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https://mathoverflow.net/questions/109889 | 9 | Let A be a noetherian ring and D(A) be the derived category of modules on A.
Recall that a dualizing complex for A is an object R in D(A) of finite injective dimension, with cohomology of finite type and such that the natural morphism of functors
$ Id \longrightarrow R\mathcal{H}om(R\mathcal{H}om(., R), R )$
is an isom... | https://mathoverflow.net/users/25309 | State of the art on a question on the existence of dualizing complex | R.Y. Sharp proved in the 80s that a Noetherian ring with a dualizing complex must be "acceptable". This is a weakening of "excellent" obtained by replacing every occurence of the word "regular" with "Gorenstein", so means (1) universally catenary, (2) Gorenstein formal fibers, and (3) Gorenstein locus open in any finit... | 8 | https://mathoverflow.net/users/460 | 109899 | 63,177 |
https://mathoverflow.net/questions/109895 | 2 | I'm interested in having especially simple embedded resolutions of a closed subscheme $Z \subset \mathbb{A}^{n}$ in characteristic zero.
Hironaka gives the existence of a smooth variety $M$ and a proper map $p: M \rightarrow \mathbb{A}^{n}$ that is an isomorphism away from $D=p^{-1}(Z)$ and $D$ has simple normal cro... | https://mathoverflow.net/users/6254 | Simple normal crossings divisor with connected intersections | Yes this is always possible. What you do is you start with a simple normal crossing resolution $M$. Let $D\_1, \ldots, D\_k$ be the irreducible components of the exceptional divisor. Then, by the simple normal crossing assumption, for any set of indices $I$ of size $n$, the intersection $\cap\_{i \in I} D\_i$ is a fini... | 3 | https://mathoverflow.net/users/8914 | 109910 | 63,180 |
https://mathoverflow.net/questions/109909 | 4 | All functors are derived and all categories are bounded derived categories of coherent sheaves. Suppose that we have got an inclusion of a smooth divisor $j:D\rightarrow X$ in a smooth projective variety. Is it true that $$j^\*j\_\*F=F\otimes j^\*j\_\*O\_D?$$
| https://mathoverflow.net/users/nan | Derived push-forward and pull back. | The answer is no. The simplest example that I know is $X = P^3$, $D = P^1\times P^1$, $F = O(0,1)$. In this case $j^\*j\_\*O\_D = O\_D \oplus O\_D(-2)[1]$ and hence $F\otimes j^\*j\_\*O\_D = F \oplus F(-2)[1]$, while $j^\*j\_\*F$ fits into a triangle
$$
F(-2)[1] \to j^\*j\_\*F \to F
$$
which is not split. To see this n... | 8 | https://mathoverflow.net/users/4428 | 109911 | 63,181 |
https://mathoverflow.net/questions/109885 | 5 | Let $X$ be a finite CW complex then with one zero cell. Then (up to homotopy) the two skeleton of X is the same as a group presentation, via the Cayley complex construction. For a while I had been searching for some planar description of the second homotopy group, which would allow a concrete combinatorial description ... | https://mathoverflow.net/users/919 | Second homotopy groups of 3-complexes and Fenn's spiders. | I am not sure (see the notes ["Diagrams and groups"](https://old.i2m.univ-amu.fr/~short/Papers/crmshort.pdf) by Hamish Short) but I think these are related to Igusa's pictures.
There is a nice paper by Loday on the idea of homotopical syzygies (J.-L. Loday, 2000, [Homotopical Syzygies](http://irma.math.unistra.fr/~l... | 6 | https://mathoverflow.net/users/3502 | 109926 | 63,185 |
https://mathoverflow.net/questions/109912 | 5 | Suppose I have some random variable X with probability distribution P(.;theta). Suppose I have a single sample x from this distribution.
Does it make any sense to ask for the derivative of x with respect to theta?
I believe it does not, but am having trouble convincing some of my colleagues.
| https://mathoverflow.net/users/10791 | Derivatives through random variables? | It does not because $x$ is just some element of your state space. You could conceivably choose $x$ as your sample point for any or all of the $\theta$'s. So what it would make sense to differentiate with respect to $\theta$ is $$\mathbb{P}\_{\theta}(X=x)$$ as long as you have a discrete distribution. For a continuous d... | 2 | https://mathoverflow.net/users/11332 | 109933 | 63,189 |
https://mathoverflow.net/questions/109892 | 3 | Suppose I have a bounded set $\Omega$ with a Lipschitz boundary. Is it possible to equip this boundary with a Riemannian distance function $d$ that satisfies: $\lambda |x-y| \leq d(x,y) \leq \mu|x-y|$ for all points in $\partial \Omega$? If not, what is the smallest regularity of the boundary one needs for this propert... | https://mathoverflow.net/users/24644 | Equipping Lipschitz Boundaries with Riemannian Metric | By compactness, there exists $r>0$ so that if points in $S=\partial \Omega$ are within Euclidean distance $\le r$, then they belong to an open subset $U\subset {\mathbb R}^n$ where $S$ is a graph of a Lipschitz function.
Let $f: D\to {\mathbb R}$ be an $L$-Lipschitz function defined on a domain in ${\mathbb R}^{n-1}... | 1 | https://mathoverflow.net/users/21684 | 109934 | 63,190 |
https://mathoverflow.net/questions/109930 | 0 | One week ago, I asked a question on math.stackexchange.com (<https://math.stackexchange.com/questions/209120/a-question-on-intuitionistc-propositional-logic>). But nobody answered my question. So I present it here:
In the Kripke's semantics of intuitionistic propositional logic, the frames are all partially ordered f... | https://mathoverflow.net/users/24928 | A question on intuitionistic propositional logic | Let $F,G$ be the two frames. Let $\beta$ be the frame formula of $F$ (using notation from the Chagrov and Zakharyaschev book you mention in the MSE question, $\beta=\beta^\sharp(F,\bot)$). Since $\beta$ is refutable in $F$, it is also refutable in $G$, hence there exists a generated subframe $H\subseteq G$ and a surjec... | 0 | https://mathoverflow.net/users/12705 | 109938 | 63,193 |
https://mathoverflow.net/questions/109927 | 1 | $P(n)$ and $D(n)$ are two large integers.
Suppose $R(n) = \frac{P(n)}{D(n)}$ is an integer.
I want to compute $R(n)\bmod m$.
$P(n)$ and $D(n)$ are too large to be computed but $P(n)\bmod m$ and $D(n)\bmod m$ can easily be computed.
For example, how to compute $S(x,k) = \sum\_{k=0}^{n}x^k\mod m$ for a large ... | https://mathoverflow.net/users/19106 | Computing the ratio of two large integers modulo m | As pointed out in the comments, the problem cannot be solved in the generality you state it in, but the magic words are "chinese remainder theorem". If you have an estimate on the sizes of $P$ and $Q$ you can compute them modulo $m\_1, \dotsc, m\_k, $ where the $m\_i$ are pairwise relatively prime (or just prime), and ... | 1 | https://mathoverflow.net/users/11142 | 109942 | 63,196 |
https://mathoverflow.net/questions/109920 | 3 | Let $G$ be a reductive algebraic group. Let $X$ be a $G$-variety and consider any closed subvariety $Z$ of $X$. Since any $g\in G$ acts as an automorphism, we know that $g.Z$ is again a closed subvariety of $X$. This yields an action of $G$ on the free module of cycles of $X$ which should induce an action of $G$ on the... | https://mathoverflow.net/users/9947 | Intersection theory for $G$-varieties - an action on the chow ring? | If you are interested in intersection theory of varieties with $G$-actions, then you want to study equivariant intersection theory. This theory exploits the $G$-action in a way that leads to deeper invariants than ordinary intersection theory. The three references I would recommend if you are first learning the subject... | 4 | https://mathoverflow.net/users/16002 | 109943 | 63,197 |
https://mathoverflow.net/questions/109904 | 7 | Let us define the following sequence of polynomials for every two non-negative integers $i,d$: $$s\_i^d(w)=\sum\_{j=0}^{d+1} (-1)^j {d+1\choose j} (j+1)^i w^{d+1-j}.$$
**Conjecture:** *The sequence* $\{s\_d^d(w),s\_d^{d-1}(w),\dots,s\_d^0(w)\}$ *is a Sturm sequence.*
An easy corollary of this conjecture is that:
**... | https://mathoverflow.net/users/23862 | Prove that a particular polynomial sequence is a Sturm sequence | This is just a partial answer, but anyway:
( I use $s[n,k]$ for $s^n\_k$ since I just copy-pasted from Mathematica:
I do not know if this helps, or if you already know, but you have the recurrence
$$s[n- 1,k + 1] = s'[n, k] - x \cdot s'[n - 1, k]$$
Note, by induction, we assume $s[n,k]$ and $s[n-1,k]$ to have inter... | 7 | https://mathoverflow.net/users/1056 | 109946 | 63,199 |
https://mathoverflow.net/questions/109813 | 5 | Let $X$ be a Kahler manifold and $Z\subset X$
be a smooth hypersurface. How to compute the Hodge
diamond of the double covering $Y\to X$ ramified
over $Z$? (And what I have to know? Would the map
$H^\*(X)\to H^\*(Z)$ be enough?)
P.S. I tried the Gysin sequence, but it looks
like there are many loose ends.
| https://mathoverflow.net/users/9833 | The Hodge numbers of a covering | The answer goes back to Esnault and Viehweg:
$$H^q(Y,\Omega\_Y^p)= H^q(X,\Omega\_X^p)\oplus H^q(X,\Omega\_X^p(\log Z)\otimes L^{-1})$$
where $L$ is the anti-invariant part of the direct image of $O\_Y$ to $X$ under the
natural $\mathbb{Z}/2$ action.
| 7 | https://mathoverflow.net/users/4144 | 109953 | 63,203 |
https://mathoverflow.net/questions/109962 | 5 | Is the following true?
Let $R$ be a commutative ring and $S$ be an $R$-algebra. Let $\mu: S\otimes\_R S\to S$ be the multiplication.
Let $I$ be the kernel of $\mu$, and $J\subseteq I$ be the smallest ideal containing $1\otimes s-s\otimes 1$ for all $s\in S.$
Then $J=I.$
| https://mathoverflow.net/users/25784 | Generators of the kernel of multiplication of a ring | Assuming by $J \subset I$ you mean $J\subset S\otimes\_R S$, yes. Let $ab=cd$. Then we must show that $a\otimes b-c\otimes d\in J$. But
$a\otimes b-c\otimes d=a\otimes b+ab\otimes 1 - c\otimes d -cd\otimes 1$
$=a \otimes 1(1\otimes b-b \otimes 1) -c \otimes 1 (1 \otimes d -d \otimes 1)\in J$
An identical argumen... | 6 | https://mathoverflow.net/users/18060 | 109965 | 63,210 |
https://mathoverflow.net/questions/109967 | 15 | Let $\Gamma = \pi\_1(S)$ denote the fundamental group of a compact surface $S$ of genus $g>1$.
Given a representation $\rho : \Gamma \to \mathrm{PSL}(2,\mathbb{C})$, specified by matrix representatives for the images of a fixed generating set, is there an algorithm to answer either of the following questions?
1) Is... | https://mathoverflow.net/users/8196 | Algorithm to test for discrete or quasi-Fuchsian subgroups of PSL(2,C) | To make the question precise, you should probably specify what model of computation you want to answer the question in? For example, if the matrix entries are given by algebraic numbers, then I think you can just work with a Turing machine. I suppose you could also work with an oracle that gives you as many decimal pla... | 12 | https://mathoverflow.net/users/1345 | 109974 | 63,214 |
https://mathoverflow.net/questions/109957 | 4 | Hi. I have only able to find the definition of $Pic(R)$ for a commutative ring $R.$ Which is the isomorphism classes of projective $R$-modules of rank $1,$ and the product given by $[A][B]=[A\otimes\_R B].$
How can i adapt this definition for the non-commutative case?
are there some books of articles related to thi... | https://mathoverflow.net/users/26901 | what is the definition of the Picard group of a (non necessarilly commutative) Ring? | The definition I appreciate most comes from Morita theory: you consider unital (just for simplicity, something slightly weaker will also work) rings and the bimodules between them. Using the tensor product this makes almost a category: the tensor product of bimodules (over the ring in the middle) is associative up to a... | 9 | https://mathoverflow.net/users/12482 | 109980 | 63,216 |
https://mathoverflow.net/questions/109903 | 6 | Hello,
Is it true that if I have a scheme $X$ which is, say, Noetherian, of finite Krull dimension, and semi-separated (intersection of two open affines is again open affine), then I can find a locally closed embedding of it into a scheme of the same type, which is, in addition, regular?
I ask just of curiosity, or... | https://mathoverflow.net/users/2095 | Embedding of a scheme into a regular scheme | A noetherian regular scheme is universally catenary (Matsumura, 14.B, 16.D), so any subscheme of a regular scheme is universally catenary.
But there are affine noetherian schemes (integral of dimension $2$) which are not universally catenary. See an example of Nagata in Matsumura, 14.E, or a slightly simpler one in ... | 11 | https://mathoverflow.net/users/3485 | 109982 | 63,217 |
https://mathoverflow.net/questions/109792 | 3 | Solve the following nonlinear equations for $v$ and $w$
$Avv^TAw=\lambda\_1v+\lambda\_2w$
$Aww^TAv=\lambda\_1w+\lambda\_2v$
$v^Tw=w^Tv=0$
$v^Tv=w^Tw=1$
where $\lambda\_1, \lambda\_2, \lambda\_3$ are real. $A$ is a symmetric matrix.
How would you generalize to the case
$Avv^TAw+Bvv^TBw=\lambda\_1v+\lambd... | https://mathoverflow.net/users/27073 | Nonlinear matrix equation | First of all, note that $w^TAv=v^TAw$ is a scalar.
Here is an idea that should greatly simplify the equation:
Your equations say that $Aw$ and $Av$ are both contained in $U=\operatorname{span}(v,w)$, therefore $U$ is an invariant subspace of $A$. You can get all two-dimensional invariant subspaces by taking $U=\ope... | 4 | https://mathoverflow.net/users/1898 | 109984 | 63,219 |
https://mathoverflow.net/questions/109818 | 4 | Let $n \in \mathbf{N}$ be a natural number and $v\_1,\cdots,v\_n$ a set of basis vectors in $\mathbb{R}^n$. How does one find the matrix $g \in \mathbf{GL}\_n(\mathbb{Z})$ orthogonalizing these best possible? That is, such that $\max \{|(g v\_i,g v\_j)|/ ||g v\_i|| ||g v\_j|| : i \not= j \in \{1,\cdots,n\}\} \in [0,1]$... | https://mathoverflow.net/users/8380 | Most orthogonal lattice basis | It seems you are asking about [Lattice reduction](http://en.wikipedia.org/wiki/Lattice_reduction), this subject goes back to 19-th century to Hermite, Minkowski and others, but it is of certain interest up to current days, since it is used in commercial GPS navigation and there is academic research on application to mu... | 1 | https://mathoverflow.net/users/10446 | 109985 | 63,220 |
https://mathoverflow.net/questions/109949 | 4 | It's possible to extend the well known cross product between two vectors in $\mathbb{R}^3$ to $n-1$ vectors in $\mathbb{R}^n$.
Let $\vec{v\_1}, \vec{v\_2}, \dots, \vec{v}\_{n-1} \in \mathbb{R}^n$ and $\vec{e}\_1 = (1, 0, 0, \dots, 0)^T, \vec{e}\_2 = (0, 1, 0, \dots, 0)^T, \dots, \vec{e}\_n = (0, 0, \dots, 0, 1)^T$ be... | https://mathoverflow.net/users/27335 | How to efficiently compute the generalized cross product? | Regarding your question 2: the approach I'd take is computing the first determinant from a RQ factorization of the leading $(n-1)\times(n-1)$ matrix, and then each other by replacing in turn each row of $R$ with $(last row)Q$ and re-orthogonalizing manually with $O(n)$ Givens transformations on the left. In this way yo... | 3 | https://mathoverflow.net/users/1898 | 109995 | 63,225 |
https://mathoverflow.net/questions/109990 | 1 | Wikipedia states that the Haar measure on a compact group is a mean (and that every compact group is amenable). But, obviously, the Haar mesure on the group of unit quaternions cannot be defined on every subset, else the Banach-Tarski paradox would not happen. What am I missing?
| https://mathoverflow.net/users/17164 | Are all compact groups amenable ? | The definition of a mean for a locally compact group is *not* a finitely additive measure defined on the entire power set of $G$---that definition is only correct for discrete $G$. See [the wikipedia article](http://en.wikipedia.org/wiki/Amenable#Definition_for_locally_compact_groups) for the correct definition. Indeed... | 5 | https://mathoverflow.net/users/1463 | 109997 | 63,226 |
https://mathoverflow.net/questions/110007 | 2 | Does somebody have some interesting examples of Courant algebroids which are not exact? By exact I mean one which is of the form $TM\oplus T^\star M$ with the standard bracket twisted by a closed 3-form $H$.
Thank you!
| https://mathoverflow.net/users/14806 | Courant algebroids which are not exact | The answer depends on what you mean by ``interesting."
For example the paper "On the Geometric Structure of Hamiltonian Systems
with Ports" by
Jochen Merker, J Nonlinear Sci (2009) 19: 717–738
DOI [10.1007/s00332-009-9052-3](http://link.springer.com/article/10.1007%2Fs00332-009-9052-3), may be considered as dealing ... | 2 | https://mathoverflow.net/users/25355 | 110013 | 63,237 |
https://mathoverflow.net/questions/110014 | 10 | I'm wondering if anyone knows how to *construct* hyperbolic 3-manifolds whose fundamental group is RFRS. Clearly the recent work of Agol, Wise, etc. says that such manifolds are abundant, and in particular present in every commensurability class. But how do you construct examples?
The only examples of RFRS manifolds ... | https://mathoverflow.net/users/8183 | Examples of 3-manifolds with RFRS fundamental group | Agol's original paper on RFRS gives a nice short proof that the fundamental group of any manifold which also happens to be a finite-index subgroup of your favourite right-angled reflection group is RFRS. (Sketch proof: watch how loops bounce off the mirrors!)
So take your favourite right-angled reflection group $\Gam... | 11 | https://mathoverflow.net/users/1463 | 110016 | 63,239 |
https://mathoverflow.net/questions/110009 | 1 | Let $n\in\omega$. Call a real $x$ $n$-special if $x\in L\_{\omega\_{n}^{CK,x}}$.
Are the $n$-special reals bounded in $L\_{\omega\_{1}}$? That is, is there $\beta<\omega\_{1}$
such that $L\_{\beta}$ contains all $n$-special reals?
| https://mathoverflow.net/users/27352 | A question about relativized admissibility theory | Let me assume first that by $\omega\_1$ you meant
$\omega\_1^L$, that is, the $\omega\_1$ of $L$. In this case, no, even the $1$-special reals are not
bounded. To see this, note first that there are unboundedly many
$\gamma\lt\omega\_1^L$ with $\gamma$ countable in $L\_{\gamma+1}$.
(Any stage where a new real is added... | 3 | https://mathoverflow.net/users/1946 | 110018 | 63,240 |
https://mathoverflow.net/questions/110017 | 0 | This question relates to one on topology and C^\*-algebras that was asked two days ago, namely at the link: [C\*-algebras with no nontrivial endomorphisms](https://mathoverflow.net/questions/109772/c-algebras-with-no-nontrivial-endomorphisms) [link text](https://mathoverflow.net/questions/109772/c-algebras-with-no-nont... | https://mathoverflow.net/users/13292 | injective *morphisms of C(D), the C^* algebra of continuous functions on the closed unit disk D | Assume $D$ lies in $\mathbb{R}^2$ and define $f:D\rightarrow D$ by $f(x):=2x$ if $\|x\|\leq 1/2$ and $f(x):=\frac{x}{\|x\|}$ if $\|x\|\geq 1/2$. Then, $f$ is onto and continuous, but it is not injective. It is easy to see that $f^\ast :C(D)\rightarrow C(D)$ is an injective $^\ast$-homomorphism, but it is not onto.
| 2 | https://mathoverflow.net/users/nan | 110021 | 63,242 |
https://mathoverflow.net/questions/109981 | 0 | why the z-ideals in C(X) are basically the sums of intersections of maximal ideals?
| https://mathoverflow.net/users/27342 | a questions about the sums of intersections of maximal ideals | Let $\mathcal{M}\_f$ be the intersection of all maximal ideals in $C(X)$ which contains $f$. it is easy to show that $$\mathcal{M} \_f=\Big(g \in C(X): Z(f) \subseteq Z(g)\Big)$$
Now Let $I$ be a $z-$ideal of $C(X)$. It means if $f \in I$ and $Z(f)\subseteq Z(g)$, then $g \in I$.
It suffices to show that for all $f... | 0 | https://mathoverflow.net/users/23317 | 110026 | 63,244 |
https://mathoverflow.net/questions/110027 | 1 | Let a,b,c denote the lengths of the sides and let A,B,C denote the corresponding opposite angles of
a triangle. In the Euclidean plane we have the law of sines. a/sin(A)=b/sin(B)=c/sin(C). A recent
article pointed out that these 3 equal ratios are also all equal to the diameter of a circle whose
circumference contains ... | https://mathoverflow.net/users/4423 | A question concerning a well known "law" about triangles. | I am amazed with your statement that "a recent article pointed...". I suspect his fact was known to the Greeks about 2000 years ago, and it is certainly mentioned in every serious trigonometry textbook.
On your question. The corresponding quantity for the spherical triangle even has a name:
"modulus of the triangle",... | 8 | https://mathoverflow.net/users/25510 | 110033 | 63,247 |
https://mathoverflow.net/questions/110022 | 3 | Let $A$ be an infinite dimensional faithful Banach algebra and let $\mathcal U$ be a free ultrafilter. Is the ultrapower $(A)\_{\mathcal U}$ faithful?
| https://mathoverflow.net/users/27066 | Is an ultrapower of a faithful Banach algebra always faithful? | (This merely expands on my comment above.)
Let $A$ be a Banach algebra and let $L: A\to {\mathcal B}(A)$ be defined by $L(a)(x)=ax$. The map $L$ is a norm-decreasing homomorphism but it need not have closed range. (For instance, take $A=\ell^p$ with pointwise multiplication, for any $1\leq p <\infty$.)
So, let us s... | 7 | https://mathoverflow.net/users/763 | 110055 | 63,256 |
https://mathoverflow.net/questions/109624 | 7 | Hi all,
I am interested in the following question (which is quite similar to one I posed a long while ago): Let $P\_{N}(t)=\underset{k=-N}{\overset{N}{\sum}}c\_{k}e^{ikt}$ be a unit norm trigonometric polynomial, we look at it as a function of $L^{2}\left(\mathbb{T}\right)$.
I'd like to find a direct proof to the fact... | https://mathoverflow.net/users/16040 | Norm concentration of trigonometric polynomials - Uncertainty principle | Let's talk about algebraic polynomials (just multiply by $z^N$). Let $P$ be a polynomial of degree $N$. Let $\max|P|=1$ and assume that this maximum is attained at some point $p$. Take the disk $D$ of radius $10C/N$ centered at $p$. Note that $z^{-N}P(z)$ satisfies the maximum principle in the complement of the unit di... | 3 | https://mathoverflow.net/users/1131 | 110067 | 63,261 |
https://mathoverflow.net/questions/110029 | 13 | Suppose I have a smooth manifold $M$ and a smooth Morse function $f$ on $M$. Is there a standard way to replace $M$ with a cell complex and $f$ with a discrete Morse function such the resulting homology (via discrete Morse theory) is the same as that of $M$?
My question is motivated by a situation where I know the cr... | https://mathoverflow.net/users/8639 | Discrete Morse function from smooth one | This is a rapidly developing area, and there are many short-cuts if all you want to do is compute the homology of sub-level sets of $f$. To answer your main question, as Liviu has already mentioned: **there is no standard way**. However, you should be able to compute the homology you desire as follows.
My impression ... | 4 | https://mathoverflow.net/users/18263 | 110068 | 63,262 |
https://mathoverflow.net/questions/105731 | 6 | I should recall the notion of maximal subring of a commutative unitary ring $R$.
>
> **Def:** A commutative ring $S$ is called a maximal subring of $R$ if $S \subset R$ and if $T \subset R$ constitute a commutative ring with the restricted addition and multiplication of $R$ and also $S\subsetneq T$ then we could de... | https://mathoverflow.net/users/23317 | functional subrings | Hi,
Yes, C(X) has very different maximal subrings. First note that the real line, say R, has uncountable many maximal subrings, see for example A. Azarang and O.A.S. Karamzadeh works about the existence of maximal subrings in Fields and Commutative Rings, Hence for any fixed maximal ideal M\_x of C(X), C(X) contains ma... | 1 | https://mathoverflow.net/users/27374 | 110072 | 63,265 |
https://mathoverflow.net/questions/110075 | 1 | I often heard about this division but always in a non-formal manner. What constitutes it? Is it a limit operation? Or a fundamental distinction between countable and uncountable sets? And what math subjects can be accurately attributed to one or another type?
I really find this division very weird.
| https://mathoverflow.net/users/27375 | What constitutes the division between discrete and non-discrete math? Are there any math subject where it's being blurred? | There is no real division. It is for convenience. An example in which you can test out ideas on this is with the theory of finite topological spaces. (NB every thing is finite so the answer to your last part is clear in this context.) Finite topological spaces when discrete are finite sets (full stop nothing more can b... | 2 | https://mathoverflow.net/users/3502 | 110082 | 63,269 |
https://mathoverflow.net/questions/110051 | 2 | I have two (not unrelated) questions. Let me first give a short introduction.
Introduction
------------
For a general overview of the setup I refer to the introduction (§1) of [Zhang].
Let $k$ be a number field or a function field of a curve. Let $X/k$ be a curve, and assume that $X$ has a semistable model. Let $... | https://mathoverflow.net/users/21815 | Chow group of a (particular) motive [+ reference request] | One can define the Chow groups of a Chow motive $M$ by the formulas
$$ \newcommand{\Hom}{\mathrm{Hom}}\newcommand{\L}{\mathbb{L}} \newcommand{\Ch}{\mathrm{CH}} \Ch^s(M) = \Hom(M,\L^s) $$
and
$$ \Ch\_s(M) = \Hom(\L^s,M).$$
Here $\L = H^2(\mathbf P^1).$ A reference for this definition is e.g. Scholl's text in the first ... | 3 | https://mathoverflow.net/users/1310 | 110087 | 63,273 |
https://mathoverflow.net/questions/110093 | 1 | A time ago I was intrigued by the following remark:
"... one ends up with a (non-sparse) system of equations in
about 10000 real variables. One important practical point is
that solving such systems is now well within the capabilities
of a standard desctop PC;...". The above is a quote from
Notices of AMS 55 no. 9 ... | https://mathoverflow.net/users/9833 | What software one needs to solve a big linear system on a small computer? | Linear systems of 10,000 equations in 10,000 unknowns can easily be solved in a few seconds using double precision floating point arithmetic on typical consumer grade PC's and even laptop computers.
If you're writing your main program in a compiled language like C or Fortran, you consider using the LAPACK and BLAS l... | 6 | https://mathoverflow.net/users/9022 | 110094 | 63,276 |
https://mathoverflow.net/questions/110062 | 12 | And what else can be said, if so?
([Original math.SE post](https://math.stackexchange.com/questions/214125/start-with-a-topological-group-take-the-meet-of-the-two-uniformities-and-take))
In more detail: Say $(G,\mathscr{T})$ is a topological group. It has a left uniformity $\mathscr{L}$ and a right uniformity $\mat... | https://mathoverflow.net/users/5583 | Start with a topological group, take the meet of the two uniformities, and take the topology. Is the result again a topological group? [xpost from math.SE] | This is most definitely not my field of expertise (so be kind!), but Section 1.8 of the book "Topological Groups and Related Structures, an introduction to topological algebra" by Arhangel'skii and Tkachenko deals with these sorts of questions.
The book is available online at [SpringerLink](http://www.springer.com/ma... | 5 | https://mathoverflow.net/users/18128 | 110102 | 63,282 |
https://mathoverflow.net/questions/110104 | 2 | We know that, if $p$ is a prime number and $k$ is a natural number, then there is just one finite field $F$, such that $|F|=p^{k}$.
How about finite commutative local ring?
Let $R$ and $S$ be local rings with maximal ideal $m\_{1}$ and $m\_{2}$, respectively, such that $|R|=|S|$ and $|m\_{1}|=|m\_{2}|$. Is it true ... | https://mathoverflow.net/users/8725 | Finite commutative local ring | No, the cardinalities alone are not enough to identify a local commutative ring. For example, you can take $R = \mathbb F\_p[x]/\langle x^3 \rangle$ and $S = \mathbb F\_p[x,y] / \langle x^2, xy, y^2\rangle$, which both have cardinality $p^3$ with maximal ideal of cardinality $p^2$.
A more refined invariant you could ... | 8 | https://mathoverflow.net/users/8914 | 110105 | 63,283 |
https://mathoverflow.net/questions/110103 | 2 | Hello!
My question is about the realization of homology class.
The definition of the realizaion of homology class is: for manifold M and a homology class $z\in H\_k(M)$, k is an integer. If we find a k-dimensional manifold N and a map $f:N \rightarrow M$ such that $f\_\* [N]=z$, $[N]$ is the fundamental class, then w... | https://mathoverflow.net/users/25054 | Geometry Realization of Homology Class | Duplicate of [this question](https://mathoverflow.net/questions/1489/cohomology-and-fundamental-classes), which has a very good answer by Eric Wofsey.
| 3 | https://mathoverflow.net/users/11142 | 110106 | 63,284 |
https://mathoverflow.net/questions/110110 | 1 | Assume that $M\subset R^n$, $n\ge 3$, is a boundary of an open bounded set $D$ containing $0$, which is starlike w.r.t. 0, meaning that each ray $[0,x]$ from $x\in M$ to $0$ meets $M$ only once. Is $M$ smooth almost everywhere?
| https://mathoverflow.net/users/26543 | Boundary of star-shaped domain | The unit circle is a boundary of an open bounded set $D$ containing $\mathbf{0}$,
which is starlike w.r.t. $\mathbf{0}$, and which is smooth almost everywhere.
When $f$ is [the original Weierstrass function](http://en.wikipedia.org/wiki/Weierstrass_function#Construction), $\;\; \left\{\begin{bmatrix} \lef... | 3 | https://mathoverflow.net/users/nan | 110112 | 63,286 |
https://mathoverflow.net/questions/110108 | 5 | Given a reduced expression for the longest word $w\_0$ in the Weyl group of $\mathfrak{g}=\mathfrak{n}^+\oplus\mathfrak{h}\oplus{n}^-$, one obtains a convex ordering on the set of positive roots, $\beta\_1<\ldots<\beta\_N$. Using the braid group action we get (divided powers of) root vectors $E\_{\beta\_i}^{(n\_i)}$ in... | https://mathoverflow.net/users/4366 | Convex PBW bases | If by Leclerc, you mean his paper "Dual canonical bases, quantum shuffles and q-characters", then his Lemma 37 is not for a general convex order, but only for those that come from Lyndon words. And if you mean a different Leclerc paper, I'd like to know which you have in mind.
I know two proofs of this fact.
One re... | 3 | https://mathoverflow.net/users/425 | 110120 | 63,290 |
https://mathoverflow.net/questions/110126 | 2 | Let $f:X\to Y$ be an algebraic fibre space, and $\dim Y=1$. What can I say about the numerical relation of the fibres?
| https://mathoverflow.net/users/3525 | When are the fibres numerically equivalent? | They are algebraically equivalent, which is stronger. Consider the subvariety on $X \times Y$ which is the inverse image of the diagonal on $Y \times Y$ under $(f,id)$. This is flat, because it's a hypersurface in a flat morphism, so all its fibers, which are the fibers of $f$, are algebraically equivalent.
This is o... | 3 | https://mathoverflow.net/users/18060 | 110127 | 63,294 |
https://mathoverflow.net/questions/110078 | 1 | Let $0< \alpha< n$, $1 < p < q < \infty$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. Then:
$ \left \| \int\_{\mathbb{R}^n} \frac{f(y)dy}{|x-y|^{n-\alpha} } \right\|\_{L^q(\mathbb{R}^n)}\leq$ $C\left\| f\right\| \_{L^p(\mathbb{R^n})}$.
| https://mathoverflow.net/users/27379 | Sobolev-type inequality. | This is the standard Hardy-Littlewood-Sobolev inequality(or the theorem of fractional integration).A more direct approach is write
$$
\int{f(x-y)|y|^{\alpha-n}dy}=\int\_{|y|<R}+\int\_{|y|\ge R}
$$
For the second term on the RHS,using Holder inequality,and easy to see that it's dominated by $\|f\|\_{L^p}R^{-\frac{q}{n... | 3 | https://mathoverflow.net/users/23078 | 110130 | 63,295 |
https://mathoverflow.net/questions/110129 | 2 | Let J a smooth group scheme over a smooth connected base S.
I assume, that over an open subset U of S, J is a torus, do I have that J is abelian?
| https://mathoverflow.net/users/27398 | smooth group scheme genrically abelien | Yes. Consider the map: $J \times\_S J \to J$ that sends $(a,b)$ to $aba^{-1}b^{-1}$. The pullback of the zero section along this map is a closed set. It is exactly the set of commuting pairs. It includes the inverse image of $U$, which is nonempty open.
Since $S$ is smooth and connected it is irreducible, so since $J... | 4 | https://mathoverflow.net/users/18060 | 110131 | 63,296 |
https://mathoverflow.net/questions/110046 | 1 | Let $K$ be a number field, $d$ a positive integer and $S$ a finite set of places of $K$.
By Cebotarev, there exists a finite set of finite places $T$ disjoint from $S$ such that the conjugacy classes of geometric Frobeni $F\_v$ ($v\in T$) fill up $\mathrm{Gal}(K^\prime/K)$ for any $K^\prime/K$ Galois of degree at mos... | https://mathoverflow.net/users/22189 | A question on Cebotarev's density theorem | **Edit:** I understand that you are happy with a $T$ which depends upon $L$ and works simultaneously for all $L'/L$ of degree bounded by $d$ and unramified outside of $S\_L$. If you are looking for *a uniform* $T$ which does the job for *all possible $L$ at once*, then the answer is no, as explained in David's comment.... | 1 | https://mathoverflow.net/users/18238 | 110132 | 63,297 |
https://mathoverflow.net/questions/109515 | 11 | Take an infinite hexagonal lattice (or equivalently, an equilateral triangular lattice), with unit spacing between the closest lattice point pairs, and draw a disc of radius $r$ centered on a lattice point at $(0, 0)$. Let $N(r, hex)$ denote the number of hexagonal lattice points at coordinates $(a, b)$ s.t. $(a^2 + b^... | https://mathoverflow.net/users/27203 | The Gauss circle problem on a hexagonal lattice | Lax and Phillips (J. Funct. Anal. vol 46 (1982), 280--350) showed, for any crystallographic group $\Gamma$ in the Euclidean plane, that
$$
N(r;x,x\_0)= \frac{\pi r^2}{|F|} + O(r^{2/3} (\log r)^{1/2}),
$$
as $r\to+\infty$, where $|F|$ denotes the volume of the fundamental domain of $\Gamma$, $x,x\_0\in\mathbb R^2$ and $... | 14 | https://mathoverflow.net/users/20052 | 110143 | 63,301 |
https://mathoverflow.net/questions/109248 | 9 | Let $A\_f$ be the abelian variety over $\mathbf{Q}$ arising as a $\mathbf{Q}$-simple factor of the Jacobian $J\_0(N)$ of the modular curve associated to a normalized newform $f$ of weight $2$ on the congruence subgroup $\Gamma\_0(N)$. There is a piece $\mathbf{T}\_f$ of the Hecke ring acting on $A\_f$. Such a ring is a... | https://mathoverflow.net/users/4800 | On simple factors of modular jacobians: endomorphism ring and simplicity of mod p reduction | I found a reference for the first question in the following PhD thesis :
J. Wilson, *[Curves of genus 2 with real multiplication by a square root of 5](http://eprints.maths.ox.ac.uk/00000032/01/wilson.pdf)*
The answer is yes. This is a consequence of the following general fact about abelian varieties (see Prop 2.5.... | 6 | https://mathoverflow.net/users/6506 | 110146 | 63,303 |
https://mathoverflow.net/questions/110148 | 5 | I was reading Postnikov's paper [TOTAL POSITIVITY, GRASSMANNiANS, AND NETWORKS][1] when I came across the definition of the totally nonnegative Grassmannian $Gr\_{kn}^{tnn} \subset Gr\_{kn}$ as the quotient
$Gr\_{kn}^{tnn} = GL\_k^{+}\backslash Mat^{tnn}\_{kn}$,
where
$Mat^{tnn}\_{kn}$ is the set of real $k\times n$... | https://mathoverflow.net/users/39184 | On totally nonnegative Grassmannian | There are two issues here:
1. You can represent a $k$-subspace of $V$ by a $k \times n$ matrix whose rows are a basis for the subspace, but this way of representing the subspace is not unique, since your subspace has many different bases. The usual way to handle this is to consider cosets of $k\times n$ matrices unde... | 11 | https://mathoverflow.net/users/23408 | 110150 | 63,304 |
https://mathoverflow.net/questions/44166 | 9 | In "The Boardman-Vogt resolution of operads in monoidal model categories," the authors construct factorizations of sufficiently nice operad maps $P\to Q$ into a cofibration followed by a weak equivalence $P\rightarrowtail W(H,Q)\_P\stackrel{\sim}{\to}Q$. This construction depends on a choice of interval $H$. When $P$ i... | https://mathoverflow.net/users/3075 | How does Berger-Moerdijk's relative Boardman-Vogt work? | It turns out that the construction as written in the paper is incorrect, but there is a fix known to the authors. Instead of collapsing $P\otimes H\otimes P$ to $P$, one only collapses $I\otimes P\otimes H\otimes P\otimes I$ (where the $I$ factors are included in $H$ with length 1, not 0). This gets rid of the problem ... | 8 | https://mathoverflow.net/users/3075 | 110163 | 63,307 |
https://mathoverflow.net/questions/110154 | 4 | **Question:** Suppose $a(x,y)\in C^\infty([0,1]\times [0,1])$ and suppose
$$\sup\_{\lambda>1} \bigg|\lambda\int\_0^1 e^{\lambda x} a(x,1/\lambda)dx\bigg|<\infty.$$
Is $a(x,0)=0$, $\forall x\in[0,1]$?
**Comments:** Here are some side comments about a special case. Suppose $a$ is independent of $y$, then the res... | https://mathoverflow.net/users/27404 | Quantitative Weierstrass Approximation and Paley-Wiener for the Laplace Transform | Unless I misunderstand the question, the answer is that $a(x,0)$ can be pretty much anything it wants.
Take any smooth $f(x)$ supported on $[0,1-\delta]$. Put
$$
a(x,y)=f(x)-[y(e^{1/y}-1)]^{-1}\int\_{0}^{1}f(t)e^{t/y}\\,dt.
$$
It looks like a $C^\infty$ function to me because the exponent $e^{\delta/y}$ in the deno... | 1 | https://mathoverflow.net/users/1131 | 110170 | 63,310 |
https://mathoverflow.net/questions/110167 | 2 | I need a reference/proof for "main conjecture of Iwasawa theory => refinement of Herbrand-Ribet ($v\_p(B\_{p-i}) = v\_p(|A\_i|)$, where $A\_i$ denotes the $i$-th eigenspace of the Galois group acting on the ideal class group mod $p$)".
| https://mathoverflow.net/users/nan | main conjecture of Iwasawa theory implies Herbrand-Ribet | The statement and proof are given in Mazur–Wiles starting at page 216.
| 4 | https://mathoverflow.net/users/1021 | 110171 | 63,311 |
https://mathoverflow.net/questions/110149 | 1 | Assume $\mathbf{G}$ is a simple adjoint algebraic group over an algebraic closure $\overline{\mathbb{F}\_p}$ of the finite field of characteristic $p > 0$ and $u \in \mathbf{G}$ is a unipotent element. Denote by $\mathfrak{B}\_u$ the variety of all Borel subgroups of $\mathbf{G}$ containing $u$. Assume now that $\mathc... | https://mathoverflow.net/users/22846 | Action of Frobenius on the Étale Cohomology of the variety of Borel subgroups for an arbitrary local system | First, note that you only get an induced $F^\* $ map on cohomology if you fix a map $F^\* \mathcal F \to \mathcal F$. Note that you get this for free if $\mathcal F$ is defined over $\mathbb F\_p$.
The Frobenius action on the cohomology group of a sheaf is the main object of study in etale cohomology over a finite fi... | 3 | https://mathoverflow.net/users/18060 | 110174 | 63,312 |
https://mathoverflow.net/questions/110180 | 5 | Is the stable homotopy category idempotent complete? I have not been able to prove it, and the proof for abelian groups seems to strongly rely on looking at elements.
Thanks,
Jon
| https://mathoverflow.net/users/11546 | Is the stable homotopy category idempotent complete? | Yes, this is a standard fact. Given a self-map $e\colon X\to X$, we write $e^{-1}X$ for the telescope of the sequence $X\xrightarrow{e}X\xrightarrow{e}X\xrightarrow{e}\dotsb$ (constructed as the cofibre of a suitable self-map of $\bigvee\_{i=0}^\infty X$). If $e$ is idempotent, one can check that the natural map $X\to ... | 14 | https://mathoverflow.net/users/10366 | 110182 | 63,317 |
https://mathoverflow.net/questions/110185 | 2 | Let T be a first order theory whose set of axioms is one of the standard finite sets of axioms for an
arbitrary group. Is it possible to have an axiomatizable first order extension of T which would
characterize just those groups that are finite, simple and "sporadic"? In other words I am asking
whether the properties ... | https://mathoverflow.net/users/4423 | A question about formulating first order axioms for group theory. | Because there are only finitely many sporadic simple groups, you can write, for each of them, a first-order sentence characterizing it up to isomorphism, and then you can form the disjunction of those sentences. But you don't want to build into your sentences our knowledge of the sporadic groups, and then you run into ... | 4 | https://mathoverflow.net/users/6794 | 110191 | 63,320 |
https://mathoverflow.net/questions/110189 | 1 | Hi,
i have two questions that seem to bother me lately. Maybe you could help me and/or point me in any related literature.
1) Assume hermitian matrix $H \in \mathcal{C}^{n \times n}$ that has rank $r$. How many real valued parameters are needed to describe it?
2) Now assume that you also have the knowledge that $... | https://mathoverflow.net/users/17246 | Number of parameters needed to specify a Hermitian matrix of rank r. | The answer to 1) is simple. An Hermitean matrix is determined by its eigenvalues and
eigenspaces. Assign $r$ distinct eigenvalues ($r$ real parameters). Their
eigenspaces are orthogonal lines. One line depends on $n-1$ complex parameters, the next line
must be perpendicular to the first, so it depends on $n-2$ complex ... | 5 | https://mathoverflow.net/users/25510 | 110195 | 63,323 |
https://mathoverflow.net/questions/110186 | 3 | In a previous question: ([The Gauss circle problem on a hexagonal lattice](https://mathoverflow.net/questions/109515/the-gauss-circle-problem-on-a-hexagonal-lattice)) I asked for an analytic approximation for the number of lattice points in or along the contour of a circle centered on a lattice point in an $A\_2$ hexag... | https://mathoverflow.net/users/27203 | An exact counting solution for the number of points within a circle of radius $r$ centered on a lattice point in a $A_2$ hexagonal lattice | This Mathematica formula reproduces the numbers in the OEIS and should be self explanatory:
>
> n[r\_] := Sum[
> 1 + 2 Floor[Sqrt[r^2 - 3 x^2]], {x, -Floor[r/Sqrt[3]],
> Floor[r/Sqrt[3]]}] +
> Sum[2 Floor[
> Sqrt[r^2 - 3 x^2] + 1/2], {x, -Floor[(r/Sqrt[3]) + 1/2] + 1/2,
> Floor[(r/Sqrt[3]) + 1/2] - 1/2}]
> ... | 2 | https://mathoverflow.net/users/20186 | 110201 | 63,327 |
https://mathoverflow.net/questions/103319 | 13 | The following dynamical system on polynomials comes mostly from idle curiosity, but I hope it is of some interest.
**Background** Fix some natural number $n$. Let $P$ be the quotient of the polynomial ring $\mathbb{Z}[x\_1,\ldots,x\_n]$ by the ideals $\lbrace(x\_j^2=x\_j)~|~1\leq j \leq n\rbrace$.
Let $\Sigma(n)$ d... | https://mathoverflow.net/users/18263 | Symmetric group action on squarefree polynomials | Interesting things happen if you reduce this dynamical system modulo some integer $N$.
This makes sense because your map $s$ is compatible with reduction modulo $N$, and your map $e$ is $\mathbf{Z}$-linear, so $f$ induces a map $f\_N : P\_N \to P\_N$ with $P\_N = P \otimes \mathbf{Z}/N\mathbf{Z}$.
In the case $N=2$... | 2 | https://mathoverflow.net/users/6506 | 110202 | 63,328 |
https://mathoverflow.net/questions/108233 | 2 | Let $K$ be a compact subset in $\mathbb{R}^n$ with $m(K)=0$, Suppose $supp\hat{u}\subset K$ for some $u\in L^p$,where $2\leq p\leq \frac{2n}{n-1}$,can we get $u\equiv 0$ ?
*Motivation:* If $K$ is a compact non-degenerate hypersurface,then it's well known that $u(x)\leq C|x|^{-\frac{n-1}{2}}$,hence $u\in L^p$ for any ... | https://mathoverflow.net/users/23078 | Fourier transform and spectrum of PDOs in $L^p$ | I discussed this problem with fedja (Overflow pen-name),
and he explained to me that the answer is no. As fedja is apparently busy, I post the answer:-) One cannot
improve the exponent $p=2$ in any dimension. The reason is that for every $p>2$ there exists a distribution on
the line whose support has Lebsegue measure ... | 2 | https://mathoverflow.net/users/25510 | 110204 | 63,330 |
https://mathoverflow.net/questions/110208 | 20 | I have a really hard time "feeling" what it means for a group to fail to be linear. Vaguely, I'd like to know how one should think about such groups. More precisely:
What are some interesting examples of groups that aren't linear?
Are there general constructions that one can use to cook up a group or a family of gr... | https://mathoverflow.net/users/22221 | Understanding groups that are not linear | Consider the class of finitely generated linear groups. Such groups $G$ satisfy certain well-known restrictions, for instance:
1. Every such $G$ is residually finite (Malcev, 1940). Thus, most Baumslag-Solitar groups, e.g.
$$
\langle a, b| a b^2 a^{-1} =b^3\rangle
$$
are not linear. This is the simplest example of ... | 40 | https://mathoverflow.net/users/21684 | 110213 | 63,335 |
https://mathoverflow.net/questions/110092 | 8 | The Ising model on $\mathbb{Z} / 2d\mathbb{Z}$ gives to the configuration $x=(x\_0, \ldots, x\_{2d-1}) \in \{-1,+1\}^{2d}$ a probability proportional to $\exp\\big(\beta \sum\_i x\_ix\_{i+1} \\big)$. The Gibbs sampler with block updates is a Markov chain $X\_k$ that evolves on the set of such configurations and updates... | https://mathoverflow.net/users/1590 | Ising model on a cycle | This is not so surprising, and is related to the lack of phase transition in the one dimensional Ising model.
Consider first why the mixing time might be large. If $\beta$ is very high, and we start with a configuration where half the circle is + and half -, it will take a fairly long time for the chain to converge t... | 7 | https://mathoverflow.net/users/9422 | 110222 | 63,343 |
https://mathoverflow.net/questions/84592 | 1 | I think this is an easy question, but I need some time to introduce it. I need to apply Yumiko Hironaka's computations on local densities of hermitian forms (see [1](http://www.sciencedirect.com/science/article/pii/S0022314X98922375)).
I would have liked to create the new tag [local-densities], but I don't have enough ... | https://mathoverflow.net/users/20052 | Local densities of hermitian forms | Several months later, I have the answer to my question. I would like to share here.
Such as Maurice Mischler (Local densities of hermitian forms, in Contemporary Mathematics vol 272 (2000), 201--208) mentioned in his abstract, Hironaka compute the local density only in the inert prime case ($p\mathcal O$ is a prime i... | 4 | https://mathoverflow.net/users/20052 | 110226 | 63,346 |
https://mathoverflow.net/questions/110235 | 0 | Some reference say that if rank($A$)=rank($A^2$),then the geometric and algebraic multiplicities of the eigenvalues $\lambda=0$ are equal;that is,all the Jordan blocks correspondint to $\lambda=0$ (if any) in the Jordan form of A are 1\*1.
I don't know how to check out this conclusion.Could you help me ? Thank you!
... | https://mathoverflow.net/users/25007 | a question about the Jordan form | Take one Jordan block with zero eigenvalue. Suppose it is of size mxm.
It has zeros everywhere except on the diagonal next to the main, total m-1 ones,
so the rank is m-1. Now square this matrix (squaring the whole matrix is equivalent to
squaring all its Jordan blocks. The result is the matrix which has zero everywher... | 1 | https://mathoverflow.net/users/25510 | 110236 | 63,350 |
https://mathoverflow.net/questions/110140 | 2 | Hallo,
I am reading the paper "Hyperkaehler structures on the total space of holomorphic cotangent bundles" by D.Kaledin and I am asking if it is possible to embedd a real-analytic Kähler manifold, isometrically, as a special Lagrangian in a Calabi-Yau manifold. Acctualy what I am looking for is the following: Start ... | https://mathoverflow.net/users/27402 | Isometric embedding of a Kaehler manifold as a special Lagrangian in a Calabi-Yau manifold | **Disclaimer:** I am not sure what kind of "explanation" you are looking for. I would guess that you are after the observation (due to Hitchin), that complex Lagrangian submanifolds become special Lagrangian after rotating the complex structure.
**Observation:**
Let $X$ be a hyperkaehler manifold. Let $\{I,J,K\}$ be ... | 2 | https://mathoverflow.net/users/6278 | 110246 | 63,352 |
https://mathoverflow.net/questions/110238 | 3 | I have a circle of radius $r$, and I wish to place this circle of a $Z^2$ integer lattice or an $A\_2$ hexagonal lattice s.t. I maximize the number of lattice points within or along the contour of the circle.
Trivially, for $r < \frac{R}{2}$, where $R$ is the smallest spacing between lattice points, it will be optim... | https://mathoverflow.net/users/27203 | Maximizing the number of lattice points in a circle of radius $r$ placed on a lattice | I want to say that by symmetry, the only relevant centers are lattice points, midpoints of square edges, and centers of squares, and that they each are optimal for varying values of $r > \sqrt(2)$.
If we take $r = 1.55$, then the circle centered at a lattice point contains $9$ lattice points, the circle centered on a... | 2 | https://mathoverflow.net/users/19029 | 110248 | 63,354 |
https://mathoverflow.net/questions/110239 | 12 | By Gauss's Theorem, every positive integer $n$ is a sum of three triangular numbers;
these are numbers of the form $\frac{m(m+1)}2$. Clearly
$$ n = \frac{m\_1^2+m\_1}2 + \frac{m\_2^2+m\_2}2 + \frac{m\_3^2+m\_3}2, $$
so multiplying through by $4$ and completing the squares gives
$$ 8n+3 = (2m\_1+1)^2 + (2m\_2+1)^2 + (2... | https://mathoverflow.net/users/27430 | Is there an algorithm for writing a number as a sum of three squares? | A representation of $n$ as a sum of three triangular numbers is equivalent to representing $8n+3$ as a sum of three odd squares. The question of computing representations as a sum of three squares has been much discussed here, see [Efficient computation of integer representation as a sum of three squares](https://matho... | 7 | https://mathoverflow.net/users/11142 | 110253 | 63,356 |
https://mathoverflow.net/questions/110252 | 6 | Let $f: M \rightarrow \mathbb{R}^3$ be an immersion of a surface $M$. For pedagogical purposes (i.e., I'm teaching a class!) I am looking for an expression for the scalar Laplace-Beltrami operator $\Delta$ applied to a real function $\phi$ on $f(M)$ that:
1. explicitly depends on the immersion $f$,
2. does not rely o... | https://mathoverflow.net/users/1557 | Surface Laplace-Beltrami without coordinates, exterior calculus? | You probably will disallow this, but the following recipe does work:
First, let $\nabla\phi:M\to\mathbb{R}^3$ be the (unique) vector-valued function that satisfies
$$
d\phi(X) = \nabla\phi\cdot df(X)\qquad\text{and}\qquad \nabla\phi\cdot N = 0.
$$
for all vector fields $X$ on $M$. Then $\Delta\phi:M\to\mathbb{R}$ is ... | 4 | https://mathoverflow.net/users/13972 | 110254 | 63,357 |
https://mathoverflow.net/questions/110243 | 3 | If $G$ is an affine group scheme over a field $k$, then we have the forgetful functor $\omega$ from the category ${\rm Rep}\_k(G)$ of finite representations of $G$ to the category of finite dimensional $k$-vector spaces. Let $S$ be a $k$-scheme, with the structure map $p: S\to k$. Then ${\rm Aut}\_S(\omega)$ (the tenso... | https://mathoverflow.net/users/18380 | automorphism of fibre functors | Yes they are.
In fact the formation of ${\rm Aut}\_S(\omega)$ commutes by definition with base change, so you can reduce to $S=k$, and $S\to k={\rm id}$.
Then the fact that the $k$-group morphism $G\to {\rm Aut}\_k(\omega)$ is an isomorphism is the reconstruction theorem in Tannaka theory, see Proposition 2.8 in th... | 6 | https://mathoverflow.net/users/11682 | 110262 | 63,361 |
https://mathoverflow.net/questions/110257 | -1 | We have four random variables say W,X,Y,Z where W and X has the same distribution and Y, Z also has the same distribution. Bad news is EX and EY may not exist but E(W+Z) is zero. Could we conclude that E(X+Y) is zero? ( I know if EX and EY where defined we used linearity and it is obvious, also we know nothing more abo... | https://mathoverflow.net/users/27415 | Expectation where linearity does not hold | Consider the following example. $W, X, Y, Z$ all have standard Cauchy distribution, but $X$ and $Y$ are independent while $Z = -W$. Then $X+Y$ also has a Cauchy distribution so its expected value does not exist, while $W+Z=0$ identically.
EDIT: With the additional assumption that $E[X+Y]$ exists, I believe you'll fin... | 4 | https://mathoverflow.net/users/13650 | 110267 | 63,366 |
https://mathoverflow.net/questions/104322 | 18 | Recently I've been studying the problem of integer representation as sum of three squares. Most of the articles that I've found study the function $r\_m(n)$ which counts the number of representations of $n$ as the sum of $m$ squares. However, this is not what I am interested in. What I'm looking for is an efficient way... | https://mathoverflow.net/users/22733 | Efficient computation of integer representation as a sum of three squares | This problem is discussed in my paper with Rabin, Randomized algorithms in number theory,
Commun. Pure Appl. Math. 39, 1985, S239 - S256. We give an algorithm that, assuming a couple of reasonable conjectures, will produce a representation as a sum of three squares in polynomial time.
| 25 | https://mathoverflow.net/users/44797 | 110277 | 63,372 |
https://mathoverflow.net/questions/110250 | 7 | This question is motivated by Mariano's comment on this question:
[transcendence degree of subring of polynomial ring](https://mathoverflow.net/questions/110060/transcendence-degree-of-subring-of-polynomial-ring)
Suppose $k$ is a field and the subring $R$ of the polynomial ring $k[x\_1,...,x\_r]$ is generated by h... | https://mathoverflow.net/users/24759 | Is a polynomial ring integral over this subring ? | There is a nice lemma that relates ideals and subalgebras in graded rings that can be applied to the problem:
>
> **Lemma:** Let $A=\bigoplus\_{n \ge 0}A\_n$ be a graded ring that is commutative (or graded commutative). Then for homogeneous elements $p\_i \in A\; (i \in I)$ of positive degree are equivalent:
>
>... | 7 | https://mathoverflow.net/users/10194 | 110282 | 63,374 |
https://mathoverflow.net/questions/110211 | 31 | Let $X$ be a finite set of $n$ elements, and consider a binary operation $\odot: X \times X \rightarrow X$. There are $n^{n^2}$ such binary operations, as the $n \times n$ table entries can each
be filled with one of $n$ elements of $X$. My question is:
>
> How many of the $n^{n^2}$ binary operations are associativ... | https://mathoverflow.net/users/6094 | How many binary operations are associative? | Here is a guide to the intuition. I will not swear that the numerics are exact, but I will bet
that the numerical truth is not far off.
Look at the diagonal for the multiplication table of a (labeled) groupoid on $n>3$ elements.
Of the n^n possibilities, only one of them is idempotent, so with one exception aa=b wil... | 5 | https://mathoverflow.net/users/3493 | 110283 | 63,375 |
https://mathoverflow.net/questions/110273 | 9 | The Fontaine-Mazur conjecture predicts that an $l$-adic Galois representation of a number field is 'geometric' if it is unramified outside a finite set of primes and is De Rham for primes lying over $l$. Now, what happens if one forgets about the latter restriction; are there any counterexamples, and is there any (gene... | https://mathoverflow.net/users/2191 | Can one ignore primes lying over $l$ in the Fontaine-Mazur conjecture? Counterexamples? | To complete Kevin's good answer: the number of $\ell$-adic representations (up to isomorphism) of a number field $K$ is countable, since so are varieties over a $K$. On the other hand, we know by Mazur's theory of deformations that representations of the type you consider
that is, of the Galois group of the maximal ext... | 12 | https://mathoverflow.net/users/9317 | 110288 | 63,376 |
https://mathoverflow.net/questions/110156 | 7 | Let $R[f](p,t)$ denote the Radon transform of smooth function $f(x) \colon \mathbb{R}^n \to \mathbb{R}$ with compact support in $\mathbb{R}^n\_+$:
$$
R[f](p,t) = \int\limits\_{x \cdot p = t} f(x) dx.
$$
In field of mathematical economics $p$ and $t$ may have meaning of prices and then be positive. Hence there arises a... | https://mathoverflow.net/users/17896 | Inversion of Radon transform by incomplete data: specific case | I second the recommendations in the comments which suggest looking at the tomography literature and at the characterization of the problem as an ill-posed inverse problem. I, in particular, recommend these two references.
The first one contains a detailed exposition of the recovery of $f$ in the special case $n=2$ (c... | 4 | https://mathoverflow.net/users/25326 | 110289 | 63,377 |
https://mathoverflow.net/questions/110293 | 9 | Given a finitely presented group $G$, how many necessary conditions do people know for $G$ to be isomorphic to the fundamental group of some closed connected 3-manifold? (e.g. residually finite)
| https://mathoverflow.net/users/27253 | Judging whether a finitely presented group is a 3-manifold group? | Apologies for the shameless self-promotion, but as you ask for *necessary* conditions, you seem to want a list of theorems of the form 'If G is a 3-manifold group then G has property P'.
Aschenbrenner, Friedl and I have given what I think is a pretty exhaustive list in our survey paper [here](http://arxiv.org/abs/120... | 15 | https://mathoverflow.net/users/1463 | 110311 | 63,385 |
https://mathoverflow.net/questions/110309 | 8 | Consider the abelian (Grothendieck) category $\mathcal{C} := \mathrm{Fun}(\{0<1\},\mathrm{Ab}) = \mathrm{Mor}(\mathrm{Ab})$. Objects are morphisms $(A \to B)$ of abelian groups, morphisms are commutative diagrams. Equivalently, this is the category of abelian sheaves on the Sierpinski space.
**Question.** How do inje... | https://mathoverflow.net/users/2841 | Injective objects in Mor(Ab) | I will use notation $A\_0 \to A\_1$ for objects of $\mathrm{Mor}(\mathrm{Ab})$.
EDIT: previously I claimed something stronger (that I can produce lifting properties in the functor category without factorizations), but I am not so sure about it.
The following is a lot more general than necessary, but I think this ad... | 3 | https://mathoverflow.net/users/12547 | 110319 | 63,386 |
https://mathoverflow.net/questions/110315 | -1 | let $G$ be a finite non-abelian simple group.If there exist $p$ and $q$ which are different prime numbers of $|G|$ such that $n\_p(G)=n\_q(G)$?
| https://mathoverflow.net/users/27449 | Question on the equal Sylow number in finite non-abelian simple group | Not sure if this question really qualifies for MO.
Anyway, the answer very much depends on the group $G$. In most cases $n\_p(G)\ne n\_q(G)$ for distinct prime divisors of the group order. However, there are infinitely many examples where equality occurs: If $r$ is an odd prime, then $n\_p(\text{PSL}(2,r))=r(r+1)/2$ ... | 2 | https://mathoverflow.net/users/18739 | 110320 | 63,387 |
https://mathoverflow.net/questions/110301 | 5 | I'm trying to use the language of strata to organize $K$-types of irreducible smooth representations of $GL(2)$ (and then hopefully prove things). Unfortunately, I'm still new to it, so I might be making some mistakes.
My main reference is the Bushnell-Henniart book. For anyone who doesn't have access to it and wants... | https://mathoverflow.net/users/6753 | Strata of K-types appearing in irreducible representations of p-adic GL(2) | I will try to answer the question as far as I have understood it. Please comment.
Clifford's theorem
Let $G$ be a finite group. Let $H$ be a normal subgroup of $G$. The groups $G$ resp. $G/H$ act on the irreducible representations of $H$ via conjugation on $H$.
1. Let $\rho$ be an irreducible representation of $G... | 3 | https://mathoverflow.net/users/10400 | 110322 | 63,388 |
https://mathoverflow.net/questions/110281 | 6 | This seems like a natural question to ask, but I've not seen it discussed in my reading around (limited to Easton's paper, the third edition of Jech's *Set theory* and a small handful of articles). What *do* you get when forcing with a proper class of conditions that you don't get when forcing with, say, a set of condi... | https://mathoverflow.net/users/4177 | Proper class forcing vs forcing with a set of conditions bigger than one's model | The idea behind the remark quoted in the question was that, in situations ordinarily treated with proper-class forcing (e.g., Easton's theorem), the work can be transcribed rather routinely into a Feferman-style set theory (ZFC plus a constant $\kappa$ for an ordinal and axioms saying, one formula at a time, that $V\_\... | 4 | https://mathoverflow.net/users/6794 | 110323 | 63,389 |
https://mathoverflow.net/questions/110317 | 1 | In M. Reid Canonical 3-folds I found this proposition:
If $\phi:Y\rightarrow X$ is a proper morphism with both $X$ an $Y$ normal and such that $f$ is étale in codimension 1 then
1) if $X$ has canonical singularities so does $Y$
2) if $Y$ has canonical singularities and $X$ is Gorenstein, then $X$ has canonical si... | https://mathoverflow.net/users/6949 | Doubt about normality and rational singularities | First, I don't think Miles Reid is dealing with pairs in that paper.
Second, I think Miles wants $f$ to be etale in codimension 1 *on $Y$*. Thus a blowup is not allowed unless the blowup is a small map. You need every divisor on $Y$ to really have image of as divisor on $X$. You can see Miles Reid using this in his p... | 4 | https://mathoverflow.net/users/3521 | 110325 | 63,390 |
https://mathoverflow.net/questions/110279 | 3 | Suppose that $R:=k[x\_0,\dots,x\_n]$ and $I$ is an ideal. Is there any relation between finding the minimal generators of $I$ and the graded betti numbers of the module $R/I$?
| https://mathoverflow.net/users/23874 | Betti Numbers and number of generators | To support J.C. Ottem's answer, let me present one example.
Let $R = \mathbb{C}[x,y]$ and $I = (x,y^2)R$. What is the minimal graded free resolution of $R/I$, equivalently $I$? That is,
$0 \rightarrow R(-3) \stackrel{d\_1}{\rightarrow} R(-1) \oplus R(-2) \stackrel{d\_0}\rightarrow R \rightarrow R/I \rightarrow 0 $... | 1 | https://mathoverflow.net/users/22388 | 110333 | 63,392 |
https://mathoverflow.net/questions/110338 | 3 | From the Thesis of B.Hassett I seem to understand that a smooth cubic 4-fold $X$ containing a $\mathbb{P}^2$ should contain also a octic K3, but I cannot see a natural way by which this K3 octic could appear inside $X$. How do you see that?
One wild guess is: via linkage of a quartic 3-fold that contains a quadric su... | https://mathoverflow.net/users/4096 | octic K3s inside cubic 4-folds | The $\mathbb{P}^{2}$ contained in your cubic fourfold $X$ is cut out by linear forms (say) $L\_{1},L\_{2},$ and $L\_{3}.$ Since the homogeneous ideal of $X$ is contained in the homogeneous ideal generated by $L\_{1},L\_{2}$ and $L\_{3},$ there exist quadrics $Q\_{1},Q\_{2},$ and $Q\_{3}$ such that $X=\{L\_{1}Q\_{1}+L\_... | 10 | https://mathoverflow.net/users/5496 | 110341 | 63,396 |
https://mathoverflow.net/questions/110332 | 9 | Hi all.
I've had this idea - a conjecture in the field of Number Theory - for a few years now.
The conjecture is rather simple, as were the logical steps that I made in order to infer it, so I would have assumed that it had already been suggested in the past. Nevertheless, I have not been able to find any piece of ev... | https://mathoverflow.net/users/27456 | A conjecture in Number Theory | Even conjecture 1. is false: For $S=\{(3n+2)/(2n+1)\;|\;n=9, 12, 14, 27, 41\}$ your product is $8$.
| 39 | https://mathoverflow.net/users/18739 | 110348 | 63,399 |
https://mathoverflow.net/questions/110345 | 75 | Consider a complex power series $\sum a\_n z^n \in \mathbb C[[z]]$ with radius of convergence $0\lt r\lt\infty$ and suppose that for every $w$ with $\mid w\mid =r$ the series $\sum a\_n w^n $ converges .
We thus obtain a complex-valued function $f$ defined on the closed disk $\mid z\mid \leq r$ by the formula $f(z)=... | https://mathoverflow.net/users/450 | Does a power series converging everywhere on its circle of convergence define a continuous function? | I searched all over for an answer to this question back in my student days. I found the answer in a paper by Sierpinski, "Sur une série potentielle qui, étant convergente en tout point de son cercle de convergence,représente sur ce cercle une fonction discontinue ", which is featured in his collected works, [see here](... | 66 | https://mathoverflow.net/users/8923 | 110350 | 63,400 |
https://mathoverflow.net/questions/110321 | 3 | I am interested in the following problem: Let $P$ be a $n\times n$ complex finite matrix such as $PP^\dagger =W$. Given $W$, what can I say about the spectrum of $P$?
This matrix "square-root" has of course no unique solution, for if $P$ is a solution, $PU$ is also a solution if $U$ is unitary. If we consider the sp... | https://mathoverflow.net/users/27455 | Effects of unitarian multiplication into the spectrum of a finite matrix. | By specifying $PP^{\dagger}=W$ you are prescribing the singular values $s\_n$ ($n=1,2,\ldots N$) of the $N\times N$ matrix $P$. These are just the positive square roots of the eigenvalues of the Hermitian, nonnegative matrix $W$. So your question can be rephrased as, what is the relation between the eigenvalues $\lambd... | 7 | https://mathoverflow.net/users/11260 | 110352 | 63,401 |
https://mathoverflow.net/questions/110285 | 7 | $\def\mc{\mathcal} \def\sm{\wedge}$
This question stems from the Goerss-Hopkins paper [Moduli Problems for Structured Ring Spectra](http://www.math.northwestern.edu/~pgoerss/spectra/obstruct.pdf). Let me begin by attempting to summarize the relevant framework -- this comes from the beginning of Chapter 3, on page 92.
... | https://mathoverflow.net/users/303 | On the natural (bigraded) homotopy groups of a simplicial object in a model category | Dear Aaron,
For question 1, you're correct in that all objects are cogroup objects. This isn't necessary for the original version from Bousfield's paper--only one of the suspensions is--and there you do need to make sure that your objects you're using to define the model structure really are cogroup objects. (You can... | 4 | https://mathoverflow.net/users/360 | 110354 | 63,402 |
https://mathoverflow.net/questions/110346 | 2 | If the probability distribution function of two sequences of random variables have the same weak limit and one of the sequences satisfies a Large deviation principle, then does it imply that the other one also satisfies a LDP with the same rate function? Here is a more precise version of my question:
Let
$$ X\_n: (... | https://mathoverflow.net/users/4463 | If two probability distributions have the same weak limit and one of them satisfies Large Deviation Principle, what can we say about the other? | No and no. To see why the limit need not exist, suppose $\mu = \delta\_0$ and $X^\prime\_n = \frac{(-1)^n}{n}$ a.s. Then $\liminf \frac{-1}{n}\log \mu^\prime\_n(0,\infty) = 0$ while $\limsup \frac{-1}{n}\log \mu^\prime\_n(0,\infty) = \infty$. Even if the limit exists, it need not be the same limit. Suppose $\mu = \delt... | 3 | https://mathoverflow.net/users/26459 | 110355 | 63,403 |
https://mathoverflow.net/questions/110351 | 8 | I've been trying to read Gross' paper on Heegner points on $X\_0(N)$ and I am stuck on a few details. The definition he is working with is that a heegner points is a pair $y=(E,E')$, where $E$ and $E'$ are elliptic curves admitting an isogeny that has cyclic kernel of order $N$ and where $E$ and $E'$ both have complex ... | https://mathoverflow.net/users/27464 | Heegner Points and Binary Quadratic Forms | I think the following facts, which you can find in Cox's book *Primes of the Form $x^2+ny^2$*, will alleviate your confusion. First off, if ${\mathfrak a}=[\alpha,\beta]$ is a proper ideal of ${\mathcal O}$ then one can show that
$$ f(x,y) := \frac{N(\alpha x-\beta y)}{N{\mathfrak a}} $$
is a primitive binary quadratic... | 6 | https://mathoverflow.net/users/430 | 110358 | 63,406 |
https://mathoverflow.net/questions/110330 | 0 | I have the following question:
Let $f$ be an analytic function satisfying the functional equation: $f(z)=u(z)f(a-z)$ where $a$ is a real constant. Let $g$ be another function satisfying the same functional equation. In this case I asking if $f=g$.
Thank you in advance.
| https://mathoverflow.net/users/25947 | functional equation | In the solution by Robert Israel, the uniqueness question (which was asked) is not addressed.
Of course one cannot conclude that $f=g$, because if $f$ satisfies the equation,
and $p$ is any function such that $p(z)=p(a-z)$ then $fp$ satisfies the equation.
And this is the general description of all solutions because ... | 2 | https://mathoverflow.net/users/25510 | 110363 | 63,407 |
https://mathoverflow.net/questions/110364 | 7 | Are there any examples of two large cardinal axioms $AX$ and $AY$, in the language of first order $ZFC$, which satisfy the following conditions.
1. Each of them defines a unique cardinal number - $C(AX)$ for $AX$
and $C(AY)$ for $AY$ - not like the axiom of measurable cardinals which defines a whole collection of car... | https://mathoverflow.net/users/4423 | A question about the comparability of large cardinals. | Let AX = "there exists a least strongly compact cardinal", and let AY be "there exists a least whatever cardinal" (there are many possibilities for "whatever", e.g., "inaccessible with a measurable below it" if you want to be modest).
C(AX) = the least strongly compact, C(AY) = the least whatever, say the least inac... | 14 | https://mathoverflow.net/users/14915 | 110366 | 63,408 |
https://mathoverflow.net/questions/110381 | 1 | Let $A$ be a commutative Noetherian ring, $R$ be a standard graded algebra over $A$, $M$ be finitely generated graded $R$-module. Let $R\_{+}$ be the irrelevant ideal. The Castelnuovo-Mumford regularity of $M$ or regularity for short is defined to be : $\text{reg}(M):=\text{max}\lbrace a(H\_{R\_+}^{i}(M))+i|i\ge 0\rbra... | https://mathoverflow.net/users/27041 | Filter-regular sequence and regularity | I've posted an answer on [M.SE](https://math.stackexchange.com/questions/219265/equivalent-definiton-of-castelnuovo-mumford-regularity/219302#219302).
| 1 | https://mathoverflow.net/users/23950 | 110411 | 63,421 |
https://mathoverflow.net/questions/110414 | 4 | I'm having difficulties to prove that the subset of even numbers is not first-order definable in $(\mathbb{N},<)$. Any hint is welcomed!
More generally, what are usual techniques in order to prove that a subset is not FO-definable. I know one using isomorphism.
What are other methods?
| https://mathoverflow.net/users/41060 | Methods for proving non FO definability | One method is to use quantifier elimination. Note that 0 and $S(x)=x+1$ are definable in $M=(\mathbb N,<)$. The expanded structure $M'=(\mathbb N,<,0,S)$ is a model of the theory $T$ of a discrete linear order with zero and successor. You can show that every formula is in $T$ equivalent to an open formula (it suffices ... | 9 | https://mathoverflow.net/users/12705 | 110419 | 63,426 |
https://mathoverflow.net/questions/110412 | 0 | Let $S$ be a semigroup. The power semigroup of $S$ is the set $P(S)=2^S\setminus\lbrace\varnothing\rbrace $ with the operation $$AB=\lbrace ab\ |\ a\in A,\ b\in B\rbrace.$$
This operation is associative so the algebraic structure is a semigroup.
This semigroup can have a zero element. I've found the following examp... | https://mathoverflow.net/users/20803 | When does a power semigroup have a zero, and what can the zero be? | You are right. More simply: $I$ is a zero in $P(S)$ if and only if $I$ is an ideal of $S$ and a subgroup of $S$. Indeed, if $I$ is a zero, it must be a subgroup and an ideal as you wrote yourself. On the other hand, if $I$ is an ideal and a subgroup, then for every $x$ in $S$, $Ix$ is a left ideal of $S$ contained in $... | 6 | https://mathoverflow.net/users/nan | 110421 | 63,427 |
https://mathoverflow.net/questions/109976 | 14 | What can be said about the set of primes $p$ for which it is proven that an infinite group with all non-trivial proper subgroups cyclic of order $p$ doesn't exist? Specifically, what is the largest such $p$ (say $p\_0$)? All I could find in the literature is $p\_0\le 10^{75}$, but I admit I didn't look very far...
| https://mathoverflow.net/users/17164 | Tarski monster groups: for which primes they don't exist? | The largest known prime for which existence of Tarski monster is not known is $997$, see Adyan, S. I.; Lysënok, I. G. Groups, all of whose proper subgroups are finite cyclic. Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), no. 5, 933--990; translation in Math. USSR-Izv. 39 (1992), no. 2, 905–957. There are currently no metho... | 9 | https://mathoverflow.net/users/nan | 110422 | 63,428 |
https://mathoverflow.net/questions/110410 | 9 | The standard way to define integration on a smooth manifold is to use partitions of unity, to extend to the case where the form you're integrating isn't supported on just one coordinate patch. Of course, in the analytic/holomorphic case, we don't have partitions of unity. So how do we do integration?
Furthermore, how... | https://mathoverflow.net/users/5279 | How does the lack of partitions of unity affect the structure of analytic/holomorphic manifolds? | (1) A holomorphic manifold is also (or "can also be viewed as") a smooth manifold, and that lets you define integration. To put it another way, you *do* have partitions of unity, just not holomorphic ones.
(2) Even before you get to tangent bundles, there are well-known cases where local things can't be patched glob... | 5 | https://mathoverflow.net/users/6794 | 110425 | 63,430 |
https://mathoverflow.net/questions/110423 | 1 | A *quasi-kernel* in a directed graph D is an independent subset of vertices $S$ so that for every $v \in V(D)-S$ either $v->s$ for some $s \in S$ or $v->w->s$ for some $w \in V(D)-S, s \in S$.
**Equivalently**, a set $S$ is a quasi-kernel if $d(s,t) \geq 2$ for every $s,t \in S$ and $d(v,s) \leq 2$ for every $v \in V... | https://mathoverflow.net/users/22051 | Is number of quasi-kernels NP-hard? | Here's a reduction that works even if the directed graphs are required to be simple. I'll argue that there is a polynomial time algorithm which, given a graph $G$, outputs a directed graph $G'$ such that the number of independent sets in $G$ is the number of quasi-kernels in $G'$. Hence the number of quasi-kernels is N... | 2 | https://mathoverflow.net/users/408 | 110431 | 63,433 |
https://mathoverflow.net/questions/110428 | 3 | Suppose you have a ring homomorphism $(-)':\mathbb{C} \to \mathbb{C}$, which is an involution such that $\sum\_i a\_i a\_i' = 0 \Leftrightarrow \forall i \ a\_i=0$, where this sum is finite.
Must this be conjugate to complex conjugation, in the sense that there exists some bijection $\phi: \mathbb{C} \to \mathbb{C}$ ... | https://mathoverflow.net/users/799 | Nondegenerate involutions on the complex numbers -- always just conjugation? | Yes. Your involution is a field automorphism, and since it is of order $2$, its fixed field $F$ is a subfield of $\mathbb C$ such that $[\mathbb C:F]=2$. (This incidentally implies that $F$ is a real-closed field, hence the condition on sums is automatically satisfied, you don’t have to assume it.) In particular, both ... | 5 | https://mathoverflow.net/users/12705 | 110433 | 63,435 |
https://mathoverflow.net/questions/110407 | 15 | Wikipedia claims the following:
>
> In mathematics, an action of a group G on a topological space X is cocompact if the quotient space X/G is a compact space or, equivalently, if there is a compact subset K of X such that
> the image of K under the action of G covers X.
>
>
>
My question is: Isn't this wrong... | https://mathoverflow.net/users/8590 | Characterization of cocompact group action | I agree with @Bugs that some extra assumptions are needed, although I do not have a counter-examples either.
Here is an argument assuming that $X$ is locally compact. For each $y\in Y=X/G$ choose (arbitrarily) a point $y'\in X$ which projects to $y$ (you'd need axiom of choice here). Now, for each $y'$ pick an open ... | 9 | https://mathoverflow.net/users/21684 | 110438 | 63,437 |
https://mathoverflow.net/questions/101796 | 5 | I have a question about whether Ryan Budney's question:
[Torus knots in Euclidean space -- a symmetry argument](https://mathoverflow.net/questions/72308/torus-knots-in-euclidean-space-a-symmetry-argument)
can be extended to links. He asks:
Suppose you have a $(p,q)$ torus knot $K$ in $\mathbb{R}^3$ fixed by a sub... | https://mathoverflow.net/users/9101 | Euclidean symmetries of torus links in R^3 | A pretty nice way to interpet your question is two consider the action of the symmetry group on $\mathbb{S}^3$. This is done for knots in [Boileau, Boyer, Cebanu, and Walsh](http://www.tufts.edu/~gwalsh01/papers/BBoCWaGT.pdf) section 3, but we can get enough of it to work to answer your question. The point is that the ... | 2 | https://mathoverflow.net/users/27453 | 110440 | 63,438 |
https://mathoverflow.net/questions/110455 | 0 | Let $A=H(D(0,1))$ the ring of holomorphic functions on the open unity disc.
I consider the function $f$:
$$f (t)=\sum f\_{i}t^{i} \in A[[t]]$$
I suppose that the $t$-adic valuation of it is less or equal than $r$, so the first $r$ functions, $f\_{1},f\_{2},...,f\_{r}$ do not simultaneously vanish.
Do I have tha... | https://mathoverflow.net/users/27398 | Laurent series with analytic coefficients | The answer is no. Let $r=2$, $f\_0,f\_1$ never simultaneously vanish.
You want $(f\_0+f\_1t+...)(g\_0+g\_1t+...)=t^2$. For this we need
$g\_0=g\_1=0$ and $f\_0g\_2=1$, which is impossible if $f\_0$ vanish somewhere.
Can you use TeX in your writing?
| 0 | https://mathoverflow.net/users/25510 | 110463 | 63,450 |
https://mathoverflow.net/questions/110427 | 10 | Is there an "elementary" (say ultrafilter-free) proof of the following fact: if $G$ is a compact (Hausdorff) topological group, if $g \in G$ is any element from this group, and if $P$ is a polynomial with integer coefficients without constant term, then the identity element of $G$ is a limit point of the sequence $n \m... | https://mathoverflow.net/users/25235 | The identity element of a compact group is a limit point of any "polynomial sequence" | If $\mathcal{H}$ is a van der Corput set of positive integers, then the closure of $\{ g^h\mid h\in\mathcal{H} \}$ contains the identity element. This generalizes the statement in the original post, because $\{ P(n)\mid n>0 \}$ is a van der Corput set for $P\in\mathbb{Z}[x]$ without a constant term.
By Terry Tao's r... | 7 | https://mathoverflow.net/users/11919 | 110464 | 63,451 |
https://mathoverflow.net/questions/110441 | 6 | Let $K$ be a field and let $Q$ be a quaternion algebra over $K$. Then it is well-known that the class $[Q]$ of $Q$ in $Br(K)$ has order $2$. One can show this by constructing an explicit isomorphism $Q \otimes\_K Q \cong M\_2(K)$. My question is about the converse.
>
> Does there exist a field $K$ and a division a... | https://mathoverflow.net/users/5101 | Brauer group elements of order $2$ | I have a vague memory that examples of $2$-torsion classes which are not exactly quaternion algebras can be found in one of the papers on Merkurjev's Theorem in "Applications of algebraic K-theory to algebraic geometry and number theory, Part 2", Contemporary Math. 1986 (either the one of Merkurjev, or the one of Wadsw... | 2 | https://mathoverflow.net/users/27494 | 110465 | 63,452 |
https://mathoverflow.net/questions/110457 | 17 | In his algebraic K-Theory book Bass gives the following property on a ring $R$ and a number $n$:
For every $n$ elements $v\_1, \ldots, v\_n$ that generate the unit ideal there are numbers $r\_1, \ldots r\_{n-1}$ such that $v\_1 + r\_1 v\_n, v\_2 + r\_2 v\_n, \ldots, v\_{n-1} + r\_{n-1} v\_n$ also generate the unit id... | https://mathoverflow.net/users/27272 | Bass' stable range condition for principal ideal domains | EDIT 3 : Sorry for editing this old answer one more time, but I want to also point out for future readers that there is a proof that Dedekind domains have stable range $2$ which is very similar to my proof for PID's in Satz K.13 of the book *Algebra* by Jantzen and Schwermer. The whole Appendix K of that book is a love... | 13 | https://mathoverflow.net/users/317 | 110472 | 63,455 |
https://mathoverflow.net/questions/110467 | 4 | Consider a random matrix $\mathbf{A} \in \mathbb{C}^{N \times N}$ of rank $m$ with $m < N$ that follows the Wishart distribution ( <http://en.wikipedia.org/wiki/Wishart_distribution> ).
I have a feeling that any submatrix that has $m$ columns is going to have rank $m$ with probability 1. Might be obvious to some of y... | https://mathoverflow.net/users/27495 | Probability of a submatrix to be full rank in a N x N Random Matrix of rank m. | Your matrix $A = X^T X$ where $X$ is a random $m \times N$ matrix with a continuous distribution having a density. An $m \times m$ submatrix of $A$ is $Q^T A P = (XQ)^T XP$ where $P$ and $Q$ are $N \times m$ matrices each consisting of $m$ columns of the $N \times N$ identity matrix. $XP$ and $XQ$ are $m \times m$ subm... | 7 | https://mathoverflow.net/users/13650 | 110474 | 63,457 |
https://mathoverflow.net/questions/110461 | 11 | I would like to know a simple proof of isometric injectivity of $L\_\infty$. The proof I've found in Topics in Banach space theory. F. Albiac, N. Kalton uses two deep result.
* $L\_\infty$ as commutative unital $C^\*$ algebra is isometrically isomorphic to $C(K)$ for some compact $K$.
* Every $C(K)$ space which is a... | https://mathoverflow.net/users/19593 | Direct proof of injectivity of $L_\infty$ | Write $L\_\infty$ as the closure of a net (directed by inclusion) of finite dimensional $\ell\_\infty$ spaces. Compose the operator into $L\_\infty$ with norm one projections onto these subspaces and extend. Use weak$^\*$ compactness of the unit ball of $L\_\infty$ to pass to a limit of a subnet of these operators.
$$
... | 16 | https://mathoverflow.net/users/2554 | 110475 | 63,458 |
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