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https://mathoverflow.net/questions/24280 | 4 | $(A, \mathfrak{m})$ a Noetherian local ring, $a\in\mathfrak{m}$ a *zero divisor*. Then is it true that $\mbox{dim}\ A/(a) = \mbox{dim}\ A$ ?
| https://mathoverflow.net/users/5292 | Converse of Principal Ideal Theorem | No. Let A be the ring k[x,y,z]/(xz,yz) localized at (x,y,z). Then the dimension of A is 2, but the dimension of A/(x) is one. Note that xz = 0 in A.
The geometric picture is this: In 3-space, take the union of the horizontal z=0 plane with the vertical line x = y = 0, and look at the local ring at the origin. Then re... | 15 | https://mathoverflow.net/users/5094 | 24285 | 15,942 |
https://mathoverflow.net/questions/24265 | 62 | Let $s\_n = \sum\_{i=1}^{n-1} i!$ and let $g\_n = \gcd (s\_n, n!)$. Then it is easy to see that $g\_n$ divides $g\_{n+1}$. The first few values of $g\_n$, starting at $n=2$ are $1, 3, 3, 3, 9, 9, 9, 9, 9, 99$, where $g\_{11}=99$. Then $g\_n=99$ for $11\leq n\leq 100,000$.
Note that if $n$ divides $s\_n$, then $n$ div... | https://mathoverflow.net/users/1243 | What is the limit of gcd(1! + 2! + ... + (n-1)! , n!) ? | This is so close to the Kurepa conjecture which asserts that $\gcd\left(\sum\_{k=0}^{n-1}k!,n!\right)=2$ for all $n\geq 2$, which was settled in 2004 by D. Barsky and B. Benzaghou "Nombres de Bell et somme de factorielles". So what they proved is that $K(p)=1!+\cdots+(p-1)!\neq -1\pmod{p}$ for any odd prime $p$. This g... | 25 | https://mathoverflow.net/users/2384 | 24286 | 15,943 |
https://mathoverflow.net/questions/24020 | 5 | I am reading the book 'surface in 4-space' about the unknotting conjecture (Page 97): a 2-knot (2-sphere in 4-sphere) is trival if and only if the fundamental group of the exterior is infinite cyclic.
It said that in TOP category, Freedman proved the statement is true. I don't know why it is also true for general su... | https://mathoverflow.net/users/5980 | Is there a knotted torus in 4-sphere whose complement's fundamental group is infinite cyclic ? | I have been avoiding addressing this since almost all that I know about the question is in that book. I don't recall exactly, but I think that Kawauchi showed that a torus with the fundamental group of the complement being Z is topologically unknotted. Recent work of Hillman <http://arxiv.org/pdf/1003.5408>
addresses s... | 1 | https://mathoverflow.net/users/36108 | 24295 | 15,948 |
https://mathoverflow.net/questions/24318 | 30 | If there are not, then would it be easier to say that 2 objects are identical as ordered fields as opposed to being isomorphic as ordered fields? Or is the word isomorphism used to emphasise the fact that the objects are different as sets?
| https://mathoverflow.net/users/4692 | Are there situations when regarding isomorphic objects as identical leads to mistakes? | Inside of the complex numbers there are lots of examples of distinct fields which are isomorphic. For instance, there are three subfields of the form ${\mathbf Q}(\alpha)$ where $\alpha^3 = 2$: take for $\alpha$ any of the three complex cube roots of 2 and you get a different subfield. What are the consequences of trea... | 42 | https://mathoverflow.net/users/3272 | 24320 | 15,959 |
https://mathoverflow.net/questions/10039 | 11 | **Originally asked by [Ali Dino Jumani](https://mathoverflow.net/users/2866/ali-dino-jumani); EXTENSIVELY EDITED by David Speyer**. The previous version was a bit confused, but Steven Sivek and Graham, in the comments, figured out what was going on.
---
G. C. Shephard, in his paper "[Twenty Problems on Convex Pol... | https://mathoverflow.net/users/2866 | Characterizing faces of 3-dimensional polyhedra. (Related to Victor Eberhard's Theorem [1890]:) | Eberhard Theorem
----------------
Consider a simple 3-polytope P, (so every vertex has 3 neighbors). If $p\_k$ is the number of faces of P which are k-gonal, Euler's theorem implies that
$$(\*) ~~\sum\_{k \ge 3} (6-k)p\_k=12.$$
Note that 6-gonal faces do not contribute to the LHS.
One way to think about it is t... | 20 | https://mathoverflow.net/users/1532 | 24340 | 15,972 |
https://mathoverflow.net/questions/24345 | 3 | Let $G$ be a reductive algebraic group and $\varrho$ a representation of $G$ in $GL(n)$. Is it true that $\varrho$ is completely reducible? Moreover, how are related the representations of the Lie algebra $\mathfrak{g}$ of $G$ with the one of $G$?Finally, the centre of the identity component of $G$ consists of semisimp... | https://mathoverflow.net/users/4821 | Representations of reductive Lie group | You need to be over a field of zero characteristic and your representation needs to be rational, i.e. matrix entries need to be algebraic functions on $G$. Then it is completely reducible, see any book on algebraic groups, e.g., Jantzen or Humphreys.
You can always differentiate, so a differential of a map $G\rightar... | 1 | https://mathoverflow.net/users/5301 | 24349 | 15,977 |
https://mathoverflow.net/questions/24321 | 1 | Let $\Lambda$ be an $n$ dimensional sublattice of the integer lattice $\mathbb{Z}^n$. The quotient $\mathbb{Z}^n/\Lambda$ has order $\sqrt{\det{\Lambda}}$.
What is the best/standard way to compute a set of coset representatives for this quotient?
Edit: I initially forgot to take the square root of $\det{\Lambda}$,... | https://mathoverflow.net/users/5378 | Computing a set of coset representatives for $\mathbb{Z}^n / \Lambda$ | As KConrad suggested (why only in the comments?), Smith's normal formal is your best bet. Its running time is insensitive to $m=|{\mathbb Z}^n/\Lambda |$ (unless you need to use arbitrary long entries in your matrix) and behaves as $n^3$.
You may also try coset enumeration, whose running time is usually unbounded but... | 3 | https://mathoverflow.net/users/5301 | 24351 | 15,978 |
https://mathoverflow.net/questions/24361 | 6 | Let $X$ be a topological space and let $\mathcal{F}$ and $\mathcal{G}$ be two sheaves over $X$.
Of course, if one has a morphism $f : \mathcal{F} \to \mathcal{G}$ such that for all $x\in X$, $f\_x : \mathcal{F}\_x \to \mathcal{G}\_x$ is an isomorphism, then it is known that $f$ itself is an isomorphism.
My question... | https://mathoverflow.net/users/2330 | Are two sheaves that are locally isomorphic globally isomorphic ? | Definitely not. If $X$ is a ringed space, then a $\mathcal{O}\_X$-module $F$ is called locally free of rank $1$, if $X$ is covered by open subsets $U\_i$ such that $F|\_{U\_i}$ is free of rank $1$ over $\mathcal{O}\_{U\_i}$. The correspond to line bundles on $X$. Line bundles form a group, called the Picard group of $X... | 15 | https://mathoverflow.net/users/2841 | 24364 | 15,987 |
https://mathoverflow.net/questions/24358 | 2 | I need a reference which states which of the "normal properties of vector spaces" carry over to free $\mathbb{Z}$-modules.
Especially I am interested in things like: If you have a linear map between two free $\mathbb{Z}$-modules and you choose a basis for its kernel, can you choose a basis of a complementary space so... | https://mathoverflow.net/users/3816 | free Z-modules: Bases etc. | What carries over?
As Peter pointed out, a submodule of a free $\mathbb{Z}$-module though
free need not have a complement. Indeed each submodule of a free
$\mathbb{Z}$-module is free, but a quotient module need not be, for instance
$\mathbb{Z}/2\mathbb{Z}$. Also a $\mathbb{Z}$-module is free if and oly if
it is proje... | 3 | https://mathoverflow.net/users/4213 | 24371 | 15,994 |
https://mathoverflow.net/questions/24307 | 17 | Henry Segerman and I recently considered the following question:
Given a fixed area $A < \pi$ and two fixed points in the upper half-plane model for hyperbolic $2$-space, what is the locus of points which give rise to a hyperbolic triangle of the given area?
We found it a fun exercise in hyperbolic geometry to show... | https://mathoverflow.net/users/6040 | Locus of equal area hyperbolic triangles | Nice problem! After googling "hyperbolic triangles of equal area on a fixed base" I found the paper "Extremal properties of the principal Dirichlet eigenvalue for regular polygons in the hyperbolic plane" by Karp and Peyerimhoff. Their Theorem 7 looks like the result you ask for. In a footnote KP refer to a 1965 book o... | 8 | https://mathoverflow.net/users/1650 | 24380 | 15,999 |
https://mathoverflow.net/questions/24350 | 12 | As I understand it, mathematics is concerned with correct deductions using postulates and rules of inference. From what I have seen, statements are called true if they are correct deductions and false if they are incorrect deductions. If this is the case, then there is no need for the words true and false. I have read ... | https://mathoverflow.net/users/4692 | What does it mean for a mathematical statement to be true? | Tarski defined what it means to say that a first-order statement is true in a structure $M\models \varphi$ by a simple induction on formulas. This is a completely mathematical definition of truth.
Goedel defined what it means to say that a statement $\varphi$ is *provable* from a theory $T$, namely, there should be a... | 24 | https://mathoverflow.net/users/1946 | 24385 | 16,003 |
https://mathoverflow.net/questions/24406 | 0 | Suppose $A\_0$ is an abelian variety over $\mathbb{C}$, $E$ is a CM field ,denote $A= A\_0\otimes\_Q E$, is there an isomorphism $ H\_1(A\_0\otimes\_Q E,Q)=H\_1(A\_0,Q)\otimes\_Q E$?how it comes?
| https://mathoverflow.net/users/3945 | homology of abelian variety ? | If $A$ over $\mathbb{C}$ is an abelian variety of dimension $g$, then for all $i \in \mathbb{N}$, $H\_i(A,\mathbb{Q}) \cong H^i(A,\mathbb{Q}) \cong \mathbb{Q}^{ {2g \choose i}}$.
Taking $i = 1$, the conclusion you are asking about is true if and only if $\operatorname{dim} A\_0 \otimes\_\mathbb{Q} E = [E:\mathbb{Q}]... | 3 | https://mathoverflow.net/users/1149 | 24409 | 16,017 |
https://mathoverflow.net/questions/24411 | 11 | Ok the question is pretty dumb: suppose you have a torus $T^n=\mathbb{R}^n/\mathbb{Z}^n$ and a vector $\bar{v}=(v\_1,\ldots,v\_n)\in\mathbb{R}^n$.
Consider the torus $T\_{\bar{v}}$ given by the closure of the one parameter group in $T^n$ generated by $\bar{v}$:
$T\_{\bar{v}}=\overline{ \{t\cdot\bar{v}\mod\mathbb{Z}... | https://mathoverflow.net/users/3465 | sub-tori of a torus, generated by 1-dimensional subgroup | The key to solving both problems is the use of the following two facts: 1) Any
closed subgroup of $T^n$ is the intersection of the kernels of characters of
$T^n$, i.e., continuous group homomorphisms $T^n \rightarrow S^1$. 2) Any
continuous homomorphism $T^n \rightarrow S^1$ is of the form $(\overline
x)\mapsto e^{2\pi... | 5 | https://mathoverflow.net/users/4008 | 24415 | 16,022 |
https://mathoverflow.net/questions/24419 | 6 | Let $L$ be the poset (ordered by set inclusion) that is the power set of some set $X$.
A *state* is a function $s:L \rightarrow [0,1]$ satisfying
i) for {$p\_1,p\_2,...$}, $p\_i \in L$ a
pairwise orthogonal (i.e. $p\_i \leq p\_j'$
where $a'$ is the complement of $a$)
countable sequence, $\bigvee\_i p\_i$
exists... | https://mathoverflow.net/users/5887 | Correspondence between functions on a set and "states" on its power set | Your correspondence is equivalent to the existence of a [real-valued measurable cardinal](http://en.wikipedia.org/wiki/Measurable_cardinal), a large cardinal concept equiconsistent with the existence of a measurable cardinal.
First, note that if $\kappa$ is a measurable cardinal, then there is a 2-valued measure $\m... | 3 | https://mathoverflow.net/users/1946 | 24420 | 16,025 |
https://mathoverflow.net/questions/24424 | 14 | I could not find any references about this fact. I apologize if this is completely trivial, but is the product of two Baire spaces, or for that matter of finitely many of them a Baire space? Now is a countable product of Baire spaces a Baire space?
What about an uncountable product of Baire space? This fact seems to... | https://mathoverflow.net/users/3859 | Products of Baire spaces | See:
Cohen, Paul E.
Products of Baire spaces.
Proc. Amer. Math. Soc. 55 (1976), no. 1, 119--124.
$ $
>
> MathSciNet review by Douglas Censer: A topological space is said to be Baire if any countable intersection of its dense open sets is dense. Assuming the continuum hypothesis (CH), J. C. Oxtoby [Fund. M... | 12 | https://mathoverflow.net/users/1149 | 24427 | 16,030 |
https://mathoverflow.net/questions/24426 | 13 | Mac Lane defines a subcategory as a subset of objects and a subset of morphisms that form a category. But the first rule of category theory is that you do not talk about equality of objects. Up to equivalence, the definition becomes a faithful functor. This is a useful concept, but I don't think it fits the name. I don... | https://mathoverflow.net/users/4639 | What do people mean by "subcategory"? | Do you want the notion of "subcategory" to be invariant under categorical equivalence? If so, then "pseudomonic" functors are the right thing: faithful, and full on isomorphisms.
But I don't think one would want this any more than the notion of "inclusion" of topological spaces to be invariant under homotopy equivale... | 6 | https://mathoverflow.net/users/4183 | 24438 | 16,035 |
https://mathoverflow.net/questions/24447 | 6 | Is there a way to dissect any tetrahedron into a finite number of orthoschemes?
I know that for a tetrahedron which only has acute angles, one can take the center of the inscribed circle and project the center on all the faces and edges and connect it with the vertices to get the orthoschemes. This however does not w... | https://mathoverflow.net/users/1612 | Dissecting a tetrahedron into orthoschemes | Yes, this is known (12 is always enough). Interestingly, in higher dimension it is open whether every simplex in $\Bbb R^d$ can be dissected into finitely many orthoschemes (also called path-simplices). This is called Hadwiger's conjecture. See [this survey](http://dare.uva.nl/document/162823) for results and refs to p... | 8 | https://mathoverflow.net/users/4040 | 24449 | 16,043 |
https://mathoverflow.net/questions/24448 | 2 | There is a proof of [Mittag-Leffler's theorem](http://en.wikipedia.org/wiki/Mittag-Leffler_theorem) with an explicit construction of a holomorphic function with the prescribed poles with prescribed order and residues, for a countable discrete set of points. I do not remember the reference; but my memory from my graduat... | https://mathoverflow.net/users/6031 | Domains of holomorphy in the complex plane | Let $\zeta\_k$ be a countable dense sequence of points in the boundary and consider $f(z) = \sum \frac{1}{2^k} \frac{1}{z-\zeta\_k}$. The sum is plainly uniformly convergent on any subset of finite distance from the boundary, in particular on any compact subset of the interior.
| 8 | https://mathoverflow.net/users/373 | 24454 | 16,046 |
https://mathoverflow.net/questions/24373 | 6 | Background
----------
Recall that a functor $G\colon A\to X$ is called
[monadic](http://ncatlab.org/nlab/show/monadic+functor) if it has a
left adjoint $F$ for which the Eilenberg--Moore comparison functor $K\colon A\to
X^{\mathbb{T}}$ is an equivalence of categories, where $\mathbb{T}$ is
the monad in $X$ defined by... | https://mathoverflow.net/users/2734 | If G is monadic and the comparison functor is an equivalence that is not an isomorphism, does G create limits? | Mac Lane-Moerdijk slipped up; they really should have said "reflects limits". Now it's true that the forgetful functor from the literal category of algebras to the underlying category creates limits (according to the definition of "creates" in CWM), but this notion doesn't transfer across equivalences, as you have obse... | 2 | https://mathoverflow.net/users/2926 | 24466 | 16,053 |
https://mathoverflow.net/questions/24453 | 8 | This is a foundational doubt I have. How does singular homology H\_n capture the number of n-dimensional holes in a space?
We disregard the case of $H\_0$ as it has the very satisfactory explanation that it is the direction sum of $\mathbb Z$ over the path-connected components of the space.
Now, handwaving aside, ... | https://mathoverflow.net/users/6031 | How does singular homology H_n capture the number of n-dimensional "holes" in a space? | The "hole detection" is rather in the very definition of homology. Consider, for example, $H\_2$: it is morally the set of closed surfaces in you space modulo those that bound a $3$-dimensional body, and if a surface is not the boundary of any $3$-dimensional body then surely there must be a hole entrapped in it, no?
... | 6 | https://mathoverflow.net/users/1409 | 24468 | 16,055 |
https://mathoverflow.net/questions/24430 | 7 | The real question is both more serious and somewhat longer than the title.
For the definition of the rational Cherednik algebra attached to a complex reflection group $W$, see for instance 5.1.1 of Rouquier's paper [here](http://arxiv.org/abs/math/0509252). It is a flat family of algebras depending on some parameters... | https://mathoverflow.net/users/nan | Why are rational Cherednik algebras so... rational? | The phenomenon seems the same as for affine Kac-Moody algebras, where rational level is where everything special happens, or quantum groups, where roots of unity are the exceptional locus (and of course these examples are related). The parameter for Cherednik algebras is an additive/Lie algebra type parameter, like the... | 6 | https://mathoverflow.net/users/582 | 24470 | 16,057 |
https://mathoverflow.net/questions/24490 | 5 | Hi!
I've always read that on a complex manifold (obviously not kahler), with a given
hermitian metric on tangent bundle, the chern connection and the levi civita connection
on the underlying real bundle could be different.
Please can someone give me an explicit example of this fact?
Thank you in advance
| https://mathoverflow.net/users/4971 | chern connection vs levi-civita connection | You need a non-Kahler complex manifold. Then the Chern connection will have nontrivial torsion. And the torsion corresponds to the non-closed Kahler form of the metric.
| 16 | https://mathoverflow.net/users/2555 | 24491 | 16,070 |
https://mathoverflow.net/questions/24493 | 6 | Let $G$ be a compact connected topological group and let $H$ be a subgroup of $G$. Suppose that $H$ is measurable with respect to the normalised Haar measure $\mu$ on $G$. Do we necessarily have $\mu(H)=0$ or $\mu(H)=1$?
Maybe this is well--known, I ask it just out of curiosity. The question is related to [this](http... | https://mathoverflow.net/users/5952 | Measurable subgroups. | Don't we still have this: if $A$ is measurable of positive measure, then $A A^{-1}$ contains a neighborhood of the identity...? So: a measurable subgroup of positive measure itself contains a neighborhood of the identity, and thus by connectedness is all of $G$.
| 10 | https://mathoverflow.net/users/454 | 24495 | 16,071 |
https://mathoverflow.net/questions/24492 | 3 | Hi,
I'm actually physics student and I've not been able to understand how the following integration has been performed:
>
> [...]
>
>
> The fields e and A have support on these discrete structures. The $su(2)$-valued 1-form
> field $e$ is represented by the assignment of an $e \in su(2)$ to each 1-cell in $\Delt... | https://mathoverflow.net/users/2597 | Delta distribution as an integral... | Whoever performed that integration is using the following fact from Fourier analysis:
The "delta function supported at the position 0" is the Fourier transform of the constant function 1.
$\delta(x) = \frac{1}{2\pi}\int\_{\mathbb{R}} 1 e^{ikx}dk$
You can prove this by approximating the delta-function with a sequ... | 7 | https://mathoverflow.net/users/35508 | 24498 | 16,073 |
https://mathoverflow.net/questions/24432 | 8 | Let $X$ be a Banach space, $X' = \mathcal{L}(X, \mathbb{K})$ its dual space. Denote by $\mathcal{B}(X)$ the $\sigma$-algebra of Borel sets and denote by $\sigma(X')$ the $\sigma$-algebra which is generated by all sets of the form $u^{-1}(C)$ for $u \in X'$ and $C \in \mathcal{B}(\mathbb{K})$.
For $X$ separable we hav... | https://mathoverflow.net/users/2258 | Borel(X) = \sigma(X') for X non-separable | If $I$ is uncountable, then in space $l^2(I)$ no countable set of functionals separates points. Consequently, for any set $A$ in the sigma-algebra generated by these functionals [the Baire sets for the weak topology, see reference below], if $0 \in A$, then an entire subspace is contained in $A$. So all elements of thi... | 8 | https://mathoverflow.net/users/454 | 24499 | 16,074 |
https://mathoverflow.net/questions/24481 | 11 | To put this in context: I am in the process of developing a package for Macaulay 2 (a commutative algebra software,) called "Permutations", which will add permutations as a type of combinatorial object that M2 will handle, hopefully integrating nicely with the (still in development) Posets package, the (still in develo... | https://mathoverflow.net/users/5002 | Computing Bruhat Order Covering Relations | My M2 permutation code is here:
<http://www.math.cornell.edu/~allenk/permutation.m2>
It's got a bunch of specialized stuff about Rothe diagrams and Fulton's essential set, but at the end it's got a BruhatLeq, for strong Bruhat order (so, not what you're computing).
If you want covering relations in strong Bruhat or r... | 8 | https://mathoverflow.net/users/391 | 24505 | 16,076 |
https://mathoverflow.net/questions/24487 | 4 | Let $\Lambda$ denote Connes's cyclic category. It is an extension of the simplex category $\Delta$ (of nonempty finite linearly ordered sets) obtained by adding an automorphism of order $n+1$ to the object $\textbf{n}$.
**Question:** Suppose $X: \Lambda^{op} \to Top$ is a cyclic space. What is a description of the ho... | https://mathoverflow.net/users/4910 | Homotopy colimits of cyclic spaces | The homotopy theory of cyclic spaces is equivalent to that of spaces over $BS^1$ (Dwyer-Hopkins-Kan). The colimit over the simplicial category is as you say a space $X$ with $S^1$ action, and the colimit over the cyclic category is the quotient (Borel construction) $X/S^1$ as a space over $BS^1$.
| 8 | https://mathoverflow.net/users/582 | 24507 | 16,077 |
https://mathoverflow.net/questions/24503 | 16 | This is quite possibly a stupid question, but it is pretty far from what I normally do, so I wouldn't even know where to look it up.
If $X$ is a projective variety over an algebraically closed field of arbitrary characteristic and $Y\subset X$ a smooth divisor. Under which conditions can I contract $Y$ to a point, i.... | https://mathoverflow.net/users/259 | Contracting divisors to a point | For a smooth $Y$, a necessary condition for contractibility is that the conormal line bundle $N\_{Y,X}^\\*$ is ample. It is also sufficient for contracting to an algebraic space. The reference is *Algebraization of formal moduli. II. Existence of modifications.* by M. Artin.
$Y$ can be contracted to a point on an al... | 28 | https://mathoverflow.net/users/1784 | 24516 | 16,082 |
https://mathoverflow.net/questions/24521 | 11 | Fix a finite set of primes $S$ and an additional prime $p$. Let $K$ be the maximal extension of $\mathbb{Q}$ that is unramified outside $S$ and $\infty$ and totally split at $p$. Is the extension $K$ finite?
My intuitive guess would be no, but the simple constructions (based on class field theory) I tried so far do n... | https://mathoverflow.net/users/6074 | Maximal extension almost everywhere unramified and totally split at one place | Nope.
I'm lacking a reference in front of me at the moment (see NSW's Cohomology of Number Fields, or Gras's Class Field Theory -- I'll update with a precise reference later), but there are remarkably clean formulas for the generator and relation ranks for the Galois group of the maximal $\ell$-extension of $\mathbb... | 11 | https://mathoverflow.net/users/35575 | 24523 | 16,086 |
https://mathoverflow.net/questions/24528 | 12 | Let *G* be a semisimple algebraic group.
Following work of Matsumoto [1], Brylinski and Deligne [2] constructed a central extension of the functor
*G* : *Rings* → *Groups* by the second algebraic *K*-theory functor.
Plugging in ℂ((t)) into those functors, we get the well known central extension $\widetilde{G\big... | https://mathoverflow.net/users/5690 | Are centrally extended p-adic groups defined over F_1? | First a small thing. I am pretty sure we don't have $K\_2(\mathbb C((t)))=\mathbb
C^\*$, we have a surjective residue homomorphism $K\_2(\mathbb C((t)))\rightarrow
\mathbb C^\*$ but, I believe, with a non-trivial kernel. In any case, we can look
at the induced central extension and then the rest of what you say is
OK. ... | 10 | https://mathoverflow.net/users/4008 | 24535 | 16,093 |
https://mathoverflow.net/questions/24506 | 15 | In model theory, two structures $\mathfrak{A}, \mathfrak{B}$ of identical signature $\Sigma$ are said to be *elementarily equivalent* ($\mathfrak{A} \equiv \mathfrak{B}$) if they satisfy exactly the same first-order sentences w.r.t. $\Sigma$. An astounding theorem giving an algebraic characterisation of this notion is ... | https://mathoverflow.net/users/4915 | Ultrafilters arising from Keisler-Shelah ultrapower characterisation of elementary equivalence | Under the Continuum Hypothesis, your solution space is *all* nonprincipal ultrafilters. This is because under CH, the ultrapower $M^N/U$ of a mathematical structure $M$ of size at most continuum does not actually depend on the (nonprincipal) ultrafilter $U$. One can see this by using the fact that the ultrapower will b... | 10 | https://mathoverflow.net/users/1946 | 24538 | 16,095 |
https://mathoverflow.net/questions/24540 | 3 | e.g. is $\widehat{\mathbf{SET}}$ locally small?
| https://mathoverflow.net/users/6082 | is the presheaf category of a locally small category locally small? | No, not in general. Todd gave a useful general answer and here is a concrete counter-example.
Let $\mathbf{C}$ be a category whose objects are sets and where the only morphisms are the identities. Consider the functors $F, G : \mathbf{C} \to \mathbf{Set}$ given by $F(X) = 1 = \lbrace 0 \rbrace$ and $G(X) = 2 = \lbrac... | 3 | https://mathoverflow.net/users/1176 | 24551 | 16,102 |
https://mathoverflow.net/questions/24550 | 10 | Let $H$ and $K$ be affine group schemes over a field $k$ of characteristic zero. Let $\varphi:H\to Aut(K)$ be a group action. Then we can form the semi-direct product $G = K\ltimes H$.
**Problem**: Describe the tannakian category $Rep\_k(G)$ in terms of $Rep\_k(K)$, $Rep\_k(H)$ and $\varphi$.
If $\varphi$ is the tr... | https://mathoverflow.net/users/1985 | Tannakian description of a semi-direct product | The algebraic stack BG is the quotient of BK by the action of H, induced by its action on K. So we can describe coherent sheaves on it (aka reps of G) via descent from BK - i.e. reps of G are H-equivariant sheaves on BK, or H-equivariant objects in Rep K. Now this is not yet the answer you want, since it involves Rep K... | 8 | https://mathoverflow.net/users/582 | 24553 | 16,103 |
https://mathoverflow.net/questions/24524 | 6 | Consider two (distinct) octahedral diagrams i.e. diagrams mentioned in the octahedron axioms of triangulated categories (with four 'commutative triangular faces' and four 'distinguished triangular faces'). Is it true than one can extend to a morphism of such diagrams:
1. any morphism of one of the 'commutative faces' o... | https://mathoverflow.net/users/2191 | Can one extend a morphism of commutative triangles to a morphism of octahedral diagrams? | It is true in a Heller triangulated category aka $\infty$-triangulated category (although strictly speaking one only needs a 3-triangulation for octahedra) that any morphism between the bases of octahedra (by which I mean the 3 objects and two composable morphisms from which the octahedron is built) extends to a morphi... | 8 | https://mathoverflow.net/users/310 | 24558 | 16,107 |
https://mathoverflow.net/questions/24556 | 9 | It is shown in Lang's Algebra (and many other books I assume) that:
if A if a principal entire ring, then A is a factorial ring.
The proof uses Zorn's Lemma. Is this theorem equivalent to the axiom of choice?
| https://mathoverflow.net/users/4002 | Factorial Rings and The Axiom of Choice | Lang uses Zorn's lemma only in the step that nonzero nonunits in a PID admit irreducible factorizations (not the uniqueness of irreducible factorizations, once we know such factorizations exist). The way he uses Zorn's lemma, I think, is excessive. What follows is how I work out the existence of irreducible factorizati... | 19 | https://mathoverflow.net/users/3272 | 24559 | 16,108 |
https://mathoverflow.net/questions/24399 | 5 | Let $k$ be a field, $K/k$ a separable quadratic extension,
and $D/K$ a central division algebra of dimension $r^2$ over $K$
with an involution $\sigma$ of second kind
(i.e. $\sigma$ acts non-trivially on $K$ and trivially on $k$).
Does there exist a field extension $F/k$ such that $L:=K\otimes\_k F$
is a field, and $D\... | https://mathoverflow.net/users/4149 | Splitting of a division algebra with an involution of second kind | I answer my own question. The answer is *yes*.
Since there are no non-trivial division algebras over finite fields,
we may assume that $k$ and $K$ are infinite.
Let $H=${$h\in D\ |\ h^\sigma=h$} denote the $k$-space of Hermitian elements of $D$.
Consider the embedding $D\hookrightarrow M\_r(\bar K)$ induced
by an iso... | 5 | https://mathoverflow.net/users/4149 | 24561 | 16,110 |
https://mathoverflow.net/questions/24513 | 3 | I need to determine the minimal polynomial for a quotient in (1).
(1) B = C / A
C is known as a root of a 36th degree polynomial and A is known as a root of a 24th degree polynomial.
However I have not been able to succeed in recovering the coefficients nor the degree of the polynomial for B.
Any suggestions? I... | https://mathoverflow.net/users/6046 | Question on determining the minimal polynomial for an algebraic quotient | Let $F$ be the polynomial for $A$, let $G$ be the polynomial for $C$. Consider the resultant of $x^{24}F(y/x)$ and
$G(y)$. This will be a polynomial whose roots are all the numbers of the form $\gamma/\alpha$, where $\gamma$ (resp., $\alpha$) runs through the roots of $G$ (resp., $F$). The resultant is the determinant... | 4 | https://mathoverflow.net/users/3684 | 24568 | 16,117 |
https://mathoverflow.net/questions/24569 | 8 | I am teaching a combinatorics class in which I introduced the notion of a "mass formula". My terminology is inspired by the [Smith–Minkowski–Siegel mass formula](http://en.wikipedia.org/wiki/Smith%25E2%2580%2593Minkowski%25E2%2580%2593Siegel_mass_formula) for the total mass of positive-definite quadratic forms of a giv... | https://mathoverflow.net/users/1450 | Seeking reference for the enumerative "mass formula" concept | I do call such things "mass formulas", but then again I am a number theorist, and one of my colleagues is a quadratic form theorist who specializes in such things. So this is mostly an expression of my specific mathematical culture.
I do not think that it is a standard term, at least not the only standard term. For ... | 6 | https://mathoverflow.net/users/1149 | 24571 | 16,118 |
https://mathoverflow.net/questions/24573 | 37 | If $k$ is a characteristic $p$ field containing a subfield with $p^2$ elements (e.g., an algebraic closure of $\mathbb{F}\_p$), then the number of isomorphism classes of supersingular elliptic curves over $k$ has a formula involving $\lfloor p/12 \rfloor$ and the residue class of $p$ mod 12, described in Chapter V of S... | https://mathoverflow.net/users/121 | Is there a nice proof of the fact that there are (p-1)/24 supersingular elliptic curves in characteristic p? | One argument (maybe not of the kind you want) is to use the fact that the wt. 2 Eisenstein series on $\Gamma\_0(p)$ has constant term (p-1)/24.
More precisely: if $\{E\_i\}$ are the s.s. curves, then for each $i,j$,
the Hom space $L\_{i,j} := Hom(E\_i,E\_j)$ is a lattice with
a quadratic form (the degree of an isogen... | 31 | https://mathoverflow.net/users/2874 | 24582 | 16,125 |
https://mathoverflow.net/questions/24585 | 16 | There seems to be a few papers around with Erdős written as Erdös. For example:
MR0987571 (90h:11090) Alladi, K.; Erdös, P.; Vaaler, J. D. Multiplicative functions and small divisors. II. J. Number Theory 31 (1989), no. 2, 183--190. (Reviewer: Friedrich Roesler) 11N37
>
> Would it be incorrect to cite such papers... | https://mathoverflow.net/users/2264 | If Erdős is published as Erdös in a paper, which do I cite? | We cite papers to show our respect to the authors and to help our readers find stuff. For the second purpose, I suspect most people would just type in names without diacritical marks, and most search facilities would find what you're looking for based on the letters alone, so it doesn't really matter. But for the first... | 33 | https://mathoverflow.net/users/3684 | 24587 | 16,128 |
https://mathoverflow.net/questions/24497 | 5 | The following seems to be a question related to standard calculus, but I am not quite sure
where to look for an answer.
Suppose $f,g:\mathbb{N} \to \mathbb{C}$ are such that the have the same asymptotical behaviour, i.e. $f(n)/g(n) \to 1$ as
$n \to \infty$. Of course, suppose that one of the sums $\sum\_{n=0}^\infty ... | https://mathoverflow.net/users/3757 | Non-absolute convergence of series with asymtotically equal coefficients | Following Theo Johnson--Freyd's suggestion, I am making my above comment an answer:
Just subtracting one series from the other, it seems that you need $\sum\_{n=0}^{\infty} (f(n)−g(n))$ to converge. Writing $$f(n)/g(n)=1+\delta\_n,$$ $$\text{so that } \qquad \qquad\sum\_{n=0}^{\infty} (f(n)−g(n))=\sum\_{n=0}^{\infty}... | 2 | https://mathoverflow.net/users/2874 | 24595 | 16,134 |
https://mathoverflow.net/questions/24593 | 1 | I'm reading the book 'A course in Modern Mathematical Physics' by
'Szekeres' and encountered a problem in interpreting the proof of the
following corollary of Schur's lemma.
The corollary and the proof in the book is as follows (I mark my problem
area's in the proof with (<1>) and (<2>) :
Corollary 4.2. Let $T : G ... | https://mathoverflow.net/users/5338 | Problem with the proof of a corollary of Schur's lemma | What Emerton says is of course correct: an irreducible representation of a finite group is necessarily finite-dimensional. Indeed, the same holds for any continuous irreducible representation of a compact group on a Hilbert space.
It seems to me that you can get away with milder assumptions: $G$ can be any group so ... | 3 | https://mathoverflow.net/users/1149 | 24599 | 16,138 |
https://mathoverflow.net/questions/24594 | 15 | Are there general surveys or introductions to the homotopy groups of spheres? I'm interested especially in connections to low-dimensional geometry and topology.
| https://mathoverflow.net/users/nan | Survey articles on homotopy groups of spheres | While my Algebraic Topology book and my unfinished book on spectral sequences (referred to in other answers to this question) contain some information about homotopy groups of spheres, they don't really qualify as a general survey or introduction. One source that fits this bill more closely is Chapter 1 of Doug Ravenel... | 32 | https://mathoverflow.net/users/23571 | 24624 | 16,153 |
https://mathoverflow.net/questions/24570 | 11 | Given a smooth vector bundle $E$ with *non-compact* base, let
$\Gamma(E)$ be the space of $C^\infty$ sections equipped with *compact-open* $C^\infty$-topology.
1. I have heard that $\Gamma(E)$ is not locally-contractible. Why not?
2. Is $\Gamma(E)$ contractible? Visibly any section can be joined to the zero section ... | https://mathoverflow.net/users/1573 | What's wrong with compact-open topology on the space of maps? | (For the more specific question)
Yes for $k = \infty$ if $M$ can be exhausted by a countable number of compact sets, no otherwise. However, the failure is due to a lack of completeness (for $k \ne \infty$) or size issues (if $M$ can't be exhausted) rather than anything else and thus the local contractibility still ho... | 5 | https://mathoverflow.net/users/45 | 24625 | 16,154 |
https://mathoverflow.net/questions/24608 | 8 | Hi. Are there nice/simple examples of cyclic extensions $L/K$ (that is, Galois extensions with cyclic Galois group) for which $L$ cannot be written as $K(a)$ with $a^n\in K$?
Thanks.
| https://mathoverflow.net/users/416 | Cyclic extensions | Dear Justin, your question is subtly ambiguous.
**First interpretation:** Is there a cyclic extension $L/K$ of degree $n$ that cannot be written $L=K(a)$ with $a\in L$ and $a^n\in K ?$.
**Answer** Yes. For example, let $p$ be a prime number and $n$ an integer.The extension
$\mathbb F\_p\subset \mathbb F\_{p^{p^n}}... | 9 | https://mathoverflow.net/users/450 | 24628 | 16,156 |
https://mathoverflow.net/questions/24626 | 5 | The symmetric product of a variety $M$ is the quotient of $M^n/S\_n$ where $S\_n$ is the symmetric group permuting components of n-fold product $M^n$. IF $M$ is an affine plane $C^k$ over complex numbers, the coordinate ring of the symmetric product is the invariant polynomials in $R:=C[x^1\_1,...,x^1\_k, x^2\_1,...,x^... | https://mathoverflow.net/users/6093 | What is the coordinate ring of symmetric product of affine plane? | Those invariant polynomials are called *multisymmetric functions*. There are several papers on them; you could start with J. Dalbec, Multisymmetric functions, Beiträge Algebra Geom. 40(1) (1999), 27-51 <http://www.emis.de/journals/BAG/vol.40/no.1/b40h1dal.ps.gz>.
| 5 | https://mathoverflow.net/users/4790 | 24629 | 16,157 |
https://mathoverflow.net/questions/24548 | -5 | Hi
On page 98 "Stochastic differential equations" of Øksendal, 6th edition,
the author writes that $$\int\_{0}^{u}\Big(\int\_{0}^{t}\frac{\partial}{\partial t}f(s,t)dR\_{s}\Big)dt=\int\_{0}^{u}\Big(\int\_{s}^{u}\frac{\partial}{\partial t}f(s,t)dt\Big)dR\_{s}$$
could some one please tell me why? thanks so much fo... | https://mathoverflow.net/users/5136 | Another question on Øksendal's book | i am also having some problems with oksendal sometimes but here i can help u: Its basically Fubini theorem where one can change the order of integration: on the left the inner integrand is being integrated over 0<=s<=t and outer integral over 0<=t<=u. When u combine these two inequalities you get 0<=s<=t<=u. So on the ... | 0 | https://mathoverflow.net/users/6114 | 24634 | 16,162 |
https://mathoverflow.net/questions/24609 | 5 | $\exists p, q \in \mathbb{P}: p^4+1 = 2q^2$? I suspect there is some simple proof that no such p, q can exist, but I haven't been able to find one.
Solving the Pell equation gives candidates for p^2=x and q=y, with x=y=1 as the first candidate solution and subsequent ones given by x'=3x+4y, y'=2x+3y; chances of a pri... | https://mathoverflow.net/users/6089 | Are there primes p, q such that p^4+1 = 2q^2 ? | This is not my solution, but I don't remember where I learned it.
Square both sides, subtract $4p^4$, and divide by 4 to obtain
$({p^4-1\over 2})^2=q^4-p^4$.
However, $z^2=x^4-y^4$ has no solutions in non-zero integers.
This is Exercise 1.6 in Edwards's book on Fermat's Last Theorem.
The proof uses the representati... | 20 | https://mathoverflow.net/users/nan | 24636 | 16,163 |
https://mathoverflow.net/questions/24603 | 3 | It is well known and easy to prove that any Denjoy counterexample in the circle is approximated by homeomorphisms of the circle which have the same rotation number and are transitive (in particular, they are conjugated to the rotation).
My question is the following. Given a $C^1$ denjoy counterexample $f:S^1\to S^1$... | https://mathoverflow.net/users/5753 | Denjoy counterexamples $C^1$ close to conjugates of rotations. | Dear Raphael,
perhaps I misunderstood the question, but in order to get your sequence $f\_n$, one can $C^1$-approximate $f$ by $C^2$-diffeomorphisms $g\_n$ and then compose each $g\_n$ with an adequate small rotations $R\_{\epsilon\_n}$ to adjust the rotation number (so that $f\_n=R\_{\epsilon\_n}\circ g\_n$ do the j... | 5 | https://mathoverflow.net/users/1568 | 24637 | 16,164 |
https://mathoverflow.net/questions/24649 | 0 | As part of working with dot products of vectors in an algebraic field, having previously solved the algebraic quotient, I need to find the algebraic square root as shown in (1)
Let $r\_i$ be a root of polynomial P and $s\_j$ be a root of polynomial Q i.e., P($r\_i$)=0, Q($s\_j$)=0.
I seek to find a third polynomial... | https://mathoverflow.net/users/6046 | Algebraic square root question | **Lemma 1:** If $P(x)$ is an integer polynomial with root $r$, then $P(x^2)$ is an integer polynomial with root $\sqrt{r}$.
**Lemma 2:** If $P(x)$ is an integer polynomial with root $r$, then $P(\sqrt{x})P(-\sqrt{x})$ is an integer polynomial with root $r^2$.
**Lemma 3:** If $P(x)$ is an integer polynomial with roo... | 4 | https://mathoverflow.net/users/290 | 24651 | 16,174 |
https://mathoverflow.net/questions/20355 | 22 | I am looking for a good book to study probability. My advisor suggested the "Probability" by Leo Breiman. I am reading it now, it seems rather a dense book, so I would like to ask you guys advice on which book you guys often start with for Probability.
| https://mathoverflow.net/users/5136 | Book for probability | I would definitely go for "Probability" by Jim Pitman. It is a very good book for learning Probability Theory, one of the best text books I have encountered in my studies.
| 9 | https://mathoverflow.net/users/1539 | 24664 | 16,181 |
https://mathoverflow.net/questions/24680 | 6 | The question is in the title exactly as I want to ask it, but let me provide some background and motivation.
Many of the properties of fields studied in the algebraic theory of quadratic forms are manifestly *elementary* properties in the sense of model theory: that is, if one field has this property, then any other ... | https://mathoverflow.net/users/1149 | If two fields are elementarily equivalent, what can we say about their Witt rings? | As you point out, one cannot hope that the Witt ring, up to isomorphism, be an elementary invariant of a field. The strongest statement which I might conjecture would be that if $K \preceq L$ is an elementary extension of fields, then $W(K) \to W(L)$ is an elementary extension of rings. If this statement were true, the... | 8 | https://mathoverflow.net/users/5147 | 24685 | 16,191 |
https://mathoverflow.net/questions/24688 | 16 | Is there an efficient way to sample uniformly points from the unit n-sphere? Informally, by "uniformly" I mean the probability of picking a point from a region is proportional to the area of that region on the surface of the sphere. Formally, I guess I'm referring to the Haar measure.
I guess "efficient" means the al... | https://mathoverflow.net/users/1042 | Efficiently sampling points uniformly from the surface of an n-sphere | Generate $X\_1, X\_2, \ldots, X\_n$ independent, normally distributed random variables See [wikipedia](http://en.wikipedia.org/wiki/Normal_distribution#Generating_values_from_normal_distribution) for information on how to do this given some standard source of randomness, for example uniform(0,1) random variables.
The... | 24 | https://mathoverflow.net/users/143 | 24690 | 16,194 |
https://mathoverflow.net/questions/24659 | 6 | I'm a computer programmer who's caught on to denotational semantics. I mostly work with Ruby, JavaScript and C, but I know a little Haskell and ML. I've taken my first steps towards reasoning about what my software *means*, but my knowledge of domain theory is weak. DCPOs, chains, new notation – can you recommend a coh... | https://mathoverflow.net/users/6100 | Resources for learning domain theory? | The [book recommended by jef](https://mathoverflow.net/a/24677/49269) is the domain-theory bible. It may be a bit overwhelming for a beginner. For an easier and more compressed introduction I recommend that you have a look at Abramsky and Jung's chapter on domain theory from the Handbook of Logic in Computer Science. I... | 4 | https://mathoverflow.net/users/1176 | 24702 | 16,203 |
https://mathoverflow.net/questions/24622 | 9 | My general question concerns what we can learn about an arbitrary, three-dimensional convex polytope (or convex hull of an arbitrary polytope) strictly from the surface areas of its two-dimensional projections on a plane as it 'tumbles' in 3-space (i.e. as it rotates along an arbitrary, shifting axis).
If it's helpf... | https://mathoverflow.net/users/3248 | Characterizing a tumbling convex polytope from the surface areas of its two-dimensional projections | You cannot recover a convex polytope from its projection areas, even if you know the whole function (unit vector) $\mapsto$ (area of projection along this vector). There exist two different polytopes $P\_1$ and $P\_2$ such that for every unit vector $v\in\mathbb R^3$, the areas of the two projections along $v$ are equa... | 10 | https://mathoverflow.net/users/4354 | 24707 | 16,206 |
https://mathoverflow.net/questions/24579 | 166 | I saw a while ago in a book by Clifford Pickover, that whether the [Flint Hills series](https://mathworld.wolfram.com/FlintHillsSeries.html) $\displaystyle \sum\_{n=1}^\infty\frac1{n^3\sin^2 n}$ converges is open.
I would think that the question of its convergence is really about the density in $\mathbb N$ of the seq... | https://mathoverflow.net/users/6085 | Convergence of $\sum(n^3\sin^2n)^{-1}$ | As Robin Chapman mentions in his comment, the difficulty of investigating
the convergence of
$$
\sum\_{n=1}^\infty\frac1{n^3\sin^2n}
$$
is due to lack of knowledge about the behavior of $|n\sin n|$ as $n\to\infty$,
while the latter is related to rational approximations to $\pi$ as follows.
Neglecting the terms of the... | 132 | https://mathoverflow.net/users/4953 | 24712 | 16,209 |
https://mathoverflow.net/questions/24718 | 7 | By a graph I will understand an undirected graph without multiple edges or loops. By a morphism of graphs I will understand a map $f$ between the underlying sets of vertices, such that if $x$ and $y$ are adjacent, then $f(x)$ and $f(y)$ are either adjacent or equal.
Let $G$ be a finite graph. One can realise $G$ as a... | https://mathoverflow.net/users/5952 | Singular homology of a graph. | You talk about morphisms from $\Delta\_\bullet$ to a graph $G$.
I presume a morphism from $\Delta\_n$ to $G$ is just an embedding
of $\Delta\_n$ in $G$, that is an $(n+1)$-clique in $G$ with a
labelling of its vertices from $0$ to $n$. It seems to me that
defining (co)homology in this way will be the same as the standa... | 3 | https://mathoverflow.net/users/4213 | 24728 | 16,215 |
https://mathoverflow.net/questions/24700 | 1 | My question concerns the existence of a nice (deterministic?) method/algorithm for calculating the distribution of surface areas for two-dimensional projections of an arbitrary polytope (or convex approximation of a polytope). Less optimistically, a method of finding the minimum, maximum, and perhaps, mean surface area... | https://mathoverflow.net/users/3248 | Calculating the surface area distribution of two-dimensional projections for a polytope | If you take the arrangement of planes determined by the faces of your polytope, then the combinatorial structure of the projection is constant throughout all the viewpoints within one cell of the arrangement. An alternative viewpoint is to partition $S^2$ by these planes moved to the center of that sphere. Within each ... | 3 | https://mathoverflow.net/users/6094 | 24730 | 16,217 |
https://mathoverflow.net/questions/24726 | -3 | Given E[v|X=x]=g[x] and the pdf of X (f[x]), how to calculate E[v|x>=x0]? The pdf of V or the joint pdf of V,X are unknown. My guess is that this problem has no solution.
| https://mathoverflow.net/users/3899 | Conditional expectation | So if $p(v,x)$ is the unknown pdf,
$f(x) = \int p(v,x) \mathrm{d}v$
$g(x) = E[v|X=x] = \int v \ P[v|X=x] \mathrm{d}v = \int v \ \frac{p(v,x)}{f(x)} \mathrm{d}v$
and then
$E[v|X \ge x\_0] = \frac{1}{P[X \ge x\_0]} \int\_{x\_0}^\infty E[v|X = x] P[x] \mathrm{d}x = \frac{ \int\_{x\_0}^\infty f(x)g(x) \mathrm{d}x}... | 2 | https://mathoverflow.net/users/5789 | 24731 | 16,218 |
https://mathoverflow.net/questions/24713 | 10 | Being far from analysis, I recently learned about the [Invariant subspace problem](http://en.wikipedia.org/wiki/Invariant_subspace_problem) and came up with the following (perhaps simple or well-known) question.
Let $H$ be a separable complex Hilbert space and $T:H\to H$ a bounded operator. Assume that the spectrum o... | https://mathoverflow.net/users/4354 | Are operators with trivial spectrum nilpotent in a sense? | Perhaps I am missing something. Isn't the example you mention quasinilpotent and and maps the finitely non zero sequences onto themselves?
I am not an expert on invariant subspaces, but I think it is widely believed among experts that the answer to the question is the same for quasinilpotent operators as for general ... | 6 | https://mathoverflow.net/users/2554 | 24739 | 16,221 |
https://mathoverflow.net/questions/24733 | 17 | I was in Paris recently for a meeting about motives or motifs, and since I'm too jet lagged
for real work let me ask the following somewhat frivolous question. The word "motif" is
usually translated as "motive" in English. However, I wonder if this is really the best choice. "Motive" has, for me, a primarily psychologi... | https://mathoverflow.net/users/4144 | Motives versus Motifs | Dear Donu, here are Grothendieck's own words:
"Contrary to what occurs in ordinary
topology, one finds oneself confronting
a disconcerting abundance of different
cohomological theories. One has the
distinct impression (but in a sense that
remains vague) that each of these theories
“amount to the same thing”, that
the... | 16 | https://mathoverflow.net/users/450 | 24749 | 16,228 |
https://mathoverflow.net/questions/24754 | 29 | Let $(G, n)$ be a pair such that $n$ is a natural number, $G$ is a finite group which is abelian if $n \geq 1$. It is well-known that $\pi\_n(K(G,n)) = G$ and $\pi\_i (K(G,n)) = 0$ if $i \neq n$.
Also it is known that these spaces $K(G,n)$ play a very important role for cohomology. For any abelian group $G$, and any ... | https://mathoverflow.net/users/6031 | (Co)homology of the Eilenberg-MacLane spaces K(G,n) | Computing the integral cohomology of $K(\pi,n)$'s is feasible but a bit tricky. In fact the only reference I know is [exposé 11 of H. Cartan's seminar, year 7](http://archive.numdam.org/ARCHIVE/SHC/SHC_1954-1955__7_1/SHC_1954-1955__7_1_A11_0/SHC_1954-1955__7_1_A11_0.pdf). I'd be interested if there are other sources th... | 19 | https://mathoverflow.net/users/2349 | 24759 | 16,235 |
https://mathoverflow.net/questions/24758 | -1 | I think it is helpful to keep a hoard of helpful sites pertaining to unsolved problems and pay a regular/casual visit to them.
Please when you answer, add a comment on the content of the site for faster surfing.
Thanks in advance.
| https://mathoverflow.net/users/5627 | List of sites related to the Riemann Hypothesis and recent developments? | <http://aimpl.org/pl>
Quote
>
> The AIM Problem Lists are part of the Bibliographic Knowledge Network (BKN) project funded by a Cyber Enabled Discovery and Innovation (CDI) grant from the National Science Foundation. Major partners in the project are UC Berkeley, Harvard, Stanford, and AIM.
>
>
>
It curren... | 2 | https://mathoverflow.net/users/3948 | 24761 | 16,237 |
https://mathoverflow.net/questions/24715 | 8 | The question I ask is in the title. This should be quite well-known, and in fact probably I am going to get the response that it is the definition. To convey my confusion, I have to convey my understanding of what is a differential form and what is the contangent bundle. To simplify things, we assume that our whole set... | https://mathoverflow.net/users/6031 | Why are order-k differential forms sections of the kth exterior power of the cotangent bundle? | You have to work a bit to get those two definitions to agree, but it is all standard lore in differential geometry. Both of the references in the comments - Spivak and Madsen & Tornehave - are good and should have what you need, the latter a bit more useful in my opinion. But I am writing this answer because in neither... | 10 | https://mathoverflow.net/users/4362 | 24767 | 16,239 |
https://mathoverflow.net/questions/24764 | 5 | A finite groups of $\mathrm{GL}\_n(\mathbb C)$ of exponent $m$ necessarily have order $C$ verifying $C\leqslant m^n$ and $n! m^n$ divides $C$, but this condition is not sufficient, for instance $\mathrm{GL}\_3(\mathbb C)$ has no subgroup of exponent $7$ and order $42$.
So can we determine fully the possible orders o... | https://mathoverflow.net/users/3958 | Finite subgroups of GL_n(C) | The paper:
Herzog, Marcel; Praeger, Cheryl E. "On the order of linear groups of fixed finite exponent."
J. Algebra 43 (1976), no. 1, 216–220.
[MR424960](http://www.ams.org/mathscinet-getitem?mr=424960)
[DOI:10.1016/0021-8693(76)90156-3](http://dx.doi.org/10.1016/0021-8693(76)90156-3)
contains the important bound, i... | 12 | https://mathoverflow.net/users/3710 | 24768 | 16,240 |
https://mathoverflow.net/questions/24773 | 38 | This is a question that I have seen asked passively in comments relating to the separation of category theory from set theory, but I haven't seen it addressed in full.
I know that it's possible to formulate category theory within set theory while still being albe to construct the useful things one would want from cat... | https://mathoverflow.net/users/3664 | Why do categorical foundationalists want to escape set theory? | I don't agree that this is what (most) categorists who are interested in foundations are doing.
It is true that Lawvere in the mid-60's (and perhaps to this day) wanted to develop a theory of categories independent of a theory of sets, but I don't think that represents the main thrust of modern-day categorical work ... | 62 | https://mathoverflow.net/users/2926 | 24783 | 16,250 |
https://mathoverflow.net/questions/24403 | 8 | Let $\varphi:\mathcal S^{d-1}\longrightarrow \mathbb R\_{>0}$ be a strictly positive function describing the boundary $\varphi(\mathbf x)\mathbf x,\mathbf x\in\mathbb S^{d-1}$ of a $d-$dimensional closed convex set $C$ with barycenter at the origin.
We consider the decomposition $\varphi=\oplus\_{\lambda\in \mathbf{Spe... | https://mathoverflow.net/users/4556 | A variation on "Hearing the shape of a drum" for polytopes. | The short answer is that there are no particular constraints on the spectral decomposition of the function $\varphi$, as long as a basic convexity condition is satisfied.
>
> **Lemma.**.Assume that $\varphi\in C^2(\mathbb S^{d-1},\mathbb R)$ satisfies for all $x\in \mathbb S^{d-1}$, $i$, $j=1,\dots,d$, and some $r... | 4 | https://mathoverflow.net/users/5371 | 24786 | 16,253 |
https://mathoverflow.net/questions/24787 | 1 | If $f\_n=1\_{(n,n+1)}(x)$, where $1\_A(x)$ is the indicator function. Why is $f\_n
\rightarrow0$? Same is true for $f\_n=1\_{(n,\infty)}$.
i just dont get it.
i thought $f\_n$ was always 0 for all n so i think $f\_n\rightarrow1$ but its not
the case. i try to reason it by the integral which is 1 for all n
but then ... | https://mathoverflow.net/users/6114 | strick inequality for Fatou theorem | This is just the definition of convergence for sequences of functions : $\forall x \in \mathbb R, \lim\_{n \to+\infty} f\_n(x) =0$, which is of course the case here, all sequences $(f\_n(x))$ being stationary.
| 4 | https://mathoverflow.net/users/5659 | 24788 | 16,254 |
https://mathoverflow.net/questions/24678 | 15 | Dear everyone,
(i) Who is the father of the adjective “syntomic” in algebraic geometry?
(ii) And why did he choose to introduce a new term for what we already know from EGA IV.19.3.6 and SGA 6.VIII.1.1 as “flat, locally of finite presentation, and local complete intersection”?
Thanks!
| https://mathoverflow.net/users/307 | Why “syntomic” if “flat, locally of finite presentation, and local complete intersection” is already available? | Mazur gives the following beautiful justification, which explains the “syn-” in “syntomic” as well.
>
> Dear Thanos,
>
>
> Thanks for your question. I'm thinking of “ local complete
> intersection” as being a way of cutting out a (sub-) space from an
> ambient surrounding space; the fact that it is flat over th... | 25 | https://mathoverflow.net/users/307 | 24790 | 16,255 |
https://mathoverflow.net/questions/24789 | 0 | Assume that you have a set S of having 2^2 elements first, let S={0,1,2,3}
Then the desired 2 partitions would be
1-{{0,1},{2,3}}
2-{{0,2},{1,3}}
3-{{0,3},{1,2}}
If S={0,1,2,3,4,5,6,7} having 2^3 elements then similarly the 2 partitions would be
1- {{0,1},{2,3},{4,5},{6,7}}
2- {{0,2},{1,3},{4,6},{5,7}}
3- {{0,3},... | https://mathoverflow.net/users/6113 | Is there any direct approach to generate discrete 2 partitions of a set of having 2^n elements for a given n ? | I presume you want $2^n-1$ partitions of ${0,\ldots,2^n-1\}$
into parts of size two so that each possible pair occurs
exactly once in a partition. You can do this as follows.
Write the numbers in question base $2$. Then each can be represented
by $n$ binary digits going from $00\cdots0$ to $11\cdots 1$. Define
an opera... | 1 | https://mathoverflow.net/users/4213 | 24791 | 16,256 |
https://mathoverflow.net/questions/24737 | 0 | I would like to draw an octahedral diagram in my paper; I would prefer to present it as the 'upper hat' + the 'lower hat' (as it is common in the texts on triangulated categories). Could anyone tell me where I can find this diagram (certainly, the 'upper hat part' is sufficient) written down in latex. Maybe, someone co... | https://mathoverflow.net/users/2191 | The (upper hat of) an octahedral diagram in (la)tex | Mikhail,
Here's an upper cap in xy-pic
\xymatrix{
X'\ar[rd]^{[1]}\ar[dd]^{[1]} & & Z\ar[ll] \\
& Y\ar[ru]\ar[ld] & \\
Z'\ar[rr]^{[1]} & & X\ar[lu]\ar[uu]
}
at least it's enough of one to get you started. Note that I didn't construct it by hand.
I have a script for building these kinds of diagrams visually:
... | 4 | https://mathoverflow.net/users/4144 | 24801 | 16,264 |
https://mathoverflow.net/questions/24780 | 5 | Is there a general method for determining the domain of dependence of (higher-order) PDEs? I would be plenty happy with a reference to a paper or textbook I could look at; despite much reading around, I couldn't find this problem addressed for anything besides first- and second-order PDEs.
If there's a simple answer ... | https://mathoverflow.net/users/5789 | Is there a general method for determing the domain of dependence of (higher-order) PDEs? | While I am aware of some facts about domain of dependence properties for hyperbolic PDEs, I don't think most of them will be useful for you. The problem is that what you consider as hyperbolic (in your footnote) is too large of a class of equations for the notion to be useful: an illustration is the Heat equation. It i... | 7 | https://mathoverflow.net/users/3948 | 24802 | 16,265 |
https://mathoverflow.net/questions/24309 | 8 | I decided to spend this summer working through exercises in Hartshorne, and I found myself frustrated by the way I was solving one of them, specifically IV.2.3 on pp. 304-305. No, I'm not asking for a solution to the problem --- I almost have the whole thing solved, as far as I can tell --- what worries me is that I th... | https://mathoverflow.net/users/5281 | Dual Curves in Fancy Language | You make your solution feel "fancier", you could start with a more coordinate free approach to the dual curve. For example, try to identify the line bundle on your curve which embeds it into the dual plane. E.g.: let $X \subset \mathbb{P}^2$ be your curve. Then define $X' \subset X \times \mathbb{P}^{2\*}$ to be the se... | 3 | https://mathoverflow.net/users/397 | 24809 | 16,269 |
https://mathoverflow.net/questions/24755 | 3 | One aspect of category theory that caught my eye is that it can give simultaneously prove the 1st isomorphism theorem for groups/rings/fields/vector spaces/... Yet whenever I look up works on category theory, it takes hundreds of pages to reach this theorem. Is it possible to prove this result using substantially less ... | https://mathoverflow.net/users/4692 | Reference request for category theory works which quickly prove the theorem which generalises the 1st isomorphism theorem for groups/rings/... | If you want a categorical proof that encompasses nonabelian groups and rings-without-identity, then you need to work with concepts such as "normal subobject" that are (I think) not really part of the category theory that most mathematicians routinely encounter. You may also find that a proof is not particularly enlight... | 6 | https://mathoverflow.net/users/121 | 24810 | 16,270 |
https://mathoverflow.net/questions/24669 | 2 | I'm going to ask a very vague question, and then give specifics for the version I particularly care about. I'm interested in answers at all levels of vagueness.
At the most vague version, I am in the following situation. I have some particular (smooth) function $A(t\_0,q\_0,t\_1,q\_1)$, where $t\_0,t\_1$ are real var... | https://mathoverflow.net/users/78 | What kind of uniqueness can I conclude for solutions to a simple functional equation? | Here's a vague answer then: there's [an old 1978 paper in Aequationes Mathematicae](https://doi.org/10.1007/BF01818567 "Baron, K., Sablik, M. On the uniqueness of continuous solutions of a functional equation of n-th order. Aequat. Math. 17, 295–304 (1978). https://zbmath.org/?q=an:0393.39002") mentioning the problem o... | 1 | https://mathoverflow.net/users/469 | 24817 | 16,272 |
https://mathoverflow.net/questions/24824 | 1 | I stumbled upon [this](http://haroonsaeed.wordpress.com/2006/04/25/self-describing-numbers-and-sequence/) number sequence while surfing the web. And I generated the next terms with my pc, and I was amazed to see, only the numbers 1 to 3 come up in megabytes of output. The sequence describes the previous number. The fir... | https://mathoverflow.net/users/4338 | Self-describing number sequence | If a 4 existed in the sequence, as such:
... 41 ...
That would mean that there were 4 one's in the previous sequence, as such:
... 1111 ...
So what does THAT describe? Basically it's saying "there's 1 occurrence of 1, followed by 1 occurrence of 1." Since you're describing consecutive numbers, that would ACTUA... | 6 | https://mathoverflow.net/users/3275 | 24826 | 16,278 |
https://mathoverflow.net/questions/24842 | 3 | If the group of rational points of $E$, which is finitely generated by the Mordell-Weil Theorem, has $g$ generators of infinite order, then the Birch-Swinnerton-Dyer conjecture gives
$L\_E(s)$ has a zero of order $g$ at $s=1$.
Assuming the BSD conjecture, is it possible to (and if so how) to construct such $L\_E(s)... | https://mathoverflow.net/users/695 | BSD conjecture and L functions with zeroes of order g | I'm not entirely sure what you mean by your question. Here are two remarks:
1. If you *assume* BSD, then to "construct" $L\_E$ you just need to give the curve $E$. There are (many) elliptic curves /$\mathbb{Q}$ whose ranks have been computed, and are (say) equal to 3 or 4.
2. If one wants an example without assuming ... | 2 | https://mathoverflow.net/users/1594 | 24846 | 16,289 |
https://mathoverflow.net/questions/24845 | 14 | Let $G$ be a simple Lie group and let $G(\mathbb{C}((t)))$ be its loop group.
The Lie algebra $\mathfrak{g}[[t]][t^{-1}]$ has a well known central extension
(see e.g. [Wikipedia](http://en.wikipedia.org/wiki/Affine_Lie_algebra)) given by the cocycle
$c(f,g) = Res\_0\langle f,dg\rangle$. Here, $\langle\ ,\ \rangle ... | https://mathoverflow.net/users/5690 | Explicit cocycle for the central extension of the algebraic loop group G(C((t))) | For $SL\_2$ a cocycle is given by
$$
\sigma(g,h)=\left( \frac{x(gh)}{x(g)} , \frac{x(gh)}{x(h)} \right)
$$
where for $g=\left(\begin{array}{ll} a & b \\ c & d\end{array}\right)\in SL\_2(\mathbb{C}((t)))$, we define $x(g)=c$ unless $c=0$ in which case $x(g)=d$. $(\cdot,\cdot)$ is the tame symbol.
I'll see if I can com... | 8 | https://mathoverflow.net/users/425 | 24847 | 16,290 |
https://mathoverflow.net/questions/24841 | 1 | Does there exists a connected graph G which is a subgraph of two graphs H and H' for which
* G, H and H' have the same vertex set,
* H is minimally 2-connected (i.e. deleting any edge from H makes is not 2-connected),
* H' is 3-connected and
* H is not a subgraph of H'?
I arrived at this question when thinking abou... | https://mathoverflow.net/users/2264 | A minimally 2-connected graph H and a 3-connected graph H' both contain G, does H' contain H? | Fedja's comment definitely earns the grant. A minimal example is to let G be a star with four leaves {1,2,3,4}, let H be the bowtie obtained from G be adding the edges 13 and 24, and let H' be the wheel obtained from G by adding the cycle 1234.
| 4 | https://mathoverflow.net/users/2233 | 24848 | 16,291 |
https://mathoverflow.net/questions/24719 | 28 | Recently I tried to learn class field theory, but I find it is difficult. I have read the book "Algebraic Number Theory" by J. W. S. Cassels and A. Frohlich. In the book, the approach to class field theory is cohomology of groups. Although I have learned cohomology of groups, I find that those theorems in the book are ... | https://mathoverflow.net/users/6104 | Suggestions for good books on class field theory | When you are first learning class field theory, it helps to start by getting some idea of what the fuss is about. I am not sure if you have already gotten past this stage, but if not, I recommend B. F. Wyman's article "What is a Reciprocity Law?" in the American Mathematical Monthly, Vol. 79, No. 6 (Jun. - Jul., 1972),... | 21 | https://mathoverflow.net/users/3106 | 24852 | 16,293 |
https://mathoverflow.net/questions/24836 | 4 | Four red vectors are given, one per quadrant, $[0,90^\circ)$,
$[90^\circ,180^\circ)$, etc.
A rigid *star* of six green vectors separated by $60^\circ$
can be positioned at
$(\theta,
\theta+60^\circ,
\theta+120^\circ,
\theta+180^\circ,
\theta+240^\circ,
\theta+300^\circ)$.
The goal is to spin the green star so that th... | https://mathoverflow.net/users/6094 | Centralizing four red vectors in six green sectors | Replace each of your red vectors by its value modulo 60. You are then seeking to find a choice for $\theta$ that is as far as possible (mod 60) from any of the red vectors. The best choice for theta is to put it into the largest gap (mod 60) between red vectors, so the worst choice for the red vectors is for them to be... | 4 | https://mathoverflow.net/users/440 | 24867 | 16,304 |
https://mathoverflow.net/questions/24864 | 28 | This is to do with high dimensional geometry, which I'm always useless with. Suppose we have some large integer $n$ and some small $\epsilon>0$. Working in the unit sphere of $\mathbb R^n$ or $\mathbb C^n$, I want to pick a large family of vectors $(u\_i)\_{i=1}^k$ which is almost orthogonal in the sense that $|(u\_i|u... | https://mathoverflow.net/users/406 | Almost orthogonal vectors | Matt, to get $k$ points, you only need $n\ge C \epsilon^{-2} \log k$. See <http://en.wikipedia.org/wiki/Johnson%E2%80%93Lindenstrauss_lemma> or Google "Johnson-Lindenstrauss lemma".
| 23 | https://mathoverflow.net/users/2554 | 24873 | 16,309 |
https://mathoverflow.net/questions/24874 | 8 | Suppose one tries to formalize first-order logic. How much "strength" is required to do this?
Strength can mean in various senses:
1. The fragment of ZFC needed to codify first-order logic.
2. Which system of 2nd-order arithmetic is needed to codify first-order logic. (reverse mathematics)
3. The fragment of PA nee... | https://mathoverflow.net/users/nan | What is the reverse mathematics of first-order logic and propositional logic? | In general, "reverse mathematics" refers to work with subsystems of second-order arithmetic only; it does not include ZFC. Assuming this is actually what you meant, everything is in Stephen Simpson's book *Subsystems of second-order arithmetic*.
All the basic syntactics of formulas can be done in a primitive recursiv... | 11 | https://mathoverflow.net/users/5442 | 24882 | 16,316 |
https://mathoverflow.net/questions/24889 | 1 | More precisely, let $M$ be a subspace $\mathbb R^n$ with the following properties:
* $M$ is a topological manifold of dimension $n-1$.
* M is compact.
Does there exist a homological characterization of when the following happens:
* $\mathbb R^n \backslash M$ has two components, the bounded one being "inside" and ... | https://mathoverflow.net/users/6031 | Does there exists a (possibly homological) characterization of the Jordan curve property in all dimensions? | More genreally, the number (finite or not) of the connected components of the complement set of a compact subset $M\subset {\mathbb R}^n$, which is the rank of $H\_0({\mathbb R}^n\setminus M)$, is a homotopic invariant for compact subspaces of ${\mathbb R}^n$, by duality in homology.
| 0 | https://mathoverflow.net/users/6101 | 24892 | 16,322 |
https://mathoverflow.net/questions/24711 | 14 | The beginning of homological algebra involves lots of diagram chasing, for proving most of the theorems. This gets repetitive after a while. To make things more interesting and satisfactory, one would like to remove the mechanical dredge work as far as possible.
One possible approach, as advocated in Lang's Algebra ... | https://mathoverflow.net/users/6031 | Ways of formulating homological algebra without diagram chasing | One other way you can prove all diagram chasing results (that I know of) is to first prove that there is a spectral sequence associated with a double complex (actually, there are two!). This involves some diagram chasing, but you only do it once. I guess you could also argue that you still have to chase the spectral se... | 11 | https://mathoverflow.net/users/1 | 24907 | 16,330 |
https://mathoverflow.net/questions/24903 | 5 | The Lakes of Wada partitions the unit square in to three regions, all of whom share a common boundary. The Wikipedia entry (<http://en.wikipedia.org/wiki/Lakes_of_Wada>) gives a construction approach, and a picture of a first few steps of the construction.
Is there a good algorithm available somewhere to explicitly l... | https://mathoverflow.net/users/102 | Algorithms for the Lakes of Wada | Wada lakes can be obtained following a recipe given by Plykin.
An algorithm is explained in an online [article](http://www.ams.org/notices/200601/fea-coudene.pdf) in the Notices of the AMS. You only need to iterate a single explicit function to get the picture. So, this can be done using any fractal generator (`chaospr... | 8 | https://mathoverflow.net/users/6129 | 24911 | 16,334 |
https://mathoverflow.net/questions/24891 | 3 | Ok, imagine having a finite line segment from point (a) to point (b) in $R^2$ . I'm not familiar with mathematical terminology of this kind, but let me state that the line we began with is the geometric interpretation of $A^1$. The geometric interpretation of $A^2$ is a square with sides $A^1$. We could go on by saying... | https://mathoverflow.net/users/93724 | The Root of a Line | Here's my proposal for the square root of a line segment.
It's the Cantor set obtaining by repeatedly
splitting the intervals into four, and removing the two middle pieces.
When you take the cartesian square of that space, you obtain something whose projection
is exactly an interval:
[alt text http://www.staff.sci... | 4 | https://mathoverflow.net/users/5690 | 24924 | 16,343 |
https://mathoverflow.net/questions/24932 | 3 | Apparently, there is the following fact:
>
> The set of homeomorphism classes of connected manifolds has the same cardinality $c$ as that of $\mathbb R$.
>
>
>
I find it to be interesting; but would be happier to see a proof, and would be grateful for a reference somewhere.
| https://mathoverflow.net/users/6031 | Counting connected manifolds | Upper bound (assuming the manifolds are second countable): every manifold admits a complete metric, and the "set" of isometry classes of complete separable metric spaces is of cardinality continuum. Indeed, every such a space is a completion of a countable metric space, and there is only continuum of metrics on a count... | 13 | https://mathoverflow.net/users/4354 | 24939 | 16,352 |
https://mathoverflow.net/questions/24936 | 11 | This question arose out of my attempts to understand [another question](https://mathoverflow.net/questions/24453/). The most popular construction for the chain complex for defining singular homology uses the $n$-simplex.
But it is also possible to use other spaces. For example, one can use the $n$-cubes instead, as d... | https://mathoverflow.net/users/6031 | Why the choice of the simplex for defining homology? | There is a specific reason that singular homology with simplices is simpler than singular homology with cubes. You would like the homology of a point to be "trivial" according to the Eilenberg-Steenrod axioms. (That is, $H^0 = \mathbb{Z}$ and the others are trivial.) However, if you look carefully at the chain complex ... | 22 | https://mathoverflow.net/users/1450 | 24944 | 16,356 |
https://mathoverflow.net/questions/24930 | 9 | This is for the sake of completeness(for references, understanding).
I ask for references for proofs that:
-There is exactly one differentiable(ie $C^\infty$) structure on $\mathbb R$, upto diffeomorphism.
-Ditto for $\mathbb R^2$.
-Ditto for $\mathbb R^3$.
| https://mathoverflow.net/users/6031 | Differentiable structures on R^3 | UPDATE : I found some precise references that answer the OP's question and fill in some details in my original answer. See the end for them.
I don't have a precise reference for the cases of $n=2$ or $n=3$, though I suspect that they could be found in Moise's "Geometric Topology in Dimensions 2 and 3".
However, I d... | 17 | https://mathoverflow.net/users/317 | 24950 | 16,361 |
https://mathoverflow.net/questions/24849 | 12 | Let $N \in \mathfrak{M}\_n(\mathbb{C})$ nilpotent, such that there exists $X \in \mathfrak M\_n(\mathbb{C})$ with $X^2=N$ (take for instance $n>2$ and $N(1,n)=1$; $N(i,j)=0$ otherwise).
Denote by $\mathcal{S}\_N$ the set of $X \in \mathfrak M\_n(\mathbb{C})$ such that $X^2=N$.
Is $\mathcal{S}\_N$ connected or path-co... | https://mathoverflow.net/users/3958 | (Path) connected set of matrices? | This is a memorial to an incorrect solution that used to be here. Unfortunately, I can't delete it since it was accepted.
| 2 | https://mathoverflow.net/users/2349 | 24956 | 16,364 |
https://mathoverflow.net/questions/24392 | 7 | Sorry the title is a bit vague. Let A be a C\*-algebra, and let x and y be positive elements in A. Is it true that $$ \|x-y\|^2 \leq \|x^2-y^2\|? $$
Well, yes. But the proof I have is a bit of a hack, so I wonder if anyone has a "nice" proof, or a reference?
Aside: if $A=C\_0(X)$ then this reduces to the inequality $... | https://mathoverflow.net/users/406 | Simple inequality in C*-algebras | Theorem 1.5 of [this 1987 paper](https://dspace.library.uvic.ca:8443/dspace/handle/1828/1506) by J. Phillips says that if $f:[0,\infty)\to [0,\infty)$ is a continuous operator monotone function and $a$ and $b$ are positive operators on a Hilbert space, then $\|f(a)-f(b)\|\leq f(\|a-b\|)-f(0)$. I think that the proof is... | 9 | https://mathoverflow.net/users/1119 | 24963 | 16,368 |
https://mathoverflow.net/questions/24970 | 38 | This was going to be a comment to [Differentiable structures on R^3](https://mathoverflow.net/questions/24930/differentiable-structures-on-r3), but I thought it would be better asked as a separate question.
So, it's mentioned in the previous question that $\mathbb{R}^4$ has uncountably many (smooth) differentiable st... | https://mathoverflow.net/users/3329 | Exotic differentiable structures on R^4? | I once heard Witten say that topology in 5 and higher dimensions "linearizes". What he meant by that is that the geometric topology of manifolds reduces to algebraic topology. Beginning with the Whitney trick to cancel intersections of submanifolds in dimension $d \ge 5$, you then get the [h-cobordism theorem](http://e... | 36 | https://mathoverflow.net/users/1450 | 24973 | 16,374 |
https://mathoverflow.net/questions/24974 | 3 | Of course if two morphisms of complexes are homotopic their induced maps coincide, but I'm wondering about the converse: if the induced maps on the cohomologies coincide, when does that imply that the morphisms are homotopic?
I've played around with it a bit and I think it might be true for complexes of projective mo... | https://mathoverflow.net/users/2503 | Coinciding induced maps | The answer is no. Consider the complex over the integers $A$ which is $\mathbb Z$ in degree $0$ and $1$ and the only non-trivial differential being multiplication by $2$ and let $B$ be the same complex shifted once to the left (so that it is $\mathbb Z$ in degrees $-1$ and $0$). We have a map of complexes $A \to B$ whi... | 7 | https://mathoverflow.net/users/4008 | 24977 | 16,377 |
https://mathoverflow.net/questions/24967 | 17 | The physical meaning of a Lie group symmetry is clear: for example, if you have a quantum system whose states have values in some Hilbert space $H$, then a Lie group symmetry of the system means that $H$ is a representation of some Lie group $G$. So you want to understand this Lie group $G$, and generally you do it by ... | https://mathoverflow.net/users/290 | What is the physical meaning of a Lie algebra symmetry? | I'm not sure which sort of examples of Lie algebras without the corresponding groups you have in mind, but here is a typical example from Physics.
Many physical systems can be described in a hamiltonian formalism. The geometric data is usually a symplectic manifold $(M,\omega)$ and a smooth function $H: M \to \mathbb... | 9 | https://mathoverflow.net/users/394 | 24979 | 16,378 |
https://mathoverflow.net/questions/24960 | 53 | Over the years, I have heard two different proposed answers to this question.
1. It has something to do with parabolic elements of $SL(2,\mathbb{R})$. This sounds plausible, but I haven't heard a really convincing explanation along these lines.
2. "Parabolic" is short for "para-Borelic," meaning "containing a Borel s... | https://mathoverflow.net/users/3106 | Why are parabolic subgroups called "parabolic subgroups"? | It appears that neither of the answers is fully correct. There is a great book, "Essays in the history of Lie groups and algebraic groups" by Armand Borel, when it comes to references of this type. To quote from chapter VI section 2:
>
> ...There was no nice terminology for the subgroups $P \_I$ with lie algebra th... | 25 | https://mathoverflow.net/users/2384 | 24987 | 16,385 |
https://mathoverflow.net/questions/24984 | 3 | According to Serre's book 'Galois cohomology', Galois chomology group are always torsion, but it seems to me that H^1(k, End\_{Z\_l}(T\_l(A)))=coker(Frob-1) on End\_{Z\_l}(T\_l(A)), which has the same Z\_l rank as End\_{k}(T\_l(A))
So maybe End\_{Z\_l}(T\_l(A)) is not a discrete galois module. And why is the Tate modul... | https://mathoverflow.net/users/3848 | what is the first Galois cohomology group of the Galois module End(T_l(A)) for some abelian variety A over a finite field k and l some prime number different from the characteristic of the base field? | In general, if $G$ is a profinite group and $M$ a continuous discrete $G$--module, then $H^i(G,M)$ is torsion for $i>0$. This applies in particular to Galois cohomology, i.e. when $G$ is a Galois group.
Tate modules are *not* discrete Galois modules, and their cohomology will usually not be torsion. The same goes for... | 0 | https://mathoverflow.net/users/5952 | 24993 | 16,390 |
https://mathoverflow.net/questions/17508 | 5 | As I mentioned before, I'm a novice at deformation theory. I was wondering if the definition of versality in deformation theory is related to the versality in moduli spaces:
### Deformation theory
"Moduli of Curves" defines a versal deformation space as a deformation $\phi: X \rightarrow Y$ such that for any other ... | https://mathoverflow.net/users/3238 | Versality in deformation theory vs. versality in moduli spaces | There is an easy answer to your question (without stacks) which has not been given yet: Yes.
The deformation space of a curve $X\_0$ is just a local model of a moduli space of all curves near a special member $X\_0$.
Your definition of versality for deformations says that, $Y$ is over-parametrizing
local deforamtio... | 7 | https://mathoverflow.net/users/5714 | 25000 | 16,396 |
https://mathoverflow.net/questions/25003 | 3 | The construction of tangent bundle on a $C^\infty$ manifold, as for example in the book of Warner, uses the existence of double derivatives. Of course the tangent space for a point is first constructed in case of an open set in a Euclidean space and then the whole setup is glued up. But then in the neighborhood of a po... | https://mathoverflow.net/users/6031 | What is the order of differentiability of the tangent bundle of a C^2- manifold? | The tangent bundle of a $C^1$ manifold exists: it's a $C^0$ manifold.
Similarly, for every $n$, the tangent bundle of a $C^n$ manifold is a $C^{n-1}$ manifold.
| 5 | https://mathoverflow.net/users/5690 | 25005 | 16,399 |
https://mathoverflow.net/questions/25008 | 10 | Given a finite field $K$, what are the possible degrees of a polynomial $p\in K[x]$ such that $x\longmapsto p(x)$ is one-to-one?
Such a polynomial has clearly not degree $0$ and it cannot be of degree two except for
$x\longmapsto (\alpha(x))^2$ for $\alpha$ an affine bijection of a field of characteristic $2$.
Are ... | https://mathoverflow.net/users/4556 | Degrees of a polynomial $p$ such that $x\mapsto p(x)$ is one-to-one | Such things are referred to as ‘permutation polynomials’ and if you do a search, you'll find a whole menagerie of non-stupid classes which is constantly expanding. One simple result going back to Dickson provides something converse to damiano's [observation](https://mathoverflow.net/questions/25008/given-a-finite-field... | 17 | https://mathoverflow.net/users/3143 | 25017 | 16,406 |
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