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https://mathoverflow.net/questions/15209 | 17 | Let $f: \mathbb{Z}\_p \rightarrow \mathbb{C}\_p$ be any continuous function. Then Mahler showed there are coefficients $a\_n \in \mathbb{C}\_p$ with
$$
f(x) = \sum^{\infty}\_{n=0} a\_n {x \choose n}.
$$
This is known as the Mahler expansion of $f$. Here, to make sense of $x \choose n$ for $x \not\in \mathbb{Z}$ de... | https://mathoverflow.net/users/1018 | When does a p-adic function have a Mahler expansion? | It is false for the valuation ring in any nontrivial finite extension of $\mathbb{Q}\_p$. The coefficients of the Mahler expansion of a continuous function $\mathcal{O} \to \mathbb{C}\_p$ are determined by its restriction to $\mathbb{Z}\_p$ (they are given as $n$-th differences of the sequence of values on nonnegative ... | 19 | https://mathoverflow.net/users/2757 | 15212 | 10,194 |
https://mathoverflow.net/questions/15220 | 49 | Let $K$ be a Galois extension of the rationals with degree $n$. The [Chebotarev Density Theorem](http://en.wikipedia.org/wiki/Chebotarev%27s_density_theorem) guarantees that the rational primes that split completely in $K$ have density $1/n$ and thus there are infinitely many such primes. As Kevin Buzzard pointed out t... | https://mathoverflow.net/users/2000 | Is there an "elementary" proof of the infinitude of completely split primes? | By the primitive element theorem, $K=\mathbb{Q}(\alpha)$ for some nonzero $\alpha \in K$, and we may assume that the minimal polynomial $f(x)$ of $\alpha$ has integer coefficients. Let $\Delta$ be the discriminant of $f$. Since $K/\mathbb{Q}$ is Galois, a prime $p \nmid \Delta$ splits completely in $K$ if and only if t... | 71 | https://mathoverflow.net/users/2757 | 15221 | 10,199 |
https://mathoverflow.net/questions/15235 | 4 | More precisely, is there a map of schemes $X$ --> $Y$ such that $f$ gives a homeomorphism between $X$ and a closed subset of $Y$, but the corresponding map on sheaves is not surjective?
| https://mathoverflow.net/users/7 | Homeomorphism onto a closed subset of a scheme that isn't a closed immersion | Yes, for example if $K \subset L$ is an inclusion fields, then the induced map
Spec $L \to $ Spec $K$ is a homeomorphism (both source and target are single points),
but the induced map on sheaves is the given inclusion of $K$ into $L$, which is
surjective only if $K = L$.
For another example, let $X'\to Y$ be a clos... | 8 | https://mathoverflow.net/users/2874 | 15236 | 10,208 |
https://mathoverflow.net/questions/15238 | 5 | If we have a linear recurrence sequence where each term depends on all previous terms, say
$a\_n = \sum\_{k=0}^{n-1} \binom{n}{k} a\_k, \quad a\_0 = 1$
is there any way to estimate the growth of a\_n in terms of a Big-O notation?
I suppose the growth must be super-exponential, because if $a\_1, \ldots, a\_{n-1}$ ... | https://mathoverflow.net/users/3736 | How to estimate the growth of a recurrence sequence | A very powerful way to estimate the growth of a recurrence is to look at the analytic properties of the generating function that it implies. In this case we should take the exponential generating function $f(x) = \sum\_{n \ge 0} \frac{a\_n}{n!} x^n$, giving the identity
$$2f(x) = e^x f(x) + 1$$
hence $f(x) = \frac{... | 16 | https://mathoverflow.net/users/290 | 15239 | 10,210 |
https://mathoverflow.net/questions/15251 | 5 | Motivation/example. Consider $K = \mathbb{Q}(\sqrt[3]{2})$. This is a number field with ring of integers $O\_K = \mathbb{Z}[\sqrt[3]{2}]$. We have a norm map $N\_{K/\mathbb{Q}}$ which maps $x + y\sqrt[3]{2} + z\sqrt[3]{4}$ to $x^3 + 2y^3 + 4z^6 - 6xyz$; restricting to $\mathcal{O}\_K$ gives of course the same form. Usi... | https://mathoverflow.net/users/1107 | homogeneous forms as norms | Just a remark : It is certainly interesting to study the equation $N\_{L|K}(x)=y$, where $L|K$ is an extension of number fields, and there must be an extensive literature on the subject. Hasse proved that if such an equation is solvable everywhere locally (at every place of $K$), and if $L|K$ is *cyclic*, then there is... | 4 | https://mathoverflow.net/users/2821 | 15253 | 10,218 |
https://mathoverflow.net/questions/15265 | 3 | As someone who mostly does symbolic computation, I've always been puzzled by the fascination mathematicians seem to have with Lp(R) (for p<∞)? To be more precise, there are **no** non-trivial polynomials in that space and, to me, polynomials are not only the simplest functions, they are the building blocks of most ever... | https://mathoverflow.net/users/3993 | Polynomials and L^p(R) | You are referring to $L^p(\mathbb{R}, \mathcal{B}, \mu)$ in the case that $\mathbb{R}$ is endowed with Lebesgue measure $\mu$. Consider instead the measure $\nu$ given by $d\nu = f d\mu$, where $f$ is in the [Schwartz space](http://en.wikipedia.org/wiki/Schwartz_space) and $f$ does not take the value zero. Because the ... | 13 | https://mathoverflow.net/users/1847 | 15267 | 10,229 |
https://mathoverflow.net/questions/15082 | 22 | Artin has a theorem (10.1 in Laumon, Moret-Bailly) that if $X$ is a stack which has separated, quasi-compact, representable diagonal and an fppf cover by a scheme, then $X$ is algebraic. Is there a version of this theorem that holds for fpqc covers?
| https://mathoverflow.net/users/28 | fpqc covers of stacks | It is false. I'm not sure what the comment about algebraic spaces has to do with the question, since algebraic spaces do admit an fpqc (even \'etale) cover by a scheme. This is analogous to the fact that the failure of smoothness for automorphism schemes of geometric points is not an obstruction to being an Artin stack... | 38 | https://mathoverflow.net/users/3927 | 15269 | 10,230 |
https://mathoverflow.net/questions/15261 | 4 | Given a locally free sheaf $M$ on $\mathbb{P}^2$ with $h^0(M)=1$.
Is it true that we have $h^2(M)=0$ in this case?
I got this idea from Friedman's book "Algebraic Surfaces and holomorphic vector bundles".
In Chapter 4, p.109, Ex. 4 he wrote: $h^0(Hom(V,V))=1$, by Serre duality $h^2(Hom(V,V))=h^0(Hom(V,V)\otimes K)=0$... | https://mathoverflow.net/users/3233 | Serre duality and low dimensional cohomology groups | 1) A locally free sheaf $M$ on $\mathbb{P}^2$ with $h^0(M)=1$ need not satisfy $h^2(M)=0$. For example, if $M=\mathcal{O} \oplus \mathcal{O}(-n)$ for $n>3$, then $h^0(M) = 1 + 0 = 0$, but $M^\vee = \mathcal{O} \oplus \mathcal{O}(n)$ and $K=\mathcal{O}(-3)$, so Serre duality shows that $h^2(M) = h^0(M^\vee \otimes K) = ... | 15 | https://mathoverflow.net/users/2757 | 15274 | 10,232 |
https://mathoverflow.net/questions/15273 | 10 | Hi,
I came across the space $BU\_\otimes$ when struggling with twisted K-theory. Segal proved that this is an H-space, right? I have read a dozen times by now that the group $[X, BU\_\otimes]$ consists of the vector bundles of virtual dimension 1. I can clearly see the semi-group structure there, but what does the i... | https://mathoverflow.net/users/3995 | BU with tensor product H-space structure | I'll write $U(X)=[X,BU\_\otimes]$. So $U(X)\subset K(X)=[X,Z\times BU]$ is the multiplicatively closed subset corresponding to "virtual bundles of rank 1".
Likewise, I'll write $I(X)=[X,BU]\subset K(X)$ for the ideal of "virtual bundles of rank 0".
There's a bijection $a\mapsto 1+a$ from $I(X)$ to $U(X)$.
Here's the ... | 14 | https://mathoverflow.net/users/437 | 15275 | 10,233 |
https://mathoverflow.net/questions/15271 | 0 | (I removed my motivation because it may be misleading :) )
Let $A$ be a noetherian commutative ring and let $M \neq 0$ be a finitely generated zero-dimensional (i.e. $\mathrm{dim} \ \mathrm{Supp}(M) = 0$) $A$-module. Then the submodule $0 < M$ has primary decomposition $0 = \bigcap\_{\mathfrak{p} \in \mathrm{Supp}(M)... | https://mathoverflow.net/users/717 | Primary decomposition of zero-dimensional modules | If $M$ is finitely generated and has $0$-dimensional support, then $M$ is in fact supported at finitely many maximal ideals (a $0$-dimensional closed subset of the Spec of a Noetherian ring is just a finite union of maximal ideals), and one has the isomorphism
$M = \oplus\_{\mathfrak p} M\_{\mathfrak p} = \oplus\_{\mat... | 6 | https://mathoverflow.net/users/2874 | 15283 | 10,239 |
https://mathoverflow.net/questions/15297 | 0 | In more rigorous language:
" **V**: a vector space having an uncountable base
**S**: The set of subspaces of **V** that have countable dimension.
Can we construct explicitly a chain in the poset **S** (ordered by inclusion), such that this chain has NO upper bound in **S**? "
Apparently, this chain must have uncoun... | https://mathoverflow.net/users/4000 | How do we construct (in a vector space) a chain of countable dimensional subspaces that can only be bounded by an subspace of uncountable dimension? | The other answers asked you first to well-order the whole vector space (or a basis for it), and those answers are perfectly correct, but perhaps you don't like well order the whole space. So let me describe a construction that appeals directly to the Axiom of Choice.
Let V be your favorite vector space having uncoun... | 7 | https://mathoverflow.net/users/1946 | 15302 | 10,254 |
https://mathoverflow.net/questions/15309 | 28 | [Qiaochu Yuan](https://mathoverflow.net/users/290/qiaochu-yuan) in his [answer](https://mathoverflow.net/questions/15292/why-cant-there-be-a-general-theory-of-nonlinear-pde/15298#15298) to [this question](https://mathoverflow.net/questions/15292/why-cant-there-be-a-general-theory-of-nonlinear-pde) recalls a [blog post]... | https://mathoverflow.net/users/1409 | Simulating Turing machines with {O,P}DEs. | [ODEs are good enough.](https://doi.org/10.1007/11494645_21) Your comment got me started digging.
| 28 | https://mathoverflow.net/users/1847 | 15310 | 10,261 |
https://mathoverflow.net/questions/15031 | 10 | ### Background
The (rational) Calogero-Moser system is the dynamical system which describes the evolution of $n$ particles on the line $\mathbb{C}$ which repel each other with force proportional to the cube of their distance. If the particles have (distinct!) position $q\_i$ and momentum $p\_i$, then the Hamiltonian ... | https://mathoverflow.net/users/750 | Is the 'massive' Calogero-Moser system still integrable? | The paper "Meromorphic Parametric Non-Integrability, the Inverse Square Potential" by E. J. Tosel, proves almost what was claimed in the comments. Except for Jacobi's theorem:
>
> The 3-body problem on a line with *arbitrary masses* and
> inverse square potential is completely integrable with rational first integr... | 7 | https://mathoverflow.net/users/2384 | 15315 | 10,265 |
https://mathoverflow.net/questions/15322 | 8 | I'm reading J.P. May's [Concise Course in Algebraic Topology](http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf "Concise Course in Algebraic Topology"), and I'm having a lot of trouble visualizing how things work in Chapter 8, "Based cofiber and fiber sequences". Of course this is pretty basic stuff, but it'... | https://mathoverflow.net/users/303 | visualizing what's going on in based homotopy theory, et al. | Another book with pictures of reduced suspension etc. is Ronnie Brown's Topology and Groupoids.
see <http://www.bangor.ac.uk/~mas010/topgpds.html>. Which is also excellent for non-based stuff.
Don't believe all you hear about the unbased case being grotty! It is beautiful, but it is possibly easier to learn Alg. Top.... | 6 | https://mathoverflow.net/users/3502 | 15324 | 10,269 |
https://mathoverflow.net/questions/15314 | 3 | Sorry for the impreciseness of the title. It is merely meant for an analogy.
Exchange of limiting operations and integrations are basically derived from Lebesgue's dominated convergence theorem. For instance, let $f: \mathbb{R}^2 \to \mathbb{R}$ be Borel measuable. Let $f(\cdot, u) \in C^k(I)$ for some open set $I$ a... | https://mathoverflow.net/users/3736 | "exchange" of real analyticity and integration | It is rather hard to work with the coefficients of the Taylor expansion directly but, fortunately, one does not have to. Real analyticity is actually analiticity in a strip (of varying width) around the real line and it is quite hard to imagine the situation when $f(x)$ can be computed/estimated on the line but not nea... | 6 | https://mathoverflow.net/users/1131 | 15329 | 10,271 |
https://mathoverflow.net/questions/15327 | 23 | There are already a lot of discussion about the motivation for prime spectrum of commutative ring. In my perspective(highly non original), there are following reasons for the importance of prime spectrum.
1. $\text{Spec}(R)$ is the minimal spectrum containing $\text{Spec}\_{\rm max}(R)$ which has good functoriality ... | https://mathoverflow.net/users/1851 | What properties "should" spectrum of noncommutative ring have? | I know almost nothing about noncommutative rings, but I have thought a bit about what the general concept of spectra might or should be, so I'll venture an answer.
One other property you might ask for is that it has a good categorical description. I'll explain what I mean.
The spectrum of a commutative ring can be ... | 23 | https://mathoverflow.net/users/586 | 15330 | 10,272 |
https://mathoverflow.net/questions/15333 | 20 | I was wondering: is there a good place to find exercises in Hodge theory? Mostly computations and proving small (preferably nifty) theorems, is what I have in mind. Something roughly like the Problems in Group Theory book by Dixon, or many other such problem books.
The biggest difficulty I'm having is that I don't ha... | https://mathoverflow.net/users/622 | Exercises in Hodge Theory | One suggestion: "Period mappings and Period Domains", by Carlson, Muller-Stach, and Peters, in the Cambridge studies in advanced mathematics series. It's a very nice read, and each chapter comes with examples and problems.
| 6 | https://mathoverflow.net/users/3545 | 15338 | 10,277 |
https://mathoverflow.net/questions/12322 | 9 | An ideal $\mathfrak{a}$ is called irreducible if $\mathfrak{a} = \mathfrak{b} \cap \mathfrak{c}$ implies $\mathfrak{a} = \mathfrak{b}$ or $\mathfrak{a} = \mathfrak{c}$. Atiyah-MacDonald Lemma 7.11 says that in a Noetherian ring, every ideal is a finite intersection of irreducible ideals. Exercise 7.19 is about the uniq... | https://mathoverflow.net/users/71 | Atiyah-MacDonald, exercise 7.19 - "decomposition using irreducible ideals" | Here is a solution to the hint:
First of all, note that since all the ideals in question contain $\mathfrak a$, we may replace
$A$ by $A/\mathfrak a$, and so assume that $\mathfrak a = 0$; this simplifies the notation somewhat.
Next, the condition that $\mathfrak b\_1 \cap \cdots \cap \mathfrak b\_r = 0$ is equival... | 14 | https://mathoverflow.net/users/2874 | 15340 | 10,279 |
https://mathoverflow.net/questions/15336 | 9 | **Background/motivation**
It is a classical fact that we have a natural isomorphism $Sym^n (V^\*) \cong Sym^n (V) ^\ast$ for vector spaces $V$ over a field $k$ of characteristic 0. One way to see this is the following.
On the one hand elements of $Sym^n (V^\*)$ are symmetric powers of degree n of linear forms on $V... | https://mathoverflow.net/users/828 | Is $Sym^n (V^*) \cong Sym^n (V)^\ast$ naturally in positive characteristic? | The answer is no (and well-known to people working in the representation theory of algebraic groups in positive characteristic). In fact for $V$ finite dimensional and of dimension $>1$ the two vector spaces are not
isomorphic as $GL(V)$-modules ($GL(V)$ is either considered naively as an abstract group when the field... | 18 | https://mathoverflow.net/users/4008 | 15344 | 10,282 |
https://mathoverflow.net/questions/15354 | 18 | This questions is inspired by an exercise in Hungerford that I have only partially solved. The exercise reads: "A domain is a UFD if and only if every nonzero prime ideal contains a nonzero principal ideal that is prime." (For Hungerford, 'domain' means commutative ring with $1\neq 0$ and no zero divisors).
One dire... | https://mathoverflow.net/users/3959 | Characterizations of UFD and Euclidean domain by ideal-theoretic conditions | Dear Arturo,
The exercise in question is actually a theorem of Kaplansky. It appears as Theorem 5 on page 4 of his *Commutative Rings*. [I was not able to tell easily whether the result appears for the first time in this book.] The proof is reproduced in Section 10 of an expository article I have written [but probabl... | 19 | https://mathoverflow.net/users/1149 | 15356 | 10,290 |
https://mathoverflow.net/questions/15357 | 8 | The question is self-contained finite group theory but the motivation requires more background.
The finite groups I am interested in are the groups $Sp(n,F\_3)$. For $n$ even these are the usual symplectic groups over the field with three elements. For $n$ odd these are the odd symplectic groups. These are a semi-dir... | https://mathoverflow.net/users/3992 | From symmetric groups to symplectic groups? | Without working it out completely, I can't give a yes or no answer.
But here's a start: Let $G = Sp(2n)$ (a group of $2n$ by $2n$ matrices, in the notation I use) and let $K = Sp(2n-2) \ltimes H$, where $H$ is the appropriate Heisenberg group. Let's work over any finite field of odd characteristic -- I'd guess that ... | 4 | https://mathoverflow.net/users/3545 | 15358 | 10,291 |
https://mathoverflow.net/questions/15284 | 13 | In my meaning, a **direct sum** in a category should really be called a "biproduct". If $X,Y$ are objects, then a direct sum $X \oplus Y$ is an object $Z$ along with isomorphisms $\hom(Z,A) = \hom(X,A) \times \hom(Y,A)$ and $\hom(A,Z) = \hom(A,X) \times \hom(A,Y)$ for all objects $A$. A direct sum is unique up to canon... | https://mathoverflow.net/users/78 | How cavalier can I be when demanding a category have direct sums? | The answer to the first question is yes. If A and B have a direct sum A ⊕ B in C, then there are inclusions iA : A → A ⊕ B, iB : B → A ⊕ B and projections pA : A ⊕ B → A, pB : A ⊕ B → B such that pAiA = 1, pBiB = 1, and iApA + iBpB = 1. Conversely, the existence of such maps in an Ab-enriched category make A ⊕ B a dire... | 15 | https://mathoverflow.net/users/126667 | 15364 | 10,295 |
https://mathoverflow.net/questions/15366 | 287 | I wonder if anyone else has noticed that the market for [expository papers](https://www.grammarly.com/blog/expository-writing/#:%7E:text=Expository%20writing%2C%20as%20its%20name,or%20attempting%20to%20persuade%20them.) in mathematics is very narrow (more so than it used to be, perhaps).
Are there any journals which ... | https://mathoverflow.net/users/1149 | Which journals publish expository work? | I'm not too familiar with [Expositiones Mathematicae](https://www.sciencedirect.com/journal/expositiones-mathematicae), but have you given them a look?
**EDIT:** The article I happened to have seen, which made me think that Expo Math might be along the lines Pete Clark was looking for, is [this paper](https://arxiv.o... | 73 | https://mathoverflow.net/users/763 | 15369 | 10,297 |
https://mathoverflow.net/questions/15373 | 4 | Suppose $K$ is a (not necessarily algebraically closed) field, and $G\_1$ and $G\_2$ are *split* semisimple algebraic groups over $K$ which become isomorphic over $\bar{K}$, the algebraic closure of $K$. Are $G\_1$ and $G\_2$ isomorphic over $K$? What about if the $G$s are reductive?
It seems like this should follow ... | https://mathoverflow.net/users/3513 | If split algebraic groups are potentially isomorphic, are they isomorphic? | The answer is yes, for arbitrary split connected reductive groups over any field. The main point is that the Existence, Isomorphism, and Isogeny Theorems (relating split connected reductive groups and root data) are valid over any field. One reference is SGA3 near the end (which works over any base scheme), but in Appe... | 8 | https://mathoverflow.net/users/3927 | 15378 | 10,302 |
https://mathoverflow.net/questions/12117 | 11 | I apologize if these questions seem naive or loaded.
Is there an analogous theory of highest weights for irreducible finite-dimensional representations of Lie algebras of algebraic group (or perhaps group schemes) over a non-algebraically closed field (resp. a "nice" ring, say a Dedekind domain).
Are there analogo... | https://mathoverflow.net/users/3247 | An arithmetic highest weight theory? | Johnson, you have one of the foremost experts in the world on such matters (over general fields) just upstairs from your office. Make use of that.
Many of the basic constructions work for split groups over fields, but proving good properties (such as irreducibility and classification results) requires being over a f... | 6 | https://mathoverflow.net/users/3927 | 15381 | 10,305 |
https://mathoverflow.net/questions/15392 | 6 | I am trying to prove a version of quantum Schur-Weyl duality. I hope to be able to generalize the proof of the Schur-Weyl duality between $U\_q(\mathfrak{gl}\_n)$ and the Hecke algebra $H\_r$. So I am looking for a good reference for this with a careful proof. It would also be nice to see a proof that uses the quantum ... | https://mathoverflow.net/users/3318 | Reference for quantum Schur-Weyl duality | This goes back to Jimbo I think. A reference is:
"A q-difference analogue of $U(\mathfrak g)$, Hecke algebra and the Yang-Baxter equation'', Lett. Math. Phys. 11 (1986).
It has been much studied though, so there are lots of subsequent papers, some of which might be closer to what you are looking for? For example [thi... | 7 | https://mathoverflow.net/users/1878 | 15395 | 10,313 |
https://mathoverflow.net/questions/15396 | 4 | Let be $d$ a positive integer, $\Omega=\mathbb{R}^{\mathbb{Z}^d}$ and fix $R\geq 2$. We define weighted Banach spaces
$$ \Omega\_p:=\left\{ x\in \Omega\left| \left[\sum\_{i\in\mathbb{Z}^d}\frac{|x\_i|^R}{(1+|i|)^p}\right]^{\frac{1}{R}} < \infty\right. \right\},\ \ \ \ \ \ \ p>d. $$
My question is why the embedding... | https://mathoverflow.net/users/2386 | Embeddings of Weighted Banach Spaces | This is a special case of a much more general phenomenon, so I'm writing an answer which deliberately takes a slightly high-level functional-analytic POV; I think (personally) that this makes it easier to see the wood for the trees, even if it might not be the most direct proof. However, depending on your mathematical ... | 5 | https://mathoverflow.net/users/763 | 15406 | 10,320 |
https://mathoverflow.net/questions/15400 | 9 | There is an ubiquitous pattern of questions concerning assumedly any kind of mathematical object or structure: groups, graphs, numbers, categories, and so on. It goes like this (informally):
>
> Can a class of objects or structures of a given kind *X* that is characterized by some "external condition" *Y* be define... | https://mathoverflow.net/users/2672 | An ubiquitous pattern of questions | There are three sibling theorems of logic which guarantee that such characterizations are bound to happen.
* [The Craig Interpolation Theorem](http://en.wikipedia.org/wiki/Craig_interpolation)
* [The Robinson Joint Consistency Theorem](http://en.wikipedia.org/wiki/Robinson%27s_joint_consistency_theorem)
* [The Beth D... | 16 | https://mathoverflow.net/users/2000 | 15413 | 10,324 |
https://mathoverflow.net/questions/15424 | 1 | Is the tr.deg of Q\_p over Q 1? and what about C over Q?
| https://mathoverflow.net/users/1238 | What is the transcendence degree of Q_p and C over Q? | In both cases the transcendence degree is the cardinality of the continuum. CH is not needed.
This is a corollary of the following result: let $K$ be any infinite field, and let $L/K$ be any extension. Then
$\# L = \operatorname{max} (\# K, \operatorname{trdeg}\_K L)$.
To prove this, in turn it suffices to estab... | 9 | https://mathoverflow.net/users/1149 | 15428 | 10,334 |
https://mathoverflow.net/questions/15447 | 13 | Is there any standard way to read (in English) the Legendre symbol (a|p) (<http://en.wikipedia.org/wiki/Legendre_symbol>), say, similar to "a choose b"
which is used for the binomial coefficients?
| https://mathoverflow.net/users/3635 | Is there a standard way to read the Legendre symbol? | I say "a on b" for the Legendre/Jacobi/Kronecker symbol. This works because, as an American, I say "a over b" for an ordinary fraction.
| 14 | https://mathoverflow.net/users/1149 | 15449 | 10,343 |
https://mathoverflow.net/questions/15435 | 2 | Just a minor curiosity that's flitted across my mind, but that's (part of) what this site's for, right?:
Is it possible for Hom(a, -) and Hom(b, -) to both be monadic functors from C to Set, for non-isomorphic objects a and b in C? Ideally, the answer would come with either a nice example or an outline of a nice proo... | https://mathoverflow.net/users/3902 | If a category is "monadic", is it necessarily so in a unique manner? | You can certainly have non-equivalent monadic functors. Here's one example: Let $\mathcal{V}\_k$ be the category of $k$-vector spaces. For a vector space $V$, let $H\_V: \mathcal{V}\_k\to \mathcal{V}\_k$ be the functor
$$
H\_V(W) = hom\_k(V,W).
$$
Such a functor is always monadic, as long as $V$ is non-zero and finite... | 6 | https://mathoverflow.net/users/437 | 15454 | 10,348 |
https://mathoverflow.net/questions/15448 | 8 |
>
> What are the derivations of the algebra of *continuous* functions on a topological manifold?
>
>
>
A *supermanifold* is a locally ringed space (X,O) whose underlying space is a *smooth* manifold X, and whose sheaf of functions locally looks like a graded commutative algebra which is an exterior algebra over ... | https://mathoverflow.net/users/184 | Derivations of C(X)? or Why Must Supermanifolds be Smooth? | Here's a proof from definition. (I don't think it has anything to do with compactness.) Let us show that derivation
$\delta:C(X)\to C(X)$ vanishes for any topological manifold $X$. Indeed, $\delta(1)=0$ as usual, so it suffices to show that whenever $f\in C(X)$ vanishes at $x\in X$, so does
$\delta(f)$. For this, it i... | 14 | https://mathoverflow.net/users/2653 | 15458 | 10,352 |
https://mathoverflow.net/questions/15440 | 32 | There is the following theorem:
"A space $X$ is the inverse limit of a system of discrete finite spaces, if and only if $X$ is totally disconnected, compact and Hausdorff."
A finite discrete space is totally disconnected, compact and Hausdorff and all those properties pass to inverse limits. I guess the other direc... | https://mathoverflow.net/users/3969 | Which spaces are inverse limits of discrete spaces ? | These are the completely ultrametrizable spaces.
Recall that a d:E2→[0,∞) is an ultrametric if
1. d(x,y) = 0 ↔ x = y
2. d(x,y) = d(y,x)
3. d(x,z) ≤ max(d(x,y),d(y,z))
As usual, (E,d) is a complete ultrametric space if every Cauchy sequence converges.
Suppose E∞ is the inverse limit of the sequence En of discre... | 42 | https://mathoverflow.net/users/2000 | 15465 | 10,358 |
https://mathoverflow.net/questions/15466 | 10 | If you have a lattice $L \subset \mathbb{C}$, you can compute the following numbers:
$
G\_4(L) = \sum\_{\omega \in L, \omega \neq 0} \frac{1}{\omega^4}, \quad G\_6(L) = \sum\_{\omega \in L, \omega \neq 0} \frac{1}{\omega^6}.
$
By a 'lattice', I just mean a closed discrete additive non-trivial subgroup of $\mathbb{C... | https://mathoverflow.net/users/401 | What do the numbers G_4 and G_6 of a lattice actually measure? | As far as I know $G\_4$ and $G\_6$ don't have a direct geometric interpretation of the type you are looking for. Rather, they appear as coefficients in the algebraic equation for
$\mathbb C/\Lambda.$ More precisely, the pair
$(\mathbb C/\Lambda, dz)$ consisting of the complex torus $\mathbb C/\Lambda$ and the everywhe... | 9 | https://mathoverflow.net/users/2874 | 15468 | 10,360 |
https://mathoverflow.net/questions/15441 | 2 | In the paper hep-th/9712042v2, p. 20, the following setup is given:
A complex manifold M and an n+1-dimensional vector bundle V on it. V has an underlying real bundle $V\_{\mathbb{R}}$ with a flat connection $\nabla$. Also, there is a holomorphic inclusion of a line bundle $L$ in $V$.
Finally, we have the section
$... | https://mathoverflow.net/users/3170 | Immersion with respect to a connection? | Just read further down the same page: "The immersion condition in (b) states that $m \to L\_m$ is an immersion into the 2n + 1 dimensional projective space of a local trivialization of $P(V\_R)\_C$."
You can also say it without reference to the flatness of the connection: the section is transverse to the horizontal dis... | 3 | https://mathoverflow.net/users/2991 | 15471 | 10,363 |
https://mathoverflow.net/questions/15178 | 3 | Let $K$ be a field, $V$ an $n-$dimensional $K$-vector space and $q: V \to K$ a quadratic form of Witt index $r$. Let $G:=SO(q)$ denote the special orthogonal group associated to $q$.
Then $G$ is an algebraic $K$-group of $K-$rank $r$.
If $q'$ is isometric to $q$, clearly $G':=SO(q')$ is isomorphic to $G$. But this is... | https://mathoverflow.net/users/3380 | Algebraic groups of relative rank 1 | Fleshing out the Galois cohomology approach suggested by Brian Conrad leads to a clean answer for all $n$, for all fields $K$ of characteristic not $2$, and for all nondegenerate quadratic forms of rank $n$ over $K$. The answer is exactly what moonface claimed:
>
> Given quadratic forms $q$ and $q'$, the algebraic ... | 7 | https://mathoverflow.net/users/2757 | 15473 | 10,364 |
https://mathoverflow.net/questions/8262 | 18 | Let $L$ be a language on a finite alphabet and let $L\_n$ be the number of words of length $n$. Let $f\_L(x) = \sum\_{n \ge 0} L\_n x^n$. The following are well-known:
* If $L$ is regular, then $f\_L$ is rational.
* If $L$ is unambiguous and context-free, then $f\_L$ is algebraic.
Does there exist a natural family ... | https://mathoverflow.net/users/290 | Is there a natural family of languages whose generating functions are holonomic (i.e. D-finite)? | I vaguely remember that the paper by Marni Mishna and Mike Zabrocki [Analytic aspects of the shuffle product](http://arxiv.org/abs/0802.2844) sheds some light on the subject.
| 6 | https://mathoverflow.net/users/3032 | 15481 | 10,367 |
https://mathoverflow.net/questions/15219 | 17 | Let $G$ be a permutation group on the finite set $\Omega$. Consider the Markov chain where you start with an element $\alpha \in \Omega$ chosen from some arbitrary starting probability distribution. One step in the Markov chain involves the following:
* Move from $\alpha$ to a random element $g\in G$ that fixes $\alp... | https://mathoverflow.net/users/2384 | Markov chain on groups | This is a classical question which is now well understood, I believe. Start with this famous paper: <http://tinyurl.com/yfyrg63> Although this question is really not about standard r.w. on groups, a survey by Saloff-Coste gives an excellent introduction and literature review: www.math.cornell.edu/~lsc/rwfg.pdf
| 9 | https://mathoverflow.net/users/4040 | 15483 | 10,369 |
https://mathoverflow.net/questions/14863 | 28 | An [alternating permutation](https://arxiv.org/abs/0912.4240) of {1, ..., n} is one were π(1) > π(2) < π(3) > π(4) < ... For example: (24153) is an alternating permutation of length 5.
If $E\_n$ is the number of alternating permutations of length n, the $\sec x + \tan x = \sum\_{n \geq 0} E\_n \frac{x^n}{n!}$ is the ... | https://mathoverflow.net/users/1358 | Random Alternating Permutations | This is very easy - I often teach this in my combinatorics classes. You start with this Pascal type triangle which you need to precompute:
<http://mathworld.wolfram.com/Seidel-Entringer-ArnoldTriangle.html>
(read Arnold's paper referenced there if the pattern is unclear or my survey paper with Postnikov "Increasing Tre... | 21 | https://mathoverflow.net/users/4040 | 15487 | 10,372 |
https://mathoverflow.net/questions/14527 | 15 | The [Robinson-Schensted correspondence](https://en.wikipedia.org/wiki/Robinson%E2%80%93Schensted%E2%80%93Knuth_algorithm) is a bijection between elements of the symmetric group and ordered pairs of standard tableaux of the same shape.
Some simple operations on tableaux correspond to simple operations on the group: sw... | https://mathoverflow.net/users/66 | What bijection on permutations corresponds under RS to transpose? | When you conjugate diagrams and apply RSK or other Young tableau bijections, the answer is typically bad, with some rare exceptions. The right way to think of RSK is to think of row length being continuous while columns still integer (see e.g. my "Geometric proof of the hook-length formula" paper and refs therein).
... | 10 | https://mathoverflow.net/users/4040 | 15490 | 10,375 |
https://mathoverflow.net/questions/15478 | 6 | Can you offer some examples of such rings, other than $\frac{k[x,y]}{(x^{2}, xy)}$. Thanks.
| https://mathoverflow.net/users/3976 | Examples of one-dimensional non-Cohen Macaulay rings | In dimension $1$, Cohen-Macaulay just mean *unmixed*, so all the associated primes have the same dimension. Thus the easiest way to cook up a non-CM ring of dimension $1$ is: Pick your favorite regular ring (say $A=k[x,y,z]$). Take an ideal of dimension $1$, say $I=(x,y)$. Take another ideal of dimension $0$, say $J=(x... | 10 | https://mathoverflow.net/users/2083 | 15491 | 10,376 |
https://mathoverflow.net/questions/15486 | 6 | Recall that a *group* is an associative, unital monoid $G$ such that the map $(p\_1,m) : G \times G \to G\times G$ is an isomorphism of sets. Here $p\_1$ is the first projection and $m$ is the multiplication, so the map is $(g\_1,g\_2) \mapsto (g\_1,g\_1g\_2)$.
My question is a basic one concerning the definition of ... | https://mathoverflow.net/users/78 | How strict can I be in the definition of "2-group"? | You can always do this. Take any $b$ and define $d = p\_2 b$. Then $b' = (p\_1, d)$ is equivalent to the original $b$. To see this note that
$$(p\_1, m) \circ b = (p\_1b, m \circ (p\_1 b, d)) \simeq id = (p\_1, p\_2) $$
The first component shows $p\_1 b \simeq p\_1$. We use this transformation $\times id$ to show ... | 4 | https://mathoverflow.net/users/184 | 15495 | 10,377 |
https://mathoverflow.net/questions/15493 | 8 | Is a closed morphism with proper fibres proper?
| https://mathoverflow.net/users/nan | morphism closed + fibres proper => proper? | The answer is no. Consider an integral nodal curve $Y$ over an algebraically closed field, normalize the node and remove one of the two points lying over the node. Then you get a morphisme $f : X\to Y$ which is bijective (hence homeomorphic), separated and of finite type, and the fibers are just (even reduced) points. ... | 29 | https://mathoverflow.net/users/3485 | 15503 | 10,382 |
https://mathoverflow.net/questions/15444 | 178 | Define an "eventual counterexample" to be
* $P(a) = T $ for $a < n$
* $P(n) = F$
* $n$ is sufficiently large for $P(a) = T\ \ \forall a \in \mathbb{N}$ to be a 'reasonable' conjecture to make.
where 'reasonable' is open to interpretation, and similar statements for rational, real, or more abstractly ordered sets fo... | https://mathoverflow.net/users/3623 | Examples of eventual counterexamples | It was once conjectured that factors of $x^n-1$ over the rationals had no coefficient exceeding 1 in absolute value. The first counterexample comes at $n=105$.
| 161 | https://mathoverflow.net/users/3684 | 15506 | 10,384 |
https://mathoverflow.net/questions/15494 | 16 | The question on games and mathematics that appeared recently on mathoverflow
([Which popular games are the most mathematical?](https://mathoverflow.net/questions/13638/which-popular-games-are-the-most-mathematical))
reminded me of a problem I encountered some time ago : starting with the insane
dream of completely solv... | https://mathoverflow.net/users/2389 | Mathematical solution for a two-player single-suit trick taking game? | Yes, it has been studied by Johan Wästlund in *A solution of two-person single-suit whist*, which gives an efficient algorithm to compute the value of a position in this game (Theorem 10.1). He has also studied the more general situation of multiple suits, but with the restriction that each suit is split evenly between... | 21 | https://mathoverflow.net/users/126667 | 15530 | 10,400 |
https://mathoverflow.net/questions/15407 | 5 | Suppose two positive holomorphic line bundles $L\_1 \to X\_1, L\_2\to X\_2$ over two projective complex manifold $X\_1, X\_2$ have isomorphic ring of sections $R=R\_1=R\_2$ where $R\_i=\oplus\_{m=0}^\infty\Gamma(X\_i,mL\_i)$. Isomorphism as graded ${\mathbb C}$- algebras.
Is there any relationship betweeen $X\_1$ and... | https://mathoverflow.net/users/nan | Relationship between Line Bundles with isomorphic ring of sections | To expand on the answer above: as B. Cais says, if the line bundles are ample (which I think follows from positivity by Kodaira), we have a canonical isomorphism $\mathrm{Proj} R\_i\cong X\_i$. Thus, if the graded rings $R\_i$ are isomorphic, then the induced map of Proj's gives an isomorphism $R\_1\cong R\_2$ carrying... | 4 | https://mathoverflow.net/users/66 | 15537 | 10,406 |
https://mathoverflow.net/questions/15496 | 9 | Let *K* be an ordered field. Define the *n*-sphere:
$$S^n(K) := \{ (x\_1,x\_2,\dots,x\_n+1) \in K^{n+1} \mid \sum\_{i=1}^{n+1} x\_i^2 = 1 \}$$
A set of vectors $v\_1, v\_2, \dots, v\_r \in S^n(K)$ is *orthonormal* if the dot product of any two of them is zero. An *orthonormal basis* is an orthonormal set of cardina... | https://mathoverflow.net/users/3040 | Spheres over rational numbers and other fields | Any orthonormal set extends to an orthonormal basis, over any field
of characteristic not $2$. This is a special case of [Witt's theorem](http://en.wikipedia.org/wiki/Witt%27s_theorem).
**EDIT:** In response to Vipul's comment: The proof of Witt's theorem is constructive, and leads to the following recursive algorith... | 16 | https://mathoverflow.net/users/2757 | 15540 | 10,408 |
https://mathoverflow.net/questions/12955 | 8 | I'm just looking for an example of an integral, rectifiable varifold, which has no locally bounded first variation.
---
### Recapitulation
for every $m$-rectifiable varifold $\mu$ exists a $m$-rectifiable set $E$ in $\mathbb R^n$, meaning $E=E\_0 \cup \bigcup\_{k\in\mathbb N} E\_k$ with $\mathcal H^m(E\_0)=0$ a... | https://mathoverflow.net/users/3538 | Example for an integral, rectifiable varifold with unbounded first variation | Something is strange here: it seems like for the sawtooth curve (the Lipschitz curve that goes up and down with slope $1$), the first variation is just the sum of $\delta$-measures at the turning points times the unit bisector vectors, so we can have fixed length and arbitrarily large first variation (just make turning... | 6 | https://mathoverflow.net/users/1131 | 15549 | 10,415 |
https://mathoverflow.net/questions/15546 | 2 | I have an inclusion of topological spaces (actually manifolds with corners) $X \to Y$. I can show that for every $x \in X$ there is a neighborhood of $x$ in $Y$ of the form $U \times V$. Also, the intersection of $U \times V$ with $X$ is $U \times V'$.
In fact, $V$ is a neighborhood of the origin in $[0,1)^k$. Also, ... | https://mathoverflow.net/users/1676 | Simple question of topological cofibration | The inclusion of a CW subcomplex $K$ into a CW complex $L$ is a cofibration. Briefly, we can extend a homotopy from the $n-1$-skeleton to the $n$-skeleton by projecting $e\times I$ to $e\times\{0\}\cup\partial E\times I$ for any $n$-cell $e$ not in $L$.
| 4 | https://mathoverflow.net/users/2349 | 15553 | 10,418 |
https://mathoverflow.net/questions/15438 | 42 | On one of my exams last year, we were given a problem (we chose five or six out of eight problems) on an exam, the goal of which was to prove the Bruhat decomposition for $GL\_n(k)$. I was one of the two people to choose said problem. I gave a very long convoluted argument which although correct was really inelegant. I... | https://mathoverflow.net/users/1353 | A slick proof of the Bruhat Decomposition for GL_n(k)? | You are asking to compute the double quotient $B\backslash G /B.$ This is the same as computing $G\backslash (G/B \times G/B)$. A point in $G/B$ is a full flag on $k^n$. So you
are trying to compute the set of pairs $(F\_1,F\_2)$ of flags, modulo the simultaneous action
of $G$.
Another way to think about $G/B$ is tha... | 54 | https://mathoverflow.net/users/2874 | 15554 | 10,419 |
https://mathoverflow.net/questions/15550 | 8 | I'll explain the problem but what I am looking for is a few suggested methods to approach this problem.
You don't need to know what a microarray but if you are interested look here [link text](http://en.wikipedia.org/wiki/DNA_microarray)
The info below is simplified, not addressed to a micro-biologist (I am not one)
... | https://mathoverflow.net/users/4045 | MicroArray, tesing if a sample is the same with high variance data. | In general, the approach of using additional measurements of other values (not the one you are interested in directly) was useful in many problems in the past, so it sounds like a good idea in your case as well. Here are a few things to look at:
Binary classification
---------------------
If the task is to determin... | 1 | https://mathoverflow.net/users/3035 | 15557 | 10,421 |
https://mathoverflow.net/questions/15558 | 8 | Do you know any explicit constructions of families of irreducible finite dimensional representations of the three string braid group?
I will describe what I already know and then invite you to suggest alternatives. This is somewhat open-ended so I will also pose some more precise questions.
---
This problem is ... | https://mathoverflow.net/users/3992 | Representations of the three string braid group | Bruce,
I can give you representants for an open piece of the moduli space (3,3) resp. (3,2,1). I'll add them in Mathematica-form so that you can plug them into any braid you like to work on. The variables a,b,d,x,y,z are the coordinates of the moduli space and l stands for a third root of unity (i dont know how to tell... | 15 | https://mathoverflow.net/users/2275 | 15563 | 10,424 |
https://mathoverflow.net/questions/15460 | 10 | In his article where he stated what we know as Eisenstein's irreducibility criterion (which actually was first proved by Schönemann, as was Scholz's reciprocity law and Hensel's Lemma), he claimed that the result also holds in the following case: assume that
$$ f(x) = x^m + a\_{m-1}x^{m-1} + \ldots + a\_0, $$
where the... | https://mathoverflow.net/users/3503 | Variants of Eisenstein irreducibility | Basically all such criteria boil down to some argument involving the Newton polygon as Kevin Buzzard mentions in the comments. While something as general as your statement has trivial counter examples the following generalization holds:
>
> Let $R$ be a unique factorization domain and $f(x) =a\_nx^n+\cdots +a\_0\in... | 11 | https://mathoverflow.net/users/2384 | 15564 | 10,425 |
https://mathoverflow.net/questions/15560 | 9 | Hello,
I wonder if the techniques introduced in Neemans paper:
"The Grothendieck duality theorem via Bousfield's techniques and Brown representability "
can be used to establish Verdier duality. More precisely:
Consider the unbounded, derived category $D(M)$ of $\mathbb{Q}$ vector spaces on a compact complex manifo... | https://mathoverflow.net/users/2837 | Verdier duality via Brown representability? | The category of sheaves of $\mathbb{Q}$ vector spaces on $M$ is a Grothendieck abelian category. It follows that the derived category of such, $D(M)$ in your notation, is a well generated triangulated category. A proof of this can be found in Neeman's paper "On The Derived Category of Sheaves on a Manifold". In particu... | 9 | https://mathoverflow.net/users/310 | 15567 | 10,428 |
https://mathoverflow.net/questions/14762 | 1 | Hi,
probably this is a fairly newbie question, but is it possible that the a generic Levy process explodes (i.e. tends to infinity for finite time t with positive probability)? If yes, could you please provide an example, if no point me to the proof.
| https://mathoverflow.net/users/3160 | Exploding Levy processes | Hi,
As remarked by Leonid Kovalev a Lévy process doesn't explodes as far as I know, nevertheless as you mention generic Levy Process for which I don't know any definition, you may be thinking of them as diffusions driven by a Lévy process.
Then looking at some SDEs, you can have some cases where explosion time is ... | 1 | https://mathoverflow.net/users/2642 | 15575 | 10,432 |
https://mathoverflow.net/questions/14959 | 5 | Let $E/F$ be a quadratic extension of number fields, and let $V$ be an $n$-dimensional Hermitian space over $E$.
Let $\tilde{G} := GU(V)$ and $G := U(V)$. Suppose that $(\pi, V\_{\pi})$ is an irreducible cuspidal representation of $G.$
Is there an irreducible cuspidal representation $(\tilde{\pi}, V\_{\tilde{\pi}})... | https://mathoverflow.net/users/3186 | extending cusp forms | I believe the answer should be yes, by some version of the following sketch of an argument:
(Note: by restriction of scalars, I regard all groups as being defined over $\mathbb Q$,
and I write ${\mathbb A}$ for the adeles of $\mathbb Q$.)
We are given $V\_{\pi} \subset Cusp(G(F)\backslash G({\mathbb A}\_F)).$
Let... | 3 | https://mathoverflow.net/users/2874 | 15581 | 10,436 |
https://mathoverflow.net/questions/15569 | 14 | How does one show that if $U \subseteq \mathbb{C}^n$ is nonempty and Zariski open then $U$ is also dense in the analytic topology on $\mathbb{C}^n$?
| https://mathoverflow.net/users/4048 | Zariski open sets are dense in analytic topology | It is enough to show that the complement of $U$ has empty interior. Also, that complement is contained in the zero set $Z$ of a non-constant polynomial $f$, so it is enough to show that $Z$ does not contain open sets.
If $z\in Z$ is a point in the interior of $Z$, then the Taylor series of $f$ at $z$ is of course ze... | 23 | https://mathoverflow.net/users/1409 | 15582 | 10,437 |
https://mathoverflow.net/questions/15514 | 0 | I have n sectors, enumerated 0 to n-1 counterclockwise. The boundaries between these sectors are infinite branches (n of them).
These branches meet at certain points (junctions). Each junction is adjacent to a subset of the sectors (at least 3 of them).
By specifying what sectors my junctions are adjacent to, I ca... | https://mathoverflow.net/users/1056 | Describe a tree by junctions | I realized that taking the dual of my trees, I always get an n-gon,
where some chords, the faces in the dual are my junctions. The bijection is now trivial.
| 1 | https://mathoverflow.net/users/1056 | 15599 | 10,446 |
https://mathoverflow.net/questions/15584 | 22 | In the very first chapter Hartshorne proposes the following seemingly trivial exercise (ex. I.2.17(ii)):
>
> Show that a strict complete intersection is a set theoretic complete intersection.
>
>
>
Here are Hartshorne's definitions:
>
> A variety $Y$ of dimension $r$ in $\mathbb{P}^n$ is a (strict) complet... | https://mathoverflow.net/users/828 | Minimal number of generators of a homogeneous ideal (exercise in Hartshorne) | Dear Andrea: Hartshorne was right, but we need to do some work. Let $\mu(I)$ be the minimal number of generators of $I$, and $\mu\_h(I)$ be the minimal number of a homogenous system of generators of $I$. Let $R=k[x\_1,\dots,x\_n]$ and $\mathfrak m=(x\_1,\dots,x\_n)$. Suppose $\mu\_h(I)=m$ and $f\_1,\dots, f\_m$ is a mi... | 21 | https://mathoverflow.net/users/2083 | 15605 | 10,450 |
https://mathoverflow.net/questions/15591 | 13 | Let $X$ be an infinite set, and let $G$ be the symmetric group on $X$. I want to understand $G$ by putting a topology on it, without imposing any more structure on $X$. What 'interesting' possibilities are there and what is known about them?
In particular, I have heard of the pointwise convergence topology (an open n... | https://mathoverflow.net/users/4053 | Topologies on an infinite symmetric group | It is known that there are exactly two separable group topologies on $S\_\infty$ (i.e., on the group of permutations of a countable set): one is antidiscrete and the other one is topology of pointwise convergence. This is a statement of Theorem 6.26 in
[here](http://arxiv.org/abs/math/0409567). Hence Polish topology on... | 9 | https://mathoverflow.net/users/896 | 15616 | 10,456 |
https://mathoverflow.net/questions/15596 | 10 | All the statements of the [Archimedean property](http://en.wikipedia.org/wiki/Archimedean_property) with which I am acquainted fundamentally uses ℕ -- more than as a totally ordered semi-group, really being the 'standard model' of the naturals. It is a fundamental ingredient in showing that the reals are (up to isomorp... | https://mathoverflow.net/users/3993 | Is there a version of the Archimedean property which does not presuppose the Naturals? | It is not surprising that some versions of the Archimedean property concern subsets of the order rather than merely elements. The reason is that the Archimedean property is provably not expressible in a first order manner.
This is because the structure of the reals R, as an ordered field, say, (but one can add any s... | 18 | https://mathoverflow.net/users/1946 | 15620 | 10,460 |
https://mathoverflow.net/questions/15628 | 41 | Let $G$ be a (maybe Lie) group, and $M$ a space (perhaps a manifold). Then a principal $G$-bundle over $M$ is a bundle $P \to M$ on which $G$ acts (by fiber-preserving maps), so that each fiber is a $G$-torsor (a $G$-action isomorphic, although not canonically so, to the action of $G$ on itself by multiplication). A ma... | https://mathoverflow.net/users/78 | What is the classifying space of "G-bundles with connections" | There is a stupid answer which is equivalence classes of G-bundles with connection on M are the same as homotopy classes of maps $M \to BG$. That is as long as two G-bundles with connection are considered equivalent if they have the same underlying principal bundle. This isn't meant to be a serious answer, just point o... | 18 | https://mathoverflow.net/users/184 | 15633 | 10,469 |
https://mathoverflow.net/questions/15479 | 12 | Constructing the Kummer K3 of an Abelian surface $A$, we have an obvious 22-dimensional collection of classes in $H^2(K3, \mathbb{Z})$ given by the 16 (-2)-curves (which by construction do not intersect each other), and the pushforward-and-pullback of the six classes generating $H^2(A, \mathbb{Z})$. However, this is cl... | https://mathoverflow.net/users/1703 | What classes am I missing in the Picard lattice of a Kummer K3 surface? | The lattice $L\_{K3}=H^2(K3,\mathbb Z)$ is $2E\_8+3U$, with $E\_8$ negative definite and $U$ the hyperbolic lattice for the bilinear form $xy$. It is unimodular and has signature $(3,19)$.
The 16 (-2)-curves $E\_i$ form a sublattice $16A\_1$ of determinant $2^{16}$. It is not primitive in $L\_{K3}$. The primitive lat... | 13 | https://mathoverflow.net/users/1784 | 15637 | 10,473 |
https://mathoverflow.net/questions/15614 | 6 | Max-flow and linear programming are two big hammers in algorithm design: each are expressive enough to represent many poly-time solvable problems. Some problems are obvious applications of max-flow: like finding a maximum matching in a graph. What I'm looking for are examples of problems that can be solved via clever e... | https://mathoverflow.net/users/3028 | Interesting applications of max-flow and linear programming | Determining whether a sports team has been mathematically eliminated from qualifying for the playoffs is a cute application of max-flow min-cut:
<http://www.cs.princeton.edu/courses/archive/spr03/cs226/assignments/baseball.html>
| 12 | https://mathoverflow.net/users/2233 | 15638 | 10,474 |
https://mathoverflow.net/questions/15641 | 7 | I need a reference (or a short proof) for the following statement:
Suppose a closed manifold $N$ is the result of a surgery (along an embedded sphere) on a closed manifold $M$. Then the difference $\sum dim H\_i(N) - \sum dim H\_i(M)$ (the homology is taken with coefficients in a field) is at most 2.
| https://mathoverflow.net/users/2823 | Surgery and homology: a reference request | To say that a smooth, closed manifold $N$ is obtained by surgery along a (framed) sphere in $M$ is to say that there is a cobordism $P$ from $M$ to $N$ and a Morse function $f\colon P\to [0,1]$, with $f^{-1}(0)=M$, $f^{-1}(1)=N$, and exactly one critical point $c$. By Morse theory, $H\_\*(P,M)$ is then 1-dimensional, g... | 12 | https://mathoverflow.net/users/2356 | 15647 | 10,480 |
https://mathoverflow.net/questions/15643 | 3 | Reading Wojtaszczyk's Banach spaces for analysts, I'm trying to understand his proof that the space of all continuous linear functionals on $(X^\star,\sigma(X^\star, X))$ is $X$.
To show the $ \subseteq$ part, he says let $\varphi$ be any linear functional on $X^\star$ continuous in $\sigma(X^\star, X)$. Then $\{x^\s... | https://mathoverflow.net/users/2586 | characterization of continuous functionals in weak-star topology | Yes to your first question. As for the second, regard the $x\_j$-s as linear functionals on $X^\*$. If you have $x\_j(x^\*)=0$ for all $j$, then every multiple of $x^\*$ is in the the first set you have in your second paragraph; i.e., $|\phi(tx^\*)| <1$ for all $t$ and hence $\phi(x^\*)=0$. Thus $\phi$ is a linear comb... | 7 | https://mathoverflow.net/users/2554 | 15649 | 10,482 |
https://mathoverflow.net/questions/15658 | 41 | I have heard the claim that the derived category of an abelian category is in general additive but not abelian. If this is true there should be some toy example of a (co)kernel that should be there but isn't, or something to that effect (for that matter, I could ask the same question just about the homotopy category). ... | https://mathoverflow.net/users/361 | How do I know the derived category is NOT abelian? | The following nicely does the trick I think...
**Lemma** Every monomorphism in a triangulated category splits.
Proof: Let $T$ be a triangulated category and suppose that $f\colon x\to y$ is a monomorphism. Complete this to a triangle
$x \stackrel{f}{\to} y \stackrel{g}{\to} z \stackrel{h}{\to} \Sigma x$
then... | 48 | https://mathoverflow.net/users/310 | 15662 | 10,492 |
https://mathoverflow.net/questions/15666 | 6 | If K is a finite extension of $\mathbb Q\_p$ for some prime number $p$, (possibly need $p \neq 2$), $L\_1$ and $L\_2$ totally ramified abelian extension of $K$, $ \pi\_1, \pi\_2$ are respectively the uniformizer that generates each field. Is it true that $ L\_1 L\_2$ is totally ramified iff $Nm\_{L\_1 / K}(\pi\_1)$, $N... | https://mathoverflow.net/users/2701 | When is the composition of two totally ramified extension totally ramified? | Let me give an elementary answer in the case of abelian exponent-$p$ extensions of $K$, where $K$ is a finite extension of $\mathbb{Q}\_p$ containing a primitive $p$-th root $\zeta$ of $1$. This is the basic case, and Kummer theory suffices.
Such extensions correspond to sub-$\mathbb{F}\_p$-spaces in $\overline{K^\ti... | 4 | https://mathoverflow.net/users/2821 | 15668 | 10,496 |
https://mathoverflow.net/questions/15611 | 23 | In his answer to a question about simple proofs of the
Nullstellensatz
([Elementary / Interesting proofs of the Nullstellensatz](https://mathoverflow.net/questions/15226/elementary-interesting-proofs-of-the-nullstellensatz)),
Qiaochu Yuan referred to a really simple proof for the case of an
uncountable algebraically cl... | https://mathoverflow.net/users/2734 | To prove the Nullstellensatz, how can the general case of an arbitrary algebraically closed field be reduced to the easily-proved case of an uncountable algebraically closed field? | These logic/ZFC/model theory arguments seem out of proportion to the task at hand. Let $k$ be a field and $A$ a finitely generated $k$-algebra over a field $k$. We want to prove that there is a $k$-algebra map from $A$ to a finite extension of $k$. Pick an algebraically closed extension field $k'/k$ (e.g., algebraic cl... | 23 | https://mathoverflow.net/users/3927 | 15672 | 10,499 |
https://mathoverflow.net/questions/15673 | 25 | In his very nice article
>
> Peter Roquette,
> History of valuation theory. I. (English summary) Valuation theory and its applications, Vol. I (Saskatoon, SK, 1999), 291--355,
> Fields Inst. Commun., 32, Amer. Math. Soc., Providence, RI, 2002
>
>
>
Roquette states the following result, which he attributes to... | https://mathoverflow.net/users/1149 | An unfamiliar (to me) form of Hensel's Lemma | What you say at the beginning of your post is right: Hensel-Kurschak's lemma may be deduced from some refined version of Hensel's lemma. Actually, it's what Neukirch does in Algebraic Number Theory (see chapter II, corollary 4.7). His proof relies on the following (see 4.6)
Hensel's lemma: Let $(K,|.|)$ be a complete... | 16 | https://mathoverflow.net/users/4069 | 15679 | 10,503 |
https://mathoverflow.net/questions/15664 | 52 | Here is the criteria for a "perfect" graph editor:
* it should be able to perform an automated, but controllable layout
* one is able to make "manual" enforcements to nodes and edges locations when you need it (or at least such fine automated layout so you don't need "manual" enforcements)
* one could add some math s... | https://mathoverflow.net/users/3315 | What is the best graph editor to use in your articles? | I already had about 30 pages of graphs typeset with xymatrix for my thesis before discovering tikz; but was so impressed by it that I was happy to rewrite them all. As well as (imho) looking better, it gave me cross-platform compatibility - xypic seems to need pstricks, so on the mac with TeXshop (which uses pdflatex, ... | 30 | https://mathoverflow.net/users/987 | 15690 | 10,508 |
https://mathoverflow.net/questions/14506 | 3 | Are the Pearson and Spearman rank correlation coefficients related in a specific way for uniform random variables? Specifically, is the relationship $\rho\_{spearman} = 2\*\sin(\frac{\pi}{6}\rho\_{pearson})$? If so, why?
| https://mathoverflow.net/users/3875 | Are the Pearson and Spearman rank correlation coefficients related in a specific way for uniform RVs? | This formula is from Pearson 1907, see e.g.
Rank Correlation and Product-Moment Correlation
Author(s): P. A. P. Moran
Source: Biometrika, Vol. 35, No. 1/2 (May, 1948), pp. 203-206
Johan
| 2 | https://mathoverflow.net/users/4070 | 15695 | 10,512 |
https://mathoverflow.net/questions/15696 | 4 | The usual way of getting a category of metric spaces is to take metric spaces as objects, and the *nonexpansive* maps (ie, functions $f : A \to B$ such that $d\_B(f(a), f(a')) \leq d\_A(a, a')$) as morphisms.
However, for my purposes I'd like to use the Banach fixed point theorem to get a category with a trace struc... | https://mathoverflow.net/users/1610 | "Category" of Nonempty Metric Spaces and Contractive Maps? | Presumably you don't want to allow arbitrary non-expansive maps, otherwise you could simply take that.
One thing that you could do artificially is to take the subcategory of "metric spaces + nonexpansive maps" generated by the strict contractions. This is a bit like adding the unit in to a non-unital ring. That may s... | 3 | https://mathoverflow.net/users/45 | 15699 | 10,515 |
https://mathoverflow.net/questions/15703 | 29 | What did Newton himself do, so that the "Newton polygon" method is named after him?
| https://mathoverflow.net/users/454 | Newton and Newton polygon | The Newton polygon and [Newton's method](http://en.wikipedia.org/wiki/Newton%27s_method) are closely related. The following theorem was first proven by Puiseux:
>
> if $K$ is an algebraically closed field of characteristic zero, then the field of [Puiseux series](http://en.wikipedia.org/wiki/Puiseux_expansion) over... | 23 | https://mathoverflow.net/users/2384 | 15708 | 10,520 |
https://mathoverflow.net/questions/15685 | 14 | From [Model Theory](http://en.wikipedia.org/wiki/Model_theory) article from wikipedia : "A theory is satisfiable if it has a model $ M\models T$ i.e. a structure (of the appropriate signature) which satisfies all the sentences in the set $T$". In [structure definition](http://en.wikipedia.org/wiki/Structure_%28mathemat... | https://mathoverflow.net/users/3811 | Is it necessary that model of theory is a set? | You seem to believe that it is somehow contradictory to have a set model of ZFC inside another model of ZFC. But this belief is mistaken.
As Gerald Edgar correctly points out, the Completeness Theorem of first order logic asserts that if a theory is consistent (i.e. proves no contradiction), then it has a countable ... | 36 | https://mathoverflow.net/users/1946 | 15713 | 10,523 |
https://mathoverflow.net/questions/15707 | 8 | I'm trying to obtain a bound for the order of some finite groups, and part of it comes down to the order of a permutation group of degree $n$ that is nilpotent. I imagine these have to be much smaller than the full symmetric group, and a bound that is sub-exponential in $n$ would seem reasonable (given that permutation... | https://mathoverflow.net/users/4053 | Largest possible order of a nilpotent permutation group? | The paper of Vdovin mentioned by Steve shows that the nilpotent subgroups of the symmetric groups of maximal order are either the Sylow 2-subgroups P(n) of Sym(n), or P(n-3) x Alt(3) when n = 2(2k+1)+1.
Vdovin, E. P. "Large nilpotent subgroups of finite simple groups."
Algebra Log. 39 (2000), no. 5, 526-546, 630; tra... | 7 | https://mathoverflow.net/users/3710 | 15716 | 10,525 |
https://mathoverflow.net/questions/15592 | 23 | It seems that I have the needed example, but I want it to be simple and self-explaining...
>
> Construct a nontrivial complete metric space $X$ with intrinsic metric which has no nontrivial minimizing geodesics.
>
>
>
**Definitions:**
* A metric $d$ is called *intrinsic* if for any two points $x$, $y$ and an... | https://mathoverflow.net/users/1441 | Intrinsic metric with no geodesics | Well, the unit ball in $c\_0$ is almost what you want (there is no unique shortest curve between points). All we need now is to enhance "bypasses" and to give disadvantage to "straight lines". This can easily be done by taking the distance element to be $(2+\sum\_n 2^{-n}x\_n)^{-1}\|dx\|\_\infty$, which is never less t... | 13 | https://mathoverflow.net/users/1131 | 15720 | 10,528 |
https://mathoverflow.net/questions/15687 | 38 | In Toen's and Vezzosi's article *From HAG to DAG: derived moduli stacks* a kind of definition of DAG is given. I am not an expert and can't see what's the relation between DAG and the motivic cohomology ideas of Voevodsky (particularly his category $DG$). It would be nice if somebody could explain that to me.
I addit... | https://mathoverflow.net/users/4011 | What is DAG and what has it to do with the ideas of Voevodsky? | There is a very general nice pattern here:
Let $C$ be a category of test spaces on which we want to model more general spaces. Then
* a "very general space" modeled on $C$ is an object in the [gros](http://ncatlab.org/nlab/show/petit+topos) [∞-topos](http://ncatlab.org/nlab/show/(infinity%2C1)-topos) $Sh\_{(\infty... | 25 | https://mathoverflow.net/users/381 | 15721 | 10,529 |
https://mathoverflow.net/questions/15717 | 3 | This is my first question. It appeared while solving a research problem in cryptography. I am computer science student, so I apologize for lack of mathematical rigor in this question. Thanks for any help.
Consider the Riemann zeta function at s = +1. It diverges, but the expression for the function is $\zeta(1) = \li... | https://mathoverflow.net/users/4074 | Truncated product of $\zeta(1)$? | Formula (8) on [this page](http://mathworld.wolfram.com/MertensConstant.html) gives the result
$$\prod\_{p \le n} \frac1{1-p^{-1}} = e^\gamma \log n \,(1 + o(1)).$$
| 11 | https://mathoverflow.net/users/126667 | 15725 | 10,533 |
https://mathoverflow.net/questions/15701 | 15 | Somehow [this question](https://mathoverflow.net/questions/15444/the-phenomena-of-eventual-counterexamples) made me think of instances of small exceptions in general, and I remembered the statement I heard once that $S\_5,A\_6,S\_6,A\_7,A\_8,S\_8$ are the only instances of symmetric/alternating groups that are not quot... | https://mathoverflow.net/users/1306 | Symmetric groups which are not quotients of Z/2Z*Z/3Z | Yes, this is well known. In fact, there is a name for such groups - this is a (2,3)-generation property. And yes, by now there is a conceptual understanding why all sufficiently large finite simple groups have this property - the basic ideas are outlined in [this](http://www.ams.org/mathscinet/search/publdoc.html?r=1&p... | 10 | https://mathoverflow.net/users/4040 | 15730 | 10,537 |
https://mathoverflow.net/questions/15727 | 7 | When A is an abelian group with trivial G-action (G being a discrete group) we get that Hn(G,A)≅Hn(BG,A). Is there a similar connection between group cohomology and topological cohomology if A is a non-trivial ℤG-module? What *can* we say in that case?
| https://mathoverflow.net/users/3238 | Group cohomology vs. topological cohomology in the case of non-trivial action | This is an example of twisted cohomology. In general, for a generalized cohomology theory (spectrum) E and a space X you can talk about E-twists over X. This is a certain structure on X. Given a particular twist $\tau$, you can then form the $\tau$-twisted cohomology $E^\tau(X)$. This was the subject of a [recent ArXiv... | 10 | https://mathoverflow.net/users/184 | 15735 | 10,540 |
https://mathoverflow.net/questions/15731 | 29 | I am curious to collect examples of equivalent axiomatizations of mathematical structures. The two examples that I have in mind are
1. **Topological Spaces.** These can be defined in terms of open sets, closed sets, neighbourhoods, the Kuratowski closure axioms, etc.
2. **Matroids**. These can be defined via indepen... | https://mathoverflow.net/users/2233 | Cryptomorphisms | The phenomenon that I think you have in mind has a name: **cryptomorphism**. I learned the name from the writings of Gian-Carlo Rota; Rota's favorite example was indeed matroids. Gerald Edgar informs me that the name is due to Garrett Birkhoff.
I think modern mathematics is replete with cryptomorphisms. In my class t... | 24 | https://mathoverflow.net/users/1149 | 15754 | 10,552 |
https://mathoverflow.net/questions/15604 | 5 | I am looking for a (already-studied or interesting) class of Matroids such that
- Class of Gammoids are contained in it
One example would be Strongly-base-orderable Matroids. I would also be grateful if someone knows a class of Matroids such that
* Class of Gammoids are contained in it AND
* It is contained in the ... | https://mathoverflow.net/users/4057 | What interesting class of Matroids are there that contains the class of Gammoids? | The way I understand your question is as follows: you have a property which you believe holds for all matroids (these are rare); you can prove it for gammoids, but your proof does not extend to a certain bigger class of "strongly-base-orderable" (SBO) matroids. I take it you have an example of a SBO matroid where your ... | 4 | https://mathoverflow.net/users/4040 | 15756 | 10,553 |
https://mathoverflow.net/questions/15755 | -2 | Let $U$ is a complete lattice with least element 0.
*Weak partitioning* is a collection $S$ of nonempty subsets of $U$ such that $\forall x\in S: x\cap\bigcup(S\setminus\{x\})=0$.
*Strong partitioning* is a collection $S$ of nonempty subsets of $U$ such that $\forall A,B\in PS:(A\cap B=\emptyset \Rightarrow \bigcup... | https://mathoverflow.net/users/4086 | Weak partitioning vs. strong partitioning | If the lattice $U$ satisfies the meet distributive law
$$x \wedge \bigvee\_{i \in I} y\_i = \bigvee\_{i \in I} x \wedge y\_i$$
where $(y\_i)\_{i \in I}$ is an arbitrary collection of elements of $U$, then "weak partitioning" implies "strong partitioning." More precisely, you only need the above to hold when the right h... | 6 | https://mathoverflow.net/users/2000 | 15765 | 10,559 |
https://mathoverflow.net/questions/15751 | 2 | Let A is a complete lattice.
I call a subset $S$ of A *filter-closed* when for every filter base $T$ in $S$ we have $\bigcap T\in S$. (A *filter base* is a nonempty, down directed set.)
I call a subset $S$ of A *chain-closed* when for every non-empty chain $T$ in $S$ we have $\bigcap T\in S$.
**Conjecture** $S$ i... | https://mathoverflow.net/users/4086 | Filter-closed vs. chain-closed | Indeed, your conjecture is correct.
**Theorem.** If L is a complete lattice and S is a subset of L, then S is chain-closed iff S is filter-closed.
Proof. Clearly filter-closed implies chain-closed, since every chain is a filter base. Conversely, suppose that S is chain-closed, and that A is a filter base contained... | 6 | https://mathoverflow.net/users/1946 | 15769 | 10,562 |
https://mathoverflow.net/questions/13812 | 1 | $$f(a,x)=\sum\_{\tau=-\infty}^{\infty}\frac{\exp\left(2\pi i\tau x\right)}{(\tau+a)^{p+1}}$$
Can I apply Euler-Maclauren formula to this sum?
where $a\in(0,0.5)$, p is a natural number, and $x$ is a real number
| https://mathoverflow.net/users/3589 | Sum and interpolation of hurwitz zeta functions | Well, if $p$ is an integer, you should realize that $\frac{1}{(\tau+a)^{p+1}}$ can be obtained by integrating $Q(x)e^{-2\pi iax}$ against $e^{-2\pi i\tau x}$ where $Q(x)$ is the (unique) polynomial of degree $p$ satisfying $Q^{(m)}(0)=e^{-2\pi ia}Q^{(m)}(1)$ for $m<p$ and $Q^{(p)}=\frac{(2\pi i)^{p+1}}{e^{-2\pi ia}-1}$... | 1 | https://mathoverflow.net/users/1131 | 15782 | 10,571 |
https://mathoverflow.net/questions/15781 | 14 | Given a number field $K$, when is its Hilbert class field an abelian extension of $\mathbb{Q}$? I am going to be on the road soon, so pleas don't be offended if I don't respond quickly to a comment.
| https://mathoverflow.net/users/4092 | Given a number field $K$, when is its Hilbert class field an abelian extension of $\mathbb{Q}$? | The genus class field of an extension $K/F$ is defined to be the largest extension $L/K$ with the following properties:
1. $L/K$ is unramified
2. $L$ is the compositum of $K/F$ and an abelian extension $A/F$.
Thus the quick answer to your question is: the Hilbert class field of $K$ is abelian over ${\mathbb Q}$ if ... | 18 | https://mathoverflow.net/users/3503 | 15794 | 10,581 |
https://mathoverflow.net/questions/15804 | 9 | Let $\mathbb{P}$ be a probability measure on some probability space $(\Omega,\mathcal{A})$. Are there conditions on the $\sigma$-algebra $\mathcal{A}$ such that for every real number $c\in [0,1]$ we find a set $A\in\mathcal{A}$ with $\mathbb{P}(A)=c$.
It is like the intermediate value theorem for continuous functions.... | https://mathoverflow.net/users/4097 | When does a probability measure take all values in the unit interval? | A measure space $(\mathbb{P},\Omega,\mathcal{A})$ is atomless if for all $A\in\mathcal{A}$ with $\mathbb{P}(A)>0$ there exists $B\subset A, B\in\mathcal{A}$ such that $0<\mathbb{P}(B)<\mathbb{P}(A)$. Now according to a theorem of Sierpinski, the values of an atomless measure space form an interval. In particular, for p... | 17 | https://mathoverflow.net/users/35357 | 15808 | 10,588 |
https://mathoverflow.net/questions/15795 | 17 | This question was inspired by and is somewhat related to [this question](https://mathoverflow.net/questions/14944/have-people-successfully-worked-with-the-full-ring-of-diferential-operators-in-ch/).
In his article "Crystals and the de Rham cohomology of schemes" in the collection "Dix exposes sur la cohomologie des s... | https://mathoverflow.net/users/259 | The Infinitesimal topos in positive characteristic | There is a paper of Ogus for 1975, "The cohomology of the infinitesimal site",
in which he shows that if $X$ is proper over an algebraically closed field $k$ of char. $p$, and embeds into a smooth
scheme over $k$, then the infinitesimal cohomology of $X$ coincides with etale cohomology with coefficients in $k$ (or more... | 24 | https://mathoverflow.net/users/2874 | 15811 | 10,590 |
https://mathoverflow.net/questions/15812 | 1 | Following the [blow](https://mathoverflow.net/questions/15685/is-it-necessary-that-model-of-theory-is-a-set). I will try to ask question in order to check if I well understand what was pointed. I decide to ask another question, because mathoverflow is not projected to be good environment for discussion, so it would be ... | https://mathoverflow.net/users/3811 | Theory interpreted in non-set domain of discourse may be consistent? | It appears that you yearn to study various first-order theories, but do not want to be constrained by any requirement that your models, or domains of discourse, be sets. There are several ways to take such a proposal.
On the one hand, many mathematicians have yearnings similar to yours, and this has led them to try ... | 6 | https://mathoverflow.net/users/1946 | 15813 | 10,591 |
https://mathoverflow.net/questions/15612 | 33 | Suppose a compact convex body $P \subset \Bbb R^3$ has only polygonal orthogonal projections onto a plane. Does this imply that $P$ is a convex polytope?
This question occurred to me when I was making exercises for [my book](http://www.math.ucla.edu/~pak/book.htm). I figured this is probably easy and well known, but ... | https://mathoverflow.net/users/4040 | Do plane projections determine a convex polytope? | Theorem 4.1 of [this paper by Klee](http://www.ams.org/mathscinet-getitem?mr=105651) says yes. Moreover, the result generalizes to higher dimensions for projections of arbitrary dimension $\ge 2$.
| 21 | https://mathoverflow.net/users/1044 | 15815 | 10,593 |
https://mathoverflow.net/questions/15802 | 9 | A short while ago, Dvir proved the Kakeya conjecture over finite fields. Does this have any implications for restriction theorems over finite fields? I am aware only of implications going in the opposite direction (Mockenhaupt-Tao). Is there anything new known about restriction theorems over finite fields?
| https://mathoverflow.net/users/398 | Restriction theorems over finite fields | Not directly. The key connection between restriction and Kakeya in Euclidean settings is that thanks to Taylor expansion, a surface in Euclidean space looks locally flat, and so the Fourier transform of measures on that surface are a superposition of Fourier transforms of very flat measures, which by the uncertainty pr... | 13 | https://mathoverflow.net/users/766 | 15816 | 10,594 |
https://mathoverflow.net/questions/15805 | 4 | Given any CAT(0) space $X$, we can define a map $s:X\times X\times [0;1]\rightarrow X$, such that $s(x,y,-)$ is the constant speed geodesic from $x$ to $y$ . Any isometry $f$ of $X$ is compatible with that map in the sense, that $s(f(x),f(y),t)=f(s(x,y,t))$. Then one can ask, whether any self-homeomorphism of $X$, whic... | https://mathoverflow.net/users/3969 | Are isometries the only geodesic preserving maps in a CAT(0)-space? | The map which you call "geodesic preserving" is usually called "affine".
It seems that affine maps to the real line are well understood even for general length space.
For your later edit: you may always take two spaces which admit self-similar maps and consider map on the product which move each coordinate with diffe... | 5 | https://mathoverflow.net/users/1441 | 15822 | 10,597 |
https://mathoverflow.net/questions/15800 | 20 | Which $6j$-symbols for quantised enveloping algebras are known explicitly?
The quantum $6j$-symbols for $sl(2)$ are well-known. The references are
[Masbaum and Vogel](http://projecteuclid.org/euclid.pjm/1102622100) and
[Frenkel and Khovanov](http://projecteuclid.org/euclid.dmj/1077242324).
What is known for other... | https://mathoverflow.net/users/3992 | Calculating 6j-symbols (aka Racah-Wigner coefficients) for quantum groups | We can also use partitions ($k$) (symmetric powers) instead of ($1^k$) on one or two of the edges. This still gives just scalars, but includes the full story for sl(2).
This problem seems to be equivalent to the problem of computing the exchange operator
in the tensor product of two (quantum) symmetric or exterior ... | 14 | https://mathoverflow.net/users/3696 | 15825 | 10,600 |
https://mathoverflow.net/questions/15830 | 11 | "Let Top be the category of topological spaces." If I see a definition like this, in which homeomorphic (isomorphic in the category) spaces are not identified together, then for each given topological space, how many times does it show up as an object in Top? Once? Countably many times? Uncountably many times? Is there... | https://mathoverflow.net/users/1874 | Confusion over a point in basic category theory | Well, such categories are actually proper classes. For instance, spaces that are homeomorphic to $\mathbb{R}$ will be given on ANY set of the same cardinality. So how many there are, depends on how many such sets there are...but there are quite a lot of sets. In fact, more than any individual set worth. (Though on the ... | 17 | https://mathoverflow.net/users/622 | 15831 | 10,605 |
https://mathoverflow.net/questions/15821 | 4 | I have a set *S* of real numbers, and I would like to create a new set *R* with exactly n real numbers (not necessarily from the set) that represent it best.
What I mean by best?
Well, I have query that asks for given point what is the closest point from S to that point, and when I ask same thing for set R I would ... | https://mathoverflow.net/users/4102 | Approximating a set with fixed number of elements | This is the $k$-center problem (or in your notation, the $n$-center problem). you're given a set $S$ of points, and you want to find a set $R$ of $n$ points such that the set of balls of radius $r$ around each point in $R$ cover all of $S$, and $r$ is minimized.
Your metric space is the line, so this problem is rela... | 5 | https://mathoverflow.net/users/972 | 15838 | 10,609 |
https://mathoverflow.net/questions/15767 | 6 | Let V be a complex affine variety given as the vanishing set of a set of polynomials with integral coefficients. I have 3 questions.
1)
Under what assumption will the dimension of V over C remain the same as the dimension of V computed over some finite field like Z\_P, assuming that the prime P does not divide any ... | https://mathoverflow.net/users/4089 | Concerning the dimension of a complex variety modulo a prime | To make more precise the answer of Felip. You have a scheme $X=Spec(A)$ over $\mathbb Z$, where $A$ is a finitely generate $\mathbb Z$-algebra such that its generic fiber $X\_{\mathbb Q}$ (just consider your polynomials as polynomials with rational coefficients) gives $V$ by field extension $\mathbb{C}/\mathbb{Q}$. Of ... | 6 | https://mathoverflow.net/users/3485 | 15843 | 10,613 |
https://mathoverflow.net/questions/15443 | 3 | Why if one have an $\varepsilon$-expansive homeomorphism $T:X \rightarrow X$ ($X$ a compact metric space) and a given partition $D$ of $X$ which has diameter smaller than $\varepsilon$ the sequence of refined partitions $D\_n = \bigvee\_{i = -n}^n T^{-i} D$ has diameter converging to zero ?
Recall that a $\varepsilon... | https://mathoverflow.net/users/4031 | Partitions and Expansiveness | Fix $\delta>0$ and let
$$
A\_n=\{(x,y): d(x,y)\ge\delta, d(T^ix,T^iy)\le\varepsilon,|i|\le n\}
$$
Sets $A\_n$ are closed, nested and, by expansiveness, $\cap A\_n=\varnothing$. Hence $A\_N=\varnothing$ for some $N=N(\delta)$. Clearly, $diam D\_N\le\delta$.
| 3 | https://mathoverflow.net/users/2029 | 15854 | 10,618 |
https://mathoverflow.net/questions/15841 | 76 | This question is about the space of all topologies on a
fixed set X. We may order the topologies by refinement, so
that τ ≤ σ just in case every τ open set is open in σ.
Equivalently, we say in this case that τ is *coarser*
than σ, that σ is *finer* than τ or that
σ *refines* τ. (See [wikipedia on comparison of
topolog... | https://mathoverflow.net/users/1946 | How do the compact Hausdorff topologies sit in the lattice of all topologies on a set? | **This is a community wiki of the answers in the comments.**
* The compact Hausdorff topologies do not generally form a maximal antichain. If X is infinite, split X into two infinite halves and put the discrete topology on one half and the indiscrete topology on the other half. (Comment by François G. Dorais) **Adden... | 39 | https://mathoverflow.net/users/2000 | 15857 | 10,620 |
https://mathoverflow.net/questions/15600 | 3 | There is an algorithm that give us cuboids in $\mathbb{R}^3$, say $Q\_1,Q\_2,\ldots$, such that $\cup\_{i=1}^{\infty} Q\_i$ is the simplex with vertices $(0,0,0), (1,0,0) , (0,1,0), (0,0,1)$, and the $Q\_i$'s are almost disjoint (i.e $\lambda(Q\_i\cap Q\_j)=0$ if $i\neq j$)?
In $\mathbb{R}^2$ there are many easy way... | https://mathoverflow.net/users/4056 | How to fill a simplex with almost disjoint cuboids? | OK, since we finally have figured out what Andres is asking and since 600 characters is a bit too restrictive, I'll post this as an answer.
The following Asymptote code will draw the filling except I used the size 4 simplex instead of size 1 one here:
```
size(400);
import three;
import graph3;
pen[] q={red,gre... | 3 | https://mathoverflow.net/users/1131 | 15866 | 10,627 |
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