parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/131579 | 6 | Let $V:=\oplus\_{j\in\mathbb{Z}}V\_j$ be a graded $\mathbb{F}$-vector space over
the field $\mathbb{F}$. The graded tensor product of graded vector spaces is given
by
$V \otimes W:= \oplus\_{j\in \mathbb{Z}}\oplus\_{p+q=j}V\_p\otimes V\_q$
and for the graded vector space $\mathbb{F}[j]$, which is $\mathbb{F}$ in deg... | https://mathoverflow.net/users/21302 | Natural Isomorphism of $S(V[1])$ and $(\bigwedge V)[n]$ | You have a detailed proof (in a much more general context but easy to read) in Proposition I.4.3.2.1 of Illusie, Complexe Cotangent et Déformations I, Springer LNM 239.
| 8 | https://mathoverflow.net/users/34259 | 131634 | 72,939 |
https://mathoverflow.net/questions/131364 | 42 | This is a somewhat imprecise question, as I am not sure how exactly how to formalise how to do mathematics "without" a certain key tool, but hopefully the intent of the question will still be clear.
Let $G$ be a finite group. Traditionally, a character $\chi: G \to {\bf C}$ on $G$ is defined as being the trace of a f... | https://mathoverflow.net/users/766 | How much of character theory can be done without Schur's lemma or the Artin-Wedderburn theorem? | I spent a long time writing an answer to this question, but MO did not believe I was a human being ( I did mis-spell one of the test words, but everyone deserves a second chance, I think ), so it seems to have disappeared. I am not sure I have the energy to do it again right now, but here (in precis, though not precise... | 23 | https://mathoverflow.net/users/14450 | 131640 | 72,940 |
https://mathoverflow.net/questions/131632 | 5 | Let $C$ be a nonsingular projective curve defined over $\mathbb{Q}$, which does **not** admit a map of degree 1 or 2 to $\mathbb{P}^1$ or to an elliptic curve. It is then a consequence of Corollary 3 of [1] that $C$ possesses only finitely many quadratic points; i.e., the set
$$\Gamma\_C := \left\{p \in C : [\mathbb... | https://mathoverflow.net/users/5744 | Can one bound the Quadratic Points on Curves? | Hi Barinder!
As far as I know there is not an algorithm to do so. See for instance the following paper of Harris and Silverman:
<http://www.ams.org/journals/proc/1991-112-02/S0002-9939-1991-1055774-0/>
Especially in the proof of Corollary 3 you can really get a clear picture of the argument. Basically if your cur... | 5 | https://mathoverflow.net/users/3384 | 131641 | 72,941 |
https://mathoverflow.net/questions/131648 | 3 | What is the number of ways to parenthesize $n$ elements using applications of operators of *arbitrary* arities larger than or equal to $2$? For example, for $n=3$, there are $3$ ways:
$$
abc, a(bc),(ab)c
$$
and for $n=4$ there are 11 ways:
$$
abcd,\ ab(cd),\ a(bc)d,\ (ab)cd ,\ a(bcd), (abc)d,$$
$$
a(b(cd)),\ a((bc)d),\... | https://mathoverflow.net/users/34320 | Even more generalized Catalan numbers | A little bit of programming and a look up in the OEIS tells me that this is the sequence A001003, the solution to [Schroeder's second problem](http://oeis.org/A001003), see also [Wikipedia](http://en.wikipedia.org/wiki/Schr%25C3%25B6der%25E2%2580%2593Hipparchus_number).
According to the page in OEIS, the asymptotic f... | 6 | https://mathoverflow.net/users/10423 | 131656 | 72,947 |
https://mathoverflow.net/questions/131494 | 8 | Let $$J\_1=\int\_0^1\frac{1}{\sqrt{1-t\_2}}dt\_2,$$
$$J\_2=\int\_0^1 \int\_0^{t\_2}\frac{1}{\sqrt{1-t\_2}}(\frac{1}{\sqrt{1-t\_3}}+\frac{1}{\sqrt{t\_2-t\_3}})dt\_3dt\_2,$$
$J\_3=\int\_0^1 \int\_0^{t\_2}\int\_0^{t\_3}\frac{1}{\sqrt{1-t\_2}}(\frac{1}{\sqrt{1-t\_3}}+\frac{1}{\sqrt{t\_2-t\_3}})(\frac{1}{\sqrt{1-t\_4}}+\fra... | https://mathoverflow.net/users/23381 | Rate of growth of an explicit integral | The growth rate is exponential. Specifically, I will prove the bound $J\_n \leq \frac{(n+1)^{n} 2^n}{(n+1)!}$, which grows like $(2e)^n$. I can also show that $J\_n \geq 2^n$. I would guess that neither of these are the true rate of growth.
I'll also give an explicit recursion for the $J\_n$, which makes it clear tha... | 10 | https://mathoverflow.net/users/297 | 131659 | 72,948 |
https://mathoverflow.net/questions/130851 | 1 | For a given partition $[n\_{1},...,n\_{k}]$ of $N \in \mathbb{N}$ there exists a corresponding nilpotent orbit variety $O\_{[n\_{1},...,n\_{k}]}$ in $\mathfrak{gl}(N)$ which can be represented by a set of polynomial equations relating the conditions on matrices in $\mathfrak{gl}(N)^{\text{nilp}}$. I was wondering if an... | https://mathoverflow.net/users/20343 | Sage or Magma Implementation of Nilpotent Orbit Varieties | I don't know how this would help "constructing local weak Neron models for the varieties given as output of the program", but the defining equations of the nilpotent varieties are known. They were conjectured by Tanisaki and proved by Weyman in
[The equations of conjugacy classes of nilpotent matrices](http://www.am... | 2 | https://mathoverflow.net/users/5740 | 131661 | 72,949 |
https://mathoverflow.net/questions/131652 | 1 | Is there a decent way to describe the canonical module of the ring $\frac{\mathbb{C}[x,y,z]}{x^2-yz}$? I am not necessarily looking for an explicit description of the canonical module, but I would appreciate any and all suggestions for describing its structure.
| https://mathoverflow.net/users/25358 | Canonical Modules | As Graham points out, this ring is Gorenstein so the canonical module is *isomorphic* to the ring itself. For most hypersurfaces, this is all you can say. However, I think in this case one can say slightly more.
The ring is also toric $k[x,y,z]/(x^2 - yz) = k[ab, a^2, b^2]$. Thus we have a canonical way to identify t... | 8 | https://mathoverflow.net/users/3521 | 131662 | 72,950 |
https://mathoverflow.net/questions/131660 | 3 | Let $X$ be a connected, integral curve over a field $k$, and let $Y \rightarrow X$ be a finite etale cover. Corresponding to this cover there is a finite extension of function fields $k(Y): k(X)$.
Question: is it possible to characterize the finite extensions of $k(X)$ which arise in this way in a purely field theor... | https://mathoverflow.net/users/19367 | Recognizing etale covers on the level of function fields | Yes, this is possible. For simplicity, compare first to number theory: given a finite field extension $E / \mathbf Q\_p$ you want to know whether $\mathcal O\_E / \mathbf Z\_p$ ($\mathcal O\_E$ is the integral closure of $\mathbf Z\_p$ in $E$) is etale. This is precisely the case if the original extension is unramified... | 2 | https://mathoverflow.net/users/18116 | 131664 | 72,951 |
https://mathoverflow.net/questions/87026 | 1 | Here is the text of Exercise:
2 a) Let $X$ be an *ordered* set. Show that the set of intervals
$\left[x, \rightarrow\right[$ (resp. $\left]\leftarrow, x\right]$)
is a base of topology on $X$; this topology is called the *right* (resp. *left*) topology of $X$. In the right topology, any intersection of open sets i... | https://mathoverflow.net/users/20821 | Possible errata in Nicolas Bourbaki's General Topology -I, Chapter 1 Exercise 2 ? | Bourbaki was right :-) On the other hand, let $(X\ \le)$ be a partially ordered set. In general the family
$$\mathbf B\ \ :=\ \ \{\ ]x,\rightarrow[\ :\ x\in X\ \}$$
is NOT a topological base for any topology in $X$. One reason is trivial: no minimal element belongs to any member of $B$; thus if there ... | 8 | https://mathoverflow.net/users/8385 | 131665 | 72,952 |
https://mathoverflow.net/questions/131657 | 28 | QUESTION
--------
They had plenty of time to adopt the theory of categories. They had Eilenberg, then Cartan, then Grothendieck. Did they feel that they have established their approach already, that it's too late to go back and start anew?
I have my very-very general answer: *World* is *Chaos*, *Mathematics* is a *... | https://mathoverflow.net/users/8385 | Why did Bourbaki's Élements omit the theory of categories? | One thing to keep in mind is that Bourbaki started in the 1930s, so in some sense simply *too early* to include category theory right from the start on, and foundational matters were rather fixed early on and then basically stayed like this. Since (I think) the aim was/is a *coherent* presentation (as opposed to merely... | 32 | https://mathoverflow.net/users/nan | 131667 | 72,954 |
https://mathoverflow.net/questions/131658 | 4 | Suppose I have an irreducible representation of a simple Lie algebra, say $\mathfrak{sl}(n)$ or $\mathfrak{so}(n)$ i.e., $A$ and $D$ type. Given such a representation, $\Gamma\_\lambda$, indexed by its highest weight $\lambda = (\lambda\_1, \dots, \lambda\_n)$, what is known about the decomposition of the symmetric pow... | https://mathoverflow.net/users/25794 | Decomposition into irreducibles of symmetric powers of irreps. | First I'd recomend that you use more careful language in describing the groups and representations, including the precise notion of *rank*. The subject is complicated enough as is, so for instance you need to refer to the Lie algebras of special orthogonal groups as having Lie type $B\_n$ or $D\_n$ in the respective ca... | 8 | https://mathoverflow.net/users/4231 | 131668 | 72,955 |
https://mathoverflow.net/questions/131524 | 4 | I am an amateur mathematician, and certainly not a set theorist, but there seems to me to be an easy way around the reflexive paradox: Add to set theory the primitive $A(x,y)$, which we may think of as meaning that $x$ is allowed to belong to $y$ and the axiom
$\forall x,\forall y, x\in y \rightarrow A(x,y)$
and mo... | https://mathoverflow.net/users/34276 | Avoiding reflexive paradox in set theory | As The User says in the comments, you still have a problem, aesthetically at least -- in order to prevent the existence of "silly" models, you need some axiom asserting that $\in^\*$ isn't too big. As is, a model in which $\in^\*$ always holds between any $x$ and $y$ satisfies your axioms; this means that your separati... | 6 | https://mathoverflow.net/users/8133 | 131672 | 72,957 |
https://mathoverflow.net/questions/131675 | 2 | Given an even integer N, what is the minimum set of primes such that any even number $x \leq N$ can be expressed as the sum of two primes in the set?
Goldbach's conjecture said Every even integer greater than 2 can be expressed as the sum of two primes.
[http://en.wikipedia.org/wiki/Goldbach's\_conjecture](http://en.... | https://mathoverflow.net/users/34326 | Given an even integer N, what is the minimum set of primes such that any even number x <= N can be expressed as the sum of two primes from the set? | It is reasonable to believe that there is a constant $c$ such that that there exists a set of prime numbers containing at most $c \sqrt{x \log x}$ elements below $x$ such that each even number is the sum of two primes from this set. So the (lower) bound of $\sqrt{x}$ you mentioned is likely not too far away from the tr... | 5 | https://mathoverflow.net/users/nan | 131677 | 72,959 |
https://mathoverflow.net/questions/131674 | 6 | Any closed form for series like $$F(x)=\sum\_{p=2}^{\infty}x^p,\quad p\text{ is prime}$$ or $$F(x)=\sum\_{i=0}^{\infty}x^{i!}\quad ?$$
More generally, we can obtain a power series from decimal expansion of a number $r$ (with $0< r<1$) by replacing $$\left(\frac{1}{10}\right)^i$$ with $$x^i$$ like $$\frac{1}{3}=3\left... | https://mathoverflow.net/users/14024 | Any closed form for series like $F(x)=\sum\limits_{p=2}^{\infty}x^p,$ where $p$ is prime? | See this blog post: <https://uniformlyatrandom.wordpress.com/tag/power-series/>
which contains a proof of the result by Fatou:
A function whose power series expansion has integer coefficients and radius of convergence 1 is either rational (in $\mathbb{Q}(x)$) or transcendental (over $\mathbb{Q}(x)$).
If $r$ is ra... | 9 | https://mathoverflow.net/users/21090 | 131678 | 72,960 |
https://mathoverflow.net/questions/88302 | 7 | Let $\mathbb{V}$ be a representation of a Lie group $G$ and let $P \to M$ be a principal $G$-bundle with a principal connection. If $\mathbb{V}$ is finite-dimensional, then one can associate to this data an associated vector bundle $P\times\_G \mathbb{V}$ with linear connection. I thought that basically the same constr... | https://mathoverflow.net/users/6818 | Associated vector bundles of infinite rank and induced connections | If you use the cocycle description of principal bundles (for finite dimensions) as in 18.7.4 of [1](http://www.mat.univie.ac.at/~michor/dgbook.pdf), you can describe the associated bundle using convenient calculus, since then you can flip coordinates freely.
Inducing connections is then described in 19.8 (for associat... | 6 | https://mathoverflow.net/users/26935 | 131694 | 72,967 |
https://mathoverflow.net/questions/131614 | 2 | I had tried to help someone on [math.StackExchange](https://math.stackexchange.com/questions/400270/how-prove-that-1-tra2-sum-1-le-i-le-j-le-3a-ij-a-ji2-4) to prove the identity:
$$ (1-Tr(A))^2+\sum\_{1\le i\le j\le 3}(a\_{ij}-a\_{ji})^2=4$$
I guess you could argue the left hand side is independent of basis. Then w... | https://mathoverflow.net/users/1358 | Symmetric sums and Representations of SO(3) | First, I think there is a typo in your question. Maybe you mean $\sum\_{1\le i\le j\le 3}(a\_{ij}-a\_{ji})^2 =4\sin^2\phi$?
Anyway, I think it is easier to work out these identities in $\mathsf{SU}(2)$, the double cover of $\mathsf{SO}(3)$. Up to conjugation, $A=\mathrm{diag}(e^{i\phi},e^{-i\phi})$ in $\mathsf{SU}(2)... | 4 | https://mathoverflow.net/users/12218 | 131695 | 72,968 |
https://mathoverflow.net/questions/131683 | 8 | What are the current best asymptotic bounds on $\pi^{-1}(x)$, where $\pi(x)$ denotes the prime counting function (number of primes at most $x$)?
In other words, I am curious about the state of the art for estimating the $n^{th}$ prime. From the prime number theorem, it seems clear that $\pi^{-1}(x)=\Theta(x \log x)$.... | https://mathoverflow.net/users/29961 | Asymptotic bounds on $\pi^{-1}(x)$ (inverse prime counting function) | Here is how we invert the prime counting function $\pi(x)$ to estimate the n-th prime $p\_n$. Let $g(u),u \ge 2$ be positive, continuous increasing function and let $f(x)$ be defined by
$$
x = \int\_{2}^{f(x)} \frac{du}{g(u)}
$$
Then
$$
f(x) = \int\_{2}^{x} g(f(u))du + f(2)
$$
Take $g(u) = \ln u$ so our first ... | 6 | https://mathoverflow.net/users/23388 | 131697 | 72,969 |
https://mathoverflow.net/questions/131670 | 22 | QUESTION
--------
I wanted to introduce and develop the complex logarithm from scratch. As the result I've arrived a couple of months ago at the following identity after which the road to complex logarithm is wide open:
$$\frac{a-b}{a+b}\cdot\frac{b-c}{b+c}\cdot\frac{c-a}{c+a}\quad +\quad \frac{a-b}{a+b}+\frac{b-c}... | https://mathoverflow.net/users/8385 | Anything special (historical?) about surface $x\cdot y\cdot z\ +\ x+y+z=0$? | The corresponding projective surface
$$S: xyz + (x + y + z)w^2=0 \subset \mathbb{P}^3,$$
is a singular cubic surface - singular cubic surfaces are special. It has three singularities, each of which has singularity type $A\_1$ (this means that locally each has the shape $xy + w^2 =0$). These are the points $(1:0:0:0),(... | 18 | https://mathoverflow.net/users/5101 | 131704 | 72,971 |
https://mathoverflow.net/questions/131708 | 34 | $\newcommand{\RR}{\mathbb{R}}$The present question arises from some confusion on my part regarding the precise statement of the strong Whitney embedding theorem for non-compact manifolds.
The strong Whitney embedding theorem is usually stated as follows.
>
> **Theorem:** If $M$ is a smooth $n$-dimensional manifol... | https://mathoverflow.net/users/21095 | Strong Whitney embedding theorem for non-compact manifolds | Regarding question 1, yes you can always ensure the image is closed. You prove the strong Whitney by perturbing a generic map $M \to \mathbb R^{2m}$ to an immersion, and then doing a local double-point creation/destruction technique called the Whitney trick. So instead of using any smooth map $M \to \mathbb R^{2m}$, st... | 26 | https://mathoverflow.net/users/1465 | 131710 | 72,973 |
https://mathoverflow.net/questions/131577 | 3 | Suppose $Y$ is a pair of pants with a hyperbolic structure and $\gamma\_i; i = 1, 2, 3$ are the geodesic boundaries of length $l\_i; i=1, 2, 3$ respectively. Now consider a essential simple arc $\sigma$ in $Y$ with end points on a same boundary component of $Y$ and $l$ denote the length of unique geodesic in the homoto... | https://mathoverflow.net/users/23358 | Hyperbolic pair of pants. | If you keep the lengths of two cuffs fixed (say equal to L) and let the third one go to infinity (say equal to R), then the length $l$ of the ortho-geodesic with endpoints on the third cuff goes to zero. There is a trigonometric formula to see this. Cut the pants into two right-angled hexagons then further into four pe... | 5 | https://mathoverflow.net/users/38319 | 131718 | 72,975 |
https://mathoverflow.net/questions/131464 | 0 | **Proposition**: If $F$ is a field, let $F[x]$ be the ring of all polynomials whose coefficients are in $F$. The fraction field of $F[x]$, denoted $F(x)$, is defined to be the ratios $r(x) = f(x)/g(x)$ for $f(x),g(x)\in F[x]$ with $g(x) \not= 0$. The $F$-automorphisms$\dagger$ of $F(x)$ are linear-fractional changes of... | https://mathoverflow.net/users/25437 | Relations between automorphisms of field of rational functions and Mobius Transfomation | Here is a relationship between your question and conformal mappings. For $X$ and $Y$ arbitrary Riemann surfaces, any $\mathbb{C}$-homomorphism between their fields of meromorphic functions comes from a unique holomorphic map $f:X\to Y$. In particular, $\mathbb{C}$-isomorphisms come from biholomorphisms, i.e. Riemann su... | 3 | https://mathoverflow.net/users/38319 | 131727 | 72,979 |
https://mathoverflow.net/questions/131422 | 8 | Consider a a sequence of non-negative functions $(f\_n)\_n$, bounded in $L^1([-1,1])$ and weakly$-\star$ converging in $\mathscr{M}^1([-1,1])$ to some $f\in L^1([-1,1])$. What I mean by this convergence is that for any continuous function $\varphi\in\mathscr{C}^0([-1,1])$,
\begin{align\*}
\int\_{-1}^1 f\_n \varphi \... | https://mathoverflow.net/users/27767 | Counterexample of non-negative sequence weakly converging in $\mathscr{M}^1$ but not $L^1$ | Let's build a "fat Cantor set". Start with $A\_0 = [0,1]$ with measure $\alpha\_0=1$.
Then remove a short open interval centered at $1/2$, leaving a set $A\_1 \subset A\_0$ of measure $\alpha\_1 < \alpha\_0$. So $A\_1$ is made up of $2$ closed intervals of length $\alpha\_1/2$. Let $B\_1$ be the removed interval with l... | 4 | https://mathoverflow.net/users/454 | 131728 | 72,980 |
https://mathoverflow.net/questions/131702 | 0 | For any positive integer $N$ and real number $a > 0$. Define $f(k) = \frac{1}{N-k} (1+\frac{a}{k})$. The problem is to find a positive integer $k$ that minimizes $f(k)$.
It is quite easy to solve the problem by finding the minimum $k$ satisfying $f(k)\leq f(k+1)$, this will give a solution $k^\* = \lceil \sqrt{aN+a^... | https://mathoverflow.net/users/34318 | Updated: finding an integer $k$ that minimizes $1/(N-k) (1+1/k)$ | **Part 1.** (the proof that rounding functions may be different).
This is not true in general. Observe that $\frac{1}{2}>\left(\sqrt{a^2+Na}-a\right)-\left(\sqrt{a^2+Na+\frac{1}{4}}-a-\frac{1}{2}\right)>0.$ So it is enough to choose integer $m,$ to satisfy the
$$\sqrt{a^2+Na}-a>m+\frac{1}{2}$$
and
$$\sqrt{a^2+Na+\fra... | 3 | https://mathoverflow.net/users/17503 | 131730 | 72,982 |
https://mathoverflow.net/questions/131719 | 11 | What are some techniques and theorems of analytic number theory that have proved useful outside of number theory?
| https://mathoverflow.net/users/136 | Spinoffs of analytic number theory | Look at the list of examples of zeta-functions on Wikipedia, and not all of them are in number theory.
Here are some specific applications of the idea of a zeta-function in other areas of mathematics.
1. If $G$ is a finitely generated group, let $a\_n$ be the number of subgroups of index $n$ and consider $\zeta\_G... | 17 | https://mathoverflow.net/users/3272 | 131735 | 72,984 |
https://mathoverflow.net/questions/131203 | 0 | It is known that $[L^p(0,T;H)]^\* = L^q(0,T;H^\*)$.
If $p=q=2$ and $H$ is a Hilbert space, is there an easier proof to show that the spaces are **isometric**? The proof that I know for the general case uses some sort of epsilon argument, but there must be an easier way when we have access to Riesz maps?
| https://mathoverflow.net/users/34177 | Dual space of Bochner space: is there an easier proof to show they're isometric? | Uhm, I might be underestimating the problem, but I believe that if $H$ is a Hilbert space, then so is $L^2(0,T;H)$. Isometry then follows directly from Riesz Representation Theorem.
| 0 | https://mathoverflow.net/users/24482 | 131737 | 72,986 |
https://mathoverflow.net/questions/131747 | 4 | This is a somewhat simple question: consider a complex manifold $M$ and its canonical bundle $\omega\_X$. It is clear that in $H^2(X,\mathbb{R})$,
$$c\_1(\omega\_X) = - c\_1(T\_X)$$
(Obvious using Chern-Weil theory).
Does this remain true in $H^2(X,\mathbb{Z})$ ? If not, is there a way to relate the two ?
| https://mathoverflow.net/users/25566 | First Chern class of canonical bundle ? | Yes. This is true for every vector bundle. By functorialuty, it is sufficient to check on just the infinite Grassmanian. But its integral cohomology is torsion-free, so Chern-Weil works.
| 3 | https://mathoverflow.net/users/18060 | 131749 | 72,991 |
https://mathoverflow.net/questions/131746 | 9 | Let $A(x\_1,...,x\_n)$ be an $n\times n$ matrix field over $R^n$.
I am interested in the partial derivative determinant of $A$ in respect to $x\_i$. In can be shown that:
$\frac{\partial{\det(A)}}{\partial{x\_i}} = \det(A)\cdot\sum\_{a=1}^{n}{\sum\_{b=1}^{n}{ A^{-1}\_{a,b} \cdot \frac{ \partial{A\_{b,a}} }{ \parti... | https://mathoverflow.net/users/8446 | Derivative of a determinant of a matrix field | your identity follows simply by using $\log({\rm det}\; A)= {\rm tr}\; (\log A)$, so
$$\frac{\partial}{\partial x\_i}{\rm det}\;A=
\frac{\partial}{\partial x\_i} \exp({\rm tr}\;\log A)=
({\rm det}\;A) \frac{\partial}{\partial x\_i}{\rm tr}\;\log A=
({\rm det}\;A)\;{\rm tr}\;\left(A^{-1}\frac{\partial}{\partial x\_i}... | 11 | https://mathoverflow.net/users/11260 | 131755 | 72,994 |
https://mathoverflow.net/questions/131473 | 0 | Let $s \in R^{n}$ (meaning $s$ is $n \times 1$ vector), where $n$ is the dimension of the vector. The ideal sliding term, $\nu$ is taken to be:
\begin{equation}
\nu = \frac{s}{\|s\|}
\end{equation}
where $\| . \|$ is taken to be the Euclidean norm. When $n = 1$, the above equation represents a step function but wit... | https://mathoverflow.net/users/34257 | Approximating higher dimension step function | A first pass to get continuity: for $||s||\geq \epsilon$, $f(s) = s/||s||$. For $||s||<\epsilon$,
$f(s) = \frac{s}{||s||} (e^{1/\epsilon^2-1/||s||^2})$
If you want differentiability you just need to fiddle with the $< \epsilon$ function to have a derivative of 0 in the increasing $||s||$ direction at $||s||=\epsilo... | 0 | https://mathoverflow.net/users/3669 | 131756 | 72,995 |
https://mathoverflow.net/questions/131758 | 12 | Let $\mathbb{R}^\mathbb{R}$ be the set of functions $\mathbb{R}\to\mathbb{R}$ patially ordered by eventual domination. Obviously, every ordinal below $\omega\_1$ can be embedded in $\mathbb{R}^\mathbb{R}$ using only constant functions.
What is the least ordinal than cannot be embedded in $\mathbb{R}^\mathbb{R}$?
| https://mathoverflow.net/users/34258 | What is the least ordinal than cannot be embedded in $\mathbb{R}^\mathbb{R}$? | Let me get things started with some simple observations.
Note that given any countable sequence of functions $f\_n$, we can
by diagonalization construct a function eventually dominating all
of them, $f(x)=\max\_{n\leq x}f\_n(x)$. It follows that we may by
transfinite recursion construct an embedding of $\omega\_1$ in... | 14 | https://mathoverflow.net/users/1946 | 131764 | 73,000 |
https://mathoverflow.net/questions/131585 | 38 | In recent work with Michael Albert and Nik Ruškuc, a family of words has arisen which is counted by the Catalan numbers. I've looked at Richard Stanley's [Catalan exercises in EC2](http://www-math.mit.edu/~rstan/ec/catalan.pdf) and his [Catalan addendum](http://www-math.mit.edu/~rstan/ec/catadd.pdf), but I don't see an... | https://mathoverflow.net/users/2663 | A family of words counted by the Catalan numbers | Below my modified answer containing a complete bijection between the above sequences and Dyck paths:
Let $a = (a\_1,\ldots,a\_n)$ be a sequence of $n$ integers. $a$ satisfies Property $A$ if it satisfies your two conditions above. This is
1. $a\_{k+1} \geq a\_k−1$,
2. If $a$ contains an $i \geq 1$, then the first $... | 15 | https://mathoverflow.net/users/21291 | 131767 | 73,002 |
https://mathoverflow.net/questions/131761 | 2 | I've made this question on math.stackexchange.com (also offering a bounty) but I did not receive any answer:
I'm looking for a reference of the following fact:
given a (countable?) amenable group $G$ and a (skew) field $K$, the following are equivalent:
(1) the group ring $K[G]$ is a domain;
(2) $K[G]$ is a (le... | https://mathoverflow.net/users/24891 | Amenable group rings embeddable in skew fields | See here: Lück, Wolfgang, L2-invariants: theory and applications to geometry and K-theory.
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 44. Springer-Verlag, Berlin... | 2 | https://mathoverflow.net/users/nan | 131768 | 73,003 |
https://mathoverflow.net/questions/131766 | 13 | From what I had read, group characters can be "glued" together in a topological fashion and there is something to this effect in the paper by [Dijkgraaf and Witten](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-129/issue-2/Topological-gauge-theories-and-group-cohomology/cmp/1104180750... | https://mathoverflow.net/users/1358 | Dijkgraaf-Witten TQFT vs. Representation Theory? | Strictly speaking, this answer is not about the 3d TQFT which you mention in your question, but rather a 2d version of Dijkgraaf-Witten theory (described in Section 2 of Freed-Hopkins-Lurie-Teleman).
To every finite group $G$, there is a 2d TQFT $Z\_G$ which assigns to a closed orientable surface $\Sigma$ the follow... | 16 | https://mathoverflow.net/users/7762 | 131774 | 73,007 |
https://mathoverflow.net/questions/131709 | 3 | There is a homological Chern character $ch\_\ast \colon K\_\ast(X) \to H\_\ast(X)$ for $X$ a smooth, compact manifold.
I found only one definition of it (in the paper "K-Homology and Index Theory" by Baum, Douglas), where it is defined (i) by the usual cohomological Chern character map and Poincare duality between $H... | https://mathoverflow.net/users/13356 | Definition of the homological Chern character | I like this question! I think this problem (the Chern character of K-homology) has been studied and solved by Alain Connes in his paper "Noncommutative differential geometry" in 1985. He indeed used cyclic cohomology etc. Unfortunately he uses analytic description of K-homology instead of dual algebras but they are ess... | 2 | https://mathoverflow.net/users/24965 | 131784 | 73,015 |
https://mathoverflow.net/questions/131703 | 0 | Let $X$ be a smooth and projective variety over a field of characteristic zero. Let $Y$ be a normal variety, with finite quotient singularities (an orbifold!) and let $\pi: Y \to X$ be a finite morphism, ramified along a simple normal crossings divisor $D$. Assume that the singularities of $Y$ lie over the singular loc... | https://mathoverflow.net/users/34336 | blow-ups and singularities | I think this is true. Taking a point $x\in X$, and base change your setting up to its formal neighborhood, we can assume $K(X)\subset K(Y)$ is a Galois extension with finite abelian group.
Now we can put a root stack structure $\mathcal{X}\to X$ which is branched over $D$ such that the morphism $Y\setminus\pi^{-1}(V)... | 2 | https://mathoverflow.net/users/10083 | 131791 | 73,018 |
https://mathoverflow.net/questions/131785 | 3 | Let $(X,||\cdot||\_1)$ be a normed space and $Y$ a linear subspace of $X$. Let $||\cdot||\_2$ be a norm on $X$ which is equivalent to $||\cdot||\_1$ on $Y$. Does there exist a norm on $X$ that coincides with $||\cdot||\_2$ on $Y$ and is equivalent to $||\cdot||\_1$ on the entire space $X$?
This smells of some versio... | https://mathoverflow.net/users/34356 | Extension of equivalent norms | Yes. Suppose $\|y\|\_2 \le A \|y\|\_1$ for $y \in Y$. Let $U$ be the convex hull of $1/A$ times the unit ball of $X$ and the $\|\cdot \|\_2$-unit ball of $Y$. Then $U \cap Y$ is the $\|\cdot \|\_2$-unit ball of $Y$. Use the gauge functional of $U$.
| 4 | https://mathoverflow.net/users/2554 | 131815 | 73,028 |
https://mathoverflow.net/questions/131810 | 16 | Is it possible to generalize functions like $x^y, \ln x, \sin x, \arctan x$ to [surreal numbers](http://mathworld.wolfram.com/SurrealNumber.html) or [surcomplex numbers](http://en.wikipedia.org/wiki/Surcomplex_number)? Which of their properties and relations (e.g. usual trig identities) will still hold in this case? Is... | https://mathoverflow.net/users/34324 | Is it possible to generalize functions like $x^y, \ln x, \sin x, \arctan x$ to surreal numbers or surcomplex numbers? | The following is taken from a [sci.math](http://www.ics.uci.edu/~eppstein/cgt/surreal.html) post by Nicolau C Saldanha. Much more detail is available there. It is important to note that Nicolau does not define functions extensionally, i.e. two functions which agree everywhere but are defined differently are considered ... | 6 | https://mathoverflow.net/users/13832 | 131816 | 73,029 |
https://mathoverflow.net/questions/131567 | 1 | I wanna prove following equation
$ \sum\_{i=1}^n \prod\_{k=1,k\neq i}^n \prod\_{j=1,j\neq k}^{n+1}(x\_j - x\_k) = -\prod\_{i=1}^n \prod\_{j=1,j\neq i}^n (x\_j - x\_i) $
I have verified several numbers such as $n=2,3,4$, and try to prove it using mathematical induction, however I can't extend the 2 $\sum \sum$. Some g... | https://mathoverflow.net/users/34288 | Help me on proof of an equation. | $\sum\_{i=1}^n \prod\_{k=1,k\neq i}^n \prod\_{j=1,j\neq k}^{n+1}(x\_j - x\_k) = -\prod\_{i=1}^n \prod\_{j=1,j\neq i}^n (x\_j - x\_i)$
Denoting $A\_i=\prod\_{j=1,j\neq i}^n (x\_j - x\_i)$ we have
$$\sum\_{i=1}^n \prod\_{k=1,k\neq i}^n (x\_{n+1}-x\_k)A\_k = -\prod\_{i=1}^n A\_i$$
From here
$$\sum\_{i=1}^n \frac{\pr... | 1 | https://mathoverflow.net/users/18814 | 131824 | 73,031 |
https://mathoverflow.net/questions/131823 | 3 | I am looking for all finite groups $G$ such that for each subgroup $H$ of $G$ and each automorphism $\sigma$ of $H$ there exists an automorphism $\psi$ of $G$ whose restriction to $H$ is $\sigma$. Is there any reference on this problem?
| https://mathoverflow.net/users/27831 | Groups with special automorphism group | These groups are called the groups of *injective type*.
It is known that nonabelian finite groups of injective type have even order.
See the following papers:
<http://dx.doi.org/10.1017/S0017089512000031>
<http://dx.doi.org/10.1142/S0219498807002235>
| 5 | https://mathoverflow.net/users/30062 | 131829 | 73,033 |
https://mathoverflow.net/questions/131830 | 1 | Let $j:\mathbb{A}^{n}\rightarrow\mathbb{A}^{n}$ an open immersion over a field $k$. Is it an isomorphism?
| https://mathoverflow.net/users/27398 | open immersion between affine spaces | If $U \cong \mathbb A^n$ is the image of $j$, then the long exact sequence of a pair shows that $\mathbb A^n \setminus U$ has no cohomology in any degree, which is only possible if $U = \mathbb A^n$.
More generally (using heavier machinery), no variety $X$ is isomorphic to a Zariski open subset $U \subsetneq X$. The ... | 4 | https://mathoverflow.net/users/1310 | 131831 | 73,034 |
https://mathoverflow.net/questions/117664 | 2 | I need to know about this non-linear logarithmic fast diffusion equation for a function $u(x,t)$ of one space variable $x$ and time $t$:
$$ u\_t = (\ln u)\_{xx}$$
which is to run on an interval $ a \leq x \leq b $ with periodic boundary conditions
$$ u(a,t) = u(b,t) $$
$$ u\_x(a,t) = u\_x(b,t) $$
for $t \geq 0$ and an... | https://mathoverflow.net/users/30302 | The logarithmic fast diffusion equation in one space variable with periodic boundary conditions. | Maybe this book is a good choice: <http://ukcatalogue.oup.com/product/9780198569039.do#.UaC1zPER0uU>
In the web of the author (<http://www.uam.es/personal_pdi/ciencias/jvazquez/coursejlv.html>) you can download the index, preface and introductory chapter.
I hope this helps you even if it's a late response.
| 1 | https://mathoverflow.net/users/33135 | 131837 | 73,036 |
https://mathoverflow.net/questions/92726 | 8 | Villani gives the following formula to find the gradient of a function $F$ of a probability density function $\rho$ in the Wasserstein space :
$$\nabla\_W F(\rho) = -\nabla.(\rho \nabla \frac{\delta F}{\delta \rho})$$
where, if $F$ is given as $F(\rho)=\int U(\rho) dx$, then $\frac{\delta F}{\delta \rho} = U'(\rho)$.
... | https://mathoverflow.net/users/8646 | derivative in the Wasserstein space | The dirac distribution $\delta\_x$ is not smooth enough to carry out this computation, which only works for functionals $F(\rho)=\int U(\rho)$...
One thing though: a displacement interpolation is nothing but a geodesic in the metric space of probabilities (endowed with the Wasserstein distance). If you look for examp... | 5 | https://mathoverflow.net/users/33741 | 131851 | 73,042 |
https://mathoverflow.net/questions/131857 | 0 | I apologize if this question is considered too mathematically imprecise. My understanding of Special and
General Relativity comes from reading books which attempt to explain them to non-expertsin these fields.
Let C denote a clock (such as an atomic clock) fixed at the earth's surface. It seems that the rate at
which a... | https://mathoverflow.net/users/4423 | A question about time in Special and General Relativity. | It is possible, but not in special relativity (because of the answer given by Youloush). Use a 2-dimensional space-time diffeomorphic to $\mathbb{R}^2$ and define a pseudo-riemannian metric (using coordinates for the tangent space from the flat case):
$g\_{t,x}((a,b), (c,d))=(1+x^4)\cdot a\cdot c-b\cdot d$
The coor... | 3 | https://mathoverflow.net/users/33842 | 131864 | 73,050 |
https://mathoverflow.net/questions/131853 | 3 | Consider the sequence S(x) = 2^x - 1. This sequence has two interesting properties:
a) If the GCD of S(x) and S(y) is S(gcd(x,y)), and
b) For any prime p, S(p-1) is divisible by p.
Property a follows from the fact that if 2^x is 1 mod n and 2^y is 1 mod n, 2^(gcd(x,y)) is 1 mod n. Property b follows from Fermat's... | https://mathoverflow.net/users/9712 | Sequences satisfying gcd(S(x), S(y)) = S(gcd(x,y)) | A *strong divisibility sequence* is a sequence of positive integers $(a\_n)\_{n\ge1}$ with the property that $\gcd(a\_n,a\_m)=a\_{\gcd(n,m)}$. See <http://en.wikipedia.org/wiki/Divisibility_sequence>.
(Strong) divisibility sequences tend to arise from algebraic groups, including sequences such as $a^n-b^n$, Fibonacci (... | 8 | https://mathoverflow.net/users/11926 | 131865 | 73,051 |
https://mathoverflow.net/questions/131863 | 9 | Is there any elementary (i.e. without using analytical methods like the theory of Riemann surfaces or more elaborate results from differential geometry) way to show that the universal covering of the compact oriented surface of genus $g>0$ is homeomorphic to $\mathbb R^2$?
| https://mathoverflow.net/users/14233 | Universal covering of compact surfaces | You can build a certain covering space of the surface $S$ rather explicitly as a nested union of closed discs $D\_1 \subset D\_2 \subset D\_3 \subset \cdots$, each contained in the interior of the next, from which it follows that the union is $\mathbb{R}^2$ and, being simply connected, is therefore the universal coveri... | 13 | https://mathoverflow.net/users/20787 | 131866 | 73,052 |
https://mathoverflow.net/questions/129345 | 20 | For integers $n$ such that $\:3< n\:$,$\:$ what is known about the following 2-player game:
Player\_1 and Player\_2 take turn choosing points on $\mathbb{R}^2$ that were not previously chosen, with Player\_1 going first. $\:$ If there is ever a segment with $n$ points chosen by Player\_1 and zero points chosen by Pla... | https://mathoverflow.net/users/nan | $n$-in-a-row game on $\mathbb{R}^2$ |
>
>
> ```
>
> The points on the lines and the lines through the points...
> The lines should be many with numerous joints,
> the points should be few and should sit at right places.
> What else? We can use high-dimensional spaces,
> if we do not mind doubling enemy's strength.
> It would not be bad if we bound the ... | 12 | https://mathoverflow.net/users/1131 | 131870 | 73,053 |
https://mathoverflow.net/questions/129053 | 8 | Suppose that $\lambda\_1,\lambda\_2,\lambda\_3$ are partititions of $n$. When do there exist permutations $\sigma\_1,\sigma\_2,\sigma\_3 \in S\_n$ such that
(1) $\sigma\_1\sigma\_2\sigma\_3$ is the identity;
(2) the $\sigma\_i$ generate a transitive subgroup of $S\_n$; and
(3) the cycle type of $\sigma\_i$ is $\l... | https://mathoverflow.net/users/33520 | Permutations of prescribed cycle types that multiply to the identity | The answer is no. No such triples exist for $n = 4$, $\lambda\_1 = (3, 1)$, $\lambda\_2 = \lambda\_3 = (2, 2)$. (Indeed, in $S\_4$, the double transpositions and the identity form the Klein four subgroup.)
For other results on this problem, see for example <http://arxiv.org/abs/math/0508434>.
| 2 | https://mathoverflow.net/users/33520 | 131879 | 73,057 |
https://mathoverflow.net/questions/130158 | 10 | The Geometric Langlands correspondence introduced by Drinfeld and Laumon conjectures a 1 to 1 correspondence between
(A) local systems on a projective smooth curve over a field
and
(B) (Hecke eigen-)perverse sheaves on the algebraic stack of vector bundles of rank n on the curve (or more generally G-bundles).
The ... | https://mathoverflow.net/users/12770 | On Geometric Langlands Correspondence | **EDIT** Few days ago a [survey by A. Parshin](http://arxiv.org/abs/1307.1878) appeared in arxiv.
I think it is the best place to look on the higher-dimensional Langlands.
From the abstract:
>
> A brief survey is given of the classical Langlands correspondence
> between n-dimensional representations of Galois grou... | 4 | https://mathoverflow.net/users/10446 | 131884 | 73,060 |
https://mathoverflow.net/questions/130363 | 2 | I have a simple question. I read that given a vector space $N\_{\mathbb{R}}$ over $\mathbb{R}$,
we can define a convex polytope in the following way:
$$P:= \Big\{ \sum\_{u\in S} \mu\_u u \,\Big| \, \mu\_u \geq 0 , \sum\_{u\in S} \mu\_u =1 \Big\} \subset N\_{\mathbb{R}}$$
with $S$ finite.
What is the definition of... | https://mathoverflow.net/users/31524 | non-convex Polytope definition | (This should be a comment, but don´t know how to post it as such)
The words "polytope" and "polyhedron" can mean different things... and different people mean different things when they say "regular polytope".
Take a look at Grünbaum´s paper:
Are your polyhedra the same as my polyhedra?, Discrete and Computationa... | 5 | https://mathoverflow.net/users/13095 | 131885 | 73,061 |
https://mathoverflow.net/questions/131881 | 5 | Let $p$ be a prime other than 5 or 7. Are $A\_p$ and $S\_p$ the only subgroups of $S\_p$ that contains a $p$-cycle and a double transposition?
As for $p = 5$, the dihedral group $D\_{10}$ contains a 5-cycle and a double transposition. For $p = 7$, the group $PSL\_3(\mathbb{F}\_2)$ (acting on the projective plane of o... | https://mathoverflow.net/users/33520 | Transitive subgroup of $S_p$ containing a $p$-cycle and a double transposition | I believe that it was already known to Jordan that if $n \geq 8$ and $G$ is a primitive subgroup of $S\_n$ containing a double transposition, then $G$ contains $A\_n$. As a subgroup of $S\_p$ containing a $p$-cycle is primitive, the answer to your question is ``yes".
| 12 | https://mathoverflow.net/users/36466 | 131886 | 73,062 |
https://mathoverflow.net/questions/131889 | 2 | Is this true?
Let $G\neq A\_5$ be a finite simple non-abelian group. Then $G$ has a cyclic subgroup
of order $2p$ and a subgroup isomorphic to the dihedral group of order $2p$, for some prime $p$.
| https://mathoverflow.net/users/27831 | a question about finite simple non-abelian groups | For $f>1$ let $G = {\rm SL}\_2({\bf F}\_{2^f})$
(a.k.a. $L\_{\phantom.2}(2^f)$ in ATLAS notation). Then $G$ is simple
and each element has exponent either $2$ or a factor of $2^f \pm 1$.
Hence $G$ has no cyclic subgroup of order $2p$ for any prime $p$
(not even $2$). For $f=2$ we recover the example of $A\_5$.
| 8 | https://mathoverflow.net/users/14830 | 131893 | 73,066 |
https://mathoverflow.net/questions/131888 | 10 | There are many examples throughout mathematics of abstracting the formal properties of a "familiar" structure, but then having a theorem stating that all models of the abstract axioms embed into one of the original "familiar" structures. Examples include
* every group is a subgroup of a symmetric group;
* every small... | https://mathoverflow.net/users/18702 | Embedding Theorem for topological spaces, and in general | I think this question is closely related to reflective subcategories and the adjoint functor theorems. Adjoint functors do not guarantee you to have real “embeddings”, however, usually this part is easy to prove.
Let me restate some results given by Joseph Van Name:
The category of compact Hausdorff spaces is a ref... | 2 | https://mathoverflow.net/users/33842 | 131902 | 73,071 |
https://mathoverflow.net/questions/131895 | 1 | hi everyone. I want to know some example of frobenius groups. I know S\_3 is a frobenius group, and if we have a semidirect produt of 2 groups then we have frobenius groups. but I want more examples of frobenius groups. Are dihedral groups , frobenius?
thanks
| https://mathoverflow.net/users/34394 | frobenius group | Just to reiterate standard facts about Frobenius groups: a finite group $G$ is said to be a Frobenius group if $G$ contains a proper non-identity subgroup $H$ such that $H \cap gHg^{-1} = 1$ for all $g \in G \backslash H.$ This hypothesis is exactly equivalent to the permutation-group-theoretic definition of Frobenius ... | 4 | https://mathoverflow.net/users/14450 | 131905 | 73,072 |
https://mathoverflow.net/questions/131922 | 13 | * How many subsets of $\mathbb{R}$ are order isomorphic to $\mathbb{Q}$?
* How many subsets of the [long line](http://en.wikipedia.org/wiki/Long_line_(topology)) $\omega\_1\times[0,1)$ are order isomorphic to $\mathbb{Q}$?
I can see that results in both cases are between $\mathfrak{c}$ and $\mathfrak{2^c}$.
| https://mathoverflow.net/users/34387 | How many subsets of $\mathbb{R}$ are order isomorphic to $\mathbb{Q}$? | There are continuum many countable subsets of the continuum (because $\mathfrak{c}^{\aleph\_0}=2^{\aleph\_0}$). Thus the answer is $\mathfrak{c}$. See [this question](https://math.stackexchange.com/questions/191006/number-of-countable-subsets-of-mathbbr).
| 2 | https://mathoverflow.net/users/33842 | 131924 | 73,081 |
https://mathoverflow.net/questions/131878 | 3 | Suppose that $X$ and $Y$ are smooth complex algebraic varieties, and that $f:X\rightarrow Y$ is an etale morphism in the sense that $d\_xf:T\_xX\rightarrow T\_{f(x)}Y$ is an isomorphism for all $x\in X$. Must $f$ be (at least locally) an open immersion? Certainly, difficulties can arise if the differential fails to be ... | https://mathoverflow.net/users/25358 | Inverse Function Theorem in Algebraic Geometry | Maybe this "answer" is too far away from schemes to be what is desired. But here it goes anyway.
I believe in general that for complex affine varieties $X,Y$, that a morphism $f:X\to Y$ is étale iff it is a local analytic isomorphism in the analytic topology. When $X,Y$ are smooth, it is enough to just check that the... | 2 | https://mathoverflow.net/users/12218 | 131934 | 73,084 |
https://mathoverflow.net/questions/131906 | 10 | I am looking for a counter example which shows, that a full rational 2D CFT (with respect to a given chiral subtheory) is not characterized by its modular invariant partition function. People tell me such an example exists, but noone could point out a concrete example or reference to me.
---
In the framework of F... | https://mathoverflow.net/users/10718 | Example for non equivalent rational full CFTs with same modular invariant (partition function) | I apologize since my answer will involve shameless self-promotion. You can find one example
of this kind on page 36 of [my slides](http://pages.uoregon.edu/vostrik/talks/beijing.pdf). In this example one Frobenius algebra is commutative
and another is not Morita equivalent to a commutative algebra.
| 10 | https://mathoverflow.net/users/4158 | 131935 | 73,085 |
https://mathoverflow.net/questions/131933 | 23 | Is it possible to construct an infinite subset of $\Bbb R$ that is not order isomorphic to any proper subset of itself?
| https://mathoverflow.net/users/34033 | Is it possible to construct an infinite subset of $\Bbb R$ that is not order isomorphic to any proper subset of itself? | There is no such countably infinite set, but there is such a set with size $2^{\aleph\_0}$. These are classic results of Dushnik & Miller [[*Concerning similarity transformations of linearly ordered sets*](http://www.ams.org/journals/bull/1940-46-04/S0002-9904-1940-07213-1/home.html), Bull. Amer. Math. Soc. 46 (1940), ... | 15 | https://mathoverflow.net/users/2000 | 131939 | 73,088 |
https://mathoverflow.net/questions/131928 | 4 | Hi everyone.
I'm pondering the following question: I have a Coxeter group $(W,S)$ of type $A\_{n-1}$, i.e. the symmetric group $W=Sym(n)$ with the neighbour transpositions as generating set $S=\lbrace (1,2),(2,3),\ldots,(n-1,n)\rbrace$.
There is a natural bijection between subsets $I\subseteq S$ and decompositions ... | https://mathoverflow.net/users/3041 | Combinatorics of index sets multiplicities in characters of symmetric groups | It can't be true in general that $m\_{\mu,I}$ is always 0 or 1. That is because $K\_{\mu,1^n}=f^\mu$, the number of standard Young tableaux of shape $\mu$. The maximum value of $f^\mu$ for $\mu\vdash n$ is approximately $\sqrt{n!}$, which is much bigger than $2^n$, the number of sets $I\subseteq\lbrace 1,\dots,n\rbrace... | 10 | https://mathoverflow.net/users/2807 | 131959 | 73,098 |
https://mathoverflow.net/questions/131958 | 0 | I have a unitary element $u\in C(\mathbb{T},M\_{n}(\mathbb{C}))$ such that $Spec(u)=\mathbb{T}$. Does there exist a unitary $v\in C(\mathbb{T},\mathbb{C})$ such that $Spec(uv)\subsetneqq\mathbb{T}$?
| https://mathoverflow.net/users/33724 | Unitary with full spectrum | No. For example, let
$$ u(z) = \left(\begin{array}{cc} 1 & 0 \newline 0 & z \end{array}\right). $$
To see that for any $v \in C(\mathbb T, \mathbb C)$, the spectrum of $uv$ is $\mathbb T$, observe that
$$ uv(z) = \left(\begin{array}{cc} g(z) & 0 \newline 0 & h(z) \end{array}\right), $$
where the number of times that $g... | 2 | https://mathoverflow.net/users/22052 | 131965 | 73,102 |
https://mathoverflow.net/questions/61007 | 10 | Recall that for a DVR A with fraction field F and residue field k, there is a "localization" fiber sequence in algebraic K-theory,
$$K(k) \rightarrow K(A) \rightarrow K(F).$$
In Remark 5.17 of his "Higher Algebraic K-theory: I" paper, Quillen gives an explicit description of the corresponding boundary map $\partial... | https://mathoverflow.net/users/3931 | Explicit description of boundary map in algebraic K-theory | Apparently there is no proof in the literature. Let me provide a proof here, in case it's helpful. Actually, first let's give the set-up, then a precise statement, then the proof. To tell the truth, the set-up takes the most time. So this fact is harder to state than to prove, which may be why it's not so useful in pra... | 21 | https://mathoverflow.net/users/3931 | 131967 | 73,104 |
https://mathoverflow.net/questions/131969 | 5 | I was wondering what people would normally mean by a simple abelian variety $A$ where $A$ is defined over a field $k$ that is not algebraically closed.
The definition I found in for example on the bottom of page 29 in Langs book Abelian Varieties states:
>
> An abelian variety $A$ is simple if $0$ and $A$ are the... | https://mathoverflow.net/users/23501 | Simple abelian varieties over non algebraically closed fields. | You're right: "simple abelian variety $A/K$" is ambiguous when $K$ is not algebraically closed. One should say **geometrically (or absolutely) simple** or **K-simple** to emphasize which is meant.
If you do encounter "simple abelian variety $A/K$" in the literature, then in my experience it usually means "geometrical... | 9 | https://mathoverflow.net/users/1149 | 131971 | 73,106 |
https://mathoverflow.net/questions/131978 | 9 | Is there a way to classify incompressible surfaces in $\Sigma \times [0,1]$ where $\Sigma$ is any closed surface? I know of the [Hatcher-Thurston](http://www.math.cornell.edu/~hatcher/Papers/2-bridgeKnots.pdf) classification of incompressible surfaces in 2-bridge knot exteriors,
but I wonder if a classification of inc... | https://mathoverflow.net/users/34415 | Is there a way to classify incompressible surfaces in $\Sigma \times [0,1]$ ? | An incompressible and boundary incompressible connected surface is isotopic to either (a) a vertical annulus or (b) a horizontal surface. A vertical annulus is of the form $\alpha \times I$ where $\alpha$ is an essential simple closed curve. A horizontal surface is of the form $\Sigma \times \{t\}$.
Here is a sketch... | 10 | https://mathoverflow.net/users/1650 | 131979 | 73,107 |
https://mathoverflow.net/questions/131976 | 0 | Hi everyone,
I am looking for an example of finite group $G$ such that (a) the number of elements of order $2$ group $G$ is $p(p+1)/2$ or $p(p-1)/2$ ($p$ is a prime divisor of order $G$) (b) the number of Sylow $p-$subgroups of $G$ is $1$ and order of a Sylow $p-$subgroup $G$ is $p^2$ (c) for every prime divisor $r$ ... | https://mathoverflow.net/users/34416 | Example of a finite group | The group $G = C\_2 \times C\_2 \times C\_3 \times C\_3$ has this property (for $p=3$).
| 3 | https://mathoverflow.net/users/12858 | 131980 | 73,108 |
https://mathoverflow.net/questions/131985 | 9 | Is there a way to axiomatize [non-abelian] free groups in first-order logic using the language of groups (which contains the binary operation symbol $\cdot$, and the constant symbol $e$)?
Is there one particular axiom, or even a schema, from which we can prove that $G$ is a free group? (Regardless to the cardinality ... | https://mathoverflow.net/users/7206 | First-order axiomatization of free groups | The free groups cannot be axiomized by first order axioms. If the free groups were axiomatizable by first order axioms, then the ultraproduct of free groups would be a free group. However, the group $\mathbb{Z}$ is free, but for every non-principal ultrafilter $\mathcal{U}$ on $\mathbb{N}$, the ultrapower $\mathbb{Z}^{... | 28 | https://mathoverflow.net/users/22277 | 131986 | 73,110 |
https://mathoverflow.net/questions/131993 | 5 | First of all, I am not an expert in neither classical, nor $p$-adic functional analysis, but anyway, I stumbled over the following lately:
Let $\varphi:\mathbb{Q}\_p\rightarrow\mathbb{C}$.
Canonically, I would define a derivative of such a function in the following way:
$\varphi'(x):=\lim\limits\_{h\rightarrow 0} ... | https://mathoverflow.net/users/33434 | Bruhat-Schwartz functions and derivatives in p-adic numbers | There is no good (nontrivial) notion for a derivative on functions on a totally disconnected group $G$ with values in $\mathbb{R}$ or $\mathbb{C}$. Assume (wlog) that $G$ is compact, then there exists a countable, ordered set of open, normal subgroups $(N\_i)$ with the projective limit of $G/N\_i$ being canonically iso... | 10 | https://mathoverflow.net/users/10400 | 131999 | 73,115 |
https://mathoverflow.net/questions/132000 | 3 | All varieties will be projective and over $\mathbb{C}$.
If $S$ is any surface in $\mathbb{P}^3$ of degree 2 that posseses an ordinary double point, it follows easily that $S$ is projectively isomorphic to the cone:
$$
x^2 + y^2 + z^2 = 0.
$$
Is there a similar standard form for quartics that possess an ordinary quadr... | https://mathoverflow.net/users/29657 | What can a quartic surface in $\mathbb{P}^3$ with an ordinary quadruple point look like? | Any irreducible hypersursurface in $X \subset \mathbb{P}^n$ of degree $d$ with an isolated singular point $p \in X$ of multiplicity $d$ is a cone over a hypersurface $Y$ of degree $d$ in $\mathbb{P}^{n-1}$.
In fact, take any point $q \in X$ different from $p$. Then the line $\overline{pq}$ has at least $d+1$ intersec... | 3 | https://mathoverflow.net/users/7460 | 132002 | 73,116 |
https://mathoverflow.net/questions/131963 | 2 | I apologize for the clumsy wording of the title-- what I'm looking for is a notion of an integer-valued dimension $d\_{\epsilon}$, which we parametrize by a real positive number $\epsilon$, of, say, a connected subset $A$ of your favorite normed linear space.
The number $\epsilon$ is our 'coarseness' parameter, for w... | https://mathoverflow.net/users/34409 | A notion of a 'coarse', parametrized dimension of an object, where the parameter determines how finely we can distinguish (say) a very thin rod from a line | A few observations first:
1. I assume in your motivating example that the coarseness parameter $\epsilon$ is smaller than $1$, otherwise you can't even tell that cylinder apart from a point.
2. Anton's comment is exactly along the right lines. For those who have not seen it before:
**Definition**: a metric space $(... | 0 | https://mathoverflow.net/users/18263 | 132006 | 73,117 |
https://mathoverflow.net/questions/131809 | 32 | This is something I found in trying to work on [Vince Vatter's excellent question](https://mathoverflow.net/questions/131585/a-family-of-words-counted-by-the-catalan-numbers). I have no solution, but a much more precise conjecture.
Recall that a rooted planar tree is a rooted tree where, for every vertex, the childre... | https://mathoverflow.net/users/297 | A double grading of catalan numbers | It appears that the zeta map which I used to answer [Vince Vatter's initial question](https://mathoverflow.net/questions/131585) and which I describe in [my answer there](https://mathoverflow.net/questions/131585#131767), see also page 50 of [Jim Haglund's book](http://math.upenn.edu/~jhaglund/books/qtcat.pdf), indeed ... | 16 | https://mathoverflow.net/users/21291 | 132013 | 73,119 |
https://mathoverflow.net/questions/132008 | 8 | Let $(X,d)$ be a metric space equipped with a probability measure $\mu$ (defined on the Borel $\sigma$-algebra on the topology induced by the metric $d$). I am interested in the different values that the following can take
$\sup\lbrace\mu(A):A\subset X\text{ is measurable and }\mu(A)\leq1/2\rbrace~~~~~~~~~~(\ast)$.
... | https://mathoverflow.net/users/nan | Supremum of measure of sets of measure less or equal to 1/2. | **Short version**: the set $\{\mu(B):B\in\mathcal{B}\}$ is a closed set for any probability space $(X,\mathcal{B},\mu)$. For atomic spaces this follows from an elementary topological argument, and for non-atomic spaces it is a closed interval by a classical (and easy) result of Sierpinski.
**Longer version with detai... | 6 | https://mathoverflow.net/users/11009 | 132023 | 73,123 |
https://mathoverflow.net/questions/132016 | 0 | Let $n$ - dimension $\geq 3$.
Consider a compact manifold (M,g). Let $\epsilon\_0$ denote the injectivity radius of $(M,g)$. Let $B\_\epsilon(0)$ denote a geodesic ball of radius $\epsilon < \epsilon\_0$.
Consider the Green's function on $B\_\epsilon(0)$ ( i.g. verifies that $\Delta G = \delta\_y$ and $G=0$ on the ... | https://mathoverflow.net/users/34430 | Estimates for Green's function | I suggest you look at:
<http://www.jstor.org/stable/2374588>
| 0 | https://mathoverflow.net/users/34154 | 132032 | 73,126 |
https://mathoverflow.net/questions/132039 | 1 | With $N$ integers, how many different GCDs can you make by taking subsets of them? More formally:
Let $S$ be a set of non-negative integers.
Define $G(S)$ as $\{gcd(T) : T \subseteq S \}$
Define $f(N)$ as $\max\_{|S| = N} \{|G(S)|\}$
For example:
* $f(0) = 1$
* $f(1) = 2$
* $f(2) = 4$ (e.g. $S = \{4, 6\}$, $... | https://mathoverflow.net/users/9714 | What is the maximum number of distinct GCDs you can make out of a set of integers? | Yes, it is possible for each subset to have a different gcd, giving $2^{N}$ distinct gcds.
Let $\mathcal{P}$ be the set consisting of the first $N$ odd primes and let $P$ be the product of the first $N$ odd primes. Then if $S = \lbrace \frac{2P}{q} : q\in \mathcal{P}\rbrace$, each subset $T\subset S$ has a unique gcd... | 12 | https://mathoverflow.net/users/9455 | 132042 | 73,129 |
https://mathoverflow.net/questions/81308 | 10 | It is well known that the Riemann Hypothesis is true iff $$\lim\_{N \rightarrow \infty}d\_N=0$$,where
$$d\_{N}^{2}=\inf\_{A\_N}\frac{1}{2 \pi}\int\_{-\infty}^{\infty}\vert 1-\zeta A\_N(1/2+it)\vert^2\frac{dt}{1/4+t^2}$$
and the inf is over all the Dirichlet Polynomials of length N.I'm very interested in finding a m... | https://mathoverflow.net/users/18286 | Question about Nyman-Beurling-Baez-Duarte Equivalent for Riemann Hypothesis | In this very nice recent paper by Bettin, Conrey and Farmer
<http://arxiv.org/abs/1211.5191>
it is proven that assuming some very reasonable hypothesis concerning $\zeta(s)$
that the optimal minimizing polynomial is in fact
$$
\sum\_{n \leq x} \frac{\mu(n)}{n^s} \cdot \bigg ( 1 - \frac{\log n }{\log x} \bigg )
$$
... | 13 | https://mathoverflow.net/users/34443 | 132061 | 73,137 |
https://mathoverflow.net/questions/132063 | 5 | Suppose I have a real unitary matrix $U$ and a unit vector $\mathbf{x}, \|\mathbf{x}\|\_2 = 1$. What is the solution to the following problem?
$$
\widehat{\mathbf{x}} = \arg\max\_{\mathbf{x}, ~\|\mathbf{x}\|\_2 = 1}\|U\mathbf{x}\|\_1.
$$
| https://mathoverflow.net/users/34445 | Maximal norm-1 projection | One solution is to take $x$ to be proportional to the column sums of $U$. Indeed, we have that $$\widehat{x} = \arg \max\_{||y||\_{\infty} = 1, ||x||\_2 = 1} y^T U x $$ which suggests we should pick $x$ proportional to $U^T \widehat{y}$ where $$\widehat{y} = \arg \max ||U^T y||\_2 $$ where the maximum is taken over $y$... | 3 | https://mathoverflow.net/users/9316 | 132071 | 73,139 |
https://mathoverflow.net/questions/132067 | 2 | Ampleness and acyclicity are related; for example, large tensor powers of an ample sheaf are acyclic and the converse holds for line bundles. The tensor product of two ample sheaves is ample; this provokes the question of whether the tensor product of two acyclic sheaves is acyclic. If this is false, are there modest a... | https://mathoverflow.net/users/7935 | Is the tensor product of two acyclic sheaves on a scheme acyclic? | No. For example, on projective space $\mathbb{P}^k$ the line bundles $\mathcal{O}(-1),\ldots, \mathcal{O}(-k)$ are acyclic, but the line bundles $\mathcal{O}(-n)$ with $n>k$ are not. It is still possible for "negative" (or anti-ample) bundles to be acyclic, and their high tensor powers are not acyclic (as they have top... | 12 | https://mathoverflow.net/users/7399 | 132074 | 73,140 |
https://mathoverflow.net/questions/131953 | 5 | Given three real, symmetric matrices $A\succ0$ and $B$, $C⪰ 0$.
How can it be shown that:
$$\det(A^2+AB+AC) \leq \det(A^2 +BA +AC+BC) ? \qquad (\star)$$
Where $A^2$ is symmetric and positive definite. Eigenvalues of $BA$, $AC$, and $BC$ are all $> 0$, but symmetry is lost.
Thank you!
| https://mathoverflow.net/users/34406 | Determinant of non-symmetric sum of matrices | Because the original question has changed so much, I am writing a new answer.
The key point to recognize is that you are trying to prove a submodularity property. Indeed, we see that we may equivalently prove
\begin{equation\*}
\log\det(A) + \log\det(A+B+C) \le \log\det(A+B) +\log\det(A+C).
\end{equation\*}
One comm... | 6 | https://mathoverflow.net/users/8430 | 132078 | 73,143 |
https://mathoverflow.net/questions/132073 | 11 | I believe it is known that if I is a set of non-measurable cardinality, then any homomorphism $Z^I\to Z$ factors through a finite power. Here $Z$ is the group of integers. Can anyone give a reference for this?
| https://mathoverflow.net/users/24338 | Homomorphisms from powers of Z to Z | This runs under the name **Łoś-Eda Theorem**. A reference is the book Paul C. Eklof, Alan H. Mekler, *Almost Free Modules* (2002):
Call a set $I$ *$\omega$-measurable* if its cardinality is greater or equal to the first measurable cardinal. This is equivalent to $I$ being uncountable and supporting a non-principal c... | 12 | https://mathoverflow.net/users/10194 | 132083 | 73,146 |
https://mathoverflow.net/questions/132082 | 8 | Hallo, I have the following question:
Let $(X,d)$ be a complete metric space. Is then $(X,\operatorname{dist})$ also complete? Here by $\operatorname{dist}$ I mean the metric induced by $d$ by: $\operatorname{dist}(x,y)=\inf(L(\gamma))$, where the infimum is taken over all paths joining $x,y$ and $L(\gamma)=\sup(\sum\_... | https://mathoverflow.net/users/34453 | complete metric space | The intrinsic metric induced by a complete metric space is always a complete extended metric. To prove this, we shall write $d^{\sharp}$ for the extended metric $dist(x,y)=\inf(L(\gamma))$.
Also, I want to remark that $d^{\sharp}$ is in general not a metric since we may have $d^{\sharp}(x,y)=\infty$ even when $X$ is... | 8 | https://mathoverflow.net/users/22277 | 132085 | 73,147 |
https://mathoverflow.net/questions/132070 | 2 | Consider $B=(B\_t)\_{t\geq 0}$ real $\mathcal F\_t$ - brownian motion starting at zero, in a probability space $(\Omega, \mathcal F, (\mathcal F\_t)\_{t\geq 0}, \mathbb P)$. Then, consider a new real $\mathcal F\_t$ - brownian motion $\tilde{B}=(\tilde{B\_0})\_{t\geq t}$ independent of $B$ as weel as a process $H=(H\_t... | https://mathoverflow.net/users/24538 | Conditional law of an Ito's stochastic integral | Hi,
If you know $B$ on the entire trajectory then $H\_t$ is no more than a "scaled"-Wiener integral (as the integrand become deterministic).
As $\int \_0^t f(B\_s) ~d \tilde B\_s \mathbf 1\_{{ \int \_0^t f^2(B\_s) ~ds>0}}$ is then a normal random variable with null expectation and variance equal to $\int \_0^t f(B\... | 4 | https://mathoverflow.net/users/2642 | 132088 | 73,149 |
https://mathoverflow.net/questions/132101 | 12 | I have a homogeneous ideal $I=\langle f\_1,\ldots,f\_r\rangle$ of the polynomial ring $\mathbb C[X\_1,\ldots,X\_n]=:R$ where each of the $f\_i$ is actually over $\mathbb Z$. My computations are usually performed over $\mathbb Q$. As an additional note, the $f\_i$ are all homogeneous of the same degree. I also have a mo... | https://mathoverflow.net/users/9947 | Ideal Membership without Certificate? | This should really be a comment. But since I dont't have enough reputation I have to write it here.
If you only want to check ideal membership you "just" have to compute a Gröbner basis and check if all S-polynomials reduce to $0$. (the above mentioned computer algebra systems check ideal member ship in this way)
I... | 7 | https://mathoverflow.net/users/22448 | 132106 | 73,156 |
https://mathoverflow.net/questions/132112 | 3 | Hello!
I have just finished my master's course on which I focused on category theory, because I found it a very interesting subject, but my background, as an undergraduate student, was on functional analysis and analytic topology. So I did not arrive to category theory from a "normal" path, i.e logic, set theory, th... | https://mathoverflow.net/users/34468 | Is a background in Category Theory enough for starting a PhD in Category Theory? | If you're like Bill Lawvere you will have no trouble doing everything with categories. Also, it may seem like you've gone some way in learning math, but trust me, there's much more to come. So don't worry about your background, what course you might not have taken, or whether one view of a subject is better than anothe... | 16 | https://mathoverflow.net/users/1176 | 132114 | 73,160 |
https://mathoverflow.net/questions/132107 | 3 | Let $X$ be a topological space and $A$ a sheaf of noncommutative associative algebras over a fixed field $k$. My questions are:
1) Does the category of modules over A have enough injective?
2) If we have the affirmative answer of question (1), then suppose $M$ is a quasi-coherent sheaf over $A$ then the first cohom... | https://mathoverflow.net/users/34466 | Sheaf cohomology in non-commutative setup | For the first question, the answer is yes, since the category of sheaves of $A$-modules is a Grothendieck Abelian Category. The least obvious condition one has to check is that this category has a generator - to construct it, just use
$\bigoplus\_{U\subseteq X} (j\_U)\_!(A|\_U)$,
where $U$ ranges over all open sets... | 8 | https://mathoverflow.net/users/2363 | 132117 | 73,161 |
https://mathoverflow.net/questions/132119 | 2 | Fix $n>0$ and $X\subseteq\mathbb{R}^n$. A function $f:X\longrightarrow\mathbb{R}$ is **linear** if it is of the form
$$
f(\bar{x})=a\_1x\_1+\ldots+a\_nx\_n+b
$$
for some $a\_i,b\in\mathbb{R}$.
Suppose we have linear functions $f\_1,\ldots,f\_t$ and $g\_1,\ldots,g\_t$ with the following property:
For any $\bar{x}\i... | https://mathoverflow.net/users/38253 | Arrangements of hyperplanes | For a counterexample, take $n=4$, take $t=2$, and take $X$ to be the standard set of four unit basis vectors in $\mathbb R^4$. At each of the four points $x\in X$, assign values to the two $f$'s and two $g$'s so as to defeat, at that point, one of the four possible inequalities $f\_i(x)\leq g\_j(x)$, while still satisf... | 2 | https://mathoverflow.net/users/6794 | 132122 | 73,162 |
https://mathoverflow.net/questions/132123 | 2 | The question is in the title: if I'm not mistaken, what misses to prove that all automorphic L-functions belong to the Selberg class is a proof of the Ramanujan Conjecture. But the Selberg class is expected to be the natural class of L-functions that satisfy the analogue of the Riemann Hypothesis.
Moreover, Ramanuja... | https://mathoverflow.net/users/13625 | Would a proof of Ramanujan Conjecture together with other known results about automorphic L-functions imply the Grand Riemann Hypothesis? | No. GRH lies much deeper than the Ramanujan's conjecture for automorphic forms for $Gl\_n$ (which of course is already pretty deep). For example, Langlands functoriality would imply Ramanujan's conjecture (by considering symmetric power of an automorphic form),
but it is not expected to imply GRH.
The analogy you o... | 12 | https://mathoverflow.net/users/9317 | 132125 | 73,164 |
https://mathoverflow.net/questions/132104 | 0 | Suppose both $A$ and $B$ are matrices with dimensions $n \times m$ and $n \times p$ respectively. Assume $m + p < n $ and $A \neq 0$ and $B \neq 0$. Further, assume Im($B$) $\cap$ Im($A$) = {0}. Then, it means that columns of $B$ are independent of columns of $A$. Hence, I can claim the following:
\begin{equation}
rank... | https://mathoverflow.net/users/34257 | Reference (or proof) for the following identity in Linear Algebra | I conjecture that you didn't ask what you meant to ask. The reason for my conjecture is that you wrote "assume Im$(B)\not\subseteq$ Im$(A)$ ... it means that columns of $B$ are independent of columns of $A$." In fact, it doesn't mean that at all. The assumption Im$(B)\not\subseteq$ Im$(A)$ allows some (though not all) ... | 5 | https://mathoverflow.net/users/6794 | 132126 | 73,165 |
https://mathoverflow.net/questions/132116 | 5 | Dear all,
Consider the finite simple Ree groups $G=^2\hspace{-1mm}G\_2(3^{2n+1})$ where $n$ is a positive integer. I would like to know the orders of conjugacy class representatives of $G$ and from there to know the number of conjugacy classes of $3$-singular elements of $G$ (an element is 3-singular if its order is ... | https://mathoverflow.net/users/32259 | Conjugacy classes of the Ree groups | To amplify Geoff's brief comment, the most standard source for details about (most of) the ordinary characters of a Ree group of type $G\_2$ (specified by an odd power of $3$ at least $27$ which we call $q$ here but is sometimes written $q^2$) is the old paper by H.N. Ward based on his thesis. This is now freely availa... | 7 | https://mathoverflow.net/users/4231 | 132138 | 73,169 |
https://mathoverflow.net/questions/132134 | 11 | The Temperley-Lieb algebra has the same generators as the $S\_n$ group algebra, and the same commuting relations, but its other relations are different. A nice diagrammatic interpretation can be seen in [the Wikipedia article](http://en.wikipedia.org/wiki/Temperley-Lieb_algebra).
>
> Is there a standard analogue fo... | https://mathoverflow.net/users/391 | Temperley-Lieb algebras for other Weyl groups? | First I don't think you have stated the defining relations correctly. [AK: you were quite right, and I've edited the question.]
It sounds as though you should start with the work of Richard Green.
The following papers seem relevant.
<http://arxiv.org/abs/q-alg/9712018>
<http://arxiv.org/abs/math/0102003>
<http:... | 8 | https://mathoverflow.net/users/3992 | 132140 | 73,170 |
https://mathoverflow.net/questions/132141 | 2 | Let $S$ be the spheres spectrum and $X$ a finite CW-complex. Then $X\times S \to X$ is the trivial fibered spectrum over $X$.
The spaces of fiber preserving equivalences $X\times S \to X\times S$ is (up to homotopy) given by the mapping space
maps$(X,G)$,
where $G=$hocolim$\strut\_{n\to\infty} (\Omega^nS^n)\_{\pm... | https://mathoverflow.net/users/4500 | Auto equivalences of the spheres spectrum as a ring spectrum | Contractible. The sphere spectrum is initial as an $E\_{\infty}$-ring.
| 15 | https://mathoverflow.net/users/7721 | 132144 | 73,172 |
https://mathoverflow.net/questions/131598 | 7 | Let $(X,\mu)$ be a $G$-space, i.e. a measure space with a Borel quasi-invariant $G$-action. Say that $X$ is amenable (equivalently, that the action is amenable) if there is a $G$-fixed point in every affine space over $X$ with an $\alpha$-twisted action, where $\alpha$ is a corresponding cocycle.
If $X$ is an amenabl... | https://mathoverflow.net/users/33576 | Characterization of amenable actions | Here is a sketch of an argument to show these properties are equivalent which is based on the proof of the same property for amenable groups. To see this I will use the fact that an action $G \curvearrowright (X, \mu)$ is amenable if and only if there is a $G$-equivariant mean from $L^\infty(X, \mu) \overline \otimes L... | 7 | https://mathoverflow.net/users/6460 | 132147 | 73,174 |
https://mathoverflow.net/questions/12586 | 7 |
>
> Let $I$ be an infinite set. There is a homomorphism of abelian groups $\mathbb{Z}^{(I)} \to \hom(\mathbb{Z}^I,\mathbb{Z})$ which sends the basis element $e\_i$ to the projection $p\_i$. If $I$ is countable, it's a famous result of Specker1 that this is actually an isomorphism. But what happens when $I$ is uncount... | https://mathoverflow.net/users/2841 | Dual of $\mathbb Z^I$ for uncountable $I$ | With regard to Mariano's answer, I believe some clarification is in order. A closely related question was asked by Michael Barr and answered by user Ralph [here](https://mathoverflow.net/questions/132073/homomorphisms-from-powers-of-z-to-z/132083#132083). In brief, the homomorphism named in Martin's question is in fact... | 9 | https://mathoverflow.net/users/2926 | 132161 | 73,181 |
https://mathoverflow.net/questions/131825 | 13 | Here is a link on the internet: <https://www.dropbox.com/s/su3uak2a057yrqv/YitangZhang.pdf>
Can someone teach me how to use trivial estimation to reach (6.1) on page 24? Namely, how to impose $(d,P\_0)<D\_1$ and replace $\theta$ by $\Lambda$ with acceptable O-term?
Thanks a lot!
| https://mathoverflow.net/users/34366 | A technical question related to Zhang's result of bounded prime gaps | Let me remind some of notations for readers:
$$\mathcal{L}=\log x,$$
$$k\_0=3.5\times 10^6, \varpi=\frac{1}{1168},$$
$$D\_0=\exp(\mathcal{L}^{\frac{1}{k\_0}}), P\_0=\prod\_{p < D\_0}p,$$
$$D\_1=x^{\varpi},P=\prod\_{p < D\_1}p,$$
$$D=x^{\frac{1}{4}+\varpi}, D\_2 = x^{\frac{1}{2}-\epsilon}$$
Here, the effect of repla... | 10 | https://mathoverflow.net/users/21090 | 132163 | 73,183 |
https://mathoverflow.net/questions/132011 | 12 | Is it true that the equation $10^{n}-9m^{3}=1$ has only one positive integer solution, namely $n=m=1$? I can't find the answer. This has an equivalent description that the repunits $R\_n = 11\dots1$ are not cubic numbers.
| https://mathoverflow.net/users/34427 | Can repunits be perfect cubes? | Rather to my surprise I find an entirely elementary proof that
$R\_1=1$ is the only (decimal) repunit cube, using nothing beyond
quadratic reciprocity (namely the formula for the Legendre symbol $(-5/p)$).
Let us first dispose of the case that $n$ is even, say $n=2k$.
This is routine: write the equation $10^n - 9m^3 ... | 18 | https://mathoverflow.net/users/14830 | 132170 | 73,186 |
https://mathoverflow.net/questions/132047 | 5 | I would like to prove that for any family of balls $\{B(c\_i,r\_i)\}\_i \subset \mathbb{R}^d$ such that $\{c\_1, \dots, c\_n\} \subset \bigcap\_i B(c\_i,r\_i) $ and $\forall i, r\_i \geq 1$, there exists a ball of radius $1-\frac{\theta\_d}{2}$ included in the intersection $\bigcap\_i B(c\_i,r\_i)$
where $\theta\_d/2 =... | https://mathoverflow.net/users/36290 | Biggest ball included in an intersection of balls | OK, let's try.
First, $n=d+1$ (Helly's theorem).
Second, if the balls of radii $r\_i-\theta\_d$ do not have a common point, there is $\theta<\theta\_d$ such that the balls of radii $r\_i-\theta$ have exactly one common point, say, the origin. Also, we can ignore the balls such that $0$ is not on their boundary. Thu... | 5 | https://mathoverflow.net/users/1131 | 132176 | 73,189 |
https://mathoverflow.net/questions/132173 | 1 | This is essentially a request for counterexamples, since I know so few $n$-buds (or as some might say, formal group law $n$-chunks). One notices that the only $1$-bud of maximal degree 1 is the additive one, $x+y$, which also happens to be a formal group law. Also, the only 2-buds that I know of over a ring $R$ are of ... | https://mathoverflow.net/users/11546 | Is every (one dimensional) n-bud of total degree n also a formal group law? | If $f(x, y)$ is a formal group law then so is $g(f(g^{-1}(x), g^{-1}(y))$ where $g$ is an invertible (under composition) formal power series. This suggests a strategy for writing down $n$-buds, namely pick a polynomial $g$, a group law $f$, and a polynomial approximation $h$ to $g^{-1}$ and then compute $g(f(h(x), h(y)... | 4 | https://mathoverflow.net/users/290 | 132181 | 73,192 |
https://mathoverflow.net/questions/132175 | 18 | Let $\kappa$ be an infinite cardinal. For a subset $A \subseteq \mathbb{R}$, we say that $A$ is $\kappa$-dense if $|A \cap (a, b)| = \kappa$ for every interval $(a, b)$. By Cantor, any two $\aleph\_0$-dense sets are order-isomorphic.
If the CH holds, then there are many non-isomorphic $\aleph\_1$-dense sets. However... | https://mathoverflow.net/users/33768 | Can all $\aleph_2$-dense subsets of $\mathbb{R}$ be isomorphic? | The consistency of *All $\aleph\_2$-dense sets are order-isomorphic* is problem 15.14 in [Arnold Miller's list of problems](http://www.math.wisc.edu/~miller/res/problems.pdf) (which he keeps up-to-date), and there is no indication there that any progress has been made on it. That's at least strong evidence that it's st... | 6 | https://mathoverflow.net/users/11445 | 132185 | 73,193 |
https://mathoverflow.net/questions/132184 | 1 | Suppose $Z\_{1}, Z\_{2}, \dots, $ are independent and identically distributed random variables with mean 0 and variance 1. Put $X\_{nk}=\sigma\_{nk} Z\_{k}$ for $n=1,2, \dots$ and $k=1, 2, \dots, r\_{n}$, where $r\_{n} \to \infty$ as $n \to \infty$ and $\sum\_{k=1}^{r\_{n}} \sigma^2\_{nk}=1$. If $\max\_{1\le k \le r\_{... | https://mathoverflow.net/users/34483 | The Lindeberg Condition | $\newcommand{\E}{\mathbf{E}}\renewcommand{\P}{\mathbf{P}}\DeclareMathOperator{\var}{Var}$
Yes. If $s\_n=\max\_{1\le k \le r\_n} \sigma\_{nk}$ then we have
$$\E\big[|X\_{nk}|^2 I\_{|X\_{nk}|>\eta}\big] \le \sigma\_{nk}^2 \E\big[|Z\_k|^2 I\_{|Z\_k|>\eta/s\_n}\big]$$
hence
$$\sum\_{k=1}^{r\_n}\E\big[|X\_{nk}|^2 I\_{|X\_{n... | 1 | https://mathoverflow.net/users/1061 | 132188 | 73,194 |
https://mathoverflow.net/questions/132179 | 2 | Let $l\_1$ and $l\_2$ be two lines in $G(1,n)$, the Grassmannian of lines in n dimensional projective space.
Suppose that their Plucker embeddings has dot product zero. Namely if $(x\_1, x\_2, \cdots, x\_N)$
and $(y\_1, y\_2, \cdots, y\_N)$ are their Plucker coordinates, $\sum\_{i=1}^N x\_i y\_i = 0$.
Is there anyth... | https://mathoverflow.net/users/3208 | Two lines with orthogonal Plucker embedding | In a real vector space you can say the following. The lines you refer to correspond to planes $P$, $Q$ in $\mathbb{R}^{n+1}$. The dot product of the Plucker coordinates is zero if and only if $P$ intersects the orthogonal complement of $Q$ nontrivially. This doesn't mean the lines are orthogonal. In fact they need not ... | 3 | https://mathoverflow.net/users/5499 | 132189 | 73,195 |
https://mathoverflow.net/questions/131981 | 1 | Is there an analogue of the [Beck-Fiala theorem](http://en.wikipedia.org/wiki/Beck%E2%80%93Fiala_theorem) for linear or hereditary discrepancies of hypergraphs?
| https://mathoverflow.net/users/22051 | Beck-Fiala for other discrepancies | If you take a subsystem, then the degree condition of the Beck-Fiala is still satisfied, so it is also trivially true for hereditary discrepancy. From Lovasz-Spencer-Vesztergombi, we know that the linear discrepancy can be bounded by twice the hereditary discrepancy, so almost the same bound applies for that as well.
... | 2 | https://mathoverflow.net/users/955 | 132190 | 73,196 |
https://mathoverflow.net/questions/132174 | 6 | This looks extremely easy, but then again it's late at night...
Let $k$ be a commutative ring with unity. An element $a$ of a $k$-algebra $A$ is said to be *transcendental* over $k$ if and only if every polynomial $P\in k\left[X\right]$ (with $X$ being an indeterminate) such that $P\left(a\right)=0$ must satisfy $P=0... | https://mathoverflow.net/users/2530 | Are roots of transcendental elements transcendental? | $\def\F{{\mathbb F}}$
This is just a proof of Qiaochu's example.
Let $k=\F[Y,Z]/(Y^2,YZ,Z^2)$, $a=\overline{Y}$, $b=\overline{Z}$. In the ring $k[X]$ we have $I=(aX^3-b)=\{aP(X)X^3-bP(X): P(X)\in \F[X]\}$, since $a(aX^3-b)=b(aX^3-b)=0$. So each element of this ideal contains both even and odd powers of $X$. Thus in $... | 6 | https://mathoverflow.net/users/17581 | 132192 | 73,197 |
https://mathoverflow.net/questions/132187 | 2 | A complex number is said to be hypertranscendental if the one is not a zero of any entire function with all rational Maclaurin coefficients. Does there exist a transcendental number which is not hypertranscendental?
| https://mathoverflow.net/users/32273 | Do there exist transcendental numbers which are not hypertranscendental? | Let $z$ be an arbitrary complex number. Since $\mathbb{Q}[i]$ is dense in $\mathbb{C}$, we can choose a sequence of complex numbers $a\_n$ such that $|a\_n|< \frac{1}{n!}$ and $a\_{n+1} - \frac{a\_n}{z} \in \mathbb{Q}[i]$. Define an entire function
$ f(t) = \sum\_{n=0}^{\infty} a\_n t^n $
Then the function $g(t) = ... | 12 | https://mathoverflow.net/users/5499 | 132195 | 73,198 |
https://mathoverflow.net/questions/132045 | 2 | Let $X$ be a smooth projective variety over a field $k$ and $D^b(X)$ its bounded derived category. Let $\bar{X}$ the base change to $\bar{k}$. Let $A$ be a triangulated subcategory of $D^b(X)$ that $\bar{A} $ is admissible inside $D^b(\bar{X})$ over $\bar{k}$. Is $A$ admissible also inside $D^b(X)$ over $k$? (admissibl... | https://mathoverflow.net/users/4096 | admissible subcategories over non algebraically closed fields | I think the answer should be yes, but the statement should be more accurate. First, in the definition of $\bar{A}$ you first extend scalars and then take the triangulated hull and add all direct summands. consequently, if you want the statement to be true you should add all direct summands to $A$ as well (or assume tha... | 2 | https://mathoverflow.net/users/4428 | 132200 | 73,201 |
https://mathoverflow.net/questions/132197 | -1 | I am interested in the intersection graphs of $\binom{X}{2}$, i.e. the set of all 2-element subsets of a (finite) set $X$.
[**Motivation**: One can represent every simple graph with $n$ vertices by an assignment of 0 or 1 to the vertices of the intersection graph of $\binom{[n]}{2}$. This is a somehow more "natural" ... | https://mathoverflow.net/users/2672 | Intersection graphs of 2-element subsets | The usual name of your graph is $L(K\_n)$, the **line graph** of the complete graph $K\_n$. The line graphs of complete graphs have also been called **triangular** graphs, as I just learned moments ago from the Wikipedia article <http://en.wikipedia.org/wiki/Line_graph> .
| 5 | https://mathoverflow.net/users/nan | 132203 | 73,204 |
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