parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/51703 | 18 | Let ZFC$^{\text{fin}}$ be ZFC minus the axiom of infinity plus the negation of the axiom of infinity. It is well-known that ZFC$^{\text{fin}}$ is biinterpretable with Peano Arithmetic. In this sense one could argue that ZFC$^{\text{fin}}$ is PA couched in the language of set theory (ie one nonlogical binary relation, $... | https://mathoverflow.net/users/2361 | Is Robinson Arithmetic biinterpretable with some theory in LST? | This is a refinement of the answer provided by Andres Caicedo.
For weak arithmetics, such as Robinson's $Q$, it is not bi-interpretability, but rather the weaker notion of *mutual interpretability* that turns out to be the "right" notion to study [See [here](http://www.math.osu.edu/~friedman/pdf/Tarski1,052407.pdf) f... | 12 | https://mathoverflow.net/users/9269 | 66289 | 40,888 |
https://mathoverflow.net/questions/66172 | 4 | Hello, I encountered the following system of linear first-order differential equations:
$y'(z)=A(z) y(z)$
where
$y(z): R \rightarrow R^2$ and
$A(z)=\begin{pmatrix}
0 & B Cos(\alpha z + \Phi\_b) \cr
A Cos(\alpha z + \Phi\_a) & 0 \end{pmatrix}$
All quantities are real and constant if not explicitly variable. (P... | https://mathoverflow.net/users/15405 | Homogeneous linear differential equation system with simple periodical coefficient matrix | OK. First of all, change $(y\_1,y\_2)$ to $(y\_1,\sqrt{B/A}y\_2)$ and the time $z$ to $t=\alpha z+\frac{\Phi\_A+\Phi\_B}{2}$. Then we'll get the system with the matrix
$$
A(t)=\begin{pmatrix}
0 & C\cos (t + \Phi) \cr
C \cos(t- \Phi) & 0 \end{pmatrix}
$$
where $C=\alpha^{-1}\sqrt{AB}$ and $\Phi=(\Phi\_B-\Phi\_A)/2$. ... | 3 | https://mathoverflow.net/users/1131 | 66293 | 40,891 |
https://mathoverflow.net/questions/66265 | 0 | We know that any submodule of a quotient module $\frac{M}{N}$ is of the form $\frac{K}{N}$, where $K$ is a submodule of $M$ containing $N$.
Now here is a question: Let $\cal F$ and ${\cal G}$
be quasi coherent sheaves on a scheme $X$. Is any quasi coherent subsheaf of a quotient sheaf $\frac{\cal F}{{\cal G}}$ of the f... | https://mathoverflow.net/users/13615 | Subsheaf of quotient of quasi coherent sheaves | With all due respect to abelian categories the pedestrian answer is this: Let $\alpha:\mathscr {F\to F/G}$ be the natural morphism and $\mathscr{M\subseteq F/G}$ a subsheaf. Then setting $\mathscr{H=\alpha^\* M}$ gives you $\mathscr{M\simeq H/G}$.
| 1 | https://mathoverflow.net/users/10076 | 66295 | 40,893 |
https://mathoverflow.net/questions/65420 | 10 | This is something all introductory texts seem to avoid proving, and many even avoid stating.
We consider untyped $\lambda$-terms on some countably infinite alphabet. If $x$ is a variable and $p$ is an $\alpha$-equivalence class of a term, then we define an $\alpha$-equivalence class $p\left[s/x\right]$ as follows:
... | https://mathoverflow.net/users/2530 | Why is alpha-equivalence in untyped $\lambda$-calculus substitutive? | I think this is proved in H. B. Curry and R. Feys. Combinatory Logic, Volume I. North-Holland Co., Amsterdam, Theorem 2a on page 95. The proof, with all the auxiliary results and some consequences, occupies pages 96-103.
| 5 | https://mathoverflow.net/users/10875 | 66296 | 40,894 |
https://mathoverflow.net/questions/66283 | 3 | Let $A$ be a commutative noetherian ring.
Let $K\_{parf}(A)$ be the full subcategory of the homotopy category $K(A)$ of $A$-modules whose objects are bounded complexes of finitely generated projective $A$-modules. The functor $K\_{parf}(A) \longrightarrow D(A)$ is fully faithful and we denote its essential image by ... | https://mathoverflow.net/users/4333 | Perfect complexes and RGamma(X,F) without mentioning derived categories | For question 1 the answer is yes, in fact the question is tautological. As for question 2 (I assume that $R\Gamma\_c$ stands for the sections with compact support which in the case of proper $X$ coincide with $R\Gamma$), of course you can use any standard complex, for example you can use Cech complex.
| 2 | https://mathoverflow.net/users/4428 | 66298 | 40,895 |
https://mathoverflow.net/questions/66195 | 1 | We are given a $n$-partite weighted graph $G$. Each partition has $n$ vertices, some of which may be isolated. Each partition must contain at least one non-isolated vertex. Let us number the vertices in some $i^{th}$ partition as $V\_{i1},V\_{i2},...,V\_{in}$. Now each non-isolated vertex $V\_{ij}$ has a set of $n-1$ n... | https://mathoverflow.net/users/39663 | "n-partite n-clique" with added conditions | Take $n$ prime and connect $(i,j)$ with $(k,l)$ if $(i-k)(j-l)\equiv 1\mod n$. Clearly, each vertex has one neighbor in each row and column except its own. Also, if $xy=1$ and $zt=1$ in $\mathbb Z\_n$, then there is no reason to expect that $(x-z)(y-t)=1$ (actually, if $-3$ is not a quadratic residue modulo $n$ (say, $... | 5 | https://mathoverflow.net/users/1131 | 66309 | 40,901 |
https://mathoverflow.net/questions/66299 | 14 | I actually looked at one of my [Questions](https://math.stackexchange.com/questions/8884/summing-frac1e2-pi-1-frac2e4-pi-1-frac3e6-pi-1-cd) (posted at **MATH.SE**) again and found a formula which actually *Ramanujan had discovered*.
>
> **Ramanujan**: If $\alpha$ and $\beta$ are positive numbers such that $\alpha \... | https://mathoverflow.net/users/1483 | Ramanujan's Incorrect formula | I believe this formula is true, provided the $\alpha$ in the second sum is changed to a $\beta$, as suggested by Todd Trimble's comment. Let
$$ P(x) = \prod\_{n=1}^\infty \frac{1}{1-x^n} $$
be the generating function for the number of partitions of a non-negative integer $n$. Dedekind proved that $P$ satisfies the tran... | 44 | https://mathoverflow.net/users/7709 | 66317 | 40,905 |
https://mathoverflow.net/questions/66313 | 7 | It's known that from every operad arises a cartesian monad whose algebras are the algebras for the operad. Leinster proved that there are different operads from which arise the same monad, in this way he proved that operads cannot be identified with monads. Now I'm wondering: is it true that every cartesian monad arise... | https://mathoverflow.net/users/14969 | Monad arising from operad | The answer is no, and it is explained in Appendix C of Leinster's book *Higher Operads, Higher Categories*.
Briefly, a 'plain operad', for Leinster, is a non-symmetric operad in Set; a T-operad for T the free-monoid monad. Leinster shows that if T is a cartesian monad then a T-operad is the same thing as a cartesian ... | 7 | https://mathoverflow.net/users/4262 | 66324 | 40,909 |
https://mathoverflow.net/questions/66321 | 3 | Consider a poset $P$ and suppose that every finite subset admits a supremum. Call an ideal $I$ of $P$ *minimal infinite* if it is infinite and every ideal properly contained in $I$ is finite. I am interested in the following "compactness property": Every infinite ideal of $P$ contains a minimal infinite ideal.
Questi... | https://mathoverflow.net/users/15394 | A compactness property of posets | (I will write "semilattice" for "Posets in which every finite subset has a supremum".)
This is a partial answer only. I am not aware of a name for this property, but there are many examples in "mathematical nature" with this property:
1. (All finite semilattices. Not a good example.)
2. The natural numbers
3. Let V... | 3 | https://mathoverflow.net/users/14915 | 66326 | 40,910 |
https://mathoverflow.net/questions/66335 | 7 | Hello!
Let $E$ and $F$ be two vector bundles and let $f:E\rightarrow F$ be a bundle map. Then the kernel of $f$ is not always a subbundle of $E$. Does somebody have a simple example? Does there exist any simple conditions for which the kernel is a subbundle?
Thank you!
| https://mathoverflow.net/users/14806 | Kernel of a bundle map | I think this may be an example:
Consider the map from $S^1 \times \mathbb{R}^2 \rightarrow S^1 \times \mathbb{R}^2$ given by the linear map $f\_\theta: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ that takes the basis vectors $e\_1, e\_2$ to $e\_1, e\_2^\theta$, where $e\_2^\theta$ is $e\_2$ rotated by $\theta$. This shoul... | 8 | https://mathoverflow.net/users/6936 | 66336 | 40,914 |
https://mathoverflow.net/questions/66332 | 12 | Motivation
----------
A *topological vector space* is a vector space over a (topological) field, *K*, that carries a topology such that addition and scalar multiplication are continuous maps, e.g., all normed vector spaces. Since $\mathbb{C} \approx \mathbb{R}^2$ as topological spaces (with their standard topologies)... | https://mathoverflow.net/users/15432 | Is the Characteristic of a Field Detectable from the Topology of a Topological Vector Space? | I think all non-archimedean locally compact fields are homeomorphic: Their rings of integers are compact, metric and totally disconnected and hence are all homeomorphic (to the Cantor set). The same is true for the units in those rings. The field is then topologically the disjoint union of the ring and a countable numb... | 17 | https://mathoverflow.net/users/4008 | 66337 | 40,915 |
https://mathoverflow.net/questions/66323 | 2 | Let $X$ be a variety and let $\mathcal{F}=(\mathcal{F}\_n)\_{n\geq 0}$ be a (constructible) $\ell$-adic sheaf on $X$. Let $K\_n$ be the object in the derived category $D(X,\mathbf{Z}/\ell^{n+1})$ of sheaves of $\mathbf{Z}/\ell^{n+1}$-modules defined by $\mathcal{F}\_n$, i.e., the complex with $\mathcal{F}\_n$ concentra... | https://mathoverflow.net/users/4333 | For an l-adic sheaf (F_n), why is the complex F_n of finite Tor dimension? | I think you have (slightly) misread your sources. If you take *Rapport*
for instance (which is the one I am familiar with) Deligne never make this claim
(as far as I can see). As a typical example consider 4.11 (proof of 2.30) where
he assumes that $(K\_n)$ is a torsion free $\mathbb Z\_\ell$-complex in which case
$K\_... | 2 | https://mathoverflow.net/users/4008 | 66340 | 40,918 |
https://mathoverflow.net/questions/66319 | 2 | Consider this setting:
$Y=X+N$
where $N$ is a Gaussian standard random variable and $X$ is another arbitrarily distributed r.v. You can think of this $X$ as a message being transmitted over an AWGN channel the output of which is the r.v. $Y$. I am wondering if anybody can introduce me some good resources on the con... | https://mathoverflow.net/users/15433 | Minimum Mean Square Error (MMSE) and Mutual Information (I) | This must be extremely relevant
"Mutual Information and Minimum Mean-square Error in Gaussian Channels" by Dongning Guo, Shlomo Shamai (Shitz), and Sergio Verdu
<http://arxiv.org/pdf/cs/0412108>
| 2 | https://mathoverflow.net/users/2763 | 66344 | 40,920 |
https://mathoverflow.net/questions/66285 | 27 | Let's work over the field $\mathbb{C}$ of complex numbers, and let $X\subset \mathbb{P}^n$ be a projective variety. Let $\tilde{X}\subset \mathbb{P}^n$ be any small open neighborhood of $X$, in the complex topology.
>
> Question: Do there exist many algebraic varieties $\subset \tilde{X}$ of dimension equal to the ... | https://mathoverflow.net/users/6074 | Accumulation of algebraic subvarieties: Near one subvariety there are many others (?) | The answer is yes (as in the case of Sasha's answer we use ramified covers)
**Proof.** Let $X$ be any variety in $\mathbb CP^n$. Take a section $s\_m$ of $O(m)$ on $X$ such that $s\_m$ is not equal to $m$-th tensor power $s^{\otimes m}$ of any section $s$ of $O(1)$ restricted on $X$. Now, let $s\_m^{\frac{1}{m}}$ be ... | 20 | https://mathoverflow.net/users/943 | 66347 | 40,921 |
https://mathoverflow.net/questions/66343 | 19 | Is it known whether any element of trace 0 in the reduced $C^\*$-algebra of a non-abelian free group, is a limit of sums of (additive) commutators?
| https://mathoverflow.net/users/14497 | Commutators in the reduced C*-algebra of the free group | This is true. Perhaps it is known whether this property for a group $G$ is equivalent to $C^\*\_\lambda G$ having a unique trace? In any case, the same proof of Powers which shows unique trace for free groups, can be adapted to this question.
First note that if $F = F(x\_1, x\_2, \ldots, x\_n)$ is the free group on $... | 20 | https://mathoverflow.net/users/6460 | 66348 | 40,922 |
https://mathoverflow.net/questions/66288 | 7 | Let $G$ be a finite group and $\sigma$ an automorphism of $G$. Suppose $p$ is a prime and $\sigma$ has prime order $q \neq p$. It's easy to see that $\sigma$ fixes a Sylow $p$-subgroup of $G$ if $\sigma$ is fixed-point free or if $q$ does not divide the order of $G$.
Suppose that $q$ does divide the order of $G$. Wha... | https://mathoverflow.net/users/15428 | Sylow subgroups invariant under an automorphism | This is a very general question, perhaps too general for a definitive answer, and as Jack Schmidt pointed out (in a now deleted comment), it is already a delicate question when $\sigma$ is an inner automorphism. There are bad examples (a little different from Jack's) for all symmetric groups of prime degree greater tha... | 6 | https://mathoverflow.net/users/14450 | 66349 | 40,923 |
https://mathoverflow.net/questions/65630 | 6 | I'm undergraduate student in probability theory (and its applications). There are lots of different and definitely good text on standard, functional analysis-based approach, but I'm interested in alternative approaches - maybe, some more algebraic variants. Could you name some ideas/papers/texts about this? I'm especia... | https://mathoverflow.net/users/12135 | Alternative approaches to probability theory | Yuri, you may have a look at the introductory probability book by Henk Tijms, Understanding Probability. An excellent book having the nice feature that simulation is used throughout to develop probabilistic intuition.
| 0 | https://mathoverflow.net/users/15443 | 66358 | 40,928 |
https://mathoverflow.net/questions/66316 | 8 | Let $g$ be a Riemannian metric on the $d$-dimensional flat space $\mathbb R^d$, and consider the usual Lagrangian $$L(x, \dot x) = \tfrac 1 2 g\_{ij}(x) \dot x^i \dot x^j.$$ Let $\hat g := \sqrt g$ denote the square root of the metric $g$, implicitly defined by the formula $\hat g\_{ai} \hat g\_{bj} \delta^{ab} = g\_{i... | https://mathoverflow.net/users/238 | Calculating the geodesic equation for a particular set of phase-space coordinates | The answer that you want, namely div $U\_g$, is not going to expressible in terms of a geometrically invariant quantity (such as, say, the scalar curvature of $g$) because it depends on the underlying coordinates $x$ in which you have presented the metric. For example, if $\det g(x)$ is constant (which is a coordinate ... | 4 | https://mathoverflow.net/users/13972 | 66364 | 40,930 |
https://mathoverflow.net/questions/66268 | 2 | I have encountered the following property. Can anybody tell me if it already exists in literature and/or is equivalent/similar to other well-known properties?
>
> **Property:** $(X,d)$ metric space. For any open ball $B\subseteq X$ and for any distinct $x,y\in B$, there exist two disjoint open balls $B\_1\ni x$ and... | https://mathoverflow.net/users/13809 | A density condition for metric spaces | An open continuous injective map can also be described as a homeomorphism with an open subset. The stated property is obviously equivalent to: every open set in every open ball $B$ in $X$ contains an open set that is homeomorphic with $B$. If the metric is bounded, $B$ can be replaced by $X$ without loss of generality ... | 1 | https://mathoverflow.net/users/10819 | 66365 | 40,931 |
https://mathoverflow.net/questions/66360 | 4 | Let $P$ be the joint distribution of two random variables $X$ and $Y$, that both have support on $(0,1)$ (I am also interested in the case where $X$ takes values on $k$-dimensional simplex, but I would be happy to start with the simple case).
Now, suppose that for all $n,m \in \mathbb{N}$, we have:
$E[X^nY^m] = E[X... | https://mathoverflow.net/users/8019 | Sufficient conditions for independence based on moments | By your assumptions, for all polynomial $Q$ and $R$, you have :
$$
E(Q(X)R(Y))=E(Q(X))E(R(Y)).
$$
Then, as $X$ and $Y$ are bounded, by Stone-Weierstrass theorem, you get the same equality for all continuous maps $Q$ and $R$. Using for example results about the Fourier transform you get independence of $X$ and $Y$.
Yo... | 6 | https://mathoverflow.net/users/12088 | 66367 | 40,933 |
https://mathoverflow.net/questions/66371 | 2 | Hello,
I read that if $A\subset D$ is the heart of a bounded t structure, D a triangulated category, then D=D^b(A).
Have not been able to find a proper reference for this. Can anyone confirm this result?
Thanks!
| https://mathoverflow.net/users/15449 | bounded t structure | You're never going to find a reference because it's false. Just take $D$ to be the stable homotopy category of Postnikov pieces, i.e. spectra with finitely many non-trivial homotopy groups. Take $D\_{\geq 0}$ (resp. $D\_{\leq 0}$) the full subcategory of spectra with non-trivial homotopy groups concentrated in non-nega... | 12 | https://mathoverflow.net/users/12166 | 66378 | 40,937 |
https://mathoverflow.net/questions/66377 | 138 | It is often said that "Differentiation is mechanics, integration is art." We have more or less simple rules in one direction but not in the other (e.g. product rule/simple <-> integration by parts/u-substitution/often tricky).
There are all kinds of anecdotes alluding to this fact (see e.g. this [nice one from Feynma... | https://mathoverflow.net/users/1047 | Why is differentiating mechanics and integration art? | One relevant thing here is that you are referring to differentiating and integrating within the class of so-called [elementary functions](http://en.wikipedia.org/wiki/Elementary_function), which are built recursively from polynomials and complex exponential and logarithmic functions and taking their closure under the a... | 144 | https://mathoverflow.net/users/2926 | 66381 | 40,939 |
https://mathoverflow.net/questions/66389 | 0 | Let $G$ be an ample $\mathbb{Q}$-divisor on a smooth variety $X$. Let $D$ be a $\mathbb{Q}$-divisor linearly equivalent to $G$. Let $f: Y\to X$ be a common log resolution of $G$ and $D$. We define the multiplier ideal of a divisor $G$ as $I(G)=f\_\*O\_Y(K\_{Y/X}-[f^\*G])$. Are the multiplier ideals $I(G)$ and $I((1-t)G... | https://mathoverflow.net/users/2348 | Stability of multiplier ideals under small perturbations | I don't think so. Take for example $X=\mathbb{P}^2$, and $G$ and $D$ to be distinct lines.
then $I(G)=\mathcal{O}\_X(-G)$ while $I((1-t)g+tD)=\mathcal{O}\_X$ for every small $t>0$.
Maybe you might want to look at the multiplier ideal associated to the linear series $|G|$.
| 3 | https://mathoverflow.net/users/6430 | 66392 | 40,945 |
https://mathoverflow.net/questions/66346 | 0 | Hi everyone,
I will be too happy if anybody help me find a solution for the following problem.
In fact, I have a big problem that I could not solve it for weeks.
Assume that we have we have two independent zero mean Gaussian random variables, $X$ and $Z$ and we define the new random variable $Y$ as $Y=X+Z$. I defin... | https://mathoverflow.net/users/15385 | Conditional distribution of the modulus of the output of AWGN channel given the modulus of the input | Yes, $Y\_M$ is independent of $X\_s$ - it doesn't matter whether or not you condition on $X\_M$.
You essentially give the answer yourself - conditioning on $X\_M$, you get that $Y | X\_M$ is a symmetric r.v., with $f(Y|X\_M) = \frac{1}{2\sigma\_Z}[\varphi(\frac{Y-X\_M}{\sigma\_z})+\varphi(\frac{Y+X\_M}{\sigma\_z})]$ wh... | 1 | https://mathoverflow.net/users/1778 | 66403 | 40,951 |
https://mathoverflow.net/questions/65483 | 5 | I have a question concerning classical invariant theory. Consider binary $n$-forms (i.e. all homogeneous polynomials of degree $n$ of two variables) over the field of complex numbers. Clearly, the group $GL(2,C)$ acts on the space of all such forms by changes of the variables. A classical relative invariant is a polyno... | https://mathoverflow.net/users/15234 | Classical invariant theory: absolute rational invariants and $GL(2)$-orbits | The space of non-degenerate binary forms is an affine variety, since it is the complement of a hypersurface in an affine space. The reductive group $GL\_2(\mathbb{C})$ acts on it with finite stabilizers, so the quotient is again affine, and its elements are distinguished by regular functions, which lift to functions of... | 3 | https://mathoverflow.net/users/2349 | 66407 | 40,954 |
https://mathoverflow.net/questions/66372 | 15 | Background:
I want to consider relative group cohomology: the construction is as follows. I have a subgroup $H\subseteq G$ (and note that I *don't* want to assume that $H$ is normal in $G$), and a $\mathbb Z[G]$-module $M$. Then we have the standard chain complxes $C^\ast(G;M)$ and $C^\ast(H,M)$, and there is a natur... | https://mathoverflow.net/users/35353 | How do I get the correct long exact sequence for relative group cohomology in terms of derived functors? | Your argument is correct, and you do get that the relative group cohomology in your terms is a shift of these Ext-groups.
One reason why the dimension-shifting behavior occurs is that, with original gradings, you can't possibly have that $H^i(G,H;M)$ are the derived functors of $H^0(G,H;M)$. In fact, $C^0(G,H;M)$ is ... | 11 | https://mathoverflow.net/users/360 | 66409 | 40,956 |
https://mathoverflow.net/questions/66379 | 5 | Let $G$ be a group. A [$G$-torsor](http://ncatlab.org/nlab/show/torsor) is a set $X$ together with an action of $G$ such that for all $x,y \in X$ there is exactly one $g \in G$ such that $gx=y$. This looks like a group which has forgotten its identity (no pun intended).
Usually it is assumed that $X$ is nonempty (or ... | https://mathoverflow.net/users/2841 | What about the empty torsor? | If you have any category that admits finite products, you have a notion of $G$-pseudotorsor, namely an object $X$ equipped with a map $act: G \times X \to X$, such that
1. $act \circ (id\_G, act) = act \circ (m\_G,id\_X): G \times G \times X \to X$ and $act \circ (e\_G, id\_X) = id\_X: X \to X$ (i.e., it is an action... | 7 | https://mathoverflow.net/users/121 | 66416 | 40,962 |
https://mathoverflow.net/questions/66417 | 5 | In Pedersen's *Analysis Now*, for example, you learn that a bounded operator on Hilbert space $T: \mathcal{H} \to \mathcal{H}$ is compact if and only if the image $T(B)$ of the unit ball is compact. It is pretty easy to see therefore that any bounded operator which carries the unit sphere to a compact set is compact. D... | https://mathoverflow.net/users/15438 | Detect compactness on unit sphere | It may fail to be closed. If *T* is injective and compact on an infinite dimensional Hilbert space, then $0$ is in the closure of $T(S)$, and not in it.
| 8 | https://mathoverflow.net/users/6101 | 66420 | 40,964 |
https://mathoverflow.net/questions/66394 | 11 | Given a Lie group G and an infinite dimensional Hilbert space $\mathcal{H}$. In the literature I have only encountered the two following notions of a representation $\pi$ of G on $\mathcal{H}$ :
1) $\pi$ : G $\to$ $B(\mathcal{H})$ is a group homomorphism (consider only the invertible bounded linear operators...) and ... | https://mathoverflow.net/users/13338 | Norm continuous infinite dimenisonal representation of a Lie group | Your question also makes sense for a Banach space and that is where the following results also apply. The Hilbert space case is then only a particular case where you can get some more information about the infinitesimal generators.
So let $G$ be a connected Lie group (the discrete pieces are sort of uninteresting...)... | 10 | https://mathoverflow.net/users/12482 | 66425 | 40,966 |
https://mathoverflow.net/questions/66418 | 6 | Could you tell me the physics background of p-Laplacian equation? Thank you!
Actually, I know nothing about this. But I am curious about the original of these PDEs or where they come from. Could you introduce some related books or articles to me?
| https://mathoverflow.net/users/15462 | On the physics background of p-Laplacian equation | It is used to model Newtonian and non Newtonian Fluids. You generally have different values for $p$ depending on the type of fluid:
* $p=2$ (Newtonian) for fluids like water and air
* $p>2$ for viscous ("sticky") fluids
* $p<2$ for non-viscous ("runny") Fluids
**Edit:** I just realized that this alone probably does... | 13 | https://mathoverflow.net/users/14517 | 66426 | 40,967 |
https://mathoverflow.net/questions/66431 | 3 | First I would like to say that geometry is far away from my domain.
I have encountered a problem that has a geometric formulation and I don't even know if this is a difficult or an easy problem.
So here it is, let us be given an n-dimensional ellipsoid such that it is centered at the origin, its algebraic equation... | https://mathoverflow.net/users/2642 | Parametrization of the intersection of an ellipsoid with a sphere | First, you can find an irrational parametrization of the (n-2)-dimensional intersection surface as follows. Projecting along the first coordinate axis to the hyperplane x1=0, you get an ellipsoid in $R^{n-1}$. Now parametrize the new ellipsoid and invert the projection map; you will get a parametrization of the initial... | 6 | https://mathoverflow.net/users/14639 | 66437 | 40,974 |
https://mathoverflow.net/questions/66401 | 24 | Hello,
Learning some Alg.geometry and Sheaf theory, I got used to the notion that cohomology arises naturally as a derived functor of some sort.
This has led me thinking, singular cohomology, from algebraic topology, was never defined (In all books i've checked) as a derived functor, but just by giving cycles and bou... | https://mathoverflow.net/users/14105 | Singular Homology/Cohomology as a derived functor? | On seeing the question, I knew that I had seen something like this. Rinehart had a TAMS paper:
1972 called SINGULAR HOMOLOGY AS A DERIVED FUNCTOR. His interpretation of derived functor is, of course, that of his era but it does give a setting that may be adapted to a more modern treatment. Following up on David's reply... | 8 | https://mathoverflow.net/users/3502 | 66441 | 40,977 |
https://mathoverflow.net/questions/66427 | 9 | The elements of a regular sequence in $k[x\_1, \ldots, x\_n]$ are algebraically independent over $k$ (see for example Matsumura ex. 16.6), and so for a length n regular sequence $(f\_i)$ of homogeneous elements, $k(x\_1, \ldots, x\_n)$ will be algebraic over $k(f\_1, \ldots, f\_n)$.
My question is whether it's also t... | https://mathoverflow.net/users/15464 | Is $k[x_1, \ldots, x_n]$ always an integral extension of $k[f_1, \ldots, f_n]$ for a regular sequence $(f_1, \ldots, f_n)$? | No. For example, the element $y \in k[x,y]$ is not integral over $k[x-1, xy]$. (To see this, write $k[x-1,xy] = k[x,xy]$ and reduce modulo $x$).
| 8 | https://mathoverflow.net/users/7721 | 66445 | 40,979 |
https://mathoverflow.net/questions/66444 | 2 | This came up in a practical problem (physics).
In the following, we work with real numbers only, and consider every vector to be normalized to 1.
To find how "similar" two vectors are (actually, two lines passing through the origin, I don't care about the direction), one can use the scalar product. If the two lines... | https://mathoverflow.net/users/8776 | Generalization of scalar product to k-dimensional subspaces (as opposed to 1-dimensional subspaces, i.e. vectors) | Let $V$ be an $n$-dimensional real vector space with an inner product $\langle \cdot,\cdot \rangle$. Then $\Lambda^k (V)$, the $k^{\text{th}}$ exterior product of $V$, inherits an induced inner product defined as follows.
If $\alpha = v\_1 \wedge \cdots \wedge v\_k$ and $\beta = w\_1 \wedge \cdots \wedge w\_k$, then ... | 5 | https://mathoverflow.net/users/6871 | 66447 | 40,981 |
https://mathoverflow.net/questions/66461 | 0 | 1. The V.I. definition goes doublepoint=overpass-underpass (or was it the other
way around? If it's 50:50, I score 0 always :-). Would it lead anywhere
to define doublepoint=overpass+underpass? (Even if it's equivalent.
As I wouldn't have to guess then :-) I ask because I hate arrows on knots.
2. Stoimenow tabulated va... | https://mathoverflow.net/users/11504 | Some Vassiliev Invariant Questions | 1. No. You can always mirror-reflect your definition across the plane that defines over/under.
2. I think so. Take a look in the [Chmutov, Duzhin, Mostovoy survey on the arXiv.](https://arxiv.org/abs/1103.5628) On page 92 they have the first ten non-trivial Vassiliev invariants, computed on sufficiently-many knots. If ... | 4 | https://mathoverflow.net/users/1465 | 66479 | 40,998 |
https://mathoverflow.net/questions/66474 | 9 | An old [MO answer](https://mathoverflow.net/questions/1365/is-there-a-universal-group-object-answered-yes/1477#1477) by Noah Snyder makes a claim I don't completely understand, but mostly because I don't know any examples. The answer claims that in some examples of (things that one would want to call) group objects $G$... | https://mathoverflow.net/users/290 | The definition of a group object is wrong? | $\newcommand{\cat}[1]{\mathbf{#1}}\newcommand{\id}{\mathrm{id}}$I like your definition of an antimorphism (which, following Ben Webster's suggestion, I will call a "heteromorphism") and I'll raise you one: if $\cat{C}$ comes with an autoequivalence $F$, an $F$-heteromorphism is by definition a morphism $X \to F(X)$. Wh... | 4 | https://mathoverflow.net/users/6545 | 66483 | 41,001 |
https://mathoverflow.net/questions/66459 | 7 | Hi! I'm trying to understand why a hyperbolic n-manifold has finite mapping class group if $n \geq 3 $. In books I'm reading it's said it's a consequence of Mostow's rigidity theorem:
"If M and N are complete hyperbolic manifolds with finite total volume, any isomorphism of fundamental groups is realized by a unique is... | https://mathoverflow.net/users/13715 | Why is the mapping class group of hyperbolic manifolds finite? | In dimension three, this was proven by [Gabai, Meyerhoff, and N. Thurston](http://www.ams.org/mathscinet-getitem?mr=1973051). This was proven orginally for Haken 3-manifolds by Hatcher. Gabai extended this to hyperbolic 3-manifolds satisfying a certain technical condition, which was then verified for all closed hyperbo... | 10 | https://mathoverflow.net/users/1345 | 66485 | 41,002 |
https://mathoverflow.net/questions/66484 | 8 | For a manifold $M$, define the mapping class group $Mod(M)$ to be the set of self-diffeomorphisms of $M$, modulo isotopy. In symbols, $Mod(M) = \pi\_0 Diff(M)$. Of course, every self-diffeomorphism gives an automorphism of $\pi\_1(M)$, well-defined up to conjugacy because of basepoint issues. Thus we have a homomorphis... | https://mathoverflow.net/users/8183 | Generalizations of Dehn-Nielsen-Baer | In general it's not injective, nor surjective. It's just a map.
For example, if $M = S^n$, the sphere. $\pi\_0 Diff(M)$ is the group of homotopy $(n+1)$-spheres, but $Out(\pi\_1 M)$ is trivial. So you have surjectivity but not injectivity provided exotic spheres exist in that dimension. There's lots of much more com... | 5 | https://mathoverflow.net/users/1465 | 66487 | 41,004 |
https://mathoverflow.net/questions/66463 | 7 | Hi,
Let $\Gamma$ be a free subgroup of rank 2 in $\mathbb{G}\_m^2(\mathbb{Q})$. For all but finitely many primes p we can reduce $\Gamma$ modulo p. Let $S$ be the of primes for which $\Gamma$ does not reduce modulo p, and for any $p$ not in $S$, let $\gamma\_p$ be the size of $\Gamma \mod p$. My question is what is k... | https://mathoverflow.net/users/421 | Estimating the size of reduction of rational points on $\mathbb{G}_m^2$ | Presumably "exceptional" means primes where either one of the generators of $\Gamma$ is 0 or $\infty$ mod p, or where $\Gamma$ mod $p$ has rank smaller than $2$. The following reference is possibly relevant to your question, although we consider a somewhat different sum. We give an upper bound (that should be fairly sh... | 5 | https://mathoverflow.net/users/11926 | 66492 | 41,007 |
https://mathoverflow.net/questions/58767 | 15 |
>
> Let $\phi$ be a real smooth superharmonic function on unit disc $D$ in $\mathbb C$; i.e. $\triangle \phi\le 0$.
> Then there is a curve $\gamma$ from the center of $D$ to its boundary such that
> $$\int\limits\_\gamma e^\phi<\infty.$$
>
>
>
The question came from my failed answer [to this question](https:... | https://mathoverflow.net/users/1441 | Curve integral of exponent of superharmonic function. | I'll assume that $\varphi=-\psi$ where $\psi$ is subharmonic and not too weird (say, with isolated non-degenerate critical points; it seems like you can always add something bounded to achieve it but I haven't checked the details). Take any piece of some level curve of $\psi$ inside the disk parameterized by length $\e... | 5 | https://mathoverflow.net/users/1131 | 66501 | 41,012 |
https://mathoverflow.net/questions/66482 | 5 | The notion of nilpotency passes nicely from groups to semigroups. Define $q\_1(x,y)=xy$ and $$q\_{i+1}(x,y,z\_1,\cdots,z\_i)=q\_i(x,y,z\_1,\cdots,z\_{i-1})z\_iq\_i(y,x,z\_1,\cdots,z\_{i-1})$$ inductively for all $x,y,z\_1,z\_2,\cdots$.
A. I. Malcev proved that nilpotent groups of class $c$ are described as the groups... | https://mathoverflow.net/users/13070 | Do all finitely generated nilpotent semigroups have polynomial growth? | The answer is "no", see Theorem 6.1 in Shneerson, L. M.
Relatively free semigroups of intermediate growth.
J. Algebra 235 (2001), no. 2, 484–546. The relatively free semigroup in the variety $xyzyx=yxzxy$ ("nilpotent" of class 2) with at least 3 generators has intermediate growth ($\sim 2^{\sqrt{n}}$).
| 9 | https://mathoverflow.net/users/nan | 66502 | 41,013 |
https://mathoverflow.net/questions/59067 | 20 | Let $X$ be a smooth algebraic variety over an algebraically closed field $k$ of characteristic $p>\dim X$. The motivation for my question comes from the following results.
>
> **1.** If $X$ is *Frobenius split* (the $p$-th power map $\mathcal{O}\_X \to F\_\* \mathcal{O}\_X$ admits n $\mathcal{O}\_X$-linear splittin... | https://mathoverflow.net/users/3847 | Frobenius splitting and derived Cartier isomorphism | Keeping the notation of the question $X$ is a smooth variety over an algebraically closed field $k$ of characteristic $p>\dim X$. Just following along from Decomposition of the de Rham Complex by V Srinivas, we see that the obstruction to lifting the pair $(X,F)$ to a pair $(X^{(2)}, F^{(2)})$ where $X^{(2)}$ is a lift... | 9 | https://mathoverflow.net/users/14672 | 66514 | 41,016 |
https://mathoverflow.net/questions/66500 | 41 | I've heard many people say that class field theory is the same as the Langlands conjectures for GL\_1 (and more specifically, that local Langlands for GL\_1 is the same as local class field theory). Could someone please explain why this is true?
My background is as follows: I understand the statements of class field ... | https://mathoverflow.net/users/1355 | Why is Class Field Theory the same as Langlands for GL_1? | What you are looking for is the correspondence between algebraic Hecke characters over a number field $F$ and compatible families of $l$-adic characters of the absolute Galois group of $F$. This is laid out beautifully in the first section of Laurent Fargues's notes [here](http://www.math.u-psud.fr/~fargues/Motifs_abel... | 25 | https://mathoverflow.net/users/7868 | 66515 | 41,017 |
https://mathoverflow.net/questions/66516 | 4 | Suppose $A$ and $B$ are contractible pointed open subspaces of $X$. The Mayer-Vietoris sequence implies that the boundary morphism $\delta: H\_n(X)\to H\_{n-1}(A\cap B)$ is an isomorphism.
I wonder, if $x$ is a $n-1$-cycle of $A\cap B$, is there an explicit algorithm to write down its $n$-cycle preimage $\delta^{-1} ... | https://mathoverflow.net/users/nan | The preimage of the boundary morphism in the Mayer-Vietoris sequence | Take a contraction of $x$ in $A$. That gives an $n$-chain $y\_A$ living in $A$, whose boundary is $x$. Similarly, a contraction of $x$ in $B$ gives an $n$-chain $y\_B$, again with boundary equal to $x$. The desired preimage is $y\_A - y\_B$.
| 11 | https://mathoverflow.net/users/5010 | 66517 | 41,018 |
https://mathoverflow.net/questions/66469 | 4 | Hi,
On page 120, chapter 4, proposition 4.2.7 in Hubbard's Teichmuller Theory book, volume 1, he proves :
Let $U,V$ be open in $C, f:U \to V $ be a homeomorphism and the restriction of $f$ on $U \backslash l$ is $K$ quasiconformal, where $l$ is a line in $U$. Then $f$ is $K$-q.c. in $U$.
FYI: his definition 4.1.... | https://mathoverflow.net/users/6953 | A quick and elementary question from Hubbard's Teichmuller Theory : Volume I | All that Hubbard has shown up to that point is that the functions $f\_z$ and $f\_{\bar{z}}$ are locally in $L^2(R)$, i.e. they are defined almost everywhere and are square integrable in $R$ as functions. But all we know at this point is that they are the distributional derivatives of $f$ away from $l$. What he shows af... | 7 | https://mathoverflow.net/users/5751 | 66518 | 41,019 |
https://mathoverflow.net/questions/66527 | 6 | Assume that $R$ is a commutative ring with a ring compatible ordering and let $A$ and $B$ be symmetric $n\times n$ matrices with entries in $R$ such that $\sum x\_iA\_{ij}x\_j\geq 0$ and $\sum x\_iB\_{ij}x\_j\geq 0$ for all $x=(x\_1,\ldots,x\_n)\in R^n$. Is it true that $\operatorname{tr}(AB)=\sum A\_{ij}B\_{ji}\geq 0$... | https://mathoverflow.net/users/15488 | Positive matrices matrices over commutative rings | Hi Joakim.
Just to clarify the notions: an ordered ring is a commutative ring $R$ (say with $1 \ne 0$) with a distinguished subset $P$, called the positive elements, such that $P + P \subseteq P$ and $P \cdot P \subseteq P$ and $R = -P \cup \{0\} \cup P$ is a *disjoint* union. Then one defines $a \le b$ iff $b-a \in ... | 3 | https://mathoverflow.net/users/12482 | 66532 | 41,024 |
https://mathoverflow.net/questions/66552 | 3 | I am looking for a reference for what is stated in Srinivasan's book "Representations of Finite Chevalley Groups", which is apparently a special case of Leray spectral sequence. I'll quote the interesting part :
>
> Let $\phi$ be a torsion sheaf on $X$,
> $f : X \to Y$ a morphism of schemes.
> Then we have a Ler... | https://mathoverflow.net/users/15404 | Special case of Leray spectral sequence | The assumption on the fibers is equivalent to that $R^qf\_! \phi$ vanishes unless $q = q\_0$. Hence there is only one nonzero row in the $E\_2$ page of the spectral sequence, which happens so often that there is a special name for this situation: the spectral sequence *collapses* at $E\_2$. Collapsing is stronger than ... | 3 | https://mathoverflow.net/users/1310 | 66556 | 41,037 |
https://mathoverflow.net/questions/66551 | 2 | Let $k$ be an algebraically closed of characteristic $\neq 2,3$. The $j$-invariant induces a bijection
$\{\text{elliptic curves over } k\}/\cong \longrightarrow k.$
Perhaps this is a silly question: I'm curious if there is any meaningful geometric interpretation of the induced ring structure on the set of isomorphi... | https://mathoverflow.net/users/2841 | Structure on the set of elliptic curves via $j$-invariant | The normalization of the j-invariant is a historical accident: it was chosen to have a zero of order 3 at a cube root of 1. There is no really good reason for this: one could equally well normalize it to have a double zero at i, or a zero at some integer of an imaginary quadratic field of class number 1. So it is unlik... | 12 | https://mathoverflow.net/users/51 | 66560 | 41,039 |
https://mathoverflow.net/questions/65729 | 143 | [This](http://www.zib.de/de/mathematik-kalender.html?oid=27084144 "link") talk is about a theory of "perfectoid spaces", which "compares objects in characteristic p with objects in characteristic 0". What are those spaces, where can one read about them?
Edit: A bit more info can be found in Peter Scholze's [seminar ... | https://mathoverflow.net/users/451 | What are "perfectoid spaces"? | Update: The lecture notes of the CAGA lecture series on perfectoid spaces at the IHES can now be found online, cf. <http://www.ihes.fr/~abbes/CAGA/scholze.html>.
It seems that it's my job to answer this question, so let me just briefly explain everything. A more detailed account will be online soon.
We start with a... | 252 | https://mathoverflow.net/users/6074 | 66563 | 41,041 |
https://mathoverflow.net/questions/66521 | 3 | My question is regarding Broer's paper "Line bundles on the cotangent bundle of the flag variety" (see <http://www.springerlink.com/content/t41418q436524515/>). Given the Springer resolution, and its projection to the flag variety, $p: \mathcal{T} = G \times\_{B} \mathfrak{u} \rightarrow G/B$; the aim is to prove that ... | https://mathoverflow.net/users/2623 | Vanishing cohomology of line bundles on the Springer resolution | For your first question, let $i : T \hookrightarrow X$ be the inclusion; then $\mathcal O\_T$ becomes a $\mathcal O\_X$-module by identifying $\mathcal O\_T$ with $i\_\* \mathcal O\_T$, an $\mathcal O\_X$-module. The isomorphism you ask about comes from the following general principle: For any rational $B$-modules $M$ ... | 3 | https://mathoverflow.net/users/1528 | 66574 | 41,048 |
https://mathoverflow.net/questions/64581 | 5 | Hi,
referring to the Riemann-Siegel approximate functional equation for Riemann's Zeta
$ \zeta(s) = \sum\_{n\leq x}\frac{1}{n^s} \ + \ \chi(s) \ \sum\_{n\leq y}\frac{1}{n^{1-s}} \ + \ O(x^{-\sigma}+ \ |t|^{\frac{1}{2}-\sigma}y^{\sigma - 1}) $
would anybody know of a similar result applying to the Dirichlet Eta fu... | https://mathoverflow.net/users/15020 | Approximate functional equation for Dirichlet eta, does any exist? | First of all, if you want just a single sum up to T, then just like with the zeta function you have an approximation:
$$\eta(s)\sim\sum\_{n=1}^{T/2\pi}\frac{(-1)^{n-1}}{n^s}$$
It turns out additionally that the full approximate functional equation (aka Riemann-Siegel formula) holds for really anything where you have ... | 1 | https://mathoverflow.net/users/9849 | 66576 | 41,049 |
https://mathoverflow.net/questions/66539 | 2 | Typically the non-empty feasible space of a SDP has some curved boundary which is why the feasible space has infinitely many extreme points. Is it ever possible to have a SDP whose non-empty feasible region is the convex hull of a finite (exponential) number of rank-$1$ matrices? If yes, how do we characterize such a S... | https://mathoverflow.net/users/39663 | Feasible space of SDP | In this case isn't the convex hull simply a simplex? In this case you are looking for SDPs that just reduce to linear programs.
As an aside, curved boundaries are in a way most natural for SDPs: for example it is known that any convex algebraic region whose boundary is everywhere either strictly convex or positively ... | 3 | https://mathoverflow.net/users/9849 | 66577 | 41,050 |
https://mathoverflow.net/questions/66580 | 2 | Suppose we have an unending source of random numbers from the interval [0,1] coming from the uniform distribution. Think of each number as the result of an experiment.
Fix a value $s$ (say, $s=100$).
We are going to create a random function $f\_s(t)$ for $t \in \mathbb{N}$ from this sequence of random numbers.
We... | https://mathoverflow.net/users/1355 | Standard Deviation of Averages of Random Numbers | Let $Y$ the be mean of $t$ samples of the distribution. $Y$ has mean $1/2$ and variance $1/(12 t)$. Then $f\_s(t)$ is the sample standard deviation of $s$ draws from $Y$. As the sample standard deviation is a consistent estimator, $f\_s(t)$ converges in probability to $\sqrt{1/(12 t)}$.
| 1 | https://mathoverflow.net/users/9896 | 66585 | 41,055 |
https://mathoverflow.net/questions/65900 | 6 | Okay, so an automorphic form $f$ on a reductive group $G/ \mathbb{Q}$ and arithmetic subgroup $\Gamma$ is a smooth function satisfying the following conditions:
(a) invariance with respect to left $\Gamma -$ translations.
(b) Right $K -$ finiteness.
(c) Annihilated with respect to a finite co-dimension ideal of ... | https://mathoverflow.net/users/14812 | A stupid question about Automorphic forms | These conditions have technical/subtle interactions. It probably suffices to think about automorphic forms on a Lie group, and not think of the interaction with finite places.
For example, on each K-isotype, the Casimir element is an elliptic operator, but it is not elliptic without specifying the behavior under K. ... | 16 | https://mathoverflow.net/users/15629 | 66598 | 41,062 |
https://mathoverflow.net/questions/66588 | 13 | Hello,
I have two questions that have been bugging me recently. The first is about the number of spanning forests in a graph and the second is about enumerating these with edge labels.
Q1: I am aware of Kirchhoff's Matrix-Tree theorem regarding the number of spanning trees in a graph. I was wondering if there is a ... | https://mathoverflow.net/users/15493 | Number of spanning forests in a graph | Two points:
(1) There is a result which is very similar to this. Let $G$ be a graph. Let $A$ be the matrix whose rows and columns are indexed by the vertices of $G$, where $A\_{ij}$ is negative the number of edges from $i$ to $j$, for $i \neq j$, and where $A\_{ii} = \lambda\_i + \deg\_i(G)$. Then the coefficient of ... | 11 | https://mathoverflow.net/users/297 | 66606 | 41,067 |
https://mathoverflow.net/questions/66615 | 0 | Let $ f : X \to S, g : Y \to S$ be the scheme morphisms, and $ X \times\_S Y$ be the product of shemes. Let $ z \in X \times\_S Y$ , and $ x=p(z) , y=q(z) ,$ where $ p: X \times\_S Y \to X, q: X \times\_S Y\to Y $ are projections. Suppose $ s=f(x)=g(y) \in S$ and $x\in U \subset X, y\in V \subset Y, s\in W \subset S, f... | https://mathoverflow.net/users/15139 | An element in the product of schemes | Let $Z=\{z\}$. Then $p$ and $q$ induce morphisms $Z\to U$ and $Z\to V$ compatible with the structure maps $U\to W$ and $V\to W$. Then by the definition of a product there exists a *unique* morphism $Z\to U\times\_W V$ compatible with these morphisms. You can repeat the same argument for $X,Y,S$ and get a unique morphis... | 3 | https://mathoverflow.net/users/10076 | 66622 | 41,076 |
https://mathoverflow.net/questions/66599 | 5 | We know that 5 points in general position determine a conic section uniquely. I wonder if there is a generalization. Consider the equation
$F(x\_1,\dots, x\_n) =0$ where $F$ is a polynomial of degree $d$ (over $R$). Is there a
(sharp) bound on the number of points $(x\_1,...,x\_n)$ (in general position) which uniquel... | https://mathoverflow.net/users/15028 | How many points determine an algebraic surface ? | The answer has essentially been given by J. C. Ottem in a comment. I just put it here (with a couple details) so that the question is "answered".
The space of degree $d$ polynomials in n+1 variables has dimension $\binom{n+d}{d}$ (coefficient count), so hypersurfaces of degree d in $\mathbb{P}^n$ are parameterized by... | 4 | https://mathoverflow.net/users/1939 | 66623 | 41,077 |
https://mathoverflow.net/questions/66611 | 1 | Let $X\subseteq\mathbb{C}P^n$ be a smooth subvariety. Let $C\subseteq X$ be an algebraic curve
and let $[C]\in H\_2(X;\mathbb{R})$ be the homology class it determines.
Does there exist a continuous map $u:\Sigma\longrightarrow X$ satisfying properties (1),(2) and (3) below?
(1) $\Sigma$ is a genus zero nodal curve (tha... | https://mathoverflow.net/users/15512 | homology class of an algebraic curve versus homology class of a genus zero holomorphic curve | A complex analytic space $X$ is called Brody hyperbolic if any holomorphic map $\mathbb C\to X$ is constant. There are many spaces that are Brody hyperbolic ([see this paper for instance](http://arxiv.org/abs/0902.3741)) and thus what you ask will not happen. (In fact, according to Kobayashi's conjecture all general hy... | 3 | https://mathoverflow.net/users/10076 | 66624 | 41,078 |
https://mathoverflow.net/questions/66617 | 2 | Suppose we're given a dominant morphism of irreducible algebraic varieties over an algebraically closed field $k$, $f:X \rightarrow Y$, then why do there exist subvarieties $Z$ of $X$ with $\dim Z=\dim Y$ and $f(Z)$ dense in $Y$? One idea I've heard is to resrict to the affine case, so $X=\text{Spec }A$ and $Y=\text{Sp... | https://mathoverflow.net/users/13139 | Subvarieties that dominate a dominant morphism | More generally, let $f:X\to Y$ be a dominant morphism of finite type of schemes, with $Y$ irreducible. Let $y$ be the generic point of $Y$. Then the fiber $X\_y$ is a nonempty scheme of finite type over $\kappa(y)$. Let $z\in X\_y$ be a closed point. By Hilbert's Nullstellensatz, $\kappa(z)$ is a finite extension of $\... | 3 | https://mathoverflow.net/users/7666 | 66629 | 41,083 |
https://mathoverflow.net/questions/66561 | 6 | I suspect that they knew that the $L-$function is defined only for $Re(s) \gt 3/2$. Did they attempt to evaluate the $L-$function at $s=1$ by plugging $s=1$ in the infinite product $\prod\_p (\frac{1}{1-a\_pp^{-s}+p^{1-2s}})$? I think that it gives $\prod (\frac{N\_p}{p})^{-1}$, which does not make sense under the usua... | https://mathoverflow.net/users/nan | How did Birch and Swinnerton Dyer arrive at their conjecture? | For what it's worth, here are some historical comments.
Both Birch and S-D spoke in Cambridge a few weeks ago, on the history of their conjecture. To my surprise, both of them emphasized the role not of the $L$-function, but of the Tamagawa number, in their comments on how it all came about.
Tamagawa had introduced... | 40 | https://mathoverflow.net/users/1384 | 66636 | 41,087 |
https://mathoverflow.net/questions/66650 | 11 | Like many of my questions, this question is actually aimed at $p$-adic analysis.
One of the main obstacles in doing analysis $p$-adically ist that the $\mathbb{Q}\_p$ is totally disconnected.
From previous answers and reading I learned that one tries to circumvent these problems
and the result are things like "rigid ... | https://mathoverflow.net/users/3757 | Connectifications? | After seeing wood's last comment (comment #2 under his question), I've decided to add a few words (a bit too many for a comment) which hopefully make clear the force of Qiaochu's answer.
Generally speaking, the categorical meaning of "completion" refers to taking a left adjoint of a *full* inclusion of categories; i... | 13 | https://mathoverflow.net/users/2926 | 66662 | 41,100 |
https://mathoverflow.net/questions/66656 | 5 | My graduation thesis was about stability theorems for $C\_0$-semigroups (see the Wikipedia article for the definitions: <http://en.wikipedia.org/wiki/C0-semigroup>). I would like to know if there is some applicability of the stability theorems I know in this field. The only applications I found for my thesis were about... | https://mathoverflow.net/users/13093 | $C_0$-semigroups applications | The first theorem (proved by Richard Datko in the Hilbert space case and by Pazy in the Banach space case) is very useful to establish controllability for systems governed by hyperbolic differential equations. Examples include the boundary control of plates, rods, and other elastic structures. There is a book by Jack L... | 6 | https://mathoverflow.net/users/7352 | 66665 | 41,102 |
https://mathoverflow.net/questions/66669 | 24 | Where can I find a proof (in English) of the Krylov-Bogoliubov theorem, which states if $X$ is a compact metric space and $T\colon X \to X$ is continuous, then there is a $T$-invariant Borel probability measure? The only reference I've seen is on the [Wikipedia](https://en.wikipedia.org/wiki/Krylov%E2%80%93Bogolyubov_t... | https://mathoverflow.net/users/8382 | Proof of Krylov-Bogoliubov theorem | First, fix $x \in X$ and let $\mu\_1 := \delta\_x$ be the Dirac measure supported at $x$. Then define a sequence of probability measures $\mu\_n$ such that for any $f \in C^0 (X)$,
$$ \int\_X f(y) \mathrm{d} \mu\_n (y) = \frac{1}{n} \sum\_{k=0}^{n-1} \int\_X f \circ T^k (y) \mathrm{d} \mu\_1 (y). $$
Apply the Banach-Al... | 29 | https://mathoverflow.net/users/11266 | 66671 | 41,104 |
https://mathoverflow.net/questions/66679 | 8 | It is hoped that in the future with the advent of quantum computing that fundamental operations on a computer will have arbitrarily high precision. Moreover, that even with such high precision, computation times with be realistic. An important concept in Numerical Analysis is ill-conditioning.
For instance it is com... | https://mathoverflow.net/users/2011 | Are there ill-conditioned problems in infinite precision arithmetric? | Ill-conditioning isn't a concept that depends on the precision that you use to compute the solution. "A small change in the data turns into a large change of the solution" isn't a concept that involves actual computation. Think of it as "the map data $\mapsto$ solution has a large Lipschitz constant".
The error on th... | 16 | https://mathoverflow.net/users/1898 | 66682 | 41,111 |
https://mathoverflow.net/questions/66685 | 11 | I'm looking for the answer of who first proved modularity of CM curves? That is if $E$ is an elliptic curve over $\mathbb{Q}$ which has complex multiplication then there's a non-constant morphism from $X\_0(N)$ to $E$ for some $N$ (not necessarily the conductor). The possible names that I've thought of are Hecke, Deuri... | https://mathoverflow.net/users/2784 | Historical question about modularity of CM curves | Dear Victor,
I believe it was Shimura in the paper *On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields*, Nagoya Math. J. 43 (1971), 199–208.
Regards,
Matthew
| 12 | https://mathoverflow.net/users/2874 | 66686 | 41,112 |
https://mathoverflow.net/questions/66684 | 4 | Are there any well-known continuous area-preserving maps between fat convex polygons and fat rectangles? Specifically, given a fat convex polygon $C$, is there a natural way to choose a fat rectangle $R$ and a continuous mapping $f:C\rightarrow R$ such that area of sub-regions is preserved? I've avoided giving a precis... | https://mathoverflow.net/users/11828 | Area-preserving map between rectangles and fat polygons | There are lots of ways to do this; without more constraints (or indication of what is desired), it's hard to pick a best one. But here's one.
We'll exploit the fact that for any two triangles, there is a unique affine linear map that takes one to the other (with vertices in a specified order); if the two triangles have... | 8 | https://mathoverflow.net/users/5010 | 66689 | 41,114 |
https://mathoverflow.net/questions/66699 | 28 | Usual mathematical formulation of a 2d (closed) TQFT is as a functor from the category of 2-dim cobordisms between 1-dim manifolds to the category of vector spaces (satisfying various properties.)
For example, a pair of pants (a morphism from $S^1$ to $S^1 \times S^1$) is mapped to a linear map $f:V\to V\otimes V$; s... | https://mathoverflow.net/users/5420 | Why is a 2d TQFT formulated as a functor? | Mathematicians have sometimes defined TQFTs in the way Yuji suggests. Indeed, Getzler and Kapranov define the notion of "modular operad" for precisely this purpose (it formalizes the relations between $f\_k$, $f\_l$ and $f\_{k+l-2}$, as well as between $f\_k$ and $f\_{k-2}$). Earlier, Kontsevich and Manin axiomatized G... | 36 | https://mathoverflow.net/users/14681 | 66705 | 41,126 |
https://mathoverflow.net/questions/66700 | 2 | Recently in my investigations I faced a problem related to amenable groups.
I have no idea if my question is suitable for this site, but after ask some collegues in my department I decided to try to find some help here.
Suppose that $G$ is an amenable group having a mean
$ m\in \left(L^{\infty}(G,\mathbb{R}) \right... | https://mathoverflow.net/users/nan | Have a general mean of an amenable group a kind of Fubini's property ? | Can't you argue as follows: given $f\in L^\infty(G\times G)$ for any $a\in L^1(G)$ define $f\_a$ by $$ f\_a(s) = \int\_G f(s,t) a(t) \ dt. $$ Then $f\_a\in L^\infty(G,\mathbb R)$ with $\|f\_a\| \leq \|f\| \|a\|$. Thus the map $$L^1(G) \rightarrow \mathbb R, \quad a\mapsto m(f\_a) $$ is bounded and linear, and so there ... | 2 | https://mathoverflow.net/users/406 | 66719 | 41,134 |
https://mathoverflow.net/questions/66720 | 0 | I'm trying to compute Neron-Severi lattices of some K3 surfaces. They have elliptic fibrations with multiple sections. Setting one section to be the identity section, I can write down a Weierstrass equation, and compute the singular fibers and Mordell-Weil group. In cases where Mordell-Weil is finite, I can work out NS... | https://mathoverflow.net/users/4192 | Neron-Severi Lattice of Elliptic K3 | Usually, the really hard part of this problem is to find a basis for the Mordell-Weil group, not to calculate the N\'eron-Severi group. You suggest that somehow you are capable to find a basis for the Mordell-Weil group.
For reasons of presentation I prefer to work with a Weierstrass equation in $\mathbb{P}(4,6,1,1)$... | 3 | https://mathoverflow.net/users/8621 | 66723 | 41,135 |
https://mathoverflow.net/questions/66715 | 5 | The complete homogeneous symmetric polynomials are defined as
$$
h\_k (x\_1, \dots,x\_n) = \sum\_{1 \leq i\_1 \leq i\_2 \leq \cdots \leq i\_k \leq n} x\_{i\_1} x\_{i\_2} \cdots x\_{i\_k}.
$$
For example,
$$
h\_3(x\_1,x\_2,x\_3) = x\_1^3 + x\_2^3 + x\_3^3 + x\_1^2 x\_2 + x\_1^2 x\_3 + x\_2^2 x\_3 + x\_2^2 x\_1 + x\_3^2 ... | https://mathoverflow.net/users/5091 | Monotonicity of complete homogeneous symmetric polynomials | I claim that, if $h\_k > h\_{k+1}$, then $h\_j > h\_{j+1}$ for all $j \geq k$. This includes both your assertions above. (Note that $h\_0$ should be considered to be $1$.)
More precisely, I will prove that, for any positive reals $x\_j$, we have
$$\frac{h\_0}{h\_1} \leq \frac{h\_1}{h\_2} \leq \frac{h\_2}{h\_3} \leq \... | 14 | https://mathoverflow.net/users/297 | 66725 | 41,137 |
https://mathoverflow.net/questions/66708 | 5 | I would like to solve the general problem of determining a linear recurrence relation that fits a given integer sequence of length $n$, or stating that none exists (with fewer than $n/2-k$ coefficients for some reasonable fixed $k$, perhaps 2, to ensure that I'm not overfitting). Actually, I'd like to find the smallest... | https://mathoverflow.net/users/6043 | Determining a recurrence relation | This problem comes up often in coding theory, and it can be solved efficiently by the Berlekamp-Massey algorithm (Wikipedia has pseudo-code). This is more or less equivalent to using continued fractions, although many expositions don't present it that way: given a sequence $a\_0,a\_1,\dots,a\_N$, look at the rational f... | 12 | https://mathoverflow.net/users/4720 | 66726 | 41,138 |
https://mathoverflow.net/questions/66681 | 46 | A lot of notions in differential geometry have direct meaning in Physics. For example:
* A Riemannian metric is a way to encode distances on a manifold and in Physics it is the gravitational field. The curvature of the Levi-Civita connection gives the strength of the gravitation in a certain sense,
* A principal $G$-... | https://mathoverflow.net/users/14806 | Classical geometric interpretation of spinors | As far as I know, this sort of structure was first invoked by Dirac in order to take a square root of the Laplacian, and this he was doing in order to write down Lorentz invariant Klein-Gordon equations. It is a useful exercise to try to solve the equation $D^2 = \Delta$ on a Euclidean space $V$ for a first order opera... | 26 | https://mathoverflow.net/users/4362 | 66734 | 41,141 |
https://mathoverflow.net/questions/66729 | 1 | Can we find a Zariski-dense subset $U$ of the affine plane over the complex number field, such that any subset $U^{\prime}$ of $U$ has its closure either the whole affine plane or finite number of points, depending the cardinal of $U^{\prime}$ is finite or not?
| https://mathoverflow.net/users/5661 | On Zariski Dense Subsets | Yes. Consider the set $U$ of pairs $(x,e^x)$ where $x \in \mathbb{Z}$ and let $U' $ be an infinite subset of $U.$ Then $U'$ is not contained in the vanishing of any nonzero polynomial in $\mathbb{R}[X,Y].$ As the vanishing of any polynomial $F\in\mathbb{C}[X,Y]$ is contained in the vanishing of some polynomial in $\mat... | 5 | https://mathoverflow.net/users/13816 | 66736 | 41,142 |
https://mathoverflow.net/questions/66721 | 2 | The 0-1 knapsack problem is known to be NP-complete, and the greedy approach by Dantzig (based on choosing on the basis of density or value/weight) can be shown to be suboptimal using counterexamples. Many of the counterexamples seem to rely on a worst case feature where the weights of the elements are huge and compara... | https://mathoverflow.net/users/15311 | Greedy approach to 0-1 Knapsack problem in specific instances | I don't think this works.
Suppose you have a counterexample to the greedy heuristic, with a knapsack of size $S$ and all elements of density at most $D$ and weight at most $W$. Now add a large number $N$ of objects of weight $W$ and density $1000D$, and consider a knapsack of size $S+NW$. Clearly the $N$ 'dense' obje... | 4 | https://mathoverflow.net/users/1898 | 66739 | 41,144 |
https://mathoverflow.net/questions/58188 | 37 | The Diophantine equation
$$x^3+y^3+z^3=3$$
has four easy integer solutions: $(1,1,1)$ and the three permutations of $(4,4,-5)$. Elsenhans and Jahnel wrote in 2007 that these were all the solutions known at that time.
>
> Are any other solutions known?
>
>
>
By a conjecture of Tyszka, it would follow that if th... | https://mathoverflow.net/users/7252 | Are nontrivial integer solutions known for $x^3+y^3+z^3=3$? | Just noticed this question. I agree with L.H.Gallardo that the problem is old (see e.g. Problem D5 in UPINT = *Unsolved Problems in Number Theory* by R.K.Guy), but not that it is hopeless: the usual heuristics suggest that the number of solutions with $\max(|x|,|y|,|z|) \leq H$ should be asymptotic to a multiple of $\l... | 34 | https://mathoverflow.net/users/14830 | 66740 | 41,145 |
https://mathoverflow.net/questions/66722 | 6 |
>
> **Question 1.** Is it known that for some free Burnside groups the word problem is undecidable?
>
>
>
Provided that the answer is negative, what about the following easier question.
>
> **Question 2.** Is there a known example of a finitely generated (and preferably finitely presented) group $G$ and an i... | https://mathoverflow.net/users/2631 | word problem in free Burnside groups (and other torsion groups) | Concerning question 1: for free Burnside groups of odd exponent $n\geq 665$ the decidability was shown by S.I.Adian. Lysionok proved the same in the case of even exponent
$n=16k\geq 8000$. The corresponding deciding procedure is just Dehn's algorithm.
It will be interesting to know how to solve the word problem for ... | 9 | https://mathoverflow.net/users/10443 | 66746 | 41,148 |
https://mathoverflow.net/questions/66732 | 10 | Let $M$ be a smooth manifold (over $\mathbb R$) and $N \hookrightarrow M$ a closed embedding. Locally near any point in $N$, I can find coordinates $x^1,\dots,x^{\dim M}$ on $M$ so that $N$ is the vanishing locus of $x^{1+\dim N},\dots,x^{\dim M}$. My question is essentially whether I can do this globally. More precise... | https://mathoverflow.net/users/78 | Is every closed embedded codimension-n submanifold cut out by a section of a rank-n vector bundle? | The answer in general is no. Take e.g. $M=\mathbb{R}^n$. There are no non-trivial vector bundles on it. So, if $N$ is the zero locus of a transversal section of a bundle $E$ (which is necessarily trivial), then using the exact sequence $$0\to T N\to T M|\_N\to \nu\_M N\to 0,$$ where $\nu\_M N$ is the normal bundle of $... | 13 | https://mathoverflow.net/users/2349 | 66747 | 41,149 |
https://mathoverflow.net/questions/66743 | 2 | Hello,
This question is a follow up to Sasha's comment in [Duals and Tensor products](https://mathoverflow.net/questions/56255/duals-and-tensor-products/56326#56326).
In the comment there, it is claimed that the given a ring $A$ and modules $M$, $N$, there
is an isomorphism
$$
RHom(M,A)\otimes^L RHom(N,A) = RHom(M\... | https://mathoverflow.net/users/10580 | dual modules in the derived category | In order to get the first isomorphism you probably need $M$ and $N$ to be perfect, i.e. they must have a projective resolution of finite length by projective modules of finite type. In that case $RHom(-,A)$ behaves as the usual duality functor on finite-dimensional vector spaces.
As for your second question, $Hom(M,... | 4 | https://mathoverflow.net/users/12166 | 66752 | 41,150 |
https://mathoverflow.net/questions/66753 | 5 | So, in chapter 2, section 2 of Hartshorne, (prop 2.3), he describes how if $\varphi : A\rightarrow B$ is a homomorphism of rings, then you get a morphism of (affine schemes):
$\newcommand{\Spec}{\textrm{Spec }}$
$\newcommand{\oO}{\mathcal{O}}$
$\newcommand{\mf}[1]{\mathfrak{#1}}$
$(f,f^\sharp) : (\Spec B, \oO\_{\Spec... | https://mathoverflow.net/users/15242 | morphisms of affine schemes question | I'm not sure what it is that you read in Hartshorne that suggested that $(f\_\*\mathcal O\_{\mathrm{Spec} B})\_{\mathfrak p}$ is equal to $(\mathcal O\_{\mathrm{Spec} B})\_{\mathfrak q}$, since this is not true.
My suggestion is that you consider two illustrative cases:
1. Let $A = k$ (a field) and $B = k\times k$... | 8 | https://mathoverflow.net/users/2874 | 66754 | 41,151 |
https://mathoverflow.net/questions/66674 | 1 | Let x(t) be a Markov process. We define the stochastic process y(t) such that :
```
y(t) = x(f(t))
f : T -> T
```
T is the parameter set of the process x(t).
If we know that f is bijective, is y(t) a Markov process ?
| https://mathoverflow.net/users/15525 | A bjection between two stochastic processes | If f is continuous, differentiable and strictly increasing, the answer is yes. It is easy to see that y(t\_n) is conditionally independent of y(t\_1), y(t\_2), y(t\_{n-2}) given y(t\_{n-1}), as this is the equivalent property enjoyed by x(t) when it is Markov. x(\tilde{t}\_n}) is independent of x(\tilde{t}*1),...,x(\ti... | 1 | https://mathoverflow.net/users/15544 | 66757 | 41,153 |
https://mathoverflow.net/questions/66716 | 4 | Let $1\leq p,q,r\leq \infty$ such that $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$. Let $S\_p$ denote the Schatten space. For any $x\in S\_p$ and any $y\in S\_q$ we have
$$
||xy||\_{S\_r} \leq ||x||\_{S\_p}||y||\_{S\_q}
$$
(noncommutative Hölder's inequality).
Does it exists necessary and sufficient conditions on $x,y$ in ... | https://mathoverflow.net/users/5210 | equality in noncommutative Hölder inequality | The necessary and sufficient condition is that $|x|^p$ and $|y^\*|^q$ are proportional. This can be deduced from [Dixmier's paper](http://archive.numdam.org/article/BSMF_1953__81__9_0.pdf) (although it is not clearly stated that way there; it is based on Proposition 8). It probably also appears in more modern treatment... | 8 | https://mathoverflow.net/users/3698 | 66762 | 41,155 |
https://mathoverflow.net/questions/66738 | 4 | (related question : [most general way to generate pairwise independent random variables?](https://mathoverflow.net/questions/7998/most-general-way-to-generate-pairwise-independent-random-variables))
Let $X\_1,X\_2,X\_3,X\_4$ be four random variables with standard Bernoulli distribution (i.e. $P(X\_i=0)=P(X\_i=1)=\fra... | https://mathoverflow.net/users/2389 | Is there a good explanation for this fact on pairwise independent variables? | This solution is ugly, sorry.
Proceed by contradiction and assume that 0000, 0001 and 0010 all have probability zero. Every event where one specifies two coordinates and one leaves free the two remaining ones has probability exactly 1/4 because this event involves only the joint distribution of two independent Bernou... | 3 | https://mathoverflow.net/users/4661 | 66766 | 41,157 |
https://mathoverflow.net/questions/66774 | 6 | Let $V\_X$ be a vector bundle of rank $r>1$ on a smooth (connected) projective variety of dimension $r$. Let s be the global holomorphic section, whose zero locus is a zero dimensional subscheme $Z\subset X$. Is there some result that the section is determined (up to scaling) by its zero locus? At least for some nice v... | https://mathoverflow.net/users/2900 | when is a section of vector bundle determined by its zero locus? | Take a look at [On sections with isolated singularities of twisted bundles and applications to foliations by curves](http://www.mathjournals.org/mrl/2003-010-005/2003-010-005-008.pdf1.2.) by Campillo and Olivares.
There you will find the following result.
>
> **Theorem.** Let $X$ be a projective manifold of dime... | 3 | https://mathoverflow.net/users/605 | 66778 | 41,164 |
https://mathoverflow.net/questions/66776 | 20 | Although, beyond any doubts, $ZFC$ is by and large the predominantly accepted theory of sets, there have been a few attempt to establish some serious competitors in town.
I just quote two of them (there are several more): $NF$ by Quine and Alternative Set Theory by Petr Vopenka. I think those attempts are epistemolo... | https://mathoverflow.net/users/15293 | Alternative Arithmetics | Recall that $NFU$ is the Quine-Jensen system of set theory with a universal set; it is based on weakening the extensionality axiom of Quine's $NF$ so as to allow *urelements*.
Let $NFU^-$ be $NFU$ plus "every set is finite". As shown by Jensen (1969), $NFU^-$ is consistent relative to $PA$ (Peano arithmetic). $NFU^-... | 15 | https://mathoverflow.net/users/9269 | 66780 | 41,165 |
https://mathoverflow.net/questions/66781 | 1 | The usual modular parametrization says that if one takes a modular form $f$ which is a newform for $\Gamma\_0(N)$ (i.e. a form in the new subspace which is a normalized eigenfunction for the Hecke operators), then one can associate to $f$ an abelian variety $A\_f$ of dimension $d$, where $d$ is the degree of $K$ over $... | https://mathoverflow.net/users/4685 | Question on the abel map and modular parametrization | No, the abelian variety $A\_f$ is typically absolutely simple, and hence does not factor
(even up to isogeny, and even over $\overline{\mathbb Q}$ or $\mathbb C$) as a product of elliptic curves. What will happen is that if you integrate $f(z)dz$ over the homology of $X\_0(N)$, you will get a finitely generated, but *n... | 7 | https://mathoverflow.net/users/2874 | 66783 | 41,167 |
https://mathoverflow.net/questions/66792 | 14 | Let R be a local ring with maximal ideal m and residue field k. Is k ever flat over R? What conditions are needed on R?
Sorry, it's not a very profound question. It came up in a derived functor calculation.
| https://mathoverflow.net/users/14220 | is residue field ever flat over its local ring? | Consider the exact sequence $0 \to \mathfrak m \to R \to k \to 0.$ Tensoring with $k$ gives
$0 \to \mathfrak m/\mathfrak m^2 \to k = k \to 0.$ Thus if $k$ is flat over $R$, then
$\mathfrak m = \mathfrak m^2$. If furthermore $R$ is Noetherian, this implies that
$\mathfrak m = 0$, and hence that $R = k$.
Conclusion: Fo... | 35 | https://mathoverflow.net/users/2874 | 66794 | 41,173 |
https://mathoverflow.net/questions/66809 | 6 | In the definition of an $n$-dimensional stratified pseudomanifold one demands the following filtration
$X=X\_n \supset X\_{n-1}=X\_{n-2} \supset X\_{n-3}\supset ... \supset X\_0 \supset X\_{-1}=\emptyset$. where all $X\_i$'s are closed subspaces of X.
Why is it required that $X\_{n-1}=X\_{n-2}$?
| https://mathoverflow.net/users/15565 | Stratified pseudomanifold | I think the reason must be that a pseudomanifold $V$ has singular locus $\Sigma V$ of codimension 2 or greater. (Stratifications of varieties are obtained by letting $X\_{k-1}$ be the singular locus of $X\_k$, or some refinement of that to get the Whitney conditions.) This codimension condition is reflected topological... | 2 | https://mathoverflow.net/users/2926 | 66813 | 41,182 |
https://mathoverflow.net/questions/66793 | 6 | Can someone point me to a modern treatment of the Veblen-Young characterization of projective spaces of dimension greater than $2$ as $P(V)$ for some vector space $V$?
[**Added**: see [here](http://en.wikipedia.org/wiki/Projective_space#Axioms_for_projective_space) for a statement of the Veblen-Young Theorem. The art... | https://mathoverflow.net/users/1409 | Reference on the Veblen-Young characterization of projective spaces | The 2-volume book by Veblen and Young is an obvious reference for the original proof. A modern reformulation of their result is the work of Tits classifying buildings of rank at least 3 in terms of algebraic groups over fields.
| 6 | https://mathoverflow.net/users/51 | 66815 | 41,184 |
https://mathoverflow.net/questions/66823 | 1 | Dear all,
I would like to know whether the following claim is true. In particular, if it is true, then I would like to know if there is some textbook that contains the statement and maybe even the proof:
**Claim**
$\forall n \in \mathbb{N}, \ \exists k\_n > 0$ such that
for any real numbers $A\_1, \ldots, A\_n... | https://mathoverflow.net/users/15570 | Can you control the amplitudes of a finite collection of sine curves just by controlling the amplitude of their superposition? | Yes, this is true. It follows from the fact that the average
$$\lim\_{T\to\infty} {1\over T}\int\_0^T\cos(\lambda\_i t+\phi\_i)\sum\_j A\_j\cos(\lambda\_j t+\phi\_j)\,dt$$
is $A\_i/2$. Clearly the average is bounded by the supremum of the function.
| 5 | https://mathoverflow.net/users/12120 | 66830 | 41,190 |
https://mathoverflow.net/questions/66827 | 7 | *Background*
Let $R$ be a (as always commutative) ring. Consider the $2$-category $\text{Cat}\_{c\otimes}(R)$ of cocomplete $R$-linear tensor categories. There are various reasons why $\text{Cat}\_{c\otimes}(R)$ can be seen as a categorified version of the category of $R$-algebras. For example, coproducts are categor... | https://mathoverflow.net/users/2841 | Example of a locally presentable $2$-category | Your 2-category can be described as commutative algebras over "R-mod" in the setting of cocomplete categories (at least this is one meaning of the word "R-linear", which does not imply that the underlying category is abelian. If you want to stick to abelian categories the answer will be the same but require some more w... | 9 | https://mathoverflow.net/users/7721 | 66835 | 41,193 |
https://mathoverflow.net/questions/66824 | 1 | Hi there,
I am trying to teach a friend about higher-dimensions and I have explained them in the following two manners:
1. A higher dimension, e.g. the 56th is the set of all 56-tuples.
2. A fractional dimension, e.g. 3/4 means that when size doubles, volume increases at a factor of 2^3/4
To me, it is obvious tha... | https://mathoverflow.net/users/15571 | Is the notion of fractional dimension compatible with considering a dimension a set of n-tuples? | The notion of "dimension" occurs in various contexts and its meaning is not always the same. There are:
(1) Topological dimension. The topological dimension ${\rm dim}(X)$ of a space $X\ne\emptyset$ is an integer $\geq0$ or $\infty$. It is defined inductively or by means of coverings: An $X$ is, e.g., two-dimensional... | 3 | https://mathoverflow.net/users/8050 | 66843 | 41,198 |
https://mathoverflow.net/questions/66829 | 7 | Let $A/k$ be an abelian variety with real multiplication by some ring of integers $\mathcal O \subset F$. Let $n$ be an integer prime to the characteristic of $k$.
We have the standart $e\_n$ pairing $A[n] \times A^\vee[n] \to \mu\_n$.
It satisfies $e\_n(a \cdot x, y) = e\_n(x, a \cdot y)$ for any $a \in \mathcal O$.... | https://mathoverflow.net/users/5273 | O-linear Weil-pairing on abelian varieties with real multiplication | Yes, it's just a linear algebra stuff. Probably, it's more convenient to look at the Weil pairing between the $\ell$-adic Tate modules of $A$ and its dual, taking into account that these modules are free $O\otimes Z\_{\ell}$-modules of the same rank.
From the linear algebra point of view the situation is as follows.
... | 7 | https://mathoverflow.net/users/9658 | 66848 | 41,201 |
https://mathoverflow.net/questions/66812 | 19 | The wikipedia page on [Srinivasa Ramanujan](http://en.wikipedia.org/wiki/Srinivasa_Ramanujan) gives a very strange formula:
>
> **Ramanujan:** If $0 < a < b + \frac{1}{2}$ then, $$\int\limits\_{0}^{\infty} \frac{ 1 + x^{2}/(b+1)^{2}}{ 1 + x^{2}/a^{2}} \times \frac{ 1 + x^{2}/(b+2)^{2}}{ 1 + x^{2}/(a+1)^{2}} \times... | https://mathoverflow.net/users/1483 | Ramanujan's eccentric Integral formula | This is one of those precious cases when Ramanujan himself provided (a sketch of) a proof. The identity was published in his paper ["Some definite integrals"](http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper11/page1.htm) (*Mess. Math.* 44 (1915), pp. 10-18) together with several related formulae.
It might... | 22 | https://mathoverflow.net/users/5371 | 66849 | 41,202 |
https://mathoverflow.net/questions/66833 | 1 | This is a bit of a bizzare question, but I'm going to ask it anyway.
If $X={\rm Spec}R$ is an affine variety, and $\mathfrak{m}$ a closed point, then the localization $R\to R\_{\mathfrak{m}}$ gives a morphism
$f:{\rm Spec}R\_{\mathfrak{m}}\to{\rm Spec}R$,
yielding an adjunction between ${\rm Qcoh} R$ and ${\rm ... | https://mathoverflow.net/users/10570 | Local to global flatness question | This is of course true, for any semi-separated scheme (i.e. the diagonal is affine), or maybe you assume $X$ is separated if you like, and you can take any point (not necessarily closed). The reason that the sheaf is flat is that ${\rm Spec}(O\_{X,x})\to X$ is affine and flat. In general if $f: X\to Y$ is flat and affi... | 3 | https://mathoverflow.net/users/11964 | 66850 | 41,203 |
https://mathoverflow.net/questions/66855 | 6 | It is a standard fact that everyt real $n$-dimensional algebra is a subalgebra of $M\_n(\mathbb R)$.
The transposition map, operating in $M\_n(\mathbb R)$, is an involutive ($(A^t)^t=A$) antiautomorphism (an $\mathbb R$-linear isomorphism satisfying $(AB)^t=B^tA^t$). This makes $M\_n(\mathbb R)$ a real \*-algebra. T... | https://mathoverflow.net/users/4118 | Matrix representation of real *-algebras | I think that if you consider $\mathbb C$ with the ∗-structure under which every element is satisfies $ z^\*=z $, then this ∗-algebra cannot be embedded into $M\_n(\mathbb R)$.
In other words, it is not a real $C^\*$-algebra.
The reason is that the spectrum of the element $i$ is purely imaginary, and that's incompatib... | 8 | https://mathoverflow.net/users/5690 | 66861 | 41,207 |
https://mathoverflow.net/questions/66863 | 0 | Inspired by a [recent problem](https://mathoverflow.net/questions/66708/determining-a-recurrence-relation) for linear recurrence relations I have the following question (which may be too much to hope for). The [Catalan numbers](http://en.wikipedia.org/wiki/Catalan_number) (just to give a specific example) have generati... | https://mathoverflow.net/users/8008 | Determining a generating function (of a restricted form) | In effect you're asking whether a given power series $y(x) = \sum\_{n=0}^\infty a\_n x^n$ satisfies a quadratic equation $a(x) y^2 + b(x) y + c(x) = 0$ with $a,b,c$ polynomials of low degree in $x$, say degree less than $\delta$. This is a linear system in $3\delta$ variables (the coefficients of $a,b,c$), so it's easy... | 8 | https://mathoverflow.net/users/14830 | 66866 | 41,210 |
https://mathoverflow.net/questions/66857 | 3 | Given a system of multivariate, polynomial equations, is there a way to determine if it has a solution in a given field (for instance the set of all reals). I don't care what the solution is, I just want to know if one exists or not.
I understand that one approach would be to determine a Groebner Basis (or lack ther... | https://mathoverflow.net/users/15577 | Existence of a real-valued solution to system of multivariate polynomial equations | The answer over the **reals** is a resounding yes: not only can you decide if the system has solutions, but you can even produce them (over other fields, not so much, e.g. $\mathbb{Q}$).
The decision problem was solved by [Tarski](http://en.wikipedia.org/wiki/Tarski-Seidenberg_theorem) (and later Seidenberg). Though... | 13 | https://mathoverflow.net/users/8212 | 66870 | 41,212 |
https://mathoverflow.net/questions/66871 | 1 | Say $A, B : H \supset D \to H$ are essentially self-adjoint operators on the dense common domain $D$. $H$ is some Hilbert space. Does it hold that $A + B$ is also essentially self-adjoint? If not, can you please give me a counterexample?
| https://mathoverflow.net/users/13338 | Addition of essentially self-adjoint operators | Let $D=H^4(0,1)\cap H^2\_0(0,1)$, $Au=u''''$, $Bu=-u''''+u''$.
| 6 | https://mathoverflow.net/users/12120 | 66875 | 41,214 |
https://mathoverflow.net/questions/66873 | 16 | I am working a combinatorial question. I need to decide when the following function
$$\frac{m(7m^2 - 22m +7)}{ \sqrt{ (m^2- 2m + 9)(9m^2 - 2m +1) } } $$takes integer values with following
assumption: 1) m takes all positive integer values; 2) $(m^2- 2m + 9)(9m^2 - 2m +1)$ is a perfect square (= $n^2$ for some integer ... | https://mathoverflow.net/users/15582 | How to decide this function takes integer values? | An elementary approach is enough here. If the given ratio is an integer, then any prime $p$ which divides $m^2 -2m+9$ divides either $m$ or $7m^2-22m+7$. If $p$ divides $m$ then clearly $p =3$. If $p$ divides $7m^2-22m+7$ then $p$ divides $$[7m^2-14m+63 - (7m^2-22m+7)] = 8m+56. $$
Hence either $p=2$ or $p$ divides $m+7... | 18 | https://mathoverflow.net/users/14450 | 66879 | 41,216 |
https://mathoverflow.net/questions/66595 | 13 | Given a rooted tree $T$ and an integer $k \geq 1$, let $N\_k(T)$ be the number
of subtrees of $T$ containing the root and having exactly $k$ nodes (take $N\_k(T)=0$ if $T$ has less than $k$ nodes).
Next, fix an integer $d \geq 2$, and let $T\_d$ be the infinite $d$-ary rooted tree (every node has $d$ children). It i... | https://mathoverflow.net/users/3401 | Counting subtrees of a random tree ("random Catalan numbers") | Let $p\_n$ be the offspring distribution for $B$ and define $q\_n = \sum\_{m\ge n} p\_m (m)\_n$, where $(m)\_n = m!/(m-n)!$ is the descending factorial.
(new paragraph to appease MO latex scripts). Then the expected number of copies of a rooted tree $\theta$ inside the Galton Watson tree $\mathcal{T}$ is given by $\... | 8 | https://mathoverflow.net/users/9422 | 66883 | 41,218 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.