parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/60902 | 6 | I'm trying to draw on the computer a curve that keeps always the same distance(given as parameter) from a given curve. I know the formula for the given curve. I tried moving perpendicular to the first derivative but in some cases when the curve is sharp there are a lot of points creating some problems. This problem hap... | https://mathoverflow.net/users/13211 | Formulas for equidistant curves | If you just want a good easy picture, the simplest thing to do, as sugggested by Charles Matthews' comment, is to draw lots of equal radius circles centered about points on the curve. Your eye and brain will see the envelope. You could also draw lots of disks, or just use a computer drawing program such as Adobe Illust... | 13 | https://mathoverflow.net/users/9062 | 60910 | 37,752 |
https://mathoverflow.net/questions/60905 | 3 | Dear Reader:
Let $K(k)$ and $E(k)$ be elliptic integrals of respectively the first and second kind, where $k$ is the elliptic modulus and $k'=\sqrt{1-k^2}$ is the complementary elliptic modulus.
I happened to encounter the following numerical "fact" (when solving an engineering problem regarding energy conversion)... | https://mathoverflow.net/users/14192 | approximate equation involving elliptic integrals | Given the Legendre relation, your question is equally about K - E. This is a difference of hypergeometric function values (see <http://en.wikipedia.org/wiki/Elliptic_integral> for all of this). You seem to be setting a condition on *k* that also is simpler when read out of the Legendre relation, on E and K'. I would th... | 1 | https://mathoverflow.net/users/6153 | 60914 | 37,754 |
https://mathoverflow.net/questions/60912 | 9 | This question is about dense sphere packings in euclidean space $\mathbb R^n$. By a sphere packing I understand any arrangement of mutually disjoint solid open spheres in $\mathbb R^n$, all of the same radius. The density of a packing is
$$\mathrm{lim}\_{R \to \infty}\frac{\mathrm{vol }(B(0,R) \cap \mathrm{spheres})}{\... | https://mathoverflow.net/users/5952 | Dense sphere packings which are not lattice packings | In ten dimensions the best packing known is the Best packing, which is not a lattice packing. Marc Best found a nonlinear $40$-element binary code of block length $10$ and minimal Hamming distance $4$, and one can turn it into a sphere packing in $\mathbb{R}^{10}$ by centering spheres at all the points in $\mathbb{Z}^{... | 24 | https://mathoverflow.net/users/4720 | 60930 | 37,762 |
https://mathoverflow.net/questions/60935 | 2 | I've been taking a look at simple Lie algebras for particle physics and I've found myself wondering about the following question. It can be shown that the adjacent roots in a root diagram corresponding to a simple Lie algebra must always lie at either 0 ,30, 45, 60 or 90 degrees (if the rank $\geq2$). In all cases but ... | https://mathoverflow.net/users/14200 | A question on the root systems of simple Lie algebras in the 90 degree case | If you look only at a *simple* Lie algebra, no two "adjacent" simple roots in the Dynkin diagram can form a right angle: being joined by at least one edge forces a different angle. In the simple case there is no ambiguity about relative lengths of roots, but of course in a direct sum of simple Lie algebras the differen... | 6 | https://mathoverflow.net/users/4231 | 60936 | 37,765 |
https://mathoverflow.net/questions/60945 | 4 | This is a continuation of [my previous question](http://mathoverflow.net/questions/60781).
Let $G$ be a connected reductive group over an algebraically closed field $k$ of characteristic 0.
We assume that $\mathrm{Pic}\ G=0$.
This is the same as to say that the derived group $G^{\mathrm{der}}$ of $G$ (which is semisimp... | https://mathoverflow.net/users/4149 | Diagonalizable subgroups in a simply connected group | The answer seems to be no, according to II, 5.8 (and following material) in the Springer-Steinberg lecture notes (Lect. Notes in Math 131, 1970), though I might be overlooking something in your question. Of course, sometimes this type of embedding is possible as pointed out in those lecture notes, but there are problem... | 5 | https://mathoverflow.net/users/4231 | 60946 | 37,769 |
https://mathoverflow.net/questions/60925 | -7 | (This problem appeared in face of me trying to generalize [my theory](http://www.mathematics21.org/algebraic-general-topology.html) of (binary) funcoids to the theory of $n$-ary funcoids (I call them "multifuncoids") for arbitrary $n$.)
Let $I$ is some indexing set.
By filters I will mean (not necessarily proper) f... | https://mathoverflow.net/users/4086 | Special infinitary relations and ultrafilters | For $a$ an $I$-indexed family of filters and $S$ an $I$-indexed family of subsets of $U$ such that $U\smallsetminus S\_i\notin a\_i$ for every $i\in I$, define the restricted product $\prod^Sa$ by
$$\left(\prod\nolimits^Sa\right)R\Leftrightarrow\left(\prod a\right)R\land\{i\in I:R\_i\ne S\_i\}\text{ is finite.}$$
This ... | 22 | https://mathoverflow.net/users/12705 | 60947 | 37,770 |
https://mathoverflow.net/questions/60884 | 3 | Let $C$ be a category with finite limits; that is, for any finite category $D$ and functor $F:D\to C$ the category $\mathrm{Cone} F$ of cones over $F$ is inhabited and has terminal objects (we could turn the morphisms around and call the limit an *initial* obect, but never mind).
That is, define
$$ (A,\phi): \mathrm{... | https://mathoverflow.net/users/1631 | Limits are terminal objects in another category; (when) are they colimits of (another diagram)? | I first of all feel like rewriting the question, to give it a less cluttered look. Let $D$ be a finite category, and let $F: D \to C$ be a functor, where $C$ is finitely complete. Let $C^D$ denote the category of functors $D \to C$, and let $\Delta: C \to C^D$ denote the diagonal functor. Then, as usual, define the cat... | 11 | https://mathoverflow.net/users/2926 | 60958 | 37,773 |
https://mathoverflow.net/questions/60875 | 13 | Is there a standard notation (perhaps $A \stackrel{\leftarrow}{=} B$) meaning "in all situations where $B$ is defined, $A$ is defined and equals $B$"?
The kind of situation in which such a notation would be useful is the teaching of formulas like $$\lim\_{x \rightarrow a} (f(x)-g(x)) = \lim\_{x \rightarrow a} f(x) - ... | https://mathoverflow.net/users/3621 | conditional equality symbol | Freyd and Scedrov, in their book *Categories, Allegories*, use for this 'directed equality' a peculiar symbol that they call a *Venturi tube* and that looks a bit like $\mathrel{>=}$, so that $x \mathrel{>=} y$ means *if $x$ is defined then so is $y$ and $x=y$*. You can find some discussion at the nLab page on [Kleene ... | 6 | https://mathoverflow.net/users/4262 | 60966 | 37,778 |
https://mathoverflow.net/questions/60938 | 11 | Given an $n \times n$ vandermonde matrix $V$ which is invertible, is any $(n-1) \times (n-1)$ submatrix of $V$ invertible also?
I think the answer is yes, but I don't know how to prove.
| https://mathoverflow.net/users/10705 | Is any $(n-1)\times (n-1)$ submatrix of an $n \times n$ Vandermonde matrix invertible? | This is an elaboration of Thierry's answer: If $0 \leq a\_0 < a\_1 < \cdots < a\_{d-1}$ is any sequence of nonnegative integers, then the determinant $\det \left( x\_j^{a\_i} \right)$ is equal to $\prod\_{i < j}(x\_i - x\_j) \cdot s\_{\lambda}(x\_1, \ldots, x\_d)$ where $\lambda = (a\_0, a\_1-1, a\_2-2, \ldots, a\_{d-1... | 18 | https://mathoverflow.net/users/297 | 60971 | 37,782 |
https://mathoverflow.net/questions/60976 | 4 | Let $G$ be a group, $S$ its finite set of generators. Denote $G^+$ the subsemigroup of $G$ generated by $S$. Let $g\in G$, $H<G$ the subgroup generated by $g$ and let $g\_1,...,g\_k\in G$ be a finite collection. Suppose $G^+\subset g\_1H\cup\cdots\cup g\_kH$. Is it true that $H$ is of finite index in $G$?
**Edit:** a... | https://mathoverflow.net/users/8699 | Group with virtually cyclic positive part. | Restating the question as I understand it:
Let $G$ be a group, $S\subset G$ a finite set of generators, $H$ a cyclic subgroup of $G$. Suppose that there is a finite collection of elements $g\_j\in G$ such that every product of finitely many elements of $S$ belongs to one of the cosets $g\_jH$. Is it true that $H$ has... | 6 | https://mathoverflow.net/users/6666 | 60980 | 37,785 |
https://mathoverflow.net/questions/60981 | 8 | I was pondering the fact that maybe the classical hard complexity-theoretic questions are undecidable, not because they are so themselves, but because some set-theoretic foundations makes the complexity-theoretic foundations shaky.
My thoughts was that perhaps something like the Continuum hypothesis makes P vs NP und... | https://mathoverflow.net/users/10316 | Are problems in complexity theory dependent on set theory? | The statement that P=NP can be expressed in first-order arithmetic, and that part of mathematics is unaffected by the known methods of proving set-theoretic independence results (forcing, inner models).
| 6 | https://mathoverflow.net/users/6794 | 60983 | 37,787 |
https://mathoverflow.net/questions/60960 | 2 | Hi,
I solved for a Poisson equation with finite elements, using piecewise linear basis functions on 2d triangles.
Now, I want to evaluate the following expressions:
$$ \int\_\Omega \Delta u ~dx$$
and
$$ \int\_\Omega (\Delta u)^2 ~dx$$
I want to evaluate these expressions using my approximated solution $u$ which ... | https://mathoverflow.net/users/8646 | integration of a laplacian | Since your approximate solution is piecewise linear, it is $H^1$ but not $H^2$. Therefore your calculation is impossible. You can do to things to overcome the difficulty:
* Either use higher-order elements,
* or post-process your approximate solution $u$. This means constructing a smoother $\bar u$ using some convolu... | 2 | https://mathoverflow.net/users/8799 | 60988 | 37,790 |
https://mathoverflow.net/questions/60984 | 0 | Hi,
I've a question regarding the Petersson operator.
We have the following definition and the lemma.
Definition
Let $k, m \in \mathbb{Z}$ and $\phi: \mathbb{H} \times \mathbb{C} \rightarrow \mathbb{C}$.
For
$\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \in \Gamma\_1$ und $\begin{pmatrix} \lambda & \mu ... | https://mathoverflow.net/users/12308 | Modular form - Petersson operator | I think the definition you want for $\phi|(M,X)$ is $(\phi |\_{k,m} M) |\_m X$.So your left hand side involving $\phi |(M,X) | (M',X')$ is the composition of four of these operators. Then the problem isn't too bad given your (i)-(iii).
| 2 | https://mathoverflow.net/users/2627 | 60989 | 37,791 |
https://mathoverflow.net/questions/60851 | 14 | Apologies if my question seems overly naive, but I haven't seen/heard/read any good answers.
What is modern computability theory "really" about? The study of feasible(even *remotely* feasible) algorithms falls under the domain of theoretical and non-theoretical computer science. There is, of course, the a posteriori... | https://mathoverflow.net/users/14183 | Is modern computability theory "really" about algorithms? | I think it's important to take a historical perspective. There was a time not so long ago when computers as we know them now did not exist. At that stage, coming up with a precise definition of an algorithm or of a Turing machine was a major advance, allowing one to build the earliest modern computers and begin the rev... | 17 | https://mathoverflow.net/users/3106 | 60993 | 37,795 |
https://mathoverflow.net/questions/60943 | 1 | What is the estimation for the positive root of the following equation
$$
ax^k = (x+1)^{k-1}
$$
where $a > 0$ (specifically $0 < a \leq 1$).
Could you point out some reference related to the question?
| https://mathoverflow.net/users/12981 | Root estimation | If we write the equation as $ax= (1+1/x)^{k-1}$ it is clear that there is exactly one positive solution (here and below I assume that $k$ is a real number larger than 1).
I. With the substitution $u=1/x$ the equation writes $a=u(1+u)^{k-1}$, and the RHS has a formal series inverse at $0$. The [Lagrange inversion for... | 10 | https://mathoverflow.net/users/6101 | 60997 | 37,798 |
https://mathoverflow.net/questions/60977 | 1 | Can anyone tell me whether the weight diagram associated with a given semi-simple Lie algebra is unique to that algebra? I feel morally certain that it is but I just can't seem to get it out. Any pointers gratefully received.
| https://mathoverflow.net/users/14200 | Weight diagrams and semi-simple Lie algebras | According to the comments in this recent MO question: [Can A Simple Lie Algebra Be Determined By Weights of its Representation](https://mathoverflow.net/questions/59484/can-a-simple-lie-algebra-be-determined-by-weights-of-its-representation), the answer to your question is no with simple examples afforded by the defini... | 1 | https://mathoverflow.net/users/12301 | 61003 | 37,800 |
https://mathoverflow.net/questions/60987 | 18 | Given a fiber bundle $S\hookrightarrow M \rightarrow S^1$ with $M$ (suppose compact closed connected and oriented) 3-manifold and $S$ a compact connected surface, it follows form the exact homotopy sequence that $\pi\_1(S)\hookrightarrow \pi\_1(M)$. Does this imply that the "fiber" of a 3-manifold which fibers over $S^... | https://mathoverflow.net/users/12952 | Fibers of fibrations of a 3-manifold over $S^1$ | There are simple examples with $M = F \times S^1$ for $F$ a closed surface of genus $2$ or more. Choose a nonseparating simple closed curve $C$ in $F$, then take $n$ fibers $F\_1,\cdots,F\_n$ of $F\times S^1$, cut these fibers along the torus $T=C\times S^1$, and reglue the resulting cut surfaces so that $F\_i$ connect... | 28 | https://mathoverflow.net/users/23571 | 61014 | 37,808 |
https://mathoverflow.net/questions/59822 | 4 | This question is a follow up question to [this](https://mathoverflow.net/questions/57235/automorphisms-of-sl-n-mathbbz/57236#57236) question. So my question is:
For which rings $R$ (commutative, with unit) (and which integers $n$) is $Out(SL\_n(R))$ a torsion group? A consequence of Theorem A and B in [O'Meara The au... | https://mathoverflow.net/users/3969 | When is Out$(SL_n(R))$ a torsion group ? | This question has been studied even for non-commutative associative rings. See, for example Golubchik, I. Z.; Mikhalëv, A. V. Isomorphisms of the general linear group over an associative ring. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1983, no. 3, 61–72. They prove that if the ring $R$ contains $1/2$ (that is $2$ is inve... | 4 | https://mathoverflow.net/users/nan | 61017 | 37,809 |
https://mathoverflow.net/questions/61016 | 3 | The title is fairly self explanatory. Let $\mathcal{A}$ be a small category , $X$ be an object of $\widehat{\mathcal{A}}$. Then the question is, are the categories, $\widehat{\mathcal{A}\downarrow X}$, and $\widehat{A}\downarrow X$ equivalent? I am 99% sure that this is true. Furthermore, if their is a reference for th... | https://mathoverflow.net/users/14167 | Are the categories, $\widehat{\mathcal{A}}\downarrow X$ and $\widehat{\mathcal{A}\downarrow X}$ equivalent. | By $\widehat{\mathcal{A}}$ you must mean the category of presheaves on $\mathcal{A}$. I take it that $X$ denotes a presheaf, and that $\mathcal{A}/X$ is the comma category $y\_{\mathcal{A}} \downarrow X$ where $y\_{\mathcal{A}}$ is the Yoneda embedding.
Another reference for this fact is Tom Leinster's book *Higher ... | 4 | https://mathoverflow.net/users/2926 | 61019 | 37,811 |
https://mathoverflow.net/questions/61005 | 2 | If we have a family of hyperelliptic curves of genus g, then I know that this family can be expressed as a double cover of the quadric surface branched along a certain curve. In the book [Moduli of Curves, Harris-Morrison, p.293] they say that such a family should be a covering of quadric surface branched along a curve... | https://mathoverflow.net/users/14181 | Families of hyperelliptic curves and double covers of quadric surface | Very concretely -- the double cover of P^1\_{x,y} x P^1\_{z,w} is obtained by adjoining a square root of f(x,y), where f is a homogenous form of degree 2g+2 in x,y. But where are the coefficients of f? They themselves are homogeneous forms in z,w, of some degree which Harris-Morrison are calling d. You should think of ... | 3 | https://mathoverflow.net/users/431 | 61020 | 37,812 |
https://mathoverflow.net/questions/60845 | 4 | Hi everyone,
I am dealing with a family of curves like f:X---->S, with X and S (and hence f) projective. In fact S, is the projective line but I don't know this helps or not. The problem is that, X is not a smooth surface. It has only one singular point which lies on a singular fiber of f. Now, can I take a semi-stab... | https://mathoverflow.net/users/14181 | Stable reduction | At least in the contexts that I am familiar with, a family $X \to S$ with $S$ a smooth
curve is called semistable at a closed point $s$ if it has a local model (local in the analytic topology if we are working over $\mathbb C$, or local in the etale topology in general) of the form Spec $k[X\_1,\ldots,X\_n,t]/(X\_1\ldo... | 13 | https://mathoverflow.net/users/2874 | 61025 | 37,816 |
https://mathoverflow.net/questions/60951 | 10 | Hi all,
Given a variety $X$ over the real numbers, we can consider the singular cohomology of the space $X(\mathbb{C})$, with coefficients in $\mathbb{Q}$, say. The action of complex conjugation on $X(\mathbb{C})$ induces an action on these cohomology groups. How do you compute/describe this action in concrete exampl... | https://mathoverflow.net/users/14204 | Galois action on Betti cohomology? | Suppose that $X$ is smooth and projective. Then Hodge theory gives a decomposition
of $H^i(X(\mathbb C),\mathbb C)$ into the direct sum of $H^{p,q}$'s (with $p + q = i$).
Now there are two complex conjugations acting on $H^i$ with $\mathbb C$ coefficients: the
"trivial action" just coming from conjugating the coeffi... | 19 | https://mathoverflow.net/users/2874 | 61026 | 37,817 |
https://mathoverflow.net/questions/61055 | 9 | I am currently thinking about a problem, and I feel that by knowing more about elliptic curves over extensions of $\mathbb{F}\_q(T)$, for $q$ a power of $p$ say, might lead to insight. I am also inspired by the following sentence of Silverman in his Advanced Topics (introduction to Chapter 3):
>
> "Thus conjectures... | https://mathoverflow.net/users/13741 | Elliptic Curves over Global Function Fields | Douglas Ulmer wrote up expository notes for his short course at PArk City on precisely this topic:
<http://arxiv.org/abs/1101.1939>
This might be a good place to start.
| 13 | https://mathoverflow.net/users/392 | 61064 | 37,836 |
https://mathoverflow.net/questions/60814 | 6 | The following fact is quite standard and does not have a very long proof:
$(\\*)$ If $u$ is harmonic on $B\_1(0)\setminus \{0\}$ and uniformly bounded, then $u$ in fact extends to a harmonic function on the whole ball.
Some googling reveals that such statements are in fact true for large classes of elliptic operato... | https://mathoverflow.net/users/4345 | Removable Singularities for Elliptic Equations | The proof I had in mind was actually simpler and appears to work only in dimension greater than 2. Here is a sketch for a second order self-adjoint elliptic operator $Pu = \partial\_i(a^{ij}\partial\_ju)$. Suppose $u$ satisfies $Pu = 0$ on $B\backslash\{0\}$. We want to show that $u$ is a weak solution on $B$. It suffi... | 5 | https://mathoverflow.net/users/613 | 61066 | 37,837 |
https://mathoverflow.net/questions/61061 | 9 | Actually I am an undergraduate student, but I want to study Heisenberg groups over arbitrary field.
Firstly, why is this group important? I know that the Heisenberg group is important in the field of quantum physics, but in mathematics how is it? And secondly I want to know the current studies of the classifications... | https://mathoverflow.net/users/13453 | About representation theory of Heisenberg group | Here is one important way in which the Heisenberg group is important:
$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\PGL{PGL}$Let $F$ be a local field of characteristic not equal to 2 (so for example, one of your fields $\mathbb{Q}\_p$ or $\mathbb{R}$ or $\mathbb{C}$ above), and let $\psi$ be a nontrivial unitary a... | 9 | https://mathoverflow.net/users/8891 | 61073 | 37,843 |
https://mathoverflow.net/questions/61058 | 6 | Suppose we consider a rigid extension field $F$, i.e., $\text{Aut}(F) = 1$ over the complex numbers $\mathbb C$. What is the minimal cardinality of $F$? In particular it should hold that in this case $|F| > |\mathbb C|$.
Moreover, if we replace $\mathbb C$ with any other algebraically closed field, what can one say i... | https://mathoverflow.net/users/13321 | What is the size of the smallest rigid extension field of the complex numbers? | Firstly, it may be worth linking to [this related question](https://mathoverflow.net/questions/22897/) on MO.
Pröhle proved that all fields of characteristic 0 can be embedded in a rigid field - see ["Does a given subfield of characteristic zero imply any restriction to the endomorphism monoids of fields?"](http://w... | 9 | https://mathoverflow.net/users/3143 | 61082 | 37,848 |
https://mathoverflow.net/questions/61048 | 4 | This is my first MO question, so please go easy on me if you think this is too vague.
Is there anything to say about the collection of chain complexes with flat homology? Is there a name for them, or a different characterization of them? Maybe there's a way to build them out of some other simpler objects?
Specifica... | https://mathoverflow.net/users/14220 | objects in the derived category with flat homology | This is my first Math Overflow answer, so hopefully people will go easy on me as well. Everything I'm about to say is based on the following paper of Mark Hovey and Keir Lockridge:
<http://arxiv.org/abs/1001.0902>
You are thinking about $\mathcal{D}(R)$, the derived category of $R$. This category is equivalent to $\... | 4 | https://mathoverflow.net/users/11540 | 61092 | 37,853 |
https://mathoverflow.net/questions/61075 | 5 | This is (a variation on) exercise 20.4 in Jech's "Set Theory." Let $j : V \to M$ witness $\lambda$-supercompactness of $\kappa$, and consider the normal measure $U$ on $P\_{\kappa}(\lambda)$ consisting of those $x$ such that $j[\lambda] \in j(x)$. (How do you make the left quotation mark symbol to denote 'j-image-of-la... | https://mathoverflow.net/users/7521 | Normal measures on $P_{\kappa }(\lambda )$ extend the club filter | Suppose $\kappa$ is $\lambda$-supercompact for some $\lambda \geq \kappa$, and let $j: V \rightarrow M$ be an elementary embedding with critical point $\kappa$ such that $j(\kappa) > \lambda$ and $M^{\lambda} \subseteq M$ for some inner model $M$. First, observe that $V$ and $M$ agree on $P\_{\kappa}\lambda$ because $M... | 3 | https://mathoverflow.net/users/11318 | 61101 | 37,858 |
https://mathoverflow.net/questions/57925 | 1 | In several complex variables , to determine the pseudoconvexity of a domain in $C^n$ is very important . There are various criterion to decide whether a domain is pseudoconvex . In particular ,if the domain is defined by a $C^\infty$ defining function $\phi>0$ , then 'Levi pseudoconvex ' is equivalent to the following ... | https://mathoverflow.net/users/4437 | how to prove the relationship between pseudoconvexity and the monge-ampere matrix? | Since $\phi$ is always nonzero inside $\Omega$ , so this matrix has precise one negative eigenvalue and n positive eigenvalues is equivalent to $-\partial\bar{\partial}log\phi$ is non-negative , but which means that $-log\phi$ is a plurisubharmonic exhaustion function for the domain $\Omega$ , i.e. $\Omega$ is pseudoco... | 0 | https://mathoverflow.net/users/4437 | 61104 | 37,861 |
https://mathoverflow.net/questions/61107 | 1 | If any recursively enumerable language can be reduced by a mapping reduction to a language $L$, then $L$ is called $RE$ complete. In that case, $L$ must be in $RE\setminus R$. But are there languages in $RE\setminus R$ which are not $RE$ complete? Can anyone give an example of such a language? I'm sure this is well kno... | https://mathoverflow.net/users/7599 | Is there a language in $RE \setminus R$ which is not $RE$-complete? | Examples of such languages are not easy to describe, and I think no
"naturally-occurring" example is known. However, Muchnik and Friedberg
found examples in 1957, and Friedberg's example is [here](http://www.ncbi.nlm.nih.gov/pmc/articles/PMC528418/).
| 1 | https://mathoverflow.net/users/1587 | 61108 | 37,862 |
https://mathoverflow.net/questions/61112 | 2 | Spectrum decomposition can be regarded as the generalizations of the following fact that:
Every Hermitian matrix $A$ can be decomposed into $A=U^{\*}\Lambda U$,where $U$ is a unitary matrix
Singular vector decomposition can be expressed as Every Matrix $A\_{mn}$ can be decomposed in to $A=U\Lambda V^{\*}$, where $U$,... | https://mathoverflow.net/users/11966 | Is there any conclusions generalized Singular Value Decomposition into Hilbert Space | The simplest generalization is that a "compact self adjoint linear operator" on a Hilbert space can be diagonalized in terms of (an infinite number of) eigenvalues and eigenfunctions that are elements of the Hilbert space. This can be extended to non-compact but still self adjoint operators, but it's more complicated b... | 3 | https://mathoverflow.net/users/9022 | 61114 | 37,865 |
https://mathoverflow.net/questions/61126 | 1 | "Formule de Weyl et de Demazure et Theoreme dc Borel-Weil-Bott pour les algebres de Kac-Moody generates"
by O.Mathieu.
I even do not know whether he published on a Mathematical Journal or not.
Can anyone tell me how to find this article?
Thank you
| https://mathoverflow.net/users/1851 | How can I get this paper? | The correct reference is O. Mathieu, Formules de Demazure-Weyl, et généralisation du théorème de Borel-Weil-Bott, *C.R. Acad. Sci. Paris* **303** (1986), 391-394 [MR 87m:17036](http://www.ams.org/mathscinet-getitem?mr=862200).
| 3 | https://mathoverflow.net/users/532 | 61129 | 37,871 |
https://mathoverflow.net/questions/61124 | 2 | We call a subset $A = \{a\_1, a\_2, a\_3, \dots\}$ of $\mathbb N$ with $a\_1 < a\_2 < \dots $ transparent if $a\_{k+1} - a\_k$ goes to $\infty$ as $k \rightarrow \infty$. Is the following true? For every finite set $P$ of prime numbers, the set $A\_P := \{p\_1 \cdots p\_s : p\_i \in P\}$ is transparent.
| https://mathoverflow.net/users/14233 | Subsets of $\mathbb N$ with a finite number of prime factors | Proving that $a\_{k+1}-a\_k\to \infty$ reduces to proving that $$a\_{k+1}-a\_k=n$$ has finitely many solutions for every $n$. If you let $S$ be the set of prime divisors of $n$ union $P$, then this follows from the [S-unit](http://en.wikipedia.org/wiki/S-unit) theorem.
| 9 | https://mathoverflow.net/users/2384 | 61133 | 37,874 |
https://mathoverflow.net/questions/61084 | 10 | Suppose $L/K$ is a Galois extension of number fields, with Galois group $G\_{L/K}$. Write $\mathrm{Br}(L)^{G\_{L/K}}$ for the subgroup of central simple algebras $A/L$ which are Galois-invariant; equivalently, these are the algebras such that for $v$ a place of $K$ and $w$ a place of $L$ with $w|v$, the local invariant... | https://mathoverflow.net/users/1464 | Examples of Galois-invariant central simple algebras which aren't base change? | [big edit]
(1) Let $L/K$ be as above. Take any non-identity element $\sigma \in G\_{L/K}$, and let $F=L^{<\sigma>}$ be the fixed field of the cyclic subgroup generated by $\sigma$. By your comment above about the vanishing of $H^3$, every galois invariant CSA of $L$ is a base change of a CSA of $F$. Hence, there are ... | 12 | https://mathoverflow.net/users/2024 | 61142 | 37,877 |
https://mathoverflow.net/questions/61116 | -2 | It's easy to see that the descending central series of a group induces a graded Lie algebra .(see for example Serre's Harvard lectures or Magnus-Solitar book). I think in general this can be complicated, but this should be well-known:
>
> What is the structure of this Lie algebra for the symmetric group?
>
>
>
... | https://mathoverflow.net/users/9163 | Lie algabra of symmetric group | For any $n>1$, the lower central series for the symmetric group is $S\_n > A\_n\geq A\_n \geq A\_n \geq \cdots$, so the Lie ring formed by the sum of successive quotients is the group $\mathbb{Z}/2\mathbb{Z}$, equipped with the Lie bracket that is identically zero.
If you want to gain intuition for this construction ... | 4 | https://mathoverflow.net/users/121 | 61144 | 37,878 |
https://mathoverflow.net/questions/61149 | 6 | Given a global field $F$, we can construct the ring of adeles. Given a general locally compact ring $R$, when is it isomorphic to the ring of adeles of some global field $F$ and how can I find $F$ in $R$?
| https://mathoverflow.net/users/10400 | When is a ring the ring of adeles of some global field | Iwasawa gave a characterisation, assuming you are given a subfield F, discrete and such that the quotient is compact. The other conditions are R a semisimple locally compact commutative topological ring with 1 (shared with F). Then R is the adele ring of the global field F.
**Edit**: I believe it is known that you ca... | 14 | https://mathoverflow.net/users/6153 | 61150 | 37,881 |
https://mathoverflow.net/questions/61053 | 7 | Let $X=\mathbb P(\mathcal E)$, where $\mathcal E$ is a locally free sheaf of rank $n+1$ on $Y$, an integral scheme of finite type over an algebraically closed field $k$. I'm trying to show that $\text{Pic }X\cong \text{Pic }Y\times \mathbb Z$. The only small point I'm stuck on is showing that every invertible sheaf on ... | https://mathoverflow.net/users/13139 | Picard Group of Projective Bundle over an Integral scheme | Thanks to Piotr Achinger for the idea to consider the euler characteristic. I was looking for an answer that doesn't use fancy machinery beyond what's presented in the main text in Hartshorne (so no generalized Riemann-Roch). Here is one based on his suggestion:
Denote by $\mathcal F$ the line bundle $\mathcal M\otim... | 2 | https://mathoverflow.net/users/13139 | 61154 | 37,883 |
https://mathoverflow.net/questions/61155 | 7 | I am looking for an integral domain $A$ with the following properties:
1. $A$ is not integrally closed
2. $A$ has a quotient field $K$ that is algebraically closed and that has characteristic 0
3. There is an integral element $x\in K$ (*over* $A$) such that $A[x]$ is integrally closed.
Can someone help to tell me i... | https://mathoverflow.net/users/1245 | Non-normal domain with algebraically closed fraction field | Try this: Let $B\_0$ be the ring of real algebraic integers, and let $B=B\_0[1/2]$, so the ring of real algebraic numbers integral except possibly at $2$. But $B[i]$ is equal to the ring of algebraic numbers integral except possibly at $2$, and this is integrally closed. And so we take $A=B[3i]$, not integrally closed,... | 11 | https://mathoverflow.net/users/11417 | 61163 | 37,886 |
https://mathoverflow.net/questions/61161 | 3 | I found this question on another forum, and after processing it a bit, I didn't find a good answer. The question is:
>
> Is the sum of two closed operators closable? If not, give an example of two closed operators such that their sum is not closable.
>
>
>
| https://mathoverflow.net/users/13093 | Sum of two closed operators closable | I might change your question a little bit: Let $X=L^2[0,1]$, $Af:=f''$ with $D(A)=H^2[0,1]$ and let $Bf=f'(0)\cdot\mathbb{1}$, with $D(B)=H^2[0,1]$. Then $B$ is not closable (easy exercise from the definition), but $B$ is relatively $A$-bounded with $A$-bound zero. Hence, $A+B$ is closed (see Kato, Thorem IV.1.1).
T... | 4 | https://mathoverflow.net/users/12898 | 61166 | 37,887 |
https://mathoverflow.net/questions/61141 | 23 | Let $n$ be a positive intger. Is the following true? For continuous maps $f: \mathbb S^n \rightarrow \mathbb S^n$ and $g: \mathbb S^n \rightarrow \mathbb R^n$, there exists a point $x \in \mathbb S^n$ such that $g(x) = g(f(x))$.
| https://mathoverflow.net/users/14233 | Generalization of Borsuk-Ulam | This is false for all $n \geq 2$, but true for $n=1$.
A map $g \colon X \to Y$ of topological space is said to be **free** if there is a map $f \colon X \to X$ such that $g(x) \neq g(f(x))$ for all $x\in X$. This definition appears to be due to Hopf, and was given in
>
> H. Hopf, *Freie Überdeckungen und freie Ab... | 19 | https://mathoverflow.net/users/430 | 61172 | 37,891 |
https://mathoverflow.net/questions/60105 | 5 | This question is aimed at a better understanding of GIT's "categorical quotients", which are defined as coequalizers of group actions $G\times X\rightrightarrows X$ in the category of schemes. See also Anton's ~~currently unanswered~~ question about [surjectivity of coequalizers](https://mathoverflow.net/questions/63/c... | https://mathoverflow.net/users/84526 | Is being a coequalizer a target-local property in schemes? (answered: no, and no) | Let me start with a remark [EDITED for clarity after Andrew's comments]. Given $h:X\to Y$, the following are equivalent:
(1) $h$ is the coequalizer of some $W\rightrightarrows X$,
(2) $h$ is the coequalizer of $X\times\_Y X\rightrightarrows X$.
In other words, being a coequalizer is equivalent to being an effe... | 7 | https://mathoverflow.net/users/7666 | 61192 | 37,905 |
https://mathoverflow.net/questions/61099 | 2 | As usual $\sigma\_k(n)$ denotes the sum of the $k$-th powers of the positive divisors of an integer $n.$
Note that $k$ is also an integer so that it may be negative.
There are no odd perfect numbers known but there are many numbers $N$ such that
$$
\sigma(N) \equiv 2 \pmod{4}
$$
i.e.; many numbers
of the form:
$$
N... | https://mathoverflow.net/users/11016 | Solutions of a exponential diophantine equation involving the $\sigma$ function | Note that
$$z = \frac{p^{4k+1}}{\sigma(p^{4k+1})} = \frac{1-p^{-1}}{1-p^{-(4k+2)}} > 1-p^{-1}.$$
Clearly, for a fixed p, the larger is $k$ the closer is $z$ to this lower bound.
Since the lower bound is greater than $z\_0$ (for any prime $p\geq 5$) and grows with $p$, it is beneficial to take $p$ as small as possible... | 2 | https://mathoverflow.net/users/7076 | 61193 | 37,906 |
https://mathoverflow.net/questions/61199 | 2 | It is well known result of linear optimization theory that, if a finite set $S$ of linear inequalities in real variables $x\_1,x\_2, \ldots ,x\_n$ implies a linear inequality $i$ in $x\_1,x\_2, \ldots ,x\_n$, then $i$ can be written as a positive linear combination of the inequalities in $S$.
I wonder if an analogous... | https://mathoverflow.net/users/2389 | Complex version of Farkas' lemma | I think, yes. Without loss of generality, all $b\_i$'s and $d$ are equal to 1. Assume that the vector $c=(c\_1,\dots,c\_n)$ does not lie in a convex hull of vectors $(wa\_{i,1},\dots,wa\_{i,n})$ for all $i=1,\dots,n$ and all $|w|=1$. Then there is a linear real functional, which separates $c$ from this convex hull. It ... | 3 | https://mathoverflow.net/users/4312 | 61201 | 37,909 |
https://mathoverflow.net/questions/61180 | 5 | Hello Everyone,
I am not sure if this question is okay for this site, in case its not feel free to close it.
However, I would love to have it answered. Here goes my question.
>
> A graph $G=(V,E)$ has a perfect matching if and only if for every $U \subseteq V$ the number of connected components with an odd number... | https://mathoverflow.net/users/12018 | Motivation behind Tutte's 1-factor theorem | One way to come up with the characterization is to just run Edmond's matching algorithm (which is pretty natural). At the end of the algorithm, a matching $M$ and a subset of vertices $U$ has been explicitly constructed that gives equality in the Tutte-Berge formula (which implies Tutte's theorem). Of course, this is n... | 4 | https://mathoverflow.net/users/2233 | 61203 | 37,911 |
https://mathoverflow.net/questions/61195 | 12 | The center of a category $C$ is defined to be $\text{Z}(C) := \text{End}(1\_C)$; here $1\_C$ is the identity functor $C \to C$. See [this](https://mathoverflow.net/questions/41789/what-is-the-center-of-qcohx) question for an important application of the center. Now [this](http://ncatlab.org/nlab/show/center+of+an+abeli... | https://mathoverflow.net/users/2841 | Functoriality of the center of a category | You can make $Z(F)(s)$ one object at a time by just choosing, for each object $d$, an object $c$ and an iso $e:F(c)\to d$, and conjugating $s$ by $e$. This is independent of choices.
| 12 | https://mathoverflow.net/users/6666 | 61208 | 37,914 |
https://mathoverflow.net/questions/61197 | 3 | Let $R$ be a ring (assumed to be commutative and with unit). What is the center of the category of $R$-algebras, i.e. $\text{Z}(R\text{-Alg})$? This is a commutative monoid. See [this](https://mathoverflow.net/questions/61195/functoriality-of-the-center-of-a-category) recent question for the definition of the center. A... | https://mathoverflow.net/users/2841 | Center of the category of $R$-algebras | It's a standard lemma that the gcd of the binomial coefficients $(k,n-k)$ (for $0\lt k\lt n$) is $p$ when $n=p^r$ for some $r>0$ (with $p$ prime) and $1$ in all other cases. It follows that for any $f(t)\in R[t]$ with properties as described, there is a finite splitting $R=\prod\_{i=1}^mR\_i$, where for each $i$ either... | 3 | https://mathoverflow.net/users/10366 | 61209 | 37,915 |
https://mathoverflow.net/questions/61204 | 4 | Define the following incidence structure of rank three. The points are the elements of $\mathbb{Z}\_7=$ {$0,\ldots,6$}. The lines of type 1 are the triples $(x,x+1,x+3)$ modulo $7$. The lines of type 2 are the triples $(x,x+1,x+5)$ modulo 7. Define the incidence relation as follows. A point is incident to a line of typ... | https://mathoverflow.net/users/12039 | What is the automorphism group of this geometry? | Your geometry has the property that each of its rank 2 restrictions is a Fano plane. In particular, the type-preserving automorphism group (let's call it $G$) is a subgroup of the automorphism group of the Fano plane, which is $PSL(3,2)$. The group $G$ has the property that the pointwise stabilizer of any two points is... | 9 | https://mathoverflow.net/users/12858 | 61212 | 37,918 |
https://mathoverflow.net/questions/61096 | 4 | I am looking for some references about irreducible representations of the Weyl Group over simple Lie Groups, both classical and exceptional ones. In particular I want to know the dimensions and the character tables of the irreps.
I heard of those Mackey theory for a while (which should be helpful for the Weyl group o... | https://mathoverflow.net/users/14226 | weyl group representations | Let me add a useful reference book, probably no longer in print but found in many libraries: R.W. Carter, *Finite Groups of Lie Type: Conjugacy Classes and Complex Characters*, Wiley-Interscience, 1985. The book includes a lot of information about the irreducible representations of Weyl groups (though not with complete... | 7 | https://mathoverflow.net/users/4231 | 61215 | 37,921 |
https://mathoverflow.net/questions/61198 | 6 | I have a short question concerning the spectral theory of automorphic forms. What is the main property of the unipotent group $N$, which consist of matrices in the form \begin{pmatrix} 1 & t \\ 0 &1 \end{pmatrix} in $GL(2)$, which provides that the cuspidal spectrum decomposes discretely?
The background: Consider $G ... | https://mathoverflow.net/users/10400 | Why is the cuspidal spectrum discrete? | This result is due to Gelfand, Graev, Piatetski-Shapiro and has a short proof. I suggest you read Bump: Automorphic forms and representations, Prop. 3.2.3, pp. 285-289.
Let me switch to $G=\mathrm{PGL}\_2(\mathbb{R})$ and $\Gamma=\mathrm{PGL}\_2(\mathbb{Z})$ for simplicity.
The idea is to consider right convolutions... | 6 | https://mathoverflow.net/users/11919 | 61220 | 37,924 |
https://mathoverflow.net/questions/61218 | 1 | Given a galois extension of number fields $L/K$ of even degree, set $n\_0=\text{lcm} (\{[L\_v:K\_v] : v \in M\_K \})$ ($L\_v$ is any completion corresponding to a place dividing $v$).
Does $2$ divide $n\_0$?
This comes up in [this question](https://mathoverflow.net/questions/61084/examples-of-galois-invariant-centr... | https://mathoverflow.net/users/2024 | Can an even degree galois extension complete p-adically to an even galois extension | If I'm not mistaken, the slightly stronger result is true, that the lcm of orders of Frobenius elements must be even (forget ramified primes, that is). Isn't that a corollary of the Chebotaryov density theorem (there is a Frobenius in every conjugacy class, and some such class contains elements of even order)?
| 5 | https://mathoverflow.net/users/6153 | 61222 | 37,926 |
https://mathoverflow.net/questions/61125 | 1 | Hi everyone, I know that this system dont have analytical solutions. I want to get numerical solutions, but in function of some constants $A\_i$. Mathematica can help me, but if somebody have idea? This equations describe a physical model
$$ A\_6 x + A\_4 (y')^2 - 2 A\_2 x'' - A\_3xy'' + A\_4yy'' = 0$$
$$ A\_5 - A... | https://mathoverflow.net/users/14234 | system of two second order differential equations | You can reduce the number of parameters quite a bit, for starters. Set
$$x = \alpha u , \; y = \beta v, \; t = \gamma \tau
$$
and write $\dot w = \frac{d}{d \tau} w, \ddot w = \frac{d^2}{d\tau^2} w$. By choosing the constants $\alpha, \beta, \gamma$ properly, you should be able to nondimensionalize the system to so... | 2 | https://mathoverflow.net/users/7352 | 61228 | 37,928 |
https://mathoverflow.net/questions/61077 | 5 | This question is related to [this one](https://mathoverflow.net/questions/61075/normal-measures-on-p-kappa-lambda-extend-the-club-filter). The setup is as follows:
In $V$, $\kappa$ is supercompact and $\mathbb{P} = \mathrm{Coll}(\kappa, \aleph\_2)$ is the Levy collapse. $G$ is $(V,P)$-generic, and in $V[G]$, $C \subs... | https://mathoverflow.net/users/7521 | Does a generic normal measure extend the club filter? | With the help of Jason's answer to the question I linked to, I think I might be able to solve this one. The key is to show that $P\_{\kappa}^{V[G]}(\lambda) \in M[G\times H]$. Let's simply denote this set by $X$. An element $x$ of $X$ can be regarded as a function $x : \omega\_1 \to \lambda$, which will be a subset of ... | 2 | https://mathoverflow.net/users/7521 | 61237 | 37,932 |
https://mathoverflow.net/questions/61235 | 5 | Greetings
As a necessity to go forward with physics, I find myself in the need to learn about manifolds. Being an engineering student, I don't have the chance to study topology in all its glory.
So, can any one point me to the right direction, i mean things I must know first and the materials to fill the gaps before ... | https://mathoverflow.net/users/14255 | Background to learn about manifolds | That depends on your mathematical background and the level of abstraction you want.
The minimum amount of concepts you should be familiar with, ranges from topological spaces and its basic properties (elements of point-set topology) to multivariable calculus (covering implicit and inverse function theorems), passing ... | 19 | https://mathoverflow.net/users/10867 | 61240 | 37,935 |
https://mathoverflow.net/questions/61230 | 18 | Does there exist an infinite finitely generated group $G$ together with a *finite* group $B$ of automorphisms of $G$ such that
1. The non-identity elements of $B$ are not inner automorphisms of $G$;
2. For every element $g \in G, g\neq 1,$ the finite set $g^B$ is a generating set for $G$?
It looks like a [Tarski Mo... | https://mathoverflow.net/users/14250 | Example of a finitely generated infinite group with a non-inner automorphism of finite order | I am pretty sure this example can be constructed. It might be not easy, though, since you need to have finite order automorphisms of the Tarski monsters without non-trivial fixed points. I do not remember if that has been done for Tarski monsters. For other monsters it was done, I think, by Ashot Minasyan in [Groups wi... | 11 | https://mathoverflow.net/users/nan | 61247 | 37,940 |
https://mathoverflow.net/questions/56829 | 8 | Let $L = \sum\_{i,j=1}^n -\frac{\partial}{\partial x^i} (a^{ij}(x)\frac{\partial}{\partial x^j}) + \sum\_{i=1}^n b^i(x) \frac{\partial}{\partial x^i} + c(x)$ be a second order elliptic operator with smooth coefficients, $\Omega$ a bounded open domain with smooth boundary in $\mathbb{R}^n$, and $f$ be a function in $L^p... | https://mathoverflow.net/users/4345 | Proof of L^p Elliptic Regularity | I stumbled across the book *Second Oder Elliptic Equations and Elliptic Systems* by Yah-Ze Chen which appears to contain an answer to my question. It is available on google books here
<http://books.google.com/books?id=eQcbiPQPweQC&pg=PA49&dq=strong+solution+dirichlet+problem+Lp&hl=en&ei=byKiTeXXO6GG0QGG7dGgBQ&sa=X&o... | 5 | https://mathoverflow.net/users/4345 | 61250 | 37,942 |
https://mathoverflow.net/questions/61254 | 4 | I am interested in computing the sum of squares of determinants of principal minors. Let $A$ be an $n\times n$ positive semidefinite matrix and $A\_S$ be a principal minor of $A$ indexed by the set $S \subseteq \{1,\ldots,n\}$. The classical result (without squares) is:
$\sum\_{S \subseteq \{1,\ldots,n\}} \det(A\_S) ... | https://mathoverflow.net/users/14259 | Sum of squares of determinants of principal minors | The identity you mention does generalize to sums of powers, but I don't know if it can give you anything computationally efficient. Given a set $X\subset \mathbb R$, let $D(X^n)$ denote all $n\times n$ diagonal matrices with diagonal elements from $X$. Then if you take $X\_k=\{1,\omega,\cdots,\omega^{k-1}\}$ the $k$-th... | 7 | https://mathoverflow.net/users/2384 | 61258 | 37,947 |
https://mathoverflow.net/questions/61245 | 3 | Let $A$ be $p\times p$ symmetric positive definite with distinct eigenvalues and $x\_p\in \mathbb{R}^p$ and consider the problem
Minimize $x'Ax + b'x$
Subject to $x'x=1$
Most of the information I've found is is either very general/theoretical or specific to linear constraints, although I'm largely flitting around... | https://mathoverflow.net/users/12064 | Optimizing a quadratic restricted to the sphere | Your problem has been studied extensively in the context of trust region methods for optimization, and there are a number of algorithms that have been developed.
See for example:
W. W. Hager, Minimizing a quadratic over a sphere. SIAM Journal on Optimization, 12:188-208, 2001.
Hager's paper gives a lemma that c... | 5 | https://mathoverflow.net/users/9022 | 61262 | 37,950 |
https://mathoverflow.net/questions/61263 | 16 | Consider the space of newforms $S^{\mathrm{new}}\_k(\Gamma\_1(q))$ of weight $k$ and level $q$ for the congruence subgroup $\Gamma\_1(q)$ of $\mathrm{SL}\_2(\mathbb{Z})$; for simplicity's sake, let's assume that $q$ is prime. Then for $k \geq 2$, it is known via Riemann-Roch that
$$\dim S^{\mathrm{new}}\_k(\Gamma\_1(q)... | https://mathoverflow.net/users/3803 | Modular Forms of Weight One | The formula for the dimension of $S\_k$ when $k \geq 2$ can be thought of as a Riemann--Roch calculation, applied to an appropriately chosen line bundle on the modular curve. The point is that when $k \geq 2$, this line bundle is positive, so positive that the $H^1$-term in Riemann--Roch vanishes. Thus the dimension of... | 22 | https://mathoverflow.net/users/2874 | 61266 | 37,953 |
https://mathoverflow.net/questions/61160 | 5 | Let $X$ be a smooth hypersurface in $\mathbb{P}^n$ and let $\mathfrak{P}$ be the completion of $\mathbb{P}^n$ along $X$.
Let $\mathcal{R}$ denote the category of reflexive sheaves on $\mathbb{P}^n$ which are bundles outside a finite set of points in $X$ (the finite set is not fixed, it could/will be different for di... | https://mathoverflow.net/users/6425 | Grothendieck group of vector bundles | I will first assume that, by $K\_0$, you mean the quotient of the free abelian group generated by isomorphism classes of objects in your category, modulo the relation
$$\sum\_i(-1)^i [F\_i] = 0 \quad (A)$$
for every exact sequence $0 \to F\_0 \to F\_1 \to \dots \to F\_{n-1} \to F\_n \to 0$.
Well, since the functor... | 2 | https://mathoverflow.net/users/7437 | 61272 | 37,956 |
https://mathoverflow.net/questions/61143 | 3 | Suppose $K$ is a centrally symmetric, strictly convex body in $\mathbb{R}^2$. Let denote the curvature and the support function of $\partial K$, boundary of $K$, respectively with $\kappa$ and $s$. If $m\le\frac{\kappa}{s^3}(K)\le M$ for some positive numbers $m$ and $M$, does it mean there are ellipsoids $E\_1$ and $E... | https://mathoverflow.net/users/12145 | minimal maximal ellipsoids | Yes this is true. Let me handle the inner ellipse, the outer one is similar.
For brevity, denote $\kappa/s^3$ by $a$. It is easy to see that
$$
a = \frac{\dot\gamma\wedge\ddot\gamma}{(\gamma\wedge\dot\gamma)^3}
$$
where $t\mapsto \gamma(t)$ is any counter-clockwise parametrization of the boundary curve. For an ellip... | 4 | https://mathoverflow.net/users/4354 | 61276 | 37,958 |
https://mathoverflow.net/questions/61123 | 4 | For each odd prime $p$, choose an arbitrary $x\_p \in \mathbb Q(\zeta\_p) \smallsetminus \mathbb Q$, where $\zeta\_p = e^{\frac{2\pi i}{p}}$. Is it always true that the set $\{x\_p : p \in \mathbb P\}$ is linearly independent over $\mathbb Q$?
| https://mathoverflow.net/users/14233 | Cyclotomic fields and $Q$-linearly independence | I think everything is clear now by George Lowther's [comment](https://mathoverflow.net/questions/61123/cyclotomic-fields-and-q-linearly-independence#comment153670_61123), reproduced here with two minor typos corrected:
>
> $\mathbb Q(\zeta\_m) \cap \mathbb Q(\zeta\_n) = \mathbb Q$ also follows from $[\mathbb Q[\zet... | 1 | https://mathoverflow.net/users/14233 | 61298 | 37,971 |
https://mathoverflow.net/questions/61295 | 6 | Let $\mathbb P$ be the set of prime numbers.
Is there a non constant polynomial $f \in \mathbb Z[X]$ such that the set $$I\_f := \{ \textstyle\frac{z}{p} : z \in \mathbb Z, p \in \mathbb P, p \mid f(z) \}$$
is dense in $\mathbb R$?
(I suppose that the following much stronger statement holds: Every polynomial $f \i... | https://mathoverflow.net/users/14233 | Distribution of zeros of a polynomial mod. a prime | The set $I\_f$ is dense for quadratic polynomials which are irreducible over $\mathbb{Q}$, by the work of Duke-Friedlander-Iwaniec (Ann. of Math. 141 (1995), 423-441) and Toth (IMRN 2000, No. 14, 719-739). In fact we have uniform distribution even when $p$ is restricted to a reduced residue class of any modulus.
| 6 | https://mathoverflow.net/users/11919 | 61305 | 37,974 |
https://mathoverflow.net/questions/61294 | 3 | Hi. I have been struggling with this question for a while now.
Given a domain $\Omega \subset \mathbb{C}^n$, $\Omega^\prime = \Omega \cap \mathbb{R}^n$, and an analytic function $f: \Omega \longrightarrow \Omega$ such that $f\_{|\Omega^\prime}$ is a (real) constant, under which assumptions on $\Omega$ and/or $\Omega^... | https://mathoverflow.net/users/14179 | Uniqueness of analytic continuation on a domain of C^n. | Without loss of generality, assume that your $f\_{|\Omega^\prime}\equiv 0$. The knowledge of this restriction is sufficient to conclude that $f$ and all of its derivatives vanish on $\Omega'$. The subset of $\Omega$ on which $f$ and all of its derivatives vanish is both closed and open (clopen). Since $\Omega$ is a dom... | 5 | https://mathoverflow.net/users/2622 | 61310 | 37,976 |
https://mathoverflow.net/questions/61314 | 5 | We say that a semigroup $S$ has solvable power problem if there is an algorithm that takes as input an element $s \in S$ and decides whether or not there exist $m,n \in \mathbb{N}$ with $m \neq n$ and $s^m=s^n$.
>
> Does anybody know an "easy" (like finitely presented with relatively few relations) example of a se... | https://mathoverflow.net/users/8434 | an example of a semigroup with solvable word problem but unsolvable power problem | The only known way to construct this example (say, in the case of groups, the case of semigroups is similar) is the following. First consider the free Abelian group $F$ with free generators $a\_1,a\_2,...$. Pick a recursively enumerable non-recursive set $I$ and impose relations $a\_n^{m!}=1$ if $n$ is the $m$th number... | 6 | https://mathoverflow.net/users/nan | 61321 | 37,982 |
https://mathoverflow.net/questions/61316 | 2 | Hi all,
I heard a claim that if I have a matrix $A\in\mathbb R^{n\times n}$ such that $A^n \to 0 \ (\text{for }n\to\infty )$
(that is, every entry of $A^n$ converges to $0$ where $n\to \infty$)
then $I-A$ is invertible.
anyone knows if there is a name for such a matrix or how (for general knowledge) to prove this ... | https://mathoverflow.net/users/13743 | invertability of a matrix | The matrices you are looking for are exactly those that have spectral radius (the max. of the absolute value of the eigenvalues) strictly less than one.
I do not know whether there is a more specific name.
(A matrix such that a finite power would be exactly the zero-matrix would be called nilpotent; but this is a dif... | 4 | https://mathoverflow.net/users/nan | 61322 | 37,983 |
https://mathoverflow.net/questions/61327 | 4 | Pausing to speak metaphorically for a bit, I have always thought of partial orders as being something like a simplified version of topological spaces. (In the finite case at least, all topological spaces are partial orders). This leads to thinking about continuous maps as a parallel concept to the idea of a poset homom... | https://mathoverflow.net/users/4642 | Is it reasonable to define `poset homotopy' as a `natural transformation of posets'? | This is an interesting line of questions, but I think it doesn't quite work as stated. First off, your notion of "homotopy" is not an equivalence relation (as far as I understand it), so it won't agree with a topological notion.
But there are also other issues; basically, any notion of "poset homotopy classes" along ... | 4 | https://mathoverflow.net/users/5010 | 61332 | 37,989 |
https://mathoverflow.net/questions/61336 | 3 | I know there are manifolds (with or without boundary) $A$ and $B$ such that $A\times C$ is homeomorphic to $B\times C$ but $A$ is NOT homeomorphic to $B$.
My question is (in the Diffeomorphism Category)
Is there any infinitely many $A\_i$, with same dimension of course, which are pairwise non-diffeomorphic, but $A... | https://mathoverflow.net/users/3922 | Existence of sequence of examples of braking 'Cancellation law in homeomorphic products' | The answer to 2 is yes, there is such an example. In
>
> McMillan, D. R., Jr., *Some contractible open $3$-manifolds*. Trans. Amer. Math. Soc. **102** (1962), 373–382.
>
>
>
there is a construction of uncountably many topologically distinct, contractible (open) $3$-manifolds $M\_\alpha$ such that $M\_\alpha \t... | 3 | https://mathoverflow.net/users/430 | 61341 | 37,993 |
https://mathoverflow.net/questions/58099 | 10 | Let $C$ be a complex curve. Recall that a Higgs bundle on $C$ is a vector bundle $E$ on $C$ equipped with a morphism $E \to E \otimes K\_C$. The space of (stable) Higgs bundles is much studied, and is in particular known to be smooth. Moreover there is a "nonabelian Hodge theorem" giving a diffeomorphism between the mo... | https://mathoverflow.net/users/4707 | What is known about Higgs bundles with sections? | There is a fundamental difference between the case of Higgs bundles (where the section lies in a twisted adjoint representation) and the case of a section of the bundle itself (where the section is in the vector representation). In the former, the notion of stability is rigid, whereas in the latter the definition of st... | 5 | https://mathoverflow.net/users/9867 | 61355 | 38,000 |
https://mathoverflow.net/questions/61350 | 8 | I am trying to determine the behavior of the following series as $n\to\infty$. Let $0<\mu<1$ be fixed and for every positive integer $n\geq 1$, consider the function $f\_n(t)$ of a real variable $t$ defined by the series $\sum\_{k=0}^\infty\mu^k(1-\mu^kt)^n$. I want to determine how $f\_n(t)$ behaves as $n\to\infty$ fo... | https://mathoverflow.net/users/14289 | Determining the asymptotic behavior of a series | This is my third response. I claim that for $0 < t < 1$ we have the uniform bounds
$$ \liminf\_{n\to\infty}\ nt f\_n(t) = \frac{-1}{\log\mu}+O(1),$$
$$ \limsup\_{n\to\infty}\ nt f\_n(t) = \frac{-1}{\log\mu}+O(1),$$
where $O(1)$ denotes quantities that are absolutely bounded. First of all,
$$ f\_n(t) = \int\_{0-}^\infty... | 9 | https://mathoverflow.net/users/11919 | 61358 | 38,001 |
https://mathoverflow.net/questions/61034 | 14 | Many know the TV game *Countdown*, whose French version *Des chiffres et des lettres* has lasted since 1965.
The rules of the count are as follows: you are given natural integers $n\_1,\ldots,n\_6$ and a target $N$. You are free to employ the four operations $+,\times,-,\div$. You may employ each $n\_j$ at most once.... | https://mathoverflow.net/users/8799 | Optimal Countdown | We can prove that $\log N\_k \sim k \log k$ as follows:
If we want to combine a set of $k$ numbers using the four arithmetic operations, we can think of inputting the numbers (in any order) along with the operations into an RPN calculator. There are $k!$ ways of ordering the numbers, $C\_{k-1} = \frac1{k}{2k-2 \choos... | 9 | https://mathoverflow.net/users/8252 | 61362 | 38,002 |
https://mathoverflow.net/questions/61360 | 1 | Hi,
This question is motivated by a statistical genetics model.
Let $(x\_1,y\_1)$, .., $(x\_N,y\_N), ... $ be i.i.d. bi-variate Gaussian random variables.
The $x\_i,y\_i$'s are standard Gaussians, $x\_i, y\_i \sim N(0,1)$, and
$corr(x\_i,y\_i) = \rho$ for some $\rho \in (0,1)$.
Let $X\_N = \max(x\_1, .., x\_N)$ ... | https://mathoverflow.net/users/1778 | Maximums of two correlated Gaussian processes | The main contribution to the correlation between $X\_N$ and $Y\_N$ is the event that the same $i$ maximizes $x\_i$ and $y\_i$. If $\rho$ is fixed, this event is asymptotically unlikely. (Given the value of $X\_N=x\_i$ we have that $E y\_i=\rho X\_N$, which is not large enough to make $y\_i$ maximal.) For essentially th... | 4 | https://mathoverflow.net/users/9422 | 61363 | 38,003 |
https://mathoverflow.net/questions/61354 | 3 | This is a question in general sense, but answers about specific examples are also welcome.
Why do we need the notion of stability of objects in a category. If we've a subcategory of stable objects, what can we do with this subcategory. What is the general philosophy of the stability data.
In case of vector bundles... | https://mathoverflow.net/users/9534 | notion of stability in a category | It is well know that general families of objects in categories behave badly. However, imposing a stability gives you well behaved families. In particular, stable objects are parametrized by moduli spaces, not just by some $\infty$-stacks. So, the answer is that you need a stability if you want to have a moduli space of... | 4 | https://mathoverflow.net/users/4428 | 61371 | 38,008 |
https://mathoverflow.net/questions/61396 | 2 | Given a finite group and a normal subgroup, does there always exist an irreducible complex representation, whose kernel is this normal subgroup?
Sorry, just it was just mentioned that this is a duplicate. See [Which finite groups have faithful complex irreducible representations?](https://mathoverflow.net/questions/5... | https://mathoverflow.net/users/10400 | Does every normal subgroup appear as a kernel of an irreducible representation? | take the canonical quotient $q:G\rightarrow G/H$ and $\lambda :G/H\rightarrow U(n)$ to be the left-regular representation of $G/H$, where $n=|G/H|$, then the composition is the map that you are looking for, unless I am missing something...
**Edit:** if you consider irreducible representations, then the answer is no f... | 7 | https://mathoverflow.net/users/8699 | 61398 | 38,020 |
https://mathoverflow.net/questions/61401 | 2 | Hi,
I would like to approximate any 2d conformal mapping, as a sum of elementary conformal mappings. So I would have some basis, a conformal mapping with some parameters, and by adding several ones of those, I would mimic any conformal mapping. I don't know a-prioris the map I'm going to approximate, I just want to be... | https://mathoverflow.net/users/14308 | Approximation of conformal mapping as a sum of elementary conformal mappings | That depends a lot on the metric in which you wish to approximate.
For entire functions you have e.g. Taylor series. For holomorphic functions you can add simple poles.
If the function can be defined only on some arbitrary domain you'll need many more basis elements.
| 1 | https://mathoverflow.net/users/9422 | 61403 | 38,024 |
https://mathoverflow.net/questions/61392 | 3 | Suppose $ X \to \mathbb{P}^1 $ is an elliptic surface with section, with Weierstrass model defined over $ \mathbb{Q} $. If $ \sigma: \mathbb{P}^1 \to X $ is a torsion section with order $n$, then for generic $ t \in \mathbb{P}^1 $, $ \sigma(t) $ is a point of order $n$ in the fiber $ X\_t $. Now it seems to me that the... | https://mathoverflow.net/users/4192 | Mordell-Weil Group of Elliptic Surface | 1):
Suppose we work over an arbitrary field $K$ and suppose $\pi: X\to \mathbb{P}^1$ is an elliptic fibration with a torsion section of order $n$. Then you obtain a natural classification map $\mathbb{P}^1 \to X\_1(n)$.
If this map is dominant then $g(X\_1(n))=0$ and hence $n\leq 12$. (You do not need Mazur's theore... | 4 | https://mathoverflow.net/users/8621 | 61416 | 38,028 |
https://mathoverflow.net/questions/61404 | 2 | Please consider the case where I have 'N' rods of length L (and width W) placed on a one- or two-dimensional surface with dimensions [0, A] in 1D, and [ [0, A], [0, B] ] in 2D. For the two-dimensional case, L and W are << A or B.
As a function of the number of rods N, and the relative dimensions of the rods and the ... | https://mathoverflow.net/users/3248 | Intersection probability for 'N' fixed-length rods in one- or two-dimensions | To compute the expected number of intersections, use the fact that expectation is additive. The expected number of intersections is just $\binom{N}{2}p$ where $p$ is the probability of an intersection.
To compute the probability of an intersection you can make your life easier (with essentially no cost to the accura... | 3 | https://mathoverflow.net/users/11054 | 61418 | 38,029 |
https://mathoverflow.net/questions/61413 | 5 | Hi, consider a scheme $X$ of finite type over $\mathbb{R}$ (or $\mathbb{C}$). In Hartshorne's appendix B on 'transcendental methods' it is shortly mentioned how to assign a reasonable topological space to $X$ and he says that one can use the 'same glueing data'. My question is two-parted:
>
> Does this construction... | https://mathoverflow.net/users/14310 | Topological space associated to a real or complex scheme | Dear mustafa-kava, Amnon Neeman has written a rather down-to-earth [book](http://books.google.fr/books?id=VacrAAAAYAAJ&q=neeman+amnon&dq=neeman+amnon&hl=frei=93ykTdmNNsiv8gOA1L25Dw&sa=X&oi=book_result&ct=result&resnum=2&ved=0CEAQ6AEwAQ) *Algebraic and Analytic Geometry* dedicated to a proof of Serre's celebrated GAGA t... | 6 | https://mathoverflow.net/users/450 | 61419 | 38,030 |
https://mathoverflow.net/questions/61382 | 2 | Let $X$ be a (generalised) flag variety over an algebraically closed field $k$ of characteristic zero, that is to say, $X$ is a projective variety which is a homogeneous space for some algebraic group $G$. Any such $X$ can be realised as a quotient $X=G/P$, where $P$ is a parabolic subgroup (please correct me if this i... | https://mathoverflow.net/users/5101 | Rationality of flag varieties | As indicated by Piotr, the answer to your question is yes. One helpful source is the Springer graduate text by Borel, *Linear Algebraic Groups*; see AG.13.7 and Chapter 14, especially 14.14. For the study of generalized flag varieties you may as well assume the group $G$ is *reductive* because the unipotent radical is ... | 7 | https://mathoverflow.net/users/4231 | 61422 | 38,033 |
https://mathoverflow.net/questions/61406 | 3 | I'm interested in fundamental group of smooth part of log fano varieties.
**Question.1**
Is there an example of a non-Gorenstein terminal Fano 3-fold whose smooth part is simply connected?
Actually, I'm interested in classification of terminal fano 3-folds which are not necessarily Gorenstein.
I think it's optimi... | https://mathoverflow.net/users/12361 | Fundamental group of smooth part of log fano varieties | **Example for Q1.** Consider $\mathbb CP^3/\mathbb Z\_2$ with the action $(x:y:z:t)\to (x:y:z:-t)$. The quotient has one terminal non-Goernstein singularity $(0:0:0:1)$ and the complement to this singularity is isomorphic to a line bundle ($O(2)$) over $\mathbb CP^2$, so it has $\pi\_1=0$.
| 3 | https://mathoverflow.net/users/943 | 61426 | 38,036 |
https://mathoverflow.net/questions/61423 | 2 | I've found lots of (more or less precise) definitions of the Alexandrov curvature, but I'm mainly interested in that of "Alexandrov curvature bounded below". Could anyone give me that or give me a reference?
Thanks in advance,
Valerio
| https://mathoverflow.net/users/13809 | Alexandrov curvature of a compact length space | From "[A.D. Alexandrov spaces with curvature bounded below](http://iopscience.iop.org/0036-0279/47/2/R01;jsessionid=D6F567EBB64F297A7AAD9AC0AF206739.c1)",
Burago, Y. and Gromov, M. and Perel'man, G.,
*Russian Mathematical Surveys*, 47, 1992, p.5:
>
> A locally complete space $Μ$ with intrinsic metric is called a
>... | 3 | https://mathoverflow.net/users/6094 | 61427 | 38,037 |
https://mathoverflow.net/questions/60507 | 3 | Let $N$ and $N'$ be regular neighborhoods of a subpolyhedron $P$ in a closed PL manifold $M$, and suppose that $t$ is a free PL involution on $M$ such that each of $\partial N$, $\partial N'$ is invariant under $t$. Does there exist an equivariant PL isotopy of $M$ taking $N$ onto $N'$?
**Edit:** *Beware that $N$ is... | https://mathoverflow.net/users/10819 | uniqueness of regular/tubular neighborhood with equivariant boundary | I learned from Piotr Akhmetiev that $S^6$ contains two smoothly embedded $5$-spheres invariant under the antipodal involution that are not equivariantly PL isotopic, and the reference is [Lopez de Medrano's "Involutions on Manifolds"](http://www.zentralblatt-math.org/zmath/search/?an=0214.22501). Of course they are bou... | 2 | https://mathoverflow.net/users/10819 | 61431 | 38,041 |
https://mathoverflow.net/questions/61229 | 7 | Suppose we are given a positive integer $k$. Let $K\_k$ denote the complete (undirected and simple) graph with vertices $1, 2, \dots, k$. The set of edges of $K\_k$ is the set $E\_k = \{ \{ x,y \} \mid \ 1 \leq x < y \leq k\}$.
A valuation of $K\_k$ is a function $\omega: E\_k \rightarrow \mathbb Z$. A splitting of $... | https://mathoverflow.net/users/14233 | Splitting an evaluated complete graph | The answer is NO, at least for $n=4$. I show here that $\eta(4) \leq 7$. So
let $\omega$ be a valuation on $K\_7$ ; we will show that it is $4$-valid.
First, we need some notation : for $A \cup B$ a nontrivial partition
of $V=\lbrace 1,2,3, \ldots ,7\rbrace$, let
$$
s(A,B)=\\sum\_{(x,y) \in A \times B} \omega(\{ x,... | 2 | https://mathoverflow.net/users/2389 | 61435 | 38,044 |
https://mathoverflow.net/questions/61433 | 4 | Let $X^n \subset \mathbb{P}^N$ to be a toric projective variety. Is $X$ a local complete intersection? Is being a local complete intersection an intrinsic property, independent of embedding?
| https://mathoverflow.net/users/5259 | Are projective toric varieties, locally complete intersection? | No, there are projective toric varieties that are not even Gorenstein (and hence not l.c.i). In fact the Gorenstein property translates into a geometric property of the defining polytope, namely that it is reflexive under taking the dual polytope.
| 8 | https://mathoverflow.net/users/3996 | 61438 | 38,045 |
https://mathoverflow.net/questions/61439 | 0 | Let $f∈Q[X]$ and not constant or of the form $(x−a)^n$. Suppose:
$f\_1:=\frac{f}{gcd(f,D^2f)}$
and
$f\_2:=\frac{f\_1}{gcd(f\_1,Df\_1)}$
where $Df$ stands for the formal derivative.
Is it true that $gcd(f\_2,Df\_2)=gcd(f\_2,D^2f\_2)=1$ ?
| https://mathoverflow.net/users/12844 | Generating polynomials that are co-prime to their first and second derivatives | No.
Let
$$f(x)=(x-3x^3)(x+1/3).$$
Then $f''(-1/3)=0$ and
$$f\_1(x)=x-3x^3.$$
Since $f\_1$ and $f\_1'$ have no common roots, $f\_2=f\_1$. But $gcd(f\_2,D^2f\_2)=x$.
| 4 | https://mathoverflow.net/users/12120 | 61444 | 38,048 |
https://mathoverflow.net/questions/61377 | 4 | How can I prove that the following 2 prehomogeneous vector spaces are not isomorphic?
1)$(GL\_n,\Lambda\_1\oplus \Lambda\_1,\mathbb{C}^n \oplus \mathbb{C}^n)$
2)$(GL\_n,\Lambda\_1\oplus \Lambda\_1^\*,\mathbb{C}^n \oplus \mathbb{C}^n)$
where $\Lambda\_1$ is the standard representation of $GL\_n$ on $\mathbb{C}^n$.
in th... | https://mathoverflow.net/users/4821 | Prehomogeneous vector spaces | Your two representations of $GL\_n$ are not isomorphic, because one of them contains $\Lambda\_1$ with multiplicity 2 and the other with multiplicity 1, $\Lambda\_1^\*$ and $\Lambda\_1$ being non-isomorphic. More generally, if $V$ and $W$ are finite-dimensional rational representations of $G=GL\_n$ with $V$ irreducible... | 5 | https://mathoverflow.net/users/5740 | 61450 | 38,051 |
https://mathoverflow.net/questions/61447 | 7 | Please don't get put off by the length, all the questions are quite simple, but given the quasi-mathematical context I tried to be precise with the formulation. The more mathematically interesting title question (and the one that's most important for my purposes) is the last one, so if anything please take a look at th... | https://mathoverflow.net/users/13753 | Is this a proper application of the Lowenheim-Skolem Theorem to a proper class? | The result you state is *not* provable in ZFC, and cannot be formalized without going to a stronger theory.
We cannot apply Lowenheim-Skolem (LS) to proper classes; the issue is that we do not have a truth predicate, so the construction of hulls cannot be formalized "from within". Note that partial truth predicates ... | 10 | https://mathoverflow.net/users/6085 | 61455 | 38,055 |
https://mathoverflow.net/questions/61443 | 26 | It's known that [every position of Rubik's cube can be solved in 20 moves or less](http://cube20.org/). That page includes a nice table of the number of positions of Rubik's cube which can be solved in k moves, for $k = 0, 1, \ldots, 20$. (Some of the entries of the table are approximations, but they're good enough for... | https://mathoverflow.net/users/143 | What relationship, if any, is there between the diameter of the Cayley graph and the average distance between group elements? | Your facts about $S\_n$ are actually all facts about Coxeter groups with the generating set given by simple reflections: the distribution is symmetric around the mean, which is half the diameter. It even has a unique local maximum (assuming you allow the floor and roof of the mean to be "one maximum" even if they're di... | 19 | https://mathoverflow.net/users/66 | 61490 | 38,082 |
https://mathoverflow.net/questions/61505 | 1 | Let $X$ be a quasiprojective algebraic variety over $\mathbb{C}$ with structure sheaf $\mathcal{O}\_X$ and let $G$ be a finite group which acts freely on $X$. The quotient $Y = X/G$ is thus a quasiprojective variety in a canonical way; let $\mathcal{O}\_Y$ be its structure sheaf. Assume that $c : G \rightarrow \mathbb{... | https://mathoverflow.net/users/14335 | Constructing invertible sheaves out of actions of finite groups | This is classical, can be found in Mumford: Abelian varieties, II.7 (i think) group actions on varieties (however without the twist, you mention, but i think it is no poblem), and the main reason (again without the twist) is that $\pi^{\*}\mathcal{F}\cong \mathcal{O}\_{X}$. You can use this to conclude...
| 1 | https://mathoverflow.net/users/12847 | 61506 | 38,090 |
https://mathoverflow.net/questions/61498 | 9 | Let $H$ be a quaternionic algebra over ${\bf Q}$, and let $R$ denote a maximal ${\bf Z}$-order in $H$. Is there a theorem on the structure of the units in $R$ analogous to the Dirichlet unit theorem? Is there an analogous theorem for the $S$-units?
| https://mathoverflow.net/users/14328 | Units in quaternionic algebras | If $H$ is definite, then the group of units of $H$ is finite. If $H$ is indefinite, then the group of units is a pretty chunky group; it embeds as a cocompact discrete subgroup of $SL(2, R)$, and the rank of its abelianization (which, if I remember correctly, can be interpreted geometrically as twice the genus of an as... | 12 | https://mathoverflow.net/users/2481 | 61512 | 38,093 |
https://mathoverflow.net/questions/61509 | 17 | I have a doubt which I hope the MO community can quickly resolve.
The associative grassmannian is an eight-dimensional homogeneous space $G\_2/SO(4)$. It can be identified with the space of quaternion subalgebras of the octonions. It has a $G\_2$-invariant riemannian metric making it a rank-2 riemannian symmetric spa... | https://mathoverflow.net/users/394 | Are the associative grassmannian and the quaternionic projective plane diffeomorphic? | According to
Characteristic Classes and Homogeneous Spaces, I
A. Borel and F. Hirzebruch,
Section 17, they are not even homotopy equivalent: $G\_2 / SO(4)$ has mod $2$ homology in degree 2, whereas $\mathbb{HP}^2$ does not.
| 18 | https://mathoverflow.net/users/318 | 61521 | 38,097 |
https://mathoverflow.net/questions/45574 | 9 | Martin's remarkable cone theorem in the theory of determinacy says the following:
>
> Suppose $A\subseteq \omega^\omega$ is Turing invariant and determined. If $\forall x\exists y(x\le\_T y\& y\in A)$ then $A$ contains a cone.
>
>
>
Let me explain what this means: $A$ is Turing invariant iff $\forall x\in A\f... | https://mathoverflow.net/users/6085 | Martin's cone theorem and recursion theory | I would like to add another example.
Given a sentence $\phi$ from partial order language, then for any Turing degree $x$, either $D(\leq x)\models \phi$ or $D(\leq x)\models \neg\phi$. By the BD, there is a Turing degree $x\_{\phi}$ so that either for all $y\geq\_T x\_{\phi}$, $D(\leq y)\models \phi$ or for all $y\ge... | 8 | https://mathoverflow.net/users/14340 | 61523 | 38,099 |
https://mathoverflow.net/questions/61518 | 2 | Hi,
Where goes a characterictic $0$ person, in order to learn about the local harmonic analysis for local fields in characteristic $p$? Is there nice and conscise reference for the local fields in positive characteristic. Something conscise like Sally's survey, which is for characterictic zero only: <http://www.sprin... | https://mathoverflow.net/users/10400 | Harmonic Analysis for Function Fields | Harmonic analysis dealing with complex-valued functions on a local field $K$ depends very little on the characteristic of $K$ and can be given in the general case, for an arbitrary $K$. See
I. M. Gelfand, M. I. Graev, and I. I. Piatetski-Shapiro, Representation Theory and Automorphic Functions, Saunders, Philadelphi... | 5 | https://mathoverflow.net/users/12205 | 61526 | 38,102 |
https://mathoverflow.net/questions/61513 | 5 | Hello. I'd like to consider the open unit ball in an infinite dimensional Hilbert space and ask when can we fit infinitely many open balls of radius $r<1$ inside.
For example, when $r=1/(1+\sqrt2)$, we can pick an orthonormal basis $(x\_1,...)$ for our Hilbert space and put the centers of the balls at $(1-r)x\_i = \s... | https://mathoverflow.net/users/5312 | Critical Radius for Infinite Dimensional Sphere Packing | Your value of $r$ is the best.
Equivalently, $\rho=\sqrt 2$, where $\rho$ is the sup, over all infinite sequence $(x\_i)$ in the unit ball of a Hilbert space, of $\inf\_{i\neq j} |x\_i-x\_j|$.
Here is a proof, by contradiction. Assume that $\rho>\sqrt 2$ and pick a sequence $(x\_i)$ such that $\inf\_{i\neq j} |x\_i... | 9 | https://mathoverflow.net/users/10265 | 61527 | 38,103 |
https://mathoverflow.net/questions/61533 | 6 | Let $K$ be a number field, $Z\_K$ its ring of integers, and $p$ a rational prime number. Then $A\_p = Z\_K/(p)$ is a finite ${\mathbb F}\_p$-algebra. Using ideal arithmetic in $Z\_K$ and the Chinese remainder theorem it is easily checked that $A\_p$ is the direct sum of finite fields if $p$ is unramified, and has addit... | https://mathoverflow.net/users/3503 | Decomposition of finite algebras over finite fields | The general form of the answer is easy to anticipate: a finite commutative ring is artinian. It will be a product of finite local rings, of characteristic that is a prime number. So your question is really which local rings you get in this case. I believe a certain amount can be done by linear algebra.
| 4 | https://mathoverflow.net/users/6153 | 61534 | 38,106 |
https://mathoverflow.net/questions/39436 | 1 | Let J be any C\*-algebra and K be the C\*-algbra of compact operators on a separable, infinite dimensional Hibert space.
How to show $K\_0(M(J\otimes K))=0$, where M denotes Multiplier algebra
| https://mathoverflow.net/users/9401 | k_0 group for $M(J\otimes K)=0$ | This is a variation of an argument that is called the Eilenberg swindle. It can be found for example in Blackadars book "K-theory for operator algebras" (see Proposition 12.2.1) and works as follows:
Note that, since $J \otimes K$ is an essential ideal in $M(J) \otimes\_{\alpha} M(K) = M(J) \otimes\_{\alpha} B(H)$, ... | 3 | https://mathoverflow.net/users/3995 | 61537 | 38,109 |
https://mathoverflow.net/questions/61368 | 3 | Depth is defined as the distance to the boundary of a set, i.e., $\operatorname{depth}(x, C) = \operatorname{dist}(x, \mathbb{R}^n \backslash C)$.
Let $C$ be a convex set that contains the origin. I believe that
$$\operatorname{depth}(0, C) = \min\_{\|u\| = 1} \ \max\_{v \in C} \ u \cdot v$$
I also believe the followi... | https://mathoverflow.net/users/7224 | A min-max formula for depth of the origin in a convex set | My office is quieter this morning, so let's try again. I'll assume that $C$ is compact with $0 \in \operatorname{int} C$. Let $B\_r$ denote the ball of radius $r$ centered at $0$. Then
$\operatorname{depth} (0,C) = \max \{ r > 0 \mid B\_r \subseteq C \}$.
The function $h\_C(u) = \max\_{v\in C} (u \cdot v)$ is called ... | 5 | https://mathoverflow.net/users/1044 | 61541 | 38,112 |
https://mathoverflow.net/questions/61540 | 6 | It's well known that there are Banach spaces which has a unique *isometric* predual-- for example, any von Neumann algebra. As other questions on here (for example, [Isomorphisms of Banach Spaces](https://mathoverflow.net/questions/1380/isomorphisms-of-banach-spaces) ) have shown, there are also Banach spaces $E$ with... | https://mathoverflow.net/users/406 | Unique preduals up to (nonisometric) isomorphism? | No. Take $E=F$ and perturb the identity on $E^\*$ by a rank one operator that is not weak$^\*$ continuous; such an operator existing is equivalent to $E$ being non reflexive.
| 9 | https://mathoverflow.net/users/2554 | 61544 | 38,115 |
https://mathoverflow.net/questions/61409 | 23 | It is well known that there is no general formula for the solution of the quintic. Of course, what this really means is that there is no general formula that only involves addition, subtraction, multiplication, division, and the extraction of $n$-th roots. Indeed, if one is allowed to use the Bring radical, that is, so... | https://mathoverflow.net/users/6856 | Using higher-order Bring radicals to solve arbitrary polynomials | The answer to the second question is "no". For a family of polynomials $p\_t$ depending polynomially on a complex parameter, as in the polynomials satisfied by your $B\_r(t)$, define its *Galois group* to be the group of permutations of the roots you see by moving around the branch points. (Assume that the roots of $p\... | 6 | https://mathoverflow.net/users/5010 | 61547 | 38,116 |
https://mathoverflow.net/questions/61545 | 9 | Recall that an Eilenberg-Maclane space $K(G, n)$ is characterized by $\pi\_i(K(G,n)) = G$ if $i=n$ and is trivial otherwise. (Of course $G$ should be abelian if $n>1$.)
I'm aware that computing $H^j(K(G,n), \mathbb Z)$ for general $j$ and $n$ is not so easy (see, e.g., [here](https://mathoverflow.net/questions/24754/... | https://mathoverflow.net/users/284 | Low degree cohomology of Eilenberg-MacLane space K(G,2)? | For a finite cyclic group G, in the range you ask for you get cohomology groups
$$\mathbb{Z}, 0, 0, G \cong Ext(G, \mathbb{Z}), 0.$$
One sees this by for example computing the Leray--Serre spectral sequence for
$$K(G, 1) \to \* \to K(G,2).$$
| 10 | https://mathoverflow.net/users/318 | 61550 | 38,118 |
https://mathoverflow.net/questions/61528 | 7 | Is the direct image of a constant sheaf a constant sheaf? I'm not an expert on sheaf theory and can't find this anywhere
| https://mathoverflow.net/users/13707 | Is the direct image of a constant sheaf a constant sheaf? | Notation: $f:X \to Y$ is the map we're pushing forward along, and $F$ is our sheaf on $X$. In general the stalks of $f\_\*F$ at different points will not be isomorphic. For instance if $f$ misses the point $y \in Y$ and your space is sufficiently separated then the stalk of $f\_\*F$ at $y$ will be 0 while it will be no... | 11 | https://mathoverflow.net/users/1123 | 61554 | 38,122 |
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