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https://mathoverflow.net/questions/145083 | 1 | How to prove that any finite set in the affine plane is realizable as intersection of TWO plane curves?
Thanks.
| https://mathoverflow.net/users/41166 | Finite number of points as intersection of two plane curves | What you want to show is that a radical ideal $I$ in $k[X,Y]$ must be equal to the radical of a two-generated ideal.
The image of $I$ in $k(X)[Y]$ is certainly principal (because $k(X)[Y]$ is a PID). So we can write $s(X)I\subset (f)$ for some $f\in k[X,Y]$.
Let $A=k[X]/(s)$ and let $B$ be $A$ mod nilpotents. Let $... | 1 | https://mathoverflow.net/users/10503 | 145093 | 78,624 |
https://mathoverflow.net/questions/145088 | 1 | Let $f$ be a non-negative function supported and integrable on the positive real axis, such that
$$\int\_0^\infty f(x+y)p(y) dy = c[p] f(x), $$
where $c[p]$ a number (functional) dependent on function $p$.
1)Suppose the above equation holds for every non-negative function $p$ supported and integrable on the positive ... | https://mathoverflow.net/users/32660 | An Integral Functional Equation | EDIT
1)
It is clear that for the equation to have a solution $f$, we must have $c[\delta\_{a + b}] = c[\delta\_{a}]c[\delta\_{b}]$ for all $a,b > 0$, where $\delta\_a$ is the Dirac measure in $a$. This means that we must have $c[\delta\_x] = e^{kx}$ for some $k$, so (if $c$ is linear) we have $c[p] = \int\_0^\infty... | 1 | https://mathoverflow.net/users/38295 | 145099 | 78,625 |
https://mathoverflow.net/questions/144816 | 15 | With the recent interest in homotopy type theory as a foundation for mathematics, it seems natural to develop algebra within the framework of type theory. So far, I can't find much literature regarding this.
I know of Danielsson and Coquand's result that isomorphism implies equality, and work by Spitters and van der ... | https://mathoverflow.net/users/2143 | What is the status of (universal) algebra in type theory? | It largely depends on how general you want to make your algebra; in particular, do you want to look just at structures on $n$-types, for some finite $n$, or consider algebraic structures on all types?
**The universal algebra of 0-types should look much like classical universal algebra;** this is what the Danielsson–C... | 10 | https://mathoverflow.net/users/2273 | 145100 | 78,626 |
https://mathoverflow.net/questions/145096 | 6 | I cite from Wikipedia:
Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields
All of these properties have a well-known *geometric* interpretation, except perhaps "(norm) Euclidean domain". Do you know one?
| https://mathoverflow.net/users/nan | geometric interpretation of "Euclidean domain" | It is a theorem of Pierre Samuel that if $R$ is normal and of finite type over a field $k$, then the following are equivalent:
1) The map taking $x$ to ($1$ plus the $k$-dimension of $R/(x)$) is a degree map that makes $R$ euclidean.
2) $R$ is a PID and there exists a fractional ideal whose divisor has odd degree a... | 5 | https://mathoverflow.net/users/10503 | 145128 | 78,636 |
https://mathoverflow.net/questions/145001 | 6 | It is well known the success of lattice dualities (as Pontryagin duality for abelian groups, Stone duality for Boolean algebras and Priestley duality for distributive lattices) to solve algebraic problems using their representation in the correspondent topological space.
Natural duality theory provides a general fra... | https://mathoverflow.net/users/30115 | Topological Problems Solved by Lattice Duality | The idea that these dualities are only used in the direction of proving algebraic results using topological spaces is not correct. Any sort of completion or compactification process (that I can think of at least) can be constructed using special knids of filters or ultrafilters on lattices. These constructions include ... | 4 | https://mathoverflow.net/users/22277 | 145129 | 78,637 |
https://mathoverflow.net/questions/145132 | 4 | The regularity of $\omega\_{1}$ is one of the most well known facts of set theory. But it seems that in order to prove this simple fact we need the "full power" of mathematics! For example by an interesting result of Gitman, Hamkins, Johnstone if we delete the axiom of power set from $ZFC$ then it is possible that $\om... | https://mathoverflow.net/users/nan | On wild behavior of $\omega_{1}$ in the absence of some essential axioms of $ZFC$ | There is an ambiguity in this kind of question, where you refer to ZFC-A for an axiom A, since what the theory is that you refer to depends on the particular way that you axiomatize ZFC. For example, different axiomatizations that are equivalent in the presence of axiom A might no longer be equivalent when you omit A, ... | 6 | https://mathoverflow.net/users/1946 | 145136 | 78,641 |
https://mathoverflow.net/questions/145118 | 8 | Let $A(X)$ denote the Waldhausen's algebraic K-theory of a space $X$, and let $n$ be **odd**.
1. Are the rational homotopy groups of $A(S^n)$ known?
2. Is the group $\pi\_{2k}(A(S^n))$ finite for all positive $k\ll n$?
A reference (or proof sketch) would be appreciated.
EDIT: I found that the answers are stated ... | https://mathoverflow.net/users/1573 | Algebraic K-theory of odd-dimensional spheres | Let $\tilde A(X)$ be the reduced functor, i.e., the homotopy fiber of the map $A(X) \to A(\ast)$. Since $A(\*)$ is rationally a product of $K(Q,4j+1)$ for $j \ge 1$, we may as well study $\tilde A(X)$ instead.
The rational homotopy of $\Omega \tilde A(\Sigma Y)$ was studied in
G. Carlsson, R. Cohen, T. Goodwillie,... | 10 | https://mathoverflow.net/users/8032 | 145140 | 78,643 |
https://mathoverflow.net/questions/144996 | 5 | Letting $a\_1,a\_2,\cdots,a\_r$ be integers which are larger than or equal to $2$, let us define
$$[a\_1,a\_2,\cdots,a\_r]=\cfrac{1}{a\_1-\cfrac{1}{a\_2-\cfrac{1}{\ddots-\cfrac{1}{a\_r}}}}$$
(Note that the **negative signs** are used)
Also, let $X, Y, Z$ be positive integers which satisfy
$$Z\lt X+Y,\ Z\gt X,\ Z\g... | https://mathoverflow.net/users/34490 | About two 'negative' continued fractions whose sum equals $1$ | The answer is **YES** even for numbers $\alpha$, $\beta$ of the form $\alpha=\frac X{Z\_1}$, $\beta=\frac Y{Z\_2}$. Suppose we look for convergents $\bar\alpha$, $\bar\beta$ to $\alpha$, $\beta$ such that $\bar\alpha+\bar\beta=1$.
If both numbers $\alpha$, $\beta$ are not less when $1/2$ an answer is trivial: $\bar\a... | 6 | https://mathoverflow.net/users/5712 | 145151 | 78,646 |
https://mathoverflow.net/questions/145174 | 23 | I'm trying to write mathematics in English and I'm clearly missing something : linking words. I'm writing "so, we get", "Observe that" too many times and I'm afraid to use some expressions : "it implies" sound weird for me (and I don't know if I'm right here...) for example.
So I'm looking for good AND bad linkings ... | https://mathoverflow.net/users/37238 | Writing Mathematics : Linking words | I can recommend you the booklet *Writing Mathematical Papers in English*, Jerzy Trzeciak, EMS Publishing, <http://www.ems-ph.org/books/book.php?proj_nr=34>.
It contains many lists of useful expressions to borrow from.
| 28 | https://mathoverflow.net/users/1898 | 145176 | 78,653 |
https://mathoverflow.net/questions/144891 | 7 | If we order all the partitions of a integer in a lexicographic order, how can we compute the position of each partition in this order without having to explicitly list all other partitions that precedes it. How is the modification done when we restrict the maximum part and the number of partitions? I would like to use ... | https://mathoverflow.net/users/26012 | Computing the lexicographic indices of integer partition | Lexicographic order seems more complex than reverse lexicographic order.
In reverse lex order, it becomes straightforward:
define p(n,k) as the number of partitions of n with largest part k (alternatively with no more than k parts). It has the well known recursion p(n,k)=p(n,k-1)+p(n-k,k).
Then, for any partition, s... | 6 | https://mathoverflow.net/users/35587 | 145186 | 78,657 |
https://mathoverflow.net/questions/144886 | 3 | I have an engineering back ground. Due to work, I came across this problem
\begin{align}
&\max\_{\lambda,y\_i\in \mathbb{R}}~\lambda \\\
s.t.~&\left(\mathbf{A}\_0+\sum\_{i=1}^{K}y\_i\mathbf{A}\_i\right)-\lambda\mathbf{I}\geq 0
\end{align}
where $\mathbf{A}\_i$ are all hermitian matrices. We are seeking $\lambda$ and ... | https://mathoverflow.net/users/27249 | Fast algorithm for maximizing smallest eigenvalue of linear combination of hermitian matrices | Two iterative algorithms that solve LMI problems is the ellipsoid algorithm and interior-point methods.
Both are described in sections 2.3 and 2.4 of Stephen Boyd's book "Linear Matrix Inequalities in System and Control Theory" [1] and in the references therein.
See also [2] for already implemented solvers. In part... | 2 | https://mathoverflow.net/users/22389 | 145191 | 78,661 |
https://mathoverflow.net/questions/145150 | 6 | I am constantly working with hermitian matrices without multiplicity in their spectrum. Since this hypothesis appear in several important problems, for instance perturbation theory, I looked in the literature for an accepted terminology but found nothing. Does anyone know a reference where these matrices, or their set,... | https://mathoverflow.net/users/20341 | A name for matrices with only simple eigenvalues? | Since your matrices are Hermitian, this is the same as *non-derogatory* ... see <http://en.wikipedia.org/wiki/List_of_matrices> for terminology *derogatory matrix*.
| 8 | https://mathoverflow.net/users/454 | 145200 | 78,663 |
https://mathoverflow.net/questions/144964 | 6 | I was wondering if the following result is known (or if there's a nice short proof without treewidth/brenchwidth related theorems): as the title says, suppose you have a graph without a big clique minor but which has a big grid minor; then, show that it either has an induced wall or the line graph of an induced wall as... | https://mathoverflow.net/users/16321 | No big clique minor but a big grid minor | The structure of graphs with big tree-width but no big clique minor is partially described by the Flat-wall theorem (Theorem 9.8 of GM XIII). With Ken Kawarabayashi and Robin Thomas, we recently posted a new proof of this result - <http://arxiv.org/abs/1207.6927>.
I don't know of your proposed theorem having appeared... | 5 | https://mathoverflow.net/users/20940 | 145201 | 78,664 |
https://mathoverflow.net/questions/145196 | 1 | Let $A$ be a separable (non-unital) C\*-algebra. Let $\tau: A\to\mathbb{C}$ a tracial state. Consider the strict topology on $A$, i.e. $a\_n\to a$ iff $a\_nx\to ax$ and $xa\_n\to xa$ for all $x\in A$. It is clear that this is the same as the usual norm-topology when $A$ has a unit.
My question is:
>
> When is $\t... | https://mathoverflow.net/users/29404 | When are tracial states continuous in the strict topology? | Go to the GNS representation. There is a cyclic vector $v$ such that $\tau(x) = \langle \pi(x)v,v\rangle$ for all $x \in A$. Now if $\|a\_nx\| \to 0$ in $A$ for every $x \in A$ then $\|\pi(a\_nx)v\|\to 0$ for all $x \in A$. Since $v$ is cyclic and $(a\_n)$ is bounded, this implies that $\pi(a\_n)v \to 0$, so $\tau(a\_n... | 3 | https://mathoverflow.net/users/23141 | 145205 | 78,665 |
https://mathoverflow.net/questions/144837 | 1 | Let $G$ be a real Lie group. Then the complexification $G\_\mathbb{C}$ of $G$ is the unique complex Lie group equipped with a map $φ:G\to G\_\mathbb{C}$ such that any map $G\to H$ where $H$ is a
complex Lie group, extends to a holomorphic map $G\_\mathbb{C}\to H$. If $\mathfrak{g}$ and
$\mathfrak{g}\_\mathbb{C}$ are ... | https://mathoverflow.net/users/nan | Para-Complexification of Lie Groups | Include $\mathfrak{g} \to \mathfrak{g}[e]$ by $A \mapsto A-Ae$, and call the image $\mathfrak{g}^{(1,0)}$.
Include $\mathfrak{g} \to \mathfrak{g}[e]$ by $A \mapsto A+Ae$, and call the image $\mathfrak{g}^{(0,1)}$.
Linear algebra: we can write every element of $\mathfrak{g}[e]$ uniquely as a sum of a $(1,0)$ with a $(... | 1 | https://mathoverflow.net/users/13268 | 145207 | 78,666 |
https://mathoverflow.net/questions/145097 | 17 | Deligne's Weil I has been published under the title "La conjecture de Weil: I" in 1974, and Weil II in 1980. So did Deligne know in 1974 that there would be a Weil II, and can one explain the period between the two publications?
| https://mathoverflow.net/users/nan | Deligne Weil II | To complete Carlo's answer, I think that one thing that can explain the long
gap (in addition of the amount of difficult material in Weil II) is that Deligne
felt the need to consolidate his result of Weil I before going further.
It should be reminded that Weil I was criticized from various directions for relying on ... | 27 | https://mathoverflow.net/users/9317 | 145211 | 78,668 |
https://mathoverflow.net/questions/145215 | 5 | Let $T \subseteq \mathbb R$ be a closed set of real numbers. Let $X := C(T, \mathbb R)$ denote the Fréchet space of continuous real-valued functions on $T$. The topology on $X$ is generated by seminorms $\|x\|\_K := \sup\_{t \in K} |x(t)|$ for any compact $K \subseteq T$.
Let $X^\*$ denote the dual space of continuo... | https://mathoverflow.net/users/238 | Dual of the space of continuous functions | It is the space of compactly supported Radon measures. See Nicolas Bourbaki, *Intégration*, chapter 4, page 156 in Springer’s 2007-edition.
It seems to me that the space spanned by evaluation functionals is dense in the weak-\*-topology (given any finite set of continuous functions choose—using compactness of the sup... | 7 | https://mathoverflow.net/users/33842 | 145216 | 78,671 |
https://mathoverflow.net/questions/145219 | 5 | Suppose I had T points in the interval $[0,1]$. Call them $e\_1, \dots, e\_T$.
Question 1:
What is a good nontrivial bound on the geometric mean of $$\{|e\_i - e\_j| : 1 \leq i < j \leq T \}, $$ as a function of $T$, independent of our choice of $e\_i$?
Question 2:
Suppose we can choose $i$ that minimizes the geome... | https://mathoverflow.net/users/41530 | Maximal geometric mean of distances between points on an interval | The extremal $e\_i$ for Question 1 are probably well-known:
they are $0$, $1$, and $(1+r)/2$ where $r$ ranges over the roots of the
[Gegenbauer
polynomial](http://en.wikipedia.org/wiki/Gegenbauer_polynomials) $C\_{T-2}^{(3/2)}$.
The product of the $|e\_i-e\_j|$ is then a power of $2$ times
the square root of $\mathop... | 5 | https://mathoverflow.net/users/14830 | 145226 | 78,673 |
https://mathoverflow.net/questions/145206 | 9 | When [the survey *Axiomatic Stable Homotopy*, Neil Strickland, 2004](http://ams.math.uni-bielefeld.de/mathscinet/search/publdoc.html?pg1=IID&s1=339774&vfpref=html&r=7&mx-pid=2061852) was written the category of motivic spectra was not investigated from the point of view of axiomatic stable homotopy, as considered e.g. ... | https://mathoverflow.net/users/1220 | Category of motivic spectra | Here's a [direct link](http://www.math.rochester.edu/people/faculty/doug/otherpapers/axiomatic.pdf) to the book by Hovey–Palmieri–Strickland.
The category of motivic spectra is known to satisfy the axioms of Definition 1.1.4 in the book when the base is a countable field of characteristic zero. Axioms (c) and (e) are... | 18 | https://mathoverflow.net/users/20233 | 145227 | 78,674 |
https://mathoverflow.net/questions/145230 | 2 | Is there any analogue of continuous martingale quadratic variation for the discrete case? If so, are there any theorems which characterize simple random walk using quadratic variation - similar to Levy Characterization of Brownian motion. Thanks
| https://mathoverflow.net/users/19126 | Quadratic variation for discrete Martingale | Certainly; if $M\_n$ is a discrete-time $L^2$ martingale, then its quadratic variation $\langle M \rangle\_n$ is the unique predictable increasing process such that $\langle M \rangle\_0 = 0$ and $M\_n^2 - \langle M \rangle\_n$ is a martingale. The existence and uniqueness follow from [Doob's decomposition](http://en.w... | 5 | https://mathoverflow.net/users/4832 | 145237 | 78,678 |
https://mathoverflow.net/questions/145190 | 10 | Suppose we have an arbitrary probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a sequence of real random variables $X\_n:\Omega\to\mathbb{R}$ such that the pushforward measures $(X\_n)\_\*(\mathbb{P})$ converge weakly to a probability $\tilde{\mathbb{P}}$.
Can we always construct a random variable $X:\Omega\t... | https://mathoverflow.net/users/36952 | Limit of pushforward measures of random variables is "represented" by a random variable | Yes I think so. We may assume that $\mathcal F$ is the $\sigma$-algebra generated by sets of the form $X\_n^{-1}(a,\infty)$ as $n$ runs over the positive integers and $a$ runs over the reals as $\mathcal F$ already contains these sets. In so doing, we might be making $\mathcal F$ smaller, but that only makes the proble... | 7 | https://mathoverflow.net/users/11054 | 145244 | 78,682 |
https://mathoverflow.net/questions/145236 | 3 | In Hartshorne's Algebraic geometry, page 158 gives the definition of the trace of $\mathfrak d$ on $Y$, where $i: \, Y \to X$ is a closed immersion of nonsingular projective varieties over an algebraically closed field $k$ and $\mathfrak d$ is a linear system on $X$. Let $\mathfrak d $ corresponds to an invertible shea... | https://mathoverflow.net/users/41541 | Geometric interpretation of the trace of a linear system in Hartshorne's Algebraic Geometry | Usually, in order to have a better geometrical intuition, it is useful to identify an invertible sheaf $\mathscr{L}$ with a line bundle on $X$.
Then, if $i \colon Y \to X$ is a closed immersion of smooth projective varieties, $i^\* \mathscr{L}$ is (essentially by definition) the restriction of the line bundle $\mathc... | 4 | https://mathoverflow.net/users/7460 | 145245 | 78,683 |
https://mathoverflow.net/questions/145240 | 2 | I'm wondering if there are any general conditions on a subdomain of $GL(n)$, which would guarantee that the map from a matrix to its matrix of eigenvectors is a diffeomorphism.
For example, given a matrix with distinct eigenvalues, does there exist a neighborhood around it such that this map is diffeomorphic? Any re... | https://mathoverflow.net/users/41543 | Given a subdomain of GL(n), when is the map from matrices to their matrices of eigenvectors a diffeomorphism? | For example, look at the vector space $S(n)$ of symmetric $x\times n$ real matrices and the action of the orthogonal group $O(n)$ by conjugation:
$$
\ell:O(n)\times S(n)\to S(n),\quad \ell(U,X)=UXU^{-1}
$$
Let $\Sigma$ be the linear subspace of diagonal matrices. Then each $O(n)$-orbit hits $\Sigma$ orthogonally, and $... | 1 | https://mathoverflow.net/users/26935 | 145254 | 78,687 |
https://mathoverflow.net/questions/145247 | 5 | The question is motivated by Toni's question "Approximation of infinite set in generic extension" (see [Approximation of infinite set in generic extension](https://mathoverflow.net/questions/145069/approximation-of-infinite-set-in-generic-extension)).
Before I state the question, let me add some remarks. In what foll... | https://mathoverflow.net/users/11115 | Adding large sets not containing countable ground model sets | The answer to your original question (now question 1) is no, this is impossible, and the GCH and measurable cardinals are not involved.
**Theorem.** There is no cardinal-preserving forcing extension $V[G]$ with a set $C\subset\aleph\_3$ having finite intersection with every countable set in $V$.
Proof. Suppose th... | 4 | https://mathoverflow.net/users/1946 | 145261 | 78,689 |
https://mathoverflow.net/questions/145220 | 6 | This is almost certainly a silly question, but:
I am currently reading Moschovakis' article "Kleene's amazing second recursion theorem" (<http://www.math.ucla.edu/~ynm/papers/1602-002-1.pdf>) and there is a footnote in it which confuses me.
In footnote 10, on page 195, Moschovakis writes:
>
> . . . [T]he oper... | https://mathoverflow.net/users/8133 | Nondeterministic Turing machines and the recursion theorem | Moschovakis by "computable" means: can be computed "by values" (see the last paragraph on the previous page). Where "computed by values" refers to the fact that we do not have access to the definition of a function, but can only ask for values of the function on its arguments.
There is, however, much deeper context ... | 7 | https://mathoverflow.net/users/13480 | 145269 | 78,693 |
https://mathoverflow.net/questions/145271 | 1 | Can anyone provide me with an example of two non-homeomorphic locally-compact Hausdorff spaces $X$ and $Y$, such that $C(X)$ and $C(Y)$ are isomorphic as Banach algebras. Clearly, the Gelfand--Naimark theorem tells us that the two algebras $C(X)$ and $C(Y)$ cannot be isomorphic as $C^\*$-algebras.
| https://mathoverflow.net/users/12653 | Banach Isomomorphic Cts Fucntion Algebras for two Non-Homeomorphic Top Spaces? | There are none.
To be more precise, for locally compact spaces $X$ you have to consider $C\_0(X)$, the space of continuous functions which vanish at $\infty$; you can reduce this case to that of compact $X$ by considering the 1-point compactification and removing the constants by $f\mapsto f- f(\infty).1$.
Now, for... | 2 | https://mathoverflow.net/users/26935 | 145288 | 78,701 |
https://mathoverflow.net/questions/145275 | 10 | Let $H$ be a subgroup of a group $G$ and $x$ be an element of $G$.
If $G$ is finite, $xH\subset Hx$ implies $xH=Hx$ since $xH$ and $Hx$ have the same finite cardinality (that of $H$).
What happens when $G$ and $H$ are infinite? Can a left coset be strictly included in a right coset?
| https://mathoverflow.net/users/41567 | Does a left coset inclusion in a right coset imply they are equal? | In the free group on two generators $x, y$, if we take $H$ to be the subgroup generated by elements of the form $x^n y^m x^{-n}$ for $n \geq 0$, then $x H x^{-1}$ is strictly contained in $H$ because $y \notin x H x^{-1}$.
| 13 | https://mathoverflow.net/users/2926 | 145291 | 78,702 |
https://mathoverflow.net/questions/128741 | 4 | Let $A(k,n)$ be the set of $\{0,1\}$ matrices of order $n$ with all their line sums equal to $k$.
Conjecture number 5 on the list from [Minc's book](http://books.google.co.il/books/about/Encyclopedia_of_Mathematics_and_Its_Appl.html?id=gnT2LCvciqUC&redir_esc=y), attributed to Ryser, says that if $A(k,n)$ contains in... | https://mathoverflow.net/users/22051 | Is Ryser's conjecture on permanent minimizers still open? | My, i.e. hyperbolic polynomials, approach falls a bit short of proving the conjecture: first of all, the bound from my inequality
is $$k^n G(k)^{n-k} \frac{k!}{k^k},$$
where $G(k) = (\frac{k-1}{k})^{k-1}$; this bound is integer only if $k=1, 2, n$. So it can't be the minimum of permanents of integer matrices.
Nobod... | 6 | https://mathoverflow.net/users/41577 | 145295 | 78,703 |
https://mathoverflow.net/questions/145301 | 8 | Suppose $G$ is semisimple Lie group, $\mathfrak{g}$ is its Lie algebra, $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$, and $W$ is the correspondent Weyl group.
Chevalley restriction theorem says that there is isomorphism of algebras of invariants $\mathbb{C}[\mathfrak{g}]^{G}\simeq \mathbb{C}[\mathfrak{h}]^... | https://mathoverflow.net/users/32741 | Chevalley restriction theorem for exterior algebras | See: [arXiv:dg-ga/9406006](http://arxiv.org/abs/dg-ga/9406006) and [arXiv:math.DG/9506223](http://arxiv.org/abs/math/9506223).
* Edit:
See lemma 3.3 in the first paper: $C^\infty$ and polynomial (by Solomon, cited in the paper)
$W$-invariant differential forms on $\mathfrak h$ correspond to horizontal $G$-invariant... | 6 | https://mathoverflow.net/users/26935 | 145303 | 78,706 |
https://mathoverflow.net/questions/145145 | 20 | The irreducible but solvable octic,
$$x^8-x^7+29x^2+29=0\tag{1}\label{1}$$
was first mentioned by Igor Schein in this 1999 [sci.math post](http://www.math.niu.edu/%7Erusin/known-math/99/galois_8) ([Wayback Machine](https://web.archive.org/web/20110530123122/http://www.math.niu.edu/%7Erusin/known-math/99/galois_8)).... | https://mathoverflow.net/users/12905 | On the solvable octic $x^8-x^7+29x^2+29=0$ | To answer your second question, there are soluble octics with the same Galois group involving
other $p$th roots of unity. Take $p\equiv 1\mod 7$, and $K={\mathbb Q}(\alpha)$
the unique degree 7 extension of ${\mathbb Q}$ in ${\mathbb Q}(\zeta\_p)$. E.g. take
$\alpha=\sum\_i \zeta\_p^i$ where $i$ ranges over all seventh... | 10 | https://mathoverflow.net/users/3132 | 145305 | 78,708 |
https://mathoverflow.net/questions/145153 | 3 | In what follows, a ‘$ \* $-representation’ always means a *non-degenerate $ \* $-representation*.
---
Let $ (\mathscr{A},G,\alpha) $ be a $ C^{\*} $-dynamical system, and let $ \pi: \mathscr{A} \to B(\mathcal{H}) $ be a faithful $ \* $-representation of $ \mathscr{A} $ on a Hilbert space $ \mathcal{H} $. Define a... | https://mathoverflow.net/users/nan | ‘Non-Induced’ Left Regular Representations of $ C^{*} $-Dynamical Systems | The answer is no in two ways. Let $G$ be a discrete group and consider the trivial action of $G$ on the reduced group $\mathrm{C}^\*$-algebra $\mathrm{C}^\*\_{\mathrm{r}}G$. The pair of the right regular representation $\rho\colon \mathrm{C}^\*\_{\mathrm{r}}G\to B(\ell\_2G)$ and the left regular representation $\lambda... | 4 | https://mathoverflow.net/users/7591 | 145307 | 78,709 |
https://mathoverflow.net/questions/145279 | 3 | In the book of Volpert on *Traveling wave solutions of Parabolic Systems* (AMS), one reads "*the following assertion is readily proved and we shall not discuss it in detail*". The same result is tacitely assumed in Evans book on *Partial Differential Equations* when dealing with traveling wave solutions of the bistable... | https://mathoverflow.net/users/41568 | Regularity of a nonlinear ODE [Traveling wave solutions of parabolic systems] | @Josh: I don't have any precise reference in mind for that, maybe in the Coddington-Levinson? but as far as I remember it is mostly for linear ODE's
Note first that reversing time $t\to -t$ is equivalent to changing $\sigma\to-\sigma$, so you only need to study one side (say $t\to+\infty$). The case $\sigma=0$ is a b... | 2 | https://mathoverflow.net/users/33741 | 145311 | 78,711 |
https://mathoverflow.net/questions/145289 | 7 | **Is the following identity true ?**
$$\int\_0^\infty \frac{b(x)}{B(x)} dx \quad \overset{?}{=} \quad \int\_0^\infty \frac{x!}{x^x} dx$$
*where*
$$b(x) = \sum\_{n=1}^\infty \frac{n^x}{n^n} \qquad \text{and} \qquad B(x) = \sum\_{n=1}^\infty \frac{n^x}{n!}$$
**NOTE:** *A short sketch of the demonstration proving ... | https://mathoverflow.net/users/39602 | Integral Identity Involving Bell Numbers | It seems that the integral on the left-hand side exceeds $2.57$,
so it's not even close to the numerical value $2.5179\ldots$ of the
integral on the right-hand side.
I told **gp**:
```
b(x) = suminf(n=1,n^x/n^n)
B(x) = suminf(n=1,n^x/n!)
r(x) = b(x)/B(x)
intnum(x=0,25,r(x))
```
and got $2.5793$+. Since the inte... | 8 | https://mathoverflow.net/users/14830 | 145313 | 78,712 |
https://mathoverflow.net/questions/145310 | 3 | I have a way of constructing (something like) a 2-category from a category with products $\mathcal{C}$.
**My question**: Is this a correct construction, and if so, does it have a name?
Define $\mathcal{C'}$ to consist of one object. The 1-morphims are objects of $\mathcal{C}$, the identity is the terminal object of... | https://mathoverflow.net/users/3676 | Create 2-category from a cartesian category | Yes, and more generally this works for any monoidal category. In fact a monoidal category is precisely a (weak) 2-category with one object, in the same way that a monoid is precisely a category with one object. (Weak 2-categories are the "correct" notion of 2-category; what you have in mind is a strict 2-category.)
... | 7 | https://mathoverflow.net/users/290 | 145320 | 78,716 |
https://mathoverflow.net/questions/145312 | 1 | Let $X=L^1\left([0,1]^3\right)$,
for numerical purpose, what are the possible basis function for $X$?
In finite element method, the basis functions are tooth functions, or polynomial functions.
Is there a generalization of this idea to product space?
I ask because I would like to numerically find some ball $B(f... | https://mathoverflow.net/users/nan | What are the basis functions for a product space? | **Edit:** Note that I understood the question for $L^1([0,1]^3)$, c.f. Robert Israels comment.
First, the term "basis" for general Banach spaces, especially ugly spaces such as $L^1$, can be complicated. As far as I remember $L^1$ is one of the spaces which do not possess an "unconditional" basis. On $L^2$ the situat... | 1 | https://mathoverflow.net/users/9652 | 145329 | 78,722 |
https://mathoverflow.net/questions/145338 | 1 | Let $X$ be a complete and separable metric space, let $G(X) \subset C([0,1],X)$ be the space of continuous curves from $[0,1]$ to $X$ with constant speed, i.e.
$$ d(f(t),f(s)) = |t-s| d(f(0), f(1)). $$
Let $S : X^2 \rightarrow 2^{G(X)}$ be such that at every couple of points $(x,y)$ it associates a curve starting at $x... | https://mathoverflow.net/users/41591 | Open set of geodesics implies the set of starting points is open | It need not be even Borel. Take $X = \mathbb{R}^2$ with $d((x\_1,x\_2),(y\_1,y\_2)) = \max(|x\_1-y\_1|,|x\_2-y\_2|)$. Then it is easy to construct a selection $S$ with $S^{-1}(A)$ not Borel for some open set $A$. (For example select for a suitable non-Borel set of doubles of points a non-Euclidean geodesic and for the ... | 2 | https://mathoverflow.net/users/11716 | 145340 | 78,727 |
https://mathoverflow.net/questions/145342 | 3 | Mostly I see a partition of a set $A$ defined as a collection of non-empty disjoint sets whose union is $A$.
I see one reference that allows empty sets to be included in the partition: (Potter, M. *Set theory and its philosophy*, 2004, Oxford University Press, p. 130). **Definition.** A collection $B$ of subsets of $... | https://mathoverflow.net/users/41592 | Terminology for a partition of a set which includes empty sets | In Bourbaki's terminology the elements of a partition may be empty (cf. E.II.4.7).
| 0 | https://mathoverflow.net/users/11025 | 145343 | 78,728 |
https://mathoverflow.net/questions/145292 | 10 | For many basic random processes (like [fractional Brownian motion](http://en.wikipedia.org/wiki/Fractional_Brownian_motion)) [Hurst exponent](http://en.wikipedia.org/wiki/Hurst_exponent) "H" and [fractal dimensions](http://en.wikipedia.org/wiki/Fractal_dimension) ( Hausdorf = Minkoswki typically) "D"
are realated by s... | https://mathoverflow.net/users/10446 | Relationship between fractal dimension and Hurst exponent |
>
> In principle, fractal dimension and Hurst exponent are independent of
> each other: fractal dimension is a local property, while the
> long-memory dependence characterized by the Hurst exponent is a global
> characteristic. For self-affine processes, the local properties are
> reflected in the global ones, re... | 10 | https://mathoverflow.net/users/11260 | 145344 | 78,729 |
https://mathoverflow.net/questions/145294 | 0 | Let $X$ be a smooth hypersurface of degree $d$in $\mathbb{P}^n$ for some $n \ge 3$ and $d>0$. Let $Z\_1, Z\_2$ be codimension $1$ closed subschemes in $X$ which are locally of complete intersection in $X$. Suppose that the natural morphism $\pi:H^0(\mathcal{O}\_X(d)) \to H^0(\mathcal{O}\_{Z\_1}(d))$ is surjective. Assu... | https://mathoverflow.net/users/32151 | A Castelnuovo-Mumford regularity type statement | No, that is not true. Is there some reason you want that? Maybe if you explain more, then the MO community could help you find a theorem to help. Anyway, take $X$ to be a smooth quadric surface in $\mathbb{P}^3$. Let $e\geq 0$ be an integer, e.g., $e=d=2$. Take $Z\_1$ to be the union of $e+1$ disjoint lines contained i... | 1 | https://mathoverflow.net/users/13265 | 145346 | 78,730 |
https://mathoverflow.net/questions/145277 | 1 | Suppose that $F, G: Q\rightarrow R{\rm -Mod}$ be two covariant functors where $Q$ is an abelain category and $R$ is a commuatative ring. Also let $\eta: F\rightarrow G$ be a natural transformation such that $\eta\_v: F(v)\rightarrow G(v)$ is a pure monomorphism for every object $v$ of $Q$. Hence $\eta\_v^+:G(v)^+\right... | https://mathoverflow.net/users/38585 | Pure monomorphism of functors- | If you take $R$ to be, say, a finite field, so that all epimorphisms and monomorphisms of $R$-modules are pure and split, and + is just vector space duality, and for simplicity restrict to functors that take values in finite-dimensional vector spaces, then your question reduces to asking whether a monomorphism (or epim... | 4 | https://mathoverflow.net/users/22989 | 145350 | 78,732 |
https://mathoverflow.net/questions/145044 | 20 | Let $\mathcal{H}=\left\{ h\_{i}\right\} \_{i=1}^{k}$ be an admissible set, and define $$\pi\_{\mathcal{H}}(x)=\left|\left\{ n\leq x\ :\ \exists\ i,j\leq k,\ i\neq j\ \text{such that both }n+h\_{i},\ n+h\_{j}\ \text{are prime}\right\} \right|.$$ The [Hardy-Littlewood](http://en.wikipedia.org/wiki/First_Hardy%E2%80%93Lit... | https://mathoverflow.net/users/12176 | Quantitative lower bounds related to Zhang's theorem on bounded gaps | In the main theorem of [this recent paper of Pintz](http://arxiv.org/abs/1305.6289), Zhang's method is used to show that (for $k$ large enough), there are $\gg\_{\mathcal H} \frac{x}{\log^k x}$ values of $n \le x$ such that two of the $n+h\_i$ are prime and the remaining $n+h\_j$ are almost prime (have $O\_k(1)$ prime ... | 18 | https://mathoverflow.net/users/766 | 145355 | 78,735 |
https://mathoverflow.net/questions/100807 | 9 | While reading [this paper of Kollár](http://arxiv.org/abs/1206.5721), the following question came up. If $f:X\to Y$ is a fiber space (i.e. surjective holomorphic map with connected fibers) with $X,Y$ smooth projective manifolds, with $K\_X\sim\_{\mathbb{Q}} 0$, and with $f$ a submersion everywhere, does there exist a f... | https://mathoverflow.net/users/13168 | Calabi-Yau fiber space without singular fibers implies finite quotient of product? | I am not familiar with this topic, but I remember seeing [this recent paper](http://arxiv.org/abs/1309.3773). It claims that if the map has no singular fiber, it is a fiber bundle (all the fibers isomorphic). Moreover, if the total space is algebraic, it can be a product family after a suitable base change, as you clai... | 5 | https://mathoverflow.net/users/41604 | 145362 | 78,739 |
https://mathoverflow.net/questions/145354 | 4 | I understand that for infinite-area hyperbolic surfaces, there are no $L^2$-eigenfunctions of the Laplace-Beltrami operator but there are a lot resonances.
But I am confused about the notion of resonances for finite-area hyperbolic surfaces. Are there resonances which are not eigenvalues?
| https://mathoverflow.net/users/41600 | Eigenvalues vs resonances | I presume by a "resonance" you mean a scattering resonance, so a pole of the determinant of the scattering matrix. On finite-area hyperbolic surfaces all scattering resonances correspond to eigenvalues, but not the other way around. The "cusp form" spectrum, which
includes all embedded eigenvalues, does not give rise t... | 3 | https://mathoverflow.net/users/11260 | 145371 | 78,743 |
https://mathoverflow.net/questions/145363 | 3 | Let $f \colon \mathbb R^2 \to \mathbb R^2$ be a function from the Schwartz class and $f(0) \neq 0$. Define it's projection $g(x) = \langle f(x), \frac{x}{|x|} \rangle \frac{x}{|x|}$, where $\langle a, b\rangle = a\_1 b\_1 + a\_2 b\_2$, $a = (a\_1,a\_2)$, $b=(b\_1,b\_2)$. Then $g$ has the same rate of decay at infinity ... | https://mathoverflow.net/users/17896 | Decay of the Fourier transform of a function with a discontinuty at zero | You can always do something cheap like isolating singularity by a cut-off with small support, or, if you want to get true asymptotics, subtract something with known Fourier transform and the same singularity so that the difference behaves better. Let me show a (crude) way to apply the first method.
If $D^2h\in L^1$, ... | 7 | https://mathoverflow.net/users/1131 | 145376 | 78,745 |
https://mathoverflow.net/questions/145375 | 2 | Let $\eta$ be an $\omega$-word over $X = \{0,1\}$ and let $F\_k(\eta)$ denote the factors of $\eta$ of length $k$. Define the following $\omega$-languages
$$
L\_k := \{ \xi : F\_k(\xi) = X^k \} = \{ \xi : \xi \mbox{ has every finite word of length k as an infix} \}.
$$
Then define
$$
L := \bigcap\_{k=1}^{\infty} L\_k... | https://mathoverflow.net/users/37580 | Proof that the $\omega$-language consisting of all words containing every finite word as a factor is not rational/regular | Indeed, your language $L$ is not Büchi-recognizable. To see this, suppose that we have a finite state Büchi automata $M$ that recognizes all the strings in $L$. That is, $M$ is a finite state automata, and when we run $M$ on any string $s\in L$ we visit an accepting state of $M$ infinitely many times. Fix any particula... | 3 | https://mathoverflow.net/users/1946 | 145378 | 78,747 |
https://mathoverflow.net/questions/145392 | 5 | Is there a model of set theory such that:
1. AC holds,
2. Every ordinal definable set is measurable,
3. Every ordinal definable set of sets of $\mathbb{R}^2$ whose projection on the first (or second axis) is ordinal definable contains a definable member.
| https://mathoverflow.net/users/38200 | Measurable and definable sets | The answer is no, there is no such model. And we don't even need AC.
To see this, note first that statement 3 is equivalent to the assertion that every real number is ordinal definable. The reason is that the set of reals that are ordinal definable is itself definable, and so the complement is also definable, but co... | 4 | https://mathoverflow.net/users/1946 | 145396 | 78,753 |
https://mathoverflow.net/questions/145405 | 1 | I apologize for asking the same question twice, since my last question was not really understood and there seems to be a problem preventing me from comment of editing the question.
Is there a model of ZFC such that:
1. Every OD set of reals is measurable.
2. Every OD ${\hspace{.03 in}\it family}$ of sets of reals c... | https://mathoverflow.net/users/38200 | Definable families of sets of reals | The answer to this question also is negative. (I take statement 2 to be only about non-empty families.)
First, your statement 2 implies that every real is ordinal definable, since the set of singletons of non-OD reals is definable, but can't have any OD member.
In particular, your statement 2 implies that there i... | 2 | https://mathoverflow.net/users/1946 | 145406 | 78,754 |
https://mathoverflow.net/questions/145394 | 0 | I was wondering what is the best tail bound for
\begin{equation\*}
\mathbb{P}\bigg\{\sum\_{k=1}^n X\_k^4>(1+t)3n\bigg\}\le ?
\end{equation\*}
where $X\_k$ are i.i.d. $\mathcal{N}(0,1)$.
| https://mathoverflow.net/users/34919 | concentration of sums of fourth moment of normals | What counts as "best"? The smallest tail bound is of course
$$
(2\pi)^{-n/2} \int\_{\{(x\_1,\dotsc,x\_n) : \sum x\_i^4 > (1+t)3n\} } e^{-\sum x\_i^2 / 2} dx\_1 \dotsb dx\_n.
$$
Presumably you want something simpler. Using standard concentration inequalities,
$$
\mathbb{P} \left\{ \|X\|\_4 > \mathbb{E}\|X\|\_4 + s \ri... | 2 | https://mathoverflow.net/users/1044 | 145414 | 78,756 |
https://mathoverflow.net/questions/145284 | 19 | It is well-known that Lie groups are, under nice conditions, essentially determined by their Lie-algebras. What's the corresponding statement for algebraic groups over fields of finite characteristic?
| https://mathoverflow.net/users/41571 | Algebraic Groups in Characteristic p | Since the question is somewhat open-ended, it may be useful to add further comments to what Dietrich and Marguax have said. [Also, it's important to correct Dietrich's first sentence: While any finite dimensional Lie algebra over $\mathbb{C}$ is the Lie algebra of some Lie group, it need not be the Lie algebra of an al... | 15 | https://mathoverflow.net/users/4231 | 145431 | 78,761 |
https://mathoverflow.net/questions/145416 | 2 | Let $S$ be a connected noetherian normal scheme of dimension 0 or 1 (i.e. $S$ is a connected Dedekind scheme).
Let $f:G\to G'$ be a morphism of semi-abelian schemes over $S$. In their book on Neron models, Bosch-Lutkebohmert-Raynaud define that $f$ is an isogeny if for all points $s\in S$ the base changed morphism $f... | https://mathoverflow.net/users/41627 | Degree of isogenies between (semi-)abelian schemes | I am going to assume that $S$ is Noetherian. By limit arguments, one should always be able to reduce to that case. Also I am going to assume that your semi-Abelian schemes are smooth over $S$; I believe this is part of the usual definition.
In fact, $f$ should be flat, which implies constancy of the degree of $f\_s$.... | 1 | https://mathoverflow.net/users/13265 | 145432 | 78,762 |
https://mathoverflow.net/questions/145428 | 8 | The prime number theorem tells us that , if $\pi\left(x\right)$ denotes the number of primes less than or equal to $x$, we have $$\pi\left(x\right)\sim\frac{x}{\log x}.$$
In a similar manner considered $1\leq a \leq q$ with $(a,q)=1$ and defined $\pi\left(x,a,q\right)$ the number of primes less than or equal to $x$ con... | https://mathoverflow.net/users/41635 | On the prime number theorem in arithmetic progression | For $x\leq\phi(q)$ the estimate $\pi(x,a,q)\ll\frac{1}{\phi(q)}\frac{x}{\log x}$ would imply $\pi(x,a,q)\ll\frac{1}{\log x}$, i.e. $\pi(x,a,q)=0$ for large $x$ which is clearly false. So a bound you envision can only hold for $x$ slightly above $\phi(q)$. On the other hand, for any $\epsilon>0$, the Brun-Titchmarsh ine... | 11 | https://mathoverflow.net/users/11919 | 145443 | 78,765 |
https://mathoverflow.net/questions/145412 | 8 | Denote $B(X,Y)$ the Banach space of bounded operators between Banach spaces $X$ and $Y$.
When $X$ and $Y$ are both finite dimensional, it follows from the formula
$$\|u\|\_{B(X,Y)} = \sup\_{\|x\|\_X < 1,\|\xi\|\_{Y^\*}< 1} \xi(u(x))$$
and the Hahn-Banach separation theorem that that the dual Banach space of $B(X,Y)$ ... | https://mathoverflow.net/users/10265 | Dual Banach space of $B(X,Y)$ when $X$ is finite dimensional | 1. If $X=\ell\_1^N$, then $B(X,Y)\equiv \ell\_\infty^N(Y)$ and the result is easy.
2. If the unit ball of $X$ has $N$ extreme points, then $X$ is a quotient of $\ell\_1^N$ and you can deduce what you want from (1).
3. The general case follows from (2) by approximation.
Notice that this is basically Dean's proof of lo... | 7 | https://mathoverflow.net/users/2554 | 145444 | 78,766 |
https://mathoverflow.net/questions/145434 | 3 | I imagine the dynamic of $SL(2,\mathbb{Z}$) on $\mathbb{C}^2$ has been studied. Does one know if it is recurrent or ergodic (with respect to the Lebesgue measure) ? Is there any explicit description of the orbits ?
Thank you for your help,
Selim
| https://mathoverflow.net/users/25511 | Dynamic of $SL_2(\mathbb{Z}$) on $\mathbb{C}^2$ | Since matrices from $SL(2,\mathbb Z)$ are real, the action on $\mathbb C^2$ is the same as the diagonal action on the product of two copies of $\mathbb R^2$. Now, the action of $SL(2,\mathbb Z)$ on $\mathbb R^2$ is well-known to be conjugate to its action on the space $H$ of horocycles in the hyperbolic plane $\mathbb ... | 10 | https://mathoverflow.net/users/8588 | 145449 | 78,769 |
https://mathoverflow.net/questions/145456 | 2 | If we denote the parabolic subgroup of $SL(2,\mathbb{C})$ by $P$, then we have the well known isomorphism $SL(2,\mathbb{C})/P \simeq S^2$, where $S^2$ is the two sphere. Now the compact real form of $SL(2,\mathbb{C})$ is given by $SU(2,\mathbb{C})$, and it gives us the alternative presentation $SU(2,\mathbb{C})/S^1 \si... | https://mathoverflow.net/users/12653 | Split real form of $SL(2,\mathbb{C})$ description of the two sphere? | That seems to be impossible. First of all, I assume that you meant to write $\textbf{SL}(2,\mathbb{R})/L \cong S^2$, since otherwise the dimensions don't work out. Denote by $L\_0$ the connected component of $L$ containing the identity element. Then $L\_0$ is a connected subgroup of $\textbf{SL}(2,\mathbb{R})$. Thus th... | 6 | https://mathoverflow.net/users/13265 | 145457 | 78,773 |
https://mathoverflow.net/questions/145484 | 8 | Pick integers $a, b \ge 2$ and let $\xi\_{a,b}$ be the sum of the series $\sum\_{n=1}^\infty a^{-b^n}$. It is known that $\xi\_{2,2}$ is transcendental: I learned a proof of this from [notes by M. Filaseta](http://www.math.sc.edu/~filaseta/gradcourses/Math785/Math785Notes5.pdf) (who attributes the basic idea to P. Erdő... | https://mathoverflow.net/users/16537 | On the irrationality measure of $\sum_{n=1}^\infty a^{-b^n}$ | This result is due originally to K. Mahler, and holds true more generally with any algebraic $a$ having $|a| > 1$ (so that the series converges absolutely). I can recommend Masser's lecture in the CIME 2000 school on diophantine approximations (LNM 1819), where the main idea of Mahler's proof is outlined as an illustra... | 8 | https://mathoverflow.net/users/26522 | 145487 | 78,784 |
https://mathoverflow.net/questions/145488 | 5 | Let $G$ be a connected, simply-connected complex semisimple linear algebraic group, and let $V$ be a finite-dimensional complex $G$-module. Is there a nice description of those $v\in V$ for which the $G$-orbit of $v$ is Zariski-closed in $V$? Another question: Which non-zero $v\in V$ have the property that the $G$-orbi... | https://mathoverflow.net/users/25358 | When are orbits of semisimple group representations closed? | There's some terminology here that might be helpful for a literature search: $v$ is said to be **semisimple** if $Gv$ is closed in $V$ and said to be **nilpotent** if $v\neq0$ and $Gv$ is *not* closed in $V\setminus\{0\}$. The set of all nilpotent $v\in V$ is called **nullcone** of $V$. If $V$ is the adjoint representa... | 7 | https://mathoverflow.net/users/430 | 145498 | 78,787 |
https://mathoverflow.net/questions/145496 | 3 | Let D in Z/3[[x]] be sum ((a\_n)(x^n)) where the sum runs over all n prime to 6 and a\_n is the mod 3 reduction of the number of ideals of norm n in the ring of integers of Q(root(-3)). (So D=x+2(x^7)+2(x^13)+2(x^19)+x^25+(higher degree terms).)
There are formal Hecke operators T\_p: Z/3[[x]]-->Z/3[[x]] for all prime... | https://mathoverflow.net/users/6214 | Are these powers of a characteristic 3 power series annihilated by certain Hecke operators? | Yes, and $D \equiv \eta(2z)^{12}$ certainly helps: it means
$$
D^{(1+3^m)/4} \equiv \eta(2z)^{3+3^{m+1}} \equiv \eta(6z) \, \eta(2\cdot 3^{m+1}z).
$$
Now $\eta = \sum\_a \chi(a) q^{a^2/24}$ where $a$ runs over integers
congruent to $1 \bmod 6$ and $\chi$ is the even Dirichlet character $\bmod\,12$.
Since $m$ is odd, sa... | 3 | https://mathoverflow.net/users/14830 | 145504 | 78,791 |
https://mathoverflow.net/questions/145509 | 15 | I will be working in ZFC, but I am *not* assuming the Continuum Hypothesis (or Martin's Axiom). I know that it is consistent with ZFC that one can cover the real line with less than continuum of meager sets. My questions are about coverings by sets which are "nicer" than just meager (for instance, their proper intersec... | https://mathoverflow.net/users/39654 | Can one cover the plane with less than continuum of lines? | For your question 1, the answer is no. If $\{X\_{\alpha}\}\_{\alpha\in A}$ cover a circle, then $|A|=2^{\aleph\_0}$ (just because each line can cover at most two points of the circle).
For the question 2, as Todd pointed out, the answer is also no.
| 37 | https://mathoverflow.net/users/14340 | 145511 | 78,793 |
https://mathoverflow.net/questions/145490 | 12 | There is a three-way correspondence between:
* Real (connected and simply connected) Lie groups of dimension $n$;
* $\mathbb R$-Lie algebras of dimension $n$;
* Formal group laws in $n$ variables over the reals.
To get from the first to the second, one only really needs the fact that manifolds have tangent spaces w... | https://mathoverflow.net/users/33042 | Is it possible to construct a formal group law from a Lie group without choosing coordinates? | Of course if one wants to avoid coordinates then one should replace formal group laws by the coordinate-invariant things that they are the coordinate-dependent versions of!
There are two dual ways to do this that I know of.
**Commutative:** First, observe that an $n$-dimensional formal group law over $k$ is precis... | 15 | https://mathoverflow.net/users/290 | 145512 | 78,794 |
https://mathoverflow.net/questions/145520 | 5 | Let $X$ be a proper scheme over a field $\Bbbk$. Let $\mathscr L\in\mathrm{Pic}(X)$ be a globally generated line bundle. If for some choice of global sections $V\subseteq\mathscr L(X)$, the induced morphism $\phi\_V:X\to\mathbb P(V)$ is finite, then $\mathscr L$ is ample. The converse also holds. However, I have search... | https://mathoverflow.net/users/9947 | Line bundle ample iff induced morphism finite, looking for reference | Robert Lazarsfeld, *Positivity in Algebraic Geometry I*, Corollary 1.2.15 page 28.
| 8 | https://mathoverflow.net/users/7460 | 145521 | 78,797 |
https://mathoverflow.net/questions/145516 | 2 | With respect to the Strong product, is the Kneser graph prime and if not how does one find a prime decomposition? Are there any references or algorithms?
| https://mathoverflow.net/users/10035 | Factors of Kneser graph | Doerfler and Imrich and (independently) MacKenzie showed that any connected graph has a unique factorization into graphs prime relative to the strong product. It follows that if a connected graph is not prime, its automorphism group is the direct product of two non-identity groups, or is a wreath product. But the autom... | 3 | https://mathoverflow.net/users/1266 | 145530 | 78,802 |
https://mathoverflow.net/questions/145517 | 4 | Let $A$ be an $n\times n$ Hermitian matrix,
with well-separated eigenvalues $\lambda\_1 > \lambda\_2 ... > \lambda\_n$,
with $|\lambda\_i-\lambda\_j|>\epsilon$, for all $i \neq j$.
Let $G$ be a Gaussian matrix,
i.e. each $G\_{i,j}$ is distributed ${\cal N}(0,1)$.
What can be said about the distribution of the eigenval... | https://mathoverflow.net/users/36272 | Distribution of the spectrum of a perturbed matrix | Since $P(G)\propto{\rm exp}\bigl(-\frac{1}{2}{\rm Tr}\,GG^{\rm T}\bigr)$ is invariant under orthogonal transformations, if $A$ is real Hermitian you can work in a basis where $A$ is diagonal, with the $\lambda\_n$'s on the diagonal. The first order correction to the $n$-th eigenvalue is then just $\delta\lambda\_{n}= f... | 2 | https://mathoverflow.net/users/11260 | 145532 | 78,803 |
https://mathoverflow.net/questions/145531 | 0 | Let $G$ be a simple graph which is a $2n$-cycle together with $n$ chords such that $G$ is $3$-regular. In other words, the set of $n$ chords is a perfect matching of $G$.
I conjecture that for every such graph $G$, there must exist at least two different $2n$-cycles in $G$. Can you prove it or give a counterexample?
... | https://mathoverflow.net/users/40096 | Does this graph contain at least two Hamiltonian cycles? | Here is a proof of Gordon's claim. We will prove something slightly stronger.
**Claim.**
Let $G$ be a $d$-regular graph with $d$ odd. Then for every $e \in E(G)$, there is an even number of Hamiltonian cycles using $e$.
*Proof.* Let $e=uv$ and let $\mathcal{H}$ be the set of Hamiltonian paths in $G$ that start at... | 3 | https://mathoverflow.net/users/2233 | 145549 | 78,806 |
https://mathoverflow.net/questions/145541 | 3 | It is well known that the blow-up of $\mathbb P^2$ in one or two points does not accept a Kahler-Einstein metric. Kahler-Einstein metrics are particular cases of constant scalar curvature Kahler metrics (cscK metrics). I was wondering if:
1) Is it known if the blow-up of $\mathbb P^2$ in one or two points accepts a c... | https://mathoverflow.net/users/1887 | Existence of constant scalar curvature Kahler metrics on projective manifolds | In general, reductiveness of the lie algebra of holomorphic vector fields is an obstruction to the existence of *cscK metrics* (constant scalar curvature Kähler), not just Kähler-Einstein metrics. In particular, the blow up of $\mathbb{P}^2$ in $1$ or $2$ points cannot admit a cscK metric in any Kähler class. I'm not s... | 6 | https://mathoverflow.net/users/22294 | 145550 | 78,807 |
https://mathoverflow.net/questions/145551 | 0 | Let $A(n), B(n) \in \mathbb{Z}[n]$ be polynomials, not both constant, such that $4A^3(n) + 27B^2(n)$ is not the zero polynomial and the polynomial (in variables $x, y$) $$y^2 - x^3 - A(n)x - B(n) \in \mathbb{C}(n)[x, y]$$
has no zeroes in $\mathbb{C}(n) \times \mathbb{C}(n)$. Let $K$ be a number field. Furhter, let $... | https://mathoverflow.net/users/10591 | One parameter families of elliptic curves over rings of integers of number fields | Let $E\_n$ denote your elliptic curve. It's probably easier to ask for an integer $n\_0$ such that the Mordell-Weil groups $E\_{n\_0}(\mathbb{Q})$ and $E\_{n\_0}(K)$ coincide. There has been a fair amount of attention given to the question of elliptic curves $E/K$ and extensions $L/K$ such that both $E(K)$ and $E(L)$ h... | 2 | https://mathoverflow.net/users/11926 | 145558 | 78,809 |
https://mathoverflow.net/questions/145542 | 3 | Fix an odd prime $p$ and an integer $r\geq 1$. Fix a Dirichlet character $\chi$ of primitive conductor $p^r$. Let $f\in S\_2(p^r,\chi)$ be a normalized newform of weight $2$, level $p^r$ and nebentype $\chi$. Assume $f$ is ordinary at $p$.
Let $K\_{f,p}$ be the finite extension of $\mathbb{Q}\_p$ generated by the fou... | https://mathoverflow.net/users/37837 | Ordinary representations associated to modularforms: Étale submodules versus étale quotients | The usual normalizations for Galois representation attached to an eigencusp form $f$ is either to take the étale cohomology or the usual Tate module of the abelian variety cut out in the Jacobian of the modular curve by the morphism $\lambda\_{f}$ sending the Hecke operator $T(\ell)$ to $a\_\ell$ such that $T(\ell)f=a\... | 2 | https://mathoverflow.net/users/2284 | 145573 | 78,815 |
https://mathoverflow.net/questions/145482 | 9 | It is well known that $SU(3)$ is the unique, non-trivial, principal $SU(2)$- bundle over $S^5$. To my knowledge the way this is proven is by using the following fact:
>
> If $G$ is a Lie group and $H$ a closed subgroup, then $G \to G/H$ is a (locally trivial) principal $H$-bundle.
>
>
>
In our case, $SU(3)$ ... | https://mathoverflow.net/users/34766 | Local trivializations of the non-trivial $SU(2)$-bundle over $S^5$ | The problem seems to be one of systematically extending $z=(z\_1,z\_2,z\_3)\in S^5$ to an orthonormal basis of $\mathbb{C}^3$. To this end, let $e\_1,e\_2,e\_3$ be the standard basis vectors for $\mathbb{C}^3$. For $j\in\{1,2,3\}$, let $$U\_j:=\{z\in S^5:\{z,e\_1,e\_2,e\_3\}\setminus\{e\_j\}\text{ is a basis of }\mathb... | 1 | https://mathoverflow.net/users/25358 | 145584 | 78,817 |
https://mathoverflow.net/questions/145578 | 1 | In general there is no relation between automorphism groups of subgraphs and the main graph. However, this question is about vertex transitive graphs.
Given vertex transitive $G$ and $H$ such that $|\mathcal{V}(G)|<|\mathcal{V}(H)|$.
If $\mathcal{Aut}(G)\supset\mathcal{Aut}(H)$, is $G\leq H$?
If $G\leq H$, is $\... | https://mathoverflow.net/users/10035 | On homomorphisms between vertex transitive graphs | The answer to the first question is certainly no. The automorphism group of the Petersen graph $H$ is $\mathfrak S\_5$ so is equal to the automorphism group of the complete graph $G=K\_5$, which has strictly fewer vertices. Both these graphs are vertex transitive and yet the Petersen graph has no $5$-clique, so $G$ is ... | 3 | https://mathoverflow.net/users/2284 | 145586 | 78,818 |
https://mathoverflow.net/questions/145593 | 7 | The question is exactly that of the title. Suppose $\varphi\in V$ is an $\mathcal{L}\_{\infty\omega}$-sentence, and $W$ is an inner model of $V$ such that $\varphi\in W$. Is the statement
>
> $\varphi$ has a model
>
>
>
absolute between $V$ and $W$? It is clearly upwards absolute, by induction on rank, so I'm... | https://mathoverflow.net/users/8133 | Is the consistency of $\mathcal{L}_{\infty\omega}$-sentences absolute? | Here is a somewhat easier counterexample.
Let $I$ be a countable set, which is uncountable in $W$. Let $c\_n$ and $d\_\alpha$ be constant symbols, for $n\in\mathbb{N}$ and $\alpha\in I$. Consider the formula $\varphi$ that asserts that all the $d\_\alpha$'s are different, but that every $d\_\alpha$ is equal to some ... | 7 | https://mathoverflow.net/users/1946 | 145595 | 78,821 |
https://mathoverflow.net/questions/145599 | 4 | Almost-huge cardinals are characterizable in terms of coherent towers of supercompactness measures, with a certain property of the direct limit model (see Kanamori's book). A useful large cardinal hypothesis between almost-huge and huge is the assertion that a cardinal has an almost-hugeness towers of unbounded height.... | https://mathoverflow.net/users/11145 | Is "ultracompact" taken? | I think that the super-almost-huge terminology is fairly widely used with this meaning, so that $\kappa$ is super-almost-huge if for every $\theta$ there is an almost hugeness embedding $j:V\to M$ with critical point $\kappa$ and $j(\kappa)\gt\theta$.
This terminology aligns with the similar terminology for [superhu... | 4 | https://mathoverflow.net/users/1946 | 145601 | 78,823 |
https://mathoverflow.net/questions/145518 | 4 | Let $(R, \mathfrak{m})$ be a Gorenstein local ring of characteistic $p>0$. Let $x\_1,...,x\_d$ be a system of parameters of $R$. Let $I$ be an $\mathfrak{m}$-primary ideal containg $(x\_1,...,x\_d)$. Then there exists an ideal $J$ containng $(x\_1,...,x\_d)$ such that
$(x\_1,...,x\_d):J = I$.
Question. Is it true tha... | https://mathoverflow.net/users/17901 | Colon property of Gorenstein rings | I am pretty sure that you already know a result of Hochster-Huneke, Corollary 4.3 in "Tight Closure, Invariant Theory, and the Briancon-Skoda Theorem", Journal of the AMS, 1990, for regular local rings.
I don't think it is true in general. Let $R = \mathbb{Z}/3[X,Y]\_{(X,Y)}/(X^4)$. Let $x,y$ denote the images of $X... | 4 | https://mathoverflow.net/users/22388 | 145609 | 78,826 |
https://mathoverflow.net/questions/145596 | 2 | I apologize if the questions are too elementary for the forum, but I am not an expert in this fields.
1. How many (rooted or unrooted) binary trees with $n$ vertices are there up to isomorphism?
2. How many maximal planar graphs with $n$ nodes are there up to isomorphism?
3. How many graphs with $n$ nodes and valenc... | https://mathoverflow.net/users/41219 | Counting graphs up to isomorphism | For question 1, I believe you refer to unordered trees, for which a summary of the available information is given page 4 of "The CRT is the scaling limit of unordered binary trees" by Marckert and Miermont, Random Structures & Algorithms, 2011. Also see Sloane A001190, for the Etherington-Wedderburn numbers, and Otter ... | 5 | https://mathoverflow.net/users/34435 | 145613 | 78,829 |
https://mathoverflow.net/questions/145082 | 3 | I'm reading the article [Generalized arithmetic intersection numbers](http://www.google.de/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CDAQFjAA&url=http://www.math.uni-hamburg.de/home/kuehn/crellekuehn.ps&ei=CuhfUvaABIORtQaC9YGYCQ&usg=AFQjCNGZBuM6_L5dUQnU2AI-dmmSfMFP_A&sig2=YNPwkGrYhS_Kp0F2yNwZ5w&bvm=bv.54176721,d.Ym... | https://mathoverflow.net/users/14549 | Section on $\mathcal{X}(1)$ induced by a cusp of $X(1)$ | This answer is just an expansion of the hints given by Damian Rössler and David Loeffler in the comments. If I made any mistake please point it out!
In the notation of the question we have that $X(\Gamma(1))$ has a rational smooth, connected and compact model $X(\Gamma(1))\_\mathbb{Q}$. By Belyi's Theorem this is tru... | 2 | https://mathoverflow.net/users/14549 | 145614 | 78,830 |
https://mathoverflow.net/questions/145603 | 0 | Is there something like the Chow lemma for complex spaces (stating that every compact complex space is birational to a projective variety, or some variant of this, like: every proper morphism of complex spaces is birational to a projective morphism)?
| https://mathoverflow.net/users/nan | Chow lemma for complex spaces | As pointed out in the comments, if a complex compact manifold $X$ of dimension $n$ is bimeromorphic to a projective variety, its field of meromorphic functions must have transcendence degree $n$ (the maximum possible) -- one says that $X$ is a *Moisezon manifold*. Conversely, a deep theorem of Moisezon asserts that any... | 4 | https://mathoverflow.net/users/40297 | 145628 | 78,835 |
https://mathoverflow.net/questions/145638 | 5 | Let $f:X\to Y$ be a finite cover of smooth algebraic varieties, branched along a divisor $R\subset Y$. Let $E$ be a vector bundle on $Y$. What is the relation between the chern classes of $E$ and the chern classes of $F:=f\_\*(f^\*E)$?
| https://mathoverflow.net/users/4096 | chern classes of push-pulled vector bundles | You need to know what is the splitting of $f\_\* \mathscr{O}\_X$, which depends on the cover.
Let us consider the double cover case, which is the easiest one. Then $$f\_\* \mathscr{O}\_X = \mathscr{O}\_Y \oplus L^{-1},$$ where $L$ is a line bundle on $Y$ such that $R \in |2L|$. Then, given any vector bundle $E$ on $Y... | 7 | https://mathoverflow.net/users/7460 | 145643 | 78,840 |
https://mathoverflow.net/questions/145637 | 11 | What's the best record toward Selberg's eigenvalue conjecture:
a Maass form on $\Gamma\_0(N)$ has eigenvalue greater than or equal to 1/4?
| https://mathoverflow.net/users/2666 | Best record toward Selberg's eigenvalue conjecture? | \[\frac{1}{4} - \left(\frac{7}{64}\right)^2 = \frac{975}{4096} \approx 0.238037\ldots\]
as far as I know. Established in an appendix by Kim and Sarnak to Kim's "[Functoriality for the exterior square of $GL\_{4}$ and the symmetric fourth of $GL\_{2}$](http://www.ams.org/mathscinet-getitem?mr=1937203)", Journal of the A... | 16 | https://mathoverflow.net/users/nan | 145645 | 78,842 |
https://mathoverflow.net/questions/145635 | 4 | As far as I understand it, by the work of Lafforgue (cf. Laumon, "Cohom. of Drinfeld ... II", Thm 12.4.1) there is a Galois representation associated to an irreducible cuspidal automorphic representation $\pi$ , if $\pi$ is Steinberg at some place $\infty$.
>
> Do we expect Galois representations also if some of th... | https://mathoverflow.net/users/41749 | To which automorphic forms/rep's over a function field can we associate a Galois representation? | Let $X$ be a smooth projective geometrically irreducible curve over $\mathbb F\_{q}$ a finite field and $F$ its global field. Let $\mathbf G/F$ be a split connected reductive group and $\widehat{\mathbf G}$ its Langlands dual. Let $\pi$ be an irreducible cuspidal automorphic representation of $\mathbf G(\mathbb A\_{F})... | 4 | https://mathoverflow.net/users/2284 | 145652 | 78,847 |
https://mathoverflow.net/questions/145639 | 1 | Let $S^{d-1}=\{(x\_1,\cdots,x\_{d})\in {\mathbb R}^{d}|{x\_1}^2+\cdots+{x\_d}^2=1\}$, and let us call the intersection of any $d-1$ dimensional subset which passes through the origin and $S^{d-1}$ '**a great circle**'. Also, let us call one of two components gotten by cutting $S^{d-1}$ by a great circle '$d-1$ dimensio... | https://mathoverflow.net/users/34490 | Finding the probability that $n$ points which are randomly selected from $d-1$ dimensional spherical surface are 'semi-spherical' | This is answered in painstaking detail [here.](http://www.mathpages.com/home/kmath327/kmath327.htm)
| 2 | https://mathoverflow.net/users/11142 | 145654 | 78,849 |
https://mathoverflow.net/questions/88847 | 1 | [Note] I've asked this question at <https://dsp.stackexchange.com/questions/1434/capacity-of-channels-with-pilots> with NO answers.
Question is about "channel capacity" <http://en.wikipedia.org/wiki/Channel_capacity> I guess it is pretty elementary for experts, however I have not seen it discussed in e.g. Proakis boo... | https://mathoverflow.net/users/10446 | Capacity of channels with pilots? | This is a classical problem which goes under the name "non-coherent detection".
Here are some facts:
1. Capacity of the block fading channel is not achieved using pilots.
2. Capacity of the channel is not achieved using Gaussian data.
3. Capacity of the channel is generally unknown.
4. Only capacity expansions in the... | 2 | https://mathoverflow.net/users/41759 | 145660 | 78,851 |
https://mathoverflow.net/questions/145659 | 4 | I want to find the p.d.f of the n-th order statistics from a set of independent, but NOT identically distributed random variables $X\_1, \dots, X\_n$ (the p.d.f. of the $X\_i$'s is at hand)
| https://mathoverflow.net/users/17243 | Order statistics of independent NOT identically distributed random variables | It is Bapat-Beg theorem: <http://en.wikipedia.org/wiki/Bapat%E2%80%93Beg_theorem>
Not suprissingly, the permanent shows up. This is why I know it.
| 4 | https://mathoverflow.net/users/41577 | 145661 | 78,852 |
https://mathoverflow.net/questions/145672 | 14 | I have a question about indefinite lattices.
QUESTION: Let $\Lambda\times\Lambda\rightarrow {\Bbb Z}$ be a lattice,
that is, ${\Bbb Z}^n$ with a non-degenerate integer quadratic form,
not necessarily unimodular, and $G:=O(\Lambda)$ the group of
(integer) isometries. Denote the set of all vectors $v\in\Lambda$ such t... | https://mathoverflow.net/users/3377 | orbits of automorphism group for indefinite lattices | It is in Kneser's book Quadratische Formen. For each r, there are only finitely many classes of representations of r by the lattice.
| 6 | https://mathoverflow.net/users/29241 | 145686 | 78,861 |
https://mathoverflow.net/questions/145678 | 2 | Suppose $X\sim \mathrm{Binomial}(n,p),$ i.e. $X$ takes values in $\{0,1,2,\ldots, n\}$ and $P(X=i) = {n\choose i} p^i(1-p)^{n-i}.$
I am looking for a good estimate for $\mathbb{E}\log(X+\alpha).$ I am thinking of $\alpha$ as some fixed number, for example, $\alpha = 1,$ $p$ as a fixed number between 0 and 1, eg. $p=\... | https://mathoverflow.net/users/7576 | Expected value of logarithm of a binomial random variable | Just expand $ln(X+\alpha)$ as a Taylor series about $X=np$ and then do the binomial sum term by term. Use tail bounds on the binomial distribution to show that the error terms are meaningful. Without much checking, I got
$$\mathbb E(\ln(X+\alpha)) = \ln(pn+\alpha) - \frac{1-p}{2pn} + O(1/n^2),$$
which ought to be preci... | 9 | https://mathoverflow.net/users/9025 | 145690 | 78,862 |
https://mathoverflow.net/questions/97857 | 4 | Suppose you have some storage medium of a given size M, and can make some kind of backup on another medium of size B with M > B. You can choose the scheme to determine the contents of the backup.
After you made that partial backup, an adversary (or a random process) will make a number of changes to your original medi... | https://mathoverflow.net/users/16793 | Partial backups | Your question is very similar to the extended idea of *erasure-resilient codes* discussed here:
[Y. M. Chee, C. J. Colbourn, A. C. H. Ling, *Asymptotically optimal erasure-resilient codes for large disk arrays,* Discrete Appl. Math. **102** (2000) 3–36](http://www.sciencedirect.com/science/article/pii/S0166218X990022... | 2 | https://mathoverflow.net/users/27829 | 145692 | 78,863 |
https://mathoverflow.net/questions/145445 | 0 | Suppose $X\_1$ and $X\_2$ are two nice metric spaces, e.g. two Riemannian manifolds, and let $G\_i=Isom(X\_i)$. Then $G\_1\times G\_2\subset Isom(X\_1\times X\_2)$.
Suppose $X\_1\times X\_2$ is not compact and let $\Gamma\subset G\_1\times G\_2$ act cocompactly on $X\_1\times X\_2$. In general the projections of $\Ga... | https://mathoverflow.net/users/40911 | discrete subgroups of the isometries of a product | This is presumably a hard question, see e.g. <http://download.springer.com/static/pdf/747/art%253A10.1007%252FBF02698916.pdf?auth66=1382759591_9b0973395ce39560e6d6963480bf665c&ext=.pdf>
| 0 | https://mathoverflow.net/users/39082 | 145700 | 78,865 |
https://mathoverflow.net/questions/145676 | 1 | I asked a shorter version of this question at [math.stackexchange.com](https://math.stackexchange.com/questions/532560/inverses-of-two-argument-functions-with-respect-to-one-argument) four days ago but it hasn't gotten any answers or comments.
Consider a function $f : A \times B \to C$ and two inverses, each with res... | https://mathoverflow.net/users/10828 | Inverses of two-argument functions with respect to one argument | Surjective and monotonic implies surjective and injective (i.e. bijective) but not vice versa. For example if $A=B=C$ is finite (and well ordered) then there is a unique surjective increasing map (the identity) and a unique decreasing one. So (for size more than two) there would not be enough maps to have them all be m... | 1 | https://mathoverflow.net/users/8008 | 145712 | 78,868 |
https://mathoverflow.net/questions/145717 | 1 | Consider an algebraic number $\alpha$, which can be taken to be an
integer. With $\deg\alpha$ a prime number, one can easily arrange that
to be such that all powers $\alpha^n$ to be of the same degree as $\alpha$.
I would like to know how to get numbers with the same property without any restriction on its degree. If... | https://mathoverflow.net/users/22878 | Algebraic numbers abhorrent to cyclotomic fields | The second question is rather easy.
One can construct Galois extensions $F/\mathbb Q$ with Galois groups nonabelian and simple. Then they are certainly linearly disjoint from all cyclotomic extensions.
For the first question, let $F = \mathbb Q(\sqrt[15]{2})$. The element $\alpha = \sqrt[15]{2}$ has its minimal poly... | 4 | https://mathoverflow.net/users/38052 | 145722 | 78,872 |
https://mathoverflow.net/questions/136148 | 12 | This question is inspired by an old [question](https://mathoverflow.net/questions/80792/given-an-integer-polynomial-is-there-a-small-prime-modulo-which-it-has-a-root) of Greg Kuperberg, about how small is the first prime $p$ which makes a given monic polynomial $P$ with integral coefficient have a (simple) root modulo ... | https://mathoverflow.net/users/9317 | Least prime $p$ such that an irreducible polynomial of degree $n$ has no root modulo $p$? | Let $K$ be the number field obtained by adjoining some root of $f(x)$ to
${\Bbb Q}$, and let $d\_K$ be its discriminant. Let $k$ be the degree of $K$, and let
$\zeta\_K(s) = \sum\_{n=1}^{\infty} a(n) n^{-s}$ denote the Dedekind zeta function, and write
$-\frac{\zeta\_K^{\prime}}{\zeta\_K}(s) = \sum\_{n=1}^{\infty} \... | 11 | https://mathoverflow.net/users/38624 | 145732 | 78,875 |
https://mathoverflow.net/questions/111278 | 4 | Is the local fundamental group of an elliptic singularity virtually solvable ? Here (the terminology is sometimes divergent) an elliptic singularity is a (germ of) normal surface $(X,x)$ such that $X$ is Gorenstein ($K\_X$ is Cartier) and $R^1\pi\_\* \mathcal{O}\_Y=\mathbb{C}\_x$ where $\pi:Y\to X$ is a resolution of t... | https://mathoverflow.net/users/25017 | local fundamental group of elliptic singularities | Well it is not always the case: elliptic singularities of surfaces can have very complicated local fundamental group. For instance, the following example
$$x^2+y^3+z^7=0$$
has a finite index subgroup which the fundamental group of a curve of genus 3 curve. To see this, note that it can be realized as a finite quotient ... | 4 | https://mathoverflow.net/users/41796 | 145733 | 78,876 |
https://mathoverflow.net/questions/145725 | 3 | I have obtained as the expression for some quantity the following gargantuan formula:
$$ \frac{k^8 + 3k^7 + 8k^6 + 3k^5 - 16k^4 - 32k^3 + 63k^2 - 34k + 6}{k^6 + 3k^5 + 6k^4 - 24k^2 + 21k - 5}$$.
What I really need is a (very) good lower bound on it, that will hopefully be a more manageable expression.
>
> Is t... | https://mathoverflow.net/users/22051 | Finding a simpler "local" lower bound for a rational function | $$k^2+1.98-2.8/k$$
lower bound, error $<0.02$ for all $k>5$
| 6 | https://mathoverflow.net/users/11260 | 145735 | 78,878 |
https://mathoverflow.net/questions/145723 | -1 | This might be too easy but I cannot proof it easily. Any reference or hint will be great.
**Q: Suppose P is a poset in which every chain is finite and $\Delta P$ is the poset complex associated to it. (i.e. faces of $\Delta P$ are the chains of the poset.). Suppose I reverse the order in the poset and suppose it aga... | https://mathoverflow.net/users/9485 | Poset complex of reverse ordering | Let $f:P\to \bar{P}$ is the antiisomorphism. It maps a chain $a\_1<\ldots< a\_n$ into $a\_n<\ldots a\_1$ so it maps every simplex into itself.
| 1 | https://mathoverflow.net/users/18814 | 145739 | 78,880 |
https://mathoverflow.net/questions/145731 | 4 | The algebra $\mathrm{End}\_{\mathbb{Q}}(A)=\mathrm{End}(A)\otimes\mathbb{Q}$ of endomorphisms of an abelian variety (defined over $\mathbb{C}$) is well understood and in particular the following is true : if $A$ is simple, the latter is a skew field and its center is (possibly a quadratic extension of) a totally real n... | https://mathoverflow.net/users/41796 | Endomorphism of complex tori | The answer to your question is "no". Oort and Zarhin (*Endomorphism algebras of complex tori*, Math. Ann. **303** (1995), 11–29) showed that any finite-dimensional division algebra over $\mathbb Q$ arises as $\mathrm{End}\_{\mathbb{Q}}(T)$ for a simple complex torus $T$.
| 6 | https://mathoverflow.net/users/430 | 145751 | 78,882 |
https://mathoverflow.net/questions/145709 | 4 | Dear Math Overflowers,
I'm looking for references on the parabolic Harnack inequality for **distributional** solutions of the heat equation on the whole space
$$
\partial\_t u=\Delta u\quad\text{and}\quad u(0)=u\_0\quad\Leftrightarrow\quad u(t,x)=\int\limits\_{R^d}\Gamma\_t(x-y)u\_0(y)dy.
$$
Here $\Gamma\_t(z)=\frac{... | https://mathoverflow.net/users/33741 | Reference request: harnack inequality for distributional solutions of the heat equation | Without knowing how the initial data is "spread," we can't get either pointwise or $L^p$ estimates independent of the measure of the initial support. Take for example the building block solution $v$ with initial data $\chi\_{B\_{\epsilon}}$. It follows from the representation formula that at time $t = \epsilon$, $v \le... | 4 | https://mathoverflow.net/users/16659 | 145755 | 78,884 |
https://mathoverflow.net/questions/145740 | 7 | Let $A$ be a square real or complex matrix. We’ll call $A$ *special* if among its rows (or among its columns) there are two identical ones, different from the zero vector, *(Added:) and if it has no nilpotent Jordan block.*
Obviously, if $A$ is special, then so are the powers $A^2,A^ 3,\dots$. I am wondering if the ... | https://mathoverflow.net/users/29783 | Powers of singular matrices and pairs of identical rows | Here is a non nilpotent example.
Take $A = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 1 & -1 \\ 3 & 2 & 0 \end{pmatrix}$. Then $A$ is not special, but $A^2 = \begin{pmatrix} 8 & 5 & 1 \\ 0 & 0 & 0 \\ 8 & 5 & 1 \end{pmatrix}$ is. Moreover, $A^{r+2} = 3^r\cdot A^2$ for any $r\geq 1$, so $A$ is not nilpotent.
---
Note that... | 5 | https://mathoverflow.net/users/39640 | 145759 | 78,887 |
https://mathoverflow.net/questions/75144 | 29 | A Sunday question for specialists of eigenverieties:
In their important paper "the eigencurve", Coleman and Mazur globalized the earlier construction of Coleman's families, constructing a beautiful eponymous rigid analytic space
that parametrizes all systems of Hecke eigenvalues of finite-slope overconvergent modular... | https://mathoverflow.net/users/9317 | Global applications of eigenvarieties | This is something I've told people privately for a while, and now have enough ingredients written up (jointly with Liang Xiao) to claim on the web somewhere: the validity of the parity conjecture is constant in $p$-adic analytic families. (Fine print: the family must be symplectic self-dual, and be equipped with a sort... | 7 | https://mathoverflow.net/users/367 | 145769 | 78,891 |
https://mathoverflow.net/questions/145688 | 12 | I have a difficult problem.
I have a very large, non-orthogonal matrix $A$ and need to project the vector $y$ onto the subspace spanning the columns of $A$. If this were a small matrix, I would use Gram-Schmidt or just compute $A(A^TA)^{-1}A^Ty$. Unfortunately, $A$ is just too big to do this in a feasible amount of t... | https://mathoverflow.net/users/41771 | How to project a vector onto a very large, non-orthogonal subspace | Depending on how many rows, maybe you could use some statistics results.
When I was doing **Principal Component Analysis** I ended up having a matrix $X\_{75\times 375000}$ and I could still find a good vector space that described well these vectors (and projections).
That's the whole point of Principal Component A... | 5 | https://mathoverflow.net/users/41818 | 145777 | 78,894 |
https://mathoverflow.net/questions/145790 | -2 | Are there known examples of compact infinite dimensional manifolds?
The word "manifold" is important.
| https://mathoverflow.net/users/41823 | Are there examples of compact infinite dimensional manifolds? | The empty space is a manifold of any dimension.
No, seriously, let's assume that "manifold" means a Hausdorff space in which every point has an open neighborhood homeomorphic to an open subset of a topological vector space. If the manifold is compact and nonempty then the vector space must be locally compact. As far ... | 9 | https://mathoverflow.net/users/6666 | 145792 | 78,899 |
https://mathoverflow.net/questions/145800 | 2 | Let $A\ge B>0$ be real constants. I say that a function $f:[0,1]\rightarrow[0,1]$ satisfies the *$(A,B)$-condition* if for all $p\in [0,1]$, the expression
$$q(A-Bp-Bf(q))$$
is maximized (not necessarily uniquely) at $q=f(p)$. More precisely, this means that for all $p,q$ we have
$$f(p)(A-Bp-Bf(f(p)))\ge q(A-Bp-Bf(q))$... | https://mathoverflow.net/users/10503 | A Recursive Maximization Problem | Here is the sketch of how one can work this problem out:
first, we can assume that $a=\frac{A}{B}>1,$ and the problem is equivalent to
$$f(p)(a-p-f(f(p)))\ge q(a-p-f(q)).$$
Replacing $q\to f(q),$ we can rewrite the inequality as follows:
$$(f(p)-f(q))(a-p)\ge f(p)f(f(p))-f(q)f(f(q)).$$
Switch $q$ and $p$ and add the... | 1 | https://mathoverflow.net/users/17503 | 145801 | 78,901 |
https://mathoverflow.net/questions/145770 | 38 | One of the major motivations of Homotopy Type Theory is that it naturally builds in higher coherences from the beginning. One important setting where higher coherence requirements get annoying is higher category theory. It's easy to talk about $\infty$-groupoids in HoTT, they're just types and you build them as higher ... | https://mathoverflow.net/users/22 | How do you define (infinity,1) categories in Homotopy Type Theory? | This is an important open problem. There are several imaginable approaches, including but not limited to:
1. Mimic some commonly used homotopical definition of $(\infty,1)$-category inside HoTT. The most likely candidate seems to be complete Segal spaces, since they have a space of objects rather than a set of object... | 33 | https://mathoverflow.net/users/49 | 145802 | 78,902 |
https://mathoverflow.net/questions/145780 | 3 | Let $X$ be a smooth scheme and $Z\subset X$ a smooth subscheme. Consider the blow-up
$$\pi:\widetilde{X}:=Bl\_{Z}X\rightarrow X$$
of $X$ along $Z$.
**What is the relation between the cohomology of the tangent bundle $T\_{\widetilde{X}}$ and the cohomology of the tangent bundle $T\_{X}$ of $X$ ?**
| https://mathoverflow.net/users/14514 | Cohomology of tangent bundles | There is an exact sequence
$$
0 \to \pi^\*\Omega\_X \to \Omega\_{\tilde X} \to i\_\*\Omega\_{E/Z} \to 0,
$$
where $E$ is the exceptional divisor and $i:E \to \tilde X$ is its embedding.
Dualizing it one gets
$$
0 \to T\_{\tilde X} \to \pi^\*T\_X \to i\_\*T\_{E/Z}(E) \to 0.\qquad(\*)
$$
On the other hand, since $E = P(N... | 11 | https://mathoverflow.net/users/4428 | 145809 | 78,905 |
https://mathoverflow.net/questions/145767 | 2 | Let $M$ be a hyperbolic $3$-manifold and $X$ a negatively curved, simply connected space with geometric boundary $\partial X$.
If $\rho:\pi\_1M\rightarrow Isom(X)$ does not fix a point in $\partial X$, then by Korevaar-Schoen (or Corlette-Donaldson, Labourie) there exists a $\rho$-equivariant harmonic map
$$H^3=\wide... | https://mathoverflow.net/users/39082 | Are harmonic maps quasiconformal at the boundary of hyperbolic spaces? | I will assume, in addition, that $M$ is compact and that the target space is again ${\mathbb H}^n$, $n\ge 1$. Now, here are some examples to ponder:
1. Suppose that the image of $\rho$ is a Schottky group (the limit set is a Cantor set $C$), then the boundary map $f$ sends $S^2$ to $C$. Can such a map be continuous?
... | 2 | https://mathoverflow.net/users/21684 | 145810 | 78,906 |
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