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https://mathoverflow.net/questions/215502 | 3 | Is it true that the space of hyperfunctions on $\mathbb{R}^n$ supported at 0 coincides with the space of Schwartz distributions supported at 0?
More explicitly, is it true that any hyperfunction supported at 0 is a finite linear combination of various partial derivatives of the delta-function?
If the answer is yes... | https://mathoverflow.net/users/16183 | Hyperfunctions supported at a point | No, this is not true. Schwartz distributions with compact support inovlve only finitely many
differentiations (every such distribiution is a derivative of some order of a usual, integrable function). Hyperfunctions may involve infinitely many differentiations.
For example, in dimension $1$ you can take any entire funct... | 4 | https://mathoverflow.net/users/25510 | 215555 | 101,810 |
https://mathoverflow.net/questions/215579 | 2 | Let $X$ be a non-empty set and let $F: X \to {\cal P}(X)$ be a function with the following property:
* for $A \subseteq X$ we have $|A| \leq |\bigcup F(A)|$.
Does this imply that there is an injective function $f: X \to X$ such that $f(x)\in F(x)$? (Note: it is clear that the above condition is necessary for this, ... | https://mathoverflow.net/users/nan | Injective subset function | The answer is Yes if $X$ is finite (and possibly also Yes if $X$ is infinite and has $F(x)$ is finite for all $x\in X$, but I don't know off the top of my head about that). The reason why the answer is positive is that this is a reformulation of [Hall's Marriage Theorem](https://en.wikipedia.org/wiki/Hall's_marriage_th... | 2 | https://mathoverflow.net/users/8628 | 215580 | 101,820 |
https://mathoverflow.net/questions/215573 | 15 | Let $(M^n,g)$ be a smooth Riemannian manifold. Consider the square of the distance function
$$dist^2\colon M\times M\to \mathbb{R}$$
given by $(x,y)\mapsto dist^2(x,y)$. It is easy to see that this function is infinitely smooth near the diagonal.
Now fix a point $a\in M$. **Consider the Taylor series of $dist^2$ at $... | https://mathoverflow.net/users/16183 | Square of the distance function on a Riemannian manifold | Fix a point $x\_0 \in M$. Then let $x = \exp\_{x\_0}(t v)$ and $y=\exp\_{x\_0}(t w)$, with $v,w \in T\_{x\_0}M$.
Then we have the following formula for the distance squared between two geodesic emanating from $x\_0$
$$ d^2(\exp\_{x\_0}(t v),\exp\_{x\_0}(t w)) = |v-w|^2t^2-\frac{1}{3}R(v,w,w,v) t^4 + O(t^5)$$
whre... | 31 | https://mathoverflow.net/users/13915 | 215581 | 101,821 |
https://mathoverflow.net/questions/199852 | 7 | Can a finite (by finite I mean when the projection $1$ is finite) von Neumann algebra be strongly morita equivalent to a properly infinite von Neumann algebra?
(Strong morita equivalence is the same as Morita equivalence as a ring for $C^\*$-algebras according to the theorem on page 253 of <http://www.sciencedirect.... | https://mathoverflow.net/users/22191 | Can a finite von Neumann algebra be strongly morita equivalent to a properly infinite von neumann algebra? | Assume $A$ and $B$ are two unital $C^\*$-algebras which are strongly Morita equivalent.
This means that there exists a Hilbert $A$-module $H$ such that $B$ is the algebra of compact operators on $H$. But as $B$ is unital it implies that the unit of $B$ has to act as the identity of $H$ (because $B$ contains all "rank... | 7 | https://mathoverflow.net/users/22131 | 215583 | 101,822 |
https://mathoverflow.net/questions/215587 | 1 | In this question we try to improve some parts of [this post](https://mathoverflow.net/questions/209694/differential-and-pre-differential-of-jacobi-identity) as follows:
>
> What is an example of a manifold $M$ and a Lie algebra $L$ (with the same dimension) such that $M$ **does not** admit an smooth $(1,2)$ tensor ... | https://mathoverflow.net/users/36688 | Manifold_Lie algebra compatibility | A simple example is to let $M=S^2$ and let $L$ be the nonabelian Lie algebra of dimension $2$. If such an $\alpha$ existed, its range would be a rank-1 subbundle $L\subset TS^2$, but this cannot exist for topological reasons.
The general obstruction is whether the manifold $M$ admits an $\mathrm{Aut}(L)$-structure, a... | 5 | https://mathoverflow.net/users/13972 | 215588 | 101,823 |
https://mathoverflow.net/questions/215590 | 0 | I have a follow-up question to [On the Riesz representation theorem](https://mathoverflow.net/q/215523/56920) .
Let $V$ be a subspace of a Hilbert space, and let $V^\times$ be the space of all antilinear functionals on $V$, equipped with the weak-\* topology.
Let $\Psi$ be a weak-\* continuous antilinear functional ... | https://mathoverflow.net/users/56920 | On the Riesz representation theorem II | Yes, a more general version of this is a well-known theorem in functional analysis. Specifically, let $W$ be any vector space and let $V$ be any vector space of functionals on $W$ which separates points of $W$. Equip $W$ with the weak topology with respect to $V$. Then every continuous functional on $W$ is in $V$.
To... | 3 | https://mathoverflow.net/users/75 | 215593 | 101,825 |
https://mathoverflow.net/questions/215605 | 6 | (This question is crossposted from [MSE](https://math.stackexchange.com/questions/1363270/when-are-direct-products-exact-in-the-category-of-quasi-coherent-sheaves), since there the question did not recieve any attention whatsoever.)
I would like to know if there is a description (or at least some sufficient condition... | https://mathoverflow.net/users/60903 | When are direct products exact in the category of quasi-coherent sheaves? | A counterexample showing that direct products in the category of quasi-coherent sheaves over the projective line $\mathbb P\_k^1$ over a field $k$ are not exact functors can be found in the paper "The stable derived category of a Noetherian scheme", by H. Krause, <http://arxiv.org/abs/math.AG/0403526> , Example 4.9 (at... | 8 | https://mathoverflow.net/users/2106 | 215622 | 101,833 |
https://mathoverflow.net/questions/215621 | -4 | I'm having difficult with the following question :
A. Show that the number of partitions of n where in each one of them the even numbers appears at most once equals to the number of partitions of n in which every number appears at most three times .
| https://mathoverflow.net/users/78517 | patitions of the number n | For **A** consider the generating function;
$$\dfrac{(1+x^2)(1+x^4)(1+x^6)\dots}{(1-x)(1-x^3)(1-x^5)\dots}$$
This reads as '$0$ or $1$ of any even number, and any number of any odd numbers'.
We make progress by multiplying top and bottom by $(1+x)$ to give:
$$\dfrac{(1+x)(1+x^2)(1+x^4)(1+x^6)\dots}{(1+x)(1-x)(1... | 1 | https://mathoverflow.net/users/70355 | 215626 | 101,834 |
https://mathoverflow.net/questions/139472 | 6 | Following on from some of myprevious MO questions on finite group theory...
$\newcommand{\Irr}{\operatorname{Irr}}\newcommand{\Conj}{\operatorname{Conj}}\newcommand{\AMZL}{{\rm AM}\_{\rm Z}}$
Let $G$ be a finite group, $\Irr$ the set of irreducible characters (working over $\mathbb C$) and $\Conj$ the set of conjugac... | https://mathoverflow.net/users/763 | A constant associated to the character table of a finite group | Let $G= \operatorname{SL}(2,5)$ and $N= \mathbf{Z}(G)$, so that $G/N = \operatorname{PSL}(2,5)\cong A\_5$. If my code works correctly, then
$$ \alpha\_{G/N} = 842/75 > 6661/600 = \alpha\_G. $$
My implementation of alpha (for character tables) in GAP:
```
alpha:= function( tbl )
local clssizes, n, kg, phi, i... | 5 | https://mathoverflow.net/users/10266 | 215633 | 101,839 |
https://mathoverflow.net/questions/215617 | 2 | Let $P$ be a polynomial in several variables, and let $P(D)$ be the corresponding differential operator. Obviously, $P(D)$ is a local operator, in the sense that I need only to know the function $u$ in a neighborhood of a point $x$ in order to evaluate $P(D)(u)(x)$.
If $P$ is not a polynomial, we can still define $P(... | https://mathoverflow.net/users/56892 | Locality of homogeneous pseudo-differential operator | A linear operator $P\colon C\_c^\infty(X)\to \mathcal D'(X)$ with kernel $K\in\mathcal D'(X\times X)$ is local if and only if $\operatorname{supp}K\subseteq \Delta\_X$, where $\Delta\_X$ is the diagonal in $X\times X$. If $P$ happens to be a pseudodifferential operator, then its kernel $K$ is conormal with respect to $... | 3 | https://mathoverflow.net/users/69194 | 215640 | 101,841 |
https://mathoverflow.net/questions/215603 | 1 | Motivated by [the answer to this question](https://mathoverflow.net/questions/215587/manifold-lie-algebra-compatibility) we ask:
>
> Is it true to say that for every real analytic tensorial Lie algebra structure $\alpha$ on $\chi^{\infty}(S^2)$, all fibers are necessarily Abelian Lie algebra? In the other word:Assu... | https://mathoverflow.net/users/36688 | Does an analytic tensorial Lie structure on $S^2$ gives a fiberwise Abelian Lie algebra structure? | No. A skew-symmetric bilinear map $V\times V\to V$ satisfies the Jacobi identity automatically if $\dim V=2$ since the Jacobi identity is skew-symmetric in its arguments. So if you take a generic skew-symmetric $\alpha$, then on most fibres it will create a two-dimensional non-Abelian Lie algebra, and on some fibres it... | 2 | https://mathoverflow.net/users/1306 | 215646 | 101,844 |
https://mathoverflow.net/questions/212388 | 4 | I know that given two matrices $A$ and $B$, estimating the eigenvalues of $A + B$ by the eigenvalues of $A$ and $B$ is generally a non-easy problem. In particular, there are some results for matrices that commute (multiplicatively!), hermitian matrices etc.
In this case $B=\operatorname{diag}(1, 0,\dots,0)$ and the s... | https://mathoverflow.net/users/69931 | Eigenvalues of the sum of two matrices, where one is $B=\operatorname{diag}(1, 0,\dots,0)$ | The question is to know if the $0$ eigenvalue of $A$ can become negative when we add $B$. We can write specific results only if $B$ is a small perturbation of $A$; it is easier to assume that $A$ is fixed and $B=diag(x,\cdots,0)$ with a small positive $x$.
Let $U$ be the matrix obtained from $A$ deleting its first co... | 0 | https://mathoverflow.net/users/9091 | 215647 | 101,845 |
https://mathoverflow.net/questions/215216 | 9 | I have been reading Hida's book "p-Adic automorphism forms on Shimura varieties" and I don't understand a point.
He first describes p-adic modular forms of tame level N as functions on the Igusa tower which can be interpreted as functions on triples $(E,\phi\_{p^\alpha},\phi\_N)$ where $\phi\_{p^\alpha}: \mu\_{p^\alph... | https://mathoverflow.net/users/69558 | Definition of p-adic modular forms | It's not a pushforward of a differential (or maybe it can be described as one, but it doesn't have to be).
Rather there is a pullback map from differentials on the elliptic curve to differentials on $\mu\_{p^\alpha}$. However, this map is an isomorphism mod $p^\alpha$, so you can apply the inverse to reverse the proc... | 3 | https://mathoverflow.net/users/18060 | 215657 | 101,852 |
https://mathoverflow.net/questions/215652 | 3 | Let $f: \mathcal{X}\to X$ be a morphism from a smooth DM-stack $\mathcal{X}$ to its coarse moduli space $X$. Assume that $X$ is also smooth. Is it true that $Lf^\*$ is fully faithful and induces an equivalence of $D^b(coh(X))$ with an admissible subcategory of $D^b(coh(\mathcal{X}))$?
| https://mathoverflow.net/users/30850 | Derived pullback of the coarse moduli morphism | If both $\mathcal{X}$ and $X$ are locally Noetherian and regular, then $f$ is flat. Then $Lf^\*$ is the usual pullback $f^\*$. If $\mathcal{X}$ is tame, then the natural transformation $$\theta:\text{Id} \Rightarrow f\_\*f^\*,$$ is a natural isomorphism. However, when $\mathcal{X}$ is not tame, this can fail. For insta... | 6 | https://mathoverflow.net/users/13265 | 215659 | 101,854 |
https://mathoverflow.net/questions/215672 | 1 | Let ${\cal L}$ be defined as in [this question](https://mathoverflow.net/questions/215582/how-big-is-the-lattice-of-all-functions). Is there a surjective lattice homomorphism $f: {\cal L}\to \mathbb{N}^\mathbb{N}$, where $\mathbb{N}^\mathbb{N}$ is the set of all functions, ordered point-wise?
| https://mathoverflow.net/users/8628 | Is there a surjective lattice homomorphism $f: {\cal L}\to \mathbb{N}^\mathbb{N}$? | $\mathcal L$ has a cofinal $\omega$-chain and $\mathbb N^{\mathbb N}$ does not.
| 5 | https://mathoverflow.net/users/75735 | 215676 | 101,857 |
https://mathoverflow.net/questions/193660 | 7 | Let $k$ be a field and $A$ a nonnegatively graded ring over $k$. Assume $A\_0 = k.$ We have a bigrading on $\operatorname{Ext}(k,k)$ (one corresponding to homological degree, one corresponding to the grading on $A$). We assume $A$ is Koszul, i.e., that $\operatorname{Ext}^{ij}(k,k)=0$ whenever $i\neq j$.
Following B... | https://mathoverflow.net/users/51424 | Compatibility of two definitions of Koszul dual | Let $A$ be a quadratic graded algebra over a field $k$ with finite-dimensional components $A\_n$ and $A\_0=k$. Then the construction of the quadratic dual algebra $A^!$ involves setting $A^!\_1$ to be the dual $k$-vector space to $A\_1$ and the subspace of quadratic relations in $A^!$ to be the orthogonal complement to... | 8 | https://mathoverflow.net/users/2106 | 215683 | 101,860 |
https://mathoverflow.net/questions/215682 | 0 | I want to prove that there exists $f:[0,1] \to [0,1]$ such that $f(0)=0$,
$$
\frac{d w(y-f(y))}{d y} = g(y) \frac{d v(f(y))}{d y}, \forall y \in [0,1],
$$
where $w:[0,1] \to [0,1]$ and $v:[0,1] \to [0,1]$ are continuous strictly increasing functions with $w(0)=v(0)=0$ and $v(1)=w(1)=1$, and $g:[0,1] \to (0,\infty)$ is ... | https://mathoverflow.net/users/49831 | How to prove that a non-linear differential equation has a solution | Before attempting anything complicated (such as the theory of Differential Algebraic Equations), I would try to put it in a standard first-order ODE form. By the chain rule:
$$w'(y-f(y))(1-f'(y))=g(y) v'(f(y)) f'(y)$$
Now, since $w'$, $v'$, and $g$ are positive, you can write an explicit ODE:
$$f'(y)=\frac{w'(y-f(y))}{... | 0 | https://mathoverflow.net/users/77927 | 215686 | 101,862 |
https://mathoverflow.net/questions/215687 | 10 | let $x=\sum\_{i=1}^{\infty}\delta\_i2^{-i},\ \delta\_i\in\{0,1\}$.
Is there an algorithm that converts the sequence $(\delta\_0,\ \delta\_1,\ ...)$ of the binary digits of $x$ to the sequence $[a\_0;a\_1,\ ...]$ of its continued fraction representation?
| https://mathoverflow.net/users/31310 | Continued Fractions from Digit Streams | Yes, there is. The algorithm is due to [Bill Gosper](http://perl.plover.com/yak/cftalk/INFO/gosper.txt) - he is considering the more general problem of doing linear fractional transformations with continued fractions - adding $2^{-i}$ is a special case. See also [Liardet and Stambul, 1998](http://www.sciencedirect.com/... | 12 | https://mathoverflow.net/users/11142 | 215688 | 101,863 |
https://mathoverflow.net/questions/214944 | 5 | For any Young tableau, one can form the Young symmetrizer. I'm naturally interested in young symmetrizers coming from standard tableaux, but I'm forced to look at Young symmetrizers of non-standard tableaux. Is there a known way of writing such a Young symmetrizer as a linear combination of Young symmetrizers of standa... | https://mathoverflow.net/users/41283 | Relations among Young symmetrizers of non-standard tableaux | Let $T$ be any (not necessarily standard) tableau of shape $\lambda$. Define $c\_\lambda(T)=a\_\lambda(T) b\_\lambda(T)$ as usual. Let $C\_i(T)$ denote the ith column of $T$. I Equation 2.4 in
<https://www3.nd.edu/~craicu/papers/yngsymm.pdf>
says that for two columns $C\_i(T)$, $C\_j(T)$ with $|C\_i(T)|\le |C\_j(T)... | 2 | https://mathoverflow.net/users/41283 | 215698 | 101,866 |
https://mathoverflow.net/questions/160728 | 4 | Suppose we have a non- tempered distribution $u\in \mathcal D'(\mathbb R^d)\backslash \mathcal S'(\mathbb R^d)$. Is it possible to have $\partial\_{x\_1}...\partial\_{x\_d}u \in \mathcal S'(\mathbb R^d)$ where the derivative is taken in the sense of Schwartz distributions? I cannot find an example nor prove the convers... | https://mathoverflow.net/users/46773 | The derivative of a non-tempered distribution can be tempered? | $e^x+e^y$ defines an obviously non tempered distribution on $\mathbb{R}^2$, whose *mixed* derivative $\partial\_x\partial\_y(e^x+e^y)=0$ ...
| 3 | https://mathoverflow.net/users/75422 | 215699 | 101,867 |
https://mathoverflow.net/questions/215171 | 4 | I was hesitant about posting this question here, but since it deals with a [partially unanswered question already on this site](https://mathoverflow.net/a/137024/73622) I figured that this would be the best place for it. I apologise in advance if not.
The question deals with explicitly calculating the ramification di... | https://mathoverflow.net/users/73622 | Ramification divisor on curves in weighted projective space | The curve $C\_7$ passes through the points $P = (0 : 0 : 1)$ and $Q = (0: 1: 0)$. You can see this by looking at what monomials can be used to define $C$. The group of order $6$ acting on $\mathbf{P}^2$ fixes these two points, hence the map $C\_7 \to C$ has ramification order $6$ at $P$ and $Q$. There are $6$ points on... | 4 | https://mathoverflow.net/users/60618 | 215700 | 101,868 |
https://mathoverflow.net/questions/215693 | 3 | Let $(P,\leq)$ be a poset with more than $1$ point such that the [interval topology](https://mathoverflow.net/questions/215673/properties-of-the-interval-topology-of-the-lattice-of-functions) $\tau\_i(P)$ is path-connected. Does this imply that $[0,1]$ order-embeds into $P$?
| https://mathoverflow.net/users/8628 | Path-connected interval topologies | No. If $P$ is any set and $<$ is the empty relation then the interval topology is just the cofinite topology. Now if $P$ has size $\mathfrak c$ then $P$ with the cofinite topology is path-connected just because any biyection $[0,1] \to P$ is continuous.
| 6 | https://mathoverflow.net/users/17836 | 215702 | 101,870 |
https://mathoverflow.net/questions/215697 | 4 | * A monoid is the same as a category with a single object.
* A monoidal category is the same as a bi-category with a single object.
* A commutative monoid is the same as a bi-category with a single object and a single 1-morphism (This is essentially the Eckmann-Hilton argument).
* A braided monoidal category is the sam... | https://mathoverflow.net/users/76299 | The "$\infty$"-column in the periodic table of n-categories | The result that $E\_n$-monoidal $(\infty,m)$-categories are equivalent to pointed $(\infty,n+m)$-categories with a single object, 1-morphism, ..., and $n$-morphism (and the more general one that $E\_n$-algebras in an $E\_n$-monoidal $\infty$-category $\mathcal{V}$ are equivalent to pointed $(\infty,n)$-categories enric... | 12 | https://mathoverflow.net/users/1100 | 215705 | 101,871 |
https://mathoverflow.net/questions/215706 | 4 | I am currently doing some inspection on the orthogonal polynomials with respect to the lognormal distribution. Does anyone already work on that or know some cool references?
All the best,
Pierre-O.
| https://mathoverflow.net/users/78551 | Orthogonal polynomials with respect to the lognormal distribution | Orthogonal polynomials with respect to the lognormal distribution go by the name of [Stieltjes-Wigert polynomials](https://en.wikipedia.org/wiki/Stieltjes%E2%80%93Wigert_polynomials). Two recent studies of their properties:
* [Global Asymptotics of
Stieltjes-Wigert Polynomials](http://arxiv.org/abs/1302.5193)
* [Vari... | 4 | https://mathoverflow.net/users/11260 | 215716 | 101,874 |
https://mathoverflow.net/questions/215738 | 11 | I have two 3D rotations about the origin, represented as
$3 \times 3$ orthogonal matrices $M\_1$ and $M\_2$
(specified by numerical entries),
and I would like to interpolate (and compute)
a continuous sequence of rotations
between $M\_1$ and $M\_2$.
Ideally this interpolation would follow a geodesic on SO($3$).
Assum... | https://mathoverflow.net/users/6094 | Geodesics on SO(3) | If you are assuming the use of the bi-invariant metric, then the geodesics are right/left translations of the one parameter sub-groups $O(t)=O(0) \exp(tG)$ where $G \in \mathfrak{so}(3)$ (i.e. it is anti-symmetric and traceless) and $O(0)\in SO(3)$.
The curve you want will have $O(0)=M\_1$ and thus be of the form $O(... | 9 | https://mathoverflow.net/users/41654 | 215740 | 101,881 |
https://mathoverflow.net/questions/215744 | 3 | is there a way to describe the convex hull of the set of permutation matrices with exactly one non-trivial cycle?
or maybe I should ask for the convex hull of cycle matrices :
let $(i\_{1},..,i\_{k})$ be a cycle then $A$ is a cycle matrix if the entries $(i\_{1},i\_{2})$ ...$(i\_{k},i\_{1})$ are $1/k$, and all the ... | https://mathoverflow.net/users/78574 | convex hull of the set of permutations with one cycle | Yes. It's exactly the collection of non-negative matrices subject to:
- The entries sum to 1;
- The diagonal entries are 0;
- For each $i$, the $i$th column and $i$th row have equal sums.
The proof is by induction on the number of non-zero entries.
Suppose that matrices satisfying the above conditions with at most $... | 0 | https://mathoverflow.net/users/11054 | 215752 | 101,888 |
https://mathoverflow.net/questions/215739 | 30 | Let's say you are a prospective mathematician with some addled ideas about cardinality.
If you assumed that the natural numbers were finite, you'd quickly vanish in a puff of logic. :)
If you thought that natural numbers and reals had the same cardinality - measure theory would almost surely break down, and your as... | https://mathoverflow.net/users/76572 | Why should we care about "higher infinities" outside of set theory? | $\newcommand\ZFC{\text{ZFC}}$Perhaps it would be useful to mention that set theorists have, of course,
studied numerous weaker set theories, including some extremely
weak theories, which do not give rise to higher cardinalities. One
may interpret your question as: to what extent do these weak set
theories serve as a fo... | 29 | https://mathoverflow.net/users/1946 | 215758 | 101,889 |
https://mathoverflow.net/questions/215757 | 0 | Let $P,Q$ be posets and endow them with the [interval topology](https://mathoverflow.net/questions/215673/properties-of-the-interval-topology-of-the-lattice-of-functions) $\tau\_i(P)$ and $\tau\_i(Q)$ respectively. Is it true that if $f: P\to Q$ is order-preserving, then it is continuous, and vice versa?
| https://mathoverflow.net/users/nan | Does order-preserving equal continuous? | Both implications are incorrect in general.
Consider the real interval $I=[0,1]$. Then the interval topology on $I$ is just the topology coming from the Euclidean metric on $I$. Let $f: I\to I$ be defined as $f(1) = 1$ and $f(x) = 0$ for $x\in I\setminus \{0\}$. Then $f$ is clearly order-preserving, but not continuou... | 1 | https://mathoverflow.net/users/8628 | 215760 | 101,891 |
https://mathoverflow.net/questions/215724 | 3 | Let $X$ be a reasonable topological space (say one that has the homotopy type of a finite CW complex) and consider a subset $\Gamma$ of $X \times X$ so that the projection $p:\Gamma \to X$ onto the first factor is surjective (making no assumptions on the other projection $q$):
$$ X \stackrel{p}{\gets} \Gamma \stackre... | https://mathoverflow.net/users/18263 | Classifying space for homology endomorphisms supported on a graph? | One obvious condition is that $H\_\*f$ must be a composition of a right inverse of $H\_\*p$ with $H\_\*q$, since $f$ will be the composition of a section of $p$ with $q$.
| 2 | https://mathoverflow.net/users/78588 | 215778 | 101,898 |
https://mathoverflow.net/questions/215403 | 5 | Let $X\_1,\dots,X\_n$ be a set of independent samples of a distribution $\mu$ on the unit square, let $\hat\mu\_n$ be the empirical distribution on the points $X\_1,\dots,X\_n$, and let $W\_1(\mu,\hat\mu\_n)$ denote the $1$-Wasserstein distance between $\hat\mu\_n$ and $\mu$. Are there any simple, concrete bounds on $\... | https://mathoverflow.net/users/70190 | Practical bounds for the Wasserstein distance in 2 dimensions | Sorry, I don't have the reputation to comment.
1. Most of the bounds in the first paper you cite are completely explicit as you go through the proofs, and can be used to obtain a bound of the form you want. Of course, the constants they write down are a little more complicated than the ones you normally get in e.g. ... | 3 | https://mathoverflow.net/users/78589 | 215781 | 101,901 |
https://mathoverflow.net/questions/215742 | 6 | Let $\Gamma$ be a discrete group. We can form two $C^\*$-algebras: the universal (or full) and reduced, to be denoted by $C^\*\_u(\Gamma)$ and $C^\*\_r(\Gamma)$ (respectively). Both of them are completions of the group algebra $\mathbb{C}\Gamma$ but with respect to different norms: universal norm is defined as the supr... | https://mathoverflow.net/users/24078 | Kernel of the natural map between group $C^*$-algebras | (Caleb Eckhardt already answered, but this was too long for a comment)
This is indeed true for general $C^\*$-algebra but use the fact that $\tau$ is a trace and not just a state.
An element of the maximal algebra is zero in the reduced algebra if it acts trivially on the regular representation, which is the GNS re... | 3 | https://mathoverflow.net/users/22131 | 215783 | 101,902 |
https://mathoverflow.net/questions/215784 | 4 | Is the following result true? If it is, could you plese give me a reference for it? Thanks in advance!
---
Let $(G, \mu)$ be any compact abelian group with Haar measure $\mu$ (The case I am interested in is $G=\mathbb{T}^L$ for some countable discrete group $L$), suppose $\phi$ is any measurable symmetric 2-cocyc... | https://mathoverflow.net/users/9305 | symmetric measurable 2-cocycles on compact abelian groups vanish? | One way to see this is to note that $T$ splits from any locally compact abelian group (D.L. Armacost, The Structure of Locally Compact Abelian Groups, 6.16). If the cocycle is commutative, then the associated extension
$$
0 \to T \to E \to G\to 0
$$
is a short exact sequence of abelian groups. Since $T$ splits, the coc... | 3 | https://mathoverflow.net/users/9672 | 215785 | 101,903 |
https://mathoverflow.net/questions/215789 | 7 | Let $X$ be a smooth projective curve. How do I construct a coherent sheaf $\mathcal{F}$ on $\text{Pic}^n X$ (i.e., the component of the Picard scheme of $X$ parametrizing line bundles of degree $n$) such that$$\mathbb{P}(\mathcal{F}) := \text{Proj}\,\text{Sym}(\mathcal{F})$$equals the $n$th symmetric power $\text{Sym}^... | https://mathoverflow.net/users/nan | Construction of coherent sheaf such that $\text{Proj}\,\text{Sym}\,(\mathcal{F}) = \text{Sym}^n X$ | There is an obstruction, as alluded to in my comments. If $X$ has genus $0$, for instance (perhaps not the case you are most interested in), then there exists such a sheaf when $n$ is odd if and only if $X$ is isomorphic to $\mathbb{P}^1\_k$, where $k$ is your field. So for a conic over $\mathbb{R}$ having no real poin... | 7 | https://mathoverflow.net/users/13265 | 215794 | 101,906 |
https://mathoverflow.net/questions/215788 | 4 | As [before](https://mathoverflow.net/questions/214873/bateman-horn-conjecture-continued), consider the "singular series", which shows up in the Bateman-Horn conjecture: for an irreducible polynomial $f,$ this is equal to
$$
s(f) = \prod\_p \frac{1-\frac{n\_f(p)}p}{1-\frac1p},
$$
which is obviously the value at $1$ of... | https://mathoverflow.net/users/11142 | Bateman-Horn, continued even further | See the following two papers of Nobushige Kurokawa, both appearing in Proc. Japan Acad. Ser. A:
"On Some Euler Products II" (volume 60, 1984, 365-368, esp. Proposition 1)
"Special Values of Euler Products and Hardy-Littlewood Constants" (volume 62, 1986, 25-28), where he writes $Z(s,f)$ for your $L\_f(s)$ and $C(f)... | 8 | https://mathoverflow.net/users/3272 | 215797 | 101,907 |
https://mathoverflow.net/questions/215673 | 3 | Let $(P,\leq)$ be a poset. The *interval topology* $\tau\_i(P)$ on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\leq x\}$ and $\uparrow x = \{y\in P: y\geq x\}$.
Define ${\cal L}$ as in [this question](https://mathoverflow.net/que... | https://mathoverflow.net/users/8628 | Properties of the interval topology of the lattice of functions | $\mathcal L$ is not connected or Hausdorff, but if you delete its countably
many isolated points the result is a closed, connected, convex sublattice (which is
still not Hausdorff).
To see this, call a function $f\in \mathbb N^{\mathbb N}$ *passive* at $n$ if
$f(n+1)=f(n)$, and *aggressive* at $n$ if $f(n+1)=f(n)+1$.... | 4 | https://mathoverflow.net/users/75735 | 215801 | 101,908 |
https://mathoverflow.net/questions/215800 | 14 | Let $Y$ be an orientable, smooth 3-manifold and let $X=Y\times S^{1}$. My question is that: when does $X$ admits a Riemannian metric with positive scalar curvature?
An obvious case is when $Y$ itself admits a psc metric. Are there any other case?
It was proved by Gromov and Lawson that any 3-manifold which contains... | https://mathoverflow.net/users/44651 | For a 3-manifold $Y$, when does $Y\times S^{1}$ admits a Riemannian metric with positive scalar curvature? | It is a theorem of Schoen and Yau (see [Jonathan Rosenberg's survey](http://www.math.umd.edu/~jmr/psc2006.pdf), theorem 1.10) that no closed aspherical 4-manifold is psc. That means that the three-manifold should be non-aspherical, which narrows it down to a very small list (especially if you insist on integer homology... | 8 | https://mathoverflow.net/users/11142 | 215803 | 101,909 |
https://mathoverflow.net/questions/214857 | 5 | I am reading V. Vapnik's "Statistical Learning Theory". The author layouts following two statistical laws related to empirical CDF. I am looking for reference about proofs on these two laws.
Let $F\_l(x)$ be the empirical CDF from $l$ i.i.d samples drawing from same distribution with CDF $F(x)$.
Iterated Logarith... | https://mathoverflow.net/users/39428 | Reference to iterated logarithm law and Smirnov law of empirical CDF | For Smirnov's result I think the easiest approach (at least on a "hand waving" level) is via empirical processes:
As long as $F$ is continuous then it suffices to consider uniform distributions.
Let $\Delta\_n(t) = n^{1/2}(F\_n(t) - t)$.
Then the Smirnov Cramer von Mises statistic you wrote is
$$
\int\_0^1 \Delta... | 1 | https://mathoverflow.net/users/57941 | 215804 | 101,910 |
https://mathoverflow.net/questions/215775 | 5 | Let $M = \{ G(x) = 0 \} \subseteq \mathbb{P}^4$ be a [quintic Calabi-Yau](https://en.wikipedia.org/wiki/Quintic_threefold) and $\mathbf{e} \in H^2(M, \mathbb{Z})$ such that $\int\_M \mathbf{e}^3 = 5$. Then as $t \gg 1$:
$$
\int\_M
e^{n \mathbf{e}}
e^{-\frac{t\mathbf{e}}{2\pi i}}
\left( 1 + \frac{5}{6} \mathbf{e}^2... | https://mathoverflow.net/users/1358 | Asymptotic $\int_M \mathrm{exp}[\mathbf{e}\left(n -\frac{t}{2\pi i}\right)] \left( 1 + \frac{5}{6} \mathbf{e}^2 \right)^{1/2} $ on quintic Calabi-Yau | OK, let's see if I can put my money where my commenting mouth is. Let me say at the outset that I have no idea where such an integral comes from, but I claim that it doesn't matter to answer the question.
First of all, $\mathbf e$ is nothing mysterious: it is the 2-form dual to a hyperplane section of $M$. The equati... | 10 | https://mathoverflow.net/users/75616 | 215811 | 101,912 |
https://mathoverflow.net/questions/215816 | 4 | Let $f$ be integrable over the interval $(0, 1)$, and
$$I\_n = \int\_0^{1} x^n f(x) \, \mathrm{d}x.$$
Suppose $f(x) = f(1-x)$; we can then show that
$$I\_n = \sum\_{k=0}^{n} \binom{n}{k} (-1)^k \, I\_{k}, $$
by letting $u = 1-x$ and using the binomial theorem.
Thanks to a [paper](http://gs1.dlut.edu.cn/newVe... | https://mathoverflow.net/users/70508 | On the search for an explicit form of a particular integral | High precision numerical computations suggest:
$$c\_1=73/5760 $$
$$c\_2=3625/580608$$
$$c\_3=5233001/1393459200$$
| 5 | https://mathoverflow.net/users/12481 | 215835 | 101,920 |
https://mathoverflow.net/questions/215817 | 20 | Brouwer famously proved, using principles motivated by intuitionistic choice sequences, that every function $\mathbb{R}\to \mathbb{R}$ is continuous. In *Sheaves in geometry and logic* (section VI.9), MacLane and Moerdijk exhibit a topos (the topos of sheaves on any sufficiently nice small full subcategory $\mathbf{T}\... | https://mathoverflow.net/users/49 | Brouwer's theorem for the Cauchy reals | Following Mike's suggestion, I post my comment as an answer.
Brouwer's theorem that all functions $\mathbb{R} \rightarrow \mathbb{R}$ are continuous holds in the effective topos. For example, this appears as Theorem 3.3.8 in van Oosten, *Realizability: An Introduction to its Categorical Side*. In fact it follows as a... | 9 | https://mathoverflow.net/users/30790 | 215842 | 101,925 |
https://mathoverflow.net/questions/215845 | 1 | I have a question. The automorphism group of the linear groups $GL(n,q)$, the group of linear transformations of $V = \mathbb{F}\_q^n$, and $SL(n,q)$, the subgroup of $GL(n,q)$ consisting of elements of determinant $1$, are well understood. Here $q$ is a prime power $p^r$ and $n$ is any positive integer.
My question ... | https://mathoverflow.net/users/5710 | Automorphism group of the affine groups AGL(n,q), ASL(n,q) | This is just a brief answer. The only autmorphisms other than the ones that you know about already arise from elements of $H^1(H,V)$ (with $H = {\rm SL}(n,q)$ or ${\rm GL}(n,q)$), where the corresponding automorphisms induce the identity on $V$ and on $G/V \cong H$, but map a complement $H$ of $V$ to a complement that ... | 5 | https://mathoverflow.net/users/35840 | 215849 | 101,927 |
https://mathoverflow.net/questions/215850 | 0 | $\mathbb Z$ acts on the lattice $\mathbb Z \times \mathbb Z$ by adding an element to itself n times.
I am studying some function arising from symplectic geometry which happens in my case to be naturally defined on $\mathbb Z \times \mathbb Z$ and is constant on each orbit of the $\mathbb Z$-action, thus it lives ob ... | https://mathoverflow.net/users/14105 | Does $\mathbb Z \times \mathbb Z$ mod the obvious $\mathbb Z$ action have more structure than just a set? | The set $(\mathbb Z\times \mathbb Z)\setminus \{(0,0)\}$ modulo the equivalence relation $(x,y)\sim(nx,ny)$, for $n\in \mathbb Z\_{\not =0}$, is known as $\mathbb Q\mathbb P^1$, the projective line over $\mathbb Q$. It's an algebraic variety over the rationals (or rather, it's more correct to say that it's the $\mathbb... | 8 | https://mathoverflow.net/users/5690 | 215857 | 101,929 |
https://mathoverflow.net/questions/215768 | 12 | Let $R$ be a commutative Noetherian ring and $I\subset R$ an ideal that is irreducible in the sense that if $I = J\_1 \cap J\_2$, then $I=J\_1$ or $I=J\_2$. Is (the ideal generated by) $I$ irreducible in the polynomial ring $R[x]$?
The answer to the question is "yes" since the number of irreducible components has a h... | https://mathoverflow.net/users/5495 | Is an irreducible ideal in $R$ also irreducible in $R[x]$? | I prove your question for (not necessarily Noetherian) commutative ring. Irreducible ideals in non-Noetherian ring are complicate (see [this question](https://mathoverflow.net/questions/87870/is-the-radical-of-an-irreducible-ideal-irreducible/88215#88215)). For Noetherian ring, see [our paper](http://arxiv.org/abs/1405... | 9 | https://mathoverflow.net/users/17901 | 215864 | 101,933 |
https://mathoverflow.net/questions/215664 | 36 | [EDITED mostly to report on the answer by **Kevin Costello**
(and to improve the **gp** code at the end)]
I thank Nicolas Dupont for the following question
(and for permission to disseminate it further):
>
> I have a playlist with, say, $N$ pieces of music.
> While using the shuffle option (each such piece is pl... | https://mathoverflow.net/users/14830 | The coupon collector's earworm | For the asymptotic case: Let $t\_1=n \log n - Cn$ and $t\_2 = n \log n + Cn$, where $C$ is slowly tending to infinity. It is a [classic result](https://stats.stackexchange.com/questions/7774/what-is-a-tight-lower-bound-on-the-coupon-collector-time) that as $C$ tends to infinity the probability all coupons are collected... | 22 | https://mathoverflow.net/users/405 | 215873 | 101,938 |
https://mathoverflow.net/questions/215868 | 5 | This may be a silly question but I don't know the answer.
I know the construction of (equivariant) K-spectrum $KU\_G$ and the periodicity of (equivariant) K-theory. But I don't know its structure maps and how they are constructed.
Can anybody show me the answer or some reference on this? Thanks very much.
| https://mathoverflow.net/users/1537 | The structure map of topological K-theory | That is a very reasonable question. By a $G$-spectrum $KU\_G$, $G$ a compact Lie group, one should mean a genuine $G$\_spectrum, so suitably indexed on representations of $G$. One gets the structure maps by use of equivariant Bott periodicity. A good sketch of how this goes, without full details, is given on pages 146-... | 5 | https://mathoverflow.net/users/14447 | 215879 | 101,942 |
https://mathoverflow.net/questions/215862 | 15 | Is there an example of a function $f:(a,b)\times(c,d)\to\mathbb{R}$, which is real analytic in its domain, integrable in the second variable, and such that the function
$$ g:(a,b)\to\mathbb{R},\qquad g(x) = \int\_c^d f(x,y) dy$$
is not real-analytic on $(a,b)$?
**Edit:** What about an example of bounded $f$ satisfyi... | https://mathoverflow.net/users/78621 | Can integration spoil real-analyticity? | $$\int\_0^1 \sqrt{x^2+y}\; dy = \dfrac{2}{3} \left((x^2+1)^{3/2} - |x|^3\right)$$
for $x \in (-1,1)$.
| 27 | https://mathoverflow.net/users/13650 | 215880 | 101,943 |
https://mathoverflow.net/questions/215852 | 12 | $\newcommand{\Z}{\mathbb{Z}}$
Suppose that $n$ is a large and known integer (say, with 100 digits) and that you are given access to a function
$$f(x) = x^a + x^b$$
with unknown exponents $a,b \in \Z/n$. You are allowed to evaluate $f$ on any cyclic ring $\Z/q$ with a solution to $x^n = 1$, where $x$ and $q$ are of ... | https://mathoverflow.net/users/1450 | Learning the exponents in a sum of two modular roots of unity | If you can compute discrete logarithms, there's an easy solution:
$$ f(\zeta) = \zeta^a + \zeta^b $$
$$ f(\zeta^2) = (\zeta^a)^2 + (\zeta^b)^2 $$
is a system of two equations in the quantities $\zeta^a$ and $\zeta^b$, allowing you to solve for $(\zeta^a, \zeta^b)$. There will be 2 solutions, but that just reflects ... | 9 | https://mathoverflow.net/users/nan | 215884 | 101,946 |
https://mathoverflow.net/questions/215838 | 3 | We consider the $n$-dimenstional finite vector space $\mathbb{F}\_2^{n}$ over the finite field of two elements. For a subset $A\subseteq \mathbb{F}\_2^{n}$ of even size $|A|=2m$ and a linear form $l\in(\mathbb{F}\_2^{n})^\*$ let us say that *$l$ bisects $A$*, if
$$|\{a\in A\ |\ l(a)=0\}|=|\{a\in A\ |\ l(a)=1\}|\,\,\,\,... | https://mathoverflow.net/users/50982 | Number of linearly bisected subsets in finite vector space $F_2^n$ | The probability that a random subset is bisected by a fixed linear subspace is
$$ \frac{ \sum\_{k=0}^{2^{n-1}}\binom{2^{n-1}}{k}^2}{2^{2^n-1}} = \frac{\binom{2^n}{2^{n-1}} }{2^{2^n-1} } $$
Using the asymptotic
$$\binom{N}{N/2} \approx \frac{ C 2^{N}} { \sqrt N} $$
for some constant $C$, we obtain
$$\approx \... | 3 | https://mathoverflow.net/users/18060 | 215888 | 101,947 |
https://mathoverflow.net/questions/215887 | 10 | Let $F\to E\to B$ be a fibration with $B$ simply-connected. Suppose all differentials in the cohomology Serre spectral sequence (corresponding to the above fibration) are zero maps. Then as a graded module,
$$
H^\*(E)\cong H^\*(F)\otimes H^\*(B).
$$
**Question 1:** as cohomology rings with cup products, do we still ... | https://mathoverflow.net/users/65800 | cup product and Steenrod operations in Serre spectral sequence | 1) No in general. A counterexample is the projective space bundle associated to a vector bundle. For a rank $n$ vector bundle, the fiber, $\mathbb C \mathbb P^{n-1}$, has cohomology ring $\mathbb Z[x]/(x^n)$ with $x$ in degree $2$. But in the total space, the usual lift of $x$ satisfies the equation
$$\sum\_{i=0}^n (... | 13 | https://mathoverflow.net/users/18060 | 215889 | 101,948 |
https://mathoverflow.net/questions/215891 | 1 | Let $\mu$ be a singular Borel probability measure on $[0, 1)$, and $f\in L^2(\mu)$. Estimate
$$\sup\_n\left|\int\_0^1 \left(1-e^{i\alpha\_n x}\right) f(x)d\mu(x)\right|$$
where $\alpha\_n\in\mathbb R$ and $|\alpha\_n|\leq L<1$.
This is my approach:
$$\sup\_n\left|\int\_0^1 \left(1-e^{i\alpha\_n x}\right) f(x)d\mu(x)... | https://mathoverflow.net/users/69931 | Estimate $\sup_n\left|\int_0^1 \left(1-e^{i\alpha_n x}\right) f(x)d\mu(x)\right|$ | This is more like comment, but too long.
1. You probably want to estimate, not to calculate.
2. $\mu$ is singular --- is this relevant?
3. You're doing almost right, although there should be absolute value under the integral:
\begin{gathered}
\left|\int\_0^1 \left(1-e^{i\alpha\_n x}\right) f(x)d\mu(x)\right|\leq \lef... | 1 | https://mathoverflow.net/users/8146 | 215892 | 101,949 |
https://mathoverflow.net/questions/215645 | 6 | Let $L$ be a line bundle over complex elliptic curve, $\deg L = k>0$. Theta functions
$$
\theta\_s(z;\tau)\_k=\sum\_{r\in \mathbb{Z}} e^{\pi i [(\frac{s}{k} + r)^2 k \tau + 2kz(\frac{s}{k}+r)]}, \hspace{20pt} s=0,...,k-1$$
form a basis of $H^0(E, L)$. I'm especially interested in the case $k=3$ when theta functions em... | https://mathoverflow.net/users/21029 | Derivatives of theta functions at zero | An eta-product identity for $k=3$, $s=1$ similar to that given in my comment above may be found in ["Some eta-identities arising from theta series"](http://www.mscand.dk/article/viewFile/12299/10315) by Günter Köhler (Math. Scand. **66**, 1990, p. 146, identity (3)):
$$
\frac{\eta^2(\tau)\eta^2(4\tau)}{\eta(2\tau)}=\su... | 6 | https://mathoverflow.net/users/41291 | 215893 | 101,950 |
https://mathoverflow.net/questions/188644 | 8 | For the purposes of this question, define the following properties of convex sets in the plane:
* A set is **$R$-fat** (for $R\geq 1$) if it contains a disc of side-length $x$ and is contained in a disc of side-length $R\cdot x$, for some positive $x$.
* A set is **$R$-cuttable** if it can be cut (using a straight li... | https://mathoverflow.net/users/34461 | Fixed point of fatness | Still, Pietro Majer is right --- there is no such $R\_0$.
Let $S$ be the convex hull of the unit circle centered at the origin and the point $A=(2R-1,0)$. Then it is $R$-fat, with the unit disk inside it and the smallest enclosing disk with radius $R$ and center $(R-1,0)$.
Now assume that $S$ is cut into two convex... | 3 | https://mathoverflow.net/users/17581 | 215901 | 101,951 |
https://mathoverflow.net/questions/215905 | 4 | Can I get a reference or some insight into the following? Suppose a particle moves by Brownian motion, starting from a point $P$ in $\mathbf{R}^{n}$. What can we say about the distribution of the time for $P$ to cross a coordinate hyperplane?
| https://mathoverflow.net/users/3394 | Time for brownian motion to cross a coordinate plane | Similar to what @zhoraster points out in the comments, this can be boiled down to 1-dimensional Brownian motion because projecting the $n$-dimensional Brownian motion $X(t)$ onto the normal direction to the hyperplane gives a Brownian motion $B(t)$. So the question becomes about a the time it takes for $B(t)$ to reach ... | 3 | https://mathoverflow.net/users/32961 | 215912 | 101,957 |
https://mathoverflow.net/questions/215825 | 6 | In [Keller: A-infinity algebras in representation theory, Proposition 1(b)], Keller states that for an associative algebra the $Ext$-algebra of the simples is generated by $Ext^1(S,S)$ as an $A\_\infty$-algebra.
Are there any other conditions on a module $M$ known to prove that $Ext^\*(M,M)$ is generated by $Hom(M,M... | https://mathoverflow.net/users/15887 | $Ext$-algebra generated by $Hom$ and $Ext^1$ as $A_\infty$-algebra? | Suppose $M=M\_1 \oplus \dots \oplus M\_r$ with $M\_i$ indecomposable for $1 \leq i \leq r$. Let $\mathcal F (M\_1, \dots, M\_r)$ denote the category of modules which admit a filtration with subquotients in the $M\_i$. The crucial assumption in Keller's proof is that $\mathcal F(M\_1, \dots, M\_r)$ is closed under syzyg... | 6 | https://mathoverflow.net/users/18756 | 215918 | 101,962 |
https://mathoverflow.net/questions/198782 | 5 | I heard that the following theorem is proved recently by Foreman-Magidor, which answers a famous old open question:
>
> **Theorem.** It is consistent, relative to the existence of large cardinals, that there is a singular strong limit cardinal $\kappa,$ such that the tree property hold at both $\kappa^+$ and $\kapp... | https://mathoverflow.net/users/11115 | Tree property and singular strong limit cardinals | The following theorem from Sinapova's paper ''[THE TREE PROPERTY AT THE FIRST AND DOUBLE
SUCCESSORS OF A SINGULAR](http://homepages.math.uic.edu/~sinapova/Single%20and%20double%20successors.pdf)'' answers your main question.
**Theorem.** Suppose that $\langle\kappa\_{n}:n<\omega\rangle$ is an increasing sequence of s... | 2 | https://mathoverflow.net/users/38866 | 215926 | 101,964 |
https://mathoverflow.net/questions/215923 | 53 | Similarly is the complement of any countable set in $\mathbb R^3$ simply connected?
Reading around I found plenty of articles discussing the path connectedness $\mathbb R^2 \setminus \mathbb Q^2$ and even an approach using cofiltered limits to approach that problem, but I am not read enough in that literature to see ... | https://mathoverflow.net/users/78672 | Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected? | Yes, the complement of any countable set in $\mathbb{R}^3$ is simply connected, by the Baire category theorem.
Say your set is $X = \{x\_1, x\_2, ... \}$, and let $y$ be any point in $\mathbb{R}^3 \setminus X.$
Let $f:S^1 \rightarrow \mathbb{R}^3 \setminus X$, and consider the space of homotopies $h:S^1 \times [0,1... | 72 | https://mathoverflow.net/users/1227 | 215930 | 101,965 |
https://mathoverflow.net/questions/215902 | 6 | I'm asking a question about Lie group representation.
Let $G$ be a Lie group, not necessarily connected. Let $\Omega$ be an element in the center of the universal enveloping algebra $U(\mathfrak{g})$ of the Lie algebra $\mathfrak{g}$ of $G$, so $\Omega$ commutes with every vector $X\in\mathfrak{g}$.
Let $\rho$ be a u... | https://mathoverflow.net/users/78655 | Does an element in the center of universal enveloping algebra becomes a scalar in irreducible representations? | No. Let $G$ be $O\_2(\mathbb{R})$, so the Lie algebra is one dimensional, and the center of the universal enveloping algebra is the symmetric algebra of the Lie algebra. Then the usual 2-dimensional representation (which is irreducible) does not have elements of the Lie algebra acting by scalars.
You need $\Omega$ to... | 8 | https://mathoverflow.net/users/121 | 215931 | 101,966 |
https://mathoverflow.net/questions/215802 | 4 | Let $X$ be a smooth projective variety (over the field of complex numbers). Let $T$ be strong generator of $D^b(X)$ : this means that every object in $D^b(X)$ can be obtained in a given finite number of steps from $T$ using the following operations : shift, direct sums, taking direct summands and cones.
Let $F,G : D^... | https://mathoverflow.net/users/37214 | Vanishing natural transformation and strong generator | This isn't really an answer to the question, but an example to show how badly things can go wrong.
Let $M$ be any indecomposable object of $D^b(X)$. The functor $\operatorname{Hom}(M,-)$, from $D^b(X)$ to vector spaces, has a subfunctor consisting of all maps from $M$ that are not split monomorphisms. Let $S\_M$ be t... | 4 | https://mathoverflow.net/users/22989 | 215958 | 101,978 |
https://mathoverflow.net/questions/215950 | 6 | Are there a finite group $G$ and a field $\mathbb{F}$ such that $\gcd(3,|G|)=1$ and the group algebra $\mathbb{F}[G]$ contains a zero divisor whose support is of size $3$?
Recall that the support of an element $\alpha$ of $\mathbb{F}[G]$ is the set $\{x\in G \;|\; \alpha(x)\neq 0\}$.
This question is related to the... | https://mathoverflow.net/users/19075 | Zero divisors with support of size 3 in group algebras of finite groups | Multiplication by $\sum\_{g\in G}g$ annihilates any element whose sum of coefficients is zero. It remains to choose such element with support of size 3, which is possible whenever $\mathbb F\neq GF(2)$.
| 11 | https://mathoverflow.net/users/17581 | 215959 | 101,979 |
https://mathoverflow.net/questions/215956 | 8 | In their paper, Darmon, Diamond and Taylor remarked the following :
(the previous paragraph of Section 2.2 (p. 55), <https://www.math.wisc.edu/~boston/ddt.pdf>)
If $\rho : G \rightarrow GL\_2(\mathbb{C})$ is irreducible, then $\text{ad}^0 \rho$ is either irreducible or the direct sum of two representations (one is a ... | https://mathoverflow.net/users/78695 | Remark on Fermat's Last Theorem by Darmon, Diamond and Taylor | The assocated $\mathbb{C}G$-module, say $M$, is $2$-dimensional, and
${\rm End}\_{\mathbb{C}}(M)$ is $4$-dimensional. The action of $g \in G$ on
${\rm End}\_{\mathbb{C}}(M)$ is via conjugation by $\rho(g)$. Then ${\rm End}\_{\mathbb{C}}(M)$ decomposes as a direct sum of two modules under this this action as $S \oplus... | 12 | https://mathoverflow.net/users/14450 | 215967 | 101,983 |
https://mathoverflow.net/questions/215960 | 2 | Let $X^n$ be an $n$-dimensional Alexandrov space with curvature bounded below. A point $x\in X$ is called regular if the space of directions $\Sigma\_x$ is isometric to the standard sphere $S^{n-1}$.
**QUESTION 1. Is it true that the set of regular points has full Hausdorff measure?**
(Rmk: Theorem 10.9.13 in the B... | https://mathoverflow.net/users/16183 | Set of regular points in an Alexandrov space with curvature bounded below | "Yes" to both questions.
For the second, take the projection to the tangent plane and note that its bi-Lipschitz in a small neighborhood of $x$ with constants as close to 1 as you want.
For the first one, see in 10.6 in "Alekandrov's Space with Curvature bounded from below" by Burago, Gromov and Perelman.
[In fac... | 7 | https://mathoverflow.net/users/1441 | 215978 | 101,986 |
https://mathoverflow.net/questions/215963 | 0 | We consider the following continuous-time nonlinear evolution problem
\begin{equation}
\begin{cases} \dot{y}(t)=Ay(t)+F(y(t),u(t)),\quad t\geq0\\y(0)=f\in\mathcal{X}\end{cases}
\end{equation}
where $A$ designs a linear operator of domain $D(A)$ defined on a Banach space $\mathcal{X}$ generating a $C\_0$-semigroup $T(t... | https://mathoverflow.net/users/24060 | Solutions of a nonlinear evolution problem | The problem is locally well-posed, i.e., the problem admits a unique
solution $y\in C([0,T];{\mathcal X})$ for some (in general small) $T>0$. In addition, it holds that $\dot{y}\in L^\infty((0,T);{\mathcal X})$ if $f\in D(A)$.
This can be proven by standard arguments. Indeed, denote the nonlinearity by $F$
(as $f$ i... | 1 | https://mathoverflow.net/users/69194 | 216000 | 101,992 |
https://mathoverflow.net/questions/215346 | 3 | The all-wise Wikipedia suggest (<https://en.wikipedia.org/wiki/Sobolev_space#Extension_by_zero>) that for $s > 1/2$ any function from $H^s\_0(\Omega)$ can be extended by zero
to whole $\mathbb{R}^n$ (with the extension being in $H^s(\mathbb{R}^n)$)
under some mysterious condition, namely $s \neq \mbox{integer} + 1/2$.
... | https://mathoverflow.net/users/78417 | Fractional Sobolev spaces and extension by zero | The answer is blowin' in the wind... unless $\Omega$ is Lipschitz domain, in which case the proof can be found in the book of`W. McLean "Strongly Elliptic Systems and Boundary Integral Equations".
I would also recommend the paper "Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains"... | 2 | https://mathoverflow.net/users/78417 | 216009 | 101,996 |
https://mathoverflow.net/questions/209703 | 8 | Does there exist a nonconstant, real analytic function $f \colon \mathbb{R} \to \mathbb{R}$ such that $f$ is periodic with period 1 and whose Maclaurin coefficients are all rational?
(The function $\sin x$ satisfies all the conditions except that its period isn't 1.)
I can't recall exactly what I was thinking about... | https://mathoverflow.net/users/7434 | Does there exist a nonconstant, periodic, real analytic function with period 1 and rational Maclaurin coefficients? | Yes, such $f$ exist, and can even be taken to be entire,
and with a preassigned finite initial segment $c\_0,c\_1,\ldots,c\_d$
of the sequence of $x^k/k!$ coefficients.
Indeed if $f(x) = \sum\_{k=0}^\infty a\_k \sin^k (2\pi x)$
then $f$ is entire provided $a\_k \rightarrow 0$ quickly enough,
say $a\_k \ll 1/k!$; and ... | 12 | https://mathoverflow.net/users/14830 | 216010 | 101,997 |
https://mathoverflow.net/questions/216003 | 6 | Let $S$ be a symmetric subset of a group $G$ containing the identity, and let $S^n$ be the set of all products of $n$ elements of $S$. If $S^3\subset gS$ for some translate $gS$ of $S$ then it follows that $S^2=S^3$. My question is, if $S^2\subset gS$ does it follow that $S=S^2$?
My question is about the growth of a ... | https://mathoverflow.net/users/78716 | The growth of a subset of a group | The answer is yes. In fact, it can be proved in a totally elementary way.
Because $1 \in S^2 \subset gS$, we have $g^{-1} \in S$ and thus $g \in S$. Then since $S^2 \subset gS \subset S^2$, it follows that $$S^2 = gS.$$ By symmetry of $S$, we obtain $S^2 = (S^2)^{-1} = (gS)^{-1} = S g^{-1}$. Hence $S \subset S^2 = g... | 5 | https://mathoverflow.net/users/36579 | 216017 | 101,998 |
https://mathoverflow.net/questions/216028 | 0 | I want to write a survey on the subject 'Sign changes for coefficients of symmetric power $L$-functions'. So, I browse the Web and I got some papers. I read it and I gave special interest to the paper namely 'Sign change of Hecke eigenvalues' of Kaisa Matomaki and Maksym Radziwill. In this paper, I found this sentence ... | https://mathoverflow.net/users/76102 | Question about sign change of Hecke eigenvalues | A sign change of a sequence $(a\_n)\_{n=1}^\infty$ means an $n$ such that $a\_na\_{n+1}<0$, i.e. $a\_n$ and $a\_{n+1}$ are of opposite sign. You can have many positive and negative terms with very few sign changes. For example, up to $x$ you can have $a\_n>0$ for $n\leq x/2$ and $a\_n<0$ for $n>x/2$, which means $x/2$ ... | 3 | https://mathoverflow.net/users/11919 | 216031 | 102,003 |
https://mathoverflow.net/questions/215927 | 6 | Let $R$ be a $k$-algebra (not necessary commutative) and let $\mathbf{D}(R)$ be its derived category (right modules). I'm interested in the class of objects $V$ of $\mathbf{D}(R^{op})$ having the following property (P):
For any (infinite) product $\prod\_{i\in I} M\_{i}\in \mathbf{D}(R)$ the natural map
$$ [\prod\_{i... | https://mathoverflow.net/users/73923 | Interchanging the tensor product with infinite product | The class of objects with property (P) is a thick subcategory of $\mathbf{D}(R^{op})$ (i.e., a triangulated subcategory closed under taking direct summands), and contains $R$, so it contains all perfect complexes, since the category of perfect complexes is the thick subcategory generated by $R$.
I think that the clas... | 5 | https://mathoverflow.net/users/22989 | 216037 | 102,004 |
https://mathoverflow.net/questions/216038 | 7 | Is there a small finite (perhaps of cardinality two or three) collection of cubic polynomials $p\_1, \dotsc, p\_k \in \mathbb{Z}[x]$ such that for every prime $p$ at least one of these is irreducible?
| https://mathoverflow.net/users/11142 | Irreducible cubics modulo primes | No. It follows from [Chebotarev's density theorem](https://en.wikipedia.org/wiki/Chebotarev's_density_theorem) that for any polynomials $p\_1,\dots,p\_k\in\mathbb{Z}[x]$ there are infinitely many primes that split all these polynomials. Simply, apply this theorem for the number field $K$ generated by the roots of $p\_1... | 11 | https://mathoverflow.net/users/11919 | 216044 | 102,007 |
https://mathoverflow.net/questions/216024 | 2 | Let $G\_i$ be connected finite simple undirected graphs with diameter $d\_i$ for $i=1,2$. Assume that $G\_1\times G\_2$ is connected. (Here $G\_1\times G\_2$ denotes the [categorical product](https://en.wikipedia.org/wiki/Tensor_product_of_graphs).)
In terms of $d\_1, d\_2$ what is the minimum and maximum value that ... | https://mathoverflow.net/users/8628 | Maximum and minimum diameter of categorical graph product | As Prof. Royle said, there is such upper bound which happen in many times. But, there is a good paper which this diameter exactly determined and maybe it is useful for you.
The paper is: "On the diameter of the Kronecker product graph" by "Fu-Tao Hu" and "Jun-Ming Xu".
you can find this paper in the Arxiv.
Also, ... | 3 | https://mathoverflow.net/users/19885 | 216048 | 102,009 |
https://mathoverflow.net/questions/216043 | 17 | Let us have a triangle ABC in the Cartesian plane and consider the following transformation of this triangle:
On the ray AB starting at A, select a point B' so that so that |AB'|=|AC|. Likewise,
on the ray BC starting at B, select a point C' so that BC' so that |BC'|=|AB|,
and on the ray CA starting at C select a ... | https://mathoverflow.net/users/78726 | A problem in elementary geometry | Let O be the point of intersection of the angle bisectors of the initial triangle ABC (the incenter). Then the length of OB' is equal to the length of OC, |OC'|=|OA| and |OA'|=|OB|. Now the statement of the problem will follow from the following fact: given point O and three length x,y,z, consider all triangles EDF tha... | 16 | https://mathoverflow.net/users/78738 | 216056 | 102,010 |
https://mathoverflow.net/questions/216042 | 8 | In the book [The homology of iterated loop spaces](http://www.math.uchicago.edu/~may/BOOKS/homo_iter.pdf), the homology Hopf algebra
(1)
$$
H\_\*(\Omega^n \Sigma^n X;\mathbb{Z}\_p)
$$
for primes $p\geq 2$ is obtained on p. 226, Thm. 3.2. In particular, the homology Hopf algebra
(2)
$$
H\_\*(\Omega^nS^n;\mathbb{Z... | https://mathoverflow.net/users/65800 | cohomology of iterated loop space on spheres | As is known from J.W. Milnor, J.C. Moore, "On the structure of Hopf algebras" Ann. of Math. (2), 81 : 2 (1965) pp. 211–264, the algebra structure of an underlying algebra of a Hopf algebra is quite limited, so that it suffices to study $p$-th powers to determine the algebra structure. Dually, it suffices to
study the ... | 7 | https://mathoverflow.net/users/43326 | 216059 | 102,012 |
https://mathoverflow.net/questions/216062 | 2 | Assume $A$ and $B$ are commutative algebras with $1$, $B = A[z] = A[Z]/(h(Z))$,
$Z$ an indeterminate.
The first comment in [this question](https://mathoverflow.net/questions/214355/is-it-possible-to-generalize-a-result-of-wang#comment533604_214355) says that, if $A$ is noetherian, then
$pd\_{B\otimes\_A B}(B) \in \{0... | https://mathoverflow.net/users/72288 | Projective dimension of a quotient ring | I will try to answer to both questions together but the second one only in a few very particular cases. I'm sorry for not having complete answers.
If $\phi :A \to B$ is flat and the rings $A$, $B$ and $B\otimes\_AB$ noetherian, then $\phi$ is smooth (or regular) if and only if fd$\_{B\otimes\_AB}(B)<\infty$ and in t... | 1 | https://mathoverflow.net/users/36672 | 216068 | 102,017 |
https://mathoverflow.net/questions/213200 | 6 | Let consider a binary word $x\_1 \ldots x\_n$ (finite sequence of elements of $\{0,1\}$.
I want to construct a polynomial $P$ that interpolates the points $(i, x\_i)$ for $i \in \{1\ldots n\}$ , such that:
* $P(x) \neq 0,1 $ for all $x \in [0,n]$ and $x \neq 1,\ldots n$
* $P(x) \in [0,1]$ for all $x \in [0,n]$
* $\... | https://mathoverflow.net/users/70925 | Polynomial interpolation of binary word signal | There was a related question:
[Finite interpolation by a nondecreasing polynomial](https://mathoverflow.net/questions/16673/finite-interpolation-by-a-nondecreasing-polynomial) with a satisfactory answer. The last two papers cited there allow to find the required polynomial.
The results of the first paper
[E. Passow... | 1 | https://mathoverflow.net/users/17581 | 216074 | 102,021 |
https://mathoverflow.net/questions/216065 | 4 | I've been reading [this paper](http://arxiv.org/pdf/math/9907044v1.pdf), in which authors prove that not all ITTM-computable functions $\Bbb R\rightarrow\Bbb R$ are 1-tape-computable, but if we put some restriction on the *output* of the function (e.g. restrict it to be $\Bbb N$), then they all are. I asked myself, "ho... | https://mathoverflow.net/users/30186 | Is every pair of writable reals one-tape-ITTM-computable? | How nice to hear that you are reading that paper. The paper appeared as:
* J. D. Hamkins and D. E. Seabold, [Infinite Time Turing Machines With Only One Tape](http://jdh.hamkins.org/onetape/), Mathematical Logic Quarterly, vol. 47, iss. 2, pp. 271-287, 2001.
It seems to me that your question is answered negatively ... | 2 | https://mathoverflow.net/users/1946 | 216076 | 102,022 |
https://mathoverflow.net/questions/216047 | 9 | Write $\partial := d/dt$, fix $m, n > 0$, and let$$F = \partial^n + f\_1(t)\partial^{n-1} + \dots + f\_{n-1}\partial + f\_0,\text{ }G:= \partial^m + g\_1(t)\partial^{m-1} + \dots + g\_{m-1}\partial + g\_0$$be a pair of ordinary differential operators with rational coefficients (i.e. with coefficients in $\mathbb{C}(t)$... | https://mathoverflow.net/users/nan | Commuting ODE's implies existence of nonzero vanishing two variable polynomial? | See Mulase's [notes.](https://www.math.ucdavis.edu/~mulase/texfiles/nato.pdf) The result whereof you speak is the Theorem of Burchnall-Chaundy.
| 4 | https://mathoverflow.net/users/11142 | 216092 | 102,026 |
https://mathoverflow.net/questions/216113 | 6 | Let $\mathbb{N}$ denote the positive integers and let $S = \{n^2: n\in \mathbb{N}\}$. For any positive integer $k$ we define $$\text{sq}(k) = |\{F\subseteq S: F\neq \emptyset, F\text{ is finite and } k = \sum\_{n\in F} n\}|.$$
Questions:
1. Is the function $\text{sq}:\mathbb{N}\to \mathbb{N}\cup\{0\}$ surjective?
2... | https://mathoverflow.net/users/78764 | Sums of unique squares | The generating function is
$$
\sum\_{n\geq 0} \text{sq}(n) z^n = \prod\_{k\geq 1} (1+z^{k^2}).
$$
Using complex integration you can use this to get an asymptotic formula for $\text{sq}(n)$. This involves quite some work, but the path is well described in Andrews, The theory of partitions, chapter 6. You will arrive at... | 14 | https://mathoverflow.net/users/37555 | 216117 | 102,039 |
https://mathoverflow.net/questions/216090 | 8 | The question seems like it should be known, but I was not able to find it anywhere.
How many binary strings of length $n$ are required so that for every $k$ positions in these strings, all $2^k$ possible subsequences occur?
For example, suppose $n=3$, and $k=2$. We want a set of binary strings of length $3$ so th... | https://mathoverflow.net/users/78756 | Repeats of all binary strings of length k | Orthogonal arrays were mentioned in another answer, but you are not requiring that each $k$-tuple occurs exactly once in every set of $k$ columns, but rather that each $k$-tuple occurs *at least once* in every set of $k$ columns.
What you are looking for is called a [covering array](http://math.nist.gov/coveringarray... | 7 | https://mathoverflow.net/users/48859 | 216129 | 102,044 |
https://mathoverflow.net/questions/216072 | 5 | Consider integers $r \geq 1$ and $k \geq 1$ and consider the following game:
We start with $r$ tokens and at each round we choose $i \in \{1,...,r\}$ tokens to bet (if we have $N<r$ tokens we can't bet more than $N$). If we chose to bet $i$ tokens then we flip a coin $X\_i$ with $P(X\_i=0) = (1-i/r)^k$ and $P(X\_i=1)... | https://mathoverflow.net/users/78340 | Optimisation of betting strategy | You can reduce the problem from optimizing on the space of strategies to analyzing the single-step deviations from what you believe is the optimal strategy. This is a common reduction.
Let $p\_k(n)$ be the probability that the game ends if you bet one token at a time, starting with $n$ tokens. At the boundary, $p\_k(... | 2 | https://mathoverflow.net/users/2954 | 216132 | 102,046 |
https://mathoverflow.net/questions/216127 | 1 | By a constructive characterisation I mean a theorem giving a list of base graphs and a list of operations such that every graph in a given class is generated from the base graphs by applying some sequence of these operations and every intermediate graph is also in the class.
For 3-regular graphs I found the paper "In... | https://mathoverflow.net/users/62562 | Are constructive characterisations of k-regular (simple) graphs known? | It's a good question for someone with a better memory than mine!
There are two papers giving such constructions for regular multigraphs without loops. One is Ding and Chen, Generating r-regular graphs, Discrete Applied Mathematics 129 (2003) 329–343. The other is a preprint of Carstens and Steffen, see [here](https:/... | 5 | https://mathoverflow.net/users/9025 | 216134 | 102,048 |
https://mathoverflow.net/questions/216135 | 5 |
>
> Let $G$ be a finitely generated residually finite group with positive
> rank gradient, and let $F\_2$ be the free group on $2$ elements. Must
> there be an embedding $i \colon F\_2 \to G$ ?
>
>
>
A group $G$ is called residually finite if the intersection of all of its finite index subgroups is trivial.
... | https://mathoverflow.net/users/38889 | Is there a nonabelian free group inside a group of positive rank gradient? | In
Denis Osin, Rank gradient and torsion groups.
Bull. Lond. Math. Soc. 43 (2011), no. 1, 10–16,
the following theorem is proved
>
> **Theorem** For every prime $p$, there exists a finitely generated infinite residually finite $p$-group with positive rank gradient.
>
>
>
<http://arxiv.org/abs/0905.1322>
... | 11 | https://mathoverflow.net/users/8176 | 216138 | 102,050 |
https://mathoverflow.net/questions/216137 | 1 | Let $A$ be a subcomplex of a CW complex $X$, let $Y$ be a CW complex, and let $f: A \to Y$ be a cellular map. What is the relationship between $H\_\*(X, A)$ and $H\_\*(Y \cup\_f X, Y)$? Is there a similar relationship between $\pi\_\*(X, A)$ and $\pi\_\*(Y \cup\_f X, Y)$?
Apologies in advance if this question is sort... | https://mathoverflow.net/users/78773 | Relationship between $H_*(X, A)$ and $H_*(Y \cup_f X, Y)$? $\pi_*(X, A)$ and $\pi_*(Y \cup_f X, Y)$? | Expanding my comment:
The natural map $H\_\*(X,A)\rightarrow H\_\*(X\cup\_f Y, Y)$ is an isomorphism, this is the so-called "excision axiom".
The map $\pi\_i(X,A)\rightarrow \pi\_i(X\cup\_f Y, Y)$ is an isomorphism through $i<n+m$, where $n,m$ are the connectivities of the maps $A\rightarrow X$, $A\rightarrow Y$, c... | 5 | https://mathoverflow.net/users/39747 | 216143 | 102,051 |
https://mathoverflow.net/questions/215756 | 0 | Let $(P,\leq)$ be a poset with more than $1$ point such that the [interval topology](https://mathoverflow.net/questions/215673/properties-of-the-interval-topology-of-the-lattice-of-functions) $\tau\_i(P)$ is path-connected and $T\_2$. Does this imply that $[0,1]$ order-embeds into $P$?
(This is a follow-up question t... | https://mathoverflow.net/users/8628 | Path-connected Hausdorff interval topologies | There is a counterexample as a subset of the real plane endowed with the natural partial order.
Take the graph $\Gamma$ of the standard continuous monotone map $f:C\to[0,1]$ of the Cantor set $C\subset[0,1]$ onto $[0,1]$. The map $f$ is defined anlytically by $f:\sum\_{n=1}^\infty \frac{2x\_n}{3^n}\mapsto \sum\_{n=1}... | 2 | https://mathoverflow.net/users/61536 | 216144 | 102,052 |
https://mathoverflow.net/questions/216156 | 8 | For a Riemannian manifold $M$ with metric $g$ and Laplace-Beltrami operator $-\Delta\_{g}$, what conditions on $M$ guarantee that $-\Delta\_{g} u(x)$ measures the difference between $u(x)$ and the average of $u$ over a geodesic ball (or sphere) centered at $x$? More precisely, what conditions on $M$ guarantee that
$$... | https://mathoverflow.net/users/78627 | Laplace-Beltrami and averaging | This holds in complete generality. I sketch the case in which, instead of the ball $B(x,h)$, you consider the metric sphere $S(x,h)$. In particular (here $\dim M = n$ and the Laplacian is $\Delta = \mathrm{div}\circ \mathrm{grad}$):
$$ (\Delta u)(x) = \lim\_{h\to 0} \frac{2n}{h^2}\frac{1}{|S(x,h)|}\int\_{S(x,h)} [u(y... | 8 | https://mathoverflow.net/users/13915 | 216157 | 102,057 |
https://mathoverflow.net/questions/216161 | 1 | When I was reading the paper of Winfried Kohnen, Yuk-Kam Lau and Igore E. Shparlinski (ON THE NUMBER OF SIGN CHANGES OF HECKE EIGENVALUES OF NEWFORMS), I found this result (which is Theorem 2 of the paper):
There are absolute constants $\eta <1$ and $A>0$ such that,
$$S\_{f}^{\pm}(x+x^{\eta})- S\_{f}^{\pm}(x)>0,$$ whe... | https://mathoverflow.net/users/76102 | Need an explanation of a deduction | The authors' formulation is a bit sloppy, but the argument is correct and straightforward.
Below, a "sign change" will mean a sign change of the sequence $(\lambda\_f(n))$ restricted to $\gcd(n,N)=1$.
The first display guarantees that in any interval $[x,x+x^\eta]$ with $x\geq (kN)^A$, there are $n^-$ and $n^+$ co... | 3 | https://mathoverflow.net/users/11919 | 216168 | 102,063 |
https://mathoverflow.net/questions/216154 | 7 | Let $M$ be any smooth manifold (could be unorientable - I think). Let $E,F \to M$ be two complex vector bundles. Let $S$ be any compact space, and let $D\_s:E\to F,s\in S$ be a continuous family elliptical pseudo-differential operator. Since $D\_s$ is a Fredholm operator (choosing the right Sobolev spaces to represent ... | https://mathoverflow.net/users/4500 | Parametrized Atiyah-Singer index theorem | Since Thomas is satisfied with my comment, I will post it as an answer to close: this result is in the 4th Atiyah-Singer paper: jstor.org/stable/1970756
| 5 | https://mathoverflow.net/users/6948 | 216169 | 102,064 |
https://mathoverflow.net/questions/216176 | 1 | I just came across the following problem:
Let us consider the unit corner of the n-cube
$$
\Delta^n = \left\{(t\_1,\cdots,t\_n)\in\mathbb{R}^n\mid\sum\_{i = 1}^{n}{t\_i} \leq 1 \mbox{ and } t\_i \ge 0 \mbox{ for all } i\right\}.
$$
Let $P$ be a polytope in $\Delta\_c^n$ generated as the intersection of $m$ half-space... | https://mathoverflow.net/users/78788 | approximate diameter of polytopes in high dimensions | This is not a definitive answer.
For the Euclidean distance, it is NP-hard to approximate the diameter within a constant. The best that can be achieved is a factor of $O(\sqrt{n/\log n})$:
>
> Brieden, Andreas. "Geometric optimization problems likely not contained in APX." *Discrete and Computational Geometry* 28... | 0 | https://mathoverflow.net/users/6094 | 216189 | 102,072 |
https://mathoverflow.net/questions/215444 | 0 | I was reading "From Rotating Needles to Stability of Waves: Emerging Connections between Combinatorics, Analysis, and PDE" by Terence Tao, which is a Notice of the American Mathematical Society Vol. 48, No 3.
There is a mention on energy estimates being fixed time estimates at a specified time, but requiring a lot o... | https://mathoverflow.net/users/nan | Reference request for the focussing example | The example is almost trivial. For argument sake fix the number of spatial dimensions to be 3. If you consider radially symmetric solutions to the wave equation, one observes that if
$$ u\_{tt} = u\_{xx} + u\_{yy} + u\_{zz} $$
is radially symmetric (meaning that the dependence on $x,y,z$ is only through the depend... | 3 | https://mathoverflow.net/users/3948 | 216194 | 102,074 |
https://mathoverflow.net/questions/216158 | 4 | While doing some computations on a *compact* Riemannian manifold I have reached the following expression:
$$ \Delta\_y \big( Ric\_y (\exp\_y ^{-1} x, \exp\_y ^{-1} x) \big) (x)$$
where $\Delta\_y$ is the Laplacian with respect to $y$, $Ric\_y$ is the Ricci tensor in $y$ and $\exp\_y : U\_y \subset TyM \to M$ is the... | https://mathoverflow.net/users/54780 | The Laplacian of an expression involving the Ricci tensor | In general, I don't think so. If my interpretation is correct (see my comment above), then if you take an Einstein manifold with $Ric = \lambda g$ you immediately see that the best you can expect is $O(r^{2})$. In fact, if you take the standard sphere $\mathbb{S}^n$, the Taylor expansion is precisely $r^2 + \ldots$ whi... | 3 | https://mathoverflow.net/users/3948 | 216196 | 102,076 |
https://mathoverflow.net/questions/216210 | 10 | I am trying to track down a copy of Graeme Segal's 1999 lecture notes on topological field theory. These are sometimes referred to as the "Stanford lectures" or something similar.
For many years these unpublished notes were available at Duke's mathematics website via the link: <http://www.cgtp.duke.edu/ITP99/segal/>... | https://mathoverflow.net/users/184 | Segal's 1999 Stanford lecture notes on TQFT, where to find them? | For dead links, sometimes <http://archive.org> helps.
In this case they might be these:
<http://web.archive.org/web/20000901075112/http://www.cgtp.duke.edu/ITP99/segal/>
| 9 | https://mathoverflow.net/users/12481 | 216213 | 102,081 |
https://mathoverflow.net/questions/216208 | 11 | In the paper [A faithful linear-categorical action of the mapping class group of a surface with boundary](http://arxiv.org/abs/1012.1032v2) it is claimed that it was (as of 2014) still unknown that mapping class group is linear, but the authors provide a faithful linear-categorical action of the mapping class group of ... | https://mathoverflow.net/users/62647 | Current status of the linearity of mapping class group | Yes. There [is the paper by Bigelow, and our own Ryan Budney](http://arxiv.org/pdf/math/0010310.pdf), who do this for genus 2. I believe that is still the last word.
| 6 | https://mathoverflow.net/users/11142 | 216219 | 102,083 |
https://mathoverflow.net/questions/216214 | 9 | Let $X$ be a topological space. Then its chain complex $C\_{\*}(X)$ is naturally a coalgebra (as explained here [Does homology have a coproduct?](https://mathoverflow.net/questions/415/does-homology-have-a-coproduct)). In particular if $X$ is simply connected we have that the homology of the cobar construction $\Omega ... | https://mathoverflow.net/users/41970 | bar construction and loop space cohomology | You should read about the Eilenberg--Moore spectral sequence. John McCleary's book "A User's Guide to Spectral Sequences" is a good place to start. Another good reference is Larry Smith's paper in Transactions of the AMS "Homological algebra and the Eilenberg--Moore spectral sequence". In particular, the answer to your... | 11 | https://mathoverflow.net/users/8103 | 216220 | 102,084 |
https://mathoverflow.net/questions/216184 | 30 | What would be a study path for someone in the level of Hartshorne's Algebraic Geometry to understand and study inter-universal Teichmuller (IUT) theory? I know that it heavily relies on anabelian geometry and earlier works of Mochizuki, but what's the order to study those material? I think I had seen somewhere a comple... | https://mathoverflow.net/users/24541 | A road to inter-universal Teichmuller theory | According to [Mochizuki himself](http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTeich%20Verification%20Report%202014-12.pdf), the essential prerequisites for the IUTeich papers are:
* [Semi-graphs of Anabelioids](http://www.kurims.kyoto-u.ac.jp/~motizuki/Semi-graphs%20of%20Anabelioids.pdf) (sections 1 to 6)
* [The Geome... | 28 | https://mathoverflow.net/users/43108 | 216226 | 102,087 |
https://mathoverflow.net/questions/215136 | 9 | Let $G\_k(\mathbb{R}^n)$, $n\geq k$ and $G\_k(\mathbb{R}^\infty)$ be the finite-dimensional and infinite-dimensional grassmannians respectively. Their cohomology rings are expressed in terms of universal Stiefel-Whitney classes
$$
H^\*(G\_k(\mathbb{R}^\infty);\mathbb{Z}\_2)=\mathbb{Z}\_2[w\_1,w\_2,\cdots,w\_k],$$
$$
H^... | https://mathoverflow.net/users/65800 | Steenrod operations on cohomology of grassmannians | As in Prasit's comment, the action of the Steenrod squares on the Stiefel-Whitney classes of any vector bundle are given by Wu's formula
$$
Sq^i(w\_j) = \sum\_{t=0}^i \binom{j+t-i-1}{t} w\_{i-t} w\_{j+t}.
$$
Here $w\_k$ is of course $0$ if $k$ exceeds the dimension of the bundle. Together with the [Cartan formula](http... | 11 | https://mathoverflow.net/users/8103 | 216229 | 102,089 |
https://mathoverflow.net/questions/216202 | 4 | Let $X$ be a smooth projective variety. Given two vector bundles $V\_1$ and $V\_2$ such that $[V\_1]=[V\_2]\in K^0(X)$, can one expect that $V\_1$ and $V\_2$ can be connected by a family of vector bundles? Or are there any counterexamples?
For example, if we have a short exact sequence of vector bundles
$$0\rightar... | https://mathoverflow.net/users/10333 | Connectedness of moduli of vector bundles | Maybe the simplest example of the phenomenon mentioned by Jason is the case of rank 2 vector bundles on $\mathbb{P}^3$ with $c\_1$ even, studied in [this paper](http://gdz.sub.uni-goettingen.de/download/PPN356556735_0035/PPN356556735_0035___LOG_0016.pdf) of Atiyah and Rees. Besides $c\_1$ and $c\_2$ which encode the $K... | 5 | https://mathoverflow.net/users/40297 | 216233 | 102,090 |
https://mathoverflow.net/questions/216215 | 11 | How much of category theory should I know to view schemes, sheaves and cohomology concepts as concrete cases of abstract categorical concepts? Is there a textbook of category theory for AG people?
| https://mathoverflow.net/users/75682 | Category theory for Algebraic Geometry | Paolo Aluffi's Algebra Chapter 0 develops abstract algebra using Category theory from the very beginning. The exposition is very clear and teaches upto and including the derived functor approach to cohomology. The category theory developed here should be more than enough to study sheaves and schemes eventually.
In a... | 17 | https://mathoverflow.net/users/26538 | 216250 | 102,100 |
https://mathoverflow.net/questions/216241 | 18 | I am sorry to ask a very vague question, but:
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> What are good ways to define the curvature of a finite metric space?
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The best way I can think of is: the curvature of a finite metric space $M$
is the infimum of the real $k$ such that there is a geodesic metric space $X$ which is $Cat(k)$ and $M$ embeds ... | https://mathoverflow.net/users/9317 | Curvature of a finite metric space | It is not a problem to define, Alexandrov's comparison inequalities make sense for all metric spaces. Check (3+1) and (2+2) point comparison in [our book](http://arxiv.org/abs/1903.08539). However, it is not clear what to do with these spaces (I do not know anything interesting about them unless they have length metric... | 10 | https://mathoverflow.net/users/1441 | 216258 | 102,103 |
https://mathoverflow.net/questions/207757 | 27 | Herstein's little book "Noncommutative Rings" has a chapter called Commutativity Theorems in which he proves results like Jacobson's theorem: if a ring (associative with identity, please) has the property that for each element $x$ there is an integer $n(x) > 1$ such that $x^{n(x)} = x$ then the ring is commutative. Thi... | https://mathoverflow.net/users/3272 | What are applications of commutativity theorems for rings? | The first profit I made of any kind off of mathematics was made off of a commutativity theorem.
In 1983, I was a first-year graduate student at Berkeley enrolled in T. Y. Lam's course Noncommutative Ring Theory. After he proved Jacobson's Theorem he reflected on how marvelous and simple the semantic proof was over a ... | 19 | https://mathoverflow.net/users/75735 | 216260 | 102,105 |
https://mathoverflow.net/questions/216242 | 7 | Let $\text{char}\,k = 0$ and $n \ge 2$. What is the easiest way to see that $k[x\_1, \dots, x\_n]$ is a free $k[x\_1, \dots, x\_n]^{S\_n}$-module with basis$$x\_2^{m\_2}x\_3^{m\_3} \dots x\_{n-1}^{m\_{n-1}} x\_n^{m\_n},\text{ }m\_2 \in [0, 1],\text{ }m\_3 \in [0, 2], \dots,\text{ }m\_n \in [0, n-1]?$$
| https://mathoverflow.net/users/78773 | Free $k[x_1, \dots, x_n]^{S_n}$-module? | It is enough to show that the given generators span $k[x\_1, \ldots, x\_n]$ as a $k[x\_1, \ldots, x\_n]^{S\_n}$ module. Once we've shown this, it is easy to see that $k(x\_1, \ldots, x\_n)$ is dimension $n!$ as a $k(x\_1, \ldots, x\_n)^{S\_n}$ vector space, so $n!$ vector which span must be linearly independent.
**No... | 4 | https://mathoverflow.net/users/297 | 216269 | 102,110 |
https://mathoverflow.net/questions/215755 | 1 | I have a rather basic question for which (surprisingly!) I cannot find a short and clear answer anywhere:
I'm currently looking at the Newman Penrose (NP) formalism (I use primarily Chandrasekhar's "Mathematical Theory of Black Holes").
My question is: what exactly is Minkowski spacetime in NP formalism? That is, w... | https://mathoverflow.net/users/51137 | Minkowski spacetime in Newman Penrose formalism | The one thing which I think you are missing is that
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> Unlike the Riemann curvature, the spin coefficients are not tensors (they are not co- or contra-variant).
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In particular, the spin coefficients heavily depend on the choice of the tetrad.
In Minkowski space you can construct "accelerated frames";... | 2 | https://mathoverflow.net/users/3948 | 216275 | 102,114 |
https://mathoverflow.net/questions/216262 | 3 | I am currently working on a problem where I have to minimize a $m$-strongly convex function
$$f ~: ~\mathbb{R}^n \rightarrow \mathbb{R}^+$$
over a bounded integer lattice,
$$L = \mathbb{Z}^n \cap [-10,10]^n \\$$
Let us denote a minimizer of $f$ over $L$ as $$x^\*\_{\text{int}} \in \underset{x \in L}{\text{argmin}} ~... | https://mathoverflow.net/users/49673 | Bounding the difference in the value of a strongly convex function at its integer minimum and other integer points | Unfortunately, no. Here is an example for $n=1$ (1-dimension). For parameters $m>0$, $b\in\mathbb{R}$ define:
$$f(x) = (m/2)(x-b)^2 $$
For any $b \in \mathbb{R}$, this function $f$ is strongly convex with modulus $m$.
Now fix $\epsilon \in (0, 1/2)$ and define $b = 1/2 - \epsilon$. Then the minimizer of $f(x)$ ov... | 3 | https://mathoverflow.net/users/73850 | 216279 | 102,116 |
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