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https://mathoverflow.net/questions/84275 | 1 | Say $\boldsymbol{\beta}$ is a random n-vector having the multivariate normal distribution with mean $\boldsymbol{b}$ and covariance matrix $\boldsymbol{S}$. And let $\boldsymbol{x}\_1$ and $\boldsymbol{x}\_2$ be two row vectors with n elements each. Then we know the distribution of $\frac{exp(\boldsymbol{x\_1\beta})}{1... | https://mathoverflow.net/users/18137 | Find the joint distribution | If $A$ is the matrix with rows $x\_1$ and $x\_2$, then $A \beta$ has a bivariate normal distribution with
mean $A b$ and covariance matrix $A S A^T$. From that you can get the joint distribution
of your two random variables.
| 3 | https://mathoverflow.net/users/13650 | 84283 | 50,288 |
https://mathoverflow.net/questions/84266 | 15 | \begin{equation}
\sigma(n) < e^\gamma n \log \log n
\end{equation}
In 1984 Guy Robin proved that the inequality is true for all n ≥ 5,041 if and only if the Riemann hypothesis is true (Robin 1984).
>
> I could not get a hold of his paper. I am trying to understand how did he derived the inequality. Can anyone can... | https://mathoverflow.net/users/2865 | On Robin's criterion for RH | I have requested [a pdf of Robin 1984](http://zakuski.utsa.edu/%7Ejagy/Robin_1984.pdf) from campus scanning service. One highlight of the article that really should be mentioned is this:
For $n \geq 3,$ we have
$$ \sigma(n) \; < \; \; e^\gamma \; n \log \log n \; + \; \frac{ \; 0.64821364942... \; \; n \; }{\log \log... | 11 | https://mathoverflow.net/users/3324 | 84285 | 50,290 |
https://mathoverflow.net/questions/84216 | 7 | Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which
$\mathcal{F}\_t$ is filtration satisfying general conditions.
$W\_t$ is
a standard Brownian motion.
Let $Y\_t$ be a martingale given by
$$Y\_t = \int\_0^t \sigma\_r d W\_r$$
where $\sigma\_t$ is a bounded $\mathcal{F}\_t$ measurable process.
The question... | https://mathoverflow.net/users/5656 | A non-degenerate martingale | Here's an explicit construction that gives a counterexample.
For simplicity let $c=0$ (not important).
First, let $\alpha>0$ and consider the probability that a standard Brownian
motion started at 0 hits 0 at some time in the interval
$(\alpha t, \alpha t+t)$.
Then (1) this probability does not depend on $t$ (by Bro... | 8 | https://mathoverflow.net/users/5784 | 84289 | 50,293 |
https://mathoverflow.net/questions/84286 | 3 | We know that $DS$ graphs are such connected graphs that determinable by their adjacency spectrum.
Suppose $DS(n)$ and $G(n)$ show the number of $DS$ graphs and all graphs with $n$ vertices,respectively.
$1)$ Do we have any good approximation for $DS(n)$(even if $n$ be sufficiently large)?
$2)$ what is the behavio... | https://mathoverflow.net/users/19885 | Estimation of DS graph growth | As far as I know, the computation of these values up to 11 vertices by van Dam and Haemers is still the best result. No asymptotics are known.
| 6 | https://mathoverflow.net/users/9025 | 84291 | 50,295 |
https://mathoverflow.net/questions/84274 | 2 | Let $K=F\_q((X))$, from Artin-Schreier theory, for a cyclic extension $L/K$ of degree p, we have $L=K(x)$, $x^p-x=\alpha$. So when $L/K$ is totally ramified, could we find some $x$ s.t the corresponding $\alpha$ has valuation -1? or $\alpha$ has some bounded valuation independent of $L$? Thanks!
| https://mathoverflow.net/users/20167 | Totally ramified p-extension over $F_q((X))$ | Artin-Schreier theory gives a bijection between the set of all degree-$p$ cyclic extension of a field $K$ of prime characteristic $p$ and the set of all $\mathbf{F}\_p$-lines in the $\mathbf{F}\_p$-space $K^+/\wp(K^+)$, where $\wp$ is the endomorphism $x\mapsto x^p-x$ of the additive group $K^+$ (thought of as an $\mat... | 3 | https://mathoverflow.net/users/2821 | 84298 | 50,298 |
https://mathoverflow.net/questions/84303 | 16 | When trying to see if a number of the form $n^8-n^4+1$ can be divisible by the square of a prime, I found that it can indeed. The first few values for $n$ are
412, 786, 1417, 1818, 2430, 2640, 2809, 2822, 2899 ...
and the first few such primes $p$ (in increasing order) are
73, 97, 193, 241, 313, 337, 409, 433, ..... | https://mathoverflow.net/users/12961 | p^2 dividing n^8-n^4+1 | Your conjecture is true in the light of the following statements.
**Proposition 1.** A prime $p$ has the form $x^2+24y^2$ if and only if $p\equiv 1\pmod{24}$.
**Proposition 2.** A prime square $p^2$ divides $\Phi\_{24}(n)$ for some $n$ if and only if $p\equiv 1\pmod{24}$.
**Proof of Proposition 1.** The four equi... | 27 | https://mathoverflow.net/users/11919 | 84304 | 50,300 |
https://mathoverflow.net/questions/84308 | 10 | As we know, if $f$ is a computable function, then every pre-image of a r.e. set under $f$ is also a r.e. set, i.e. $f^{-1}(X)$ is a r.e.set if $X$ is a r.e. set. So I want to know that if a function satisfies that every pre-image of a r.e. set is also a r.e. set, then can we conclude that $f$ is a computable function? ... | https://mathoverflow.net/users/15572 | Can we represent computable functions by r.e. sets ? | If your hypothesis has a degree of uniformity, so that we may uniformly enumerate $f^{-1}X$ from an enumeration of $X$, that is, if given an index $e$ for a c.e. se $W\_e$, then we may compute an index for $f^{-1}W\_e$, then the answer is yes. To compute $f(n)$, simply consider indices for the singleton sets $W\_{e\_m}... | 10 | https://mathoverflow.net/users/1946 | 84311 | 50,301 |
https://mathoverflow.net/questions/84309 | 23 | I am a training Algebraic Geometer, and am currently trying to prepare a course on category theory. I would be really thankful if people could tell me about "real-math" applications of theorems about monads which they came across during their career.
| https://mathoverflow.net/users/20177 | Why are monads useful? | The abstract algebra of monads is similar to the algebra of monoids, so that constructions on monoids often suggest similar constructions on monads. This applies in particular to bar constructions.
A pretty striking application in its time was the use of bar constructions on monads to construct deloopings. Around 19... | 20 | https://mathoverflow.net/users/2926 | 84318 | 50,307 |
https://mathoverflow.net/questions/84317 | 5 | Given a finite-dimensional semisimple Lie algebra $\frak g$, take an irreducible representation $V$, and let $ann(V)$ the annihilator of $V$ in $U(\mathfrak g)$. Such ideals are called primitive ideals. Then the variety defined by the associated graded ideal $gr(ann(V))$ of $gr U(\mathfrak{g})=S\mathfrak g$ is known to... | https://mathoverflow.net/users/5420 | Primitive ideals of the universal enveloping algebras of affine Lie algebras | I'm not aware of any reasonable analogue of the nilpotent variety (or related theory of associated varieties) in this infinite dimensional setting. But you may get some inspiration from the work of Joseph, including his book (which has many citations listed on MathSciNet):
Anthony Joseph, *Quantum groups and their pr... | 2 | https://mathoverflow.net/users/4231 | 84319 | 50,308 |
https://mathoverflow.net/questions/84078 | 3 | Let $X \rightarrow Y$ be a birational projective morphism between smooth varieties over $\mathbb{C}$.
I think that the exceptional locus $E \subset X$ of $f$ is codimension $1$.
Assume that $\dim X = \dim Y =3$.
**Question** Is $E$ normal crossing?
If you know counterexample, please let me know.
| https://mathoverflow.net/users/12390 | Exceptional locus of a projective birational morphism between smooth varieties | The answer is no. Here's an example.
Fix $Y$ and a closed point $p \in Y$. Blow up $p$. This is clearly smooth. Call the resulting scheme $X'$ and let $F$ denote the exceptional divisor of $f' : X' \to Y$. I am actually going to assume that $Y = \mathbb{A^3}$ in my mind.
Here's the idea then, within $X'$, choose a ... | 5 | https://mathoverflow.net/users/3521 | 84322 | 50,310 |
https://mathoverflow.net/questions/84228 | 4 | Let $n>15$ be an integer. Suppose also $n=\sum\_{i=1}^n ic\_i$, where $c\_i$ are non-negative integers. Assume further that $c\_1<4$.
Is the following inequality true?
$$\frac{n!}{\prod\_{i=1}^{n}i^{c\_i}\prod\_{i=1}^{n}c\_i!}>n(n-1)(n-2)(n-3)$$.
Motivation: The left hand side of the inequality is the size of the c... | https://mathoverflow.net/users/19075 | Factorial-inequalities | Note that for integers $i > 2$ and. $h > 0$, and except for the case $(i,h) = (3,1)$, one has
$i^h(h!) < 2^{\lfloor ih/2 \rfloor}(\lfloor ih/2 \rfloor !)$. We can account for the exception and bound
from above the denominator of the left hand side of the posted inequality by
$(3)2^{(n - c\_1)/2}(((n - c\_1)/2)!)(c\_... | 1 | https://mathoverflow.net/users/3568 | 84331 | 50,315 |
https://mathoverflow.net/questions/84306 | 1 | * What are some of interesting results that arise from using difference equations in number theory , Combinatorics or any other field ?
| https://mathoverflow.net/users/18240 | What are other applications of difference equations in other branches of mathematics ? | Hrushovski used the model theory of difference fields to give another proof of the Manin-Mumford conjecture.
| 3 | https://mathoverflow.net/users/2290 | 84338 | 50,321 |
https://mathoverflow.net/questions/84353 | 0 | Naive question, why is the equivalence relation necessary in the definition of composition of profunctors: <http://en.wikipedia.org/wiki/Profunctor>. Or if it is not necessary what is the advantage of adding the equivalence relation?
Edited: The remarks on bilinearity, and coherence answered my question. But let's s... | https://mathoverflow.net/users/20196 | on composition of profunctors | Profunctor composition is not strictly associative even after putting on the equivalence relation. One is in a bicatgeory setting here. The equivalence relation is among other things to make identity profunctors work. You can think of profunctors as bimodules and composition as tensor product. The equivalence relation ... | 3 | https://mathoverflow.net/users/15934 | 84363 | 50,333 |
https://mathoverflow.net/questions/84367 | 0 | I'm reading the proof of Lemma 4.1 [1] which says:
"Let $F = K(x,y), y^q = f(x)$, where $q$ is a prime different from characteristic of $K$.
Let $Z := Gal(F/K(x))$ and we have $Z < G < Aut(F/K)$ Then:
$Z < Z(G) \iff$ $Z$ is normal in $G$ and for all $\sigma \in G$ there exists $0 \neq B\_{\sigma} \in K(x)$ with $\s... | https://mathoverflow.net/users/6776 | The image of generator under an automorphism of a cyclic function field | OK, finally, I think I got it, but it is not that trivial to simply be omitted from the proof (If I complicated it and there's is a straight forward way to see it please tell me):
We have $\sigma(y)^q \in K(X)$. Expanding $\sigma(y)^q$, we see that every term in the expansion has the form $y^{\sum s\_i i}B\_i^{s\_i}$... | 0 | https://mathoverflow.net/users/6776 | 84368 | 50,335 |
https://mathoverflow.net/questions/84365 | 4 | People haven't been rushing in to answer [this question](https://math.stackexchange.com/questions/94149/distribution-of-shapes-of-delaunay-triangles) I asked on stackexchange yesterday.
The way I phrased it initially was this: Does anyone know the probability distribution of the shapes of Delaunay triangles in a cons... | https://mathoverflow.net/users/6316 | Distribution of shapes of Delaunay triangles | See t[his paper of R. E. Miles](http://dl.dropbox.com/u/5188175/milesrand2.pdf) (he has plenty of related results for points on the sphere, etc, etc, mathscinet will tell you more). The results you want are in section 9 (p. 112, and thereabouts). (the paper is: On the homogeneous planar Poisson point process, R. E. Mil... | 7 | https://mathoverflow.net/users/11142 | 84378 | 50,343 |
https://mathoverflow.net/questions/84254 | 2 | For me, a simplicial groupoid is a simplicial object in ${\mathbf{Grpd}}$. I am more general than Goerss-Jardine in this definition.
Do you have examples simplicial groupoids that occur in nature? Here's what I have got:
1. Given a simplicial group $G$ acting on a simplicial set $X$, the action groupoid $X//G$ is a... | https://mathoverflow.net/users/6301 | Examples of Simplicial Groupoids in Nature | See some papers following
Ehlers, P.J. and Porter, T. Varieties of simplicial groupoids. I. Crossed
complexes. *J. Pure Appl. Algebra* 120~(3) (1997) 221--233.
(which you may already have).
As another example, given a double groupoid, it's simplicial nerve in one direction is a simplicial groupoid in your sens... | 1 | https://mathoverflow.net/users/19949 | 84390 | 50,348 |
https://mathoverflow.net/questions/84392 | 5 | Some weeks ago I was asked to solve one ODE. I tried all methods I know, but couldn't crack this equation. Also I tried to use Matlab's dsolve function - without any result.
$y' + \frac{2x}{y} = x^2$
Does anyone have any suggestions on this?
P.S. This is not for homework or anything like this, just want to know i... | https://mathoverflow.net/users/20208 | Any help on one ODE | If you mean a (real) analytical solution with $y(0)=0$, then the answer is 'no'. If you write this as the problem of looking for integral curves of $\omega = (2x-x^2y)\ dx + y\ dy$ in the $xy$-plane, you'll immediately see that the origin is an isolated singular point of elliptic type (i.e., the eigenvalues of the line... | 15 | https://mathoverflow.net/users/13972 | 84397 | 50,350 |
https://mathoverflow.net/questions/84405 | 0 | Hi
Here there are two graphs for two functions from $R^2\mapsto R$.
Is there similar graph for the absolute value of a complex variable function $f:C\mapsto C$ that has the same point (like saddle point or transition). I know some functions that have the point $(x,y,|f(x+iy)|)$ on that such that in one direction it i... | https://mathoverflow.net/users/11733 | graph of the size of a complex function | For any nonconstant analytic function $f$, if $f'(p) = 0$ but $f(p)$ and $f''(p)$ are nonzero, then the graph of $|f(z)|$ will have a saddle point at $p$.
| 1 | https://mathoverflow.net/users/13650 | 84410 | 50,355 |
https://mathoverflow.net/questions/84409 | 1 | Let $X=2^\omega = \lbrace 0,1 \rbrace^{\mathbb{N}}$ and $T\colon X \to X$ the (left) shift map. The space $\mathcal{M}$ of $T$-invariant Borel probability measures is convex with $\mathcal{M}^e$, the set of all ergodic measures, exactly the extremal points. Moreover, the set $\mathrm{C}(\mathcal{M}^e)$ of convex combin... | https://mathoverflow.net/users/8382 | Shift invariant measures that are(n't) convex combinations of ergodic measures | The ergodic decomposition theorem states that any shift-invariant Borel probability measure, $\mu$, can be *uniquely* expressed in the form
$$
\mu=\int\_{\mathcal M^e(X)}\nu\,d\eta(\nu).
$$
This means that $C(\mathcal M^e)$ is the set of measures that are (finite) convex combinations of ergodic measures and $\mathcal... | 7 | https://mathoverflow.net/users/11054 | 84411 | 50,356 |
https://mathoverflow.net/questions/84379 | 4 | I would like to know if there are in the literature explicit computations of the volume of complex hyperbolic manifolds.
More precisely, let $\mathcal O$ be an imaginary quadratic number field, and let $\Gamma$ be the set of "integer" elements in $U(n,1)$, that is
$$
\Gamma = U(n,1) \cap M(n+1,\mathcal O).
$$
How ... | https://mathoverflow.net/users/20052 | volume of complex hyperbolic manifolds | This answer belongs to the second author of this [paper](http://arxiv.org/abs/1107.5281) (=:[ES]).
First, let $\Gamma\_0 = SU(n, 1) \cap M(n + 1, \mathcal O)$ and $M\_0 = H\_\mathbb{C}^n / \Gamma\_0$. By (1) and (28) in [ES], we obtain that
$$
\mathrm{vol}(M\_0) = \frac{(4 \pi)^n}{(n + 1)!} |d\_{\mathcal O}|^s \left... | 5 | https://mathoverflow.net/users/20052 | 84416 | 50,359 |
https://mathoverflow.net/questions/84414 | 22 | Here is a basic question. When does $H^1\_{et}(X,\mathbb{Z})$ vanish? Using the exact sequence of constant etale sheaves $0\rightarrow\mathbb{Z}\rightarrow\mathbb{Q}\rightarrow\mathbb{Q}/\mathbb{Z}\rightarrow 0$, it is enough to show that $H^1\_{et}(X,\mathbb{Q})$ vanishes. It is known, for instance by 2.1 of Deninger'... | https://mathoverflow.net/users/100 | Etale cohomology with coefficients in the integers | The standard example is a copy of $\mathbb A^1\_k$, where $k$ is an algebraically closed field, with two points glued. In algebraic terms, $X = \mathop{\rm Spec}R$, where $R := k[x,y]/(y^2 - x^3 + x^2)$. Consider the finite morphism $\pi\colon \mathbb A^1 \to X$, which yields an exact sequence
$$
0 \to \mathbb Z\_X \to... | 25 | https://mathoverflow.net/users/4790 | 84417 | 50,360 |
https://mathoverflow.net/questions/84420 | 5 | Assume that a square, symmetric matrix $A$ can be factored into $A=LDL^T$ where $L$ is unit lower triangular and $D$ is diagonal. For indefinite $A$, $D$ may have $2x2$ blocks on the diagonal. How much information about the spectrum of $A$ can we obtain from $D$?
For example, it is known by Sylvester's law of matrix ... | https://mathoverflow.net/users/1074 | Spectral properties of the LDL^T matrix factorization | Well, for the closely related Cholesky factorization, there is the following:
Fast Accurate Eigenvalue Computations Using the Cholesky Factorization (1997) (by Roy Matthias), which says that the eigenvalues are very close to the squares of the diagonal elements of the Cholesky factor. (the paper is available on CiteS... | 4 | https://mathoverflow.net/users/11142 | 84422 | 50,362 |
https://mathoverflow.net/questions/84421 | 4 | Let $i:T^{2}\rightarrow S^{1}\times S^{2}$ be an embedding map.
If $i(T^{2})$ is inseparable in $S^{1}\times S^{2}$, then $S^{1}\times S^{2}-i(T^{2})\cong (O\cup K)^{c}$. Here $(O\cup K)^{c}$ is a two components link complement in $S^{3}$ such that $O$ is a trivial knot and $K$ can be any unlinked with $O$.
| https://mathoverflow.net/users/14500 | what is the meaning of "inseparable" in this case | I think you mean "non-separating" rather than "inseparable" (or at least I'm not familiar with that terminology), which means as you say that when you cut along the submanifold, you get a connected space. This follows because $i(T^2)$ must be compressible, by the loop theorem since it cannot be $\pi\_1$-injective (I'm ... | 5 | https://mathoverflow.net/users/1345 | 84424 | 50,364 |
https://mathoverflow.net/questions/84394 | 4 | [I already posted this question on stackexchange a while ago, but did not get any response: http://math.stackexchange.com/questions/93437/ideal-class-groups-and-extension-of-number-fields]
Let $(X, \mathcal{O}\_X)$, $(Y, \mathcal{O}\_Y)$ be schemes and $ f:X \to Y$ be a morphism. Suppose $f^\#:\mathcal{O}\_Y \to f\_\... | https://mathoverflow.net/users/5181 | Ideal class groups and extension of number fields | if your aim is really to relate $Pic(Y)$ to $Pic (X)$ it is probably a good idea to pybass the use of $\mathcal Q$ and to consider instead the more powerful technique of introducing the relative Picard scheme (functor) $Pic\_{X/Y}$. There are plenty of references, among them :
Néron models.
Bosch; Lütkebohmert ; Ra... | 2 | https://mathoverflow.net/users/11682 | 84438 | 50,371 |
https://mathoverflow.net/questions/84433 | 1 | For an algebraicaly closed field $k$ are all finite subsets of Affine $n$-space $A^{n}\left(k\right)$
algebraic sets (here for $n>1$), and if so, for a given finite set $X\subset A^{n}\left(k\right)$
what set of polynomials in $k\left[x\_{1},\dots,x\_{n}\right]$ has
$X$ as its common zero set? This probably has an answ... | https://mathoverflow.net/users/20221 | Are all Finite Subsets of Affine n-space Algebraic sets, and related question | Let $I\subset \mathbb Q[x\_1,...,x\_n]$ be the ideal generated by the polynomials $P\_1,...,P\_k$ and $A$ the $\mathbb Q$-algebra $A=\mathbb Q[x\_1,...,x\_n]/I$.
You are interested in the scheme $V=Spec(A)\subset \mathbb A^n\_\mathbb Q= Spec(\mathbb Q[x\_1,...,x\_n])$ and its $k$-points for $k$ an extension field of... | 4 | https://mathoverflow.net/users/450 | 84439 | 50,372 |
https://mathoverflow.net/questions/84425 | -1 | Let $\lambda\_1 (\cdot)$ be the larger absolute value
eigenvalue of a $2\times2$ matrix and $\lambda\_2 (\cdot)$
the smaller absolute value eigenvalue of a $2\times2$ matrix, i.e.
$|\lambda\_1 (\cdot)| \ge |\lambda\_2 (\cdot)|$.
Is it true that
$$\Big||\lambda\_1 (A+B)|-|\lambda\_1 (A)|\Big|^{1/3}+\Big||\lambda\_2 (A+... | https://mathoverflow.net/users/20216 | On an eigenvalue inequality | The alleged inequality is **false**, even if you restrict $A$ and $B$ to be positive definite matrices. Consider the following,
$$ A = \begin{bmatrix}
1.2281 & 0.6361\\\\
0.6361 & 1.9690
\end{bmatrix},\quad\quad
B = \begin{bmatrix}
3.7829 &-0.6021\\\\
-0.6021 & 0.4002
\end{bmatrix}.
$$
Then, we have the following:
... | 7 | https://mathoverflow.net/users/8430 | 84447 | 50,375 |
https://mathoverflow.net/questions/34387 | 14 | This may be an easy exercise but I am not getting it. Let $\mathbf F\_q$ be a finite field with $q$ elements and $\mathbf F\_{q^2}$ be its degree two extension. Define an automorphism $\sigma$ of $\mathbf F\_{q^2}$ by $\sigma (x) = x^q$. For any matrix $A = (a\_{ij}) \in M\_n(\mathbf F\_{q^2})$, let $A^{\star} = (a\_{j... | https://mathoverflow.net/users/7386 | Order of finite unitary group | This question from 2010 was just listed as "active", apparently because someone (not myself) downvoted it today. Anyway, rather than prolong the previous list of comments, I'll offer an explicit answer by pointing to an online resource that slightly predates Steinberg's 1968 AMS Memoir and is still a useful way to get ... | 7 | https://mathoverflow.net/users/4231 | 84449 | 50,376 |
https://mathoverflow.net/questions/84381 | 17 | Direct to the point.
Since now I've looked a lot of presentations of $\infty$-categories, but it seems that the only way to do explicit computations on these objects is via model categories. Is that so, or maybe there are other way of doing computations in $\infty$-categories, and if there such different ways what are ... | https://mathoverflow.net/users/14969 | Computations in $\infty$-categories | (This is an answer to a question below from Akhil Mathew; he wanted examples of
``explicit computations'' since all he knew were classical 1950s calculations and
abstract theory. My answer is too long for a comment and too short to do justice
to the question.)
That is terrible!!! I don't know where to begin, since t... | 22 | https://mathoverflow.net/users/14447 | 84456 | 50,380 |
https://mathoverflow.net/questions/84442 | 3 | Are there any restrictions known on a (complex reducible) projective variety $Y$ that can be presented as $Z\cap H$, where $Z$ is a normal (or just irreducible) closed subvariety of a smooth projective $P$, and $H$ is a smooth hyperplane section of $P$ ($Y$ is fixed, and everything else varies)? I am mostly interested ... | https://mathoverflow.net/users/2191 | Which (reducible) projective varieties could be presented as 'relatively smooth' hyperplane sections of irreducible (normal) ones? | Any *normal* projective scheme appears that way. Let $Y$ be normal projective and consider an embedding $Y\subseteq \mathbb P^n$ given by a *complete* linear system. Or more generally an embedding such that $Y\subseteq \mathbb P^n$ is projectively normal. (For the fact that this is indeed more general see [Hartshorne, ... | 4 | https://mathoverflow.net/users/10076 | 84457 | 50,381 |
https://mathoverflow.net/questions/84458 | 9 | Topological spaces have diagonal maps $X \rightarrow X \times X$ and $X \rightarrow X \wedge X$, and suspension spectra also have diagonal maps $\Sigma^\infty X \rightarrow \Sigma^\infty(X \wedge X) \cong (\Sigma^\infty X) \wedge (\Sigma^\infty X)$. What about general spectra? (i.e. symmetric spectra, S-modules, or any... | https://mathoverflow.net/users/1874 | do spectra have diagonal maps? | No. Let $[X,A]$ be the set (in fact abelian group) of homotopy classes of maps from one spectrum to another. If $A$, $B$, and $C$ are spectra, any natural map of sets $[X,A]\times [X,B]\to [X,C]$ is induced by a map $A\times B\to C$. Since $A\times B=A\coprod B$, this amounts to two maps $A\to C$ and $B\to C$ inducing ... | 14 | https://mathoverflow.net/users/6666 | 84462 | 50,384 |
https://mathoverflow.net/questions/84440 | 6 | Assume $V$ is a handlebody and $C$ be a simple closed curve contained in the interior of $V$.
As Sam said, there exists some simple closed curve such that every dehn surgery along it produces a handlebody. So I assume $C$ can not be isotopic to a simple closed curve in $\partial V$.
Obviously, trivial dehn surgery ... | https://mathoverflow.net/users/18496 | Dehn surgery on handlebody | There's an extensive literature on this and more general questions.
First, let's consider a more precise formulation of the question.
Let $H$ be a handlebody, and let $K\subset H$ be a knot. Let's
assume that $\partial H \subset H-K$ is incompressible. Otherwise,
$H=H\_0 \natural H\_1$ the boundary connect sum of t... | 14 | https://mathoverflow.net/users/1345 | 84463 | 50,385 |
https://mathoverflow.net/questions/84471 | 3 | Given a polynomial $p(x\_1,\ldots, x\_k)$ in $k$ variables with maximum degree $n$, and $x\_1,\ldots, x\_k \in [0,1]$. Suppose $\max\_{x \in [0,1]^k} p(x) = 1$, can we get an upper bound on the probability $\mathbb{P}(|p(x)| < \epsilon)$ where we treat the product space $[0,1]^k$ as a natural probability space? Using t... | https://mathoverflow.net/users/4923 | bounding the probability that a polynomial is near 0 | I believe that the result you want follows from the results in [this very cool paper.](http://arxiv.org/pdf/math/0108212v2) (see, in particular page 9). There are probably improvements since then...
| 2 | https://mathoverflow.net/users/11142 | 84475 | 50,389 |
https://mathoverflow.net/questions/84460 | 8 | I was wondering whether the various model structures on the category of small categories are combinatorial. I think that the ones I know are at least cofibrantly generated. In order to be combinatorial, the category of small categories must be locally presentable. Is this true or false?
Intuitively, every small cate... | https://mathoverflow.net/users/12166 | Is the category of small categories locally presentable? | As Mike says, it's locally finitely presentable because $Cat$ is (equivalent to) the category of models of a finite limit sketch. A quick way of describing this is to say that a category is (or is the nerve of) a simplicial set such that certain squares in the combinatorial definition of simplicial set (via faces, dege... | 13 | https://mathoverflow.net/users/2926 | 84477 | 50,390 |
https://mathoverflow.net/questions/84481 | 6 | Marker Theorem 3.1.4 says:
Suppose $T$ is a theory in a language with at least one constant symbol.
Then an $L$-formula $\phi(x)$ is $T$-equivalent to a quantifier-free formula iff, whenever $M$ and $N$ are models of $T$, $A \subseteq M$ and $A \subseteq N$, then $M \models \phi(a)$ iff $N \models \phi(a)$ for any $a$ ... | https://mathoverflow.net/users/5651 | Does model-complete in a language with a constant symbol imply EQ? | Per JDH's suggestion, I'll turn my earlier comment into an answer.
---
Assuming $T$ to be model-complete, then whenever $M$, $N$ and $A$ are all models of $T$, it would certainly follow from $A \subseteq M$ and $A \subseteq N$ that $M \models \phi(a)$ iff $N \models \phi(a)$ for any $a$ from $A$ (as whenever one ... | 7 | https://mathoverflow.net/users/4137 | 84488 | 50,394 |
https://mathoverflow.net/questions/84443 | 6 | I am not familiar with newforms, so this may not make any sense.
OEIS sequence [A116418](https://oeis.org/A116418) is Expansion of a newform level 18 weight 3 and character [3]
Numerical evidence suggest that up to $10^5$
$$ \text{A116418}[n] \equiv \sigma(3n+1) \pmod 3$$
>
> What is the complexity of computing... | https://mathoverflow.net/users/12481 | Complexity of computing expansion of a newform level 18 weight 3 and character [3] - OEIS A116418 | [More a comment than an answer, but too long for the comment space]
Call this form
$$
\varphi := \frac{\eta(q^3)^2 \eta(q^6)^3 \eta(q^9)^2}{\eta(q^{18})}
= q - 2q^4 - 4q^7 + 6q^{10} + 8q^{13} \cdots.
$$
The listing of coefficients in the OEIS is correct as far as it goes (checked
with copy-and-paste to **gp**). The f... | 8 | https://mathoverflow.net/users/14830 | 84489 | 50,395 |
https://mathoverflow.net/questions/56103 | 20 | A referee asked me to include a reference or proof for the following classical fact. It's not hard to prove, but I'd prefer to just give a reference -- does anyone know one?
Let $X$ be a nice space (eg a smooth manifold, or more generally a CW complex). The topological Picard group $Pic(X)$ is the set of isomorphism ... | https://mathoverflow.net/users/317 | First Chern class of a flat line bundle | I noticed that someone voted this up today. Since this might indicate that someone else is interested in the answer, I thought I'd remark that Oscar Randal-Williams and I worked out a proof of this when I visited him earlier this year. A version of this proof can be found in Section 2.2 of my paper
The Picard group o... | 10 | https://mathoverflow.net/users/317 | 84492 | 50,397 |
https://mathoverflow.net/questions/84493 | 1 | Let $m$ be a fixed integer.
I want to count number of prime triplet $(p,q,r)$ such that $p < q < r < 2p$ with $m$ divides $p-1, q-1$ but not $r-1$ and the product $pqr$ is an $l$ digit integer.
| https://mathoverflow.net/users/15864 | Counting the number of prime triplet | Under your assumptions $p,q,r$ are all about size $x= 10^{l/3}$. The congruence conditions are basically independent so you'd get about $(x/\log x)^3(\phi(m)-1)/\phi(m)^3$. There may be a constant in front to account for the inequalities among the primes and the fact that you want exactly $l$ digits. This should be OK ... | 4 | https://mathoverflow.net/users/2290 | 84498 | 50,398 |
https://mathoverflow.net/questions/84374 | 57 | This is a part of my answer to [this question](https://mathoverflow.net/questions/84310/generalizing-euclids-proof-of-the-infinity-of-primes/84342#84342) I think it deserves to be treated separately.
**Conjecture** Let $A$ be the set of all primes from $2$ to $p>19$. Let $q$ be the next prime after $p$. Then $q$ d... | https://mathoverflow.net/users/nan | Every prime number > 19 divides one plus the product of two smaller primes? | This answer is a **heuristic** along the lines of Joro's.
We use $p,q,r$ to denote primes. Let $S(p,a)$ denote the number of pairs of primes $(q,r)$ with $q,r\leq p$ and $p|(qr+a)$. We are interested in the case $a=1$, but in general by the orthogonality relations of the characters we have $$ S(p,a)=\frac{1}{\phi(p)}... | 29 | https://mathoverflow.net/users/12176 | 84501 | 50,399 |
https://mathoverflow.net/questions/84500 | 7 | The well-known *transfer map* in group (co)homology can be defined with only homological algebra, or with algebraic topology via classifying spaces (group cohomology of $G$ is isomorphic to ordinary cohomology of $BG$). Here I fix a group $G$ with finite-index subgroup $H$, and field $k=\mathbb{Z}\_p$. Let $p:\tilde{Y}... | https://mathoverflow.net/users/12310 | The Norm Map in (group) cohomology via classifying spaces | First let me describe the norm in degree 0. Let $k$ be a commutative ring, given $\alpha\in H^0(BH,k)$, which corresponds to a an $H$ equivariant homotopy class of a map $EH\_+\rightarrow k$ where $k$ has the trivial $H$ action. W can take the $n$th smash power of this map to obtain
\begin{equation\*} (EH\_+)^{\wedge ... | 5 | https://mathoverflow.net/users/8818 | 84506 | 50,402 |
https://mathoverflow.net/questions/84508 | 4 | This is a followup to [Spaces of Finite Subspaces](https://mathoverflow.net/questions/84245/spaces-of-finite-subsets). Just for convenience, $\exp\_nX$ is the space whose underlying set is the set of nonempty subsets $S\subseteq X$ with $|S|\le n$.
As Alex Suciu pointed out in his answer to the previous post (refere... | https://mathoverflow.net/users/12310 | Spaces of Finite Subsets - homeomorphism type | Yes, the manifold question is completely answered [here](http://arxiv.org/pdf/0902.3773v2). Namely $\exp\_n X^k$ (where $X^k$ is a $k$-manifold) is a manifold *if and only if* $k=1, n=3$ or $k=n=2.$ This is Theorem 1.3 in the referenced paper. **EDIT** Also, of course if $n=1,$ though the authors overlook this...
**E... | 4 | https://mathoverflow.net/users/11142 | 84514 | 50,406 |
https://mathoverflow.net/questions/84399 | 1 | Say I have a family of linear spaces, and that I can solve all Schuber problems of that family (that is, how many members of the family pass through a set $S$ of linear spaces,
where we consider all possible $S$). How can I go from these solutions to the cycle
class of the family in the corresponding Grassmanian. I wou... | https://mathoverflow.net/users/19945 | Schubert problems to cycle class in Grassmanian | For simplicity, I am supposing your family is a pure-dimensional (though not necessarily irreducible) subvariety $V$ in the Grassmannian.
As you probably know, given a fixed $i$, the classes $[X\_\lambda]$ of Schubert subvarieties $X\_\lambda$, where $\lambda$ is a partition which has $i$ boxes and fits inside a $k \... | 2 | https://mathoverflow.net/users/3077 | 84530 | 50,413 |
https://mathoverflow.net/questions/84532 | 18 | For a smooth 3-manifold $M$, is the natural map from the space of diffeomorphisms of $M$ to the space of homeomorphisms of $M$,
$${\sf Diff}(M) \longrightarrow {\sf Top}(M),$$
a weak homotopy equivalence? Equivalently, is the space of smooth structures on a topological 3-manifold contractible? (This is as opposed t... | https://mathoverflow.net/users/2327 | Diffeomorphisms vs homeomorphisms of 3-manifolds | $\mathsf{Diff}(S^3)$ is homotopy equivalent to $O(4)$, by Hatcher's solution to the Smale Conjecture: Hatcher, Allen E. A proof of the Smale conjecture, Diff(S3)≃O(4). Ann. of Math. (2) 117 (1983), no. 3, 553–607.
Cerf proved that the Smale conjecture implies that $\mathsf{Diff}(M) \to \mathsf{Homeo}(M)$ is a weak e... | 26 | https://mathoverflow.net/users/1335 | 84541 | 50,416 |
https://mathoverflow.net/questions/84536 | 1 | See [Smooth manifold with non-trivial inertia group? (wrt homotopy spheres)](https://mathoverflow.net/questions/79991/smooth-manifold-with-non-trivial-inertia-group-wrt-homotopy-spheres) for the definition of $\Theta\_n$ and inertia subgroups.
I'm wondering what can be said about Lie groups. If $M^n$ is an n-dimensio... | https://mathoverflow.net/users/17812 | The inertia subgroup of `$\Theta_n$` for Lie groups | There is no natural Lie group structure on a connected sum of a Lie group and an exotic sphere. Where would it possibly come from?
Also, the implication you are trying to derive is wrong.
Note that $S^3$ is a Lie group and hence so is a product of several $S^3$s.
I'm not an expert on surgery theory but it's well-kno... | 6 | https://mathoverflow.net/users/18050 | 84548 | 50,423 |
https://mathoverflow.net/questions/84550 | 0 |
>
> For each non-negative integer $n$, what antichain(s) in $\{0,1\}^n$ with the pointwise partial order:
> $\;\;$ 1. $\;$ have the most elements
> $\;\;$ 2. $\;$ minimize the maximum of its elements' sum of coordinates, among those satisfying (1)
> ?
>
>
>
The obvious candidate is the set of ... | https://mathoverflow.net/users/nan | Subset-Free Codes | Your "obvious candidate" is right. this is Sperner's Theorem (not to be confused with Sperner's Lemma).
| 4 | https://mathoverflow.net/users/6794 | 84551 | 50,424 |
https://mathoverflow.net/questions/84553 | 1 | Let
$$
L=\mathbb{Q}(\sqrt{-1})\otimes\_\mathbb{Q} \mathbb{Q}\_3
$$
where $\mathbb{Q}\_3$ denotes de $3$-adic rational numbers.
Then $L$ is a quadratic extension of the local field $\mathbb{Q}\_3$.
Furthermore, the valuation ring of $L$ is $B:=\mathbb{Z}[\sqrt{-1}] \otimes \mathbb{Z}\_3$.
It is known that the norm ... | https://mathoverflow.net/users/20052 | Norm map and units in local rings | One approach is by Hensel's Lemma: consider $N(x+i)=x^2+1=-1$, for example. Since $1^2+1^2=-1 \mod 3$ while $2\cdot 1=2\not= 0 \mod 3$, the equation $x^2+1=-1$ has a solution in $\mathbb Z\_3$. Hensel's lemma gives a sequence of integers approaching the solution. Note that there are many things in $\mathbb Q\_3(i)$ wit... | 5 | https://mathoverflow.net/users/15629 | 84554 | 50,426 |
https://mathoverflow.net/questions/84557 | 4 | Hello all,
If you have a theorem $\vdash \alpha \rightarrow \beta$ and a theorem $\vdash \gamma$ then if $\alpha$ is a sub-expression of $\gamma$, and this sub-expression has an even number of negations within $\gamma$ (and it is not within a xor or boolean equality), then a new theorem can be obtained by replacing $... | https://mathoverflow.net/users/5917 | Is there any literature about inner-replacement rule? | I don't know a name for the particular inference you indicate, but its feature that it operates "deeply" within the formulas at hand rather than at the root of their parse trees brings to mind current proof-theoretic work in *deep inference*. Perhaps check out Alessio Guglielmi's [Deep Inference in One Minute](http://a... | 4 | https://mathoverflow.net/users/4137 | 84564 | 50,430 |
https://mathoverflow.net/questions/84538 | 4 | Suppose we have a primal problem
$
\min\_x f(x), s.t. h\_i(x)=0,
$
where $h\_i$ are all affine, and $f$ is convex.
Then its Lagrangian is
$\min\_x \max\_{z\_i} f(x) + \sum\_i z\_i h\_i(x)$
and the dual problem is
$\max\_{z\_i} \min\_x f(x) + \sum\_i z\_i h\_i(x)$
The KKT condition (sufficient and necessa... | https://mathoverflow.net/users/20263 | Lagrangian duality | At the level of generality you asked about, the answer is **no**, the claim is not correct. Of course, your case of interest may rule out counterexamples like the one below.
It can happen that the primal is bounded below but does not achieve its optimum, whereas the dual does. For example take $f(x\_1,x\_2) = \exp(x\... | 3 | https://mathoverflow.net/users/5963 | 84565 | 50,431 |
https://mathoverflow.net/questions/84562 | 4 | Are there known families of distance-regular graphs with girth larger than 4 where for given vertex/edge count there are more than one non-isomorphic instances? The following is what I have found so far (but none of them satisfies the criteria):
* Hadamard graphs (non-isomorphic instances exist for $n \geq 64$ but gi... | https://mathoverflow.net/users/20266 | Families of distance-regular graphs with large girth | The point-line incidence graph of a finite projective plane is distance regular with
diameter three and girth six. In fact a bipartite graph with diameter three and girth six
is necessarily the incidence graph of a projective plane. Next, the point-line incidence
graph of a generalized quadrangle with parameters $(s,s)... | 8 | https://mathoverflow.net/users/1266 | 84566 | 50,432 |
https://mathoverflow.net/questions/84549 | 4 | Is there an example of a smooth projective hypersurface in $\mathbb{P}^n\_k$ ($k=\overline{k}$) that does not contain any projective toric varieties (**edit**: of positive dimension)? Or is it the case that every such hypersurface will contain a projective toric variety (**edit**: of positive dimension)?
| https://mathoverflow.net/users/16046 | Do projective hypersurfaces contain projective toric varieties? | As Alexander Woo said in a comment, toric varieties are rational. Now, it turns out that projective hypersurfaces have strong hyperbolicity-type properties. This properties have been established by several authors in the last decades.
First, in 1986 Clemens showed that if $X$ is a generic hypersurface
of degree $d \g... | 14 | https://mathoverflow.net/users/9871 | 84567 | 50,433 |
https://mathoverflow.net/questions/84503 | 4 | Hello,
this is a math forum, I know, but my question is about classical mechanics. I am looking for a general (but simple proof) of the very intuitive idea physicists have about the following problem.
We consider a particle in $\mathbb{R}^d$ evolving in a potential $V$ and with a friction coefficient $\gamma$. The ... | https://mathoverflow.net/users/15792 | mechanics: convergence to an equilibrium point | Consider the total energy
\begin{equation}
E = x'^2/2 + V(x)
\end{equation}
and assume that $V$ is bounded below and $V(x) \rightarrow \infty$ as $||x||\rightarrow \infty$
(i.e., V is radially unbounded). Since
\begin{equation}
E' = -\gamma x'^2 < 0, \quad \forall x' \neq 0,
\end{equation}
it follows from LaSalle's in... | 4 | https://mathoverflow.net/users/12400 | 84578 | 50,439 |
https://mathoverflow.net/questions/84527 | 4 | Suppose we have a lattice path in 2D starting at the origin, in which north, south, east, and west steps are allowed. For a given path $L$, let $\max(L)$ be the maximum value of the $y$ coordinate achieved by $L$ over the length of its path. For example, the path NENWSWW would have $\max(L) = 2$.
>
> What is known... | https://mathoverflow.net/users/9716 | Maximum vertical distance for a lattice path when NSEW steps are allowed | Anthony Quas's comments gave me the ideas I needed to answer the question. (Thanks, Anthony!) Here's the solution in case anyone else is interested. The answer turns out to be that the number of $n$-step paths with max height $y$ is $\binom{2n}{n+y} + \binom{2n}{n+y+1}$.
First, let $(X\_n,Y\_n)$ denote the final posi... | 4 | https://mathoverflow.net/users/9716 | 84579 | 50,440 |
https://mathoverflow.net/questions/84531 | 9 | The Kullback-Leibler Divergence (KLD) of two PMF's $P(x)$ and $Q(x)$ is $D(P||Q)=\sum\_x P(x)\log(P(x)/Q(x))$, with the provisos that $0\cdot \log (0/p)=0$ and $p\cdot \log (p/0)=+\infty$ whenever $p>0$.
It is known that KLD is continuous at $(P,Q)$ if $Q$ is *strictly positive over all $x$'s*. What can be said other... | https://mathoverflow.net/users/20262 | Lower semicontinuity of Kullback-Leibler divergence | In addition to the conventions you have mentioned, it is also assumed that $0\log(0/0)=0$.
With these conventions, I think, in the finite case, it is always true that $$\lim\_{n\to \infty} D(P\_n||Q\_n)=D(P||Q)$$ As you said, if $Q(x)>0$ for all $x$, its immediate from the Dominated Convergence theorem.
The problem i... | 5 | https://mathoverflow.net/users/7699 | 84582 | 50,442 |
https://mathoverflow.net/questions/84555 | 14 | The Mazur-Ulam theorem says that any surjective isometry of normed vector spaces is affine. This argument doesn't seem to apply to Minkowski space (of special relativity) since the metric is indefinite. How would one show that the Poincaré group consists of affine maps? This seems really standard but I can't seem to fi... | https://mathoverflow.net/users/20264 | Why are isometries of Minkowski space necessarily linear? | Let's fix notation and define the bilinear form $\eta: \mathbb{R}^4 \times \mathbb{R}^4 \to \mathbb{R}$ by:
$\eta((x,y,z,t),(x',y',z',t')) = xx'+yy'+zz'- tt'$
Given a map $T:\mathbb{R}^4 \to \mathbb{R}^4$ which fixes $0$ and preserves $\eta$ we want to show that $T$ is linear.
Let $e\_1,e\_2,e\_3,e\_4$ be the can... | 11 | https://mathoverflow.net/users/7631 | 84595 | 50,450 |
https://mathoverflow.net/questions/84597 | 2 | Suppose a (linear algebraic) group $G$ acts on a variety $X$ and that $U$ is a $G$-invariant open subvariety. My question is: under what conditions is the restriction functor
$i^\*: Vect^G(X) \rightarrow Vect^G(U)$
an equivalence of categories of $G$-equivariant vector bundles? Obviously not in general, but we do h... | https://mathoverflow.net/users/1713 | When is restriction an equivalence of categories of equivariant vector bundles? | The statement is true if $X$ is regular of dimension 2 (an in very few other cases, I would guess). Anyway, this certainly applies to your example.
The point is that every locally free sheaf on $U$ has an extension to a reflexive sheaf on $X$, which in this case is locally free. Then you need to show that the extensi... | 2 | https://mathoverflow.net/users/4790 | 84599 | 50,452 |
https://mathoverflow.net/questions/84605 | 10 | Let $w$ be a word in letters $x\_1,...,x\_n$. A value of $w$ is any word of the form $w(u\_1,...,u\_n)$ where $u\_1,...,u\_n$ are words. For example, $abaaba$ is a value of $x^2$. A word $u$ is called unavoidable if every infinite word in a finite alphabet contains a value of $u$ as a subword. There is a nice character... | https://mathoverflow.net/users/nan | Ubiquitous Zimin words | Yes. However, the initial application is related to semigroup varieties, so it is likely very boring to you.
In studying the hyperidentity for associativity, one can look at its representation on algebras of type <2>, a.k.a. groupoids or magmas or sets with one binary operation. Such an algebra is hyperassociative if... | 9 | https://mathoverflow.net/users/3402 | 84606 | 50,455 |
https://mathoverflow.net/questions/84523 | 3 | Give $x \in (0,0.5)$, how can we compute the asymptotic result of $\sum\_{k=n}^{\infty} {k+n \choose n} x^{k}$ as $n \rightarrow \infty$? Thanks.
| https://mathoverflow.net/users/20258 | An asymptotic question | Concerning the case $x\rightarrow \frac{1}{2}$ we have :
$f(1/2,n)= 2^n+\binom{2n}{n} 2^{-n}$ (nearly $2^n (1+1/\sqrt{\pi n}))$
found at page 247 of *Concrete Mathematics* Graham, Knuth, and Patashnik. See too equation (5.20) and the discussion in ['partial sum involving factorials'](https://math.stackexchange.co... | 0 | https://mathoverflow.net/users/16380 | 84607 | 50,456 |
https://mathoverflow.net/questions/84364 | 9 | Can anyone suggest a simple and efficient way (preferably embodied in computer code) to compute the average height function for lozenge tilings of an $a,b,c,a,b,c$ semiregular hexagon? I prefer to scale my height functions so that the height-change along each tile-edge is 1 and so that the lowest possible height of any... | https://mathoverflow.net/users/3621 | computing average height-functions for lozenge tilings | David Speyer made a blog post a while back saying how to do this:
<http://sbseminar.wordpress.com/2009/10/21/rhombus-tilings-and-an-over-constrained-recurrence/>
It wasn't the main point of the blog post, but he does say how to use Kuo's graphical condensation method to compute such things. In short, if $H(a,b,c)$ ... | 6 | https://mathoverflow.net/users/20281 | 84609 | 50,458 |
https://mathoverflow.net/questions/78497 | 14 | Same way as the two dimensional tilings by rhombi come from minimal surfaces in a $D$ dimensional cubical lattice as mentioned in [this answer](https://mathoverflow.net/questions/78302/rhombus-tilings-with-more-than-three-directions/78372#78372), one can consider higher dimensional zonotopes tiled by rhombic polytopes.... | https://mathoverflow.net/users/2384 | Arctic regions in higher dimensional zonotopes | Basically nothing is known beyond what you've said, and it doesn't look like that's going to change anytime soon.
There is at least one piece of relevant numerical work which I know of: J. Linde, the first author in the paper you mentioned, posted some graphs on his website which suggest that the height function isn'... | 12 | https://mathoverflow.net/users/20281 | 84611 | 50,459 |
https://mathoverflow.net/questions/84621 | 6 | I asked my friend (a Set Theorist) this question and he said that every model of ZFC thinks it is the Standard Model. But, I'm not sure it is so simple. First, because I don't know how a Universe could test whether or not it is the Standard Model. And second, because I think that various models of ZFC could have differ... | https://mathoverflow.net/users/nan | Is there a model of Set Theory which thinks it is the Standard Model, i.e. is there a Universe U such that U $\models$ U=V? | One of the main lessons of set theory is that many of our familiar set-theoretic concepts, such as countability, uncountability, well-orderedness, ill-foundedness, even finiteness, depend on the set-theoretic context in which they are considered. We know that different models of set theory can disagree about whether a ... | 9 | https://mathoverflow.net/users/1946 | 84623 | 50,465 |
https://mathoverflow.net/questions/84524 | 15 | Let $a, b, n$ be positive integers. Assume that $\gcd(a,b,n)=1$.
It seems that one can prove that there exist two integers
$c$ and $d$ bounded from above by $( \log n )^{O(1)}$ such that
$ \gcd (ac + bd, n) =1$. However the only proof I can see is
by a complicated exclusive-inclusive argument.
I am wondering whether it... | https://mathoverflow.net/users/3208 | gcd of three numbers | The answer is yes, there exists $c$ and $d$, even with $c = 1$, and
$d \ll (\log(n))^{O(1)}$. This follows from a result of Iwaniec.
It suffices to assume that $(a,b) = 1$.
Suppose that $(n,b) = e$, which implies that $(a,e) = 1$.
Since $b$ is divisible by $e$, it follow that $(a+bk,e) = 1$
for all integers $k$. Le... | 8 | https://mathoverflow.net/users/nan | 84628 | 50,467 |
https://mathoverflow.net/questions/84622 | 5 | Consider a full binary tree with $k>10$ levels. Let the *lengths* of individual edges in this tree be i.i.d. random variables with finite moments. Then total lengths of the $2^{k-1}$ source-to-sink paths in this tree are approximately Gaussian by the CLT, regardless of the edge-length distribution. We are interested in... | https://mathoverflow.net/users/17596 | Is the maximum tree-path length distributed lognormally (in the limit) ? | Unless I misunderstood your question, this can be entirely rephrased in terms of branching random walks. This goes as follows: at time 0 there is 1 individual at position 0. Each individual gives birth to two descendants, whose position is the position of the parent plus a jump, where all jumps are i.i.d. random variab... | 3 | https://mathoverflow.net/users/19649 | 84642 | 50,469 |
https://mathoverflow.net/questions/84615 | 3 | Do we have any good sources(lecture notes or books) for learning about $Ihara$ Coefficient?
Is there any relation between $Ihara$ Coefficient and the eigenvalues of graphs?
Thanks for any help.
| https://mathoverflow.net/users/19885 | learning sources about Ihara Coefficient | Following Igor Rivin's suggestion I found the thesis of Matthew Horton (student of Stark and Terras) pretty interesting : ['Ihara zeta functions of irregular graphs'](http://escholarship.org/uc/item/3ws358jm).
| 5 | https://mathoverflow.net/users/16380 | 84647 | 50,472 |
https://mathoverflow.net/questions/84634 | 0 | Let $X\_1,…,X\_n$ are exchangeable of random variables, and $n$ is an even number.
$S\_k=X\_1+\dots+X\_k$. $M\_k=X\_{n/2}+\dots+X\_{n/2+k}$.
I want to prove:
$$\Pr(\max\_{1 \le k \le n}{|S\_k|>\epsilon}) \le \\Pr(\max\_{1 \le k \le n/2}{|S\_k|>\epsilon/2}) + \Pr(\max\_{1 \le k \le n/2}{|M\_k|>\epsilon/2})$$
---... | https://mathoverflow.net/users/8379 | Help prove a maximal inequality | You can prove it by using the fact that the following holds always:
$\max\_{1 \le k \le n}|S\_k| \le \max\_{1 \le k \le n/2}|S\_k| + \max\_{1 \le k \le n/2}|M\_k|$
If the left hand side is larger than $\epsilon$ then one of the right hand terms is larger than $\epsilon/2$.
This also shows that the inequality is v... | 4 | https://mathoverflow.net/users/7631 | 84650 | 50,474 |
https://mathoverflow.net/questions/84638 | 5 | Recall that a (combinatorial) simplicial complex $X$ is said to be $n$-dimensional if it contains at least one face of dimension $n$ and no faces of dimension $n+1$. Further, an $n$-dimensional finite simplicial complex $X$ is said to be *pure* if every face $\delta$ of dimension $k\lt n$ is a face of some $n$-dimensio... | https://mathoverflow.net/users/1353 | Extending the definition of "pure of dimension n" from simplicial complexes to simplicial sets? | I [EDIT: almost] would follow the suggestion you made in your comment.
[EDIT: below the line was my original answer, which is wrong, as pointed out by Karol in the comments. Here is an answer which is less functorial, but I hope more correct.]
* It would be nice to define pure dimension internal to categories othe... | 3 | https://mathoverflow.net/users/4177 | 84651 | 50,475 |
https://mathoverflow.net/questions/84630 | -1 | If one forces with measure algebra, then every formula $\phi(\tau\_{1},\tau\_{2},...,\tau\_{k})$ where $\tau\_{i}$'s are names, has a truth value, the measure of which is a real number. It is just a curiosity that is the "set"(may I say set?) of measures of all truth values equals to $[0,1]$? If fixed a name for a rand... | https://mathoverflow.net/users/18692 | distibution of truth values of all formulas on [0,1] | To answer your latter question, if by a name for a random real you mean the canonical name for the real determined by the generic filter over the measure algebra, which is known as a *random* real, then the answer is yes. This is true because for any notion of forcing $\mathbb{B}$, then the boolean value $[\![\check b\... | 3 | https://mathoverflow.net/users/1946 | 84654 | 50,477 |
https://mathoverflow.net/questions/84662 | 2 | For a natural number $n\geq 1$, let $PF(n)$ denote the number of prime factors (with multiplicity) of $n$. For example, since $48=2\*2\*2\*2\*3$, we have $PF(48)=5$.
For any natural number $N\geq 1$, define $$E(N)=\frac{\sum\_{k=1}^N PF(k)}{N},$$ the expected value for the number of prime factors for an integer betwe... | https://mathoverflow.net/users/2811 | What is the growth rate for divisibility of integers | This Theorem 430, on page 355 of Hardy and Wright, that he "average order" of $\Omega(n)$ is $\log \log n.$ Then they point out, formula 22.10.2, that
$$ \sum\_{n \leq x} \; \Omega (n) \; = \; x \log \log x + B\_2 x + o(x) $$ and say how to find the constant $$B\_2 = B\_1 + \sum\_p \; \frac{1}{p(p-1)}. $$
Previously , ... | 7 | https://mathoverflow.net/users/3324 | 84666 | 50,484 |
https://mathoverflow.net/questions/84668 | 5 | What is the Classifying Space of the Discrete Heisenberg Group? Which paper/book contains a detailed proof?
Thank you for your time.
| https://mathoverflow.net/users/15770 | The Classifying Space of the Discrete Heisenberg Group | As was said by Andy, the classifying space of the discrete Heisenberg group $\Gamma$ is $B\Gamma=G/\Gamma$, where $G$ is the 3-dimensional Heisenberg group over the reals. Due to the central extension
$$0\rightarrow\mathbb{Z}\rightarrow\Gamma\rightarrow\mathbb{Z}^2\rightarrow 0$$
you may view $B\Gamma$ as a circle bund... | 11 | https://mathoverflow.net/users/14497 | 84676 | 50,488 |
https://mathoverflow.net/questions/84685 | 0 | Let $A$ be a local CM ring, and $M$ a maximal CM $A$-module. Is it true that $\operatorname{Supp}M=\operatorname{Spec}A$ ? This suspicion stems from such statements as:
* If $\omega$ is a canonical module of $A$, then $\omega\_{\mathfrak{p}}$ is a canonical module of $A\_{\mathfrak{p}}$ for every $\mathfrak{p}\in\ope... | https://mathoverflow.net/users/5292 | Are maximal Cohen-Macaulay modules supported everywhere? | No. $R = k[x,y]/(xy)$, $M = R/(x)$.
The zero module has infinite depth and support of dimension $-\infty$, so should not be considered MCM.
| 3 | https://mathoverflow.net/users/460 | 84688 | 50,493 |
https://mathoverflow.net/questions/84695 | 4 | Let $A$ be a local CM ring, and $\omega$ a canonical module of $A$. Here are two properties of $\omega$ from Bruns & Herzog:
* $\omega\_{\mathfrak{p}}$ is a canonical module of $A\_{\mathfrak{p}}$ for every $\mathfrak{p}\in\operatorname{Spec}A$.
* $\mu\_i(\mathfrak{p},\omega)=\delta\_{i}^{\operatorname{ht}\mathfrak{p... | https://mathoverflow.net/users/5292 | Why are canonical modules supported everywhere? | See (1.7) on page 87 of [*Some basic results on canonical modules*](https://projecteuclid.org/journals/kyoto-journal-of-mathematics/volume-23/issue-1/Some-basic-results-on-canonical-modules/10.1215/kjm/1250521612.full). For a local CM ring condition (b) there holds.
| 5 | https://mathoverflow.net/users/16046 | 84700 | 50,498 |
https://mathoverflow.net/questions/84667 | 12 | By the Barratt-Priddy-Quillen theorem, the space $B \Sigma\_\infty^+$ is the infinite loop space $\Omega^\infty \Sigma^\infty S^0$. I'm curious about a "high-concept" reason that $B \Sigma\_\infty^+$ (and "more generally" $BGL\_\infty^+(R)$ for a ring $R$) should be infinite loop spaces (well, almost that). For instanc... | https://mathoverflow.net/users/344 | $\mathcal{I}$-functors and infinite loop spaces | You are looking at the telescope of maps $BG\_n\longrightarrow BG\_{n+1}$ where the coproduct
of the $G\_n$ (thought of as categories) has a structure of permutative category. The group
completion property of infinite loop space machines defined on permutative categories shows
easily that there is a canonical map from ... | 9 | https://mathoverflow.net/users/14447 | 84713 | 50,504 |
https://mathoverflow.net/questions/84745 | 8 | (I'm not sure if I should post this here rather than at [Theoretical Computer Science](https://cstheory.stackexchange.com/), I've found a lot of type theory related questions on MathOverflow)
I'm working in Martin-Löf type theory with inductive types. Everything I say below for booleans should be understood with the ... | https://mathoverflow.net/users/10217 | Reduction rules for inductive types | Your second reduction is called a commutative conversion. You can read about it in Girard, Taylor and Lafont, Proofs and Types, p. 80, for example. The congruence relation with commutative conversions for coproducts is has normal forms, see for instance:
Normalization by evaluation for typed lambda calculus with copr... | 8 | https://mathoverflow.net/users/2004 | 84753 | 50,519 |
https://mathoverflow.net/questions/84732 | 2 | Surely one could formulate the following question more generally, but as I am primarily concerned with abelian schemes, I will choose this setting:
Let $A$ be an abelian scheme over a locally noetherian base $S$. Each fibre over a point $s$ of $S$ is an abelian variety over the residue field $k(s)$ of $s$.
Now is t... | https://mathoverflow.net/users/18183 | Fibrewise properties of abelian schemes | (1)
---
I've just realized that I did not read the question carefully, so an edit is in order.
(a) $\mathscr G$ is flat over $S$
---------------------------------
So, let's see the real question: Let $\mathscr F\to \mathscr G$ be a morphism of coherent sheaves on $A$ with kernel $\mathscr K$ and cokernel $\mathsc... | 3 | https://mathoverflow.net/users/10076 | 84758 | 50,521 |
https://mathoverflow.net/questions/84734 | 9 | In what follows, $2$-categories will be strict, and "$2$-functor" will mean "strict $2$-functor". (Please mention which terminological conventions you are using when answering.) I guess that the answer to my question is rather standard for people working daily with $2Cat$ as a $2$- or $3$-category, and I think I may kn... | https://mathoverflow.net/users/5587 | How do various notions of natural transformation relate to various notions of homotopy in $2Cat$? | For strict transformations between strict 2-functors, you can just use the cartesian product $\Delta\_1\times A$.
For lax transformations between strict 2-functors, this is what the lax version of the Gray tensor product does. The nLab page is mostly about the pseudo version (which corresponds to pseudo natural trans... | 7 | https://mathoverflow.net/users/49 | 84759 | 50,522 |
https://mathoverflow.net/questions/84624 | 12 | I am interested in the irreducible unitary representations of the orthogonal groups $O(p,q)$. By $O(p,q)$ I mean the real Lie groups which preserve the quadratic form of signature $(p,q)$ in $\mathbb{R}^n$, $n = p+q$ dimensions. Special cases of interest in physics are the conformal group O(4,2), the deSitter group O(4... | https://mathoverflow.net/users/2365 | unitary irreps of O(p,q) | The unitary dual of Spin(n,1) is known for all n (Hirai, 1962, see Math Reviews MR0696689).
This gives the unitary dual of the identity component of SO(n,1) (which is a quotient
of Spin(n,1)). The unitary dual of any group can be deduced readily from that of its identity
component. Also SL(2,C)=Spin(3,1), and SL(2,R)... | 14 | https://mathoverflow.net/users/6030 | 84762 | 50,524 |
https://mathoverflow.net/questions/84511 | 12 | While studying calculus of variations, there is one question that I feel is missing in the texts I'm reading:
What can we say about the existence and/or uniqueness of solutions to Euler-Lagrange equations? Is there any general result?
I tried to google it, but found nothing.
| https://mathoverflow.net/users/7519 | Results about existence/uniqueness of solution to Euler-Lagrange equations? | The so called direct method of the calculus of variations provides one such existence and uniqueness result.
Here is the gist of it. Suppose that $X$ is a reflexive Banach space, e.g. a Hilbert space or a space of the form $L^p(\Omega)$, $p\in (1,\infty)$, $\Omega$ open subset of some Euclidean space. We are given a ... | 8 | https://mathoverflow.net/users/20302 | 84763 | 50,525 |
https://mathoverflow.net/questions/84757 | 8 | Let $n$ and $n$ are positive integers, $b>1$.
Express $n$ in $b$-basis
$$n = a\_kb^k + \cdots + a\_1b + a\_0.$$
We consider the polynomial
$$f\_{b,n}(X) = a\_kX^k + \cdots + a\_1X + a\_0 \in \mathbb{Z}[X].$$
Question 1: Let $p$ is a prime number. Then, is it true that $f\_{2,p}(X)$ is an irreducible polynomial?
I h... | https://mathoverflow.net/users/17901 | irreducible polynomial with repect to prime number | The answer is yes to both questions: it is a theorem of Brillhart, Filaseta, and Odlyzko
(see corollary 2, p.1058):
<http://cms.math.ca/10.4153/CJM-1981-080-0>
| 12 | https://mathoverflow.net/users/nan | 84768 | 50,529 |
https://mathoverflow.net/questions/84773 | 9 | I am preparing a course on profinite groups, to be delievered to early graduate students. The first part of the course will discuss the equivalent characterizations of profinite groups. I will first define a profinite group as a Hausdorff, compact topological group such that the open subgroups form a base for the neigh... | https://mathoverflow.net/users/20332 | Topological examples of profinite groups | If you view the Cantor space as a sequence space, then the isometry group is profinite (using the usual sort of metric that words are close if they have a long common prefix). The automorphism group of a locally finite rooted tree is profinite or more generally the stabilizer of a vertex is profinite in the automorphis... | 13 | https://mathoverflow.net/users/15934 | 84774 | 50,532 |
https://mathoverflow.net/questions/84772 | 5 | It is well known that there are bounded sequences with divergent Cesàro mean, i.e., a bounded $a\_n$ for which given $$c\_N := \frac{1}{N}\sum\_{n=1}^N a\_n,$$ the sequence $(c\_N)\_{N\geq1}$ has no limit. A simple example is $a\_n = (-1)^{\lfloor \log\_2 n\rfloor}$, for which the $(2^{n+1}-1)$-th term of the sequence ... | https://mathoverflow.net/users/9211 | Bounded sequences with divergent Cesàro mean | Choose any bounded infinite sequence $\{ b\_m \}\_{m \geq 0}$ of integers that is not eventually stationary, and let $a\_n = b\_{\lfloor \log (\log (n+2)) \rfloor}$. When $m$ is a large integer, and $N+2$ is almost $e^{e^m}$, the Cesàro mean $c\_N$ will be very close to $b\_{m-1}$.
If you want arbitrary iterated Cesà... | 5 | https://mathoverflow.net/users/121 | 84782 | 50,537 |
https://mathoverflow.net/questions/84705 | 32 | There are various frameworks around which enlarge the category of rings to include more exotic objects such as the 'field with one element,' $\mathbb{F}\_1$. While these frameworks differ in their details, there are certain things this should be true of any object that deserves to be called $\mathbb{F}\_1$. For example... | https://mathoverflow.net/users/4910 | Is the moduli space of curves defined over the field with one element? | This "answer" will basically restate the comments of Marty and Jason Starr.
Any variety covered by schemes of the form $\mathrm{Spec}(\mathbf Z[M\_i])$, or any torified variety, is rational. And indeed Severi conjectured at one point that $M\_g$ is rational for any $g$! But we know a lot about the Kodaira dimension ... | 33 | https://mathoverflow.net/users/1310 | 84785 | 50,538 |
https://mathoverflow.net/questions/84792 | 1 | Hi,
Consider the set $\lambda P\_n \subset \mathbb{Q}\_p$, where $\lambda \in \mathbb{Q}\_p^{\times}$ and $P\_n$ is the set $\lbrace x \in \mathbb{Q}\_p \mid \exists y \in \mathbb{Q}\_p x= y^n\rbrace$. Is there a way to measure $\lambda P\_n \cap B(r)$ for $r$ in $\mathbb{R}\_{>0}$ the ball of radius $r$ (with Haar me... | https://mathoverflow.net/users/nan | The set of $p$-adic numbers of some fixed $n^ {\rm th}$-power residue | What you are denoting $P\_n$ is more usually denoted $\mathbf{Q}\_p^{\times n}$, the image of the endomorphism $(\ )^n$ of $\mathbf{Q}\_p^\times$.
To compute the quotient $\mathbf{Q}\_p^\times/\mathbf{Q}\_p^{\times n}$, recall that there is an exact sequence
$$
1\to\mathbf{Z}\_p^\times\to\mathbf{Q}\_p^\times\to\mat... | 3 | https://mathoverflow.net/users/2821 | 84798 | 50,543 |
https://mathoverflow.net/questions/84672 | 4 | Recall that given two strict ω-categories $A$ and $B$, their lax Gray tensor product $A\otimes B$ is sent to the Verity-Gray tensor product of their associated complicial sets $N\_{\operatorname{Strat}}(A) \otimes N\_{\operatorname{Strat}}(B)$ by the stratified nerve of Verity. Further, given any strict ω-category $A$,... | https://mathoverflow.net/users/1353 | Why is the Street nerve of the Gray tensor product $[1]\otimes [1]$ isomorphic to $[1]\times [1]$ | Alright, I figured it out.
Here's the problem: The Verity tensor product of complicial sets is obtained as follows:
$$A\otimes\_{\operatorname{Cs}} B = L\_{\operatorname{Cs}} (\iota\_{\operatorname{Cs}}(A) \otimes\_{\operatorname{Strat}} \iota\_{\operatorname{Cs}}(B)),$$
where $$L\_{\operatorname{Cs}}:\operatorna... | 3 | https://mathoverflow.net/users/1353 | 84802 | 50,545 |
https://mathoverflow.net/questions/84805 | 0 | Assume we have a colored Gaussian process $z\_t$, with an autocorrelation function $cov(z\_t,z\_s)$ given by an analytical function $\alpha(t,s)$ (if it helps, one can assume that $\alpha(t,s) = \kappa e^{-w(t-s)}$). Consider now a process defined by
$Z\_t := \int\_0^t z\_s ds$
Now, my 3 questions:
* is $Z\_t$ we... | https://mathoverflow.net/users/1580 | Integrated colored Gaussian noise | Q1: Rather yes. If we assume that function $w$ is not crazy, e.g. $C^1$, then there exists a continuous version of $z$ and the integral can be computed path-by-path using the Lebesgue (or even Riemann) integral theory.
Q2:Under assumption above $Z$ has paths of the finite variation. Hence there exists the Stieltjes i... | 1 | https://mathoverflow.net/users/1302 | 84808 | 50,548 |
https://mathoverflow.net/questions/84811 | 3 | I am looking for
1. varieties without $\mathbf{Q}$-rational points where the absence can be explained by the Brauer-Manin obstruction, but not by the absence of adelic points
2. varieties without $\mathbf{Q}$-rational points where the absence cannot be explained by the Brauer-Manin obstruction
Thank you in advance!... | https://mathoverflow.net/users/nan | Brauer-Manin obstruction and Hasse principle | Everything you want (and a whole lot more) is here: A. N. Skorobogatov, Torsors and rational points. Cambridge Tracts in Mathematics, 144. Cambridge University Press, 2001.
| 13 | https://mathoverflow.net/users/2290 | 84814 | 50,552 |
https://mathoverflow.net/questions/84812 | 5 | I want to know about reference of formulas for
$$
L(s,D)=\sum\_{n=1}^\infty \left(\frac{D}{n}\right)\,n^{-s}
$$
for $s$ a positive integer number and $D$ a fundamental discriminant. For $s=1$ we have the *Dirichlet class number formula*.
I would like to have some reference for $s\geq2$. Thanks.-.
| https://mathoverflow.net/users/20052 | Values of Dirichlet L-funcions at natural numbers | Let $\chi$ be any Dirichlet character modulo $q$, and let $m$ be a positive integer. Then Theorem 4.2 of Washington's *Introduction to Cyclotomic Fields* states that
$$ L(1-m,\chi) = - \frac{q^{m-1}}{m} \sum\_{a=1}^q \chi(a)B\_{m}(\tfrac{a}{q}). $$
Here $B\_m(x)$ is the usual Bernoulli polynomial, defined by
$$ \... | 13 | https://mathoverflow.net/users/3659 | 84817 | 50,553 |
https://mathoverflow.net/questions/84820 | 4 | The Busy-Beaver trick provides a nice example of non-computable functions (let say from $\mathbb{N}$ to $\mathbb{N}$) which grows faster than any computable functions. But what can we say when we do not restrict ourselves to computable functions. My question could be, given a countable sets of functions, can we always ... | https://mathoverflow.net/users/14490 | What about the fastest-growing non-computable function ? | An easy diagonalization shows that for every countable family of functions $g\_n:\mathbb{N}\to\mathbb{N}$, there is a function $f$ eventually exceeding any one of them. Just let $f(n)=\sup\_{k\leq n}g\_k(n)+1$.
Although it may seem difficult to extend this idea to uncountable families of functions, the fact is that ... | 14 | https://mathoverflow.net/users/1946 | 84822 | 50,555 |
https://mathoverflow.net/questions/84801 | 8 | A virtually-$\mathbb{Z}$ group $G$ admits either a epimorphism onto $\mathbb{Z}$ or a epimorphism onto $D\_\infty$.
So what happens if one replaces $\mathbb{Z}$ by another group $F$ (like the free group or $\mathbb{Z}^n$). My first guess would be that any virtually-$F$ group $G$ maps surjectively onto one of the gro... | https://mathoverflow.net/users/3969 | Analogues of the dihedral group | Here's an idea for a proof that the modular group $\Gamma=\mathbb{Z}/2\*\mathbb{Z}/3$, which is, of course, virtually free, doesn't surject a group of the form $F\rtimes H$. I don't have time to work out the details.
First, I think it's plausible that the only reduced, non-trivial graph-of-groups decomposition for $\... | 8 | https://mathoverflow.net/users/1463 | 84829 | 50,557 |
https://mathoverflow.net/questions/84842 | 10 | I was helping a friend prepare for his intro abstract final and he mentioned the professor had once asked the question: name a group and an automorphism that takes $3/4$ of the elements of the group to their own inverses (for instance, the dihedral group $D\_4$ of order $8$, with identity automorphism). I tried to figu... | https://mathoverflow.net/users/20349 | Finite groups with automorphism mapping $a/b$ of the elements of $G$ to their own inverses? Case $a/b=3/4$? | This may be a well-known chestnut? (well-known to those that know it well, that is)
The fraction can never be between 3/4 and 1. To prove this, suppose $\phi\colon G\to G$ is an automorphism of $G$ that sends more than 3/4 of the elements of $G$ to their inverses. Let $S=\lbrace g\in G\colon \phi(g)=g^{-1}\rbrace$.
... | 18 | https://mathoverflow.net/users/11054 | 84844 | 50,566 |
https://mathoverflow.net/questions/84858 | 2 | I am interested in the distance between two conjugacy classes in a group like $SO(n)$. However let's consider $U(n)$ for simplicity. My conjecture is that the Hausdorff distance between the conjugacy classes of $\Lambda\_1$ and $\Lambda\_2$, both of which are diagonal matrices, is given by the distance of the eigenvalu... | https://mathoverflow.net/users/4923 | Riemannian Hausdorff distance between two conjugacy classes in a compact Lie group | Diagonalization gives a map $f\colon U(n)\to \mathbb{T}^n/S\_n$.
The map $f$ is also the projection to the orbit-space of the $U(n)$-action by conjugacy on $U(n)$.
Hence $f$ is a submetry;
i.e., $f(B\_r(M))=B\_r(f(M))$ for any matrix $M$.
Hence your statement follows.
| 5 | https://mathoverflow.net/users/1441 | 84862 | 50,574 |
https://mathoverflow.net/questions/84783 | 12 | I am interested in regular graphs in which every edge lies in a triangle.
For 3-regular graphs, only the complete graph $K\_4$ has this property, so there's not much to see here.
For 4-regular graphs, there are more graphs, including some infinite families, but few enough and slowly-growing enough that I have some ... | https://mathoverflow.net/users/1492 | 4-regular graphs with every edge in a triangle | yes, it feels like one could proof my characterization along the following lines:
* If the graph does not contain $K\_4^-$, then it is a line graph of a 3-regular graph.
* Now continue with a copy of $K\_4^-$. The two vertices of degree three must each be incident to another edge. If these two edges meet, then we can... | 7 | https://mathoverflow.net/users/12487 | 84875 | 50,579 |
https://mathoverflow.net/questions/84867 | 0 | Let $T\_1,T\_2,....T\_n$ be numbers such that
$T\_k= k$ no. of digits in decimal expansion of an irrational number, say $\alpha$, starting from $(\frac{k(k-1)}{2}+1)^{th}$ digit in the decimal expansion. e.g. for $\pi$, according to this definition, we have
$T\_1=0.1, T\_2=0.41, T\_3=0.592$ and so on.
Question: Und... | https://mathoverflow.net/users/17614 | Under what condition will this set contain a limit point of [0,1)? | In the customary language *a limit point* means *limit of a subsequence*. If so, the condition just means that the sequence $T\_k$ does not converge to $1$, and this reflects on the form of the decimal expansion of $\alpha$ quite in a simple way. Of course, it could be one of the billions of impossible, yet not very in... | 0 | https://mathoverflow.net/users/6101 | 84879 | 50,581 |
https://mathoverflow.net/questions/84824 | 24 | Recently, as part of some joint research, Tom Roby was led to a curious operation on strings of L's and R's which he calls "bounce-reading": We start by reading the string at the left. When the symbol we have just read is an L, the next symbol we read is the leftmost unread symbol; but when the symbol we have just read... | https://mathoverflow.net/users/3621 | an operation on binary strings | Well, it has now, since I just sank my morning into studying it. I sure am a sucker for a naive combinatorics problem. Here's what I know, or can conjecture:
* The map you describe is a bijection on words of length $n$, because it's easy to write down its inverse. I've included python code below.
* Let $B\_L$ be the ... | 19 | https://mathoverflow.net/users/20281 | 84880 | 50,582 |
https://mathoverflow.net/questions/84856 | 0 | For $T: \mathbb{R} \mapsto \mathbb{{R}\_{+}}$, we have $\{ {T}^{n}(\theta)\ mod \ 1\} \subset [0,1)$. (where ${T}^{n}(\theta)$ means applying $T$ $n$ times on $\theta$, not the $n$th power of $T(\theta)$)
**Question 1:** Does there exist (preferably elementary) $T$ such that $\{{T}^{n}(\theta)\ mod \ 1\}$ is dense in... | https://mathoverflow.net/users/75935 | Modulo dynamics on [0,1) | 1. Interval Exchange Transformations provide a vast class of further examples.
2. Let $f:[0,1] \to [0,1]$ be increasing, then the conjugation $f^{-1} \circ T \circ f$ is another example for $T$ being an example.
3. Let $\mathbb{T} = \mathbb{R}/\mathbb{Z}$. Then for $f:\mathbb{T} \to \mathbb{T}$ twice continuously diffe... | 2 | https://mathoverflow.net/users/3983 | 84891 | 50,589 |
https://mathoverflow.net/questions/84853 | 2 | Consider first the torus group $\mathbb{T}^k$. A natural $L^2$ basis is given by the 1-dimensional complex representations: $(\theta\_1, \ldots, \theta\_k) \mapsto e^{i \sum\_j c\_j \theta\_j}$ for integral $c\_j$'s. This basis has also the nice property that each element in it is point-wise bounded by $1$ in absolute ... | https://mathoverflow.net/users/4923 | L^2 basis of class functions on a compact Lie group that are point-wise small | There is a sharp comparison of sup norm to $L^2$ norm on compact topological groups $K$. (I saw a proof of the version of this for orthogonal groups or spheres in Stein-Weiss "Fourier Analysis on Euclidean Spaces", but the argument succeeds generally. E.g., section 7 of <http://www.math.umn.edu/~garrett/m/mfms/notes_c/... | 2 | https://mathoverflow.net/users/15629 | 84892 | 50,590 |
https://mathoverflow.net/questions/84898 | 7 | I have two questions about Cayley graphs. Any answers will be appreciate.
1) Do we have any Cayley graph that has Petersen graph as its induced subgraph?
2) Suppose $Cay(G,S)$ be a Cayley graph that $G$ is a finite group. Can we characterize any induced subgraphs of $Cay(G,S)$?
Thanks for any answer and guidance.... | https://mathoverflow.net/users/19885 | Cayley graphs and its subgraphs | If $X$ is a vertex-transitive graph and the stabilizer of a vertex has order $m$, then the lexicographic product of $K\_m$ by $X$ is a Cayley graph. We get the lexicographic product here by replacing each vertex of $X$ by $K\_m$ and, where two vertices of $X$ are adjacent, join
each vertex in one $K\_m$ to each vertex ... | 12 | https://mathoverflow.net/users/1266 | 84899 | 50,593 |
https://mathoverflow.net/questions/84726 | 0 | I have several questions on Lindelöf property.
If every point countable open cover of $X$ has a countable subcover (**Condition A**), does $X$ have Lindelöf property? How far is having **Condition A** from Lindelöf property?
**A space $X$ is called $\omega\_1$-Lindelöf if every $\omega\_1$-sized open cover of $X$ c... | https://mathoverflow.net/users/18465 | some questions on Lindelöf property | A space is Lindelöf iff it satisfies condition A and is metaLindelöf (every open cover has a point-countable refinement), so one could say the difference is metaLindelöfness.
| 1 | https://mathoverflow.net/users/2060 | 84905 | 50,596 |
https://mathoverflow.net/questions/84902 | 4 | Can someone point out the gap in this argument. Consider a simply-connected Lie group with the (-)-connection. This connection is flat and so the sectional curvatures are zero. Then, by the Cartan-Hadamard theorem and simple-connectedness, the Lie group must be diffeomorphic to ${\Bbb R}^n$. However, I don't think that... | https://mathoverflow.net/users/14454 | Cartan-Hadamard Theorem | As Emerton pointed out, you need to be careful about the connection. Cartan-Hadamard theorem is a statement involving the curvature of the Levi-Civita connection determined by some metric.
If $G$ is a Lie group equipped with a **bi-invariant** metric $h$, then this metric induces a metric $\langle-,-\rangle$ on the L... | 7 | https://mathoverflow.net/users/20302 | 84908 | 50,599 |
https://mathoverflow.net/questions/84857 | 8 | I'm not a mathematician but I working on a problem that feels like it an example of a more general kind of problem and I'm hoping that someone might be able to point me in the right direction.
The problem is trying to find a convex combination of different ways of ranking n items that satisfy certain constraints. Le... | https://mathoverflow.net/users/13456 | Equitable Allocation of Individuals to Positions | This is not a complete answer but too long for a comment. You wrote,
>
> I originally thought this could be framed as a linear programming problem, where the goal is to find weights for each of the n! possible orderings. Maybe this would work, but it would be computationally infeasible.
>
>
>
Linear programs w... | 5 | https://mathoverflow.net/users/11828 | 84913 | 50,603 |
https://mathoverflow.net/questions/84904 | 4 | Without appealing to a guess-and-check approach, how might I select a pair of random points inside of a sphere of radius $R$ s.t. the points always a distance $d \leq R$ apart? Can the selected points be 'random' in the sense that a large number of such pairs can be split into two equal-sized populations that independe... | https://mathoverflow.net/users/20364 | Selecting two random points inside a sphere which are a fixed distance apart | From an algorithmic point of view, a very simple and computationally effective way to produce points is exactly the guess and check that you say you don't want. [ You don't say, but I'm assuming you're working in 3 dimensions? As the dimensionality increases, so the methods that I'm talking about become progressively w... | 5 | https://mathoverflow.net/users/11054 | 84914 | 50,604 |
https://mathoverflow.net/questions/84910 | 3 | We define the affine Grassmannian to be the quotient $Gr = GL\_n(\mathbb{C}((t)))/GL\_n(\mathbb{C}[[t]])$ where $\mathbb{C}((t))$ is the field of formal Laurent series and $\mathbb{C}[[t]]$ is the ring of formal power series. (The affine Grassmannian can be defined more generally, but here we restrict to a special case... | https://mathoverflow.net/users/20366 | What is the Bruhat decomposition of the affine Grassmannian? | There are two discrepancies: first, the "strict" affine Bruhat decomposition applies to SL(n), not GL(n); second, the decomposition you're wanting, in the *strict* case, would be $G=\bigcup\_w IwI$, that is, with Iwahori on both sides. (This would make it a disjoint union.)
To adapt to GL(n), the affine Weyl group ne... | 3 | https://mathoverflow.net/users/15629 | 84916 | 50,605 |
https://mathoverflow.net/questions/84881 | 5 | Is there a nonunimodular group for which Wiener's Tauberian theorem is true?
Is a locally compact topological group whose volume grows polynomially with radius always unimodular?
| https://mathoverflow.net/users/17822 | Wiener Tauberian Theorem for nonunimodular group | The answer to the second question is yes. Let us show that a non-unimodular, locally compact group $G$ cannot have polynomial growth. Let $\mu$ be left Haar measure, $\Delta$ be the modular function, so that $\mu(Ag)=\mu(A)\Delta(g)$ for $A$ a Borel subset in $G$. Now take for $A$ a compact neighborhood of identity, an... | 6 | https://mathoverflow.net/users/14497 | 84919 | 50,606 |
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