflow.net/questions/307116 | 0 | Assume that $K(t,s)$ is a (1) symmetric, (2) continuous, and (3) positive definite kernel on $[0,1] \times [0,1]$. The spectral decomposition of $K(t,s)$ is:
$$
K (t,s) = \sum\_{i=1}^\infty \lambda\_i \phi\_i (t) \phi\_i (s)
$$
with $\{ \phi\_i (t): i \in \mathbb{N}\}$ orthonormal and $\lambda\_1 > \lambda\_2 > \cdots ... | https://mathoverflow.net/users/112304 | Does $K^{1/2} (t,s)$ inherit the continuity of $K(t,s)$? | **No**.
For example, the Green function for $-\Delta$ (the 1-D Laplace operator) in the interval $(0, 1)$ with Dirichlet boundary condition is continuous (it is $\min(x (1 - y), (1 - x) y)$), but the Green function for the square root of this operator is unbounded: it has logarithmic singularity near the diagonal. Th... | 2 | https://mathoverflow.net/users/108637 | 307128 | 133,836 |
https://mathoverflow.net/questions/307058 | 3 | The following is taken from Dvoretzky, 1972, ASYMPTOTIC NORMALITY FOR SUMS OF DEPENDENT RANDOM VARIABLES, Equation 4.6.
$$\{X\_{n,k}\}\_{n=0,1,...;k=0,1...,k\_n}$$
is a (triangular) array of r.v.'s /w
$$E[X\_{n,k}|\mathcal{F}\_{n,k-1}]=0$$
$$\sum\_{k=1}^{k\_n}E[X\_{n,k}^2|\mathcal{F}\_{n,k-1}]=1$$
$$\lim\_n\sum\_{... | https://mathoverflow.net/users/126914 | Lindeberg implies convergence of max of conditional variances in L1 | Let $u:=\epsilon$, $E\_{k-1}Z:=E(Z|\mathcal F\_{n,k-1})$,
$$A\_{n,k}:=E\_{k-1}X\_{n,k}^2,\quad B\_{n,k}:=E\_{k-1}X\_{n,k}^2\,1\{|X\_{n,k}|\le u\},\quad C\_{n,k}:=E\_{k-1}X\_{n,k}^2\,1\{|X\_{n,k}|> u\},$$
$$A\_n:=\max\_k A\_{n,k},\quad B\_n:=\max\_k B\_{n,k},\quad C\_n:=\max\_k C\_{n,k}.$$
We need to check that $EA\... | 2 | https://mathoverflow.net/users/36721 | 307131 | 133,838 |
https://mathoverflow.net/questions/307088 | 11 | I am actually interested in the $\infty$-categorical case, but the same question is meaningful in the $1$-categorical situation as well.
A nice property of presentable $\infty$-categories is that if you have one such $\mathcal{C}$ and you have a full subcategory $\mathcal{D}\subseteq\mathcal{C}$ of it, then:
>
> ... | https://mathoverflow.net/users/50409 | What are the reflective subcategories of the category of presentable categories? | Some ideas, building off of Simon Henry and Ivan di Liberti's remarks:
1. **$Pr^L$ is in fact essentially a large (not huge) category (in either the ordinary or $\infty$ context).**
That is, let $\lambda$ be the size of (a skeleton of) the universe of small sets (i.e. "small" means $\lambda$-small, and $\lambda$ is... | 9 | https://mathoverflow.net/users/2362 | 307132 | 133,839 |
https://mathoverflow.net/questions/307126 | 7 | Let us say a metric probability space $(X,\rho,\mu)$ has property (\*) if:
the support of $\mu$ is contained in a separable subspace of $X$.
Questions:
1. Is there a standard name for this property?
2. Is it true that continuum+choice implies property (\*) -- is there a reference?
3. Is it true that if (either? bot... | https://mathoverflow.net/users/12518 | Non-separable metric probability space | Apparently, a relevant source here is H. J. Keisler and A. Tarski, From accessible to inaccessible cardinals. Fundamenta mathematicae, vol. 53 (1964), pp. 225--308, a review of which is given at <https://philpapers.org/rec/KEIFAT> .
I don't have the paper by Keisler and Tarski at this point with me. I found a refere... | 6 | https://mathoverflow.net/users/36721 | 307139 | 133,842 |
https://mathoverflow.net/questions/303882 | 9 | Consider all knots with fixed genus $g\ge 2$ (I am considering the classical 3-genus). Do there exist infinite families of genus $g$ knots with arbitrarily large volume?
The answer seems like it should definitely be yes, but I can’t seem to find any references.
| https://mathoverflow.net/users/111700 | Knots of fixed genus with arbitrarily large volume | The result of Brittenham for genus 1 knots pointed out by Sam Nead was generalized to all genus in Theorem 8.2 of this paper:
*Stoimenow, A.*, [**Realizing Alexander polynomials by hyperbolic links**](http://dx.doi.org/10.1016/j.exmath.2009.06.003), Expo. Math. 28, No. 2, 133-178 (2010). [ZBL1196.57009](https://zbmat... | 10 | https://mathoverflow.net/users/1345 | 307144 | 133,844 |
https://mathoverflow.net/questions/306684 | 5 | This question is a [follow up](https://mathoverflow.net/questions/306653/minimize-spectral-norm-under-diagonal-similarity).
Let $A$ be a real square matrix of size $n \times n$. How to determine the minimum spectral norm under diagonal similarity, i.e.,
$$
s(A) = \inf\_{D} \lVert D^{-1} A D\rVert\_2,
$$
where $D$ ... | https://mathoverflow.net/users/51478 | Numerical minimization spectral norm under diagonal similarity | Here is a better, more direct solution.
This problem can be cast as a *Generalized Eigenvalue Problem* as is shown by
Boyd, El Ghaoui, Feron, and Balakrishnan on page 39 (§3.3) of [Linear Matrix Inequalities in System and Control Theory](https://web.stanford.edu/~boyd/lmibook/lmibook.pdf):
$$
s(A) = \inf \left\{\g... | 4 | https://mathoverflow.net/users/8430 | 307145 | 133,845 |
https://mathoverflow.net/questions/307117 | 8 | $\newcommand{\Hdr}{H\_{\mathrm{dRh}}}$
$\newcommand{\spec}[1]{\mathrm{spec}(#1)}$
$\require{amsmath}$
Let $A = k[x\_1,\ldots,x\_n]$ the polynomial ring over a field $k$ of characteristic zero and $I \subseteq A$ an ideal of $A$. Let
$Y= V(I) = \spec{A/I}$ and $X = \spec{A}$.
According to R. Hartshorne,
On the de R... | https://mathoverflow.net/users/21940 | Definition of algebraic de Rham cohomology of non-smooth affine variety | (Synthesis of answers from comments, posted as community-wiki answer for convenience.)
1. If $k = \mathbb{C}$ then algebraic de Rham cohomology, defined a la Hartshorne using the completion of $X$ along $Y$, is *always* isomorphic to singular cohomology. This is a theorem of Hartshorne, stated as Theorem 1.6 of his 1... | 13 | https://mathoverflow.net/users/2481 | 307146 | 133,846 |
https://mathoverflow.net/questions/307115 | 6 | Let $p$ be an odd prime, $g$ a primitive root of $p$ and $A=\{1, 2, \ldots, p-1 \}$.
Obviously, $\sigma\_g(p)=\begin{pmatrix}
1 & 2 & \ldots & {p-1} \\
g^1\pmod{p} & g^2\pmod{p} & \ldots & g^{p-1}\pmod{p}
\end{pmatrix}$ is a permutation of $A$.
I observed that "almost always" $\sigma$ is a product of cycles who... | https://mathoverflow.net/users/38851 | Can a primitive root-permutation of $A=\{1, 2, \ldots, p-1 \}$ be a cycle of length $p-1$ only for finitely many $p$? | As pointed out in the comments, primitive roots with a single cycle appear to be rather common. The standard heuristic argument suggests that there are infinitely many such $p$, and a bit more.
The share of $N$-cycles among all permutations on $N$ symbols is $1/N$. By the Borel-Cantelli lemma, given an infinite sequ... | 3 | https://mathoverflow.net/users/5740 | 307148 | 133,848 |
https://mathoverflow.net/questions/306542 | 11 | As an example, consider the following Poisson bracket on ${\mathbb R}^n$:
$$\{x\_i, x\_{i+1}\} = x\_ix\_{i+1}(x\_i+x\_{i+1}),\\
\{x\_i, x\_{i+2}\} = x\_ix\_{i+1}x\_{i+2}.$$
The indices are taken modulo $n$, and the "distant" variables commute. The Jacobi identity holds but does not look obvious. This bracket appears in... | https://mathoverflow.net/users/98590 | Is there a classification of polynomial Poisson brackets? | I don't think we have anything like a classification. In the book *Poisson Structures* (2013) by Laurent-Gengoux, Pichereau, and Vanhaecke, it is written
>
> "For higher order Poisson structures, starting with quadratic Poisson structures, there
> is no general theory and there is no immediate interpretation, as ... | 10 | https://mathoverflow.net/users/51668 | 307152 | 133,850 |
https://mathoverflow.net/questions/306144 | 8 | Consider following program:
1. Generate random 3-manifold embedded in $R^4$.
2. Perform its triangulation.
3. Put it to [Regina](https://en.wikipedia.org/wiki/Regina_(program)) and calculate what manifold it is.
Assuming that we have good algorithm for random submanifolds in point 1. then we can conclude which 3-ma... | https://mathoverflow.net/users/nan | Random 3-manifolds in $R^4$ | Regarding A: As far as I know, there's only some special cases and no big familiy of interesting examples known.
Regarding C: Yes, Regina has a fairly good Python interface. We don't have every feature of the C++ library implemented in Python, but quite a bit is.
Regarding D: Perhaps the most sensible way to defi... | 5 | https://mathoverflow.net/users/1465 | 307158 | 133,852 |
https://mathoverflow.net/questions/307157 | 5 | For any topological space $(X,\tau)$ we define a *matching* to be a collection of non-empty and pairwise disjoint open sets. We define the *matching number* $\nu(X,\tau)$ to be the smallest cardinal $\kappa$ such that for every matching ${\cal M}\subseteq \tau$ we have $|{\cal M}|\leq \kappa$.
Recall that $D\subseteq... | https://mathoverflow.net/users/8628 | "König's theorem" for $T_2$-spaces? | What you are calling the "matching number" of $X$ is usually called its *Souslin number* -- the smallest cardinal bounding the size of any collection of pairwise disjoint open subsets of $X$.
What you are calling the "vertex covering number" of $X$ is usually called its *density*.
Thus your question is how the Sous... | 11 | https://mathoverflow.net/users/70618 | 307161 | 133,853 |
https://mathoverflow.net/questions/253902 | 3 | Suppose that $w=\sqrt{R^2 + s^2 -2Rs\cos\theta}$ with $R\ge s>0$, that $p$ is a positive integer and that $j$ is an integer with $0\le j\le p$. Let $I$ and $K$ denote the modified Bessel functions of the first and second kind respectively. I want to prove the following identity (actually I would be happy with the case ... | https://mathoverflow.net/users/458 | A Bessel function integral identity involving $\int_0^\pi \frac{K_{j-1/2}(w)}{w^{j-1/2}}\sin^{2p-1}(\theta)\, d\theta$ | I have proved this Bessel identity in my paper [The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials](https://arxiv.org/abs/1708.03227). It is Theorem 4 which is proved in Section 3.
| 4 | https://mathoverflow.net/users/458 | 307171 | 133,856 |
https://mathoverflow.net/questions/249456 | 21 | I want to prove the following identity calculating the integral of an exponential over an even dimensional sphere in terms of functions $\chi\_i(R)$ and $\tilde\psi\_i(s)$ (described below) which are essentially modified spherical Bessel functions.
First set up some notation. Let $p>0$ be an integer and $R>0$. Let $... | https://mathoverflow.net/users/458 | Proof of an identity involving $\int \exp(-|x-s|)dx$ over an even sphere | Following on from the answer of Sam Dolan, I generalized my conjecture to the following, which I prove as Theorem 4 in my paper [The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials](https://arxiv.org/abs/1708.03227). [The statement in the question is case where $j=0$.]
**Theorem.** For $0... | 4 | https://mathoverflow.net/users/458 | 307173 | 133,858 |
https://mathoverflow.net/questions/228485 | 5 | Consider the sequence $(\varphi\_i)$ of reverse Bessel polynomials which begins as follows.
\begin{align\*}
\varphi\_0&=1\\
\varphi\_1&=x\\
\varphi\_2&=x^2 + x\\
\varphi\_3&=x^3 + 3x^2 + 3x\\
\varphi\_4&=x^4 + 6x^3 + 15x^2 + 15x
\end{align\*}
In general we have
$$\varphi\_0=1;\qquad\varphi\_i = \sum\_{k=1}^{i} \frac... | https://mathoverflow.net/users/458 | Evaluation of Hankel determinants for the reverse Bessel polynomials | There is a combinatorial formula for the Hankel determinant $H\_n= \det\left([\varphi\_{i+j}]\_{i,j=0}^{n}\right)$ in terms of weighted sums of disjoint collections of Schröder paths (and also for the offset version). This is given in Theorem 26 of my paper [The magnitude of odd balls via Hankel determinants of reverse... | 2 | https://mathoverflow.net/users/458 | 307177 | 133,859 |
https://mathoverflow.net/questions/307181 | 1 | A family of residue classes $a\_i (\mod n\_i)$ with $2\leq n\_1\leq\cdots\leq n\_r$ is called a covering system of congruences if every integer belongs to at least one of the residue classes, that is, every integer satisfies at least one of the congruences $a\_i (\mod n\_i)$. The known examples are:
$0 (\mod 2),\ 0 (... | https://mathoverflow.net/users/56947 | constructing a covering system of congruences? | Here's one way to construct those covers.
Every integer is $0\bmod2$ or $1\bmod2$. Keep one of those congruences, doesn't matter which one, say, $0\bmod2$, and go to work on the other one.
Every integer that's $1\bmod2$ is either $1\bmod4$ or $3\bmod4$. Keep $1\bmod4$, and go to work on $3\bmod4$ (as before, and ... | 2 | https://mathoverflow.net/users/3684 | 307183 | 133,861 |
https://mathoverflow.net/questions/307185 | 0 | I am currently studying parameter dependent symbols, $s(t,x,\xi)$, where $t\in [0,1],x\in \Omega, \xi \in \mathbb{R^n}$. I wanted to know how the low regularity (for example, $s$ is just continuous w.r.t. $t$) of symbol w.r.t. parameter affects further study of symbols and the corresponding operators.
To be specific,... | https://mathoverflow.net/users/102092 | References for the study of parameter dependent symbols $s(t,x,\xi)$ having low regularity in parameter($t$) | There exists a vast literature on this topic. The general philosophy is that one can compensate for less regular coefficients in time by a greater regularity of the initial data. For analytic data, typically you only need $L^1\_{loc}$ in time coefficients; Hölder coefficients are suited for problem posed in Gevrey clas... | 1 | https://mathoverflow.net/users/7294 | 307186 | 133,862 |
https://mathoverflow.net/questions/307203 | 3 | Is there an example of compact complex manifold, which is formal, but does not admit complex structure satisfying $dd^c$-lemma?
| https://mathoverflow.net/users/81055 | Formal complex manifold without dd^c | The Hopf surface $S^1 \times S^3$ is formal but does not admit a complex structure satisfying the $dd^c$ lemma. Indeed, a closed four-manifold admitting a complex structure satisfying the $dd^c$ lemma (equivalently the $\partial \bar \partial$ lemma) admits a Kähler structure and thus has even first Betti number.
| 6 | https://mathoverflow.net/users/104342 | 307214 | 133,870 |
https://mathoverflow.net/questions/307200 | 11 | In some related MO questions like [The set-theoretic multiverse as a (bi)category](https://mathoverflow.net/questions/75836/the-set-theoretic-multiverse-as-a-bicategory) it is discussed how one might represent *the multiverse* (see [The set-theoretic multiverse](https://arxiv.org/abs/1108.4223)) in a category theoreti... | https://mathoverflow.net/users/57517 | Set-theoretical multiverses and their representation as functors? Why *the* multiverse? | Of course we have been investigating a wide variety of
multiverse concepts, and in this sense, yes, we have not just one,
but many, multiverses.
But to be sure, much of this multiverse analysis has been inspired by
the philosophical idea that there is or might be a unique grand
multiverse for mathematics: the actual ... | 8 | https://mathoverflow.net/users/1946 | 307219 | 133,874 |
https://mathoverflow.net/questions/307061 | 6 | Let $p$ be a prime, $K$ be a finite extension of $\mathbb{Q}$ and $K\_{\infty}$ be a cyclotomic $\mathbb{Z}\_p$-extension of $K$ i.e. Gal$(K\_{\infty}/K) \cong \mathbb{Z}\_p$, the group of $p$-adic integers under addition.
Then
>
> What can we say about the prime decomposition in $K\_{\infty}$ for any prime of $\m... | https://mathoverflow.net/users/44637 | Splitting of primes in cyclotomic $\mathbb{Z}_p$-extension | For any $\mathbb{Z}\_p$-extension the ramification is concentrated among the primes above $p$. Those that are ramified are totally ramified.
For the cyclotomic $\mathbb{Z}\_p$-extension all places above $p$ are totally ramified. That is because you obtain it as a subextension of $\bigcup\_n K(\mu\_{p^n})$. For all un... | 5 | https://mathoverflow.net/users/5015 | 307227 | 133,875 |
https://mathoverflow.net/questions/307164 | 3 | Let $G$ be a locally compact Hausdorff (second countable) groupoid with Hausdorff (second countable) unit space $X$.
Assume $G$ is étale, i.e., the source and range maps of $G$ are local homeomorphisms.
We say that $G$ is *proper* if the map $(s,r)\colon G \to X\times X$ is proper, i.e., the preimage of a compact subs... | https://mathoverflow.net/users/89722 | Slice theorem for proper groupoids | It looks like the result suggested by Praphulla Koushik [here](https://mathoverflow.net/questions/303183/proper-and-etale-groupoid-is-locally-a-translation-groupoid) is enough in the étale case.
Indeed, the image of the map in my question can be rewritten as the image of $$r\colon s^{-1}(U)\to X.$$
Since $r$ is a loca... | 3 | https://mathoverflow.net/users/89722 | 307229 | 133,876 |
https://mathoverflow.net/questions/307228 | 4 | The usual formulation of the Plancherel theorem one writes $f(1)$ as an integral over the dual $\widehat G$. The support of the measure is the set of representations which weakly occur in $L^2(G)$. But this theorem doesn't give any multiplicities. For instance, let $D$ be a discrete series representation of $G=SL\_2({\... | https://mathoverflow.net/users/nan | Multiplicities in Plancherel theorem for SL2(R) | These multiplicities are all 1. Abstract Plancherel theorem (in e.g. Dixmier’s C\*-algebras, 18.8.1):
$L^2(G)=\int\_{\hat G}^\oplus H\_\pi\otimes H\_\pi^\*\,d\mu(\pi)$, for Plancherel measure $\mu$ on any type I loc. compact unimodular $G$.
| 5 | https://mathoverflow.net/users/19276 | 307232 | 133,877 |
https://mathoverflow.net/questions/307062 | 14 | By KL divergence I mean $D(P||Q) = \int dP \log(\frac{dP}{dQ})$. I am looking for the conditions under which this strong convexity is true and possible references. I could not find an answer for infinite dimensions. I am specifically interested in the case where the dimension is uncountable.
Note that this strong con... | https://mathoverflow.net/users/127114 | Is KL divergence $D(P||Q)$ strongly convex over $P$ in infinite dimension | $\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}$
Take any probability measures $P\_0,P\_1$ absolutely continuous with respect (w.r.) to $Q$.
We shall prove the following:
>
> **Theorem 1.** For any $t\in(0,1)$,
> \begin{align\*} \De:=(1-t)H(P\_0)+tH(P\_1)-H(P\_t)
> \ge\frac{(1-... | 9 | https://mathoverflow.net/users/36721 | 307251 | 133,883 |
https://mathoverflow.net/questions/307244 | 3 | I'm looking for a general equation/function z = f(x, y, radius1, radius2, p1, p2) for an atoric surface. p1 and p2 could be either eccentricity or conic constant values. Can anyone help me with that?
In case it is not clear what kind of surface it is, it's similar to a toric surface (so with spherical cross sections ... | https://mathoverflow.net/users/127205 | Atoric equation | For reference, here is the description of the atoric surface described in [Spectacle lenses incorporating atoric surfaces](https://patents.google.com/patent/US7111937).
The profile lies in the $x$–$z$ plane and is parameterised by two variables $u,v$:
$$x(u,v)=u+a(u)q(v),\;\;y(u,v)=v,\;\;z(u,v)=p(u)+\gamma(u)q(v)$$
$... | 2 | https://mathoverflow.net/users/11260 | 307255 | 133,884 |
https://mathoverflow.net/questions/307262 | 0 | I came across this point in a paper recently and I'm having difficulty seeing why it's true. Any explanations or hints would be appreciated.
>
> For any prime $\mathfrak{p}$ of $\mathbb{F}\_q [t]$ such that $\mathfrak{p}$ is not the pole of $t$, it is the case that for some $m\_0 \in \mathbb{Z}\_{>0}$, for all posi... | https://mathoverflow.net/users/126815 | Understanding a valuation property of function fields | Let $m\_0$ be the degree of $\mathfrak p$, so the splitting field of $\mathfrak p$ is the extension of $\mathbb F\_q$ of degree $m\_0$. The set of roots of $t^{p^m}-t$ contains this field extension if and only if $m\_0$ divides $m$, and this is equivalent to $\mathfrak p$ dividing $t^{p^m}-t$. The claim now follows sin... | 4 | https://mathoverflow.net/users/18739 | 307267 | 133,887 |
https://mathoverflow.net/questions/307266 | 14 | We can rigorously talk about Top, the category of all topological space, and also FTop, the category of all finite topological space. So I thought, we can define a category FTop', where we “mod out by homeomorphism” between objects in FTop. i.e. define the equivalence relation ~ in FTop as X ~ Y if they are homeomorphi... | https://mathoverflow.net/users/74664 | Why “modding out the homeomorphism” in the category Top makes no rigorous sense? | There are a couple of things to watch out for.
First, the collection of all spaces which are homeomorphic to a given space $X$ forms a [proper class](https://en.wikipedia.org/wiki/Class_(set_theory)), and so we cannot just naively use set theory to form it. Nevertheless this is not a true obstacle, since we can emplo... | 32 | https://mathoverflow.net/users/1176 | 307273 | 133,889 |
https://mathoverflow.net/questions/307257 | 1 | Let $n$ be a positive integer number and $P$ be a point in a plane. Let $A\_1$, $A\_2$, $\cdots$, $A\_m$ be $m$ points in the plane, we take modulo $m$ for $A\_j$ (it is mean $A\_{m+i}=A\_{i}$ for $i=1, 2, \cdots$).
Now we [rotate](https://en.wikipedia.org/wiki/Rotation_(mathematics)) $P$ clockwise above $A\_1$ with ... | https://mathoverflow.net/users/122662 | Chain rotation of a point | А composition of rotations is ether rotation or translation (a special case of Chasles' theorem) depending on the total rotation angle which is $k\pi /n$, so $P\_k=R\_0^{k\pi /n}(O\_k)$ (rotation with some center $O\_k$) for $k\not\equiv 0\pmod {2n}$ and $P\_k=T\_{v\_k}(P)$ (translation) for $k\equiv 0\pmod{ 2n}$ (This... | 2 | https://mathoverflow.net/users/5712 | 307282 | 133,893 |
https://mathoverflow.net/questions/306960 | 3 | *Premise*
Let $K$ be a field of characteristic zero and $f\in K[X\_1,\dots,X\_m]$. By Hironaka's theorem, there exists a log resolution (over $K$) of the ideal $(f)$. Let $\{(N\_i,\nu\_i)\}\_i$ be the numerical data of a fixed log resolution. The quantity $$ lct\_K(f):=\min\_{i}\frac{\nu\_i}{N\_i} $$ does not depend... | https://mathoverflow.net/users/127070 | Comparisons of log canonical thresholds | *CORRECTION ON THE PREVIOUSLY GIVEN ANSWER*
As remarked in the comments, Hironaka's construction behaves well under extension of the base field, that is a log-resolution for $(X\_L,D\_L)$ can be obtained via base change from a log-resolution of $(X\_K,D\_K)$ for any field extension $K\hookrightarrow L$. Also, an irre... | 1 | https://mathoverflow.net/users/127070 | 307305 | 133,896 |
https://mathoverflow.net/questions/307201 | 8 | I wish to show that a function which is "essentially constant" (defined shortly) can't be a good classifier (machine learning). For this i need to estimate the "complexity" of such a class of functions.
So let $\mathcal X$ be an abstract set (we may assume has metric structure, e.g $\mathbb R^d$). Given $0 < \alpha \... | https://mathoverflow.net/users/78539 | VC dimension, fat-shattering dimension, and other complexity measures, of a class BV functions | I think the family $\mathcal{H}\_\alpha$ is ill-suited for your purposes, because it is too rich.
For example in the case $\mathcal{X} = [0,1]$ (or anything atomless probability measure space) one can find arbitrarily many $h\_\tau\in\mathcal{H}\_\alpha$ such that any two of them are at distance $\alpha/2$. This mean... | 3 | https://mathoverflow.net/users/4961 | 307307 | 133,897 |
https://mathoverflow.net/questions/307258 | 12 | $$a\_0 = 1, \ \ a\_{n+1} = \ 1+\frac{n \*a\_n}{n+a\_n} , \ \ n=0,1,2,3,4,...$$
I have built the above recursive formula. Some terms of sequence are:
1, 1, 3/2, 13/7, 73/34, 501/209, 4051/1546, 37633/13327, 394353/130922, 4596553/1441729, 58941091/17572114, 824073141/234662231,... Τhe numerators of the fractions are ... | https://mathoverflow.net/users/116870 | A recursive formula | For $n\geq 1$, let $p\_n$ be the $(n+1)$-th term of [A000262](https://oeis.org/A000262), and let $q\_n$ be $n$-th term of [A002720](https://oeis.org/A002720). Then, according to the description of these two sequences (more precisely by the contributions of Dennis P. Walsh and Paul Berry)
$$p\_n=\sum\_{k=0}^{n-1}\binom{... | 15 | https://mathoverflow.net/users/11919 | 307310 | 133,898 |
https://mathoverflow.net/questions/307274 | 8 | Let us consider the first-order logic extended with the least fixed point operator (FO+LFP). That is, together with the usual first-order formulas, we also have formulas of the form:
$$\mu X[\overline{y}] . \phi(X, \overline{y})$$
where $X$ (must occur positively in $\phi$) is a "predicate" variable of arity equal ... | https://mathoverflow.net/users/13480 | Expressive power of FO with $\mu$ | Your intuition is right, and the way to formalize it is by Moschovakis's stage comparison theorem.
Suppose $\psi$ is an (FO+LFP)-formula which starts with a $\mu$ operator. So $\psi$ has the form $\mu X[y].\phi(X,y)$. We can define the semantics of $\psi$ in a structure $M$ by transfinite induction. Set $X^\*\_0 = \... | 5 | https://mathoverflow.net/users/2126 | 307326 | 133,903 |
https://mathoverflow.net/questions/307209 | 36 | The renowned mathematician, Paul Erdős, has published more than 1500 papers in various branches of mathematics including discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. A complete list of his published works is available in an [archive ... | https://mathoverflow.net/users/82843 | How to find Erdős' treasure trove? | Following my private communication with [Ron Graham](https://en.wikipedia.org/wiki/Ronald_Graham), he kindly shared all what he knew about Erdős' mathematical diaries with me. Here is part of our discussion which I am allowed to share on MathOverflow for the benefit of the public mathematical community. At Ron's reques... | 17 | https://mathoverflow.net/users/82843 | 307330 | 133,905 |
https://mathoverflow.net/questions/307289 | -1 | The Madsen-Weiss Theorem, as described by Hatcher, states that there is an isomorphism $H\_\*( \mathscr{C}\_{\infty})\cong H\_\*(\Omega\_0^{\infty}AG^+\_{\infty,2})$ where $\Omega\_0^{\infty}AG^+\_{\infty,2}$ denotes the base point path-connected component of $\Omega^{\infty}AG^+\_{\infty,2}$ for the direct limit $\mat... | https://mathoverflow.net/users/98896 | Generalizing the Madsen-Weiss Theorem via the scanning map $\mathscr{C}(M,\mathbb{R}^{\infty})\to\Omega^{\infty}AG^+_{\infty,d}$ | The sequence of papers
>
> S. Galatius, O. Randal-Williams, *Stable moduli spaces of high-dimensional manifolds*. Acta Math. 212 (2014), no. 2, 257–377. ([DOI: 10.1007/s11511-014-0112-7](https://doi.org/10.1007/s11511-014-0112-7), [projecteuclid](https://projecteuclid.org/euclid.acta/1485801812))
>
>
> S. Galatiu... | 9 | https://mathoverflow.net/users/318 | 307340 | 133,911 |
https://mathoverflow.net/questions/307256 | 5 | Let $G$ be a connected graph with vertices $V(G)$. A *bramble* of $G$ is a set of connected subgraphs $H\_1,\ldots,H\_n$ such that for each $i$ and $j$, $H\_i$ *touches* $H\_j$; that is, either $H\_i$ intersects $H\_j$ in a vertex, or there is an edge in $G$ that connects a vertex of $H\_i$ to a vertex of $H\_j$. The *... | https://mathoverflow.net/users/127215 | Can the bramble number and the strict bramble number of a graph be equal? | Indeed, for every connected graph $G$ with at least two vertices, we have $sBr(G)<Br(G)$. This follows from a theorem by Seymour and Thomas and its proof (Theorem 12.4.3 in [Reinhard Diestel, Graph Theory, Springer GTM 173, 5th edition 2016](http://diestel-graph-theory.com/basic.html)). The theorem states that the tree... | 3 | https://mathoverflow.net/users/24076 | 307343 | 133,912 |
https://mathoverflow.net/questions/307349 | 0 | Assume that $M$ is a submanifold of $\mathbb{R}^n$ and is equipped with a Riemannian metric such that the parallel transports associated with corresponding LC conection preserve the inner products of tangent spaces which they inherit from the standard metric of $\mathbb{R}^n$.
Does this imply that $(M,g)$ is embedded... | https://mathoverflow.net/users/36688 | A sufficient condition for isometrically embedding of manifolds in the Euclidean space they have already sat | Consider $M = \mathbb R^1$ with the standard inner product, embedded into $\mathbb R^1$ by the multiplication by $2$ map $x \mapsto 2x$. Then distance is not preserved, but parallel transport remains the same.
| 2 | https://mathoverflow.net/users/52918 | 307351 | 133,917 |
https://mathoverflow.net/questions/307355 | 3 | Let $n \in \mathbb{N}$. Define the *radical* $R(n)$ of $n$ by
$$\displaystyle R(n) = \prod\_{p | n} p.$$
In other words, $R(n)$ is the largest square-free number which divides $n$.
For an integer $k \geq 2$ we say that an integer $n$ is $k$-free (generalization of square-free) if $p | n$ implies that $p^k \nmid ... | https://mathoverflow.net/users/10898 | Number of $k$-free integers of bounded radical | If $m=p\_1\cdots p\_{\ell}$ is square-free, then the $k$-free integers $n$ that have $m$ as a radical are given by
$$
\prod\_{j=1}^{\ell} p\_j^{a\_j}
$$
with $1\le a\_j \le (k-1)$. Clearly there are $(k-1)^{\ell}$ such integers $n$. Therefore the problem amounts to evaluating
$$
\sum\_{\substack{m \le x \\ m \text... | 3 | https://mathoverflow.net/users/38624 | 307358 | 133,918 |
https://mathoverflow.net/questions/307362 | 0 | I'm wondering if it is possible to run a set of numbers ('target numbers') through a function and get out a number that when queried against in some way (with a 'target' number as all/part of the query) would return a true / false as to whether or not that number exists in the output of the function. Tough for me to ve... | https://mathoverflow.net/users/127278 | Derive unique denominators from number | I assume that you care about the members of the list and not their order or possible repeats.
Consider the function $f(S)=\sum\_{s \in S}2^s.$ That is in some sense the most efficient possible function which gives a unique output to every set of numbers. Every positive integer is a possible output.
The function $g(... | 1 | https://mathoverflow.net/users/8008 | 307364 | 133,919 |
https://mathoverflow.net/questions/307123 | 3 | I am reading [Differentiable Stacks and Gerbes](https://arxiv.org/abs/math/0605694) by Kai Behrend and Ping Xu.
They define gerbe over a stack as follows.
>
> Let $\mathfrak{X}$ be a differentiable stack. An $\mathfrak{S}$-stack $\mathfrak{R}$ endowed with a morphism $F:\mathfrak{R}\rightarrow \mathfrak{X}$ is ca... | https://mathoverflow.net/users/118688 | Understanding definition of gerbe over a stack | I am trying to write down what does it mean to say those two maps $\mathcal{D}\rightarrow \mathcal{C}$ and $\mathcal{D}\rightarrow \mathcal{D}\times\_{\mathcal{C}}\mathcal{D}$ to be epimorphisms.
I am not very comfortable to use the definition of epimorphism as in [Differentiable Stacks and Gerbes](https://arxiv.org... | 1 | https://mathoverflow.net/users/118688 | 307366 | 133,920 |
https://mathoverflow.net/questions/307159 | 12 | The question is in the title:
>
> **Q1:** Is there a topological space $X$ containing a copy of the real line and having the property that all the nonempty open subsets of $X$ are homeomorphic?
>
>
>
Let us say that $X$ is a *homeomorphic open set space*, or a *hoss* for short, if all the nonempty open subset... | https://mathoverflow.net/users/70618 | Can the real line be embedded in a space $X$ such that all the nonempty open subsets of $X$ are homeomorphic? | A metrizable example can be constructed as follows. In the plane consider the subset $$\Xi:=\big\{(x,\tfrac{2k+1}{2^n}):k,n\in\mathbb Z,\;x\in\mathbb R\setminus \tfrac1{2^n}\mathbb Z\big\}.$$
It is clear that $\Xi$ contain (countably many) topological copies of the real line.
There are at least two ways of proving ... | 8 | https://mathoverflow.net/users/61536 | 307368 | 133,922 |
https://mathoverflow.net/questions/307345 | 1 | I have a set of polynomials $f\_1, \dots, f\_m \in \mathbb{Z}[x\_1, \dots, x\_n]$ and I am interested in finding if these polynomials have a common root inside either $\mathbb{C}[x\_1, \dots, x\_n]$ or $\overline{\mathbb{F}\_p}[x\_1, \dots, x\_n]$ for some prime $p$. One way to do this is to calculate the Gröbner basis... | https://mathoverflow.net/users/121829 | Finding a characteristic for which the zero-locus of an ideal is not empty | A nice algorithm (with very readable pseudocode) is given by Susan Margulies in her [thesis.](https://www.usna.edu/Users/math/margulies/papers/thesis.pdf) There are a number of published papers (joint with her advisor Jesus de Loera, not sure which one is the best published reference). She had implemented the algorithm... | 1 | https://mathoverflow.net/users/11142 | 307370 | 133,924 |
https://mathoverflow.net/questions/307375 | 5 | I have already found two definitions for a Baer group.
1. $G$ is a Baer group if it is generated by all cyclic subnormal subgroups.
2. $G$ is a Baer group if every cyclic subgroup is subnormal.
I want to prove the equivalence of the two definitions. Obviously, (2) implies (1). Please help me with the converse.
| https://mathoverflow.net/users/114477 | Prove that if a group $G$ is generated by all cyclic subnormal subgroups, then every cyclic subgroup is subnormal | **Claim**. If group is generated by cyclic subnormal groups, then every finitely generated subgroup is subnormal.
*Proof*. Let $A, B < G$ be f. g., nilpotent, and subnormal. We want to prove that $C := \langle A, B \rangle$ has this properties. (Then every cyclic will be subnormal, because every subgroup of nilpotent... | 3 | https://mathoverflow.net/users/81055 | 307385 | 133,927 |
https://mathoverflow.net/questions/307387 | 2 | Let $(M^n,g)$ be a Riemannian manifold with non-empty smooth boundary $\partial M$. For any two points $x,y\in M$, the distance between $x$ and $y$ may be defined as
$$ d(x,y)=\inf\_\gamma Length(\gamma), $$
where the infimum is taken over all $C^1$ curves lying in $M$. Can we prove there exists a path in the closure $... | https://mathoverflow.net/users/119968 | The existence of length-minimizing path between two points in a Riemannian manifold with boundary | As stated the question is unclear. What do you mean by the "closure"? Metric completion? If so, what do you mean by piecewise $C^1$? Do you know the answer when the boundary is empty? The boundary does not seem to matter here.
The standard result of this type is the Hopf-Rinow theorem which implies that any two poin... | 8 | https://mathoverflow.net/users/1573 | 307391 | 133,929 |
https://mathoverflow.net/questions/304881 | 4 | Let $f\_1,\ldots,f\_r \in \mathbb{R}[x\_1,\ldots,x\_n]$ be $r$ homogeneous polynomials of the same odd degree $d$, where $d \in \{3,5,7,\ldots\}$.
>
> For which values of $r,n,d$ there exists a **real** solution $a=(a\_1,\ldots,a\_n) \in \mathbb{R}^n-\{(0,\ldots,0)\}$
> to the system of $r$ inequalities $f\_i \geq... | https://mathoverflow.net/users/72288 | Solutions to a system of homogeneous equations (inequalities) | I cannot say much about $\geq0$. For $=0$ one can prove the following statement: if $r\leq n-1$ then then there is an $a\in \mathbf R^n\setminus\{0\}$ such that $f\_i(a)=0$ for all $i$.
Indeed, each polynomial $f\_i$ defines a vanishing set $Z(f\_i)$ in real projective $(n-1)$-space $\mathbf P^{n-1}(\mathbf R)$. Sinc... | 1 | https://mathoverflow.net/users/85592 | 307395 | 133,930 |
https://mathoverflow.net/questions/307378 | 0 | Let $p\_1, p\_2, t\_1, t\_2, a \in \mathbb{C}$ be constants. Consider the following plane complex curve in $\mathbb{C}^2$ ($c\_1, c\_2$ are indetermniates)
\begin{align}
& {p\_1}^2 {p\_2}^2 c\_1 {t\_1}^2 t\_2 + {p\_1}^2 p\_2 {c\_1}^2 c\_2 {t\_1}^2 t\_2 + {p\_1}^2 p\_2 c\_2 {t\_1}^4 + {p\_1}^2 c\_1 {c\_2}^2 {t\_2}^3 \\
... | https://mathoverflow.net/users/11877 | How to classify a plane complex curve? | A computation with magma reveals that this curve is isomorphic, as a genus two curve over the function field $\mathbb{C}(p\_1,p\_2,t\_1,t\_2,a)$, to the hyperelliptic curve given by the Weierstrass equation
$$y^2=f(x):=4t\_2^6x^6 + 4p\_1t\_1^2t\_2^4x^5 + (p\_1^2t\_1^4t\_2^2 - 4p\_1t\_1^2t\_2^4a -
4p\_2t\_1^2t\_2^4)x^... | 2 | https://mathoverflow.net/users/24442 | 307399 | 133,933 |
https://mathoverflow.net/questions/307392 | 4 | If $n$ is in the range of the Euler totient function, certain multiples of $n$ are likewise guaranteed to be totient values. The simplest nontrivial example of this is that, if $n$ is in the range of totient, so is $2n$:
Write $n = \phi(k)$.
If $k$ is odd, then $2n = \phi(4k)$. If $k$ is even, then $2n = \phi(2k)... | https://mathoverflow.net/users/12610 | The range of the Euler totient function and multiplication by 28 | The answer is: *no*.
Here is the smallest counterexample: take $n = 29\cdot7645373 = 221715817 = 29\cdot 197^3$.
Then $\phi(n) = 212983792$ and $28\cdot\phi(n)=5963546176$, but the last number is not totient of any number.
I found this using the following [pari](https://pari.math.u-bordeaux.fr/) one-liner:
``... | 10 | https://mathoverflow.net/users/39495 | 307400 | 133,934 |
https://mathoverflow.net/questions/307397 | 0 | Let $C$ be the following curve in $\mathbb{C}^2$.
\begin{align}
& 11664\, {c\_1}^3\, {c\_2}^2 + 536544\, {c\_1}^3\, c\_2 + 6170256\, {c\_1}^3 + 67068\, {c\_1}^2\, {c\_2}^2 + 1542564\, {c\_1}^2\, c\_2 \\
& + 3085128\, c\_1\, {c\_2}^2 - 32393844\, c\_1\, c\_2 + 3085128\, c\_1 + 17739486\, {c\_2}^2 + 6941538\, c\_2 = 0.
\... | https://mathoverflow.net/users/11877 | How to find the normal form of an elliptic curve? | First, you need a nonsingular point of the curve. The only singular points are the two points at infinity, so the point $(0,0)$ is safe.
Now, you can use magma to, first, define the affine curve, then its projective closure $CP$, and then call the function EllipticCurve(CP, P);
where $P$ is the point $CP![0,0,1]$. I... | 4 | https://mathoverflow.net/users/24442 | 307404 | 133,935 |
https://mathoverflow.net/questions/307323 | 4 | Is there $C > 0$ such that the inequality
$$
\prod\_{n\in\mathbb{N}} p(n)^{a\_n} \leq p\left(C\prod\_{n\in\mathbb{N}} n^{a\_n}\right)
$$
holds for all finitely supported sequences $(a\_n)$ with $a\_n\geq 0$ and $\sum\_n a\_n = 1$ and polynomials with nonnegative coefficients $p\in \mathbb{R}\_+[X]$?
Is it even possib... | https://mathoverflow.net/users/27013 | A kind of exponential concavity for polynomials? | The example by Philipp Lampe can be modified to prove that no such $C$ exists:
Suppose $k$ is an arbitrary positive integer and take $p(x)=x(x+k^2-1)$. Now choose $a\_1=a\_n=\frac{1}{2}$ for some $n\geq 1$ and we get that $C$ must satisfy
$$\sqrt{p(1)p(n)}\le p(C\sqrt{n})$$
$$\iff \sqrt{k^2n(n+k^2-1)}\le C\sqrt{n}(C\... | 5 | https://mathoverflow.net/users/2384 | 307413 | 133,937 |
https://mathoverflow.net/questions/307285 | 4 | I want to understand the framing dependence of the Khovanov-Rozansky homology, and as its first step, I am trying to understand the framing dependence of the HOMFLY polynomial (i.e. quantum $sl(n)$ invariants). (Of course, both of them can be defined in a framing independent manner, but as I am mostly interested in qua... | https://mathoverflow.net/users/45553 | Framing dependence of HOMFLY polynomial | It turned out that it was a very simple calculation to compute $\langle \square,\square+2\rho\rangle$ for $sl(n)$ using Cartan matrices. For $sl(n)$, $\langle \square,\square+2\rho\rangle = 2-\frac{1}{n}$, and hence $$t\_{\square\square} = q^{2-\frac{1}{n}}.$$ Note, by the way, that the convention used here is $q=e^{\f... | 1 | https://mathoverflow.net/users/45553 | 307414 | 133,938 |
https://mathoverflow.net/questions/307403 | 19 |
>
> **Question.** *Suppose $m>n$ are positive integers. Is there a one-to-one $f: \Bbb{R}^m \to \Bbb{R}^n$ such that the graph $\Gamma\_f$ of $f$ is closed in* $\Bbb{R}^{m+n}$?
>
>
>
**Remark 1.** The answer to the above question is well-known to be negative by basic results in topological dimension theory if th... | https://mathoverflow.net/users/9269 | Can an injective $f: \Bbb{R}^m \to \Bbb{R}^n$ have a closed graph for $m>n$? | There is no such function.
Suppose $f: \mathbb R^m \rightarrow \mathbb R^n$ is an injective function with $\Gamma\_f$ closed in $\mathbb R^{m+n}$. For each $i \in \mathbb N$, let $K\_i = f^{-1}([-i,i]^n)$.
I claim that each $K\_i$ is closed and nowhere dense in $\mathbb R^m$.
$K\_i$ is closed because it is the ... | 20 | https://mathoverflow.net/users/70618 | 307419 | 133,942 |
https://mathoverflow.net/questions/307401 | 7 | Let $k\_0$ be a field of characteristic 0, and let $k$ is a fixed algebraic closure of $k\_0$.
Write $G={\rm Gal}(k/k\_0)$.
Let $A\_0$ be a finitely generated commutative $k\_0$-algebra with a unit.
Then $A:= A\_0 \otimes\_{k\_0} k$ is a finitely generated commutative $k$-algebra with a unit.
The Galois group $G $ ac... | https://mathoverflow.net/users/4149 | Infinite Galois descent for finitely generated commutative algebras over a field | As an alternative to my first answer, here is an easier (but less powerful) approach that uses the finite generation in a more essential way:
Let $k \to \ell$ be a Galois field extension with Galois group $G$. Let $B$ be an $\ell$-algebra with an $\ell$-semilinear action $G \times B \to B$ with open stabilisers. Assu... | 5 | https://mathoverflow.net/users/82179 | 307433 | 133,948 |
https://mathoverflow.net/questions/307434 | 7 | Let $k$ be a field, let $X/k$ be a nodal curve (which means $X\_{\overline{k}}$ is a connected reduced proper curve with at worst nodal singularities.)
Does there always exists a proper flat family $\mathcal{X}/\mathrm{Spec}(k[[t]])$ with special fiber $X$ and smooth generic fiber? (Is there a reference for this?)
| https://mathoverflow.net/users/nan | Smoothen a nodal curve | Yes. One reference I know is Brian Conrad's appendix to Matt Baker's paper, "[Specialization of linear systems from curves to graphs](https://projecteuclid.org/euclid.ant/1513797305)". See, in particular, Theorem B.2 therein.
| 4 | https://mathoverflow.net/users/37821 | 307437 | 133,949 |
https://mathoverflow.net/questions/307252 | 7 | Let $\mathbf{B}$ and $\mathbf{B'}$ be strict bicategories and $F: \mathbf{B} \to \mathbf{B'}$ a weak functor which preserves horizontal composition strictly (i.e. $Ff \* Fg = F(f \* g)$ natural in f and g.)
Does this imply $F$ preserves identities strictly, i.e. $1\_{F\_a} = F(1\_a)$?
With the unit axiom for weak f... | https://mathoverflow.net/users/118675 | Is a weak functor which strictly preserves horizontal composition and which runs between strict bicategories automatically strict? | It seems to me that the simplest possible counterexample works. Restrict first to the case when $\mathbf{B}$ and $\mathbf{B}'$ have one object, so we are talking about a strong monoidal functor $F:C\to D$ between strict monoidal categories. Now let $C$ be the terminal strict monoidal category, with one object $I$, only... | 4 | https://mathoverflow.net/users/49 | 307441 | 133,951 |
https://mathoverflow.net/questions/307439 | 1 | Problem: Let $a\_1\dots a\_k$ be integers in $Z\_{n}$ such that $n=k(k-1)+1$ and that the list of differences $a\_i-a\_j \bmod n$ is unique to $i,j$ (for $i\neq j$). Such a set exists for $k=1\dots6,8$. No such set exists for $k=7$? (Verified by exhaustive search, but I can't figure out why that case would be different... | https://mathoverflow.net/users/103753 | Perfect Difference Set of order 7^2-7+1=43 | Ok, thanks to @Seva giving me the right name for these, I found out that:
* The sets I'm describing are Perfect Difference sets.
* The one-dimensional version I'm working with are equivalent to projective planes.
* <https://en.wikipedia.org/wiki/Projective_plane> says that there is no plane for $j=6$ $(j^2+j+1)$, whi... | 1 | https://mathoverflow.net/users/103753 | 307457 | 133,957 |
https://mathoverflow.net/questions/307325 | 7 | I want to find a visual proof of the following fact:
>
> For any convex figure in the plane there is a sequence of Steiner's symmetrizations that makes it arbitrary close to a circular disc.
>
>
>
All proofs I know require some integral estimates. I would prefer a more visual proof (even if it is more involved... | https://mathoverflow.net/users/1441 | Visual proof of convergence for Steiner's symmetrization | (1) Let
$$ C = {\rm conv}\ B\bigcup \{x\}$$ where $B$ is a closed ball of center $o$ and
$x$ is not in $B$.
Consider arc $A\subset B$, $$ \partial B \bigcap {\rm Int}\ C $$
Note that by any Steiner symmetrization $S\_L$, $A$ is still in the interior ${\rm Int}\ S\_L(
C)$.
If $x\_1$ is close to $x$ s.t. $\overrigh... | 4 | https://mathoverflow.net/users/36572 | 307463 | 133,958 |
https://mathoverflow.net/questions/307455 | 5 | This is a linear algebra question that came up in my research, and I feel like there ought to be either a simple proof or a simple counterexample, but I have been unable to find either.
Assume $V$ and $W$ are real finite-dimensional inner product spaces, so there is an induced inner product on $\text{Hom}(V,W)$ defin... | https://mathoverflow.net/users/22948 | invertible endomorphisms on a space of linear maps between finite-dimensional vector spaces | It does not follow that $\Phi$ is invertible.
Consider $V=W=\mathbb{R}^2$ together with the standard inner product. With respect to the standard basis we identify $\operatorname{Hom}(V,W)$ with the space of $2\times 2$ real matrices. We define the map $\Phi$ by
\begin{align\*}
\operatorname{Mat}\_{2,2}(\mathbb{R})&\t... | 9 | https://mathoverflow.net/users/296 | 307466 | 133,959 |
https://mathoverflow.net/questions/307465 | 2 | Let $\newcommand{\GG}{\mathbf{G}}\newcommand{\g}{\mathfrak{g}}\GG$ be a connected semisimple algebraic group over the algebraically closed field $k=\overline{\mathbb{F}\_q}$, and let $\g$ be its Lie-algebra. Let $x\in\g$ be a semisimple element. My question is the following-
>
> Assuming that the characteristic of ... | https://mathoverflow.net/users/14443 | Connectedness of centralizers of semisimple Lie-algebra elements under the action of a semisimple algebraic group | This is true for $p$ not a torsion prime of $G$; it is exactly Theorem 3.14 (p. 88) of [Steinberg - Torsion in reductive groups](http://www.sciencedirect.com/science/article/pii/0001870875901255) ([MSN](http://mathscinet.ams.org/mathscinet-getitem?mr=354892)). In fact, as you probably know, it's always good to look in ... | 7 | https://mathoverflow.net/users/2383 | 307467 | 133,960 |
https://mathoverflow.net/questions/307452 | 1 | Let $H$ be a separable Hilbert space and $L^{1}(H)$ be the space of trace class operators on $H$.
Let $f:\mathbb{R}\to L^{1}(H)$ be a measurable function that is $t\to \operatorname{tr}(f(t))$ is Borel measurable.
>
> Q. Suppose that $\int \operatorname{tr}(f(t))d\mu$ is finite. Can we conclude that $\int \oper... | https://mathoverflow.net/users/84390 | Compatibility of the absolute value with the integration | What if $H$ is two dimensional, $\mu$ = Lebesgue measure and
$$
f(t) = \begin{bmatrix} 1 & 0\\0 &-1\end{bmatrix}
$$
so that $\mathrm{tr}(f(t))=0$
Then I guess
$$
|f(t)| = \begin{bmatrix} 1 & 0\\0 &1\end{bmatrix}
$$
so that $\mathrm{tr}\big(|f(t)|\big) = 2$ is not integrable.
| 2 | https://mathoverflow.net/users/454 | 307468 | 133,961 |
https://mathoverflow.net/questions/307461 | 16 | Let $A$ be a matrix and $x$ a fixed vector. How can we determine whether or not there exists a permutation matrix $P$ such that $APx=0$? Does this problem reduce to anything well-understood?
| https://mathoverflow.net/users/41669 | Determining if some permutation of a vector satisfies a system of linear equations | Let's see if I can convince everyone that this problem is NP-complete.
First: it is in NP because a permutation $P$ can be guessed and checked in polynomial time.
I'll restate the problem: Given a vector $x$ and a vector space $V$ (the null-space of $A$), is there a permutation of $x$ that lies in $V$?
I'll take ... | 23 | https://mathoverflow.net/users/9025 | 307470 | 133,962 |
https://mathoverflow.net/questions/307471 | 5 | Let $\pi:\mathcal{C} \to \Delta$ be a family of projective curves of genus $g \ge 2$ over the unit disc $\Delta$, smooth over the punctured disc $\Delta\backslash \{0\}$ and central fiber $\pi^{-1}(0)$ is an irreducible nodal curve with exactly one node. For $t$ close to $0$, denote by $\delta\_t$ the vanishing cycle o... | https://mathoverflow.net/users/32151 | Degeneration of smooth curves and Picard-Lefschetz formula | Because the symplectic form on $H\_1(C\_t, \mathbb Z)$ is a perfect pairing, it suffices to check that there is a group homomorphism $H\_1(C\_t,\mathbb Z) \to \mathbb Z$ that sends $ \gamma$ to $1$, which follows if $\gamma$ is not divisible by any $n>1$ in $H\_1(C\_t,\mathbb Z)$.
Because $\gamma$ is defined as the g... | 8 | https://mathoverflow.net/users/18060 | 307477 | 133,964 |
https://mathoverflow.net/questions/266795 | 1 | Assume that $X$ is a compact metric space and $G$ is compact
totally disconnected group. And $X$ has isometric free $G$-action
i.e. $gx=x\Rightarrow g=e$.
>
> Then the following holds $${\rm dim}\ X/G={\rm dim}\ X-{\rm
> dim}\ G\cdot o$$ where $o\in X$ and dimension is Hausdorff dimension ?
>
>
>
Thank you... | https://mathoverflow.net/users/36572 | Dimension of quotient of compact totally disconnected group action | It may happen that
$$\dim\_H X/G<\dim\_H X-\dim\_H (G\cdot o);$$
the following example is almost identical to Example 7.8 in "Fractal geometry" by Kenneth Falconer which provides spaces $X$ and $Y$ such that
$$\dim\_H (X\times Y)>\dim\_HX+\dim\_HY.$$
Denote by $W(\varepsilon)$ the two-point space with distance $\vare... | 2 | https://mathoverflow.net/users/1441 | 307485 | 133,965 |
https://mathoverflow.net/questions/307373 | 12 | Yesterday I observed (and proved) the following odd fact, which I found very surprising. I'm very curious to know if this was known by some people, or if it follows from some other more general fact, or if anyone had any kind of comments about it. (for example "this is false" would be a very helpful comment ! ). I hope... | https://mathoverflow.net/users/22131 | On a surprising property of free theories | This is a consequence of [first-order unification](https://en.wikipedia.org/wiki/Unification_(computer_science)). The main issue is the existence of (co)equalizers, since (co)products are trivially seen to exist. Here is a brief explanation of how to translate calculating (co)equalizers into the language of unification... | 4 | https://mathoverflow.net/users/2000 | 307495 | 133,969 |
https://mathoverflow.net/questions/307498 | 1 | In Witten's paper [Three Dimensional Gravity Revisited](https://arxiv.org/pdf/0706.3359.pdf) and [Quantization of Chern-Simons Theory with Complex Gauge Group](https://projecteuclid.org/download/pdf_1/euclid.cmp/1104202513), he used a fact that for a principal $G$-bundle, the quantization of the Chern number is reduced... | https://mathoverflow.net/users/120604 | Why is the Chern Number Invariant under A Continuously Shrinking of the Structure Group? | I don't quite understand the quantization part of the question.
However, the characteristic numbers are always given by pairing products of the (characteristic) cohomology classes with the fundamental class of the manifold $M$, say. Such classes are pullbacks of classes in the cohomology of $BG$, where $G$ is the st... | 3 | https://mathoverflow.net/users/8032 | 307500 | 133,970 |
https://mathoverflow.net/questions/307502 | 2 | Let $X\_1, \ldots, X\_n$ be independent Bernoulli random variables. Then $Pr[X\_i=1]=Pr[X\_i=0]=1/2$. Let $X = (X\_1, \ldots, X\_n)$ and $v \in \{0,1\}^n$, $Y=v \cdot X$, $Z=Y-1$. Let
\begin{align}
\mu\_1(x) = Pr[Y=x], \ \mu\_2(x)=Pr[Z=x], \ x \in [n]=\{1, \ldots, n\}.
\end{align}
Are there some method to estimate th... | https://mathoverflow.net/users/11877 | How to estimate a total variation distance? | For the TV-distance, we have
\begin{align}
d\_{TV}= \frac{1}{2} \sum\_{x=0}^k |d\_x|
\end{align}
(with the summation actually beginning at $x=0$), where
\begin{multline\*}
d\_x:=P(S\_k=x) - P(S\_k=x+1)=\frac1{2^k}\,\Big(\binom kx-\binom k{x+1}\Big) \\
=\frac1{2^k}\,\frac{k!}{(x+1)!(k-x)!}\,(2x-(k-1))
\end{multlin... | 3 | https://mathoverflow.net/users/36721 | 307506 | 133,972 |
https://mathoverflow.net/questions/307487 | 8 | I want to find some algebraic models relating string topology to Hochschild and cyclic homologies. If the space $X$ is simply-connected and we are working over rational numbers, we can use Sullivan model or Quillen model of $X$ to do the job. See [nLab](https://ncatlab.org/nlab/show/Sullivan+model+of+free+loop+space) p... | https://mathoverflow.net/users/91245 | Algebraic models of non-simply connected spaces in string topology | In my paper with Zeinalian (<https://arxiv.org/abs/1612.04801>) we prove that the coHochschild complex of the dg coalgebra of singular chains on a path connected (possibly non-simply connected) space calculates the homology of the free loop space of $X$.
In my paper with Saneblidze (<https://arxiv.org/abs/1712.02644... | 11 | https://mathoverflow.net/users/5450 | 307508 | 133,973 |
https://mathoverflow.net/questions/307482 | 34 | Is there a place to watch ICM 2018 plenary lectures (and other lectures if possible)?
Here is the official Youtube channel of the ICM but they don't seem to be posting the lectures.
<https://www.youtube.com/channel/UCnMLdlOoLICBNcEzjMLOc7w>
**Update:** The public lectures of Etienne Ghys, Cedric Villani, and Ingr... | https://mathoverflow.net/users/824 | ICM 2018 lecture videos | **Update 2018.09.22.** Go to the website icm2018.org, and use the menu icon (three horizontal bars) to get a menu list that has ICM videos on it. That takes you to a YouTube channel which has some of the plenary and invited lecture videos being uploaded. If there is a problem accessing or playing the videos please let ... | 20 | https://mathoverflow.net/users/3402 | 307515 | 133,975 |
https://mathoverflow.net/questions/307507 | 3 | I want to know if it is possible to express the operation
$$
\nabla \phi \times (\nabla \times \mathbf A)
$$
as the divergence of second order tensor field $T$. Here $ \phi$ is a scalar field and $\mathbf A$ is a solenoidal vector field ($\nabla \cdot \mathbf A=0$)
I have used all possible identities and finally... | https://mathoverflow.net/users/127354 | Curl as a divergence... Is it possible? | Let me expand my comment that I think this is only possible if the Laplacian of $\phi$ vanishes.
The object we are considering is
$$\mathbf V\equiv\nabla\phi\times(\nabla\times \mathbf A)=(\nabla \mathbf A)·\nabla \phi - (\nabla\phi)·\nabla \mathbf A.$$
Writing this in components,
$$V\_i=\sum\_j(\partial\_i A\_j) (\... | 7 | https://mathoverflow.net/users/11260 | 307522 | 133,979 |
https://mathoverflow.net/questions/307526 | 13 | This may be an easy question or it may be related to a well known open problem in Computer Science.
Let $\alpha>0$. We say that $\alpha$ is computed in time $T(n)$ if there is a Turing machine which for every $n>0$ written in binary produces a finite binary approximation of $\alpha$ with error bounded by $\frac 1{2^... | https://mathoverflow.net/users/nan | Real numbers with given complexity | Yes. First use the [Time Hierarchy Theorem](https://en.wikipedia.org/wiki/Time_hierarchy_theorem) to find a problem in $DTIME(2^{2\cdot 3^{2^n}}) = DTIME({(2^{3^{2^n}})}^2)$ but not in $DTIME(2^{3^{2^n}})$. Let $f(k)$ be the answer (1 for yes, 0 for no) to the k-th instance of this problem. Here the k-th instance is go... | 11 | https://mathoverflow.net/users/45707 | 307530 | 133,982 |
https://mathoverflow.net/questions/307531 | -3 | Can you prove that there are infinite palindromic primes that when squared give a palindromic number?
| https://mathoverflow.net/users/127375 | Is it true that there are infinite palindromic primes that when squared give palindromic number? | There are in fact more than those 4, and they have [their own page on OEIS](https://oeis.org/A225603). Two conjectures, then, would be:
* are they infinitely many? I suspect so.
* are $2$ and $3$ the only ones formed with *decimal* digits other than $0$ and $1$? I suspect so.
| 8 | https://mathoverflow.net/users/54780 | 307535 | 133,984 |
https://mathoverflow.net/questions/307383 | 4 | There is a notion of Gerbe over a Manifold and a notion of Gerbe over a stack. Given a manifold $M$, there is a way to associate a stack $\underline{M}$ with it and this gives an embedding of category of smooth manifolds into category of Categories fibered in groupoids which can be found in [Orbifolds as Stacks](https:... | https://mathoverflow.net/users/118688 | Is a gerbe over a manifold is a special case of a gerbe over a stack? | Yes. The category of manifolds embeds fully faithfully into the 2-category of stacks on Mfld, essentially by Yoneda, and the site structures are likewise compatible, so when restricted to the special case of your stack being a manifold, the definitions are equivalent.
| 3 | https://mathoverflow.net/users/4177 | 307538 | 133,985 |
https://mathoverflow.net/questions/307533 | 5 | Theorem 9.3.1 in Hall's group
theory says: Let $G$ be a solvable group and $|G|=m\cdot n$, where $%
m=p\_{1}^{\alpha \_{1}}\cdot \cdot \cdot p\_{r}^{\alpha \_{r}}$, $(m,n)=1$. Let $%
\pi =\{p\_{1},...,p\_{r}\}$ and $h\_{m}$ be the number of $\pi -$Hall subgroups
of $G$. Then $h\_{m}=q\_{1}^{\beta \_{1}}\cdot \cdot \cdo... | https://mathoverflow.net/users/127363 | Question on Hall's theorem | I think the answer is yes. The number of Sylow $p$-subgroups of $G$ is $[G:N\_{G}(P)],$ where $P$ is a Sylow $p$-subgroup of $G$. This number is unchanged
if we pass to $G/O\_{p}(G),$ so we might as well suppose that $O\_{p}(G) = 1.$ Then since $G$ is $p$-solvable, we have $O\_{p^{\prime}}(G) = N \neq 1.$
The Schur-... | 4 | https://mathoverflow.net/users/14450 | 307550 | 133,987 |
https://mathoverflow.net/questions/307537 | 4 | Let $X= x\_1 + x\_2 + \ldots + x\_m$, $Y=y\_1 + y\_2 + y\_3 + \ldots + y\_n$, and $Y' = y'\_1 + y'\_2 + \ldots + y'\_n$, where
* Each $x\_i$ is a Bernoulli variable which takes value $1$ with probability $p\_i>0$.
* Each $y\_i$ is a Bernoulli variable which takes value $1$ with probability $q\_i>0$.
* For $i>2$, eac... | https://mathoverflow.net/users/51176 | $Pr(A>B)$, where $A$ and $B$ are sum of Bernoullies | Both $A>A'$ and $A'>A$ are possible.
Let $S:= Y\_3+Y\_4+\cdots Y\_n$.
Assuming independence, we have
\begin{align}
\triangle A := A-A' &=
\mathbb{P}(Y\geq X) - \mathbb{P}(Y'\geq X) \\
&=
\sum\_a\mathbb{P}(X=a)\big[\mathbb{P}(Y\geq a) - \mathbb{P}(Y'\geq a)\big] \\
&=
\sum\_a\mathbb{P}(X=a)\Big[\sum\_{b\geq a}\m... | 4 | https://mathoverflow.net/users/23297 | 307557 | 133,988 |
https://mathoverflow.net/questions/307493 | 5 | It is well-known that if a topological space $Y$ is
locally compact (not necessarily Hausdorff),
then the map
$$
\operatorname{Hom}(X \times Y, Z) \to
\operatorname{Hom}(X, Z^Y)
$$
(here we use the compact-open topology for $Z^Y$)
is bijective for arbitrary topological spaces $X, Z$.
**Does the converse of this hold?**... | https://mathoverflow.net/users/119076 | Exponential law w.r.t. compact-open topology | The answer is yes. Let $C(Y,Z)$ denote the compact-open topology on $Z^Y$ and let $E(Y,Z)$ denote the exponential topology on $Z^Y$. Let $S$ denote the Sierpinski space. In the following, we will identify $S^Y$ with the set of open subsets of $Y$ in the natural way.
**Claim:** If $Y$ is core-compact and $C(Y,S)$ refi... | 2 | https://mathoverflow.net/users/2362 | 307565 | 133,989 |
https://mathoverflow.net/questions/307568 | 13 | Let $D(\mathbb R) $ be the set of all differentiable functions $f: \mathbb R \to \mathbb R$. Then obviously $D(\mathbb R)$ forms a semigroup under usual function composition. Can we characterize (up to semigroup isomorphism) all finite subsemigroups of $D(\mathbb R)$ which do not contain any constant function ?
| https://mathoverflow.net/users/127387 | Semigroup of differentiable functions on real line | **Yes, we can!** ;-)
In fact there are only **two** finite subsemigroups of $D(\mathbb{R})$ which do not contain constant functions.
---
Every finite subsemigroup of $D(\mathbb{R})$ necessarily contains an idempotent $f$, i.e. a function $f$ such that $f \circ f = f$. Let us examine the properties of $f$.
Let... | 8 | https://mathoverflow.net/users/108637 | 307586 | 133,993 |
https://mathoverflow.net/questions/307588 | 0 | Consider the following optimization problem in positive integers $n\_1, n\_2, n\_3$.
$$\begin{array}{ll} \text{maximize} & n\_1(n\_2+n\_3)\\ \text{subject to} & n\_1+n\_2+n\_3 = N\end{array}$$
If $n\_1, n\_2, n\_3$ were reals, the solution would be $n\_1 = \frac N2$ and $n\_2 = n\_3$. However, in my problem, $n\_1,... | https://mathoverflow.net/users/120111 | A simple quadratic integer optimization problem | It is readily apparent that:
1. if $N \le 2$, the problem is infeasible
2. if $N = 3$, the optimal solution is [1,1,1]
3. If $N = 4$, the optimal solution is [2,1,1]
4. If $N \ge 5$ and odd, then the optimal solutions are
a) $n\_1 = \lfloor N/2 \rfloor$, $n\_2$ and $n\_3$ being any combination of positive integers ... | 2 | https://mathoverflow.net/users/75420 | 307592 | 133,995 |
https://mathoverflow.net/questions/307551 | -5 | Galatius and Randal-Williams proved the following generalized Mumford conjecture in their joint paper, ["*Stable Moduli spaces of High Dimensional Manifolds*"](https://arxiv.org/pdf/1201.3527.pdf). For each characteristic class of oriented $2n$-dimensional vector bundles $c\in H^{2n+k}(BSO(2n))$, one can define the ass... | https://mathoverflow.net/users/98896 | Computing $H^*(BDiff(W_{\infty},D^{\infty});\mathbb{Q})$ via Mumford-Morita-Miller classes | As suggested by Dr. Hatcher, the dimension of the manifold $W\_g$, $2n$, is fixed and one considers embedded submanifolds of $\mathbb{R}^N$ diffeomorphic to $W\_g$. Then one lets $N$ go to infinity via the natural inclusion of $\mathbb{R}^N$ in $\mathbb{R}^{n+1}$, so submanifolds of $\mathbb{R}^N$ are regarded as subma... | 0 | https://mathoverflow.net/users/98896 | 307598 | 133,998 |
https://mathoverflow.net/questions/307596 | 1 | Let $E$, $F$ be two complex Hilbert spaces and $\mathcal{L}(E)$ (resp. $\mathcal{L}(F)$) be the algebra of all bounded linear operators on $E$ (resp. $F$).
The algebraic tensor product of $E$ and $F$ is given by
$$E \otimes F:=\left\{\xi=\sum\_{i=1}^dv\_i\otimes w\_i:\;d\in \mathbb{N}^\*,\;\;v\_i\in E,\;\;w\_i\in F
\... | https://mathoverflow.net/users/113054 | The tensor product of two bounded operators | I think this is easy. It follows from the simple fact that if $E\_0$ is a dense linear subspace of $E$ then $T(E\_0)$ is dense in $T(E)$. Thus Im$(T)\otimes$ Im$(S) =$ Im$(T \otimes S) = T\hat{\otimes} S(E\otimes F)$ is dense in Im$(T\hat{\otimes} S)$, and it is easy to see that Im$(T)\otimes$ Im$(S)$ is contained in t... | 4 | https://mathoverflow.net/users/23141 | 307599 | 133,999 |
https://mathoverflow.net/questions/307593 | 5 | Suppose $P(\lambda, i)$ is the probability that a Poisson random variable with average $\lambda$ is equal to $i$, i.e. $\frac{\lambda^i}{e^{\lambda}i!}$
I think the following system of equations always has solution in $x$ and $y$, non-negative real numbers, for any $\alpha>0$ and $k\in \mathbb{N}\_+$
\begin{cases}
... | https://mathoverflow.net/users/127403 | Existence of Solution, System of Equations | Let $a:=\alpha$ and
\begin{equation\*}
F\_k(x,y):=\sum\_{j=0}^\infty \frac{x^j}{j!}\frac{y^{k+j}}{(k+j)!}\,e^{-x-y},
\end{equation\*}
assuming the standard convention $0^0:=1$.
We have to consider the existence of a solution in $x$ and $y$ of the system
\begin{equation\*}
a=F\_k(x,y)=F\_{k+1}(x,y). \tag{1}
\end{eq... | 6 | https://mathoverflow.net/users/36721 | 307606 | 134,000 |
https://mathoverflow.net/questions/307154 | 11 | Let $\mathcal{L}$ be the first order language of ZFC set theory, and let $\mathcal{L}\_{\infty,\infty}$ be the usual infinitary extension of the language allowing arbitrary long disjunctions/conjunctions and quantifications.
What happens if one replaces the usual axiom schema of replacement for ZFC by a new schema ov... | https://mathoverflow.net/users/3199 | Set theory bootstrapping | Let me describe how I understand the question. You want to consider
assertions in the infinitary language
$\newcommand\L{\mathcal{L}}\L\_{\infty,\infty}$, and assert
instances of replacement for formulas in this language.
Thus, our meta-theoretic context should have considerable
set-theoretic resources, in order to h... | 12 | https://mathoverflow.net/users/1946 | 307624 | 134,005 |
https://mathoverflow.net/questions/307575 | 2 | Suppose that we have an uncountable collection $C\_\alpha$ of disjoint Cantor Sets contained in the closed unit interval $I$. Suppose we have ordered the indices $\alpha \in [0,1]$ as well. Then is it possible for the collection to satisfy the following property: If $x\_n \in C\_{\alpha\_n}$ and $x\_n \rightarrow y$ wi... | https://mathoverflow.net/users/110965 | Topologically Ordered Families of Disjoint Cantor Sets in $I$? | Such an enumeration does not exist. To derive a contradiction, take any family of Cantor sets $(C\_\alpha)\_{\alpha\in A}$ in $[0,1]$, indexed by an uncountable subset $A\subset[0,1]$.
Let $\mathcal K[0,1]$ be the space of non-empty compact subsets of $[0,1]$, endowed with the Hausdorff metric $d\_H$. It is well-kno... | 5 | https://mathoverflow.net/users/61536 | 307633 | 134,007 |
https://mathoverflow.net/questions/307597 | 7 | Dear Colleagues and Friends,
Please let me know if you are aware of any references to the following question.
The classical result of Atiyah, Patodi and Singer tells us that if $W$ is a compact oriented Riemannian 4-manifold with boundary $M$ and, moreover, if we assume that near M the metric is isometric to a pr... | https://mathoverflow.net/users/39331 | Atiyah-Patodi-Singer for manifolds with cusps | This type of questions has been investigated systematically by Melrose in the framework of '*c*-calculus', where $c$ stands for the cusp. The basic idea, if I recall correctly is to blow up the boundary with the cusp three times (two blow ups for the boundary, an extra blow-up for the cusp on the boundary) and work wit... | 2 | https://mathoverflow.net/users/18850 | 307634 | 134,008 |
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