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https://mathoverflow.net/questions/34782 | 1 | I feel that the answer to the title quesiton is "yes". However, I tried using different bounds on such least common multiples to prove this with no luck. Any input on this is highly appreciated.
| https://mathoverflow.net/users/8215 | Does $\frac{\mbox{lcm}(1,2,\dots,n+1)}{\mbox{lcm}(1,2,\dots,n)}\to\infty$? | The answer is "no", the limit does not exist, because (now I'm just collecting the comments already made...) consider the series $(2p\_n-1)\_n$, where $p\_1,p\_2,\dots$ denote all primes. We have
$$\operatorname{lcm}(1,\dots,2p\_n) = \operatorname{lcm}(1,\dots,2p\_n-1),$$
since $p\_n$ is smaller than $2p\_n$ (so $p\_n... | 4 | https://mathoverflow.net/users/8153 | 34786 | 22,497 |
https://mathoverflow.net/questions/34788 | 2 | Let $$f(k) := \frac{2k-1}{k}\bigl(1-\sum\limits\_{i\lt k}\frac{i\ f(i)}{k+i-1}\bigr)$$ for $k\in\mathbb{N}^{+}$.
So $f(1) = 1$, $f(2) = 3/4$, $f(3) = 35/72$, etc.
(This function arises when calculating an upper bound for the worst-case behaviour of the first-fit algorithm for dynamic storage allocation.)
Numerica... | https://mathoverflow.net/users/8217 | Approximating a recursively-defined function | Rewriting the recursion yields
$$1 = \sum\limits\_{1\leq i \leq k} \frac{i\ f(i)}{k+i-1}.$$
By assuming $f(k)=\frac{1}{k \ln 2} (1+o(1))$ --- where the little o-Notation $o(1)$ simply means that it is smaller than every constant for large $k$ --- and inserting this, we get
$$\sum\limits\_{1\leq i \leq k} \frac{i\ f(i)}... | 3 | https://mathoverflow.net/users/8153 | 34791 | 22,500 |
https://mathoverflow.net/questions/34799 | 2 | For a smooth function $f:M\to \mathbb{R}$ one usually defines the degeneracy and index of a critical point $p\in M$ in terms of the eigenvalues of the Hessian matrix $(\partial^2 f/\partial x\_i\partial x\_j)$.
On the other hand, if we have a Riemannian metric $g$ we can define the Hessian tensor $H(f,g)=\nabla df$.... | https://mathoverflow.net/users/5323 | Index of a Morse function via the Hessian tensor | The Hessian tensor (defined at a critical point of a function) is a symmetric bilinear form on the tangent space of the manifold at that point. For a symmetric bilinear form on a finite-dimensional vector space (in characteristic different from $2$) there is always a basis such that the corresponding symmetric matrix i... | 11 | https://mathoverflow.net/users/6666 | 34800 | 22,506 |
https://mathoverflow.net/questions/34798 | 6 | The category of topological spaces has a forgetful functor to set which commutes with both small limits and colimits (it has both a left and a right adjoint). Moreover Set is a [Grothendieck topos](http://ncatlab.org/nlab/show/Grothendieck+topos) and in any Grothendieck topos colimits are stable under pull-backs in the... | https://mathoverflow.net/users/184 | Are finite colimits of topological spaces stable under pull-back? | In *compactly generated weak hausdorf spaces* (CGWH), proclusions are compatible with base change. (A *proclusion* $f:X\to Y$ is a map of topological spaces which is homeomorphic to a quotient map.) That is, if $f:X\to Y$ is a proclusion between CGWH-spaces, and $B\to Y$ any map between CGWH-spaces, then $A=X\times\_Y ... | 9 | https://mathoverflow.net/users/437 | 34805 | 22,509 |
https://mathoverflow.net/questions/34795 | 2 | Consider torsion free modules over the germ of a fixed isolated algebraic hypersurface singularity {$f=0$}$\subset\mathbb{C}^n$.
There are natural functors (using categories of finitely generated modules):
modules over $\mathbb{C}[x\_1,..,x\_n]\_{(x\_1,\dots,x\_n)}/(f)$--> modules over $\mathbb{C}${$x\_1,..,x\_n$}$/... | https://mathoverflow.net/users/2900 | Algebraic, analytic, formal modules | The most relevant result I am aware of is the following: suppose $R$ is a local Gorenstein ring, essentially of finite type over a field, with an isolated singularity (so certainly include your case) then the stable categories of maximal Cohen-Macaulay modules over $R$ and the completion $\hat R$ are equivalent up to d... | 4 | https://mathoverflow.net/users/2083 | 34812 | 22,514 |
https://mathoverflow.net/questions/34811 | 11 | This question was inspired by [Math puzzles for dinner](https://mathoverflow.net/questions/29323/math-puzzles-for-dinner/29399#29399).
The arrow compatibility conditions in that problem can be considered an attempt to discretize the notion of a continuous vector field.
The [Hairy-Ball Theorem](http://en.wikipedia.o... | https://mathoverflow.net/users/4345 | The Discrete Hairy-Ball Theorem | By interpolating "legal configuration" if it were to exist, I think you can obtain a nowhere vanishing, continuous vector field on $S^2$. Suppose that for some $n$ there is a legal configuration on the $n\times n \times n$ cube. For each 1x1 face, put a vertex at the center. Then connect vertexes whose 1x1 face's touch... | 3 | https://mathoverflow.net/users/1540 | 34821 | 22,520 |
https://mathoverflow.net/questions/34818 | 4 | In model theory for *standard* first-order logic, one constructs a single model, a reduced product, from a collection of first-order models, together with an index set and a filter on the index set.
In model theory for *modal* first-order logic using Kripke frames, one constructs a single model containing substructur... | https://mathoverflow.net/users/8224 | Modal models as reduced products? | I am not sure whether this is what you are driving at, but every normal modal logic can be described by algebraic models. An algebraic model is a modal algebra (Boolean algebra with additional modal operations) together with an ultrafilter and an assignment of elements of the modal algebra to the propositional variable... | 5 | https://mathoverflow.net/users/7743 | 34822 | 22,521 |
https://mathoverflow.net/questions/34806 | 46 | This question is inspired in part by [this answer](https://mathoverflow.net/questions/26613/papers-that-debunk-common-myths-in-the-history-of-mathematics/30272#30272) of Bill Dubuque, in which he remarks that the fairly common belief that Kummer was motivated by FLT to develop his theory of cyclotomic number fields is ... | https://mathoverflow.net/users/2874 | What was the relative importance of FLT vs. higher reciprocity laws in Kummer's invention of algebraic number theory? | When Kummer started working on research problems,he tried to solve what became known as "Kummer's problem", i.e., the determination cubic Gauss sums (its cube is easy to compute). Kummer asked Dirichlet to find out whether Jacobi or someone else had already been working on this, and to send him everything written by Ja... | 52 | https://mathoverflow.net/users/3503 | 34830 | 22,525 |
https://mathoverflow.net/questions/34279 | 3 | Let $f(x)=\Re(\sum\_{k=1}^n a\_k e^{i\lambda\_k x})$ for $0 < \lambda\_1 < \lambda\_2 < \dots < \lambda\_n$ and some complex $a\_1$, $a\_2$, $\dots$, $a\_n$. What is the best (in some sense) estimate for $\inf\_{[-M,M]} f(x)$ for large $M$ (in particular, for $M=+\infty$). For example, is it true that $\inf f(x)\leq -C... | https://mathoverflow.net/users/4312 | one-side estimates for quasi-trigonometric polynomial | Let's do the case of full line, which is nice and clean. Without loss of generality, $\lambda\_1=1$. Then you can ignore all non-integer $\lambda$s because if $f=g+h$ where $g$ includes all integer frequences and $h$ all non-integer ones, then $\inf \Re g$ is the same as the infimum of $\lim\_{T\to+\infty}\Re \frac 1{2... | 4 | https://mathoverflow.net/users/1131 | 34837 | 22,530 |
https://mathoverflow.net/questions/34827 | 0 | Earlier I saw a transcription request which was accepted on MO: [Transcription of an interview of Kazuya Kato](https://mathoverflow.net/questions/27621/transcription-of-an-interview-of-kazuya-kato)
But the request I have here is on the discussion about Grisha Perelman on Russian television “Moscow time” on 30.04.2010... | https://mathoverflow.net/users/5627 | Transcription of a panel discussion about G.Perelman | Dear Unknown,
I don't think anything useful could be gained from this "panel discussion". It's from a talk show "Let them speak", one of the yellowest shows ever on the Russian TV. The panel, whose learned opinion you wish to know, included:
An actor
A mathematician (Polishchuk, Rostislav F.,
two papers are pres... | 21 | https://mathoverflow.net/users/7983 | 34841 | 22,534 |
https://mathoverflow.net/questions/34832 | 14 | [I'm not familiar with the terminology, so when I write P (resp. NP) set, I mean a subset of the integers whose membership function is a decision problem in P (resp. NP).]
Is it correct to say that a set is NP if and only if it is the image of a P set under a polynomial time function?
This seems fairly clear to me:... | https://mathoverflow.net/users/3676 | Correct to characterise NP set as P-time image of P set? | No. In particular, the images of sets in $P$ under polynomial time reductions contain many more sets than $NP$. Below I show that the halting problem is among them.
Let {$M\_i$} be a list of all Turing machines.
Let $f$ be a mapping from {$M\_i$} $\times$ {0,1}\* $\times$ {0,1}\* to {$M\_i$} $\times$ {0,1}\* , such ... | 17 | https://mathoverflow.net/users/2618 | 34844 | 22,536 |
https://mathoverflow.net/questions/34848 | 24 | There exist topological manifolds which don't admit a smooth structure in dimensions > 3, but I haven't seen much discussion on homotopy type. It seems much more reasonable that we can find a smooth manifold (of the same dimension) homotopy equivalent to a given topological manifold. Is this true, or is there a counter... | https://mathoverflow.net/users/8188 | Are topological manifolds homotopy equivalent to smooth manifolds? | It is false for compact manifolds in 4 dimensions. Freedman showed that there is a compact simply connected topological 4-manifold with intersection form E8, but Donaldson showed that there is no such smooth manifold.
| 28 | https://mathoverflow.net/users/51 | 34854 | 22,542 |
https://mathoverflow.net/questions/34853 | 5 | Let $X$ be a Hausdorff, locally compact, and totally disconnected topological space, which I call an $\ell$-space, and write $A = C^{\infty}\_C(X)$ for the algebra of locally constant complex-valued functions on $X$ with compact support (under pointwise multiplication). The algebra $A$ is nonunital when $X$ is noncompa... | https://mathoverflow.net/users/3544 | sheaves of modules on an $\ell$-space | You probably know about this already, but this is discussed in the great paper of Bernstein and Zelevinsky, "Representations of the group GL(n, F), where F is a non-Archimedean local field" to some extent. They define an "l-sheaf", which is a sort of intermediate concept between sheaves (which make sense on any space) ... | 2 | https://mathoverflow.net/users/6545 | 34855 | 22,543 |
https://mathoverflow.net/questions/34861 | 54 | An [answer](https://mathoverflow.net/questions/34843/what-is-realistic-mathematics/34851#34851) to a recent question motivated the following question:
>
> (how) is category theory actually
> useful in actual physics?
>
>
>
By "actual physics" I mean to refer to areas where the underlying theoretical principle... | https://mathoverflow.net/users/1847 | (How) is category theory actually useful in actual physics? | Fusion categories and module categories come up in topological states of matter in solid state physics. See the research, publications, and talks at [Microsoft's Station Q](http://stationq.ucsb.edu/).
| 32 | https://mathoverflow.net/users/22 | 34862 | 22,546 |
https://mathoverflow.net/questions/34843 | 45 | This post is partially about opinions and partially about more precise mathematical questions. Most of this post is not as formal as a precise mathematical question. However, I hope that most readers will understand this post and the nature of the question.
I will first try to explain what I would call *Realistic Mat... | https://mathoverflow.net/users/8176 | What is Realistic Mathematics? | When Solovay showed that ZF + DC + "all sets of reals are Lebesgue measurable" is consistent (assuming ZFC + "there is an inaccessible cardinal" is consistent), there was an expectation among set-theorists that analysts (and others doing what you call realistic mathematics) would adopt ZF + DC + "all sets of reals are ... | 61 | https://mathoverflow.net/users/6794 | 34863 | 22,547 |
https://mathoverflow.net/questions/34860 | 3 | I tried writing the full motivation for this question, but it turned out to be too long a detour for a question that is really quite specific. So I will only sketch the motivation this time.
### Sketch of motivation
I'm taking a $G$-Galois cover of $\mathbb{P}^1\_{\overline{\mathbb{C}(t)}}$ which is defined (not in... | https://mathoverflow.net/users/2665 | A computation of ramification | I think there is no vertical ramification. Denote by $D={\mathbb C}[[t]]$ and $K\_0$ its fraction field. Your curve is defined over $K\_0$ by an affine equation $z^n+f(y)=0$ with $f(Y)\in D[Y]$ monic. Let $U$ be the affine scheme associated to $D[y,z, {{1}\over{f(y)}}]/(z^n-f(y))$. It is clearly finite and étale over t... | 3 | https://mathoverflow.net/users/3485 | 34870 | 22,551 |
https://mathoverflow.net/questions/34865 | 6 | Let P be a convex polygon with area A(P), and to each side of P, attach the largest area triangle possible that lies entirely within P. Must the sum S(P) of the areas of these triangles always satisfy $S(P) \geq 2A(P)$? If P is a rectangle, then $S(P) = 2A(P)$.
Supposing this is true, is there a lower bound with some... | https://mathoverflow.net/users/35336 | Elementary problem about triangles inside a convex polygon | (This should be a comment, but I can't comment yet)
Indeed, the problem is from an IMO. You can look at the solutions posted [here](http://www.artofproblemsolving.com/Forum/viewtopic.php?p=572824&sid=16048d121b1aca0c9211681c2b9bbfae#p572824)
| 5 | https://mathoverflow.net/users/1105 | 34871 | 22,552 |
https://mathoverflow.net/questions/34874 | 2 | If you visit this [link](http://www.springerlink.com/content/ug8h1563j3484211/), you'll see at the top of the PDF view. Basic properties of finite abelian groups:
Every quotient group of a finite abelian group is isomorphic to a subgroup.
If the above statement true, it would make some proofs in Serge Lang's Algebr... | https://mathoverflow.net/users/8237 | Any factor group of a finite abelian group is isomorphic to some subgroup | The result you are interested in is Theorem 19 on page 8 of
[http://alpha.math.uga.edu/~pete/4400algebra2point5.pdf](http://alpha.math.uga.edu/%7Epete/4400algebra2point5.pdf)
As I explain there, this fact is a kind of duality statement, but it lies deeper than the fact that passage to the dual group takes injection... | 6 | https://mathoverflow.net/users/1149 | 34877 | 22,556 |
https://mathoverflow.net/questions/34878 | 3 | Hello,
I was wondering if there is a nice counterexample to the following question.
Suppose $X$ is a CW-complex which is not simply connected and there is a point $x\in X$ such that $X-x$ is contractible. Is $X$ homotopy equivalent to a wedge of circles? Maybe we do not even need the CW-complex condition.
| https://mathoverflow.net/users/5450 | X not simply connected and X-x contractible | Take a disconnected space $Y$ that isn't homotopically trivial, for example the disjoint union of two circles, and let $X$ be its suspension. Let $x$ be one of the two "vertices" of the suspension. $X$ isn't simply connected because there's a loop that starts at $x$, goes through one component of $Y$ to get to the othe... | 12 | https://mathoverflow.net/users/6794 | 34880 | 22,557 |
https://mathoverflow.net/questions/34869 | 3 | A problem involving a bit of group theory to solve. Actually it may well be thoroughly solved already, but I'm not sure how to go about finding the solution. I call it the Scale Problem, and it is essentially about determining which rational numbers give rise to feasible scales for composing music, particularly music u... | https://mathoverflow.net/users/8235 | Looking for a musical group | Darij Grinberg answered the question in a comment, but I thought a more geometric interpretation might be helpful.
The group of positive rational numbers under multiplication is isomorphic to an infinite sum of copies of $\mathbb{Z}$, where the set of primes forms a natural basis. If you choose a positive integer $N$... | 3 | https://mathoverflow.net/users/121 | 34883 | 22,558 |
https://mathoverflow.net/questions/34884 | 2 | If $M\_{g}$ is the moduli space of Riemann surface of genus $g$, $M^1\_{g}$ is the moduli space of Riemann surface of genus $g$ with one boundary, how can we show that the natural map:
$M^1\_{g} \rightarrow M\_{g}$
the fiber over a point $S \in M\_{g}$ is the sphere bundle associated to the tangent bundle of Rieman... | https://mathoverflow.net/users/2391 | Question related to the moduli space of Riemann surfaces and a fibration | I'll give you references for the appropriate fact about the mapping class group. Let $Mod\_{g,b}^p$ be the mapping class group of a genus $g$ surface with $b$ boundary components and $p$ punctures $\Sigma\_{g,b}^p$. There are really two fact. The first is the Birman exact sequence (by the way, I believe this was Joan B... | 4 | https://mathoverflow.net/users/317 | 34886 | 22,560 |
https://mathoverflow.net/questions/34879 | 2 | $A$ a Gorenstein ring, $M\neq 0$ a finite $A$-module with finite injective dimension. According to Bruns, this implies that $M$ has finite projective dimension. How do I see that?
| https://mathoverflow.net/users/5292 | Modules over a Gorenstein ring | Here is a proof which may not be the best but demonstrates some standard techniques:
Since $R$ has finite inj. dim. one can replace $M$ by a high syzygy, so one can assume $M$ has full depth. Thus one can kill a full regular sequence for both $M$ and $R$ and (as finiteness of proj. or inj. dim are not affected) assum... | 5 | https://mathoverflow.net/users/2083 | 34888 | 22,562 |
https://mathoverflow.net/questions/34889 | 13 | Hi,
Is there any language $L$ know to be complete for $NP \cap co-NP$, i.e. any language $L^{\prime} \in NP\cap co-NP$ reduces in polynomial-time to $L$ and it is known that $L\in NP\cap co-NP$?
Thanks
| https://mathoverflow.net/users/1612 | A language complete for NP intersection co-NP | $NP \cap coNP$ is not known to have complete languages, but I don't know of any consequences as strong as what Marcos claims. Juris Hartmanis and his students worked extensively on this problem in the early 80's. Two references I know are:
>
> Michael Sipser: On Relativization and the Existence of Complete Sets. IC... | 21 | https://mathoverflow.net/users/2618 | 34895 | 22,565 |
https://mathoverflow.net/questions/34891 | 16 | If $X$ is a complex Abelian variety of dimension $g$, then
* The canonical sheaf is trivial
* $\dim {\rm H}^i(X; \mathcal{O}\_X) = \binom{g}{i}$.
When $g =1,2$, then any connected, projective nonsingular $X$ satisfying the above two must be an Abelian variety. Is this true for higher $g$? If not, what other condit... | https://mathoverflow.net/users/321 | Characterizations of complex Abelian varieties (especially 3-folds) among projective nonsingular varieties? | A result of Kawamata (Kawamata, Yujiro, Characterization of abelian varieties. Compositio Mathematica, 43 no. 2 (1981), p. 253-276) implies that, under your assumptions, $X$ is birational to an abelian variety (in fact you just need the Kodaira dimension of $X$ to be zero and the irregularity to be equal to the dimensi... | 22 | https://mathoverflow.net/users/4344 | 34897 | 22,566 |
https://mathoverflow.net/questions/34904 | 3 | X and Y are irreducible curves, and f:X-->Y a morphism. Let E be a vector bundle over X.
When does there exist a vector bundle F over Y such that f\*F=E and when will it be unique?
| https://mathoverflow.net/users/8234 | When does a vector bundle descend? | If $f$ does not factor through a point then $f$ is flat. So, you can use the usual descent condition --- let $p\_1,p\_2:X\times\_Y X \to X$ be the projections. Then $E$ descends if there is an isomorphism $p\_1^\*E \cong p\_2^\*E$ satisfying the cocycle condition on the triple fiber product. Each such isomorphism gives... | 3 | https://mathoverflow.net/users/4428 | 34910 | 22,571 |
https://mathoverflow.net/questions/34701 | 1 | Given a constant $k \in \mathbb N$, and a set of $p$ multi-variate polynomials {$P\_j:\mathbb N^n\to \mathbb Z$}$\\_{j=1...p}$, with $P\_j \not\equiv 0$.
Is the following true:
There exists $n$ sets $S\_1,\dots,S\_n$, and $p$ constants $b\_1,\dots,b\_p$
with $S\_i \subset \mathbb N$, $|S\_i| = k$, $b\_j \in${-1,1... | https://mathoverflow.net/users/5650 | Finding regions where multi-variate polynomials are positive | The claim is true. A proof idea is as follows: we first pick sufficiently large numbers for S1, then we pick sufficiently large numbers for S2 given S1, and continue this until we decide all of S1, …, Sn.
Filling in a little more detail, the proof will be as follows. We prove the claim by mathematical induction on n.... | 1 | https://mathoverflow.net/users/7982 | 34918 | 22,577 |
https://mathoverflow.net/questions/34923 | -3 | I've read that the set of real numbers *R* is uncountable. It was proved by contradiction. A number *x* that is not in the *f(n)* side was constructed.
Ultimately it was said that "*x* is not *f(n)* for any *n* because it differs from *f(n)* in the *n*th fractional digit.
I can't understand the last part. I understand... | https://mathoverflow.net/users/8074 | Uncountable Sets | Assume the interval $[0,1)$ is countable. Then we can write down all numbers like that:
* $a\_0 = 0.a\_{0,0}a\_{0,1}a\_{0,2}\dots$
* $a\_1 = 0.a\_{1,0}a\_{1,1}a\_{1,2}\dots$
* $a\_2 = 0.a\_{2,0}a\_{2,1}a\_{2,2}\dots$
* $\dots$
Now we construct a new number $b=0.b\_0b\_1b\_2\dots$ in the following way. We look at $a... | 2 | https://mathoverflow.net/users/8153 | 34926 | 22,582 |
https://mathoverflow.net/questions/34927 | 14 | If $X$ is a based space, then we have $\pi\_1(X) \cong \pi\_0(\Omega X)$. This is to say we can identify elements in the fundamental group of $X$ with path components of the first loop space of $X$. My question is this: do all the path components of $\Omega X$ necessarily have the same homotopy type?
I ask because wh... | https://mathoverflow.net/users/4466 | Are the path components of a loop space homotopy equivalent? | Does the following work?
Let $A$ and $B$ be components of $\Omega X$ and assume that $A$ is
the component containing the trivial path based at $x\_0$. Let $f$
be any element of $B$. Then $f$ is a path from $x\_0$ to $x\_0$.
There's a map $\phi:A\to B$ obtained by following a path in
$A$ by $f$.
Similarly there's $\ps... | 15 | https://mathoverflow.net/users/4213 | 34928 | 22,583 |
https://mathoverflow.net/questions/34921 | -1 | (I started working on this problem after trying to get any "interesting" pattern out of the number that Gowers randomly wrote while answering:[What is realistic mathematics?](https://mathoverflow.net/questions/34843/what-is-realistic-mathematics).)
The number was 123871205412470874297947938271423698765734564756028492... | https://mathoverflow.net/users/5627 | Taming this Conway-type sequence | You have to be a bit more precise - for instance what happens if there are more than nine of a particular digit ? Regardless, a cursory literature search comes up with
[an article](http://www.jstor.org/pss/2974579) by Sauerberg and Shu which studies the Conway sequence as well as ones similar to yours, which are called... | 3 | https://mathoverflow.net/users/3143 | 34929 | 22,584 |
https://mathoverflow.net/questions/34931 | 4 | Hi,
I recently came across the famous Buffon's needle problem (<http://en.wikipedia.org/wiki/Buffon%27s_needle>), and there is no doubt that the problem as well as its answer are elegant.
However, the problem I have in mind, is slightly modified. The original problem deals with parallel lines separated by fixed distanc... | https://mathoverflow.net/users/8245 | Buffon's needles revisited | This is solved in *Fifty Challenging Problems in Probability with Solutions*, the 1965 book by Frederick Mosteller (Problem #54). A very good book to have around in any case.
| 3 | https://mathoverflow.net/users/8212 | 34933 | 22,586 |
https://mathoverflow.net/questions/34938 | 44 | If one iterates the map $z \mapsto z^2 + c$ there is obviously a simple formula for the sequence one gets if $c=0$. Less obviously, there is also a simple formula when $c = -2$ (use the identity $2 \cos(2x) = (2\cos(x))^2 - 2)$. Are there any other values of $c$ for which one can solve this recurrence explicitly? (For ... | https://mathoverflow.net/users/51 | When does iterating $z \mapsto z^2 + c$ have an exact solution? | No, there are no others.
Analytically, one can show that if a Julia set contains an analytic arc, it is in fact a straight line or a circle (up to conjugation). For the class $z^2+c$, $0$ and $-2$ are the only values where this occurs.
This does not quite imply that there are no closed-form solutions of the recurr... | 38 | https://mathoverflow.net/users/3993 | 34939 | 22,588 |
https://mathoverflow.net/questions/34934 | 20 | My question is of a logical nature and concerns what I perceive to be two different types of mathematical independence.
Suppose we have a (sufficiently strong) axiomatic theory $T$. Gödel's Incompleteness Theorems state that:
1. $T$ is not a complete theory. That is, there is a sentence (expressible in the language... | https://mathoverflow.net/users/7154 | Logically independent but true sentences | The division you see has to do with the level of conservativity of the theories in question. On the one hand, the theory ZFC + Con(ZFC) is not $\Pi^0\_1$-conservative over ZFC since Con(ZFC) is a $\Pi^0\_1$ sentence which is not provable from ZFC. On the other hand, CH is $\Pi^0\_1$-conservative over ZFC since ZFC + CH... | 20 | https://mathoverflow.net/users/2000 | 34941 | 22,590 |
https://mathoverflow.net/questions/34935 | -1 | So it occurred to me recently that 21 kinda stands out among among its nearby neighbors in that there are several sets of consecutive positive integers that can be summed to equal 21 -- namely 1+2+3+4+5+6, 6+7+8, and 10+11. So I started thinking - "I wonder if all numbers can be written as a summation of two or more co... | https://mathoverflow.net/users/6898 | Summation of consecutive positive integers | The number of ways of writing *n* as a sum of [more than one] consecutive positive integers
is equal to the number of odd divisors of *n* [that are greater than one].
| 8 | https://mathoverflow.net/users/5690 | 34942 | 22,591 |
https://mathoverflow.net/questions/34763 | 27 | Let D ⊂ ℂ be the closed unit disc in the complex plane, and let C be a continuously embedded path in D between the points -1 and 1. The curve C splits D into two halfs $D\_1$ and $D\_2$.
Let *f* : D→ℂ be a continuous function that is holomorphic on the interiors of $D\_1$ and $D\_2$.
Is *f* then necessarily holomo... | https://mathoverflow.net/users/5690 | Continuous + holomorphic on a dense open => holomorphic? | [Denjoy](http://www.numdam.org/item?id=BSMF_1932__60__27_0) makes a detailed study of this question, and in particular constructs counterexamples where the curve C is the graph of a continuous function. Apparently, the construction works for curves which are 'very' non rectifiable, i.e., the local variation is infinite... | 18 | https://mathoverflow.net/users/7294 | 34955 | 22,598 |
https://mathoverflow.net/questions/34971 | 4 | As observed on [Mathworld](http://mathworld.wolfram.com/RelativelyPrime.html), "Amazingly, the probabilities for random pairs of integers and Gaussian integers being relatively prime are the same as the asymptotic densities of squarefree integers of these types" (respectively, 6/π2 and 6/π2K where K is Catalan's consta... | https://mathoverflow.net/users/4336 | Coprimality and squarefree numbers | To spell out what Robin said at a more basic level (blame me if this is inapt): Consider a given prime p. The chance that a "random integer" does **not** divide by $p^2$ is $1-1/p^2$. So heuristically the probability that the random integer is square free is the product of $1-1/p^2$ over all primes. Making that precise... | 7 | https://mathoverflow.net/users/8008 | 34975 | 22,607 |
https://mathoverflow.net/questions/34982 | 18 | There is a surjective continuous map $[0;1]\rightarrow [0;1]^2$ ("space filling curve"). Using such a map one can easily get space filling curves for all finite dimensional cubes.
So my question is: Is there a space filling curve of the Hilbert cube
$[0;1]\rightarrow [0;1]^\mathbb{N}$ ?
| https://mathoverflow.net/users/3969 | Are there space filling curves for the Hilbert cube? | Ah, it's called the "Hahn–Mazurkiewicz theorem".
<https://en.wikipedia.org/wiki/Peano_space>
And it apparently appears in the Willard text.
| 15 | https://mathoverflow.net/users/1465 | 34983 | 22,611 |
https://mathoverflow.net/questions/34989 | 5 | It is well known that satisfiability of Horn formulae can be checked in polynomial time using unit propagation.
But suppose we relax the condition for horn clauses from at most one un-negated literals to two un-negated literals. Then is it possible to prove that satisfiability of such a formula can be checked in time... | https://mathoverflow.net/users/8246 | Horn clauses and satisfiability | I think 3SAT can be reduced to your problem, since
($a\_1$ OR $a\_2$ OR $a\_3$) AND ($b\_1$ OR $b\_2$ OR $b\_3$) AND ($c\_1$ OR $c\_2$ OR $c\_3$) AND ...
is satisfiable iff
(NOT $A\_1$ OR $a\_2$ OR $a\_3$) AND ($A\_1$ OR $a\_1$) AND (NOT $B\_1$ OR $b\_2$ OR $b\_3$) AND ($B\_1$ OR $b\_1$) AND (NOT $C\_1$ OR $c\_2$... | 6 | https://mathoverflow.net/users/2530 | 34991 | 22,613 |
https://mathoverflow.net/questions/34651 | 11 | Wada Lakes are three disjoint open subsets of $\mathbb R^2$ with common boundary. Originally they were constructed by hand, but they also arise naturally in the real life, that is, theory of dynamical systems. (See, for example, this <http://www.math.cornell.edu/~hubbard/pendulum.pdf> paper of John Hubbard.)
From dyn... | https://mathoverflow.net/users/2029 | Permute Wada Lakes keeping the coastline intact? (still open in dim >2) | In dimension 2, the answer is also "no".
Recall the classical construction of the [Wada lakes](http://en.wikipedia.org/wiki/File:Lakes_of_Wada.jpg).
In the linked picture, one sees little "straits" connecting the red/blue/green regions at stage $n$ with the extra windy strip that is added at stage $n+3$. For example,... | 5 | https://mathoverflow.net/users/5690 | 34993 | 22,614 |
https://mathoverflow.net/questions/34944 | 12 | Coming up with examples of $D\_8$-covers of $\mathbb{C}(x)$ is easy. For example:
$Quot(\mathbb{C}(x)[y,z]/(y^2=x(x-7), z^4=(y+\sqrt{-6})^2(y-\sqrt{-6})^2(y+\sqrt{-10})(-y+\sqrt{-10})^3))$
defines a $D\_8$ extension of $\mathbb{C}(x)$. If you view this field extension as an extension of the corresponding smooth pro... | https://mathoverflow.net/users/2665 | Realizing D_8 as a Galois group over C(x) with prescribed decomposition groups | As you say, the fixed field by $\langle a\rangle$ is $\mathbb{C}(u)$ with $u^2=-5x/2(x-7)$, and the fixed field by $\langle a^2,b \rangle$ is similarly $\mathbb{C}(v)$ with $v^2=7(x-2)/2(x-7)$ (constants chosen so that $u^2+v^2=1$). So their compositum, the fixed field by $\langle a^2\rangle$, has genus zero and is $\m... | 5 | https://mathoverflow.net/users/5480 | 34998 | 22,616 |
https://mathoverflow.net/questions/34631 | 31 | Let $k$ be a field complete with respect to a non-archimedean absolute value, and $X$ a separated algebraic space of finite type over $k$.
If $X$ is a scheme then $X(k)$ inherits a natural (Hausdorff) topology from $k$ (uniquely determined by functoriality, compatibility with open immersion, closed immersions, fiber... | https://mathoverflow.net/users/3927 | Structure on $X(k)$ for separated finite type alg. space $X$, for complete valued $k$. | Assume $k$ is a field and $F$ is a contravariant functor from $k$-schemes of finite type to sets. For each $X/k$ and $x\in F(X)$ we get a "characteristic map" $X(k)\to F(k)$ by pulling back $x$. Now if $k$ is a topological field, we can define a topology on $F(k)$, which is the finest making all these maps (for all pai... | 22 | https://mathoverflow.net/users/7666 | 35000 | 22,618 |
https://mathoverflow.net/questions/34967 | 4 | Let $V$ be the Banach space of bounded sequences of reals with the sup norm. Does there exists a subset $B$ of $V$ such that
* Linear Independence: For all functions $c$ in $\mathbb{R}^B$, if $\sum\_{b \in B} c(b) \cdot b = 0$, then $c$ is identically zero.
* Spanning Set: For all vectors $v$ in $V$, there exists a f... | https://mathoverflow.net/users/nan | Basis for L_infty(R) | The space $\ell^\infty\_R$ does not have even an M-basis; i.e., a biorthogonal set $(x\_t,x\_t^\*)$ such that the span of the $x\_t$ is dense and the $x\_t^\*$ are total (Lindenstrauss, late 1960s IIRC), so it has nothing like a Schauder basis. Later I proved [PAMS 26. no. 3 467-468 (1970)] that $\ell^\infty$ also does... | 10 | https://mathoverflow.net/users/2554 | 35004 | 22,621 |
https://mathoverflow.net/questions/34873 | 4 | Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra (or the corresponding Lie group). For definiteness, I'll take $\mathfrak{g}$ to be of type $A\_n$, that is, $\mathfrak{g} = \mathfrak{sl}\_{n+1}(\mathbb{C})$, but my question applies to semisimple Lie algebras of arbitrary Lie type. Consider the ... | https://mathoverflow.net/users/7932 | Restriction map for Lie algebra/Lie group cohomology associated to a complex semisimple Lie algebra and a semisimple Lie-subalgebra | The cohomology rings of Lie groups are not just rings but also Hopf algbras with a coproduct Δ coming from the group structure. Picking primitive (Δ(x) = x⊗1 + 1⊗x) homogeneous generators gets close to giving a canonical set of generators up to scalars (though there is still some ambiguity for Dn which can have 2 gener... | 4 | https://mathoverflow.net/users/51 | 35007 | 22,623 |
https://mathoverflow.net/questions/35003 | 2 | A state lottery draws *p* numbers out of a grid of *n* numbers. Players participate by filling in *p* numbers into a grid, at unit cost. They can sumbit as many grids as they like.
The lottery pays out when a player's grid matches at least *q* numbers with the outcome of the drawing, the "winning combination" *pT*. H... | https://mathoverflow.net/users/8275 | What is the minimum set of combinations C(p,n) required to guarantee q<p matches with a target combination (pT,n) | Hmmm... I recently saw a paper dealing exactly with this. You may like to learn that this problem is known in the litterature as the "lottery number".
I was not able to find this paper back, though I found one from 2008 whose introduction contains several interesting references :
[A note on a symmetrical set coveri... | 1 | https://mathoverflow.net/users/1715 | 35018 | 22,630 |
https://mathoverflow.net/questions/34996 | 11 | I guess the answer is that this unknown.
Maybe this implies some "lowness" result on NP relative to BPP?
| https://mathoverflow.net/users/8268 | Does EXP $\in$ P/poly imply NP=RP? | Note that it would suffice to prove $EXP \subseteq P/poly$ implies $NP = BPP$ (since the latter implies $NP = RP$).
I am pretty sure this is not known. $NP \neq RP$ does not seem to imply any circuit lower bounds for $EXP$.
Note that $EXP \subseteq P/poly$ does imply $P \neq NP$. If $EXP \subseteq P/poly$ then ther... | 12 | https://mathoverflow.net/users/2618 | 35020 | 22,632 |
https://mathoverflow.net/questions/34314 | 13 | Let C be an nxn matrix, then the polynomials in C (with appropriate coefficients) form an algebra of commuting matrices. I feel that I should know if the converse is true but I do not. So my first question (adding several conditions) is: Suppose A and B are symmetric integer matrices with AB=BA, must there be a matrix ... | https://mathoverflow.net/users/8008 | When is an algebra of commuting matrices (contained in one) generated by a single matrix? | Thanks for the answers. Just to wrap up a bit, here are a few examples.
1. Sometimes an ACM (algebra of commuting matrices) is sure to be generated by one of its members
2. Other times it has dimension too large to possibly be (embedded in) an ACM with a single generator.
3. An ACM might be generated by $2$ matrices,... | 5 | https://mathoverflow.net/users/8008 | 35024 | 22,634 |
https://mathoverflow.net/questions/35021 | 13 | Call **a computable function** a total function $\mathbb{R} \to \mathbb{R}$, for which there exists a Turing machine outputting arbitrary close approximation to $f(x)$ given arbitrary close approximation to $x$.
1. Obviously not every computable function is differentiable (for example, absolute value).
For arbitrary ... | https://mathoverflow.net/users/158 | Differentiability of computable functions | John Myhill gave an example of *a recursive function defined on a compact interval and having a continuous derivative that is not recursive* [Michigan Math. J. 18 (1971), 97-98, [MR0280373](http://www.ams.org/mathscinet-getitem?mr=280373)]. However, Pour-El and Richards have shown that if a recursive function defined o... | 22 | https://mathoverflow.net/users/2000 | 35025 | 22,635 |
https://mathoverflow.net/questions/35032 | 11 | Prompted by Vinay Deolalikar's purported proof of P != NP, I've been reading up on Descriptive Complexity for some background material.
The major successes of Descriptive Complexity include Fagin's result that $NP=SO\exists$ (that is, the class NP is equal to the class of models of a second-order existential query o... | https://mathoverflow.net/users/5534 | Descriptive complexity theoretic-characterizations of P and NP | Remember that this is finite model theory and it is quite different from logic on infinite structures, e.g. satisfiability of first-order formulas on finite structures is $\Sigma\_1$, whereas the same question is $\Pi\_1$ for general structures (where the formula is satisfiable iff it does not lead to (i.e. prove) cont... | 11 | https://mathoverflow.net/users/7507 | 35035 | 22,640 |
https://mathoverflow.net/questions/35039 | 2 | Let $n$ be a positive integer.
>
> Let $S \subseteq \mathbb{R}^n$. Is the Hausdorff dimension of the boundary of $S$ always smaller than the Hausdorff dimension of $S$?
>
>
>
I have not found anything concerning those questions in some looked up books, I was not able to prove one of the statements, and I faile... | https://mathoverflow.net/users/8153 | Hausdorff dimension: subset of $\mathbb{R}^n$ vs. boundary of this subset | "Smaller" in the sense of $\le$ ... If $S$ is closed and has Hausdorff dimension $< n$, then $S$ has empty interior, so (as noted by Joel) $S$ is its own boundary, and thus we have equality for the two dimensions. And of course if (perhaps not closed) set $S$ has dimension $n$, then the boundary could have any dimensio... | 6 | https://mathoverflow.net/users/454 | 35046 | 22,647 |
https://mathoverflow.net/questions/35044 | 0 | Given a matrix $A$, each element $A\_{i,j} \geq 0$, find the vector $\vec x$ that maximizes the minimum element in $\vec b$ ($\vec b = A \vec x$). Note that this is not a linear equation system as I don't know $\vec b$.
Extra contraints on the solution are $x\_i \geq 0$, and $\sum x\_i = 1$.
Is this possible to sol... | https://mathoverflow.net/users/7049 | For Ax = b, x and b unknown vectors, how do I solve the x that maximizes min(b_i)? | This is a linear program. Put down the contraints $r \leq b\_i \ \forall i$, together with all the linear constraints you have above, and maximize $r$. All the constraints are linear, so linear programming will do this. There are tons of linear programming packages (Mathematica and MATLAB have decent ones), and there s... | 11 | https://mathoverflow.net/users/2294 | 35047 | 22,648 |
https://mathoverflow.net/questions/35049 | 11 | It's fairly easy to write down families of elliptic curves over $\mathbb{Q}(t)$ such that almost every (i.e. when the "height" of $t$ is sufficiently large) curve in the family has positive rank over $\mathbb{Q}$. One can do this by constructing the fibration so it has sections /$\mathbb{Q}$ a priori, or by fiddling ar... | https://mathoverflow.net/users/1464 | Families of genus 2 curves with positive rank jacobians | My guess for some examples is the family of (genus 2) hyperelliptic curves y2=degree 6 poly in x passing through n "randomly chosen" rational points (for n=2,3,4,5, or 6). The family of such curves has dimension 7-n, and I would guess that if a hyperelliptic curve has n "random" rational points on it then the Q rank of... | 4 | https://mathoverflow.net/users/51 | 35053 | 22,651 |
https://mathoverflow.net/questions/35054 | 5 | I am interested in a characterization of the creation and annihilation operators that is in some sense invariant under $O(n)$ rotations of $\mathbb{R}^n$:
Background
----------
The Harmonic Oscillator on $\mathbb{R}^n$ is the differential operator
$$ H := \sum\_{k=1}^n \left[x\_k^2-\frac{\partial^2}{\partial x\_... | https://mathoverflow.net/users/1540 | Characterizing the harmonic oscillator creation and annihilation operators in a rotationally invariant way | Let me try to answer this question. I apologise if my notation is slightly different, since I will work in some more generality, since the equivariance properties of the creation and annihilation operators are actually more transparent, I believe, relative to the the general linear group instead of the orthogonal group... | 4 | https://mathoverflow.net/users/394 | 35064 | 22,653 |
https://mathoverflow.net/questions/35063 | 5 | A monoidal category has (among other things) a pair of natural transformations called the left and right unitors
$$ \lambda\_A:I\otimes A \cong A $$
$$ \rho\_A:A\otimes I \cong A $$
The categories I work with on a daily basis have $\lambda\_I=\rho\_I:I\otimes I \cong I$, but I think that is mostly a consequence of... | https://mathoverflow.net/users/2361 | Must the left and right unitors of a monoidal category coincide at the neutral object? | In *Categories for the Working Mathematician* MacLane included $\lambda\_I = \rho\_I$ as one of three diagrams involving the associator and the two unitors that must commute (the other two being the usual pentagon and triangle diagrams) as the axioms defining monoidal categories.
It was however proven to follow from ... | 6 | https://mathoverflow.net/users/1797 | 35071 | 22,657 |
https://mathoverflow.net/questions/35094 | 2 | Let $B\_i^p$ be a family of sets, where $p\in \mathbb{N}$ and $i \in I$, $I$ being a directed set, and such that, for every $i$, we have a descending chain of inclusions
$$
\dots \supset B\_i^{p-1} \supset B\_i^p \supset B\_i^{p+1} \supset \dots
$$
Question: is the following implication true?
$$
\bigcap\_p B\_i^p... | https://mathoverflow.net/users/1246 | Colimit of intersections | Unfortunately, there seems to be a counterexample: Let $B^p\_i$ be the interval $(p-i,\infty)$ in the integers $\mathbb{Z}$, for natural numbers $p$ and $i$. In this case, we have $B^{p+1}\_i\subset B^p\_i$, and for any fixed $i$, we have $\bigcap\_p B^p\_i=0$, since $p$ runs out to infinty. But for fixed $p$, the limi... | 2 | https://mathoverflow.net/users/1946 | 35098 | 22,670 |
https://mathoverflow.net/questions/34251 | 8 | What is the dual group of the additive group of rational numbers equipped with the standard topology inherited from $\mathbb R$? As a group, this dual group is isomorphic to $\mathbb R$ (see the answer of Ekedahl given below), but it should be equipped with the topology of uniform convergence on compact subsets of $\ma... | https://mathoverflow.net/users/7886 | The dual group of $\mathbb Q$ | In fact, uniform convergence on compact subsets of $\mathbb{Q}\subset\mathbb{R}$ induces the usual topology on its group of (continuous) characters $\mathbb{R}\simeq\{t\mapsto\exp(ixt)\}\_{x\in\mathbb{R}}$.
Namely, consider $K=\{0\}\cup\{1/n,n\geq 1\}$. For $x\in\mathbb{R}$, the corresponding character is uniformly $... | 0 | https://mathoverflow.net/users/6451 | 35105 | 22,675 |
https://mathoverflow.net/questions/35057 | 1 | Let $A$ be a Hopf algebra dually paired with a quasi-triangular Hopf algebra $B$. If $x$ is some fixed element of $A$, then we can define a linear map
$$
P\_x: A \to \mathbb{C}
$$
by setting
$$
P\_x:a \mapsto \langle R,x \otimes a \rangle.
$$
Let us take the case $A = SL\_q(2)$, $B = U\_q({\mathfrak sl}\_2)$, and l... | https://mathoverflow.net/users/1095 | Map constructed from the coquasitriangular structure of SLq(2) which appears not to respect the standard commutation relations | Dear John,
I tried to follow your computation until the first place where I couldn't understand a step. This comes at:
>
> However,
> $$
> P\_{u^2\_1}(u^1\_1u^1\_2) = \langle R, u^2\_1 \otimes u^1\_1u^1\_2 \rangle = \sum\_z \langle R,u^2\_z \otimes u^1\_1 \rangle \langle R, u^z\_1 \otimes u^1\_2 \rangle,
> $$
> ... | 2 | https://mathoverflow.net/users/1040 | 35111 | 22,677 |
https://mathoverflow.net/questions/35087 | 14 | **Question:** Is the tangent bundle of the long line $L$ homeomorphic to $L\times\mathbb R$? I'd guess that the answer doesn't depend on choice of differentiable structure, but maybe it does.
**Motivation:** One night at dinner, someone brought up a puzzle involving infinitely many prisoners standing in a line, and ... | https://mathoverflow.net/users/300 | Is the tangent bundle of the long line $L$ homeomorphic to $L\times\mathbb R$? | The long line has the property that there is no continuous self map $f:L\to L$ such that $f(x)>x$ (or $f(x)<x$) for all $x\in L$. Indeed, $f^n(0)$ is an increasing sequence, hence converges to some $x$, which is a fixed point. So if there were an everywhere nonzero tangent vector field (for some differentiable structur... | 11 | https://mathoverflow.net/users/6451 | 35113 | 22,678 |
https://mathoverflow.net/questions/35090 | 3 | Normally a 2-sided error randomized algorithm will have some constant error $\varepsilon < 1/2$. We know that we can replace the error term for any inverse polynomial. And the inverse polynomial can be replaced for an inverse exponential. Say that we have an algorithm $A$ with $\varepsilon\_A=1/p(n)$ for some polynomia... | https://mathoverflow.net/users/7692 | Practical use of probability amplification for randomized algorithms | There seems to be some confusion in your notation. When you says error $\epsilon$, do you mean the probability of failure is $\epsilon$, which means that the algorithm outputs the wrong answer with probability $\epsilon$. In this case, if $\epsilon$ is polynomially small, that's great. If you run it a few times you sho... | 5 | https://mathoverflow.net/users/8075 | 35119 | 22,683 |
https://mathoverflow.net/questions/35112 | 3 | Mini introduction
-----------------
Suppose $U \subset \mathbb R^n, V \subset \mathbb R^m$ are two open sets. If we take http://en.wikipedia.org/wiki/Distributions\_space#Test\_function\_space">test functions $f\_i \in \mathfrak D (U),~g\_i \in \mathfrak D (V)$ for $1 \leq i \leq n$, then $f\_1(x)g\_1(y) + \dots + f\... | https://mathoverflow.net/users/7095 | Cartesian product of test function spaces | This is true.
By a partition of unity, the proof can be reduced to the case when the test functions have their supports in a unit cube and the result follows from a more or or less straightforward manipultation with the corresponding Fourier series.
See, for instance, Theorem 4.3.1 in ["Introduction to the Theory... | 3 | https://mathoverflow.net/users/5371 | 35120 | 22,684 |
https://mathoverflow.net/questions/35110 | 1 | Let $A$ be an algebra over $\mathbf{C}$ with an involution operator, and let $\|\cdot\|\_1$ and $\|\cdot\|\_2$ be two $equivalent$ operator norms, making $A$ into a $\*$-Banach algebra (we denote them as $A\_1$ and $A\_2$). The obvious morphism $A\_1\to A\_2$, $a\mapsto a$ is bounded by definition.
Should it also be... | https://mathoverflow.net/users/4807 | Completely equivalent operator norms on $*$-Banach algebras. | A priori, it does not make sense to talk about complete boundedness, since there are no specified operator space structure on $A\_1$ and $A\_2$.
In general, an infinite-dimensional Banach space can carry many incomparable operator space structure. Most prominently, there is the minimal and the maximal operator space ... | 4 | https://mathoverflow.net/users/8176 | 35126 | 22,686 |
https://mathoverflow.net/questions/35097 | 7 | I'm looking for references on the structure which can be roughtly described as follows: given a (braided or symmetric) monoidal category $C$, I want to consider a simplicial set $N(\mathbf{B}C)$ with a single vertex, an edge for every object of $C$, a triangle with edges $X,Y,Z$ for every morphism $\varphi:Z\to X\otime... | https://mathoverflow.net/users/8320 | Nerves of (braided or symmetric) monoidal categories | If you want to capture the structure of the category together with its monoidal structure, you may need a $k$-fold simplicial set for $k>1$, i.e., a functor from $(\Delta^{op})^k$ to sets. One of the simplicial coordinates encodes the composition law in the category, another encodes the monoidal structure, and the rest... | 9 | https://mathoverflow.net/users/121 | 35128 | 22,687 |
https://mathoverflow.net/questions/34300 | 9 | I'm doing computations in the integral group ring of a discrete group,
in particular the discrete Heisenberg group. In this case elements
are integral combinations of monomials $x^k y^m z^n$, where the
generators $x$, $y$, and $z$ satisfy $xz=zx$, $yz=zy$, and $yx=xyz$.
Is there mathematical software for doing calculat... | https://mathoverflow.net/users/8112 | Mathematical software for computing in integral group rings of discrete groups? | You can do this with [GAP](http://www.gap-system.org/). The example below assumes that you have the [polycyclic](http://www.gap-system.org/Packages/polycyclic.html) package installed.
First, you tell GAP which group you want to work with. Luckily, Heisenberg groups are polycyclic, and the polycyclic package provides ... | 13 | https://mathoverflow.net/users/8338 | 35135 | 22,691 |
https://mathoverflow.net/questions/30299 | 7 | Let $X$, $Y$ be complex projective varieties of dimension $n$, let $f:X \rightarrow Y$ be a surjective finite morphism of degree $d$ and let $B$ be a big line bundle on $Y$.
Is that true that vol($f^\*B$)=d $\cdot$ vol($B$)?
(I know that if $B$ is not only big but also nef then the formula is true by Lazarsfeld's P... | https://mathoverflow.net/users/7276 | volume of big line bundles under finite morphisms | Yes, this is true, even in a slightly more general context (the varieties can be defined over any field $k$, and the morphism only needs to be generically finite and surjective).
The main parts of the argument are given in the books of Lazardfeld (Positivity in Algebraic geometry) and Debarre (Higher-dimensional Alge... | 6 | https://mathoverflow.net/users/8333 | 35137 | 22,693 |
https://mathoverflow.net/questions/35139 | 8 | Fix a prime p, and look at elliptic curves in some family (e.g. all elliptic curves ordered by height). How often do the Fourier coefficients a\_p occur? Are there any conjectures?
| https://mathoverflow.net/users/5730 | Fourier coefficients for elliptic curves on average | There is a very clear discussion of some results in this direction in section 1 of Lenstra's paper ["Factoring Integers with Elliptic Curves"](http://www.jstor.org/stable/1971363). I'll attempt to summarize.
There are finitely many (about $2p$) isomorphism classes elliptic curves over $\mathbb{F}\_p$. Most sampling m... | 8 | https://mathoverflow.net/users/297 | 35145 | 22,697 |
https://mathoverflow.net/questions/35150 | 9 | Sum Equal Product
There are many articles in re this issue on the Web, although many are restricted to special cases (e.g., John Cook’s sum of tangents = product of tangents), and even some involving mixed numbers.
There is one excellent article in re: sum equal product written by two Polish mathematicians which i... | https://mathoverflow.net/users/8288 | Sum Equals Product | Let the positive, unequal integers be $a\_1 < a\_2 < \cdots < a\_k$ with $a\_1+a\_2+\cdots+a\_k = a\_1a\_2\cdots a\_k = n$. Obviously if $k = 1$ all positive integers work; suppose from now on that $k > 1$. Note that $a\_k \ge k$. Then we have $a\_1 + a\_2 + \cdots + a\_k < ka\_k$, while $a\_1a\_2\cdots a\_k \ge (k-1)!... | 18 | https://mathoverflow.net/users/8345 | 35152 | 22,701 |
https://mathoverflow.net/questions/35142 | 0 | I've been coming across $\mathbb{Q}$-lattices in $\mathbb{R}^n$ in my reading, and I'm having trouble understanding the definitions. Connes and Marcolli define it as a lattice $\Lambda \in \mathbb{R}^n$ together with a homomorphism $\phi : \mathbb{Q}^n / \mathbb{Z}^n \to \mathbb{Q} \Lambda / \Lambda$. Moreover, two $\m... | https://mathoverflow.net/users/8341 | Q-lattices and commensurability, any insight into the definition and intuition? | The condition $\mathbb Q\Lambda\_1=\mathbb Q\Lambda\_2=:X$ means that we have
$\Lambda\_1,\Lambda\_2\subseteq X$ and then we have $\Lambda\_1,\Lambda\_2\subseteq
\Lambda\_1+\Lambda\_2\subseteq X$. This means that we have quotient maps
$$X/\Lambda\_1\rightarrow X/(\Lambda\_1+\Lambda\_2){\rm\quad and\quad }X/\Lambda\_... | 3 | https://mathoverflow.net/users/4008 | 35154 | 22,703 |
https://mathoverflow.net/questions/34753 | 2 | As nobody seems to be able to give any kind of answer to that problem over there at math.stackexchange I post this question here:
How can I show with a heuristic argument based on a Taylor expansion that for Stratonovich stochastic calculus the chain rule takes the form of the classical (Newtonian) one?
The intuiti... | https://mathoverflow.net/users/1047 | Taylor expansion to show that for Stratonovich stochastic calculus the chain rule takes the form of the classical one | Hi,
Well you can have a look at the book of Kloeden and Platen "Numerical Solution of Stochastic Differential Equations" where the derivation of Taylor expansion for diffusion is derived based on iterated Wiener Itô (or Stratanovitch) Integrals.
Best Regards
| 1 | https://mathoverflow.net/users/2642 | 35166 | 22,712 |
https://mathoverflow.net/questions/35121 | 11 | Good afternoon,
There is an example of a Riemannian metric on $S^2 \times S^2$ of nonnegative sectional curvature that is not a product metric. I know there is one; however, I cannot find a specific reference. Any suggestions?
| https://mathoverflow.net/users/8334 | A Riemannian metric on $S^2 \times S^2$ of nonnegative curvature that is not a product | A discussion of nonnegatively curved metrics on $S^2\times S^2$ can be found in the [survey](https://aimath.org/WWN/nnsectcurvature/WilkingSurvey.pdf) by B. Wilking, see page 26 and last paragraph on page 25. In particular, there is a one parametric family of metrics of nonnegative curvature on $S^2\times S^2$ which a... | 11 | https://mathoverflow.net/users/1573 | 35167 | 22,713 |
https://mathoverflow.net/questions/35180 | 33 | Let $X,Y$ be finite-dimensional differentiable manifolds, and let's assume that they are connected. In fact, in applications I would like both $X$ and $Y$ to be riemannian manifolds.
Let $C^\infty(X,Y)$ denote the space of smooth maps $f: X \to Y$. I'm interested in, say, the connected components, fundamental group,.... | https://mathoverflow.net/users/394 | Topology of function spaces? | Homotopy theory of function spaces is a healthy subfield of homotopy theory. See e.g. recent [Oberwolfach report](http://www.sju.edu/~smith/pdf_files/OWR_2009_19.pdf) from a meeting on the subject.
If you share what $X,Y$ you are interested in, I may be able to say more, even though I am by no means an expert.
ED... | 13 | https://mathoverflow.net/users/1573 | 35183 | 22,722 |
https://mathoverflow.net/questions/35151 | 11 | Many complexity theorists assume that $P\ne NP.$ If this is proved, how would it impact quantum computing and quantum algorithms? Would the proof immediately disallow quantum algorithms from ever solving NP-Complete problems in Quantum Polynomial time?
[According to Wikipedia](http://en.wikipedia.org/wiki/QMA), quant... | https://mathoverflow.net/users/8347 | What impact would P!=NP have on the characterization of BQP? | David is right about one thing. Scott had a discussion about this on his blog and I was also involved.
On the one hand, many complexity theorists simply also assume that BQP does not contain NP, just as they assume that P does not contain NP. The evidence for the former is not as dramatic as that for the latter, but ... | 12 | https://mathoverflow.net/users/1450 | 35185 | 22,723 |
https://mathoverflow.net/questions/35184 | 4 | Let AC0 be the set of decision problems solvable by a logspace-uniform family of constant-depth, polynomial-width boolean circuits with unbounded fanin. Let BPAC0 be the modification of AC0 allowing it to use 0-ary random-bool gates, such that the probability of giving the wrong answer is less than 1/3. Is it known whe... | https://mathoverflow.net/users/nan | Use of randomness in constant parallel time | Not sure what you mean by "0-ary random-bool gates", but I think you mean: take a circuit $C$ with $n$ *real* inputs and $poly(n)$ *extra* inputs. For each input $x$ of length $n$, the "probabilistic" circuit $C$ is said to output $b$ on $x$ iff when we attach $x$ to the real inputs, and put a uniform random input on t... | 4 | https://mathoverflow.net/users/2618 | 35188 | 22,724 |
https://mathoverflow.net/questions/35191 | 15 | Suppose we have a function $f : \mathbb{R}^N \rightarrow \mathbb{R}$ which, given a vector, returns the value of its smallest element. How can I approximate $f$ with a differentiable function(s)?
| https://mathoverflow.net/users/5223 | A differentiable approximation to the minimum function | If signs aren't a big deal, use the generalized mean formula
$$
\left(\frac{1}{n}\sum x\_i^k\right)^{1/k}
$$
for $k\to -\infty$.
| 12 | https://mathoverflow.net/users/947 | 35193 | 22,728 |
https://mathoverflow.net/questions/35198 | 36 | Let us define the nth smooth homotopy group of a smooth manifold $M$ to be the group $\pi\_n^\infty(S^k)$ of smooth maps $S^n \to S^k$ modulo smooth homotopy. Of course, some care must be taken to define the product, but I don't think this is a serious issue. The key is to construct a smooth map $S^n \to S^n \lor S^n$ ... | https://mathoverflow.net/users/4362 | Smooth homotopy theory | Yes, the map you mention is an isomorphism. I think the main reason people rarely address your specific question in literature is that the technique of the proof is more important than the theorem. All the main tools are written up in ready-to-use form in Hirsch's Differential Topology textbook.
There are two steps ... | 28 | https://mathoverflow.net/users/1465 | 35200 | 22,733 |
https://mathoverflow.net/questions/35203 | 1 | I want to express the following sentence in first order logic.
There are naturals numbers that can not be expressed as one natural number raised to the power of another natural number other than one.
Under normal circumstances this is very simple. I am wondering if the the exponent function involved here can be e... | https://mathoverflow.net/users/8246 | Exponent function as uninterpreted function in first order logic | The function $f(m,n)=m^n$ is primitive recursive, so expressible in
first-order arithmetic: there is a formula in three free variables
$F(m,n,p)$ over the language of first-order arithmetic
which is valid in Peano arithmetic for numerals $m$, $n$ and $p$ iff $p=m^n$.
Logic texts (e.g. Boolos and Jeffrey) will prove t... | 5 | https://mathoverflow.net/users/4213 | 35205 | 22,737 |
https://mathoverflow.net/questions/35217 | 7 | Hello.
I study [primitive recursive arithmetic](http://en.wikipedia.org/wiki/Primitive_recursive_arithmetic) and have the following questions.
1) Is it possible to express in the PRA that Ackermann function is total?
2) If yes, is such expression decidable in the PRA ?
Can u suggest some literature on this topi... | https://mathoverflow.net/users/8381 | Ackermann function in the Primitive recursive arithmetic | You can express the totality of any computable function in PRA, using Kleene's T predicate, which is primitive recursive. So if you pick any index $e$ for the Ackermann function, the formula $(\forall n)(\exists t) T(\underline{e}, n, t)$ is already in the language of PRA.
However, you cannot prove the totality of t... | 15 | https://mathoverflow.net/users/5442 | 35219 | 22,744 |
https://mathoverflow.net/questions/35223 | 17 | I started thinking about this question because of this discussion:
<http://sbseminar.wordpress.com/2010/08/10/negative-value-added-by-journals/>
about how journals often change a paper (for the worse) after acceptance.
Here's my question: Since it doesn't make sense to number every single equation (especially if ... | https://mathoverflow.net/users/6871 | Should one use "above" and "below" in mathematical writing? |
>
> Since it doesn't make sense to number every single equation...
>
>
>
I used to number only equations I referred to, but then someone pointed out the following. When you write a paper and make it public, you are de facto allowing other people to talk about your paper and simply because you in your paper see n... | 73 | https://mathoverflow.net/users/394 | 35229 | 22,748 |
https://mathoverflow.net/questions/35224 | 4 | Given an even dimensional manifold, the mapping class group acts on middle dimensional cohomology (or homology) and this action preserves the intersection form. For manifold of dimension $4k+2$, the action symplectic, while it is orthogonal for manifold of dimension $4k$.
In dimension 2, it is well-known that any in... | https://mathoverflow.net/users/2183 | Action of the mapping class group on middle-dimensional cohomology | Without other assumptions, the answer is an easy no. For instance, if $N^3$ is a homology 3-sphere with infinite fundamental group, then $M^6 = N^3 \times S^3$ does not have very many lifts of automorphisms of its middle homology, because there exists a degree one map $S^3 \to S^3$ but no map $S^3 \to N^3$ with non-zer... | 7 | https://mathoverflow.net/users/1450 | 35230 | 22,749 |
https://mathoverflow.net/questions/35194 | 13 | Let $E\to X$ be a principal $U(N)$-bundle over a (nice) topological space $X$. It is well known that vanishing of the Chern classes of $E$ is not a sufficient condition for $E$ to be trivial, the simplest example being probably the nontrivial $U(2)$-bundle over $S^5$. However one may wonder what happens for a specific ... | https://mathoverflow.net/users/8320 | Complex vector bundles with trivial Chern classes on k-tori | As the cohomology of $(S^1)^n$ is torsion free every stable bundle on $(S^1)^n$ is
determined by Chern classes (this also follows from the $K$-theory Künneth
formula) so just as for the spheres it is an unstable problem. As for the
unstable problem unless I have miscalculated, if $(S^1)^5\rightarrow S^5$ is a
degree $1... | 15 | https://mathoverflow.net/users/4008 | 35232 | 22,751 |
https://mathoverflow.net/questions/35209 | 4 | Let $\xi\_t$ be a zero-mean gaussian process on $[0,1]$ with covariance operator $C$.
I would like to better understand the relation between the covariance operator and the regularity of the trajectories.
I already know that
>
> **Theorem** (Kolmogorov) If there exists $\alpha>1,C\geq 0$ and
> $\epsilon>0$ su... | https://mathoverflow.net/users/6531 | Relation between regularities of the trajectory of a mean zero gaussian process and its covariance operator | Suppose we already know that the process is continuous with probability one, so that the process takes values in the Banach space $X = C([0,1])$ with distribution $\mathbb P$. The covariance operator $C : X^\* \to X$ is then a map from the dual space $X^\*$ to $X$. The support of the Gaussian measure $\mathbb P$ is the... | 2 | https://mathoverflow.net/users/238 | 35234 | 22,753 |
https://mathoverflow.net/questions/20664 | 21 | In an exercise in his algebraic topology book, Munkres asserts that $\mathbf{C}P^n$ is triangulable (i.e., there is a simplicial complex $K$ and a homeomorphism $|K| \rightarrow \mathbf{C}P^n$). Can anyone provide a reference or a proof?
| https://mathoverflow.net/users/4194 | Why is complex projective space triangulable? | I will present a triangulation of $\mathbb{CP}^{n-1}$. More specifically, I will give an explicit regular CW structure on $\mathbb{CP}^{n-1}$. As spinorbundle says, the first barycentric subdivision of a regular CW complex is a simplicial complex homeomorphic to the original CW complex.
---
Recall that to put a r... | 8 | https://mathoverflow.net/users/297 | 35241 | 22,756 |
https://mathoverflow.net/questions/35228 | 2 | Let $p$ be a prime number and $\zeta\_{p^n}$ a primitive $p^n$-th root of unity. Find $f \in \mathbf Q\_p[[X]]$ fulfilling $f(\zeta\_{p^n}-1)=1/p^n$ for all sufficiently large $n$.
| https://mathoverflow.net/users/8380 | Power series $f$ such that $f(\zeta_{p^n}-1)=1/p^n$ for almost all $n \geq 0$ | As Brian pointed out, the principal parts problem always has a solution on the open unit disk, and Lazard's 1962 article "Les zéros des fonctions analytiques d’une variable sur un corps valué complet" gives a nice proof which is also rather explicit.
Your problem has additional symmetries so one can be a bit more ex... | 10 | https://mathoverflow.net/users/5743 | 35245 | 22,759 |
https://mathoverflow.net/questions/35246 | 19 | In this post, let $I=[0,1]$.
Something about the definition of homotopy in algebraic topology (and in particular in the study of the fundamental group) always puzzled me. Most books on the fundamental group often begin with the basic notion of a homotopy of curves (or more generally, continuous functions between topo... | https://mathoverflow.net/users/7392 | The definition of homotopy in algebraic topology | There is a natural topology on the function space called the
[compact-open topology](http://en.wikipedia.org/wiki/Compact-open_topology).
In extreme levels of generality, your two definitions are different (they are the same for eg locally compact spaces). Let me give a more general discussion of this.
Let $X$, $Y... | 19 | https://mathoverflow.net/users/317 | 35249 | 22,762 |
https://mathoverflow.net/questions/35263 | 1 | `Ax = b`. I need a way to analyze a square matrix `A` to see if its solution vector `x` will always be positive when `b` is positive.
This question arises from solving the radiosity equation:
<http://www.siggraph.org/education/materials/HyperGraph/radiosity/images/slide11.jpg> (noob alert)
I'm interested to know ... | https://mathoverflow.net/users/1585 | How can I characterize the type of solution vector that comes out of a matrix? | If and only if all the entries of $A^{-1}$ are non-negative.
**Proof:** If $(A^{-1})\_{ij}$ is negative, and $b$ is $1$ in the $j$-th coordinate and very small in every other, then $A^{-1} b$ is negative in the $i$-th component.
On the other hand, if every entry of $A^{-1}$ is non-negative, then clearly $b$ posit... | 5 | https://mathoverflow.net/users/297 | 35266 | 22,766 |
https://mathoverflow.net/questions/35253 | 25 | A well known theorem of Cartan states that every free homotopy class of closed paths in a compact Riemannian manifold is represented by a closed geodesic (theorem 2.2 of Do Carmo, chapter 12, for example). This means that every closed path is homotopic to a closed geodesic (through a homotopy that need not fix base poi... | https://mathoverflow.net/users/4362 | Do free higher homotopy classes of compact Riemannian manifolds have preferred representatives? | A theorem of Sacks and Uhlenbeck tells us
<http://www.jstor.org/pss/1971131>
that we are still in the good shape for free homotopy $2$-classes. Namely, any such class can be represented by a collection of minimal spheres joined by geodesic segments. I would guess that in the case when you consider $\pi\_n(M^{n+1})$ som... | 13 | https://mathoverflow.net/users/943 | 35271 | 22,767 |
https://mathoverflow.net/questions/35274 | 3 | My loose question is like this: what would you say about an equivalence of categories where both are concrete categories, and the equivalence functor is induced from a set-theoretic bijection at the level of objects? It should be something like "equivalence of categories induced by a natural isomorphism over the catego... | https://mathoverflow.net/users/3040 | A functor that comes from a morphism in a bigger category | If I understand your question right, the term you want is *an equivalence (or isomorphism) over* **Set**. Concretely, this means: it's an equivalence in which the categories have (forgetful) functors to **Set**, the functors of the equivalence commute down to **Set**, and the natural transformations are identities on u... | 3 | https://mathoverflow.net/users/2273 | 35278 | 22,772 |
https://mathoverflow.net/questions/35236 | 27 | Background
==========
The complexity classes [BPP](http://en.wikipedia.org/wiki/BPP), [BQP](http://en.wikipedia.org/wiki/BQP), and [QMA](http://en.wikipedia.org/wiki/QMA) are defined semantically. Let me try to explain a little bit what is the difference between a semantic definition and a syntactic one. The complexi... | https://mathoverflow.net/users/7507 | Is there a syntactic characterization for BPP, BQP, or QMA? | No, I don't think any syntactic characterization is known for BPP, BQP or QMA. (BPP might turn out to be P, and then we'd have such a characterization of course.)
In particular we don't know any languages that are complete for either of these classes. A lot of people believe that classes like QMA do not even have com... | 16 | https://mathoverflow.net/users/8075 | 35300 | 22,786 |
https://mathoverflow.net/questions/35287 | 4 | hello.
I am looking for tensor manipulation software that would allow me:
* declare indices
* declare results of contraction (or simplification rules)
* allow algebraic simplifications and expansion
* index renaming
So far I have found Maxima to satisfy my requirement more or less, <http://maxima.sourceforge.net/... | https://mathoverflow.net/users/5925 | Indexed tensor manipulation CAS | I think that everything in your list (except the Python interface) can be found in Kasper Peeters' [Cadabra](http://cadabra.phi-sci.com/).
As for a Python interface, there are two directions:
1. [It is planned](http://cadabra.phi-sci.com/ideas.html) to add an interface layer to Cadabra to either Maxima or SymPy - ... | 2 | https://mathoverflow.net/users/358 | 35311 | 22,793 |
https://mathoverflow.net/questions/35156 | 22 | If X is a variety over the complex numbers, one reasonable thing to do is to consider the associated analytic space $X\_{an}$ and to take the topological Euler characteristic of that.
Is there a purely algebraic way to obtain this number?
If X is non-singular then one might define it as the integral of the top Cher... | https://mathoverflow.net/users/3701 | How do you define the Euler Characteristic of a scheme? | If $X$ is a smooth variety over the complex numbers, you can use the topological Euler characteristic of the support $X(\mathbb{C})$ of the scheme. The most efficient way to compute it is to use **Chern classes** and the **Poincaré-Hopf theorem**: the Euler characteristic is the degree of the top Chern class. Chern cla... | 17 | https://mathoverflow.net/users/4046 | 35316 | 22,795 |
https://mathoverflow.net/questions/35260 | 11 | I am trying to do a measurement uncertainty calculation. I have a gaussian distributed phase angle (theta) with a mean of 0 and standard deviation of 16.6666 micro radians. The variance is the square of the standard. The formula for the measurment uses cos(theta) in the calculation. I need to know the mean, the varianc... | https://mathoverflow.net/users/8391 | Resultant probability distribution when taking the cosine of gaussian distributed variable | I wrote out the first few terms in the power series for $ \cos \theta $
and then the first few terms of the series for $ \cos^2 \theta .$
I used your hypothesis of normal distribution, the mean of $ \theta $ is $ \mu = 0$ while the
variance is some $ \sigma^2 .$
Then I looked up the expected values of $ \theta^2, \;... | 13 | https://mathoverflow.net/users/3324 | 35318 | 22,796 |
https://mathoverflow.net/questions/35320 | 29 | The Perron-Frobenius theorem says that the largest eigenvalue of a positive real matrix (all entries positive) is real. Moreover, that eigenvalue has a positive eigenvector, and it is the only eigenvalue having a positive eigenvector.
Now suppose we want to construct a positive rational matrix with a particular Perro... | https://mathoverflow.net/users/8410 | Perron-Frobenius "inverse eigenvalue problem" | The answer to a sharper question involving integers, rather than rationals, is affirmative.
>
> Let $\lambda$ be a positive real algebraic integer that is greater in absolute value than all its Galois conjugates ("Perron number" or "PF number"). Then $\lambda$ is the Perron–Frobenius eigenvalue of a positive intege... | 30 | https://mathoverflow.net/users/5740 | 35322 | 22,798 |
https://mathoverflow.net/questions/35301 | 8 | By the Schwartz–Zippel lemma, "Is this arithmetic formula identically zero?" is in coRP $\subseteq$ BPP $\subset$ P/poly, with the second inclusion by Adleman's theorem. By basically following the proof, but using the improved error bound that comes from the original algorithm only having one-sided error, one gets an a... | https://mathoverflow.net/users/nan | P/poly algorithm for polynomial identity testing | The Schwartz-Zippel lemma is very fast, only one evaluation of the formula at one random point. There's nothing better known that minimizes time and error as well as Schwartz-Zippel. But Schwartz-Zippel requires a lot of randomness in each repetition: a fresh new point of n elements.
Have you tried some of the polyno... | 10 | https://mathoverflow.net/users/2618 | 35324 | 22,800 |
https://mathoverflow.net/questions/35337 | 3 | I am giving a "non-technical" seminar in which I would like to give an elementary introduction to the Hausdorff dimension and measure.
For simplicity, I was hoping to give a more intuitive introduction using 'squared paper': suppose you draw your figure on squared paper with squares of size $2^{-n}$, and let $M(n)$ t... | https://mathoverflow.net/users/1898 | Simple definition of the Hausdorff measure using squared paper | The discussion at <http://en.wikipedia.org/wiki/Box-counting_dimension> (Minkowski–Bouligand dimension) seems fairly thorough.
| 5 | https://mathoverflow.net/users/6153 | 35338 | 22,807 |
https://mathoverflow.net/questions/35339 | 6 | It's well known that there are no non-constant polynomials with integer coefficients whose values at integer points are primes. Could this result be generalized to the case of prime powers?
The question is whether there exists a polynomial $p(x) \in \mathbb{Z}[x]$ with degree at least one such that for all $x \in \ma... | https://mathoverflow.net/users/7079 | Polynomial with prime powers values | With integer coefficients the answer is surely no: let $f(x)\in\mathbb{Z}[x]$ a degree $d$ polynomial, and let $a\in \mathbb{Z}$ be such that $f(a)\neq0,1,-1$. Then there exist a prime $p$ such that $f(a)=0 \mod p$. Now consider the sequence $x\_n=f(a+np)$. Since $x\_n=0\mod p$, by the assumption on the values of $f$, ... | 9 | https://mathoverflow.net/users/8320 | 35342 | 22,809 |
https://mathoverflow.net/questions/35334 | 25 | Is there a version of Stokes' theorem for vector bundle-valued (or just [vector-valued](https://en.wikipedia.org/wiki/Vector-valued_differential_form)) differential forms?
Concretely: Let $E \rightarrow M$ be a smooth vector bundle over an $n$-manifold $M$ equipped with a connection. First of all, is there an $E$-val... | https://mathoverflow.net/users/8415 | Integration and Stokes' theorem for vector bundle-valued differential forms? | For a general vector bundle, I there is no "$E$-valued integration" as you put it. You are trying to add up elements in the fibres of $E$, but since the fibres over different points are not the same vector space you can't add their elements.
For the trivial bundle $M \times \mathbb{R}^k$ - and with a fixed choice of ... | 42 | https://mathoverflow.net/users/380 | 35345 | 22,811 |
https://mathoverflow.net/questions/35335 | 1 | Suppose I have a category C. And there are two objects X and Y with no morphisms between them. I've checked up "quotient category" on wikipedia, but there I can only make isomorphic objects with morphisms between them.
Is there a categorial notion available that I can use in this case?
| https://mathoverflow.net/users/nan | Collapsing objects in a category | Short answer: yes. There is a special class of 2-categorical limits called iso-inserters that does the trick. The paper to check is M. Kelly's "Elementary observations on 2-categorical limits", Bull. Austr. Math. Soc. 39 (1989), 301-317.
Instead of explaining what these are let me go about explaining how you would ma... | 3 | https://mathoverflow.net/users/2562 | 35352 | 22,815 |
https://mathoverflow.net/questions/35351 | 1 | I have run into this problem (or something similar to it) a few times now and I am wondering if the answer is known.
Given an vector $s$ of integers let $d(s)$ be the minimum difference between any two integers in $s$, that is
$$d(s) = \min\_{i,j \in s} |i - j|.$$
For $s$ a vector of length $m$ from $\lbrace 1,2,\dot... | https://mathoverflow.net/users/5378 | Minimum differences in vectors of naturals | The number of $m$-subsets of {$1,2,\ldots,n$} with distance at least $k$ between any pair is
$n - (k-1)(m-1) \choose m$.
Proof: for any subset of size $m$ of the first $n-(k-1)(m-1)$ integers, you can get a subset $S$ of the first $n$ with $d(S)\geq k$ by just adding $k-1$ consecutive integers after each of the first... | 7 | https://mathoverflow.net/users/2294 | 35353 | 22,816 |
https://mathoverflow.net/questions/35360 | 7 | This question is inspired by [Project Euler's Problem 47](http://projecteuler.net/index.php?section=problems&id=47).
Let *m* and *n* be positive integers. Consider the following four (related) statements:
1. There exists *m* consecutive integers, each of which has at least *n* distinct prime factors.
2. There exist... | https://mathoverflow.net/users/4747 | Consecutive integers with many prime factors | Answer for the first question. Let $w\_k$ be the product $p\_{nk+1} p\_{nk+2} \cdots p\_{nk+n}$ where $p\_i$ is $i$-th prime. So each $w\_k$ is equal to product of $n$ distinct primes and $w\_k$ and $w\_q$ are coprime for $k \ne q$. Now take $t$ such that $t \equiv -k \pmod {w\_k}$ for $k = 1, 2, \cdots, m$. Such $t$ e... | 7 | https://mathoverflow.net/users/7079 | 35364 | 22,824 |
https://mathoverflow.net/questions/35355 | 8 | Given an abelian group $G$, one can form the endomorphism ring $\mbox{End}(G)$ by letting $\alpha+\beta=\alpha(x)+\beta(x)$, and $\alpha\beta=\alpha(\beta(x))$, where $\alpha$ and $\beta$ are endomorphisms. Clearly, composition distributes over addition, and addition is commutative, so $\mbox{End}(G)$ is a ring. My que... | https://mathoverflow.net/users/6856 | Criterion for an abelian group to have a commutative endomorphism ring | Amongst the finitely generated abelian groups, those with commutative endomorphism ring are exactly the cyclic groups.
Torsion abelian groups with commutative endomorphism rings are exactly the locally cyclic groups, that is, the subgroups of Q/Z. They were classified in:
* Szele, T.; Szendrei, J.
"On abelian group... | 16 | https://mathoverflow.net/users/3710 | 35366 | 22,825 |
https://mathoverflow.net/questions/35356 | 3 | Let $G$ be a (topologically) simple Hausdorff topological group. Let $H$ be a dense subgroup of $G$. Now throw away the topology. What restrictions are known on the structure of $H$ as an abstract group? I imagine not much can be said if $G$ has a very coarse topology, but I am particularly interested in the case where... | https://mathoverflow.net/users/4053 | Topological simplicity and dense subgroups | In many cases a simple topological group contains a nonabelian free group which is dense. This is true for Lie groups and for many examples of locally compact totally disconnect groups. In many of these cases every limit group can be found as a dense subgroup. Now, I do not rememebr the details, but you can find more i... | 4 | https://mathoverflow.net/users/5034 | 35371 | 22,827 |
https://mathoverflow.net/questions/29869 | 23 | This question came up when I was doing some reading into convolution squares of singular measures. Recall a function $f$ on the torus $T = [-1/2,1/2]$ is said to be $\alpha$-Hölder (for $0 < \alpha < 1$) if $\sup\_{t \in \mathbb{T}} \sup\_{h \neq 0} |h|^{-\alpha}|f(t+h)-f(t)| < \infty$. In this case, define this value,... | https://mathoverflow.net/users/7165 | Density of smooth functions under "Hölder metric" | In our PDE seminar, we met the same kinds of questions, and
we think the answer is "WRONG". The smooth functions is NOT
dense in Hölder spaces.
An example is,
$$f(x) = |x|^{1/2} \quad x \in (-1,1)$$
it is easy to check that $f$ is $1/2$-Hölder continuous.
For details,
for any $g \in C^{1}((-1,1))$, then the deriv... | 24 | https://mathoverflow.net/users/8424 | 35372 | 22,828 |
https://mathoverflow.net/questions/35382 | 12 | Inspired by the party game [Mafia](http://en.wikipedia.org/wiki/Mafia_(party_game)), in particular those situations where nobody is clearly innocent or guilty and the group wants to decide on someone random to eliminate.
Suppose n people each have their own personal random number generator (a machine which generates ... | https://mathoverflow.net/users/1060 | Untrustworthy people picking a random number | The group agrees on a strong hash function H. Each person in sequence generates a number x and reveals H(x). Then each person in sequence reveals x. If all hashes can be verified, then the first bit of sum(x) mod 1 is used.
| 10 | https://mathoverflow.net/users/2003 | 35386 | 22,835 |
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