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https://mathoverflow.net/questions/219189 | 3 | Let $\pi: M \to N$ be a smooth submersion from a bounded open subset of $\mathbb{R}^m$ onto $ N \subset \mathbb{R}^n$, $m \geq n$. Further, let $M$ be given a probability measure $\mu$. Then the map induces a probability measure $\nu$ on $N$. The goal is to compute the latter's density with respect to Lebesgue measure,... | https://mathoverflow.net/users/4923 | Reference request for a result regarding density of induced probability measure under a submersion | Answering my own question: Corollary 1.4 of this online note seems to address exactly the problem: <http://www3.nd.edu/~lnicolae/Coarea.pdf>
As expected it is intimately related to the co-area formula (Theorem 1.3). The latter is proved using Fubini's theorem locally, partition of unity, and implicit function theorem... | 1 | https://mathoverflow.net/users/4923 | 219278 | 102,922 |
https://mathoverflow.net/questions/219267 | 15 | According to [this paper](http://arxiv.org/pdf/math/9612208v1.pdf) of Chaz Schlindwein, any countable support iteration of semi-proper forcings is semi-proper. This seems like a breakthrough simplification, and I wonder why it is not more well-known. Before diving into the arguments I want to ask the forcing community ... | https://mathoverflow.net/users/11145 | Who needs RCS iterations? | I think that this claim is false, at least in the way that I understand "countable support". The following was explained to me by Menachem Magidor in the 1990s; it may be folklore, and I suspect it is the reason for introducing RCS in the first place.)
Consider the following countable support iteration $(P\_\alpha, ... | 15 | https://mathoverflow.net/users/14915 | 219279 | 102,923 |
https://mathoverflow.net/questions/219076 | 5 | I asked this [question at MSE](https://math.stackexchange.com/questions/1438481/a-question-on-galois-theory) but I did not receive an answer. So I ask it at MO:
We denote the field of rational numbers by $\mathbb{Q}$. The Galois group of a polynomial $f$ is denoted by $Gal(f)$. The commutator subgroup of a group $G$ ... | https://mathoverflow.net/users/36688 | Is there a $\mathbb{Q}$-linear map $T$ over $\mathbb{Q}[x]$ such that for all polynomials $Gal(T(f))$ $\simeq$ The commutator subgroup of $Gal(f)$? | The answer is no. First we see that $T$ is injective: Let $0\ne g\in\mathbb Q[X]$ with $T(g)=0$. Then $\textrm{Gal}(f-\alpha g)'=\textrm{Gal}(f)'$ for all $f\in\mathbb Q[X]$ and $\alpha\in\mathbb Q$. Pick $\beta\in\mathbb Q$ with $g(\beta)\ne0$, and $f\in\mathbb Q[X]$ of degree $n$ larger than $\deg g$ and $2$ such tha... | 17 | https://mathoverflow.net/users/18739 | 219290 | 102,926 |
https://mathoverflow.net/questions/219292 | 8 | I am trying to understand the behaviour of
$$\int^\infty\_{-\infty}\frac{\xi(1-it)}{\xi(1+it)}h(t)\frac{dt}{t}$$
where $h$ is a Schwartz function on $\mathbb R$, and $\xi(s)$ the completed Riemann zeta function. Clearly it is the quotient of zeta functions that is the most difficult to study.
One knows certain things... | https://mathoverflow.net/users/48554 | Behaviour of $\zeta(1-it)/\zeta(1+it)$? | For zeta itself, there is a clear result, that eventually the argument of zeta on the line $\Re(s)=1$ becomes very regular. Some more sophisticated things around this are in
D. Hejhal, "On a result of G. Polya concerning the
Riemann $\xi$-function", J. D'analyse Math. 55 (1990), 60-95.
Also, (5.14) of that paper r... | 7 | https://mathoverflow.net/users/15629 | 219294 | 102,928 |
https://mathoverflow.net/questions/219264 | 28 | (Crossposted from [math.stackexchange](https://math.stackexchange.com/questions/1450641/mochizukis-evidence-against-langlands) by suggestion)
On page 12 of Shinichi Mochizuki's "[On the Verification of Inter-universal Teichmuller Theory: A Progress Repor](http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTeich%20Verificati... | https://mathoverflow.net/users/nan | Mochizuki's "phenomena in number theory" outside the scope of Langlands | As noted by Lucia, large parts of number theory are completely "beyond the scope of the Langlands program": most of analytic number theory, obviously, is, but also many important, active and beautiful subfields of algebraic number theory -- for a list of examples, see for instance the list of publications of Bjorn Poon... | 30 | https://mathoverflow.net/users/9317 | 219306 | 102,933 |
https://mathoverflow.net/questions/219311 | 18 | A set $X\subseteq\mathbb{R}$ is *strong measure zero* if, for every sequence $(\epsilon\_i)\_{i\in\mathbb{N}}$ of positive reals, there is a sequence $(I\_i)\_{i\in\mathbb{N}}$ of open intervals covering $X$ such that $\mu(I\_n)<\epsilon\_n$.
Looking at various computability theoretic questions around strong measure ... | https://mathoverflow.net/users/8133 | Two strengthenings of "strong measure zero" | Strategically strong measure zero is equivalent to countable. To prove the nontrivial direction, suppose $X$ is strategically strong measure zero and $s$ is a winning strategy for player II. Consider the tree of finite plays in which player I plays only rational numbers and player II follows $s$. For any partial play $... | 20 | https://mathoverflow.net/users/6794 | 219317 | 102,938 |
https://mathoverflow.net/questions/219322 | 2 | It seems the real solution of the following equation $$t=u^2+u\log(u) $$ has no closed form in view of the output of
```
Reduce[t == u^2 + u*Log[u], u, Reals]
```
The question arises about the asymptotics of that solution as $t$ tends to infinity.
| https://mathoverflow.net/users/35959 | Asymptotics of solution of transcendental equation | Your equation has the form
$$u=\sqrt{t-u\log u}.$$
First obvious approximation is $u\_0(t)=\sqrt{t}$, second approximation is
$$u\_1(t)=\sqrt{t-u\_0(t)\log u\_0(t)}=\sqrt t-\frac{\log t}4+O\left(\frac{\log^2t}{\sqrt t}\right).$$ Next one is $u\_2(t)=\sqrt{t-u\_1(t)\log u\_1(t)}.$
| 4 | https://mathoverflow.net/users/5712 | 219324 | 102,939 |
https://mathoverflow.net/questions/219323 | 1 | Given $m,n\in\Bbb N$ with $m<n$, how many pythagorean triples $p^2+r^2=q^2$ satisfy $$m\leq p<r\leq n?$$
Is there a way to give a sharp estimate?
| https://mathoverflow.net/users/nan | Sharply Estimating Pythagorean Triples | See [my preprint](http://arxiv.org/abs/1211.6509). Page 12 (this has since been published, but preprint is easiest to access).
| 1 | https://mathoverflow.net/users/11142 | 219328 | 102,942 |
https://mathoverflow.net/questions/219334 | 4 | I am a beginner trying to understand Dan Quillen's idempotent map $\xi:MU\_{(p)} \to MU\_{(p)}$ described in his classic paper 5-page paper [On formal group laws...](http://www.ams.org/journals/bull/1969-75-06/S0002-9904-1969-12401-8/S0002-9904-1969-12401-8.pdf)
Let $F$ be a formal group law over a ring $R$ and and $... | https://mathoverflow.net/users/80739 | very basic question on p-typical group law and MU and BP | For the first question: In the universal (torsion-free) case, this can be found in Ravenel's book "Complex Cobordism and
Stable Homotopy Groups of Spheres" starting on page 345. [The book](http://www.math.rochester.edu/people/faculty/doug/mybooks/ravenel.pdf) is available online on its author's website.
| 2 | https://mathoverflow.net/users/69949 | 219340 | 102,947 |
https://mathoverflow.net/questions/219305 | 9 | Consider the following question:
Let $X$ be a simply connected, symplectic 4-manifold. Does there exists a smoothly embedded Klein bottle $K\subset X$ such that the following conditions are both satisfied?
(1) The self intersection number of $K$ is 0 (which means that the normal bundle has a non-where vanishing secti... | https://mathoverflow.net/users/44651 | Essential Klein bottle in simply connected symplectic 4 manifolds | Let $X$ be the algebraic variety over $\mathbb{R}$ obtained from the projective plain $\mathbb{P}^2$ by blowing up one point $P \in \mathbb{P}^2(\mathbb{R})$. Since $X$ is a smooth projective algebraic variety the space of complex points $X(\mathbb{C})$ admits a Kahler structure and hence the underlying $4$-dimensional... | 8 | https://mathoverflow.net/users/51164 | 219346 | 102,949 |
https://mathoverflow.net/questions/219336 | 5 | I am wondering if the following assertion is true:
Let $f,g:\mathbb{R}\_+\rightarrow [0,1]$ be completely monotone functions on $\mathbb{R}\_+^\*$, that is, $(-1)^n f^{(n)}(x)\geq 0$ and $(-1)^n g^{(n)}(x)\geq 0$ for any $x>0$ and any $n\in \mathbb{N}$. Assume that $f(0)=g(0)$ and $\lim\_{x\rightarrow \infty} f(x) =... | https://mathoverflow.net/users/54263 | Order between two completely monotone functions? | No. Let's take $g(x)=\epsilon e^{-\epsilon x}+(1-\epsilon)e^{-\alpha x}$,
$$
f(x) = \int\_{\epsilon}^{1+\epsilon} e^{-tx}\, dt = \frac{1}{x}e^{-\epsilon x}(1-e^{-x}) .
$$
Then clearly $f\le g$ near infinity, and near zero, $f(x)\simeq 1-(1/2+\epsilon)x$, $g(x)\simeq 1-(\epsilon^2+(1-\epsilon)\alpha)x$. I also want $f\l... | 6 | https://mathoverflow.net/users/48839 | 219348 | 102,950 |
https://mathoverflow.net/questions/219339 | 3 | I want to estimate $f\_p(e/p)$ as $p\to \infty$ for the series
$$
f\_p(x) = \sum\_{k=1}^p k! x^k
$$
I am sure this question in the right context is easy but I haven't found the information on how to compute it.
| https://mathoverflow.net/users/12337 | Asymptotic Estimation of the Factorial Power Series | Denote $k=p-m$. Then $$k!x^k=(p!x^p)e^{-m}\cdot\frac{p^m}{p(p-1)\dots (p-m+1)}.$$
For any fixed $m$ the fractions tends to 1. It follows that $$\lim \frac{\sum\_{m=0}^N (p-m)!x^{p-m}}{p! x^p}=\sum\_{m=0}^N e^{-m},$$
hence $$\liminf \frac{\sum\_{m=0}^{p-1} (p-m)!x^{p-m}}{p! x^p}\geq \sum\_{m=0}^\infty e^{-m}=\frac{e}{e-... | 4 | https://mathoverflow.net/users/4312 | 219351 | 102,951 |
https://mathoverflow.net/questions/219355 | 5 | This is a general question: As we know there are a lot of vanishing theorems like Fujita vanishing, kodaira Nakano vanishing, vanishing for big nef line bundle, Kollár vanishing, etc. Those just for projective guys.
My question is: are there some important vanishing problems for general complete algebraic manifold? A... | https://mathoverflow.net/users/63996 | vanishing theorem in algebraic geometry | I guess I could have just told you in person, but anyway, yes your specific question has a positive answer. To see this, use Chow's theorem and resolution of singularities to find a birational map $\pi:\tilde X\to X$ with $\tilde X$ smooth and projective. Now apply Kollár vanishing twice (or really Grauert-Riemenschnei... | 5 | https://mathoverflow.net/users/4144 | 219359 | 102,955 |
https://mathoverflow.net/questions/217792 | 65 | I know the question "how to study math" has been asked dozens of times before in many variations, but (I hope) this one is different.
My goal is to study derived algebraic geometry, where derived schemes are built out of simplicial commutative rings rather than ordinary commutative rings as in algebraic geometry (the... | https://mathoverflow.net/users/52914 | Derived algebraic geometry: how to reach research level math? | I propose the following plan, assuming a basic background in scheme theory and algebraic topology. I assume that you are interested in derived algebraic geometry from the point of view of applications in algebraic geometry. (If you are interested in applications to topology, you should replace part 2) of the plan by Lu... | 81 | https://mathoverflow.net/users/2503 | 219361 | 102,956 |
https://mathoverflow.net/questions/219349 | 1 | Let $X$ be a smooth proper variety of dimension greater than one defined over a number field. Let $A\_i$ be a sequence of pairwise disjoint finite closed subsets of $X$ indexed by natural numbers defined over number fields such that $|A\_{i}| \rightarrow \infty$ as $i \rightarrow \infty$. Suppose that any closed subvar... | https://mathoverflow.net/users/80578 | Constancy on large subsets | If $X$ is proper then $f$ is locally constant so it's possible to have non-constant $f$ iff $X$=$Y\cup Z$ such that $Y$ and $Z$ are each non-empty unions of connected components of $X$, $Y$ and $Z$ are disjoint, and each $A$$i$ is contained in $Y$ or is contained in $Z$.
I give 2 examples with $X$ = affine plane over... | 0 | https://mathoverflow.net/users/59248 | 219363 | 102,957 |
https://mathoverflow.net/questions/219344 | 12 | **I-** Is the following statement still a conjecture [see this article](http://arxiv.org/pdf/0711.1946.pdf) ?
**Conjecture (?)**
Let $M$ be a simply connected compact oriented $d$-manifold (smooth), then $HH^{\ast}(C^{\ast}(M))$ the Hochschild cohomology of cochain complexes associated to $M$ is isomorphic as a Gerst... | https://mathoverflow.net/users/79949 | Gerstenhaber conjecture for free loop space | I. To my knowledge over any field this conjecture is still open.
II. Over a field of characteristic zero, this conjecture is true, and we have more, we have a BV-isomorphism:
Félix, Yves; Thomas, Jean-Claude
"Rational BV-algebra in string topology."
Bull. Soc. Math. France 136 (2008), no. 2, 311–327.
| 9 | https://mathoverflow.net/users/27816 | 219381 | 102,962 |
https://mathoverflow.net/questions/219399 | 5 | Suppose that $T$ is an operator from a Banach space $X$ to a Banach space $Y$. Let $1<p<q<\infty$. If $TS$ is compact for any operator $S:l\_{p}\rightarrow X$, is $TR$ compact for any operator $R:l\_{q}\rightarrow X$?
| https://mathoverflow.net/users/41619 | A question on compact operators with domain $l_{p}$ | Yes. If $R:\ell\_q \to X$ and $TR$ is not compact then there is a normalized block basis $u\_n$ of the unit vector basis for $\ell\_q$ s.t. $TRu\_n$ is bounded away from zero. Let $V:\ell\_p \to \ell\_q$ be the bounded linear operator that maps $e\_n$ to $u\_n$ and set $S=RV$. Then $TS$ is not compact.
| 4 | https://mathoverflow.net/users/2554 | 219401 | 102,973 |
https://mathoverflow.net/questions/219391 | 12 | Consider any Banach space $X$, and let $Y$ be any dense subspace, then does it necessarily exist a closed linear operator $T$ defined on $X$, such that the domain of $T$ is exactly $Y$, i.e., $D(T)=Y$? Perhaps this looks not like a problem of research level, however, I can't solve it myself and can't find any reference... | https://mathoverflow.net/users/35702 | Existence of closed operators with arbitrary dense domain of a given Banach space | You remarked in comments that $T$ should be an unbounded operator from $X$ to $X$.
If $X$ is separable then $D(T)$ has to be a Borel set in $X$.
Note that $X \times X$ is a separable Banach space under a norm such as $\|(x\_1, x\_2)\| = \|x\_1\| + \|x\_2\|$. The graph $\Gamma(T) = \{(x, Tx) : x \in D(T)\}$ is by as... | 5 | https://mathoverflow.net/users/4832 | 219402 | 102,974 |
https://mathoverflow.net/questions/219366 | 21 | *This is a crossposting of <https://math.stackexchange.com/questions/1446602/anti-random-reals>, which has not gotten any answers; after thinking about the problem, I've become more convinced that it belongs here instead.*
For a function $f: \mathbb{N}\rightarrow\mathbb{R}\_{>0}$ and a set $X\subseteq \mathbb{R}$, an... | https://mathoverflow.net/users/8133 | Antirandom reals | Let A be a c.e. set with the property that there are infinitely many stages $s$ such that $A\_s\upharpoonright s = A\upharpoonright s$. Call such a stage "strongly true". Noncomputable $A$ with infinitely many strongly true stages can be produced using a movable marker construction.
I claim that any such $A$ is antir... | 13 | https://mathoverflow.net/users/68448 | 219406 | 102,975 |
https://mathoverflow.net/questions/219414 | 17 | Let $p$ be a prime congruent to $1$ mod. 8.
If $p= 17$ one has : $p+ 8 = 5 ^2$.
If $p= 41$ one has : $p+ 8 = 7 ^2$.
If $p= 73$ one has : $p+ 8 = 9 ^2$.
If $p= 89$ one has : $p+ 32 = 11 ^2$.
If $p= 97$ one has : $p+ 128 = 15 ^2$.
If $p= 113$ one has : $p+ 8 = 11 ^2$.
If $p= 137$ one has : $p+ 32 = 13 ^2$.... | https://mathoverflow.net/users/39552 | Does the equation $241+2^{2s+1}=m^2$ have a solution? | To answer your first question: there is indeed no $s$ such that
$241+2^{2s+1}$ is a perfect square. -- **Proof:** $2^{2s+1}$ is always
congruent to either $2$, $8$ or $32$ modulo $63$, which makes
$241+2^{2s+1}$ congruent to either $21$, $54$ or $60$ modulo $63$.
However none of these values is a quadratic residue modu... | 38 | https://mathoverflow.net/users/28104 | 219422 | 102,979 |
https://mathoverflow.net/questions/219425 | 5 | I have read around (e.g. in [Wikipedia](https://en.wikipedia.org/wiki/Antichain)) that if the ascending (descending) chain condition holds, all upper (lower) sets are the upper (lower) closure of an antichain, but I cannot find a proof.
More precisely, let $(P, \leq)$ be a poset, $A(P)$ be the set of antichains on $... | https://mathoverflow.net/users/37129 | ACC (DCC) implies upper (lower) sets are upper (lower) closure of antichains? | $\newcommand\P{\mathbb{P}}$Let $U$ be any upper set in a partial
order $\langle\P,\leq\rangle$, which satisfies the [descending
chain condition](https://en.wikipedia.org/wiki/Ascending_chain_condition), which asserts that every descending sequence of
points terminates. In other words, the order is a [well-founded relat... | 3 | https://mathoverflow.net/users/1946 | 219426 | 102,981 |
https://mathoverflow.net/questions/219432 | 14 | I am reading [this paper](http://arxiv.org/pdf/0712.0160v3.pdf) and at the top of page 5 the author makes reference to categories consisting of flat complex vector bundles over $X$ where $X$ is an arbitrary topological space. However, the only notion of flat vector bundle which I have seen uses a connection on the vect... | https://mathoverflow.net/users/70461 | Is there a notion of "flat vector bundle over a topological space"? | A flat vector bundle over a topological space is a bundle whose transition functions can be taken to be locally constant; equivalently, over a path-connected space, it's the same data as a principal $G$-bundle ($G = GL\_n(\mathbb{R})$ or $GL\_n(\mathbb{C})$ as appropriate) where $G$ is given the discrete topology. Over... | 21 | https://mathoverflow.net/users/290 | 219437 | 102,985 |
https://mathoverflow.net/questions/219449 | 7 | Here is an observation (thanks to OEIS):
$$\sum\_{i=0}^\infty \frac{i^k}{i!}= B\_k e,$$ where $B\_k$ is the $k$-th Bell number. I might be having reading comprehension issues, but I don't see this formula in the OEIS notes. I assume this is very well known - can someone point me at a reference or a simple proof?
| https://mathoverflow.net/users/11142 | Another formula for Bell numbers | That is essentially [Dobinski's formula](https://en.wikipedia.org/wiki/Dobinski%27s_formula).
| 12 | https://mathoverflow.net/users/2954 | 219451 | 102,992 |
https://mathoverflow.net/questions/219446 | 5 | Let $p$ be a prime number, and denote by $\mathbb{F}\_p$ the field with $p$ elements.
Is there a classification of the maximal subgroups of $G = \mathrm{SL}\_3(\mathbb{F}\_p)$ ?
I am interested in the indices of these subgroups in $G$ and in their ranks (minimal cardinality of a generating set).
| https://mathoverflow.net/users/38889 | Maximal subgroups of special linear groups over finite fields | The original references are:
H.H. Mitchell.
Determination of the ordinary and modular ternary linear groups.
Trans. Amer. Math. Soc. 12 (1911), 207-242.
R.W. Hartley.
Determination of the ternary collineation groups whose coefficients
lie in the $\mathrm{GF}(2^n)$.
Ann. of Math. 27 (1925/6), 140-158.
Our book is
... | 9 | https://mathoverflow.net/users/35840 | 219452 | 102,993 |
https://mathoverflow.net/questions/219471 | 7 | Let $A$ and $B$ be two square matrices with complex entries.
Let $\lambda\_1, \ldots, ,\lambda\_n$ be the Eigenvalues of $A$ and
$\mu\_1, \ldots, ,\mu\_m$ be the Eigenvalues of $B$.
Then the Eigenvalues of the Kronecker product are exactly the products $\lambda\_i \cdot \mu\_j$.
Is there an analogue for the sums of E... | https://mathoverflow.net/users/36563 | "Additive version" of Kronecker product | **Federico** already mentioned the keyword. The precise answer may be found among others as [Theorem 13.16, of this book](http://www.siam.org/books/textbooks/OT91sample.pdf). (That theorem makes a restriction to real matrices, but that is not necessary).
| 7 | https://mathoverflow.net/users/8430 | 219474 | 102,999 |
https://mathoverflow.net/questions/219478 | 4 | Given a $\mathbb{C}$-scheme $S$, two $S$-schemes $X$ and $Y$ that are flat over $S$ and a coherent sheaf of $O\_Y$-modules $F$.
Assume we have a (faithfully) flat $S$-morphism $\pi: X \rightarrow Y$ and we know that $\pi^{\*}F$ is a coherent sheaf of $O\_X$-modules that is flat over $S$.
Can we conclude that $F$ is... | https://mathoverflow.net/users/70593 | Can relative flatness of a sheaf be tested using (faithfully) flat morphisms? | You can assume less: $\pi$ faithfully flat, $F$ quasi-coherent, no assumptions on structure morphisms. Write down the functors and see immediately their how faithfulness and exactness depend on each other's.
If you need a reference, EGA IV$\_2$ 2.2.11 (iii).
| 3 | https://mathoverflow.net/users/10941 | 219483 | 103,002 |
https://mathoverflow.net/questions/219433 | 7 | Let $\{X\_{1},X\_{2},\cdots,X\_{n}\}$ be a random sample of size $n$. Denote $(X\_{(1)},X\_{(2)},\cdots,X\_{(n)})$ to be its descending order statistics. Define gap $g\_{i}(n)$ to be $g\_{i}(n)=X\_{(i)}-X\_{(i-1)},1\leq i\leq n$. My question is what is the limiting distribution of $g\_{i}(n)$ as $n\to\infty$ or after s... | https://mathoverflow.net/users/18420 | Gaps between descending order statistics | If the $X\_i$ are independent and uniformly distributed on $[0,1]$, the (finite) point process $\sum\_1^n \delta\_{nX\_i}$ converges in law to a Poisson point process on $\mathbb R^+$ with unit intensity, so the scaled gaps $ng\_i(n)$ are exponentially distributed (density $e^{-x}$) in the limit.
More generally, if $... | 2 | https://mathoverflow.net/users/75422 | 219490 | 103,003 |
https://mathoverflow.net/questions/219459 | 5 | (In what follows, Freiling's Axiom of Symmetry is simply the following:
($A\_{\aleph\_0}$) :( $\forall$$f$: $\mathbf R$ $\rightarrow$$\mathbf R\_{\aleph\_0}$)($\exists$$x\_1$,$x\_2$)($x\_2$$\notin$$f$($x\_1$) $\land$ $x\_1$$\notin$$f$($x\_2$)), where $\mathbf R$ are the reals, $\mathbf R\_{\aleph\_0}$ is the set of a... | https://mathoverflow.net/users/20597 | A question regarding a common critique of Freiling's Axiom of Symmetry | Claim: The principle $A\_{null}$ follows from continuum is RVM.
Proof: Let $m$ be a total extension of Lebesgue measure. Suppose every vertical section of $A \subseteq \mathbb{R}^2$ is Lebesgue null. First check that $A$ is $m \otimes m$-null. Put $B = \{(y, x) : (x, y) \in A\}$. Then $B$ is also $m \otimes m$-null. ... | 5 | https://mathoverflow.net/users/2689 | 219496 | 103,005 |
https://mathoverflow.net/questions/219498 | 4 | Let $X \subseteq \mathbb{A}^n$ be algebraic, and let $x$, $y \in \mathbb{A}^n - X$. How do I see that there exists $f \in I(X)$ with $f(x) \neq 0$ and $f(y) \neq 0$.
| https://mathoverflow.net/users/nan | Exists $f \in I(X)$ such that $f(x) \neq 0$, $f(y) \neq 0$ | Given an affine scheme $S=\operatorname {Spec (R)} $ and a closed subscheme $T=V(J)\subset S$, the restriction mapping $$\mathcal O(S)=R\to \mathcal O(T)=R/J$$ is obviously surjective.
Applying this to $S=\mathbb A^n, T=X\cup \{x,y\}$ and taking for $f\_0\in \mathcal O(T)$ the function equal to $0$ on $X$ and $1$ on... | 7 | https://mathoverflow.net/users/450 | 219504 | 103,007 |
https://mathoverflow.net/questions/219501 | 5 | Let $X$ be a Banach space. My question is: $X$ contains no copy of $l\_{1}$ if and only if any operator from $X$ to $l\_{1}$ is compact? I guess that the necessary part may be true. But is the sufficient part true? At least, the sufficient part is true for $X=c\_{0},l\_{p}(1<p<\infty)$.
| https://mathoverflow.net/users/41619 | A question on characterizing a Banach space containing no copy of $l_{1}$ | Since weakly compact operators into $\ell\_1$ are compact, and since by a result of Kadec and Pelczynski every non-weakly compact operator into $\ell\_1$ fixes a copy of $\ell\_1$, we have that if $X$ contains no copy of $\ell\_1$ then every operator from $X$ to $\ell\_1$ is compact.
However, the converse is not tru... | 3 | https://mathoverflow.net/users/848 | 219506 | 103,008 |
https://mathoverflow.net/questions/219529 | 7 | Let $f:[0,1]^2\rightarrow \mathbb{R}$ be a twice continuously differentiable function with the property that for all $x\in [0,1]$, there is an interval $I\_x\subset [0,1]$ such that $f(x,y)=0$ for all $y\in I\_x$. Does it follow that there must exist an open ball in $[0, 1]^2$ where $f$ is identically $0$?
| https://mathoverflow.net/users/80842 | On the zero set of a $C^2$ function on $[0,1]^2$ | Yes. Each $I\_x$ contains some segment $[p,q]$ with rational endpoints. Let $A(p,q)$ be a set of corresponding $x$. It is impossible by Baire category theorem that each $A(p,q)$ is nowhere dense. Thus there exist $p,q$ and a segment $\Delta$ of positive length such that $A(p,q)$ is dense in $\Delta$. Hence $f$ does van... | 13 | https://mathoverflow.net/users/4312 | 219530 | 103,012 |
https://mathoverflow.net/questions/219522 | 8 | Let us consider dgCat, the "collection" of all small dg-categories. In [On differential graded categories](https://atlas.mat.ub.edu/grgta/articles/Keller.pdf) and [Lectures on dg categories](http://www.mi.ras.ru/~akuznet/dgcat/Toen%20Lectures%20on%20DG-categories.pdf) the authors state that they form a category, i.e. d... | https://mathoverflow.net/users/24965 | Is dgCat a category or a 2-category? | The model structure on the category of dg-categories presents an $(\infty,1)$-category DGCat. This structure is essentially provided by the existence of mapping spaces (or mapping $\infty$-groupoids) between dg-categories.
To see why DGCat admits the further structure of an $(\infty,2)$-category, it is sufficient to ... | 10 | https://mathoverflow.net/users/2503 | 219532 | 103,013 |
https://mathoverflow.net/questions/219512 | 1 | Let $ a $, $ b $ and $ c $ be real constants such that $ \Delta \stackrel{\text{df}}{=} a c - b^{2} > 0 $. The *Beltrami Equations* are defined as the following system of PDE’s on the domain $ \Bbb{R}^{2} $:
$$
u\_{x} = \frac{1}{\sqrt{\Delta}} (b v\_{x} + c v\_{y}), \qquad
u\_{y} = - \frac{1}{\sqrt{\Delta}} (a v\_{x} +... | https://mathoverflow.net/users/50614 | Are solutions of the Beltrami Equations necessarily smooth? | As Deane Yang suggested you can regard the first order Beltrami system as an elliptic system, see Chapter 7.3 in the following book
Astala, Kari; Iwaniec, Tadeusz; Martin, Gaven Elliptic partial differential equations and quasiconformal mappings in the plane. Princeton Mathematical Series, 48. Princeton University Pr... | 4 | https://mathoverflow.net/users/26608 | 219533 | 103,014 |
https://mathoverflow.net/questions/219491 | 7 | Let $p$ be a prime number, and let $\mathbb{F}\_p$ be the unique field of cardinality $p$.
What is $\max \{d(H) : H \leq \mathrm{SL}\_3(\mathbb{F}\_p)\}$?
Here we denote by $d(G)$ the smallest cardinality of a generating set of a group $G$.
| https://mathoverflow.net/users/38889 | What is the maximal possible rank of a subgroup of a special linear group mod a prime? | I am very confident that, for $q=p^e$ with $p$ prime, the answer is $2e$ when $q$ is even, and $2e+1$ when $q$ is odd. There is an elementary abelian subgroup $P$ of rank $2e$ with
$$P=\left( \begin{array}{ccc}1&a&b\\0&1&0\\0&0&1\end{array} \right).$$
When $q$ is odd, all elements of $P$ are inverted by the diagonal ma... | 5 | https://mathoverflow.net/users/35840 | 219535 | 103,015 |
https://mathoverflow.net/questions/219505 | 5 | In [Wise1](http://www.gidon.com/dani/tl.cgi?athe=pspapers/CycHNN.ps) Wise shows that hyperbolic graphs of free groups with cyclic edge groups are subgroup separable. In [Hsu-Wise](https://docs.google.com/open?id=0B45cNx80t5-2dFpkTTlvSXREaFk) these are shown to be cubulated, and by Agol they're virtually special, so qua... | https://mathoverflow.net/users/38698 | Local quasiconvexity in graphs of free groups with cyclic edge groups | You're right that hyperbolic graphs of free groups with cyclic edge groups are locally quasiconvex. This can be proved by combining subgroup separability with results about combination of quasiconvex subgroups, such as the [one](http://arxiv.org/abs/0709.2113) proved in the relatively hyperbolic context by Eduardo Mart... | 1 | https://mathoverflow.net/users/1463 | 219547 | 103,020 |
https://mathoverflow.net/questions/219527 | 2 | Here is the problem. Suppose I have a positive definite matrix-valued function $A\colon \mathbb{R}^n\to \mathbb{R}^{n\times n}$. Then we know that there is a matrix-valued function $B\colon \mathbb{R}^n\to \mathbb{R}^{n\times n}$ such that $A=B^TB$, where $B^T$ is the transpose of $B$. I wonder if $A\in C^k$, what is t... | https://mathoverflow.net/users/26608 | Regularity of decomposition of matrix-valued function | I recommend that you apply the Cholesky algorithm leading to the matrix $B$, see <https://en.wikipedia.org/wiki/Cholesky_decomposition>. It will follow that the smoothness class of $B$ will be the class obtained from the smoothness class of $A$ by applying several times the square root operation to positive functions (... | 2 | https://mathoverflow.net/users/29557 | 219549 | 103,022 |
https://mathoverflow.net/questions/219542 | 4 | Several important papers on Lévy processes are referring to the following paper:
>
> K. Urbanik and WA Woyczynski, *A random integral and Orlicz spaces*,
> Bulletin de l'Académie Polonaise des Sciences, Série des sciences
> mathématiques, astronomiques et physiques, 15 (1967), p. 161-169
>
>
>
I couldn't fin... | https://mathoverflow.net/users/39261 | Reference request: Urbanik's work on random integrals and Orlicz spaces | It doesn't seem to be available on the internet. I uploaded it here though:
<http://dropcanvas.com/tpwrx>
| 4 | https://mathoverflow.net/users/54263 | 219553 | 103,024 |
https://mathoverflow.net/questions/219552 | 5 | Consider the following variant of [Freiling's Axiom of Symmetry, $\mathsf{AS}$](https://en.wikipedia.org/wiki/Freiling%27s_axiom_of_symmetry), which will be denoted $A\_{< 2^{\aleph\_0}}$:
* given any function $f$ from $\mathbb{R}$ into the families of of subsets of $\mathbb{R}$ of size $< 2^{\aleph\_0}$ there are $x... | https://mathoverflow.net/users/20597 | A variant of Freiling's Axiom of Symmetry and a weak form of the Continuum Hypothesis in models where all sets of reals are Lebesgue measurable | Assuming $AD$, a version of the continuum hypothesis holds: every set of reals is either countable or of size continuum (this is a consequence of the *perfect set property*). So assuming $AD$, your $A\_{<2^{\aleph\_0}}$ and Freiling's $AS$ are equivalent. In particular, both can hold in Solovay's model, so the answer t... | 3 | https://mathoverflow.net/users/8133 | 219554 | 103,025 |
https://mathoverflow.net/questions/218878 | 2 | Consider a closed connected Riemannian manifold $M$, together with the associated Hilbert space $L^2(M)$ defined with respect to the Riemannian volume density. Let $-\Delta$ be the positive Laplacian $H^2(M)\to L^2(M)$, and let $\{E\_j\}\_{j\in \mathbb N}$ be the sequence of its eigenvalues, ordered in a non-decreasing... | https://mathoverflow.net/users/58125 | Spectral geometry: asymptotic sequences of subspaces of $L^2(M)$ and the geometry of $M$ | Let me explain how to canonically produce a family of $\Delta$-invariant finite dimensional subspaces of $C^\infty(M)$ that completely determines the geometry of $M$.
First, I need to introduce some notation. $\newcommand{\bR}{\mathbb{R}}$ Set $m:=\dim M$. Denote by $g$ the Riemann metric on $M$.
Choose an orthono... | 1 | https://mathoverflow.net/users/20302 | 219574 | 103,033 |
https://mathoverflow.net/questions/219500 | 7 | Take for example the measure $\mu(n)=n^2$ on $\{1, \ldots, N\}$ and a random variable $X$ distributed according to the probability obtained by normalizing $\mu$.
Does there exists a constant $K>0$ such that $H(X \mid X \in C) \leq K.H(X)$ for every subset $C \subset \{1, \ldots, N\}$ and every $N$ ?
I know how to ... | https://mathoverflow.net/users/21339 | Estimating entropy conditional to an event | For $\mu(n)=n^2$, a quick calculation (\*) gives $H(X)\ge (1-\mathcal o(1))\log N$. On the other hand, a probability carried by a finite set $C$ has entropy at most $\log |C|$ (entropy of the equiprobable measure), so that $H(X|X\in C)\leq\log N$. This should be enough for your inequality, with $K\to 1$ for large $N$.
... | 3 | https://mathoverflow.net/users/75422 | 219584 | 103,035 |
https://mathoverflow.net/questions/219581 | 4 | This question came to my mind this afternoon while trying to figure out a possible way to tackle de Polignac's conjecture, which states that every even positive integer can be written as the difference of two consecutive primes in infinitely many ways, and is of course related to [About a possible generalization of Gre... | https://mathoverflow.net/users/13625 | Green-Tao theorem for 1-central numbers | The statement in the original post is true. This follows from a [recent result of Pintz](http://arxiv.org/abs/1509.01564), which is a common generalization of the Green-Tao theorem and the Maynard-Tao theorem:
**Theorem.** Let $m\geq 2$ be an integer. Then for any sufficiently large admissible set $\mathcal{H}$, ther... | 20 | https://mathoverflow.net/users/11919 | 219586 | 103,036 |
https://mathoverflow.net/questions/219541 | 2 | In this question I ask for a generalization of [What is the maximal possible rank of a subgroup of a special linear group mod a prime?](https://mathoverflow.net/questions/219491/what-is-the-maximal-possible-rank-of-a-subgroup-of-a-special-linear-group-mod-a/219535#219535)
Let $p\_1, \dots, p\_r$ be $r$ distinct odd p... | https://mathoverflow.net/users/38889 | The maximal possible rank of a subgroup of a product of special linear groups | I should answer the question! In my answer to the [earlier question](https://mathoverflow.net/questions/219491) I sketched a proof that the maximal rank of a subgroup of ${\rm SL}(3,p)$ with $p$ prime is $3$, except for when $p=2$, when it is $2$. So $3r$ is an upper bound on ranks of subgroups of the group $G$ in the ... | 2 | https://mathoverflow.net/users/35840 | 219593 | 103,038 |
https://mathoverflow.net/questions/219590 | 55 | Today I started reading Maddy's [Believing the axioms](http://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf). As I knew beforehand, it includes some discussion of ZFC axioms. However, I really hoped for a more extensive discussion of axiom of foundation/regularity.
Apparently, the reason why we usually take it i... | https://mathoverflow.net/users/30186 | Why should we believe in the axiom of regularity? | Regularity (aka Foundation) can be seen philosophically as an axiom of *restriction*. It is not necessarily saying “all the things you consider as sets must be well-founded”. It can be read saying “for the purposes of this set theory, we restrict our universe of discourse to just the well-founded objects”. It’s clarify... | 61 | https://mathoverflow.net/users/2273 | 219595 | 103,039 |
https://mathoverflow.net/questions/219572 | 6 | Let $A$ be an abelian variety over a field $k$ of characteristic $0$. How do I prove, without using transcendental methods, that if $\nabla$ is an integrable connection on a vector bundle $L$ on $A$ then $(L, \nabla)$ comes from a $G$-bundle with connection for some abelian algebraic subgroup $G \subset GL(n)$? And the... | https://mathoverflow.net/users/nan | $(L, \nabla)$ comes from a $G$-bundle with connection for some abelian algebraic subgroup $G \subset GL(n)$? | Here is one approach using the Fourier transform for $D$-modules on an abelian variety due to Laumon.
Let $A^\flat$ be the moduli space of rank one local systems on $A$, it is the universal extension of the dual abelian variety $A^\vee$ by a vector space. Note that L is a D-module such that the underlying O-module i... | 4 | https://mathoverflow.net/users/2653 | 219597 | 103,040 |
https://mathoverflow.net/questions/219596 | 6 | I'm not an expert in combinatorics, but it sometimes comes up in my research with students in computer science (which is already pretty far away from my speciality of abstract homotopy theory). I just managed to reduce a problem to the question in the title, namely: given an integer $k$, what is the smallest $n$ such t... | https://mathoverflow.net/users/11540 | Given k, what is the minimum n such that n choose n/2 is greater than k? | This [note by Kessler and Schif](http://u.math.biu.ac.il/~schiff/Papers/prepap3.pdf)f gives pretty extensive asymptotics for the central binomial coefficients. I am quite sure that any sort of exact formula is hopeless.
| 1 | https://mathoverflow.net/users/11142 | 219598 | 103,041 |
https://mathoverflow.net/questions/219623 | 6 | If $G$ is the real points of a semisimple algebraic group and $\rho:G\to GL(n,\mathbb R)$ is continuous representation. Is $\rho$ an algebraic morphism?
| https://mathoverflow.net/users/21929 | Do representations of real semisimple algebraic group have to be algebraic? | If $G$ is a real simple algebraic group and $\rho$ is a finite dimensional continuous representation, and if $G$ is not the group of complex points and *$G(\mathbb C)$ is simply connected*, then $\rho$ is indeed algebraic. This follows, for example, by going to the Lie algebra level and using classification of lie alge... | 7 | https://mathoverflow.net/users/23291 | 219626 | 103,055 |
https://mathoverflow.net/questions/219630 | 19 | This question occurred to me while thinking on another one here, [Name for an operation on matrices?](https://mathoverflow.net/q/219582/41291)
Can one define *in an invariant way* a binary operation on finite-dimensional vector spaces - let us denote it somehow suggestively by $(V,W)\mapsto V^{\otimes W}$ - with the ... | https://mathoverflow.net/users/41291 | Exponentiation of vector spaces? | Okay, so you can get pretty close as follows: I still don't think $V^{\otimes W}$ makes sense, but riffing off of your comment, we can make sense of $(1 \oplus V)^{\otimes W}$ (where $1$ denotes the $1$-dimensional vector space). The guiding intuition is the binomial expansion
$$(1 + V)^W = 1 + {W \choose 1} V + {W \... | 16 | https://mathoverflow.net/users/290 | 219634 | 103,058 |
https://mathoverflow.net/questions/219644 | 6 | Is it true that if $ E,F$ are two topological vector spaces (or say Banach spaces) over $\mathbb{R}$ such that they have nonempty open subsets $U\subset E, V\subset F$ which are homeomorphic, then the two vector spaces are isomorphic? If false, then what can we say if the two open subsets are $\mathcal{C}^1$-diffeomorp... | https://mathoverflow.net/users/57544 | Two vector spaces with homeomorphic open subsets are isomorphic? | This is false. All separable Banach spaces, for example, are homeomorphic.Indeed, there is a considerable body of work on when topological vector spaces are homeomorphic (see Bessaga and Pelczynski "Selected Topics in infinite-dimensional Topology" for starters).
| 9 | https://mathoverflow.net/users/80774 | 219646 | 103,061 |
https://mathoverflow.net/questions/219601 | 3 | $X$ follows Exponential $(\lambda)$. Can we split $X$ into two independent r.v.'s, i.e.,
do there exist functions $g$ and $h$ such that $g(X)$ and $h(X)$ are independent for any fixed $\lambda$? $g(X)$ and $h(X)$ can have non-exponential distribution.
The context. The above can be thought as the parallel of the nor... | https://mathoverflow.net/users/69932 | splitting exponential random variable into independent components | One curious property of [Gamma distributions](https://en.wikipedia.org/wiki/Gamma_distribution) is their relation with the [Dirichlet distributions](https://en.wikipedia.org/wiki/Dirichlet_distribution). If a random variable $Y \sim \mathrm{Gamma}(\alpha, \gamma)$ is independent of a random vector $(\pi\_1, \dots, \pi\... | 3 | https://mathoverflow.net/users/80885 | 219648 | 103,063 |
https://mathoverflow.net/questions/99855 | 3 | We can define p-adic Bernoulli polynomials by using q-integral on $\mathbb{Z}\_p$ and Taekyun Kim's method.
But how can we define p-adic poly-Bernoulli numbers and polynomials by using integral on $\mathbb{Z}\_p$?
| https://mathoverflow.net/users/nan | p-adic poly-Bernoulli numbers | I think what you are looking for is:
$$B\_n^{k}(x)=\frac{Li\_k (1-e^{-t})}{t}\int\_{\mathbb{Z}\_p}(x+y)^ndy$$
for $n\geq0$, and $Li\_k$ the polylogarithmic function.
See for example, "[Poly-Bernoulli Polynomials and Their Applications](http://www.m-hikari.com/ijma/ijma-2014/ijma-29-32-2014/kimIJMA29-32-2014-2.pdf... | 1 | https://mathoverflow.net/users/43108 | 219660 | 103,069 |
https://mathoverflow.net/questions/219652 | 7 | Hilbert space satisfies the following condition: if two triangles $\triangle ABC$, $\triangle A\_1B\_1C\_1$ have equal sides lengths: $|AB|=|A\_1B\_1|$, $|BC|=|B\_1C\_1|$, $|AC|=|A\_1C\_1|$ they also have equal medians lengths: $|AM|=|A\_1M\_1|$, where $M=(B+C)/2$, $M\_1=(B\_1+C\_1)/2$. Are there other Banach spaces wi... | https://mathoverflow.net/users/4312 | Equality of triangles in normed spaces | Firstly, notice that you may apply your condition to the triangles $ABM$ and $A\_1B\_1M\_1$ and iterate this process. So, e.g.. for every points $X$ and $X\_1$ on $BC$ and $B\_1C\_2$ with $BX/BC=B\_1X\_1/B\_1C\_1=k/2^n$ with integer $k,n$ we obtain $AX=A\_1X\_1$. Since the binary rationals are dense on $[0,1]$, the sam... | 1 | https://mathoverflow.net/users/17581 | 219675 | 103,074 |
https://mathoverflow.net/questions/219637 | 10 | Let $\pi:X \to S$ be a morphism of schemes (I can assume that $\pi$ is sufficiently nice, e.g. proper and flat, but certainly not smooth).
>
> Does there exist a scheme $I\_{X/S}$ which parametrises the irreducible components of the fibres of $\pi$?
>
>
>
In the case where $S = \mathrm{Spec}(k)$ for some field... | https://mathoverflow.net/users/5101 | Scheme of irreducible components | There is a definition of a functor of irreducible components in [that paper](https://perso.univ-rennes1.fr/matthieu.romagny/articles/composantes.pdf). If $\pi$ is finitely presented with geometrically reduced fibres, then the functor is representable by an étale algebraic space. As mentioned by Martin Bright, it is not... | 6 | https://mathoverflow.net/users/17988 | 219687 | 103,080 |
https://mathoverflow.net/questions/216055 | 8 | Does anyone have a suggestion for the best computer program to perform calculations in the 2nd Weyl algebra?
| https://mathoverflow.net/users/76665 | Computer algebra system for Weyl algebra computations | Weyl algebra computations are implemented in
[Macaulay 2](http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.8.2/share/doc/Macaulay2/Dmodules/html/) (package D-modules). Here is a [manual.](http://www.math.kobe-u.ac.jp/~taka/2007/knx/leykin-2002-m2.pdf)
| 5 | https://mathoverflow.net/users/11260 | 219691 | 103,081 |
https://mathoverflow.net/questions/219669 | 1 | Fractional Sobolev space $H^s\_p(\mathbb R), s>0, 1<p<\infty$ is a space of tempered distributions $f$ that satisfy $F^{-1}((1+|\xi|^2)^{s/2} F(f)) \in L\_p(\mathbb R)$.
Here, $F$ denotes the Fourier transform.
Such spaces are known to admit Littlewood-Paley characterisation, i.e. one can esitmate the norm $\|\phi\|\_{... | https://mathoverflow.net/users/80894 | Fractional Sobolev spaces on the circle with a Littlewood-Paley characterisation | Yes.
A sledge hammer with which you can hit this is Stein's *Topics in Harmonic Analysis related to the Littlewood-Paley Theory*, published by the Annals of Math Studies series of the PUP. The main thing you are looking for the Theorem 2, the "square function theorem" for decompositions of functions on compact Lie g... | 2 | https://mathoverflow.net/users/3948 | 219692 | 103,082 |
https://mathoverflow.net/questions/219689 | 5 | Cross-posted from [M.SE](https://math.stackexchange.com/questions/1454095/short-primitive-vectors-in-a-lattice-in-mathbbz2).
Given $a,n$ coprime positive integers, let $L = \{(x,y)\in \mathbb{Z}^2, ax=y(n)\}$ be the lattice of all points satisfying $ax=y\pmod{n}$.
I want to find an order-of-magnitude bound on the s... | https://mathoverflow.net/users/70060 | Average of Short Character Sum over All Dirichlet Characters Mod n | Suppose $n=2m$ and $a=m+1$. Clearly $(a,m)=1$. Then there are no $x,y$ odd with $ax \equiv y$ mod $n$, $x<m$, $y<m$. Indeed in this case
$$y \equiv ax= mx+x \equiv m+x \mod 2m$$
Clearly this cannot be satisfied for $0<x<m$, $0<y<m$.
You can try to write the sum instead using additive characters instead and you wi... | 7 | https://mathoverflow.net/users/18060 | 219694 | 103,083 |
https://mathoverflow.net/questions/219672 | 10 | The category of $C^{\*}$ algebras is denoted by $\mathcal{A}$.
Is there a functor $\mathcal{F}$ on $\mathcal{A}$ which send each object $A\in \mathcal{A}$ to its center $Z(A)$. In the other words, can we extend the maping $A\mapsto Z(A)$ on objects to a functor on this category?
| https://mathoverflow.net/users/36688 | Is this a functor on the category of $C^{*}$ algebras? | Here is another attempt at proving no such functor exists — I apologize to Chris and to Manny if something like this is already in the papers which they cite.$\newcommand{\Mat}{{\bf M}}\newcommand{\Cplx}{{\bf C}}\newcommand{\Cst}{{\rm C}^\*}$
Let $\Mat\_2$ denote the algebra of $2\times 2$ complex-valued matrices, an... | 10 | https://mathoverflow.net/users/763 | 219696 | 103,084 |
https://mathoverflow.net/questions/215632 | 6 | By a result of Laczkovich ('[Analytic subgroups of the reals](http://dx.doi.org/10.1090/S0002-9939-98-04241-5)' Proc AMS Vol 126 (1998)), any non-open analytic subgroup of a Polish locally compact group can be covered by countably many closed Haar null sets.
Is this result of Laczkovich true for any Polish (not nece... | https://mathoverflow.net/users/61536 | Can each non-open analytic subgroup of a Polish abelian group be covered by countably many closed Haar null subsets? | The answer to both problems (1 and 2) is negative: the Polish group $G=\mathbb Z^\omega$ contains a dense meager Borel subgroup $H$ (which can be written as the difference $H=A\setminus B$ of two $F\_\sigma$-sets in $G$) which cannot be covered by countably many closed Haar-meager subsets of $G$. Such subgroup $H$ is c... | 2 | https://mathoverflow.net/users/61536 | 219716 | 103,090 |
https://mathoverflow.net/questions/219715 | 8 | The GAGA theorem is a celebrated elaboration of the idea that complex analytic and complex algebraic geometry are equivalent, at least for smooth projective varieties/manifolds.
I am aware *why* this is a theorem about projective varieties; historically the two classes of varieties people cared about were projective ... | https://mathoverflow.net/users/39713 | What is the role of projective spaces in GAGA? | The Serre comparison theorems are valid for complete (= proper) varieties over ${\Bbb C}$, with no relation to projective space. See [this talk](http://archive.numdam.org/ARCHIVE/SHC/SHC_1956-1957__9_/SHC_1956-1957__9__A2_0/SHC_1956-1957__9__A2_0.pdf) by Grothendieck (Séminaire Cartan 9 (1956-1957), Exposé No. 2).
| 16 | https://mathoverflow.net/users/40297 | 219718 | 103,091 |
https://mathoverflow.net/questions/219724 | 5 | Let $X$ be a finite dimensional (possibly compact) Alexandrov space with curvature $\geq K$. **Is it true that its boundary is again Alexandrov space with curvature bounded from below? If yes, is the curvature at least $K$?**
| https://mathoverflow.net/users/16183 | Is the boundary of Alexandrov space again an Alexandrov space? | This is an open problem.
It is a special case of the following question:
>
> Is it true that every extremal subset is again an Alexandrov space?
>
>
>
The answer to this question is "No". Petrunin has constructed a counterexample in codimension three, [here](http://library.msri.org/books/Book30/files/petru... | 5 | https://mathoverflow.net/users/68708 | 219731 | 103,094 |
https://mathoverflow.net/questions/219727 | 1 | Is there an infinite topological space $(X,\tau)$ with the following property?
There is an open cover ${\cal U}^\*$ such that
* $X\notin {\cal U}^\*$;
* every finite subset $F\subseteq X$ is contained in some member of ${\cal U}^\*$ and
* for every subcover ${\cal V} \subseteq {\cal U}^\*$ there is $x\in X$ such th... | https://mathoverflow.net/users/8628 | Open cover not containing a certain subcover | As a counterexample, consider any topological space $X$ which is not metacompact.
We recall that a topological space $X$ is *metacompact* each each open cover of $X$ has a point-finite refinement. By Theorem 3.5 in the survey "Covering properties" by D.Burke in the Handbook of Set-Theoretic Topology (1984), a topolo... | 1 | https://mathoverflow.net/users/61536 | 219732 | 103,095 |
https://mathoverflow.net/questions/219708 | 9 | Is it possible to find matrix solutions to the following :
$$\left(\sum\_1^m M\_k x\_k\right)^n=\left(\sum\_1^m x\_k^n\right)I\_d$$
where $M\_k$ are the desired $d \times d$ matrices (no restriction on $d$) and $x\_i$ are indeterminate variables;
For n=2 the gamma matrices satisfing $M\_i M\_j + M\_j M\_i = 2\delta\_{i... | https://mathoverflow.net/users/16739 | a generalization of gamma matrices | Yes, these are called Generalized Clifford Algebras. The earliest reference I could find was [an article by Yamazaki](http://hdl.handle.net/2261/6042) from 1964.
An explicit construction is given by [Morris](http://qjmath.oxfordjournals.org/content/18/1/7)
For example, with $m=2$ and $n=3$, we find as a solution $M... | 3 | https://mathoverflow.net/users/3897 | 219738 | 103,097 |
https://mathoverflow.net/questions/219733 | 10 | I would like to know two things: one, whether the best quantative bounds in the Prime Number Theorem are still basically those given by the Vinogradov-Korobov zero-free region? and two, whether there are any elementary proofs substantially different from the Erdős/Selberg proofs?
I realise this is probably trivial to... | https://mathoverflow.net/users/36212 | Quantitative and elementary proofs of the Prime Number Theorem | The best known error term in PNT is to my knowledge Big-Oh of
$$x\exp \left(-C \frac{ (\log x)^{\frac35}}{(\log \log x)^{\frac15}} \right),$$ and for $C$ one can take $-0.2098$, given in:
>
> K. Ford. Vinogradov’s integral and bounds for the Riemann zeta
> function. Proc. London Math. Soc., 85(3):565–633, 2002.
>... | 7 | https://mathoverflow.net/users/nan | 219739 | 103,098 |
https://mathoverflow.net/questions/219729 | 4 | Suppose $M$ is a symplectic toric manifold. This means there is a compact torus
$T$ that has a Hamiltonian action on $M$, with moment map $\mu:M \to \mathfrak t^\*$, and $\dim(M)=2\dim(T)$. Can one tell from the moment polytope $\mu(M)$ whether $M$ is Fano?
| https://mathoverflow.net/users/15197 | Condition on moment polytope for a toric manifold to be Fano | The basic answer is "yes, *of course,* because the toric variety is uniquely determined by the polytope. But no, because it's the wrong polytope for the question of Fanoness."
The question is whether the anticanonical class, not the given class $[\omega]$, is ample. Translate the polytope to contain $0$ in the interi... | 3 | https://mathoverflow.net/users/391 | 219745 | 103,100 |
https://mathoverflow.net/questions/219741 | 6 | Let $\mathbf{Poset}$ denote the category of partially ordered sets and order-preserving maps. Does $\mathbf{Poset}$ have quotients?
| https://mathoverflow.net/users/8628 | Quotients of posets | (There may be some users who think this would have been better asked at Mathematics StackExchange, but I'll go ahead and answer because there are several ways of looking at it.)
The answer is "of course". See [The Joy of Cats, p. 119](http://katmat.math.uni-bremen.de/acc/acc.pdf#119). The coequalizer of two maps $f,... | 9 | https://mathoverflow.net/users/2926 | 219750 | 103,103 |
https://mathoverflow.net/questions/219744 | 6 | Let $X$ and $Y$ be separable complete metric spaces (if necessary, they may be assumed to be compact). Let $R\subset X\times Y$ be a closed subset such that the projection of $R$ to $X$ is onto.
**Is it true that there exists a Borel measurable map $f\colon X\to Y$ such that $(x,f(x))\in R$ for every $x\in X$?**
I... | https://mathoverflow.net/users/16183 | Existence of a measurable map between metric spaces | For a compact space $Y$ the answer is affirmative, but in general case of Polish space $Y$ it is negative.
Results yielding nice selections of relations $R$ are known in Descriptive Set Theory as Uniformization Theorems, see Section 18 of the standard textbook [A.Kechris, Classical Descriptive Set Theory, Springer, ... | 7 | https://mathoverflow.net/users/61536 | 219751 | 103,104 |
https://mathoverflow.net/questions/219749 | 5 | NB the original question asked about $\beta\mathbb{D}$ rather than the corona, hence some of the initial comments.
---
Is there a group operation on $\beta \mathbb{D} \setminus \mathbb{D}$ extending complex multiplication on $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$?
| https://mathoverflow.net/users/34686 | Is $\beta \mathbb{D}\setminus \mathbb{D}$ a group? | No, because $\beta\mathbb D$ and $\beta\mathbb D\setminus \mathbb D$ are not Dugundji compact (and not supercompact) whereas all compact topological groups are Dugundji compact according to a result of Uspenskii
<https://www.researchgate.net/publication/266010994_Topological_groups_and_Dugundji_compacta>
and supercompa... | 9 | https://mathoverflow.net/users/61536 | 219753 | 103,105 |
https://mathoverflow.net/questions/219693 | 2 | I've been having trouble with finding the curvature computations of globally symmetric spaces of rank $1$.
More specifically, I need to use results about the eigenvalues of the operator $R:T\_pM \rightarrow T\_pM$; $X \mapsto -[[X,c],c]$ for a given $c \in T\_pM$. The fact is, using the book of Besse "Manifolds all ... | https://mathoverflow.net/users/48745 | Curvature computations of globally symmetric spaces of rank $1$ | I assume that you are interested in globally compact symmetric spaces of compact type. The curvature depends on the choice of a Riemannian metric.
The vector $c$ in your formula should be a unit vector with respect to that metric. Depending on the metric you choose, in the case of complex or quaternionic projective spa... | 2 | https://mathoverflow.net/users/70808 | 219757 | 103,108 |
https://mathoverflow.net/questions/218737 | 16 | Over a commutative ring $R$, a finite type locally free (weak sense) module for which the rank function is locally constant is projective.
If we notice that for each minimal prime $p$ of the ring, the rank function is constant on $V(p)$, the adherence of $p$ in the Zariski topology (because if $p\subset q$ the rank a... | https://mathoverflow.net/users/3333 | Every finitely generated flat module over a ring with finitely many minimal primes is projective | There are two questions here. First of all, yes, the argument is fine; secondly, yes, there are rings with finitely many minimal primes, but infinitely many associated primes. So all together, the criterion is slightly more general than the one by Raynaud-Gruson mentioned in the question, but the proof is much easier. ... | 5 | https://mathoverflow.net/users/15782 | 219759 | 103,109 |
https://mathoverflow.net/questions/219667 | 9 | Let $f$ be an arbitrary weight 1 newform. We know by Serre-Deligne that there is an odd 2-dimensional irreducible Artin representation $\rho$ such that $L\_f(s)=L(\rho,s)$.
I was wondering how much can we tell about $\rho$ from $f$ alone, **without computing $\rho$.**
(For instance, one could make complete use of n... | https://mathoverflow.net/users/43108 | Type of a modular form | EDIT : I wanted to add details after the OP's comment, and I realized I forgot one case (the dihedral case $D\_{2n}$ with $n$ even). Below is a slightly corrected version.
---
You can determine the type of the representation by looking at the density of primes $p$ such that $a\_p = 0$. If that density is $\geq 1/... | 6 | https://mathoverflow.net/users/9317 | 219761 | 103,110 |
https://mathoverflow.net/questions/219755 | 4 |
>
> Let $(\Omega, \mathcal{F}, \mathbb{P})$ be the unit interval with Lebesgue measure on the Borel subsets. Then we can find independent random variables $X\_1, X\_2, X\_3, \dots$ defined on $(\Omega, \mathcal{F}, \mathbb{P})$, each normal mean zero, variance $1$.
>
>
>
This is a special case of the Borel isomo... | https://mathoverflow.net/users/nan | How to construct i.i.d. standard normal random variables on $\Omega = [0, 1]$ with the Lebesgue measure | If we can find independent random variables $U\_j$ uniform on $[0,1]$ we can transform them to $\mathcal N(0,1)$. If $Y$ is uniform on $[0,1]$, let $D\_j$ be its $j$'th decimal place, i.e. $Y = \sum\_{j=1}^\infty 10^{1-j} D\_j$ with
$D\_j \in \{0,1,\ldots, 9\}$. If $p\_j$ is the $j$'th prime, let
$U\_j = \sum\_{k=1}^\... | 7 | https://mathoverflow.net/users/13650 | 219762 | 103,111 |
https://mathoverflow.net/questions/219650 | 5 | Let $X=\mathbb{G}\_Q(l,p)$ be the isotropic Grassmannian, where $l\leq p-2$. Let $q=p-l$. Let $W^P$ be the set of minimal length representatives. Let $\tilde{\mathcal{Q}}(l,p)$ be the set of partition pairs $(\alpha,\lambda)$ with type attached to them where $\alpha$ is a partition of shape $q\times l$ and $\lambda$ is... | https://mathoverflow.net/users/66288 | Parametrization of Schubert varieties in isotropic Grassmannians by partitions | Let $w$ be a minimal length representative with representation as in the question. Then we can define a map $\psi\colon W^P\to W^P$ by setting $\psi(w)=w'$ where $$w'=(p+1-u\_t,\ldots,p+1-u\_1,\overline{p+1-u\_l},\ldots,\overline{p+1-u\_{t+1}}.p+1-u\_p,\ldots,\widehat{p+1-u\_{l+1}})\,.$$ This is in an involution, in pa... | 0 | https://mathoverflow.net/users/66288 | 219767 | 103,114 |
https://mathoverflow.net/questions/219763 | -3 | Let $A$ be a Banach algebra. Is there a Banach algebra $B$ and a non-trivial closed ideal $I$ of $B$ such that $\frac{B}{I}\cong A$?
| https://mathoverflow.net/users/27066 | Quotient of a Banach algebra | Sure. Let $K$ be any compact Hausdorff space that contains at least two points and for $B$ take the space of $A$ valued continuous functions on $K$. Take any $p$ in $K$. Let $I$ be the ideal of all functions in $B$ that vanish at $p$.
| 3 | https://mathoverflow.net/users/2554 | 219769 | 103,116 |
https://mathoverflow.net/questions/219760 | 4 | (this question [did not get any answers](https://math.stackexchange.com/questions/1451101/normal-form-of-currents) on math.SE, so I am reposting it here)
Let $M$ be an $n$-dimensional manifold. Then the space of currents $\mathcal D^k(M)$ of degree $k$ on $M$ is the space of continuous linear functionals on the spac... | https://mathoverflow.net/users/80741 | normal form of currents? | It may be helpful to look at a simple example. $\newcommand{\bR}{\mathbb{R}}$ Assume for simplicity that $M=\bR^n$. Fix a Radon measure $\mu$ on $\bR^n$. It defines a $0$-dimensional current on $\bR^n$. (I define the dimension of a current to be $n-\deg$.) If $\mu$ is not absolutely continuous with respect to the Lebes... | 3 | https://mathoverflow.net/users/20302 | 219770 | 103,117 |
https://mathoverflow.net/questions/219201 | 4 | Let $\Lambda(x,y)$ be the count of totatives of $x$ that are less than or equal to $y$.
I am asking for the following result to be verified, (particularly the final proposal), I have found no counterexamples and believe the reasoning to be correct.
The first part of this is an interpretation of $\phi(x)$ and the se... | https://mathoverflow.net/users/41928 | Relative-totient function (2nd attempt) | I will not comment on the soundness of the approach, but I will render a subjective opinion: I don't like it. One of the reasons is that I have found most people do not have a good understanding of probability (I think I have a better than average understanding, but not a good one), and that when one tries such an argu... | 3 | https://mathoverflow.net/users/3206 | 219782 | 103,120 |
https://mathoverflow.net/questions/219773 | 1 | maybe this question is trivial and, then this is the reason I've never seen this written.
The motivation is to define internal $\infty$-groupoids (that are preferably) Kan fibrant and to see if Kan fibrancy is really a necessary ontological assumption. As I understand there are non-Kan fibrant models of smooth $\inf... | https://mathoverflow.net/users/40883 | Is there an operad that codifies groupoids? | The question (under the relaxed notion of operad in the original post) will be settled negatively if we show that the category $\mathrm{Grpd}$ (internal groupoids in $\mathrm{Set}$) isn't monadic over $\mathrm{Set}$ for *any* choice of underlying functor $U: \mathrm{Grpd} \to \mathrm{Set}$. (Note: I am using $\mathrm{G... | 9 | https://mathoverflow.net/users/2926 | 219784 | 103,121 |
https://mathoverflow.net/questions/219774 | 10 | It is well known that for each $m\in\mathbb{N}$
$$\lim\_{N\to\infty}\frac1N\sum\_{n=1}^Ne^{2\pi i\sqrt{nm}}=0$$
My question is whether there is some uniformity in the variable $m$.
More precisely, is it true that
$$\lim\_{N\to\infty}\frac1N\sum\_{n=1}^Ne^{2\pi i\sqrt{nN}}=0?$$
I would be particularly interested in a po... | https://mathoverflow.net/users/18698 | Bounding exponential sum with square roots | I write $e(x)=e^{2\pi i x}$. Here is a naive bound via [Kuzmin's estimate](http://www.math.harvard.edu/~elkies/M259.02/kmv.pdf), which says that if you have a sequence $c\_n$ with monotonic differences $\delta\_n$ all in some interval $[k+\epsilon, k+1-\epsilon]$, then the entire exponential sum is small:
$$|\sum\_n ... | 5 | https://mathoverflow.net/users/70060 | 219789 | 103,122 |
https://mathoverflow.net/questions/219740 | 2 | Let $L$ be a lattice and let $\textbf{DM}(\cdot)$ denote the [Dedekind-MacNeille completion](http://en.wikipedia.org/wiki/Dedekind%E2%80%93MacNeille_completion).
Is there a lattice $L$ that is not a quotient of $\textbf{DM}(L)$? And what if we generalise this question to posets: is every poset $P$ a quotient of $\tex... | https://mathoverflow.net/users/8628 | Are lattices quotients of their Dedekind-MacNeille completion? | In order to finally kill every possible version of this question: No, even if $P$ is finite. Let $P$ be two incomparable elements. Then $DM(P)$ is the diamond poset $0 < p,q < 1$. The only maps of posets from $DM(P)$ to $P$ are to send everything to $0$ or everything to $1$. In particular, there is no surjection $DM(P)... | 4 | https://mathoverflow.net/users/297 | 219795 | 103,124 |
https://mathoverflow.net/questions/219807 | 12 | Ordinary homology and cohomology factor through chain complexes via singular homology and cohomology. What about other (co)homology theories?
That is, for each spectrum $E$, do we have a lift in the following diagram?
$\begin{array}[ccc]
& \mathsf{HoTop} & \overset{E}{\to} & \mathsf{GrAb} \\
& \underset{?}{\searrow... | https://mathoverflow.net/users/2362 | Does every (co)homology functor (in particular, stable homotopy) factor through chain complexes? | No (I mean, not in a triangulated way), otherwise any generalized homology theory of a mod 2 Moore space would be 2-torsion, but this is not true for mod 2 stable homotopy groups (it's well known that you get a cyclic group of order 4). For positive results under extra hypotheses see:
[Heller, A., 1966. Extraordinary... | 17 | https://mathoverflow.net/users/12166 | 219812 | 103,133 |
https://mathoverflow.net/questions/219645 | 4 | I think this is a well knowing result but I can't find any reference,
Let $(E,q)$ be a vector bundle with a non degenerated quadratic form $q:E\rightarrow E^\*$ with trivial determinant, suppose that $q$ is symmetric (resp. skew-symmetric).
My question: Is there an equivalent of the stack of such couples with the... | https://mathoverflow.net/users/66528 | Vector bundle with a perfect pairing and ($\mathbb Z/2$, $SL_r$)-bundle | Let me answer my last question (last comment), if we give a $\rho:G\rightarrow G$ an automorphism of $G$, let us denote by $$E^\rho=E\times^GG$$ where the action of $G$ on itself is given by $\rho$, then generally $E^\rho$ is not isomorphique to $E$. It is when $\rho$ is an interior automorphism. For exemple taking $\r... | 2 | https://mathoverflow.net/users/66528 | 219817 | 103,136 |
https://mathoverflow.net/questions/219600 | 8 | Let $\pi$ be an irreducible cuspidal representation of a semi-simple $p$-adic group $G$. It is well-known that the character of $\pi$ is concentrated in the set of compact elements in $G$.
What is the best reference for this result?
Also, if there is a really simple proof of this result, I would be glad to learn it!
... | https://mathoverflow.net/users/3891 | Characters of cuspidal representations | This is an old result due to Deligne:
Le support du caractère d'une représentation supercuspidale.
C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 4, Aii, A155–A157.
| 4 | https://mathoverflow.net/users/4767 | 219819 | 103,138 |
https://mathoverflow.net/questions/219830 | 2 | I am trying to calculate the entries of the Riemann curvature tensor $R^m\_{\phantom{m}ijk}$ for the metric $g\_{ij}$.
The Riemann-Christoffel tensor is given as
\begin{align}
R^m\_{\phantom{m}ijk} = \frac{\partial}{\partial x^j}{m \brace i\,\,k} - \frac{\partial}{\partial x^k}{m \brace i\,\,j} +
{n \brace i\,\,k}... | https://mathoverflow.net/users/80965 | Calculating the Riemann Christoffel tensor for a diagonal metric | I managed to find a post on Physics StackExchange :
<https://physics.stackexchange.com/questions/179393/any-tips-on-evaluating-riemann-tensor>
which details what I feared. The process it self is too tedious. The best solution is to use a symbolic package which can do all these calculations, like for example the pyt... | 1 | https://mathoverflow.net/users/80965 | 219834 | 103,144 |
https://mathoverflow.net/questions/219833 | 0 | I asked this question on math.stack but I got no answer, so I try here.
Let $\phi(t)$ be a solution for the nonlinear Schroedinger equation\begin{equation}
i\partial\_t\phi(t)=-\Delta\phi(t)+(V\*|\phi|^2)\phi(t)
\end{equation}
inside the Hilbert space $L^2(\mathbb{R}^d)$. I know $\phi(t)$ is also supposed to live in ... | https://mathoverflow.net/users/48303 | Norm of derivative of rank one projector | I haven't checked the paper, but $\nabla p(t)$ most likely mean the operator $\psi \mapsto \nabla \phi \cdot \langle \phi | \psi\rangle$ and since it looks like by assumption $\phi$ has $L^2$ norm one (since you are using it do define a projector), the equality follows.
---
To be more detailed, $\|\nabla p(t) \p... | 1 | https://mathoverflow.net/users/3948 | 219836 | 103,146 |
https://mathoverflow.net/questions/219405 | 7 | Let $\{g(\cdot-k),k\in\mathbb Z\}$ be a Riesz basis, and let $\varphi\in L^2(\mathbb R)$ be a function defined by its Fourier transform
$$\hat{\varphi}(\xi)=\frac{\hat{g}(\xi)}{\Gamma(\xi)},$$
where
$$\Gamma(\xi)=\left(\sum\_k |\hat{g}(\xi+2k\pi)|^2\right)^{1/2}$$
Then $\{\varphi(\cdot-k),\ k\in\mathbb Z\}$ is an ortho... | https://mathoverflow.net/users/69931 | Derive an orthonormal system by Riesz basis $\{g(\cdot-\lambda_k),\ \lambda_k\in\mathbb R, \ k\in\mathbb Z\}$ | This is an answer to the second version: a function $f$ such that $f(⋅−k)$ form an orthonormal basis with the same span as $(g(⋅−λ\_k))$ exists only in very special cases.
Assume it exists, then on the Fourier transform side span $(g-\lambda\_k)$ is the space of functions of the form $\phi\cdot \hat f$ where $\phi$ is ... | 1 | https://mathoverflow.net/users/71040 | 219841 | 103,148 |
https://mathoverflow.net/questions/219723 | 2 | Is there an example of an unital commutative semi-simple Banach algebra which it is not amenable?
| https://mathoverflow.net/users/27066 | Semi-simple Banach algebra | Some examples off the top of my head:
1) the disc algebra
$$ A({\mathbb D}) = \{ f \in C(\overline{\mathbb D}) \mid \hbox{$f$ is analytic on $\mathbb{D}$} \} $$
equipped with the supremum norm.
2) $C^1{([0,1])}$, the space of continuously differentiable functions $[0,1]\to {\mathbb C}$ equipped with the norm
$$ \Ve... | 4 | https://mathoverflow.net/users/763 | 219842 | 103,149 |
https://mathoverflow.net/questions/219676 | 11 | I believe I know how to prove the following results. I also know to whom to cite fancy-shmancy results that have these as a very special case. **My question is:** what are the correct citations for these versions? **Pre-question:** Are the results correctly stated?
**Version 1**
Let $\mathrm{Man}\_d$ denote the cat... | https://mathoverflow.net/users/78 | Reference request: sheaves on the site of d-manifolds | Version 1 is a simple corollary of the existence
of [differentiable good open covers](http://ncatlab.org/nlab/show/good+open+cover#DifferentiablyGoodOpenCover), i.e., covers whose finite intersections are diffeomorphic to R^n.
This shows that sheaves on the site of n-manifolds and etale maps are equivalent to sheaves... | 1 | https://mathoverflow.net/users/402 | 219874 | 103,162 |
https://mathoverflow.net/questions/219722 | 6 | It seems strange to me that all we can find about Schilder's theorem in the literature is on a finite interval of Brownian path.
If we equip the space of continuous function starting from $0$, defined on $\mathbb{R}\_+$ with the topology of uniform convergence on compacts. Can we have the similar large deviation pri... | https://mathoverflow.net/users/80925 | Large deviation for Brownian path on $[0,\infty)$ | It is not in the uniform topology but with a topology tapered off at infinity it is correct. It is done that way in the book of Deuschel and Stroock on large deviations.
| 4 | https://mathoverflow.net/users/35520 | 219878 | 103,164 |
https://mathoverflow.net/questions/219887 | 2 | Let $g(u,\delta)=E[f(x)]$ where the expectation is over $N(u,\delta^2)$.
Is there a characterization what function $g(u,\delta)$ can be produced this way? Is there a procedure solve the inverse problem, that is, given
$g(u,\delta)$, the procedure should output $f(x)$?
This seems like a fundamental question. But I d... | https://mathoverflow.net/users/44539 | What function is a Gaussian integral | I think we can give a quick characterization adding some smootheness and boundness hypothesis via Stochastic calculus. They are exactly the solutions to the heat equation.
First of all we have $$g(u,\delta)=E[f(u+\delta Z)]$$ where $Z$ is a standard gaussian so necessarily $g(u,0)=f(u)$ so the function $f$ is fully d... | 4 | https://mathoverflow.net/users/30889 | 219891 | 103,167 |
https://mathoverflow.net/questions/219855 | 6 | If $a\_1,a\_2,\dots$ are IID random bits (correction as per Anthony Quas: these "bits" are $+1$ and $-1$ with equal probability), then with probability 1, the set of natural numbers $n$ such that $a\_1+a\_2+\dots+a\_n \leq 0$ has lower density 0 and upper density 1, so it has no density in the ordinary sense. Still, I ... | https://mathoverflow.net/users/3621 | Generalized density functions on the natural numbers | So I think a logarithmic average will do the trick for you. If you define $Y\_n$ to be the sign of $a\_1+\ldots+a\_n$, then calculations with Brownian motion in place of random walk suggest the covariance of $Y\_n$ and $Y\_m$ with $m<n$ is approximately $(1/2\pi)\arctan\sqrt{m/(m-n)}$. Now define $S\_N=(1/\log N)(Y\_1/... | 7 | https://mathoverflow.net/users/11054 | 219893 | 103,169 |
https://mathoverflow.net/questions/219895 | 3 | Excuse me if the question is not appropriate for Mathoverflow. I havs asked it in math.stackexchange, but did not get any response. And so, I dared to put it here. I am trying to learn free probability from scratch, mostly by myself. I am trying to prove the following result.
>
> If $\mu$ and $\nu$ are compactly s... | https://mathoverflow.net/users/651 | Existence of free operators, independent and with given distributions | Consider the C$^\*$-algebras $\mathcal A = L^\infty(\mu)$ and $\mathcal B = L^\infty(\nu)$. They are C$^\*$-probability spaces when equipped with the states $\phi(c) = \int c \;\mu$ and $\psi(c) = \int c \; d\nu$.
Then the elements $a = z$ and $b=z$ in $\mathcal A$ and $\mathcal B$ have distributions $\mu$ and $\nu$ re... | 3 | https://mathoverflow.net/users/76593 | 219903 | 103,172 |
https://mathoverflow.net/questions/219728 | 9 | If $G,H$ are simple, undirected graphs, we say that they are in a *hom(omorphism)-relation* if there is a graph homomorphism from $G$ to $H$ or from $H$ to $G$.
For any graph $G$ let $L(G)$ denote its [line graph](https://en.wikipedia.org/wiki/Line_graph).
If $G,H$ each have at least one edge, and they are in a hom... | https://mathoverflow.net/users/8628 | Graph homomorphisms and line graph | Well, this question turned out more interesting than I thought at first.
If there is a homomorphism $f:G \to H$ then at first sight, it seems that the natural induced mapping on edges would be a homomorphism from $L(G) \to L(H)$, but this turns out to be false.
In fact, the whole thing turns out to be false, in tha... | 8 | https://mathoverflow.net/users/1492 | 219905 | 103,173 |
https://mathoverflow.net/questions/219853 | 2 | Let *P* be a permutation group with some generating set *S* and let *W* be the word acceptor automaton of *P*, if I know the value of *k* (*k-fellow-traveller property* of CayleyGraph CG(P,S)).
I realized that the number of states *W* changes in function of *k*, when value of *k* is "low", the total of states of *W* i... | https://mathoverflow.net/users/80975 | k-fellow traveler property and automatic structur | The first point to note is that it is not so much the value of $k$ in the $k$-fellow-traveller property that determines the number of states of $W$, but rather the number $d$, say, of *word-differences*. These are the labels of the shortest words in the Cayley-graph that connect corresponding vertices in fellow-travell... | 2 | https://mathoverflow.net/users/35840 | 219910 | 103,175 |
https://mathoverflow.net/questions/219907 | -1 | Say that $G'$ is a graph re-weighted from $G$ using the rule: $w'
(u, v) = w(u, v) − f(u) + f(v)$, where $f$ always produce the positive results for any nodes. Can we prove
that the shortest path $P\_G$ from $s$ to $t$ remains the shortest path from $s$ to $t$ in $G'$?
| https://mathoverflow.net/users/80058 | Shortest path problem | Yes, because any path from $s$ to $t$ of weight $w$ in $G$ will have weight $w - f(s) + f(t)$ in $G'$. This is the Edmonds-Karp trick for solving minimum-cost flow problems, see Section 2 of the paper "Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems".
(I would post this as a comment, but ... | 5 | https://mathoverflow.net/users/16139 | 219923 | 103,180 |
https://mathoverflow.net/questions/219797 | 2 | Let $(M,g)$ be a closed Einstein manifold of dimension $m>2$ and
$$
\mathrm{Ricc}(g)=\lambda g,
$$
$h$ a symmetric $2$-covariant tensor, $\Delta=\nabla^\*\nabla$ the Laplacian on functions as well as the rough or connection Laplacian on symmetric $2$-covariant tensors, and $\delta\_g$ the divergence on symmetric $2$-co... | https://mathoverflow.net/users/43445 | Does this PDE only have the trivial solution? | There are lots of nontrivial solutions for any negative $\lambda$. Here's how to construct them all.
First, we can decompose an arbitrary symmetric $2$-tensor $h$ as $h=fg+u$, where $f$ is a scalar function and $u$ is trace-free. It follows that $\operatorname{tr} h = mf$, and $\delta\_g\delta\_g h = -\Delta f + \del... | 11 | https://mathoverflow.net/users/6751 | 219935 | 103,186 |
https://mathoverflow.net/questions/219931 | 14 | For any set $B\subseteq \mathbb{N}$ one can associate the formal series
$$f\_B(z) = \sum\_{b\in B}z^b$$
and obtain
$$f\_B(z)^k = \sum\_{n\geqslant 0} r\_{B,k}(n)z^n,$$
where $r\_{B,k}(n) = |\{(x\_1,\cdots,x\_k)\in B^k : x\_1+\cdots+x\_k=n\}|$. As it is characteristic of the Circle Method, one them have
$$ r\_{B,k}(n) =... | https://mathoverflow.net/users/74026 | Erdös-Turán via Hardy-Littlewood circle method? | Your question probably doesn't have a definitive answer, as it is unlikely that one can prove that the `circle method' (it has evolved in many directions taking in tools from a very broad area of mathematics, so it is likely difficult to define exactly what the circle method entails) can never be used to prove the Erdö... | 6 | https://mathoverflow.net/users/10898 | 219936 | 103,187 |
https://mathoverflow.net/questions/219641 | 13 | The literature has definitions (seemingly plural, though they might be linked) of a $q$-deformed Virasoro algebra. But is there any link of these to a $q$-deformed Heisenberg algebra? (Classically there is an expression for $L\_n$ in terms of a normal ordered quadratic in the generators of the Heisenberg algebra.) I wo... | https://mathoverflow.net/users/29625 | q-Virasoro and q-Heisenberg algebras | The main sources are Awata et al or Frenkel-Reshetikhin. In <http://arxiv.org/pdf/q-alg/9507034v5.pdf> section 4, you can see the q,t case. You can also look at <http://arxiv.org/pdf/q-alg/9505025v1.pdf> where the introduction gives more references to how this relates to the undeformed case. This gives the classical Vi... | 6 | https://mathoverflow.net/users/69850 | 219940 | 103,189 |
https://mathoverflow.net/questions/219913 | 9 | Following the ideas of S.-S. Chern, J. L. M. Barbosa associated a holomorphic curve in $\mathbb{C}P^m$ to a minimal immersion of the 2-sphere into the $2m$-sphere in [his 1972 paper](http://projecteuclid.org/euclid.bams/1183534303). However, his construction seems to be very unclear because of heavy use of coördinates.... | https://mathoverflow.net/users/35080 | Minimal immersions of the 2-sphere | I don't know exactly where it appears in the literature, but a coordinate-free interpretation is certainly known, based on the idea of Penrose's twistor construction. The main points can be summarized as follows:
First, everything is equivariant under the group $\mathrm{SO}(2m{+}1)$: One has $S^{2m} = \mathrm{SO}(2m{... | 10 | https://mathoverflow.net/users/13972 | 219959 | 103,192 |
https://mathoverflow.net/questions/219957 | 9 | I have heard it said in some differential geometry talks that "the generic situation in such and such case is real analytic". My question is, is the generic smooth manifold also real analytic in some sense of genericity? I do not have a specific purpose in asking this question, just curious. Thanks!
Edit after Robert... | https://mathoverflow.net/users/81039 | Is a manifold generically real analytic (with generic real analytic metric)? | Your two final questions are not the same, in spite of the "That is,..." that starts the second one, but, in the interpretations that make the most sense to me, the answer to the first is 'no' and the answer to the second is 'yes'.
For the second question, you know that a smooth manifold $M$ has a compatible real-ana... | 20 | https://mathoverflow.net/users/13972 | 219969 | 103,196 |
https://mathoverflow.net/questions/219974 | 5 | Suppose that $A$ and $B$ are topological spaces homotopy equivalent to finite cell complexes, and let $B^A = \mathrm{maps}(A,B)$ denote the space of maps from $A$ to $B$. **Is it there a formula for the Euler characteristic $\chi(B^A)$ of this mapping space in terms of, say, the Euler characteristics of $A$ and $B$ (or... | https://mathoverflow.net/users/78 | What is the Euler characteristic of a mapping space? | This Euler characteristic usually won't be well-defined. For example, take $A = S^1$ and $B = S^3$. Then the mapping space $[A, B]$ is the free loop space $L S^3$, which decomposes as a product
$$L S^3 \cong S^3 \times \Omega S^3$$
because $S^3$ has a Lie group structure. This makes the rational cohomology of $L S^... | 25 | https://mathoverflow.net/users/290 | 219975 | 103,197 |
https://mathoverflow.net/questions/219866 | 3 | I had posted [this](https://math.stackexchange.com/questions/1460855/a-basic-question-on-local-cohomology) question on stackexchange but did not get any response, hence putting it up on mathoverflow.
Let $X$ be a smooth, projective variety, $i:X \hookrightarrow \mathbb{P}^n$ a closed immersion for some $n>0$, $U \sub... | https://mathoverflow.net/users/46578 | A basic question on local cohomology | You've actually made way too many assumptions! All you need is the following setup: $f\colon X \to Y $ a map of topological spaces, $Z $ a subspace of $X $, and $U $ an open neighborhood in $X$ of $Z $ in which $Z $ is closed. Let $f'=f|\_U $. If $F $ is any sheaf on $X $, recall that
$$\Gamma\_Z (X, F) := \ker (F (U)... | 3 | https://mathoverflow.net/users/36720 | 219976 | 103,198 |
https://mathoverflow.net/questions/219970 | 2 | Let E(3) be 3-dimensional Euclidean space. I have submitted the following question to Mathstackexchange and other mathematical websites, but have never received any responses-not even rejections on the grounds that it was inappropriate. If C is a subset of E(3) that is a simple closed curve, does C always contain 4 pai... | https://mathoverflow.net/users/4423 | A question about simple closed curves in 3-dimensional Euclidean space | Take $C(0), C(1/4), C(1/2), C(3/4)$. If they are coplanar, then we are done. If they are not, they should form a tetrahedron, and we can choose a plane $X$ which completely divides the four points so that $C(0), C(1/2)$ lie on one side, and $C(1/4), C(3/4)$ lie on the other side. Then in each interval $(0, 1/4), (1/4, ... | 13 | https://mathoverflow.net/users/36579 | 219977 | 103,199 |
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