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https://mathoverflow.net/questions/212515 | 3 | My question is extremely simple to state: I am looking for a characterization of multivariate complex polynomials $f$ such that $f(Sing(f))=\{0\}$. My motivation is that I recently read somewhere that any polynomial only possessing one degenerate fiber defines an isotrivial family away from that fiber. If that question... | https://mathoverflow.net/users/43593 | Polynomials with Unique Critical Value | Whoops, figured it out, at least in the case of complex polynomials.
We're looking for the set of $f$ such that $Z(f)=f^{-1}(0)\subseteq Z(j(f))$, where $j(f)$ denotes the Jacobian ideal of $f$. However, a result of Saito shows that this is true if and only if $f$ is weighted homogeneous. Does anyone know of an analo... | 1 | https://mathoverflow.net/users/43593 | 212516 | 100,989 |
https://mathoverflow.net/questions/212510 | 1 | Let $X\_n = Bl\_{p\_1,...,p\_n}\mathbb{P}^2$ be the blow-up of $\mathbb{P}^2$ in $n$ **general points** $p\_1,...,p\_n\in\mathbb{P}^2$.
Let $f\_i:\mathbb{P}^{2}\dashrightarrow\mathbb{P}^1$ be the linear projection from the point $p\_i\in\mathbb{P}^2$. Then $f\_i$ lifts to a fibration $\widetilde{f}\_i:X\_n\rightarrow... | https://mathoverflow.net/users/nan | Fibrations on blow-ups of $\mathbb{P}^2$ | As Mohan indicated, this is not always going to be true. Let $n$ be $4$. The group $\text{Aut}(\mathbb{P}^2)$ acts transitively on the set of ordered $4$-tuples $(p\_0,p\_1,p\_2,p\_3)$ such that no $3$ of these points are collinear. Thus, up to projective linear transformation, assume the points are $p\_0=[1,1,1]$, $p\... | 1 | https://mathoverflow.net/users/13265 | 212520 | 100,990 |
https://mathoverflow.net/questions/212517 | 0 | I would like to know if there are infinitely many $k$ for which $$\sigma(k)/k=n^p$$ such that $m=k{n}^{p-1}$ with $m,n>0$ and $p$ is an odd prime?
Note: $\sigma(\frac{m}{{n}^{p-1}})$ is the sum of divisors function of $\frac{m}{ {n}^{p-1}}$ such that $m=k{n}^{p-1}$.
**Edit 2**: I edited the question as it has the s... | https://mathoverflow.net/users/74330 | Are there infinitely many $k$ for which $\frac{\sigma(k)}{k}=n^p$ and $p$ is an odd prime? | The question can be reformulated as:
>
> **Question.** Are there infinitely many $k$'s such that $\sigma(k)/k$ is a square?
>
>
>
If the answer is yes, then (trivially) there are infinitely many [multiperfect numbers](https://en.wikipedia.org/wiki/Multiply_perfect_number). This consequence is not known uncondi... | 8 | https://mathoverflow.net/users/11919 | 212521 | 100,991 |
https://mathoverflow.net/questions/212183 | 2 | I have a vector $\mathbf{x}$ of size $m\cdot n$ of zeros and ones, i.e., $\mathbf{x}\in\{0,1\}^{m\cdot n}$ and a matrix $\mathbf{A}$ of size $\left(m\cdot n+m+n+1\right)\times\left(m\cdot n\right)$ of reals, i.e., $\mathbf{A}\in\mathbb{R}^{\left(m\cdot n+m+n+1\right)\times\left(m\cdot n\right)}$.
The vector $\mathbf... | https://mathoverflow.net/users/52953 | Complexity of sparse matrix-vector multiplication? | The problem is essentially equivalent to asking for the complexity of calculating the vector
$$ \mathcal{V} := \{ \mathcal{A}\_0\cap \mathcal{x}, \mathcal{A}\_1\cap \mathcal{x},\ ...,\ \mathcal{A}\_{mn+m+n}\cap\mathcal{x} \}$$
where the $\mathcal{A}\_i$ are the sets of indices corresponding to the non-zero entries of t... | 2 | https://mathoverflow.net/users/31310 | 212530 | 100,993 |
https://mathoverflow.net/questions/212531 | -2 | A metric space $(X,d)$ is said to be *bounded* if there is $r\in\mathbb{R}$ such that for all $x,y\in X$ we have $d(x,y) \leq r$.
A *self-isometry* is a map $\iota:X\to X$ such that for all $x,y\in X$ we have $d(x,y) = d(\iota(x), \iota(y))$.
Does there exist a bounded metric space with a non-surjective self-isomet... | https://mathoverflow.net/users/8628 | Bounded metric spaces with non-surjective self-isometry | Let $B$ be the unit ball in the sequence space $\ell^2$. Send $(x\_0, x\_1,\dots)$ to $(0,x\_0,x\_1,\dots)$.
| 2 | https://mathoverflow.net/users/76536 | 212532 | 100,994 |
https://mathoverflow.net/questions/212465 | 5 | Related to the n-conjecture.
We are looking for Weierstrass form and map from it of the genus one curve:
$$ y^{10} z^{30} - 8000 y^{4} z^{20} + 12800000 z^{20} + 1600 y^{2} z^{10} - 64=0 $$
It is of degree $40$ and has only five monomials.
---
**Added** We suspect there are finite number of rational points ... | https://mathoverflow.net/users/12481 | Weierstrass form of genus one $y^{10} z^{30} - 8000 y^{4} z^{20} + 12800000 z^{20} + 1600 y^{2} z^{10} - 64=0$ | (a) The given curve is irreducible over $\mathbb Q$, but reducible over $\mathbb Q(\sqrt[5]{2})$. So it does not have a Weierstrass form, and each rational point has to be a singularity. The curve however does not have rational singularities (there are two on the projective completion though).
(b) From the latest co... | 13 | https://mathoverflow.net/users/18739 | 212534 | 100,996 |
https://mathoverflow.net/questions/212536 | 1 | Let $L$ be an complete lattice. A lattice homomorphism $f: L\to L$ is said to be *join-incomplete* if there is an infinite set $S \subseteq L$ such that $f(\bigvee\_L S) > \bigvee\_L f(S).$
Is it true that if there is a join-incomplete lattice homomorphism $f:L\to L$, then there is also a join-incomplete lattice homo... | https://mathoverflow.net/users/76042 | Reducing join-incomplete lattice homomorphisms to homomorphisms with co-domain ${\bf 2}$ | No. The counterexample to [this](https://mathoverflow.net/questions/211919/simplyfing-join-incomplete-lattice-endomorphisms/211922#211922) question also works here.
| 1 | https://mathoverflow.net/users/75735 | 212541 | 100,997 |
https://mathoverflow.net/questions/212528 | -2 | We are given a large, round table with $n$ seats. It is easy to see that whenever $p\geq \text{int}(\frac{n}{2}) + 1$ people are seated, at least $2$ people will sit next to each other (here $\text{int}(x)$ denotes the largest integer less or equal than $x$).
Given $n$ seats at a round table, let $s(n)$ be the smalle... | https://mathoverflow.net/users/8628 | Minimum number of people such that 2 can be expected to sit next to each other | Some partial answer. Once you fix the first guy then you can see if $k$ persons are seated in the table with distance greater and $1$ then this give by counting the spaces between each one this gives you a partition of $n-k$ in $k$ parts. So the cases that you don't want to count are in correspondence with the ways of ... | 2 | https://mathoverflow.net/users/34575 | 212544 | 100,999 |
https://mathoverflow.net/questions/212548 | 2 | I am a graduate student with good mathematical maturity (I took advanced courses like category theory, commutative algebra...). I want to study algebraic geometry from Mumford's red book. I find it difficult to keep motivated. Is there any class notes supplements Mumford?
| https://mathoverflow.net/users/75682 | Supplementary notes to Mumford's The Red Book of Varieties and Schemes | You could have a look at Vakil's "Foundations of algebraic geometry" ? [Available on his Stanford homepage]
Eisenbud&Harris' "Geometry of schemes" is quite good to get the points too (pun intended)
| 7 | https://mathoverflow.net/users/12664 | 212551 | 101,000 |
https://mathoverflow.net/questions/212570 | -2 | May we find the common term for recursive sequence? if yes that how to find the common term of recursive sequence such: 1 2 1 3 1 2 1 4 1 2 1 3 1 2 1 5 1 2 1 3 1 2 1 4 1 2 1 3 1 2 1 6 ...
* in a pyramidal form:
```
4
3 3
2 2 2 ... | https://mathoverflow.net/users/76556 | Recursion, Common Term, Combinatorics | A good place where one can answer questions like this easily is the Online Encyclopedia of Integer Sequences, oeis.org .
Your sequence is <https://oeis.org/A001511> .
| 3 | https://mathoverflow.net/users/44797 | 212573 | 101,005 |
https://mathoverflow.net/questions/212271 | 7 | Kazhdan-Lusztig theory gives a correspondence between irreducibles of the affine Hecke algebra for a simply connected linear algebraic group $G$ and certain homological data extracted from the Steinberg variety (for $q$ not a root of unity). I want to see this correspondence explicitly for $SL\_2$ by (1) looking at gen... | https://mathoverflow.net/users/6059 | 1-dimensional representations of the affine Hecke algebra for $SL_2$ | It turns out I was confused about the statement on the geometric side. I'm using Ginzburg's [paper](http://arxiv.org/abs/math/9802004) as a reference. One doesn't just take the (co)homology of the fibers for each orbit representative. The statement (page 44) is really that the irreducibles (for a given central characte... | 3 | https://mathoverflow.net/users/6059 | 212588 | 101,014 |
https://mathoverflow.net/questions/212580 | 3 | Let $\mathcal C$ be a cocomplete category. Consider the following two conditions:
1. $\mathcal C$ is locally presentable.
2. The Yoneda embedding $$\mathcal C \hookrightarrow \{\text{continuous functors } \mathcal C^{\mathrm{op}} \to \mathrm{SET}\}$$ is an equivalence of categories. (By the Yoneda lemma, it suffices ... | https://mathoverflow.net/users/78 | Does the following characterize local presentability? | The following argument is from Martin Brandenburg's comment to the question linked to by Omar Antolin-Camarena. As you say, by the Yoneda lemma condition #2 holds iff every continuous functor $C^{op} \to \text{Set}$ is representable, or equivalently has a left adjoint. By the special adjoint functor theorem, it suffice... | 5 | https://mathoverflow.net/users/290 | 212594 | 101,018 |
https://mathoverflow.net/questions/212591 | 4 | Given a model of $\sf ZFC$, and an infinite ordinal $\alpha$. Can we prove that there is always a cardinal $\kappa$, and a forcing $\Bbb P$, such that:
1. $\Bbb P$ does not add sets of rank $\leq\alpha$.
2. $\Bbb P$ adds sets to $\kappa$.
3. $\Bbb P$ does not collapse cardinals.
Any two of the three can be easily d... | https://mathoverflow.net/users/7206 | Can we always add sets without collapsing cardinals or adding [very] bounded sets? | As was confirmed in the comments, if $0^\sharp$ does not exist then SCH must hold. If we now let $\lambda$ be the least $\beth$-fixed point above $\alpha$, we can consider the forcing $\mathbb{P}=\mathrm{Add}(\lambda^+,1)$. This has $\lambda^{++}$-cc precisely because SCH holds, ensuring $2^\lambda=\lambda^+$. It follo... | 7 | https://mathoverflow.net/users/1058 | 212599 | 101,021 |
https://mathoverflow.net/questions/212597 | 14 | Suppose $\kappa$ is strongly inaccessible and every stationary subset of $\kappa$ reflects. Must $\kappa$ be Mahlo?
Remarks:
1. It is possible for every stationary subset of $\kappa$ to reflect, but $\kappa$ is only weakly inaccessible (and not strongly inaccessible).
2. If $V=L$ then the answer is "yes", and in fa... | https://mathoverflow.net/users/26319 | Does stationary reflection imply Mahloness? | Not necessarily. Here's a counterexample, but I'm sure it is a ridiculous overkill in consistency strength:
Suppose $\kappa$ is the least inaccessible limit of supercompact cardinals. Then $\kappa$ is not Mahlo. If $S \subseteq \kappa$ is stationary, then there is a stationary $T \subseteq S$ such that $T$ concentrat... | 15 | https://mathoverflow.net/users/11145 | 212600 | 101,022 |
https://mathoverflow.net/questions/212584 | 0 | Given two abelian monoidal categories ${\cal C,D}$ (where the monoidal operation is bilinear) and an additive monoidal functor $F:{\cal C} \to {\cal D}$. Will $F$ always admit an adjoint?
| https://mathoverflow.net/users/42100 | Adjointable Abelian Monoidal Functor | Certainly not in general. For instance, let $F$ be the inclusion of the category of finite-dimensional vector spaces (over some fixed field) into the category of all vector spaces.
If you want $\mathcal{C}$ to be complete and cocomplete, here's another counterexample. Let $U$ be a large (i.e., proper-class sized) ab... | 4 | https://mathoverflow.net/users/75 | 212607 | 101,025 |
https://mathoverflow.net/questions/212364 | 26 | *This is a purely idle question, which emerged during a conversation with a friend about what is (not) known about the space of compact metric spaces. I originally asked this question at math.stackexchange (<https://math.stackexchange.com/questions/1356066/global-structure-of-the-gromov-hausdorff-space>), but received ... | https://mathoverflow.net/users/8133 | On the global structure of the Gromov-Hausdorff metric space | To answer the main question -- there are no nontrivial self-isometries of $\mathcal{GH}$.
I can give a proof of this, but as it is getting rather long, I will state some facts in $\mathcal{GH}$ without proof for now, and will come back and provide provide proofs or references.
First, a bit of notation. Let ${\rm di... | 23 | https://mathoverflow.net/users/1004 | 212608 | 101,026 |
https://mathoverflow.net/questions/212626 | 0 | Let $X$ be a non-reduced noetherian scheme over $\mathbb{Z}$ or $\mathbb{C}$. Assume that $X^{red}$ is quasi-projective and separated, does the same hold for $X$ ?
(By the way, projective implies a priori separated, but this is not true for quasi-projective, right?)
any reference is welcome
thanks
| https://mathoverflow.net/users/48866 | quasi-projective and separated as topological properties | No. If the reduction of a morphism is quasiprojective (or even projective), then the morphism itself need not be quasiprojective. See EGA II.5.3.5.
(Concerning your other question: Quasiprojective morphisms are separated. See EGA II.5.3.1.)
| 3 | https://mathoverflow.net/users/11025 | 212633 | 101,033 |
https://mathoverflow.net/questions/212635 | 10 | Let $G=(V,E)$ be a graph (loops and multiple edges are permitted). Three following systems of dependent sets in $E$ define matroids:
1) Set $A\subset E$ is dependent if $A$ contains cycle. This is a cycle matroid, maybe the most popular matroid.
2) Set $A\subset E$ is dependent if $A$ contains two different cycles ... | https://mathoverflow.net/users/4312 | Matroids similar to the cycle matroid | The first two examples fall neatly into the class of [$(k,l)$-sparsity matroids](https://en.wikipedia.org/wiki/Dense_graph#Sparse_and_tight_graphs) (see the references in this Wikipedia article and references contained within). The cycle matroid (1) is the (1,1)-sparsity matroid on $G$ and the second matroid you descri... | 6 | https://mathoverflow.net/users/353 | 212638 | 101,034 |
https://mathoverflow.net/questions/212636 | 9 | [*a repost from SE due to the lack of response*]
Given a group $G$, let $A$ be a $G$-module and let $N\trianglelefteq G$.
If I understand it correctly, the superscript "G/N" in the third term of the standard [inflation-restriction](https://en.wikipedia.org/wiki/Inflation-restriction_exact_sequence) exact sequence
$$
... | https://mathoverflow.net/users/12961 | The term $H^1(N,A)^{G/N}$ in the inflation-restriction exact sequence | You do not need LHS spectral sequence for this action.
The functors $H^\*(N,-)$ are the derived functors of $(-)^N:G\text{-mod}\rightarrow G/N\text{-mod}$,
so they will carry a structure of $G/N$-modules.
Explicitly, on the level of cocycles, it can be described as follows:
if you have a one cocycle $f:N\rightarrow A$... | 13 | https://mathoverflow.net/users/41644 | 212640 | 101,035 |
https://mathoverflow.net/questions/212639 | 0 | For a topological group $(G,\mathcal T)$ and a subgroup $H\le G$, we say $\mathcal T$ and $H$ are permutable if for every neighborhood $U$ of $1$, there is a neighborhood $V$ of $1$ with $VH\subseteq HU$.
Do you have an example of an infinite (preferably nonabelian) Hausdorff topological group $(G,\mathcal T)$ such t... | https://mathoverflow.net/users/47958 | A group topology which commutes with closed subgroups | I will give a class of metric examples.
Let $G\_1$ be an abelian group with a metric $d\_1$.
Let $G\_2$ be a discrete nonabelian group, with metric $d\_2(x,y)=1$ for all $x\not=y$.
Let $G:= G\_1\times G\_2$, with $d( (x\_1, x\_2), (y\_1, y\_2)) = d\_1(x\_1,y\_1) + d\_2(x\_2, y\_2)$.
Let $H$ be any subgroup o... | 0 | https://mathoverflow.net/users/14915 | 212645 | 101,037 |
https://mathoverflow.net/questions/212634 | 7 | Let $S$ be a smooth quasi-projective curve over the complex numbers. Let $P$ be a closed point in $S$. Let $f:\mathcal X \to S$ be a polarized family of smooth projective connected varieties. To this family and the point $P$ we can associate the Kodaira-Spencer map
$$ Tan\_S(P) \to H^1(X,T\_X).$$
Here $X $ is the ... | https://mathoverflow.net/users/76588 | Isotrivial families with non-zero Kodaira spencer map | The most general (most useful) formulation of the Kodaira-Spencer map uses the triangle of transitivity of cotangent complexes and relative duality. Even for such mildly singular varieties as nodal curves, the following is not sufficient (I recommend Section 6 of K. Behrend and B. Fantechi, "The Intrinsic Normal Cone",... | 6 | https://mathoverflow.net/users/13265 | 212660 | 101,040 |
https://mathoverflow.net/questions/212606 | 6 | Math Overflow seems to have a dearth of low dimensional topology, but this seems like an interesting question. Let $L$ be a $2$-component link in $S^3$. Suppose that there is a framed arc joining the two components such that surgery along that arc yields an unknot. (In other words, fatten the arc to a long thin rectang... | https://mathoverflow.net/users/9417 | Surgery along an arc connecting the components of a $2$-component link gives the unknot | One source of restrictions is the Montesinos trick: if you take the branched double cover of $L$, then a small neighborhood of the framed arc lifts to a solid torus because it intersects $L$ in two small arcs. Replacing those small arcs with the ones coming from the surgery produces another solid torus in the new branc... | 6 | https://mathoverflow.net/users/428 | 212661 | 101,041 |
https://mathoverflow.net/questions/212650 | 3 | Given any unital non-commutative subalgebra $\mathcal M$ of $L(\mathbb F\_2)$ is it true that $\mathcal M' \bigcap L(\mathbb F\_2)^\mathcal U = \mathbb C I$ for any free ultrafilter $\mathcal U$?
| https://mathoverflow.net/users/76593 | A relative property gamma and $L(\mathbb F_2)$ | Hah, should have gone down the hall at UVa first. The following negative answer was given to me by Scott Atkinson:
The hyperfinite II$\_1$ factor $\mathcal R$ is the minimal object among II$\_1$ factors. As such, there is a copy of $\mathcal R$ sitting as a subalgebra of $L(\mathbb F\_2)$. Thus by an abuse of notatio... | 2 | https://mathoverflow.net/users/76593 | 212662 | 101,042 |
https://mathoverflow.net/questions/212659 | 6 | I was recently introduced to Connes' Embedding Conjecture (CEC) which states:
>
> Every separable type $II\_{1}$ factor is embeddable into $R^{\omega}$. Where $\omega$ is a generic free ultrafilter on $\mathbb{N}$, $R$ is a hyperfinite type $II\_{1}$ factor, and $R^{\omega}$ is the ultrapower of $R$ with respect to... | https://mathoverflow.net/users/22763 | Connes Embedding Conjecture and Fusion Categories | For fusion categories, there is a hyperfinite embedding result which does not require passing to ultraproducts.
If $\mathcal{C}$ is a unitary fusion category, then by results of Popa [MR1055708] (see also [MR3028581, Theorem 4.1]), there is a finite index, depth two subfactor, which we could reasonably denote $R\sub... | 3 | https://mathoverflow.net/users/351 | 212665 | 101,045 |
https://mathoverflow.net/questions/212666 | 8 | What is the consistency strength of the following statement:
>
> $\kappa$ is a strong cardinals and it is indestructible under $Col(\kappa, <\theta),$ where $\theta> \kappa$ is some fixed inaccessible cardinal.
>
>
>
**Remark 1.** By Laver's result, the above statement is consistent relative to the existence o... | https://mathoverflow.net/users/11115 | Consistency strength of being strong cardinal and indestructible under collapses | Apter and I proved that the only way to force this situation by
any forcing that resembles anything like the Laver preparation is
to begin with a supercompact cardinal (or a partially supercompact
cardinal for $\theta$ fixed). Specifically, in our article [A. W.
Apter and J. D. Hamkins, “Indestructible weakly compact c... | 9 | https://mathoverflow.net/users/1946 | 212668 | 101,047 |
https://mathoverflow.net/questions/212614 | 8 | A [locale](http://ncatlab.org/nlab/show/locale) is a sort of "formal topological space", which "may not have enough points to separate its open sets". For instance, there is a "locale of all real numbers that are both rational and irrational", which of course has no points at all (since there are no such numbers), but ... | https://mathoverflow.net/users/49 | Detecting positive endomaps of the formal reals | I think the answer is no. Work in a recursive context; eg the effective topos.
Consider a (uniformly continuous) function $f:R\to R$ such that $f(x)>0$ for all $x$, but we don't have a positive infimum. This is standard (use the Kleene tree).
Now consider the embedding of pointwise spaces into locales; the work by Palm... | 4 | https://mathoverflow.net/users/25122 | 212669 | 101,048 |
https://mathoverflow.net/questions/212649 | 1 | I was confused by the following question for a long time:
Does there exists a transcendental entire function $f$ such that
$J(f)\cap J(f')=\emptyset$ ?
where $J(f)$, ($J(f')$) is the Julia set of $f$ $(f').$
Edit: Following the same style, one may also ask
Does there exists a transcendental entire function $f$ ... | https://mathoverflow.net/users/11966 | A question on $J(f)$ and $J(f')$ | In my opinion, the question is completely arbitrary. There is no reason to expect relationships between the dynamics of $f$ and that of its derivative. Their relationship will even change under affine conjugacy.
This being said, the answer to the question is positive in that such a function $f$ does indeed exist; i.e... | 5 | https://mathoverflow.net/users/3651 | 212672 | 101,049 |
https://mathoverflow.net/questions/212675 | 4 | There are many equivalent definitions for stability, one of them being that being unstable is equivalent to the existence of a formula having the order property.
While intuitively it makes sense that having a lot of types (which is the definition that Shelah uses) would lead to a lot of models , the order property is... | https://mathoverflow.net/users/nan | History of unstable formulas | I disagree with the claim that the order property isn't intuitive.
When I took my first model theory class, types were introduced to me as a generalization of Dedekind cuts. So the very first examples we looked at were linear orders. Moreover, the construction of theories with exactly $n$ distinct countable models (f... | 4 | https://mathoverflow.net/users/8133 | 212680 | 101,051 |
https://mathoverflow.net/questions/212698 | 2 | I consider $\Sigma:=S^2$, as a unit radius sphere in $\mathbb R^3$ so that it has an induced metric, measure etc on it. Is the following statement true?
For a large constant $K$, there exist constants $\epsilon(K)$ and $c(K)$ such that the following is satisfied: for any
smooth positive real-valued function $f$ on $\... | https://mathoverflow.net/users/15197 | Is there a curve on a surface where an integrable function is pointwise bounded? | In short: There is an integral bound but no pointwise bound.
**Integral bounds:**
Fix any $\epsilon>0$.
Let us denote the circle of radius $\epsilon$ centered at $x\in\Sigma$ by $A\_x=A\_x(\epsilon)$, and let $\sigma=\sigma(\epsilon)$ be the (dimension one Hausdorff) measure of this circle (independent of $x$).
For a... | 2 | https://mathoverflow.net/users/55893 | 212704 | 101,061 |
https://mathoverflow.net/questions/212697 | 7 | Let $(M,g)$ be a smooth manifold equipped with a metric tensor $g$, and $f\in C^\infty(M)$ a regular function (i.e., with nowhere vanishing differential).
Denote by $\mathrm{Hess}\_g(f):=\nabla df$ the Hessian tensor of $f$ with respect to the metric $g$, and by $N\_f:=f^{-1}(\{0\})$ the one-codimensional submanifold... | https://mathoverflow.net/users/22606 | What does it mean that the Hessian is proportional to the metric? | It is known (say, Y. Tashiro, Complete Riemannian manifolds and some vector fields, Trans.Amer.Math.Soc. 117(1965) 251–
275; I am not sure that Tashiro is the first who proved it and there were many later papers which independently prove the same result later. The proof is pretty straightforward.) that the existence of... | 8 | https://mathoverflow.net/users/14515 | 212705 | 101,062 |
https://mathoverflow.net/questions/212677 | 3 | Let $\{X\_i\}$ denote an i.i.d. sequence of Bernoulli variables with parameter $p$. A Bernoulli factory is a procedure that generates events with probability $f(p)$ using the observations $\{X\_i\}$, where $f$ is a known function and $p$ is assumed to be unknown.
Is there some general result, or bound, on the number... | https://mathoverflow.net/users/40432 | Number of samples needed as input to Bernoulli factory | Two references come to mind:
<http://arxiv.org/abs/math/0309222>
<http://arxiv.org/abs/math/0304143>
The gist of it is, if I remember it right, that fast implementation is possible iff f is real-analytic.
| 2 | https://mathoverflow.net/users/9430 | 212717 | 101,065 |
https://mathoverflow.net/questions/212727 | 2 | Is it known that there exist no coprime positive integers $A$, $B$ and $C$ such that $A^3+B^4=C^3$? This is a particular case of [Beal's Conjecture](https://en.wikipedia.org/wiki/Beal%27s_conjecture).
| https://mathoverflow.net/users/20757 | Particular case of Beal's Conjecture | You can rewrite the diophantine equation in the form $C^3-A^3 = B^4,$ that is $C^3+ (-A)^3=B^4.$
Now, Bruin proved that integer solutions to $x^3+y^3 = z^n$ with $n \in \{4, \, 5\}$ and $xyz \neq 0$ satisfy $\textrm{gcd}(x, \, y, \, z) >1$, see
N. Bruin, [On powers as sums of two cubes](http://link.springer.com/c... | 12 | https://mathoverflow.net/users/7460 | 212729 | 101,073 |
https://mathoverflow.net/questions/212735 | 4 | It is "well-known" that the stack of polarized varieties is an algebraic stack with quasi-compact and separated diagonal.
In particular, if $(X,L)$ and $(Y,M)$ are polarized schemes over a scheme $S$, the Isom-scheme $$Isom((X,L),(Y,M))$$ is separated and quasi-compact over $S$.
I had the feeling that these Isom-sc... | https://mathoverflow.net/users/76588 | Are automorphism groups of polarized varieties of finite type | Breaking with my habit of writing everything in comments, this is Section 2.1 of my article with de Jong.
MR2745688 (2012e:14073) Reviewed
Starr, Jason(1-SUNYS); de Jong, Johan(1-CLMB)
Almost proper GIT-stacks and discriminant avoidance. (English summary)
Doc. Math. 15 (2010), 957–972.
14J10 (14L15) ... | 5 | https://mathoverflow.net/users/13265 | 212737 | 101,075 |
https://mathoverflow.net/questions/212722 | 2 | Let $C \in Cat$ be an $(\infty,1)$-category.
Let $P$ be the partially ordered subset of full subcategories of $C$.
Is there a (canonical?) functor from the nerve of $P$ to $Cat$? I think the answer must be yes, but I can only define such a functor on $0$ and $1$ simplices of the nerve.
| https://mathoverflow.net/users/76625 | Do the full subcategories have a simple structure in higher category theory? | Fully faithful functors are, in particular, monomorphisms in $\mathrm{Cat}\_\infty$. So let $P'$ be the full $\infty$-subcategory of $(\mathrm{Cat}\_\infty)\_{/C}$ spanned by inclusions of full subcategories. Because these inclusions are monomorphisms, $P'$ is equivalent to the poset $P$, so your desired functor comes ... | 4 | https://mathoverflow.net/users/437 | 212738 | 101,076 |
https://mathoverflow.net/questions/207489 | 7 | Is there someone who can show me how I can prove this conjecture? Or at least show me how to do the first implication ?
>
> **Conjecture**: Assume $\alpha,\beta, \lambda \in [0,\infty)$. Then every positive solution of the difference equation :
> $$z\_{n+1}=\frac{\alpha+z\_{n}\beta +z\_{n-1}\lambda}{z\_{n-2}},\q... | https://mathoverflow.net/users/51143 | Boundedness of solutions of a difference equation | **Note** :This is comments for the answer related to the question and not lead to the Answer
of the above question .and may it is helpful to be constrictive for a complet proof
The discussion of energy functions is at best very difficult to understand and at worst incorrect or incomplete. To illustrate this consi... | 0 | https://mathoverflow.net/users/74330 | 212739 | 101,077 |
https://mathoverflow.net/questions/212716 | 2 | In a recent paper by Bourgain, Sarnak, Gamburd [[1](https://www.math.ias.edu/files/Bourgain_Gamburd_Sarnak_GenSel.pdf)] talks about subgroups of $SL(2,\mathbb{Z})$.
>
> Let $\Lambda$ be a finitely generated non-elementary subgroup of $SL(2,\mathbb{Z})$ with Hausdorff dimension $\delta(\Lambda) > \frac{1}{2}$ ...
>... | https://mathoverflow.net/users/1358 | Hausdorff Dimensions of Limit set of subgroups of SL(2,Z) | **Added later**: One key word you might pursue is [Patterson-Sullivan theory](http://link.springer.com/chapter/10.1007/978-0-8176-4653-0_14#page-1), which among other things equates the Hausdorff dimension of the limit set of a geometrically finite group action on hyperbolic space and the so-called "exponent of converg... | 8 | https://mathoverflow.net/users/20787 | 212747 | 101,080 |
https://mathoverflow.net/questions/212746 | 3 | Consider: a vector field $X=\nabla \phi$ on a compact, semi-simple, connected matrix Lie group $G$ where $\phi$ as a smooth scalar field on $G$ possessing only a single maxima which topologically is a point, and $\nabla$ is the gradient w.r.t to the bi-invariant metric (unique up to a constant factor which makes no dif... | https://mathoverflow.net/users/41654 | Do the values of the differential of a function on a Lie group with a single maximum span the Lie algebra? | Sure. The metric is a red herring here: this is the same as asking if the translates of the differential $d\phi$ span $\mathfrak{g}^\*$. Phrased this way, the answer is easy to see. If they didn't span, then there would a non-zero element of the perpendicular to the smaller subspace they span, that is, a non-zero vecto... | 8 | https://mathoverflow.net/users/66 | 212750 | 101,081 |
https://mathoverflow.net/questions/212560 | 5 | A graph is *vertex transitive* if $x \mapsto y$ by an automorphism.
A graph is *generously vertex transitive* if $x \mapsto y \mapsto x$ by an automorphism.
Simple facts:
* GVT $\rightarrow$ unimodular. Just plug in the definition of unimodular to see why.
* Cayley $\not \rightarrow$ GVT. Free product $\mathbb Z\... | https://mathoverflow.net/users/76548 | A generously vertex transitive graph which is not Cayley? | Take the graph product $G = P \times \mathbb{Z}$ of the Petersen graph with the infinite path graph. This is clearly infinite, finite-degree, and generously vertex-transitive.
Then we have two distinguishable types of edge, namely $P$-edges (ones which belong to $5$-cycles) and $\mathbb{Z}$-edges (ones which do not).... | 5 | https://mathoverflow.net/users/39521 | 212751 | 101,082 |
https://mathoverflow.net/questions/212764 | 25 | I'm a layman in mathematics, so please excuse me in advance for anything in this question that may be inappropriate :D. Well: Four years ago, I was reading (and working to solve the puzzles on) Winkler's *Mathematical Puzzles: A Connoisseur's Collection*, and in the section "Unsolved Puzzles", there was the problem "Sq... | https://mathoverflow.net/users/76639 | Why is it so hard to prove Toeplitz' conjecture? | Let me elaborate on Sam Hopkins' comment.
The main reason that makes this and other problems on continuous curves so hard is that a "simple closed curve" or "Jordan curve", i.e. a non-self-intersecting continuous loop in the plane, can be a horrible object, for instance a nowhere differentiable curve such as the Koch... | 29 | https://mathoverflow.net/users/7460 | 212768 | 101,090 |
https://mathoverflow.net/questions/212741 | 22 | $\newcommand{\Prb}[1]{\mathcal{P}\_{#1}}$
I have the following number theory problem, related to Odlyzko's improvement on Shor’s factoring algorithm (see [this cstheory.sx question](https://cstheory.stackexchange.com/q/32077/2368) for details).
>
> Let $N$ be a (large) integer and $1≤B≤N$ a bound. If I chose the in... | https://mathoverflow.net/users/4780 | Lower bounding the probability that $\gcd(t,N)≤B$, for a random $t$ and fixed (large) $N$ | In fact one can prove a stronger result -- namely the probability is bounded away from zero, so long as $B \ge (\log N)^\delta$ for some $\delta >0$. This is best possible, by taking $N$ to be the product of the first few primes. Since $\phi(N/d) \ge \phi(N)/d$, our probability is bounded below by
$$
\frac{\phi(N)}{N... | 14 | https://mathoverflow.net/users/38624 | 212770 | 101,092 |
https://mathoverflow.net/questions/212748 | 8 | We call a matrix $M \in \mathbb{R}^{d \times d}$ transitive if it satisfies the following:
>
> For any three vectors $u, v, w$ in $\mathbb{R}^d$. If $u^T M v > 0$ and $v^T M w > 0$ then $u^TMw > 0$.
>
>
>
**EDIT**: The case when $d = 1$ is simple. For $d > 1$ the following is true:
a) M must be Posi... | https://mathoverflow.net/users/29887 | Existence and characterization of transitive matrices? | I assume from the wording of the question (positive semidefinite, Cholesky decomposition) that you intend $M$ to be symmetric. Write $M$ as $N^TN$ and let $V\leq \mathbb R^d$ be the range of $N$. For $u, v, w\in \mathbb R^d$ let $x=Nu, y=Nv, z= Nw$. Then your condition reduces to
$$
x^Ty>0 \;\&\; y^Tz>0 \Rightarrow x^... | 5 | https://mathoverflow.net/users/75735 | 212773 | 101,094 |
https://mathoverflow.net/questions/136136 | 6 | Fix $n \times n$ real symmetric positive definite matrices $A$ and $B$. Fix vectors $x$ and $y$ in $\mathbf{R}^n$. I want to compute the following bilinear products quickly: $\{x^T (A+mB)^{-1} y\}\_{m=0}^M$.
A naive but practical method would involve inverting each matrix individually, thus requiring $O(Mn^3)$.
Are... | https://mathoverflow.net/users/36779 | Algorithm to quickly compute the individual inverses of a linear sequence of matrices | Here is a faster way (perhaps not the nicest from a numerical stability point of view).
\begin{eqnarray\*}
x^T(A+mB)^{-1}y &=& x^T(A^{1/2}(I+mA^{-1/2}BA^{-1/2})A^{1/2})^{-1}y\\
&=& x^TA^{-1/2}(I+mZ)^{-1}A^{-1/2}y\\
&=& p^T(I+mZ)^{-1}q.
\end{eqnarray\*}
Now let $Z=U\Lambda U^T$, so that we need to compute
\begin{equa... | 4 | https://mathoverflow.net/users/8430 | 212780 | 101,097 |
https://mathoverflow.net/questions/212781 | 4 | Fix a commutative ring $R$. There's a contravariant functor from finite sets to finite $R$-algebras sending $X$ to $R^X$. Viewed as a covariant functor $\text{set}^{op}\to R\text{-alg}$, this functor preserves finite limits and colimits; this is easy to verify directly by checking finite products ($R^{\coprod\_i X\_i} ... | https://mathoverflow.net/users/1474 | Reference for (co)limit-preserving functor $X\mapsto R^X$ | Maybe this is more complicated than you are looking for, but you can get your right adjoint to exist without any Noetherian hypothesis by extending the domain to the category of all Boolean algebras. Specifically, let $\text{Bool}$ be the category of Boolean algebras and $R\text{-Alg}$ be the category of $R$-algebras (... | 5 | https://mathoverflow.net/users/75 | 212783 | 101,099 |
https://mathoverflow.net/questions/212784 | 11 | The following is a [re-post from MSE](https://math.stackexchange.com/questions/1354791/integral-formula-for-int-0-inftye-3-pi-x2-sinh-pi-x-sinh-3-pi) because I did not get any answer even after offering a bounty.
Towards the end of G. N. Watson's (one of the joint authors of famous book "A Course of Modern Analysis")... | https://mathoverflow.net/users/15540 | Integral formula for $\int_{0}^{\infty}e^{-3\pi x^{2}}((\sinh \pi x)/(\sinh 3\pi x))\,dx$ by Ramanujan | This relation seems to be part of the story of Ramanujan's mock theta function. The identity you discuss is a specialization of transformation properties of a mock theta function evaluated at $\tau =i$. See this paper of [Zwegers](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.164.7621&rep=rep1&type=pdf) for ... | 21 | https://mathoverflow.net/users/38624 | 212788 | 101,102 |
https://mathoverflow.net/questions/212758 | 4 | The curvature tensor, $R\_{ab}{}^c{}\_d$, can be obtained from a connection which not necessarily is a metric connection.
By construction it is antisymmetric in the first two indices, since roughly speaking $[\nabla\_a , \nabla\_b] \simeq R\_{ab}{}^\bullet{}\_\bullet$. I'm assuming the connection has vanishing torsio... | https://mathoverflow.net/users/25356 | Symmetries of non-Riemannian curvature tensor | Part of the confusion is that your positional notational convention is not the standard one. In most books, what you are writing as $R\_{ab}{^c}\_d$ would be written as $R{^c}\_{dab}=-R{^c}\_{dba}$ (though the letters used would probably be different).
That said, the condition $R{^c}\_{cab}=0$ is the condition that t... | 8 | https://mathoverflow.net/users/13972 | 212794 | 101,105 |
https://mathoverflow.net/questions/212625 | 3 | As you may know term "polygon" might mean few different things
and its meaning has to guessed from context.
By some reason I have to use few of these meaning in one place.
So I converge to the following convention:
Polygon is a cyclically ordered set of points.
Then I could define its *sides*;
I can say then if th... | https://mathoverflow.net/users/1441 | Terminology for polygons | [Hull](https://en.wikipedia.org/wiki/Convex_hull)? .
| 4 | https://mathoverflow.net/users/43108 | 212795 | 101,106 |
https://mathoverflow.net/questions/212791 | 1 | I appreciate if you could help me to find a reference (and a proof).
Combining Dirichlet theorem on primes in arithmetic progression with Chebotarev densitiy theorem, we know that given two positive integers $a$ and $m$ with $\gcd(a,m)=1$ the proportion of primes $p$ with $p\cong a\mod m$ is $1/\varphi(m)$, where $\v... | https://mathoverflow.net/users/45242 | reference on Dirichlet theorem on primes in arithmetic progression | If $n$ is sufficiently large compared to $m$, then we have
$$\frac{|\{p\mid p\textrm{ prime with }p\leq n \textrm{ and }p\equiv a\mod m\}|}{|\{p\mid p\textrm{ prime with }p\leq n \}|}\geq \frac{1}{2\varphi(m)}.$$
This is, of course, a consequence of (the quantitative form of) Dirichlet's theorem on arithmetic progressi... | 6 | https://mathoverflow.net/users/11919 | 212798 | 101,107 |
https://mathoverflow.net/questions/212799 | 5 | [I was reading this article](https://en.wikipedia.org/wiki/Pseudogroup) on wikipiedia and was interested by the apparent link between Homological Algebra and PDEs. What is an accessible reference which showcases the link between these topics?
| https://mathoverflow.net/users/36886 | soft: Reference/ Suggested Read: Homological Algebraic techniques in PDEs | First, a critique. Both homological algebra and PDEs are far from being monolithic subjects. So, a well-posed question about the relation between the two should be quite specific. Though you don't explicitly say it, your question suggests that you are interested in the homological algebra methods that appeared in the w... | 14 | https://mathoverflow.net/users/2622 | 212822 | 101,118 |
https://mathoverflow.net/questions/212816 | 7 | As mentioned in Terry Tao's comment to [this question](https://mathoverflow.net/q/44466/5810), it is constructively known
that there are primes between sufficiently large cubes. $\:$ [According to wikipedia](https://en.wikipedia.org/wiki/Prime_gap#Upper_bounds),
"there exists a constant $\: \theta < 1 \:$ such ... | https://mathoverflow.net/users/nan | Are there effective small intervals in which primes are dense? | One can read off such an effective result from the [work of Dudek](http://arxiv.org/abs/1401.4233). Indeed, examining his Section 3.3, we can get the following explicit lower bound for $x>\exp(8\times 10^{14})$ and $h=3x^{2/3}$:
$$ \sum\_{x<p\leq x+h}\log p>h-\tfrac{1}{2}(1-10^{-3})h-\tfrac{1}{2}(1-10^{-3})h-0.0008h. $... | 7 | https://mathoverflow.net/users/11919 | 212824 | 101,119 |
https://mathoverflow.net/questions/212776 | 1 | I'm reading the classic paper "[Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one](http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1992__76_/PMIHES_1992__76__165_0/PMIHES_1992__76__165_0.pdf)" by Gromov-Schoen. In Section 6, they define the notion of F-connectedness as follo... | https://mathoverflow.net/users/76649 | Understanding the definition of an F-connected simplicial complex | Here's a write-up of the comments, giving an example as asked for.
We assume that for two simplices in $X$ to be "adjacent" means that their intersection is nonempty.
Let $X$ be the quotient of a pair of 2-simplices by identifying a vertex on one with a vertex on the other to a single vertex $V$. Each 2-simplex is... | 0 | https://mathoverflow.net/users/20787 | 212825 | 101,120 |
https://mathoverflow.net/questions/212815 | 1 | The question is coming from Definition 1.5 in Deligne's Expose X in SGA 7 on intersection theory.
Let $X$ be a connected regular scheme of dimension $2$ and $Y \subset X$ a reduced divisor that admits a proper morphism $Y \rightarrow \mathrm{Spec}(k)$ for some field $k$. Suppose also that $D$ and $E$ are two effectiv... | https://mathoverflow.net/users/70964 | Tor-amplitude [0, 1] in the setting of intersection theory on a regular surface? | You did not explicitly ask the following, but it is related to your second question, and it comes up in practice quite often. So I would like to state this at any rate. Let $X$ be a locally Noetherian scheme. Let $D\subset X$ be a closed subscheme such that there exists a complex $E^\bullet$ of (finite rank) locally fr... | 2 | https://mathoverflow.net/users/13265 | 212827 | 101,121 |
https://mathoverflow.net/questions/212733 | 1 | If we consider crystallographic root systems, then for each $k$ such that $n \leq k \leq d-1$ where $d$ is the Coxeter number, it seems to be the case that there is exactly one root of height $k$ with nonzero coefficients with respect to each simple root.
It seems to work if you check it case by case, but I would be... | https://mathoverflow.net/users/15482 | roots in a root system which have nonzero coefficients with respect to each simple root | This does not hold in general. In type E6, for example, there are roots for which the "middle" simple root has multiplicity 2 and so does one of the two other simple roots corresponding to a vertex adjacent to it, while all the other simple roots have multiplicity one.
| 3 | https://mathoverflow.net/users/468 | 212836 | 101,125 |
https://mathoverflow.net/questions/212830 | 6 | Does anyone recognize this problem? There are $2p$ equations and $2p$ unknowns, and it feels like a classic, but I've never encountered it before:
Given $d\_1, d\_2, \ldots d\_p$, find $a\_1, a\_2, \ldots a\_p$ and $n\_1, n\_2, \ldots n\_p$:
$$ \left[ \begin{array}{ccc}
d\_1\\
d\_2\\
d\_3\\
\vdots\\
d\_{2p}
\end{arra... | https://mathoverflow.net/users/17087 | Deceptive linear algebra problem | Let $$f(X) = (X - a\_1) \cdots (X - a\_p) = X^p + e\_1 X^{p-1} + \dots + e\_p.$$ Using $f(a\_i) = 0$ for each $i$, we get
\begin{equation} \begin{bmatrix} d\_{p+1} & d\_p & \cdots & d\_1 \\ d\_{p+2} & d\_{p+1} & \cdots & d\_2 \\ \vdots & \vdots & \ddots & \vdots \\ d\_{2p} & d\_{2p-1} & \cdots & d\_p \end{bmatrix} \be... | 10 | https://mathoverflow.net/users/36579 | 212839 | 101,127 |
https://mathoverflow.net/questions/212848 | 15 | Let me quickly explain what I mean with my question.
Let $(\kappa\_i)\_{i\in I}$ be a collection of cardinal numbers, indexed by elements of some set $I$. We can try to define $\sum\limits\_{i\in I}\kappa\_i$ as follows: take collection $(A\_i)\_{i\in I}$ of sets such that $|A\_i|=\kappa\_i$, and let $A=\{(a,i):i\in ... | https://mathoverflow.net/users/30186 | Does "cardinal arithmetic is well-defined" imply axiom of choice? | First let me answer the first question. Yes. It is possible. You can find the proof in Jech "The Axiom of Choice" in Theorems 11.2 and 11.3. Also you might be interested in reading Pincus' paper,
>
> *David Pincus*, [**Cardinal representatives**](http://www.ams.org/mathscinet-getitem?mr=366666), *Israel J. Math.* *... | 17 | https://mathoverflow.net/users/7206 | 212849 | 101,129 |
https://mathoverflow.net/questions/212858 | 2 | A definition of multiplication on the projective $1$-points $(a:b)$ of $P\_K^1$ with $a$ and $b$ elements of a field $K$ ( e.g. the real or rational numbers ) can be given by mimicking the multiplication of complex numbers.
Is it known whether there a way to define an addition between projective $1$-points ?
| https://mathoverflow.net/users/49056 | Can the projective line be provided with a ring structure? | The topological space $\mathbb{P}^1\_{\mathbb{R}}$ is a circle, so it has an abelian group structure just defined by adding angles but this is not algebraic, not compatible with the multiplication you describe, and it can't be extended to $\mathbb{P}^1\_{\mathbb{C}}$ (which is a sphere and thus has no topological group... | 9 | https://mathoverflow.net/users/66 | 212859 | 101,135 |
https://mathoverflow.net/questions/212868 | 0 | Given the elliptic curve $E:y^2=x^3-4x+4$.
(a) How to prove that the group of rational points $E(\mathbb{Q})$ is generated by $P=(2,2)$.
(b) If we consider the piece of curve on the region $0<x<2, 0<y<2$ with the aid of Magma we find the points
\begin{eqnarray\*}
7P &=& \left(\dfrac{10}{9},\dfrac{26}{27}\right),\\... | https://mathoverflow.net/users/76689 | A particular interesting elliptic curve | (a) This isn't really a MathOverflow level question. See for example Cremona's book for a description of how to do a 2-descent, which is probably all that's needed for this curve.
(b) There is a real analytic isomorphism $E(\mathbb R)\to\mathbb R/\mathbb Z$, i.e., the real locus is isomorphic to the circle group. The... | 6 | https://mathoverflow.net/users/11926 | 212874 | 101,142 |
https://mathoverflow.net/questions/212840 | 7 | Let $u\_n = x + n\alpha \pmod 1$ with $\alpha$ irrational. We know that $(u\_n)\_{n \geq 0}$ is dense in $\mathbb{R}/\mathbb{Z}$ (equivalently $(u\_n)\_{n \geq 0}$ visits every open interval infinitely many times).
Now, let $(\epsilon\_n)\_{n \geq 0}$ be a sequence of independent uniform Bernoulli random variables on... | https://mathoverflow.net/users/21339 | Random suborbits of a rotation | For your initial question I think I have an answer. WLOG assume $x=0$.
~~Assume that $\alpha$ has a *universal* binary expansion, i.e., the one which contains every finite 0-1 word as a factor. (This is a generic property in $\alpha$.) Then the sequence $2^n\alpha\bmod1$ is clearly dense in $(0,1)$. (Alternatively, y... | 2 | https://mathoverflow.net/users/8131 | 212877 | 101,143 |
https://mathoverflow.net/questions/212882 | 2 | Let $f$ be a faithful state on a $C^\*$-algebra $A$, i.e. $f(a^\*a)=0$ implies $a=0$. in general, call a mapping $T:A \to B$ between $C^\*$-algebras faithful if $T(a^\*a)=0$ implies $a=0$. How to prove that if $f$ is faithful then the map $f \otimes id: A \otimes A \to A$ is faithful? I think that it should be simple h... | https://mathoverflow.net/users/24078 | Faithul map and (minimal) tensor product of $C^*$-algebras | See p. 434 in:
>
> D. Avitzour, [Free products of C\*-algebras](http://www.ams.org/journals/tran/1982-271-02/S0002-9947-1982-0654842-1/S0002-9947-1982-0654842-1.pdf), *Trans. Amer. Math. Soc.* **271** (1982), 423–435.
>
>
>
Actually a more general statement holds true; namely tensor products of c.p. faithful m... | 7 | https://mathoverflow.net/users/15129 | 212884 | 101,146 |
https://mathoverflow.net/questions/212832 | 6 | I met the following two combinatorial concepts during a study outside of combinatorics. I am wondering if there are common terminologies in combinatorics.
1. A finite graph $G$ has the following property:
>
> For any vertex $a$ and edge $\{b,c\}$ of $G$, there is an edge connecting them: there is at least one of... | https://mathoverflow.net/users/40262 | Terminology in combinatorics | This is only an answer to your first question.
The graphs with your property can be characterised by forbidding as induced subgraph $\overline{P}\_3$. By Andreas Blass's comment, your property implies that the complement of the graph is a disjoint union of cliques. These graphs are called complete multipartite graph... | 9 | https://mathoverflow.net/users/38267 | 212885 | 101,147 |
https://mathoverflow.net/questions/212567 | 4 | Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk, and let $\Omega$ be a closed 2-form on $E$ such that it is non-degenerate fiberwise.
For any $x \in E$, there is a decomposition $TE\_x=TE\_x^h \oplus TE\_x^v$, $TE^v=\ker d\pi$ and $TE\_x^h=\{v \in TE\_x: \Omega(v,w)=0,\forall w \in TE\_... | https://mathoverflow.net/users/60693 | Non-degenerate periodic orbits in the boundary of Lefschetz fibration over a disk | This is very rough and probably not too precise but here is what I
think.
For a two-form $\omega$ on a manifold $M$ denote by
$\text{null}(\omega\_x)$ the vector subspace of the tangent space
$T\_xM$ defined as follows:
$$\text{null}(\omega\_x) = \left\{ \, W \in T\_xM \,\, | \,\, \omega\_x(W,v) = 0
\, \text{ for all... | 3 | https://mathoverflow.net/users/75853 | 212887 | 101,148 |
https://mathoverflow.net/questions/212888 | 0 | Take the generalized flag variety $SO(2n,\mathbb{C})/B$, considered as the moduli of isotropic flags (according to the form $\langle e\_i, e\_{2n+1-j}\rangle=\delta\_{ij}$)
$$F\_1\subset F\_2\subset\cdots F\_n\subset\mathbb{C}^{2n}$$
with $\dim(F\_i)=i$ and $\dim(F\_n\cap\mathrm{Span}(e\_1,\ldots,e\_n))$ even.
Let $P... | https://mathoverflow.net/users/3077 | The weird projection from SO(2n)/B to maximal isotropic grassmannian | The other projection goes to the other Lagrangian subspace containing $F\_{n-1}$. If you look at $F\_{n-1}^\perp/F\_{n-1}$, this is a 2-dimensional space with symmetric non-degenerate bilinear form. Thus, it has exactly 2 isotropic lines (there are coordinates where the matrix of the form is $\begin{bmatrix}0 & 1\\ 1&0... | 3 | https://mathoverflow.net/users/66 | 212889 | 101,149 |
https://mathoverflow.net/questions/212853 | 2 | Let $A$ and $B$ unital $C^\ast$-algebras, $f:A\to B$ a linear, bounded map such that $f(a^\*)=f(a)^\*$ for all $a\in A$, $f(1\_A)=1\_B$ and $f(a)f(b)=0$ for all $a,b\in A\_{sa}$ with $ab=0$. Follows $f(a^2)=f(a)^2$ for all $a\in A$?
I have tried to proof this first for self-adjoint elements $a\in A$ using the continu... | https://mathoverflow.net/users/75338 | map between $C^*$-algebras with special properties, why is $f(a^2)=f(a)^2 $ for all $a\in A$? | Yemon Choi pointed out in a comment to [this related question](https://mathoverflow.net/questions/91700/properties-of-orthogonality-preserving-c-p-maps-between-c-algebras) that we can conclude $f$ is a \*-homomorphism under the stronger assumption that $ab = 0$ implies $f(a)f(b) = 0$ for arbitrary (not just self-adjoin... | 2 | https://mathoverflow.net/users/23141 | 212894 | 101,150 |
https://mathoverflow.net/questions/212908 | 2 | Let's define for a positive integer $n$: $$a(n) = \min \{|n! - 2^m| : m \in \mathbb N \}.$$ Does there exist a good asymptotic lower bound for the values $a(n)$ for large $n$? In particular, is the following conjecture true?:
>
> Conjecture: For all positive $k$, $a(n)$ is bigger than a constant multiple of $2^{((\... | https://mathoverflow.net/users/51663 | Minimum distance between factorials and powers of 2 | Let $2^r$ be the highest power of 2 dividing $n!$. If $m$ minimizes your expression, it's easy to see that $m \geq r$. So your expression, which is non-zero for $n \geq 3$, will be divisible by and hence at least $2^r$. Now your conjecture (and more) follows from (e.g.)
$$r = \sum\_{k=1}^{\infty}{\lfloor\frac{n}{2^k}\r... | 2 | https://mathoverflow.net/users/73430 | 212910 | 101,155 |
https://mathoverflow.net/questions/212902 | 5 | I'm trying to understand why a certain action of a Lie Group is hamiltonian.
Let $(M,g)$ be a geodesically complete Riemannian manifold.
Then there exists a canonical one-form on the cotangentbundle $T^\*M$ given by $\lambda\_0 : T( T^\*M) \to R, (v, \alpha) \mapsto \alpha(\pi\_\*v)$, with $\pi\_\*$ the derivative of... | https://mathoverflow.net/users/75382 | Action generated by geodesic flow is hamiltonian | $\phi$ is the Hamiltonian flow corresponding to the ``Hamiltonian'' $E:TM\rightarrow\mathbb{R}$ given by $E(V) = \frac{1}{2}g(V,V)$. This is pretty standard, and you can read about it in most references on symplectic geometry, for example [Foundations of Mechanics](http://authors.library.caltech.edu/25029/) (in particu... | 3 | https://mathoverflow.net/users/17945 | 212914 | 101,156 |
https://mathoverflow.net/questions/212916 | 1 | First, let $A$ be a finitely generated *free* abelian group, and $s$ an automorphism of order $2$ of $A$. Set $G=\{1,s\}$. Then we know that $A$ is a sum of indecomposable $G$-lattices $A\_i$, where each $A\_i$ is either $(\mathbb{Z},1)$, or $(\mathbb{Z},-1)$, or $(\mathbb{Z}+\mathbb{Z},\sigma)$, where $\sigma$ interch... | https://mathoverflow.net/users/4149 | Involutive automorphisms of a finite abelian p-group | There is a big difference between the case $p=2$ and $p\neq 2$.
If $p\neq 2$ then $A$ will be the direct sum of $(\mathbb{Z}/p^n,1)$ and $\mathbb{Z}/p^n,-1)$ for some $n$.
In this case all the indecomposables are also irreducible.
If $p=2$ and $A$ is 2-elementary abelian, then $A$ will split as the direct sum of $(\... | 3 | https://mathoverflow.net/users/41644 | 212917 | 101,157 |
https://mathoverflow.net/questions/212912 | 4 | Are there non compact smooth manifolds which have the property that every Riemannian metric is geodesically convex?
Note that a manifold for which every Riemannian metric is complete must be compact.
(In particular, it can be proved that every metric has a conformally equivalent one which is bounded, and the claim ... | https://mathoverflow.net/users/46290 | Smooth manifolds for which every metric is geodesically convex | No = there are no non compact smooth manifolds which have the property that every Riemannian metric is geodesically convex.
Note that any non compact $n$-manifold $M$
contains a closed subset homeomorphic to $[0,\infty)\times D^{n-1}$
Indeed choose a bounded metric on the manifold $M$
and a minimizing geodesic $\gamm... | 5 | https://mathoverflow.net/users/1441 | 212919 | 101,158 |
https://mathoverflow.net/questions/212921 | 3 | I am interested in examples of dual Banach spaces $X$ with the Schur property (weakly convergent sequences in $X$ are norm convergent) like $\ell\_1$.
The Lorentz spaces $d(w,1)$ [Lindenstrauss and Tzafriri. Classical Banach spaces I. Sequence spaces. Section 4.3] are candidates because they admit a predual and are her... | https://mathoverflow.net/users/39421 | On the Lorentz sequence space $d(w,1)$ | No. The unit vector basis $(e\_n)$ is unconditional and symmetric but not equivalent to the unit vector basis for $\ell\_1$, hence $(e\_n)$ converges weakly to zero.
| 2 | https://mathoverflow.net/users/2554 | 212923 | 101,160 |
https://mathoverflow.net/questions/212929 | 14 | For sake of simplicity, let's say that we live in $G = SL(2, \mathbb{C})$. Every conjugacy class of diagonalizable matrices $$[A] := \{gAg^{-1} \mid g \in G\}$$ can be identified with its set of eigenvalues $\{\lambda, \lambda^{-1}\}$. As such, the collection of diagonalizable matrices in $SL(2, \mathbb{C})$ can be ide... | https://mathoverflow.net/users/nan | The conjugacy classes of diagonalizable $2 \times 2$ matrices can be identified with their eigenvalues, what about pairs? | You have a geometric invariant theory question here: your space is the space of the pairs of matrices, and the action is given by conjugation. One possible way to deal with this is to study the ring of invariant functions. In case of one matrix it really boils down to the ring generated by the coefficients of the chara... | 11 | https://mathoverflow.net/users/41644 | 212934 | 101,162 |
https://mathoverflow.net/questions/212942 | 7 | Does there exits a smooth proper algebraic space $X$ over $\mathbb C$ with "large" fundamental group such that no finite etale cover of $X$ is a scheme?
By "large" fundamental group I mean that $X$ has infinitely many non-trivial finite etale covers (of increasing degree).
Equivalently, does there exist a Moishezon... | https://mathoverflow.net/users/76723 | Does there exist an algebraic space with large fundamental group but no finite etale covers by schemes | You can get one class of examples by modifying the Hironaka construction, cf. Appendix B, Examples 3.4.1 and 3.4.2 of Hartshorne's "Algebraic Geometry". What is perhaps surprising is that Hartshorne begins with a scheme -- the scheme from Example 3.4.1 -- that has a fixed point free $\mathbb{Z}/2\mathbb{Z}$-action, and... | 9 | https://mathoverflow.net/users/13265 | 212945 | 101,164 |
https://mathoverflow.net/questions/212939 | 5 | Let $H$ be a simple algebraic group of type $\mathbf{G}\_2$ over $\mathbb{C}$.
Let $\rho$ be the standard 7-dimensional complex representation
$$ \rho\colon H=\mathbf{G}\_2\to \mathrm{SO}\_7.$$
We consider the corresponding homomorphism
$$H\to \mathrm{Spin}\_7=G.$$
We regard $H$ as a subgroup of $G$.
Let $T\_H\subset H... | https://mathoverflow.net/users/4149 | Matrix from a homomorphism of simply connected groups | Like Jason I'd be more inclined to look at this kind of embedding in a geometric or conceptual way. But for your purposes a concrete description seems needed. For this it might be simpler to work in the Lie algebra setting. I don't see immediately what the group viewpoint does for you, since the centerless group $G\_2$... | 4 | https://mathoverflow.net/users/4231 | 212952 | 101,166 |
https://mathoverflow.net/questions/212745 | 2 | Below, is the definition of a **prime-partitionable** integer taken from W. Holsztynski, R. F. E. Strube, [Paths and circuits in finite groups](http://www.sciencedirect.com/science/article/pii/0012365X78900596), Discr. Math. 22 (1978) 263-272 and is apparently the same in W. T. Trotter, Jr. and Paul Erdős, [When the Ca... | https://mathoverflow.net/users/76635 | Proofs needed for observations regarding prime-partitionable numbers | Let's assume a limited (and unproved) version of Linnik's theorem: There is a prime $q$ of the form $kp + 1$ for $k \leq (p-2)$ and $p$ a prime. Experimentally this is true, and can be proved for many primes, but at present not all. With this in hand, the proofs of the conjectures are exercises:
* 1) note that any nu... | 2 | https://mathoverflow.net/users/3206 | 212955 | 101,168 |
https://mathoverflow.net/questions/212891 | 4 | **EDIT:** As mentioned in my answer below, I was mistaken in thinking Dirichlet convolution distributes over ordinary convolution. I'm leaving this question here for reference.
---
I keep stumbling on the same algebraic structure, and I have no clue how to understand or characterize it at all. It's basically the ... | https://mathoverflow.net/users/24611 | A particularly "natural" algebraic structure with three commutative, pairwise-distributive operations | Ack - Dirichlet convolution does NOT distribute over ordinary convolution. I went wrong somewhere way in the beginning of this process and need to backtrack.
See below, where sequences are indexed starting from 1:
$
[2,0,0,0,...] \ast\_D \left( [1,0,0,0,...] \ast\_C [1,0,0,0,...] \right) \\
= [2,0,0,0,...] \ast\_D ... | 2 | https://mathoverflow.net/users/24611 | 212956 | 101,169 |
https://mathoverflow.net/questions/212984 | 19 | Are there topological spaces $X,Y$, each having more than $2$ points, satisfying the following two properties?
* $X\not\cong Y$, and
* there is a bijection $\varphi: X\to Y$ such that for all $x\in X$ the spaces $X\setminus \{x\}$ and $Y\setminus \{\varphi(x)\}$ are homeomorphic.
| https://mathoverflow.net/users/8628 | Non-homeomorphic spaces such that taking away a point makes them homeomorphic | If $X$ and $Y$ are Hausdorff then the one easily sees that the homeomorphism can not be induced by $\varphi$. But you did not request that.
Let $C\_0$ denote a disjoint union of countably many copies of the Cantor set. It is easily seen that the Cantor set $C$ is homeomorphic to the one point compactification of $C\_... | 19 | https://mathoverflow.net/users/16678 | 212990 | 101,179 |
https://mathoverflow.net/questions/212631 | 6 | Assume that I have a finitely presented algebra $A$ over the complex numbers (by which I mean that $A$ is generated over $\mathbb{C}$ by finitely many elements $x\_1,...,x\_n$ subject to finitely many relations). Let now $y\_1,\ldots, y\_m$ be a finite subset of elements of $A$. Is there an algorithm which gives a pres... | https://mathoverflow.net/users/41644 | presentations of subalgebras | If we modify your first question only slightly, then the answer to the question is no, there is no algorithm. By Theorem 1 of the paper G. Baumslag, W. W. Boone and B. H. Neumann, Some unsolvable problems about elements and subgroups of groups, Math. Scand. 7, 191-201 (1959), let $G$ be a finitely presented group for w... | 4 | https://mathoverflow.net/users/3199 | 213002 | 101,183 |
https://mathoverflow.net/questions/212998 | 15 | A classical result states that all the irreducible representations of a finite group over $\mathbb{C}$ are characters if and only if $G$ is abelian. I would like to know what happens if we consider a field different from $\mathbb{C}$. Clearly as long as the field is algebraically closed and its characteristic does not ... | https://mathoverflow.net/users/38889 | Which finite groups have no irreducible representations other than characters? | A group satisfies this property if and only if it is an extension of a $p$ group by an abelian group. The reason is the following: Assume that indeed all the irreducible representations of $G$ are one dimensional. This means that all elements of the form $x-1$ where $x$ belongs to $[G,G]$ are in the Jacobson radical, b... | 12 | https://mathoverflow.net/users/41644 | 213004 | 101,185 |
https://mathoverflow.net/questions/212989 | 5 | (This is [reposted](https://math.stackexchange.com/questions/1376961/three-and-a-half-basic-questions-on-the-weil-restriction-of-scalars) from mathstackexchange, where it received no answer so far.)
I am currently trying to get familiar with the Weil Restriction functor.
For a finite field extension $L|K$ it assoc... | https://mathoverflow.net/users/76748 | Three and a half basic questions on the Weil restriction of scalars | It is best not to use the word (quasi-projective) "variety" in this context because (as alluded to in Scott Carnahan's comment to the question posed) Weil restriction does not preserve geometric integrality or even geometric irreducibility or geometric reducedness or non-emptiness in general (allowing $L/K$ that is pos... | 9 | https://mathoverflow.net/users/70739 | 213012 | 101,187 |
https://mathoverflow.net/questions/213026 | 3 | The following question has been [asked](https://math.stackexchange.com/questions/1377795/free-groups-normal-supplements-of-the-commutator-subgroup) about a week ago on MathUnderflow (no answers).
Let $F$ be a free group and let $N$ be a normal subgroup of $F$ such that
\begin{equation\*} \tag{\*}
F = [F,F] N.
\end... | https://mathoverflow.net/users/76758 | Commutator subgroups as normal supplmements | The equation $F = [F,F]N$ is equivalent to $G = [G,G]$ for $G:=F/N$. Thus, if $N$ is the kernel of a homomorphism of $F$ onto a perfect group, then the equation will hold.
On the other hand, if $N$ is a proper verbal subgroup of a free group $F$, then $F/N$ is never perfect, so $F = [F,F]N$ cannot hold in this case. ... | 5 | https://mathoverflow.net/users/75735 | 213029 | 101,191 |
https://mathoverflow.net/questions/213035 | 4 | We say that an abelian category is **semisimple** if every object is a semisimple object, which is to say, a direct sum of finitely many simple objects.
Let $({\cal C},\otimes,\*)$ be a semisimple monoidal category with duals. If $V$ is a simple object in ${\cal C}$, then can $V \otimes V$ ever contain $V^\*$ as a s... | https://mathoverflow.net/users/2612 | Semisimple monoidal category with duals | Yes it can.
Example: $\mathcal{C} = Rep-S\_3$, and $V$ the two dimensional irreducible representation.
Then $V\otimes V$ splits as the direct sum $V\oplus \mathbb{1}\oplus \chi$ where $\mathbb{1}$ is the trivial representation and $\chi$ is the sign representation. The dual of $V$ is $V$ itself, which is a direct summa... | 3 | https://mathoverflow.net/users/41644 | 213036 | 101,193 |
https://mathoverflow.net/questions/213013 | 1 | Steinberg has a beautiful theorem counting $F$-stable maximal tori in a reductive group. Here's the version of the result that you can find as Theorem 3.4.1 of Carter's *Finite groups of Lie type*:
>
> **Theorem (Steinberg)**: Let $G$ be a connected reductive group, $F$ a Frobenius endomorphism of level $q$, and $\... | https://mathoverflow.net/users/801 | Generalization of a theorem of Steinberg | Concerning your Question 1, the formulation looks somewhat out of focus. Keep in mind that there is for each Lie type of simple algebraic group a well-organized classification of parabolic subgroups and their Levi subgroups (up to conjugacy). So the centers are in principle not impossible to find, and their dimensions ... | 1 | https://mathoverflow.net/users/4231 | 213039 | 101,194 |
https://mathoverflow.net/questions/213022 | 60 | Out of sheer curiosity I have been reading Stewert and Tall's "Algebraic Number Theory and Fermat's Last Theorem" (2001). As it contains various bits of history, I found out to my own shame that I was not even aware of the fact that there was another serious attack at FLT in the 80s - that of Y. Miyaoka.
Now, we usu... | https://mathoverflow.net/users/1849 | What were the main ideas and gaps in Yoichi Miyaoka's attempted proof (1988) of Fermat's Last Theorem? | Here's some information from Barry Cipra's June 1988 article "Fermat's Theorem remains unproved" in *Science* magazine.
>
> Parshin showed that the arithmetical version of a certain inequality involving geometric invariants of surfaces—an inequality that Miyaoka proved for the geometric case in 1974—would lead by a... | 45 | https://mathoverflow.net/users/3106 | 213043 | 101,198 |
https://mathoverflow.net/questions/92557 | 15 | I know that it is impossible to axiomatize the multiplicative structures of rings, called $R$-semigroups. Is anything known about the first-order axiomatizability of the class of abelian groups which are additive groups of some ring? I don't want to restrict the meaning "ring" here. I would like to know whether this qu... | https://mathoverflow.net/users/20803 | Is the class of additive groups of rings axiomatizable? | The answer to the question in the title is No. You can prove this from the work of Wanda Szmielew on the elementary properties of abelian groups. This answer works for any kind of nonzero, bi-additive, binary multiplication (associative or not, commutative or not, unital or not).
In particular, an abelian group $A$ i... | 16 | https://mathoverflow.net/users/75735 | 213052 | 101,202 |
https://mathoverflow.net/questions/213055 | 6 | Let $K$ be some number field, $\mathcal O\_K$ denote its ring of integers, and let $n$ be a positive integer. Take $\alpha \in \mathcal O\_K$, and consider the quantity $r\_{n,K}(\alpha)$, which denotes the number of solutions to the equation
$$
\alpha = x\_1^2 + x\_2^2 + \ldots + x\_n^2
$$
with $x\_1, x\_2, \ldots, x\... | https://mathoverflow.net/users/22733 | Number of representations as sums of squares in rings of integers of number fields | There are some results known in the $n = 4$ case. In particular, in 1928 Gotzky (see Mathematische Annalen volume 100 pages 411-437) proved a formula for the number of representations of a totally positive integer $\alpha \in \mathcal{O}\_{K}$ as a sum of four squares of elements in $\mathcal{O}\_{K}$ for $K = \mathbb{... | 3 | https://mathoverflow.net/users/48142 | 213060 | 101,203 |
https://mathoverflow.net/questions/213073 | 0 | Let $(X,\tau), (Y,\sigma)$ be $T\_2$-spaces such that there are injective lattice homomorphisms $f: \tau\to \sigma$ and $g:\sigma\to \tau$.
Does this imply that $(X,\tau)\cong (Y,\sigma)$?
| https://mathoverflow.net/users/nan | $T_2$-spaces such that the lattices of open sets can be embedded into each other | No, $T\_2$-spaces with this property aren't necessarily homeomorphic.
Let $\mathbb{N} = \{0,1,2,\ldots\}$ be the set of non-negative integers and let $\tau = {\cal P}(\mathbb{N})$ be the discrete topology, and define $$\sigma = {\cal P}(\mathbb{N}\setminus\{0\}) \cup \{U\subseteq \mathbb{N}: 0\in U\text{ and } \mathb... | 3 | https://mathoverflow.net/users/8628 | 213075 | 101,208 |
https://mathoverflow.net/questions/213078 | 5 | Do Fredholm operators show up in K-theory? Why or why not? The idea of infinite Grassmannians classifying vector bundles is pretty straightforward, but why would adding in additive inverses and what not give you this? Is it a generalization of some nice finite-dimensional concept? Does it have a deep connection to (fin... | https://mathoverflow.net/users/76743 | Fredholm operators in $K$-theory? | A classical connection is the Atiyah-Jänich Theorem, see
Klaus Jänich: Vektorraumbündel und der Raum der Fredholm-Operatoren. Math. Ann. 161 (1965) 129–142.
Let ${\mathcal{F}}$ be the space of Fredholm operators with the operator norm, and $X$ any compact space. To a map $F\colon X\to {\mathcal{F}}$ one can assoc... | 13 | https://mathoverflow.net/users/39082 | 213081 | 101,210 |
https://mathoverflow.net/questions/213058 | 4 | Suppose $B\_t$ is a standard Brownian motion. Does there exist $c < \infty$ such that with probability one$$\limsup\_{t \to \infty} {{B\_t}\over{\sqrt{t \log t}}} \le c?$$I need to know whether or not this is true for something I am doing, but I do not do probability by trade...
| https://mathoverflow.net/users/nan | Brownian motion, exists $c < \infty$? | As already stated in the comments it follows from the law of iterated logarithm for Brownian motion.
Write
$\displaystyle{\limsup\_{t\rightarrow \infty} \frac{B\_t}{\sqrt{t \log(t)}} = \limsup\_{t\rightarrow \infty} \frac{B\_t}{\sqrt{2t\log(\log(t))}} \frac{\sqrt{2\log(\log(t))}}{\sqrt{\log(t)}}} = 0$
as $\displa... | 4 | https://mathoverflow.net/users/74045 | 213086 | 101,211 |
https://mathoverflow.net/questions/213068 | 3 | I have read several other questions about wild representation type but may I ask..what actually can be done and what have been done (such as partial results) about classifying indecomposable representations of the quiver $\mathbb{F\_2}$?
Especially is it possible to find infinite (perhaps even countable) sets of con... | https://mathoverflow.net/users/73892 | Is it possible to classify the indecomposable representations of the wild quiver with one vertex and two arrows using infinite sets of parameters? | The representation theory of the quiver with one vertex and two loops (that is, the problem of classifying finite dimensional representations of the free algebra $\mathbb{C} \langle x,y \rangle$ upto isomorphism) is 'in principle' accessible via the tools of geometric invariant theory.
First one fixes the dimension $... | 10 | https://mathoverflow.net/users/2275 | 213087 | 101,212 |
https://mathoverflow.net/questions/212980 | 7 | When do partial derivatives $p\_x$, $p\_y$ of a polynomial over $\mathbb{C}$ not have any common factor? Is there a general approach for any number of variables, aka when is the variety defined by the partial derivatives vanishing (the would-be singular points) a finite point set? I.e. when in $\mathbb{C}[x, y, \dots]$... | https://mathoverflow.net/users/76743 | When do partial derivatives $p_x$, $p_y$ of a polynomial over $\mathbb{C}$ not have any common factor? | In the case of two variables, my example is the only one. Let $g$ be an irreducible polynomial that divides both $p\_x,p\_y$. On the curve $g=0$, we have $dp = p\_xdx+p\_ydy=0$ so $p$ is constant, say $p=c$ on this is curve, so $g$ divides $p-c$. Let's write $p-c = gh$ for some $h$. Then, again on $g=0$, $0=p\_x = g\_x... | 5 | https://mathoverflow.net/users/2290 | 213097 | 101,218 |
https://mathoverflow.net/questions/213111 | 3 | Let $A$ be a $C^\ast$-algebra. Consider the canonical embedding $A\to A^{\*\*},\; a\mapsto i(a)$, such that $i(a)(a^\*)=a^\*(a)$ for all $a\in A$. Here is $A^{\*\*}$ considered as a Banach space. It's well known that $i(A)$ is weak$\*$-dense in $A^{\*\*}$. I want to endow $A^{\*\*}$ with an inner multiplication, a norm... | https://mathoverflow.net/users/75338 | short question about biduals of $C^\ast$-algebras | I'm not sure what you mean by "the canonical embedding is not multiplicative" if you're only taking $A^{\*\*}$ as a Banach space. A natural way to give $A^{\*\*}$ multiplicative structure is via the [Arens product](http://www.encyclopediaofmath.org/index.php/Arens_multiplication), which does make the canonical embeddin... | 4 | https://mathoverflow.net/users/23141 | 213116 | 101,222 |
https://mathoverflow.net/questions/213117 | 1 | I have the following integral:
$$\int\_0^1 \frac{P\_3(t)}{\sqrt{1-k^2 P\_3^2(t)}}dt$$
where $P\_3(t)$ is a third-degree polynomial with all coefficients different from zero and $k$ a generic constant. This integral, once known coefficients and the constant $k$, can be integrated numerically, but I wonder if, in some re... | https://mathoverflow.net/users/69931 | Analytic approximation $\int_0^1 \frac{P_3(t)}{\sqrt{1-k^2 P_3^2(t)}}dt$ | You could expand in a series in powers of $k$:
$$ \dfrac{P(t)}{\sqrt{1-k^2 P(t)^2}} = \sum\_{j=0}^\infty \dfrac{(2j)! 4^{-j}}{j!^2} k^{2j} P(t)^{2j+1} $$
| 4 | https://mathoverflow.net/users/13650 | 213119 | 101,223 |
https://mathoverflow.net/questions/213102 | 1 | Is it true than an aperiodic, ergodic, minimal and equicontinuous dynamical system on a compact metric space is totally ergodic ?
According to some results I found in some books, a rotation on a compact metric group is equicontinous, and it is minimal ~~and totally ergodic~~ whenever it is ergodic.
| https://mathoverflow.net/users/21339 | Do ergodicity, minimality and equicontinuity on a compact space imply total ergodicity? | The answer is no in this generality! If you consider the classical odometer (i.e. addition by 1 on 2-adic integers) then its second power (addition by 2) is not minimal. This second power preserves the numbers starting with 0 (the "even numbers") and the numbers starting with 1 (the "odd numbers"). But of course this e... | 2 | https://mathoverflow.net/users/15342 | 213127 | 101,228 |
https://mathoverflow.net/questions/213118 | 3 | It is reported in [this paper](http://people.mpim-bonn.mpg.de/zagier/files/doi/10.4153/CMB-1985-044-8/fulltext.pdf) by Zagier, as well as in Sage, that the elliptic curve $E=197A1$ has congruence number 10. (Since $E$ has prime conductor, a theorem of Ribet ensures that the congruence number is equal to the modular deg... | https://mathoverflow.net/users/10547 | Congruence Number of 197A1 | I am writing an answer to expand slightly on my comment. One of Zagier's definitions of the congruence number is the largest positive integer $r$ so that
there is a cusp form $g$ with integer coefficients (and not necessarily a Hecke eigenform) that is orthogonal to $f$ (under the Petersson inner product).
This is re... | 5 | https://mathoverflow.net/users/48142 | 213132 | 101,230 |
https://mathoverflow.net/questions/213071 | 6 | The subcategory of hypercomplete objects in an ∞-topos is a left-exact-reflective subcategory by the remarks after 6.5.2.8 of *Higher topos theory*. Can it ever happen that this subcategory is also *coreflective*?
| https://mathoverflow.net/users/49 | Can hypercomplete objects be coreflective? | Let $\mathcal{X}$ be the $\infty$-category of $1$-excisive functors from (pointed) spaces to (unpointed) spaces: equivalently, the $\infty$-category of pairs $(X, E)$ where $X$ is a space and $E$ is a local system of spectra on $X$. Then $\mathcal{X}$ is an $\infty$-topos (for example, it's a left exact localization of... | 12 | https://mathoverflow.net/users/7721 | 213135 | 101,231 |
https://mathoverflow.net/questions/213105 | 0 | I am currently reading a paper on symplectic geometry: [Periodic orbits for Hamiltonian systems in cotangent bundles](http://arxiv.org/abs/math/9201297) by Christopher Golé.
It deals with the question how the stability properties of a sequence of (periodic) points or fixed points can be related to the second derivati... | https://mathoverflow.net/users/75290 | Steps in paper on sympl. geometry unclear | It just looks like a basic application of the chain rule to the immediately preceding equation. Maybe it's the notation that's confusing you; the previous equation has the form
$$
(\partial\_2F)(x,y) + (\partial\_1 G)(y,z) = 0.
$$
Considering the left hand side as a function of three independent variables $(x,y,z)$, an... | 1 | https://mathoverflow.net/users/17945 | 213147 | 101,234 |
https://mathoverflow.net/questions/213145 | 1 | Suppose $\kappa$ is an infinite cardinal. Let $\cal S$ be a collection of pairwise non-isomorphic complete lattices on the ground set $\kappa$. What cardinality can $\cal S$ have at most? Is the answer different if we restrict ourselves to complete *distributive* lattices?
| https://mathoverflow.net/users/8628 | Cardinality of pairwise non-isomorphic complete lattices on an infinite cardinal $\kappa$ | There is a trivial upper bound of $2^\kappa$. In fact, this bound can already be achieved for complete totally ordered sets. For instance, given any subset $A\subseteq\kappa+1$, let $L\_A$ be obtained from $\kappa+1$ by replacing each element of $A$ with a copy of $\mathbb{Z}\cup\{-\infty,\infty\}$. Then $L\_A\cong L\_... | 8 | https://mathoverflow.net/users/75 | 213148 | 101,235 |
https://mathoverflow.net/questions/213151 | 2 | It is well-known, that if $A = \mathrm{id} + S$ is a bounded operator on a separable Hilbert Space $H$ with $S$ trace-class, then there is a well-defined notion of determinant, e.g. in terms of the singular value decomposition of $S$.
Now let $H$ and $K$ be two separable, infinite-dimensional Hilbert Spaces. To defin... | https://mathoverflow.net/users/16702 | Infinite Determinant between different Hilbert Spaces | It seems this implies that $V^{-1}U$ can be diagonalised isn't it ?
Indeed: $V^{-1}U$ is a unitary, let just call it $W$. If $1-W$ is trace class it is in particular compact, moreover it is normal (as $W$ is unitary) hence it can be diagonalised in some Hilbert basis which obviously also diagonalize $W$.
The detemi... | 2 | https://mathoverflow.net/users/22131 | 213152 | 101,236 |
https://mathoverflow.net/questions/213091 | 19 | Let $\mathfrak{g}$ be the adjoint representation of a simple Lie algebra (which is not of type $A$). Then the space of intertwiners between the third exterior power of $\mathfrak{g}$ and the third symmetric power of $\mathfrak{g}$ has dimension one. I have checked on a case-by-case basis using LiE that these two repres... | https://mathoverflow.net/users/50658 | What is the homomorphism between the third exterior and third symmetric power of the adjoint representation of a simple Lie algebra? | There *is* a natural homomorphism that I am pretty sure is not zero in general that can be described most easily using the dual language of the Lie algebra $\frak g$:
Recall that there is a differential $\mathrm{d}:{\frak{g}}^\ast\to \Lambda^2({\frak{g}}^\ast)$ that extends uniquely to a degree-one, square-zero deri... | 25 | https://mathoverflow.net/users/13972 | 213163 | 101,238 |
https://mathoverflow.net/questions/213168 | 2 | The subcategory of hypercomplete objects in an ∞-topos is a left-exact-reflective subcategory by the remarks after 6.5.2.8 of *Higher topos theory*, i.e. its inclusion functor has a left-exact left adjoint. Can this left adjoint ever have a further left adjoint, so that the hypercomplete objects would be an [essential ... | https://mathoverflow.net/users/49 | Can hypercompletion be an essential localization? | Jacob's [answer](https://mathoverflow.net/a/213135/49) to my last question actually answers this one too: if the hypercomplete objects are coreflective as well as reflective, and the coreflector is *the same functor* as the reflector, then of course the reflector also has a left adjoint, namely the inclusion functor.
... | 2 | https://mathoverflow.net/users/49 | 213169 | 101,240 |
https://mathoverflow.net/questions/213165 | 8 | Let $K/\mathbb{Q}$ be an algebraic extension, and let $E\_1,E\_2/\mathbb{Q}$ be elliptic curves. Is it possible that the Mordell-Weil rank of $E\_1(K)$ is finite while that of $E\_2(K)$ is infinite?
| https://mathoverflow.net/users/38889 | Ranks of elliptic curves depend only on the field? | The answer is yes, at least if you assume that Tate-Shafarevich conjecture.
Let $E$ be an elliptic curve over a number field $k$. Under some mild hypothesis (see Corollary 1.10 of <http://arxiv.org/abs/0904.3709>) there exist quadratic twists of $E$ which have trivial Mordell-Weil rank. In other words, there exists q... | 13 | https://mathoverflow.net/users/51164 | 213173 | 101,244 |
https://mathoverflow.net/questions/213072 | 18 | **EDIT:** According to some comments on this post I revise the title to remove the misunderestanding.
Assume that $M$ is a Riemannian manifold of dimension $n$. The natural Laplace operator associated to the metric is denoted by $\Delta$.
>
> Are there $n$ vector fields $X\_{1},X\_{2}, \ldots, X\_{n}$ such that $... | https://mathoverflow.net/users/36688 | Can the Laplace operator on $n-$ manifolds be represented as a sum of $n$ second order derivational operators | As Raziel wrote, the local question is whether one can find a local basis of orthonormal vector fields that are divergence-free.
It's true that, in dimension $2$, this can only be done if the metric is locally flat, which is the local obstruction. This is because this is an overdetermined problem; one has two equati... | 22 | https://mathoverflow.net/users/13972 | 213189 | 101,251 |
https://mathoverflow.net/questions/213188 | 3 | I have a simple question regarding complex geometry: **is there an analog for the Stokes Theorem for the Dolbeault Operator $\bar{\partial}$?** For instance, suppose that $M$ is a closed complex manifold and I am looking for some identity like
$$\int\_M \bar{\partial}(\cdots)=\cdots$$
For it to make sense, I suppos... | https://mathoverflow.net/users/62367 | Stokes-like Theorem for Dolbeault Operator | If $M$ is an $n$-dimensional complex manifold, then $M$ is a $2n$-dimensional smooth manifold, so you should integrate a $2n$-form (note that $\bar{\partial}$ of a $(0, n-1)$-form is an $n$-form).
Consider the expression $\int\_M\bar{\partial}\alpha$ where $\alpha$ is a complex $(2n-1)$-form. As $\mathcal{E}^{2n-1}(... | 3 | https://mathoverflow.net/users/21564 | 213190 | 101,252 |
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