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https://mathoverflow.net/questions/128976 | 5 | Can you make the category whose objects are pairs of spaces $(X,A)$, and morphisms the obvious diagrams, into a model category? Of course I want this to be done in a meaningful way, that is, agreeing with the adjoint functors $X\mapsto (X,\emptyset)$ and $(X,A)\mapsto X$?
There might be some intuitive reason that it ... | https://mathoverflow.net/users/24021 | Can one make the category of pairs of topological spaces a model category? | The answer has to be no. There's just no good homotopy theoretic way to talk about subspaces, because up to homotopy every map is an inclusion (via the mapping space construction). So even before you run into the completeness issue Karol raises you have a more fundamental issue of what the homotopy category would be. I... | 8 | https://mathoverflow.net/users/11540 | 129137 | 71,808 |
https://mathoverflow.net/questions/129050 | 11 | Schützenberger promotion, studied (for example) in [Richard Stanley, *Promotion and Evacuation*, 2009](http://www-math.mit.edu/~rstan/papers/evac.pdf), is a permutation of the set of all linear extensions of a finite poset. Since one can identify the linear extensions of a poset with saturated chains of order ideals in... | https://mathoverflow.net/users/2530 | How exactly does Schützenberger promotion relate to Striker-Williams promotion? | When a tableau has two rows (of any length, skew is ok), there is a nice bijection between linear extensions on the poset defined by the shape of the tableau and order ideals on another related poset. This is what Theorem 4.12 in Promotion and Rowmotion says. Once we move to three rowed tableaux, though, it's not so cl... | 12 | https://mathoverflow.net/users/33546 | 129140 | 71,809 |
https://mathoverflow.net/questions/73548 | 10 | The Hahn-Banach theorem implies that Lebesgue measure can be extended give a "measure" on *all* subsets of [0,1], but this measure is only guaranteed to be *finitely* additive. It might magically turn out that this measure is countably additive, but this can only happen if the continuum is a real-valued measurable card... | https://mathoverflow.net/users/11145 | extensions of lebesgue measure | The answer is no. A proof can be found [here](http://www.math.wisc.edu/~akumar/INDUCED_IDEALS.pdf).
| 3 | https://mathoverflow.net/users/2689 | 129154 | 71,814 |
https://mathoverflow.net/questions/129111 | 1 | I'm looking for a reference for an inequality related to the "fundamental inequality" about entropy and rate of escape of random walks (on the Cayley graph of a group). Namely,
$\textbf{Question}$: Under which conditions (on the measure $P$), do the following inequalities hold:
$$
l\_P^2 \leq h\_P \leq v l\_P.
$$
... | https://mathoverflow.net/users/18974 | Fundamental inequality of entropy in random walks | You got all the references and some of the notions you mention wrong. First about the "fundamental inequality". It was first obtained by Guivarc'h in late 70s (although in a somewhat different form), however, by mid-80s it was obvious to all specialists in the area (in particular, for technically more involved Brownian... | 2 | https://mathoverflow.net/users/8588 | 129159 | 71,815 |
https://mathoverflow.net/questions/129145 | 3 | **Motivation.** In my research I have a situation where a monoid $M$ is acting by nice cellular maps on a contractible cell complex and so the augmented chain complex is a resolution of the trivial module over $\mathbb ZM$. Now I can show by a somewhat messy argument that this resolution is projective. I have a much sm... | https://mathoverflow.net/users/15934 | Resolutions chain homotopic to projective ones | Here is a conceptual answer that tells you that although resolutions
chain homotopy equivalent to projective resolutions need not
be projective, they are perfectly acceptible to homological algbra.
As with any form of algebraic homotopy theory, it is very nice
to develop homological algebra in the context of model c... | 3 | https://mathoverflow.net/users/14447 | 129169 | 71,818 |
https://mathoverflow.net/questions/128906 | 0 | I am thinking of a version of HNN extensions as follows:
Assume $H,K$ are subgroups of a group $G$ and $\phi:H\to K$ is an isomorphism. We define
$ G\_{\phi,n}$ to be the group generated by $G$ and $x\notin G$ satisfying the conditions $x h x^{-1}=\phi(h)$ and $x^n=1$.
I was wondering if such a construction has ... | https://mathoverflow.net/users/nan | Torsion version of HNN extensions. | It has been considered by Sergey V. Ivanov in "On HNN-extensions in the class of groups of large odd exponent". Groups, rings, Lie and Hopf algebras (St. John's, NF, 2001), 39–53, Math. Appl., 555, Kluwer Acad. Publ., Dordrecht, 2003. Particular cases of HNN extensions in the class of torsion groups were also used in o... | 3 | https://mathoverflow.net/users/nan | 129175 | 71,821 |
https://mathoverflow.net/questions/129144 | 2 | What can be said about a discrete finitely generated subgroup $G$ of $PSL(2,\mathbb C)$ whose
nontrivial elements are parabolic? If $G$ is geometrically finite, one can show that $G$ must be elementary so the real question is: Can $G$ be geometrically infinite?
| https://mathoverflow.net/users/1573 | Purely parabolic Kleinian groups | Suppose that $G$ is a (not necessarily discrete) nonelementary group of isometries of the hyperbolic $n$-space. Then pairs of fixed points of loxodromic elements of $G$ are dense in $L(G)\times L(G)$, where $L(G)$ is the limit set of $G$. See Lemma 3.24 in "Hyperbolic Manifolds and Discrete Groups". I am quite sure tha... | 2 | https://mathoverflow.net/users/21684 | 129176 | 71,822 |
https://mathoverflow.net/questions/129166 | 5 | **Definition 1.** For each $n\in\mathbb{Z}^+$, the $n$-dimensional **Hamming cube** is the set of ordered $n$-tuples of $\lbrace 0,1\rbrace$, denoted by $\lbrace 0,1\rbrace ^n$.
**Definition 2.** The binary operation that turns $\lbrace 0,1\rbrace ^n$ into a group is $\oplus$ (**XOR**), which is bitwise addition redu... | https://mathoverflow.net/users/31084 | Bounded Hamming distance | I think you're assuming $x \not= y$ when you say "for any $x, y \in S$." In any case, your question seems like a mix of coding theory and design theory.
If you find the case when $a = b = \frac{n}{2}$ interesting, a $q$-ary code $\mathcal{C} \subset \mathbb{F}\_q^n$ is said to be *equidistant* if for any pair $\bolds... | 8 | https://mathoverflow.net/users/27829 | 129179 | 71,824 |
https://mathoverflow.net/questions/129157 | 1 | Let $X,Y$ be varieties over $\mathbb{C}$, and $D(X),D(Y)$ be the derived categories of bounded complex of coherent sheaves. Let $U \subset X, V \subset Y$ be open subvarieties， and let $X-U, Y-V$ have codimensions $c\_1, c\_2$ respectively. Assume $U $ is isomorphism to $V$. I heard a result that says: when the codimen... | https://mathoverflow.net/users/29730 | Derived equivalence of two varieties which are isomorphism over certain open subvarieties | In your assumption the varieties $X$ and $Y$ are birational. In particular, there is a smooth variety $Z$ with two proper birational maps $p:Z\to X$ and $q:Z\to Y$ which are isomorphisms over $U$ and $V$ (for $Z$ you can take an appropriate resolution of singularities of the graph of your birational isomorphism).
The... | 6 | https://mathoverflow.net/users/4428 | 129183 | 71,827 |
https://mathoverflow.net/questions/129161 | 0 | Is there any correspondence between Lie rings and Lie groups such that if one proves for a Lie algebra $g$ with ideal $I$ that $g/I$ is a Lie algebra, then the same result holds automatically for the quotient ring?
Thanks in advance AB
| https://mathoverflow.net/users/33483 | Quotient of Lie rings and quotient of Lie groups! | The question is not posed in a clear way but, if I am interpreting correctly, it is enough to recall that the universal enveloping algebra $U(g/I)$ of the Lie algebra quotient $g/I$ is isomorphic to $U(g)/B$, where $B$ is the two-sided ideal of $U(g)$ generated by $I$. (Of course, here $g$ is identified with its isomor... | 2 | https://mathoverflow.net/users/14653 | 129186 | 71,828 |
https://mathoverflow.net/questions/129007 | 3 | Hello. I thank for your answer, in advance.
Let $G$ be a finite group and $G$ has an element of order $n$ such that $\pi(n)=\pi(G)$
where $\pi(n)$ denote the set of prime divisors of $n$ and $\pi(G)$ denote the set of prime divisor of $|G|$. What can be said about the structure of the group? I know in nilpotent group... | https://mathoverflow.net/users/33508 | element of order n such that $\pi(n)=\pi(G)$, where $\pi(n)$ denote the prime divisors of $n$ | There is a pathological example that pretty much demonstrates that the existence of such an element gives no significant information about the group:
>
> **Example 1.** Let $H$ be any finite group, and let $\pi(H)=\{p\_1,\dots, p\_k\}$. Now let $C$ be a cyclic group of order $p\_1\cdot p\_2\cdots p\_k$ with generat... | 7 | https://mathoverflow.net/users/801 | 129191 | 71,833 |
https://mathoverflow.net/questions/86339 | 7 | consider a Turing machine with a set of states $s\_n$ and alphabet symbols $a\_n$. now consider a "run sequence" generated from a starting input in the following sense. the run sequence is defined as the sequence of state-symbol pairs that ensue in the computation. call the $i$th ensuing state $s'\_i$ and the $i$th ens... | https://mathoverflow.net/users/20793 | compression of a Turing machine run sequence | A *run sequence* $r = [s'\_1a'\_1,s'\_2a'\_2,s'\_3a'\_3,...]$, of a Turing machine $M$ on input $x$ is a highly compressible string:
$$K(r) = K(\langle M, x, z \rangle) + c \leq |\langle M \rangle| + | x | + \log z + c' $$
where $z$ is the number of steps of the run sequence.
| 4 | https://mathoverflow.net/users/12875 | 129195 | 71,835 |
https://mathoverflow.net/questions/126710 | 5 | The resolution of the Diophantine equation $$m! = n(n+1)$$ was asked on [M.SE](https://math.stackexchange.com/questions/350637/find-the-positive-integer-solutions-of-m-nn1). My intuition says that this cannot be solved by elementary means - apologies if I am mistaken.
I felt that the following path was promising, con... | https://mathoverflow.net/users/32875 | What analytic tools can provide a lower bound for this Diophantine equation? | Here is another comment, which is not really an answer, but too long for a comment.
If we have a plane algebraic curve $C$, given by, say $F(x,y)=0$, then we are interested
in the set $C(\mathbb{Z})$ of integral points on $C$. This set by itself does not have any useful
additional structure, but we might embed the curv... | 2 | https://mathoverflow.net/users/32332 | 129198 | 71,836 |
https://mathoverflow.net/questions/129196 | 0 | Let $\pi$ be a supercuspidal representation of $G =GL\_2(F)$ for a non-archimedean local field $F$, then there exists a maximal subgroup $K$ of $G$, which is compact modulo the center, and a representation $\rho$ of $K$ such that $\pi = Ind\_K^G \rho$.
It is possibly to show that $tr\; \sigma( \phi) \neq 0 $ iff $\s... | https://mathoverflow.net/users/10400 | Orbital integrals of pseudo coefficients of supercuspidal reps | In fact $\phi$ is not only a (multiple of a) pseudo coefficient, but is a (multiple of a) coefficient of $\pi$. See e.g. Carayol's article "Représentations cuspidales du groupe linéaire", Ann. ENS.
Now to answer your main question, there is indeed a lot of such computations in e.g. the series of papers written by Bu... | 2 | https://mathoverflow.net/users/4767 | 129203 | 71,837 |
https://mathoverflow.net/questions/129081 | 5 | Let $G$ be a reductive group over a non-archimedean field $F$ with reisdue field $f$.
*Edit: The statements only make sense modulo tensoring by one-dimensional representations.*
>
> Are the unitary, square-integrable representation (modulo tensoring by one-dimensional reps) of $G(F)$, which are not supercuspidal... | https://mathoverflow.net/users/10400 | Steinberg reps of reductive groups over local fields vs finite fields | For a general $G$, it is false that a general square integrable representation has a fixed non zero vector under the first congruence subgroup (even after a suitable twisting by a character) (there is a counter-example for e.g. ${\rm GL}(4)$). So in general you have to restrict to "level $0$" square integrable represen... | 6 | https://mathoverflow.net/users/4767 | 129205 | 71,838 |
https://mathoverflow.net/questions/129109 | 4 | The [variational bicomplex](http://ncatlab.org/nlab/show/variational+bicomplex) seams to provide a modern formulation of the variational problem in terms of modern differential geometry. In particular the bigraded complex of differential forms $\Omega^{p,q} (J^\infty E)$ with $D = \delta + d$ on the infinite jet bundle... | https://mathoverflow.net/users/17047 | Extending the variational bicomplex to Hamiltion or Hamiltion-Jacobi formalism | I'm not sure that there is a right answer to your question, since what is and what is not a satisfactory generalization of symplectic geometry can be quite subjective and in the end only can judge what you find satisfactory. I myself find what you call the "standard approach" (at least in my understanding after your cl... | 3 | https://mathoverflow.net/users/2622 | 129206 | 71,839 |
https://mathoverflow.net/questions/129200 | 1 | Let $F$ be a finite field extension of $\mathbb Q$, with ring of integers $\mathcal O\_F$. Let $G=CL(\mathcal O\_F)$ be the class group of $\mathcal O\_F$ and let $M$ be a minimal set of generators of $G$ (i.e. a subset of $G$ of minimal cardinality which generates $G$).
Question: a) Can one find a set $M$ such that ... | https://mathoverflow.net/users/32209 | Generators of class groups | The group $Cl(O\_F)$ is the Galois group of an abelian extension of $F$ (the Hilbert class field of $F$). Every element of the Galois group of a finite Galois extension $E/F$ is represented by a prime ideal of $F$ (the Cebotarev density theorem). So the answer to (a) is yes. This is proved in Serge Lang's book on algeb... | 4 | https://mathoverflow.net/users/23291 | 129207 | 71,840 |
https://mathoverflow.net/questions/129142 | 15 | The main [Robertson-Seymour Theorem](https://en.wikipedia.org/wiki/Robertson%E2%80%93Seymour_theorem) states that finite graphs form a [well-quasi-ordering](https://en.wikipedia.org/wiki/Well-quasi-ordering) under the [graph minor](https://en.wikipedia.org/wiki/Graph_minor) relation. In other words, in every infinite s... | https://mathoverflow.net/users/7252 | Weakest choice principle required for Robertson-Seymour Graph Minor Theorem? | As far as I can see, the Robertson–Seymour theorem is provable in plain ZF.
First, the restricted version of the theorem for graphs whose vertices are natural numbers can be written as a $\Pi^1\_1$ second-order arithmetical sentence, hence its provability in ZF follows from its provability in ZFC by the Schoenfield a... | 11 | https://mathoverflow.net/users/12705 | 129211 | 71,843 |
https://mathoverflow.net/questions/129119 | 7 | Hi,
Suppose one has an incompletely specified $2^n \times 2^n$ matrix over some fixed finite field $\mathbb{F}\_{p^k}$. In fact, one knows that the diagonal entries are zero and all other entries are non-zero. Graphically:
$$\left( \begin{array}{ccccc} 0 & ? & \cdots & ? & ? \\\ ? & 0 & \cdots & ? & ? \\\ ? & ? & \... | https://mathoverflow.net/users/5152 | Dimension of incomplete matrix over finite fields. | It is true. Let $N=2^n$ be the dimension, and $q=p^k$ be the size of the field. Let $v\_i$ be the $i$'th column vector. Define multiplication of vectors coordinate-wise, i.e., $uu'$ is vector whose $i$'th coordinate is $u\_i u'\_i$. Similarly, for a vector $u$ define $u^r$ be the result of raising all the elements of $... | 10 | https://mathoverflow.net/users/806 | 129217 | 71,847 |
https://mathoverflow.net/questions/129100 | 3 | Let $E=\mathfrak{so}(n,\mathbb{C})$ be the Lie algebra of antisymmetric complex matrices. We consider the action of the complex orthogonal group $SO(n,\mathbb{C})$ on $E$ by conjugation. Is there a nice description of the orbits ? Something similar to the fact the antisymmetric matrices are orthogonaly similar to matri... | https://mathoverflow.net/users/8887 | Reduction of antisymmetric complex matrices | In the wider setting of simple Lie algebras over $\mathbb{C}$, you are looking for the adjoint orbits of a Lie algebra of type B or D (odd or even orthogonal case).
While this can be viewed concretely as a problem in linear algebra, the more uniform treatment in terms of Jordan-Chevalley decomposition is probably more ... | 4 | https://mathoverflow.net/users/4231 | 129219 | 71,848 |
https://mathoverflow.net/questions/124981 | 2 | A few years ago, when I was working on first-passage percolation problems, I thought about the following problem. Recently it came back to my mind.
Consider, for some $\delta=n^{-1}>0$, the grid $\delta\mathbb{Z} \times \delta\mathbb{Z}$ and associate with each edge $e$ an independent random weight $w(e)$. The edge w... | https://mathoverflow.net/users/30264 | Continuum limit of first-passage percolation paths | Due to what's written in the text of Benjamini, I would expect it to converge to a straight line, (unless, perhaps, if your law has too heavy log-tails, I'm not sure): see
<http://www.wisdom.weizmann.ac.il/~itai/randomplanar2.pdf>, section 2: *«Large balls converge after rescaling to a convex
centrally symmetric shape.... | 1 | https://mathoverflow.net/users/31371 | 129228 | 71,853 |
https://mathoverflow.net/questions/129227 | 4 | **Fact** : Consider the inclusion $V^{n-1} \rightarrow M^n$ where $M$ is a closed orientable simply
connected positively curved manifold.
Then connectivity lemma implies that the inclusion is $(n-1)$-connected so that
$M$ is homeomorphic to a sphere.
**Situation** : As you know ther exists a $S^3$-action on $M=... | https://mathoverflow.net/users/36572 | Positively curved manifold with a codimension 1 totally geodesic submanifold. | $\mathbb{CP}^n$ (with $n>1$) does indeed not have any codimension $1$ totally geodesic manifold; neither does $\mathbb{CH}^n$. You can probably find a proof in Goldman's book on complex hyperbolic geometry.
(**Added later:** this is true even locally: there are no *open* codimension $1$ totally geodesic manifold in $... | 6 | https://mathoverflow.net/users/4961 | 129236 | 71,855 |
https://mathoverflow.net/questions/126532 | 3 | If $\mathcal F$ is a constructible sheaf (say of $\mathbb C$-modules) on a (real) manifold concentrated in degree $0$ and $i\colon Z \hookrightarrow X$ is a submanifold, can I say anything about $H^j(i^!\mathcal F)$ for $j > \operatorname{codim}\_X Z$? Specifically, if $\mathcal F$ is constructible on $\mathbb R^n$ and... | https://mathoverflow.net/users/459 | Vanishing of !-restriction of constructible sheaves | I think the vanishing you want holds. For simplicity consider the case where $i$ is the inclusion of $Z = \{ z \}$ a point. (One should be able to reduce to this case by taking a normal slice.)
The question is local so we can replace $X$ by a small neighbourhood $U$ of $z$. Let $j$ denote the inclusion of $U - \{ z \... | 7 | https://mathoverflow.net/users/919 | 129244 | 71,859 |
https://mathoverflow.net/questions/129239 | 2 | As we all know, for any homogeneous space $G/H$ we have that the equivariant vector bundles over $G/H$ are characterized by the representations of $H$. Thus, for the the complex projective line $CP^1 \simeq SU(2)/U(1)$, it must hold that all its line bundles are indexed by the integers $L\_k$, for $k \in Z$, and more g... | https://mathoverflow.net/users/11206 | Are all Equivariant Bundles of a Total Flag Manifold Constructable from Line Bundles? | I think perhaps the confusion stems from the following.
It is true that the category of $G$-equivariant $G$-bundles on $G/H$ is equivalent to the category of representations of $H$. The flag variety $\mathcal F l\_n$ can be realized as either $U(n)/U(1)^n$, or as $GL\_n(\mathbb C)/B\_n$, where $B\_n$ is the group of ... | 7 | https://mathoverflow.net/users/7762 | 129247 | 71,861 |
https://mathoverflow.net/questions/129148 | 2 | Let M be your favorite moduli stack over the field of complex numbers.
Is it reasonable to expect M to be a Deligne-Mumford stack?
I know this is true for the moduli space of curves of genus g, ppav's and K3 surfaces. I'm just wondering what I should expect when considering other moduli stacks.
| https://mathoverflow.net/users/30342 | are moduli stacks deligne-mumford stacks in general | If the objects under consideration all have finite automorphism groups, you should expect your stack to be Deligne-Mumford. Otherwise, it isn't Deligne-Mumford; but that is no cause for alarm.
| 8 | https://mathoverflow.net/users/299 | 129250 | 71,862 |
https://mathoverflow.net/questions/128751 | 2 | Let $g(x):\mathbb{R}\_{\geq0}\rightarrow\mathbb{R}$ be real analytic s.t. $g(0)\neq 0$ and $g(x)=O(x^{-2})$ as $x\rightarrow\infty$. I think the following integral should be bounded as $\lambda\rightarrow0$, but would like to have a proof:
$$
J(\lambda) = \int\_0^{\infty}dy \sin\left(y\right)\log(y)g(\lambda y)
$$
I ... | https://mathoverflow.net/users/30947 | Boundedness of an Oscillating Integral | Yes it is bounded. Assume throughout that $\lambda<1$ and split
$$J(\lambda)=\int\_0 ^1 \sin(y) \log(y) g(\lambda y)dy + \int\_1 ^\infty \sin(y) \log(y) g(\lambda y)dy=:J\_0(\lambda)+J\_1(\lambda) .$$
In $[0,1]$ we have that $0<\lambda y <\lambda <1$ so that $|g(\lambda y)|\lesssim\_g 1$ and this is best possible since... | 5 | https://mathoverflow.net/users/881 | 129255 | 71,863 |
https://mathoverflow.net/questions/129152 | 3 | **Background**
Let $a,b,c,d$ be nonnegative constants and consider the map $T\colon [0,1]\times[0,1] \rightarrow [0,1]\times[0,1]$ defined by
$$
T(x,y) := \left( \frac{1}{1 + ax + by}, \frac{1}{1 + cx + dy} \right), \quad (x,y) \in [0,1] \times [0,1].
$$
I'm interested in this map (and other similar ones) in the ... | https://mathoverflow.net/users/31451 | Uniqueness of fixed points for rational transformations | I deleted the other answer in favor of a more elementary approach outlined below. Some details are missing, but they can be filled in easily.
$\newcommand{\R}{\mathbb{R}} \newcommand{\nlsum}{\sum\nolimits}$
We'll show that under fairly mild assumptions, your map has a unique fixed point towards which it converges.
... | 2 | https://mathoverflow.net/users/8430 | 129260 | 71,865 |
https://mathoverflow.net/questions/129254 | 3 | I'm trying to understand an assertion that appears in Luck's book on $L^2$-invariants. I believe that it's elementary functional analysis, but I can't seem to figure it out.
Here goes. Let
$$T^n = \{\text{$(z\_1,\ldots,z\_n)$ $|$ $|z\_i|=1$ for $1 \leq i \leq n$}\} \subset \mathbb{C}^n$$
be the $n$-torus and let $G$ ... | https://mathoverflow.net/users/33575 | Certain bounded linear operators on L^2 of a torus | Yes, this is standard functional analysis. You're starting out the right way.
I'll sketch the argument. First observe that $\phi$ is invariant for the ${\bf Z}^n$ action if and only if it commutes with the multiplication operators $M\_h$ for $h(z\_1, \ldots, z\_n) = z\_1^{l\_1}\cdots z\_n^{l\_n}$. By taking linear co... | 6 | https://mathoverflow.net/users/23141 | 129268 | 71,869 |
https://mathoverflow.net/questions/129264 | 7 | Let $a$ be an integer, $p$ a prime (much) greater than $a$, and $\chi$ a Dirichlet character.
There is an abundant literature on the sums
$$S(\chi,a)=\sum\_{i=1}^a \chi(i),$$
called short (or incomplete) character sums, and their average in various sense. Yet I have not found in this literature an answer to the followi... | https://mathoverflow.net/users/9317 | short character sums averaged on the character | **Updated answer:** Here's an unconditional lower bound of exactly the right order $\sqrt{a}$, however I need $a$ to be comparable to $p$. It is known that (for any $a$)
$ \frac{1}{p-1} \sum\_{\chi} |S(\chi, a)|^{2k} \ll\_{k} p^{k} $
[See: H. Montgomery, R. Vaughan, [Mean values of character sums](http://www.ams.or... | 11 | https://mathoverflow.net/users/630 | 129269 | 71,870 |
https://mathoverflow.net/questions/129174 | 26 | If $n$ is odd then $S^{n-1}$ doesn't admit a nowhere-vanishing vector field, and if $n$ is even then there does exist one (Hairy Ball Theorem). We can then ask, on $S^{n-1}$, *what is the maximum number $k(n)$ of linearly independent vector fields?* Rewriting $n=2^{4a+b}(2s+1)$, Adams computes $k(n)=2^b+8a-1$.
In par... | https://mathoverflow.net/users/12310 | Vector fields on $(4n+1)$-spheres | *Thanks Misha for the reference! (Just rewriting it here to complete this thread).*
It seems that an off-the-cuff calculation (what I refer to as "down-to-earth") probably isn't going to suffice; there is some intricate stuff going on in the proofs involving the homotopy groups of the rotation groups of the spheres... | 2 | https://mathoverflow.net/users/12310 | 129274 | 71,872 |
https://mathoverflow.net/questions/129286 | 2 | Let $X,Y$ be two elliptic curves over $K:=K(R)$ which have good models over $R$, Char($K$)=$0$, where $R$ complete DVR with algebraically residue field $k$.
If $L$ is a finite extension of $K$, such that $X \otimes L \cong Y \otimes L$.
My question is:
Is there $X \cong Y$ ?
| https://mathoverflow.net/users/5274 | morphism between two elliptic curves over a local field | I assume that by "good models" you mean models $\mathscr{X}$, $\mathscr{Y}$ which are elliptic $R$-curves. These are pointed curves of genus 1, and in particular stable curves. This implies that the sheaf
$$I:=\underline{\rm Isom}\_{\text{$R$-ell. curves}}(\mathscr{X},\mathscr{Y})$$
is a scheme, finite and unramified ... | 9 | https://mathoverflow.net/users/7666 | 129289 | 71,876 |
https://mathoverflow.net/questions/129265 | 6 | Let $(M,\times)$ be a monoid with zero. Let $\Sigma(M,\times)$ be the set of binary operations $+$ on $M$ such that $(M,+,\times)$ is a ring. Let $\sim$ be an equivalence relation on $\Sigma(M,\times)$ defined by $$+\_1\sim+\_2\iff(M,+\_1,\times)\cong(M,+\_2,\times).$$
Let's denote the quotient set $\Sigma(M,\times)/... | https://mathoverflow.net/users/20803 | For a monoid with zero $M$, how many additive operations on $M$ can there be making $M$ a ring? | I think the answer to (2) is yes (which also answers (1) and (4), of course).
First, note that $R=\mathbb{Z}/4\mathbb{Z}$ and $S=\mathbb{F}\_2[x]/(x^2)$ have isomorphic multiplicative monoids (via the map $0\mapsto 0$, $1\mapsto 1$, $2\mapsto x$, $3\mapsto 1+x$).
Call this monoid $M$.
Then the direct product $M^n... | 12 | https://mathoverflow.net/users/22989 | 129296 | 71,880 |
https://mathoverflow.net/questions/129300 | 9 | **For example**
*3 consecutive primes in arithmetic progression*
```
3,5,7 distance 2
151,157,163 distance 6
```
*4 consecutive primes in arithmetic progression*
```
251,257,263,269 distance 6
1741,1747,1753,1753 distance 6
76543,76561,76579,76597 distance 18
```
**Are there five consecutive primes in ar... | https://mathoverflow.net/users/33595 | Are there five consecutive primes in arithmetic progression? | Yes for the lengths you ask for this is known to exist.
The minimal example is $9843019+ 30 n$ for $n=0,1,2,3,4$ (taken from the page at the end).
A more common way to phrase this would be to ask about (five) *consecutive* primes in arithmetic progression.
Indeed, it is conjectured that there are arbitrarily l... | 13 | https://mathoverflow.net/users/nan | 129304 | 71,881 |
https://mathoverflow.net/questions/129261 | 11 | I will denote by $\Lambda$ the ring of symmetric functions, and by $\mathrm{QSym}$ the ring of quasisymmetric functions (both in infinitely many variables $x\_1$, $x\_2$, $x\_3$, ..., both over $\mathbb Z$). See [Victor Reiner, *Hopf algebras in combinatorics*](http://www.math.umn.edu/~reiner/Classes/HopfComb.pdf), cha... | https://mathoverflow.net/users/2530 | Plethysm of $\mathrm{QSym}$ into $\mathrm{QSym}$: can it be defined? | As per @darij's suggestion regarding my above comment: Loehr and Remmel's [A computational and combinatorial exposé of plethystic calculus](http://link.springer.com/article/10.1007%2Fs10801-010-0238-4) is a good resource for a combinatorial point of view of plethysm; I think it clarifies what one should do when both f... | 5 | https://mathoverflow.net/users/32309 | 129308 | 71,884 |
https://mathoverflow.net/questions/129242 | 14 | An effective Cartier divisor on a scheme $X$ (a closed subscheme of $X$ that is locally cut out by one equation that is not a zero-divisor) is always a regular embedding of codimension 1 (in the stalks, it is cut out by one equation that is not a zero-divisor). The converse is true if $X$ is locally Noetherian, using N... | https://mathoverflow.net/users/299 | Example of codim 1 regular embedding that is not an effective Cartier divisor? | Let $(A,m,k)$ be a discrete valuation ring. Consider the subring $R\subset A^\mathbb{N}$ of converging sequences (for the discrete topology on $A$), i.e. ultimately constant sequences. We have a morphism $\varphi:R\to A$ sending any sequence to its limit. Put $\mathfrak{p}=\varphi^{-1}(m)$: this is the maximal ideal of... | 19 | https://mathoverflow.net/users/7666 | 129311 | 71,885 |
https://mathoverflow.net/questions/129279 | 1 | In the Laplacian comparision theorem by Yau and Schoen, the proof assume a $n$ dimensional space form $N$ with sectional curvature $-k^{2}$. And we have a separate $M$ with $Ric<-(n-1)k^{2}$. But after reading the proof several times and scratching my head for hours, I do not understand what is the role of $N$ at here.... | https://mathoverflow.net/users/18850 | Confusion with the Laplacian comparision theorem | The form $-k^2$ is just a way of saying that the sectional curvature is nonpositive. There is a nice, modern and detailed discussion of this Laplacian comparison result in P. Petersen, "Riemannian Geometry" (second edition), Springer-Verlag, 2006. The result you want is the first inequality in Lemma 34, page 268. There... | 2 | https://mathoverflow.net/users/11211 | 129318 | 71,889 |
https://mathoverflow.net/questions/129298 | 3 | I'm trying to understand the notion of a naked singularity on a more mathematical level (intuitively, it's a singularity "one can see and poke with a stick", but I'm having troubles on how to actually show it).
Based on what (little) is written in Choquet-Bruhat's, a naked singularity is the one for which we can exte... | https://mathoverflow.net/users/33594 | naked singularity and null coordinates | The formal definition of a naked singularity can be found in standard GR books (Wald, Hawking & Ellis, probably others). However, here's a slightly crude, rule of thumb way of checking whether your spacetime has a singularity and whether it is naked. There is a singularity if an affinely parametrized causal curve (past... | 4 | https://mathoverflow.net/users/2622 | 129322 | 71,890 |
https://mathoverflow.net/questions/129319 | 3 | Let $M$ and $N$ be finite dimensional smooth manifolds and $p: E \rightarrow N$ a finite rank vector bundle. $M$ can be assumed to be compact if necessary but I would prefer to work without this assumption.
As described e.g. in Kriegl's and Michor's book, $C^\infty(M,N)$ is an $\infty$-dimensional manifold. Now I wou... | https://mathoverflow.net/users/33600 | Constructing a special infinite-dimensional vector bundle | In "Convenient Setting" the construction that you want is done in 42.17.
There is also the old book, where 10.10 is relevant and your construction (for the special case $E=TN$) is done in (too many) details in 10.12:
* Peter W. Michor: Manifolds of differentiable mappings. Shiva Mathematics Series 3, Shiva Publ., ... | 7 | https://mathoverflow.net/users/26935 | 129323 | 71,891 |
https://mathoverflow.net/questions/129293 | 0 | Let $p=2^a-1$ be a Mersenne prime and so $a$ is an odd prime if $p>7$. We know that if $p=7$, then $(p^2+1)/2$ is equal to $5^2$.
Can we prove that if $p>7$ , then $(p^2+1)/2$ is not equal to the powers of a prime number $t$, i.e. there exists no prime number $t$ such that $(p^2+1)/2=t^\beta$ for some $\beta>0$?
One ... | https://mathoverflow.net/users/31045 | Factorization of $p^2+1$ where $p$ is a Mersenne prime | If $p=2^a-1$ and $(p^2+1)/2=t^b$, then $2^{2a-1}-2^a+1=t^b$. If $a \equiv 1 \pmod 4$ and $b$ is even then we get an integral point on the curve $y^2=2x^8-2x^4+1$, so there are only finitely many such $p$. Likewise, we can deal with $a \equiv 3 \pmod 4$, using another curve. For $b>1$ odd, you get a curve that depends o... | 1 | https://mathoverflow.net/users/2290 | 129324 | 71,892 |
https://mathoverflow.net/questions/127808 | 5 | I have [asked this question on math.se](https://math.stackexchange.com/questions/356925/a-basis-of-the-symmetric-power-consisting-of-powers), but did not get an answer - I was quite surprised because I thought that lots of people must have though about this before:
Let $V$ be a complex vector space with basis $x\_1,\... | https://mathoverflow.net/users/9947 | A basis of the symmetric power consisting of powers | I would look up a book on the calculus of finite differences in a multivariate setting.
The claim here is to show that for any multi-index $\alpha=(\alpha\_1,\ldots,\alpha\_n)$
of length $k$ one can express the multiple derivative at zero
$$
\left(\frac{\partial}{\partial t}\right)^\alpha \
(t\_1x\_1+\cdots+ t\_n x\_n... | 8 | https://mathoverflow.net/users/7410 | 129327 | 71,895 |
https://mathoverflow.net/questions/129321 | 22 | If $M$ is a connected smooth manifold, then it is easy to show that there is a sequence of connected compact smooth submanifolds with boundary $M\_1\subseteq M\_2\subseteq\cdots$ such that $M=\bigcup\_{i=1}^\infty(M\_i)^\circ$.
I would guess it should also be true that if $M$ is a connected **topological** manifold t... | https://mathoverflow.net/users/35353 | Does a *topological* manifold have an exhaustion by compact submanifolds with boundary? | Since topological manifolds of dimension $\le 3$ are smoothable, the question is about manifolds of dimension $\ge 4$. Kirby and Siebenmann proved for $n\ge 6$ that every topological $n$-manifold admits a handle decomposition; this was extended to $n=5$ by Freedman and Quinn (I think, it is Quinn's paper "Ends of maps,... | 30 | https://mathoverflow.net/users/21684 | 129331 | 71,899 |
https://mathoverflow.net/questions/129292 | 4 | I have two bags: one filled with red marbles, and one filled with blue marbles. I would like to fill a bin with only $k$ red marbles and no blue marbles. However, I can only sample (with replacement) from the bag of red marbles with probability $p$ and from the bag of blue marbles with probability $(1-p)$. Thus, the ex... | https://mathoverflow.net/users/33592 | Filling a bin with one type of element when uniformly selecting from a set of two (with bias) | One strategy is to start removing marbles whenever you have drawn a blue marble. Let $a(p,n)$ be the expected number of removals under this strategy before reaching a collection of $n$ red marbles.
$$a(p,n) = \frac{1}{p} a(p,n-1) + \frac{1-p}{p} \bigg(\frac{n+1}{2} - \frac{1}{n} \sum\_{k=0}^{n-1} a(p,k) \bigg)$$
I... | 2 | https://mathoverflow.net/users/2954 | 129332 | 71,900 |
https://mathoverflow.net/questions/129310 | 4 | Does anyone have a link to a copy of Beilinson-Bernstein's "Localisation de g-modules", in which they prove the Beilinson-Bernstein theorem? I can't find it anywhere.
| https://mathoverflow.net/users/30576 | Reference request - localisation de g-modules | [http://gallica.bnf.fr/ark:/12148/bpt6k6226873r/f29](http://gallica.bnf.fr/ark%3A/12148/bpt6k6226873r/f29)
| 8 | https://mathoverflow.net/users/19276 | 129333 | 71,901 |
https://mathoverflow.net/questions/129257 | 7 | I'm trying to understand the proof of the Beilinson-Bernstein localisation theorem at the moment, but there's just one point where I'm having a mental block, and was wondering if anybody could clarify things for me.
Specifically, it's this (not quite general) theorem which I'm trying to prove:
Let $G$ be a semisimp... | https://mathoverflow.net/users/30576 | A question about the proof of Beilinson-Bernstein localisation | This may not be the most efficient way to get to the result, but here's how I would think about it:
The universal enveloping algebra can be identified with right $G$-invariant differential operators on the group $G$, via the map sending an element of the Lie algebra to the corresponding left translation vector field ... | 3 | https://mathoverflow.net/users/66 | 129334 | 71,902 |
https://mathoverflow.net/questions/129230 | 2 | Suppose $u(t,\omega), t \geq 0$ is a continuous time stochastic process with smooth paths so $\frac{d}{dt} u(t,\omega)$ exists for all $\omega$. Suppose you know the distribution of $\frac{d}{dt} u(t,\omega)$ and the initial distribution of $u$. Can you use this information to determine the distribution of $u(t,\omega)... | https://mathoverflow.net/users/33566 | Distributions of Time Derivatives of Stochastic Processes | I would think the answer is "no" to both questions.
Let me abbreviate $v(t)=du/dt$, so $u(t)=u(0)+\int\_0^t v(t')dt'$. Now ask for the expectation of $u^2(t)$. You'll need to know how $v(t')$ and $v(t'')$ are correlated for any $t',t''$ in the interval $(0,t)$. That information is not given.
Similarly, in the othe... | 0 | https://mathoverflow.net/users/11260 | 129336 | 71,904 |
https://mathoverflow.net/questions/129344 | 1 | Let $X$ be a complex algebraic variety. Is it true that $X$ contains a dense affine subvariety?
| https://mathoverflow.net/users/25358 | Dense Affine Subvarieties of Algebraic Varieties | If $X$ is irreducible, then any open subset is dense, and since $X$ can be covered by open affines, any one of these will do. If $X$ isn't irreducible, take an affine dense subvariety of each component, and take their union.
Edited to add: As commenters have noted, you'll want to be sure the affine dense subvarieties... | 6 | https://mathoverflow.net/users/10503 | 129346 | 71,908 |
https://mathoverflow.net/questions/129364 | 50 | Philosophically why should proving that $\gamma$ is irrational (let alone transcendental) be so much harder than proving $\pi$ or $e$ are irrational?
| https://mathoverflow.net/users/16557 | Why is it hard to prove that the Euler Mascheroni constant is irrational? | There are number theorists who understand this subject much better than I do. However, I feel obliged to post an incomplete answer quickly before people have a chance to close this question.
---
There are a lot more connections known between $\pi$ and $e$ and other numbers than between $\gamma$ and other numbers.... | 54 | https://mathoverflow.net/users/2954 | 129367 | 71,914 |
https://mathoverflow.net/questions/129374 | 1 | Suppose $Y$ is a Banach space and $X$ is a **finite-dimensional** subspace of $Y$. Further assume $T:X \rightarrow X$ is a linear operator which is power bounded from above **and below**, in other words there is $0 < c \leq 1 \leq C < \infty$ such that $c \cdot \|x\| \leq \|T^n(x)\| \leq C \cdot\|x\|$ for all $n \in \m... | https://mathoverflow.net/users/12978 | Extension of power bounded operators over a finite subspace | A finite-dimensional subspace is complemented, so we can write $Y = X \oplus Z$ for some closed subspace $Z$. There are constants $m$ and $M$ such that for all $x \in X$ and $z \in Z$,
$m (\|x\| + \|z\|) \le \|x + z\| \le M (\|x\| + \|z\|)$. Extend $T$ to $\overline{T}$ so that $\overline{T} = I$ on $Z$. Then $\overlin... | 3 | https://mathoverflow.net/users/13650 | 129380 | 71,918 |
https://mathoverflow.net/questions/129368 | 4 | I have found many references to Poincaré and Borel in relation to their work on asymptotic series, but so far, every source I can get my hands on is very old, hence hard to read (this is not true in general, but in this case, texts that predate Oh notation tend not to be clear).
>
> Can you explain the idea behind ... | https://mathoverflow.net/users/13923 | Asymptotic series | Web pages of
>
> Michael Berry: <http://www.phy.bris.ac.uk/people/berry_mv/dingle.html>
>
>
>
and
>
> John Boyd: <http://www-personal.umich.edu/~jpboyd/>
>
>
>
both include helpful published and publicly available sources.
| 4 | https://mathoverflow.net/users/nan | 129387 | 71,921 |
https://mathoverflow.net/questions/129350 | 19 | Hello everybody! I would be interested in knowing, what the reason is for investigating coherent sheaves on complex manifolds. By definition a sheaf $F$ on a complex manifold $X$ is coherent, when it is locally finitely presented (i.e. every point possesses a neighborhood $U$, such that $F \mid U$ is the image of $O\_X... | https://mathoverflow.net/users/29973 | Reason for studying coherent sheaves on complex manifolds. | First let me note that your definition of coherent sheaf is misleading: it implies that $\mathcal O\_X$ is coherent by definition, whereas in reality coherence of $\mathcal O\_X$ is a very deep theorem due to Oka .
The correct definition is that a sheaf $\mathcal F$ of $\mathcal O\_X$-Modules is coherent if :
1)... | 34 | https://mathoverflow.net/users/450 | 129390 | 71,923 |
https://mathoverflow.net/questions/129362 | 5 | Some colleagues and I were wondering if there is a citation out there which shows there are no exceptional eigenvalues, $\lambda$, of classical weight 1/2 Maass forms on $\Gamma\_0(4)$, which is to say $\lambda=1/4+t^2>1/4$, so $t \in \mathbb{R}\_{>0}$.
A resource where somebody has computed the lowest eigenvalue fo... | https://mathoverflow.net/users/7505 | No Exceptional Eigenvalues of Weight 1/2 Maass Forms on $\Gamma_0(4)$? | A weight $1/2$ Hecke-Maass form of eigenvalue $(1-s^2)/4$ on $\Gamma\_0(4)$ has a Shimura lift to a weight $0$ even Hecke-Maass form of eigenvalue $1/4-s^2$ on $\Gamma\_0(1)$, see e.g. the proof of Theorem 1.5 in Baruch-Mao: A generalized Kohnen-Zagier formula for Maass forms (manuscript [here](http://math.newark.rutge... | 4 | https://mathoverflow.net/users/11919 | 129391 | 71,924 |
https://mathoverflow.net/questions/129384 | 2 | For a problem in group Theory I need some information about the Mersenne primes:
Let $p=2^a-1>7$ be a Mersenne prime and so $a$ is an odd prime. Is it true that $p^2+1$ is square free. i.e.
if there exists a prime number $q$ such that $q$ divides $(p^2+1)/2$, then $q^2$ does not didivde $(p^2+1)/2$?
Also is there ... | https://mathoverflow.net/users/31045 | Is it true that $p^2+1$ is square free if $p>7$ is a Mersenne prime | No, this is not true. -- For example $p := 2^{2203}-1$ is a Mersenne prime
(cf. <http://en.wikipedia.org/wiki/Mersenne_prime>), but $p^2+1$ is divisible by $5^2 = 25$.
Edit: To answer D. Burde's question: $p := 2^{11213}-1$ is a Mersenne prime as well, and
$p^2+1$ is divisible by $13^2 = 169$.
| 8 | https://mathoverflow.net/users/28104 | 129393 | 71,925 |
https://mathoverflow.net/questions/129372 | 0 | I read somewhere that 3SAT can be used to solve Integer Factorization.
If that is true, could someone teach me a simple example of generating the 3SAT by using a small number? Let's say you are given the number 6, then the factors are 2 and 3.
| https://mathoverflow.net/users/33614 | Generating 3SAT circuit for Integer factorization example | <https://cstheory.stackexchange.com/questions/6755/fast-reduction-from-rsa-to-sat> answers this question directly.
| 2 | https://mathoverflow.net/users/nan | 129398 | 71,927 |
https://mathoverflow.net/questions/129365 | 5 | Is there a standard normal form for homogeneous cubic polynomials in $\mathbb{R}[x\_1, x\_2, x\_3]$? Or, put another way, is there a nice way to describe the orbit space of the natural (diagonal) action of $GL(3, \mathbb{R})$ on $Sym^3(\mathbb{R}^3)$?
(This is related to my previous question [Normal form for trace-fr... | https://mathoverflow.net/users/18048 | Normal forms for homogeneous cubic polynomials in $\mathbb{R}[x_1, x_2, x_3]$ | Hi, Jeanne! It may help to have some geometric explanation of the normal forms over $\mathbb{R}$. The standard account is this:
If the projective cubic curve $F(x\_1,x\_2,x\_3)=0$ is nonsingular (the 'generic' case), it has exactly three real flexes, they are distinct and lie on a line. One can make a linear change t... | 10 | https://mathoverflow.net/users/13972 | 129404 | 71,929 |
https://mathoverflow.net/questions/129397 | 4 | I need a few examples of graphs that are strongly regular as well as rigid, i.e., have only the trivial automorphism. Any references to relevant literature would be appreciated. Thanks.
| https://mathoverflow.net/users/39663 | Rigid Strongly Regular Graphs | At Ted Spence's web page <http://www.maths.gla.ac.uk/~es/srgraphs.php> you can find all the strongly regular graphs on 25 and 26 vertices, and these include examples of asymmetric graphs. (No srg on fewer than 25 vertices is asymmetric.)
You get more examples as Latin square graphs. Take an $n\times n$ Latin square (... | 7 | https://mathoverflow.net/users/1266 | 129409 | 71,930 |
https://mathoverflow.net/questions/129389 | 1 | Hi, I have some repeated measures data, one measurement a day for three days in a row, and the measured variable looks normally distributed. I have two groups, the "really ill" and the "not ill after all", and want to use ROC curves to see if it's a good idea to use my measured variable as a test for this illness. Does... | https://mathoverflow.net/users/17176 | ROC curve with repeated measures | Without any further information, there's no way to specify a "canonical" way to combine your three measurements into a single quantity (which is what you need to do to create a single ROC curve that takes into account all the data simultaneously). If you have no further theoretical insight into the nature of your measu... | 0 | https://mathoverflow.net/users/3106 | 129428 | 71,938 |
https://mathoverflow.net/questions/129353 | 0 | Recall that an abelian group $G$ is $Z$-torsionless if for all $a\in G$ ($\neq 0$) there is a homomorphism of $\phi\in Hom(G,Z) = G^\*$ so that $\phi(a)\neq 0$
Suppose $S$ is a subgroup of *torsionless* (edit) abelian $A$ so that $A/S$ is torsionless, and $T$ is a large summand of $S$, say $S=T\oplus \langle s \rangl... | https://mathoverflow.net/users/17764 | example re torsionless quotients of abelian groups | I think your question is equivalent to asking whether there is an extension $0\to\mathbb{Z}\to B\to C\to 0$ of abelian groups where $C$ is torsionless but $B$ isn't.
Given such an extension, take $A$ to be a free abelian group with a surjective map $A\to B$, let $T$ be the kernel of this map, and let $S$ be the kern... | 2 | https://mathoverflow.net/users/22989 | 129429 | 71,939 |
https://mathoverflow.net/questions/129432 | 0 | A graph is called *matching-covered* if every edge is containd in a perfect matching. (Such graphs are also sometimes called "elementary", e.g. in Chapter 4 of "Matching Theory" by Lovasz & Plummer). It is well-known that for bipartite graphs this is equivalent to the existence of [ear decompositions](http://en.wikiped... | https://mathoverflow.net/users/22051 | Counting matchings in a bipartite matching-covered graph | Yes, it is still just as hard. Given an arbitrary bipartite graph, in polynomial time you can remove every edge that is not in a perfect matching (test one edge at a time), thus reducing the problem to a matching-covered graph.
| 1 | https://mathoverflow.net/users/9025 | 129434 | 71,941 |
https://mathoverflow.net/questions/129370 | 4 | I am interested in finding the derivation of the duplication, triplication and quintuplication formulae for Klein’s j-invariant, which are equations (13) – (24) of the corresponding page (Klein’s j-invariant) of MathWorld:
<http://mathworld.wolfram.com/KleinsAbsoluteInvariant.html>
Yes, I've asked the MathWorld tea... | https://mathoverflow.net/users/33612 | j-invariant duplication, triplication and quintuplication formulae... how? | One can explain this conceptually in terms of "modular equations".
It's been known since the late 19th century that for any integer $N$, there is some polynomial $\Phi\_n(X, Y)$ with the property that $\Phi\_n(J(\tau), J(N\tau)) = 0$ for all $\tau$ in the upper half-plane. This is the "classical modular equation". Fo... | 5 | https://mathoverflow.net/users/2481 | 129440 | 71,944 |
https://mathoverflow.net/questions/129437 | 5 | In a topos which is not Boolean topos, can we use proof by contradiction?
| https://mathoverflow.net/users/26296 | Proof by contradiction in a topos | It depends on what examples you have in mind when you say "proof by contradiction". This topic has come up a number of times recently at MO, but I recommend to your attention the useful [blog post](http://math.andrej.com/2010/03/29/proof-of-negation-and-proof-by-contradiction/) by Andrej Bauer, which explains that ther... | 11 | https://mathoverflow.net/users/2926 | 129443 | 71,946 |
https://mathoverflow.net/questions/129452 | 5 | Tracing through a trail of references I found myself needing something proven in Appendix A of "The Stable Category and Generalized Thom Spectra" by Gaunce Lewis (I believe this was his thesis at UChicago under Peter May, from 1978). My library doesn't seem to have a copy and I can't find anything online. Does anyone k... | https://mathoverflow.net/users/11540 | Does anyone know where I can get a copy of Gaunce Lewis's thesis? | Peter has been so kind to scan it a while ago. Maybe he has already put it on his webpage. Here is another link that should work.
<http://www.math.ku.dk/~xvd217/Compactly%20Generated%20Spaces-MAY.pdf>
| 7 | https://mathoverflow.net/users/24727 | 129455 | 71,952 |
https://mathoverflow.net/questions/129407 | 5 | I've encountered an interesting problem but can solve it only partially:
>
> Prove that random graph $G\sim G\left(n,\frac cn\right)$, $c=const$, almost surely is isomorphic to some unit distance graph on a plane if $c$ is sufficiently small and almost surely won't be ismorphic to any unit distance graph on a plane... | https://mathoverflow.net/users/26959 | Random graphs nonisomorphic to unit distance graphs | The almost sure asymptotic chromatic number of $G$ goes to $\infty$ with $c$, see for example the precise result by Achlioptas and Naor in Annals of Math. 2005.
The chromatic number of a unit-distance graph (and in fact of the whole plane) is bounded above by $7$, see e.g. the math coloring book by Soifer (this is si... | 6 | https://mathoverflow.net/users/4961 | 129457 | 71,953 |
https://mathoverflow.net/questions/129470 | 4 | Consider a tree $(T, <\_T)$ of height $\omega\_1$, with countable levels. One can view $T$ as a forcing poset by calling a condition $s\in T$ stronger than $t\in T$ if $t <\_T s$.
My question is: when is $T$ proper, as a forcing poset?
If this is too vague, consider the following. The ccc trees are exactly the Susl... | https://mathoverflow.net/users/11233 | Which $\omega_1$-trees are proper? | Concerning your final question, it is consistent that there is a proper normal $\omega\_1$-tree that is not Suslin. If $T$ is a Suslin tree, then we may build a new tree $T^+$, consisting of the all-zero branch, together with nodes branching off (at the first one), followed by a copy of $T$. This is proper as a notion ... | 4 | https://mathoverflow.net/users/1946 | 129471 | 71,959 |
https://mathoverflow.net/questions/129479 | 0 | Let $R$ be a complete DVR with algebraically closed residue field $k$ and fractional field $K$ , $PGL(2)$ the automorphic group of projective line over $\overline K$.
My question is:
When $H^{1}(Gal(\overline K/K), PGL(2))=0$ ? Is this group trivial if $Char(K) \neq 2$ ?
| https://mathoverflow.net/users/20956 | a question of Galois cohomology | The claimed triviality holds (the nonabelian cohomology set is not a group though), and I don't think you need $Char(K) \neq 2$. To argue this, I will use the long exact nonabelian cohomology sequence of the central extension $1 \rightarrow \mathbf{G}\_m \rightarrow GL\_2 \rightarrow PGL\_2 \rightarrow 1$, a segment of... | 8 | https://mathoverflow.net/users/5498 | 129482 | 71,965 |
https://mathoverflow.net/questions/129478 | 2 | Let $\mathbb{S}$ be a closed and bounded convex body in 2-D with some non-empty intersection with positive quadrant and let it also contain origin. Let $c>0$ be the right-most point on the x-axis such that $(c,y)\in \mathbb{S}$ for some $y$. Define the function
\begin{align}
f(x)=\max\_{(x,y)~\in~\mathbb{S}}y ~~,x\in[... | https://mathoverflow.net/users/27249 | Does the Border (Boundary) Points of a convex body make a concave function? | As Wlodzimierz points out, the answer is trivially yes. Maybe it would help to recall that $S$ is compact, so that for each $x$ there exists $y$ such that $(x,y) \in S$ and $f(x) = y$. So if $f(x\_1) = y\_1$ and $f(x\_2) = y\_2$ then the point $\frac{1}{2}((x\_1, y\_1) + (x\_2, y\_2)) = (\frac{x\_1 + x\_2}{2}, \frac{y\... | 2 | https://mathoverflow.net/users/23141 | 129484 | 71,966 |
https://mathoverflow.net/questions/129458 | 0 | Suppose that $X$ and $Y$ are complex algebraic varieties with $Y$ affine. Suppose that $f:X\rightarrow Y$ is a variety morphism. I am interested in finding conditions on the $\mathbb{C}$-algebra morphism $f^\*:\mathbb{C}[Y]\rightarrow\mathbb{C}[X]$ under which the image of $f$ is open. I would appreciate any and all re... | https://mathoverflow.net/users/25358 | Images of Variety Morphisms | Assuming $\mathbb C[X]$ is the ring of global sections of $X$: This is not possible. The strongest possible condition is that the morphism be an isomorphism. But that is not sufficient. Choose $Y$ to be $\mathbb A^3$. For $X$, first blow up $\mathbb A^3$ at the origin, then remove a line passing through the origin. The... | 9 | https://mathoverflow.net/users/18060 | 129491 | 71,970 |
https://mathoverflow.net/questions/128747 | 3 | I have recently become aware of the following neat statement.
Consider a convex polygon $P$ in the real plane with integral vertices. If we associate with every integral point $(a,b)$ the monomial $x^ay^b$ we then arrive at the Laurent polynomial $S(P)$ equal to the sum of corresponding monomials over all the integra... | https://mathoverflow.net/users/19864 | Formalization (and background) of a formula, concering the integral points of a polygon. | As the other answer pointed out, this is Brion's theorem, and you can probably find several articles devoted to it. There are many proofs in the literature, the original one using equivariant K-theory and toric geometry, but there are more elementary proofs as well.
My perspective is that Brion's formula gives you an... | 3 | https://mathoverflow.net/users/2384 | 129493 | 71,972 |
https://mathoverflow.net/questions/129494 | 0 | Consider the following diagram which lives in the category of $R$-modules.
$$
\begin{array}{ccccccccc}
0 & \xrightarrow{i} & A & \xrightarrow{f} & B & \xrightarrow{q} & C & \xrightarrow{d} & 0 \newline
& & \downarrow & & \downarrow & & \downarrow & & \newline
0 & \xrightarrow[j]{} & A & \xrightarrow[g]{} & E & \xrigh... | https://mathoverflow.net/users/11290 | Is this square a push-out square? | The right square is a pushout square as soon as the first downward arrow is an epimorphism, and we do not even need $f$ and $g$ to be monomorphisms. This is true in any abelian category.
Since diagrams are not easy to draw, let me label the downward arrows as $a : A \to A'$ (which we assume is an epimorphism but not... | 3 | https://mathoverflow.net/users/11640 | 129512 | 71,978 |
https://mathoverflow.net/questions/129509 | 6 | I've come across the following question, which I think must be easy for experts: is there a complex elliptic curve $E$ with an isogeny of degree 2 to itself?
Of course one can ask the same question for isogenies whose degree is not a square, or for higher dimensional abelian varieties etc.
| https://mathoverflow.net/users/10610 | elliptic curve with a degree 2 isogeny to itself? | Expanding on Francois's answer, $E$ has an endomorphism of degree 2 if and only if its endomorphism ring $R=\operatorname{End}(E)$, which is an order in an imaginary quadratic field, has an element of norm 2. There are exactly three such orders, namely $\mathbb{Z}[i]$, $\mathbb{Z}[\sqrt{-2}]$, and $\mathbb{Z}[(1+\sqrt{... | 13 | https://mathoverflow.net/users/11926 | 129515 | 71,980 |
https://mathoverflow.net/questions/129501 | 1 | I believe the following problem is related to something called the "voter model" in statistics. This is not my area of expertise so please forgive me if the answers turn out to be well known.
Consider a vector $V$ of length $n$ where the elements are integers chosen from $[m]$. At each (discrete) time step we apply t... | https://mathoverflow.net/users/nan | Transition time in finite voter model | From any position, the expected number of steps of type $2$ before the vector is completely constant is at most $(n-1)^2$, with equality if all positions have different values. See ["A balls-and-colours problem."](https://mathoverflow.net/questions/41939/a-balls-and-colours-problem)
I don't agree with the description... | 2 | https://mathoverflow.net/users/2954 | 129524 | 71,986 |
https://mathoverflow.net/questions/129528 | 6 | I am sorry to bother you with this question but I can't figure out this myself (and Mathematics Stack Exchange didn't help).
Is the category of Harish-Chandra Modules of $PSL\_2(\mathbb{R})$ equivalent to the category of Harish-Chandra Modules of $SL\_2(\mathbb{R})$ with even $K$-types?
The motivation of this quest... | https://mathoverflow.net/users/32972 | Harish-Chandra modules of $\mathrm{PSL}_2(\mathbb{R})$ | Yes, the central character of the even $K$-types is trivial, and of the odd ones is the sign character.
Also, this can easily be seen from the classification of irreducible representation on Hilbert spaces.
More generally, the category of rep on Hilbert spaces of a locally compact group $G$ with trivial central cha... | 6 | https://mathoverflow.net/users/10400 | 129529 | 71,989 |
https://mathoverflow.net/questions/129525 | 1 | I'm transcribing parts of Harm van der Lek's thesis 'The homotopy type of complex hyperplane complements' and due to it being written in 1983 the typesetting isn't very detailed. In latex, how should the picture \*\* be written (just the part that comes after 'This pregallery is denoted:')? Are the $S\_i$'s and $K\_i$'... | https://mathoverflow.net/users/33654 | Notation of a pregallery | It seems to me that the $K$'s and $S$'s should be the same size. They play the same role as the edges and vertices in a path in a graph. The reason they are written on different levels is just to make the alternating structure easier to see. In particular, the K's are not exponents, not are the S's subscripts.
In LaT... | 3 | https://mathoverflow.net/users/6794 | 129532 | 71,991 |
https://mathoverflow.net/questions/129505 | -1 | x is a normal distributed variable. then what is the expectation of ln(1+e^x).
i simulated this distribution and find that when x is N(0, 100), the mean of this function is around 4.1, and when x is N(-10, 100), the mean is around 0.9.
can anyone tell me the exact expectation of this function, when x is normally d... | https://mathoverflow.net/users/33648 | expectation of ln(1+e^x) | It is easy to obtain lower bounds on the expectation by noting that
$$
\ln(1+e^x) \geq \max\{0,x\}
$$
which simplifies the expectation extremely and leads to the result for $X\sim \mathcal{N}(\mu,\sigma)$
$$
E(\ln(1+\exp(X))) \geq E(\max\{0,X\}) = \frac{\sqrt{\sigma}}{\sqrt{2\pi}} e^{-\frac{\mu^2}{2\sigma}} + \mu \... | 0 | https://mathoverflow.net/users/13400 | 129535 | 71,993 |
https://mathoverflow.net/questions/129541 | 13 | I've been investigating the Milnor hypersurfaces, and every reference seems to point to the paper by Milnor, "On the cobordism ring $\Omega\_\*$ and a complex analogue II". Despite my best efforts, I cannot seem to find it. Was this paper ever published? If not, is there a draft that is available?
| https://mathoverflow.net/users/4517 | Does the paper "On the cobordism ring $\Omega_*$ and a complex analogue II" exist? | Here's what John Milnor writes about this, in his collected works:
>
> The projected Part II of this paper
> was never written. In fact I am
> chagrined to discover that I have
> never published any details about some
> of the announced results which were
> intended to appear in it. I was very grateful when Th... | 23 | https://mathoverflow.net/users/11260 | 129542 | 71,995 |
https://mathoverflow.net/questions/129543 | 3 | I was told singular quartic algebraic surfaces in $\mathbb{P}^3\_{\mathbb{C}}$ have been completely classified and their singularities have been described.
Can anyone provide me with a resource where this classification is described, in particular the possible combination of singularities on such surfaces?
A search p... | https://mathoverflow.net/users/29657 | Reference for the classification of (singular) degree 4 surfaces in $\mathbb{P}^3_{\mathbb{C}}$? | Look at the classical book by C. M. Jessop [*Quartic surfaces with singular points*](http://archive.org/details/cu31924062545383) (1916).
The paper by C. Segre *Etude des différentes surfaces du 4-eme ordre à conique double ou cuspidale*, Math. Ann. **24** (1884), no. 3, 313–444 is also useful.
At any rate, I do n... | 2 | https://mathoverflow.net/users/7460 | 129544 | 71,996 |
https://mathoverflow.net/questions/129498 | 6 | There is a tension between (1) interpreting proper class talk in set theory as talk about first-order formulas and satisfaction; and (2) taking it to be an interesting and non-trivial result that there is no (non-trivial) elementary embedding from V into V and/or taking it to be an open question whether there can be su... | https://mathoverflow.net/users/17968 | The Kunen inconsistency and definable classes | My perspective on this issue is that there are a variety of ways to take the claim of the Kunen inconsistency, and we needn't pick a particular one as the only right one. Rather, we gain a fuller perspective of the result by understanding the full robust context including all of the interpretations.
* Kunen proved hi... | 10 | https://mathoverflow.net/users/1946 | 129549 | 71,997 |
https://mathoverflow.net/questions/129472 | 1 | This is a construction [Definition 6.1] given in the paper D-equivalence and K-equivalence by Kawamata.
>
>
> >
> > Let $X$ be a normal quasiprojective variety such that the canonical divisor $K\_X$ is a $\mathbb{Q}-$Cartier divisor. Each point $x \in X$ has an open neighborhood $U\_x$ such that $m\_xK\_X$ is a
... | https://mathoverflow.net/users/29730 | Uniqueness of the canonical etale coverings | I recommend reading about "root stacks" as studied by Charles Cadman. Given an invertible sheaf, e.g., $\mathcal{O}\_X(mK\_X)$, and given an integer $m$, there is an Artin stack $\mathcal{R}\to X$ parameterizing $m^{\text{th}}$ roots of the pullback of this invertible sheaf. This stack is a $\mu\_m$-gerbe over $S$, hen... | 3 | https://mathoverflow.net/users/13265 | 129550 | 71,998 |
https://mathoverflow.net/questions/129571 | 1 | How do we construct Seifert fibration on mapping torus of surface with monodromy a periodic mapping class. I know that the fiber of the Seifert fibration has to be transverse to the surface fiber of the fibration over $S^1$ but I do not know how we do the construction. Under which conditions can we do the same construc... | https://mathoverflow.net/users/17492 | construct Seifert fibration on mapping torus of surface with monodromy a periodic mapping class | Just use the suspension flow of the periodic diffeomorphism $f: S\to S$ in the periodic mapping class. Then all flow lines will be periodic (i.e., circles) and you are done; the base will be the quotient $S/f$.
For the second question, the answer is yes; again, just suspend the invariant multicurve.
| 4 | https://mathoverflow.net/users/21684 | 129576 | 72,009 |
https://mathoverflow.net/questions/40145 | 45 | What is known about irrationality of $\pi e$, $\pi^\pi$ and $e^{\pi^2}$?
| https://mathoverflow.net/users/9550 | Irrationality of $ \pi e, \pi^{\pi}$ and $e^{\pi^2}$ | I believe most such questions are still very far from being resolved.
Apparently, it is not even known if $\pi^{\pi^{\pi^\pi}}$ is an integer (let alone irrational).
| 48 | https://mathoverflow.net/users/33664 | 129577 | 72,010 |
https://mathoverflow.net/questions/129575 | 44 | Does an existence of [large cardinals](https://en.wikipedia.org/wiki/Large_cardinal) have implications in more down-to-earth fields like number theory, finite combinatorics, graph theory, Ramsey theory or computability theory? Are there any interesting theorems in these areas that can be proved based on assumptions of ... | https://mathoverflow.net/users/33664 | Does an existence of large cardinals have implications in number theory or combinatorics? | Harvey Friedman has recently produced some results in this area. See for example Friedman, *[Invariant Maximal Cliques and Incompleteness](https://cpb-us-w2.wpmucdn.com/u.osu.edu/dist/1/1952/files/2014/01/MaxClique100711-19p50cc.pdf)*, 2011. There is also a draft of a text book titled [*Boolean Relation Theory and Inco... | 25 | https://mathoverflow.net/users/30790 | 129582 | 72,012 |
https://mathoverflow.net/questions/129581 | 4 | **Disclaimer:** this is not research-level, but I've read some non research-level questions/answers on quasinilpotent operators here, some of them involving renowned users. So I thought I'd give it a try. I've already asked two persons who know functional analysis very well without success. I apologize in advance if a ... | https://mathoverflow.net/users/35324 | On the self-adjoint part of a quasinilpotent operator | I think the Volterra operator is a counterexample. Consider $L^2[0,1]$ and define
$$
(Vf)(x) = \int\_0^x f(t) dt.
$$
A calculation shows that
$$
(V^\*f)(x) = \int\_x^1 f(t) dt.
$$
So $V+V^\*$ is positive, since it is the orthogonal projection onto the constant functions. But $V$ is well-known to be quasinilpotent, sin... | 5 | https://mathoverflow.net/users/13360 | 129584 | 72,014 |
https://mathoverflow.net/questions/129546 | 2 | Consider a discrete even torus $G=(V,E)$, i.e. the graph on $\lbrace 0,1,\dots,n-1 \rbrace^2$, $n$ even, where two vertices are connected by an edge only if they differ by 1 in only one coordinate, modulo $n$.
$G$ is a bipartite graph. Call $O$ and $E$ the two sets into which the vertex set $V$ is partitioned (consis... | https://mathoverflow.net/users/13388 | Peculiar vertex-isoperimetric inequality on the discrete torus (and generalization) | The case $n=2$ is the usual discrete cube. Thinking of this as the power set of $[n]$, Harper's theorem tells us that initial segments of the simplicial order (ordering by set size then lexicographically) minimise the vertex boundary. If we want to minimise over odd or even sets then the best possible result we could h... | 3 | https://mathoverflow.net/users/25485 | 129588 | 72,015 |
https://mathoverflow.net/questions/129536 | 9 | It is well-known that many modular forms can be expressed as infinite products. For instance, the most famous one is probably the expansion
$$\Delta(q) = q \prod\_{n=1}^\infty (1-q^n)^{24}$$
for the discriminant cusp form of weight $12$ and level $1$. Another example is the cusp form of weight $2$ and level $11$
... | https://mathoverflow.net/users/6779 | Where do the product expansions of modular forms come from? | Many "natural" examples of automorphic infinite products (also known as Borcherds products) can be explained using the singular theta lift of Harvey-Moore and Borcherds. These examples have the property that the exponents of the product are coefficients of a modular form.
For example, the 24 in the exponents of the p... | 2 | https://mathoverflow.net/users/121 | 129596 | 72,017 |
https://mathoverflow.net/questions/129554 | 0 | Suppose we have polynomials $f$ and $g$ over a field $F$ and for some $a\in F$ we know $(x-a)^m$ divides $f\circ g$, i.e. $f\circ g$ has a zero of order $m$ at $a$. Then we have that $(x-a)^{m-n}$ divides $(f\circ g)^{(n)}$, the $n$'th derivative of $f\circ g$. Now suppose we know that $g'$ is relatively prime to $x-a$... | https://mathoverflow.net/users/33661 | Zeros of compositions of polynomials and derivatives | This doesn't work in general. It only works if we differentiate $f$ along the direction of $g$, that is, apply the multivariate chain rule to deduce results like
$\sum\_{i=1}^l g\_i' \frac{df}{dx\_i}=0$
$\sum\_{i=1}^l\sum\_{j=1}^l g\_i' g\_j' \frac{d^2f}{dx\_idx\_j} +\sum\_{i=1}^l g\_i'' \frac{df}{dx\_i}$
We cann... | 2 | https://mathoverflow.net/users/18060 | 129598 | 72,018 |
https://mathoverflow.net/questions/129600 | 16 | This question is inspired from this [one](https://mathoverflow.net/questions/129143/verifying-the-correctness-of-a-sudoku-solution), where it is asked what is the minimum number of checks needed to verify that a Sudoku solution is correct. Let
$$
E=\{r\_1, \dots, r\_9\} \cup \{c\_1, \dots, c\_9\} \cup \{b\_1, \dots,... | https://mathoverflow.net/users/2233 | Is there a Sudoku matroid? | By some sort of strange mathematical cosmic entanglement, it appears that François Brunault answered his own question in the other [thread](https://mathoverflow.net/questions/129143/verifying-the-correctness-of-a-sudoku-solution) while I was writing this question.
The answer is indeed **yes.** Feel free to click on ... | 10 | https://mathoverflow.net/users/2233 | 129601 | 72,020 |
https://mathoverflow.net/questions/129595 | 1 | **Background**: A polygonal billiards table $P$ with rational angles gives rise to a flat structure $S(P)$ in a standard way, described [here](http://www.math.psu.edu/tabachni/prints/masur.pdf). Curves of constant argument on $S(P)$ which start and end at a zero of the $1$-form are called *saddle connnections*, and are... | https://mathoverflow.net/users/13832 | Computing saddle connections in flat structures | I am pretty sure this is unknown. The interesting case is when everything lies in some number field. Then for any $L$ it is possible (with a computer) to compute all saddle connections in a given direction of length at most $L$, but in general, one does not know how to rule out that longer saddle connections exist.
T... | 2 | https://mathoverflow.net/users/16143 | 129615 | 72,025 |
https://mathoverflow.net/questions/129616 | 4 | In a finite group what is relationship between the number of Sylow $p$-subgroups with the number of elements of order a multiple of $p$?
Is there any reference for my question?
| https://mathoverflow.net/users/33677 | Relationship between the number of Sylow subgroups with element orders in finite group | One relationship is that the number of $p$-singular elements ( that is, elements whose order is divisible by $p$) is divisible by the number of Sylow $p$-subgroups of $G$. This is a consequence of a theorem Frobenius, together with Sylow's theorem, though I don't recall seeing the fact stated in print.
Let $P$ be a S... | 6 | https://mathoverflow.net/users/14450 | 129619 | 72,027 |
https://mathoverflow.net/questions/129531 | 3 | **EDIT**: André Henriques has commented below that the correct separability condition is not weak-\* separability as I have written below, but separability of the predual.
This post came out a bit long, but if you're familiar with the topic, you can probably just skim through most of it - I've put the questions in b... | https://mathoverflow.net/users/33647 | When does a $W^*$-algebra have a standard Borel spectrum? | [The category of commutative von Neumann algebras is contravariantly equivalent to the category of measurable spaces.](https://mathoverflow.net/questions/23408/reference-for-the-gelfand-neumark-theorem-for-commutative-von-neumann-algebras) Assuming the axiom of choice, isomorphism classes of objects in the above two ca... | 5 | https://mathoverflow.net/users/402 | 129625 | 72,029 |
https://mathoverflow.net/questions/129594 | 6 | The definition I know of for a cuspidal automorphic representation of, say, $G=\mathrm{GL}\_2$ over a number field $F$ (relative to a choice of compact open subgroup $K\_f$ of $G(\mathbf{A}\_F^\infty)$ and a maximal compact subgroup $K\_\infty$ of $G(F\otimes\_\mathbf{Q}\mathbf{R})$) is: an irreducible subquotient (equ... | https://mathoverflow.net/users/4351 | Cuspidal automorphic representations as the space of $K$-finite vectors in a unitary cuspidal automorphic representation. | **Yes.** All these viewpoints are equivalent. We write "rep" for irreducible representation.
Edit on request:
An analytic cuspidal rep is subrepresentation $\pi$ of $L\_0^2(G(F) \backslash G(A), \omega)$ for a central character $\omega$.
An algebraic cuspidal rep is an irreducible Harish-Chandra module and an irr... | 7 | https://mathoverflow.net/users/10400 | 129630 | 72,030 |
https://mathoverflow.net/questions/129514 | 3 | For two $n \times n$ nonnegative definite Hermitian matrices $A$ and $B$ over the real number field $\mathbb R$:
>
> **Question1:Is there always a
> nonsingular matrix $P$ over the same
> field $F$ which makes $P^{H}AP$ and
> $P^{H}BP$ both diagonal?**
>
>
>
If the answer is yes, in which part of the matrix... | https://mathoverflow.net/users/25437 | Diagonalize the simultaneous matrices and its background | You might be interested to look at Section 20 on Simultaneous diagonalization of a pair of Hermitian forms of the following book:
>
> **Prasolov, V. V.
> [Problems and theorems in linear algebra.](http://staff.math.su.se/mleites/books/prasolov-1994-problems.pdf)
> Translations of Mathematical Monographs, 134. Am... | 4 | https://mathoverflow.net/users/30062 | 129649 | 72,036 |
https://mathoverflow.net/questions/129653 | 3 | What are the Stiefel-Whitney classes of complex projective spaces?
| https://mathoverflow.net/users/33699 | Stiefel-Whitney class | It follows from the Whitney sum formula that $c(T\_{\mathbf{C}P^n}) = (1+z)^{n+1}$, where $z \in H^2(\mathbf{C}P^n,\mathbf{Z})$ is the generator corresponding to the canonical bundle. (I assume you meant the Chern classes; for the Stiefel-Whitney of a complex vector bundle restricted to a real vector bundle, $w\_{2i}$ ... | 5 | https://mathoverflow.net/users/nan | 129654 | 72,037 |
https://mathoverflow.net/questions/129647 | 6 | If $G$ is an infinite group, is there necessarily an unbounded left-invariant metric on $G$?
| https://mathoverflow.net/users/33695 | Unbounded metrics on groups | The answer is no. See Thm 1.2 of <http://homepages.math.uic.edu/~rosendal/PapersWebsite/Property(OB)10.pdf>. There is a property discussed in the intro of this paper which is equivalent to all left invariant metrics are bounded. It is known that certain large permutation groups have this property.
| 8 | https://mathoverflow.net/users/15934 | 129674 | 72,044 |
https://mathoverflow.net/questions/129665 | 1 | Let $K$ be a number field with unit rank $r$, and consider the usual logarithmic map $L:K^{\ast}\rightarrow\mathbb R^r$ for which $L(\mathcal O\_K^{\times})$ is a lattice of rank $r$. Given a set $\varepsilon\_1,\ldots,\varepsilon\_r$ of fundamental units in $\mathcal O\_K$, we can consider the Euclidean lengths of the... | https://mathoverflow.net/users/27318 | Short basis for the unit group of a number field | The geometry of numbers should say that there is a basis for the units whose lengths are bounded by an explicit function of the regulator $R\_K$. So this reduces to the question of upper bounds for $R\_K$, and of course $R\_K$ does not depend on the choice of basis.
Since $R\_K$ is logarithmic, one might hope for a ... | 2 | https://mathoverflow.net/users/11926 | 129678 | 72,045 |
https://mathoverflow.net/questions/129569 | 3 | Let $n$ be a positive integer and let $X\_n$ be an $n\times n$ random matrix whose entries are iid standard gaussian random variables. I am interested in the distribution of the average singular value of $\frac1{\sqrt{n}}X\_n$.
Let $\alpha\_n$ be the expected value of the average of the singular values of $\frac1{\sq... | https://mathoverflow.net/users/4878 | Non-asymptotic results for bulk of random Wishart matrix | You ask for the average of the singular values of the Wishart matrix. I'm pretty sure there is no closed form expression valid for any $n$. If instead you would ask for the average of *the square* of the singular value, then the answer is very simple, this is just unity independent of $n$.
More generally, if $X\_{n}$... | 3 | https://mathoverflow.net/users/11260 | 129681 | 72,048 |
https://mathoverflow.net/questions/129672 | 3 | " Every curve of the n-th order is in a flat space of n dimensions or less "
and
" If there be a system of $n + m + 1$ quantities $x$ connected by $n + m - 1$ homogeneous equations ; and if this system be such that, upon the addition to it of one equation more, linear and homogeneous in the quantities os and havin... | https://mathoverflow.net/users/33704 | Why these two propositions are equivalent ? | In modern terms, a curve of degree $n$ in some projective space of dimension $n+m$ is actually contained in a projective subspace of dimension $n$. For a proof, project from a point on the curve to a hyperplane and argue by induction on $n$. The second paragraph is just recasting the definition of curve of degree $n$ a... | 7 | https://mathoverflow.net/users/2290 | 129689 | 72,050 |
https://mathoverflow.net/questions/129688 | 3 | Consider a probability measure $\mu$ on, let's say, $\mathbb R$.
Is there a necessary and sufficient condition so that $\mu$ has compact support $Supp(\mu)$ ?
I agree this question is too vague, and it may tempting to answer it by quoting the definition of compactness for the support, so let me be more precise.
... | https://mathoverflow.net/users/15517 | A sufficient condition for a probability measure to have compact support | The Paley-Wiener theorem as it is presented in W.Rudin's "Functional analysis" (Theorem 7.23), will it be satisfactory for you? In your situation: $\mu$ has compact support if and only if $F\_\mu$ can be extended as an entire function to $\mathbb C$, and for some $C>0$ and $r>0$
$$
|F\_\mu(z)|\le C\cdot e^{r\cdot |{\rm... | 8 | https://mathoverflow.net/users/18943 | 129690 | 72,051 |
https://mathoverflow.net/questions/129645 | 9 | The following is an old result of Erdős and Turán (American Mathematical Monthly, 1934):
Given a set of $2^n + 1$ distinct positive integers, all of its two-term sums cannot be composed of the same $n$ primes. (An integer $m$ is said to be composed of primes $p\_1, p\_2, \dots$ is every prime factor of $m$ is one of ... | https://mathoverflow.net/users/4558 | Composing two-term sums from the same primes | This problem and generalizations of it are discussed in the following papers:
1. P. Erdős, A. Sárközy, C. Stewart, [On prime factors of subset sums](http://www.ams.org/mathscinet-getitem?mr=1260108). J. London Math. Soc. (2) 49 (1994), no. 2, 209–218.
2. C. Stewart, [On prime factors of integers which are sums or shi... | 9 | https://mathoverflow.net/users/630 | 129695 | 72,054 |
https://mathoverflow.net/questions/129613 | 5 | The [Sz.-Nagy dilation theorem](http://en.wikipedia.org/wiki/Contraction_%28operator_theory%29#Dilation_theorem_for_contractions) says that for a Hilbert space $H$ with nonexpansive operator $T$, there is a larger space $H'$ containing $H$ and a unitary operator $U$ on $H'$ such that for all $x \in H$ and all $n$, $T^n... | https://mathoverflow.net/users/12978 | Sz.-Nagy dilation for uniformly convex Banach spaces | The Akcoglu-Sucheston-Peller dilation theorem gives indeed a *characterization* of operators on a $L^p$-space with an isometric dilation on a $L^p$-space. These operators are the contractively regular operators. And it is well-known that we can find contractive operators without this property (some 2x2 matrix on $\ell^... | 6 | https://mathoverflow.net/users/33709 | 129709 | 72,058 |
https://mathoverflow.net/questions/129703 | 3 | It is known, that the étale fundamental group of a normal connected scheme equals the galois group of the maximal unramified extension of its function field.
This is not true for integral schemes in general. The point is somehow, that one cannot construct flat finite covers from a function field extension as one does... | https://mathoverflow.net/users/32576 | The étale fundamental group in the non-normal case | Take a nodal cubic $C$, that is a projective line of which you identify two points $0$ and $\infty$.
This curve has connected étale covers of any degree: take $n$ copies of the projective line,
numbered circularly, and identify the $\infty$ of each of them with the $0$ of the next one.
This is a $\mathbf Z/n\mathbf Z$-... | 11 | https://mathoverflow.net/users/10696 | 129711 | 72,059 |
https://mathoverflow.net/questions/129704 | 22 | It is known that the consistency strength of $\sf ZFC+\rm Con(\sf ZFC)$ is greater than that of $\sf ZFC$ itself, but still weaker than asserting that $\sf ZFC$ has a transitive model. Let us denote the axiom "There is a transitive model of $T$" by $\rm St(\sf ZFC)$.
If $M$ is a transitive model of $\sf ZFC$ of size ... | https://mathoverflow.net/users/7206 | If ZFC has a transitive model, does it have one of arbitrary size? | No. Because the construction of the inner model $L$ is absolute for transitive models of ZFC the ordinal heights of transitive models of ZFC are precisely the ordinals $\alpha$ such that $L\_\alpha$ is a model of ZFC. If $\alpha\_0,\alpha\_1$ are the first two ordinals such that $L\_{\alpha\_0}$ and $L\_{\alpha\_1}$ ar... | 21 | https://mathoverflow.net/users/2000 | 129715 | 72,061 |
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