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https://mathoverflow.net/questions/105216 | 3 | Let $(E,\mathscr E)$ be a measurable space and $Q:E\times \mathscr E\to\Bbb [-1,1]$ be a signed bounded kernel, i.e. $Q\_x(\cdot)$ is a finite measure on $(E,\mathscr E)$ for any $x\in E$ and $x\mapsto Q\_x(A)$ is a measurable function for any set $A\in \mathscr E$.
For any fixed $x$, let the measure $Q^+\_x$ be a po... | https://mathoverflow.net/users/11768 | Is positive part of the kernel measurable? | I will assume that $E$ is [standard Borel](http://en.wikipedia.org/wiki/Borel_space#Standard_Borel_spaces_and_Kuratowski_theorems).
A kernel is just a measurable map into the space of measures (with a measurable structure generated by evaluations on sets), so it is sufficient to show that the map $\mu \mapsto \mu^+$ ... | 3 | https://mathoverflow.net/users/22758 | 105275 | 60,907 |
https://mathoverflow.net/questions/105279 | 0 | Context
-------
Let $t\_1, .., t\_n$ be jointly independent boolean random variables.
Let $X, Y$ be monotone functions (i.e. $\forall i: t\_i \geq t'\_i$ implies $X(t\_1, ... t\_i ..., t\_n) \geq X(t'\_1, ... t'\_i, ..., t'\_n)$.
Then, $E(XY) \geq E(X)E(Y)$
This is easily proved by induction on $n$:
* noramli... | https://mathoverflow.net/users/25872 | Continuous variant of KFG Inequality | There are many variants of the FKG inequality. For example, consider ${\mathbb R}^n$ as a lattice with partial order $x \le y$ if $x\_i \le y\_i$ for all $i$, and let $X\_i$ be independent random variables. Then for any increasing functions $f$, $g$ on ${\mathbb R}^n$,
$E[f(X\_1,\ldots,X\_n) g(X\_1,\ldots,X\_n)] \ge E... | 0 | https://mathoverflow.net/users/13650 | 105283 | 60,912 |
https://mathoverflow.net/questions/105292 | 7 |
>
> My question has to do with distance spheres in $\mathbb CP^{n+1}$. I am interested in knowing what is the radius $r$ of a distance sphere $S(r)$ around a point that makes it a *minimal submanifold* of $\mathbb CP^{n+1}$. It is known that, in this type of geometric situation, the distance spheres have constant mea... | https://mathoverflow.net/users/15743 | Minimal distance spheres in complex projective spaces | A clearer and more correct way to do the second calculation is to look at the volume of the preimage of $S\_r^{2n+1}$ in standard $S^{2n+3}$. The preimage is a certain torus $T\_r$, and one has
$$\mathop{Vol} T\_r = 2\pi\mathop{Vol} S\_r^{2n+1}.$$
At the same time,
$$\mathop{Vol} T\_r \propto (\cos r)(\sin r)^{2n+1},$$... | 5 | https://mathoverflow.net/users/1450 | 105300 | 60,913 |
https://mathoverflow.net/questions/105303 | 4 | Given an elementary abelian $p$-group $G$, it's well known that it can be seen as a vector space over $\mathbb{Z}\_p $.
But, does someone have an idea about possible sources where I can find proofs that use this fact?
Examples: linear independence of elements in a group $G $ modulo $\phi(G) $ ($\phi(G)$ is the Frat... | https://mathoverflow.net/users/25080 | Linear Independence & Group Theory | One example is in the $\sigma$-reduction of a finite group. Denote by $\sigma(|G|)$ the number of prime divisors of $|G|$, and let $\sigma(G)$ denote the maximum number of prime divisors of the order of an element of a group $G$. We call a group $H$ $\sigma$-reduced if every prime $p$ appears in at most one chief facto... | 3 | https://mathoverflow.net/users/25494 | 105311 | 60,915 |
https://mathoverflow.net/questions/105299 | 0 | I asked this question in MSE here: <https://math.stackexchange.com/questions/184524/diametrically-opposite-points-go-to-diametrically-opposite-points-under-stereogr> but I didn’t get an answer. I really need to solve this problem and believe me this is not a homework.
Suppose $P\_1$ and $P\_2$ are two diametrically o... | https://mathoverflow.net/users/23980 | Diametrically opposite points go to diametrically opposite points under stereographic projection | Let $C$ be a round circle in $R^2$ centered at $p$. Let $g$ be a rotation fixing $p$; then, as a Moebius transformation of $R^2\cup \infty$, $g$ has two fixed points $p, \infty$. Let $g^\sigma$ be the congutate of $g$ by $\sigma$. Then $g^\sigma$ is again a Moebius transformation of $S^2$ which preserves the circle $C'... | 3 | https://mathoverflow.net/users/21684 | 105314 | 60,917 |
https://mathoverflow.net/questions/105304 | 23 | If $f, g\in \mathbb C[a,b]$ are polynomials in two variables, are there easy criteria that allow to see if $f(x,y)-g(t,z)\in \mathbb C[x,y,t,z]$ is irreducible?
Thank you very much,
best
| https://mathoverflow.net/users/6949 | Criteria for irreducibility of polynomial | There has been a lot of work on establishing when polynomials of the form
$$f(x\_1, \dots, x\_r) - g(y\_1, \dots, y\_s)$$
are reducible (over the complex numbers, but also over other fields). (Negating them, or special cases thereof would thus yield criteria, in the sense of conditions, when such a polynomial is irred... | 42 | https://mathoverflow.net/users/nan | 105323 | 60,921 |
https://mathoverflow.net/questions/105318 | 0 | Notation abuse warning: I will use the E7 series irrep names. You'll soon see why.
In the general Lie algebra, the irrep you "start" with is $J$, the
adjoint. From the series for $J\bigotimes{J}=1+J+A...$ ($1$ is the 1-dimensional irrep) you get $A$, the antisymmetric one. (The quantum dimensions up to here are all ... | https://mathoverflow.net/users/11504 | "Fictive" irreps of the enveloping general Lie algebra | The (quantum) dimensions are all given in Vogel's papers (all but one unpublished). There may be some errors in my paper you refer to as I confused Vogel's parameters with the values of the quadratic Casimir.
I don't follow your "silly part". The representation $V$ is not defined for $E\_8$ and I don't see how you ge... | 1 | https://mathoverflow.net/users/3992 | 105326 | 60,922 |
https://mathoverflow.net/questions/104642 | 10 | **Motivation:** There have been the *instanton* (anti-self dual connection) solutions to the Yang-Mills equation $d\_A^\ast F\_A=0$ which extremize the YM energy $\int\_M|F\_A|^2$, leading to the Donaldson invariants and even a Floer homology. There have been the *monopole* (connection + spinor) solutions to the Seiber... | https://mathoverflow.net/users/12310 | Floer homology and Invariants for Einstein Field Equations? | Let me clarify a couple of issues from the previous answers/comments:
1) The linearization of $Rc-\tfrac{1}{2}Rg$ has mixed signs, and for this reason the equation $\partial\_t g=-(Rc-\tfrac{1}{2}Rg)$ is bad, a kind of coupled backwards/forwards heat equation, i.e. no short time existence.
2) The linearization of $... | 11 | https://mathoverflow.net/users/22029 | 105334 | 60,924 |
https://mathoverflow.net/questions/105333 | 7 | Suppose that $X = \mathbb{P}^n\_k$ and $G$ is a coherent sheaf on $X$.
**Question:** Is there a way to determine *some* integer $n\_0$ such that $H^1(X, G \otimes O\_X(n)) = 0$ for all $n \geq n\_0$? (obviously $n\_0$ depends on $G$)
Of course, the higher cohomologies would be very interesting as well, but perhaps... | https://mathoverflow.net/users/3521 | Effective Serre Vanishing | Take $n\_0$ as (Castelnuovo–Mumford) regularity of $G$ minus $1$.
This generalizes to higher cohomology: for $H^i$ take the regularity minus $i$
Let $M$ be a module representing $G$.
Then, the (Castelnuovo–Mumford) regularity of $M$ is an upper bound of the regularity of $G$.
So for an implementation you can take the... | 7 | https://mathoverflow.net/users/25523 | 105335 | 60,925 |
https://mathoverflow.net/questions/105080 | 10 | Let $(k,|.|)$ be a complete non-archimedean valued field. Let $D$ be the open unit disc over $k$. (Anything I write could be adapted to the case of an open annulus.) The ring $\mathcal{O}(D)$ of analytic functions on $D$ admits an explicit description: it consists of the series $\sum\_{n\ge 0} a\_n T^n$ such that for a... | https://mathoverflow.net/users/4069 | Analytic elements in non-archimedean geometry | Vladimir Berkovich has indicated to me that the answer is no: on a general curve, functions need not restrict to analytic elements. Let me copy here his argument.
The algebra of analytic elements $\mathcal{H}(D)$ can be defined as the completion of the inductive limit of the $k$-affinoid algebras $\mathcal{A}\_X$ wit... | 5 | https://mathoverflow.net/users/4069 | 105336 | 60,926 |
https://mathoverflow.net/questions/105330 | 4 | I am working on a very large dataset of a single [DAG](http://en.wikipedia.org/wiki/Directed_acyclic_graph) whose vertices have a low branching factor. I need to generate all possible (simple) paths starting from the source and write them to a file.
My question is: what is [computational complexity](http://en.wikiped... | https://mathoverflow.net/users/22197 | Generation of All Path in a Directed Acyclic Graph | The worst case is exponential in the number of nodes. Consider an [Ordered Binary Decision Diagram](https://en.wikipedia.org/wiki/Binary_decision_diagram) (OBDD) which is a DAG with outdegrees 2. Let it be on $n$ boolean variables and on $m$ nodes. Each assignment of the variables corresponds to a path to either True o... | 5 | https://mathoverflow.net/users/12481 | 105340 | 60,928 |
https://mathoverflow.net/questions/105337 | 5 | A triangle is regular, provided it is equilateral, or, also, equiangular. How these conditions generalize to characterizations of regularity of simplices?
In particular, it turns out that
>
> a simplex, all of whose facets meet with the same angle, is regular.
>
>
>
I think I have a proof this fact, but it's ... | https://mathoverflow.net/users/6101 | Regularity of simplices | I am not sure what you mean by a direct proof, but here is a reasonably straightforward argument. Consider your equi-angular simplex $S$, and consider the link $L(v)$ of a vertex $v$ (the intersection of a small sphere centered at $v$ with your simplex, scaled so that the sphere has radius $1.$ The dihedral angles of $... | 7 | https://mathoverflow.net/users/11142 | 105341 | 60,929 |
https://mathoverflow.net/questions/105339 | 5 | Let $M$ a right simple module and $N$ be a left simple module over a ring $R$. I'm seeking a kind of Schur's lemma, with $\mathrm{Hom}\_R (M,N)$ replaced by $M \otimes\_R N$. So my questions are:
Can we describe $M \otimes\_R N$ explicitly?
In particular, for a fixed $M$, is $N$ such that $M \otimes\_R N \neq 0$ un... | https://mathoverflow.net/users/22758 | Tensor product of simple modules | Sasha's statement is true for any pair of modules.
The center of $R$ is a commutative ring $S$. Since the endomorphisms of a simple module are a division algebra, whose center is a field, the action of $S$ on every simple module factors through some field, so the action of $R$ of course factors through an algebra ove... | 6 | https://mathoverflow.net/users/18060 | 105344 | 60,931 |
https://mathoverflow.net/questions/105269 | 33 | Can someone describe explicitly an abelian group $A$ such that the extension $$0 \to \mathbb{Z} \to A \to \mathbb{Q} \to 0$$ doesn't split ?
Background: The Stein-Serre theorem (Hilton, Stammbach: A course in homol. algebra, Theorem 6.1) states that if $A$ is abelian of countable rank (=maximal number of linear inde... | https://mathoverflow.net/users/25869 | Non-split extension of the rationals by the integers | Nice question. For a prime $p$ let $\mathbb{Z}\_{(p)} = \lbrace \frac{a}{b}\in \mathbb{Q}\mid p \nmid b\rbrace$ and $\mathbb{Z}[p^{-1}] = \lbrace \frac{a}{p^n}\in \mathbb{Q}\mid n \ge 0 \rbrace$. Then
$$A := \lbrace (x,y) \in \mathbb{Q} \times \mathbb{Z}[p^{-1}] \mid x-y \in \mathbb{Z}\_{(p)} \rbrace$$
has the desir... | 25 | https://mathoverflow.net/users/10194 | 105345 | 60,932 |
https://mathoverflow.net/questions/101700 | 5 | In an attempt to understand a bit better large cardinals, I have been thinking along the following lines, which could be summarized under the slogan
>
> **Talk about cardinals without the
> (ambient) set theory**
>
>
>
the class ON is first-order axiomatizable, and thus it looks like I can carve out of ON the... | https://mathoverflow.net/users/15293 | Large cardinals without the ambient set theory? | *But anyway, my point is this: just look at ON and totally forget that is the spine of V, and see it as a system of ordered numbers. Now axiomatize what you see, much in the same way as you axiomatize its initial segment N. – Mirco Mannucci Jul 8 at 22:21*
*for instance, one could add to the theory an equivalence rel... | 4 | https://mathoverflow.net/users/22247 | 105347 | 60,933 |
https://mathoverflow.net/questions/105346 | 0 | I am sorry if this is a very stupid question, but I am new in the topic and I was not able to find either a solution or relevant literature. (I am so new that I really don't know even what the right tags are!)
I have a system of two linear recurrences. In particular, the first recurrence is $x\_n=998x\_{n-1}+995y\_{n... | https://mathoverflow.net/users/13809 | A system of linear recurrences | The limit of $x\_n/(x\_n+y\_n)$ is not 1/2 but
$$331667 + 332\sqrt{995001}\over 663334 + 665\sqrt{995001}$$
The limit of $x\_n/y\_n$ is not 1 but
$$331667 + 332\sqrt{995001}\over 331667 + 333\sqrt{995001}$$
| 4 | https://mathoverflow.net/users/10503 | 105349 | 60,935 |
https://mathoverflow.net/questions/1714 | 178 | I know of two good mathematics videos available online, namely:
1. Sphere inside out ([part I](https://www.youtube.com/watch?v=BVVfs4zKrgk) and [part II](https://www.youtube.com/watch?v=x7d13SgqUXg))
2. [Moebius transformation revealed](https://www.youtube.com/watch?v=0z1fIsUNhO4)
Do you know of any other good math... | https://mathoverflow.net/users/416 | Best online mathematics videos? | I have compiled a list (1500+) of math videos at <http://pinterest.com/mathematicsprof/> . If anyone is aware of others, please send them to me.
| 46 | https://mathoverflow.net/users/25910 | 105360 | 60,940 |
https://mathoverflow.net/questions/105358 | 3 | Suppose we have a first order theory $T$ in a language which contains the identity symbol $=$, constants $a\_1$, $a\_2$, $a\_3$, ... $b\_1$, $b\_2$, $b\_3$, ... and no relation or function symbols. For natural numbers $i \neq j$, suppose that $a\_i \neq a\_j$ is contained in $T$, and let us suppose that $T$ has no mode... | https://mathoverflow.net/users/25908 | a theory of identity with constants | The answer is no.
Here is a simplified counterexample (my earlier post was more
complicated).
Let $T$ be the theory asserting $a\_i\neq a\_j$ and also the
assertions, for every particular $k$ and $j$, that if $b\_1=a\_k$, then $b\_{k+1}\neq a\_j$.
This theory is consistent, since we can let $b\_1\neq a\_k$ for an... | 9 | https://mathoverflow.net/users/1946 | 105362 | 60,942 |
https://mathoverflow.net/questions/105353 | 1 | It is well known that isoperimetric inequalities on a hypercube are closely related to influences, but all the theorems I'm aware of deal with monotone sets. Now suppose we have an arbitrary set $X \subset \{0, 1\}^n$, and let us color all vertices of a hypercube that lie in $X$ in black, others in white. The boundary ... | https://mathoverflow.net/users/25905 | Hypercube isoperimetric inequality for non-increasing events | $|D^+| - |D^-|$ can be expressed in terms of the level counts.
Let $S\_k$ be the set of vertices of the cube with $k$ coordinates equal to $1$. Let $c\_k = |X \cap S\_k|$.
Let $d\_k$ be the contribution to $|D^+| - |D^-|$ from the edges between $S\_k$ and $S\_{k+1}$.
Weight each edge $+1$ if it is white-black, $0$... | 1 | https://mathoverflow.net/users/2954 | 105366 | 60,945 |
https://mathoverflow.net/questions/105370 | 9 | In a symplectic manifold $(X^{2n},\omega)$, a hypersurface $Y\subset X$ has *contact-type* if there is a contact form $\lambda$ such that $d\lambda=\omega|\_Y$. Recall that a contact form is a 1-form with $\lambda\wedge(d\lambda)^{n-1}>0$, i.e. the opposite of a foliation. For example, a starshaped hypersurface has con... | https://mathoverflow.net/users/12310 | When does a hypersurface have contact-type? | Not all hypersurfaces in $\mathbb{R}^{2n}$ are of contact type.
Weinstein, in the paper: "On the hypothesis of Rabinowitz' periodic orbit theorems", where he defined the concept of contact type, gives a criterion. If $H^1(\Sigma)=0$, and $\Sigma$ is of contact type, then the characteristic line bundle comes with a di... | 6 | https://mathoverflow.net/users/12156 | 105374 | 60,951 |
https://mathoverflow.net/questions/105354 | 5 | Dear all,
It is clear that if $f:R\mapsto R$ is a continuous function, than $< f, \delta\_x >=f(x)$. Now, if $f$ is only semicontinous, can we say that $< f, \delta\_x >=f(x)$? I think this is true at all continuous points of $f$. But when $f$ has a jump at $x$, can we properly define this inner product? Does anyone ... | https://mathoverflow.net/users/36814 | Can we calculate the inner product of a semicontinous function with the Dirac delta function? | This is an ill posed problem. The Dirac $\delta$ is a continuous linear map from the (locally convex) space of continuous function on $\mathbb{R}^n$ to $\mathbb{R}$. You ask if it admits an extension to the larger set of semicontinuous functions. First, semicontinuity is not a linear condition: the sum of a lower semic... | 5 | https://mathoverflow.net/users/20302 | 105378 | 60,954 |
https://mathoverflow.net/questions/105371 | 4 | I stumbled upon this isolated singularity of a Calabi-Yau fourfold:
\begin{equation}
x\_1x\_2+x\_3x\_4+x\_5^2=0
\end{equation}
as a hypersurface in $\mathbb{C}^5$.
Clearly, I can resolve this by a simple blow-up. This does not seem to be a crepant resolution however.
So my question is:
Is there some simple criterio... | https://mathoverflow.net/users/9271 | Crepant resolution of isolated fourfold singularity | That particular singularity only can have a crepant resolution if it has a small resolution.
Terminal singularities
----------------------
If a variety has terminal singularities (like yours does), the then every further blow-up also has discrepancies $> 0$. (In other words, whether or not the variety has terminal ... | 10 | https://mathoverflow.net/users/3521 | 105382 | 60,957 |
https://mathoverflow.net/questions/105384 | 1 | If $A$ is a matrix and $D$ is a diagonal matrix, is there some special name for $DAD$?
| https://mathoverflow.net/users/22051 | name for a matrix operation | it is called "diagonal congruence" [here](http://dx.doi.org/10.1080/03081080600872327). This makes sense, at least when $D$ is real, since it is a [congruence](http://en.wikipedia.org/wiki/Matrix_congruence). "Conjugate" sounds more like $D^{-1}AD$ or $\overline{A}$ to me.
| 11 | https://mathoverflow.net/users/1898 | 105386 | 60,959 |
https://mathoverflow.net/questions/105390 | 1 | Suppose $x \in \mathbb{R}^n$, $B,U \in \mathbb{R}^n\times\mathbb{R}^n$ and $U$ a unitary matrix. Define $g\_{U}(x) = || BUx||$ where $||.||$ is some norm or norm-ish function on $\mathbb{R}^n$ (not unitarily invariant obviously). How can we choose $U$ in the unitary matrices to minimize $g$? Or what kind of results are... | https://mathoverflow.net/users/8916 | Results for minimizing the norm w.r.t a unitary matrix | In short: Householder transformations.
More specifically: for $||\cdot||\_\infty$, first think about the case where $B$ is the identity. Then you simply need to rotate $x$ such that it points along some axis -- this way all of the "mass" is concentrated in a single component. More explicitly, you can construct $U$ as... | 1 | https://mathoverflow.net/users/1557 | 105393 | 60,961 |
https://mathoverflow.net/questions/95418 | 4 | Let $X$ be a smooth connected quasi-projective curve over $\mathbf{Q}$. Let $U$ be the pro-unipotent etale fundamental group of $X$ over $\mathbf{Q}\_p$.
$U^1 = U$ and let $U^n =[U,U^{n-1}]$.
Let $n\geq 1$.
Why is $U^n/U^{n+1} \cong \mathbf{Q}\_p(n)^{r\_n}$ for some positive integer $r\_n$?
I "know" this is tr... | https://mathoverflow.net/users/22189 | A question about the Tannakian etale fundamental group of a curve | Al ulrich says, this is not true in general. If $X$ is non-compact (i.e. affine), then the Lie algebra of U is isomorphic to the free Lie algebra on $H^1\_\mathrm{et}(X,\mathbb{Q}\_p)^\vee$. Hence in the case of $\mathbb{P}^1\setminus \{0,1,\infty \}$, where the first etale cohomology is $\mathbb{Q}\_p(-1)^{\oplus 2}$,... | 7 | https://mathoverflow.net/users/13647 | 105394 | 60,962 |
https://mathoverflow.net/questions/105400 | 11 | **Property of any odd number of nonnegative integers:**
Given $x\_1 \leq \cdots \leq x\_{2n + 1}$ with each $x\_i \in \mathbb{Z}\_{\geq 0}$, suppose that for any $x\_i$ we remove, the remaining numbers can be assigned to disjoint multisets $A$ and $B$ such that $|A| = |B| (= n)$ and $\sum\_{x \in A} A = \sum\_{x \in ... | https://mathoverflow.net/users/22971 | Extending an assignment property from Q to R (or C) | Yes, the result can be deduced from the rational case. Let $x\_1, \ldots, x\_{2n+1}\in \mathbb C$ be given. Fix a basis of the $\mathbb Q$-vector space spaned by the $x\_i$'s and write the coordinates of $x\_i$ in this basis as $(x\_i^1, \ldots, x\_i^r)$.
Now the same property holds for each family $x\_1^k,\ldots,x\_{2... | 13 | https://mathoverflow.net/users/19534 | 105404 | 60,966 |
https://mathoverflow.net/questions/97851 | 7 | Let $K$ be an imaginary quadratic field and $E$ an elliptic curve with CM by the maximal order of $K$, such that $E$ is defined over the Hilbert class field $H$. Is it known whether there is a bound (independent of the degree of $H$) on the order of a $H$-rational torsion point on $E$?
| https://mathoverflow.net/users/16858 | Uniform bounds for the order of a rational torsion point on CM elliptic curves | Hey, I probably should have answered this one some time ago. It was proved in 1989 by J.L. Parish that the order of an $H$-rational torsion point is 1,2,3,4 or 6, and this also can be deduced from work of either Silverberg or Prasad-Yogananda. In any case the statement you want is at the beginning of section VI in the ... | 5 | https://mathoverflow.net/users/3384 | 105411 | 60,971 |
https://mathoverflow.net/questions/105425 | 8 | I'm trying to find the eigenvector corresponding to the second smallest eigenvalue of a large $(4,000,000 \times 4,000,000)$ matrix $M$. $M$ is a Laplacian matrix, and it has the following structure: $M = D - AA'$, where $D$ is a diagonal matrix and $A$ is a large sparse matrix. $A$ has dimension $4,000,000 \times 10,0... | https://mathoverflow.net/users/17073 | Finding the smallest eigenvalues of a large, but structured, matrix | Well, if you add $c I$ to your matrix, for some reasonable value of $c,$ it will become nonsingular. As for the left inverse, since your matrix is sparse, to compute the backward iteration you can use the conjugate gradient method, which will be very fast (there are fancier "Krylov space" methods, but you need not go t... | 4 | https://mathoverflow.net/users/11142 | 105426 | 60,981 |
https://mathoverflow.net/questions/104035 | 1 | Hi, I am interested in learning a bit more about this space. I have exhausted all the books available at my disposal, and none of them explain much of the basics for me. Here's a definition of this space.
The seminorm is
$$[u] = \sup\_{(x,t), (y,s) \in Q} \frac{|u(x,t) - u(y,s)|}{(|x-y|^2 + |t-s|)^{\frac{\alpha}{2}}}... | https://mathoverflow.net/users/25266 | Basic questions about parabolic Holder space | In this PhD thesis you can find all necessary information about parabolic Holder Space
<http://www.google.fr/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&cad=rja&ved=0CEgQFjAD&url=http%3A%2F%2Fpeople.maths.ox.ac.uk%2Fjoyce%2Ftheses%2FBehrndtDPhil.pdf&ei=UPk3UMG4Iue70QWh0oDIBQ&usg=AFQjCNEWZ_LHmQEZsgNr47KxPWmDD8lk-g&sig2=owL... | 2 | https://mathoverflow.net/users/nan | 105427 | 60,982 |
https://mathoverflow.net/questions/105438 | 41 | Suppose $f$ is a $C^\infty$ function from the reals to the reals that is never negative. Does it have a $C^\infty$ square root? Clearly the only problem points are those at which $f$ vanishes.
| https://mathoverflow.net/users/24338 | Square root of a positive $C^\infty$ function. | The answer is "no". This is covered in great detail here:
<http://www.math.polytechnique.fr/~bony/BBCP_jfa.pdf>
| 41 | https://mathoverflow.net/users/7311 | 105442 | 60,994 |
https://mathoverflow.net/questions/105388 | 2 | Let $f:R\rightarrow R$. If there exists the finite limit $$\lim\_{(x,y) \rightarrow a \atop x\neq y} \frac{f((y)-f(x)}{y-x}$$ then obviously there is a finite derivative $f'(a)$ and is equal this limit.
What about similar problem for higher order divided differences?
May is it true that existence of finite $$\lim\_... | https://mathoverflow.net/users/25921 | Higher order divided differences and derivatives | The answer is yes:
Assume we have a $\delta$ such that for $|x\_i-a|<\delta$, we have $|[x\_0,...,x\_n;f]-\lim| < \epsilon$. Assume WLOG that $\lim = 0$ (by subtracting off a polynomial of degree $n$). Then if $y\_0, ..., y\_{n-1}$ and $z\_0, ..., z\_{n-1}$ are in the $\delta$-ball around $a$, we can show
$|[y\_0,.... | 1 | https://mathoverflow.net/users/2363 | 105452 | 60,997 |
https://mathoverflow.net/questions/105399 | 22 | I've a slightly technical question about the Yangian which I'm hoping an expert out there can answer.
Recall that the Yangian $Y(\mathfrak{g})$ is a Hopf algebra quantizing $U(\mathfrak{g}[z])$. Drinfeld, in his quantum groups paper, explains that the algebra of integrals of motion of certain integrable lattice model... | https://mathoverflow.net/users/14681 | Question about the Yangian | It seems that subalgebra in the question is the so-called "Bethe subalgebra" of the Yangian.
It is not immediate for me to recognize the connection with definition given in the question and the definition I'll give below - but I am sure that it should be simple and well-known. Can someone clarify this? However, the s... | 21 | https://mathoverflow.net/users/10446 | 105460 | 61,001 |
https://mathoverflow.net/questions/105457 | 14 | There are several ways to compute the classical integral
$$
\int\_{\mathbb R}e^{-x^2}dx=\sqrt{\pi}.
$$
Probably, best known are
(1) squaring the integral with subsequent change
of (now two) variables to the polar form, and
(2) the reducing to the Gamma-function at $1/2$.
I am interested though in a "complex" anal... | https://mathoverflow.net/users/4953 | Complex evaluation of a classical (real) integral | Yes! For a long time that was thought impossible, but then it was found how to do it using a parallelogram as a contour.
Desbrow, Darrell
On evaluating $\int\_{-\infty}^\infty e^{ax(x-2b)}dx$ by contour integration round a parallelogram.
Amer. Math. Monthly 105 (1998), no. 8, 726–731.
According to Desbrow, ... | 15 | https://mathoverflow.net/users/454 | 105462 | 61,003 |
https://mathoverflow.net/questions/105458 | 2 | Is there, e.g. in $\mathbb R^4$ a simple curve that does not contain the origin and intersects every subspace of dimension 2?
Sorry if the question is too easy, but I just cannot figure it out.
In three dimensions such a curve exists, but I cannot imagine four dimensions. Is it possible to somehow lift-up the Peano-c... | https://mathoverflow.net/users/955 | Can a simple curve intersect every subspace of dim 2 and avoid the origin? | Consider closed space filling curve $\theta:\mathbb S^1\to\mathbb S^3$.
You can choose $\theta$ so that $\theta^{-1}(x)$ is finite for any $x\in\mathbb S^3$
and $|\theta^{-1}(x)|=1$ for all but countable set in $\mathbb S^3$.
Then it is easy to find a function $\rho:\mathbb{S}^1\to\mathbb R\_+$
so that the curve $\ga... | 8 | https://mathoverflow.net/users/1441 | 105465 | 61,005 |
https://mathoverflow.net/questions/105473 | 1 | Let $E$ be an elliptic curve over an algebraically closed field $k$ of characteristic $p$. Is there any nice computation for the group $H^1(E,\alpha\_p)$ and $H^1(E,\mathbb{G}\_a)$? The cohomology is taken in the flat topology.
When these groups are trivial? and is there any way to describe them? I am not asking for ... | https://mathoverflow.net/users/18380 | 1st-flat cohomology group for elliptic curves | $\mathbb G\_a$ is a smooth group scheme, so the flat cohomology is the same as the etale cohomology. It is also a quasicoherent sheaf, so the etale cohomology is the same as the Zariski cohomology, which is a $1$-dimensional vector space over $k$.
For $\alpha\_p$, you can use the exact sequence $0 \to \alpha\_p \to \... | 7 | https://mathoverflow.net/users/18060 | 105476 | 61,010 |
https://mathoverflow.net/questions/105471 | 1 | The Knaster-Kuratowski-Mazurkiewicz Lemma is the continuous analogue of Sperner's lemma. I wonder if the following, more general version is true.
Let S be the standard simplex spanned by the standard orthonormal basis for $\mathbb R^{n+1}$, so S equals the convex hull of $(e\_i:i\in [n+1])$. Assume we have n+1 closed... | https://mathoverflow.net/users/955 | Is this stronger Knaster-Kuratowski-Mazurkiewicz Lemma true? | $\def\conv{\mathop{\rm conv}}\def\aff{\mathop{\rm aff}}\let\eps\varepsilon$It seems that you may set $n=k+t$. Consider the sets $C\_1,\dots,C\_t$. If they have a nonempty intersection, we are done. Otherwise, by KKM they do not cover $S\_t=\conv\{e\_i\colon i\in[t]\}$. Take a point $s\in S\_t$ which is not covered. Sin... | 1 | https://mathoverflow.net/users/17581 | 105479 | 61,011 |
https://mathoverflow.net/questions/105482 | 0 | Can there be two different elliptic curve $E\_{1}$ and $E\_{2}$ and two different rational points $P\_{1}$ and $P\_{2}$ such that $P\_{1}, P\_{2} \in E\_{1}$ and $P\_{1}, P\_{2} \in E\_{2}$ but $P\_{1} + P\_{2}$ is a different point for $E\_{1}$ and $E\_{2}$. If so, is it easy for find an example?
Or given two differ... | https://mathoverflow.net/users/25952 | Can two different elliptic curves have rational points in common | Yes. Let $E\_1$ be the curve defined by the equation $y^2=x(x-1)(x-2)$ and $E\_2$ be the curve $y^2=x(x-1)(x-3)$. Let $P\_1=(0,0)$ and $P\_2=(1,0)$. These points are certainly on both curves.
On $E\_1$, $P\_1+P\_2=(2,0)$ but on $E\_2$ it's $(3,0)$. (Assuming that the point at infinity is the identity.)
In fact, the... | 10 | https://mathoverflow.net/users/18060 | 105484 | 61,013 |
https://mathoverflow.net/questions/105480 | 5 |
>
> For which irrational numbers $\xi$ does there exist a constant $A$ such that $\left|\frac{p}{q}-\xi\right|<\frac{1}{Aq^2}$ (where $p/q$ is a rational number) has only finitely many solutions?
>
>
>
Background
----------
I apologize if this is a terribly elementary question—my background is in analysis, not... | https://mathoverflow.net/users/25951 | Are there lower bounds on the quality of a rational approximation? | Here are some standard facts that you can find in many textbooks, e.g. in Cassels: An introduction to diophantine approximation.
1. There exists an $A>0$ such that $\left|\frac{p}{q}-\xi\right|<\frac{1}{Aq^2}$ has only finitely many rational solutions $\frac{p}{q}$ if and only if the continued fraction expansion of $... | 5 | https://mathoverflow.net/users/11919 | 105487 | 61,015 |
https://mathoverflow.net/questions/103229 | 5 | I know that if there are enough Hermitian elements in a Banach algebra, then the Banach algebra is stellar. In particular, I'm interested in the two spaces $B(L^1(S^1,\Sigma,\mu))$ the space of bounded linear operators on Lebesgue integrable functions of the circle and $B(ba(\Sigma))$ the space of bounded linear operat... | https://mathoverflow.net/users/25361 | When is a Banach Algebra $C^\star$ | This is not an answer to the question posed in your first paragraph (which I think is asking for more than you need, and more importantly, more than you can really hope for). However, for the specific purpose outlined in your second paragraph, the following paper might be helpful:
>
> H. König, A functional calculu... | 2 | https://mathoverflow.net/users/763 | 105492 | 61,017 |
https://mathoverflow.net/questions/105475 | 12 | Let $X$ be a scheme and let $G$ be an abstract group acting on $X$ by scheme automorphisms. I'm happy to assume finiteness conditions on $X$ (such as locally Noetherian) and on $G$ (such as $G$ is finite) as necessary.
I understand that it is possible to enlarge the category of schemes in such a way that there is a g... | https://mathoverflow.net/users/25949 | Quasi-coherent sheaves on $X/G$ | I think this question is a good one, but don't expect an encyclopedic answer — MO is not an encyclopedia. Here are some answers, with the disclaimer that I'm a category theorist but not an algebraic geometer.
To question A, by and large the 21st perspective will probably say that it is definitely a *stack*, in some n... | 5 | https://mathoverflow.net/users/78 | 105496 | 61,018 |
https://mathoverflow.net/questions/105513 | 2 | Let $G$ be a finite group and $N$ be a normal subgroup of G.
Suppose that $\chi \in Irr(G)$. If $\theta, \lambda \in Irr(N)$ satisfy $[\chi\_{N}, \theta] > 0$ and $[\chi\_{N}, \lambda] > 0$, is it true that $\theta(1) = \lambda(1)$?
On the other hand, are irreducible constituents of $\chi\_{N}$ unique?
| https://mathoverflow.net/users/25961 | Irreducible constituent in a normal subgroup | Yes, it is true ( the irreducible constituents of the restriction of an irreducible character
to a normal subgroup all have equal degree). This is part of Clifford's theorem. It actually applies not just to complex irreducible characters or representations, but to irreducible representations over any field. In the case... | 6 | https://mathoverflow.net/users/14450 | 105517 | 61,024 |
https://mathoverflow.net/questions/105515 | 8 | I have seen in many textbooks on analysis that the Archimedean property of reals is a consequence of the completeness axiom. However I am not convinced that we need to use such a powerful axiom (as the completeness axiom) to prove a very basic property like Archimedean Property. To me it looks simple enough as follows:... | https://mathoverflow.net/users/15540 | Archimedean Property of Real Numbers | One cannot prove the Archimedean property for the reals by appealing only to first-order algebraic truths of the ordered real field and the subring of the integers sitting inside it. The reason is that those statements are all first-order expressible in the structure $\langle\mathbb{R},{+},{\cdot},0,1,\lt,\mathbb{Z}\ra... | 14 | https://mathoverflow.net/users/1946 | 105519 | 61,026 |
https://mathoverflow.net/questions/105550 | 1 | Can one let me know about the functional equation of the alternating zeta function similar to the well known for the rieman function.
| https://mathoverflow.net/users/25947 | Functional equation of the alternating zeta function | Since one has
$$\sum\_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^s} = (1 -2^{1-s}) \zeta (s)$$
you get a Functional equation directly from the one for $\zeta$.
Note: This is function is also called [Dirichlet eta function](http://en.wikipedia.org/wiki/Dirichlet_eta_function) ; the linked Wikipedia page also has the equatio... | 6 | https://mathoverflow.net/users/nan | 105554 | 61,047 |
https://mathoverflow.net/questions/105540 | 1 | I'm having some issue in understanding the channel capacity.
$C=max\_{p(x)}I(X, Y)$
In particular the practical side. For example (an exercise), if I toss a fair coin and I transmit the result in a binary channel. What's the channel capacity?
I'm trying to imagine the channel graph. I believe is something like th... | https://mathoverflow.net/users/25965 | Channel capacity of a coin flip | If you channel is that you can decide if the cion shows head or tail and the pass the coin to your friend with that side up, then the channel has capacity of one bit. This corresponds to your first computation.
If you channel is that you throw the coin to you friend on such a way that it is head with 50% probability... | 0 | https://mathoverflow.net/users/2097 | 105556 | 61,048 |
https://mathoverflow.net/questions/105543 | 8 | I need to know what are the efficient algorithms to find all the irreducible polynomials of a given degree, say $d$ over a given finite field, say $\mathbb{F}\_{p^n}.$
One way is to factorize the polynomial $x^{p^{dn}}-x$, which is the product of all irreducible polynomials whose degree divides $d$, using factorizati... | https://mathoverflow.net/users/23980 | Algorithms to find irreducible polynomials of a given degree | The last word on the second question is this [paper of Couveignes and Lercier.](http://arxiv.org/abs/0905.1642) The question is highly nontrivial.
| 5 | https://mathoverflow.net/users/11142 | 105558 | 61,049 |
https://mathoverflow.net/questions/105557 | 3 | I've been using the Reidemeister-Schreier process (detailed in e.g. Holt et al. - *Handbook of Computational Group Theory*) to find the presentations of various modular subgroups. For example, this process tells us that the presentation of the principal congruence subgroup $\Gamma\left(4\right)$ is (before simplificati... | https://mathoverflow.net/users/25565 | Simplifying presentations of modular subgroups | If I understand your presentation correctly, you have determined that $\Gamma(4)$ is a free group on $5$ generators. This is not surprising. The modular curve $X(4)$ is $\mathbb P^1$ with six cusps and no elliptic points, so its fundamental group is the free group on $5$ generators. You appear to have found one set of ... | 5 | https://mathoverflow.net/users/18060 | 105565 | 61,052 |
https://mathoverflow.net/questions/105514 | 2 | Suppose we have a finitely presented group $G$ with a concrete presentation and a subgroup $H$, generated by a finite set of elements from $G$. How to find the presentation for $H$?
If $H$ has finite index we can use Reidemeister-Schreier procedure.
But what if $H$ has infinite index? What methods exist?
Are there s... | https://mathoverflow.net/users/8381 | Presentations of infinite index subgroups | The following answer will only be about decidability results or, more often, undecidability results. Of course, if you have a particular group in mind, it may be that something positive can be said. I will leave others to talk about practical algorithms.
Let $G=\langle X\mid R \rangle$ be a finitely presented group, ... | 4 | https://mathoverflow.net/users/1463 | 105568 | 61,054 |
https://mathoverflow.net/questions/20497 | 73 | I have just learned [here](https://mathoverflow.net/questions/20430/is-there-an-explicit-example-of-a-complex-number-which-is-not-a-period) that we know numbers that are not periods; is it known meanwhile that the ring of periods is not a field? I know that it is conjectured that $1/\pi$ is not a period, but the existe... | https://mathoverflow.net/users/3503 | Is it known that the ring of periods is not a field? | I think the questions were about unconditional proofs or counter examples. I don't have an answer to any of those questions but I think it is still interesting to understand how the yoga of motives suggests natural answers to theses questions. Even though this may seem trivial to people familiar with the subject.
Le... | 24 | https://mathoverflow.net/users/1985 | 105576 | 61,057 |
https://mathoverflow.net/questions/105572 | 19 | What are examples (general features) of the finite groups $G$, such that every irrep (irreducible representation) is contained (as constituent) in the representation induced from trivial representation of some non-trivial subgroup? (I allow subgroups to vary - I mean take all subgroups, induce reps from trivial charact... | https://mathoverflow.net/users/10446 | Finite groups such that every irrep can be induced from trivial irrep of a subgroup ? | EDITED IN RESPONSE TO COMMENTS BY DAVID SPEYER AND F. LADISCH: Examples of finite groups which fail to have the desired property, which are effectively exhaustive, are the finite groups which occur as Frobenius complements. These are the finite groups which admit a (necessarily faithful) representation in which every n... | 16 | https://mathoverflow.net/users/14450 | 105578 | 61,058 |
https://mathoverflow.net/questions/105575 | 17 | Since the work of Serre in the early 50's on homotopy groups of spheres, it is known that the homotopy group $\pi\_k(S^n)$ is finite, except when $k=n$ (in which case the group is $\mathbb{Z}$), or when $n$ is even and $k=2n-1$ (in which case the group is the direct sum of $\mathbb{Z}$ and a finite group). As a consequ... | https://mathoverflow.net/users/349 | Finiteness of stable homotopy groups of spheres | I agree with Ryan that Serre's proof can be viewed as perfectly conceptual, but here is a modern version. Accept from Serre that the homotopy groups of spheres are finitely generated. Let
$k\colon S^n \longrightarrow K(\mathbf{Z},n)$ be the canonical map. We know how to rationalize
spaces and maps. The rationalization... | 26 | https://mathoverflow.net/users/14447 | 105583 | 61,062 |
https://mathoverflow.net/questions/105586 | 1 | A result of Serre-Tate states that we can canonically lift an ordinary abelian variety over a perfect field $k$ of positive characteristic to an abelian scheme over the ring of Witt vectors of $k$ and that we can also lift an endomorphism of the variety uniquely to an endomorphism of the canonical lifting.
In the ca... | https://mathoverflow.net/users/25975 | Serre-Tate canonical lifts for finite fields | As you just said, the canonical lift is an abelian scheme over the ring of Witt vectors $W(k)$. Now, if $k$ is finite of characteristic $p$, $W(k)$ is the ring of integers of the unramified extension (say $K$) of $\mathbb{Q}\_p$ whose residue field is $k$. The canonical lift has a generic fiber which is an abelian vari... | 6 | https://mathoverflow.net/users/2290 | 105589 | 61,063 |
https://mathoverflow.net/questions/105577 | 14 | The analytic rank of the Mordell elliptic curve $y^2=x^3-86069^5$ indicates that it has rank 2. However, deriving a set of generators, and hence the regulator, is proving to be a little bit of an intractable problem.
I'm posting this question in the hope that someone may have investigated this curve already and, as s... | https://mathoverflow.net/users/25232 | Rational Points on $y^2=x^3-86069^5$ | This particular curve, which I'll call $E$, may be quite challenging. The analytic rank is probably 2, but as far as I know, the only way to prove this is to show that the algebraic rank is not 0. Assuming the analytic rank is 2 and the full BSD formula, the product of the Regulator and the order of the Tate-Shafarevic... | 25 | https://mathoverflow.net/users/4872 | 105591 | 61,065 |
https://mathoverflow.net/questions/104965 | 24 | After reading [this MO post](https://mathoverflow.net/questions/104575/under-exactly-what-extra-conditions-if-any-is-a-connected-hausdorff-manifold), I am wondering:
**Is every (connected) Hausdorff Banach manifold a regular space?**
Though unjustified, page 53 of [this paper](http://www.maik.ru/full/rusmath/97/10... | https://mathoverflow.net/users/22971 | Non-regular Connected Hausdorff Banach Manifold | Apparently the answer is **no**, not every connected Hausdorff Banach manifold is regular, not even when it is modeled on a separable Hilbert space.
I quote (verbatim) from J. Margalef-Roig, E. Outerelo-Dominguez, *Differential Topology*, North Holland Mathematics Studies 173, 1992, page 44f.
>
> It is well known... | 17 | https://mathoverflow.net/users/11081 | 105595 | 61,069 |
https://mathoverflow.net/questions/105601 | 11 | Hello,
A flat principal $G$-bundle over $X$ is determined by its holonomies, which are (after picking a trivialization) group homomorphisms $\pi\_1(X)\rightarrow G$. The fiber of the bundle is not canonically identified with $G$, so these maps are only determined up conjugation by $G$. Equivalently these are gauge tr... | https://mathoverflow.net/users/5312 | Characterizing flat 2-connections by their holonomy | Let $\mathcal{P}$ be a principal 2-bundle with structure 2-group a crossed module $t:H \to G$. Then, the holonomy of a connection on $\mathcal{P}$ around a surface $\Sigma$ is a well-defined element $$\mathrm{Hol}\_{\mathcal{P}}(\Sigma) \in H/[G,H],$$ where $[G,H]$ is the normal subgroup of $H$ generated by all element... | 12 | https://mathoverflow.net/users/3473 | 105607 | 61,072 |
https://mathoverflow.net/questions/105596 | 0 | I've posted this already in stats.stackexchange. I'm not sure what the rules are for cross-posting but mathoverflow seems to be more active.
Suppose we have data $x\_i, i=1,2,3,...n$ that are *dependent* and identically distributed with marginal $f(\cdot|\alpha)$. If we model this with the likelihood
$
L = c(F(x\_1... | https://mathoverflow.net/users/8916 | marginal parameter estimation in copula with copula (dependence) parameter known | The maximization by parts algorithm for maximum likelihood estimation is capable of exactly handling that situation
[Peter X.-K Song, Yanqin Fan & John D Kalbfleisch (2005), Maximization by Parts in Likelihood Inference, Journal of the American Statistical Association.](http://pubs.amstat.org/doi/pdf/10.1198/01621450... | 1 | https://mathoverflow.net/users/25326 | 105610 | 61,074 |
https://mathoverflow.net/questions/90527 | 11 | I would like to know the historical reason why the letter H is used for Sobolev spaces. In particular, why not S? It would be interesting to know the same for the letter W.
| https://mathoverflow.net/users/824 | History of Sobolev space notations | The history of both the name "Sobolev space" and the notation (which changed over the years), has been well described by J. Naumann [[link to pdf file](http://edoc.hu-berlin.de/series/mathematik-preprints/2002-2/PDF/2.pdf)]. The history of the name is particularly amusing:
*These spaces, at least in the particular ca... | 12 | https://mathoverflow.net/users/11260 | 105620 | 61,078 |
https://mathoverflow.net/questions/105624 | 8 | Say we are working in a category which has all binary products. I guess that the identity transformation is the only natural transformation from $\times$ to $\times$. Is this really the case? If yes, how can I prove this? If not, does it help to assume that the category is cartesian closed and has all coproducts? What ... | https://mathoverflow.net/users/25527 | Is there only one natural transformation from the product functor to itself? | The key issue is how many natural transformations there are from the identity functor on $C$ to itself. Chris Schommer-Pries observed that there are many such transformations for $C = Vect$, one for each scalar.
A product functor $\prod: C \times C \to C$ is right adjoint to the diagonal functor $\Delta: C \to C \ti... | 20 | https://mathoverflow.net/users/2926 | 105630 | 61,082 |
https://mathoverflow.net/questions/105632 | 3 | Given a binary quadratic form with negative discriminant, such as $3x^{2} + y^{2} = k$, is there an efficient algorithm to compute a value $k$ (or all the values) for which the form has exactly $n$ integer solutions?
| https://mathoverflow.net/users/25996 | Number of solutions of a binary quadratic form. | You should clarify what you mean by "efficient algorithm" and "to compute". In general you can express the number of representations as a linear combination of Dirichlet coefficients of certain Hecke $L$-functions. The Hecke characters are the unramified Hecke characters of $\mathbb{Q}(\sqrt{-d})$, where $-d$ is the di... | 7 | https://mathoverflow.net/users/11919 | 105633 | 61,083 |
https://mathoverflow.net/questions/105612 | 2 | Imagine I have an $(p\_1, ..., p\_N) \in P$ points, on a two-dimensional plane, patterned in a rectangular or hexagonal lattice arrangement in a circle of radius $R\_c$, with a spacing between the points of $r\_s$.
Let $C\_P$ be the centroid of the $P$ points. If I randomly select some subset of $k$ points from $P$,... | https://mathoverflow.net/users/25881 | The distance between the centroid of $P$ points and the centroid of a subset of the points | I might not be understanding the question, but the centroid is just the mean of the sample, so for mildly large samples from a mildly large set it will be (bivariate) normally distributed, and all the statistics are easy to compute.
| 1 | https://mathoverflow.net/users/11142 | 105634 | 61,084 |
https://mathoverflow.net/questions/105474 | 2 | Hello all,
Assume we have a sequence of **quasiconcave** functions (in $X$) denoted by $f\_{i,j}(X)$ for $i,j = 1,\ldots,n$. Denote by $F(X)$ the $n\times n$ matrix whose $(i,j)$ entry is the function $f\_{i,j}(X)$.
Assuming that $F\succ0$ (positive definite for all $X$), I want to prove (or disprove) that the fun... | https://mathoverflow.net/users/25265 | Proving that a specific function is quasiconvex | It's not true.
Consider the $2 \times 2$ matrix $$F(X) = \pmatrix{f(X) & 0\cr 0 & f(X-2)\cr}$$
where $f$ is an even function, everywhere $> 0$, and decreasing on $[0,\infty)$. Take $a = (1,1)^T$.
Then $g(X) = a^T F(X)^{-1} a = 1/f(X) + 1/f(X-2)$.
In particular $g(0) = g(2) = 1/f(0) + 1/f(2)$ while $g(1) = 2/f(1)$.
... | 1 | https://mathoverflow.net/users/13650 | 105638 | 61,087 |
https://mathoverflow.net/questions/105629 | 5 | Hi,
let $T>0$, $\Omega\subset\mathrm{R}^n$ be a bounded smooth domain and suppose
$$u\in L^2(0,T;W^{1,2}(\Omega))\cap L^\infty((0,T)\times\Omega))\ \text{and } \partial\_tu\in L^2(0,T;W^{-1,2}(\Omega)),$$
where $W^{-1,2}(\Omega)=(W^{1,2}\_0(\Omega))^\*$. Is there a result stating that from those regularities one can... | https://mathoverflow.net/users/25995 | Continuity with values in L^2 | In the case $\Omega=\mathbb{R}^n$ we have
\begin{equation}
L^2([0,T];W^{1,2}(\Omega))\cap W^{1,2}([0,T];W^{-1,2}(\Omega))\hookrightarrow BUC([0,T];X),
\end{equation}
where $X$ is given via real interpolation:
\begin{equation}
X=\big(W^{1,2}(\Omega),W^{-1,2}(\Omega)\big)\_{1/2, 2}=B^0\_{2,2}(\Omega).
\end{equation}
Th... | 2 | https://mathoverflow.net/users/8707 | 105640 | 61,088 |
https://mathoverflow.net/questions/105622 | 11 | If $G$ is a finite group and $\lbrace x\_{g} \rbrace\_{g\in G}$ are commuting formal variables, then one can form a matrix whose $(g,h)$ entry is $x\_{gh^{-1}}$. The determinant of this matrix is a polynomial with integer coefficients and is called the *group determinant*. Considering the factorization of this determin... | https://mathoverflow.net/users/6269 | Is there a Dedekind-Frobenius group determinant for infinite groups? | Such a thing is defined in Steve Humphries' very interesting paper:
Humphries, Stephen P.(1-BYU)
Cogrowth of groups and the Dedekind-Frobenius group determinant.
Math. Proc. Cambridge Philos. Soc. 121 (1997), no. 2, 193–217
I am not sure it is as general as you might like, but...
| 7 | https://mathoverflow.net/users/11142 | 105642 | 61,089 |
https://mathoverflow.net/questions/105639 | 6 | I'm looking for an explanation of the following result:
If D is a maximal order in a definite (i.e. ramified at infinity) quaternion algebra B over $\mathbb{Q}$, and $\phi : B\otimes\mathbb{R}\rightarrow\mathbb{H}$ is an isomorphism over $\mathbb{R}$ (where $\mathbb{H}$ is the Hamiltonian quaternions over $\mathbb{R}... | https://mathoverflow.net/users/3345 | Volumes of fundamental domains of maximal orders in definite quaternion algebras over Q | As in the Iwasawa-Tate style treatment of zeta functions of number fields and residue of first pole as volume of idele class group (with the non-compact "ray" removed), this volume is essentially the residue of the leading pole of the zeta function of the quaternion algebra, and this zeta function factors as zeta and a... | 7 | https://mathoverflow.net/users/15629 | 105643 | 61,090 |
https://mathoverflow.net/questions/105387 | 4 | The unitary dual (unitary irreducible represenations) is determined for every connected noncompact semisimple Lie group of real rank one. I would like to have a reference for the particular case $SO\_0(n,1)$. I know the paper of Baldoni Silva-Barbasch ("The unitary spectrum for real rank one", Invent. math. 72) and the... | https://mathoverflow.net/users/20052 | comprehensive presentation of the unitary dual of $SO_0(n,1)$ | Though the notes by Collingwood aren't aimed directly at the unitarity question for groups of real rank one, his survey 9.1 toward the end fills in some of the ideas with a lot of references to the literature up to then. In this treatment your group occurs in the framework of its 2-fold universal covering $\mathrm{Spin... | 2 | https://mathoverflow.net/users/4231 | 105666 | 61,102 |
https://mathoverflow.net/questions/105614 | 9 | I have the following problem: Let $A, B\subset R^3$, $A$ is homeomorphic to a ball, while $B$ is a standard Euclidean ball. Can it happen that the fundamental group of $A\setminus B$ is a perfect group? I am interested in answers for $A$ and $B$ both closed and open, so in fact this is 4 questions.
I am aware of dis... | https://mathoverflow.net/users/25989 | Can the fundamental group of an intersection of a homeomorphic image of a ball with a complement of a ball in $R^3$ be perfect? | Consider a smooth properly embedded surface $P\subset \mathbb{R}^3$. Then $\mathbb{R}^3= X\cup Y$, where $X\cap Y=P$ and $X, Y$ are properly embedded submanifolds with $\partial X=\partial Y=P$. By Mayer-Vietoris, we have an exact sequence $0=H\_2(\mathbb{R}^3)\to H\_1(P)\to H\_1(X)\oplus H\_1(Y)\to H\_1(\mathbb{R}^3)=... | 10 | https://mathoverflow.net/users/1345 | 105672 | 61,105 |
https://mathoverflow.net/questions/105603 | 3 | In his book *On Numbers and Games* (pg. 38) John Horton Conway makes the following remark, "But the collection of all gaps is not even a Proper Class, being an illegal object in most set theories." Is it true that the collection of all gaps is an "illegal object" in most set theories, and in what set theories is this c... | https://mathoverflow.net/users/20597 | A question regarding a remark of John Horton Conway | The gaps in $No$ are discussed in some detail in the Postscript to my paper “The Absolute Arithmetic Continuum and its Peircean Counterpart,” in New Essays on Peirce’s Mathematical Philosophy, edited by Matthew Moore, Open Court Press, 2010, pp. 235-282, which may be downloaded from <http://www.ohio.edu/people/ehrlich/... | 8 | https://mathoverflow.net/users/18939 | 105675 | 61,107 |
https://mathoverflow.net/questions/105684 | 4 | Consider two ultrafilters, $U$ and $V$, on the same cardinal $\kappa$. Let $D(U, V)=\lbrace X\subseteq \kappa: X\in U-V\rbrace$; clearly $D(U, V)$ is a lattice under $\subseteq, \cap, \cup $ since the intersection of two $U$- or $V$-large sets is $U$- or $V$-large, and the union of two $U$- or $V$-small sets is $U$- or... | https://mathoverflow.net/users/8133 | Lattice of differences between ultrafilters | I've got it!
**Theorem.** The lattices of the form
$D(U,V)$ admit a complete classification by the isomorphism classes of $U$ and $V$ and the question of whether $U\neq V$.
The point is that the lattice isomorphism class of $D(U,V)$, when
$U\neq V$, determines and is determined by isomorphism classes of $U$ and $... | 4 | https://mathoverflow.net/users/1946 | 105700 | 61,119 |
https://mathoverflow.net/questions/105697 | 6 | Let $S$ be any positive semi-definite symmetric matrix (Hermitian psd matrices work as well). The **Hadamard inequality** is that
$$\det S\le\prod\_{i=1}^n s\_{ii}.$$
My question is whether there are some other upper bounds of $\det S$ in terms of a partial sum
$$\sum\_{\sigma\in F}\epsilon(\sigma)\prod\_{i=1}^n s\_{i... | https://mathoverflow.net/users/8799 | Hadamard-like inequalites for positive definite symmetric matrices | If I interpret correctly the terms and notations, this seems to be a result of Schur. See p.4 [here](http://math.ecnu.edu.cn/~zhan/papers/ZhanICCM.pdf).
| 9 | https://mathoverflow.net/users/22051 | 105707 | 61,122 |
https://mathoverflow.net/questions/105702 | 3 | Does anyone know a good reference for the constructions of a Greens functions fur the Sturm-Liouville Boundary Value Problem.
| https://mathoverflow.net/users/19874 | Good reference for the construction of a Greens functions fur the Sturm-Liouville | My choice would be **Boundary Value Problems and Green's Functions** by Ivar Stakgold. It have an introduction to *distribution theory* and them apply it to finding Green's functions.
It includes:
* ODE
* PDE with initial conditions
* PDE with boundary conditions.
I found a preview [here](http://cam.ucsd.edu/~mho... | 3 | https://mathoverflow.net/users/25356 | 105709 | 61,124 |
https://mathoverflow.net/questions/105716 | 3 | Let $G(k,n)$ be the Grassmannian of complex $k$-planes in $\mathbb{C}^n$. Then for $k\_1+k\_2=k$ and $n\_1+n\_2=n$, $G(k\_1,n\_1)\times G(k\_2,n\_2)$ is a submanifold of $G(k,n)$. So the cohomology class of it should be written as a linear combination of Schubert classes. Is there a method to compute the coefficients?
... | https://mathoverflow.net/users/11846 | product of two sub-Grassmannians | The subvariety of $k$-planes whose intersection with a fixed $n\_1$-plane has dimension at least $k\_1$ is the closure of a specific Schubert cell. This is the intersection of that subavariety with a similar one, the space of $k$-planes whose intersection with a fixed $n\_2$-plane has dimension at least $k\_2$.
If th... | 4 | https://mathoverflow.net/users/18060 | 105723 | 61,125 |
https://mathoverflow.net/questions/105720 | 5 | Let $C$ and $D$ be plane cubic curves over the real numbers. Suppose that the real loci in the projective plane of $C$ and $D$ consist of two connected components. Denote $C\_0$ and $D\_0$ the bounded component (which is often referred as "oval") of $C$ and $D$, and denote $C\_1$ and $D\_1$ the unbounded component of $... | https://mathoverflow.net/users/26020 | Real intersections of plane cubic curves | The answer is $9$. Choose a curve $C$ with the required topology. Choose $8$ points $x\_1$, $x\_2$, ..., $x\_8$ on $C\_1$. The conditions of passing through the $x\_i$ impose $8$ linear conditions on the $10$ dimensional space of cubics, so we can find a second cubic $E$ passing through the $x\_i$ and not proportional ... | 10 | https://mathoverflow.net/users/297 | 105728 | 61,127 |
https://mathoverflow.net/questions/105727 | 17 | Let $k$ be a field and $q\in k^{\*}$. The quantum plane $k\_{q}[x,y]$ is the algebra $k\langle x,y\rangle/\langle xy=qyx \rangle$ (i.e. the quotient of the free non-commutative $k$-algebra on two variables $x$ and $y$ modulo the ideal given).
>
> Question: For $q,r\in k^{\*}$ and $q\neq r$, when is $k\_{q}[x,y]$ i... | https://mathoverflow.net/users/13215 | Isomorphisms of quantum planes | This answer feels so glib I'm quite worried it's wrong, but anyway:
Write $D\_q$ for the full ring of fractions of $k\_q[x,y]$. By Alev-Dumas, "Sur le corps des fractions de certaines algebres quantiques", Corollary 3.11c, we know that for $q$, $r$ non-roots of unity, $D\_q \cong D\_r$ iff $r = q^{\pm1}$.
It's clea... | 16 | https://mathoverflow.net/users/22460 | 105738 | 61,134 |
https://mathoverflow.net/questions/105687 | 7 | It is well known that the universal covering of a complete Kahler manifold with constant bisectional curvature is $\mathbb{C}^n$, $\mathbb{B}^n$ or $\mathbb{CP}^n$. I need original paper(s) that prove this theorem.
| https://mathoverflow.net/users/15466 | Kahler manifolds with constant bisectional curvature | This is theorem 7.9 in the book of Kobayashi-Nomizu "Foundations of Differential Geometry Vol.II". There the authors attribute it to [Hawley](http://dx.doi.org/10.4153/CJM-1953-007-1) and [Igusa](http://dx.doi.org/10.2307/2372709) independently. These are probably the first papers where this result was proved.
Of cou... | 7 | https://mathoverflow.net/users/13168 | 105743 | 61,135 |
https://mathoverflow.net/questions/105747 | 22 | Let $k$ be an algebraically closed field (in my application, it is characteristic zero, but this probably doesn't matter so much), and let $P: k \to k$, $Q: k \to k$ be polynomials of one variable. Then $P(x)-Q(y)$ is a polynomial of two variables $x,y$. Generically, one expects this polynomial to be irreducible, but t... | https://mathoverflow.net/users/766 | When is P(x)-Q(y) irreducible? | THis is answered in great detail by @quid in [this question.](https://mathoverflow.net/questions/105304/criteria-for-irreducibility-of-polynomial/105323#105323)
| 17 | https://mathoverflow.net/users/11142 | 105750 | 61,139 |
https://mathoverflow.net/questions/105746 | 2 | Context
-------
I'm studying a classical results of Erdos and Lovasz, on colorings of the real line.
The theorem to be proved is as follows:
Let $m, k$ be two positive integers satisfying:
$$e(m(m-1)+1)k\left(1-\frac{1}{k}\right)^m \leq 1$$
Then, for any set $S$ of real numbers with $|S| = m$ (note, $S$, has ... | https://mathoverflow.net/users/26025 | Technique: Compactness => (Finite -> Reals) | Our situation: We have the assertion we want to prove, let us call it $P\_A$ , depending on a set $A$. We know already $A$ is true for all finite $A$ and now want to show it for all infinite sets, too.
[If I understood correctly you are not inetrested in further details for the proof of finite sets, so I say we know it... | 2 | https://mathoverflow.net/users/nan | 105751 | 61,140 |
https://mathoverflow.net/questions/105752 | 6 | For a fixed elliptic curve $E$ over $\mathbb{Z}[1/N]$, is there a non-trivial upper bound in terms of $x$ for the number of primes $p \leq x$ with $p \nmid N$ for which $E(\mathbb{Z}/p^2\mathbb{Z}) \cong (\mathbb{Z}/p\mathbb{Z})^2$? Anything less than $\frac{x}{(\log x)^{2}}$ would be very useful for me.
**Some thoug... | https://mathoverflow.net/users/17907 | E an elliptic curve over Z[1/N], how many p such that E(Z/p^2) = (Z/p)^2? | I don't know how easy it will be to prove anything along these lines, but the condition is that the Serre-Tate parameter of $E$ at $p$ satisfies $q \equiv 1 \mod p^2$. It's natural to assume that $(q-1)/p \mod p$ is random as $p$ varies, just because I know of no reason why it should be otherwise. If that's the case th... | 3 | https://mathoverflow.net/users/2290 | 105759 | 61,143 |
https://mathoverflow.net/questions/105755 | 6 | For a while I've been reading J.E.Humphreys's book "Representations of semisimple Lie algebras in the BGG category $\mathcal O$" under the impression that any [module in $\mathcal O$](http://en.wikipedia.org/wiki/Category_O) has a finite generating set composed of highest weight vectors. Now I've realised that I'm lack... | https://mathoverflow.net/users/19864 | Module in category O not generated by a finite set of HWVs. | This is false. Consider the contragradient dual on a Verma module $V\_\lambda$. This is a module given by linear functions on $V\_\lambda$ which kill all but finitely many weight spaces. This module has no highest weight vectors of weight other than $\lambda$; any non-zero vector $\xi$ of weight $< \lambda$ is non-zero... | 5 | https://mathoverflow.net/users/66 | 105768 | 61,149 |
https://mathoverflow.net/questions/105771 | 10 | By a theorem of Specker, the group $\mathrm{Hom}(\prod\_{\aleph\_0} \mathbb{Z},\mathbb{Z})$ is isomorphic to $\bigoplus\_{\aleph\_0}\mathbb{Z}$ and is in particular a free abelian group. I wonder, if this generalizes to all cardinals:
**Question:** Is it true that $\mathrm{Hom}(\prod\_{\kappa} \mathbb{Z},\mathbb{Z})... | https://mathoverflow.net/users/10194 | Is the dual of the product of infinite cyclic groups a free abelian group ? | If there are no measurable cardinals, or just if there are no measurable cardinals $\leq\kappa$, then the answer to your question is yes, and in fact all homomorphisms from $\prod\_\kappa\mathbb Z$ to $\mathbb Z$ are linear combinations of the $\kappa$ projection maps. If, on the other hand, $\kappa$ or some smaller ca... | 15 | https://mathoverflow.net/users/6794 | 105772 | 61,151 |
https://mathoverflow.net/questions/105769 | 29 | I'm exploring differentiation under the integral sign (I want to be much faster and more assured in doing this common task). So one thing I'm interested in is good counterexamples, where both expressions
$\frac{d}{dx} \int f(x,y)dy$
and
$\int \frac{\partial}{\partial x} f(x,y)dy$
exist at some value of x bu... | https://mathoverflow.net/users/26031 | Counterexamples to differentiation under integral sign? | This is an interesting question... which appears to be about "calculus", but which asks for a better answer than could be given in "calculus". And, in fact, I would advocate turning the question around so that the answer to "Can we interchange?" is "Yes, with suitable interpretation...", rather than "Sometimes, but som... | 18 | https://mathoverflow.net/users/15629 | 105775 | 61,153 |
https://mathoverflow.net/questions/105758 | 9 | In working with the classification of stable vector bundles on $\mathbb{P}^2$, I've found that I need to answer a fairly basic question from analysis/point set topology. Here it is.
Suppose $f:\mathbb{Q}\to \mathbb{Q}$ is
1. strictly increasing,
2. not bounded above or below,
3. a local homeomorphism (with the top... | https://mathoverflow.net/users/7399 | Homeomorphism of the rationals | First construct the Cantor set: An uncountable closed set whose complement is a dense open set containing all the rationals. There are a bunch of ways to do this.
Pull back the regular Cantor set along a homeomorphism $\mathbb R\to \mathbb R$ that sends all the rationals to the rationals not in the Cantor set, possib... | 3 | https://mathoverflow.net/users/18060 | 105780 | 61,156 |
https://mathoverflow.net/questions/105767 | 1 | Hi there,
In my studies I come up with this nonconvex optimization problem
argmin |Ax|\_2+lamda\*|x|\_1 subject to x'x=1
where cost function is nonsmooth but convex and the constrant in nonconvex.
I tries subgradient projection method for convex constraints but the global solution is not my desired solution.
My questio... | https://mathoverflow.net/users/26030 | non convex optimization | You can have a look of these papers：
1. Jonathan H. Manton, Optimization algorithms exploiting unitary constraints.
2. Zaiwen Zai and Wotao Yin, A feasible method for optimization with orthogonality constraints.
Wish these studies can help you.
| 1 | https://mathoverflow.net/users/26009 | 105799 | 61,163 |
https://mathoverflow.net/questions/105803 | 5 | (Berger, 1958) Let M be a closed n-manifold with sec ≥ 1 and injp > π/2 for some p ∈ M, then M is (n − 1)-connected and hence a homotopy sphere.
I don't quite understand the "hence".Must a n-1 connected manifold be a homotopy sphere?
After we get M is n-1 connected,how can we prove M is a homotopy sphere?
| https://mathoverflow.net/users/24637 | Berger sphere theorem | By Hurewicz, (n-1)-connected implies vanishing of the first n-1 homology groups. Since the manifold is closed and (by simple connectedness) also orientable, we have $H\_n={\mathbb Z}$. Of course the higher homology groups vanish. Thus the manifold is a simply connected homology sphere, hence by the converse of Hurewicz... | 10 | https://mathoverflow.net/users/39082 | 105805 | 61,167 |
https://mathoverflow.net/questions/105621 | 2 | Let $C$ be an algebraic curve. One of the easiest examples of stabilty functions is
$$Z:Coh(C)/ \{ 0 \} \rightarrow \overline{\mathbb{H}};\ \ \ \ Z(E):=-deg(E)+i\cdot rk(E).$$
This induces the classical $\mu$-stability on vector bundles of given rank on $C$. I wonder what happens if one modifies this, for instance ... | https://mathoverflow.net/users/4096 | Moduli spaces of vector bundles and stability conditions | First : I don't know how to construct a structure of "moduli space" on the set of
semi-stable objects for a stability condition which is no longer a GIT stability
condition (precisely because then I have no GIT construction).
However, the question has also a sense for the set of semi-stable objects.
The general pict... | 1 | https://mathoverflow.net/users/25309 | 105809 | 61,170 |
https://mathoverflow.net/questions/105790 | 0 | Let $\mu\_{n}$ be the unit measure over $S^{n-1}$,and consider the convolution operator$$Tf=\mu\_{n}\ast f,\quad f\in \mathcal{S}$$
then,it's well-known that T can be extend to a bounded operator on $L^{1}$.My question is whether it's bounded from $H^{1}$ to $H^{1}$ ?
The question is equivallent to say that whether $... | https://mathoverflow.net/users/23078 | Convolution operators defined by compactly supported distribtion | The Hardy space $H^1(\mathbb R^n)$ is equal to
{$u\in L^1(\mathbb R^n),\forall j, R\_ju\in L^1(\mathbb R^n)$},
where
$R\_j$ are the Riesz operators (Fourier multiplier $\xi\_j/\vert\xi\vert$)
and the following norm is equivalent to the $H^1$ norm:
$$
\Vert u \Vert\_{H^1}=\Vert u \Vert\_{L^1}+\sum\_{1\le j\le n}\Vert R... | 3 | https://mathoverflow.net/users/21907 | 105813 | 61,173 |
https://mathoverflow.net/questions/105124 | 12 | Does anyone know of a nice simple example of a space $X$ with an odd-dimensional integral cohomology class $a\in H^{2k+1}(X;\mathbb{Z})$ whose square is nonzero?
I once thought that the one-dimensional generator $a\in H^1(K;\mathbb{Z})\cong\mathbb{Z}$ in the cohomology of the Klein bottle had $a^2\in H^2(K;\mathbb{Z}... | https://mathoverflow.net/users/8103 | Wanted: Odd-dimensional integral cohomology class whose square is nonzero | As Ralph is being modest, I have decided to make his comment into a CW answer.
Recall that the short exact coefficient sequence $0\to \mathbb{Z}\to \mathbb{Z}\to \mathbb{Z}/2\to 0$ leads to a long exact sequence
$$
\cdots \to H^\ast(X;\mathbb{Z})\to H^\ast(X;\mathbb{Z})\stackrel{\rho}{\to} H^\ast(X;\mathbb{Z}/2)\stac... | 9 | https://mathoverflow.net/users/8103 | 105817 | 61,175 |
https://mathoverflow.net/questions/105806 | 13 | It is well-known that the total space of the cotangent bundle $T^\*X$ of a given smooth manifold $X$ admits a [symplectic form](http://en.wikipedia.org/wiki/Symplectic_manifold) $\omega$. It is actually exact: $\omega=d\lambda$. The $1$-form $\lambda$ is called the [Liouville form](http://en.wikipedia.org/wiki/Liouvill... | https://mathoverflow.net/users/7031 | What structure on the second order cotangent bundle ? | It's not completely clear to me what form of answer you would accept. In one sense, the answer is 'the structure on $T^\ast\_2M$ is the pseudogroup structure that is induced by prolongation of the pseudogroup $\mathrm{Diff}(M)$, but this kind of tautological answer is not useful.
If you are asking how one would *char... | 14 | https://mathoverflow.net/users/13972 | 105829 | 61,182 |
https://mathoverflow.net/questions/105824 | 9 | The question does not mean sphere eversion is intuitive to me! In fact, it is just the opposite and that is the purpose of this question.
Recently, I was reading about [Smale's paradox](http://en.wikipedia.org/wiki/Smale%27s_paradox), the problem of sphere eversion (turning a sphere inside out). The wiki article is ... | https://mathoverflow.net/users/7333 | What is the 'non-intuitive' part in sphere eversion (turning inside out)? | Watch [Outside In](http://www.youtube.com/watch?v=wO61D9x6lNY&feature=gv) (something we should all do anyway, to commemorate Bill Thurston's passing).
To understand the mathematics behind sphere eversions, you should first get a good intuition for the concepts of [immersion](http://en.wikipedia.org/wiki/Immersion_(ma... | 15 | https://mathoverflow.net/users/8103 | 105830 | 61,183 |
https://mathoverflow.net/questions/105753 | 17 | Hey
Is there a universal property for the smash product (of pointed spaces or pointed CW-complexes or something of that ilk)? I've seen the smash product of spectra defined with a universal property in terms of the smash product of pointed-spaces, but I was wondering if there was just some simple universal property y... | https://mathoverflow.net/users/24021 | Universal Property of the Smash Product (of pointed spaces) | $X \wedge Y$ represents maps from $X \times Y$ that are base-point-preserving separately in each variable, just as the tensor product represents maps that are linear separately in each variable.
| 30 | https://mathoverflow.net/users/1100 | 105833 | 61,184 |
https://mathoverflow.net/questions/105851 | 4 | Let $\mathfrak A = (A, \cdot)$ be a semigroup (written multiplicatively). We say that $\mathfrak A$ is *linearly orderable* if there exists a total order $\le$ on $A$ such that $ac < bc$ and $ca < cb$ for all $a,b,c \in A$ with $a < b$ (note strict inequalities).
Some examples of linearly orderable semigroups are: th... | https://mathoverflow.net/users/16537 | Strictly totally ordered semigroups - Looking for references | Corollary 3.4 of Gilmer's book on commutative semigroup rings states a commutative semigroup is totally orderable (in your sense) iff torsion-free and cancellative.
| 3 | https://mathoverflow.net/users/15934 | 105875 | 61,203 |
https://mathoverflow.net/questions/105868 | 8 | Let $f(x)$ be a continuous probability distribution in the plane. It is obvious that if $X$ and $X'$ are two independent random samples from $f$, then $\mathbf{E}(\|X - X'\|) \leq 2 \mathbf{E}(\|X\|)$ by the triangle inequality. Can this upper bound be made tighter if we assume that $f$ is rotationally symmetric about ... | https://mathoverflow.net/users/26064 | Expected distance between two points in the plane | Using the pareto distribution $f(x) = \frac{\alpha}{x^{\alpha+1}}$ ($x > 1$) , the ratio $\frac{E(||X-X'||)}{E(||X||)}$ approaches a $2$ as $\alpha$ tends to 1.
To find such a distribution, consider that all else equal, you want to maximize the difference $||X||-||X'||$ since the angle between the two is independent... | 5 | https://mathoverflow.net/users/8737 | 105879 | 61,206 |
https://mathoverflow.net/questions/105885 | 2 | I have the following question,
Is it possible to get somehow a compact Hausdorff space $X$ which is second-countable from a unital commutative C\*-algebra. If it is possible, what should we assume for our C\*-algebra.
Gelfand-Naimark theorem gives us $C(X)$, where $X$ is a compact Hausdorff space, but I'm asking how... | https://mathoverflow.net/users/16872 | Assumptions on a commutative C*-algebra to get a nice C(X) - space | The space $X$ is second countable if and only if $C(X)$ is separable for the norm. This is proved, for example, as Theorem 2.4 of the little article ["Notes on the Separability of C\* algebras" by Chun-Yen Chou](http://journal.taiwanmathsoc.org.tw/index.php/TJM/article/view/1786). Actually, the short proof given there ... | 4 | https://mathoverflow.net/users/6269 | 105890 | 61,212 |
https://mathoverflow.net/questions/105870 | 9 | Short version:
>
> Over which fields is the (appropriate version of the) "Sylvester law of inertia" valid?
>
>
>
Long version:
Let $V$ be a finite dimensional vector space over the field $\Bbbk$ of characteristic different from $2$ (so quadratic forms are the same as symmetric bilinear forms). Consider the ... | https://mathoverflow.net/users/4721 | Over which fields is the Sylvester law of inertia valid? | OK, I now have a complete answer; I'll delete the other shortly. The answer to question 2 is yes and easily so; I wonder if I am missing something. If one quadratic form is $\sum a\_i x\_i^2$, and other is $\sum b\_i y\_i^2$, and $a\_i = b\_i u\_i^2$ for $u\_i \neq 0$, then clearly the two quadratic forms are equivalen... | 7 | https://mathoverflow.net/users/297 | 105892 | 61,214 |
https://mathoverflow.net/questions/105893 | 3 | Let $M$ be a meet-semilattice with a least element $0$. Suppose there is an order-reversing involution $a \mapsto -a$ on $M$ such that for all $a, b \in M$, $a \wedge b = 0$ if and only if $b \le -a$. Then $M$ is a Boolean algebra.
Is there a paper/book I can cite for this claim? The proof I have is just ploughing th... | https://mathoverflow.net/users/4053 | A characterisation of Boolean algebras | I have a pretty simple proof. As Tom has already observed, we have a bounded lattice. Now one characterization of Boolean algebra is a (bounded) lattice such that for every element $a$ there is an element $-a$ such that
$$a \wedge b \leq c \qquad iff \qquad a \leq -b \vee c$$
In category-speak, notice that the po... | 5 | https://mathoverflow.net/users/2926 | 105901 | 61,220 |
https://mathoverflow.net/questions/105904 | 8 | First let $L^{\bullet}$ be a pro-nilpotent differential graded Lie algebra (dgla). We have the set of Maurer-Cartan elements in $L^{\bullet}$ ($MC(L^{\bullet})$) which are $\alpha \in L^1$ such that it satisfies the Maurer-Cartan equation
$$
\partial \alpha+ \frac{1}{2}[\alpha,\alpha]=0.
$$
We have a definition of gaug... | https://mathoverflow.net/users/24965 | How to define the equivalence of Maurer-Cartan elements in an $L_{\infty}$-algebra? | To add a bit to what Damien says, addressing your question on how to generalise the gauge approach (which is equivalent to the approach outlined by Damien, as proved by several people):
You can view gauge symmetries in DGLAs via solving the differential equation
$$
\frac{d\alpha}{dt}=-\partial\xi-[\alpha,\xi],
$$
w... | 9 | https://mathoverflow.net/users/1306 | 105911 | 61,224 |
https://mathoverflow.net/questions/105878 | 23 | I am studying orbifold fundamental group (or more generally orbifold homotopy groups). In a nutshell, my questions is: what are they intuitively? In what follows I give definitions and more precise questions. My definition of orbifold fundamental group is via classifying space of groupoid, which is explained in the nex... | https://mathoverflow.net/users/25713 | How should one understand orbifold fundamental groups? | The definition of an orbifold in terms of a groupoid is flexible and technically useful and gives very clean definitions, but it's not so close to geometric intuition. Perhaps it is once you've mastered the art of thinking simplicially, which I probably haven't. I tend to think of orbifolds like this: the simplest orbi... | 27 | https://mathoverflow.net/users/1310 | 105913 | 61,226 |
https://mathoverflow.net/questions/105907 | 0 | I am trying to understand what would be the logical reasons behind our assumption that our physical space is equivalent to $\mathbb{R}^3$ or 'physical straight line' is equivalent to $\mathbb{R}$.
$\mathbb{R}$ is a basically an algebraically constructed set, which is nothing but the completion of $\mathbb{Q}$, the s... | https://mathoverflow.net/users/23980 | Why is the physical space equivalent to $\mathbb{R}^3$ | this is basically a question on the granularity of space, which is an active topic of research in physics: space appears to be continuous, but does it actually come in discrete chunks on some very small length scale (Planck length)? there are some attempts to formulate (quantum) mechanics in discrete space-time; loop-q... | 2 | https://mathoverflow.net/users/11260 | 105915 | 61,227 |
https://mathoverflow.net/questions/105914 | 0 | Context
-------
I'm working on a problem involving Lovasz Local Lemma, for proving that there exists a graph with a certain property.
What I need to prove:
---------------------
There exists some constant $c$, and functions $p,a$ (which can depend on $n$) s.t.
$$ \frac{p^3}{a} \leq \left(1- \left[\frac{1-p}{(1-... | https://mathoverflow.net/users/26075 | How to Rigorize an inequalities argument | Take, say, $a=2p^3$, $p=1/2\sqrt{n}$. Then
$$(1-p)/(1-a)^n=\exp(-p+na+O(p^2+na^2))=\exp(-1/4\sqrt{n}+O(1/n)).$$
You need lower estimate of the RHS, hence upper estimate of $(1-p)/(1-a)^n$, you mau use $\exp(-1/5\sqrt{n})$ for large enough $n$. Next, use an estimate like $1-p\geq e^{-2p}$ for $0 < p < 1/100$, and for ... | 0 | https://mathoverflow.net/users/4312 | 105924 | 61,232 |
https://mathoverflow.net/questions/105922 | 16 | The Cuntz algebra $\mathcal{O}\_{\infty}$ is the universal $C^\*$-algebra generated by countably infinitely many isometries $s\_i$ satisfying the relations $s\_i^\*s\_j = \delta\_{ij}$ (there is no condition about the sum $\sum\_i s\_is\_i^\*$ in this case, since the sum would be infinite).
There is another way to d... | https://mathoverflow.net/users/3995 | Von Neumann algebra associated to the infinite Cuntz algebra | The von Neumann algebra $M$ generated by $\mathcal O\_\infty$ is all of $B(\mathcal F(H))$.
Indeed, if $a$ belongs to its commutant, let me prove that $a$ is a multiple of the identity. First since for all $v \in H$, $s\_v^\* (a \Omega)= a (s\_v^\* \Omega)=0$, we have that $a \Omega=\lambda \Omega$ for some $\lambda... | 21 | https://mathoverflow.net/users/10265 | 105930 | 61,236 |
https://mathoverflow.net/questions/105781 | 8 | The Jacobson-Morozov theorem that any nilpotent element $e$ in the Lie algebra of a simple algebraic group $G$ can be embedded in an $\mathfrak{sl}\_2$-triple, has a restriction (in terms of the Coxeter number) on the characteristic of the underlying field (assumed to be algebraically closed). This restriction is also ... | https://mathoverflow.net/users/26032 | Failure of Jacobson-Morozov in positive characteristics | The uniqueness can break down very badly in positive characteristic. Supose $G=SL\_p$ where $p$ is the characteristic of the base field. Take a regular nilpotent element $e$ in $\mathfrak{g}=\mathfrak{sl}\_p$. Then there is a nilpotent element $f\in\mathfrak{g}$ such that $e$, $f$ and $h=[e,f]$ form an $\mathfrak{sl}\_... | 17 | https://mathoverflow.net/users/24386 | 105935 | 61,237 |
https://mathoverflow.net/questions/105918 | 1 | Let $g:[0,\infty) \rightarrow \mathbb{R}$ be an increasing function. Is there a way to construct an entire function $f(z)$ such that $f(x)=g(|x|)$ for all real $x$?
| https://mathoverflow.net/users/10035 | Real function to entire functions | You should keep in mind analytic continuation. Let $f:\mathbb R\rightarrow\mathbb R$
be a $C^\infty$ function ; if there exists an entire function extending $f$, then
(1) It is unique by analytic continuation,
(2) The function $f$ must be real-analytic.
Of course (2) is not sufficient: think about the real-analy... | 5 | https://mathoverflow.net/users/21907 | 105936 | 61,238 |
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