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https://mathoverflow.net/questions/125988 | 2 | Let $T$ be a topos, and $F \in T$, $T/F$ a localization of $T$. So we have a natural morphism $i: T/F \longrightarrow T$.
My questions are:
1.What are the definitions of $i\_{\ast}$ and $i^{\ast}$ without using site ?
2.Dose there exist a site $S$ such that $Sh(S) \cong T$ and there is an $U \in S$ which $U^{\sim... | https://mathoverflow.net/users/5274 | questions of localization of topos | $i^\*$ is the functor which send an object $X \in T$ to $X \times F$ with the natural projection as map into $F$.
$i\_\*$ is a little harder to describe, if $p: Y \rightarrow F$ is an object of $T/F$, then $i\_\*(Y)$ is the sub-object of $[F,Y]$ (the internal hom object) which corresponds to map $f$ from $F$ to $Y$ s... | 6 | https://mathoverflow.net/users/22131 | 125990 | 70,385 |
https://mathoverflow.net/questions/125853 | 7 | Let $H$ be a Hilbert space and $G$ be a finitely generated group. Let $\pi:G\rightarrow GL(H)$ be a representation.
A map $c:G\rightarrow H$ is called *cocycle* if $c(gh)=π(g)c(h)+c(g)$ for all $g,h$ in the group $G$. A cocycle $c$ is *proper* if the map $g\mapsto c(g)$ is proper, i.e., for every constant $K$ the num... | https://mathoverflow.net/users/8699 | Unbounded representations of groups | Q1 and Q2 have a positive answer for all countable groups (conversely a discrete uncountable group cannot bear any proper function). Let $\mu$ be a proper function from $G$ to the positive reals, and view it as a discrete measure on $G$. Assume in addition that $\mu$ grows reasonably, and more precisely satisfies an eq... | 10 | https://mathoverflow.net/users/14094 | 125993 | 70,388 |
https://mathoverflow.net/questions/125969 | 1 | Let $S$ denote the category of projective (left) $R$-modules with isomorphisms for arrows. We have that
>
> $BS^{-1}S \sim B \text{GL}(R)^+ \times K\_0(R)$
>
>
>
In proving this, in Srinivas' algebraic K-theory text, the following is casually written:
>
> $H\_p(BS^{-1}S) \cong H\_P(BS^{-1}S^0) \times K\_0(... | https://mathoverflow.net/users/19313 | homology of $B S^{-1} S$ computation in the proof that $+ = Q$ | If you want to apply Kunneth, then you should take into account that $H\_0 X$ is the free abelian group on $\pi\_0 X$, so you have to replace your use of $K\_0R$ by the free abelian group on $K\_0R$. Then the Tor vanishes, and we see that Srinivas' formula is incorrect.
Simpler than applying Kunneth would be to think... | 2 | https://mathoverflow.net/users/15247 | 126000 | 70,394 |
https://mathoverflow.net/questions/125944 | 27 | Let $f$ be a homogeneous polynomial of degree $d$ in $n$ variables. Restricted to the unit sphere $S^{n-1}$, it might or might not be a Morse function.
If $f$ is a Morse function of degree $1$, you get everyone's favorite Morse function on the sphere, with two critical points, corresponding to the minimal CW decompos... | https://mathoverflow.net/users/18060 | How many polynomial Morse functions on the sphere? | I've been trying to answer this question for several years and it turned out to be really hard, even for the $2$-sphere. Below I will discuss this case.
First of all one should ask what is the number $m(k)$ of topological types of (stable) Morse functions on $S^2$ with precisely $k$ saddle points. (such a function ha... | 29 | https://mathoverflow.net/users/20302 | 126001 | 70,395 |
https://mathoverflow.net/questions/126003 | 3 | Hi!
Let $f\in C^{2,\alpha}\left( \mathbb{C}^{m}\setminus \overline{B\_{R}},\mathbb{R} \right)$ with $m\geq 2$, $R>0$ and s.t. $f$ has an expansion of type
$$f=1+\mathcal{O}\left( \frac{1}{|z|} \right)$$
Suppose, moreover, that
$$\partial\overline{\partial}f=\sum\_{i,j=1}^{m}\left(\partial\_{i}\overline{\partial}\_{j... | https://mathoverflow.net/users/4971 | Does $\partial\overline{\partial}f=0$ imply $f\equiv c$ for particular kind of $f$? | Yes, this is true. The point is that $\partial\bar\partial f = 0$ on the complement $C$ of a ball in $\mathbb{C}^n$ ($n\ge 2$) implies that $f = h\_+ + \overline{h\_-}$ for some functions $h\_\pm$ that are holomorphic on $C$. (They are unique up to adding a constant to one and subtracting it from the other.) Now Hartog... | 6 | https://mathoverflow.net/users/13972 | 126011 | 70,398 |
https://mathoverflow.net/questions/123301 | 3 | I am working on an optimization problem which I am stuck on towards the end.
Essentially, I have two probability density functions in $\mathbb{R}^2$, call them $q(x,y)$ and $p(x,y)$, now I define the objective functional to be:
$C(p,q) = \int p(x,y) \ln\left[\frac{p(x,y)}{q(x,y)}\right] dxdy \cdots (1)$
Now assu... | https://mathoverflow.net/users/31338 | Probability Density Optimization | Just differentiate $L(p,q)$ with respect to $p(x,y)$, assuming regularities, to get
$$\frac{\partial}{\partial p(x,y)}L(p,q)=1+\ln p(x,y)-\ln q(x,y)+\lambda\_1 I\_{[X^x\_d, \infty)} + \lambda\_2 I\_{[X^y\_d, \infty)}+\mu$$ and equate to zero (the necessary condition in Lagrange multiplier method).
Now using the con... | 2 | https://mathoverflow.net/users/7699 | 126014 | 70,399 |
https://mathoverflow.net/questions/126002 | 8 | I have heard there is some fairly recent result showing that whenever theories $T$ and $T'$ have the same consistency strength, then each can interpret the other. I suppose it refers to first order theories, and I do not know exactly what kind of interpretability it uses or what measure of consistency strength.
Can a... | https://mathoverflow.net/users/38783 | Interpretability and consistency strength | John Steel spoke at the EFI series at Harvard concerning his ideas on [The triple helix](http://logic.harvard.edu/EFI_Steel_TripleHelix.pdf), which has to do in part with the interplay of large cardinal strength and the arithmetic interpretability hierarchy. This is not exactly the claim you mention, but there is a fam... | 3 | https://mathoverflow.net/users/1946 | 126020 | 70,401 |
https://mathoverflow.net/questions/126016 | 8 | Hi, everyone. I am interested in the dehn filling and Hyperbolic 3-manifold.
Suppose M be an orientable compact 3-manifold with one torus boundary and int(M) admit a
hyperbolic structure. Thurston proved that almost all the dehn fillings produced
hyperbolic 3-manifolds.
My question is:
Among all the hyperboli... | https://mathoverflow.net/users/18496 | Do different Dehn fillings produce homeomorphic 3-manifolds ? | This phenomenon is called "cosmetic surgery."
If $K$ is an amphichiral knot in the $3$--sphere with exterior $M\_K$, then $M\_K(p/q) \cong - M\_K(-p/q)$. So if $p/q$ is a hyperbolic filling slope, then this is the example you seek. Note that the homeomorphism is orientation-reversing. If there is an orientation-pres... | 13 | https://mathoverflow.net/users/1335 | 126023 | 70,403 |
https://mathoverflow.net/questions/125965 | 5 | From a result Obtained by O. Schreier and B. L. van der Waerden [Math. Sem. Univ. Hamburg 6, 303- 322 (1928)], one can show that for two fields $\mathbb F$ and $\mathbb G$, and integers $n>m>2$, the only group homomorphism $\mathrm{SL}\_n(\mathbb F) \to \mathrm{SL}\_m(\mathbb G)$ is the trivial homomorphism.
My quest... | https://mathoverflow.net/users/3958 | Morphisms $\mathrm{SL}_n(\mathbb Z) \to \mathrm{SL}_m(\mathbb Z)$ | I'll try to flesh out the first comment to give an answer: for all $n\ge 4$ there is a representation $\rho:A\_{n+1}\to \rm SL\_n(\bf Z)$ whose image normally generates $\rm SL\_n(\bf Z)$ (because it injects into all $\mathrm{PSL}\_{n-1}(\mathbf{Z} / N)$ for $N\ge 1$ and $\rm SL\_n(\bf Z)$ has CSP). But for $n\ge 5$ th... | 8 | https://mathoverflow.net/users/32210 | 126027 | 70,404 |
https://mathoverflow.net/questions/126025 | 5 | As is well known, the conjugacy classes of the free group $F\_2$ are parametrised by cyclically reduced words, up to cyclic permutation. In particular, it's easy to tell whether two elements of $F\_2$ are conjugate.
What about the *automorphism* classes of $F\_2$? For $u,v\in F\_2$ write $u\sim v$ if there is an auto... | https://mathoverflow.net/users/20598 | Automorphism classes of the free group | "Let $w\_1$ and $w\_2$ be elements of a free group $F$. Then it is decidable
whether there is an automorphism of $F$ carrying $w\_1$ into $w\_2$."
(R.C.Lyndon, P.E.Schupp, Combinatorial Group Theory, Chapter I, Prop.4.19)
Is this an answer on your question? I think it's hard to get something more specific.
| 5 | https://mathoverflow.net/users/18814 | 126028 | 70,405 |
https://mathoverflow.net/questions/126037 | 4 | Are there any reasonably natural algebras whose product (bracket) almost, but does not quite, satisfy the Jacobi relation?
A priori it doesn't matter whether the bracket is anti-symmetric.
The question is deliberately vague about "almost, but does not quite", just to see if this strikes any chord. It can mean that... | https://mathoverflow.net/users/32668 | Almost-Lie Algebras? | See Section 2.3 of the lecture notes called *Geometric Models for Noncommutative Algebras* by Ana Cannas da Silva and Alan Weinstein. There they define an "almost Lie algebra" to be something with an antisymmetric bracket but which does not necessarily satisfy Jacobi. In Section 3.2 they connect this to the notion of "... | 3 | https://mathoverflow.net/users/703 | 126040 | 70,411 |
https://mathoverflow.net/questions/125923 | 5 | Let $f(x)=1+x+x^2+ \ldots + x^n$. There is a theorem saying
"For 'most' $n$, $f'(x)$ is irreducible". (Ref: matwbn.icm.edu.pl/ksiazki/aa/aa90/aa9023.pdf)
$f'(x)$ has the property that its coefficients form an arithmetic progression. So I wonder if any generalizations applies for such polynomials. For the easiest ca... | https://mathoverflow.net/users/32631 | irreducible polynomials with arithmetic progression coefficients | I've heard from Zhi-Wei Sun that he recently considered this question. In a post a few days ago to OEIS [Least integer b>2n+1 such that the numbers written as [1,3,...,2n-1,2n+1] and [2n+1,2n-1,...,3,1] in base b are both prime.](http://oeis.org/A218465) He gives the first of what he conjectures are infinitely many bas... | 4 | https://mathoverflow.net/users/8008 | 126046 | 70,414 |
https://mathoverflow.net/questions/126038 | 6 | I have been learning some representation theory and have some questions about the following pattern:
**Instance 1:** If we have a finite group $G$ and a field $k$, a representation of $G$ over $k$ consists of a finite dimensional $k$-vector space $V$ and a homomorphism $G \to GL(V)$. We have a Hopf algebra $kG$ and... | https://mathoverflow.net/users/4002 | Naive question about the representation theory of algebraic groups and hopf algebras | If $G$ is an affine algebraic group, its coordinate ring $\mathcal O(G)$ is a Hopf algebra. The multiplication is the usual (commutative) pointwise multiplication of functions. The comultiplication is pullback under the map $m:G\times G \to G$ given by the group structure (this is only cocommutative if $G$ is abelian).... | 10 | https://mathoverflow.net/users/7762 | 126047 | 70,415 |
https://mathoverflow.net/questions/95129 | 20 | Suppose that two compact topological manifolds with boundary have homeomorphic interiors. Can we conclude that the two manifolds are homeomorphic? What happens in the smooth category?
| https://mathoverflow.net/users/23193 | Manifolds with homeomorphic interiors | Gjergji Zaimi's answer gives a strong positive conclusion: the product of the boundaries with $\mathbb{R}$ are necessarily homeomorphic. I just want to add a couple of explicit examples illustrating that the boundaries themselves need not be homeomorphic.
The first example is given by Milnor in his article "[*Two com... | 17 | https://mathoverflow.net/users/21095 | 126049 | 70,416 |
https://mathoverflow.net/questions/126026 | 2 | I have a basic question which I am not able to figure out. If we do a Gluck twist on a nullhomologous 2-sphere in a 4-manifold, it is said that it does not change its intersection form. But as far as I understand the Gluck twist changes the framings and knotting of the link
components which represent the second homolo... | https://mathoverflow.net/users/31475 | Gluck twist on four-manifolds | Basically this is Lefschetz duality. Chopping out the neighbourhood of the 2-sphere gives a 4-manifold $X$ with boundary $Y = S^2 \times S^1$. You can think about the intersection form here and how this will change after the surgery.
The point of $S^2$ being null-homologous is that if you look at the long exact seque... | 3 | https://mathoverflow.net/users/3923 | 126062 | 70,421 |
https://mathoverflow.net/questions/126060 | 5 | If I want to apply for a postdoctoral job, can I mention the name of my recommenders in my cover letter just to bolster my application, particularly when I am sure that the people who will read my application will know the recommenders personally? I know that they will actually read the recommendation letter themselves... | https://mathoverflow.net/users/6953 | On mentioning recommenders' names in cover letter for postdoctoral applications | I have made my answer CW, because I believe the question should be.
My answer should carry very little weight, because I have never served on a postdoc committee: I am in my final months as a graduate student, and my experience is only as a (successful, thankfully) postdoc applicant in the US.
What I did, and have ... | 12 | https://mathoverflow.net/users/78 | 126065 | 70,424 |
https://mathoverflow.net/questions/126071 | 23 | We can define the algebra of quaternions $\mathbb H$ over any field $k$, and depending on the arithmetic of $k$ it is either a division algebra or a matrix algebra.
We can also define the algebra of octonions $\mathbb O$ over any field $k$, and if over $k$ the $8$-ary quadratic form $Q=x\_0^2+x\_1^2+x\_2^2+x\_3^2+x\_... | https://mathoverflow.net/users/1409 | The octonions on a bad day | $\DeclareMathOperator\Tr{Tr}
$Suppose that $k$ is a field with $\operatorname{char}(k) \neq 2$. Let's agree that an "octonion algebra" over $k$ is an 8-dimensional unital $k$-algebra $A$, endowed with a quadratic form $N: A \rightarrow k$, whose associated bilinear form $T(x,y) = N(x+y) - N(x) - N(y)$ is nondegenerate,... | 23 | https://mathoverflow.net/users/3545 | 126073 | 70,427 |
https://mathoverflow.net/questions/126039 | 4 | Gauss's Circle Problem [[1](http://mathworld.wolfram.com/GausssCircleProblem.html)] is to find the number of lattice points inside the boundary of a circle with a given radius and center at the origin. I'm interested in the "reversed" version of this problem. Given that I know the number of lattice points, what is the ... | https://mathoverflow.net/users/32669 | Reverse Gauss's Circle Problem | The number of nodes for radius $r$ is $N\_r \approx \pi r^2$ so you can invert to $r\_N\approx\sqrt{\frac{N}{\pi}}$. If you wanted to derive the squared distance $r^2$ from $N$, you would have trouble but the square roots are bunched together so maybe that is accurate enough. This does leave some amount of uncertainty:... | 5 | https://mathoverflow.net/users/8008 | 126082 | 70,432 |
https://mathoverflow.net/questions/126059 | 2 | I need to show one of the two following equivalent results. If true, it must be a simple proof but I do not seem to be able to make it work. Thank you in advance.
1) Consider a continuous symmetric bilinear form $B$ on a Hilbert space. Let P be a closed subspace of that Hilbert space over which the form is coercive, ... | https://mathoverflow.net/users/30954 | Coercive Symmetric Bilinear form on a Hilbert space | This is not true. In $R^2$ consider $B(x,y) = x\_1y\_1 + x\_2y\_2 - x\_1y\_2 - y\_1x\_2$. Then $B(e\_1,e\_1) = B(e\_2,e\_2) = 1$ but $B(e\_1+e\_2,e\_1+e\_2)=0$. For 1) take $P=span(e\_1)$, $y=e\_2$, for 2) take $v\_k=e\_k$, $k=1,2$.
| 2 | https://mathoverflow.net/users/24179 | 126087 | 70,435 |
https://mathoverflow.net/questions/126057 | 5 | Hi,
Can one define a Fubini-Study metric/Kaehler metric on the projective space of an infinite dimensional Hilbert space, i.e. using the formula $\partial \bar{\partial} \log |Z|^2$?
This should be very well-known to the experts. Anyhow I don't have much experience with infinite dimension and worried that something ... | https://mathoverflow.net/users/21491 | Fubini-Study metric for an infinite dimensional Hilbert space | Maybe it's easier to see that the definition extends if you **don't** use a formula:
The unit sphere inherits a Riemannian metric from the Hilbert space in the standard manner and since it is invariant under the circular symmetry $(e^{i\theta},x) \mapsto e^{i\theta} x$, it will
project down to a Riemannian metric on ... | 3 | https://mathoverflow.net/users/21123 | 126097 | 70,440 |
https://mathoverflow.net/questions/125971 | 11 | I'm looking a reference (or quick proof) for the following fact, which doesn't appear in the standard sources I've consulted (for instance, Milnor's book on algebraic k-theory).
Let $\phi : G \rightarrow G'$ be a homomorphism between perfect groups. Set $A = H\_2(G)$ and $A' = H\_2(G')$, and let
$$1 \longrightarrow A... | https://mathoverflow.net/users/29685 | Functorality of universal central extension | Two sources come to mind, though there may be more recent ones. These are both available online now.
1) A concise exposition of central extensions is given by Steinberg in Section 7 of his 1967-68 Yale lectures on Chevalley groups, distributed in mimeographed form and currently linked on his UCLA homepage [here](http... | 11 | https://mathoverflow.net/users/4231 | 126100 | 70,441 |
https://mathoverflow.net/questions/125863 | 6 | Hi,
Let $f : X \rightarrow Y$ a projective morphism of quasi-projective algebraic varieties over $\mathbb{C}$. Assume that $X$ is smooth, that $Y$ is normal and that:
$$\textbf{R} f\_\* \mathcal{O}\_X = \mathcal{O}\_Y $$
(here $ \textbf{R} f\_\* $ denotes the derived push-forward of $f$). Let $E$ be a Cartier d... | https://mathoverflow.net/users/31278 | Relatively numerically trivial divisor | Hi,
So the result is true and a proof has been nicely explained to me by Yoshinori Namikawa.
First, let's recall the following "classical" fact (whose proof can be found as lemma $7.7$ in the book *Fourier-Mukai and Nahm transforms in Geometry and Mathematical Physics*, for instance):
**Lemma** :
Let $f : Y \righ... | 2 | https://mathoverflow.net/users/31278 | 126108 | 70,442 |
https://mathoverflow.net/questions/126107 | 3 | In most references, only a principal G-bundle is called universal (ie every other bundle can be pullbacked from this one, or an equivalent definition).
Does it make sense to speak of a universal F-G bundle ? (ie a bundle with fiber F and structure group G)
Specifically, during a course, we defined the universal G... | https://mathoverflow.net/users/32691 | Confusions over the definitions of universal bundle and characteristic class | If $B$ is a principal $G$-bundle, $B \times\_G F$ is an $F-G$ bundle. Conversely, if $C $ is an $F -G$ bundle, $Hom (F,C)$ (structure-preserving Homs) is a principal bundle. So the two notions are equivalent.
| 4 | https://mathoverflow.net/users/18060 | 126110 | 70,443 |
https://mathoverflow.net/questions/125678 | 4 | * I want to know how one can write down a CFT such that its conserved currents will satisfy some chosen (affine) Lie algebra $G$.
On the few pages leading up to page 192 [in here](http://srv2.fis.puc.cl/~mbanados/Cursos/Cuerdas/LustTheisen%20.pdf) one can see see the analysis of the CFT obtained in the compactified ... | https://mathoverflow.net/users/2678 | CFTs corresponding to affine Lie algebras | You can construct Wess-Zumino-Novikov-Witten model starting with finite-dimensional subalgebra of your affine Lie algebra. The action of this model is written in terms of field $g:\mathbb{C}\cup\left\{\infty\right\}\sim S^2 \to G$, where $G$ is a Lie group, such that $\mathfrak{g}$ is finite dimensional subalgebra and ... | 5 | https://mathoverflow.net/users/140 | 126117 | 70,447 |
https://mathoverflow.net/questions/11289 | 29 | Let $X$ be a projective algebraic variety over a algebraic closed field $k$. Let $\mathcal{F}$ be a coherent sheaf on $X$. We know that $H^0(X, \mathcal{F})$ is the vector space of global sections of $\mathcal{F}$. This gives us a geometric illustration of $H^0$. For example, let $I\_D$ be the ideal sheaf of a hypersur... | https://mathoverflow.net/users/2348 | Geometry meaning of higher cohomology of sheaves? | Let's start way back. The invention of schemes moved algebraic geometry away from thinking about varieties as embedded objects. However, embedding an abstract scheme into projective space has a lot of advantages, so if we can do that, it's useful. And even if we cannot embed our scheme into projective space, but we can... | 20 | https://mathoverflow.net/users/10076 | 126119 | 70,448 |
https://mathoverflow.net/questions/126104 | 22 | *This is a crosspost from [math.stackexchange](https://math.stackexchange.com/questions/345623/difference-between-parallel-transport-and-derivative-of-the-exponential-map)*
Given a Riemannian manifold $M$, let $c(t) = \exp\_p(tX)$ be the geodesic emanating from $p \in M$ with initial value $X$. Let $t\_0$ be small en... | https://mathoverflow.net/users/16702 | Difference between parallel transport and derivative of the exponential map | To understand the realationship between $Q$ and $P$, it suffices to study how they act on an orthornormal(o.n.) basis. Here is the detailed argument:
Let $J\_i(t)$ be the Jacobi field along $c$, with $J\_i(0)=0$ and $J\_i'(0)=e\_i$, where $e\_1, \cdots, e\_{n-1}, X$ is an o.n basis of $T\_p(M)$. Then one can show tha... | 8 | https://mathoverflow.net/users/1190 | 126121 | 70,449 |
https://mathoverflow.net/questions/126106 | 27 | Let $R$ be a finitely generated ring with identity, $M\_n(R)$ the set of $n\times n$ matrices. Are there any nontrivial ring homomorphisms $M\_{n+1}(R)\rightarrow M\_n(R)$? This should be an elementary question in abstract algebra. But even if $R$ is a field, I couldn't get a quick (negative) proof. Any comments are we... | https://mathoverflow.net/users/1546 | Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$? | According to the Amitsur-Levitzki theorem, $n \times n$ matrices over a commutative ring satisfy a polynomial identity of degree $2n$ and none of smaller degree. So there can be no injective ring homomorphism $M\_{n+1}(R) \to M\_n(R)$, which at least rules out the case when $R$ is a field.
| 34 | https://mathoverflow.net/users/35840 | 126124 | 70,450 |
https://mathoverflow.net/questions/126076 | 6 | Let $F$ be an infinite field and $f$ a homogeneous form on $F$ such that $f$ has no non-trivial zero in $F$. Let $F'$ be a finite extension of $F$ such that $f$ has a non-trivial zero in $F'$. Is it true there exists a simple extension of $F$ of the form $F(\alpha)$ contained in $F'$ which contains a non-trivial zero o... | https://mathoverflow.net/users/32151 | Simple field extension and rational points | I believe the following is a negative example: Let $s,t,u,v$ be variables over $\mathbb F\_p$. Set $F=\mathbb F\_p(s,t,u,v)$ and $F'=F(\sigma,\tau)$ with $\sigma^p=s$, $\tau^p=t$.
Set $$f(X,Y,Z)=(X^p-sZ^p)u+(Y^p-tZ^p)v.$$
Then $f(\sigma,\tau,1)=0$.
We show that any solution of $f=0$ over $F'$ has this form up to ... | 5 | https://mathoverflow.net/users/18739 | 126130 | 70,454 |
https://mathoverflow.net/questions/125966 | 2 | Let $S$ be a (small) symmetric monoidal category and $X$ a (small) category on which $S$ acts. $\pi\_0(S) = \pi\_0(BS)$ is naturally an abelian monoid, with $[A] + [B] := [A+B]$, where $[A]$ denotes the path component containing the 0-cell, i.e., object $A$.
>
> $\pi\_0(S)$ acts on $H\_p(BX, \mathbb Z)$. How is thi... | https://mathoverflow.net/users/19313 | Path components of a monoidal category acting on homology | The action of $S$ on $X$ induces an action of $BS$ on $BX$. One way to see it is to do it at the level of $p$ simplices and construct a map $B\_pS\times B\_pX\to B\_pX$ for all $p$. For any category $C$, the $p$ simplices $B\_pC$ are the functors $[p]\to C$. Since $S$ acts on $X$, we have a functor $S\times X\to X$. Ma... | 4 | https://mathoverflow.net/users/10707 | 126131 | 70,455 |
https://mathoverflow.net/questions/126138 | 1 | Let $X$ be an ordered set. A *down-set* (also called a *lower set* or an *order ideal*) of $X$ is a subset $D$ of $X$ such that for every $x, y \in D$, if $x \in D$ and $y \leq\_X x$, then $y \in D$. The *down-set lattice* $\mathcal{O}(X)$ of $X$ is the set of all down-sets of $X$ ordered by inclusion.
Let $P$ and $Q... | https://mathoverflow.net/users/32701 | Order-isomorphic down-set lattices | If $\mathcal{O}(P)$ and $\mathcal{O}(Q)$ are order-isomorphic, then $P$ and $Q$ are isomorphic as well. Furthermore, not only can we recover the poset $P$ from the lattice $\mathcal{O}(P)$, but the mapping $P\mapsto\mathcal{O}(P)$ gives us a duality between the category of all posets and certain kinds of lattices calle... | 1 | https://mathoverflow.net/users/22277 | 126140 | 70,462 |
https://mathoverflow.net/questions/126158 | 38 | For ordered fields, we have a “mother of all ordered fields”, the surreal numbers $\mathbf{No}$, a proper-class “field” which includes (an isomorphic copy of) every other ordered field as a subfield. So, I wondered: does a similar mother object exist for other kinds of object, like groups? That is, is there some “meta-... | https://mathoverflow.net/users/11576 | A “mother of all groups”? What kind of structures have "mother of all"s? | The surreal numbers exhibit much stronger universal properties
than you have
mentioned, for they also exhibit very strong homogeneity and [saturation](http://en.wikipedia.org/wiki/Saturated_model)
properties. For example, every automorphism of a set-sized elementary substructure of
the surreals extends to an automorphi... | 50 | https://mathoverflow.net/users/1946 | 126173 | 70,474 |
https://mathoverflow.net/questions/126061 | 5 | Fix two positive integers $d, e$ and assume $d>e$. Is it true that a general degree $e$ curve which lies in a complete intersection of a degree $e$ and a smooth degree $d$ surface in $\mathbb{P}^3$ is a plane curve (in the sense the radical ideal of the curve contains a linear polynomial)?
There is another way of fo... | https://mathoverflow.net/users/32151 | Complete intersection space curves | The answer is no in general. The simplest example is for $e=3$ (because a line and a conic in $\mathbb{P}^3$ are plane curves). There are plenty of twisted cubic curves lying on a smooth cubic surface in $\mathbb{P}^3$ and all lie in a complete intersection: take a general quartic passing through the twisted cubic.
I... | 3 | https://mathoverflow.net/users/23758 | 126180 | 70,477 |
https://mathoverflow.net/questions/126164 | 7 | I am reading Landsberg's "Tensors: Geometry and Applications". Here he mentions tensor formulation of Strassen's algorithm and shows that the rank of Strassen's matrix multiplication tensor is $7$ and $7$ is the lower bound for any such tensor and hence one needs $7$ multiplications for $2 \times 2$ matrix multiplicati... | https://mathoverflow.net/users/10035 | Strassen's algorithm | This $7$ is an absolute lower bound. The result is due to Hopcroft and Kerr "On minimizing the number of multiplications necessary for matrix multiplication." SIAM J. Appl. Math. (1971) and Winograd "On multiplication of $2\times 2$ matrices." Linear Algebra and Appl. (1971). [The former assume that entries of the matr... | 11 | https://mathoverflow.net/users/nan | 126181 | 70,478 |
https://mathoverflow.net/questions/126149 | 3 | Let $\overline{M\_g}$ be the moduli stack of stable curves of genus $g$. Let $H\_g$ be the moduli stack of smooth hyperelliptic curves and $\overline{H\_g}$ its compactification whose stack structure is given by the Hurwitz stack parameterizing degree two admissible covers in the sense of Harris--Mumford. There is a na... | https://mathoverflow.net/users/32337 | normal bundle of hyperelliptic locus | At least in characteristic not equal to $2$ (so that double covers are tamely ramified), there is a nice stack parameterizing "twisted stable maps" introduced by Abramovich, Corti and Vistoli (and studied further by Abramovich, Olsson and Vistoli in positive characteristic). The deformation theory is discussed in those... | 2 | https://mathoverflow.net/users/13265 | 126187 | 70,481 |
https://mathoverflow.net/questions/126193 | 6 | we know Darboux theorem for higher-symplectic geometry is not correct in general,
but is there any Darboux like theorem for non-degenerate 3-forms in 6-manifolds?
| https://mathoverflow.net/users/nan | Darboux like theorem for non-degenerate 3-forms in 6-manifolds | This depends on what you mean by 'Darboux-like'. It is certainly not true that a closed nondegenerate 3-form on a 6-manifold is necessarily locally equivalent to one of the 'flat' models, so there is no direct analog of the Darboux' theorem in this case.
As I remark in my article "Remarks on the geometry of almost c... | 13 | https://mathoverflow.net/users/13972 | 126197 | 70,485 |
https://mathoverflow.net/questions/126191 | 7 | Consider the direct image functor $f\_\*: Sh(X) \rightarrow Sh(Y)$, let $X$ and $Y$ be topological spaces, let $f: X \rightarrow Y$ be a continuous map, let $G \in Sh(X)$ be a sheaf. I was reading this course on sheaves:
<http://bit.ly/14IBBTZ>.
On page 1 they define this subpresheaf $f\_!G \subset f\_\*G$ then on ... | https://mathoverflow.net/users/22191 | Are subfunctors of left exact functors also left exact? | Here's a counterexample with additive functors on abelian categories. If $A$ is an abelian group, let $F(A)$ denote the subgroup of elements that are divisible by $2$. It is easy to see that $F:Ab\to Ab$ is an additive functor, and $F$ is a subfunctor of the identity. But $F$ is not left exact because it does not prese... | 21 | https://mathoverflow.net/users/75 | 126198 | 70,486 |
https://mathoverflow.net/questions/126171 | 3 | Consider a binary composition $\star:\Bbb R^2\_{>0}\rightarrow \Bbb R\_{>0}:(x,y)\mapsto x\star y$ with the following properties.
(Commutativity)$\quad x\star y=y\star x\;$for all $x,y\in\Bbb R\_{>0};$
(Associativity)$\quad(x\star y)\star z=x\star(y\star z)\;$for all $x,y,z\in\Bbb R\_{>0};$
(Continuity)$\quad x\m... | https://mathoverflow.net/users/7458 | Which compositions have these sum-like and product-like properties on the positive reals? | As remarked in comments under the question, an example would be $x \star y = \sqrt{x^2 + y^2}$. Indeed, if $f$ is any strictly monotonic bijection on $\mathbb{R}\_{>0}$ (either monotone increasing or monotone decreasing), then $x \star y = f^{-1}(f(x) + f(y))$ or $x \star y = f^{-1}(f(x)f(y))$ furnish examples which ad... | 5 | https://mathoverflow.net/users/2926 | 126202 | 70,488 |
https://mathoverflow.net/questions/126203 | 14 | If we have a [Jacobi PDE system](https://mathoverflow.net/questions/112173/under-which-conditions-jacobi-pde-system-can-be-represented-to-symplectic-monge-a) with conservation law $\theta \in \Omega^1(M)$ such that $d \theta$ is non-degenerate 2-form , then we know this fact that it can be written as symplectic 2D Mong... | https://mathoverflow.net/users/nan | conservation law and generalized Symplectic Monge-Ampere equation arising from 3-variables | I think that you should be careful to define your terms, but let me guess: A *Jacobi PDE system* for three unknown functions $h\_1,h\_2,h\_3$ of three independent variables $x\_1,x\_2,x\_3$ is a set of three partial differential equations, each of which can be written as a linear combination of the minors (of any rank,... | 5 | https://mathoverflow.net/users/13972 | 126204 | 70,489 |
https://mathoverflow.net/questions/126199 | 3 | There are three well known model structures in the category of spaces - the Quillen's structure, the Strom's structure and the mixed structure.
I was wondering if there is some other nice structures. In particular, for a fixed $n\in\mathbb{N} $, is there a model structure in Top such that the weak equivalences are n-eq... | https://mathoverflow.net/users/18017 | Top - Model structures | A different way of getting a model structure which captures "truncation" data was asked about [here](https://mathoverflow.net/questions/112069/a-fibrant-objects-structure-on-top/114916#114916), and I answered it. There a $\pi\_n$ weak equivalence is an isomorphism for $t\leq n$ but we make no mention of surjection in t... | 2 | https://mathoverflow.net/users/11540 | 126206 | 70,490 |
https://mathoverflow.net/questions/125706 | 5 | Let $M$ be a lattice polygon on a plane (i.e. its vertices are integer points $(i,j)\in\mathbb Z^2$).
Let us define lattice width in a direction $v=(m,n)\in\mathbb Z^2$ as $w\_v(M)=\max\limits\_{x,y\in M} v\cdot(x-y)$.
Suppose the $minimal$ lattice width of $M$ equals $d$. It is clear that the area of $M$ should be... | https://mathoverflow.net/users/4298 | Area of a lattice polygon in terms of its width | $\alpha=3/8$ is sharp according to, say [this](http://www.math.illinois.edu/~z-furedi/PUBS/furedi_barany_local-diam.pdf) article, the authors refer to
[L. Fejes-Toth and E. Makai, Jr., On the thinnest non-separable lattice of convex
plates, Studia Sci. Math. Hungar. 9 (1974), 191–193.]
| 3 | https://mathoverflow.net/users/4312 | 126224 | 70,499 |
https://mathoverflow.net/questions/119115 | 25 | It is well-known that a fiber bundle under some mild hypothesis is a fibration, but I don't know any examples of fiber bundles which aren't (Hurewicz) fibrations (they should be weird examples, I think, because if the base space is paracompact then the bundle is a fibration).
Does anybody know an example?
Thanks!
... | https://mathoverflow.net/users/30709 | Example of fiber bundle that is not a fibration | $\newcommand{\RR}{\mathbb{R}}
\newcommand{\To}{\longrightarrow}
\newcommand{\id}{\mathrm{id}}$The example described in [Tom Goodwillie's answer](https://mathoverflow.net/questions/106497/non-trivial-vector-bundle-over-non-paracompact-contractible-space/106563#106563) to a related mathoverflow question essentially solve... | 21 | https://mathoverflow.net/users/21095 | 126228 | 70,501 |
https://mathoverflow.net/questions/126033 | 3 | Consider a sequence of maps between $R$ modules (where $R$ is a ring with unity) $$\cdots \rightarrow M\_{n+1} \xrightarrow{d\_{n+1}} M\_{n} \xrightarrow{d\_{n}} M\_{n-1} \rightarrow \cdots$$ such that $\ker(d\_n) \subseteq \text{im}(d\_{n+1})$ for all $n$ (so the sequence is not necessarily a complex, and, if it is, i... | https://mathoverflow.net/users/nan | Sequences of maps between modules such that $\ker(d_n) \subseteq \text{im}(d_{n+1})$ |
>
> Fair enough. But I still think perhaps there are interesting things one might be able to say about the category of such objects (if one were to form the category of these sequences of R modules, call it C, in the natural way). For instance, is there any relationship between C and the category of complexes of R-mo... | 2 | https://mathoverflow.net/users/25437 | 126230 | 70,502 |
https://mathoverflow.net/questions/126236 | 0 | Let $G$ and $H$ be locally compact totally disconnected abelian groups, and $f:G\rightarrow H$ a surjective open map. Let $Y\subseteq G$ be a discrete subgroup in the subspace topology. Is it true that the image $f(Y)$ is also discrete in the subspace topology? If so, how can one prove it?
| https://mathoverflow.net/users/32746 | Is the image of discrete set under an open map discrete? | No, that's false. You didn't say that $f$ is a homomorphism, but the answer is still no if we require this.
Let $G = {\bf Z} \times C\_2 \times C\_3 \times C\_5 \times \cdots$ be the product of the infinite cyclic group and the cyclic groups of all prime orders. Let $H = C\_2 \times C\_3 \times C\_5 \times \cdots$ an... | 5 | https://mathoverflow.net/users/23141 | 126240 | 70,506 |
https://mathoverflow.net/questions/126243 | 2 | Let $f$ be an analytic function verfifying
$f(s)=\epsilon f(2-s)$
where $\epsilon=\pm 1$. The expression of Hasse-Weil L-function $f$ is
$$f(s)=N^{s/2}(2\pi)^{-s}\Gamma(s)\sum\_{n=1}^{\infty}\frac{a\_{n}}{n^{s}}$$
where $N$ is an integer and $\Gamma(s)$ is the gamma function.
Let $r$ be an integer. I have a... | https://mathoverflow.net/users/25947 | The Hasse-Weil L-function and some equations | No, that is impossible. The $k$-th derivative of a L function has necessarily infinitely many zeros. So you can choose $s\_j$ and $t\_j$ inductively such that the products give distinct zeros of $f^j$. Moreover, if one of $s\_j$ is zero you can't say anything clever either, but I assume that you simply have forgotten t... | 1 | https://mathoverflow.net/users/10400 | 126247 | 70,507 |
https://mathoverflow.net/questions/126205 | 4 | There are several different definitions of relative entropy, and some of them are not equivalent. Following is the definition we will use in this question.
Let $M$ be a closed manifold and $\mathcal{P}$ the set of Borel probability measures on $M$. Given a reference measure $\omega\in \mathcal{P}$ (usually the normal... | https://mathoverflow.net/users/11028 | Convexity and semicontinuity of the relative entropy function | If $M$ is complete and separable, then $E(\mu|\omega)$ is lower semicontinuous in $\mu$ on the set of all probability measures on $M$ with respect to the weak convergence of probability measures, see Theorem 1 in section III of this [paper](http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1055416).
Once we have l... | 2 | https://mathoverflow.net/users/7699 | 126248 | 70,508 |
https://mathoverflow.net/questions/126237 | 1 | Consider the two-dimensional lattice $G=(\mathbb{Z}^2,\mathbb{E}^2)$, where edge set $\mathbb{E}^2$ is given by the pairs of nearest neighbors in the $\ell^1$ norm in $\mathbb{Z}^2$.
Let $V\subset\mathbb{Z}^2$ an infinite subset and suppose that $G[V]$, the induced subgraph, is connected.
Let $\Lambda\_n=([-n,n]\time... | https://mathoverflow.net/users/2386 | Independent bond percolation on upper density zero subgraphs of the square lattice can have a non-trivial critical point ? | This is not true without additional assumptions on your subset. E.g. consider $$V = \{(x,y) \in \mathbb Z^2 : x>0, |y| < \sqrt x \}.$$ It clearly has zero asymptotic density, but on the other hand its critical point is equal to $1/2$ like that of $\mathbb Z^2$ itself. (You can prove that it percolates for every $p>1/2$... | 6 | https://mathoverflow.net/users/9430 | 126249 | 70,509 |
https://mathoverflow.net/questions/126244 | 11 | Let $V\models\sf ZFC$, and let $V[r]$ be a generic extension obtained by adding one Cohen real, or equivalently $\omega$ Cohen reals.
It is clear that $\Bbb R^{V[r]}$ and $\Bbb R^V$ have the same cardinality, and that $\Bbb R^{V[r]}\setminus\Bbb R^V$ also have that cardinality.
Furthermore any generic real must be ... | https://mathoverflow.net/users/7206 | The transcendence degree of $\mathbb R$ after adding a Cohen | The transcendence degree of $\mathbb{R}^{V[c]}$ over $\mathbb{R}$ is $2^{\aleph\_0}$, the largest it could be. The reason is that adding a single Cohen real $c$ adds a family of continuum many Cohen reals, which are finitely-mutually generic. That is, in the extension, there is a family of continuum many reals, such th... | 10 | https://mathoverflow.net/users/1946 | 126253 | 70,510 |
https://mathoverflow.net/questions/126227 | 2 | Dear all,
When working with a group theory problem, I come up with the equation:
$$p^x-1=2^y,$$ where $p$ is a prime and $x,y$ are positive integers. I would like to show that this equation has infinitely many solutions $(p,x,y)$ ($p$ is also a variable!) but not sure if this is correct. So my question is:
Is it ... | https://mathoverflow.net/users/32259 | A Diophantine equation involving prime powers. | I only answer the (newly) added question (the others being adressed in comments):
>
> Are there infinitely many primes of the form $a2^n + 1$, where $a$ is a fixed number?
>
>
>
Certainly *not* for each $a$. More precisely, Sierpiński (1060) showed that *there exist* infinitely many odd $a$ such that *all num... | 3 | https://mathoverflow.net/users/nan | 126258 | 70,514 |
https://mathoverflow.net/questions/126251 | 9 | Suppose one has a finite number of distances $d\_1,\ldots,d\_k$ on the Euclidean plane all of which metricize the usual Euclidean topology.
Define for each pair of points $x$ and $y$ in the plane
$$d(x,y) = \inf\left\lbrace d\_{i\_1}(x\_0,x\_1) + \cdots d\_{i\_l}(x\_{l-1},x\_l) \right\rbrace$$
where the infimum is ta... | https://mathoverflow.net/users/7631 | Infimum of a finite number of distances in the plane | It can be zero.
Take the standard metric on $\mathbb R^3$ and the one given in [this example](https://mathoverflow.net/questions/125283/on-lipschitz-embeddability-of-certain-compact-metric-spaces-into-mathbbrn/125295#125295) by S. Ivanov.
Below I give simplification of his example which works in your case.
**Simpli... | 3 | https://mathoverflow.net/users/1441 | 126267 | 70,518 |
https://mathoverflow.net/questions/126182 | 5 | In the paper E. Mehlum, Appell and the apple (nonlinear splines in space), Technical
Report No. 1676 (1981), Central institute for industrial research, Oslo (reproduced in the book Mathematical Methods for Curves and Surfaces, pages 365–384, Vanderbilt University Press, 1995) we can find the following interesting ident... | https://mathoverflow.net/users/32389 | Identity involving Fresnel integrals | It seems easier to integrate over the unit square, giving
$$ C^2(x) + S^2(x) = \int\_0^x \int\_0^x \cos(y^2-z^2) \;\mathrm{d}y\;\mathrm{d}z. $$
Using the Taylor series for cosine, and then normalizing the integral by setting $s = xy$, $t = xz$, gives
$$ C^2(x) + S^2(x) = \sum\_{n=0}^\infty \frac{(-1)^n}{(2n)!} x^... | 6 | https://mathoverflow.net/users/7709 | 126276 | 70,520 |
https://mathoverflow.net/questions/125075 | 1 | Suppose $A$ is an augmented commutative algebra over a field $k$. What is the relation between Hochschild homology $H\_n(A,k)$ and Kahler differential $\Omega\_{A/k}$? The same question is also asked about $H^n(A,k)$ and $\Omega\_{A/k}$. Here $k$ is considered as the trivial $A$-bimodule via the augmentation.
| https://mathoverflow.net/users/25701 | Hochschild (co)homology and Kahler differentials | Xingting, I do not know the answer for your question. Nevertheless, the concepts involved in such questions are nicely developed in the following book (expressed in BibTex format):
@book {MR2640631,
AUTHOR = {Majadas, Javier and Rodicio, Antonio G.},
TITLE = {Smoothness, regularity and complete intersection},
SERI... | 0 | https://mathoverflow.net/users/30397 | 126278 | 70,521 |
https://mathoverflow.net/questions/126275 | 12 | Let $M^n$ be a smooth manifold equipped with a nondegenerate Lagrangian $L:TM\to\mathbb R$, $L=L(x,y)$, $x\in M$, $y\in T\_xM$. The stationary points of the corresponding integral functional on curves are the solutions of the Euler-Lagrange equation, which in coordinates reads
$$
\frac d{dt} \frac{dL}{dy\_i}(x(t),\dot... | https://mathoverflow.net/users/4354 | Invariance of the l.h.s. of Euler-Lagrange equation | There is a coordinate-free description using only natural objects on $TM$. Here is one way to do it.
First, consider the basepoint submersion $\pi:TM\to M$. For each $v\in TM$, the linear map $\pi'(v):T\_v(TM)\to T\_{\pi(v)}M$ is surjective, and the $\pi$-fiber through $v$ is equal to $T\_{\pi(v)}M$, a vector space. ... | 16 | https://mathoverflow.net/users/13972 | 126298 | 70,529 |
https://mathoverflow.net/questions/126290 | 4 | Let $C$ be a curve in $\mathbb{P}^3$, possibly non-reduced. Assume, there exists a smooth surface in $\mathbb{P}^3$ containing $C$. Is it true that for $d \gg 0$, a generic element of $I\_d(C)$ defines a smooth surface in $\mathbb{P}^3$?
| https://mathoverflow.net/users/32151 | Existence of smooth surfaces containing a curve | I hope this is not a homework exercise, but I do not recall this from the standard textbooks.
This is already false for planar double lines. Let $\mathbb{P}^3$ have homogeneous coordinates $[T\_0,T\_1,T\_2,T\_3]$. Let $S$ be the plane $Z(T\_3)$. Let $C$ be the curve $Z(T\_2^2,T\_3)$ with induced reduced curve $L=Z(T\... | 5 | https://mathoverflow.net/users/13265 | 126303 | 70,532 |
https://mathoverflow.net/questions/126284 | 10 | I came across the following simple question: what odd integer squares have exactly 3 ones in their binary expansion?
After looking at it for a while I convinced myself that the only solutions to $r^2 = 1+ 2^m + 2^n$ (with $n > m \ge 3$) are $n=2m-2$ (the "trivial" case) and $(m,n) = (4,5),(4,9)$ (the "sporadic" cases... | https://mathoverflow.net/users/2784 | Binary expansion of squares | This was solved in a paper of Szalay (in Indag. Math. in 2002), using lower bounds for approximations to $\sqrt{2}$ by rationals with denominators of the form $2^k$ (obtained by Beukers using Pad\'e approximations to $\sqrt{1-z}$).
| 17 | https://mathoverflow.net/users/7302 | 126319 | 70,539 |
https://mathoverflow.net/questions/126318 | 3 | Let $L$ be the first-order language with binary function symbol $+$, unary function symbol $E$. Let $T$ be the set of sentences over this language that are true in the natural numbers, with $+$ interpreted in the usual way, and $E(n$) interpreted as $2^n$. Is the set $T$ recursive?
| https://mathoverflow.net/users/32769 | Is the structure $(\omega,+,2^n)$ undecidable? | As Marty explained in [this answer](https://mathoverflow.net/questions/103896/beyond-presburger-arithmetic/103914#103914), this question is the central topic of the paper [On the expansion $\langle \mathbb{N},+,2^x\rangle$ of Presburger arithmetic](http://www.math.osu.edu/~friedman.8/pdf/0AppB072710.pdf), by Françoise ... | 4 | https://mathoverflow.net/users/1946 | 126321 | 70,541 |
https://mathoverflow.net/questions/126151 | 9 | Let $\theta>0$ be irrational, and define
$$\zeta\_\theta(s) = \sum\_{n=1}^\infty \frac{\lfloor n \theta \rfloor}{n^s}.$$
This converges to an analytic function on $\Re(s)>2$. Does $\zeta\_\theta(s)$ have a meromorphic continuation to $\mathbb{C}$?
Some motivation: I'm interested in zeta functions associated to distan... | https://mathoverflow.net/users/32705 | Meromorphic continuation of a Dirichlet series associated to an irrational number | Yes, the conjecture is true -- $\zeta\_\theta(s)$ extends to a meromorphic function on $\mathbb{C}$ for all real quadratic irrationals $\theta$ with poles at $s=2$, $s=1$ and in the infinite set of vertical lines $\lbrace -2n+i\mathbb{R}\colon n\in\mathbb{Z}\_{\ge0}\rbrace$. I'm not sure about the more tentative conjec... | 7 | https://mathoverflow.net/users/1004 | 126327 | 70,543 |
https://mathoverflow.net/questions/126286 | 6 | In thinking about a [MathOverflow question](https://mathoverflow.net/questions/126045/sum-of-digits-in-different-bases) pertaining to numbers whose decimal and binary digit sums are equal, I found myself asking:
>
> Are there any solutions in
> non-negative integers $(a,b,c,d)$ to
> the equation
>
>
> $$2^a + ... | https://mathoverflow.net/users/15837 | How often do two powers of 2 equal two powers of 10 (when summed)? | There's an elementary way of solving this (and similar equations). Let's start with
$$
1+2^n=5^a(1+10^m)
$$
which you want to solve in positive integers $n$, $a$, $m$. Clearly $m < n$ and $a < n$. As wccanard points out, $2$ is a primitive root modulo $5^a$ and so we obtain that $n$ is divisible by $2 \cdot 5^{a-1}$. ... | 13 | https://mathoverflow.net/users/4140 | 126331 | 70,545 |
https://mathoverflow.net/questions/126322 | 5 | How could we work out a grammar if we know the language? How could we work out a grammar if we know the language that is restricted to a special kind like CFL or CSL? For example, we know $$L=\{a^nb^nc^n \mid n \in \mathbb{N}\}$$ How can we get the grammar? Is there any algorithm?
**PS**: Language here means at least... | https://mathoverflow.net/users/14024 | How to work out a grammar if we know the language? | **Theorem.** There is no computable procedure which, given as
input a Turing machine program $e$ that enumerates a c.e. set that
happens to be a context-free language, outputs a context-free
grammar for that language.
Proof. Let us denote by $W\_e$ the set enumerated by program $e$.
Suppose that there were such a com... | 7 | https://mathoverflow.net/users/1946 | 126335 | 70,546 |
https://mathoverflow.net/questions/126328 | 5 | I know that the space of all complex 1-tori (elliptic curves) is modeled by $SL(2, \mathbb{R})$ acting on the upper half plane. There are many explicit formulas for this action.
Similarly, I have been told that in the higher dimensional cases, the symplectic group $Sp(2n, \mathbb{R})$ acts on some such space to give ... | https://mathoverflow.net/users/22781 | Moduli Spaces of Higher Dimensional Complex Tori | The fact that all $1$-dimensional tori are projective means care is sometimes needed in making analogies with higher dimensional tori. This is one of those times. The natural 'moduli space' of all $d$-dimensional complex tori constructed by Will is not very nice when $d>1$. For example, the action of $GL\_{2d}(\mathbb ... | 7 | https://mathoverflow.net/users/430 | 126336 | 70,547 |
https://mathoverflow.net/questions/126333 | 7 | A spin structure on a Riemannian bundle of rank >2 is the lift of the structure group from $\text{SO}(n)$ to its universal cover $\text{Spin}(n).$ It may also be defined in the case $n=2$ as the lift of the structure group to a double cover of $\text{SO}(2)$, which is of course not the universal cover.
So what about... | https://mathoverflow.net/users/19860 | is there an anyon structure analogous to spin structure for rank 2 bundle? | I know nothing about the physics you have in mind, but I can tell you about the topology. The classifying space $BSO(2)$ is a $K(\mathbb{Z},2)$, so oriented 2-plane bundles are in bijection with classes in $H^2(B,\mathbb{Z})$. For any $n$, the $n$-sheeted cover $SO(2)\to SO(2)$ corresponds to multiplication by $n$ on $... | 5 | https://mathoverflow.net/users/75 | 126341 | 70,548 |
https://mathoverflow.net/questions/126350 | 1 | What is the correct terminology for the following property of a simplicial set $X\_\bullet$:
>
> For every $k\geq 0$, every map $\partial\Delta^k\to X\_\bullet$ can be extended to a map $\Delta^k\to X\_\bullet$.
>
>
>
| https://mathoverflow.net/users/35353 | How to call a simplicial set where every boundary of a simplex can be filled? | This is just a contractible Kan complex. It's equivalent to the same condition where you replace the pairs $(\Delta^k,\partial\Delta^k)$ with *all* pairs $(A,B)$ where $B\subset A$, since $A$ can be built from $B$ by iteratively filling in simplices whose boundaries are already filled in. In particular, any such $X$ wi... | 7 | https://mathoverflow.net/users/75 | 126353 | 70,553 |
https://mathoverflow.net/questions/126352 | 6 | Given a positive real number $l$. Does there exist a closed hyperbolic surface $X$ so that injectivity radius not less than $l$?
| https://mathoverflow.net/users/23358 | injectivity radius of hyperbolic surface | Buser in 1992 gives a lower bound on the Bers constant for surfaces of genus $g$, and it goes to infinity as $g$ goes to infinity. So this means there's compact hyperbolic surfaces whose injectivity radius is arbitrarily large, but you have to go to high genus to realize them.
P. Buser. *Geometry and spectra of comp... | 7 | https://mathoverflow.net/users/1465 | 126356 | 70,554 |
https://mathoverflow.net/questions/126337 | 1 | Let $\Lambda$ be an $n$-dimensional lattice in $\mathbb R^n$ and let $\cal B$ be the set of all bases that generate $\Lambda$. For a basis
$\mathbf{B}=[\mathbf{b}\_1, ... ,\mathbf{b}\_n]\in {\cal B}$, define $\mathbf{B}^\dagger = [\mathbf{b}^\dagger\_1, ... ,\mathbf{b}^\dagger\_n]$ to be the Gram-Schmidt orthogonaliza... | https://mathoverflow.net/users/22171 | Lattice basis with Gram-Schmidt vectors of increasing length | $\def\b{{\bf b}}$
No, not any lattice has such a basis.
Notice that $||\b\_n^\dagger||$ is the distance from $\b\_n$ to the hyperplane $H\_{n-1}=\langle \b\_1,\dots,\b\_{n-1}\rangle $; thus it is not greater than the minimal length of a lattice vector outside $H\_{n-1}$. Moreover, they can be equal only in the case ... | 2 | https://mathoverflow.net/users/17581 | 126360 | 70,556 |
https://mathoverflow.net/questions/126340 | 2 | Let $E$ be an elliptic curve and $E\_n$ be its quadratic twist by $n$. Let $\phi\_n: X\_0(N\_n) \to E\_n$ be the normalized modular parametrization of $E\_n$ ($\infty \to O$)
For some particular curves $E$, it seems (based on computing some examples in sage) to always be the case that $\phi\_n(0) = O$ when the rank o... | https://mathoverflow.net/users/32344 | Possible reasons why the image of 0 in the modular parametrization would always be O for a family of quadratic twists of elliptic curves | Note: this gives a plausible explanation, not a whole solution.
Hi, the BSD-quotient is $L(E\_n,1)/\Omega\_n = Sha\cdot\prod\_p c\_p(E\_n)/|T|^2$. I think your question is mostly equivalent to asking why this quotient is an integer when twisting by $n$? The torsion is of size 4 (maybe there is an exceptional twist wh... | 1 | https://mathoverflow.net/users/32778 | 126363 | 70,558 |
https://mathoverflow.net/questions/126368 | 8 | After an extensive unsuccessful search: I need a reference (preferably a book) for the Donsker's invariance principle for Riemannian manifolds. Thanks.
| https://mathoverflow.net/users/23509 | Reference needed: Donsker's Invariance Principle for Riemannian Manifolds | The generalization of Donsker's theorem from $N$-dimensional Euclidean space to general Riemannian manifolds has been worked out by Erik Jørgensen, [The Central Limit Problem for Geodesic Random Walks](http://link.springer.com/content/pdf/10.1007%2FBF00533088).
>
> The purpose of the present work is to
> consider ... | 6 | https://mathoverflow.net/users/11260 | 126371 | 70,559 |
https://mathoverflow.net/questions/125900 | 5 | A function $f$ is called *holonomic* if it satisfies some linear differential equation with polynomial coefficients $$p\_n(x) f^{(n)}(x)+\dots+p\_1(x)f'(x)+p\_0(x)f(x)=0.$$ Now if $f,g$ are holonomic then so are their sum and product. To obtain a differential equation for $h=fg$, first observe that
$$ h^{(k)} = \sum\_{... | https://mathoverflow.net/users/23862 | Constructing a linear ODE for a product of two holonomic functions without introducing additional singularities | Hi Dima,
The operator you obtain by the algorithm you describe has minimal possible order. You pay for the minimality of the order by having (in general) a nonminimal degree of the polynomial coefficients. You can turn the minimal-order operator into a minimal-degree operator if you are willing to pay the price of a... | 4 | https://mathoverflow.net/users/32783 | 126373 | 70,561 |
https://mathoverflow.net/questions/126351 | 5 | The 'hereditarily countable names' are as defined in Shelah's Proper and Improper Forcing, Chapter 3 Definition 4.1. Let $\mathbb{P}$ be a proper forcing notion and $\dot{Q}$ a $\mathbb{P}$-name such that $\Vdash\_{\mathbb{P}}$ "$\dot{Q}$ is a proper forcing notion with set of elements $\check{\kappa}$ and maximal elem... | https://mathoverflow.net/users/29231 | Hereditarily Countable Names and Proper Forcing | Counterexample: Let $\kappa $ be uncountable and let $\mathbb P= \kappa$ be an antichain (with a special weakest element $0\_{\mathbb P}$, and let $\mathbb Q$ be forced to be the same forcing. Let $\sigma$ be the $\mathbb P$-name of the generic element of $\mathbb P$.
Now note that each "hereditarily countable" name... | 6 | https://mathoverflow.net/users/14915 | 126376 | 70,563 |
https://mathoverflow.net/questions/126343 | 4 | I revised the question. In smooth ergodic theory, a diffeomorphism is said to be conservative (I), if it preserves the Lebesgue measure. So for some of us, conservativity is just short for measure-preserving.
On the other hand, we can define the conservative part $C\_f$ for general measure-class preserving maps (see ... | https://mathoverflow.net/users/11028 | The relations between conservative part and conservativity | Several comments
(1) There is no need to require invariance or finiteness of the measure in order to define the Hopf decomposition - it makes sense for any quasi-invariant measure.
(2) There is no need to evoke metric spaces, homeomorphisms, manifolds, etc. The Hopf decomposition is defined entirely in the measure ... | 1 | https://mathoverflow.net/users/8588 | 126377 | 70,564 |
https://mathoverflow.net/questions/126379 | 1 | I would like to know that whether this paper " Groups having three complex irreducible character degrees by Thomas Noritzsch" has a corrigendum?
| https://mathoverflow.net/users/32784 | Reference needed | According with MathSciNet and Zentralblatt, it seems that there is no author's corrigendum for this paper. However, a corrected version of his results can be found in the following paper:
M. Lewis - J. Riedl:
Affine semi-linear groups with three irreducible character degrees,
J. Algebra 246 (2001), no. 2, 708–720.
... | 6 | https://mathoverflow.net/users/14653 | 126382 | 70,568 |
https://mathoverflow.net/questions/126272 | 3 | Let $(X,\mathcal U)$ be a uniform space and let $U\in \mathcal U$. Is this statement true?
$$\forall V\in \mathcal U, \exists W\in \mathcal U, U\circ W\subseteq V\circ U$$
I think if the above statement is true then we can easily prove:
$$\forall V\in \mathcal U, \exists W\in \mathcal U, W\circ U\subseteq U\circ V$$
... | https://mathoverflow.net/users/31968 | For any entourage $U,V$ there's an entourage $W$ such that $U\circ W\subseteq V\circ U$ | The second statement follows from the first one by passing to inverses (i.e. reflecting along the diagonal).
Now consider the uniform structure on $\mathbb{R}$ consisting of all subsets of $\mathbb{R}^2$ that contain an open neighborhood of the diagonal.
Just to avoid confusion with left and right, I want to use t... | 1 | https://mathoverflow.net/users/3969 | 126386 | 70,571 |
https://mathoverflow.net/questions/126339 | 10 | I'm looking for a reference (or quick proof) of the following fact. Fix some $n \geq 3$ and some $\ell \geq 2$. Set $\Gamma\_n(\ell) = \text{ker}(\text{SL}\_n(\mathbb{Z}) \rightarrow \text{SL}\_n(\mathbb{Z}/\ell \mathbb{Z}))$. Next, set
$$\Gamma\_n'(\ell) = \{\text{$A \in \Gamma\_n(\ell)$ $|$ for all diagonal entries... | https://mathoverflow.net/users/29685 | Generators for a certain congruence subgroup of SL(n,Z) | There is a Theorem of Tits which says that the group generated by $e\_{ij}^l$ has finite index in $SL\_n({\mathbb Z})$ if $n\geq 3$. The paper is a Comptes rendus announcement (generating systems of congruence subgroups, CR. ACad. Sci 283 (1976), no. 9 A693-A695 (see the following review <http://www.ams.org/mathscinet/... | 3 | https://mathoverflow.net/users/23291 | 126389 | 70,574 |
https://mathoverflow.net/questions/126295 | -4 | Let $A$ and $H$ be closed subgroups of a $\sigma$-compact locally compact group $G$. Assume further that $A$ is abelian. Is the group $AH$ locally compact subgroup in the subspace topology?
| https://mathoverflow.net/users/32746 | Is the product of closed subgroups in a locally compact group locally compact? | The is not true in general. For example, if $G=\mathbb R$, $A=\mathbb Z$, and $H=h\mathbb Z$ for some $h\in\mathbb R\smallsetminus\mathbb Q$, then $AH$ (that is, $\mathbb Z+h\mathbb Z$) is a countable dense subgroup of $\mathbb R$, and as such it is not locally compact.
Furthermore, as pointed out by Misha, the produ... | 1 | https://mathoverflow.net/users/12705 | 126390 | 70,575 |
https://mathoverflow.net/questions/105946 | 4 | I am interested in the following variant of the usual Isotopy Extension Theorem:
$\textbf{Question}:$Let $K$ be a graph (1-complex) embedded on a surface $S$, and $i$ be an isotopy of $K$. Does $i$ extend to an isotopy of $S$ (generally called an ambient isotopy or a diffeotopy) ? If $i$ fixes a set of points $V\subs... | https://mathoverflow.net/users/6325 | Extensions of non-smooth isotopies of not-submanifolds on surfaces | The answer is yes, see the appendix of
<http://www.math.jussieu.fr/~lerouxf/RECHERCHE/TEXTES/0-LE%20ROUX-These-97.pdf>
This reference contains statement and proofs in the case of the 2-sphere, see Theorem A.3.1.
The proof can be adapted for other surfaces.
Frederic
| 1 | https://mathoverflow.net/users/32785 | 126394 | 70,576 |
https://mathoverflow.net/questions/126395 | 12 | Who invented projective space $\mathbb{P}^n$ as an extension of the usual [affine space](http://en.wikipedia.org/wiki/Affine_space%20affine%20space) $\mathbb{A}^n$?
Who was the first person to consider projective closure of
[plane affine algebraic curves](http://en.wikipedia.org/wiki/Algebraic_curve%20plane%20affine%... | https://mathoverflow.net/users/22481 | who invented projective space $\mathbb{P}^n$? | The idea of projective space goes back to the study of perspective in painting. The first formalization known is due to G. Desargues, with the book *Brouillon Projet d'une atteinte aux événements des rencontres du Cône avec un Plan* (Rough draft for an essay on the results of taking plane sections of a cone) published ... | 26 | https://mathoverflow.net/users/6348 | 126399 | 70,578 |
https://mathoverflow.net/questions/126383 | 4 | Consider the hyperplane $H=\{f\in L^\infty: \int f = 0\}$ of $L^\infty = L^\infty[0,1]$. My question is:
**1. What is the Banach-Mazur distance between $H$ and $L^\infty$? Are there "natural" isomorphisms between these two spaces?**
Since $L^\infty$ is 1-injective, we have the lower bound
$$
inf\{\|P\|: \mbox{$P$... | https://mathoverflow.net/users/27566 | On hyperplanes of $L\infty$ | IIRC, if $K$ is a compact Hausdorff space that has no isolated points, then every projection from $C(K)$ onto a hyperplane has norm at least two. Maybe Dan Amir proved this? Anyway, in your situation the proof goes like this. A projection $P$ from $L^\infty$ onto $H$ has the form $Pf = f -(\int f) g$, where $\int g = 1... | 5 | https://mathoverflow.net/users/2554 | 126402 | 70,579 |
https://mathoverflow.net/questions/126397 | 0 | I find this concept in Kollar and Mori's book {\em Birational Geometry of Algebraic Varieties}, but cant search the precise definition in the book or google. Can you tell me the definition? Thanks for any comments or references.
| https://mathoverflow.net/users/3525 | What is the definition of a sufficiently ample line bundle? | There are many theorems of the form
>
> **Theorem Frame**
> If something holds and $\mathscr L$ is an ample line bundle, then there exists an $n\_0\in \mathbb N$ such that for all $n\geq n\_0$, something else holds with $\mathscr L^{\otimes n}$ in it.
>
>
>
You should think of things like
* a sheaf being g... | 3 | https://mathoverflow.net/users/10076 | 126405 | 70,580 |
https://mathoverflow.net/questions/126407 | 3 | I am looking at p-adic distributions, and in this case p-adic measures. To say that $\mu$ is a distribution means that the arguments of $\mu$ are compact open subsets of $\mathbb{Z}\_p$, $\mu$ is finitely additive, and the values $\mu$ takes are in $\mathbb{C}\_p$. To say that $\mu$ is a measure means that $\mu$ is a d... | https://mathoverflow.net/users/32788 | Trivial p-adic measures | Yes. Suppose that $\mu$ of some set is nonzero. Then $\mu$ of some interval of the form $a+p^n \mathbb Z\_p$ is zero. Then it must be nonzero modulo $p^k$ for some $k$. By finite additivity:
$\sum\_{t=0}^{p-1} \mu(a+p^{n}t+ p^{n+1} \mathbb Z\_p) = \mu(a+p^n \mathbb Z\_p) \not\equiv 0$ modulo $p^k$.
So for some $t$,... | 5 | https://mathoverflow.net/users/18060 | 126416 | 70,581 |
https://mathoverflow.net/questions/126415 | 3 | Suppose I have $n$ electric point charges in, say, two dimensions. Is there any algorithm (and I have a hunch that it might be related to the Fourier transform) to compute the net forces that act on each point charge in less than $O(n^2)$, preferably something like $O(n \log n)$? Thanks!
An approximation might be goo... | https://mathoverflow.net/users/16793 | Using Fourier Transform to speed up calculation of forces following an inverse square law | The FFT is an important part of the [fast multipole method](http://en.wikipedia.org/wiki/Fast_multipole_method), which is probably what you would want to use.
| 3 | https://mathoverflow.net/users/1847 | 126417 | 70,582 |
https://mathoverflow.net/questions/126396 | 7 | I have a recurrence
$$f(i,j) = 1+ \frac{N-i}{N} f(i,j-1) + \frac{i}{N} f(i-1, j)$$
$$f(i,0) = 0$$
$$f(0,j) = j$$
I would like to compute $f(N,M)$ in terms of N and M. The system is defined for $0\leq i\leq N$ and $0\leq j\leq M$ where $N$ and $M$ are non-negative integers.
I am familiar with techniques for so... | https://mathoverflow.net/users/nan | Methods for solving two variable recurrence | You have $f(0,j) = 1 + f(0,j-1)$ so $f(0,j) = j$. I get
$$f(1,j) = j + 1 - \left(\frac{N-1}{N}\right)^j $$
$$ f(2,j) = j + 2 - 2 \frac{(N-1)^{j+1}}{N^{j+1}} - 2 \frac{(N-2)^j}{N^{j+1}}$$
$$ f(3,j) = j + 3 - 3 \frac{(N-1)^{j+2}}{N^{j+2}} - 6 \frac{(N-2)^{j+1}}{N^{j+2}} - 9 \frac{(N-3)^j}{N^{j+2}}$$
$$ f(4,j) = j... | 11 | https://mathoverflow.net/users/13650 | 126432 | 70,584 |
https://mathoverflow.net/questions/126307 | 5 | Hi everyone,
I am trying to go through parts of Saint-Donat's 1974 paper 'Projective Models of K3-surfaces', and have been stuck on a few claims for a while now - I'd appreciate some help explaining them.
Here is the set-up: $X$ is a (smooth, projective, algebraic) K3 surface. For a divisor $D$ on $X$ the Riemann-R... | https://mathoverflow.net/users/11071 | Rookie questions about k3's | As you said, it's a consequence of Riemann-Roch and the equation $h^1(D)=h^0(D,\mathcal O\_D)-1$, which follows from taking cohomology of
$$0\to \mathcal O\_X(-D)\to \mathcal O\_X\to \mathcal O\_D\to 0$$
for an effective divisor $D$.
Applying Riemann-Roch to $D=D'+\Delta$, we get
$$h^0(D)-h^1(D) = \frac{1}{2}D^2+2 = ... | 3 | https://mathoverflow.net/users/30554 | 126438 | 70,589 |
https://mathoverflow.net/questions/126425 | 7 | Suppose $X$ and $Y$ are two $CW$ complexes and $f:X\rightarrow Y$ is a continuous surjection such that fiber of each point (i.e. $f^{-1}(y)$ for each $y\in Y$) is contractible. Does it implies that $X$ and $Y$ are homotopy equivalent.
PS-1:By Whitehead's Theorem it will be enough to show that $f$ induces an isomorphi... | https://mathoverflow.net/users/9485 | Homotopy equivalence from contractibility of fiber | In his paper
MR0087106 (19,302f)
Smale, Stephen
A Vietoris mapping theorem for homotopy.
Proc. Amer. Math. Soc. 8 (1957), 604–610.
Smale proved the following theorem:
**Theorem** : Let $X$ and $Y$ be connected, locally compact separable metric spaces. Assume also that $X$ is locally contractible. Consider a pro... | 12 | https://mathoverflow.net/users/317 | 126449 | 70,593 |
https://mathoverflow.net/questions/126447 | 2 | Any results or concise introduction about nonassociative algebra that even does not satisify Power associativity?
| https://mathoverflow.net/users/14024 | Any results or concise introduction about nonassociative algebra that even does not satisify Power associativity? | K.Zhevlakov, A.Slin'ko, I.Shestakov, A.Shirshov, Rings that are nearly associative. Academic Press, 1982 (Chapt.1)
P.Cohn, Universal algebra, Harper and Row, 1965 (Chapt.7, Sec.5 "Linear algebras")
| 1 | https://mathoverflow.net/users/18814 | 126450 | 70,594 |
https://mathoverflow.net/questions/126391 | 1 | For any number $x$, I define the "density" $d(x)$ to be the ratio of the number of "one" digits and the number of all digits in the binary expansion of $x$. For example, $d(13)=\frac{3}{4}$ and $d(9)=\frac{2}{4}$.
I run some experiments and have the following observation.
* $d(x^2)\le0.87$ for all $x \in [2,10^7]$.... | https://mathoverflow.net/users/7020 | The ratio of one digits and all digits in the binary expansions of the square numbers | This answer adresses the first of the two questions (and was originally written, except for minor changes, for a slightly vaguer version of the question; thus the material that might not seem completely fitting now, except that it could be helpful for the second question, so I leave it).
It is a result of Lindström (... | 5 | https://mathoverflow.net/users/nan | 126453 | 70,596 |
https://mathoverflow.net/questions/126444 | 2 | I have an impression that there is linkage or relation between singulariry of algebraic variety and continued fraction when I read some book on resolution of singularity or algebraic geometry.Could any one give some reference for that?
| https://mathoverflow.net/users/14024 | Linkage between singularities of algebraic varieties and continued fractions | Continued fractions appear naturally in the resolution of quotient singularities of surfaces (and presumably in higher dimensions as well). From a topological point of view, a neighborhood of the singularity is the cone on a lens space L(p,q), and a particular continued fraction for q/p gives an explicit piece of a smo... | 4 | https://mathoverflow.net/users/3460 | 126463 | 70,599 |
https://mathoverflow.net/questions/119590 | 2 | Hello,
I am reading the paper
Futaki; Ono; Wang
Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds.
J. Differential Geom. 83 (2009), no. 3, 585–635.
For your convenience I repeat the setting.
Let $(M^{2n+1}, \eta, \xi, g)$ be a compact Sasakian manifold and $\pi\_\alpha: U\_\alpha \ri... | https://mathoverflow.net/users/19545 | Normalized Hamiltonian holomorphic vector fields on Sasakian manifolds | The equation
$$\int (\Delta^h u \cdot v - u \Delta^h v)e^h (1/2 d\eta)^n \wedge \eta$$
already gives a proof. $u$ and $v$ are eigenvalues of $\Delta^h$, so the integrand is zero.
This covers the case of $c\_1(\mathcal{F})=ad\eta,\ a>0,$ but I'm pretty sure the proof works for general Sasakian manifolds. We have ... | 2 | https://mathoverflow.net/users/10749 | 126465 | 70,600 |
https://mathoverflow.net/questions/123613 | 1 | Dear Experts,
I'm a graduate student, dealing with group-theory.
In my current research, I used the bound "Alexander Gruber" wrote about in this post:
[See Here](https://mathoverflow.net/questions/108581/number-of-normal-subgroups-in-a-p-group)
(Actually, I have just found out that the bound is known and already ... | https://mathoverflow.net/users/31956 | Normal subgroups In a p-group [Reference?] | For general Hall Algebra,
Ian MacDonald's *Symmetric Functions and Hall Polynomials*
<http://books.google.com.hk/books/about/Symmetric_Functions_and_Hall_Polynomials.html?id=srv90XiUbZoC>
For general reference about the number of vector subspaces over a finite p-field,
I suggest that you can google the keyword "... | 0 | https://mathoverflow.net/users/25437 | 126469 | 70,602 |
https://mathoverflow.net/questions/126464 | 11 | I am trying to find out the essence of what a determinant is. Besides, in finite dimensions, determinant is the kind of numerical invariant that determines the invertibility of a linear operator, but what about infinite case? Is there a similar invariant? Why if not?
| https://mathoverflow.net/users/27040 | Is there an analog of determinant for linear operators in infinite dimensions as that of finite dimensions? | We can define the Fredholm determinant on operators on Hilbert space which differ from the identity by a trace-class operator. This satisfies
$$\det(\exp(T)) = \exp(\text{Tr}(T))$$ for trace-class operators $T$.
See <http://en.wikipedia.org/wiki/Fredholm_determinant>
| 18 | https://mathoverflow.net/users/13650 | 126472 | 70,605 |
https://mathoverflow.net/questions/126477 | 7 | Hello!
Let $(M,g)$ be a Riemannian manifold and $-\Delta$ the Laplacian on M (acting on smooth functions). Then the resolvent $R(\xi)$ of $-\Delta$ is a compact operator.
Is it possible to find for every $\epsilon>0$, a point in the resolvent set $\xi$, s.t. $\Vert R(\xi) \Vert\leq \epsilon$?
Maybe it is very ea... | https://mathoverflow.net/users/21870 | Resolvent of Laplacian | Sure, since $-\Delta$ is a positive operator, by the spectral theorem it can be realized as multiplication by a positive function $f(x)$ on some $L^2(X)$ space. Then $R(\xi)$ is multiplication by $\frac{1}{\xi - f(x)}$ and its operator norm is the sup norm of this function. If $\xi < 0$ then the function $\frac{1}{\xi ... | 9 | https://mathoverflow.net/users/23141 | 126482 | 70,609 |
https://mathoverflow.net/questions/126466 | 16 | Suppose that $\mathcal{T}$ is an abstract $2$-category we know is equivalent to the $2$-category of Grothendieck topoi via some equivalence $$\phi:\mathcal{T} \to \mathfrak{Top},$$ and let $E$ be an object of $T$. Can we recover the underlying category $\phi(E)$ without using $\phi$?
I am asking because often proper... | https://mathoverflow.net/users/4528 | Are there non-categorical notions in topos theory? | There is a Grothendieck topos $\textbf{Set}[\mathbb{O}]$ with the following universal property: for all Grothendieck toposes $\mathcal{E}$, the category $\textbf{Geom}(\mathcal{E}, \textbf{Set}[\mathbb{O}])$ of geometric morphisms $\mathcal{E} \to \textbf{Set}[\mathbb{O}]$ and "geometric transformations" (a misnomer; t... | 18 | https://mathoverflow.net/users/11640 | 126486 | 70,613 |
https://mathoverflow.net/questions/124991 | 33 | The title of this post expresses what I really want, which is to learn how to wield the internal logic of a topos more effectively. However, to bring it down to earth, I'll ask a few basic questions about the topos $\mathcal{Grph}$ of directed graphs, whose underlying category is
$$\mathcal{Grph}:={\bf Set}^{\mathcal{... | https://mathoverflow.net/users/2811 | What can be expressed in and proved with the internal logic of a topos? | In general the internal language of a topos can only express those statements that make sense in every topos. In essence, this limits you to something like bounded Zermelo set theory, without global membership.
The right way to use the internal language of a *particular* topos, such as your topos of directed graphs, ... | 31 | https://mathoverflow.net/users/1176 | 126492 | 70,618 |
https://mathoverflow.net/questions/126498 | 4 | Let $X$ be a locale or a topological space. $I$ denote the unit interval of the real numbers, and $X^I$ the space of functions from $I$ to $X$ (The locale exponential if $X$ is a locale or the set of function endowed with the open-compact topology if $X$ is a topological space.)
In both case, we consider the map $X^I... | https://mathoverflow.net/users/22131 | On the openness of the map $X^I \to X \times X.$ | Here are what I believe are the standard definitions:
A topological space $X$ is *locally path-connected* if for every $x\in X$ and every open neighbourhood $U$ of $x$, there exists a smaller open neighbourhood $V\subseteq U$ of $x$ such that $V$ is path-connected. Equivalently, path components of open subsets are op... | 3 | https://mathoverflow.net/users/8103 | 126500 | 70,621 |
https://mathoverflow.net/questions/126519 | 31 | Every mathematician knows what "simplify" means, at least intuitively. Otherwise, he or she wouldn't have made it through high school algebra, where one learns to "simplify" expressions like $x(y+x)+x^2(y+1+x)+3(x+3)$.
But is there an accepted rigorous "mathematical" definition of "simplify" not just for algebraic e... | https://mathoverflow.net/users/7089 | Is there a "mathematical" definition of "simplify"? | In full generality, there provably isn't any method for complete simplification (i.e., bringing an expression into a canonical simplest form). Simplifying should have two key properties: it should be algorithmic, and simplifying two different expressions for the same thing should give the same simplified form. If you h... | 36 | https://mathoverflow.net/users/4720 | 126522 | 70,630 |
https://mathoverflow.net/questions/126474 | 17 | The MSRI is organising [a programme with the above title](http://www.msri.org/web/msri/scientific/programs/show/-/event/Pm8996) from Aug 11, 2014 to Dec 12, 2014. Here is a short description from their website :
>
> The branches of number theory most
> directly related to the arithmetic of
> automorphic forms ha... | https://mathoverflow.net/users/2821 | New Geometric Methods in Number Theory and Automorphic Forms | Knowing the organizers well and working in the field, I can try an answer, but this is nothing more than an educated guess.
First, the breakthroughs in question include
(i) The construction and study of Galois representations attached to self-dual cohomological automorphic forms for $Gl\_n$ (satisfying local-globa... | 29 | https://mathoverflow.net/users/9317 | 126523 | 70,631 |
https://mathoverflow.net/questions/126529 | 8 | Let $A=K[[X\_1,\dots,X\_n]]$ where $K$ is a field. Let $M$ be a finitely generated torsion-free $A$-module, such that
1. for all $k$, the $A[1/X\_k]$-module $M[1/X\_k]$ is free of rank $d$;
2. for every $i \neq j$, we have $M = M[1/X\_i] \cap M[1/X\_j]$.
Does this imply that $M$ is free?
It certainly does if $n=... | https://mathoverflow.net/users/5743 | A criterion for freeness over a local ring | No, this is false as soon as $n ≥ 3$. A second syzygy $M$ of the residue field $K$ gives a counterexample: each $M[1/X\_i]$ is projective, hence free, and it is reflexive, so the second condition is satisfied. On the other hand the projective dimension of $K$ as a module is $n$, so $M$ can not be free.
| 8 | https://mathoverflow.net/users/4790 | 126533 | 70,634 |
https://mathoverflow.net/questions/126269 | 9 | Let $\Omega$ be a standard atomless probability space, we can assume $\Omega=(0,1)$ with Lebesgue measure. A bijection $f:\Omega/A\_1\to\Omega/A\_2$ is almost automorphism, if $P(A\_1)=P(A\_2)=0$, $f(A)$ is measurable if and only if $A$ is, and $P(f(A))=P(A)$ in this case. An almost automorphism preserves a (real-value... | https://mathoverflow.net/users/31472 | Random variables invariant under almost automorphisms. | Fix for convenience $\Omega = [0,1)$ with Lebesgue measure $\mu$. We exhibit a family of continuum-many Borel functions $X\_\alpha \colon [0,1) \to \mathbb{R}$, indexed by $\alpha \in (0, 1/2)$, such that
* Each $X\_\alpha$ has only trivial almost automorphisms, in the sense that each almost automorphism $f$ preservi... | 4 | https://mathoverflow.net/users/14913 | 126544 | 70,641 |
https://mathoverflow.net/questions/126542 | 3 | Assume that $X\_1,\ldots,X\_r\subseteq\mathbb P^n$ are irreducible, reduced hypersurfaces in complex projective space, each of the same degree $d$. In other words, $X\_i=Z\_\ast(f\_i)$ for certain irreducible, homogeneous polynomials $f\_i\in\mathbb C[X\_0,\ldots,X\_n]\_d$. Let $X:=X\_1\cap\cdots\cap X\_r$ be their int... | https://mathoverflow.net/users/9947 | The Hilbert function of an intersection | It sure does help to assume complete intersection. It's nicer to describe the Hilbert series $H(t) := \sum\_k h(k) t^k$ than the Hilbert function $h(k)$. For the polynomial ring itself, it's $H(t) = 1/(1-t)^{n+1}$. For the complete intersection, it's $(1-t^d)^r/(1-t)^{n+1}$ (with an obvious generalization if the degree... | 5 | https://mathoverflow.net/users/391 | 126551 | 70,644 |
https://mathoverflow.net/questions/126408 | 2 | Let $\tilde{\mathbb N}$ be the Abelian semigroup (under addition) given by $\mathbb N\cup\{0,\infty\}$, and let $S\_n$ be the Abelian monoid $\tilde{\mathbb N}^{2^n}$ under point-wise addition. Introduce the transition maps $\{\phi\_n:S\_{n+1}\to S\_n\}\_{n\in\mathbb N}$ defined by
$$\phi\_{n-1}(n\_1,n\_2,n\_3,n\_4,\... | https://mathoverflow.net/users/16023 | Projective limit construction of a semigroup | Let ACom be the class of all finite aperiodic commutative monoids (aperiodic means each subgroup is trivial, i.e. the monoid satisfies some identity of the form x^n=x^{n+1}); let pro-ACom be the class of all projective limits of monoids in ACom. The class pro-ACom has free objects on each compact totally disconnected s... | 1 | https://mathoverflow.net/users/32831 | 126557 | 70,646 |
https://mathoverflow.net/questions/126558 | 7 | It is well-known that $Hilb^n(X)$, the hilbert scheme of $n$ points on a smooth projective surface $X$, is isomorphic to $M\_X(1,\mathcal O\_X,n)$, the moduli space of rank one semistable sheaves with trivial determinant and second chern class $n$. The canonical morphism in one direction sends a subscheme $Z\subset X$ ... | https://mathoverflow.net/users/13139 | Hilbert scheme of points on a surface as moduli space of semistable sheaves | By definition semistable sheaves are torsion-free. Any torsion-free $F$ includes into its double dual, $F\to F^{\ast\ast}$. The double dual is a reflexive sheaf, so any singularities occur in codimension 3. In the surface case, we conclude $F^{\ast\ast}$ is a line bundle with trivial determinant, so must be $\mathcal O... | 10 | https://mathoverflow.net/users/7399 | 126563 | 70,647 |
https://mathoverflow.net/questions/126561 | 0 | By filters I will mean filters on some set $\mho$.
I define product of an infinite family of filters in two ways. I feel (by analogy with properties of Tychonoff product vs box product of topological spaces) that these two products are different, but don't know how to find a counter-example supporting their non-equal... | https://mathoverflow.net/users/4086 | Two different products of filters | If all the $a\_i$ are principal ultrafilters on sets with at least two elements, then $\Pi\_1$ will also be principal, since it concentrates on the singleton that picks out the base of each $a\_i$. But $\Pi\_2$ will not contain any singleton set (if the index set is infinite), since the $\Pi\_{i}B\_i$ will all contain ... | 3 | https://mathoverflow.net/users/1946 | 126564 | 70,648 |
https://mathoverflow.net/questions/126560 | 1 | Suppose I have an Artin stack $\mathfrak M$ locally of finite-type over $\mathbb C$ with presentation $M\rightarrow \mathfrak M$. Suppose further that $\mathfrak M$ "represents" (in the stack sense) some set of objects which are bounded, i.e. roughly there is a family of such objects $\mathcal F$ over a scheme $S$ of f... | https://mathoverflow.net/users/13139 | Finite-type Artin Stack over $\mathbb C$ | Consider the fiber product $S \times\_{\frak M} M$; its projection onto $S$ is smooth and surjective, hence open. Since $S$ is quasi-compact, there exists a quasi-compact open subset $U$ of $S \times\_{\frak M} M$ surjecting onto $S$. The image of $U$ in $M$ is quasi-compact; let $V$ be a quasi-compact open subscheme o... | 5 | https://mathoverflow.net/users/4790 | 126565 | 70,649 |
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