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https://mathoverflow.net/questions/67304 | 12 | This is inspired by
[Does "all points rational" imply "constant" for this "cubic" curve over an arbitrary field?](https://mathoverflow.net/questions/67126/does-all-points-rational-imply-constant-for-this-cubic-curve-over-an-arbit) .
Say $K/F$ is a finite separable extension of fields. Assume $F$ is infinite (or el... | https://mathoverflow.net/users/1384 | Image of projective 1-space contained in projective 1-space over a smaller field? | This proof was inspired by an idea of Francois Brunault and appears piecemeal in the comments to the question; it's probably not materially different from David's proof. Choose some $F$-basis $\{a\_1,\ldots,a\_n\}$ for $K$ with $a\_1=1$, and let $p:\mathbf{A}^n\_K\to\mathbf{A}^1\_K$ be the map $(X\_1,X\_2,\ldots,X\_n)\... | 7 | https://mathoverflow.net/users/7868 | 67460 | 41,524 |
https://mathoverflow.net/questions/67470 | 1 | Consider the set $S=\{2,3,\ldots\}$ equipped with the operation $n\cdot m=n^m$.
>
> **Question:** Do there exist a mean on $S$ which is left and right invariant with respect to $\cdot$?
>
>
>
Thanks in advance,
Valerio
| https://mathoverflow.net/users/13809 | Mean on the natural numbers which is invariant with respect to the power | Posting it as an answer, since OP confirmed I did understand the question right.
Such a mean cannot exist. In fact, if it would, we could denote it by $M$ and compute:
$M\left(S\right)=M\left(2^S\right)+M\left(3^S\right)+M\left(S\setminus 2^S\setminus 3^S\right)$ (since the sets $2^S$, $3^S$ and $S\setminus 2^S\set... | 3 | https://mathoverflow.net/users/2530 | 67472 | 41,528 |
https://mathoverflow.net/questions/67473 | 14 | For two sets A and B. Suppose|2^A| = |2^B| (cardinality of power sets of A and B), does |A|=|B| ?
(It is easy to see that|A|=|B| if we assume generalized continuity hypothesis. Do we have the same result without it?)
| https://mathoverflow.net/users/15711 | Equality of Cardinality of Power Set | The answer is that if the axioms of set theory are consistent, then you cannot prove that conclusion. Although it seems very reasonable to expect that a smaller
set must have strictly fewer subsets, which is another way
of stating your property, in fact this property is
independent of ZFC.
(The fact that many people ... | 27 | https://mathoverflow.net/users/1946 | 67474 | 41,529 |
https://mathoverflow.net/questions/67348 | 2 | Here is a definition of holomorphic convexity taken from the notes of Eyssidieux:
**Defintion.** A complex analytic space $S$ is holomorphically convex if there is a proper holomorphic morphism $\pi: S\to T$ with $\pi\_\*O\_S=O\_T$ such that $T$ is a Stein space. $T$ is then called *Cartan-Remmert* reduction of $S$.
... | https://mathoverflow.net/users/13441 | A basic question on the definition of Cartan-Remmert reduction and holomorphic convexity | A full statement of the Cartan-Remmert reduction includes also a universal property which should answer your question (you will get uniqueness up to a unique isomorphism ):
in the Encyclopedia of Math. Sciences (several complex variables, vol. 7) you will find the following:
Let $X$ be a holomorphically convex space.... | 2 | https://mathoverflow.net/users/15673 | 67479 | 41,532 |
https://mathoverflow.net/questions/67483 | 4 | I'm wondering if there's any sort of Ramsey relation that allows for the tuples to be of arbitrary infinite size $\mu$? This $\mu$ is below some strongly compact cardinal, so I'm not worried about large cardinal hypotheses.
| https://mathoverflow.net/users/15713 | Is there Ramsey Theorem for infinitary tuples? | Infinite exponent partition relations are inconsistent with the axiom of choice, so in ZFC, this phenomenon does not exist, but nevertheless, in the context of $ZF+\neg AC$ there is a robust theory. See for example [Andres Caicedo's discussion](http://caicedoteaching.files.wordpress.com/2009/04/580-partition21.pdf), [t... | 5 | https://mathoverflow.net/users/1946 | 67484 | 41,534 |
https://mathoverflow.net/questions/67436 | 25 | Is there any sequence $a\_n$ of nonnegative numbers for which $\displaystyle\sum\_{n \geq 1}a\_n^2 <\infty$ and
$$\sum\_{n \geq 1}\left(\sum\_{k \geq 1}\frac{a\_{kn}}{k}\right)^2=\infty\quad?$$
See also <https://math.stackexchange.com/questions/42624/double-sum-miklos-schweitzer-2010>
| https://mathoverflow.net/users/15702 | Is there any sequence $a_n$ of nonnegative numbers for which $\sum_{n \geq 1}a_n^2 <\infty$ and $\sum_{n \geq 1}\left(\sum_{k \geq 1}\frac{a_{kn}}{k}\right)^2=\infty$? | Yes, such sequences exist.
In effect the problem concerns the linear operator, call it $T$, that maps any sequence $(a\_n)$ to the sequence whose $n$-th term is $\sum\_{k\geq1} a\_{kn}/k$. The problem asks whether there exists $a$ such that the $l^2$ norm $\|a\|\_2$ is finite but $\|Ta\|\_2 = \infty$. We show that su... | 37 | https://mathoverflow.net/users/14830 | 67487 | 41,536 |
https://mathoverflow.net/questions/67485 | 7 | Recall the following corollary to the proper and smooth base change theorems:
>
> Let $\pi: X \to S$ be a proper, smooth morphism. Then the direct images $R^i \pi\_\* \mathcal{F}$ are locally constant constructible for any l.c.c. sheaf $\mathcal{F}$ (with torsion prime to the characteristic of the residue fields) o... | https://mathoverflow.net/users/344 | Can proper-smooth base change be used to show that varieties cannot be lifted to characteristic zero? | There is an example due to Hirokado of a Calabi-Yau threefold in characteristic 3 with third Betti number zero which implies that it cannot be lifted to characteristic zero. See: Hirokado, Masayuki -
A non-liftable Calabi-Yau threefold in characteristic 3.
Tohoku Math. J. (2) 51 (1999), no. 4, 479–487.
| 12 | https://mathoverflow.net/users/519 | 67491 | 41,538 |
https://mathoverflow.net/questions/67493 | 4 | Let $\varphi : U \rightarrow X$ be a holomorphic mapping of some open set $U\subseteq\mathbb{C}$ into a complex $n-$dimensional manifold $X$. If we know that this mapping is diffeomorphic onto its immage does it follow that it is also biholomorphic onto its image ?
| https://mathoverflow.net/users/15715 | diffeomorphic, holomorphic, biholomorphic | Yes. Since the map is a diffeomorphism onto its image it means that the induced map on tangent spaces (thought of as $C^{\infty}$ manifolds) is an injection. But the tangent space as a complex manifold is the same space with an added complex structure, so it is still an injection. The (holomorphic) implicit function th... | 5 | https://mathoverflow.net/users/519 | 67494 | 41,539 |
https://mathoverflow.net/questions/67504 | 7 | I suspect I will show my ignorance here, but this 'theorem' I would consider to be intuitively sensible, but I cannot find anything similar by looking through a few books or on the web. If would seem true in principal, but it probably needs some modification to how I have formulated it below. I was wondering if anyone ... | https://mathoverflow.net/users/7381 | Jordan Curve Theorem for Manifolds | As pointed out by Francesco, part (1) is false in general; however, it is true when the first Betti number of $M$ is 0. Part (2) is correct. All this follows easily from Alexander duality, stating that if $d$ is the dimension of $M$, we have $\mathrm H\_{d-1}(S^n) \simeq \mathrm H^{1}(M, M \smallsetminus S^n)$.
Of co... | 8 | https://mathoverflow.net/users/4790 | 67508 | 41,541 |
https://mathoverflow.net/questions/67507 | 5 | Here the base field is the field of complex numbers.
| https://mathoverflow.net/users/15720 | Is any K3 surface of degree 8 in P^5 the complete intersection of quadrics? | *Any* $K3$ surface $S$ of degree $8$ in $\mathbb{P}^5$ is contained in $3$ linearly independent quadrics. It can be seen that in the *general* case $S$ is a complete intersection of $3$ quadrics. However, there are some special cases where this is not true, but they can be completely described.
The point is that $S$... | 11 | https://mathoverflow.net/users/7460 | 67510 | 41,542 |
https://mathoverflow.net/questions/67509 | -1 | Let $S$ be a symmetric set of generators of the (finite) group $G$. Having a bi-invariant metric $d$ on $S$ (meaning that whenever $s,t,g,gs,gt,sg,tg\in S$, then $d(s,t)=d(sg,tg)=d(gs,gt)$), is it always possible to extend $d$ to a bi-invariant metric on $G$?
**Update:** It has been shown below by Lucasz that the ans... | https://mathoverflow.net/users/13809 | Extending a bi-invariant metric from a set of generators to the whole group. | No, it's not always possible. Take $G$ to be the cyclic group of order $20$, let $g$ be its generator. Let $S =$ { $g,g^3,g^{17},g^{19}$ }. Define $d$ on $S$ by putting $d(g,g^3)=2$ and all the other distances equal to $1$.
Now, the pair $(g,g^3)$ is not in the same "$(S\times S)$-orbit" as $(g^{17},g^{19})$, and th... | 4 | https://mathoverflow.net/users/2631 | 67513 | 41,543 |
https://mathoverflow.net/questions/67511 | 2 | I wonder if we can characterize weak measurability of a function taking values in a Banach space using sequence of step functions (functions that have finite range) just like how we define strong measurability?
More specifically, a function $f:\Omega\mapsto X$ defined on a measure space $(\Omega,\Sigma,\mu)$ and tak... | https://mathoverflow.net/users/15719 | Characterization of Weakly measurable functions | If there is a sequence of step functions such that $\phi\_n\to f$ weakly a.e., then $f$ is almost separably valued. But if it is weakly measurable and almost separably valued, it is strongly measurable.
| 4 | https://mathoverflow.net/users/12120 | 67514 | 41,544 |
https://mathoverflow.net/questions/67458 | 28 | In an appendix to his [notes](http://faculty.tcu.edu/gfriedman/notes/ih.pdf) on intersection homology and perverse sheaves, MacPherson writes
>
> Why do we want to consider only spaces $V$ that admit a decomposition into manifolds? The intuitive answer is found by considering the group of all self homeomorphisms of... | https://mathoverflow.net/users/430 | A "meta-mathematical principle" of MacPherson | A precise version of this statement is the Bing-Borsuk conjecture
that a homogeneous ENR is a manifold.
[Here](http://arxiv.org/abs/0811.0886) is a recent survey,
generally in the direction of my answer.
There is a candidate counter-example, due to Bryant, Ferry,
Mio, and Weinberger, but they can't show it is homogeneo... | 25 | https://mathoverflow.net/users/4639 | 67525 | 41,551 |
https://mathoverflow.net/questions/67527 | 1 | A binary De Bruijn sequence of index $n$ is a circular sequence $S=a\_1 a\_2 \dots a\_{2^n},$ with $a\_i \in \{0,1\},$ and such that each of the $2^n$ binary $n$-uples occurs exactly once in $S.$
Is there an infinite binary sequence $b\_1 b\_2 b\_3 \dots$ such that $b\_1 \dots b\_{2^n}$ is a binary De Bruijn Sequence... | https://mathoverflow.net/users/122383 | Nested De Bruijn Sequences | No. It may as well start with 0 in which case it is forced to be 01 for order 2 and then be 0110 for order 4. In order to include 111 it could only continue to be 01101110 or 01100111 but both of those have 011 twice and miss 000.
| 2 | https://mathoverflow.net/users/8008 | 67529 | 41,553 |
https://mathoverflow.net/questions/67531 | 4 | Before I ask my question, let me give you a mini-preamble: in 2006, during an animated discussion on feasibility, ultrafinitism, and what else on FOM, I introduced (informally, and to speak the tuth, quite vaguely) a seemingly new notion: **UNUTTERABILITY** (see [here](http://cs.nyu.edu/pipermail/fom/2006-July/010657.h... | https://mathoverflow.net/users/15293 | Natural numbers of great kolmogorov complexity | Any theory containing $I\Delta\_0+\mathit{EXP}+B\Sigma\_1$ and having a universal evaluator for your terms (which $I\Delta\_0+\mathit{SUPEXP}$ does, if you stick to the arithmetical language and exponentiation) proves that there exist numbers with arbitrary large Kolmogorov complexity of terms. In fact, considering ter... | 4 | https://mathoverflow.net/users/12705 | 67535 | 41,557 |
https://mathoverflow.net/questions/67219 | 2 | The Dedekind zeta function of an abelian extension $E$ of $\mathbb{Q}$ factors as a product of Artin L function $L(s, \chi)$, where the product runs over all irreducible representations $\chi$ of $Gal(E : \mathbb{Q})$ .
Question: What is known for irreducible representation $ \sigma$ of $G(F) = Gal(\overline{Q}, F)$.... | https://mathoverflow.net/users/10400 | Decomposition of Artin L functions | You should define what you mean by a decomposition of an Artin $L$-function. If you assume standard conjectures of Langlands and Selberg, then the Artin $L$-function of an irreducible representation of $G(\mathbb{Q})$ is a primitive function in the Selberg class, hence it has no nontrivial decomposition there (or among... | 2 | https://mathoverflow.net/users/11919 | 67541 | 41,560 |
https://mathoverflow.net/questions/67528 | 2 | Let $X$ be a $n$ dimensional complex manifold with complex structure $I$ and assume one has a diffeomorphism $f : \mathbb{C} \rightarrow X$ of some open set $U$ in $\mathbb{C}$ into its image $f(U)$. Also $f$ is holomorphic (from this it follows that $f$ is a biholomorphism). Is then $f(U)$ a complex $1$ dimensional su... | https://mathoverflow.net/users/15724 | biholomorphism complex manifold induced structure | Provided I understand the question this time, the answer is no.
**Example.** There are plenty of contrexamples. Let us construct one, so that the closer of $F(U)$ is a complex manifold of complex dimension $2$. To do this, take a complex algebraic torus $T^2$ of dimension $2$ and consider a linear map from $\mathbb C... | 4 | https://mathoverflow.net/users/943 | 67546 | 41,563 |
https://mathoverflow.net/questions/67543 | 1 | Cauchy-Schwarz inequality of determinants:
for $A\_{n\times k}$, $B\_{n\times k}$, and $B'B$ non-singular, we have
$|A'B|^2\leq |A'A||B'B|$
I was wondering what's the sufficient and necessary conditions for the equality to hold.
I know a sufficient condition:
when $\exists C\_{k\times k}$, such that $A=BC$, ... | https://mathoverflow.net/users/15729 | Sufficient and necessary conditions on equality of Cauchy-Schwarz inequality of determinants | The standard argument goes as follows: Let $M=B'A(A'A)^{-1}A'B$ and $N=B'(I-A(A'A)^{-1}A')B$. Then it is easy to check that $M$ and $N$ are positive definite and that the Cauchy-Schwarz inequality is equivalent to the following (true) inequality
$$
|M+N| \ge |M|
$$which has equality if and only if $N$ is the zero matri... | 3 | https://mathoverflow.net/users/3996 | 67547 | 41,564 |
https://mathoverflow.net/questions/67549 | 8 | After learning about the fundamental group, and proving that $\mathbb{R}^n$ minus any countable set is path-connected, I started wondering if the fundamental group of $\mathbb{R}^2-\mathbb{Q}^2$ is known. Does anyone know whether or not it is? Or how one might go about determining it?
| https://mathoverflow.net/users/36720 | Fundamental group of R^2-Q^2 | The space $X = \mathbb{R}^2 - \mathbb{Q}^2$ is not [semilocally simply connected](http://en.wikipedia.org/wiki/Semi-locally_simply_connected), and so in a sense the fundamental group is a poor measure of the homotopy 1-type of $X$. It is an exercise in Hatcher's book Algebraic Topology that this group is uncountable.
... | 21 | https://mathoverflow.net/users/4177 | 67552 | 41,566 |
https://mathoverflow.net/questions/67550 | 6 | Let S be the set of all ordinal-definable real numbers and let A(S) be the statement that S is
denumerably infinite. If ZFC is consistent, it has been proved (by A. Levy) that it remains
consistent if we adjoin A(S) to it as a new axiom-so let us assume that this has been done.
Now S has a natural well-ordering inherit... | https://mathoverflow.net/users/4423 | A question about large denumerable ordinal numbers | The class HOD consisting of the hereditarily ordinal definable sets is transitive proper class inner model of ZFC, and your assumption A(S) amounts to the assertion that $\mathbb{R}^{\text{HOD}}$, the set of reals of this model, is a countable set in the ambient set-theoretic universe V. The shortest enumeration of thi... | 8 | https://mathoverflow.net/users/1946 | 67554 | 41,567 |
https://mathoverflow.net/questions/67536 | 1 | Hello?
I have some questions in the group theory.
I know that the intersection of the lower central series of a finitely generate free group is trivial.
So I wonder whether every nontrivial subgroup of the free group containsu a term or not.
I've tried, but coudn't have shown or found a counter example.
It may be false... | https://mathoverflow.net/users/15728 | Any subgroup of f.g. free group with finite index contains a term of lower central series? | The answer is "no" in both cases.
The terms of the lower central series of a group are verbal subgroups. If we let $\gamma\_c(G)$ denote the $c$th term of the lower central series of $G$, then for any groups $G$ and $K$ and any group homomorphism $\varphi\colon G\to K$, we have $\varphi(\gamma\_c(G))=\gamma\_c(\varph... | 4 | https://mathoverflow.net/users/3959 | 67556 | 41,569 |
https://mathoverflow.net/questions/67551 | 4 | Suppose I have an $H$-space $H$ and a topological group $G$, such that for *compact* spaces $X$ there is a natural equivalence of group valued functors
$[X, H] \to [X, G]$
Now $H$ is a (non-finite) CW-complex (in my case it is $BU\_{\otimes}$), but $G$ is some really huge not even locally compact space. Can I someh... | https://mathoverflow.net/users/3995 | Weak homotopy equivalence of $H$-spaces | If $H$ and $G$ are spaces having abelian fundamental groups (for all basepoints), and if there is a natural isomorphism between the sets $[X,H]$ and $[X,G]$ for finite CW complexes $X$, then $H$ and $G$ are weakly homotopy equivalent.
Proof: Wlog $H$ and $G$ are CW complexes. $H$ is the direct limit of all of its fin... | 8 | https://mathoverflow.net/users/6666 | 67559 | 41,571 |
https://mathoverflow.net/questions/67565 | 3 | Frequently in the literature on Hecke algebras for the symmetric group and their generalisations, one encounters references to Young's seminormal form and Young's orthogonal form. I have a good understanding of the seminormal form, which gives simple formulae for the actions of simple transpositions on Specht modules f... | https://mathoverflow.net/users/15632 | Difference between orthogonal form and seminormal form | The only difference between the two is rescaling the basis vectors, i.e. conjugating by a diagonal matrix.
For instance with the representations of the symmetric group, the usual choice for the seminormal representation would be to have matrices of the form
$$ \begin{pmatrix} -1/k & 1 \\ 1 - 1/k^2 & 1/k \end{pmatri... | 4 | https://mathoverflow.net/users/362 | 67566 | 41,574 |
https://mathoverflow.net/questions/67562 | 1 | (ZF + Countable Choice)
Let $\langle X,\mathcal{T} \hspace{.06 in} \rangle$ be a second-countable Hausdorff space. Let $\mu$ be a Borel measure on $X$.
Let $\langle I,\leq\_I \rangle$ be a [directed set](http://en.wikipedia.org/wiki/Directed_set), and let $\{\mu\_i : i\in I\}$ be a collection of Bor... | https://mathoverflow.net/users/nan | Weak convergence of measures on non-metrizable spaces | Allowing infinite measures will defeat equivalence of 1 and 2. Say Hausdorff measures of various dimensions $<1$ on $\mathbb R$. All nonempty open sets have measure $\infty$. But closed sets can have interesting values.
On the other hand, for finite measures, then of course 1 and 2 are equivalent, by taking compleme... | 1 | https://mathoverflow.net/users/454 | 67577 | 41,578 |
https://mathoverflow.net/questions/67571 | 7 | Let $K$ be a number field and let $X$ be a smooth projective geometrically connected curve over $K$.
There exists a finite field extension $L/K$ such that $X\_L=X\otimes\_K L$ has semi-stable reduction, i.e., there exists a semi-stable arithmetic surface $\mathcal{X}$ over the ring of integers $O\_L$ with generic fib... | https://mathoverflow.net/users/4333 | Can we bound the minimal degree of a field extension required to obtain semi-stable reduction | I think the answer to Question 1 is yes. One may use the fact that a curve has semistable reduction iff its Jacobian does and apply Grothendieck's theorem which says that an abelian variety has semistable reduction (over a local field) iff the representation of the intertia group on the Tate module is unipotent (it is ... | 7 | https://mathoverflow.net/users/519 | 67579 | 41,580 |
https://mathoverflow.net/questions/67581 | 10 | The question is: Which countable linear orders are $\aleph\_0$-categorical?
I have a bit of progress on this:
Define a *discrete tuple* to be a set of elements, ordered discretely, such that if $a$ and $b$ are in the tuple, and $c$ is between $a$ and $b$ in the structure, then $c$ is part of the tuple.
Then if th... | https://mathoverflow.net/users/15735 | Which countable linear orders are $\aleph_0$-categorical? | In the paper below Rosenstein gives a *complete characterization* of $\aleph\_0$-categorcial theories of linear order.
Rosenstein, Joseph G.
$\aleph \_0$-categoricity of linear orderings.
Fund. Math. 64 1969 1–5.
You can also find the result, and many other gems, in Rosenstein majestic [text on linear orders](htt... | 12 | https://mathoverflow.net/users/9269 | 67584 | 41,582 |
https://mathoverflow.net/questions/67582 | 16 | Consider the unit sphere $\mathbb{S}^d.$ Pick now some $\alpha$ (I am thinking of $\alpha \ll 1,$ but I don't know how germane this is). The question is: how many spherical caps of angular radius $\alpha$ are needed to cover $\mathbb{S}^d$ completely? There is an obvious bound coming from dividing the volume of the sph... | https://mathoverflow.net/users/11142 | covering by spherical caps | There exist coverings such that each point is covered at most $400 d \log d$ times, and you can improve this bound a little if you look at the covering density, i.e., the average number of times each point is covered. See the "Covering the sphere by equal spherical balls" by Boroczky and Wintsche (available at <http://... | 20 | https://mathoverflow.net/users/4720 | 67587 | 41,583 |
https://mathoverflow.net/questions/67593 | 0 | How is the following formula derived which yields the probability that the sum of the squares of n random draws from the closed interval [-1,1] is less than one?
formula: (1/2^n)\*pi^(n/2)/(n/2)!
| https://mathoverflow.net/users/15740 | Probability formula derivation | Volume of the ball divided by volume of the cube.
| 6 | https://mathoverflow.net/users/12120 | 67596 | 41,587 |
https://mathoverflow.net/questions/67537 | 7 | For a compact quantum group $C\_q[G]$, it was shown by Woronowicz that $C\_q[G]$ contains a dense Hopf algebra generalising the algebra of representations of $G$. I am interested in the other way around, ie given a Hopf algebra $H$ (say a Drinfeld--Jimbo algebra if it makes things easier) can it always be completed to ... | https://mathoverflow.net/users/1095 | Compact Quantum Groups from Hopf Algebras | I think this was solved in the paper:
MR1310296 (95m:16029)
Dijkhuizen, Mathijs S.(NL-MATH); Koornwinder, Tom H.(NL-AMST-CS)
CQG algebras: a direct algebraic approach to compact quantum groups. (English summary)
Lett. Math. Phys. 32 (1994), no. 4, 315–330.
They show that a Hopf $\*$-algebra $A$ is the max... | 12 | https://mathoverflow.net/users/406 | 67599 | 41,589 |
https://mathoverflow.net/questions/67600 | 6 | Hello mathematicians,
i'm looking for explicit computations of expressions like
$$
\sum\_{\substack{0\leq i,j,k<n\\i\neq j\neq k \neq i}}\zeta\_n^{ip^{k\_1}+jp^{k\_2}+kp^{k\_3}}
$$
and its generalizations, where $p$ is a prime, $n$ an integer (not assumed prime with $p$) and $\zeta\_n$ is a primitive $n$-th root of th... | https://mathoverflow.net/users/3680 | Sum of products of p-th powers of roots of 1 and monomial symmetric functions | The evaluation of the Schur function $s\_{\lambda}$ at the $n$th-roots of unity is the coefficient of $s\_{\lambda}$ in the expansion in the Schur basis of the plethysm $h\_k[p\_n]=p\_n[h\_k]$ of the complete sum $h\_k$ with the power sum $p\_n$, when $k\cdot n=|\lambda|$ (and $0$ if $|\lambda|$ is not a multiple of $n... | 10 | https://mathoverflow.net/users/6768 | 67607 | 41,595 |
https://mathoverflow.net/questions/67595 | 13 | To say I am a novice in $K$-theory is to overstate my experience with the field. I've been reading the various wiki articles so as to have some preparation before jumping in, and I couldn't answer the following question to myself:
I understand that $K$-theory had started with the Grothendieck-Riemann-Roch in mind, an... | https://mathoverflow.net/users/5756 | Why was it reasonable to ask what the higher K-groups are? | The idea of considering higher K-groups comes from topology, and is due to Atiyah, Bott, and Hirzebruch. Atiyah and Hirzebruch defined topological K theory and observed that Bott periodicity says that $K(X)$ is more or less the same as $K(S^2X)$. This suggested to them defining a generalized cohomology theory of period... | 11 | https://mathoverflow.net/users/51 | 67609 | 41,597 |
https://mathoverflow.net/questions/67610 | 6 | This is a [question on math.se](https://math.stackexchange.com/questions/43749/computing-the-dimension-of-representations-in-a-reducible-induced-representation) that got no answers.
1) Is there a relatively general method of computing the dimensions of representations in a reducible induced representation?
An expli... | https://mathoverflow.net/users/2024 | Computing the dimensions of representations in a reducible induced representation | Certainly for the finite groups of Lie type there is a general technology for treating induced characters from parabolics, based on Deligne-Lusztig theory. Even without getting fully into that story, Lusztig's methods yield (recursively) the degrees of irreducible characters. The 1985 book by Roger Carter *Finite Group... | 5 | https://mathoverflow.net/users/4231 | 67612 | 41,598 |
https://mathoverflow.net/questions/67622 | 12 | Could you suggest me a basic reading list on the Springer resolution? Is there a textbook I can refer to? Or do I need to start with the original paper?
Unfortunately googling for "Springer" and "resolution" was not very helpful so far, due to the existence of a certain publisher called Springer which, from some stra... | https://mathoverflow.net/users/5420 | Literature on the Springer resolution | I would highly recommend chapter 3 from Chriss and Ginzburg's textbook "Representation Theory and Complex Geometry" (really, the entire book is worthy of recommendation).
In a very similar vein, I would also recommend Ginzburg's article "Geometric methods in the representation theory of Hecke algebras and quantum gro... | 13 | https://mathoverflow.net/users/916 | 67626 | 41,603 |
https://mathoverflow.net/questions/67585 | 4 | Say $\mathbb{C}^d \subset Y^{N-k} \subset \mathbb{C}^N$ are closed imbeddings of complex analytic subvarieties of the indicated dimension, **$Y$ is not smooth**. At a point $y \in Y$, a generic, sufficiently small polydisc $\mathbb{D}^k \ni y$ will satisfy $\mathbb{D}^k \cap Y = y$. I would like to do this continuously... | https://mathoverflow.net/users/4707 | Transversals to singular subvarieties | I don't think this can be done.
It seems to me that your $\widetilde U$ would be a submanifold of $\mathbb C^N$, so it should be a local complete intersection and then $U$ would be a local complete intersection in $Y$. However, that does not have to be the case.
Let's say that $N=3$, $d=1$, $k=1$. Or even more sp... | 2 | https://mathoverflow.net/users/10076 | 67628 | 41,605 |
https://mathoverflow.net/questions/67611 | 4 | Let $V$ be a finite-dimensional irreducible representation (complex or $\ell$-adic) of a group $G$ (compact Lie group or algebraic group etc.). Does there always exist a linear character $\rho$ of $G$, such that $V\otimes\rho$ is a self-dual irrep. of $G?$ Namely $V\otimes\rho\simeq(V\otimes\rho)^\*.$ If not, is there ... | https://mathoverflow.net/users/370 | self-dual representations | If you want a representation $V$ to be self-dual up to a character, then either $S^2V$ or $\Lambda^2V$ (considered as representations of $G$) should have a 1-dimensional summand (corresponding to the isomorphism $V \to V^\*\otimes\chi$). But as it was mentioned by Jim there are a lot of representations $V$ for which bo... | 12 | https://mathoverflow.net/users/4428 | 67633 | 41,606 |
https://mathoverflow.net/questions/42327 | 8 | Is there a category theoretic definition for the Fourier transform using only its universality properties? I am not looking for the most general definition -- one that works only in some special settings will do. I am looking for a simple definition that will make precise my (possibly incorrect) intuition that Fourier ... | https://mathoverflow.net/users/4048 | Universal definition of Fourier transform | For me, the "traditional Fourier Transform" is a **change of basis** of the algebra of functions from a *group* to some chosen field: from the canonical basis to something sometimes called the Fourier Basis. Because the Transform is constructed using the representation theory of the group, it has "natural" generalisati... | 9 | https://mathoverflow.net/users/12793 | 67639 | 41,609 |
https://mathoverflow.net/questions/66103 | 15 | Is there a known example of a countable discrete group G whose full group C\*-algebra C\*(G) is not quasidiagonal?
Let us recall that a separable C\*-algebra A is quasidiagonal if it admits a faithful
\*-representation $\pi:A \to L(H)$ on a separable Hilbert space with the property
that there is an increasing sequen... | https://mathoverflow.net/users/15392 | Discrete groups G whose full C*-algebra C*(G) is not quasidiagonal? | I don't know much about the subject, but isn't this an example? There is a discrete property (T) group $G$ without nontrivial finite dimensional representations [2, Remark in the last page]. Then, its max C\*-algebra contains a non-QD algebra as its direct summand, namely $C^\*(G)(1-z)$ for the Kazhdan projection $z$ (... | 5 | https://mathoverflow.net/users/9942 | 67646 | 41,611 |
https://mathoverflow.net/questions/67656 | 3 | Let $X$ be a projective surface and let $x\in X$ be a smooth point. Consider the blow up $Bl\_{x}X$ of $X$ in $x$, and let $E$ be the exceptional divisor. Suppose we know that $E$ is the only (-1)-curve in $Bl\_{x}X$, so that $\alpha(E) = E$ for any $\alpha\in Aut(Bl\_{x}X)$. Then any automorphisms $\alpha\in Aut(Bl\_{... | https://mathoverflow.net/users/14514 | Lifting automorphisms on a Blow-up surface | Let $\sigma:X\to X$ be an automorphism fixing a point $x\in X$. Then since the inverse image of $x$ is a Cartier divisor in both $Bl\_x X$ and $Bl\_x \sigma(X)$, by the universal property of blow-ups, we get an automorphism $\tilde{\sigma}$ of the blow-up $Bl\_x X$. Since $\tilde{\sigma}$ restricts to $\sigma$ on the c... | 9 | https://mathoverflow.net/users/3996 | 67657 | 41,618 |
https://mathoverflow.net/questions/67653 | 1 | I am trying to solve the system of differential equations
dx/dt = (y-x)/(x+y), dy/dt = -y/(x+y), where x and y are functions of t, and x(0)=0, y(0)=c (positive constant). I would like to find x and y explicitly in terms of t.
This problem is related to my previous question "Probability problem with solution involvin... | https://mathoverflow.net/users/15754 | System of ordinary differential equations | Continuing Michael's answer: Solve $dy/dx = y/(x-y), y(0)=c$ to get (in terms of the Lambert W function) $y(x) = \operatorname{e} ^{W(-x/c) + \mathrm{log} (c)}$. Substitute this into the original equations to get a single equation:
$$
\frac{d x (t)}{d t} = \frac{\operatorname{e} ^{\Bigl(W \Bigl(-\frac{x (t)}{c}\Bigr) +... | 1 | https://mathoverflow.net/users/454 | 67660 | 41,620 |
https://mathoverflow.net/questions/67658 | 9 | Suppose $Y$ is a smooth hypersurface in projective space $\mathbb{P}^n$, $X = \mathbb{P}^n - Y$ is the hypersurface complement. Is there a general method to compute cohomology of $X$? In particular, for small n, is there any examples or references?
| https://mathoverflow.net/users/8932 | Cohomology of Hypersurface complement | This is **very** computable, using several methods. I assume you are over $\mathbb{C}$ and
that by cohomology
you mean singular cohomology, but other choices are also just as straight forward.
Let $P=\mathbb{P}^n$.
Then by the Gysin sequence
$$ \ldots H^i(P)\to H^i(X)\to H^{i-1}(Y)\to H^{i+1}(P)\ldots $$
you can basica... | 14 | https://mathoverflow.net/users/4144 | 67662 | 41,621 |
https://mathoverflow.net/questions/67590 | 5 | Given a factor *M* (=von Neumann alg. with center ℂ), let us write *BIM* for the ⊗-*C*\*-category of *M*-*M*-bimodules.
>
> Which ⊗-*C*\*-categories can one faithfully embed into *BIM*?
>
>
>
⓵ Are there ***necessary*** conditions for a ⊗-*C*\*-category to be representable in *BIM*?
⓶ Are there ***sufficien... | https://mathoverflow.net/users/5690 | representing tensor-C*-categories in BIM | I don't know about necessary conditions, but here are some results concerning sufficient conditions:
* In MR1749868, Hayashi and Yamagami realize amenable $C^\*$-tensor categories in the category of bifinite (Jones index) bimodules of the hyperfinite $II\_1$-factor.-
* In arXiv:0811.1764v4, Stefaan Vaes and Sébastien... | 5 | https://mathoverflow.net/users/351 | 67669 | 41,626 |
https://mathoverflow.net/questions/67641 | 2 | Consider a doubly non-negative matrix $A$ of order $n$. $A$ is completely positive if and only if $A$ can be factorized into $BB^{T}$ where all entries in $B$ are non-negative. $B$ is $n\times k$. The smallest possible value of $k$ is the cp-rank of $A$. If $r$ is the rank of $A$ then $k\geq r$. Consider such a factori... | https://mathoverflow.net/users/39663 | Condition for doubly non-negative matrices to be completely positive | If I understand correctly, the anser is yes. A completely positive $n\times n$ matrix can always be viewed as the gram matrix of some vectors in the nonnegative orthant of some $R^k$ and vice versa. The smallest such $k$ is another way of defining the cp-rank.
The existence of an $n\times n$ matrix $A$ whose cp-rank ... | 1 | https://mathoverflow.net/users/5963 | 67678 | 41,629 |
https://mathoverflow.net/questions/67640 | 8 | I wonder that whether there exists a version of the inverse function theorem for smooth maps from a smooth manifolds with boundary to a smooth manifold without boundary? More precisely, whether the following assertion is true?
Let $M$ be a smooth manifold with boundary $\partial M$ and $N$ be a smooth manifold withou... | https://mathoverflow.net/users/14462 | Inverse function theorem for manifolds with boundary as the domain | Note that, the question being local you can work in local charts. Also, recall that, by definition of manifold with boundary, and by definition of smooth maps between manifolds with boundary, you can assume w.l.o.g. that $f$ is the restriction to $U:=V\cap H$ of a $C^1$ map $\tilde f$ defined on a nbd $V$ of $x:=0\in\m... | 4 | https://mathoverflow.net/users/6101 | 67682 | 41,633 |
https://mathoverflow.net/questions/67618 | 3 | Let $S, S\_1$ be subsets of the positive numbers $\mathbb{N}$. We say (as usual) that $S$ is multiplicative closed
if $x \in S$ and $y \in S$ implies $xy \in S.$ We say also that $S\_1$ is arithmetic closed if
$x \in S\_1$ and $y \in S\_1,$ and $\gcd(x,y)=1,$ implies $xy \in S\_1.$
Let $T$ be a subset of the positive... | https://mathoverflow.net/users/11016 | Arithmetic closed subsets | It does not matter much if one includes 1 or not. In the case that $T$ is finite and the members are square free you have the question: Given a family $T$ of subsets of a finite set $U$, what can we say about the smallest subset of the power-set $2^U$ which is closed under disjoint unions? I'm not sure how much there i... | 2 | https://mathoverflow.net/users/8008 | 67685 | 41,635 |
https://mathoverflow.net/questions/67055 | 2 | Is it possible to construct an algorithm which takes as input a pushdown automaton $M$ along with the information that the language accepted by this automaton $L(M)$ is a deterministic context-free language and outputs a deterministic pushdown automaton $N$ which accepts precisely the language accepted by $M$?
An equ... | https://mathoverflow.net/users/15615 | Given a PDA M such that L(M) is in DCFL construct a DPDA N such that L(N) = L(M) | Undecidable, see here: <https://cstheory.stackexchange.com/questions/6947/given-a-pda-m-such-that-lm-is-in-dcfl-construct-a-dpda-n-such-that-ln-lm>
| 0 | https://mathoverflow.net/users/15615 | 67692 | 41,640 |
https://mathoverflow.net/questions/67524 | 6 | Just my curiosity... Are there proofs the following fact, which does not involve Hall's matching theorem:
A group $\Gamma$ is amenable if and only if it does not admit a paradoxical decomposition.
**Def:** A group $\Gamma$ has a **paradoxical decomposition** if there are pairwise disjoint subsets $F\_1,\ldots, F\_n... | https://mathoverflow.net/users/8699 | a paradoxical decomposition of a group | According to a note in Grigorchuk's and Sunic's Self-Similarity and Branching in Group Theory, there is a proof not using the matching theorem in the book The Banch-Tarski Paradox by Stan Wagon.
By the way, you have to mention that $\Gamma$ is also the union of $E\_1,...,E\_m,F\_1,...,F\_n$.
| 3 | https://mathoverflow.net/users/5048 | 67697 | 41,643 |
https://mathoverflow.net/questions/67695 | 1 | I am afraid to continue to ask trivial things but really I do not know how to proceed, so I ask the experts:
A specially multiplicative function, is a function
$f$ from positive integers to complex numbers,
that satisfies:
$$
f(n)f(m) = \sum\_{d \mid \gcd(m,n)} f(\frac{mn}{d^2})g(d)
$$
for all $m,n$, where $g$ is a c... | https://mathoverflow.net/users/11016 | Explicit formula for completely multiplicative functions $a,b$ such that $\tau = a * b$; $\tau$ is Ramanujan's tau function, and $*$ is the Dirichlet convolution. | I don't understand your definition of "specially multiplicative"; if $m$ and $n$ are coprime, it gives $f(m)f(n)=mn$, which is not true of the sum of divisor function...
However, the Ramanujan $\tau$-function satisfies $\tau(m)\tau(n)=\sum\_{d\mid (m,n)}{\mu(d)\tau(mn/d^2)}$, which might be what you mean?
In any ca... | 7 | https://mathoverflow.net/users/20038 | 67698 | 41,644 |
https://mathoverflow.net/questions/67704 | 8 | It is easy to show, using the axiom of Zorn, that there exists a transcendence basis for $\mathbb{R}/\mathbb{Q}$, i.e. a set $S$, algebraically independent over $\mathbb{Q}$, such that $\mathbb{R}/\mathbb{Q}(S)$ is an algebraic extension.
What can we say about $T=\mathbb{R} - \mathbb{Q}(S)$? It is easy to show that $... | https://mathoverflow.net/users/6779 | What is a "best" transcendence basis for R/Q ? | No, you cannot choose $T$ so nicely. For cardinality reasons, we can choose some $s\in S$ with $s\notin \overline{\mathbb{Q}}$. Now, consider the element $\sqrt{s}$. I'll leave it to you to consider how large this makes $T$.
| 8 | https://mathoverflow.net/users/3199 | 67705 | 41,648 |
https://mathoverflow.net/questions/67709 | 8 | In his "Algebraic Geometry", Hartshorne proves that for any ringed spaces $(X,\mathcal O\_X)$, category $Mod(X)$ of sheaves of $\mathcal O\_X$-modules has enough injectives. If we take $\mathcal O\_X$ to be constant *sheaf* of rings $\underline{\mathbb Z}\_X^{\natural}$ (i.e. sheaf associated to a constant presheaf $\u... | https://mathoverflow.net/users/15292 | Sheaves of $\mathbb Z$-modules = sheaves of abelian groups | $\def\sh#1{\mathcal{#1}}\def\csheaf#1{\underline{#1}}\def\on#1{\operatorname{#1}}$First of all, if $X$ is an irreducible scheme (or any such topological space), then all of its open subsets are connected and there are no complications such as you describe. However, the complications dry up under close examination no ma... | 10 | https://mathoverflow.net/users/6545 | 67710 | 41,649 |
https://mathoverflow.net/questions/67671 | 4 | Hi everybody!
I am looking for results about how to bound from above the number of prime factors of the order of a non-abelian simple group $S$ in terms of, say, the index of a subgroup of $S$. I call $\omega(x)$ the number of prime factors of the integer $x$. For alternating groups everything is very simple, and we ... | https://mathoverflow.net/users/5710 | Number of prime factors of the order of a finite non-abelian simple group | To help searching: ω(|*G*|) = |π(*G*)|, and I see the latter usually.
A finite simple group *G* with |π(*G*)| = 1 must be cyclic of order *p*. By Burnside's *p**a**q**b* theorem, if |π(*G*)|=2, then *G* is not simple.
The finite simple groups with |π(*G*)| = 3 were handled in several specific cases are handled by B... | 15 | https://mathoverflow.net/users/3710 | 67715 | 41,650 |
https://mathoverflow.net/questions/67723 | 2 | Is it true that an irreducible component of a Cohen-Macaulay variety is also Cohen-Macaulay? If not, then in what cases does this fact hold?
| https://mathoverflow.net/users/15764 | Irreducible component of a Cohen-Macaulay variety | Start with your favorite example of an affine irreducible variety $X$ that is not Cohen-Macaulay. Embed $X$ in $\mathbb A^N$, and call $c$ its codimension. Now take $c$ general polynomials that contain $X$: their intersection $Y$ has codimension $c$, and contains $X$. Then $Y$ is a complete intersection (hence Cohen-Ma... | 12 | https://mathoverflow.net/users/4790 | 67725 | 41,655 |
https://mathoverflow.net/questions/67722 | 4 | What is vector multiplet and hypermultiplet moduli space associated to IIA/B string theory (or in general to a N = 2 Supersymmetric theory) ?
The vector multiplet moduli space is special Kahler while hypermultiplet moduli is hyperkahler. It seems that the vector multiplet is the moduli of the Calabi - Yau on which th... | https://mathoverflow.net/users/9534 | vector multiplet/hypermultiplet moduli space of String Theory | The answer depends on precisely which string theory you study, and whether you consider compact or non-compact Calabi-Yaus. Let me focus on the case of compact Calabi-Yaus.
First, let's consider Type IIB string theory on a compact Calabi-Yau threefold $X$.
The "easy" part of the answer is that the vector multiplet ... | 11 | https://mathoverflow.net/users/580 | 67734 | 41,660 |
https://mathoverflow.net/questions/67706 | 6 | ### The DGA
For $k$ some field, let $R$ be a $k$-algebra, and let $r\in R$.
Define a differential graded algebra $\mathbf{R}\_r$ as follows. As a graded algebra, it is isomorphic to $R\langle t\rangle$, the free $k$-algebra over $R$ with non-central variable $t$ added. The algebra $R$ has degree zero, and $t$ has d... | https://mathoverflow.net/users/750 | A potential resolution of $R/r$ | This complex (in simultaneously a more general setting, where you have several elements $t\_1,\ldots,t\_p$, and a more special setting, because only the case of $R$ being a free algebra was studied then) was introduced by Shafarevich [E. S. Golod, I. R. Shafarevich, “On the class field tower”, Izv. Akad. Nauk SSSR Ser.... | 6 | https://mathoverflow.net/users/1306 | 67738 | 41,661 |
https://mathoverflow.net/questions/67699 | 4 | Disclaimer: I know very little about Shimura varieties.
Some Shimura varieties have a contractible universal covering space, for instance $A\_g$ itself. Are there any nice necessary and/or sufficient conditions implying this?
| https://mathoverflow.net/users/1310 | When does a Shimura variety have contractible universal cover? | Connected Shimura varieties are quotients $S=X/\Gamma$, where $X$ is a Hermitian symmetric space without compact factor and $\Gamma$ is a discrete subgroup acting properly discontinuously on $X$. If $\Gamma$ acts without fixed points, then $X\to S$ is a universal covering. (In the general case, the universal covering o... | 11 | https://mathoverflow.net/users/10696 | 67740 | 41,662 |
https://mathoverflow.net/questions/67739 | 6 | I want to ask a question about a statement that I found on the paper: Principal Eigenvalues for Problems With indefinite Weight Function in $R^N$.
The statement is the following:
Suppose that $g:\mathbb{R}^2\to\mathbb{R}$ is a $C^{\infty}$ function which changes sign on $\mathbb{R}^2$ and there exist constants $K... | https://mathoverflow.net/users/2386 | How to prove this Poincare Inequality | Just assume that $g$ is a bounded function with $\int\_B g < 0$, and positive somewhere in th ball $B$ . Then the set
$$\Big \{u\in H^1(B)\, : \|u\|\_2= 1\, ,\, \int\_B g u^2\ge 0 \Big \}$$
is not empty and does not contain constant functions. By weak compactness $\|\nabla u\|\_2$ attains a non-zero minimum on it, and... | 8 | https://mathoverflow.net/users/6101 | 67742 | 41,664 |
https://mathoverflow.net/questions/67721 | 1 | Hartshorne gave the exact sequences in Chapter II.8 and just say they followed from the affine case. But this view should consider the compkex glueing construction. I am disturbed completely. How should I thunk this? Thanks!
| https://mathoverflow.net/users/3525 | Question about the exact sequences of sheaves of relative differentials | I think the usual arguments "by gluing" can be replaced by more conceptual, global arguments. Namely, the sheaf of differentials has itsself a universal property, classifying global derivations. Actually this works for every morphism of (locally) ringed spaces. Then you can just write down the proof for rings also for ... | 2 | https://mathoverflow.net/users/2841 | 67744 | 41,665 |
https://mathoverflow.net/questions/67736 | 16 | I am looking for a good text book on Matroid theory. Ideally, one that might be better suited to engineers than pure mathematicians...but any book that is well written/organized would do.
I have just started looking into this area and am working through some survey papers (Wison's "An Introduction to Matroid Theory" ... | https://mathoverflow.net/users/4677 | Good introductory text book on Matroid Theory? | My first recommendation would be Oxley's [Matroid Theory](http://ukcatalogue.oup.com/product/9780199603398.do). The second edition was just released this year (19 years after the original), so this is a very 'modern' textbook.
Another option would be Welsh's [Matroid Theory](http://rads.stackoverflow.com/amzn/click/0... | 10 | https://mathoverflow.net/users/2233 | 67753 | 41,669 |
https://mathoverflow.net/questions/67756 | 3 | Let G be a group and g and h be elements in G. If g commutes with all conjugates of g and h commutes with all conjugates of h, can one conclude that gh commutes with all conjugates of gh?
Thanks!!
| https://mathoverflow.net/users/15770 | Question about some element in a group commutes with its all conjuagates. | No; the smallest counterexamples are given by the groups SmallGroup(54,5) and SmallGroup(54,6) (in GAP's SmallGroups library); these are groups of the form
$$G\_1 = ((3 \times 3) : 3) : 2 \text{ and } G\_2 = (9 : 3) : 2.$$
In both cases, there are 15 elements with the given property (i.e. that they commute with all the... | 9 | https://mathoverflow.net/users/12858 | 67759 | 41,673 |
https://mathoverflow.net/questions/67727 | 2 | Hello! I wonder how hard is it to implement more or less general symbolic integration algorithm (number of lines in a certain language)? Maybe someone here did this or knows some good blog posts devoted to the subject.
And how hard is it to implement an algorithm to decide whether a given function could be integrated... | https://mathoverflow.net/users/14251 | The easiest symbolic integration method to try implementing. | Perhaps pmint, The Poor Man's Integrator,
<http://www-sop.inria.fr/cafe/Manuel.Bronstein/pmint/index.html>
is of interest to you (<100 lines Maple).
The page contains also references to the literature and examples.
Moreover, searching for Poor Man's Integrator turns up a couple of related and recent online disc... | 5 | https://mathoverflow.net/users/nan | 67767 | 41,677 |
https://mathoverflow.net/questions/67763 | -2 | Hi,
is the space $C\_0(\mathbb{R}^m)$, $m \in \mathbb{N}$ of continuous functions with compact support separable? If yes: where can I find a proof for that?
Please note: this is not a duplicate of this question [(click)](https://mathoverflow.net/questions/54026/separability-of-a-certain-space-of-continuous-function... | https://mathoverflow.net/users/15771 | Separability of continuous functions with compact support | Each $C([-N,N]^m)$ is separable, and so is the subspace $X\_N\subseteq C([-N,N]^m)$ consisting of functions which vanish on the boundary $\partial[-N,N]^m$. Take a countable dense subset of each $X\_N$, and take the union over all integers $N$. This is then a countable dense subset of $C\_0(\mathbb R^m)$.
| 5 | https://mathoverflow.net/users/35353 | 67769 | 41,678 |
https://mathoverflow.net/questions/67521 | 9 | There are various "[concentration-of-measure](http://en.wikipedia.org/wiki/Concentration_of_measure)" theorems,
the best known that due to Lévy,
which is this (informally): the volume of a sphere $S^d$ in $d$ dimensions is largely
concentrated around an $\epsilon$-tubular neighborhood of an equitorial hyperplane $H$.
... | https://mathoverflow.net/users/6094 | Concentration of measure for arbitrary convex bodies? | There are many results, and an active research industry, along these lines. In general the Euclidean ball is the best-behaved convex body in this respect, and just how similar an arbitrary convex body is depends on how you try to make your question more precise.
Here's one very general result, a special case of what is... | 11 | https://mathoverflow.net/users/1044 | 67770 | 41,679 |
https://mathoverflow.net/questions/67772 | 7 | Can structure theorem for modules be extended to modules over UFDS , to modules over Neotherian rings ? if yes then can one get the statement and reference?
Since operations on matrices with coefficients as polynomials in several variables some extension seems possible .
| https://mathoverflow.net/users/15700 | structure theorem for modules | Let me start by something classical: extending the classical result for PIDs, by Steinitz's Theorem (1912) all finitely generated modules over Dedekind domains are characterized, see <http://en.wikipedia.org/wiki/Dedekind_domains> and scroll down.
Beyond Dedekind domains things get complicated, but there is considera... | 6 | https://mathoverflow.net/users/nan | 67778 | 41,683 |
https://mathoverflow.net/questions/67776 | 1 | This is just a stupid question about a good terminology. I'm interested in sequences $a\_n$ with a growth that can be bounded by an arbitrarily small positive power of $n!$, i.e. for every $\epsilon > 0$ there should be a constant $c$ with $|a\_n| \le c (n!)^{\epsilon}$. I wanted to call this growth "sub-factorial" but... | https://mathoverflow.net/users/12482 | Notation for growth $a_n \le c (n!)^\epsilon$ | In the need of a name, I would also go for "sub-factorial growth". In this case I see no danger of confusion; "sub-linear" and "sub-exponential growth" are already used in analogous meaning (even if not completely standard). Also note that in your condition you can replace $n!$ with $n^n$ (which unfortunately didn't gi... | 1 | https://mathoverflow.net/users/6101 | 67788 | 41,688 |
https://mathoverflow.net/questions/67762 | 15 | Does there exist an integer $N$ such that every set of $\geq N$ points in $\mathbb R^4$
contains six distinct points which are vertices of two intersecting triangles?
More generally, given dimensions $d\_1,\dots,d\_k$ such that generic affine subspaces
of $\mathbb R^d$ of dimensions $d\_1,\dots,d\_k$ intersect in a p... | https://mathoverflow.net/users/4556 | An Erdős-Szekeres-type question | The answer to the first question is yes (with $N = 7$). Consider a $2$-dimensional simplicial complex $K$ with 7 vertices and with all possible triangles. Assume there are 7 points in $\mathbb R^4$. You can construct a linear mapping $f \colon |K| \rightarrow \mathbb R^4$ ($|K|$ denotes the geometric realization of the... | 13 | https://mathoverflow.net/users/15650 | 67800 | 41,695 |
https://mathoverflow.net/questions/67751 | 10 | (Note: I asked this question a few days ago on [math.stackexchange](https://math.stackexchange.com/questions/44754/seeking-rationale-for-hadamards-finite-part-of-a-divergent-integral) but didn't get any responses. I've therefore decided to post it here instead.)
I have a problem justifying throwing away the divergent... | https://mathoverflow.net/users/7486 | Rationale for Hadamard's finite part of a divergent integral | This can be viewed as a meromorphic continuation in s of the distribution which is integration against |x-t|^s in t. M. Riesz (1938)first observed that this is a meromorphic continuation of convergent integrals, Gelfand-Shilov (1958) formalized this in the context of Schwartz' distributions. Gelfand-Graev's volume I di... | 12 | https://mathoverflow.net/users/15629 | 67802 | 41,697 |
https://mathoverflow.net/questions/67747 | 9 | Ok, so this might be a really naive question (and clearly related to [Special values of Artin L-functions](https://mathoverflow.net/questions/41358/special-values-of-artin-l-functions)).
The Stark conjecture postulates that all Artin L-functions has a transcendental (over $\mathbb{Q}$) leading coefficient at $s=1$ (... | https://mathoverflow.net/users/2147 | Special values of Artin L-function | Ok so I think I have worked out which integers are critical. Let $F$ be a finite Galois extension of $\mathbf{Q}$ and $\rho:\operatorname{Gal}(F/\mathbf{Q}) \to GL(V)$ be an irreducible Artin representation, with $\rho \neq 1$.
First I should mention that there is a functional equation relating $L(\rho,s)$ and $L(\ov... | 8 | https://mathoverflow.net/users/6506 | 67804 | 41,699 |
https://mathoverflow.net/questions/67809 | 7 | $\def\mc#1{\mathcal#1}\def\seq#1{\langle#1\rangle}\def\bbR{\mathbb R}\def\gt{>}\def\dom{{\rm dom\ }}$In some instances, I have seen an appeal to the concept of "the double" of a smooth manifold with non-empty boundary. [Wikipedia](http://en.wikipedia.org/wiki/Double_%28manifold%29) gives a pure nonsense for this: "Prec... | https://mathoverflow.net/users/12643 | The double of a smooth manifold with boundary? | I believe that the usual remedy is a collar. That is, for any smooth manifold there is a suitable diffeomorphism from a neighborhood of $\partial M$ to $[0,1)\times \partial M$, or in other words a smooth embedding $[0,1)\times \partial M\to M$ that is "the identity" on the boundary. This allows you to glue along the b... | 19 | https://mathoverflow.net/users/6666 | 67813 | 41,705 |
https://mathoverflow.net/questions/67663 | 1 | Here is an unsolved problem for me in Kaplansky's "Commutative rings" p. 103, no. 18.
Let $R$ be a Cohen-Macaulay ring. Let $T$ be a ring containing $R$ and suppose that as an $R$-module it is free and finitely generated. Show that $T$ is also a Cohen-Macaulay ring.
| https://mathoverflow.net/users/13351 | Cohen Macaulay, free and finitely generated module | Let $\mathfrak p\in\mathrm{Spec}T$ and $\mathfrak q=\mathfrak p\cap R\in\mathrm{Spec}R$. Since $T$ is finite and hence integral over $R$, the dimension of $T\_{\mathfrak p}$ is the same whether considered as a ring, a module over itself or a module over $R\_{\mathfrak q}$. It will be denoted by $\dim T\_{\mathfrak p}$.... | 3 | https://mathoverflow.net/users/10076 | 67816 | 41,708 |
https://mathoverflow.net/questions/65810 | 7 | Recently, I have been learning about nef line bundles. I know that when $X$ is projective or Moishezon, a line bundle $L$ over $X$ is said to be nef iff $$L.C=\int\_{C}c\_{1}(L)\ge 0$$ for every curve $C$ in $X$.
Demailly gave a definition of nefness that works on an arbitrary compact complex manifold, i.e., a line b... | https://mathoverflow.net/users/11850 | Are these two definitions of nef-ness equivalent for Moishezon manifolds? | Yes, the equivalence is true (the second notion used to be called "metric nef" by some). This was an open problem for quite some time until it was solved in
M. Paun "Sur l'effectivité numérique des images inverses de fibrés en droites" Math. Ann. 310 (1998), no. 3, 411–421, see the Corollaire on page 412.
| 4 | https://mathoverflow.net/users/13168 | 67821 | 41,710 |
https://mathoverflow.net/questions/67812 | 28 | Of course not.
But after reading a bit, some points make me believe it should be:
Let $S$ be a nice$^{\\*}$ surface defined over $Spec\ \mathbb{Z}$.
1. The Brauer group $Br(S\otimes \bar{\mathbb{Q}})$ is an abelian divisible group,
2. It is also a $Gal(\bar{\mathbb{Q}}/\mathbb{Q})$ module,
3. For good primes ther... | https://mathoverflow.net/users/2024 | Is the Brauer group of a surface an elliptic curve? | This is not a general answer to your question, but evidence of the intriguing connection between Brauer groups of surfaces and elliptic curves. Let $X$ be a K3 surface over the complex numbers $\mathbb{C}$. Then, the rank of $H^2(X,\mathbb{Z})$ is $22$, and the Hodge numbers are $h^{0,2}=h^{2,0}=1$ and $h^{1,1}=20$. If... | 16 | https://mathoverflow.net/users/100 | 67823 | 41,712 |
https://mathoverflow.net/questions/67029 | 8 | Let $B\_t(0)$ denote the $n$ dimensional ball of radius $t$ centered at the origin. Does there exist a $\phi\in C(\mathbb{R}^n)$ function with the properties:
$
\phi (x) =
\begin{cases}
1&x\in B\_r(0)
\\\
0&x\not\in B\_{r+3}(0)
\end{cases}
$
and for any real-valued function $f\in \mathcal{H}^\tau(\mathbb{R^n})$ ($\... | https://mathoverflow.net/users/2011 | Extension theory with bump function | **Short answer: yes.**
Let $\psi\_\epsilon(x):=\frac{1}{\epsilon^n}\exp{\epsilon^2/(\epsilon^2-|x|^2)}$ for $|x|<\epsilon$, and $\psi\_{\epsilon}(x)=0$ for $|x|\geq \epsilon$. Set $\epsilon=2$, and define
$\phi$ is the convolution of $C\phi\_{\epsilon}$ with the characteristic function of $B\_{r+3/2}(0)$, that is,
... | 4 | https://mathoverflow.net/users/14740 | 67825 | 41,713 |
https://mathoverflow.net/questions/67757 | 0 | It is a known result that if $B$ is an $n$ braid over a disk, then $B$ naturally induces an isomorphism between the fundamental group of a disk with n points removed and the fundamental group of the space $D\times [0,1]-B$, where $D$ is a disk. My question is, in which book/paper can I find a proof of this result?
| https://mathoverflow.net/users/15770 | Where can one find reference proving that Braid group induces isomorphism between punctured disk and the complement of the braid? | The isomorphism of fundamental groups comes from a diffeomorphism of spaces: $D\times[0,1]\setminus B$ is diffeomorphic to the product of an n-punctured disk with $[0,1]$. To see this, note that you can untangle the braid by sliding the ends of the braid along the surface of $D\times[0,1]$. This sliding is not a braid ... | 2 | https://mathoverflow.net/users/9417 | 67835 | 41,720 |
https://mathoverflow.net/questions/67808 | 2 | I am reading the Functional Analysis book of Conway, one question from the book is find a closed subspace M of $l^{\infty}=l^{\infty}(\mathbb{N})$ with the property that $l^{\infty}/M$ is separable. I have found a solution for this but here is my question :
1. Is it true that every non-separable normed space $X$ alwa... | https://mathoverflow.net/users/5136 | Separable quotients of non-separable Banach spaces? | Your question is the famous "separable quotient problem", as Ady mentioned. From here on, "space" means "infinite dimensional Banach space". A space $X$ has a separable quotient provided $X^\*$ has a reflexive subspace (obvious), a subspace isomorphic to $c\_0$ (Rosenthal and me), or $\ell\_1$ (Hagler and me). A result... | 14 | https://mathoverflow.net/users/2554 | 67836 | 41,721 |
https://mathoverflow.net/questions/67831 | 5 | Given a monoidal category $M$ one can consider its Drinfeld centre $Z(M)$. Objects of the Drinfeld centre are pairs $(m, \alpha)$ where $m$ is an object and $\alpha$ is an isomorphism $\alpha: - \otimes m \to m \otimes -$ satisfying some "obvious" conditions.
A simple and important example of a monoidal category is t... | https://mathoverflow.net/users/919 | Morita invariance of Drinfeld centre | To expand on Noah's answer, Müger shows that if two fusion categories are Morita equivalent, in the sense that there is an invertible bimodule between them, then their Drinfeld centers are equivalent. In fact, the reverse implication is also true: this is Theorem 3.1 in [a different Etingof-Nikshych-Ostrik paper](http:... | 9 | https://mathoverflow.net/users/396 | 67837 | 41,722 |
https://mathoverflow.net/questions/15038 | 9 | What is the Kullback-Leibler divergence of two Student's T distributions that have been shifted and scaled? That is, $\textrm{D}\_{\textrm{KL}}(k\_aA + t\_a; k\_bB + t\_b)$ where $A$ and $B$ are Student's T distributions.
If it makes things easier, $A$ could be a Gaussian. (That is, it could have infinite degrees of ... | https://mathoverflow.net/users/634 | Kullback-Leibler divergence of scaled non-central Student's T distribution | I guess I can't leave comments just yet so this will have to be in the form of an answer:
Assuming that the distributions have the same number of degrees of freedom ($n$), I think the answer will look something like (with some abuse of notation)
$\mathcal{H}(k\_aA + t\_a \mid k\_bB + t\_b) = \log(\frac{k\_b}{k\_a})... | 2 | https://mathoverflow.net/users/15752 | 67841 | 41,725 |
https://mathoverflow.net/questions/67843 | 2 | I have been working with p-adically closed fields and there are two results that are used time and times again in what I am reading, but I cannot find any references where they are proved...
The first is that p-adically closed fields have a finite number of algebraic extensions of a given degree, the other that all a... | https://mathoverflow.net/users/15587 | Algebraic extensions of p-adic closed fields | Both the results you ask about, in the case of $p$-adic fields, follow from Krasner's lemma.
See for example, Proposition 3 and 4 of Lang, Algebraic Number Theory, p.43,44.
The main point
is that two irreducible polynomials which are sufficiently close in the $p$-adic topology have roots which generate the same exte... | 5 | https://mathoverflow.net/users/519 | 67845 | 41,727 |
https://mathoverflow.net/questions/67824 | 7 | I was studying the axioms of a category, and noted that one axiom says there is an element $1\_X\in Hom(X,X)$ for any object $X$ which serves as the identity. Why is this axiom necessary? What happens if I drop this axiom?
---
Background: I can define the category of affine holomorphic symplectic varieties, by sa... | https://mathoverflow.net/users/5420 | Why does Hom need an identity in the definition of the category? | Your structure can be described as a "category without identity", which has been given the names "semicategory" and "semigroupoid" presumably due to independent discoveries.
Some Googling suggests the term "semicategory" came first, in [a 1972 TAMS paper by Mitchell](http://www.ams.org/journals/tran/1972-167-00/S0002... | 13 | https://mathoverflow.net/users/121 | 67856 | 41,735 |
https://mathoverflow.net/questions/67873 | 2 | Since I haven't received a satisfactory answer to my initial question I'm going to ask a somewhat weaker one...
This time we say $X$ is a Vitali set in the closed interval $[0, 1]$ with respect to $\mathbb{Q}$ if $X$ is a selector of the partition of $[0, 1]$ canonically associated with the equivalence relation $x \... | https://mathoverflow.net/users/15666 | Follow up question on union of disjoint Vitali sets... | No, there does not exist such a $V$.
Let $W = V \cup (V \oplus r)$, and suppose $W = F \Delta Q$. Note that for
$s \in {\mathbb Q}$, $W \cap (W \oplus s)$ is nonempty if and only if $s$ or $s-r$ or $s+r$ is an integer. But if $F$ contained an interval of positive length, $W \cap (W \oplus s)$ would be nonempty for a... | 3 | https://mathoverflow.net/users/13650 | 67878 | 41,746 |
https://mathoverflow.net/questions/67882 | 9 | The classical Krein-Rutman theorem states that any positive compact linear endomorphism $T:X \to X$ on a Banach space $X$ with positive spectral radius $r(T)$ has an eigenvalue $r(T)$ with a positive eigenvector. Papers and textbooks seem to write off the theorem as "standard" and "well-known", but I have not been able... | https://mathoverflow.net/users/8452 | The classical Krein-Rutman theorem | ["Topological Vector Spaces"](http://books.google.com/books?id=9kXY742pABoC&printsec=frontcover&dq=topological+vector+spaces&cd=1#v=onepage&q&f=false) by Helmut Schaefer contains a thorough treatment of the classical Krein-Rutman theorem for compact positive operators in an ordered Banach space along with several gener... | 9 | https://mathoverflow.net/users/5371 | 67887 | 41,749 |
https://mathoverflow.net/questions/67888 | 0 | why if G is an abelian p-group not divisible then exists an element g in G which is not divisible by p?
thanks
| https://mathoverflow.net/users/15802 | abelian p-group not divisible | As Pace said, but with more detail:
If $G$ is an abelian $p$-group, then for any $g$ in $G$, the order of $g$ is a power of $p$, say $p^k$. Thus for any integer $n$ coprime with $p$, $n$ is a unit (mod $p^k$), so for some $m$, $nm=1$ mod $p^k$. So $n(mg)=(nm)g=(ap^k+1)g=g+a(p^kg)=g+0=g$. Thus $g$ is divisible by $n$.... | 0 | https://mathoverflow.net/users/15735 | 67890 | 41,750 |
https://mathoverflow.net/questions/67852 | 2 | How are polynomials called that are positive for all positive, real arguments, e.g., xy + z?
How can one determine if a polynomial has this property?
| https://mathoverflow.net/users/15794 | Positive polynomials | A matrix $A$ such that $x'Ax\geq 0$ for all $x$ is called positive semidefinite. A matrix $A$ such that $x'Ax\geq 0$ for all $x\geq 0$ (componentwise) is called copositive. While positive semidefiniteness is easy to test algorithmically, the weaker condition of copositivity is co-NP-complete to test.
Up to some detai... | 5 | https://mathoverflow.net/users/5963 | 67891 | 41,751 |
https://mathoverflow.net/questions/67883 | 1 | I was studying Theorem 4.4.1 from John H. Hubbard's Teichmuller Theory, vol I, Theorem 4.4.1 ( P. 129 ) which states :
Let $X,Y$ be two hyperbolic Riemann surfaces with hyperbolic metrics $d\_X,d\_Y$ respectively and let $K\geq 1 $.Then there exists a function ( homeomorphism of positive real numbers) $\delta\_K:(0,\... | https://mathoverflow.net/users/6953 | Two questions from Hubbard's Teichmuller theory book Vol I, P. 130 , Thm 4.4.1, ( QC maps ) | 1) As you said the first point is covered in prop. 4.4.6. anyway.
2) Concerning the second point, you can actually use that same proposition 4.4.6:
first the notation $d\_{D\_{r}}(0,z)$ means that you are considering the distance between $0$ and $z$ in the hyperbolic metric associated to the disk $D\_{r}$ centered ... | 2 | https://mathoverflow.net/users/15673 | 67894 | 41,753 |
https://mathoverflow.net/questions/67893 | 8 | Let $\mathcal{M}$ be a compact Riemannian manifold and let $\Delta$ be the (scalar) Laplace-Beltrami operator on $\mathcal{M}$. Then $\Delta$ has a discrete spectrum and if we order its **distinct** eigenvalues $\lambda\_i$ by magnitude then some very simple examples suggest that the magnitude of $\lambda\_i$ might be ... | https://mathoverflow.net/users/1557 | Growth of Laplacian eigenvalues on a compact domain? | [Weyl's formula](http://en.wikipedia.org/wiki/Hearing_the_shape_of_a_drum#Weyl.27s_formula):
$$N(R)=\frac{1}{(4{\cdot}\pi)^{d/2}{\cdot}\Gamma\left(\frac d2+1\right)}{\cdot}V{\cdot}R^{d/2}+o(R^{d/2}).$$
where $d$ --- dimension, $V$ --- volume, $N(R)$ --- number of eigenvalues $\le R$.
It works for any compact Riemanni... | 13 | https://mathoverflow.net/users/1441 | 67895 | 41,754 |
https://mathoverflow.net/questions/67900 | 4 | Suppose that S is a compact convex subset of the Euclidean plane E whose interior is non-empty.
If p is a point of E such that every straight line in E which passes through p bisects the area
of S, is S necessarily centro-symmetric with respect to p?
| https://mathoverflow.net/users/4423 | A question about bisecting plane convex sets | Yes: the condition is equivalent to : any straight line through $p$ intersects $S$ in a segment whose midpoint is $p$, that is the convex is center-symmetric.
$$\*$$
**Details:** Assume $p=0\in\mathbb{C}$ and let $H \_ +$ and $H \_ -$ be resp. the upper and lower half-planes. Consider the difference of the areas of t... | 3 | https://mathoverflow.net/users/6101 | 67901 | 41,757 |
https://mathoverflow.net/questions/67897 | 14 | My question may be absolutely elementary and is probably answered in 19th century. A reference or a short clear argument would be highly appreciated.
Let $V\_1, \ldots V\_n$ be finite dimensional vector spaces over the same field (may assume complex numbers). What are $GL(V\_1)\times \ldots \times GL(V\_n)$-orbits on... | https://mathoverflow.net/users/5301 | Invariants and orbits of $n$-tensors | Here is a start, suppose that $V\_i$ is $\mathbb C^{k\_i}$ (and restricting to $k\_1,k\_2,\dots,k\_n, n\geq 2$). The tuples $(k\_1,k\_2,\dots,k\_n)$ for which the action of $GL\_{k\_1}\times\cdots\times GL\_{k\_n}$ on $\mathbb{C}^{k\_1}\otimes \cdots\otimes \mathbb{C}^{k\_n}$ has only finitely many orbits are $(k,l),(2... | 8 | https://mathoverflow.net/users/2384 | 67904 | 41,758 |
https://mathoverflow.net/questions/67910 | 7 | Is it true that for any point on any compact Riemann surface there exists a global holomorphic one-form, which does NOT have a zero at that point.
| https://mathoverflow.net/users/15805 | Zeros of holomorphic one-forms on Riemann surface | On $\mathbb P^1$, there is no non zero holomorphic $1$-form, on any elliptic curves, the holomorphic forms are "constant" (the canonical bundle is trivial), so never vanish if they are not identically zero.
As for the other surfaces, namely if $g(X) \geqslant 2$, then $|K\_X|$ has no base point (cf Hartshorne, IV, le... | 13 | https://mathoverflow.net/users/5659 | 67911 | 41,761 |
https://mathoverflow.net/questions/67838 | 6 | Let $X=(V,E)$ be a finite, connected graph on $n$ vertices, endowed with its graph metric $d$. The average squared distance of $X$ is $avg(d^2)=\frac{1}{n(n-1)}\sum\_{x,y\in V,x\neq y} d(x,y)^2$; it satisfies the obvious bound $avg(d^2)\leq diam(X)^2$, where $diam(X)$ is the diameter of $X$.
Now assume that $X$ is v... | https://mathoverflow.net/users/14497 | Average squared distance vs diameter in vertex-transitive graphs | For Random 3-regular graphs \lambda is asymptotically 1. I suspect that for Ramanujan graphs it will be 1 as well?
Regarding random graphs papers to look are papers citing Bollobas and de la Vega, Combinatorica 2
(1982), 125-134.
<http://www.stanford.edu/class/msande337/notes/the%20diameter%20of%20random%20regular%20... | 2 | https://mathoverflow.net/users/15809 | 67921 | 41,764 |
https://mathoverflow.net/questions/67924 | 10 | I'm trying to find information on the eigenvalues of an $n \times n$ matrix A such that
$A = D + J$
Where $D$ is some complex valued diagonal matrix, and $J$ is an matrix consisting of all $1$'s.
When $D$ has identical values, the problem is equivalent to finding the eigenvalues of $J$.
So my question is this... | https://mathoverflow.net/users/15811 | Eigenvalues of the sum of a diagonal and a unit matrix | There is a formula. Recall that $\det(I-AB)=\det(I-BA)$ for any matrices
$A$ and $B$ such that both products $AB$ and $BA$ are defined. Now
$$
\det(tI-D-J) = \det(tI-D) \det(I-(tI-D)^{-1}J).
$$
If $u$ is the column vector with each entry equal to 1 then $J=uu^T$ and
$$
\det(I-(tI-D)^{-1}J) = \det(I-(tI-D)^{-1}uu^T) =... | 14 | https://mathoverflow.net/users/1266 | 67935 | 41,769 |
https://mathoverflow.net/questions/67929 | 1 | Joel David Hamkins in an answer to my question [Countable Dense Sub-Groups of the Reals](https://mathoverflow.net/questions/67259/countable-dense-sub-groups-of-the-reals) points out that "one can find an uncountable chain of countable dense additive subgroups of $\mathbb{R}$ whose subset relation has the order type of ... | https://mathoverflow.net/users/15666 | Cardinality of the set of countable dense subgroups of the reals up to isomorphism. | The result of Joel quoted above certainly shows a complicated structure, but does not actually provide continuum many non-isomorphic subgroups since there is no reason why $G$ being a subgroup of $H$ should imply that $G$ and $H$ are not isomorphic. Indeed, considering two groups generated by the rationals and two infi... | 5 | https://mathoverflow.net/users/13878 | 67936 | 41,770 |
https://mathoverflow.net/questions/67889 | 5 | Let $n$ be a growing integer parameter, and suppose that $X\_1,\dotsc,X\_n$ are independent Bernoulli random variables with the probabilities of success $p\_i:={\mathsf P}(X\_i=1)$. If $X=X\_1+\dotsb+X\_n$ then, trivially, ${\mathsf
E}(\sqrt X)\le\sqrt{np}$, where $p=(p\_1+\dotsb+p\_n)/n$. When can one expect ${\mathsf... | https://mathoverflow.net/users/9924 | The expectation of $\sqrt{B(n,p)}$ | Having carefully checked the things, I am not really interested in the case where $pn\to 0$; and, when $pn\gg 1$, the observations of camomille and Ori resolve the problem, leading to ${\mathsf E}(\sqrt X)\gg\sqrt{np}$. Thanks to all those who have replied!
| 0 | https://mathoverflow.net/users/9924 | 67938 | 41,772 |
https://mathoverflow.net/questions/67902 | 0 | Can anyone point me to any proofs (pref. interesting ones!) that make good (or bad) use of the Finite or Infinite Priority Injury Argument?
**Edit:** I would suppose that my question could be put this way too. Are priority injury method proofs limited to recursion theory, or have they been used elsewhere?
Motivatio... | https://mathoverflow.net/users/15756 | Proofs that use Infinite/Finite Priority Injury Method | The original proof of Borel determinacy by Donald Martin (1975, Annals of Mathematics) used a priority argument; I haven't read that paper, but Martin cited the complexity of the original proof as a motivation in the paper where he published his second, "purely inductive", proof (1985, Proc. Sympos. Pure Math. 42). The... | 5 | https://mathoverflow.net/users/5442 | 67946 | 41,775 |
https://mathoverflow.net/questions/67943 | 9 | Let $(X,d)$ be a separable metric space with Borel measure $\mu$. Let $f:X \times X \to \mathbb{R}$ be Borel measurable with respect to the product measure on $X \times X$, and let $g(x)=\operatorname{ess sup}\_{y \in X} f(x,y)$. Is $g(x)$ necessarily measurable? (Is there some argument that can be pieced together usin... | https://mathoverflow.net/users/15815 | Measurability of essential supremum of function of two variables | You are right. For each $n$ choose a set of measure less than $1/n$ on the complement of which $f$ is continuous. Now take the actual sup on each vertical section of this restricted function. This yields a measurable function $f\_n$ for each $n$ defined on $X$. The sup of the increasing sequence of $f\_n$ will also be ... | 6 | https://mathoverflow.net/users/13878 | 67949 | 41,776 |
https://mathoverflow.net/questions/67445 | 6 | Is the Ricci curvature of the compact symplectic group $Sp(n)$ bounded below by $cn$ for some constant $c > 0$ independent of $n$?
For $O(n)$ and $U(n)$ I know many references which state such a bound on Ricci curvature, although none of them actually include complete proofs. (Pointers to such proofs for $O(n)$ and $... | https://mathoverflow.net/users/1044 | Ricci curvature of the symplectic group | The groups $SO(n)$, $SU(n)$, and $Sp(n)$ all have Ricci tensor equal to a constant times the metric tensor. (Note: contrary to what I wrote in the question and what one may find stated in several places in the literature, this is false for $U(n)$. This is easy to see from Claudio's answer: if a Lie group has nontrivial... | 7 | https://mathoverflow.net/users/1044 | 67951 | 41,778 |
https://mathoverflow.net/questions/67932 | 7 | There are numerous examples of models of computation in which all programs halt, for example primitive recursion.
Are there (non-trivial) examples of models in which only some programs halt, but the halting problem is still decidable?
Does the decision procedure need to lie outside of the original model itself?
... | https://mathoverflow.net/users/9896 | Models of computation with decidable halting problem? | Joel Hamkins points out that the decision procedure for any reasonable notion of "computability" is not going to be solvable by a function that is "computable" within that notion.
Here is a contrasting example of a nontrivial model of computation in which the halting problem is solvable *in the usual sense of comput... | 5 | https://mathoverflow.net/users/5442 | 67954 | 41,780 |
https://mathoverflow.net/questions/67898 | 0 | Hello,
I have a question about trace measurable operators and I think it's not a hard one. However, I'm quite confused because I cannot prove it.
Let $\mathcal{M}$ be a semi-finite von Neumann algebra with a faithful normal semi-finite trace $\tau$. Let $T$ be a $\tau$- measurable operator (densely defined closed (... | https://mathoverflow.net/users/15777 | trace measurable operators | Like Matthew said, this doesn't look like it has to do with measurability at all. Those two inequalities follow from the operator inequalities
`\[
|T|\,E_{(s,\infty)}(|T|) \geq s\,E_{(s,\infty)}(|T|), \ \ \
|T|\,E_{[0,s]}(|T|) \leq s\,E_{[0,s]}(|T|).
\]`
In turn, these inequalities follow from the corresponding inequ... | 4 | https://mathoverflow.net/users/3698 | 67967 | 41,783 |
https://mathoverflow.net/questions/67960 | 2 | Can anyone give me a hint for an algorithm to find a simple cycle of length 4 (4 edges and 4 vertices that is) in an undirected graph, given as an adjacency list? It needs to use $O(v^3)$ operations (v is the number of vertices) and I'm pretty sure that it can be done with some kind of BFS or DFS.
The algorithm only ... | https://mathoverflow.net/users/15816 | Cycle of length 4 in an undirected graph | Oh, and there is another way, with the BFS you mentioned. Iteratively, do a BFS from each node. By slightly modifying the BFS algorithm, you can instead of computing the distances from your source vertex to any other, remember the number of shortest paths from your source vertex to any other.
If there is a vertex at ... | 4 | https://mathoverflow.net/users/1715 | 67970 | 41,785 |
https://mathoverflow.net/questions/67791 | 4 | Hello, I have a question which is related to a partial order in a set of self-adjoint operators.
Let $\mathcal{M}$ be a semifinite von Neumann algebra with a faithful semi-finite normal trace $\tau$. Let $T$ and $S$ be two self-adjoint operators (possibly unbounded) $\tau$-measurable (here probably the assumption tha... | https://mathoverflow.net/users/15777 | Partial order - Unbounded normal operators affiliated with von Neumann algebra. | I assume you are following the proof in Fack-Kosaki (if you are not, we are talking here about Proposition 2.2 and 2.5 there).
Note that there is no need for absolute value bars since both $T,S$ are positive.
The key fact is that $E\_{(s,\infty)}(T)\wedge E\_{[0,s]}(S)=0$ (to be proven afterwards). Using this, we... | 3 | https://mathoverflow.net/users/3698 | 67974 | 41,787 |
https://mathoverflow.net/questions/67975 | 1 | Given the seemingly broad definition of NP, it is very interesting that one can prove that any member of NP can be reduced in polynomial time to any member of NPC. (I guess this is true by definition of NPC, so let me restate my question.) How does one prove that no NP problem exists for which there is no polynomial-ti... | https://mathoverflow.net/users/15822 | Proof that any NP problem can be reduced (in P time) to any problem in NPC? | This is Cook-Levin theorem, look it up on Wikipedia.
| 1 | https://mathoverflow.net/users/9896 | 67977 | 41,788 |
https://mathoverflow.net/questions/67978 | 2 | First, I am by no means well-versed on cohomology so I apologize if this is too elementary.
I have been going through some basics of etale cohomology, with my ultimate goal being an understanding of some basic applications. I have gone through the Kummer and Artin-Schreier sequences, and wanted to get an idea for how... | https://mathoverflow.net/users/3261 | Additive form of Hilbert 90 for schemes? | $H^1\_{ét}(X, \mathbf{G}\_a) = H^1\_{Zar}(X, \mathcal{O}\_X)$, see Milne, Étale Cohomology III.§3. (as for the étale cohomology of any coherent sheaf)
| 6 | https://mathoverflow.net/users/nan | 67980 | 41,790 |
https://mathoverflow.net/questions/67972 | 0 | In Kharazishvili's "Nonmeasurable Sets and Functions" there is the following theorem:
>
> There exists a subset $X$ of $\mathbb{R}$ which is a Vitali set and a Bernstein set.
>
>
>
The proof is as follows:
>
> Let $\alpha$ denote the first ordinal of cardinality the continuum. Let {$x\_{\xi} : \xi < \alpha... | https://mathoverflow.net/users/15666 | Difference between a partial selector and a selector... | This might depend on exactly what choices you make during the construction. It is easy to choose the $x\_\beta$'s in such a way that $X\setminus X'$ is not a null set but in fact is another Bernstein set and thus is not disjoint from any set of positive measure. To see this, imagine doing Kharazishvili's construction w... | 0 | https://mathoverflow.net/users/6794 | 67981 | 41,791 |
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