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https://mathoverflow.net/questions/128533 | 4 | Let $\sigma\_m (n)=\sum\_{d|n}d^m$, and $d(n)=\sigma\_0(n)$ as stated in the title.
My question is: Does the q-series $f(q)=\sum\_{n\ge 1} d(n)q^n$ gives something like a modular form (i.e. with some kind of identity under the action of $SL\_2$) when $q=e^{2\pi i z}$?
When $d(n)$ is replaced by $\sigma\_{2m-1} (n)$... | https://mathoverflow.net/users/32631 | q-series related to d(n)=# of divisors of n | The coefficient $d(n)$ is the limit of the $n$-th Hecke eigenvalue of the (nonholomorphic) Eisenstein series $E(z,s)$ as $s\to 1/2$. The limit of the Eisenstein series itself is zero, hence it is more natural to consider the $(\partial/\partial s)E(z,s)$ at $s=1/2$, which is precisely
$$ \sqrt{y}\log y+4\sqrt{y}\sum\_{... | 7 | https://mathoverflow.net/users/11919 | 128535 | 71,535 |
https://mathoverflow.net/questions/127892 | 6 | Most Heegaard diagrams contain many rectangles, for instance from loops that circle around one of the handle disks. You can always `twist' a Heegaard diagram to get more and more rectangles (as in page 5 of [this paper](https://web.math.princeton.edu/~szabo/clay.pdf)). Can one reverse this process to eliminate rectangl... | https://mathoverflow.net/users/27933 | Untwisting Heegaard diagrams | I'm sorry to say that this condition is quite rare in practice. If the splitting has high enough distance in the curve complex, any pair of curves from the two disk sets will intersect a lot, resulting in many rectangles. Furthermore, high-distance splittings are "generic."
The word "generic" can be made precise in t... | 8 | https://mathoverflow.net/users/8183 | 128536 | 71,536 |
https://mathoverflow.net/questions/128497 | 0 | Let $X$ be a nodal curve (over $\mathbb{C}$) consisting of two components $C$ and $E$ where $C$ is hyperelliptic of genus three and $E$ is an elliptic curve. The inverse image of the node on $C$ is not an Weierstrauss point. Now let $L$ be a line bundle on $X$ such that $\deg L|\_C =2$ and $h^0(C, L|\_C)=2$ and $L|\_E ... | https://mathoverflow.net/users/32337 | line bundle on a nodal curve coming degeneration of smooth ones | Let $\pi:\mathcal{X} \to T$ be a versal family of stable curves over $T$ such that the fiber over $t\_0\in T$ equals $X$. By unobstructedness of deformations of stable curves, $T$ is smooth of dimension $3(4)-3=9$.
I do not claim that the relative Picard functor is representable, but there a family $(T'\to T,\mathca... | 1 | https://mathoverflow.net/users/13265 | 128542 | 71,538 |
https://mathoverflow.net/questions/128508 | 2 | A theorem of Giraud says that gerbes over a scheme $X$ bounded by a sheaf of Abelian groups $A$ are classified by elements of the etale cohomology group $H^2(X,A)$. Similar statements hold in other categories (differential, topological) as well.
One of the points of view on gerbes (emphasized, for example, by Hitchin... | https://mathoverflow.net/users/2234 | gluing gerbes over a spectrum of a field | I don't think so. I think a gerbe bound by $A$ over the spectrum of a field $k$ gives a cohomology class $\eta\in H^2(k,A)$, and the gerbe trivializes over an extension $k'/k$ if and only if this cohomology class trivializes. Now take $k=\mathbb{R}$, $A=\mathbf{G}\_{m,\mathbb{R}}$, then $H^2(k, A)={\rm Br}(\mathbb{R})=... | 1 | https://mathoverflow.net/users/4149 | 128543 | 71,539 |
https://mathoverflow.net/questions/128112 | 1 | Let $G$ be a Lie group and $H$ a closed Lie subgroup. Is there an explicit way to construct a local cross section of $H$ in $G$ so that $\pi: G\to G/H$ is a fiber bundle?
| https://mathoverflow.net/users/12577 | Cross section for closed Lie subgroup in a Lie group | You might consider your problem in a generalized setting. Let $H$ be a Lie group and $X$ an $H$-manifold on which $H$ acts freely and properly. There is a unique manifold structure on $X/H$ for which $X\rightarrow X/H$ is a principal $H$-bundle (and in particular, a fiber bundle). If $H$ is a closed subgroup of a Lie g... | 3 | https://mathoverflow.net/users/25358 | 128546 | 71,541 |
https://mathoverflow.net/questions/128441 | 9 | Let $\mathbb T^2=(S^1)^2$ be the 2-torus, for convenience. $\def\Imm{\operatorname{Imm}}$
Consider the Frechet manifold of immersion $\Imm(\mathbb T^2, \mathbb R^3)$ and the smooth mapping
$R:\Imm(\mathbb T^2,\mathbb R^3)\to C^\infty(\mathbb T^2,\mathbb R^3)$ which is given by
$$
R(f) := \partial\_x f\times \partial\_y... | https://mathoverflow.net/users/26935 | Representing immersions from a surface into 3-space | I may have to enter this as a sketch and fill in details later, but I thought that I'd go ahead and get the main ideas out there.
The first thing to notice is that the given problem is equivalent to the problem of solving $\rho(f)=g$ where $\rho:\mathrm{Imm}(T^2,\mathbb{R}^3)\to\Omega^2(T^2,\mathbb{R}^3)$ is the diff... | 15 | https://mathoverflow.net/users/13972 | 128552 | 71,545 |
https://mathoverflow.net/questions/128537 | 6 | Suppose $X$ is a topological space and $k$ some discrete coefficient field. Let's define the category of "$\infty$-local systems on $X$" to be DG representations of the ring $C\_\*(\Omega X,k)$ of chains of loops on $X$. (I think this is equivalent, at least when $\text{char}(k)=0$, to locally constant sheaves where "s... | https://mathoverflow.net/users/7108 | What is a higher derived constructible sheaf | In the appendix of "Higher Algebra" (<http://www.math.harvard.edu/~lurie/papers/HigherAlgebra.pdf>), Jacob Lurie describes constructible sheaves on a stratified space as representations of an exit path $\infty$-category. So one option is to take representations of this exit path $\infty$-category valued in the stable $... | 6 | https://mathoverflow.net/users/7762 | 128553 | 71,546 |
https://mathoverflow.net/questions/128400 | 2 | **Overall problem:** Sample i.u.d. from $\{1,\dots,n\}$. What is (a good lower bound for) the probability of getting the values $1$ and $2$ before either you get a number you have seen before or you have sampled $\lceil \sqrt{n} \rceil$ numbers? In particular, how can we prove it is at least $1/2n$?
I have formulated... | https://mathoverflow.net/users/45564 | Probability of event occurring before either of two stopping conditions | This particular problem can be solved by dividing it into two parts.
First, the probability that the first $\sqrt{n}$ numbers chosen are all distinct. Second, the probability that a random subset of $\{1,..,n\}$ of size $\sqrt{n}$ contains $1$ and $2$. The first probability is $\prod\_{k=1}^{\sqrt{n}} (1-\frac{k}{n})... | 2 | https://mathoverflow.net/users/1061 | 128560 | 71,549 |
https://mathoverflow.net/questions/128544 | 1 | The original problem I'm looking at is: given a bound on the operator norm of $\Lambda A \Lambda,$ where $\Lambda, A$ are positive definite matrices and $\Lambda$ is diagonal, what is the tightest bound on the operator norm of $A \Lambda^2.$
My starting point is the fact that these two matrices have the same eigenval... | https://mathoverflow.net/users/2586 | relationship between numerical and spectral radii for product of positive definite matrices? | This was too long for a comment--I'm not sure if this helps you or not, but it seems like there can't be a bound independent of the sizes of the matrices. Work in size $n\times n$ and consider
$$
\Lambda = diag(1,0,0\dots 0)
$$
(This isn't positive definite, of course, but it's the limiting case of $\Lambda = diag(1,\... | 1 | https://mathoverflow.net/users/13360 | 128562 | 71,551 |
https://mathoverflow.net/questions/128524 | 3 | It is known that there is a model structure on category of dg algebras (non-commutative over arbitrary commutative ring). In particular it is complete and co-complete category. My question is how to construct limits and co-limits. I'm especially interested in co-equalizers.
| https://mathoverflow.net/users/23011 | Coequalizer in category of dg-algebras | There is a general bit of category theory that was applied to
ring spectra in EKMM ([83] on my website) and I'll refer
to that for details. Unless I am missing something, the
discussion surely specializes just as well to dg algebras over a
commutative ring $R$. I'll outline the recipe it gives
for constructing all co... | 5 | https://mathoverflow.net/users/14447 | 128568 | 71,553 |
https://mathoverflow.net/questions/128514 | -1 | Let $\Omega$ be a domain in $\mathbb{R}^3$, does there exist a vector-valued Sobolev-type space, or maybe space in other sense, $V$, satisfying the following:
1) $S:=\lbrace v\in V:\|v\|\_{L^\infty(\Omega)}\leq 1\rbrace$ is bounded in $V$;
2) $V$ has one degree of regularity, or maybe weaker, so that the divergen... | https://mathoverflow.net/users/33350 | A suitable Sobolev-type space | I think that such a space will not exists. In particular your requirements (1) and (2) contradicts in the following sense: (1) says basically, that the norm in $V$ is weaker than the $L^\infty(\Omega)$-norm, whereas (2) requires that you can control the divergence by the norm in $V$. Imho, this is not possible.
| 3 | https://mathoverflow.net/users/32507 | 128578 | 71,555 |
https://mathoverflow.net/questions/128564 | 6 | Let $C$ be a fixed category, and let $T\_1$ and $T\_2$ be two Grothendieck (pre)topologies on $C$. We say $T\_1$ is subordinate to $T\_2$ if every covering in $T\_1$ has a refinement in $T\_2$. We say $T\_1$ and $T\_2$ are equivalent if $T\_1$ is subordinate to $T\_2$ and $T\_2$ is subordinate to $T\_1$. We know that i... | https://mathoverflow.net/users/26081 | Can Inequivalent Topologies Have Same Sheaves/Cohomology? | I would think so.
Two pretopologies are equivalent when they generate the same topology, that is, when the have the same sieves. If $A$ is an object of $C$, a sieve on $A$ is a subfunctor of the functor $h\_A$ represented by $A$; and a subfunctor $S$ of $h\_A$ is a sieve with respect to a topology $T$ if and only if... | 5 | https://mathoverflow.net/users/4790 | 128583 | 71,558 |
https://mathoverflow.net/questions/128593 | 5 | Hi.
What are the simplest examples to have in mind of non-isomorphic smooth (complex) algebraic curves with isomorphic jacobian variety?
| https://mathoverflow.net/users/33392 | Simplest complex curves with isomorphic Jacobian | It is not easy to give equations of non-isomorphic Riemann surfaces whose Jacobians are isomorphic as (unpolarized) Abelian varieties.
Some explicit examples can be found in the paper by Everett W. Howe [*Infinite Families of Pairs of Curves Over $\mathbb{Q}$ with Isomorphic Jacobians*](http://jlms.oxfordjournals.org... | 5 | https://mathoverflow.net/users/7460 | 128600 | 71,565 |
https://mathoverflow.net/questions/128053 | 7 | This is the third question in a series whose purpose has been to flesh out an example of the optimality of the p-Lebesgue differentiation theorem for Sobolev functions. This theorem says that for $f \in W^{1,p}\_{loc}(\mathbb{R}^N;\mathbb{R})$,
$\lim\_{r\to0} \frac{1}{r^{N+p}}\int\_{B(0,r)} |f(x+h)-f(x)-\nabla f(x)h... | https://mathoverflow.net/users/28090 | Optimality of p-Lebesgue Differentiation Theorem for Sobolev Functions | It depends on the values of $p$ and $N$ - sometimes there will be counterexamples and sometimes there will not. For instance, any function $f \in W^{1,q}$ for $q>N$ is classically differentiable almost everywhere, and hence the above result is true for all $1\leq p < \infty$. When $1\leq q < N$, then the result holds f... | 5 | https://mathoverflow.net/users/33395 | 128604 | 71,566 |
https://mathoverflow.net/questions/128606 | 2 | Let $K$ be a discrete valuation field with valuation $v:K\rightarrow \mathbb Z\cup \{\infty\}$ which is normalized by $v(\pi)=1$ for a prime element $\pi$. Let $v:\overline K\rightarrow \mathbb Q\cup\{\infty\}$ be an extension of the normalized discrete valuation $v$ to a separable closure of $K$. For a rational number... | https://mathoverflow.net/users/33398 | ramification of discrete valuation field | Yes, there is. Inertia is a subgroup of the decomposition group which by definition preserves the extended valuation. Consequently, the action of $I$ on $\overline{K}$ preserves both $\mathfrak{m}^r$ and $\mathfrak{m}^{r+}$, hence induces the action on the quotient. The induced action on the quotient preserves the $k =... | 3 | https://mathoverflow.net/users/5498 | 128608 | 71,567 |
https://mathoverflow.net/questions/128607 | 4 | It is well-known that any symmetric monoidal category is equivalent to a strict symmetric monoidal category. The construction of this strict monoidal category is rather technical and it appears to me that there is a significantly easier way of obtaining an even better result. Of course this means almost certainly that ... | https://mathoverflow.net/users/18744 | "Wrong" strictification of symmetric monoidal categories | There are a couple of issues:
The first thing is that **isomorphism classes don't form a category**.
You may force them into being a category by picking a representative in each isomophrpism class, and then looking at the full subcategory that they span.
But this construction **depends** on the choice of represent... | 15 | https://mathoverflow.net/users/5690 | 128610 | 71,568 |
https://mathoverflow.net/questions/128522 | 4 | I am interested in references and suggestions concerning the use of stratifications in topology to inductively compute topological invariants. I would appreciate a fairly introductory reference on the subject matter. Also, my stratification consists of orbits of an action of a connected, simply-connected complex semisi... | https://mathoverflow.net/users/25358 | Stratifications and Cohomology Computations | As *some guy on the street* hinted, any "elementary" inductive approach is a spectral sequence in disguise. You have not indicated what topological invariants you have in mind. If Betti numbers suffices for your needs, then in some instances the spectral sequences become relatively simple.
One such instance comes fr... | 0 | https://mathoverflow.net/users/20302 | 128623 | 71,575 |
https://mathoverflow.net/questions/128491 | 2 | Let's say that a discretely ordered ring has rank 1 if it has elements greater than any integer, and for any two such elements $x<y$ there is an integer $n$ such that $x^n>y$.
Question: Let $f(\bar{x})$ be a polynomial in any number of variables with integer coefficients that has a zero in at least one discretely ord... | https://mathoverflow.net/users/5229 | Zeros of polynomials in discretely ordered rings | The answer is no. The argument below is essentially due to Kaye [1] (Lemmas 2.1 and 5.8). Put $$p(a,x,y)=x^2-2axy+y^2-1.$$
**Lemma 1:** In any discretely ordered ring (DOR), if $p(a,x,y)=0$, $0< x\le y$, and $a>0$, then $y\le 2ax\le x+y$ and $p(a,2ax-y,x)=0$.
**Proof:** Straightforward computation.
**Lemma 2:** L... | 2 | https://mathoverflow.net/users/12705 | 128624 | 71,576 |
https://mathoverflow.net/questions/128596 | 7 | Let $K$ be a finite extension of the $p$-adic numbers. $G\_K$ be its absolute Galois group and $I\_K$ the inertia subgroup. Are finite index subgroups of $I\_K$ closed in its profinite topology?
By a result of [Nikolov/Segal](http://www.sciencedirect.com/science/article/pii/S1631073X03003492) it suffices to show, tha... | https://mathoverflow.net/users/448 | Are finite index subgroups of inertia closed? | Here is another way to see that the wild inertia group $P\_K=\mathrm{Gal}(\bar K|T)$ (where $T$ is the maximal tamely ramified extension of $K$ (a finite extension of $\mathbf{Q}\_p$) in $\bar K$, an algebraic closure of $K$, is not finitely generated (as a profinite group). One doesn't need class field theory, only Ku... | 8 | https://mathoverflow.net/users/2821 | 128631 | 71,580 |
https://mathoverflow.net/questions/128613 | 3 | In Atiyah and Bott's paper "The Moment Map and Equivariant Cohomology", they say that for any exact sequence of modules over $\mathbb{C}[u\_1,...,u\_l]$
$$D \to E \to F,$$
we have that
Supp $E \subset$ Supp $D \cup$ Supp $F$,
where Supp$E$ is defined to be $\cap V\_f$ where $f$ runs over all polynomials such that $... | https://mathoverflow.net/users/21180 | Support of a module over a polynomial algebra | If I understand the definitions properly, the following should work:
Let $D \xrightarrow{\alpha} E \xrightarrow{\beta} F$ be exact and suppose $\text{Supp} E \not\subseteq \text{Supp}D \cup \text{Supp}F$. Hence there is $a \in \text{Supp} E$ such that $a \notin \text{Supp}D$ and $a \notin \text{Supp} F$. This means:... | 3 | https://mathoverflow.net/users/10194 | 128632 | 71,581 |
https://mathoverflow.net/questions/128618 | 8 | Let KL denote König's Lemma (for trees over $\mathbb{N}$), and RT(3) denote the
Infinite Ramsey Theorem for triples over $\mathbb{N}$ (notation as in Simpson's
book *Subsystems of second order arithmetic*, 2ed, 2009, see definitions below).
>
> **Question:** Is there a simple direct derivation of KL from RT(3)?
>
... | https://mathoverflow.net/users/33399 | Deriving Konig's Lemma directly from Infinite Ramsey's Theorem for triples | Here is an attempt...
The function $\Delta:[\mathbb{N}^{< \omega}]^2 \rightarrow \mathbb{N}$ is given by $\Delta(a,b)$ is the largest natural number $i$ such that $\langle a(0), a(1) \ldots a(i-1)\rangle = \langle b(0), b(1) \ldots b(i-1)\rangle$. If $a =b$ then this is the length of $a$, and if $a$ is an initial seg... | 7 | https://mathoverflow.net/users/3462 | 128641 | 71,586 |
https://mathoverflow.net/questions/127509 | 9 | Consider a knot to be a diagram in a plane--- i.e. a drawing of a finite connected planar graph (loops and multiple edges allowed) whose vertices are 4-valent with cyclic ordering for the incident edges of each vertex, two non-consecutive of which are called "over" and two "under" (with an additional connectivity condi... | https://mathoverflow.net/users/2051 | Diagrammatic proof of unique prime decomposition of knots | I am fairly certain that there's no **known** diagrammatic proof of the uniqueness of prime decompositions. That answers the "who is it due to?" question.
Of course, a diagrammatic argument may still be out there, waiting to be discovered. But, echoing Ryan's comment, I expect that line of argument to be very difficu... | 6 | https://mathoverflow.net/users/8183 | 128644 | 71,589 |
https://mathoverflow.net/questions/128569 | 31 | Is there a model where Dedekind reals and Cauchy reals are different? I'd appreciate if someone can refer me to any related work in case such a model exists.
| https://mathoverflow.net/users/33090 | A Model where Dedekind Reals and Cauchy Reals are Different | In Lubarsky and Rathjen, *[On the Constructive Dedekind Reals](http://math.fau.edu/lubarsky/Dedreals.pdf)*, Proceedings of LFCS '07, Lecture Notes in Computer Science #4514, they construct a model in which the Cauchy reals form a set, but the Dedekind reals are a proper class. Certainly they are different in this case!... | 13 | https://mathoverflow.net/users/30790 | 128646 | 71,590 |
https://mathoverflow.net/questions/128574 | 3 | Let $M$ be a complete Riemannian manifold of finite volume whose sectional curvatures $\kappa$ satisfy $a \leq \kappa \leq 0$ for some $a \leq 0$. Let $\tilde{M}$ be the universal cover of $M$. The space $\tilde{M}$ has a visual boundary $\tilde{M}(\infty)$ (homeomorphic to a sphere). Let $X \subset \tilde{M}(\infty)$ ... | https://mathoverflow.net/users/33385 | Visual boundaries of universal covers of finite-volume nonpositively curved manifolds | Here is the proof in the case of locally symmetric spaces; you will have to combine it with "rank rigidity theorem" and Igor's comments to get a complete proof.
Let $G$ be a semisimple Lie group, $K\subset G$ a maximal compact subgroup, $B\subset G$ a Borel subgroup; $X=G/K$ is the symmetric space of $G$. Let $r$ de... | 5 | https://mathoverflow.net/users/21684 | 128647 | 71,591 |
https://mathoverflow.net/questions/128645 | 2 | Let $X$ be a smooth, projective and geometrically connected curve of genus $g\geq 3$ defined over $\mathbb C$. Suppose the function field $k(X)$ of $X$ takes the form $k(X)=\mathbb C(x)[y]$, where $$y^a=f(x)$$ with $a\geq 3$ an integer and with $f\in \mathbb C[x]$ a separable polynomial of degree $d=\deg(f)\geq 3$.
... | https://mathoverflow.net/users/33407 | Relation of degree and genus of superelliptic curves | $2g=da+ma\_1-2a-d-m+2$, where $a=a\_1m,$ and $m$ is the g.c.d ($d,a$).
Edit.
Let $\chi(S)=2-2g$ be the Euler characteristic. Hurwitz formula gives
$$\chi(S)=2a-r,$$
where $r$ is the ramification: a branch point of order $k$ contributes
$k-1$ to $r$. As your polynomial has $d$ simple roots, these roots
contribute $(a-... | 5 | https://mathoverflow.net/users/25510 | 128654 | 71,597 |
https://mathoverflow.net/questions/128589 | 4 | Hello!
From the book "Einstein manifolds" by Arthur L. Besse (at section 7.B), Lie groups $Sp(n)$, $Sp(n)\cdot U(1)$, $SU(2n)$ and $U(2n)$ constitute the complete list of Lie subgroups of $U(2n)$ acting transitively and effectively on the sphere $S^{4n-1}$.
Consider natural inclusions of $Sp(n)$, $Sp(n)\cdot U(1)$... | https://mathoverflow.net/users/33387 | Transitive action on the sphere | Yes, there is always such an $M$.
To see this, first note that saying two representations of $G$ on a vector space $V$ are equivalent is the same as saying the two images of $G$ in $Gl(V)$ are conjugate by an element of $Gl(V)$. A theorem due to Mal'cev which can be found in
>
> Mal'cev. On semisimple subgroups o... | 7 | https://mathoverflow.net/users/1708 | 128655 | 71,598 |
https://mathoverflow.net/questions/106035 | 4 | Let $p$ be a prime and $q=p^n$. Let $\mathbb F\_{q^2}$ be a field with $q^2$ elements and $\sigma$ its authomorhism of order two. A $m$ by $m$ matrix $A$ over $\mathbb F\_{q^2}$ is *hermitian* if $A^\sigma$ coincides with the traspose of $A$.
Do you know a reference to the following result?
**Theorem**. The ring of... | https://mathoverflow.net/users/10482 | Generating of the matrix ring by two hermitian matices | The statement is false if $m=2$.
For $m>2$, let $\alpha$ be a generator of the multiplicative group of $\mathbb F\_{q^2}$. Then the matrices $\alpha E\_{1,2}+\sigma(\alpha) E\_{2,1}$ and $\sum\_{i=1}^{m-1} (E\_{i,i+1}+E\_{i+1,i})$ generate the ring of m by m matrices over $\mathbb F\_{q^2}$.
| 1 | https://mathoverflow.net/users/10482 | 128664 | 71,604 |
https://mathoverflow.net/questions/128665 | 6 | We know now that hyperbolic 3-manifolds virtually embed into right-angled Artin groups as quasiconvex subgroups. Also, quasiconvexity depends on the generating set.
I have been constructing a space at infinity for right-angled artin groups where the boundaries of quasiconvex hyperbolic subgroups will embed in a natur... | https://mathoverflow.net/users/27933 | Do quasi convex hyperbolic subgroups remain quasi convex after adding redundant generators? | First, one should decide what quasiconvexity (qc) means in the context of subgroups $H\subset G$, where $G$ is merely a semihyperbolic group, e.g., a RAAG. (I am assuming that generating sets are fixed.) Here are eight competing definitions:
1. $H$ is qc in $G$ if there exists a constant $C$ so that every geodesic in... | 14 | https://mathoverflow.net/users/21684 | 128669 | 71,605 |
https://mathoverflow.net/questions/128675 | 2 | Is there an analog of Fourier series in the function field setting based on the Carlitz exponential? I mean, something like:
Let $\Omega$ be the completion of an algebraic closure
of $\mathbb F\_q\left(\left(\frac1T\right)\right)$ and $\exp\_{\mathcal C}(z)=\sum\_{n\ge0}\frac{z^{q^n}}{D\_n}$ be the Carlitz exponenti... | https://mathoverflow.net/users/33128 | Fourier expansions in function fields | Yes, there is and is called $t$-expansion rather than $q$-expansion - may be, though, you might want to replace your $\Omega$ by
$$
\widehat{\bar{K\_\infty}}\setminus K\_\infty
$$
where I denote by $K\_\infty$ your completion $\mathbb{F}\_q\big(\big(\frac{1}{T}\big)\big)$. This is the content of the paper *On the coeff... | 3 | https://mathoverflow.net/users/18238 | 128677 | 71,607 |
https://mathoverflow.net/questions/128676 | 30 | Is there a simple answer to the question "what happens to the continued fraction expansion of an irrational number when you add 1/2?" A closely related question is "what happens to such an expansion when you multiply by 2?"
Remarks: I'm not sure what qualifies for an answer. The motivation comes from wanting to bette... | https://mathoverflow.net/users/10774 | What is the effect of adding 1/2 to a continued fraction? | [Gosper](http://www.inwap.com/pdp10/hbaker/hakmem/cf.html#item101b) showed how to produce the continued fraction for
$$\frac{axy+bx+cy+d}{exy+fx+gy+h}$$
and even more complicated expressions given the continued fractions for $x$ and $y$. As a special case, he covers fractional linear transformations of $x$, which ... | 34 | https://mathoverflow.net/users/2954 | 128679 | 71,608 |
https://mathoverflow.net/questions/128681 | 6 | Let $d(n)$ denote the number of divisors of $n$. Is it known that the series
$$\sum\_{p \text{ prime}} \frac{1}{d(p-1)}$$
diverges?
This would follow immediately from the Sophie Germain Conjecture. Indeed, if there are infinitely many primes of the form $2p+1$ ($p$ a prime), then infinitely many terms of the series a... | https://mathoverflow.net/users/6779 | A divergent series related to the number of divisors of of p-1 | Since every divisor $k$ of $p-1$ either satisfies $k\le \sqrt{p-1}$ or $\frac{p-1}{k}\le \sqrt{p-1}$, we have $d(p-1) \le 2\sqrt{p-1}$. If we let $p\_n$ denote the $n$th prime, then since $p\_n-1 \le n^2$ for all $n$ (an easy consequence of the prime number theroem...), we have
$\sum\_{n=1}^N \frac{1}{d(p\_n-1)} \ge ... | 9 | https://mathoverflow.net/users/2363 | 128682 | 71,610 |
https://mathoverflow.net/questions/128673 | 3 | $\textbf{Some definitions:}$
Let $G$ be an algebraic group (for me that is the complex points of an affine algebraic group). We say $G$ is reductive if its unipotent radical (maximal connected normal unipotent subgroup) $R\_u(G)$ is trivial and we say $G$ is semisimple if its radical (maximal connected normal solvable ... | https://mathoverflow.net/users/22221 | Is there an almost-direct product decomposition for disconnected reductive algebraic groups? | A solvable reductive group has a very simple structure - the component group is solvable, and the connected component of the identity is an algebraic torus. Unfortunately it is not really possible to decompose it further, because the conjugation action would need to decompose as well, but there are many indecomposable ... | 3 | https://mathoverflow.net/users/18060 | 128684 | 71,611 |
https://mathoverflow.net/questions/128540 | 14 | In [the paper by Greene, Nijenhuis and Wilf](http://www.math.upenn.edu/~wilf/website/Probabilistic%20proof.pdf), an algorithm is proposed for generating uniformly random Young tableaux of shape $\lambda$. The algorithm is to uniformly randomly pick a starting cell, and then do a hook walk algorithm until it terminates ... | https://mathoverflow.net/users/934 | Generating Random Young Tableaux: A peculiar probability identity | This is a technical question for those very familiar with the proof. Basically, GNW show by induction that the quantity on the l.h.s. has a product formula, with some hook numbers inside. Since the probabilities add up to 1, this gives a recurrence relation for the number of standard Young tableaux (SYT), mimicking the... | 11 | https://mathoverflow.net/users/4040 | 128690 | 71,615 |
https://mathoverflow.net/questions/128692 | 3 | Let $Z$ be a $\Delta$-generated space (a colimit of simplices -not sure that this hypothesis is important but it is the framework I am working in). The set of continuous maps $Z^{[0,1]}$ from $[0,1]$ to $Z$ is equipped with the *$\Delta$-kelleyfication* of the compact-open topology (the internal hom of the category, se... | https://mathoverflow.net/users/24563 | Topological question about right-lifting property and the evaluation map | $\newcommand{\into}{\hookrightarrow}$It seems that if $Z$ has the indiscrete topology, then the evaluation map $ev\_0 : Z^I \to Z$ has the right lifting property with respect to any map. That provides a simple counter-example to the question.
So I will assume for the remainder of my answer that $Z$ is $T\_1$, i.e. si... | 1 | https://mathoverflow.net/users/21095 | 128706 | 71,622 |
https://mathoverflow.net/questions/128711 | 4 | Due to Cerf, there's exist a certain homotopy between two Morse functions. It is said that this homotopy is "generic". What is a precise definition of the property to be "generic" in this case?
| https://mathoverflow.net/users/24417 | What does it mean that homotopy is generic? | Intuitively, for the homotopy to be generic has the same meaning as for the Morse function itself to be generic: the set of points where it fails to be a submersion is as simple as possible. Practically, this means that for all but a finite number of times (values of the $t$ parameter) the function $H(x,t)$ is a Morse ... | 6 | https://mathoverflow.net/users/20787 | 128714 | 71,624 |
https://mathoverflow.net/questions/128519 | 2 | Is the secondary polytope of a simplicial polytope necessarily simplicial?
| https://mathoverflow.net/users/19642 | Secondary Polytope Simplicial? | The secondary polytope of a simplicial polytope is not neccessarily either simplicial or simple. The secondary polytope of the cyclic 4-polytope with 8 vertices in given as an example in by Billera, Filliman, and Sturmfels in "Constructions and complexity of secondary polytopes" - Adv. Math. 83 (1990), no. 2, 155–179. ... | 8 | https://mathoverflow.net/users/19642 | 128715 | 71,625 |
https://mathoverflow.net/questions/128712 | 2 | Let $A\_g$ be the moduli space of principally polarised abelian varieties of dimension $g$ over the complex numbers. (EDIT: I mean the coarse moduli space.) Is this smooth?
Since $A\_g$ is the quotient of Siegel upper half space by $\mathrm{Sp}\_{2g}(\mathbb{Z})$ and this group has torsion elements, it seems likely t... | https://mathoverflow.net/users/1046 | Is the moduli space of ppAVs smooth? | The answer is no, it is not smooth for any $g \geq 2$. For $g \geq 3$ the singular locus is precisely the locus of PPAVs with automorphism group greater than $\pm \mathrm{id}$, as proven in Oort, Frans: "Singularities of coarse moduli schemes". For $g=2$ there is IIRC a unique singular point which is in $M\_2$ (the ope... | 7 | https://mathoverflow.net/users/1310 | 128718 | 71,626 |
https://mathoverflow.net/questions/128666 | 1 | Let $\Omega$ be a measurable set having finite Lebesgue measure. Let $p\geq 1$ and $u\in
L^p(\Omega)$. Is it true that the minimum value of the real function
$$
c\in
\mathbb{R}^n\mapsto\int\_\Omega |u-c|^p \mathrm{d}\mu
$$
is achieved when $c$ is the average of $u$? I'm interested in the case $p\neq 2$. Indeed for $p=2... | https://mathoverflow.net/users/33410 | variational characterization of the average of an $L^p$ function | Let us try an easy counter-example:
$\Omega=[0,1]^2, \ u(x)=x\_1x\_2\ $ so that $\ u\_\Omega=\frac{1}{4}$ and $p=4$. Then the integral $\int\_\Omega |x-c|^4dx$ is a non-negative polynomial in $c$ and has a (local) minimum whenever its derivative (in $c$) vanishes. Thus the local minima of the integral can be found w... | 1 | https://mathoverflow.net/users/881 | 128720 | 71,628 |
https://mathoverflow.net/questions/128716 | 8 | During courses on geometry it is sometimes necessary to draw a triangle on the blackboard that can easily be recognized as a *general triangle*. It must not be rectangular and must not have two or more equal angles. Further all angles should be less than $\pi/2$. Has anybody optimized this old problem of geometry-teach... | https://mathoverflow.net/users/nan | What is the best *general triangle*? | The book by "Humor in der Mathematik" by Friedrich Wille (from the 1970s or 1980s) contains the tongue-in-cheek theorem "Up to similarity, there is a unique general triangle". (Google book search for "Friedrich Wille" "Allgemeines Dreieck")
"General" is defined as "all angles must differ from each other, and from 90 ... | 24 | https://mathoverflow.net/users/14915 | 128726 | 71,633 |
https://mathoverflow.net/questions/128738 | 2 | Let $G$ be either a finite group or a Lie group. Let $r\_1$ and $r\_2$ be two linear representations of $G$. $r\_1$ is an ordinary representation but $r\_2$ may be projective. Is there a criterion to decide the following question: given $G$, $r\_1$, $r\_2$, does there exist an ordinary linear representation $R$ such th... | https://mathoverflow.net/users/33431 | Inequivalence of group representations preserved under tensor product? | If $G$ is a compact connected Lie group, then the ring of characters is an integral domain (it is the ring of Weyl group invariants in the co-ordinate ring of a maximal torus). Let $r\_1,r\_2, R, T\_1, T\_2$ be as in the question. Assume that $r\_1,r\_2$ are ordinary irreducible representations
If $t\_1,t\_2$ are the... | 3 | https://mathoverflow.net/users/23291 | 128745 | 71,638 |
https://mathoverflow.net/questions/128743 | 13 | Let $D$ denote a [divergent series](https://en.wikipedia.org/wiki/Divergent_series) and let $C$ denote a [convergent series](https://en.wikipedia.org/wiki/Convergent_series).
Furthermore, let $s : $ { Series } $\to$ $\mathbb{C}$ be a regular, linear divergent series operator, which is either one of these operators: ... | https://mathoverflow.net/users/93724 | Is there an algebra for divergent series summation operators? | I'll try to offer as much knowledge as I can on this topic, which might not be that much.
Firstly your linear form $s$ is defined on a strict linear subspace $L$ of the vector spaces of all formal series. Secondly you seem to mix formal numeric series and formal power series, which are distinct rings.
1) 2) By def... | 8 | https://mathoverflow.net/users/24309 | 128752 | 71,640 |
https://mathoverflow.net/questions/128746 | 5 | The *divisor bound* asserts that for a large integer $n\in \mathbb{Z}$, the number of divisors of $n$ is at most $n^{o(1)}$ as $n\rightarrow\infty$. See [here](http://terrytao.wordpress.com/2008/09/23/the-divisor-bound/) for a discussion of proofs using elementary analysis.
This [MO topic](https://mathoverflow.net/q... | https://mathoverflow.net/users/7831 | The divisor bound for binary quadratic forms | Let $r\_Q^\*(n)$ be the number of primitive representations of $n$ by $Q$. Let $\mathcal{C}$ be
a set of representatives of the classes of binary quadratic forms of discriminant $\Delta$. We can assume that $Q\in\mathcal{C}$. It is known that $\sum\_{Q'\in\mathcal{C}}r\_{Q'}^\*(n)$ equals the number of residue classes... | 6 | https://mathoverflow.net/users/11919 | 128757 | 71,641 |
https://mathoverflow.net/questions/128753 | 0 | Hey,
Is there a local existence/uniqueness theory for a system of equations of the form
$$\sum\_{i=1}^n a\_{ij}\dot u\_i(t) + b\_{ij}u\_i(t) = f\_j(t)\qquad\text{for $j=1,...,n$}$$
where the $f\_j$ are in $L^2(0,T)$, the $b\_{ij}$ are in $L^\infty(0,T)$ and the $a\_{ij}$ are in some Lebesgue space (not necessarily co... | https://mathoverflow.net/users/33101 | Is there existence and uniqueness theory of this system of ODE? | If the matrix $A(t)=(a\_{ij}(t))$ is invertible, e.g. if there exists $B(t)\in L^\infty$, with
$B(t) A(t)=Id$, then you get a linear non-characteristic system of type
$$
\dot u +C(t) u=g(t),\quad C\in L^\infty.
$$
However without invertibility, you may be in deep trouble, even in the scalar case: consider
for $\nu>0$ t... | 0 | https://mathoverflow.net/users/21907 | 128762 | 71,645 |
https://mathoverflow.net/questions/128638 | 1 | Gallarati studied contact of surfaces in $\mathbb{P}^3$, that is surfaces $V,W \subset \mathbb{P}^3$ such that $V.W = qD$ with $q$ an integer that is at least 2 and $D$ some curve.
I would like to read his results, contained in the 1960 paper "Ricerche sul contatto di superficie algebriche lungo curve". However, this... | https://mathoverflow.net/users/29657 | Modern (english) version of 1960 Italian paper by Gallarati? | I think Gallarati's construction is explained in modern terms in this paper: <http://math.mit.edu/~ssam/papers/Catanese.pdf>.
(Disclaimer: I've never looked at Gallarati's paper, so I'm not sure whether it has all been reworked in the above reference or not).
| 3 | https://mathoverflow.net/users/10610 | 128764 | 71,646 |
https://mathoverflow.net/questions/128768 | 5 | Let $G$ be a topological group. In some cases, e.g. when $G$ is discrete or when the spaces $G^n$ are locally contractible and the coefficients are discrete, the cohomology of the classifying space $BG$ is the group cohomology of $G$. So, for simplicity, let us assume that $G$ is discrete.
My question: is there a nic... | https://mathoverflow.net/users/18116 | Truncation of BG? | There are several functorial models for $BG$, see for example [Adem, Milgram: Cohomology of Finite Groups, Chapter II] where
$$BG = \coprod\_{i=0}^\infty \sigma^i \times G^i/(\text{relations})$$
with $\sigma^i=\lbrace (x\_1,...,x\_i)\mid 0\le x\_1 \le \cdots \le x\_i \le 1\rbrace$ the standard $i$-simplex.
Now assu... | 7 | https://mathoverflow.net/users/10194 | 128774 | 71,651 |
https://mathoverflow.net/questions/128773 | 12 | An exact (small) category $P$ is an environment in which we make sense of the "put-together"-edness of objects via (short) exact sequences. It seems like the K-theory of an exact category encodes the high order relations of how objects fit together, but I can't see how the $Q$-construction is the natural medium for thi... | https://mathoverflow.net/users/19313 | What is the Q-construction, metaphysically? | There's an interesting motivation in the paper by G. Segal:
*K-homology theory and algebraic K-theory.* K-theory and operator algebras (Proc. Conf., Univ. Georgia, Athens, Ga., 1975), pp. 113–127. Lecture Notes in Math., Vol. 575, Springer, Berlin, 1977
It is well-known that the space of **Fredholm operators** giv... | 8 | https://mathoverflow.net/users/8032 | 128776 | 71,652 |
https://mathoverflow.net/questions/128775 | 6 | I am interested in the complexity of convex functions, specifically the "doubling dimension" of the class of convex functions defined on a compact subset of Euclidean space, when compared using the $L^\infty$ metric.
Formally, let $F$ be the class of convex functions mapping $[0,1]^n$ to $[0,1]$. For $f,g \in F$, de... | https://mathoverflow.net/users/33442 | What is the doubling dimension of convex functions? | No the space is not doubling.
Take a strongly convex function, say $f(x)=x^2$.
It is sufficient to show that there is $N(\varepsilon)$ which goes to $\infty$ as $\varepsilon\to0$, such that $\varepsilon$-neighborhood of $f$ contains $N(\varepsilon)$ points on distance $>\varepsilon$ from each other.
To see this tak... | 4 | https://mathoverflow.net/users/1441 | 128782 | 71,656 |
https://mathoverflow.net/questions/128780 | 11 | I am wondering if there are any rigorous results telling that some dynamical system hits infinitely many primes (except for the case when orbits are just arithmetic progressions). To make it specific, is there a polynomial $f(x)$ such that its iterations $f^n(x)$ are prime for infinitely many $n$ given that the integer... | https://mathoverflow.net/users/25905 | Integer dynamics hitting infinitely many primes | I'm fairly certain that nothing along these lines is known.
Until the late 1990's there were no known `elementary' polynomial sparse sequences that contained infinitely many primes(\*). In 1998 Friedlander and Iwaniec proved that the sequence of integers of the form $x^2 +y^4$ contains infinity many primes. The numbe... | 14 | https://mathoverflow.net/users/630 | 128783 | 71,657 |
https://mathoverflow.net/questions/128786 | 30 | Inscribe an $n$-ball in an $n$-dimensional hypercube of side equal to 1, and let $n \rightarrow \infty$. The hypercube will always have volume 1, while it is a fun folk fact (FFF) that the volume of the ball goes to 0.
I first learnt of this in relation to Gromov. In the story I heard, he used to ask incoming student... | https://mathoverflow.net/users/13923 | History of the high-dimensional volume paradox | Brian Hayes wrote a column on the volume of the $n$-sphere for *American Scientist* a couple of years ago, available [online here.](https://www.americanscientist.org/article/an-adventure-in-the-nth-dimension) It includes a bit of history, with bibliography, toward the end, which might be of help here.
**Added 4/26/13... | 23 | https://mathoverflow.net/users/15837 | 128789 | 71,660 |
https://mathoverflow.net/questions/128788 | 3 | Suppose $G$ is a linear Lie group (i.e. $G$ admits a finite dimensional faithful representation) and $G$ has finitely many connected components. Let $G\_0$ be the identity component of $G$. If $N$ is a normal subgroup of $G\_0$, is $N$ necessarily normal in $G$?
| https://mathoverflow.net/users/27253 | Normal subgroup of the identity component of a linear Lie group is normal in the whole group? | No. Take $H$ to be a connected group, and $G=(\prod \_{i=1}^n H)\rtimes S\_n$. Where $S\_n$ acts by permuting the factors. Then $G^0=\prod \_{i-1}^n H$, and each of the factors $H$ is a normal subgroup in $G^0$ which is not normal in $G$.
| 10 | https://mathoverflow.net/users/23291 | 128792 | 71,662 |
https://mathoverflow.net/questions/111794 | 10 | Consider the classical Nim game with total number of stones being odd. Then the first players wins, of course, what follows from the general description of winning positions. But is there some shorter (independent of full theory) explanation of this fact, maybe with implicit strategy or whatever?
| https://mathoverflow.net/users/4312 | Nim game for odd number of stones | The following strategy-stealing argument is similar to the one in my answer to ["An unfair game involving an odd number of pieces of chocolate."](https://mathoverflow.net/questions/127116/an-unfair-game-involving-an-odd-number-of-pieces-of-chocolate)
Inductively assume that all positions with a smaller odd number of ... | 6 | https://mathoverflow.net/users/2954 | 128795 | 71,665 |
https://mathoverflow.net/questions/128799 | -1 | Any grammar for the language
$$L =a^p,\text{ $p$ is prime and }p\in \mathbb{N}?$$
Is such a grammar related to any question of number theory like RH or the conjecture of twin primes?
| https://mathoverflow.net/users/14024 | Any grammar for the language $L =a^p$, $p$ is prime number of $\mathbb{N}$ | $L$ is not context free, so has no context-free grammar describing it, but it is decidable, so there is an unrestricted grammar for it (there should also be a context-sensitive grammar, but I haven't thought too hard about that).
I can't see why this would have any bearing on number theoretic questions like the Riema... | 4 | https://mathoverflow.net/users/24952 | 128801 | 71,667 |
https://mathoverflow.net/questions/128772 | 4 | For positive integers $m$ and $n$, consider a regular polytope in ${\mathbb R}^{m+n+mn}$ with $2^{m+n}$ vertices, corresponding to each $\sigma \in \{-1,1\}^{m+n}$ as follows. The first $m+n$ coordinates are $\sigma\_i$, and the last $mn$ are the products $\sigma\_i \sigma\_j$ for $i = 1 \ldots m$, $j = m+1 \ldots m+n$... | https://mathoverflow.net/users/13650 | A regular polytope | Your polytope is the marginal polytope of the complete bipartite graph $K\_{m,n}$ and almost a cut
polytope. Let $G = (V,E)$ be an undirected graph. A cut is a bipartition of the vertex set into two sets. For each cut, an edge is either cut (its vertices are not in the same block of the partition), or not cut (its vert... | 8 | https://mathoverflow.net/users/5495 | 128809 | 71,669 |
https://mathoverflow.net/questions/128614 | 5 | Let $F$ be a finite connected set in a graph (soon to be the Cayley graph of a group) and $\mathrm{Ex}\_x^F$ be the function on the vertices in $F^c$ which are neighbour to vertices in $F$ defined as follow $\mathrm{Ex}\_x^F(y)$ is the probability that the first time a random walker starting at $x$ exits $F$ is through... | https://mathoverflow.net/users/18974 | Probabilities of a random walk exiting a set | Say that a graph $G$ has the Liouville property if all bounded (discrete-)harmonic functions on it are constant.
* If it is possible to couple random walks on $G$ started from any two starting points in such a way that they almost surely coincide after some (random) time, then the graph is Liouville. The reason for t... | 7 | https://mathoverflow.net/users/9430 | 128810 | 71,670 |
https://mathoverflow.net/questions/128739 | 3 | Let $f$ be a modular form on $\Gamma\_0(4)$ of weight $k\in\tfrac 12\mathbb{Z}$ with trivial Nebentypus.
Is it true that if you twist $f$ by $\tfrac 12$, i.e. look at the function $g$ with $g(\tau)=f(\tau+\tfrac 12)$, this is again a modular form of the same weight $k$ on $\Gamma\_0(4)$, but this time with Nebentypus... | https://mathoverflow.net/users/33434 | Modular Forms on $\Gamma_0(4)$ with Nebentypus | **EDIT** : In what follows the weight $k$ is assumed to be an integer.
The matrix $\begin{pmatrix} 1 & 1/2 \\ 0 & 1 \end{pmatrix}$ normalizes the group $\Gamma\_0(4)$, so in your case $g$ is still a modular form on $\Gamma\_0(4)$ with trivial Nebentypus.
More generally if you start with $f$ on $\Gamma\_0(N)$ and t... | 4 | https://mathoverflow.net/users/6506 | 128815 | 71,672 |
https://mathoverflow.net/questions/128763 | 7 | Suppose $G$, $H$ are finite groups and $M$ is a module over $G\times H$.
**Question:** Is the exponent of $H^i(G\times H,M)$ a divisor of $lcm(|G|,|H|)$ for $i> 0$ ?
The Künneth formula answers the question affirmatively if $M$ is trivial or, more generally, if one of the groups acts trivially on $M$. But I don't... | https://mathoverflow.net/users/27895 | Exponent of the cohomology of a product of groups | For any finite group $\Gamma$, if $I$ is the augmentation ideal of ${\mathbb Z}\Gamma$, then $H^1(\Gamma,I)\cong {\mathbb Z}/|\Gamma|{\mathbb Z}$, which gives a counterexample if you take $\Gamma = G\times H$ for any $G$ and $H$ whose orders are not coprime.
| 5 | https://mathoverflow.net/users/22989 | 128820 | 71,674 |
https://mathoverflow.net/questions/128803 | 4 | Say that a triple of real numbers $(a,b,c)$ is a realizable triple if there are matrices $A,B\in SL\_2(\mathbb{R})$ such that $tr (A)=a$, $tr (B)=b$, and $tr (AB)=c$. Question: what is the shape of the non-realizable set?
This is surely known, but I couldn't find an answer by myself, nor a reference on the www. It's ... | https://mathoverflow.net/users/20108 | tracial triples | Let $x, y, z$ be traces of $A, B, AB$ respectively. Define
$$
k(x,y,z)= x^2+y^2 + z^2 -xyz -2.
$$
Then a triple of real traces $(x, y, z)$ is realizable in $SL(2,R)$, unless it is realizable in $SU(2)$, the latter happens if and only if $x, y, z\in [-2,2]$ and $k(x,y,z)\le 2$. See Goldman's paper "Topological component... | 3 | https://mathoverflow.net/users/21684 | 128833 | 71,680 |
https://mathoverflow.net/questions/128701 | 2 | I have a Hilbert space of quantum density matrices written in the Glauber-Sudarshan P representation - ie. we have coherent states $|\alpha \rangle$ and we write density matrices as
$$ \rho = \int d^2\alpha \ P(\alpha) |\alpha\rangle \langle\alpha|.$$
The states $|\alpha\rangle$ form an overcomplete set and are not a... | https://mathoverflow.net/users/14976 | Existence of a projection operator onto a classical set of density matrices |
>
> Can we construct a linear projection operator P onto C?
>
>
>
No. The range of any linear operator will be a linear subspace.
>
> If not, is there a nonlinear projection operator and if so how would one construct it?
>
>
>
Yes, if $K$ is a closed convex subset of a Hilbert space $H$ there is a stand... | 1 | https://mathoverflow.net/users/23141 | 128839 | 71,684 |
https://mathoverflow.net/questions/128812 | 3 | Let $X$ be a topological space. For every point $x \in X$ let $R\_x$ be a local ring. Under what (necessary / sufficient / necessary and sufficient) conditions is there a sheaf ${\cal O}\_X$ such that $(X,\mathcal{O}\_X)$ is a locally ringed space with $\mathcal{O}\_{X,x} \cong R\_x$ for all $x \in X$?
If $X$ is disc... | https://mathoverflow.net/users/2841 | Families of local rings coming from a locally ringed space | Here is another necessary condition for the existence of such a sheaf:
>
> For all $x \in X$ and for all $f \in R\_x$, there exists a neighborhood $U$ of $x$ such that the $f$ is contained in the image of the canonical homomorphism $$\varprojlim\_{y \in U} R\_y \to R\_x.$$
>
>
>
For suppose that such a sheaf $... | 2 | https://mathoverflow.net/users/778 | 128849 | 71,691 |
https://mathoverflow.net/questions/128853 | 15 | Hi,
Is there a simple example of an (affine) algebraic variety $X$ over $\mathbb C$ where
the $H^\*\_{dR}(X/\mathbb C) = H^\*(\Omega^\bullet\_{A/\mathbb C})$ differs from the singular cohomology $H^\*\_{sing}(X(\mathbb C)^{an},\mathbb C)$?
Such an example has to be singular (by a theorem of Grothendieck), but I am ... | https://mathoverflow.net/users/36285 | algebraic de Rham cohomology of singular varieties | A likely candidate for this would be a non-Du Bois singularity.
Du Bois, following Deligne's ideas, constructed a filtered complex of sheaves with coherent cohomology sheaves that gives a resolution of the constant sheaf $\mathbb C$ for any reduced finite type scheme over $\mathbb C$. This complex agrees with the de... | 17 | https://mathoverflow.net/users/10076 | 128855 | 71,693 |
https://mathoverflow.net/questions/128825 | 1 | Let $G$ be a compact Lie group. What can we say about subgroups of $G$ isomorphic to $Z\_n$? For example, $G$ is $SU(2)$, or $SO(3)$ and $n$ is $2$ or $3$ and so on.
A naive idea is that if $x\in G$ is the image of the generator, then such subgroups are in one to one correspondence with the solution of equation $x^3=... | https://mathoverflow.net/users/27205 | finite abelian subgroup of a compact lie group | It's clear that you might start by looking inside a maximal torus of the given compact (say connected) Lie group. But given the long history of such problems, naive methods are unlikely to get very far with this type of question. Two useful sources are (1) a Bourbaki seminar report by Serre
[here](http://www.numdam.org... | 2 | https://mathoverflow.net/users/4231 | 128857 | 71,694 |
https://mathoverflow.net/questions/128864 | 5 | I just filled a gap in my education by learning about the Cayley-Menger theorem, and the Cayley-Menger determinant:
If $P\_0, \dots, P\_n$ are $n+1$ point in $\mathbb{R}^n$, and $d\_{i,j} = |P\_i - P\_j|$ is the Euclidean distance from $P\_i$ to $P\_j$, we first form the $n+1 \times n+1$ matrix of squares of the dist... | https://mathoverflow.net/users/2784 | The Cayley Menger Theorem and integer matrices with row sum 2 | Let us first consider an $n\times n$ matrix with zero diagonal and $n(n-1)$ indeterminates.
Every term (monomial) of the determinant corresponds to a permutation matrix $P$ with zero diagonal (i.e., an integer nonnegative matrix with trace 0 and row and column sums 1).
Let us now consider a *symmetric* $n\times n$ ma... | 9 | https://mathoverflow.net/users/30800 | 128867 | 71,697 |
https://mathoverflow.net/questions/128883 | 8 | This is definitely not a research level question. I believe this is "common sense" among homotopists, however after "extensive" googling for 2 days I could not find a proof of it online or in standard textbooks (Hatcher, Milnor-Stasheff, Husemoller). I asked on [mathstackexchange](https://math.stackexchange.com/questio... | https://mathoverflow.net/users/18850 | Why $\Omega X$ and $BG$ are adjoint functors? | Are you willing to accept a proof of the adjunction in the homotopy category?
I think the more natural way to do that is to prove first that $\Omega B G \simeq G$ (homotopy equivalence) and $B \Omega X \simeq X$ (note that $X$ needs to be connected for this second statement to be true, so you have to restrict your ca... | 7 | https://mathoverflow.net/users/9481 | 128890 | 71,707 |
https://mathoverflow.net/questions/128866 | 10 | Dear all, I am teaching a course in Riemannian geometry, and I would like to prove some comparison theorems in the next lessons, building on the well-known theory of Jacobi fields, and of Rauch comparison Theorem for Jacobi fields. I would like to stress the fact that several arguments can work in the case of geodesic ... | https://mathoverflow.net/users/6206 | Alexandrov angles in Riemannian manifolds | Your equality is two inequalities.
To show the upper bound you can use the triangle inequality --- come closer to $p$ along the geodesic and apply the local estimates.
(This is the "first variation inequality" it holds in any metric space where angles defined.)
The lower bound follows since the injectivity radius a... | 8 | https://mathoverflow.net/users/1441 | 128896 | 71,710 |
https://mathoverflow.net/questions/128888 | 3 | Consider the action of $\mathbb{Z}/p^\times$ the units of $\mathbb{Z}/p$ on the classifying space $B\mathbb{Z}/p$ by left multiplication on the $n$-simplices
$$
\alpha\cdot (g\_1,...,g\_n)=(\alpha g\_1,...,\alpha g\_n).
$$
Here $B\mathbb{Z}/p$ denotes the usual model as the realization of the simplicial set whose $n$-... | https://mathoverflow.net/users/19409 | Quotients of classifying spaces | $\newcommand{\Ext}{\operatorname{Ext}}$$\newcommand{\To}{\longrightarrow}$$\newcommand{\dash}{\text{-}}$$\newcommand{\sSet}{\mathrm{sSet}}$$\newcommand{\ZZ}{\mathbb{Z}}$For convenience, I will denote by $G$ the group $(\ZZ/p)^\times$, and by $X$ the simplicial set $B(\ZZ/p)$ for $p$ a prime. I will prove later in this ... | 6 | https://mathoverflow.net/users/21095 | 128905 | 71,714 |
https://mathoverflow.net/questions/128907 | 6 | According to the main theorem of CM, for every abelian variety $A$ associated to
a CM field $K$, one obtains a certain unramified abelian extension of the reflex field $K^\times$ given by the field of moduli of $A$. Similarly, the fields of moduli of ideal section points generate certain ramified extensions. At many pl... | https://mathoverflow.net/users/33479 | CM fields and Hilberts 12th problem | Beginning with the work of Taniyama, Shimura, and Weil in the late fifties,
the theory of elliptic curves and elliptic modular curves has been
successfully generalized to higher dimensions. In this theory, an elliptic
curve with complex multiplication by an imaginary quadratic field is replaced
by an abelian variety w... | 6 | https://mathoverflow.net/users/33481 | 128915 | 71,717 |
https://mathoverflow.net/questions/128918 | 1 | From Wikipedia article on the Feit-Thompson Theorem proving a conjecture of Burnside that groups of odd order are solvable: "The attack on Burnside's conjecture was started by Michio Suzuki (1957), who studied CA groups; these are groups such that the Centralizer of every non-trivial element is Abelian. In a pioneering... | https://mathoverflow.net/users/25762 | Why didn't finite group theorists consider groups where all centralizers of non-identity elements are solvable? | Groups in which the centralizer of every involution is solvable were classified by D. Gorenstein and various co-authors. Also J. G. Thompson classified finite groups such that the normalizer of every non-identity solvable subgroup is solvable. Results of this kind were in some ways more general than the problem you ask... | 10 | https://mathoverflow.net/users/14450 | 128919 | 71,718 |
https://mathoverflow.net/questions/128901 | 1 | This is a follow up question to [this one](https://mathoverflow.net/questions/120421/).
If $X$ is a metric space, denote by $C\_u(X)$ the $C^\ast$-algebra of all bounded, uniformly continuous functions on $X$ (with the sup-norm).
>
> Do we have $C\_u(X\_1 \times X\_2) = C\_u(X\_1) \hat{\otimes} C\_u(X\_2)$?
>
> ... | https://mathoverflow.net/users/13356 | Tensor product of C*-algebras of bounded, uniformly continuous functions on metric spaces | The spectrum of the $C^\ast$-algebra of bounded, uniformly continuous functions on a uniform space is the Samuel compactification. So your query can be restated in the form:
is the Samuel compactification of a product naturally identifiable with the product of the Samuel compactifications of the individual spaces. This... | 2 | https://mathoverflow.net/users/26013 | 128924 | 71,719 |
https://mathoverflow.net/questions/128910 | 0 | We know that for each elliptic curve over rationals, we can define the Dirichlet series of the Hasse–Weil $L$-function, i.e., the function associated with an elliptic curve over rationals.
Then my **question** is:
Consider a motivic $L$-functions $f$, can we find a set of elliptic curves over rationals associated wit... | https://mathoverflow.net/users/25947 | Can we find a set of elliptic curves over rationals associated with $f$?. | This has already had some votes to close, but I'll see if I can answer it anyway...
The answer is "no". There are lots of motivic L-functions that are not elliptic curve L-functions, just because there are lots of motives that are not $H^1$ of an elliptic curve! For instance, the L-function attached to a modular form... | 4 | https://mathoverflow.net/users/2481 | 128926 | 71,721 |
https://mathoverflow.net/questions/128922 | 7 | Hello
I was wondering if anybody can direct me to a paper or a book regarding the volume of $Gr(2,4) $ or generic complex Grassmanian manifolds of order $k$. My own heuristic method seems not to work! It is based on the adaption of the same procedure one has to follow for finding the volume of complex projective spac... | https://mathoverflow.net/users/33483 | Volume of Gr(2,4) | Check section 9.1.2 of [these notes](http://www3.nd.edu/~lnicolae/Lectures.pdf) There I compute the volumes of *real* Grassmannians. A similar computation works in the complex case.
**Update** Using the description $\mathrm{Gr}\;(k, N)\cong U(N/U(k)\times U(N-k)$ and a bi-invariant metric on $U(N)$, this induces bi-i... | 10 | https://mathoverflow.net/users/20302 | 128928 | 71,723 |
https://mathoverflow.net/questions/128891 | 5 | I had completed a paper describing the $q$-Catalan numbers, which is the $q$-analog of the Catalan numbers.
The $n$-th Catalan numbers can be represented by:
$$C\_n=\frac{1}{n+1}{2n \choose n}$$
and with the recurrence relation:
$$C\_{n+1}=\sum^n\_{i=0}C\_i C\_{n-i}\ \ \ \ \ \forall n\geq 0$$
Now, for the $q$... | https://mathoverflow.net/users/33472 | What does the $q$-Catalan Numbers count? | As Vasu commented already: there is not "the" q-analogue of the Catalan numbers. And indeed, you're mixing two different here.
* Your first q-Catalan numbers defined by the $q$-binomials is MacMahon's q-Catalan numbers which is (and I don't actually know many others) the **major index generating function** on Dyck pa... | 10 | https://mathoverflow.net/users/21291 | 128932 | 71,724 |
https://mathoverflow.net/questions/128874 | 5 | We know in differential geometry, given a $C^k$ manifold for $k>1$, the tangent space at a point in this manifold is parametrized by curves passing through this point modulo certain equivalence relation. The tangent space is given by the velocity vectors to these curves at this point.
Intuitively, I would think this ... | https://mathoverflow.net/users/32151 | Tangent space in Algebraic geometry and Differential geometry | If p is a nonsingular point, then we can define a tangent vector as an equivalence class of (nonsingular) curves, just as in the differentiable case. In fact, being nonsingular is equivalent to every tangent vector being tangent to a curve.
In this case, the intersection of all curves in the equivalence class is a zero... | 13 | https://mathoverflow.net/users/4164 | 128933 | 71,725 |
https://mathoverflow.net/questions/128903 | 27 | The edit or [Levenshtein](http://en.wikipedia.org/wiki/Levenshtein_distance) distance between two strings is the minimum number of single symbol insertions, deletions and substitutions to transform one string into another. For example $$\operatorname{E}(01010,00100)=2.$$
Let $E\_n$ be a random variable giving the edi... | https://mathoverflow.net/users/nan | Expected edit distance | The only rigorous bound I am aware of is due to Gonzalo Navarro\*
$$c\geq 1-{\rm e}/\sqrt{\sigma},$$
for an alphabet of $\sigma$ characters. Obviously, for the binary string ($\sigma=2$) this bound is ineffective. Navarro also mentions a large-$\sigma$ conjecture $c=1-1/\sqrt{\sigma}$, which for the binary string w... | 12 | https://mathoverflow.net/users/11260 | 128941 | 71,727 |
https://mathoverflow.net/questions/128920 | 3 | I am working on the discrete theory of compressible fluids dynamics, i.e., numerically solving and simulating the compressible fluids , we are interested in the way using discrete exterior calculus, my question is: Is there any work on the discrete theory (especially using discrete exterior calculus) of compressible fl... | https://mathoverflow.net/users/2391 | The discrete theory of compressible fluids dynamics | There's unpublished work by Gay-Balmaz and Pavlov, *Variational Discretization of Compressible Fluids*, described [here](http://d-pavlov.com/resources/Research_Statement.pdf), with an instructive summary of the difficulties involved in extending the discrete theory from incompressible to compressible fluids.
| 1 | https://mathoverflow.net/users/11260 | 128943 | 71,729 |
https://mathoverflow.net/questions/128934 | 8 | **Question.** Given a positive-definite $n \times n$ matrix $A = (a\_{ij})$ with eigenvalues
$$
\lambda\_1 \leq \cdots \leq \lambda\_n ,
$$
is there a sharp upper bound for the product $\lambda\_2 \cdots \lambda\_n$ in terms of the quantity
$$
\|A\|\_\infty := \max\_{1 \leq i, j \leq n} |a\_{ij}| ?
$$
A classic ineq... | https://mathoverflow.net/users/21123 | A spectral inequality for positive-definite matrices | $\newcommand{\trace}{\operatorname{trace}}$
The result below mentions a reasonably improved inequality.
Let $m = \frac{\trace(A)}{n}$, and $s^2= \frac{\trace(A^2)}{n}-m^2$. Then, [Wolkowicz and Styan](http://orion.math.uwaterloo.ca/~hwolkowi/henry/reports/bndseigs80.pdf) (*Linear Algebra and its Applications*, 29:47... | 11 | https://mathoverflow.net/users/8430 | 128953 | 71,731 |
https://mathoverflow.net/questions/128946 | 5 |
>
> Let Bor($X$) = class of all borel subsets of $X$. Cohen algebra is defined as Bor(X) modulo the ideal of meager sets.
>
>
>
The Cohen algebra has a combinatorial : it is the unique atomless complete Boolean algebra with a countable dense subset, (i.e it is a completion of a countable dense Boolean subalgebra... | https://mathoverflow.net/users/33492 | Cohen algebra (generalization) | Regarding question $1$, it seems that you want to know whether
you've got the unique complete c.c.c. Boolean algebra with density
$\kappa$. The answer is no.
On the one hand, the forcing notion $\text{Add}(\omega,\omega\_1)$
to add $\omega\_1$ many Cohen reals is c.c.c. and has density
$\omega\_1$. This is another wa... | 7 | https://mathoverflow.net/users/1946 | 128956 | 71,734 |
https://mathoverflow.net/questions/128937 | 0 | Let $X\_n$ be a random variable distributed on $A\_n:=\{1, \ldots, n\}$ and $g\_n\colon A\_n \to A\_n$ such that $\Pr\big(X\_n \neq g\_n(X\_n)\big) \to 0$. Putting $Y\_n=g(X\_n)$ then by Fano's inequality $$\frac{H(X\_n\mid Y\_n)}{\log n} \to 0,$$ which can be written $$\frac{H(X\_n\mid Y\_n)}{H(X\_n)} \to 0 \qquad (\a... | https://mathoverflow.net/users/21339 | order of convergence of the conditional entropy | Let $n=k+k^k$ and let $X$ take each element $1\le j\le k$ with probability $1/k-1/k^2$. Let the remaining $k^k$ elements have probability $1/k^{k+1}$.
Let $g(i)=\min(k+1,i)$.
We now have $\mathbb P(X\ne Y)=\mathbb P(X>k)=k^k/k^{k+1}=1/k$, which converges to 0 as required.
We have
$$
\begin{split}
H(X)&=-k(1/k-1/... | 2 | https://mathoverflow.net/users/11054 | 128967 | 71,740 |
https://mathoverflow.net/questions/101707 | 4 | 2-cocycles of a given Hopf algebras $H$ no longer form a group, but a groupoid between different Doi twists of the Hopf algebra $H,L$. The subgroup of "lazy" 2-cocycles precisely preserve the underlying Hopf algebra. They are usually presented as $H$-$L$- resp. $H$-$H$-Bigalois Objects.
Now we know from Schauenburg, ... | https://mathoverflow.net/users/22709 | Bigalois Groupoid Of Drinfel'd Group Double | For a finite dimensional Hopf algebra H, $Bigal(D(H))=Bigal(H\otimes H^\*)$, becuase $^{D(H)}\mathcal{M}\cong\_{\otimes}\ ^{H\otimes H^\*}\mathcal{M}$.
If $H=kG$ (Gfinite group $G$), then you need first the description of all Bigalois objects of $\mathcal{O}\_k(G)$. A description using other name was done by Davydov... | 2 | https://mathoverflow.net/users/6517 | 128972 | 71,743 |
https://mathoverflow.net/questions/128965 | 2 | Let $X$ be an affine algebraic variety over $\mathbb{C}$, and let $G$ be a semisimple complex linear algebraic group acting by variety automorphisms with finitely many orbits. The decomposition of $X$ into orbits is sometimes called a stratification of $X$. However, I have encountered many different definitions of "str... | https://mathoverflow.net/users/25358 | Thom-Gysin Sequences and Stratifications | I haven't read Kirwan's thesis but I think this is what you are after. If $U \subset V$ is an open subvariety (no conditions on the variety $V$) then there is a long exact sequence (Thom-Gysin)
$$ H^\bullet\_c(U) \to H^\bullet\_c(V) \to H^\bullet\_c(V \setminus U) \to H^{\bullet+1}\_c(U). $$
Let's now declare a stra... | 1 | https://mathoverflow.net/users/1310 | 128978 | 71,744 |
https://mathoverflow.net/questions/128961 | 9 | A [Toeplitz matrix](http://en.wikipedia.org/wiki/Toeplitz_matrix) or diagonal-constant matrix is a matrix in which each descending diagonal from left to right is constant.
>
> What is the probability that a random $n \times n$ binary Toeplitz matrix is
> invertible over $\mathbb{R}$ and what is the probability th... | https://mathoverflow.net/users/nan | Probability of random (0,1) Toeplitz matrix being invertible | See: E. Kaltofen and A. Lobo, [On rank properties of Toeplitz matrices over finite fields](http://www4.ncsu.edu/~kaltofen/bibliography/96/KaLo96_issac.pdf).
In Proc. 1996 Internat. Symp. Symbolic Algebraic Comput. (ISSAC'96)
The probability of a random Toeplitz matrix over a finite field of order $q$ being non-singu... | 7 | https://mathoverflow.net/users/630 | 128979 | 71,745 |
https://mathoverflow.net/questions/128791 | 12 | Nonstandard analysis is a useful tool which can be used to prove a number of results in analysis.
Question
--------
>
> Can it also be used to prove results in computable or constructive analysis?
>
>
> If so, what are some examples? (They don't need to be ground-breaking.)
>
>
>
Motivation
----------
Th... | https://mathoverflow.net/users/12978 | Can nonstandard analysis be used to prove results in constructive or computable analysis? | Nonstandard Analysis (NSA) can be used to prove results in computable/constructive analysis; The central notion is $\Omega$-invariance, defined as follows.
[As usual, the set $N$ consists of the standard/finite/natural numbers; The set ${^{\star}}N$ is an end-extension of $N$ and $\Omega={^{\star}}N\setminus N$ cons... | 14 | https://mathoverflow.net/users/33505 | 128991 | 71,752 |
https://mathoverflow.net/questions/128974 | 11 | The group mentioned in the title, $\langle x,y,z|xyzx^{-1}y^{-1}z^{-1}=1\rangle$, is in between the torus fundamental group $\langle x,y|xyx^{-1}y^{-1}=1\rangle$ and the two-holed torus fundamental group $\langle x,y,z,w|xyzx^{-1}y^{-1}z^{-1}w^{-1}=1\rangle$.
It is not the hexagonal presentation of the torus because... | https://mathoverflow.net/users/27933 | Why isn't $\langle x,y,z|xyzx^{-1}y^{-1}z^{-1}\rangle$ a hyperbolic surface group? | I think Lee's and Steve's comments pretty much answer this question. Let me try to summarize, and clear up a couple of misconceptions that seem to be lurking. For convenience, I'll denote your group by $G$.
**The map $G\to F\_2\times F\_2$.**
Actually, I don't think the map $G\to F\_2\times F\_2$ that you describe ... | 8 | https://mathoverflow.net/users/1463 | 129005 | 71,759 |
https://mathoverflow.net/questions/128831 | 14 | It is known (cf. Lurie's book [*Higher Topos Theory*](http://www.math.harvard.edu/~lurie/papers/highertopoi.pdf) for instance) that higher ($\infty$-) category, in particular topological higher ($\infty$-) groupoids are "better" defined as *weak Kan complexes*, aka *quasi-categories*. Let me recall that a simplicial sp... | https://mathoverflow.net/users/20005 | Infinity-categories vs Kan complexes | I suspect your confusion arises in part because homotopies of paths are continuous maps $I^2 \to X$, while 2-morphisms in $\pi\_{\lt \infty} X$ are continuous maps $\Delta^2 \to X$. That is, 2-morphisms are not strictly the same as homotopies of paths.
The dictionary between the two structures is not too bad:
1. A ... | 12 | https://mathoverflow.net/users/121 | 129009 | 71,761 |
https://mathoverflow.net/questions/129012 | 0 | Let's assume that $ f(x) = \sum\_{n=1}^\infty a\_n x^n $ has a radius of convergence $1$
and that
$ \lim\_{x\to 1^-} f(x) = +\infty. $
Does it imply that power series $ g(x) = \sum\_{n=1}^\infty (1-x) a\_n x^n $
is uniformely convergent
on $[0,1]$?
Thanks for any help, this one has killed me, I've been trying t... | https://mathoverflow.net/users/33509 | Uniform convergence of $g(x) = \sum_{n=1}^\infty (1-x) a_n x^n$ | I suspect that putting in a double pole at $x=1$ will show this implication to be false. In other words, take $f(x) = x/(1-x)^2 = \sum\_{n=1}^\infty nx^n$. Then $g(x) = (1-x)f(x)$ still satisfies $\lim\_{x\to1^-} g(x) = \infty$, which should rule out uniform convergence. (Don't know why you're starting your power serie... | 0 | https://mathoverflow.net/users/5091 | 129013 | 71,763 |
https://mathoverflow.net/questions/128993 | 2 | Hello, members.
I have a problem for the following problem
when I derive an optimization algorithm for stochastic singular systems
$$S(k+1)=A(k)S(k)A^{T}(k)+R(k)+F(k)S(k+1)F^{T}(k)$$
where $R(k)>=0$
So, how to calculate $S$, is there analytic solution or numerical solution to $S$?
This problem is different from the... | https://mathoverflow.net/users/25957 | On solution of a discrete-time equation | you'll want to solve this equation iteratively, considering $S(k)$ as known and $S(k+1)$ as unknown; for $F(k)$ invertible you then have a Sylvester equation, of the form $F^{-1}(k)S(k+1)-S(k+1)F^{T}(k)=C(k)$, which has a unique solution iff $F^{-1}(k)$ and $F(k)$ have no common eigenvalue. The [Wikipedia page](http://... | 1 | https://mathoverflow.net/users/11260 | 129019 | 71,764 |
https://mathoverflow.net/questions/129014 | 0 | Consider a minuscule representation (in a semisimple case). This gives a decomposition:
g=p\oplus n. Is n commutative?
(Sorry for a stupid question - I am not an expert and it's Sunday and I have no one to ask)
| https://mathoverflow.net/users/nan | Commutativity of nilpotents in minuscule case | $\mathfrak{n}$ is not well defined, it is better to speak about $\mathfrak{u}$, the unipotent radical of $\mathfrak{p}$. It is commutative in the COminuscule case.
| 1 | https://mathoverflow.net/users/4428 | 129020 | 71,765 |
https://mathoverflow.net/questions/129031 | 5 | $\newcommand{\CC}{\mathbb{C}}$
$\newcommand{\ZZ}{\mathbb{Z}}$
$\newcommand{\PP}{\mathbb{P}}$
$\newcommand{\QQ}{\mathbb{Q}}$
$\newcommand{\hH}{\mathcal{H}}$
$\newcommand{\eE}{\mathcal{E}}$
$\newcommand{\dD}{\mathcal{D}}$
$\newcommand{\aA}{\mathcal{A}}$
$\newcommand{\oO}{\mathcal{O}}$
$\newcommand{\Tate}{\text{Tate}}$
$\... | https://mathoverflow.net/users/15242 | equivalence between katz and classical modular forms | On $\omega\_{E/R}$: Yes. I don't remember if this is always free. But the definition means exactly what it says $E/R$ is an elliptic curve, and $\omega$ is a basis for $\omega\_{E/R}$. From this we can conclude that $\omega\_{E/R}$ has a basis. I'd also like to point out that we can always make $\omega\_{E/R}$ free by ... | 3 | https://mathoverflow.net/users/18060 | 129035 | 71,771 |
https://mathoverflow.net/questions/101739 | 5 | In Chriss and Ginzburg's book "Representation Theory and Complex Geometry" as well as the paper "Geometric Methods in Representation Theory of Hecke Algebras and Quantum Groups", the group algebra $\mathbb{C}[W]$ and the universal enveloping algebra $U(sl\_n)$ have been realized as the cohomology ring $H^\bullet(Z)$ fo... | https://mathoverflow.net/users/24965 | What kind of algebra has geometric realization as in "Geometric Methods in Representation Theory of Hecke Algebras and Quantum Groups" | To answer your last question, a geometric realization of $U(\mathfrak{g})$ and its (irreducible highest weight) representations is given for any symmetric Kac-Moody algebra by the work of Nakajima (see his 1998 paper in Duke Math Journal).
I doubt that a complete answer to your general question exists. However, in re... | 6 | https://mathoverflow.net/users/29738 | 129039 | 71,773 |
https://mathoverflow.net/questions/129037 | 4 | This question has some overlap with previous ones but doesn't seem to have a well-documented answer. I recall some literature (mostly involving Lie groups and hermitian symmetric pairs, etc.) which concerns *maximal* parabolic subalgebras of a simple Lie algebra $\mathfrak{g}$ over the field $\mathbb{C}$ or related par... | https://mathoverflow.net/users/4231 | Criterion for nilradical of a maximal parabolic subalgebra to be abelian? | Denote by $\mathfrak{l}$ the Levi factor of the parabolic, so that $\mathfrak{p} = \mathfrak{l} \oplus \mathfrak{n}$, and note that this is a splitting as $\mathfrak{l}$-modules. Also denote by $\mathfrak{n}\_-$ the nilradical of the opposite parabolic subalgebra; this is the dual of $\mathfrak{n}$ via the Killing form... | 6 | https://mathoverflow.net/users/703 | 129043 | 71,777 |
https://mathoverflow.net/questions/129015 | 3 | There is an abundant literature, and even here on MO no shortage of questions, on the question of the smallest prime primitivee root modulo $q$ (where $q$ is a prime, or more generally
an odd prime power, so that $(\mathbb Z/q \mathbb Z)^\ast$ is cyclic, and has generators, called *primitive root*). I wonder if there i... | https://mathoverflow.net/users/9317 | Least non primitive root | Trivially, any upper bound for the least prime quadratic residue modulo $p$ is also an upper bound for the least prime non-primitive root modulo $q$. I can't recall what's been proved about the latter problem assuming GRH (probably a power of $\log q$), but that will form a good conjectural upper bound.
As for a conj... | 3 | https://mathoverflow.net/users/5091 | 129047 | 71,780 |
https://mathoverflow.net/questions/128149 | 4 | Suppose I have a collection of polynomials with multiple variables (more polynomials than variables, say), and I'm given noisy versions the values of these polynomials at a certain unknown point. I would like to solve for this point in a stable manner. What is known about this problem?
Certainly, the notion of stabil... | https://mathoverflow.net/users/29873 | Stability in algebraic geometry | Clearly if $F$ is not injective, then as $||e||$ goes to $0$, $||\hat{x}-x||$ can approach a constant, or alternately $\hat{x}$ can be undefined.
But if $F$ is injective, and the Jacobian of $F$ is of full rank at $x$, then $||\hat{x}-x||=O(x)$. This is because $\hat{x}$ certainly converges to $x$ as $e$ goes to $0$... | 1 | https://mathoverflow.net/users/18060 | 129052 | 71,784 |
https://mathoverflow.net/questions/129075 | 14 | What can one say about the order of the torsion group of an elliptic curve defined over the compositum of all quadratic extensions of $\mathbb{Q}$ ?
| https://mathoverflow.net/users/30999 | Order of torsion group | I think the papers you should have a look at are those two by Fujita, great material and very well-written (in my humble opinion):
1) Y. Fujita, Torsion subgroups of elliptic curves with non-cyclic torsion over ${\mathbb Q}$ in elementary abelian 2-extensions of ${\mathbb Q}$, Acta Arith. 115 (2004) 29–45. MR2102804 ... | 8 | https://mathoverflow.net/users/24859 | 129079 | 71,787 |
https://mathoverflow.net/questions/129086 | 9 | I am trying to read Harish-Chandra's book on automorphic forms on Semisimple Lie groups, and he keeps referring to Borel's Paris lecture notes. Does anyone have an online version of these notes or know how I could get it? *Ensembles fondamentaux pour les groupes arithmétiques et formes automorphes*, Lectures at Institu... | https://mathoverflow.net/users/27791 | Borel's Paris Lectures | The Paris lectures, along with others he gave later, were spliced together into a publication: *Introduction aux groupes arithmetiques* (softcover, Hermann, Paris, 1969). As his nominal assistant at IAS in 1968-69, I tried to help with the splicing process but didn't manage to clean up all the inconsistent notation and... | 13 | https://mathoverflow.net/users/4231 | 129089 | 71,791 |
https://mathoverflow.net/questions/129083 | 3 | Let $a\_0,a\_1,\dots$ be the sequence satisfying
$$
\left(\sum\_{n=0}^\infty a\_n x^n\right)\left(\sum\_{n=0}^\infty \frac{x^n}{n+1}\right)=1.
$$
This means that $a\_0=1$ and $a\_{n+1}=-\sum\_{j=0}^n\frac{a\_j}{n+2-j}$.
One gets $a\_1=-\frac12$ and $a\_3=-\frac{13}{720}$.
The first and most important question is:
Is ... | https://mathoverflow.net/users/nan | Sign of coefficients | These numbers (with alternating signs) are called Bernoulli numbers of the second kind or Cauchy numbers. Two proofs that these numbers are negative can be found in <http://people.brandeis.edu/~gessel/homepage/slides/analysis-nec.pdf>.
Another reference is
[Merlini, Donatella; Sprugnoli, Renzo; Verri, M. Cecilia,... | 7 | https://mathoverflow.net/users/10744 | 129094 | 71,792 |
https://mathoverflow.net/questions/129096 | 7 | In its asymmetric version, the [Mahler conjecture](https://en.wikipedia.org/wiki/Mahler_conjecture) states that if $K \subset \mathbb{R^n}$ is a convex body containing the origin as an interior point and
$$
K^\* := \{y \in \mathbb{R}^n : \langle y, x \rangle \leq 1 \mbox{ for all } x \in K \}
$$
is its polar body, the... | https://mathoverflow.net/users/21123 | A question on the Mahler conjecture | No, it is not known that the minimum is unique, but it is believed to be. In fact, [this paper by Kim and Reisner](http://arxiv.org/abs/1001.0217) proves that the simplex is (modulo linear equivalence) a strict local minimum; thus the whole conjecture would follow from uniqueness of *local* minima.
| 5 | https://mathoverflow.net/users/1044 | 129105 | 71,796 |
https://mathoverflow.net/questions/129099 | 2 | I'm looking for a simple identity for the formula:
$$
\sum\_{n = 0}^{p} \binom{p}{n} \cdot n! \cdot x^n
$$
In words, I have $p$ "players" who can choose to play or not (every player is represented by a unique id). Those who chose to play are lined up in all possible orders. Then every playing player picks an elemen... | https://mathoverflow.net/users/11998 | Identity of binomial series with factorial. | Darij's comment...
Tthe truncated exponential series:
$$
e\_p(z) = \sum\_{n=0}^p\frac{z^n}{n!}
$$
Your sum is
$$
p!x^pe\_p(1/x)
$$
| 1 | https://mathoverflow.net/users/454 | 129112 | 71,797 |
https://mathoverflow.net/questions/129107 | 7 | For the projective space, we have the very well known Euler sequence
$$0 \to \Omega\_{P^n\_k} \to \mathcal{O}\_{P^n\_k}(-1)^{n+1} \to \mathcal{O}\_{P^n\_k} \to 0.$$
Are there any generalizations to other rational homogeneous spaces?
| https://mathoverflow.net/users/14385 | Euler Sequence on Homogeneous Spaces | Here is how it works for the (complex) Grassmannian. I will leave you the pleasure to extend this point of view to others homogeneous spaces (for instance complete and incomplete flag manifolds).
Firs of all, let me give a slightly different point of view for the Euler exact sequence on the projective space. Let $V$ ... | 10 | https://mathoverflow.net/users/9871 | 129114 | 71,798 |
https://mathoverflow.net/questions/129106 | 10 |
>
> Does anyone know of a pair of different links which the HOMFLY polynomial does not distinguish, but HOMFLY homology does? Or does there exist such a pair of links?
>
>
>
I'm assuming there does exist such a pair, but have never seen it. I've been looking for this for a few days and have had no luck finding ... | https://mathoverflow.net/users/31116 | Links which HOMFLY homology distinguish but the HOMFLY polynomial does not. | Although $5\_1$ and $10\_{132}$ cannot be distinguished by Jones, Alexander and (uncolored) HOMFLY-PT polynomials, their HOMFLY homologies do tell them. (See the [review](http://arxiv.org/abs/1211.6075) by Gukov-Saberi.)
In addition, some mutant pairs can be distinguished by Khovanov homology. (See the [paper](http:/... | 11 | https://mathoverflow.net/users/17644 | 129120 | 71,801 |
https://mathoverflow.net/questions/129025 | 13 | Is there a practical way to compute the permanent of a large ($91 \times 91$) $(0,1)$ matrix?
I have tried to use the [matlab function written by Luke Winslow](http://www.mathworks.com/matlabcentral/fileexchange/36224-matrix-permanent-using-ryser-algorithm/content/permanentRyser.m) which works great for smaller matri... | https://mathoverflow.net/users/22051 | Computing a large permanent | The answer is unfortunately probably no, but there are a few things you could try.
There are algorithms that run in time polynomial in the *value* of the permanent, meaning that the permanent can be computed quickly if its value is small, but that is not going to help you for the incidence matrix of a 9x9 projective ... | 8 | https://mathoverflow.net/users/3106 | 129125 | 71,802 |
https://mathoverflow.net/questions/129062 | 3 | I place some number of coins, $(c\_1, ..., c\_N) \in C$ on a table, where each coin is originally tails up. Let's call the "tails" state $0$ and the "heads" state $1$. I then perform the following procedure some number of times $R$ with the goal of having only a specific subset of $M$ coins, $T \subset C$, left on the ... | https://mathoverflow.net/users/33521 | Optimizing a stochastic "flip and prune" procedure for selecting a subset of coins | First, suppose $|T| = 0$. The number of times you toss every coin for each coin to turn up heads is the maximum of $N$ IID geometric random variables with success probability $1-p$. (There are two conventions for geometric random variables differing by $1$. Some people count failures before the first success, but here ... | 1 | https://mathoverflow.net/users/2954 | 129131 | 71,805 |
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