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https://mathoverflow.net/questions/277623 | 6 | I want to compute an integral like this
$$\frac{\int\_y g(y) e^{-\beta f(y)} \text{d} y } {\int\_y e^{-\beta f(y)} \text{d} y}$$
where $f(y)$ is not necessarily convex and the dimension $d$ of $y$ is large.
This problem can be viewed as to integrate a function with respect to a density function whose normalized ... | https://mathoverflow.net/users/112817 | Does MCMC overcome the curse of dimensionality? | You need a **global** convexity to enjoy the optimal convergence rate, otherwise even local convexity will almost surely(not in probabilistic sense) lead to the worst rate you pointed out.
MCMC(Markov Chain Monte Carlo) does not overcome the curse of dimensionality. Quite the contrary, Bayesians are working very hard... | 5 | https://mathoverflow.net/users/25437 | 277677 | 123,166 |
https://mathoverflow.net/questions/277673 | 0 | Suppose that I have the following recursion for all $a\_k >1$: $a\_{k+1}^2 \leq a\_k^2 - a\_k$. Then one can see that the following inequality is true (proved via induction): $a\_k \leq a\_0 - \frac{k}{2}$. Now if I update my recursion as $a\_{k+1}^2 \leq a\_k^2 - a\_k +\frac{1}{2}$, the inequality, $a\_k \leq a\_0 - \... | https://mathoverflow.net/users/16976 | upper bound involving recursions | Rewrite $a\_{k+1}^2 \leq a\_k^2 - a\_k + \frac12$ as $4a\_{k+1}^2 \leq (2a\_k-1)^2 + 1$. Then
$$\frac{2a\_{k+1}}{2a\_k-1} \leq \left(1+\frac{1}{(2a\_k-1)^2}\right)^{1/2} \leq 1 + \frac{1}{2(2a\_k-1)^2} \leq 1 + \frac{1}{2(2a\_k-1)}.$$
Hence, $a\_{k+1} \leq a\_k - \frac{1}{4}$, implying that $a\_k \leq a\_0 - \frac{k}{4... | 1 | https://mathoverflow.net/users/7076 | 277694 | 123,170 |
https://mathoverflow.net/questions/277633 | 7 | The counit of an adjunction is split monic precisely when the right adjoint is full. Can anything nice be said about when the counit is monic (ideally in terms of the induced comonad)?
| https://mathoverflow.net/users/10368 | When are counits monic? | As noted in a comment (in the dual case), if the left adjoint is fully faithful then this property has a name: a coreflective subcategory whose counits are mono is called *monocoreflective* (the dual of "epireflective"). This is of course a property of the induced comonad (since a coreflection is the same as an idempot... | 7 | https://mathoverflow.net/users/49 | 277696 | 123,172 |
https://mathoverflow.net/questions/277645 | 4 | In these days I am thinking to a problem on accessible categories. I have a faithful functor $F : \mathcal{K} \to \text{Set}$ and I want to know if it preserves directed colimits. More generally I want to know if such a functor exists.
This was just context and motivations. Now comes the question.
For a diagram $D ... | https://mathoverflow.net/users/104432 | How to compute (co)continuity of functors | The "reason" that continuity of abelian (and also stable/triangulated) functors can be detected by zeroness of objects is that more generally, invertibility of morphisms in an abelian/stable/triangulated category is detected by zeroness of objects. That is, $f:A\to B$ is an isomorphism exactly when its kernel and coken... | 8 | https://mathoverflow.net/users/49 | 277697 | 123,173 |
https://mathoverflow.net/questions/277684 | 8 | For $\Omega\subset \mathbb{R}^3$ a region with $|\Omega| = |B\_1|$, let
$$
C(\Omega) = \int\_\Omega\int\_\Omega \frac{dxdy}{|x-y|}
$$
denote the Coulomb (or gravitational, etc) energy.
Poincaré is credited with an incomplete proof that (\*) $C(\Omega) \leq C(B\_1)$. I know the (now standard) proof using symmetrizati... | https://mathoverflow.net/users/1540 | Poincare's argument for maximizing the Coulomb energy | H. Poincaré, [Sur une théorème de M. Liapounoff relatif a l’équilibre d’une masse fluide,](http://gallica.bnf.fr/ark:/12148/bpt6k30607/f621.item.r=Poincare) Comptes Rendus de L’Academie des Sciences **104**, 622–625 (1887).
As discussed by [G.C. Evans,](http://www.ams.org/journals/bull/1941-47-10/S0002-9904-1941-0754... | 6 | https://mathoverflow.net/users/11260 | 277704 | 123,175 |
https://mathoverflow.net/questions/277656 | 3 | Let $G\_0$ be a connected semisimple algebraic group defined over an *algebraically closed* field $k\_0$. Let $k\supset k\_0$ be a larger algebraically closed field.
We write $G=G\_0\times\_{k\_0} k$ for the base change of $G\_0$ from $k\_0$ to $k$.
>
> **Question.** Let $H\subset G$ be a *spherical subgroup* of $G... | https://mathoverflow.net/users/4149 | Is any spherical subgroup conjugate to a subgroup defined over a smaller algebraically closed field? | The morphism $G/H\to\textrm{Spec}\,k$ is defined and flat over over some finitely generated $k\_0$-subalgebra of $k$. Thus, the question amounts to whether a flat family of homomogeneous spherical $k\_0$-varieties is generically trivial in the étale toplogy. In characteristic zero, this is true by Thm. 3.1 of Alexeev, ... | 3 | https://mathoverflow.net/users/89948 | 277708 | 123,178 |
https://mathoverflow.net/questions/275857 | 1 | As we know, that the halting problem of Turing machines is undecidable. given some restriction on $TM$ set of Turing Machines, we get a subclass $TM\_s$, halting problem of what subclasses of $TM$ can be decidable?
Does there exist a subclass $TM\_{max}$ halting problem of which is decidable such that any decidable ... | https://mathoverflow.net/users/14024 | Halting problem about subclass of Turing Machines | Let $K$ denote the set of halting Turing machines (TMs). You are asking (1) for what kind of sets $A$ of TMs is $A \cap K$ decidable, and (2) is there a maximal such $A$. Question 2 has been answered in the comments, I think, but to go over it again: Since $F \cap K$ is decidable for every finite $F$, any maximal such ... | 4 | https://mathoverflow.net/users/112863 | 277710 | 123,179 |
https://mathoverflow.net/questions/277723 | 4 | Let $X$ be a smooth projective variety and $W \subset X$ a smooth, projective subvariety. Let $\pi:\tilde{X} \to X$ be the blow-up of $X$ along $W$. Let $E$ be the exceptional divisor of $\pi$ and $i:E \hookrightarrow \tilde{X}$ the natural closed immersion. Let $\alpha \in H^k(\tilde{X})$ such that $i^\*\alpha=0$ in $... | https://mathoverflow.net/users/58203 | Gysin map and blow up | Yes. Put $U:=\widetilde{X}\smallsetminus E=X\smallsetminus W$. By the open-closed exact sequence, $\alpha $ comes from a class in $H^k\_c(U)$. But the map $H^k\_c(U)\rightarrow H^k(\widetilde{X})$ factors through $H^k(X)$, hence $\alpha =\pi ^\*\beta $ for some class $\beta $ in $H^k(X)$. Applying $\pi \_\*$ gives $\be... | 5 | https://mathoverflow.net/users/40297 | 277728 | 123,183 |
https://mathoverflow.net/questions/272514 | 2 | Consider the following Cauchy problem
$$u\_t - \Delta u + f(x,u,u\_x) = 0, \quad (t,x) \in (0,T) \times \mathbb{R}^n,$$
with initial condition $u(0,x) = g(x), \ x \in \mathbb{R}^n.$
Suppose that $u \in C^2((0,T) \times \mathbb{R}^n)$ is a solution of the initial value problem.
Can we obtain (under some reasonable... | https://mathoverflow.net/users/110835 | Solution to semilinear heat equation at $t=0$: $u_t(0,x) - \Delta u(0,x) + f(x,u,u_x)= 0 \ ?$ | The question is vague, because the OP does not mention in which sense the PDE is to be satisfied at initial time.
In addition, the assumption that $u$ is $C^2$ for $T>0$ is very weak, if not useless. For let me assume that $f$ is a smooth function with $f(\cdot,0,0)\equiv0$, and that the initial data $g$ is $L^1\cap ... | 0 | https://mathoverflow.net/users/8799 | 277730 | 123,184 |
https://mathoverflow.net/questions/277738 | 0 | Let $F$ and $H$ be normed spaces and let $E$ be a locally convex space.
Let $T:F\to H$ and $S:H\to E$ be linear operators, such that $\|T\|= 1$, $S$ is an injective semi-embedding (i.e. $S\overline{B}\_{H}$ is closed in $E$), and $ST$ is weakly compact.
>
> Does it follow that $T$ is weakly compact?
>
>
>
The ... | https://mathoverflow.net/users/53155 | Semi-embeddings and weak compactness | No. Let $S$ be the formal inclusion from $L\_\infty(0,1)$ into $L\_1(0,1)$ and let $T$ be the identity on $L\_\infty(0,1)$.
| 1 | https://mathoverflow.net/users/2554 | 277744 | 123,187 |
https://mathoverflow.net/questions/277781 | 3 | A complex manifold admits an almost complex structure, $J$, which satisfies
$$
{J^i}\_j{J^j}\_k=-{\delta^i}\_k,
$$
and a Hermitian metric, $g$, which satisfies
$$
g\_{st}{J^s}\_i{J^t}\_j=g\_{ij}. \tag{1}
$$
A hyperkahler manifold, on the other hand, admits an almost quaternionic structure which satisfies
$$
{(J\_u)^... | https://mathoverflow.net/users/99595 | How does the concept of a hermitian metric generalize to a hyperkahler manifold? | Here are some small facts about Quaternionic Geometry. First let us leave metrics aside, for the moment. There are 2 analogues of complex manifolds in the quaternionic world, namely triholomorphic manifolds and quaternionic manifolds. They are very different. Triholomorphic manifolds admit a 2-sphere of complex structu... | 2 | https://mathoverflow.net/users/81645 | 277784 | 123,202 |
https://mathoverflow.net/questions/277803 | 8 | The Kazhdan-Lusztig Conjecture (which is actually a theorem) gives the character of some irreducible modules of a (say) simple complex Lie algebra $\mathfrak{g}$ in terms of characters of Verma modules. More precisely, following the notations of *Representations of Semisimple Lie Algebras in the BGG Category O*, by Hum... | https://mathoverflow.net/users/61018 | Kazhdan-Lusztig theorem for composition factors of Verma modules | It's valid for integral weights of the form $\lambda-2\rho$, with $\lambda$ antidominant (these are the anti-dominant weights that have a dominant weight $w\_0\lambda$ in their orbit under the dot action). The proof is that if you tensor the simples $L\_w(-2\rho)$ and Vermas $M\_w(-2\rho)$ with the simple represenation... | 4 | https://mathoverflow.net/users/66 | 277806 | 123,207 |
https://mathoverflow.net/questions/277776 | 1 | Let $(M,g)$ be a Riemannian manifold and $V$ a unit Killing vector field on it. Under what condition on curvature tensor the following equation hold:
$$\nabla\_VQ=0,$$
where $Q$ is the Ricci operator defined as $g(QX,Y)=\rho(X,Y)$.
**Update1:** Einstein metrics that admit a unit Killing vector field satisfies the abo... | https://mathoverflow.net/users/90655 | Under what condition the covariant derivative of Ricci operator along Killing vector field vanish? | If $V$ is a Killing vector, then necessarily its Lie derivative annihilates all curvature tensors, including the Ricci operator, namely
$$
\mathcal{L}\_V Q = \nabla\_V Q + Q J\_V - J\_V Q = 0 ,
$$
where $J\_V$ is the ("Jacobian") endomorphism on tangent vectors defined by $\nabla\_U V = J\_V U$. Thus, the condition th... | 2 | https://mathoverflow.net/users/2622 | 277811 | 123,210 |
https://mathoverflow.net/questions/277800 | 4 | I know that the Clifford torus in $S^3$ is Willmore. I can also think about $S^1$-equivariant tori and Hopf tori as Willmore in the $3$-sphere.
Does anyone know if there are more Willmore tori in $S^3$? Any source, idea or comment is very welcome.
| https://mathoverflow.net/users/84866 | Willmore tori in $S^3$ | Yes, there are many examples . First of all, there are infinitely many minimal tori (immersed, not embedded ) in the 3-sphere.
There are also minimal surfaces with embedded planar ends of genus one.
Also, there is the Babich-Bobenko paper in which Willmore tori with umbilical lines are constructed.
| 5 | https://mathoverflow.net/users/4572 | 277814 | 123,212 |
https://mathoverflow.net/questions/277700 | 3 | The number of integer points with a given norm in the integer grid $\mathbb{Z} \times \mathbb{Z}$ may be calculated via the generating function
$$\theta\_3(q)^2= \left(\sum\_{n \in \mathbb{Z}} q^{n^2}\right)^2 = 1+4 q + 4 q^2 + 4 q^4 + 8 q^5 + \cdots,$$
which is a so called (squared) Jacobi function. Similarly the gen... | https://mathoverflow.net/users/44293 | "Sparse" Theta Series | Won't the following argument show that the difference between successive exponents can never be bounded away from zero no matter how clever you try to be in selecting $(a,b)$?
The idea is to consider pairs of integers $(n,m)$ such that $na-mb$ is close to zero.
1. If the ratio $a/b$ is rational then we can find int... | 3 | https://mathoverflow.net/users/15503 | 277832 | 123,217 |
https://mathoverflow.net/questions/277579 | 13 | The [Steinberg representation](https://en.wikipedia.org/wiki/Steinberg_representation) is a remarkable irreducible representation of a reductive algebraic group over a finite field or local field, or a group with a BN-pair. It is analogous to the 1-dimensional sign representation ε of a Coxeter or Weyl group that takes... | https://mathoverflow.net/users/10446 | Steinberg representation for sporadic simple groups? | The approach that I have taken to generalizing the Steinberg module to finite groups other than groups of Lie type is that, in general, the object we should consider is a chain complex, rather than just a module. For every finite group G and for every prime p there is a finite chain complex of projective modules, canon... | 17 | https://mathoverflow.net/users/112925 | 277843 | 123,221 |
https://mathoverflow.net/questions/277722 | 16 | A Grothendieck topos $\mathcal{E}$ is equivalent to the category of sheaves on some site $Q$. We say a sheaf $X\colon Q^{\text{op}}\to\mathsf{Set}$ is *constant* if it is the sheafification of a constant presheaf, i.e. one that factors through the terminal map $Q^{\text{op}}\to \{\*\}$.
But what if we forget the site... | https://mathoverflow.net/users/2811 | Characterize constant objects in the internal language of a topos? | For the first question: no, the notion of "constant object" doesn't depend on the site. The reason is in Anton's comment: every Grothendieck topos comes with a unique geometric morphism $p : \mathcal{E} \leftrightarrows \mathrm{Set}$, and the constant objects are those in the essential image of $p^\* : \mathrm{Set} \to... | 16 | https://mathoverflow.net/users/49 | 277844 | 123,222 |
https://mathoverflow.net/questions/277766 | 2 | I have three equations:
${m \choose 2} + nk = {x \choose 2}$
${n \choose 2} + mk = {y \choose 2}$
$x + y = m + n + k$
$m, n, k, x, y$ are natural numbers. I want to deduce from this 3 equations either $x = y$ or $m = n$. From where I got these equations makes me sure that this is only possible if $x = y$ and $... | https://mathoverflow.net/users/84871 | How to deduce an equation from this 3 Diophantine equations with 5 variables? | The first two equalities imply $x>m$ and $y>n$ so one can substitute $x=m+X$, $y=n+Y$ and $k=X+Y$, with still $X,Y \in \mathbb N$:
${X \choose 2}=nX+nY-mX\tag{1}$
${Y \choose 2}=mX+mY-nY\tag{2}$
From (1) follows: $\quad m=n+n\frac{Y}{X}-\frac{1}{X}{X \choose 2}$,
then eliminate $m$ from (2): $\quad {Y \choose 2... | 8 | https://mathoverflow.net/users/2480 | 277850 | 123,224 |
https://mathoverflow.net/questions/275796 | 1 | Sequence A002846 in <https://oeis.org/A002846> (OEIS) gives, for each positive integer $n$, the number $a(n)$ of ways of transforming a set of $n$ indistinguishable objects into $n$ singletons via a sequence of $n-1$ refinements, i.e. the number of ways of transforming a set of $n$ unlabelled objects into $n$ singleton... | https://mathoverflow.net/users/85541 | OEIS Sequence A002846 and properties of matrix inverses | Actually, this turned out to be embarrassingly simple. Let $M$ be such an $n\times{}n$ matrix and set $\mathbf{i}\_n$ to be the column matrix with entries 1 to $n$ and $\mathbf{1}\_n$ to be the column matrix with entries all equal to 1. By construction, each row of the matrix represents a partition of $n$. The first el... | 4 | https://mathoverflow.net/users/85541 | 277859 | 123,228 |
https://mathoverflow.net/questions/277854 | 4 | For $X = (x\_1^T,\ldots,x\_N^T)^T \in \mathbb{R}^{Nm \times 1}$, where $x\_i \in \mathbb{R}^{m \times 1}$ for $i \in \{1,\ldots,N\}$, $A \in \mathbb{R}^{r \times Nm}$, and $r \geq Nm$, I want to obtain a closed form solution of
$$\max\_{X \in \mathbb{R}^{Nm}}\{\|AX\|: \|x\_i\|^2=1\}$$
where $\|\cdot\|$ denotes the... | https://mathoverflow.net/users/112930 | Maximize the Euclidean norm of a matrix times a vector on unit sub-spheres | You cannot hope for anything like a closed form solution, or even an exact efficient algorithm for this problem, because it is [NP-hard](https://en.wikipedia.org/wiki/NP-hardness). The reduction is from the [max-cut](https://en.wikipedia.org/wiki/Maximum_cut) problem. Let's look at the special case $m=1$. Let $G = (V, ... | 7 | https://mathoverflow.net/users/35733 | 277861 | 123,229 |
https://mathoverflow.net/questions/277888 | 1 | Suppose that the matrix $M$ has non-positive off-diagonal elements. The matrix $M$ is
said to be an M-matrix if $M$ is non-singular and each entry of $C:=M^{−1}$
is non-negative.
I think I've proved the following implication: If $M$ is *symmetric*, then
$$ c\_{ij}=0 \Longrightarrow m\_{ij}=0 \quad (\textrm{for all ... | https://mathoverflow.net/users/85966 | zero patterns of M-matrices and their invereses | I haven't encountered this exact result, but a closely related one is a complete characterization of the possible zero patterns for inverse M-matrices in <https://doi.org/10.1016/j.laa.2009.03.022>.
It seems to me that this one can be proved with the same techniques (decomposition into irreducible blocks + keeping tr... | 1 | https://mathoverflow.net/users/1898 | 277893 | 123,242 |
https://mathoverflow.net/questions/277890 | 1 | I am interested in studying the following problem:
\begin{align}
\sup\_{\mu \in \mathcal{D} } \int\_{\mathbb{R}} f(x) d\mu(x)
\end{align}
where $\mu$ is a probability measure. Assume that $\mathcal{D}$ is closed, covex and compact (in weak^{\*} topology).
We know that if $f(x)$ is continuous and bounded then $\mu \t... | https://mathoverflow.net/users/69661 | Maximizing linear function (not necessarily continuous) over a compact, closed and convex domain | The answer is yes.
Call ${\mathcal E}$ the set of extreme points of ${\mathcal D}$.
It is a consequence of [Choquet's theorem](https://en.wikipedia.org/wiki/Choquet_theory) that for every $\mu\in{\mathcal D}$ there exists a probability measure $\nu=\nu\_\mu$ on ${\mathcal E}$ such that $\mu=\int\_{\mathcal E}\rho d\nu... | 1 | https://mathoverflow.net/users/18698 | 277896 | 123,243 |
https://mathoverflow.net/questions/272583 | 8 | From "[The multiple facets of the associahedra](http://www.claymath.org/library/academy/LectureNotes05/Lodaypaper.pdf)" by Loday:
>
> Let us consider the formal power series
>
>
> $$f(x) = x+a\_1 x^2 +a\_2 x^3 + \cdots+ a\_n x^{n+1} + \cdots$$
>
>
> and let
>
>
> $$ g(x) = x+b\_1 x^2 + b\_2 x^3 + \cdots + b\_... | https://mathoverflow.net/users/12178 | Loday's characterization and enumeration of faces of associahedra (Stasheff polytopes) | I believe that the proof to which Loday is referring is the one that appears for Proposition 13.11.7 in [*Algebraic Operads*](https://books.google.com/books?id=s9Usr1BK4pwC&pg=PA550&lpg=PA550&dq=associahedron+inverse+power+series&source=bl&ots=uKg6O09Kb5&sig=Ih2h3Bd4qNXURDcfVSkGg0COrY4&hl=en&sa=X&ved=0ahUKEwjtyIz967vVA... | 3 | https://mathoverflow.net/users/104932 | 277902 | 123,247 |
https://mathoverflow.net/questions/277915 | 2 | I see this page [Ordinary least square and random projection](https://mathoverflow.net/questions/158270/ordinary-least-square-and-random-projection), and I am thinking that how $L^2$ integrable random variables be regarded as projections over a defined filtration sequence $\mathcal{F\_n}$ ?
It seems that a vector spa... | https://mathoverflow.net/users/112960 | Square integrable conditional expectations as projections | No. Vector space structure is not enough, we actually need a compatible lattice structure to make things work. To apply the conditional expectation operator $E(\bullet\mid Y)$ onto the Hilbert space consisting of $L^2(\mathcal{X})$ random variables defined on the probability space $\mathcal{X}$, it requires us to consi... | 4 | https://mathoverflow.net/users/25437 | 277917 | 123,253 |
https://mathoverflow.net/questions/277874 | 3 | Assume that $G$ is an affine reductive algebraic group (I am mostly interested in the case $GL\_n$) over an algebraically closed field $K$ of characteristic zero. Assume also that $G$ acts on an affine variety $X$ with finite stabilizers. I would like to ask how does the isomorphism type $Stab\_G(x)$ varies with respec... | https://mathoverflow.net/users/41644 | variations of finite stabilizer in the action of an algebraic group on an affine variety | If all stabilizers are finite then all orbits are closed. Hence Luna's slice theorem applies to every orbit which means that one can reduce to the case that $G$ is finite. This yields a lot of regularity:
* The action is proper.
* All sets $X\_H$ are locally closed.
* There is a unique $H$ (the generic isotropy group... | 3 | https://mathoverflow.net/users/89948 | 277936 | 123,261 |
https://mathoverflow.net/questions/277764 | 6 | For two non-empty finite sets $A,B$ in the real line define the $\ell\_1$-distance $d\_1(A,B)$ between $A$ and $B$ as the smallest Lebesgue measure of a closed subset $\Gamma\subset \mathbb R$ such that $A\cup B\subset\Gamma$ and each connected component of $\Gamma$ intersects both sets $A$ and $B$. It is clear that su... | https://mathoverflow.net/users/61536 | A quick algorithm for calculating the $\ell_1$-distance between two finite sets on the real line? | While it doesn’t seem to be possible to beat the $O(n\log n)$ bound for general input, the problem can be solved in time $O(n)$ if $A\cup B$ is given sorted.
This builds on the ideas in fedja’s and Aaron Meyerowitz’s answers. As implicit in the other posts, I am assuming unit-cost real RAM or a similar model, so that... | 3 | https://mathoverflow.net/users/12705 | 277940 | 123,262 |
https://mathoverflow.net/questions/277329 | 11 | In a 1981 [talk](http://rene.guitart.pagesperso-orange.fr/textespublications/guitart81theosketches.pdf), René Guitart cites Albert Burroni as having given "A first interesting example of a mixed sketch...for the category of topological spaces" in 1970. This was apparently done in Burroni's Université de Paris thesis, a... | https://mathoverflow.net/users/43000 | What was Burroni's sketch for topological spaces? | If you read french, you can look at page 6 [here](https://webusers.imj-prg.fr/~rene.guitart/docpp/jaburroni02.pdf). It is far from formal but it gives a good idea of the mixed sketch. In my understanding, there is a unique inductive cone in that mixed sketch, which comes from the canonical representation of the filter ... | 10 | https://mathoverflow.net/users/38838 | 277945 | 123,265 |
https://mathoverflow.net/questions/277954 | 2 | Consider a set of distinct vectors {$a,b,c,d,...$} in a subspace $C$ of $R^n$.
$C = \pi (C)$ where $\pi : R^n \rightarrow R^n$ defined by $\pi (x\_1,x\_2,...,x\_n)=(x\_2,...,x\_n,x\_1)$
How can I prove the following?
>
> For any two vectors $a$ and $b$ belong to $C$, $a \circledast b \in C$ ($\circledast$ : cir... | https://mathoverflow.net/users/112983 | circular convolution over Invariant subspace | If we write the standard basis vectors as $e\_1,\dots,e\_n$, with the cyclic convention that $e\_{n+1}=e\_1$, note that $\pi(e\_i)=e\_{i-1}$ and that $\pi$ is linear, so $\pi(\sum b\_i e\_i)=\sum b\_{i+1}e\_i$. By induction, $\pi^{\circ j}(\sum b\_i e\_i)=\sum b\_{i+j} e\_i$.
$c=a\*b=\sum\_{ij} a\_j b\_{i-j} e\_i=\sum\... | 3 | https://mathoverflow.net/users/13268 | 277958 | 123,268 |
https://mathoverflow.net/questions/277942 | 1 | For a knot complement on 3-sphere, there's at least one Lens-space Dehn filling (1/0-slope). Is it also true for any 1-cusp 3-manifold?
| https://mathoverflow.net/users/112978 | Is there any 1-cusp 3-manifold without Lens space Dehn filling? | This question is not really research-level, but I suppose it might be hard to find explicitly in the literature.
"Most" 1-cusped hyperbolic 3-manifolds will not admit a lens space space filling (in an imprecise sense). The simplest obstruction is homology: if $H\_1(M,\partial M;\mathbb{Q})\neq 0$ ($M$ has "cuspidal ... | 6 | https://mathoverflow.net/users/1345 | 277968 | 123,272 |
https://mathoverflow.net/questions/277838 | 4 | Anders Kock's two texts on [synthetic differential geometry](https://ncatlab.org/nlab/show/synthetic+differential+geometry) (SDG) are a great place to get geometric intuition, especially when it comes to jets. Unfortunately, he doesn't seem to cover semi-holonomic jets in his text, and I find the usual definition of se... | https://mathoverflow.net/users/56938 | Semi-holonomic jets in synthetic differential geometry | According to Liebermann's [*Introduction to the theory of semi-holonomic jets*](https://eudml.org/doc/226286) p.177:
>
> a local section $s:U\subset M\to J^1E$ is said to be *adapted at $x\in U$* if $s(x)=j^1\_x(\beta \circ s)$ where $\beta:J^1 E\to E$ is the target map; then the jet $j^1\_x(s)$ is called *semi-hol... | 3 | https://mathoverflow.net/users/745 | 277972 | 123,273 |
https://mathoverflow.net/questions/270330 | 8 | Let $Q=(Q\_0,Q\_1,h,t)$ be a quiver consisted of a pair of finite sets $Q\_0$(vectors),and $Q\_1$ (arrows) supplied with two maps $h : Q\_1 → Q\_0$ (head) and $t : Q\_1 → Q\_0$ (tail ). This definition allows the underlying graph to have multiple edges and (multiple) loops.
Then the path algebra of $Q$ is the graded ... | https://mathoverflow.net/users/83928 | A question about the quivers with potentials | Let $B$ be an $n\times n$ skew-symmetrizable matrix with integer coefficients. The first question seems to be "how to 'realize' $B$ through a 'path algebra'?". There are at least three different approaches to this question (Demonet,Dlab-Ringel,Geiss-Leclerc-Schröer). The three approaches seem to fit in the following ge... | 6 | https://mathoverflow.net/users/112993 | 277973 | 123,274 |
https://mathoverflow.net/questions/277925 | 2 | Let $T$ be a first-order theory which eliminates imaginaries, and let $A$ be an algebraically closed set in a model of $T$. Let $Gal\_L(T[A])$ be the Lascar group of the theory $T[A]$, which is $T$ with constants added for $A$. Is $Gal\_L(T[A])$ trivial? I would think not, but I don't know a counterexample.
| https://mathoverflow.net/users/2362 | Is the Lascar group over $A$ trivial when $T=T^{eq}$ and $A = acl(A)$? | There are several notions of strong type in model theory. $a$ and $b$ have the same:
* Lascar strong type over $A$, if they are equivalent under every bounded $A$-invariant equivalence relation.
* Kim-Pillay strong type (aka compact strong type) over $A$, if they are equivalent under every bounded $A$-type-definable ... | 4 | https://mathoverflow.net/users/2126 | 277983 | 123,276 |
https://mathoverflow.net/questions/277947 | 7 | Some motivation: The matrix $M$ is *Butson Hadamard* if the entries of $M$ are $k^{\textrm{th}}$ roots of unity (for some $k$), and distinct pairs of rows are orthogonal under the usual Hermitian inner product. I am interested in classifying the Butson Hadamard matrices for which some power is a real scalar matrix. (Th... | https://mathoverflow.net/users/27513 | Pairs of roots of unity whose real part satisfies a polynomial identity | We have
$$ \frac{\zeta\_1+\frac{1}{\zeta\_1}}{\zeta\_2+\frac{1}{\zeta\_2}}=\sqrt{2}. $$
By applying Galois group automorphisms we can assume that
$\zeta\_2=e^{2\pi i/n}$ for some $n$, where we can take either value
of $\sqrt{2}$. Solving for $\zeta\_1$ gives
$$ \sqrt{2}\zeta\_1=\zeta\_2+\frac{1}{\zeta\_2}
\pm\sqrt... | 10 | https://mathoverflow.net/users/2807 | 277986 | 123,278 |
https://mathoverflow.net/questions/277338 | 9 | Is there a (canonical) Riemannian submersion from the complex hyperbolic space $\mathbb C\mathbb H^n$ into the hyperbolic space $\mathbb H^n$?
In the affirmative case, what can we say about the geometry of the fibers or about the O'Neill tensors (T and A)?
| https://mathoverflow.net/users/112690 | Riemannian submersions from complex hyperbolic space into the hyperbolic space | I don't have a complete answer, but here are a few remarks about this that you may find interesting or useful:
The OP didn't specify exactly what was meant by 'complex hyperbolic space' $\mathbb{CH}^n$ and 'hyperbolic space' $\mathbb{H}^n$, in the sense that the sectional curvatures of the two spaces weren't specifie... | 11 | https://mathoverflow.net/users/13972 | 277996 | 123,284 |
https://mathoverflow.net/questions/278021 | 0 | Guys this problem really bothers me (I don`t know how to prove it) please help:
What is the maximum number of points in convex position on a $n\times m$ grid?
(My guess would be $2\*(m+n)-4$.)
| https://mathoverflow.net/users/113022 | Maximum number of points in convex position on a grid | Your guess is correct. There are at most two vertical sections which contain more than 2 points. So, totally we have at most $m+m+(n-2)2=2m+2n-4$ points. The example is the perimeter of the grid.
| 2 | https://mathoverflow.net/users/4312 | 278022 | 123,289 |
https://mathoverflow.net/questions/278028 | 7 | Let $\kappa\geq \aleph\_0$ be a cardinal, and suppose that ${\cal U}$ is a non-principal ultrafilter on $\kappa$. We regard ${\cal U}$ as a poset $({\cal U}, \subseteq)$.
Suppose that there are posets $P, Q$ such that ${\cal U} \cong P\times Q$. Does this imply one of $P, Q$ consists of one point only?
| https://mathoverflow.net/users/8628 | Are free ultrafilters as posets product-irreducible? | No. Every nonprincipal ultrafilter $U$, considered as a partial under $\subseteq$, is a nontrivial product order. To see this, suppose that $U$ is a nonprincipal ultrafilter on $\kappa$.
Partition $\kappa=A\sqcup B$ into two sets with $A\in U$ and $B$ nonempty.
Every $X\in U$ can be written as
$X=(X\cap A)\sqcup (X\ca... | 11 | https://mathoverflow.net/users/1946 | 278031 | 123,293 |
https://mathoverflow.net/questions/278034 | 4 | Let $l$ be a prime number and $q$ be a power of a prime number different from $l$. Recall that a Weil number (relative to $q$) is an element $x$ of $\overline{\mathbb{Q}\_l}$ that is an algebraic integer and such that for any embedding $\sigma:\overline{\mathbb{Q}}\_l\to\mathbb{C}$ one has $|\sigma(x)|=q^{w/2}$ for som... | https://mathoverflow.net/users/10707 | Weight of Weil numbers in the residue field of $\overline{\mathbb{Q}}_l$ | Presumably by a map $\phi: \overline{W} \rightarrow \mathbb{Z}/m$ that is "compatible with the weight function" you would require that Weil numbers $W^0$ of weight zero lie in the kernel of $\phi$. But every non-zero element the residue field of $\overline{\mathbb{Q}}\_l$ lifts to a root of unity of $\overline{\mathbb{... | 3 | https://mathoverflow.net/users/113030 | 278038 | 123,295 |
https://mathoverflow.net/questions/278013 | 35 | I will word this question in terms of linear operators acting on $\mathbb{C}^n$ for simplicity. Feel free to provide an answer in terms of more general Hilbert spaces if you think it makes more sense that way.
The standard norm induced by the inner product on $\mathbb{C}^n$ is the Euclidean norm $ \sqrt{\langle x, x\... | https://mathoverflow.net/users/113018 | What is the intuition for the trace norm (nuclear norm)? | Another answer is that $M\_n$, the space of $n\times n$ complex matrices, carries an *operator norm* where the norm of a matrix is its norm as a linear operator from $\mathbb{C}^n$ to itself (giving $\mathbb{C}^n$ euclidean norm). For some of us, this is the most natural and useful norm on $M\_n$.
With operator norm,... | 19 | https://mathoverflow.net/users/23141 | 278040 | 123,296 |
https://mathoverflow.net/questions/278045 | 2 | As in my other question, it is assumed that the (total) function describing a given notation is denoted as $address:p\rightarrow \Bbb{N}$ and assumed to be bijective.
Suppose we are given two notations $N\_1$ and $N\_2$ for some $p \in \omega{\_C}{\_K}$(Church-Kleene). Denote the mapping from $N\_1$ to $N\_2$ as $P{... | https://mathoverflow.net/users/112385 | Mapping between Notations | Your question is about the oracle strength needed to compute an
isomorphism between two isomorphic computable well-orders. In
general, $0''$ is not necessarily enough to compute such an isomorphism, unless the order-types are sufficiently small, and the general
phenomenon is that the strength needed to compute the isom... | 8 | https://mathoverflow.net/users/1946 | 278049 | 123,298 |
https://mathoverflow.net/questions/278053 | 4 | Let $\mathbb{R}^\infty$ be the product of countably many real lines.
Assume that a finitely generated group $\Gamma$ acts on $\mathbb{R}^\infty$ (linearly and continuously) and there is a nonempty convex compact $\Gamma$-invariant subset $K$.
Is it true that $\Gamma$ has a fixed point in $K$?
(In my case, the gro... | https://mathoverflow.net/users/1441 | Fixed point of a group action | I don't think so. Let $\Gamma$ be the free group with 2 generators. It has a natural action on the boundary $\partial\Gamma$ and therefore on the space of probability measures on the boundary as well. The latter action is affine and has no fixed points (no invariant measures on the boundary). Let us now realize this ac... | 8 | https://mathoverflow.net/users/8588 | 278054 | 123,299 |
https://mathoverflow.net/questions/278067 | 18 | I would like to learn more about Cartan Geometry ("les espaces généralisés de Cartan"). I ordered Rick Sharpe's book "Differential Geometry: Cartan's generalization...", which would take a long time to arrive though. In the mean time, can someone recommend possibly some online lecture notes, or some online papers conta... | https://mathoverflow.net/users/81645 | Online References for Cartan Geometry | There is a [series of four recorded lectures](https://www.youtube.com/watch?v=fTlRYcyISfA) by Rod Gover introducing conformal geometry and tractor calculus. Tractor bundles are natural bundles equipped with canonical linear connections associated to $(\mathfrak{g}, H)$-modules. Tractor connections play the same role in... | 11 | https://mathoverflow.net/users/56938 | 278069 | 123,305 |
https://mathoverflow.net/questions/278066 | 1 | Consider a Riemannian manifold $\mathcal{M}$ modeled on a (possibly infinitely dimensional) Hilbert space. Suppose that $\{p\_i\} \subset \mathcal{M}$ and $p\_i \to p$ .
We say that a sequence of covectors $v^\*\_i \in T^\*\_{p\_i}\mathcal{M}$ is weakly converges to $v \in T\_p^\*\mathcal{M} $
if for every smooth fu... | https://mathoverflow.net/users/53059 | Weak convergence and bounded sequence on a Riemannian manifold | Sorry, I did not read it carefully enough. Here is the revised version.
The answer is yes, if it is a strong Riemannian manifold modelled on Hilbert space (strong: generated the topology on each tangent space).
Namely, since $p\_i \to p$ we may assume without loss that all $p\_i$ are in one chart, so we are in a Hi... | 4 | https://mathoverflow.net/users/26935 | 278084 | 123,310 |
https://mathoverflow.net/questions/194399 | 10 | I have seen computations of the Dolbeault cohomology groups on compact Kahler manifolds using Hodge theory.
I have never seen the computation of Dolbeault cohomology for simple domains in $\mathbb{C}^n$, aside from showing that they are trivial (for domains of holomorphy).
For example, I would like to see a computa... | https://mathoverflow.net/users/1106 | Computing Dolbeault cohomology of some simple domains | $\def\CC{\mathbb{C}}\def\cO{\mathcal{O}}$Here is a computation of the Dobault cohomology of $X:=B(\infty) \setminus B(0) = \CC^2 \setminus \{ (0,0) \}$. I think that balls of finite radius should be behave basically the same way, but the details will be messier and it sounds like you just want to see an example.
Set... | 8 | https://mathoverflow.net/users/297 | 278087 | 123,311 |
https://mathoverflow.net/questions/275214 | 7 | While studying the behaviour of umbilic points on Weingarten surfaces I discovered that the following combinatorial identity must be true.
For all $l,m\in{\mathbb N}$ with $l\geq m-1\geq0$ the following holds:
$
\sum\_{k=m-1}^l(-1)^{k+m}\frac{k+2}{k+1}{l \choose k}{k+1 \choose m}=
\left\{\begin{array}{ccl}
0&if& l>... | https://mathoverflow.net/users/21179 | Is there a simple proof of the following Identity for $\sum_{k=m-1}^l(-1)^{k+m}\frac{k+2}{k+1}{\binom l k}\binom{k+1}m$? |
>
> We obtain for $l,m\in\mathbb{N}$ with $0\leq m-1 \leq l$:
> \begin{align\*}
> \color{blue}{\sum\_{k=m-1}^{l}}&\color{blue}{(-1)^{k+m}\frac{k+2}{k+1}\binom{l}{k}\binom{k+1}{m}}\\
> &=\frac{1}{m}\sum\_{k=m-1}^l(-1)^{k+m}(k+2)\binom{l}{k}\binom{k}{m-1}\tag{1}\\
> &=\frac{1}{m}\binom{l}{m-1}\sum\_{k=m-1}^l(-1)^{k+m}... | 7 | https://mathoverflow.net/users/51505 | 278088 | 123,312 |
https://mathoverflow.net/questions/278042 | 5 | Let $G$ be a finite group, and consider the composition $X \* Y$ on $\mathbb{C}G$ defined by $$(\sum\_g \alpha\_g u\_g) \* (\sum\_g \beta\_g u\_g) = \sum\_g \alpha\_g \beta\_g u\_g.$$ This composition can be reformulated as a *convolution* using the Fourier transform.
*Question*: Let $X,Y \in \mathbb{C}G$ be positive... | https://mathoverflow.net/users/34538 | A Schur-like product theorem on groups | The answer is positive. For ease of notation, let me write $a=\sum\_g \alpha\_g u\_g$ and $b = \sum\_g \beta\_g u\_g$.
Let's equip $\mathbb{C}G$ with the positive inner product induced by the standard basis. In terms of the usual normalized trace on a group algebra, this inner product is given by $(x,y)\mapsto\mathrm... | 4 | https://mathoverflow.net/users/27013 | 278089 | 123,313 |
https://mathoverflow.net/questions/278082 | 0 | Let $E$ be an elliptic curve defined over a fixed number field $F$. Note that there exists a unique complex number $\tau$ in the upper half plane so that
$E(\mathbb{C})\simeq\mathbb{C}/L\_\tau$
where $L\_\tau=\mathbb{Z}+\mathbb{Z}\tau$. From this, we can find an equation for $E$ of the form
$y^2=4x^3-g\_2(\tau)x-... | https://mathoverflow.net/users/44005 | Equation of elliptic curve and field of definition | First, $\tau$ is certainly not unique, it is only well-defined up to linear fractional conjugation by elements of $\text{SL}\_2(\mathbb Z)$. Second, to answer your question, no, $g\_2(\tau)$ and $g\_3(\tau)$ need not be in $F$. What is true is that there exists a complex number $u\in\mathbb C^\*$ such that $u^2g\_2(\ta... | 7 | https://mathoverflow.net/users/11926 | 278091 | 123,314 |
https://mathoverflow.net/questions/278033 | 13 | **Preliminary:** I believe the notion of *primitive recursive* functions on ordinals is standard and unproblematic (the main difference with the finite case is that one needs to introduce a $\sup$ or $\limsup$ in definition of primitive recursion). If there is any doubt, I refer to the notion defined in either one of t... | https://mathoverflow.net/users/17064 | About primitively recursively recognizable ordinals | Call $\alpha\in
On$ *p.r.closed*, if every *p.r. set function* $f$ is total on $L\_{\alpha}$.
(Note (1) If $\alpha^\ast$ is the least p.r. closed ordinal $> \alpha$ then it is the next image of a point in the $\omega$'th Veblen function, so in any case is much smaller than the next admissible. Hence
(2) the p.r.clos... | 6 | https://mathoverflow.net/users/6942 | 278097 | 123,316 |
https://mathoverflow.net/questions/278074 | 5 | This is a related question to the one I posted on MO earlier:
[Is there a simple proof of the following Identity for $\sum\_{k=m-1}^l(-1)^{k+m}\frac{k+2}{k+1}{\binom l k}\binom{k+1}m$?](https://mathoverflow.net/questions/275214/is-there-a-simple-proof-of-the-following-identity-for-sum-k-m-1l-1km?noredirect=1#comment6... | https://mathoverflow.net/users/21179 | Is there a simple proof of the following binomial Identity (part 2)? | Following the hint @darijgrinberg stated in the comment section with respect to the *beauty* inside the square brackets we focus on the sum and
>
> we obtain
> \begin{align\*}
> \color{blue}{\sum\_{k=m+1}^{l+1}}&\color{blue}{(-1)^{k+m}\binom{l+1}{k}
> \left[(1-(2k+1)(m+2))\frac{2m+2}{2k+1}\binom{k}{m+1}\right.}\\
... | 3 | https://mathoverflow.net/users/51505 | 278105 | 123,321 |
https://mathoverflow.net/questions/251996 | 16 | Today I read about Gromov's definition of minimal volume for smooth manifolds.
$$\min {\rm Vol}(M):=\inf\_{|K\_g|\leq1}\{{\rm Vol}(M,g)\}.$$
Gromov's conjecture states that for every closed simply connected odd-dimensional manifold $\min {\rm Vol}(M)=0$. Is the Gromov conjecture still open? Can anybody give an exam... | https://mathoverflow.net/users/90655 | Is the Gromov conjecture still open? | According to [wikipedia](http://en.wikipedia.org/wiki/Minimal_volume), the conjecture is still open. I do not see at all how RF can help here. (Apart from dimension 3 when the statement is of course a corollary of Perelman's geometrization theorem.) There is no positivity assumption on curvature(s) of $M$ in this conje... | 11 | https://mathoverflow.net/users/21684 | 278107 | 123,322 |
https://mathoverflow.net/questions/278072 | 1 | For a complete non-compact Riemannian manifold with sectional curvature positive, it is diffeomorphic to $\mathbb{R}^n$ by known result. Choose a point $p$ on the manifold, is it possible that the distance function $d(p, \cdot)$ has a sequence of critical points going to the infinity of the manifold?
| https://mathoverflow.net/users/100486 | complete non-compact Riemannian manifolds and critical points | Yes, it is a classical result.
Let $q\_1,\dots,q\_n$ be a sequence of critical points such that $$|q\_{n+1}-p|\ge 2\cdot |q\_n-p|.$$
By Toponogov's comparison,
$$\measuredangle [p\,^{q\_i}\_{q\_j}]\ge \tfrac\pi3.$$
Hence we get a bound on $n$.
| 6 | https://mathoverflow.net/users/1441 | 278108 | 123,323 |
https://mathoverflow.net/questions/278095 | 4 | Let $(X,\tau)$ be a topological space. If $A\subseteq X$ we define the following equivalence relation on $X$: $$\sim\_A = \{(x,y) \in X^2: x=y \text{ or }\{x,y\}\subseteq A \}.$$
Let $(X,\tau)$ be an infinite space with the following property:
>
> If $A\subseteq X$ and $|A|<|X|$ then $X\cong X/\sim\_A$.
>
>
>
... | https://mathoverflow.net/users/8628 | Spaces that are invariant under some contractions | For a countable space $X$ the discussed property means that $X$ is homeomorphic to $X/A$ for any finite subset $A\subset X$. Let us call this property *quotient-homogeneous*.
There are many quotient-homogeneous countable spaces which do not fall into the categories (1)-(3) of Dominic van der Zypen:
4) any countabl... | 2 | https://mathoverflow.net/users/61536 | 278111 | 123,326 |
https://mathoverflow.net/questions/278114 | 5 | Suppose that a conference paper already has a MathSciNet review.
If later that paper appears with the same abstract in a journal, what is the point of making a new review?
Also, wouldn't it make more sense to first ask the person who made the review for the conference version?
(In a way, of course it's better to have d... | https://mathoverflow.net/users/955 | What is the need for MathSciNet reviewing again the journal versions of conference papers? | At the moment most of those decisions come from me, at least for computer science papers (those with a 68 [class](http://www.ams.org/msc/msc2010.html) as primary). The practice of having proceedings and final versions of papers is not exclusive to computer science, but this is where it is most common.
I've found more... | 17 | https://mathoverflow.net/users/6085 | 278115 | 123,327 |
https://mathoverflow.net/questions/277438 | 2 | Does there exist a complete, non-compact Riemannian manifold $(M^n, g)$, which has only one ray starting from one fixed point $p\in M^n$?
| https://mathoverflow.net/users/100486 | Rays on non-compact complete Riemannian manifolds | Recall that a *ray* is a geodesic defined on $[0,\infty)$ that minimizes the distance between any of its points.
A capped round cylinder has required properties: From any point sufficiently far away from the cap there is only one ray going along the axis of the cylinder. (This is because no ray passed through the ca... | 5 | https://mathoverflow.net/users/1573 | 278116 | 123,328 |
https://mathoverflow.net/questions/278073 | 11 | I've been recently looking into extending the Grothendieck construction to the case of a strict $\omega$-functor $F$ from a strict $\omega$-category $X$ to $\omega-\operatorname{Cat}$. By some recent developments (a paper of Dimitri Ara and Georges Maltsiniotis on defining the lax/oplax join and slice for strict $\omeg... | https://mathoverflow.net/users/1353 | Does Dyer's Thesis prove that the sheaf/fibration equivalence fails in dimension n>2? | On glancing at the paper, I think that Dyer is just confused. If anything, the place where something new and different happens is $n=2$, not $n=3$, but that case was explained by Hermida.
Dyer's proposition 5.3.2 says (in the case $k=n=2$ that he proves) that every 2-cell factors as a "composition" of a cartesian arr... | 9 | https://mathoverflow.net/users/49 | 278119 | 123,329 |
https://mathoverflow.net/questions/278121 | 5 | Let $G$ be a finite abelian group, and let $R$ be a ring (commutative with 1). In particular I'm interested in the case where $R$ is $\mathbb{Z}/n\mathbb{Z}$ for some $n$. By functoriality, every automorphism $\alpha\in\text{Aut}(G)$ induces an automorphism of the group algebra $R[G]$, and hence an automorphism of the ... | https://mathoverflow.net/users/88840 | If $G$ is a finite abelian group, and $R$ a ring, then does every automorphism of $G$ induce the identity on the group $R[G]^\times/G$? | No.
Let $G=\mathbb{Z}\_8=\langle w\rangle$ and set $v=2-w^4+(1-w^4)(w+w^{-1})$. This is a (normalized) unit in $\mathbb{Z} G$ by 10.8 in S. K. SEHGAL, Units in Integral Group Rings (with an appendix by A. Weiss), Pitman Monographs and Surveys in Pure and Applied Math.69, Longman Scientific & Technical, Harlow, 1993.
... | 6 | https://mathoverflow.net/users/41862 | 278123 | 123,330 |
https://mathoverflow.net/questions/278016 | 8 | Consider the following integral
$$
\int\_0^{\infty}\frac{e^{-x}-1}{x^{2+\frac{A}{\log b-5/6}}}\frac{1}{\log(b/x)-i\pi/2}\,dx
$$
where $A>0$ and $b>0$. I am interested in the small $b$ asymptotics of this integral. Any ideas on how to proceed?
| https://mathoverflow.net/users/42072 | How to get this integral's asymptotics? | The integral in question is
\begin{equation\*}
I:=\int\_0^\infty f(x)\,dx, \tag{1}
\end{equation\*}
where
\begin{equation\*}
f(x):=\frac{e^{-x}-1}{x^{2-\varepsilon}}\frac{1}{\ln(b/x)-i\pi/2},\quad
\varepsilon:=-\frac{A}{\ln b-5/6}\downarrow0.
\end{equation\*}
Let
\begin{equation\*}
C:=C\_\varepsilon:=A-\tfrac56... | 3 | https://mathoverflow.net/users/36721 | 278128 | 123,332 |
https://mathoverflow.net/questions/278120 | 7 | Recall that a $\sigma$-space is a topological space such that every $F\_{\sigma}$-set is $G\_{\delta}$-set.
$X$ - $\sigma$-set, if $X$ is a $\sigma$-space and it is subset of real line $R$.
Let $F$ be a Borel isomorphism (Baire isomorphism order $\alpha$) from a $\sigma$-set $X$ onto a topological space $Z$. Does ... | https://mathoverflow.net/users/112417 | Are σ-sets preserved by Borel isomorphisms? | Under some set-theoretic assumptions the answer to this question is negative.
Namely, if there exists a $Q$-set $X$, then $X$ is a $\sigma$-set which is Borel isomorphic to a (hereditarily normal compact) topological space $Y$ which is not a $\sigma$-space.
A topological space $X$ is called a $Q$-*space* if each s... | 5 | https://mathoverflow.net/users/61536 | 278133 | 123,333 |
https://mathoverflow.net/questions/278130 | 26 | 1. Do you think the letter $\wp$ has a name? It may depend on community - the language, region, speciality, etc, so if you don't mind, please be specific about yours. (Mainly I'd like to know the English names, if any, but other information is welcome.) If yes, when and how did you come to know it? When, how, and how o... | https://mathoverflow.net/users/56062 | The letter $\wp$; Name & origin? | Apparently first introduced by Weierstrass in [Winter 1862/63 lectures](http://archive.org/stream/encyklomath202encyrich#page/n272/mode/1up) published by H. A. Schwarz (1881, [1885](https://archive.org/stream/acq9098.0001.001.umich.edu#page/10), [1892](https://archive.org/stream/fomundezumderfun00weierich#page/10), [18... | 32 | https://mathoverflow.net/users/19276 | 278138 | 123,334 |
https://mathoverflow.net/questions/278135 | 10 | It's well known that $SL\_2(\widehat{\mathbb{Z}})$ contains $SL\_2(\mathbb{Z})$ as a dense and finitely generated subgroup. However, $GL\_2(\mathbb{Z})$ is not dense in $GL\_2(\widehat{\mathbb{Z}})$, since $GL\_2(\mathbb{Z})$ is contained in the closed subgroup of matrices with determinant $\pm 1$, which is very far fr... | https://mathoverflow.net/users/15242 | Does $GL_2(\widehat{\mathbb{Z}})$ contain a dense finitely generated subgroup? | The answer is no. If $GL\_2(\widehat{\mathbb Z})$ is topologically finitely generated, then so is the quotient $\widehat {\mathbb Z}^\*$ (quotient via the determinant map). The latter has quotient $\widehat {\mathbb Z}^\*/ (\widehat{\mathbb Z}^\*)^2=\prod \_p {\mathbb Z}\_p^\*/({\mathbb Z}\_p^\*)^2$, where the product ... | 22 | https://mathoverflow.net/users/23291 | 278139 | 123,335 |
https://mathoverflow.net/questions/278143 | 7 | Fix an integer $n\ge 2$ and suppose that ${\cal L}$ is a set of [lines](https://en.wikipedia.org/wiki/Line_(geometry)#In_Euclidean_space) in $\mathbb{R}^n$. Is there a set $M\subseteq \mathbb{R}^n$ with the following properties?
1. $M$ intersects all the elements of ${\cal L}$, but
2. for all $m\in M$, the set $M\set... | https://mathoverflow.net/users/8628 | Meeting a set of lines in $\mathbb{R}^n$ | If we have a set $\mathcal L$ of lines in $\mathbb R^n$ such that $|\mathcal L|=\mathfrak c$, we can get the set $M$ with the desired properties using transfinite induction.
* Take any well-ordering of the set of lines $\mathcal L=\{l\_\alpha; \alpha<\mathfrak c\}$ (such that $\alpha\ne\beta$ implies $l\_\alpha\ne l\... | 4 | https://mathoverflow.net/users/8250 | 278147 | 123,337 |
https://mathoverflow.net/questions/277884 | 2 | According to Wikipedia, [Agmon's inequality](https://en.wikipedia.org/wiki/Agmon%27s_inequality) provides a bound on the $L^\infty$ norm of a $H^2$ function on a (regular) subset of $\mathbb{R}^3$. In the book of JC Robinson et al. "The Three-Dimensional Navier-Stokes Equations: Classical Theory" in Thm. 1.20 the same ... | https://mathoverflow.net/users/2675 | Proof of Agmon's inequality in $\mathbb{R}^3$ | We note that $u$ is Hölder continuous,
and we may write for any $M>0$ (to be chosen later),
$$
u(x)=\int\_{\vert \xi\vert\le M} e^{2π i x\xi}\hat u(\xi)(1+\vert \xi\vert) (1+\vert \xi\vert)^{-1}d\xi
+\int\_{\vert \xi\vert\ge M} e^{2π i x\xi}\hat u(\xi)\vert \xi\vert^2
\vert \xi\vert^{-2}d\xi,
$$
so that using Cauchy-S... | 5 | https://mathoverflow.net/users/21907 | 278148 | 123,338 |
https://mathoverflow.net/questions/227073 | 12 | Let $Y$ be a Banach space isomorphic to $\ell\_p$, $1<p<\infty$. Is it true that any finite subset of $\ell\_p$ is isometric to some finite subset of $Y$?
It seems to me that it is an interesting question. It can be regarded as a special case of a recent question [Under which conditions is it possible to find points ... | https://mathoverflow.net/users/37822 | Is $\ell_p$ $(1<p<\infty)$ finitely isometrically distortable? | We (me and the author posing the question) have answered this question in the negative - for each $p \neq 2$ we have found a space $X$ such that $X$ is isomorphic to $\ell\_p$ but there are 5 element subsets of $\ell\_p$ that do not embed isometrically into $X$. Our paper is on arxiv here: <https://arxiv.org/abs/1708.0... | 5 | https://mathoverflow.net/users/61129 | 278149 | 123,339 |
https://mathoverflow.net/questions/278141 | 3 | Let $\Sigma$ be a closed connected orientable surface of genus $g \geq 2$. I have been told that every diffeomorphism $\phi: \Sigma \times S^1 \to \Sigma \times S^1$ extends to a diffeomorphism $\Phi : \Sigma \times D^2 \to \Sigma \times D^2$. I would love to know a reference for this fact and I would love to know if a... | https://mathoverflow.net/users/99414 | Disk bundles over surfaces - extending automorphisms of the boundary over the whole space | The answer to your first question is yes. Suppose that $M$ is a closed Seifert fibered space with (a) vertical tori and (b) hyperbolic base orbifold. Suppose that $f \colon M \to M$ is a self-diffeomorphism. Then $f$ is isotopic to a fiber-preserving diffeomorphism. See the proof of Proposition 2.4 in Hatcher's notes o... | 1 | https://mathoverflow.net/users/1650 | 278151 | 123,340 |
https://mathoverflow.net/questions/278153 | 7 | I need to understand the structures of $\pi\_{4n}(S^{2n})$ and $\pi\_{r}(S^{2n};\mathbb{Z}\_k) (r\geq 4n-1)$, where the latter group is the homotopy group with coefficient defined as $[P^r(k), S^{2n}]$. Does anyone know their computations and any references are available?
| https://mathoverflow.net/users/113079 | Homotopy groups of even-dimensional spheres | The $2$-primary parts of the first few groups are as follows:
$$\begin{array}{ll}
n & \pi\_{4n}(S^{2n}) \\
1 & \mathbb{Z}\_2 \\
2 & \mathbb{Z}\_2^2 \\
3 & \mathbb{Z}\_2\\
4 & \mathbb{Z}\_2^4\\
5 & \mathbb{Z}\_2\oplus\mathbb{Z}\_4\\
6 & \mathbb{Z}\_2^2\\
7 & \mathbb{Z}\_2\oplus\mathbb{Z}\_8\\
8 & \mathbb{Z}\_2^... | 14 | https://mathoverflow.net/users/10366 | 278155 | 123,341 |
https://mathoverflow.net/questions/277946 | 1 | I'll start with a motivating example and only then proceed to the question.
Consider a list of total packages of milk that were purchased on 9 consecutive days on a given store,
$z\_1,\ldots,z\_9 = 1,0,0,2,0,1,0,1,3$
Assume I have an algorithm that predicts $\hat{z}\_{10}=2$. I'm interested in assessing the conf... | https://mathoverflow.net/users/56735 | Conformal prediction for the case of single tailed events | Marco, thanks for contacting me by email.
It is possible to modify your nonconformity measure $A(B,z)=|\bar z\_B - z|$ to solve your problem: it needs to be made "one-sided", so that only large values are regarded as strange. Namely, you can use $A(B,z)=z-\bar z\_B$. You suggest using $A(B,z)=z$ instead; this is a sm... | 2 | https://mathoverflow.net/users/113083 | 278160 | 123,344 |
https://mathoverflow.net/questions/278146 | 1 | Let $\Omega$ be an open subset of $\mathbb{R}^n$ and take a family of continuous compactly supported functions $f\_n$ on $\Omega$ normalized to one (in the $L^2$ sense).
Then, these functions span a subspace $V$ of $L^2$. Now, consider two $L^2$ functions $g$ and $h$ with non-intersecting support. Assume that both $g... | https://mathoverflow.net/users/112877 | Compactly supported functions and projections | I think V is large means that V tends to $L^2(\Omega)$, i.e. you can select enough elements $f\_n$ so that $V\_n = span\{f\_1,\cdots,f\_n\}$ and the space span{$f\_1,\cdots,f\_{\infty} $} forms a complete basis of $L^2(\Omega)$. and (it should means that , otherwise you could add infinite k with support has no intersec... | 0 | https://mathoverflow.net/users/105975 | 278183 | 123,353 |
https://mathoverflow.net/questions/277476 | 1 | Suppose we have $n$ strings (or vectors) $a\_1, a\_2, \dots, a\_n \in A^m$, where $A$ is an arbitrary set satisfies $|A| \geq n$. And we limit their pairwise [hamming distance](https://en.wikipedia.org/wiki/Hamming_distance) by
$$ d(a\_i, a\_j) \geq d\_{ij} \mathrm{\ for\ all\ } i \neq j$$
with given $d\_{ij}$. Then we... | https://mathoverflow.net/users/22954 | Minimize average bitwise entropy with given pairwise hamming distance | First decompose the total distance $D$ by component: $$D = \sum\_{i,j\in [n]} d(a\_i,a\_j) = \sum\_{l \in [m]} \sum\_{i,j\in [n]} \mathbf{1}\{a\_{il} \neq a\_{jl}\}.$$ Let $\delta\_l = \frac{1}{n^2} \sum\_{i,j\in [n]} \mathbf{1}\{a\_{il} \neq a\_{jl}\}$.
Compute the collision entropy of $X\_l$:
$$H\_2(X\_l) = - \log \l... | 2 | https://mathoverflow.net/users/76368 | 278188 | 123,356 |
https://mathoverflow.net/questions/278051 | 11 | Let $K$ be a finite simplicial complex with an admissible action of a finite group $G$.
(*Terminology:* By an action of a group $G$ on $K$ I mean an action by simplicial automorphisms. The action is called admissible if for all $g\in G$ and all simplices $\sigma\in K$ the equality $g\sigma=\sigma$ implies that $gv=v$... | https://mathoverflow.net/users/2578 | Fixed-point-free group action on a finite, contractible, 3-dimensional simplicial complex | **EDIT (2021-02-26)**: The authors of <https://arxiv.org/abs/2102.11458>, <https://arxiv.org/abs/2102.11459> claim that they've proven that every action of a finite group on a finite and contractible 2-complex has a fixed point. It follows (see the reasoning below) that **the answer to Q2 is negative** (i.e. every acti... | 4 | https://mathoverflow.net/users/2578 | 278190 | 123,358 |
https://mathoverflow.net/questions/278145 | 14 | The Gauss–Bonnet theorem in differential geometry is an important statement about surfaces which connects their geometry to their topology and has very important applications to Riemann surface theory.
The generalized Gauss–Bonnet theorem (Gauss-Bonnet-Chern) in dimension $n=4$, for a compact oriented manifold states t... | https://mathoverflow.net/users/90655 | What are the important geometric-topological consequences of 4-dimensional version of Gauss-Bonnet-Chern theorem? | One can find some applications in Arthur Besse's book "Einstein manifolds", $\S6\ D$.
**Proposition.** If a closed manifold $M^4$ admits an Einstein metric, then $\chi(M)\ge 0$. Moreover $\chi(M)=0$ iff $M$ admits a flat metric.
*Proof.* Since $Rc(g)=\lambda g$, we have $|Rc|^2=4\lambda^2=r^2/4$. This implies $\chi... | 17 | https://mathoverflow.net/users/40950 | 278198 | 123,361 |
https://mathoverflow.net/questions/278191 | 8 | Let $I$ be an ideal in a regular ring $R$. Suppose $I$ can be generated by $n$ elements. Let $P$ be an associated primes of $I$. Is it true that the height of $P$ is bounded above by $n$?
Remark: (1) The question has an affirmative answer when $P$ is a minimal associated prime of $I$ by Krull's principal theorem.
(... | https://mathoverflow.net/users/17901 | Height of associated primes in regular rings | For a regular ring $R$ and an ideal $I$ generated by $n$ elements, the embedded primes of $I$ can, indeed, have height strictly larger than $n.$ For instance, let $R$ be $k[x,y,u,v].$ Let $I$ be the ideal generated by $3$ elements,
$$ I = \langle xu^2,yv^2,x^2u-y^2v\rangle.$$ The congruence class of the element $xyuv$ ... | 6 | https://mathoverflow.net/users/13265 | 278204 | 123,362 |
https://mathoverflow.net/questions/278165 | 2 | The Riemann hypothesis is equivalent to
$\forall n\geqslant2,\: \sigma(n)<H\_n+e^{H\_n}\log{H\_n}$,
where $\sigma(n)$ is the divisor sum of $n$ and $H\_n$ is the nth harmonic number.
For large $n$, $H\_n$ is small. The only $n\geqslant 2$ I have found for which $\sigma(n)>e^{H\_n}\log{H\_n}$ are 2, 3, 4, 6, 12, 2... | https://mathoverflow.net/users/nan | Is any $n>60$ known to have a divisor sum greater than $e^{H_n}\log({H_n})$, where $H_n$ is the nth harmonic number? | According to Lemma 3.1 of [Lagarias's paper](https://arxiv.org/abs/math/0008177) in which he introduces his criterion for RH, one has $$ e^{H\_n} \log H\_n \geq e^{\gamma} n \log \log n$$ for any $n \geq 3$. This means that any $n$ which violates your inequality would also yield a counterexample to Robin's inequality $... | 7 | https://mathoverflow.net/users/nan | 278210 | 123,364 |
https://mathoverflow.net/questions/278209 | 13 | This question is a duplicate of [an existing MO question](https://mathoverflow.net/questions/9143/analytic-density-of-the-set-of-primes-starting-with-1), but that other MO question has an accepted answer that does not actually answer the question, and I'm not sure how to fix that other than by re-asking the question.
... | https://mathoverflow.net/users/3106 | Existence of relative Dirichlet density of primes starting with 1 | If a set of primes has a logarithmic density, then it also has a Dirichlet density, and the two agree --- and vice versa! For the direction you want (logarithmic -> Dirichlet), this follows by partial summation. Suppose $P$ is a set of primes with logarithmic density $\delta$, and let $S(t) = \sum\_{p \le t,~p \in P} 1... | 11 | https://mathoverflow.net/users/16510 | 278215 | 123,365 |
https://mathoverflow.net/questions/277991 | 20 | Let $C$ be a symmetric monoidal category.
* Recall that a *dual* for $X \in C$ is an object $X^\vee$ and maps $\eta: I \to X \otimes X^\vee$ and $\varepsilon: X^\vee \otimes X \to I$ (where $I$ is the monoidal unit) satisfying the triangle identities.
* Let's say that $X$ is *self-dual* if there is a dual $X^\vee$ fo... | https://mathoverflow.net/users/2362 | If a $\otimes$-idempotent object has a dual, must it be self-dual? | EDIT: It looks like I was too hasty to delete this answer. I think I have fixed the gap that Anton noted in the comments and in his answer. As Anton pointed out in his answer, the trace loop that I dropped out at the end is nontrivial, but it can be removed from any diagram where an "X" appears in another connected com... | 6 | https://mathoverflow.net/users/112641 | 278217 | 123,366 |
https://mathoverflow.net/questions/278202 | 26 | Serre's finiteness theorem says if $n$ is an odd integer, then $\pi\_{2n+1}(S^{n + 1})$ is the direct sum of $\mathbb{Z}$ and a finite group. By looking at the table of homotopy groups, say on [Wikipedia](https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres#Table_of_homotopy_groups), one empirically observes that i... | https://mathoverflow.net/users/98447 | A refinement of Serre's finiteness theorem on unstable homotopy groups of spheres | $P(\eta) = [i\_n,i\_n] \circ \eta$, where $[i\_n,i\_n]: S^{2n-1} \rightarrow S^n$ is the Whitehead product of the identity map with itself. So you are asking if this composite is null.
I don't know if this is an easy problem. One sufficient condition is that $[i\_n,i\_n]$ be divisible by 2, but, ha, ha, this is now k... | 27 | https://mathoverflow.net/users/102519 | 278219 | 123,367 |
https://mathoverflow.net/questions/278206 | 9 | Working in ZF, it's well-known that for any $n \ge 2,$ the claim that there is a $\Sigma\_n$ well-ordering of the universe is equivalent to the axiom $V=HOD.$ It seems natural to believe there should be a similar theorem for $n=1,$ perhaps that there being a $\Sigma\_1$ well-ordering of the universe is equivalent to $V... | https://mathoverflow.net/users/109573 | Is $V=L$ equivalent to there being a $\Sigma_1$ well-ordering of the universe? | No, see the paper [On $ Σ\_1$ Well-Orderings of the Universe](http://www.jstor.org/stable/2272956?seq=1#page_scan_tab_contents) by Harrington and Jech.
| 11 | https://mathoverflow.net/users/11115 | 278222 | 123,369 |
https://mathoverflow.net/questions/241926 | 7 | Let $\mathcal{A}$ be an affinoid algebra over a complete non-archimedean field $K$. We have two objects we can investigate, namely the algebro-geometric spectrum $X = \operatorname{spec} \mathcal{A}$ and the non-archimedean analytic space $X^{an} = \operatorname{sp} \mathcal{A}$. The former has an etale theory from the... | https://mathoverflow.net/users/48362 | Comparison between analytic etale cohomology and algebraic etale cohomology for affinoids | Not sure if this is still relevant, but the desired comparison in the case of "principal interest" to you can be proved in two different ways.
On one hand, you can deduce it from some [recent results of Achinger](https://arxiv.org/abs/1701.03197), which imply that both cohomologies in question are unchanged if you r... | 2 | https://mathoverflow.net/users/1464 | 278226 | 123,370 |
https://mathoverflow.net/questions/278232 | 1 | Let $G$ be a finite $p$-group. Suppose that $G$ has only one characteristic subgroup(nontrivial proper). What possible structures does $G$ have?
| https://mathoverflow.net/users/97247 | Characteristic subgroups of $p$-groups | Have a look at :
S. P. Glasby, P. P. Pálfy, Csaba Schneider, p-Groups with a unique proper non-trivial characteristic subgroup, Journal of Algebra, Volume 348,Pages 85-109
<http://www.sciencedirect.com/science/article/pii/S0021869311005552>
| 9 | https://mathoverflow.net/users/22377 | 278245 | 123,373 |
https://mathoverflow.net/questions/278225 | -1 | We know that functions in $\mathcal{LP}$-class, and only these, are uniform limits, on compact subsets of $\mathbb{C}$, of polynomials with only real zeros.
Question:
1. Does it means that these function in $\mathcal{LP}$-class has only real zeroes?
2. Let $\phi(x)$ be an entire function but not in $\mathcal{LP}$-c... | https://mathoverflow.net/users/110052 | A simple question about the zeros of an Entire Function in LP-class | To the first question the answer is yes: the limit of functions with all zeros on
a closed set has zeros on the same closed set, if this limit is not identically zero.
To the second question the answer is no: $e^{z^2}$ and $ze^{z^2}$ are not in LP, but have only real zeros.
| 1 | https://mathoverflow.net/users/25510 | 278249 | 123,375 |
https://mathoverflow.net/questions/278263 | 6 | There have been a number of posts on related questions, such as:
[Geometric interpretation of Cartan's structure equations](https://mathoverflow.net/questions/61878/geometric-interpretation-of-cartans-structure-equations),
[What is the geometric significance of Cartan's structure equations?](https://mathoverflow.net/qu... | https://mathoverflow.net/users/81645 | Cartan's Structure Equations VS Cartan's Method of Equivalence | You are right that structure equations are the result of applying the method of equivalence. You don't start with an adapted coframe. (Some people might say that you do, but that is not quite what the method does.) You start with the bundle of all adapted coframes. You then construct on it the bundle of all adapted pse... | 3 | https://mathoverflow.net/users/13268 | 278271 | 123,382 |
https://mathoverflow.net/questions/278264 | 5 | Let $M$ be a matroid on the set $\{1,\dots,m\}$. The Bergman fan $\tilde{B}(M)$ is defined in literature to be the set in $\mathbb{R}^{m}$ consisting of the vectors $v=(w\_1,\dots,w\_m)$ such that for every circuit $c\in C(M)$, the minimum $\min\_{i\in c}w\_i$ is attained at least twice.
It is then mostly left impli... | https://mathoverflow.net/users/64302 | Definition of the Bergman fan | From [Matroid polytopes, nested sets and Bergman fans](https://arxiv.org/abs/math/0411260v1), Feichtner and Sturmfels, just above Proposition 2.5 on the bottom of page 4:
>
> Two vectors $w,w' \in \mathbb{R}^n$ are considered equivalent for the matroid $M$ if $M\_w = M\_{w'}$. The equivalence classes are relatively... | 5 | https://mathoverflow.net/users/111804 | 278274 | 123,383 |
https://mathoverflow.net/questions/278227 | 4 | It is well known that braid groups and knot groups share many common properties. For example, they have the same $H\_1$ and they are both residually finite and (hence) Hopfian. On the other hand, we know that $B\_2$ and $B\_3$ are isomorphic to the knot groups of unknot and trefoil knot respectively. My question is, fo... | https://mathoverflow.net/users/27111 | Braid group and knot group | This is a simple addendum to Danny's answer (because I can't use the citation thing in comments): Homology of braid groups has been computed by a number of people, and [an extensive survey](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.488.4621&rep=rep1&type=pdf) is:
*Vershinin, Vladimir V.*, [**Homology o... | 3 | https://mathoverflow.net/users/11142 | 278277 | 123,385 |
https://mathoverflow.net/questions/278233 | 9 | It's a classical fact that the commutative power series ring $\mathbb{Z}\_p[[x\_1,x\_2]]$ is isomorphic to the completed group algebra $\mathbb{Z}\_p[[\mathbb{Z}\_p\times\mathbb{Z}\_p]]$, the isomorphism sending generators $a\_1,a\_2$ of $\mathbb{Z}\_p\times\mathbb{Z}\_p$ to $1+x\_1,1+x\_2$.
In $\mathbb{Z}\_p[[x\_1,x... | https://mathoverflow.net/users/88840 | Which elements of $1+(x_1,x_2)\subset\mathbb{Z}_p[[x_1,x_2]]^\times$ are in $\langle 1+x_1,1+x_2\rangle$? | The usual power series for $\log(1+x)$ determines an injection from $1+(x\_1,x\_2)\subset\mathbb{Z}\_p[[x\_1,x\_2]]$ into $\mathbb{Q}\_p[[x\_1,x\_2]]$. A power series $f\in 1+(x\_1,x\_2)$ is in $\langle 1+x\_1,1+x\_2\rangle$ if and only if there are $\alpha\_1$, $\alpha\_2\in\mathbb{Z}\_p$ such that
\begin{align\*}
\lo... | 5 | https://mathoverflow.net/users/5263 | 278311 | 123,402 |
https://mathoverflow.net/questions/278255 | 9 | I am curious if the following topological problem is decidable.
Let $M,N$ be two closed manifolds. Given a group isomorphism $p: \pi\_1(M)\to \pi\_1(N)$, is there a homeomorphism
$\phi: M\to N$ such that $\phi\_\*=p$?
EDIT. (Thanks for helpful comments.) There are some relevant results: "Algorithmic aspects of home... | https://mathoverflow.net/users/9833 | A homeomorphism with a prescribed action on the fundamental group - decidable or not? | This is undecidable. For a construction, see the first page of Nabutovsky and Weinberger, "[Algorithmic aspects of homeomorphism problems](https://books.google.com/books?hl=en&lr=&id=AVMbCAAAQBAJ&oi=fnd&pg=PA245&ots=jxTQBoauiy&sig=FpaqRpITrgqFR5VaxpPJrmNsTZ4#v=onepage&q&f=false)", Rothenberg Festschrift 1998.
| 7 | https://mathoverflow.net/users/59302 | 278312 | 123,403 |
https://mathoverflow.net/questions/278305 | 2 | Given a sufficiently large positive integer $n$, I would like to know the asymptotic bound in $n$ of the following summation:
$$
\sum\_{i=0}^{n}{\frac{P(n, i)}{n^i}} =
\sum\_{i=0}^{n}{\frac{n!}{(n-i)! \cdot n^i}.}
$$
Is it $O(n)$, $O(\log n)$, $\dots$?
| https://mathoverflow.net/users/113159 | Finding the asymptotic bound of a summation | If you don't need very precise information, then the quick answer is that $n^{1/2}$ is about right.
I want to use [Stirling's formula](https://en.wikipedia.org/wiki/Stirling%27s_approximation) in the version
$$
An^{n+1/2}e^{-n} \le n! \le Bn^{n+1/2}e^{-n} .
$$
This shows that
$$
\frac{n!}{(n-j)!n^j}\simeq e^{-j}\left... | 5 | https://mathoverflow.net/users/48839 | 278315 | 123,405 |
https://mathoverflow.net/questions/277565 | 3 | Are there infinite cardinals $\kappa < \lambda$ such that here is a collection ${\cal A}$ of subsets of $\lambda$ with the following properties:
1. $|{\cal A}| = 2^\lambda$, and
2. $A\neq B\in {\cal A}$ implies $|A\cap B|\leq \kappa$.
| https://mathoverflow.net/users/8628 | Set systems with "small" intersections | Here's a quick observation: suppose $2^\kappa=\kappa^+$. Then we can mimic the construction of an almost disjoint set of sets of natural numbers of size continuum: to each map $p: \kappa^+\rightarrow 2$, we associate the set of *small approximations* $A\_p=\{p\upharpoonright\alpha:\alpha<\kappa^+\}$. By $2^\kappa=\kapp... | 2 | https://mathoverflow.net/users/8133 | 278318 | 123,406 |
https://mathoverflow.net/questions/278244 | 8 | I've come across a family of posets which appear to have a couple of remarkable enumerative properties, and I'm wondering whether anyone has seen these before.
Take $n\geqslant3$, and let $\preccurlyeq$ be a partial order on the set $\{1,\dots,n\}$. Say that $\preccurlyeq$ has property M if:
1. There is a unique wa... | https://mathoverflow.net/users/6771 | Has anyone seen these posets before? | I claim:
A poset with property M has precisely two maximal elements.
Precisely one maximal element is greater than every non-maximal element. Call this the supermaximal element.
Removing the supermaximal element (if $n>3$) leaves a poset with property M.
So there is a unique way to build any example from the $n... | 8 | https://mathoverflow.net/users/22989 | 278321 | 123,408 |
https://mathoverflow.net/questions/275158 | 11 | Suppose I have to put $N$ points $x\_1, x\_2, \ldots, x\_N$ on the circle $S^1$ of length 1 so as to achieve the largest minimum separation (packing radius). The optimal solution is the equally spaced arrangement $x\_k = k/N \bmod 1$, which achieves $\min\_{i<j}\|x\_i-x\_j\| = 1/N$.
However, suppose I didn't know in ... | https://mathoverflow.net/users/20186 | Sequential addition of points on a circle, optimizing asymptotic packing radius | I will show $\mu = \tfrac{1}{\log 4}$.
---
I first prove the upper bound $\mu \leq \tfrac{1}{\log 4}$. Fix a positive integer $r$. For $0 \leq k \leq r-1$, let $N\_k$ be $2^{k/r} N$ rounded to the nearest integer, where $N$ is a large parameter. Considering the $N\_{r-1}$ intervals present at time $N\_{r-1}$, let... | 6 | https://mathoverflow.net/users/297 | 278322 | 123,409 |
https://mathoverflow.net/questions/278323 | 2 | Please improve the title if it is not suitable.
Does there exist a finite group $G$ with the following properties?
$G$ has a minimal subgroup say $L$ isomorphic to $\mathbb{Z}\_{p}$ and a maximal subgroup say $M$ isomorphic to $\mathbb{Z}\_{pq}$ containing properly $L$ ($p$ and $q$ are some distinct primes), such t... | https://mathoverflow.net/users/97247 | Finite nonabelian groups with abelian maximal subgroups | How about $G = {\rm SL}(2,3) = Q\_8.C\_3$ with $|L|=3$ and $|M|=6$?
There are similar examples of the form $N.C\_p$, with $N$ an extraspecial $q$-group exponent $q$, where the $C\_p$ subgroup acts irreducibly on $N/Z(N)$.
| 4 | https://mathoverflow.net/users/35840 | 278330 | 123,411 |
https://mathoverflow.net/questions/278335 | 4 | Consider a Banach space $E$ and a discrete set $X$. For an operator $T$ on $\ell^2(X)$ I can consider and induced operator $T'$ on the Bochner-Lebesgue space $\ell^2(X;E)$ of $E$-valued square-summable functions. This operator is defined by the same matrix representation, but simply acting on a larger Banach space. If ... | https://mathoverflow.net/users/110128 | Norm of "tensoring" with the identity | No. This is Proposition 4.1 in Kwapien, "Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients." Studia Math. 44 (1972), 583–595. [MathSciNet](http://www.ams.org/mathscinet-getitem?mr=341039) See [Studia Math Archive](http://pldml.icm.edu.pl/pldml/element/bwmeta1.elem... | 7 | https://mathoverflow.net/users/406 | 278344 | 123,415 |
https://mathoverflow.net/questions/278296 | 2 | I know there is a spectral sequence for a variety with normal crossing singularities $X$ which gives a tool for making the computation of the mixed hodge structure computable. How can I compute the mixed hodge structure for a projective variety which has non-normal crossing singularities? For example, if I have
$$
X = ... | https://mathoverflow.net/users/78824 | How can I compute the mixed hodge structure for three copies of $\mathbb{P}^1$ intersecting at one point? | Let me expand my previous comment, and place it in a more general context. Suppose that $X$ is an singular connected but possibly reducible projective curve. Let $\tilde X\to X$ be the normalization. $\tilde X$ is a disjoint union of the normalizations of the irreducible components of $X$.
Let us say that there $n$ suc... | 2 | https://mathoverflow.net/users/4144 | 278355 | 123,419 |
https://mathoverflow.net/questions/278284 | 19 | Here a combinatorics problem. I offer 30 euro for a proof and 100 bounty points for a counterexample:
Let $n \geq 2$.
An $n$-*Kupisch series* is a list of $n$ numbers $c:=[c\_1,c\_2,...,c\_n]$ with $c\_n=1$, $c\_i \ge 2$ for $i \neq n$ and $c\_i-1 \leq c\_{i+1}$ for all $i=1,...,n-1$ and setting $c\_0:=c\_n$.
The num... | https://mathoverflow.net/users/61949 | Combinatorics problem related to Motzkin numbers with prize money I | UPDATE
------
The promised writeup, proving both conjectures, is now finished. It will be available on the arxiv on Friday, in the meantime it is available on the [arXiv](https://arxiv.org/abs/1811.05846).
FindStat's original answer
--------------------------
I have the following refined conjecture, which hopeful... | 8 | https://mathoverflow.net/users/113201 | 278376 | 123,429 |
https://mathoverflow.net/questions/278375 | 19 | **Edit:** Shortly after this post it was rightly pointed out by @AntonPetrunin that the measure $\mu$ may not be unique. @R W then showed how one can construct a metric space where the limiting measure is not unique. Hence @R W answered the following question in the negative: Does the following construction always prod... | https://mathoverflow.net/users/113199 | Does every compact metric space have a canonical probability measure? | As Anton has already mentioned, one can only claim that if the sequence $\mu\_{\epsilon\_n}$ associated to a certain sequence of minimal $\epsilon\_n$-nets $Y\_n$ converges, then it will also converge to the same limit for for any other sequence of $\epsilon\_n$-nets. However, a different sequence $(\epsilon\_n)$ may p... | 16 | https://mathoverflow.net/users/8588 | 278386 | 123,434 |
https://mathoverflow.net/questions/278329 | 6 | After some months wandering through examples of algebraic groups in the theory of automorphic forms and number theory, I wonder why so many efforts are spent in understanding $GSp(4)$ (local newforms, liftings, correspondences, etc.) without any mention to $PSp(4)$.
Explicit examples and computable settings are rare... | https://mathoverflow.net/users/43737 | $GSp(4)$ vs $PSp(4)$ | If we think about classical modular forms, one typically works on SL(2). One could also work on PSL(2), but one would like to write down congruence subgroups in terms of matrices, so one often phrases things for SL(2). Moreover, if one wants to deal with nebentypus, one is forced to work with SL(2).
However, if we wa... | 7 | https://mathoverflow.net/users/6518 | 278388 | 123,435 |
https://mathoverflow.net/questions/278328 | 7 | In some cases the relations between a global group $G$ (over the adeles $\mathbb{A}$ of a field $F$) and its local components $G\_v$ (where $v$ are the places of $F$) are well known. Obviously a group determines its local components, so the question aims at, given a family of local groups , understanding whether or not... | https://mathoverflow.net/users/43737 | Which groups can have $GSp(4)$ as local component? | In addition to the relatively boring extension/restriction of scalars for $GSp(2n,\widetilde{k})$ for an extension field $\widetilde{k}$ of the ground field $k$... :
The Galois twist groups often denoted by $Sp^\*(p,q)$ (or $GSp^\*(p,q)$...) over ground field $\mathbb R$, defined via non-degenerate quaternion-symmetr... | 3 | https://mathoverflow.net/users/15629 | 278389 | 123,436 |
https://mathoverflow.net/questions/278391 | 9 | It is known that there are non-amenable groups not containing $F\_2$, the free group on two generators; for example, Olshanskii's group. But does every non-amenable group contain a 2-generated non-amenable subgroup?
| https://mathoverflow.net/users/95282 | Does every non-amenable group contain a 2-generated non-amenable subgroup? | The answer is "no". There are non-amenable Golod-Shafarevich groups where every 2-generated subgroup is finite. See my answer [here.](https://mathoverflow.net/questions/137678/non-amenable-groups-with-arbitrarily-large-tarski-number/137715#137715)
| 15 | https://mathoverflow.net/users/nan | 278394 | 123,437 |
https://mathoverflow.net/questions/278332 | 6 | Let $(X,\tau)$ be a topological space. With $T\_2(\tau)$ we denote the collection of $T\_2$-topologies on $X$ that contain $\tau$.
Is there an example of a topology $\tau$ such that the partially ordered set $(T\_2(\tau),\subseteq)$ contains no minimal members?
| https://mathoverflow.net/users/8628 | Topologies with no minimal $T_2$ topologies above them | *There exists a topology $\tau$ with two non-isolated points on a countable set $X$ such that the poset $T\_2(\tau)$ does not have minimal elements.*
To construct such topology $\tau$, take any Hausdorff $(\omega\_1,\omega\_1)$-gap on $\omega$, which is a pair $\big((A\_\alpha)\_{\alpha\in\omega\_1},(B\_\alpha)\_{\al... | 4 | https://mathoverflow.net/users/61536 | 278398 | 123,438 |
https://mathoverflow.net/questions/278393 | 11 | If a statement $P$ has a ZFC proof of length $n$, must it also have a cut-free ZFC proof of length polynomial in $n$?
By a cut-free ZFC proof, I mean a proof in sequent calculus without cut rule of some sequent "finite set of ZFC axioms $\implies P$ ". This is arguably nonstandard and different from a cut-free proof ... | https://mathoverflow.net/users/113213 | Cut-free proofs in ZFC | Yes, one can eliminate cuts polynomially for proofs in ZFC, PA, and similar theories that have nontrivial axiom schemata allowing arbitrary formulas.
For PA, replace a cut
$$\color{blue}{\dfrac{\Gamma\implies A,\Delta\qquad\Gamma,A\implies\Delta}{\Gamma\implies\Delta}}$$
with a cut-free subderivation
$$\dfrac{\dfrac{... | 13 | https://mathoverflow.net/users/12705 | 278403 | 123,440 |
https://mathoverflow.net/questions/278410 | 2 | Let $L: L^2(\mathbb{R}) \supseteq Dom(L) \rightarrow L^2(\mathbb{R})$ be a densely defined closed operator. Assume that the resolvent admits an integral kernel (Greens function) $G$, i.e. for $z\in \mathbb{C}\setminus \sigma(L)$ and every $f\in L^2(\mathbb{R})$ the resolvent satisfies
$$ ((L-z)^{-1}f)(x) = \int\_{\ma... | https://mathoverflow.net/users/91098 | How to determine the spectrum from the diagonal Green's function | For the Sturm–Liouville operators with discrete spectrum, $G(z,x,y)$ is the Stieltjes transform of the discrete measure $$\mu\_{x,y}(ds) = \sum\_{n = 1}^\infty \varphi\_n(x) \overline{\varphi\_n(y)} \delta\_{-\lambda\_n}(ds) ,$$ that is, $$G(z,x,y) = \int\_\mathbf{R} \frac{1}{z - s} \, \mu\_{x,y}(ds) = \sum\_{n = 1}^\i... | 4 | https://mathoverflow.net/users/108637 | 278413 | 123,442 |
https://mathoverflow.net/questions/278418 | 13 | This question initially arose out of a question in asymptotic matroid theory. The matroid question has since been answered in a different way, but the extremal set theory question remains unanswered and may be of interest in its own right.
>
> **Question.** Let $\mathcal{F}$ be a set family such that for all disti... | https://mathoverflow.net/users/2233 | Set family $\mathcal{F}$ such that for all $A,B,C \in \mathcal{F}$ both $A \cap B \not \subseteq C$ and $C \not \subseteq A \cup B $ | $f(n)$ grows exponentially in $n$ as may be seen by a simple probabilistic argument: choose $M$ subsets $A\_1,\dots,A\_M$ at random ($M$ to be specified later) independently. Then for given indices $i,j,k$ the probability that $A\_i\cap A\_j\subset A\_k$ is $(7/8)^n$ (there should be no element which does not belong to... | 11 | https://mathoverflow.net/users/4312 | 278419 | 123,443 |
https://mathoverflow.net/questions/278363 | 3 | This question may be seen as a follow up of [this one](https://mathoverflow.net/questions/277803/kazhdan-lusztig-theorem-for-composition-factors-of-verma-modules/277806).
Let $\mathfrak{g}$ be a rank $r$ simple complex Lie algebra, and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. We call $\alpha\_i$ ... | https://mathoverflow.net/users/61018 | Kazhdan-Lusztig Theorem for a special class of integral weights | The answer to the question at the end is certainly yes, following the pattern in the previous question. But it's not helpful to emphasize the partial ordering of weights here. Instead, note that $\lambda$ might well be *regular* relative to the dot-action of the Weyl group. For example, in the case of $\mathfrak{sl}\_3... | 3 | https://mathoverflow.net/users/4231 | 278422 | 123,445 |
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