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https://mathoverflow.net/questions/29593 | 7 | It is well discussed in many graph theory texts that it is somewhat hard to distinguish non-isomorphic graphs with large order. But as to the construction of all the non-isomorphic graphs of any given order not as much is said. So, it follows logically to look for an algorithm or method that finds all these graphs.
... | https://mathoverflow.net/users/5627 | Non-isomorphic graphs of given order. | Acknowledging Timothy’s comment, let me answer the question.
For a diagrammatic list of the non-isomorphic graphs (all in pdfs):
1. <http://keithbriggs.info/images/g4.pdf>
2. <http://keithbriggs.info/images/g5.pdf>
3. <http://keithbriggs.info/images/g6.pdf>
4. <http://keithbriggs.info/images/g7.pdf>
5. <http://keit... | 3 | https://mathoverflow.net/users/5627 | 29875 | 19,505 |
https://mathoverflow.net/questions/29851 | 2 | The question is about commutator in integral forms. Let $A$ an associative algebra over a field of characteristic zero, $x\in A$ and $k\in \mathbb Z$, we denote $x^{(k)}=\frac{x^{k}}{k!}$.
How to calculate brackets $[x\_\beta^{(n)}, x\_\alpha^{(m)}]$? Can I proceed doing the calculation for $[x\_\beta^{n}, x\_\alpha^{m... | https://mathoverflow.net/users/40886 | hyperalgebras (positive characteristic) | As I understand, your question is about relations on the generators of the Garland integral form for the hyper loop algebra of $\mathfrak g$. Maybe the following papers will help you:
H.Garland, The arithmetic theory of loop algebras, J. Algebra 53 (1978), 480--551.
D. Jakelic, A. Moura, Finite-dimensional represen... | 3 | https://mathoverflow.net/users/7174 | 29899 | 19,517 |
https://mathoverflow.net/questions/29546 | 7 | String topology studies the algebraic structure of the homology of the free loop space $LM = Map(S^1,M)$ of a oriented closed manifold. One aspect of this structure is that the pair $(H\_\ast(LM;\mathbb{Q}),H\_\ast(M;\mathbb{Q}))$ forms a open-closed HCFT with positive boundary (work of Godin). This means that there ar... | https://mathoverflow.net/users/798 | Can string topology be a open-closed TCFT with the full set of branes? | No, the conjecture is not false. The mistake is the following: I said "On the other hand, Costello has proven a classification theorem of open-closed TCFT." He proves a classification of open TCFT's and gives a construction of an open-closed TCFT from an open TCFT. String topology can't be obtained by this specific con... | 4 | https://mathoverflow.net/users/798 | 29901 | 19,519 |
https://mathoverflow.net/questions/29913 | 4 | Given a map between two manifolds that induces an isomorphism on integral cohomology in the top dimension, it follows from naturality of the cup product and Poincaré duality and universal coefficient theorem that all maps on cohomology in every dimension are injective with torsion free cokernel.
Is there an example w... | https://mathoverflow.net/users/6960 | map of manifolds inducing iso on top cohomology, but not surjective on one other cohomology group | For any orientable manifold M of dimension n, map M to $S^n$ by sending some suitable neighborhood of a point homeomorphically to $\mathbb{R}^n$ and the rest to $\infty$.
| 10 | https://mathoverflow.net/users/4183 | 29916 | 19,528 |
https://mathoverflow.net/questions/29578 | 2 | Given a conflict graph $G = (V, E)$, a man has to transport a set $V$ of items/vertices across the river. Two items are connected by an edge in $E$, if they are conflicting and thus cannot be left alone together without human supervision. The available boat has capacity $b\geq 1$, and thus can carry the man together wi... | https://mathoverflow.net/users/7094 | Monadic Second Order (MSO) logic on graphs | Maybe there is a solution. But, for that I assume there is an upper bound in the number of rounds needed, say n, and that the value b is fixed upfront. Then, there is the following EMSO formula,
$\exists L\_{1} \exists B\_{1} \exists R\_{1} ... \exists L\_{n} \exists B\_{n} \exists R\_{n} \phi(L\_{1},B\_{1},R\_{1} ..... | 4 | https://mathoverflow.net/users/7176 | 29918 | 19,530 |
https://mathoverflow.net/questions/29921 | 5 | Consider $M$, a positive definite matrix. Let $M^{(1)}$ be the diagonal matrix which agrees with $M$ on the diagonal ($M\_{ii}=M^{(1)}\_{ii}$). We have that $M^{(1)}$ is positive definite because it is diagonalizable and it has non-negative eigenvalues.
What about general bands? Let $M^{(b)}$ be the restriction of $M... | https://mathoverflow.net/users/6154 | Are the banded versions of a positive definite matrix positive definite? | No. The matrix
$M = \begin{bmatrix}5 & 4 & 4 \\\\ 4 & 5 & 4 \\\\ 4 & 4 & 5\end{bmatrix} = \begin{bmatrix}2 & 2 & 2\end{bmatrix}\begin{bmatrix}2 \\\\ 2 \\\\ 2\end{bmatrix} + I$
is positive definite, but
$\begin{bmatrix}1 & -\sqrt{2} & 1\end{bmatrix}M^{(2)}\begin{bmatrix}1 \\\\ -\sqrt{2} \\\\ 1\end{bmatrix} = 20 -... | 10 | https://mathoverflow.net/users/5963 | 29925 | 19,532 |
https://mathoverflow.net/questions/29935 | 3 | Let $k$ be a field, $L$, $H$ extension fields of $k$, and $G=L\otimes\_k H$. I wonder why (I want to know the proof but I can't find) the prime ideal of $G$ must be maximal, and its properties:
a) if $L$ is separable over $k$, then $G$ is reduced.
b) if $L$ is algebraic and purely inseparable over $k$, then $G$ has... | https://mathoverflow.net/users/3945 | question about tensor of two fields | For (a): a field extension $L/k$ is separable is $L$ is a separable algebra, that is, if $L\otimes K$ is a semisimple algebra for all field extensions $K/k$. In particular, if $L$ is separable over $k$ and $K$ is an extension of $k$, then $L\otimes K$ will have no nilpotent elements because it is semisimple.
For deta... | 6 | https://mathoverflow.net/users/1409 | 29936 | 19,539 |
https://mathoverflow.net/questions/29825 | 1 | Assume I have a quantile function for an arbitrary probability distribution for random variable **x**.
Would the **x**-value corresponding to the 99th percentile be the same as the **x**-value corresponding to a p-value of 0.01 (one-sided test, right tail)?
**Details for my specific problem:**
I have fitted a gamm... | https://mathoverflow.net/users/7152 | What is the relationship between quantile functions and p-values | First I'll address your initial question without taking into account the details of the specific problem. The answer is "yes" if, and only if, the probability distribution is that of a test statistic, where the null hypothesis will be rejected if the test statistic is too big.
When you get into the details of your sp... | 1 | https://mathoverflow.net/users/6316 | 29943 | 19,542 |
https://mathoverflow.net/questions/29949 | 40 | In short, my question is:
>
> What is the shortest computer program for which it is not known whether or not the program halts?
>
>
>
Of course, this depends on the description language; I also have the following vague question:
>
> To what extent does this depend on the description language?
>
>
>
He... | https://mathoverflow.net/users/6950 | What is the shortest program for which halting is unknown? | There is a 5-state, 2-symbol Turing machine for which it is not known whether it halts. See
<http://en.wikipedia.org/wiki/Busy_beaver>.
| 59 | https://mathoverflow.net/users/2807 | 29955 | 19,550 |
https://mathoverflow.net/questions/29907 | 6 | Consider a connected symplectic manifold $(M, \omega)$ of dimension $m=2n$. A few preliminary reminders (mostly to fix the notation): A vector field $X$ is symplectic if its flow preserves the symplectic form, ie. $L\_X \omega = 0$, where $L\_X$ denotes Lie derivative with respect to $X$. The Cartan formula shows that ... | https://mathoverflow.net/users/4747 | Which tensor fields on a symplectic manifold are invariant under all Hamiltonian vector fields? | Any symplectic linear transformations in $T\_xM$ is locally realizable as a Hamiltonian vector field, thus for questions 1 and 2, one can profitably use representation theory of the symplectic group.
**FACT** (*Lefschetz decomposition*) Let $W$ be a $2n$-dimensional symplectic vector space, $\bigwedge^\ast W$ its ext... | 14 | https://mathoverflow.net/users/5740 | 29976 | 19,563 |
https://mathoverflow.net/questions/29978 | 1 | Do there exist nonconstant real valued functions $f$ and $g$ such that the expression:
$$f(x) -v/g(x)$$
is maximized at $x = v$ for all positive real $v$?
| https://mathoverflow.net/users/7192 | Do there exist nonconstant functions such that... | Take $f(x)=(x+1)e^{-x}$ and $g(x)=e^x$, then $f(x)-v/g(x)=(x+1-v)e^{-x}$ and the derivative with respect to $x$ is $(v-x)e^{-x}$.
| 6 | https://mathoverflow.net/users/5735 | 29985 | 19,569 |
https://mathoverflow.net/questions/24697 | 3 | Let $M$, $N$ be two modules over ring $A$. If $M\oplus M\cong N\oplus N$, can we conclude $M\cong N$? In the case that $M$, $N$ are completely decomposable (e.g. finite-length module by Krull-Schmidt Theorem), it is easy to show this must be true. Does the general case also hold?
| https://mathoverflow.net/users/6103 | Isomorphism between direct sum of modules | There are even counterexamples in the case $A = {\mathbb Z}$: at the end of B. Jónsson’s paper “On direct decompositions of torsion-free abelian groups,” *Math. Scand.* **5** (1957), 230–235, an example is given of torsion-free, finite-rank abelian groups $B \not\cong C$ such that $B \oplus B \cong C \oplus C$.
A fu... | 9 | https://mathoverflow.net/users/6521 | 29986 | 19,570 |
https://mathoverflow.net/questions/29992 | 3 | Let $X$ be a simplicial set. Let $X\to \Delta^n$ be a right fibration (has the right lifting property with respect to right horn inclusions), and let $$\Delta^{\{n-i\}}\hookrightarrow \Delta^{\{n-i,\dots, n\}}\hookrightarrow \Delta^n$$ (for a fixed $i: 0\leq i\leq n$) be the obvious inclusion maps.
Then why is the i... | https://mathoverflow.net/users/1353 | Why is the induced map between pullbacks (of inclusions) by a right fibration a deformation retract? | In topology, if a map $Y\to B$ is some sort of fibration then you would think that $B$ being equivalent to a subspace $A$ would imply that $Y$ is equivalent to $Y\_A=Y\times\_BA$. If fibration means Serre fibration and equivalent means weakly, then you might want to use homotopy groups and the Whitehead Theorem. But wh... | 4 | https://mathoverflow.net/users/6666 | 30003 | 19,579 |
https://mathoverflow.net/questions/29508 | 15 | I have a sequence of centered independent random variables $X\_i$ that are all
bounded by one in absolute value. They are not identically distributed, though.
I would like to know if the **central limit theorem** is still true
for such a sequence. Putting $S\_n= X\_1+...+X\_n$, do we have
$$
c\_n = P(\ {S\_n\over\sig... | https://mathoverflow.net/users/7082 | Is there a central limit theorem for bounded non identically distributed random variables? | *Theorem* (Billingsley, "probability and measure", example 27.4)
Let X\_i a sequence of independent, uniformly bounded random variables with zero mean, such that $\sigma(S\_n)$ goes to infinity with n. Then $S\_n/\sigma(S\_n)$ converges in law to the normalized Laplace-Gauss distribution.
This follows from the *Lin... | 10 | https://mathoverflow.net/users/6129 | 30022 | 19,593 |
https://mathoverflow.net/questions/29993 | 30 | What's wrong with defining the rank of a finitely generated module over any (commutative) ring to be just the smallest number of generators? All books I know define rank only locally this way. But why not define it globally?
| https://mathoverflow.net/users/5292 | Rank of a module | Since your profile says you are interested in Algebraic Geometry, here are geometric considerations that might appeal to you.
Consider a projective module $P$ of finite type over a commutative ring $A$. It corresponds to a locally free sheaf $\mathcal F $ over $X=Spec(A)$. The rank of $\mathcal F $ at the prime ideal... | 43 | https://mathoverflow.net/users/450 | 30024 | 19,594 |
https://mathoverflow.net/questions/30025 | 1 | Let $F=\mathrm{GF}\left(p^k\right)$ be any finite field. Let $G$ be the group of all affine permutations on $F$ (i.e. permutations of form $x\mapsto ax+b$). Then the set of all functions from $F$ to $\bar{F}$ is a linear representation of $G$, where $g(f)(x)=f(gx)$.
What are all sub-representations of this representa... | https://mathoverflow.net/users/4246 | Sub-representations of the affine group | As Victor explained consider the functions $X^m$ where $X^m(\alpha)=\alpha^m$. As $m$ runs between $0$ and $p^k-1$, these functions form a basis of your space of functions. This is a nice wavy basis, i.e., its elements span one-dimensional subrepresentations under the multiplicative group.
Now you have to take the ad... | 3 | https://mathoverflow.net/users/5301 | 30036 | 19,598 |
https://mathoverflow.net/questions/30031 | 25 | I made a passing comment under Max Alekseyev's cute answer to [this question](https://mathoverflow.net/questions/29926/3n-2m-pm-41-is-not-possible-how-to-prove-it) and Pete Clark suggested I raise it explicitly as a different question. I cannot give any motivation for it however---it was just a passing thought. My only... | https://mathoverflow.net/users/1384 | Proving non-existence of solutions to $3^n-2^m=t$ without using congruences | I believe this is closely related to the conjecture by Brenner and Foster [here](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-101/issue-2/Exponential-Diophantine-equations/pjm/1102724775.full). They ask if an exponential Diophantine equation of the form $$\sum \epsilon\_i p\_i^{m\_i}=t$$ whe... | 6 | https://mathoverflow.net/users/2384 | 30037 | 19,599 |
https://mathoverflow.net/questions/30030 | 26 | I've noticed that the Galois groups associated to Galois field extensions $L$ of a given field $K$ seem remarkably like a sheaf, with the field extensions taking the place of open set, and the Galois group $\mathrm{Gal}(L/K)$ of a field extension $L/K$ is the sections over $L/K$. This makes sense because we have a natu... | https://mathoverflow.net/users/1355 | Galois Group as a Sheaf | You might introduce a Grothendieck topology on the Galois subextensions of $L/K$; then the Galois group is indeed a sheaf.
I just want to remark the following: There is a natural homeomorphism between $Gal(L/K)$ and $Spec(L \otimes\_K \overline{K})$. Namely, if $\sigma \in Gal(L/K)$, then the kernel of $L \otimes\_K ... | 14 | https://mathoverflow.net/users/2841 | 30038 | 19,600 |
https://mathoverflow.net/questions/30042 | 6 | Let $\mathfrak{g} \subset \mathfrak{gl}\_n$ be one of the classical real or complex semisimple Lie algebras. If $g \in \mathfrak{g}$, then $g$ has a Jordan decomposition $g = g\_s + g\_n$ with $g\_s$ semisimple and $g\_n$ nilpotent, and $[g\_s,g\_n]=0$.
The elements $g\_s,g\_n$, which a priori are just in $\mathfrak... | https://mathoverflow.net/users/379 | Jordan decomposition in a classical group | Give the proof in Humphreys' "Linear Algebraic Groups". It is essentially a context-free version of the argument you give, and hinges only on the fact that if $\rho\_g$ is right-translation by $g$ in $k[\operatorname{GL}\_n]$ and $I$ is the ideal defining $G$ in $\operatorname{GL}\_n$, then $g \in G$ if and only if $\r... | 2 | https://mathoverflow.net/users/6545 | 30050 | 19,606 |
https://mathoverflow.net/questions/30035 | 34 | Recall that the scalar curvature of a Riemannian manifold is given by the trace of the Ricci curvature tensor. I will now summarize everything that I know about scalar curvature in three sentences:
* The scalar curvature at a point relates the volume of an infinitesimal ball centered at that point to the volume of th... | https://mathoverflow.net/users/4362 | Some questions about scalar curvature | The Kazdan-Warner theorem goes a long way toward answering the first and second questions.
(For notes typed up by Kazdan, see <http://www.math.upenn.edu/~kazdan/japan/japan.pdf>.)
Here's what is says (taken almost verbatim from the notes, page 93): Divide the class of all closed manifolds (edit: of dimension > 2. S... | 27 | https://mathoverflow.net/users/1708 | 30054 | 19,608 |
https://mathoverflow.net/questions/30051 | 2 | Let $S$ be a locally noetherian scheme, $Y$ a locally noetherien $S$-scheme and $X$ an abelian scheme over $S$. It is known that the map between groups $Hom(Y,X) \to Hom(Pic(X/S),Pic(Y/S)), f \mapsto f^\*$ is quadratic, i.e. we have
$(f+g+h)^\* - (f + g)^\* - (f + h)^\* - (h + h)^\* + f^\* + g^\* + h^\* = 0$.
Howev... | https://mathoverflow.net/users/2841 | Picard functor is not linear | Let $E$ be an elliptic curve; take $f = g = \mathrm{id}\_E$. Then $f+g$ is multiplication by $2$, and has degree $4$; hence if $L$ is an invertible sheaf on $E$, the sheaf $(f+g)^\*L$ has degree $4 \deg L$, while $f^\*L \otimes g^\*L = L^{\otimes 2}$ has degree $2\deg L$.
| 5 | https://mathoverflow.net/users/4790 | 30056 | 19,610 |
https://mathoverflow.net/questions/30058 | 2 | Let $G$ be a finite abelian group. When $\prod\_{g\in G\setminus 1} (1-g)$ vanishes in (say, complex) group algebra of $G$?
It is easy to see that for cyclic group $G$ such product does not vanish, since $G$ may be embedded into a (complex) field.
Oh, it looks like it is obvious: for non-cyclic $G$ there is no exa... | https://mathoverflow.net/users/4312 | vanishing of certain product in group algebra | I think the answer is "if and only if the group $G$ is not cyclic". Why?
1) An element of $\mathbb C\left[G\right]$ is zero if and only if it acts as zero on each irreducible representation of $G$ (since $\mathbb C\left[G\right]$ is the direct sum of the endomorphism rings of the irreducible representations).
2) An... | 4 | https://mathoverflow.net/users/2530 | 30063 | 19,614 |
https://mathoverflow.net/questions/30064 | 5 | Hi,
I was wondering how much (if anything) $\mathcal{L}\_{PA}$ can express about individual nonstandard elements in a nonstandard model of PA. For instance, presumably it can say that each has $k$-many predecessors, for each $k\in\mathbb{N}$. But:
(a) I can't see that there is any way that the type of one element i... | https://mathoverflow.net/users/7209 | Are the types of nonstandard natural numbers within a Z-chain identical? | Since the nonstandard numbers believe that every other number is even and every other number is odd, a fact that is expressible in the language you mention, it follows that the types are not the same for every two elements in a $Z$-chain. In fact, more is true: any two nonstandard natural numbers in a common $Z$-chain ... | 13 | https://mathoverflow.net/users/1946 | 30076 | 19,619 |
https://mathoverflow.net/questions/29942 | 4 | You can map whole numbers to combinations when taking them in order. For example, 13 choose 3 would look like:
```
0 --> (0, 1, 2)
1 --> (0, 1, 3)
2 --> (0, 1, 4)
etc...
```
Given a particular combination, such as `(0, 3, 9)`, is there a way to determine which whole number maps to it (26, in this case), short of w... | https://mathoverflow.net/users/1646 | Order of a combination when mapping them to whole numbers | Let $N(n;a\_1,\dots,a\_k)$ where $0\leq a\_1 < a\_2 < \dots < a\_k < n$ be the order number of $(a\_1,\dots,a\_k)$ as a combination from ($n$ choose $k$).
Since there are exactly $\binom{n-1}{k-1}$ combinations with $a\_1 = 0$, we have a recurrence:
if $a\_1 = 0$, then
$$ N(n;a\_1,\dots,a\_k) = N(n-1;a\_2-1,\dots,a... | 5 | https://mathoverflow.net/users/7076 | 30077 | 19,620 |
https://mathoverflow.net/questions/30072 | 10 | I believe that there is no common theory for finding roots of polynomial sum. In my case I have
$$P\_{n}(x)+AQ\_{n}(x)$$.
I am wondering how roots of this sum depend on $A$?
| https://mathoverflow.net/users/3589 | roots of sum of two polynomials | If they are complex polynomials or can be treated as such, then you could apply [Rouche's theorem](http://en.wikipedia.org/wiki/Rouch%C3%A9%27s_theorem), where the location of the zeros is determined by the dominant polynomial within the sum. (*"Walk the dog on the leash"*)
Possibly related: you could use the [Wronsk... | 9 | https://mathoverflow.net/users/5372 | 30086 | 19,628 |
https://mathoverflow.net/questions/30087 | 2 | The motivation behind this is to find the points of intersection between a ray and a level set of a potential function $g$, built in terms of a basic potential function $f$ (the building is explained later). This is a problem in ray tracing where the level set is known as a meta-ball. Normally, the basic potential func... | https://mathoverflow.net/users/3121 | Root Finding for Raytracing (Ray and Meta-Ball Intersection) | While this isn't precisely high mathematics, I've done a fair bit of metaball modeling in the past. They're not a basic potential functions in the sense you use ($f$ is only non-zero on a bounded interval), but either the classic ease-in ease-out function $f(x) = 1+3x^2-2x^3 (x\leq1)$, $f(x) = 0 (x\geq1)$ (with $r^2$ b... | 5 | https://mathoverflow.net/users/7092 | 30095 | 19,632 |
https://mathoverflow.net/questions/30088 | 1 | Does the notion of $\omega$-monoid exist, analogous to the notions of $\omega$-groupoid and $\omega$-category? If so, some references would be appreciated.
This is an attempted rephrasing of question: [Chain/Hierarchy of Monoids](https://mathoverflow.net/questions/24723/chain-hierarchy-of-monoids). My application dom... | https://mathoverflow.net/users/2620 | $\omega$-monoids | Sure. A monoid is the same as a (pointed) category with a single object.
So an $n$-monoid is the same as a pointed $n$-category with a single object.
These usually go by names like $A\_\infty$-algebras (mostly if they are linear) or similar.
If you want *strict* $\infty$-monoids, then the notion of a strict $\o... | 3 | https://mathoverflow.net/users/381 | 30102 | 19,636 |
https://mathoverflow.net/questions/30026 | 4 | In my recent studies (fourier multipliers on weighted Lp spaces) I have to deal with this kind of transformation: if w is a measurable function on $R^n$, define
$w^\*(x)=\sup\_y \frac{w(x+y)}{w(y)}$.
Has anyone seen this transformation before? Has this interesting properties?
In particular I'm interested in functions s... | https://mathoverflow.net/users/4928 | Is anything known about $w^*(x)=\sup_y w(x+y)/w(y)$ for measurable functions w on $R^n$ | Sort of a random thought: if $w = w^\*$, then $w$ must be positive. So take logs on both side. Since log is monotonic, it commutes with sup. Then we have $a = \log w$, and $$a(x) = \sup\_y a(x+y) - a(y)$$
which implies that
$$ a(x) + a(y) \geq a(x+y)$$
so $a$ must be sub-additive. Helge's comment tells us that $a(0) =... | 4 | https://mathoverflow.net/users/3948 | 30103 | 19,637 |
https://mathoverflow.net/questions/30112 | 9 | Background
----------
I came up with this while trying to find a sort of high-level exposition of the exterior algebra of a vector space. Let $V$ be a vector space of dimension $n$ over $\mathbb{C}$, and let $k \in \mathbb{N}$. One picture of $\Lambda^k(V)$, the $k^{th}$ exterior power of $V$, is as the space of tota... | https://mathoverflow.net/users/703 | (Elementary?) combinatorial identity expressing binomial coefficients as an alternating sum over permutations. | $n(n-1)...(n-(k-1))$ is the number of injective functions from a set of size $k$ to a set of size $n$. We can count these using inclusion-exclusion: first include all such functions, of which there are $n^k$. Then, for each transposition $(ij)$ in $S\_k$, exclude all the functions such that $f(i) = f(j)$, of which ther... | 16 | https://mathoverflow.net/users/290 | 30114 | 19,641 |
https://mathoverflow.net/questions/30113 | 19 | The classifying space of the nth symmetric group $S\_n$ is well-known to be modeled by the space of subsets of $R^\infty$ of cardinality $n$. Various subgroups of $S\_n$ have related models. For example, $B(S\_i \times S\_j)$ is modeled by subsets of $R^\infty$ of cardinality $i + j$ with $i$ points colored red and $j$... | https://mathoverflow.net/users/4991 | Geometric model for classifying spaces of alternating groups | $n$ linearly independent points in $R^\infty$ together with an orientation of the $n$-plane which they span.
| 22 | https://mathoverflow.net/users/284 | 30115 | 19,642 |
https://mathoverflow.net/questions/29982 | 18 | Let $n\geq 2$ be a positive integer. For the purposes of this definition, let a *colored graph* be a finite undirected graph in which each edge is colored with one of $n$ colors so that no vertex is incident with two edges of the same color. (Without loss of generality, we suppose that every vertex is incident with *ex... | https://mathoverflow.net/users/2559 | Has this notion of product of graphs been studied? | If you wanted this for directed graphs, I would say this:
As Andy Putman suggests, look at Stallings Inventiones paper "The topology of finite graphs."
A directed graph colored with $n$ colors admits an obvious map to a colored oriented wedge of $n$ circles, $X$ say.
Given two directed colored graphs, the fiber p... | 7 | https://mathoverflow.net/users/1335 | 30118 | 19,645 |
https://mathoverflow.net/questions/30120 | 3 | Hi I'm currently learning Hamiltonian and Lagrangian Mechanics (which I think also encompasses the calculus of variations) and I've also grown interested in functional analysis. I'm wondering if there is any connection between functional analysis and Hamiltonian/ Lagrangian mechanics? Is there a connection between func... | https://mathoverflow.net/users/7223 | Functional Analysis and its relation to mechanics | (1) Depends on what you mean by Hamiltonian and Lagrangian mechanics.
If you mean the classical mechanics aspect as in, say, Vladimir Arnold's "Mathematical Methods in ..." book, then the answer is no. Hamiltonian and Lagrangian mechanics in that sense has a lot more to do with ordinary differential equations and sy... | 5 | https://mathoverflow.net/users/3948 | 30127 | 19,649 |
https://mathoverflow.net/questions/30130 | 5 | Let $K$ be a field, $\alpha\in\bar{K}$, and $L/K$ a finite extension. How can we determine whether $\alpha\in L$, preferably in as much generality as possible?
Of course, there may be special cases where this is easy, e.g. $K\subset\mathbb{R}$ and $\alpha\in\mathbb{C}\setminus\mathbb{R}$. Another trick, using the fie... | https://mathoverflow.net/users/1916 | Methods of showing an element is / is not in a field | Zev, I think in complete generality there is no good answer to the question. Just think about how to decide if a cubic is reducible. In principle it's easy: find a root! But unless the field has some special structure that makes it feasible to search for roots this may be easier said than done. This can be done on fini... | 11 | https://mathoverflow.net/users/3272 | 30134 | 19,654 |
https://mathoverflow.net/questions/30136 | 0 | I was reading a blog post on [a simple derivation of the cross product](http://behindtheguesses.blogspot.com/2009/04/dot-and-cross-products.html). I learned how to determine the area of a [parallelogram enclosed by two vectors $A$ and $B$](http://4.bp.blogspot.com/_D1sP-NndkqU/SfDq13JeaCI/AAAAAAAAHHc/Q_IyEv4AXkY/s1600-... | https://mathoverflow.net/users/7229 | Why does the area function of a parallelogram have a nonintuitive geometric solution? | It is better to see this property if you stay with parallelograms and use the Cavalieri Principle. Here is [page](http://matematica-para-todos.wikispaces.com/Propriedade+de+Determinante+por+Cavalieri) with an animation showing what I mean (clink on the link and accept).
There is also a demonstration of the first form... | 0 | https://mathoverflow.net/users/7231 | 30137 | 19,656 |
https://mathoverflow.net/questions/30144 | 2 | This question is related to [Degree of sum of algebraic numbers](https://mathoverflow.net/questions/26832/degree-of-sum-of-algebraic-numbers). Forgive me if
this is a dumb question, but are there two algebraic numbers $a$ and $b$ of degree $3$ and $6$ respectively, such that the sum $a+b$ has degree $12$ ?
Intuitivel... | https://mathoverflow.net/users/2389 | algebraic numbers of degree 3 and 6, whose sum has degree 12 | The splitting field $K$ of $x^6-2$ has degree 12 over the rationals. Its Galois group is the dihedral group
$D\_{12}$. This group has subgroups $A$ and $B$, where $A$ has order 2, $B$ has order 4, and $B$ does not contain
$A$; the subgroup generated by $A$ and $B$ together is all of $D\_{12}$. Now let $E$ and $F$ be ... | 17 | https://mathoverflow.net/users/3684 | 30145 | 19,661 |
https://mathoverflow.net/questions/30147 | 1 |
>
> **Possible Duplicate:**
>
> [Primes P such that ((P-1)/2)!=1 mod P](https://mathoverflow.net/questions/16141/primes-p-such-that-p-1-21-mod-p)
>
>
>
Motivation comes from comments in [this question](https://mathoverflow.net/questions/30101/composite-pairs-of-the-form-n-1-and-n1), and it is interesting in ... | https://mathoverflow.net/users/2024 | Primes p such that p | ((p-1)/2)! + 1 | Yes, this follows from the analytic class number formula.
See
<http://www.math.niu.edu/~rusin/known-math/97/sign> .
**Added** I have now found a reference. This is a theorem of Mordell:
L. J. Mordell,
The congruence $(p - 1/2)! \equiv \pm 1 (\operatorname{mod} p)$,
*American Mathematical Monthly*, **68** (1961), 1... | 4 | https://mathoverflow.net/users/4213 | 30148 | 19,663 |
https://mathoverflow.net/questions/30143 | 0 | Let $A$ and $B$ be closed sets (subsets of $\mathbb{R}^m$ and $\mathbb{R}^n$, say), and let $f : A \times B \rightarrow \mathbb{R}$ be a continuous function.
Consider the function $g : A \rightarrow B$ defined by
$g(x) = \underset{y \in B}{\operatorname{arg}\max} f(x,y)$
assuming some tie-breaking strategy for $f... | https://mathoverflow.net/users/nan | Is there a name for the "projection" of a function under argmax? | If you do not use any tie breaking strategy, you simply get the *argmax-correspondence*. That's the name I know. If B is compact, so maximizers actually exist, the argmax-correspondence has nonempty and compact values and is [upper hemicontinuous](http://en.wikipedia.org/wiki/Hemicontinuity#Upper_hemicontinuity). This ... | 0 | https://mathoverflow.net/users/35357 | 30155 | 19,666 |
https://mathoverflow.net/questions/30186 | 6 | This question is based on the following phrase:
"In a sense, $\textrm{Spec} \ \mathbf{Z}$ looks topologically like a 3-dimensional sphere viewed as the Hopf fibration over $\mathbf{S}^2$."
See page 88 of *Algebraic Geometry II* by Shafarevich.
I find this remark very interesting but I can't seem to parse it.
I... | https://mathoverflow.net/users/4333 | The ring of integers looks like the 3-dimensional sphere viewed as the Hopf fibration | Various pieces of exposition and references are to be found - [here](http://golem.ph.utexas.edu/category/2007/10/this_weeks_finds_in_mathematic_18.html), [here](http://math.ucr.edu/home/baez/week257.html), [here](http://www.ucl.ac.uk/~ucahmki/baez13.12.pdf), and [here](http://golem.ph.utexas.edu/category/2009/04/aftern... | 5 | https://mathoverflow.net/users/447 | 30187 | 19,688 |
https://mathoverflow.net/questions/30179 | 9 | Let $M$ be a compact space and $f,g:M \to M$ whose non-wandering sets satisfy $\Omega(f)=\Omega(g)=M$. Can we have $\Omega(f \circ g)=M$?
Or more specifically, if $\Omega(f)=M$, can we have $\Omega(f^n)=M$ for any positive integral number $n$?
Any reference would be helpful. Thanks!
| https://mathoverflow.net/users/3926 | Is the composition of non-wandering maps still non-wandering? | To the first question the answer is negative. There are two homeomorphisms of the circle with irrational rotation number such that their composition is Morse-Smale (in fact, you can multiply two $2\times 2$ matrices with complex eigenvalues to get one hyperbolic matrix, the action on the proyective space does the trick... | 10 | https://mathoverflow.net/users/5753 | 30211 | 19,705 |
https://mathoverflow.net/questions/30004 | 2 | On page 43 of the pdf given as reference 2 at <http://en.wikipedia.org/wiki/AKS_primality_test>, the authors mention that this can be done in almost cubic time with Newton's method, although I can't figure out how this would work.
I do know about almost-linear time multiplication.
(this is theoretical enough that I'm... | https://mathoverflow.net/users/nan | Checking if a positive integer is a power other than a first power | I don't know how to do it in cubic time, but I suppose that, to use Newton's Method, you could do the following:
Find floor(log2*n*), and this is the largest "power" that it can be. Then, define:
$f\_k(x) = x^k - n$ where *n* is your number and *k* is the floor value, and iterate Newton's Method until you get a number ... | 0 | https://mathoverflow.net/users/1982 | 30216 | 19,709 |
https://mathoverflow.net/questions/30217 | 1 | This might be a very silly question, but I just wanted to make sure I have all the right steps.
Suppose we have a univariate continuous random variable $X$, with some pdf and cdf ${{f}\_{X}}(x)$ and ${{F}\_{X}}(x)$, respectively. Now look at the transformation $Y = X + k$, with $k\in \mathbb{R}$. Then, the cdf and pd... | https://mathoverflow.net/users/7254 | Does the translation of a random variable preserve its distribution type? | Traditionally, the distributions of r.v.'s $X$ and $Y$ are said to be of the same type if there are constants $a>0$ and $b\in\mathbb{R}$ such that the distribution of $aX+b$ coincides with that of $Y$, see, e.g., p.31 in "A modern approach to probability theory" by Bert Fristedt,Lawrence F. Gray (look for it at google ... | 2 | https://mathoverflow.net/users/2968 | 30228 | 19,714 |
https://mathoverflow.net/questions/30191 | 6 | Does anybody know about software that exactly calculates the tree-width of a given graph and outputs a tree-decomposition? I am only interested in tree-decompositions of reasonbly small graphs, but need the exact solution and a tree-decomposition. Any comments would be great. Thanks!
| https://mathoverflow.net/users/6726 | Software for Tree-Decompositions | For general graphs there are no good algorithms known, as the problem of determining the treewidth of a graph is NP-hard. So if your graphs are not from some special class, and instances are small, then a brute force search over all decompositions of small width is a reasonable approach.
As a previous answer suggeste... | 5 | https://mathoverflow.net/users/7252 | 30230 | 19,715 |
https://mathoverflow.net/questions/3428 | 1 | I am looking for a way to partially "grid" the surface of a sphere to have certain nice properties which will be defined precisely below.
* The areas should be "almost equal".
* It should be possible to calculate in constant time what grid cell any point belongs to (including the boundaries, see below)
* I want to e... | https://mathoverflow.net/users/494 | Grid with nice mathematical properties | Unfortunately, the sphere is not a "developable surface". This fact has annoyed map-makers for more than a millennium.
I find your focus on "cells" fascinating.
Most people seem fixated on trying to get points on the globe to correspond with points on a flat image, and don't seem concerned about dividing it up into a... | 5 | https://mathoverflow.net/users/7234 | 30232 | 19,717 |
https://mathoverflow.net/questions/30248 | 11 | The title basically says it all.
Is there a group with more than one element that is isomorphic to the group of automorphisms of itself?
I'm mainly interested in the case for finite groups,
although the answer for infinite groups would still be somewhat interesting.
| https://mathoverflow.net/users/nan | Is there a non-trivial group G isomorphic to Aut(G)? | The automorphism group of the symmetric group $S\_n$ is (isomorphic to) $S\_n$ when $n$ is different from $2$ or $6$. In fact, if $G$ is a **complete** group you can ascertain that $G \simeq \mathrm{Aut}(G)$. The reverse implication needn't hold, though.
| 29 | https://mathoverflow.net/users/1593 | 30250 | 19,729 |
https://mathoverflow.net/questions/30224 | 1 | I was not sufficiently clear on my last attempt at asking a similar (but not identical) question. Tom Goodwillie mentioned (in the accepted answer) that the question can be reduced to this one and mentioned a fact of which I was unaware, but after trying to prove this for the past day, I have returned to ask for a sket... | https://mathoverflow.net/users/1353 | Proof Sketch: The pullback of the inclusion of the 0th vertex into the standard n-simplex by a right fibration is a deformation retract (450 point bounty if answered by 2am EST) | Harry, let $Y$ be the fiber of $X\to\Delta^n$ over the 0th vertex. The sense in which $Y$ is going to be a deformation retract of $X$ is going to be the following: There is a map $\Delta^1\times X\to X$ such that (1) on $1\times X$ it's the identity and (2) on $\Delta^1\times Y$ it's the constant homotopy (i.e. project... | 4 | https://mathoverflow.net/users/6666 | 30260 | 19,736 |
https://mathoverflow.net/questions/30290 | 2 | Let $A$ be a $C$-algebra, where $C$ is a commutative ring with $1$, and $M$ be a finitely generated left $A$-module.
Question: Is it true that we can always find a positive integer $n$, a $C$-subalgebra $B$ of $M\_n(A)$ and an ideal $J$ of $B$ such that $B/J$ is isomorphic to $End(M)\ ?$ If not, what other condition... | https://mathoverflow.net/users/6941 | Algebra of endomorphisms of f.g. modules as subquotients of matrix algebras | Let $e\_1, \dots, e\_n$ be a basis for $A^{\oplus n}$ and let $m\_1, \dots, m\_n$ be generators for $M$ so that your map $f: A^{\oplus n} \twoheadrightarrow M$ has $f(e\_i) = m\_i$. Now, suppose $\varphi \in \mathrm{End}\_A(M)$.` For each $i$, choose $a\_{i,j} \in A$ such that $$\varphi(m\_i) = \sum\_j a\_{i,j} m\_j.$$... | 2 | https://mathoverflow.net/users/6401 | 30295 | 19,753 |
https://mathoverflow.net/questions/30292 | 16 | One can view a random walk as a discrete process whose continuous
analog is diffusion.
For example, discretizing the heat diffusion equation
(in both time and space) leads to random walks.
Is there a natural continuous analog of discrete self-avoiding walks?
I am particularly interested in self-avoiding polygons,
i.e.,... | https://mathoverflow.net/users/6094 | Random walk is to diffusion as self-avoiding random walk is to ...? | In 2D the scaling limit is believed to be SLE with parameter 8/3. This was conjectured by Lawler, Schramm and Werner and, to the best of my knowledge, still remains open.
| 15 | https://mathoverflow.net/users/2968 | 30297 | 19,755 |
https://mathoverflow.net/questions/30302 | 20 | If an algebraic variety $X$ over a field characteristic p is given by equations $f\_i(x\_1,...,x\_k) = 0$, we can consider the variety $X^{(p)}$ obtained by applying p-th powers to all the coefficients of all $f\_i$'s.
Frobenius morphism, as I understand it, is a morphism $X \to X^{(p)}$,
given on points as raising all... | https://mathoverflow.net/users/2260 | Geometric vs Arithmetic Frobenius | Geometric and arithmetic Frobenius live in a Galois group, they are different from the Frobenius morphism. The Galois group of a finite field of cardinality $q$ has a canonical generator $x \mapsto x^q$; this is the arithmetic Frobenius element. Its inverse, i.e., $x \mapsto x^{1/q}$, is the geometric Frobenius element... | 16 | https://mathoverflow.net/users/1729 | 30309 | 19,758 |
https://mathoverflow.net/questions/30237 | 10 | Somewhere, I don't remember where, I saw a beautiful 3D figure of part a CAT(0) simplicial complex. I am thinking and hoping that this was some finite piece of an affine building of type A2, presumably in characteristic 2. But I'm very frustrated now that I just can't remember exactly what I saw or where I saw it. It w... | https://mathoverflow.net/users/1450 | Looking for figure of part of an A2 affine building | Check out <http://dean.clas.uconn.edu/teitelbaum/colorout.gif>
I made this image long ago for the cover of the AMS Notices.
See <http://www.ams.org/notices/199510/teitelbaum.pdf>
| 6 | https://mathoverflow.net/users/3394 | 30319 | 19,764 |
https://mathoverflow.net/questions/30328 | 11 | Does anyone know where to find an English translation of Riemann's Habilitation Thesis concerning trigonometric series? The German title of the work is "Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe" and the English title is "On the representation of a function as a Trigonometric Series". So... | https://mathoverflow.net/users/5431 | English translation of Riemann's Habilitation Thesis | This is XII in Riemann's Werke (1876; 1892) and has been published in English in
Bernhard Riemann, *Collected papers.*
Translated from the 1892 German edition by Roger Baker, Charles Christenson and Henry Orde. Kendrick Press, Heber City, UT, 2004 ISBN: 0-9740427-2-2; 0-9740427-3-0
[MR2121437 (2005m:01028)](http://w... | 11 | https://mathoverflow.net/users/5740 | 30336 | 19,774 |
https://mathoverflow.net/questions/30329 | 1 | In answer to the question [Demystifying complex numbers](https://mathoverflow.net/questions/30156/demystifying-complex-numbers), Charles Matthews suggests "finding the points at twice the distance from (-1, 0) that they are from (1, 0)." as a motivation for complex numbers.
Suppose you want to find these points in hy... | https://mathoverflow.net/users/3537 | Points at twice the distance from (-1, 0) that they are from (1, 0) in hyperbolic geometry | This is a bit basic for MO, but I'll present my solution.
I don't understand what the OP's notation is so I'll
use my favourite model for hyperbolic space, the Poincaré
upper half plane ('cos I like modular forms).
In the upper half plane model the distance satisfies
$$d(a+bi,c+di)=\cosh^{-1}\frac{(a-c)^2+b^2+d^2}{2b... | 4 | https://mathoverflow.net/users/4213 | 30337 | 19,775 |
https://mathoverflow.net/questions/30334 | 2 | Let define procedure for converting second order theory to first order:
1. Take any second order theory with equality
2. Invent sort Bool' and new fresh constants F' and T', of sort Bool'
3. Create fresh sort CC
4. Replace each proposition P(x) (where x : XX) except equality to P'(x) == T' where P' : XX -> Bool'
5. R... | https://mathoverflow.net/users/7257 | Lowering order of theory | The answer to your question is that it depends on what semantics you want to use for higher-order logic.
* If you use *full higher-order semantics*, then you cannot reduce your theory to a first-order theory. In these semantics, the higher-order quantifiers range over *all* objects of the appropriate type, and so a m... | 4 | https://mathoverflow.net/users/5442 | 30341 | 19,777 |
https://mathoverflow.net/questions/23724 | 8 | This question is the two-dimensional analogue of [Etale coverings of certain open subschemes in Spec O\_K](https://mathoverflow.net/questions/22883/etale-coverings-of-certain-open-subschemes-in-spec-o-k)
There I asked if one could characterize in a way certain covers of $\textrm{Spec} \ O\_K$. As Cam Mcleman answered... | https://mathoverflow.net/users/4333 | Covers of the projective line over Z and arithmetic Grauert-Remmert | Regarding Q3: For any scheme $X$ of finite type over $\mathbb{C}$ the Riemann-Existence Theorem (See SGA1 XII.5) says that the category of finite étale coverings of $X$ is equivalent to the category of finite covering spaces of the associated analytic space $X^{an}$. This implies that the finite quotients of the topolo... | 2 | https://mathoverflow.net/users/259 | 30347 | 19,780 |
https://mathoverflow.net/questions/30243 | 4 | Let $F$ be a function field of "transcendental degree one" over its full constant field $K$. Let $x \in F \backslash K$. We know the divisor of $(x) = (x) - (1/x)$ in $K(x)$. Could you please give me an algorithm to compute the places over two above places in $F$ and the ramification degrees.
If this setting is too a... | https://mathoverflow.net/users/6776 | Computing places over x in F/K(x) | For $x\in F\setminus K$, the degree of $x$ is the degree of the field extension $F/K(x)$. For example, in the $F$ corresponding to your curve, the degree of $x$ is $2$, since the extension $F/K(x)$ is the simple extension corresponding to $y^2 + xy + x^3 + 1 = 0$. Similarly, the degree of $y$ is $3$, since $F/K(y)$ is ... | 2 | https://mathoverflow.net/users/2490 | 30348 | 19,781 |
https://mathoverflow.net/questions/30265 | 10 | Let $G$ be a connected reductive group over a (perfect, why not) field $F$. Let $m$, $pr\_1$, $pr\_2$ denote the multiplication, first, and second projection maps from $G \times G$ to $G$.
Then I'm pretty sure that I can prove the following fact: if $L$ is a line bundle on $G$, then $m^\ast(L)$ is isomorphic to $pr\_... | https://mathoverflow.net/users/3545 | Reference for Pic(G) and central extensions. | I can give a reference only for the second part of the question,
namely, about central extensions.
It was answered by Colliot-Thélène in 2008, not 30 years ago!
Colliot-Thélène's paper *Résolutions flasques des groupes linéaires connexes*, J. für die reine und angewandte Mathematik (Crelle) 618 (2008), 77--133,
contai... | 3 | https://mathoverflow.net/users/4149 | 30349 | 19,782 |
https://mathoverflow.net/questions/30020 | 3 | 1-In his article written in German "Über unerreichbare Kardinalzahlen" (On inaccessible cardinals), inside Fund. Math. 1938 (pages 68-89), Alfred Tarski states his axioms A and A' as follows.
Axiom A: "For every set x, there exists a set y satisfying the four following conditions:
* A1: x is a member element of y;
* ... | https://mathoverflow.net/users/30395 | About Tarski's axioms a and A' and around (1) | First, I note that you appear to be missing a *not* in A4,
and you should say that "if $z$ and $y$ are *not*
equipollant", for otherwise we could take $z=y$ and thereby
deduce $y\in y$, contrary to the Foundation axiom.
With this correction, both your axioms are equivalent in
ZFC to the assertion that there is a prop... | 2 | https://mathoverflow.net/users/1946 | 30356 | 19,785 |
https://mathoverflow.net/questions/30358 | 5 | Related to [A000679](http://www.oeis.org/A000679) (Number of groups of order $2^n$), how many non-Abelian groups of order $2^n$ are there?
| https://mathoverflow.net/users/2391 | Number of non-Abelian groups of order $2^n$ | It is true that there is no known formula for the number of isomorphism classes of groups of order $n$, but there is a very nice asymptotic formula for $p$-groups.
In particular, the number of isomorphism classes of groups of order $p^n$ grows as $p^{\frac{2}{27}n^3 + O(n^{8/3})}$. This function grows very rapidly, a... | 13 | https://mathoverflow.net/users/4558 | 30364 | 19,789 |
https://mathoverflow.net/questions/30345 | 16 | On page 283 of Max Karoubi’s book, “K-theory,” he states that for any compact Hausdorff space $X$, the Chern character determines an isomorphism from the group $K^0(X) \otimes Q$ to $H^{even}(X; Q)$, the direct sum of the even-dimensional Cech cohomology groups of X with rational coefficients. In particular, this theor... | https://mathoverflow.net/users/7262 | What is an example of a compact smooth manifold whose K-theory and Cech cohomology are not isomorphic? | For $X=\mathbb RP^4$ the groups $K^0(X)$ and $H^{even}(X)$ respectively are $\mathbb Z\oplus \mathbb Z/4$ and $\mathbb Z\oplus \mathbb Z/2\oplus \mathbb Z/2$. More generally, $K^0(\mathbb RP^{2k})\cong \mathbb Z\oplus \mathbb Z/2^{k}$. These computations of real and complex $K$-groups of real and complex projective spa... | 22 | https://mathoverflow.net/users/6666 | 30365 | 19,790 |
https://mathoverflow.net/questions/30353 | 15 | While digging through old piles of notes and jottings, I came across a question I'd looked at several years ago. While I was able to get partial answers, it seemed even then that the answer should be known and in the literature somewhere, but I never knew where to start looking. So I thought I'd ask here on MO if anyon... | https://mathoverflow.net/users/763 | Approximating operators on Banach spaces by bounded operators on a proper dense subspace | Q1: Yes. You ask ``If $X$ is a countable dimensional dense subspace of the Banach space $Y$, are the operators on $Y$ which leave $X$ invariant dense in the operators on $Y$?" Use Mackey's argument for producing quasi-complements (just a biorthogonalization procedure, going back and forth between a space and its dual) ... | 11 | https://mathoverflow.net/users/2554 | 30366 | 19,791 |
https://mathoverflow.net/questions/30066 | 6 | M a finitely generated module over a commutative ring A. I can't think of an example of two maximal linearly independent subsets of M having different cardinality. I know that they all have the same cardinality if A is integral domain. Any suggestions are welcome!
| https://mathoverflow.net/users/5292 | Cardinality of maximal linearly independent subset | I found an old paper by Lazarus (Les familles libres maximales d'un module ont-elles le meme cardinal?, Pub. Sem. Math. Rennes 4 (1973), 1-12) which contains the the following result: Let A be a commutative ring with unit and M an A-module. In the following situations, maximal linearly independent subsets of M have the... | 10 | https://mathoverflow.net/users/5292 | 30369 | 19,794 |
https://mathoverflow.net/questions/30377 | 2 | Let $G$ be a finite Abelian group with endomorphism ring $End(G)$. I am interested in the probability $P(\phi(g\_1) = g\_2)$ for fixed $g\_1,g\_2 \in G$ and a uniformly chosen endomorphism $\phi(\cdot)$ from $End(G)$. Essentially, I want to understand where the set of endomorphisms will take each element $g \in G$. I r... | https://mathoverflow.net/users/2878 | Image of a fixed element under a random endomorphism in an Abelian group | For a group $G=\mathbb{Z}\_{p^r}^k$ this is quite straightforward.
Let $g$ have order $p^r$ in $G$ (if not then we are effectively working
in $\mathbb{Z}\_{p^s}$ where $s < k$). Applying an automorphism of $G$
we can assume that $g=(1,0,\ldots,0)$. Endomorphisms of $G$ correspond
to matrices over $\mathbb{Z}\_{p^r}$. T... | 2 | https://mathoverflow.net/users/4213 | 30378 | 19,800 |
https://mathoverflow.net/questions/30288 | 20 | I came across the concept of a hyperring in two recent papers by Connes and Consani ([From monoids to hyperstructures: in search of an absolute arithmetic](http://arxiv.org/abs/1006.4810) and [The hyperring of adèle classes](http://arxiv.org/abs/1001.4260)). It's a weakening of the ring concept, but where the addition ... | https://mathoverflow.net/users/447 | What are hypergroups and hyperrings good for? | While I don't know much about hyperstructures other than hypergroups, I know it is hard to study the history behind them because of the non-consistent terminology attributed to these objects by different authors in different periods. I will say something about hypergroups and hopefully some specialist can come and give... | 16 | https://mathoverflow.net/users/2384 | 30388 | 19,804 |
https://mathoverflow.net/questions/30387 | 3 | An old question that occurred to me again recently: are there any explicit formula known for sequences of irreducible polynomials $g\_{p^n}(X)$ in $Z/pZ[X]$ such that for the finite field with $p^n$ elements
$F\_{p^n} = (Z/pZ)[X] / (g\_{p^n}).$
Such polynomials exist, as anyone who's studied algebra knows, but I've... | https://mathoverflow.net/users/5621 | Explicit representations of finite fields | There is no known simple explicit formula for an irreducible polynomial
of given degree $n$ over $\mathbb{F}\_p$. However there has been a lot of work
on explicit irreducible polynomials for certain families of $n$, notably
$n$ of the form $rq^s$ where $r$ is fixed and $q$ is a fixed prime. For one
modest contribution ... | 3 | https://mathoverflow.net/users/4213 | 30389 | 19,805 |
https://mathoverflow.net/questions/30381 | 13 | What is authoritative canonical formal definition of function?
For example,
According to [Wolfram MathWorld](http://mathworld.wolfram.com/Function.html),
$$isafun\_1(f)\;\leftrightarrow\;
\forall a\in f\;(\exists x\exists y \;\langle x,y\rangle = a)
\; \wedge \;
\forall x\forall y\_1\forall y\_2\;((\langle x,y\_1\r... | https://mathoverflow.net/users/7257 | Definition of Function | The fact is that different subject areas of mathematics use different definitions for this basic concept. The Bourbaki definition is quite common, particularly in many of the areas well-represented here on MO, but other areas use the ordered-pair definition.
For example, if you open any set-theory text, you will fin... | 29 | https://mathoverflow.net/users/1946 | 30397 | 19,810 |
https://mathoverflow.net/questions/30307 | 24 | I heard or have read the following nice explanation for the origin of the convention that one uses (almost) always $x,y,z$ for variables. (This question was motivated by question
[Origin of symbol \*l\* for a prime different from a fixed prime?](https://mathoverflow.net/questions/30081/origin-of-symbol-l-for-a-prime-di... | https://mathoverflow.net/users/4556 | Explanation why $x,y,z$ are always variables | You'll find details on this point (and precise references) in Cajori's *History of mathematical notations*, ¶340. He credits Descartes in his *La Géometrie* for the introduction of $x$, $y$ and $z$ (and more generally, usefully and interestingly, for the use of the first letters of the alphabet for known quantities and... | 28 | https://mathoverflow.net/users/1409 | 30414 | 19,821 |
https://mathoverflow.net/questions/28519 | 26 | Hi,
I'm starting to prepare a graduate topics course on Complex and Kahler manifolds for January 2011. I want to use this course as an excuse to teach the students some geometric analysis. In particular, I want to concentrate on the Hodge theorem, the Newlander-Nirenberg theorem, and the Calabi-Yau theorem.
I have ... | https://mathoverflow.net/users/6871 | References for "modern" proof of Newlander-Nirenberg Theorem | There is a proof due to Malgrange which can be found in Nirenberg's, Lectures on Linear Partial Differential Equations. I am not sure that one can call the proof modern, but it is the simplest proof that I know.
| 13 | https://mathoverflow.net/users/7300 | 30420 | 19,823 |
https://mathoverflow.net/questions/30392 | 17 | A bit unsure if my use/mention of proprietary software might be inappropriate or even frowned upon here. If this is the case, or if this kind of experimental question is not welcome, please let me know and I'll remove it quickly.
I was experimenting on the distribution of eigenvalues of random matrices via the follow... | https://mathoverflow.net/users/7294 | An experiment on random matrices | The distribution of the bulk of the spectrum is an example of the [circular law](http://mathworld.wolfram.com/GirkosCircularLaw.html). For the model you selected (where each entry is uniformly chosen at random from an interval), the law was first proven [by Bai](http://www.ams.org/mathscinet-getitem?mr=1428519) (at lea... | 21 | https://mathoverflow.net/users/766 | 30433 | 19,831 |
https://mathoverflow.net/questions/30436 | 4 | What kind of 3-manifolds can arise as hypersurfaces $\{ f(x,y,z,w) = 0\} \subset \mathbb{R}^4$? Can they have nontrivial H1 or H2?
| https://mathoverflow.net/users/1358 | What kind of 3-manifolds arise has hypersurfaces in R^4? | A simple construction that bears on the narrow version of John's question: If $M$ is a closed $n$-manifold that embeds in $\mathbb{R}^{n+1}$ (which can only happen if $M$ is orientable), then $M \times S^k$ embeds in $\mathbb{R}^{n+1+k}$. Thicken $M$ in $\mathbb{R}^{n+1}$, then cross with $I^k$ in the new dimensions, a... | 5 | https://mathoverflow.net/users/1450 | 30444 | 19,838 |
https://mathoverflow.net/questions/30425 | 12 |
>
> Let $n>1$. Which smooth manifolds admit a diffeomorphism $f$ of order $n$?
>
>
>
For a closed orientable surface $S\_g$ of genus $g$ and $n=2$ the answer is in the affirmative, since $S\_g$ can be embedded in $\mathbb{R}^3$ in a symmetric way. A similar argument gives a positive answer for $n=3$: $S\_g$ can ... | https://mathoverflow.net/users/4698 | Which manifolds admit a diffeomorphism of order $n$? | The Nielsen Realisation Problem asks when a (finite) subgroup of the mapping class group (the group of isotopy classes of diffeomorphisms) of a surface can be realised as a group of diffeomorphisms. Kerckhoff proved in the 80s that **every** finite subgroup of the mapping class group can be realised. (For infinite subg... | 12 | https://mathoverflow.net/users/4910 | 30445 | 19,839 |
https://mathoverflow.net/questions/30434 | 3 | A friend is looking for a clean proof of the following inequality of Bernstein: If $f: R \to R$ is a bounded function whose Fourier transform has compact support, then
$ \|f'\|\_{\infty} \le C \| f \|\_{\infty} $
where $C$ only depends on the support of the Fourier transform. Any reference would be very much appreciat... | https://mathoverflow.net/users/3635 | A proof of the Bernstein inequality | The Fourier transform of $f'(x)$ is $i\xi\hat{f}(\xi)$, which has the same support as $\hat{f}(\xi)$. So we can write $i\xi\hat{f}(\xi)$ = $i\xi\hat{f}(\xi)\phi(\xi)$, where $\phi(\xi)$ is a smooth bump function depending on the support of $\hat{f}$, that is equal to one on the support of $\hat{f}$. Taking inverse Four... | 12 | https://mathoverflow.net/users/2944 | 30447 | 19,841 |
https://mathoverflow.net/questions/30357 | 9 | Does anyone know of any lower bounds on the number of nonzero digits that appear in powers of 2 when written to base 3? (Other than the easy "If it's more than 8 it has to have at least 3.") I know there's been some stuff done with powers of 2 written to base 3, but I can't seem to find anything that quite answers this... | https://mathoverflow.net/users/5583 | Lower bound on # of nonzero digits in ternary expansions of powers of 2? | A nontrivial lower bound can be found in a paper of Cam Stewart (see <http://www.math.uwaterloo.ca/PM_Dept/Homepages/Stewart/Jour_Books/J-reine-ange-Math-1980.pdf>). He proves, more generally, for fixed bases a and b for which $\log a/\log b$ is irrational, that the sum of the number of nonzero digits in the base a and... | 13 | https://mathoverflow.net/users/7302 | 30449 | 19,843 |
https://mathoverflow.net/questions/30453 | 17 | Possibly the correct answer to this question is simply a pointer towards some recent literature on Tannaka-Krein-type theorems. The best article I know on the subject is the excellent
* André Joyal and Ross Street, An introduction to Tannaka duality and quantum groups, *Category Theory, Lecture Notes in Math*, 1991 v... | https://mathoverflow.net/users/78 | Does the Tannaka-Krein theorem come from an equivalence of 2-categories? | Your question is dealt with (in a slightly more general setting) in section 11 of Daniel Schäppi's paper, [Tannaka duality for comonoids in cosmoi](https://arxiv.org/abs/0911.0977). Specializing to your setting, he shows that there is a biadjunction (a weak 2-categorical form of adjunction) between the 2-category of $k... | 5 | https://mathoverflow.net/users/396 | 30456 | 19,845 |
https://mathoverflow.net/questions/30457 | 10 | Does every Banach space admit an equivalent strictly convex norm (i.e. such a norm, that a unit sphere does not contain segments)?
| https://mathoverflow.net/users/4312 | Strictly convex equivalent norm |
>
> Every separable Banach space has an equivalent norm which is both strictly convex and smooth. For certain nonseparable spaces, in particular, $\ell\_{\infty}(\Gamma)$ with $\Gamma$ uncountable, there may be no equivalent strictly convex or smooth norm.
>
>
>
[Link.](http://books.google.co.uk/books?id=1A7ppE... | 11 | https://mathoverflow.net/users/5371 | 30458 | 19,846 |
https://mathoverflow.net/questions/30455 | 9 | A definition of wedge sum can be found here:
<http://en.wikipedia.org/wiki/Wedge_sum>
My professor has claimed that wedge sums of path connected spaces X and Y are well-defined up to homotopy equivalence, independently of choice of base points x0 and y0. Base point here means the points that are identified under th... | https://mathoverflow.net/users/7305 | The Wedge Sum of path connected topological spaces | A counterexample is shown on the cover of the paperback edition of the classic textbook Homology Theory by Hilton and Wylie. This can be viewed on the amazon webpage for the book. The example consists of the wedge of two copies of a cone, the cone on the sequence 1/2, 1/3, 1/4, ... together with its limit point 0. With... | 19 | https://mathoverflow.net/users/23571 | 30465 | 19,850 |
https://mathoverflow.net/questions/30474 | 0 | I am looking for reference giving the original definition of prime cycles of coherent sheaves on noetherian scheme. Was it in EGA? I googled, but could not find proper reference.
Thanks in advance
| https://mathoverflow.net/users/1851 | What is Grothendieck associated points(Prime cycles) of coherent sheaves on noetherian scheme? | EGA IV$\_4$, 3.1.1 for any quasi-coherent sheaf on any (pre)scheme: the commutative algebra definition on stalks. So one needs 3.1.2 and 3.1.3 there to get useful alternative formulations of this definition in the locally noetherian case. You can find it by looking for "Point associe" or "Cycle premier associe a un Mod... | 4 | https://mathoverflow.net/users/6773 | 30475 | 19,854 |
https://mathoverflow.net/questions/30480 | 28 | This question is asked from a point of complete ignorance of physics and the standard model.
Every so often I hear that particles correspond to representations of certain Lie groups. For a person completely ignorant of anything physics, this seems very odd! How did this come about? Is there a "reason" for thinking th... | https://mathoverflow.net/users/5309 | Particle Physics and Representations of Groups | The "chronology" isn't clear to me, and having looked through the literature it seems much more convoluted than it should be. Although it seems like this is basically how things were done since the beginning of quantum mechanics (at least, by the big-names) in some form or another, and was 'partly' formalized in the '3... | 11 | https://mathoverflow.net/users/3329 | 30485 | 19,860 |
https://mathoverflow.net/questions/30497 | 7 | What is the étale fundamental group of projective space over an algebraically closed field?
In char = 0 it is trivial (Lefschetz principle), as well as in dimension 1 (Riemann-Hurwitz).
| https://mathoverflow.net/users/6960 | étale fundamental group of projective space | It is a birational invariant (for smooth proper connected schemes over a field, ultimately due to Zariski-Nagata purity of the branch locus), and its formation is compatible with products (for proper connected schemes over an algebraically closed field), so we can replace projective $n$-space with the $n$-fold product ... | 11 | https://mathoverflow.net/users/6773 | 30499 | 19,867 |
https://mathoverflow.net/questions/30494 | 6 | Suppose an operator $T$ is bounded on $L^2$ and also bounded from $L^{1}$ to $L^{1}$-weak. Then by Marcinkewicz interpolation one gets that $T$ is bounded on every $L^{p}$ for p between 1 and 2. Precise versions of the theorem (see the book of L.Grafakos) give an estimate of the norm of $T$ on $L^{p}$, and of course th... | https://mathoverflow.net/users/7294 | A puzzling question on real interpolation | (Note: I don't have my copy of B&L handy, so I'm sort of doing this from memory.)
The problem is that $(L\_{p\_0.q\_0},L\_{p\_1,q\_1})\_{\theta,q} = L\_{p\_\theta,q}$ *under equivalent norm*. There's no saying what the constant of equivalency are. Now, if you look at $T: \{X\_1,X\_2\} \to \{Y\_1,Y\_2\}$ with norm $\{... | 8 | https://mathoverflow.net/users/3948 | 30503 | 19,869 |
https://mathoverflow.net/questions/30491 | 3 | Is there any remotely efficient way to determine whether a graph can be disconnected by the removal of fewer than k edges, or even one that has a lower asymptotic complexity than just trying each set of k-1 edges?
If it helps, you can assume the graph is k-regular that k is much smaller than the number of vertices.
... | https://mathoverflow.net/users/nan | determining k-edge-connectivity of a graph | Try:
<http://portal.acm.org/citation.cfm?id=122416>
and the references there.
| 2 | https://mathoverflow.net/users/1618 | 30507 | 19,870 |
https://mathoverflow.net/questions/30509 | 1 | In Munkres (Chapter 3, Section 23, p. 148), Munkres shows that if a subspace $Y$ of a space $X$ is not connected, then there are two disjoint open subsets $A,B$ such that the union of $A$ and $B$ contains $Y$. What doesn't make sense is that a separation of $Y$ only requires two open subsets of $Y$ which are disjoint, ... | https://mathoverflow.net/users/1355 | Definition of Connected Subspace | Per your comment, I think you misunderstood what Munkres is trying to say.
>
> If Y is subspace of X, a separation of Y is a pair of disjoint nonempty sets A and B whose union is Y, neither of which contains a limit point of the other. The space Y is connected if there exists no separation of Y.
>
>
>
I read ... | 3 | https://mathoverflow.net/users/3948 | 30510 | 19,872 |
https://mathoverflow.net/questions/30487 | 11 | My advisor mentioned to me that he talked to Witten last summer on representation theory, and Witten told him that unitary representations of Kac-Moody algebra are important to working physicists. But he did not explain in details to me.
My question is Why?
Second question: I know there are a lot of people devoti... | https://mathoverflow.net/users/1851 | Why do Physicists need unitary representation of Kac-Moody algebra? | As others have mentioned, the reasons lie indeed in two-dimensional conformal field theory and in string theory.
The propagation of string on a compact Lie group $G$ is described by the Wess-Zumino-Witten model, whose dynamical variables are maps $g:\Sigma \to G$ from a riemann surface $\Sigma$ to $G$. The quantisati... | 9 | https://mathoverflow.net/users/394 | 30515 | 19,875 |
https://mathoverflow.net/questions/20946 | 35 | Given a polynomial equation $x^n+a\_{n-1}x^{n-1}+\cdots+a\_1x+a\_0=0$, where $n$ is even and all the coefficients $a\_i$ are real, what is the best way to determine whether it has a real root or not?
I know [Sturm's theorem](http://en.wikipedia.org/wiki/Sturm%27s_theorem), but I am wondering if it's possible to deter... | https://mathoverflow.net/users/3350 | Criteria to determine whether a real-coefficient polynomial has real root? | There is indeed an easy way to check if a univariate poly with real coefficients has a real root, without computing the roots.
Note that the answer for odd degree polynomials is always yes. For an even degree polynomial $p(x)$ do the following:
1. Compute the Hermite form of the polynomial. This is a symmetric matr... | 49 | https://mathoverflow.net/users/50854 | 30533 | 19,888 |
https://mathoverflow.net/questions/30538 | 1 | I want to generate a code on n bits for k different inputs that I want to classify. The main requirement of this code is the error-correcting criteria: that the minimum pairwise distance between any two encodings of different inputs is maximized. I don't need it to be exact - approximate will do, and ease of use and sp... | https://mathoverflow.net/users/942 | Algorithm for generating a size k error-correcting code on n bits | My answer that I gave on StackOverflow:
The problem of finding the exact best error-correcting code for given parameters is very hard, even approximately best codes are hard. On top of that, some codes don't have any decent decoding algorithms, while for others the decoding problem is quite tricky.
However, you're ... | 4 | https://mathoverflow.net/users/1450 | 30541 | 19,890 |
https://mathoverflow.net/questions/30544 | 2 | Lusztig's theory of character sheaves gives a geometric
way to obtain character tables of finite groups of Lie type (coming
from reductive groups). I am interested to know if there is a similar
theory for other kinds of algebraic groups. Thanks so much.
| https://mathoverflow.net/users/6277 | Is there a version of character sheaves for non reductive algebraic groups? | Boyarchenko and Drinfeld have developed a theory of character sheaves for unipotent groups. See [*A motivated introduction to character sheaves and the orbit method for unipotent groups in positive characteristic*](https://arxiv.org/abs/math/0609769) for an introduction (as well as subsequent papers by Boyarchenko and ... | 5 | https://mathoverflow.net/users/396 | 30547 | 19,892 |
https://mathoverflow.net/questions/30548 | 3 | I haven't seen this mentioned in Jech or Kunen, sorry if these are basic questions but I am not understanding these points.
If $\kappa$ is regular uncountable then in ZFC every stationary subset of $\kappa$ is the disjoint union of $\kappa$ stationary sets. So this holds for every successor $\gamma^+$. we have strict... | https://mathoverflow.net/users/3859 | Counting stationary and c.u.b sets | For every infinite cardinal $\kappa$, the number of club subsets of $\kappa$ is $2^\kappa$, fully as large as it could possibly be. Since every club set is stationary, this means also there are fully $2^\kappa$ many stationary sets also.
To see that there are this many club sets, observe that there are $\kappa$ many... | 5 | https://mathoverflow.net/users/1946 | 30550 | 19,893 |
https://mathoverflow.net/questions/30542 | 5 | As a complex affine variety or projective variety, is it possible it is a manifold with boundary?
| https://mathoverflow.net/users/2391 | Is it possible a variety be a manifold with boundary | No, this is not possible (unless you allow the boundary to be empty). If $X$ is a complex algebraic variety (affine or projective, this doesn't matter), there are two possibilities. If $X$ is smooth (as an algebraic variety), then $X$ is a smooth manifold with empty boundary. Otherwise let $Y$ be the singular locus of ... | 7 | https://mathoverflow.net/users/4384 | 30553 | 19,895 |
https://mathoverflow.net/questions/30549 | 4 | Is Deligne-Mumford space could also be defined in the complex geometry context? I check wiki, it says we can similarly define Riemann surface with nodes and stability condition, I am wondering if there is any reference providing more details about this aspect. Thanks!
| https://mathoverflow.net/users/2391 | Deligne-Mumford space defined in complex geometry category | I'm not sure where to point you for full details of this, but quite a few details are in some old research announcements of Bers. See his papers
MR0361051 (50 #13497)
Bers, Lipman
Spaces of degenerating Riemann surfaces. Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973)... | 5 | https://mathoverflow.net/users/317 | 30554 | 19,896 |
https://mathoverflow.net/questions/30555 | 7 | Hello all, I would appreciate comments on the following question:
A main theorem of symmetric functions might be formulated: Let k be a field of char. 0. Then $k[x\_1,...,x\_n]^{S\_n} = k[s\_1,...,s\_n]$, i.e. symmetric polynomials can be written as polynomials in the elementary symmetric polynomials. Moreover, $s\_1... | https://mathoverflow.net/users/2095 | Symmetric polynomials theorem | I hope the following works. Let $A=k[x\_1,\ldots,x\_n]^{S\_n}$, and let $B=k[s\_1,\ldots,s\_n]$. The polynomial algebra $k[x\_1,\ldots,x\_n]$ is an integral extension of $B$, and hence, a fortiori, $A$ is integral over $B$. I think your argument proves that $A$ and $B$ have the same fraction field. However, since $B$ i... | 6 | https://mathoverflow.net/users/4384 | 30558 | 19,898 |
https://mathoverflow.net/questions/30557 | 7 | Suppose $U'\cup U''=X$ is an open cover $U$ of a topological space $X$ and $F$ is a sheaf on $X$ with values in abelian groups. There is a special instance of the Grothendieck spectral sequence relating Cech to sheaf cohomology:
$$E\_2^{p,q}=\tilde{H}^p(U,H^q(-,F))\Rightarrow H^{p+q}(X,F)$$
I would like to see, how... | https://mathoverflow.net/users/7316 | Grothendieck spectral sequence and Mayer-Vietoris sequence | Recall that the Cech-to-derived functor spectral sequence is constructed as follows. We start with a sheaf $F$ and an open cover $\mathfrak{U}$. Then we can write the Cech resolution of the sheaf; take an injective (or Godement or...) resolution thereof to get a double complex. Let $C^{\ast,\ast}$ be the resulting comp... | 10 | https://mathoverflow.net/users/2349 | 30564 | 19,900 |
https://mathoverflow.net/questions/28829 | 4 | I sometimes read $\int\_{E(\mathbf{R})} \frac{dx}{2y + a\_1x + a\_3}$ and sometimes $\int\_{E(\mathbf{R})} |\frac{dx}{2y + a\_1x + a\_3}|$. Furthermore, one has to choose an orientation on $E(\mathbf{R})$.
So what's the correct definition for the constant appearing in the BSD conjecture?
| https://mathoverflow.net/users/nan | How is the period of an elliptic curve defined exactly? | The comments above give already the answer, but for the sake of completeness let us be a bit more precise.
Let $E/\mathbb{Q}$ be an elliptic curve. Let $\Omega^{+}$ be the smallest positive element in the period lattice $\Lambda$. Then the conjecture of Birch and Swinnerton-Dyer predicts that
$$
\frac{L^{\*}(E,1)}{... | 5 | https://mathoverflow.net/users/5015 | 30568 | 19,903 |
https://mathoverflow.net/questions/30572 | 9 | Sorry if this is an easy one, I'm a little rusty on my group theory. My first guess was that it's simply the inverse limit of the Aut($\mathbb{Z}/p^i\mathbb{Z})$, with the map when $i\leq j$ given by taking $\sigma\in$ Aut$(\mathbb{Z}/p^j\mathbb{Z})$ to the map $\tilde{\sigma}:\mathbb{Z}/p^i\mathbb{Z}\rightarrow\mathbb... | https://mathoverflow.net/users/1916 | What is the automorphism group of the additive group of the p-adic integers? | First the $p$-adic integers are finitely generated (actually cyclic) pro-$p$ group therefore from a result of Serre all automorphisms are continuous. Now as it cyclic it is enough to see what happens to $1$. It has to go to another generator, i.e. any element of the form $a\_0+a\_1p+a\_2p^2+\cdots$, where $0 \leq a\_i ... | 11 | https://mathoverflow.net/users/5034 | 30574 | 19,908 |
https://mathoverflow.net/questions/30559 | -1 | Let (δ;U) is a proximity space.
I will call a set A connected iff for every partition {X,Y} of the set A holds X δ Y.
Is the following true? (I need a proof or a counter-example.)
**Conjecture** If S is a collection of connected sets and ∩S≠∅ then ∪S is connected.
Note that instead of proximity we may consider ... | https://mathoverflow.net/users/4086 | Union of proximally connected sets | Let $\{X,Y\}$ be a partition of the union of the family $S$, and let p be a point in the intersection of $S$ (which you've assumed is nonempty). Without loss of generality, p is in $X$. But, since a partition can't contain the empty set, $Y$ contains some point q from some set $A$ in the family $S$. Then $\{A\cap X, A\... | 3 | https://mathoverflow.net/users/6794 | 30576 | 19,909 |
https://mathoverflow.net/questions/30581 | 11 | Fix a prime number $p$. Suppose that I have a valuation $v\_p: \mathbb{Q} \to \mathbb{Q}$ on the rationals $\mathbb{Q}$. That is, $v\_p( p^n(\frac{a}{b})) = p^{-n}$ where each of $a,b$ is not divisible by $p$.
How can I extend $v\_p$ to $v$ on the reals $\mathbb{R}$ such that $v|\_\mathbb{Q} = v\_p$? I am looking fo... | https://mathoverflow.net/users/7328 | Extension of valuation | As you point out, it follows on general grounds that there is an extension of $v\_p$ to a valuation on ${\mathbb R}$ (in fact, there are uncountably many such extensions), but it is impossible to give an "explicit" description. Indeed, not only will any such extension by discontinuous with respect to the usual Euclidea... | 15 | https://mathoverflow.net/users/5147 | 30582 | 19,914 |
https://mathoverflow.net/questions/30599 | 3 | Let A be an artin ring which is also a finitely generated algebra over Z.
Show that $|A|<\infty$.
If A would have been a field then I know how to prove it. I know that A is a product of local rings, so I could restrict the question to Local artin rings that are finitely generated algebra over Z. But how does this h... | https://mathoverflow.net/users/7332 | Finitely-generated algebra over Z | Take $A$ local (you already reduced to it), with $m$ the max. ideal. I claim that $A/m$ is a finite field. Suppose first that it has char. 0. Then we get injections $\mathbb Z \to \mathbb Q \to A/m$. By Zariski's lemma, $\mathbb Q \to A/m$ is finite, since it is of finite type.
Now (unfortunately I don't have it on me... | 6 | https://mathoverflow.net/users/1729 | 30603 | 19,924 |
https://mathoverflow.net/questions/30598 | 1 | I recently learned about Dirichlet problems and was wondering if there were similar solutions in the case where only few temperature points are known instead of a continuous temperature boundary.
For instance, say the temperature is known at the three points of an equilateral triangle, and is assumed to be at a stead... | https://mathoverflow.net/users/7168 | Point boundary problems | I think you may have gotten things backwards.
The point of differential equations is to describe macroscopic (global) phenomena via microscopic (local) physical laws, as differential operators are strictly local objects. Solving a differential equation one often finds families of solutions, which can live in various... | 3 | https://mathoverflow.net/users/3948 | 30605 | 19,925 |
https://mathoverflow.net/questions/30529 | 3 | What's the name for a digraph such that for each pair of vertices $u,v$, there is either a path from $u$ to $v$ or a path from $v$ to $u$? I'd call it just connected, since this is an intermediate property between weak and strong connectivity, and is in fact equivalent to the existence of a path containing all vertices... | https://mathoverflow.net/users/7320 | Digraph intermediate connectivity | Just `connected' is fine. For example, [Wikipedia](http://en.wikipedia.org/wiki/Connectivity_%28graph_theory%29) and [Tutte](http://books.google.com/books?id=uTGhooU37h4C&lpg=PP1&dq=graph%2520theory&pg=PA132#v=onepage&q&f=false) agree. However, since "the number of systems of terminology presently used in graph theory ... | 5 | https://mathoverflow.net/users/840 | 30615 | 19,931 |
https://mathoverflow.net/questions/30597 | 2 | I'm working on (yet) an(other) exercise from Mosher & Tangora's "Cohomology Operations and Applications to Homotopy Theory". This one is about the homotopy exact couple, which is defined for a complex $K$ by $D\_{p,q}=\pi\_{p+q}(K^p)$ and $E\_{p,q}=\pi\_{p+q}(K^p,K^{p-1})$. So that we have relative Hurewicz, we assume ... | https://mathoverflow.net/users/303 | How can I prove that the derived couple of the homotopy exact couple is an invariant? | Let me start by making a definition: an $n$-skeleton of a space $X$ is an
$n$-equivalence
$X\_n \to X$, where $X\_n$ is an $n$-dimensional (at most) CW complex
($X$ itself need not be a CW complex). Obviously, $n$-skeleta are not unique,
but any two $n$-skeleta for the same space factor through one another: there ar... | 3 | https://mathoverflow.net/users/3634 | 30617 | 19,933 |
https://mathoverflow.net/questions/20879 | 4 | A morphism of varieties over $\mathbb{C}$, $f:V\to W$ is proper if it is universally closed and separated. One way to check properness is the [valuative criterion](http://books.google.com/books?id=3rtX9t-nnvwC&lpg=PP1&dq=hartshorne&pg=PA101#v=onepage&q&f=false).
What other methods do we have for determining if a morp... | https://mathoverflow.net/users/622 | When is a morphism proper? | Assume $V$ and $W$ are quasiprojective. Let $i:V\to X$ be a locally closed embedding with $X$ projective (for instance $X$ could be $P^n$). Consider the induced map $g:V\to X\times W$; this is also a locally closed embedding. Then $f$ is proper iff $g$ is a closed embedding, or equivalently if $g(V)$ is closed.
As fo... | 9 | https://mathoverflow.net/users/4164 | 30627 | 19,941 |
https://mathoverflow.net/questions/30629 | 13 | I believe that I once saw a statement that every compact, smooth Calabi-Yau manifold in dimension at least 3 is algebraic, but I can remember neither the reference nor the proof (which would have been quite short) and I might just be confusing this with something else. Is it true?
| https://mathoverflow.net/users/7343 | Are Calabi-Yau manifolds in dimension >= 3 algebraic? | It depends a little bit on your definition of CY. If you're using a good one, it will imply that the Hodge numbers $h^{0,p} = 0$ for $p \neq 0,d$ (see, for example, Prop. 5.3 of Joyce's <http://arxiv.org/abs/math/0108088>). This implies that $H^2(X) \cong H^{1,1}(X)$. Since the Kaehler cone is an open set in $H^{1,1}(X... | 15 | https://mathoverflow.net/users/947 | 30634 | 19,944 |
https://mathoverflow.net/questions/30633 | 15 | Background
----------
An ordinal $\alpha$ is called a *recursive ordinal* if there is a recursive well-order $R$ on $\mathbb{N}$ such that ordertype($\mathbb{N},R) = \alpha$. For example, $\omega\cdot 2$ is a recursive ordinal because the ordering of $\mathbb{N}$ as 0, 2, 4, 6, 8, ... 1, 3, 5, 7, ... is computable an... | https://mathoverflow.net/users/6649 | Sneaky Recursive Non-Well-Orders | In the classic paper [Recursive pseudo-well-orderings](http://www.ams.org/journals/tran/1968-131-02/S0002-9947-1968-0244049-7/S0002-9947-1968-0244049-7.pdf), [TAMS 131 (1968), 526–543](http://www.ams.org/journals/tran/1968-131-02/S0002-9947-1968-0244049-7/home.html), Joe Harrison showed that one can in fact do much bet... | 15 | https://mathoverflow.net/users/2000 | 30644 | 19,950 |
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