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https://mathoverflow.net/questions/55328 | 1 | I have some trading data in the form of (exchange rate, volume, time) tuples. I'm trying to estimate the rate of change of the exchange rate. Of course the trade data is non-uniformly sampled.
Also, the function is rather noisy, so the estimation has to be robust.
So what are good ways of estimating the derivative ... | https://mathoverflow.net/users/180 | Estimating the derivative of a noisy, non-uniformly sampled function | The magic words are: "Kalman filter" (this solves this problem in a relatively simple setting, there are a number of extensions, many of them proprietary). The wikipedia article on Kalman filtering, and the references therein, is a good place to start.
| 1 | https://mathoverflow.net/users/11142 | 55377 | 34,584 |
https://mathoverflow.net/questions/55392 | 15 | In his *Set Theory. An Introduction to Indepencence Proofs*, Kunen develops $ZFC$ from a platonistic point of view because he believes that this is pedagogically easier. When he talks about the intended interpretation of set theory he says such things as, for example, that the domain of discourse $V$ is the collection ... | https://mathoverflow.net/users/6466 | Intended interpretations of set theories | While Kunen takes for universe the collection of all hereditary sets, Marc proposes to restrict the universe to those hereditary sets which are first-order definable without parameters (Marc's "provable sets"). To make this more mathematical, let me rephrase the question as follows:
>
> Suppose $M$ is a model of ZF... | 12 | https://mathoverflow.net/users/2000 | 55394 | 34,592 |
https://mathoverflow.net/questions/55390 | 7 | Let $F$ be a finite extension of $Q\_2$ (2-adic field) or $F\_2((x))$ (function field). Let $E/F$ be a separable extension of degree $2$.
1. What is the image of the norm map $N\_{E/F}$?
2. In particular - is it true that the index $[F^{\star} : N\_{E/F}(E^{\star})]$ depends only on the ramification $e(E|F)$?
| https://mathoverflow.net/users/7386 | Image of norm map for local field | As KConrad points out, you perhaps mean to say that $F$ is a *finite* extension of $\mathbf{Q}\_2$ or of $\mathbf{F}\_2((x))$, and that the quadratic extesnions $E|F$ is *separable* (and hence galoisian) in the second case.
With this interpretation of the question, $N\_{E|F}(E^\times)$ is a closed subgroup of index $... | 7 | https://mathoverflow.net/users/2821 | 55399 | 34,596 |
https://mathoverflow.net/questions/55397 | 11 | Is the number $\sum\_{n=1}^\infty \frac{1}{2^{n^2}}$ known to be transcendental?
Is there a survey with up-to-date transcendence results?
| https://mathoverflow.net/users/2631 | Is this number already known to be transcendental? Is there a survey about up-to-date trascendence results? | I have checked with [*Introduction to Algebraic Independence Theory*](http://books.google.com.ua/books?id=liYae-vUZs4C&printsec=frontcover&dq=introduction+to+algebraic+independence+theory&source=bl&ots=6Kf4QYSDlS&sig=bJ2TfJ5Fi54ktqf56svrBYIkXyk&hl=en&ei=2BNZTfDtAcWSOq_PpPoE&sa=X&oi=book_result&ct=result&resnum=1&ved=0C... | 26 | https://mathoverflow.net/users/5371 | 55400 | 34,597 |
https://mathoverflow.net/questions/55404 | 15 | I have been searching for a version of the isoperimetric inequality which is something like:
$P(\Omega) - 2\sqrt{\pi} Vol(\Omega)^{1/2} \geq \pi (r\_{out}^2 - r\_{in}^2)$ where $r\_{out}$ and $r\_{in}$ are the inner and outer radius of a given set. There are of course details which I am missing such as what kind of s... | https://mathoverflow.net/users/8755 | Stronger version of the isoperimetric inequality | A classical result along these lines is [Bonnesen's inequality](http://en.wikipedia.org/wiki/Bonnesen%27s_inequality), which states
$$
L^2 - 4\pi A \ge \pi^2 (r\_{out} - r\_{in})^2,
$$
where $L$ is the length and $A$ is the enclosed area of a simple planar closed curve. There are many other results along these lines, c... | 16 | https://mathoverflow.net/users/1044 | 55412 | 34,603 |
https://mathoverflow.net/questions/55332 | 5 | Most mathematicians seem to be contented with the fact, that abstract structures cannot be directly modelled as such in a set theory without ur-elements. What seems to me the standard stance: Set theory works pretty well without ur-elements, and it's simpler without them.
Given an abstract (uninterpreted) theory ther... | https://mathoverflow.net/users/2672 | Concrete models of abstract structures | I take your question to be about what we might call the
*structuralist* perspective, the view that we specify
mathematical objects and structures by their defining
structural features, ignoring any internal or otherwise
irrelevant structure that an instantiation of the object
might exhibit. You perceive a tension betwe... | 13 | https://mathoverflow.net/users/1946 | 55417 | 34,606 |
https://mathoverflow.net/questions/55408 | 7 | If we have an extension of groups (say algebraic groups or group schemes) $1\to F\to P\to G\to 1$, then $P$ is a principal $F$-bundle over $G$ (is it locally trivial?). How about going in the opposite direction?
>
> **Question.** Let $F$ and $G$ be groups. $P$ be a principal $F$-bundle over $G$. When does $P$ carry... | https://mathoverflow.net/users/3847 | Principal bundles over groups | There is the Grothendieck theory of bitorsors (see e.g. SGA 7, Exp VII) which gives an abstract answer to this question. The key point is that when you have an actual group extension you do not just have a torsor but a bitorsor, i.e., $P$ is an $F$-torsor in two ways given by left and right multiplication. The group st... | 4 | https://mathoverflow.net/users/4008 | 55419 | 34,608 |
https://mathoverflow.net/questions/53126 | 15 | Let $G$ be a finite group. Then the irreducible complex representations of $G$ come in three sorts: real, complex and symplectic=quaternionic. The type of an irreducible character $\chi$ can be read of from the Frobenius-Schur indicator
$$ s\_2(\chi) = \frac{1}{|G|}\sum\_{g\in G} \chi( g^2 ) \in \{ 1,0,-1 \}. $$
Now t... | https://mathoverflow.net/users/10266 | Finite groups in which every character has real values: grading the representations | As Zoltan suspected in his answer, the statement in my question is not true. I have now found counterexamples: Let $q$ be an odd prime power. Then $SL(2,q)$ contains a group isomorphic to the quaternion group $Q\_8$, which yields a (semiregular) action of $Q\_8$ on $V=(\mathbb{F}\_q)^2$.
Let $G = V \rtimes Q\_8$ be th... | 9 | https://mathoverflow.net/users/10266 | 55420 | 34,609 |
https://mathoverflow.net/questions/55371 | 7 | Suppose $(R,m)$ is a regular, local ring. Let $x\_1,x\_2,...,x\_n$ be a regular system of parameters. Let $I$ be an ideal generated by squarefree monomials in the $x\_i$'s. Is $I$ a radical ideal? The motivation for this is the polynomial ring in finitely many indeterminates over a field (which although not local is re... | https://mathoverflow.net/users/12800 | Are squarefree monomial ideals on a regular system of parameters in a regular local ring radical? | ADDED: here is a proof of the statement you need (namely the square free monomial ideal $I$ is a intersection of primes generated by subsets of parameters) without using the modularity property. We will use induction on $N=$ the total numbers of times the parameters appear in the generators of $I$. For example if $I=(x... | 4 | https://mathoverflow.net/users/2083 | 55422 | 34,611 |
https://mathoverflow.net/questions/55294 | 18 | I've been looking at various general strategies for proving that some category is triangulated, and Lurie manages to prove that a huge class of interesting examples of categories that we know about are triangulated in his book *Higher Algebra* (formerly DAG I-IV and VI).(EDIT: here's a link to the [book](http://www.mat... | https://mathoverflow.net/users/6936 | Proof that the homotopy category of a stable $\infty$-category is triangulated | Alright, here's a proof and construction: Suppose we're given a $2$-simplex $X\to Y\to Z$ in $\mathcal{C}$. We have a lemma:
Every $2$-simplex in $\Delta^2\to \mathcal{C}$ can be (right Kan-)extended by zeroes via the map $\Delta^2\hookrightarrow \Delta^1\vee \Delta^3$, which gives diagrams of the form:
$$0\leftar... | 15 | https://mathoverflow.net/users/1353 | 55426 | 34,614 |
https://mathoverflow.net/questions/55415 | 10 | Primitive recursive functions are syntactically constructible in the sense that from a set of "axioms" we can build every function in the set $PR$. This basicly means that we can build a machine that prints the definition for every function in $PR$.
Now, we can build hierarchies in the set $PR$ by adding some semanti... | https://mathoverflow.net/users/nan | Syntactically capturing complexity classes | If you are interested in characterizations using recursion operators like in the definition of primitive recursive functions or the Grzegorczyk hierarchy, Cobham characterized P (or rather, FP, as a class of functions on binary strings) as the closure of a handful of initial functions under composition and limited recu... | 6 | https://mathoverflow.net/users/12705 | 55433 | 34,620 |
https://mathoverflow.net/questions/55435 | 7 | I've recently become interested in the elementary theory of groups due to Sela and Myasnikov-Kharlampovich's work with free groups. I'd like a good introduction to the field of the elementary theory of groups, and in particular I'd like a reference to contain examples of group properties that cannot be read from a grou... | https://mathoverflow.net/users/8434 | need a good reference for introduction to elementary theory of groups | I learned a lot from reading Bestvina and Feighn's article [Notes on Sela's work: Limit groups and Makanin-Razborov diagrams](http://arxiv.org/abs/0809.0467). It's not a broad introduction to elementary theory, but it does express some of Sela's ideas quite succinctly. You may need a background in geometric group theor... | 5 | https://mathoverflow.net/users/1463 | 55440 | 34,622 |
https://mathoverflow.net/questions/55437 | 11 | The eigencurve $\mathcal{E}$ is a rigid-analytic space parametrizing certain $p$-adic families of modular forms and associated Galois representations. By constructing an auxiliary reduced rigid curve locally isomorphic to $\mathcal{E}$ it was shown that $\mathcal{E}$ is a curve. There is a morphism of rigid spaces $f: ... | https://mathoverflow.net/users/12235 | Consequences of the geometric properties of the eigencurve | The eigencurve is an honest moduli space---it parametrises families of finite slope overconvergent modular eigenforms (or more precisely, of systems of overconvergent finite slope Hecke eigenvalues)---but I know of no "natural" properties of p-adic modular forms that one can deduce from any geometric structure, other t... | 11 | https://mathoverflow.net/users/1384 | 55441 | 34,623 |
https://mathoverflow.net/questions/55438 | 7 | Let $\mathfrak{g}$ be a $k$-Lie algebra, and $Q: \bigwedge^2 \mathfrak{g}^\* \rightarrow k$; define $U\_Q(\mathfrak{g})$ to be the quotient of the full tensor algebra over $\mathfrak{g}$ by the ideal generated by elements of the form $x\otimes y - y \otimes x -[x,y] - Q(x,y)$. This definition does not depends properly ... | https://mathoverflow.net/users/12830 | "Twisted" universal enveloping algebra? | These algebras were considered by Ramaiengar Sridharan a long time ago. See [Sridharan, R. Filtered algebras and representations of Lie algebras. Trans. Amer. Math. Soc. 100 1961 530--550. [MR0130900](http://www.ams.org/mathscinet-getitem?mr=MR0130900) (24 #A754)]
If the map you are using to twist is not a Chevalley-... | 10 | https://mathoverflow.net/users/1409 | 55444 | 34,625 |
https://mathoverflow.net/questions/55442 | 8 | Given a symplectic manifold $(X, \omega)$ and a group $G$ acting on $X$ preserving the symplectic form, we define the moment map $\mu : X \to \mathfrak{g}^\*$ so that
$$
\langle d\mu(v), \xi\rangle = \omega\big(\xi^\*(x), v\big)
$$
where $\xi \in \mathfrak{g}$, $\xi^\*$ is the vector field generated by $\xi$, and $v \i... | https://mathoverflow.net/users/1703 | Why can we define the moment map in this way (i.e. why is this form exact)? | Both answers are "No."
There are well-known obstructions to the existence of an equivariant momentum mapping arising from the action by symplectomorphisms of a group $G$ on a symplectic manifold. They can be phrased in many ways, but if $G$ is connected and its Lie algebra is semisimple, for example, the obstructions... | 17 | https://mathoverflow.net/users/394 | 55446 | 34,626 |
https://mathoverflow.net/questions/55454 | 9 | Let $X$ be a smooth projective curve over a field $k$. We let $\omega$ be the canonical
line bundle of $X$ and we denote by $F$ the field of $k$-valued rational functions on $X$.
(1) When $k$ is algebraically closed then $\omega$ is a dualizing sheaf for $X$. From there it is
easy to prove Riemann-Roch for regular (... | https://mathoverflow.net/users/11765 | Is there a Riemann-Roch for smooth projective curves over an arbitrary field? | Yes. There is a Riemann-Roch for smooth projective curves over arbitrary fields. It was proved by the German school of function fields in the 30's. From (2) I deduce that you've been reading Weil's "Basic Number Theory". Anyway, the proof that Weil gives there is a shortened version of a proof he gave of the full theor... | 18 | https://mathoverflow.net/users/2290 | 55461 | 34,632 |
https://mathoverflow.net/questions/55455 | 4 | Hi,
Does anyone have an idea about an exact or approximate formulae for the following summation?
$$
\sum\_{j=1}^n \frac{j^k}{(j-1)!}
$$
where k is a positive integer (the denominator of the j^th term is of course $\Gamma(j)$).
| https://mathoverflow.net/users/12981 | Summation of an expression | **REVISED ANSWER.**
In retrospect, deriving the approximation is quite easy. Indeed,
$$
\sum\limits\_{j = 1}^n {\frac{{j^k }}{{(j - 1)!}}} = e\sum\limits\_{j = 1}^n {e^{ - 1} \frac{{j^{k + 1} }}{{j!}}} \approx e\sum\limits\_{j = 0}^\infty {e^{ - 1} \frac{{j^{k + 1} }}{{j!}}} = eB\_{k + 1}.
$$
(For large $k$, you may ... | 10 | https://mathoverflow.net/users/10227 | 55464 | 34,634 |
https://mathoverflow.net/questions/55450 | 7 | Let $f$ be a positive real-analytic function on the closed unit disk. Consider the eigenvalue problem $\Delta \phi = \lambda f \phi$,
with $\phi = 0$ on the boundary. There exists a sequence of eigenvalues $\lambda\_n$. Now suppose $f$ depends real-analytically on a parameter $t$ for $t$ in some interval containing $0$... | https://mathoverflow.net/users/12669 | dependence of eigenvalues on parameters | In your case I think you can apply Rellich's theorem, that is Theorem VII.3.9 in Kato's book (p.392 in my edition). The result states that, whenever you have a family of selfadjoint operators with compact resolvent, depending analytically on a real parameter on some open interval of the reals, with a common domain inde... | 7 | https://mathoverflow.net/users/7294 | 55469 | 34,638 |
https://mathoverflow.net/questions/55468 | 3 | Suppose $\mathcal{L}$ is a bounded linear operator and I have the solution to Eigenvalue problem
$\mathcal{L} \phi + \lambda \phi = 0$
wish to solve the following PDE
$\left(-\partial\_t + \mathcal{L}\right)u = 0$.
If the spectrum of $\mathcal{L}$ is continuous or discrete, then a general solution to the PDE is... | https://mathoverflow.net/users/12983 | question about mixed spectrum of a linear operator $\mathcal{L}$ | Well, I have to confess that I am not sure about your goals. In which space are you working? But there is always a way to represent the solution as a generazied exponential function, called operator semigroup. If your operator $\mathcal L$ is indeed bounded, then you can represent the solution via Dunford-Riesz calculu... | 2 | https://mathoverflow.net/users/12898 | 55472 | 34,641 |
https://mathoverflow.net/questions/55459 | 4 | Tunnel's result on the congruent number problem hinges on the fact that there are modular forms with fourier coefficients related to the values $L(E\_n,1)$.
Is there an interesting function that has coefficients related to $L'(E\_n,1)$ instead? (for a reasonable definition of "interesting" and "related")
This is in... | https://mathoverflow.net/users/2024 | Tunnel like theorem: is there an interesting function with fourier coefficients related to $L'(E_n,1)$ instead of $L(E_n,1)$? | Yes. Bruinier and Ono have shown in their paper "Heegner divisors, L-functions and harmonic weak Maass forms" that the vanishing or nonvanishing of central derivatives of twisted L-functions like this is related to the algebraicity properties of coefficients of a certain harmonic weak Maass form. You should also look a... | 4 | https://mathoverflow.net/users/1464 | 55475 | 34,644 |
https://mathoverflow.net/questions/55462 | 6 | Suppose $f: X\rightarrow S$ is of finite type, S is Noetherian. Now X=Spec B is affine, but the morphism f is not an affine morphism. S is not affine (or really f does not factor through any affine subscheme on S), nor is S of finite type over $\mathbb Z$. Now my question is, can we have an immersion of $X\rightarrow \... | https://mathoverflow.net/users/1877 | Immerse an affine schemes into $A^n_S$ | As noticed by Mattia, you have to suppose $f$ affine. Now under the further assumption that $S$ is separated and $X$ is affine, there exists a closed immersion as you want (Mattia's arguments then work): for any affine open subset $V$ of $S$, the canonical morphism
$$f^{-1}(V)=X\times\_S V\to X\times\_{\mathbb Z} V$$ ... | 8 | https://mathoverflow.net/users/3485 | 55478 | 34,647 |
https://mathoverflow.net/questions/55480 | 9 | Is there a specific formula/method to find geodesics for a Homogeneous space? (excluding general methods applicable to arbitrary riemannian manifold)
| https://mathoverflow.net/users/12573 | Geodesics for a Homogeneous Space? | I assume you mean riemannian homogeneous space and you are talking about geodesics relative to the Levi-Civita connection.
If so, you can always try to find geodesics which are **homogeneous**; that is, geodesics which correspond to the orbits of one-parameter subgroups.
Let $M = G/H$ and let $\mathfrak{g} = \mathf... | 11 | https://mathoverflow.net/users/394 | 55482 | 34,649 |
https://mathoverflow.net/questions/49303 | 36 | The story of the analogy between knots and primes, which now has a literature, started with an unpublished note by Barry Mazur.
I'm not absolutely sure this is the one I mean, but in his paper, Analogies between group actions on 3-manifolds and number fields, Adam Sikora cites
B. Mazur, Remarks on the Alexander pol... | https://mathoverflow.net/users/10909 | Mazur's unpublished manuscript on primes and knots? | This showed up in my snail-mail today, so I'm sharing the wealth:
<http://ifile.it/rodc5is/mazur.pdf>
| 23 | https://mathoverflow.net/users/10909 | 55488 | 34,654 |
https://mathoverflow.net/questions/55453 | 4 | Guys,
I have N < 2^n randomly generated n-bit numbers stored in a file the lookup for which is expensive. Given a number Y, I have to search for a number in the file that is at most k hamming dist. from Y. Now this calls for a C(n 1) + C(n 2) + C(n 3)...+C(n,k) worst case lookups which is not feasible in my case. I t... | https://mathoverflow.net/users/12980 | finding numbers at k hamming distance | If you're willing to live with approximations, then the standard approach to near-neighbor search (or in your case fixed radius search) in a Hamming space is by using [locality-sensitive hashing](http://www.mit.edu/~andoni/LSH/). Your case is even simpler because you know the radius you're concerned with. Alternatives ... | 4 | https://mathoverflow.net/users/972 | 55494 | 34,660 |
https://mathoverflow.net/questions/36451 | 6 | Let $V$ be a smooth algebraic variety defined over complex numbers. Suppose that $G$ admits a free discrete action on $V$ so that $V/G$ is compact and Kahler (or algebraic). Is it ture that $G$ is virtually abelian (i.e., contains an abelian subgroup of finite index)?
If we don't ask $V/G$ to be algebraic (Kahler) th... | https://mathoverflow.net/users/943 | Smooth algebraic varieties with smooth Kahler quotients. | The following article <http://arxiv.org/abs/1102.2762> of BENOÎT CLAUDON, ANDREAS HÖRING, AND JÁNOS KOLLÁR answers positively the question, provided $V/G$ is projective, $V$ is quasiprojective and $\pi\_1(V)=0$. Moreover, assuming abundance conjecture, they prove under the above conditions that $V$ is biholomophric to ... | 5 | https://mathoverflow.net/users/943 | 55495 | 34,661 |
https://mathoverflow.net/questions/54788 | 7 | Let C be a cyclic subgroup of S\_n.
How does the representation $Ind\_C^{S\_n}\rho$, where $\rho$ is some representation of $C$, decompose into irreducible components?
Is there are a way to know which components appear with multiplicity 1?
| https://mathoverflow.net/users/4246 | Decomposition of induced representations in S_n | There is a combinatorial way to decompose $Res\_C^{S\_n}S^\lambda$ for an irreducible $S\_n$-module $S^\lambda$. We use the notion of the "major index" for a standard tabuleau of shape $\lambda$. If $C=C\_n$, the result is obtained by [Kraśkiewiz-Weyman](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&c... | 8 | https://mathoverflow.net/users/12988 | 55506 | 34,669 |
https://mathoverflow.net/questions/55432 | 5 | Consider the statement: "ZFC is consistent". Normally this is considered at first sight as a statement in the metatheory. But if we follow Kunen's (informally) description of what the metatheory is (i.e., finitistic reasoning), there seem to be some problems to place that statement precisely at the metatheoretical leve... | https://mathoverflow.net/users/12976 | Should consistency be considered as a concept in the metatheory? | Most people regard induction and recursion as finitistic. As Andrej Bauer commented:
>
> Induction and recursion are rules which tell you how to do something. They do not presuppose any kind of infinity or anything like that.
>
>
>
As a consequence, the standard weak metatheory used in practice is [Primitive R... | 7 | https://mathoverflow.net/users/2000 | 55507 | 34,670 |
https://mathoverflow.net/questions/55517 | 0 | Does anybody know any example of a semistable and nonsimple sheaf on a K3 surface $S$?
| https://mathoverflow.net/users/33841 | Semistable sheaves on a K3 which are not simple | The trival sheaf $\mathcal O\_S\oplus\mathcal O\_S$ is one: it has non-trivial global endomorphism algebra $gl\_2(k)$ and is semi-stable.
| 6 | https://mathoverflow.net/users/8726 | 55519 | 34,675 |
https://mathoverflow.net/questions/55458 | 23 | I am very interested in proofs. I have taken an undergraduate course
called "Logic and Set Theory" which I found very interesting, but ultimately
unsatisfying. My biggest disappointment has to do with the language in which
proofs are expressed. It seems to me that we have all of the symbols necessary
to express a proof... | https://mathoverflow.net/users/2377 | Writing "Semi-Formal" Proofs | The question becomes interesting when it is interpreted as a technical question about the extent to which we can have a semi-formal language somehow in-between the truly formal proofs, which are largely unreadable by humans, and the informal proofs used by professional mathematicians.
In fact, there has been some tr... | 43 | https://mathoverflow.net/users/1946 | 55523 | 34,679 |
https://mathoverflow.net/questions/55526 | 23 | I know the definition of $K\_X$ on a normal, singular variety, but I don't have a good set of examples in my mind. What's an example of a variety where $K\_X$ is $\mathbb Q$-Cartier but not Cartier? Are there any conditions under which an adjunction formula lets me compute the canonical class of a singular divisor?
| https://mathoverflow.net/users/12992 | Example of a variety with $K_X$ $\mathbb Q$-Cartier but not Cartier | **Note:** I added an addendum below in response to *quinque*'s comment and the subsequent discussion of the issue (s)he raised on math.stackexchange (see the link in *quinque*'s comment).
---
An easy way to produce a $\mathbb Q$-Cartier but not Cartier canonical divisor is by a quotient. For instance for the quot... | 22 | https://mathoverflow.net/users/10076 | 55529 | 34,682 |
https://mathoverflow.net/questions/55527 | 17 | I'm trying to understand better the mathematical notion of elliptic cohomology. Note that I only know the physics definition of the elliptic genus given in Witten's [paper](http://projecteuclid.org/euclid.cmp/1104117076).
Let $X$ be a Calabi-Yau manifold. (The elliptic genus can be defined for any $X$ with less prop... | https://mathoverflow.net/users/5420 | Virasoro action on the elliptic cohomology | There is an extensive math literature on related constructions. The key word is "chiral de Rham complex", introduced by Malikov, Schechtman and Vaintrob [here](http://arxiv.org/abs/math/9803041) and further developed in many many papers, starting with work of Gorbounov-Malikov-Schechtman. The connections to elliptic ge... | 19 | https://mathoverflow.net/users/582 | 55540 | 34,688 |
https://mathoverflow.net/questions/55539 | 4 | We are taking a random walk on the set of natural numbers. If we are at $M$, then with probability 1/4, we stay at $M$, with probability 5/12 we move to some random number less than or equal to $M/2$, and with probability 1/3, we move to a random greater than $M$ but less than or equal to $(3M+1)/2$. Is it true that al... | https://mathoverflow.net/users/3960 | A random walk on natural numbers | Yes. The possibility of staying at $M$ is irrelevant, so let's ignore it, so that the probability of an increase is 4/9 and the probability of a decrease is 5/9. For the moment, let's also ignore the actual sizes of these increases and decreases and let $X\_n$ equal the total number of increases in the first n steps mi... | 16 | https://mathoverflow.net/users/11771 | 55541 | 34,689 |
https://mathoverflow.net/questions/16771 | 11 | Let $f(x)=e^{i\phi(x)}$ define a function from $[0,1]$ to the complex unit circle through the real smooth function $\phi(x)$. Also, this function is periodic: $\phi(0)=\phi(1)=0\text{ mod }2\pi$ and has bounded derivatives for $x\in[0,1]$: $$\vert d\phi(x)/dx\vert\le \omega.$$ Consider the (truncated) Fourier transform... | https://mathoverflow.net/users/1837 | Lower bounds on (truncated) Fourier transform of functions of constant modulus and bounded derivative | It is an interesting problem which is related to some recent work of mine. The reason for why I started to work on similar problems is because connections to a problem of Ramachandra on Dirichlet polynomials, connections to the nordic school of Hardy classes of Dirichlet series (Hedenmalm, Saksman, Seip, Olsen, Olofsso... | 9 | https://mathoverflow.net/users/10811 | 55543 | 34,690 |
https://mathoverflow.net/questions/55510 | 3 | Let $M$ be a submanifold of $\mathbb R^n$. Call $M$ *locally convex* if locally $M$ is contained in the boundary of a convex domain of $\mathbb R^n$.
Is there any known condition that is equivalent to local convexity?
Some special cases are easy to treat. If the second fundamental form with respect to some normal d... | https://mathoverflow.net/users/1626 | Submanifolds lying on the boundary of a convex domain | First another example (elaborating Petrunin's comment) of a curve in $\mathbb R^3$ that is not locally convex: consider $x \mapsto (x, x^3, x^5)$. From the first derivative, any support plane at the origin needs to contain the $x$-axis. From the third derivative, if there's a supporot plane it needs to
be the $xy$-plan... | 5 | https://mathoverflow.net/users/9062 | 55544 | 34,691 |
https://mathoverflow.net/questions/55496 | 3 | Suppose $(X,\omega)$ is a closed symplectic manifold. Let $H$ denote a time-dependent Hamiltonian, all of whose critical points are non-degenerate, and fix an $\omega$-compatible time dependent family of almost complex structures.
Let $\mathcal{M}$ denote the set of all *finite energy* Floer flow lines. That is, maps... | https://mathoverflow.net/users/9052 | Length of Floer flow lines | In your symplectically aspherical setting, bounds on length will indeed exist.
Suppose one has a sequence of solutions $u\_n$ to Floer's equation, of bounded energy, and a sequence of points $t\_n\in S^1$ with lengths $l(u\_n(\cdot,t\_n))\to \infty$. Gromov-Floer compactness tells us that after passing to a subseque... | 7 | https://mathoverflow.net/users/2356 | 55561 | 34,698 |
https://mathoverflow.net/questions/55560 | 8 | Let $X\subset \mathbb P^3$ be a smooth variety and $D \subset X$ be an effective, reduced, irreducible divisor. My question is the following.
If I know the defining equations of $X$ and $D$ then is there any software that can compute $H^2(X, T\_X(-\log D))$?
| https://mathoverflow.net/users/12969 | Computing H^2(X, T_X(-\log D)) | Computing cohomology of the logarithmic tangent sheaf $T\_X(-\log D)$ is a usually easier once you know the cohomology of the sheaves $T\_X, \quad T\_X(-D)$ and $N\_D$, the normal bundle of $D$ in $X$. To get $H^i(X,T\_X(-\log D))$ one can use the exact sequences
$$
0 \to T\_X(-D) \to T\_X(-\log D) \to T\_D \to 0
$$or
... | 3 | https://mathoverflow.net/users/3996 | 55562 | 34,699 |
https://mathoverflow.net/questions/55555 | 19 | Markovich and Saric proved the following remarkable theorem. Let $S$ be a compact surface of genus at least $2$ and let $MCG(S)=\pi\_0(Homeo^{+}(S))$ be the mapping class group of $S$. There is then no right inverse to the natural projection $Homeo^{+}(S) \rightarrow MCG(S)$. This should be contrasted with the solution... | https://mathoverflow.net/users/13001 | Realizing braid group by homeomorphisms | The problem is much easier, but still interesting and nontrivial, if you replace homeomorphisms by diffeomorphisms, so it might be worth to study this case.
Logically, because any action by diffeomorphisms is an action by homeomorphisms and if no action by homeomorphisms exists, then certainly there cannot be any actio... | 12 | https://mathoverflow.net/users/9928 | 55563 | 34,700 |
https://mathoverflow.net/questions/55565 | 10 | Assume $X$ is smooth "simply connected" complex projective variety and $Y\subset X$ a smooth hyperplane section. ( $Y= X\cap H$, $H\subset \mathbb{P}^n$).
Let's $NE(X)$ be the cone of effective 1-cycles modulo numerical-equivalence. Let's $\mathfrak{K}\_X$ be the Kahler cone.
I have couple of question on these con... | https://mathoverflow.net/users/5259 | Question on Kähler/ample cone, cone of curves.... | **EDIT** Added a concrete example for the answer for Q4.
By a result of Kleiman (somewhere in SGA6 and also in Lazarsfeld's book around 1.1.20) a numerically trivial line bundle has a power that's in ${\rm Pic}^\circ X$ which is trivial if $X$ is simply connected, so on such an $X$ a numerically trivial line bundle i... | 7 | https://mathoverflow.net/users/10076 | 55568 | 34,704 |
https://mathoverflow.net/questions/55575 | 14 | The set of continuous homomorphisms from a torus ${\mathbb T}^n = ({\mathbb R}/{\mathbb Z})^n \to {\mathbb R}/{\mathbb Z}$ can be identified with ${\mathbb Z}^n$ if we assign to each $k = (k\_1, \ldots k\_n) \in {\mathbb Z}^n$ the character $x \mapsto k \cdot x$.
The fundamental group of homotopy classes of loops ${\... | https://mathoverflow.net/users/7193 | Why is the dual of a torus the same as its fundamental group? | The two are naturally dual lattices. The fundamental group of a torus $T$ can be canonically identified with the group (known as the cocharacter lattice) of $\it homomorphisms$ from the circle group to $T$, or equivalently the kernel of the (universal cover=exponential map) homomorphism from the Lie algebra $t$ to $T$.... | 15 | https://mathoverflow.net/users/582 | 55577 | 34,708 |
https://mathoverflow.net/questions/55579 | 4 | Hi all,
I have a system of 6th-order polynomial equations in 4 variables $q\_1, q\_2, q\_3, q\_4$ (i.e. polynomials with all the terms such as $q\_1^6, q\_2^6, q\_2^4 q\_3^2$):
$P\_k(q\_1, q\_2, q\_3, q\_4) = 0$ with $k=2,\dots,N$
I don't have any good guess of the q\_i. So, Newton method and its variant won't wo... | https://mathoverflow.net/users/12993 | Numerical solution for a system of multivariate polynomial equations | It really depends on the kind of system you have: do you have any reasons, for instance, to believe that the number of solutions is finite?
If the system is zero-dimensional (which essentially means that you have a finite number of solutions over the complexes), then the [rational univariate representation](http://en... | 6 | https://mathoverflow.net/users/8212 | 55580 | 34,710 |
https://mathoverflow.net/questions/36867 | 5 | Hello,
Thanks for reading my question ! Could anybody give me some references ( books, papers containing elementary results etc ) on the eigen values and eigenspectra of NON-compact Riemann surfaces. I studied the compact cases briefly and want to know the analogues or further results for the non-compact ones.
Also... | https://mathoverflow.net/users/6953 | Books about the spectra of non-compact Riemann surfaces | If your French is alright, Nicolas Bergeron has written a beautiful book, *Le spectre des surfaces hyperboliques*, available [here](http://www.math.jussieu.fr/~bergeron/Travaux_files/bergeron.pdf), which covers many of these topics in detail.
| 4 | https://mathoverflow.net/users/1464 | 55586 | 34,714 |
https://mathoverflow.net/questions/55594 | 4 | I have read in some number theory books and in some online resources that it is known that there exist infinitely many irregular primes (a fact apparently proven quite some time ago, around 1915 by K. L. Jensen according to the Wikipedia entry).
I haven't been able to find any reference, either in books or in the in... | https://mathoverflow.net/users/4170 | What is known about the conjectured infinitude of regular primes ? | 1. and 2.: I am not aware of any ideas in this direction.
2. The most obvious comment is the fact that there are Bernoulli numbers, and that
their numerators must have prime factors. If these prime factors are large enough,
they will produce irregular primes.
For proving the existence of regular primes you have to s... | 6 | https://mathoverflow.net/users/3503 | 55598 | 34,723 |
https://mathoverflow.net/questions/55306 | 1 | Let $a\le b$ and $k\ge 0$ be given and fixed. Let furthermore $x$ and $y$ denote two different elements of a Hilbert space $H$. Suppose $u:\mathbb{R}\rightarrow H$ is a $C^k$-embedding connecting $x$ and $y$, s.t. the derivatives up to order $k$ vanish at infinity and $f:\mathbb{R} \rightarrow \mathbb{R}$ a given $C^k$... | https://mathoverflow.net/users/3509 | continuity of extension of maps along curves | Some rough ideas to construct a continuous map. I'm not sure that
there aren't any obstructions this might run into.
First, it's nice to have an (explicit) construction of the map
$F:u,f \mapsto \tilde{f}$ to be able to say anything about continuity.
An explicit construction of a tubular neighborhood of $u([a,b])$ ... | 0 | https://mathoverflow.net/users/3928 | 55603 | 34,725 |
https://mathoverflow.net/questions/55602 | 16 | A *complex-oriented cohomology theory* $E^\*$ is a multiplicative cohomology theory with a choice of Thom class $x\in\tilde{E}^2(\mathbb{C}P^\infty)$ for the universal complex line bundle (which can be used to define generalised Chern classes for all complex vector bundles).
A *real-oriented cohomology theory* $F^\*$... | https://mathoverflow.net/users/8103 | Which cohomology theories are real- and complex-orientable? | A real-orientable ring spectrum $F$ admits a ring map from $MO$, and there is a straightforward ring map $MU\to MO$, so $F$ is also complex orientable. Moreover, $MO$ is a wedge of $H/2$'s, so $MO\wedge F$ is also a wedge of $H/2$'s, and $F$ is a retract of $MO\wedge F$ (by using the ring structure etc) so it is again ... | 17 | https://mathoverflow.net/users/10366 | 55605 | 34,727 |
https://mathoverflow.net/questions/55550 | 2 | Let $k$ be an algebraically closed field, and consider some subscheme $X\subset \mathbb{P}\_k^n$. Let $x$ be a closed point of $X$, and $H$ a general hyperplane containing $x$. There is a regular map $\phi:\mathbb{P}^n\setminus{\{x\}}\rightarrow \mathbb{P}^{n-1}$ gotten by projecting from the point $x$.
My question: ... | https://mathoverflow.net/users/12730 | General hyperplane sections and projection from a point | **EDITED** to match clarifications in the question and in Sándor's answer.
The question is equivalent to asking whether the tangent cone at $x$ of the hyperplane section coincides with the hyperplane section of the tangent cone:
Blow up $x$, and denote $\tilde{\mathbb{P}}^n$, $\tilde X$, $\tilde H$ the resulting va... | 3 | https://mathoverflow.net/users/1939 | 55612 | 34,733 |
https://mathoverflow.net/questions/55384 | 21 | I'm looking over a paper, "Primes represented by quadratic polynomials in two variables" [1] which attempts to characterize the density of the primes in two-variable quadratic polynomials. Its apparent improvement over previous works is that it covers not just quadratic forms plus a constant but quadratic forms plus li... | https://mathoverflow.net/users/6043 | Primes represented by two-variable quadratic polynomials | The modern reference work on the subject seems to be [1], but it spends only a page and a half on the subject of primes in multivariate quadratic polynomials (pp. 396-397). More than half this space is devoted to Iwaniec's 1974 result. The balance mentions Sarnak's application to the Problem of Apollonius and a result ... | 7 | https://mathoverflow.net/users/6043 | 55614 | 34,735 |
https://mathoverflow.net/questions/55585 | 20 | Hi! I'm new here. It would be awesome if someone knows a good answer.
Is there a good lower bound for the tail of sums of binomial coefficients? I'm particularly interested in the simplest case $\sum\_{i=0}^k {n \choose i}$. It would be extra good if the bound is general enough to apply to $\sum\_{i=0}^k {n \choose i... | https://mathoverflow.net/users/13006 | Lower bound for sum of binomial coefficients? | First, what the Stirling bound or Stanica's result give is already a $(1+O(n^{-1}))$ approximation of $\binom nk$, hence the only problem can be with the sum. I don't know how to do that with such precision, but it's easy to compute it up to a constant factor by approximating with a geometric series:
$$\sum\_{i\le k}... | 17 | https://mathoverflow.net/users/12705 | 55617 | 34,736 |
https://mathoverflow.net/questions/55625 | 11 | How can one solve the following recurrence relation?
$f(n) = 1 + \frac{1}{2^n} \sum\_{k = 0}^n {{n}\choose{k}} f(k)$
$f(0) = 0$
As it happens, I can show $f(n) = \Theta(\log n)$ through other means (see below). But I'd like to know how to solve the recurrence "directly".
The recurrence relation comes from the f... | https://mathoverflow.net/users/5873 | Coin flipping and a recurrence relation | The exponential generating function of the sequence $(f(n))$ is
$$
\sum\_{n\ge0}f(n)\frac{s^n}{n!}=\mathrm{e}^{s}\sum\_{k\ge0}(1-\mathrm{e}^{-s/2^k}).
$$
Not sure that this formula helps to recover the asymptotic behaviour of $f(n)$ when $n\to\infty$, though, even if it yields a few equivalent expressions of every $f(... | 11 | https://mathoverflow.net/users/4661 | 55628 | 34,741 |
https://mathoverflow.net/questions/55633 | 5 | I have reduced a knotty research problem to the following reasonable looking form:
Given any two integers $a$ and $b$, show that there are natural numbers $x\_1,x\_2,x\_3$ and a (probaby negative) integer $n$, where $-3n < x\_1+x\_2+x\_3$, satisfying:
$x\_1x\_2x\_3=-n^3-an-b,$ and
$x\_1x\_2+x\_1x\_3+x\_2x\_3=a+3... | https://mathoverflow.net/users/4078 | Diophantine problem | Following up Charles Matthews' idea, [Maclaurin's inequality](http://en.wikipedia.org/wiki/Maclaurin%27s_inequality) gives
$$\frac{x\_1 + x\_2 + x\_3}{3} \ge \sqrt{ \frac{3n^2 - 2n + a}{3} } \ge \sqrt[3]{ n^3 + an - b}.$$
The second inequality in particular expands out to an inequality of the form $-54n^5 + \text{l... | 3 | https://mathoverflow.net/users/290 | 55645 | 34,753 |
https://mathoverflow.net/questions/55620 | 7 | Suppose $M$ is a "hyperbolic" matrix in $GL(n,\mathbb Z)$, i.e., that its characteristic polynomial $p$ is irreducible over $\mathbb Z$ and has no roots of modulus 1.
Is there a closed description of the set of elements of $GL(n,\mathbb Z)$ which commute with $M$?
I have a vague recollection that it is somewhat s... | https://mathoverflow.net/users/8131 | Commuting matrices in GL(n,Z) | This has very little to do with being hyperbolic; the key point is that the characteristic polynomial is irreducible. It is convenient to "close in" on $\mathbb{Z}$ be thinking about easier rings.
---
Let $M$ be a matrix in $GL(n, \mathbb{C})$ with distinct eigenvalues. Then I claim that the set of $n \times n$ m... | 12 | https://mathoverflow.net/users/297 | 55646 | 34,754 |
https://mathoverflow.net/questions/55606 | 12 | The permutation groups $A = PSL(2,7)$ with its natural action on the projective line $\mathbb{P}^1(\mathbb{F}\_7)$ and $B = A\Gamma L(1,8)$ with its natural action on the affine line $\mathbb{F}\_8$ have the interesting property that $A$ and $B$ are non-isomorphic, but the point stabilizers $A\_x$ and $B\_y$ are isomor... | https://mathoverflow.net/users/12858 | Non-isomorphic two-transitive permutation groups with isomorphic point stabilizers | There are lists of known finite 2-transitive groups, and for examples other than those of affine type (i.e. those with a regular normal elementary abelian subgroup), it should not be hard to show that there are no more examples.
But I would expect there to be examples of affine type, and by searching through the Magm... | 10 | https://mathoverflow.net/users/35840 | 55654 | 34,758 |
https://mathoverflow.net/questions/55593 | 1 | How many n-element semilattices there are?
For example, for n-element partially ordered set we can figured out, that there are $2^{n\*(n-1)}$ possible sets.
And can I find all possible n-element semilattices? Or maybe there are some useful rule to generate all that kind of sets? [I'm trying do something](https://st... | https://mathoverflow.net/users/12234 | Semilattices with n elements | By adding a top element to a (meet) semilattice we get a lattice. Hence the number of $n$-element semilattices is the number of $(n+1)$-element lattices. This number is treated in
A006966 of Sloane.
| 4 | https://mathoverflow.net/users/2807 | 55660 | 34,763 |
https://mathoverflow.net/questions/55649 | 11 | Hi all,
I am doing a work in collaboration with other mathematicians about phase transition in the Ising model and we need to know if exponential upper bounds exist for the number of lattice animals with boundary of size $n$.
To be precise, consider the square lattice $\mathbb{Z}^2$ as graph where the edges are p... | https://mathoverflow.net/users/2386 | Exponential bounds for the number of lattice animals with a given boundary. | I think the following (or something close) proves Leandro’s point:
Let $n=5k$ for some odd integer $k$, and let $B\_k$ be the subgraph of $\mathbb{Z}^2$ with vertex set $[1\dots k]\times[1\dots k]$ (and all edges between these vertices from the lattice $\mathbb{Z}^2$). Let K be any subset of $[2\dots k-1]\times[2\dot... | 6 | https://mathoverflow.net/users/8201 | 55661 | 34,764 |
https://mathoverflow.net/questions/55647 | 17 | This is a generic question, a good answer to it may be a reference to a corresponding paper\textbook, but any useful comments would be okay too.
Let $\mathfrak{g}$ be a (simple) Lie algebra and $U\_q(\mathfrak{g})$ be its q-deformation of its universal enveloping algebra. For example, for $\mathfrak{g} = \mathfrak{su... | https://mathoverflow.net/users/5550 | Relationship between "different" quantum deformations | There is certainly a way to quantize the algebra of functions on a Lie group in a way that is compatible with the $q$-deformation of the universal enveloping algebra of its Lie algebra. The standard way to do it is this: let $G$ be a simple, connected, simply connected complex Lie group with Lie algebra $\mathfrak{g}$.... | 14 | https://mathoverflow.net/users/703 | 55662 | 34,765 |
https://mathoverflow.net/questions/55666 | 13 | Let $A$ be an absolutely simple abelian variety over a number field $K$. Assume that, for some prime $p$, the Tate module $T\_p A$ has a submodule of rank one, invariant under the absolute Galois group of $K$. Does it follow that $A$ is has CM?
For elliptic curves, I guess this follows from Serre's open image theorem... | https://mathoverflow.net/users/2290 | Is an abelian variety with a Galois invariant, rank one submodule of its Tate module, CM? | Yes. This follows from the main result of [the following paper](https://projecteuclid.org/journals/duke-mathematical-journal/volume-54/issue-1/Endomorphisms-and-torsion-of-abelian-varieties/10.1215/S0012-7094-87-05410-X.short) of Zarhin.
---
MR0885780 (88h:14046)
Zarhin, Yu. G.
Endomorphisms and torsion of abelia... | 14 | https://mathoverflow.net/users/1149 | 55670 | 34,769 |
https://mathoverflow.net/questions/55674 | 16 | I have several questions concerning some properties of algebraic numbers. The first concerns the folowing statement:
*Given algebraic integers $\alpha$ and $\beta$ they have a unique greatest common divisor modulo asociates. ie there is an algebraic integer $\delta$ with $\delta \vert \alpha$ and $\delta \vert \beta... | https://mathoverflow.net/users/10670 | Greatest common divisor of algebraic integers | I give a proof of this result -- actually the stronger result that the GCD of any two algebraic integers may be expressed as a linear combination -- in $\S 23.4$ of [these commutative algebra notes](http://alpha.math.uga.edu/%7Epete/integral.pdf).
The required input from algebraic number theory is nontrivial -- namel... | 20 | https://mathoverflow.net/users/1149 | 55675 | 34,772 |
https://mathoverflow.net/questions/55672 | 11 | I was browsing through the litterature, hoping to find sufficient and necessary conditions for a smooth manifold to have finite-dimensional de Rham cohomology, but I can't find any satisfactory answer. I wonder if anyone has ever encountered a paper, or a book, answering (possibly in part) the question. I am especially... | https://mathoverflow.net/users/13027 | Finite-dimensionality for de Rham cohomology | We can restrict your problem to the case of open manifolds. It turns out that "finite dimensional" integral singular homology (i.e., finitely generated in each degree) is almost the same thing as the manifold being the interior of a compact manifold with boundary. For example, if $M$ is a 1-connected and open manifold ... | 17 | https://mathoverflow.net/users/8032 | 55678 | 34,774 |
https://mathoverflow.net/questions/55681 | 4 | Let $\mathbb{F}$ be a cubic field, i.e, $\mathbb{F} = \mathbb{Q}(\alpha)$ where $\alpha$ is a root of a cubic irreducible polynomial over $\mathbb{Q}$, satisfying $disc(\mathbb{F}/\mathbb{Q})$ is a prime or a power of a prime, say $q$. By standard number theory, we know that $\mathbb{F}$ is ramified only at $q$. My que... | https://mathoverflow.net/users/12312 | Splitting of primes in cubic fields with limited ramifications. | Yes, certainly. As long as p is tamely ramified (which is the case for you once p > 3) then the power of p in the discriminant is e-1, where e is the ramification degree. By requiring that the discriminant is exactly p, you are forcing e-1 = 1, i.e. you are restricting yourself to the second of your two options.
| 7 | https://mathoverflow.net/users/431 | 55684 | 34,776 |
https://mathoverflow.net/questions/55685 | 10 | The title pretty much summarizes the question: does every $p$-group have a *faithful* unipotent representation (with coefficients in $\mathbb{F}\_p$ or some finite extension thereof)?
| https://mathoverflow.net/users/11142 | faithful unipotent representations of (finite) $p$-groups | I guess you mean finite $p$-groups. Then every finite-dimensional representation of a finite $p$-group $G$ is unipotent in characteristic $p$. (Indeed, all eigenvalues of every element of $G$ are $p$-power roots of unity and therefore must be equal to $1$. This means that every element of $G$ acts as a unipotent operat... | 11 | https://mathoverflow.net/users/9658 | 55687 | 34,778 |
https://mathoverflow.net/questions/55688 | 1 | Let $G=(V,E)$ be an isoradial graph. In other words the graph can be imbedded into the plane such that each face (plaquette) can be circumscribed a circle of radius 1 with the circle's center belonging to the closure of the face. Let $Z$ be the set of circle centers. Then for any walk $C$ on $G$ that starts at some set... | https://mathoverflow.net/users/934 | Winding number bijection on graphs | Are two walks considered to be the same if they use the same edges but in a different order? If not, walks ABCDAECFA and ABCFAECDA could have the same winding number around every point in the plane, yet they'd be different walks.
EDIT: And what if your graph is a tree? Then the set of centers is empty, and the walks... | 1 | https://mathoverflow.net/users/3684 | 55696 | 34,780 |
https://mathoverflow.net/questions/55699 | 7 | For $n > 1$, $2n$-dimensional sphere $S^{2n}$ does not admit symplectic structures. Then how about the product with a manifold? Are there any results about the symplectic structures on $M \times S^{2n}$?
| https://mathoverflow.net/users/11846 | Symplectic structures on $M \times S^{2n}$ | From the [Künneth theorem](http://en.wikipedia.org/wiki/K%25C3%25BCnneth_theorem#Singular_homology_with_coefficients_in_a_field) you can check that there is no class $\omega\in H^2(M^{2d}\times S^{2n};\mathbb{R})$ such that $\omega^{d+n}\neq 0$. (This is an excellent thing for you to work out for yourself.) Since a sym... | 14 | https://mathoverflow.net/users/250 | 55700 | 34,783 |
https://mathoverflow.net/questions/55663 | 8 | Let $G$ be the abolute Galois group of $\mathbb Q\_p$, let $\delta\_1, \delta\_2: G\rightarrow L^{\times}$ be continuous characters, where $L$ is a finite extension of $\mathbb Q\_p$. Assume that $\delta\_1\delta\_2^{-1}$ is neither trivial nor the cyclotomic character then
$Ext^1\_{G}(\delta\_2, \delta\_1)$ is one di... | https://mathoverflow.net/users/13024 | When is an extension of characters de Rham? | Dear Vytas,
lemma 6.5 of my 2002 inventiones paper says that if $V$ is any de Rham representation all of whose HT weights are at least 1, then any extension of $Q\_p$ by $V$ is itself de Rham. This holds for reps of $G\_K$ where the residue field $k$ of $K$ can be any perfect field (not merely finite).
If $k$ is fi... | 8 | https://mathoverflow.net/users/5743 | 55705 | 34,786 |
https://mathoverflow.net/questions/55704 | 13 | This semester I am teaching a graduate course in commutative algebra, and I have been taking the occasion to try to look at the proofs of some the results in my basic source material (Matsumura, Eisenbud, Bourbaki...) which I skipped as a little too complicated the first $N$ times around.
Recently I got the chance to... | https://mathoverflow.net/users/1149 | Example of a projective module which is not a direct sum of f.g. submodules? | For question two the example that is given most frequently seems to be that of the ring $R$ of continuous real valued functions on $[0,1]$ and the ideal of all functions $f$ which vanish on some interval $[0,\epsilon(f)]$ where $\epsilon(f)\in (0,1)$. This ideal is countably generated and projective but not a direct su... | 14 | https://mathoverflow.net/users/2384 | 55707 | 34,788 |
https://mathoverflow.net/questions/55618 | 5 | Hello,
I'm interested in the distribution of the trace of an inverse-Wishart matrix $W\_n^{-1}(I,n)$, where $I$ is $n\times n$ identity matrix. More precisely, I seek for an asymptotic estimate (when $n\to\infty$) for a function $f(n)$ such that $Pr[Tr(W)< f(n)]>2/3$, say.
What I've learned so far:
1. I know the ... | https://mathoverflow.net/users/13015 | Distribution of trace of inverse-Wishart matrix $W_n(I,n)$ | I guess, I've done it.
My idea is to detach two eigenvalues, use the smallest eigenvalue estimation for them and then apply the known result for the expectation of the rest of the matrix.
Let $G$ be $n\times n$ matrix with entries being pairwise independent standard Gaussians. Then let $W=G^TG$ and I am interested ... | 1 | https://mathoverflow.net/users/13015 | 55717 | 34,793 |
https://mathoverflow.net/questions/55741 | 3 | These feel like basic enough questions, but I don't know where to find the answer.
Let $X\_1,X\_2,X\_3,\dots$ be a supermartingale such that $|X\_{n+1} - X\_n| < K$ for all $n$ ($K$ fixed). Does the event $X\_n \rightarrow +\infty$ necessarily have probability zero? What if we also have the condition that given $X\_1... | https://mathoverflow.net/users/4053 | Supermartingales and convergence | For a martingale $M\_n$ with bounded increment, then, almost surely :
* either $M\_n$ converges to a finite limite.
* or $\limsup M\_n=\infty$ and $\liminf M\_n=-\infty$.
Sketch of the proof. One can assume that $M\_0=0$. Let $T$ be the first time at which the martingale goes below $-A$. Then $M\_{n \wedge N}$ is a... | 3 | https://mathoverflow.net/users/12088 | 55745 | 34,808 |
https://mathoverflow.net/questions/55746 | 6 | Fix a universe $\mathcal{U}$. Call a category $\mathcal{U}$-complete if every diagram indexed by a $\mathcal{U}$-small category has a limit, and a functor $\mathcal{U}$-continuous if it preserves $\mathcal{U}$-small limits. Usually, when one fixes a universe, one calls this simply complete and continuous.
Now assume ... | https://mathoverflow.net/users/2841 | Colimits in a bigger universe | No. I'll answer for the case of colimits as follows: Consider for example the category $O$ of ordinals (as a poset), and adjoin a terminal object $T$, making a larger category $C$. Then this terminal object is a (large) colimit over the diagram $O\to C$. However, a cocontinuous functor $C\to D$ can send $T$ anywhere th... | 4 | https://mathoverflow.net/users/1474 | 55747 | 34,809 |
https://mathoverflow.net/questions/55736 | 5 | This is a problem I encountered in Martin Isaacs' 'Finite Group Theory'. It's located at the end of Chapter II which deals with subnormality, and the particular paragraph is concerned with a couple of not so well-known results which I quote for reference:
(In what follows $F$ is the Fitting subgroup)
**Theorem (Zen... | https://mathoverflow.net/users/nan | A problem in Finite Group Theory | You were almost there:
Since $N$ and $F(G)$ are two normal subgroups intersecting trivially, they commute. But now take a non-trivial element $a \in A \cap F(G)$; then by the previous observation, $a$ commutes with every element of $N$. But this contradicts the fact that $C\_A(N) = 1$.
| 1 | https://mathoverflow.net/users/12858 | 55751 | 34,812 |
https://mathoverflow.net/questions/55733 | 0 | In [this ArXiv paper](http://arxiv.org/abs/hep-ph/9908459) by Wilk and Wlodarczyk (published in Physical Review Letters), equation 16 has essentially the following definition of a function:
$$\text{f(x)=}\frac{c}{2Dx^2}\exp[\int^x\_0 \frac{\mu}{t^2}+\frac{1-2\alpha}{t} dt]$$
They claim that with the normalization con... | https://mathoverflow.net/users/13036 | Can you interpret this divergent integral? | The trouble (as was already explained to you) lies in the starting point $t=0$ of the integral in the exponential. Fortunately, W+W are only interested in steady solutions of equation (15) of their arXiv preprint and these can be written as the function in their equation (16) provided one replaces the starting point $t... | 8 | https://mathoverflow.net/users/4661 | 55755 | 34,814 |
https://mathoverflow.net/questions/55749 | 10 | Suppose you are given a closed subvariety $V$ of projective space $\mathbb{P}^n\_k$. Let's say we fix the degree and the dimension of $V$. Can we then bound the arithmetic genus of $V$, or does no such bound exist?
| https://mathoverflow.net/users/1107 | Is there a bound on arithmetic genus of a variety in projective n-space in terms of dimension and degree? | I don't know about an explicit bound, but a bound exists in theory for arbitrary varieties. This follows from the
fact that the set of cycles, and in particular subvarieties, in $\mathbb{P}^n$ of fixed degree $d$ and dimension $N$ are parameterized by a Chow variety $Chow\_{d,N}$ (it needn't be irreducible, but it is c... | 8 | https://mathoverflow.net/users/4144 | 55764 | 34,822 |
https://mathoverflow.net/questions/55718 | 23 | While preparing a course in complex analysis, I stumbled over a remark in Dudziak's book on removable sets, namely that any totally disconnected $K \subset\subset {\mathbb C}$ must have a connected complement; a remark, "that verifying the reader may find one of those exercises in 'mere' point-set topology that is a we... | https://mathoverflow.net/users/13034 | complement of a totally disconnected closed set in the plane | A proof of the statement using knowledge that has withstood the test of time is to
cite the [Alexander duality theorem](http://en.wikipedia.org/wiki/Alexander_duality), for which you can find multiple modern sources. The relevant form of Alexander duality states that the if $X$ is a compact subset of $S^n$, then the re... | 26 | https://mathoverflow.net/users/9062 | 55767 | 34,823 |
https://mathoverflow.net/questions/55742 | 6 | Let $M$ be a Riemannian manifold such that its isometry group $G=\textrm{Iso}(M)$ is a Lie group, and let $\Gamma$ be a subgroup of $G$.
>
> 1) What does the phrase "$\Gamma$ is a cocompact group of isometries of $M$" mean? Does it mean that the quotient *space* $M/\Gamma$ is compact, or does it mean that the *cos... | https://mathoverflow.net/users/4721 | Terminology: "cocompact" | 1) from wikipedia "In mathematics, an action of a group G on a topological space X is cocompact if the quotient space X/G is a compact space."
2) Let $\bar{M}$ be a noncompact manifold, $\Gamma$ its fundamental group, and $M$ its universal cover. Given a riemannian metric on $\bar{M}$, we can lift the metric to $M$ a... | 6 | https://mathoverflow.net/users/6129 | 55771 | 34,825 |
https://mathoverflow.net/questions/55693 | 12 | I'm looking for a reference for the following fact: given two Riemann surfaces and an identification of their boundaries, once I topologically glue the surfaces together there exists a *unique* conformal structure on my new surface that is compatible with the conformal structures I started with.
| https://mathoverflow.net/users/361 | Conformal Welding Reference | See MR1966191 (2005e:30012)
Hamilton, D. H.(1-MD)
Conformal welding. Handbook of complex analysis: geometric function theory, Vol. 1, 137–146, North-Holland, Amsterdam, 2002.
30C35
and other papers by David Hamilton.
| 5 | https://mathoverflow.net/users/11142 | 55781 | 34,834 |
https://mathoverflow.net/questions/55787 | 3 | Let $f : \mathbb{Z}/N\mathbb{Z} \rightarrow \mathbb{C}$ be a function. Is its Fourier transform typically defined by
$$\widehat{f}(\xi) = \sum\_{x \in \mathbb{Z}/N \mathbb{Z}} f(x) e^{- 2 \pi i x \xi /N},$$
or the same expression with $\frac{1}{N}$ in front?
I recently encountered both in material on additive com... | https://mathoverflow.net/users/1050 | Choice of normalization of the finite Fourier transform | The "right" category in which to define the Fourier transform in general is that of a locally compact abelian (LCA) group $G$, equipped with a Haar measure $\mu\_G$. Once one fixes the Haar measure $\mu\_G$, there is a natural dual measure $\mu\_{\hat G}$ on the Pontryagin dual group $\hat G$, such that all the usual F... | 17 | https://mathoverflow.net/users/766 | 55791 | 34,838 |
https://mathoverflow.net/questions/46777 | 9 | (I posted this [question on Math.SE](https://math.stackexchange.com/questions/8415/asymptotic-difference-between-a-function-and-its-binomial-average) a few weeks ago. I got a few comments, but nothing definite, and so I thought I would try MO.)
The origin of this question is the identity
$$\sum\_{k=0}^n \binom{n}{k} ... | https://mathoverflow.net/users/9716 | Asymptotic difference between a function and its "binomial average" | The function $f(n)$ must be $\Theta (\log n)$. **Update:** As Didier Piau points out in the comments, we can say something stronger: $\frac{f(n)}{\log\_2 n} \to L$ as $n \to \infty$. See the update at the end of the argument.
Suppose, for some positive $L$ (the negative case is similar), $$\lim\_{n \to \infty} \left... | 5 | https://mathoverflow.net/users/9716 | 55795 | 34,842 |
https://mathoverflow.net/questions/55769 | 28 | Infinite dimensional constructions, such as spaces of diffeomorphisms, spectra, spaces of paths, and spaces of connections, appear all over topology. I rather like them, because they sometimes help me to develop a good mental picture of what is going on. Emmanuel Farjoun, in the first lecture of an Algebraic Topology c... | https://mathoverflow.net/users/2051 | Are infinite dimensional constructions needed to prove finite dimensional results? | Any compact Lie group admits an embedding into a large linear group $GL\_n (\mathbb{C})$.
I think it is no question that this is a genuinely finite-dimensional and important statement. The proof is via the Peter-Weyl theorem; essentially one has to show that there are enough finite-dimensional representations. How is... | 30 | https://mathoverflow.net/users/9928 | 55798 | 34,845 |
https://mathoverflow.net/questions/55754 | 4 | Recently I became curious about the following question:
Let $V$ be a finite dimensional vector space over $k$ and let $A\_1, \cdots, A\_n: V \rightarrow V$ be a set of commuting maps. Question: describe the structure of $V$ as a module over $k[x\_1, \cdots, x\_n]$ where $x\_i$ acts by $A\_i$.
Since $V$ has a finite... | https://mathoverflow.net/users/7041 | Generalization of Jordan Decomposition for Several Commuting Operators | It is well known that representation theory of a (even *commutative*) Artinian $k$-algebra $R$ can be wild (meaning that one can embed $\mod(A)$ into $\mod(R)$ for any finite-dimensional, not necessarily commutative, $k$-algebra $A$). The easiest example is $k[x,y]/(x^2,xy^2,y^3)$, see Example 4 in [this paper](http://... | 3 | https://mathoverflow.net/users/2083 | 55810 | 34,853 |
https://mathoverflow.net/questions/55807 | 3 | I would like to prove the following fact, which I learned from a previous MO question.
Let $S\_\cdot,T\_\cdot\in\mathbf{sSET}$ be simplicial sets, and assume that $T\_\cdot$ is Kan. Then there is a weak equivalence
$$
|\underline{\mathbf{sSET}} (S\_\cdot,T\_\cdot)|\simeq \underline{\mathbf{TOP}} (|S\_\cdot|,|T\_\cdot... | https://mathoverflow.net/users/11300 | Inner hom and geometric realization. | The unit map $T\to \mathrm{Sing}|T|$ is far from being a trivial fibration; it's actually injective. Did you mean to say it's a trivial *cofibration*?
It *is* a weak equivalence for any simplicial set $T$ (so the map is actually a trivial cofibration), and this would complete your proof. Unfortunately, this seems to ... | 8 | https://mathoverflow.net/users/437 | 55816 | 34,859 |
https://mathoverflow.net/questions/54402 | 14 | Could anyone point me to a place where I could find Deligne's letter to Piatetskii-Shapiro from 1973? It is cited for example in Berkovich's "Vanishing cycles for formal schemes II".
| https://mathoverflow.net/users/10701 | Deligne's letter to Piatetskii-Shapiro from 1973 | I have typeset Deligne's letter, and placed the result here:
<http://www.math.ias.edu/~jaredw/DeligneLetterToPiatetskiShapiro.pdf>
I have made some minor edits so that the text reads more naturally to a native speaker of English. Also I made a few annotations where I truly believe there is an error in the original.... | 23 | https://mathoverflow.net/users/271 | 55817 | 34,860 |
https://mathoverflow.net/questions/55821 | 1 | When I am reading one paper, I have met the following statement:
It is impossible to define a $Z\_{2}\times Z\_{2}$ action on a connected closed curve on a compact Riemann surface.
The claim is equivalent to say that the existence of two fixed point free involutions on a circle is not true.
The author just took t... | https://mathoverflow.net/users/13054 | The antipodal action on a connected one dimensional manifold | Assuming you want *fixed-point free* actions...
Since $\pi\_1(S^1)$ is cyclic, all connected coverings of $S^1$ have cyclic group of covering transformations. Now, if $\mathbb Z\_2^2$ acted without fixed points on $S^1$, the corresponding quotient map would be a covering $S^1\to S^1$ of group $\mathbb Z\_2^2$, which ... | 5 | https://mathoverflow.net/users/1409 | 55825 | 34,865 |
https://mathoverflow.net/questions/55801 | 13 | One possibility is the integration by substitution formula. But this doesn't explain why $\int\_1^{\infty}x^{-3}dx$ is easy to evaluate (and rational) whilst $\sum\_1^{\infty}n^{-3}$ is unknown (and seemingly not a simple combination of well known numbers). It is true that $x^{-3}$ has a simple inverse derivative whils... | https://mathoverflow.net/users/4692 | Is there a reason why integrals are so much easier to evaluate than sums? | Well, some things are easier in one place, and other things are easier in the other. The many assertions in answers that integrals of $x^n$ are easier than the corresponding sums is misleading. In the continuous world, $x^n$ is a very natural object. In the discrete world, however, the natural object is the falling fac... | 33 | https://mathoverflow.net/users/935 | 55828 | 34,868 |
https://mathoverflow.net/questions/55832 | 9 | Hi all,
Can someone please explain the idea and the main steps in a random real forcing?
- what makes it (the new real) different from adding a Cohen real?
- is there a good reference for it?
Thanks.
| https://mathoverflow.net/users/10708 | What is random real forcing? | This is relatively easy to answer. One of main differences between Cohen forcing and random real forcing is that random real forcing does not add an unbounded real.
That is, every function $f:\omega\to\omega$ in the forcing extension is bounded by
function $g:\omega\to\omega$ in the ground model, in the sense that ... | 14 | https://mathoverflow.net/users/7743 | 55836 | 34,873 |
https://mathoverflow.net/questions/55831 | 0 | Given a number of length $l$, the sum of the digits range from $0$ to $9l$
For each sum, there are $x$ permutations of digits. find the sequence of length $9l+1$ that solves $x$.
I found a function that solves this problem without expanding the polynomial equation $l$ times:
$f(n \in S,l,b) = \sum\limits\_{i=0}^{... | https://mathoverflow.net/users/10246 | How many permutations for each sum of digits of a number of length l? (solved but asking for insight) | Your sequences are simply the coefficients of the polynomials
$$(1 + x + x^2 + \dots + x^9)^L .$$
(This is a straightforward application of the theory of generating functions.)
| 7 | https://mathoverflow.net/users/12858 | 55837 | 34,874 |
https://mathoverflow.net/questions/41086 | 2 | Many theorems have the form : Premise(es) implies Conclusion(s)
**Example A of wrongness**:
There are many examples in which a theorem is stated without mentioning that part of the premise is not necessary to reach the conclusion.
Usually it is simple (and much better) to add a remark stating that the result is... | https://mathoverflow.net/users/3005 | Theorems true but wrong. | A classic example for B is the theorem (proved using Gauss' lemma) usually stated as: if $R$ is a unique factorization domain, then so is the ring of polynomials $R[x]$. Now, $R$ is a UFD iff it is a GCD-domain with no infinite descending chain of proper divisors (ACCP), and
* if $R$ is a GCD-domain, then so is $R[x]... | 6 | https://mathoverflow.net/users/12705 | 55840 | 34,876 |
https://mathoverflow.net/questions/55833 | 10 | Fix a prime $p$ and consider everything mod $p$. Steenrod operations arise somehow from the loss of information passing from the singular complex of a space to its cohomology ring. Are they exactly this gap, i.e. can I get the singular complex back from the cohomology ring of a space and its structure as a module over ... | https://mathoverflow.net/users/2625 | Singular complex = cohomology ring + Steenrod operations? | No. For instance, Massey products on the cohomology are extra information that neither the ring structure nor the Steenrod operations see. The complement of the Borromean rings, for example, and the complement of three unlinked circles in $R^3$ have the same cohomology ring and Steenrod operations, but cannot be chain ... | 20 | https://mathoverflow.net/users/4183 | 55844 | 34,879 |
https://mathoverflow.net/questions/55692 | 4 | Dear all,
Thank you for your time reading this post. I am a student in computer science so this viewpoint of the second fundamental form may be interesting to you.
I would like to understand the second fundamental form of an affine (or projective) variety of dimension $m$ in affine (or projective) space $\mathbb{A}... | https://mathoverflow.net/users/8012 | viewing the second fundamental form as a tensor | I believe the definition of the second fundamental form for a projective variety is explained very nicely in
Griffiths, Phillip; Harris, Joseph Algebraic geometry and local differential geometry. Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 3, 355–452.
| 2 | https://mathoverflow.net/users/613 | 55852 | 34,884 |
https://mathoverflow.net/questions/55846 | 5 | Let $G$ be an algebraic connected subgroup of $GL\_n(\mathbb{C})$ and let $\chi : G \to \mathbb{C}^\*$ a character. Consider $d\_e\chi : \mathfrak{g} \to \mathbb{C}$ the differential of $\chi$ at the identity of $G$. Is it true that $d\_e\chi(\nu)=0$ if $\nu$ is a nilpotent element?If not, is it true under some assumpt... | https://mathoverflow.net/users/4821 | Differential of a nilpotent or semisimple element | The question should be formulated more precisely. The assumption seems to be that $G$ is an *algebraic* group and that $\chi$ is a morphism of algebraic groups (?) If so, assuming as we may that $\chi$ is nontrivial, $\chi$ induces an isomorphism of a 1-dimensional quotient of $G$ (a torus) onto the image . The corresp... | 6 | https://mathoverflow.net/users/4231 | 55857 | 34,888 |
https://mathoverflow.net/questions/55858 | 3 | I'm trying to show that all possible splitting fields occur for a class of cubic polynomials, so have started by looking at the discriminants.
Clearly, given that they are products of squares, all discriminants are congruent to 0 or 1 mod 4. And it is easy to show that all numbers of this form appear as discriminants... | https://mathoverflow.net/users/4078 | Which numbers appear as discriminants of cubics? | There are two different types of cubic fields: cyclic cubics and cubics with Galois group $S\_3$.
Cyclic cubic fields are, by Kronecker Weber, subfields of cyclotomic fields. There is no
cyclic cubic field with discriminant $5^2$ since the field of $5$th roots of unity does not have a cubic subfield; the cyclic cubi... | 19 | https://mathoverflow.net/users/3503 | 55862 | 34,891 |
https://mathoverflow.net/questions/55867 | 5 | Hey everyone,
I was wondering if anyone knows what the canonical divisor of the Hilbert scheme $Hilb^n P^2$ is --$Hilb^n P^2$ is the Hilbert scheme of degree-n zero dimensional subschemes of the projective plane $P^2$. Any references?
Many thanks in advance.
| https://mathoverflow.net/users/12828 | The canonical divisor of the Hilbert scheme $Hilb^n P^2$? | There is an easy formula for the canonical divisor on the Hilbert scheme of $n$ poin ts on any any smooth projective surface $X$. Let's first fix some notations. Denote by $X^{n}$ the $n-$fold product with projections $pr\_i X^{n}\to X$. We can consider line bundles of the form
$$
L^{[n]}=pr\_1^\* L \otimes\cdots \otim... | 7 | https://mathoverflow.net/users/3996 | 55870 | 34,895 |
https://mathoverflow.net/questions/55859 | 6 | Let $\mathcal{U}$ be a universe and $\mathcal{U}^+$ a universe with $\mathcal{U} \in \mathcal{U}^+$. Denote by $\text{Cat}(\mathcal{U})$ the $\mathcal{U}^+$-category of all $\mathcal{U}$-categories, and by $\text{Cat}\_c(\mathcal{U})$ the full subcategory consisting of $\mathcal{U}$-cocomplete categories, i.e. in which... | https://mathoverflow.net/users/2841 | Universal cocompletion without leaving our universe | you can generalizing the $Ind(\mathcal{C})$ construction (see SGA.4-I) for general small diagrams (no neccessarly directed), using the general criterion that characterizes the purpose of a diagram $ D: I \to \mathcal{C} $ in terms of connecting category paragraph
$ X \downarrow D, \ X \in \mathcal{C} $). In this way y... | 3 | https://mathoverflow.net/users/6262 | 55873 | 34,897 |
https://mathoverflow.net/questions/55353 | 10 | Let $A$ be any bialgebra (associative, unital, etc.) over a ring $k$. Then among other things it has a counit $\epsilon : A \to k$, and hence an *augmentation ideal* $I = \ker \epsilon$, which is a Hopf ideal. Any ideal determines a filtration
$$ A \supseteq I \supseteq I^2 \supseteq \dots$$
and hence an *associated gr... | https://mathoverflow.net/users/78 | An explicit description of $\operatorname{gr}(k \cdot G)$ for the filtration induced by the augmentation ideal? | Daniel Quillen has answered this in
Quillen, Daniel G., *On the associated graded ring of a group ring.* J. Algebra 10 1968 411–418.
From the Mathematical Reviews by J. Knopfmacher:
"Let $KG$ denote the group algebra of a group $G$ over a field $K$ of characteristic $p$, and let $KG$ be filtered by the powers o... | 6 | https://mathoverflow.net/users/8176 | 55876 | 34,898 |
https://mathoverflow.net/questions/55874 | 2 | This is my second question on supermanifolds, the previous one is at
[Morphisms between supermanifolds R^{0|1}→R^{0|1}](https://mathoverflow.net/questions/54927/morphisms-between-supermanifolds-r01r01)
I've learn the difference between homomorphism and internal-hom of supermanifolds. Also, I know that the homomorp... | https://mathoverflow.net/users/7341 | How to caculate the internal hom of supermanifolds? | (Sorry for not answering this on the previous post, you asked this question before. By the way, you didn't say which paper you are reading...)
$Map(\mathbb R^{0|0},M)=M$. Trivial.
$Map(\mathbb R^{0|1},M)=\Pi T M$. ($TM$ is the tangent bundle of $M$, and $\Pi$ reverses grading of a vector bundle. So $\Pi T M$ denote... | 2 | https://mathoverflow.net/users/1090 | 55886 | 34,905 |
https://mathoverflow.net/questions/55864 | 26 | Consider the following game, played by two players,
called Q and A, in a time frame t = 1, 2, ....
At every time point i, Q mentions some $Q\_i \subset \mathbb{R}$,
after which A mentions $A\_i$ such that either $A\_i = Q\_i$ or
$A\_i = Q\_i^c$.
Define $C(i) = \bigcap\_{k \lt i} A\_i$ and
$C(\infty) = \bigcap\_{k \i... | https://mathoverflow.net/users/5901 | Game involving 'asking questions about a real' | Neither player has a winning strategy, see below.
However, when the game is restricted so that Q can only play sets with the [Baire property](http://en.wikipedia.org/wiki/Property_of_Baire), then A has a winning strategy. Note that sets with the Baire property form a $\sigma$-algebra which includes all analytic sets,... | 11 | https://mathoverflow.net/users/12705 | 55889 | 34,908 |
https://mathoverflow.net/questions/55891 | 5 | For any $A$-algebra $B$ ( commutative ring with 1 ), we have the existence of $\Omega\_{B/A}$, the module of relative differentials of $B$ over $A$, which can be defined by an universal property. In the case $A = k$ being a field and $B = k[[X]]$ being the formal power series ring over $k$, $\Omega\_{B/A}$ is not alway... | https://mathoverflow.net/users/5482 | module of differentials of formal power series ring and of its field of quotiens | This is not a direct answer to your question, but a few comments about the topic.
First, in general it is not true that $\Omega^1\_{B/A}$ is a free module. In fact, assuming $A \to B$ is flat and finitely generated, $\Omega^1\_{B/A}$ is locally free if and only if $A \to B$ is a smooth ring map.
If in your situatio... | 4 | https://mathoverflow.net/users/3759 | 55895 | 34,912 |
https://mathoverflow.net/questions/55890 | 14 | We play a game. I shuffle a deck of cards and start dealing them face up. After any card you can say "stop", at which point I pay you 1 dollar for every red card dealt and you pay me 1 for every black card dealt. What is your optimal strategy, and how much would you play to pay this game?
---
Clearly the game is ... | https://mathoverflow.net/users/11727 | Card game / options pricing / Brownian bridge question | I think the article [Optimal stopping of a Brownian bridge](https://doi.org/10.1239/jap/1238592123) by Ekström and Wanntorp ([preprint](http://www2.math.uu.se/%7Ehenrik/artiklar/HW5.pdf)) would give you the answer for the limiting case as $n\to\infty$.
| 7 | https://mathoverflow.net/users/13073 | 55905 | 34,918 |
https://mathoverflow.net/questions/55913 | 5 | One always sees the definition of a contact manifold $(X,\xi)$ as an odd dimensional manifold with a hyperplane distribution $\xi$ which can locally be expressed as $\xi = \ker \alpha$ for a 1-form $\alpha$. But in fact, it seems that in every example I know of, one always assumes that $\xi$ is **cooriented**, and henc... | https://mathoverflow.net/users/12998 | Contact manifolds that are not cooriented | As you say, the primary focus of research is on cooriented contact structures, but there is still some interest in non-coorientable contact structures (and it would become even more frustrating for those of us who are looking for information on the general case if the definition ruled out such).
For instance, [David... | 7 | https://mathoverflow.net/users/353 | 55921 | 34,926 |
https://mathoverflow.net/questions/55937 | 3 | Let $A$ be an algebra over some field $k$. Let $K\_P(A)$ be the Grothendieck group of the category of projective $A$-modules and $K\_F(A)$ the category of finite dimensional $A$-modules. I've been told there are examples where $K\_P(A)$ and $K\_F(A)$ have different rank, but I've never seen an example.
Does anyone h... | https://mathoverflow.net/users/4366 | When are PIMS and Irreducibles not in correspondence? | Let $A=k[x]$. Then $K\_P(A)$ has rank 1 (I assume that you consider only finitely generated projective modules) and $K\_F(A)$ has infinite rank.
| 4 | https://mathoverflow.net/users/4158 | 55938 | 34,938 |
https://mathoverflow.net/questions/55942 | 1 | Let $\mathcal{C}$ be any (perhaps small) category. Is there a (co)complete category $\mathcal{C}'$ and an inclusion $\mathcal{C}\hookrightarrow\mathcal{C}'$ which is universal among (co)complete categories containing $\mathcal{C}$? Perhaps something akin to the Yoneda embedding into the category of presheaves over $\ma... | https://mathoverflow.net/users/8157 | (Co)completions? | The Yoneda $\mathcal{C}\to Psh(\mathcal{C})$ is initial among pairs (D,F) consisting of a cocomplete category $D$ and a functor $F:\mathcal{C}\to D$. That is, it's the initial object of the comma category $\mathcal{C}\downarrow\_{Cat} CoComp$, where $CoComp$ is the category of cocomplete categories with colimit preserv... | 1 | https://mathoverflow.net/users/1353 | 55943 | 34,941 |
https://mathoverflow.net/questions/55959 | 10 | I have noticed that there are quite a few publications, many of them recent, on trying to determine the supremum of the gaps (normalized) between zeros of $\zeta \left(\frac{1}{2} + i t \right)$. Several make use of Wirtinger's inequality. I've been studying some of these papers in an attempt to broaden my knowledge of... | https://mathoverflow.net/users/8955 | Importance of large gaps between zeros of zeta function? | Here are two papers that show a connection between the spacing of the zeros of the zeta function and the class number problem for imaginary quadratic fields:
Conrey, J. B., and H. Iwaniec, “Spacing of zeros of Hecke L-functions and the
class number problem.”
Montgomery, H. L., and P. J. Weinberger, “Notes on small ... | 8 | https://mathoverflow.net/users/2384 | 55961 | 34,950 |
https://mathoverflow.net/questions/55953 | 15 | Hello,
Suppose $A$ is an Abelian variety of dimension $g$ over a number field $k$. Then using height functions one can show that there are non-torsion points in $A(\bar k)$. This looks like an overkill. Is there an easy, elementary way to see this?
Thanks!
Ramin
| https://mathoverflow.net/users/10458 | Torsion points in Abelian varieties over number fields | Let's take a page from Silverman's book, VII.3. Let $\mathfrak{p}$ be one of the primes of good reduction of $A$. Let $K/k$ be any extension, and let $\mathfrak{P}$ be a prime of $K$ above $\mathfrak{p}$. The reduction map $A(K)\to A(\mathcal{O}\_K/\mathfrak{P})$ becomes injective when you restrict to torsion points of... | 12 | https://mathoverflow.net/users/271 | 55965 | 34,954 |
https://mathoverflow.net/questions/55930 | 7 | is it possible to explicitly parametrise all the t-structures
on the derived category of finitely generated abelian groups?
| https://mathoverflow.net/users/11786 | t-structures on the derived category of finitely generated abelian groups | I guess the answer is the following. Take arbitrary subset $S$ of the set of all prime numbers. Let $A\_S = <Z,\{Z/pZ\}\_{p \not\in S},\{Z/pZ[-1]\}\_{p \in S}>$. Then $A\_S$ is a heart of the t-structure which is obtained by a simple tilting from the standard one. The claim is that any bounded t-structure is a shift of... | 7 | https://mathoverflow.net/users/4428 | 55967 | 34,955 |
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