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https://mathoverflow.net/questions/162583 | 21 | 99-Graph: Is there a graph with 99 vertices in which every edge (i.e. pair of joined vertices) belong to a unique triangle and every nonedge (pair of unjoined vertices) to a unique quadrilateral?
| https://mathoverflow.net/users/nan | Is there a graph with 99 vertices in which every edge belong to a unique triangle and every nonedge to a unique quadrilateral? | First we will prove the graph is regular.
Let $x,y$ be two non-adjacent vertices, and let $a,b$ be their common neighbours. Define $X$ to be the neighbourhood of $x$ other than $a,b$, and $Y$ to be the neighbourhood of $y$ other than $a,b$.
Considering the edge $ax$, there is a unique vertex $u\in X$ adjacent to b... | 28 | https://mathoverflow.net/users/9025 | 162592 | 85,244 |
https://mathoverflow.net/questions/162598 | -2 | **Edit** according to the comment of user36931 I remove the "motivation" from the previous version and I add an statement to the first question
We consider the following two classes of smooth maps on $\mathbb{R}^{n}.$ ( $n$ is not necessarily even):
**$HP\_{1}$**: $\;$A smooth map $f:\mathbb{R}^{n} \to \mathbb{R^{n... | https://mathoverflow.net/users/36688 | Holomorphic maps on $\mathbb{R}^{n}$ (for n not necessarily even) | This is an extended comment. The first property $HP\_1$ simply says that $f$ is conformal.
It is not true that holomorphic maps of $C^n=R^{2n}$ are conformal, except when $n=1$.
In fact there are very few conformal maps in $R^n$ for $n\geq 3$: they are only Mobius transfomrations (compositions of reflections).
| 7 | https://mathoverflow.net/users/25510 | 162602 | 85,247 |
https://mathoverflow.net/questions/162591 | -2 | I want to know whether an automorphism group of a simple Lie algebra over $GF(2)$, acts transitively on non-zero elements of Lie algebra or not? How can I check this property?
| https://mathoverflow.net/users/40491 | Action of automorphism group on Lie algebra | Surely not. For example some elements are $\mathrm{ad}$-nilpotent and others are $\mathrm{ad}$-diagonable.
| 2 | https://mathoverflow.net/users/327 | 162603 | 85,248 |
https://mathoverflow.net/questions/162499 | 4 | Let $f \in \mathbb{R}[x,y]$ be a polynomial satisfying the following conditions:
(i) $f(\mathbb{R}^2) \subset [a,\infty)$ where $a>0$;
(ii) $f$ is non-degenerate, in the sense that there isn't a non-singular change of variables that turns $f$ into a function of one variable.
(iii) The leading homogeneous part has... | https://mathoverflow.net/users/4140 | Convergence of the Double Integral of a Polynomial Reciprocal | No. Take $f(x) = 1 + y^2 + (xy-1)^2$. Let
$$D = \{ (x,y) : 0 \leq x,\ 0 \leq y \leq 2,\ xy \leq 2 \}.$$
Then $f(x,y) \leq 3$ on $D$, so the integral of $1/f$ is bounded below by $(1/3) \mathrm{Area}(D)$. But the area of $D$ is infinite.
| 3 | https://mathoverflow.net/users/297 | 162608 | 85,251 |
https://mathoverflow.net/questions/162549 | 2 | Let $A, B$ be two convex and closed subsets of $\mathbb{R}^n$. We would like to the minimum distance between these two sets. i.e., we want to find a solution for the following problem.
$$ \min \{||x-y||~ ~\hbox{for}~ x\in A, y\in B\}$$.
I would like to see if there is a method to find a solution for this problem in ... | https://mathoverflow.net/users/44534 | Distance between two sets | You are trying to solve what is known as a *best approximation problem*.
1. von Neumann's alternating projections **does not** work here (as might have been perhaps suggested above)
2. You can use [Dykstra's projection algorithm](http://en.wikipedia.org/wiki/Dykstra%27s_projection_algorithm), which will find the des... | 2 | https://mathoverflow.net/users/8430 | 162610 | 85,252 |
https://mathoverflow.net/questions/162613 | 13 | Given the set of \*-homomorphisms between two $C^\*$-algebras $A$ and $B$, we may define a metric on it by setting $d(f,g):= \sup\_{0<\|a\|\le 1}\|f(a)-g(a)\|$. Could it be true that, for each \*-homomorphism $f\colon A \to B$ there exists such $\epsilon >0$ that if $d(f,g)<\epsilon$, then there exists a homotopy betwe... | https://mathoverflow.net/users/4807 | Is the space of *-homomorphisms between two $C^*$-algebras locally path connected | Let $X$ be compact. Then $\mathrm{Hom}(C(X),\mathbb{C}) = X$, and in that case the metric you give is the discrete metric, which is [as noted by Vahid Shirbisheh below] locally path connected.
[In light of Vahid's comment, with a silly mistake corrected by Eric:]
Suppose that $B$ is commutative; we may then assume ... | 6 | https://mathoverflow.net/users/327 | 162627 | 85,260 |
https://mathoverflow.net/questions/162606 | 3 | I am trying to prove that for any divisible torsion $\mathbb{Z}$-module $V$,
this map
$$f:\mathbb{Q}/\mathbb{Z}\otimes\_E\text{Hom}(\mathbb{Q}/\mathbb{Z},V)\longrightarrow V\mbox{ defined by }
f((q+\mathbb{Z})\otimes g)=g(q+\mathbb{Z})$$
is an isomorphism, where tensor is taken over the ring $E=\operatorname{End}(\mat... | https://mathoverflow.net/users/49234 | Divisible torsion $\mathbb{Z}$-modules | First, consider an element $v\in V$. As $V$ is torsion we can choose $n$ such that $n!v=0$. For $k\leq n$ put $u\_k=(n!/k!)v$. Then choose $u\_k$ for $k>n$ inductively with $ku\_k=u\_{k-1}$ (which is possible because $V$ is divisible). There is then a unique homomorphism $\phi\colon\mathbb{Q}/\mathbb{Z}\to V$ such that... | 4 | https://mathoverflow.net/users/10366 | 162638 | 85,264 |
https://mathoverflow.net/questions/162634 | 1 | Denote by $HC(A,M)$ the Hochschild homological complex of an algebra $A$ with coefficients in an $A$-bimodule $M$, and let $B\rightarrow A$ be an $R$-flat extension of $R$-algebras, for some $CRing$ $R$.
On page 11 of this article:
<http://arxiv.org/pdf/1103.4377v2.pdf>
the author defines a filtration $G\_n^p$ of the... | https://mathoverflow.net/users/36886 | Jacobi-Zariski exact sequence question | As observed in that paper, $G\_n^p=G\_n^{p+1}$ for $p\geq n$, so that the colimit, which is simply the union, is equal to $G\_n^n$. It is clear that $G\_n^n=CH\_n(\mathscr A,M)$, in the notation of the paper. This shows the claim you mention in each degree.
| 2 | https://mathoverflow.net/users/1409 | 162646 | 85,267 |
https://mathoverflow.net/questions/161467 | 4 | For integral polytopes, it is conjectured (T. Hibi), that if the [$h^\*$-vector](http://en.wikipedia.org/wiki/Ehrhart_polynomial#Ehrhart_Series) is symmetric, then it is also unimodal (increasing, then non-decreasing).
A non-integral polytope do not, in general, have a polynomial [Ehrhart function](http://en.wikipedi... | https://mathoverflow.net/users/1056 | On a conjecture by Hibi regarding h-vectors | The conjecture is false for integral polytopes in dimension $\ge 6$; see Mircea Mustata and Sam Payne, Ehrhart polynomials and stringy Betti numbers, Math. Ann. 333 (2005), no. 4, 787-795 ([arXiv version](http://arxiv.org/abs/math/0504486)), and Sam Payne, Ehrhart series and lattice triangulations, Discrete Comput. Geo... | 6 | https://mathoverflow.net/users/3193 | 162652 | 85,270 |
https://mathoverflow.net/questions/161451 | 4 | Definition: A "$k$-chain" is a multi-graph obtained from a path of length $k$ by duplicating every edge.
Note that the number of paths between two endpoints of a $k$-chain is $2^k.$
**Question:** Let $G$ be a simple graph on $n$ nodes
and let $s$ and $t$ be two nodes of $G.$
Suppose that number of (simple) paths f... | https://mathoverflow.net/users/48547 | Do graphs with large number of paths contain large chain minor? | No, there is no such bound.
Consider the graph $G\_n$ whose vertices are pairs $(k,\nu) \in \{1, 2, \dots, n\} \times \{0, 1\}$, with edges between $(k, \nu)$ and $(k+1, \mu)$ for every $k \in \{0, 1, \dots, n-1\}$ and every $\nu, \mu \in \{0, 1\}$. Take $s = (n, 0)$ and $t = (n, 1)$. Then $\left|G\_n\right|$ grows l... | 5 | https://mathoverflow.net/users/44291 | 162668 | 85,274 |
https://mathoverflow.net/questions/162655 | 7 | Does there exist a Ricci flat Riemannian or Lorentzian manifold which is geodesic complete but not flat? And is there any theorm about Ricci-flat but not flat?
I am especially interset in the case of Lorentzian Manifold whose sign signature is (- ,+ ,+ , + ). Of course, the example is not constricted in Lorentzian c... | https://mathoverflow.net/users/43941 | Can anyone give an example of Ricci flat Riemannian or Lorentzian Manifold that is not flat? | All riemannian manifolds with holonomy contained in $SU(n) \subset SO(2n)$, $Sp(n) \subset SO(4n)$, $G\_2 \subset SO(7)$ and $Spin(7) \subset SO(8)$ are Ricci-flat. There are plenty of non-flat examples; e.g., those with holonomy *precisely* those groups.
In the Lorentzian setting, you could consider a subclass of lo... | 20 | https://mathoverflow.net/users/394 | 162672 | 85,276 |
https://mathoverflow.net/questions/162671 | 12 | Frankl's union-closed conjecture states that if $F$ is a finite union-closed family of sets (i.e. a family that is closed under taking unions), then there must be an element that belongs to at least half the sets.
1) Does anyone know an example of a finite union-closed family $F$ such that the set $\mathcal{A}(F)$ of... | https://mathoverflow.net/users/49268 | A strengthening of Frankl's union-closed conjecture? | The simplest example I found is the family
$$ \emptyset, \{1\}, \{1,2,3\}, \{1,2,4\}, \{1,2,3,4\} $$
This is a union-closed family with 5 elements. 1 appears in 4, 2 appears in 3, 3 and 4 both appear in 2, but $\{1,2\}$ is not in the family.
| 8 | https://mathoverflow.net/users/385 | 162691 | 85,281 |
https://mathoverflow.net/questions/162677 | 9 | In [Selinger, P. *A survey of graphical languages for monoidal categories* (New Structures for Physics, Springer, 2011, 813, 289-233)](http://www.mscs.dal.ca/~selinger/papers/graphical.pdf%E2%80%8E), it is stated that:
>
> Lemma 4.17 ([23, Prop. 7.2]). A braided monoidal category is autonomous if and only
> if it ... | https://mathoverflow.net/users/49270 | Why is a braided left autonomous category also right autonomous? | Here's a hint: what you should probably use here is the method of string diagrams (due to Joyal and Street, but by now ubiquitous). In other words, draw a picture in terms of tangles; you will see two crossings, and you want to apply a series of Reidemeister II and III moves to make those crossings cancel out; these mo... | 12 | https://mathoverflow.net/users/2926 | 162693 | 85,283 |
https://mathoverflow.net/questions/162690 | 4 | In Proposition 3.1. in [this article](http://link.springer.com/article/10.1007%2FBF02100612#page-1) by John Franks, he applies the Nielsen-Thurston classification of surface homeomorphisms to a homeomorphism $ \ f:M \rightarrow M$ of an **open** surface $M$ which is a $k$-times punctured sphere with $k \geq 3$. Think a... | https://mathoverflow.net/users/49283 | Nielsen-Thurston classification of homeomorphisms for open surfaces? | The theories for an open surface $S$ and for a compact surface with boundary $\overline S$ whose interior is identified with $S$ are the same. The inclusion of $S$ into $\overline S$ defines an isomorphism of mapping class groups. The case of a compact surface with boundary is covered in all of the early sources that I... | 5 | https://mathoverflow.net/users/20787 | 162699 | 85,285 |
https://mathoverflow.net/questions/162705 | 3 | A subset $A\subset\mathbb{R}$ is negligible if for each $\epsilon>0$ there exists a sequence $(I\_n)$ of intervals such that $A\subset\cup\_n I\_n$ and $\sum\_n \vert I\_n \vert \leq \epsilon$. Let us say that $A$ is ultra-negligible if for any sequence $(\epsilon\_n)$ of positive numbers, there exists a sequence $(I\_... | https://mathoverflow.net/users/49268 | Countable vs. ultra-negligible sets | The sets you refer to as "ultranegligible" are known as the [strong measure zero sets](https://mathoverflow.net/questions/63497/cohen-reals-and-strong-measure-zero-sets), and the assertion that every strong measure zero set is countable is known as the [Borel conjecture](http://en.wikipedia.org/wiki/Borel_conjecture), ... | 5 | https://mathoverflow.net/users/1946 | 162711 | 85,289 |
https://mathoverflow.net/questions/162625 | 4 | Is there any 'guess' as to how the Riemann zeta function $\zeta(\sigma+it)$ (or its modulus) behaves to leading order as $t\rightarrow\infty$, for fixed $\sigma$ in the critical strip? Obviously this can't be known for sure until at least the Lindelof Hypothesis is solved, but have people come up with *either* a good g... | https://mathoverflow.net/users/21052 | leading-order behaviour of riemann zeta function? | An asymptotic result is *much* stronger than a Big Oh bound, and no results like the ones you hope for can be true: In Titchmarsh's "Theory of the Riemann Zeta Function", Theorem 11.9 shows that for fixed $\sigma\_0$ in the interval $(1/2,1]$, the values of $\log(\zeta(\sigma\_0+it))$, $t>0$ are dense in the complex pl... | 7 | https://mathoverflow.net/users/6756 | 162713 | 85,290 |
https://mathoverflow.net/questions/160162 | 21 | Let $p:F\_2^n\rightarrow F\_2$ be a multivariate polynomial, let's say of degree 3. (Both the degree and the order of the field could probably be replaced by other constants without affecting this question.)
I'm interested in the following computational problem: we're given $p$ as input, simply by a listing of its $O... | https://mathoverflow.net/users/2575 | Minimum number of variables on which a multivariate polynomial depends? | Just an update: with help from Dan Shepherd and Greg Kuperberg, I now understand how to solve this problem in deterministic, classical polynomial time, *even when the degree $d$ is as large as the field characteristic or larger* (the case that originally interested me, and that David Speyer's excellent answer doesn't t... | 8 | https://mathoverflow.net/users/2575 | 162721 | 85,294 |
https://mathoverflow.net/questions/162650 | 8 | I'm interested in calculating all of the zeroes of the first derivative of the Riemann $\zeta$ function up to an arbitrary height. I know that (on the RH), all of these zeroes will have real part $\ge \frac{1}{2}$. I am curious if there are equally strong upper bounds.
According to Titchmarsh, there is a constant c (... | https://mathoverflow.net/users/49249 | Sharpest bound on the zero free region of $\zeta^{\prime}$? | If $s=\sigma+it$ with $\sigma >1$ then note that
$$
\Big| \frac{\zeta^{\prime}}{\zeta}(s) \Big| =\Big| \sum\_{p} \frac{\log p}{p^s-1} \Big|
\ge \frac{\log 2}{|2^s-1|} - \sum\_{p\ge 3} \frac{\log p}{|p^{s}-1|}
\ge \frac{\log 2}{2^{\sigma}+1} - \sum\_{p\ge 3} \frac{\log p}{p^{\sigma}-1}.
$$
Thus
$$
\Big|\frac{\zet... | 12 | https://mathoverflow.net/users/38624 | 162725 | 85,296 |
https://mathoverflow.net/questions/29169 | 18 | The Dushnik–Miller [dimension of a partial order](http://en.wikipedia.org/wiki/Order_dimension) $(P,{\leq})$ is the smallest possible size $d$ for a family ${\leq\_1},\ldots,{\leq\_d}$ of total orderings of $P$ whose intersection is ${\leq}$, i.e. $x \leq y$ iff $x \leq\_i y$ holds simultaneously for all $i = 1,\ldots,... | https://mathoverflow.net/users/2000 | Subposets of small Dushnik-Miller dimension | Elyse Yeager and I have constructed examples for an upper bound. Basically, if you understand the dimension of subposets of $P$, then you understand the dimension of the subposets of the lexicographic power $P^k$. Starting with an appropriate standard example then gets you a sublinear upper bound on $F\_d(n)$; this bou... | 5 | https://mathoverflow.net/users/49301 | 162730 | 85,298 |
https://mathoverflow.net/questions/162703 | 9 | Consider a large, fixed $M>2$. For each $n$, let $\alpha\_n$ denote the smallest algebraic integer of degree at most $n$, all of whose Galois conjugates lie in the real interval $(0,M)$.
Is there anything known on the rate at which $\alpha\_n$ decays to $0$ as $n\to\infty$ ?
The exponential lower-bound $\alpha\_n... | https://mathoverflow.net/users/49288 | How small can a totally positive integer be? | $\alpha\_n$ can be exponentially small once $M$ is large enough,
say $M \geq 6$.
For $m > 0$ let $\tau\_m$ be the (monic, degree-$m$) polynomial
such that $\tau\_m(z+z^{-1}) = z^m + z^{-m}$; in other words,
$\tau\_m(x) = 2 T\_m(x/2)$ where $T\_m$ is the $m$-th Čebyšev polynomial.
Then $\tau\_m$ has all its roots real... | 12 | https://mathoverflow.net/users/14830 | 162732 | 85,300 |
https://mathoverflow.net/questions/161998 | 9 | Let A be the infinite Hankel matrix with the coefficient
$$A\_{kj}=e^{(-t(k+j)^2)}-e^{(-t(k+j+2)^2)},$$ with $t$ a nonnegative real number.
Is $A$ in trace class with a norm bounded by an absolute constant?
It is not hard to see A is in trace class with a constant depending on t by either a result of J. S. Howlan... | https://mathoverflow.net/users/48977 | Is this Hankel matrix in trace class | I think that the answer is yes.
**Edit:** as noticed in the comments, the answer would be *obviously yes* if the matrix $A$ was positive, since in this case its trace norm (denoted $\|A\|\_1$) would be equal to its trace which is less than $1$. But $A$ is not positive: when $t$ goes to $0$, $A\_{i,j}/4t$ goes to to $... | 6 | https://mathoverflow.net/users/10265 | 162738 | 85,303 |
https://mathoverflow.net/questions/162557 | 5 | A projective smooth variety $X$ is weak Fano if $-K\_X$ is nef and big. We say that $X$ is log Fano is there exists a divisor $D$ such that $-(K\_X+D)$ is ample and $(X,D)$ is Kawamata log terminal.
Is it true that weak Fano implies log Fano? What is an example of a variety which is log Fano but not weak Fano?
| https://mathoverflow.net/users/nan | Weak Fano and Log fano varieties | That is true. Basically it is a consequence of the following fact:
*Let $D$ be a nef and big divisor on an irreducible projective variety $X$. Then there exist an effective divisor $E$ and a rational number $0 <\epsilon\ll 1$ such that $D-\epsilon E$ is ample.*
Proof: Let $D$ be a nef and big divisor. Since $D$ is ... | 2 | https://mathoverflow.net/users/14514 | 162742 | 85,305 |
https://mathoverflow.net/questions/162651 | 19 | Usually in set theory, Borel sets are much more nicely behaved than arbitrary sets of reals. One reason for this is Borel determinacy, which immediately yields measurability, Baireness, and the perfect set property for Borel sets; another reason (not totally distinct from the first) is that Borel sets enjoy a number of... | https://mathoverflow.net/users/8133 | Pathological behavior of Borel sets? | Joel speaks on the case where the real numbers are a countable union of countable sets. The Feferman-Levy model is a strange model indeed.
However, I find the Truss construction to be even weirder. Truss repeated the construction of Solovay by starting from an arbitrary limit cardinal, and he proves that the prefect ... | 14 | https://mathoverflow.net/users/7206 | 162749 | 85,307 |
https://mathoverflow.net/questions/162737 | 6 | Let $F$ be a field and let $B$ be an $F$-algebra. The *degree* of $B$ over $F$ is the smallest positive integer $\deg\_F B = d \geq 1$ such that every element of $B$ satisfies a (monic) polynomial of degree $d$ over $F$ if such an $m$ exists, otherwise $\deg\_F B = \infty$.
An algebra of finite degree over $F$ need n... | https://mathoverflow.net/users/4433 | Infinite dimensional simple algebras of finite degree | Yes $D$ must be finite-dimensional over $F$. This follows from a Theorem of Kaplansky's that I found in Herstein's monograph "Noncommutative Rings". The first step is to show that an algebraic algebra of bounded degree satisfies a polynomial identity (see Lemma 6.2.3 in Herstein), i.e., is a P.I. algebra. Note, however... | 9 | https://mathoverflow.net/users/11791 | 162750 | 85,308 |
https://mathoverflow.net/questions/162748 | 5 | Let $M$ be a compact connected manifold with an $S^1$-action. Suppose that $S^1$ has a fixed point in $M$. Is it true that $\pi\_1(M)=\pi\_1(M/S^1)$?
I is there some reference or a short proof of this fact?
**PS**. I am sorry for amending the question. In reality I only want to know that the kernel of the map $\pi\... | https://mathoverflow.net/users/13441 | Fundamental group of a manifold with an $S^1$-action | The answer to the amended question still seems to be **NO** For a source of counterexamples, check out [Frank Raymond's 1968 paper on circle actions on 3-manifolds](http://www.ams.org/journals/tran/1968-131-01/S0002-9947-1968-0219086-9/S0002-9947-1968-0219086-9.pdf).
(it seems that the fundamental group of the quotie... | 6 | https://mathoverflow.net/users/11142 | 162755 | 85,310 |
https://mathoverflow.net/questions/162709 | 0 | Let X be a cadlag Lévy process with $X\_0=0$ and let $p$ be a real number in $[1,\infty)$. Then, the following are equivalent.
1): $X$ is $L^p$-integrable.
2): $X^\*\_t= \mathop{\sup}\_{0\leq s\leq t} |X\_s|$ is $L^p$ integrable.
I would like to see a reference such that I can refer to it this theorem.
| https://mathoverflow.net/users/44534 | Running supremmum of a Levy process | See Theorem 25.18 p168 in "Lévy Processes and Infinitely Divisible Distributions" by K-I Sato
| 2 | https://mathoverflow.net/users/46773 | 162772 | 85,314 |
https://mathoverflow.net/questions/162630 | 13 | Let $\mathbb{G}$ be a reductive group defined over a number field $K$, let $Z$ be its center, and let $\mathbb{A}:=\mathbb{A}\_K$ be the ring of adeles of $K$. Reasonably, we care about the $\mathbb{G}(\mathbb{A})$-representation: $L^2(\mathbb{G}(K)\backslash \mathbb{G}(\mathbb{A}))$. It naturally contains the sub-repr... | https://mathoverflow.net/users/49242 | What is the intuition behind the definition of cuspidal representations? | In addition to Paul Garrett's answer, I address your last paragraph in a special example:
Strong approximation gives a homeomorphism
$SL\_2(Z) \backslash H \cong Z(A) GL\_2(Q) \backslash GL\_2(A) / \prod\_p GL\_2(Z\_p) \times O(2)$.
Lets $f$ corresponds to $\tilde{f}$. This translates
$$ \int\_{0}^1 f( y + t)\; d... | 2 | https://mathoverflow.net/users/10400 | 162777 | 85,316 |
https://mathoverflow.net/questions/162665 | 1 | Let $R\_\alpha f(t) = \int\_0^t (t-s)^{-\alpha} f(s)\,ds$ the fractional integration operator. If $f \in L\_q(0,1)$ for some $q>2$ then $R\_{1/2} f$ is (even Hölder) continuous on $[0,1]$.
My question is what happens in the endpoint: does $R\_{1/2}$ map $L\_2(0,1)$ to $C([0,1])$?
thank you very much, Bernhard
| https://mathoverflow.net/users/49265 | Is the fractional integral of order 1/2 of an L_2 function continuous | The answer is no as the following example shows: Let
$$
F(t)= \begin{cases} a\_nt^{-\frac 12+b\_n},& t\in [t\_{n+1},t\_n),\quad n\geq 1,\\
0,& \text{otherwise,}
\end{cases}
$$
where $0< t\_{n+1}< t\_n<1$, $n\geq 1$. Then $F\in L^2(0,1)$ provided
$$
\sum\_{n=1}^\infty \frac{a\_n^2}{2b\_n}(t\_n^{2b\_n}-t\_{n+1}^{... | 3 | https://mathoverflow.net/users/47586 | 162790 | 85,319 |
https://mathoverflow.net/questions/161944 | 26 | Was Vinogradov's first proof of the three-prime theorem effective?
Reasons for my question: Vinogradov presented his proof in 1937 in a monograph; the English translation by K.F. Roth and A. Davenport is based on the second version of the monograph, from 1947. There is no doubt that the proof in the second version is... | https://mathoverflow.net/users/398 | Was Vinogradov's 1937 proof of the three-prime theorem effective? | Concerning question (b). It seems the full version of Borozdkin's proof has never appeared as a normal article. In 1939 he got $C=e^{e^{e^{41.96}}}$ in his unpublished PhD thesis (see <http://cheb.tsput.ru/attachments/451_tom13_v2_Kasimov.pdf> ). The bound was further improved by him in 1956 to $C=e^{e^{16.038}}$ and a... | 7 | https://mathoverflow.net/users/32389 | 162799 | 85,323 |
https://mathoverflow.net/questions/162795 | 9 | Would it be possible to enlighten me (or even better give a reference) about enough projectives (injectives) in functor categories?
Here is a precise question. Let $C$ be a small category, whose total morphism set has cardinality $\alpha$. Let $A$ be an abelian category with enough projectives (dually, injectives) an... | https://mathoverflow.net/users/5301 | Projectives and Injectives in Functor Categories | For each object $c$ in $\mathcal{C}$, let $c^\* : [\mathcal{C}, \mathcal{A}] \to \mathcal{A}$ be evaluation at $c$. It is an exact functor, so if a left adjoint $c\_! : \mathcal{A} \to [\mathcal{C}, \mathcal{A}]$ exists, $c\_!$ will preserve projective objects. Assume $\mathcal{C}$ has $\le \alpha$ morphisms and $\math... | 14 | https://mathoverflow.net/users/11640 | 162801 | 85,324 |
https://mathoverflow.net/questions/115657 | 29 | In topology the spheres $S^n$ are the "simplest" closed manifolds, and they are like "Dirac's delta at $n$" for (reduced) cohomology groups. Furthermore they are boundaries of the simplest compact manifolds-with-boundary, i.e. the disks $D^{n+1}$, which are contractible. And $S^{n}$ is obtained by glueing two copies of... | https://mathoverflow.net/users/4721 | What is the algebraic geometry version of the spheres? | To expand on Tom Goodwillie's answer:
a precise definition of "motivic sphere" would be
$$S^{p,q} = \big( \Delta^{p-q} / \partial \Delta^{p-q} \big) \wedge \big( \bigwedge^q \mathbb{G}\_m\big)$$
which you can interpret as the $q$-fold smash product of the multiplicative group $\mathbb{G}\_m$ smashed with a $(p-q)$-di... | 11 | https://mathoverflow.net/users/956 | 162805 | 85,325 |
https://mathoverflow.net/questions/162807 | 4 | Consider a finite alphabet $\{0,1, \ldots, n-1\}$. Let $\Sigma\_n = \mathop{\prod}\limits\_{j=1}^{\infty}\{0, \ldots n-1\}$ be the set of infinite one sided sequences and $\prec$ the lexicographic order defined in $\Sigma\_n$. Given $a, b \in \Sigma\_n$ with $a \prec b$ it is possible to define the *lexicographic subsh... | https://mathoverflow.net/users/10518 | Graph presentation of Lexicographic shifts | Just as the $\beta$-shift arises as the coding space for the transformation $x\mapsto \beta x$ (mod 1), so the shifts you describe arise as coding spaces for $x\mapsto \alpha + \beta x$ (mod 1). The characteristic sequences $a,b$ correspond to the codings of 0 and 1, respectively.
In particular, you can describe the ... | 7 | https://mathoverflow.net/users/5701 | 162814 | 85,328 |
https://mathoverflow.net/questions/162798 | 4 | I do not know much about mixed Hodge modules. I would like to ask: Let $X$ be a smooth connected algebraic complex variety, with a chosen point. Could one describe smooth mixed Hodge modules on $X$ as some representations of the fundamental group? Of course, I ask because I have in mind other cases - for example, smoot... | https://mathoverflow.net/users/2095 | Smooth mixed hodge modules - representations of fundamental group? | The short answer is no. The best you can say is the following: there is a functor rat from mixed Hodge modules to perverse sheaves, and for smooth $X$, $\mathrm{rat}(\mathscr M)$ is a local system iff $\mathscr M$ is smooth. But a smooth mixed Hodge module contains more information than its underlying local system (thi... | 5 | https://mathoverflow.net/users/1310 | 162817 | 85,329 |
https://mathoverflow.net/questions/162724 | 5 | This is a refined version of my earlier question [Convex extensibility of combination of two lines](https://mathoverflow.net/questions/161059/convex-extensibility-of-combination-of-two-lines).
>
> Is there a smooth function $f:[0,1]\times [0,1]\rightarrow\mathbb R$ such
> that for all $x\in [0,1]$, $$ f(x,1)=x,\qq... | https://mathoverflow.net/users/4600 | Smooth convex extensibility of combination of two line segments | A cheap convex solution on $\mathbb{R}^2$ is
$$f\_0(x,y):= \big(x+3y-2\big)\_+ -y \, ,$$
which also verifies $f\_0(x,y)=-y$ for all $(x,y)$ in the rectangle $$[-1/4, 5/4]\times [-1/4,1/4]=\big([0,1]\times\{0 \}\big)\;{\bf +}\; [-1/4,1/4]^2 \subset\{x+3y-2\le0\} \, , $$
and $f\_0(x,y)=x+2(y-1)$ in the rectangle $$[-... | 3 | https://mathoverflow.net/users/6101 | 162826 | 85,332 |
https://mathoverflow.net/questions/162756 | 5 | Let $Q$ be a random variable taking as its values the set of $n \times k$ real matrices with orthogonal columns, and whose distribution is the Haar measure on the Stiefel manifold $O(n)/O(n-k)$. This random variable can be realized computationally by taking a random $n \times k$ matrix $A$ whose entries are drawn from ... | https://mathoverflow.net/users/7378 | Reference request: results on the asymptotic distribution of singular values related to a random orthogonal matrix | On a closer look at the paper mentioned above, it turns out that the joint pdf of the singular values $\lambda\_1, \ldots, \lambda\_k$ is actually classically known from Muirhead's *Aspects of Multivariate Statistical Theory* and is reproduced in equation (1) of the [paper](http://www-math.mit.edu/~edelman/homepage/pap... | 2 | https://mathoverflow.net/users/7378 | 162827 | 85,333 |
https://mathoverflow.net/questions/152660 | 5 | Let $F\_{2}$ be the free group with two generators.
Then $F\_{2}=\{\text{odd words}\}\sqcup\{\text{even words}\}$. This gives us a $Z\_{2}$ graded structure for $C^{\*}\_{red} (F\_{2})$, in a natural way.
My question:
>
> Is there another $Z\_{2}$ graded structure for $C^{\*}\_{red} (F\_{2})$ which is not grad... | https://mathoverflow.net/users/36688 | $Z_{2}$- graded structures for $C_{red} ^{*} (F_{2})$ | It is well-known that $K\_1(C^\*\_{\rm red}(F\_2))={\mathbb Z^2}$ with generators given by $[u]$ and $[v]$, where $F\_2=\langle u,v\rangle$.
Now, the automorphism of order two associated with the even-odd grading is multiplying each generator by $-1$, which is homotopic to the multiplication by $1$. Hence it is trivi... | 6 | https://mathoverflow.net/users/8176 | 162841 | 85,335 |
https://mathoverflow.net/questions/162836 | 46 | I'm trying to understand why [John Nash's 1950 2-page paper](http://www.pnas.org/content/36/1/48.full) that was published in PNAS was such a big deal. Unless I'm mistaken, the 1928 paper by John von Neumann demonstrated that all n-player non-cooperative and zero-sum games possess an equilibrium solution in terms of pur... | https://mathoverflow.net/users/31429 | Why was John Nash's 1950 Game Theory paper such a big deal? | I think von Neumann dealt with the case $n=2$, and it was by no means obvious how to extend the concept of equilibrium for the general case and prove that it always exists. More precisely, $n$ players before Nash were reduced to the $n=2$ case by partioning the players into two groups in all possible ways. Once you reg... | 38 | https://mathoverflow.net/users/11919 | 162843 | 85,336 |
https://mathoverflow.net/questions/162844 | 10 | We all know the notation $\aleph\_\lambda$ for the $\lambda$th (or, I guess, $\lambda+1$st) infinite cardinal number; in particular $\aleph\_0$ is the cardinality of the the set of natural numbers $\mathbb N$.
Out of curiosity:
>
> Is it the case that historically, the Hebrew letter $\aleph$ (aleph) was chosen be... | https://mathoverflow.net/users/4600 | $\aleph$ looks like $\mathbb N$? | According to not necessarily reliable internet sources, Georg Cantor "told his colleagues and friends that he was proud of his choice of the letter aleph to symbolize the transfinite numbers, since aleph was the first letter of the Hebrew alphabet and he saw in the transfinite numbers a new beginning in mathematics: th... | 18 | https://mathoverflow.net/users/39754 | 162845 | 85,337 |
https://mathoverflow.net/questions/162842 | 3 | Suppose $f(x)\in\mathbf Z[x]$ is nonconstant. I would like to know if either of the following statements is true.
1. If $a$ and $b$ are coprime integers (probably with some additional restriction), then there exist infinitely many primes $p$ such that $p\equiv a\bmod b$ and $f(x)$ has a root modulo $p$.
2. It is poss... | https://mathoverflow.net/users/46987 | Congruences among primes modulo which a given polynomial has roots | Here's a survey of the possible things that can happen. In regards to your first question, given any polynomial $f(x)$, there is a positive integer $M$ so that if $\gcd(b,M) = 1$, then there are infinitely many primes $p \equiv a \pmod{b}$ for which $f(x)$ has a root modulo $p$. (One can take for $M$ the modulus of the... | 9 | https://mathoverflow.net/users/48142 | 162857 | 85,342 |
https://mathoverflow.net/questions/162863 | 11 | Take $E$ to be an elliptic curve over $\mathbb{Q}$, and consider the coefficients $a\_E(p)$of the dirichlet series $L(E,s)$ so that $E(\mathbb{F}\_p)=p+1-a\_E(p)$.
The Sato-Tate conjecture (now proven) explains the distribution of $a\_E(p)/\sqrt{p}$.
**Question:** For a prime $l>0$, is there an $l$-adic analogue o... | https://mathoverflow.net/users/4181 | Is there a p-adic Sato - Tate conjecture? | The answer is yes, and it is much simpler than the true Sato-Tate conjecture.
Let $\rho: G\_{\mathbb Q} \rightarrow GL\_2(\mathbb Z\_\ell)$ be the Galois representation attached to $E$ on its Tate module, and let $\Gamma$ be the image of $\rho$. That's a compact group, hence it has an Haar measure $\mu\_\Gamma$. Now th... | 17 | https://mathoverflow.net/users/9317 | 162864 | 85,346 |
https://mathoverflow.net/questions/162839 | 6 | Let $Y\subset\mathbb{P}^n$ be a smooth variety of codimension two. Consider the blow-up $X = Bl\_Y\mathbb{P}^n$ of $\mathbb{P}^n$ along $Y$, and let $E$ be the exceptional divisor over $Y$. Then $E$ has a structure of $\mathbb{P}^1$-bundle over $Y$. The anticanonical divisor is
$$-K\_X = (n+1)H-E.$$
I would like to co... | https://mathoverflow.net/users/nan | Top self-intersection of exceptional divisors | So you want to compute the intersection numbers $(H^p\cdot E^q)$, $p+q=n$.
Let me start with some notation. Let $b: X\rightarrow \mathbb{P}^n$ be the blowing up, $i:E \hookrightarrow X$ the embedding, $p:E\rightarrow Y$ the projection. Write one factor $E$ as $i\_\*1$ (in $CH(X)$, say) and use the projection formula ... | 7 | https://mathoverflow.net/users/40297 | 162867 | 85,348 |
https://mathoverflow.net/questions/159677 | 2 | By lattice points, I will always mean points in $\mathbb{Z}^n$ and all polytopes here are convex *rational* polytopes.
If $P$ is an integral polytope, the counting function for the number of lattice points inside $nP$ is a polynomial $p(n)$.
This is the Erhart polynomial, and if $P$ does not have integral vertices,
$... | https://mathoverflow.net/users/1056 | Relation between non-integral polytopes, integrally closed polytopes and polynomial Erhart quasi-polynomials | A combination of the polytopes introduced in Exercise 2.24 of [*Polytopes, Rings, and K-Theory*](http://math.sfsu.edu/gubeladze/publications/kripo.html) by W. Bruns and J. Gubeladze (see also Section 3 of [this paper](http://arxiv.org/abs/1307.7422)) with a 3-dimensional version of the polygons in [this paper by T. McA... | 2 | https://mathoverflow.net/users/3193 | 162871 | 85,350 |
https://mathoverflow.net/questions/162848 | 6 | A non-negative sequence $\{a\_n\}$ is sub-additive if $a\_{m+n}\leq a\_m + a\_n.$ Fekete's lemma says that for any non-negative sub-additive sequence:
$$\lim\_{n\to\infty} \frac{a\_n}{n} = \inf\_{n} \frac{a\_n}{n}.$$
Of course, one way to show this would be to show that $\frac{a\_n}{n}$ is non-increasing, but I hav... | https://mathoverflow.net/users/7576 | Regarding sub-additive sequences and Fekete's lemma | Here is another "natural" example. Fix an integer $b \ge 2$ and let $s\_b(n)$ denote, for each $n \in \mathbf N^+$, the sum of the $b$-digits of $n$. Then $s\_b$ is subadditive: This comes, e.g., from the first of the identities mentioned [here](https://mathoverflow.net/questions/161960/looking-for-a-reference-to-a-cla... | 2 | https://mathoverflow.net/users/16537 | 162877 | 85,353 |
https://mathoverflow.net/questions/162888 | 11 | A common approach to forcing is to use countable transitive model $M \in V$ with $\mathbb{P} \in M$ and take a $G \in M$ (which always exists) to form a countable transitive model $M[G]$. Another approach takes $M$ to be countable such that $M \prec H\_\theta$ for sufficiently large $\theta$ (and hence may not be trans... | https://mathoverflow.net/users/43354 | Forcing with Nontransitive Models | All standard forcing machinery works when forcing over such $M$ because they satisfy a large enough fragment of $ZFC$, namely $ZFC$ without the powerset axiom. The purpose of forcing over such models is rarely to transfer results to $V$, although something like this can be done in the following way. Suppose that $M\pre... | 9 | https://mathoverflow.net/users/5984 | 162892 | 85,357 |
https://mathoverflow.net/questions/149022 | 4 | This question might have a formulation in higher dimensions, but for now let's deal with the 2 dimensional Radon transform:
$\newcommand{\R}{\mathbb{R}}$
$$
Rf(\varphi,s)=\int\_{-\infty}^\infty f(s\theta+t\theta^\perp)ds,
$$ where $\theta=[\cos(\varphi),\sin(\varphi)]^\intercal\in S^1$ and $s\in\Bbb{R}$. As $t\in\R$ v... | https://mathoverflow.net/users/38536 | Interpretation of the integral "with respect to a plane wave" in terms of Radon transform | Sure you can't - but somehow you can. Obviously, $x\mapsto h(\theta\cdot x)$ is not an integrable function (if not $\equiv 0$) since it is constant along lines perpendicular to $\theta$. However, if $f$ is not $L^2$ but $L^1$, then you can view the integral as duality pairing: If $h$ is bounded, i.e. $h\in L^\infty$, t... | 4 | https://mathoverflow.net/users/9652 | 162893 | 85,358 |
https://mathoverflow.net/questions/162895 | 6 | I am looking for a paper titled *The Cauchy integral* by M. Privalov (Saratov, 1919).
The paper is written in Russian. It is being cited (for instance) by A. Kolmogorov in the paper *Sur les fonctions harmoniques conjuguées et les séries de Fourier* (Fundamentals Math. **7**, pp. 23-28 (1925)), which is written in Fr... | https://mathoverflow.net/users/9762 | Looking for paper: The Cauchy integral by M. Privalov | This book can be found on the internet using the right search terms: Привалов, интеграл Kоши (= Privalov, integral Cauchy). This quickly leads you to the book (it's not an article) at <http://bookre.org/reader?file=729538> and press the button marked >| to advance a page or use the page drop-down button to go immediate... | 9 | https://mathoverflow.net/users/3272 | 162898 | 85,360 |
https://mathoverflow.net/questions/162875 | 11 | The Kronecker-Weyl theorem asserts the following: fix real numbers $\theta\_1,\dots,\theta\_d$, and consider the infinite ray $t(\theta\_1,\dots,\theta\_d)$ $(t\in\Bbb R)$ inside the $d$-dimensional torus $(\Bbb R/\Bbb Z)^d$. Then there exists a subtorus $A$ such that the limiting distribution of $t(\theta\_1,\dots,\th... | https://mathoverflow.net/users/5091 | Reference for Kronecker-Weyl theorem in full generality | Why not just prove the result from scratch? It's only a couple of pages and involves only basic Fourier analysis for locally compact abelian groups.
Let
$$\mathbb{T}^n = \left\{(z\_1,\ldots,z\_n) \in \mathbb{C}^n : |z\_l| = 1 \text{ for all $1 \leq l \leq n$}\right\}$$
be the $n$-torus. Let $t\_1, \ldots, t\_n$ be ar... | 18 | https://mathoverflow.net/users/3803 | 162929 | 85,367 |
https://mathoverflow.net/questions/162838 | 1 | I have the PDE
$$u\_t(t) - \Delta f(u(t)) = 0$$
in $H^{-1}(\Omega)$ where $f$ is a nonlinear function.
Define $F(s) = \int\_0^s f(s)$. Note that if $u\_t(t) \in L^2(\Omega)$,
$$\frac{d}{dt}F(u(t)) = f(u(t))u\_t(t).$$
We could test the PDE with $f(u(t))$ (i.e multiply by it and integrate) and if $u\_t \in L^2(\Omeg... | https://mathoverflow.net/users/49672 | Getting an a priori bound on a nonlinear gradient term in PDE; how to adapt trick from $L^2$ case to $H^{-1}$ case? | Is f continuous with respect to someoing appropriate norm? If so, then I think the argument goes something like the following. Since $$u\_t - \Delta f(u(t)) = 0$$ in the $H^{-1}$ sense then $$\langle u\_t , v \rangle = \langle \Delta f(u(t)), v \rangle \; \forall v\in H\_0^1$$ Take $v = f\_\epsilon(u(t)) = \phi\_\epsil... | 2 | https://mathoverflow.net/users/49404 | 162931 | 85,369 |
https://mathoverflow.net/questions/162920 | 8 | Let $L/K$ be an extension of fields of characteristic zero. I want to prove that if $L^\times/K^\times$ is a torsion group (i.e. for every element $\alpha \in L$, some power of $\alpha$ lies in $K$), then $L=K$.
Reducing the the case where $L$ is of the form $L=K(\sqrt[p]{\alpha})$ seems appropriate. I have started s... | https://mathoverflow.net/users/37644 | $L^\times / K^\times$ torsion $\Rightarrow L = K$? | This was proved by I. Kaplansky, "A theorem on division rings", Canadian J. Math. 3 (1951), 290-292, see this [link](http://cms.math.ca/cjm/v3/cjm1951v03.0290-0292.pdf).
| 12 | https://mathoverflow.net/users/49410 | 162939 | 85,374 |
https://mathoverflow.net/questions/162941 | 1 | THE PROBLEM:
Let $U$ be a uniform distribution and $U\_{n}$ be its nth empirical distribution. Suppose $t\in (0,1)$ and $n\in \mathbb{N}$ are constants. What's the explicit expression to
$$E\{U\_{n}^{-}(t) - t\}^{2}?$$
THE CONTEXT:
The generalized inverse of distribution function $F$, or quantil function, is define... | https://mathoverflow.net/users/49357 | A calculation involving a uniform random variable quantile | The "explicit expression" will be by $n$ different cases depending on where $t$ lies.
For $n=0$, the answer is 0.
For $n=1$,
$$
\mathbb E[(U^-\_n(t)-t)^2] = \mathbb E [(U-t)^2]=\int\_0^1 (x-t)^2 \,dx = \frac13 - t + t^2.$$
For $n=2$, we will have to split into cases depending on whether $t\ge 1/2$ or not,
and get $\m... | 1 | https://mathoverflow.net/users/4600 | 162946 | 85,378 |
https://mathoverflow.net/questions/162926 | 9 | I think I sound stupid but I have heard a lot about Algebraic Geometry as a subject and wish to study it without actually studying abstract algebra. I have never studied abstract algebra since I am a physics student. I have always studied mathematics as part of math physics course. Means I know basics of analysis and s... | https://mathoverflow.net/users/49406 | Algebraic Geometry for non-mathematician | I would highly recommend first covering the book [An Invitation to Algebraic Geometry](http://rads.stackoverflow.com/amzn/click/0387989803) by Smith et al. This requires very little abstract algebra as prerequisite (only understanding of very basics about rings) and gives a very well-written introduction to algebraic g... | 7 | https://mathoverflow.net/users/1355 | 162950 | 85,381 |
https://mathoverflow.net/questions/162868 | 10 | It is well known that for a given polynomial $f \in \mathbb{Z}[x]$ the number of primes $p$ s.t. $f$ has a root modulo $p$ is infinite. In fact, one can even write down a formula for the density of such primes (e.g. D. Berend, Y. Bilu "Polynomials with roots modulo every integer").
Now suppose that $f = \sum\_{i=0}^... | https://mathoverflow.net/users/25905 | Uniform bound for the number primes $p$ s.t. a polynomial has a root modulo $p$ | As Joel has pointed out this kind of question is delicate, and in fact even in the case $d=2$ no such result is known (for any fixed $C>0$). To see this, suppose that $p$ is a prime for which the quadratic character $\chi(n) = (\frac{n}{p})$ has a Siegel zero. That is there is a $\beta >1-\epsilon/\log p$ with $L(\beta... | 7 | https://mathoverflow.net/users/38624 | 162957 | 85,385 |
https://mathoverflow.net/questions/162953 | 7 | In the course of investigating a conjecture about a "strange duality" for sections of line bundles on various models of moduli of sheaves on $\mathbb P^2$, another student and I reduced one special case to the following (conjectural, to me) isomorphism
$$
\operatorname{Sym}^k V \cong \left[\bigwedge^k (\mathbb C^k \oti... | https://mathoverflow.net/users/19088 | Are SL(n) Invariants of this wedge product isomorphic to a symmetric product? | Yes, this is correct. Let me hit it with a more general statement in case that becomes useful for further generalizations.
There is a general formula (Cauchy identity) for the action of $GL(V) \times GL(W)$ on the exterior power $\bigwedge^n(V \otimes W)$. This is written as
$\bigoplus\_{|\lambda|=n} S\_\lambda(V) ... | 9 | https://mathoverflow.net/users/321 | 162962 | 85,389 |
https://mathoverflow.net/questions/162975 | 4 | I have strong feeling that the function,
$$
f\_\alpha (x) = \sum\_{n=0}^\infty \frac{x^n}{n!\Gamma(1+n\alpha)},
$$
is a known special function (here $\Gamma(x)$ is the usual extension of the factorial). Is this the case?
| https://mathoverflow.net/users/12337 | Recognize this sum | This is an entire function of order $1/\alpha$ when $\alpha>1$. So for irrational $\alpha$
it cannot satisfy any linear differential equation with polynomial coefficients.
If $0<\alpha<1$, the order is $1$ but the type is minimal, so again it cannot
satisfy any such equation. This excludes most special functions.
(But ... | 11 | https://mathoverflow.net/users/25510 | 162978 | 85,392 |
https://mathoverflow.net/questions/162974 | 2 | Is anyone aware of a result which states that convergence of random variables in $\mathbb L^p$ are preserved under conditioning on sub-sigma fields?
I'm new to probability/measure theory, and trying to get a handle on the idea of combining $\mathbb L^p$ spaces with conditioning. I was trying to tackle this with mart... | https://mathoverflow.net/users/49420 | Convergence of random variables in LP preserved under conditioning on sub sigma field | This is an immediate consequence of the [conditional Jensen inequality](http://en.wikipedia.org/wiki/Jensen%27s_inequality#Proof_3_.28general_inequality_in_a_probabilistic_setting.29).
| 1 | https://mathoverflow.net/users/4832 | 162979 | 85,393 |
https://mathoverflow.net/questions/162985 | 4 | I have a question, but not sure how to prove this.
We are given $n$ points in the Euclidean plane such that there exists no disk of radius $a$ which contains all of the points.
Conjecture: There must exist three of these points which are not contained in a disk of radius $a$.
Any idea about how to prove this?
T... | https://mathoverflow.net/users/49425 | Points contained in a disk | There is a classical result of convex geometry, called Helly's theorem (<http://en.wikipedia.org/wiki/Helly%27s_theorem>). It states that if you have $n$ convex subsets of $R^d$ and any $d+1$ of these convex subsets have nontrivial intersection, then all of them have nontrivial intersection.
For your question you jus... | 12 | https://mathoverflow.net/users/33286 | 162996 | 85,396 |
https://mathoverflow.net/questions/163003 | 7 | I am trying to get a better understanding of "real" $C^\*$-algebras. I encountered them in the paper
[D. Voiculescu, Dual algebraic structures, J. Operator Theory 17(1987), 85-98](http://www.theta.ro/jot/archive/1987-017-001/1987-017-001-004.pdf),
which cites
[G.G. Kasparov, Hilbert $C^\*$-modules: theorems of St... | https://mathoverflow.net/users/36090 | Literature on "real" $C^*$-algebras | Real operator algebras, Bing-Ren Li, Pub. Co. Pte. Ltd, 2003
(and the bibliography in that book).
Also the other operator algebra book by the same author has parts about real operator algebras (together with the traditional complex ones)
An older book is by [Goodearl](http://www.getcited.org/pub/102337023), and a cla... | 4 | https://mathoverflow.net/users/46855 | 163013 | 85,400 |
https://mathoverflow.net/questions/161195 | 3 | (*I originally asked this question on Math.SE [here](https://math.stackexchange.com/q/716247/13487). As suggested on meta.MathOverflow ([posting an unanswered Math.SE question on MathOverflow](https://meta.mathoverflow.net/q/1569/36720)), I've waited about a week before reposting it here. Note that although the origina... | https://mathoverflow.net/users/36720 | local cohomology mayer-vietoris sequence | (*This is an elaboration on Vinteuil's answer. I did not accept Vinteuil's answer due to the lack of detail.*)
So, it seems that I made a very foolish error: Consider the long exact sequence of $H\_J$ that I refer to in the 4th-to-last line of my original post. Written out, it is
$$\Gamma\_J(M)\to \Gamma\_J(M/\Gamma\... | 2 | https://mathoverflow.net/users/36720 | 163017 | 85,402 |
https://mathoverflow.net/questions/162983 | 2 | Suppose I have a diagonal $n \times n$ matrix $\Gamma$ with positive entries, and a fixed $n \times k$ matrix $P$ with $P^\intercal P = I$ (here, $k \leq n$). I'm interested in knowing whether the function
$$ f(\Gamma) = \log\det\left(P^\intercal \, \Gamma \,P\right) $$
is always concave as a function of the the entrie... | https://mathoverflow.net/users/48584 | Question about the log-det function | Log det is known to be concave as a function on the positive semidefinite cone (there are many proofs, the one I like best is by way of Chandler Davis' theorem (see [this preprint](http://arxiv.org/abs/math/0208223), or Davis' original paper cited there). Since your set is a linear subspace, the affirmative answer foll... | 5 | https://mathoverflow.net/users/11142 | 163019 | 85,403 |
https://mathoverflow.net/questions/163014 | -3 | I need to create a equation to distribute points for users in the following game:
There are x users that play a game.
If only one of them hit he gets max points.
If all of them hit each gets min points.
Everything between should be proportional.
Number of users, max points and min points are variables.
| https://mathoverflow.net/users/49440 | Simple equation to distribute points in a game | We can assume that $n \geq 2$, where $n$ is the total number of players; the one-player game can you figure out yourself.
Let $\alpha = (hits-1)/(n-1)$, where $hits$ is the total number of hits.
Note that $\alpha=0$ if only one player hits, and $\alpha=1$ if all player hits.
Then, using a weighted mean, each player... | -2 | https://mathoverflow.net/users/1056 | 163020 | 85,404 |
https://mathoverflow.net/questions/162673 | 3 | Is there any linear map that lowers the number of variables of functions, namely a map that maps a function of several variables to functions of one variable and at the same time the original function can be reconstructed with its image through an explicit form.
For example, $$T: C(\mathbb{R}^n)\rightarrow C(\mathb... | https://mathoverflow.net/users/43172 | tranforms that lowers the number of variables of a function | As your question is not completely precise, this might not really an answer but at least it might be an occasion to modify your question in a more precise way.
If you enlarge the point of view from continuous functions to Borel functions, then the problem is easy: all Polish spaces with no isolated points are Borel-i... | 2 | https://mathoverflow.net/users/46855 | 163022 | 85,405 |
https://mathoverflow.net/questions/163018 | 1 | Let ${f\_x}(m) = \sum\limits\_{\left. p \right|m} {{f\_x}(p)}$ be a
strongly additive function on positive integer number $m$, where $p$ is a prime number. Set
$${f\_x}(p) = \left\{ {\begin{array}{\*{20}{c}}
{0,}\\
{1,}\\
2,
\end{array}} \right.\begin{array}{\*{20}{c}}
{{\rm{ }}p < \ln \ln x{\rm{ }}\ or \ {\rm{ }}p \g... | https://mathoverflow.net/users/48365 | How to prove $\mathop {\lim }\limits_{x \to \infty } \sum\limits_{{f_x}(p) = 1} {\frac{1}{p}} = \ln 2$ for $p \le x$? | It looks like a partial summation-style problem. So consider the case when $f(p) = 1$
\begin{align}
\sum\limits\_{\log \log x \leq n < (\log \log x)^2} 1\_{\mathbb{P}}(n)n^{-1} &= \pi((\log \log x)^2)(\log \log x)^{-2} - \pi(\log \log x)(\log \log x)^{-1} \\
&+ \int\_{\log \log x} ^{((\log \log x)^2} \frac{1\_{\mathb... | 6 | https://mathoverflow.net/users/49438 | 163024 | 85,406 |
https://mathoverflow.net/questions/163021 | 3 | Let $C\_1,C\_2$ be smooth, projective curves of genera $g\_1,g\_2 \geq 2$. Assume that a group $G$ of order $(g\_1 - 1)(g\_2 - 1)$ acts on $C\_1$ and $C\_2$ such that $C\_1/G \cong \mathbb{P}^1$ and $C\_2/G \cong \mathbb{P}^1$. Assume that $G$ acts freely on $C\_1 \times C\_2$, then the quotient $S = (C\_1 \times C\_2)... | https://mathoverflow.net/users/1107 | Quotient of product of curves | Yes.
By a result of Freitag, in general one has $q(S) = g(C\_1/G)+g(C\_2/G)$. Look [here](http://arxiv.org/abs/0805.1424), Corollary 3.6.
| 3 | https://mathoverflow.net/users/7460 | 163030 | 85,407 |
https://mathoverflow.net/questions/163007 | 14 | I have seen several questions here on MO regarding large numbers, (uparrow notation, etc.), and different way to construct and compare such numbers.
I am curious what the applications are for the study of such numbers, what is the motivation behind such constructs? There are plenty of constructs that produce large nu... | https://mathoverflow.net/users/1056 | Applications of really large numbers | For the conceptual picture that you seem to be looking for, you probably want to think in terms of *fast-growing functions* as the primary object of interest, and very large numbers as evaluations of these functions at specific values of interest. One major reason for the interest in fast-growing functions is that they... | 10 | https://mathoverflow.net/users/3106 | 163038 | 85,411 |
https://mathoverflow.net/questions/163039 | 14 | In section 4.1 of Lurie's DAG VIII, he implies the existence of an $E\_\infty$-ring spectrum $A$ such that the coconnective truncation $\tau\_{\leq 0} (A)$ does not admit the structure of an $E\_\infty$-ring spectrum. What is an explicit example? Is it possible to give an intuitive explanation for why this fails? Also,... | https://mathoverflow.net/users/49450 | Truncations of E_infinity algebras | In general the issue is that the natural map $A \to \tau\_{\leq 0} A$ often cannot be compatible with a ring structure, because on the level of homotopy groups or homology groups it acts as a quotient map that destroys all elements in positive degree. This is often not compatible with the multiplicative structure.
A ... | 16 | https://mathoverflow.net/users/360 | 163045 | 85,412 |
https://mathoverflow.net/questions/123693 | 14 | Adding a single Cohen real makes the set of reals from the ground model strong measure zero (see [this question](https://mathoverflow.net/questions/63497/cohen-reals-and-strong-measure-zero-sets)).
The notion of strong measure zero sets has its dual concept in the category branch -- strongly meager sets. A set $X\sub... | https://mathoverflow.net/users/31944 | Random reals and strongly meager sets | As I have written above the affimative answer itself was known to many people including T. Bartoszyński. The following proof is due to T. Weiss (my advisor).
Proof. We follow closely the proof and notation of Lemma 3.2.42 from [1]. Let $A$ be a Borel measure zero set in $M[r]$, where $r$ is a random real over $M$. Th... | 2 | https://mathoverflow.net/users/31944 | 163057 | 85,415 |
https://mathoverflow.net/questions/163028 | 3 | Given an ergodic Markov chain $(X\_n)\_{n\geq 1}$ in $R^d$with $\pi$ as the invariant distribution of the transition kernel, under good conditions we have that the empirical occupation measure converges to $\pi$, i.e.:
\begin{align}
\forall A\in \mathcal{B}(R^d), S^A\_N = \frac{1}{N}\sum\_{k=1}^{N}1\_{X\_k \in A} \to... | https://mathoverflow.net/users/49445 | convergence rate of occupation measure of ergodic Markov Chain | Here is a finite time bound on $||S\_N-\pi||\_1$:
<https://dl.dropboxusercontent.com/u/3198145/chernoff-dkw.pdf>
Theorem 1.3
| 0 | https://mathoverflow.net/users/12518 | 163058 | 85,416 |
https://mathoverflow.net/questions/163029 | 3 | In the paper "A generalization of Kodaira-Ramanujam's vanishing theorem", Kawamata states a covering lemma (Lemma 5) which is
>
> Let $X$ be a non-singular projective variety, and $D$ be a divisor with normal crossing on $X$. Let $m\_i$ be any positive integers attached to the irreducible components $D\_i$ of $D$. ... | https://mathoverflow.net/users/29730 | A covering lemma of Kawamata | I'm afraid that what you would like is not going to happen.
The problem with achieving (1) is that the procedure requires a step when you take a multiple of an ample line bundle that is very ample and then that multiple will contribute to the degree of the map. So, this degree will depend on $X$. I suppose you may be... | 5 | https://mathoverflow.net/users/10076 | 163059 | 85,417 |
https://mathoverflow.net/questions/161551 | 1 | Let $\mathbf{S}$ be a $m\times n$ matrix, with $m < n$. We define a subdimensional polytope as the space of $n$-dimensional vectors $\mathbf{x}$ that satisfy the following equation:
$$\mathbf{S}\cdot\mathbf{x}=0$$
subject to the inequalities:
$$\mathbf{a}\le\mathbf{x}\le\mathbf{b}$$
where $\mathbf{a}$ and $\mat... | https://mathoverflow.net/users/16615 | What is the functional form of the projections of a subdimensional polytope? | Assuming that $S$ is full rank, let $u$ be some unit vector not orthogonal to the null space of $S$, and let $\Omega\_t =\lbrace x-t u~:~x\in\Omega,~ (x-tu)\cdot u = 0\rbrace$. then you are interested in the $(n-m-1)$-dimensional volumes
$f(t)=|\Omega\_t|$. Note that convexity of $\Omega$ implies that for $0\le\lambda\... | 2 | https://mathoverflow.net/users/20186 | 163063 | 85,420 |
https://mathoverflow.net/questions/163055 | 8 | The Hasse principle is perhaps an at-first naive generalization of the Chinese remainder theorem; that if a linear equation can be solved modulo $p$ for any prime $p$, then it can be solved in the integers. The first significant result is the Hasse-Minkowski theorem, which asserts that the local to global principle hol... | https://mathoverflow.net/users/10898 | Hasse principle and Brauer-Manin obstruction for forms of large degree | Let $X$ be a non-singular hypersurface of degree $d$ in $\mathbb{P}^n$ over a number field $k$. If $d > n+1$, then $X$ has *general type*. The Bombieri-Lang conjectue therefore predicts that the rational points on $X$ are not Zariski dense. So in general, one should expect very few rational points and perhaps one would... | 9 | https://mathoverflow.net/users/5101 | 163064 | 85,421 |
https://mathoverflow.net/questions/163043 | 4 | Let $E \in D^{b}\_{c}(X,\overline{\mathbb{Q}}\_{l})$ where $X$ is a $k$ scheme of finite type for a field $k$.
Let $Y\rightarrow X$ a finite flat surjective morphism such that $f^{\*}E$ is perverse and irreducible.
Do we have that $E$ is perverse and irreducible?
| https://mathoverflow.net/users/27398 | flat descent for perverse sheaves | $f\_\*$ for a finite morphism preserves perversity, so $f\_\* f^\* E$ is going to be perverse. $f\_\* f^\* E= E \otimes f\_( \overline{\mathbb Q\_l}$. $f\_\* (\overline{\mathbb Q\_l})$ has the constant sheaf as a summand, (by averaging), so $E$ is a summand of a perverse sheaf, hence is perverse.
Since inseperable mo... | 5 | https://mathoverflow.net/users/18060 | 163069 | 85,425 |
https://mathoverflow.net/questions/162194 | 1 | (Sorry if this is a noob question. I'm a mathematician learning statistics.)
I would like to know if it's sound (or advisable) to test many p-values against the continuous uniform distribution using Kolmogorov-Smirnov or Anderson-Darling.
For example, suppose I want to demonstrate that Rock-Paper-Scissors is a fair... | https://mathoverflow.net/users/41416 | Can I test many p-values with KS or AD | I think what you're looking for is the higher criticism statistic, which was suggested originally by Tukey: he compares the number of $\alpha$-level significant detections to the expected number of such detections you should expect under the joint null. Higher criticism has some good mathematical properties, as shown b... | 1 | https://mathoverflow.net/users/48584 | 163070 | 85,426 |
https://mathoverflow.net/questions/163077 | 10 | It may well be a trivial question but I was wondering if there is any relation between $K$-groups and ultrapowers of $C^\*$-algebras. For instance, if $A$ is a $C^\*$-algebra does $K\_0(A^U)$ depend on the choice of a free ultrafilter $U$? What if $A$ is a von Neumann algebra with a trace and $A^U$ is the tracial ultra... | https://mathoverflow.net/users/49468 | K-theory of ultrapowers | If $R$ is type II finite AW\* or W\* factor, then $K\_0(R^U) $ is naturally order isomorphic to the reals, as $R^U$ is again a type II finite AW\* or W\* factor. More generally, if $A$ is a C\* algebra with stable range 1, then $l^{\infty}(A)$ (the algebra of bounded sequences of elements of $A$) has the interesting pr... | 13 | https://mathoverflow.net/users/42278 | 163079 | 85,430 |
https://mathoverflow.net/questions/162764 | 6 | I noticed and employed (without a problem) an approximation for Stirling's number of the second kind found on Wikipedia (<http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind>), in particular this approximate expression: (<http://upload.wikimedia.org/math/8/a/f/8afeed15dd40295320cf974418895cc2.png>):
$n \... | https://mathoverflow.net/users/49313 | Understanding a particular approximation for Stirling's number of the second kind | I cannot provide you with a reference, but I can sketch with my own derivation (I have a draft somewhere...).
First, let $X$ be a **zero-truncated multinomial** $(n,k)$ random variable (we throw $n$ ball in $k$ urns and discard the results having empty urns). Its probability mass function is
$$P\_X({\bf x})=\frac{... | 5 | https://mathoverflow.net/users/5428 | 163085 | 85,432 |
https://mathoverflow.net/questions/162989 | 3 | everyone, I want to ask is there any result in the literature
similar to the following:
Let $ X=\mathbb{P}^1\backslash \{0,1,\infty\}$, then $X$ is defined over $\mathbb{Z}$. Let $X\_{\mathbb{Q}}$ denote the its generic fiber, and for any prime number $p$, $X\_{\mathbb{Q}\_p}:= X\_{\mathbb{Q}}\times Spec(\mathbb{Q}... | https://mathoverflow.net/users/4504 | representation of algebraic fundamental group of projective line minus three point | Yes, I think so. The geometric $\pi\_1$ is a normal subgroup of $\pi\_1$, and by (1) consists of upper triangular unipotent matrices. So the whole representation must live in the normalizer, which is all upper-triangular matrices. In other words, the representation is an extension of two one-dimensional representations... | 4 | https://mathoverflow.net/users/18060 | 163089 | 85,434 |
https://mathoverflow.net/questions/163091 | 5 | I have two very related questions:
If $f(N)$ is the number of square-free integers in the interval $[1, N]$, it is well known that $$f(N) \sim \frac{6}{\pi^{2}} N.$$
My first question is, if we impose the additional condition that the integer is not divisible by any prime smaller than $ N^{1/k}$, for some fixed in... | https://mathoverflow.net/users/29979 | Square-free integers not divisible by any "small" primes | Forgetting the squarefree condition for a moment, the number of integers up to $N$ that are not divisible by any primes less than $N^{1/k}$ is asymptotic to
$$
\omega(k) \frac N{\log N} \sim e^\gamma \omega(k) N \prod\_{p\le N^{1/k}} \bigg( 1-\frac1p \bigg),
$$
where $\omega$ is the [Buchstab function](http://mathworld... | 8 | https://mathoverflow.net/users/5091 | 163094 | 85,437 |
https://mathoverflow.net/questions/163098 | 3 | What is the current status of representations of $GL\_n(F)$ (and other algebraic groups)?
When $F$ is a local field, the representations of $GL\_n(F)$ are classified by [Bernstein and Zelevinsky](http://en.wikipedia.org/wiki/Bernstein%E2%80%93Zelevinsky_classification) in terms of cuspidal representations. What abou... | https://mathoverflow.net/users/11877 | What is the current status of representations of $GL_n(F)$ (and other algebraic groups)? | Let $G$ be reductive over the field $F$.
When $F$ is finite, there is Deligne-Lusztig theory.
When $F$ is archimedean, there is Langlands classification. (Knapp's book "Representation theory of semisimple groups tries to develop it).
When $F$ is non-archimedean and $G$ is $GL(n)$, $SL(n)$, then Bushnell and Kutz... | 8 | https://mathoverflow.net/users/10400 | 163100 | 85,439 |
https://mathoverflow.net/questions/163108 | -1 | For some reason I need some irreducible polynomial $f$ on $\mathbb{F}\_{2}[x]$ where $\deg f \in [10^3,10^6]$. Could someone give information about this? Thx.
| https://mathoverflow.net/users/22954 | Irreducible polynomial on $\mathbb{F}_{2}[x]$ | You might take a look at the paper "A fast algorithm for testing reducibility of trinomials mod 2 and some new primitive trinomials of degree 3021377'' by Richard P. Brent, Samuli Larvala and Paul Zimmerman published in Mathematics of Computation in 2003 (pages 1443-1452) (I think the article is free online [here](http... | 4 | https://mathoverflow.net/users/48142 | 163110 | 85,442 |
https://mathoverflow.net/questions/163083 | 5 | Knaster's pseudo-arc and Hilbert cube are topologically homogeneous continua. The easier question is: do these spaces admit a topological group structure? (I am sure that the answer is negative). Thus the harder question is, do they admit a structure of a quotient of a topological group divided (left or right) by a clo... | https://mathoverflow.net/users/8385 | When does topological homogeneity imply algebraic homogeneity? Pseudo-arc and Hilbert cube | As it was mentioned in the comments, the pseudo-arc and the Hilbert cube have the fixed point property so they cannot be homeomorphic to a topological group.
On the other hand it was proved by G.S. Ungar in "*On all kinds of homogeneous spaces*" (TAMS, 1975), that any homogeneous compact metric space is homeomorphic ... | 5 | https://mathoverflow.net/users/17836 | 163116 | 85,443 |
https://mathoverflow.net/questions/163117 | 12 | Let $r \ge 3$ be a fixed integer. I'm interested in primes p such that no integer in the interval $(-\sqrt{p}, \sqrt{p})$, except $1$ (and $-1$ if $r$ is even), is an r-th root of unity modulo p.
The naive heuristic that $r$-th roots of unity should be "randomly distributed" suggests that there should be infinitely m... | https://mathoverflow.net/users/2481 | Roots of unity near 1 in Z / p Z | OK, thinking a little more clearly about this... (hopefully)
Say $p\le N$ fails to have the property you want. Then $p | n^r-1$ for some
$|n| < \sqrt{p} \le \sqrt{N}$. There are only $O(\sqrt{N})$ integers of the form $n^r-1$ with $n$ in this range, and each has only $O(\log{N})$ prime factors.
So there are only $O(\... | 16 | https://mathoverflow.net/users/16510 | 163123 | 85,447 |
https://mathoverflow.net/questions/108433 | 38 | Three good answers were received — by Alex Gavrilov, Bjørn Kjos-Hanssen, and Terry Tao — and the bounty has been awarded (somewhat arbitrarily) to Alex Gavrilov.
The answers are summarized below; because they are open-ended and technically subtle, the question has been flagged for conversion to community Wiki.
Tha... | https://mathoverflow.net/users/11394 | For which Millennium Problems does undecidable -> true? | As Harry Altman pointed out, for a conjecture *undecidable -> true*
means that it can be formulated as a $\Pi\_1^0$ statement. To put it simply, if the conjecture
is false, one can prove this by an explicit (finite) calculation.
I would leave Yang–Mills and Navier–Stokes to someone more familiar with mathematical ph... | 17 | https://mathoverflow.net/users/9833 | 163127 | 85,450 |
https://mathoverflow.net/questions/163120 | 7 | I found an old MOF post about representations attached to p-adic modular forms: [Representations attached to p-adic modular forms](https://mathoverflow.net/questions/108350/representations-attached-to-p-adic-modular-forms) and I have some follow up questions on the same topic.
If we have a classical form of weight k,... | https://mathoverflow.net/users/49492 | Properties of representations attached to p-adic modular forms | There are two subtleties regarding how to formulate this question.
Firstly, there are several notions of "p-adic modular form". There's Hida's ordinary p-adic modular forms (a very small space); there's Coleman's overconvergent p-adic modular forms (a much bigger space); and there's Serre and Katz's space of p-adic m... | 8 | https://mathoverflow.net/users/2481 | 163128 | 85,451 |
https://mathoverflow.net/questions/163130 | 1 | I need an estimate of the form
$$ \|v\|\_{L^p} \le C \|(K-\Delta- c|x|^{-2})^s v\|\_{L^p} $$
where $K>0$ can be large if necessary, $c$ is positive but below the Hardy constant $(n-2)^2/4$, where $n$ is the space dimension. The power $s>0$ can be equal to 1 or another integer if this simplifies the proof.
This is tri... | https://mathoverflow.net/users/7294 | $L^p$ estimate for (powers of) a Laplacian with inverse square potential | See Lemma 5.3 in the paper
V.F. Kovalenko, M.A. Perelmuter, Yu.A. Semenov, Schrödinger Operators with $L^{l/2}\_w (R^l)$-Potentials, J. Math. Phys., Vol. 22, No. 5, 1981, pp. 1033-1044
You can use the similar approach: Neumann series and norm of the operator $|x|^{-2}(-\Delta)^{-1}$ in $L^p$.
| 3 | https://mathoverflow.net/users/40206 | 163133 | 85,452 |
https://mathoverflow.net/questions/163142 | 2 | It is an undergraduate exercise to show that the generating function for the sequence of unrestricted integer partitions $p(n)$ is the celebrated infinite product
$$\prod\_{k=1}^{\infty}\frac{1}{1-x^k}$$
More work has to be done if one wants to get asymptotic estimates of the function $p(n)$ (see for instance the p... | https://mathoverflow.net/users/46573 | An infinite product: combinatorial interpretation | This is called the MacMahon function, and counts plane =(3d, confusingly) partitions.
<http://en.wikipedia.org/wiki/Plane_partition>
| 10 | https://mathoverflow.net/users/1102 | 163144 | 85,455 |
https://mathoverflow.net/questions/163138 | 2 | In the category of C\*-algebras and $\*$-homomorphisms, a morphism is monic precisely when it is injective, and epic precisely when it is surjective (see [Mono- and epi-morphisms for C\*-algebras](https://mathoverflow.net/q/58416/10368)). Is this still true in the category of C\*-algebras and completely positive maps?
... | https://mathoverflow.net/users/10368 | When are completely positive maps monic/epic? | Yes. Every positive map $f$ is self-adjoint: $f(x^\*)=f(x)^\*$ for every $x$. Hence, if $f(a)$=0 and $a=b+ic$ with $b,c$ self-adjoint, then $f(b)=0=f(c)$. Suppose for a contradiction that $b\neq0$. Then, for $b=b\_+-b\_-$ with positive $b\_+,b\_-$, one has $f(b\_+)=f(b\_-)$. The cp maps $g\_{\pm}\colon{\mathbb C}\to A$... | 7 | https://mathoverflow.net/users/7591 | 163145 | 85,456 |
https://mathoverflow.net/questions/163156 | 4 | The motivation for this question is the Selmer curve, given by
$$\displaystyle 3x^3 + 4y^3 + 5z^3 = 0.$$
One can show that this curve has no rational integer solutions, despite having a solution modulo $p$ for any prime $p$ and a solution over $\mathbb{R}$ (in other words, the Selmer curve fails the Hasse principle). H... | https://mathoverflow.net/users/10898 | Fermat surface known to have very few rational integer solutions | These answers:
[Are most curves over Q pointless?](https://mathoverflow.net/questions/138581/are-most-curves-over-q-pointless/138592#138592)
[Are most cubic plane curves over the rationals elliptic?](https://mathoverflow.net/questions/11349/are-most-cubic-plane-curves-over-the-rationals-elliptic?rq=1)
both mentio... | 3 | https://mathoverflow.net/users/2290 | 163162 | 85,464 |
https://mathoverflow.net/questions/163143 | 1 | Is there a characterization of elliptic curves over $\mathbb Q$ whose conductor is a square? Does this property have a geometric meaning?
| https://mathoverflow.net/users/46987 | Elliptic curves with square conductor | Square conductor implies that all reduction is additive. (The converse is not quite true, since at 2 and 3, the conductor can have higher valuation, up to valuation 5 for $p=2$ and up to 3 for $p=3$. So maybe a better question is: what does it mean for the conductor to be powerful?) A geometric implication is that ever... | 4 | https://mathoverflow.net/users/11926 | 163164 | 85,465 |
https://mathoverflow.net/questions/162173 | 35 | This is a question first I asked in SE but since there was no suggestion or solution, I decide to put it here.
Consider an $n\times n \times n$ Cube containing $n^3$ unit cubes. Is it possible to place numbers $1$ to $n^3$ in the unit cubes so that the numbers in any two adjacent unit cubes are co-prime? (Two unit cu... | https://mathoverflow.net/users/49035 | Placing numbers $1,2,\ldots,n^3$ in a cube so that numbers of any two adjacent unit subcube are coprime | Here is a proof that for any fixed dimension $d$, there is a computable $n\_0(d)$ such that for all $n\ge n\_0(d)$ we can place the numbers $1, ..., n^d$ in a $d$-dimensional cube of side length $n$ such that any two adjacent numbers are coprime.
The main idea is to use the following lemma:
**Lemma 1.** If $G = (V,... | 15 | https://mathoverflow.net/users/2363 | 163165 | 85,466 |
https://mathoverflow.net/questions/163181 | 32 | There are two classes of maps $S^1\to S^1$ for which I know how to define the winding number:
• **Continuous maps:**
Using the [unique path lifting property](http://en.wikipedia.org/wiki/Homotopy_lifting_property) of the universal covering map $\mathbb R\to S^1$, any continuous map $\gamma:S^1\to S^1$
can be lifted ... | https://mathoverflow.net/users/5690 | For which maps $S^1\to S^1$ is the winding number defined? | A class of maps including both continuous and $H^{1/2}$, where an extension is available, is *Vanishing Mean Oscillation*, [VMO](http://en.wikipedia.org/wiki/Vanishing_mean_oscillation#The_space_VMO). This has been treated by several authors starting I think with Haïm Brezis. You can find quite a lot googling "degree t... | 30 | https://mathoverflow.net/users/6101 | 163182 | 85,471 |
https://mathoverflow.net/questions/163180 | 9 | I am interested in any sources that can be helpful for learning the representation theory of real reductive groups.
I am currently reading Wallach book, but I feel that I don't understand the subject properly and want to find an alternative.
| https://mathoverflow.net/users/45262 | Learning representation theory of real reductive lie groups | First note that there is the book of Vogan (Representation Theory if real reductive groups) which discusses the case of $SL\_2(\mathbb{R})$ on a very basic level. I think this is a good start. In my opinion the remainder of the book is not very accessible.
However for the whole theory I would recommend that you first... | 6 | https://mathoverflow.net/users/32972 | 163184 | 85,473 |
https://mathoverflow.net/questions/163174 | 0 | I am trying to calculate the Laplacian and Adjacency matrix of a graphs for positive and negative weights. If a graph be simple with only non-negative weight it is easier. But in my graph I have some negative weights and loops.
Please give me some references and hints if available.
Thank you for your help.
| https://mathoverflow.net/users/36977 | Laplacian matrix of a graph with negative weights | The most natural definition of Laplacian matrix is to me $\mathcal L=\mathcal I\mathcal I^T$, where $\mathcal I$ is the incidence matrix of an arbitrary orientation of the graph; or more generally $\mathcal L=\mathcal I\mathcal M\mathcal I^T$, where $\mathcal M$ is the diagonal matrix whose entries are the edge weights... | 2 | https://mathoverflow.net/users/26039 | 163187 | 85,474 |
https://mathoverflow.net/questions/163185 | 7 | Let $G$ be a complex affine reductive algebraic group, $B\subseteq G$ a Borel with maximal torus $T$ and unipotent radical $U$. Let $w\in\operatorname N\_G(T)$ be a representative of the longest Weyl element. I am wondering whether the big open Bruhat cell $BwB\subseteq G$ is a principal open set, i.e. whether there is... | https://mathoverflow.net/users/9947 | Is the big cell a principal open set? | This is true if $G$ is (semi-simple) simply-connected, because then $\mathrm{Pic}(G)=(0)$, which means that $\mathbb{C}[G]$ is factorial; however, it is false for the simplest non simply-connected example, namely $G=\mathrm{PGL}(2)$. Indeed $G$ is the complement of the quadric $ad-bc=0$ in $\mathbb{P}^3$; this implies ... | 15 | https://mathoverflow.net/users/40297 | 163189 | 85,475 |
https://mathoverflow.net/questions/163173 | 1 | **NOTATION**: $O\_x$ -- the product of all odd primes $\le x$.
E.g. $O\_7=3\cdot 5\cdot 7 = 105$.
**QUESTION**: Are the three ordered pairs $\ (d\ p)=(1\ 3)\ \ (2\ 3)\ \ (4\ 5)\ $ the only solutions of the equation:
$$|O\_p-2^d|=1$$
in natural numbers $d$, and odd primes $p$?
(I don't know an answer).
**MOTIVAT... | https://mathoverflow.net/users/8385 | Powers of $2$ and the products of initial odd primes | There is no solution for $O\_n=2^d-1$ with $n \geq 7$.
If 5 divides $2^d-1$ and 7 divides $2^d-1$, then 9 divides $2^d-1$. [Because 4 divides $d$ and 3 divides $d$; 6 divides $d$ and hence $2^{d}-1$ is divisible by 9.]
| 9 | https://mathoverflow.net/users/10926 | 163190 | 85,476 |
https://mathoverflow.net/questions/162503 | 3 | given the following functional
$h(\rho) = c \|\rho\|\_{3} - \int\_{\mathbb R^3} \, dx \frac{\rho(x)}{|x|} $ with $\rho>0$ , $\|\rho\|\_1 = 1$ and obviously $\rho\in L^1(\mathbb R^3)$.
Can I see somehow that there exists a minimizer?
If I know that, I can easily derive it with the Variational principle and by incl... | https://mathoverflow.net/users/47482 | Existence of Minimizer of $h(\rho) = c \|\rho\|_{3} - \int_{\mathbb R^3} \, dx \frac{\rho(x)}{|x|} $ | Here are a few clues that might help.
Note that the $L^3$ conjugated exponent is $3'=3/2$. For given $R>0$ let $B\_R=B\_R(0)$ be the ball of radius $R$ centered at the origin. Using spherical coordinates with $dx\sim r^2 dr$ in dimension $3$ you can compute explicitly
$$
\left|\frac{1}{|x|}\right|\_{L^{3/2}(B\_R)}=CR... | 2 | https://mathoverflow.net/users/33741 | 163193 | 85,477 |
https://mathoverflow.net/questions/163170 | 6 | Suppose $f(x)$ is a polynomial of degree 4 with integer coefficients and nonzero discriminant. Let $C$ be the hyperelliptic curve of genus 1 defined by $y^2=f(x)$. If we assume that $C$ has a rational point, then $C$ can be given the structure of an elliptic curve $E$. Now let $d$ be a squarefree integer. Thinking of $... | https://mathoverflow.net/users/46987 | Quadratic twist of an elliptic curve given by non-Weierstrass model | Let's assume the characteristic of the ground field $k$ is not $2$.
If $C$ is of the form $y^2=f(x)$ with $f$ a separable quartic, and $E$ is the Jacobian of $C$ (hence $E$ is an elliptic curve), then the Jacobian of the twist $C\_d$ is $E\_d$. (This is not hard to deduce from the construction of the Jacobian of a ge... | 6 | https://mathoverflow.net/users/17907 | 163201 | 85,480 |
https://mathoverflow.net/questions/163172 | 4 | We konw that in quantum harmonic oscillator $H=a^\dagger a$, $a^\dagger$, $a$, $1$ will span a Lie algebra, where $a, a^\dagger$ is annihilation and creation operator, $H$ is the Hamiltonian operator. The algebraic relation is following
$$[H,a^\dagger\ ]= a^\dagger$$
$$[H,a]=-a$$
$$[a,a^\dagger]=1$$
$$[H,1]=[a,1]=[a^... | https://mathoverflow.net/users/43941 | Lie group about the quantum harmonic oscillator | Note that this a $4$-dimensonal solvable Lie algebra and $a$, $a^\dagger$, $1$ span an ideal isomorphic to the $3$-dimensional Heisenberg algebra. So one realization is obtained by taking
$$ a=\left(\begin{array}{ccc}0&1&0\\0&0&0\\0&0&0\end{array}\right),\quad
a^\dagger=\left(\begin{array}{ccc}0&0&0\\0&0&1\\0&0&0\end{a... | 4 | https://mathoverflow.net/users/15155 | 163203 | 85,481 |
https://mathoverflow.net/questions/162205 | 2 | For $X$ and $Y$ finite sequences of finite trees, let us say that $X$ is everywhere contained in $Y$ ($X\subseteq\_{ec}Y$) iff, for every $y\in Y$, there is some $x\in X$ such that $x$ is a minor of $y$.
Is $\subseteq\_{ec}$ a well-quasiordering on the set of finite sequences of finite trees? (I.e.: Does every infinite... | https://mathoverflow.net/users/49044 | A variant of Kruskal's theorem | To reformulate your question, given a wqo $(A,\leq)$, here finite trees with the minor ordering, you ask whether $({\mathcal P}\_\mathrm{fin}(A),\sqsubseteq)$ the set of finite subsets of $A$ with a `minoring' ordering is also a wqo: $X\sqsubseteq Y$ iff $\forall y\in Y.\exists x\in X.x\leq y$.
This is does not hold ... | 3 | https://mathoverflow.net/users/36985 | 163214 | 85,485 |
https://mathoverflow.net/questions/163222 | 11 | It is well-known that if $G$ is a discrete group, then $BG=K(G,1)$. I'm interested in comparing classifying spaces of topological groups with the classifying spaces of the same groups but equipped with the discrete topology. They will be quite different in general, that much is clear.
For instance, $S^1$ can be thoug... | https://mathoverflow.net/users/49520 | Eilenberg-MacLane Spaces of "large" groups | Let me just point out that if you're interested in, say, homology, then discrete $S^{1}$ is not as complicated as it might seem. The resulting invariants will be huge, of course, but one should be able to compute them explicitly.
The point is that $S^{1} = \mathbb{R} / \mathbb{Z}$ and $\mathbb{R} \simeq \bigoplus \m... | 11 | https://mathoverflow.net/users/16981 | 163232 | 85,494 |
https://mathoverflow.net/questions/163224 | 1 | This is just a notation question.
If I cannot find a preexisting name, I would try one-sided convex, or something of the sort.
| https://mathoverflow.net/users/49537 | Is there a name for functions f(x,y) that are only convex in x (and continuous in y). | I have not heard the name of such the function (I deal with convex functions)
A function which is continuous wrt. x and wrt. y is called separately convex. Maybe something in this direction.
A function continuous wrt. x and measureble wrt. y is called Caratheodory function. This is another hint.
| 2 | https://mathoverflow.net/users/47248 | 163238 | 85,498 |
https://mathoverflow.net/questions/163240 | 3 | **My question is highlighted in bold at the end.**
$\mathrm{\underline{Background}}$
Consider a product of i.i.d. $d\times d$ random matrices $A\_{i}$
(with $\mathbb{E}\log\left\Vert A\_{i}\right\Vert <\infty$) acting
on a non-zero vecor $X$, i.e.
$$
A\_{n}\cdots A\_{1}X.
$$
The Lyapunov exponents are used to descr... | https://mathoverflow.net/users/48466 | Lyapunov Exponents for independent-nonidentically distributed matrices? | Yes, it is true. However, in your question you mix up a number of things (to begin with, it is Kesten, not Keston). The Multiplicative Ergodic Theorem is *not* the same as the Furstenberg-Kesten theorem. You can find answers to all your questions in [this article](http://www.ams.org/mathscinet-getitem?mr=947327).
| 4 | https://mathoverflow.net/users/8588 | 163243 | 85,500 |
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