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https://mathoverflow.net/questions/86818 | 8 | Let $G$ be a finite simple graph and let $C(G)$ be the flag complex associated to $G$ (the set of vertices of $C(G)$ is the vertex set of $G$ and the set of all cliques of $G$ are its simplexes).
Are there characterizations of contractibility of $C(G)$ ONLY in terms of the graph theoretical properties of $G$?
| https://mathoverflow.net/users/19075 | A flag complex is contractible iff the underlying graph is....? | It is known that every induced subcomplex of the flag complex of a graph is contractible iff the graph is chordal (no induced cycles of length 4 or more). I doubt a necessary and sufficient condition that is purely graph theoretic for contractibility of just the flag complex is possible because the barycentric subdivis... | 6 | https://mathoverflow.net/users/15934 | 86823 | 51,565 |
https://mathoverflow.net/questions/86800 | 13 | I am curious about how the Heegaard genus changes after a finite covering.
Is there anyone constructing an closed hyperbolic 3-manifold $N$ such that
the Heegaard genus of a finite covering of $N$ is less than the Heegaard genus of $N$?
Thank you!
Note: Heegaard genus of a 3-manifold means the minimal genus ... | https://mathoverflow.net/users/18496 | Heegaard splitting of covering hyperbolic manifold. | There are examples like this. Check out section 4.5 of Shalen's paper "Hyperbolic volume, Heegaard genus and ranks of groups." It's here: <http://arxiv.org/abs/0904.0191>
He gives a reference for a genus 3 example by Alan Reid and a sketch of a technique for producing examples by Hyam Rubinstein. Shalen also conjectu... | 14 | https://mathoverflow.net/users/5413 | 86831 | 51,568 |
https://mathoverflow.net/questions/86828 | 4 | For $z\in\mathbb{C}$ with real part greater than $1$ the sum $$\sum\_{p}{\frac{1}{p^z}},$$ where the sum is taken over all primes $p$, converges absolutely. It is also well known that the same sum with $z=1$ does not converge. Now my question is if there are $y\in\mathbb{R}$ such that
$$\sum\_{p}{\frac{1}{p^{1+iy}}}$$
... | https://mathoverflow.net/users/20934 | Convergence of the summation 1/p^(1+iy) (over all primes p with y a nonzero real number) | This is always convergent for any real $y \neq 0$. This follows from the fact that
the related integral
$$\int\_2^\infty \frac{x^{iy-1}}{\log x} dx $$
is convergent (to see this use the substitution $t=\log x$ ), and say the prime number theorem with some weak error term, in fact
$$ \pi(x)=\frac x {\log x} \left( 1+O ... | 4 | https://mathoverflow.net/users/10811 | 86836 | 51,571 |
https://mathoverflow.net/questions/86824 | 0 | Let $X$ be a noetherian scheme over base $S$ and $Y$ a closed subscheme of $X$ with arrow $j$ into $X$, $F,G$ two quasicoherent modules on $Y$. With $\boxtimes$ denote the exterior tensor product bifunctor (i.e.: pullback via the projections and tensor on the product scheme) on $Y$ resp. $X$.
Does one have a canonica... | https://mathoverflow.net/users/18183 | Exterior Product of module sheaves | This is indeed true, even more general: Let $i : Y \to X$, $j : Y' \to X'$ closed immersions of $S$-schemes and $F \in \mathrm{Qcoh}(Y)$, $G \in \mathrm{Qcoh}(Y')$. Then there is a canonical isomorphism $i\_\* F \boxtimes\_{X,X'} j\_\* G \cong (i \times j)\_\* (F \boxtimes\_{Y,Y'} G)$.
Proof: The commutative diagram
... | 2 | https://mathoverflow.net/users/2841 | 86842 | 51,574 |
https://mathoverflow.net/questions/86817 | 5 | Are there any results on the number of subgraphs in a labeled tree (or a general labeled graph)? I would also be happy to know any results on the number of subgraphs in an unlabeled tree. Cayley's formula says how many different trees I can form given n vertices, but it doesn't seem to relate to the problem of counting... | https://mathoverflow.net/users/20928 | Counting the number of subgraphs in a given labeled tree | The following algorithm should efficiently calculate the answer for the number of subtrees of a labeled graph.
Let $(T, r)$ be a labeled, rooted tree with root $r$. We first calculate the number of subtrees containing $r$. Call this value $N\_1(T, r)$. If $r\_1, \dots, r\_k$ are the neighbors of $r$ and $T\_1,\dots, ... | 3 | https://mathoverflow.net/users/20940 | 86845 | 51,576 |
https://mathoverflow.net/questions/86844 | 0 | Assume $0 < a\_i \leq 1$ for $i = 1, 2 \ldots n$. I am interested in the random sum $X = \sum\_i a\_i X\_i$ where $X\_i$ are iid random Bernoulli variables with some mean $p \in (0, 0.5)$. I would like to know if one can relate $P(X \leq 1)$ to $P(X \leq \delta)$ for some $\delta < 1$. Specifically, what are the tighte... | https://mathoverflow.net/users/5873 | Lower bound on sum of independent random variables | There is no such bound which depends only on $\delta$: if you take all $a\_i=1$ and $p=1/2$, then for any $\delta<1$ the ratio between $\mathbb{P}(X\le \delta)$ and $\mathbb{P}(X\le 1)$ is $1/(n+1)$.
| 2 | https://mathoverflow.net/users/1061 | 86846 | 51,577 |
https://mathoverflow.net/questions/86815 | 1 | Suppose I have a family of countable state-space, discrete-time Markov chains, indexed by a parameter $r \in \mathbb{R}$. The state space is the same for all values of $r$; the transition probabilities depend on $r$ continuously. Is there some result like:
If the chain obtained for $r=r\_0$ is positive recurrent, the... | https://mathoverflow.net/users/17883 | Continuous family of Markov chains | No it's not true. Consider Markov chains on $\mathbb N$ where the only transitions are from $n$ to $n\pm 1$.
For $n>1$, set $P\_{n,n+1}=r+\frac12(1-\frac1n)$ and $P\_{n,n-1}=-r+\frac12(1+\frac1n)$.
For $r=0$, solving the detailed balance equation, we get $\pi\_{n+1}n(n+2)=\pi\_n(n-1)(n+1)$ which has solutions $\pi\... | 4 | https://mathoverflow.net/users/11054 | 86847 | 51,578 |
https://mathoverflow.net/questions/86848 | 2 | Is there any known (symbolic) method that solves a system of equations/inequalities that have trigonometric functions on the left-hand side of the system?
Ex) Find $x,y,\theta \in \mathbb{R}$ that satisfy
\begin{align}
2x + y + 3\cos(\theta) - 2\sin(\theta) \le& 0 \\\
x - y + 4\cos(\theta) + 2\sin(\theta) \le& 0
\... | https://mathoverflow.net/users/20941 | Solving a system of equations/inequalities that have trigonometric functions on the left-hand side | Using, e.g., the sin function, one can write a system of inequalities in a given variable $x$ that is satisfied if and only if $x$ is an integer. Therefore, an algorithm for solving inequalities of the kind you asked about would give an algorithm of finding all integer solutions to an arbitrary system of inequalities. ... | 1 | https://mathoverflow.net/users/5229 | 86851 | 51,579 |
https://mathoverflow.net/questions/86797 | 4 | This question is inspired by the discussion in MO questions "[Local minimum from directional derivatives in the space of convex bodies](https://mathoverflow.net/questions/86653)" and "[Bodies of constant width?](https://mathoverflow.net/questions/86742)" about generalized notions of minimum widths and constant widths. ... | https://mathoverflow.net/users/20186 | Generalized widths and reverse Urysohn inequalities | This is hardly a direct answer to your question, but a new paper—at least tangentially
relevant—by HaiLin Jin and Qi Guo
addresses the question of how assymetric can a constant-width body be. In
"Asymmetry of Convex Bodies of Constant Width"
([*Discrete & Computational Geometry* Vol. 47, No. 2, Mar. 2012, 415-423](http... | 1 | https://mathoverflow.net/users/6094 | 86857 | 51,581 |
https://mathoverflow.net/questions/86765 | 2 | In [1, page 7], the author says.
>
> Kolmogorov showed that if the function $$f(x) = \sum\_{n=1}^{\infty} \frac{\cos 3^n x}{3^n}$$ has a finite or infinite generalized derivative on a set of positive measure, then the function is *nonmeasurable*.
>
>
>
Where can I find a proof/explanation of this result (and/... | https://mathoverflow.net/users/8382 | Kolmogorov's example of a measurable function not (generally) differentiable | A translation in english is in "Selected Works of A.N. Kolmogorov I" : ["On the possibility of a general definition of derivative, integral and summation of divergent series"](http://books.google.fr/books?id=ikN59GkYJKIC&pg=PA33) (page 33 and 34).
| 3 | https://mathoverflow.net/users/16380 | 86862 | 51,584 |
https://mathoverflow.net/questions/86735 | 8 | Maybe this is not a research level question. I post it because I heard that the path integral can be rigorous by Brownian motion. But my knowledge of probability is so limited.
If $$L=\frac{1}{2}(-\frac{d^2}{dx^2}+x^2),$$
we know that $Sp(L)=\{1/2,3/2,5/2,...\}$. So we get
$$\mathrm{Tr}[e^{-L}]=\frac{1}{2\sinh1/2}.$... | https://mathoverflow.net/users/16326 | Path integral and harmonic oscillator | Take a look at Appendix A of the 2nd edition of Glimm & Jaffe's book. They give a rigorous construction of the measure you're after aka, the Ornstein-Uhlenbeck measure, which is the cylinder measure you get by taking the continuum limit of the Euclidean signature harmonic oscillator). The key point is that the cylinder... | 8 | https://mathoverflow.net/users/35508 | 86865 | 51,585 |
https://mathoverflow.net/questions/86864 | 4 | Let $\lbrace P\_n(z)\rbrace\_{n\in\mathbb N\_0}$ be a family of polynomials defined by a generating function $g(t,z)=\sum\limits\_{n=0}^\infty P\_n(z)t^n$ or by a contour integral $P\_n(z)=\frac1{2\pi i}\oint\frac{g(t,z)}{t^{n+1}}dt$.
Are there known sufficient conditions on $g$ or on the $P\_n$ themselves that guarant... | https://mathoverflow.net/users/29783 | When can a family of polynomials get a weight function to be made orthogonal? | Favard's theorem characterizes this in terms of the three-term recurrence. Suppose the polynomials $P\_n$ are normalized so that they are monic. Then they are orthogonal polynomials with respect to some Borel measure if and only if there are constants $\alpha\_n$ and $\beta\_n$ such that $P\_n(x) = (x+\alpha\_n) P\_{n-... | 10 | https://mathoverflow.net/users/4720 | 86867 | 51,587 |
https://mathoverflow.net/questions/86807 | 9 | Let $(X,d)$ be compact metric space of curvature greater than $-1$ (in the sense of comparison triangles), assume that its Hausdorff dimension is $2$. Then a result of Perelman says that $X$ is a 2-dimensional manifold.
Claim
-----
$(X,d)$ can be Gromov-Hausdorff approximated by a sequence of Riemannian surfaces $(... | https://mathoverflow.net/users/8887 | Smoothability of compact Alexandrov surfaces with curvature bounded from below | *Edit: Addressing Igor's comment I'd like to correct the references I gave. The correct reference for the exact argument I sketch should be the original book by Alexandrov ["Intrinsic Geometry of Convex Surfaces"](http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=CNO&s1=29518)(Chapter 7, section 6, Lemmas 1-3).... | 11 | https://mathoverflow.net/users/18050 | 86868 | 51,588 |
https://mathoverflow.net/questions/86454 | 3 | Let $G=PSL(n,q)$ be the projective linear group over $\mathbb{F}\_q$ and let $\sigma$ be an **outer** automorphism of $G$. (The description of outer automorphism group of $PSL(n,q)$ is well-known, see for example [Wilson's
book](http://books.google.com.ar/books?id=lYMAg_Sj7hUC&printsec=frontcover&dq=finite+simple+group... | https://mathoverflow.net/users/17845 | Sizes of twisted conjugacy classes of $PSL(n,q)$ | My inclination at first is to be skeptical: Is there any numerical evidence?. The setting of the question is perhaps nonstandard, since for finite groups of Lie type the starting point for this kind of twisting has more often been the ambient algebraic group. Much of this is influenced by papers and lecture notes of Sp... | 2 | https://mathoverflow.net/users/4231 | 86870 | 51,590 |
https://mathoverflow.net/questions/86674 | 13 | The following question is Problem 1.1.2.c in Thurston's book "Three-dimensional geometry and topology". I have not managed to solve it despite quite a bit of effort.
One can obtain a 2-dimensional torus $T$ by identifying the sides of a hexagon in an appropriate way (see, for example, [here](http://en.wikipedia.org/w... | https://mathoverflow.net/users/20887 | Embedding torus in space such that its 6-fold symmetry extends | Lurking here on MO, I've noticed that unanswered questions get bumped to the top periodically. Since this question was answered by Ryan Budney in the comments, I've decided to write his answer here (marked "community wiki" so I get no reputation points) to prevent this from happening.
The answer is no for both $\math... | 17 | https://mathoverflow.net/users/20887 | 86880 | 51,592 |
https://mathoverflow.net/questions/86889 | 4 | Is the non-principal ultraproduct of finite fields $\prod\_p \mathbb{F}\_p/\sim$ a nonstandard model of the rationals $\mathbb{Q}$?
EDIT: Can we realize $\mathbb{Q}^\*$ as an ultraproduct?
| https://mathoverflow.net/users/nan | Ultraproduct $\prod_p \mathbb{F}_p/\sim$ and $\mathbb{Q}^*$ | It is easy to see that at least one of $-1,2,-2$ is a square in that field: the set of primes where neither $-1$ nor $2$ is a quadratic residue is contained in the set of primes where $-2$ is a quadratic residue.
| 7 | https://mathoverflow.net/users/2035 | 86893 | 51,596 |
https://mathoverflow.net/questions/86750 | 6 | According to Rajendra Bhatia in his book Fourier Series, Weyl's Equidistribution Theorem states that if $x$ is an irrational number, then for every subinterval $[a, b]$ of $(0, 1)$ we have
$\lim\_{n \rightarrow \infty} \frac{1}{N}\operatorname{card}\{k : 1 \leq k \leq N, \tilde{(kx)} \in [a, b]\} = b - a$ where $\til... | https://mathoverflow.net/users/15666 | Weyl's Equidistribution Theorem and Measure Theory | This is a very interesting question, which actually asks about the interplay between equidistribution (or harmonic analysis if you would like to call it that way) and ergodic theory.
As Vaughn mentioned, for any $L^{p}$ function ($p\geq 1$), the pointwise ergodic theorem would imply that for Lebesgue almost every poi... | 3 | https://mathoverflow.net/users/8857 | 86907 | 51,600 |
https://mathoverflow.net/questions/86891 | 3 | Hi everybody,
let $X$ be a smooth projective variety of dimension $n$ and let $f:\mathbb{P}^1 \rightarrow X$ be a morphism.
A theorem of Grothendieck says that the vector bundle $f^{\*}T\_X$ splits as a sum of line bundles, hence we can write $f^{\*}T\_X \cong \bigoplus\_{i=1}^{n}\mathcal{O}\_{\mathbb{P}^1}(a\_i)$ ... | https://mathoverflow.net/users/15415 | Decomposition of $f^{*}T_X$ for a morphism $f:\mathbb{P}^1 \rightarrow X$ | The obvious place to look for these kind of issues is János Kollár's [Rational curves on algebraic varieties](http://books.google.com/books?id=oqW3GabJLjgC&printsec=frontcover#v=onepage&q&f=false).
As Jason and JC point out, this is not true as stated. However, there is indeed something resembling this that might be ... | 6 | https://mathoverflow.net/users/10076 | 86917 | 51,606 |
https://mathoverflow.net/questions/86905 | 2 | Is it possible to show that there is a simple formula, preferably existential, that characterizes a nonstandard model of the ring of integers among the elements of $\prod\_p \mathbb{F}\_p/\mathcal{U}$?
Thank you
| https://mathoverflow.net/users/nan | Defining $\mathbb{Z}^*$ in $\prod_p \mathbb{F}_p/\mathcal{U}$ (or pseudo-finite fields) | There is no first-order definable subring of the ultraproduct $\Pi\_p\mathbb{F}\_p/U$ satisfying the theory of $\mathbb{Z}$. Indeed, every definable subset of $\Pi\_p\mathbb{F}\_p/U$ containing $1$ and closed under addition is the whole of $\Pi\_p\mathbb{F}\_p/U$, regardless of the complexity of the definition. To see ... | 8 | https://mathoverflow.net/users/1946 | 86918 | 51,607 |
https://mathoverflow.net/questions/86906 | 9 | I'm just asking because I'm curious.
I was seeking references on the following problem, that a friend exposed to me last holidays :
Problem
-------
Given $n$ red points and $n$ blue points in the plane in general position (no 3 of them are aligned), find a pairing of the red points with the blue points such that th... | https://mathoverflow.net/users/8887 | Complexity of matching red and blue points in the plane. | The Ghosts and Ghostbusters problem can be solved in $O(n\log n)$ time, which is considerably faster than the $O(n^2\log n)$-time algorithm suggested by CLRS.
The [ham sandwich theorem](https://en.wikipedia.org/wiki/Ham_sandwich_theorem) implies that there is a line $L$ that splits both the ghosts and the ghostbuster... | 11 | https://mathoverflow.net/users/6710 | 86922 | 51,611 |
https://mathoverflow.net/questions/86932 | 16 | First-time here... I hope my question isn't silly or anything... anyway...
Consider the category of chain complexes and chain maps. We can also define chain homotopies between chain maps. Does this form a 2-category? I am able to construct vertical and horizontal composition of chain homotopies but am unable to prove... | https://mathoverflow.net/users/20961 | A 2-category of chain complexes, chain maps, and chain homotopies? | Part of the problem is that you're not using a convenient definition of chain homotopy. I'll use a differential going down in degree. Let $I$ be the [interval object](http://ncatlab.org/nlab/show/interval+object) in the category of chain complexes; that is, $I$ is $\text{span}(0, 1)$ in degree $0$ and $\text{span}(e)$ ... | 9 | https://mathoverflow.net/users/290 | 86933 | 51,614 |
https://mathoverflow.net/questions/84682 | 1 | We have financial some data (500-1000 samples), which is not normally distributed (well known fact from the literature). I have some ideas to do parametric transformations of this data (using some other data) to produce "adjusted" series. My goal is to find a transformation that makes the series normally distributed (w... | https://mathoverflow.net/users/3160 | Normality tests | The Anderson-Darling test is considered one of the best tests for normality, I think.
| 1 | https://mathoverflow.net/users/15411 | 86934 | 51,615 |
https://mathoverflow.net/questions/86894 | 17 | Let $\Sigma$ be a compact connected oriented surface and $p:\tilde{\Sigma}\to\Sigma$ a finite regular cover.
Consider the set $\Gamma$ of simple closed curves on $\tilde{\Sigma}$ obtained as a connected component of $p^{-1}(\gamma)$ where $\gamma$ is a simple curve in $\Sigma$.
My question is: is it true that $\G... | https://mathoverflow.net/users/14547 | Homology generated by lifts of simple curves | As far as I know, this is open.
In fact, I think the following weaker question is open.
Let $\Theta$ be the set of loops $\gamma$ in $\widetilde \Sigma$ such that the image of $\gamma$ in $\Sigma$ is not a filling curve. If $\Sigma$ is not a pair of pants, is $H\_1(\widetilde \Sigma ; \mathbb{Z})$ generated by $\T... | 17 | https://mathoverflow.net/users/1335 | 86938 | 51,617 |
https://mathoverflow.net/questions/86923 | 14 | This question is about the elements in a localization $S^{-1} A$ of a commutative ring $A$. Is it possible to derive $\frac{a}{1} = 0 \in S^{-1} A \Rightarrow \exists s \in S : sa = 0$ only using the universal property of $S^{-1} A$? In order to make this question clear enough I will have to digress a little bit.
**E... | https://mathoverflow.net/users/2841 | Elements in a localization - category theoretic approach | If you want to understand $S^{-1}A$ for any $S\subset A$, you may write $S$ as a filtered union of its finite subsets $S\_i$, and it is clear from the universal properties that
$$S^{-1}A=\varinjlim\_i \ S\_i^{-1}A$$
Therefore it is sufficient to consider the case where $S$ consists of a finite set of elements of $A$. I... | 12 | https://mathoverflow.net/users/1017 | 86939 | 51,618 |
https://mathoverflow.net/questions/86930 | 3 | Considering the path algebra of the quiver $\mathbb{A}\_n$, it is well known its Auslander-Reiten quiver with the canonical orientation of $\mathbb{A}\_n$, that is, with all the arrows from, say, left to right. I can find as examples in several texts the AR-quivers of $\mathbb{A}\_n$ with other orientations.
QUESTION... | https://mathoverflow.net/users/20947 | An algorithm for constructing the AR-quiver of a path algebra corresponding to a change in the orientation. | The algorithm for constructing the AR-quiver of any orientation of $A\_n$ is the same as the algorithm for constructing the AR-quiver of the orientation you describe. Start out by figuring out all the irreducible maps between the indecomposable projectives. Then write out the cokernels, and then the cokernels of the ne... | 1 | https://mathoverflow.net/users/5323 | 86940 | 51,619 |
https://mathoverflow.net/questions/86915 | 3 | Let $M$ be a von Neumann algebra and $\varphi$ be a normal weight on it,
and let $A$ be its definition subalgebra. We still denote $\varphi$
the extension to $A$ as a linear positive functional.
It is known that $\varphi$ is lower-ultraweakly-semicontinuous on $M^+$ (the positive elements
of $M$).
Questions:
* Is... | https://mathoverflow.net/users/20756 | Continuity of a weight on its definition domain in a von Neumann algebra | 1) if $A=M$ the assertion is indeed classical: normal states are exactly those ultraweakly continuous. But consider the case where $M=B(H)$ and $\varphi$ is the trace. Then the definition subalgebra $A$ is exactly the trace-class operators. Let $\{p\_k\}\subset A$ be a maximal net of pairwise orthogonal projections of ... | 5 | https://mathoverflow.net/users/3698 | 86941 | 51,620 |
https://mathoverflow.net/questions/71014 | 5 | I was using the built-in functions for Root Systems in SAGE, and I noticed that the Cartan Matrices for Type $B\_n$ and type $C\_n$ are interchanged from what I thought they would be, i.e. following the Plates in the back of Bourbaki's *Lie Groups and Lie Algebras, vol. 4-6*.
Are there different conventions for choo... | https://mathoverflow.net/users/339 | Cartan Matrices of type B and C. | This question (which I overlooked for a long time) reflects a natural notational confusion but is easy to answer. The Cartan integers themselves are unambiguous for each root system, but the meaning of the two *indices* used in writing $c\_{i,j}$ is conventional and is reversed in some sources.
For types $B,C$ that rev... | 10 | https://mathoverflow.net/users/4231 | 86960 | 51,630 |
https://mathoverflow.net/questions/86947 | 6 | For a (bounded) double complex (of abelian groups or vector spaces) one can consider two spectral sequences that converge to the cohomology of the totalization: one can first compute either the cohomology of rows, or the cohomology of columns. Suppose that one of these spectral sequences degenerates at $E\_1$ (i.e. the... | https://mathoverflow.net/users/2191 | On two spectral sequences for the cohomology of a double complex | There is a basic way to see whether things like this should be true. Any bounded double complex of vector spaces over a field $k$ is (noncanonically) the direct sum of complexes of the following two sorts:
**Squares:** $$\begin{matrix} k & \rightarrow & k \\ \uparrow & & \uparrow \\ k & \rightarrow & k \end{matrix}$$... | 28 | https://mathoverflow.net/users/297 | 86971 | 51,634 |
https://mathoverflow.net/questions/80519 | 3 | Let $\rho$ be an irreducible representation of a group $N$, and let $G,H$ be groups with $N$ of finite index in $H$ and $H$ normal in $G$. Let $\pi=\rho^H$ be the induced representation of $\rho$ to $H$; I'd like to understand the isotropy of $\pi$ in $G$, that is,
$I\_G(\pi)=\lbrace g\in G:\pi^g\sim \pi\rbrace$, thos... | https://mathoverflow.net/users/10481 | Isotropy (aka inertia) of induced representation | The "guess" is wrong; here is a counterexample. Take $G$ to be dihedral of order 16. Let $H$ be one of the two copies of the dihedral group of order 8 in $G$, and let $N$ be one of the two copies of the Klein fours group in $H$. Let $\pi$ be the unique irreducible character of degree $2$ of $H$, and let $\rho$ be one o... | 8 | https://mathoverflow.net/users/9694 | 86982 | 51,642 |
https://mathoverflow.net/questions/86991 | 3 | Let us first remark that all of this takes place on the boundary of $\Delta^n$. The question that I wanted to solve that led to the question in the title is as follows: Let $f:\Delta^{n-1}\hookrightarrow\Delta^n$ be an injective map. What I would have liked to do is to extend that map to a map, $\tilde{f}:\partial\Delt... | https://mathoverflow.net/users/14167 | What is the chromatic number of the graph whose vertices dimension $n-2$ subsimplicies of $\Delta^n$ and an edge between two vertices is given if the two associated $n-2$ vertices are contained in the same $n-1$ subsimplex? | It is known that the complete graph $K\_{2m}$ has a 1-factorization
(e.g., <http://en.wikipedia.org/wiki/Graph_factorization>). This means
that its set of edges can be written as a disjoint union of $2m-1$
complete matchings $M\_1, \dots, M\_{2m-1}$, each with $m$ edges. If $e$
is an edge of $K\_{2m}$ regarded as a 2-e... | 5 | https://mathoverflow.net/users/2807 | 86999 | 51,648 |
https://mathoverflow.net/questions/86949 | 2 | Let $X$ be a complex irreducible quasi-projective variety, $f:X\longrightarrow\mathbb{P}^N$ a morphism, $H\subset\mathbb{P}^N$ a hyperplane, $Z:=f^{-1}(H)$ which is irreducible, $Y\subset X$ a irriducible closed subset.
Clearly we have $f(Y)\cap H=f(Y\cap Z)$, is it true that $\overline{f(Y)}\cap H=\overline{f(Y\cap Z)... | https://mathoverflow.net/users/15606 | A simple question on the closure of the image of a morphism | Actually, $\overline{f(Y)}\cap H$ and $\overline{f(Y\cap Z)}$ don't even have to be of the same dimension:
Let $X=Y=\mathbb A^2$ with coordinates $x,y$ and $f:X\to \mathbb P^2$ the morphism $(x,y)\mapsto [x:xy:1]$. Further let $x\_0,x\_1,x\_2$ denote the homogenous coordinates on $\mathbb P^2$ and let $H=Z(x\_0)$. T... | 2 | https://mathoverflow.net/users/10076 | 87000 | 51,649 |
https://mathoverflow.net/questions/86965 | 4 | The [proof of the Wigner Semicircle Law](http://www.aimath.org/conferences/ntrmt/talks/Mezzadri3.pdf) comes from studying the GUE Kernel
\[ K\_N(\mu, \nu)=e^{-\frac{1}{2}(\mu^2+\nu^2)} \cdot \frac{1}{\sqrt{\pi}} \sum\_{j=0}^{N-1}\frac{H\_j(\lambda)H\_j(\mu)}{2^j j!} \]
The eigenvalue density comes from setting $\mu = \... | https://mathoverflow.net/users/1358 | Traceless GUE : Four Centered Fermions | Traceless GUE was studied by Tracy and Widom in their paper "On the distributions of the lengths of the longest monotone subsequences in random words", Probab. Theory Relat. Field 119, 350-380 (2001). In Section 4.4 of that paper they basically show (actually for the largest eigenvalues, but the same argument applies t... | 6 | https://mathoverflow.net/users/13034 | 87019 | 51,658 |
https://mathoverflow.net/questions/87024 | 1 | A coherent sheaf $V$ on a say noetherian scheme is called reflexive if the canonical map $V \rightarrow \mathcal Hom\_{\mathcal O\_X}(\mathcal Hom\_{\mathcal O\_X}(V,\mathcal O\_X),\mathcal O\_X)$ is an isomorphism of sheaves.
In principle, one can define this notion also for quasicoherent sheaves, and this is what m... | https://mathoverflow.net/users/18183 | Criterions for Reflexiveness of sheaves and a special case | I'll add another example to the mix even in the ring setting (as opposed to the sheaf setting).
Fix a DVR $(R, \langle x \rangle)$. Then the fraction field $K(R) = \bigcup\_n (x^{-n}R)$, is an ascending union of free (reflexive) modules. But clearly $K(R)$ is not itself reflexive.
On the other hand, under mild condit... | 4 | https://mathoverflow.net/users/3521 | 87035 | 51,668 |
https://mathoverflow.net/questions/87022 | 5 | Let $S\_{g,1}$ be a orientable compact surface of genus $g$ with one boundary component and $\Gamma\_{g,1}$ the mapping class group.
By $F\_n$ I denote the free group on $n$ generators.
One obtains a representation $\rho: \Gamma\_{g,1} \rightarrow Aut(F\_{2g})$.
What is the kernel of $\rho$?
| https://mathoverflow.net/users/20990 | Kernel of the representation of the mapping class group to $Aut(F_n)$ | The representation is faithful, since a mapping class is determined by its action on the fundamental group of the surface. A surface is a $K(\pi,1)$, so given any element $Aut(S\_{g,1})$, one obtains a (pointed) map $\varphi:S\_{g,1}\to S\_{g,1}$ which is unique up to homotopy. Now one needs to know that two homotopic ... | 7 | https://mathoverflow.net/users/1345 | 87043 | 51,671 |
https://mathoverflow.net/questions/87045 | 7 | I have several related questions, i do not know which one is more important to me, i think it would depend on their answers.
1. Is it true that the Euler characteristic of a finite connected aspherical simplicial 2-complex cannot be greater than 1?
2. If $A$ is a finite simplicial 2-complexe that retracts by deformat... | https://mathoverflow.net/users/20995 | Is the Euler characteristic of aspherical connected 2-complexes at most 1? (No!) What can be said about subcomplexes of 2-complexes deformation retractible onto graphs. | This answer concerns Question 1. Take a free group on $2$ generators and $k$ very long relations at random. Then, with high probability, the representation satisfies a small cancellation condition and the associated 2-complex $X$ is aspherical. Moreover, if this happens you also have $\chi(X) = k-1$.
A negative answe... | 11 | https://mathoverflow.net/users/8176 | 87051 | 51,672 |
https://mathoverflow.net/questions/86985 | 8 | In Lurie's "A Survey of Elliptic Cohomology", he writes on page 14 that if $A$ is an ordinary commutative ring considered as an $E\_\infty$-ring, then $A$-module spectra are the same thing as objects of the derived category of $A$-modules. This is mysterious to me. On the one hand, to an $A$-module spectrum $M$ we migh... | https://mathoverflow.net/users/303 | If $A\in \mbox{Rings}\subset E_\infty\mbox{-rings}$, what is the equivalence between objects of $\mathcal{D}(\mbox{Mod}_A)$ and $A$-module spectra? | This question already has been answered in the comments.
(Tilson) We regard a commutative ring as an $E\_\infty$ spectrum via the EM functor $H$. This is definitely what Jacob is doing. One could also use associative rings and $A\_\infty$ spectra for what follows.
(Wilson) Many of the correspondences between algeb... | 8 | https://mathoverflow.net/users/8818 | 87058 | 51,675 |
https://mathoverflow.net/questions/87071 | 0 | Let $C$ be the set of points $(a,b,c,d) \in \mathbb{C}^4$ which satisfy
1) $ \left|a\right|^2+\left|c\right|^2=\left|b\right|^2+\left|d\right|^2 =1 $.
2) $ a\bar{b}+c\bar{d}=0 $
There is a (component-wise) $S^1$ action on $C$ and let $S$ be the quotient ($S$ is a 3-manifold).
Is $S$ orientable or not ?
Thanks... | https://mathoverflow.net/users/5259 | Is this manifold orientable? | Identifying your points with matrices $M$ with column vectors $(a,b)^T$ and $(c,d)^T$, your equations come from the components of $M M^\dagger=I$ where $M^\dagger$ denotes the conjugate transpose. So $C$ is $U(2)$ and $S$ is $SU(2)$. Since $SU(2)$ is diffeomorphic to $S^3$, it is orientable.
| 5 | https://mathoverflow.net/users/19731 | 87077 | 51,684 |
https://mathoverflow.net/questions/87083 | 2 | The free group on two generators $F\_2=\langle x,y|\rangle$ is the fundamental group of $\mathbb P^1(\mathbb C)\setminus\{0,1,\infty\}$. Now, there are plenty of galois covers of this space whose galois group is not solvable. Thus the "maximal solvable cover" (i.e. the limit over all galois covers with solvable galois ... | https://mathoverflow.net/users/35353 | Explicit element in free group which is killed by every solvable quotient | There are no such elements -- the intersection of the derived series of a free group is trivial. In fact, even more is true -- the intersection of the lower central series of a free group is trivial. This is a theorem of Magnus, and by now there are many proofs. The classical one is in the final chapter of Magnus-Karas... | 20 | https://mathoverflow.net/users/317 | 87084 | 51,688 |
https://mathoverflow.net/questions/87085 | 2 | This is from CM Rings (Bruns & Herzog) p 64 2.1.20. Show that a one dimensional Noetherian local ring has a maximal Cohen Maucaulay module.
| https://mathoverflow.net/users/13874 | maximal Cohen-Macaulay module | Take $R/\mathfrak{p}$, where $\mathfrak{p}$ is a minimal prime ideal of $R$.
| 3 | https://mathoverflow.net/users/16046 | 87086 | 51,689 |
https://mathoverflow.net/questions/87070 | 38 | Suppose a finite-dimensional Lie group $G$ is given. Does there exist a connected manifold $M$ and a Riemannian metric $g$, such that $G$ is the **full** isometry group of $(M,g)$?
For example if I try to do this for a connected $G$, then I often get a bigger group as the full isometry group, which includes e.g. the ... | https://mathoverflow.net/users/20999 | Can every Lie group be realized as the full isometry group of a Riemannian manifold? | The article of de Groot is the one cited here: [What kind group can be realized as a Isometry group of some space?](https://mathoverflow.net/questions/24255/what-kind-group-can-be-realized-as-a-isometry-group-of-some-space)
That every compact group is the full isometry group of a compact Riemannian manifold is shown ... | 36 | https://mathoverflow.net/users/11142 | 87101 | 51,693 |
https://mathoverflow.net/questions/87039 | 7 | It seems that when we talk about the $p$-adic uniformization, we typically mean those uniformized by either the Drinfel'd upper spaces (for which we think of the examples of Mumford curves and some local studies of Shimura varieties), or the Tate uniformizaation of abelian varieties with multiplicative reduction (which... | https://mathoverflow.net/users/9246 | $p$-adic uniformization not from the Drinfel'd spaces? | There is another type of uniformization introduced in Mochizuki's book *Foundations of $p$-adic Teichmüller theory*. It uses curves equipped with nilpotent indigenous bundles.
I don't see what local contractibility has to do with non-existence of simply connected spaces. The finite étale covers of the affine line and... | 3 | https://mathoverflow.net/users/121 | 87103 | 51,695 |
https://mathoverflow.net/questions/87047 | 1 | Although it is known the solution to the first two questions, somebody may have different nice answers, so I include them:
1. Given two circles in the plane, there is (at least) a line which is tangent to both of them.
2. Given three spheres in the space, there is a plane which is tangent to all of them.
3. In genera... | https://mathoverflow.net/users/20947 | Tangent lines to 2 circles, tangent planes to 3 spheres, and so on. | This answers expands on my comment on the original question.
**1.** *Given two circles in the plane, there is (at least) a line
which is tangent to both of them*:
this is not true unless we allow lines with complex coefficients
(and even then there's an exception, see below). In the real plane,
two circles have:
$\... | 4 | https://mathoverflow.net/users/14830 | 87111 | 51,698 |
https://mathoverflow.net/questions/87108 | 7 | Is there a systematic/standard way of extracting the nef cone of a toric variety $X$ from its fan (or polytope)? Can I tell from the basis of $A^1(X)$ induced by the one-skeleton, based on coefficients, when a given divisor class is nef?
In particular I am working working with blowups of $\mathbb{P}^n$. I am uncertai... | https://mathoverflow.net/users/17350 | nef Cone of a Toric Variety | You can use the fact that a divisor class $D$ on a toric variety is nef if and only if it has non-negative intersection with the finitely many classes of torus invariant curves. If you are using the torus invariant divisors as the basis for $A^1(X)$, then these numbers are easy to compute combinatorially and this will ... | 12 | https://mathoverflow.net/users/3996 | 87122 | 51,703 |
https://mathoverflow.net/questions/87091 | 2 | I was wondering if anyone knows where I can find a formula for the dimension of an irreducible module of highest weight $\Lambda$ expressed only in terms of the Young diagram corresponding to $\Lambda$.
Thank you.
| https://mathoverflow.net/users/21007 | Dimension of $\mathfrak{sl}_n$ modules | One influential older source to consult is the concise Springer Lecture Notes No. 682 by Gordon James *The Representation Theory of the Symmetric Groups* (1978). Section 26 applies the symmetric group theory to general linear groups, using the language of "Weyl modules" and partitions. See in particular his Theorem 26.... | 3 | https://mathoverflow.net/users/4231 | 87139 | 51,714 |
https://mathoverflow.net/questions/87147 | 3 | Let $U$ be the set of unipotent upper triangular matrices and $B$ the upper triangular matrices of $GL\_3$. How could I describe $GL\_3/U$ ? Using coordinates, in a projective or an affine space.
For example, I already know the identification of $SL\_2/U$ with $\mathbb{A}^2 \setminus (0,0)$ and the identification of ... | https://mathoverflow.net/users/15404 | Description of $GL_3/U$ | Let $V$ be the basic (3-dimensional) representation of $GL(3)$. Then $SL(3)/U$ is the set of all pairs $x\in V, y\in V^\*$ where $x$ and $y$ are non-zero and $(x,y)=0$.
The quotient $GL(3)/U$ is non-canonically product of the above by $C^{\times}$. Canonically,
you need to choose non-zero $x\_i\in \Lambda^i(V)$ (for... | 4 | https://mathoverflow.net/users/3891 | 87165 | 51,723 |
https://mathoverflow.net/questions/84074 | 24 | Godel's undecidable sentences in first-order arithmetic were guaranteed to be true, by construction. But are there examples of specific sentences known to be undecidable in first-order arithmetic whose truth values *aren't* known? I'm thinking, by contrast, of the situation in set theory: CH is undecidable in ZFC, but ... | https://mathoverflow.net/users/3092 | undecidable sentences of first-order arithmetic whose truth values are unknown | **Update.** I've improved the argument to use only the consistency of $T$. (2/7/12): I corrected some over-statements previously made about Robinson's Q.
---
I claim that for every statement $\varphi$, there is a variant way
to express it, $\psi$, which is equivalent to the original
statement $\varphi$, but which... | 22 | https://mathoverflow.net/users/1946 | 87168 | 51,726 |
https://mathoverflow.net/questions/87174 | 8 | Suppose $X$ and $Y$ are spectra (or homotopy classes thereof) such that $X$ is p-local and $Y$ is q-local, for primes $p\neq q$. Is it indeed true then, and if so how would one show that $[X,Y]\_\ast=0$?
Thanks!
| https://mathoverflow.net/users/11546 | Absence of Maps Between p-local and q-local spectra | The rational Eilenberg-Mac Lane spectrum $H\mathbb{Q}$ is $p$-local for every prime $p$, but certainly $[H\mathbb{Q}, H\mathbb{Q}]\_\*\neq 0$.
If you replace "$p$-local" and "$q$-local" with "$p$-complete" and "$q$-complete", then your conclusion does hold.
| 12 | https://mathoverflow.net/users/437 | 87179 | 51,731 |
https://mathoverflow.net/questions/87188 | 25 | Is the property of not containing the free group on two generators invariant under quasi-isometry? Amenability is, so if there is a counterexample it is also a solution to the von Neumann-Day problem (which of course already has a solution).
| https://mathoverflow.net/users/10774 | Is the property of not containing $\mathbb{F}_2$ invariant under quasi-isometry? | It is a famous open problem. Akhmedov in MR2424177 claimed he could prove that the answer is "no". No proof exists, so I guess he discovered a gap in his argument.
| 23 | https://mathoverflow.net/users/nan | 87194 | 51,738 |
https://mathoverflow.net/questions/60968 | 4 | Is the circle externally tangent to the three excircles of an irregular non-Euclidean triangle internally tangent to the incircle of the triangle, the tangent point being a generalized Feuerbach point? In Euclidean plane geometry, the circle externally tangent to the excircles of an irregular triangle is internally tan... | https://mathoverflow.net/users/14207 | Is there a generalized Feuerbach point for an irregular non-Euclidean triangle? | Akopyan pointed out to me that Hart proved this. Hart's 1861 article can be found at <http://books.google.com/books?id=y9xEAAAAcAAJ&pg=PA260#v=onepage&q&f=false>. Akopyan's article in translation appears at <http://arxiv.org/pdf/1105.2153.pdf>. My dynamic illustration appears at <http://demonstrations.wolfram.com/NonEu... | 2 | https://mathoverflow.net/users/14207 | 87195 | 51,739 |
https://mathoverflow.net/questions/87157 | 9 | I know the definition of *absolute continuity* if there is a function $f:(a,b)\rightarrow R$.
I wonder what is an analogy of this concept if we have a function $f:A\rightarrow R$, where $A\subset R^{n}$ is open set.
| https://mathoverflow.net/users/15946 | Absolute continuity on $R^{n}$ | I guess it may depend on exactly which property of absolutely continuous functions you think is most important to keep, or to put it another way, exactly which definition you prefer in one dimension. For me the most commonly useful property of absolutely continuous functions is that they map sets of Lebesgue measure ze... | 10 | https://mathoverflow.net/users/5701 | 87198 | 51,741 |
https://mathoverflow.net/questions/87201 | 4 | Undoubtedly, these terms play essential roles in (pure) mathematics. My problem is that I have feelings what they mean in different fields, such as, differential geometry (abstract manifolds vs. embedded ones), algebraic geometry (more down to earth, the study of Riemann surfaces and algebraic curves) when our objects ... | https://mathoverflow.net/users/13351 | Intrinsic vs. Extrinsic | Intrinsic properties are those which are invariant under isomorphism, whatever that notion happens to mean in the category under consideration.
Edit: I guess I would say also that an extrinsic property of an object is not a property of the object itself but a property of the object together with some other data, for ... | 7 | https://mathoverflow.net/users/703 | 87203 | 51,743 |
https://mathoverflow.net/questions/87202 | 12 | While doing some work in geometric representation theory I have come across the following
sequence of polynomials in two variables $(q,x)$ which I would like to denote
by $n!\_{q,x}$. For small $n$ these polynomials look as follows:
$2!\_{x,q}=x+q$
$3!\_{x,q}=x^3+x^2(q^2+q)+x(q^2+q)+q^3$
$$
4!\_{x,q}=x^6+x^5(q^3... | https://mathoverflow.net/users/3891 | $(q,x)$-analog of $n!$ | I was hesitating to write an answer since I don't have references at hand but let me mention that if you denote your polynomials $P\_n(x,q)$ and look at $Q(x,q)=x^{\binom{n}{2}}P\_n(x^{-1},q)$ then (my guess is that) you are looking at:
$$Q(x,q)=\sum\_{\pi\in S\_n}x^{maj(\pi)}q^{inv(\pi)}=\sum\_{\pi\in S\_n}x^{maj(\pi)... | 17 | https://mathoverflow.net/users/2384 | 87209 | 51,746 |
https://mathoverflow.net/questions/87214 | 2 | Hi fellows,
I was wondering. Is the axiom of choice used to show that $\mathbb{R}$ is complete? If yes, is there a way to construct monotonic bounded sequences that do not converge?
Thanks in advance!
| https://mathoverflow.net/users/20103 | Axiom of choice and convergence | The standard construction(s) of $\mathbb R$ do not use the Axiom of Choice. Therefore one cannot construct a bounded monotone sequence that does not converge.
Maybe, it worths to say that there are also constructions of $\mathbb R$ that makes use of AC. See for instance <http://en.wikipedia.org/wiki/Construction_of_... | 3 | https://mathoverflow.net/users/13809 | 87216 | 51,751 |
https://mathoverflow.net/questions/87219 | 1 | Let $R$ be a local ring, $m$ its maximal ideal and $k:= R/m$ its residue field.
Suppose that a finite cyclic group $G= \mathbb{Z}/ m \mathbb{Z}$ has a linear nontrivial action on $R$.
Let $R^G$ be a ring of invariant elements of $R$ by this action.
Let $E:= E\_R(k)$ be an injective hull of $k$
**Question 1** Is... | https://mathoverflow.net/users/12390 | Injective hulls of residue fields of a local ring and its ring invariants by finite group action | In the case of a hypersurface of dimension $d$, or any Gorenstein singularity of dimension $d$, $E \cong H^d\_{\mathfrak{m}}(R)$ (of course, this isomorphism is up to multiplication by a unit). $G$ should act on $H^d\_{\mathfrak{m}}(R)$ directly (you should even be able to do this explicitly via Cech cohomology). This ... | 1 | https://mathoverflow.net/users/3521 | 87226 | 51,755 |
https://mathoverflow.net/questions/87049 | 3 | I would like to know if there are explicit formulas for the Hodge-Deligne structure or the Hodge-Deligne polynomials for quotients X/G for finite groups G acting on a (smooth, projective) scheme X.
The only formula that I know is lemma 2.6 on this paper
<http://arXiv.org/abs/math/0701642v1>
by Munoz, Ortega and ... | https://mathoverflow.net/users/11060 | Are there any known formulas about the Hodge-Deligne structure of quotients by actions of groups? | It's hard to give a very precise answer to such a general question. But perhaps I
can try to complement the answers already given with some more specific examples/tricks.
As Sándor and algori have pointed out, $X/G$ is essentially smooth for what you seem
to be after.
In principle there are many known formulas, althoug... | 4 | https://mathoverflow.net/users/4144 | 87231 | 51,757 |
https://mathoverflow.net/questions/87228 | 4 | Consider the Poincare half plane model for the n-dimensional hyperbolic space $\mathbb{H}^n$.
$\mathbb{H}^n$ can be constructed out of $\mathbb{R}^{n-1}$ by crossing it with $(0;\infty)$ and equpping the product with the following metric:
Let $\gamma=(\gamma\_1,\gamma\_2)$ be a path $[0;t]\rightarrow \mathbb{R}^{n-1... | https://mathoverflow.net/users/3969 | Hyperbolizing geodesic spaces | This construction is called "parabolic cone" and indeed it turns CAT(0) spaces into CAT(-1). See [this paper](http://www.math.uiuc.edu/~sba/wp.pdf) of Alexander and Bishop.
| 7 | https://mathoverflow.net/users/1441 | 87233 | 51,759 |
https://mathoverflow.net/questions/87224 | 15 | Let $\mathcal{A}$ be a $C^\ast$-algebra. Consider vector space of matrices of size $n\times n$ whose entries in $\mathcal{A}$. Denote this vector space $M\_{n,n}(\mathcal{A})$. We can define involution on $M\_{n,n}(\mathcal{A})$ by equality
$$
[a\_{ij}]^\*=[a\_{ji}^\*],\qquad\text{where}\quad [a\_{ij}]\in M\_{n,n}(\mat... | https://mathoverflow.net/users/19593 | Matrices with entries in a $C^*$-algebra | For $x=(x\_i)\_{i=1}^n, y=(y\_i)\_{i=1}^n \subseteq A$ define $(x,y) = \sum\_i x\_i y\_i^\* \in A$, and set $\|x\| = \|(x,x)\|^{1/2}$.
>
> Lemma: We have that $(x,y)^\* (x,y) \leq \|x\|^2 (y,y)$ the order in the C$^\*$-algebra sense.
>
>
> Proof: (Copied from Lance's Hilbert C$^\*$-module book). Wlog $\|x\|=1$. F... | 15 | https://mathoverflow.net/users/406 | 87251 | 51,768 |
https://mathoverflow.net/questions/87250 | 1 | Hi
I have a proof for a Lemma which splits into an odd and even case.
The proof for the even case was already published by someone else in a different context and the proof for for the odd case is very similar (but not trivial) to the even case proof.
So how should I now proceed about the odd case proof?
Is it ok... | https://mathoverflow.net/users/44243 | Reusing Parts of a Proof | Mathematics often progresses by small changes in already extant work. In fact I like to work by writing and rewriting, trying to make things clear, in the first place to me.
You need only say, for example, that the proof for the even case given by X can with some non trivial modifications also work for the odd case.... | 5 | https://mathoverflow.net/users/19949 | 87252 | 51,769 |
https://mathoverflow.net/questions/86896 | -1 | Hi everyone.
I'd like to refer you to two papers by Arian Berdellima on odd perfect numbers:
More Properties About Odd Perfect Numbers
<http://mpra.ub.uni-muenchen.de/31587/1/MPRA_paper_31587.pdf>
Perfect numbers - a lower bound for an odd perfect number
<http://mpra.ub.uni-muenchen.de/31218/1/MPRA_paper_3121... | https://mathoverflow.net/users/10365 | Question Re: Arian Berdellima's Papers On Odd Perfect Numbers | Dear Arnie,
In the first paper, "More properties about the odd perfect numbers", there is a flow in the statement of Little Fermat Theorem. The modular congruences that I have used should also satisfy p=1 (mod q) and q(i)=1(mod q(j)) in order for the results to be applicable. In the second paper there is no error as ... | 3 | https://mathoverflow.net/users/21058 | 87253 | 51,770 |
https://mathoverflow.net/questions/87220 | 4 | I am an arithmetic geometry graduate student, and I find myself needing to learn about factorisation in orders in division algebras. I know something aout algebraic number theory and commutative algebra, but very little about noncommutative rings.
Let $R$ be a Dedekind domain and $K$ its field of fractions.
The follo... | https://mathoverflow.net/users/1046 | Prime ideals in maximal orders (1- and 2-sided) | No. The problem is that $\Lambda$ might have too few units. Here is an example that illustrates this point.
Let $\Lambda$ be the ring of Hurwitz quaternions. This is the subring of the usual quaternions $\mathbb{R} \oplus \mathbb{R} i \oplus \mathbb{R}j \oplus \mathbb{R}k$, freely generated as an abelian group by the... | 4 | https://mathoverflow.net/users/6827 | 87262 | 51,776 |
https://mathoverflow.net/questions/86652 | 0 | Hi all, I was wondering whether you could help me understand a part in a proof that is assumed to be an "obvious detail" by the author, but that I can't wrap my head around. Essentially, as far as I can understand, there are two values $A$ and $B$, and we want to prove that:
$\Bigg|A-B \; \Bigg| > \frac{1}{p}$
for... | https://mathoverflow.net/users/18322 | Problem with making an estimate when values of many variables are unknown? | If we truly have (as you say in the comments) that $0 \leq \alpha,\beta, A, B \leq 1$ then
$$
\frac2p \leq |\alpha A - \beta B| \leq 1
$$
implying that $p \geq 2$.
This in turn means that $\frac{A}{3p^2} \leq \frac1{3p^2} \leq \frac1{6p}$.
Letting
$$
\epsilon=\beta-\alpha
$$
and using
$$
\left| \beta \cdot A - \b... | 0 | https://mathoverflow.net/users/20665 | 87266 | 51,777 |
https://mathoverflow.net/questions/81710 | 3 | Let $S/k$ be a smooth variety and $A/S$ be an abelian scheme. Let $Z \hookrightarrow S$ be a reduced closed subscheme of codimension $\geq 2$.
I want to show that in this situation, $H^i\_Z(S, A) = 0$ for $i \leq 2$ (étale cohomology).
Note that we have a long exact sequence
$\ldots \to H^i\_Z(S, A) \to H^i(S, ... | https://mathoverflow.net/users/nan | vanishing of étale cohomology groups with small support with values in an abelian scheme | This can be proved for $\mathcal{A} = \mathrm{Pic}\_{\mathcal{C}/S}$ using [vanishing of cohomology sheaves with supports and values in the multiplicative group](https://mathoverflow.net/questions/87156/vanishing-of-cohomology-sheaves-with-supports-and-values-in-the-multiplicative-gr) and the Leray spectral sequence (m... | 0 | https://mathoverflow.net/users/nan | 87267 | 51,778 |
https://mathoverflow.net/questions/87238 | 23 | I've come across several references to MK (Morse-Kelley set theory), which includes the idea of a proper class, a limitation of size, includes the axiom schema of comprehension across class variables (so for any $\phi(x,\overline y)$ with $x$ restricted to sets, there a class $X=(x : \phi(x,\overline y))$).
I have se... | https://mathoverflow.net/users/15735 | Morse-Kelley set theory consistency strength | Let me give an easier (sketch of an) answer to the part of the question about proving Con(ZFC) in MK. Unlike Emil's answer, the following does not cover the case of arbitrary finitely axiomatized subtheories of MK. Intuitively, there's an "obvious" argument for the consistency of ZFC: All its axioms are true when the v... | 24 | https://mathoverflow.net/users/6794 | 87268 | 51,779 |
https://mathoverflow.net/questions/86853 | 16 | What is the relation between Lafforgue's result on Langlands
and Frenkel-Gaitsgory-Vilonen ? ( <http://arxiv.org/abs/math/0012255> , <http://arxiv.org/abs/math/0204081> )
Does one imply other ? If not why ?
More technical:
Do FGV work only with unramified Galois irreps (Seems Yes) ? If Yes, is it difficult to cover ... | https://mathoverflow.net/users/10446 | What is the relation between L. Lafforgue and Frenkel-Gaitsgory-Vilonen results on Langlands correspondence ? | Let me try to answer. [FGV] is only about unramified representations of the Galois group
but they prove a stronger fact in this case (existence of certain "automorphic sheaf"). Lafforgue's result doesn't follow from there for several reasons:
a) Formally [FGV] use Lafforgue, but this was actually taken care of by a l... | 13 | https://mathoverflow.net/users/3891 | 87273 | 51,782 |
https://mathoverflow.net/questions/87248 | 6 | I would like to know if it is always possible to find a one-dimensional ideal in a local commutative ring... actually I am interested in finding a curve through a point on a scheme (locally). If the ring is of finite dimension it should be obvious, but does anybody know about the situation in more general rings?
| https://mathoverflow.net/users/18305 | Are there one-dimensional ideals in any local ring | Given your comment about curves I suppose by a one dimensional ideal you mean an ideal such that the ring mod this ideal is one-dimensional.
The answer is this: if you assume your ring to be noetherian yes, if not no.
Let $(A,m)$ be a local ring.
**Case 1**: $A$ is noetherian. Take the set of prime ideals other t... | 17 | https://mathoverflow.net/users/10076 | 87275 | 51,783 |
https://mathoverflow.net/questions/87279 | 9 | Let $K \subseteq{\mathbb{S}}^3=∂\mathbb{D}^4$ be a non-trivial slice knot, i.e. $K$ bounds a slice disk $\Delta$ in $\mathbb{D}^4$. Let $N(\Delta)$ be a regular neighborhood of $\Delta$ in $\mathbb{D}^4$.
>
> What is known about $\pi\_i(\mathbb{D}^4-N(\Delta))$, for $i\geq2$?
>
>
>
For what it's worth, $\pi\_... | https://mathoverflow.net/users/14006 | Higher homotopy groups of slice disk complement | The homotopy groups can be pretty big things. For example, your $D^4 - N(\Delta)$ class of spaces contains the class of all $2$-knot complements -- simply remove a 4-ball neighbourhood of $S^4$ that intersects the $2$-knot in an unknotted disc.
$2$-knot complements have fairly complicated homotopy groups. For exampl... | 9 | https://mathoverflow.net/users/1465 | 87280 | 51,786 |
https://mathoverflow.net/questions/87276 | 2 | Let $R\subseteq\mathbb{R}^2$. Consider the set of all "horizontal sections" $H\_R =$ {$ Rb|b\in\mathbb{R}$} where $Rb=${$ a\in\mathbb{R} | (a,b)\in R$}. Similarly consider the set of "vertical sections" of $R$, $V\_R =${$ aR|a\in\mathbb{R}$} where $aR=${$ b\in\mathbb{R} | (a,b)\in R$}. Now define the equivalence relati... | https://mathoverflow.net/users/20947 | An equivalence relation on the power set of the plane. | The equivalence class of the closed unit disk $ \{(x,y): x^2 + y^2 \le 1 \}$
consists of sets $S = \{(x,y) \in [-1,1] \times [-1,1]: |y| \le f(|x|)\}$ where
$f$ is a decreasing homeomorphism from $[0,1]$ onto $[0,1]$.
| 3 | https://mathoverflow.net/users/13650 | 87288 | 51,788 |
https://mathoverflow.net/questions/87285 | 6 | I am currently work on a senior project trying to prove for semisimple Lie groups, $R(T)$ is a free module over $R(G)$ by computing an explicit basis for all the A,B,C,D cases. The canoical reference is a paper by Pittie( H.V. Pittie: Homogeneous vector bundles on homogeneous spaces, Topology II (1972) 199-203), but I ... | https://mathoverflow.net/users/5175 | Reference Request: Steinberg's 1975 paper "On a paper of Pittie"(retrieved) | To supplement Barry's citations, I'd point out that the journal *Topology* was at that time managed by a company which eventually gave up on it after editors resigned partly in protest against the high prices charged. While the online rights now belong to the ScienceDirect conglomerate, it's expensive to access. This c... | 5 | https://mathoverflow.net/users/4231 | 87296 | 51,790 |
https://mathoverflow.net/questions/87281 | 4 | Is it true that the Moore spectrum for the group $\mathbb{Z}\_{(p)}$ can be constructed by smashing $\mathbb{S}$ with $q^{-1}\mathbb{S}$ for each $q\neq p$ (here both $q$ and $p$ are primes). It seems we might wish to show this by showing that $[\mathbb{S},\mathbb{S}\_{(p)}\wedge H\mathbb{Z}]\_\ast\cong[\mathbb{S},H\ma... | https://mathoverflow.net/users/11546 | A Model for the Moore Spectrum of $\mathbb{Z}_{(p)}$ | This is true, if you define an infinite smash product as a colimit of finite smash products.
If you define a Moore spectrum for the abelian group $A$ to be a spectrum $X$ such that $X\wedge H\mathbb Z=HA$, then obviously $\mathbb S$ is a Moore spectrum for $\mathbb Z$. An arbitrary abelian group can be obtained from ... | 7 | https://mathoverflow.net/users/20233 | 87299 | 51,792 |
https://mathoverflow.net/questions/87291 | 5 | I find myself in the following situation:
I have a sequence of first quadrant spectral sequences, let's call them $ E(n)\_{p,q}^\* $, each convergent to $E(n)\_{p,q}^\infty$, with spectral sequence morphisms $E(n)\_{\*,\*}^\* \to E(n-1)\_{\*,\*}^\*$, so we have an inverse directed system of spectral sequences.
Each... | https://mathoverflow.net/users/17353 | Inverse limit of spectral sequences | The inverse limit of directed systems of locally finite dimensional graded vector spaces is an exact functor. That is why it takes directed systems of spectral sequences to spectral sequences of the inverse limits.
In the case of the first quadrant spectral sequences, the notion of convergence does not involve taking... | 11 | https://mathoverflow.net/users/2106 | 87313 | 51,796 |
https://mathoverflow.net/questions/87309 | 22 | I have come across a sequence of representations $V\_n$ of the symmetric group $S\_{n+2}$ which has the property that restricting the action $S\_n \subset S\_{n+2}$ gives the regular representation:
$$ Res^{S\_{n+2}}\_{S\_n} V\_n = \mathbb{Q}S\_n. $$
In other words, there is some natural way to give the regular rep of ... | https://mathoverflow.net/users/9068 | An n!-dimensional representation of the symmetric group S_{n+2} | 1. Yes your series of representations looks (except for the first term - but there must be the law of small numbers lurking around) like the sign representation times the *Whitehouse module*, see, e.g. [these slides by Richard Stanley](http://www-math.mit.edu/~rstan/transparencies/whouse.ps). Basically, the Whitehouse ... | 19 | https://mathoverflow.net/users/1306 | 87315 | 51,797 |
https://mathoverflow.net/questions/87323 | 3 | Consider the Hilbert space $L^2[0, 1]$ of square integrable functions on $[0, 1]$.
Saying more carefully, there is one subtle detail: this space is defined as factor space by the functions which are have norm zero, in particular if we change value of $f(x)$ at one point, then in this factor space the two functions will... | https://mathoverflow.net/users/10446 | How to extend evaluation at a point from continuous maps to square-integrable ones? | We can't extend the evaluation at $x$ to a continuous linear functional on $L^2[0,1]$, because it is not continuous on $C([0,1])$ wrto the $L^2$ norm. For instance $f\_n(t):=(1-nt)\_+$ has $f\_n(0)=1$ and $\|f\_n\|\_2\to 0$.
| 4 | https://mathoverflow.net/users/6101 | 87327 | 51,802 |
https://mathoverflow.net/questions/87326 | 3 | Is it possible to show that there is a simple formula, preferably existential, that characterizes a nonstandard model of the ring of integers among the elements of $\prod\_p \mathbb{F}\_p(t)/\mathcal{U}$ where $\mathcal{U}$ is a nonprincipal ultrafilter over the prime numbers?
Thank you
| https://mathoverflow.net/users/nan | Defining $\mathbb{Z}$ in $\prod_p \mathbb{F}_p(t)/\mathcal{U}$ | No, such a formula does not exist. If $R$ is a definable subset of $\prod\_p\mathbb F\_p(t)/\cal U$ which contains $1$ and is closed under addition, then by the argument Joel David Hamkins gave in [his answer](https://mathoverflow.net/questions/86918#86918) to your previous question, we must have $R\supseteq S:=\prod\_... | 4 | https://mathoverflow.net/users/12705 | 87334 | 51,804 |
https://mathoverflow.net/questions/87336 | 0 | Let $A$ be an unbounded self-adjoint operator with spectrum $\sigma(A)=\mathbb R$ in a Hilbert space $\mathcal H$. Let $P$ be a bounded operator in $\mathcal H$ satisfying $P\ge1$ and
$$
{\rm Domain}(AP)
\equiv\big[\varphi\in\mathcal H:P\varphi\in{\rm Domain}(A)\big]
={\rm Domain}(A).
$$
Finally, suppose that the opera... | https://mathoverflow.net/users/21080 | Spectrum of the operator PAP, with A self-adjoint and P strictly positive | The only reasonable condition I can think of is that $P = g(A)$ where $g \ge 1$ is a bounded continuous function on $\mathbb R$.
It is instructive to consider the case where $A$ is the multiplication operator $(A f)(x) = x f(x)$ on $L^2({\mathbb R})$ and $P = g(A)$ for some bounded measurable function $g \ge 1$. Then... | 0 | https://mathoverflow.net/users/13650 | 87349 | 51,809 |
https://mathoverflow.net/questions/87343 | 6 | For a closed smooth $n$-manifold ($n\ne 4$), the Lipschitz structure is unique by the result of Sullivan. How about the Alexandrov spaces?
At first I was thinking it is trivially true by induction, since locally distance ball in an Alexandrov space is homeomorphic to cone over it's space of directions, which is also ... | https://mathoverflow.net/users/3922 | Is the Lipschitz unique for close Alexandrov space other than dimension 4? | First, I want to point out that unlike for manifolds there is no notion of a geometric Lipschitz structure for general Alexandrov spaces as they are not locally modeled on a fixed space topologically. One can still ask whether you can have two Alexandrov spaces which are homeomorphic but not bi-Lipschitz homeomorphic b... | 8 | https://mathoverflow.net/users/18050 | 87353 | 51,812 |
https://mathoverflow.net/questions/86805 | 2 | Coproducts of modules over an algebraic monad $\Sigma$ are described in Section 4.16.14/15 in Durov's [thesis](http://arxiv.org/abs/0704.2030). It is claimed there that for $\Sigma$-modules $M,N$, the set $M \coprod N$ generates $M \oplus N$ and that this implies that every element of $M \oplus N$ may be written as $t(... | https://mathoverflow.net/users/2841 | Coproducts of modules over an algebraic monad | I think the point is that an operation in an algebraic theory (even a noncommutative one) need not preserve the order of its inputs. There is a binary operation in the theory of groups which takes the input $(g,h)$ to the product $h g$. More generally, the symmetric group on $n$ letters acts on the set of $n$-ary opera... | 2 | https://mathoverflow.net/users/49 | 87355 | 51,813 |
https://mathoverflow.net/questions/87350 | 17 | The Algebraic Hartogs Lemma states that in a Noetherian normal scheme, a rational function that is regular outside a closed subset of codimension at least two, is in fact regular everywhere.
In a research problem I was working on recently, I was (following suggestions by my advisor) using this to prove that a particu... | https://mathoverflow.net/users/5094 | A Relative Algebraic Hartogs Lemma | Actually, a very similar statement can be found in the paper [Reflexive pull-backs and base extension](http://www.ams.org/journals/jag/2004-13-02/S1056-3911-03-00331-X/) by Brendan Hassett and myself. See Proposition 3.5. Indeed you do not need normality, only that the fibers are $S\_2$ and the sheaf does not need to b... | 14 | https://mathoverflow.net/users/10076 | 87360 | 51,815 |
https://mathoverflow.net/questions/87358 | 2 | Suppose that $Y/k$ is a an algebraic variety over a field $k$ of characteristic zero and
that $Y\subseteq X$ is a closed embedding into a smooth variety over $k$. Then the completion
of the de Rham complex of $X/k$ along $Y$ is independent (up to quasi-isomorphism) of $X$.
This is proven in Hartshorne's paper on de Rha... | https://mathoverflow.net/users/36285 | algebraic de Rham cohomology functoriality | I don't think they are homotopic in general. If I remember correctly, the argument of independence is as follows: consider $p\_1, p\_2 : X\times X\to X$. Then maps $g\_1, g\_2 : X'\to X$ induce a map $g : X'\to X\times X$. Since $g = (f, f)$ when restricted to $Y'$, $g$ induces a map $g^\* : \hat{\Omega^{\*}\_{X\times ... | 2 | https://mathoverflow.net/users/36285 | 87370 | 51,821 |
https://mathoverflow.net/questions/81539 | 12 | Let W be a Coxeter group with Coxeter generators S. The corresponding 0-Hecke monoid H(W) has generating set S, the braid relations of W and the relations that each element of S is an idempotent. If one specializes the Hecke algebra associated to W to q=0 one gets the monoid algebra of H(W) (replace generators by their... | https://mathoverflow.net/users/15934 | Is the following construction of the 0-Hecke monoid (well) known? | Here is one such reference:
Representation and classification of Coxeter monoids
by S. V. Tsaranov
European Journal of Combinatorics, Volume 11 Issue 2, Mar. 1990
<http://dl.acm.org/citation.cfm?id=84891>
| 6 | https://mathoverflow.net/users/21089 | 87371 | 51,822 |
https://mathoverflow.net/questions/87368 | 2 | Reading through a text on computability I came across a c.e. set defined as follows: Let $K=\lbrace x \in W\_x \rbrace$ and let $f$ be a computable function. Then there exists $n \in \omega$ such that $W\_n=\lbrace x \in K : x \leq f(n) \rbrace$.
My question is how can we find such an index $n$? My first thought was ... | https://mathoverflow.net/users/20921 | Is this c.e. set obtained via the Recursion Theorem? | Taking the point of view (as in Rogers's book) that an index for a c.e. set $A$ amounts to a program that converges on exactly those inputs that are in $A$, here's an informal description of the program $g(k)$: "On input $x$, first check whether $x\leq f(k)$; if not, diverge (say by going into a loop). If so, start run... | 7 | https://mathoverflow.net/users/6794 | 87374 | 51,824 |
https://mathoverflow.net/questions/87364 | 1 | I was looking up some stuff when I stumbled across S unit equations. It seems to me that they are quite helpful in number theory, as given in this paper.
<http://faculty.nps.edu/pstanica/research/fiboprimeProcAMS.pdf>
Here, the authors prove that there are only a finite number of Fibonacci numbers that are the sum ... | https://mathoverflow.net/users/17614 | A good introduction to S unit equations | Lang: Fundamentals of diophantine geometry,Ch. 8, or Bombieri and Gubler: Heights in Diophantine Geometry Ch. 5.
| 5 | https://mathoverflow.net/users/2290 | 87375 | 51,825 |
https://mathoverflow.net/questions/87199 | 5 | I have a time-inhomogeneous Galton-Watson binary branching process over a finite number of generations $n$. In each generation $i$, there is a probability $p\_i$ of a child surviving; so each node has 2 children with probability $p\_i^2$, 1 child with probability $2 p\_i (1-p\_i)$, and zero children with probability $(... | https://mathoverflow.net/users/9896 | Branching process survival probability | A new answer for the new version of the question.
Under the constraint that $p\_i$ are decreasing and $\mu\_n\ge 1$, the minimal survival probability is obtained when all $p\_i=1/2$. You can see this by showing that for any level $j$, if you fix all the $p\_i$ except for $p\_j$ and $p\_{j+1}$ then the minimum is obta... | 2 | https://mathoverflow.net/users/1061 | 87381 | 51,828 |
https://mathoverflow.net/questions/87271 | 9 | For a category $C$, let $C-Set$ denote the category of set-valued functors $\delta\colon C\to Set$. Given categories $C$ and $D$, and a functor $F\colon C\to D$, composition with $F$ yields a functor that I'll denote by $$\Delta\_F\colon D-Set\longrightarrow C-Set.$$ The functor $\Delta\_F$ has both a left adjoint, whi... | https://mathoverflow.net/users/2811 | When do functors induce monadic adjunctions to presheaf categories | I will try to give an answer to your first question.
The functor $\Delta\_F$ verifies automatically nearly all the conditions of the monadicity theorem:
* it is a right adjoint;
* it is a left adjoint with cocomplete domain, and thus coequalizers exist in the source and are preserved by $\Delta\_F$.
It remains to... | 9 | https://mathoverflow.net/users/21095 | 87391 | 51,831 |
https://mathoverflow.net/questions/87347 | 27 | The group of Higman:
$
\langle
\
a\_0, a\_1, a\_2, a\_3 \ | \
a\_0 a\_1 a\_0^{-1}=a\_1^2,
\ a\_1 a\_2 a\_1^{-1}=a\_2^2,
\ a\_2 a\_3 a\_2^{-1}=a\_3^2,
\ a\_3 a\_0 a\_3^{-1}=a\_0^2
\ \rangle .
$
Is it simple? What is actually known about it except the fact that it does not have non-trivial homomorphims into a finit... | https://mathoverflow.net/users/8699 | The Higman group | Higman's group is not simple. Indeed, if you look at Higman's paper, you will see that his group is an amalgamated product of two groups $K\_{1,2}=\langle a\_1, a\_2, b\_2\mid a\_1^{-1}a\_2a\_1=a\_2^2, a\_2^{-1}b\_2a\_2=b\_2^2\rangle$ and $K\_{3,4}=\langle a\_3, a\_4, b\_4\mid a\_3^{-1}a\_4a\_3=a\_4^2, a\_4^{-1}b\_4a\_... | 23 | https://mathoverflow.net/users/nan | 87395 | 51,833 |
https://mathoverflow.net/questions/87369 | 2 | The MacWilliams identities relate the weight enumerators of a code and its dual code. I treated the version for linear codes in my combinatorics class, but felt unsatisfied because I didn't have a use for them (and judging from the feedback, some students shared this feeling).
So... do you know of any neat applicatio... | https://mathoverflow.net/users/4438 | What are the key applications of the MacWilliams identities in coding theory? | There is a very nice proof that there is no projective plane with order 6 mod 8 in Assmus, E. F., Jr.; Maher, David P. Nonexistence proofs for projective designs. Amer. Math. Monthly 85 (1978), no. 2, 110–112. This uses the weight enumerator.
| 5 | https://mathoverflow.net/users/1266 | 87399 | 51,835 |
https://mathoverflow.net/questions/87407 | 0 | Let (R,m) be a CM ring of dimension d, and M a finite R- module. Is pd M always finite?
| https://mathoverflow.net/users/13874 | pd finite for finite module over local CM ring? | No. Put $k=R/m$. Then $k$ is of finite projective dimension if and only if $R$ is regular. This is the famous theorem of Auslander-Buchsbaum, Serre (see for instance Bruns-Herzog Cohen-Macaulay rings theorem 2.2.7 for a proof). So the residual field of a non-regular CM ring will give a counter-example.
| 4 | https://mathoverflow.net/users/2284 | 87411 | 51,840 |
https://mathoverflow.net/questions/87410 | 10 | Here's a challenge for elliptic curve descenders/programmers. It seems no public software or public tables can determine if the rank is zero for the following curves (over rational x,y):
```
y^2 = x^3 - 9122*x + 106889
y^2 = x^3 - x^2 - 42144*x + 66420
y^2 = x^3 - x^2 - 168615*x + 21827700
y^2 = x^3 - 210386*x +... | https://mathoverflow.net/users/20757 | Specific Elliptic Curves: Rank | Your elliptic curves $E$ all (provably) satisfy $L(E,1) \neq 0$, so by Kolyvagin's theorem , they have rank $0$. You can prove that $L(E,1) \neq 0$ by using the command *ellanalyticrank* in Pari/GP (there are similar commands in Magma and Sage).
By the way, your first elliptic curve has conductor 223960, so is likely... | 20 | https://mathoverflow.net/users/6506 | 87412 | 51,841 |
https://mathoverflow.net/questions/87413 | 1 | Let $p:\mathbb{R}^n\to[0,1)^n$ be the map defined by $p(x\_1,\ldots,x\_n)=(\{x\_1\},\ldots,\{x\_n\})$, where $\{\cdot\}$ is the fractional part operator. Experimentation suggests that if $S \subseteq \mathbb{R}^n$ is semialgebraic, then the closure of the image set $p(S)$ is semialgebraic. (By a "semialgebraic" subset ... | https://mathoverflow.net/users/5229 | Is the closure of a semialgebraic set mod 1 also semialgebraic? | For $n=2$, the set $S=\{(x,y)\in \mathbb R^2| xy=1\}$ provides a counterexample.
For $n=1$, the statement is true.
| 3 | https://mathoverflow.net/users/5690 | 87419 | 51,844 |
https://mathoverflow.net/questions/87424 | 1 | Let $f$ be a $C^1$ functional on a Hilbert space $X$, and $Y$ a closed subspace of $X$.
Suppose the restriction of $f$ on $Y$ has a critical point $x\_0 \in Y$.
Q: Is $x\_0$ a critical point of $f$?
| https://mathoverflow.net/users/14361 | Critical points in Hilbert space | Especially in Calculus of Variations and Mechanics, a submanifold $Y$ of a manifold $X$ is usually called "a natural constraint" for a functional $f$ on $X$, if the special circumstance that you are considering does happen: constrained critical points of $f$ on $Y$ are free critical points: $\operatorname{crit} (f\_{|Y... | 3 | https://mathoverflow.net/users/6101 | 87431 | 51,848 |
https://mathoverflow.net/questions/87427 | 22 | I'm trying to understand where the definition of simplicial sheaf on a space/site comes from.
For a presheaf $F$ of sets on a topological space $X$, the sheaf condition can be viewed as saying that $F$ is a 'local object' with respect to the maps of presheaves $colim(\coprod U\_{ij} \underrightarrow{\rightarrow} \co... | https://mathoverflow.net/users/21028 | Necessity of hypercovers for sheaf condition for simplicial sheaves | Actually, for simplicial sheaves, and to be more accurate infinity sheaves of infinity groupoids, you do not need hypercovers. Your "naive" idea about multiple intersections (actually fibered products) is correct. If you instead use hypercovers, you get the notion of a \*hyper\*sheaf. Both infinity sheaves and hypershe... | 16 | https://mathoverflow.net/users/4528 | 87436 | 51,852 |
https://mathoverflow.net/questions/87433 | 2 | In an article by Lubich, I came across a decomposition for points on the straight line between two points lying in an embedded submanifold $M$ of $R^{n}$.
To be precise, it is proposed that for $X$, $\tilde X \in M$ and $\tau$ small enough, any point $X + \tau(\tilde X -X)$ may be decomposed into $$ X + \tau(\tilde X... | https://mathoverflow.net/users/21107 | Decomposition of straight line between points on a manifold | In fact, any point $\tilde x$ near $x\in M$ can be decomposed like that. You can see this as follows: Locally $M\subset \mathbb{R}^n$ can be given as a graph: w.l.o.g. $x=0$ and there is a small ball $U$ around $x$ and a function $f\colon U\cap V\to W$ where $V\oplus W=\mathbb{R}^n$ is an decomposition into orthogonal ... | 2 | https://mathoverflow.net/users/4572 | 87441 | 51,855 |
https://mathoverflow.net/questions/87421 | 1 | Hello
I am trying to find the asymptotic expansion for $t\_1,t\_2 \to 0+$ of the function
$$ F\_{w\_0,\tau}(t\_1,t\_2) = \sum\_{w \in \mathbb{Z}\tau+\mathbb{Z}} w \ \operatorname{exp}(- |w|^2t\_1 - |w\_0-w|^2t\_2) $$
where we may assume that $w\_0 \in \mathbb{Z}\tau +\mathbb{Z}$. The obvious first step would pro... | https://mathoverflow.net/users/21102 | Asymptotic expansion of Theta type sum | You can use the [Poisson summation formula](http://dlmf.nist.gov/1.8#P8). Your summand is essentially a Gaussian, $\sim \exp(-Q(m,n))$, where $w=m+\tau n$ and $Q$ is a quadratic polynomial in $m$ and $n$. The Poisson summation formula converts this sum into a sum with a different essentially Gaussian summand, $\sim \ex... | 1 | https://mathoverflow.net/users/2622 | 87443 | 51,856 |
https://mathoverflow.net/questions/87442 | 4 | This question came up in a practical situation at work...
Given a set S of size n, what is the minimum number of subsets of S of size k s.t. each pair of elements of S occurs in the same subset at least once?
| https://mathoverflow.net/users/21110 | minimum number of subsets? | If you want each $l$-tuple of elements to occur at least once then you can do this with $(1 + o(1))\binom{n}{l}/\binom{k}{l}$ sets, which is within $1 + o(1)$ of optimal by a simple counting argument. This is proven using something called the Rödl Nibble. Most likely when $l = 2$, which is the situation you're interest... | 7 | https://mathoverflow.net/users/5575 | 87444 | 51,857 |
https://mathoverflow.net/questions/51533 | 4 | Is there a low-tech way to compute the chern class of the theta bundle, i.e. the bundle associated to theta divisor? Here, the theta divisor is the analytic set of holomorphic line-bundles in $Pic\_{g-1}(\Sigma)$ which possesses a non-trivial holomorphic section. By low-tech I mean without much knowledge about algebrai... | https://mathoverflow.net/users/4572 | Riemann's theorem on theta | The locus of holomorphic line bundles in $Pic\_{g-1}(\Sigma)$ with a nontrivial holomorphic section is equivalently characterized as the image of $u\_{g-1} : Sym^{g-1} \Sigma \to Pic\_{g-1}(\Sigma)$ under the Abel-Jacobi map. In section 4 of chapter 1 of the book of Arbarello-Cornalba-Griffiths-Harris, the theta diviso... | 4 | https://mathoverflow.net/users/83 | 87447 | 51,859 |
https://mathoverflow.net/questions/87346 | 2 | To use classical Ito formula
\begin{equation}
f(t,B\_t) - f(0,B\_0) = \int\limits\_0^t f'\_s(s,B\_s)ds + \frac 12\int\limits\_0^t f''\_{xx}(s,B\_s)ds + \int\limits\_0^t f'\_x(s,B\_s)dB\_s
\end{equation}
$f(t,x)$ needs to be $C^{1,2}([0,\infty)\times\mathbb R)$.
Is there any possibility to use it if $f(t,x)$ is piece... | https://mathoverflow.net/users/19988 | Ito formula for discontinuous function | If $f$ is continuous in $t$ that still works. If not, you need to add the term $\sum\_i \Delta\_t f(t\_i,B\_{t\_i})$ to the right hand side where $\Delta\_t f(t,B\_t) = f(t\_+,B\_t)-f(t\_-,B\_t)$ and the sum goes over those $i$ where $t\_i \le t$. I guess.
| 1 | https://mathoverflow.net/users/21111 | 87451 | 51,860 |
https://mathoverflow.net/questions/87449 | 1 | We say that a matrix $A\in M\_n(\mathbb{C})$ is a conditionaly negative definite matrix if it is hermitian and if for all complex numbers $c\_1,\ldots,c\_n$ such that $c\_1+\cdots +c\_n=0$ we have
$$
\sum\_{j,k=1}^{n}c\_j\overline{c\_k}a\_{jk}\leq 0.
$$
I'm interested by non-trivial families $(A\_n)\_{n\in \mathbb{N}... | https://mathoverflow.net/users/5210 | references for families of conditionaly negative definite matrices | Please have a look at: [my answer here](https://mathoverflow.net/questions/44777/how-to-characterize-real-square-matrices-a-such-that-vav-0-for-all-real-vec/44789#44789)---there you will find several references, from which you can gather a list of nontrivial cnd matrices (especially, the nontrivial ones that arise from... | 3 | https://mathoverflow.net/users/8430 | 87453 | 51,861 |
https://mathoverflow.net/questions/87429 | 4 | In view of Mariano Suárez-Alvarez's answer I see how badly phrased my question was, and decided to rewrite it. The drawback is that some comments of Martin Brandenburg are now incomprehensible, but I thought it would suffice to say here that Martin made some legitimate constructive criticisms to the original wording of... | https://mathoverflow.net/users/461 | Generators of a certain ideal | A polynomial $f\in K[\underline Y]$ is in the kernel of your map iff $f$ is zero in the quotient $$\frac{k[X,Y]}{\bigl((X\_i-X\_j)Y\_{i,j}-1:1\leq i<j\leq n\bigr)}.$$In other words, your kernel is the intersection of the ideal in the denominator with the ring $k[Y]$, $$\ker\varepsilon=k[Y]\cap\bigl((X\_i-X\_j)Y\_{i,j}-... | 4 | https://mathoverflow.net/users/1409 | 87464 | 51,863 |
https://mathoverflow.net/questions/87461 | 3 | This question was originally asked in stackoverflow (<https://math.stackexchange.com/questions/103941/every-positive-polynomial-is-above-a-completely-q-factorized-positive-polynomial>) but as it has remained without further feedback for a week I migrate it here.
Let $P$ be a unitary polynomial with rational coefficie... | https://mathoverflow.net/users/2389 | Every positive polynomial with rational coefficients is above a completely Q-factorized nonnegative polynomial ? | The function has no real roots, so all its roots are complex numbers, in conjugate pairs. Thus we can factor it into terms of the form $((x-a\_i)^2+b\_i)$. First consider the product of all the $(x-a\_i)^2$. This will be strictly smaller than the original polynomial. The difference will have degree $2d-2$. Our challeng... | 7 | https://mathoverflow.net/users/18060 | 87466 | 51,865 |
https://mathoverflow.net/questions/87455 | 3 | I was wondering whether it is not very difficult to see the following:
for a nonconstant irreducible polynomial $p(x) \in \mathbb{Q}[x]$ does there always exist a polynomial $q(x) \in \mathbb{Q}[x]$ of degree at least $2$ such that the composition $p(q(x))$ is irreducible in $\mathbb{Q}[x]$?
Thank you,
Albertas
| https://mathoverflow.net/users/10591 | Irreducibility of compositions of polynomials | Yes, there does. In fact, we will show that a polynomial $q(x) = x^2 + d$ works for some $d \in \mathbb{Q}$. The conclusion will follow from [Hilbert's Irreducibility Theorem](http://en.wikipedia.org/wiki/Hilbert%27s_irreducibility_theorem) once we show that $p(x^2 + d)$ is irreducible (as a polynomial in two variables... | 7 | https://mathoverflow.net/users/5498 | 87467 | 51,866 |
https://mathoverflow.net/questions/87459 | 6 | If $K$ is an ordered field, let $B$ be the subring comprising the $x \in K$ such that $|x| \le n$ for some $n \in N$, and let $I$ be the ideal of $B$ comprising the infinitesimal elements (i.e. the $x \in K$ such that $|x| \le 1/n$ for every non-zero $n \in N$). Then $I$ is a maximal ideal and the order on $K$ induces ... | https://mathoverflow.net/users/8187 | "Archimedeanising" an ordered field | Yes, the map is sometimes called the standard part map, and it turns out that one map define it in general for a group in an o-minimal structure. See [here](http://math.haifa.ac.il/kobi/Wroclaw_proc.pdf), for example. Regarding the splitting, if it should only respect the additive group structure, I think it exists sin... | 5 | https://mathoverflow.net/users/10174 | 87469 | 51,868 |
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