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https://mathoverflow.net/questions/134771 | 3 | Let $\mathcal{K} = \mathbb{C}((t)), \mathcal{O}=\mathbb{C}[[t]]$, $G$ be a reductive group, and $\text{Gr}\_G=G(\mathcal{K})/G(\mathcal{O})$; there is a left action of $G(\mathcal{O})$ on $\text{Gr}\_G$. Let $X\_\*(T)=\text{Hom}(\mathbb{C}^{\times},T)$ (note that there is a natural embedding of $X\_\*(T)$ inside $G(\ma... | https://mathoverflow.net/users/2623 | Dimension of orbits in affine Grassmanian | The same way you find the dimension of an orbit in any homogeneous space. The orbit is $G(\mathcal{O})/(G(\mathcal{O})\cap t^{\lambda}G(\mathcal{O})t^{-\lambda})$, so you just have to calculate the corresponding quotient $\mathfrak{g}[[t]]/(\mathfrak{g}[[t]]\cap \operatorname{Ad}\_{t^\lambda}\mathfrak{g}[[t]])$ for Lie... | 6 | https://mathoverflow.net/users/66 | 134843 | 74,352 |
https://mathoverflow.net/questions/134682 | 1 | Let $R=k[v,x,y,z]/I$, with $I=\langle v^2,z^2,xy,vx+xz,vy+yz,vx+y^2,vy-x^2\rangle$,and let
$f:R^2 \rightarrow R^2$ denote the map given by the matrix
$$M=\begin{pmatrix}
v & y \\
x & z
\end{pmatrix}$$
I guess that there is a module $N$ such that $\operatorname{coker}f \cong N\oplus k$, but I don't know how to prove i... | https://mathoverflow.net/users/5775 | Decomposition of a quotient module | **Edited to add:** Well, now I feel embarrassed to have gotten an answer accepted which is absolute garbage, so I think I should offer an actual answer in addition to the indirect proof in a comment I made above (i.e. the Betti numbers of the cokernel $A$ of your matrix are 2,2,4,11,32,95,..., while those of the residu... | 1 | https://mathoverflow.net/users/460 | 134854 | 74,354 |
https://mathoverflow.net/questions/134776 | 4 | Let $A\in \mathbb{R}^{n\times n}$ and $u$ and $v$ be independent random $\{-1,1\}^n$-vectors. (i.e., each coordinate of $u$ is $1$ with prob. $1/2$ and $-1$ with prob. 1/2 and the coordinates of $u$ are independent with each other)
**Question.** Is there a tail bound for $\Pr\{u^T A v > t\}$?
I am hoping something... | https://mathoverflow.net/users/nan | bilinear form tail bound | You can write the bilinear form as a quadratic form, by letting x = (u,v) and rewrite the matrix A.
| 4 | https://mathoverflow.net/users/35837 | 134865 | 74,356 |
https://mathoverflow.net/questions/134813 | 3 | Let $K$ be a non-empty compact convex subset of a Banach space $E$, and let $f : K \longmapsto K$ be a continuous function. Fix $u\_0 \in K$, and define by recurrence $u\_{n+1} = \frac{1}{n+1} \sum\_{j=0}^n f(u\_j)$ ; or equivalently $u\_{n+1} - u\_n = \frac{1}{n+1} (f(u\_n) -u\_n)$.
>
> Is it always true that $(u\... | https://mathoverflow.net/users/21724 | A recurrent sequence related to the Brouwer fixed-point theorem | A counterexample in dimension $2$: take $K$ the unit closed disk of $\mathbb{C}$, and $f:K\to K$ the map
$$f(z)=e^{i\pi/4}\frac{z}{|z|}\min(2|z|,1)$$
for $z\neq0$, and $f(0)=0$, which is the only fixed point of $f$. Then, starting by $u\_0\in K$ with $|u \_ 0|\ge1/2$ produces a sequence with $|u\_n|\ge 1/2$. (Reason: ... | 4 | https://mathoverflow.net/users/6101 | 134866 | 74,357 |
https://mathoverflow.net/questions/134844 | 11 | Can one prove that Poincaré duality groups cannot have intermediate growth?
| https://mathoverflow.net/users/2029 | Growth of Poincaré duality groups | The question is well beyond of what is currently known about Poincare duality groups and groups of intermediate growth. The only known case is of PD(2) groups, since they are virtually surface groups. The answer is unclear already for PD(3) groups (conjecturally, they are 3-manifold groups, so they should not have inte... | 13 | https://mathoverflow.net/users/21684 | 134879 | 74,358 |
https://mathoverflow.net/questions/134822 | 4 | Theorem 3.3.4 in Davies' *Heat Kernels and Spectral Theory* begins with ``on-diagonal'' lower bounds for the heat kernel $K$ of $H$, (i.e. $K = e^{-Ht}$), where $H$ is a uniformly elliptic operator acting on $L^{2}(\mathbb{R}^{N})$. That is, Davies has already proved
$$K(t,x,y) \geq C t^{-N/2} \qquad \text{ when } \qq... | https://mathoverflow.net/users/12968 | On-diagonal to off-diagonal heat kernel lower bounds, Davies' argument | This is a classic trick for heat kernel proofs:
The idea is that if $f(t,x)$ is a solution to the heat equation with initial data $f\_0(x)$, then $g(t,x) := f(t+s,x)$ is a solution to the heat equation with initial data $g\_0(x) = f(s,x)$.
Rephrasing that $f(t,x)$ is a solution to the heat equation as:
$$
f(t,x) = ... | 4 | https://mathoverflow.net/users/1540 | 134913 | 74,360 |
https://mathoverflow.net/questions/134823 | 1 | Is there an approximation for the maximum difference between P(n) and n x ln(n) as a function of n, where P(n) denotes the nth prime number?
In other words, given D(n) = Max(|P(n) - n x ln(n)|), is there a known formula for D(n)?
Thanks
| https://mathoverflow.net/users/27456 | The maximum difference between the nth prime number and n x ln(n) | There are several good inequalities for this difference, e.g.,
$$
n \log n + n(\log \log n -1)<P(n) < n \log n + n\log \log n
$$
for all $n\ge 6$, which can be derived from the prime number theorem.
Better estimates are given (among others) in the thesis of Dusart - see <http://www.unilim.fr/laco/rapports/1998/R1998_06... | 4 | https://mathoverflow.net/users/32332 | 134922 | 74,361 |
https://mathoverflow.net/questions/133399 | 4 | **Setup.** Let casino generate a color: black or red with equal probability.
Let client try to guess the color. If guess is correct - he earns 1 coin from casino, if not - he gives one to casino. If he loses all his coins he stops the game.
Assume that client comes to casino with just 1 coin, while casino has infin... | https://mathoverflow.net/users/10446 | Casino does not win, while clients do lose ? Prob_loss(T) = 1 - .8/sqrt(T)? | By the reflection method, the number of ways to reach $a$ wins and $b \le a$ losses without the losses ever exceeding the number of wins is ${a+b \choose b}-{a+b \choose b-1}$, and the probability is the count times $2^{-(a+b)}$. The probability that you have not lost by time T is the sum of these probabilities so that... | 6 | https://mathoverflow.net/users/2954 | 134923 | 74,362 |
https://mathoverflow.net/questions/134733 | 10 | I am looking for a generalized central limit theorem for non-square integrable stationary sequences. More precisely I suspect that when $(X\_j)\_{j\geqslant 1}$ is a stationary sequence such that $X\_i$ belongs to the domain of attraction of the $\alpha$-stable law then
$$n^{1/\alpha}\sum\_{i=1}^n X\_i$$
converges ... | https://mathoverflow.net/users/1302 | Generalized central limit theorem | One of the most recent results in this area is
Katarzyna Bartkiewicz, Adam Jakubowski, Thomas Mikosch, Olivier Wintenberger, *Stable limits for sums of dependent infinite variance random variables*, Probability Theory and Related Fields 150, 3 (2011) 337-372
which is available [here](http://arxiv.org/abs/0906.2717... | 7 | https://mathoverflow.net/users/17118 | 134931 | 74,364 |
https://mathoverflow.net/questions/134902 | 9 | Given a smooth genus 2 curve $C$, it is canonically a two-fold cover of a $\mathbb{P}^1$, branched at six points. Allowing stable curves allows degenerations in two directions: the six points are allowed to collide (in certain ways), and $\mathbb{P}^1$ can "break" into a degenerate conic in $\mathbb{P}^2$ (here a union... | https://mathoverflow.net/users/344 | Examples of families of stable genus 2 curves | As Akhil points out, you can have both types of behavior simultaneously. For instance, consider a curve $C$ that is a union of two irreducible components $C'$ and $C''$ that intersect transversally at a single point $p$ (smooth on each component). Let $C'$ be a smooth, genus $1$ curve. Let $C''$ be a nodal genus $1$ cu... | 4 | https://mathoverflow.net/users/13265 | 134933 | 74,365 |
https://mathoverflow.net/questions/134824 | 10 | I am trying to understand quasi-isomorphisms in an exact category as defined via the mapping cylinder. I would like to know whether these form a category of weak equivalences in the sense of Waldhausen.
Some background.
Recall that in an exact category in the sense of Quillen we have a good notion of a long exact s... | https://mathoverflow.net/users/31603 | Quasi-isomorphisms in exact categories | If $\mathcal E$ is karoubian, i.e. idempotent complete, then $q\_1=q\_2$ on bounded complexes. This was noticed in 1.11.8 of Thomason-Trobaugh's paper, see also 1.11.5. On unbounded complexes, $q\_1\subsetneq q\_2$ even in the karoubian case, e.g. in the category $\mathcal E$ of free $\mathbb Z/4$-modules, the complex
... | 3 | https://mathoverflow.net/users/12166 | 134935 | 74,367 |
https://mathoverflow.net/questions/134924 | 7 | **Motivation**
For a fixed group $G$, consider the category of $G$-groups, whose objects consist of (not necessarily abelian) groups $A$ together with an action $G\to \mathrm{Aut}(A)$. Now we can construct a nonabelian $H^1$, a functor from $G$-groups to pointed sets by mimicking the abelian construction. This constr... | https://mathoverflow.net/users/1437 | Does the forgetful G-Groups to Groups (nonabelian) have a left adjoint? | Yes. I believe there's a general theorem to the effect that if $T\_1 \to T\_2$ is any morphism of [Lawvere theories](http://en.wikipedia.org/wiki/Lawvere_theory), then the corresponding functor $\text{Prod}(T\_2, \text{Set}) \to \text{Prod}(T\_1, \text{Set})$ between the categories of models has a left adjoint (I think... | 6 | https://mathoverflow.net/users/290 | 134936 | 74,368 |
https://mathoverflow.net/questions/134928 | -1 | Suppose A is a covariance matrix whose diagonal elements are same, i.e. $A\_{1,1}=A\_{2,2}=\cdots=A\_{N,N}$, can we conclude that A is full rank?
Suppose the absolute values of the off-diagonal elements in A are all smaller than the diagonal elements in A.
Thanks!
| https://mathoverflow.net/users/35853 | Rank of covariance matrix whose diagonal elements are same | No. Let $a,b,c,d$ be complex numbers of modulus $1$, chosen so that $a\neq c$, $b\neq d$, and $ab^\*\neq cd^\*$. Form the column vectors $v=(1,a,b)^T$ and $w=(1,c,d)^T$. Then the matrix
$$
vv^\* +ww^\*=\begin{pmatrix} 2 & a^\*+c^\* & b^\*+d^\* \\ a+c & 2 & ab^\*+cd^\* \\ b+d & a^\*b+c^\*d & 2 \end{pmatrix}
$$
has rank ... | 0 | https://mathoverflow.net/users/13360 | 134941 | 74,369 |
https://mathoverflow.net/questions/134727 | 16 | **Motivation**:
I am trying to see for what class of Banach spaces the following result is true:
>
> There exists an increasing sequence of finite dimensional subspace {$V\_n$} of a Banach space X (with some property) and corresponding projections $P\_n: X \to V\_n$ such that
>
> a) $\cup V\_n$ is dense in $X$ ... | https://mathoverflow.net/users/31548 | What (classes of) Banach spaces are known to have Schauder basis? | The property you define is usually called the $\pi$ property.
For a good expository article, read Casazza's contribution in the Handbook of the Geometry of Banach Spaces, vol. 1.
Most of the classical separable Banach spaces are known to have a Schauder basis. The book of Albiac and Kalton is good place to start. S... | 11 | https://mathoverflow.net/users/2554 | 134950 | 74,372 |
https://mathoverflow.net/questions/134921 | 4 | For a number field F, let E/F be an elliptic curve with CM by a quadratic field K. Let $\rho\_\ell: \text{Gal}\_F \to \text{Aut}(T\_{\ell}E)$ be the $\ell$-adic representation associated to E for some prime $\ell$, and let $\mathcal{G}\_{\ell}$ denote its image. According to Theorem 5 in Serre's "Groupes de Lie $\ell$-... | https://mathoverflow.net/users/16120 | Abelian image of l-adic representation | There are no restrictions on $l$. What's going on is that if the $CM$ endomorphisms are defined over $KF$ but not over $F$, there must be elements in the $\operatorname{Gal}(KF|F)$ that conjugate these endomorphisms to different endomorphisms. Since the endomorphisms act faithfully on the Galois representation, the onl... | 2 | https://mathoverflow.net/users/18060 | 134953 | 74,374 |
https://mathoverflow.net/questions/134974 | 3 | It is known that for $SU(N)$
$$
\int \chi\_{\mu\_1}(UV\_1)\chi\_{\mu\_2}(U^{-1}V\_2)\, dU = \delta\_{\mu\_1\mu\_2}\frac{\chi\_{\mu\_1}(V\_1V\_2)}{\dim(\mu\_1)}
$$
where $dU$ is Haar measure on $SU(N)$ normalized such that with respect to it $\operatorname{Vol}(SU(N))=1$; and $\chi\_{\mu}(U)$ means the trace of $U$ in ... | https://mathoverflow.net/users/35745 | Integration over special unitary group | Do the Weingarten formulas help at all? See Theorem 2.5, and the surrounding discussions, in
<http://arxiv.org/abs/0903.5143>
| 1 | https://mathoverflow.net/users/763 | 134976 | 74,381 |
https://mathoverflow.net/questions/134944 | 3 | Consider a Poisson process with arrival rate $\lambda$ arrivals per unit time. Given a window of time $W$ and a total of $k$ events, what is the upper bound of the probability that no three events happen in that window? Said another way, if events are numbered $1,2,\ldots,k$, what is the upper bound of $$P(\operatornam... | https://mathoverflow.net/users/35875 | Probability that no three events happen in a pre-defined window | Let $G\_j$ be the event that $\operatorname{gap}(j,j+2)>W$. In terms of the interarrival times $X\_i$ we can write this as the event that $X\_{j+1}+X\_{j+2}>W$. The independence of the interarrival times implies that $G\_j$ is independent of $G\_{k}$ for $j-k>1$. This will make the calculation very simple. All we need ... | 1 | https://mathoverflow.net/users/11152 | 134977 | 74,382 |
https://mathoverflow.net/questions/133620 | 13 | **Motivation**: The Mandlebrot set is a simply connected set with an infinitely complex boundary, but CAN one move from interior to the exterior of this topological space by just crossing over a finite set of points?
**Application**: I am modeling the leakage of emission of EM waves, through a fractal forest where di... | https://mathoverflow.net/users/34859 | Analysis of the boundary of the Mandelbrot set | Your notation is unusual, and I am not sure whether I entirely understand it.
I shall take your question to mean: *Can every point of the Mandelbrot $M$ set be connected to $\infty$ by a path that intersects the boundary $\partial M$ in only finitely many points?*
This is connected to a very famous conjecture, name... | 23 | https://mathoverflow.net/users/3651 | 134985 | 74,386 |
https://mathoverflow.net/questions/134980 | 6 | I am studying De Giorgi's proof of Holder continuity of solutions of elliptic equations with bounded measurable coefficients.
This is the translation of the original paper
[De Giorgi paper](http://books.google.co.uk/books?id=80wlU42UC9IC&pg=PA149&lpg=PA149&dq=on%20the%20analyticity%20of%20extremals%20regular%20multip... | https://mathoverflow.net/users/35796 | Caccioppoli-Leray Inequality for De Giorgi's theorem proof | I made a trip to the library and scanned the relevant pages from Miranda's 1955 book:
[page 152-153](http://ilorentz.org/beenakker/MO/Miranda_1.jpg) and [page 154-155](http://ilorentz.org/beenakker/MO/Miranda_2.jpg)
the references are:
[3] J. Leray, J.Math. pures et appl. **17**, 89-104 (1938)
[8] R. Cacciopoli... | 11 | https://mathoverflow.net/users/11260 | 134987 | 74,388 |
https://mathoverflow.net/questions/134990 | 8 | By the prime number theorem we know that the "average gap" between the first $n$ primes is $\ln p\_n$. I would like to know the density of consecutive prime pairs whose gap is less than the average gap between the first $n$ primes? In other words, does the following limit exist and what is its value?
$$
\lim\_{n \to ... | https://mathoverflow.net/users/23388 | Density of prime pairs whose gap is less than the average gap | It is conjectured (see e.g. Goldston-Pintz-Yildirim <http://arxiv.org/abs/1103.5886>) that
$$
\lim\_{n \to \infty}\frac{1}{n}\#\Big\{m \le n : \alpha<\frac{p\_{m+1} - p\_m}{\ln p\_m} < \beta \Big\} = \int\_\alpha^\beta e^{-t} dt,
$$
and thus in particular your value should be $1-e^{-1}$. (GPY proved that this is po... | 20 | https://mathoverflow.net/users/10811 | 134993 | 74,389 |
https://mathoverflow.net/questions/135000 | 12 | In the paper
Deligne, Pierre; Illusie, Luc (1987), "*Relèvements modulo $p^{2}$ et décomposition du complexe de De Rham*", Inventiones Mathematicae **89** (2): 247–270, doi:10.1007/BF01389078
I found the following
>
> **Theorem.** Let $X$ be a smooth and projective scheme over a field $k$ with $\textrm{char}(k)... | https://mathoverflow.net/users/14514 | Vanishing theorems in positive characteristic | In the book *Lectures on Vanishing Theorems* by Esnault and Viehweg (Birkhäuser 1992) this is stated as an open problem. See in particular Problem 11.7 page 132.
However, they are able to prove the result when $\dim X=2$, since in this case one can perform the embedded resolution of singularities for curves on surfac... | 10 | https://mathoverflow.net/users/7460 | 135001 | 74,391 |
https://mathoverflow.net/questions/134997 | 8 | I hope this question has not been asked before.
I would like to know which Ideas led (Deligne), Kazhdan and Lusztig believe, that Kazhdan–Lusztig polynomials can be expressed via intersection cohomology groups of Schubert varieties?
Is there nowadays more insight why intersection cohomology groups (or perverse shea... | https://mathoverflow.net/users/32972 | Kazhdan-Lusztig Polynomials and Intersection Cohomology | First I'd comment that there are quite a few questions on MO related to this one, but apparently not quite identical. (It's hard to search the site efficiently.)
In any case I won't attempt a detailed answer but will rather suggest a reference.
While Kazhdan and Lusztig themselves must have interesting views on this ... | 8 | https://mathoverflow.net/users/4231 | 135006 | 74,392 |
https://mathoverflow.net/questions/135012 | 14 | How to prove (or to disprove) that all the roots of the polynomial of degree $n$ $$\sum\_{k=0}^{k=n}(2k+1)x^k$$ belong to the disk $\{z:|z|<1\}?$ Numerical calculations confirm that, but I don't see any approach to a proof of so simply formulated statement. It would be useful in connection with an irreducibility proble... | https://mathoverflow.net/users/35959 | Do all the roots of the polynomial lie in the unit disk? | Let $f$ denote your polynomial. Roots of the polynomial
$$g(x)=\sum\_{k=0}^nx^{2k+1}=x\frac{x^{2(n+1)}-1}{x^2-1}$$
are $0$ and roots of unity. By the [Gauss–Lucas theorem](http://en.wikipedia.org/wiki/Gauss%E2%80%93Lucas_theorem), the roots of $g'(x)=f(x^2)$ lie in their convex hull, and a fortiori in the disk $\{z:|z|... | 34 | https://mathoverflow.net/users/12705 | 135018 | 74,396 |
https://mathoverflow.net/questions/134781 | 16 | Suppose $S\subset\mathbb{R}$ is dense without interior point, and for every open interval $I,J\subset\mathbb{R}$, $I\cap S$ is homeomorphic to $J\cap S$.
Is $S\times S$ homeomorphic to $S$?
By Luzin scheme, if $S$ is the set of rationals or irationals , I can see this statement is true.
| https://mathoverflow.net/users/35739 | Question about product topology | In the following [paper](http://www.ams.org/journals/proc/1996-124-08/S0002-9939-96-03561-7/S0002-9939-96-03561-7.pdf), van Engelen constructs a strongly homogeneous, zero dimensional, Borel, dense subspace $Y$ of $2^{\mathbb{N}}$ such that while $Y$ does not admit a topological group structure, $Y^2$ does. I haven't r... | 5 | https://mathoverflow.net/users/2689 | 135030 | 74,402 |
https://mathoverflow.net/questions/135050 | 1 | **QUESTION** Do there exist integers $u\ x\ A\ B$ such that $x\ne 0$, and the following two equalities hold:
* $ x^2 + (x-u)^2\ =\ A^2$
* $ x^2 + (x+u)^2\ =\ B^2$
**?**
**REMARK** I have a family of pairs of quadruples $S\ T\subseteq\mathbb Z^2$, parametrized by $(u\ x)$, such that $S\ T$ ha... | https://mathoverflow.net/users/8385 | Pythagorean triples related to non-isometric equidistant plane quadruples | It's not even possible for the *product* of $x^2 + (x-u)^2$ and $x^2 + (x+u)^2$ to be a square unless $x=0$ or $u=0$: that product is $4x^4+u^4$, and
the elliptic curve $4x^4+u^4 = y^2$ is isomorphic to $Y^2 = X^3 - X$,
for which Fermat already proved that the obvious rational points are
the only ones. For your appli... | 11 | https://mathoverflow.net/users/14830 | 135054 | 74,412 |
https://mathoverflow.net/questions/134539 | 7 | Let's consider a fibering hyperbolic 3-manifold obtained as a mapping torus over some hyperbolic surface with pseudo-Anosov monodromy, and let's suppose that the surface is punctured at the singular points of the two invariant foliations.
Ian Agol has introduced a canonical layered ideal triangulation of such manifol... | https://mathoverflow.net/users/24768 | Comparing different layered structures for fibered 3-manifolds: example request. | The two triangulations (or cellulations) do not coincide in general.
Context:
In fact, given a centrally symmetric convex shape $S$ in the plane, one could at any time $t$ on the Teichmüller line look for maximal copies of $S$ (up to scaling and translation, but not rotation) disjoint from the singularities. Such max... | 5 | https://mathoverflow.net/users/36005 | 135061 | 74,413 |
https://mathoverflow.net/questions/135010 | 4 | Let $K$ be a char $0$ field with algebraic closure $\bar K$ and absolute Galois group $G$. Let $\mathcal U$ be an ultrafilter on $\mathbb N$ and $F=\bar K^\mathbb N/\mathcal U$ be the ultrapower of $\bar K$ on $\mathcal U$, which has a natural action of $G$.
What can be said of the field $F^G$ of $G$-invariant points... | https://mathoverflow.net/users/25887 | Galois action on ultrapowers | This is quite trivial but not enough for a comment: $F^G$ is not always $K^\mathbb N/\mathcal U$.
Let $x\_1,x\_2,\dots$ be elements of $K$ such that no product of finitely many of them is a perfect square in $K$. For instance, if $K=\mathbb Q$ we can take them to be the primes.
Our element of $\bar{K}^\mathbb N/\ma... | 3 | https://mathoverflow.net/users/18060 | 135062 | 74,414 |
https://mathoverflow.net/questions/134960 | 1 | Let $\cal A$ be a Banach algebra, $\cal U$ be a free ultrafilter, and $\phi$ be a character. Let ${(w\_{\alpha})}\_{\alpha}$ be a net in $(\cal A)\_{\cal U}$, and suppose that for every $(a\_i)\in (\cal A)\_{\cal U}$ we have
$$\lim\_{\alpha}\|a\_i w\_{\alpha}-\phi(a\_i)w\_{\alpha}\|=0,$$
so$$\lim\_{\cal U} \lim\_{\alph... | https://mathoverflow.net/users/27066 | When can we "displace" an ultrafilter limit with another limit? | With hindsight, this question is really too elementary for MO, in my opinion. However, since I am not on MSE, here is a sketch of what I think is a counter-example (but I think there is some merit to the argument that questions at this level should not be encouraged on MO). Also, **I will not be happy if I learn later ... | 2 | https://mathoverflow.net/users/763 | 135070 | 74,419 |
https://mathoverflow.net/questions/134853 | 6 | Let $a\_1,a\_2 \in \mathbb{Q}:a\_1≥1,a\_2≥1$. What should be the minimum value of $x\in\mathbb{R}$: $n∈[1,x]$ to ensure that $4k−3≤na\_1≤4k−1$ such that $k∈N$ and $4l−3≤na\_2≤4l−1$ such that $l∈N$ for all $a\_1, a\_2$?
Numerical computations suggest the answer to this is $3$ but I'm out of ideas how to prove this for... | https://mathoverflow.net/users/33333 | An easy-to-state elusive combinatorial problem | The bound is $3$. For readability, I'll change $(a\_1, a\_2)$ to $(x,y)$. Without loss of generality, $x \geq y$. We break into cases.
**Case 1 (the main case):** $y \geq 7/3$. In this case, there exists an integer $\ell$ such that $y \leq 4 \ell-3 < 4 \ell-1 \leq 3y$. As $r$ ranges from $(4 \ell-3)/y$ to $(4 \ell-1)... | 4 | https://mathoverflow.net/users/297 | 135071 | 74,420 |
https://mathoverflow.net/questions/135063 | 4 | I have a set of unique strings (with an alphabet size of >=4) of equal length to represent the vertices (not all possible strings are represented in the graph). I have an edge between a pair of vertices if the Hamming distance between the two strings is 1.
I have an interest in graphs and how they can be use in my bi... | https://mathoverflow.net/users/36006 | What type of Hamming Graph is this? | Do you have all $k^n$ strings of length $n$ using your $k$ symbol alphabet or a selection such as all consecutive 30 letter strings found in the DNA of some individual. The former is a Hamming graph but the latter is "subgraph of a Hamming graph" and I'm not sure how easy it is to say if a given graph can be realized i... | 4 | https://mathoverflow.net/users/8008 | 135074 | 74,423 |
https://mathoverflow.net/questions/135082 | 9 | Is there any integer $n\ge 2$ such that $2^n\equiv 3 \bmod n$? I understand that $n$ must be an odd non-prime. I checked up to a million with no success (but $2^n\equiv 5 \bmod n$ and $2^n\equiv 7 \bmod n$ have solutions).
| https://mathoverflow.net/users/20757 | Range of $2^n$ mod $n$ | The smallest such $n$ is $n=4700063497$. A few others are known. J. Crump found $n=8365386194032363$ in 2000. Max Alekseyev found $n=3468371109448915$. Joe Crump found $n=10991007971508067$. Some information on these is at <http://www.cs.ucla.edu/~klinger/newno.html>
| 14 | https://mathoverflow.net/users/3684 | 135083 | 74,424 |
https://mathoverflow.net/questions/114886 | 7 | Let $X\_i$ ($i=1,...,n$) be a sequence of independent and identically distributed random variables. Denote $\mu=\mathbb{E}[X\_i]$ and $S\_n=\frac{1}{n}\sum\_{i=1}^nX\_i$. This question concerns the tail probability $\Pr[S\_n \ge \mu+\epsilon]$, where $\epsilon>0$ is a constant.
In the case that $|X\_i|$ is bounded wi... | https://mathoverflow.net/users/29510 | Sufficient Condition for Exponential Decay in Chernoff Bound (Large Deviations) | Note that Cramer's theorem applies in dimension 1 w/out any moment assumptions, but of course the rate function may vanish (for this version see theorem 2.2.3 in Dembo-Zeitouni). So the question is equivalent to the question ``when does the rate function vanish'' (the rate function is $\sup\_{\lambda\in R}
(\lambda x-\... | 6 | https://mathoverflow.net/users/35520 | 135091 | 74,426 |
https://mathoverflow.net/questions/135093 | 6 | What do I call two sequences $a, b$ such that $\lim\_{n\to\infty} |a\_n - b\_n| = 0$? Or what do I call two functions $f, g$ such that $\lim\_{x\to c} |f(x) - g(x)| = 0$? (For my purposes, these are essentially the same thing, and I will happily extend a term for one to the other.)
This seems like such a straightforw... | https://mathoverflow.net/users/8508 | Terminology for sequences/functions that approach each other | You could say $ a $ and $ b $ are mutually asymptotic. Example: "Since $ n -1/n$ and $n+1/n $ are mutually asymptotic. .." And usually even drop the word "mutually".
| 4 | https://mathoverflow.net/users/4600 | 135096 | 74,427 |
https://mathoverflow.net/questions/135086 | 3 | The magic eye optical illusions create stereographic pictures by taking two rectangles and slightly shifting the patterns, so that when you cross your eyes to overlap them, the subtle differences correspond to height.
This motivates my question: given a continuous function $f$ from the unit square to the interval $[-... | https://mathoverflow.net/users/27933 | A mathematical version of the Magic Eye optical illusion | Yes. For example $g\_2(x,y)=(x+1+\frac{1}{2}f(x,y),y)$ and $g\_1(x,y)=(x-2-\frac{1}{2}f(x,y),y)$.
| 3 | https://mathoverflow.net/users/17836 | 135098 | 74,428 |
https://mathoverflow.net/questions/135077 | 12 | I posted this on [math.stack.exchange](https://math.stackexchange.com/) but didn't get a helpful response, so please let me try it here.
Let $D$ be a small category and $F:D\to sSets$ a functor.
There is a bisimplicial set indicated by
$$
...\begin{array}{c}\to \\ \to\\\to\end{array}\coprod\_{d\_1\to d\_0}F(d\_1)\... | https://mathoverflow.net/users/31205 | Three questions on $\operatorname{hocolim}$ | **First question**$\newcommand{\op}[1]{{#1}^{\mathrm{op}}}$$\newcommand{\sSet}{\mathrm{sSet}}$$\newcommand{\Grpd}{\mathrm{Grpd}}$$\newcommand{\Cat}{\mathrm{Cat}}$$\newcommand{\NN}{\mathbb{N}}$$\newcommand{\sres}{\mathrm{sres}}$$\newcommand{\hocolim}{\operatorname{hocolim}}$$\newcommand{\diag}{\operatorname{diag}}$$\new... | 14 | https://mathoverflow.net/users/21095 | 135103 | 74,429 |
https://mathoverflow.net/questions/135106 | 2 | How do I go about diagonalizing such a matrix.
I ask because I need to sort out the following problem:
Let $D$ be the quaternion algebra over $\mathbb{Q}$ with $i^2 = -1, j^2 = -11, ij=-ji=k$.
Let $\mathfrak{O} = \langle 1,i,\frac{1+j}{2},\frac{i+k}{2}\rangle\_{\mathbb{Z}}$ be a maximal order.
Consider the H... | https://mathoverflow.net/users/21698 | Diagonalization of Quaternion Hermitian matrices | Note that $U=\pmatrix{1&t\cr 0&1}$ and $X=\pmatrix{a&b\cr c&d}$ has $UX\bar U^T=\pmatrix{?&b+dt\cr c&d}\pmatrix{1&0\cr -t&1}=\pmatrix{?&b+dt\cr c-dt&?}$ (for $t$ with trace 0). So just make $t=(j+3k)/11$ in your example, killing the terms off the diagonal. Maybe I transposed this, but you can work it out.
| 1 | https://mathoverflow.net/users/36061 | 135109 | 74,432 |
https://mathoverflow.net/questions/134967 | 1 | Assume we have a Riemannian manifold $M$ embedded isometrically in $\mathbb{R}^n$ (It may not be closed or open). Let $V(i)$ be a smooth orthonormal vector field on $M$ (which can be extended to a basis in $T(\mathbb{R}^n)$ at every point).
Given that a smooth function $f$ on $M$ can be extended to $\mathbb{R}^n$ smoot... | https://mathoverflow.net/users/35911 | extension of smooth function | No: Take a spiral in $\mathbb R^2$ like $\mathbb R\_{>0}\ni t\mapsto t.e^{it}$ with the induced metric, $g(x,y)=x$. You cannot change $g$ such that its gradient is tangent to the spiral: You have problems at 0.
| 2 | https://mathoverflow.net/users/26935 | 135122 | 74,436 |
https://mathoverflow.net/questions/135120 | 7 | Let $\kappa$ be an infinite cardinal.
We know that the notion of being closed and unbounded (club) in $\kappa$ is strictly stronger than being stationary in $\kappa$.
My question: Do you know any notion that is strictly between being a club and being a stationary set?
| https://mathoverflow.net/users/13694 | Notion strictly between stationary and club | Here are a few intermediate notions:
* "Being in the club filter" is strictly stronger than stationary and strictly weaker than club.
* More generally, "being in $F$", for some fixed normal filter $F$ on $\kappa$. By normality, $F$ contains the club filter and so all such sets are stationary, but needn't be club.
* "... | 12 | https://mathoverflow.net/users/1946 | 135123 | 74,437 |
https://mathoverflow.net/questions/135125 | 5 | Let $B:=B\_1\cap B\_2\cap...\cap B\_n$, where each $B\_j$ is a *reflexive* Lebesgue space or Sobolev space (such as $L^4$, $H^1$, etc.) on a domain in $\mathbb{R}^d$. Then $B$ is a Banach space endowed with the norm
$$\|\cdot\| = \|\cdot\|\_{B\_1}+...+\|\cdot\|\_{B\_n}.$$
Let $\{f\_n\}$ be a sequence in $B$ and $f\in... | https://mathoverflow.net/users/4119 | Weak convergence in the intersection of Lebesgue spaces or Sobolev spaces | The answer to both questions is "yes". One way of seeing this is to observe that $B$ embeds isometrically into $B\_1 \oplus\_1 B\_2 \dots \oplus\_1 B\_n$ via $b \mapsto (b,b,\dots, b)$. Another way is to use the (obvious) fact that a sequence in a Banach space converges weakly to $x$ if and only if for every subsequenc... | 6 | https://mathoverflow.net/users/2554 | 135129 | 74,440 |
https://mathoverflow.net/questions/135132 | 14 | Compact Hausdorff spaces are algebras of the ultrafilter monad on Set.
Is the category of Tychonoff spaces also monadic over Set?
| https://mathoverflow.net/users/1261 | Is there a monad on Set whose algebras are Tychonoff spaces? | No. In fact any full subcategory of $\mathbf{Top}$ that contains all the discrete spaces cannot be monadic over $\mathbf{Set}$ unless it contains only discrete spaces. Indeed, for any such subcategory $\mathcal{C}$, the forgetful functor $\Gamma : \mathcal{C} \to \mathbf{Set}$ has a left adjoint $\Delta : \mathbf{Set} ... | 20 | https://mathoverflow.net/users/11640 | 135136 | 74,444 |
https://mathoverflow.net/questions/134694 | 5 | In an attempt to resolve a question posed by Cain in his paper on Automaton Semigroups (open problem 6.12), I would like to know if there exists a finite semigroup $S$ satisfying the following properties:
1. $S$ is self-dual (anti-isomorphic to itself)
2. $S\neq S^2$
3. $S^2$ is a band
4. $S$ has a faithful left-regu... | https://mathoverflow.net/users/35564 | Existence of a possible counterexample in automaton semigroups | I believe I have an example but you should check the details of whether it works. Maybe GAP can be used.
Let $X=\{1,2,3,4,5\}$ and $X'=\{1',2',3',4',5'\}$. Let $a,b\colon X\to X$ be given by $$a=\begin{pmatrix} 1 & 2 & 3& 4 &5\\ 2& 3& 3 &4& 5 \end{pmatrix}\qquad b=\begin{pmatrix} 1 & 2 & 3& 4 &5\\ 4& 5& 4 &4& 5 \end... | 3 | https://mathoverflow.net/users/15934 | 135158 | 74,450 |
https://mathoverflow.net/questions/135163 | 8 | Let $X$ be a Banach space, and $(P\_t)\_{t \ge 0}$ a strongly continuous semigroup of bounded operators on $X$. Using the uniform boundedness principle, it's simple to prove that there are constants $M, \omega$ such that $$\|P\_t\| \le M e^{\omega t} \quad (\*)$$ for all $t$. Moreover, if $M=1$ you can get a contractio... | https://mathoverflow.net/users/4832 | Strongly continuous semigroups that cannot be contractions | There are many examples constructed with weighted shifts. The following is a Hilbert space example.
Let us consider the Hilbert space $L^2\big((0,1),\mu\big)$, where $\mu$ denotes the measure defined by
$$\mu(A):=2\lambda(A\cap(0,\tfrac12))+\lambda(A\cap(\tfrac12,1)).$$
for all Lebesgue measurable sets A. Here $\lamb... | 11 | https://mathoverflow.net/users/12898 | 135165 | 74,452 |
https://mathoverflow.net/questions/135146 | 3 | Let $d$ be a prime number. Is the polynomial $x^d+x+1$ a primitive polynomial? In other words I need the minimal polynomial of $\alpha$ in $F\_{2^d}=F\_{2}(\alpha)$.
Thank you.
| https://mathoverflow.net/users/27932 | primitive polynomial in $F_2$ | See <http://oeis.org/A073639> for the list of n such that x^n+x+1 is primitive modulo 2.
The sequence begins [2,3,4,6,7,15,22,60,63,127,153,471,532,865,900,1366]
You will also find some pertinent references there.
I found this using Maple's function "Primitive(f(x)) mod p" which returns true if f(x) is primitive mod ... | 7 | https://mathoverflow.net/users/36085 | 135166 | 74,453 |
https://mathoverflow.net/questions/135148 | 16 | Before I start my question, I should probably mention that this question might not be the right question to ask here, but I tried academiabeta, and stackoverflow, but without getting any to-the-point responses. I also noticed that there were some questions of similar interest asked here, but mine is little different an... | https://mathoverflow.net/users/35936 | Switching from pure mathematics (e.g. geometry) to more applied areas (e.g imaging) after Ph.D., as postdoc and chance of getting such a postdoc? | There is a newish emerging area, `applied algebraic topology'. It includes Topological Data Analysis (see work by Gunnar Carlsson and others). There is also new work again in Applied Algebraic Topology looking at Configuration spaces and probablity theory. Look for research assistant jobs at Post Doc level on specific ... | 7 | https://mathoverflow.net/users/3502 | 135169 | 74,455 |
https://mathoverflow.net/questions/135114 | 3 | As we known, the integer ring R of 256-th cyclotomic field is not a Principal Ideal Domain.
And rational prime 257 is split completely in R.
Suppose prime ideal P of R is an arbitrary ideal lying over 257.
**My question**: Is P a principal ideal in R?
If it is, how to find one of its generator? Or, is there an a... | https://mathoverflow.net/users/36063 | Is the prime ideal principal which is in 256-th cyclotomic ring lying over 257? | Let $K/\mathbf{Q}$ be the degree $32$ field of $64$th roots of unity, and let $L/K$ be the field of $256$th roots of unity. The class number of $K$ is $17$, and the degree of $K$ is just small enough to be able to compute the unit group, etc. According to pari, if $\mathfrak{p}$ is a prime of norm $257$ in $K$, then $\... | 9 | https://mathoverflow.net/users/36099 | 135176 | 74,458 |
https://mathoverflow.net/questions/135156 | -2 | I am studying papers about " Prime graph" , for example "Prime graph components of finite groups" [ williams], " Groups with complete prime graph connected components" [ Lucido and moghaddanfar]. But I do not understand some of scientific terms in papers to relate this matter. For example, concepts like torus, maximal ... | https://mathoverflow.net/users/33508 | Maximal torus and application in prime graph | We know that each torus is a maximal abelian subgroup and so the prime divisors of $|T|$ are mutually adjacent in the prime graph of $G$.
So knowing the order of maximal abelian subgroups of a simple group give many adjacency in the prime graph.
Only one need to determine the adjacency between the characteristic o... | 1 | https://mathoverflow.net/users/31045 | 135179 | 74,459 |
https://mathoverflow.net/questions/135184 | 11 | Question
========
>
> For which pairs $M,N$ of compact metric spaces does there exist a metric space $K$ along with isometric embeddings $i:M \to K$ and $j:N \to K$ so that the Hausdorff distance between $i(M)$ and $j(N)$ in $K$ equals the Gromov-Hausdorff distance between $M$ and $N$ exactly?
>
>
>
We always... | https://mathoverflow.net/users/18263 | For which metric spaces is Gromov-Hausdorff distance actually achieved? | Unless I am missing something, the distance is achieved for compact metric spaces. Take isometries $i\_n :M \to K\_n$ and $j\_n : N \to K\_n$ that give you the Gromov-Hausdorff distance up to $1/n$. Let $K$ be an ultraproduct of the $K\_n$ and $i$, $j$ the isometries from $M$, $N$ into $K$ induced by $i\_n$, $j\_n$. Th... | 10 | https://mathoverflow.net/users/2554 | 135190 | 74,463 |
https://mathoverflow.net/questions/135186 | 1 | Let $(X\_t)\_{t \in \mathbb{N}}$ be a discrete-time Markov process on the finite state space $\mathcal{X}$, with transition matrix $T$. Suppose $f: \mathcal{X} \to \mathcal{Y}$ is a (deterministic) function such that $(f(X\_t))\_{t \in \mathbb{N}}$ is a Markov process on the state space $\mathcal{Y}$. What can we say a... | https://mathoverflow.net/users/7967 | When is a coarsening of a Markov chain Markov? |
>
> Are there any other cases?
>
>
>
Sure there are. Say, $(X\_t)\_{t\in\mathbb{N}}$ is the same as a chain $(Y\_t)\_{t\in\mathbb{N}}$ except that it has two copies $a\_0,a\_1$ of a certain state $a$ of $(Y\_t)\_t$. Every time $(Y\_t)\_t$ has a transition to $a$, $(X\_t)\_t$ flips a coin to decide which of $a\_0... | 4 | https://mathoverflow.net/users/23297 | 135194 | 74,465 |
https://mathoverflow.net/questions/135189 | 29 | When dealing with moduli spaces of, say connections or metrics, I am using the notions of Frechet spaces/manifolds/groups. I have become familiar with Banach manifolds (I think), but Frechet manifolds less so. In my studies, we want to apply the *Inverse and Implicit Function Theorems*, which work on Banach spaces, but... | https://mathoverflow.net/users/12310 | Intuition for failure of Implicit Function theorem on Frechet Manifolds | The usual proof of the inverse function theorem in the setting of Banach spaces uses the Banach fixed point theorem. We cannot make sense of the Banach fixed point theorem in a Fréchet space, since a Fréchet space is merely metrizable: there is no preferred choice of metric. I do not know exactly how crucial the Banach... | 31 | https://mathoverflow.net/users/21375 | 135202 | 74,470 |
https://mathoverflow.net/questions/135142 | 10 | Let $\pi: R\to S$ be a local morphism of Henselian local rings. Let $f: R \to \hat{R}$ and $g: S \to \hat{S}$ be their completions. Let $\mathcal F$ be a constructible $l$-adic sheaf on $\operatorname {Spec} S$ for $l$ invertible in $R$. Is the natural base change map $f^\*R^i\pi\_\* \mathcal F \to R^i \hat{\pi}\_\* g^... | https://mathoverflow.net/users/18060 | Etale cohomology of the completion of a Henselian local ring | You probably meant to assume $R$ and $S$ are noetherian. The answer is "no" to the initial hypergeneral part of the question. EDIT: In the 2nd half (below the long line), I now give a proof of an affirmative answer to the added part involving maps of affine spaces.
Counterexamples to the initial hypergeneral part can... | 14 | https://mathoverflow.net/users/35361 | 135205 | 74,471 |
https://mathoverflow.net/questions/135157 | 2 | Can we say something like monotonicity, growth rate and oscillation of the Fourier transform of a banded function $f$ with support $[0, N]$
$$\mathcal{F}f(\xi) = \int\_{0}^N f(x)e^{-ix\xi}dx.$$
Of course generally $\mathcal{F}f(\xi)$ is complex. I guess it is impossible for the real part or imaginary part $\mathcal{F}f... | https://mathoverflow.net/users/36081 | Non-global oscillation of banded Fourier transform | There is a paper related to your question:
MR2072747
Ostrovskii, I. V.; Ulanovskii, A.
Non-oscillating Paley-Wiener functions.
J. Anal. Math. 92 (2004), 211–232.
The Paley-Wiener theorem completely describes the Fourier transforms of functions
with bounded support. Because of the importance of these functions in si... | 1 | https://mathoverflow.net/users/25510 | 135215 | 74,477 |
https://mathoverflow.net/questions/133375 | 20 | For a fixed finite alphabet $A=\{a,b,...\}$, write $x \sim\_n y$ if the two words $x$ and $y$ have the same (scattered) subwords of length at most $n$. The relation $\sim\_n$ is a congruence of finite index and [SS83] asks what is the number of congruence classes. Has there been any progress on this question? I cannot ... | https://mathoverflow.net/users/34866 | congruence on words: having the same (scattered) subwords of length at most n | I could generalize my earlier proof for the $k=2$ case and show that, for any fixed $k$, $C\_{n,k}$ is in $2^{O(n^k)}$ hence simply exponential.
When some given $n$ is understood, we say that $x\in A^\*$ is *minimal*
if $x$ has minimal length inside its $\sim\_n$-class. Since $\sim\_n$ is a
congruence, all factors of... | 4 | https://mathoverflow.net/users/34866 | 135217 | 74,479 |
https://mathoverflow.net/questions/135191 | 2 | Let $X$ be a connected smooth scheme over a field and let $VB(X)$ be the exact category of vector bundles (i.e. locally free $\mathcal{O}\_X$-module whose rank is finite at every point) over $X$.
Let the abelian group $\tilde K\_0(X)$ denote the kernel of the rank morphism from the zeroth algebraic K-theory group $K\... | https://mathoverflow.net/users/31205 | Can this reduced version of algebraic K-theory be identified with this direct limit? | It's already false for $\mathbb P^1$. Let $L$ be a line bundle with no global sections except zero. Then $L\oplus L^{-1}$ is not a trivial bundle because it is not generated by global sections, and this remains true when you add a trivial bundle to it, but it is trivial in your reduced $K^0$.
| 2 | https://mathoverflow.net/users/6666 | 135222 | 74,480 |
https://mathoverflow.net/questions/135228 | 11 | Let $\operatorname{ZFC}^{-}$ be the theory of $\operatorname{ZFC}$ minus the axiom of foundation and define the proper classes $G$ and $V$ as follows:
$G:=$ The proper class of all sets.
$V:=$ The proper class of Von neumann cumulative heirachy.
And let the statements $G=V$ and $|G|=|V|$ be:
$G=V~:~~\forall x~... | https://mathoverflow.net/users/nan | How big is the proper class of all sets? | Both the statement are true, which is fine because we talk about relative consistency here.
If $\sf ZFC$ is consistent then we know how to generate a model of $\sf ZFC^-+\it G\neq V$.
On the other hand, it is consistent that we have a proper class of atoms: sets of the form $x=\{x\}$, with global choice. Then by ta... | 7 | https://mathoverflow.net/users/7206 | 135229 | 74,484 |
https://mathoverflow.net/questions/135231 | 5 | This should be obvious but I'm not seeing it:
The $\mathfrak T$ be a triangulated category with coproducts and with a compact generator $A$ (that is, the functor $\mathfrak T(A,\\_)$ preserves coproducts and the localizing subcategory $\langle A\rangle$ of $\mathfrak T$ generated by $A$ is all of $\mathfrak T$.)
Fo... | https://mathoverflow.net/users/1291 | Is this a description of the $\aleph_1$-localizing subcategory generated by a compact generator? | I don't think what you say is true. Let $k$ be a field, $V$ a $k$-vector space of uncountable dimension and $R=k\oplus V$ the $k$-algebra where $V$ is a square-zero ideal. Consider $\mathfrak T=D(R)$ the derived category of $R$, and $A=R$. Take a non-trivial vector $0\neq v\in V$. The complex
$$B=\cdots\rightarrow 0\r... | 7 | https://mathoverflow.net/users/12166 | 135249 | 74,491 |
https://mathoverflow.net/questions/135181 | 9 | Let $g$ be a positive integer, and let $G$ be a commutative group with the following constraint on its torsion subgroup: there is an injection $G[\operatorname{tors}] \hookrightarrow (\mathbb{Q}/\mathbb{Z})^{2g}$. Must there be subgroups $G\_1,\ldots,G\_g$ of $G$ such that
(i) $G = G\_1 \times \ldots \times G\_g$ (i... | https://mathoverflow.net/users/1149 | Direct product decomposition for infinite abelian groups with constrained torsion | The answer is negative.
Define $S=\bigoplus\_p C\_p$, where $C\_p$ is cyclic of order $p$ and $p$ ranges over prime numbers. Fix an integer $n\ge 1$. Define $P=\prod\_p C\_p$, and consider a maximal $\mathbf{Z}$-free family in $P^n/S^n$, generating a free abelian group (of continuum rank) $L/S^n$. Fix a prime $q$ and... | 3 | https://mathoverflow.net/users/14094 | 135254 | 74,492 |
https://mathoverflow.net/questions/135234 | 0 |
>
> Let $p$ be a prime number. Is there a prime number $q$ such that $p$ is a primitive root of $q$?
>
>
>
| https://mathoverflow.net/users/36141 | on prime numbers which are primitive roots of a prime | Gupta and R. Murta showed that there were infinitely many primes $p$ for which there are infinitely many $q$. Heath-Brown generalized this to show that there are at most two prime numbers $p$ for which there are not infinitely many $q$ that they are primitive roots modulo $q$.
Heath-Brown, D. R.(4-OXM)
Artin's conjec... | 2 | https://mathoverflow.net/users/2043 | 135256 | 74,494 |
https://mathoverflow.net/questions/135233 | 3 | Are there any good way to understand $k$-polarized Abelian surfaces? I am aware that if $A \cong \mathbb{C}^2/\Gamma$ is $k$-polarized, the lattice $\Gamma$ can be taken of the form
$$
\begin{bmatrix}
1 & 0 & \tau\_1 & \tau\_2\\
0 & k & \tau\_3 & \tau\_4
\end{bmatrix}
$$
over $\mathbb{Z}$ (think of $\mathbb{C}^2\cong ... | https://mathoverflow.net/users/50973 | Intuitive meaning of $k$-polarized Abelian surface? | It is well known that any polarized abelian variety is isogenous to a principally polarized one. More precisely, given any polaized abelian variety $(A, \, L)$ there exists a principally polarized abelian variety $(B,\, \Theta)$ and an isogeny $u \colon A \longrightarrow B$ such that $L=u^\* \Theta$.
Then any polariz... | 5 | https://mathoverflow.net/users/7460 | 135266 | 74,499 |
https://mathoverflow.net/questions/135272 | 10 | If $G$ is a discrete or topological group, $G$ is a closed subgroup of $EG$, and normal iff $G$ is abelian, according to Segal, *Cohomology of topological groups*, Symposia Mathematica IV (1970) (a reference I found from two answers by Chris Schommer-Pries: [Classifying Space of a Group Extension](https://mathoverflow.... | https://mathoverflow.net/users/19860 | Why isn't $BG$ a group, for $G$ not abelian? | The classifying space functor may be a monoidal functor out of $\text{Grp}$, but nonabelian groups aren't group objects in $\text{Grp}$. (The group objects in $\text{Grp}$ are precisely the abelian groups. This is also a corollary of the Eckmann-Hilton argument.)
| 23 | https://mathoverflow.net/users/290 | 135276 | 74,506 |
https://mathoverflow.net/questions/135246 | 5 | Let $\mathcal{C}$ be a small category. Then, its *Segal subdivision* $\text{sd }\mathcal{C}$ is a new category whose objects are morphisms of $\mathcal{C}$, and a morphism from $f:x \to y$ to $g: w \to z$ is a pair of morphisms $a: x \to w$ and $b: y \to z$ in $\mathcal{C}$ so that $f = b\circ g \circ a$. It is easy to... | https://mathoverflow.net/users/18263 | How does one Segal-subdivide a 2-category? | Let me make a proposal. I won't check that the property you demand about nerves is satisfied. I suspect it holds, but the checking may require some work, so if I did it I would be tempted to do something with it. Since you may really want to do something with it, I leave it to you ;-)
Given a $2$-category $\mathcal C... | 5 | https://mathoverflow.net/users/12166 | 135291 | 74,516 |
https://mathoverflow.net/questions/135287 | 6 | Let $\mathcal{F}$ be the smallest family of (simple, finite) graphs such that:
1. The empty graph is in $\mathcal{F}$.
2. If $G \in \mathcal{F}$, then $G\_{\bullet}$, the graph obtained from $G$ by adding another vertex $v$ connected by an edge to each vertex in $G$, is also in $\mathcal{F}$.
3. If $G,H \in \mathcal{... | https://mathoverflow.net/users/25028 | Does the family of graphs closed under taking unions and adding dominating vertices have a name? | It appears that you are looking at the family of [*quasi-threshold graphs*](http://www.graphclasses.org/classes/gc_781.html). These are the graphs which can be built, starting from $K\_1$, by repeatedly taking the disjoint union of two quasi-threshold graphs, or the join of a quasi-threshold graph with a single new ver... | 9 | https://mathoverflow.net/users/2663 | 135298 | 74,519 |
https://mathoverflow.net/questions/135224 | 2 | Let $G$ be a finite subgroup of $SL(n,\mathbb{C})$. Let $G$ act on $\mathbb{C}^{n}$ with the action induced by $SL(n,\mathbb{C})$ and the induced action of $G$ on the unit sphere $S^{2n-1}$ is free. My question is the following: is there a $N>0$ s.t.there exist a $G$-invariant holomorphic map $i\_{N}$
$$i\_{N}:\math... | https://mathoverflow.net/users/4971 | Embeddings of of quotient singularities | Yes. By Hilbert's theorem, the ring of $G$-invariant functions on $\mathbb C^n$ is finitely generated. Say the number of generators is $N$, then this gives a $G$-invariant holomorphic map to $\mathbb C^N$.
If one defines the quotient $\mathbb C^n/ G$ as the spectrum of this ring, then the map is clearly an embedding,... | 3 | https://mathoverflow.net/users/18060 | 135299 | 74,520 |
https://mathoverflow.net/questions/135294 | 11 | Ed Sandifer writes on Euler's wonderful work on prime numbers: The sum of the series of reciprocals of prime numbers $\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+...$ is infinitely large, and is infinitely less than the harmonic series $1 +\frac{1}{2}+\frac{1}{3}+...$. Moreover, the first sum is almost the logarithm of the sec... | https://mathoverflow.net/users/nan | An Euler-proof that cannot be repaired? | This result of Euler is the last theorem in <http://eulerarchive.maa.org/docs/originals/E072.pdf>, and if you prefer English to Latin look at the last theorem in <http://eulerarchive.maa.org/docs/translations/E072en.pdf>. I don't know why Sandifer says the proof can't be repaired. Euler's goal is to show $\sum\_{p} 1/p... | 16 | https://mathoverflow.net/users/3272 | 135301 | 74,522 |
https://mathoverflow.net/questions/135319 | 4 | First let me make a definition. Let $M$ be a smooth manifold and
$S \subset M $ a topological subspace of $M$. We say that $S$ has
"dimenion" at most $k$ if $S$ is a subset of
$$ X\_1 \cup X\_2 \ldots \cup X\_n $$
such that each of the $X\_i$ is a smooth manifold of dimension less than or
equal to $k$. Of course... | https://mathoverflow.net/users/4463 | Is it impossible for the dimension of a topological space to increase under a smooth map? | I guess "smooth manifold" means "smooth submanifold".
In particular a curve which pass through a given point infinitely many times is not a smooth submanifold.
If yes then it is easy to construct an example.
Take a smooth curve $\gamma\colon(0,1)\to \mathbb R^2$ which pass through $(0,0)$ infinitely many times under ... | 3 | https://mathoverflow.net/users/1441 | 135329 | 74,534 |
https://mathoverflow.net/questions/135218 | 10 | What apps can be found in the Apple app store, Google Play, Blackberry World etc. showcasing specific mathematics research?
(Edit - Since the first version of the question got closed, examples should showcase specific work of specific mathematicians. Tools such as Wolfram Alpha, bibliography managers, arXiv app, etc.... | https://mathoverflow.net/users/4600 | Math research in the app store | Akio Kawauchi, Ayaka Shimizu, and Kengo Kishimoto have created an app called [Region Select](http://www.sci.osaka-cu.ac.jp/math/OCAMI/news/gamehp/etop/gametop.html), based on a knot theory paper by Shimizu. She found an algorithm to unknot a knot diagram by "region crossing change" moves, and then they realized that it... | 10 | https://mathoverflow.net/users/2051 | 135331 | 74,536 |
https://mathoverflow.net/questions/135344 | 14 | Let $(M,d)$ and $(N,\rho)$ be metric spaces. A function $f: M \to N$ is *Lipschitz* if there exists some constant $\kappa \geq 0$ so that $\rho(f(x),f(y))$ is smaller than $\kappa d(x,y)$ for all points $x,y \in M$.
If $f$ and $g$ are two Lipschitz maps from $M$ to $N$, a *Lipschitz homotopy* between them is a Lipsch... | https://mathoverflow.net/users/18263 | When does a pair of homotopic Lipschitz functions fail to admit a Lipschitz homotopy? | There is no "justice" in the world of arbitrary metric spaces:-)
Let your first space $M$ be the ordinary circle with its ordinary metric, and the second
space $N$ be homeomorphic to a cylinder with two border circles. Let the metric
be usual on the border circles, but very rough (bumpy) on the middle part of the cylin... | 13 | https://mathoverflow.net/users/25510 | 135349 | 74,545 |
https://mathoverflow.net/questions/135241 | 7 | I asked this question on [Math.Stack](https://math.stackexchange.com/questions/430687/the-irreducible-corepresentations-of-the-eight-dimensional-kac-paljutkin-quantum) but have not had any answers.
**Question**
What are the irreducible corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group, $A$?
T... | https://mathoverflow.net/users/35482 | The Irreducible Corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group | The Kac-Paljutkin Quantum Group $A$ is self-dual, i.e. the dual space $A^\*$ (which is of course again finite quantum group = finite-dimesional $C^\*$-Hopf algebra, with the multiplication being the dual of the coproduct of $A$ and the coproduct the dual of the multiplication of $A$) is isomorphic to $A$.
This must h... | 7 | https://mathoverflow.net/users/36090 | 135351 | 74,546 |
https://mathoverflow.net/questions/135347 | 23 | Consider that the question does not concern the origin of the ideas of equivalence relation and equivalence class. It exactly concerns **the origin of the terms** "equivalence relation" and "equivalence class". It seems that the terms weren't in use at least until 1903 where Russell writes:
>
> Peano has defined a ... | https://mathoverflow.net/users/29316 | Who introduced the terms "equivalence relation" and "equivalence class"? | Von Neumann uses "equivalence class" in [*Zur Prüferschen Theorie der idealen Zahlen*, Acta Sci. Math. (Szeged) **2** (1926) 193-227](http://pub.acta.hu/acta/showCustomerVolume.action?id=5089&dataObjectType=volume), p. 197 (viewable after free registration):
>
> Wir nennen $R$ und $S$ *äquivalent*, in Zeichen: $R\s... | 21 | https://mathoverflow.net/users/19276 | 135375 | 74,552 |
https://mathoverflow.net/questions/135352 | 1 | Let $(M,\omega)$, be pre-symplectic, then can we say, we have a foliation of $M$, with tangent spaces $ker\omega$.What can we say about its trajectories. ?
| https://mathoverflow.net/users/nan | pre-symplectic and foliation and its trajectories | If your definition of pre-symplectic involves constant rank and $\ker\omega$ defines a smooth distribution of hyperplanes in the tangent bundle, then an application of the Frobenious theorem shows that you have a foliation tangent to $\ker \omega$. I'm not sure that one can say more than that in general.
| 1 | https://mathoverflow.net/users/2622 | 135389 | 74,557 |
https://mathoverflow.net/questions/133732 | 15 | What is the tightest upper bound we can establish on the central binomial coefficients $ 2n \choose n$ ?
I just tried to proceed a bit, like this:
$ n! > n^{\frac{n}{2}} $
for all $ n>2 $. Thus,
$ \binom{2n}{n} = \frac{ (n+1) \ldots (2n) }{n!} < \frac{\left(\frac{\sum\_{k=1}^n (n+k) }{n}\right)^n }{n^{n/2}} = \... | https://mathoverflow.net/users/34960 | Upper limit on the central binomial coefficient | Here's a way to motivate and refine the argument that
Péter Komjáth attributes to Erdős.
Start by computing the ratio between the $n$-th and $(n-1)$-st
central binomial coefficients:
$$
{2n \choose n} \left/ {2(n-1) \choose n-1} \right.
= \frac{(2n)! \phantom. / \phantom. n!^2}{(2n-2)! \phantom. / \phantom. (n-1)^2}
... | 16 | https://mathoverflow.net/users/14830 | 135396 | 74,560 |
https://mathoverflow.net/questions/135401 | 6 | Let $C$ be a category. For the purposes of this question, I would like to avoid cases where the answer might be "no" simply because $C$ is "too large", and so I will ask that $C$ has a set of **generators**, i.e. a set of objects $\{X\_\alpha\}\_{\alpha \in A}$ such that for any two morphisms $f,g : Y \to Z$ that are n... | https://mathoverflow.net/users/78 | Do cocontinuous SET-valued functors separate points? | No. Let $C$ be any category with a zero object. If $F : C \to \text{Set}$ is a cosheaf, then in particular it sends the zero object to the empty set. But since the zero object is also the terminal object, every object in $c$ is equipped with a morphism $c \to 0$, from which it follows that $F(c)$ is equipped with a mor... | 7 | https://mathoverflow.net/users/290 | 135409 | 74,565 |
https://mathoverflow.net/questions/135410 | -1 |
>
>
> >
> > I need the english translation of the article (or book) by jean-Pierre Serre in French on the topic Homologie singulière des espaces fibrés
> >
> > I am interested in understanding about this conjecture: there exist infinitely many geodesics between two points on a closed Riemannian manifold.
> > ... | https://mathoverflow.net/users/36244 | English translation of book by Jean-Pierre Serre? | this is Serre's Ph.D. thesis; it has been translated into english but only the introduction is available for free:
<http://www.worldscientific.com/doi/suppl/10.1142/8444/suppl_file/8444_chap01.pdf>
the remaining 100 pages are behind a £ 20 paywall:
<http://www.worldscientific.com/doi/pdf/10.1142/9789814401319_000... | 7 | https://mathoverflow.net/users/11260 | 135411 | 74,566 |
https://mathoverflow.net/questions/135412 | 2 | Ref: <http://link.springer.com/article/10.1007%2FBF02399204>
My question is about the proof of Theorem 0(b). On p.213, we see the expression
$\Delta\_{\alpha\ast\beta}(x;q,a)=\frac{1}{\varphi(q)}\sum\_{\chi\neq\chi\_0}\bar{\chi}(a)\left(\sum\_m\alpha\_m\chi(m)\right)\left(\sum\_n\beta\_n\chi(n)\right).$
I suppose... | https://mathoverflow.net/users/36274 | Question about BFI's "Primes in arithmetic progressions to large moduli" | I think you just want to forget the $n\le x$ conditions in the $\Delta$ definition of (1.1). Once they start at the bottom of p206 to switch to Motohashi's formulation, then the $n$-restriction on any $\Delta$-sum is better seen as coming from the support of the $\alpha$ and $\beta$. Prior to that, they needed to intro... | 2 | https://mathoverflow.net/users/36061 | 135413 | 74,567 |
https://mathoverflow.net/questions/135403 | 10 | It is well-know that the Bessel function has the asymptotic expansion $J\_n(\omega) \sim \left( \frac 2 {\pi \omega} \right)^{1/2} \left( \cos \left(\omega -\frac 1 2 n \pi - \frac 1 4 \pi\right) - \sin \left(\omega -\frac 1 2 n \pi - \frac 1 4 \pi\right) \frac {4n^2 - 1} {8 \omega} ...\right)$.
We also have the integr... | https://mathoverflow.net/users/36266 | Is there something wrong with Hörmander's theorem on stationary phase method | Looking closely again, I am pretty sure [my comment](https://mathoverflow.net/questions/135403/is-there-something-wrong-with-hormanders-theorem-on-stationary-phase-method#comment348904_135403) gives the answer. Note that Hörmander uses the notation $D = - i \partial$. So in particular
$$ Du = -i (n\pi i) u $$
when ... | 9 | https://mathoverflow.net/users/3948 | 135421 | 74,570 |
https://mathoverflow.net/questions/135425 | 28 | This might be well known for algebraic topologist. So I am looking for an explicit example of a 4 dimensional manifold with fundamental group isomorphic to the rationals $\mathbb Q$.
| https://mathoverflow.net/users/1190 | Example of 4-manifold with $\pi_1=\mathbb Q$ | Since any compact manifold has the homotopy type of a finite CW-complex (see this MathOverflow question: [Are non-PL manifolds CW-complexes?](https://mathoverflow.net/questions/36838/are-non-pl-manifolds-cw-complexes)) and $\mathbb{Q}$ is not finitely presented, the manifold $X$ you are looking for is necessarily non-c... | 58 | https://mathoverflow.net/users/7460 | 135427 | 74,574 |
https://mathoverflow.net/questions/135429 | 4 | Is there an analog of Fermat’s Two Squares theorem over polynomial rings $\Bbb Z[x]$ by considering a version for non-negative polynomials?
Fermat two squares says primes $p$ is sum of two squares iff $-1$ is quadratic residue mod $p$. Is there a sum of two squares formulation over polynomial rings using quadratic r... | https://mathoverflow.net/users/10035 | Fermat’s Two Squares for polynomials | $\def\QQ{\mathbb{Q}}$Yes. From [this answer](https://mathoverflow.net/a/82294/297), I learn of the following result of [Davenport, Lewis and Schinzel](http://www.ams.org/mathscinet-getitem?mr=163880):
>
> If $f(x)$ is a polynomial with integer coefficients such that every
> arithmetic progression contains an integ... | 5 | https://mathoverflow.net/users/297 | 135437 | 74,576 |
https://mathoverflow.net/questions/135417 | 13 | Given a simplicial operad one can form its category of operators. This is a simplicial category with a functor to the category of finite pointed sets which is a bijection on objects and whose hom-spaces have a particular product decomposition. Assuming the spaces in the operad are fibrant we can apply the coherent nerv... | https://mathoverflow.net/users/8818 | Correspondence between operads and $\infty$-operads with one object | The answer to Q1 is indeed yes. The construction you describe (category of operators followed by coherent nerve) gives a functor from the category of fibrant colored simplicial operads to the category of $\infty$-operads (in the sense of Lurie). This functor preserves weak equivalences and induces an equivalence on the... | 14 | https://mathoverflow.net/users/36302 | 135441 | 74,578 |
https://mathoverflow.net/questions/135204 | 5 | Is there a standard name or standard symbol for the binary operation that combines $x$ and $y$ to give $xy/(x+y)$, or equivalently $1/(1/x+1/y)$? (At least the expressions are equivalent if we ignore the case where $x$ or $y$ is 0.)
One possible name that comes to mind is "harmonic sum", but some might say that the h... | https://mathoverflow.net/users/3621 | Name and notation for a binary operation | I like Richard Stanley's suggested notation best, and I plan to use the term "harmonic sum" (unless someone points out existing notation and/or terminology).
| 1 | https://mathoverflow.net/users/3621 | 135445 | 74,580 |
https://mathoverflow.net/questions/135443 | 5 | Let $S$ be the spectrum of a discrete valuation ring and $f:X\rightarrow S$ be a relative projective curve with generic fiber smooth and special fiber semistable. How much differ the sheaf $R^2f\_{\operatorname{et},\*}\mathbb{G}\_m$ (computed in the étale topology) from $R^2f\_{\operatorname{Zar},\*}\mathbb{G}\_m$ (com... | https://mathoverflow.net/users/36304 | $R^2f_{\operatorname{et},*}\mathbb{G}_m$ vs $R^2f_{\operatorname{Zar},*}\mathbb{G}_m$ | For a smooth morphism, see Grothendieck, Dix exposés, Groupe de Brauer III, Corollaire 3.2.
| 4 | https://mathoverflow.net/users/nan | 135446 | 74,581 |
https://mathoverflow.net/questions/135467 | 7 | Suppose $\chi$ is a regular cardinal, and $N$ is an elementary submodel of the structure $\langle H(\chi),\in\rangle$, where $H(\chi)$ is those sets hereditarily of cardinality less than $\chi$.
Can we characterize when the transitive collapse of $N$ is $H(\lambda)$ for some $\lambda<\chi$?
The motivation is idle c... | https://mathoverflow.net/users/18128 | A question on elementary submodels | This hypothesis is equivalent to an instance of $\kappa$, the least point not in $N$, having some degree of [extendibility](http://en.wikipedia.org/wiki/Extendible_cardinal). After all, if $j$ is the inverse of the collapse, then we have an elementary embedding $j:H(\lambda)\to H(\chi)$ with critical point $\kappa$.
... | 8 | https://mathoverflow.net/users/1946 | 135468 | 74,590 |
https://mathoverflow.net/questions/135464 | 7 | Suppose
$$
P \subseteq \{1,2,\dots,N\},\quad |P| = K
$$
We calculate the differences as: $$d=p\_i-p\_j\mod N,\quad i\ne j$$
Now let $a\_d$ denote the number of occurrence of $d$ (for $d = 1, 2, \dots , N −
1$), then we have a set $$A=\{a\_1,a\_2,...,a\_{N-1} \}$$
As you know given $P$ and $N$, it is easy to build $A$. ... | https://mathoverflow.net/users/36047 | Difference Sets | A keyword in this area is *[homometric](http://en.wikipedia.org/wiki/Homometric_structures)*, and a key paper this one:
>
> Joseph Rosenblatt and Paul D. Seymour.
> "The Structure of Homometric Sets."
> *SIAM. J. on Algebraic and Discrete Methods*, **3**, 343-350, 1982.
> ([PDF download](http://epubs.siam.org/do... | 13 | https://mathoverflow.net/users/6094 | 135471 | 74,591 |
https://mathoverflow.net/questions/135463 | 7 | Suppose that we have a finite group $G$. Its classifying space $BG$ is not naturally pointed, so if we would like to consider it as a spectrum there are two possible approaches. The first is to take any point in the classifying space and let it be the basepoint when constructing the suspension spectrum; the second is t... | https://mathoverflow.net/users/1378 | Reconstructing a group from deformed classifying space | Let $X\_+$ be a space $X$ together with a disjoint base-point. Since $\Sigma^\infty(BG\_+)=\Sigma^\infty(BG)\vee S$, where $S$ is the sphere spectrum, there's not much difference between the two spectra you propose. In Example 5.2 of Martino–Priddy's 'Stable homotopy classification of $BG^{\hat{}}\_p$' you can find non... | 8 | https://mathoverflow.net/users/12166 | 135476 | 74,594 |
https://mathoverflow.net/questions/135462 | 5 | It is shown in Remark 7.1.12 of (a newer version of) Mark Hovey's book *Model Categories* that, in a stable model category, homotopy pullback squares coincide with homotopy pushout squares. The argument goes as follows: given a square $S$ of the form
$$
\begin{matrix}
W & \to & X\\
\downarrow\_h & & \downarrow\_g\\
Z &... | https://mathoverflow.net/users/1291 | Homotopy limit-colimit diagrams in stable model categories | This is not true. For example, take $W = X = Y = 0$, and the map $Z \rightarrow V$ to be an isomorphism. Then you've got a homotopy colimit diagram, but it is only a homotopy limit diagram if $Z$ is weakly contractible.
| 9 | https://mathoverflow.net/users/7721 | 135478 | 74,595 |
https://mathoverflow.net/questions/135355 | 2 | An oft-cited theorem of Serre states that there are infinitely many geodesics between any two points in a closed Riemannian manifold. Could someone please provide an intuitive sketch of the proof?
| https://mathoverflow.net/users/36244 | Proof of a theorem of Jean-Pierre Serre on geodesics of closed Riemannian manifolds | The question having been reopened, I'll re-post my comment as an answer: There is a paper by Nabutovsky and Rotman, available at <http://www.math.toronto.edu/alex/morseoct12.pdf> in which they give "a somewhat modernized sketch of the proof of Serre’s theorem given by A.Schwarz...."
| 13 | https://mathoverflow.net/users/15837 | 135479 | 74,596 |
https://mathoverflow.net/questions/135480 | 1 | Let $f: X \rightarrow X'$ be a morphism of schemes, and let $\mathcal{I}^{\bullet}$ be a complex of $\mathcal{O}\_X$-modules. There are two spectral sequences (well, more than that, but these are the two I care about) abutting to the hypercohomology $\mathbb{H}^n(X, \mathcal{I}^{\bullet})$. The first is the second spec... | https://mathoverflow.net/users/31771 | Comparison between $E_2$-terms of Leray and "second hypercohomology" spectral sequences | In the second spectral sequence you write $R^q f\_\ast I^\bullet$. This is correct but $R^qf\_\ast$ must be considered as the $q$th *hyper-derived functor* applied to the complex $I^\bullet$, it is not the ordinary derived functor $R^qf\_\ast$ applied termwise. See my answer to [Construction of the spectral sequence of... | 5 | https://mathoverflow.net/users/1310 | 135494 | 74,598 |
https://mathoverflow.net/questions/135489 | 8 | Is the cohomology of the genus 2 mapping class group (that is, the cohomology of the moduli stack $M\_2$ of genus 2 curves) known? I'd be interested in references. The rational cohomology is known to be trivial, but I am interested in torsion phenomena, especially at the prime $2$.
| https://mathoverflow.net/users/344 | Cohomology of the genus 2 mapping class group | The possible automorphism groups of a curve of genus two are known very classically, this is usually attributed to Bolza. In his list we find only three prime factors in the orders of the groups: 2,3 and 5. So this is the only possible torsion.
I haven't read Benson and Cohen, "Mapping class groups of low genus and t... | 9 | https://mathoverflow.net/users/1310 | 135499 | 74,600 |
https://mathoverflow.net/questions/135492 | 0 | Does there exist for arbitrary $\alpha$, $0<\alpha<1$, a measurable subset $A$ of the closed unit interval $[0,1]$ such that Lebesgue measure $m(A)=\alpha$ and the following "homogeneity" condition is satisfied: for any subinterval $[a,b]\subseteq[0,1]$ one has: $m(A\cap[a,b])=\alpha\cdot (b-a)$?
| https://mathoverflow.net/users/2164 | homogeneous subset of [0,1] of arbitrarily small Lebesgue measure | It turned out that such sets do not exist, due to the Lebesgue density theorem, as prompted by Nate Eldridge. Applied to the unit interval, the Lebesgue density theorem states that
If $A$ is a measurable subset of $[0,1]$, then
$$
\lim\_{\epsilon\to 0} \frac{m(A\cap (x-\epsilon,x+\epsilon))}{2\epsilon} = 1
$$
for a... | 2 | https://mathoverflow.net/users/2164 | 135500 | 74,601 |
https://mathoverflow.net/questions/135076 | 17 | We are given rational numbers $[c\_1, c\_2, \ldots, c\_n]$ and $v$ from the interval $[0,1]$.
Consider the $n$-fold integral
$$
J = \int\_{\theta\_1 \in I\_1, \theta\_2 \in I\_2 \ldots, \theta\_n \in I\_n} d\theta\_n\ldots d\theta\_2 d\theta\_1
$$
whose intervals are defined by
$$
I\_j = \begin{cases}
[0,1] & j = ... | https://mathoverflow.net/users/6495 | An NP-hard $n$ fold integral | The integral essentially asks for the probability that, for $n$ independent "events" uniformly distributed in $[0,1]$, at least one happens after $c\_n$, at least two happen after $c\_{n-1}$, etc (thinking of the unit interval as time). Let $P\_n(c\_1,\dots,c\_n) = n!\cdot J$ denote this probability (the integral $J$ a... | 11 | https://mathoverflow.net/users/14302 | 135502 | 74,602 |
https://mathoverflow.net/questions/135497 | 4 | Let $f : X \to Y$ be a continuous map of [complete](http://nlab.mathforge.org/nlab/show/complete+space) topological vector spaces. Suppose that $A \subseteq X$ and $B \subseteq Y$ are proper, dense linear subspaces, and that the restriction map $f : A \to B$ is an isomorphism. Is the original map an isomorphism?
| https://mathoverflow.net/users/238 | Isomorphisms between topological vector spaces | Robertson & Robertson: [Topological Vector Spaces](http://books.google.hu/books?id=mV44AAAAIAAJ). Chapter VI (Completeness), Proposition 6 and Corollary 1.
If $E$ and $F$ are separated l.c. spaces, $t:E\to F$ continuous, linear, then there is a unique continuous linear extension $\hat t: \hat E \to \hat F$. Here $\ha... | 4 | https://mathoverflow.net/users/12898 | 135503 | 74,603 |
https://mathoverflow.net/questions/135458 | 8 | Let $N \ge 1, k \ge 2$ be integers, and $M\_k(\Gamma\_1(N))$ the space of weight k modular forms of level $\Gamma\_1(N)$. Let $\mathbb{T}$ be the $\mathbb{Z}$-subalgebra of $\operatorname{End} M\_k(\Gamma\_1(N))$ generated by the diamond operators $\langle d \rangle$ and the Hecke operators $T\_\ell$ for $\ell \nmid N$... | https://mathoverflow.net/users/2481 | Omitting primes from a Hecke algebra | It seems to me that the proof you describe is fine except maybe at the prime 2, the problem being that the argument you are sketching (presumably) relies on expressing the missing operators via traces of Frobenius morphisms modulo arbitrary powers of maximal ideals of the Hecke algebra, but this works only for Hecke op... | 7 | https://mathoverflow.net/users/2284 | 135507 | 74,605 |
https://mathoverflow.net/questions/65250 | 7 | Let $\omega\_1$ be smallest uncountable ordinal. I am trying to understand the possible "large" subspaces of $C[0,\omega\_1]$, namely those which are isomorphic to the whole space. Therefore I have the following question:
Does every subspace of $C[0,\omega\_1]$ isomorphic to $C[0,\omega\_1]$ contain a complemented co... | https://mathoverflow.net/users/15129 | Subspaces isomorphic to $C[0, \omega_1]$ | Yes, it can deduced from Lemma 1.2 combined with Proposition 2 of
>
> D. Alspach and Y. Benyamini, [Primariness of spaces of continuous functions on ordinals](http://link.springer.com/content/pdf/10.1007/BF02761606.pdf). *Israel J. Math*. **27** (1977), 64–92.
>
>
>
Another proof can be found [here (Cor. 1.12... | 3 | https://mathoverflow.net/users/15129 | 135511 | 74,606 |
https://mathoverflow.net/questions/135333 | 8 | Let $S$ be an Enriques surface over complex numbers. It is known that $S$ admits an elliptic fibration over $\mathbb{P}^1$ with $12$ nodal singular fibers and $2$ double fibers. How can I see this well-known fact? Is it possible to explicitly construct such a fibration via the Enriques lattice $NS(S)\cong U\oplus E\_8$... | https://mathoverflow.net/users/50973 | A question on an elliptic fibration of the Enriques surface | A complete and detailed treatment of elliptic pencils on Enriques surfaces (included the result you are quoting) can be found in the book by Barth-Hulek-Peters-Van de Ven *Compact Complex Surfaces* (Ergebnisse der Mathematik und ihrer Grenzgebiete **4**, Springer).
See in particular Chapter VIII, Section 17 "Elliptic... | 2 | https://mathoverflow.net/users/7460 | 135519 | 74,611 |
https://mathoverflow.net/questions/135509 | 5 |
>
> What is the largest possible number of subsets of a $4n$-element set $X$, such that each subset contains precisely $2n$ elements, and such that each of the pairwise intersections of the subsets has precisely $n$ elements?
>
>
>
The best bounds I have for the maximal size $N$ of such a set system are $\lfloor... | https://mathoverflow.net/users/7252 | Better bounds for exact-intersection Erdős–Ko–Rado system? | Your upper bound seems wrong to me.
Let first $n=1$, $X=\{1,2,3,4\}$ and $A\_i=\{1,i\}$ for $i=2,3,4$. Then we have $3>2\cdot 1-1$ subsets of size $2$ intersecting each other in $1$ point.
More generally, for any integer $k>1$ Sylvester constructed a Hadamard matrix of order $2^k$, that is a matrix with $1$ and $-1... | 3 | https://mathoverflow.net/users/4961 | 135522 | 74,612 |
https://mathoverflow.net/questions/87886 | 2 | It is a well-known and cute exercise in algebraic graph theory to show that $K\_{10}$ cannot be written as the edge-disjoint union of three copies of the Petersen graph $P$. Indeed, the graph $G$ whose edge-set is the complementary of the two copies of $P$ in $K\_{10}$ is a $3$-regular *bipartite* graph. When I taught ... | https://mathoverflow.net/users/2284 | Decomposition of $K_{10}$ in copies of the Petersen graph | The general picture is as follows. Let $\mathcal P\_{i}$ be an edge-disjoint decomposition of $K\_{10}$ in 3 3-regular graphs with $\mathcal P\_{1,2}$ isomorphic to the Petersen graph. Then there exists two eigenvectors common to all three $\mathcal P\_{i}$: the indicatrix of $K\_{10}$ and one associated with the eigen... | 0 | https://mathoverflow.net/users/2284 | 135529 | 74,615 |
https://mathoverflow.net/questions/135524 | 5 | Does the nerve functor $\operatorname{Cat}\to \operatorname{sSet}$ from small categories to simplicial sets preserve weak equivalences?
If $f\colon C\to D$ and $g\colon D\to C$ are functors of small categories such that $\alpha\colon f\circ g\sim 1\_D$ and $\beta\colon g\circ f \sim 1\_C$, then we have a pair of map... | https://mathoverflow.net/users/7341 | The nerve of categories preserves weak equivalence? | Such questions are typically framed in terms of Classifying Spaces, but the answer is yes. It follows from, for instance, from Proposition 2.1 in Graeme Segal's article
*Classifying spaces and spectral sequences*. Publications Mathématiques de l'IHÉS, 34 (1968), p. 105-112, available [here](http://www.numdam.org/item?... | 6 | https://mathoverflow.net/users/18263 | 135530 | 74,616 |
https://mathoverflow.net/questions/135537 | 4 | I am currently reading Chang and Keisler's model theory textbook, and there is something that I don't seem to be able to understand about their proof (through the Omitting Types Theorem) that every countable model $\mathfrak A = \langle A, E\rangle$ of $ZF$ has an end-elementary extension $\mathfrak B = \langle B, F\ra... | https://mathoverflow.net/users/36380 | Replacement axiom and existence of end-extensions (Chang and Keisler's Model Theory) | The negation of the second formula is logically equivalent to
$$\forall x\in a\,\exists y\,\forall z\,(\phi(x,z)\to z\in y).$$
Using collection (and union), this implies
$$\exists y\,\forall x\in a\,\forall z\,(\phi(x,z)\to z\in y),$$
which is equivalent to the negation of the first formula.
| 4 | https://mathoverflow.net/users/12705 | 135538 | 74,618 |
https://mathoverflow.net/questions/135432 | 24 | In 1978, in a series of unpublished conferences in Montréal, A. Joyal announced a remarkable theorem that unified several completeness theorems for fragments of first order logic, as well as first order logic itself. Given a coherent category $T$, we have the evaluation functor $e\_T: T \to \mathcal{S}et^{\mathcal{M}od... | https://mathoverflow.net/users/12976 | On Joyal's completeness theorem for first order logic | The following is based on vague memories from long ago, so it comes with no guarantees. If it turns out to be totally wrong, then someone will surely say so and I'll delete it.
I think the poset $P$ that you want, to replace the category $M=\mathcal Mod\_c(T)$, consists of finite composable sequences of morphisms of ... | 11 | https://mathoverflow.net/users/6794 | 135566 | 74,629 |
https://mathoverflow.net/questions/135562 | 4 | Let $G$ be a finite group of Lie type , and let $P$ be a parabolic subgroup of $G$ containing a proper Borel subgroup $B$. Let $s$ be a semisimple element of $G$. I want to count the number of conjugates of $P$ that contain $s$. But first I need the answer of the following question.
Assume that $s$ is contained in $... | https://mathoverflow.net/users/16049 | counting the parabolic subgroups that contain a fixed semisimple element | You must have meant to assume $s \in B$, since $s$ lies in the identity component of its centralizer (in the ambient connected semisimple group that you did not name) and so lies in every Borel subgroup of that identity component.
Consequently, your question can only have an affirmative answer if $s \in B\_s$, which... | 6 | https://mathoverflow.net/users/35361 | 135585 | 74,632 |
https://mathoverflow.net/questions/132311 | 0 | Let $$G=\langle x, y, z\mid xyx^{-1}=zy, xzx^{-1}=z, yz=zy\rangle,$$ denote $l^1(G)^{\times}$ to be the set of units in $l^1(G)$, which we have considered as a ring with multiplication defined by the usual convolution, i.e., $(\sum\_{g\in G}\lambda\_gg)(\sum\_{h\in G}\mu\_hh)=\sum\_{g, h\in G}\lambda\_g\mu\_hgh$.
Can... | https://mathoverflow.net/users/9305 | Find a special element in group algebra | Omit this problem, we can take $l=z+\frac{1}{2}(-z-1)x+\frac{1}{4}x^2$, then $l^{-1}=\sum\_{n\geq 0, m\geq 0}\frac{1}{2^{n+m}}x^{n+m}z^{-m-1}\in l^1(G)$.
| 1 | https://mathoverflow.net/users/9305 | 135599 | 74,634 |
https://mathoverflow.net/questions/135588 | 2 | Let $q$ be a large prime and $e$ an integer such that $GCD(e,q-1)=1$. Let $p(x)$ be a polynomial of degree $e^n$ with coefficients in $\mathbb Z\_q$ such that there exists a progression of polynomials (the "composition")
* $p\_i(x) = a\_i(p\_{i-1}(x))^e+b\_i$, $a\_i, b\_i \ne 0$ with
* $p\_0(x) = x$ and
* $p(x) = p\_... | https://mathoverflow.net/users/36408 | Algorithm for representing a polynomial as a composition of lower degree polynomials | As in my comment, I will assume that $p(x)$ and all $p\_i(x)$ are monic. I will also assume that $\gcd(e,q)=1$. All of these hypotheses can be removed easily. To make the problem nontrivial, assume that $e>1$. The only issue is to determine $b\_n$; if that can be done, then $p\_{n-1}(x)$ is the unique monic $e$-th root... | 4 | https://mathoverflow.net/users/30412 | 135602 | 74,635 |
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