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https://mathoverflow.net/questions/97876 | 7 | Does anyone know if the following holds?
Conjecture: Any two commuting elements in a reductive algebraic group G over C of rank>1 lie in a proper parabolic subgroup of G.
To make things easier, you can assume that these elements are semi-simple.
Note that if G is simply-connected then the centralizer of any semi-... | https://mathoverflow.net/users/23935 | commuting elements in a reductive group | Consider images in $\mathrm{PGL}\_n$ of the matrices $A$ and $B$, where $A$ is the diagonal matrix whose $i^{\rm th}$ diagonal entry is $\omega^i$, where $\omega$ is a primitive $n^{\rm th}$ root of $1$, and $B$ is the permutation matrix associated with the cycle $(1\, 2\, \ldots\, n)$. These two elements are semisimpl... | 13 | https://mathoverflow.net/users/4790 | 97886 | 57,105 |
https://mathoverflow.net/questions/97888 | 1 | Gauss's inequality is for unimodal distributions, concerning distance from the mode.
A similar result is Vysochanskiï–Petunin inequality, which is for the distance from the mean rather than the mode. Chebyshev's inequality generalizes Vysochanskiï–Petunin inequality by concerning distance from the mean without requi... | https://mathoverflow.net/users/5142 | Generalization of Gauss's inequality for not necessarily unimodal distributions? | Chebyshev's inequality is crude: The probability of being more than $k$ standard deviations away from the mean is at most $1/k^2$. A similar coarse estimate applies to the distance from the mode or any other point. Let $r\_x^2$ be the average of the square of the distance from $x$, which can be computed as $(x-\mu)^2 +... | 2 | https://mathoverflow.net/users/2954 | 97889 | 57,106 |
https://mathoverflow.net/questions/84072 | 2 | How do I compute the following?
$$ \mathrm{Prob}\left( \sum\_{i=1}^N x\_i >1\right) = ?$$
where $x\_i \sim \mathrm{Log}\mathcal{-N}(\mu\_i, \sigma\_i^2)$.
AFAIK, we do not know how the sum of log-normal distributions is distributed, so what I have in mind is to pick an appropriate interval for $x$, sample unifo... | https://mathoverflow.net/users/5223 | probability computation involving sum of log-normal random variables | The following paper: "The Distribution of Products of Beta, Gamma and Gaussian Random Variables"
Author(s): M. D. Springer and W. E. Thompson (on JSTOR)
gives a solution for the problem in the case of mean-zero independent normal random variables, using the Mellin transform.
In this case the density is given using ... | 2 | https://mathoverflow.net/users/6494 | 97891 | 57,107 |
https://mathoverflow.net/questions/97877 | 5 | Does anyone know a reference for the 2-dimensional version of the Schoenflies theorem? To be precise, I'd like a reference for the fact that every continuous, 1-1 map $S^1\rightarrow \mathbb{R}^2$ extends to a homeomorphism $\mathbb{R}^2 \rightarrow \mathbb{R}^2$. The discussions of the Jordan Curve Theorem that I can ... | https://mathoverflow.net/users/4042 | Reference request: 2-dimensional Schonflies theorem | Thomassen's paper on triangulating surfaces addresses this as well.
See: [Triangulating surfaces](https://mathoverflow.net/questions/17578/triangulating-surfaces)
| 2 | https://mathoverflow.net/users/23938 | 97895 | 57,109 |
https://mathoverflow.net/questions/97893 | 5 | The Turing equivalence relation on $\cal P(\mathbb{N})$ is defined by $A\equiv\_T B$ iff $A\leq\_T B$ and $B\leq\_T A$. This is a countable Borel equivalence relation on the polish space $\cal P(\mathbb{N})$. And as stated, the answer to my question is yes since a theorem by Feldman and Moore says: for any countable Bo... | https://mathoverflow.net/users/6342 | Is the Turing equivalence relation the orbit equiv. relation of the action of a countable group? | These [slides of Simon Thomas](http://www.math.cmu.edu/~eschimme/Appalachian/athens2.pdf) for the Appalachian Set Theory Seminar 2007 are directly engaged with aspects of this question, particularly the question of which countable Borel equivalence relations arise as free Borel actions of countable groups or are Borel ... | 5 | https://mathoverflow.net/users/1946 | 97896 | 57,110 |
https://mathoverflow.net/questions/97861 | 1 | I have some question about a special type of hypersurfaces in manifolds. Let $M$ be a compact Riemannian manifold of nonpositive sectional curvature with convex boundary. We call two totally-geodesic immersions $\phi\_1: M\_1 \rightarrow M$ and $\phi\_2: M\_2 \rightarrow M$ of closed manifolds $M\_1, M\_2$ of nonpositi... | https://mathoverflow.net/users/20818 | Properties of hypersurfaces in manifold of non-positive curvature. | First of all, in order for Claim 1 to be true you need to assume that $F$ has codimension 1 in $\tilde{M}$. Second, by "set of all flats" you presumably mean "union of all flats". Now, let's prove Claim 1: Take the union $U$ of all "parallel strips" of the form $F\times [0, a]\subset X$, where $F$ is identified with $F... | 1 | https://mathoverflow.net/users/21684 | 97905 | 57,111 |
https://mathoverflow.net/questions/97901 | 3 | The story of how the name 'orbifold' came about is pretty well-documented, but I can't find any explanation as to why Satake originally named orbifolds 'V-manifolds'. The 'manifold' part is clear enough; it's the 'V' I'm curious about.
Does the 'V' stand for anything, or is it just a random letter of the alphabet? (o... | https://mathoverflow.net/users/nan | What does the 'V' in 'V-manifold' stand for? | Satake (in his PNAS paper where V-manifolds are introduced and in his Journal of the Mathematical Society of Japan paper where Gauss-Bonnet theorem for V-manifolds is proven) never explains the origin of the name. If I were to guess, I, as temp, would say "V" stands for "virtual" (since, for instance, in topology and g... | 8 | https://mathoverflow.net/users/21684 | 97908 | 57,113 |
https://mathoverflow.net/questions/97880 | 4 | I am currently writing my master's thesis and I was wondering if the [rank distribution conjecture](http://math.stanford.edu/~fthorne/ec-ranks.pdf) was ever formally written down. Recall that it says that:
Half of all elliptic curves have rank $0$, half have rank $1$, and the rest have rank $\geq 2$.
I am looking f... | https://mathoverflow.net/users/22095 | Reference for Rank Distribution Conjecture. | Goldfeld made the conjecture for the family of quadratic twists of a given curve; it would be interesting to check the paper to see if he wrote anything about the more general case (I don't have it at hand). For the latter, the possibility that the average rank might be 1/2 was certainly folklore quite a while before K... | 2 | https://mathoverflow.net/users/21428 | 97913 | 57,117 |
https://mathoverflow.net/questions/97909 | 10 | How do we formally prove that the fundamental group of any Lie group is always commutative?
| https://mathoverflow.net/users/14210 | Commutativity of the fundamental group of any Lie Group | As Vahid says, it is true for any topological group. Here is a proof. I'm sure there are nicer, more conceptual ones out there, but here goes.
Let $G$ be your topological group. Take two loops $\sigma$ and $\gamma$ in $G$, based at the identity of $G$, which we will denote by $e$. Let $\sigma \cdot \gamma$ be the con... | 19 | https://mathoverflow.net/users/703 | 97916 | 57,120 |
https://mathoverflow.net/questions/97932 | 1 | Let $\mathcal A$ and $\mathcal B$ be abelian categories (with enough injectives and countable products) and $$F:D^+(\cal A) \rightarrow D^+(\cal B)$$
be a triangulated functor. I am interested whether there exists a dg-lift, by which I mean dg-functor
$$\tilde F:C^+(inj \cal A)\rightarrow C^+( inj \cal B)$$ between t... | https://mathoverflow.net/users/2837 | When do dg-lifts exist? | The question, stated in terms of abelian categories, is more subtle, but you could extend your question to differential graded algebras, i.e. instead of the derived category of an abelian category such as the category of modules over a ring, consider the derived category of a differential graded algebra. In tha case th... | 3 | https://mathoverflow.net/users/12166 | 97937 | 57,131 |
https://mathoverflow.net/questions/97939 | 12 | How many disks with radius 1/2 are needed to cover a disk with radius 1? It certainly cannot be done with less than 5 small disks, and some non-rigorous drawings of mine suggest it can be done with 7 small disks. Can it be done with less?
Slightly more generally, can anybody point me in the direction of any work on ... | https://mathoverflow.net/users/23951 | covering disks with smaller disks | You'll need seven disks with $r=0.5$ to cover a disk with $r=1$. See <http://mathworld.wolfram.com/DiskCoveringProblem.html>.
| 9 | https://mathoverflow.net/users/12357 | 97943 | 57,134 |
https://mathoverflow.net/questions/97954 | 2 | I just want to find some standard reference to the following result: let $(a\_k)\_k$ be the sequence of coefficients of a lacunary Fourier series which converges to an $L\_1(T)$ function in the sense of tempered distributions; then $(a\_k)\_k \in \ell\_2$.
Can someone help on this? And make the statement more precise... | https://mathoverflow.net/users/21553 | convergence of the coefficients of lacunary series | This seems to be stated, with copious references, and plenty of related results in Jean Pierre Kahane's survey Lacunary Taylor and Fourier series, Bulletin AMS 1964
| 1 | https://mathoverflow.net/users/11142 | 97959 | 57,141 |
https://mathoverflow.net/questions/97927 | 6 | If we have a rewrite system for primitive recursive functions, which simplifies each term according to how the function was defined, then what is the computational complexity of this calculation? That is, what is the coplexity of the normalization procedure? I have heard a claim that for a closed term calculating the v... | https://mathoverflow.net/users/23947 | computational complexity of primitive recursive functions | I find some aspects of this question difficult to understand. For example, the part about "calculating the value of the function requires transfinite induction up to $\varepsilon\_0$" seems to conflate methods of proof (like transfinite induction) with methods of computation. Also, much (if not all) of the question app... | 7 | https://mathoverflow.net/users/6794 | 97965 | 57,145 |
https://mathoverflow.net/questions/97941 | 2 | Let $G$ be a countable discrete group. Given a vector $\xi \in l^{2}(G)$, is there any way to naturally construct a [positive definite function](http://en.wikipedia.org/wiki/Positive-definite_function_on_a_group) on $G$ using $\xi$?
This question is rather vague and open-ended so I've made this CW, but I'd be *very* ... | https://mathoverflow.net/users/6269 | Positive definite functions on G from Hilbert space vectors? | I'm assuming you mean positive definite in the sense of <http://en.wikipedia.org/wiki/Positive-definite_function_on_a_group>. Well, consider the representation of $G$ on $\ell^2(G)$ by $\left(U\_g v\right)\_h = v\_{g^{-1} h}$. Then you might take the positive definite $L(\ell^2(G))$-valued function $F$ on $G$ defined b... | 1 | https://mathoverflow.net/users/13650 | 97967 | 57,147 |
https://mathoverflow.net/questions/97897 | 1 | Suppose we have a joint distribution $P(D,X,L) = P(D|X)P(X|L)P(L)$. Here, D and L are discrete but X is a continuous random variable. I want to compute $P(D=d)$. How do I do this numerically? The fact that $X$ is continuous confuses me.
| https://mathoverflow.net/users/5223 | Marginalizing over discrete and continuous random variables | I'm assuming that your "continuous" is actually "absolutely continuous", i.e. $X$ has a density.
$$P(D=d) = E[P(D=d|X)] = \int dx \ P(D=d|X=x) f\_X(x) = \int dx \sum\_\ell \ P(D=d|X=x) f\_{X|L}(x|\ell) P(L=\ell)$$
where the sum is over the possible values of $L$, and $f\_{X|L}(x|\ell)$ is the conditional density of... | 1 | https://mathoverflow.net/users/13650 | 97970 | 57,149 |
https://mathoverflow.net/questions/97971 | 5 | I was wondering if anybody knows a good reference book or exposition for Haar measures over profinite groups (with some concrete examples and computations)?
| https://mathoverflow.net/users/nan | Haar measure for profinite groups (reference needed) | Bill Casselman has [a very nice set of notes.](http://www.math.ubc.ca/~cass/research/pdf/Profinite.pdf)
| 5 | https://mathoverflow.net/users/11142 | 97973 | 57,150 |
https://mathoverflow.net/questions/97858 | 5 | This question is a refined version of [Representations of infinite dimensional Lie algebras as vector fields on manifolds](https://mathoverflow.net/questions/97659/representations-of-infinite-dimensional-lie-algebras-as-vector-fields-on-manifold)
I'm interested in the finite dimensional homogeneous spaces of $Diff(S^... | https://mathoverflow.net/users/17660 | Finite dimensional homogeneous spaces of $Diff(S^1)$ | I'm afraid that you are mistranslating Cartan's notation. The first 3-dimensional example that Cartan gives is this: Every diffeomorphism of the line, written in the form $X = f(x)$, can be 'lifted' to a diffeomorphism of 3-space:
$$
X = f(x),\qquad Y = \frac{y}{f'(x)},\qquad Z = \frac{z}{f'(x)} - \frac{f''(x)}{f'(x)^2... | 5 | https://mathoverflow.net/users/13972 | 97980 | 57,153 |
https://mathoverflow.net/questions/97988 | 4 | In 1987 Roggenkamp and Scott published a solution of the integral isomorphism problem for $p$-groups, i.e. if $G,H$ are $p$-groups and $\mathbb{Z}[G] \cong \mathbb{Z}[H]$ as rings then $G \cong H$.
However, in practice I guess it is at least as hard to show that two group rings aren't isomorphic than to show that th... | https://mathoverflow.net/users/17734 | Applications of the Roggenkamp-Scott theorem ? | The *Annals* paper by Roggenkamp and Scott was certainly a landmark in the ongoing study of the isomorphism problem for integral group rings of finite groups, which apparently goes back to the thesis work of Graham Higman and later related work by Richard Brauer. Zassenhaus refined and extended the underlying problem o... | 6 | https://mathoverflow.net/users/4231 | 98000 | 57,158 |
https://mathoverflow.net/questions/98002 | 4 | Charles Hermite have created a method using elliptic functions to solve fifth degree polynomial, to get around the theory of Galois. Can someone explain me it and give a simple example?
Tank you.
| https://mathoverflow.net/users/23968 | How to solve a fifth degree polynomIal | What about the topic on Quintic Functions in [Wikipedia](http://en.wikipedia.org/wiki/Quintic_function)?
| 2 | https://mathoverflow.net/users/22714 | 98005 | 57,160 |
https://mathoverflow.net/questions/97990 | 1 | If the function $f:R\to R$ is of BV class then it has at most countably many discontinuity points (since it can be represented as a sum of two monotonic function).
I am interested to know whether the BV function $f:R^2\to R$ has at most countable many curves along which it is discontinuous?
Thanks a lot in advance... | https://mathoverflow.net/users/23967 | cardinality of discontinuity curves of BV function | The answer seems to be negative.
For a counterexample, let $T$ be a continuous infinite binary tree in the unit disk, where one finds the branching nodes as one moves away from the origin and toward the boundary of the unit disk. I imagine that one starts at the origin and finds the branching nodes of the tree as one... | 2 | https://mathoverflow.net/users/1946 | 98008 | 57,162 |
https://mathoverflow.net/questions/96776 | 8 | Consider a smooth $k$-dimensional foliation of the unit ball $B$ of $\mathbb{R}^n$, all of whose leaves are diffeomorphic to $k$-disks.
**Question**: Is there a leaf whose $k$-volume is at least $\omega\_k$?
Here $\omega\_k$ denotes the volume of the unit ball of $\mathbb{R}^k$.
*Particular cases*. If $k=1$, the... | https://mathoverflow.net/users/23288 | Existence of a large leaf in a foliation of the ball | I think one might be able to prove this without using the heavy duty machinery from the geometric measure theory that Anton mentioned by modifying the proof of the main result of the following paper of Gromov ["Isoperimetry of Waists and Concentration of Maps"](http://www.springerlink.com/content/1lnx26ecfnxkxtva/).
... | 8 | https://mathoverflow.net/users/18050 | 98015 | 57,165 |
https://mathoverflow.net/questions/90507 | 4 | Hi, I was thinking about the following question ; I will appreciate it if somebody can give me a full or partial answer or can at least cite any reference(s)/ papers etc :
By $ \bar{P} $ , we denote a topological pair of pants ( that is, a 2-sphere with three open disks removed ) with boundary, and by $P$, we would m... | https://mathoverflow.net/users/6953 | Characterization of the moduli space of the pair of pants in terms of the modules of the extremal ring domains | I think the map $F$ is injective, but not surjective.
There is a unique conformal map of the interior of the pants to the complement of 3 slits in $\mathbb{RP}^1\subset \mathbb{CP}^1$, up to the action of $PGL\_2(\mathbb{R})$. If we parameterize the slits as $[a\_0,a\_1],[a\_2,a\_3],[a\_4,a\_5]$, where $a\_0 < a\_1 ... | 3 | https://mathoverflow.net/users/1345 | 98016 | 57,166 |
https://mathoverflow.net/questions/97787 | 4 | Let $G$ be a finite abelian group, $n$ a positive integer and let $G^n$ denote the direct product of $n$ copies of $G$. We say an element of $G^n$ is *full* if it acts as a nonidentity element of $G$ in each of the factors of $G^n$.
Now consider the following random process. Sample a full group element $(g\_1,g\_2,\l... | https://mathoverflow.net/users/1171 | Generating a group by randomly sampling generators | Let's get a lower bound by considering sampling *with* replacement, for which we can even get the exact answer. If we sample $k$ full elements, what is the probability that they agree at exactly $l$ locations? For $G = \mathbb{Z}/2 \times \mathbb{Z}/2$, the answer is
$$
p(k,l) = {n \choose l} \left(\left(\frac{1}{3}\ri... | 0 | https://mathoverflow.net/users/1171 | 98017 | 57,167 |
https://mathoverflow.net/questions/97709 | 2 | Recall from [my previous question](https://mathoverflow.net/questions/97636/representability-of-hom-sheaves-of-various-moduli-spaces) the definition of a local hom-sheaf of a stack of categories. I am interested in stacks of categories such that the underlying stack of groupoids is a moduli stack.
In [his very useful... | https://mathoverflow.net/users/4177 | Carving out subsheaves of local hom-sheaves of stacks of categories | Taking equalizers, or limits, is also the standard algebraic geometry way. So, for example, say that $X \to S$ and $Y \to S$ are finitely presented and proper, with $X\to S$ flat, and have sections $S \to X$ and $S \to Y$. You want to consider the subsheaf $H'$ of $H :=\underline{Hom}\_S(X, Y)$ of morphisms preserving ... | 2 | https://mathoverflow.net/users/4790 | 98021 | 57,169 |
https://mathoverflow.net/questions/98013 | 7 | Given a Regular language (represented as a black box to which one can apply inputs and get 0/1) Is there an algorithm that can find a finite deterministic automaton that produces that language?
| https://mathoverflow.net/users/23970 | Is there an algorithm that can "reverse engineer" a Regular Expression? | Without additional assumptions there is no such algorithm. In finite time an alleged algorithm could only test for finitely many inputs, but since there are infinitely many regular languages which match any given finite number of test cases, the algorithm cannot work.
In machine learning this is a common situation (g... | 13 | https://mathoverflow.net/users/1176 | 98022 | 57,170 |
https://mathoverflow.net/questions/97863 | 5 | Let $n \ge 4$ be a natural number.
Consider the quotient map from the double cover $2 \cdot A\_n$ of the alternating group $A\_n$ to $A\_n$. For any conjugacy class in $A\_n$ of size $r$, the inverse image in $2 \cdot A\_n$ has size $2r$. This inverse image is either a single conjugacy class or "splits" as a union of... | https://mathoverflow.net/users/3040 | Conjugacy class splitting in double cover of alternating, symmetric group | I believe the answer is that the class of an element of $g \in A\_n$ becomes a single class when lifted to the 2-fold central covering group of $A\_n$ if and only if $g$ has even order and $g$ has (at least) two cycles of the same length (including cycles of length 1).
The class of an element of $g \in S\_n$ lifts to... | 5 | https://mathoverflow.net/users/35840 | 98026 | 57,172 |
https://mathoverflow.net/questions/97984 | 7 |
Given the Chromatic Convergence Theorem, can we state this globally as some convergence in the category of categories? That is, we have the subcategories of finite spectra in each stable homotopy category of $E(n)$-localized spectra, say $\mathcal{L}\_n^{fin}$. Is it correct that there should be "inclusion" functors $... | https://mathoverflow.net/users/11546 | Chromatic convergence of E(n)-localized homotopy categories | You have inclusion and localization functors. The inclusion functors go in the direction you indicate (any $E(n)$-local spectrum is also $E(n+1)$-local), and the localization functors are left adjoints, goint in the oposite direction. In your heuristics, I would consider the localization functors because of the followi... | 4 | https://mathoverflow.net/users/12166 | 98033 | 57,175 |
https://mathoverflow.net/questions/40277 | 4 | To state my question, I first have to explain what I mean by "using operations and their compositions only". In short, it means that I am looking for theorems from clone theory (or on operations and relations in general) that can still be stated if we consider the operations as morphisms in an abstract category that ha... | https://mathoverflow.net/users/8590 | Theorems (from clone theory) that can be stated only by using operations and their composition. | You are probably aware of it, but what you describe are precisely all statemtents that soley depend on the abstract clone behind the clone. Or, equivalently speaking, that hold for the Lawvere theory that is behind the clone (looking at it as the model of this particular Lawvere theory).
An example for such a stateme... | 4 | https://mathoverflow.net/users/23978 | 98034 | 57,176 |
https://mathoverflow.net/questions/97976 | 18 | Similar to [this topic](https://mathoverflow.net/questions/28268/do-you-read-the-masters), what are the easiest foundational French texts for someone learning the language? My intuition would be Cauchy and Lebesgue, but I have no idea where to start or which of their works are readily available.
| https://mathoverflow.net/users/23959 | Approachable French masters | Since you mention Lebesgue, I would recommend the following two classics, which build on his lectures at the Collège de France :
*[Leçons sur l'intégration](http://archive.org/stream/leonssurlintgra00lebegoog#page/n8/mode/2up)*
*[Leçons sur les séries trigonométriques](http://archive.org/stream/leonssurlessrie01le... | 9 | https://mathoverflow.net/users/6506 | 98042 | 57,178 |
https://mathoverflow.net/questions/98027 | 4 | There are some pretty simple methods to do uniform sampling on the surface of high-dimensional spheres or cubes.
Are there any methods that sample uniformly on the surface of a high-dimensional polytope?
Given a high-dimensional polytope like this:
$\| A \mathbf{x} \|\_{\infty} = 1$
where $A$ is a given $m \times... | https://mathoverflow.net/users/23975 | Is it possible to sample uniformly on the surface of a high-dimensional polytope? | Your question is ambiguous: from the title, it seems that you want to sample uniformly from the interior of the convex polytope. This is a heavily studied area, see for example [this question](https://mathoverflow.net/questions/9854/uniformly-sampling-from-convex-polytopes) or [this paper by Diaconis, Lebeau, Michel](h... | 2 | https://mathoverflow.net/users/11142 | 98044 | 57,179 |
https://mathoverflow.net/questions/98031 | 0 | Consider a locally profinite group $G$, i.e. a locally compact, totally disconnected topological group. Suppose it admits an open maximal compact subgroup named $K$.
It is known that $G$ admits as a neighborhood basis of the identity element a collection $\{ K\_i \}$ of open compact subgroups, but what can we say about... | https://mathoverflow.net/users/20228 | Neighborhood basis of the identity in a locally profinite group | Check out [Casselman's notes.](http://www.math.ubc.ca/~cass/research/pdf/Profinite.pdf) They are quite enlightening, and just came up yesterday for [another question.](https://mathoverflow.net/questions/97971/haar-measure-for-profinite-groups-reference-needed)
| 0 | https://mathoverflow.net/users/11142 | 98052 | 57,184 |
https://mathoverflow.net/questions/98043 | 18 | Let $S=\mathbb{C}[x\_1,x\_2,\dots,x\_n]$ be a polynomial ring. Let $n \geq 3$. Let $h\_a$ denotes the complete homogeneous symmetric polynomial of degree $a$.
$$ h\_a=\text{ sum of all monomials of degree } a.$$
For example: for $n=3$ and $a=2$, one has: $$h\_2=x\_1^2+x\_2^2+x\_3^2+x\_1x\_2+x\_1x\_3+x\_2x\_3.$$
**Ques... | https://mathoverflow.net/users/23981 | Is a complete homogeneous symmetric polynomial irreducible? | EDIT : prompted by Will Sawin's comment, the argument now works for every $n \geq 3$. Thanks !
The polynomial $h\_a(x\_1,\ldots,x\_n)$ is irreducible for every $a \geq 1$ and $n \geq 3$.
Recall that if $h\_a = FG$ with $F$ and $G$ non constant then $F$ and $G$ have to be homogenous. By Bézout's theorem, the hypersu... | 17 | https://mathoverflow.net/users/6506 | 98054 | 57,186 |
https://mathoverflow.net/questions/98050 | 3 | Consider the following impartial combinatorial game played with finite graphs: A move removes two adjacent vertices; and of course all edges connected with them. The game then continues with the new graph. Often this turns out to be disconnected, so we end up in a sum of smaller games. Let us choose the normal play rul... | https://mathoverflow.net/users/2841 | The game of removing two vertices in a graph | Per Martin's request, here is a more detailed version of my comment. Given a graph $G$, the [independence game](http://www.jstor.org/stable/3619902?seq=1) is the game in which two players take turn in removing a vertex and all of its neighbors. The game terminates when a player can make no further move, and it is a win... | 4 | https://mathoverflow.net/users/8761 | 98055 | 57,187 |
https://mathoverflow.net/questions/98075 | 2 | Let $S=\mathbb{C}[x\_1,\dots,x\_n]$ be a polynomial ring and $p\_a=x\_1^a+\cdots+x\_n^a$ be a power sum symmetric polynomial in $S$. Let $n \geq 3$.
Question: To show $p\_m,p\_{2m}, \dots,p\_{nm}$ forms a regular sequence in $\mathbb{C}[x\_1,\dots,x\_n]$ for any $m \in \mathbb{N}$. Or equivalently let $I=\langle p\_m... | https://mathoverflow.net/users/23981 | Regular sequence of power sum symmetric polynomials in polynomial ring. | I believe this is true for a very simple reason. See lemma 2.2 in Conca, Krattenthaller and Watanabe.
A non-trivial common zero point of $p\_{m},p\_{2m},\dots, p\_{nm} $ in $n$ variables exists iff there is a non-trivial common zero of $p\_{1},p\_2,\dots,p\_{n}$, but this is absurd. (Every symmetric polynomial would ... | 4 | https://mathoverflow.net/users/2384 | 98076 | 57,198 |
https://mathoverflow.net/questions/98082 | 13 | Suppose that a mathematician such as Bugs Bunny answers one of my math questions here on MathOverflow, and then I use the idea in a research paper. How should I acknowledge such a distinguished mathematician in my paper, submitted to a reputable journal and all that?
| https://mathoverflow.net/users/23993 | Etiquette question: how to acknowledge Bugs Bunny? | Look right underneath where your question is posted. Click on the "cite" link. It will pop up a text box, from which you can copy citation data for this post in either bibtex or amsrefs format. So you can directly cite the MO thread.
| 21 | https://mathoverflow.net/users/703 | 98083 | 57,200 |
https://mathoverflow.net/questions/98093 | 2 | First, apologies for the title. This is an odd question, and I couldn't come up with a simple title for it.
---
My question is as follows.
Given a positive integer $k$, determine a set of properties $S$ such that exactly $k$ positive integers satisfy all the properties in $S$, subject to the following condition... | https://mathoverflow.net/users/23976 | Generating a set of integer passwords that can be securely authenticated | What about this simple solution: the "safe" contains the $k$ public keys of some RSA pair, the users own each one his/her private key. Standard public/private key authentication methods can now be used. Your set $S$ contains the single property "the key is the private key associated to either one of these $k$ public ke... | 3 | https://mathoverflow.net/users/1898 | 98098 | 57,207 |
https://mathoverflow.net/questions/98096 | 4 | One classical Mertens' theorem tells us that $$\prod\_{p \leq n} (1-\frac{1}{p})^{-1} = e^\gamma \log n + \mathcal{O}(1).$$
It is now very natural to ask, whether we have some good estimate to $$\prod\_{p \leq n} (1-\frac{1}{p^s})^{-1}$$ for, let's say, $s > 1$ real. Of course the limit is $\zeta(s)$ for growing $n$, a... | https://mathoverflow.net/users/13763 | Generalization of Mertens' theorem | I detail. It suffices to study the "tail" $P(X) = \prod\_{p > X} (1- \frac{1}{p^s})^{-1} $. Using $-\log(1-y) = y + O(y^2)$, we get for real $s>1$
$$ \log P(X) = \sum\_{p > X} \frac{1}{p^s} + O \left( \frac{1}{X^{2s-1}} \right) = \int\_{Y>X} \frac{d\pi(Y)}{Y^s} + O \left( \frac{1}{X^{2s-1}} \right) $$
$$= \int\_{Y>X} ... | 9 | https://mathoverflow.net/users/21724 | 98106 | 57,210 |
https://mathoverflow.net/questions/98103 | 3 | For any p-adic local field K, all 1-dim **semi-stable** Galois repn: $G\_K \to Q\_p^{\*}$ are just $Q\_p(n)\otimes \mu$, where $Q\_p(n)$ is the Tate twist of cyclotomic character, and $\mu$ an unramified charater.
**My question** is what if we replace the coefficient field to $E \neq Q\_p$?
In fact, at the end of t... | https://mathoverflow.net/users/24003 | 1-dimensional semi-stable Galois representations with coefficients | If $E \neq Q\_p$ then there may be more $1$-dimensional crystalline representations that the ones you mention. By Lubin-Tate theory, every character of $G\_K$ can be written as an unramified character times a character of $O\_K^\times$ (after making proper choices and identifications). The algebraic characters of $O\_K... | 5 | https://mathoverflow.net/users/5743 | 98115 | 57,213 |
https://mathoverflow.net/questions/98118 | 2 | What would you do if you see a notion which you used to work with has two names and the same name you use for this notion is used for another notion too. For example, after the work of J.B. Bost, A. Connes, (Hecke algebras, type III factors and phase transition with spontaneous symmetry breaking in number theory. Selec... | https://mathoverflow.net/users/nan | When a name is used for two different notions! | To start with the obvious, in a given work you should conform to the usage that the majority of your readers expect. A second principle is that you should work out some reasonable system for your own work and follow that consistently. In cases where you expect readers from both camps, you will need to frequently draw y... | 3 | https://mathoverflow.net/users/1266 | 98120 | 57,216 |
https://mathoverflow.net/questions/98107 | 10 | Is it possible to deform the metric $g$ of a closed Riemannian manifold $(M,g)$ satisfying $\mathrm{Ricci}(M,g) > 0$ and $\mathrm{sec}(M,g) \geq 0$ to a metric $g\_1$ satisfying $\mathrm{sec}(M,g\_1) > 0$?
I understand that this question is ludicrous, at best (for instance, an affirmative would prove that $S^n\times ... | https://mathoverflow.net/users/20557 | Metric deformations from non-negative to positive curvature | As Benoît Kloeckner points out this is false for non simply connected manifolds with $RP^2\times RP^2$ being a counterexample (by Synge's theorem). For simply connected manifolds this is a well known open problem.
BTW, Sha-Yang examples are only known not to admit nonnegative sectional curvature for connected sums o... | 14 | https://mathoverflow.net/users/18050 | 98128 | 57,220 |
https://mathoverflow.net/questions/98121 | 0 | Are there any three dimensional subalgebras of GA(n) where GA(n) is the geometric algebra corresponding to $R^n$? If yes, what about for GA(2)?
Edit: a geometric algebra is a Clifford algebra.
| https://mathoverflow.net/users/13491 | Three dimensional subalgebras of Clifford Algebras | I see from Wikipedia that "geometric algebra" is another name for a Clifford algebra, which I suppose explains the clifford-algebras tag. Since you are working over the real numbers, you need to specify what the signature of the quadratic form is, as the isomorphism type of the Clifford algebra strongly depends on the ... | 2 | https://mathoverflow.net/users/703 | 98137 | 57,225 |
https://mathoverflow.net/questions/77679 | 3 | Consider an infinite, upper-triangular Toepliz-matrix, with first row $x\_1,x\_2,\dots,x\_n.$
Then there is a sequence of determinants obtained from the $m \times m-$ sub-matrices
with upper left corner at first row, and $k^{th}$ column (thus $x\_k$ is in the upper left corner of the sub-matrix).
Thus, for each $n$ ... | https://mathoverflow.net/users/1056 | Find recurrence in Pascal-like triangle of polynomials | I found an explicit way to express all these recurrences.
Put $t \mapsto -t$ in this question,
[Constructing new polynomials by product of roots](https://mathoverflow.net/questions/88853/constructing-new-polynomials-by-product-of-roots)
and you have the characteristic polynomial for the recurrence.
This is actually... | 3 | https://mathoverflow.net/users/1056 | 98138 | 57,226 |
https://mathoverflow.net/questions/98111 | 2 | Let $G$ be a group, $a \in G$, $|a|$ is infinite. If $|\langle a \rangle ^G : \langle a \rangle |$ is finite ($>1$). I want know whether or not that $|x|$ is infinite for every non-trivial element in $\langle a \rangle^G$?
We hope to investigate the influence of normal closure, so any reference about this situation w... | https://mathoverflow.net/users/22049 | About normal closure of cyclic subgroup | No. Consider the semi-direct product $(\mathbf{Z}\times \mathbf{Z}/2)\rtimes \mathbf{Z}/2$, where the rightmost factor acts by $(1,0)\mapsto (1,1)$ and fixes $(0,1)$. Then the normal closure of $\mathbf{Z}\times 0\times0$ is $\mathbf{Z}\times\mathbf{Z}/2\times 0$.
| 4 | https://mathoverflow.net/users/5513 | 98142 | 57,228 |
https://mathoverflow.net/questions/98130 | 1 | Let there be $m$ points in the Euclidean space $\mathbb R^n$. Randomly choose $k$ distinct pairs of these $m$ points, and assign a random (positive) value for the Euclidean distance between each of these $k$ pairs.
Determine the maximum value of $k$ as a function of $n$ and $m$ such that, for any random choice of $k... | https://mathoverflow.net/users/23976 | Determining the maximum number of distance relationships that can be defined between points in Euclidean space | The following 6 distances between 4 points $a,b,c,d$ can not be realized in a Euclidean space of any dimension: $d(a,b)=d(b,c)=d(a,c)=1$ and $d(a,d)=d(b,d)=d(c,d)=0.51$, although all triangle inequalities are satisfied and even strict.
Adding any number of points and assigning any set of other distances to be equal t... | 6 | https://mathoverflow.net/users/4354 | 98143 | 57,229 |
https://mathoverflow.net/questions/74287 | 3 | Just to include something that starts to answer my own question [Topological Quantum Computation Lecture notes](http://books.google.com/books?id=jkzVwceFaAsC&dq=zhenghan+wang+topological+quantum+computation&source=gbs_navlinks_s) covers a lot of the Mathematics of the Fractional Quantum Hall effect, or topological quan... | https://mathoverflow.net/users/17532 | Fractional Quantum Hall Effect - Mathematics | Here is [a review](http://arxiv.org/abs/1203.3268) of FQHE for mathematicians.
| 1 | https://mathoverflow.net/users/17787 | 98165 | 57,241 |
https://mathoverflow.net/questions/98149 | 6 | This is a natural generalization of [this question](https://mathoverflow.net/questions/76193/cyclotomic-polynomials-coprime-to-a-fixed-polynomial).
Let $f$ be a monic irreducible polynomial over $\mathbb Z$. Let $S\_f$ be the set of natural numbers $n$ such that one of the three equivalent conditions hold:
a. There... | https://mathoverflow.net/users/18060 | The resultant of an arbitrary polynomial and a cyclotomic polynomial | For an algebraic number $\gamma$ which is not a root of unity, Baker's theorem gives
a bound (uniform in $n$) of the form $|\gamma^n - 1| > n^{-C}$ for some constant $C$
(only depending on $\gamma$).
In particular, if $s\_n:=\prod |\alpha^n\_i - 1|$, then Baker's method gives the following estimate uniform in $n$ (fo... | 7 | https://mathoverflow.net/users/nan | 98176 | 57,247 |
https://mathoverflow.net/questions/97929 | 16 | Referring to [this](https://math.stackexchange.com/questions/147377/other-functional-equations-for-zetas) question I asked on math.SE.
I am posting a more generalized question here, for answers and further inquiry.
>
> For the Riemann zeta function, we know of the standard functional equation that relates $\zeta(s)... | https://mathoverflow.net/users/2865 | Certain functional equations for the Riemann Zeta function? | Equations of this type are known. You may see, for example, the classical
book "Primzahlen" by Landau paragraph 67. "Continuation of the zeta function by partial integration"
There it is proved the formula
$$ (s-1)(\zeta(s)-1)-1=-\frac{(s-1)s}{2!}(\zeta(s+1)-1)-
\frac{(s-1)s(s+1)}{3!}(\zeta(s+2)-1)-\cdots$$
$$\cdots-... | 16 | https://mathoverflow.net/users/7402 | 98177 | 57,248 |
https://mathoverflow.net/questions/98184 | 6 | I have a difficulty with hyperbolic geometry.
Let $\mathbb{H}^{2}$ be a 2-dimensional hyperbolic plane.
(i.e., upper half plane in $\mathbb{R}^{2}$ with a metric $\frac{ds}{y}$)
(or, upper half plane in $\mathbb{C}$ with a metric $\frac{|dz|}{\textrm{Im}(z)}$ )
You may have heard about pseudosphere in $\mathb... | https://mathoverflow.net/users/24057 | A question about embedding hyperbolic space onto pseudosphere | You should be looking at the theory of Bäcklund transformations for surfaces of Gaussian curvature $K=-1$. There is a large literature on this, and there are many examples of pseudospherical immersions that are not surfaces of revolution. You should especially look at the work of Chuu-Lian Terng in this area. She, toge... | 7 | https://mathoverflow.net/users/13972 | 98186 | 57,254 |
https://mathoverflow.net/questions/94624 | 5 | Somewhat nebulous question: there are many well known "special" values of
the Jones polynomial, especially those at roots of unity. I always run into
one that has unlink value $\phi$ (golden mean) and writhe factor $(-1)^{1/5}$.
Is there something special about it (maybe it's "at the intersection" of
the Lie groups A1 ... | https://mathoverflow.net/users/11504 | Jones(unlink)=phi | In some sense this is the smallest possible quantum group, so it's perhaps not surprising that it comes up often. In fact, if you have only 2 objects, then there are very few possibilities, see Ostrik's paper <http://arxiv.org/abs/math/0203255>.
| 6 | https://mathoverflow.net/users/22 | 98202 | 57,259 |
https://mathoverflow.net/questions/98218 | 3 | Consider complex affine space $\mathbb{C}^n$ and let $U$ be a Zariski open subset of $\mathbb{C}^n$. By a celebrated result of Deligne, the cohomology $H^i(U)$ has a canonical Hodge structure. In particular, $H^i(U)$ has a weight filtration and a subalgebra of pure classes (since a cohomology class can't have lower wei... | https://mathoverflow.net/users/66 | Is it written anywhere that open subvarieties of affine spaces have "completely impure" cohomology? | Maybe I'm missing something, but I think this should be a simpler proof. Let $j \colon U \hookrightarrow \mathbf P^n$ be the natural compactification, and let $k > 0$. Then $W\_k H^k(U,\mathbf Q)$ is the image of $j^\ast \colon H^k(\mathbf P^n,\mathbf Q) \to H^k(U,\mathbf Q)$ (Hodge II, Corollaire 3.2.17). But $j^\ast$... | 13 | https://mathoverflow.net/users/1310 | 98220 | 57,268 |
https://mathoverflow.net/questions/98212 | 2 | I have a totally ordered group $(\mathbb{R};\leq,\oplus,0,-)$ with the reals as base set satisfying monotonicity, i.e.
for all $x,y,z$ we have that if $y\leq z$ then $x\oplus y \leq x\oplus z$, and I am wondering if this already suffices to show that the group is archimedean. (i.e., that for all $a,b$ with $a \leq b$ t... | https://mathoverflow.net/users/24039 | Non-archimedean group over the reals | I guess that the notation $(\mathbb R;\leq,\oplus,0)$ is intended to say that your ordered group is the set of reals with its standard ordering $\leq$ but with a possibly strange group operation $\oplus$ (whose identity element is nevertheless the standard 0). In this case, archimedianness can be proved as follows. Con... | 9 | https://mathoverflow.net/users/6794 | 98221 | 57,269 |
https://mathoverflow.net/questions/98228 | 0 | I'm working with maximum a posteriori estimation and managed to show that every probability density function that is continuous in all $R^n$ always has at least one global maximum. I've search around and asked a few fellow engineers and professors but am not sure if this is widely known. This can actually be extended t... | https://mathoverflow.net/users/24041 | Is it known that every PDF continuous in all $R^n$ has a maximum? | Take $n=1$ and put a triangle with height $2^m$ and width $2^{-2m}$ at each integer $m=0,1,2,\dots$
| 5 | https://mathoverflow.net/users/19276 | 98237 | 57,275 |
https://mathoverflow.net/questions/98229 | 19 | In other words:
>
> What is $\mathrm{Ext}\_{\mathcal{A}}^{4,t}(\mathbb{Z}/2,\mathbb{Z}/2)$?
>
>
>
If the 4-line is not known, how much is known about it?
Here, $\mathcal{A}$ is the 2-primary Steenrod algebra, $4$ is the homological degree corresponding to the Adams filtration, and $t$ is the internal grading... | https://mathoverflow.net/users/288 | Is the 4-line of the E_2 term of the classical Adams spectral sequence known? | The 4-line is determined by Wen-Hsiung Lin in "$Ext\_A^{4,\*}({\bf Z}/2,{\bf Z}/2) $ and $Ext\_A^{5,\*}({\bf Z}/2,{\bf Z}/2) $", Topology and its Applications (2008) vol 155 no.5 pp 459-496.
He gives a basis for the indecomposable elements in $Ext\_A^{4,\*}$ and generators and relations for the quotient of $Ext\_A^{s... | 23 | https://mathoverflow.net/users/4648 | 98254 | 57,282 |
https://mathoverflow.net/questions/98241 | 2 | A projector $P$ is a Hermitian matrix satisfying $P^2=P$. For any two projectors, it is easy to show that there exists a unitary matrix $U$ such that both $U^\*PU$ and $U^\*QU$ are block-diagonal matrices, each block is of size at most $2$.
For arbitrary $k$ projectors, there is no simultaneous decomposition such th... | https://mathoverflow.net/users/2738 | Simultaneous decomposition of three projectors | One way to put these questions is to ask for a classification of finite-dimensional $\*$-representations of a universal $C^\*$-algebra. In the first case it is the universal $C^\*$-algebra generated by two projections, i.e. the unital $C^\*$-free product of $\mathbb C^2$ with itself. You observed that the maximal dimen... | 5 | https://mathoverflow.net/users/8176 | 98255 | 57,283 |
https://mathoverflow.net/questions/98227 | 2 | I asked the same question on stackexchange, but I didn't get an answer, so I am reposting it here in hope of one (or an appropriate reference to a textbook or otherwise). I am assuming all groups finite. Suppose $A$ is an elementary abelian $2$-group and $C$ is a cyclic group of odd order acting fixed-point-freely on $... | https://mathoverflow.net/users/nan | Product of an elementary abelian group and a cyclic group of coprime order | I think that one reason that you got no responses on stackexchange was that your question was too vague, and you should ask more specific questions, like you have done here.
The answer to your question is no. Let $G$ be the subgroup of $S\_8$ generated by the permutations
$$(1,2)(3,4), (5,6)(7,8), (1,2,3)(5,6,7).$$... | 7 | https://mathoverflow.net/users/35840 | 98258 | 57,284 |
https://mathoverflow.net/questions/98243 | 2 | Let $G$ be a graph. I've heard that, if we use nauty to canonically label $G$ on two different platforms, it's possible to obtain distinct labels. However, I've never actually seen this occur.
The nauty user guide ([pdf](http://cs.anu.edu.au/~bdm/nauty/nug.pdf)) writes:
>
> Beginning at version 2.1, the
> canoni... | https://mathoverflow.net/users/2264 | An example of when nauty, on two different platforms, gives different canonical labels for the same input graph? | It's only possible if there's a bug, and since there are no bugs it doesn't happen.
| 7 | https://mathoverflow.net/users/9025 | 98264 | 57,287 |
https://mathoverflow.net/questions/98272 | 0 | Let $A$ be an artinian ring. It is well known that for an element $x$ in $R$ the right annihilator $Ann\_r(x)$ is non trivial (i.e. contains a nonzero element ) if and only if the left annihilator $Ann\_l(x)$ is non trivial.
If we assume that the ring $A$ is finite is true that the cardinality of $Ann\_r(x)$ equals ... | https://mathoverflow.net/users/20272 | Annihilators in artinian ring | No. Let $k$ be a finite field. Let $A$ be the non-commutative $k$ algebra generated by $x$ and $y$ subject to $x^2=0$, $y^2=0$, $yx=0$. So $\dim\_k A = 4$, with basis $1$, $x$, $y$, $xy$. Look at the element $y$: Its left annihilator has basis $(y, xy)$ and its right annihilator has basis $(x, y, xy)$.
| 3 | https://mathoverflow.net/users/297 | 98275 | 57,291 |
https://mathoverflow.net/questions/98266 | 1 | I encounter the following claim in my research for which I couldn't get a solution for a long time. I asked a more general version of the [question](https://math.stackexchange.com/questions/145811/a-problem-with-ell-p-norm) at math.stackexchange which did not attract much attention there. Following is a particular vers... | https://mathoverflow.net/users/7699 | An inequality with $\ell_p$ norm | The answer is no, for any $n\ge 3$ and any $C^1$-smooth strictly convex norm on $\mathbb R^n$. (Here "strictly convex" means that the triangle inequality is strict for any two non-collinear vectors. This is equivalent to the following: the norm restricted to any affine subspace not containing 0 is a strictly convex fun... | 3 | https://mathoverflow.net/users/4354 | 98279 | 57,292 |
https://mathoverflow.net/questions/98262 | 15 | What are the simplest numerical examples of *even* dihedral (resp. tetrahedral, resp. octahedral, resp. icosahedral) representations
$\rho:\mathrm{Gal}(\bar{\mathbf{Q}}|\mathbf{Q})\to\mathrm{GL}\_2(\mathbf{C})$
and their associated Maaß forms $f\_\rho\ $ ? The word *simplest* can be taken to mean that the conduct... | https://mathoverflow.net/users/2821 | The simplest even Artin representations of degree 2 and the corresponding Maaß forms | I'll leave the dihedral and octahedral case to others, but for the tetrahedral ($A\_4$) and icosahedral ($A\_5$) case, I can give some answer.
For the tetrahedral case, the smallest conductor is 163. See my question: [Does anyone want a pretty Maass form?](https://mathoverflow.net/questions/22908/does-anyone-want-a-p... | 9 | https://mathoverflow.net/users/3545 | 98290 | 57,295 |
https://mathoverflow.net/questions/98286 | 1 | Here is one of the classical versions of the nullstellensatz: Let $K$ be a field and let $\mathfrak{m}$ be a maximal ideal of the polynomial ring $K[T\_1,\ldots,T\_n]$. Then $K[T\_1,\ldots,T\_n]/\mathfrak{m}$ is a finite extension of $K$.
I am interested in a noncommutative version of this theorem. To be more precise... | https://mathoverflow.net/users/7139 | A Version of Nullstellensatz for Rings of Dİfferential Operators | I don't think it's true. Namely, Stafford showed ("Non-holonomic modules over Weyl algebras and enveloping algebras," Inventiones Mathematicae, 1985, Volume 79, Number 3, Pages 619-638) that if you choose $\lambda\_2, \dots, \lambda\_n$ linearly independent over $\mathbb{Q}$ then the element $x\_1 + (-\partial\_1)\big(... | 6 | https://mathoverflow.net/users/2628 | 98292 | 57,296 |
https://mathoverflow.net/questions/97850 | 1 | So let $\Gamma\subseteq SL\_2(\mathbf{Z})$ be a finite index subgroup (not necessarily a congruence subgroup). Recall that we have an action of $SL\_2(\mathbf{R})$ on
$\mathbb{H}=\{z\in\mathbf{C}:\Im(z)>0\}$ by moebius transformations and therefore of $\Gamma$. If $\tau\in\mathbb{H}$ and $[a,b,c,d]=\gamma\in SL\_2(\ma... | https://mathoverflow.net/users/11765 | On the algberaicity of the universal elliptic curve associated to a torsion free subgroup | Here is an alternative take on Donu's argument: Removing the image $\sigma$ of a section of
$E\_\Gamma$ allows one to regard a fiber of $E\_{\Gamma}$ as a once-punctured torus $S$.
(In order to construct a section, use the standard upper half-space model of the Teichmuller space of the tori, so that each marked torus... | 3 | https://mathoverflow.net/users/21684 | 98295 | 57,298 |
https://mathoverflow.net/questions/98288 | 0 | Let $z\_i \in \mathbb{C}\:$ for $i=1,\dots, n\;$ be complex numbers, all with absolute value $|z\_i|\le 1\;$.
Prove (or disprove) that there exists a choice of signs $s\_i \in \{\pm 1\}$ such that
$$\left|\sum\_{i=1}^n s\_i\cdot z\_i\right| \le \sqrt{2}.$$
[My interest in this problem is purely for fun. I couldn't ... | https://mathoverflow.net/users/5542 | Bounding a signed sum of complex numbers | What follows is not an answer, but is too long for a comment.
This problem and its natural higher-dimensional generalization is connected with the recent MO questions [Covering a unit ball with balls half the radius](https://mathoverflow.net/questions/98007/covering-a-unit-ball-with-balls-half-the-radius) and [coveri... | 4 | https://mathoverflow.net/users/21724 | 98297 | 57,299 |
https://mathoverflow.net/questions/98291 | 19 | Forgive the naiveness of this question. Whatever an $n$-vector space exactly is, one expects that the basic example of fully extended $n$-dimensional tqft is a symmetric monoidal functor $Cob\_n\to n$-Vect. Now, whatever an $n$-vector space exactly is, one expects $(n-1)$-Vect to be the based loop space of $n$-Vect. Th... | https://mathoverflow.net/users/8320 | Are fully extended TQFTs generalized cohomology theories? | A fully extended tqft is not quite a generalized homology theory... But almost. You can find a preliminary reference here (notes of a talk by Hiro Tannaka at the mit Talbot workshop):
<http://math.mit.edu/conferences/talbot/2012/notes/14_Tanaka_FactorizationHomology(hiro).pdf>
The precise statements you might be inter... | 14 | https://mathoverflow.net/users/7031 | 98298 | 57,300 |
https://mathoverflow.net/questions/98260 | 5 | Let $A,B\in\mathbb{R}^{n\times n}$. Suppose that $B$ is nonsingular and that there exists $m$ reals pairwise distinct $\lambda\_{1},\cdots,\lambda\_{r},\cdots,\lambda\_{m}$ such that
$$B^{-1}A=\begin{pmatrix}
\lambda\_{1}& 1&&&&&&\cr
&\lambda\_{1}&\ddots&&&&&\cr
&&\ddots&&&\LARGE{0}&&\cr
&&&\lambda\_{r}&1&&&\cr
&&&&\l... | https://mathoverflow.net/users/24060 | A question about matrices with more details | Let
$$A=\pmatrix{1&0\cr 0&0},\quad B=\pmatrix{0&1\cr 0&0}.$$
Then $A^2=A$, $AB=B$, $BA=0$, $B^2=0$. It follows that
$$A\prod (A+t\_iB)B=B,$$
$$B\prod (A+t\_iB)A=0.$$
| 13 | https://mathoverflow.net/users/12120 | 98304 | 57,303 |
https://mathoverflow.net/questions/97758 | 17 | Let $(a\_{ij})$ be a $n\times n$ symmetric matrix such that $a\_{ij}\geq 0$ for all $i,j$ and $a\_{ii}=0$ for all $i$. Under which conditions on the $a\_{ij}$'s can one find $n$ vectors $v\_1,\ldots,v\_n\in{\mathbb R}^n$ such that for all $i,j$ the area of the parallelogram spanned by $v\_i$ and $v\_j$ equals $a\_{ij}$... | https://mathoverflow.net/users/23904 | Prescribing areas of parallelograms (or 2x2 principal minors) | In the case where the matrix $(a^2\_{ij})\\_{i,j=1,\ldots,n}$ is nonsingular, then the problem reduces to the condition that it has a single positive eigenvalue. In fact, we have the following for any $n\times n$ nonzero symmetric matrix $A$ with nonnegative components and zero diagonal (I'll remove the square from $a\... | 7 | https://mathoverflow.net/users/1004 | 98309 | 57,306 |
https://mathoverflow.net/questions/89460 | 7 | Let $K$ be a $p$-adic field, that is a complete discrete valuation ring of characteristic $0$ with a perfect residue field $k$ of characteristic $p > 0$ (to simplify one could also take $K$ to be a finite extension of $\mathbb{Q}\_p$). Let $\mathbb{C}\_K$ be the completion of a (fixed) algebraic closure $\overline{K}$ ... | https://mathoverflow.net/users/5498 | Is there an integral version of Faltings' isomorphism in p-adic Hodge theory between etale and Hodge cohomologies | Dear Kestutis,
your question is *integral* in two ways: first of all, you would like to consider schemes over a whole DVR instead of the generic fiber only. Secondly, you would like to have a comparison theorem between two $\mathcal{O}\_\overline{K}$-modules and not only between $K$-vector spaces. The two are tightly... | 11 | https://mathoverflow.net/users/18238 | 98324 | 57,317 |
https://mathoverflow.net/questions/98284 | 1 | Let $\mathcal{E}$ and $\mathcal{F}$ be two coherent sheaves on a polarized projective variety $(X,\mathcal{O}\_X(1))$.
Denote by $E=\Gamma\_\*(\mathcal{E})=\oplus\_{k\in\mathbb{Z}}\mathcal{E}(k)$,
$F=\Gamma\_\*(\mathcal{F})$, and $R= \Gamma\_\ast(\mathcal{O}\_X)$.
Is it true that
$$\oplus\_{k\in\mathbb{Z}}Ext^i\_... | https://mathoverflow.net/users/2348 | Ext modules of coherent sheaves and associated modules | I assume you mean to ask whether:
$$\oplus\_{k\in\mathbb{Z}}Ext^i\_{\mathcal{O}\_X}(\mathcal{E}, \mathcal{F}(k))=Ext^i\_R(E, F)?$$
For this question, the answer is no, set $\mathcal{E} = \mathcal{O}\_X$. Choose $\mathcal{F}$ such that $H^i(X, \mathcal{F}) \neq 0$ for some $i > 0$. Then obviously
$Ext^i\_R(R, F) = 0$.... | 1 | https://mathoverflow.net/users/3521 | 98325 | 57,318 |
https://mathoverflow.net/questions/96531 | 9 | Does anyone have a copy of Mazur's unpublished notes:
Arithmetic in the geometry of symmetric spaces
that they are willing to post? Thank you!
| https://mathoverflow.net/users/23587 | Request: Mazur' "Arithmetic in the geometry of symmetric spaces" | Dear modular1,
I sent the link to your question to [Barry Mazur](http://www.math.harvard.edu/~mazur/), and he has very kindly sent the paper to me. I've got it scanned and sent it back to him, so if you ask him gently, he'll send the pdf file to you.
I'm happy to have helped you in a small way, because there is a ... | 9 | https://mathoverflow.net/users/2821 | 98335 | 57,320 |
https://mathoverflow.net/questions/98338 | 4 | I am a software developer who self study maths as my hobby, I have had taken a linear algebra course in my undergraduate/graduate study, but you know, math books written in China are such a rubbish that I just throw them away. So I want to buy some renowned textbook on linear algebra for me to study. I skimmed through ... | https://mathoverflow.net/users/24087 | Can anyone recommand a good textbook for self-learning linear algebra? | Strang and Kunze&Hoffman are good choices. You can also take up Halmos's Finite Dimensional Vector Spaces for a more abstract approach.
As for solutions, I would be actually a bit wary of a mathematical textbook that includes them. Most good ones don't (but not all, one notable exception is Spivak's fantastic Calcul... | 2 | https://mathoverflow.net/users/22051 | 98341 | 57,322 |
https://mathoverflow.net/questions/98060 | 8 | (This question is a spin-off from [this other question](https://mathoverflow.net/questions/97624/the-first-odd-degree-2-artin-representation-for-which-the-artin-conjecture-was-pr/97637#97637), and is largely inspired by it.)
Let $f \in S\_1(\Gamma\_1(800),\chi)$ be the weight-one "icosahedral" eigenform constructed b... | https://mathoverflow.net/users/5744 | Constructing an icosahedral weight 2 eigenform? | (This answer is partly a precis of an answer I gave to Barinder in person, which I am posting here in case anyone else is interested.)
The Deligne--Serre lifting lemma is a completely general statement about endomorphisms of free modules over discrete valuation rings; there is no need to assume that the residue field... | 7 | https://mathoverflow.net/users/2481 | 98352 | 57,326 |
https://mathoverflow.net/questions/98343 | 6 | A long time ago I've seen a paper considering, given $\ell$ fixed, estimates for
$$
\sum\_{n \leq x, (n, \ell) = 1} 1
$$
Of course, this is easy to estimate with a trivial error term of $O(\varphi(l))$. However in the paper I am looking for the authors
attempted obtaining better bounds, using some Fourier analysis ... | https://mathoverflow.net/users/23998 | Number of integers coprime to l | It is easy to explain "how the Fourier analysis meshed in". Namely, using the standard notation for the Möbius function, the Euler's totient function, and the integer / fractional part functions, your sum can be written as
$$ \sum\_{n\le x} \sum\_{d\mid(n,l)} \mu(d) = \sum\_{d\mid l} \mu(d) \lfloor x/d \rfloor
= x \s... | 7 | https://mathoverflow.net/users/9924 | 98357 | 57,328 |
https://mathoverflow.net/questions/98334 | 6 | Suppose $X$ and $Y$ are pointed Kan complexes. Is their smash product $X\wedge Y$ also a Kan complex?
I expect the answer is probably no, but it would be nice to see a counterexample.
| https://mathoverflow.net/users/49 | Smash product of Kan complexes | The basic idea is the map $X \times Y \to X \wedge Y$ is a bijection on simplices except over the basepoint. We can construct an extension problem that doesn't lift to an extension problem on the cartesian product.
Let $X$ and $Y$ both be the standard model for $E\mathbb{Z}/2$, which is the nerve of a groupoid with t... | 9 | https://mathoverflow.net/users/360 | 98362 | 57,329 |
https://mathoverflow.net/questions/98367 | 2 | Let $X$ be an $n\times n$ symmetric matrix. Suppose $\lambda\_1(X)\geq \lambda\_2(X) \geq \cdots \geq \lambda\_n(X)$ are eigenvalues of $X$. Let $r$ be any integer with $1\leq r\leq n$. It is well-known that $\sum\_{i=1}^r \lambda\_i(X)$ is convex. Now, my question is: Is the following function convex?
$$\sum\_{i=1}^r ... | https://mathoverflow.net/users/11870 | A sum of eigenvalues | The answer is yes, because your function (let me call it $f$) is the maximum of convex functions. As such, it is convex. The formula :
$$f(X)=\max\left(0,\max\_{1\le r\le n}\sum\_{j=1}^r\lambda\_j(X)\right).$$
| 3 | https://mathoverflow.net/users/8799 | 98371 | 57,333 |
https://mathoverflow.net/questions/98344 | 5 | I have currently started a new research line aiming to prove a mapping between a 2-symbol Turing machine and the one dimensional Ising model. The connection is seen by recognizing that a set of symbols on the tape of the machine is indeed a configuration of a one-dimensional Ising model.
The relevance of this connect... | https://mathoverflow.net/users/19520 | Turing machines and Ising model | Of course there is a huge literature on Turing machines, probabilistic Turing machines and so on, including thousands of research articles and hundreds of books.
So let me consider only the final question, *Are probabilistic Turing machines equally as powerful computationally as deterministic Turing machines?*
The... | 8 | https://mathoverflow.net/users/1946 | 98374 | 57,335 |
https://mathoverflow.net/questions/98359 | 2 | Let $G$ be a connected reductive group (although I don't think this is relevant here), $B$ a Borel subgroup containing a maximal torus $T$ and $U$ the associated unipotent radical ($U^-$ the opposite unipotent subgroup). Finally, let $v \in W$ and $\dot{v}$ a lift of $v$ in $N\_G(T)$.
How do you prove that there is a... | https://mathoverflow.net/users/15404 | Double coset isomorphism | I assume you mean that there is an isomorphism of varieties
$$ (U \dot v \cap \dot v U^-) \times B \to B \dot v B : (x,y) \mapsto xy.$$
Note that it is actually somewhat more natural to write the isomorphism as
$$ (U \cap \dot v U^- \dot v^{-1}) \times B \to B \dot v B : (x,y) \mapsto x \dot v y.$$
The proof of this fa... | 1 | https://mathoverflow.net/users/12858 | 98377 | 57,336 |
https://mathoverflow.net/questions/98365 | 13 | One of the most important constructions in ZF+$\lnot$AC is Hartogs number, defined as:
$$\aleph(X)=\min\lbrace\alpha:|\alpha|\nleq|X|\rbrace$$
We can prove that this ordinal always exists in the following way:
Consider every well-ordered subset of $X$, $\langle W,\prec\rangle$, for every $x\in W$ we can take $W\_... | https://mathoverflow.net/users/7206 | Hartogs number and the three power sets | Hi Asaf,
I thought about this a while ago. Of course, the question had been asked and solved before. Digging through the FOM archives for Spring 2009, I found (April 28, 2009; I fixed a typo in what follows):
>
> In a message dated Jan. 28, I asked whether Sierpinski's ZF result that
> $\aleph(X) < \aleph({\math... | 15 | https://mathoverflow.net/users/6085 | 98381 | 57,339 |
https://mathoverflow.net/questions/98339 | -1 | Dear All,
I’d appreciate very much if you could address the following question:
Given two composable functions [domain (one) = codomain (other)]: the unique function ‘i’ with empty set E as domain and a set X as codomain, and a function ‘f’ with the set X as domain and a set Y as codomain [domain (f) = codomain (i)... | https://mathoverflow.net/users/23869 | Indicating Dots in Graphs | The composition is defined by the fact that it is from 0 to Y; it is unique anyway. All you need is domain and codomain, and you have both.
| -1 | https://mathoverflow.net/users/20031 | 98396 | 57,345 |
https://mathoverflow.net/questions/98395 | 0 | Does someone know whether the order of the automorphism group of a general p-group of order $p^n$ is bounded from above by $(p^n)^2 $? (Every element can possibly be transferred to one of other $p^n$ elements)...
If this fact is incorrect, is it possible to deduce a bound on the order of such an automorphism group i... | https://mathoverflow.net/users/20568 | Automorphism Group of a p-group (finitely generated) | For a specific counterexample to the fact that $p^{2n}$ is not an upper bound, take the elementary abelian group of order $p^3$. It has automorphism group of order $(p^3-1)(p^3-p)(p^3-p^2)$ (pick a basis; the first basis vector can go to any nonzero vector; the second to any vector not in the linear span of the first; ... | 2 | https://mathoverflow.net/users/3959 | 98398 | 57,347 |
https://mathoverflow.net/questions/98366 | 73 | If $X$ is a nonsingular algebraic (or analytic) variety over $\mathbb C$ or $\mathbb R$ then it is certainly $C^\infty$ over the reals.
The converse is false for a silly reason : in the real or complex affine plane with coordinates $x,y$ the variety $x^2=0$ is singular since it is not reduced, but set-theoretically ... | https://mathoverflow.net/users/450 | When is a singular point of a variety ($\mathcal{C}^\infty$-) smooth? | *NB: This answer is directed to the questions about the real case, not the complex case, which was already treated by Francesco. Added 5 July 2021: Because of some questions I have received over the years since this was written, I have realized that there are a few places where the logic and flow are not completely tra... | 53 | https://mathoverflow.net/users/13972 | 98402 | 57,349 |
https://mathoverflow.net/questions/95205 | 5 | Let $K/{\mathbb Q}$ be an extension of degree $d$. Let $S$ be the set of primes $p$ which split completely in $K$. What can one say about the analytic properties of
$$
\zeta\_{K, S}(s) : = \prod\_{p \in S} \frac{1}{1-p^{-s}}.
$$
More generally, one can define a similar partial Euler product for any splitting type, an... | https://mathoverflow.net/users/10458 | A question about partial Euler products | This question turned out to be not too difficult. Please see
<http://www.math.uic.edu/~rtakloo/euler-product.pdf>
for a (casual) writeup of an answer.
Thanks for your comments and hints.
| 6 | https://mathoverflow.net/users/10458 | 98407 | 57,351 |
https://mathoverflow.net/questions/98384 | 2 | Hello to everyone.
What the question means is that different ways of
expressing the same relation between the data and unknown variables produce
really weird fit results:
The problem:
I have the unknown variables $x\_1,x\_2,x\_3,x\_4,x\_5,x\_6$
and the observations $\vec{q}\_i=\Big(q\_i(1),q\_i(2),q\_i(3)\Big), i=... | https://mathoverflow.net/users/24097 | My overdetermined linear system gives both bad and good estimates. Why ? | In using least squares, you normally want the residuals from the various equations to be indpendent of each other. If your $q\_{i}$ vectors are imprecise measurements, then this will introduce correlation between the residuals in the pair of equations invovling $q\_{i}$.
When you say that you get a "good fit" using ... | 1 | https://mathoverflow.net/users/9022 | 98408 | 57,352 |
https://mathoverflow.net/questions/98420 | 0 | Hi,
All iterative solvers I've been able to find are for a system Ax = b where b is a vector. Does anyone know of general iterative solvers for AX = B where X, B are matrices, or more specifically finding the inverse A^-1? (Assuming A is large and sparse.)
Thanks!
| https://mathoverflow.net/users/24111 | Recommendations for sparse iterative solvers to find a matrix inverse | Unless your matrices have a special structure, both $A^{-1}$ and $A^{-1}B$ are dense (full), general matrices without any special properties. Even storing them in memory will be prohibitive.
The short answer is: don't do it. :)
The long answer is: look for structure in your matrices, and exploit it. If there is no ... | 2 | https://mathoverflow.net/users/1898 | 98422 | 57,356 |
https://mathoverflow.net/questions/98421 | 1 | I'm working on an application for which I want to take the set $C$ of all the possible $k$-combinations of elements in $M$ (with $||M|| = m$), and cover $C$ with the sets of $k$-combinations of subsets $N\_i$ of $M$, with $||N\_i|| = n < m$ $\forall$ $N\_i$
So there are $m \choose k$ combinations to cover, and each ... | https://mathoverflow.net/users/24072 | how to cover set of k-combinations of M with subsets of M | The magic words are "design" (if the minimum can be realized) and "covering design" otherwise. There is a vast literature and no algorithm.
| 4 | https://mathoverflow.net/users/1266 | 98423 | 57,357 |
https://mathoverflow.net/questions/98427 | -1 | How does one generally show syntactical statements in PA about primitive recursive functions or relations?
For example something like:
Let $A$ be a prim. rec. relation such that $n\in A$ for every numeral $n$. Show $PA \vdash \forall x A(x)$ (where $A$ denotes the relation as well as the formula representing the r... | https://mathoverflow.net/users/24112 | Syntactical statements in PA about primitive recursive functions | Sometimes you can't prove what you asked for. For example, it is routine to give a primitive recursive definition of the predicate $A(x)$ formalizing in a natural way "$x$ is not the Gödel number of a proof in PA of a contradiction." Then, for each natural number $n$, this predicate holds of $n$ (because PA is consiste... | 2 | https://mathoverflow.net/users/6794 | 98433 | 57,362 |
https://mathoverflow.net/questions/98431 | 4 | Has there been any progress on the [Bouniakowsky conjecture](http://mathworld.wolfram.com/BouniakowskyConjecture.html)? In particular, has anyone been able to prove something for a particular polynomial - or for a class of them?
(I can't seem to find anything, but that could be due to the fact that there seem to be m... | https://mathoverflow.net/users/19313 | Progress on Bouniakowsky's Conjecture | It now goes by the name "Schinzel's Hypothesis H", which has a [Wikipedia entry](http://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H). A quantitative form is known as the "Bateman-Horn Conjecture", which also has a [Wikipedia entry](http://en.wikipedia.org/wiki/Bateman-Horn_conjecture).
Short answer: no progress, ... | 8 | https://mathoverflow.net/users/935 | 98437 | 57,365 |
https://mathoverflow.net/questions/98445 | 0 | Hello,
I've read that if $N\_0$ has a topological generator $\gamma$, then its continuous cohomology on some vector space can be computed by the complex
$$V \overset{\gamma-1}{\longrightarrow} V$$
If someone knows why...?
Thanks !
edit : $N\_0$ is isomorphic to $\mathbb{Z}\_p$ and
by topological generator I mean ... | https://mathoverflow.net/users/24114 | Cohomology of a group with a topological generator | Well, if you consider the definition of cohomology, you see that $H^0(N\_0,V)$ are the fixed point, namely $v\in V$ such that $gv=v$ for all $g\in N\_0$ (which, I suppose, is your group). If the group were cyclic, all $g$ would be of the form $\gamma^k$ for some $k\in\mathbb{Z}$ so $gv=v$ for all $g$ if and only if $\g... | 2 | https://mathoverflow.net/users/18238 | 98453 | 57,372 |
https://mathoverflow.net/questions/98413 | 5 | I know this not, everybody else seems to. This is from page 215 of Cassels, Rational Quadratic Forms, formula 4.1, or SPLAG, page 389 formula (36). quote:
>
> If $f$ and $g$ are forms of
> determinant $d$ in the same genus,
> then they are rationally equivalent by
> some transformation whose denominator
> is pr... | https://mathoverflow.net/users/3324 | Matrix version of number theoretic integral lattice claim | Write $P$ in Smith normal form as $SDT$, where $S$ and $T$ have determinant $\pm 1$ and $D$ is diagonal with diagonal entries $d\_1 | d\_2 | d\_3 | \cdots | d\_n$. We may replace $(F, G, P)$ by $(S^{-T} F S^{-1}, T^{T} G T, D)$ and therefore assume $P$ is diagonal with entries $d\_i$ as stated. Note that this replaceme... | 4 | https://mathoverflow.net/users/297 | 98454 | 57,373 |
https://mathoverflow.net/questions/98356 | 0 | In my hunt for spurious "alternatives" to the $E\_7$ family I always encounter
"fake" solutions. They turn out to be mostly $E\_7$ family solutions disguised
by $q\rightarrow{i\*q}$. The effect is that the dimension (at $q=1$) comes out totally
wrong. Generalizing a bit:
Take a random quantum dimension, say $d(G\_2)... | https://mathoverflow.net/users/11504 | Pseudo-dimensions of quantum Lie groups | 1. One explanation: for small values of n one gets integer quantum dimensions because the fusion category associated with small n is either degenerate or at least has very few simple objects. Getting q-dimension 0 means the weight is on a reflection of the outer wall of the Weyl alcove, so this is pretty common when n ... | 2 | https://mathoverflow.net/users/6355 | 98455 | 57,374 |
https://mathoverflow.net/questions/98452 | 1 | Let $\mathcal{A}$ and $\mathcal{B}$ be two sub-$\sigma$-algebras in a measure space. To each one, there is a conditional expectation associated, respectively $E^\mathcal{A}$ and $E^\mathcal{B}$. Given the two $\sigma$-algebras, we can form a third one, $\sigma(\mathcal{A},\mathcal{B})$, generated by both, and consequen... | https://mathoverflow.net/users/23871 | Conditional expectation and algebraic expressions | Clearly not. Let the measure space be the uniform measure on {$1,2,3,4$}. $A$ allows you to discern whether the number is greater or less than $2.5$ or not. $B$ allows you to discern whether the number is 0 or 1 mod $2$. Let $f$ be $x^2-5x+6$, then $E^Af=1$, $E^B f=1$, $E^{A,B}f=f$. One can't write $f$ as any polynomia... | 2 | https://mathoverflow.net/users/18060 | 98456 | 57,375 |
https://mathoverflow.net/questions/98364 | 2 | Hi,
I need a reference for the following result:
>
> Let $S$ be a scheme and let $X$ be an algebraic torus over $S$. Then the functor $F\_X :S'\mapsto Hom\_{S'}(X\times S',\mathbb{G}\_M\times S')$ is representable by an $S$- group scheme $Y$ which is locally etale isomorphic to $\mathbb{Z}^n$. Furthermore, the si... | https://mathoverflow.net/users/421 | Reference request for Cartier Duality of algebraic tori | I believe you are looking for SGA3 Exp. X, Corollary 5.7.
| 5 | https://mathoverflow.net/users/121 | 98462 | 57,379 |
https://mathoverflow.net/questions/98418 | 9 | I have spent some time being confused by the nature of global methods in number theory. It seems that there are in some sense (for my purposes) three levels at which algebraic number theorists operate: local (at one prime), everywhere local (at all primes simultaneously, including the infinite ones) and global (actuall... | https://mathoverflow.net/users/8080 | What are the truly 'global methods' in number theory? | The following example is perhaps expressed a bit loosely but it is morally correct, and it is fundamental.
Suppose that for each prime $p$ we have chosen a decomposition group $G\_p=\mathrm{Gal}(\bar{\mathbf{Q}}\_p|\mathbf{Q}\_p)\ $ of $G=\mathrm{Gal}(\bar{\mathbf{Q}}|\mathbf{Q})$. If we are given a global character ... | 6 | https://mathoverflow.net/users/2821 | 98468 | 57,383 |
https://mathoverflow.net/questions/98466 | 3 | Given an elliptic curve $E$ defined over $\mathbb{Z}$, and a prime $p$, I know that Hasse's theorem gives, *when $p$ is a good prime*, a relation between the number of solutions over $\mathbb{F}\_{p^n}$ and the number of solutions over
$\mathbb{F}\_p$ (for this, the coefficients of the equation are reduced mod $p$).
... | https://mathoverflow.net/users/24121 | How many points are there on an elliptic curve reduced at a bad prime? | If the reduction is additive, there are $p+1$ points including one singular point. If it is split multiplicative it is $p$ and if non-split multiplicative, then it is $p+2$. See Washington "Elliptic curves, Number Theory and Cryptography ", section 2.10 on page 59.
... and see François comment below for $n>1$.
| 5 | https://mathoverflow.net/users/5015 | 98469 | 57,384 |
https://mathoverflow.net/questions/98448 | 5 | I have read some articles on locally compact quantum groups and the Fourier Transform on them. I wonder why we define the Fourier transform as an operator valued functions from $L^1(\mathbb{G})$ to $L^\infty(\widehat{\mathbb{G}})$.
Why do not we used the intrinsic group $Gr(\mathbb{G})$ which has been defined by Mehr... | https://mathoverflow.net/users/24115 | Fourier transform on locally compact quantum groups | My view is that one should treat LCQGs as a self-dual category; so there is no reason to prejudice, for a classical group $G$, the commutative case (leading to $L^1(\G)$) over the co-commutative (leading to $A(G)$).
The co-commutative is nice from the point of view of intrinsic groups-- this goes back to Takesaki and... | 5 | https://mathoverflow.net/users/406 | 98470 | 57,385 |
https://mathoverflow.net/questions/98403 | 1 | I'm currently facing the problem of computing chern classes for Varieties. More precisely the product of such varieties.
Let $C\_i$ be a variety in $\mathbb{CP}^2$ given by the Weierstraß $\wp$-map.
I want to construct a product of 3 such varieties. Nothing fancy, just $C\_1\times C\_2 \times C\_3$ and calculate it's... | https://mathoverflow.net/users/14517 | Computing chern classes for products of varieties | It appears that you are assuming that your varieties $C\_{i}$ are smooth (you seem to assume that since you are talking about the tangent *bundle*). In this case each $C\_{i}$ is an elliptic curve (I guess, this is what you meant by "toric") and so $C\_{1}\times C\_{2}\times C\_{3}$ is a three dimensional abelian varie... | 4 | https://mathoverflow.net/users/439 | 98481 | 57,393 |
https://mathoverflow.net/questions/98474 | 8 | Let $S=\mathbb{C}[x\_1,x\_2,\dots,x\_n]$ be a polynomial ring. Let $e\_a$ denotes the elementary symmetric polynomials of degree $a$ in $S$.
For $n=2$:
$e\_1=x\_1+x\_2$;
$e\_2=x\_1x\_2$.
For $n=3$:
$e\_1=x\_1+x\_2+x\_3$;
$e\_2=x\_1x\_2+x\_1x\_3+x\_2x\_3$,
$e\_3=x\_1x\_2x\_3.$
In general for any $n$ a... | https://mathoverflow.net/users/23981 | Is an elementary symmetric polynomial an irreducible element in the polynomial ring? | For $\alpha\neq n$, the symmetric polynomial is of the form $f\cdot x\_n + g$ where $f,g$ are non-zero elements of $A={\mathbb C}[x\_1,...,x\_{n-1}]$ with no common factor.
Thus $${\mathbb C}[x\_1,...,x\_n]/(e\_\alpha)=A[x\_n]/(f x\_n+g)=A[g/f]\subset K$$
where $K$ is the quotient field of $A$. It follows that ${\m... | 11 | https://mathoverflow.net/users/10503 | 98486 | 57,397 |
https://mathoverflow.net/questions/98405 | 14 | For $0<\theta<\frac{1}{2}$, denote by $C\_\theta$ the Cantor set with dissection ratio $\theta$, i.e. the Cantor set obtained from dissection parttern $(\theta, 1-2\theta,\theta)$. It is known that $C\_\theta$ carries a uniform measure $\mu\_\theta$ which is usually called Cantor measure. And it is not hard to show tha... | https://mathoverflow.net/users/13776 | Fourier decay rate of Cantor measures | The first part of the question is easy: the power decay is typical. Let's look, say, at $\theta\in[1/3,1/2)$. Take any $\xi>0$ and consider the sequence $\nu\_k=\pi^{-1}\xi\theta^k$ up to the moment it goes below $1$. Let $m$ be the number of terms in this sequence. Let $n\_k$ be the nearest integer to $\nu\_k$. The po... | 9 | https://mathoverflow.net/users/1131 | 98488 | 57,398 |
https://mathoverflow.net/questions/98504 | 0 | In the paper "Fibonacci Series Modulo m" by D.D. Wall (found [here](http://www.jstor.org/discover/10.2307/2309169?uid=3739728&uid=2129&uid=2&uid=70&uid=4&uid=3739256&sid=47699053016367)), there is a table in the Appendix listing values for the function $k(p)$. This function is defined as the period of the Fibonacci num... | https://mathoverflow.net/users/13490 | Fibonacci Numbers Modulo m | Your program, being based on floating-point approximations, may run into problems with roundoff error. It's much easier to calculate Fibonacci numbers mod $p$ directly, using the recurrence $a\_{n+1} = a\_n + a\_{n-1} \mod p$.
The Fibonacci numbers mod $113$ have period $76$ and the Fibonacci numbers mod $181$ have p... | 7 | https://mathoverflow.net/users/13650 | 98505 | 57,403 |
https://mathoverflow.net/questions/98497 | 1 | Is it known whether or not the endomorphism rings (or ring spectra) of the localized sphere spectra in $L\_nSp$ and $L\_{K(n)}Sp$ are Noetherian or not? Are they well understood? Perhaps, in the vein of making it an algebraic stable homotopy category, we should instead look at $[L\_{K(n)}F(n),L\_{K(n)}F(n)]$ for some f... | https://mathoverflow.net/users/11546 | Properties of endmorphism rings of E(n),K(n)-localized spheres | Firstly, using the universal property of localisation we see that $[L\_nS,L\_nS]\_\*=\pi\_\*(L\_nS)$ and similarly for $L\_{K(n)}$, so we can just talk about the homotopy rings.
Next, $\pi\_\*(L\_1S)$ is closely related to the image of $J$ and is described completely in papers by Bousfield and Ravenel. There is a cop... | 7 | https://mathoverflow.net/users/10366 | 98509 | 57,406 |
https://mathoverflow.net/questions/98500 | 5 | Let $k$ be a field. Consider the ring $A=k[x][[t]]$ of formal power series in a variable $t$ over the polynomial ring $k[x]$. This ring contains the ring $B=k[[t]][x]$ of polynomials in the variable §x§ over the power series ring $k[[t]]$.
I want to understand the intersection of $A$ with the fraction field
$F=\mat... | https://mathoverflow.net/users/11599 | power series over a polynomial ring | Equals. $k[[t]]$ is a unique factorization domain, so $k[[t]][x]$ is a unique factorization domain. Thus we can assume that $g$ and $f$ share no common factors. Let $p$ be a prime dividing $g$ that is not a unit mod $t$. Then since its Newton polygon is flat, and the coefficient of some power of $x$ is nonzero mod $t$,... | 5 | https://mathoverflow.net/users/18060 | 98512 | 57,407 |
https://mathoverflow.net/questions/97924 | 2 | For a [Wigner-Ville quasi-probability distribution](http://en.wikipedia.org/wiki/Wigner_quasi-probability_distribution) $W(q,p,t)$ and Hamiltonian $H(q,p)$, we can write a quantum Liouville equation:
$\frac{\partial W(q,p,t)}{\partial t} = -\{\{W(q,p,t) , H(q,p )\}\}$
where $\{\{a, b\}\}$ is the [Moyal bracket](htt... | https://mathoverflow.net/users/23944 | Moyal brackets / quantum Liouville equation for non-Wigner representations? | I got an answer to this offline. The answer is that the form looks the same for all representations with a suitable redefinition of the star product. For example, in the Husimi Q representation, we can define a new kind of star product as follows:
$
f\circledast g= f \exp \left ( {{\frac{\hbar }{2}}(\stackrel{\leftar... | 1 | https://mathoverflow.net/users/23944 | 98518 | 57,409 |
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