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https://mathoverflow.net/questions/316473 | 2 | This is a more elegant version of the original version which can be found below; it is based on a suggestion by [Peter Mueller](https://mathoverflow.net/users/18739/peter-mueller).
Let $\mathbb{N}$ denote the set of positive integers and for $n\in\mathbb{N}$ let $\text{Mat}(n\times n,\mathbb{Z})$ be the set of intege... | https://mathoverflow.net/users/8628 | Set of integer matrices $A$ such that $(A^k)_{k\in\mathbb{N}}$ is eventually periodic | Of course. The eigenvalues of this matrix (over $\mathbb{C}$) may only be zeros and roots of unity (whose minimal polynomial is of degree at most $n$, as they are roots of the characteristic polynomial), and blocks of size bigger than one in the Jordan normal form may only correspond to the eigenvalue zero.
Note als... | 6 | https://mathoverflow.net/users/1306 | 316482 | 137,314 |
https://mathoverflow.net/questions/316484 | 3 | Ariyan Javanpeykar said [here](https://mathoverflow.net/questions/308474/atlas-of-gerbe-over-stack?noredirect=1#comment789174_308474) in comments that,
>
> If the diagonal is representable, then isn't any morphism $S\rightarrow \mathcal{X}$ with $S$ a scheme representable?
>
>
>
I could not find the statement ... | https://mathoverflow.net/users/118688 | Diagonal is representable then any morphism is representable | I guess it is correct (and may be rendered in a simpler way). Ideed, let $\delta:\mathcal{X}\rightarrow \mathcal{X}\times\_{T}\mathcal{X}$ be the diagonal map. If it is representable then every morphism $u : S → \mathcal{X}$ is representable. For $v : V → \mathcal{X}$ another morphism with $V$ a scheme, we have that
$$... | 5 | https://mathoverflow.net/users/6348 | 316487 | 137,315 |
https://mathoverflow.net/questions/316480 | 9 | What is known about weak systems of axiomata that allow one to prove [van der Waerden's theorem](https://en.wikipedia.org/wiki/Van_der_Waerden%27s_theorem)?
van der Waerden's theorem can be used to show that there are infinitely many primes (see below). Is this proof ever not pointless — i.e. is there an axiom system... | https://mathoverflow.net/users/131893 | van der Waerden's theorem in Reverse Mathematics | There is a powerful combinatorial theorem, known as the [Hales–Jewett theorem](https://en.wikipedia.org/wiki/Hales%E2%80%93Jewett_theorem), which readily implies van der Waerden's theorem. On the other hand, the paper below by Matet exhibits *primitive recursive upper bounds* for the Hales–Jewett function $HJ(m,n)$, an... | 15 | https://mathoverflow.net/users/9269 | 316489 | 137,316 |
https://mathoverflow.net/questions/315940 | 6 | Let $G<\mathrm{GL\_n}$ be a simple linear algebraic group defined over a finite field $K$. Let $\mathfrak{g}$ be its Lie algebra. Assume $\mathfrak{g}$ is simple.
Is it necessarily the case that there is no subspace $\mathfrak{v}\subset \mathfrak{g}$ with $0<\dim(\mathfrak{v})<\dim(\mathfrak{g})$ such that $\mathfra... | https://mathoverflow.net/users/398 | "Almost-ideals" in the (simple) Lie algebra of an algebraic group? | You are looking for the following theorem of Steinberg:
Let q=|K|. An irreducible algebraic representation of $G(\overline{K})$ of highest weight λ remains irreducible when restricted to G(K) if $\langle \lambda,a^\vee\rangle <q$ for all simple coroots $\alpha^\vee$.
The reference is [here](https://projecteuclid.or... | 2 | https://mathoverflow.net/users/425 | 316493 | 137,318 |
https://mathoverflow.net/questions/316469 | 4 | Let $G$ be a simply connected simple algebraic group over a field $k$.
Let $\theta\colon G\to G$ be an involution of $G$ over $k$ (an automorphism of order 2).
Let $H=(G^\theta)^0$, the identity component of the fixed point subgroup $G^\theta$ of $\theta$ in $G$.
We say that $H$ is a *symmetric subgroup* of $G$ and tha... | https://mathoverflow.net/users/4149 | Symmetric subgroups of simple algebraic groups over finite fields | The classification of automorphisms of order $2$ in finite and algebraic groups of Lie type over finite fields of odd characteristic, and the structure of their centralizers, are quite well-known. Calculations and tables are available in (chapters 2 and 4 especially of) D. Gorenstein, R. Lyons, and R. Solomon, {\it The... | 10 | https://mathoverflow.net/users/99221 | 316513 | 137,321 |
https://mathoverflow.net/questions/316458 | 3 | Let $A$ be a finite dimensional local Frobenius algebra with simple module $k$ and an indecomposable non-projective module $M$ (that is also finite dimensional).
Question:
>
> Is there an example of such an $M$ such that every map $f: \Omega^n(M) \rightarrow k \rightarrow \Omega^{n-1}(M)$ is zero in the stable mo... | https://mathoverflow.net/users/61949 | Question on $\operatorname{Ext}$ in a local Frobenius algebra | Let $A$ be the six-dimensional algebra $\langle x,y\mid x^2=y^2=0, xyx=yxy\rangle$, and $M$ the three dimensional uniserial (right) module $A/yA$.
Then $\Omega M\cong M$, and so $\Omega^n M\cong M$ for all $n\in\mathbb{Z}$.
Up to scalar multiplication, there is a unique map $M\to M$ that factors through $k$, and th... | 4 | https://mathoverflow.net/users/22989 | 316514 | 137,322 |
https://mathoverflow.net/questions/316510 | 3 | Let $\{a\_n\}\_{n=1}^\infty$ and $\{b\_m\}\_{m=1}^\infty$ be two sequences of points in $\mathbb{C}$ such that
$$
f(z)=\prod\_{n=1}^\infty\left(1-\frac{z}{a\_n}\right)\quad\mbox{and}\quad g(z)=\prod\_{m=1}^\infty\left(1-\frac{z}{b\_m}\right)
$$
are entire functions of finite exponential types $0<A\_f<\infty$ and $0<A\_... | https://mathoverflow.net/users/89313 | Exponential type of a product of entire functions | On the first question: Your inequality is incorrect for exponential type functions. Take $f(z)=e^z,\; g(z)=e^{-z}$, both have exponential type 1. These examples are of course not of the form of infinite product that you request.
But if $f$ is defined by an infinite product as you wrote, then (with usual understanding... | 4 | https://mathoverflow.net/users/25510 | 316520 | 137,323 |
https://mathoverflow.net/questions/316441 | 4 | Let's consider in an $\infty$-topos, we have an object $X$ of homotopy dimension $\leq n$ (in the sense of Lurie HTT), let $f: A\to B$ be an $n$-equivalence morphism. Can we conclude that $f$ induces bijection on homotopy sets
$$
f\_\*: [X, A]\xrightarrow{\cong}[X, B]?
$$
This is a finite analog of Whitehead Theorem, s... | https://mathoverflow.net/users/42571 | A finite Whitehead Theorem for $\infty$-topos | Let $\mathcal{X}$ be the $\infty$-topos in question containing an object $X \in \mathcal{X}$. I assume that by $X$ having homotopy dimension $\leq n$ you mean that the $\infty$-topos $\mathcal{X}\_{/X}$ has homotopy dimension $\leq n$ in the sense of Def. 7.2.1.1 of HTT, and that your notion of an $n$-equivalence is th... | 5 | https://mathoverflow.net/users/51164 | 316522 | 137,324 |
https://mathoverflow.net/questions/316521 | 0 | A graphical representation of a planar graph divides the plane into *regions* or *meshes* (as they are called in certain applications, e.g. in circuit theory).
Yes, the above fact is intuitive, but what I couldn't find in common textbooks on graph theory is a formal definition of mesh related to the definition of gra... | https://mathoverflow.net/users/84320 | On the formal definition of mesh or region for a planar graph | If the graph is already drawn on the plane, the regions are connected components of the set complement of the union of the curves presenting the edges. Sometimes you may define the regions combinatorially. For example, if you have a convex polytope, any face is bounded by a simple cycle such that removing it from the g... | 2 | https://mathoverflow.net/users/4312 | 316523 | 137,325 |
https://mathoverflow.net/questions/316505 | 16 | Let $\omega=\sum\_{i=1}^n dx\_i\wedge dy\_i\in\bigwedge^2(\mathbb{R}^{2n})^\*$ be a standard symplectic form. The following result is due to Lefschetz:
>
> For $k\leq n$, the Lefschetz operator
> $L^{n-k}:\bigwedge^k(\mathbb{R}^{2n})^\*\to
> \bigwedge^{2n-k}(\mathbb{R}^{2n})^\*$ defined by $$
> L^{n-k}\alpha=\alp... | https://mathoverflow.net/users/121665 | The Lefschetz operator | There is an elementary proof in our 2003 book *Exterior Differential Systems and Euler-Lagrange Partial Differential Equations* (Bryant, et al, University of Chicago Press). It does not use any representation theory and is not 'brute force'; it only takes a couple of paragraphs using elementary facts about exterior alg... | 29 | https://mathoverflow.net/users/13972 | 316527 | 137,327 |
https://mathoverflow.net/questions/316531 | 2 | Let $M$ be an $n \times m$ matrix over $\mathbb{F}\_2$ with no repeated columns, and suppose that $m \leq 2^{n-1}$: *i.e.*, it is possible to have a matrix with fewer rows that still has $m$ unique columns.
Is it always possible to find such a smaller matrix by taking a linear combination of the rows of $M$?
For ex... | https://mathoverflow.net/users/69010 | Matrices with distinct columns | It is the same as asking whether given $m\le 2^{n-1}$ distinct vectors $v\_j$ in $\mathbb F\_2^n$ you can find $n-1$ vectors $w\_1,\dots,w\_{n-1}$ such that for each $j\ne k$, we have $w\_\ell\cdot(v\_j-v\_k)\ne 0$ for some $\ell$. You are clearly in trouble if $v\_i-v\_j$ run over the entire $\mathbb F\_2\setminus\{0\... | 4 | https://mathoverflow.net/users/1131 | 316537 | 137,331 |
https://mathoverflow.net/questions/316511 | 2 | Is there a set $B$ about which it provably cannot be decided whether it is [computable](https://en.wikipedia.org/wiki/Recursive_set) in $\mathsf{ZFC}$?
| https://mathoverflow.net/users/8628 | Meta-incomputability | An explicit example: $B=$ the set of all theorems of ZFC.
| 5 | https://mathoverflow.net/users/4600 | 316558 | 137,340 |
https://mathoverflow.net/questions/316552 | 6 | **Short version:**
>
> Why is the projective dimension of a graded module the same as the projective dimension of its underlying ungraded module?
>
>
>
**Longer version:**
Let $G$ be a commutative group, let $R$ be a $G$-graded commutative ring, and let $M$ be a $G$-graded $R$-module. The category of $G$-gr... | https://mathoverflow.net/users/11025 | Projective dimension of graded modules | The other inequality follows from Schanuel's lemma.
For $M$ as is in your question, consider a truncated resolution of projective $G$-graded modules
$$Q\_{n-1}\xrightarrow{f}Q\_{n-2}\to\dots\to Q\_1\to Q\_0\to M.$$
Then
$$U(Q\_{n-1})\xrightarrow{f}U(Q\_{n-2})\to\dots\to U(Q\_1)\to U(Q\_0)\to U(M)$$
is a truncated pr... | 4 | https://mathoverflow.net/users/86006 | 316559 | 137,341 |
https://mathoverflow.net/questions/316538 | 5 | I have a question about the statement of Lemma 5.5.2.1 in Lurie's [`Higher Topos Theory'](http://www.math.harvard.edu/~lurie/papers/HTT.pdf).
``Let $S$ be a small simplicial set, let $f: S\rightarrow \mathcal{S}$ be an object of $\mathcal{P}(S^{op})$, and let $F: \mathcal{P}(S^{op})\rightarrow \widehat{\mathcal{S}}$... | https://mathoverflow.net/users/131918 | A question about HTT Lemma 5.5.2.1 | The middle term should be $\mathcal{P}(S^{op})^{op}$. The Yoneda embedding gives a functor $S^{op} \to \mathcal{P}(S^{op})$. The functor corepresented by $f \in \mathcal{P}(S^{op})$ is given by $Hom\_{\mathcal{P}(S^{op})}(-,f)$, which is a functor $\mathcal{P}(S^{op})^{op} \to \widehat{\mathcal{S}}$.
As a reality che... | 3 | https://mathoverflow.net/users/360 | 316561 | 137,342 |
https://mathoverflow.net/questions/316546 | 5 | Definable subsets of $\mathbb N$ in the language of Presburger arithmetic are exactly the eventually periodic sets and quantifier free part corresponds to Integer Programming with linear inequalities and variations lead to mixed integer linear programming, convex integer programming with convex constraints. What about ... | https://mathoverflow.net/users/10035 | What are the definable sets in Skolem arithmetic? | $\def\mr{\mathrm}$As it happens, quantifier elimination for Skolem arithmetic came up recently in my research. The concise description is that every formula $\phi(x\_1,\dots,x\_k)$ is in $(\mathbb N^{>0},{\cdot})$ equivalent to a Boolean combination of formulas expressing
$$\tag1\bigl|\{p\in\mathbb P:\psi(v\_p(x\_1),\d... | 5 | https://mathoverflow.net/users/12705 | 316563 | 137,343 |
https://mathoverflow.net/questions/316582 | 8 | Recall that a tree on $\omega\_1$ is called Suslin if every chain and antichain are countable. If every level is countable and there are no cofinal branches, then it is called Aronszajn (in particular Suslin is Aronszajn). Moreover, an Aronszajn tree is called special if it is the countable union of antichains.
So sp... | https://mathoverflow.net/users/7206 | Destroying Suslin, nothing special | Chapter IX of Proper and Improper Forcing addresses this issue.
Shelah proves that Souslin's Hypothesis does not imply every Aronszajn tree is special, and he does this by investigating weak notions of specialness that are still incompatible with Souslinity. He shows that there are forcings that "specialize" Aronszaj... | 10 | https://mathoverflow.net/users/18128 | 316589 | 137,347 |
https://mathoverflow.net/questions/316446 | 4 | Let us assume our base field $k$ has characteristic zero.
From a series of papers by Borel and Siebenthal it is known that there is an embedding of groups
$A\_2 \times A\_2 \times A\_2$ into $E\_6$.
This gives a map
$H^1(k, A\_2 \times A\_2 \times A\_2) \rightarrow H^1(k, E\_6)$.
Let us consider the map
$A\... | https://mathoverflow.net/users/51251 | Constructing algebraic groups of type E6 with split Tits algebras | 1. Yes, it definitely makes sense, but you should be slightly more accurate. The actual group sitting inside $E\_6$ is indeed semisimple of type $A\_2+A\_2+A\_2$, but it is $(SL\_3\times SL\_3\times SL\_3)/\mu\_3$ if your $E\_6$ is simply connected and $(SL\_3\times SL\_3\times SL\_3)/\mu\_3^2$ if it is adjoint. So not... | 4 | https://mathoverflow.net/users/5107 | 316596 | 137,349 |
https://mathoverflow.net/questions/316593 | 7 | I would be very grateful for a reference
to the following results (which are, I think, true,
though I never saw it in the literature).
Let $G\subset GL(n,{\Bbb C})$ be $U(n)$,
abd $A\in GL(2n,{\Bbb R})$ an endomorphism which satisfies
$AGA^{-1}=G$. Then $A\in {\Bbb R}^\* \times U(n)$.
Let $G\subset GL(n,{\Bbb C})$ ... | https://mathoverflow.net/users/3377 | Stabilizer of Sp(n) and U(n) in GL(n) | First, let me fix a misunderstanding: $\mathrm{Sp}(n)$ does not sit in $\mathrm{GL}(n,\mathbb{C})$, but in $\mathrm{GL}(2n,\mathbb{C})$, so I'll assume that you mean, for the second part that $A$ lies in $\mathrm{GL}(2n,\mathbb{C})$.
These both follow immediately from the facts that all the automorphisms of
$\mathrm... | 13 | https://mathoverflow.net/users/13972 | 316597 | 137,350 |
https://mathoverflow.net/questions/316240 | 0 | I'm looking for any known exact results about inscribing simple convex bodies inside a convex polytope. The most famous is the [Löwner-John ellipsoid](https://en.wikipedia.org/wiki/John_ellipsoid), but as far as I understood in general there is no closed form solution to this problem. Are there some particular cases, w... | https://mathoverflow.net/users/97557 | Closed form solutions for maximal subsets of convex polytopes | By Ivan's request, I describe here two algorithms for finding the maximum-size ball contained in a given $n$-dimensional convex polytope $P$. The first algorithm, requiring central symmetry of $P$, provides a solution in one step only, the second one is recursive, and may not produce the solution in finitely many steps... | 2 | https://mathoverflow.net/users/36904 | 316601 | 137,353 |
https://mathoverflow.net/questions/316605 | 1 | Suppose we have a function $F: [a,b]^n \to \mathcal{M}\_{n \times n }(\mathbb{R})$ where $\mathcal{M}\_{n \times n }(\mathbb{R})$ is the space of $n \times n$ real matrices, a compact set $B \subset \mathbb{R}^n$, a constant $C \in \mathbb{R}$. What kind of $F$ satisfies the following condition
$$\Vert F(x\_1) y\_1 - F... | https://mathoverflow.net/users/55926 | A question on a special "metric" | Taking $y\_1=y\_2=y$, we see that $F(x\_1)y=F(x\_2)y\,$ for all $x\_1,x\_2$ in $[a,b]^n$ and for all $y\in B$, and hence for all $y\in \text{span}\,B$. In particular, if $B$ spans $\mathbb R^n$, then the matrix $F(x)$ does not depend on $x$, and in such a case the problem becomes "tirvial", as you said.
| 0 | https://mathoverflow.net/users/36721 | 316608 | 137,354 |
https://mathoverflow.net/questions/316595 | 9 | Let $V\subset \mathbb{A}^n$ be an irreducible affine variety. The set of singular points of $V$ is a subvariety $W$ of $V$; denote its components by $W\_i$. How may we bound $\sum\_i \deg(W\_i)$ in terms of $\deg(V)$ and $n$?
I am satisfied if we can find a proper subvariety $Z$ of $V$ containing $W$
and a bound for ... | https://mathoverflow.net/users/398 | Degree of the variety of singular points | **Edit.** As the OP points out, for his purpose it suffices to take the zero locus of a single (nonzero) partial derivative. So the OP produces a proper closed subset of $V$ containing the singular locus and having degree bounded by $(\text{deg}(V)-1)\text{deg}(V)$. Although this is not what the OP asks, there are case... | 5 | https://mathoverflow.net/users/13265 | 316615 | 137,356 |
https://mathoverflow.net/questions/316280 | 7 | I was wondering... Is every symplectic connection $\nabla$
on some symplectic manifold $(M,ω)$ the Levi-Civita connection of some metric $g$ on $M$? What about the local statement?
| https://mathoverflow.net/users/131790 | Symplectic connections are (locally) Levi-Civita connections | I think I've got the answer, and it is "no", at least for the global question.
When I started to try to understand the problem, I realized that to get some structure that is invariant by the connection we can (have to) fix the structure at some point and use parallel transport through picewise smooth paths to extend ... | 2 | https://mathoverflow.net/users/131790 | 316624 | 137,361 |
https://mathoverflow.net/questions/316632 | 2 | Dirichlet's Arithmetic Progression Theorem states that:
>
> Given $a, b\in\mathbb{Z^+}$ with $(a,b)=1$, then $a+kb$ is prime for an infinite number of $k\in\mathbb{Z^+}.$
>
>
>
For any given $a$ and $b$ let $K\_{a,b}=\{k\mid a+kb \text{ is prime}\}$.
Also consider another Dirichlet-Valid AP $c+jd$. Restrict ... | https://mathoverflow.net/users/70355 | Extension of Dirichlet's Arithmetic Progression Theorem | Consider the arithmetic progressions $2+3\mathbb N$ and $1+5\mathbb N$ and observe that if $2+3k$ is prime, then $k$ is odd. On the other hand, if $1+5k$ is prime, then $k$ should be even. So, for any $k\in\mathbb N$ the numbers $2+3k$ and $1+5k$ cannot be simultaneously prime.
The same contradiction could be attaine... | 7 | https://mathoverflow.net/users/61536 | 316633 | 137,363 |
https://mathoverflow.net/questions/315061 | 3 | For droplet interactions in low-Reynolds number flow, solutions are available when the underlying flow can be written as linear compositions of strain and rotation, see [Batchelor & Green (1972a)](https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/hydrodynamic-interaction-of-two-small-freelymovi... | https://mathoverflow.net/users/131234 | Difference between linear and parabolic velocity profiles in Stokes flows of two fluids | I will give a sketchy answer below to close this thread for now. Maybe experts in this field can give a better, more definite answer in the future.
The short answer is this: the linearity of the Stokes flow, combined with a linear velocity distribution in the far field, allows the manipulation of known solutions in s... | 1 | https://mathoverflow.net/users/131234 | 316636 | 137,364 |
https://mathoverflow.net/questions/316642 | 6 | [Nonexistence of boundary between convergent and divergent series?](https://mathoverflow.net/questions/49415/nonexistence-of-boundary-between-convergent-and-divergent-series)
I'm hoping the following is true:
>
> Suppose $a\_i $ is a positive sequence and $\sum\_i a\_i < \infty.$ Then there exists a positive sequ... | https://mathoverflow.net/users/131958 | On convergent series - in the spirit of Abel and Dini | This is not correct. Take $a\_n=1/n^2$.
Let us show that $b\_n$ with
required properties does not exist. Consider the set
$$E=\{ n: b\_n\geq 1/n\}.$$
As $\sum b\_n<\infty$, we have $$\sum\_E1/n<\infty.$$
Now on $N\backslash E$ we have $b\_n<1/n,$ so $1/b\_n>n$ and as $\sum a\_n/b\_n<\infty$,
we conclude that $$\sum... | 15 | https://mathoverflow.net/users/25510 | 316645 | 137,365 |
https://mathoverflow.net/questions/316649 | 10 | I'm studying category theory by myself and I just came across this sentence from [Wikipedia](https://en.wikipedia.org/wiki/Adjoint_functors):
*An adjunction between categories C and D is somewhat akin to a "weak form" of an equivalence between C and D, and indeed every equivalence is an adjunction. In many situations... | https://mathoverflow.net/users/nan | Upgrade adjunction to equivalence | Let $\mathcal C$ and $\mathcal D$ be two categories, and let $F\colon\mathcal C\longrightarrow \mathcal D$ and $G\colon\mathcal D\longrightarrow\mathcal C$ be two functors, with $F$ left adjoint to $G$. Then there are natural transformations $F\circ G\longrightarrow \operatorname{Id}\_{\mathcal D}$ and $\operatorname{I... | 14 | https://mathoverflow.net/users/2106 | 316665 | 137,373 |
https://mathoverflow.net/questions/316661 | 3 | Consider a vector $p =(p\_1,...p\_n)$, $p\_i>0$, $\sum p\_i = 1$
and a matrix $M\_{ij}$, which is [doubly stochastic](https://en.wikipedia.org/wiki/Doubly_stochastic_matrix): $\sum\_i M\_{ij} = 1, \sum\_j M\_{ij} = 1, M\_{ij} > 0$.
**Question 1** Just apply matrix M to a vector $p$ , i.e. $q = Mp$ (i.e. $q\_i = \sum\... | https://mathoverflow.net/users/10446 | Apply doubly stochastic matrix M to a probability vector, then entropy increases? | This is a particular case of the following general principle. On the one hand, a bistochastic matrix $M$ has the property that for every non-negative vector (say, a probability vector), $Mp\succ p$. On another hand, the order $\succ$ can be defined by $x\succ y$ iff $\sum\_if(x\_i)\le\sum\_if(y\_i)$ for every convex fu... | 4 | https://mathoverflow.net/users/8799 | 316667 | 137,375 |
https://mathoverflow.net/questions/198204 | 3 | Let $k, n \in \mathbb{N}$, $k = (1 - \epsilon)n$ where $1 >\epsilon > 0$.
I want to find $f: \{0,1\}^k \to \{0, 1\}^n$
such that:
1) $f(a) \not= f(b)$ if $a \not=b $
2) for any $x \in \{0,1\}^n$ $V\_x(k/2) \cap Im(f) \le 2^{k(1 - \delta)}$ for some $\delta > 0$, where $V\_x(k/2)$ is a full-sphere with center $x$ a... | https://mathoverflow.net/users/31356 | PRNG and coding theory | Consider the Reed Solomon code with parameters $[n,k,d=n-k+1]$ over the field $\mathbb{F}\_q.$ Take $q=2^m,$ and note that the usual choice is to choose $n=q-1,$ for $q-$ary symbols. Therefore the *binary* length of this code is $N=nm=(2^m-1)m$ while its binary "input length" is $K=km.$
This code is MDS, its full wei... | 1 | https://mathoverflow.net/users/17773 | 316677 | 137,379 |
https://mathoverflow.net/questions/316641 | 4 | Let $M$ be a Kähler manifold. The complex structure on it naturally gives rise to the real analytic structure. I wonder if there exist Kähler manifolds such that the associated symplectic $2$-form $\omega$ is $C^\infty$-smooth but not real analytic.
| https://mathoverflow.net/users/88385 | Symplectic form on a Kähler manifold can be not real analytic? | The answer is positive for any Kähler manifold.
Consider first surfaces. Take a compact Riemann surface $\Sigma$, then any symplectic form on it is associated to a Kähler form, so the answer is yes, since there are plenty of non-analytic $2$-forms.
More generally, for any Kähler manifold $M$ take any Kähler form $\... | 7 | https://mathoverflow.net/users/943 | 316678 | 137,380 |
https://mathoverflow.net/questions/314898 | 8 | Does there exist a compact connected Lie group $G$ acting smoothly as isometries on the standard sphere $S^n$ for some $n\ge 3$, so that no totally geodesic hypersphere $S^{n-1}$ is $G$-invariant, but there exists embedded $G$-invariant diffeomorphic $S^{n-1}$?
By $G$-invariant, I mean the submanifold is a union of o... | https://mathoverflow.net/users/nan | Lie group actions on $S^n$ with some invariant hypersphere but no totally geodesic ones | The answer below is by no means complete, but at least it treats the case when $G$ is a torus, and proposes an idea of what might be tried in the general case. The following fact is useful:
*Remark.* $G$ preserves a geodesic sphere on $S^n$ iff it fixes a point on $S^n$.
**Statement 1.** If the compact Lie group is... | 3 | https://mathoverflow.net/users/943 | 316680 | 137,381 |
https://mathoverflow.net/questions/316430 | 6 | This is, essentially, a geometrically rendered version of [the question](https://mathoverflow.net/q/315973/9924) I asked a week ago, with the emphases slightly shifted; it seems more natural and appealing (to me, at least) in this form.
Let $p\ge 3$ be a prime number. Suppose we are given $N$ lines $l\_1,\dotsc,l\_N... | https://mathoverflow.net/users/9924 | Covering the finite plane with lines | Let $C>0$ be any fixed number. Take $p-3$ horizontal lines and $p-b$ vertical lines where $p\gg b\gg C$. If we want to stay within $2p+C-3$ lines, we should be able to cover some $3\times b$ rectangular configuration by at most $b+C$ lines of any prescribed slopes $a\_1,\dots, a\_{b+C}\ne 0$. Notice that we have $3b$ p... | 4 | https://mathoverflow.net/users/1131 | 316684 | 137,384 |
https://mathoverflow.net/questions/316625 | 5 | Let $H$ denote the irreducible component of $\text{Hilb}^{3t+1}\mathbb{P}^3$ whose general member corresponds to a non-singular twisted cubic. Let $C$ be a subscheme lying in the boundary of $H$ and assume it lies in a surface $S \subseteq \mathbb{P}^3$.
Then why is it possible that we can find families $C\_R, S\_R \... | https://mathoverflow.net/users/131951 | Flat limit (of twisted cubic) contained in surfaces | As @abx and @ulrich explain in the comments, the original question is equivalent to a question about constancy of Hilbert functions for the universal family restricted over the irreducible component $H$. The Hilbert function is constant on $H$, as I explain below. I believe the OP is confused because, at this point in ... | 4 | https://mathoverflow.net/users/13265 | 316689 | 137,387 |
https://mathoverflow.net/questions/316423 | 10 | I understand that in the number field / function field analogy, the ideles $\mathbb I\_K$ of a number field $K$ are supposed to be analogous to the Picard group of a function field.
**Question:** Is this more than an analogy? Is there an actual geometric setting in which the ideles parameterize "line bundles" over a ... | https://mathoverflow.net/users/2362 | Are the ideles literally a Picard group? | As explained in the comments, I disagree with this analogy. Nonetheless, there is a way you can realize the idele class group (not the ideles) as a group of line-bundle-like objects under the tensor product.
First, some context.
Suppose we have a number field $K$ with ring of integers $O$.
We have a surjective ma... | 16 | https://mathoverflow.net/users/7935 | 316695 | 137,390 |
https://mathoverflow.net/questions/316628 | 5 | I am familiar with Barnette's Lower Bound Theorem on the number of facets a $d$-dimensional simplicial convex polytope with $n$ vertices can have. Is there a similar result for a general (i.e. not necessarily simplicial) $d$-polytope with $n$ vertices?
| https://mathoverflow.net/users/131952 | What is the lower bound for the number of facets that a general convex $d$-polytope with $n$ vertices can have? | A lower bound on the number of facets can be obtained from McMullen's upper bound theorem. It says that, for a given number of vertices, neighborly polytopes maximize the number of faces in all dimensions (among all polytopes, simplicial or not). In particular, a $d$-polytope $P$ with $m$ vertices has at most $F(d,m)$ ... | 3 | https://mathoverflow.net/users/98590 | 316700 | 137,391 |
https://mathoverflow.net/questions/316713 | 13 | Let $\mathcal{O}$ be the BGG category $\mathcal{O}$ with respect to a finite dimensional semisimple Lie algebra $\mathfrak{g}$ and its Borel subalgebra $\mathfrak{b}$ (as define in this [book](https://bookstore.ams.org/gsm-94/) by Humphreys). If $\mathfrak{h}$ is the Cartan subalgebra of $\mathfrak{g}$ contained in $\m... | https://mathoverflow.net/users/130461 | Tilting Objects in BGG Categories $\mathcal{O}$ | Words change their meanings.
The original meaning of “tilting module” is that of Happel and Ringel in the representation theory of finite dimensional algebras, which requires the projective dimension to be one, and a “generating” condition (the third condition in your link).
Fairly soon, it was recognized that gene... | 17 | https://mathoverflow.net/users/22989 | 316717 | 137,395 |
https://mathoverflow.net/questions/316719 | 5 | Let G be the free profinite group on 2 generators, $A=G/[G,[G,G]],B=G/[[G,G],[G,G]]$, then what is the structure of the groups $A$ and $B$?
I heard that $A$ is isomorphic to the group of such ($3\times 3$ below) matrices with entries in $\hat{\mathbb{Z}}$, is this right and why?
$$
\begin{pmatrix}
1 & \* & \*\\
0 & 1... | https://mathoverflow.net/users/108486 | Linear representation of the free metabelian / 2-step nilpotent profinite groups on 2 generators | The group $B$, the free pro-metabelian group, has the following description, due to Jorge Almeida. I’ll do it for an arbitrary finite set $|X|$ of cadinality at least $2$. Consider $\widehat{\mathbb Z}^X$, the free pro-abelian group on $X$. Then we can consider the edge set of its Cayley graph $E=\widehat{\mathbb Z}^X\... | 7 | https://mathoverflow.net/users/15934 | 316724 | 137,398 |
https://mathoverflow.net/questions/316723 | 3 | I remember reading (without proof) that for $\Gamma$ a profinite, pro-$p$ group, the following are equivalent:
1) Every open subgroup $\Gamma\_0$ is topologically finitely generated.
2) The abelianization of every open subgroup $\Gamma\_0$ is finitely generated (as a $\mathbb Z\_p$ module .
3) Every open subgroup... | https://mathoverflow.net/users/58001 | For a pro-p, profinite group, abelianization being finitely generated is the same as being topologically finitely generated | Following TKe's reference, here is the solution:
Suppose $\mathscr I$ is a set of elements in $G$, a pro-$p$ group.
Denote by $G^\*$ the group generated by $G^p$ and $[G,G]$ together. Note that $H^1(G,\mathbb F\_p) = \operatorname{Hom}(G,\mathbb F\_p) = \operatorname{Hom}(G/G^\*,\mathbb Q\_p/\mathbb Z\_p)$. Then, t... | 2 | https://mathoverflow.net/users/58001 | 316725 | 137,399 |
https://mathoverflow.net/questions/316714 | 8 | To motivate this question, I'm going to try and explain some background notions. This won't be absolutely necessary for experts, but I want to be vaguely honest about where this question comes from. I also want to say that this is the type of question that I would ask at a coffee break during a conference about this su... | https://mathoverflow.net/users/49247 | Beilinson-Drinfeld quantization and stable bundles | Answer is Yes for all. These commuting differential operators can be defined,
on the moduli space of stable bundles and written in theta functions terms (but it might not be illuminating or suggestive). In GL(1) case it is just "free system"
just $H\_i = \partial\_i^2$ on the Jacobian.
I'll give refrences below.
So... | 6 | https://mathoverflow.net/users/10446 | 316733 | 137,401 |
https://mathoverflow.net/questions/316367 | 18 | Has the 3-tag system investigated by Emil Post $(0\to00, 1\to1101)$ been solved? Is there a decision algorithm to determine which starting strings terminate, which end up in a cycle, and which (if any) grow without bound?
Also, what cycle structures are there? Setting $a=$ '00' and $b=$ '1101', the only cycles I kno... | https://mathoverflow.net/users/38744 | Is Post's tag system solved? | Here are the two irreducible repeating patterns that Liesbeth de Mol discovered, together with a third high-period irreducible repeating pattern discovered by Rich Schroeppel:
* $b^3 a^5 b^5$ (period 40);
* $a b^2 a b^3 a^3 b^3 a^2 b^2 a^4 b^2$ (period 66);
* $a b^3 a b a b a^2 b^2 a b^9 a^2$ (period 282);
This was... | 7 | https://mathoverflow.net/users/39521 | 316737 | 137,403 |
https://mathoverflow.net/questions/316524 | 8 | Ariyan Javanpeykar said [here](https://mathoverflow.net/questions/308474/atlas-of-gerbe-over-stack?noredirect=1#comment789174_308474) in comments that,
>
> $X\times\_{\mathcal{X}}X$ being a scheme is equivalent to representability of $X\rightarrow \mathcal{X}$.
>
>
>
Context is as in [this question](https://ma... | https://mathoverflow.net/users/118688 | $2$-fiber product is a scheme then map of stacks is representable | This is not true even if $\mathcal X$ is an Artin stack. For example, let $G$ be a smooth group scheme over the base $T$, and let $\mathbf BG$ be its classifying stack (the category of $G$-torsors fibered over $Sch/T$). Then $T\times\_{\mathbf BG}T=G$ is a scheme. However, the atlas $T\to \mathbf BG$ is not representab... | 10 | https://mathoverflow.net/users/20233 | 316738 | 137,404 |
https://mathoverflow.net/questions/316736 | 4 | On page $10$ of the survey article *[Algebraic stacks](https://arxiv.org/abs/math/9911199)*, by T. Gomez (arXiv:math/9911199), we have following result
>
> If a stack has an object with an automorphism other than the identity, then the stack can not be represented by a scheme.
>
>
>
Is the converse true?
Su... | https://mathoverflow.net/users/118688 | Stack being represented by a scheme/manifold | If all objects of a stack have trivial automorphism groups then it is equivalent to a sheaf, as pointed out by Daniel Litt in the comments. Pick your favourite non-representable sheaf as a counterexample to the claim that the stack is representable. For instance, on the site of manifolds, defined to be Hausdorff, parac... | 5 | https://mathoverflow.net/users/4177 | 316741 | 137,405 |
https://mathoverflow.net/questions/316701 | 4 | Let $X$ be a Banach space.
By Banach-Alaoglu and Krein-Milman Theorems, one can show that if $X$ is a dual space, then $X$ must have at least one extreme point of the closed unit ball.
I am interested in its converse.
More precisely,
>
> Question: Let $X$ be a Banach space.
> If the closed unit ball of $X$ has... | https://mathoverflow.net/users/42411 | If the closed unit ball of Banach space has at least one extreme point, must the Banach space the be a dual space? | Every separable Banach space $X$ can be equivalently renormed so that every point in the unit sphere is an extreme point: Take an injective bounded linear operator $T$ from $X$ into $\ell\_2$ and use $|x| := \|x\|\_X + \|Tx\|\_2$. Of course, there are many separable Banach spaces that are not isomorphic to a separable ... | 10 | https://mathoverflow.net/users/2554 | 316744 | 137,407 |
https://mathoverflow.net/questions/316743 | 35 | Let ${\cal P}$ be the set of prime numbers. Define a subset ${\cal P}'=\{p\_1,p\_2,p\_3,\cdots\}$ of ${\cal P}$ by setting $p\_1=2$ and defining $p\_{n+1}$ to be the smallest element of ${\cal P}$ dividing $1+p\_1\cdots p\_n$. Is there any obstruction to ${\cal P}'={\cal P}$ ?
| https://mathoverflow.net/users/131996 | Does the algorithm of the Greeks produce all prime numbers? | According to [Booker - A variant of the Euclid-Mullin sequence containing every prime](https://arxiv.org/abs/1605.08929), as of 2016, this question remains open.
>
> One of the central questions in this area was posed by Mullin [6] in 1963: Does the Euclid–Mullin sequence contain every prime number? Despite a compe... | 33 | https://mathoverflow.net/users/44191 | 316745 | 137,408 |
https://mathoverflow.net/questions/316749 | 0 | The title is the first sentence of Hermann Weyl's 1930 essay,
"[Levels of Infinity](http://peterpesic.com/levels-of-infinity/)."
He focuses on
>
> "the distinction between actuality and potentiality, between
> *Being and Possibility*."
>
>
>
He opines
>
> "the impossibility of grasping the continuum as a... | https://mathoverflow.net/users/6094 | "Mathematics is the science of the infinite" | I think there is a fairly straightforward answer to this question coming from [reverse mathematics](https://en.m.wikipedia.org/wiki/Reverse_mathematics). According to Wikipedia, "finitistic reductionism" is represented by the system WKL${}\_0$, and reverse math people know a lot about what mathematics can be done in WK... | 4 | https://mathoverflow.net/users/23141 | 316751 | 137,409 |
https://mathoverflow.net/questions/316739 | 6 | I have seen this result in several places without an English reference:
There exist infinitely many primes $p$ such that $p-1=2q\_1q\_2$ where $q\_1$ and $q\_2$ are prime numbers with $q\_1,q\_2>p^{1/4}$.
There is a French reference (E. Bombieri. Le Grand Crible dans la Theorie Analytique des Nombres. Asterisque 18... | https://mathoverflow.net/users/81443 | Reference Request for a result on divisors of $p-1$ | If you allow $(p-1)/2$ to be prime, not just a product of two primes exceeding $p^{1/4+\epsilon}$, then the result is contained in somewhat stronger form in Heath-Brown: Artin's conjecture for primitive roots (Quart. J. Math. Oxford 37 (1986), 27-38). See Lemma 1 in that paper, and apply it with $k=1$, $u=3$, $v=16$. T... | 5 | https://mathoverflow.net/users/11919 | 316753 | 137,410 |
https://mathoverflow.net/questions/316756 | 6 | Let $J$ be the hyperplane divisor in $\mathbb{C}P^2$, and let $i:C \hookrightarrow \mathbb{C}P^2$ be the closed immersion of a smooth generic curve of degree 2. We know that $C\simeq \mathbb{C}P^1$, and let $H$ be the hyperplane divisor in $C$. So loosely $H \sim \frac{1}{2}J|\_C$. The question is, what is the projecti... | https://mathoverflow.net/users/121526 | Resolution of a torsion sheaf | Write the equation of $C$ as $\ XY-Z^2=0$, and consider the homomorphism $u: \mathcal{O}\_{\mathbb{P}^2}(-1)^2\rightarrow \mathcal{O}\_{\mathbb{P}^2}^2$ given by the matrix $\begin{pmatrix}
X & Z\\Z& Y
\end{pmatrix}$. It is invertible outside $C$, and has rank 1 at every point of $C$. Thus $u$ is injective, and its cok... | 8 | https://mathoverflow.net/users/40297 | 316759 | 137,412 |
https://mathoverflow.net/questions/316772 | 3 | Following reference <https://pdfs.semanticscholar.org/e86e/8d7a267a29b9ad4ca112828109adfec55e8b.pdf> claims integer coprimality is in $NC$ and it also has one citation. Is this claim valid?
| https://mathoverflow.net/users/10035 | Is coprimality in $NC$? | Probably it's not valid:
[On ECCC, Comment #3 to TR98-009](https://eccc.weizmann.ac.il/report/1998/009/) | 11th May 2001 12:33
Comment on parallel complexity of coprimality
Referee Report of SICOMP submission
...
**Recommendation**
The paper is not acceptable for publication. The algorithm contains a
f... | 3 | https://mathoverflow.net/users/35419 | 316776 | 137,416 |
https://mathoverflow.net/questions/316764 | 0 | Let $X$ be an irreducible hypersurface defined by a polynomial $f$ of degree $5$ in $\mathbb{P}^3$. Let the homogeneous co-ordinates is given by $[x, y, z, w]$ and let $H$ be a hyperplane given by $w= 0$. Assume that the intersection $X \cap H$, a curve of degree $5$ in the plane $H$, is given by an equation of the for... | https://mathoverflow.net/users/130022 | singularity of a hypersuface in $\mathbb{P}^3$ | Consider the blowup $BL\_C\mathbb{P}^3$ of $\mathbb{P}^3$ along the conic
$$
C = \{ w = g = 0 \} \subset \mathbb{P}^3.
$$
Let $H$ be the pullback of the class of a hyperplane in $\mathbb{P}^3$ and $E \subset Bl\_C\mathbb{P}^3$ is the exceptional divisor. Then the linear system of quintics in $\mathbb{P}^3$ singular al... | 3 | https://mathoverflow.net/users/4428 | 316786 | 137,419 |
https://mathoverflow.net/questions/316777 | 8 | It is a known result that based point maps $f:S^2\to S^2$ are classified by their degree. That is, by the induced map at $\pi\_2$-level $f\_{\*2}:\pi\_2(S^2)\to\pi\_2(S^2)$ (the subindex $\*2$ just means it is the induced map by $f$ at $\pi\_2$-level). However, I am interested in knowing which is the relationship (if t... | https://mathoverflow.net/users/104774 | Relationship between induced maps at homotopy groups level for maps $f:S^2\to S^2$ | I'll write $n\_d$ for the degree $n$ map on $S^d$, and $\eta$ for the Hopf map $S^3\to S^2$. It is well-known that $\eta\_\*\colon\pi\_3(S^3)\to\pi\_3(S^2)$ is an isomorphism, so that $\pi\_3(S^2)=\{\eta\circ n\_3:n\in\mathbb{Z}\}$, and $\eta\circ n\_3$ is just $n$ times $\eta$ with respect to the standard abelian grou... | 16 | https://mathoverflow.net/users/10366 | 316788 | 137,421 |
https://mathoverflow.net/questions/316782 | 2 | I have this inequality $$\frac{1}{a}\exp\bigl\{-\frac{4}{h^2}\bigr\} \geq \frac{1}{f}$$ where $$ a \leq \Bigl(\pi^{d/2}\Gamma(\frac{1}{2}d+1)^{-1} + 1\Bigr) \left(\frac{h^{d+1}}{2} \Gamma \left(\frac{d+1}{2} \right) + h^d \left(\frac{d}{2}\right)^\frac{d}{2} \exp\Bigl\{-\frac{d}{2}\Bigr\} \right) $$
I tried to lowerbou... | https://mathoverflow.net/users/128868 | Complicated bound after using Stirling's approximation | I understand from the reference to Stirling that you are looking for a large-$d$ approximation of
$$a\_{\rm max}= \Bigl(\pi^{d/2}\Gamma(\frac{1}{2}d+1)^{-1} + 1\Bigr) \left(\frac{h^{d+1}}{2} \Gamma \left(\frac{d+1}{2} \right) + h^d \left(\frac{d}{2}\right)^\frac{d}{2} \exp\left(-\frac{d}{2}\right)\right).$$
With some... | 3 | https://mathoverflow.net/users/11260 | 316790 | 137,422 |
https://mathoverflow.net/questions/257792 | 10 | Does there exist a (pure) shellable simplicial complex $\Delta$ with the
following property? There is some facet $F$ of $\Delta$ such that no
shelling can begin with $F$.
This condition is easily seen to be equivalent to the existence of a
shellable simplicial complex $\Delta'$ with at least two facets, such
that for... | https://mathoverflow.net/users/2807 | Shellable simplicial complex with restriction on shellings | The answer to your question is "yes". Such complexes were ~~first~~ exhibited by Hachimori in his PhD thesis. See the clear explanation on his webpage:
[http://infoshako.sk.tsukuba.ac.jp/~hachi/math/library/nonextend\_eng.html](http://infoshako.sk.tsukuba.ac.jp/%7Ehachi/math/library/nonextend_eng.html)
The $f$-ve... | 6 | https://mathoverflow.net/users/19729 | 316793 | 137,424 |
https://mathoverflow.net/questions/316815 | 3 | Is it true that any smooth manifold of dimension $n$ can be smoothly embedded into $\mathbb{R}^{2n+1-a(n)}$ where $a(n)$ is the number of appearances of digit "1" in the binary expansion of $n$?
| https://mathoverflow.net/users/131924 | Making immersions from immersion conjecture into embeddings | The following result was proved by Ralph Cohen [1]:
>
> **Theorem.** Let $\alpha(n)$ be the number of ones in the binary representation of $n$, and $M^n$ is a compact $C^\infty$,
> $n$-manifold, $n>1$. Then there is a smooth immersion of $M^n$ to
> $\mathbb{R}^{2n-\alpha(n)}$.
>
>
>
Whether, there is an embe... | 1 | https://mathoverflow.net/users/121665 | 316817 | 137,433 |
https://mathoverflow.net/questions/316774 | 10 | I have asked this in [MSE](https://math.stackexchange.com/questions/3016990/what-are-the-commutative-rings-r-for-which-a-otimes-bbb-z-b-a-otimes) but there was no reply. Feel free to close if inappropriate.
>
> Let $R$ be commutative ring, what can we say about the rings $R$ such that $A \otimes\_{\Bbb Z} B \cong ... | https://mathoverflow.net/users/97321 | Classify commutative rings $R$ such that $A \otimes_{\Bbb Z} B = A \otimes _{R} B$ | Fernando and Pierre-Yves in the comments are right; $R$ has this property (the version where the canonical map is an isomorphism, as YCor says in the comments) iff it is a [solid ring](https://mathoverflow.net/questions/95160/solid-rings-and-tor), meaning the multiplication map $m : R \otimes\_{\mathbb{Z}} R \to R$ is ... | 8 | https://mathoverflow.net/users/290 | 316821 | 137,435 |
https://mathoverflow.net/questions/316784 | 4 | In [these notes](http://wwwf.imperial.ac.uk/~anskor/SCHLANK-SKOROBOGATOV.pdf) the following theorem is stated, among other things.
Let $X$ be a pointed connected geometrically unibranch scheme over
$\mathbb{C}$. Then Artin-Mazur etale homotopy type of $X$ is equivalent to the profinite completion of the homotopy type... | https://mathoverflow.net/users/131924 | Etale homotopy type of non-unibranch scheme over $\mathbb{C}$ | If $X$ is not geometrically unibranch, the etale homotopy type might not be profinite anymore, but Artin and Mazur proved (in Chapter 12, Theorem 12.9) that its profinite completion agrees with the profinite completion of $X(\mathbb{C})$.
Maybe this is not the best possible result, but it's not clear how to improve ... | 6 | https://mathoverflow.net/users/3847 | 316823 | 137,436 |
https://mathoverflow.net/questions/316775 | 6 | In "Floer Homology groups in Yang-Mills theory", Donaldson says that if we take an $U(2)$-vector bundle $E$ and we construct the bundle $\mathfrak{g}\_E$ of trace-free, skew adjoint automorphisms of $E$, ($\mathfrak{g}\_E \subset E^\*\otimes E$) then we have the following relation
$$w\_2(\mathfrak{g}\_E) = c\_1(E) \ \t... | https://mathoverflow.net/users/99042 | Characteristic classes of the bundle of trace free, skew adjoint endomorphisms | As a real vector bundle, $E^{\ast} \otimes E$ decomposes as the direct sum of two copies of the bundle $\mathfrak{su}(E)$ of trace-free skew-adjoint endomorphisms and two copies of the trivial bundle. This follows from the corresponding decomposition of $V^{\ast} \otimes V$ where $V$ is the defining $2$-dimensional rep... | 6 | https://mathoverflow.net/users/290 | 316825 | 137,437 |
https://mathoverflow.net/questions/296287 | 1 | Let $X$ be the Cantor set, and let $g$ be a minimal homeomorphism of $X$. Let $h$ be a homeomorphism in the topological full group of $g$, that is, for every $x \in X$, there is a neighbourhood of $x$ such that $h$ restricts to a power of $g$ on that neighbourhood. Write $\langle h \rangle$ for the group generated by $... | https://mathoverflow.net/users/4053 | Topological full groups and minimal orbit closures | Having worked on this problem, I can now say the answer is yes, $h$ has minimal orbit closures. Actually YCor's comment is spot on: analogous to Keane's interval exchange transformations, the space decomposes into finitely many clopen $\langle h \rangle$-invariant pieces such that on each piece, the restriction of $h$ ... | 1 | https://mathoverflow.net/users/4053 | 316842 | 137,443 |
https://mathoverflow.net/questions/316844 | 9 |
>
> Can one construct an embedding of $\mathbb{S}^1$ into $\mathbb{R}^3$
> so that **every** orthogonal projection onto a two dimensional plane
> is a unit disc?
>
>
>
It is easy to construct an embedding of $\mathbb{S}^1$ into $\mathbb{R}^3$ so that one orthogonal projection is a unit disc: take a Peano-type ... | https://mathoverflow.net/users/121665 | A wild embedding of $\mathbb{S}^1$ into $\mathbb{R}^3$ | No.
Assume it is possible, that is, there is an embedding $f\colon\mathbb{S}^1\to \mathbb R^3$ such that any projection of $f(\mathbb{S}^1)$ is a unit disc.
By [this answer](https://mathoverflow.net/a/39131/1441), the convex hull of the image $f(\mathbb{S}^1)$ is a unit ball.
Further note that every extreme point l... | 14 | https://mathoverflow.net/users/1441 | 316845 | 137,444 |
https://mathoverflow.net/questions/316840 | 1 | One form of de Franchis theorem for algebraic curves is the following: let $X$ be an algebraic curve (defined over $\mathbb{C}$ say) with genus $g > 1$. Then there are only finitely many (isomorphism classes of) curves $Y$ with genus $g' > 1$ such that there is a non-constant map $f : X \rightarrow Y$.
My question is... | https://mathoverflow.net/users/10898 | A question regarding de Franchis theorem | Let $Y$ be an arbitrary fixed curve of genus $h>1$ and $\mathscr L$ a fixed ample line bundle. Further let $m\in\mathbb N$ fixed such that $\mathscr L^{\otimes m}$ is very ample and let $D\_\lambda\subseteq Y$ be general members of the linear system corresponding to $\mathscr L^{\otimes m}$.
Finally, let $X\_\lambda\t... | 3 | https://mathoverflow.net/users/10076 | 316854 | 137,447 |
https://mathoverflow.net/questions/316868 | 9 | Does there exist a closed topological manifold supporting two non-diffeomorphic smooth structures both of which admit a compatible complex structure? Also the same question, but for symplectic structure.
| https://mathoverflow.net/users/131924 | Two homeomorphic non-diffeomorphic complex manifolds | Yes there are. There exists infinitely many proper elliptic surfaces that are homeomorphic and no two are diffeomorphic. Each of them carries a Kähler structure. For details see Corollary 3.3.23 of [this book](https://www3.nd.edu/~lnicolae/swnotes.pdf).
| 10 | https://mathoverflow.net/users/20302 | 316871 | 137,453 |
https://mathoverflow.net/questions/316870 | 1 | Let $G=(V,E)$ be an infinite, simple, undirected graph, such that for all $v\in V$ we have $\text{deg}(v) \geq \aleph\_0$. Given an integer $k\geq 1$, is there always $E^{(k)}\subseteq E$ such that $(V, E^{(k)})$ is $k$-regular, that is, every vertex has exactly $k$ neighbors?
| https://mathoverflow.net/users/8628 | Does $G$ with $\delta(G)\geq \aleph_0$ contain $k$-regular sub-edge-sets? | The complete bipartite graph $K\_{\aleph\_0,\aleph\_1}$ does not contain a $k$-regular spanning subgraph for any positive integer $k$.
| 2 | https://mathoverflow.net/users/43266 | 316873 | 137,454 |
https://mathoverflow.net/questions/316867 | 6 | Call two primes 2-power-twins if their difference is (can you guess?) a power of 2.
For example, 11 and 19 are 2-power-twins.
>
> Is there a 2-power-twinless prime?
>
>
>
I would imagine that this is doable the following way.
If I take a prime of the form $3k+1$, then I know that adding an odd power of 2 or ... | https://mathoverflow.net/users/955 | Is there a 2-power-twinless prime? | Erdos proved that there is an arithmetic progression of odd numbers, none of which can be expressed as a sum of a power of two and a prime. I believe one can arrange for such an arithmetic progression to satisfy the hypotheses of Dirichlet's Theorem on primes in arithmetic progression, so that means there are infinitel... | 9 | https://mathoverflow.net/users/3684 | 316877 | 137,455 |
https://mathoverflow.net/questions/316858 | 0 | Consider an urn containing red, blue and green balls (the situation is the same illustrated in [this](https://mathoverflow.net/q/316277/124302) post).
Let $X$ be the non-negative, integer-valued random variable defined as the number of trials (of one ball at a time, with replacement) in correspondence of which we get... | https://mathoverflow.net/users/124302 | A problem related to the comparison of two integer-valued random variables | As in the linked answer, for each $j=1,2,3$, let $p\_j$ denote the probability of getting a red, blue, green ball (respectively) in one trial.
For $x=0,1,\dots$ and $j=1,2,3$, let $N\_{x,j}$ denote the number of trials with outcome $j$ among the first $x$ trials.
Note that $X\ge2$.
For $x=2,3,\dots$
\begin{equatio... | 1 | https://mathoverflow.net/users/36721 | 316882 | 137,458 |
https://mathoverflow.net/questions/310147 | 3 | I've recently come across many results discussing the differentiation of the Moreau envelope defined by
\begin{equation}
e(f)(x)\triangleq \min\_{h \in H} \|h-x\|^2 + f(h),
\end{equation}
where $f$ is a convex functional on a separable Hilbert space $H$.
Examples of results on differentiability are [here](http://jou... | https://mathoverflow.net/users/36886 | Differentiability of the Moreau envelope | This is not an answer, but a few thoughts on the subject:
As you know, the Moreau enveloppe of a convex lower semicontinuous function is always Fréchet differentiable, with its gradient being $1$-lipschitz continuous.
Actually, its gradient can be expressed as follows: $\nabla e(f)(x) = x - \text{prox}\_f (x)$, where... | 5 | https://mathoverflow.net/users/73608 | 316886 | 137,459 |
https://mathoverflow.net/questions/316829 | 9 | Let $\mathcal O(\Omega)$ be the algebra of functions holomorphic on the open set $\Omega\subset\mathbb C$. For $\gamma$ a simple compact curve in $\mathbb C$ consider the linear operator given by path-integrating against the Cauchy kernel $$ \Gamma : \mathcal O(\mathbb C) \longrightarrow \mathcal O(\mathbb C\setminus \... | https://mathoverflow.net/users/24309 | Cauchy path integral as a linear operator: kernel and image? | Such an integral over a simple (non-closed) curve is called the Cauchy type
integral. It is convenient to define
$$F(z)=\frac{1}{2\pi i}\int\_\gamma \frac{f(\zeta)d\zeta}{\zeta-z},\quad z\not\in\gamma.\tag{1}$$
The curve $\gamma$ is oriented, so for our function $F$ defined in $C\backslash\gamma$ and $z\in\gamma$ dif... | 8 | https://mathoverflow.net/users/25510 | 316889 | 137,461 |
https://mathoverflow.net/questions/316747 | 1 | Let's denote a sentence $P$ as "***weak Godel sentence of theory $T$***", if and only if
$$[\neg (T \vdash P) \wedge \neg (T\vdash \neg P)] \wedge [Con(T)=Con(T+P) \wedge Con(T)=Con(T+ \neg P)] $$
In English this is: $P$ is independent of $T$ and the addtion of $P$ or $\neg P$ to $T$ doesn't prove the consistency of ... | https://mathoverflow.net/users/95347 | Can we have a theory $T$ that is complete for simple sentences in the language of $T$ that are weaker than $ Con(T)$? | Unless I'm missing something, every sentence is equivalent to a simple sentence. To see this, take a sentence $\varphi$ and produce an equivalent sentence $\psi$ by the following procedure: Let $x$ be a variable not occuring in $\varphi$. Replace each atomic sub-formula $\chi$ of $\varphi$ by $\chi \wedge x = x$ if $\c... | 2 | https://mathoverflow.net/users/83901 | 316912 | 137,470 |
https://mathoverflow.net/questions/316864 | 0 | Let $\mu$ be a probability measure on $I=[0,1]$, absolutely continuous with respect to Lebesgue measure. Denote by $T$ the "doubling angle map" on $I$, where $T(x)=2x \text{ mod }1$. Is it true, in general, that
$\mu\left(T^{-n}\left[0,\frac12\right)\right)$ converges to $\frac12$?
| https://mathoverflow.net/users/101832 | Evolution of a density under the doubling angle map | Write $d\mu(x)=f(x)\,d\lambda(x)$ where $\lambda$ is Lebesgue measure. Then you're asking about $\int \mathbf 1\_{[0,\frac 12]}\circ T^n\,d\mu = \int \mathbf 1\_{[0,\frac 12]}\circ T^n f(x)\,d\lambda(x)$. Since $T$ is mixing with respect to the invariant measure $\lambda$, this converges to $\int \mathbf 1\_{[0,\frac 1... | 2 | https://mathoverflow.net/users/11054 | 316918 | 137,473 |
https://mathoverflow.net/questions/316915 | 2 | It is well known that there is a notion of Brownian motion on smooth Riemannian manifolds.
>
> I am wondering if there is a more general notion of Brownian motion on finite dimensional Alexandrov spaces with curvature bounded below.
>
>
>
Such spaces are roughly Riemannian “manifolds” with metric and topolog... | https://mathoverflow.net/users/16183 | Is there Brownian motion on Alexandrov spaces? | Yes, there is a natural Brownian motion on an Alexandrov space.
In the following paper:
>
> *Kuwae, Kazuhiro; Machigashira, Yoshiroh; Shioya, Takashi*, [**Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces**](http://dx.doi.org/10.1007/s002090100252), Math. Z. 238, No. 2, 269-316 (2001). [ZBL1001.5301... | 4 | https://mathoverflow.net/users/4832 | 316924 | 137,475 |
https://mathoverflow.net/questions/316875 | 8 | Let $M$ be a manifold and $V$ be an oriented vector bundle. It's well known that if the Euler class of $V$ is non zero, then $V$ can't have a non-vanishing section. The converse is not true, see [Vanishing of Euler class](https://mathoverflow.net/questions/31376/vanishing-of-euler-class) , or the answer by John Klein b... | https://mathoverflow.net/users/13441 | Can one disjoin any submanifold in $\mathbb R^n$ from itself by a $C^{\infty}$-small isotopy? | Here's a counter example. Take any embedding of $\mathbb{C} P^2$ in $\mathbb{R}^7$ (such embeddings exist by
*Steer, B.*, [**On the embedding of projective spaces in Euclidean space**](http://dx.doi.org/10.1112/plms/s3-21.3.489), Proc. Lond. Math. Soc., III. Ser. 21, 489-501 (1970). [ZBL0206.25501](https://zbmath.or... | 9 | https://mathoverflow.net/users/8103 | 316935 | 137,478 |
https://mathoverflow.net/questions/316909 | 3 | Can anyone provide an example of such a module? or show that no such module exists? For semisimple rings, we have co-Hopfian if and only if finitely generated. Perhaps the fact that QF rings (commutative) are finite products of local artinians (with simple socle) has something to lend to the situation.
| https://mathoverflow.net/users/8027 | example of a non-finitely generated co-Hopfian module over a commutative QF ring | Let $R$ be the four-dimensional algebra $k[x,y]/(x^2,y^2)$, where $k$ is an infinite field.
For each $\lambda\in k$, $M\_\lambda=R/(x-\lambda y)R$ is a two-dimensional module with one-dimensional radical. If $\lambda\neq\mu$, then every homomorphism $M\_\lambda\to M\_\mu$ has image contained in $\text{rad}(M\_\mu)$ a... | 2 | https://mathoverflow.net/users/22989 | 316947 | 137,482 |
https://mathoverflow.net/questions/316898 | 10 | Good morning,
I've came across this question, which has been puzzling me for some days. Suppose we are given a vector-valued function $v:\mathbb{R}^n\to \mathbb{R}^n$, $v(x)=\left( v\_1(x),\dots, v\_n(x) \right)$ for every $x\in \mathbb{R}^n$.
Given a real function $\lambda:\mathbb{R}^n\to \mathbb{R}$, we are inter... | https://mathoverflow.net/users/132086 | Can a vector-function $v:\mathbb{R}^n\to \mathbb{R}^n$ be an eigenvector of its own Jacobian matrix? | I am surprised that Ramesh's answer was voted up. Its first paragraph does not convince me. Here is my analysis:
Consider an integral curve of $v$, that is a curve $t\mapsto X(t)$ so that $\dot X=v(X)$. Then
$$\frac{d}{dt}v=(\dot X\cdot \nabla)v=\lambda v.$$
This shows that $t\mapsto v(X(t))$ keeps a constant directi... | 6 | https://mathoverflow.net/users/8799 | 316951 | 137,483 |
https://mathoverflow.net/questions/295605 | 9 | Let $X$ be a finite type scheme over $\mathbb{C}$ and let $ \mathcal{X} \to Spf(\mathbb{C}[[x]])$ be a formal deformation of $X$. **Which of the following assumptions (or combinations thereof) are sufficient to imply that this deformation is convergent?** (i.e. that it comes from some flat analytic family $\tilde{\math... | https://mathoverflow.net/users/22810 | When is a formal deformation convergent? | **EDIT.** New version, addressing questions in the comments.
(1) Affine and smooth implies what you want.
Indeed, suppose $\mathcal{X}$ is smooth and that $H^1(X\_0, T\_{X\_0/\mathbb{C}}) = 0$ where $X\_0/\mathbb{C}$ is the special fiber and where $T\_{X\_0/\mathbb{C}}$ is the tangent bundle. This is of course sat... | 4 | https://mathoverflow.net/users/3847 | 316964 | 137,486 |
https://mathoverflow.net/questions/316968 | 3 | Suppose that $\mathbb H$ is a separable Hilbert space and $X\_1,\ldots,X\_n$ are independent zero mean $\mathbb H$-valued random elements such that $\|X\_i\|\le s$ for each $1\le i\le n$, where $\|\cdot\|$ is the norm of $\mathbb H$. Denote $S\_n=X\_1+\ldots+X\_n$.
>
> Is it true that
> $$
> P(\|S\_n\|\ge t)\le Ce... | https://mathoverflow.net/users/46211 | Hoeffding's inequality for Hilbert space valued random elements | This is a special case of more general [Theorem 3.5](https://projecteuclid.org/euclid.aop/1176988477), which holds for martingales in $2$-smooth Banach spaces. Note that (i) any Hilbert space is $2$-smooth, with the $2$-smoothness constant $D=1$ and (ii) the consecutive sums of independent zero-mean random vectors cons... | 4 | https://mathoverflow.net/users/36721 | 316974 | 137,488 |
https://mathoverflow.net/questions/316954 | 26 | I find the following mind-boggling.
Suppose that runner $R\_1$ runs distance $[0,d\_1]$ with average speed $v\_1$. Runner $R\_2$ runs $[0,d\_2]$ with $d\_2>d\_1$ and with average speed $v\_2 > v\_1$. I would have thought that by some application of the [intermediate value theorem](https://en.wikipedia.org/wiki/Interm... | https://mathoverflow.net/users/8628 | Runner's High (Speed) | The constant is $2$. Let $n=\lfloor d\_2/d\_1 \rfloor \geq 1$, and let $t\_k$ be the time which
the long distance runner takes to arrive at the distance $kd\_1$ from the origin,
$1\leq k\leq n$.
Proving by contradiction,
suppose that on every interval $[(j-1)d\_1,jd\_1], j=1,...,k$ the average speed of the long dist... | 17 | https://mathoverflow.net/users/25510 | 316976 | 137,489 |
https://mathoverflow.net/questions/316965 | 1 | Here is a question about decomposition of measures in singular parts and in positive and negative parts.
$\newcommand{\RR}{\mathbb{R}}$
Let $\Omega\_{1/2}$ be compact subsets of $\RR^d$ equipped signed measures $\alpha\_{1/2}$. Let $\lambda$ denote the Lebesgue measure on $\Omega\_{1/2}$. We decompose
$$
\alpha\_i = ... | https://mathoverflow.net/users/9652 | Positive part of "outer sums" of measures | We have
$$
\alpha\_1\oplus\alpha\_2 = (f\_1+\eta\_1^+)\times\lambda + \lambda\times\alpha\_2 -\eta\_1^-\times\lambda , \quad\quad\quad\quad (1)
$$
and if we drop the $\eta\_1^-$ from $\alpha\_1$, then the last term is missing. This last term, however, is singular with respect to the first two terms: it's supported by a... | 2 | https://mathoverflow.net/users/48839 | 316979 | 137,491 |
https://mathoverflow.net/questions/316963 | 3 | **Motivation:** The Oka's coherence theorem tells us that the structure sheaf of a complex manifold is coherent. Taking into account the fact that coherence is a local property stable under finite direct sums, we obtain that sheaves of sections of holomorphic vector bundles (i.e. locally free sheaves of modules of fini... | https://mathoverflow.net/users/132126 | Is a factorial scheme with Noetherian stalks locally Noetherian? | Here are two negative results in this direction.
>
> A ring whose stalks are noetherian and factorial need not be coherent.
>
>
>
A counterexample was constructed by Harris and Nagata; see Example on p. 51 in S. Glaz, *Commutative coherent rings,* Lecture Notes in Math. 1371, Springer, Berlin, 1989.
>
> A ... | 2 | https://mathoverflow.net/users/11025 | 316999 | 137,498 |
https://mathoverflow.net/questions/316994 | 4 | Let $(X,\omega)$ be a symplectic 4-manifold such that $\omega$ has a rational cohomology class. I am interested in Donaldson divisors (surfaces) $D$ in $(X,\omega)$ whose complement is a 1-handle body. Is there any obstruction for existence of such $D$? How about a transverse pair of them: Is there any obstruction for ... | https://mathoverflow.net/users/5259 | Complement of Donaldson divisors in dimension 4 | It seems to me that there is a kind of obstruction. Indeed, Donaldson divisors are usually surfaces of high genus, so they are of negative Euler characteristic. On the other hand the majority of known symplectic $4$-manifolds have positive Euler characteristics. There exists a folklore (or maybe Gompf's) conjecture sta... | 3 | https://mathoverflow.net/users/943 | 317000 | 137,499 |
https://mathoverflow.net/questions/316986 | 3 | Let $D(0,1)$ be the Skorohod space with the Skorohod topology, i.e. the space of real-valued càdlàg-functions on $[0,1]$ with topology induced by the metric
$$d(f,g) = \inf\_{\varphi \in \Lambda} \left\{ \lVert \varphi - \operatorname{Id} \rVert\_{\infty} \lor \lVert f - g \circ \varphi \rVert\_{\infty} \right\},$$
whe... | https://mathoverflow.net/users/56931 | Continuous embedding of the Skorohod space D(0,1) into L^2(0,1) | I think the following works, but please check me.
Note first that cadlag functions are measurable and bounded, so $D(0,1) \subset L^2(0,1)$.
Suppose $f\_n \to f$ in the Skorokhod metric. Note that $f$ has at most countably many discontinuities, and $f\_n(t) \to f(t)$ for each continuity point $t$ of $f$ ((\*), see ... | 3 | https://mathoverflow.net/users/4832 | 317001 | 137,500 |
https://mathoverflow.net/questions/316820 | 0 | Let $S$ be the regular n-dimensional simplex. We create a graph of where the vertices are m-faces and two vertices are connected if there exist a common (m-1)-face which they share.
Then the number of vertices is $binomial(n,m)$ and the graph $G$ is regular with $m(n-m)$ vertices.
Prove or disprove:
1) The diameter... | https://mathoverflow.net/users/nan | Some questions about a family of regular undirected graphs | As for your question 2), according to <https://arxiv.org/pdf/0806.2074.pdf>, a regular graph is periodic if and only if its eigenvalues are distinct, which [is the case](https://www.emis.de/journals/ELA/ela-articles/articles/vol17_pp154-167.pdf) for the Johnson graphs.
| 1 | https://mathoverflow.net/users/69010 | 317003 | 137,501 |
https://mathoverflow.net/questions/316992 | 2 | Given $X\_i = A\_i - B\_i$ where $A\_i\sim \text{ Exp}(\alpha)$ and $B\_i \sim \text{ Exp}(\lambda)$. Define $S\_k = \sum\_{i=1}^k X\_i$ with $S\_0 = 0$, and
$$M\_n = \max\_{1\leq k \leq n} S\_k.$$
Is it possible to calculate the quantity $\mathbb{P}(M\_n \leq x)$ explicitly? I have tried it and the result is written ... | https://mathoverflow.net/users/49551 | Maximum of sums of iid $X_i$'s where $X_i$ is the difference of two exponential r.v | This seems to be problem 21.8 in Takacs’ book, chapter 2, [available online](https://www.eurandom.tue.nl/reports/index.htm). Solutions are given there as well.
| 1 | https://mathoverflow.net/users/nan | 317006 | 137,503 |
https://mathoverflow.net/questions/316987 | 9 | Let $M$ denote a closed manifold. Let $\Gamma(TM\setminus 0) $ denote the space of non-vanishing sections of $TM$. Note that the diffeomorphism group $\text{Diff} (M)$ acts on $\Gamma(TM\setminus 0) $ via $f. \phi(x) =Df\_{f^{-1}(x)}(\phi(f^{-1}(x))$. I'm interested in the induced action on $\pi\_0( \Gamma(TM\setminus ... | https://mathoverflow.net/users/89741 | Action of diffeomorphism group on non-vanishing vector fields | In general, the action of $\pi\_0(Diff(M))$ on $\pi\_0(\Gamma(TM\backslash 0))$ is not faithful. One can find hyperbolic homology 3-spheres with non-trivial isometry group. By a theorem of [Gabai](https://mathscinet.ams.org/mathscinet-getitem?mr=1895350), the mapping class group is the isometry group. But the action on... | 7 | https://mathoverflow.net/users/1345 | 317013 | 137,506 |
https://mathoverflow.net/questions/316953 | 9 | Let $c(n)=\frac{1}{n+1}\binom{2n}{n}$ be the Catalan number. It seems that a product $\prod\_{n\in I} c(n)$, where $I\subset\mathbb N\_{>1}$, is never a Catalan number. Is this a (known) fact?
| https://mathoverflow.net/users/3032 | Products of Catalan numbers | It should be possible here to mimic the same argument that Erdos uses in his paper ["On some divisibility properties of $\binom{2n}{n}$"](https://cms.math.ca/10.4153/CMB-1964-047-5).
Suppose that $c(n)=c(a\_1)c(a\_2)\cdots c(a\_k)$ and $n$ is large enough (not sure what constant is exactly needed here, but checking a... | 7 | https://mathoverflow.net/users/2384 | 317022 | 137,509 |
https://mathoverflow.net/questions/316997 | 6 | To keep the notations simple I'll restrict my attention to the complete flag variety although the question should be equally valid for partial flag varieties. Consider $G=SL\_n(\mathbb C)$ with Borel $B\subset G$, Cartan $T\subset B$ and flag variety $F=G/B$. The Plücker embedding $$F\subset\mathbb P(\wedge^1\mathbb C^... | https://mathoverflow.net/users/19864 | Irreducibility of Gelfand-Serganova strata | The strata need not be irreducible.
Quoting page 2 of Knutson, Lam, Speyer (<https://arxiv.org/abs/0903.3694v1>): "the strata can have essentially any singularity [Mn88]. In particular, the nonempty ones need not be irreducible, or even equidimensional."
They are referencing ["Mnëv's universality theorem"](https://... | 5 | https://mathoverflow.net/users/25028 | 317048 | 137,514 |
https://mathoverflow.net/questions/317059 | 6 | Let $G(F)$ be a reductive $p$-adic group. A result of Bernstein says that we can correspond each smooth irreducible representation to a “cuspidal pair” where it is embedded, and at most finitely many smooth irreducible representations correspond to the same pair.
My question is: Can we distinguish between them in a ... | https://mathoverflow.net/users/119736 | P-adic representations corresponding to the same cuspidal pair | I don't think anything quite so simple should work in general, but something close to this should (conjecturally).
Consider the cuspidal pair in GL(2) consisting of the trivial representation of the Borel subgroup. Parabolically induce this and you get a length two indecomposable representation. Its unique sub is the... | 3 | https://mathoverflow.net/users/128771 | 317061 | 137,518 |
https://mathoverflow.net/questions/317031 | 1 | I only want to know whether a construction that I use appears in literature and maybe has a name already.
Let $V$ be a $\mathbb Q$ vector space of dimension $d\in\mathbb N$.
A subset $C\subset V$ is called a proper cone with $d$ sides if there are linearly independent $\alpha\_1,\dots,\alpha\_d\in \mathrm{Hom}(V,{\ma... | https://mathoverflow.net/users/nan | Basis of cone lattice | I think that $\alpha\_1, \dots \alpha\_d$ are called lattice ray generators and $E$ is the set of lattice points in the fundamental domain. It is a special case of Gordan's lemma: the case when the cone is simplicial.
| 0 | https://mathoverflow.net/users/132176 | 317070 | 137,521 |
https://mathoverflow.net/questions/317065 | 11 | Suppose we have a (co)filtered digaram $\dots \rightarrow X\_{2}\rightarrow X\_{1}$ of topological space. Is is true that the natural map $\pi\_{0}[\lim X\_{i}]\rightarrow \lim \pi\_{0}(X\_{i})$ is an isomorphism ?
| https://mathoverflow.net/users/82229 | homotopy and (co)filtered limits | This is not true, for two distinct reasons.
1. The first is that the inverse system of spaces may not behave well homotopy-theoretically. If $X\_n = [n, \infty) \subset \Bbb R$, then the limit of $\dots \to X\_2 \to X\_1 \to X\_0$ is empty. However, on path components it is the constant system $\dots \to \* \to \* \t... | 30 | https://mathoverflow.net/users/360 | 317071 | 137,522 |
https://mathoverflow.net/questions/316630 | 18 | Let say I have two different sites $(\mathcal{C},I)$ and $(\mathcal{D},J)$ for an ordinary topos $\mathcal{T}$. I.e.
$$Sh(\mathcal{C},I) \simeq \mathcal{T} \simeq Sh(\mathcal{D},J)$$
And we want to know if one also have an equivalence of the $\infty$-topos of sheaves of spaces on these sites:
$$ Sh\_{\infty}(\ma... | https://mathoverflow.net/users/22131 | Equivalences of categories of sheaves vs categories of $\infty$-Stack | I just found an example, so I thought it would be good to post it here, but if anyone knows other examples, or a more general way to construct some I would be interested to see them as well.
This example comes relatively immediately from the example of $1$-site producing a non Hypercomplete $\infty$-topos in the pape... | 12 | https://mathoverflow.net/users/22131 | 317085 | 137,527 |
https://mathoverflow.net/questions/317084 | 10 | Let $(M,g)$ be a Riemannian compact manifold without boundary, and $\Delta$ is the Laplace-Beltrami operator on $M$. Is there any result on the elliptic regularity like this:
For any $u\in H^1(M)$, and $f\in L^2(M)$ such that $\Delta u = f$ (in the sens of distributions), Then $u \in H^2(M)$.
If there is a nice refer... | https://mathoverflow.net/users/124904 | Elliptic regularity on compact manifold without boundary | This follows from the following regularity estimate for the flat Laplacian case (which is, I believe, proved in Warner's book using Fourier series on a torus but also in most standard texts on elliptic PDEs): Given a bounded open domain $\Omega \subset \mathbb{R}^n$, there exists $C>0$ such that for any function (or ev... | 11 | https://mathoverflow.net/users/613 | 317091 | 137,530 |
https://mathoverflow.net/questions/317029 | 0 | Consider $T=k[x\_1,\ldots,x\_n]$ ( $k$ alg. closed and of char $k=0$), and consider the ideal $$I=(x\_1,x^{a\_2}\_2,\ldots,x^{a\_n}\_n)$$
with $2\leq a\_2 \leq\ldots\leq a\_n$. I want to prove that $$\sum\_{i=0}^t HF(T/I,i)=\prod\_{i=2}^n a\_i.$$
I know that $I$ is a complete intersection ideal, so $T/I$ is Artinian a... | https://mathoverflow.net/users/126594 | Relation between Hilbert function and complete intersection ideals | The sum of values of the Hilbert function is equal to the vector space dimension of the algebra:
$$
\sum\_{i=0}^{\infty} HF(T/I,i) = \dim(T/I).
$$
In this case, the vector space $T/I$ has a basis consisting of monomials $x\_1^{b\_1} \dotsm x\_n^{b\_n}$ with $0 \leq b\_i \leq a\_i-1$ for each $i$ (where I am taking $a\... | 0 | https://mathoverflow.net/users/88133 | 317093 | 137,531 |
https://mathoverflow.net/questions/317100 | 24 | This question is motivated by one of the problem set from this year's [Putnam Examination](https://www.maa.org/programs-and-communities/member-communities/maa-awards/putnam-competition-individual-and-team-winners). That is,
>
> **Problem.** Let $S\_1, S\_2, \dots, S\_{2^n-1}$ be the nonempty subsets of $\{1,2,\dots... | https://mathoverflow.net/users/66131 | A Putnam problem with a twist | $\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\set}[1]{\left\{ #1 \right\}}
\newcommand{\abs}[1]{\left| #1 \right|}
\newcommand{\tup}[1]{\left( #1 \right)}
\newcommand{\ive}[1]{\left[ #1 \right]}
\newcommand{\suml}{\sum\limits}
\newcommand{\sumS}{\suml\_{S \in P}}
\newcommand{\prodl}{\prod\limits}
\newcommand{\prodS}{\prod... | 27 | https://mathoverflow.net/users/2530 | 317105 | 137,534 |
https://mathoverflow.net/questions/317111 | 6 | I'm interested in affinely connected spaces, on which a metric is not necessarily defined, i.e. $(\mathcal{M},\Gamma)$. Since (as a physicist) my goal is to consider a generalized model of gravity, I restrict myself to the space of possible connections compatible with a symmetry group, by requiring that the Lie derivat... | https://mathoverflow.net/users/25356 | Is a symmetric, parallel (0,2)-tensor a metric? | You don't need parallelism: a symmetric tensor of type $\binom{0}{2}$ is positive definite if and only if it is a Riemannian metric, and has signature $(1,n-1)$ (where $n$ is the dimension of your manifold) if and only if it is a Lorentzian metric, and has signature $(p,q)$ if and only if it is a $(p,q)$ pseudo-Riemann... | 7 | https://mathoverflow.net/users/13268 | 317113 | 137,536 |
https://mathoverflow.net/questions/317109 | 2 | I have a surjective smooth map with surjective differential between two balls $\phi:B^{2n}\rightarrow B^{2k}$. Fix an integrable almost complex structure $J$ on $B^{2n}$. Assume that $\mathrm{Ker}\:d\phi$ is preserved by the action of $J$.
For any point $q \in B^{2k}$, I can find a point $p\in B^{2n}$ satisfying $\ph... | https://mathoverflow.net/users/132193 | Pushing forward a complex structure by submersion | In general, even if the fibers of $\phi$ are connected and the map is proper, there need not be an almost complex structure on $B^{2k}$ such that the differential of $\phi$ is complex linear. (I assume that you meant to assume that the kernel of the differential of $\phi$ is preserved by $J$, not that $J$ acts triviall... | 2 | https://mathoverflow.net/users/13972 | 317114 | 137,537 |
https://mathoverflow.net/questions/317126 | 7 | I heard that there are no 3 nonisomorphic simple groups of the same order.
>
> Question: Is there an elementary proof of this?
>
>
>
In case this is not the case, here a modified question:
>
> Question: Is there an elementary proof that there are not $m$ nonisomorphic simple groups of the same order with $... | https://mathoverflow.net/users/61949 | Simple groups of the same order | No, there are no known proofs of any results of this type that do not rely on the complete classification of finite simple groups
In particular, the result of Pyber (1993) giving an upper bound on the number of isomorphism classes of finite groups of order $n$ (see Jack Schmidt's answer to [this question](https://mat... | 14 | https://mathoverflow.net/users/35840 | 317128 | 137,540 |
https://mathoverflow.net/questions/317133 | 4 | A celebrated theorem of Nagata and [subsequent refinements](https://stacks.math.columbia.edu/tag/0ATT) to schemes and algebraic spaces say that over a not-completely-monstrous base scheme, any separated morphism can be openly immersed in a proper morphism.
The construction of such a compactification is involved and I... | https://mathoverflow.net/users/69037 | What is the topological/smooth analogue of Nagata compactification | The required condition for smooth manifolds (you can see it as an analogue of the finite type condition of Nagata's theorem) is that the 'ends' of the manifold aren't too complicated. The results of Browder-Levine-Livesay (<https://www.jstor.org/stable/2373259>) and others say that if M has f.g. homology and is 'connec... | 3 | https://mathoverflow.net/users/5279 | 317136 | 137,543 |
https://mathoverflow.net/questions/317158 | 5 | Let's say we have a homogeneous space $H\backslash G$.
Is it possible to tell whether this homogeneous space admits a conformally flat metric just from its group structure?
I am particularly interested in a situation when $H\backslash G$ is maximally-noncompact, i.e. $H$ is a maximally compact subgroup of $G$.
I... | https://mathoverflow.net/users/132222 | Conformally flat homogeneous spaces | A complete classification of homogeneous conformally flat Riemannian manifolds is given [here](https://link.springer.com/article/10.1007/BF01812986), namely, [Alekseevskiĭ, D. V.; Kimelʹfelʹd, B. N. Classification of homogeneous conformally flat Riemannian manifolds. Mat. Zametki 24 (1978), no. 1, 103–110, 143].
In... | 4 | https://mathoverflow.net/users/1573 | 317165 | 137,557 |
https://mathoverflow.net/questions/317152 | 22 | This might be a load of old nonsense.
I have always had it in my head that if $f:X\to Y$ is an injection, then $f$ has some sort of "canonical factorization" as a bijection $X\to f(X)$ followed by an inclusion $f(X)\subseteq Y$. Similarly if $g:X\to Y$ is a surjection, and if we define an equivalence relation on $X$ ... | https://mathoverflow.net/users/1384 | are quotients by equivalence relations "better" than surjections? | It seems that you've got factorization of maps covered, so let me address the question of why canonical quotient maps and canonical inclusions are "better".
Given a set $X$, in general there is a proper class of injections $Y \to X$. However, many of these are isomorphic, where injections $i : Y \to X$ and $j : Z \to... | 14 | https://mathoverflow.net/users/1176 | 317166 | 137,558 |
https://mathoverflow.net/questions/317132 | 4 | Marsden's Identity states that for every $\tau$ in $\mathbb{R }$:
$$(\cdot -\tau)^{k-1}=\sum\_j\Psi\_{j,k}(\tau)B\_{j,k,t} \, ,$$
with $\Psi\_{j,k}=(t\_j-\tau)\times...\times(t\_{j+k-1}-\tau)$.
Following de Boor's notation we have that $B\_{j,k,t}$ stands for the $j-th$ B-spline of order $k$ defined over the knot... | https://mathoverflow.net/users/132216 | Marsden's Identity and B-splines | I found the solution after some research, hence I'll post it here in case anyone have curiosity:
Marsden's Identity states that for all $\tau$ in $\mathbb{R}$ it holds that:
$$(\cdot -\tau)^{k-1}=\sum\_j\Psi\_{j,k}(\tau)B\_{j,k} \, ,$$
It's straightforward to show that $((\cdot-\tau\_j)^{k-1}:j=1,...,k)$ , $\tau\_... | 2 | https://mathoverflow.net/users/132216 | 317186 | 137,565 |
https://mathoverflow.net/questions/317212 | 1 | Let $(X,o)$ be an isolated normal surface singularity. Denote by $U:=X\backslash \{o\}$. I am looking for conditions on $(X,o)$ under which there exists a holomorphic section $\omega \in H^0(U, \Omega^2\_U)$ which does not vanish anywhere on $U$. If I understand correctly, this holds if $(X,o)$ is Gorenstein. Is there ... | https://mathoverflow.net/users/32151 | Existence of meromorphic 2-forms over normal surface singularities | This is equivalent to the Gorenstein condition. If $\omega\_X$ denotes the dualizing module, then $\omega\_X|U=\Omega^2\_U$. So, you get a nowhere vanishing section of $\omega\_X|U$. Since $\omega\_X$ has depth 2, this section extends to a nowhere vanishing section of $\omega\_X$ and thus $(X,o)$ is Gorenstein.
| 4 | https://mathoverflow.net/users/9502 | 317216 | 137,572 |
https://mathoverflow.net/questions/317222 | 11 | Is a scheme being Noetherian equivalent to the underlying topological space being Noetherian and all its stalks being Noetherian?
| https://mathoverflow.net/users/52982 | Is a scheme Noetherian if its topological space and its stalks are? | This is false. The easiest counterexample I could come up with is the following "affine line with embedded points at *every* closed [rational] point":
**Example.** Let $k$ be an infinite field, let $R = k[x]$, and for each $\alpha \in k$ let $R\_\alpha = R[y\_\alpha]/((x-\alpha)y\_\alpha,y\_\alpha^2)$. Then $R\_\alph... | 25 | https://mathoverflow.net/users/82179 | 317224 | 137,576 |
https://mathoverflow.net/questions/317232 | 8 | Does there exist a closed Moishezon manifold that does not have the homotopy type of the analytification of a smooth proper complex variety (I think we know that every closed Moishezon manifold is bimeromorophic to the analytification of a smooth proper complex variety, so for example fundamental groups have to be the ... | https://mathoverflow.net/users/nan | Moishezon manifold vs proper complex variety | **Edit:** I just realised that the OP asked for proper, not projective varieties. As it stands, it is still possible that Oguiso's Moishezon Calabi-Yau threefold is homotopy equivalent to a proper varitey, so the question remains open.
---
The only example I am aware of is due to Oguiso, *[Two remarks on Calabi-Yau... | 4 | https://mathoverflow.net/users/15782 | 317235 | 137,578 |
https://mathoverflow.net/questions/316852 | 2 | Let $X(t)$ be a martingale w.r.t. filtration generated by Brownian motion $B(t)$. There is a well-known theorem that states that there is a unique adapted process $H(t)$ such that
$$ X(t) = \int\_0^t H(s)dB(s).$$
Let's define step process $$M(t)=2 \sum\_{j=1}^{\lfloor t \rfloor} 1\_{\{ B(j)-B(j-1) > 0 \}} - \lfloor t ... | https://mathoverflow.net/users/116749 | Martingale representation theorem for symmetric random walk | I've caught myself into a trap here: the reason why I couldn't find a martingal representation is that $M(t)$ is not a martingale. For example:
$$ \mathbb{E}[M(1) | M(1/2)] = 2 \mathbb{P}(B(1) > 0 |B(1/2)) - 1. $$
And if $B(1/2) \neq 0$, which is true a.s., $\mathbb{E}[M(1) | M(1/2)] \neq M(1/2).$
| 0 | https://mathoverflow.net/users/116749 | 317242 | 137,580 |
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