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https://mathoverflow.net/questions/220577 | 13 | The journal [Discrete Mathematics](http://www.journals.elsevier.com/discrete-mathematics/) contains a lot of short notes (i.e., less than 7 journal pages). What are some other journals that publish short notes in discrete mathematics? I've looked at other journals, but most of them seem to contain primarily long papers... | https://mathoverflow.net/users/79906 | Which journals publish short notes in discrete mathematics? | There are various *general* journals that are focused on "short papers." There a seven pages paper would fit just fine.
Examples include, Proceedings of the AMS, Bulletin of the LMS, Archiv der Mathematik. For more you could read [Which journals publish 1-page papers](https://mathoverflow.net/questions/39686/which-j... | 8 | https://mathoverflow.net/users/nan | 220579 | 103,444 |
https://mathoverflow.net/questions/220148 | 5 | Let $A\_{\alpha}\subset B(H)$ be a bunch of unital C\*-algebras acting on a Hilbert space $H$ given together with their character spaces $M(A\_{\alpha})$'s. A very nice theorem of Stephen C. Power identifies the character space of $C=C^{\ast}(\cup\_{\alpha}A\_{\alpha})$ the C\*-algebra generated by the union of $A\_{\a... | https://mathoverflow.net/users/81087 | Power's Theorem for irreducible representations | An important point to remember about Power's result is that if $\varphi$ is a character of $C$ then $\varphi|\_{A\_\alpha}$ will be a character of $A\_\alpha$. This does not happen for irreducible representations.
In particular, consider $C = \overline{\cup M\_{2^n}}$, the CAR algebra. Any non-zero irreducible repre... | 3 | https://mathoverflow.net/users/76593 | 220580 | 103,445 |
https://mathoverflow.net/questions/220062 | 34 | I wonder if it is possible to show, without using the Schmidt subspace/Roth theorem/Baker's bounds on linear forms in logarithms or other very deep results, that, in a number field, not all integral elements are sums of two units/most of them are not.
The known results that state that there are few elements that are ... | https://mathoverflow.net/users/10591 | Simple argument regarding sums of two units in a number field? | $\newcommand\p{\mathfrak{p}}$
$\newcommand\OL{\mathcal{O}}$
$\newcommand\P{\mathfrak{P}}$
Here is a solution which is essentially an elaboration on Felipe's answer.
Instead of working with squares, consider working with $m$th powers instead.
**Lemma:** If $(1 - v u^m)$ is exactly divisible by a prime $\p$ of $\OL\... | 18 | https://mathoverflow.net/users/81311 | 220587 | 103,446 |
https://mathoverflow.net/questions/220583 | 2 | Classical electromagnetism (with no sources) follows from the actions$$S = \int d^4x\left(-{1\over4}F\_{\mu\nu}F^{\mu\nu}\right),\text{ where }F\_{\mu\nu} = \partial\_\mu A\_\nu - \partial\_\nu A\_\mu.$$The Lagrangian for $A\_\mu$, including a gauge fixing term, is$$\mathcal{L} = -{1\over4}F^2 - {\lambda\over2}(\partia... | https://mathoverflow.net/users/nan | Gauge field quantization, electromagnetism | the answer is on page 189 of [Field Quantization](https://books.google.nl/books?id=VvBAvf0wSrIC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false) by Greiner & Reinhardt (their $\zeta$ is your $\lambda$):
| 1 | https://mathoverflow.net/users/11260 | 220589 | 103,448 |
https://mathoverflow.net/questions/220578 | 4 | How can one describe the orbits of the Lie group $G=\mathrm{SU}(2,1)$ in its Lie algebra $\mathfrak{g}=\mathfrak{su}(2,1)$ with respect to the adjoint representation?
| https://mathoverflow.net/users/4149 | Orbits in the adjoint representation of $SU(2,1)$ | A systematic method to do this for any real reductive group is given in Propositions 2.10 through 2.13 of Vogan's course [*The method of coadjoint orbits for real reductive groups*](http://www.ams.org/mathscinet-getitem?mr=1737729) (available from [his web page](http://www-math.mit.edu/~dav/paper.html)). As he says, yo... | 4 | https://mathoverflow.net/users/19276 | 220591 | 103,449 |
https://mathoverflow.net/questions/220595 | 9 | I have been reading about the topic motivated by a problem I read that asked for the first three digits of the sum of the LIS lengths in all permutations of length $n$. It is easy to see that we are really interested in the expected value of the LIS length in a random permutation of the first $n$ positive integers. A [... | https://mathoverflow.net/users/81313 | State of the art in the expected length of the Longest Increasing Subsequence of a random permutation | @user61318 and @JosephORourke, thanks for the advertisement for my book.
@chubakueono, as far as I know the answer to your questions are no and no. Pages 148-149 in my book have the state of the art (essentially the formula you wrote, along with some additional background) and the relevant references. However, in a b... | 13 | https://mathoverflow.net/users/78525 | 220614 | 103,459 |
https://mathoverflow.net/questions/220610 | 1 | I am facing the following problem. I have a function which is defined through a discrete sum of Gaussians
$$F\_M(t) = 2\sum\limits\_{n=1}^{M}e^{-t^2 \sigma^2 n^2}\times \sum\limits\_{k=n}^{M}p\_k p\_{k-n} + \sum\limits\_{k=0}^{M}p\_k^2$$
where $\sigma$ is just a real parameter and $p\_k$ are weights
$$\sum\limits\_{k=0... | https://mathoverflow.net/users/57699 | Discrete summation of Gaussian functions. Decay time problem | the decay time in your "simple case" is well approximated by the large-$M$ limit [\*]
$$\lim\_{M\rightarrow\infty}M\sigma\tau=2.84$$
here is a plot of
$$f\_M(s)=\left.\frac{F\_M(t)}{1-F\_M(\infty)}\right|\_{t=s/(M\sigma)}$$
for $M=5,10,100$, that shows the half-time $s\approx 3$ is quite accurate already for no... | 1 | https://mathoverflow.net/users/11260 | 220635 | 103,467 |
https://mathoverflow.net/questions/217969 | 18 | Let $X$ be a scheme over $\mathbb{C}$.
1. When does the topological space $X\left(\mathbb{C}\right)$ of $\mathbb{C}$-points have the homotopy type of a finite CW-complex?
2. When does the topological space $X\left(\mathbb{C}\right)$ of $\mathbb{C}$-points have the *weak* homotopy type of a finite CW-complex, (i.e. w... | https://mathoverflow.net/users/4528 | Homotopy types of schemes | Any scheme which is separated of finite type, has at least a triangulation, hence is, in particular, a CW-complex. In fact, by a theorem of [Lojasiewicz](http://www.numdam.org/numdam-bin/fitem?id=ASNSP_1964_3_18_4_449_0), this is true for any semi-algebraic set (one can even get this for subanalytic sets, by a result o... | 23 | https://mathoverflow.net/users/1017 | 220653 | 103,474 |
https://mathoverflow.net/questions/219737 | 8 | Fix some theory $T$.
Let $p$ be a type over some Model M and let $q$ be some global extension of $p$.
Note:
The number of global coheirs of $p$ is bounded by the number of ultrafilters on $M$.
Also if $T$ is stable, it follows that $q$ is a non-forking extension of $p$ if and only if $q$ is a coheir of $p$.
If $... | https://mathoverflow.net/users/47687 | Does non-stablity imply that there is a difference between non-forking and coheir extension | In general there are TP2 theories in which every global type non-forking over a small model is finitely satisfiable in it. An example of this is constructed in "On non-forking spectra", Artem Chernikov, Itay Kaplan, Saharon Shelah, <http://arxiv.org/abs/1205.3101>, Section 5.3.
Restricting to NTP2 theories, bounded n... | 4 | https://mathoverflow.net/users/25726 | 220655 | 103,475 |
https://mathoverflow.net/questions/220672 | 5 | I am very confused by the following and would appreciate any help.
Let $\mu\_p \subset \mathbb{G}\_m$ be the $p$-torsion subgroup scheme of the multiplicative group over $\mathbb{Z}\_p$. I would like to compute the Lie algebra of $\mu\_p$ (at the identity section) to make sure that I understand Lie algebras well. I h... | https://mathoverflow.net/users/63877 | Lie algebra and base change | The quotation you have heard is false because over an affine base $S = {\rm{Spec}}(k)$ for a commutative ring $k$ and a $k$-group scheme $G$, the Lie algebra is the linear dual ${\rm{Hom}}\_k(e^{\ast}(\Omega^1\_{G/k}),k)$ and that generally does not commute with non-flat base change when $G$ is not $k$-smooth. You have... | 10 | https://mathoverflow.net/users/81332 | 220680 | 103,485 |
https://mathoverflow.net/questions/220673 | 6 | I would like to know if there are any known examples of families of complex manifolds for which the Frölicher spectral sequence of one fibre degenerates on the $E\_m$ page and the spectral sequence of a different fibre degenerates on the $E\_n$ page for $m \neq n$.
Since the spectral sequence degenerates on the $E\_1... | https://mathoverflow.net/users/11084 | Does the degeneracy of the Frölicher spectral sequence vary in families? | For example, in Corollary 4.7 in "[Invariant complex structures on 6-nilmanifolds: classification, Frölicher spectral sequence and special Hermitian metrics](http://arxiv.org/pdf/1111.5873.pdf)" by Manuel Ceballos, Antonio Otal, Luis Ugarte, Raquel Villacampa (JGA), a family of complex non-Kähler structures such that t... | 6 | https://mathoverflow.net/users/29341 | 220691 | 103,489 |
https://mathoverflow.net/questions/220627 | 1 | Why is every l-adic Galois representation
$$G\_{\mathbb{Q}\_p}\rightarrow GL\_n(\mathbb{Q}\_{l})$$
conjugate to one over the l-adic integers?
$$G\_{\mathbb{Q}\_p}\rightarrow GL\_n(\mathbb{Z}\_{l})$$
| https://mathoverflow.net/users/70751 | Why is every l-adic Galois representation conjugate to one over the l-adic integers? | Recall that all ${\mathbb Z}\_l$-lattices in the ${\mathbb Q}\_l$-vector space ${\mathbb Q}\_l^n$ are conjugate in ${\rm GL}\_n ({\mathbb Q}\_l )$. Since the ${\rm GL}\_n ({\mathbb Q}\_l )$ stabilizer of the standard lattice ${\mathbb Z}\_l^n$ is ${\rm GL}\_n ({\mathbb Z}\_l )$, all lattices in ${\mathbb Q}\_l^n$ have ... | 4 | https://mathoverflow.net/users/4767 | 220692 | 103,490 |
https://mathoverflow.net/questions/220676 | 13 | Let $M$ be a manifold. The total Stiefel-Whitney class of $M$ is defined to be the Stiefel-Whitney class of the tangent bundle $TM$
$$
w(M)=1+w\_1(TM)+w\_2(TM)+\cdots
$$
I want to find references for
$$
w(SO(n)/SO(k)),(i.e. w(V\_{n-k}(\mathbb{R}^n))), \\
w(U(n)/U(k)),(i.e. w(V\_{n-k}(\mathbb{C}^n))), \\
w(Sp(n)/Sp(k)),... | https://mathoverflow.net/users/65800 | References for Stiefel-Whitney class of Stiefel manifolds and Grassmannians | The Stiefel manifolds are all parallelizable for $n-k\ge2$, so their total Stiefel-Whitney classes are equal to $1$. A reference is Theorem 3.1 of [this paper](http://www.ams.org/journals/tran/1975-213-00/S0002-9947-1975-0431194-X/home.html) by Kee Yuen Lam.
For the finite Grassmannians, things are a little more comp... | 9 | https://mathoverflow.net/users/8103 | 220706 | 103,496 |
https://mathoverflow.net/questions/157413 | 4 | Let $S$ be a smooth projective surface (I am mostly intrested in the case when $S$ is a product of curves, say $S=\mathbb{P}^1 \times \mathbb{P}^1$ but probably this is not important).
Consider a family of curves $X \subset S \times T$ parametrised by a variety $T$ of dimension 2 (the fibres $X\_t$ are distinct). De... | https://mathoverflow.net/users/2234 | base points of multiplicity $>1$ | I guess I post an answer to close the issue.
The statement is true in characteristic 0, but not in positive characteristic. Conterexamples in the latter case are easy to come by an $S=\mathbb{A}^1 \times \mathbb{A}^1= \mathbb{A}^2$, say, using curves with projections on one of the $\mathbb{A}^1$ factors everywhere ra... | 0 | https://mathoverflow.net/users/2234 | 220707 | 103,497 |
https://mathoverflow.net/questions/220721 | 1 | Assume $X$ and $Y$ are noetherian schemes over $\mathbb{C}$ and there is a proper and faithfully flat morphism $f: X\rightarrow Y$.
Assume the canonical morphism $F\xrightarrow{\sim} f\_{\*}f^{\*}F$ is an isomorphism for all $F\in Coh(X)$.
Can we say anything about the canonical morphism $f^{\*}f\_{\*}f^{\*}F\right... | https://mathoverflow.net/users/70593 | About the canonical morphism from $f^{*}f_{*}f^{*}F$ to $f^{*}F$ | In short: always.
Indeed, given a functor $F : \mathcal{C} \to \mathcal{D}$ left adjoint to $G : \mathcal{D} \to \mathcal{C}$, the triangle identities say that the composites of the canonical morphisms
$$F X \to F G F X \to F X$$
$$G Y \to G F G Y \to G Y$$
are identities for all $X$ in $\mathcal{C}$ and all $Y$ in $... | 5 | https://mathoverflow.net/users/11640 | 220723 | 103,502 |
https://mathoverflow.net/questions/220718 | 1 | Given a set $X$, a function $x \colon \mathbb{R} \to X$ is *periodic* if there exists $\tau>0$ such that $x(t+\tau)=x(t)$ for all $t \in \mathbb{R}$; and if $\tau$ is the smallest positive number with this property, we say that $\tau$ is the *least period* of $x$.
There obviously exist non-constant periodic functions... | https://mathoverflow.net/users/15570 | Can a (non-measurable) autonomous flow have a non-trivial periodic orbit without a minimal period? | Two questions were asked
Concerning question I:
Here's an application for my favorite bijection!
Consider R the group of reals with addition and its subgroup Q of rationals.
The group R mod Q has the same cardinality as R does (I guess you need the axiom of choice here).
Let $\phi$ be a bijection from R mod Q to R.
L... | 2 | https://mathoverflow.net/users/58307 | 220726 | 103,503 |
https://mathoverflow.net/questions/220662 | 21 | In Weil cohomology, a nice curve has cohomology up to degree 2, or more generally a nice $n$-dimensional variety has cohomology up to degree $2n$.
I know that this was motivated at least in part by a desire to extend cohomology of complex manifolds to algebraic varieties over other fields, and since complex curves ar... | https://mathoverflow.net/users/nan | Why should curves be two-dimensional? | This is a very subtle question, I think. First of all, the sense in which an $n$-dimensional algebraic variety $X$ acts as if it is "cohomologically $2n$-dimensional" is quite complicated--for example, unless one uses some notion of cohomology with compact support, an affine $n$-dimensional variety looks $n$-dimensiona... | 26 | https://mathoverflow.net/users/6950 | 220730 | 103,504 |
https://mathoverflow.net/questions/220709 | 8 | Assume that $X$ is a path-wise connected Hausdorff space, and assume that its fundamental group is non-trivial. Does it always exist a simple curve in $X$ which is non-null-homotopic?
Such curve does exist if we further assume that $X$ possesses a universal cover.
| https://mathoverflow.net/users/43681 | Existence of a non-null-homotopic simple closed curve | No. The harmonic archipelago is a compact $2$-dimensional counterexample, embedded in $\mathbb R^3$.
Let $S\_n$ denote the planar circle with center on the $x$-axis, and whose intersection with the $x$-axis is
$$
\left\{\Big(\frac{1}{n+1},0\Big), \Big(\frac{1}{n},0\Big)\right\}.
$$ Take the closure of the union of t... | 13 | https://mathoverflow.net/users/17029 | 220733 | 103,507 |
https://mathoverflow.net/questions/220536 | 8 | We know that every symmetric fusion category (SFC) gives rise to data
$N^{ij}\_k$ that describe the fusion of simple objects:
$i\times j = N^{ij}\_k k$, and the data $\theta\_i =\pm 1$ that describe the twist of simple objects.
My question is what are the conditions on the data $N^{ij}\_k,\theta\_i$ such that the dat... | https://mathoverflow.net/users/17787 | Conditions on the fusion data of symmetric fusion category | The information in the $\theta$'s is very weak, for example if a FC admits a symmetric structure with $\theta\_i=-1$ for at least one $i$, it also admits a symmetric structure with $\theta\equiv 1$. Even, it is possible to have several non-equivalent symmetric structures with $\theta\equiv 1$ over a fixed fusion catego... | 6 | https://mathoverflow.net/users/6517 | 220742 | 103,510 |
https://mathoverflow.net/questions/220345 | 10 | In my endless fiddling with formulas I discovered one that fills in the blanks in a generic formula I saw in a paper, but I'm wondering if maybe it's already known and the paper was just mentioning the form of it casually. The formula I saw, which has undetermined constants, expresses a Schubert polynomial in $n$ varia... | https://mathoverflow.net/users/62135 | Reduction formula for Schubert polynomials | This is essentially Theorem 4.6.1 in (<http://arxiv.org/pdf/alg-geom/9703001v1.pdf>) Schubert polynomials, the Bruhat order, and the geometry of flag manifolds by Bergeron and Sottile.
| 2 | https://mathoverflow.net/users/62135 | 220753 | 103,516 |
https://mathoverflow.net/questions/220755 | 2 | Why do we call the cone of curves(effective one cycles) on a variety $X$ as $NE(X)$, what does $NE$ stand for?
| https://mathoverflow.net/users/nan | Where does the name $NE(X)$ come from? | As far as I know, the notation appears first in Mori's landmark paper of 1982. He uses $N(X)$ for the group of 1-cycles modulo **N**umerical equivalence (tensored with $\mathbb{R}$), and then, quite naturally, $NE(X)$ for the convex cone spanned by classes of **E**ffective cycles in $N(X)$.
| 8 | https://mathoverflow.net/users/40297 | 220766 | 103,520 |
https://mathoverflow.net/questions/220770 | 12 | It is classically known that every positive integer is a sum of at most four squares of integers, i.e. every sum of squares of integers is a sum of four squares of integers. Now consider a symmetric $n\times n$ matrix $M$ with integer entries which can be written as $M= Q^{\rm T} Q$ for an $m \times n$ matrix $Q$ with ... | https://mathoverflow.net/users/36563 | "Pythagoras number" for integral matrices | Here is an answer (see the last point). It differs from what I had been claiming in my first post. There I was saying that any positive bilinear module $\Lambda$ over $\mathbf Z[\frac 12]$ was representable by some euclidean module $\mathrm{I}\_n\otimes\mathbf Z[\frac 12]$. This is true (with $n\leq \text{rk}(\Lambda)+... | 7 | https://mathoverflow.net/users/39552 | 220773 | 103,525 |
https://mathoverflow.net/questions/220376 | 2 | When applying Block's Theorem on the structure of differentiably simple rings to Lie algebras most authors require an algebraically closed field, but I can see no reference to algebraic closure in Block's paper. What extra does the algebraic closure give? I apologise for asking such a basic (and perhaps easy) question ... | https://mathoverflow.net/users/37902 | Where does the algebraic closure enter into Block's Theorem? | Block's theorem does not require the base field $k$ to be algebraically closed but one has to be careful when $k$ is imperfect. Then $k$ will admit field extensions of the form $K=k(a)$ with $a\not\in k$ and $a^p\in k$. Note that $K$ will become a truncated polynomial ring in one variable over a field extension $L=k(b)... | 4 | https://mathoverflow.net/users/24386 | 220778 | 103,527 |
https://mathoverflow.net/questions/220739 | 1 | I was validating the percentage of cases where the generic two parameter polynomial for Galois group ${A}\_{4}$ is valid. We have
\begin{equation\*}
{f}^{{A}\_{4}} \left({x, \alpha, \beta}\right) = {x}^{4} - \frac{6\, A}{B}\, {x}^{3} - 8\, x + \frac{1}{{B}^{2}} \left({9\, {A}^{2} - 12 \left({{\alpha}^{3} - {\beta}^{... | https://mathoverflow.net/users/62471 | Generic polynomial for alternating group ${A}_{4}$ is not correct | The Galois group is a subgroup of $A^4$ if and only if the discriminant is a perfect square. If you change $x^3$ to $x^2$ you get the discriminant to be:
$$\frac{1728^2 \left(b^2+3 b+9\right)^2 \left(a^3 (2 b+3)-3 a^2 \left(b^2+3 b+9\right)+\left(b^2+3 b+9\right)^2\right)^2}{\left(a^3-3 a \left(b^2+3
b+9\right)+2 b^3+... | 3 | https://mathoverflow.net/users/11142 | 220782 | 103,528 |
https://mathoverflow.net/questions/220451 | 0 | Given a Hilbert Manifold $M$ does there exist a smooth map into some very large Hilbert space taking geodesics to straight lines?
| https://mathoverflow.net/users/36886 | Embedding Riemmanian Manifold Linearly | No this is not possible, for example let $M$ be such that there exists a compact geodesic $\gamma: [0,1]\rightarrow M$ (for example $M=S^n$). n this case if such a map $\phi:S^n\rightarrow V$ did exist (where $V$ is a linear space) then $\phi\circ\gamma: [0,1] \rightarrow V$ must also be compact, however a line in $V$ ... | 0 | https://mathoverflow.net/users/36886 | 220783 | 103,529 |
https://mathoverflow.net/questions/220700 | 8 | I'm looking for a book/paper where the conformal compactification of Kerr spacetime is calculated. I've seen plenty of reference for the Minkowski, but none (explicitly calculated) for Kerr.
Thank you.
PS: prior to writing this post, I was already referred to "Large scale structure of spacetime" by Hawking and Elli... | https://mathoverflow.net/users/51137 | Conformal compactification of Kerr spacetime | See Section 8 of Pretorius and Israel, "Quasi-spherical light cones of the Kerr geometry". <http://arxiv.org/abs/gr-qc/9803080> The paper contains quite a bit more of course. But I want to point out that in the expression you find there, $R^2$ is fairly reasonable as a quantity in the usual (say, Boyer-Lindquist) coord... | 5 | https://mathoverflow.net/users/3948 | 220789 | 103,532 |
https://mathoverflow.net/questions/220740 | 7 | I've encountered a strange situation while thinking about modular curves... Consider the modular curve $Y(3)$ parametrizing elliptic curves with a symplectic basis for their 3-torsion. This curve has degree 12 over the $j$-line $Y(1)$. Let $y\in Y(1)$ be a $\mathbb{Q}$-rational point, then its fiber in $Y(3)$ has degre... | https://mathoverflow.net/users/15242 | confounding riddle about fine moduli schemes and twists of elliptic curves | $\newcommand{\QQ}{\mathbb{Q}}$
$\newcommand{\ZZ}{\mathbb{Z}}$
Okay, so the solution appears to be this (Thanks to Ari Shnidman, Joseph Silverman, nfdc23, and eric for their comments)
Fix an $N\ge 3$. Let $y\in Y(1)$ be a $\QQ$-point, then the fiber $Y(N)\_y$ of $Y(N)$ above $y$ is a $\QQ$-algebra $A$ of degree $d\_... | 2 | https://mathoverflow.net/users/15242 | 220808 | 103,543 |
https://mathoverflow.net/questions/220813 | 13 | The Posner-Robinson theorem states that, if $X$ is noncomputable, there is some $G$ such that $X\oplus G=G'$; that is, even though genuine jump inversion only works above $0'$, *every* (nontrivial) $X$ is "almost" the jump of something. There are a number of extensions and variations of the Posner-Robinson theorem; I'm... | https://mathoverflow.net/users/8133 | Woodin on Posner-Robinson for the hyperjump and sharp |
>
> [MR2449474 (2009j:03067)](http://www.ams.org/mathscinet-getitem?mr=2449474) Woodin, W. Hugh. *A tt version of the
> Posner-Robinson theorem*. Computational prospects of infinity. Part II.
> Presented talks, 355–392, Lect. Notes Ser. Inst. Math. Sci. Natl.
> Univ. Singap., 15, World Sci. Publ., Hackensack, NJ, ... | 13 | https://mathoverflow.net/users/6085 | 220816 | 103,546 |
https://mathoverflow.net/questions/220832 | 0 | Let $T$ be an operator from a Banach space $X$ into a Banach space $Y$ and $1\leq p<\infty$. If $ST$ is compact for any operator $S$ from $Y$ into $l\_{p}$, Is $T(X)$ separable? Or under what conditions on $X,Y$, this question is true?
| https://mathoverflow.net/users/41619 | On the separability of operator range | No. Consider the identity operator on $\ell\_r(S)$ with $p<r<\infty$ and $S$ uncountable. Or consider any weakly compact operator with non separable range into a $C(K)$ space or an $L\_1$ space and use the Dunford-Pettis property of these spaces.
| 1 | https://mathoverflow.net/users/2554 | 220835 | 103,554 |
https://mathoverflow.net/questions/220828 | 12 | Let $\pi$ be a generic irreducible admissible representation of $GL\_n(L)$, where $L$ is a $p$-adic field, $R$ is its ring of integers, and $\mathfrak{p}$ is its prime ideal. The *conductor* of $\pi$ has the following definition (follwoing Jacquet, Pietetski-Shapiro, and Shalika): fix a non-negative integer $r$, and le... | https://mathoverflow.net/users/30726 | Growth of dimension of fixed spaces in $GL_n(\mathbb{Q}_p)$-representations | For question 1, you are asking about the theory of oldforms (or oldvectors). I think the canonical reference for $\mathrm{GL}\_n$ is "[Oldforms on $\mathrm{GL}\_n$](https://www.jstor.org/stable/2374790)" by Mark Reeder. In particular, he discusses how to find a basis of $\pi^{K(r')}$ if one already has a basis of $\pi^... | 11 | https://mathoverflow.net/users/3803 | 220836 | 103,555 |
https://mathoverflow.net/questions/220797 | 4 | Let $X\_t$ be a Brownian motion or a Brownian Bridge on a (\edit: compact) Riemannian manifold. Let $T>0$ be given.
The question is: Does there exists a constant $C>0$ such that for all partitions $0 = \tau\_0 < \tau\_1 < \dots < \tau\_N \leq T$, we have
$$ \mathbb{E}\left[ \exp \left(\sum\_{j=1}^N d(X\_{\tau\_{j-1}... | https://mathoverflow.net/users/16702 | Exponential of approximate quadratic variation of Brownian motion | Yes, there is a bound like that for $T \le \mathrm{const}$. I'll do the case of Brownian motion, since the Brownian bridge reduces to it.
The proof consists of two stages: proving the bound in hyperbolic space and reducing the general case to it.
In order to redue the general case to the constant curvature case we ... | 2 | https://mathoverflow.net/users/22758 | 220838 | 103,556 |
https://mathoverflow.net/questions/220834 | 4 | Given that, with integers $t \geq 1$ and $q \geq 3,$ there are solutions to $$ x^2 - q x y + y^2 = - t q $$
with integers $x,y \geq 1,$ I was able to show that
$$ q \leq 1 + \frac{324}{25} t^2. $$
If there is any solution $(x,y),$ there are infinitely many, as this is an indefinite quadratic form in $(x,y).$ Hurwitz... | https://mathoverflow.net/users/3324 | optimal bound in diophantine representation question | $x^2+y^2=kq$ for some positive integer $k$. Since $x\ge1$ and $y\ge1$, this implies $xy\ge\sqrt{kq-1}$. Then $$-tq=x^2-qxy+y^2\le kq-q\sqrt{kq-1}$$ which says $-t\le k-\sqrt{kq-1}$, $\sqrt{kq-1}\le t+k$, $kq-1\le t^2+2kt+k^2$, $$q\le{t^2\over k}+2t+k+{1\over k}$$ If $k=1$, this gives $q\le t^2+2t+2$, as desired. In gen... | 3 | https://mathoverflow.net/users/3684 | 220844 | 103,559 |
https://mathoverflow.net/questions/220539 | 10 | Let $\mathcal{C}$ and $\mathcal{D}$ be two equivalent categories. Furthermore, assume $\mathcal{C}$ is enriched over a monoidal category $(\mathcal{M}, \otimes)$. Can one use the equivalence to enrich $\mathcal{D}$ over $(\mathcal{M}, \otimes)$?
| https://mathoverflow.net/users/81287 | Enriching categories and equivalences | I think most category theorists would answer "yes, obviously", and not bother to write down a proof. But presumably that isn't sufficiently convincing, since you ask the question, so let me try to make it a bit more explicit with some big words. (-:
An $M$-enriched category with set of objects $A$ is equivalently a l... | 9 | https://mathoverflow.net/users/49 | 220851 | 103,561 |
https://mathoverflow.net/questions/220786 | 11 | Working in $\sf ZFC$, is it provable, or at least consistent (say, over $L$), that you have $\aleph\_1$ forcings, $\Bbb P\_\alpha$ such that:
1. $\Bbb P\_\alpha$ is c.c.c.
2. $\Bbb P\_\alpha$ adds a real which determines the generic.
3. For every countable $A\subseteq\omega\_1$, and $\alpha\notin A$ the finite suppor... | https://mathoverflow.net/users/7206 | Can you have many independent reals? | First add $\omega\_1$ Cohen reals, then partition this set of Cohen reals into $\omega\_1$ disjoint sets $A\_i$ each of size $\omega\_1$. Let $P\_i$ be a sigma centered forcing whose generic is a real coding a meager set covering $A\_i$. Then, it is easy to check that the family $\{P\_i : i < \omega\_1\}$ is as require... | 7 | https://mathoverflow.net/users/2689 | 220853 | 103,562 |
https://mathoverflow.net/questions/220746 | 3 | Any submersion $f: M → N$ defines a foliation of M whose
leaves are the connected components of the fibres of $f$. Foliations
associated to the submersions are called simple foliations. The foliations
associated to submersions with connected fibres are called *strictly
simple*. A simple foliation is strictly simple pre... | https://mathoverflow.net/users/81421 | When are simple foliations strictly simple? | By Tsemo's answer, only one direction is still open. Let $B$ be the leaf space with the quotient topology, and let $g\colon M\to B$ and $p\colon B\to N$ be the natural continuous maps with $f=p\circ g$. Then $B$ is second countable. Assume that $B$ is Hausdorff, too. If we show that $p$ is a local covering, then $B$ in... | 0 | https://mathoverflow.net/users/70808 | 220855 | 103,564 |
https://mathoverflow.net/questions/220491 | 5 | Let a $3$-dimensional subspace $V$ of $\mathbb{R}^4$ be $$V=\{(x\_1,x\_2,x\_3,x\_4)\in\mathbb{R}^4\mid\sum\_{i=1}^4x\_i=0\}.$$ The alternating group $A\_4$ acts on $V$ by
$$\sigma(x\_1,x\_2,x\_3,x\_4)=(x\_{\sigma(1)},x\_{\sigma(2)},x\_{\sigma(3)} ,x\_{\sigma(4)})$$ for any $\sigma\in A\_4$.
Since $V$ is linearly iso... | https://mathoverflow.net/users/65800 | triviality of Whitney sums of a vector bundle | There is a simple general argument showing that $\xi$ is trivial: Take any homogeneous space $G/H$ and any representation $V$ of $G$ and restrict the representation to $H$. Then the homogeneous vector bundle $G\times\_H V\to G/P$ is a trivial as a vector bundle. A trivialization can be written down explicitly. It is in... | 5 | https://mathoverflow.net/users/64141 | 220859 | 103,566 |
https://mathoverflow.net/questions/220845 | 15 | I know the following theorems by Serre:
1, The 2-dim l-adic representation associated to a non-CM elliptic curve is open.
2, The 2-dim l-adic representation associated the weight-12 cusp form $\Delta$ has open image (even before Deligne's construction of 2-dim l-adic representations).
So is there any general theo... | https://mathoverflow.net/users/42690 | When is the image of a 2-dim l-adic representation associated to a modular form open | This is more subtle than it looks. I asked exactly the same question some years back (see [here](https://mathoverflow.net/questions/24076/when-do-the-galois-reps-of-modular-forms-have-open-image)); but I'm not going to flag this question as duplicate, because the answer that was given to my question at the time, which ... | 18 | https://mathoverflow.net/users/2481 | 220865 | 103,568 |
https://mathoverflow.net/questions/220863 | 2 | Let $k$ be a field of characteristic $0$ (not necessarily algebraically closed), and let $A=k[x^1,\ldots,x^m]/(f\_1,\ldots,f\_N)$ be an affine variety which is a complete intersection; i.e. $\dim(A)=m-N$. (For definiteness, let's assume that $m-N\geq 3$.)
Is it always possible to find an affine hypersurface $B=k[x^1,... | https://mathoverflow.net/users/15488 | Singularities of complete intersections of affine varieties with hypersurfaces | This follows from Bertini's Theorem: [see for example here](https://www.encyclopediaofmath.org/index.php/Bertini_theorems).
Note that your variety does not need to be projective. Let $\overline{k}$ be the algebraic closure of $k$. You know that there is an open dense subset of hyperplanes in $(\mathbb{P}\_{\overline{k}... | 1 | https://mathoverflow.net/users/36563 | 220871 | 103,571 |
https://mathoverflow.net/questions/220870 | 3 | Because of my interest in [this question](https://mathoverflow.net/q/108530), I listed the subgroups of ${\frak S}\_n$ for $1\le n\le4$. I found that the number of subgroups are, respectively, $1,2,6,24$. It might be a coincidence, or it could reveal a deep connection.
>
> Is it always the case that ${\frak S}\_n$ ... | https://mathoverflow.net/users/8799 | The number of subgroups of ${\frak S}_n$ | As Francesco Polizzi mentions, the answer is no alredy for ${\frak S}\_4$: there are $30$ subgroups, but $4!=24$.
Here are some more (very small) calculations:
\begin{array}{|c|c|c|c|}
\hline
\mathrm{group}& \mathrm{\# subgroups} & n! \\ \hline
{\frak S}\_1 & 1 &1\\ \hline
{\frak S}\_2 & 2 &2\\ \hline
{\frak S}\_3 ... | 14 | https://mathoverflow.net/users/43108 | 220874 | 103,572 |
https://mathoverflow.net/questions/217994 | 3 | Suppose I have two simplicial based topological spaces $X\_\bullet$ and $Y\_\bullet$, and the degeneracy maps of each satisfy the based homotopy extension property (but not necessarily the unbased homotopy extension property. That is, they may not be degreewise well-pointed spaces). Will a simplicial map which is a deg... | https://mathoverflow.net/users/45846 | Geometric Realizations of Simplicial Based Spaces | This is going to be the same example the one from this previous question: <https://mathoverflow.net/a/171423/360>
Let $A = \Bbb N$ and $B = \{0,1,1/2,1/3,1/4,\dots\}$, with the map $f: X \to Y$ given as follows:
$$
f(n) = \begin{cases}
1/n & n > 0\\
0 & n = 0
\end{cases}
$$
Let $g: X\_\bullet \to Y\_\bullet$ be the m... | 3 | https://mathoverflow.net/users/360 | 220880 | 103,575 |
https://mathoverflow.net/questions/24076 | 8 | Suppose *f* is a newform (with coefficients generating some number field E), and $\rho\_{f,\lambda}: {\rm Gal}(\overline{\mathbb{Q}} / \mathbb{Q}) \to {\rm GL}\_2(E\_\lambda)$ the associated Galois rep (for some prime $\lambda$ of E). When does $\rho$ have open image in ${\rm GL}\_2(E\_\lambda)$?
This clearly isn't t... | https://mathoverflow.net/users/2481 | When do the Galois reps of modular forms have open image? | The answer of TSG is not correct in all cases. For a complete and detailed answer see this later [question](https://mathoverflow.net/questions/220845/when-is-the-image-of-a-2-dim-l-adic-representation-associated-to-a-modular-form?answertab=votes#tab-top) and the answer of David Loeffler (the very OP of the current ques... | 4 | https://mathoverflow.net/users/9317 | 220888 | 103,578 |
https://mathoverflow.net/questions/208849 | 6 | For a given real quadratic field $K$, the group of units of its ring of integers is $\mathcal{O}\_K^{\times}\cong(\pm1)\times \mathbb{Z}$ by the Dirichlet unit theorem. For each $\mathcal{O}\_K$, pick the fundamental unit as $\epsilon >1$, then $\epsilon=m\sqrt{d}+n$, where $m,n>0$ are integers or half integers. Now fo... | https://mathoverflow.net/users/31134 | Counting fundamental units of real quadratic fields | First, let's count
$$ v(x) = \sum\_{1 < \mu < x} 1 $$
where $\mu$ ranges over every unit greater than $1$ of every real quadratic field.
Setting $\mu = m \sqrt{d} + n$, we require $n^2 - m^2 d = \pm 1$. Given that equation, the inequalities are equivalent:
$$ 1 < \mu < x \Longleftrightarrow 1 < n < \frac{x^2 \p... | 4 | https://mathoverflow.net/users/nan | 220901 | 103,583 |
https://mathoverflow.net/questions/220895 | 7 | I was investigating the idea of fractional derivatives and devised the following definition. WHich definition is it equivalent to and can I have a reference for it?
$$\frac{d^n}{dx^n}f(x) = \lim\_{h \to 0} \frac{\sum\_{i = 0}^\infty (-1)^i\binom{n}{i} f(x - ih)}{h^n} $$
| https://mathoverflow.net/users/75293 | A definition of the fractional derivative | For $\alpha\in (0,1)$ the derivative of order $\alpha$ of $f(x)$ is defined to be (see Section I.5.5 in the first volume of the book *Generalized Functions* by Gelfand and Shilov)
$$\frac{d^\alpha}{dx^\alpha} f(x):=\frac{1}{\Gamma(1-\alpha)} \int\_0^xf'(\xi)(x-\xi)^{-\alpha} d\xi. $$
One can define derivatives of a... | 13 | https://mathoverflow.net/users/20302 | 220911 | 103,586 |
https://mathoverflow.net/questions/220858 | 28 | First of all, I am neither a physicist nor a mathematician. And I am afraid that mathoverflow is not a suitable place for my question, but having asked similar questions on math SE it is obvious that this question is not appropriate for math.SE.
As far as I have searched in mathematical physics literature, historical... | https://mathoverflow.net/users/81462 | How and why did mathematicians develop spin-manifolds in differential geometry? | I can't give a comprehensive history (if you don't get that here, you might try [hsm.se]---a lot of mathematicians are active on that site), nor can I explain how or why the theory of spin manifolds first emerged. But I think I can say something about how and why spin manifolds became important.
The pre-history is an... | 18 | https://mathoverflow.net/users/4362 | 220913 | 103,587 |
https://mathoverflow.net/questions/63100 | 2 | Hello
I am trying to derive the second equation displayed in section 7.1 (or p. 41) of this [article](http://ta.twi.tudelft.nl/users/vuik/numanal/kort_afst.pdf) or equation (6.3) of this [book](http://books.google.co.uk/books?id=0dyagVg20XQC&pg=PA182&dq=6.3+conditional+sampling+copula&hl=en&ei=AvrOTayfGMuBhQfV3-HyDA... | https://mathoverflow.net/users/9404 | Proof of conditional copula relation to the marginal copulas | Have a look at:
<http://publications.rwth-aachen.de/record/59254/files/04_198.pdf>
in Section 2.2, the derivation is done for you.
| 1 | https://mathoverflow.net/users/81494 | 220926 | 103,593 |
https://mathoverflow.net/questions/220912 | 6 | An operator ideal $\mathfrak J$ is a class of continuous operators. Namely, for every pair of complex Banach spaces, $\mathfrak X,\mathfrak Y$, we have that $\mathfrak J(\mathfrak X,\mathfrak Y) \subseteq \mathfrak L(\mathfrak X,\mathfrak Y)$ is a closed two-sided ideal, which means
\begin{align\*}
1.& \ \ \ A,B\in \m... | https://mathoverflow.net/users/76593 | Is every ideal part of an operator ideal? | The answer is yes; take
$$ \mathfrak{I}(\mathfrak{Y},\mathfrak{Z} ) = {\rm span}\{ T \in \mathfrak{L}(\mathfrak{Y},\mathfrak{Z}) \mid \exists U \in \mathfrak{L}(\mathfrak{Y},\mathfrak{X}) , \exists V \in \mathfrak{L}(\mathfrak{X},\mathfrak{Z}) , \exists S \in \mathfrak{I}(\mathfrak{X}) , T= VSU \} $$
More generall... | 5 | https://mathoverflow.net/users/848 | 220929 | 103,595 |
https://mathoverflow.net/questions/220923 | 15 | While trying to prove some identities for generating functions, I ended up needing to show that
$$\sum\_{p=1}^n \exp\left(\frac{i\pi p l}{2m}\right)\prod\_{\substack{k=1\\k\neq p}}^n\frac{1}{\sin\left(\frac{\pi (k-p)}{2m}\right)} \stackrel{?}{=} 0$$
for integers $m \geq 1$, $2\leq n\leq 2m$, and $l = -n+2,-n+4,\ld... | https://mathoverflow.net/users/47484 | Why does $\sum_{p=1}^n \exp\left(\frac{i\pi p l}{2m}\right)/\prod_{k=1,k\neq p}^n\sin\left(\frac{\pi (k-p)}{2m}\right)$ vanish? | Let us consider the case when $n$ is odd. Let $$P(x):=\exp\left(\frac{ixl}{2}\right)\left(\exp\left(\frac{ix}{2}\right)-\exp\left(\frac{-ix}{2}\right)\right),$$
and notice that the degree of $P$ as an exponential polynomial equals $$\max\left(\left|\frac{l+1}{2}\right|,\left|\frac{l-1}{2}\right|\right)\leq\frac{n-1}{2}... | 16 | https://mathoverflow.net/users/2384 | 220933 | 103,597 |
https://mathoverflow.net/questions/220930 | 0 | Suppose we have a pullback of topological spaces (CW-complexes) $B\rightarrow A\leftarrow C$ which I will denote by $D$.
**Assumptions**
1. The induced map $D\rightarrow C$ is a trivial fibration
2. The map $f:B\rightarrow A$ has weakly contractible fibers i.e., for any $a\in A$ we have $f^{-1}(a)\simeq \ast$
3. Th... | https://mathoverflow.net/users/21369 | fiber, homotopy fiber of spaces | Counterexample:
Let $A=[0,1]$ with the usual topology, $B=C=[0,1]^{\delta}$ (this means the discrete topology). The maps $f:B \to A$ and $g:C \to A$ are the identity. Then $f^{-1} (t) = \*$ for all $t \in A$, but $f$ is not a homology isomorphism, and $g$ is surjective in homology. Furthermore, the pullback $D$ is th... | 6 | https://mathoverflow.net/users/9928 | 220942 | 103,599 |
https://mathoverflow.net/questions/220946 | 3 | Let $f$ be a modular function (that is, a meromorphic modular form of weight 0) holomorphic on $\mathcal{H}$ which is invariant under $\Gamma\le SL\_2(\mathbb{Z})$ (not necessarily congruence!), and not invariant under any larger group.
Is $\mathbb{C}(j)[f]$ precisely the function field $\mathbb{C}(\Gamma)$ of the mo... | https://mathoverflow.net/users/15242 | Does a modular function primitive for $\Gamma$ generate the function field of $\mathcal{H}/\Gamma$? | I think the answer to your question is yes.
If $\Gamma$ is a finite index subgroup of $\Gamma\_1 = \mathrm{PSL}\_2(\mathbb{Z})$, then there exists a finite index normal subgroup $\tilde{\Gamma}$ of $\Gamma\_1$ such that $\tilde{\Gamma} \subset \Gamma \subset \Gamma\_1$. Now the extension of function fields $\mathbb{C... | 4 | https://mathoverflow.net/users/6506 | 220962 | 103,606 |
https://mathoverflow.net/questions/220952 | 4 | Is there a clean proof that the $L\_n$, localization at $E(n)$, is simply rationalization (i.e. $L\_0$) on Eilenberg-MacLane spectra? Eric Peterson asked this [here](http://chat.stackexchange.com/transcript/message/19941162#19941162), but I haven't seen an answer.
| https://mathoverflow.net/users/nan | Localization at the Johnson-Wilson spectrum and rationalization | First, recall that any rational spectrum is a wedge of suspended copies of $H\mathbb{Q}$. It follows that $H\mathbb{Q}$ is a retract of $E(n)\mathbb{Q}$ and so is $E(n)$-local. From this we see that the canonical map $H\to H\mathbb{Q}$ factors uniquely through $L\_nH$.
Over $\pi\_\*(E(n)\wedge H/p)$ we have a formal ... | 11 | https://mathoverflow.net/users/10366 | 220963 | 103,607 |
https://mathoverflow.net/questions/220966 | 7 | Given $c\in (0,1)$ and a graph $G=(V,E)$ such that any subset $U\subset V$ contains an independent subset of cardinality at least $c|U|$. Does it allow to bound the chromatic number $\chi(G)$ by the constant depending only on $c$? If yes, what is the best constant?
UPDATE. For $c\geq 1/2$ we get that $G$ is bipartite... | https://mathoverflow.net/users/4312 | Chromatic numbers of nowhere dense graphs | Take the [Kneser graph](https://en.wikipedia.org/wiki/Kneser_graph) $K(2k+r,k)$, defined as the graph where the vertices are the $k$-element subsets of $\{1,2,\dots,2k+r\}$, and there is an edge between two vertices if the corresponding sets are disjoint. It is not hard to prove that we can take any $c < \frac{1}{2+\fr... | 9 | https://mathoverflow.net/users/2384 | 220967 | 103,609 |
https://mathoverflow.net/questions/220968 | 7 | I begin by clarifying that the "higher-dimensional" in my question refers to analogues of Artin L-functions over higher dimensional base schemes than $\mathrm{Spec}(\mathbb{Z})$.
Now for the set-up. This will be very similar to the set-up of Chapter 9 of Serre's book "Lectures on N\_X(p)".
Let $R$ be a finite type fl... | https://mathoverflow.net/users/5101 | Higher-dimensional Artin L-functions | According to Serre in "Zeta and L-functions" (MR0194396), these L-functions have been defined by Artin himself. Moreover, it seems that in the formula
$$
L(s,\rho) = \prod\_{\substack{\text{closed points} \\ x \in X}}\mathrm{det}\left(I - \frac{\rho(\mathrm{Frob}\_x)}{N(x)^s}\right)^{-1}
$$
one should define $\rho(\mat... | 3 | https://mathoverflow.net/users/6506 | 220985 | 103,613 |
https://mathoverflow.net/questions/220817 | 5 | let $\mathbb{H}$ be the hyperbolic plane and let $k(t,x,y)$ be the associated heat kernel.
I am wondering, if for any fixed $y\in M$ and $\epsilon >0$ the function $u\_t(x):=k(t,x,y)$ is continuous in $S:=\lbrace x\in M : d(x,y)>\epsilon \rbrace$ uniformly in $t\in(0,\infty)$?
For instance in Euclidean space $\math... | https://mathoverflow.net/users/21870 | Is the heat kernel for the hyperbolic plane uniformly continuous in $t\in(0,\infty)$? | It follows from the interior Schauder estimates for parabolic equations. Namely, put for simplicity's sake $y=0$, $k(t,x)=k(t,x,0)$ and for negative values of $t$ continue $k$ by zero: $k(x,t)=0$, $t<0.$ For $r>0$ denote $B\_r(x\_0)=\{x\in \mathbb{H}^2|d(x,x\_0)<r\}$ a ball on the plane and $B\_r=B\_r(0)$. Then $k(x,t)... | 4 | https://mathoverflow.net/users/14551 | 220988 | 103,614 |
https://mathoverflow.net/questions/220811 | 8 | In the study of partial differential equations, it is often considered enough to analyze the principal symbols and their characteristic variety (see for example, <http://www.sciencedirect.com/science/article/pii/S0001870802000993>, and <https://en.wikipedia.org/wiki/Symbol_of_a_differential_operator>). In the first lin... | https://mathoverflow.net/users/46856 | Characteristic Variety of the Principal Symbol solves PDE system? | (*This really should be a comment, but it was too long.*)
Part of the problem is that when you're dealing with a *system* of (linear) PDE's, the principal symbol is not the "right" object. Instead, you want to look at the characteristic ideal. It's defined as follows:
Let $D$ be the ring of linear partial differen... | 6 | https://mathoverflow.net/users/36720 | 220990 | 103,615 |
https://mathoverflow.net/questions/220994 | 2 | Good evening, I'm trying to find an asymptotic of this sum:
$$\sum\_{j=0}^n (-1)^j {n \choose j} (n - j)^n = n^n - {n \choose 1} (n - 1)^n + {n \choose 2} (n - 2)^n + ... + (-1)^n {n \choose n} (n - n)^n $$
I think there is no close form. But I don't know how to calculate an asymptotics than. Maybe I should try to ... | https://mathoverflow.net/users/81528 | Asymptotic of a sum involving binomial coefficients | Separate the different roles that $n$ plays in this sum, and look at
$$\sum\_{j=0}^n (-1)^j \binom{n}{j} (x-j)^n.$$
If $f(x)=f\_n x^n + (\mbox{lower order terms})$ is any polynomial of degree $n$, then $\sum\_{j=0}^n (-1)^j \binom{n}{j} f(x-j)$ is $n! f\_n$. So your sum is equal to $n!$, and asymptotics are given by St... | 6 | https://mathoverflow.net/users/297 | 220996 | 103,618 |
https://mathoverflow.net/questions/220998 | 2 |
>
> Let $G$ be a graph with disjoint copies of $K\_{1,3}$. Prove that if there are uncountably many copies of $K\_{1,3}$ in $G$, then $G$ is not planar.
>
>
>
I have a proof of this statement by contradiction i.e. assuming it is planar with uncountably many copies of $K\_{1,3}$ in $G$, but I am not satisfied. I ... | https://mathoverflow.net/users/nan | Planarity of infinite graphs | This is a 1928 theorem of R. L. Moore "[Concerning Triods in the Plane and the Junction Points of Plane Continua](http://www.jstor.org/stable/85527)" PNAS Vol. 14, No. 1 (Jan. 15, 1928), pp. 85-88. Greg Kuperberg gave a [nice proof](https://mathoverflow.net/a/27248/297) of it here on MO.
| 3 | https://mathoverflow.net/users/297 | 221002 | 103,620 |
https://mathoverflow.net/questions/220943 | 5 | This question might be too conceptual.
Congruences between modular forms (due to Shimura, Hida, etc) are really amazing. I know that the eigencurve construction are closely related to these relations. The basic reference is "The Eigencurve" by Coleman and Mazur. Besides, I think "A brief introduction to the work of ... | https://mathoverflow.net/users/42690 | Congruences between modular forms and the eigencurve construction | Here's a theorem about congruences between modular forms:
*Theorem*. Let $f$ be a (normalised) eigenform of weight $k$ and level $\Gamma = \Gamma\_1(N) \cap \Gamma\_0(p)$. Then for any $r$, and any $k'$ sufficiently close (\*) to $k$, there exists an eigenform of weight $k'$ and level $\Gamma$ that is congruent to $f... | 5 | https://mathoverflow.net/users/2481 | 221008 | 103,624 |
https://mathoverflow.net/questions/220969 | 10 | Let $f$ be a modular form -- more specifically, a normalized new eigenform which is not of CM type.
We say $f$ has **extra twists** if there exists some $\sigma \in \operatorname{Aut}( \mathbf{C})$ such that the Galois conjugate $f^\sigma$ is equal to the twist of $f$ by some non-trivial Dirichlet character, so $a\_n... | https://mathoverflow.net/users/2481 | Do "most" modular forms have no extra twists? | If a newform $f \in \mathcal{S}\_k^{\mathrm{new}}(\Gamma\_0(N),\varepsilon)$ has an inner twist by some $\sigma \in \operatorname{Aut}(\mathbb{C})$, then $f^{\sigma}$ is a newform of the same level as $f$. Moreover, if $\varepsilon$ is trivial, then so is the nebentypus of $f^{\sigma}$ (see (3.8) of [Ribet's paper](htt... | 11 | https://mathoverflow.net/users/3803 | 221020 | 103,629 |
https://mathoverflow.net/questions/220796 | 41 | The Lubin-Tate theory gives an amazingly clean and streamlined way of constructing the subfield (usually denoted) $F\_\pi\subset F^\mathrm{ab}$ for a local field $F$ fixed by the Artin map associated to the prime element $\pi$ (i.e. such that $F^{\mathrm{ab}}=F\_\pi\cdot F^{\mathrm{un}}$ with the usual notations). The ... | https://mathoverflow.net/users/44812 | Motivating Lubin-Tate theory | Sorry I didn’t see this earlier. My memory is vague, and probably colored by subsequent events and results, but here’s how I recall things happening.
Since I had read and enjoyed Lazard’s paper on one-dimensional formal group (laws), which dealt with the case of a base field of characteristic $p$, I decided to look a... | 86 | https://mathoverflow.net/users/11417 | 221032 | 103,634 |
https://mathoverflow.net/questions/221039 | 1 | I am interested in using degree theory to examine some semilinear problems.
But instead of just looking for solutions lets assume i am looking for a certain class of solutions; for instance lets consider just stable solutions. Is there some way to adjust the usual degree theory so that it can be applied in the restric... | https://mathoverflow.net/users/66623 | degree theory for elliptic equations; special solutions | The whole point of degree theory is its topological invariance. Now consider a saddle node bifurcation where you have a change from no solution to a stable and an unstable solution. This example should be enough to convince you that a "degree" which counts only stable solutions cannot exist.
| 2 | https://mathoverflow.net/users/12120 | 221042 | 103,638 |
https://mathoverflow.net/questions/221043 | 2 | Before stating my question I would like to provide afew motivating examples:
**Examples:**
1. In the category of Finitely-generated-projective $R$-modules, we have that:
$M^{\vee}:=Hom\_R(M,R)$ satisfies: $Hom\_R(M^{\vee},R)\cong M$.
2. If $G$ is a locally compact abelian topological group over the circle group $T$... | https://mathoverflow.net/users/36886 | Creating Duals in A Category | Here is a construction that covers the first example but not, I think, the other two. Suppose $C$ is a [closed](http://ncatlab.org/nlab/show/closed+monoidal+category) symmetric monoidal category with unit object $1$ and that $c$ is a [dualizable object](http://ncatlab.org/nlab/show/dualizable+object) in $C$. Then the d... | 5 | https://mathoverflow.net/users/290 | 221044 | 103,639 |
https://mathoverflow.net/questions/221047 | 6 | Let $a,b \in \mathbb{N} \ \ s.t. \ \ a > b$ have $\gcd(a,b) =1$. We can define the Hirzebruch-Jung modified euclidean algorithm as follows:
Let $e\_i \in \mathbb{N} >2$, and $ r\_k \in \mathbb{N}$ where $0\le r\_k<r\_{k-1}$. Then we can do the following procedure (analogous to the standard Euclidean algorithm):
$$ \b... | https://mathoverflow.net/users/41585 | Motivation for Hirzebruch-Jung Modified Euclidean Algorithm | I think the motivation for the Hirzebruch-Jung algorithm is not the algorithm itself, but the fact the it yields *directly* a very interesting continued fraction expansion. It is those, now called **Hirzebruch-Jung continued fractions**, that have a wide number of applications. A bit more on that later.
Using your no... | 4 | https://mathoverflow.net/users/43108 | 221052 | 103,642 |
https://mathoverflow.net/questions/221035 | 3 | Let $M$ be a manifold and $\pi : E \to M$ a rank $n$ vector bundle on $M$. We can define a connection on $E$ in two ways:
* We can specify the covariant derivatives $\nabla\_X s$ or
* We can choose a connection form $\Phi \in \Omega^1({\rm Fr} \, E) \otimes \mathfrak{gl}\_n$ where ${\rm Fr} \, E$ is the frame bundle... | https://mathoverflow.net/users/4002 | A question about curvature for linear connections | There are many textbooks on Differential geometry answering your question in detail, for example the Kobayashi-Nomizu book, or in not so much detail, e.g. Roe's "Elliptic operators...".
Concerning question 1: Basically, $\Omega$ and $R$ are the same: $R$ is a 2-form on $M$ with values in the endomorphism bundle of $... | 2 | https://mathoverflow.net/users/4572 | 221055 | 103,643 |
https://mathoverflow.net/questions/221059 | 1 | I am trying to find a Poisson bracket on an algebra, and need to find a solution to a system of equations. The system of equations is very complicated, with more than 10000 equations and 60 variables.
The following is a part of the system of the equations:
\begin{align}
& 6 z\_{2} z\_{10} - 12 z\_{4} + 6 z\_{3} z\_{1... | https://mathoverflow.net/users/11877 | How to solve this system of equations? | It seems this is a system of quadratic equations, so try to write it as an equation $z^T M z = 0$ with $z = (z\_0,...,z\_{\text{whatever}})$ and a symmetric matrix $M$ (it's always possible to choose $M$ symmetric). If the linear terms aren't typos, you can include them via $z = (1,z\_0,...,z\_{\text{whatever}})$.
As... | 7 | https://mathoverflow.net/users/81400 | 221065 | 103,646 |
https://mathoverflow.net/questions/221036 | 1 | Which class of reflexive spaces $X$ having the property: if a sequence $(x\_{n})\_{n}\subset B\_{X}$ converges to $x$ weakly and $\|x\_{n}\|\rightarrow 1$, then the norm of $x$ must be 1. Of course, the classical sequence spaces $l\_{p}$ do not have this property.It seems that this condition is too strong.
| https://mathoverflow.net/users/41619 | A question about weak convergence on the unit ball of a reflexive space | I think the following space which is called Grothendieck space may be partial answer for your question.
<https://en.wikipedia.org/wiki/Grothendieck_space>
| -1 | https://mathoverflow.net/users/11966 | 221070 | 103,647 |
https://mathoverflow.net/questions/221056 | 7 | Let $G$ be a graph in which any two odd cycles have a common vertex. It is easy to see that $\chi(G)\leq 5$ (choose minimal odd cycle $C$, use two colors for $G\setminus C$ and three colors for $C$). And this is sharp as $K\_5$ shows.
May we improve this bound to $\chi(G)\leq 4$ assuming additionally that $G$ does not ... | https://mathoverflow.net/users/4312 | Graphs in which any two odd cycles have a common vertex | The claim on graphs without $K\_5$ is a particular case of the (still open in general) [Erdos--Lovasz Tihany conjecture](http://www.math.ucsd.edu/~erdosproblems/erdos/newproblems/DecomposeToIncreaseChromaticNumber.html). (Tihany is not a surname, but the name of a peninsula on Balaton lake in Hungary.)
This particula... | 14 | https://mathoverflow.net/users/17581 | 221071 | 103,648 |
https://mathoverflow.net/questions/221085 | 19 | Suppose we have a curvature-like tensor $R\in \wedge^2 T^\*\_p M \otimes T^\*\_p M \otimes T\_p M$ on a manifold $M$, that is $R(X,Y)Z = - R(Y,X)Z$. How does one determine whether or not this is a curvature tensor for some torsion-free affine connection?
One obvious condition is that the first Bianchi identity $R(X,Y... | https://mathoverflow.net/users/3172 | When is curvature given by a connection? | Yes, there are typically many further conditions. In dimension $n$, the space of curvature-like tensors that satisfy the first Bianchi identity are the sections of a bundle of rank $\tfrac13n^2(n^2{-}1)$, while the space of torsion-free connections is the space of sections of an affine bundle of rank $\tfrac12n^2(n{+}1... | 28 | https://mathoverflow.net/users/13972 | 221095 | 103,658 |
https://mathoverflow.net/questions/220824 | 5 | Suppose I have a set of $k$ points $\{x\_1,x\_2,\ldots,x\_k\}$ in $\mathbb{R}^n$ that I can project into $\mathbb{R}^m$ with the linear operator $\mathcal{P}$, with $\alpha, \beta, \ldots$ parameters of the projection operator. Is there a well known best method for determining the class of functions $f:(\mathbb{R}^m)^k... | https://mathoverflow.net/users/81448 | Is there a class of functions acting on a set of projected points that remain invariant under changes in projection parameters? | I can think of (at least) two ways of interpreting this question.
**First:** You are given some specific list of $k$ points $x\_1$, $x\_2$, ..., $x\_k$ in $\mathbb{R}^n$, and you want to detect whether $k$ points $y\_1$, ..., $y\_k$ in $(\mathbb{R}^m)^k$ could be a linear projection of the original $k$ points.
**S... | 3 | https://mathoverflow.net/users/297 | 221099 | 103,659 |
https://mathoverflow.net/questions/221097 | 5 | I am trying to determine a metric for measuring cluster stretch. Let $C$ be a cluster of points $P\_0, P\_1,...,P\_n$ in a two dimensional space with the same units.
I need a metric that will allow me to differentiate clusters that are long and thin from other clusters.
I imagine it like a function that would be c... | https://mathoverflow.net/users/81577 | Determining the stretch of a cluster of points | There are many *roundness* measures that have been explored for different applications, which may give you ideas.
A good source for roundness in image processing is this paper, which analyzes
and compares several different measures:
>
> Ritter, Nicola, and James Cooper. "New resolution independent measures of circu... | 8 | https://mathoverflow.net/users/6094 | 221104 | 103,662 |
https://mathoverflow.net/questions/221023 | 4 | Let $B\_t$ be a standard Brownian motion. Let$$M\_n = \max\{|B\_t - B\_{n-1}| : n - 1 \le t \le n\}.$$For which $r > 0$ is it the case with probability one, for all $n$ sufficiently large$$M\_n \le r\sqrt{\log n}?$$
| https://mathoverflow.net/users/81544 | For which $r > 0$ is it the case with probability one, for all $n$ sufficiently large $M_n \le r\sqrt{\log n}$? | The condition $r>\sqrt 2$ is also sufficient. The random variables $(M\_n)\_{n\geq1}$ are iid (in fact, we need that they are identically distributed), due to the independence of the increments and the translation invariance of Brownian motion. We shall use the first Borel-Cantelli lemma to prove that for $r>\sqrt 2$,
... | 3 | https://mathoverflow.net/users/8966 | 221105 | 103,663 |
https://mathoverflow.net/questions/221107 | 3 | Assume that $ab \neq 0$. What is $$N\_p := \text{card}\{(x, y, z, t) \in (\textbf{F}\_p)^4 : ax^4 + by^4 + z^2 + t^2 = 0\}?$$I need this result, but unfortunately I am not a number theorist. Could anyone provide me a reference/supply a computation? Thanks!
| https://mathoverflow.net/users/81544 | $N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2 = 0\}?$ | We need to distinguish according to whether $p$ is congruent to $1$ or $3$ modulo $4$, and whether $-b/a$ is or is not a fourth power modulo $p$. (Note that the case $p=2$ is trivial since $ax^4+by^4+z^2+t^2 \equiv ax+by+z+t \pmod{2}$, so this gives exactly $2^3=8$ solutions modulo $2$.)
We need to use the following
... | 8 | https://mathoverflow.net/users/17907 | 221112 | 103,664 |
https://mathoverflow.net/questions/220246 | 34 | Recall that a Kan extension is called *pointwise* if it can be computed by the usual (co)limit formula, or equivalently if it is preserved by (co)representable functors.
I have seen pointwise Kan extensions defined in many texts, such as Mac Lane's book or Borceux's book, but these texts don't typically prove any the... | https://mathoverflow.net/users/2362 | What is the point of pointwise Kan extensions? | I have always thought that pointwise Kan extensions are better than normal Kan extensions because you can actually compute them using a (co)end
| 7 | https://mathoverflow.net/users/4002 | 221142 | 103,675 |
https://mathoverflow.net/questions/221111 | 3 | I have a question about the following definition:
A probability measure $\mu$, such that the Markov semigroup $e^{Lt} \in \mathcal{L}(L^2)$ exists and is symmetric, satisfies the Sobolev inequality iff for some $p \in (2, \infty)$ and two finite constants $(a,b) \in [0,\infty)^2$, we have
$$\|f\|\_p^2 \le a \|\Gamm... | https://mathoverflow.net/users/77929 | Markov-semigroup Sobolev inequality | You can find many details and references on this inequality (and related ones) in Chapter 6 of the recent book [Analysis and Geometry of Markov diffusion operators](http://doi.org/10.1007/978-3-319-00227-9) by Bakry, Gentil and Ledoux. The constants $a$ and $b$ do indeed depend on $p$. In this book the inequalities are... | 3 | https://mathoverflow.net/users/54789 | 221143 | 103,676 |
https://mathoverflow.net/questions/221076 | 4 | Let $L(n,k)$ be the increasing $k$-tuples from $\{1,\dots,n\}$, listed in lexicographic order.
Eg, for $n=9$, $k=3$, the sequence $L(n,k)$ would be:
$$(1,2,3), (1, 2, 4), (1, 2, 5),\dots,(7, 8, 9).$$
The question is: given that a $k$-tuple $(a\_1,\dots,a\_k)$ is in position $N$ in $L(n,k)$, with $a\_k<n$, is the... | https://mathoverflow.net/users/81567 | Lexicographic order on increasing $k$-tuples | Write all $k$-tuples from $\lbrace 0,\dots,n-1\}$ in *reverse*
lex order. E.g., for $n=5$ and $k=3$ we get $012, 013, 023, 123, 014,
024, 124, 034, 134, 234$. Call the terms $x\_0,x\_1,\dots$, so for the
above example $x\_5=024$ (short for $(0,2,4)$). Now suppose that
$(a\_1,a\_2,\dots,a\_k)=x\_N$. Then $N={a\_k\choose... | 7 | https://mathoverflow.net/users/2807 | 221154 | 103,681 |
https://mathoverflow.net/questions/221144 | 1 | Consider the set $\mathcal{G}\_v$ of all finite simple graphs on a given set of $v$ vertices. Let $m={v\choose 2}$ for sake of notation. Given an identification of $\{1,\dots,m\}$ with the set of 2-element subsets of $\{1,\dots,v\}$, there is a natural bijection $\Phi$ from $\mathcal{G}\_v$ onto the $m$-hypercube, $Q\_... | https://mathoverflow.net/users/47707 | Simple Graphs and Automorphisms of the Hypercube | So let $H$ be the group of automorphisms $f$ of $Q\_m$ such that $f(x)\simeq x$ as graph for every $x\in Q\_m$ ($x$ being viewed as a graph using the bijection $\mathcal{G}\_v\to Q\_m$). If $v\le 2$, then $H$ is reduced to the trivial group.
Assume $v\ge 3$. Then $H$ contains the group $K\simeq\mathfrak{S}\_m$ induce... | 2 | https://mathoverflow.net/users/14094 | 221156 | 103,682 |
https://mathoverflow.net/questions/221147 | 0 | In this comprehensive [answer](https://mathoverflow.net/questions/45004/kahler-structure-on-flag-manifolds) to an old question, it is stated that
>
> Flag manifolds exhaust all compact homogeneous Kähler manifolds corresponding to a compact connected semi-simple Lie group.
>
>
>
Firstly, where can I find a pr... | https://mathoverflow.net/users/60986 | Classifying compact homogeneous Kähler manifolds | The comments make your second question moot unless reformulated, right?
For the first, this goes back to [Borel (1954, Thms 1 & 2)](http://www.pnas.org/content/40/12/1147.citation); more details in e.g. [Serre (1954, Thms 1,2,3 and remark following Thm 1)](http://www.numdam.org/item?id=SB_1951-1954__2__447_0), [Matsu... | 2 | https://mathoverflow.net/users/19276 | 221160 | 103,685 |
https://mathoverflow.net/questions/221168 | 6 | Let $d,m, r$ be positive integers, and define
$$
S = \left\{ (i\_1, i\_2, \dots, i\_m) \in {\bf Z}\_{+}^{m} \left | \sum\_j i\_j = d; \& \forall j, i\_j \leq r \right. \right\};
$$
Here ${\bf Z}\_+$ denotes the set of nonnegative integers (that is, including zero). So we are taking all (ordered) integer partitions of $... | https://mathoverflow.net/users/42278 | Sums of reciprocals of products of factorials | I'll just make my comment above an answer. The sum in question multiplied by $d!/m^d$ can be interpreted as the probability that when $d$ balls are thrown into $m$ boxes then each box contains no more than $r$ balls (the balls are thrown into boxes at random uniformly and independently).
This problem arose earlier on... | 6 | https://mathoverflow.net/users/38624 | 221178 | 103,691 |
https://mathoverflow.net/questions/221128 | 1 | Let $G\_n$ be the $4$-regular tree of depth $n$, that is to say the finite graph given by the ball of radius $n$ in the Cayley graph of the free group on two generators. By the root I mean the vertex at the center. If I choose vertices uniformly at random from $G\_n$, what is the probability that when I choose the root... | https://mathoverflow.net/users/30721 | Probability of paths to the boundary of a tree | It converges to a strictly positive limit.
Perhaps easiest to think about it in this way; assign every vertex an independent time which is uniform on $[0,1]$. If the vertices of $G\_n$ are added in increasing order of their times, then this is equivalent to adding them one by one uniformly as you describe. But this w... | 4 | https://mathoverflow.net/users/5784 | 221179 | 103,692 |
https://mathoverflow.net/questions/221086 | 7 | Operate in ZFC. Can we find a function-class $\phi$ whose domain is the class of ordinals such that the following properties hold?
* If $x \in \phi(\alpha)$, then either $x \in \mathbb{N}$ or there exists some ordinal $\beta < \alpha$ with $\phi(\beta) = x$;
* If $\phi(\alpha) \subseteq \phi(\beta)$, then $\alpha = \... | https://mathoverflow.net/users/39521 | Ordinal-indexed transitive antichain of sets with urelements | Adam has shown that Vopěnka's principle implies that there is no
such $\phi$ as in the question. Let me prove this conclusion under
a weaker hypothesis, a large cardinal assumption weaker than VP.
Specifically, I claim that if there is a stationary proper class
of Woodin cardinals, then there is no such $\phi$ as in th... | 6 | https://mathoverflow.net/users/1946 | 221180 | 103,693 |
https://mathoverflow.net/questions/221146 | 2 | Consider $ E(x)$ some smooth function on $ \Omega$ (some smooth bounded domain in $ R^N$) and suppose $E=0$ on $ \partial \Omega$.
Suppose one knows that there is some $C\_1,C\_2 \in R$ such that
$ x \mapsto C\_1 | \nabla E(x)|^2 + C\_2 \Delta E(x)$ is constant on $ \partial \Omega$. I am interested in what one can... | https://mathoverflow.net/users/66623 | comparing Laplacian and gradient of function on boundary | For any smooth $\Omega$ and $E$ one can perturb $E$ so that the desired conditions hold. Indeed, let $d$ be a global smooth function agreeing with the distance from $\partial\Omega$ in a neighborhood of the boundary. Let $F$ be any smooth function on the boundary, globally extended so that near the boundary it is const... | 5 | https://mathoverflow.net/users/16659 | 221184 | 103,695 |
https://mathoverflow.net/questions/221188 | 2 | I have a question:
$$\min\_x {1\over 2} \|x-t\|^2 + \lambda \|x\|\_\infty$$
where $t, \lambda$ are given constant.
I think this may be a classic problem? However, I didn't find closed form of its solution. What I tried is as follows.
I tried transforming it into a constrainted version:
$$\min\_{x,s} {1\over 2} \|x-t\... | https://mathoverflow.net/users/81629 | L-infinity-norm regularized proximity problem | This is indeed a classic problem. Recall the more general problem of computing the *prox operator* of an lsc convex function $f$, i.e.,
\begin{equation\*}
\text{prox}\_f(y) := \operatorname{argmin}\quad \tfrac12\|x-y\|^2 + f(x).
\end{equation\*}
For this prox operator, we have the well-known *Moreau decomposition*
\beg... | 6 | https://mathoverflow.net/users/8430 | 221190 | 103,699 |
https://mathoverflow.net/questions/221130 | 8 | Let $P$ be a simple convex polytope in $\mathbb{R}^d$ (that is, any vertex belongs to exactly $d+1$ facets). Given the collection of outer normals to facets of $P$, combinatorics of $P$ may be different, of course. But is this information enough to reconstruct number of faces of $P$ of all dimensions? If yes, what is s... | https://mathoverflow.net/users/4312 | $f$-vector of simple convex polytope via directions of facets | Consider a simplex $S$ in five dimensions, and one more face normal $\pi$ in general position with respect to $S$ (no hyperplane perpendicular to $\pi$ passes through more than one vertex of $S$). Intersect $S$ with a halfspace $H$ perpendicular to $\pi$, and translate $H$ continuously towards $S$, starting from a posi... | 8 | https://mathoverflow.net/users/440 | 221192 | 103,701 |
https://mathoverflow.net/questions/155920 | 3 | Consider the following problem: given a finite set $S$ of intervals on the line and a number $k$. We need to colour this set in $k$ colours so that the measure of the set of points, which are contained in two or more intervals coloured with the same colour, is minimal for given $k$. By colouring a set $S$ in $k$ colour... | https://mathoverflow.net/users/38324 | Minimize the length of intersection of the set of intervals | Let $I\_1,\ldots,I\_n$ be the intervals in $S$. For simplicity, we may assume that the intervals are open and that all $2n$ endpoints are distinct. The latter property can be ensured by changing the intervals slightly, which changes the value of the objective function by an arbitrarily small amount. Furthermore, we may... | 2 | https://mathoverflow.net/users/81636 | 221195 | 103,703 |
https://mathoverflow.net/questions/221177 | 8 | I am studying finite index subgroups of certain finitely presented groups. The particular conditions on my groups make this problem easier than I phrase it here, but I am curious about a more general answer. After the main question I will indicate a simplified case for which there is probably a simplified answer.
**S... | https://mathoverflow.net/users/14835 | What is the most efficient way to factor a matrix into a given set of generators? | As to your general question, there is a method which is better
than the inefficient solution you give. -- Namely, compute spheres of radii
$r = 1, 2, \dots$ with respect to the word metric about the identity and
about the element $m$ to be factored, until these spheres intersect
nontrivially. This way you always get t... | 4 | https://mathoverflow.net/users/28104 | 221209 | 103,709 |
https://mathoverflow.net/questions/219775 | 12 | A theory $T$ is called *near model complete* if every formula is equivalent to a Boolean combination of existential formulas mod $T$. I wonder whether there is an equivalent "semantic" definition of this notion like model completeness. (A theory is model complete if every formula is equivalent to an existential formula... | https://mathoverflow.net/users/80940 | Near model completeness | ~~I'm still thinking about your main question. Meanwhile here is my answer to your additional question.~~
**A sandwich criterion by Kueker and Turnquist**
You may have seen this before, but it's still worth stating here as it may well be the best answer possible. Proposition 2.3 in [Nearly Model Complete Theories](... | 5 | https://mathoverflow.net/users/nan | 221212 | 103,711 |
https://mathoverflow.net/questions/221214 | 14 | One can define fields using a finite list of axioms that quantify over the field itself. However, the obvious way to define algebraically closed fields involves either an infinite list of axioms, or a more logically complex framework where one can quantify over natural numbers. It would be interesting if one could avoi... | https://mathoverflow.net/users/10366 | Logical complexity of algebraically closed fields | From Dirk van Dalen's Logic and Structure: the theory of algebraically closed fields is [not finitely axiomatizable](https://books.google.com/books?id=u0wlXPHATDcC&pg=PA108&lpg=PA108&dq=algebraically+closed+finitely+axiomatizable&source=bl&ots=meDLA-Mu8t&sig=U1zZBmNrOjLCpUgqtWc6yTyKAHw&hl=en&sa=X&ved=0CEAQ6AEwBGoVChMIp... | 15 | https://mathoverflow.net/users/2926 | 221220 | 103,713 |
https://mathoverflow.net/questions/210771 | 4 | Is the importance of developing forking machinery in the way we set it up, or is it in the fact that it allows us to come up with a notion of independence via the properties of non-forking? I'm currently reading Baldwin's Fundamentals of Stability Theory and would like to know if I need to have a very deep understandin... | https://mathoverflow.net/users/nan | Non-Forking and Related Concepts | For stable theories, it's really enough to know that the forking machinery exists and what properties it has. However, there are an awful lot of properties being used all the time (in stable contexts), which may or may not be obvious depending on which characterisations you are familiar with. And there is an awful lot ... | 6 | https://mathoverflow.net/users/nan | 221223 | 103,715 |
https://mathoverflow.net/questions/220216 | 7 | Pardon if this is well known. Suppose I have a (say complex) connected reductive group $G^{\vee}$ with the $\tilde{\Delta}=\Delta\cup\{\alpha\_0\}$ being the simple roots plus the negative highest root of $G^{\vee}$. Any proper subset of $\tilde{\Delta}$ defines a connected reductive subgroup $H^{\vee}\subset G^{\vee}$... | https://mathoverflow.net/users/31327 | Equality of codimension under Lusztig-Spaltenstein induction | What you're asking for can be phrased entirely in terms of nilpotent orbits. If $\mathcal{O}$ is the nilpotent orbit of $H$ then the nilpotent orbit of $G$ you obtain via your process is the induced nilpotent orbit $\mathrm{Ind}\_H^G(\mathcal{O})$. It is well known that the codimensions of these orbits coincide. See Pr... | 4 | https://mathoverflow.net/users/22846 | 221228 | 103,719 |
https://mathoverflow.net/questions/221215 | 3 | I'm struggling to understand the explicit description of the cosimplicial simplicial set $Q^{\bullet}$ on page 76 (section 2.2.2) of Lurie's book *Higher Topos Theory*, and would be grateful if someone could elaborate on that a bit.
More concretely, we have that $$Q^n = \operatorname{Map}\_{\mathfrak{C}\left[J^n\righ... | https://mathoverflow.net/users/25477 | explicit description of the cosimplicial simplicial set $Q^{\bullet}$ | Here is how I think about this; don't know if it will help.
Start with the straightening construction: this takes a map $f\colon X\to S$ of simplicial sets to a simplicial functor $\def\St{\mathrm{St}}\def\op{\mathrm{op}}\St(f)\colon \mathfrak{C}(S)^{\op}\to s\mathrm{Set}$.
The key example is when $f=\mathrm{id}\c... | 4 | https://mathoverflow.net/users/437 | 221231 | 103,721 |
https://mathoverflow.net/questions/221034 | 8 | Suppose that we have distributions $F\_1 $ and $F\_2$. Under what conditions on $F\_1,F\_2$ is it possible to construct random variables $X\sim F\_1,Y\sim F\_2$ such that $Y=E(X|\mathscr{G})$, that is, $Y$ is the conditional expectation of $X$ with respect to some $\sigma$-field $\mathscr{G}$?
| https://mathoverflow.net/users/43062 | If $X∼F_1$, $Y∼F_2$, under what conditions on $F_1$, $F_2$ can we construct $Y=E(X\mid\mathscr{G})$ for some $\mathscr{G}$? | The condition proposed by Nate Eldredge is not only necessary, but also sufficient. This is classical Strassen's theorem:
>
> **Theorem** For integrable random variables $X$ and $Y$ the following is equivalent:
>
>
> * There exist $\hat X \overset{d}{=} X$ and $\hat Y \overset{d}{=} Y$ such that $E[\hat X | \hat ... | 7 | https://mathoverflow.net/users/8146 | 221239 | 103,723 |
https://mathoverflow.net/questions/211443 | 1 | The following definitions and Theorems come from M. Magidor's paper "On the Role of Supercompact and Extendible Cardinals in Logic" (Israel J. Math., Vol. 10, 1971):
"Definition: Logic is called $\kappa$ compact iff for every set of formulae A in this logic, if every subset of A of cardinality $\lt$$\kappa$ has a mod... | https://mathoverflow.net/users/20597 | A question regarding extendible cardinals and a result of M. Magidor | For your second question, the answer is "no." Since, with Henkin semantics, second-order logic is just re-syntacted first-order logic, of course theorem 4 fails for Henkin semantics: $L\_\kappa^2$ with Henkin semantics is compact iff $L\_{\kappa,\omega}$ is compact - that is, iff $\kappa$ has the *tree property* (which... | 2 | https://mathoverflow.net/users/8133 | 221245 | 103,727 |
https://mathoverflow.net/questions/202684 | 2 | In their paper "The Role of the Foundation Axiom in the Kunen Inconsistency" ([arXiv:1311.0814](http://arxiv.org/abs/1311.0814) [Math.LO]), Daghighi, Golshani, Hamkins, and Jerabek show that the patterns of possibility for the existence of nontrivial automorphisms and nontrivial elementary embeddings of the universe in... | https://mathoverflow.net/users/20597 | The patterns of possibility for nontrivial automorphisms and nontrivial elementary embeddings of the universe | I think there's some confusion over automorphisms/embeddings which are "internal" vs. those which are "external." Some comments which I hope clear things up:
* No model of ZF has nontrivial definable (even from parameters) automorphisms. This is a proof by transfinite induction: let $\alpha$ be least such that some e... | 5 | https://mathoverflow.net/users/8133 | 221250 | 103,728 |
https://mathoverflow.net/questions/221262 | 1 | Suppose $\mu$ and $\nu$ are finite positive measures on a measurable space $(X,\mathcal A)$. Let $\mathcal G$ be an algebra of $\mathcal A$. If $\mu$ and $\nu$ are equivalent on $\mathcal G$ in the sense that $ \mu(A)=0$ if and only if $\nu(A)=0$ for all $A\in \mathcal G$, can we conclude that they are equivalent on $\... | https://mathoverflow.net/users/34483 | Equivalent measures on algebra also equivalent on $\sigma$-algebra? | No. Let $X$ be the unit interval $(0,1]$ with its Borel $\sigma$-algebra $\mathcal{A}$. Let $\mu$ be Lebesgue measure. Enumerate the rationals in $(0,1]$ as $\{q\_n\}$ and let $\nu$ be a measure assigning mass $2^{-n}$ to the point $q\_n$. Let $\mathcal{G}$ be the algebra of all finite union of half-open intervals $(a,... | 4 | https://mathoverflow.net/users/4832 | 221263 | 103,731 |
https://mathoverflow.net/questions/221260 | 10 | This question is indirectly related to my previous question [Is an elliptic curve that is isomorphic to its Frobenius conjugate defined over $\mathbb{F}\_p$?](https://mathoverflow.net/questions/220449/is-an-elliptic-curve-that-is-isomorphic-to-its-frobenius-conjugate-defined-over/220452?noredirect=1#comment543861_22045... | https://mathoverflow.net/users/63877 | Reinterpreting Galois descent over finite fields | Yes, assuming $X$ is reduced (so that its absolute Frobenius endomorphism is schematically dominant); this really is a formal consequence of the usual descent formalism (despite whatever red herring appears with Frobenius maps not being isomorphisms).
As a warm-up, consider the case $n=1$ (so $X^{(q)} = X$ tautologi... | 12 | https://mathoverflow.net/users/81332 | 221265 | 103,732 |
https://mathoverflow.net/questions/221255 | 1 | It is unknown whether $2^n-1$ is (a Mersenne) prime for infinitely many values of $n$. Such $n$ must necessarily be prime itself due to the factorization below $$(2^{ab}-1)=(2^a-1)(2^b+2^{b-1}+\cdots+2+1).$$
I am aware that there are related results on the size of the largest prime divisor of Mersenne numbers $M\_n=... | https://mathoverflow.net/users/17773 | How often is $2^n-1$ a number with few divisors? | It is not so hard to prove that $\phi(2^p-1)/(2^p-1) \to 1$ as $p\to \infty$ along prime values. This follows from the usual formula for $\phi(n)/n$ as a product over primes along with the observation that $\sum\_{q\mid 2^p-1,~q\text{ prime}} 1/q \to 0$, as $p\to\infty$. This last fact can be seen by noting that --- fr... | 9 | https://mathoverflow.net/users/16510 | 221269 | 103,734 |
https://mathoverflow.net/questions/221251 | 6 | The question I want to ask is vague in a sense. We have examples of mapping class relations, e.g. lantern relation, chain relations, etc. For instance the latern relation on a disk with three boundary components $a$, $b$ and $c$ is: $t\_a t\_b t\_c t\_d=t\_{ab}t\_{bc}t\_{ca}$, where $d$ is the curve enclosing all three... | https://mathoverflow.net/users/31475 | mapping class group relations | I can answer your last question: there exist natural ways of embedding the fundamental group of the unit tangent bundle of a surface into the mapping class group, and the lantern relation is the image of an "obvious" relation in the unit tangent bundle group (coming from a visually obvious relation in the fundamental g... | 8 | https://mathoverflow.net/users/317 | 221271 | 103,735 |
https://mathoverflow.net/questions/221256 | 4 | Suppose $ \Omega$ is a smooth bounded domain in $ R^2$. I am interested in the regularity of solutions to
$$-\Delta u(x) = f(x) \mbox{ in } \Omega$$ with $ u=0$ on $ \partial \Omega$.
If $ f \in L^1(\Omega)$ then one just misses $ u \in C(\Omega)$. There was a result of Wente that said something like if $ f= \nabla... | https://mathoverflow.net/users/66623 | Elliptic regularity for two dimensional domains | It is true, but you must use the fact that $W^{1,1}$ is embedded in the Lorentz space $L^{2,1}$, see Helein's book, *Harmonic maps, conservation laws and moving frames*, theorem 3.3.10, you will find all the material about Hardy, Lorentz spaces in chapter 3 and more generally this book is just awsome!!
Then using the... | 2 | https://mathoverflow.net/users/9253 | 221273 | 103,736 |
https://mathoverflow.net/questions/221206 | 5 | It seems that the following assertion is widely accepted:
For $k\in\mathbb N$, $p\geq 2$, $\Omega \subset \mathbb R^n$ bounded with $\partial\Omega\in C^{k+2}$ and $f\in W^{k,p}(\Omega)$, the weak solution $u\in H^1\_0(\Omega)$ of the problem
$$
\begin{equation}
\left\{
\begin{aligned}
-\Delta u =f \text{ in } \Omega... | https://mathoverflow.net/users/40644 | Regularity up to the boundary for the Poisson problem | P. Grisvard, Elliptic Problems in Nonsmooth Domains, 1985:
Thm. 2.5.1.1 (you even need to impose less regularity on $\partial\Omega$).
| 2 | https://mathoverflow.net/users/26039 | 221275 | 103,737 |
https://mathoverflow.net/questions/201555 | 8 | Assume that $V\neq HOD$ and let $\kappa = \min \{\alpha\in On \mid \mathcal{P}(\alpha) \not\subseteq HOD\}$.
Clearly, $\kappa$ is a cardinal.
**Question:** Is it consistent that $\kappa = \aleph\_\omega$?
Note that it is consistent that $\kappa$ is a regular cardinal: start with $V=L$ and force with $Add(\kapp... | https://mathoverflow.net/users/41953 | Can the first ordinal in which $V\neq HOD$ be $\aleph_\omega$? | Assume $GCH$ and let $\kappa$ be $(\kappa+2)-$strong. Force with extender based Prikry forcing $P$ with interleaved collapses to make $\kappa=\aleph\_\omega$
and $2^{\aleph\_\omega}=\aleph\_{\omega+2}.$ Call the resulting extension $V[H].$
Also let $V[G]$ be an intermediate submodel, which just adds the Prikry sequence... | 8 | https://mathoverflow.net/users/11115 | 221277 | 103,738 |
https://mathoverflow.net/questions/175743 | 20 | Let $W\subseteq V$ be two models of $\sf ZFC$ with the same ordinals. Is the following situation consistent:
1. For every $x\in\Bbb R^V$ there is some $P\_x\in W$ such that for some $G\subseteq P\_x$ which is $W$-generic, $x\in W[G]$.
2. There is no $P\in W$ and $G\subseteq P$ that is $W$-generic such that $\Bbb R^{W... | https://mathoverflow.net/users/7206 | If all reals are generic, is the set of reals generic? | I met Woodin recently and asked him that. He came up with a solution, modulo some technical assumption which Ashutosh [showed to be consistent](https://mathoverflow.net/a/220853/7206) (although admittedly, not the same suggestion that Woodin had for solving this issue). With his kind permission, I am posting this solut... | 4 | https://mathoverflow.net/users/7206 | 221283 | 103,741 |
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