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https://mathoverflow.net/questions/78596 | 2 | Let $x$ and $p$ be real numbers with $x \ge 1$ and $p \ge 2$ . Show that $(x - 1)(x + 1)^{p - 1} \ge x^p - 1$ .
I recently discovered this result. I am sure it is known, but it is new to me. It is quite easy to prove if $p$ is an integer, even a negative one. I have a proof in the general case above, but it seems ove... | https://mathoverflow.net/users/16839 | Power function inequality | We prove strict inequality for $x>1$ and $p>2$. Add $1$ to both sides and divide by $x^p$ to get an equivalent inequality that can be written as
$$
\frac{x-1}{x} \left(\frac{x+1}{x}\right)^{p-1} + \frac1x \left( \frac1x \right)^{p-1} \geq 1.
$$
Since $p > 2$ the function $f : X \mapsto X^{p-1}$ is strictly convex upwa... | 7 | https://mathoverflow.net/users/14830 | 78605 | 47,352 |
https://mathoverflow.net/questions/74614 | 16 | The symmetric square of a topological space $X$ is obtained from the usual square $X^2$ by identifying pairs of symmetric points $(x\_1,x\_2)$ and $(x\_2,x\_1)$. Thus, elements of the symmetric square can be identified with unordered pairs $\{x\_1,x\_2\}$ from $X$ (including the degenerate case $x\_1 = x\_2$). With thi... | https://mathoverflow.net/users/2000 | Continuously selecting elements from unordered pairs | In [this paper](http://www.ams.org/mathscinet-getitem?mr=627702%20) Van Mill and Wattel proved that the existence of a continuous selection characterizes orderability in the class of compact Hausdorff spaces.
| 6 | https://mathoverflow.net/users/5903 | 78607 | 47,354 |
https://mathoverflow.net/questions/78586 | 10 | Let $G(n,p)$ denote the Erdős–Rényi model of random graph. For a given function $p = p(n)$ we say that $G \in G(n,p)$ asymptotically almost surely has property $\mathcal{P}$ if
$$\mbox{Pr}[G \mbox{ has property } \mathcal{P}] \to 1 $$ as $n \to \infty$.
The property $\mathcal{P}$ I am interested in is the following:... | https://mathoverflow.net/users/4558 | For what range of edge probability does the following property hold for random graphs? | For every $v$, we have $|N(v)|\approx np$ (I assume $p$ is not very small so the fluctuations are small enough to ignore). You want two of these such that $N(x)\cap N(y)=\{v\}$, or in other words, if you now forget about $v$ itself, you want these two sets to be disjoint.
For any specific $x$ and $y$ the probability ... | 7 | https://mathoverflow.net/users/1061 | 78612 | 47,357 |
https://mathoverflow.net/questions/78592 | 0 | A book on Quantum Mechanics states, "A unitary operator can be considered to be a complex valued function of a Hermitian operator."
Please give a hint on how to prove this assertion.
| https://mathoverflow.net/users/nan | Unitary Operator as a complex valued function | Sounds more like a homework/wikipedia problem and not suitable for here but anyways:
First one should maybe mention
[Stones Theorem](http://en.wikipedia.org/wiki/Stone%27s_theorem_on_one-parameter_unitary_groups) which says there is a one-to-one correpsondence between strongly continuous unitary one-parameter group ... | 3 | https://mathoverflow.net/users/10718 | 78615 | 47,358 |
https://mathoverflow.net/questions/78609 | 2 | Once, someone showed me a nice argument using elementary substructures for proving chain conditions about forcings. It was a basic example, maybe that Cohen forcing is ccc. I've been trying to look up this argument, but I've only found more combinatorial proofs. Can someone remind me how this argument goes or point me ... | https://mathoverflow.net/users/18654 | Substructure Argument for Chain Conditions | Say that $P=\{p \subset \kappa \times 2 : p \mbox{ is a finite function} \}$ is ordered by reverse inclusion (adding $\kappa$ Cohen reals). We show that $P$ is ccc:
Let $A \subseteq P$ be an antichain and let $M$ be a countable elementary submodel of (a sufficiently large initial segment of) the universe such that $A... | 10 | https://mathoverflow.net/users/17836 | 78622 | 47,361 |
https://mathoverflow.net/questions/78618 | 1 | I have arrived at an elementary-looking "result" via a sketchy argument. Having unsuccessfully searched for the statement and its "proof" in the literature, I would like to hear if anyone knows whether or not it is true, and if they might provide me with a reference? The statement is as follows: let $f:\mathbb{N}\right... | https://mathoverflow.net/users/10980 | Convergence of Dirichlet series | Something similar is true. Let
$$ A(x):=\sum\_{n\leq x}f(n),\qquad x>0,$$
then the first condition is equivalent to
$$ \forall\sigma>\sigma\_c:\ A(x)\ll\_\sigma x^\sigma. \tag{1}$$
This clearly implies
$$ \forall\sigma>\sigma\_c:\ \int\_0^\infty |A(x)|^2 x^{-2\sigma}\frac{dx}{x}<\infty, \tag{2}$$
which by
$$ F(s)=s\int... | 6 | https://mathoverflow.net/users/11919 | 78629 | 47,365 |
https://mathoverflow.net/questions/78569 | 3 | Background
----------
Considering a set of points $(x\_i, y\_i)$ in $\mathbb R^2$ and constraints between some triples of them, which state, whether the three points of the triple are oriented clockwise (R), counter-clockwise (L) or collinear (I), we can translate this into a set of inequalities of the form
$x\_1 y... | https://mathoverflow.net/users/13282 | A fast way to decide satisfiability of a set of simple fewnomial inequalities? | I believe your problem is NP-hard.
If you had, not just triples-constraints for *some* triples, but had given the
orientation of *every* triple, then you have specified what is known as the combinatorial *order type*
of the point configuration. (See [Handbook of Discrete and Computational Geometry](http://www.crcpress.... | 4 | https://mathoverflow.net/users/6094 | 78631 | 47,366 |
https://mathoverflow.net/questions/78623 | 7 | One of my friend (who is working in mathematics) was asking the following question. Let us take Liouville [λ(n) function](http://en.wikipedia.org/wiki/Liouville_function).
et S={ λ(1), λ(2), λ(3), ..... } . Then every finite length (say l) subsequence of S occurs infinitely many times. In other words every finite bl... | https://mathoverflow.net/users/18659 | Fluctuations of Liouville function | Hildebrand (On consecutive values of the Liouville function, Enseign. Math. (2) 32 (1986), 219–226) proved the conjecture for $l=3$, i.e. all 8 combinations $\pm 1,\pm 1,\pm 1$ occur infinitely often in the Liouville sequence. Christian Elsholtz proved very recently that all 16 combinations $\pm 1,\pm 1,\pm 1,\pm 1$ oc... | 13 | https://mathoverflow.net/users/11919 | 78632 | 47,367 |
https://mathoverflow.net/questions/78613 | 0 | I am reading the paper
G. A. Edgar, A long James space, in: *Measure Theory*, Oberwolfach 1979, Lectures Notes in Math. 794, Springer-Verlag (1980) pp. 31-37.
and I am pretty confused by the remarks after the proof of Proposition 3.
Is it clear that $J(\omega\_1)$ is of codimension 1 in $J(\omega\_1)^{\*\* }$ (vi... | https://mathoverflow.net/users/18657 | Codimension of $J(\omega_1)$ in its bidual | Bill is correct: $J(\omega\_1)$ is not of codimension $1$ in its bidual. The remarks after Proposition 3 say: (if $\eta$ is infinite) then $J(\eta)^{\*\*}$ is isometric to $\widetilde{J}(\eta+1)$, and the set-theoretic inclusion is the canonical embedding. The tilde on the $J$ means that we drop the requirement of cont... | 1 | https://mathoverflow.net/users/454 | 78633 | 47,368 |
https://mathoverflow.net/questions/78614 | 3 | Let $(M,g)$ be a closed, Riemannian manifold. Let $S(z)$ be a holomorphic family of pseudo-differential operators, with $z \in \Bbb{C}$. Let $u$ be a smooth function. Does it follow that $\lim\_{y \rightarrow z} ||S(y)u - S(z)u||\_\infty = 0$?
| https://mathoverflow.net/users/15856 | meromorphic family of pseudo-differential operators | If you set $A(y) = S(y) - S(z)$, your question is equivalent to asking if $A(z)$ is a holomorphic family of pseudo-differential operators such that $A(0) = 0$, then does $\|A(z)u\|\_\infty \rightarrow 0$ as $z \rightarrow 0$.
Here's what I think is true: Let $a(z, x, \xi)$ be the symbol of $A(z)$. Assume $a$ is a smo... | 4 | https://mathoverflow.net/users/613 | 78638 | 47,370 |
https://mathoverflow.net/questions/78627 | 13 | In ZFC we know that the continuum cannot have cofinality $\omega$.
However, in the Feferman-Levy model we have that $\frak c=\aleph\_1$, and that $\operatorname{cf}(\omega\_1)=\omega$. In fact in the Feferman-Levy model, $\aleph\_\omega^L=\aleph\_1^V$.
Is it consistent with ZF that $\frak c=\aleph\_\omega$? Does ... | https://mathoverflow.net/users/7206 | Is it consistent relative to ZF that $\frak c = \aleph_\omega$? | The answer is no. The continuum cannot be $\aleph\_\omega$, and this can be proved in ZF, that is, without using the axiom of choice. To see
this, suppose towards contradiction that $P(\omega)$ is equinumerous with
$\aleph\_\omega$. Since $P(\omega)$ is equinumerous
with $P(\omega)^\omega$, and this does not require AC... | 14 | https://mathoverflow.net/users/1946 | 78640 | 47,371 |
https://mathoverflow.net/questions/78641 | 7 | I am interested in the relation between the property of countable chain condition (ccc) and the property of separable. Could someone recommend some papers or books about this to me? thanks in advance.
| https://mathoverflow.net/users/18465 | the example of ccc but not separable | Nathan's answer seems to indicate that there are some strange spaces that are ccc but not separable. But in fact, such spaces are rather common:
All products of separable Hausdorff spaces are ccc, but if the spaces have at least two different points, then products with more than $2^{\aleph\_0}$ factors are not separa... | 13 | https://mathoverflow.net/users/7743 | 78646 | 47,376 |
https://mathoverflow.net/questions/78652 | 8 | This I read in a paper:
"The class of integrals that are elementary is very
small compared with nonelementary integrals."
What is the precise meaning of this sentence? E.g., does that mean that the former class of functions is meagre (in a suitable functional space) while the latter is not ? Is there a reference fo... | https://mathoverflow.net/users/18666 | Is the class of elementary integrals "small" ? | It is small in the same sense that the set of polynomials solvable by radicals is small. The canonical reference on the subject is probably the late, lamented Manuel Bronstein's book:
Symbolic Integration I: Transcendental Functions (Algorithms and Computation in Mathematics) (v. 1) [Hardcover]
Otherwise, look up "diff... | 7 | https://mathoverflow.net/users/11142 | 78653 | 47,378 |
https://mathoverflow.net/questions/78647 | 0 | Consider an abelian variety $X$ over a field and denote by $Z$ the first infinitesimal neighborhood of the diagonal coming with natural projections
$p\_1: Z \rightarrow X$, $p\_2: Z \rightarrow X$.
Let $Y$ be the first infinitesimal neighborhood of zero in $X$.
Then why has one an isomorphism
$X\times Y \righta... | https://mathoverflow.net/users/18665 | First infinitesimal neighborhood of diagonal on abelian variety | Consider an automorphism $X\times X \to X\times X$, $(x\_1,x\_2) \mapsto (x\_1-x\_2,x\_2)$.
It identifies the diagonal with $0 \times X$, and hence the infinitesimal neighborhood of the diagonal with the infinitesimal neighborhood of $0\times X$ which is $Y \times X$.
| 2 | https://mathoverflow.net/users/4428 | 78657 | 47,381 |
https://mathoverflow.net/questions/78659 | 2 | The Hook lenght formula gives the number of standard Young tableaux on a given diagram.
A variant gives the number of semistandard tableuax.
Does there exist a formula for counting "weighted tableaux"? By weighted tableaux I mean that there exists a vector $(a\_1,\dots,a\_n)$ and I only want to count the tableaux... | https://mathoverflow.net/users/4096 | Dimension of spaces of invariants/tableaux functions | The numbers you refer to are known as Kostka numbers. They are discussed in standard references like Fulton's Young Tableaux and Stanley's Enumerative Combinatorics. The weights of a tableaux are often referred to as their content, as well.
| 3 | https://mathoverflow.net/users/16002 | 78663 | 47,382 |
https://mathoverflow.net/questions/78661 | 9 | Hi, I don't know if this question is appropriate for Math Overflow but I was wondering if there is anything known about the following: Let
$$
S(\alpha) = \sum\_{n \leq x}\Lambda(n)e(n\alpha).
$$
Then asymptotically, how small can
$$
\inf\_{\alpha}\left|S(\alpha)\right|
$$
be relative to $x$? Also, for each $x$, if ... | https://mathoverflow.net/users/18494 | Infimums of exponential sums involving primes | Timothy,
This is likely to be a pretty difficult question I think. For a random sequence of $\pm 1$s in place of the von Mangoldt function $\Lambda(n)$ the answer is a little surprising: the infimum is basically $1/\sqrt{x}$, a result of Konyagin and Schlag. This is available here:
www.math.uchicago.edu/~schlag/pa... | 14 | https://mathoverflow.net/users/5575 | 78666 | 47,384 |
https://mathoverflow.net/questions/78665 | 1 | Could one find a counterexample that a topology space X is Tychonoff, seperable but hasn't
a $G\_\delta$-diagonal? A topology space has a $G\_\delta$-diagonal when there is a sequence
${G\_n}$ of open sets belonging to $X^2$ with the diagonal $\Delta$ = $\cap{G\_n}$.
| https://mathoverflow.net/users/18465 | $G_\delta$-diagonal | The product space $[0,1]^\kappa$ for $\aleph\_1\le\kappa\le\mathfrak c$ is compact $T\_2$ (hence Tychonoff) and separable (by the Hewitt–Marczewski–Pondiczery theorem), but it does not have a $G\_\delta$ diagonal (in fact, if a compact $T\_2$ space has a $G\_\delta$ diagonal, then its unique uniform structure has a cou... | 8 | https://mathoverflow.net/users/12705 | 78669 | 47,386 |
https://mathoverflow.net/questions/78621 | 6 | Let $G$ be a complex reductive group, and $K$ a maximal compact subgroup (such that $K\_{\mathbb{C}}=G$). By the polar decomposition theorem one has that, as manifolds, $G\cong T^\*K$. The inherited symplectic structure is compatible with the complex structure, making $G$ into a Kähler manifold.
On the other hand $G$... | https://mathoverflow.net/users/940 | Kähler structure on a complex reductive group | Isn't the answer no in the very simplest case? If $K$ is the circle group, then the Kähler structure on the cotangent bundle makes it metrically a cylinder $R \times S^{1}$. I believe this cylinder cannot be isometrically embedded in $C^n$ (apply the maximum modulus principal to the derivative of the map).
| 8 | https://mathoverflow.net/users/16193 | 78670 | 47,387 |
https://mathoverflow.net/questions/78660 | 15 | Hello,
Does someone know an explicit basis of the space of harmonic homogeneous polynomial in N variables.
When $N=3$, if I'm not mistaking Legendre polynomial allow to write an explicit basis.
Is there a known explicit basis when $N > 3$ ?
Thanks for your answers, and reference in case you know one.
| https://mathoverflow.net/users/8801 | Basis for the space of Harmonic homogeneous polynomial in N variables. | Let $K$ denote the [Kelvin transform](http://en.wikipedia.org/wiki/Kelvin_transform), and let $|\alpha|:=\sum\_{j=1}^n\alpha\_j$ denote the weight of the multi-index $\alpha\in\mathbb{N}^n$. Then, an explicit base for the space of homogeneous harmonic polynomials in $n$ variables and degreee $m$, $\mathcal H^m:=\mathca... | 19 | https://mathoverflow.net/users/6101 | 78676 | 47,390 |
https://mathoverflow.net/questions/78650 | 0 | Is it possible to construct a polynomial of degree `N`, with all of them as integer coefficient have a `root` as the given value. The root value provided is not necessarily a rational number.
For example, if the root is `28.552622898861801` we can have a polynomial of degree 10 whose one root will be the given value.... | https://mathoverflow.net/users/16487 | Polynomial of degree N with integer coefficient for a given root. | The problem can be solved by running some Integer Relation algorithm (e.g., PSLQ) on the numbers $1, r, r^2, \dots, r^N$ where $r$ is a given root.
See <http://en.wikipedia.org/wiki/Integer_relation_algorithm>
For example, here is computation in PARI/GP which gives a better result than the polynomial shown in quest... | 4 | https://mathoverflow.net/users/7076 | 78688 | 47,393 |
https://mathoverflow.net/questions/78674 | 4 | Let $U$ be a smooth variety, $m >1$ be a positive integer and $D\_{m,U} \in | {-}m K\_U|$ be a smooth irreducible divisor.
Let $\pi: V\_m:= Spec \bigoplus\_{i=0}^{m-1}
\mathcal{O}\_U(i K\_U) \rightarrow U$ be a cyclic cover determined by $D\_{m,U}$.
Assume that $|-K\_U|$ contains a smooth irreducible member $D\_U$ a... | https://mathoverflow.net/users/12390 | pullback or push forward of logarithmic differential sheaf by cyclic cover | Question 1 indeed seems right until one reads your comment. It seems right, because
we are thinking that $D\_U$ is the branch divisor. It is indeed right if you take
$D\_U=D\_{m,U}$ in place of your choice.
On the other hand, as stated, Question 1 could not be correct!! Observe that the right
hand side is independent... | 4 | https://mathoverflow.net/users/10076 | 78693 | 47,396 |
https://mathoverflow.net/questions/78702 | 3 | I'd like to see a complete proof of the simplest version of the following rough statement: "If $f/g$ is a rational function on a reduced scheme ($g$ not a zero divisor), and $f/g$ doesn't have poles in codimension $1$, then $f/g$ is a well-defined function on the normalization."
I figure this should be called the val... | https://mathoverflow.net/users/391 | Simple reference for valuative criterion of integrality? | This is corollary 11.4 in Eisenbud's book, namely *a normal domain is the intersection of its localizations at primes of codimension 1.*
| 5 | https://mathoverflow.net/users/10696 | 78703 | 47,399 |
https://mathoverflow.net/questions/78677 | 2 | I want to apologize in advance if this is blatantly trivial, but I already posted on math.stackexchange.com and got no answer at all.
Let $A$ be a Noetherian domain containing an algebraically closed field $\Bbbk$. If you want, you can also assume that $A$ is local and regular. Let $I\subseteq A$ be a radical ideal ... | https://mathoverflow.net/users/9947 | Extension of radical ideal after adjunction of roots | The answer is yes if $n$ is invertible in $A$. That $A$ contains a field doesn't matter in my proof.
What you want is $B/IB$ is reduced. We have $B/IB=(A/I)[T]/(T^n-\bar{x})$. So after replacing $A$ by $A/I$, we can suppose $I=0$, $A$ is reduced and $x$ doesn't belong to any minimal prime ideal of $A$, and we have t... | 3 | https://mathoverflow.net/users/3485 | 78713 | 47,404 |
https://mathoverflow.net/questions/78694 | 5 | Recently, a [question](https://mathoverflow.net/questions/78660/basis-for-the-space-of-harmonic-homogeneous-polynomial-in-n-variables/78676#78676) about the beautiful theory of harmonic polynomials made me aware there is something
I've wanted to know for a long time.
As is well known, for any number of variables $n... | https://mathoverflow.net/users/6101 | Symmetric basis of harmonic homogeneous polynomials | Already the desired result is false for $n = 3, m = 2$, but for simpler reasons than I suggested in the comments. In this case the polynomials you give are $x^2 - 2y^2 + z^2, xy$ and their permutations. The sum of the permutations of $x^2 - 2y^2 + z^2$ is zero, so none of its permutations can be part of a permutation-i... | 5 | https://mathoverflow.net/users/290 | 78715 | 47,405 |
https://mathoverflow.net/questions/71451 | 3 | Suppose I am inside a finite, weighted cubic graph without loops, with no information regarding its layout, including the number of vertices or distances to the adjacent vertices. I want to reach a target vertex, but I will only know where it is once I reach it.
1. Assuming that the bottleneck on search time depends ... | https://mathoverflow.net/users/4336 | Cubic graphs which are "difficult to navigate" | With respect to your question about on which graphs the algorithms performs poorly, have you considered expanders? Expanders look locally tree-like, which means that the number of vertices at distance $d$ from your starting vertex is exponential in $d$ (at least as long as the ball of radius $d$ around the start vertex... | 1 | https://mathoverflow.net/users/3806 | 78718 | 47,406 |
https://mathoverflow.net/questions/78221 | 3 | I am looking for a reference, in the form of a textbook, that contains proofs of following statements.
**NOTE:** I am NOT looking for the proofs, I am looking for a reference! Proofs of these statements are elementary. In fact, I need a reference to avoid writing the proofs. I just want to be able to refer somewhere... | https://mathoverflow.net/users/16046 | Reference for submultiplicativity of length of tensor product | Look at lemmas 45.12 and 45.13 of the following: <http://www.math.columbia.edu/algebraic_geometry/stacks-git/algebra.pdf>
This comes from the Stacks open source textbook project, you can browse their chapters here:
<http://www.math.columbia.edu/algebraic_geometry/stacks-git/browse.html>
| 1 | https://mathoverflow.net/users/11661 | 78719 | 47,407 |
https://mathoverflow.net/questions/78706 | 2 | Let $X$ be a Markov chain, with countable state space $I$ and transition probability matrix $P$. $X$ is irreducible, but need not be recurrent. Let $S$ be a fixed subset of $I$.
Define a subset $K$ of $I$ to be "nice" if there exists $\epsilon = \epsilon\_K$ such that for all $k \in K$, $P\_{kS} \geq \epsilon$. (Here... | https://mathoverflow.net/users/17883 | Probability-one event for Markov chain | If I've understood your problem correctly, an argument along these lines
may help:
---
Let ${\cal F}\_n=\sigma(X\_0,X\_1,\dots,X\_n)$ and define $S\_n=\left(X\_n\in S\right)$,
so that $S\_n\in {\cal F}\_n$.
We will use Levy's generalization of the Borel-Cantelli Lemma which states
that
$$\left( S\_n\mbox{ i.o.} \... | 5 | https://mathoverflow.net/users/nan | 78721 | 47,409 |
https://mathoverflow.net/questions/78695 | 3 | In the book "Degeneration of abelian varieties" by Faltings, Chai it reads (cf. p.81,82) as if the following holds:
take an abelian scheme $G$ over a base scheme $S$ with dual scheme $\hat{G}$ (for my applications you may always assume $S$ to be the spectrum of a field) and $P$ the Poincaré bundle on $G\times \hat{G}... | https://mathoverflow.net/users/18183 | Question about Faltings, Chai: Degeneration of abelian varieties | Let $G$ be any group scheme over $S$. Let $I\subset\mathcal{O}\_G$ be the augmentation ideal defining the identity section of $G$ over $S$. Then, by definition, the first infinitesimal neighborhood of the identity in $G$ is the closed sub-scheme of $G$ defined by the ideal $I^2$. Moreover, $I/I^2$ is a coherent sheaf o... | 4 | https://mathoverflow.net/users/7868 | 78725 | 47,413 |
https://mathoverflow.net/questions/78714 | 7 | Hi,
Where can I learn about the reduction of the Jacobians of modular curves
such as X\_0(N) and X\_1(N) at primes p dividing N?
Thanks!
| https://mathoverflow.net/users/36285 | References for bad reduction of Jacobians of modular curves? | The standard places that one learns this are (or at least, used to be): Mazur's *Eisenstein ideal* paper (which treats the case of $X\_0(N)$ for $N$ prime in great detail),
Ribet's papers (his Herbrand criterion paper, his Warsaw ICM talk, his Inventiones 100 paper,
and several others as well), Gross's *Tameness crite... | 19 | https://mathoverflow.net/users/2874 | 78738 | 47,419 |
https://mathoverflow.net/questions/78727 | 3 | Consider the integers as a first-order structure in the language *{0,+,-}* of abelian groups. I suspect that the collection of definable subsets (without parameters) of this structure is an algebra containing nZ for all natural n (i.e. all periodic sets). This is to say, it's trivial to conclude that the collection of ... | https://mathoverflow.net/users/18658 | Definable subsets of the integers as an abelian subgroup? | Of course, the definable sets are closed under finite
unions, intersections and complement, and so you will
easily get more than just the sets $n\mathbb{Z}$, since you
also get the complements of these sets and their unions and
so on. But in fact, the definable sets in your structure
are exactly the finite boolean comb... | 9 | https://mathoverflow.net/users/1946 | 78740 | 47,420 |
https://mathoverflow.net/questions/78739 | 2 | In connection with the Galois theoretic results surrounding the irreducibility of $f(x)= x^{N}-x-1$ over $\mathbb{Q}$, I've been trying to prove for a while that the discriminant of $f$ is actually squarefree as it sounded plausible. After failed attemps I started believing that this might not be actually true so I did... | https://mathoverflow.net/users/16321 | Searching for polynomials with squarefree discriminant | My two cents: Schur proved that the discriminant of
$$n!\left(\frac{x^{n}}{n!} + \ldots + \frac{x^{2}}{2!} + x+1\right),$$
is equal to $(-1)^{n(n-1)/2}(n!)^n$, which is not a perfect square as long as $n\neq 0\pmod{4}$. See [this note](http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/schurtheorem.pdf) by K. Conrad... | 7 | https://mathoverflow.net/users/2384 | 78744 | 47,424 |
https://mathoverflow.net/questions/78705 | 6 | This question is inspired by Hartshorne's exercise II.5.7 (c-d): the problem reads:
Let $0\rightarrow \mathcal{F}'\rightarrow\mathcal{F}\rightarrow\mathcal{F}''\rightarrow0$ be a short exact sequence of locally free sheaves. Then for any $r$ there is a finite filtration $F$ of $S^r(\mathcal{F})$ (the sheaf of symmetric... | https://mathoverflow.net/users/17121 | Spectral sequence of symmetric or exterior algebras? | Take the symmetric power of the complex $F'' \to F'[1]$. However, this is a very exotic point of view on the question.
ADDITION. A complex is a chain of morphisms with zero compositions. You can consider complexes in the derived category as well. In some sense those can be thought of as bicomplexes in the original ab... | 8 | https://mathoverflow.net/users/4428 | 78758 | 47,428 |
https://mathoverflow.net/questions/78757 | 0 | Consider an ultrametic space $X=K^n$ (for the norm $\| x\| =\max\_{i=1,n} (\mid {x\_1}\mid,\dots,\mid{x\_n}\mid)$ where $K$ is an ultrametric field. Let $B(1):=\lbrace x \in X \mid \|x\| \leq 1\rbrace$ be the unit ball. Equip $X$ with Haar measure. Is it possible to partition $B(1)$ into $k$ smaller balls $B\_{a\_1}(r\... | https://mathoverflow.net/users/nan | Partitioning the unit ball in an ultrametric space | If you can do the case of $K$ itself, decompose into $k$ smaller balls, then for $K^n$ you can decompose into $k^n$ smaller balls. For some ultrametric fields this works. Indeed, you say "Haar measure" so you must be assuming local compactness, and it works for any such field, since they are finite extensions of $p$-ad... | 1 | https://mathoverflow.net/users/454 | 78759 | 47,429 |
https://mathoverflow.net/questions/78751 | 5 | I am a beginner of forcing, often I read from some articles something like "$p \Vdash \dot{G}$ is $P$-generic over $\check{M}$" (where $M$ is a countable transitive model, for instance).
Q1. I learnt from Jech's book a definition of "$p \Vdash \dot{x} \in \check{M}$", but I don't know how to translate "$p \Vdash \do... | https://mathoverflow.net/users/18692 | "name" for the ground model | In the Boolean-value approach to forcing, one may introduce a new predicate symbol $\check M$ into the forcing language, and then define that $[[\tau\in\check M]]=\bigvee\_{x\in M}[[\tau=\check x]]$. That is, the Boolean value that $\tau$ is in $\check M$ is precisely the extent to which it is equal to something in the... | 7 | https://mathoverflow.net/users/1946 | 78762 | 47,431 |
https://mathoverflow.net/questions/78505 | 3 | Let $\mathcal{D} \approx \mathbb{P}^{\delta\_d}$ be the space of homogeneous
degree $d$ polynomials in three vriables, where
$\delta\_d = \frac{d(d+3)}{2}$. Let
$$ X \subset \mathcal{D} \times \mathbb{P}^2$$
be a smooth embedded complex submanifold, not necessarily closed.
Given a point $p\in \mathbb{P}^2$, we g... | https://mathoverflow.net/users/4463 | Can you ``perturb'' a submanifold to intersect transversally with any other smooth submanifold of projective space? | My first answer was wrong. The answer is no.
First observe that the embedding $X\to \mathcal D\times\mathbb P^2$ is a bit distracting. The map $X\to \mathbb P^2$ plays no role in the question of transversality to a given $\tilde H\_q$. So I think about the problem like this:
You have smooth maps
$\mathbb P^2\le... | 4 | https://mathoverflow.net/users/6666 | 78774 | 47,436 |
https://mathoverflow.net/questions/78772 | 9 | Given an isolated singularity $p$ in a hypersurface $Y$ of dimension $n$ (let say a surface in $\mathbb{P}^3$). I can intersect $Y$ with a hyperplane $H$ passing through $p$ such that it induces a singularity in a lower dimension. For example, if we start with a surface $Y$, we obtain a plane singularity on $H$.
I w... | https://mathoverflow.net/users/16409 | hyperplane sections of isolated hypersurface singularities. | This is a good strategy and it does work for many types of singularities. In fact, many times you don't even need "vicinity" data if your singularity is isolated. (In other words, being isolated is the "vicinity" data). If you do not assume that the singularities are isolated, then you need to assume something about $X... | 16 | https://mathoverflow.net/users/10076 | 78776 | 47,437 |
https://mathoverflow.net/questions/78753 | 3 | Let $\Omega$ be an open subset of the upper half-plane in the complex plane. I am considering the following problem:
(1) $\overline{\partial}u=f,$ $\textrm{Im} f=0$ on the real line for maps complex-valued maps on $\Omega.$
Here, $\overline{\partial}$ denotes the classical Cauchy-Riemann operator. Usually one cons... | https://mathoverflow.net/users/3509 | Elliptic estimates and regularity of the $\overline{\partial}$-operator with totally real boundary conditions in $W^{1,p},$ $1<p\le 2$ | I'm almost certain that Abbas-Hofer does this in the appendix in which they prove the $L^p$ estimates. If I recall correctly, the proof shows the estimate in weak $L^1$ and for $L^2$, and then uses interpolation to get $1 < p \le 2$. You then obtain the $2 < p < \infty$ by duality. If you don't have access to it, I wil... | 2 | https://mathoverflow.net/users/477 | 78780 | 47,438 |
https://mathoverflow.net/questions/78775 | 6 | Let $({\cal C},\otimes)$ be a monoidal category, $X$ an object in ${\cal C}$, and $\Psi:X \otimes X \to X \otimes X$ an isomorphism such that $\Psi$ satisfies the braid relation:
$$
(\Psi \otimes \text{id}) \circ (\text{id} \otimes \Psi) \circ (\Psi \otimes \text{id}) = (\text{id} \otimes \Psi) \circ (\Psi \otimes \tex... | https://mathoverflow.net/users/3072 | Name for an Isomorphism in a Monoidal Category that Satisfies the Braid Relation | Assuming S. Carnahan's surmise in his comment is correct, I believe the correct term for this is "Yang-Baxter" operator in a monoidal category (or, you could call an object $X$ equipped with such an automorphism $R: X \otimes X \to X \otimes X$ a [Yang-Baxter object](http://ncatlab.org/nlab/show/braid+category)). This ... | 13 | https://mathoverflow.net/users/2926 | 78781 | 47,439 |
https://mathoverflow.net/questions/67025 | 18 | Suppose $n$ is an integer and we wish to factor it. As a special case we have $n = pq$ with $p,q$ distinct primes. The problem: factoring $n$ via complex analysis tools
Background
----------
I have been interested in integer factorization for some time now. Recently I have been trying to apply generating function t... | https://mathoverflow.net/users/15493 | Factoring Integers using Complex Integrals | I can give you a negative answer and a kind of a positive answer to your question. First, similar to what Henry Cohn says, the conventional view of your construction is that it is a restatement of the factoring problem rather than a step to an algorithm. A function with a lot of oscillation is at first glance a functio... | 14 | https://mathoverflow.net/users/1450 | 78789 | 47,443 |
https://mathoverflow.net/questions/78787 | 7 | Does there exist a variety of groups $\mathfrak{V}$ such that the relatively $\mathfrak{V}$-free group of rank 2 is finite, but the relatively $\mathfrak{V}$-group of rank 3 is infinite?
(In other varieties of algebras this can occur; for example, in the variety of all lattices, the free lattice of rank 2 is finite, ... | https://mathoverflow.net/users/3959 | Varieties of groups with infinite relatively free group of rank 2 finite, infinite in rank 3 | There are varieties of semigroups like that. Take any finite inherently non-finitely based semigroup $S$, say, the 6-element Brandt monoid $B\_2^1$, and the variety $M$ given by all identities of $S$ depending on at most 2 variables. Then all 2-generator semigroups in $M$ are in $var S$, so are finite, but three-genera... | 8 | https://mathoverflow.net/users/nan | 78792 | 47,446 |
https://mathoverflow.net/questions/78793 | 5 | I am interested in quaternionic-Kahler metrics that are "as inhomogeneous as possible."
Every complete quaternionic-Kahler manifold $X$ I can remember hearing of is a discrete quotient of some $Y$, such that $Isom(Y)$ contains a nontrivial connected Lie group. Are there any known examples of complete quaternionic-Kah... | https://mathoverflow.net/users/580 | Quaternionic-Kahler metrics whose universal covers have only discrete isometry groups? | I'm not familiar with the nonpositive case (negative since you exclude hyperkahler case) but there are no such examples known for postive quaternion Kahler manifolds (i.e. those with positive scalar curvature). They are all conjectured to be symmetric spaces (conjecture of LeBrun and Salamon) and this conjecture has be... | 6 | https://mathoverflow.net/users/18050 | 78797 | 47,448 |
https://mathoverflow.net/questions/78802 | 1 | This is perhaps an easy question, but...
Let $M$ be a matroid on a ground set $E$, and let $A$ and $B$ be non-disjoint subsets of $E$ such that $M|A$ and $M|B$ are both connected. Is $M|(A\cup B)$ then necessarily connected? Clearly this is true for graphic matroids, but I can't find any results in the literature reg... | https://mathoverflow.net/users/2189 | Is a non-disjoint union of connected matroids always connected? | Yes. Let $E$ be the ground set of a matroid. Define an equivalence relation $\sim$ on $E$ by imposing that $i \sim j$ if $i$ and $j$ are in the same circuit of $E$. Then the equivalence classes of $\sim$ are the connected components of the matroid.
Any circuit of $M|\_A$ is also a circuit of $M$, so if two elements o... | 3 | https://mathoverflow.net/users/297 | 78809 | 47,453 |
https://mathoverflow.net/questions/78808 | 4 | It is known (Lindenstrauss, Tzafriri, On the complemented subspaces problem) that a real Banach space all of whose closed subspaces are complemented (i.e. have a closed supplement) is isomorphic (as a tvs) to a Hilbert space. But I am interested in complementing a special kind of subspaces: subspaces F of a Banach spac... | https://mathoverflow.net/users/336 | Complemented subspaces of Banach spaces | I believe the answer to your first question is no. The counterexample I have in mind is related to the peculiar fact (first proved by Enflo, Lindenstrauss and Pisier) that being isomorphic to a Hilbert space isn't a "three-space property". More specifically, there is an example of a Banach space $E$ with an uncomplemen... | 5 | https://mathoverflow.net/users/430 | 78812 | 47,455 |
https://mathoverflow.net/questions/78813 | 21 | What could be a reference about binomial expansions for non-commutative elements?
Specifically, where can I find a closed formula for the expansion of $(A+B)^n$ where $[A,B]=C$ and $[C,A]=[C,B]=0$?
I've found some ideas about that and also a proof using PDE's in the following website: [link](http://www.voofie.com/... | https://mathoverflow.net/users/40886 | Binomial Expansion for non-commutative setting | I don't know if you prefer a particular presentation of the formula, but this is essentially covered by the Baker-Campbell-Hausdorff formula, or actually it's dual, [Zassenhaus formula](http://en.wikipedia.org/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula#The_Zassenhaus_formula), which in your case reduces to
$... | 31 | https://mathoverflow.net/users/2384 | 78814 | 47,456 |
https://mathoverflow.net/questions/57395 | 17 | Let $P(n)$ denote the largest prime factor of $n$. For any integer $x\ge2$, define the median
$$
M(x) = \text{the median of the set }\{P(2), P(3), \dots, P(x) \}.
$$
Classical results of Dickman and de Bruijn show that the median is *roughly* $x^{1/\sqrt{e}}$. More specifically, I think that the Dickman-de Bruijn rho-f... | https://mathoverflow.net/users/5091 | Median largest-prime-factor | This is one of many questions that has been answered in the comments, so I will just summarize the answer with a CW posting: $M(x) < x^{1/\sqrt{e}}$ according to the poster Greg Martin. In a computer search, the ratio seems to converge to roughly $0.74$ for $x$ up to a million. On the other hand, it doesn't converge ve... | 7 | https://mathoverflow.net/users/1450 | 78815 | 47,457 |
https://mathoverflow.net/questions/78763 | 5 | Let $q$ denote a prime power and $\text{GL}\_n(q)$ and $\text{U}\_n(q^2)$ the general linear and unitary group, respectively. Then $\text{U}\_n(q^2)$ is naturally a subgroup of $\text{GL}\_{n}(q^2)$, so one kind of groups can be embedded into the other. Let $C(g)$ be the conjugacy class of an element $g$ in its respect... | https://mathoverflow.net/users/43085 | Embeddings of finite classical groups | I guess by $U\_n(q^2)$ you mean the general unitary group in which the field of representation has order $q^2$? That is often denoted by ${\rm GU}\_n(q)$, but I will use your notation.
In general ${\rm GL}\_n(q^2)$ embeds into $U\_{2n}(q^2)$, by acting on a totally isotropic space of dimension $n$, and it does not em... | 7 | https://mathoverflow.net/users/35840 | 78826 | 47,462 |
https://mathoverflow.net/questions/78824 | 0 | Let's consider closed simply connected manifold $M^n$ and a $a\in H^k(M)$ and $a\*\in H^{n-k}(M)$ is the dual to $a$.
Is it true that dual to $a$ is realisable as a immersed sphere or $ a\*=bc $ for some $b,c\in H^\*(M)$ ?
Edit: it is more natural to ask about possibility to decompose dual to $a$ as a product, see... | https://mathoverflow.net/users/4298 | Realisability cohomological class as product or as immersed sphere | I will construct a closed simply-connected $8$-manifold $M$ and an $a\in H^3(M;\Bbb Z)$ such that the Poincare dual $b$ of $a$ is not realizable by a map $S^5\to M$, and a Hom-dual element in $H^5(M;\Bbb Z/2)$ to the $\bmod 2$ reduction of $b$ is not a nontrivial product.
Let $K$ be the suspension over $\Bbb C P^2$.... | 4 | https://mathoverflow.net/users/10819 | 78834 | 47,466 |
https://mathoverflow.net/questions/78839 | 1 | Let H denote Hilbert space, the space of square-summable infinite sequences of real
numbers-which is infinite-dimensional and separable. Let S1,S2 denote subsets of H
such that a point p of H belongs to S1 or S2 according to whether the sum of the
abslolute values of the co-ordinates of p is convergent or divergent res... | https://mathoverflow.net/users/4423 | Questions about the topological properties of certain subsets of Hilbert space. | The unit ball of $\ell\_1$ is weakly compact in $\ell\_2$ and closed in $\ell\_2$, so $S1$ is $F\_\sigma$ and $S2$ is $G\_\delta$.
ADDED 10/22/11: $S2$ is also arcwise connected. Give $x\_i$ in $S2$ for $i=1,2$ you can
choose a partition $A \cup B$ of the natural numbers so that $1\_A x\_i$ and $1\_B x\_i$ are in $... | 6 | https://mathoverflow.net/users/2554 | 78844 | 47,473 |
https://mathoverflow.net/questions/78850 | 5 | This question loosely elaborates on an [earlier question](https://mathoverflow.net/questions/66098/sheaves-with-isomorphic-cohomology-but-not-quasi-isomorphic). It is pretty silly, but I'd like to hear some authoritative answers.
Recall that if $f:S^{\bullet}\to T^{\bullet}$ is a quasi-isomorphism of sheaves over $X$... | https://mathoverflow.net/users/1622 | Derived Equivalence of Sheaves and Homotopy | I think you are asking: when is the functor $R\Gamma$ conservative (in the derived sense - i.e. if $R\Gamma (f)$ is a quasi-isomorphism then $f$ is a quasi-isomorphism). This is equivalent to $R\Gamma$ having no kernel - i.e. if $R\Gamma (F) \cong 0$, then F $\cong 0$ (by taking cones).
If you restrict to the triangu... | 8 | https://mathoverflow.net/users/7762 | 78854 | 47,478 |
https://mathoverflow.net/questions/78855 | 2 | Sorry if this isn't the right place for this, it hasn't gotten any answers on ME. I'm reading Lang's section on field theory and he stresses that, unlike typical "universal" constructions which are determined up to unique isomorphism, algebraic closures (and by extension, their Galois groups) are determined only up to ... | https://mathoverflow.net/users/18702 | Automorphisms and Bicategories | I don't see bicategories coming into this in a useful way, but I think what you have is a consequence of two more general facts:
* The non-uniqueness of algebraic closures is a general fact about [injective hulls](http://nlab.mathforge.org/nlab/show/injective+hull) -- they are 'unique' up to *non-unique* isomorphism.... | 5 | https://mathoverflow.net/users/4262 | 78859 | 47,482 |
https://mathoverflow.net/questions/78467 | 5 | Let's consider a algebraic contact structure $P$ on $\mathbb CP^3$
and a algebraic curve $C$ degree $d$ and genus $g$. Let's assume
that contact structure has degree $p$ (see
[Polynomial contact structures on $RP^3$](https://mathoverflow.net/questions/58000/polynomial-contact-structures-on-rp3)
about algebraic contact ... | https://mathoverflow.net/users/4298 | Thom polynomial for contact algebraic structures | Let $i:C \to \mathbb P^3$ be the normalization of an irreducible curve $C\_0\subset \mathbb P^3$ of degree $d$ and geometric genus $g$.
If $\mathcal D$ is a distribution on $\mathbb P^3$ of degree $p$ then it is defined by a section $\omega$ of $\Omega^1\_{\mathbb P^3} \otimes \mathcal O\_{\mathbb P^3}(p+2)$. To com... | 3 | https://mathoverflow.net/users/605 | 78861 | 47,483 |
https://mathoverflow.net/questions/78863 | 5 | It is known that if there is a measurable cardinal then every $\Pi\_1^1$ set has the perfect set property (i.e it is either countable or contains a copy of $2^{\omega}$). Also if we have $\Pi\_1^1$-determinacy (or in other words $0^{\sharp}$) then we get that $\Sigma\_2^1$ has the perfect set property. Note the result ... | https://mathoverflow.net/users/3859 | Large cardinal axioms and the perfect set property | Solovay showed that the following are equivalent:
1. $\boldsymbol{\Sigma}^1\_2$ sets have the perfect set property
2. $\boldsymbol{\Pi}^1\_1$ sets have the perfect set property
3. $\aleph\_1^{L[a]} < \aleph\_1$ for every real $a$
You only need an inaccessible to force (3).
| 9 | https://mathoverflow.net/users/2000 | 78864 | 47,484 |
https://mathoverflow.net/questions/78796 | 18 |
>
> Let $K$ be a field and consider a power series $f(T) \in K[[T]]$. Under what conditions (on $K$ and/or on $f$) can we conclude that if $\alpha$ is a root of $f(T)$ then $\alpha$ is in fact algebraic over $K$?
>
>
>
This question is inspired by the following: In [this paper](http://math.bu.edu/people/rpollack... | https://mathoverflow.net/users/10547 | When are roots of power series algebraic? | At first I thought the question concerned the closed unit disc. Since it involves the open unit disc in ${\mathbf C}\_p$ we need a little detail to see why the Weierstrass preparation theorem for series on the closed unit disc can be used. We'll pass to a suitable finite extension of K to pull this off.
Since there ... | 19 | https://mathoverflow.net/users/3272 | 78871 | 47,485 |
https://mathoverflow.net/questions/78876 | 3 | Let $S\subset \overline{\mathbf{Q}}\subset \mathbf{C}$ be the set of solutions to the unit equation, i.e., $S$ consists of algebraic integers $a$ such that $a$ and $1-a$ are units in the ring of algebraic integers.
Let $U$ be a non-empty open subset in the Euclidean topology on $\mathbf{C}$.
Does $U$ contain infin... | https://mathoverflow.net/users/18722 | Do the solutions to the unit equation lie dense in the complex numbers | If $f\in\mathbf Z[X]$ is any monic polynomial, the solutions of $x(1-x)\cdot f(x)=1$ are solutions of the unit equation. Take some $y\in U\setminus\mathbf R$. Since the substitution $z\mapsto1/(1-z)$ leaves $S$ invariant, we may assume $|y|>1$. For $n$ given, choose $u,v\in\mathbf R$ such that $y(1-y)\cdot(y^n+uy+v)=1$... | 7 | https://mathoverflow.net/users/2035 | 78880 | 47,490 |
https://mathoverflow.net/questions/76350 | 10 | Consider the $N\times N$ matrix $$
M = \left(\begin{array} \\
0 & 1 & & 0 \\
1 & \ddots & \ddots & \\
& \ddots & \ddots & 1 \\
0 & & 1 & 0 \\
\end{array}\right)
$$
which comes from the adjacency matrix of a graph corresponding to a one-dimensional chain of $N$ nodes with dangling ends. A cartoon of this graph is $$\... | https://mathoverflow.net/users/1674 | Relationship between free probability and deterministic graphs? | I believe the relation between deterministic graphs and free probability you mentioned is not something generic. In fact, the main property of your matrix $M$ which makes connection with free probability (at the best of my knowledge) is not to be the adjacency matrix of some graph, but a Jacobi matrix related to some o... | 9 | https://mathoverflow.net/users/15517 | 78883 | 47,491 |
https://mathoverflow.net/questions/59680 | 13 | This is a reference request, since I'm sure what follows isn't new, but I can't seem to find it.
Suppose that we have a finite tree $T$ with non-negative weights on the edges. Naively, computing the path lengths (i.e., sum of the weights along the unique path) between every pair requires $O(n^3)$ steps: there are $\b... | https://mathoverflow.net/users/11978 | All-pairs shortest paths in trees? | Just do a bfs on every node. Every search gives you a fine one-to-all shortest path in the tree.
All in all $n$ times $O(n)$ = $O(n^2)$.
You can also do it in $O(n)$, if you don't mind the distances being stored implicitly (still $O(1)$ lookups): Make an [LCA datastructure](https://en.wikipedia.org/wiki/Lowest_comm... | 7 | https://mathoverflow.net/users/5429 | 78889 | 47,495 |
https://mathoverflow.net/questions/78873 | 9 | All of the models of CH which I know of also satisfy $\diamondsuit$. What is the easiest way to produce a model of CH wherein $\diamondsuit$ is false?
| https://mathoverflow.net/users/18719 | A model of CH +$\lnot \diamondsuit$ | "The easiest way" to produce a model of CH in which $\diamondsuit$ is false is to start with a model of GCH and then do a countable support iteration of length $\omega\_2$ killing off a potential $\diamondsuit$ sequence at each stage.
The forcing for doing this is straightforward: supposing $\langle A\_\alpha:\alpha<... | 10 | https://mathoverflow.net/users/18128 | 78898 | 47,501 |
https://mathoverflow.net/questions/78877 | 9 | Given an n-dimensional electrically neutral, solid metal ball (a point for n=0; a rod, n=1; a disc, n=2; a solid ball, n=3; ...), place N=(n+1)! identical ions on the ball. As one of my favorite physics professors used to say, forget the mathematics, intuitively I expect the ions to equilibrate to the vertices of an n-... | https://mathoverflow.net/users/12178 | Equilibrium configurations of ions on n-Dim balls. | It turns out to pretty hard to guess the answers to these kinds of extremal problems. The permutohedron is almost certainly not the answer in $\mathbb{R}^n$ for $n \ge 3$, for any reasonable potential function (for example, an inverse power law). Specifically, the truncated octahedron is not good at minimizing things, ... | 10 | https://mathoverflow.net/users/4720 | 78900 | 47,502 |
https://mathoverflow.net/questions/78888 | 2 | Take $U=\mathbb{D}\_2\setminus \overline{\mathbb{D}\_1}$ ($\mathbb{D}\_r$ is the open disc centered at 0 with radius $r$) and consider the space $A(U)$ of all functions on $\overline{U}$ which are holomorphic in $U$ and admit a continuous extension to $\overline{U}$ (with obvious operations and the supremum norm).
Is... | https://mathoverflow.net/users/18725 | Reversed disc algebra? | No. The spectrum of $A(U)$ is $\overline{U}$, while the spectrum of the disc algebra is $\overline{\mathbb D\_1}$, and these two spaces aren't homeomorphic.
| 4 | https://mathoverflow.net/users/430 | 78901 | 47,503 |
https://mathoverflow.net/questions/78905 | 5 | As the title says, let $k \geq 2$ be a positive integer and let $G$ be a $(k-1)$-edge-connected $k$-regular graph with an even number of vertices. Then, for every edge $e$ of the graph there is a perfect matching of $G$ containing $e$.
First, I was wondering if this is new and if there are approaches different from ... | https://mathoverflow.net/users/16321 | A k-1 edge connected k regular graph is matching covered | If you look at the math review of the following:
MR0317999 (47 #6548)
Plesník, Ján
Connectivity of regular graphs and the existence of 1-factors.
Mat. Časopis Sloven. Akad. Vied 22 (1972), 310–318.
You will see that this result was already known in 1972, and if you look at the actual paper (available for free on... | 6 | https://mathoverflow.net/users/11142 | 78914 | 47,508 |
https://mathoverflow.net/questions/76604 | 34 | This is a Banach space version of Andre Henriques' question
[Trace Question](http://mathoverflow.net/questions/76386/trab-trba)
for Hilbert spaces. Let $a:X\to Y$ and $b:Y\to X$ be bounded linear operators between Banach spaces s.t. $ba$ and $ab$ are both nuclear. Assume whatever approximation properties on $X$ a... | https://mathoverflow.net/users/2554 | tr(ab)=tr(ba), part 2. | My question has a negative answer.
**Lemma.** Suppose $X$ has the approximation property (AP), $Y$ is a subspace of $X$, and $X/Y$ fails the AP. Then there is a nuclear operator $T$ on $X$ s.t. $TX\subset Y$, $T^2=0$, and $tr(T)=1$.
Suppose you have $X$, $Y$, $T$ as in the lemma and $Y$ has the AP. Define $a:X\to Y... | 21 | https://mathoverflow.net/users/2554 | 78920 | 47,514 |
https://mathoverflow.net/questions/78899 | 0 | Let
$$ X, Y \subset \mathbb{P}^N$$
be two non singular algebraic varieties of dimensions $k$ and $l$ that
intersect transversally. Is it true that the ``dimension'' of the variety
$\overline{X} \cap \overline{Y} - X\cap Y$ is strictly less than $k+l-N$,
which is the dimension of $X\cap Y$ as a complex manifold.
Wha... | https://mathoverflow.net/users/4463 | If you take the closure of two smooth varieties and then take their intersections, is the singular locus still small? | There are already two answers pointing out why your statement cannot hold as stated, so let's see if we can fix it.
Let $X, Y\subseteq \mathbb P^N$ be two irreducible (quasi-projective) algebraic varieties of dimension $k$ and $l$ respectively. Then $\overline X,\overline Y\subseteq \mathbb P^N$ are two closed irred... | 3 | https://mathoverflow.net/users/10076 | 78923 | 47,515 |
https://mathoverflow.net/questions/78902 | 1 | Let us consider the complex projective plane $P^2$ and two distinct lines $L,L'\subset P^2$. Let us moreover consider the restriction of the natural action of $SL\_3$ to $L\cup L'$. Can you tell in what way does $SL\_3$ act on $L \cup L'$? What is the stabilizer of $L \cup L'$?
| https://mathoverflow.net/users/18728 | A weird action of SL_3 on a pair of lines | One way to describe this, that fits into various larger patterns, is as a minimal parabolic intersected with its conjugate by a simple root-reflection, and with that reflection adjoined.
In coordinates: take lines $x$-axis and $y$-axis. The upper-triangular matrices $P$ form a standard minimal parabolic. The positive... | 1 | https://mathoverflow.net/users/15629 | 78926 | 47,517 |
https://mathoverflow.net/questions/78897 | 2 | Hello,
I am trying to understand Orlov: Remarks on generators of triangulated categories.
Let E be a full subcategory of D^b(coh(P^1)). Let [E] be the smallest full subcategory of D^b(coh(P^1)) such that [E] is closed under direct summands, finite direct sums and shifts.
Now, let [E]x[E] be the full subcategory ... | https://mathoverflow.net/users/15449 | dimension of D^b(coh( P^1)) | Let me give a name to notion. If $T$ is a triangulated category and $G$ is an object, the minimal number of cones required to generate any object of $T$ starting with $G$ (and allowing for arbitrary finite sums, shifts, and splitting of summands) is called the generation time of $G$. So we want to check that the genera... | 5 | https://mathoverflow.net/users/1404 | 78931 | 47,521 |
https://mathoverflow.net/questions/78886 | 10 | Given a group scheme $X$ over $S$, where $S$ is an arbitrary locally noetherian scheme, then how does one define the Lie algebra of $X$? And how does it behave with respect to base change?
Is there any good reference for the theory of group schemes apart from Demazure/Gabriel's book about Algebraic Groups?
All of t... | https://mathoverflow.net/users/18183 | Lie Algebra of Group Scheme | To elaborate on a comment by ulrich: SGA3 Exp. 2, section 4 treats Lie algebras of arbitrary group-valued functors over an arbitrary scheme (no locally noetherian hypothesis). I'm not sure what results you want with respect to base change, but most will follow straightforwardly from some combination of Definition 1.1 a... | 9 | https://mathoverflow.net/users/121 | 78937 | 47,524 |
https://mathoverflow.net/questions/44446 | 5 | I have a question regarding non-cuspidal Hilbert modular forms. If one starts with a non-parallel weight for example, it is easy to prove that there are no Eisenstein series of any level, or as is generally stated, all forms are cuspidal. My question is what happens with mod p Hilbert modular forms? Are there (non-zero... | https://mathoverflow.net/users/4685 | Eisenstein mod p Hilbert modular forms | The partial Hasse invariants $h\_1,\ldots,h\_d$ are mod $p$ Hilbert modular forms of non-parallel weight whose $q$-expansion at each cusp is equal to 1. The forms $h\_1-1,\ldots,h\_d-1$ generate the kernel of the $q$-expansion map over $\mathbb{F}\_p$. Technically, these forms are of weight $(0,\ldots,0,p,-1,0,\ldots,0... | 4 | https://mathoverflow.net/users/5513 | 78942 | 47,527 |
https://mathoverflow.net/questions/77629 | 7 | I'm searching for a "simple" description of the basis of the Barnes-Wall lattices
in (real) dimension $2^n$, if possible in a basis of minimal vectors, so that I can
do some calculations.
Can anyone tell me where to find such a description ?
**Note :** I'm not looking for examples in fixed dimensions, like the ... | https://mathoverflow.net/users/17443 | Simple basis for Barnes-Wall lattices in dimension `$2^n$` | Henry Cohn cited a very nice definition of the Barnes-Wall lattices, but in my opinion, [this definition](http://www.cs.stevens.edu/~nicolosi/papers/isit08.pdf) that I just found in a paper by Micciancio and Nicoli is even better. (Although the two definitions are similar.) The Barnes-Wall lattice in $\mathbb{C}^{2^{n-... | 11 | https://mathoverflow.net/users/1450 | 78944 | 47,528 |
https://mathoverflow.net/questions/78956 | 3 | Is it known an explicit formula for the number of subgroups of a given exponent of a finite abelian $p$-group?
| https://mathoverflow.net/users/17565 | A question on the number of subgroups of a given exponent of a finite abelian p-group | I had to look this up as well at some point in my research. The answer is yes, and a Google search for "number of subgroups of an abelian group" leads to several downloadable papers, not all of them easy to read. The paper "On computing the number of subgroups of a finite abelian group" by T. Stehling, in *Combinatoric... | 13 | https://mathoverflow.net/users/5091 | 78960 | 47,536 |
https://mathoverflow.net/questions/78954 | 11 | Hi,
Let $f$ be a cuspidal modular form of some weight and level $N$. Then it determines
an irreducible automorphic representation $\pi = \bigotimes'\pi\_p$ of $GL\_2(\mathbf Q)$.
Let $f = \sum\_i a\_i q^i$ be its fourier expansion. Then it is known that if $p\nmid N$,
then $a\_p$ determines $\pi\_p$ (it is an unramif... | https://mathoverflow.net/users/36285 | modular form Fourier coefficients and associated automorphic representation | Jared Weinstein and I wrote a paper on how to compute $\pi\_p$: see [here](http://dx.doi.org/10.1090/S0025-5718-2011-02530-5%20).
As Olivier says, $a\_p$ will often be zero, and in fact if the central character is trivial (or has conductor coprime to $p$) this is always the case when $p^2$ divides the level of $f$. ... | 11 | https://mathoverflow.net/users/2481 | 78962 | 47,537 |
https://mathoverflow.net/questions/70913 | 9 | The paper [Polynomial-Time Quantum Algorithms for Pell's Equation and the Principal Ideal Problem](http://www.cse.psu.edu/~hallgren/pell.pdf) claims
>
> There are reductions from factoring to solving Pell’s equation, and from solving Pell’s
> equation to solving the principal ideal problem [BW89b]
>
>
>
Can't... | https://mathoverflow.net/users/12481 | Reduction from factoring to solving Pell equation | If you had a fast method for solving Pell equations $x^2 - dy^2 =1$, you can factor numbers $N$ quickly: all you have to do is compute gcd$(x-1,d)$ for $d = N, 2N, 3N, \ldots$ until
you find a factor; if the factor is not prime, repeat the procedure.
Schoof showed that you don't have to know the actual solution of th... | 5 | https://mathoverflow.net/users/3503 | 78974 | 47,542 |
https://mathoverflow.net/questions/78949 | 24 | Fermat may or may not have known that there are 3-term arithmetic progressions of squares (like $1^2, 5^2, 7^2$, and that there are no 4-term APs. Murky history aside, Keith Conrad has two pleasant expositions ([here](http://www.math.uconn.edu/%7Ekconrad/blurbs/ugradnumthy/3squarearithprog.pdf) and [here](http://www.ma... | https://mathoverflow.net/users/935 | Arithmetic Progressions of Squares | Probably not, but a proof is hopeless. Ruzsa and Gyarmati have a preprint in which they construct such a subset of size something like $N/\log \log N$.
Even the colouring version (that is, finite colour the squares, does one of the classes contain a 3-term progression) is open. A very closely-related question (Schur'... | 24 | https://mathoverflow.net/users/5575 | 78977 | 47,544 |
https://mathoverflow.net/questions/71120 | 7 | It's well-known that any compact polyhedron $P$ in $\mathbb{R}^n$ (we talk about piecewise-linear setting there, i.e. $P$ is a finite union of compact convex polytopes) can be triangulated into (geometric) simplices, although sometimes it is necessary to add "extra points" in $P$ to serve as vertices of simplices in th... | https://mathoverflow.net/users/11100 | non-rigidity of interior points in polyhedral triangulations? | The answer for arbitrary polyhedra is no. If a 4-dimensional polyhedron has a 3-dimensional Schönhardt polyhedron as one of its faces, there will need to be a new vertex added somewhere within that face, which will not be free to move in an open set.
I believe that the answer is yes in 3d and yes to higher-dimensiona... | 2 | https://mathoverflow.net/users/440 | 78982 | 47,545 |
https://mathoverflow.net/questions/78984 | 5 | I am interested in studying Riemann surfaces that are not of finite type. By a non-finite type Riemann surface, I mean a Riemann surface that is not conformally equivalent to any Riemann sub-surface of a compact Riemann surface. I have the following questions about such surfaces:-
[1]. Is there any classification the... | https://mathoverflow.net/users/36038 | Riemann surfaces that are not of finite type | It should be pointed out that your definition of finite type is not the usual one. The usual definition is that a Riemann surface is of finite type if it is conformally equivalent to a compact Riemann surface minus a finite set of points. For instance, under this usual definition, an annulus of finite modulus is not of... | 12 | https://mathoverflow.net/users/1335 | 78993 | 47,550 |
https://mathoverflow.net/questions/78999 | 6 | Let $G$ be a real connected semi-simple Lie group. Let $M$ be a finite dimensional representation of it. Are there general criteria when the continuous cohomology groups $H\_\text{cont}^q(G,M)$ vanish?
A situation of particular interest for me is $G=SO^+(n-1,1)$, namely the connected Lorentz group, and $M$ is the sta... | https://mathoverflow.net/users/16183 | Continuous cohomology of semi-simple Lie group | As [pointed out](https://mathoverflow.net/a/79001/2383) by Konrad, this follows from the generalisation of van Est's theorem from [Group cohomology and Lie algebra cohomology in Lie groups](https://mathscinet.ams.org/mathscinet-getitem?mr=59285) to the continuous case (see [Hochschild and Mostow - Cohomology of Lie gro... | 7 | https://mathoverflow.net/users/394 | 79006 | 47,557 |
https://mathoverflow.net/questions/79011 | 4 | Before me, the following was asked:
[etale fundamental group and etale cohomology of curves](https://mathoverflow.net/questions/16566/etale-fundamental-group-and-etale-cohomology-of-curves)
However, that question dealt only with projective curves.
### Question
Let $X$ be any scheme (or if you prefer something mor... | https://mathoverflow.net/users/5309 | The etale fundamental group and etale cohomology with compact support | In general, it's always true (for a connected scheme) that $H^1\_{et}(X, \mathbb{Z}/l \mathbb{Z}) = \hom(\pi\_1^{et}(X), \mathbb{Z}/l\mathbb{Z})$ (not compactly supported). Taking inverse limits over $l$ then gives the claim.
The reason this is true is that $H^1\_{et}(X, \mathbb{Z}/l\mathbb{Z}$) can be computed by Ce... | 5 | https://mathoverflow.net/users/344 | 79012 | 47,559 |
https://mathoverflow.net/questions/79003 | 2 | Fix a number field $K$ and a polynomial $F(x)\in K[x]$ of degree at least $4$. For a squarefree integer $d$, define the curve $X\_d$ over $K$ by the equation $dy^2 = F(x)$. Note that the curves $X\_d$ are isomorphic over $\overline{\mathbf{Q}}$. Therefore, they have the same (stable) Faltings height and the same genus.... | https://mathoverflow.net/users/4333 | Is there an easier argument to prove that almost all of these curves have no semi-stable reduction | There is a finite set of non-Archimedean primes $S$ of $K$ such that $F(x)\in O\_S[x]$ with leading coefficient and discriminant $\Delta$ both in $O\_S^\*$, and $O\_S$ is unramified over $\mathbb Z$.
If $d$ has an irreducible factor $\mathfrak p\notin S$, then $v\_{\mathfrak p}(d)=1$ and $F(x)$ reduces to a separabl... | 4 | https://mathoverflow.net/users/3485 | 79019 | 47,563 |
https://mathoverflow.net/questions/78975 | 3 | Is there a CCC and collectionwise normal space, that isn't paracompact?
As we know, CCC + monotone normality => Lindelöf.
CCC + collectionwise normality => paracompact?
CCC = countable chain condition
Collectionwise normality = if $X$ is a $T\_{1}$ space and for every discrete family
$\{F\_{s}\}\_{s \in S}$... | https://mathoverflow.net/users/18465 | CCC + collectionwise normality => paracompact? | Yes, there is.
Let $I = \omega\_1$ be the first uncountable ordinal, and let $P = \{0,1\}^I$ be the uncountable product of discrete spaces of 2 points. Let $S$, the so-called $\Sigma$-product be its subspace of all points that have at most countably many coordinates different from $0$.
It is well known that $S$ is ... | 10 | https://mathoverflow.net/users/2060 | 79021 | 47,565 |
https://mathoverflow.net/questions/77570 | 13 | Suppose one has a link diagram of the unknot, and applies random Reidemeister moves
until the unknot is reached.
Surely it requires an exponential number of moves, exponential in, say, the crossing number
of the original diagram?
The 2001 Hass-Lagarias paper, "[The number of Reidemeister moves needed for unknotting](ht... | https://mathoverflow.net/users/6094 | Random Reidemeister moves to unknot | This question has been fully answered (the expected number of moves is $\infty$), as detailed in an addendum to the question.
I place this community-wiki "answer" here so I can accept it and so
prevent the MO software-bot from re-asking the question.
| 5 | https://mathoverflow.net/users/6094 | 79035 | 47,574 |
https://mathoverflow.net/questions/78754 | 5 | Consider the following optimization problem:
$\max\_{\lambda\_j(X)}\sum\_{j=1}^n d\_j\lambda\_j(X)$ subject to $v\_j^TXv\_j \leq 1, X \geq 0$.
$d\_j$ are such that $d\_1 \geq d\_2 \geq \ldots \geq d\_k > 0$, $\lambda\_j(X)$ is the $j$th largest eigenvalue of the positive semidefinite matrix $X$ of dimension $n\ti... | https://mathoverflow.net/users/18693 | Maximize sum of largest eigenvalues | If I understand the question correctly, the answer is that no, optima need not occur at a unique point where some of the hyperplanes defined by tightness of the linear inequalities meet the boundary of the positive semidefinite cone.
For example, let $v\_i$ be the $i^{\text{th}}$ unit vector, so the linear constraint... | 3 | https://mathoverflow.net/users/5963 | 79038 | 47,577 |
https://mathoverflow.net/questions/79025 | 3 | Let $\mathcal{D} \approx P^{\delta\_d}$ be the space of homogeneous degree $d$
polynomials in three variables (up to scaling), where $\delta\_d = \frac{d(d+3)}{2}$. A point
$p\in \mathbb{P}^2$ gives us a hyperplane $H\_p \subset \mathcal{D}$,
i.e it is the space of degree $d$ polynomials vanishing at $p$.
Define ... | https://mathoverflow.net/users/4463 | Does passing through a point in general position cut down the dimension by one? | Q1: In general no: take $X$ to be the subvariety of $P^{\delta\_d\vee}$, the dual of $P^{\delta\_d}$, formed by all $H\_p$'s. Note that $X$ is just the image of the Veronese map $\mathbb{P}(V)\to\mathbb{P}(Sym^d(V^\vee))$ for $V=\mathbb{C}^3$. So $X$ lies on a quadric $Q$ given by $x\_i x\_j=x\_kx\_l$ with $i,j,k,l$ pa... | 2 | https://mathoverflow.net/users/2349 | 79039 | 47,578 |
https://mathoverflow.net/questions/79041 | 6 | Let $\mathfrak{g}$ be the Lie algebra of a Lie group $G$ which acts on a manifold $M$.
It is quite standard that the basic forms in $\Omega^\*(M) \otimes W(\mathfrak{g}^\*)$ form a model for the singular equivariant cohomology of $M$. However, I have never seen a proof and it is not straightforward to me. Could someone... | https://mathoverflow.net/users/5450 | Cartan-Weil model for Equivariant Cohomology | see the very nice book of [Guillemin-Sternberg (Supersymmetry and ...)](http://books.google.com/books?id=zYMp0GWLFiAC&lpg=PA248&ots=Bx2FxpUDmI&dq=guillemin%2520sternberg%2520supersymmetry&pg=PA182#v=onepage&q&f=false); it also has a reprint of Cartan's paper.
| 5 | https://mathoverflow.net/users/11786 | 79042 | 47,579 |
https://mathoverflow.net/questions/78994 | 17 | In obtaining the spectral decomposition of $L^2(\Gamma \backslash G)$ where $G=SL\_2(\mathbb{R})$, and $\Gamma$ is an arithmetic subgroup (I am satisfied with $\Gamma = SL (2,\mathbb{Z})$) we have a basis of eigenfunctions of the hyperbolic Laplacian, and orthogonal to that we have the space spanned by the incomplete E... | https://mathoverflow.net/users/16389 | Where do the real analytic Eisenstein series live? | Surely there is not a single good answer, since the question is about how to legitimize "generalized eigenvectors", and there is no single-best notion of "legitimize".
As in other answers, one interpretation of Eisenstein series is as being in the dual to "rapidly decreasing" functions. This has various weaknesses.
... | 6 | https://mathoverflow.net/users/15629 | 79043 | 47,580 |
https://mathoverflow.net/questions/79030 | 1 | I was reading "[Origins of the Calculus of Binary Relations](http://boole.stanford.edu/pub/ocbr.pdf)" by Vaughan Pratt where he says "it consists of two components, a logical or static component and a relative or dynamic component" but it seems as if it should be possible to define the "static component" purely in term... | https://mathoverflow.net/users/18757 | Calculus of Binary Relations | As I understand it, and in more modern-day terms, the question asks whether it is possible to define the operations $0, 1, \cap, \cup, \neg$ on $P(X^2)$ (operations belonging to the "static component") in terms of "dynamic" operations $\delta', \delta, \circ, \circ', (-)^{op})$ where $\circ$ is relational composition (... | 6 | https://mathoverflow.net/users/2926 | 79045 | 47,581 |
https://mathoverflow.net/questions/79047 | 1 | In 1914 Jentzsch proved that if
$$
g(z)=1+a\_1z+\ldots+a\_nz^n+\ldots
$$
has the unit circle as circle of convergence then every point of this circle is a cluster-point of zeros of partial sums
$$
s\_n(z)=1+a\_1z+\ldots+a\_nz^n.
$$
I was wodering if you could point me out an alternative English written reference for
... | https://mathoverflow.net/users/2386 | Roots of Taylor Polynomials of analytic function with finite radius of convergence | A generalization is proved, in English, in detail, in Hans-Peter Blatt, Simon Blatt, and Wolfgang Luh, On a generalization of Jentzsch's theorem, J. Approx. Theory 159 (2009), no. 1, 26–38, MR2533389 (2010d:30004).
| 4 | https://mathoverflow.net/users/3684 | 79049 | 47,583 |
https://mathoverflow.net/questions/79018 | 9 | The $2$-dimensional family of solution curves $y = u(x, \xi, \eta) \approx \eta + \xi x + \mathcal O(x^2)$ near $y = 0$ of a differential equation
$y'' = \Phi(x, y, y')$
has been called *infinitesimally Desarguian* if
\begin{equation\*}
\Phi(x, y, p) = \mathcal O(|y|^3 + |p|^3) \quad \textrm{as $(y, p) \to (0, 0)$... | https://mathoverflow.net/users/16546 | Infinitesimally Desarguian curve families | The study of dual path geometries in the plane goes back to Lie, Tresse, and Cartan. There are some comprehensible modern treatments, but there are also some careless ones, so caution is needed. There is some useful information about dual path geometries near the end of the Ivey-Landsberg book, "Cartan for Beginners".
... | 12 | https://mathoverflow.net/users/13972 | 79054 | 47,586 |
https://mathoverflow.net/questions/78987 | 0 | I am trying to derive a closed form for computing the number possible outcomes of rolling $k$ dices such that the sum is $n$.
This seems to be the problem of finding number of positive integral solution of an equation $$x\_1+x\_2+\cdots+x\_k=n$$ with $x\_1,x\_2,\cdots,x\_n \in [1,6] $,but here a solution $(a,a,a,a,\c... | https://mathoverflow.net/users/18748 | Deriving a closed form for rolling a sum $n$ with $k$ dice using stars and bars | Answer is given by the coefficient of $z^n$ in
$$(z+z^2+\dots+z^6)^k = \left(z\frac{1-z^6}{1-z}\right)^k = z^k (1-z^6)^k(1-z)^{-k}.$$
An explicit formula for this coefficient is:
$$\sum\_{i=0}^{\min(k,\lfloor (n-k)/6\rfloor)} (-1)^{n+i} \binom{k}{i} \binom{-k}{n-k-6i} = \sum\_{i=0}^{\min(k,\lfloor (n-k)/6\rfloor)} (-1... | 1 | https://mathoverflow.net/users/7076 | 79058 | 47,589 |
https://mathoverflow.net/questions/78978 | 5 | If $X$ is a normal projective variety and $D$ a divisor in it, we say that $\pi\colon (\widetilde X,\widetilde D)\rightarrow (X,D)$ is a log resolution if $\widetilde X$ is a resolution of $X$, the strict transfor $\widetilde D$ of $D$ is non-singular and $\widetilde D \cup \{ \text{exceptional divisors} \}$ have simpl... | https://mathoverflow.net/users/1887 | Log resolutions on surfaces and 3-folds in characteristic p | What you want does follow from Cutkosky's paper cited by Donu Arapura (and presumably already from results of Abhyankar, but I have not checked). One just has to combine his Theorems 1.1 and 1.2. More precisely, one can resolve singularities of $X$ using Theorem 1.1
to get $\pi\_1: X\_1 \to X$ with $X\_1$ smooth and t... | 5 | https://mathoverflow.net/users/519 | 79060 | 47,590 |
https://mathoverflow.net/questions/79067 | 8 | Is every solvable subgroup of $GL(n,\mathbb{Z})$ polycyclic?
The first solvable group that is not polycyclic is $\mathbb{Z}[1/2]\rtimes \mathbb{Z}$ (where the automorphism is given by multiplication with 2) and I do not see a way of embedding it into $GL\_n(\mathbb{Z})$ for some $n$.
| https://mathoverflow.net/users/3969 | Is every solvable subgroup of $GL(n,\mathbb{Z})$ polycyclic? | It is a theorem of Mal'cev that all solvable subgroups of $GL(n,\mathbb Z)$ are polycyclic, and a theorem of Auslander that every polycyclic group is isomorphically embeddable in $GL(n,\mathbb Z)$, for some $n$. Auslander's theorem was later reproved by Swan purely algebraically by adapting the proof of Ado's theorem.
... | 17 | https://mathoverflow.net/users/2384 | 79071 | 47,595 |
https://mathoverflow.net/questions/79065 | 1 | Hi,
I am looking for an example of a commutative algebra object in a braided monoidal category C which it can also be turned into a commutative Frobenius algebra. If you have any examples could you also tell me what the multiplication and unit are?
Thank you
Dimtris
| https://mathoverflow.net/users/18768 | Example of a commutative algebra object in a braded monoidal category C | The standard example here is where the braided tensor category is the Drinfeld center Z(C) and the algebra object is the induction of the trivial object from C to Z(C). If C is semsimple over an algebraically closed field then this can be written explicitly as $\sum\_x x \otimes x^\*$ with half braiding given by Theore... | 4 | https://mathoverflow.net/users/22 | 79076 | 47,597 |
https://mathoverflow.net/questions/79068 | 6 | Let $C$ be a symmetric monoidal category. I am interested in objects $X \in C$ such that the symmetry
$S\_{X,X} : X \otimes X \cong X \otimes X$
is equal to the identity. There are many examples of such objects, e.g. invertible sheaves. My first question is: How would you call such an object?
Now assume that $X$ ... | https://mathoverflow.net/users/2841 | Does the dual of an object with trivial symmetry also have trivial symmetry? | I believe the answer to your question is yes, without a further assumption that e is an isomorphism. The symmetry S\_{Y,Y} can be obtained from the symmetry S\_{X,X}
as follows
$Y\otimes Y \xrightarrow{c\circ c} Y\otimes Y \otimes X\otimes X \otimes Y \otimes Y \xrightarrow{id\_Y^{\otimes 2} S\_{X,X}\otimes id\_Y^{... | 5 | https://mathoverflow.net/users/1040 | 79077 | 47,598 |
https://mathoverflow.net/questions/79040 | 9 | Let $X$ be a smooth complex projective variety of dimension $n$.
Under the duality between $N\_1(X)$ and $N^1(X)$ we know that closure of cone of effective curves $\overline{NE}(X)$ is dual to closure of ample cone $\overline{Amp}(X)$.
It was proved in 2004 that the closure of cone of effective divisors $\overline{... | https://mathoverflow.net/users/5259 | Cone of movable curves | A more direct approach is the following:
Let $X$ be the projective bundle $\pi:\mathbb{P}(\mathcal{E})\to \mathbb{P}^1$ where $\mathcal{E}=\mathcal{O}\oplus \mathcal{O}(-1) \oplus \mathcal{O}(-2)$. Let $M$ be the tautological bundle of $X$. It is easily checked that the ample line divisors $H\_i$ on $X$ correspond to... | 7 | https://mathoverflow.net/users/3996 | 79078 | 47,599 |
https://mathoverflow.net/questions/78822 | 4 | I am reposting a question on math.stackexchange which did not recieve good questions.
The orginal questio is at <https://math.stackexchange.com/questions/73091/distribution-of-a-maximum>.
Randomly select $n$ numbers from ${\{1,2,\dots,m\}}$ without replacement, and order the chosen elements increasingly: $X\_1 < X\_2... | https://mathoverflow.net/users/8379 | Distribution of a maximum | If, as suggested by Ori Gurel-Gurevich, we sample from uniform distribution on $[0,1]$, then $Z$ will typically be of order $1/\sqrt{n}$.
A convenient way of generating the points $X\_1,\dots,X\_n$ in order is letting $W\_1,\dots,W\_{n+1}$ be independent exponential(1) variables with partial sums $S\_k = W\_1+\cdots... | 3 | https://mathoverflow.net/users/14302 | 79080 | 47,600 |
https://mathoverflow.net/questions/79004 | 28 | Let $H$ be an $(\infty,1)$-topos (seen as a generalization of the homotopy category of spaces).
You can define the suspension of an object $X$ as the (homotopy) pushout of $\*\leftarrow X \to \*$, hence you can define inductively the spheres $\mathbb{S}^n$ (the sphere of dimension $-1$ is the initial object of $H$ an... | https://mathoverflow.net/users/10217 | Homotopy groups of spheres in a $(\infty, 1)$-topos | * If $H$ is the terminal category (=sheaves on the empty space), then $\pi\_k^HS^n$ (notation for homotopy groups of "spheres" in $H$) is known!
* The slice category $H=\mathrm{Spaces}/B$ is an $(\infty,1)$-topos. The homotopy groups of spheres in this setting amount to the homotopy groups of the space $\mathrm{map}(B,... | 28 | https://mathoverflow.net/users/437 | 79082 | 47,601 |
https://mathoverflow.net/questions/79102 | 8 | I know this question may seem nonsensical at first but let me exlain what i have in mind:
In enriched category theory we define categories enriched over a monoidal category $(\mathcal{V},\otimes, I)$. An enriched category then is given by a set/class of objects $\mathcal C$ and a rule assigning to every pair $X,Y$ of... | https://mathoverflow.net/users/1261 | Definition of enriched caterories or internal homs without using monoidal categories. | This is exactly the notion of a [closed category](http://en.wikipedia.org/wiki/Closed_category). See Eilenberg and Kelly's article in the 1965 La Jolla proceedings (Springer 1966). I think they also describe categories enriched in a closed category.
| 12 | https://mathoverflow.net/users/4262 | 79103 | 47,612 |
https://mathoverflow.net/questions/70900 | 7 | Using the classic spherical harmonics theory, one obtains the $k$-th eigenvalue of the $n$-dimensional round sphere $S^n$ to be $k(k+n-1)$, and its multiplicity is $\binom{n+k}{k}-\binom{n+k-1}{k-1}$, see e.g. [Berger, Gauduchon,Mazet, "Le spectre d'une variété riemannienne", Lecture Notes in Mathematics, Vol. 194 Spri... | https://mathoverflow.net/users/15743 | Multiplicity of eigenvalues of the Laplacian on quaternionic projective space | These dimensions have been calculated explicitly for all the compact rank 1 symmetric spaces. See Cahn and Wolf, "Zeta functions and their asymptotic expansions for compact symmetric spaces of rank one", Commentarii Mathematici Helvetici, vol. 51 (1976), pp. 1-21.
| 6 | https://mathoverflow.net/users/18505 | 79105 | 47,614 |
https://mathoverflow.net/questions/79046 | 8 | Let $\overline{NE}(X)$ be the closure of the cone generated by the numerical classes of effective curves and $\overline{\mathrm{Mov}}(X)$ the closure of the cone of moving curves.
(Q) Is there an example of a smooth projective variety $X$ such that
* $\overline{NE}(X)$ is (finite) polyhedral, but
* $\overline{\ma... | https://mathoverflow.net/users/10076 | Effective versus movable cones of curves | As J.C. indicates in the comments, an example for Q1 can be gotten from the variety considered in [this paper](http://arxiv.org/abs/0910.5888). This isn't spelled out in the paper, so let me explain it here.
First let's change the question into its dual form. The cone of curves is dual to the nef cone, and Boucksom--... | 5 | https://mathoverflow.net/users/nan | 79108 | 47,616 |
https://mathoverflow.net/questions/79106 | 0 | Consider 1 < $p<\infty$ and an integer $k$. Does interior elliptic regularity for the Laplacian also hold in the Sobolev space $W^{k,p}$ of negative order?
More precisely I am interested in the following question: Let $u\in W^{-1,p}(R^n)$ be a distributional solution of $\Delta u=Su,$ where $S$ is smooth. Is it then... | https://mathoverflow.net/users/3509 | Elliptic regularity in Sobolev spaces of negative order | The smoothness result holds even for solutions from $\mathcal D'(\mathbb R^n)$. See, for example, Theorem IX.26 in Vol.2 of "Methods of Modern Mathematical Physics" by Reed and Simon.
| 3 | https://mathoverflow.net/users/12205 | 79111 | 47,617 |
https://mathoverflow.net/questions/79112 | 3 | For any sequence of complex numbers $(a\_n)$, an application of the Cauchy-Schwarz inequality gives
$$\left|\sum\_{m=1}^{n}a\_m\right|\leq \sqrt {n\sum\_{m=1}^{n}|a\_m|^2}.$$
Putting $a\_n=\mu(n)/\sqrt n$, one (trivially) finds that
$$\sum\_{m=1}^{n}\frac{\mu(m)}{\sqrt m}= O(\sqrt{n\log n}).$$
Is a better uncondition... | https://mathoverflow.net/users/10980 | Is Cauchy-Schwarz still the best we have on Mobius? | Summing by parts and using the bound on the Mertens function $M(n)=o(n)$ (which is equivalent to the Prime Number Theorem) one gets for your sum $S(n)=o(\sqrt n)$. Better bounds on the order of magnitude of $M(n)$ of course give a better one for $S(n)$ (see e.g. the *Handbook of Number Theory*, by J. Sándor, Dragoslav ... | 6 | https://mathoverflow.net/users/6101 | 79123 | 47,621 |
https://mathoverflow.net/questions/79109 | 14 | I have been using random matrix theory in signal processing and have some trouble *understanding* what the Stieltjes transform does.
The gist of my work is that I have an $N\times N$ true covariance matrix of the population with $k$ eigenvalues $\lambda\_i>1,\ i=\lbrace 1,\ldots,k\rbrace$, corresponding to the signa... | https://mathoverflow.net/users/nan | Intuitive understanding of the Stieltjes transform | Firstly, the equation you attribute to Silverstein (and is sometimes known as the "self-consistent equation" for the Stieltjes transform) is not exact, but only asymptotically valid in the limit $n \to \infty$. The definition given in Wikipedia is the exact formula. (Your final formula, by the way, is missing a normali... | 22 | https://mathoverflow.net/users/766 | 79129 | 47,626 |
https://mathoverflow.net/questions/79122 | 2 | Where appears for the first time the term Hodge-Tate representation.
Can i find somewhere explanation of the terminology Hodge-Tate, Derham etc. for representations and Fontaine's rings.
| https://mathoverflow.net/users/18777 | Terminology-history of p-adic representations | The notion of Hodge-Tate decomposition has been introduced by Tate, in 1967.
(The paper itself is called *$p$-divisible groups*, and it appeared
in the Proceedings of a conference on local fields that took place in Driebergen.)
There, he shows that over a $p$-adic field, the $p$-adic Tate module $T\_p(G)$ of an Abelia... | 6 | https://mathoverflow.net/users/10696 | 79131 | 47,627 |
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