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https://mathoverflow.net/questions/127818 | 1 | Consider the following controllable and observable linear system
$$\dot x=Ax+Bu, y=Cx,$$
where $x\in \mathbb{R}^n, u\in \mathbb{R}^m, y\in \mathbb{R}^p$.
The observability of $(A,C)$ tells us that Cx(t)=0 implies x(t)=0 for any trajectory of the system. Considering this, I have the following questions:
1. Why not de... | https://mathoverflow.net/users/23540 | Control a linear system to the kernal space of the output matrix. | In linear geometric control theory, the problem you posed is called the *output stabilization problem (OSP)*. See Wonham's book on linear geometric control.
The approach to solving OSP, conceptually, is as follows.
Consider the LTI system: $\dot x = A x + B u, y=Cx$. We wish to design a state feedback such that for... | 3 | https://mathoverflow.net/users/38433 | 139047 | 76,135 |
https://mathoverflow.net/questions/138958 | 4 | A quick look at the [wikipedia article](http://en.wikipedia.org/wiki/Partition_%28number_theory%29) on partitions of $n \in \mathbb{N}$ shows that the number of ordered partitions is $2^{n-1}$, and the number of unordered partitions is asymptotically $ \sim \frac{1}{4n\sqrt{3}} e^{\pi \sqrt{\frac{2n}{3}}}$.
We can w... | https://mathoverflow.net/users/20062 | Which unordered partition of $n$ gives rise to the largest number of ordered partitions? | We will prove that $\lim\_{n\to\infty} a\_i^\*/n = 1/2^{i+1}$.
Also let $s\_\*=\sum a\_i^\*$
To do this, note that if $a^\*$ is optimal, adding $1$ to $a\_i$ and $a\_j$ and subtracting one from $a\_{i+j}$ must reduce the number of unordered partitions, so
$$ \frac{ (s^\*+1) (a\_{i+j} ^\*)} { (a\_i^\*+1)(a\_j^\*+1... | 8 | https://mathoverflow.net/users/18060 | 139050 | 76,137 |
https://mathoverflow.net/questions/139036 | 6 | If $d(n)$ denotes the number of divisors of $n \in \mathbb{N}$, we may define the function $$C(n) = \frac{\log(d(n)) \cdot \log(\log n)}{\log 2 \log n}.$$
According to Wikipedia, the Swedish mathematician Carl Severin Wigert proved that $\displaystyle \limsup C(n) = 1.$ $C$ can therefore reasonably be used as a meas... | https://mathoverflow.net/users/26327 | "Ultracomposite" numbers | EDIT, 11:36 pm. The conjecture is correct. From the first result of Nicolas and Robin,
$$ C\_2(n) \leq 1.53793986 + \frac{0.56367483}{\log \log n}. $$
This bound decreases with $n.$ Furthermore this gives
$ 1.743557976 $ when $n = 5433960.$ This says that we need only confirm the value of the ratio $C\_2(n)$ for $n \le... | 7 | https://mathoverflow.net/users/3324 | 139052 | 76,138 |
https://mathoverflow.net/questions/139022 | 6 | This is a repost of [this](https://math.stackexchange.com/questions/458780/probability-that-a-random-edge-coloring-of-the-complete-graph-is-proper) math.se question that I am posting here since it received no attention there.
>
> What is the probability that a random edge coloring of $K\_n$ with
> $m \geq n$ col... | https://mathoverflow.net/users/1737 | Probability that a random edge coloring of the complete graph is proper | Let $p(n)$ be this probability. Now consider you coloured $K\_{n-1}$ successfully and add one new vertex. As the edges from the new vertex is coloured, there are simple bounds on how many colours are available. I get that the $i$-th new edge ($i$ starting at 0) has between $m-n+2-i$ and $m-n+2$ colours available. There... | 3 | https://mathoverflow.net/users/9025 | 139059 | 76,140 |
https://mathoverflow.net/questions/138979 | 4 | I am trying to pin down: who first proved that [$\mathsf{WKL}\_0$](http://en.wikipedia.org/wiki/Reverse_mathematics#Weak_K.C3.B6nig.27s_lemma_WKL0) has an $\omega$-model in which every set is of [low degree](http://en.wikipedia.org/wiki/Low_%28computability%29)? As shown in Simpson's *Subsystems of Second Order Arithme... | https://mathoverflow.net/users/4137 | Who first proved there's an $\omega$-model of $\mathsf{WKL}_0$ in which all sets are low? | The general issue here is that there are some results in Reverse Mathematics that are very little besides rephrased computability theory results. In such cases, it is common for nobody to take credit for the Reverse Mathematics result.
Another example of this is the theorem that $\mathsf{RT}^3\_2$ implies $\mathsf{A... | 4 | https://mathoverflow.net/users/5442 | 139069 | 76,145 |
https://mathoverflow.net/questions/139009 | 11 | Let $H^\*$ and $K^\*$ be two cohomology theories, and $X$ a reasonable spectrum. Here, I'm thinking that $H^\*$ is singular cohomology (and for my purposes, rational cohomology will suffice), and $K$ is a Morava K-theory.
Suppose that I understand $H^\*(X)$ and $K^\*(X)$. Is there any way to use this (or other!) info... | https://mathoverflow.net/users/4649 | Homology of localisations of spectra | I'm going to phrase in terms of $H$-homology instead of cohomology.
* If $H$ is a finite spectrum, smashing with it always commutes across the limit. More generally, if you express $H$ as a (homotopy) colimit of finite spectra -- eg, by taking a cellular filtration -- then you get a directed system of spectral sequen... | 8 | https://mathoverflow.net/users/360 | 139075 | 76,148 |
https://mathoverflow.net/questions/139090 | 3 | Why not just have arrows in the category of opens represent coverings instead of inclusions?
It seems to me like both conventions (whether presheaves are co/contra and which of the two dual orderings to use to represent spaces as thin categories) are pretty arbitrary, and switching them would simplify notation.
Is ... | https://mathoverflow.net/users/38334 | Why are (pre)sheaves defined as contravariant functors? Why not just reverse the arrows in the first place? | The category of open subsets of a space $X$ is a subcategory of the category of spaces $Y \to X$ over $X$, so the usual direction of arrows is natural in this sense.
Constructing the category of presheaves $\text{Psh}(C)$ on a category $C$ is a covariant construction. Moreover, it is the [free cocompletion](http://n... | 7 | https://mathoverflow.net/users/290 | 139091 | 76,152 |
https://mathoverflow.net/questions/139093 | 7 | Question [Is there any forcing free proof for hard independence results?](https://mathoverflow.net/questions/138790/is-there-any-forcing-free-proof-for-hard-independence-results#comment358616_138790) talks about the use of forcing for independence results such as: $Con(ZFC)\longrightarrow Con (ZFC+\neg CH) $. For that ... | https://mathoverflow.net/users/38783 | Does forcing generally go one way? | There is a wide variety of statements $S$ for which forcing can be used to prove the (relative) consistency of both $S$ and $\lnot S$ with $\mathsf{ZFC}$.
For example, Cohen developed forcing to prove the consistency of $\lnot\mathsf{CH}$, but we can also prove $\mathsf{CH}$ consistent this way: We can add to the uni... | 14 | https://mathoverflow.net/users/6085 | 139094 | 76,154 |
https://mathoverflow.net/questions/131562 | 2 | There are several occasions in the study of dynamical systems that are called phase transitions. For example consider a homeomorphism $f:X\to X$ and a potential function $\phi\in C(X,\mathbb{R})$. Consider the topological pressure $P(f,t\phi)$, where the parameter $t$ maybe viewed as a inverse-temperature of $(X,f,\phi... | https://mathoverflow.net/users/11028 | Phase transition in dynamical systems | Hermann Haken in his theory of Synergetics discussed over years the huge variety of phase transitions and particularly the non-linear phase transition phenomena. His series with Springer publishing on synergetics (<http://www.springer.com/series/712>) brings extensive literature on this matter. He approached phase tran... | 1 | https://mathoverflow.net/users/32300 | 139099 | 76,157 |
https://mathoverflow.net/questions/32899 | 3 | For a monoid $M$ and a subset $S$ of $M$, define the *syntactic congruence* $\equiv\_S$ of $S$ as the least congruence on $M$ that saturates $S$, i.e. :
$$u \equiv\_S v \Leftrightarrow (\forall x, y)[xuy \in S \leftrightarrow xvy \in S].$$
Now define the *Nerode equivalence* as the following right congruence :
$$u \s... | https://mathoverflow.net/users/7687 | What is the relation between the number syntactic congruence classes, and the number of Nerode relation classes? | Let $L$ be a regular language. Then the index of the Nerode equivalence of $L$ is the number of states $n$ of its minimal automaton and the index its syntactic congruence is the size $N$ of its syntactic monoid.
It is well known that $N \leqslant n^n$ and that this bound can be reached on a three-letter alphabet. I ... | 2 | https://mathoverflow.net/users/38236 | 139107 | 76,160 |
https://mathoverflow.net/questions/139109 | 3 | We're interested in recursive predicates $P(n)$ with RE range $R$ and non-RE complement $R^\prime$. For various $n \in R^\prime$ we may be able to prove that $n \in R^\prime$. For instance, if $P$ is the halting problem, then we can build a non-halting algorithm, figure out its index $n$, and we then know that $n \in R... | https://mathoverflow.net/users/16503 | Membership Provability in co-RE Sets | This clearly depends on what you mean by "provably."
Under one reasonable interpretation, the answer is very much "yes." The set $S\_T$ of theorems of a (consistent, recursively axiomatizable, extending $PA$) theory $T$ is r.e., but $T$ cannot prove that the complement of $S\_T$ is nonempty (Goedel's Theorem), much l... | 7 | https://mathoverflow.net/users/8133 | 139110 | 76,161 |
https://mathoverflow.net/questions/138875 | 37 | I am trying to understand why induction up to exactly $\epsilon\_0$ is necessary to prove the cut-elimination theorem for first-order Peano Arithmetic; or, as I understand, equivalently, why the length of a PA-proof with all cuts eliminated grows (in the worst case) as fast as $f\_{\epsilon\_0}$ in the fast-growing hie... | https://mathoverflow.net/users/38330 | Why do stacked quantifiers in PA correspond to ordinals up to $\epsilon_0$? | **8/14: Substantially edited in response to comments: added to 1st part, added new 2nd and 4th parts**
*There's also some discussion underneath, and a link to a [partial write-up](http://www.sas.upenn.edu/~htowsner/AckermannCutElimination.pdf) of a case of cut-elimination involving the Ackermann function*
If you ha... | 21 | https://mathoverflow.net/users/8991 | 139127 | 76,169 |
https://mathoverflow.net/questions/139133 | 5 | Let $Y$ be a smooth cubic 4-fold in $\mathbf{P}^5$. The derived category of $Y$ admits a semiorthogonal decomposition
$$D^b(Y) = \langle \mathcal{A}\_Y, \mathcal{O}\_Y, \mathcal{O}\_{Y}(1), \mathcal{O}\_{Y}(2) \rangle,$$
where $\mathcal{A}\_Y$ is the right-orthogonal to the triangulated subcategory generated by $\... | https://mathoverflow.net/users/38471 | Serre functor of orthogonal subcategory of derived category of cubic 4-fold | The first is a straightforward computation. For example, to compute $O^2$ you note that the convolution of kernels preserves exactness of the triangles, hence one has distinguished triangles
$$
\Delta\_{Y\*}O\_Y(1) \circ K\_1 \to O\_Y(2)\boxtimes O\_Y \to \Delta\_\*O\_Y(2),
$$
$$
(O\_Y(1)\boxtimes O\_Y)\circ K\_1 \to V... | 7 | https://mathoverflow.net/users/4428 | 139138 | 76,174 |
https://mathoverflow.net/questions/138991 | -1 | Let consider a smooth projective curve $X$ over $\mathbb{C}$. We consider the scheme that classifies effective divisors of degree $d$, which is isomorphic to $X^{d}/S\_{d}$ where $S\_{d}$ is the symmetric group.
We consider the map which associates to an effective divisor D of degree $d$
the sum of its multiplicities... | https://mathoverflow.net/users/27398 | effective divisors on a curve and upper semi-continuity | The space $X^d/S\_d$ has a stratification with strata $X\_\lambda$ indexed by partitions $\lambda$ of $d$. The stratum $X\_\lambda$ is in the closure of the stratum $X\_\mu$ if and only if $\lambda$ is a subpartition of $\mu$ (i.e. one can split the summands of $\lambda$ into groups such that each summand of $\mu$ is a... | 1 | https://mathoverflow.net/users/4428 | 139154 | 76,182 |
https://mathoverflow.net/questions/139159 | 8 | It is known that a group $G$ acts geometrically on $\mathbb{H}^2$ if and only if $G$ is word-hyperbolic and its boundary $\partial G$ is homeomorphic to $S^1$.
The analogous statement for $\mathbb{H}^3$ and $S^2$ is open and is a conjecture of Cannon.
I read somewhere that this fails in higher dimensions, but I ca... | https://mathoverflow.net/users/38423 | Counterexamples to analogue of Cannon conjecture in higher dimensions | There are various compact manifolds of negative curvature which are not homnotopy-equivalent to closed hyperbolic manifolds: Locally symmetric ones (complex hyperbolic, etc) as well as Gromov-Thurston and Mostow-Siu examples. Their $\pi\_1$'s are Gromov-hyperbolic, boundary is a topological sphere. The examples exist i... | 10 | https://mathoverflow.net/users/21684 | 139164 | 76,185 |
https://mathoverflow.net/questions/139131 | 4 | I'm trying to prove the following:
>
> Let $L$ be a positive holomorphic line bundle on a compact complex manifold $X$. For any hermitian holomorphic vector bundle $E$ on $X$, there is $k \in \mathbb{N}$ such that $E\otimes L^k$ is Nakano positive.
>
>
>
I'm using the decomposition given by diverietti in [this... | https://mathoverflow.net/users/21564 | If $L$ is positive, $E\otimes L^k$ is Nakano positive for some $k$ | This follows by the following general, quite elementary
**Fact.** *Suppose you have a complex vector bundle $E\to X$ on a compact manifold $X$ and two hermitian form $h\_1$ and $h\_2$ on $E$, such that the first one is positive definite. Then, there exists a constant $C\_0>0$ such that for all $C\ge C\_0$ the hermit... | 3 | https://mathoverflow.net/users/9871 | 139165 | 76,186 |
https://mathoverflow.net/questions/138275 | 37 |
>
> Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$, and set $K=\sum\_i\sqrt{X(i)Y(i)}$ so that $Z:=\frac{1}{K}\sqrt{XY}$ is also a probability measure on $\{1,2,\dots,n\}$. How can we prove the inequality
> $$H(X)+H(Y)\geq 2K^2 H(Z),$$ where $H(X)=-\sum\_{i=1}^n X(i)\log X(i)$ is the [entropy function](http:/... | https://mathoverflow.net/users/12176 | An entropy inequality | OK, having spent about 20 hours on the search of a nice proof (which extinguished my passion for beauty for the next several days at least), I'm resorting to the brute force. I will love to see someone else to avenge this pitiful defeat of mine...
As I mentioned in the comment, the key to the solution is the inequal... | 19 | https://mathoverflow.net/users/1131 | 139166 | 76,187 |
https://mathoverflow.net/questions/139158 | 6 | I am interested in the growth rate of this type of group: $G=\mathbb{Z}^2\rtimes\_{\sigma} \mathbb{Z}$, where $\sigma(a)=\begin{pmatrix}x&y\\z&w\end{pmatrix}\in SL\_2(\mathbb{Z})$, where $a$ is the generator on the right copy of $\mathbb{Z}$ and the action is just by matrix multiplication.
Here are two examples:
Fo... | https://mathoverflow.net/users/9305 | growth rate of $\mathbb{Z}^2\rtimes_{\sigma} \mathbb{Z}$? | To answer your 1st question see the paper of Milnor, "Growth of finitely generated solvable groups."
To answer your 2nd question, simply generalize your second example to higher dimensions, e.g. take $\mathbb{Z}^3\rtimes\_{\sigma} \mathbb{Z}$ where $\sigma \in SL\_3(\mathbb{Z})$ has an eigenvalue not on the unit cir... | 4 | https://mathoverflow.net/users/20787 | 139168 | 76,189 |
https://mathoverflow.net/questions/139169 | 2 | I was asked to prove that if
$$ T\_{n}^{+}(\mathbb{R}) \subseteq M\_{n}(\mathbb{R})$$
denotes the set of upper triangular matrices with positive diagonal entries, then prove that the multiplication map
$$ \mu : O\_{n}(\mathbb{R}) \times T\_{n}^{+}(\mathbb{R}) \rightarrow GL\_{n}(\mathbb{R})$$
is a homeomorphism... | https://mathoverflow.net/users/36035 | Iwasawa Decomposition for Matrices | There is an elementary argument for the Iwasawa decomposition for classical groups, and it is especially simple for $G=GL\_n(\mathbb R)$: use the fact that $K=O(n,\mathbb R)$ is transitive on the set of vectors in $\mathbb R^n$ of a given length, by rotating. Thus, given $g\in G$, first left-multiply by an element to m... | 6 | https://mathoverflow.net/users/15629 | 139178 | 76,193 |
https://mathoverflow.net/questions/139147 | 2 | Let $p$ be a rational prime number, and let $K$ be a finite extension of $\mathbb Q\_p$. Let $A$ be an abelian variety over $K$. For any rational prime number $\ell$, let $K(A[\ell])$ be the field of definition of the $\ell$-torsion points of $A$.
Suppose $A$ and $A'$ are $K$-isogenous abelian varieties over $K$.
... | https://mathoverflow.net/users/38481 | Division fields of isogenous abelian varieties over a p-adic field | Let $f:A\to A′$ be an isogeny over $K$. Then there exists an isogeny $g:A'\to A$ such that the composite of $f$ with $g$ is multiplication by the degree of $f$ (see any book on abelian varieties). It follows that, for every prime $l$ not dividing its degree, $f$ defines an isomorphism from $A[l]$ onto $A'[l]$. This iso... | 4 | https://mathoverflow.net/users/36469 | 139184 | 76,197 |
https://mathoverflow.net/questions/139201 | 4 | Consider the space $S$ of smooth free simple (non-self-intersecting) loops in the plane $\mathbb{R}^2.$ By Grayson's theorem, $S$ is connected, but is more known about its topology? Is it known to be contractible (or known NOT to be contractible)? By contrast, the set of self-intersecting loops in not connected, but of... | https://mathoverflow.net/users/11142 | space of simple loops in the plane | The space of smooth embeddings $S^1 \to \mathbb R^2$ has the homotopy-type of $O\_2$. Denote this embedding space by $Emb(S^1,\mathbb R^2)$.
The proof goes like this. Let $Emb(D^2, \mathbb R^2)$ be the space of smooth embeddings of the 2-disc in $\mathbb R^2$. There is a locally trivial fibre bundle (due to Palais) g... | 6 | https://mathoverflow.net/users/1465 | 139203 | 76,206 |
https://mathoverflow.net/questions/139170 | 1 | Let's define a graph manifold as 3-manifold which is obtained by plumbing of a circle bundle over $\Sigma \_g$ (with Euler number 0) with a circle bundle over $\Sigma\_h$ (with Euler number k). Is there any way to find an open book decomposition of this manifold explicitly?(I think this is the same as asking how to swi... | https://mathoverflow.net/users/31475 | open book decomposition of graph manifolds | One can produce open books for graph manifolds in basically the same way that one produces an open book when cabling a fibered knot. Let's begin there.
With a fibered knot you've got a torus $T$ that separates a $D^2 \times S^1$ neighborhood of the knot from a surface bundle over the circle. Near $T$, these both look... | 2 | https://mathoverflow.net/users/31434 | 139204 | 76,207 |
https://mathoverflow.net/questions/139175 | 18 | A $L\_\infty$-algebra can be defined in many different ways. One common way, that
gives the 'right' kind of morphisms, is that a $L\_\infty$-algebra is a graded cocommutative and coassociative coalgebra, cofree in the category of *locally nilpotent* differential graded coalgebras and their morphisms are coalgebra
morp... | https://mathoverflow.net/users/21965 | What is a homotopy between $L_\infty$-algebra morphisms | One standard answer\*, in which any reasonable (characteristic $0$ — I haven't thought about any other case) algebraic category can be given a simplicial structure, is the following.
Let $\mathbb Q[\Delta^k] = \mathbb Q[t\_0,\dots,t\_k,\partial t\_0,\dots,\partial t\_k] / \bigl\langle \sum t\_i = 1,\ \sum\partial t\_... | 12 | https://mathoverflow.net/users/78 | 139210 | 76,209 |
https://mathoverflow.net/questions/139207 | 1 | Suppose $Y\_1, Y\_2$ are independent real-valued random variables and whose distributions have a density with respect to the Lebesgue measure. Assume that all moments $\mathbb{E}Y\_1^a, \mathbb{E}Y\_2^b$ exist and are finite. Let $p(y\_1,y\_2)$ be any polynomial whose degree with respect to $y\_1$ is positive.
Is it... | https://mathoverflow.net/users/7576 | On polynomial functions of random variables and independence | Your argument can be extended to any polynomial.
Let $q(y\_1,y\_2)=(p(y\_1,y\_2))^2$. Then the leading coefficient in $q$, as a polynomial in $y\_1$ is a polynomial $r(y\_2)$ which is a square. Since $Y\_2$ is not supported on $r^{-1}(0)$ (which is finite), we have $\mathbb{E}\_{Y\_2}[r(Y\_2)]>0$.
In particular, th... | 1 | https://mathoverflow.net/users/1061 | 139218 | 76,213 |
https://mathoverflow.net/questions/130992 | 3 | It has been asked [here](https://mathoverflow.net/questions/9922/projections-of-finitely-presented-groups), whether a retract of a finitely presented group is again finitely presented, i.e. if $G$ is a finitely presented group and $H$ is a group which fits into a split exact sequence $1\rightarrow K\rightarrow G\righta... | https://mathoverflow.net/users/27923 | Is a retract of a group of type F_n again of this type? | I close this question by giving the answer of Misha in the comments: The statement is true because a group is of type $F\_n$ if and only if it is coarsely $n-1$-connected. Thanks to Misha for pointing this out.
| 2 | https://mathoverflow.net/users/27923 | 139229 | 76,217 |
https://mathoverflow.net/questions/139222 | 8 | While the language of ZFC set theory does not admit classes, this is not the case for NBG set theory. But the language of NBG does not allow quantification on symbols for classes, and it is known that NBG is conservative on ZFC for sets, meaning that every theorem concerning only sets provable in NBG is also provable i... | https://mathoverflow.net/users/30395 | Theorems on sets provable in KM but not in ZFC set theory | KM proves that for any class $X$, there are class club many cardinals $\delta$ that are fully [$X$-correct](http://cantorsattic.info/Correct), meaning that $\langle V\_\delta,{\in},X\cap V\_\delta\rangle\prec\langle V,{\in},X\rangle$ (and furthermore, this elementarity is expressible in KM, unlike GBC or ZFC). The reas... | 10 | https://mathoverflow.net/users/1946 | 139233 | 76,219 |
https://mathoverflow.net/questions/139215 | 3 | What are all the nondegenerate rational binary operations that are commutative and associative? (Examples: $(x,y) \mapsto x+y$, $xy+x+y$, $xy/(x+y)$.)
Feel free to re-tag if you can think of something better than "algebra".
Clarification: I intended that $x$ and $y$ denote complex numbers; that the operations be de... | https://mathoverflow.net/users/3621 | Commutative associative rational binary operations | Many examples of such operations can be constructed as follows.
Let $F$ be a meromorphic function which has a rational addition theorem.
This means that $F(u+v)=R(F(u),F(v))$ for all (complex) $u$ and $v$, where $R$ is a
rational function. Then $R$ gives you a rational commutative associative operation.
All such me... | 5 | https://mathoverflow.net/users/25510 | 139237 | 76,221 |
https://mathoverflow.net/questions/139191 | 3 | Let $R$ be a ring spectrum (in the world of EKMM $S$-modules) and let $E$ be a smashing $R$-module. Denote by $R\_E$ the $E\_\*$-localization of $R$. By a theorem of Wolbert (Theorem 2 in [Classifying modules over $K$-theory spectra](http://www.sciencedirect.com/science/article/pii/S0022404996001120)), the derived cate... | https://mathoverflow.net/users/1291 | Does Wolbert's derived equivalence between $E_*^R$-local $R$-modules and $R_E$-modules come from a Quillen equivalence? | This is an elaboration on Lennart's comment.
This can be made to come from a Quillen equivalence. Here are the ingredients you'd usually need to show it. (Sorry, I don't have my copy of EKMM handy and so I can't provide theorem numbers.) The problem is that you haven't specified $R\_E$ as an actual object yet rather ... | 3 | https://mathoverflow.net/users/360 | 139238 | 76,222 |
https://mathoverflow.net/questions/139108 | 1 | Let $S$ be an arbitrary scheme, and let $X,Y,S'$ be $S$-schemes. EGA 1, Chap 1, 3.3, gives nice properties for products of schemes with respect to base change $S'\to S$. For example (3.3.10): There exists a canonical isomorphism $$\varphi:(X\times\_SY)\_{S'}\cong X\_{S'}\times\_{S'}Y\_{S'}$$
of $S'$-schemes.
Suppose ... | https://mathoverflow.net/users/36759 | Formal base change properties of group schemes | To show that the scheme isomorphism is an isomorphism of group schemes, it suffices to check that the multiplication maps coincide, i.e., $$m\_{(X \times\_S Y)\_{S'}} = \phi^{-1} \circ (m\_{X\_{S'}} \times\_{S'} m\_{Y\_{S'}}) \circ \tau\_{23} \circ (\phi \times\_{S'} \phi),$$
where $\tau\_{23}: X\_{S'} \times\_{S'} Y\_... | 2 | https://mathoverflow.net/users/121 | 139243 | 76,225 |
https://mathoverflow.net/questions/139236 | 2 | If I have a cubic Bezier curve specified by two endpoints and two control points, how can I find an offset curve which is "parallel" to the original at some given distance, after i have determined the other 2 endpoints ? The red dots in the image <http://s11.postimg.org/n44nmqzoj/bezier_question.png> are the endpoints,... | https://mathoverflow.net/users/38518 | Paralel bezier curve | There's no simple solution for this. I think the parallel curve is usually not exactly a Bezier curve: just think of how even a circle is not a Bezier curve.
As a practical solution, computer programs simply approximate Bezier curves with polygons (made of straight line segments), and compute polygons parallel to th... | 3 | https://mathoverflow.net/users/5340 | 139246 | 76,226 |
https://mathoverflow.net/questions/139249 | 16 | In the paper "A CRITERION FOR THE NORMALITY OF UNBOUNDED OPERATORS AND APPLICATIONS TO SELF-ADJOINTNESS" by M. H MORTAD (<http://arxiv.org/pdf/1301.0241.pdf>), the author states the following theorem
Theorem 1
---------
Let $A$ and $B$ be two operators with domains $D(A)$ and $D(B)$ respectively, such that $A+B$ i... | https://mathoverflow.net/users/38526 | Perturbation of unbounded self-adjoint operators | I'm not familiar with Putnam's book, but part (2) of this theorem should be available in any of the standard references, e.g., Conway's *Course in Functional Analysis* or Reed and Simon, *Functional Analysis* vol. 1.
To understand why he requires $A$ and $B$ to be positive, think of them as multiplication operators o... | 18 | https://mathoverflow.net/users/23141 | 139251 | 76,228 |
https://mathoverflow.net/questions/138788 | 5 | This is a follow-up to [Computing Ext in Exterior algebra (related to Koszul duality)](https://mathoverflow.net/questions/134595/computing-ext-in-exterior-algebra-related-to-koszul-duality) .
Let $V = \mathbb{C} x$, $A = \Lambda^{\bullet}(V) = A\_0 \oplus A\_1$ is graded (with $A\_0 =
\mathbb{C}, A\_1 = V$). Conside... | https://mathoverflow.net/users/2623 | Computing multiplication in the Ext (for a simple example) | This is a standard computation, if I understand the question
correctly. You are interested in computing $Ext\_A(k,k)$ as an
algebra, starting
with an exterior algebra $A$ on $n$ generators $x\_1,\cdots, x\_n$
over a field $k$ (no reason to restrict to $\mathbb C$); $A$ is
graded with $k$ in degree $0$ and the $x\_i$ i... | 15 | https://mathoverflow.net/users/14447 | 139256 | 76,233 |
https://mathoverflow.net/questions/138851 | 7 | Let $h(x,y)$ be a polynomial with real coefficients. Suppose there are infinitely many integer solutions to $|h(x,y)|<1$. What can I say about $h$?
When $h$ itself has integer coefficients, a famous theorem of Siegel tells me that the curve $h(x,y)$ has geometric genus zero and either $1$ or $2$ points at infinity. T... | https://mathoverflow.net/users/297 | Siegel's theorem with real coefficients | Nice question! And it is also not hard at all if the degree is not too small. The only downside is that the answer is negative: the growth can be as fast as one wishes.
For a counterexample, we'll just use $P(x,y)=a^px^p-y^p$ with $a\in(0.4,0,6)$ and $p$ to be chosen later. Note that, given an integer $x>0$, $P(x,y)$... | 7 | https://mathoverflow.net/users/1131 | 139260 | 76,235 |
https://mathoverflow.net/questions/139258 | 10 | Let $\mathcal{T}\_\kappa$ be the set of all linear order types of cardinality $\kappa$. Let $\prec$ denote a binary relation on $\mathcal{T}\_\kappa$ representing embeddability of order types (note that $\prec$ is a quasi-order).
* What is the cardinality of a subset of $\mathcal{T}\_\kappa$ of the largest size that ... | https://mathoverflow.net/users/33664 | Maximal chains in a quasi-order of linear order types | If you only consider countable linear orders, then for example the order types $\eta+2+\eta+2+\eta+3+\eta+\cdots$ and
$\eta+3+\eta+2+\eta+3+\eta+\cdots$ are not isomorphic. (Where $\eta$ is the order type of the rationals.)
This way you can get continuum many nonisomorphic orders, any two of which are bi-embedable.... | 8 | https://mathoverflow.net/users/14915 | 139261 | 76,236 |
https://mathoverflow.net/questions/139076 | 4 | Let $\lambda\_1,\dots,\lambda\_d$ be complex numbers that constitute the spectrum of a nonnegative integer matrix, and $P\_1,\dots, P\_d$ be complex polynoms, such that the sequence $$u\_n=\Sigma\_{i=1}^d P\_i(n)\lambda\_i^n$$ takes only real nonnegative values, i.e. $u\_n\geq0, \forall n\in\mathbb N$.
I'm interested... | https://mathoverflow.net/users/21059 | A complex sequence with positive values | No.
Consider the matrix
$$\begin{pmatrix}
7 & 8 & 4 & 4 & 0 \\
0 & 7 & 4 & 4 & 0 \\
4 & 4 & 7 & 8 & 0 \\
4 & 4 & 0 & 7 & 0 \\
0 & 0 & 0 & 0 & 5 \\
\end{pmatrix}$$
The eigenvalues are $3 \pm 4i$, $11 \pm 4 \sqrt{3}$ and $5$. So
$$u\_n = (3+4i)^n + (3-4i)^n + 3 \cdot 5^n + 0 \cdot (11+4 \sqrt{3})^n + 0 \cdot (11-4 \sqr... | 1 | https://mathoverflow.net/users/297 | 139262 | 76,237 |
https://mathoverflow.net/questions/139267 | 24 | Since Boole it is known that probability theory is closely related to logic.
According to the axioms of Kolmogorov, probability theory is formulated with a (normalized)
probability measure $\mbox{Pr}\colon \Sigma \to [0,1]$ on a Boolean
$\sigma$-algebra $\Sigma$ (of events).
Realizing these data by a set $X$ (sam... | https://mathoverflow.net/users/38532 | Is there a probability theory developed in intuitionistic logic? | The title of the question is a bit of a misnomer, or at least has the potential to cause confision. "Intutionistic probability theory" means to me "theory of probability developed in intuitionistic logic". But you seem to be asking whether we can replace $\sigma$-algerbas (which are Boolean algebras) with Hetying algeb... | 23 | https://mathoverflow.net/users/1176 | 139271 | 76,241 |
https://mathoverflow.net/questions/139245 | 7 | It is a basic fact that $H^n(X, F) = 0$ if $X$ is noetherian affine, $n > 0$, and $F$ a quasi-coherent sheaf.
If $Y \to X$ is a blow-up of a smooth variety in a smooth center, then then exceptional divisor is a projective bundle over the center, and so $H^n(Y, \mathcal{O}\_Y) = H^n(X, \mathcal{O}\_X)$. (right?)
I h... | https://mathoverflow.net/users/38525 | Hn(X, OX) = 0 for X birational to a regular affine variety? | Let me give a counter-example, where $f$ is moreover proper (see Piotr Achinger's comment), but $Y$ is not normal.
Let $X$ be the affine plane over a field $k$ and let $\tilde{Y}$ be the blow-up of the origin in $X$. We choose two $k$-points $P\_1$ and $P\_2$ on the exceptional divisor, and consider the variety $Y$ ... | 8 | https://mathoverflow.net/users/2868 | 139276 | 76,243 |
https://mathoverflow.net/questions/139277 | 13 | When working over a model $V$ of $ZFC$, countably closed forcings are extremely nice:
>
> If $\mathbb{P}$ is countably closed, then $V[G]$ has no new $\omega$-sequences of elements of $V$. In particular, countably closed forcing adds no new reals.
>
>
>
This can fail miserably if $V\models ZF+\neg AC$. In part... | https://mathoverflow.net/users/8133 | How to make countably closed forcing "nice" without choice | This is actually a question I am quite concern with at the moment. In its generality, let me give a slight re-hash of what Joel said, with perhaps a slight generalization.$\newcommand{\forces}{\Vdash}$
---
We say that $\Bbb P$ is $\kappa$-closed if every decreasing sequence of length $\kappa$ (or shorter) has a l... | 12 | https://mathoverflow.net/users/7206 | 139279 | 76,245 |
https://mathoverflow.net/questions/139273 | 0 | On $\mathbb{T}^n$ with a Riemannian metric, the stable norm is defined as
$$\Vert h\Vert=\inf \sum |r\_i| \cdot \mathrm{length}(\sigma\_i),$$
where $h\in H\_1(\mathbb{T}^n,\mathbb R)$ and $\sum\_i r\_i\sigma\_i$ is a Lipschitz cycle representing $h$. The $\inf$ is taken over all such cycles.
The question is, if w... | https://mathoverflow.net/users/15214 | rational rotation vector and closed curves | Let $\mathbb T^n=\mathbb R^n\pmod {2{\cdot}\pi}$.
Consider the embedding $f\colon\mathbb T^2\hookrightarrow \mathbb T^3$
defined as
$$f\colon(u,v)\mapsto (\sin u,\cos u, v).$$
It is easy to find a metric on $\mathbb T^3$ so that for $h=(0,0,1)$,
the infimum is achieved on a geodesic which runs in the embedded $\m... | 1 | https://mathoverflow.net/users/1441 | 139284 | 76,248 |
https://mathoverflow.net/questions/139098 | 5 | I want to lower bound the expected value of the square root of a randomly chosen eigenvalue of a Wishart matrix.
To get the bound I want I need a lower bound on
$$T\_n = \int\_0^\infty\sqrt{x}e^{-x}L\_n(x)^2dx,$$
for $n>1$ where $L\_n(x)$ is the $n$-th Laguerre polynomial.
Mathematica convinced me that $T\_n > ... | https://mathoverflow.net/users/4878 | Lower bound of integral involving Laguerre polynomials | Here is a proof.
First evaluate the integral. I use the generating function for squares of Laguerre polynomials ($I\_0$ the modified Bessel function):
$$
\sum\_{n=0}^{\infty}L\_{n}(x)^2 z^n = \frac{1}{1-z} \exp\left(-\frac{2 x z}{1-z}\right) I\_{0}\left(\frac{2 x \sqrt{z}}{1-z}\right)
$$
for $|z|<1$ (All the form... | 6 | https://mathoverflow.net/users/37436 | 139300 | 76,251 |
https://mathoverflow.net/questions/139305 | 1 | Generalized complex structures were introduced by Nigel Hitchin in 2002. A generalized almost complex structure is an almost complex structure of the generalized tangent bundle which preserves the natural inner product: i.e.,
${\mathcal J}: \mathbf{TM}\oplus\mathbf{TM}^\*\rightarrow \mathbf{TM}\oplus\mathbf{TM}^\*$
suc... | https://mathoverflow.net/users/nan | The space of generalized complex structures in sense of N.Hitchin is contractible? | The space of generalized complex structures at a point of $M$ of dimension $2n$ is diffeomorphic to $O(2n,2n)/U(n,n)$ (see e.g. [arxiv:0703298](http://arxiv.org/abs/math/0703298)), hence is not even connected.
It has $4$ components, each homotopy equivalent to $SO(2n)/U(n)\times SO(2n)/U(n)$, hence non-contractible ... | 6 | https://mathoverflow.net/users/6451 | 139315 | 76,256 |
https://mathoverflow.net/questions/139268 | 7 | Given $W \subset \mathbb C$, let $S\_W$ be the set of polynomials in $\mathbb R[x]$ that vanish on $W$ and have *only nonnegative coefficients*.
>
> **Warm-up question:** It's clear that if $W$ contains a positive real number, then $S\_W = \{0\}$. Is the converse true?
>
>
>
I'm pretty sure the answer is "yes"... | https://mathoverflow.net/users/2502 | How few terms may appear in a polynomial with given (cyclotomic) roots and nonnegative coefficients? | $\def\ZZ{\mathbb{Z}}$ $\def\RR{\mathbb{R}}$ $\def\QQ{\mathbb{Q}}$ $\def\Re{\mathrm{Re}}$ $\def\Im{\mathrm{Im}}$Let $|W|=n$. I will show that $2n+1$ monomials are always sufficient, and are generically necessary. (Noam Elkies has already shown that $2n+1$ is generically sufficient.) If $W$ is closed under complex conjug... | 8 | https://mathoverflow.net/users/297 | 139317 | 76,257 |
https://mathoverflow.net/questions/139308 | 1 | Let $f,g,h$ be "nice" functions and $h(H(x))=H(h(x))=x$. Let $F(x)=H(f(h(x)))$ and $G(x)=H(g(h(x)))$. Obviously $f(g(x))=g(f(x)) \Leftrightarrow F(G(x))=G(F(x))$.
Example: $f(x)=ax, g(x)=bx$ and $F(x)=a+x,G(x)=b+x$ are related by a log transformation.
Question: Can all commuting functions $f,g$ be transformed to ... | https://mathoverflow.net/users/11504 | Are all commuting functions "the same"? | Let me rephrase your question: is every pair of commuting functions simultaneously conjugate to
a pair shifts?
(Your conditions, literally understood imply that your $h$ is a bijective map.)
The answer is evidently no.
In your example, $ax$ and $bx$ are restricted to the positive ray.
If you consider the "same" funct... | 10 | https://mathoverflow.net/users/25510 | 139325 | 76,262 |
https://mathoverflow.net/questions/139313 | 1 | Given an algebraic equation of degree $n$ of form: $$x^{n} - a\_{n-1}x^{n-1} - a\_{n-2}x^{n-2} - \dots - a\_{0} = 0$$ where each $a\_{i} \in \Bbb Q^{+}$ and atleast one positive root, how does one decide efficiently if the largest magnitude positive root lies in a cyclotomic extension?
| https://mathoverflow.net/users/10035 | Deciding if the largest absolute value real root lies in a cyclotomic extension | (1) Factoring polynomials over $\mathbb{Q}$ is easy both in theory and practice so you might as well assume that you know the minimal polynomial of the root.
(2) Let $r$ be the largest real root. The assumption that your polynomial is of the minimal form implies that all the other roots have magnitude $\leq r$ (as I ... | 3 | https://mathoverflow.net/users/297 | 139333 | 76,267 |
https://mathoverflow.net/questions/139257 | 6 | (Please, what is going on?) The following claim is from a paper which was apparently reviewed by László Erdös, Zhongwei Shen, and Bernard Heffler. Someone tell me it's true. Surely it's true. The entire paper depends on it being true.
I am looking at a 2000 article by Kazuhiro Kurata in the J. London Math. Soc.: [An ... | https://mathoverflow.net/users/12968 | Local boundedness of weak solutions of heat equations...? | I think by "standard subsolution estimate", Kurata means the following:
Let $u\ge0$ be a subsolution in $Q\_2:=B\_2(0)\times(-4,0)$, i.e. $\partial\_t u - \nabla\cdot(A\nabla u)\le 0$. Then $u$ satisfies
$$sup\_{Q\_{1/2}} u \le C\Big( \int\_{Q\_1} |u|^2 \;dxdt \Big)^{\frac{1}{2}}$$
for some constant depending only on... | 5 | https://mathoverflow.net/users/25062 | 139340 | 76,270 |
https://mathoverflow.net/questions/139319 | 11 | A [paper](http://arxiv.org/abs/0909.4559) on supersymmetry in 3-dimensions uses results on the spectra of elliptic operators on $S^3$:
>
> The eigenvalues of the vector Laplacian on divergenceless vector
> fields is $(\ell + 1)^2$ with degeneracy $2\ell(\ell+2)$ with $\ell \in \mathbb{ Z}$.
>
>
>
They came up... | https://mathoverflow.net/users/1358 | the spectrum of the Laplacian and Dirac operator on $S^3$ | Let $M=G/K$ with $G$ compact and let $(\tau,W\_\tau)$ be an irreducible representation of $K$, let $E\_\tau$ be the associated $G$-homogeneous vector bundle of $M$.
Then, the space of $L^2$-sections of $E\_\tau$ decomposes as
$$
L^2(E\_\tau) = \sum\_{\pi\in\widehat G} V\_\pi\otimes \operatorname{Hom}\_K(V\_\pi,W\_\ta... | 9 | https://mathoverflow.net/users/20052 | 139341 | 76,271 |
https://mathoverflow.net/questions/68546 | 18 | At my institution incoming graduate students must take a semester long course on pedagogy taught by current grad students. I may soon be in the position of having to teach this course and I'm looking for advice for readings to give the students. The problem is that our grad students don't teach till after they pass the... | https://mathoverflow.net/users/11540 | Teaching a pedagogy course | Thanks everyone for your advice. The course is over now and it seems to have been a success. I created a webpage which contains all the materials I used as well as the syllabus. Please feel free to make use of this material if you should ever need to
<http://dwhite03.web.wesleyan.edu/pedagogyLinks.html>
| 2 | https://mathoverflow.net/users/11540 | 139348 | 76,275 |
https://mathoverflow.net/questions/139355 | 4 | For a prime $p$, let $S$ be the set of all $p$-roots of unity in the complex plane. Now, consider the sum, $W(R)$ of the members of a set $R$ which is a proper subset of $S$. I suspect that $R \ne R'$ implies $W(R) \ne W(R')$. Is this true?
I further ask whether there is sub-exponential method for calculating $R$ fr... | https://mathoverflow.net/users/38570 | Are all sums of subsets of roots of unity unique? | The part about $W(R) \neq W(R')$ when $R \neq R'$ are proper subsets of $S$ is true:
All the $p$-th roots of unity are powers of $\omega := e^{2 \pi i / p}$, and when $p$ is prime, the minimal polynomial of $\omega$ is well-known to be $f(x):=x^{p-1} + x^{p-2} + \cdots + 1$. A relation of the form $W(R) = W(R')$ woul... | 9 | https://mathoverflow.net/users/644 | 139356 | 76,277 |
https://mathoverflow.net/questions/139364 | 7 | This is a question regarding exercise (II.2.15) in Kunen's Set Theory (2011):
The exercise reads:
"We shall see later that $ZFC\vdash $Con$(\Gamma)$ whenever $\Gamma$ is a finite subset of $ZFC$. Use this to define explicitly, in $ZFC$, a binary relation $E$ on $\omega$ such that $ZFC\vdash\varphi^{\omega,E}$ for e... | https://mathoverflow.net/users/38572 | Exercise in Kunen. Demonstrating a relative interpretation into a set | This kind of thing can be confusing, so let me try to help.
Enumerate the axioms of ZFC in some canonical manner, and let $\Gamma$ be the longest initial segment of this list of axioms that is consistent (perhaps this is all of ZFC, if ZFC is consistent, or perhaps it is the theory arises from only finitely many axio... | 11 | https://mathoverflow.net/users/1946 | 139367 | 76,282 |
https://mathoverflow.net/questions/83552 | 51 | Let G be a finite undirected connected graph. A divisor on G is an element of the free abelian group Div(G) on the vertices of G (or an integer-valued function on the vertices.) Summing over all vertices gives a homomorphism from Div(G) to Z which we call degree.
For each vertex v, let D(v) be the divisor
$d\_v v -... | https://mathoverflow.net/users/431 | What is the sandpile torsor? | **Answer:** The Pic0(*G*)-torsor structure is independent of the vertex *v* **if and only if** *G* is a planar ribbon graph.
This is the main theorem of "[Rotor-routing and spanning trees on planar graphs](http://arxiv.org/abs/1308.2677)", by [Melody Chan](https://www.math.brown.edu/~mtchan/), Thomas Church, and [Jos... | 32 | https://mathoverflow.net/users/250 | 139384 | 76,290 |
https://mathoverflow.net/questions/139381 | 3 | I am trying to grok matrix ordered operator algebras, but I am having a hard time understanding their significance from the definition. Here is the definition (or at least, one way of stating it):
Let $B$ be some complex vector space and $M\_n(B)$ the space of $n \times n$ matrixes with entries in $B$; then $B$ is sa... | https://mathoverflow.net/users/19442 | What is the significance of matrix ordered algebras? | Okay. What you have defined is a matrix ordered vector space, not a matrix ordered operator algebra. But this is actually a better place to start.
First question, give examples of such things. The standard example is to let $B$ be a linear subspace of some $M\_k(\mathbb{C})$, or more generally, a linear subspace of s... | 5 | https://mathoverflow.net/users/23141 | 139385 | 76,291 |
https://mathoverflow.net/questions/139388 | 61 | Can anyone give an example of an unnatural isomorphism? Or, maybe, somebody can explain why unnatural isomorphisms do not exist.
Consider two functors $F,G: {\mathcal C} \rightarrow {\mathcal D}$. We say that they are *unnaturally isomorphic* if $F(x)\cong G(x)$ for every object $x$ of ${\mathcal C}$ but there exists... | https://mathoverflow.net/users/5301 | Example of an unnatural isomorphism | For a simpler, but arguably more artificial, example than Mark's, take $\mathcal{C}$ to be the category with one object and two morphisms. Then the identity functor $\mathcal{C}\to\mathcal{C}$ is "unnaturally isomorphic" to the functor that sends both morphisms to the identity map.
| 49 | https://mathoverflow.net/users/22989 | 139392 | 76,294 |
https://mathoverflow.net/questions/139389 | 0 | This arose from a discussion with a friend (people involved are two engineers) who argued that every result in mathematics should be transformable into another branch. For example, he argued that Pythagoras theorem can be proved using tools of probability. Another example is that, he believed there should be a way to t... | https://mathoverflow.net/users/27249 | Can one branch of mathematics be completely learned from the perspective of another branch of mathematics? | ***Proving*** the Pythagorean theorem from the viewpoint of probability would be somewhat tough because it would require being extremely cautious to avoid a vicious circle and not to use the Descartes' description of the plane as $\mathbb R\times\mathbb R$ or anything similar anywhere (once you use it, it becomes uncle... | 6 | https://mathoverflow.net/users/1131 | 139402 | 76,301 |
https://mathoverflow.net/questions/139410 | 3 | I'm having some trouble finding literature on the developing map.
All the sources I could find on it seem to refer to thurston's definition in either:
<http://www.ucl.ac.uk/~ucahhjr/Notes/Essay.pdf>
or
<http://library.msri.org/books/gt3m/PDF/3.pdf>
I don't have much background is geometric topology, so here I'm jus... | https://mathoverflow.net/users/15242 | question about the developing map | False.
True.
It does not exist.
In his notes, before defining the developping map, Thurston specifies that $G$ has to act on the real analytic manifold $X$ by real analytic diffeomorphisms. In other words, he assumes that every $g \in G$ is defined everywhere on $X$. Hence a $(G,X)$-manifold is one such that its ... | 5 | https://mathoverflow.net/users/38319 | 139420 | 76,307 |
https://mathoverflow.net/questions/139412 | 7 | Are there examples of Zoll spheres which are not the surfaces of revolution?
| https://mathoverflow.net/users/34984 | Zoll sphere which is not a surface of revolution | You might need to be more precise in your question. As BS noted there *exist* Zoll $2$-spheres that are not surfaces of revolution, but you may have wanted to know whether there are any *explicitly known* examples. The answer to this is also 'yes'. For some examples, see Kiyohara, *Two-dimensional geodesic flows having... | 7 | https://mathoverflow.net/users/13972 | 139424 | 76,308 |
https://mathoverflow.net/questions/102993 | 6 | Soon after his famous paper "A categorification of the Jones polynomial", Khovanov
introduced a "bordered" version. His theory assingns to every oriented even tangle
a complex of (H\_n,H\_m)-bimodules, where H\_n is a certain family of rings and n,m\geq 0
depend on the number of points on the top (resp. bottom) of the ... | https://mathoverflow.net/users/5001 | Computations and applications of Khovanov's functor valued invariant of tangles | I recall discussing this with you in Budapest last summer, and I was curious about the same thing. Here are my thoughts a year later:
I haven't seen any computational examples in the literature (not that they don't exist), for Q1, but in my experience any computation using Bar-Natan's formal-pictures theory (dotted c... | 4 | https://mathoverflow.net/users/8041 | 139436 | 76,314 |
https://mathoverflow.net/questions/139435 | 9 | Let $G$ be a compact semisimple group and let $\Gamma$ be a finite subgroup of $G$. I am interested, for $(\pi,V)\in \widehat G$ (irred rep of $G$), in a formula for $\mathrm{dim} V^\Gamma$, the dimension of the invariant space of $V$ by $\Gamma$. When $\Gamma$ is the trivial group, Weyl dimension formula says that
$$
... | https://mathoverflow.net/users/20052 | Extension of the Weyl dimension formula | Probably there is no explicit formula of the type you want. In any case, it's important to look first at the most accessible special cases (even though Weyl's formulas may be kept in the background). For example, consider $G= \mathrm{SU}(2)$ and its finite subgroups, using both the McKay correspondence and the Lie theo... | 5 | https://mathoverflow.net/users/4231 | 139440 | 76,316 |
https://mathoverflow.net/questions/139272 | 4 | I will be so thankful if someone helps me with the following question. There exists finite non-abelian p-groups G (except non-abelian groups of order $p^3$) with the following properties:
1. all non-central elements have abelian centralizer.
2. cs(G) has exactly two integers, where cs(G) is the set of all sizes of co... | https://mathoverflow.net/users/38534 | p-group with abelian centralizer | Let be $G=F/F^p[F,F,F]$, with $F$ denotes the free group on $n$ generators. Then $G$ satisfies $Z(G) = \Phi (G)=G'$.
If $x \in G-Z(G)$ then $C\_G(x)= \langle x, Z(G) \rangle$ which is abelian as $C\_G(x)/Z(G)$ is cyclic. Now we have your condition (1). For (2) observe that $C\_G(x)$ has index $p^{n-1}$ in $G$, whene... | 4 | https://mathoverflow.net/users/31883 | 139468 | 76,332 |
https://mathoverflow.net/questions/86888 | 4 | By a paper of Scott Crass <http://xxx.lanl.gov/pdf/math/9903111v1.pdf>
we know that $A\_6$ (Permutation on 6 elements) is an automorphism group of $\mathbb{P}\_2$ which fix a sextic. What is the geometry of this action i.e., what are the orbits of this action explicitly.
| https://mathoverflow.net/users/13559 | Orbits of the action of $A_6$ on $\mathbb{P}_2$ | As noted in the comments, this is a classical object, credited to
[Valentiner 1889]; the [paper
by Robert Crass](http://arxiv.org/pdf/math/9903111v1.pdf) that you cite refers to that paper and also to work of
Wiman [1895] and Fricke [1926] that should contain the answer to your
question, and Crass himself collects thos... | 10 | https://mathoverflow.net/users/14830 | 139480 | 76,336 |
https://mathoverflow.net/questions/138148 | 9 | For every commutative ring $A$, let $\mathbf{Symm}\_A$ be the ring of symmetric functions over $A$. Let $\mathbf{Symm}$ without a subscript denote $\mathbf{Symm}\_{\mathbb{Z}}$.
We can define a bilinear map $\boxdot : \mathbf{Symm}\_{\mathbb{Q}} \times \mathbf{Symm}\_{\mathbb{Q}} \to \mathbf{Symm}\_{\mathbb{Q}}$ by s... | https://mathoverflow.net/users/2530 | Arithmetic product of symmetric functions: why is it integral? | About the question
"Is there a more direct combinatorial or even representation-theoretical significance of this map?"
Probably you know that the arithmetic product is an operation between representations of the symmetric group, because the species is a theory of representations of the symmetric group that gives concr... | 9 | https://mathoverflow.net/users/38623 | 139482 | 76,337 |
https://mathoverflow.net/questions/139484 | 7 | I was reading [these notes](http://www.math.jussieu.fr/~keller/publ/emalca.pdf) by Keller. On the page 19 he defines the Gerstenhaber bracket on the Hochschild cochain complex. But first of all he defines the operation $\bullet$ on the cochains by
$$
\small
(f\bullet g)(a\_1,\dots,a\_{p+q-1})=\sum\limits\_{i=0}^{p} (-... | https://mathoverflow.net/users/32741 | Is the definition of Gerstenhaber bracket related to operads? | Yes, absolutely. I really think the best place for learning this is the book by Loday and Vallette.
This example fits more generally into the following context. Let $P$ be a Koszul operad and $A$ a $P$-algebra. There is attached to $P$ a cohomology theory for $P$-algebras. The two most well known examples is that whe... | 7 | https://mathoverflow.net/users/1310 | 139487 | 76,338 |
https://mathoverflow.net/questions/139491 | 0 | Do there exist a family of graphs with the property:
$$\left|\alpha\left({G \boxtimes \bar{G}}\right) - \alpha\left({\overline{G \boxtimes \bar{G}}}\right)\right| = O(\log(N\_{G}))$$
where $G$ is the graph, $\bar{G}$ is its complement, $\boxtimes$ denotes the strong product, $\alpha(G)$ denotes the independence number ... | https://mathoverflow.net/users/10035 | A question on graphs | Since $$\alpha(\overline{K\_n \boxtimes \overline{K\_n}}) = n$$ and $$\alpha(K\_n \boxtimes \overline{K\_n}) = n$$ it follows that the family of complete graphs satisfies your assumptions.
To see the first equality observe that $K\_n \boxtimes \overline{K\_n}$ is the disjoint union of $n$ copies of $K\_n.$ Hence in t... | 3 | https://mathoverflow.net/users/1737 | 139497 | 76,343 |
https://mathoverflow.net/questions/139499 | 6 | Is there an L-function ($L\_s=\sum\_{n =1}^{\infty} \frac{a\_n}{n^s}$) having a functional equation coming from a relation of the type [1]:
$\sum\_{n =1}^{\infty} a\_n \; e^{-2\pi nx}= \frac{A}{x^k} \sum\_{n =1}^{\infty} a\_n'\; e^{-2\pi\frac{n}{Nx}}$ but not satisfying the Riemann Hypothesis?
Notice that in the abov... | https://mathoverflow.net/users/38290 | Sufficient condition for Riemann Hypothesis? | There certainly are L-functions having a functional equation and violating the analogue of RH. (I am not completely sure I understand what type of functional equation you mean by your restriction, but the ones I mention satisfy a functional equation of so to say the usual form, so I hope this is fine.)
Eppstein zeta-... | 12 | https://mathoverflow.net/users/nan | 139503 | 76,346 |
https://mathoverflow.net/questions/139463 | 4 | Consider a smooth projective variety $X$ over $\mathbb{C}$ such that $X$ has models over $\mathbb{Z}[1/N]$ and $X\_p=X\_{\mathbb{Z}[1/N]}\times \text{Spec}(\mathbb{F}\_p)$ is also a smooth projective variety.
Now let $L$ be a line bundle over $X\_{\mathbb{Z}[1/N]}$ and $L\_\mathbb{C}$ and $L\_p$ be the corresponding ... | https://mathoverflow.net/users/38612 | Properties of divisors when moving from char 0 to char p. | No, that is not true: the property of being nef, ample, etc. is not stable under specialization (although it is stable under generization). For instance, begin with $Y=\mathbb{P}^2\_{\mathbb{Z}}$, i.e., $\text{Proj} \ \mathbb{Z}[s,t,u]$ together with its natural projection $$\pi:Y\to \text{Spec}\ \mathbb{Z}.$$ Consider... | 6 | https://mathoverflow.net/users/13265 | 139509 | 76,349 |
https://mathoverflow.net/questions/138768 | 13 | Let $X\_1$ and $X\_2$ be independent Poisson distributed random variables with parameters $\lambda\_1$ and $\lambda\_2$, respectively.
Let $a = P(X\_1 > X\_2)$ and $b = P(X\_1 = X\_2)$.
**Question**: regarding $(a,b)$ as data, does it uniquely determine $\lambda\_1$ and $\lambda\_2$?
**Idea 1**: we have the expr... | https://mathoverflow.net/users/38295 | Does $P(X_1>X_2)$ and $P(X_1=X_2)$, where $X_1$ and $X_2$ are independent and Poisson distributed, uniquely determine the parameters? | Consider the function $F: (\lambda\_1,\lambda\_2) \rightarrow (a,c)$, where $a = P(X\_1 > X\_2)$ and $c = 1 - a - b = P(X\_1 < X\_2)$. We claim that $F$ is invertible, considered as a map from $(0,\infty)^2$ to the open triangular set $T = \{(x,y) \in \mathbb{R}^2; x>0, y>0, x + y < 1\}$.
**$F$ is injective**
First... | 7 | https://mathoverflow.net/users/38295 | 139519 | 76,354 |
https://mathoverflow.net/questions/139515 | 16 | The Coxeter–Dynkin diagrams tell us that in a spherical Coxeter simplex most of the dihedral angles are right. Say among $\tfrac{n{\cdot}(n+1)}2$ dihedral angles we can have at most $n$ angles which are not right.
>
> Is it possible to see this statement without classification?
>
>
>
| https://mathoverflow.net/users/1441 | why most of the angles are right | I'd be very surprised if one can do better than just reading part of the proof of the classification, which is a beautiful combinatorial argument already, in my opinion.
For example, in the proof of the classification of irreducible root systems given in Fulton and Harris (of Theorem 21.11), part (iv) says that you c... | 4 | https://mathoverflow.net/users/20281 | 139523 | 76,355 |
https://mathoverflow.net/questions/139489 | 2 | Take an example polynomial $f(x, y) = y^2 x + y^3 - x^2$. A solution to $f(x,y)=0$ exists with Puiseux series given by $y(x) = x^{2/3} - x/3 + x^{4/3}/9+\cdots$. I got this by having Mathematica directly solve $f=0$ for $y$ and then perform a Puiseux series expansion for me about $x=0$.
However, since a direct soluti... | https://mathoverflow.net/users/38630 | Determine asymptotic behavior of algebraic curves | Finding such solutions was in fact Newton's original motivation for inventing "Newton polygons". There's a nice exposition of this in Chapter 2 of "The Implicit Function Theorem: History, Theory, and Applications" by Steve Krantz and Hal Parks.
| 4 | https://mathoverflow.net/users/8112 | 139524 | 76,356 |
https://mathoverflow.net/questions/139496 | 7 | Let $\{X\_n\}\_{n\geq 0}$ be a random walk. Let us assume that $\mathbb{E}X\_1 =0$ and $\mathbb{E}X\_1^2=1$. Let also $\mathbb{E}\exp(c|X\_1|)<+\infty$ for some $c>0$ and $X\_1$ has a law with unbounded support. I conjecture that for any $A>0$
$\mathbb{P}(\forall\_{i\in \{1,2,\ldots,n\} } X\_i \geq A \sqrt{i} ) \sim... | https://mathoverflow.net/users/1302 | Random walk over a function | Here is a suggestion, I have not checked all details:
You could mimic the Gaussian computation by doing an exponential change of measure that makes the mean of the $i$th summand to be $1/\sqrt{i}$. The change of measure then will read roughly as $$\Lambda\_n=e^{\sum\_{i=1}^n (c/\sqrt{i}) X\_i-c'\sum\_{i=1}^n (1/i)},$... | 1 | https://mathoverflow.net/users/35520 | 139533 | 76,357 |
https://mathoverflow.net/questions/139528 | 5 | I'd like to ask how to prove or disprove the following identity (which I have to use on some computation regarding $\epsilon$-constant for constructing certain local Langlands parameter):
Let $\psi$ be any non-trivial additive character on $\mathbb{F}\_q$, the field of $q$ elements. Then
$$\sum\_{x\in\mathbb{F}\_{q... | https://mathoverflow.net/users/31327 | Identity on Gauss sums | I think this is just the Davenport-Hasse Theorem, which [Wikipedia](http://en.wikipedia.org/wiki/Hasse%E2%80%93Davenport_relation) calls the Hasse-Davenport lifting relation.
| 6 | https://mathoverflow.net/users/3684 | 139552 | 76,363 |
https://mathoverflow.net/questions/139540 | 6 | One can represent a quantum system by the Weyl algebra (which is a C\*-algebra). For instance, a 1 degree of freedom system can be represented by the algebra generated by $e^{\imath t Q}, e^{\imath s P}$. My guess for modeling a quantum field would be to look at C\*-algebras with an uncountably infinite number of gener... | https://mathoverflow.net/users/37661 | C*-algebras and quantum fields | The Weyl algebra construction can be done abstractly for any real vector space (even infinite-dimensional) endowed with an antisymmetric bilinear form, thanks to B. Blackadar's universal C\*-algebra construction using generators and relations ("Shape theory for C∗-algebras", [Math. Scand. **56** (1985) 249–275](http://... | 13 | https://mathoverflow.net/users/11211 | 139554 | 76,364 |
https://mathoverflow.net/questions/139553 | 4 | If $\sum\_{p\mid n} 1=\omega(n)$, with $1\leq t\leq q-1$ an integer, and $q$ an arbitrary prime.
Then what known asymptotics or bounds can be obtained for $\sum\_{n\leq x}e^{2\pi i t\omega(n)/q}$
More specifically I am looking for an estimate on,
$$\sum\_{p\leq x}\frac{1}{p}\sum\_{n\leq x}\sum\_{t=1}^{p-1}e^{2\pi... | https://mathoverflow.net/users/38626 | Help estimating an exponential sum | The sum $\sum\_{n\le x} e^{2\pi i t \omega(n)/q}$ is the sum
of a multiplicative function whose value at a prime $p$ is
$e^{2\pi i t/q}$. Asymptotics for partial sums of multiplicative
functions $f$ with $f(p)=z$, a given number, are well understood
by work of Selberg and Delange. The answer is
$$
\sim C x (\log ... | 4 | https://mathoverflow.net/users/38624 | 139558 | 76,367 |
https://mathoverflow.net/questions/139368 | 4 | Let $M$ be a closed compact Riemannian manifold, $\mathcal{F}$ be a $C^1$ foliation on $M$. Let $F(x)\in\mathcal{F}$ be the leaf containing $x$.
**Definition.** $\mathcal{F}$ is said to be a *unstable foliation*, if there exist $\lambda>1$, and a diffeomorphism $g:M\to M$ such that
- preserves $\mathcal{F}$: $\... | https://mathoverflow.net/users/11028 | Unstable Foliations | Recently, Y. Shi has constructed a partially hyperbolic diffeomorphism in a nilmanifold where the unstable foliation is not minimal (the global dynamics is Axiom A, so it has a hyperbolic attractor). The example is not yet written, but some insight can be gained from [this paper](https://www.aimsciences.org/journals/di... | 4 | https://mathoverflow.net/users/5753 | 139582 | 76,370 |
https://mathoverflow.net/questions/139547 | 1 | In symbolic dynamics, the context-free shift is the set of biinfinite concatenations of strings of the form $01^k2^k$ for $k\in\mathbb{N}\cup\lbrace 0\rbrace$. I've reduced a certain problem to finding a reasonable upper bound on the number of ways to concatenate such strings to create a string of length n. This can be... | https://mathoverflow.net/users/38658 | Estimation of number of ways to concatenate strings of the form $01^k2^k$ to create a string of length n | Here is another approach to your question that could be used more generally for context-free shifts. Let C be the language of strings $01^k2^k$. This language is a suffix code (no two words are suffixes of each other). Thus the set $C^\*$ of concatenations of elements of C is a free monoid generated freely by $C$ and s... | 2 | https://mathoverflow.net/users/15934 | 139583 | 76,371 |
https://mathoverflow.net/questions/139577 | 5 | First of all, I must clarify at the outset that I am simply asking if there is an *alternative* way to solve an already known problem. It is known that the answer to my question is yes. The problem is as follows:
Consider $(\mathbb{R}^2, g\_{H})$ with the hyperbolic metric, i.e.
$$ ds^2 = e^{2y} dx^2 + dy^2 .$$
Does... | https://mathoverflow.net/users/4463 | Can one use the continuity method to show that the two dimensional hyperbolic space can be immersed in five dimensional Euclidean space? | I think you are asking the wrong question.
By scaling you have that if you can solve for
$$ \mathrm{d}s^2 = e^{2ty} \mathrm{d}x^2 + \mathrm{d}y^2 $$
for any non-zero $t$ you can also solve for
$$ \mathrm{d}s^2 = e^{2\tilde{t}y} \mathrm{d}x^2 + \mathrm{d}y^2 $$
for any non-zero $\tilde{t}$.
Proof: Let $u'(... | 5 | https://mathoverflow.net/users/3948 | 139585 | 76,373 |
https://mathoverflow.net/questions/139575 | 7 | I use Magma to calculate the L-value, yields
E:=EllipticCurve([1, -1, 1, -1, 0]);
E;
Evaluate(LSeries(E),1),RealPeriod(E),Evaluate(LSeries(E),1)/RealPeriod(E);
Elliptic Curve defined by y^2 + x\*y + y = x^3 - x^2 - x over Rational Field
0.386769938387780043302394751243 3.09415950710224034641915800995
0.125000000000... | https://mathoverflow.net/users/38686 | BSD conjecture for X_0(17) | To expand my comment, there are at least 3 subtle ways to get BSD wrong:
1) The BSD period over ${\mathbb Q}$ is the real period $\Omega\_\infty$ when $E({\mathbb R})$ is connected ($\Delta(E)<0$) and $2\Omega\_\infty$ when it has two connected components ($\Delta(E)>0$). The same thing happens over number fields, at... | 28 | https://mathoverflow.net/users/3132 | 139590 | 76,374 |
https://mathoverflow.net/questions/139593 | 4 | I'm working on a dynamical systems problem and have arrived at a naive question in differential topology? geometric measure theory?
I have a smooth path $\gamma\colon \mathbb R\to\mathcal G\_2(\mathbb R^d)$ (that is $\gamma(t)$ is a 2-dimensional subspace of $\mathbb R^d$ for each $t$). I assume that $\mathbb R^d$ i... | https://mathoverflow.net/users/11054 | dimension of a union of grassmannians | Denote by $S^{d-1}$ the unit sphere in $\newcommand{\bR}{\mathbb{R}}$ $\bR^d$ and consider the manifold
$$ X= \bigl\lbrace (v,t)\in S^{d-1}\times \bR;\;\; v\perp \gamma(t)\;\bigr\rbrace. $$
The natural map $X\to\bR$ defines a fiber bundle with fiber $S^{d-3}$. The fiber over $t$ is the the $(d-3)$-dimensional spher... | 4 | https://mathoverflow.net/users/20302 | 139596 | 76,376 |
https://mathoverflow.net/questions/139423 | 5 | I am trying to maximize the function $$f\_s(k)=\frac{1}{2k+s}\sum\_{i=0}^k {2k+s\choose i}2^{-(2k+s)}$$ for both $\{s,k\}\in\mathbf N$, that is, for fixed $s$ what is the value of $k$ that maximizes $f\_s(k)$. I have seen previous discussions in [Sum of 'the first k' binomial coefficients for fixed n](https://mathoverf... | https://mathoverflow.net/users/23543 | Maximize a weighed combinatorial sum | Using Stirling's formula one sees that $\binom{n}{n/2+\ell}$ is about
$$
\sqrt{\frac{2}{\pi n}} 2^{n} e^{-2\ell^2/n},
$$
when $\ell$ is not too large. You can use this to evaluate your sum.
Let's look at the range when $k$ is of size about $s^2$ which is where the
maximum will be attained. Writing $n=2k+s = s^2/t... | 3 | https://mathoverflow.net/users/38624 | 139597 | 76,377 |
https://mathoverflow.net/questions/139589 | 15 | I have encountered this when I was thinking about differentiability in Banach spaces. There, for $x\in X$ we usually need functionals $u\in X^\*$ such that $|u|=1$ and $u(x)=|x|$. This is a simple consequence of Hahn-Banach theorem and enables one to convert the problem at hand into a problem in ordinary calculus. Now ... | https://mathoverflow.net/users/35800 | Does ZF imply a weak version of Hahn-Banach? | Yemon Choi suggested a counterexample in the comments which can easily be made into an actual counterexample.
It is consistent with ZF (see Asaf Karagila's answer [here](https://math.stackexchange.com/q/103476) for a few references) that the dual space of $\ell\_\infty$ is $\ell\_1$ with the usual norm. The duality ... | 12 | https://mathoverflow.net/users/29555 | 139599 | 76,378 |
https://mathoverflow.net/questions/129585 | 1 | Given an integer $n$ the number of partitions of $n$ into two colors can be represented as
$$p\_2(n)=\sum\_{k=0}^n p(k)p(n-k)$$ where $p(k)$ counts the number of ordinary partitions of $k.$ What is the distribution of $$P(k)=\frac{p(k)p(n-k)}{p\_2(n)}$$ as $n\to \infty.$
I feel as though this question has probably be... | https://mathoverflow.net/users/12337 | Distribution of colors in the number of integer partitions of n | The Hardy-Ramanujan asymptotic formula gives $p(n) \sim \exp(C\sqrt{n})/(4n\sqrt{3})$
with $C= \pi \sqrt{2/3}$. From this one sees that if $\ell$ is not too big then
$$
p(n+\ell)p(n-\ell) \sim p(n)^2 \exp(C (\sqrt{n+\ell}+\sqrt{n-\ell} -2\sqrt{n}) )
\sim p(n)^2 \exp\Big( -C \frac{\ell^2}{4n^{3/2}} \Big).
$$
Thus y... | 1 | https://mathoverflow.net/users/38624 | 139615 | 76,390 |
https://mathoverflow.net/questions/139578 | 5 | Theorem 11 (*Conway & Sloane, Sphere Packings, Lattices and Groups, 3rd Edition, pp 383, Ch 15*). If a system of putative $p$-adic symbols for each $p$ satisfies the determinant, oddity, and $p$-adic existence conditions, then there exists an integral quadratic form with these $p$-adic symbols.
The book does not prov... | https://mathoverflow.net/users/21978 | Proving the existence of an integral quadratic form | The Conway–Sloane method is getting popular, partly because they gave the version of The Mass Formula that everyone uses. Some students of [Gabriele Nebe](http://www.math.rwth-aachen.de/homes/Gabriele.Nebe/), at Aachen, have begun doing calculations with it, see [Lorch and Kirschmer - Single-Class Genera of Positive In... | 3 | https://mathoverflow.net/users/3324 | 139620 | 76,393 |
https://mathoverflow.net/questions/137061 | 6 | A nonempty subset $S$ of a group $G$ is called *small* if there is an infinite sequence of elements $g\_n$ in $G$ such that the translated sets $g\_nS$ are pairwise disjoint.
Question: Is there a group which is a (disjoint) union of three small subsets, but it is not a union of two small subsets?
Remark: Such a g... | https://mathoverflow.net/users/33828 | Partition of a group into small subsets | I believe the answer to the bonus question is negative, that is, there exist non-amenable (finitely generated) groups which are not representable as a union of finitely many small subsets. This is directly related to the problem of constructing non-amenable groups
with arbitrarily large Tarski numbers discussed [here](... | 3 | https://mathoverflow.net/users/38713 | 139623 | 76,396 |
https://mathoverflow.net/questions/139591 | 2 | I am working with hyperspectral image data in R, so I have subset an image to a region of 5000 pixels, each containing a vector 254 bands in length.
I would like to cluster this data in order to try and map regions with similar surface composition.
Due to differences in surface reflectance, if I plot two pixels, wh... | https://mathoverflow.net/users/38697 | Appropriate histogram comparison distance measure | A good place to start would be the [Wasserstein metric](http://en.wikipedia.org/wiki/Wasserstein_metric#Applications).
| 1 | https://mathoverflow.net/users/1847 | 139626 | 76,399 |
https://mathoverflow.net/questions/139619 | 2 | Suppose $(X,\Delta\ge 0)$ is a pair such that $(p^g-1)(K\_X+\Delta)$ is an Integral Weil Divisor for some $g>0$ and $X$ is a normal variety. Define $\mathcal{L}\_{e,\Delta} = \mathcal{O}\_X( (1-p^g)(K\_X + \Delta) )$ and note it is a reflexive sheaf.
We can still define the map $\phi^e: \mathcal{L}\_{e, \Delta} \to \... | https://mathoverflow.net/users/26003 | On Non F-pure ideal and Sharp F-Purity for a pair $(X, \Delta)$ where $K_X+\Delta$ is NOT $\mathbb{Q}$-Cartier | I think the answer is that we don't know.
You can certainly make the definition of $\sigma(X, \Delta)$ as that intersection. It seems a reasonable one to me. The intersection is still descending.
However, I don't think we know whether that intersection stabilizes (if I recall correctly, the usual proof doesn't wor... | 3 | https://mathoverflow.net/users/3521 | 139631 | 76,401 |
https://mathoverflow.net/questions/139608 | 8 | For the lattices of all subsets of a given set, an axiomatic characterization is known: A poset is isomorphic to a set of all subsets of some set iff it is a complete atomic boolean algebra.
The question: How to characterize the sets of filters on a set? That is, having a poset, how to check whether it is isomorphic ... | https://mathoverflow.net/users/4086 | A characterization of the poset of filters on a set | Under Stone-duality, the lattices which are isomorphic to the lattices of filters on a Boolean algebra are precisely the compact zero-dimensional frames. The lattices of filters on a complete Boolean algebra correspond to the compact zero-dimensional extremally disconnected frames. Therefore, by [this](https://mathover... | 8 | https://mathoverflow.net/users/38724 | 139636 | 76,403 |
https://mathoverflow.net/questions/134141 | 3 | [This paper](http://arxiv.org/pdf/math/0612106v2.pdf) p.4 claims:
Corollary C. Assume RH. For all large $t$ we have
$$|\zeta(\frac12 +it)| \le \exp\left(\frac38 \frac{\log{t}}{\log{\log{t}}}\right) \qquad (1) $$
$t$ a Gram points often appears counterexample to (1) according to mpmath, pari and maple. E.g. for $t... | https://mathoverflow.net/users/12481 | How is "large" defined in an equality for the modulus of Riemann zeta? | You could look at Chandee's paper (in Proc AMS) and on arxiv at <http://arxiv.org/pdf/0906.4177v1.pdf> where explicit bounds are worked out for zeta
and L-function, and it is specified when they start to hold. For example
see Corollary 1.4 there which holds for $t$ at least $\exp(\exp(10))$ which is about $10^{9500}$... | 5 | https://mathoverflow.net/users/38624 | 139638 | 76,405 |
https://mathoverflow.net/questions/139644 | 5 | In W. Neumann, J. Wahl, "Casson invariant of links of singularities",
Comment. Math. Helv.,1990, Vol. 65, Issue 1, pp 58-78 some connection between the Casson invariant and the signature is formulated. Is there any heuristic explanation, why such a connection should be? The only reason I see is the Rokhlin invariant, ... | https://mathoverflow.net/users/21697 | Casson invariant and signature | Andras Nemethi and I have considerably generalized this result to rational homology $3$-spheres that are links of certain isolated surface singularities.
A rational homology $3$-sphere $M$ which is the link of an isolated surface singularity is equipped with a canonical spin-c structure. Associated to this spin-c str... | 4 | https://mathoverflow.net/users/20302 | 139647 | 76,406 |
https://mathoverflow.net/questions/139630 | 0 | There are two hyperbolic closed 3-manifolds, but I don't know whether they are homeomorphic or not. The only thing I know is that the Cayley graphs of their fundamental groups are quasi-isometric.
My question is:
Will the manifolds be homeomorphic?
| https://mathoverflow.net/users/18496 | A question on Cayley graphs and hyperbolic 3-manifolds | $\newcommand{\HH}{\mathbb{H}}$Here is an expansion of what Anton is saying.
Suppose that $M$ is a closed hyperbolic three-manifold. It follows that the universal cover of $M$ is $\HH^3$: hyperbolic space. The covering map of $M$ comes with a deck group - namely there is an action of $\pi\_1(M)$ on $\HH^3$ so that th... | 10 | https://mathoverflow.net/users/1650 | 139658 | 76,412 |
https://mathoverflow.net/questions/139648 | 5 | Given two irreducible polynomials $f\_{u}(x),f\_{r}(x) \in \Bbb Q[x]$, can one find two polynomials or rational functions $h\_{u}(x),h\_{r}(x) \in \Bbb Q[x]$ or $\Bbb Q(x)$ respectively such that:$$f\_{u}(h\_{u}(x)) = f\_{r}(h\_{r}(x))?$$
Cross Posted: <https://math.stackexchange.com/questions/469190/on-composition-o... | https://mathoverflow.net/users/10035 | On composition of polynomials | Here is a description of all pairs $(f,g)$ of nonconstant polynomials in $\mathbf{Q}[x]$ for which there exist nonconstant $p,q\in\mathbf{Q}(x)$ such that $f(p(x))=g(q(x))$. This is a consequence of my work last summer with Alex Carney, Thao Do, Jared Hallett, Xiangyi Huang, Yuwei Jiang, Qingyun Sun, Yuhou Xia, Ben Wei... | 13 | https://mathoverflow.net/users/30412 | 139661 | 76,415 |
https://mathoverflow.net/questions/139667 | 11 | **Conjecture:** Let $p$ be a prime.
Then the group
$G := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^9, (([a,b]^4)b)^{2p} \rangle$
has a composition series of the form
${\rm PSL}(2,8) - {\rm Z}\_p - {\rm Z}\_p - {\rm Z}\_p - {\rm Z}\_p - {\rm Z}\_p - {\rm Z}\_p - {\rm Z}\_p$.
Is there any literature on this subj... | https://mathoverflow.net/users/38744 | Group theory conjecture on hurwitz groups | For some reason, people seem to be voting to close this, so I will give a quick reply. Your conjecture is true and it can be proved mainly by computer.
The group $G=\langle a,b \mid a^2, b^3, (ab)^7, [a,b]^9 \rangle$ has a homomorphism onto ${\rm PSL}(2,8)$. Let $K$ be the kernel. Then it can be shown that $K$ is nil... | 17 | https://mathoverflow.net/users/35840 | 139669 | 76,419 |
https://mathoverflow.net/questions/139670 | 4 | Is it true that a general curve of genus 3 is a plane section of an appropriate Kummer surface in $\mathbb P^3$? By Kummer surface I mean image of a principally polarized Abelian surface w.r.t. the map defined by the complete linear system $|2\Theta|$, where $\Theta$ is a theta divisor of polarization.
This assertion... | https://mathoverflow.net/users/29992 | General curves of genus 3 as plane sections of Kummer surfaces | Let $C$ be a non hyperelliptic curve of genus 3, let $f\colon C'\to C$ be an e'tale double cover and let $A$ be the Prym variety of $f$. Then $A$ is a principally polarized surface and the Abel-Prym map embeds $C'$ in $A$ as an element of $|2\Theta|$. Multiplication by $-1$ on $A$ restricts on $C'$ to the involution in... | 13 | https://mathoverflow.net/users/10610 | 139675 | 76,421 |
https://mathoverflow.net/questions/139677 | 5 | DISCLAIMER: All pointclasses considered here are boldface.
Most of the time, when doing descriptive set theory, we want the projective sets to "behave well;" for example, maybe we don't want there to be nonmeasurable projective sets, or projective well orderings of $\mathbb{R}$, etc. Generally, this means making some... | https://mathoverflow.net/users/8133 | Does the boldface class $\Delta^1_2$ have the uniformization property? (assuming $V=L$) | Unless I am mistaken, it seems to me that $\Delta^1\_2$ does have the uniformization property in $L$.
For any set $A$ in $\Delta^1\_2$, let $B$ select the $L$-least witness on each slice. So $B$ unifomizes $A$, and the graph of $B$ appears to be $\Delta^1\_2$, by the following reasoning:
* $x\oplus z\in B$ if and ... | 6 | https://mathoverflow.net/users/1946 | 139679 | 76,422 |
https://mathoverflow.net/questions/38738 | 18 | For each positive integer n we may define the convergent sum $$ s(n)=\sum\_{p}\frac{(n,p-1)}{p^2} $$
where the summation is over primes p and $(a,b)$ denotes the greatest common divisor of a,b.
It is immediate to deduce that s(n) is bounded on average:
Using $\sum \limits\_{d|a, d|b}\phi(d)=(a,b)$ and inverting th... | https://mathoverflow.net/users/9232 | A question on the prime divisors of p-1 | Your guess that $s(n)$ gets large if $\omega(n)$ is large is not correct.
It is possible for $n$ to have many primes, and for $s(n)$ still to be small.
This can be seen from some of the work in your question. As you note $s(n) =\sum\_{d|n} \phi(d) a\_d$ where $a\_d =\sum\_{p\equiv 1\pmod d} p^{-2} \ll 1/d^2$. The... | 9 | https://mathoverflow.net/users/38624 | 139696 | 76,426 |
https://mathoverflow.net/questions/139711 | 1 | Let $(a,b)\in \mathbb R^n$. We consider the following open cover of the compact line segment $[a,b]$: $$[a,b]\subset\underset{x\in [a,b]}{\bigcup}B(x,\rho\_x),$$
where for $x\in K,B(x,\rho\_x)$ is a given nonempty open ball of center $x$.
Since $[a,b]$ is compact, there exist a finite subcover such that
$$[a,b]\subse... | https://mathoverflow.net/users/24060 | Special finite subcover of a compact | The answer is no. Consider the interval $[0,2]$, and define $\rho\_x$ as follows:
1- For $x=1$, let $\rho\_x=0.5$
2- For $x\neq 1$, Let $\rho\_x$ be the distance from $x$ to $1$, so $\rho\_x=|1-x|$
Now suppose that a subcovering $B(x\_i,\rho\_i)$, $i=1\cdots,n$ of the wanted form exists. By the choice of the $\rh... | 3 | https://mathoverflow.net/users/38700 | 139718 | 76,430 |
https://mathoverflow.net/questions/123442 | 8 | The operadic butterfly is a diagram in the category of operads in vector spaces. It extends the short exact sequence relating commutative, associative and Lie operads.
$$\begin{array}{ccccc}
& Dend & & & & Dias & \newline
\nearrow & &\searrow & &\nearrow & &\searrow\newline
Zinb & & &Ass & & & \quad Leib \newline
... | https://mathoverflow.net/users/4960 | Is the operadic butterfly symmetric? | I have never seen such a functor and I doubt that it would exist. For instance, the operad Com is symmetric and one-dimensional in every arity. Whereas the operad Dend is regular (coming from a non-symmetric one) and is generated by two (non-symmetric) generators. I also now no non-trivial functor which preserves Zinb ... | 6 | https://mathoverflow.net/users/12352 | 139721 | 76,432 |
https://mathoverflow.net/questions/139719 | 0 | Please give me the proof that for a formal space Massey triple products vanish.
| https://mathoverflow.net/users/33699 | Formal Space and Rational Homotopy Theory | This is a very special case of a very general result. A quasi-isomorphism of DGAs identifies Massey products (not just triple products but matrix Massey products of all sizes). See for example Theorem 1.5 of ``Matric Massey products'', <http://www.math.uchicago.edu/~may/PAPERS/8.pdf>. But of course you must interpret y... | 5 | https://mathoverflow.net/users/14447 | 139722 | 76,433 |
https://mathoverflow.net/questions/139725 | 0 | For any positive integer n, let E(n) denote n-dimensional Euclidean Space and let L(n) denote n-dimensional Lebesgue Measure on E(n). Take n=2 for simplicity. There are uncountably many convex subsets of E(2) which are neither open nor closed in E(2). For example, if the subset D of E(2) is an open disk, then the union... | https://mathoverflow.net/users/4423 | Two questions about Convex Sets and Lebesgue Measure | The first question has an affirmative answer; see this question on math.se:
<https://math.stackexchange.com/questions/207609/the-measurability-of-convex-sets>
(This was the first Google hit for "convex set Lebesgue measurable").
| 2 | https://mathoverflow.net/users/4832 | 139727 | 76,435 |
https://mathoverflow.net/questions/139715 | 5 | Let $S \in \mathbb{Z}\langle\langle A\rangle\rangle$ be a rational series in noncommutative variables. The support of $S$ is the set of all words $u \in A^\*$ such that $(S, u) \not= 0$. It is undecidable to know whether the support of a given$^\*$ rational series is cofinite (respectively equal to $A^\*$). However, it... | https://mathoverflow.net/users/38236 | Is it decidable whether the support of a rational $\mathbb{Z}$-series is a regular language? | **Update.** It is undecidable. Here is the proof.
If $f,g\colon A^\*\to \{a,b\}^\*$ are two morphisms, then one can construct a rational Z-series over A whose support is the complement of the equalizer of f,g. This is how Post correspondence is reduced to universality of $\mathbb{Z}$-series and is based on a faithfu... | 7 | https://mathoverflow.net/users/15934 | 139729 | 76,437 |
https://mathoverflow.net/questions/139702 | 11 | Jack Silver proved that if $x$ is a real so that every $x$-admissible ordinal is a cardinal in $L$, then $0^{\sharp}$ exists.
I wonder whether various weaker or stronger versions of Silver's result have been considered in the literature. For example,
$\bf{Question \ 1.}$:
How strong is the statement that there is r... | https://mathoverflow.net/users/14340 | Concerning Silver's result | Re Q1: Sy Friedman has shown by class forcing over L that there can be consistently a real r
with the r-admissibles precisely the recursively inaccessibles. (S Friedman, "Strong Coding" APAL, vol 35,1987).
Re Q2: All $0^\sharp$-admissibles are limits (indeed fixed points in the enumeration) of Silver indiscernibles (... | 10 | https://mathoverflow.net/users/6942 | 139732 | 76,439 |
https://mathoverflow.net/questions/139733 | 4 | Let $M$ be a semi-Riemannian manifold and $G\subset Iso(M)$ a closed connected Lie subgroup which acts properly on $M$. It is known that every orbit of the action is a (closed) submanifold of $M$. My question is that is every orbit of the action a **semi-Riemannian** submanifold of $M$? precisely, for any orbit $O=G(x)... | https://mathoverflow.net/users/nan | Are all orbits semi-Riemannian submanifolds? | No!
Consider the adjoint action of $SL(2,\mathbb R)$ on its Lie algebra with the Killing form,
which is $\mathbb R^{1,2}$. The orbits are:
* Non-closed: future light cone, past light cone. 0 is in the closure of both.
* Closed: 1-sheeted hyperboloids (signature $1,1$)
* Closed: 1 sheet of the two-sheeted hyperboloi... | 3 | https://mathoverflow.net/users/26935 | 139735 | 76,440 |
https://mathoverflow.net/questions/139713 | 1 | On one hand the real locus of a complex elliptic curve is the intersection of a plane with a torus (i.e. a torus embedded in $\mathbb{C}^2$ plus infinity). And an elliptic curve has no cusps or self-intersections. But the intersection of a plane with a non-singular torus can have those.
I suppose those intersections ... | https://mathoverflow.net/users/38783 | Visualizing singular points of real loci of elliptic curves | The tangent plane to an elliptic curve at a point is a holomorphic curve, meaning it is closed under multiplication by $i$. The real plane does not intersect its product with $i$, Which is the imaginary plane. Hence they cannot be identical, which would happen at a cusp or self-intersection point.
| 2 | https://mathoverflow.net/users/18060 | 139740 | 76,442 |
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