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https://mathoverflow.net/questions/191651 | 3 | This a repost of a question I [asked](https://math.stackexchange.com/questions/1061534/trying-to-prove-a-congruence-for-stirling-numbers-of-the-second-kind) at Stack Exchange, but I got no answer so far, so I am trying here, even though it may not suit the "research level" requirement.
Proposition: When $n$ and $m$ a... | https://mathoverflow.net/users/64384 | Trying to prove a congruence for Stirling numbers of the second kind | Here is a quick generating function argument. Start with the following lemma
>
> **Lemma:** The power series $$\frac{1}{(1-x)(1-2x)\cdots (1-kx)}$$
> is even $\pmod{k+1}$ when $k+1$ is odd, and even $\pmod{\frac{k+1}{2}}$ when $k+1$ is even.
>
>
>
*Proof*: By grouping terms, when $k+1$ is odd we have $$\frac{... | 5 | https://mathoverflow.net/users/2384 | 191655 | 94,210 |
https://mathoverflow.net/questions/189183 | 9 | My colleague and I are currently teaching "true infinitesimal calculus" (TIC), in the sense of calculus with infinitesimals, to a class of about 120 freshmen at our university, based on the book by Keisler <https://www.math.wisc.edu/~keisler/calc.html>. Two of my colleagues in Belgium are similarly teaching TIC at two ... | https://mathoverflow.net/users/28128 | Which universities teach true infinitesimal calculus? | Since you mentioned University of Hawaii --
[David Ross](http://www.math.hawaii.edu/~ross/) regularly teaches nonstandard analysis and infinitesimal calculus at University of Hawaii in various forms, for instance
* [MATH 649K](http://www.math.hawaii.edu/~ross/649.htm) (a regular grad course), Spring 2008
* [MATH 6... | 3 | https://mathoverflow.net/users/4600 | 191667 | 94,216 |
https://mathoverflow.net/questions/191666 | 10 | There are many quantities in enumerative combinatorics that grow roughly exponentially, like the Fibonacci numbers, the Catalan numbers, and the factorials; indeed, most of the functions that arise in pre-19th century combinatorics -- even ones as large as the function $n^{n-2}$, which counts spanning trees of the comp... | https://mathoverflow.net/users/3621 | Cubic-exponential enumerative combinatorics | Since you mentioned Cayley's theorem for spanning trees, I believe one important example is its higher dimensional analogue due to Kalai. Indeed the number of simplicial spanning trees of the k-skeleton of an n-simplex is $$f(n)=n^{\binom{n-2}{k}} \implies \lim\_{n\to \infty}\frac{\log\log f(n)}{\log n}=k.$$
Some fur... | 15 | https://mathoverflow.net/users/2384 | 191670 | 94,217 |
https://mathoverflow.net/questions/191669 | 3 | I want to model the problem of household formation by a finite number of individuals, each of whom has preferences over sets of housemates. A collection of households is **unstable** if there is a set of individuals who can all be made happier by leaving their current households and forming a new household together.
... | https://mathoverflow.net/users/10503 | Stable Household Formation | This problem has been studied under the name hedonic coalition formation, and your stability notion is called core stability. See this [survey](http://link.springer.com/chapter/10.1007/978-3-642-35843-2_4) by Gerhard Woeginger, [arxiv:1212.2236](http://arxiv.org/abs/1212.2236).
| 7 | https://mathoverflow.net/users/12674 | 191671 | 94,218 |
https://mathoverflow.net/questions/191672 | 3 | What is the co-rank of a group $$G=\langle a\_1,a\_2,\dots,a\_h\mid a\_1^2a\_2^2\dots a\_h^2=1\rangle,$$ that is, finitely generated group with $h$ generators and one relation?
By co-rank, I mean the maximum rank $n$ of a free homomorphic image $F\_n$ of the group.
I suspect that the answer is $corank\ G=\left[\fr... | https://mathoverflow.net/users/49372 | Co-rank of a group with $a^2b^2c^2=1$ (fundamental group of non-orientable surface) | If you know that corank is the cut number, then it is $[h/2]$. Each nonseparating two-sided circle is a handle. You can remove them one by one (remove the circle and patch the two holes with disks), each time reducing $h$ by $2$.
Or, even better, remove all $r$ circles at once and patch the resulting $2r$ holes with ... | 4 | https://mathoverflow.net/users/44953 | 191688 | 94,225 |
https://mathoverflow.net/questions/191637 | 5 | Let $G$ be a finitely generated group. The amenability of $G$ is equivalent to the existence of a certain "weak measure" on $G$. Is there such a characterization for residually amenable groups as well? That is:
***Is the residual amenability of a discrete finitely generated group $G$ equivalent to the existence of a ... | https://mathoverflow.net/users/38889 | Characterizing residually amenable groups | It is easy to see that a finitely generated group is residually amenable if and only if there exists an bi-invariant ultra-metric on $G$ and a finitely additive $G$-invariant measure on open (with respect to the metric) subsets of $G$.
| 5 | https://mathoverflow.net/users/8176 | 191693 | 94,227 |
https://mathoverflow.net/questions/191682 | 5 | Let $G$ be a semi simple Lie group. I'm particularly interested in $SL(n,\mathbb{R})$. It is proved in
I. E. Segal and J. von Neumann, A theorem on unitary representations of semisimple
Lie groups, Annals of Mathematics 52 (1950), 509–517.
that measurable unitary representations of $G$ are actually continuous. Is t... | https://mathoverflow.net/users/64399 | Measurable representations of semi simple Lie groups | This is true and due to Béla von Szőkefalvi-Nagy, [*Über meßbare Darstellungen Liescher Gruppen* (1936)](http://www.digizeitschriften.de/dms/resolveppn/?PPN=GDZPPN002278286). Generalized to finite-dimensional representations of locally compact groups in A. Weil, *L'intégration dans les groupes topologiques* (1940, p. 6... | 5 | https://mathoverflow.net/users/19276 | 191695 | 94,229 |
https://mathoverflow.net/questions/191653 | 2 | Let $X$ be a scheme and let $\mathscr{I} \subset \mathscr{O}\_X$ be a quasi-coherent sheaf of ideals. Suppose that for each $x \in X$, the stalk $\mathscr{I}\_x$ is generated by an element $f\_x \in \mathscr{O}\_x$ that comes from a nonzero divisor $f \in \Gamma(U, \mathscr{O}\_X)$ for some affine open neighborhood $U$... | https://mathoverflow.net/users/53197 | Is a quasi-coherent sheaf of ideals with free stalks of rank 1 a Cartier divisor? | CW answer to get this of the unanswered list. Answered in the comments:
>
> **user74230:** The authors simply forgot to assume that I is of finite presentation as an OX-module. What ultimately matters is Lemma 6 on p. 213, so don't take that claim on p. 212 too seriously as written. When you write a 300-page book o... | 3 | https://mathoverflow.net/users/21815 | 191703 | 94,230 |
https://mathoverflow.net/questions/177785 | 2 | I've been trying to find a closed form of $\displaystyle \sum\_k{\tan{(k)}}$ that contains only elementary functions, and I think I may be onto something. But rather than reinvent the wheel, I want to ensure that this isn't already known.
So, I am specifically interested in the sum of the tangent. However, if it's no... | https://mathoverflow.net/users/24942 | What summations of elementary trig functions are known to have (elementary) closed forms? | From [this answer of mine](https://mathoverflow.net/a/42903/10059) The antidifference of tangent is:
$$\sum\_z\tan z=\sum \_{k=1}^{\infty } \left(\psi \left(k \pi -\frac{\pi }{2}+1\right)+\psi \left(k \pi -\frac{\pi }{2}\right)-\psi \left(k \pi -\frac{\pi }{2}+1-z\right)-\psi \left(k \pi -\frac{\pi }{2}+z\right)\righ... | 2 | https://mathoverflow.net/users/10059 | 191712 | 94,232 |
https://mathoverflow.net/questions/191709 | 2 | While reading Rezk's paper "A model for the homotopy theory of homotopy theory", I found a remark which contradicts a guess of mine, but I can't see where I am wrong (perhaps it might be a silly mistake, though I can't see where the following reasoning fails).
Given a category $\mathscr{C}$ and a subcategory $\mathsc... | https://mathoverflow.net/users/57280 | Is the discrete nerve of a small category a complete Segal space? | Your assertion $discnerve(\mathcal{C})=N\tilde{\mathcal{C}}$ is false.
Indeed, the category $id(\mathcal{C}^{[m]})$ (identities between chains of $m$ maps of $\mathcal{C}$), whose nerve is $discnerve(\mathcal{C})\_m$, and the category $\tilde{\mathcal{C}}^{[m]}$
(identities between chains of $m$ maps of $\tilde{\mathca... | 6 | https://mathoverflow.net/users/36625 | 191713 | 94,233 |
https://mathoverflow.net/questions/191730 | 1 | Is there exist a 4-manifold which intersection form has the following property
$$
(a,a) \neq 0\ \text{if}\ a\neq 0,
$$
and the second (or the first) Chern class (for some almost complex stucture) vanishes?
Thanks.
| https://mathoverflow.net/users/37807 | relations between intersection form and Chern classes | No. The first Chern class (reduced modulo 2) is the characteristic class of the intersection index form. Hence, your manifold has even form. On the other hand, it is also definite and, by Donaldson, must be diagonalizable. (Since you speak about an almost complex structure, I presume that the manifold is smooth.) This ... | 5 | https://mathoverflow.net/users/44953 | 191731 | 94,237 |
https://mathoverflow.net/questions/191725 | 17 | Let $X$ be a smooth projective irreducible curve defined over an algebraically closed field $\mathbb{K}$ (of arbitrary characteristic), and let $p\in X$ be a closed point. Denote by $\mathcal{O}\_p(X)$ the local ring of rational functions which are regular at $p$. Then, is it true that the completion of $\mathcal{O}\_p... | https://mathoverflow.net/users/23758 | Completion of a local ring of a curve | Let me expand and generalize my comments above. We can prove the following
>
> **Proposition.** Let $X$ be a projective scheme of dimension $n$ which is defined over an algebraically closed field $k$. If $p \in X$ is a closed, regular point and $\mathcal{O}\_{X, \, p}$ is the local ring of $X$ at $p$, then there is... | 14 | https://mathoverflow.net/users/7460 | 191737 | 94,239 |
https://mathoverflow.net/questions/191720 | 9 | The corresponence between compact Hausdorff topological spaces and commutative unital $C^\*$-algebras is rather well known: Gelfand Najmark theorem gives perfect correspondence between these categories. What is very easy: to define a structure of $C^\*$-algebra (commutative and unital) on the algebra $C(X)$ where $X$ i... | https://mathoverflow.net/users/24078 | Commutative spectral triples | From the perspective of the Gelfand–Naimark theorem, the heart of the reconstruction theorem is the following statement, Theorem 11.4 in [Connes's paper](http://arxiv.org/abs/0810.2088):
>
> Let $\mathcal{A}$ be a commutative unital complex $\ast$-algebra. Then $\mathcal{A} \cong C^\infty(X)$ for some compact orien... | 11 | https://mathoverflow.net/users/6999 | 191743 | 94,242 |
https://mathoverflow.net/questions/185901 | 2 | I would like to know the definition of Givental $J$-function of cotangent bundle of flag variety. To state my question more precisely, let us briefly recall the definition of the Givental $J$-function of a **compact** Kahler variety $X$. Let $T\_0=1,T\_1, \cdots, T\_m$ be the basis of the cohomology group $H^\*(X,\math... | https://mathoverflow.net/users/17644 | Definition of Givental $J$-function of cotangent bundle of flag variety | The parameter $m$ appears if you study everything equivariantly with respect to the
${\mathbb C}^\*$-action which acts in the standard way on the fibers of the cotangent bundle.
| 2 | https://mathoverflow.net/users/3891 | 191774 | 94,252 |
https://mathoverflow.net/questions/191754 | 3 | This question is motivated by [another question on math.stackexchange](https://math.stackexchange.com/questions/1052783/mandelbrot-and-julia-set).
From a function $g:X^k\to X$ it is possible to define an iterated function system on $X^k$ with the function $f:X^k\to X^k$ defined by
$$f(x) = (g(x), x\_1, x\_2, \ldots... | https://mathoverflow.net/users/16518 | Is there literature available on iterated function systems of the form $f^n = (g f^{n - 1}, g f^{n - 2}, \ldots)$? | *(More an answomment)*. I'd say your iteration on $X^k$ is what one gets by a standard reduction writing a $k$-order recurrence on $X$,
$$x\_{n}=g(x\_{n-1},x\_{n-2},\dots,x\_{n-k}),\qquad x\_i\in X$$
in form of a first-order recurrence on $X^k$,
$$y^{n+1}=f(y^n),\qquad y^n=(y^n\_1,\dots,y^n\_k)\in X^k$$
putting $y^n\_... | 1 | https://mathoverflow.net/users/6101 | 191782 | 94,255 |
https://mathoverflow.net/questions/102488 | 2 | I'm doing summer cryptography research and I am have been looking for a security analysis of the Guillou-Quisquater (GQ) digital signature scheme, but I have been unable to find one.
Since this is not a very common digital signature scheme I will mention the protocol.
>
> GQ: Public: $n,e,I$ has function $H$, whe... | https://mathoverflow.net/users/20343 | Is there a security analysis of the GQ digital signature scheme? | [M. Bellare, A. Palacio - GQ and Schnorr Identification Schemes: Proofs of Security against Impersonation under Active and Concurrent Attacks](http://cseweb.ucsd.edu/~mihir/papers/gq.pdf)
| 3 | https://mathoverflow.net/users/62682 | 191784 | 94,256 |
https://mathoverflow.net/questions/191775 | 2 | Consider sets of Vitali's type in models of $\mathsf{ZF}+\mathsf{GCH}$ where $V \neq L$. Are there sets of Vitali's type in both $L$ and $V \backslash L$? If so, is there any way one can distinguish the constructible sets of Vitali's type from the nonconstructible sets of Vitali's type?
By a *set of Vitali's type* it... | https://mathoverflow.net/users/20597 | Sets of Vitali's type in models of $\mathsf{ZF}+\mathsf{GCH}$ where $V \neq L$ | There are several things to say that may answer your question.
* It is consistent with ZFC$\pm$GCH that the set of all real numbers in $L$ is countable in $V$, in which case every set of reals in $L$ has measure zero.
* It can happen that $V$ and $L$ have precisely the same sets of reals, yet $V\neq L$, because they ... | 5 | https://mathoverflow.net/users/1946 | 191786 | 94,257 |
https://mathoverflow.net/questions/191785 | 7 | It is known that for a topological space with different metrics, the Hausdorff dimensions may not be equal in general.
For the case of manifolds, suppose $M$ is a $n$-manifold with a metric(distance), from <https://math.stackexchange.com/questions/931628/hausdorf-dimension-of-a-manifold-of-dimension-n>, we know that... | https://mathoverflow.net/users/62195 | The relation between Hausdorff dimension of an $n$-manifold and $n$ | In a metrizable topological space, Hausdorff dimension is always larger or equal than the topological (covering) dimension. See Theorem 6.3.10 in Edgar's book "Measure, topology and fractal geometry". In particular, for an $n$-dimensional manifold $M$, if $\rho$ is any metric compatible with the Euclidean topology, the... | 13 | https://mathoverflow.net/users/11009 | 191787 | 94,258 |
https://mathoverflow.net/questions/144451 | 3 | As far as I know, in a simply connected compact manifold, still there exists no well-known obstruction for a manifold with a **quasi-positive curvature** to be a manifold with **positive curvature**.
But Hopf's conjecture is unsolved, i.e., the conjecture: $S^2\times S^2$ has a positive curvature.
So I think that... | https://mathoverflow.net/users/36572 | Manifold with a quasi-positive curvature | For high rank Lie group, this is in fact the generalized Hopf conjecture which is still open.
| 2 | https://mathoverflow.net/users/61515 | 191795 | 94,260 |
https://mathoverflow.net/questions/191702 | 4 | This is a version of
[this question of Klim Efremenko.](https://mathoverflow.net/questions/59807)
Let $r>2$ be a natural number, say $r=3$ or $r=10$. Let $G$ be a finite group and $\rho$
be an irreducible complex representation of $G$. We consider the following minimum $m=m(r,G,\rho)$:
$$
m(r,G,\rho)= {\rm min\ rank... | https://mathoverflow.net/users/4149 | A small rank linear combination of a small number of elements of a group | This answer shows that one cannot find $(G,\rho)$ as required if $G$ is supposed to be group of Lie type defined over a large field.
Let $G$ be a group of Lie type defined over a field with $q$ elements. Let $\rho$ be an irreducible representation of $G$ which is not $1$-dimensional. Gluck (D. Gluck. Sharper characte... | 3 | https://mathoverflow.net/users/8176 | 191797 | 94,261 |
https://mathoverflow.net/questions/191791 | 13 | I'm taking a course on topology and probabily. Today, the professor remarked something along the lines of:
>
> If you look at the space of probability distributions with $0$ mean and variance $1$, equipped with convolution, then the Gaussian distribution is characterized as the fixed point of each orbit."
>
>
>
... | https://mathoverflow.net/users/53127 | Gaussian distributions as fixed points in Some distribution space | Not sure if this is what you want, but orbits in spaces of probability distributions can be thought of as simple cases of **renormalization group flows** in statistical mechanics, see e.g. the discussion in [this paper of Calvo et al](http://link.springer.com/article/10.1007%2Fs10955-010-0065-y#page-1) and its referenc... | 5 | https://mathoverflow.net/users/353 | 191823 | 94,268 |
https://mathoverflow.net/questions/191829 | 2 | Suppose we have a closed (compact without boundry) Riemannian manifold $(M,g)$, do we have uniqueness (assuming solutions exist on $[0,T]$ ) for the backward heat equation
$$\frac{\partial f }{\partial t}+\Delta\_gf=0$$
$$f(x,0)=f\_0(x),$$
where $f\_0$ is some Borel function on $M$.
| https://mathoverflow.net/users/64473 | Uniqueness of backward heat equation on closed manifold with given initial data | I'm assuming you are supposing that $f\in C^\infty(M)$ for each $t \in [0,T]$, or something along this line. I'm not sure what happens if you allow $f$ to have some singularities. Then, picking any $0 < T\_1 < T$, we can consider $f$ as solving a forwards parabolic PDE from $T\_1$ to $0$. This puts parabolic regularity... | 2 | https://mathoverflow.net/users/1540 | 191832 | 94,271 |
https://mathoverflow.net/questions/191799 | 10 | By "saturated class of morphisms" in a category $\mathcal{C}$, I mean a subcategory $\mathcal{W} \subset \mathcal{C}$ such that the image of $\mathcal{W}$ in $\mathcal{C}[\mathcal{W}^{-1}]$ consists of exactly the isomorphisms. By the universal property of $\mathcal{C}[\mathcal{W}^{-1}]$, this is equivalent to saying t... | https://mathoverflow.net/users/2362 | Example of a saturated class of morphisms which is not _obviously_ saturated? | In the [canonical model structure](http://arxiv.org/abs/0712.0617) on $\omega\mathrm{Cat}$, the weak equivalences are not defined as preimages of isomorphisms under any functor, and are not even even closely related to any such preimage the way weak equivalences of unbased spaces are. And indeed, one of the hardest par... | 6 | https://mathoverflow.net/users/49 | 191839 | 94,273 |
https://mathoverflow.net/questions/191867 | 4 | It is an easy exercise that using ruler and compass one find the focus of a given parabola.
Can one do the same using only a ruler? -- if not, why?
| https://mathoverflow.net/users/13070 | Focus of parabola using only a ruler | No, because there exists a projective map which preserves the parabola but does not preserve its focus (this is so for any conic and any point.)
| 22 | https://mathoverflow.net/users/4312 | 191869 | 94,283 |
https://mathoverflow.net/questions/191854 | 13 | Sometimes, it happens that researchers publish a new proof of an old well-known result in "basic real analysis" (I'm referring to what some American people may call "honors calculus"). For instance, we can consider [this article](http://www.jstor.org/discover/10.2307/4145046?uid=3738296&uid=2&uid=4&sid=21104830505661).... | https://mathoverflow.net/users/nan | New research and re-discovering classic results in "basic" real analysis | Using Google Scholar to search for recent American Mathematical Monthly articles containing the term "new proof" turns up some candidates. For example, Steve Roman's paper on [The Formula of Faà di Bruno](http://www.jstor.org/stable/2320788) derives the formula using the umbral calculus. The umbral calculus is a classi... | 10 | https://mathoverflow.net/users/3106 | 191879 | 94,287 |
https://mathoverflow.net/questions/191850 | 8 | The symmetric group $S\_n$ acts on $\mathbb R^n$ by permuting the coordinates and the ring of polynomial invariants is generated by the elementary symmetric polynomials. If we restrict the action to the alternating group $A\_n$ then the invariant ring is generated by elementary symmetric polynomials along with the disc... | https://mathoverflow.net/users/64495 | Polarizations generate the ring of invariants? | To complete Peter's answer, I would like to give a concrete counterexample: by computing (e.g. using GAP) the Molien series for $S\_3$ (resp. for $A\_3$) acting naturally on $\mathbb{R}^3\oplus \mathbb{R}^3$, one sees that already in degree 2 the dimension of the space of $A\_3$-invariant polynomials is bigger (by 1) t... | 3 | https://mathoverflow.net/users/11100 | 191880 | 94,288 |
https://mathoverflow.net/questions/191800 | 2 | I have a question concerning 2nd order evolution equation of the form $u''(t)+A(t)u(t) = f(t)$ in $L^2(0,T;V^\*)$, where $f\in\ L^2(0,T;H)$ holds. Under what assumptions is it possible, to guarantee a unique solution for a more general $f\in L^2(0,T;V^\*)$. I wrote down an exact setting following Wloka in his book *par... | https://mathoverflow.net/users/64457 | Regularity of solution to a hyperbolic pde | If you want f to take values in $V^\*$ rather than $H$, you can do this if you assume more temporal regularity on f. Basically, the idea is to integrate by parts in the term
$\int\_0^t \langle u',f\rangle$ in the energy estimate. You will have no trouble finding results of this type in the literature.
| 1 | https://mathoverflow.net/users/12120 | 191884 | 94,289 |
https://mathoverflow.net/questions/191885 | 8 | What is the Schur multiplier of the $n$-dimensional affine linear group $\mathrm{AGL}(n,q)$ over the Galois field with $q$ elements?
I am particularly interested in the simple case $n=1$. Computation using GAP shows that the Schur multiplier of $\mathrm{AGL}(1,q)$ is trivial for any prime power $q$ up to 19, except f... | https://mathoverflow.net/users/56827 | What is the Schur multiplier of the affine linear group AGL(n,q)? | Let me concentrate on the ${\rm AGL}(1,q)$ case for now, where $q=p^k$ is a prime power. This group is a split extension $N:C$, where $N$ and $C$ are the additive and multiplicative groups of the field of order $q$, and the action of $C$ on $N$ is by multiplication in the field. So $N$ is elementary abelian and $C$ is ... | 8 | https://mathoverflow.net/users/35840 | 191897 | 94,292 |
https://mathoverflow.net/questions/191902 | 1 | Maxim Kontsevich and Don Zagier defined the algebra of periods and conjectured that one can pass from a representation of a given period to another one using only three rules. Assuming this conjecture, can we define the automorphism group of a given period using only these three rules? Can it lead to a method to prove ... | https://mathoverflow.net/users/13625 | automorphism group of a given period | The idea that periods should be subject to a "transcendental Galois theory" has been first advanced by Grothendieck, who sketched a beautiful (but extremely conjectural) relationship with his theory of motives and motivic Galois groups. The resulting Period conjecture is very closely related to the Kontsevich-Zagier pe... | 7 | https://mathoverflow.net/users/7878 | 191912 | 94,298 |
https://mathoverflow.net/questions/191891 | 2 | An hyperoctahedral group $G$ is the wreath product of $S\_2$ and $S\_n$, where $S\_{n}$ is the symmetric group on $n$ letters, or in other words the semi-direct product $G=S\_2^n\rtimes S\_n$, w.r.t. the action on the indices. $G$ comes with a natural action on $[2]\times [n]$, so we consider it as a subgroup of $S\_{2... | https://mathoverflow.net/users/2042 | Does the hyperoctahedral group have only 3 maximal normal subgroups? | Perhaps I should expand my comment into an answer! The group $G$ in question is a split extension $V:S\_n$ of the natural permutation module for $S\_n$ over the field of order $2$ by $S\_n$.
As I said, it is well-known that, at least when $n>2$, the permutation module $V$ (over any field) for $S\_n$ over any field ha... | 7 | https://mathoverflow.net/users/35840 | 191913 | 94,299 |
https://mathoverflow.net/questions/191909 | 15 | Is there anything known about what drew the attention of ancient greek mathematicians to conic sections and, what were the models they used to study conic sections?
What I would like to know, is whether ancient mathematician actually became aware of conic sections by looking at sections of circular cones or, whether t... | https://mathoverflow.net/users/31310 | Discovery and Study of Conic Sections in Ancient Greece | **Discovery:** [Menaechmus](https://en.wikipedia.org/wiki/Menaechmus) is credited with the discovery of conic sections around the years 360-350 B.C.; he used them to solve the problem of "doubling the cube". The construction required a parabola, which he called "a section of a right-angled cone", and a hyperbola, "a se... | 15 | https://mathoverflow.net/users/11260 | 191915 | 94,300 |
https://mathoverflow.net/questions/191796 | 12 |
>
> I met with the following difficulty reading the paper [Li, Rong Xiu "The properties of a matrix order column" (1988)](http://www.cnki.com.cn/Article/CJFDTotal-ZZDZ198801008.htm):
>
>
> Define the matrix $A=(a\_{jk})\_{n\times n}$, where
> $$a\_{jk}=\begin{cases}
> j+k\cdot i&j<k\\
> k+j\cdot i&j>k\\
> 2(j+k\c... | https://mathoverflow.net/users/38620 | How do we show this matrix has full rank? | OK, let me try again, maybe I'll get it right this time. I'll show that $P$ is positive definite. This will imply the claim because if $(P+iQ)(x+iy)=0$ with $x,y\in\mathbb R^n$, then $Px=Qy$, $Py=-Qx$, and by taking scalar products with $x$ and $y$, respectively, we see that $\langle x, Px \rangle = -\langle y, Py\rang... | 16 | https://mathoverflow.net/users/48839 | 191918 | 94,302 |
https://mathoverflow.net/questions/191910 | 2 | Let $G$ be a dimension group (i.e. a directed, unperforated abelian group satisfying the Riesz interpolation property) with order unit $u\in G^{+}$. There is a canonical positive group homomorphism $\theta\colon G\to \operatorname{Aff}(S(G,u))$, where $S(G,u)$ is the state space of $G$ and $\operatorname{Aff}(S(G,u))$ ... | https://mathoverflow.net/users/39489 | When do infinitesimals split in dimension groups? | Since every countable torsion-free abelian group can arise as the underlying group of a dimension group, even with unique trace (state), the answer to the first question is no, even when $G$ is also simple. A stationary example is given by almost any $2 \times 2$ strictly positive integer matrix both of whose eigenvalu... | 3 | https://mathoverflow.net/users/42278 | 191921 | 94,304 |
https://mathoverflow.net/questions/191718 | 2 | **Notation:**
Let $k$ be a commutative local ring and let $HH^{i}(A,N)$ denote the $i^{th}$ Hochschild cohomology $k$-module of a $k$-algebra A with coefficients in an $(A,A)$-bi-module $N$.
---
If $x:=\{x\_1,...,x\_d\}$ is a maximal regular sequence in $A$ then is it true that:
$HH^{i+d}(A,N)$ vanishes *only if... | https://mathoverflow.net/users/36886 | Hochschild cohomology of commutative quotients | No, it is false for example:
Let $A:=\mathbb{C}[x]/(x^4)$ and $N:=A$ then:
$HH^{4}(A,N)\cong \mathbb{C}[x]$ (Weibel: p.304)
Contrastingly: $\mathbb{C}[x]$ ($\cong \mathbb{C}\left<x\right>$) is quasi-free (see Cuntz-Quillen's [article](http://www.ams.org/journals/jams/1995-08-02/S0894-0347-1995-1303029-0/S0894-0347-... | 2 | https://mathoverflow.net/users/64546 | 191967 | 94,311 |
https://mathoverflow.net/questions/191876 | 4 | Let $G$ be a compact Lie group. I know that the Burnside ring $A(G)$ is isomorphic to the zeroth $G$-equivariant stable homotopy $\pi^{G}\_0(S^0)$. What is the isomorphism between $A(G)$ and $\pi^{G}\_0(S^0)$ and if I consider an element of $\pi^{G}\_0(S^0)$ then how can I determine the corresponding one in $A(G)$? For... | https://mathoverflow.net/users/62800 | Isomorphism between the Burnside ring $A(G)$ and the zeroth $G$-equivariant stable homotopy $\pi^{G}_0(S^0)$ | Expanding on the answer by **user43326**, and covering the compact Lie case: The map $A(G) \to \pi^G\_0(S^0)$ indeed comes from the Pontrjagin-Thom construction. Each generator of $A(G)$ is a $G$-orbit $G/H$ (with $G/NH$ finite). Take an embedding $G/H\to V$ in some $G$-representation $V$, with normal bundle $\nu$. Thi... | 10 | https://mathoverflow.net/users/58888 | 191969 | 94,312 |
https://mathoverflow.net/questions/191961 | 3 | I not sure that my question has a research level so feel free to remove it, I'll not be offended.
Let $S^{n}$ be a sphere of dimension $n>1$ and $p<q$ two prime numbers. Is there always a minimal integer $i\_{p}$ such that $\pi\_{i\_{p}}S^{n}$ has $p$-torsion ? and is true that $i\_{p}\leq i\_{q}$ if $p<q$ ?
Thanks.... | https://mathoverflow.net/users/64541 | homotopy groups of spheres. | There is an obvious map $i\colon S^n\to K(\mathbb{Z},n)$, with fibre $F$ say. The homotopy groups of $F$ are essentially the same as those of $S^n$. The mod $p$ cohomology of $K(\mathbb{Z},n)$ is polynomial tensor exterior, with a generator $u$ in degree $n$, and other generators obtained by applying Steenrod operation... | 12 | https://mathoverflow.net/users/10366 | 191970 | 94,313 |
https://mathoverflow.net/questions/133249 | 10 | For algebraic groups or Lie groups, the subject of *Levi decompositions* tends to be surrounded by some mystery in the literature (and in an older question raised [here](https://mathoverflow.net/questions/67939/is-there-a-levi-decomposition-for-lie-group-and-algebraic-group)). While I postpone further my intention to p... | https://mathoverflow.net/users/4231 | Levi decomposition in disconnected linear algebraic group (characteristic 0)? | See G. P. Hochschild: Basic Theory of Algebraic Groups and Lie Algebras (Graduate Texts in Mathematics)-Springer (1981) VIII, Theorem 4.3
| 3 | https://mathoverflow.net/users/9401 | 191974 | 94,317 |
https://mathoverflow.net/questions/191930 | 3 | If $f\in L^p(R)$ with $1\leq p\leq 2$, then Hausdorff-Young inequality implies that the Fourier transform $\widehat{f}\in L^{p'}$, $p'$ is the dual exponent of $p$, and
$$
\|\widehat{f}\|\_{L^{p'}}\lesssim \|f\|\_{L^p}.
$$
We note that $f\in L^p$ is equivalent to $|f|\in L^p$, so we find $\widehat{|f|}\in L^{p'}$ and
$... | https://mathoverflow.net/users/23355 | Connection between the Fourier transform of f and |f| | No. The inequality $||\hat{f}||\_{L^p} \lesssim || \hat{|f|} ||\_{L^{p}}$ does not hold for $p \neq 2$. This is, perhaps, easier to see in the case of Fourier series on the circle. A sketch of a contstruction is given in my answer to another mathoverflow quesion [here](https://mathoverflow.net/questions/143342/lp-norm-... | 3 | https://mathoverflow.net/users/630 | 191978 | 94,320 |
https://mathoverflow.net/questions/190850 | 4 | recently in my researches I've come across the following operator
$$L\left(\begin{array}{c}
a\_1\\
\vdots\\
a\_n
\end{array}\right)=M\_1\left(\begin{array}{c}
\partial\_{z\_1}a\_1\\
\vdots\\
\partial\_{z\_1}a\_n
\end{array}\right)+\dotso+M\_n\left(\begin{array}{c}
\partial\_{z\_n}a\_1\\
\vdots\\
\partial\_... | https://mathoverflow.net/users/57571 | System of linear first order PDE with constant coefficients | Since you wrote the system explicitly, here's an explicit solution using Fourier transforms. Let $a(z) = \int e^{ik\cdot z} a(k)\, d^3k $ and $b(z) = \int e^{ik\cdot z} b(k)\, d^3k$. The $a(k)$ and $b(k)$ vectors will satisfy the Fourier transformed equation $iM(k) a(k) = b(k)$, where $M(k) = (M\_1 k\_1 + M\_2 k\_2 + M... | 3 | https://mathoverflow.net/users/2622 | 191979 | 94,321 |
https://mathoverflow.net/questions/191966 | 6 | For any positive $p,q\in\mathbb{N}$ there is a finite subset $S$ of $\{\frac{1}{n}:n\in\mathbb{N}, n\geq 1\}$ such that $\sum\_{s\in S} s=\frac{p}{q}$, see [this article](https://www.renyi.hu/~p_erdos/1963-18.pdf) by Paul Erdös and Sherman Stein *(Sums of distinct unit fractions. Proceedings of the American Mathematica... | https://mathoverflow.net/users/8628 | Minimum number of unit fractions to sum up a given positive rational | In fact, this is listed as an open problem in the [Wikipedia page on Egyptian Fractions](http://en.wikipedia.org/wiki/Egyptian_fraction), presumably because they do use the output size as a parameter.
| 3 | https://mathoverflow.net/users/11142 | 191984 | 94,323 |
https://mathoverflow.net/questions/191988 | 3 | Suppose $M$ is a smooth manifold and a compact Lie group $G$ acts freely on $M$, then it is well known that $M/G$ has a manifold structure.
Are there any results for the general case? (a) If the action is not free, what can we say about the local structure of the quotient? Can we define a stratification on the space... | https://mathoverflow.net/users/37354 | Local structure of the quotient of a Lie group action | Suppose one relaxes the condition that the action is free, replacing it with the condition that every point in $M$ has a finite $G$-stabilizer. In this case, the topological quotient $M/G$ carries a natural orbifold structure. The local model is therefore the quotient of $\mathbb{R}^n$ by a finite group acting linearly... | 4 | https://mathoverflow.net/users/25358 | 191990 | 94,325 |
https://mathoverflow.net/questions/191788 | 3 | Let $V\subset\mathbb{P}^5$ be the Veronese surface and let $X\subset\mathbb{P}^6$ be the cone over it. Since $X$ is $\mathbb{Q}$-factorial there are two integers $a,b$ such that $aK\_X = \mathcal{O}\_X(b)$.
Furthermore, if $f:Y\rightarrow X$ is the blow-up of the vertex then $Y$ is smooth and we may write $K\_Y = f^... | https://mathoverflow.net/users/nan | Cone over the Veronese surface | Let me start being a little nitpicking with the formulation of the question. The fact that $X$ is $\mathbb Q$-factorial does not in itself imply that such $a$ and $b$ exists. One also needs the fact that the Picard number of $X$ is $1$. This is indeed true, but perhaps should be mentioned. In fact, the Picard group of ... | 9 | https://mathoverflow.net/users/10076 | 191997 | 94,328 |
https://mathoverflow.net/questions/191981 | 4 | Let $m$ and $n$ be two positive integers and denote by $P(n,m)$ the number of partitions of $n$ into $m$ non-negative integers.
Is there an asymptotic formula for $P(n,m)$ ?? Any reference is welcome. Thanks
| https://mathoverflow.net/users/48866 | asymptotic for restricted partitions | For $m\geq n^{1/6}$ there is an asymptotic formula due to Szekeres. See <http://www.combinatorics.org/ojs/index.php/eljc/article/view/v4i2r6/pdf> for references and another proof.
| 8 | https://mathoverflow.net/users/2807 | 192003 | 94,331 |
https://mathoverflow.net/questions/191914 | 8 | A nodal projective curve in $\mathbb{CP}^2$ inherits a Kähler metric from the Fubini-Study metric, and hence a Riemannian metric. In particular, with respect to this metric, a line has constant Gaussian curvature 1; according to [Vitter](http://www.iumj.indiana.edu/docs/23067/23067.asp) (page 826), the Fermat conic $\{... | https://mathoverflow.net/users/13119 | Projective curves of constant curvature | A general result of [Hulin](https://doi.org/10.1007/BF02921947) shows that if $M$ is a compact complex manifold with a holomorphic embedding $f:M\to\mathbb{CP}^n$ such that $f^\*g\_{FS}$ is Einstein, then the Einstein constant is positive. Hence the answer to your first question is **no.**
Also, if furthermore $\math... | 9 | https://mathoverflow.net/users/13168 | 192023 | 94,337 |
https://mathoverflow.net/questions/192028 | 3 | Motivated by this [question](https://mathoverflow.net/questions/191120/the-letters-of-the-word-art) we ask:
>
> Up to homeomorphism, are there only a finite number of connected locally compact hausdorff topological space $X$ such that $X$ has an open set $U$ homeomorphic to $\mathbb{R}$ such that $X-U$ is homeomorp... | https://mathoverflow.net/users/36688 | A question in general topology | There are continuum pairwise non-homeomorphic closed $1$-dimensional subspaces $\ X\ $ of $\ \mathbb R\times[0;\infty)\ $ which contain $\ \mathbb R\times\{0\},\ $ and such that $\,\ X\,\setminus\,(\mathbb R\times\{0\})\,\ $ is homeomorphic to $\ \mathbb R$.
**The construction**: a line goes from up high gradually do... | 14 | https://mathoverflow.net/users/8385 | 192037 | 94,341 |
https://mathoverflow.net/questions/192034 | 0 | Let suppose function $g(s,t)$ satisfies partial differential equations:
$g\_{ss} g\_{tt} - g\_{st}^2=0$. It may be treated as the surface has zero gaussian curvature. I am searching for general solution of this equation locally. Are there any results for it?
| https://mathoverflow.net/users/nan | General description of surface with zero gaussian curvature | A surface of Gaussian curvature zero is locally isometric to the plane, and is said to be developable. A complete surface of Gaussian curvature zero in Euclidean three space is a cylinder (where a cylinder means the surface generated by the lines parallel to a given axis passing through a fixed curve in the subspace pe... | 6 | https://mathoverflow.net/users/9471 | 192039 | 94,342 |
https://mathoverflow.net/questions/191996 | 6 | Given the Diophantine equation,
$$x\_1^k+x\_2^k+x\_3^k = y\_1^k+y\_2^k+y\_3^k\tag1$$
there is the rather curious observation that the **smallest** positive solutions for $k=5$ or $6$ is *multi-grade*.
$$24^k+28^k+67^k=3^k+54^k+62^k,\quad k = 1,5$$
$$15^k + 10^k + 23^k = 3^k + 19^k + 22^k,\quad k = 2,6$$
Dunca... | https://mathoverflow.net/users/12905 | Density of multi-grade solutions to $x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k$ for $k = 5$ or $6$? | This is more of a long comment than an answer.
This problem is very difficult and it is not clear to me whether anyone will be able to prove the statments you are looking for. The usual way of studying such equations is to use the circle method, however you have too few variables to get the circle method to work.
I... | 9 | https://mathoverflow.net/users/5101 | 192053 | 94,351 |
https://mathoverflow.net/questions/191945 | 1 | A coproduct $\varphi: \mathbb{C}\_q[U] \to \mathbb{C}\_q[U] \otimes \mathbb{C}\_q[U]$ is given by: $x \mapsto 1 \otimes x + x \otimes 1$, where $x$ is a generator of $\mathbb{C}\_q[U]$. There is a coproduct $\mathbb{C}[U] \to \mathbb{C}[U] \otimes \mathbb{C}[U]$ which is the pull-back of the multiplication map: $U \tim... | https://mathoverflow.net/users/11877 | The coproducts $\mathbb{C}_q[U] \to \mathbb{C}_q[U] \otimes \mathbb{C}_q[U]$ and $\mathbb{C}[U] \to \mathbb{C}[U] \otimes \mathbb{C}[U]$ | The algebra $ \mathbb{C}[U] $ is a polynomial ring on generators $ x\_{ij} $ where $ i < j $. The coproduct is given (as you say) by $ \psi(x\_{ik})= \sum\_j x\_{ij} \otimes x\_{jk} $ where where interpret $ x\_{ij} = 0 $ if $ i > j $ and $ x\_{ii} = 1 $. In particular, we see that
$$
\psi(x\_{i i+1})= 1 \otimes x\_{i... | 1 | https://mathoverflow.net/users/438 | 192060 | 94,352 |
https://mathoverflow.net/questions/192061 | 5 | Is there a finitely generated non-elementary word hyperbolic group the profinite completion of which is known (or conjectured) to be rather restricted, that is: abelian, pro-$p$, virtually prosolvable, etc. ?
| https://mathoverflow.net/users/38889 | A hyperbolic group with a small profinite completion | It's a famous open question whether every word-hyperbolic group is residually finite. Kapovich--Wise showed that this is equivalent to asking whether every non-trivial word-hyperbolic group has non-trivial profinite completion. In the other direction, Agol--Groves--Manning showed that it's equivalent to asking whether ... | 12 | https://mathoverflow.net/users/1463 | 192068 | 94,354 |
https://mathoverflow.net/questions/192069 | 3 | For $\ell/k$ a finite separable extension of fields and an affine variety $X\_{/\ell}$, the Weil restriction of scalars $R\_{\ell/k}(X\_{/\ell})$ represents the functor $R\_{\ell/k}(X\_{/\ell})(S) := X\_{/\ell}(S \times\_k \ell)$, for any $k$-scheme $S$. One can show that if $X$ is a smooth affine algebraic group over ... | https://mathoverflow.net/users/64591 | Solubility of an algebraic group and Weil restriction | Let me try to deduce a positive answer from standard properties of the Weil restriction and the results presented in this paper (found by Google): <http://math.stanford.edu/~conrad/papers/appbnew.pdf>. This is probably an overkill, especially if you are only interested in perfect ground fields.
Weil restriction prese... | 1 | https://mathoverflow.net/users/5498 | 192073 | 94,356 |
https://mathoverflow.net/questions/192058 | 4 | A covering design $(v,k,t)$ is a family of subsets of $[v]$ each having $k$ elements such that given any subset of $[v]$ of $t$ elements it is a subset of one of the sets of the family. A problem is to find the minimum number of subsets such a family can have.
I am interested in the case $(v,k,2)$. It seems to be equ... | https://mathoverflow.net/users/24478 | covering designs of the form $(v,k,2)$ | **Edit:** ***The possible "gap" of sort in Caro and Yuster's proof of their upper bound has just been fixed!*** *See Ben Barber's comment below (and his joint paper with Daniela Kühn, Allan Lo and Deryk Osthus* [*on arXiv*](http://arxiv.org/pdf/1410.5750v1.pdf). *This shows Gustavsson's theorem (whose proof by Gustavss... | 8 | https://mathoverflow.net/users/27829 | 192076 | 94,358 |
https://mathoverflow.net/questions/191965 | 8 | Is there a reasonable random model for selecting a finitely presented group $G$ such that with positive probablity (or even with probability almost $1$) some of the following properties hold:
1. $G$ is residually finite.
2. $G$ is subgroup seperable (LERF).
3. The first $l\_2$ Betti number of $G$ is positive.
4. The ... | https://mathoverflow.net/users/38889 | Is residual finiteness a property of "many" finitely presented groups? | A random group at density less than 1/6 is known to be the fundamental group of a compact, non-positively curved cube complex, by Ollivier--Wise. By Agol's theorem, all such hyperbolic groups are virtually special, and hence residually finite, QCERF, etc.
| 4 | https://mathoverflow.net/users/1463 | 192098 | 94,362 |
https://mathoverflow.net/questions/192082 | 4 | Suppose $A$ is a (if necessary unital) associative ring and $I$ is a left ideal in $A$. Let $\operatorname{pd}(M)$ denote the projective dimension of a left $A$-module $M$.
Then do either of the following exist:
1. $M\in \space \_A\mathrm{Mod}$ such that $ \infty >\operatorname{pd}(M)\geq \operatorname{pd}(M/IM)$ a... | https://mathoverflow.net/users/64546 | Existence of small projective dimensioned modules | If $A$ is a commutative local artinian ring which is not a field, and $I \neq 0$ is the maximal ideal in $A$, then no such modules exist.
In this case every $A$-module has projective dimension $0$ or $\infty$, see for instance
(Bass, Hyman. [Finitistic dimension and a homological generalization of semi-primary rings... | 4 | https://mathoverflow.net/users/18756 | 192100 | 94,363 |
https://mathoverflow.net/questions/192102 | 15 | I can see the benefit of writing a mathematical monograph: you revise and organize your own work and recollect the key ideas of your own research. But this applies only to books aimed at researchers and graudate students in a very specific area on which you've worked a lot during your career.
I don't fully understand... | https://mathoverflow.net/users/nan | Research and exposition: how does writing "basic" books affect your "serious" research work? | Let's take three well know examples. Lawvere (set theory), Menger (calculus), MacLane (set theory,categories). All wrote textbooks.
All of them indicated that they did it primarily to (a) make more accessible a particular way of approaching some basic material which they would like to see more of, (b) to determine, w... | 21 | https://mathoverflow.net/users/58777 | 192111 | 94,368 |
https://mathoverflow.net/questions/192101 | 14 | I know that we can do a lot of 2-category theory, seeing 2-categories as Cat-enriched categories. Yet, I know that there are some limitations of this approach.
I also know that there are many articles which could help to understand 2-category theory... (I am only familiar with a few of the Lack's, Street's and Kelly's ... | https://mathoverflow.net/users/18017 | 2-category theory | One aspect of 2-category theory which I've sometimes found difficult or tricky is 2-limits (or variants thereof). If that is troubling you too, some of these papers (mentioned in the nLab article on 2-limits) could be helpful:
* Ross Street, *Limits indexed by category-valued 2-functors*, Journal of Pure and Applied... | 15 | https://mathoverflow.net/users/2926 | 192113 | 94,369 |
https://mathoverflow.net/questions/192089 | 3 | Let $X$ be a smooth projective variety over $\mathbb{C}$. Let $D$ be a Cartier divisor on $X$. If $C$ is a curve on $X$ such that the intersection $C\cdot D <0$, we have $C \subseteq \mathbf{B}(D)$, where $$\mathbf{B}(D):= \bigcap\_{m \in \mathbb{N}, F\in |mD|}F$$ is the stable base locus of $D$.
I want to know if th... | https://mathoverflow.net/users/29730 | Stable base locus of a divisor and negative intersection with curves | Both are false. Let $X$ be the blow-up of $\mathbb P^3$ at $8$ very general points, and let $D = 2H - \sum E\_i$ be the linear system of quadrics through the $8$ points. There is a pencil of such quadrics. The stable base locus of $D$ is a genus $1$, degree $4$ curve (obtained as the intersection of any two quadrics). ... | 7 | https://mathoverflow.net/users/nan | 192117 | 94,372 |
https://mathoverflow.net/questions/192077 | 4 | Let $G$ be a simply connected algebraic group over $\mathbb{C}$, $W$ be the Weyl group for $G$ and $W\_{aff}$ be the affine Weyl group for the loop group $G(\mathbb{C}((t)))$, $\Phi$ be the coweight lattice of $G$.
$W\_{aff} \cong W \ltimes \Phi.$
There is a simple reflection $s\_0$ in $W\_{aff}$ that does not corres... | https://mathoverflow.net/users/7780 | How to think about the simple reflection s_0 in the affine Weyl group? | You are asking several questions here, so it may be useful to separate out what is going on first in the setting of affine reflection groups. This is independent of the application to algebraic groups or loop groups.
The answer to your first question (how to write $s\_0$ in terms of the presentation) is straightforwa... | 6 | https://mathoverflow.net/users/4231 | 192119 | 94,373 |
https://mathoverflow.net/questions/192131 | 5 | Given a simple, undirected graph and a vertex $v$ of the graph, let $L\_v$ denote the set of automorphisms of the graph that fixes the vertex $v$ and each of its neighbors. When the graph is vertex-transitive, the vertex-neighborhood stabilizer $L\_v$ is independent of the choice of $v$.
I noticed that for many vert... | https://mathoverflow.net/users/62347 | Structure of the stabilizer of a vertex-neighborhood of a vertex-transitive graph | Consider the Kneser graph $K(v,k)$, with vertices the $k$-subsets of $V=\{1,\ldots,v\}$,
where the $k$-subsets are adjacent if they are disjoint. Let $\alpha=\{1,\ldots,k\}$
and let $G$ be the subgroup of the symmetric group on $V$ that fixes each element
of the complement of $\alpha$. So $|G|=k!$ and $G$ is a subgroup... | 6 | https://mathoverflow.net/users/1266 | 192133 | 94,381 |
https://mathoverflow.net/questions/191300 | 8 | Let $M$ be a compact Riemannian manifold and $\Sigma\subset M$ a closed submanifold. Given $x\in M$ we define the *distance function to* $\Sigma$ by $$d\_\Sigma(x):=\inf\{d(x,y):y\in \Sigma\},$$ where $d$ is the metric on $M$. Of course, in a small tubular neighborhood of $\Sigma$ the function $d\_\Sigma$ will be smoot... | https://mathoverflow.net/users/64231 | Distance function to a submanifold | *I'm posting this sketch just for the sake of completeness. This question was already marked as answered by Anton. But I thought of this, more geometrical, argument after the discussion that answered the question in the first time. Notice that the fact that the $\Sigma$ are assumed to be submanifolds is of no relevance... | 6 | https://mathoverflow.net/users/64231 | 192138 | 94,383 |
https://mathoverflow.net/questions/192071 | 2 | If $X$ is a smooth projective variety, Kodaira's lemma states that a big line bundle $D$ can be decomposed (as $\mathbb Q$-divisors) as $A+E$, with $A$ ample and $E$ effective.
I am wondering what the correct form of this lemma in the relative setting is, and where I can read about it. Suppose that $f : X \to Y$ is a... | https://mathoverflow.net/users/nan | Relative form of Kodaira's lemma? | The usual proof of Kodaira's lemma should work:
Let $D$ be $f$-big and let $A$ and arbitrary relatively ample and effective divisor. Then consider the short exact sequence:
$$
0\to \mathscr O\_X(mD-A) \to \mathscr O\_X(mD) \to \mathscr O\_X(mD)\left|\_A\right. \to 0
$$
In the absolute case, one observes that as $... | 2 | https://mathoverflow.net/users/10076 | 192166 | 94,394 |
https://mathoverflow.net/questions/192183 | 1 | Let $G,H$ be groups and suppose that $\varphi: G\to H$ is a bijection such that for any *proper* subgroup $G'\neq G$ of $G$ the image $\varphi(G')$ is a subgroup of $H$ and the restriction $\varphi|\_{G'}: G'\to \varphi(G')$ is a group isomorphism.
Does it follow that $G\cong H$?
| https://mathoverflow.net/users/8628 | Reconstructibility of a group from subgroups | No. Here is a counterexample:
* Take $G = \mathbb{Z}/4\mathbb{Z}$, and $H = \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/2\mathbb{Z}$.
* Take any bijection $\phi \colon G \to H$ such that $\phi(0) = (0,0)$.
There are only two proper subgroups of $G$, namely $\{0\}$ and $\langle 2 \rangle$. Both are mapped isomorphically... | 10 | https://mathoverflow.net/users/21815 | 192187 | 94,403 |
https://mathoverflow.net/questions/192178 | 5 | In [this paper](http://arxiv.org/abs/1403.4325) Ando, Blumberg, Gepner, Hopkins and Rezk define the twisted $R$-Homology of a ring spectrum $R$ together with a map $f \colon X \to R$-$Line$ to be
$$
R^f\_n(X) =
\pi\_0(map\_R(\Sigma^nR, Mf)) \cong \pi\_n(Mf)
$$
where $Mf$ is the Thom spectrum associated to the above m... | https://mathoverflow.net/users/3995 | Mayer-Vietoris sequence for twisted R-homology | Here is a sketch. First, here is the argument for untwisted homology that I want to base the argument for twisted homology off of. Every categorical thing I say below is $\infty$-categorical by default, e.g. every colimit is an $\infty$-colimit and so forth.
The untwisted $R$-homology spectrum of a space $X$ with co... | 5 | https://mathoverflow.net/users/290 | 192189 | 94,405 |
https://mathoverflow.net/questions/191715 | 24 | My question is for which varieties over local fields is "independence of l" known for
etale cohomology. Say $X/{\mathbb Q}\_p$ is a complete non-singular variety and $W\_l$ is the (complex) Weil-Deligne representation associated to its etale cohomology group $H^i(\bar X,{\mathbb Q}\_l)$. Is it known that $W\_l$ is ind... | https://mathoverflow.net/users/3132 | When is "independence of l" known? | So maybe everything I'm about to say you already know, so apologies if I'm teaching my grandmother to suck eggs.
This is discussed a bit at the end of a paper of Fontaine "Representations $\ell$-adiques potentiellement semistables" (section 2.4) where he describes a notion of independence that (kind of) doesn't depen... | 10 | https://mathoverflow.net/users/13647 | 192191 | 94,406 |
https://mathoverflow.net/questions/192169 | 4 | Let $G$ be a real Lie group and $A$ a unital Banach algebra. Let us call a map $\varphi:G\to A$ a (norm-)continuous representation, if it is continuous
$$
x\_i\to x\quad\Longrightarrow\quad ||\varphi(x\_i)-\varphi(x)||\to 0,
$$
and multiplicative
$$
\varphi(x\cdot y)=\varphi(x)\cdot\varphi(y), \quad \varphi(1)=1.
$$
... | https://mathoverflow.net/users/18943 | Are norm-continuous representations smooth? | First an attempt at a counterexample. Theorem 5.2 of the paper
* Peter W. Michor: The moment mapping for a unitary representation. Annals of Global Analysis and Geometry 8, No 3 (1990), 299--313. [(pdf)](http://www.mat.univie.ac.at/~michor/moment.pdf)
shows (in a simple way) that for a unitary representation
$\rho... | 4 | https://mathoverflow.net/users/26935 | 192194 | 94,408 |
https://mathoverflow.net/questions/192202 | 11 | Let's write $S$ for the $K(1)$-local sphere at the prime 2. Then there is a cofibre sequence
$$S \to KO \to KO$$
where I'm using $KO$ to denote the $K(1)$-localization of orthogonal K-theory, and the self map of $KO$ is given in terms of Adams operations: $\psi^3-1$. Drew Heard [reminded me](http://chat.stackexchan... | https://mathoverflow.net/users/4649 | Ring structure on the $K(1)$-local homotopy of $S^0$ at the prime 2 | It must be the case that $x^2 = 0$. One can see this in at least two ways:
* The action of $\psi^3$ on $KO$ is by a ring map, and so the homotopy fixed-point spectral sequence for its action on $KO$ is multiplicative. The $\Bbb Z/2$ lives on the 1-line of this spectral sequence, and there is nothing on the 2-line or ... | 13 | https://mathoverflow.net/users/360 | 192205 | 94,411 |
https://mathoverflow.net/questions/192203 | 2 | Let $x\neq \emptyset$ be a set and let $\text{Part}(x)$ be the collection of
all partitions of $x$. We need the following notation. Let $P \in \text{Part}(x)$
and $t\subseteq x$. We set
$$P\_{[t]} = \{p\in P:p\cap t \neq \emptyset\}.$$
We define the *tiling relation*
on $\text{Part}(x)$ by
$$ P \triangleleft Q \textrm{... | https://mathoverflow.net/users/nan | Tiling relation on the set of partitions | **Question 1.** The answer is No.
Let $x= \{1,2,3,4\}$. For $i\in \{1,2,3\}$ let
$P\_i$ be the partition that has $\{i, i+1\}$ as the only non-singleton
partition block. Then $P\_i\triangleleft P\_{i+1}$ for $i = 1,2$, but it is easy to verify that $\neg(P\_1 \triangleleft P\_3)$.
**Question 2.** I can only answer t... | 2 | https://mathoverflow.net/users/8628 | 192206 | 94,412 |
https://mathoverflow.net/questions/192224 | 3 | My question is about the proof of 8.4/2 in "Neron models." The claim is that if $f\colon X \rightarrow S$ is a proper flat morphism of finite presentation such that $H^2(X\_s, \mathscr{O}\_{X\_s}) = 0$ for every $s \in S$, then the relative Picard functor $\mathrm{Pic}\_{X/S}$ is formally smooth.
The proof proceeds ... | https://mathoverflow.net/users/53197 | Fiberwise vanishing of $H^2$ and formal smoothness of the Picard functor | I'll write $p = f \times\_S Z$, and ${\cal O}^\times$ (resp. $\bar{{\cal O}}^\times$) for the units in the structure sheaf of $X \times\_S Z$ (resp. $X \times\_S Z\_0$, viewed as a sheaf on the same space).
First, one shows that the map $p\_\*{\cal O}^\times \to p\_\* \bar{{\cal O}}^\times$ is a surjective map of she... | 6 | https://mathoverflow.net/users/360 | 192235 | 94,420 |
https://mathoverflow.net/questions/192234 | 1 | What is known about the asymptotic behavior of
$$
f(x)=\sum\_{p\le x}\frac1p-\log\log x-B\_1?
$$
Of course by Mertens we know that $f(x)=o(1),$ but has more been proved (in terms of $O$ or $\Omega\_\pm$, say)? A reference would be great.
| https://mathoverflow.net/users/6043 | Error term for prime harmonic | ["Mertens' Proof of Mertens' Theorem"](http://arxiv.org/abs/math/0504289) suggests that Mertens had an error term of $O\left(\frac1{\ln x}\right)$, though that's not tight; theorem 14 there offers an $O\left(\frac1{\ln^2x}\right)$ unconditional estimate and an $O(x^{-1/2}\ln x)$ estimate dependent on RH, with reference... | 5 | https://mathoverflow.net/users/7092 | 192236 | 94,421 |
https://mathoverflow.net/questions/192229 | 4 | Let $B(n)$ denote the Bell's number which is the number of the equivalence relation which can be defined on a set of cardinality $n$.
While I was trying to solve a problem, I reached another result;
$$B(p^k)\equiv k+1 \ mod \ p$$
Is this result evident or trivial ? Any comments and remark are welcome.
**Note:**... | https://mathoverflow.net/users/47344 | Is this property of the Bell's number evident? | This follows from the well-known Touchard's congruence ([here](http://arxiv.org/abs/0906.0696) for a random reference). Following your notation, the congruence is:
$$B(n+p^k)\equiv kB(n)+B(n+1) \mod \ p$$
Taking $n=0$:
$$B(p^k)\equiv kB(0)+B(1) \mod \ p$$
And since $B(0)=B(1)=1$,
$$B(p^k)\equiv k+1 \mod \ p$$... | 5 | https://mathoverflow.net/users/43108 | 192237 | 94,422 |
https://mathoverflow.net/questions/192225 | 2 | Is there agreement on how to interpret $r$ and $\varphi$ on a manifold if a reference point and a reference direction are given, or, put differently, does the definition of a reference point and, of a reference direction suffice to make polar functions on the manifold well defined?
While I think that the definition o... | https://mathoverflow.net/users/31310 | Polar Coordinate Systems on Manifolds | Typically such a chart is only used or only useful when there is an SO(2)$\cong$U(1) isometry of the manifold, or more generally an SO($n$) isometry for a higher dimensional manifold. If so, then it's conventional to choose the angular coordinates in the standard way on the $(n-1)$-spheres which are the orbits under th... | 5 | https://mathoverflow.net/users/26762 | 192238 | 94,423 |
https://mathoverflow.net/questions/192239 | 1 | Consider a surface of revolution of positive curvature. My question is, what are the surfaces (with boundary) in $\mathbb{R}^3$ which are isotopic to the surface of revolution, provided each member of the family is positively curved? I am trying to see if we can achieve any positively curved surface (with boundary) dif... | https://mathoverflow.net/users/64672 | Isotopy of positively curved surfaces of revolution in $\mathbb{R}^3$ | 1. The two boundary components of positively curved surface might be linked.
In this case it is not isotopic to the surface of revolution.
2. The topology of surface of revolution is either disc or cylinder.
In the first case you can construct an isotopy to a tiny cap in the disc which is isotopic to the rotationally s... | 0 | https://mathoverflow.net/users/1441 | 192243 | 94,424 |
https://mathoverflow.net/questions/192259 | 2 | Let $X$ be a set and $\omega$ be a family of its subsets. Consider the family
$\mathcal{F}$ of subsets of $X$, such that any $A\in\mathcal{F}$ has a
non-empty intersection with each element of $\omega$. Let $\tau(\omega)$
denote the following cardinal
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \tau(... | https://mathoverflow.net/users/64681 | Is there a name for this cardinal? | It appears in the Hitting Set Problem, and so it would make some sense to call $\tau(\omega)$ the [hitting set](http://www.nada.kth.se/~viggo/wwwcompendium/node149.html) cardinality.
| 1 | https://mathoverflow.net/users/4600 | 192261 | 94,429 |
https://mathoverflow.net/questions/192213 | 3 | Let $X$ be a n-dimensional complex (not necessarily compact) manifold, let $G$ be a finite subgroup of $Aut(X)$ acting by biholomorphic maps on $X$ and let $D(X)$ be the ring of differential operators on $X$. I would like to compute the Hochschild cohomology group $HH^{m}(D(X)\rtimes G,D(X)\rtimes G)$, $m\geq0$ of the ... | https://mathoverflow.net/users/48531 | Hochschild cohomology of the skew group ring D(X)#G in the complex analytic case | Here is my solution to the problem I posted yesterday.
I would make use of the fact that $D(X)$ is a locally convex algebra. Assuming that one can define a bornology on it (see <http://arxiv.org/pdf/0706.0027.pdf>) the smash product $D(X)\rtimes G$ can be made into a bornological algebra, too. Then we may use the fact ... | 1 | https://mathoverflow.net/users/48531 | 192268 | 94,431 |
https://mathoverflow.net/questions/192221 | 1 | Let $R^{1,2}$ be the Minkowski 3-space, I would like to know any references about minimal (maximal) orientable Lorentzian surfaces in $\mathbb{R}^{1,2}$, including examples and maybe general theories, say something like Weierstrass representation. Here by minimal (maximal) I simply mean that the Lorentzian surface, as ... | https://mathoverflow.net/users/41734 | Reference request: minimal (maximal) Lorentzian surfaces in $\mathbb{R}^{1,2}$ | These two books might be useful:
<http://books.google.be/books/about/An_Introduction_to_Lorentz_Surfaces.html?id=1S_YJ39DSdcC&redir_esc=y>
<http://www.worldscientific.com/worldscibooks/10.1142/7542>
(the Weierstrass representation is treated in detail in the second one)
| 2 | https://mathoverflow.net/users/61461 | 192272 | 94,432 |
https://mathoverflow.net/questions/191305 | 8 | Let $NC(n)$ denote the lattice of noncrossing partitions of $n$, and let $G$ denote the Hasse diagram of $NC(n)$ with respect to covering relations, viewed as an undirected graph.
I'm interested in the connectivity of graphs of geodesics between vertices of $G$. If $p$ and $q$ are two noncrossing partitions of $n$, l... | https://mathoverflow.net/users/8041 | For any two noncrossing partitions $p, q$ of $n$, is the graph of geodesics from $p$ to $q$ in $NC(n)$ connected? | Third attempt, this time I try to prove that $G\_{p,q}$ is connected.
Before the proof we need a simple observation about NC partitions.
**Observation.**
Draw the elements of $\{1,\ldots,n\}$ on a circle.
Any NC partition $p=(p\_1,\ldots,p\_k)$ of $\{1,\ldots,n\}$ either has a singleton part $|p\_i|=1$ or a part that... | 4 | https://mathoverflow.net/users/955 | 192273 | 94,433 |
https://mathoverflow.net/questions/186368 | 8 | Let's say we have a finite subgroup $\Gamma \subseteq SL(2,\mathbb{C})$ and consider the quotient variety $\mathbb{C}^2/\Gamma$, which will have one of the well-known ADE or du Val surface singularities and can be embedded into $\mathbb{C}^3$ as a hypersurface with a singular point at the origin. These singularities ha... | https://mathoverflow.net/users/61411 | Resolving ADE singularities by blowing up | (Maybe this is an answer, maybe it merely moves the lump around under the carpet.) M. Artin showed ["On rational singularities of surfaces"] that for any rational surface singularity (in particular, for du Val singularities) the fundamental cycle $Z$ (by definition, the smallest non-zero effective divisor $Y$ supported... | 6 | https://mathoverflow.net/users/8726 | 192277 | 94,434 |
https://mathoverflow.net/questions/192270 | 5 | Is there a reference that contains explicit examples of component crossing of Hida families at height one primes? The paper of Emerton, Pollack, and Weston addresses component crossing obtained through level raising. I am interested in examples caused by other phenomena (e.g. a CM family meeting another CM family comin... | https://mathoverflow.net/users/13158 | Examples of component crossing between families of modular forms | There is an interesting example in this [paper](http://www.math.tifr.res.in/~eghate/weight1-jntb-final.pdf) by Dimitrov and Ghate, section 7.3.
| 4 | https://mathoverflow.net/users/9317 | 192280 | 94,435 |
https://mathoverflow.net/questions/192287 | 6 | Let $\mathbb{S}$ be the Sierpinski space, that is $\mathbb{S}$ has $\{0,1\}$ as a base set, and $\tau = \{\emptyset, \{0\}, \{0,1\}\}$ as a topology.
The Sierpinski space $\mathbb{S}$ has the following property:
>
> $(\star)$ Given any topological spaces $X,Y$ and a function $f:X\to Y$, then $f$ is continuous if... | https://mathoverflow.net/users/8628 | Sierpinski-like spaces | The following are equivalent:
1. $S$ is Sierpiński-like.
2. $\mathbb S$ embeds in $S$ as a subspace.
3. $S$ is not [symmetric](https://en.wikipedia.org/wiki/T1_space#Definitions) (i.e., $R\_0$).
$2\to1$ follows from the facts that $\mathbb S$ is Sierpiński-like, and if $Y\subseteq Z$ is a subspace and $f\colon X\to... | 7 | https://mathoverflow.net/users/12705 | 192288 | 94,438 |
https://mathoverflow.net/questions/87860 | 31 | A celebrated recent theorem of [Birkar-Cascini-Hacon-McKernan](http://dx.doi.org/10.1090/S0894-0347-09-00649-3) and [Siu](http://arxiv.org/abs/math/0610740) says that the canonical ring $R(X)=\oplus\_{m\geq 0}H^0(X,mK\_X)$ of any smooth algebraic variety $X$ over $\mathbb{C}$ is a finitely generated $\mathbb{C}$-algebr... | https://mathoverflow.net/users/13168 | Example of a compact Kähler manifold with non-finitely generated canonical ring? | As pointed out by Ruadhai Dervan in the comments, [a paper by Fujino](http://arxiv.org/abs/1309.3015) contains the answer to this question: the canonical ring of any compact Kähler manifold is finitely generated. By bimeromorphic invariance of this ring, the result even holds for compact complex manifolds in Fujiki's c... | 7 | https://mathoverflow.net/users/13168 | 192292 | 94,440 |
https://mathoverflow.net/questions/192290 | 4 | In Landau, Lifshitz, "Quantum Mechanics, non-relativistic theory" in $\S18$ "The fundamental properties of Schrödinger's equation" the following is said about potential $U(x,y,z)$ in a footnote:
>
> it must be mentioned that, for some particular mathematical forms of the function $U(x,y,z)$ (which have no physical ... | https://mathoverflow.net/users/29747 | Examples of potentials for which Schrödinger equation lacks discrete points in continuous spectrum | The spectrum of an operator is always a closed set. But perhaps they are defining the "continuous spectrum" to be all points of the spectrum that are not in the point spectrum (i.e. not eigenvalues). Then there can be eigenvalues surrounded by continuous spectrum. The classic example of this in a Schrödinger operator i... | 4 | https://mathoverflow.net/users/13650 | 192293 | 94,441 |
https://mathoverflow.net/questions/103358 | 18 | Suppose $M$ is a Moishezon manifold with $c\_1(M)=0$ in $H^2(M,\mathbb{R})$. Does it follow that $K\_M$ is torsion in $\mathrm{Pic}(M)$?
This is true whenever $M$ is Kähler (and therefore projective) and was proved independently by [Bogomolov,](http://iopscience.iop.org/0025-5726/8/1/A02) [Fujiki](https://doi.org/10.... | https://mathoverflow.net/users/13168 | Moishezon manifolds with vanishing first Chern class | This problem is solved affirmatively in Theorem 1.5 in [this paper.](http://arxiv.org/abs/1401.4797)
The idea is that after some blowups we obtain a compact Kähler manifold whose canonical bundle is effective after twisting by a numerically trivial line bundle. A seminal result of [Simpson](http://www.numdam.org/ite... | 3 | https://mathoverflow.net/users/13168 | 192294 | 94,442 |
https://mathoverflow.net/questions/89798 | 36 | As observed by [Calabi](http://dx.doi.org/10.2307/1993108) a long time ago, the manifold $S^2\times S^4$ admits an almost-complex structure (obtained by embedding it in $\mathbb{R}^7$ and using the octonionic product), which however is not integrable.
Is it known whether $S^2\times S^4$ admits an integrable complex s... | https://mathoverflow.net/users/13168 | Is S^2 x S^4 a complex manifold? | This is still an open problem. See [this paper](http://arxiv.org/abs/1301.6835) for some progress, which was prompted by this MO question.
| 8 | https://mathoverflow.net/users/13168 | 192295 | 94,443 |
https://mathoverflow.net/questions/192301 | 6 | As should be clear, I would like to know if it is true that a given commmutative square of spaces (i.e. simplicial sets) is a homotopy pullback iff the induced map on each homotopy fiber is a weak equivalence. More precisely, consider the following diagram:
$$\begin{array}{c}&&&&& A& \longrightarrow & B\\
&&&&&\downarr... | https://mathoverflow.net/users/57280 | Can homotopy pullbacks of spaces be checked on fibers? | $\require{AMScd}$I don't know a reference but the proof is easy enough. Form homotopy pullback squares
\begin{CD}
Fu @>>> Ff @>>> A \\
@VVV @VVV @V{u}VV \\
\* @>>> Fg @>>> P @>>> B \\
@. @VVV @VVV @VV{g}V \\
@. \* @>>> C @>>> D
\end{CD}
so that $Ff$, $Fg$ and $Fu$ are the homotopy fibers of $f$, $g$ and $u$ (where ... | 15 | https://mathoverflow.net/users/12547 | 192303 | 94,447 |
https://mathoverflow.net/questions/192316 | 10 | After Bott periodicity is proved, one still has to compute the stable values. For the unitary group $U$, this is easy since you can get away with just $\pi\_0$ and $\pi\_1$. However, I'm having trouble doing this for the orthogonal group. It's easy to work out $\pi\_i(O)$ for, say, $0 \leq i \leq 2$, but already for $i... | https://mathoverflow.net/users/64701 | The periodic values in Bott periodicity | Bott's result is best formulated by saying that the sequence of spaces
$$
\mathbb Z\times BO,\quad O,\quad O/U,\quad U/Sp,\quad \mathbb Z\times BSp,\quad Sp,\quad Sp/U,\quad U/O
$$
are related by the property that each one is the loop space of the previous one (mod 8).
From that, you get for example that $\pi\_4(O)=... | 20 | https://mathoverflow.net/users/5690 | 192317 | 94,452 |
https://mathoverflow.net/questions/192325 | 1 | every orthonormal basis is a parseval frame. but what about the converse in the finite dimensional case?
Let's say $H$ is a n-dimensional Hilbert space and $a\_1,..,a\_n$ a parseval frame. then, of course, $a\_1,..,a\_n$ is a basis, since every frame spans the whole space (or maybe the closure in the infinite dimens... | https://mathoverflow.net/users/64704 | When is a Parseval frame an orthonormal basis? | Yes, because a Parseval frame has the property that for any $i$,
$$a\_i = \sum\_{j=1}^n \langle a\_i, a\_j \rangle a\_j$$
and since the $a\_i$ form a basis, the coefficients of $a\_i$ on both sides must be the same, which implies that all the $a\_i$ are orthogonal to one another, and also that $||a\_i||^2 = 1$.
(If ... | 4 | https://mathoverflow.net/users/24993 | 192326 | 94,457 |
https://mathoverflow.net/questions/192320 | 2 | Let $G$ be a reductive group and let $P$ be a parabolic subgroup of $G$ all defined over $\mathbb{Z}$.
Also, let $F$ be a number field, is it true (and if so, please provide a reference) that
$$ \left(G/P\right)(F) = G(F)/P(F)$$
If it is not always true, is there a criterion for $G$, $P$ and $F$ so it will be true?
| https://mathoverflow.net/users/64702 | Rational Points of a Quotient of a Reductive Group by a Parabolic Subgroup | This is true: Borel/Tits,Groupes Reductifs, IHES, 27, 1965, Theorem 4.13(a).
| 4 | https://mathoverflow.net/users/6030 | 192328 | 94,458 |
https://mathoverflow.net/questions/192246 | 1 | We have the following result by Spitzer (see (1) or Port)
$lim\_{t\to \infty}\frac{1}{t}\int\_{\mathbb{R}^{n}/B\_{r\_{0}}}P\_{x}(T\_{B\_{r\_{0}}}<t)dx=Cap(B\_{r\_{0}})=\frac{r\_{0}}{4\pi}$
By Chuancun and Rong (2) we have
$P\_{x}(T\_{B\_{r\_{0}}}<t)=\frac{2r\_{0}}{\pi |x|}\int\_{0}^{\infty}u^{-1}(1-e^{\frac{-u^{2... | https://mathoverflow.net/users/40793 | Deriving Newtonian capacity of sphere from Brownian motion | First of all, please note that your second equation should have $u^{-1}$ instead of $u$, see page 580 of the Chuancun & Rong paper you cite. So we need to evaluate (in $n=3$ dimensions):
$$Cap(B\_{r\_{0}})=\lim\_{t\to \infty}\frac{1}{t}\int\_{r\_0}^\infty 4\pi r^2\,dr\int\_{0}^{\infty}du\, \frac{2r\_0}{\pi r}\,u^{-1}... | 1 | https://mathoverflow.net/users/11260 | 193337 | 94,460 |
https://mathoverflow.net/questions/73166 | 6 | These proceedings have long been freely available on the AMS website, but now it seems we can't even find them anymore (e.g. <http://www.ams.org/publications/online-books/pspum331-index> and <http://www.ams.org/publications/online-books/pspum332-index>). Unfortunately, I don't own a copy, nor have I ever downloaded it ... | https://mathoverflow.net/users/13027 | Corvallis 1979 proceedings | Given the importance of the proceedings, I leave a link to the copy at Library Genesis:
A. Borel and W. Casselman (Editors) [Automorphic Forms, Representations and L-functions](http://libgen.in/get.php?md5=B528264153CDDD8461BE1CC50CE4000F) (1979)
| 6 | https://mathoverflow.net/users/43108 | 193340 | 94,462 |
https://mathoverflow.net/questions/192046 | 6 | An action of a group $G$ on a set $X \neq \emptyset$ is called [*transitive*](http://en.wikipedia.org/wiki/Group_action#Types_of_actions) if $\forall x,y \in X$, $\exists g \in G$ such that $g.x = y$.
It is called [*primitive*](http://en.wikipedia.org/wiki/Group_action#Types_of_actions) if it is transitive and prese... | https://mathoverflow.net/users/34538 | Are the distributive permutation groups linearly primitive? | **Yes**, a distributive permutation group is linearly primitive.
*Proof*: Let $G \subset S\_n$ be a distributive permutation group, then $G\_1 \subset G$ is a core-free subgroup and the lattice $[G\_1, G]$ is distributive. Then, by the [dual Ore's theorem](https://mathoverflow.net/q/195366/34538), there is an irr. co... | 1 | https://mathoverflow.net/users/34538 | 193342 | 94,463 |
https://mathoverflow.net/questions/192306 | 7 | The ordinary Laplacian on $\mathbb{R}^N$ behaves nicely under a [stereographic projection](http://mathworld.wolfram.com/StereographicProjection.html) onto $\mathbb{S}^N\setminus\{P\}$. (Here $P$ is either the north or south pole of the unit sphere $\mathbb{S}^N$.) Namely, letting
$$
\Omega=\frac{2}{1+\lvert x \rvert^2... | https://mathoverflow.net/users/13042 | Fractional Laplacian and stereographic projection | Unless I'm misreading, there exist such an analogue at least for some $\gamma$. See equation 2.5 and theorem 3.2 of [Fractional Laplacian in Conformal Geometry](http://www.pagines.ma1.upc.edu/~mgonzalez/Papers/Chang_Gonzalez.pdf) by Sun-Yung Alice Chang and María del Mar Gonzáles.
To spell it out, the theorem 3.2 say... | 3 | https://mathoverflow.net/users/6818 | 193344 | 94,464 |
https://mathoverflow.net/questions/191166 | 8 | In [Old Home Page of Andreas Weiermann](http://wwwmath.uni-muenster.de/u/weiermann/) Andreas Weiermann has stated the following:
>
> Quite recently I submitted a preprint about an application of the Riemann hypothesis and the ABC conjecture to independence results for publication.
>
>
>
**Question 1.** can som... | https://mathoverflow.net/users/11115 | Application of the Riemann hypothesis and the ABC conjecture to independence results | Presumably this is my task:
**Question 1. can someone give a short description of the stated work(s)?**
The work centers around the "phase transition for
Gödel incompleteness" program. The basis idea is to consider an assertion $A(f)$
depending on a function parameter $f$ so that $A(f)$ is true, $A(f)$ is PA-prova... | 15 | https://mathoverflow.net/users/62695 | 193350 | 94,466 |
https://mathoverflow.net/questions/193346 | 1 | Say $A$ is a $(n-1)\times (n-1)$ matrix and we augment it by a $n^{th}$ row and a column and get a $n \times n$ matrix $B$. Is there a nice way to relate $det(B)$ and $det(A)$ and the added row and column?
---
A close by thing I am reminded of is this,
<http://en.wikipedia.org/wiki/Matrix_determinant_lemma>
| https://mathoverflow.net/users/36554 | Are there good ways of relating a minor to the full determinant? | yes, there is the Sherman-Morrison formula
$$\det B=(\det A)(b-yA^{-1}x),$$
where $b, x$ and $y$ are blocks:
$$B=\begin{pmatrix} A & x \\ y & b \end{pmatrix}.$$
*Edit*. After Hachino's comment, one can also write
$$\det B=b\det A-y\hat Ax,$$
where $\hat A$ is the transpose of the cofactor matrix.
| 8 | https://mathoverflow.net/users/8799 | 193351 | 94,467 |
https://mathoverflow.net/questions/193330 | 3 | I asked the question on [math.stackexchange.com](https://math.stackexchange.com/questions/1090494/method-of-characteristics-of-a-system-of-first-order-pdes), but didn't get any reply. So, I asked it again here. Any suggestion or hint is welcome, and thank you for your attention.
Consider the system of first order PD... | https://mathoverflow.net/users/56654 | Method of characteristics of a system of first order pdes | First of all, your system seems to uncouple quite strongly. The first and third equations only involve the unknowns $v\_1$ and $p\_1$ and the second and fourth equations only involve $v\_2$ and $p\_2$, so you are really talking about two independent problems, so you should separate them this way.
Second, if you add ... | 6 | https://mathoverflow.net/users/13972 | 193354 | 94,469 |
https://mathoverflow.net/questions/193357 | 2 | Let $(M,g)$ be a compact Riemannian manifold. It is known there exist Gaussian estimates of the heat kernel and its derivatives acting on functions on $M$.
The kind of estimate I'm looking for could be found in the first page of the following article : <http://www.math.uni-bielefeld.de/~grigor/grad.pdf>
My question... | https://mathoverflow.net/users/nan | Heat Kernel estimate at the level of the form | There should be better references, but for a beginning, how about the following: Look at Lemme 1 in Thierry Bouche's paper *Convergence de la metrique de Fubini-Study d'un fibre lineaire positif*, Ann. Inst. Grenoble 1990. Here is the link to the paper:
<http://archive.numdam.org/ARCHIVE/AIF/AIF_1990__40_1/AIF_1990__... | 0 | https://mathoverflow.net/users/26522 | 193364 | 94,471 |
https://mathoverflow.net/questions/193363 | 8 |
>
> There is a pile of $n$ items. Every time you divide a pile into two piles, you get a score being the product of the number of items in the two piles. Show that the sum of your scores at the end is always $\binom{n}{2}$.
>
>
>
My question: What are some (preferably well-known) books/articles that discuss or a... | https://mathoverflow.net/users/65718 | Reference for puzzle on dividing piles and scoring products | This [entry](http://www.cut-the-knot.org/proofs/piles.shtml) on *Cut the Knot* gives a proof and the reference to the book *Exploring Mathematics with Your Computer* by A. Engel. I am not sure if the puzzle was invented by Engel, but hopefully the book will have an earlier reference if that's not the case (unfortunatel... | 6 | https://mathoverflow.net/users/2233 | 193370 | 94,473 |
https://mathoverflow.net/questions/193369 | 3 | Let $G$ by a hyperbolic group, and let $H \lhd G$ be a normal quasiconvex subgroup. Is it possible that $|H| = [G : H] = \infty$ ?
| https://mathoverflow.net/users/38889 | Are there quasiconvex normal subgroups? | Greenberg's theorem for hyperbolic groups, proved by Kapovich and Short, asserts that such an $H$ is finite.
| 7 | https://mathoverflow.net/users/1463 | 193371 | 94,474 |
https://mathoverflow.net/questions/185900 | 2 | **Background:** Let $(Y, \mathcal{B},\mu,T)$ be an ergodic probability system and let $G$ be a compact metrizable group with compact subgroup $H$. Given a measurable map $\rho:Y \to G$. We may define the skew product system $W\_{H,\rho}=(Y \times G/H, \mu \times m\_{G/H}, T\_{\rho})$ where the transformation is given b... | https://mathoverflow.net/users/47704 | Eigenfunction of ergodic skew product fixed by commutator? | Yes, the eigenfunctions of $X = Y\times G/H$ come from $Y\times A$, where $A$ is an abelian quotient of $G$. I'll indicate how to construct $A$, checking the required properties is then routine. Note that we may assume $H$ trivial, since eigenfunctions of $Y\times G/H$ are also eigenfunctions of $Y\times G$.
Note tha... | 1 | https://mathoverflow.net/users/24201 | 193372 | 94,475 |
https://mathoverflow.net/questions/192267 | 3 | In the context of the Minimal Model Program it can arise that we need to deal with contractions of extremal rays that are conic bundles $\pi:X\to Y$ with relative Picard number 1 (with possibly degenerate fibers). So we can consider the discriminant of $\pi$:
$$\Delta=\{y\in Y\;|\;\pi^{-1}(y)\;\text{is not a smooth c... | https://mathoverflow.net/users/31724 | Discriminant of a singular conic bundle | The OP clarified that there should be no flatness hypothesis in this question. Without any flatness hypothesis, certainly there are examples where $\Delta$ is not a divisor. (There may be flat examples as well.) First of all, for a coherent sheaf $\mathcal{F}$ on $Y$ that is reflexive of rank $2$, yet not locally free,... | 5 | https://mathoverflow.net/users/13265 | 193375 | 94,476 |
https://mathoverflow.net/questions/192045 | 3 | This is an extended question based on
[Large deviations for maximizer of random walk with drift](https://mathoverflow.net/questions/189143/large-deviations-for-maximizer-of-random-walk-with-drift).
Let $$S\_k = X\_1 + \ldots + X\_k,$$ where $X\_i$ are i.i.d. with mean $-\mu < 0$ and unit variance. Assume any nice pro... | https://mathoverflow.net/users/8369 | Maximizer of random walk with very small drift | I looked at this question a while ago, see <http://peter.windridge.org.uk/minRWepsdrift.pdf> . Minimiser is $O\_p(1/\mu^2)$ = Lemma 2.2, using Kolmogorov's max inequality etc. I think that part only uses 2 moments.
Disclaimer: those notes are not the latest version, so might contain minor mistakes and the conditions ... | 0 | https://mathoverflow.net/users/57941 | 193376 | 94,477 |
https://mathoverflow.net/questions/193358 | 2 | Here I am asking for an analogue of [Generating infinite index subgroups of a free group](https://mathoverflow.net/questions/191855/generating-infinite-index-subgroups-of-a-free-group)
Let $F$ be a nonabelian finitely generated free group, let $H \leq F$ be a finitely generated subgroup of infinite index, and let $x ... | https://mathoverflow.net/users/38889 | Conjugates and infinite index subgroups of free groups | The proof is like before. Let $\Gamma$ be the Stallings graph for $H$. Choose a word $w$ labeling a path from the base point to a vertex q where some letter or its inverse, call it a, cannot be read. If $axa^{-1}$ is reduced, then by sowing an edge a at q followed by x as a loop, we obtain a Stallings graph of an infin... | 6 | https://mathoverflow.net/users/15934 | 193380 | 94,478 |
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