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https://mathoverflow.net/questions/260835 | 0 | Let $R$ be a right Noetherian ring and $S=\{f\in R[x]\;|\;f\text{ monic}\}$. It is a result of Stafford that $S$ is a right denominator set in $R[x]$, so in particular we can localize $R[x]$ at any $f\in S$. My question is this, can we localize $R[x,x^{-1}]$ at any $f\in S$? I know,
$R[x,x^{-1}]\_f\cong R[x]\_{xf}\co... | https://mathoverflow.net/users/102861 | Right localization of $R[x,x^{-1}]$ at monic $f\in R[x]$ | I may be wrong here but does this not follow from the fact $S=\{f^n\:|\:n\geq 0\}$ is right Ore? Consider $f^{m}, h$ for some $m\geq 0$ and $h\in R[x,\,x^{-1}]$. Let $y=x^{-1}$ and note that if $h\in R[x]$ then there is nothing to prove. For simplicity, suppose $h\in R[y]$ and has degree $k$, say. Then choose $h^\prime... | 0 | https://mathoverflow.net/users/73736 | 260837 | 117,565 |
https://mathoverflow.net/questions/260821 | 8 | Let $P(z)$ be a non-null complex polynomial in $n$ variables $z=(z\_1,\dots,z\_n)$:
\begin{equation}
P(z)=\sum\_{|\alpha| \leq N} c\_{\alpha} z^{\alpha},
\end{equation}
where as usual for every $\alpha=(\alpha\_1,\dots,\alpha\_n) \in \mathbb{N}^{n}$ we set $|\alpha|=\alpha\_1+\dots+\alpha\_n$, and $z^{\alpha}=z\_1^{\al... | https://mathoverflow.net/users/99197 | Division of Distributions by Polynomials |
>
> Hörmander's result shows that ${\mathcal M}\_{\mathcal S}$ is an isomorphic copy of $\mathcal S$. Especially, this means that any linear-topological operation one performs on the side of $\mathcal S$ has its analogue on the side of ${\mathcal M}\_{\mathcal S}$. In particular, both are Fréchet spaces. Now, ${\math... | 3 | https://mathoverflow.net/users/69194 | 260840 | 117,567 |
https://mathoverflow.net/questions/260800 | 8 | We know that a supermanifold $M$ is a locally ringed space $(M,O\_M)$ which is locally isomorphic to $(U,C^\infty(U) \otimes \wedge W^\ast)$, where $U$ is an open subset of $\mathbb{R}^n$, $W$ is a finite dimensional real vector space and the above isomorphism defined in the category of $\mathbb{Z}\_2$ graded algebra i... | https://mathoverflow.net/users/104177 | What is the geometric significance of the definition of supermanifold? | The Wikipedia page on supermanifolds does a decent job of presenting different definitions and their relations. See in particular the statement of [Batchelor's theorem](https://en.wikipedia.org/wiki/Supermanifold#Batchelor.27s_theorem).
Forgetting about the "super" part, already an ordinary manifold can be described ... | 3 | https://mathoverflow.net/users/2622 | 260844 | 117,569 |
https://mathoverflow.net/questions/260853 | 17 | Andre Weil noticed as a student in 1925 that the polynomial ring $\mathbb{Z}[x]$ comes close to being a PID, and he describes this as `` the embryo of my future thesis.''
He observed that, given $f(x),g(x)\in\mathbb{Z}[x]$, the Euclidean algorithm computes a sequence of polynomials where each is a linear combination... | https://mathoverflow.net/users/38783 | Is there a name for this property Weil saw for integer polynomials? | I do not know of a name. But Weil's observation follows from properties of the resultant, and generalizes to other rings $A[x]$ ($A$ a GCD domain).
The resultant $R(f,g)$ of two polynomials is defined either as a certain determinant or as a certain product over pairs of roots of $f,g$, see [this Wikipedia page](https... | 18 | https://mathoverflow.net/users/31469 | 260858 | 117,575 |
https://mathoverflow.net/questions/260849 | 1 | Is there any kind of generalization of Vélu formulae for Jacobians?
The question technically is:
Given a hyperelliptic curve $H/\overline{\mathbb{F}}\_q$ and its jacobian $J\_H$ of dimension $2$, if $D\in J\_H$:
Is there a way to construct an isogeny $\psi:J\to{J\_D}$ with kernel $nD$ for all $n\in \mathbb{Z}$ an... | https://mathoverflow.net/users/91023 | Isogeny from kernel in higher dimensional abelian varieties | This question seems a bit confused.
If $D$ is an arbitrary point in the Jacobian then one cannot construct an isogeny with kernel the subgroup generated by $D$ -- as this subgroup is typically infinite, and the kernel of an isogeny is finite.
If however $D$ is assumed torsion, then one can certainly construct an is... | 2 | https://mathoverflow.net/users/1384 | 260861 | 117,577 |
https://mathoverflow.net/questions/260871 | 3 | I was revising some old postgraduate notes of mine in homological algebra (written during a postgrad course on the topic, I had taken more than ten ;) years ago) and I came accross the following problem: Weibel's book "*An introduction to homological algebra*" (which had been among my textbooks by that time), states th... | https://mathoverflow.net/users/85967 | On an exercise from Weibel's book on homological algebra | You are correct, the converse also holds. This is a mistake in Weibel, see [this answer](https://math.stackexchange.com/a/1405990/99887 "this answer").
| 10 | https://mathoverflow.net/users/3075 | 260887 | 117,586 |
https://mathoverflow.net/questions/260886 | 7 | I've encountered a curious identity as a codicil in some work. Is there a proof or reference?
$$\sum\_{k=-n}^n\frac{2k+1}{n+k+1}\binom{2n}{n-k}\frac{x^k}{1+x^{2k+1}}=\frac{x^n}{1+x^{2n+1}}.$$
| https://mathoverflow.net/users/66131 | A special binomial identity in need of a proof | The $k=j-1$ and $k=-j$ terms cancel, so all that's left is the $k=n$ term.
| 23 | https://mathoverflow.net/users/10744 | 260893 | 117,587 |
https://mathoverflow.net/questions/260816 | 8 | Let $M$ be a von Neumann algebra with predual $M\_\*$, and let $T\colon M\to M$ be a bounded, linear map. Let us say that $T$ is (sequentially) weak\*-norm continuous if for every net (sequence) $(a\_j)\_j$ in $M$ that weak\* converges to $a$ in $M$ we have $\lim\_j \| T(a\_j) - T(a) \| = 0$.
In particular, given $h\... | https://mathoverflow.net/users/24916 | Weak*-norm continuous operators on von Neumann algebras | Since taking the adjoint is continuous in both topologies, the set of $h$ such that the multiplication map has the said continuity property is $\ast$-closed, so we may assume that $h$ is self-adjoint -- let's take a look at its spectral decomposition. Suppose that the spectrum has a continous bit, say that $b\neq 0$ is... | 5 | https://mathoverflow.net/users/24953 | 260902 | 117,590 |
https://mathoverflow.net/questions/260768 | 3 | Given a "hybrid" Black-Scholes Hull White (BSHW) model. That is, the stock price is modelled by a Black Scholes SDE:
\begin{equation} dS(t) = \mu(t)S(t)dt + \sigma\_{S}(t)S(t)dW^{\mathbb{P}}\_{S}(t)
\end{equation}
and short rate evolution is given by a Hull-White model
\begin{equation} dr(t) = (\theta(t) - ar(t))dt +... | https://mathoverflow.net/users/104165 | Is the "hybrid" Black-Scholes Hull-White model arbitrage free? | The discounted stock price satisfies
$$
dX(t) = \big(\mu(t) - r(t)\big)X(t) dt + \sigma\_S(t) X(t) dW\_S^{\mathbb P}(t).
$$
The Girsanov density for $X$ is
$$
Z(T) = \exp\left\{\int\_0^T \nu(t)dW^{\mathbb P}\_S(t) - \frac12\int\_0^T \nu(t)^2 dt \right\}
$$
with $\nu(t) = (r(t)-\mu(t))/\sigma\_S(t)$. The problem, as u... | 2 | https://mathoverflow.net/users/8146 | 260904 | 117,591 |
https://mathoverflow.net/questions/260857 | 2 | Is it true that that ${{n}\choose{k}} p^k (1-p)^{n-k}$ is dominated by $\frac{1}{n}$, at least for $k$ sufficiently big?
EDIT: I saw that the question was absolutely not stated as I intended. The real question is: given a real number $p \in (0, 1)$, assuming $k$ fixed big enough, is there some constant $C > 0$ indepe... | https://mathoverflow.net/users/103674 | Asymptotic behaviour of binomial term | Robert's answer is correct, if you want an asymptotic answer. However, for the question as stated in the edit (that is for all $n\ge k$...) the answer is no.
Take for example the case of $p=\frac12$. For any $k$, if you take $n=2k$ then ${n \choose k}p^k(1-p)^{n-k}=\frac{(2k)!}{k!k!2^{2k}}$ which is roughly $\frac{1}... | 0 | https://mathoverflow.net/users/1061 | 260906 | 117,593 |
https://mathoverflow.net/questions/260885 | 10 | I am reading the paper *Frobenius splitting of Hilbert schemes
of points on surfaces* by Kumar and Thomsen. At the end of Lemma 11, they seem to imply that the dualizing sheaf on a Cohen-Macaulay scheme is reflexive. I am not familiar with reflexive sheaves, so I would like to ask whether this statement is true or not.... | https://mathoverflow.net/users/104222 | Is the dualizing sheaf on a Cohen-Macaulay scheme reflexive? | The answer is *yes*. Let me briefly explain why.
There are several ways to define a canonical sheaf on a projective variety $X$. One of them, that works for any *normal* $X$ is the following. Let $U$ be the smooth locus of $X$ and $i \colon U \hookrightarrow X$ be the natural inclusion. Then define $$\omega\_X:=i\_\*... | 9 | https://mathoverflow.net/users/7460 | 260907 | 117,594 |
https://mathoverflow.net/questions/260914 | 2 | I have seen stronger propositions that imply this, but both their statement and their proofs require more advanced tools than I'd like to use in my text, which is aimed at a general scientific audience.
I want to prove: *With $f\colon \mathbb R^k \to \mathbb R$ smooth (infinitely differentiable) and $x$ a non-critica... | https://mathoverflow.net/users/8948 | The level sets of a differentiable function is a manifold | Proof of the implicit function theorem in several variables calculus requires the contraction mapping theorem, so is probably not suitable for your audience.
You need to use an iterative method and take a limit. You can look for a complete proof in Spivak, Calculus on Manifolds.
If you just replace one of the coordin... | 3 | https://mathoverflow.net/users/13268 | 260915 | 117,597 |
https://mathoverflow.net/questions/260916 | 2 | I have encountered the following equality
$\arg\max\_{\text{tr}\left(\boldsymbol{S}^{H}\boldsymbol{S}\right)=1}\text{tr}\left(\boldsymbol{S}^{H}\boldsymbol{A}\boldsymbol{S}\left(\boldsymbol{S}^{H}\boldsymbol{B}\boldsymbol{S}\right)^{-1}\right)=\arg\max\_{\text{tr}\left(\boldsymbol{S}^{H}\boldsymbol{S}\right)=1}\frac{... | https://mathoverflow.net/users/103133 | equality between the ratio trace and the determinant ratio | The equality of the [arg max](https://en.wikipedia.org/wiki/Arg_max) of the trace of the ratio and the ratio of determinants follows from the fact that each of these two maximisation problems has the same solution, given by the matrix $S$ composed of column vectors $s\_k$ that solve the generalised eigenvalue problem $... | 2 | https://mathoverflow.net/users/11260 | 260918 | 117,599 |
https://mathoverflow.net/questions/260671 | 18 | Let $X$ and $Y$ be two nonsingular projective varieties defined over the complex numbers.
>
> If $X$ and $Y$ are $\mathbb{A}^1$-homotopy equivalent, then does $X$ satisfying the Hodge conjecture imply that $Y$ satisfies the Hodge conjecture?
>
>
>
| https://mathoverflow.net/users/nan | Is the Hodge Conjecture an $\mathbb{A}^1$-homotopy invariant? | Isn't this very easy? If the varieties are $\mathbb{A}^1$-homotopy equivalent, then their Voevodsky motives are isomorphic also (since there is a connecting functor making the obvious diagram commutative). So it remains to note that all the ingredients of your question are "motivic". For this purpose one may recall tha... | 15 | https://mathoverflow.net/users/2191 | 260920 | 117,600 |
https://mathoverflow.net/questions/260749 | 1 | Is there a similar theorem to max flow min cut for finding max flow in a digraph where edges are colored red and blue, and the flow is required to alternate between red and blue edges? I may also allow the assumption that the source only has red out-edges and the sink only has blue in-edges.
| https://mathoverflow.net/users/10304 | Max flow min cut for alternating flow | I realised that you can apply max flow min cut to the graph where you have two copies of each vertex from the original graph: the first one has only red out-edges and blue in-edges, while the second one has blue out-edges and red in-edges.
| 1 | https://mathoverflow.net/users/10304 | 260923 | 117,601 |
https://mathoverflow.net/questions/260559 | 5 | Let $X$ be a variety (or more generally a quasi-compact, separated scheme) and $D(X)$ be the derived category of complexes of $\mathcal{O}\_X$-modules with quasi-coherent cohomologies. Let $\mathcal{E}$ be a compact generator of $D(X)$. It is well-known that $\mathcal{E}$ is a perfect complex ([Generators and represent... | https://mathoverflow.net/users/24965 | Is the dual of a compact generator also a compact generator of the derived category of a variety? | Let me try to sketch an argument (though I am not quite sure in details).
A theorem of Neeman implies that a compact object $M$ in a compactly generated triangulated category $T$ is a generator if and only if the smallest full dense subcategegory $T$ containing $M$ equals with the subcategory of compact objects of $T... | 2 | https://mathoverflow.net/users/2191 | 260925 | 117,602 |
https://mathoverflow.net/questions/260822 | 1 | Let $\Omega$ be a non-empty open subset of $\mathbb{R}^n$ and $\mathcal{D}(\Omega)$ the usual space of test functions of distribution theory, with the usual topology $\tau$ of the inductive limit of the Fréchet spaces $\mathcal{D}\_K$, whose topology I denote with $\tau\_K$.
Consider a linear mapping $\Lambda$ from a... | https://mathoverflow.net/users/99197 | Continuous Linear Mappings on Subspaces of $\mathcal{D}(\Omega)$ | Life would be very easy if (locally convex) inductive limit would "commute" with subspaces! For a concrete example, let $P:\mathscr D'(\Omega) \to \mathscr D'(\Omega)$ be a (partial differential) operator which is not surjective but whose restriction to the smooth functions $P:\mathscr E(\Omega) \to \mathscr E(\Omega)$... | 4 | https://mathoverflow.net/users/21051 | 260929 | 117,603 |
https://mathoverflow.net/questions/257397 | 8 | Let $f$ and $g$ be analytic functions in the unit disk $D$, continuous in the closed disk and locally univalent, $f'(z)\neq 0,\; g'(z)\neq 0,\; z\in D$.
Assume that each has only finitely many singularities on $\partial D$, so that
the images of $\partial D$ are piecewise analytic.
Suppose that there exists a homeomo... | https://mathoverflow.net/users/25510 | Uniqueness theorem for conformal mapping | This is a form of the following topological question: Does the boundary value uniquely determine an immersion of the closed disk into the plane, up to precomposition with a diffeomorphism of the disk. The answer is negative
and is due to John Milnor, see Poenaru's paper "Extension des immersions en codimension 1." Sémi... | 8 | https://mathoverflow.net/users/21684 | 260936 | 117,607 |
https://mathoverflow.net/questions/260662 | 4 | On most partially ordered sets, the order-convergence topology (defined below) is often highly disconnected, often even discrete or [extremally disconnected].[1](https://en.wikipedia.org/wiki/Extremally_disconnected_space)
However, the order-convergence topology is connected for $\mathbb{R}$ and coincides with the E... | https://mathoverflow.net/users/8628 | $\mathbb{R}$ and the order-convergence topology | I will show below that for any linearly ordered set $(L, \leq)$, the order-convergence topology coincides with the usual order topology. Thus the answer to the OP´s question is no, since there are linear continua (e.g. a nowhere-separable Aronszajn continuum) that do not contain copies of $\mathbb{R}$.
Fix $a \in L$.... | 4 | https://mathoverflow.net/users/17836 | 260938 | 117,608 |
https://mathoverflow.net/questions/260931 | 2 | Let $G=(V,E)$ be any simple, undirected graph. A *dominating set* is a set $D\subseteq V$ such that for all $v\in V\setminus D$ there is $d\in D$ such that $\{v,d\}\in E$.
Is there an infinite graph $G=(V,E)$ such that there is a dominating subset $D\subseteq V$ such that for any dominating subset $D\_1\subseteq D$ t... | https://mathoverflow.net/users/8628 | Minimal dominating subsets in infinite graphs | Let $V(G)$ be the set of non-empty subsets of $\mathbb N$ and join two sets by an edge whenever they intersect. Let $D$ be the set of initial segments of $\mathbb N$. Then subsets of $D$ are dominating if and only if they are infinite.
| 4 | https://mathoverflow.net/users/25485 | 260943 | 117,610 |
https://mathoverflow.net/questions/260939 | 6 | Consider an arbitrary linear program:
$$\max \vec c \cdot \vec x$$
subject to:
$$\textbf{A}\cdot \vec x = 0, \quad \vec a \le \vec x \le \vec b$$
Assume that this program is feasible and bounded. Now suppose I perturb the bounds by small amounts:
$$\vec a' = \vec a + \delta \vec p, \quad \vec b' = \vec b + \d... | https://mathoverflow.net/users/16615 | Linear programming is continuous | The answer is no. Consider the probem
$$x+y\to\max,$$
$$x\geq 0,\; y\geq 0,$$
$$x+y\leq 1.$$
It has infinitely many solutions. One of them is $(0,1)$.
Now change the last inequality to
$$x+(1+\epsilon)y=1.$$
The new problem has a unique solution $(1,0)$ which is not close
to the solution of the first one.
One can al... | 7 | https://mathoverflow.net/users/25510 | 260947 | 117,611 |
https://mathoverflow.net/questions/260919 | 6 | In the book *Twelve sporadic groups*, Griess states
>
> If $A$ is an abelian group, $G$ acts on $A$, $z\in Z(G)$ satisfies $z-1\in$ Aut$(G)$, then $H^n(G,A)=0$ for $n\geq 0$. This is an observation of Jack McLaughlin.
>
>
>
My first simple question is *what do we mean here by* $z-1\in$ Aut$(G)$?
Second ques... | https://mathoverflow.net/users/6761 | Understanding (statement of) a theorem of Jack McLaughlin | The statement is not identical, but it is the "center kills" argument, as in 149 page 42 of [these lecture notes](http://courses.washington.edu/mnotes/groupcoho/notes.pdf).
For $g \in Z(G)$, the map $m \mapsto gm$ is a ${\mathbb Z}G$-automorphism of $A$. The map it induces on the cohomology groups $H^n(G,A)$ is the ... | 7 | https://mathoverflow.net/users/35840 | 260954 | 117,616 |
https://mathoverflow.net/questions/260959 | 1 | I saw a proof that $|p\_n - li^{-1}(n)| \leq n e^{-c \sqrt{\ln(n)}} $,
without saying anything about $c$ !
My questions is, what the explicit value of $c$ ??
It just says for some number $c$ without giving any information about $c$ !
| https://mathoverflow.net/users/95470 | An explicit value for a bound proof | An explicit version of the Prime Number Theorem required here is Theorem 1.12 on Page 42 of [Dusart's thesis](http://www.unilim.fr/laco/theses/1998/T1998_01.pdf). In particular, it follows from this theorem that
$$ |\pi(x)-\mathrm{Li}(x)|<x e^{-0.32\sqrt{\ln x}},\qquad x\geq 59. $$
Using Kadiri's explicit zero-free reg... | 18 | https://mathoverflow.net/users/11919 | 260970 | 117,621 |
https://mathoverflow.net/questions/260878 | 10 | Suppose $X \to Y$ is a finite-sheeted covering of CW-complexes. Moreover, assume that the total space is homotopy equivalent to a (closed, connected, smooth) manifold $M$. I am interested in conditions to impose on the map $X \to Y$ so that we can find a manifold $N \simeq Y$ and a covering $M \to N$ making the diagram... | https://mathoverflow.net/users/14233 | Spaces that are finitely covered by manifolds | Conditions where $N$ exists are for example when $M$ is a non-positively curved locally symmetric manifold, by a theorem of Mostow (for hyperbolic manifolds) and Margulis in general. Mostow's theorem is generalized to manifolds with hyperbolic fundamental group whose [Gromov boundary is a sphere of dimension at least 5... | 6 | https://mathoverflow.net/users/1345 | 260973 | 117,623 |
https://mathoverflow.net/questions/260972 | 4 | As stated in [this](https://mathoverflow.net/questions/89444/do-disjoint-unions-and-fiber-products-commute) question, coproducts commute with pullbacks in the category of sets.
Let $Grpd\_{\infty}$ denote the $\infty$-category of $\infty$-groupoids. Do coproducts commute with pullbacks in $Grpd\_{\infty}$?
| https://mathoverflow.net/users/102957 | Do coproducts of infinity-groupoids commute with pullbacks? | Yes, they do, if you mean what is meant in the question you linked, but it is misleading to describe this property as "coproducts commute with pullbacks", say instead "pullback along a fixed morphism preserves coproducts", or "pullback *functors* preserve coproducts", or with more jargon, "coproducts are universal".
... | 4 | https://mathoverflow.net/users/644 | 260978 | 117,624 |
https://mathoverflow.net/questions/260769 | 4 | This question is inspired by [the MO query here](https://mathoverflow.net/questions/260169/lower-bound-for-frac-sum-i-j-minf-i-f-j2-g-i-g-j2-sum-i-j-max), although it has no direct implications.
Define the family of polynomial functions
$$f\_n(x)=n^2x^{n-1}-\frac{d}{dx}\left(\frac{x^n-1}{x-1}\right),$$
and the associ... | https://mathoverflow.net/users/66131 | what is this sum of squares of algebraic functions? | A large part of the result doesn't really depend on $f\_n$. For an arbitrary sequence $(a\_n)$, define two sequences by $G\_0=0$, $H\_0=0$, and for $n\ge1$,
$$G\_n=\frac{a\_n}{n}+\sum\_{j=1}^n{\frac{a\_j}{j(j+1)}}\quad\text{and}\quad H\_n=\sum\_{k=0}^n{(G\_n-G\_k)^2}.$$
Then
$$H\_{n+1}-H\_n=G\_{n+1}^2-G\_n^2+\sum\_{k=... | 3 | https://mathoverflow.net/users/80257 | 260988 | 117,627 |
https://mathoverflow.net/questions/260783 | 26 | The Green-Tao theorem states that for every $n$, there is an arithmetic sequence of length $n$ consisting of primes.
For primes, $p$, let $P(p)$ be the maximum length of an arithmetic progression of primes whose least element is $p$.
>
> Is it known whether $P(p)=p$ for every prime?
>
>
>
(This clearly gener... | https://mathoverflow.net/users/24478 | What is the status on this conjecture on arithmetic progressions of primes? | Yes, this is unknown; it is even unknown (as GH from MO suspected in a comment) whether $P(p) \ge 3$ always. An equivalent statement to $P(p) \ge 3$ is that there exists an integer $x>0$ such $p+x$ and $p+2x$ are both prime. This is a twin-prime-like problem: nobody has ever proved a statement saying that two fixed lin... | 28 | https://mathoverflow.net/users/5091 | 260990 | 117,629 |
https://mathoverflow.net/questions/260955 | 2 | I would like to fill a detail in the answer given by Francesco Polizzi to the question ["Is the dualizing sheaf on a Cohen-Macaulay scheme reflexive?"](https://mathoverflow.net/questions/260885/is-the-dualizing-sheaf-on-a-cohen-macaulay-scheme-reflexive/260907#260907).
Let $X$ be a normal, Cohen-Macaulay scheme of di... | https://mathoverflow.net/users/104222 | Dualizing sheaf on a Cohen-Macaulay scheme | This is more or less standard material on Grothendieck and Serre Duality Theory, and there are several references available.
For instance, the statement that you want is a consequence of the results in Chapter 5 of S.Ishii's book *[Introduction to Singularities](http://www.springer.com/la/book/9784431550808)*. Look i... | 2 | https://mathoverflow.net/users/7460 | 260993 | 117,630 |
https://mathoverflow.net/questions/260991 | 0 | I have a random variable $X$ defined on $(0,\infty)$. For each $n\in \mathbb N$,
define $X\_n = X \mathbf{1}\_{0 < X \leq C\_n}$, where $C\_n$ is a monotonically increasing sequence of positive numbers such that $C\_n\to\infty$.
I want to evaluate the limint of an expectation
$$
\lim\_{n\to\infty} E[f\_n(X\_n)],
$$
w... | https://mathoverflow.net/users/80302 | Bounded convergence for expectation of random variables | Yes, it's true.
You only need basic facts about convergence in distribution (of real rvs). Both can be e.g. be found in Billingsley's book "Convergence of Probability Measures".
Let $(Y\_n)$ be a sequence of real random variables, $Y$ be a real rv.
(1) $Y\_n\longrightarrow Y$ in probability implies that
$Y\_n$ c... | 1 | https://mathoverflow.net/users/48831 | 261000 | 117,632 |
https://mathoverflow.net/questions/260660 | -1 | In the [Wikipedia article on scalar curvature](https://en.wikipedia.org/wiki/Yamabe_invariant), it is noticed that the Hölder inequality implies
$$Y(g) \geq - \left(\int\_M |R(g)|^{n/2} \mathrm{d}V\_g\right)^{n/2}$$
for the Yamabe functional $Y(g)$ and the scalar curvature $R(g)$. It is now concluded that this implies ... | https://mathoverflow.net/users/16702 | $L^{n/2}$ norm of scalar curvature | It's not obvious, but it follows by a straightforward argument using formula (1.1) in the [Bulletin survey article by Lee and Parker](https://projecteuclid.org/euclid.bams/1183553962). The case $n = 2$ is trivial, so assume $n > 2$. If a metric $\bar{g}$ is conformal to the metric $g$, then there exists a positive func... | 5 | https://mathoverflow.net/users/613 | 261011 | 117,634 |
https://mathoverflow.net/questions/260982 | 17 | It is a result of Serre (Morphismes universels et varietes d'albanese) that the Albanese (abelian) variety, i.e. an initial object for morphisms to (torsors over) abelian varieties, exists for any reduced scheme over a perfect field.
I have a counterexample for non-reduced schemes, but how about the perfectness of th... | https://mathoverflow.net/users/26735 | Albanese variety over non-perfect fields | The arguments of Serre can be in fact made to work over any separably closed field. The result in the general case can then be deduced using Galois descent. Details can be found in Section 2 and the appendix of:
Olivier Wittenberg - On Albanese torsors and the elementary obstruction.
This is in particular shows the... | 13 | https://mathoverflow.net/users/5101 | 261013 | 117,636 |
https://mathoverflow.net/questions/260994 | 0 | Suppose $R$ is $\mathbb{Z}$ graded ring and $0\neq f \in R\_1.$ Then I want to show that $R\_f \cong S[X,X^{-1}],$ where $S=(R\_f)\_0$ and $X$ transcendental over $S.$
I wanted to use the isomorphism $R\_f\cong R[X]/(Xf-1)$ but I was not able to get the result.
| https://mathoverflow.net/users/103965 | $R$ is $\mathbb{Z}$ graded ring and $0\neq f \in R_1,$ show that $R_f \cong S[X,X^{-1}]$ | It suffices to note that $A:=R\_f$ is a $\mathbb{Z}$-graded ring with an invertible element $f\in A\_1$. The $A\_0$-homomorphism $A\_0[X,X^{-1}]\to A$ sending $X$ to $f$ respects $\mathbb{Z}$-gradings and is an isomorphism in degree $0$. So it must be bijective because multiplication by powers of $X$ (resp. $f$) in $A\... | 3 | https://mathoverflow.net/users/7666 | 261020 | 117,639 |
https://mathoverflow.net/questions/261007 | 19 | Zermelo-Fraenkel set theory (with choice) is commonly accepted as the standard foundation of mathematics. It is a material set theory. For every two objects/sets $a,b$ one can ask whether $a=b$ or not. Also, one can always ask whether $a\in b$ is true or not. So $\in$ is a *global element relation*.
As an alternative... | https://mathoverflow.net/users/nan | Which kind of foundation are mathematicians using when proving metatheorems? | Your question is much more specific than your title suggests. As to the question itself, my answer is that it doesn't matter. The proof is given in mathematics, not in any formal system. A foundation for mathematics, in order to count as a foundation of mathematics, must be able to formalize most ordinary mathematical ... | 19 | https://mathoverflow.net/users/49 | 261028 | 117,642 |
https://mathoverflow.net/questions/261037 | 16 |
>
> **Question.** Can this number be expressed in terms of classical values?
> $$\sum\_{n,m=1}^{\infty}\frac1{(n^2+m^2)^{\frac32}}=1.056348517615643291\dots$$
>
>
>
**UPDATE.** I'm encouraged by Noam, Kevin and Igor's directional replies. To spice things up, I ask: is this true?
$$\sum\_{n,m=1}^{\infty}\frac1{(... | https://mathoverflow.net/users/66131 | What can be said about this double sum? | I don't know how "classical" you find these values, but here's perhaps something.
Define $E=\sum\_{m,n\in\mathbb{Z}}q^{m^2+n^2}$, which is known to be a weight 1 level 4 modular form. In fact $E$ is an eigenform for the Hecke operators, and if we write $E=\sum\_{r\geq0}a\_rq^r$ then $L(E,s)=\sum\_{r\geq1}a\_r/r^s$ eq... | 18 | https://mathoverflow.net/users/1384 | 261040 | 117,648 |
https://mathoverflow.net/questions/260975 | 5 | **Question:** Let $X$ be a scheme separated and of finite type over $\mathbb C$ with a $\mathbb C^\*$-action. Suppose that
1. The fixed point locus is proper.
2. Each orbit has proper closure.
Does this imply that $X$ is proper? Proper means proper over $\mathbb C$, of course.
**Some comments:** It would be hand... | https://mathoverflow.net/users/40862 | Properness of schemes with $\mathbb C^*$-action via properness of orbits and fixed points locus | **Edit.** I am making the answer shorter by eliminating the repeated argument.
Let $k$ be a perfect field. Let $X$ be a finite type, separated $k$-scheme. Let $$m:\mathbb{G}\_m \times\_{\text{Spec}\ k} X \to X$$ be a $k$-action of $\mathbb{G}\_m$ on $X$. Since $k$ is perfect, $m$ induces a $k$-action of $\mathbb{G}\_... | 8 | https://mathoverflow.net/users/13265 | 261041 | 117,649 |
https://mathoverflow.net/questions/261050 | 0 | [See this link](https://mathoverflow.net/questions/97574/van-den-berg-kesten-reimer-inequality) for a description of the van den Berg-Kesten-Reimer inequality. How important is the assumption that $\Omega\_i$ are finite spaces?
When Berg-Kesten state the inequality in [their 1985 paper](http://www.jstor.org/stable/32... | https://mathoverflow.net/users/47510 | Berg-Kesten-Reimer inequality on infinite spaces? | Check this out, I hope it helps.
*The van den Berg--Kesten--Reimer operator and inequality for infinite spaces*, by Arratia-Garibaldi-Hales
<https://arxiv.org/abs/1508.05337>
| 2 | https://mathoverflow.net/users/66131 | 261051 | 117,653 |
https://mathoverflow.net/questions/261036 | 22 | In a paper I'm working on, I'm tempted to write something like:
>
> Note that the argument above also proves the following result:
>
>
> **Scholium.** bla bla
>
>
>
Is this ok? Is it correct to say that a "scholium" is a "corollary of a proof"?
| https://mathoverflow.net/users/1516 | What is a "scholium"? | I am not a specialist in either etymology nor the english language (I am not a native speaker of english as well) but since the words scholium and porism have both greek origins, I thought it might be of some interest to add some info on how these words have been used in both ancient and modern greek:
The word *"por... | 27 | https://mathoverflow.net/users/85967 | 261056 | 117,655 |
https://mathoverflow.net/questions/261063 | 0 | The following problem has recently come up in the course of a project I'm working on, and I would be very appreciative if someone could point me in the right direction:
Let $X \colon I \to G$ be a smooth curve taking values in a Lie group $G$ with a surjective exponential map, and define another curve $v\colon I \to ... | https://mathoverflow.net/users/88654 | Kinematics on Lie groups | $v(t)$ is widely used, even for a proof of existence of a Lie group for a given Lie algebra (see the book of Duistermaat and Kolk).
It is also used a lot for infinite dimensional Lie groups where the exponential map is not a local diffeomorphism, in general (for diffeomorphism groups, for example). See the book "You... | 1 | https://mathoverflow.net/users/26935 | 261075 | 117,662 |
https://mathoverflow.net/questions/260808 | 1 | Let $x\_1,\ldots,x\_n$ is a sequence in a Noetherian local ring $R$. We say $x\_1,\ldots,x\_n$ is a $d$-sequence if
1) $x\_i\notin (x\_1,\ldots,\hat{x\_i},\ldots,x\_n),$
2) for all $k\geq i+1$ and all $i\geq 0,$ $(x\_0=0)$ $$((x\_0,\ldots,x\_i):x\_{i+1}x\_k)=((x\_0,\ldots,x\_i):x\_k).$$
In this paper [here](https... | https://mathoverflow.net/users/9485 | Properties of d-sequence | Huneke only states an inclusion. But you are right, one gets a slightly stronger statement, with weaker assumptions. Namely, for any ring $R$, and any sequence $x\_1,\dots,x\_n$ a of elements of $R$, consider the following two properties (both implied by property $2)$ of $d$-sequences) :
$(a)$ For any $j \leq n$, one... | 3 | https://mathoverflow.net/users/21724 | 261081 | 117,666 |
https://mathoverflow.net/questions/261064 | 10 | R. Guy, *Unsolved problems in number theory*, 3rd edition, Springer, 2004.
In this book, on page 167-168, Problem C5, *Sums determining members of a set*, discusses a question Leo Moser asked: suppose $X\subset\mathbb{Z}$ is an $n$-element set and $A$ be set of all $k$-element **sums** of subsets of $X$. Is there any... | https://mathoverflow.net/users/66131 | what is the status of this problem? an equivalent formulation? | In a recent paper [A set of 12 numbers is not determined by its set of 4-sums](http://mi.mathnet.ru/eng/znsl6308) (in Russian), Isomurodov and Kokhas identify an error in the argument of Ewell (1968) and present two distinct sets of 12 integers with the same multiset of 4-sums.
They also give a nice overview of the a... | 17 | https://mathoverflow.net/users/7076 | 261085 | 117,668 |
https://mathoverflow.net/questions/261046 | 2 | Suppose we have a principal bundle $p: P \to M$ with principal connection form $\omega: TP \to \mathfrak{g}$.
Given any arbitrary curve $\gamma: I \to P$ in the total space of the principal bundle, where $I=[0,1]$, we can use the nonabelian fundamental theorem of calculus [[R. W. Sharpe](https://books.google.co.nz/b... | https://mathoverflow.net/users/56938 | Parallel transport in a principal bundle via the nonabelian fundamental theorem of calculus | Yes, they coincide (at least if I understand you correctly).
Let $\check{\gamma}$ be a path in $M$. If we fix a lift $\gamma$ in P, then the horizontal lift $\gamma\_\omega$ of $\check{\gamma}$ is necessarily of the form $\gamma\_\omega(t) = \gamma(t) \cdot g(t)$ for some curve $g: [0,1] \to G$.
Differentiating this ... | 1 | https://mathoverflow.net/users/17047 | 261089 | 117,670 |
https://mathoverflow.net/questions/261066 | 2 | Given $\lbrace Y\_i\rbrace$ a non-homogenous Poisson process with mean density $\theta y^{-1}e^{-y}$ where $y>0$ $(\theta>0)$. I.e., the number of points of $\lbrace Y\_i\rbrace$ in $(a,b)$ with $0<a<b\le \infty$ has a Poisson distribution with mean $\int\_a ^b \theta y^{-1}e^{-y}dy$.
How can the density of $Y\_j$ be... | https://mathoverflow.net/users/95756 | Density of Non-Homogeneous Poisson Process | the desired density $\rho\_j(y)d\epsilon$ is the probability that there is an event in the interval $(y,y+d\epsilon)$, with $d\epsilon$ infinitesimal, multiplied by the independent probability that there are $j-1$ events in the interval $(y,\infty)$:
$$\rho\_j(y)=\left(\lim\_{\epsilon\rightarrow 0}\frac{1}{\epsilon}\... | 1 | https://mathoverflow.net/users/11260 | 261090 | 117,671 |
https://mathoverflow.net/questions/261087 | 5 | At the risk of this question having an easy answer, I am asking the following:
Suppose we have a measurable cardinal $\kappa$ and a (possibly regular) cardinal $\lambda>\kappa$. Is there any condition, like the existence of an appropriate ultrafilter, that implies the existence of an elementary embedding $j:V\to M$ w... | https://mathoverflow.net/users/35968 | Can we control where the critical point of an elementary embedding is mapped? | Here is a necessary and sufficient criterion: $\lambda>2^\kappa$. This is easily seen to be necessary, since if $j:V\to M$ has critical point $\kappa$, then the power set $P(\kappa)$ is contained in $M$, and from this it follows that $j(\kappa)>2^\kappa$ since $j(\kappa)$ is inaccessible in $M$. Conversely, we can hit ... | 10 | https://mathoverflow.net/users/1946 | 261104 | 117,675 |
https://mathoverflow.net/questions/261084 | 6 | Let $X$ be a topological space and $R$ be a commutative ring with unit, $D(X,R)$ is the derived category of unbounded complexes of sheaves of $R$-modules. Moreover we suppose that $X$ is a stratified space and $D\_c(X,R)\subset D(X,R)$ is the derived category of complexes of sheaves with constructible cohomology sheave... | https://mathoverflow.net/users/27816 | Picard group of derived category of sheaves | Thanks to Drew Heard's comment I was able to find answers to my questions. In his paper "Picard groups of derived categories" H. Fausk proves the following theorem (see Theorem 4.2).
**Theorem:** *Let $(\mathcal{E},\mathcal{O})$ be a commutative unital ringed Grothendieck topos with enough points such that for all po... | 6 | https://mathoverflow.net/users/27816 | 261106 | 117,676 |
https://mathoverflow.net/questions/261109 | 1 | I need the following theorem: Let $A$ and $B$ be two $n-1$ dimensional bounded smooth manifolds in $R^{n}$. Let $C$ be the set of all points in $A \cap B$ where the normals to $A$ and $B$ are unequal. Then $C$ has measure zero, in the $n-1$ dimensional Hausdorff measure.
I actually only need the cases $n \leq 4$, and... | https://mathoverflow.net/users/22344 | Measure of the intersection of two manifolds | This is Thom-Boardman transversality, and is an elementary exercise, since the
intersection is a submanifold of dimension $n-2$.
A pretty complete reference: <https://en.wikipedia.org/wiki/Transversality_(mathematics)>
| 1 | https://mathoverflow.net/users/13268 | 261110 | 117,677 |
https://mathoverflow.net/questions/261115 | 3 | Let $G$ be a spanning subgraph of $K\_{n,n}$ with minimum degree $\delta(G) \geq n/2$. It's easy to show using Hall's theorem that $G$ has a perfect matching, and the example of two disjoint copies of $K\_{\lfloor n/2 \rfloor + 1, \lceil n/2 \rceil - 1}$ side by side shows that $n/2$ is sharp. This extremal example "al... | https://mathoverflow.net/users/25485 | Number of perfect matchings in bipartite graph with given minimum degree | It's [a theorem of Marshall Hall](http://www.ams.org/journals/bull/1948-54-10/S0002-9904-1948-09098-X/) that in a bipartite graph of minimum degree $r$, where there exists a perfect matching, there will be at least $r!$ perfect matchings. In your case the conditional disappears and you get a lower bound of $\lceil\frac... | 7 | https://mathoverflow.net/users/2384 | 261121 | 117,680 |
https://mathoverflow.net/questions/261108 | 3 | Given an augmented graded associative $K$ -algebra $A$, we can construct a free resolution of $K$ given by $K\_A$ modules which gives nice combinatorial informations about the homology classes of the algebra $A$. This type of resolution is first constructed by David Anick which was given in his paper
"On the Homology o... | https://mathoverflow.net/users/104177 | Geometric significance of Anick's resolution | You might look at Ken Brown's <https://www.math.cornell.edu/~kbrown/scan/1992.0000.0137.pdf> where he gives a geometric proof of the resolution for groups given by a complete rewriting system and the reinterprets it algebraically for monoids. The same sort of thing can be done for the two-sided bar construction for alg... | 1 | https://mathoverflow.net/users/15934 | 261125 | 117,682 |
https://mathoverflow.net/questions/261045 | 2 | Let $T$ be a labeled tree with the root $v$, such that:
$(i)$ The height of the tree is $x$,
$(ii)$ the degree of the vertex $v$ is $y-2$,
$(iii)$ the degree of each vertex, except the leaves and the vertex $v$ is $y$.
Let $T(x,y)$ be the the set of subtrees of $T$ such that:
$(i)$ each subtree in $T(x,y)$ ha... | https://mathoverflow.net/users/42971 | An approximation for the number of subtrees of a tree | For $y=3$, one takes one branch of a (otherwise full) binary tree of height x and finds a recurrence of the form T(x,y)= sum over z from 1 to x-1 of T(z,3)\*T(x-1-z,3) , which smells to me like a recurrence for Catalan numbers, which would give a correspondence T(x,3) is the (x-c)th Catalan number for some small consta... | 1 | https://mathoverflow.net/users/3402 | 261127 | 117,683 |
https://mathoverflow.net/questions/261095 | 5 | Let $M,N$ be smooth oriented $d$-dimensional Riemannian manifolds, $\, f:M \to N$ a smooth map. Let $\Omega^1(M,f^\*TN)=\Gamma(T^\*M \otimes f^\*TN)$ be the space of $f^\*TN$-valued one-forms.
Let $d$ be the covariant exterior derivative associated with the pullback connection of the Levi-Civita connection on $N$ (v... | https://mathoverflow.net/users/46290 | Are all the mappings which satisfy this equation scaled isometries? | Here's a simple counterexample: Let $M=N=T^2$ (the standard torus, thought of as $\mathbb{R}^2/\mathbb{Z}^2$). Let $f:M\to N$ be the identity, and let the metrics on $M$ and $N$ be any two translation-invariant metrics on $T =\mathbb{R}^2/\mathbb{Z}^2$ (so, in particular, they don't need to be conformal). Then one easi... | 6 | https://mathoverflow.net/users/13972 | 261139 | 117,686 |
https://mathoverflow.net/questions/84866 | 4 | Background
----------
A projective representation $V\_g\in \mathrm{GL}\_n(\mathbb{C})$ of a group $G$ is characterized by $V\_gV\_h=\omega(g,h)V\_h$, where $\omega(g,h)\in\mathrm{U}(1)$ is a 2-cocycle. Changing the phase of each $V\_g$ by $\phi\_g$ induces an equivalence relation
$$
\omega(g,h) \sim \omega(g,h)\phi\... | https://mathoverflow.net/users/16710 | Identifying projective representations using "gauge-invariant" traces tr[V_g V_h V_k ... ] | Since the action on $U(1)$ is trivial, and since $U(1)$ is injective as an abelian group, the Universal Coefficients Theorem will give you an isomorphism
$$H^2(G,U(1))\cong Hom(H\_2(G,\mathbb{Z}),U(1)).$$
Now for a discrete group $G$ which is written as $F/R$ (with $F$ a free group), we have $$M:=H\_2(G,\mathbb{Z})\co... | 4 | https://mathoverflow.net/users/41644 | 261141 | 117,688 |
https://mathoverflow.net/questions/261047 | 9 | I know of Tarski monsters and the Burnside Problem. I would like to know if there is an infinite finitely generated group $G$ such that for any $g$ and $h$ in $G$, the subgroup generated by $\{g,h\}$ is finite.
I am also interested in related questions:
* For which $(m,n)$ does there exist an infinite $m$-generated... | https://mathoverflow.net/users/56097 | Finitely generated group every 2-generated subgroup of which is finite | Questions 1 and 2 are answered by Golod's theorem, see Theorem 3.3 in
the survey:
M. Ershov, [Golod-Shafarevich groups: a survey](https://arxiv.org/pdf/1206.0490v1.pdf). Internat. J. Algebra and Computation, vol. 22 (2012).
Namely, for every $d\ge 2$ there exists an infinite $d$-generated group such that every $d... | 10 | https://mathoverflow.net/users/39654 | 261144 | 117,690 |
https://mathoverflow.net/questions/261150 | 4 | Skolem proved the need for the axiom schema of Replacement by, in essence, showing that $V\_{ω+ω}$ is a model of Zermelo's set theory Z (the smallest "natural" model) and, hence, that the existence of the set $\{ω, ℘(ω), ℘(℘(ω)), ...\}$ is independent of Z (where ω is understood to be the usual von Neumann ordinal). No... | https://mathoverflow.net/users/42416 | Independence of a certain set w.r.t. Z set theory with urelements | If you start with the elements in $V\_{\omega+1}$ and a nonempty set $A$ of urelements, and then simply close under pairing, unions and power sets, you get a model of Zermelo + urelements, but where $A^\*$ doesn't exist, even when $A$ is finite, since it doesn't appear after any finitely many applications of pairing, u... | 5 | https://mathoverflow.net/users/1946 | 261155 | 117,694 |
https://mathoverflow.net/questions/261149 | 3 | Let $A=\mathbb{C}[u,x,y,w]/(uy-x^2,xw-y^2,uw-xy)$, $X=Spec A$. $A$ is a Veronese subring and from the answer of [Is there a simple method to test a local ring to be Cohen Macaulay?](https://mathoverflow.net/questions/26651/is-there-a-simple-method-to-test-a-local-ring-to-be-cohen-macaulay), we can see that $X$ is CM bu... | https://mathoverflow.net/users/48616 | Canonical sheaf of affine variety | For a toric variety, such as this cone over the 3rd Veronese of $\mathbb P^1$, the complement of the open torus orbit is an anticanonical divisor (indeed, that is the one given by the unique $T$-invariant section of the anticanonical bundle). Correspondingly, the canonical module is the submodule of $\mathcal O$ given ... | 5 | https://mathoverflow.net/users/391 | 261158 | 117,697 |
https://mathoverflow.net/questions/261091 | 5 | Let $B$ be a connected, solvable algebraic group (of dimension 2) acting on a projective variety $Y$ (of dimension $3$): for instance, let $B$ be a Borel subgroup of a reductive algebraic group $G$.
**Question1**: Is it always true that there is a $B$-invariant complete flag (i.e. codimension $1$, irreducible and $B... | https://mathoverflow.net/users/104316 | B-invariant subvarieties | Here is an argument which avoids the use of Hilbert schemes and works for any complete (irreducible) variety $Y$.
As in Jason's answer it suffices to construct a $B$-stable subset $Y\_1$ of codimension $1$, provided that $\dim Y\ge1$. For this we consider two cases:
1. $Y$ does not contain an open $B$-orbit. By Ros... | 7 | https://mathoverflow.net/users/89948 | 261172 | 117,704 |
https://mathoverflow.net/questions/261178 | 0 | Let $A=C^\*\_r(G)$ be the reduced group $C^\*$-algebra of a finitely generated group. Consider the map $M\_2(A)\to M\_2(A)$,
$$\left[\begin{array}{ll}a&b\\c&d \end{array}\right]\mapsto \left[\begin{array}{ll}tr(a)I&b\\c&tr(d) I \end{array}\right].$$
$I$ is the identity in $A$ and $tr(a)$ is the trace of $a$.
Is this ... | https://mathoverflow.net/users/104362 | Completely positive map defined by the trace | This map is not even positive, let alone completely positive.
This comes from the following lemma: A matrix $\left[\begin{array}{cc} I & a \\ a^{\ast} & b\end{array}\right]$ is positive in $M\_{2}(A)$ iff $a^{\ast}a \leqslant b$. Therefore the matrix $\left[\begin{array}{cc} I & a \\ a^{\ast} & a^{\ast}a\end{array}\r... | 5 | https://mathoverflow.net/users/24953 | 261182 | 117,708 |
https://mathoverflow.net/questions/261185 | 6 | For a Hecke L-function, if all of the local eigenvalues are roots of unity, is it an Artin L-function?
| https://mathoverflow.net/users/104370 | A question on the Hecke L-function | Whatever your answer to my question in comment, the answer to the title question is yes. Take the case of the $L$-function attached to a modular form $f$, which we assume an eigenform for almost all Hecke operators (since otherwise there is no decomposition of the L-function into Euler product and the phrase "local eig... | 12 | https://mathoverflow.net/users/9317 | 261199 | 117,712 |
https://mathoverflow.net/questions/261163 | 2 | If $(X,\tau)$ is a topological space and $x,y\in X$ we say that $x$ is *mappable to* $y$ if there is a non-constant continuous map $f:X\to X$ with $f(x) = y$.
Is there a Hausdorff space $X$ with more than one point such that whenever $x\neq y\in X$ and $x$ is mappable to $y$, then $y$ is *not* mappable to $x$?
| https://mathoverflow.net/users/8628 | Anti-symmetric mappability relation | Yes, there are such spaces. One example is referenced by Ramiro de la Vega in his answer to [this related question](https://mathoverflow.net/questions/188729/strongly-rigid-hausdorff-spaces?noredirect=1&lq=1).
As Joel points out in the comments, the property you describe, namely "$X$ has an anti-symmetric mappability... | 2 | https://mathoverflow.net/users/70618 | 261200 | 117,713 |
https://mathoverflow.net/questions/261198 | 2 | Let $f:[0,1]\to\mathbb{R}\_+$ be a convex, strictly increasing function such that $f(0)=0$ (typically, $f$ is very flat at $0$, i.e. increases very slowly). I would like to prove or disprove the following statement:
>
> For all decreasing sequence $(a\_n)\_{n\ge 0}$ of positive reals such that $\lim a\_n = 0$, ther... | https://mathoverflow.net/users/4961 | Enforcing an inequality on series | No even for $f(x)=x^2$. We should have $(b\_1p\_1+b\_np\_n)^2\geqslant (a\_1p\_1+a\_np\_n)(p\_1+p\_n)$ (take all $p\_i$'s for $i\ne 1,n$ equal to $0$). But the discriminant of the quadratic form $(b\_1p\_1+b\_np\_n)^2- (a\_1p\_1+a\_np\_n)(p\_1+p\_n)$ becomes positive for large $n$.
| 1 | https://mathoverflow.net/users/4312 | 261208 | 117,716 |
https://mathoverflow.net/questions/261021 | 6 | Several of the many notions that don't work the same way when passing to $\infty$-categories are the ones mentioned in the title. I'm trying to understand the conceptual picture around these notions in the world of higher category theory (conceptual answers will be more useful to me than model specific answers).
My q... | https://mathoverflow.net/users/22810 | Monomorphisms, epimorphisms, (co-)images and factorizations in $\infty$-categories | An **$n$-monomorphism** is a map $A\to B$ for which $Map(X,A)\to Map(X,B)$ has all homotopy fibers **$n$-truncated**, for all $X$; a space (=$\infty$-groupoid) is $n$-truncated if its homotopy groups all vanish in dimensions $>n$.
An **$n$-epimorphism** is an $n$-monomorphism in the opposite $\infty$-category.
When... | 12 | https://mathoverflow.net/users/437 | 261209 | 117,717 |
https://mathoverflow.net/questions/261197 | 2 | Let $G$ be a Lie group acting on two smooth manifolds $M$ and $N$ such that both $M/G$ and $N/G$ are smooth manifolds. Suppose that $f: M\to N$ is a smooth **proper** $G$-map. Then it induces a smooth map $\bar f: M/G\to N/G$. My question is
Is the induced map $\bar f$ **proper**?
Here, proper means preimages of c... | https://mathoverflow.net/users/46774 | Does an equivariant proper map induce a proper map between quotient spaces? | Assuming the group actions to be smooth, the quotient maps $p:M\to G\backslash M$ and $q:N\to G\backslash N$ are open. Let $C\subset G\backslash N$ be compact. For $q(n)\in C$, $n\in N$, pick a relatively compact open neighborhood $U\_n$ of $n$ in $N$. The sets $q(U\_n)$, $q(n)\in C$, form an open covering of $C$, so f... | 5 | https://mathoverflow.net/users/nan | 261210 | 117,718 |
https://mathoverflow.net/questions/261222 | 0 | Let $n=p\_1^{\alpha\_1}\cdots p\_r^{\alpha\_r}$ be the prime decomposition of the integer $n$. Define $$n' = n \sum\_{i=1}^r \frac{\alpha\_i}{p\_i}\quad\text{and}\quad\Omega(n) = \sum\_{i=1}^r \alpha\_i\quad\text{and}\quad\omega(n) = r.$$ Let $a,b$ be relatively prime, i.e., $\gcd(a,b)=1$, and let $c = a+b$. Suppose th... | https://mathoverflow.net/users/nan | A question about numbers | No. How about $a=9$ and $b=16$? Then $c=25$ so $\Omega(a)=\Omega(c)=2\leq\Omega(b)$, the gcd's are $3,16,5$ so the left hand side is 6 and the right hand side only 5.
Edit: if $a=316$ and $b=27$ then we even have the left hand side being greater than $\Omega(ab)$, answering a question in the comments.
Editedit: if ... | 2 | https://mathoverflow.net/users/1384 | 261224 | 117,720 |
https://mathoverflow.net/questions/261249 | 13 | If $G$ is a group and $H\le G$ a subgroup, let $NH$ denote the normalizer of $H$ in $G$, and let $WH = NH/H$; following May's *Concise Course*, §3.4 this I call the *Weyl group*. I have also seen the definition $WH=NH/C(H)$, where $C(H)$ is the centralizer of $H$ in $G$, from <https://groupprops.subwiki.org/wiki/Weyl_g... | https://mathoverflow.net/users/18702 | What is the universal property of the Weyl group? | The group $WH$ is the automorphism group of $G/H$ as a $G$-set. Thus, $X^H$ is the set of morphisms from $G/H$ into $X$ in the category of $G$-sets, and this has an obvious action by precomposition with automorphisms.
| 21 | https://mathoverflow.net/users/66 | 261251 | 117,729 |
https://mathoverflow.net/questions/261253 | 2 | The Vietoris-Rips complex (<https://en.wikipedia.org/wiki/Vietoris%E2%80%93Rips_complex>) is an abstract simplicial complex that can be defined from any metric space M and distance $\delta$ by forming a simplex for every finite set of points that has diameter at most $\delta$. That is, if the distance between each pair... | https://mathoverflow.net/users/83274 | Intermediate between Vietoris-Rips complex and Cech Complex | I think you need to look back at the history of these ideas and you will see another image of what is happening. A good place to start is a paper by Dowker:
C. H. Dowker, Homology Groups of Relations, Annals of Maths, 56, (1952), 84 – 95.
He showed how any relation, $R$, between two sets, say $X$ and $Y$, gave two ... | 5 | https://mathoverflow.net/users/3502 | 261264 | 117,735 |
https://mathoverflow.net/questions/261254 | 5 | This is probably well-known to representation theorists, but this doesn't imply being well-known to me.
Let $k$ be a field, and let $A$ be a $k$-algebra that is finite-dimensional as a $k$-vector space. Let $S\_1, S\_2, \ldots, S\_m$ be a complete list of pairwise non-isomorphic simple $A$-modules (up to isomorphism)... | https://mathoverflow.net/users/2530 | Is the Cartan matrix of a finite-dimensional (Hopf) algebra invertible over the rationals? | A counterexample is given by the [Taft Hopf algebra](https://en.wikipedia.org/wiki/Taft_Hopf_algebra). (I checked this at least for n=2, where this Hopf algebra also goes by a different name that I forget today).
| 6 | https://mathoverflow.net/users/425 | 261266 | 117,736 |
https://mathoverflow.net/questions/261274 | 2 | Are there any known common parametrizations of SL(3,R)? I know that it is easy to obtain a local parametrization by just exponentiating generators from the Lie algebra, but I do not know if they are global or not. How can I know if a parametrization obtained by exponentiating Lie algebra generators is a global parametr... | https://mathoverflow.net/users/64029 | Parametrization of SL(3,R) | By singular value decomposition, every matrix in $SL\_3$ factors as $UDV$ with $U$ and $V$ in $SO(3)$ and $D$ of the form $\mathrm{diag}(x,y,1/(xy))$ (or $\mathrm{diag}(e^u,e^v,e^{-u-v})$ if you prefer). Any matrix in $SO(3)$ can be factored as
$$\begin{bmatrix}
\cos \alpha & - \sin \alpha & 0 \\
\sin \alpha & \cos \al... | 7 | https://mathoverflow.net/users/297 | 261288 | 117,739 |
https://mathoverflow.net/questions/261286 | 1 | Suppose that $T$ is a bounded operator on a Hilbert space $\mathsf{H}$. Fix $\varepsilon > 0$. Assume that I have two unit vectors $\xi$ and $\eta$ such that $\xi$ belongs to the spectral subspace $\mathbb{1}\_{|T| \leqslant \varepsilon}$ (spectral projection of $|T|:=\sqrt{T^{\ast}T}$ corresponding to the interval $[0... | https://mathoverflow.net/users/24953 | Spectral projections of non-commuting operators | I'm afraid there is no such estimate. Let $\delta>0$ and let $\xi$ and $\eta$ be any linearly independent norm-one vectors in ${\mathbb C}^2$ with $\parallel \xi-\eta\parallel<\delta$. Let $S$ be the matrix with columns $\xi$ and $\eta$ and let $T=S\left(\begin{array}{cc}0&0\\ 0&1\end{array}\right)S^{-1}$. Then $T\xi=0... | 3 | https://mathoverflow.net/users/nan | 261292 | 117,740 |
https://mathoverflow.net/questions/261305 | 1 | Recall that the Calkin algebra, is the quotient $B(H)/B\_0(H)$, where $H$ is a Hilbert space and $B(H)$ and $B\_0(H)$ are the algebra of bounded and compact operators on $H$.
Let $H$ be separable and $Q:B(H)\rightarrow B(H)/B\_0(H)$ be a natural quotient map. Is it true for quotient norm that
$$\|Q(T)\|=\lim\_n\|T(... | https://mathoverflow.net/users/84700 | About the quotient norm in the Calkin algebra | Yes, because $(P\_n)$ is an approximate unit for the ideal of compact operators. If $C$ is any compact operator then $$\|T(I - P\_n)\| \leq \|(T-C)(I-P\_n)\| + \|C(I-P\_n)\| \leq \|T - C\| + \|C - CP\_n\|,$$ and $\|C - CP\_n\| \to 0$ as $n \to \infty$, so the norm in the Calkin algebra is achieved by taking the limit $... | 2 | https://mathoverflow.net/users/23141 | 261309 | 117,750 |
https://mathoverflow.net/questions/261230 | 4 | Let $f:X \to Y$ be a morphism of reasonable schemes (qcqs). Let $f^\*: D(Y) \to D(X)$ be the pullback defined on the derived unbounded categories of quasi-coherent sheaves.
>
> Question: **When does $f^\*$ commute with arbitrary products?**
>
>
>
| https://mathoverflow.net/users/22810 | When does derived pullback commute with infinite products? | The functor $f^\*: D(Y) \to D(X)$ preserves limits if $Y$ is noetherian and $f$ is proper flat with Gorenstein fibers, for example if $f$ is smooth and proper. Under these assumptions: $f^!$ is right adjoint to $f\_\*$ by properness, $f^!(\mathcal F)=f^!(\mathcal O\_Y)\otimes f^\*(\mathcal F)$ by finite tor-amplitude, ... | 7 | https://mathoverflow.net/users/20233 | 261312 | 117,752 |
https://mathoverflow.net/questions/261311 | 4 | It is a kind of folklore but I would like to see the proof of the following fact: given two smooth manifolds $M$ and $N$ if we assume that the algebras $C^{\infty}\_0(M)$ and $C^{\infty}\_0(N)$ are isomorphic (as algebras) then $M$ and $N$ are diffeomorphic.
| https://mathoverflow.net/users/98866 | Isomorphic algebras determine diffeomorphic manifolds | Theorem 2.8 on p. 30 of *Models for smooth infinitesimal analysis* by Moerdijk and Reyes. The theorem works with $C^\infty$ rings, which are a bit more general than algebras of functions. The advantage is that the proof is short.
| 8 | https://mathoverflow.net/users/25355 | 261314 | 117,753 |
https://mathoverflow.net/questions/261322 | 5 | A visible point is a point $(a, b)\in \mathbb{Z}^2,$ with $gcd(a,b)=1$. It is well-known that the number $V(N)$ of visible points with $0<a, b \leq N$ is asymptotic to $N^2/\zeta(2)$, but suppose I really want to compute it exactly. Can it be done in polynomial time (polynomial in $\log N,$ that is)?
If one wants to... | https://mathoverflow.net/users/11142 | Complexity of computing the number of visible points | There is an algorithm for computing $F(N) = \# \{ (a,b) : 1 \leq a, b \leq N, \gcd(a,b) = 1 \}$ in time $O(N^{5/6 + \epsilon})$. This relies on the algorithm of Deleglise and Rivat (see [their paper here](https://projecteuclid.org/euclid.em/1047565447)) that computes $M(x) = \sum\_{n \leq x} \mu(n)$ in time $O(x^{2/3} ... | 9 | https://mathoverflow.net/users/48142 | 261326 | 117,755 |
https://mathoverflow.net/questions/261307 | 13 | If $G$ is any group and $\gamma\_k(G)$ denotes the $k$th term in the lower central series of $G$, then the commutator bracket on $G$ endows
$$\mathcal{L}(G) = \bigoplus\_{k=1}^{\infty} \gamma\_k(G) / \gamma\_{k+1}(G)$$
with the structure of a Lie ring. A famous theorem that should probably be attributed to Magnus s... | https://mathoverflow.net/users/317 | Free groups and free restricted Lie algebras | This is Theorem 6.5 p.130 from Lazard, Michel.
Sur les groupes nilpotent et les anneaux de Lie.
Ann. Sci. Ecole Norm. Sup (3) 71 (1954) 101-190.
| 6 | https://mathoverflow.net/users/1109 | 261329 | 117,758 |
https://mathoverflow.net/questions/261302 | 8 | In model theory, the Keisler-Shelah isomorphism theorem asserts that two models of a theory are elementary equivalent if and only if they have isomorphic ultrapowers. On the other hand, assuming that $\kappa$ is strongly compact, we can ensure the existence of a $\kappa$-complete ultrafilter on every set (that is, an u... | https://mathoverflow.net/users/12976 | Does Keisler-Shelah isomorphism theorem hold for infinitary logics? | This is a very nice question.
The answer is no, it doesn't necessarily hold. Let me describe a counterexample. Consider the language of linear orders. By the pigeon-hole principle, since there are only a set of possible $L\_{\kappa,\kappa}$-theories in this language, there must be two ordinals $\alpha<\beta$ such th... | 10 | https://mathoverflow.net/users/1946 | 261331 | 117,759 |
https://mathoverflow.net/questions/261321 | 8 | This question arose today at Yevgeny Gordon's talk, *"Will nonstandard analysis be
the analysis of the future?"* at the CUNY Logic
Workshop. Here is my way of asking it.
Consider the ordered real field $\newcommand\R{\mathbb{R}}\R$ with a predicate for the natural numbers, and a
nonstandard version of it $\R^\*$, the... | https://mathoverflow.net/users/1946 | Are the definable hyper-reals, using quantifiers only over the standard reals and natural numbers, the same as the algebraic numbers? | $\newcommand{\st}{\textrm{st}}\newcommand{\bR}{{\bf R}}\newcommand{\bN}{{\bf N}}$
I think the limit of any definable convergent sequence $(a\_n)$ (including $e$) is definable by the formula
$$\varphi(x)=(\exists x'\in {\bR}^\st\, x'=x)\land \forall N\in\bN^\st\exists n\_0\in \bN^\st\forall n\in \bN^\st\_{>n\_0}(a\_n-x)... | 8 | https://mathoverflow.net/users/54415 | 261356 | 117,766 |
https://mathoverflow.net/questions/261360 | 5 | It is not difficult to show that $\ln(x)$ for real positive $x$ can be approximated by the series
$$
-\gamma+\sum\_{k=1}^\infty \frac{(-1)^{k+1}}{k!k}x^k
$$
up to an absolute error bounded by $e^{-x}/x,$ where $\gamma$ is the Euler-Mascheroni constant. I assume this must be well known and would appreciate any referenc... | https://mathoverflow.net/users/91419 | Approximating logarithm by a series for large values | The statement is equivalent to an expansion for the [Exponential Integral function](https://en.wikipedia.org/wiki/Exponential_integral#Convergent_series) $\operatorname{Ei}(x)$, so any text on special functions should do (check e.g. the references in the linked wiki article).
| 4 | https://mathoverflow.net/users/6101 | 261361 | 117,768 |
https://mathoverflow.net/questions/261344 | 5 | In a comment exchange around an answer to [Is a group scheme determined by its category of representations?](https://mathoverflow.net/q/151097/41291) there arose the issue of Tannakian reconstruction for non-affine algebraic groups (e. g. abelian varieties, in particular, elliptic curves). Some very naïve approach came... | https://mathoverflow.net/users/41291 | A "prequestion" about meromorphic representations of algebraic groups | **Edit.** Thanks to nfdc23 for pointing out a couple of corrections. As nfdc23 points out, these kinds of things are important in passing from a birational group law to a (regular) group law, so they may go back to Weil. Also, I vaguely remember something about some of this in SGA 3, so that might also be a reference.
... | 3 | https://mathoverflow.net/users/13265 | 261366 | 117,770 |
https://mathoverflow.net/questions/261374 | 14 | Ramanujan delivered his [famous congruences](https://en.wikipedia.org/wiki/Ramanujan's_congruences)
$$p(5n+4)\equiv\_50, \qquad p(7n+5)\equiv\_70, \qquad p(11n+6)\equiv\_{11}0$$
for the integer partitions with generating function $F(x)=\prod\_{k=0}^{\infty}\frac1{1-x^k}=\sum\_{n\geq0}p(n)x^n$.
Let $p\geq5$ be a pri... | https://mathoverflow.net/users/66131 | powered partition function generator: 1/2 of them are zeros? | This is true. First, note that by the Pentagonal number theorem due to Euler,
$$\frac{1}{F(x)} = \sum\_{k \in \mathbb{Z}} (-1)^k x^{\frac{k}{2}(3k-1)}.$$
For a given prime $p \ge 5$, the function $f(k)=\frac{k}{2}(3k-1),f:\mathbb{Z}\to \mathbb{F}\_p$ has only $\frac{p+1}{2}$ values in its image, one of which is $0$. ... | 22 | https://mathoverflow.net/users/31469 | 261377 | 117,775 |
https://mathoverflow.net/questions/261378 | 1 | In the proof of Lemma 3.2.2 in Chriss and Ginzburg's Representation Theory and Complex Geometry, the final step states that "the annihilator $\mathfrak{b}^\perp\subset \mathfrak{g}^\*$ gets identified with the annihilator of $\mathfrak{b}$ in $\mathfrak{g}$ with respect to the invariant form; the latter is equal to $\m... | https://mathoverflow.net/users/74343 | The annihilator of a Borel subalgebra being its nilpotent radical | As in Chriss-Ginzburg p. 130, pick a regular semisimple $h\in\mathfrak h$ (Cartan subalgebra) and write $\mathfrak g\_a$ for the eigenspace of $\text{ad}\_h$ belonging to eigenvalue $a$. Invariance of the Killing form $\langle\cdot,\cdot\rangle$ implies $$0=\langle [h,X],Y\rangle+\langle X,[h,Y]\rangle=(a+b)\langle X,Y... | 5 | https://mathoverflow.net/users/19276 | 261382 | 117,778 |
https://mathoverflow.net/questions/246896 | 4 | I am trying to compute the indefinite integral
$$
\int\_0^u {}\_2F\_1\left(\frac{1}{4},\frac{5}{4},2,1-v^2\right)^2 dv
$$
for $0<u<1$. Using [Clausen's formula](https://en.wikipedia.org/wiki/Clausen%27s_formula "Clausen's formula") for the square of the hypergeometric function, this can be written as
$$
\int\_0^u {}\_3... | https://mathoverflow.net/users/47683 | Indefinite integral of squared hypergeometric function | Using Maple to play with the equation ([details](http://dl.vwx.fr/8UAD7bnP6v0gtuD1/foo.pdf)) suggests that
$$
\int\_0^u {}\_2F\_1\left(\frac{1}{4},\frac{5}{4},2,1-v^2\right)^2 dv
= -\frac{32}{\pi} + \frac83 u \cdot {}\_3F\_2\!\left(\frac12, \frac12, \frac52, 1, 2, 1-u^2\right).
$$
You can *almost* prove this equality b... | 5 | https://mathoverflow.net/users/27491 | 261385 | 117,780 |
https://mathoverflow.net/questions/261391 | 5 | It is a well known fact that a smooth cubic surface in $P^3$ contains 27 lines. One proof proceeds by moving through the parameter space $U$ of smooth cubics until one reaches an cubic that can be understood by simple algebra, such as the Fermat cubic $V(x\_0^3 + x\_1^3 + x\_2^3 + x\_3^3)$. This is formalized by introd... | https://mathoverflow.net/users/41873 | If $X$ is a degree 3 smooth integral surface in $P^N$, $N > 3$, is it still true that it contains 27 lines? | If $X\subset \mathbb P^n$ is a smooth and non-degenerated variety then
$$
\deg X \ge 1 + \mathrm{codim} X
$$
as you can learn from [Varieties of Minimal Degree](https://www.msri.org/~de/papers/pdfs/1987-001.pdf) by Eisenbud and Harris.
Thus to understand smooth cubic surfaces, it suffices to consider
cubic surfac... | 12 | https://mathoverflow.net/users/605 | 261393 | 117,785 |
https://mathoverflow.net/questions/246986 | 5 | The Ray-Singer paper "R-torsion and the Laplacian on Riemannian manifolds" claimed that one may prove the metric invariance of analytical torsion by forming a homotopy between metric $\rho\_{0},\rho\_{1}$ using $\rho\_{\mu}=\mu\rho\_{0}+(1-\mu)\rho\_{1}$. Then one differentiate
$$
\frac{\partial }{\partial u}f(u,s)=\fr... | https://mathoverflow.net/users/18850 | On Ray-Singer's proof of the metric invariance of analytical torsion | Ray and Singer state their independence result in Thm 2.1 for closed manifolds and give a full proof. This uses Proposition 6.1, which is formulated for manifolds with boundary, and the result slightly differs from
the one used in chapter 2. However, the only difference is that $\alpha d\delta$ occurs instead of $d\del... | 5 | https://mathoverflow.net/users/70808 | 261394 | 117,786 |
https://mathoverflow.net/questions/261392 | 1 | I'm trying to understand better what it means for a monad to be finitary. I know that Lawvere theories correspond to finitary monads, but I don't really understand the definition in terms of filtered colimits.
Normal lists satisfy
$$L(X) = 1 + X\cdot L(X).$$
Now suppose that at position n in the list, we can use ... | https://mathoverflow.net/users/756 | Do "factoradic" lists form a finitary monad? | Yes, it is finitary. If you think about $X$ as a set of variable and about $T(X)$ as a set of structures of some sort, then $T : Set \to Set$ is finitary if every structure $t \in T(X)$ can use only finite number of variables (roughly speaking). Every list uses only a finite number of variables (up to its length), so d... | 0 | https://mathoverflow.net/users/62782 | 261396 | 117,787 |
https://mathoverflow.net/questions/261411 | 11 | This question might be too elementary for MO, in which case I would gladly move it to math.stackexchange.com
Consider [Tarski's axiomatization](https://en.wikipedia.org/wiki/Tarski's_axioms) of Euclidean Geometry. It is stated in the wikipedia page linked and many other places that Tarski proved this first-order the... | https://mathoverflow.net/users/9317 | What is needed to prove the consistency of Tarski's Euclidean geometry? | In 1999, Harvey Friedman [showed](https://u.osu.edu/friedman.8/files/2014/01/ConsRCF85B15D.23.99-1zxwltk.pdf) how to prove the consistency of Tarski's axioms for geometry in EFA. This is Elementary Function Arithmetic, otherwise known as $I\Delta\_0(exp)$, a subtheory of PRA with functions bounded by towers of exponent... | 16 | https://mathoverflow.net/users/nan | 261416 | 117,790 |
https://mathoverflow.net/questions/261423 | 15 |
>
> Is there a torsion-free abelian group $A$ such that $A \not \simeq A \oplus \Bbb Z \simeq A \oplus \Bbb Z \oplus \Bbb Z$ (as groups)?
>
>
>
Notice that $\Bbb Z$ is [not cancellable](https://math.stackexchange.com/questions/349826), so
$A \oplus \Bbb Z \simeq (A \oplus \Bbb Z) \oplus \Bbb Z$ doesn't imply th... | https://mathoverflow.net/users/84923 | Torsion-free abelian group $A$ such that $A \not \simeq A \oplus \Bbb Z \simeq A \oplus \Bbb Z^2$ | $\mathbb{Z}$ is cancellable for *abelian* groups. This was proved in the 1950s by Walker and Cohn (independently) and is often called "Walker's cancellation theorem". The proof is only a few lines.
So if $A$ is an abelian group with $A\oplus\mathbb{Z}\cong A\oplus\mathbb{Z}^2$, then $A\cong A\oplus\mathbb{Z}$.
| 17 | https://mathoverflow.net/users/22989 | 261426 | 117,796 |
https://mathoverflow.net/questions/261410 | 3 | Let $z\_1,z\_2,\dots,z\_n\in\Bbb{C}$ be distinct and $w\_1,w\_2,\dots,w\_n\in\Bbb{C}$ be arbitrary. Suppose $f, g$ are two polynomials of degree less than $n$ such that
$$f(z\_j)=w\_j,\qquad g(z\_j)=\bar{w}\_j \qquad\text{for $1\leq j\leq n$}.$$
Define $\Omega(z)=\prod\_{j=1}^n(z-z\_j)$. The following puzzles me.
> ... | https://mathoverflow.net/users/66131 | complex polynomials and inequalities | For $n\ge 3$, let's take $z\_j$, for $1\le j\le n$, be the $n$-th roots of unity, and $w\_j:=z\_j+4\bar{z\_j}$, so that the assumption are satisfied by $$f(z):=z+4z^{n-1}$$ $$g(z):=4z+z^{n-1}.$$ However,
$$ |g'(z\_k)|=|4+(n-1) z\_k^{-2}| \le n+3<\phantom{Z} $$ $$ \phantom{ZZZZ} <4n-5\le|1+4(n-1) z\_k^{-2}|= |f'(z\_k)|... | 7 | https://mathoverflow.net/users/6101 | 261430 | 117,798 |
https://mathoverflow.net/questions/261414 | 8 | In my research, I found two combinatorial identities. Mathematica can give the answers immediately, but I don't know how to prove them. Could someone help me? Thank you!
Here are they:
Let $\alpha$, $\beta$ be two arbitrary complex numbers, and $k$, $l$ be two positive integers, then:
\begin{align\*}
\sum\_{m=1}^... | https://mathoverflow.net/users/60695 | Two combinatorial identities | At first, I reformulate your identity (the first identity, the second is the same up to change of variables, as you note in the post).
Denote $(\alpha+\beta)/\beta=\lambda$ and divide both parts by $\binom{\lambda k}k \binom{\lambda l}l$. We get an equivalent identity
$$
\sum\_{m=0}^k m\frac{k(k-1)\dots (k-m+1)\cdo... | 6 | https://mathoverflow.net/users/4312 | 261434 | 117,800 |
https://mathoverflow.net/questions/261433 | 2 | Can we get a closed form for the following series:
$$\sum\limits\_{x=1}^c \dfrac{(x+c-1)!}{x!} {}\_2F\_1(x+c,x,x+1,z)$$
where $c$ is a positive integer and $z$ is a real number less than -1?
Any suggestions or hints are appreciated.
| https://mathoverflow.net/users/nan | Partial sum of Hypergeometric2F1 function | $$\sum\limits\_{x=1}^c \dfrac{(x+c-1)!}{x!} {}\_2F\_1(x+c,x,x+1,z)=\frac{(c-1)!}{(z-1)^{2c-1}}P\_c(z),$$
where $P\_c(z)$ is a polynomial in $z$ of degree $2c-2$, the first few are
$$P\_1(z)=1$$
$$P\_2(z)=z^2-4 z+5$$
$$P\_3(z)=z^4-6 z^3+16 z^2-24 z+19$$
$$P\_4(z)=z^6-8 z^5+29 z^4-64 z^3+97 z^2-104 z+69$$
$$P\_5(z)=z... | 4 | https://mathoverflow.net/users/11260 | 261435 | 117,801 |
https://mathoverflow.net/questions/261422 | 1 | Let $X$ be a topological vector space (not necessarily Hausdorff), with topology $\tau$, and $M, N$ linear subspaces of $X$. Let $\pi:X \rightarrow X/N$ be the quotient map, which associates to each $x \in X$ the coset $x + N \in X/N$, and consider the linear subspace $\pi(M)$ of $X/N$.
Define the map $\Phi: M /(M \c... | https://mathoverflow.net/users/99197 | Subspaces of Quotient Spaces | I develop here the suggestion given by Bill Johnson in the comment above.
Take $X=\ell^2$, and let $M$ be the subspace all the complex sequences $x=(x\_0, x\_1,x\_2, \dots) \in \ell^2$ such that $x\_{2n}=0$ for all non-negative integers $n$, and let $N$ be the subspace of all complex sequences $x=(x\_0, x\_1,x\_2, \... | 1 | https://mathoverflow.net/users/99197 | 261441 | 117,804 |
https://mathoverflow.net/questions/261436 | 5 | Let $X\neq \emptyset$. A set $L\subseteq {\cal P}(X)$ is said to be a *projective plane* on $X$ if the following conditions are met:
1. if $x\neq y\in X$ there is a unique $l\in L$ such that $x, y \in l$,
2. if $l\neq m \in L$, then we have $|l\cap m | = 1$,
3. there are four distinct elements of $X$ such that no mem... | https://mathoverflow.net/users/8628 | Non-isomorphic projective planes on $\omega$ | You ask for the number of isomorphism classes of projective planes on $\omega$. I claim that it is exactly $2^{\aleph\_0}$.
**It is at most $2^{\aleph\_0}$.**
Indeed, a projective plane on $\omega$ can be encoded by the set of $\{x,y,z\}\subset\omega$ of cardinal $3$ which are aligned (i.e., such that $z$ lies on t... | 11 | https://mathoverflow.net/users/17064 | 261442 | 117,805 |
https://mathoverflow.net/questions/254049 | 5 | Let $X$ be a set and let $\text{End}(X)$ be the set of all functions $f:X\to X$. We say that $f, g\in \text{End}(X)$ *commute* if $g\circ f = f\circ g$, and $S\subseteq \text{End}(X)$ is a *commuting subset* if any two members of $S$ commute.
Using Zorn's Lemma, a routine verification shows that for any commuting sub... | https://mathoverflow.net/users/8628 | Maximal commuting subsets of $\text{End}(X)$ | Here is a concrete way to construct examples of maximal commuting monoids of different cardinalities: As Anthony Quas said: we can get a maximal monoid of cardinality $|X|$
This can be generalized in the following way: let $G$ be an abelian group. Let $X$ be the $G$-set $G\sqcup Y$ where the action of $G$ on $Y$ is t... | 1 | https://mathoverflow.net/users/41644 | 261444 | 117,806 |
https://mathoverflow.net/questions/261138 | 1 | Let $A$ be a Poisson super algebra ($A$ is a super algebra and $A$ satisfies super Jacobi identity, super commutativity, super Leibniz rule).
Super version of the product of two tensor products is
\begin{align}
(x \otimes y)(x' \otimes y') = (-1)^{|x'||y|} x x' \otimes y y', \ x, y,x',y' \in A.
\end{align}
Are the... | https://mathoverflow.net/users/11877 | Super version of Poisson brackets of tensor products | To eliminate this from unanswered questions, let me elaborate on Gabriel's answer.
First of all, you should always remember that in the case of superalgebras all signs appear from the Koszul sign rule. Second, the tensor product of Poisson algebras is a Poisson algebra because the Poisson operad is a Hopf operad. Thi... | 2 | https://mathoverflow.net/users/1306 | 261448 | 117,808 |
https://mathoverflow.net/questions/261449 | 2 | Let $S\_n$ act on the set of boolean functions of size $n$ in the following way:
If $f$ is a boolean function and $\alpha \in S\_n$, then $g=\alpha f$ and $g(x)=f(\alpha(x))$ where $x$ is boolean vector of size $n$;
Let $p$ be the number of orbits. Since each orbit can have no more than $n!$ elements it is obvious ... | https://mathoverflow.net/users/42586 | Symmetric group acting on the set of boolean functions | The number of orbits of a two-cycle $\sigma$ acting on subsets of
$\{1,\dots,n\}$ is $3\cdot 2^{n-2}$. If follows that the number of orbits of
$\sigma$ acting on boolean functions is $2^{2^{n-1}}\cdot 3^{2^{n-2}}$.
By the Cauchy-Frobenius lemma
(a.k.a. Burnside's lemma), it is then easy to see that
$$ p =\frac{1}{n!}\... | 5 | https://mathoverflow.net/users/2807 | 261451 | 117,809 |
https://mathoverflow.net/questions/242455 | 7 | I found this question when I tried to figure out what kind of subgroups of a free abelian group behave just as well as in the finitely generated case.
Let $M$ be a free abelian group and $N$ a subgroup of $M$. Suppose that $M/N$ is a direct sum of cyclic groups. Can we always find a basis $\{x\_i\}\_{i\in I}$ of $M$ ... | https://mathoverflow.net/users/69184 | On describing a sort of "well-behaved" subgroups of a free abelian group | It is proved in
*Joel M. Cohen and Herman Gluck*, MR 254028 [**Stacked bases for modules over principal ideal domains**](http://www.ams.org/mathscinet-getitem?mr=254028), *J. Algebra* **14** (1970), 493--505,
that the answer is yes. They credit Kaplansky for asking the question. The paper also contains an example w... | 2 | https://mathoverflow.net/users/22989 | 261489 | 117,818 |
https://mathoverflow.net/questions/261490 | 2 | Let $T$ be a compact operator on a Hilbert space. Let $(\lambda\_n)$ be the sequence of its eigenvalues repeated with their algebraic multiplicity. The Theorem of Lidskii says that if $T$ is trace class, then the sum $\sum\_n\lambda\_n$ converges absolutely and equals the trace of $T$. Now my question is for the conver... | https://mathoverflow.net/users/nan | Converse for Lidskii's Theorem | No, this is not true. Take for example the following operator on $l^2(\mathbb{N})$. $$T(e\_{2n})=0,T(e\_{2n+1})=\frac{1}{\sqrt{n}}e\_{2n}$$One can check easily that $T^2=0$, so all $\lambda\_n=0$ but $\mu\_{2n}=0,\;\mu\_{2n+1}=\frac{1}{n}$. Hence $T$ is compact but not trace class.
| 12 | https://mathoverflow.net/users/89956 | 261491 | 117,819 |
https://mathoverflow.net/questions/261469 | 4 | My question is whether the following has been considered as an axiom, and if so, where I may find a discussion of it:
>
> $\Xi : \prod\_{A : \mathcal{U}} \|A\| \to A$.
>
>
>
For example, using this axiom, we can define the $n$-truncation of a type $A$ in the following way:
Let
>
> $P(A) :\equiv \sum\_{X... | https://mathoverflow.net/users/25477 | The axiom $\Xi : \prod_{A:\mathcal{U}} \|A\| \to A$ and the $n$-truncation of a type | This rule contradicts the univalence axiom.
Indeed, first note that it implies the axiom of choice:
$$ \prod\_{x : A} \| B x \| \to \| \prod\_{x : A} B x \|.$$
$\Xi$ implies that $\prod\_{x : A} \| B x \| \to \prod\_{x : A} B x$. Hence, the axiom of choice holds.
The axiom of choice implies the law of excluded middle... | 6 | https://mathoverflow.net/users/62782 | 261492 | 117,820 |
https://mathoverflow.net/questions/261478 | 9 | this is my first thread on mathoverflow, and apologies if this is a trivial question.
In Dwyer and Spalinski's *Homotopy theories and model categories*, they gave the definition of derived functors, and based on their definition, we require the source **C** of a functor *F* : **C** $\to$ **D** to be a model category,... | https://mathoverflow.net/users/95608 | Does the existence of a derived functor imply existence of model structure? | This is not quite the answer to your question as you pose it. I hope it will be useful anyway. By and large I am just expanding user337830 comments. Everything will use homological grading (what can I say, I am a homotopy theorist :)).
The relationship between derived functors in homological algebra and derived funct... | 3 | https://mathoverflow.net/users/43054 | 261502 | 117,822 |
https://mathoverflow.net/questions/261479 | 13 | Let $G$ be a reasonably nice group, say residually finite if need be.
We may consider the group algebra $\mathbb{Z}[G]$.
Let $\widehat{\mathbb{Z}[G]} := \varprojlim\_I\mathbb{Z}[G]/I$ be the profinite completion, where $I$ runs over all ideals of finite index.
Let $\widehat{\mathbb{Z}}[[\widehat{G}]] := \varprojl... | https://mathoverflow.net/users/88840 | Difference between the completed group algebra and the profinite completion of a group ring | It seems to me that these are indeed isomorphic. Namely, if $n$ is an integer and $U$ is a finite index normal subgroup of $G$, then the kernel $I\_{n,U}$ of the natural map $\mathbb ZG\to \mathbb Z/n[G/U]$ is an ideal of $\mathbb ZG$ of finite index and hence part of the inverse system defining $\widehat{\mathbb ZG}$.... | 8 | https://mathoverflow.net/users/15934 | 261504 | 117,823 |
https://mathoverflow.net/questions/261503 | 3 | Working in Martin-Löf intensional type theory, $\mathsf{ITT}$, with a universe $U$ within the theory closed under all the usual constructors, the univalence of the canonical family $\mathsf{EI}$ over $U$ can (via the Curry-Howard correspondence) be regarded as a type $\mathsf{isUnivalent}(U)$.
>
> Is $\mathsf{isUni... | https://mathoverflow.net/users/nan | Is the univalence of the canonical family over a universe small? | Let me first try to make the question precise. Suppose $U$ is a universe. The statement "$U$ is univalent" which you are asking about is
$$\prod\_{A,B : U} \mathsf{isEquiv}(\mathsf{idtoeq}\_{A,B}) \tag{1}$$
where
$$\mathsf{idtoeq}\_{A,B} : \mathsf{Id}\_U(A,B) \to \mathsf{Equiv}(\mathsf{El}(A), \mathsf{El}(B))$$
is the ... | 4 | https://mathoverflow.net/users/1176 | 261505 | 117,824 |
https://mathoverflow.net/questions/261525 | 0 | Let $A$ be a Poisson $\mathbb{C}$-algebra: $A$ has a the structure of a complex commutative algebra and at the same time it carries the structure of a Lie algebra, with Lie bracket $\{\cdot, \cdot\}$. The latter should be a biderivation of the commutative multiplication, as well as the Lie multiplication.
Just as the... | https://mathoverflow.net/users/83211 | Are simple Poisson $A$-modules finitely generated as $A$-modules? | This is definitely not true; in fact, it's easy to construct counterexamples with $A$ finite dimensional.
For any Lie algebra $\mathfrak{g}$, you can define a Poisson algebra structure on $\mathbb{C}\oplus \mathfrak{g}$ such that $(a\_1+X\_1)(a\_2+X\_2)=a\_1a\_2+a\_1X\_2+a\_2X\_1$ for $a\_i\in \mathbb{C}$ and $X\_i\i... | 1 | https://mathoverflow.net/users/66 | 261527 | 117,829 |
https://mathoverflow.net/questions/261533 | 0 | I have an infinite series of independent identically distributed random variables $\{X\_i\}\_{i=1}^\infty$ which follows extreme value type I distribution which can be found [here] (<https://en.wikipedia.org/wiki/Gumbel_distribution>), then I was wondering what is the distribution of $Y:=\underset{i\geq 1}{\max}~X\_i$?... | https://mathoverflow.net/users/97186 | The distribution of the maximum of a series of extreme value type I random variable | For any $n$,
$$\Pr(Y\le n)=\Pr(X\_i\le n\,(\forall i))=\prod\_i\Pr(X\_i\le n)=\lim\_{i\to\infty}\Pr(X\_1\le n)^i=0.$$
So $Y=\infty$ with probability 1.
| 1 | https://mathoverflow.net/users/4600 | 261542 | 117,831 |
https://mathoverflow.net/questions/261509 | 4 | A Cat(0) metric space $(X,d)$ of constant and finite local dimension is approximately flat if there exists a dense subset $U\subset X$ such that every $x\in U$ has a flat neighborhood (i.e. isometric to the euclidean space).
I'm interested in a generalization of Radon's theorem (<https://matthewhr.wordpress.com/2013... | https://mathoverflow.net/users/99511 | Generalization of Radon's theorem | In dimension 1, Radon's theorem says that for any 3 points on the real line, one of them belongs to the segment between the two others. This becomes false if one replaces the real line by the tripod (the tree with 3 leaves), which is a 1-dimensional CAT(0) space. Indeed, none of the three leaf vertices of the tripod in... | 5 | https://mathoverflow.net/users/4354 | 261545 | 117,832 |
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