parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/176402 | 1 | I have a point process defined in two parts as follows. Consider first the main process which we call $A$ which is homogeneous Poisson process with conditional intensity
$$\lambda(t) = \mu$$
For every arrival time $x\_i$ that $A$ produces we run another process which we call $B$. The conditional intensity of this p... | https://mathoverflow.net/users/45564 | An efficient method to find the MLE of the combination of two point processes | If I'm not mistaken, there are always infinitely many solutions to this problem.
Informally: Suppose you plant an average of $\mu$ (fertile) parent plants per year, and each parent yields an average of $\gamma$ (infertile) children per year. An outside observer will then see the number of plants grow as $O(T^2)$. Ho... | 2 | https://mathoverflow.net/users/56198 | 176696 | 89,019 |
https://mathoverflow.net/questions/176644 | 1 | I posted this question on math.stackexchange but got no answer, so I decided to post it here instead. Sorry about the impreciness, not professional mathematician here.
Let's assume we have a set of 2D-points, and their coordinates are in a $n \times 2$ matrix. My claim is that if that set has at least one valid symme... | https://mathoverflow.net/users/7420 | True or false: if a set of 2D points has valid symmetry axes, then at least one of them is equal to a principal component vector | It is sufficient to consider the situation where the axis in question is the first coordinate axis (if it is not, apply rotation). In this situation, the matrix of covariances has zero off-diagonal entries. To see this, reflect all your data with respect to the first axis. On the one hand, by doing so you will change t... | 2 | https://mathoverflow.net/users/2968 | 176701 | 89,022 |
https://mathoverflow.net/questions/176705 | 2 | Take a vector space $V$ (finite dimensional, over the complex numbers), let $G=SL(V)$. The group $G$ acts on $\mathbb{P}V$ and we can linearize its action to an action on the line bundle $\mathcal{O}(1)$. This gives an action on $H^0(\mathbb{P}V,\mathcal{O}(1))$. My question is
as $G$-module, is $H^0(\mathbb{P}V,\mat... | https://mathoverflow.net/users/48866 | Example of linearization for GIT | By definition $\mathbb{P} V = \mathrm{Proj}\ \mathrm{Sym}(V^\vee)$. Quasi-coherent sheaves on the $\mathrm{Proj}$ are identified with graded modules over the graded ring (modulo torsion). Under this correspondence the sheaf $\mathcal{O}(1)$ goes to the graded module $\mathrm{Sym}(V^\vee)[1]$ (the ring shifted down by 1... | 3 | https://mathoverflow.net/users/18512 | 176713 | 89,030 |
https://mathoverflow.net/questions/176637 | 4 | Let $R\_1,R\_2$ be (left and right) ore domains. Does $ Mat\_n(R\_1)\cong Mat\_m(R\_2)$ implie m=n and $q.f.(R\_1)\cong q.f.(R\_2)$?
An counter example, a proof or a reference is welcomed.
Thanks
| https://mathoverflow.net/users/55982 | Isomorphism of matrix ring over ore domain | In an non-commutative ring $A$, recall that an idempotent $\pi$ is an element with $\pi^2=\pi$. Two idempotents $\pi$ and $\pi'$ are called orthogonal if $\pi \pi' = \pi' \pi=0$.
**Claim** Let $D$ be a division algebra. Then $n$ is the maximal size of a set of nonzero mutually orthogonal idempotents in $M\_n(D)$.
*... | 3 | https://mathoverflow.net/users/297 | 176715 | 89,031 |
https://mathoverflow.net/questions/176695 | 5 | Consider the full flag variety $F\_n$ consisting of full flags in $\mathbb C^n$. There is a collection of tautological bundles on $F\_n$:
$0=U\_0\subset U\_1\subset ...\subset U\_{n-1}\subset U\_n=\mathbb C^n\otimes O\_{F\_n}$.
Recall that for $i=1,...,n-1$ the classes $\sigma\_i=c\_1(U\_i^\*)\in H^2(F\_n)$ form a... | https://mathoverflow.net/users/13441 | Non-vanishing of elements in cohomology of full Flag varieties | Yes. I claim that $(\sigma\_1 \sigma\_2 \cdots \sigma\_{n-1})^{\lfloor n/2 \rfloor} \neq 0$, which is better than anything you ask for. (Here $\lfloor x \rfloor$ is $x$ rounded down to the nearest integer.)
Recall that a basis for $H^{\ast}(FL\_n)$ is the Schubert classes $[X\_w]$, indexed by permutations $w$ in $S\... | 5 | https://mathoverflow.net/users/297 | 176717 | 89,033 |
https://mathoverflow.net/questions/176663 | 4 | Let $A\_1,\dots,A\_L$ be $N\times N$ hermitian matrices. Define the simplex
\begin{align}
\mathcal{S}=\left\{[x\_1,\dots,x\_L]\mid x\_i\geq 0,~\sum\_{i=1}^{L}x\_i=1 \right\}
\end{align}
and consider the function over the simplex
\begin{align}
f(x\_1,\dots,x\_L)=\lambda\_{min}\left(\sum\_{i=1}^{L}x\_iA\_i\right)
\end{al... | https://mathoverflow.net/users/27249 | Is this function well studied? | Yes, this function is well-studied. It can be maximized by semidefinite programming (and is in fact one of the standard examples of a non-trivial function that can be maximized by SDP).
The paper "[M. Huhtanen and O. Seiskari. Computational geometry of positive definiteness, Linear Algebra and its Applications, 437:1... | 5 | https://mathoverflow.net/users/11236 | 176718 | 89,034 |
https://mathoverflow.net/questions/176689 | 8 | Let $G$ be an Erdos-Renyi random graph (i.e. an edge ($ij$) exists with probability $0 < p < 1$ and all edges are independent). Let $L$ be the Laplacian matrix of this graph (i.e $L=D-A$, where $A$ is the adjacency matrix of the graph and $D$ is diagonal with $D\_{ii} = $ degree of node $i$). Let finally $0 = \lambda\_... | https://mathoverflow.net/users/56254 | limiting empirical spectral distribution of the Laplacian matrix on an Erdos-Renyi graph? | Maybe relevant:
"Lifshitz tails for spectra of Erdős–Rényi random graphs"
Oleksiy Khorunzhiy, Werner Kirsch, and Peter Müller
"We consider the discrete Laplace operator (the graph Laplacian) on Erdos–Rényi random graphs and show in Theorem 2.5 that the asymptotic behavior of its limiting integrated density of sta... | 4 | https://mathoverflow.net/users/39754 | 176725 | 89,036 |
https://mathoverflow.net/questions/176057 | 0 | I want to implement a minimisation on a 2D spin Ising model with 30x30 grid. The spin variables is 0,1 and the objective is to minimize the sum of products of spins. For simplicity, I only include NN pair interaction and a "triangular" interaction term. The formulation I give is as followed. And I wish to ask for sugge... | https://mathoverflow.net/users/40780 | Mixed integer programming formulation for Ising model | This problem is studied in Condensed Matter. You may wish to look at Projected Entangled Pair States - the 2D generalization of famous Density Matrix Renormalization Group / Matrix Product States ansatz. The recent development is reported in <http://arxiv.org/abs/1405.3259> but do consider references therein as well.
... | 1 | https://mathoverflow.net/users/52964 | 176726 | 89,037 |
https://mathoverflow.net/questions/176714 | 1 | For any real number $x$, let's define $Om\_{k}(x)$ as the number of positive integers $m$ below $x$ such that $\Omega(m)-\omega(m)=k$, where $\omega(n)$ is the number of distinct primes dividing $n$, and $\Omega(n)$ the total number of prime factors of $n$ counted with multiplicity. Obviously $Om\_{0}(x)$ is just the n... | https://mathoverflow.net/users/13625 | Asymptotics and error terms for an arithmetic function built upon $\omega$ and $\Omega$ functions | This problem was studied by Renyi, in
*On the density of certain sequences of integers*
Publ. Inst. Math. Belgrade **8** (1955) 157-162.
<http://elib.mi.sanu.ac.rs/files/journals/publ/14/13.pdf>
Let $d\_k = \lim\_{x\to\infty} Om\_k(x)/x$. Then the $d\_k$ are the coefficients in the following beautiful identit... | 10 | https://mathoverflow.net/users/16510 | 176729 | 89,039 |
https://mathoverflow.net/questions/173261 | 2 | Two binary forms $f, g \in k[x, y]$ are equivalent when there exists an $M \in GL\_2 (k)$ such that $f^M = g$. For simplicity we take $k$ such that $char (k) =0$ and $k=\bar k$.
The equivalence classes of binary forms are determined by the $GL\_2 (k)$-invariants and they are known for degree $d \leq 8$ (and possibly... | https://mathoverflow.net/users/nan | Equivalent binary forms | From what I have found it seems as the best way to do this is to use an algorithm of Stoll on reducing the binary form. The algorithm is based on Julia's invariant and it is very nice.
For details please see
[Reduction theory of point clusters in projective space](http://arxiv.org/abs/0909.2808), Michael Stoll,
... | 0 | https://mathoverflow.net/users/nan | 176730 | 89,040 |
https://mathoverflow.net/questions/176697 | 6 | I am looking for number fields $K$ which satisfy the following properties:
1. $[K:\mathbb{Q}]=5$.
2. The Galois closure of $K$ has Galois group $S\_5$.
3. For each prime $p$ which *ramifies* in $K$, there exists a prime ideal $\mathfrak{p}$ of $K$ of inertia degree and ramification index $1$ above $p$, i.e., we have
... | https://mathoverflow.net/users/5101 | Constructing quintic number fields with certain splitting behaviour | Since you are okay with other non-abelian Galois groups, let $F$ be a quadratic extension of $\mathbb{Q}$ with class number divisible by $5$. (There are infinitely many, see [this paper](http://www.math.snu.ac.kr/~dhbyeon/23_class57-2.pdf) and references therein). By class field theory, $F$ has an unramified $\mathbb{Z... | 5 | https://mathoverflow.net/users/297 | 176734 | 89,041 |
https://mathoverflow.net/questions/176732 | 7 | Let us consider unquantified formulas of a set theory (for example, NBG), more precisely,
the formulas, constructed from variables and the constants $\emptyset, V$ (the empty set
and the class of all sets respectively), using only the following set-theoretic symbols:
$\cup, \cap, C, \subseteq, =$ (union, intersection, ... | https://mathoverflow.net/users/5761 | Set-theoretic tautologies | $\DeclareMathOperator\pr{Prop}\let\sset\subseteq\let\nsset\nsubseteq$First, if $F$ is any valid quantifier-free formula, then $\pr(F)$ is a tautology: given a propositional assignment $e$ such that $e(\pr(F))=0$, the valuation
$$v(X)=\begin{cases}\varnothing&e(X)=0,\\V&e(X)=1\end{cases}$$
is a counter-example to $F$. T... | 12 | https://mathoverflow.net/users/12705 | 176751 | 89,045 |
https://mathoverflow.net/questions/176746 | 5 | Let $\mathbb R$ be endowed with the standard Euclidean topology and let $\widetilde {\mathbb R}$ denote the line endowed with the *discrete* topology. Let $\mu$ and $\nu$ denote the Lebesgue and counting measures on these two spaces, respectively. Define the product measure in the usual way: $$(\mu\times\nu)(E)\equiv\i... | https://mathoverflow.net/users/55976 | Product of Lebesgue and counting measures | The conclusion is still true.
Suppose $E$ is covered by the sets $A\_k\times B\_k$, $k\in\mathbb N$.
Let $M\subset\mathbb N$ be the set of such indices $k$ that $\mu(A\_k)=0$.
Let $S=\bigcup\_{k\in M}A\_k$; we obviously have $\mu(S)=0$ so that $[0,1]\setminus S$ is uncountable.
Now the set $F=\{(x,x);x\in [0,1]\set... | 5 | https://mathoverflow.net/users/55893 | 176752 | 89,046 |
https://mathoverflow.net/questions/176754 | 3 | Let $p:M\to \mathbb{R}^{n+1}$ be the closed immersed hypersurface. Is the following thing right? If there exists a point $x\_0$ in $\mathbb{R}^{n+1}$ such that $\langle x(p)-x\_0,\nu(p)\rangle>0$ for all $p\in M$, then $M$ is diffeomorphic to a sphere. Here $\nu(p)$ is the outward normal vector.
| https://mathoverflow.net/users/48506 | Is the hypersurface satisfying $\langle x-x_0,\nu\rangle>0$ diffeomorphic to sphere? | Assuming that $M$ is connected compact without boundary the answer is YES.
1. Consider a map $\pi: p \mapsto \frac{x(p)-x\_0}{||x(p)-x\_0||}$ from $M$ to $S^n$.
2. $\pi$ is an immersion since $\langle x(p)-x\_0,\nu(p)\rangle >0$.
3. $\pi$ is a covering since $\pi$ is an immersion, $M$ is a closed $n-$manifold, and $... | 1 | https://mathoverflow.net/users/38523 | 176759 | 89,048 |
https://mathoverflow.net/questions/172532 | 8 | For $n$ an integer divisible by $8$, let me denote by $E\_n$ the "usual" even non-degenerate positive definite integral symmetric bilinear form over $\mathbf Z^n$.
It is well known that in dimension 16, the only even non-degenerate positive definite integral symmetric bilinear form are (up to isometry) $E\_8 \oplus E... | https://mathoverflow.net/users/5239 | Easiest way to distinguish $E_8 \oplus E_8$ from $E_{16}$ | One of the possible tricks I was after was the following: the lattice generated by the roots (vectors of length $2$) of $E\_8 \oplus E\_8$ is $E\_8 \oplus E\_8$, whereas the lattice generated by the roots of $E\_{16}$ is $D\_{16}$ (which is of index $2$ in $E\_{16}$).
Pari/gp has a very efficient way to list the root... | 9 | https://mathoverflow.net/users/5239 | 176761 | 89,049 |
https://mathoverflow.net/questions/176733 | 10 | I am trying to prove a result for which I need the nth term of the Baker-Campbell-Hausdorff formula. I came at this particular result (which is not of significance for the question, but mentioning for context) by hypothesizing and using the first few terms of the Baker formula to verify. In order to prove my result rig... | https://mathoverflow.net/users/37893 | nth term in the Baker-Campbell-Hausdorff formula | The Dynkin formula is somewhat cumbersome. Maybe a better choice is Goldberg's version <http://projecteuclid.org/euclid.dmj/1077466673> In the commutator form Goldberg's result is reformulated in <http://www.ams.org/journals/proc/1982-086-01/S0002-9939-1982-0663855-0/> (Cyclic relations and the Goldberg coefficients in... | 10 | https://mathoverflow.net/users/32389 | 176764 | 89,051 |
https://mathoverflow.net/questions/176769 | 4 | Let $I(X,Y):=H(X)+H(Y)-H(X,Y)$ be the mutual information of the joint probability distribution $p\_{XY}$ (here $H(\cdot)$ is the Shannon entropy of its argument). I know that the mutual information is invariant IF $X'=\text{invertible\_function}(X)$ and $Y'=\text{invertible\_function}(Y)$, that is, it is invertible und... | https://mathoverflow.net/users/56299 | Invariance of mutual information | This is not true. Look at the following counterexample:
Suppose $X=X'$ and $Y'$ be a sufficient statistics of $Y$ with respect to $X$, that is we have a double Markov relation $X\to Y\to Y'$ and $X\to Y'\to Y$. The double Markov relation implies $I(X; Y')=I(X; Y)$. It also implies that either $X$ and $(Y, Y')$ are in... | 5 | https://mathoverflow.net/users/41666 | 176773 | 89,056 |
https://mathoverflow.net/questions/176578 | 11 | The fact that the Schur (that is, element wise) product of two positive definite (symmetric) matrices is positive definite immediately implies (using the convexity of the positive semi definite cone) that if $A$ is PSD, then so is $B=A\_f,$ such that $b\_{ij} = f(a\_{ij}),$ where $f$ is an analytic function all of whos... | https://mathoverflow.net/users/11142 | elementwise functions of positive definite matrix | If the matrices are real and the function you have in mind is real-valued, then you indeed get the characterization you suggested. This was first shown in "I. J. Schoenberg. Positive definite functions on spheres. Duke Math. J., 9:96-108, 1942" under the additional assumption that $f$ is continuous. As noted by UwF in ... | 7 | https://mathoverflow.net/users/11236 | 176774 | 89,057 |
https://mathoverflow.net/questions/176777 | 3 | Suppose $ \mathbb{P} := (P, {\leq\_P}, 1\_P) $ is a separative partial order. Let $ \mathbb{B} := \operatorname{RO}(\mathbb{P}) $ denote the Boolean completion.
Fix some dense embedding $ i \colon P \to B^+ $. Then $ i $ is an order-isomorphism (one-to-one, order-preserving) to its image.
Assume that $ \mathbb{P} $... | https://mathoverflow.net/users/48586 | Boolean completion (of a forcing notion) isomorphic to each of its cones | Your second statement is correct, simply because boolean completions are unique up to isomorphism.
For the stronger statement, let $b \in \mathbb{B}^+$. Let $\{ p\_\alpha : \alpha < \kappa \}$ be a maximal antichain of elements of $i[\mathbb{P}]$ below $b$. Pick some maximal antichain $\{ q\_\alpha : \alpha < \delta ... | 4 | https://mathoverflow.net/users/11145 | 176784 | 89,061 |
https://mathoverflow.net/questions/176293 | 0 | I have the following function for two matrices ${\bf A}$ and ${\bf B}$:
$f({\bf A}, {\bf B}) = \| {\bf Y - XAB} \|\_F^2 = trace\{({\bf Y - XAB)}^T({\bf Y - XAB)}\}$
where matrices ${\bf X}\_{n \times p}$ and ${\bf Y}\_{n \times q}$ are fixed, and matrices ${\bf A}\_{p \times r}$ and ${\bf B}\_{r \times q}$ are the ... | https://mathoverflow.net/users/50191 | Convexity of the Frobenius norm of the product of two matrices | $f$ is convex separately for $A$ or $B$ but is not (in general) for the couple $(A,B)$.
Proof: The Hessian of $f$ in $X,Y$ is the following QUADRATIC form:
$Q(H,K)=2(||XHB+XAK||^2+2trace((XAB)^TXHK))-4trace(Y^TXHK)$.
Then $Q(H,0)=2||XHB||^2$ and $f$ is convex for $A$ (and similarly for $B$) - One must say a litt... | 3 | https://mathoverflow.net/users/9091 | 176792 | 89,064 |
https://mathoverflow.net/questions/176786 | 8 | This is primarily a reference request. Does anyone know of any writing about algebraic K-theory spectra picking up elements in the stable homotopy groups of spheres in their Hurewicz image coming from the map $\mathbb{S}\to K(R)$? I have heard that $K\_1(\mathbb{Z})$ is generated by $\eta$ but don't have any idea of ev... | https://mathoverflow.net/users/11546 | Detection of stable homotopy by K-theory spectra | $K(\mathbb Z)$ detects the image of $J$ and Quillen's proof of the Adams conjecture about the image of $J$ is closely related to the $K$-theory of finite fields.
$S$ splits off of $K(S)$, if you are interested in such things.
The Quillen-Lichtenbaum conjecture says that $K(R)$ is pretty close to étale $K$-theory, ... | 12 | https://mathoverflow.net/users/4639 | 176830 | 89,077 |
https://mathoverflow.net/questions/176839 | 3 | What is the cardinality of the continuous dual of $C([0,1])$ (the set of continuous functions from $[0,1]\to \mathbb{R}$)?
| https://mathoverflow.net/users/29961 | Cardinality of $C^*([0,1])$ | The polynomial space $P([0,1])$ is dense in $C([0,1])$ by the Stone-Weierstrass theorem. Therefore $P$ and $C$ have the same (continuous) dual space. But since $P$ has a countable Hamel basis, its *algebraic* dual has the same cardinality as $\mathbb R^{\mathbb N}$. Thus $|C^\*|\leq|\mathbb R^{\mathbb N}|$. The evaluat... | 6 | https://mathoverflow.net/users/55893 | 176840 | 89,079 |
https://mathoverflow.net/questions/176673 | 3 | Consider the heat equation
$$ (\partial\_t + \Delta + V)u = 0$$
on a complete (open) Riemannian manifold with bounded geometry, where $V$ is a smooth and bounded potential.
Consider the semigroup generated by $A:=\Delta + V$ on the spaces $X = C\_0(M)$ or $X = L^p(M)$ for $1\leq p < \infty$. The domain of $A$ is in e... | https://mathoverflow.net/users/16702 | Decay of Solutions to the Heat equation | For $C$ It follows from the interior Schauder estimates. Namely, for $r>0$ denote $H\_r(x\_0,t\_0)=\{|x-x\_0|<r^2,\ 0<t\_0-t<r\}$. Then for $0<\alpha<1$
$$
\|u\|\_{C^{2+\alpha}(H\_r(x,t))}\le C\|u\|\_{C(H\_{2r}(x,t))}.
$$
Taking $r=t/2$ for $t\le 1$ and $r=1/2$ for $t>1$ gives $|\nabla^ku| \in CM)$, $k\le2$. Differenti... | 2 | https://mathoverflow.net/users/14551 | 176848 | 89,084 |
https://mathoverflow.net/questions/176849 | 0 | I want an example of a completely regular space which is not a normal space. I have tried a lot but am unable to construct any example
| https://mathoverflow.net/users/51004 | Topology : Study on Separation Properties | The example you are looking for is probably <http://en.wikipedia.org/wiki/Niemytzki_plane>
| 2 | https://mathoverflow.net/users/46104 | 176853 | 89,085 |
https://mathoverflow.net/questions/176851 | 3 | I'd appreciate if someone could check my reasoning. Suppose $S$ is a lightface $\Delta^1\_1$ class of reals. I want to argue that there is a computable $\Delta^0\_\alpha$ formula $\phi(Y)$, for $\alpha$ a computable ordinal and $Y$ a real free-variable, such that $X \in S \iff \phi(X)$. It seems to follow by a variant ... | https://mathoverflow.net/users/32178 | Are lightface \Delta-1-1 classes of reals describable with hyperarthmetic formulae? | I just recognized you are asking a weaker version. Then it is even simpler. By Kleene's classical result, a real is $\Delta^1\_1$ if and only if it is recursive in $0^{(\alpha)}$ for some recursive ordinal $\alpha$. Uniformly relativizing the proof, we have that a set $A$ of reals is $\Delta^1\_1$ if and only if $\Phi^... | 3 | https://mathoverflow.net/users/14340 | 176861 | 89,091 |
https://mathoverflow.net/questions/176862 | 32 | I've been calculating some Jones polynomials lately and I was just curious if there was a "physical" (or, rather, geometric) meaning to evaluating the Jones polynomial at a particular value of $t$.
For example, if I take the Jones polynomial for the (right) Trefoil knot, I have
$J(t) = t + t^3 - t^4$.
Is there s... | https://mathoverflow.net/users/47547 | The Jones polynomial at specific values of $t$ | The evaluation of the Jones polynomial at $e^{i\pi/3}$ is related to the number of [3-colourings](http://en.wikipedia.org/wiki/Fox_n-coloring) $tri(K)$ of $K$ (see also [here](http://en.wikipedia.org/wiki/Tricolorability)) as well as to the topology of the branched double cover $\Sigma(K)$:
$$tri(K) = 3\left|V^2\_K(e... | 25 | https://mathoverflow.net/users/13119 | 176866 | 89,093 |
https://mathoverflow.net/questions/176865 | 2 | Does the Diophantine equation $y^2=x^2-x^3+x^4$ have solutions other than
$x=1,y=1$? Interestingly, the Diophantine equation $y^2=x^2-x^3+x^5$ has such solutions: $x=3,y=15$, $x=5,y=55$, $x=56,y=23464$. However I don't know if there are any other solutions. The solution $x=3,y=15$ is related to the Mordell's equations ... | https://mathoverflow.net/users/32389 | Integer points on $y^2=x^2-x^3+x^4$ | The only integer solutions $(x,y)$ are $(0,0)$ and $(1,\pm 1)$.
Your equation is $y^2=x^2(1-x+x^2)$. This can only have integer solutions if $p(x)=1-x+x^2$ is a square or $y=0$. For $x>1$ we have $(x-1)^2<p(x)<x^2$, so $p(x)$ is not a square. For $x<0$ we have $x^2<p(x)<(x-1)^2$, so $p(x)$ is not a square. The only p... | 12 | https://mathoverflow.net/users/55893 | 176868 | 89,094 |
https://mathoverflow.net/questions/176869 | 16 | This is a cross-post of [this question](https://math.stackexchange.com/questions/865820/very-tight-prime-bounds) on MSE. I would not usually do this, but have decided to in this case since it has had no responses having been posted as a bounty question. I did not delete the MSE question since so many users have starred... | https://mathoverflow.net/users/45057 | Tight prime bounds | The inequality you propose would imply that
$$ |\operatorname{li}(n)-\pi(n)| \ll \frac{\sqrt{n}}{\log n}. $$
On the other hand, Littlewood (1914) showed that this relation is false, in fact he proved that the left hand side divided by the right hand side is $\Omega(\log\log\log n)$.
So the inequality you propose is f... | 27 | https://mathoverflow.net/users/11919 | 176870 | 89,095 |
https://mathoverflow.net/questions/176779 | 1 | For small values of $n$ ($2\leqslant n\leqslant 5$), the coefficients $a\_k = (-1)^k{n\choose k}{n+k\choose k}$ of the shifted Legendre polynomial $\tilde{P}\_n(x)$ satisfy the identity $\displaystyle\sum\_{k=1}^n \frac {ka\_k}{(n+k)(2k+1)} = \frac{(-1)^n}{4n^2-1}.$ Does that identity hold for all $n$? If so, how might... | https://mathoverflow.net/users/56304 | Identity involving shifted Legendre coefficients | If we write the right side of Robert's identity as
$$(-1)^n\frac{(\nu-2)(\nu-4)\cdots(\nu - 2n+2)}{(\nu+2)(\nu+4)\cdots(\nu+2n)},$$
we see that the identity is a partial fraction expansion of a proper rational function. (This is a littler simpler if we replace $\nu$ with $2\nu$.)
| 3 | https://mathoverflow.net/users/10744 | 176871 | 89,096 |
https://mathoverflow.net/questions/176822 | 17 | The [lonely runner conjecture](http://en.wikipedia.org/wiki/Lonely_runner_conjecture)
has several formulations.
They all involve a number $n$ runners running on a circular track,
each with a different speeds, and the conjecture is that each runner is eventually
"lonely" in a technical sense.
My question is essentiall... | https://mathoverflow.net/users/6094 | Not-lonely runners | I'll use Christian Remling's notation: We want to show that we can always find an empty interval of size $\epsilon\_n$ among any set of $n$ runners.
The Masked Avenger gives an argument for $\epsilon\_n =1/n$, and suggests that $\epsilon\_n=2/n$ may be close to optimal by an argument I can't follow.
Assuming the l... | 8 | https://mathoverflow.net/users/18060 | 176881 | 89,099 |
https://mathoverflow.net/questions/176873 | 7 | I'm reading [Osváth-Szabó's notes on Heegard Floer homology](http://math.mit.edu/~petero/Lectures.pdf), in particular about the surgery exact triangle.
On page 14 (numbered 42 on the document), they describe an isomorphism between the space of homology classes of Whitney triangles $\pi\_2(\mathbf{x},\mathbf{y},\mathb... | https://mathoverflow.net/users/56345 | In the definition of the Heegard Floer surgery exact triangle, what exactly is the correspondence between Whitney triangles and periodic domains? | If I had to guess, I'd go for the typo *and* a small misunderstanding.
I think that the isomorphism should indeed depend on the choice of a triangle $\psi\_0$ with $n\_z(\psi\_0) = 0$ (careful here: this has to be 0 and not 1, as you wrote above), but that it should be defined as
$$\psi\mapsto (n\_z(\psi), \mathcal... | 4 | https://mathoverflow.net/users/13119 | 176882 | 89,100 |
https://mathoverflow.net/questions/176863 | 22 | As noted in the question "[Links between Riemann surfaces and algebraic geometry](https://mathoverflow.net/questions/2704/links-between-riemann-surfaces-and-algebraic-geometry)", there are strong connections between Riemann surfaces and algebraic geometry - for example, compact Riemann surfaces are equivalent to smooth... | https://mathoverflow.net/users/16082 | History of the connection between Riemann surfaces and complex algebraic curves | The connection came from the paper by Dedekind and Weber "*Theorie der algebraischen Functionen einer Veranderlichen*", Crelle's Journal, 1882. In this paper the authors recover the theory by Riemann (including the famous Riemann-Roch theorem) by the abstract procedure of assigning a "curve" to an degree one transcende... | 19 | https://mathoverflow.net/users/6348 | 176887 | 89,102 |
https://mathoverflow.net/questions/176879 | 8 | Let $X$ be a (smooth projective geometrically connected) surface over a finitely generated field $k$. The Tate conjecture predicts that, for $l$ a prime number invertible in $k$, the Chern class map from Pic$(X) \otimes \mathbb Q\_l$ to the Galois invariants of the second etale cohomology group of $X$ is surjective (an... | https://mathoverflow.net/users/56348 | Is the Tate conjecture known for etale covers of products of curves | The answer to the last question (and therefore to the others) is yes. An étale cover of a variety $X$ is dominated by an étale Galois $G$-cover, for some finite group $G$; and this is given by a homomorphism $u:\pi \_1(X)\rightarrow G$. If $X=X\_1\times \ldots \times X\_p$, this gives homomorphisms $u\_i:\pi \_1(X\_i)\... | 8 | https://mathoverflow.net/users/40297 | 176888 | 89,103 |
https://mathoverflow.net/questions/176875 | 4 | In his striking paper of may 2013, Zhang showed the existence of an even integer $g\lt 70,000,000$ such that $g$ is a prime gap occurring infinitely often. What is the best unconditional lower bound for the number of primes $p$ below $x$ such that $p'-p=g$, with $p'$ the smallest prime strictly greater than $p$?
Tha... | https://mathoverflow.net/users/13625 | Lower bound for a prime gap occurring infinitely often | Zhang's proof can be refined to show that the number of those primes is $\gg x/\log^k x$, where $k$ is the size of the tuple whose translates contain the relevant pairs, i.e. $k=3{,}500{,}000$ in the original proof and $k=50$ in the current record by PolyMath8b. For more details see the Main Theorem in Pintz's article ... | 15 | https://mathoverflow.net/users/11919 | 176895 | 89,106 |
https://mathoverflow.net/questions/176844 | 16 | Let $\mathbb{H}^3$ be the three-dimensional hyperbolic space. Let $K$ be an imaginary quadratic number field and $\mathcal{O}\_K$ its ring of integers. Then $SL\_2(\mathcal{O}\_K)$ acts on $\mathbb{H}^3$ and the quotient
$$
M=\mathbb{H}^3 / SL\_2(\mathcal{O}\_K)
$$ is a non-compact hyperbolic manifold. I have heard se... | https://mathoverflow.net/users/56337 | Borel-Serre compactification of $\mathbb{H}^3 / SL_2(\mathcal{O}_K)$ | I do not know of any specific reference, but I will try to explain the situation a bit. For most of the stuff, one does not actually need references besides the paper of Borel-Serre defining the Borel-Serre compactification. Maybe the book of Elstrodt, Grunewald, Mennicke "Groups acting on hyperbolic space" contains so... | 14 | https://mathoverflow.net/users/50846 | 176910 | 89,114 |
https://mathoverflow.net/questions/176889 | -2 | Let a nonlinear differential equation is invariant under all diffeomorphisms, then we get lack of parabolicity?
| https://mathoverflow.net/users/56353 | Lack of parabolicity of PDE due to invariancy under diffeomorphisms? | I'll (try to) answer the question specifically in the case of Ricci flow. View the Ricci flow on a closed manifold $M$ as an initial value problem on the positive cone of positive definite symmetric two-tensors, where given an initial metric $g(0),$ we wish to find a path of metrics $g(t)$ solving,
\begin{align}
\frac{... | 1 | https://mathoverflow.net/users/49247 | 176916 | 89,115 |
https://mathoverflow.net/questions/176917 | 4 | Are there any good/modern references on the Keisler order. I have been reading Keisler's original paper, "Ultraproducts which are not Saturated", which introduces the order. However it is somewhat archaic and outdated (e.g. instead of writing $\prod\_D \mathfrak{A}$, we writes $prod$-$D$ $\mathfrak{A}$). The paper also... | https://mathoverflow.net/users/51323 | References for the Keisler Order | (4) is still wildly open; for (2) and (3), see theorem H in chapter 1 of Shelah's classification theory. I'm not sure what you mean by (1); if you meant "well-ordered" instead of "well defined," then that's also wildly open. EDIT: it seems immediate that the Keisler order is a partial order; am I missing something?
I... | 5 | https://mathoverflow.net/users/8133 | 176918 | 89,116 |
https://mathoverflow.net/questions/176893 | 2 | Suppose I have a polynomial
$$
p(x)=\sum\_{i=0}^n p\_ix^i.
$$
For simplicity furthermore assume $p\_n=1$.
As it is well known we may use Gershgorin circles to give an upper bound for the absolute values of the roots of $p(x)$. The theorem states that all roots are contained within a circle with radius
$$
r=\max\{|p\... | https://mathoverflow.net/users/14865 | Estimate maximal coefficient of a polynomial from a circle containing all roots | There are actually stronger estimates that deal with all of the roots. Let $r\_1,\ldots,r\_n$ be the roots of your polynomial (with multiplicities as appropriate). Then
$$
\max\_{0\le i\le n} |p\_i| \ge \frac{1}{4^n} \prod\_{j=1}^n \max\bigl\{|r\_j|,1\bigr\}
$$
and
$$
\max\_{0\le i\le n} |p\_i| \le 4^n \prod\_{j=1}^... | 6 | https://mathoverflow.net/users/11926 | 176919 | 89,117 |
https://mathoverflow.net/questions/176915 | 6 | This question is related to
[this one](https://mathoverflow.net/questions/141536/what-is-the-definition-of-being-smooth-for-a-function-from-a-lie-group-to-a-fr%C3%A9c). Let $G$ be a real Lie group (I should emphasize I only care about ordinary Lie groups, not Lie groups modeled on locally convex spaces or anything lik... | https://mathoverflow.net/users/4351 | $C^\infty$-vectors in general representations of Lie groups on locally convex spaces | Yes, the notions are pretty standard. Grothendieck gives a brief summary of differentiable vector-valued functions in Chapter III.8 of his Topological vector spaces.
As starting point, I would suggest Garth Warner's Harmonic Analysis on Semi-simple Lie groups, I, especially Appendix 2. There you'll find a ton of refe... | 9 | https://mathoverflow.net/users/56365 | 176921 | 89,119 |
https://mathoverflow.net/questions/146305 | 3 | I don't understand why for geometric description of a regular system, we take always the classical phase space as a symplectic manifold?
| https://mathoverflow.net/users/nan | a question about geometric quantization background | What is the classical phase space (by phase space I'll, atleast in this post, mean *classical* phase space)? It is simply the cotangent bundle $\mathrm{T}^{\*}(\mathscr{C})$ of the configuration space $\mathscr{C}$. "... in a lecture Thurston mentioned "Any manifold can be seen as the configuration space of some physic... | 0 | https://mathoverflow.net/users/nan | 176928 | 89,122 |
https://mathoverflow.net/questions/176933 | 2 | I want to read SGA7. Without considering the others SGA and EGA, Which are the textbooks for monodromy theory?
| https://mathoverflow.net/users/56372 | Reference request: SGA7 | Maybe the following book will be helpful:
<http://www.springer.com/birkhauser/mathematics/book/978-3-7643-7535-5> (The Monodromy Group, by Henryk Zoladek).
| 2 | https://mathoverflow.net/users/32389 | 176935 | 89,125 |
https://mathoverflow.net/questions/176926 | 3 | Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. Fix a basis $e\_1,\dots,e\_n$ of $\mathfrak{g}$ and let $e^1,\dots,e^n$ be its dual basis. We also use $e^i$ to denote the left-invariant 1-form on $G$ obtained from $e^i$. It seems that I can prove the following identity by brute force:
\begin{equation}\sum\_{... | https://mathoverflow.net/users/41734 | A Lie algebra identity | **Some premilinaries.**
Let $\Omega\cong\mathfrak{g}\otimes\bigwedge\mathfrak{g}^\*$ be the space of left-invariant $\frak{g}$-valued differential forms on $G$. $\Omega$ is a graded vector space, the Maurer-Cartan form $\theta=\sum e\_i\otimes e^i$ belonging to its degree $1$ part.
Extend the Killing form $(\cdot,\cd... | 3 | https://mathoverflow.net/users/17294 | 176943 | 89,129 |
https://mathoverflow.net/questions/176813 | 3 | Is the standard smooth structure of $S^{7}$ the only structure for which each of the following canonical embedding is an smooth embedding?:
1)$S^{6}\to S^{7}$
2)$S^{7}\to S^{8}$
| https://mathoverflow.net/users/36688 | The embeddings $S^{6} \to S^{7}$ and $S^{7}\to S^{8}$ | I will interpret (the negation of) your questions in the following way.
>
> Let $n>5$ and consider the topological embedding of the equator $S^{n-1}$ in a *topological* $S^n$.
>
> 1) Is there an exotic smooth structure on $S^n$ such that, for the round structure on $S^{n-1}$ the embedding is smooth?
>
> 2... | 6 | https://mathoverflow.net/users/13119 | 176946 | 89,131 |
https://mathoverflow.net/questions/176954 | 3 | Let $\mathcal M$ be a symmetric monoidal category, $S\subset \mathcal M$ a collection of objects and morphisms. I would like to construct the localization $\mathcal M \mathop{\longrightarrow}^T \mathcal M[S^{-1}]$, defined by
1. for each $x\in S$, there is $y\in\mathcal M[S^{-1}]$ such that $Tx\otimes y \cong I$;
2. ... | https://mathoverflow.net/users/18702 | Localization of symmetric monoidal category | Localising with respect to morphisms is fairly standard. Assuming that the tensor product of two morphisms in $S$ is again in $S$, the localised category should inherit a symmetric monoidal structure, just by the universal property.
Localising with respect to objects is Quillen's $\mathcal{A}^{-1}\mathcal{A}$ constru... | 5 | https://mathoverflow.net/users/10366 | 176965 | 89,138 |
https://mathoverflow.net/questions/176898 | 6 | Let $G$ be a compact group, $H$ a normal open subgroup, and $K$ a $p$-adic field (so that not all $G$-reps with coefficients in $K$ are semisimple). Let $V$ be a finite-dimensional topological $K$-vector space with continuous action of $H$, such that $V$ is irreducible as an $H$-representation. Is the induced represent... | https://mathoverflow.net/users/30726 | If $G$ is compact, $H \leq G$ open, $V$ an irreducible $H$-rep, is $\text{Ind}_H^G$ semisimple? | The answer is yes. I think it should be an exercise in any book on representation theory. Since $H$ has finite index in $G$, and $V$ is finite dimensional, so is the representation $W=Ind \_H^G (V)$ induced to $G$. In characteristic zero, this is equivalent to proving that the Zariski closure of $G$ in $GL(W)$ is reduc... | 6 | https://mathoverflow.net/users/23291 | 176971 | 89,143 |
https://mathoverflow.net/questions/143935 | 5 | I discovered today the notion of elegant Reedy category introduced in the paper *Reedy categories and the $\Theta$-construction* of Julia E. Bergner and Charles Rezk. An interesting property of such categories is the following (quoting nlab [elegant Reedy category](http://ncatlab.org/nlab/show/elegant+Reedy+category)):... | https://mathoverflow.net/users/36625 | About elegant Reedy categories | Actually, I think the statement that you quoted from the nLab is wrong. It was copied from v1 of the Bergner-Rezk paper, but v2 corrected the statement to be only about *simplicial* presheaves on $R$ (although I think the proof would probably work for presheaves on $R$ with values in any other presheaf category). I've ... | 5 | https://mathoverflow.net/users/49 | 176981 | 89,145 |
https://mathoverflow.net/questions/176991 | 4 | I want to find a example of a manifold that has positive scalar curvature but is not half conformally flat.
Does there exists such manifolds?
Thanks.
| https://mathoverflow.net/users/39997 | Example of a specific manifold | Take $S^2\times S^2$ with the standard product metric. It has positive scalar curvature, it can't be conformally flat because it's simply connected and compact and yet not the $4$-sphere, and it can't be conformally half-flat because switching orientation on one of the two factors gives an isometric manifold but switch... | 11 | https://mathoverflow.net/users/13972 | 176996 | 89,150 |
https://mathoverflow.net/questions/176994 | 6 | Let $G$ be a compact Lie group with real representation ring $RO(G)$. Recently, I have been learning about some aspects of $RO(G)$-graded cohomology theories (for a precise definition, see Chapter XIII of May's book, Equivariant Homotopy and Equivariant Cohomology). One impetus for introducing this definition is that i... | https://mathoverflow.net/users/25358 | $RO(G)$-Graded Cohomology Theories | There is no relationship in general. If $E$ is a module over the $G$-equivariant $MU$ spectrum then $E\_G^{n+V}(\text{pt})$ is naturally isomorphic to $E^{n+\text{dim}\_{\mathbb{R}}(V)}(\text{pt})$ for all complex representations $V$. The same works for real representations if $E$ is a module over the equivariant $MO$.... | 5 | https://mathoverflow.net/users/10366 | 177000 | 89,151 |
https://mathoverflow.net/questions/176997 | 7 | A well-known theorem of Grauert and Fischer states that a smooth proper family of complex manifolds is a locally trivial fibration as soon as all the fibers are isomorphic. It is also easy to obtain an algebro-geometric variant of this; cf. Lemma 1.3 in Buium's *Differential Algebra and Diophantine Geometry*.
For a c... | https://mathoverflow.net/users/26522 | When are isotrivial families split by a finite base-change? | Let me work out the case of an isotrivial family $X\rightarrow B$ of curves -- the same argument applies to abelian varieties. The point is that the moduli space $\mathscr{M}\_g(n)$ of curves $C$ of genus $g$ with a level $n$ structure (that is, a symplectic isomorphism $(\mathbb{Z}/n)^{2g}\stackrel{\sim}{\rightarrow }... | 10 | https://mathoverflow.net/users/40297 | 177001 | 89,152 |
https://mathoverflow.net/questions/176998 | 6 | Let $X$ be a real-valued random variable with $X \geq 0$ and $\mathbb E X >0$.
I would like to bound $\mathbb P(X >0)$ from below using information about the first few moments of the variable.
From Cauchy-Schwarz inequality we know $\mathbb P(X>0) \geq \frac{(\mathbb E X)^2}{\mathbb E X^2 }$. Unfortunately, this is n... | https://mathoverflow.net/users/16436 | non-negative random variable | Let $f(x) = \sum\_{j=0}^d a\_j x^j$ be any polynomial of degree $d$ such that
$f(0) = 0$ and $f(x) \le 1$ for all $x \ge 0$. Then
$$P(X > 0) \ge E[f(X)] = \sum\_{j=0}^d a\_j E[X^j]$$
Your Cauchy-Schwarz bound is the case $f(x) = 1 - (x - c)^2/c^2 = 2 x/c - x^2/c^2$ where $c = E[X^2]/E[X]$.
If you want a bound that ... | 9 | https://mathoverflow.net/users/13650 | 177003 | 89,153 |
https://mathoverflow.net/questions/176988 | 7 | I recently had to explain how the basic axioms in Simpson's *Subsystems of Second Order Arithmetic* were interpretable in Robinson's Q. Most of the axioms are actually the same, except that Simpson includes an ordering relation $\lt$ with the axioms
$$\lnot(m \lt 0)$$
and
$$m \lt n + 1 \leftrightarrow m \lt n \lor m = ... | https://mathoverflow.net/users/2000 | The definition of < in Robinson's Q | 1. I can’t give you a source from the top of my head, but I’m pretty sure that all obvious variants of the definition of ordering in Q appear *somewhere*.
2. The technical advantage of the definition in Hájek and Pudlák is that it provably satisfies
$$x\le\overline n\lor\overline n\le x$$
for every standard $n$, which ... | 3 | https://mathoverflow.net/users/12705 | 177010 | 89,158 |
https://mathoverflow.net/questions/176906 | 8 | Let $M$ be an injective von Neumann subalgebra of $B(H)$. For a completely positive map $\phi:B(H)\to B(H)$, let $Mult(\phi)$ denote be the multiplicative domain of $\phi$. For any conditional expectation (CE) $E:B(H)\to M$, the range of $E$ is contained in $Mult(E)$ because $E$ is a bimodule map. Indeed, the image of ... | https://mathoverflow.net/users/56316 | Multiplicative domains and conditional expectations | This is not true in general. For example, consider an infinite amenable group $\Gamma$ and let $M$ be the group von Neumann algebra $L\Gamma \subset \mathcal B(\ell^2\Gamma)$ endowed with its usual trace $\tau(x) = \langle x \delta\_e, \delta\_e \rangle$. For $\gamma \in \Gamma$ let $P\_\gamma$ denote the rank one proj... | 7 | https://mathoverflow.net/users/6460 | 177015 | 89,161 |
https://mathoverflow.net/questions/177014 | 3 | Please forgive me if this is not the right forum for this question.
Let $$ X = \cdots \rightarrow X\_n \rightarrow X\_{n-1} \rightarrow \cdots \rightarrow X\_0 = \ast$$ be a tower of fibrations of simplicial sets. I am trying to understand a proof of Milnor's exact sequence found in Goerss-Jardine, but I am having so... | https://mathoverflow.net/users/nan | Milnor's exact sequence and a certain proof | First take a space $X$, and try the fibration $Hom(\Delta\_1,X)\to X\times X$. The fiber over $(x,x')\in X\times X$ is the space of paths in $X$ which go from $x$ to $x'$ (which can sometimes be empty). Over a point $(x,x)$, the fiber is $\Omega X$ (based loops at $x$).
Now consider the sequence
$$
\pi\_n \Omega X \t... | 3 | https://mathoverflow.net/users/437 | 177016 | 89,162 |
https://mathoverflow.net/questions/172895 | 2 | Let $M$ be a manifold and fix a Lagrangian $L\in C^\infty(T M )$. Let $x\_1,\dots x\_n$ be local coordinates for $M$ and equip the tangent bundle and cotangent bundle with standard coordinates $(x\_1,\dots, x\_n, v\_1,\dots, v\_n)$ and $(x\_1,\dots, x\_n,\xi\_1,\dots, \xi\_n)$ respectively. The Legendre transform assoc... | https://mathoverflow.net/users/54440 | Computation with the Legendre Transform | I actually was interpreting what she was saying wrong. The coordinates $\xi$ on $T^\ast M$ are of the form $\xi=\Phi\_L(x,v)$ so that $\xi\_i=\frac{\partial L}{\partial v\_i}$ and the Legendre transform is involutive so $\Phi\_L^{-1}(\xi)=\Phi\_{L^\ast}(\xi)=v$. This makes the computation fairly straightforward.
| 0 | https://mathoverflow.net/users/54440 | 177027 | 89,167 |
https://mathoverflow.net/questions/177030 | 5 | Let $G$ be a finite group. Atiyah proved that the $K$-cohomology of $BG$ vanishes in odd degrees and in even degrees is the completion of the representation ring of $G$ at the augmentation ideal.
What is the $K$-homology of $BG$? I mean the stable homotopy groups of the complex $K$-theory spectrum smash $BG$.
| https://mathoverflow.net/users/56415 | $K$-homology of $BG$ | In "$K$-homology of universal spaces and local cohomology of the representation ring" Greenlees proves that
$$
K\_i(BG\_+) \simeq H\_J^i(R(G))
$$
for $i=0,1$, where $J$ is the augmentation ideal and $H\_J^i(R(G))$ refers to [local cohomology.](https://en.wikipedia.org/wiki/Local_cohomology) In fact Greenlees proves so... | 10 | https://mathoverflow.net/users/16785 | 177033 | 89,170 |
https://mathoverflow.net/questions/177039 | -1 | I'm trying to fit a quadratic $a\_0 + a\_1x + a\_2x^2$ by Polynomial Regression:
$$
\begin{pmatrix}
n & \Sigma x\_i & \Sigma x\_i\\
\Sigma x\_i & \Sigma x\_i^2 & \Sigma x\_i^3\\
\Sigma x\_i^2 & \Sigma x\_i^3 & \Sigma x\_i^4\\
\end{pmatrix}
\begin{pmatrix}
a\_0\\
a\_1\\
a\_2\\
\end{pmatrix}=
\begin{pmatrix}
\Sigma y\_... | https://mathoverflow.net/users/56421 | Fitting a quadratic using regression when the y-intercept needs to be 0 | If you want $a\_0$ to be $0$, then the curve you are trying to fit has equation $y = a\_1 x + a\_2 x^2$. If you derive the normal equations starting from this equation, you get
$a\_1 \sum x\_i^2 + a\_2 \sum x\_i^3 = \sum x\_i y\_i\\
a\_1 \sum x\_i^3 + a\_2 \sum x\_i^3 = \sum x\_i^2 y\_i$
Or
$\left[\begin{array}{c... | 0 | https://mathoverflow.net/users/52842 | 177046 | 89,175 |
https://mathoverflow.net/questions/177047 | 11 | *I asked this over at [math.stackexchange](https://math.stackexchange.com/questions/873489), and though a number of people were interested enough to vote up the question, I didn't get an answer -- which makes me wonder whether it isn't quite so trivial/dumb as I originally feared it was. So let me try again in this mor... | https://mathoverflow.net/users/14111 | The (un)decidability of Robinson-Arithmetic-without-Multiplication? | The theory $T'$ with axiom $Sx\ne x$ in place of $Sx\ne0$, as it was originally written, is undecidable, because the theory of groups with a distinguished nonidentity element $a$ is a conservative extension of the theory of nontrivial groups, shown undecidable by Tarski. Since this is a finite extension of $T'$ (modulo... | 21 | https://mathoverflow.net/users/12705 | 177051 | 89,178 |
https://mathoverflow.net/questions/177054 | 0 | Let $C$ be a triangulated category that is closed with respect to arbitrary small coproducts; let $D$ be some class of objects of $C$. Then it would be natural to say that $D$ generates $C$ either if
(i) There are no proper triangulated subcategories of $C$ that are closed with respect to small coproducts and contain... | https://mathoverflow.net/users/2191 | On two notions of 'generators' for a 'large' triangulated category | The obvious implication is (i) $\Longrightarrow$ (ii), of course. Indeed, the class of all $c\in C$ not satisfying (ii) is precisely the full triangulated subcategory of objects right orthogonal to the minimal triangulated subcategory satisfying (i) in $C$. Any triangulated subcategory with a nontrivial orthogonal subc... | 4 | https://mathoverflow.net/users/2106 | 177057 | 89,180 |
https://mathoverflow.net/questions/176721 | 6 |
>
> What is the analog of HKR theorem in the noncommutative world?
>
>
>
Recall that the well-known theorem by Hochschild-Kostant-Rosenberg says that for a *smooth* *commutative* algebra $A$ of finite type over a field $k$ (let's assume of characteristic zero) there is an isomorphism $$HH\_\bullet(A)\simeq \Omeg... | https://mathoverflow.net/users/32741 | Noncommutative HKR theorem | Maybe the following paper helps:
* Andreas Cap, Andreas Kriegl, Peter W. Michor, Jiri Vanzura: The Frölicher-Nijenhuis bracket in non commutative differential geometry, Acta Math. Univ. Comenianae 62(1993), 17--49 [pdf](http://www.mat.univie.ac.at/~michor/ncdg.pdf)
| 2 | https://mathoverflow.net/users/26935 | 177060 | 89,181 |
https://mathoverflow.net/questions/177065 | 1 | Let $K$ be a field and let $V$ be the set of sequences $\{v\_1,v\_2,\dots\}$ of elements of $K$. If $A=\{a\_1,a\_2,\dots\}$ is also a sequence of elements of $K$, then it defines an endomorphism of $V$ $$f\_A : \{v\_1,v\_2,\dots\} \mapsto \{a\_1 v\_1, a\_2 v\_2,\dots\}.$$
Let $(W,f)$ be a $K$-vector space with an end... | https://mathoverflow.net/users/5743 | Linear map with two "incompatible" representations | I don't think so ... let $K = \mathbb{R}$ and let $A$ and $B$ be any two disjoint sequences such that each one contains infinitely many distinct elements. Then let $\vec{v} = \{1,1,1, \ldots\} \in V$ and define $W = {\rm span}\{\vec{v}, f\_A\vec{v}, f\_A^2\vec{v}, \ldots\}$ and $f = f\_A$. The identity map is the first... | 2 | https://mathoverflow.net/users/23141 | 177076 | 89,186 |
https://mathoverflow.net/questions/177064 | 4 | I am studying the Lagrangian and Hamiltonian description of some dynamical systems. The problem with this description of the particular kind of systems I am studying, is that the Legendre Transformation is degenerate. I could not find the meaning of this on googling but I guessed that there could be multiple Hamiltonia... | https://mathoverflow.net/users/37893 | What is a degenerate Legendre Transformation? | I) We give here a possible definition for a $\mathbb{R}$-vector space $V$, and leave it to the reader to generalize the construction e.g. to a tangent bundle.
Let $C\subseteq V$ be a convex subset. Let there be given a function $L:C\to\mathbb{R}$. Define a new function $L\_H: V^{\ast} \times C \to \mathbb{R}$ as
$... | 3 | https://mathoverflow.net/users/13917 | 177077 | 89,187 |
https://mathoverflow.net/questions/177041 | 11 | If $n$ is a prime then for all $k$ with $1 \le k \le [n/2]$,
$k$ divides ${n-1 \choose 2k-1}$ because of the identity
${n-1 \choose 2k-1} \frac{n}{k}=2{n \choose 2k}$. My question is whether
an integer $n$ satisfying the divisibility conditions must be prime ?
This may be interesting because of the identity
$ \sum\_... | https://mathoverflow.net/users/1894 | A strengthening of base 2 Fermat pseudoprime | Such an $n$ must be prime. If $\frac{1}{k} \binom{n-1}{2k-1}$ is an integer for all $1 \leq k \leq \lfloor \frac{n}{2} \rfloor$, then $n$ divides $\binom{n}{2k}$
for all $1 \leq k \leq \lfloor \frac{n}{2} \rfloor$.
Suppose that $n$ is odd and squarefree and $p$ is a prime dividing $n$.
Then $p$ is odd, and so $n-p$ ... | 10 | https://mathoverflow.net/users/48142 | 177078 | 89,188 |
https://mathoverflow.net/questions/177058 | 11 | I'm looking for a reference for analogues of the Blakers-Massey triad connectivity theorem (and its higher-order generalization) for ring spectra. That is:
Suppose that $A\to A\_1$ is a $k\_1$-connected map of (associative) ring spectra and $A\to A\_2$ is a $k\_2$-connected map of ring spectra, and that the maps are ... | https://mathoverflow.net/users/6666 | Homotopy excision for structured ring spectra -- reference? | Theorems 1.4–1.11 in Ching and Harper's paper “Higher homotopy excision and Blakers-Massey theorems for structured ring spectra” ([arXiv:1402.4775](http://arxiv.org/abs/1402.4775))
give higher homotopy excision and Blakers-Massey (and their dual versions) for structured ring spectra and more generally, for algebras ove... | 9 | https://mathoverflow.net/users/402 | 177080 | 89,190 |
https://mathoverflow.net/questions/176794 | 2 | Consider the following mixed semi definite and second order programming:
$\begin{array}{l}
\mathop {{\rm{min}}}\limits\_{\bf{X}} \,{\rm{Tr}}\left( {{\bf{XA}}} \right)\\
{\rm{s}}{\rm{.t:}}\, & {\rm{Tr}}\left( {{\bf{XA'}}} \right) + \left\| {{\rm{vec}}{{\left( {\bf{X}} \right)}^H}{\bf{A''}}} \right\| \ge a\\
& {\bf{X}... | https://mathoverflow.net/users/51780 | mixed semi definite and second order programming complexity order | It took me some time, but I hope it still helps. Coming back to your problem I realized that the norm constraint is not convex (you should have the opposite inequality for convexity).
Assuming you have the opposite inequality, $ Tr(XA^{\prime})+\|vec(X)^HA\|\leq a$, you can write the problem in standard form:
\begi... | 2 | https://mathoverflow.net/users/39129 | 177081 | 89,191 |
https://mathoverflow.net/questions/177088 | 0 | I've been trying to prove this (probably very simple) result that is stated in a paper that I'm reading:
Let $G$ and $H$ be connected semisimple algebraic groups defined over a field $F$, and let $f: G \longrightarrow H$ be a separable isogeny. Then $f$ is a central isogeny.
I'm not seeing why the fact that $\mathr... | https://mathoverflow.net/users/31495 | How do I show that a separable isogeny is central? | Consider the action of $G$ on $\ker(f)$ by conjugation. We have that $\ker(f)$ is finite and smooth, and $G$ is connected, so the action of $G$ on $\ker(f)$ is trivial. Since this conjugaction action is trivial, we have that $\ker(f)$ is central.
| 7 | https://mathoverflow.net/users/18060 | 177090 | 89,193 |
https://mathoverflow.net/questions/177089 | 8 | Is there a method to find the value of the $n$-th decimal place of $\pi$ which is more efficient than having to compute all decimal places before as well?
| https://mathoverflow.net/users/56445 | Compute an arbitrary decimal place of $\pi$ | Yes, there are such algorithms.
-- See e.g. Xavier Gourdon: [Computation of the $n$-th decimal digit of $\pi$ with low memory](http://numbers.computation.free.fr/Constants/Algorithms/nthdecimaldigit.pdf).
There also is the Bailey-Borwein-Plouffe formula already mentioned by Geoff Robinson in a comment -- see <https://w... | 11 | https://mathoverflow.net/users/28104 | 177091 | 89,194 |
https://mathoverflow.net/questions/177098 | 4 | Assume $f$ and $g$ are nonnegative with
$$\int\_0^\infty f(x)dx=1=\int\_0^\infty g(x)dx
$$
and
$$\int\_0^\infty xf(x)dx<\infty > \int\_0^\infty xg(x)dx
$$
Is it true for nonnegative numbers $p$, $q$ with $p+q=1$, and $b\ge 0$ that
>
> $$
> p\int\_c^\infty xf(x)dx + q\int\_d^\infty xg(x)dx \le p\int\_b^\infty xf(x)d... | https://mathoverflow.net/users/4600 | An inequality concerning convexity and expectation | Yes, this works. Let's say $c\le b$, so $b\le d$. Then (slightly rearranging) we want to show that
$$
p\int\_c^b xf \le q \int\_b^d xg \quad\quad\quad (1).
$$
Rearranging the definition of $c,d$, we see that
$$
\int\_c^b f = q\int\_b^{\infty}(g-f),\quad\quad
\int\_b^d g = p \int\_b^{\infty}(g-f) ,
$$
and this produces ... | 5 | https://mathoverflow.net/users/48839 | 177101 | 89,198 |
https://mathoverflow.net/questions/177082 | 3 | In the appendix to Carlsson's "Equivariant stable homotopy and Segal's Burnside ring conjecture," he introduces a spectrum BG^-V associated to a G-representation V. It is like a Thom spectrum of the vector bundle over BG associated to V, but the stabilization maps come from the Thom space of (V plus a copies of the reg... | https://mathoverflow.net/users/56415 | Carlsson's spectrum BG^-V | You can define the Thom spectrum $X^V$ for any *virtual* bundle $V$, i.e. any formal difference $V-W$, where $V$, $W$ are vector bundles (or, in your case, representations of $G$ when $X=BG$. Carlsson gives an ad-hoc definition of $BG^{-V}$ (the Thom space of the negative of a representation), which is maybe not the be... | 5 | https://mathoverflow.net/users/4183 | 177105 | 89,199 |
https://mathoverflow.net/questions/177118 | 0 | Is there a manifold $M$ such that for every $x\in M$, $M-\{x\}$ is not parallelizable but there is a finite set $S\subset M$, with $\# S>1$, such that $M-S$ is parallelizable?
| https://mathoverflow.net/users/36688 | A question on parallelizability | No: any finite set $S \subset M$ can be contained in the interior of an embedded closed disc in $M$, and cutting this out gives a manifold diffeomorphic to $M - \{x\}$. So if $M - S$ were parallelisable, $M-\{x\}$ would be too.
| 3 | https://mathoverflow.net/users/318 | 177120 | 89,205 |
https://mathoverflow.net/questions/177086 | 9 | Let $H$ be a finite $p$-group, and let $G$ be a compact connected Lie group. Then
it is well-known that $[BH,BG]\cong Rep(H,G)$, where $BH$ and $BG$ are classifying spaces and $Rep(H,G)$ is the set of representations of $H$ in $G$. What happens if we assume that $H$ is any finite group -- do we also have $[BH,BG]\cong... | https://mathoverflow.net/users/56443 | Representation of finite groups in a compact Lie group | In "Maps from $B\pi$ into $X$" Quart. J. Math. Oxford Ser. (2) 39 (1988), no. 153, 117–127., Wojtkowiak proves that the natural map
$Rep(H,G)\rightarrow [BH,BG]$ is not surjective when $H=\Sigma \_3$, $G=U(2)$.
| 10 | https://mathoverflow.net/users/43326 | 177129 | 89,208 |
https://mathoverflow.net/questions/176914 | 10 | This is pretty much straightforward curiosity. Is there anything known about Newton polygons of classical modular polynomials (polynomial relations between $j(\tau)$ and $j(n\tau)$)? I understand that modular polynomial is not really a practical way of approaching modular curves, still it would be interesting to see if... | https://mathoverflow.net/users/38468 | Newton polygons of modular polynomials | The polynomial $\Phi\_n(x,y)$ has a degree of $n \prod\_{p|n } \frac{p+1}{p}$ in each variable, so the Newton polygon is contained in a square of that side length. Call this $n'$. For each fixed value of $j\_1$, there are $n'$ possible values of $j\_2$ with multiplicity, so plugging in any value of $x$ or $y$ gives a d... | 6 | https://mathoverflow.net/users/18060 | 177136 | 89,211 |
https://mathoverflow.net/questions/177138 | 2 | Let $\pi:\mathcal{X} \to B$ be a flat family of projective schemes, $B$ is irreducible. Let $\mathrm{Spec} K$ be a *generic* point on $B$. Denote by $\mathcal{X}\_K$, the pull-back of $\mathcal{X}$. This is flat, projective on $K$. It will then follow that $\mathcal{X}\_{K\_{\mathrm{red}}}$, the associated reduced sche... | https://mathoverflow.net/users/46578 | Generic vs General property of reducedness in a family of projective schemes | This is true if you assume moreover that $(\mathcal{X}\_K)\_{red}$ is *geometrically* reduced -- in particular, in characteristic $0$. First of all, note that set-theoretically $\mathcal{X}'\_b=\mathcal{X}\_b$ for all $b$ in $B$ : since $\pi $ open, $\pi (\mathcal{X}-\mathcal{X}')$ is an open subset of $B$ which does n... | 3 | https://mathoverflow.net/users/40297 | 177139 | 89,213 |
https://mathoverflow.net/questions/177141 | 3 | For a positive integer $k$ let $\gamma\_k(n)$ be the number of representations of $n$ as a sum of strictly increasing perfect $k^{\text{th}}$ powers. For example $\gamma\_k(2)=0$ for any $k$. Now is the following true?
>
> For any $x\in \mathbb{R}^\ast\_+$ there's an integer $N$ such that for all $n\ge N$
> $$\dfr... | https://mathoverflow.net/users/50068 | Arbitrarily large number of representations | Yes, the statement is true.
For any positive integer $m$ there are $2^m$ subsets of $\{1^k,2^k,\ldots,m^k\}$. Each subset has sum bounded by $m^{k+1}$, so by the pigeonhole principle
$$
\max\_{1\leq i\leq m^{k+1}} \gamma\_k(i)\geq \frac{2^m}{m^{k+1}}.
$$
This shows that
$$
\frac{\max\_{1\leq i\leq n}\gamma\_k(i)}{n}\... | 5 | https://mathoverflow.net/users/5263 | 177142 | 89,215 |
https://mathoverflow.net/questions/177049 | 1 | I am struggling with a problem like this: In dimension $n\geq 3$,
consider the following uniformly elliptic equation
$$a^{ij}(x)u\_{ij}(x)+u\_{nn}=0$$
where $i,j = 1, \dots, n-1$, and $\lambda \operatorname{id} < (a^{ij}) < \Lambda \operatorname{id}$. Are there any interior gradient estimates? More specifically, I... | https://mathoverflow.net/users/56430 | Interior gradient estimate for uniformly elliptic equations | For the constant of ellipticity $\nu=\lambda/\Lambda$ close to $1$
for elliptic equations of the form
$$\sum\_{i,j=1}^na^{ij}(x)u\_{ij}(x)=0$$ Cordes proved in
>
> Cordes, Heinz Otto. *Über die erste Randwertaufgabe bei quasilinearen Differentialgleichungen zweiter Ordnung in mehr als zwei Variablen*. (German) Mat... | 1 | https://mathoverflow.net/users/14551 | 177143 | 89,216 |
https://mathoverflow.net/questions/177124 | 2 | Definitions and notations.
==========================
Let $\mathcal{P}(X)$ the **power set** of $X$.
Let $\tau\_X\subseteq\mathcal{P}(X)$ a **topology** on X.
We call $A$ **irreducible** if every time $A=B\cup C$ with $B,C$ closed set then $(B=A)\vee(C=A)$.
We call $X$ **sober** if every non empty irreducible c... | https://mathoverflow.net/users/54610 | Why the intersection of a scott open (or \w the relatively compactness property) filter on a topology of a sober (and 2nd countable) space is compact? | A self-contained proof is in the book "Continuous lattices and domains" by G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove and D. S. Scott.
The particular place you need is Lemma II-1.19 on page 146.
| 2 | https://mathoverflow.net/users/41291 | 177145 | 89,217 |
https://mathoverflow.net/questions/177146 | 6 | I'm trying to go through the proof that all maximal compact subgroups of a semisimple Lie group $G$ are conjugate. I know that a possible proof follows the following steps:
* Take one maximal compact subgroup $K$ of $G$ (which I know that exists). Consider $G/K$. This is a $CAT(0)$ space.
* Take any other maximal com... | https://mathoverflow.net/users/56465 | Maximal compact subgroups of a semisimple Lie group are conjugate | Consider the fiber bundle
\begin{align}
K\rightarrow G\rightarrow G/K.
\end{align}
Herein we assume $G$ is connected.
Since $G/K$ is non-positively curved, it is aspherical (i.e. all its higher homotopy groups vanish). Thus, the long exact sequence of homotopy groups induced by the above fiber bundle implies the sh... | 1 | https://mathoverflow.net/users/49247 | 177148 | 89,219 |
https://mathoverflow.net/questions/177157 | 2 | I am trying to have a better understanding of how one goes , "travels" between the different formats/layouts of open books for a fixed given 3-manifold M; between the abstract type and the "actual" type. I am new to this, and I have not been able to find clear explanations, or whether there are canonical choices that h... | https://mathoverflow.net/users/56470 | Going Back-and-Forth Between Different Expressions/"Representations" for Open Books. | You are asking definitional questions about surface homeomorphisms, so I suggest you read one of the standard references. The book by Casson and Bleiler is short and, as I recall, very clear.
1) Remember that $f\_\pi$ is a mapping class, not an actual surface homeomorphism.
2) Yes, this is how things are done. The... | 1 | https://mathoverflow.net/users/1650 | 177158 | 89,223 |
https://mathoverflow.net/questions/176400 | 9 | A pentagon equation is a system of polynomial equations of degree $3$ with several variables and integer coefficients, given by a fusion ring.
A fusion ring is given by a finite set of integer matrices checking some axioms (see [here](https://drive.google.com/open?id=0B2P_JgZe-Zd0OVRjakNSUUhleUk) p 22).
A solutio... | https://mathoverflow.net/users/34538 | Are there workable algebraic geometry approaches for the pentagon equation? | I'm not an expert in algebraic geometry, but I can say something about methods for solving pentagon equations that will hopefully be of use.
The primary way to determine whether or not there is a solution to the pentagon equations is to use Groebner basis methods. However, these begin to break down very quickly for ... | 6 | https://mathoverflow.net/users/25642 | 177162 | 89,226 |
https://mathoverflow.net/questions/176394 | 3 | I would like a source for the following result, which I expect to be true (probably well known):
Let $X$ be a complex projective variety, $L$ an ample line bundle and $E$ a slope stable vector bundle with respect to $L$. Then for all line bundles $H$, there exists an $\epsilon>0$ such that $E$ is stable with respect ... | https://mathoverflow.net/users/56123 | How does one show that slope stability of a vector bundle is an open condition with respect to the polarisation? | I don't know a reference either but you can proceed as follows...
In the vector space of numerical $\mathbb{R}$-divisors, $N^1(X)\_{\mathbb{R}}$, being ample is an open condition. Take an open ball $B \subset N^1(X)\_{\mathbb{R}}$ satisfying $L \in B$. Look at the map from $N^1(X)\_{\mathbb{R}}$ to numerical $\mathbb... | 2 | https://mathoverflow.net/users/17630 | 177167 | 89,228 |
https://mathoverflow.net/questions/167548 | 6 | As I was trying to exhibit new interesting(?) path transformations of Brownian motion, I became interested in
the (random) set of times $t$ such that $B(t)=B(t+1)=0$, where $B(t)$ denotes a standard Brownian motion.
Not surprisingly, I found that this set is almost surely empty. By scaling invariance of Brownian motion... | https://mathoverflow.net/users/49268 | Properties of the algebraic self-difference set of Brownian motion zeros | Regarding Hausdorff dimension $\dim\_H$, the following holds:
$$
\dim\_H(Z-Z)=1 \quad\text{a.s.}
$$
This is essentially a consequence of Marstrand's projection theorem, the (stochastic) self-similarity of $Z$, and the well known fact that $\dim\_H Z=1/2$ (and the box counting dimension is also $1/2$).
Recall that... | 4 | https://mathoverflow.net/users/11009 | 177172 | 89,231 |
https://mathoverflow.net/questions/177173 | 5 | Let $p$ be a prime such that the free 2-generator group $B(2,p)$ of exponent $p$ is infinite. Consider the short exact sequence
$$
1\to K \to B(2,p) \to B\_0(2,p) \to 1,
$$
where $B\_0(2,p)$ is the biggest finite $2$-generator group of exponent $p$, which exists by [RBP](http://en.wikipedia.org/wiki/Burnside's_problem#... | https://mathoverflow.net/users/12961 | Properties of the Burnside kernel | The answer to question 2. is yes, since the [centralizer of elements in a free Burnside group (of exponent $p > 665$) are finite cyclic of order $p$](https://mathoverflow.net/a/79615/1345).
For question 3., the answer is no (for sufficiently large $p$ at least). [A theorem of Olshanskii](http://www.ams.org/mathscine... | 6 | https://mathoverflow.net/users/1345 | 177178 | 89,233 |
https://mathoverflow.net/questions/177179 | 3 | Is there an infinite finitely generated (non-cyclic) $p$-group $G$ such that the intersection of all nontrivial subgroups of $G$ is a nontrivial subgroup?
| https://mathoverflow.net/users/47958 | Existence of an infinite finitely generated $p$-group with nontrivial intersection of nontrivial subgroups | An example is given [here](https://mathoverflow.net/questions/102208/finitely-generated-infinite-p-group). And here are some more words to make MO happy.
| 4 | https://mathoverflow.net/users/4183 | 177182 | 89,234 |
https://mathoverflow.net/questions/177175 | 1 | Let $\mathcal{T}(S)$ denotes the Teichmuller space of a finite type surface $S$ equipped with Teichmuller metric and $\mathcal{C}(S)$ denotes the curve complex. Define a map $$\phi:\mathcal{T}(S)\rightarrow \mathcal{C}(S),$$ by taking a hyperbolic metric to one of its systols. I read somewhere that this map is coarsely... | https://mathoverflow.net/users/9485 | Reference for the result that the systol map from Teichmuller space to curve complex is coarsely Lipschitz | The proper reference is Lemma 2.4 of the paper "Geometry of the complex of curves I: Hyperbolicity" by Masur and Minsky. They use extremal length to determine the systole set. They remark on the same page that hyperbolic length would also work.
I haven't found an explicit reference for the latter. You can prove it f... | 2 | https://mathoverflow.net/users/1650 | 177187 | 89,236 |
https://mathoverflow.net/questions/177203 | 10 | I want to use the technicalities of structured ring spectra for the first time in my life, and I am not really familiar with the relevant literature. I am looking for a reference that defines algebraic K-theory for associative unital algebras in symmetric spectra.
| https://mathoverflow.net/users/6666 | Algebraic $K$-theory of algebras in symmetric spectra: reference | Tom, not precisely sure what you want. I'm guessing you want
to think of a commutative symmetric ring spectrum $R$ and then an
$R$-algebra $A$. Without the extra layer, just using an $S$-algebra $R$
(not necessarily commutative), the original source for the algebraic
$K$-theory of $R$ is Chapter VI of EKMM, but of cou... | 7 | https://mathoverflow.net/users/14447 | 177205 | 89,245 |
https://mathoverflow.net/questions/144272 | 0 | Let $E$ be an elliptic curve over $\mathbb{Q}$ and $E(\mathbb{Q})[2]=\{o,T\_1,T\_2,T\_3\}$. Let $P=2R$ be a point in $2E(\mathbb{Q})$, using $2$-division polynomial, we can compute $1/2P$, but it gives the set $\{R,R+T\_1,R+T\_2,R+T\_3\}$. How can find $R$ in this set? In other word, is there an algorithm for point hal... | https://mathoverflow.net/users/41032 | Point halving on elliptic curves over $\mathbb{Q}$ | To some extent, it depends on the magnitude of the numbers involved. But when you say that you "know" $P$, I assume that means that you can write down the coordinates of $P$ as rational numbers. So there may be hundreds, or even thousands, of digits in the numerators and denominators of the coordinates of $P$, but not ... | 2 | https://mathoverflow.net/users/11926 | 177207 | 89,246 |
https://mathoverflow.net/questions/177210 | 5 | A *left distributive algebra* is a set $A$ together with a binary operation, $\cdot$, satisfying $a\cdot(b\cdot c)=(a\cdot b)\cdot(a\cdot c)$.
One important example of left distributive algebras arises in set theory. For $\lambda$ a limit cardinal let $\mathcal{E}\_\lambda$ be the set of elementary embeddings $j: V\_... | https://mathoverflow.net/users/8133 | The word problem of the free left distributive algebra on one generator | Yes. If by the word problem for left-distributive algebras, we mean determining if two elements of a free left-distributive algebra (with any number of generators) are equal. In the Handbook of Set Theory, Theorem 2.11 in the chapter on algebras of elementary embeddings states that in ZFC left division in the free left... | 9 | https://mathoverflow.net/users/22277 | 177213 | 89,249 |
https://mathoverflow.net/questions/177206 | 14 | **The question.** Is an asymptotic equivalent known or conjectured for the number $N(d)$ of $\alpha \in \bar{\mathbb{Q}}$ with $h(\alpha) < 1$ and $[\mathbb{Q}(\alpha):\mathbb{Q}] \leq d$?
The rather crude proof of Northcott's theorem bounds $\log{N(d)}$ by $O(d^2)$. On the other hand, upon extracting $\lfloor d/m \... | https://mathoverflow.net/users/26522 | Asymptotics for algebraic numbers of height less than one | We can make a conjecture based on the function field model. Replace $\mathbb Q$ with $\mathbb F\_q(t)$, that is, the function field of $C= \mathbb P^1$. Then any element of $\overline{\mathbb F\_q(t)}$ corresponds to a (multivalued) algebraic function on $C$. This is the same as an irreducible subvariety of $\mathbb P^... | 5 | https://mathoverflow.net/users/18060 | 177219 | 89,253 |
https://mathoverflow.net/questions/177204 | 3 | Let $L/K$ be a finite Galois extension of nonarchimedean local fields, and let $A$ and $A^t$ be dual abelian varieties over $K$. Tate local duality tells us that $A^t(K)$ and $H^1(K, A)$ are Pontryagin dual locally compact Hausdorff abelian topological groups, and likewise over $L$. Under this duality, why does the dua... | https://mathoverflow.net/users/53197 | Why is the norm map dual to restriction under Tate local duality? | Note first of all that the "norm" map you speak of does not make sense unless the field extension is separable. That is, for a separable field extension $k'/k$ of finite degree and a commutative $k$-group scheme $C$ there is a canonical "norm" homomorphism $C(k') \rightarrow C(k)$ induced by the passage to Galois-invar... | 5 | https://mathoverflow.net/users/52824 | 177220 | 89,254 |
https://mathoverflow.net/questions/176983 | 11 | The common definition of Reedy category seems to be [this one](http://ncatlab.org/nlab/show/Reedy+category) that a Reedy category is a small category $R$ with two wide subcategories $R\_+$ and $R\_-$ and an ordinal-valued degree function on its objects such that
* Every nonidentity morphism in $R\_+$ raises degree,
*... | https://mathoverflow.net/users/49 | The definition of Reedy category | Is this a counterexample?
$R$ is the poset category $1\to 0\to 2$.
Nonidentity maps in $R^+$: $0\to2$, $1\to 2$.
Nonidentity maps in $R^-$: $1\to 0$.
There are no other maps in $R$.
The map $1\to 2$ admits two distinct "Reedy factorizations": $1\to1\to 2$ and $1\to 0\to 2$. (This is the only map with more tha... | 7 | https://mathoverflow.net/users/437 | 177225 | 89,257 |
https://mathoverflow.net/questions/177224 | 0 | is there a way to construct a transition matrix of a time-homogeneous, finite Markov chain such that the stationary distribution always has full support (this is equivalent to all states of the chain being positive recurrent).
I am looking for something quick that doesn't involve drawing the state-transition graph an... | https://mathoverflow.net/users/5887 | Constructing a transition matrix of a time-homogeneous, finite Markov chain with full support stationary distribution | When you write "**the** stationary distribution, I suppose you mean that the Markov chain is irreducible.
An easy sufficient condition is that all entries are strictly positive (and that the sum of each row is $1$ so it is a transition matrix).
| 3 | https://mathoverflow.net/users/13650 | 177228 | 89,258 |
https://mathoverflow.net/questions/177227 | 12 | Let $M$ be a pure motive over a field $k$. Roughly speaking, the L-function of $M$ is the product over all primes $p$ of
$$L\_p(M,s)=\det(I-Fr\_p|\_{M\_\ell^I} N(p)^{-s})^{-1}$$
where $Fr\_p$ is a Frobenius element and $M\_\ell^I$ is the $\ell$-adic realization of $M$ stable under inertia $I\_p$. On the other hand, the... | https://mathoverflow.net/users/48554 | Motivic L-function vs motivic zeta function | The idea is to rewrite the $L$-function as the sum of $\sharp [Sym^nM](F\_p)t^n$; here "the number of $F\_p$-points" of a motif is a natural homomorphism from the Grothendieck group of motives to abelian groups that extends the "usual" number of points over the field $F\_p$ for varieties. So, the $L$-function can be ob... | 8 | https://mathoverflow.net/users/2191 | 177231 | 89,260 |
https://mathoverflow.net/questions/168806 | 3 | Assume a metatheory that supports lambda-abstraction, and an object language that is merely first-order. Now let $\varphi$ denote a formula in the object language with one free variable $x$. Then we can write $\lambda x. \varphi$ in order to mean the function that accepts an expression $E$ in the object language and re... | https://mathoverflow.net/users/26080 | What do we call functions that behave like predicate symbols? | I would call such a thing *propositional function*.
| 5 | https://mathoverflow.net/users/1176 | 177238 | 89,262 |
https://mathoverflow.net/questions/177253 | 9 | Let $X$ be a smooth projective variety and let $Y\subset X$ be a smooth subvariety. Given a vector bundle $E$ on $Y$, when can $E$ be extended to a vector bundle $\tilde E$ on $X$? I.e., are there cohomology groups containing the obstructions to extending $E$?
| https://mathoverflow.net/users/56505 | Extending vector bundles from subvarieties | Without hypotheses on $Y$ there is no hope to define such obstructions, already for line bundles. A natural hypothesis is to take for $Y$ a (smooth) ample divisor in $X$, of dimension $\geq 2$. In [this paper](http://projecteuclid.org/download/pdf_1/euclid.jmsj/1239802165), Fujita gives some cohomological conditions wh... | 7 | https://mathoverflow.net/users/40297 | 177254 | 89,267 |
https://mathoverflow.net/questions/177169 | 2 | Consider the function
\begin{eqnarray}
f(x\_1,x\_2,\cdots, x\_n) = \frac{\sum\_{i}^{n}a\_ix\_i}{\sum\_{i}^{n}b\_ix\_i},
\end{eqnarray}
over the set $S = \{x := (x\_1,x\_2,\cdots, x\_n):-1 \leq x\_i \leq 1,\; \forall 1 \leq i \leq n,\; \sum\_{i}^{n}b\_ix\_i \neq 0, \sum\_{i=1}^{n}c\_i^{k}x\_i = 0, 1 \leq x\_i \leq m, ... | https://mathoverflow.net/users/56478 | Boundedness of ratio of linear functions | One can write $a=(\lambda b + c) + \bar{a}$ where $\lambda \in \mathbb{R}$, $c\in span(c^1,\ldots,c^m)$, and $\bar{a} \in \{b,c^1,\ldots,c^m\}^\perp$.
Also, One can write $b=d+\bar{b}$ where $d\in span(c^1,\ldots,c^m)$, and $\bar{b} \in \{c^1,\ldots,c^m\}^\perp$ (Note that if $\bar{b}=0$ then $S=\emptyset$, thus we c... | 2 | https://mathoverflow.net/users/53059 | 177266 | 89,273 |
https://mathoverflow.net/questions/177259 | 10 | Consider a generic $n \times n$ matrix $M$.
Define the $(n-1) \times n$ matrix $M\_q$ to be $M$ with the $q$th row omitted, and assume that $M\_q$ possesses a right inverse, $R\_q$:
$$R\_q = M\_q^T (M\_q M\_q^T)^{-1}$$
The components of $R\_q$ will be *rational* functions of the components of $M$, with the determ... | https://mathoverflow.net/users/23829 | Why does this antisymmetric product factor out a determinant? | $\let\sumnonlimits\sum
\let\prodnonlimits\prod
\renewcommand{\sum}{\sumnonlimits\limits}
\renewcommand{\prod}{\prodnonlimits\limits}
$ I will write $R$ for $R\_{q}$, because $q$ is constant. In the following, I am
going to assume that $R$ is an *arbitrary* right inverse of $M\_{q}$, rather
than the specific right inver... | 8 | https://mathoverflow.net/users/2530 | 177277 | 89,279 |
https://mathoverflow.net/questions/177278 | 4 | Consider an abstract surface $(M,g)$ embedded into $\mathbb{R}^3$ via $f:M \to \mathbb{R}^3$. Denote by $N:M \to \mathbb{R}^3$ the Gauss map (normal field) of the surface. Write the Laplace Beltrami operator as $\Delta^{(1)}$.
Define the *vector* Laplace Beltrami operator acting on the normal field as
$$\Delta^{(3)} ... | https://mathoverflow.net/users/43839 | Vector Laplace Beltrami operator of the Gauss map | This is a standard formula in curves and surfaces, and you should be able to find it in most elementary differential geometry books or derive it yourself from the structure equations. The answer is
$$
\Delta N = (4H^2{-}2K)\ N + 2\ \nabla H
$$
where $H$ and $K$ are the mean and Gaussian curvatures (respectively), and $... | 6 | https://mathoverflow.net/users/13972 | 177284 | 89,282 |
https://mathoverflow.net/questions/177245 | 6 | I'm trying to do some calculations with bimodules (over Azumaya algebras, as it happens), and I need a string diagram notation that mixes the tensor product over the base ring (a symmetric monoidal product) and the tensor product of bimodules. I haven't seen any diagrammatic calculus that allows for this mixing, but I ... | https://mathoverflow.net/users/4177 | String diagrams for bimodules over noncommutative algebras? | In general, the correct generalization of the "stringy" notation for bicategories — in which objects label regions in $\mathbb R^2$, 1-morphisms label codimension-1 defects (whose projection to $\mathbb R$ has no critical points), and 2-morphisms label codimension-2 defects, and composition is a "pushforward" of defect... | 5 | https://mathoverflow.net/users/78 | 177289 | 89,284 |
https://mathoverflow.net/questions/177275 | 2 | Is there a version of Bochner's theorem characterizing measures of positive type on a locally compact group? By a measure of positive type on the group $\Gamma$, I mean a measure $\mu$ satisfying $\int f^\* \* f~d\mu \geq 0$ for all $f \in C\_c(\Gamma)$. If so, what is a reference?
I am mainly interested in the case ... | https://mathoverflow.net/users/51431 | Bochner's theorem for measures of positive type | On $\Gamma = \mathbf R^n$, there is a Bochner theorem for not only measures but *distributions of positive type*, in Schwartz, Théorie des distributions, Chap. VII, §9:
>
> Théorème XVIII (Bochner) Pour qu'une distribution $T$ soit $>>0$, il faut et il suffit qu'elle soit transformée de Fourier d'une mesure $\mu\ge... | 4 | https://mathoverflow.net/users/19276 | 177290 | 89,285 |
https://mathoverflow.net/questions/177241 | 29 | My question refers to a folklore statement that I have now seen a couple of times, but never really precise. One avatar is:
"For spectra every cofibration is equivalent to a fibration" (e.g. in the accepted answer here <https://mathoverflow.net/a/56575/18744>),
Another one is:
"For spectra, fibrations and cofibr... | https://mathoverflow.net/users/18744 | Fibrations and Cofibrations of spectra are "the same" | One specific statement that people are likely referring to when they say things about fibrations and cofibrations being "the same" in spectra is that a homotopy pushout square of spectra is also a homotopy pullback square (considering squares with one corner trivial gives homotopy fibration and cofibration sequences). ... | 17 | https://mathoverflow.net/users/4042 | 177307 | 89,292 |
https://mathoverflow.net/questions/177296 | 2 | This is a question about characterizing Hilbert spaces in terms of quadratic forms. Let $X$ be a real Banach space and $E$ a bounded quadratic form on it, it is called positive definite if $E(x,x)\geq0$ for all $x\in X$. If moreover $E(x,x)\geq a\|x\|^2$ for some $a>0$ then $E(x,x)^\frac12$ is an equivalent norm on $X$... | https://mathoverflow.net/users/51484 | Positive definite quadratic forms on Banach spaces | If the bounded 2-form $E$ satisfies $E(x,x)\ge b |Q(x,x)|$ for any bounded 2-form $Q$ and for some $b>0$ depending on $Q$, then actually $E(x,x)\ge a\|x\|^2$ for some $a >0$.
Indeed, suppose the latter does not hold. Then for some sequence $(x\_n)\_n\subset X$ there holds $E(x\_n,x\_n)<4^{-n}\|x\_n\|^2$. Let, for an... | 3 | https://mathoverflow.net/users/6101 | 177311 | 89,294 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.