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https://mathoverflow.net/questions/188771 | 2 | How many right circular cylinders can pass through 5 general points in ℝ3 ? Edit: 0,2,4, or 6.
| https://mathoverflow.net/users/59248 | How many circular cylinders through 5 general points? | If the 5 points are generic then I believe Lemma 7.1 in arXiv: 1306.2346 is relevant to your question.
The Lemma, for the case of cylinders, says if there are two cylinders that pass through the same 5 generic points and there is an isometry that maps one of the cylinders onto the other, then the two cylinders are th... | 3 | https://mathoverflow.net/users/62562 | 188792 | 93,461 |
https://mathoverflow.net/questions/188525 | 11 | Let $P\in\mathbb{C}[X]$ be a complex polynomial of degree $n\geq 2$ with complex roots $\alpha\_1, \alpha\_2,\ldots, \alpha\_n$. My question is about the existence of a formula for the variance of the roots of $P$ in terms of the coefficients of $P$. First, let me fix some notations.
I will denote by $m=m(P)$ the ari... | https://mathoverflow.net/users/49268 | Variance of the roots of a complex polynomial | You may notice that your formula is very similar to the formula for the roots of the polynomial. That is not an accident - they are basically equally difficult. So in particular there is no formula in radicals for $n\geq 5$, and you don't want to see the formula in radicals for $n=4$.
proof: Take any formula in terms... | 10 | https://mathoverflow.net/users/18060 | 188793 | 93,462 |
https://mathoverflow.net/questions/188791 | 3 | Given a three dimensional Riemannian manifold $(M,g)$ and a surface $\Sigma \subset M$ can one categorize surfaces where the second fundamental form of $\Sigma$ is a scalar multiple of the induced metric on $\Sigma$? I mean do these surfaces exist and are they well studied?
i.e. $ h\_{ij} = f {\bar g}\_{ij}$ where $f \... | https://mathoverflow.net/users/50438 | Surfaces with specific types of second fundamental form | It's not a vague question. Such surfaces are known as *umbilic surfaces*. The generic $3$-dimensional Riemannian manifold $(M,g)$ has no umbilic surfaces whatsoever, and when they do exist, they form a finite dimensional family, in fact, of dimension at most $4$, since, if two umbilic surfaces are tangent at a point an... | 7 | https://mathoverflow.net/users/13972 | 188795 | 93,463 |
https://mathoverflow.net/questions/188796 | 11 | Let $C$ be a Jordan curve in $\mathbb{R}^2$. By the Jordan curve theorem, $\mathbb{R}^2 \smallsetminus C$ is uniquely partitioned into two connected regions $A$ and $B$ (the interior and exterior).
**Question 1:** Let $x\_0$ be an arbitrary point in $A$. Let $X$ be a planar Brownian motion starting at $x\_0$. Let
$$... | https://mathoverflow.net/users/62563 | Does Brownian motion immediately visit both sides of a Jordan curve? | As to question 2: Planar Brownian motion started at $y\_0$ will almost surely loop around $y\_0$, i.e., disconnect $y\_0$ from $\infty$ immediately, so it has to hit $A$ and $B$ immediately, too, and $\tau\_A = \tau\_B = 0$ a.s., no matter what kind of Jordan curve $C$ is. This should also imply that the answer to the ... | 11 | https://mathoverflow.net/users/26834 | 188800 | 93,465 |
https://mathoverflow.net/questions/188712 | 20 | In the following, let $G=\operatorname{GL}\_n(\mathbb{C})$ or $G=\operatorname{\mathbb PGL}\_n(\mathbb{C})$, depending on whichever has a better chance of yielding an affirmative answer. One could more generally ask the question for a complex, (reductive, affine,) algebraic group - but since this generalization is not ... | https://mathoverflow.net/users/9947 | Is there a scheme parametrizing the closed subgroups of an algebraic group? | Here is an affirmative answer in the sense of algebraic spaces under a reductivity hypothesis on the subgroups, using some hard input from SGA3. (The representability by a scheme for the functor classifying parabolic subgroups is well-documented in the literature in various settings; see section 3 of Exp. XXVI in SGA3 ... | 15 | https://mathoverflow.net/users/61939 | 188803 | 93,468 |
https://mathoverflow.net/questions/188711 | 2 | Reposting my [question from math.stackexchange](https://math.stackexchange.com/questions/580314/verma-modules-and-delta-function):
What is the relationship between Verma modules and delta functions? Here's the quote from Woit's notes on Lie theory (<http://www.math.columbia.edu/~woit/LieGroups-2012/vermamodules.pdf>):
... | https://mathoverflow.net/users/62647 | Relationship between Verma modules and delta functions | In general cases Verma modules are corresponding to D-modules on flag variety and then via Riemann-Hilbert correspondence to constructible sheaves on flag variety. Hence the suitable generalization of delta functions should be constructible sheaves.
As S. Carnathan pointed out, this is the standard topic of geometric... | 1 | https://mathoverflow.net/users/24965 | 188804 | 93,469 |
https://mathoverflow.net/questions/188780 | 12 | First a disclaimer: I am at best a part-time arithmetic geometer, so please accept my apologies when I am too naive or get something wrong.
From time to time I have tried to learn something about Shimura varieties. The friendliest example I came across so far are Shimura curves arising from quaternion algebras. By t... | https://mathoverflow.net/users/2039 | Concrete Examples of Shimura Surfaces | As Dylan says, the examples you are looking for are called Picard modular surfaces.
So far as I know, the simplest example of a Shimura surface is from a paper "Sur des fonctions de deux variables indépendantes analogues aux modulaires" of Picard (yes, that Picard) from 1883. From the formal group perspective, it's a... | 20 | https://mathoverflow.net/users/360 | 188805 | 93,470 |
https://mathoverflow.net/questions/188763 | 11 | Let $X$ be a projective variety over $\mathbb{C}$, denote by $D^b(X)$ the bounded derived category of coherent sheaves on $X$. Suppose we have a Fourier-Mukai functor $\Phi\_{X\rightarrow X}^\mathcal{P}:D^b(X)\rightarrow D^b(X)$ being an auto-equivalence on $D^b(X)$, and further assume that $\Phi\_{X\rightarrow X}^\mat... | https://mathoverflow.net/users/62555 | Fourier-Mukai functors being identity on objects | If $X$ is *smooth* and projective, then any such FM functor is in fact naturally isomorphic to the identity functor. This follows immediately from Corollary 5.23 of Huybrechts' book on Fourier-Mukai tranforms. Briefly, the idea is that the hypotheses ensure that $\mathcal{P}$ is a quasi-isomorphic to a sheaf on $X\time... | 9 | https://mathoverflow.net/users/100 | 188809 | 93,472 |
https://mathoverflow.net/questions/188808 | 20 | I know that it is possible to construct the hyperreal number system in ZFC by using the axiom of choice to obtain a non-principal ultrafilter. Would the non-existence of a set of hyperreals be consistent with just ZF, without choice? Let me be conservative, and say that by a "set of hyperreals," I just mean a set toget... | https://mathoverflow.net/users/62575 | Is non-existence of the hyperreals consistent with ZF? | The answer is yes, provided ZF itself is consistent. The reason is that the existence of the hyperreals, in a context with the [transfer principle](http://en.wikipedia.org/wiki/Transfer_principle#Transfer_principle_for_the_hyperreals), implies that there is a nonprincipal ultrafilter on $\mathbb{N}$.
Specifically, i... | 31 | https://mathoverflow.net/users/1946 | 188810 | 93,473 |
https://mathoverflow.net/questions/121913 | 23 | Consider the set of polynomials with real coefficients as a vector space with the following inner-product: $\langle f, g \rangle = \int\_{a}^{b} f(x)g(x) dx$.
Hilbert showed, in [a paper from 1894](http://link.springer.com/article/10.1007%252FBF02418278), that the norm (with respect to this inner-product) of a non-ze... | https://mathoverflow.net/users/31469 | Hilbert's Theorem on $L_2$ norm of polynomials in $\mathbb{Z}[X]$ - Explicit construction and a converse? | For Question 1: Once $b-a \geq 4$, the norm cannot get arbitrarily small.
Suppose $p(x)$ has leading term $c\_n x^n$.
The minimum of $\int\_a^b p^2(x) \, dx$
over all polynomials $p(x) = c\_n x^n + O(x^{n-1})$ with *real* coefficients
is a multiple of $P\_n(l(x))$, where $P\_n$ is the $n$-th
[Legendre
polynomial](htt... | 18 | https://mathoverflow.net/users/14830 | 188816 | 93,475 |
https://mathoverflow.net/questions/188814 | 5 | This was asked before on [stackexchange](https://math.stackexchange.com/questions/666122/kronecker-product-and-characteristic-polynomial) but no answer was given. The question is the following:
Let $A$ and $B$ be matrices in $GL(n)$ and $GL(m)$ respectively. Their tensor product $A\otimes B$ is defined explicitly by ... | https://mathoverflow.net/users/48554 | Characteristic polynomial of Kronecker/tensor product | As David Handleman observed, you need (assuming you are over a splitting field) simply the polynomial that has the products of eigenvalues as roots.
Using the resultant, you could calculate this polynomial as $\mbox{res}\_y(P\_A(y),P\_B(x/y)\cdot y^m)$. (This is a polynomial in $x$.)
For example, let $P\_A(x)=(x-2)(x... | 14 | https://mathoverflow.net/users/59303 | 188820 | 93,477 |
https://mathoverflow.net/questions/188807 | 8 | Let's consider the space $L^2[a,b]$ of functions on the interval and the norm:
$$ ||f(x)||^2 = \int\_a^b |f(x)|^2 \, dx $$
Now what if we consider only polynomials with integer coefficients: $f(x) \in \mathbb{Z}[x]$?
If we write $f(x) = \sum a\_n x^n$ with each $a\_n \in \mathbb{Z}$, this norm is a rational quad... | https://mathoverflow.net/users/1358 | minimizing an integral over integer-coefficient polynomials $\displaystyle \inf_{f \in \mathbb{Z}[x]} \int_a^b f(x)^2 \, dx $ | Thanks to **BigM** for the link to **Ofir's**
[MO Question 121913](https://mathoverflow.net/questions/121913),
which cites a 120-year-old paper of Hilbert for the result that
the integral can get arbitrarily small as long as $b-a < 4$:
>
> D. Hilbert: Ein Beitrag zur Theorie des Legendre'schen Polynoms, *Acta Math.... | 11 | https://mathoverflow.net/users/14830 | 188823 | 93,479 |
https://mathoverflow.net/questions/188446 | 3 | As the comment of Andreas Thom indicated [here](https://mathoverflow.net/questions/188344/simple-z-algebra), a separable $C^\star$ algebra $A$ can not contain a $Z^\star$ algebra.(A $Z^\star$ algebra is a $C^\star$ algebra which all elements are zero divisor). So separability is an obstruction for $A$ to **contain** a ... | https://mathoverflow.net/users/36688 | Obstructions for $C^\star$ algebras to contain a $Z^\star$ algebra | $A$ contains a $Z^\*$ $C^\*$-subalgebra iff it contains an uncountable family $\{ a\_i \}$ of nonzero positive mutually orthogonal elements. If it has such a family, then $C^\*(\{ a\_i \})$ is $Z^\*$. If it does not have such a family and $B$ is a $C^\*$-subalgebra, then any maximal family $\{ a\_n \}$ of n.p.m.o. elem... | 5 | https://mathoverflow.net/users/7591 | 188830 | 93,481 |
https://mathoverflow.net/questions/70976 | 1 | So I have finitely presented group with 2 generators. Can I solve word problem in it (check if two words X and Y are actually the same element of my group)?
| https://mathoverflow.net/users/16653 | Word problem in group of rank 2 | It is a classical result, originally by Higman-Neumann-Neumann, that every finitely presented group embeds in a 2-generator finitely presented group. In particular, every finitely presented group with unsolvable word problem embeds in a 2-generator group. Given that having solvable word problem is preserved by subgroup... | 0 | https://mathoverflow.net/users/62003 | 188838 | 93,484 |
https://mathoverflow.net/questions/188842 | 3 | Let $G$ be a connected unipotent algebraic (affine) group defined over a perfect field $k$. Then $G$ is $k$-isomorphic as an algebraic $k$-variety to the affine space $\mathbb A^n\_k$.
Is there any good reference for this result?
It is probably in Demazure, Michel; Gabriel, Pierre Groupes algébriques, but I do not ... | https://mathoverflow.net/users/10482 | Connected unipotent algebraic groups | It is, for example, explained in paragraph $8$ (Appendix) of the book [Unipotent Algebraic Groups](http://link.springer.com/book/10.1007/BFb0070517), Lecture Notes in Mathematics Volume 414, 1974, by Tatsuji Kambayashi, Masayoshi Miyanishi, and Mitsuhiro Takeuchi, which is accessible online. It is also discussed in the... | 5 | https://mathoverflow.net/users/32332 | 188847 | 93,486 |
https://mathoverflow.net/questions/188788 | 2 | Given $u\in BV(R^N)$, we say $u$ is approximate continuous at $x$ and the approximation limit is $l\in R$ if
$$ \lim\_{r\to 0}\frac{\mathcal{L}^N(B(x,r)\cap \{|u-l|>\epsilon\})}{r^N} =0 $$
for all $\epsilon>0$. (I know the approximate continuity can be defined for even just a barely measurable function. But given I am ... | https://mathoverflow.net/users/62560 | Question regarding to approximate continuity | Based on the following observation, I believe that conditions $(1)$ and $(2)$ are not equivalent.
Since the set
$$B(r)=:\{B(x\_0,r)\cap \{u>l\}\}=\bigcup\_{n}\{B(x\_0,r)\cap \{u>l+\frac{1}{n}\}\}:=\bigcup\_{n}B\_n(r),$$
and $B\_n(r)\subset B\_{n+1}(r)\subset\cdots\to B(r)$, we know that
$$\lim\_{n\to \infty}\mathc... | 1 | https://mathoverflow.net/users/26608 | 188853 | 93,487 |
https://mathoverflow.net/questions/188787 | 0 | All the following we use Evans notation.
By Green's reconstruction formula, we could represent $u$ by
$$ u(x)=\int\_\Omega-\triangle u(y)G(x,y)dy-\int\_{\partial \Omega}u(y)\partial\_\nu G(x,y)d\sigma(y) $$
where $\Omega$ is open bounded with smooth boundary and $\nu$ is the outer normal vector, $G(x,y)$ is the Gree... | https://mathoverflow.net/users/62560 | Question regarding Laplace equation under Evans setting | I think usually the standard approximation arguments gives you the result $C^2(\Omega)\cap C^1(\bar{\Omega})$ and I did not quit see how to reduce to $C^2(\Omega)\cap C^0(\bar{\Omega})$.
However, your second question is a good exercise for you to understand the properties of the Green's function. Roughly speaking, $... | 1 | https://mathoverflow.net/users/26608 | 188861 | 93,490 |
https://mathoverflow.net/questions/188872 | 0 | Let
* $X$ be compact Hausdorff topological space,
* $C(X)$ denote the algebra of complex-valued continuous functions on $X$,
* $b\in \mathbb{C}^m$,
* $\mathbf{A}\in C(X)^{m\times n}$,
* for all $x\in X$, $b\in \textrm{range}(\mathbf{A}(x))$.
Question: Does there exist an $\mathbf{x}\in C(X)^{n\times 1}$ such that ... | https://mathoverflow.net/users/62590 | $Ax=b$ in a function space | Let $$A(x)=\pmatrix{x&1\cr 0&x},$$ for $x\in [-1,1]$ and $$b=\pmatrix{1\cr 0}.$$ Then $b$ is in the range of $A(x)$ for every $x$, but for $x\neq 0$, we have $$A^{-1}(x)b=\pmatrix{1/x\cr 0},$$ so there is no continuous solution.
| 7 | https://mathoverflow.net/users/12120 | 188882 | 93,494 |
https://mathoverflow.net/questions/188874 | 7 | I'm not sure if this counts as research level, since it might just be an expression of my ignorance. But anyway.
Let $\Delta^n\_A = Spec(A[X\_0, \dots, X\_n]/\sum X\_i - 1)$ denote the standard algebraic simplex over $A.$ These assemble into the standard cosimplicial affine scheme $\Delta\_A^\bullet.$ Hence, given an... | https://mathoverflow.net/users/5181 | Computing naive algebraic singular homology | There has not been so much research on $\operatorname{Sing}(X)(A)$, and most of it concentrated on its homotopy. Nevertheless, some things are known:
* Among the first results are those by Jardine in a series of papers at the beginning of the 80s, see [the first five entries on his list of publications.](http://www.m... | 8 | https://mathoverflow.net/users/50846 | 188892 | 93,498 |
https://mathoverflow.net/questions/188900 | 7 | Burton, in "Elementary Number Theory", states that the following problems are due to Euler 1775:
>
> If $n$ is a triangular number, then so are $9n+1$, $25n+3$ and $49n + 6$.
>
>
>
R. F. Jordan in the J. of Recreational Mathematics (1991, vol.23, p.78) proves the following generalization:
>
> Let $t\_k$ be... | https://mathoverflow.net/users/8542 | Euler's Triangular Number closure properties | Dickson, History of the Theory of Numbers, Volume II, page 12, writes,
L. Euler (pp. 264-5, about 1775) noted that $9\Delta\_a+1=\Delta\_{3a+1}$, $25\Delta\_a+3=\Delta\_{5a+2}$, $49\Delta\_a+6=\Delta\_{7a+3}$, $81\Delta\_a+10=\Delta\_{9a+4}$.
The reference Dickson gives is to Opera Postuma, 1, 1862.
| 4 | https://mathoverflow.net/users/3684 | 188911 | 93,505 |
https://mathoverflow.net/questions/188895 | 1 | Let $X$ be a K3 surface and $C$ a curve on $X$. We say that $C$ is $d$-gonal if it admits a pencil of degree $d$ (and none of smaller degree).
I am wondering if there exist characterizations of $d$-gonal curves on $X$ for small $d$.
For example, if $d=2$ (that is, $C$ is hyperelliptic) then Saint-Donat proved tha... | https://mathoverflow.net/users/40038 | Characterization of $d$-gonal curves on a K3 surface | There is indeed such a characterisation. It is essentially due to Donagi-Morrison, see section 4 of the paper "Linear-systems on K3 Surfaces" in J. Diff. Geo. Let $C$ be a $d$-gonal curve on a K3 surface, and assume for simplicity that the $A \in W^1\_d(C)$ achieves the Clifford index (this will be verified in generic ... | 2 | https://mathoverflow.net/users/36622 | 188914 | 93,507 |
https://mathoverflow.net/questions/188879 | 3 | Let $\mathcal{G}\_{<\omega}$ be the set of graphs $G = (V,E)$ such that $V = \{0,\ldots,n\}$ for some $n \geq 0$ and $E \subseteq \mathcal{P}\_2(V) = \{\{a,b\} : a,b \in V \textrm{ and } a\neq b\}$. We write $\mathcal{G}\_{<\omega}/\cong$ for the set of isomorphism classes. The following defines an ordering relation on... | https://mathoverflow.net/users/8628 | Minor ordering for finite graphs | Re part (1): It's certainly not a lattice. For instance if $G$ is a four-vertex path and $H=K\_{1,3}$, a four-vertex star, then the minimal graphs that contain both $G$ and $H$ as minors are a four-vertex graph with a triangle and a tail, or a five-vertex tree. Neither is a minor of the other: the five-vertex tree has ... | 3 | https://mathoverflow.net/users/440 | 188933 | 93,510 |
https://mathoverflow.net/questions/188618 | 0 | Let $G, H$ be simple, undirected graphs without loops. We say that $G, H$ have the same homomorphism fingerprint if $|\text{Hom}(X, G)| = |\text{Hom}(X, H)|$ for all graphs $X$. (By graph homomorphisms I mean edge preserving maps; so for instance $\text{Hom}(X,K\_2)=\emptyset$ if $\chi(X) > 2$).
From Lovasz [1] we ha... | https://mathoverflow.net/users/8628 | "Homomorphism fingerprint" for graphs | In any category, if $G$ and $H$ are objects such that there exist monic maps $G\to H$ and $H\to G$, then $|\text{Hom}(X, G)| = |\text{Hom}(X, H)|$ for all $X$. There are plenty of pairs of non-isomorphic infinite graphs with this property.
| 8 | https://mathoverflow.net/users/75 | 188936 | 93,512 |
https://mathoverflow.net/questions/188948 | 3 | Consider the problem (Nonlinear Schrödinger equation)
\begin{equation}
\left\{
\begin{array}{rl}
iu\_t + \Delta u\mp u|u|^{\alpha}=0\\
u(0) =\varphi\in H^{1}(\mathbb{R}^N), \\
\end{array}\right.
\end{equation}
where $N\geq 3$ and $0<\alpha<\frac{4}{N}$.
I consider solution in the strong sense, i.e. $u(t,x)\in C^0([... | https://mathoverflow.net/users/45729 | Dichotomy for global existence or blow up for solutions of evolution problems | It is indeed a quite general fact and this is a consequence of the use of the fixed-point theorem.
You have rewritten your equation in the mild form
\begin{equation}
u(t)= e^{it\Delta}\varphi \pm i\int\_0^t e^{i(t-s)\Delta}u|u|^{\alpha}(s)ds,
\end{equation}
which allows you to solve the abstract equation $u = f +... | 4 | https://mathoverflow.net/users/62629 | 188954 | 93,516 |
https://mathoverflow.net/questions/188955 | 6 | Is it conceivable that [Frankl's union closed sets conjecture](http://en.wikipedia.org/wiki/Union-closed_sets_conjecture) is undecidable in $\mathsf{ZFC}$, or is this quite implausible, perhaps due to the "finitistic" nature of the statement, or for some other reason?
| https://mathoverflow.net/users/8628 | Decidability of Frankl's union-closed sets conjecture | The statement has [complexity $\Pi^0\_1$](http://en.wikipedia.org/wiki/Arithmetical_hierarchy), which means that it has a
single universal quantifier, quantifying over the possible
union-closed sets, and then making a simple assertion about those
objects.
Although this is a very simple level of complexity, it is the ... | 8 | https://mathoverflow.net/users/1946 | 188957 | 93,517 |
https://mathoverflow.net/questions/188952 | 7 | I'm working with the equilibrium equations in linear elasticity, which I have not worked with in the past. My engineering colleagues seem to "know" that the maximum Von Mises stress occurs on the boundary of the domain (assume the domain experiences no body force, and is loaded on part of its boundary while satisfying ... | https://mathoverflow.net/users/56831 | Is there a maximum principle for stress in continuum mechanics? | That depends on the elastic model that you deal with. Is it linear (infinitesimal displacements) or non-linear ? Is it isotropic or not ?
In general, because elasticity is a system, not an equation, there is no such scalar quantity that would obey a maximum principle. However, a linear isotropic model is governed by ... | 6 | https://mathoverflow.net/users/8799 | 188958 | 93,518 |
https://mathoverflow.net/questions/188964 | 1 | Suppose that $f(x) \in \mathbb{Z}[x]$ is an irreducible polynomial (over $\mathbb{Q}$). Let $p$ be a very large prime with respect to the coefficients of $f$. Then it is possible that $f(x)$ may factor over $\mathbb{F}\_p$, and indeed $f$ may split completely over $\mathbb{F}\_p$. My question concerns whether $f$ may h... | https://mathoverflow.net/users/10898 | A question on how polynomials split over $\mathbb{F}_p$ for large primes $p$ | If $r \ll p^{1/d}$ and $p \mid f(r)$ then (since $f(r) \neq 0$)
$f(r) = ap$ for some nonzero $a \ll 1$. Hence for each of
finitely many choices of $a$ we are asking for prime values of $f(r)/a$
as $r$ ranges over ${\bf Z}$. That's a characterization of sorts,
though (for each choice of implicit constant $C$ in $|r| \le... | 8 | https://mathoverflow.net/users/14830 | 188966 | 93,520 |
https://mathoverflow.net/questions/188969 | -3 | Definitions of ordinary functor categories and higher categories are considered with very similar algebraic and geometric methods such as graph structures and simplicial sets. I know the differences between ordinary algebra and higher algebra. But are there deeper relations or results between ordinary functor categorie... | https://mathoverflow.net/users/61384 | Relations between ordinary functor categories and higher categories | Maybe this doesn't address completly to the question, but I think it's a start.
Functor categories and higher categories are quite different objects,
the only relation that I can think of is that usually higher categories are defined either as objects in a presheaf-category satisfying a property (for instance as simp... | 1 | https://mathoverflow.net/users/14969 | 188979 | 93,522 |
https://mathoverflow.net/questions/188984 | 2 | I would like to show that when $f(x,y)$ is irreducible over $\mathbb{C}[x,y]$ then $f(x^2,y)$ is irreducible over $\mathbb{C}[x,y]$. I know that this is not true in general, for example, $f(x,y) = y^2 - x$. However, I know that the original polynomial contains only even powers of $x$. Does this extra condition allow me... | https://mathoverflow.net/users/62641 | Irreducibility after substitution | The answer is yes. (In what follows, I write $f(x^2,y)$ for your $f(x,y)$.)
1) By symmetry, if $h(x,y)$ is an irreducible factor of $g(x^2,y)$ then so is $h(-x,y)$.
2) In particular, if $h(x,y)$ is an irreducible factor of $f(x^4,y)$ then so is $h(-x,y)$.
3) Write $f(x^4,y)$ as a product of irreducible factors. ... | 7 | https://mathoverflow.net/users/10503 | 188986 | 93,526 |
https://mathoverflow.net/questions/188972 | 1 | I was wondering if someone knows how to prove the following fact (which might not be a fact ;) ):
let X being a stochastic process with almost surely continuous sample path, and such that, there exists a constant $\eta>0$ such that,
$
\forall \epsilon>0,\ \exists \mathcal{C}\_{\epsilon}>0, \ \mathbb{P}(|X\_t|>\epsi... | https://mathoverflow.net/users/62636 | Does very fast convergence in probability imply almost sur convergence for a continuous stochastic process? | Consider a sequence of independent uniform(0,1) random variables $U\_j$, and
let $$X\_t = \sum\_{j=1}^\infty \max(0,1 - 2^j |t - j - U\_j|)$$ In particular
$X\_t = 0$ outside the intervals $(j + U\_j - 2^{-j}, j + U\_j + 2^{-j})$, and so
$P(X\_t > 0) < 2^{2-t}$. But of course $X\_t$ does not converge to $0$.
| 1 | https://mathoverflow.net/users/13650 | 188991 | 93,528 |
https://mathoverflow.net/questions/188921 | 3 | My question is about the proof of Lemma D.3 on p. 147 of the book "Neron models" by Bosch, Lutkebohmert, and Raynaud. Namely, towards the end of that proof there is the sentence "That $\varphi$ satisfies the cocycle condition follows in a similar way from Lemma D.1". My question is: how precisely does this follow? Let ... | https://mathoverflow.net/users/53197 | Extending descent data from the special fiber of an extension of DVR's | This seems to be a gap in the proof mentioned above, since Lemma D.1 in its present formulation does not seem to imply that $\varphi$ is a descent datum as claimed (i.e., that it satisfies the cocycle condition).
To fix the gap, one notes that Lemma D.1 stays true with the same proof and assumptions if $R''$ is take... | 3 | https://mathoverflow.net/users/53197 | 188993 | 93,530 |
https://mathoverflow.net/questions/188977 | 24 |
>
> **Question:** Which natural numbers are of the form $a^2 - b^2 - c^2$ with $a>b+c$?
>
>
>
This question came up in (Eike Hertel, Christian Richter, *Tiling Convex Polygons with Congruent Equilateral Triangles*, Discrete Comput Geom (2014) 51:753–759), where it was shown that numbers of this form are numbers ... | https://mathoverflow.net/users/8176 | Which natural numbers are a square minus a sum of two squares? | The numbers that are not of the form $a^2-b^2-c^2$ with $b$ and $c$ positive and $a>b+c$ are precisely the idoneal numbers apart from $7$, $28$, $112$, $15$, $60$, and $240$. As noted in the problem, the paper by Hertel and Richter shows that the numbers not of this form are necessarily idoneal numbers, and a quick cal... | 23 | https://mathoverflow.net/users/38624 | 188994 | 93,531 |
https://mathoverflow.net/questions/188975 | 12 | I know only a little bit about C$^\*$-algebras and I want a to know if you know a nice apllication or the influence of them in the field of partial differential equations (it is better that it is understandable for graduate students), or maybe can explain me why they are important for pseudo-differential-operators. A s... | https://mathoverflow.net/users/nan | applications of C$^*$-algebras in the field of PDEs | I think the canonical connection between C\*-algebra and differential operators is Connes' index theorem for foliated manifolds. I don't know if that counts as PDEs but it's certainly related. Every foliated manifold $M$ has an associated C\*-algebra $A$ which is noncommutative (except in trivial cases) but in some way... | 13 | https://mathoverflow.net/users/23141 | 189010 | 93,534 |
https://mathoverflow.net/questions/188371 | 37 | In preparing some practice problems for my complex analysis students, I stumbled across the following. It is not hard to show, using Liouville's theorem, that
$$\pi\cot(\pi z)=\frac{1}{z}+\sum\_{n=1}^\infty\left(\frac{1}{z+n}+\frac{1}{z-n}\right),$$
which implies that
$$-\frac{\pi z}{2}\cot(\pi z)=-\frac{1}{2}+\sum\_{k... | https://mathoverflow.net/users/11919 | $\zeta(0)$ and the cotangent function | This is not a completely satisfactory answer. I would like a simpler one.
Nevertheless still probably a good exercise in Complex variables.
I will only sketch it.
What we want to show is equivalent to
$$\zeta(2n)=-\frac{1}{2\pi i}\int\_{C\_r}\frac{\pi z \cot(\pi z)}{2z^{2n+1}}\,dz,\qquad n\ge 0,\quad n\in{\bf Z}.\ta... | 18 | https://mathoverflow.net/users/7402 | 189028 | 93,540 |
https://mathoverflow.net/questions/188862 | 1 | **Edit:** According to the comments to the previous version of this question, I remove my essential errors in the question. I thank the commenters very much.
Let $M$ be a n dimensional manifold. For $k<n$, $G\_{k}TM$ is the space of all $k$ dimensional subspace of $T\_{x}M, x\in M$. It is a $n+k(n-k)$ dimensional ma... | https://mathoverflow.net/users/36688 | Sectional curvature as a Hamiltonian on the Grassmanization of the tangent bundle | This isn't an answer, other than a general set of comments to explain why there *is* no answer in the form that the OP wants.
First of all, when $V$ is a vector space of dimension $n$ and $k$ is an integer satisfying $0<k<n$, there is no symplectic structure on $\mathrm{Gr}\_k(V)$, the space of $k$-dimensional subspa... | 7 | https://mathoverflow.net/users/13972 | 189029 | 93,541 |
https://mathoverflow.net/questions/189011 | 5 | We know that Con(ZF) implies Con(ZFC+GCH), Con(ZF+neg(AC)) and Con(ZFC+neg(CH)). But what are some weak theories in which these relative independence results are provable? In particular, are they provable in PA and even in some interesting proper subtheories of PA? Are there similar relative independence results whose ... | https://mathoverflow.net/users/17218 | Formal systems needed to formalize relative independence results | Independence results such as the ones mentioned that can be proved using basic syntactic methods (relative interpretation, forcing) can be formalized in the theory $\mathit{PV}\_1$ (also known as $T^0\_2$, $\mathit{VP}$, and the $\forall\Sigma^b\_1$-fragment of $S^1\_2$): it has function symbols for all polynomial-time... | 14 | https://mathoverflow.net/users/12705 | 189032 | 93,543 |
https://mathoverflow.net/questions/189018 | 5 | We work in the set theory NBG with the axiom of (local choice but without global (class) choice. For every class A P(A) is the class of all sets x included in the class A.
We know that P(A) is a set iff A is a set and a proper class iff P(A) is a proper class. We also know that if A is a set there is no bijection be... | https://mathoverflow.net/users/30395 | When do we have a bijection between a proper class A and its power set class P(A)? | Yes, this is provable in NBG. To see this, let $F$ be a one-one function from $\mathcal P(A)$ into $A$. By transfinite recursion on $\in$, we define a function $G$ from $V$ to $A$ such that $G(x) = F(G[x])$. A simple induction then establishes that $G$ is one-one.
| 5 | https://mathoverflow.net/users/17968 | 189041 | 93,546 |
https://mathoverflow.net/questions/189001 | 6 | It is known that the $n^{\text{th}}$ Morava $K$-theory at a prime $p$, denoted $K(n)$, is complex oriented. In other words, it admits a theory of Chern classes, or equivalently a morphism of homotopy associative, commutative ring spectra $f:MU\to K(n)$. This complex orientation is related to the height $n$ Honda formal... | https://mathoverflow.net/users/11546 | Uniqueness of Complex Orientation of Morava K-theory | First, I claim that if we ignore the ring structure and complex orientation, then there is a unique spectrum (up to homotopy equivalence) that deserves to be called $K(n)$. I do not know whether there is a nice way to see this before setting up a lot of chromatic homotopy theory based on a particular choice of $K(n)$. ... | 17 | https://mathoverflow.net/users/10366 | 189045 | 93,548 |
https://mathoverflow.net/questions/189039 | 4 | In HTT(Higher Topos Theory) Remark7.3.1.19, it it sketched that the proper base change theorem for $\infty$-topos implies the usual proper base change theorem in (unbounded) derived category. However, I can't understand its detail.
Let $A=Ch(Ab)$ be the category of chain complexes of abelian groups endowed with a con... | https://mathoverflow.net/users/55871 | unbounded derived category of a $\infty$-topos | *Note: answer corrected thanks to the comments of Dylan Wilson and Marc Hoyois.*
Let $X$ be a topological space. Its derived category $D(X)$ is the derived category of the abelian category $Ab(Sh(X))$ of sheaves of abelian groups on $X$. As an infinity-category it may be defined as the infinity-category associated to... | 10 | https://mathoverflow.net/users/62356 | 189046 | 93,549 |
https://mathoverflow.net/questions/189048 | 6 | Let $A= (a\_{ij})\_{ij}, 1 \leq i, j \leq n$ be a symmetric $n \times n$ matrix. Suppose
(1) $a\_{ij} \geq 0$ are real numbers;
(2) The sum of each row $\sum\_{j=1}^{n} a\_{ij} = 1$ for $1 \leq i \leq n$.
Then I want to show the following: there must exists a nonzero $\prod\_{i=1}^n a\_{i, \sigma(i)}$, where $\s... | https://mathoverflow.net/users/29730 | A question about symmetric matrix | Your matrix $A$ is doubly stochastic (because of condition (2) and of its symmetry). The [Birkhoff-von Neumann theorem](http://en.wikipedia.org/wiki/Doubly_stochastic_matrix#Birkhoff_polytope_and_Birkhoff.E2.80.93von_Neumann_theorem) yields that it is a convex combination of permutation matrices. Take any permutation m... | 11 | https://mathoverflow.net/users/2530 | 189052 | 93,551 |
https://mathoverflow.net/questions/187701 | 2 | Let $X \in \mathbb{R}^{p\times p}$ be a large square matrix, consisting of i.i.d. Gaussian entries. Then it is known that the singular values of $X$ follow the Marchenko-Pastur law.
Now let's introduce an adversary, who arbitrarily selects $p/2$ rows of X and form a submatrix. Can the adversary succeed to make the si... | https://mathoverflow.net/users/8369 | Restricted singular values of random matrix | It depends what do you mean by "look very different". I assume that you mean
that the empirical measure is close to that of the MP law. The answer below
assumes this is what you meant.
Short answer: no.
Longer answer: there is an exponential (in $p$) number of ways to choose the rows. But the
concentration of the... | 2 | https://mathoverflow.net/users/35520 | 189078 | 93,562 |
https://mathoverflow.net/questions/189073 | 0 | Let $u$ be a harmonic function and we define
$$ q(r)=\int\_{\partial B(0,r)}u^2(x)\,dx $$
The question is about to prove that $q(r)$ is log-convex, i.e., I want to show $\log q(r)$ is convex function of $\log r$
I compute
$$ \frac{d^2}{(d\log r)^2} \log q(r) = r^2\frac{q''(r)}{q(r)}+r\frac{q'(r)}{q(r)}-\left(r\fra... | https://mathoverflow.net/users/62560 | Prove a function, defined by integration of a harmonic function, is log-convex | I think this is more or less an exercise, instead of a research problem, which is not suitable for posing on this site.
However, I provide an link for your problem,
<http://www.math.caltech.edu/~2010-11/3term/ma110c/HarmonicFunctions.pdf>
You could find more information by just google "subharmonic functions", he... | 2 | https://mathoverflow.net/users/26608 | 189081 | 93,563 |
https://mathoverflow.net/questions/189083 | 5 | Just wanted to know if Giroux's theorem for 3-dimensional contact manifolds can be generalized:
In contact geometry for manifolds of dimension 3 , we have Giroux's theorem , stating that for any compact, oriented manifold there is a bijection between the set of contact structures up to isotopy and the set of open boo... | https://mathoverflow.net/users/62686 | Generalization of Giroux's Theorem for Higher Dimensions? | Giroux proved that for every contact manifold $(M^{2n+1},\alpha)$, there exists an open book supporting the contact form. An open book consists of a codimension two submanifold $K^{2n-1}\subseteq M^{2n+1}$ and a fibration $M\setminus K\to S^1$ (which is "standard" in some tubular neighborhood of $K$) with fibers $F\_t$... | 6 | https://mathoverflow.net/users/35353 | 189084 | 93,564 |
https://mathoverflow.net/questions/189085 | 1 | Consider $(q\_1,q\_2,...) \in Q$ non-intersecting sets of distinct elements $(e\_{(i,1)},e\_{(i,2)},...)\in q\_i$.
How many ways can one write down an ordering of all of the $\sum\_j |q\_j|$ elements s.t.:
(1) For any two elements $e\_{(a,1)}$ & $e\_{(b,1)}$, where $a < b$, it must be the case that $e\_{(a,1)}$ com... | https://mathoverflow.net/users/62688 | The number of orderings of elements with order-of-appearance constraints | Notice that your ordering defines a partial order on $Q$ which has a tree as its Hasse diagram. The number you are trying to calculate is the number of [linear extensions](http://en.wikipedia.org/wiki/Linear_extension) of this poset, or in other words the number of increasing labelings of your tree.
When the Hasse di... | 1 | https://mathoverflow.net/users/2384 | 189087 | 93,565 |
https://mathoverflow.net/questions/189086 | -1 | Suppose $f(x)=\sum\_{|\alpha|=0}^{\infty}a\_{\alpha} x^{\alpha}$ for all $x\in\mathbb{R}^n$. Moreover we know a priori that $f$ is an algebraic function.
Is $f$ necessarily a polynomial?If not what are typical counterexamples?
| https://mathoverflow.net/users/48438 | Smooth algebraic functions | You wrote a series. If you mean that it converges for all $x\in R^n$ than this is an entire
function that is the series also converges for $x\in C^n$. An entire algebraic function is
a polynomial, indeed. This follows from the growth estimate and Cauchy's inequalities, for example.
| 4 | https://mathoverflow.net/users/25510 | 189092 | 93,567 |
https://mathoverflow.net/questions/189042 | 4 | What is the lower central series $\Gamma\_k(P\_n)$, where $P\_n$ is the pure braid group with $n$ strands? We know that $P\_n$ is generated by elements $A\_{i,j}$; do we know the generators of $\Gamma\_k(P\_n)$?
| https://mathoverflow.net/users/15770 | Lower Central Series of Pure Braid Groups? | As Jim Conant says, the structure of the lower central series of the pure braid group $P\_n$ can be deduced from the fact that $P\_n$ is an iterated semidirect product of free groups, $P\_n=F\_{n-1}\rtimes F\_{n-2}\cdots F\_2\rtimes F\_1$, though that's not quite enough: one also needs to know that the action of $P\_{n... | 10 | https://mathoverflow.net/users/17846 | 189093 | 93,568 |
https://mathoverflow.net/questions/189089 | 6 | May I humblely ask what is a good reference for conormal distributions (for student with some rudimentary pseudo-differential operator background)? I heard from my advisor that it is useful in index theory, but most of the lecture notes I read are quite opaque (like Simanca's) by using baroque notations, and it is diff... | https://mathoverflow.net/users/18850 | What is a good reference for conormal distributions? | I think the third volume of Hormander's The Analysis of Linear Partial Differrential Operators is a good reference. chapter 18 is an introduction to PsDOs, especially, in section 18.2, it's devoted to conormal distributions, which I think is very useful for people with PsDO background. He starts with the fact that the ... | 5 | https://mathoverflow.net/users/35702 | 189098 | 93,570 |
https://mathoverflow.net/questions/189094 | 7 | In my class, we proved the following condition: define the polynomial $P\_l(x)$ as
$$P\_l(x) = \sum\_{r=1}^{l-1}{\frac{1}{r}x^{l-1-r}}$$
Then if for all $a \in \mathbb{Z}/l\mathbb{Z}-\{0,1\},$ $P\_l(x)$ does not vanish (mod $l$), then the First Case of Fermat's Last Theorem(1CFLT) is true for $\mathbb{Z}$ and expo... | https://mathoverflow.net/users/22094 | Roots of a polynomial in a finite field related to Fermat's Last Theorem | This is **true**.
Let $P(x) = \sum\_{n=1}^{l-1} \frac{x^n}{n} \in {\mathbb F}\_l[x]$ (this is $x^{l-1} P\_l(x^{-1})$). Then we have $P(1-x) = P(x)$ (check that they have the same derivative and the same value at $1/2$) and $x^l P(x^{-1}) = -P(x)$.
Now if $l \equiv 1 \bmod 3$, then there is a primitive sixth root of... | 20 | https://mathoverflow.net/users/21146 | 189104 | 93,572 |
https://mathoverflow.net/questions/188462 | 4 | Recall the definition of Hardy's hierarchy:
$H\_0(n)=n+1\\
H\_{\alpha+1}(n)=H\_\alpha(n+1)\\
H\_\alpha(n)=H\_{\alpha[n]}(n)$,
where the last rule applies if $\alpha$ is limit ordinal, and $\alpha[n]$ for $\alpha<\varepsilon\_0$ are fundamental sequences (we are using their standard definitions). We can generalize ... | https://mathoverflow.net/users/30186 | Functions in "gaps" in Hardy hierarchy | The function F exists. This follows from results from subrecursive degree theory.
A possible reference is: Streamlined subrecursive degree theory.
Lars Kristiansen, Jan-Christoph Schlage-Puchta & Andreas Weiermann
Annals of Pure and Applied Logic 163 (6):698-716 (2012)
Andreas Weiermann
| 8 | https://mathoverflow.net/users/62695 | 189105 | 93,573 |
https://mathoverflow.net/questions/189130 | 1 | The answers to [this question](https://mathoverflow.net/questions/37101/computing-the-index-of-a-lie-algebra-what-is-known-beyond-the-reductive-case) indicate that there is a fairly vast literature on the index of Lie algebras. Unfortunately I was not able to find in this literature (or maybe to extract from it) an ans... | https://mathoverflow.net/users/11216 | Inequality for the index of a Lie algebra using its Levi decomposition | Your initially proposed inequality fails if $G$ is Euclid's group $\mathrm{SO}(3)\ltimes\mathbf R^3$. Indeed we have $\operatorname{ind}\mathfrak g = 2$ (generic coadjoint orbits of $G$ have dimension 4), $\operatorname{ind}\mathfrak{so}(3) = 1$ (generic coadjoint orbits of $\mathrm{SO}(3)$ have dimension 2), hence
$
\... | 6 | https://mathoverflow.net/users/19276 | 189136 | 93,586 |
https://mathoverflow.net/questions/189139 | 4 | I have a question about the proof of the Lemma 7.2 in the paper
>
> I. Juhász, P. Koszmider and L. Soukup,
> *A first countable, initially $\omega\_{1}$-compact but non-compact space*,
> Topology and its Applications **156** (2009), 1863-1879.
> doi:[10.1016/j.topol.2009.04.004](http://dx.doi.org/10.1016/j.topol... | https://mathoverflow.net/users/39086 | CCC Forcing and $\omega_1$ conditions | If there were no such $q$, then it would be forced, by all conditions, that $\{\xi\in\omega\_1:q\_\xi\in\mathcal G\}$ is countable and therefore bounded below $\omega\_1$. Consider a maximal antichain of conditions that force specific values ($\check\gamma$) for the bound. Since the antichain is countable, there are on... | 9 | https://mathoverflow.net/users/6794 | 189140 | 93,589 |
https://mathoverflow.net/questions/189124 | 2 | Let $A$ be the algebra of all bounded functions from $[0,\;1]$ to $\mathbb{C}$.
For $f\in A,\;$ $\omega\_{f}$ is the standard [oscillation function.](http://en.wikipedia.org/wiki/Mathematics_of_oscillation). Each of the following two (equivalent) norms on $A$, defines a Banach algebra structure on $A$.
$$\parallel ... | https://mathoverflow.net/users/36688 | A commutative Banach algebra with an abundance of discountinuous functions | *This is an answer to Q2, because as pointed out by Yemon, Q1 and Q3 follow easily from the fact that this algebra is isomorphic to $C(\beta [0,1]\_d)$. It is enough to prove then that if two compact spaces $X$ and $Y$ are continuous images of $\beta \kappa$ for some cardinal $\kappa$, then so is $X\times Y$.*
Let $X... | 2 | https://mathoverflow.net/users/15129 | 189144 | 93,591 |
https://mathoverflow.net/questions/186648 | 3 | I wanted to check whether $A(x,y):=\frac{xy}{x+y}$ is an associative operation in every commutative [vN regular](http://en.wikipedia.org/wiki/Von_Neumann_regular_ring) ring. Now $A(-1,A(1,1))=A(-1,\frac{1}{2})=1\neq 0 =A(0,1)=A(A(-1,1),1)$. On the other hand, $\frac{xy}{x+y}=\frac{1}{1/x+1/y}$, hence $A(x,A(y,z))=\frac... | https://mathoverflow.net/users/20781 | Is the equational theory of commutative vN regular rings decidable? | Emil Jeřábek's comment explains how to prove that the equational theory of commutative vN regular rings decidable:
>
> I know nothing about von Neumann regular rings, however such an identity (or any quantifier-free formula for that matter) fails in a field iff it fails in an algebraically closed field (by embeddin... | 2 | https://mathoverflow.net/users/20781 | 189146 | 93,592 |
https://mathoverflow.net/questions/189141 | 3 | I apologize in advance if this question is thought to be too soft or otherwise inappropriate for mathoverflow.net. Let M be the infinite set of all homeomorphism types of finite dimensional topological manifolds which are compact, connected and metrizable. Many years ago, the question of whether M was countable, was an... | https://mathoverflow.net/users/4423 | Is a particular type of question about certain infinite sets still being asked? | Here is a question of classical function theory of this sort. Let $f$ be an entire function
of one variable. Consider all factorizations $f=g\circ h$ where $g$ and $h$ are entire.
Two factorizations $f=g\_1\circ h\_1$ and $f=g\_2\circ h\_2$ are called equivalent if there is
a polynomial $L$ of degree $1$ such that $g\_... | 9 | https://mathoverflow.net/users/25510 | 189154 | 93,595 |
https://mathoverflow.net/questions/186303 | 3 | Please allow me to resort once again to the expertise of the MathOverflow community :
During research I encoutered the following infinite series :
$$\sum\_{n=-\infty}^{+\infty} \frac{u^{2n}}{1+\rho^{2n+1}},$$
where $0<\rho<1$,$-1<u<1$ and $\rho<|u|$.
Is this series related to some well-known special functions, f... | https://mathoverflow.net/users/1162 | Is this infinite series related to some well-known special functions? | This is a special case of Ramanujan's ${}\_1\psi\_1$ summation, which sums the series
$$\sum\_{-\infty}^\infty\frac{(a;q)\_n}{(b;q)\_n}\,x^n,$$
where $(a;q)\_n=(1-a)(1-aq)\dotsm (1-a q^{n-1})$. If we let $b=aq$, the series simplifies to
$$\sum\_{n=-\infty}^\infty\frac{1-a}{1-aq^n}\,x^n,$$
which is equivalent to yours. ... | 5 | https://mathoverflow.net/users/10846 | 189175 | 93,602 |
https://mathoverflow.net/questions/188639 | 3 | Let $P\colon E \to F$ be an elliptic pseudodifferential operator over $M$. Assuming that $P$ defines a finitely summable Fredholm module, we may apply the Chern-Connes character to it to get a cyclic cocycle and this may then be mapped into the homology of $M$, i.e., we have $ch\_\ast(P) \in H\_\ast(M)$.
On the other... | https://mathoverflow.net/users/13356 | Local index formula for >ungraded< elliptic operators | Meanwhile I found the answer to my question in the literature, concretely in:
Baum, Douglas, *K-Homology and Index Theory*, Proceedings of Symposia in Pure Mathematics, vol. 38 (1982), Part 1.
If $P\colon E \to E$ is elliptic and self-adjoint, then its symbol $\sigma(P) \colon \pi^\ast E \to \pi^\ast E$ is a self-a... | 0 | https://mathoverflow.net/users/13356 | 189176 | 93,603 |
https://mathoverflow.net/questions/189180 | 3 | Let $(X,\tau)$ be a topological space. We say that $A\subseteq X$ is a
1. *topological* retract if there is a continuous map $r:X\to A$ onto a subspace $A \subseteq X$ such that for all $a\in A$ we have $r(a) = a$;
2. *categorical* retract if there are continouus maps $r: X\to A$ and $f: A\to X$ such that $r \circ f ... | https://mathoverflow.net/users/8628 | Topological retraction vs categorical retraction | The two notions agree.
Clearly, if we have a topological retraction $r : X \to A$ we may take $f : A \to X$ to be the subspace inclusion.
Conversely, given a categorical retraction $r : X \to A$, $f : A \to X$ we have the corresponding topological retraction $r' : X \to A'$ where $A' = \mathsf{im}(f) \subseteq X$ ... | 16 | https://mathoverflow.net/users/1176 | 189184 | 93,606 |
https://mathoverflow.net/questions/189170 | 0 | I am interested in an article in this book: Calculation of the Amount of Information about a Random Function Contained in Another Such Function.
I tried my University library, google scholar, AMS website. But still couldn't find this book. The ISBN is 1399884948. Is there a place I can get old journal papers?
| https://mathoverflow.net/users/62729 | How to find American Mathematical Society Translations: Series 2 Volume 12 published in 1959 | here is a link to the scanned article at Google Books:
<http://tinyurl.com/pywo7dw>
| 1 | https://mathoverflow.net/users/11260 | 189185 | 93,607 |
https://mathoverflow.net/questions/189164 | 4 | Let $\mathcal L$ be a differential operator with constant coefficients and $\mathcal{L} f=0$ for some $f\in C^{\infty}(\mathbb{R}^n).$
Under what conditions on $\mathcal {L}$ the function $f$ extends to an entire function in $\mathbb{C}^n$?
| https://mathoverflow.net/users/48438 | Extension of solutions of PDEs with constant coefficients | If that property is satisfied, then "hypoelliptic analyticity" holds, which means that $\mathcal L f$ analytic implies $f$ analytic. For constant coefficient operators that property is equivalent to ellipticity, a sharp contrast with the $C^\infty$ case, where hypoellipticity was characterized by L. Hörmander and holds... | 5 | https://mathoverflow.net/users/21907 | 189187 | 93,608 |
https://mathoverflow.net/questions/189189 | 2 | Let $(p\_{n})\_{n∈ℕ}$ be the sequence of consecutive primes. In P. Ribenboim, *The New Book of Prime Number Records*, Springer-Verlag, 1995, page 185, the author says:
>
> A new conjecture by F. Firoozbakht, dating from about 1982, was communicated to me by the author; as far as I know, it remains unpublished. The ... | https://mathoverflow.net/users/25947 | Have there been any new developments in the Firoozbakht conjecture? | The status as of 2010 is summarized in [On a new property of primes that leads to a generalization of Cramer's conjecture](http://arxiv.org/abs/1010.1399) . ([Firoozbakht's](http://en.wikipedia.org/wiki/Firoozbakht%27s_conjecture) conjecture is slightly stronger than [Cramér's,](http://en.wikipedia.org/wiki/Cramer%27s_... | 5 | https://mathoverflow.net/users/11260 | 189194 | 93,611 |
https://mathoverflow.net/questions/189165 | 11 | It is known that locally one can ``code'' any set in the von Neumann universe $V$ by a set of ordinals. But can one do this globally? In other words, is there a first-order definable bijection $P(On) \longrightarrow V$, where $P(On)$ is the class of all subsets of the class $On$ of all ordinals?
Note that $P(On)$ is ... | https://mathoverflow.net/users/17218 | First-order definable bijection between $P(On)$ (or $No$) and $V$? (Is this equivalent to $V = HOD$?) |
>
> I will use $(1)$, $(2)$, and $(3)$ as defined in the EDIT to the question.
>
>
> It is known that $(1)\Rightarrow (2)$ does not reverse over ZF, indeed, by a result due to Solovay (appearing as Theorem 3.3 of my paper below) if ZF is consistent, then there is a model of ZF + $(2)$ + $\lnot$ AC, which shows that... | 11 | https://mathoverflow.net/users/9269 | 189213 | 93,619 |
https://mathoverflow.net/questions/189216 | 6 | Let $F$ be a $p$-adic field. Let $(G\_{F}, W\_{F}, I\_{F})$ denote the (absolute Galois group, Weil group, inertia group) of $F$.
Let $X/F$ be a proper smooth variety. Let $\ell$ be a prime number $\ne p$. The $\ell$-adic cohomology $H\_{\ell}^{i} = H\_{\text{ét}}^{i}(X\_{\bar{F}}, \mathbb{Q}\_{\ell})$ is naturally end... | https://mathoverflow.net/users/21815 | Is the Weil–Deligne representations coming from $\ell$-adic cohomology independent of $\ell$? | In a [recently published paper](http://www.ams.org/mathscinet/search/publdoc.html?pg1=MR&s1=3123639), Rutger Noot has studied the case of abelian varieties. In 2011, he had conducted the [PhD Thesis of Abhijit Laskar](https://hal.archives-ouvertes.fr/tel-00644861/document) where the methods are applied in a motivic set... | 4 | https://mathoverflow.net/users/10696 | 189218 | 93,620 |
https://mathoverflow.net/questions/188906 | 8 | Call a commutative unital ring $R$
* *Hermite* if for all $m, n\in \mathbb{N}$ with $m<n$, and all $f\in R^{m\times n}$ such that transpose($f$) is left invertible (with a matrix with entries from $R$ serving as a left inverse of $f$), there exists a $g\in R^{n\times (n-m)}$ such that $\ker f = g R^{(n-m)\times 1}$. ... | https://mathoverflow.net/users/62590 | Is there a ring which is not Hermite but is coherent? | Let $F$ be a field not of characteristic $2$, and let $R:=F[x,y,u,v,s,t]/(xu+yv=1,xs+yt=0)$. This ring has a reduction system given by $yt\mapsto-xs$ and $xu\mapsto 1-yv$. (Think of this as trying to get rid of $t$'s the most, then $u$'s, and so forth.)
This is a Noetherian ring, hence coherent (if I'm understanding ... | 1 | https://mathoverflow.net/users/3199 | 189225 | 93,623 |
https://mathoverflow.net/questions/188832 | 4 | EDIT: rewritting the question to linear algebra to make it more accessible.
Denote by $\Delta([n])$ the set of all probability distributions over $\{1,2,\ldots,n\}$, that is:
$$\Delta([n])=\{x\in[0,1]^n\mid \sum\_{i=1}^n x\_i=1\}$$
Let $A\in [0,1]^{n\times n}$ be a matrix, and let $x,y,z\in \Delta([n])$.
>
> **... | https://mathoverflow.net/users/47499 | (linear algebra) - Can a symmetric equilibrium achive higher social-welfare than some equilibrium with the same support? | I am the aforementioned economist.
I figured out a counterexample for the non-singular case:
Let
$$
A = \left( \begin{array}{ccc}
0 & 7 & 0 \\
7 & 0 & 0 \\
3 & 3 & 1 \end{array} \right)
$$
$$
z = \left( \begin{array}{ccc}
\frac{1}{3}, & \frac{1}{3}, & \frac{1}{3} \end{array} \right)
$$
$$
x = \left( \begin{arra... | 3 | https://mathoverflow.net/users/62762 | 189229 | 93,625 |
https://mathoverflow.net/questions/189214 | 2 | Let $X$ be a normal scheme ad $D = \sum\_id\_iD\_i\subset X$ be a $\mathbb{Q}$-divisor such thay $K\_X+D$ is $\mathbb{Q}$-Cartier. Let $f:Y\rightarrow X$ be a log resolution of the pair $(X,D)$ and let us write:
$$K\_Y = f^{\*}(K\_X+D)+\sum\_{j}b\_jE\_j-\widetilde{D}$$
where the $E\_j$'s are $f$-exceptional and $\widet... | https://mathoverflow.net/users/nan | Log Canonical pairs | I think at the very least you should require that $d\_i \leq 1$ even if this is not clear from the literature.
If you don't require that $d\_i \leq 1$, then lots of things don't work.
Problems with the given definition
----------------------------------
1. The definition of log canonical you gave is *not* indep... | 2 | https://mathoverflow.net/users/3521 | 189233 | 93,626 |
https://mathoverflow.net/questions/189235 | 0 | Suppose X and Y are two unit length vector fields on a Riemannian manifold which are orthogonal at each point. Is it true that the lie bracket of X, Y belongs to the span of the vector fields at each point in the manifold?
Is there a way to prove or disprove this without going into local coordinate expressions.
Thanks
... | https://mathoverflow.net/users/50438 | Frobenius condition | Pick any manifold of dimension 3 or higher and any two non-vanishing vector fields that do not satisfy the Frobenius condition. Define a Riemannian metric such that the vector fields are orthonormal.
| 5 | https://mathoverflow.net/users/613 | 189238 | 93,627 |
https://mathoverflow.net/questions/189161 | 15 | An colleague recently came to me with a problem concerning the scheduling of tasks in the presence of constraints (of the kind: task $x$ can't begin until task $y$ has been completed). It turned out that the problem was equivalent to that of computing the width (cardinality of maximum antichain) of a poset.
I know t... | https://mathoverflow.net/users/21690 | Fastest algorithm to compute the width of a poset | For the general case, I believe that there is nothing asymptotically faster than the $O(n^{5/2})$ algorithm that you alluded to. However, if for example the posets in question are known to have small width, then you can do better. See "Recognition algorithms for orders of small width and graphs of small Dilworth number... | 9 | https://mathoverflow.net/users/3106 | 189247 | 93,631 |
https://mathoverflow.net/questions/189109 | 3 | I am beginner in the theory of Grothendieck topologies and I have the following question.
Let $X, Y$ be schemes over an algebraically closed field $k$. Denote by $X\_{et}$ and $Y\_{et}$ the Etale sites on $X$ and $Y$, respectively. Suppose that there is a continous morphism from $X\_{et}$ to $Y\_{et}$. Is there any add... | https://mathoverflow.net/users/46578 | Morphism on schemes induced by continuous morphism of sites | There is a full and faithful embedding of the category of schemes into the $2$-category of (*edit: stricly Henselian ringed*) topoi, which sends each scheme $X$ to the topos $Sh\left(X\_{et}\right)$ (in fact, this is fully faithful even on Delign-Mumford stacks, see: <https://eudml.org/doc/90454> Theorem 50, and Defini... | 3 | https://mathoverflow.net/users/4528 | 189258 | 93,633 |
https://mathoverflow.net/questions/189008 | 4 | Consider a commutative noetherian local ring $R$ of dimension $d$ and define
$$c\_R\colon=\min\_{(x\_1,\ldots,x\_d)} \{\mathrm{length}\ R/(x\_1,\ldots,x\_d)R\mid (x\_1,\ldots,x\_d)\ \mathrm{is\ a\ system\ of parameters\ of\ }R\}.$$
1) Does $c\_R$ always give information about the ring $R$? For instance, clearly $c\... | https://mathoverflow.net/users/16046 | Minimal length of quotient by parameter ideals | Here is answer of (1), $c\_R=2$. I assume that $R$ is complete (as $\mathfrak{m}$-adic topology) and the residue field $A/\mathfrak{m}$ is infinite (if necessary).
Let $v(R) = \dim\_{R/\mathfrak{m}} \mathfrak{m}/\mathfrak{m}^2$ be the embedded dimension of $R$. Since $c\_R = 2$ we have $R$ is not a regular local rin... | 1 | https://mathoverflow.net/users/17901 | 189261 | 93,634 |
https://mathoverflow.net/questions/189230 | 3 | Computing $p$-adic orbital integral I come to the following question. My ground field $k$ is the residue field of a non-arch local field, i.e. a finite field. I am happy to put any assumption on $\text{char}(k)$ so that Springer theory behaves just like over $\mathbb{C}$. Let
$$\pi:\widetilde{\mathcal{N}}\rightarrow\ma... | https://mathoverflow.net/users/31327 | counting points on nilpotent Springer fiber | The nilpotent cone is "rationally smooth" i.e. one has $IC(\mathcal{N}) = (\mathbb{Q}\_\ell)\_{\mathcal{N}}[d]$ (where $d$ denotes the dimension of the nilpotent cone). One can find this in "Partial resolutions of nilpotent varieties" by Borho-MacPherson. In particular the trace of Frobenius on any stalk agrees with th... | 4 | https://mathoverflow.net/users/919 | 189264 | 93,636 |
https://mathoverflow.net/questions/188815 | 16 | Let $P:\Gamma(E)\rightarrow\Gamma(F)$ be an elliptic partial differential operator, with index $=0$ and closed image of codimension $=1$, between spaces $\Gamma(E)$ and $\Gamma(F)$ of smooth sections of vector bundles $E\rightarrow M$ and $F\rightarrow M$ on a compact Riemannian manifold $(M,g)$ without boundary.
**Q... | https://mathoverflow.net/users/43445 | Green's operator of elliptic differential operator | If it exists, the inverse of an elliptic operator $P$ is its Green's operator. In general, an inverse does not exist, but a parametrix does. A parametrix is an operator $Q$ such that $PQ-I$ and $QP-I$ are compact operators. (I am assuming that $P$ is an elliptic operator on a closed manifold and acts between sections o... | 10 | https://mathoverflow.net/users/nan | 189273 | 93,639 |
https://mathoverflow.net/questions/189263 | 3 | let alpha be a cardinal.
I want to look at topological spaces with the property that their topology has a basis with cardinality at most alpha. This property of a topological space certainly has a name. Could you tell how it is called?
For the case: alpha=aleph\_0, this is just "second countable", but the property ... | https://mathoverflow.net/users/58628 | Name for this topological property similar to "second countable" | A topological space $X$ is said to be of *weight* $\leq \alpha$ if it has a basis of cardinality $\leq \alpha$. So you're asking about spaces of weight at most $\alpha$.
The weight $w(X)$ of $X$ is the least cardinality of a basis. This is one of the most important cardinal characteristics of a topological space.
Y... | 8 | https://mathoverflow.net/users/62791 | 189280 | 93,644 |
https://mathoverflow.net/questions/189269 | 10 | Let's say we have a locally $\lambda$-presentable category and a pair of $\lambda$-presentable objects $A$ and $B$. Is it true that $A \times B$ is $\lambda$-presentable?
| https://mathoverflow.net/users/62782 | Cartesian product of small objects | In $\mathbf{Grp}$, the finitely presentable objects are precisely the finitely presented groups. Let $F\_2$ be the free group on two elements. Then [$F\_2 \times F\_2$ is finitely generated but not finitely presented](https://math.stackexchange.com/questions/35579/non-finitely-presented-subgroup), so the class of finit... | 10 | https://mathoverflow.net/users/11640 | 189281 | 93,645 |
https://mathoverflow.net/questions/189291 | 1 | Let $(X,\tau)$ be a Hausdorff space such that for every non-empty $A\subseteq X$ there is a continuous map $r:X\to A$ such that $r(a) = a$ for all $a\in A$. Does $\tau$ have to be discrete?
| https://mathoverflow.net/users/8628 | Hausdorff spaces such that every subset is a retract | Since a retract of a Hausdorff space is closed, such a space must be discrete.
| 5 | https://mathoverflow.net/users/3634 | 189292 | 93,647 |
https://mathoverflow.net/questions/188876 | 6 | I'm interested in a question regarding the identification of some duals of quasi-Banach spaces.
However, I'm not familiar with the quasi-Banach literature, so I'm hoping somebody can point me in the right direction regarding a specific question.
First I'll get the definitions out of the way: a quasi-Banach space is a... | https://mathoverflow.net/users/nan | Can one show that the dual of a quasi-Banach space separates points without explicitly identifying the dual? | I realised that this is actually not such a difficult question, so I'll answer it myself!
(My solution is somewhat inspired by Jochen Wengenroth's comment. Thanks Jochen)
From the definition of $|| \cdot ||\_{X\_c}$, we immediately have $||x||\_{X\_c} \leq ||x||\_X$ for all $x \in X$. So if $||\cdot||\_{X\_c}$ is a... | 1 | https://mathoverflow.net/users/nan | 189293 | 93,648 |
https://mathoverflow.net/questions/189101 | 16 | It's well known that Reinhardt cardinals are inconsistent, provided that we have access to axiom of choice, but, as far as I know, we are clueless about this when we don't assume choice. For me, the fact that Reihardt cardinals are inconsistent with choice already makes them, in a way, "wrong", that is, I'd believe per... | https://mathoverflow.net/users/30186 | What are the current views on consistency of Reinhardt cardinals without AC? | The following theorems of Woodin may be related:
**Theorem.** ($ZF$) Assume that $ZFC$ proves the $HOD$ Conjecture. Suppose $\delta$ is an extendible cardinal. Then for all $ \lambda>\delta$ there is no non-trivial
elementary embedding $j : V\_{\lambda+2}\to V\_{\lambda+2}$
Thus (assuming that $ZFC$ proves the $HO... | 15 | https://mathoverflow.net/users/11115 | 189294 | 93,649 |
https://mathoverflow.net/questions/189295 | 4 | Let $X$ be an infinite set and let $(X,\tau)$ be a topological space such that for every non-empty $A\subseteq X$ there is a continuous map $r:X\to A$ such that $r(a) = a$ for all $a\in A$. Does this imply that $\tau$ is either discrete ($\tau = \mathcal{P}(X)$) or indiscrete ($\tau = \{\emptyset, X\}$)?
| https://mathoverflow.net/users/8628 | Infinite topological spaces such that every subset is a retract | Let $X=\mathbb N$ with the Alexandrov topology corresponding to the usual order. Then if $A\subseteq X$ is nonempty,
$$r(x)=\begin{cases}\min(A)&x<A\\
\max(A\cap\{0,\dots,x\})&\text{otherwise}\end{cases}$$
defines a retraction of $X$ to $A$ (continuity follows from the fact that it is nondecreasing).
| 6 | https://mathoverflow.net/users/12705 | 189299 | 93,652 |
https://mathoverflow.net/questions/189312 | 0 | Let $\pi : E\to B$ be a fiber bundle with fiber $F$ over a finite complex $B$ whose structure group is a compact Lie group $G$. How can we determine the principal $G$-bundle associated to $\pi$? For example if we consider the bundle $S({\gamma}\_{1}\times {\gamma}\_{1})\to {BO}\_{1}\times {BO}\_{1}$ where ${\gamma}\_{1... | https://mathoverflow.net/users/62800 | Principal bundle associated to a fiber bundle | The associated principal bundle is the set of pairs $(b,i)$ so that $i$ is an isomorphism of $E\_b$ with $F$ as $G$-space.
| 5 | https://mathoverflow.net/users/13268 | 189316 | 93,658 |
https://mathoverflow.net/questions/189283 | 5 | Let $X\_{t}$ be a real, one-dimensional diffusion process satisfying the stochastic differential equation
\begin{equation}
dX\_{t} = f(X\_{t})dt + dW\_{t},
\end{equation}
where $f \in C\_{b}^{2}(R)$ is a bounded, twice continuously differentiable function with bounded derivatives. Y. Takahashi and S. Watanabe (Springer... | https://mathoverflow.net/users/62787 | Onsager-Machlup function and most probable path of a diffusion process | Certainly, if $\phi\in C\_b$ then you only care about the values of $f$ at in a neighborhood of points in the range of $\phi$. In particular, the behavior of $f$ at infinity is irrelevant.
| 2 | https://mathoverflow.net/users/35520 | 189317 | 93,659 |
https://mathoverflow.net/questions/189290 | 2 | **Question**
Let $X$ be a manifold, and $\mu\_A$, $\mu\_B$ two Riemannian metric on it which agree on an open subset $U\subset X$, i.e. $\mu\_{A\,|U} = \mu\_{B\,|U}$. Let $K\_A(t;z,w)$ resp. $K\_B(t;z,w)$ be the heat kernel associated to the metric $\mu\_A$ resp. $\mu\_B$, is then the formula
$$K\_A(t;z,z) = K\_B(t;z,... | https://mathoverflow.net/users/14549 | Is the on-diagonal heat kernel “local” with respect to the metric? | The heat kernel is a non local object: $K\_A(t,z,z) \neq K\_B(t,z,z)$.
There is a very intuitive probabilistic explanation: $K\_A(t,z,z)dz$ is the probability that a Brownian motion started at $z$ is, at time $t$, around $z$. A brownian path can go outside the domain $U$ and then come back inside the domain. This me... | 7 | https://mathoverflow.net/users/48356 | 189319 | 93,660 |
https://mathoverflow.net/questions/189311 | 4 | This [link](http://math.mit.edu/%7Ebottman/docs/Uhlenbeck_1.pdf) provides a short introduction to the contributions of Uhlenbeck about regular gauge fixing. However, I feel quite puzzled about it and I do not understand the real novelty apported by this work. More concisely, it would be helpful for me to understand
1... | https://mathoverflow.net/users/62367 | Uhlenbeck's theorem novelty | Denote by $A$ the connection and by $F\_A$ its curvature. Then
$$dA=F\_A-A\wedge A. $$
If $A$ is in Coulomb gauge we have an additional equation
$$d^\*A=0. $$
The advantage is that the operator $d\oplus d^\*$ is elliptic and now we have an equation of the form
$$ (d\oplus d^\*)A= \mbox{something}. $$
Ellipt... | 9 | https://mathoverflow.net/users/20302 | 189326 | 93,664 |
https://mathoverflow.net/questions/189323 | 58 | This is a question that occurred to me years ago when I was first learning algebraic topology. I've since learned that it's a somewhat aesthetically displeasing question, but I'm still curious about the answer.
Is it possible for a subset of $\mathbb R^2$ to have a nontrivial singular homology group $H\_2$? What abou... | https://mathoverflow.net/users/3106 | Can a subset of the plane have nontrivial $H_2$ or $\pi_2$? | Apparently the asphericity is due to Zastrow (see [Cannon-Conner-Zastrow](http://www.sciencedirect.com/science/article/pii/S0166864101000050#)).
Also apparently the result that the higher homology groups vanish is [due to Zastrow](https://mat.ug.edu.pl/~zastrow/nonanorn.ps), but his habilitation thesis never seems t... | 42 | https://mathoverflow.net/users/1345 | 189327 | 93,665 |
https://mathoverflow.net/questions/189318 | 18 | I am trying to figure out what the next calculation of the "area" (or "volume" in higher dimensional analogues) using Stokes' theorem really means. Here is my thought process:
**$2$-dimensional case:** given a closed simple piece-wise smooth curve $C$ in $\mathbb{R}^2$ you can find out the area enclosed by $C$ using... | https://mathoverflow.net/users/3461 | Using Stokes' theorem to define "area" enclosed by a curve | Yes, this is a lower bound on the area of a surface bounded by the curve. Parameterize the surface and apply Stokes' theorem. Your squared area is:
$$\sum\_{1 \leq i < j \leq n} \left( \int\_S dx\_i dx\_j \right)^2 = \int\_S \int\_S \sum\_{1\leq i < j \leq n} dx\_i dx\_j dy\_i dy\_j$$
whereas the actual squared are... | 15 | https://mathoverflow.net/users/18060 | 189335 | 93,669 |
https://mathoverflow.net/questions/189302 | 13 | Assume V=L. Is there a semi-proper notion of forcing that is not proper?
Namba forcing isn't semi-proper in L, and Prikry forcing isn't even available there.
| https://mathoverflow.net/users/18128 | Semiproper but not proper | The following is proved as Claim 2.1 in Shelah's paper ``[Forcing axiom failure for any $\lambda>\aleph\_1$](http://link.springer.com/article/10.1007%2Fs00153-003-0208-9)''
>
> **Theorem.** There is a forcing notion of size $2^{\aleph\_2}$ which is not proper but
> is $\{\aleph\_1\}$-semi proper.
>
>
>
In fac... | 7 | https://mathoverflow.net/users/11115 | 189336 | 93,670 |
https://mathoverflow.net/questions/179695 | 7 | Let $\mathfrak g$ be a simple Lie algebra.
Let $\widetilde{L\mathfrak g}$ be the universal central extension of $L\mathfrak g:=\mathfrak g[t,t^{-1}]$. Let $V\_\lambda$ be a positive energy representation of $\widetilde{L\mathfrak g}$ of level $k$ and highest weight $\lambda$.
Then the minimal energy $h\_\lambda$ of $V\... | https://mathoverflow.net/users/5690 | minimal energy of affine Lie algebra reps | This question has been answered by Victor Ostrik in the comments.
| 0 | https://mathoverflow.net/users/5690 | 189339 | 93,672 |
https://mathoverflow.net/questions/188698 | 13 | Let $G$ be a compact Lie group, and let $H$ be a closed subgroup such that $G/H$ is homotopy equivalent to a torus. Is it true that $H$ is normal and $G/H$ is isomorphic to a torus as a Lie group?
One line of attack would be to reduce to the case where $G$ is connected, then to the case where $G$ is the product of a... | https://mathoverflow.net/users/10366 | Homogeneous spaces that are homotopy tori | Reduce to the case when $G$ is connected.
Because $H$ is compact, $\pi\_0(H)$ is finite, so by the homotopy long exact sequence $\pi\_1(G)\to \pi\_1(G/H) = \mathbb Z^n$ maps onto a finite index subgroup, hence $H^1(G/H, \mathbb Z) \to H^1(G, \mathbb Z)$ is surjective. Fix an invariant metric on $G$ and, applying Hodg... | 7 | https://mathoverflow.net/users/18060 | 189340 | 93,673 |
https://mathoverflow.net/questions/189044 | 1 | Let $U \subset \mathbb{R}^2$ be a domain im $\mathbb{R}^3$ with smooth boundary. Let a point move inside $\mathbb{R}^3$ along the smooth curve $x(t)$. We denote by $\mbox{dist}(x(t), \partial U)$ the distance of $x(t)$ to the boundary $\partial U$.
Do you know any partial differential equation which $\mbox{dist}(x(t), ... | https://mathoverflow.net/users/62672 | Point moving inside smooth domain? | Let $P$ be the shortest point projection onto $\partial U$. Then
$$d(t):=dist(x(t),\partial U)=\|P(x(t))-x(t)\|\_2$$
and therefore
$$\dot d(t)=\frac{d}{dt}(\|P(x(t))-x(t)\|^2\_2)^{\frac{1}{2}}=d(t)^{-1}\langle (P-I)x(t),(DP-I)\dot{x}(t)\rangle.$$
As $DP\dot{x}(t) \perp (P-I)x(t)$ we have
$$\dot d(t)=-d(t)^{-1}\langle (... | 0 | https://mathoverflow.net/users/35593 | 189348 | 93,677 |
https://mathoverflow.net/questions/189346 | 2 | Let $S$ be a locally noetherian scheme and let $G$ be a separated and smooth $S$-group algebraic space of finite presentation. Does there exist an open sub-(group algebraic space) $G^0 \subset G$ characterized by the requirement that $G\_s^0$ be the identity component of $G\_s$ for every $s \in S$? (Note that the $S$-f... | https://mathoverflow.net/users/53197 | Relative identity component for group algebraic spaces | There is even a reference for algebraic stacks: see M. Romagny, manuscripta math. 136, 1–32 (2011), especially section 2.2.
| 6 | https://mathoverflow.net/users/7666 | 189351 | 93,678 |
https://mathoverflow.net/questions/189361 | 3 | Let $P,Q$ be partially ordered sets (posets). We consider the set $\text{Hom}(P,Q)$ of order-preserving functions $f:P\to Q$. There is a natural ordering relation on $\text{Hom}(P,Q)$ given by $f\leq g$ if and only if $f(p) \leq\_Q g(p)$ for all $p\in P$.
It is not hard to prove that if $Q$ is a lattice, then so is $... | https://mathoverflow.net/users/nan | When is the homomorphism poset between posets a lattice? | The answer is yes, and it follows from two observations:
**Observation 1**: $Q$ is a retract of $\textrm{Hom}(P,Q)$.
*Proof*: Fix $p^\ast\in P$ and define $r:\text{Hom}(P,Q)\to Q$ by $f\mapsto f(p^\ast)$. It's easy to see that $r$ is order-preserving. Moreover, define $e:Q\to \text{Hom}(P,Q)$ by $q\mapsto c\_q$ whe... | 10 | https://mathoverflow.net/users/8628 | 189363 | 93,685 |
https://mathoverflow.net/questions/189352 | 1 | This question might seem elementary but I cannot answer it.
Let M be an infinite dimensional vector space and $f\_1, f\_2, \cdots , f\_r \in M^\*$ be a set of linear independent vectors with $r \geq 2$. Here $M^\*:=Hom\_K(M, K)$ is the linear dual of $M$.
Do there exist $m \in M$ such that $f\_1(m)\neq 0$ but $f\_i... | https://mathoverflow.net/users/62813 | Infinite dimensional vector spaces | You can find $m$ by Induction on $r$. For $k\in \{2,\dots,r\}$ let $m\_k$ be such that $f\_l(m\_k)=\delta\_{lk}$ for $l\in \{2,\dots,r\}$. By linear independce of $f\_1,f\_2,\dots,f\_r$ there exist $v \in M$ with $f\_1(v)-\sum\_{l=2}^r f\_1(m\_l)f\_l(v)=1$. Define now
$$m:=v-\sum\_{l=2}^r f\_l(v)m\_l.$$
| 3 | https://mathoverflow.net/users/35593 | 189364 | 93,686 |
https://mathoverflow.net/questions/189365 | 2 | Let $P=\{P\_1,P\_2\cdots P\_n\}$ be a set of $n\geq 4$ points in the plane and $P\_iP\_j$ be the line segment connecting $P\_i$ and $P\_j$ that satisfy:
$(1)$Any three points of $P$ are not on a line;
$(2)$In the set $\{P\_1P\_2,P\_2P\_3,\cdots,P\_{n-1}P\_n,P\_nP\_1\}\setminus \{P\_iP\_{i+1}\}$,$P\_iP\_{i+1}$ only... | https://mathoverflow.net/users/58096 | a question about points and line segments in the plane | Yes, the polygon formed by the broken line $P\_1\dots P\_n$ may be triangulated by $n-3$ inner diagonals and in any triangulation there exists a triangle (at fact, at least two triangles for $n\geq 4$) of the form $P\_{i-1}P\_iP\_{i+1}$.
| 3 | https://mathoverflow.net/users/4312 | 189368 | 93,687 |
https://mathoverflow.net/questions/189354 | 2 | When defining tensor products $M\otimes\_R N$ over a commutative ring $R$ one can use a universal property with respect to bilinear maps $M\times N\rightarrow P$, for any $R$-module $P$.
On the other hand, in the general case, for noncommutative rings one has to use balanced maps $M \times N \rightarrow Z$ instead of... | https://mathoverflow.net/users/62813 | Tensor product-definition-balanced versus bilinear maps | You seem to be asking whether any **group** homomorphism $M\otimes\_RN\to P$ is $R$-linear. There is no reason why this should be true. As a huge overkill, consider $\Bbb R\otimes\_{\Bbb R}\Bbb R=\Bbb R\to\Bbb R$: there are uncountably many group homomorphisms (even $\Bbb Q$-linear maps) that are not $\Bbb R$-linear.
... | 2 | https://mathoverflow.net/users/44953 | 189369 | 93,688 |
https://mathoverflow.net/questions/189301 | 7 | I am interested in the relation between Bousfield localizations of model categories and localizations of $(\infty,1)$-categories.
According to Hirschhorn's book we can form the left Bousfield localization of a left proper cellular model category along any set of maps. According to Lurie's book we can form the (left) ... | https://mathoverflow.net/users/56531 | Localizations of model categories and $\infty$-categories | There is also an existence theorem for right Bousfield localizations of presentable $\infty$-categories.
In fact, it follows from the existence theorem for left Bousfield localizations.
Let $K$ be a collection of objects in $C$ and let $D\subset C$ be the subcategory of $K$-colocal objects.
Then $D$ is manifestly clo... | 6 | https://mathoverflow.net/users/20233 | 189378 | 93,692 |
https://mathoverflow.net/questions/189286 | 4 | Let G be a (not necessarily linear) algebraic group and H a closed algebraic subgroup. Does the, say categorical, quotient G/H exist?
If yes, where do I find a proof?
If no, do you have a counterexample?
Remark: I am aware of the following MO discussion, but it does not answer my question: [Quotient of an algeb... | https://mathoverflow.net/users/62434 | Quotient of a (non-linear) algebraic group by a closed subgroup | See SGA3, Exp. VI$\_{\rm{A}}$, Theorem 3.2(iv): for any group scheme $G$ locally of finite type over a field $k$ and any closed subgroup scheme $H$ there exists a locally finite type $k$-scheme $G/H$ equipped with a faithfully flat $k$-morphism $G\rightarrow G/H$ invariant under the right $H$-action such that $G×H→G\ti... | 6 | https://mathoverflow.net/users/61939 | 189383 | 93,694 |
https://mathoverflow.net/questions/189382 | 2 | I would like to find ALL eigenfunctions to the operator, for $f$ a real function on R+\*:
$f \rightarrow \sum\_{1}^{\infty} f(nx)$
So to find $f$ such that: $\sum\_{1}^{\infty} f(nx) = \lambda f(x)$
It is obvious that $f(x)=x^{a}$ is a solution but are they other?
May be you can advise me a reference on this su... | https://mathoverflow.net/users/38290 | Eigenfunctions of an infinite summation operator | You do not state any conditions on the functions on which your operator is defined (it involves an infinite series which must converge). So my answer will also be purely formal.
Setting $g(t)=f(e^t)$, we obtain the equation
$$\sum\_n a\_n g(t+c\_n)=\lambda g(t),$$
where $c\_n=\log n$. This is a linear equation and su... | 2 | https://mathoverflow.net/users/25510 | 189385 | 93,695 |
https://mathoverflow.net/questions/189374 | 4 | What are the orders of maximal abelian subgroups of the simple groups $F\_4(q)$ and $C\_4(q)$, where $F\_4(q)$ is an exceptional group and $C\_4(q)$ is a symplectic group?
| https://mathoverflow.net/users/30252 | orders of maximal abelian subgroups | For a finite group $G$, denote the maximum of the orders of its Abelian subgroups by $a(G)$. Then we have, for $G=F\_4(q)$ and $q$ even,
$$
q^{11}\le a(G)\le q^{17},
$$
and for $G=F\_4(q)$, $q$ odd,
$$
q^{9}\le a(G)\le q^{14}.
$$
For $G=C\_n(q)$ we have, for $n\neq 2$,
$$
a(G)=q^{\frac{n(n+1)}{2}}.
$$
For a reference,... | 8 | https://mathoverflow.net/users/32332 | 189386 | 93,696 |
https://mathoverflow.net/questions/189379 | 8 | I need a global choice function defined by a formula in (a fragment of) ZF. There is no harm in assuming V=L for my purposes. But I wonder if there are any familiar alternative ways to get this?
The comments make see I also want a weakened part of GCH. Namely the power set of $\aleph\_n,\ n\in\mathbb{N}$ should be $... | https://mathoverflow.net/users/38783 | Are there known ways to posit definable global choice in ZF without positing V=L? |
>
> Two comments/answers:
>
>
>
**(1)** By an old theorem of Roguski, for any $\Sigma\_2^{\text{ZFC}}$ sentence $\phi$, the theories $\text{ZFC} + \phi$ and $\text{ZFC + V=HOD} + \phi$ are equiconsistent.
Roguski's result appears in his paper *Extensions of models for ZFC to models for ZF+V=HOD with applicatio... | 10 | https://mathoverflow.net/users/9269 | 189393 | 93,700 |
https://mathoverflow.net/questions/189241 | 0 | I encountered the following ode in the attempt to solve the cauchy problem of Liouville equation. I have tried for a long time to give it a solution but failed.
$(K e^f h + f'h'-2h'')^2=g^2((h')^2-\frac{K}{2}h^2 e^f)$
where $K$ is a given constant, $f,g$ are known smooth functions. The problem is to solve $h$ from th... | https://mathoverflow.net/users/62766 | Solution of a second order nonlinear ode | Solving the linear first-order
$$
h'=h K^{1/2}e^{f/2}/\sqrt 2
\tag{$\ast$}$$
makes the rhs to vanish. We get then
\begin{align}
2h''-f'h'-Ke^f h&=h' K^{1/2}e^{f/2}\sqrt 2+h K^{1/2}e^{f/2}f'(\sqrt 2)^{-1}-f'h K^{1/2}e^{f/2}(\sqrt 2)^{-1}-Ke^f h
\\
&=
h' K^{1/2}e^{f/2}\sqrt 2-Ke^f h=h K^{1/2}e^{f/2}(\sqrt 2)^{-1} K^{1/2}... | 1 | https://mathoverflow.net/users/21907 | 189395 | 93,701 |
https://mathoverflow.net/questions/189388 | 0 | I feel very uncomfortable with some aspects of the theory of valuations, places, and valuation rings. Here is one of my problems : Assume that L/K is a finite Galois extension of fields, and that F is a place from K to its residual field k, whose associated valuation ring is discrete.
F extends to a place F' of L, from... | https://mathoverflow.net/users/62826 | Valuations and places - decomposition and inertia group | I think your understanding is good, maybe the terminology needs a little clarification:
|k':k| = |k'sep:k||k'pi:k|
i.e. f = fsepfpi
|D/J| = fsep
|L:K| = efg
example: K=F(T)((X)), L=F(T1/p)((X)), valuation=X-adic, k=F(T), k'=F(T1/p), e=1, f=p, fsep=1, g=1
| 1 | https://mathoverflow.net/users/59248 | 189397 | 93,702 |
https://mathoverflow.net/questions/189143 | 2 | Is it easy to write down the large deviations rate for the maximizer of a random walk with negative drift?
Let $X\_i$ be the (iid, mean $-\mu$, variance $\sigma$, arbitrarily nice tails) jumps of a random walk $S\_i$. I am interested in the location of the maximum of $S\_i$, i.e. in $\arg\max\_k S\_k$. (If $X\_i$ are... | https://mathoverflow.net/users/17883 | Large deviations for maximizer of random walk with drift | The upper bound gives the correct large deviations; that is, the answer is
$$n^{-1}\log P(\mbox{ arg max}\_k S\_k>n)\to -I(0)\,,$$
where $I(0)=-\inf\_\lambda \log E(e^{\lambda X\_1})$.
The upper bound appears in the OP. For the lower bound, just consider the event
$$ {\cal A}\_n:=\{S\_n\geq \epsilon n \}$$
and note ... | 3 | https://mathoverflow.net/users/35520 | 189417 | 93,708 |
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