parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/148644 | 4 | I've been learning about non-projective complete varieties and I am trying to get a handle on how crazy they can get.
$\textbf{Question:}$ Let $V$ be a complete threefold over $\mathbb{C}$. Given distinct points $P,Q$ on $V$, does there exist a complete surface $S\subset V$ which contains neither $P$ nor $Q$?
I wo... | https://mathoverflow.net/users/4181 | Non-projective smooth complete threefolds with a pair of points intersecting every surface | I do not know the answer to the question. However, let me point out that the answer is negative if you look for surfaces avoiding more points. More precisely, there exists a smooth complete threefold that does not satisfy Sándor's property $NC\_{10}$: it contains 10 points that cannot be simultaneously avoided by a div... | 5 | https://mathoverflow.net/users/2868 | 148865 | 79,771 |
https://mathoverflow.net/questions/148866 | 23 | I am teaching Mathematical analysis. A student asked this question. I think this is a good question, but don't know the answer.
| https://mathoverflow.net/users/27205 | Is there a function defined on real numbers which is continuous from the left, but not from the right, everywhere | Here's an elementary construction that may be more digestable by your student.
1. By definition of left continuity, for every point $x$ and every $\epsilon > 0$ there exists $\delta > 0$ such that if $y\in (x-\delta,x)$, $|f(y)-f(x)| < \epsilon$. We can upgrade this by the triangle inequality to for every $y\_1,y\_2... | 11 | https://mathoverflow.net/users/3948 | 148870 | 79,773 |
https://mathoverflow.net/questions/148851 | 2 | I am reading Friedman & Stanley *A Borel reducibility theory for classes of countable structures* (J. Symbolic Logic 54 (1989), 894–914; [MR1011177](http://www.ams.org/mathscinet-getitem?mr=1011177)) and a caret (${}^\wedge$) appears as notation in the following contexts without definition:
1. $\mathbb{Q} \times (-\i... | https://mathoverflow.net/users/6649 | Meaning of notation $\mathbb{Q}^\wedge k$, $-\infty^\wedge \mathbb{Q}$ for linear orders | It looks to me like a primitive concatenation symbol, like what might today be denoted $\sigma^\frown\eta$, and this would align with your expectation in the last paragraph. If we regard $k$ as the $k$ point order (as it is with the von Neumann ordinals), then $\mathbb{Q}$^$k$ would mean adding k points on top of the r... | 2 | https://mathoverflow.net/users/1946 | 148872 | 79,775 |
https://mathoverflow.net/questions/148877 | 3 | Consider a countable transitive model of ZFC $\mathfrak{M}$.
Let $X$ in $\mathfrak{M}$ be some definable set.
Can we define the "type" $p$ of nondefinable elements of $X$? (By type I mean the set of formulas satisfied by the nondefinable elements).
Is $p$ principal or not? (By principal I mean whether $p$ has a g... | https://mathoverflow.net/users/38200 | The type of nondefinable elements | In any structure, the non-definable elements are exactly the elements that realize the type $p(x)$, which is the type containing all assertions of the form:
$$\varphi(x)\to\exists y\neq x\ \varphi(y).$$
The reason is that if $x$ is not definable in a structure $\mathcal{M}$, then it is not the unique satisfying in... | 4 | https://mathoverflow.net/users/1946 | 148881 | 79,776 |
https://mathoverflow.net/questions/148591 | 7 | Let $E$ be a globally generated vector bundle of rank $r$ on a normal irreducible projective variety $X$. Suppose that $E$ induces a finite map
$$X \to \mathbb{G}r (H^0(E), r)$$
to the Grassmannian of rank r quotients.
Can we say that a symmetric power of $S^mE$ for $m>>0$ induces an immersion
$$X \hookrightarrow \math... | https://mathoverflow.net/users/29542 | Global sections of determinant bundle of symmetric powers | The suggestion in my comment turns out to be impossible, and this leads to counterexamples to the OP's question, already for $\text{dim}(X)=2$. Let $U$ be a $2$-dimensional vector space over a field $k$. Let $\mathbb{P}(U^\vee)$ denote $\text{Proj} \text{Sym}^\bullet\_k U$, i.e., $\mathbb{P}^1\_k$. To be very precise, ... | 7 | https://mathoverflow.net/users/13265 | 148899 | 79,781 |
https://mathoverflow.net/questions/148888 | 12 | I'm looking for a book or introductory article, that explains branching rules in representation theory of real Lie groups.
When a Lie group has a set of irreducible representations, I'd like to know how these representations decompose into irreducible representations of a subgroup.
I heard of "Symmetry, representations... | https://mathoverflow.net/users/13767 | What is a good introduction to branching rules in representation theory? | Zhelobenko has books on the subject from 1970, 1983, 1994, 2004. I'm pretty sure it's the 1970 that I saw the most concrete branching laws in.
The special cases you're interested in are *really* easy, by the way, in that the branching laws are "multiplicity-free". For example, the $SU(2)$ irrep of dimension $n+1$ bre... | 5 | https://mathoverflow.net/users/391 | 148906 | 79,786 |
https://mathoverflow.net/questions/148894 | 3 | I am interested in using a result about Newton's method, which basically states that if f is convex on $[a,b]$ and it holds $f(a)<0$ and $f(b)>0$, then the Newton iteration converges to $x^\*\in[a,b]$ with $f(x^\*)=0$ for every starting value $x\_0\in[a,b]$ with $f(x\_0)\geq 0$.
The proof is not that complicated. Un... | https://mathoverflow.net/users/38547 | Literature on root finding of convex Functions | [Global convergence of Newton's method on an interval](http://folk.uib.no/ssu029/Pdf_file/Thorlund-Petersen04.pdf), Lars Thorlund-Petersen (2004).
>
> Global convergence of Newton's method is considered in the strong
> sense of convergence for any initial value in an interval. The class
> of functions for which t... | 0 | https://mathoverflow.net/users/11260 | 148911 | 79,787 |
https://mathoverflow.net/questions/148904 | 2 | The restriction of an irreducible $\mathfrak{sl}\_n(\mathbb{C})$-module to $\mathfrak{sl}\_{n-1}(\mathbb{C})$ is described by a branching rule which says that if $L(\lambda)$ is the simple $\mathfrak{sl}\_n(\mathbb{C})$-module corresponding to a partition $\lambda$ then its restriction is a direct sum of $L(\mu)$s wher... | https://mathoverflow.net/users/6481 | Branching rule for classical Lie algebras in positive characteristic | As my comment indicated, there is currently little hope of writing down general branching rules in characteristic $p$. In fact, given the history of work on Lusztig's conjecture about formal characters of simple modules, it's unclear whether good "formulas" as such will exist in this subject; perhaps one has to settle ... | 3 | https://mathoverflow.net/users/4231 | 148912 | 79,788 |
https://mathoverflow.net/questions/47779 | 9 | How can we force the failure of $\square(\kappa)$ at an inaccessible $\kappa$, where
$\square(\kappa)$ is defined as follows: There is a sequence $(C\_i:i< \kappa)$ such that:
(1) $C\_{i+1} = \{i\}$ and $C\_i$ is closed and cofinal in $i$ if $i$ is a limit
ordinal.
(2) If $i$ is a limit point of $C\_j$, then $C\_i... | https://mathoverflow.net/users/11115 | failure of $\square(\kappa)$ at an inaccessible $\kappa$ | The consistency strength of the failure of $\square(\kappa)$ for non-weakly Mahlo inaccessible cardinal $\kappa$ (in particular, for the first strongly inaccessible) is higher than weakly compact. For example it implies that $0^\#$ exists.
<http://www.jstor.org/stable/27590333>
We know that global square holds in $L$, ... | 7 | https://mathoverflow.net/users/41953 | 148915 | 79,790 |
https://mathoverflow.net/questions/45992 | 5 | Please consider the following problem:
Given: a simple graph (without self-loops and without multiple edges) $G$ on $n$ vertices.
Task: place equidistantly the vertices of $G$ on a circle of unit radius and draw the edges of $G$ in such a way that their total length is minimized.
What is the complexity of this ... | https://mathoverflow.net/users/nan | Drawing graphs on circles | As S. Carnahan and Kristal Cantwell mentions it in the comments, this problem is NP-hard. See:
Rubber bands, convex embeddings and graph connectivity by N. Linial, L. Lovász and A. Wigderson
| 0 | https://mathoverflow.net/users/38267 | 148917 | 79,792 |
https://mathoverflow.net/questions/148905 | 3 | in my research, I need to deal with a stochastic integral with respect to a compensated poisson process, namely,
$ \int\_0^t f(X\_t) dM\_t,$
where $M(t) = N(t) - \int\_0^t \lambda(s)ds$.
The integrand $f$ is bounded, and my question is do I still need $f$ to be predictable in order to make sure such integral is agai... | https://mathoverflow.net/users/42801 | Stochastic integral with respect to discontinuous martingale | Yes, you do need the integrand $f(X\_t)$ to be predictable. If it is merely adapted you may not get a martingale.
Intuitively, the stochastic integral $\int Y\_t\,dM\_t$ tells you the profit from a stock trading strategy. $M\_t$ is the share price at time $t$, and $Y\_t$ is the number of shares you hold at time $t$. ... | 8 | https://mathoverflow.net/users/4832 | 148922 | 79,795 |
https://mathoverflow.net/questions/148869 | 3 | Suppose we're in $\mathbb R^n$, and we have a function on line segments ,$\omega(I)$, with values in $\mathbb R$. Give sufficient conditions for $\omega$ to be given by a generalized 1-form (that is, an $(n-1)$ -current), which is integrable (that is, can be evaluated on line segments). Obviously $\omega$ should be add... | https://mathoverflow.net/users/7499 | How to define a generalized differential form through its values on submanifolds | First of all, since you are looking for a *differential form*, $\omega$ should be a function defined on *oriented* segments. By additivity it can be extended to $1$-dimensional polygonal chains in $\newcommand{\bR}{\mathbb{R}}$ $\bR^n$.
In his beautiful book **Geometric Integration theory**, *H. Whitney* addresses a... | 6 | https://mathoverflow.net/users/20302 | 148923 | 79,796 |
https://mathoverflow.net/questions/148926 | 6 | My question may sound weird and I have no deep motivation behind it other than curiosity.
As is well-known, quasi-reflexive spaces have the Radon-Nikodym property hence their balls have lots of extreme points (they even have the so-called Krein-Milman property). However, can one give me an example of a quasi-reflexi... | https://mathoverflow.net/users/42808 | Quasi-reflexive spaces which are not isometric to dual spaces | Every non-reflexive Banach space can be equivalently renormed so as not to be isometrically isomorphic to a dual space.
$$
$$
Davis, William J.; Johnson, William B.
A renorming of nonreflexive Banach spaces.
Proc. Amer. Math. Soc. 37 (1973), 486–488.
| 6 | https://mathoverflow.net/users/2554 | 148927 | 79,797 |
https://mathoverflow.net/questions/148887 | 4 | Let $\Lambda^k(V)$ be the space of alternating $k$-linear tensors on $V$. Consider the map $f: \left(\mathbb{R}^n\right)^{n-k} \to \Lambda^k(\mathbb{R}^n)$ given by $\left(v\_1,v\_2, ..., v\_{n-k}\right) \mapsto \text{Det}(v\_1,v\_2,...,v\_{n-k},\cdot)$, i.e. the $k$-form you get is a partially applied determinant.
I... | https://mathoverflow.net/users/1106 | Variety determined by interior product of the determinant? | Using the determinant is something of a distraction. In completely coordinate-free terms, and after applying the observation in abx's answer, the map you're trying to understand is the wedge product map
$$V^k \ni (v\_1, ..., v\_k) \mapsto v\_1 \wedge ... \wedge v\_k \in \Lambda^k(V)$$
where $V$ is a finite-dimensiona... | 5 | https://mathoverflow.net/users/290 | 148930 | 79,798 |
https://mathoverflow.net/questions/148937 | 14 | Let $C \subset {\bf CP}^2$ be an irreducible algebraic smooth (projectively) planar curve over the complex numbers of degree $d$ (we allow finitely many points to be deleted from $C$ to make it smooth). Then a generic complex line in ${\bf CP}^2$ intersects $C$ in $d$ distinct points $z\_1,\ldots,z\_d$.
>
> **Quest... | https://mathoverflow.net/users/766 | Permuting collinear points on a curve | The answer to question 1 is yes, and is known as Harris's Uniform Position Lemma. It was proved in Harris's 1980 paper [Galois groups of enumerative problems](http://www.ams.org/mathscinet-getitem?mr=552521).
You can find a nice exposition in Chapter 9 of [Solving Polynomial Equations](http://books.google.com/books?id=... | 23 | https://mathoverflow.net/users/297 | 148939 | 79,803 |
https://mathoverflow.net/questions/148935 | 4 | Can anyone please provide me with a reference on $H^n(M\_{0,n+3},{\mathbb C})$ where $M\_{0,n+3}$ is the (affine) scheme parametrizing $n+3$ labeled distinct points on ${\mathbb C\mathbb P}^1$? I am looking for combinatorial description, dimension, weight filtration, etc.
| https://mathoverflow.net/users/38468 | On $n$-th cohomology of $M_{0,n+3}$ | $M\_{0,n+3}$ can be identified with the set $(z\_1, z\_2, \ldots, z\_n) \in \mathbb{C}^n$ so that $z\_i \neq 0$, $z\_i \neq 1$ and $z\_i \neq z\_j$ for every $i \neq j$. In other words, it is the complement of $2n+\binom{n}{2}$ hyperplanes in $\mathbb{C}^n$. The cohomology of the complement of a hyperplane arrangement ... | 10 | https://mathoverflow.net/users/297 | 148940 | 79,804 |
https://mathoverflow.net/questions/106265 | 4 | In *Fonctions L p-adiques des corps quadratiques imaginaires et de leurs extensions abéliennes*,
J. Reine Angew. Math. 358 (1985), 76–91, Roland Gillard shows the following result (I mainly follow the MR review here):
Let $K$ be an imaginary quadratic field and let $p > 3$ be a prime number which splits in K into
$... | https://mathoverflow.net/users/7443 | Vanishing of certain $\mu$-invariants attached to abelian extensions of imaginary quadratic fields | The question has now been answered by this preprint: <http://arxiv.org/abs/1311.3565>
| 1 | https://mathoverflow.net/users/7443 | 148971 | 79,815 |
https://mathoverflow.net/questions/148980 | 9 | For $n\ge 2\in\mathbb N$, let $S\_n$ be the volume of a $(n-1)$ dimensional solid which satisfies
$$\sum\_{i=1}^{n}x\_i=0, |x\_i|\le1\ (i=1,2,\cdots,n).$$
Then, here is my question.
>
> **Question** : Can we represent $S\_n$ by $n$ ?
>
>
>
**Remark** : This question has been [asked previously on math.SE](ht... | https://mathoverflow.net/users/34490 | About a solid which satisfies $\sum_{i=1}^{n}x_i=0, |x_i|\le1\ (i=1,2,\cdots,n)$ | This question (in a much more general form) is answered in [this preprint by Marichal and Mosinghoff.](http://arxiv.org/abs/math/0607715) They point out that the answer to your question actually goes back to Polya's PhD thesis.
| 11 | https://mathoverflow.net/users/11142 | 148983 | 79,819 |
https://mathoverflow.net/questions/148981 | 0 | Let $C\_1$ and $C\_2$ be two rationally equivalent curves in $\mathbb{P}^3$. Is it true that the dimension of $H^0(\mathcal{N}\_{C\_1|\mathbb{P}^3})$ equal to that of $H^0(\mathcal{N}\_{C\_2|\mathbb{P}^3})$?
| https://mathoverflow.net/users/32151 | Rational equivalence and infinitesimal deformation of curves | No, even if you assume $C\_1$ and $C\_2$ smooth irreducible. A rational equivalence class of curves in $\Bbb{P}^3$ is determined by its degree. Take for $C\_1$ a complete intersection of 2 quadrics, you get
$\dim H^0(\mathcal{N}\_{C\_1|\mathbb{P}^3})= 16$; and for $C\_2$ a plane quartic curve, you find $\dim H^0(\math... | 2 | https://mathoverflow.net/users/40297 | 148987 | 79,821 |
https://mathoverflow.net/questions/148996 | 5 | I posted this question on Math StackExchange but did not get a full answer. I hope it's not a problem if I ask again here.
Is there a way to compute explicitly the $L$−rank $\rho(\bigcup x)$ of $\bigcup x$ in terms of $\rho(x)$? I know it's necessarily $\rho(\bigcup x)\leq\rho(x)$ and that both $\rho(\bigcup x)<\rho(... | https://mathoverflow.net/users/42836 | Computing $L$-rank (constructible universe) | There is no way to compute explicitly $\rho(\bigcup x)$ in terms of $\rho(x)$, in any meaningful fashion: e.g. (working in $L$) for arbitrarily large countable $\alpha$ there are reals $x\subseteq \omega$ with $\rho(x)$ a limit or successor greater than $\alpha$ but with $\rho(\bigcup x) =\omega$ of course. (Because fo... | 9 | https://mathoverflow.net/users/6942 | 149002 | 79,825 |
https://mathoverflow.net/questions/148973 | 5 | **Edit:** This is a question related to my other post, stated in a much more concrete way I think.
I am interested in anything (ideas, references) related to the following problem:
Suppose that $A \subset \mathbb{Z}\_p$ is a set of size $\delta p$ for $\delta > \frac{1}{2}$ (i.e. relatively big). What is known abou... | https://mathoverflow.net/users/13099 | Anticoncentration of the convolution of two characteristic functions | I assume you use the counting norm to define the convolution, as otherwise the $\ell\_1$-norms of $\mu\_A$ and $\mu\_A\ast\mu\_A$ are just of different order of magnitude. Thus, $(\mu\_A\ast\mu\_A)(g)=|A|^{-2}r(g)$, where $r(g)$ is the number of representations of $g$ as a sum of two elements of $A$. We therefore have ... | 3 | https://mathoverflow.net/users/9924 | 149003 | 79,826 |
https://mathoverflow.net/questions/149007 | 3 | Let $\mathcal C$ be a category, and consider a new category $\mathcal C'$ with
$Obj(\mathcal C') := \{$pairs $(X \in Obj(\mathcal C), T \in End\_{\mathcal C}(X)) \}$
$Hom\_{\mathcal C'}((X,T\_X),(Y,T\_Y)) := \{R \in Hom\_{\mathcal C}(X,Y) : T\_Y \circ R = R\circ T\_X \}$
>
> Does this $\mathcal C'$ have a name,... | https://mathoverflow.net/users/391 | Slice-category-like terminology question | I think what you have defined is just called the **category of endomorphisms** in $\mathcal{C}$. See for instance Marian Mrozek's 1992 [paper](http://www2.im.uj.edu.pl/actamath/PDF/29-181-198.pdf) *Normal functors and retractors in categories of endomorphisms* for some properties of this category.
The oldest mention ... | 5 | https://mathoverflow.net/users/18263 | 149009 | 79,829 |
https://mathoverflow.net/questions/148993 | 2 | Let $R$ be a finite local ring (with identity) with exactly one minimal left ideal. Is it necessarily true that $R$ has exactly one minimal right ideal !?
| https://mathoverflow.net/users/41303 | A question on local rings | The answer seems to be yes. In fact, the unique minimal left $R$-ideal is also the unique minimal right $R$-ideal.
Indeed, let $M$ be the unique maximal ideal of $R$. Then we have a sequence
$$ 0=M^{r+1}\subsetneq M^r \subsetneq \ldots \subsetneq M^2 \subsetneq M \subsetneq R $$
of two-sided ideals.
Let $L$ be the ... | 3 | https://mathoverflow.net/users/39640 | 149010 | 79,830 |
https://mathoverflow.net/questions/149004 | 7 | This is a question i asked on math.stackexchange but i didn't get any answer.
Let $A$ be algebra over commutative ring $k$ and $P\_{\bullet}=(P\_i,d\_i)\rightarrow A$, $k$ projective resolution. Then we have obvious lift of multiplication $f:A\otimes A\rightarrow A$ to $F:P\otimes P \rightarrow P$. Of course there is... | https://mathoverflow.net/users/35861 | Algebra structure $Tor(A,A)$ | More generally, if $A,B,C,D $ are $k$-algebras, there's a multiplication
$$Tor\_m^k(A,B)\otimes\_k Tor\_n^k(C,D)\rightarrow Tor^k\_{m+n}(A\otimes\_k C,B\otimes\_k D)\qquad(1)$$
If you take $C=A$ and $B=D$, this becomes
$$Tor\_m^k(A,B)\otimes\_k Tor\_n^k(A,B)\rightarrow Tor^k\_{m+n}(A\otimes\_A A,B\otimes\_k B)$$
and yo... | 5 | https://mathoverflow.net/users/10503 | 149015 | 79,832 |
https://mathoverflow.net/questions/149011 | 16 | **Question:** What is the order of magnitude of the following sum?
$$ \sum\_{\substack{p<n\\\text{$p$ prime}}} \frac{1}{\log{p}} $$
**Additional information**: Since
$$ \sum\_{\substack{p<n\\\text{$p$ prime}}} \frac{1}{\log{n}} \leq \sum\_{\substack{p<n\\\text{$p$ prime}}} \frac{1}{\log{p}} \leq \sum\_{p<n} \fr... | https://mathoverflow.net/users/38267 | Order of magnitude of $\sum \frac{1}{\log{p}}$ | The contribution of the primes $p\leq n/\log ^3 n$ is clearly $O(n/\log^3 n)$. For the remaining primes we have
$$ \log n-3\log\log n <\log p<\log n,$$
$$ \frac{1}{\log p}=\frac{1}{\log n}+O\left(\frac{\log\log n}{\log^2 n}\right), $$
so by the Prime Number Theorem their contribution is
$$ \left(\frac{n}{\log n}+O\left... | 22 | https://mathoverflow.net/users/11919 | 149019 | 79,834 |
https://mathoverflow.net/questions/149018 | 12 | This might be a vague question, but I am troubled by the fact that fields do not admit a nifty categorical definition. An obvious attempt such a definition would be to say that fields are commutative groups in $\mathcal{Ab}$, using that rings are monoids. This fails because the tensor product isn't the categorical prod... | https://mathoverflow.net/users/nan | Fields aren't group objects in Ab, so what are they? | Fields are the [simple objects](http://ncatlab.org/nlab/show/simple+object) in $\text{CRing}$.
**Edit:** Some philosophical remarks. Elements having inverses is a property and not a structure, so in some sense it's not obviously a good idea to treat the inverse as extra structure. Talking about group objects instead... | 22 | https://mathoverflow.net/users/290 | 149023 | 79,836 |
https://mathoverflow.net/questions/149021 | 41 | MO has seen plenty of roadmap questions but oddly enough I haven't seen one for homotopy theory. As an algebraic geometer who's fond of derived categories I would like some guidance on how to build up some background on homotopy theory.
Does the analogue of Hartshorne exist? Are there any must-reads for stable homotopy... | https://mathoverflow.net/users/36922 | Why not a Roadmap for Homotopy Theory and Spectra? | There have been several questions previously in this vein. [This one](https://mathoverflow.net/questions/136077/an-advanced-beginners-book-on-algebraic-topology) asks for an advanced beginners book. The consensus seemed to be that it was difficult to find a one-size-fits-all text because people come in with such divers... | 23 | https://mathoverflow.net/users/11540 | 149041 | 79,846 |
https://mathoverflow.net/questions/148963 | 15 | I'm confused by the following question:
$f:X\to Y$ is a weak homotopy equivalence, that is $f\_\*:\pi\_\*(X)\to \pi\_\*(Y)$ is an isomorphism for any dimensional homotopy groups. However, for the stable homotopy groups, is the homomorphism $f\_\*:\pi\_\*^s(X)\to \pi\_\*^s(Y)$ still an isomorphism?
Any comments are we... | https://mathoverflow.net/users/37899 | Must a weak homotopy equivalence induce an isomorphism between stable homotopy groups? | Let's be precise about the question! I claim it is
not meaningful until you choose basepoints in X and
Y and restrict to based maps, since otherwise the
suspension used to define the stable homotopy groups
is ambiguous. And then you might well get different
answers for different choices of basepoint: some might
be deg... | 11 | https://mathoverflow.net/users/14447 | 149043 | 79,847 |
https://mathoverflow.net/questions/149063 | 5 | I am aware of the following examples of normal surfaces in $\mathbb P^3$ that are projectively isomorphic to their dual varieties:
1. the smooth quadric;
2. Kummer surfaces;
3. The surface with the equation $x\_0^3=x\_1x\_2x\_3$ (in homogeneous coordinates).
What else is known? The base field is algebraically close... | https://mathoverflow.net/users/29992 | Self-dual surfaces in $\mathbb P^3$ with isolated singularities | In his paper [Some invariants for conics and their applications, *Publ. RIMS (Kyoto Univ.)* **19** (1983), 1139-1151] Naruki gives an example of a self-dual quartic surface in $\mathbb{P}^3$ with three singular points of type $A\_3$ and seven points of type $A\_1$ (i.e., ordinary double points).
The paper can be dow... | 7 | https://mathoverflow.net/users/7460 | 149064 | 79,854 |
https://mathoverflow.net/questions/149045 | 4 | My question is related to Geometric Quantization.
I don't undrestand the philosophy of following assertion
>
> If $(V,\omega)$ be a symplectic vector space then the quantizations of
> $V$ corresponds to choices of Lagrangian subspaces of $V$,
>
>
>
| https://mathoverflow.net/users/nan | Quantization of symplectic vector space and choice of lagrangian subspaces | 1. The first attempt to "quantize" a dynamical variable $u$ on a symplectic manifold $(M,\omega)$, that is, to associate a linear operator $\hat u$ on the space of square summable smooth function $\psi : M \to \bf{C}$, is to define
$$
\hat u(\psi) = {\partial \psi \over \partial x} {\rm grad}\_\omega(u),
$$
where ${\r... | 14 | https://mathoverflow.net/users/11885 | 149075 | 79,856 |
https://mathoverflow.net/questions/149072 | 3 | Let $f\in L^{1} (\mathbb R) := \{f:\mathbb R \rightarrow \mathbb C \ \text {measurable functions} : \int\_{\mathbb R} | f(x)| dx < \infty \}$ and the Fourier transform of $f$,
$\hat{f} (y) : = \int \_ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R $
and $\widehat{|f|} (y) : = \int \_ {\mathbb R} |f(x)| e^... | https://mathoverflow.net/users/33018 | Let $f \in S(\mathbb R)$. Can we say $\widehat{|f|} \in L^{1}(\mathbb R)$? | If f is in S, then |f| is in $H^1$. This is enough to guarantee that its Fourier transform is in $L^1$.
| 6 | https://mathoverflow.net/users/12120 | 149078 | 79,857 |
https://mathoverflow.net/questions/149076 | 3 | This is mostly a reference question. Suppose that I have an action of (say, finite) group $G$ on an algebraic stack $X$ (in my case it is a Deligne-Mumford stack, but this shouldn't matter). As far as I understand, in this case it makes sense to talk about the fixed points stack $X^G$ (which is not a closed substack of... | https://mathoverflow.net/users/3891 | Group action on a stack and fixed points | A standard reference is
Romagny, Matthieu
Group actions on stacks and applications.
Michigan Math. J. 53 (2005), no. 1, 209–236.
<http://projecteuclid.org/euclid.mmj/1114021093>
<http://perso.univ-rennes1.fr/matthieu.romagny/articles/group_actions.pdf>
| 4 | https://mathoverflow.net/users/11682 | 149081 | 79,859 |
https://mathoverflow.net/questions/132098 | 1 | I would like to have an estimate for the series
$$P(t) = \sum\limits\_{k = 0}^\infty (e^{-t}\frac{t^k}{k!})^m,$$
where $e$ is the base of natural logarithm, $k!$ is the factorial of the integer $k$, $t$ represents the time and $t>0$, $m$ is a positive integer and $m>1$ (Obviously, $P(t)=1$ when $m=1$ since $P(t)$ is ex... | https://mathoverflow.net/users/24304 | Estimate the scale of the power series with Poisson pdf/pmf-like terms | Actually this can be solved by standard asymtotic analysis methods (with help from Mathematica):
Expand the factorial in the sum with Sterling. Then do as if $k$ is continuous and find $k=k\_{0}$ where the term in the sum is maximal (the result contains Lambert's $W$ function). Next expand quadratically around $k$ wi... | 0 | https://mathoverflow.net/users/37436 | 149089 | 79,863 |
https://mathoverflow.net/questions/148953 | 4 | Given an algebraic torus $(\mathbb{C}^\ast)^n$, what's the moduli space of complex structures? Even for $\mathbb{C}^\ast$, since it's a non-compact Riemann surface with puncture, it doesn't seem trivial for me. Does anyone know some references on this?
Many toric varieties don't have moduli, e.g. $\mathbb{CP}^2$, but... | https://mathoverflow.net/users/13244 | What is the moduli of an algebraic torus | There is just no definition of the moduli for complex structures on non-compact manifolds, but by any reasonable definition, it would be (generally) very bad space, certainly infinite-dimensional. For example, if you are interested in the bounded Stein subsets of ${\Bbb C}^n$ with smooth boundary, the boundary (more pr... | 8 | https://mathoverflow.net/users/3377 | 149094 | 79,865 |
https://mathoverflow.net/questions/145860 | 3 | Suppose $x$, $y$, $z$ are three variables satisfying $yz=zy$, $zx=xz$, $xy=yzx$.
1. Could anyone give me two (non-commutative) polynomials $f$ and $g$ in the above three variables such that the following equality holds:
$$f(x,y,z)\cdot(3z^2+zy^2x+x^2)=g(x,y,z)\cdot(3+y^2x+x^2)
$$
**Especially, I expect $f$, $g$ to ... | https://mathoverflow.net/users/9305 | noncommutative polynomials equality | This is not the answer that I expected, but at least it gives an algorithm to find some $ f, g$ such that $ f(x,y,z) a(x,y,z)= g(x,y,z) b(x,y,z)$ for given $a, b$.
See theorem 8 in the paper by Ore " Theory of non- commutative polynomials", Annals of Mathematics, vol.34, 1933, 480-508.
| 0 | https://mathoverflow.net/users/9305 | 149095 | 79,866 |
https://mathoverflow.net/questions/149093 | 3 | Let $M$ be a compact finite-dimensional manifold and $f\colon M\to M$ be a diffeomrphism. By $P\_n(f)$ we denote the number of periodic points of $f$ with period $n$, that is, the number of fixed points of $f^n$.
Katok [Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publications Mathématiques de l... | https://mathoverflow.net/users/24676 | Katok's conjecture on entropy and periodic orbits for generic $C^1$ diffeomorphisms | Nice question. I think it has not been directly adressed in the literature, but combining known results it seems that a positive answer to that question can be given. Still some details must be carried out (which I did not) to be sure.
The main point is that $C^1$-*far from homoclinic tangencies* the result must be ... | 2 | https://mathoverflow.net/users/5753 | 149097 | 79,867 |
https://mathoverflow.net/questions/148441 | 8 | Let $ (\mathscr{A},G,\alpha) $ be a $ C^{\*} $-dynamical system, and consider the twisted convolution $ \* $-algebra $ ({L^{1}}(G,\mathscr{A}),\star,^{\*}) $ defined by
\begin{align\*}
\forall \phi,\psi \in {L^{1}}(G,\mathscr{A}), ~ \forall g \in G: \quad
(\phi \star \psi)(g) & \stackrel{\text{def}}{=}
\int\_{G} \phi(x... | https://mathoverflow.net/users/nan | Extending a $ * $-Representation of $ ({C_{c}}(G,\mathscr{A}),\star,^{*}) $ to a $ * $-Representation of $ \mathscr{A} \rtimes_{\alpha} G $ | Here's the discrete group argument, when $A=\mathbb C$. Then $C\_c(G,A)$ is just the algebra of finitely supported functions $G\rightarrow\mathbb C$, which has a basis $(\delta\_g)\_{g\in G}$ say. That $\pi:C\_c(G,A)\rightarrow B(H)$ is a $\*$-representation means that firstly $\pi(\delta\_e)$ is a self-adjoint idempot... | 2 | https://mathoverflow.net/users/406 | 149102 | 79,870 |
https://mathoverflow.net/questions/149107 | 1 | Consider the following differential equation
$$F(cx) = F(x) + x F'(x)$$
for $c>1$.
1. Does this differential equation belong to a some well known class?
2. Is there a way to find all the solutions $F(\cdot)$ of this equation that are also cumulative distribution functions?
3. $F(x) = x^a$ for a properly chosen $a... | https://mathoverflow.net/users/42754 | Unusual Differential Equation for CDF | You do not tell the range of $x$. Distribution functions are usually defined on the real line, while $x^a$ on the real line is not a distribution function for any $a$.
Anyway, here is a way to solve your equation. Set $F(x)=\phi(\log x)$ and then $\log x=t$. You obtain
$$\phi(a+t)=\phi(t)+\phi'(t),$$
where $a=\log c$... | 2 | https://mathoverflow.net/users/25510 | 149111 | 79,872 |
https://mathoverflow.net/questions/139563 | 6 | Assume $u,v\in\mathbb{C}^n$ are complex vectors. I was wondering if there is a closed form expression for the following problem in terms of $u$ and $v$
\begin{equation\*}
\arg\min\_{x\in\mathbb{C}^n} \|uv^\*+vu^\*-xx^\*\|^2\_F
\end{equation\*}
F here denotes Frobenius norm.
| https://mathoverflow.net/users/34919 | Best rank one approximation | We assume that the system $u,v$ is $\mathbb{C}$-free. Let $w=uv^\*+vu^\*$. Note that if $X$ is a solution then $\exp(i\theta)X$ is also a solution. We study the minimum of $trace((w-XX^\*)^2)=trace(w^2+XX^\*XX^\*-wXX^\*-XX^\*w)=trace(w^2)+trace((X^\*X)^2)-2trace(wXX^\*)$,
that is we study the minimum of $\phi(X)=||X|... | 1 | https://mathoverflow.net/users/9091 | 149112 | 79,873 |
https://mathoverflow.net/questions/149115 | 22 | I ran across this discussion by Daniel Shanks,
>
> "Is the quadratic reciprocity law a deep theorem?."
> *Solved and Unsolved Problems in Number Theory*. Vol. 297. AMS, 2001. p.64ff.
>
>
>
which made me wonder:
>
> **Q**. Is there a theorem in some formal system whose proofs are known to be
> necessarily... | https://mathoverflow.net/users/6094 | Deep theorems and long proofs | There is a body of very interesting work surrounding the proof complexity of various formulations of the well-known pigeon-hole principle, the fact that there is no injective function from a set of size $m$ to a set of size $n$, when $m\gt n$. It turns out that the difficulty of proving this depends on how much bigger ... | 19 | https://mathoverflow.net/users/1946 | 149120 | 79,877 |
https://mathoverflow.net/questions/149105 | 13 | Are computations of the mapping class groups of small Seifert-fibred 3-manifolds recorded in some convenient location?
For most Seifert manifolds working out the mapping class group is easy-enough (at least, reducing it to a 2-dimensional mapping class group is easy) since diffeomorphisms usually are isotopic to fib... | https://mathoverflow.net/users/1465 | Mapping class groups of small Seifert-fibred 3-manifolds | The determination of mapping class groups of small Seifert manifolds was completed by M. Boileau and J.-P. Otal in a paper in Invent. Math. 106 (1991), 85-107. They give references for cases previously done:
1. Lens spaces by Bonahon and Hodgson-Rubinstein.
2. Multiple fibers of orders (2,2,n) by Asano and Rubinstein... | 15 | https://mathoverflow.net/users/23571 | 149121 | 79,878 |
https://mathoverflow.net/questions/149130 | 1 | Let $Y$ be a normal projective algebraic variety with rational singularities. Assume that there is an effective divisor $\Delta$ on $Y$ such that $\omega\_Y+\Delta$ is a Cartier divisor. Does the analog of Kawamata-Viehweg theorem hold for $\omega\_Y+\Delta$? In other words, if $L$ is a very ample line bundle on $Y$, i... | https://mathoverflow.net/users/3891 | Kawamata-Viehweg vanishing for non-Gorenstein singularities | You don't need the $\Delta$. In fact, it is true without the $\Delta$ and may or may not be true with it. It depends on the singularities of the pair $(Y,\Delta)$.
From the question I am guessing you want to work in characteristic zero.
>
> **KV Vanishing with rational singularities** Let $Y$ be a projective vari... | 7 | https://mathoverflow.net/users/10076 | 149132 | 79,880 |
https://mathoverflow.net/questions/148928 | 4 | Let $K$ be a nonarchimedean local field of residue characteristic $p \neq l$ and let $I\_K$ be the inertia subgroup of its absolute Galois group. Let $V$ an irreducible representation of $I\_K$ over $\overline{\mathbb{F}}\_l$ with finite image $I$. Suppose that $V$ is primitive, i.e., is not induced from a proper subgr... | https://mathoverflow.net/users/5498 | Pro-$l$ Sylow action in a primitive representation of inertia over $\overline{\mathbb{F}}_l$ | Yes, this could happen:
Say $p=2$, $l=3$, and take an elliptic curve $E/{\mathbb Q}\_2$ with largest possible inertia image, $I=\text{SL}(2,{\mathbb F}\_3)$. Then the $3$-adic representation $V\_3(E)$ is irreducible, $I$ acts faithfully and so has an element of order 3 on it, but it is not an induced representation ... | 5 | https://mathoverflow.net/users/3132 | 149140 | 79,881 |
https://mathoverflow.net/questions/149135 | 2 | I am trying to understand to what extent the following result of Hille is an extension of the usual theorems on differentiation under the integral sign.
>
> **Theorem** (Hille). Let $(\Omega,\Sigma,\mu)$ be a measure space and $X$ and $Y$ be Banach spaces. Let $f:\Omega \to X$ be a Bochner-integrable function and
>... | https://mathoverflow.net/users/8452 | Precise versions of "differential operators are unbounded but closed linear operators" | The question needs to be formulated more precisely. First a comment.
Any unbounded operator
$$ A: D(A)\subset E\to F $$
$E,F$ Banach spaces, can also be viewed as a bounded operator
$$ D(A)\to F $$
where $D(A)$ is equipped with the graph-norm $\vert-\vert\_G$
$$\vert x\vert\_G = \vert x\vert\_E+\vert Ax\ver... | 6 | https://mathoverflow.net/users/20302 | 149141 | 79,882 |
https://mathoverflow.net/questions/149134 | 6 | Let $f$ be a entire function (stable on $\mathbb{R}$), and $E\_{\mathbb{R}}$ its real escaping set : $$E\_{\mathbb{R}} = \{ x \in \mathbb{R} : f^{(k)}(x) \rightarrow\_{k \to \infty} \infty \} $$
We put the Lebesgue measure on $\mathbb{R}$.
>
> **Question** : If $E\_{\mathbb{R}}$ is a (measurable) null set, is i... | https://mathoverflow.net/users/34538 | Entire functions with a null real escaping set | For every continuous function $g$ on the real line and for every positive continuous function $\epsilon$ on the real line, there exists an entire function $f$ such that
$|f(x)-g(x)|<\epsilon(x)$ for all real $x$ (This is due to Carleman). So if you can construct a real continuous
function with your property then you ca... | 5 | https://mathoverflow.net/users/25510 | 149147 | 79,886 |
https://mathoverflow.net/questions/149099 | 5 | I always wondered if the fact that the quartic can be solved by a cubic can be generalized to other ***even*** degrees $n$, namely if there is an ordering of the roots $x\_i$ of form $x\_1x\_2+x\_3x\_4+\dots+x\_{n-1}x\_{n} = y$ such that $y$ is *reduced* to an algebraic number of deg $n-1$. It seems it can be if its Ga... | https://mathoverflow.net/users/12905 | Reducing 12th degree eqns (12T179) to an 11th degree eqn | There are no other examples of degree $12$ besides the one you found. Moreover, there are no examples of degrees $10$ or $14$, and in fact the only additional degrees up to $32$ in which there exist examples are $16$, $28$, and $32$, where the groups in the latter cases are precisely (in Klueners/GAP/MAGMA notation) 16... | 9 | https://mathoverflow.net/users/30412 | 149154 | 79,887 |
https://mathoverflow.net/questions/149153 | 13 | This question arose from a problem in Niven & Zuckerman's book "Introduction to the Theory of Numbers". In the chapter that the authors introduce the mobius function, the first exercise is the following:
Find a positive integer $n$ such that $\mu(n)+\mu(n+1)+\mu(n+2)=3$, i.e, $\mu(n)=\mu(n+1)=\mu(n+2)=1$. A brute-fo... | https://mathoverflow.net/users/37392 | Mobius function of consecutive numbers | General conjectures of Chowla predict cancelation in correlations of the Mobius function: e.g. in $\sum\_{n\le x}\mu(n)\mu(n+1)$ or $\sum\_{n\le x} \mu(n)^2 \mu(n+1)\mu(n+2)$ etc. These conjectures would imply an asymptotic formula for the number of solutions to $\mu(n)=\mu(n+1)=\mu(n+2)$. However the conjectures are w... | 15 | https://mathoverflow.net/users/38624 | 149162 | 79,890 |
https://mathoverflow.net/questions/149161 | 4 | A finite simplicial set is a simplicial set having only a finite number of non degenerate simplicies. It is not hard to show that every finite simplicial set has only a finite number of simplicies in each degree. My question is: does the converse hold? that is, is every simplicial set, having a finite number of simplic... | https://mathoverflow.net/users/42440 | Finite simplicial sets | Let G be a finite group viewed as a one object category. Then the nerve BG is a simplicial set with finitely many simplices in each dimension but it is not finite.
| 9 | https://mathoverflow.net/users/15934 | 149165 | 79,891 |
https://mathoverflow.net/questions/149106 | 3 | Let $f$ be a function, holomorphic in $\mathbb{C}$, and $K(f)$ its non-escaping set :
$$K(f) = \{ z \in \mathbb{C} : f^{(k)}(z) \nrightarrow\_{k \to \infty} \infty \} $$
>
> **Question** : If $K(f)$ is connected, is it also contractible ?
>
>
>
| https://mathoverflow.net/users/34538 | Contractibility of connected holomorphic dynamics? | The answer to your question is negative.
**EDIT** I have added some additional details and made some corrections.
**In the entire case**, it is possible to construct an entire function with the following properties:
(a) The Fatou set consists of a single conneted attracting basin;
(b) If $C$ is a component of the... | 4 | https://mathoverflow.net/users/3651 | 149170 | 79,894 |
https://mathoverflow.net/questions/149128 | 1 | Problem:
--------
Let $p$ and $q$ be two integers, and $q > p>0$. Does the following ODE have a general solution on some finite time interval $[0,T]$? If yes, how can I obtain the solution?
$$
\dot x(t) = (x(t))^{p/q} + u(t),
$$
where $x(0)=x\_0>0$, and $0<u(t)<b<+\infty$ for all $t$.
Based on the problem formu... | https://mathoverflow.net/users/26600 | How do I solve this nonlinear ODE with a fractional order term | I understand that your ODE is one-dimensional: in that case you can avoid Lipschitz continuity and replace it by transversality:
The autonomous equation
$$
\dot x =f(x),\quad x(0)=x\_0,
$$
has a unique local solution provided $f$ is continuous and $f(x\_0)\not=0$. Peano's theorem provides existence whereas uniqueness f... | 4 | https://mathoverflow.net/users/21907 | 149180 | 79,898 |
https://mathoverflow.net/questions/149181 | 4 | Let $f:X \to \mathbb{P}^1$ be a projective flat morphism, $X$ is a projective scheme. Let $\mathcal{F}$ be a locally free sheaf on $X$. Are the higher direct image sheaves $R^if\_\*\mathcal{F}$ locally free for all $i>0$?
| https://mathoverflow.net/users/32151 | Is the higher direct image sheaf of a locally free sheaf over $\mathbb{P}^1$ locally free? | No. For instance there is a flat, projective morphism $f:X\rightarrow \Bbb{P}^1$ such that $X\_t:=f^{-1}(t)$ is a smooth rational curve for $t\neq 0$, but $X\_0$ is a nodal plane cubic curve with an embedded point (see Hartshorne, III.9.8.4). Then $H^1(X\_t,\mathcal{O}\_{X\_t})$ is zero for $t\neq 0$, but $\ \dim H^1(X... | 15 | https://mathoverflow.net/users/40297 | 149182 | 79,899 |
https://mathoverflow.net/questions/149194 | 14 | **Question:** Is there a function $f(n) \rightarrow \infty$, such that infinitely often the interval $[n,n+\frac{f(n) \log(n)}{\log{\log(n)}}]$ does not contain a squarefree integer?
**Additional information:** If we consider the simultaneous congruences:
$$ x \equiv 0 \ (\text{mod} \ 2^2)\\ x \equiv 1 \ (\text{m... | https://mathoverflow.net/users/40458 | Consecutive non squarefree integers | Erdos has mentioned this lower bound in several places, adding always that he's never been able to improve it. For example see <http://renyi.hu/~p_erdos/1951-13.pdf> (page 107; in fact he gives an explicit constant here that he says he cannot improve), and page 8 of <http://hsb.org.hu/~p_erdos/1981-21.pdf> .
A stand... | 12 | https://mathoverflow.net/users/38624 | 149198 | 79,906 |
https://mathoverflow.net/questions/149193 | 11 | There are an infinite number of regular polygons in the plane,
five regular polyhedra,
six regular polytopes in $\mathbb{R}^4$,
and then three regular polytopes in every dimension $d > 4$.
There are eight convex [deltahedra](http://en.wikipedia.org/wiki/Deltahedron) (all faces equilateral triangles),
five in $\mathbb... | https://mathoverflow.net/users/6094 | Why do convex polytope options constrict with dimension, rather than expand? | This is a well known "dimension curse" phenomenon. It is easier to explain for spherical polyhedra. Let $P$ be a regular spherical convex polyhedron of dimension $d$. Starting with dimension $d=3$, all spherical convex polyhedra are rigid, so they are determined by their links (intersection of a vertex cones with an $\... | 14 | https://mathoverflow.net/users/4040 | 149211 | 79,911 |
https://mathoverflow.net/questions/149213 | 2 | Let $G$ be a simply connected group over an algebraically closed field $k$, and
$I:=\{(g,\gamma)\in G\times G\vert~ g\gamma=\gamma g\}$
the scheme of centralizer.
Is $I$ a Cohen-Macaulay scheme over $k$?
| https://mathoverflow.net/users/27398 | Cohen-Macaulayness of the scheme of centralizer | The answer is not known. In fact, the analogous result for the "commuting variety" $\mathcal{C}(\mathfrak{g})$ of a reductive Lie algebra $\mathfrak{g}$ ($\mathcal{C}(\mathfrak{g})=\{(x,y)\in \mathfrak{g}\times \mathfrak{g}\ |\ [x,y]=0\}$) is a classical conjecture -- even the normality of $\mathcal{C}(\mathfrak{g})$ i... | 7 | https://mathoverflow.net/users/40297 | 149220 | 79,915 |
https://mathoverflow.net/questions/149027 | 14 | Given a connected space $B$, is there always some space $X$ with $B \simeq \mathbf{B}(\mathrm{Aut}(X))$?
Here by space I mean simplicial set, by $\mathrm{Aut}(X)$ I mean the simplicial monoid of auto-equivalences of $X$ (not just strict automorphisms), and by $\mathbf{B}$ I mean the classifying space of this; but I’d... | https://mathoverflow.net/users/2273 | Is every connected space equivalent to some B(Aut(X))? | Here is the $1$-type case. I assume all spaces are of the homotopy type of CW. Let me write $haut(X)$ (resp. $haut\_\*(X)$) for the monoid of self-equivalences (resp. pointed ones) to avoid posible confusion with the group-theoretic notation. These spaces have the correct homotopy type by our assumption.
From [*All G... | 12 | https://mathoverflow.net/users/33199 | 149236 | 79,920 |
https://mathoverflow.net/questions/149242 | 3 | It is well-known that the ends of a finite-volume hyperbolic manifold are warped products $$(0,\infty)\times\_f T$$
for some euclidean manifold $T$ and $f(t)=e^{-t}$.
Question: Is there a similar result for rank 1 symmetric spaces (or even for more general negatively curved manifolds)?
| https://mathoverflow.net/users/39082 | Cusps as warped products | It may be not the answer you are hoping for but still: one can show (short tensor calculations) that, even locally, the metric of an irreducible
symmetric space different from the space of constant curvature
can not be a warped product metric.
Now, concerning arbitrary negatively curved metrics: I do not really u... | 2 | https://mathoverflow.net/users/14515 | 149246 | 79,924 |
https://mathoverflow.net/questions/149245 | 10 | Suppose $f$ is a linear map, and consider $\Lambda^k f$ as the usual exterior power of $f$ (if you prefer matrices, it is a matrix whose entries are the $k\times k$ minors of $f.$) The coefficients of the characteristic polynomial of $f$ are the traces of the various $\Lambda^k f,$ but my question is: is there a simple... | https://mathoverflow.net/users/11142 | Characteristic polynomial of exterior power | I am not sure what data about $f$ you could use other than the characteristic polynomial. The $l$th coefficient of the characteristic polynomial of $\Lambda^k f$ is the trace of $\Lambda^l (\Lambda^k f)$. You would like to write the trace of $\Lambda^l (\Lambda^k f)$ in terms of the trace of $\Lambda^i f$, $1\leq i \le... | 13 | https://mathoverflow.net/users/18060 | 149249 | 79,925 |
https://mathoverflow.net/questions/149247 | 7 | I am trying to find an estimate for the following sum:
$$
\sum\_{\substack{n \leq x \\ n \equiv k (m)}} d(n),
$$
where $d(n)$ is number of divisors of $n$. I found estimates for the case when $k$ and $m$ are coprime, but nothing explicit for general case. My expectation for this sum is that this should be known, bu... | https://mathoverflow.net/users/42944 | Sum of divisor function over arithmetic progression | In the context of the "divisor problem for arithmetical progressions",
there is the following article by Fouvry, Iwaniec and Katz, on "The divisor function over arithmetic progressions": <http://matwbn.icm.edu.pl/ksiazki/aa/aa61/aa6134.pdf>.
The authors use Fourier series techniques
and Weil’s estimate for Kloosterman ... | 6 | https://mathoverflow.net/users/32332 | 149250 | 79,926 |
https://mathoverflow.net/questions/149215 | 11 | As we know one of the most important and fundamental books in stability, simplicity, forking and ... classification theory, is Shelah's "Classification Theory" where lots of original ideas of the subject could be found.
I am studying model theory and wanted to start reading that book in order to be able to understand... | https://mathoverflow.net/users/42321 | Shelah's book on "Classification Theory" | I think Shelah's book is very good, but it is very difficult to read, in part intrinsically because of its subject matter, and in part because it was written by Shelah. Recently, Richard Elwes [reminded us](https://plus.google.com/+RichardElwes/posts/4cG2M7w3sr1) of the [review](http://www.ams.org/mathscinet-getitem?mr... | 15 | https://mathoverflow.net/users/6085 | 149259 | 79,927 |
https://mathoverflow.net/questions/149253 | 2 | It is a fact that if $\Lambda$ is a nonelementary subgroup of ${\rm PSL\_2}(\mathbb{C})$ which contains an hyperbolic transformation and moreover ${\rm tr}(g)\in\mathbb{R}/\pm 1$ for all $g\in\Lambda$ then in fact $\Lambda$ is a Fuchsian subgroup, i.e. contained in a conjugate of ${\rm SL}\_2(\mathbb{R})$. The proof I ... | https://mathoverflow.net/users/32210 | Fixed directions and Zariski density of hyperbolic groups | I will start with few observations; along the way, I will correct your question. Let $g\in SO(n,1)$ be an orientation-preserving loxodromic element. Let $U\_g\in SO(n)$ denote the rotational part of $g$. Then $U\_g$ always has a nonzero fixed vector in ${\mathbb R}^n$. Furthermore, if $n$ is even then it follows that t... | 2 | https://mathoverflow.net/users/21684 | 149273 | 79,930 |
https://mathoverflow.net/questions/149127 | 23 | Apologies if this question isn't suitable for MathOverflow; I posted it on MSE [here](https://math.stackexchange.com/questions/568961/what-motivated-rademachers-contour-along-the-ford-circles-when-he-used-the-circ) but it didn't get a response and it felt like it was on the cusp of being suitable for here.
After Rama... | https://mathoverflow.net/users/40804 | What motivated Rademacher's contour along the Ford circles? | In 1954 Hans Rademacher gave a series of lectures in which he described his reasoning step by step, see [Lectures on Analytic Number Theory](http://www.math.tifr.res.in/~publ/ln/tifr02.pdf), page 113 and following. This should give some insight into his creative process, of which he himself says: *"The path [of integra... | 16 | https://mathoverflow.net/users/11260 | 149282 | 79,936 |
https://mathoverflow.net/questions/149266 | 6 | By $\mathbb{R^N}$ I mean the real vector space with the natural componentwise addition and scalar multiplication. Certainly ZFC+(V=L) gives definable bases, but does ZFC?
| https://mathoverflow.net/users/38783 | Is definability of a basis for $\mathbb{R^N}$ independent of ZFC? | The answer, if I understand the question correctly, is negative. That is, if we understand "definable" as "can be defined from ordinals [and a real number]", or in simpler words, sets which are in $\sf HOD(\Bbb R)$. Here is a sketch of an argument I suspect is correct.
Consider Solovay's model which we get from colla... | 4 | https://mathoverflow.net/users/7206 | 149283 | 79,937 |
https://mathoverflow.net/questions/149272 | 5 | Gromov and Thurston proved in "Pinching constants for hyperbolic manifolds" that any finite ramified covering of a compact hyperbolic manifold, along a codimension $2$ totally geodesic submanifold, can be endowed with a riemannian metric of negative sectional curvature. Pansu mentions in a survey (4/TSG\_1985-1986\_*4*... | https://mathoverflow.net/users/25511 | Are ramified covering of negatively curved manifolds negatively curved? | Suppose that $M$ is a closed hyperbolic 3-manifold and $K$ is a trivial knot contained in a ball $B\subset M$. Then $M$ admits a 2-fold covering $M'\to M$ ramified over $K$ so that $M'$ is not even aspherical (it is homeomorphic to the connected sum of two copies of $M$). Therefore, $M'$ cannot admit a negatively curve... | 8 | https://mathoverflow.net/users/21684 | 149289 | 79,941 |
https://mathoverflow.net/questions/149284 | 6 | Novikov conjeture is a famous open problem in Geometric topology.It predicts that higher signature is oriented-homotopy invariant.
<http://en.wikipedia.org/wiki/Novikov_conjecture>
I am a student interested in geometric topology.My question is:
What are the possible consequences of this famous conjecture? Thank y... | https://mathoverflow.net/users/41110 | consequence of Novikov conjecture | The Novikov conjecture is known for some groups, and open for other groups. Often Novikov conjecture is implied by a more general conjecture (e.g. Borel, Farrell-Jones, or Baum-Connes) which in my amateur view are more intersting.
All these conjectures are true e.g. for $G=\mathbb Z^n$, $n>4$ and in many other cases. ... | 7 | https://mathoverflow.net/users/1573 | 149303 | 79,946 |
https://mathoverflow.net/questions/149219 | 27 | Is there any characterization of continuous functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ such that for any linearly independent set $A$ (over the rationals) $f(A)$ is also linearly independent ?
| https://mathoverflow.net/users/41303 | Continuous functions $f$ with $f(A)$ linearly independent when $A$ is independent | This may not be a complete answer to the question, but it is too long for a comment.
Given such a function $f$, the restriction $f \restriction [0,\infty)$ is a power function $\alpha x^\beta$ where $\alpha \ne 0$ and $\beta \ge 0$, and similarly the restriction $f \restriction (-\infty,0]$ is a power function of $-x... | 26 | https://mathoverflow.net/users/1682 | 149305 | 79,947 |
https://mathoverflow.net/questions/149301 | 4 | Let $X$ be a CW-complex and $X^m$ it's $m$-skeleton. I think that for any $n\geq 2$ and $1\leq r\leq n-1$ it should be possible to obtain $X^{n+r}$ directly from $X^n$ via a homotopy push-out
$$\begin{array}{ccc}
Y\vee X^{n}&\rightarrow &X^{n}\\
\downarrow&&\downarrow\\
X^{n}&\rightarrow &X^{n+r}
\end{array}$$
Were... | https://mathoverflow.net/users/12166 | Attaching cells of different dimensions at once in a CW-complex | Map the function $Y\to \*$ into the left edge to get two homotopy pushouts in a row; this would produce a cofiber sequence $Y\to X\_n\to X\_{n+r}$. But this can't be done in general.
One interesting way to see that this is not the case in general is that it would lead to cone decompositions of a generic space with l... | 3 | https://mathoverflow.net/users/3634 | 149307 | 79,948 |
https://mathoverflow.net/questions/149257 | 0 | (This is a cross-post from [here](https://math.stackexchange.com/questions/569754/relative-interior-and-dense-subsets).) Let $A,B\subseteq \mathbb R^d$ be non-empty, such that $B\subseteq \overline A.$ For $S\subseteq\mathbb R^d$ define the relative interior of $S$ by $$\text{ri}(S)=\{s\in S\mid \exists\varepsilon>0:B\... | https://mathoverflow.net/users/31306 | Relative interior and dense subsets | As noted in the comments, the case where $A\subseteq\mathbb{R}^2$ is the interior of the unit square and $B$ is a side of that square satisfies the assumptions posed, but in that case neither (1) nor (2) holds. This example is easy to generalize, of course, and perhaps will lead you towards the claim you really want to... | 1 | https://mathoverflow.net/users/20186 | 149308 | 79,949 |
https://mathoverflow.net/questions/149318 | 4 | This question is related to [Attaching cells of different dimensions at once in a CW-complex](https://mathoverflow.net/questions/149301/attaching-cells-of-different-dimensions-at-once-in-a-cw-complex) There, I didn't manage to formalize the idea I had in mind, and ended up with a question whose answer was obviously *no... | https://mathoverflow.net/users/12166 | Attaching cells of different dimensions at once in a CW-complex II | Yes. More generally, suppose you have a square
$$\begin{array}{ccc}
A&\rightarrow &B\\
\downarrow&&\downarrow\\
C&\rightarrow &D
\end{array}$$
and you want a space $Y$ between $A$ and the pullback of $B\to D\leftarrow C$ such that
$$\begin{array}{ccc}
Y&\rightarrow &B\\
\downarrow&&\downarrow\\
C&\rightarrow &D
... | 4 | https://mathoverflow.net/users/6666 | 149329 | 79,952 |
https://mathoverflow.net/questions/149320 | 4 | By similar arguments as for the proof of the golden-thompson inequality (see "Log majorization and complementary Golden-Thompson type inequalities" by T.Ando and F.Hiai) we can show that for all A,B symmetric positive definite we have
$$\|\log(A)+\log(B)\|\_{tr}\leq \|\log(A^{1/2}BA^{1/2})\|\_{tr},$$
where $\log$ is th... | https://mathoverflow.net/users/35593 | Equivalent metrics on symmetric positive definite matrices | Check out:
[Reverse inequality to Golden–Thompson type inequalities: Comparison of $e^{A+B}$ and $e^Ae^B$
Jean-Christophe Bourin, Yuki Seo (2007)](https://www.sciencedirect.com/science/article/pii/S0024379507002091), Linear Algebra and its Applications
Volume 426, Issues 2–3, 15 October 2007, Pages 312-316.
| 1 | https://mathoverflow.net/users/11142 | 149335 | 79,956 |
https://mathoverflow.net/questions/149336 | 4 | I'm working on some palindromic words right now. Those are the elements of $F\_2$, the free group on two generators, which are the same if reveresed. For example $xyx, y^2, xyxxyx$ and so on.
Can you write each element of $F'$, the derived group, as a product of palindromes of EVEN length? Does this property hold if ... | https://mathoverflow.net/users/23232 | $\left[x,y\right]$ as a product of palindromes of even length? | The set of products of palindroms of even length $P$ is a normal subgroup, because the inverse of a palindrome is a palindrome and because $xpx^{-1} = xpx\cdot x^{-1} x^{-1}$ for all generators $x$ and all $p\in P$.
In the quotient $F(S)/P$ every generator $s\in S$ squares to 1. Therefore $F(S)/P$ is a quotient of $\... | 14 | https://mathoverflow.net/users/3041 | 149345 | 79,961 |
https://mathoverflow.net/questions/149311 | 4 | Let $K$ be a field and $G=Gal(K\_s/K)$ is its absolute Galois group. Then, by Galois theory, the category of finite separable algebras over $K$ (denoted by $Sep(K)$) and the category of finite continuous $G$-sets with discrete topology are (denoted by $G-Set\_f$) anti-equivalent.
However, it is well known that the c... | https://mathoverflow.net/users/36961 | Exponentials in the opposite category of finite separable algebras | This has a very simple answer via Galois descent, and after giving it I will provide you with a broader algebro-geometric context (namely: Hom-schemes). Let $K'/K$ be a Galois extension such that $B\_{K'} := K' \otimes\_K B$ is a split $K'$-algebra (i.e., product of copies of $K'$); in other words, $K'$ is a common fin... | 12 | https://mathoverflow.net/users/39487 | 149346 | 79,962 |
https://mathoverflow.net/questions/149316 | 7 | This question is related to my question "[Forcing with c.c.c forcing notions, Cohen reals and Random reals](https://mathoverflow.net/questions/146800/forcing-with-c-c-c-forcing-notions-cohen-reals-and-random-reals)".
A natural way to answer Prikry's conjecture is to build a c.c.c. forcing notion which adds a minimal... | https://mathoverflow.net/users/11115 | A special c.c.c forcing notion and adding minimal generic reals | Groszek studied this and related forcings in *Combinatorics on ideals and forcing with trees* (JSL 52 (1987), 582–593; [MR0902978](http://www.ams.org/mathscinet-getitem?mr=902978)). (Note that Groszek allows each node $\sigma$ to have a different ideal, so yours is a special case of what Groszek denotes $L(\Sigma^+)$. ... | 5 | https://mathoverflow.net/users/2000 | 149351 | 79,963 |
https://mathoverflow.net/questions/149255 | 1 | It is known conditionally on GRH that, for all $\alpha<44.7632 \dots$, there are are only finitely many number fields $K$ satisfying $\Delta\_K ^{ 1/[K:\mathbb Q]}< \alpha$.
For which $\alpha$ can we make this effective by giving an upper bound on $[K:\mathbb Q]$ for all fields satisfying $\Delta\_K ^{ 1/[K:\mathbb Q... | https://mathoverflow.net/users/18060 | Effective bounds on root discriminant assuming GRH? | In fact, the answer is contained in one of the references in the survey Lucia mentioned. The relevant table is [here](http://www.dtc.umn.edu/~odlyzko/unpublished/discr.bound.table1).
| 2 | https://mathoverflow.net/users/18060 | 149353 | 79,964 |
https://mathoverflow.net/questions/149296 | 1 | By the Teichmuller uniqueness theorem, given a homeomorphism $f:X \rightarrow X$ where $X$ is the $n$-punctured sphere, there is a unique quasiconformal homeomorphism $g$ fixing $0$, $1$, and $\infty$ (in the same homotopy class as $f$) whose Beltrami coefficient $\mu$ has the smallest $L^{\infty}$ norm $x$.
My ques... | https://mathoverflow.net/users/3709 | A continuous version of Teichmuller uniqueness | Here are some details. First, in your setting, all the maps $g\_n$ are K-quasiconfirmal for a certain K. Thus, by the convergence property for qc maps, the sequence $(g\_n)$ has a convergent subsequence and the limit is qc. Clearly, the limit is homotopic to the maps $g\_n$. It suffices to show then that if $(g\_n)$ it... | 1 | https://mathoverflow.net/users/21684 | 149364 | 79,966 |
https://mathoverflow.net/questions/149331 | 6 | The matrix ring $k^{n\times n}$ can be realized in many ways as a quotient of a path algebra: For example choose the quiver $1\leftrightarrows 2 \leftrightarrows \cdots \leftrightarrows n-1\leftrightarrows n$ and impose the relations $i\to i\pm 1\to i = e\_i$ where $e\_i$ is the idempotent associated to the vertex $i$.... | https://mathoverflow.net/users/3041 | Homotopy of quivers | Once upon a time I noticed roughly this but didn't know what to do with it. I would rephrase as follows.
Any category $C$ has an associated "category algebra" $k[C]$ spanned by the morphisms of $C$ where the product of two morphisms is $0$ if they can't compose and their composition otherwise. This algebra is unital ... | 10 | https://mathoverflow.net/users/290 | 149368 | 79,967 |
https://mathoverflow.net/questions/149379 | 7 | Let $G$ be a connected, simply-connected complex semisimple linear algebraic group, and denote by $\mathcal{G}$ its affine Grassmannian. Fix a maximal torus $T\subseteq G$. We know that $\mathcal{G}$ has a natural stratification $\{\mathcal{G}^{\lambda}\}\_{\lambda\in\Lambda^+}$, where $\Lambda^+$ is the collection of ... | https://mathoverflow.net/users/25358 | Strata of the Affine Grassmannian | The general statement to know is, if $\mathbb G\_m$ acts on a smooth complete scheme $X$, then each Białynicki-Birula stratum is an affine bundle over its fixed-point set. This is in the original paper [B-B].
It's tempting to try to apply it to the ind-scheme $\mathcal G$, but that's infinite-dimensional, and more im... | 3 | https://mathoverflow.net/users/391 | 149381 | 79,972 |
https://mathoverflow.net/questions/149362 | 7 | For a prime gap of length at least $n$, a trivial upper bound for its first occurrence is $N=n!$ or $N=lcm(2,\dots,n)$. A bit better is $N=p\_1\cdots p\_n$ where $p\_k$ is the $k$th prime, as then $N+2,\dots,N+(p\_{n+1}-1)$ are all composite. In “real life” however, the first big gaps will occur much earlier, e.g. betw... | https://mathoverflow.net/users/29783 | Large gaps between consecutive irreducible polynomials with small heights | At the cost of having the degree be very large you can always choose a $k$-gap with coefficients in $\lbrace 0,1\rbrace$. Pick a large $n$ so that $n\equiv -j\pmod{p\_j}$, for all $1\le j\le k$. Where $p\_j$ is the $j$th prime. Let $P(x)=x^{a\_1}+\cdots+x^{a\_n}$ and choose the exponents $a\_i$ by the chinese remainder... | 9 | https://mathoverflow.net/users/2384 | 149383 | 79,973 |
https://mathoverflow.net/questions/149355 | 2 | Let $L$ be a line bundle over a compex manifold $X$, a square-root of $L$ is a line bundle $M$ such that $M^{\otimes2}=L$. My question is when the square-root of Line Bundle is unique?
| https://mathoverflow.net/users/nan | Uniqueness on square root of complex Line Bundle | Let's start with a counter-example. Take a one dimensional complex torus $X=\mathbb C/\mathbb Z\oplus \mathbb Z\tau$. Take a point which is a $2$-torsion in the group structure coming from $\mathbb C$, for instance the point $(\frac 12, \frac 12 \tau)$. The line bundle that corresponds to this point (if you don't know ... | 3 | https://mathoverflow.net/users/10076 | 149386 | 79,975 |
https://mathoverflow.net/questions/149391 | 4 | Let $\mathbb{k}$ be a field, $\mathfrak{g}$ be a finite-dimensional Lie algebra over $\mathbb{k}$.
In Bourbaki's "Lie Groups and Lie Algebras", Ch I, he defines four radical-like ideals of $\mathfrak{g}$:
1. the radical $\mathfrak{r}$, i.e. the maximal solvable ideal;
2. the radical of Killing form $\mathfrak{k}$, i.... | https://mathoverflow.net/users/43006 | On radicals of a lie algebra | If we let $\mathbb{k}$ be the field itself thought of as a Lie algebra with trivial bracket, then $\mathfrak{n}=\mathbb{k}$ since this is a nilpotent Lie algebra but $\mathbb{s}=\{0\}$, since the obvious representation of $\mathbb{k}$ be scalar multiplication is faithful. In general, these will never coincide for a nil... | 4 | https://mathoverflow.net/users/66 | 149392 | 79,976 |
https://mathoverflow.net/questions/149394 | 12 | It is known that all locally compact groups, and therefore compact groups, have a left-invariant Haar measure which is unique up to scalar constant, also a right-invariant one. Is there a strictly wider class of groups that has such a measure? What about weakening of these measures? What about hypergroups?
Thank you.... | https://mathoverflow.net/users/14555 | To what extent has the Haar measure been generalized? | For general locally compact hypergroups, the existence of a Haar measure is still an open problem as far as I know. It has been answered affirmatively for Abelian, compact, or discrete hypergroups and those arising as coset spaces from locally compact groups.
| 10 | https://mathoverflow.net/users/43014 | 149401 | 79,979 |
https://mathoverflow.net/questions/149403 | 2 | Let $R$ be a ring with identity. A matrix $A=[a\_{ij}] \in M\_n(R)$ is called a J-matrix if for any $i$, $a\_{ii} \not \in J(R)$ but for any $i \not = j$, $a\_{ij} \in J(R)$. Now suppose that every J-matrix is non-singular. Is it true that the ground ring $R$ is local ? (Note that the converse is clearly true)
| https://mathoverflow.net/users/41304 | Rings in which every J-matrix is non-singular | I might be interpreting your question wrong, but if not then I believe the answer is yes. This boils down to the fact that a ring is local if and only if the Jacobson radical $J(R)$ is maximal if and only if every element $x \notin J(R)$ is a unit. This characterization of local comes from the fact that the Jacobson ra... | 2 | https://mathoverflow.net/users/12402 | 149406 | 79,981 |
https://mathoverflow.net/questions/149313 | 3 | If I take $A^TA$, where $A$ is a full-rank random matrix (let's say with Gaussian-distributed independent entries), can I expect it to be positive-definite? It will be positive semi-definite trivially, since $x^TA^TAx = \|Ax\|^2$, so I guess it will not be positive-definite only if the random matrix $A$ is rank-deficie... | https://mathoverflow.net/users/40379 | Probability for a random positive-semidefinite matrix to not be positive-definite? | Not sure if this is what you are really after, but anyway: The paper by Rudelson and Vershynin that Igor Rivin [linked to](https://mathoverflow.net/a/149334/9652) contains lots of things which may be helpful for you. For example, for random $N\times n$ matrices ($N>n$) with iid Gaussian entries, there is Theorem 2.6 th... | 3 | https://mathoverflow.net/users/9652 | 149407 | 79,982 |
https://mathoverflow.net/questions/149217 | 1 | We consider the boundary value problem for potential in the form:
$$-\Delta u(\boldsymbol{x})=0,\quad \boldsymbol{x}\in \mathbb R^3\smallsetminus S,$$
with boundary conditions
$$\nabla u(\boldsymbol{x}) = -\delta g(\boldsymbol{x}), \boldsymbol{x}\in \partial S, \\
u(\boldsymbol{x}) \rightarrow 0,\quad as \quad |\bolds... | https://mathoverflow.net/users/42732 | Proof of regularity for bounded elliptic problem | On the first problem, if you mean
$$
-\Delta u=0\mbox{ in }\mathbb{R}^n\setminus \bar{B}\_R
$$
with $u\to 0$ as $|x|\to \infty$, that is on the exterior or a ball, not of a sphere, then regularity is standard.
Given your solution, it is trivial to prove that it is $C^\infty$ away from $B\_R$.
Then, you are looking at ... | 0 | https://mathoverflow.net/users/40120 | 149408 | 79,983 |
https://mathoverflow.net/questions/149360 | 11 | Let $\mathcal{E}$ be the minimal set of symbolic expressions (without any predefined meaning) such that
* The symbol $x$ is in $\mathcal{E}$, and
* If expressions $P,Q\in\mathcal{E}$, then the superscript expression $(P)^{(Q)}\in \mathcal{E}$.
For every expression $S\in\mathcal{E}$, define the *interpretation* of $... | https://mathoverflow.net/users/33664 | Is equivalence of functions built from nested exponentiations a decidable problem? | The problem is effectively decidable, and we will describe an algorithm. By way of preparation, we need to mention the o-minimality of the real exponential field, and Wilkie's solution to Tarski's High School Algebra problem.
The real exponential field $\mathbb{R}\_{exp}$ is the field of real numbers with an addition... | 11 | https://mathoverflow.net/users/5229 | 149442 | 79,993 |
https://mathoverflow.net/questions/149395 | 3 | Does there exist a finitely generated discrete group $G$ such that it has property (T), but for every $\varepsilon > 0$ there exists a generating set $S$ with the corresponding Kazhdan constant less than $\varepsilon$?
| https://mathoverflow.net/users/2192 | Uniform bounds on Kazhdan constants in groups | The answer is yes, by Gelander and Zuk: <http://www.ams.org/mathscinet/search/publdoc.html?pg1=INDI&s1=697297&vfpref=html&r=24&mx-pid=1910934>
| 4 | https://mathoverflow.net/users/33828 | 149444 | 79,995 |
https://mathoverflow.net/questions/149455 | 0 | Given the following PDE:
$$\partial\_t\Psi(x,t)=\partial\_{xx}\Psi(x,t)+k\partial\_x\Psi(x,t)+g(x,t)-\beta\Psi(x,t)=0$$
where:
$$g(x,t)=g\_0\left(\frac{a\_0}{2}+\sum\_{k=1}^\infty\left[a\_k\sin(kt)+b\_kcos(kt)\right]\right)$$ with:
$\Psi(0,t)=0,\Psi(L,t)=0$
and $\Psi(0,x)=\Psi\_0$, $\beta$ constant $(\beta\gt\gt1)$ and... | https://mathoverflow.net/users/21258 | Solution of a partial differential equation containing a Fourier series | Let us write the operator, with the notation $D\_x=-i\partial\_x$,
$$
L=D\_x^2-ik D\_x+\beta,\quad\text{the equation is}\quad\partial\_t\Psi+L\Psi=g.
$$
We start with noticing
$
\partial\_t-ik D\_x+\beta=e^{iktD\_x}e^{-t\beta}\partial \_t e^{t\beta}e^{-iktD\_x}
$
so that with $\Psi=e^{iktD\_x}e^{-t\beta} u$, we get the... | 0 | https://mathoverflow.net/users/21907 | 149457 | 79,999 |
https://mathoverflow.net/questions/149447 | 3 | Let $k$ be a field (I am mostly concerned with $k=\mathbb{Q}$) and let $A$ denote the group of affine transformations of $k$. In other words $A$ is (isomorphic to) the group
$$A=\left\{\left(\begin{array}{cc}x&y\\0&1\end{array}\right):x,y\in k;x\neq0\right\}$$
I am interested in injective endomorphisms of this group.
M... | https://mathoverflow.net/users/18698 | What are the endomorphisms of the group of affine transformations of a field? | The group $A = {\rm AGL}(1,k)$ is a semidirect product $N \rtimes H$, where $N$ is isomorphic to the additive group of the field, and H to its multiplicative group, and the action of $H$ on $N$ is defined by field multiplication - so it is fixed-point-free.
If $k$ does not have characteristic 2,then we have $-1 \in H... | 2 | https://mathoverflow.net/users/35840 | 149466 | 80,002 |
https://mathoverflow.net/questions/149468 | 7 | Assume that there exists a first order elliptic operator $D$ acting on functions from $\mathbb{R}^n$ to some vector space $V$. What can we conclude about $V$?
For example, is the dimension of $V$ always greater than $[n/2]$? Is $D$ necessarily some kind of modified Dirac operator? Can we somehow construct a pointwise... | https://mathoverflow.net/users/16702 | First order Elliptic operator | Here is a simple way of producing first order elliptic operators $\newcommand{\bR}{\mathbb{R}}$ $C^\infty(\mathbb{R}^n, W)\to C^\infty(\bR^n, W)$ with constant coefficients. Denote by $L(W)$ the space of linear operators $W\to W$. Consider a map $\newcommand{\si}{\sigma}$
$$\si :\bR^n\to L(W), $$
such that $\si(x)$... | 10 | https://mathoverflow.net/users/20302 | 149473 | 80,006 |
https://mathoverflow.net/questions/149453 | 0 | Let $x \in \mathbb{R}^p$ denote a $p$-dimensional data point (a vector). I have two sets $A = \{x\_1, \dots, x\_n\}$ and $B = \{x\_{n+1}, \dots, x\_{n+m}\}$, so $|A| = n$, and $|B| = m$. Given $k \in \mathbb{N^\*}$, let $d\_x^{(A, k)}$ denote the mean Euclidean distance from $x$ to its $k$ nearest points in $A$; and $d... | https://mathoverflow.net/users/21685 | Does this algorithm terminate in all scenarios? | Counterexample
==============
This algorithm does not terminate. Here is a counterexample, in dimension $p=2$, with $m=n=2$ points in each set, only looking at the $k=1$ nearest neighbours.
$$
A=\left\{
\begin{pmatrix}-2\\0\end{pmatrix},
\begin{pmatrix}0\\1\end{pmatrix}
\right\}
\qquad
B=\left\{
\begin{pmatrix}3\\0... | 2 | https://mathoverflow.net/users/25563 | 149480 | 80,010 |
https://mathoverflow.net/questions/149440 | 0 | I am interested in the behaviour of:
$\gamma\_k=\sum\_{i=0}^{k} {n \choose i}$
as n becomes large and where $k$ could potentially be a function of $n$ rather than a constant. One line of attack I can think of is to consider it as the cumulative distribution function of a Binomial Distribution and then approximating... | https://mathoverflow.net/users/43034 | Asymptotic behaviour of Binomial Sum | The book by Barbour et al (whose title escapes me at the moment) discusses much more complicated versions of this; you might want to take a look at it.
For $k$ much smaller than $n$, there is a pretty simple observation.
Since the coefficients form a strongly unimodal (aka log concave) sequence, tails decay at lea... | 1 | https://mathoverflow.net/users/42278 | 149483 | 80,012 |
https://mathoverflow.net/questions/149418 | 2 | Let $i\_1:X \hookrightarrow \mathbb{P}^n$ and $i\_2:Y \hookrightarrow \mathbb{P}^N$ be two projective schemes.
Let $f:X \to Y$ be a surjective projective morphism between smooth projective varieties over $\mathbb{C}$. Denote by $g$ the composition of $f$ with $i\_2$. Under what condition on $g$ can we conclude that th... | https://mathoverflow.net/users/32151 | Degree of a smooth projective variety | Jana, the current formulation of your question is still not right. Everything else fixed one can still choose a different $i\_1$ with which one can change the degree of $X$ (if $g$ is finite, this will not change the degree of the general fiber).
To get a sensible formulation you should do this: Indeed, degree of a ... | 8 | https://mathoverflow.net/users/10076 | 149493 | 80,015 |
https://mathoverflow.net/questions/149492 | 19 | Suppose I have a smooth manifold with a tangent bundle, and I have a connection. If this connection is curvature-free, is it guaranteed to be torsion-free? (I am not assuming a metric, just a finite-dimensional smooth manifold.)
I know that in general curvature-free connections do not exist, and that in general torsi... | https://mathoverflow.net/users/3319 | Can a manifold have a curvature-free connection that is not torsion-free? | Many manifolds have curvature-free (i.e., flat) connections on their tangent bundles. For example, any orientable $3$-manifold $M$ is parallelizable, i.e., its tangent bundle is trivial, so it carries a flat connection (in fact, many flat connections). However, nearly all of these connections will have torsion. In fact... | 31 | https://mathoverflow.net/users/13972 | 149496 | 80,017 |
https://mathoverflow.net/questions/149506 | 6 | Is there any list (incomplete of course) of knots, that have similar HOMFLY polynomials? I am mainly interested in torus knots.
| https://mathoverflow.net/users/43065 | Knots indistinguishable by HOMFLY | There is some list on Thistlethwaite's knot page, I think, see "the knot atlas". Kanenobu has shown that there are infinitely many distinct knots with the same HOMFLY polynomial:
Kanenobu, T. "Infinitely Many Knots with the Same Polynomial." Proc. Amer. Math. Soc. 97, 158-161, 1986.
For knots with "similar" HOMFLY... | 7 | https://mathoverflow.net/users/32332 | 149514 | 80,021 |
https://mathoverflow.net/questions/149361 | 7 | My question is the following: is there an analog of [Morita theorem](http://ncatlab.org/nlab/show/Morita%20equivalence#classical_morita_theorem_11) in the simplicial setting?
I mean, we can define two simplicial rings $A,B$ to be simplicially Morita equivalent is the categories of simplicial modules $Mod(A)$ and $Mod... | https://mathoverflow.net/users/32741 | Morita theorem for simplicial rings | Recall that if $A$ is a ring and $\mathcal{C}$ is a cocomplete $\mathsf{Ab}$-category, then cocontinuous $\mathsf{Ab}$-functors $F : \mathsf{Mod}(A) \to \mathcal{C}$ correspond to left $A$-module objects $(M,\theta)$ in $\mathcal{C}$ (i.e. $M \in \mathcal{C}$ and $\theta : A \to \mathrm{End}(M)$ is a ring homomorphism)... | 4 | https://mathoverflow.net/users/2841 | 149517 | 80,022 |
https://mathoverflow.net/questions/149524 | 6 | We often assume manifolds to be paracompact Hausdorff. Clearly, this implies normal.
However, there is a manifold (I mean locally Euclidean Hausdorff space) which is not paracompact. Without paracompactness, they are still regular because manifolds are locally compact, but does it imply normal?
The only example of ... | https://mathoverflow.net/users/36961 | "Abnormal" manifold | The fiberwise one-point-compactification of the tangent bundle of the long line (pick any smooth structure) is not normal.
The two distinguished sections of this $S^1$ bundle (the zero section & the section at infinity) cannot be separated by open subsets.
| 14 | https://mathoverflow.net/users/5690 | 149526 | 80,025 |
https://mathoverflow.net/questions/149527 | 9 | Does there exist infinitely many coprime pairs of integers x,d such that x, x+d, x+2d, x+4d are all square numbers?
One example would be 49,169,289,529. This is the only example I have found so far and I believe it may be the only one.
| https://mathoverflow.net/users/43036 | 4 squares almost in an arithmetic progression | The answer is yes. Here is another example,
$$
x=2021231^2, \qquad d=82153503191760.
$$
Then
$$
x+d=9286489^2, \qquad
x+2d=12976609^2, \qquad
x+4d=18240049^2.
$$
Basically, write $x=X^2$, $x+d=Y^2$, $x+2d=Z^2$, and $x+4d=W^2$. Eliminating $x$ and $d$ reduces to the two equations:
$$
X^2-2Y^2+Z^2=0, \qquad
3X^2-4Y^2+W... | 29 | https://mathoverflow.net/users/4140 | 149529 | 80,027 |
https://mathoverflow.net/questions/149507 | 1 | I was reading about the Kalman filter and I do not understand how it should be used when our measurements have a long term offset like GPS location updates do.
As I understand, the Kalman filter models our measurements by a linear combination of the state variables and additional error variables, where the errors are... | https://mathoverflow.net/users/43066 | Kalman filter with long term bias | The usual practice, I think, is to add a bias term to the state vector and use the Kalman filter to estimate that bias. The bias itself is often modeled as a random walk. Bias in gyroscope measurements is usually handled this way.
| 1 | https://mathoverflow.net/users/16042 | 149534 | 80,028 |
https://mathoverflow.net/questions/149531 | 2 | Given $n$ points $p\_1,\dots,p\_n$ in $S^2$ one gets a product link $L\_n=\{p\_1,\dots,p\_n\}\times S^1$ inside the closed 3-manifold $S^2\times S^1$, which can be looked at as a trivially framed link (by picking a tangent vector $v\_i$ at each $p\_i$ and dragging it along the factor $S^1$ to get the framing). I'm inte... | https://mathoverflow.net/users/8320 | Link surgery on $S^2\times S^1$ | The manifold you're describing has a very natural open book decomposition (see e.g. John Etnyre's [lecture notes](http://arxiv.org/abs/math/0409402)) in which the pages are the $n$-punctured spheres $(S^2 \backslash \cup\_i N(p\_i)) \times \{\*\}$ and the monodromy is trivial. We can view this as a Murasugi sum of n-1 ... | 4 | https://mathoverflow.net/users/428 | 149535 | 80,029 |
https://mathoverflow.net/questions/149539 | 7 | Is it still open whether ZFC+GCH is consistent with the statement that there are no $\aleph\_2$-Suslin trees?
| https://mathoverflow.net/users/11145 | $\aleph_2$ Suslin Hypothesis | From <http://www.users.muohio.edu/larsonpb/kly_guessing.pdf> (pdf 9 of 16):
>
> Note that the consistency of GCH + "no $\omega\_{2}$-Suslin-trees" is
> still an open question. ...this old question is now more open than
> ever.
>
>
>
As best as I can tell, the matter has not been settled in the six years sin... | 6 | https://mathoverflow.net/users/22971 | 149542 | 80,031 |
https://mathoverflow.net/questions/149476 | 6 | Let $f(z) = \sum\_{n=0}^\infty a\_nq^n$ be the fourier expansion of a (quasi-)modular form (with $q = e^{2\pi i z}$). Consider the following related functions:
$$f\_{m,k}(z) = \sum\_{n=0}^\infty a\_{mn + k}q^n$$
$$g\_{m,k}(z) = \sum\_{n=0}^\infty a\_{mn + k}q^{mn+k}$$
for, naturally, $0 \leq k < m$.
My question... | https://mathoverflow.net/users/1703 | Does the following operation on modular forms yield something modular? | I know nothing about quasimodular forms, so let me answer the question for modular forms in the classical sense.
First, your $f\_{m, k}$. If $k = 0$ this is essentially the image of $f$ under the Hecke operator $U\_m$, so it is always going to be modular of some level (and if f has level $\Gamma$, then $f\_{m, 0}$ wi... | 3 | https://mathoverflow.net/users/2481 | 149545 | 80,032 |
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