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https://mathoverflow.net/questions/216124 | 5 | Is it possible to convert a fractional problem (maximization) with objective function equal to the ratio of a concave function and convex function ? This question sound impossible but I have read this claim in wikipedia without any reference !
<https://en.wikipedia.org/wiki/Nonlinear_programming#Methods_for_solving_t... | https://mathoverflow.net/users/38361 | convert a special case of nonlinear fractional programming into a convex problem | Here is some insight into Jean's answer, showing what that final problem means. Suppose you transform the problem to finding a vector $(y,t) \in \mathbb{R}^{n+1}$ to solve:
\begin{align\*}
&\mbox{Max: } & tf(y/t) \\
&\mbox{Subject to: } & tg(y/t) \leq 1 \\
& & y/t \in S \\
& & t>0 \\
& & y \in \mathbb{R}^n
\end{align... | 2 | https://mathoverflow.net/users/73850 | 216290 | 102,120 |
https://mathoverflow.net/questions/216268 | 6 | Suppose $k>0$ is some fixed constant, and $n$ is a positive integer tending to infinity. Find $j\equiv j(n,k)$ such that
$$ \frac{\binom{n}{j}}{j!} \sim k. $$
The asymptotic expression for $(n!)^{-1}$ (<https://stackoverflow.com/questions/3084937/how-to-calculate-the-inverse-factorial-of-a-real-number>) was initiall... | https://mathoverflow.net/users/36883 | Asymptotic expression for $j$ which satisfies $\binom{n}{j}/j! \sim k$ as $n\to\infty$ | (CORRECTED EDITION)
By mucking around with expansions like Igor suggested, I found
$$ j \approx J(n,k)=
en^{1/2} - \tfrac14\ln(n) -\tfrac12\ln(2\pi k)-\tfrac14 e^2-\tfrac12.
$$
It seems good when $k$ is small but not extremally small. For example, $\binom{100}{22}/22!\approx 6.523187$ and $J(100,6.523187)\approx 21.... | 6 | https://mathoverflow.net/users/9025 | 216294 | 102,121 |
https://mathoverflow.net/questions/216296 | 8 | Are there collections of already generated *large* strongly regular graphs available to download? By *large* I mean $n \geq 200$ where $n$ is the number of vertices. I found Ted Spence's [page on srgs](http://www.maths.gla.ac.uk/~es/srgraphs.php), which contains a very nice list of srgs up to $n \leq 64$, but they are ... | https://mathoverflow.net/users/78831 | Generating (or availability of) large strongly regular graphs | [Sage(math)](https://sagemath.org) aims to have graphs from A.Brouwer's [web pages on s.r.g.'s](https://www.win.tue.nl/%7Eaeb/graphs/srg/srgtab.html) available. Namely, at least one example for each case for which the existence is known. E.g. it will be possible to do things like
```
sage: t=graphs.strongly_regular_... | 8 | https://mathoverflow.net/users/11100 | 216301 | 102,124 |
https://mathoverflow.net/questions/216292 | 10 | $\DeclareMathOperator\Spec{Spec}$I am reading the paper "Constancy of the adjoint L-invariant" by H. Hida (J. Number Th., 2011, [DOI link](https://doi.org/10.1016/j.jnt.2011.02.001)).
Correct me if I'm wrong, but I've read/heard that the arithmetic points $p \in \Spec(\mathbb{I})$ are Zariski dense (Where $\Spec(\mat... | https://mathoverflow.net/users/47195 | Arithmetic points are dense on a Hida family | Hida theory is a vast domain of research. I am assuming that that you are in the simplest and oldest setting: Hida theory for ordinary eigencuspforms for the group $\operatorname{GL}\_2$ over $\mathbb Q$ (the arguments I am giving would apply with little modification to $\operatorname{GL}\_2$ over a totally real field ... | 7 | https://mathoverflow.net/users/2284 | 216307 | 102,127 |
https://mathoverflow.net/questions/216216 | 3 | I can't understand the following example of measure of non-compactness, which was given in [this](https://www.google.com.pk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&cad=rja&uact=8&ved=0CCgQFjACahUKEwiphKzw7tfHAhXB1RQKHdgSAXg&url=http%3A%2F%2Fwww.math.ubbcluj.ro%2F~nodeacj%2Fdownload.php%3Ff%3D052appell.pdf&usg=AFQjCNEi... | https://mathoverflow.net/users/43418 | Example of measure of non-compactness | -How is it not a measure of non-compactness?
The set of Rademacher functions $R:=\{r\_n(t) : n \in N\}$ is bounded but not pre-compact (the sequence $(r\_n)$ is equivalent to the unit vector basis of $\ell\_2$) in $X$, but $\phi(R)=0$ because $|r\_n(t)|\leq 1$.
-How to show that any set $M$ which is bounded in $L\... | 4 | https://mathoverflow.net/users/39421 | 216311 | 102,129 |
https://mathoverflow.net/questions/216316 | 4 | I have seen mentioned in a talk an operation that takes a graph $G=(V,E)$ and constructs a new bipartite graph $G'=(V',E')$ such that $V' = V\times \{0,1\}$ and $E'=\{((i,1),(j,0)) : (i,j)\in E\} \cup \{((i,0),(j,1)) : (i,j)\in E\}$.
In words, one takes two copies of the vertices of the graph, and makes the edges go ... | https://mathoverflow.net/users/1898 | Method to construct a bipartite graph G' with 2n vertices from a graph G | What you are looking for is [the bipartite double cover](https://en.wikipedia.org/wiki/Bipartite_double_cover).
| 6 | https://mathoverflow.net/users/11142 | 216317 | 102,131 |
https://mathoverflow.net/questions/216309 | 4 | Assuming that we construct a link/chain using a collection of knots.
Is there a way to measure the physical strength of this chain?
| https://mathoverflow.net/users/29611 | Physical strength of a link | There is one problem along these lines that allows for some rather precise mathematical statements: *What is the condition for a hitch to hold?* A hitch is a knot around a rigid object (a pole). Because the pole is not compressible, and assumed to be much thicker than the rope, the problem whether or not the knot holds... | 11 | https://mathoverflow.net/users/11260 | 216325 | 102,135 |
https://mathoverflow.net/questions/216326 | 2 | Let $\{X\_k\}$ be an ergodic process. I know that if $f$ is a smooth real valued function then by Birkoff's ergodic theorem, $$\lim\_{n\to \infty}\frac{1}{n}\sum\_{k=1}^n f(X\_k)=\mathbb{E}(f(X\_1))\ a.s.$$ Is there any similar result for the $\limsup\_n$ or $\liminf\_n$ of the sequence $\{f(X\_k)\}$, i.e. results whic... | https://mathoverflow.net/users/64194 | Is there a way to find $\limsup$ and $\liminf$ for ergodic processes almost surely? | For ergodic $X\_n$, almost surely
$$\limsup\_n f(X\_n) = \sup\_n f(X\_n) = \sup \{a \in \mathbb{R} : Pr(f(X\_0) > a) > 0\}.$$
In other words, by recurrence, what can happen (with positive probability) will eventually happen---infinitely often.
**Update (Sep 4, 2015)**:
You seem most interested in the case of a Ga... | 1 | https://mathoverflow.net/users/12978 | 216327 | 102,136 |
https://mathoverflow.net/questions/216264 | 3 | I am currently trying to evaluate a complicated function $f$ at a point $x+a$ using a high order Taylor polynomial about the point $a$. The polynomial has the standard form:
$$\sum\_{n=1}^k \frac{x^n}{n!} f^{(n)}(a)$$
However it takes a long time to evaluate because the $n^{th}$ derivative requires me to compute th... | https://mathoverflow.net/users/49673 | Simplifying a Taylor polynomial that involves Stirling numbers of the second kind | Changing the order of summation, we get
$$ \sum\_{n=1}^k \frac{x^n}{n!} \sum\_{m=1}^n (-1)^{m-1} (m-1)!S(n,m) a^m$$
$$ = \sum\_{m=1}^k (-1)^{m-1} (m-1)! a^m \sum\_{n=m}^k \frac{x^n}{n!} S(n,m).$$
Using formula (13) from <http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html> , we further get
$$= \sum\_{m=... | 3 | https://mathoverflow.net/users/7076 | 216342 | 102,142 |
https://mathoverflow.net/questions/216343 | 1 | I have a rather simple question of number theory which I can't seem to be able to find a good reference for. I am not a specialist and I don't really know where to look. I would like to show that the following sequence
$$ \frac{1}{N^2}\sum\_{p \leq N \ p \ \mathrm{prime }}{p^2} $$ goes to infinity. Numerical computa... | https://mathoverflow.net/users/25511 | Asymptotic of the sum of squared primes | [This question appeared on Math.stackexhange](https://math.stackexchange.com/questions/49383/how-does-sum-px-p-s-grow-asymptotically-for-textres-1/49434#49434). The answer linked above uses partial summation to show that for $k>-1$ we have
$$\sum\_{p\leq x}p^{k}=\text{li}\left(x^{k+1}\right)+O\left(x^{k+1}e^{-c\sqrt{\l... | 2 | https://mathoverflow.net/users/12176 | 216355 | 102,147 |
https://mathoverflow.net/questions/215629 | 3 | A famous Steinhaus theorem says that if measurable subsets $A,B$ of a locally compact topological group $G$ have positive Haar measure, then the difference $AA^{-1}$ is a neighborhood of the unit and the sumset AB has non-empty interior in $G$.
The first part of this theorem was generalized to Polish groups by Chris... | https://mathoverflow.net/users/61536 | Is the sumset of two Haar positive closed subsets of a Polish group non-meager? | I have just realized that this my question has a simple negative answer: Denote by $\mathbb R\_+=[0,\infty)$ the half-line. Observe that the countable product of lines $G=\mathbb R^\omega$ is an Abelian Polish group and the subsemigroup $A=\mathbb R\_+^\omega$ is not Haar null in $G$ (since for each compact subset $K$ ... | 3 | https://mathoverflow.net/users/61536 | 216364 | 102,150 |
https://mathoverflow.net/questions/216335 | 4 | A density matrix is a matrix $\rho \in \mathscr{D}:=\{A \in \mathbb{C}^{n \times n}; A^\*=A; \operatorname{tr}(A)=1; A \ge 0\}.$
In Quantum Mechanics it is natural to look at a group action
$\Phi: U(n) \times \mathscr{D} \rightarrow \mathscr{D}, (U,\rho) \mapsto U\rho U^\*.$
Now, my quetion is: What is the kind o... | https://mathoverflow.net/users/77929 | Set of density matrices | This is a symmetric space when equipped by the metric discussed in [this question](https://mathoverflow.net/questions/115192/power-log-distance-between-matrices/115227#115227) (it is Riemannian, when $p=2$). The metric is clearly invariant under the action; its other properties can be checked (see the references in Suv... | 4 | https://mathoverflow.net/users/11142 | 216371 | 102,153 |
https://mathoverflow.net/questions/216365 | 6 | In a $C^\*$-algebra ${\cal A}$, a **positive** element is a one of the form $aa^\*$, for some $a \in {\cal A}$. It is known that the set of positive elements is a [cone](https://en.wikipedia.org/wiki/Cone_%28linear_algebra%29), and that for $a,b$ two non-zero elements, and $\lambda \in {\bf R}\_{>0}$, we have $a + \lam... | https://mathoverflow.net/users/57635 | Positive Elements of a $\ast$-Algebra | The answer to the second part is also no. That is, $aa^\*+bb^\*$ could be zero for nonzero $a$ and $b$. Look at the $\*$-algebra $\mathbb{C}^2$ with component-wise addition and multiplcation and $\*$ given by $(x,y)^\* = (\overline{y},\overline{x})$. It's straightforward to calculate that anything of the form $aa^\*$ w... | 8 | https://mathoverflow.net/users/351 | 216381 | 102,157 |
https://mathoverflow.net/questions/114943 | 60 | According the [the Wikipedia page](http://en.wikipedia.org/wiki/Classification_of_finite_simple_groups#Second-generation_classification), the second generation proof is up to at least nine volumes: six by Gorenstein, Lyons and Solomon dated 1994-2005, two covering the quasithin business by Aschbacher and Smith in 2004,... | https://mathoverflow.net/users/4177 | Where are the second- (and third-)generation proofs of the classification of finite simple groups up to? | There is an [interesting review by Ron Solomon](http://blogs.ams.org/beyondreviews/2015/09/02/ron-solomon-reviews-recent-work-on-finite-simple-groups/) of a paper in this area, which has been featured on the Beyond Reviews blog. In particular, he outlines the broad tactics that people are using in CFSG II, and some of ... | 36 | https://mathoverflow.net/users/4177 | 217397 | 102,167 |
https://mathoverflow.net/questions/217395 | 2 | How do I see that each $\Phi^q$ of a stable cohomology operation $\{\Phi^q\}$ is a natural homomorphism?
| https://mathoverflow.net/users/78773 | Stable cohomology operation, natural homomorphism | For any (reduced) cohomology theory $\tilde H^\*$, not necessarily ordinary, addition in
$\tilde H^r(\Sigma X)$ is induced by the pinch map $\Sigma X \to \Sigma X\ \vee \Sigma X$, using the natural isomorphism $\tilde H^r(\Sigma X \vee \Sigma X) \cong \tilde H^r(\Sigma X)\times \tilde H^r(\Sigma X)$ and functoriality.... | 5 | https://mathoverflow.net/users/14447 | 217398 | 102,168 |
https://mathoverflow.net/questions/217399 | 3 | Let $M,N$ be two manifolds of different dimensions. Suppose $M\simeq N$, i.e. $M$ is homotopy equivalent to $N$. Do the Stiefel-Whitney classes of the tangent bundles of $M$ and $N$ equal
$$
w(TM)=w(TN)?
$$
Do the Chern classes of the tangent bundles of $M$ and $N$ equal
$$
c(TM)=c(TN)?
$$
| https://mathoverflow.net/users/65800 | characteristic classes of homotopy equivalent manifolds | Wu's formula for the Stiefel-Whitney classes implies that they are invariants of homotopy type. See for example [here](http://www.map.mpim-bonn.mpg.de/Wu_class).
Chern classes are not even diffeomorphism invariant, and it is possible to have two complex structures on the *same* manifold with different Chern classes. ... | 13 | https://mathoverflow.net/users/21375 | 217400 | 102,169 |
https://mathoverflow.net/questions/215661 | 12 | Let the set $\mathcal{F}$ consist of subsets of $[n]$. Suppose that for any incomparable $A$ and $B$ in $\mathcal{F}$, we have $A \cap B \notin \mathcal{F}$. What is the largest possible size of $\mathcal{F}$?
| https://mathoverflow.net/users/48489 | What is the maximum size of a set system where the intersection of any two incomparable members is not in the set? | It was proved by Kleitman that $|\mathcal F|\le {n\choose n/2}+2^n/n$, see <http://www.sciencedirect.com/science/article/pii/0097316576900376>.
| 5 | https://mathoverflow.net/users/955 | 217422 | 102,175 |
https://mathoverflow.net/questions/217424 | -1 | Let $\mu$ be a Borel probability measure on $[0, 1)$, and $\{g\_k\}\_{k=0}^\infty$ be a Parseval frame for $L^2(\mu)$. Does
$$\sum\_{k=0}^\infty \left\|g\_k\right\|$$
converges?
| https://mathoverflow.net/users/69931 | Parseval frame, convergence of $\sum_{k=0}^\infty \left\|g_k\right\|$ | Any orthonormal basis of $L\_2(\mu)$ is a Parseval frame which will cause the series to diverge. So, in general, the answer is no.
| 0 | https://mathoverflow.net/users/76593 | 217435 | 102,179 |
https://mathoverflow.net/questions/217440 | 9 | I shall keep this to the point: Given a time domain signal say microphone recording of a conversation:
1. Laplace tranfrom of **x** is some function **X(s)** say defined in the complex plane. I like to think of this variable **s** as a measure of frequency and damping inherrent in the signal **x(t)** (If anyone has a... | https://mathoverflow.net/users/78870 | Physical interpretation of the mellin transform variable? | * *Physical interpretation:* To develop a physical intuition, this article might be informative:
[The power spectrum of the Mellin transformation with applications to scaling of physical quantities](http://scitation.aip.org/content/aip/journal/jmp/15/6/10.1063/1.1666723)
>
> The Mellin transform is used to diagon... | 7 | https://mathoverflow.net/users/11260 | 217445 | 102,184 |
https://mathoverflow.net/questions/217420 | 2 | Set $w(x) = (1 + |x|^2)^{1/2}$ with $|\cdot|$ the Euclidian norm on $\mathbb{R}^n$. For $s,\mu \in \mathbb{R}$, we define the Sobolev space
$$H\_2^{s}(\mathbb{R}^n) = \left\{f : \lVert f \rVert\_{s} := \left( \int\_{\mathbb{R}^n} |\widehat{f}(\omega)|^2 w(\omega)^{2s} \mathrm{d} \omega \right)^{1/2} < \infty \right\}$$... | https://mathoverflow.net/users/39261 | When is the identity Hilbert-Schmidt between weighted Sobolev spaces? | There is an explicit operator that maps $L^2$ isometrically onto $H^{s,\mu}\_2$ :$$I\_{s,\mu}u(x)=(1+|x|^2)^{-\mu/2}(I-\Delta)^{-s/2}u(x)$$The (inverse) Fourier transform $k\_s(x)$ of $(1+|\omega|^2)^{-s/2}$ is also well documented (Bessel functions etc), and then $I\_{s,\mu}u(x)=\int (1+|x|^2)^{-\mu/2} k\_s(x-y)u(y)\ ... | 4 | https://mathoverflow.net/users/75422 | 217447 | 102,186 |
https://mathoverflow.net/questions/217408 | 5 | Given a (real or almost complex) manifold $M$, Let the 2-nd unordered configuration space be the quotient space
$$
B(M,2)=(M\times M\setminus\ \Delta)/\ \mathbb{Z}\_2
$$
where
$$
\Delta=\{(m,m)\mid m\in M \}
$$
and $\mathbb{Z}\_2$ acts by reversing the coordinates order.
Then $B(M,2)$ is a (real or almost complex) ma... | https://mathoverflow.net/users/65800 | characteristic classes of tangent bundle of 2-nd unordered configuration space | An excellent discussion of symmetric powers in general is given by [Pavle Blagojevich](http://www.maths.ed.ac.uk/~aar/papers/blag.pdf) - one of the first things he mentions is that the $k$-th symmetric power of $\mathbb{R}P^n$ is $\mathbb{R}P^{kn},$ a result due to Arnold. This is NOT true of $\mathbb{C}P^n,$ but for $... | 0 | https://mathoverflow.net/users/11142 | 217449 | 102,188 |
https://mathoverflow.net/questions/215971 | 5 | Let $\Sigma\_g$ be a closed hyperbolic surface and $\rho\colon\pi\_1\Sigma\_g\to G$ an Anosov representation into a suitable Lie group. By definition of Anosovness, one has a $\rho$-equivariant continuous map
$$\partial\_\infty\rho\colon S^1\to G/P$$
from $S^1=\partial\_\infty H^2$ into the Furstenberg boundary $\part... | https://mathoverflow.net/users/39082 | Anosov representations and boundaries of (harmonic) maps | First of all there are isometries of symmetric spaces (of noncompact type) which have positive displacement, but no invariant geodesics. This means that if you replace surface groups by cyclic groups, there are no (in general) equivariant harmonic maps. Using this example, one can build an example of a surface group re... | 1 | https://mathoverflow.net/users/21684 | 217452 | 102,190 |
https://mathoverflow.net/questions/216322 | 6 | Let $a,b,c,d$ be four prime numbers. We set the polynomial :
$$P(X)=\frac{(1-X^{abc})(1-X^{abd})(1-X^{acd})(1-X^{bcd})(1-X^a)(1-X^b)(1-X^c)(1-X^d)}{(1-X)^2(1-X^{ab})(1-X^{ac})(1-X^{ad})(1-X^{bc})(1-X^{bd})(1-X^{cd})}$$
By numerical tests, i see that $P(X)$ always has at least one negative coefficient, how can i prove i... | https://mathoverflow.net/users/78847 | Negative coefficient in an almost cyclotomic polynomial | Suppose that $a<b<c<d$. We show that the coefficient of $X^c$ or of $X^{b+c-1}$ of $P(X)$ is negative. In order to do so, it suffices to work in the power series ring $\mathbb Q[[X]]$ modulo $X^{b+c}$. Note that $ac>b+c$ and so on, hence
\begin{equation}
P(X)\equiv\frac{(1-X^a)(1-X^b)(1-X^c)(1-X^d)}{(1-X)^2(1-X^{ab})}\... | 6 | https://mathoverflow.net/users/18739 | 217455 | 102,191 |
https://mathoverflow.net/questions/217450 | 6 | In [this question](https://mathoverflow.net/questions/216320/the-sum-of-a-series) the OP asks whether the sum
$$
f(q, \alpha) = \sum \_{k=1}^{\infty } \frac{q^k \left(q^k-1\right)^\alpha}{(q;q)\_k}
$$
is ever zero. An experiment with Mathematica indicates, to any reasonable precision that
$$f(2, 1) = \sum \_{k=1}^{\inf... | https://mathoverflow.net/users/11142 | The sum of a series, continued | $$1/(2-1)\dots (2^{k-1}-1)+1/(2-1)\dots (2^k-1)=2^k/(2-1)\dots (2^k-1),$$
thus alternating sum of expression on the right is telescopical.
| 6 | https://mathoverflow.net/users/4312 | 217456 | 102,192 |
https://mathoverflow.net/questions/217466 | 1 | According to [this paper](https://www.math.uwaterloo.ca/~nwormald/papers/girth.pdf), the probability that a random $d$-regular graph of order $n$ has no cycles of length $c\_1,c\_2,\ldots,c\_t$ is $$P=\exp\left(-\sum\_{i=1}^t\mu\_i+o(1)\right)$$ as $n\rightarrow\infty$, where $\mu\_i=\frac{(d-1)^{c\_i}}{2c\_i}$. So it ... | https://mathoverflow.net/users/79906 | Probability of having no cycles of fixed length in $d$-regular graphs | I can't comment, but $P$ is the probability that there are NO cycles of those lengths. So this probability goes down. Your intuition holds.
| 5 | https://mathoverflow.net/users/76060 | 217468 | 102,195 |
https://mathoverflow.net/questions/216204 | 6 | This [paper](http://arxiv.org/pdf/math/0110227v7.pdf), "AF-algebras and topology of 3-manifolds" seems to claim on page 5 that a [transversely] oriented measured foliation on a compact surface is given by a closed form. On the other hand, the book ["Differential Geometry of Foliations: The Fundamental Integrability Pro... | https://mathoverflow.net/users/62647 | Are all transversely oriented, transversely measured foliations given by closed forms? | The claim is true and you find it on page 319 of Farb-Margalit's book.
The point is that the transition maps between foliation charts (with the leaves as y-level sets) are of the form $(x,y)\to (f(x,y),\pm y+c)$. (This is because the transverse measure has to be preserved. So the transverse measure seems essential fo... | 2 | https://mathoverflow.net/users/39082 | 217474 | 102,199 |
https://mathoverflow.net/questions/217444 | 3 | Let $(X,\tau)$ be a topological space. We topologize $\tau$ itself in the following way. For $K\subseteq X$ compact, we set $${\cal V}\_K=\{U\in \tau: U \supseteq K\}.$$ The *compact-open* topology on $\tau$ is the topology generated by $\{{\cal V}\_K: K\subseteq X\text{ is compact}\}.$ (The reason I call this topology... | https://mathoverflow.net/users/8628 | Intersection of compact sets in the compact-open topology | The answer is yes if $X$ is locally compact. Indeed, consider the topological space $X \times \mathcal{K}$ and let $V = \{(x,U) \in X \times \mathcal{K} | x \in U\}$. We claim that $V$ is open in $X \times \mathcal{K}$. To see this, take $(x,U) \in V$ and let $W$ be an open neighborhood of $x$ such that $\overline{W}$ ... | 4 | https://mathoverflow.net/users/51164 | 217475 | 102,200 |
https://mathoverflow.net/questions/212978 | 1 | I have the following paper:
Fritz John, [*The ultrahyperbolic differential equation with four independent variables*](https://projecteuclid.org/euclid.dmj/1077490637), Duke Math. J. 4 (1938), no. 2, pages 300-322
doi:[10.1215/S0012-7094-38-00423-5](https://doi.org/10.1215/S0012-7094-38-00423-5)
Now I want to che... | https://mathoverflow.net/users/13904 | A question on theorem 1.1 of Fritz John ultrahyperbolic pde | I don't have John's paper available at the moment, but are you sure
$$ | \xi - \eta| = \sum (\xi\_i - \eta\_i)^2 $$
and not
$$ | \xi - \eta|^2 = \sum (\xi\_i - \eta\_i)^2 ? $$
For one, I don't think John would define a "norm" with the wrong scaling, and for two, because if the latter holds you have that
$$ \... | 2 | https://mathoverflow.net/users/3948 | 217484 | 102,202 |
https://mathoverflow.net/questions/216171 | 2 | Consider a smooth curve $U\_s:[0,T] \rightarrow SU(4)$ which solves:
$\frac{d U\_s}{ds} = (a + w(s)b)U\_s$
for some given $a,b \in \mathfrak{su}(4)$ (which generate $\mathfrak{su}(n)$) and a smooth real, bounded function $w: [0,T] \rightarrow \mathbb{R}$ (T>0). Also consider some given $\xi \in \mathfrak{su}(4)$. $... | https://mathoverflow.net/users/41654 | Simultaneous integral equation on $SU(n)$ | Well, I think that, in principle, this is answerable, but it may be hard to come up with an explicit example. I don't have a proof, one way or the other, about what might happen or not, but here are some thoughts that you might find useful:
First, think of this as a problem with a number of parameters: $p = (a,b,\xi,... | 4 | https://mathoverflow.net/users/13972 | 217487 | 102,204 |
https://mathoverflow.net/questions/217488 | 5 | Suppose $X \rightarrow Y$ is a map of projective schemes over a field $k$. Is $\{y \in Y: \pi^{-1}(y) \text{ is irreducible}\}$ a constructible subset of $Y$?
Note: One cannot hope to do "better" than constructible. That is, if we let H be the hilbert scheme of conics in $\mathbb P^2$ and let $\mathscr C$ be the univ... | https://mathoverflow.net/users/75970 | Is the locus of points which have irreducible fibers constructible? | I double-checked. This is Théorème 4.10, p. 36 of the following.
MR0725671 (86b:13007) Reviewed
Jouanolou, Jean-Pierre
Théorèmes de Bertini et applications. (French)
Progress in Mathematics, 42. Birkhäuser Boston, Inc., Boston, MA, 1983. ii+127 pp.
ISBN: 0-8176-3164-X
As mentioned above, you need t... | 4 | https://mathoverflow.net/users/13265 | 217494 | 102,207 |
https://mathoverflow.net/questions/216282 | 2 | A Markov chain $(X\_i)\_{i\in \mathbb{N}}$ on a measurable space $(E,\Sigma)$ is (see e.g. Revuz or Meyn/Tweedie) constructed on the following probabilty space.
$$ \Omega = \{ (x\_l)\_{l \in \mathbb{N}} \mid x\_l \in E \text{ for all } l \in \mathbb{N} \}$$
and $X\_i$ is definied as
$$X\_i ((x\_l)\_{l \in \mathbb{N}})... | https://mathoverflow.net/users/73569 | General Markov Chains on same Probability Space? | The answer is yes: both processes can be constructed simultaneously on the same probability space. This is a standard fact (true in much greater generality - look up "product measure" and "Kolmogorov extension theorem" on Wikipedia and in any graduate-level textbook on probability theory).
To add a bit of detail, the... | 2 | https://mathoverflow.net/users/78525 | 217497 | 102,208 |
https://mathoverflow.net/questions/217492 | 3 | I have a bivariate integer function where x,y are positive integers in the function $f(x,y)=5+23x+7y+30xy$. The lattice points of this function, or its range, contain a large number of values. I'm trying to see if there is any way to create a positive, increasing, infinite sequence or function which avoids the range of... | https://mathoverflow.net/users/78626 | Avoiding the range of a bivariate function or Diophantine function | By [Dirichlet's theorem](https://en.wikipedia.org/wiki/Dirichlet's_theorem_on_arithmetic_progressions), there are infinitely many positive integers $k$ such that $30k+11$ is a prime. I claim that no such $k$ is in the range of your function. Indeed, if we had $k=5+23x+7y+30xy$ for some positive integers $x$ and $y$, th... | 4 | https://mathoverflow.net/users/11919 | 217503 | 102,209 |
https://mathoverflow.net/questions/216298 | 5 | A Penrose diagram (also known as a conformal diagram or Carter-Penrose diagram) is a technique for visualizing the causal (light-cone) structure of a 3+1-dimensional manifold. Usually the diagram is two-dimensional, but it is also possible to have more dimensions. My question is whether there are definite conditions fo... | https://mathoverflow.net/users/nan | Conditions for existence of Penrose diagrams | There is no definitive answer to your specific question, so I'm going to talk around the topic and hope that it's informative.
As you've noted the classical examples all basically look like special cases. This isn't helped by, pretty much all, books and articles using the same examples. There are a few common threads... | 4 | https://mathoverflow.net/users/5993 | 217517 | 102,211 |
https://mathoverflow.net/questions/217461 | 4 | Let six points $A, A', B, B', C, C'$ lie on a conic and a cubic. Let a conic through $B, B', C, C'$ and meets the cubic again at $A\_1, A\_2$. Let a conic through $C, C', A, A'$ and meets the cubic again at $B\_1, B\_2$. Let a conic through $A, A', B, B'$ and meets the cubic again at $C\_1, C\_2$. Then six points $A\_1... | https://mathoverflow.net/users/76698 | A conjecture like Cayley–Bacharach theorem | If the cubic is smooth and given the usual group law with an inflexion as identity, the condition that given six points $P\_1,\ldots,P\_6$ on the cubic they also lie on a conic is $P\_1+\cdots+P\_6=0$. In your case
$$A+A'+B+B'+C+C'=0$$
$$A\_1+A\_2+B+B'+C+C'=0$$
$$A+A'+B\_1+B\_2+C+C'=0$$
$$A+A'+B+B'+C\_1+C\_2=0$$
Addin... | 6 | https://mathoverflow.net/users/2290 | 217522 | 102,213 |
https://mathoverflow.net/questions/217524 | 2 | Assuming Riemann Hypothesis Hardy showed primes $3\bmod 4$ are more common than primes $1\bmod 4$ <https://en.wikipedia.org/wiki/Riemann_hypothesis#Consequences_of_the_generalized_Riemann_hypothesis>.
Is there any reason to believe there are asymptotically equal number of consecutive primes that are $1\bmod 4$ and $... | https://mathoverflow.net/users/nan | Finiteness of number of consecutive primes with gap $4$ | The answer ought to be 'no', that one should not expect there to only be finitely many prime pairs of the form $(4k+1, 4k+5)$ while expecting infinitely many pairs of the form $(4k+3, 4k+7)$. While we cannot prove this, it is known that weaker version of this is true.
Consider a $k$-tuple of linear functions $(L\_1(n... | 1 | https://mathoverflow.net/users/10898 | 217528 | 102,214 |
https://mathoverflow.net/questions/217483 | 3 | Brownian motion $g\_t$ on a compact Lie group satisfies the stochastic differential equation
$$dg\_t = dB\_t \circ g\_t$$
where $B\_t$ is Brownian motion on the Lie algebra and $\circ$ denotes differential in the sense of Stratonovich. We take $g\_0 = 1$. (We assume the Lie group and its Lie algebra are embedded in... | https://mathoverflow.net/users/58478 | Brownian bridge on a Lie group as a stochastic differential equation | There is a general formula for the infinitesimal generator of the Brownian bridge on an arbitrary Riemannian manifold, and I do not see how it would become simpler in your particular case. For instance, see formula (2.2) on p.105 of
MR1027823 (90m:58227) Hsu, Pei(1-ILCC) Brownian bridges on Riemannian manifolds.
Pr... | 1 | https://mathoverflow.net/users/8588 | 217537 | 102,217 |
https://mathoverflow.net/questions/217539 | 1 | Let $M$ be a compact Riemannian manifold and $V=L^2(M)$. Let $\Delta$ be the negative-definite Laplacian. Let $f \in V$ and $x \in M$ be arbitrary, but fixed.
Is it true that ${\rm Re} \ (\Delta f) (x) \overline {f (x)} \le 0$? (This is the result of a chain of calculations that is not relevant to this question. $\rm... | https://mathoverflow.net/users/54780 | Negativity of a quadratic form on $L^2(M)$ | It's not true in this pointwise sense. Consider $M = S^1$ seen as $[0,2\pi]$ with endpoints identified, so that $\Delta f = f''$. Let $f(x) = \sin(x) + 2$. Then $\bar{f} \Delta f > 0$ everywhere on $(\pi, 2\pi)$. (Notice that $f$ is a sum of two eigenfunctions of $\Delta$, with different eigenvalues.)
(In general, th... | 3 | https://mathoverflow.net/users/4832 | 217543 | 102,218 |
https://mathoverflow.net/questions/217542 | 11 | Is it consistent that there are regular cardinals $\kappa < \lambda$, such that $\lambda$ is a successor cardinal and for every coloring $d\colon[\lambda]^2\to\kappa$ there is some $A\subseteq\lambda$ of cardinality $\lambda$ and $\eta < \kappa$ such that $\forall x,y\in A$, $d(x,y)< \eta$?
| https://mathoverflow.net/users/41953 | Partition relation at successor cardinal | If $\lambda\nrightarrow[\lambda]^2\_\lambda$ then there is no such $\kappa<\lambda$: modify the coloring that witnesses this by defining it to be $0$ on pairs of ordinals sent to values above $\kappa$. (The original coloring takes on all values on any unbounded subset of $\lambda$, hence the modified coloring takes on ... | 11 | https://mathoverflow.net/users/18128 | 217551 | 102,221 |
https://mathoverflow.net/questions/214936 | 3 | I am studying the construction of the motivic stable homotopy category of schemes $\mathbf{SH}(S)$ following Riou's paper [Categorie homotopiquement stable d'un site suspendu avec intervalle](http://www.math.u-psud.fr/~riou/doc/sh.pdf) (click to see the paper) and I have a problem due to my weakness in homotopy theory.... | https://mathoverflow.net/users/12204 | Triangulated structure on $\mathbf{SH}(S)$: $\mathbb{P}^1$-suspension versus classical suspension | This question is answered by Hoyois' first comment: suspension is always $\underline{\phantom{a}}\wedge S^1$. One should write $\mathbb{P}^1$-suspension otherwise.
| 2 | https://mathoverflow.net/users/12204 | 217568 | 102,223 |
https://mathoverflow.net/questions/215989 | 7 | The following might be very well known for people who works with model categories, but I do not find the answer.
Let $A$-be a ring. Denote $\mathbf{Ch}\_+(A)$ the category of positive degree cochain complexes of $A$-modules (complexes where differential rises the degree and which are zero in negative degree). There i... | https://mathoverflow.net/users/12204 | Injective model structure on sheaves of bounded complexes of $A$-modules | David White's comment led me to adequate references to answer this question. Thank you very much
**Question:** *Is there an analogue model structure in $\mathbf{Ch}\_+(\mathbf{Shv}\_X(A))$ to the injective model structure on $\mathbf{Ch}\_+(A)$?* More concretely, if we consider quasi-isomorphisms as weak equivalences... | 4 | https://mathoverflow.net/users/12204 | 217569 | 102,224 |
https://mathoverflow.net/questions/217566 | 0 | Assume that for some smooth bounded open subset $\Omega$ in $\mathbb{R}^n$ and some $u\in H^{1,2}(\mathbb{R}^n)$ we know that $\chi\_{\Omega} u\in H^{1,2}(\mathbb{R}^n)$. Is it then true that $u\in H^{1,2}\_{0}(\Omega)$? Here $\chi\_{\Omega}$ denotes the characteristic function which is one on $\Omega$ and zero elsewhe... | https://mathoverflow.net/users/79956 | Characterization of Sobolev spaces | It is true that
$$
H\_0^{1,2}(\Omega)= \bigl\{u\bigr|\_\Omega\bigm| u\in H^{1,2}(\mathbb R^n),\, \operatorname{supp}u\subseteq\overline\Omega\bigr\}.
$$
Indeed, the inclusion $\subseteq$ is obvious. To see the inclusion $\supseteq$, notice that, for $u\in H^{1,2}(\mathbb R^n)$, taking the trace $u\bigr|\_{\partial\Ome... | 2 | https://mathoverflow.net/users/69194 | 217571 | 102,225 |
https://mathoverflow.net/questions/217565 | 32 | I am looking for a mathematical precise introductory book on general relativity. Such a reference request has already been posted in the physics stackexchange [here](https://physics.stackexchange.com/questions/363/getting-started-self-studying-general-relativity). However, I'm not sure whether some physicists know what... | https://mathoverflow.net/users/27923 | Modern mathematical books on general relativity | [Curvature in Mathematics and Physics](http://store.doverpublications.com/0486478556.html) (2012), by Shlomo Sternberg, based on an earlier book
*Semi-Riemann Geometry and General Relativity* [free [download](http://www.math.harvard.edu/~shlomo/docs/semi_riemannian_geometry.pdf) from the author's website] covers much o... | 12 | https://mathoverflow.net/users/11260 | 217576 | 102,228 |
https://mathoverflow.net/questions/217580 | 3 | Consider a compact manifold $M$ with boundary and corner. As an example, we could have the cube $\{(x\_1, x\_2,..x\_n) \in \mathbb{R}^n : x\_i \in [0,1]\}$. We could very well define the Laplacian $\Delta$ on such an $M$ with Dirichlet or Neumann boundary conditions. In such situations, are the eigenfunctions of the La... | https://mathoverflow.net/users/79963 | Eigenfunctions of the Laplacian on singular spaces | There is an extensive literature on elliptic problems on domains with corners. Grisvard's book (Elliptic Problems in Nonsmooth Domains) is a good place to start.
| 4 | https://mathoverflow.net/users/12120 | 217586 | 102,230 |
https://mathoverflow.net/questions/217557 | 3 | Let A be a finite subset of $\mathbb{N}$, $\mathbb{R}$, or a sufficiently small subset of $\mathbb{F}\_{p}$.
Do we have a lower bound of the form $|A|^{1+\delta}$ on the following quantity:
$$\max (|\{a+b : a,b \in A\}|, |\{a^2+b^2 : a,b \in A\}| ) ? $$
In other words, is either the sumset of $A$ or the sumset of... | https://mathoverflow.net/users/630 | Is the sumset or the sumset of the square set always large? | A more general result than what you want appears as Theorem 1 in <http://arxiv.org/pdf/1002.2554>. (A slightly weaker result had appeared before as Theorem 3.1 in <http://arxiv.org/pdf/0909.5471>).
Curiously, it is open if at least one of $A^2+A^2$ and $A^3+A^3$ is necessarily large.
| 6 | https://mathoverflow.net/users/806 | 217590 | 102,233 |
https://mathoverflow.net/questions/217594 | 2 | Let $\mathbb{N}$ be the set of all positive integers and let $P(n),Q(n)$ be a pair of general recursive mappings of $\mathbb{N}$ into itself such that for all pairs $h,k$ of distinct positive integers, the absolute value of $(P(k)/(Q(k))-(P(h)/Q(h))$ does not exceed $(1/k)+(1/h).$
We have defined a general recursive... | https://mathoverflow.net/users/4423 | Is every computable real primitively recursively computable? | No. Counterexample: consider a real number $0<x<1$ whose binary expansion is
$$0.x\_1x\_2\dots$$
where
$$x\_{\langle i,j\rangle}=f\_i(j)$$where $f\_i$ is the $i$th function in a fixed computable list of the primitive recursive $\{0,1\}$-valued functions.
If $P$ and $Q$ for this $x$ are primitive recursive then we sho... | 6 | https://mathoverflow.net/users/4600 | 217595 | 102,234 |
https://mathoverflow.net/questions/217426 | 1 | A major argument against Freiling's Axiom of Symmetry is the following (this from the wikipedia article of the same name):
"The naive probabalistic notion used by Freiling tacitly assumes that there is a well-behaved way to associate a probability to any subset of the reals. But the mathematical formalization of the ... | https://mathoverflow.net/users/20597 | A question regarding the relation between Freiling's Axiom of Symmetry and real-valued measurable cardnals | Real-valued measurable cardinals (RVM) are equiconsistent with 2-valued measurable cardinals. I believe this is due to Solovay and Kunen. (Solovay for the forcing direction, Kunen for the inner model $L[U]$.) They contradict CH, so their existence is not implied by standard large cardinals.
I claim that Freiling's ar... | 11 | https://mathoverflow.net/users/11145 | 217601 | 102,237 |
https://mathoverflow.net/questions/217592 | 15 | Long story short, I personally find Macdonald's [celebrated book](http://rads.stackoverflow.com/amzn/click/0198504500) *Symmetric Functions and Hall Polynomials* somewhat difficult to read for various reasons. I also know for a fact that I'm not the only one to hold that opinion =)
On the other hand, Macdonald polyno... | https://mathoverflow.net/users/19864 | An introduction to Macdonald polynomials other (better?!) than SFHP | I recommend reading Macdonald's volume University Lecture Series Vol 12 *Symmetric functions and orthogonal polynomials*.
It is a rather short introduction to Macdonald polynomials for the symmetric group and for general finite Weyl groups, but everything is very well explained, organized, and self-contained. After r... | 10 | https://mathoverflow.net/users/21291 | 217603 | 102,239 |
https://mathoverflow.net/questions/217582 | 4 | The compound Poisson distribution is defined as(see [Levy processes and infinitely divisible distributions](http://rads.stackoverflow.com/amzn/click/1107656494) page: 18):
Let $c>0$ and $\sigma$ be a measure on $\mathbb{R}$ with $\sigma(\{0\})=0$, a compound distribution $\mu$ on $\mathbb{R}$ is called a *compound Po... | https://mathoverflow.net/users/79962 | The Levy measure of the compound Poisson distribution | This requirement is not necessary, but convenient.
The measure $\sigma$ is the distribution of jumps. If $\sigma(\{0\})>0$, then the jump is $0$ with positive probability, meaning that there is in fact no jump.
With the restriction $\sigma(\{0\})=0$, the parameters $\sigma$ and $c$ are uniquely determined by the ... | 3 | https://mathoverflow.net/users/8146 | 217604 | 102,240 |
https://mathoverflow.net/questions/215996 | 10 | My question is partially inspired by the following discussion:
[Topology on the Unitary Dual](https://mathoverflow.net/questions/134312/topology-on-the-unitary-dual)
Let me remind/explain how the Fell topology is defined (at least I recall the definition which I saw): let $G$ be a locally compact group. One first... | https://mathoverflow.net/users/24078 | Fell topology vs. convergence of matrix coefficients | In your inequality you need to evaluate the $\pi$'s somewhere!
Unless I am mistaken unpacking [Fell (1962), Theorem 2.2 and Remark following](http://www.ams.org/mathscinet-getitem?mr=150241), $[\pi\_j]\to[\pi]$ means that for every choice of an $\varepsilon>0$, a compact set $K\subset G$, an integer $n$, and vectors $... | 5 | https://mathoverflow.net/users/19276 | 217606 | 102,241 |
https://mathoverflow.net/questions/217439 | 4 | Recall that the Hilbert space version of von Neumann's mean ergodic theorem says the following.
Let $\{F\_n\}\_{n=1}^\infty$ be a right Følner sequence of a countable discrete amenable group $\Gamma$ and $\pi:\Gamma\to B(H)$ be a unitary representation of $\Gamma$ on a Hilbert space $H$. Then
$$\lim\_{n\to\infty}\fra... | https://mathoverflow.net/users/7360 | The converse of von Neumann's mean ergodic theorem | If $\Gamma$ is a property (T) group with infinite conjugacy classes, e.g., $\Gamma = PSL\_3(\mathbb Z)$, then $\Gamma$ will have such a sequence. The ``spectral gap'' criterion for property (T) shows that there exists $c > 0$ and $F \subset \Gamma$ finite such that for any unitary representation $\pi$ we have $\left\| ... | 4 | https://mathoverflow.net/users/6460 | 217616 | 102,247 |
https://mathoverflow.net/questions/201817 | 11 | If I understand correctly, $n$-th arithmetic Chow group of arithmetic variety $X$ is defined as a quotient of the group of pairs of the form $(\sum\limits\_in\_iZ\_i, g)$ where $Z := \sum\limits\_in\_iZ\_i$ is an algebraic $n$-cycle in X, and $g$ is an $n$-current on $X(\mathbb{C})$ satisfying the equation $\partial \b... | https://mathoverflow.net/users/62647 | Equivalence of various definitions of arithmetic Chow groups | I will start discussing the relationship between first and third definition.
The key point to understand the different definitions of arithmetic Chow groups is to understand the equation $\partial \bar \partial g-\delta\_Z=\omega$. In this equation appears the second order differential operator $\partial \bar \partia... | 13 | https://mathoverflow.net/users/79986 | 217625 | 102,252 |
https://mathoverflow.net/questions/217585 | 2 | This is a follow up question to the insightful answer by Theo Johnson-Freyd to the question [105221](https://mathoverflow.net/questions/105221/quantum-groups-not-via-presentations).
This answer explained that the quantised enveloping algebra, $U\_q(\mathfrak{g})$, (defined by a presentation due to Jimbo) could not be... | https://mathoverflow.net/users/50658 | Deforming the category of representations of a simple Lie algebra? | I haven't thought about this stuff in a while, but here's what I can remember. (Perhaps I should just read [my notes for a talk on the subject](http://www.math.northwestern.edu/~theojf/CatQuantTalk.pdf) instead.) Before I begin, let me emphasize that nothing I'm about to say is due to me --- much of it dates from my ma... | 2 | https://mathoverflow.net/users/78 | 217626 | 102,253 |
https://mathoverflow.net/questions/217545 | 6 | Take the familiar Universal Coefficient Theorem for ordinary homology with $\mathbb{Z}$-coefficients and ordinary cohomology with coefficients in some abelian group $A$:$$0\rightarrow \text{Ext}\_\mathbb{Z}^1(H\_{i-1}(X;\mathbb{Z}),A)\rightarrow H^i(X;A)\rightarrow \text{Hom}\_\mathbb{Z}(H\_i(X;\mathbb{Z}),A)\rightarro... | https://mathoverflow.net/users/51107 | generalized universal coefficient sequence | There is more than one possible generalization. The most common is the *universal coefficient spectral sequence*. Given a (homotopy) commutative ring spectrum $E$ and a spectrum $X$, there is under certain conditions a spectral sequence
$$ Ext^{p,q}\_{E\_\*}(E\_\*(X), E\_\*) \Rightarrow E^{q-p}(X) $$
This is true for e... | 11 | https://mathoverflow.net/users/2039 | 217639 | 102,258 |
https://mathoverflow.net/questions/217598 | 18 | Let $X$ be a smooth projective rationally connected variety over $\mathbb C$. Let $C$ be a curve class in $X$ such that rational curves equivalent to $C$ connect every two points.
Let $f: Y \to X$ be a finite morphism from a normal variety $Y$ to $X$, generically of degree $d$. Let $D$ be the branch divisor of $f$. (... | https://mathoverflow.net/users/18060 | Are rationally connected varieties robustly simply connected? | Denote the curve class by $\beta$. Let $M$ be the normalization of a closed subvariety of $\overline{\mathcal{M}}\_{0,2}(X,\beta)$ such that the restricted evaluation morphism, $$\text{ev}|\_M = (\epsilon\_1,\epsilon\_2):M\to X\times X,$$ is surjective and generically finite. Denote by $m$ the degree of $\text{ev}|\_M$... | 12 | https://mathoverflow.net/users/13265 | 217652 | 102,263 |
https://mathoverflow.net/questions/217651 | 3 | All rings considered are commutative with $1$. In my study of arbitrary chains of prime ideals of a ring $R$, I have noticed two possiblities for the cardinality of a chain when $\dim(R)=\infty$:
1. All chains of primes are finite, but their lengths are unbounded.
2. R contains an uncountable chain of prime ideals.
... | https://mathoverflow.net/users/69591 | Question regarding arbitrary chains of prime ideals | See [Example Tag 01IY](http://stacks.math.columbia.edu/tag/01IY) of the Stacks project.
| 3 | https://mathoverflow.net/users/80000 | 217657 | 102,264 |
https://mathoverflow.net/questions/217642 | 2 | I want to learn Multi dimensional symbolic dynamics. can you point to any recent thesis containing a good exposition or lecture notes?
| https://mathoverflow.net/users/75682 | Multi dimensional symbolic dynamics | Klaus Schmidt and I wrote a survey about multi-dimensional dynamics in 2002, which you can find at: *Symbolic and algebraic dynamical systems*, Handbook of dynamical systems, Vol. 1A, 765–812, North-Holland, Amsterdam, 2002. A lot has happened since, but it would be a good starting point.
| 4 | https://mathoverflow.net/users/8112 | 217666 | 102,269 |
https://mathoverflow.net/questions/217596 | 2 | We know that for if $X$ is a stochastic integral of the form below -
$X\_t = \int\_0^t v(s,\omega) db(s,\omega)$.
then we can use time change formula to claim that
$X\_t = W\_{\alpha(t)}$ where $W$ is a different brownian motion with a changed clock.
This holds true, under mild conditions, even when $b$ is a d-d... | https://mathoverflow.net/users/78761 | Time Change of a Brownian motion | I think you're right and the new clock $\alpha\_t$ is proportional to $\int\_0^t |v(s,X\_s)|^2\ ds$, if the $b\_i$ are i.i.d.. It needs a proof, but to my intuition the result is clear enough: a stochastic integral $\int\_0^t \lambda\_s\cdot db\_s$, where the vector-valued process $\lambda\_t$ is adapted (continuous, e... | 1 | https://mathoverflow.net/users/75422 | 217668 | 102,271 |
https://mathoverflow.net/questions/217546 | 7 | I wanted to understand or at least to know if what follows make sense.
Given a connected toplogical space $X$, I want to associate a scheme. In the following way.
For a space $X$ and $A(X)$ the Sullivan minimal model, first I associate the differential graded commutative ring $A(X)$. Then I associate to $A(X)$ the c... | https://mathoverflow.net/users/79949 | Schematization of a topological space | If $X$ has mild finiteness conditions (its even-dimensional cohomology is noetherian and its odd-dimensional cohomology is finitely generated as a module over it), then it is possible to completely determine the spectrum (as defined by Balmer) of the category of (perfect) modules over the cochain algebra $C^\*(X; \math... | 8 | https://mathoverflow.net/users/344 | 217670 | 102,272 |
https://mathoverflow.net/questions/217674 | 8 | Is there a group $G$ and subgroups $H\_1, \dots, H\_n \leq G$ for some $n \in \mathbb{N}$, such that $[G : H\_i] = \infty$ for each $1 \leq i \leq n$, and $$G = \bigcup\_{i=1}^n H\_i \ \ ?$$
| https://mathoverflow.net/users/38889 | Can a group be a union of finitely many subgroups of infinite index? | It may well be a lot easier than this, but it follows from the answer to the weaker question [Can a group be a finite union of (left) cosets of infinite-index subgroups?](https://mathoverflow.net/questions/17396/can-a-group-be-a-finite-union-of-left-cosets-of-infinite-index-subgroups?rq=1) that it's not possible.
| 12 | https://mathoverflow.net/users/22989 | 217679 | 102,275 |
https://mathoverflow.net/questions/217680 | 2 | Suppose we're given four positive integers $a$, $b$, $c$, $d$ such that $a$ and $b$ are coprime, and $c$ and $d$ are coprime. Is there a non-negative integer $k$ such that both $ak+b$ and $ck+d$ are prime numbers? How about a special case when $b=d=1$?
This seems like something that should hold, some kind of a genera... | https://mathoverflow.net/users/1716 | Primes in simultaneous arithmetic progressions | This is a special case of [Dickson's conjecture](https://en.wikipedia.org/wiki/Dickson's_conjecture) (for suitable $a,b,c,d$, cf. Wojowu's comment below your post), and as Felipe Voloch remarked, it is an open problem. There have been some striking recent advances towards this conjecture, check out the work [Green-Tao]... | 4 | https://mathoverflow.net/users/11919 | 217682 | 102,276 |
https://mathoverflow.net/questions/198851 | 2 | In a 1998 paper by Hjorth, Kechris, and Louveau, there was a definition given of a "potentiality class." That is, given an invariant equivalence relation $E$ on a standard Borel space $X$, we say $E$ is (for example) *potentially $\Pi^0\_3$* if, for some Polish topology on $X$ yielding the same Borel sets, $E$ is a $\P... | https://mathoverflow.net/users/15735 | Potentiality classes and Borel reductions | It is not necessarily true that Borel equivalence relations that are potentially in the same pointclass are Borel bireducible. For example, consider the orbit equivalence relations of the logic action of $S\_{\infty}$ on the standard Borel space of torsion-free abelian groups of rank $n$. Then these orbit equivalence r... | 2 | https://mathoverflow.net/users/33039 | 217684 | 102,277 |
https://mathoverflow.net/questions/217476 | 8 | Definitions
-----------
Let $A$ be an Artin algebra (for instance, take $A$ to be a finite dimensional algebra over some field) and label the isomorphism classes of simple $A$-modules by the elements of a partially ordered set $(\Lambda, \leq)$.
Denote the simple $A$-modules (up to isomorphism) by $L\_{\lambda}$, ... | https://mathoverflow.net/users/36805 | Jordan-Hölder-like statements for modules with $\Delta$-filtrations over a quasihereditary algebra | Let $M$ be a module with a $\Delta$-filtration. Let $N$ be a $\Delta$-semisimple submodule of M such that $M/N$ still has a $\Delta$-filtration -- I prove that there is a unique maximal module with this property (and hope that my justification is not immensely stupid).
This will be a consequence of the following Lemm... | 2 | https://mathoverflow.net/users/36805 | 217686 | 102,278 |
https://mathoverflow.net/questions/217672 | 4 | Let $p$ be a large prime. I would like to say that the multi-set $[1,p^{1-\varepsilon}]^2 = \{ab \mod p: a, b \in [1,p^{1-\varepsilon}]\}$ is close to uniformly distributed, i.e. that every nonzero residue class mod $p$ occurs with almost equal frequency.
Quantitatively, is it true that (almost?) every nonzero resid... | https://mathoverflow.net/users/70060 | Well-distribution of square of an interval $[1,p^{1-\varepsilon}]$ modulo a prime $p$ | Here is a more explicit version of Noam Elkies's comment.
**Theorem.** Let $p>2$ be a prime number. Let $\mathcal{U},\mathcal{V}\subseteq\{1,2,\dots,p-1\}$ be two intervals, and let $r\in\{1,2,\dots,p-1\}$
be a nonzero residue modulo $p$. Then
$$\Biggl|\sum\_{\substack{u\in\mathcal{U},\ v\in\mathcal{V} \\ uv\equiv r ... | 5 | https://mathoverflow.net/users/11919 | 217691 | 102,280 |
https://mathoverflow.net/questions/217698 | 16 | Let $r\_3(n) = \left|\{(a,b,c)\in {\mathbb Z}^3 :\, a^2+b^2+c^2=n \}\right|$. I am looking for the maximum asymptotic size of $r\_3(n)$. That is, the maximum number of representations that a number can have as a sum of three squares.
Gauss proved that $r\_3(n) = \frac{A\sqrt{n}}{\pi}\sum\_{m-1}^\infty\left(\frac{-n}{... | https://mathoverflow.net/users/17509 | Many representations as a sum of three squares | The formula for $r\_3(n)$ essentially connects this with a class number of an imaginary quadratic field, or (apart from the $\sqrt{n}$ scaling) with the value of an $L$-function at $1$. So your question may be reformulated as asking how large can $L(1,\chi\_{d})$ be as $d$ runs over negative fundamental discriminants (... | 19 | https://mathoverflow.net/users/38624 | 217699 | 102,284 |
https://mathoverflow.net/questions/217509 | 6 | Let $S^m$ be the $m$-sphere and $$F(S^m,2)/\mathbb{Z}\_2=\{(a,b)\mid a,b\in S^m, a\neq b\}/(a,b)\sim (b,a)$$ be the $2$-nd unordered configuration space on $S^m$. How to compute the total Stiefel-Whitney class of the tangent bundle of $F(S^m,2)/\mathbb{Z}\_2$
$$
w(TF(S^m,2)/\mathbb{Z}\_2)?
$$
Is $F(S^m,2)/\mathbb{Z}\... | https://mathoverflow.net/users/65800 | Stiefel-Whitney class of unordered configuration space | The configuration space $F(S^m, 2)/\mathbb Z\_2$ is diffeomorphic to the total space of the $m$-dimensional vector bundle $E=\gamma^\perp$ over $\mathbb RP^m$, where $\gamma\subset \mathbb R^{m+1}$ is the tautological 1-dimensional line bundle and $\gamma^\perp$ is its orthogonal complement.
To define a diffeomorphis... | 4 | https://mathoverflow.net/users/40950 | 217702 | 102,286 |
https://mathoverflow.net/questions/215196 | 21 | The Jones polynomial can be computed from the representation theory of $\mathcal{U}\_q(\mathfrak{sl}(2))$. The Alexander polynomial has an analogous description in terms of the representation theory of the superalgebra $\mathcal{U}\_q(\mathfrak{gl}(1|1))$ (see, for example, Sartori <http://arxiv.org/abs/1308.2047>).
... | https://mathoverflow.net/users/8041 | Why, in terms of quantum groups, does the knot determinant appear as an evaluation of both the Jones and Alexander polynomials? | You haven't specified what sorts of explanations you would find "nice," so the following collection of thoughts might not be helpful, but here goes (or it might be helpful, but contain only things you already know). [By the way, this answer only attempts to "explain" why $V\_K(i) = \Delta(-1)$ -- it does not address th... | 2 | https://mathoverflow.net/users/492 | 217706 | 102,287 |
https://mathoverflow.net/questions/217675 | 3 | Fix a discount rate $r>0$, and let $m,v,f:\mathbb{R} \rightarrow \mathbb{R}$ be bounded measurable functions of locally bounded variation, with $v$ globally bounded below by some strictly positive constant i.e. $v(x) > c > 0$
Given a standard Brownian motion B and a starting value x, I know the SDE
$dX\_t = m(X\_t) ... | https://mathoverflow.net/users/78761 | Continuity of expected payoff from a diffusion | This is just a partial answer addressing the continuity: $U$ is continuous, even uniformly continuous. Below is the idea.
Denote $\tau\_x^y = \inf\{ t\ge 0: X\_t = y\,|\, X\_0 = x\}$, then $\tau\_x^y<\infty$ a.s. in view of the uniform non-degeneracy of $v$ and boundedness of $m$, so, using the strong Markov property... | 2 | https://mathoverflow.net/users/8146 | 217714 | 102,289 |
https://mathoverflow.net/questions/217687 | 6 | I previously put this forward on the math.stackexchange community with little luck:
I am looking for a comprehensive account of the properties and applications of the Barnes G-Function. Everything from recurrence relations, proof of the infinite product representation, functional relations, e.t.c. Ideally with proof... | https://mathoverflow.net/users/78870 | Modern comprehensive account of the Barnes G-Function | Perhaps one of the most recent overviews of the Barnes G-function (with some new results) is [Contributions to the Theory of the Barnes Function](http://arxiv.org/abs/math/0308086) (2003). It is not a self-contained treatise, but it does contain many pointers to the literature.
| 5 | https://mathoverflow.net/users/11260 | 217717 | 102,291 |
https://mathoverflow.net/questions/217730 | 34 | For a year I have been giving lectures on a (probalby) new way to present an explicit sphere eversion. These lectures include a review of many other explicit eversions that have been described, as text, drawings or even computer generated movies, since Smale proved his theorem.
During these lectures, I have been freque... | https://mathoverflow.net/users/58307 | Classifiying sphere eversions | Answer Summary
--------------
The fundamental group of the space of immersions of $S^2$ into $\mathbb{R}^3$ is
$$ \pi\_1 Im(S^2, \mathbb{R}^3) \cong \mathbb{Z}/2 \times \mathbb{Z}$$
This means that there are infinitely many different sphere eversions (where "different" mean not homotopic in the path space of immers... | 40 | https://mathoverflow.net/users/184 | 217734 | 102,295 |
https://mathoverflow.net/questions/217710 | 8 | It is well-known that the Littlewood-Richardson coefficient $c^{\nu}\_{\lambda \mu}$ is the number of times the irreducible representation $V\_\lambda \bigotimes V\_\mu$ of the product of symmetric groups $S\_{|\lambda|} × S\_{|\mu|}$ appears in the restriction of the representation $V\_{\nu}$ of $S\_{|\nu|}$ to $S\_{|... | https://mathoverflow.net/users/32968 | Littlewood-Richardson-Type Rule for Restriction from $S_{2n}$ to $S_{2(n-t)} \times (S_2 \wr S_t)$ | For the trivial representation the problem is much simplified. Let $1\_G$ denote the trivial character of a group $G$. It is well known that $1\_{S\_2 \wr S\_t}\!\!\uparrow^{S\_{2t}} = \sum\_{\mu \vdash t} \chi^{2\mu}$ where $2\mu$ is the partition obtained from $\mu$ by doubling the length of each part.
By two appl... | 6 | https://mathoverflow.net/users/7709 | 217739 | 102,298 |
https://mathoverflow.net/questions/217724 | 5 | I would like to know how to cite this theorem (which has a quite surprising consequence):
>
> A model category $\mathcal{M}$ is right proper if and only if for any
> weak equivalence $f:A\to B$, the Quillen adjunction
> $\Sigma\_f:\mathcal{M}/ A \leftrightarrows \mathcal{M} / B:f^\*$ is a Quillen equivalence.
>
... | https://mathoverflow.net/users/24563 | Characterization of right properness using slice categories | This is Proposition 2.7 in Rezk's [*Every Homotopy Theory of Simplicial Algebras Admits a Proper Model*](http://arxiv.org/abs/math/0003065v1)
| 4 | https://mathoverflow.net/users/11540 | 217748 | 102,300 |
https://mathoverflow.net/questions/217732 | 1 | Given a hermitian, but not necessarily positive, sparse matrix $C = (c\_{ij}) \in \mathbb{C}^{n \times n}$ and $n \ggg 1$ ($n \approx 2^{100}$) with eigenvalues $\lambda\_1 \le \lambda\_2 \le \dots \le \lambda\_n$.
Is there an upper bound for
$\mathrm{tr} \sqrt{C^\dagger C} = \sum\_i |\lambda\_i| $? Preferable in te... | https://mathoverflow.net/users/80030 | Upper bound for sum of absolute values of eigenvalues of Hermitian matrix | Set $$H = \left( \begin{matrix} 1 & 1 \\ 1 & -1 \end{matrix} \right).$$
Then, the matrix $H\_k := H^{\otimes k}$ is a $2^k \times 2^k$-matrix with entries $\pm 1$ (with respect to the natural basis). Set $n:=2^k$. It is easy to see that $H\_k^2=n$.
It follows that the sum of absolute values of eigenvalues of $H\_... | 1 | https://mathoverflow.net/users/8176 | 217751 | 102,301 |
https://mathoverflow.net/questions/217727 | 1 | Let $X, Y$ be topological spaces and $f,g: X\to Y$ continuous. Then we say that $f, g$ are *similar* if for all $V\subseteq Y$ open we have either
* $f^{-1}(V) = g^{-1}(V) = \emptyset$, or
* $f^{-1}(V) \cap g^{-1}(V) \neq \emptyset$.
Note that this relation is reflexive and symmetric (but not necessarily transitive... | https://mathoverflow.net/users/8628 | Similarity graph for continuous maps between Hausdorff spaces | Fix a graph $G=(V,E)$ where $E \subset [V]^2$.
Give the discrete topology to both $X:=V \cup E \cup \{\infty\}$ and $Y:=2$. For each $v \in V$ let $f\_v:X \to Y$ be the (automatically continuous) map defined by $f\_v^{-1}(1)=\{v\}\cup\{e\in E: v\in e\}$.
The graph induced in $\{f\_v :v\in V\}\subseteq \mathrm{Hom}... | 1 | https://mathoverflow.net/users/17836 | 217753 | 102,302 |
https://mathoverflow.net/questions/217750 | 2 | Let $\mathrm{r}\mathscr{O}$ be the family of open domains (regular open sets) of a topological space $\langle X,\mathscr{O}\rangle$, that is:
$$A\in\mathrm{r}\mathscr{O}\iff A=\mathrm{int(\mathrm{cl(A)})}.$$
Define $A$ to be *well inside* $B$:
$$\tag{df $\Subset$}
A\Subset B\iff \mathrm{cl}(A)\subseteq B,
$$
Call a... | https://mathoverflow.net/users/22019 | Special filters in the algebra of regular open sets of a topological space | The answer is no. Consider the case of a discrete space $X$, where every set is open and hence also regular open. In this case, the well-inside relation coincides with the ordinary inclusion relation. Your question asks whether the filter generated by an inclusion chain in $P(X)$ is maximal, and the answer is that this... | 4 | https://mathoverflow.net/users/1946 | 217758 | 102,303 |
https://mathoverflow.net/questions/217731 | 3 | I've recently been learning about the special values of symmetric square $L$-functions of modular forms.
If $f$ is a cuspidal modular eigenform (of some weight $k \ge 2$) then its symmetric square $L$-function is the $L$-function given by $\prod\_{\text{$\ell$ prime}} D\_\ell(\ell^{-s})^{-1}$, where for almost all $\... | https://mathoverflow.net/users/2481 | Which values of symmetric square $L$-functions are critical? | I have not yet checked Hida's normalization, but I think you're right about the critical range.
The critical range of the symmetric square L-function is studied in C.G. Schmidt's paper "p-adic measures attached to automorphic representations of GL(3)".
Using the notation of this paper (see bottom of page 603), for ... | 3 | https://mathoverflow.net/users/6506 | 217768 | 102,306 |
https://mathoverflow.net/questions/217676 | 2 | Can someone please explain why blow up in a Symplectic toric manifold corresponds to chopping off a corner in the Delzant polytope?
| https://mathoverflow.net/users/21488 | Blowup and Delzant Polytope | You must restrict to the case of blowing up at a fixed point of the torus action, otherwise the manifold is no longer toric, and the remainings are nonsense. Naively one just replaces the corner with a $\mathbb{P}^{n-1}$. Note that onece this is done, the $T^n$-action will have degenerate orbits when acting on this exc... | 1 | https://mathoverflow.net/users/43423 | 217770 | 102,307 |
https://mathoverflow.net/questions/217763 | 4 | I am looking for the best upper bounds on the bit complexity for testing the singularity of an integer $n\times n$ matrix, where each integer is represented with $k$ bits.
I know the fast method for computation of the determinant in
[Storjohann,
The shifted number system for fast linear algebra on integer matrices]... | https://mathoverflow.net/users/56920 | testing singularity of integer matrices | This is addressed in [Storjohann's paper on computing Smith Normal Form.](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.26.8839&rep=rep1&type=pdf)
| 2 | https://mathoverflow.net/users/11142 | 217773 | 102,309 |
https://mathoverflow.net/questions/217747 | 13 | Let $S$ be a finite simple group. All representations below are over the complex numbers.
Let
* $d\_0(S)$ be the smallest dimension of a faithful representation of $S$,
* $d\_1(S)$ be the smallest dimension of a faithful representation of some central extension of $S$, and
* $d\_2(S)$ be the smallest dimension of a... | https://mathoverflow.net/users/14094 | Linear occurrences of finite simple groups | Seems that $S = O\_{2n}^\pm(2)$ are examples of this for $n=5$, and probably for all $n \geq 5$. Such $S$ is a Jordan-Hölder factor of the automorphism group of the extraspecial group $2^{1+2n}\_\pm$, so $d\_2 \leq 2^n$. But the Schur multiplier is trivial, so $d\_0 = d\_1$, and the ATLAS of Conway et al. reports minim... | 10 | https://mathoverflow.net/users/14830 | 217778 | 102,310 |
https://mathoverflow.net/questions/217711 | 24 | Set
$$
g(x)=\sum\_{k=0}^{\infty}\frac{1}{x^{2k+1}+1} \quad \text{for} \quad x>1.
$$
Is it true that
$$
\frac{x^{2}+1}{x(x^{2}-1)}+\frac{g'(x)}{g(x)}>0 \quad \text{for}\quad x>1?
$$
The answer seems to be positive. I spent several hours in proving this statement but I did not come up with anything reasonable. Mayb... | https://mathoverflow.net/users/50901 | Curious inequality | I am not sure at all, please double check.
Denote $u(t)=(1+t)^{-1}$, it is a decreasing convex function on $[0,1]$, but $u(e^s)$ is concave on $(-\infty,0]$. As Neil Strickland notes, what we have to prove is that $(x-1/x)g(x)$ increases, or, if we denote $y=1/x$, that $(1/y-y)g(1/y)$ decreases. We have $$(1/y-y)g(1/... | 13 | https://mathoverflow.net/users/4312 | 217784 | 102,314 |
https://mathoverflow.net/questions/217788 | 5 | The ordinary Cauchy completion $\overline{C}$ of a small category $C$ satisfies a number of conditions: Every idempotent in $\overline{C}$ splits, there's an equivalence of categories $[C^{op}, Set] \simeq [\overline{C}^{op}, Set]$, etc...
There's also a notion of Cauchy completion for enriched categories, my questio... | https://mathoverflow.net/users/21136 | Enriched Cauchy completions and underlying categories | It works the other way around --- you should have tried to answer your question by yourself before you posted it here :-)
1. Yes, there is an equivalence $[X^{op}, V] \simeq [\overline{X}^{op}, V]$. You may find more details in "Basic Concepts of Enriched Category Theory" by M. Kelly (Chapter 5.5).
2. No. The name "C... | 4 | https://mathoverflow.net/users/13480 | 217793 | 102,316 |
https://mathoverflow.net/questions/215636 | 12 | QUESTION: Let $\Lambda\times\Lambda\rightarrow {\Bbb Z}$ be a lattice,
that is, ${\Bbb Z}^n$ with a non-degenerate integer quadratic form, not
definite, not necessarily unimodular, $n>2$. I want to show that for each $N>0$
there exists a 2-dimensional primitive sublattice $\Lambda\_0\subset \Lambda$,
also not definit... | https://mathoverflow.net/users/3377 | 2-dimensional sublattices with all vectors having very big square (in absolute value) | For the record: OP **Misha Verbitsky** writes that
"[$\Lambda$ of] rank $\geq 6$ or $\geq 7$ is a usual assumption
in these kind of applications", in which case
Ekaterina Amerik's suggestion of using
$N\cdot H$ (for some fixed indefinite rank-$2$ form $H$)
surely yields the easiest construction;
but there are also pri... | 3 | https://mathoverflow.net/users/14830 | 217800 | 102,320 |
https://mathoverflow.net/questions/209477 | 7 | I've encountered the following problem that I'm sure someone more topologically inclined can answer:
Say that a homotopy of maps $f:X\times[0,1)\to Y$ between two compact smooth manifolds $X$ and $Y$ is good (I don't know the exact term for it if there is one) if the homotopy is constant in time outside of some coord... | https://mathoverflow.net/users/58787 | Are maps homotopic with respect to a uniform number of local homotopies | Yes, you can always connect $f\_0$ and $f\_1$ by a finite sequence of good homotopies supported in any given open cover (and in fact, this sequence of good homotopies can be chosen to be homotopic to $f$ relative to $X\times \{0,1\}$). By pulling back the $U\_\alpha$ to an open cover of $X\times [0,1]$ via $f$, it suff... | 3 | https://mathoverflow.net/users/75 | 217802 | 102,321 |
https://mathoverflow.net/questions/217815 | 5 | Recall that a group $G$ satisfies *max* (or is said to be *Noetherian*) if all its proper subgroups are finitely generated. Similarly $G$ satisfies *max-n* if all its normal subgroups are normal closures of finite subsets. Note that property *max* is "closed with respect to extension", i.e if $N \unlhd G$ and $G \backs... | https://mathoverflow.net/users/70809 | Is being Noetherian a quasi-isometry invariant for f.g. groups? | For finitely *presented* groups, you're close to two well known open questions/conjectures.
**Question 1:** If a finitely presented group is Noetherian, is it virtually polycyclic?
(This question is FP11 [here](http://www.sci.ccny.cuny.edu/~shpil/gworld/problems/probFP.html), where it's attributed to S. Ivanov.)
... | 8 | https://mathoverflow.net/users/1463 | 217818 | 102,325 |
https://mathoverflow.net/questions/217817 | 3 | Is there an $N \in \mathbb{N}$ such that every finite index subgroup of $\mathrm{SL}\_3(\mathbb{Z})$ has a generating set of size $N$?
| https://mathoverflow.net/users/38889 | Is there a bound on the rank of finite index subgroup of SL_3(Z)? | No. Counterexamples are constructed by Sury and Venkataramana in (the answer is positive for *principal* congruence subgroups, but not otherwise):
MR1239806 (95a:20051) Reviewed
Sury, B.(6-TIFR-SM); Venkataramana, T. N.(6-TIFR-SM)
Generators for all principal congruence subgroups of SL(n,Z) with n≥3. (English summar... | 7 | https://mathoverflow.net/users/11142 | 217820 | 102,327 |
https://mathoverflow.net/questions/217827 | 1 | I was looking at the following interesting [question](https://mathoverflow.net/questions/215067/equivariant-almost-complex-structures-on-the-full-flag-manifolds) about the number of equivariant almost complex structures on the full flag manifold of $SU(3)$, and I began to wonder how many equivariant symplectic structur... | https://mathoverflow.net/users/76104 | Unique Equivariant Symplectic Structure for the Full Flag Manifold of $SU(3)$? | If $G$ acts on $M$ (both compact and finite-dimensional) preserving the symplectic form, and $M$ is simply-connected, the action is Hamiltonian. Then $M$ maps symplectomorphically to a coadjoint orbit of $G$. This gives a $rank(G)$-dimensional family of symplectic forms, not a unique one.
You should consider the $SU(... | 4 | https://mathoverflow.net/users/391 | 217831 | 102,330 |
https://mathoverflow.net/questions/217845 | 7 | What is $$ \limsup\_{n \to \infty} \frac{\log(\mathrm{lcm}(1,2, \dots, n))}{n} \ \ ?$$
| https://mathoverflow.net/users/38889 | Asymptotics of the least common multiple of the first natural numbers | It is well-known that $\operatorname{lcm}(1,\ldots,n) = e^{\psi(n)}$, where $\psi$ is the Chebyshev's function. Since $\psi(x) = x + o(x)$, as $x \to +\infty$, (a form of the Prime Number Theorem) it follows that actually $\lim\_{n \to +\infty} \frac{\log(\operatorname{lcm}(1,\ldots,n))}{n} = 1$. (See Part 1 of G. Tene... | 17 | https://mathoverflow.net/users/nan | 217847 | 102,338 |
https://mathoverflow.net/questions/217839 | 5 | The situation is this. I have a space $X$ which is second countable, compact, and Hausdorff (it's a modified form of a type space, though I don't think that matters here). It has size continuum. It may or may not have isolated points.
Must $X$ have a discrete subset of size continuum?
The obvious inductive construc... | https://mathoverflow.net/users/15735 | Must uncountable compact Hausdorff spaces have large discrete subsets? | Without even using compactness (or metrisability), a subspace of a second-countable space is second-countable. There is no discrete, uncountable, second-countable space.
| 11 | https://mathoverflow.net/users/nan | 217858 | 102,344 |
https://mathoverflow.net/questions/206862 | 4 | A colleague and I are interested in idempotent relations from $I=[0,1]$ to $I$ - relations such that $R\circ R(x)=R(x)$ for all $x\in I$. Specifically, the graphs of the relations we care about must be closed subsets of the square.
Is there any work in the literature addressing these objects from a topological point ... | https://mathoverflow.net/users/73785 | Idempotent relations on the unit square with closed graphs | <https://www.researchgate.net/publication/281110530_Destruction_of_metrizability_in_generalized_inverse_limits>
We worked out the details to get what we needed in that paper. Specifically, if $f$ is an idempotent upper-semicontinuous continuum-valued function from $I$ to $I$ (equivalently, idempotent with a graph whi... | 1 | https://mathoverflow.net/users/73785 | 217866 | 102,348 |
https://mathoverflow.net/questions/217850 | 1 | We know that there exists a polynomial the Fatou set $F(P)$ is connected, which is just an attracting basin for infinity.
I have a question: Given a rational function $R$
such that $F(R)$ is connected, is this always true that $F(R)$ is just a attracting basin?
If the question is not, I wondered whether there e... | https://mathoverflow.net/users/11966 | classification of rational map with exactly only one Fatou component | Clearly if $F(R)$ is connected, then $F(R)$ consists of a single attracting or parabolic basin.
Both cases can occur. Indeed, if you consider the slice $\operatorname{Per}\_1(1)$ of quadratic rational maps having a fixed point with multiplier one, then this slice has an analog of the Mandelbrot set inside it, which ... | 1 | https://mathoverflow.net/users/3651 | 217874 | 102,353 |
https://mathoverflow.net/questions/217875 | 2 | I am not an expert in elliptic curves at all, so my question may naive and/or obvious. Let $E$ be an (affine) elliptic curve defined over a finite (or perfect) field of characteristic $p$. Since its module of derivations is free of rank one over its coordinate ring $R$, we can choose a generator (unique up to a constan... | https://mathoverflow.net/users/22873 | The $p$-th power of the invariant derivative on an elliptic curve in characteristic $p$ | $\delta^p = A\delta$ where $A$ is the Hasse invariant. In particular $\delta^p = 0$ if and only if $A=0$, i.e. $E$ is supersingular.
| 8 | https://mathoverflow.net/users/2290 | 217876 | 102,354 |
https://mathoverflow.net/questions/217878 | 1 | I am trying to characterize matrices with a certain property :
Define $U$ as an $n \times n$ matrix (over C or R; you can also assume
that it is unitary or orthogonal if it helps). Now take $n$
unknowns and transform by $U$ : $y=Ux$; I want the product
$y\_1 y\_2 \cdots y\_n = \sum c\_i x\_i^n$; so basically I don't
wa... | https://mathoverflow.net/users/16739 | characterize certain type of matrices | I'm guessing that these don't have a name, as they are fairly degenerate matrices.
If more than two of the $c\_i$ are nonzero, then no matrices satisfy these criteria. This follows since the Fermat curve
$$X^n + Y^n = Z^n$$
is irreducible and a decomposition as you describe would make it a union of lines.
If two of... | 3 | https://mathoverflow.net/users/69868 | 217897 | 102,363 |
https://mathoverflow.net/questions/217718 | 5 | As far as I have known, for irreducible admissible representation $\pi$ of $p$-adic group $G$, the matrix coefficient is defined as follows:
For $v\in \pi$ and $w \in \pi ^\vee$, the contragredient representation of $\pi$, there is a canonical pairing $B:\pi \times \pi ^\vee \to \mathbb{C}$ such that $B(v,w):=w(v)$.
... | https://mathoverflow.net/users/35898 | On the definition of matrix coefficient | Given a representation π you can define the complex conjugate representation by taking the same vector space on which the new mult by a complex number is the old multiplication by the conjugate of the complex number. An invariant Hermitian form on π gives an identification of the contragredient of π with the complex co... | 3 | https://mathoverflow.net/users/23291 | 217898 | 102,364 |
https://mathoverflow.net/questions/217835 | 4 | Let $k$ be a field of prime characteristic $p$ and $\Sigma\_n$ be the symmetric group.
If I have a concrete $k[\Sigma\_n]$-module $M,$ how to compute the direct summand corresponding to the principal block of the group algebra $k[\Sigma\_n]$?
Is there a formula for the central idempotent $e$? If I had such a formula... | https://mathoverflow.net/users/23310 | Explicit description of the principal block of the symmetric group | If you just want to find the summand which lies in the principal block, that's easy: it's the submodule where the semi-simple part of every central element acts with the same eigenvalue as on the trivial module. That is, let $x=\sum a\_g g$ be a central element of $k\Sigma\_n$ (so $a\_g$ is constant on conjugacy classe... | 5 | https://mathoverflow.net/users/66 | 217900 | 102,366 |
https://mathoverflow.net/questions/217861 | 4 | The following problem seems like a very simple and natural one, but I am not familiar with any existing work on it; in particular I am hoping to prove it is NP hard:
Let $G$ be a complete weighted graph that satisfies a triangle inequality, let $k \geq 2$ be an integer, and let $r>0$. My goal is to partition the vert... | https://mathoverflow.net/users/70190 | Complexity of a very simple graph partitioning problem | For $k=2$, it's polynomial (in fact quadratic in the number of vertices). Draw an edge (in an auxiliary graph) between two vertices iff their distance is greater than $r$ (marking that these two vertices must be in the same part). Then you can partition the graph into two sets if and only if the auxiliary graph is disc... | 2 | https://mathoverflow.net/users/12487 | 217901 | 102,367 |
https://mathoverflow.net/questions/217885 | 1 | Before I get started, let me say for complete disclosure this question came up while I was solving a problem from <https://projecteuler.net/>.
I've been trying to find a non-recursive representation of the following function-
$$ S(n) = \sum\_{i=1}^{n} \frac{S(n-i)-1}{i!} $$
And I have not been able to find a metho... | https://mathoverflow.net/users/80092 | How to solve a complex recursive relation | With $S(0)=0$ one has
$$
S(n)=-\sum\_{k=1}^n\frac{F\_k}{k!}
$$
where $F\_k$ are the [Fubini numbers](http://oeis.org/A000670) (also known as [ordered Bell numbers](https://en.wikipedia.org/wiki/Ordered_Bell_number)). The proof is contained in my comments above, given that the exponential generating function for these n... | 2 | https://mathoverflow.net/users/41291 | 217909 | 102,370 |
https://mathoverflow.net/questions/217904 | 3 | It is claimed that if there are two weakly compact cardinals, then there is a generic extension in which $\aleph\_2$ and $\aleph\_4$ have the tree property. Assuming one knows Mitchell's forcing, what is the argument? If you just iterate Mitchell's forcing twice, then why does the second stage preserve the tree propert... | https://mathoverflow.net/users/11145 | tree property at $\aleph_2$ and $\aleph_4$ | Starting from $\omega$-many weakly compact cardinals, you can for example force the tree property at all $\aleph\_{2n}$'s, $0<n<\omega.$ See theorem 5.1 of ["The tree property at the $\aleph\_{2n}$'s and the failure of SCH at $\aleph\_\omega$''](http://logika.ff.cuni.cz/radek/papers/Friedman_Honzik_treeprop_revised.pdf... | 4 | https://mathoverflow.net/users/11115 | 217912 | 102,372 |
https://mathoverflow.net/questions/217914 | 5 | First I will introduce some notation and definitions.
Fix a level $N$ (take $N=1$ if it makes things easier) and a prime $p$. Let $k$ be a finite field of characteristic $p$ and let $\mathcal{C}$ be the category of complete noetherian local rings $(R, \mathfrak{m}\_R)$ together with an isomorphism $R/\mathfrak{m}\_R ... | https://mathoverflow.net/users/15899 | Smoothness of Hecke algebras |
>
> It follows from deformation theory of Galois (pseudo-)representations that $\mathbb T(\Lambda)$ is a complete noetherian semilocal ring. The maximal ideals correspond to mod $p$ modular representations.
>
>
>
The fact that $\mathbb T(\Lambda)$ is a semi-local ring follows from the existence of the natural su... | 2 | https://mathoverflow.net/users/2284 | 217918 | 102,377 |
https://mathoverflow.net/questions/217838 | 3 | Given $G \in SU(4)$, what are the level sets of the function $F:SU(n)\rightarrow \mathbb{R}$ defined by $F(V) = |tr(G^{\dagger}V)|^2$?
Can they be written only in terms of abstract linear maps, not in terms of the components of $V$ and $G$ in some basis?
Cross-posted after no answer on MSE.
| https://mathoverflow.net/users/41654 | Level sets on $SU(4)$ | First of all, the $G$ is a red herring. Multiplying on the left by $G$ (which is an isometry of $\mathrm{SU}(4)$ with respect to any left-invariant metric), we have $F(GV) = \bigl|\mathrm{tr}(V)\bigr|^2$, so it suffices to understand what is going on when $G=I$. So, from now on, set $G=I$.
Second, the function $F(V) ... | 7 | https://mathoverflow.net/users/13972 | 217928 | 102,382 |
https://mathoverflow.net/questions/217868 | 7 | I am computing the Galois groups of quintics using the theorems from Ryan Kavanagh paper "On Irreducible Rational Quintics" using the decic resolvent ${P}\_{10} \left({x}\right) = \prod\limits\_{1 \le i < j \le 5} \left({x - \left({{\alpha}\_{i} + {\alpha}\_{j}}\right)}\right)$. The relevant part is the second part of ... | https://mathoverflow.net/users/62471 | Theorems of the Galois groups of quintics appears not to work for the ${F}_{20}$ group determination | Alas, I have insufficient reputation to comment, so I'll submit my comment as an answer.
Thanks to everybody for the feedback; I'll look at working it into the paper and trying to salvage it. I wrote it during my third year of undergraduate studies for my Galois theory class, and to be honest, I never expected anybod... | 7 | https://mathoverflow.net/users/80115 | 217932 | 102,384 |
https://mathoverflow.net/questions/217922 | 7 | I'm playing with exponential sums...
If $q$ is an odd prime and $a$ an integer such that $q \nmid a$, then the following formula for the Gaussian sum is known
$$\sum\_{x=0}^{q-1} e\_q(ax^2) = \left(\frac{a}{p}\right) i^{\left(\frac{q-1}{2}\right)^2} \sqrt{q} ,$$
where $e\_q(z) := \exp\left(\frac{2\pi i z}{q}\right)$ ... | https://mathoverflow.net/users/nan | The number of solution of $x_1^2 + \cdots + x_k^2 \equiv \lambda \bmod q$ | Yes, this is standard. More generally, we have the following
**Theorem.** Let $p$ be an odd prime, and let $a\_1,\dots,a\_k\in\mathbb{F}\_p^\times$. Then the number of solutions of the equation $a\_1x\_1^2+\cdots +a\_kx\_k^2=1$ in $\mathbb{F}\_p$ equals
\begin{align\*}
p^{k-1}-\left(\frac{a\_1\dots a\_k}{p}\right)p^{... | 10 | https://mathoverflow.net/users/11919 | 217933 | 102,385 |
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