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https://mathoverflow.net/questions/290464 | 0 | In Fourier theory, the pair composed of a variable and its Fourier transform is called *conjugate variables,* and one crucial property between the two is the uncertainty relation. This relation tells us that e.g. if one variable/function has a bounded (or compact?) support, its Fourier transform *cannot* have a bounded... | https://mathoverflow.net/users/115841 | Does the uncertainty relation of Fourier transforms also extend to linear operators? | Yes, the Heisenberg uncertainty principle literally is the Fourier uncertainty relation. For simplicity consider the one-dimensional case.
The Hilbert space of a spinless one-dimensional particle is $L^2(\mathbb{R})$ and position is represented by $Q = M\_x$, the operator of multiplication by $x$. Whereas momentum is... | 3 | https://mathoverflow.net/users/23141 | 290470 | 128,055 |
https://mathoverflow.net/questions/290471 | 2 | Who first used the word "homomorphism" to describe a link between two similar structures ?
Following this, who specialized this concept with the words:
- "Isomorphism";
- "Endomorphism";
- "Automorphism" ?
Gérard Lang
| https://mathoverflow.net/users/30395 | Who first used the word "Homomorphism"? | According to Jeff Miller's [Earliest known uses of Some of the Words of Mathematics](http://jeff560.tripod.com/mathword.html "Earliest Known Uses of Some of the Words of Mathematics"),
>
> HOMOMORPHISM is found in English in 1935 in the Duke Mathematical Journal [OED].
>
>
>
A [Google search](https://books.go... | 3 | https://mathoverflow.net/users/35609 | 290473 | 128,057 |
https://mathoverflow.net/questions/290340 | 6 | [Sagan and Savage](http://users.math.msu.edu/users/sagan/Papers/Old/cib-pub.pdf) gave a combinatorial interpretation of a polynomial generalization of [Fibonomial coefficients](https://en.wikipedia.org/wiki/Fibonomial_coefficient). Their proof uses the recurrence relation for the Lucas polynomials that generalize the F... | https://mathoverflow.net/users/113161 | Direct bijections for $s,t$-Fibonomial identities | I'm glad to see you're interested in this question. A number of people have thought about it including Art Benjamin and myself. The person who seems to have gotten the furthest is Curtis Bennett who came up with a new way to view the rectangular tilings in terms of tilings of a triangle with a lattice path through it w... | 10 | https://mathoverflow.net/users/119492 | 290477 | 128,059 |
https://mathoverflow.net/questions/290474 | 8 | For a prime number $p$, let $\Phi(p)$ be the subset of $\{ 1, 2, \ldots, p-1 \}$ consisting of primitive roots modulo $p$. (Thus $\# \Phi(p) = \phi(p-1)$, where $\phi$ denotes the totient.)
I am curious about the distribution of $\Phi(p)$ as $p$ varies. I imagine that this is classic analytic number theory, but I'm ... | https://mathoverflow.net/users/3545 | Distribution of primitive roots, as p varies | Let me adjust your quantity slightly as
$$\tilde D\_p(f) := \frac{1}{\phi(p-1)}\sum\_{x \in \Phi(p)} f \left( \frac{x}{p} \right),$$
and let me initially impose the natural condition $f(0)=f(1)$. Then I claim that
$$\lim\_{p\to\infty}\tilde D\_p(f)=\int\_0^1 f(t)\,dt.$$
It suffices to show the claim when $f(t)=e^{2\p... | 14 | https://mathoverflow.net/users/11919 | 290478 | 128,060 |
https://mathoverflow.net/questions/279513 | 6 | I've been looking over Lurie's DAG X, and he introduces a combinatorial construction called the *twisted arrow construction* for simplicial sets that generalizes the following ordinary categorical notion:
The Yoneda embedding $\mathcal{C}\to \widehat{\mathcal{C}}$ is adjunct under the hom-tensor adjunction to the fun... | https://mathoverflow.net/users/1353 | Twisted-arrow construction for 2-categories | While it is more explicit and combinatorial, this probably isn't precisely what you're looking for. It does seem relevant to your question, though. A year or so ago I worked out a 2-categorical analogue of the $(\infty,1)$ twisted arrow construction (I'm working with $(\infty,1)$-categories arising as Joyal fibrant rep... | 3 | https://mathoverflow.net/users/89763 | 290497 | 128,065 |
https://mathoverflow.net/questions/290220 | 7 | Define $A=(a\_n)$ and $B=(b\_n)$ as follows: $a\_0=1$, $a\_1=2$, $b\_0=3$, $b\_1=4$, and $$a\_n=a\_0b\_{n-1}+a\_1b\_{n-2}$$ for $n \geq 2$, where $A$ and $B$ are increasing and every positive integer occurs exactly once in $A$ or $B.$ Can someone prove that $\lim\_{n \to \infty} a\_n/n = 4$?
Here are first terms and... | https://mathoverflow.net/users/61426 | Limit associated with complementary sequences | *[edit 12/01/2018: a remark]*.
The sequence $a\_n-4n$ being unbounded also rules out any polynomial recurrence of the form
$$a\_{n+r+1}=P(a\_{n+1},\dots,a\_{n+r})$$
for $P\in\mathbb{C}[x\_1,\dots,x\_{r}]$.
Otherwise $b\_n:=a\_{n+1}- a\_{n}$ would also satisfy a polynomial recurrence
$$ b\_{n+s+1}=Q(b\_{n+1},\dots,b... | 6 | https://mathoverflow.net/users/6101 | 290502 | 128,066 |
https://mathoverflow.net/questions/290495 | 3 | On the Wikipedia page regarding the [Painlevé transcendents](https://en.wikipedia.org/wiki/Painlev%C3%A9_transcendents#History) it says:
>
> Émile Picard pointed out that for orders greater than 1, movable essential singularities can occur, and found a special case of what was later called Painleve VI equation
>
>... | https://mathoverflow.net/users/43020 | Reference request - existence of movable essential singularities | Picard discusses differential equations with fixed singularities in the end of Chapter V (pp. 291-300). The special case of Painleve VI that he discovered is
written in the end on p. 299. Example of a second order equation whose solutions
have movable essential singularities is given on p. 291. He does not care to writ... | 8 | https://mathoverflow.net/users/25510 | 290504 | 128,067 |
https://mathoverflow.net/questions/290461 | 7 | Let $D<0$ be a fundamental discriminant and consider the theta series $$\vartheta\_Q(\tau)=\sum\_{v\in\mathbb{Z}^2} q^{Q(v)}$$
associated to a quadratic form $Q$ of discriminant $D$. It appears to be true that $\vartheta\_Q=\vartheta\_Q|U(|D|)$, where $U$ is the usual Atkin U-operator, defined by its action on $q$-seri... | https://mathoverflow.net/users/49340 | (Binary) Theta functions and Atkin's U-operator | I think you want to show that in each ideal class the number of ideals of norm a is equal to the number of ideals of norm $a \cdot |D|$. This should follow directly from the following 2 facts.
(1). All primes $p$ dividing $|D|$ ramify in the ring of integers of $\mathbb{Q}(\sqrt{D})$.
(2). The unique ideal of norm ... | 7 | https://mathoverflow.net/users/6214 | 290512 | 128,071 |
https://mathoverflow.net/questions/290522 | 47 | This might be a very naive question. But what is quantum algebra, really?
[Wikipedia](https://en.wikipedia.org/wiki/Quantum_algebra) defines quantum algebra as "one of the top-level mathematics categories used by the arXiv". Surely this cannot be a satisfying definition. The arXiv admins didn't create a field of math... | https://mathoverflow.net/users/36146 | What is quantum algebra? | Quantum algebra is an umbrella term used to describe a number of different mathematical ideas, all of which are linked back to the original realisation that in quantum physics, one finds noncommutativity. The areas now encompassed by the term "quantum algebra" are not necessarily directly or obviously related to each o... | 50 | https://mathoverflow.net/users/13215 | 290525 | 128,075 |
https://mathoverflow.net/questions/290528 | 8 | I had asked [this](https://math.stackexchange.com/questions/2601034/in-a-non-compact-metric-space-topologically-transitivity-need-not-imply-onto) question on [Mathematics Stack Exchange](https://math.stackexchange.com/) yesterday but it got no response so I'm asking here.
---
Let $X$ be a compact metric space and... | https://mathoverflow.net/users/119514 | In a non-compact metric space, topological transitivity need not imply onto | By [Birkhoff's theorem](http://home.iitk.ac.in/~chavan/linear_dynamics.pdf), a bounded linear operator on a Banach space is topologically transitive if and only if it is [hypercyclic](https://en.wikipedia.org/wiki/Hypercyclic_operator). Charles Read has developed a whole machinery for constructing non-surjective, hyper... | 11 | https://mathoverflow.net/users/15129 | 290529 | 128,076 |
https://mathoverflow.net/questions/290432 | 4 | Consider the Hilbert space $L^2\_w$ with scalar product $\langle f,g\rangle\_w =\int\_0^\infty f(x)g(x)w(x)dx$ where the weight $w$ is the density function of a log-normal distribution
$$ w(x)=\frac{1}{\sqrt{2\pi}\sigma x}e^{-\frac{(\ln(x)-\mu)^2}{2\sigma^2}},$$
for some $\mu\in\mathbb{R}$ and $\sigma>0$.
The log-norm... | https://mathoverflow.net/users/119453 | Closure of polynomials in $L^2_w$ with log-normal weight function | By squaring this is the same question as is $X$ in the span of $1, X^2, X^4,...$. I will produce a function that has the property that $E(FX^{2j}) = 0, E(FX) \ne 0$. Set $Z = log(X)$ which is a normal r.v. Regarding $F$ as a function of Z, $E(FX^{2j}) = 0$ is equivalent to $E(F(Z+2j) ) = 0$ since $X^{2j} e^{ - 2j^2} $ ... | 0 | https://mathoverflow.net/users/nan | 290530 | 128,077 |
https://mathoverflow.net/questions/290516 | 8 | Let $k=\mathbb{F}\_q$ where $q$ is a prime power of odd cardinality.
**Where could I find explicit models of all irreducible cuspidal (complex) representations of $GL\_n(k)$ for $n\ge 3$?**
I understand that the characters of such representations was constructed by J.A. Green “The characters of the finite general l... | https://mathoverflow.net/users/13466 | Reference request: Models of cuspidal representations of GL(n,k) where k is a finite field | For the finite groups GL$\_n(\mathbb{F}\_q)$ there is an early paper by Lusztig well worth checking out [*here*](https://mathscinet.ams.org/mathscinet-getitem?mr=0382419).
This predates his broader work on finite groups of Lie type with Deligne (1976), where they found a way to construct virtual characters using $\ell$... | 7 | https://mathoverflow.net/users/4231 | 290537 | 128,080 |
https://mathoverflow.net/questions/290182 | 11 | Kuperberg's [Knottedness is in $\mathsf{NP}$, modulo GRH](https://arxiv.org/pdf/1112.0845.pdf) provides a certificate that a knot $K$ given by a knot diagram on $c$ crossings is not trivial. The certificate is a prime $p$, along with a solution in $\mathbb{Z}/p$ to a system of $m$ polynomial equations on $n$ variables ... | https://mathoverflow.net/users/8927 | How hard is it to guess Kuperberg's certificate of knottedness? | This question is probably very open if we want to be rigorous. With the roughest heuristics, the chance $p$ divides a polynomial value $f\_i$ is $1/p$, and so the chance of recovering $p$ is $1/p^n$ where $n$ is the number of polynomials. Now if we sum over all primes and $n>2$ the expected number of solutions is less ... | 3 | https://mathoverflow.net/users/6084 | 290551 | 128,086 |
https://mathoverflow.net/questions/290555 | 0 | Where I can find a proof of the following statement:
,,We have two categories: $C$ and $C'$. If they are equivalent, then geometric realizations of its nerves are homotopy equivalent"?
| https://mathoverflow.net/users/119526 | Homotopy equivalence of nerves | Geometric realization is a functor, so a functor $C \to C^\prime$ induces a morphism of realizations. Since a natural transformation is a functor $C \times (\bullet \to \bullet) \to C^\prime$, any natural transformation induces a homotopy. So you only need an adjunction for an equivalence of realizations. More generall... | 3 | https://mathoverflow.net/users/10605 | 290557 | 128,089 |
https://mathoverflow.net/questions/290532 | 7 | Let $C$ be a curve over $k$ and $w\_C$ it's dualizing sheaf.
If $dim\_k H^0(C, \mathcal{O}\_C) =1$ and $g:= H^0(C, \mathcal{O}\_C)$ the arithmetic genus one easy computes $$deg(w\_C) = 2g-2$$ where $deg$ is the map $deg:Pic(C) \to \mathbb{Z}, \mathcal{L} \mapsto \chi(\mathcal{L}) - \chi(\mathcal{O}\_C)$.
Fothermore, ... | https://mathoverflow.net/users/108274 | Relation Degree of Dualizing Sheaf and Euler Characteristic | Another explanation for the equality $\mathrm{deg}(\omega\_C) = -e(C)$, complementing that contained in Javier-Alvarez's MSE exposition linked by Francesco Polizzi, is *Hodge theory*. In short, Hodge theory gives an instance and a precise statement of how topological invariants are related to algebraic/holomorphic inva... | 3 | https://mathoverflow.net/users/37821 | 290558 | 128,090 |
https://mathoverflow.net/questions/289943 | 0 | 1. Is the following schema equivalent to the axiom schema of Replacement over the rest of axioms of $ZF$ [equality axioms, full versions of Pairing, Union and Power; Infinity, Foundation, Extensionality].
Scheme: If $\phi$ is a formula in which only symbols $``x",``z"$ occur free, then the following sentence is an ax... | https://mathoverflow.net/users/95347 | Equivalents of Replacement under removal of Extensionality? | I think I have the solution to this in my head, It is to interpret in $\text{ZF-Ext.}$ (with the last version of replacement) the theory $\text{ZFA}$ which in turn interpret $\text{ZF}$.
The crux of the proof is to use Marcel Crabbe` approach to interpreting equality and membership used in the equalization of $\text... | 1 | https://mathoverflow.net/users/95347 | 290567 | 128,091 |
https://mathoverflow.net/questions/290352 | 1 | Let $V\subseteq\mathbb{P}^5$ be the Veronese surface and $C\_V\subseteq\mathbb{P}^6$ the cone over it. Let $F\subseteq\mathbb{P}^5$ be the scorll $\mathrm{Proj}\_{\mathbb{P}^1}(\mathcal{O}(2)+\mathcal{O}(2))$ embedded with the relative $\mathcal{O}(1)$, and $C\_F$ the cone over it. If we cut $C\_V$(resp. $C\_F$) with a... | https://mathoverflow.net/users/117078 | An example of simultaneous Du Val resolution | Let's start with the second set of questions, because that partially explains what happens in the first.
**Why [is] the singularity at the vertex of $C\_V$ the same as $\mathbb C^3/(x∼−x)$?**
Well, the Veronese is the image of the map taking the degree $2$ monomials in $3$ variables, so its homogenous coordinate ri... | 1 | https://mathoverflow.net/users/10076 | 290593 | 128,098 |
https://mathoverflow.net/questions/290582 | 7 | Given a $n\times n$ symmetric random matrix whose diagonal elements are all fixed as $1$. In addition, there are $k$ $1$s will be randomly scattered in upper triangular (of course, the corresponding places in the lower-triangle will be filled with $1$, and $2k < n^2-n$). All other elements are independent uniform rando... | https://mathoverflow.net/users/70424 | Bound for largest eigenvalue of symmetric matrices of uniform random variables over $[0,1]$ and fixed $1$s along diagonal and scattered $1$s | The diagonal elements just shift the spectrum (and the top eigenvalue) by $1$. So we may assume they are $0$.
You are essentially dealing with a symmetric matrix whose entries above the diagonal are iid, sum of a Bernoulli $\{0,1\}$ of parameter (=mean) $q=2k/n^2$ and of a uniform random variable on $[0,1]$. The mea... | 3 | https://mathoverflow.net/users/35520 | 290598 | 128,099 |
https://mathoverflow.net/questions/290412 | 32 | I was going over P. Scholze's [paper on $p$-adic Hodge Theory for rigid analytic varieties](http://www.math.uni-bonn.de/people/scholze/pAdicHodgeTheory.pdf).
This question is around the "**Poincaré Lemma**" in the paper.
Throughout, let $X$ be a proper smooth rigid analytic variety over $\mathbf{Q}\_p$, of pure dim... | https://mathoverflow.net/users/nan | $p$-adic Hodge Theory for rigid spaces, after P. Scholze | Let me start with the second question first:
The usual de Rham complex is not locally acyclic in positive degrees, in any of the topologies (analytic (= of rational subsets), étale, pro-étale, ...). The problem is that rigid-analytic spaces are not "locally contractible". For example, on the annulus
$$
\mathbb T = \{... | 43 | https://mathoverflow.net/users/6074 | 290614 | 128,104 |
https://mathoverflow.net/questions/290611 | 2 | Let $X$ be a smooth quasi-projective variety over a field $k$, with pure dimension $d$, $K/k$ an arbitrary field extension.
* For any algebraic cycle $\eta$ of codimension $1$ on $X\_K$ ($\eta\in Z^1(X\_K)$), does there exist an algebraic cycle $\xi\in Z^1(X)$ (ie. defined over $k$) such that $\xi\_K-\eta$ is effecti... | https://mathoverflow.net/users/nan | Effective cycles of codimension 1 and field extensions | The answer to the both questions is no. Consider $X=\mathbb{A}^1$, $k=\mathbb{Q}$, and $K=\mathbb{C}$ (any transcendental extension will do here). Let $\eta=\{\pi\}$. The cycles on $X\_K$ of the form $\xi\_K$ are precisely the finite Galois-stable linear combinations of elements of $\bar{\mathbb{Q}}\subset\mathbb{C}$, ... | 5 | https://mathoverflow.net/users/5263 | 290619 | 128,107 |
https://mathoverflow.net/questions/290626 | 11 | While doing some estimates for PDEs I came across the following equation:
>
> $$
> y'(t) = \alpha(t) + \left( \int\_0^t y(\tau) \, d\tau\right)^\gamma, \qquad t \in [0,1]
> $$
>
>
>
where $\alpha \colon [0,1] \to \mathbb R$ is some given function and $\gamma \ge 1$ is a fixed real number. I have a non-negativ... | https://mathoverflow.net/users/100976 | An (hopeless) integro-differential equation | If I define $f(t)=\int\_0^t y(\tau)d\tau$, I need to solve
$$f''(t)=\alpha(t)+f(t)^\gamma.$$
For $\alpha\equiv 0$ this has the implicit solution
$${({\gamma}+1) f(t)^2 \left(c\_1 {\gamma}+c\_1+2 f(t)^{{\gamma}+1}\right)^2 \, \_2F\_1\left(\frac{1}{2},\frac{1}{{\gamma}+1};1+\frac{1}{{\gamma}+1};-\frac{2 f(t)^{{\gamma}+... | 15 | https://mathoverflow.net/users/11260 | 290628 | 128,110 |
https://mathoverflow.net/questions/208514 | 25 | This question is a crosspost of the second part of [this MSE question](https://math.stackexchange.com/questions/1284179/derived-functors-homotopical-vs-homological-approach).
In my first course in homological algebra, derived functors were defined in terms of universal $\delta$-functors. In the text *Homotopy Limits ... | https://mathoverflow.net/users/69037 | Derived functors - homotopical vs homological approach | **EDIT** Corrected a couple of inaccuracies and mistakes, added some references.
For the sake of clarity, let me work with non-negatively graded cochain complexes, and analyze the case of a left exact covariant functor, to prove that its right derived functor is a universal (covariant) cohomological $\delta$-functor.... | 14 | https://mathoverflow.net/users/119308 | 290629 | 128,111 |
https://mathoverflow.net/questions/290543 | 4 | **Question 1.** Can a reflexive Delzant polytope of some dimension contain a $2$-face with more than $11$ edges?
**Motivation.** I would like more generally to get an answer to the following question:
**Question 2.** Suppose $X$ is a smooth Fano variety with a $\mathbb C^\*$-action. Let $Y\subset X$ be the connecte... | https://mathoverflow.net/users/13441 | 2-faces of reflexive Delzant polytopes | Haase and Melnikov have proved [here](https://arxiv.org/abs/math/0406485) that every lattice polytope can be realized as a face of some reflexive polytope. They do this by an iterative procedure that increases the dimension by one until the polytope becomes reflexive. At each step the new polytope is a wedge over a fac... | 5 | https://mathoverflow.net/users/2384 | 290632 | 128,112 |
https://mathoverflow.net/questions/290631 | 5 | Let $W$ be the space of continuous functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $\lim\_{x\rightarrow \pm \infty} f(x)=0$, and consider the sup-norm topology on $W$.
**Problem.** does there exist $f\in W$ such that the set of translations of $f$ (i.e., the set of functions $f\_i(x)=f(x+i)$, $i\in \mathbb... | https://mathoverflow.net/users/81443 | Set of translations of a real function having a dense linear span | Yes. This was proved by [Atzmon and Olevski.](https://www.sciencedirect.com/science/article/pii/S0021904596901069)
| 7 | https://mathoverflow.net/users/3675 | 290636 | 128,113 |
https://mathoverflow.net/questions/290576 | 2 | Let's consider the following differential equation on $\mathbb{R}$:
$$-u''(x)+u(x)-V(x)u(x)=\lambda u(x),$$ where $\lambda<1$ and $V$ is a bounded.
We consider only that solution $u(x) \in C^1$ which decays exponentially as $x\to +\infty$.
I am interested in the behaviour of $\frac{u'(x)}{u(x)}$ as $\lambda \to -\i... | https://mathoverflow.net/users/95174 | the asymptotic behaviour of function as $\lambda \to -\infty$ | First of all, I think you should really absorb the stray $u$ on the left-hand side by either $Vu$ or $\lambda u$, so consider
$$
-u''+Vu =\lambda u ,
$$
under the same assumptions. Next, instead of talking about an exponentially decaying solution (whose existence you'd have to prove first), I think it would be much bet... | 3 | https://mathoverflow.net/users/48839 | 290647 | 128,119 |
https://mathoverflow.net/questions/290642 | 1 | I just came across the notion of [ends](https://en.wikipedia.org/wiki/End_(topology)) of a space, and I wonder if the following are equivalent for $G$ a locally finite connected graph:
1. There exists an infinite path $v\_1,v\_2,\dots$ in $G$ which hits every vertex at least once but not infinitely many times
2. $G$ ... | https://mathoverflow.net/users/36505 | Characterizing 1-ended graphs | I'll prove the direction that 2 implies 1.
Using that $G$ is connected and 1-ended, you can obtain an increasing sequence of connected, finite diameter subgraphs
$$G'\_0 \subset G'\_1 \subset G'\_2 \subset \cdots
$$
whose union is $G$, such that for all $i$, all but one component of the subgraph $G \setminus G'\_i$ h... | 4 | https://mathoverflow.net/users/20787 | 290651 | 128,122 |
https://mathoverflow.net/questions/290434 | 3 | Some of the second order ODE can be considered as Euler-Lagrange equations for an appropriate Lagrangian. However this is true not for arbitrary second order equation. But some of important equations of physics do satisfy this property which is called the least action principle.
I am interested in the following ODE i... | https://mathoverflow.net/users/119471 | Can one obtain this ODE as an Euler-Lagrange equation? | The Euler-Lagrange equations for the *time-dependent* Lagrangian $L(\mathbf{x}, \dot{\mathbf{x}}, t) = \frac{m\dot{\mathbf{x}} \cdot \dot{\mathbf{x}}}{2}e^t$ are precisely the ODEs you're after: $0 = \frac{\mathrm{d}}{\mathrm{d}t}\big( me^t \dot{\mathbf{x}} \big) = m e^t(\ddot{\mathbf{x}} + \dot{\mathbf{x}})$. For a so... | 2 | https://mathoverflow.net/users/43324 | 290652 | 128,123 |
https://mathoverflow.net/questions/290505 | 6 | This post was inspired by the [Square-Sum Problem](https://youtu.be/G1m7goLCJDY) presented in Numberphile by Matt Parker.
He asked about Hamiltonianness for $n=2$, and we ask about connectedness for all $n \in \mathbb{N}^\*$.
Given $n \in \mathbb{N}^\*$, let $\mathcal{G}\_n$ be the graph $(\mathbb{N}^\*,\{ \{a,b... | https://mathoverflow.net/users/34538 | Is the nth-power-sum graph connected? | Yes, the graph $\mathcal{G}\_n$ is connected, and its diameter is at most $n2^n$. To see this, write $s$ for $n2^{n-1}$, and fix any two vertices $a,b\in\mathbb{N}^\*$. By [Wright's solution](http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002375354) of Waring's problem with proportionality conditions, for a large p... | 6 | https://mathoverflow.net/users/11919 | 290658 | 128,125 |
https://mathoverflow.net/questions/290577 | 5 | Emerson and Meyer's Paper "Dualizing the Coarse Assembly Map" (2006) states the following Proposition (5.1):
*Let $X = [0,\infty)$ be the ray with its Euclidean metric coarse structure.
Then the reduced K-theory of the Higson compactification $\eta X$ of $X$ is uncountable.*
The authors state that this is proved in... | https://mathoverflow.net/users/78729 | Finding a proof within a paper: reduced $K$-theory of Higson compactification of $[0,\infty)$ is uncountable | It seems that Keesling's 1994 paper "The One-Dimensional Cech Cohomology of the Higson Compactification and Its Corona" contains the required result (Corollary 1). It may have been mis-cited.
| 2 | https://mathoverflow.net/users/78729 | 290668 | 128,130 |
https://mathoverflow.net/questions/290633 | -2 | Is there a meaningful way to transform logical equations (for instance $a \implies b$; $b \land a \implies c$ etc.) into geometrical representation in spaces such $\mathbb R^n$, $\mathbb C^n$ or manifolds?
I am curious if there is a way to analyze logic equations via transforming them into geometrical objects. Sorry ... | https://mathoverflow.net/users/119575 | Representing logic formulae in manifolds | One possible interpretation of your somewhat vague question is discussed in Steve Vickers' book 'Topology via Logic'. The interpretation is more on the level of open sets than `big things' like manifolds, but the discussion in the early parts of the book may be useful for you to help you reformulate your question at a ... | 3 | https://mathoverflow.net/users/3502 | 290670 | 128,132 |
https://mathoverflow.net/questions/290667 | 4 | Every semigroup containing an ideal subgroup is called a homogroup. Let $(S,\cdot)$ be homomgroup, hence it contains an ideal $I$ that is also a subgroup. It is easy to see that $I$ is the least ideal, a maximal subgroup of $S$, and its identity (denoted by $e\_I$) is a central idempotent of $S$. Now,
(1) Is the idea... | https://mathoverflow.net/users/40520 | Some questions about homogroups | The answers to these problems are the following:
(1) Yes: the ideal subgroup $I$ is unique. Indeed, if $H$ is another ideal subgroup, then $HI\subset H\cap I$, $H\cap I$ is a subgroup of $H$ and $I$, so $e\_I=e\_H$ and $HI\subset H\cap I\subset H\cup I\subset HI$ implies $H=I$.
(2,3,4) No: the semigroup $S=\{0,1\}... | 6 | https://mathoverflow.net/users/61536 | 290673 | 128,133 |
https://mathoverflow.net/questions/290650 | 3 | Suppose $C$ is a (non-singular) compact Riemann surface of genus $g$ and with $n$ (distinct) marked points $p\_1,\ldots,p\_n$. If we assume the stability condition ($2-2g-n<0$), then it is proved in "Geometry of Algebraic Curves (vol 2)" by Arbarello et al that the cohomology group $H^1(C,T\_C(-p\_1-\ldots-p\_n))$ para... | https://mathoverflow.net/users/110236 | Deformation Theoretic Interpretation of $H^1(C,T_C(-2p))$ | Given a collection of positive numbers $\{n\_i\}\_{i=1}^N$, you can consider the moduli space $\mathcal{M}$ parametrizing compact Riemann surfaces $C$ of genus $g$ together with $N$ marked points $p\_1,\dots,p\_N$ and an $(n\_i-1)$-jet of a coordinate at $p\_i$ for every $i$. For instance, a $0$-jet of a coordinate at ... | 5 | https://mathoverflow.net/users/18512 | 290683 | 128,136 |
https://mathoverflow.net/questions/290680 | 6 | Let $\mu(n)$ the Möbius function, we define $F:[0,1]\to[0,1]$ as $$F(x)=\sum\_{n=1}^\infty\frac{\mu(n)}{n}x^n.\tag{1}$$
For a function of this kind (I presume that this continuous function has image $[0,1]$) was defined, for example in last paragraph of page 986, what is a periodic point, and its corresponding order ... | https://mathoverflow.net/users/nan | What about of periodic points of $\sum_{n=1}^\infty\frac{\mu(n)}{n}x^n$, $0<x<1$, where $\mu(n)$ is the Möbius function? | I claim $F(x)<x$ for all $0<x<1$. From there it follows that the only periodic point is the fixed point $0$.
Let $m(x)=\sum\_{n\leq x}\frac{\mu(n)}{n}$. We have (see e.g. equation (5) [here](https://terrytao.wordpress.com/2009/08/30/an-elementary-inequality-involving-the-mobius-function/)) $m(x)\leq 1$ for all $x$, w... | 17 | https://mathoverflow.net/users/30186 | 290688 | 128,137 |
https://mathoverflow.net/questions/290640 | 0 | Let $\epsilon \in [0, \infty[$. Consider the following operator on $L^2(\mathbb{R})$:
\begin{equation}
H(\epsilon) = -\frac{d^2}{dx^2} + x^2 + \epsilon |x|.
\end{equation}
How does one show that the lowest eigenvalue of $H(\epsilon)$, denoted by $\lambda\_1(\epsilon)$ satisfies:
>
> \begin{equation}
> \epsilon \... | https://mathoverflow.net/users/115381 | Limit (at infinity) for the lowest eigenvalue of a perturbed harmonic oscillator | Consider two operators $L\_1w=-w''+U(x)w$ with eigenvalues $\lambda\_k$ and $L\_2w=-w''+V(x)w$
with eigenvalues $\mu\_k$. If $U\geq V$ then $\lambda\_k\geq \mu\_k$.
To prove this consider the Rayleigh ratio:
$$R\_j(w)=\frac{\int \overline{w}L\_jw}{\int |w|^2}.$$
The smallest eigenvalue is the minimum of the Rayleigh ra... | 2 | https://mathoverflow.net/users/25510 | 290691 | 128,138 |
https://mathoverflow.net/questions/290395 | 5 | Let $m$,$l$ be coprime integers where $m,l\geq 2$. For any integer $a$ and positive base $b \ (b\geq 2)$, let
$
[a]\_b
$ denote the element of $\{0,\ldots, b-1\}$ that satisfies the equivalence
$[a]\_b \equiv a \bmod b$.
For any integer $n$, one can write
$$
nl[l^{-1}]\_m - nm[(-m)^{-1}]\_l = n,
$$ as Bézout's Ident... | https://mathoverflow.net/users/17592 | When does the following congruence identity hold? | Each integer number $n$ can be uniquely expressed in the form $n=lx+my$, where $0\le x\le m-1$ and $y\in \mathbb{Z}.$ This artificial numeral system is more suitable for the riven problem because for $n=lx+my$
$$l[nl^{-1}]\_m - m[n(-m)^{-1}]\_l = lx -m[-y]\_l. $$
The last expression is equal to $n$ iff $-l<y\le 0.$ So ... | 3 | https://mathoverflow.net/users/5712 | 290695 | 128,139 |
https://mathoverflow.net/questions/290677 | 3 | Let $X$ be a smooth projective $k$-scheme, $k$ being a number field. Let $\mathcal{O}\_k$ be the ring of integers of $k$.
Fix a large enough category of schemes $\text{Sch}/k$ containing $X$, and consider the big étale site $(\text{Sch}/k)\_{\rm Ét}$. The functor of points $h\_X$ of $X$ is a sheaf on $(\text{Sch}/k)\... | https://mathoverflow.net/users/nan | Néron models vs integral models | Note that sheaf pushforward and preheaf pushforward agree, so this a question about categories, not sites.
>
>
> >
> > **Lemma.** If $X$ is a finite type $k$-scheme with $\dim X > 0$, then $j\_\*h\_X$ is not representable by a finite type algebraic space over $\mathcal O\_k$.
> >
> >
> >
>
>
>
*Proof (ske... | 3 | https://mathoverflow.net/users/82179 | 290700 | 128,141 |
https://mathoverflow.net/questions/290708 | 2 | Consider the Johnson-Lindenstrauss lemma in the case where we can assume the $n$ input points $x\_i$ in $\mathbb{R}^d$ are actually located on the sphere
$$S^{d-1}(r):=\{u=(u\_1,\ldots,u\_{d}): u\_1^2+\cdots+u\_d^2=r\},$$
for some $r>0.$
How does this impact the conclusion of the theorem? In particular, can the uppe... | https://mathoverflow.net/users/17773 | Johnson-Lindenstrauss Lemma on $S^{d-1}$ | Given that the worst-case configurations require dimension projection proportional to the one obtained by linear embeddings
<http://people.seas.harvard.edu/~minilek/publications/papers/jl_tight.pdf>
it's not possible to gain by restricting the points to the sphere. Furthermore, I believe in the proof of this paper ... | 1 | https://mathoverflow.net/users/39129 | 290713 | 128,144 |
https://mathoverflow.net/questions/290714 | 7 | This question is closely related to [What is the geometric object corresponding to a subalgebra in a polynomial ring](https://mathoverflow.net/questions/101176/what-is-the-geometric-object-corresponding-to-a-subalgebra-in-a-polynomial-ring).
There, it is asked, given a subalgebra of an algebra $S \subset R$ over a fiel... | https://mathoverflow.net/users/94086 | Geometric object corresponding to subalgebra generated by an ideal? | The image of the composition $S \to R \to R/I$ is equal to the constants $k \subset R/I$. This indicates that the morphism of the zero locus $V(I) = Z \to \mathrm{Spec}(R) \to \mathrm{Spec}(S)$ factors through the structure morphism $Z \to \mathrm{Spec}(k)$, and so the image must be a point. Therefore the morphism corr... | 7 | https://mathoverflow.net/users/70019 | 290715 | 128,145 |
https://mathoverflow.net/questions/290731 | 6 | Is the category of commutative group algebraic spaces (commutative group objects in algebraic spaces) locally of finite type over a field, an abelian category?
I would benefit from a reference
| https://mathoverflow.net/users/nan | Commutative group algebraic spaces | The intervention of algebraic spaces is a red herring because algebraic space groups $G$ locally of finite type (and quasi-separated!) over a field $k$ are necessarily schemes. (I am assuming you are only interested in the quasi-separated case, though no motivating context for the question was given.) This is Lemma 4.2... | 10 | https://mathoverflow.net/users/81332 | 290735 | 128,154 |
https://mathoverflow.net/questions/290734 | 0 | I found an interesting question on quora and need help in solving this question. I've just started understanding permutations but could not understand as to how I can come up with a general formula for this problem.
Consider all permutations of the numbers 1 to n. A good permutation is one where for any number i at pos... | https://mathoverflow.net/users/119627 | Permutations which avoid consecutive entries of the form (m,m+1) | This is [A000255](https://oeis.org/A000255), and can be described in several ways. Perhaps the easiest to see is the recursion
$$a(n+1)=na(n)+(n-1)a(n-1)$$
Note that my indexing is different from the OEIS, in my notation $a(n)$ counts the good permutations of $[n]$. To see the recursive identity justify that a good per... | 1 | https://mathoverflow.net/users/2384 | 290736 | 128,155 |
https://mathoverflow.net/questions/290738 | 3 | Consider $\mathcal{M}\_{0,n}$, the moduli space of genus zero curves with $n$-punctures.
Using a combination of the Kodaira-Spencer map, Riemann-Roch, and Serre Duality, I have calculated the dimension of $\mathcal{M}\_{0,n}$ to be $n-3$.
Using such heavy theorems, I have lost intuition as to why adding a punctur... | https://mathoverflow.net/users/117411 | Why does adding a puncture to the moduli space of genus zero curves increase its dimension by $1$? | Let me interpret $\mathcal{M}\_{0,n}$ as the moduli space of genus zero curves with $n$ marked points, as opposed to punctures. This is just a psychological thing and should not make a difference to what follows.
That $\dim(\mathcal{M}\_{0,n}) = 0$ for $n = 0,1,2,3$ is very classical: this amounts to saying that all ... | 5 | https://mathoverflow.net/users/37821 | 290740 | 128,156 |
https://mathoverflow.net/questions/290712 | 3 | In Joe Harris's book "Algebraic Geometry: A First Course", we find the following proposition:
Let $f:X\longrightarrow Y$ be a dominant rational map, for $X$ and $Y$ be two varieties.
Proposition 7.16. The general fiber of the map $f$ is finite if and only if the inclusion $f^\*$ expresses the field $K(X)$ as a fi... | https://mathoverflow.net/users/29836 | Number of points in a general fiber for a dominant rational map | You cannot characterize the subset $Z \subseteq Y$ of points over which the fibre is not finite by looking at the field extension, since by the function fields $K(Y)$ and $K(Y-Z)$ are the same. In fact, the function field is a *birational* invariant, not a biregular one.
If you want to figure out what "general" means... | 3 | https://mathoverflow.net/users/7460 | 290749 | 128,159 |
https://mathoverflow.net/questions/289597 | 5 | Given an elliptic curve group with a generator $G$ where $G$ has a prime order, p.
Given a point $P=aG$ for some unknown $a$. Is it possible to efficiently calculate $Q=a^{-1}G$ without a discrete log operation?
With a discrete log, the problem is simple: first calculate $a$, then $a^{-1} = a^{p-1} $ mod $p$.
But ... | https://mathoverflow.net/users/119111 | Elliptic curves: for $P = aG$ for some $a$, what is $Q = a^{-1}G$? | The name of the problem is `the Inverse Diffie-Hellman problem'. It is as hard as solving the computational Diffie-Hellman problem. A proof can be found in chapter 21, p.448-449 of Mathematics of [Public Key Cryptography by Steven Galbraith (2012)](https://www.math.auckland.ac.nz/~sgal018/crypto-book/crypto-book.html).... | 2 | https://mathoverflow.net/users/119111 | 290754 | 128,162 |
https://mathoverflow.net/questions/290729 | 2 | Since today is the [Chow](https://mathoverflow.net/questions/290690/codimension-restrictions-on-intersections)-[variety day](https://mathoverflow.net/questions/290682/on-a-class-of-loci-in-chow-varieties), I'm going to ask my question here.
Suppose I have a smooth projective variety $X$ over a field of characteristic... | https://mathoverflow.net/users/nan | Pull-back of algebraic cycles | **Counterexamples.** Here are examples showing that each of the properties above can fail. Let $X$ be a smooth cubic surface in $\mathbb{P}^3\_k$, where $k$ is a field. Let $H$ be a smooth hyperplane section of $X$. This is a smooth, geometrically connected, projective curve of genus $1$ (a plane cubic).
The Fano sc... | 7 | https://mathoverflow.net/users/13265 | 290758 | 128,163 |
https://mathoverflow.net/questions/287861 | 2 | Let $p:A \to S$ be a projective abelian scheme, where $S$ is some smooth scheme over a base field $k$. Then we have the Kodaira-Spencer morphism
$$
\kappa : T\_{S/k} \to R^1p\_\*T\_{A/S}
$$
where $T\_{S/k}$ (resp. $T\_{A/S}$) denotes the dual module of $\Omega^1\_{S/k}$ (resp. $\Omega^1\_{A/S}$).
Let $\text{Lie}\_SA$... | https://mathoverflow.net/users/nan | Identification of cohomology sheaf in the definition of the Kodaira-Spencer morphism for abelian schemes | Posting my comment as an answer:
Since $T\_{A/S}$ is trivial locally on $S$, we have $T\_{A/S} = p^\* p\_\* T\_{A/S} = p^\* {\rm Lie}\_S A$. By the projection formula, we get
$$ R^1 p\_\* T\_{A/S} = R^1 p\_\* p^\* {\rm Lie}\_S A = (R^1 p\_\* \mathcal{O}\_A)\otimes {\rm Lie}\_S A. $$
| 3 | https://mathoverflow.net/users/3847 | 290760 | 128,164 |
https://mathoverflow.net/questions/290590 | 3 | In the literature are there some concept of geometric version of Morse or Picard Lefschets theory? That is the comparison of level sets as Riemannian submanifold not merely as topological manifolds.
In particular is there a complete classification of Polynomials $P(z,w): \mathbb{C}^2 \to \mathbb{C}$ such that all reg... | https://mathoverflow.net/users/36688 | Geometric Morse theory ( and its complex analogy) | Here is an answer for the last question: The regular level set $L\_c := \{(z,w) \in \mathbb{C}^2 : P(z,w) = c \}$ of $P(z,w)=z^2 + w^2$ is not isometric to the level set $L\_{2c}$. To see this, notice that $L\_c$ and $L\_{2c}$ are complete Riemannian manifolds. The map $(z,w) \to (\sqrt{2}z,\sqrt{2}w)$ restricts to a h... | 3 | https://mathoverflow.net/users/43122 | 290765 | 128,167 |
https://mathoverflow.net/questions/290772 | 15 |
>
> Why aren't there any "crowdsourced" projects for mathematical textbooks?
>
>
>
Every year, many mathematicians put a lot of effort into crafting their own lecture notes or writing textbooks (or also research monographs). Why aren't there any open-source "crowdsourced" cooperative efforts towards book writing... | https://mathoverflow.net/users/nan | Polymath-type projects for textbooks? | The [Homotopy Type Theory](https://homotopytypetheory.org/book/) book is massively collaborative and regularly updated (through version control). It's also the definitive textbook in its field, and I think Voevodsky himself contributed to it. In particular, it is mentioned that:
>
> We have released the book under ... | 13 | https://mathoverflow.net/users/39521 | 290781 | 128,174 |
https://mathoverflow.net/questions/290786 | 11 | In their paper "Uniformity of rational points", Caporaso, Harris, and Mazur asserted the following:
"The Geometric Lang conjecture has been proved for all surfaces with $c\_1^2 > c\_2$ ([B]), and has recently been announced for all surfaces ([LM])."
The referece [LM] in that paper (link here: <http://www.ams.org/jo... | https://mathoverflow.net/users/10898 | Geometric Lang conjecture - reference | abx's comment was made while I was writing this, but I am posting it as an answer anyway.
There has not been a proof of this conjecture of Lang, which remains a wide open problem. Lu and Miyaoka's paper to which Caporaso, Harris and Mazur refer has to be [this one](https://www.intlpress.com/site/pub/files/_fulltext/j... | 9 | https://mathoverflow.net/users/26522 | 290788 | 128,176 |
https://mathoverflow.net/questions/290679 | 5 | For finite-dimensional (non-weak) Hopf C\*-algebras it is known that the antipode is always involutive, as claimed e.g. in <https://arxiv.org/pdf/1007.5283.pdf>. I couldn't find the same statement for weak Hopf C\*-algebras, however. Are there counterexamples?
| https://mathoverflow.net/users/115363 | Are there examples of finite-dimensional weak Hopf C*-algebras with non-involutive antipode? | According to [this paper](https://www.sciencedirect.com/science/article/pii/S002212369993522X) of Nikshych-Vainerman, a finite-dimensional weak Kac algebra is precisely a finite-dimensional [weak Hopf](https://en.wikipedia.org/wiki/Weak_Hopf_algebra) ${\rm C}^{\star}$-algebra with an involutive antipode ($S^2 = id$), a... | 4 | https://mathoverflow.net/users/34538 | 290792 | 128,177 |
https://mathoverflow.net/questions/290789 | 3 | Let $d \in \mathbb{N}$ and $\Omega$ be a bounded domain of $\mathbb{R}^d$.
Consider $m,n,p,q \in \mathbb{N}$ and $T>0$.
Is the space $W^{m,p}([0,T],W^{n,q}(\Omega))$ compactly embedded in any space of continuous functions (such as $C^k([0,T],C^l(\Omega))$ with $k,l\in\mathbb{N}$ ) ?
I guess it is true if $n=m$ and ... | https://mathoverflow.net/users/112416 | Compact embedding for Sobolev space involving time | The paper ["Compact embeddings of vector-valued Sobolev and Besov spaces" by Amann](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.629.8063&rep=rep1&type=pdf) should probably cover your questions.
| 3 | https://mathoverflow.net/users/85906 | 290796 | 128,178 |
https://mathoverflow.net/questions/290795 | 2 | Given Banach spaces $X$, $Y$ and a bounded operator $T:X\to Y$ with non-closed range, a perturbation argument shows that there exists an infinite-dimensional closed subspace $M$ of $X$ such that the restriction of $T$ to $M$ is compact.
Let $J:L\_\infty(0,1)\to L\_1(0,1)$ denote the natural inclusion. Is it posible ... | https://mathoverflow.net/users/39421 | Compact restrictions of the inclusion of $J:L_\infty(0,1)\to L_1(0,1)$ | Take the functions which are constant on the intervals $[1/(n+1),1/n)$ for all positive integers $n$.
| 3 | https://mathoverflow.net/users/4312 | 290797 | 128,179 |
https://mathoverflow.net/questions/290625 | 4 | I was looking at this paper [“Islands in Sea” and “Lakes in Mainland” phases and related transitions simulated on a square lattice](https://drive.google.com/file/d/1onBqzq50ApMzyvgMYekvjXdUXRbuhoZN/view?usp=sharing) on Percolation theory. The concept of phase transition used here seems to be a bit different compared to... | https://mathoverflow.net/users/nan | Critical Exponents for Island Mainland Transition (Percolation Theory) | The problem considered in the [Island-Mainland paper](https://arxiv.org/abs/1612.04522) is site-percolation on a two-dimensional square lattice, with one modification of the conventional problem: two squares of the same colour are considered connected if they are nearest-neighbor (they share an edge) or next-nearest-ne... | 2 | https://mathoverflow.net/users/11260 | 290804 | 128,182 |
https://mathoverflow.net/questions/290711 | 4 | I am asking a question which looks very elementary to experts.
Let $F$ be a number field and $\mathbb{A}\_F$ its adele ring. Let $\omega$ be a unitary central character of $GL\_2(\mathbb{A}\_F)$,
$X\_{GL\_2}=GL\_2(F) \backslash GL\_2(\mathbb{A}\_F)$, and
$C^{\infty}\_{\omega}(X\_{GL\_2})=\{f:X\_{GL\_2} \to \math... | https://mathoverflow.net/users/29422 | Projection onto locally constant function | It is clear from the definition and right-invariance of the measure $dh$ that $Pf$ is left-invariant under $SL\_2(\mathbb{A}\_F)$. As Paul Garrett kindly explained, $Pf$ is also left-invariant under $GL\_2(F)$, because $f$ is left-invariant under $GL\_2(F)$ and $GL\_2(F)$ normalizes the coset space $SL\_2(F)\backslash ... | 4 | https://mathoverflow.net/users/11919 | 290805 | 128,183 |
https://mathoverflow.net/questions/290793 | 2 | Let $(\cal{C},\otimes)$ and $(\cal{D},\odot)$ be two monoidal categories. Moreover, assume that $\cal{C}$ and $\cal{D}$ are abelian and semisimple. Let $X,Y$ be two simple objects in $\cal{C}$, and let
$$
X \otimes Y \cong Z\_1 \oplus \cdots \oplus Z\_k,
$$
be their decomposition into simple objects. Is there a name fo... | https://mathoverflow.net/users/81477 | A non-monoidal functor that respects fusion rules | I don’t think there’s a well-established name, but they’re called “quasi monoidal functors” in <https://arxiv.org/abs/1711.00645>
| 1 | https://mathoverflow.net/users/22 | 290817 | 128,186 |
https://mathoverflow.net/questions/290346 | 6 | For analytic function $f:\mathbb{C}\to\mathbb{C}$ with
>
> $$f(z)=-\dfrac{\log(1-z)}{z}$$
>
>
>
I want to prove that it's a convex map in the unit disk $\mathbb{D}$, i.e. that $f$ maps the unit disk conformally onto a convex domain. I know $${\bf Re}\left(1+z\dfrac{f''(z)}{f'(z)}\right)>0$$
for $z\in\mathbb{D... | https://mathoverflow.net/users/103210 | Prove $f(z)=-\frac{\log(1-z)}{z}$ is convex in $\mathbb{D}$ | The normalization of f, (f-1)/2 is the image of the fundamental convex function z/(1-z) under the operator H(g) = (2/z)\* primitive g.
It is a theorem of Libera (Goodman, Univalent Functions 1983, vol 2, page 156) that while H doesn't preserve the S class (schlicht), it does preserve the classes CV (convex S), ST (star... | 0 | https://mathoverflow.net/users/119661 | 290820 | 128,187 |
https://mathoverflow.net/questions/290822 | 5 | Let $\Phi\_m(x)$ and $\Phi\_n(x)$ be two different cyclotomic polynomials. Then $\Phi\_m(x)$ and $\Phi\_n(x)$ are coprime, so there are two polynomials $s(x), t(x)$ with, say, rational coefficients such that $s(x)\Phi\_m(x)+t(x)\Phi\_n(x)=1$.
**Question.** Can one find these $s$ and $t$ with integer coefficients?
... | https://mathoverflow.net/users/nan | Cyclotomic polynomials. | If this were true, then you would prove that $\Phi\_m(x)$ and $\Phi\_n(x)$ are coprime after reduction modulo $p$, which is far from true. For instance, $\Phi\_4(x)=x^2+1$ and $\Phi\_2(x)=x+1$ are not coprime modulo $2$. (Even $\Phi\_2(x)$ and $\Phi\_1(x)$ are not coprime modulo $2$, of course.)
| 16 | https://mathoverflow.net/users/1306 | 290823 | 128,188 |
https://mathoverflow.net/questions/290826 | 16 | My naive [question](https://mathoverflow.net/questions/290822/cyclotomic-polynomials) may actually lead to something interesting.
Let $\Phi\_m(x)$, $\Phi\_n(x)$ be cyclotomic polynomials, $m<n$. These polynomials are relatively prime and so there are polynomials $s(x), t(x)$ with rational coefficients such that $s\P... | https://mathoverflow.net/users/nan | Cyclotomic polynomials 2 | [This note](https://projecteuclid.org/euclid.rmjm/1181071615) (see theorem 2.2) attributes this result to
>
> Diederichsen, Fritz-Erdmann,
> "Über die Ausreduktion ganzzahliger Gruppendarstellungen bei arithmetischer Äquivalenz", Abh. Math. Sem. Hansischen Univ. 13, (1940). 357–412
>
>
>
but also points out t... | 10 | https://mathoverflow.net/users/2384 | 290831 | 128,190 |
https://mathoverflow.net/questions/290842 | 8 | Where do I find a proof of the fact that over global function fields of characteristic $p>0$, finiteness of the Tate-Shafarevich group of an abelian variety is equivalent to finiteness of its $\ell$-primary torsion part for some (arbitrary) prime $\ell$ ($\ell = p$ allowed)?
What is the main idea?
One knows the Tat... | https://mathoverflow.net/users/nan | Sha finiteness vs $\ell$-primary torsion | One shows that the finiteness of one $\ell$-primary component of Sha is equivalent to the Birch-Swinnerton-Dyer conjecture. $\mathrm{rk} A(K) = \mathrm{ord}\_{s=1}L(A/K,s)$ is independent of $\ell$ and a non-zero rational number (the special $L$-value) has only finitely many prime divisors.
The references are:
1. P... | 8 | https://mathoverflow.net/users/nan | 290845 | 128,194 |
https://mathoverflow.net/questions/290778 | 2 | This is a follow-up to [this question](https://mathoverflow.net/questions/290082/recognition-of-finite-simple-groups-by-number-of-sylow-p-subgroups).
Let $G$ and $G'$ be two finite simple groups of the following structures:
1- $A\_{p}$, for some primes $p$;
2- $PSL\_{p}(q)$, for some prime $p$ and some prime powe... | https://mathoverflow.net/users/97247 | Recognition of finite simple groups by number of Sylow p-subgroups (2) | ${\rm PSL}(2,31)$ and ${\rm PSL}(2,32)$ both have $496$ Sylow $3$-subgroups.
I think that when you ask question like this, you should provide at least some kind of reason, however tenuous, as to why you might expect the answer to be yes. Conjectures in group theory that are made purely on the basis that one is unable... | 6 | https://mathoverflow.net/users/35840 | 290856 | 128,197 |
https://mathoverflow.net/questions/290716 | 14 | In 1995, Robert Thomason published “Symmetric monoidal categories model all connective spectra” in TAC. On page 2, he argues that symmetric monoidal categories are more convenient than “May’s coordinate-free spectra” for various reasons, one of which is that a symmetric monoidal structure is easier to obtain (this was ... | https://mathoverflow.net/users/11540 | Status of Thomason's idea for a symmetric monoidal model of stable homotopy - from his last paper | I think that the best thing in this direction is the paper "Permutative categories, multicategories and algebraic K-theory" by Elmendorff and Mandell.
I have only skimmed this so I may well not be understanding it correctly.
Anyway, we want to consider symmetric monoidal categories, whose monoidal structure shoul... | 5 | https://mathoverflow.net/users/10366 | 290862 | 128,200 |
https://mathoverflow.net/questions/290861 | 7 | Let $F$ be a complex Hilbert space. We recall that an operator $S\in\mathcal{B}(F)$ is said to be hyponormal if $S^\*S\geq SS^\*$ (i.e. $\langle (S^\*S-SS^\*)z,z \rangle\geq 0$ for all $z\in F$).
>
> Assume that $S$ is hyponormal operator. Is $\omega(S)=\|S\|?$, with $\omega(S)$ denotes the numerical radius of $S$ ... | https://mathoverflow.net/users/116483 | numerical radius of hyponormal operator | Yes, the proof can be found in Stampfli, Joseph G., *Hyponormal operators*, Pacific J. Math. (12), no. 4 (1962), 1453--1458.
The proof is actually very simple. First of all, hyponormality may be stated as $\|Sx\|\geqslant \|S^{\ast}x\|$ for any $x\in F$. It follows that $\|Sx\|^2 = \langle Sx, Sx \rangle = \langle x,... | 8 | https://mathoverflow.net/users/24953 | 290866 | 128,201 |
https://mathoverflow.net/questions/290846 | 13 | In a footnote to the list of known Mersenne prime numbers which can be found [here](https://www.mersenne.org/primes/), we read that the "ranking" therein is a provisional one since not all possible exponents between $37 \, 156 \, 667$ and $77 \, 232 \, 917$ have been eliminated/tested.
If we may infer from this that ... | https://mathoverflow.net/users/1593 | Question on the 50th (known) Mersenne prime number | As Jan Grabowski notes, all the primes that you mention have been discovered by GIMPS. GIMPS draws a distinction between *testing* and *double-checking*. When a Lucas–Lehmer test is performed on a Mersenne number and delivers a verdict that the number is prime, the computation is immediately verified. Only after the ve... | 17 | https://mathoverflow.net/users/3106 | 290868 | 128,203 |
https://mathoverflow.net/questions/290837 | 1 | Suppose that $R$ is a ring, that $M$, $N$ and $L$ are right $R$-modules, and that $N$ is an $R-R$-bimodule.
Is there any formula for $Ext^1 (M\otimes\_R N,L)$ or generally for $Ext^n (M\otimes\_R N,L)?$
Thanks in advance.
| https://mathoverflow.net/users/114580 | What is $Ext^1 (M\otimes_R N,L)$? | I assume from how the question is phrased that the OP is hoping for a kind of Hom-Tensor duality for Ext. The first comment below shows that in general
$$\operatorname{Ext}^n(M\otimes\_R N,L) \not \cong \operatorname{Ext}^n(M,\operatorname{Hom}\_R(N,L))$$
Instead, what you have are the well-known Ext-Tor relations.... | 2 | https://mathoverflow.net/users/11540 | 290869 | 128,204 |
https://mathoverflow.net/questions/290851 | 4 | I want to know properties of the following sum: $$\sum\_{j=0}^{p-1} \omega^{\beta j^2}= ~? $$
where $p$ is a prime, and $\omega^p=1$, is a $p$th root of unity (and $\beta$ is an integer between $0$ and $p-1$). If anyone has any references related to this, let me know.
Also, if someone can tell me about the sum:
$$\s... | https://mathoverflow.net/users/119673 | Summation formulas involving roots of unity to various powers | Your first sum is a special Gauss sum. For its value in general, see Corollary 9.16 in Montgomery-Vaughan: Multiplicative number theory I. Your second sum can also be expressed in terms of Gauss sums (associated with primitive characters modulo $p$), and in particular its absolute value is at most $(n-1)\sqrt{p}$, assu... | 9 | https://mathoverflow.net/users/11919 | 290874 | 128,206 |
https://mathoverflow.net/questions/290824 | 2 | I am looking for a reference that gives the definition and has summarized the dual Coxeter number for superalgebras, especially for $\mathfrak{u}(m|n)$ (the Lie algebra of unitary supergroup $U(m|n)$).
| https://mathoverflow.net/users/64606 | Dual Coxeter Number for Superalgberas | The dual Coxeter number for basic Lie superalgebras is given in the table on page 16 of [Kostant’s cubic Dirac operator of Lie superalgebras](https://core.ac.uk/download/pdf/25269566.pdf). Also, the generalization of the Freudenthal-de Vries strange formula holds for Lie superalgebras as well (equation (38) of the abov... | 2 | https://mathoverflow.net/users/64606 | 290879 | 128,209 |
https://mathoverflow.net/questions/290813 | 9 | An informal definition of a logical truth is a sentence that's true in virtue of its form alone: $\phi$ is logically true iff all substitutions of $\phi$ that leave its logical vocabulary alone are true.
We might try to formulate a version of this idea in modal logic. Let $\mathcal{L}$ be a modal language, and let $... | https://mathoverflow.net/users/78564 | Substitutional modality | $\def\ml{\mathrm{ML}}\let\LOR\bigvee\let\ET\bigwedge$The question you asked is a variant of Problem 42 in Friedman [1]. It also has an intuitionistic analogue, Problem 41, which asks if there exists a set $V$ of propositional formulas such that
* $A\land B\in V$ iff $A\in V$ and $B\in V$,
* $A\lor B\in V$ iff $A\in V... | 8 | https://mathoverflow.net/users/12705 | 290882 | 128,210 |
https://mathoverflow.net/questions/290881 | 7 | Wikipedia [article](https://en.wikipedia.org/wiki/Totally_bounded_space) on totally bounded spaces states "... the completion of a totally bounded space might not be compact in the absence of choice." Where is the axiom of choice used, and do you need it for metric spaces or only for general uniform spaces?
| https://mathoverflow.net/users/91419 | Totally bounded spaces and axiom of choice | The issue here is that a metric space might not have non-trivial (read: not eventually constant) Cauchy sequences. For example, if the underlying space is a Dedekind finite set.
Indeed it is consistent that there is a dense subset of $[0,1]$ which is Dedekind finite. As a space with the inherited metric it is complet... | 12 | https://mathoverflow.net/users/7206 | 290883 | 128,211 |
https://mathoverflow.net/questions/290888 | 8 | I have read (mainly in the articles of Atiyah) that the partition function is the simplest topological invariant of a quantum field theory.
I have an arithmetic geometry background and know statistical physics quite well, but am a beginner in QFT. A reference for details on the above statement in mathematical languag... | https://mathoverflow.net/users/85544 | Reference request: Partition function as a topological invariant of a QFT | This is one case where you might just want to go back to the original article that proved that a partition function is metric-independent and hence a topological invariant: Edward Witten, [Quantum field theory and the Jones polynomial](https://projecteuclid.org/euclid.cmp/1104178138) (1989) --- see page 361.
For a sk... | 2 | https://mathoverflow.net/users/11260 | 290889 | 128,214 |
https://mathoverflow.net/questions/290870 | 4 | I asked this problem on [MSE](https://math.stackexchange.com/questions/2554069/convergence-stability-of-sde-that-depends-on-an-ergodic-process) some while ago, but it has stubbornly resisted any attempts at solving it.
Maybe there is someone here who can either close the gap in one of the existing answers or has an in... | https://mathoverflow.net/users/69603 | Almost sure stability of a scalar, nonautonomous, nonlinear SDE | Given a realization of the Ornstein-Uhlenbeck process $X\_t$, the SDE $$
d Y\_t = Y\_t (1- Y\_t) X\_t (dt + d V\_t) \tag{1}
$$ is scalar, nonautonomous, and nonlinear. Note that (1) has two fixed points at $0$ and $1$, which are asymptotically stable in the following sense.
>
> Theorem. For almost all $Y\_0 \in (0,... | 3 | https://mathoverflow.net/users/64449 | 290890 | 128,215 |
https://mathoverflow.net/questions/290568 | 7 | Given a set $A \subseteq \omega^\omega$, let $G\_A$ denote the Gale-Stewart game with payoff set $A$ (so player $I$ wants the real built over the course of play to be in $A$ and player $II$ wants it not to be). Notice that the set of (not necessarily winning) strategies for player $I$ is simply the set of functions map... | https://mathoverflow.net/users/114946 | On the topological complexity of the set of winning strategies for Gale-Stewart Games | Abstractly, $W(A)$ is of the form $\forall^{\mathbb R} A \vee \forall^{\mathbb R} \lnot A$; in fact, if $I$ has a winning strategy for $A$, then $W(A)$ is $\forall^{\mathbb R} A$, and if $II$ has a winning strategy for $A$, then $W(A)$ is $\forall^{\mathbb R} \lnot A$. As Joel points out, though, $W(A)$ may be of much ... | 3 | https://mathoverflow.net/users/114509 | 290912 | 128,225 |
https://mathoverflow.net/questions/290863 | 1 | I propose the following problem (Maybe it has a trivial solution):
Let $n$ be a positive integer such that $$n\equiv1 \pmod 4.$$
Then the problem is to find a rational $x$ as a function of $n$ such that
$$ \dfrac{3n+3x+n^{2}}{12} \quad\text{and}\quad \dfrac{n(n+3)(3n+3x+n^{2})}{36x}$$
are both strictly positive in... | https://mathoverflow.net/users/74668 | Finding a solution for this system of two diophantine equations (depending on a parameter) | I think we can completely solve the problem. Since $\displaystyle\frac{3n+3x+n^2}{12}$ must be an integer, $y:=3x$ must be an integer. Put also $N:=n(n+3)$. Then we want
$$\frac{N+y}{12},\text{ and }\frac{N(N+y)}{12y}$$ to be positive integers.
Therefore we need $N+y=12r$ **(1)** and $y=\displaystyle\frac{N^2}{12k-... | 2 | https://mathoverflow.net/users/1234 | 290919 | 128,227 |
https://mathoverflow.net/questions/290921 | 9 | Let $\lambda$ be a singular cardinal of countable cofinality.
Is there necessarily a sequence $\{A\_\alpha\mid\alpha<\lambda^+\}$ of countable subsets of $\lambda$, such that $\alpha<\beta$ if and only if $A\_\alpha\setminus A\_\beta$ is finite?
In other words, we know that for $\omega$, there is an uncountable seq... | https://mathoverflow.net/users/7206 | "Towers" on singular cardinals with countable cofinality | For $\lambda > 2^{\aleph\_0}$, there is no such sequence.
Suppose $\lambda > 2^{\aleph\_0}$. Because $2^{\aleph\_0}$ cannot have countable cofinality, there is some $\kappa < \lambda$ with $2^{\aleph\_0} < \kappa$. Consider the sequence $\{A\_\alpha \cap A\_\kappa \mid \alpha < \kappa\}$. For each particular $\alpha$... | 12 | https://mathoverflow.net/users/70618 | 290925 | 128,228 |
https://mathoverflow.net/questions/290520 | 2 | Assume sequence $(X\_1,X\_2, X\_3, \ldots)$ is a first-order Markov sequence of real random variables where $X\_i \in \mathcal{X}$ for some alphabet $\mathcal{X}$ of finite size $k$. Define emperical conditional probability $p\_{X\_i| X\_{i-1}}(x\_{i}|x\_{i-1})$ after $n$ samples as
$$\hat{p}\_{X\_{i}|X\_{i-1}}(x\_{i}|... | https://mathoverflow.net/users/119130 | Concentration of emperical conditional probability | A simple answer is the minimum convergence rate of the nominator and the denominator, to their respective true probabilities.
If you consider $\frac{a\_n}{b\_n}$ where $a\_n \to a$ and $b\_n \to b$ with some convergence rates $f\_1(n)$ and $f\_2(n)$,then for sufficiently large $n$ (at which $|b\_n - b|\leq \epsilon$... | 2 | https://mathoverflow.net/users/74156 | 290935 | 128,233 |
https://mathoverflow.net/questions/290550 | 4 | Let $\Gamma^a{}\_{bc}=\Gamma^a{}\_{cb}$ be a symmetric connection whose curvature is $$R^a{}\_{bcd}=\partial\_c\Gamma^a{}\_{bd}-\partial\_d\Gamma^a{}\_{bc}+\Gamma^a{}\_{ec}\Gamma^e{}\_{bd}-\Gamma^a{}\_{ed}\Gamma^e{}\_{bc}.$$
What conditions can we observe locally on $\Gamma^a{}\_{bc}$ or $R^a{}\_{bcd}$ to conclude t... | https://mathoverflow.net/users/119252 | Locally Riemannian Connection | Perhaps I can offer some information and comment on this problem. An essential part of the problem is how to interpret terms such as 'observe', 'accessible', 'identify', as the OP wants to know how to write down a *computable* criterion for a torsion-free connection to be the Levi-Civita connection of a Riemannian metr... | 15 | https://mathoverflow.net/users/13972 | 290936 | 128,234 |
https://mathoverflow.net/questions/290943 | 2 | Let $R$ be a Dedekind domain and $A, B$ be finitely generated projective $M\_n(R)$-modules. Is it true that
$A\oplus M\_n(R)\cong B\oplus M\_n(R)\:\:\Rightarrow\:\:A\cong B$?
Here, the isomorphism is over $M\_n(R)$. Here are my thoughts so far:
If $P(R)$ is the set of isomorphism classes of finitely generated pro... | https://mathoverflow.net/users/102861 | Does $A\oplus M_n(R)\cong B\oplus M_n(R)$ imply $A\cong B$? $R$ Dedekind domain | Under Morita equivalence, $M\_n(R)$ is associated to $R^n$, and your question is equivalent to: For $A, B$ finitely generated projective $R$-modules, does
$$A \oplus R^n \cong B \oplus R^n\quad \Rightarrow\quad A \cong B?$$
Then you can simply use the classification of finitely generated $R$-modules, and the answer is... | 5 | https://mathoverflow.net/users/50609 | 290945 | 128,237 |
https://mathoverflow.net/questions/290941 | 4 | **Notation**
1. $\le$ is used for the subgroup relation;
2. $P$ means polynomial time in input size;
3. $\Omega = \{1,2,3,\cdots,n\}$ is a input domain;
4. $\mathrm{Sym}(\Omega)$ means the symmetric group on $\Omega$;
5. $G = \langle A \rangle $ means the subgroup $G$ generated by the subset $A$ of $\mathrm{Sym}(\Om... | https://mathoverflow.net/users/111831 | Is the Normal centralizer problem in P? | Yes. This is [Proposition 7.3](https://books.google.com/books?id=zuNgdF0RXmAC&pg=PA159) of
Eugene M. Luks.
[Permutation groups and polynomial-time computation](http://ix.cs.uoregon.edu/~luks/dimacs.pdf).
Pages 139-175 of:
Larry Finkelstein and William M. Kantor, editors.
Groups and Computation, Volume 11 of Amer. Ma... | 20 | https://mathoverflow.net/users/35840 | 290946 | 128,238 |
https://mathoverflow.net/questions/290922 | 7 | Is there a skyscraper group scheme?
Let $S$ be a DVR. Is there a group scheme $\mathcal{G}$ over $S$ which is generically {1} trivial i.e, identity group, but at the closed point some nontrivial group $G$?
For example: Let $C$ be a curve over $S$ whose generic fibre is smooth of genus $g$ with no automorphism but t... | https://mathoverflow.net/users/nan | "skyscraper group scheme" | Yes.
Let $\mathbb{A}^{1,2}$ be the affine line with a double origin. Consider the natural map $\mathbb{A}^{1,2}\to \mathbb{A}^1$. (To define this map, let $0\_1$ and $0\_2$ be the origins in $\mathbb{A}^{1,2}$. The above map sends any $x\neq 0\_1, 0\_2$ to $x$. It sends $0\_1$ and $0\_2$ to the origin in $\mathbb{A}^... | 3 | https://mathoverflow.net/users/4333 | 290949 | 128,239 |
https://mathoverflow.net/questions/290217 | 8 | It is a theorem of Baumgartner and Laver that iterating Sacks forcings of weakly compact length gives rise to the tree property at $\omega\_2$. Natural questions (at least for me) are: do we get stronger tree properties at $\omega\_2$ if we start with larger cardinals? In particular, if the length is strongly compact, ... | https://mathoverflow.net/users/23835 | Iterated forcing and the super tree property at $\omega_2$ | The answer to the supercompact case is yes. More specifically, in the forcing extension obtained by iterating Sacks forcing of supercompact length, the super tree property at $\omega\_2$ holds. This follows from the following:
1) Countable support iteration of Sacks forcing satisfies $\omega\_1$-approximation property... | 5 | https://mathoverflow.net/users/23835 | 290957 | 128,242 |
https://mathoverflow.net/questions/290750 | 4 | Please see the definition of Hardy spaces on the unit disc [here](https://math.stackexchange.com/questions/2586370/if-a-function-belongs-to-the-hardy-space). Let $0<p\leq\infty$. Let $f\in H^p$ with $\|f-1\_e\|\_p<1$ (Where $1\_e$ Is the constant function one). Then is $f$ an **outer function**?
| https://mathoverflow.net/users/119639 | Regarding outer functions | This is true for $p=\infty$, but is not true for $p=2$. Let $g$ be an inner function. Then, the optimal polynomial approximant $p\_{n}^\*$ when minimizing $\|pg-1\|\_{2}$ over degree $n$ polynomials, $p$, satisfies $\|p\_{n}^{\*}g-1\|\_{2}^2=1-|g(0)|^2$ (see <http://shell.cas.usf.edu/~dkhavins/files/GenInnerFinal.pdf>,... | 1 | https://mathoverflow.net/users/118731 | 290962 | 128,244 |
https://mathoverflow.net/questions/290958 | 2 | Let $R$ be a $k-$algebra and $M,N$ two irreducible $R-$modules, isomorphic as vector spaces. If we know that for every $r\in R$ we have the same eigenvalues on $M$ and $N$ (with multiplicities) is it true that $M\simeq N$?
I have two thoughts:
1. Denoting $St(m)=\{ rm=m \mid r\in R\}$, if we have $St(m)=St(n)$ for ... | https://mathoverflow.net/users/119736 | Isomorphism of irreducible R-modules | If $k$ is a field and $M$ and $N$ are finite-dimensional, then the answer is yes. That is part of (one version of) the Brauer-Nesbitt theorem. Having the same eigenvalues with the same algebraic multiplicities is equivalent to having the same characteristic polynomials.
For convenience define the characteristic polyn... | 5 | https://mathoverflow.net/users/3041 | 290963 | 128,245 |
https://mathoverflow.net/questions/290993 | 6 | Is it possible to find two matrices $A$ and $B$, so that there does not exists a product of matrices $A$,$A^{-1}$,$B$,$B^{-1}$ that is equal to $Id$, under the condition that the product is irreducible, that is, it is is not trivial as a word (e.g. $AA^ {-1}B^{-1}B$).
If there is no simple answer I would be happy with ... | https://mathoverflow.net/users/101335 | Trivial product of two matrices? | If I understand correctly, you will find the answer in a good exposition of the [Banach-Tarski paradox](https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox#A_sketch_of_the_proof). Finding two rotation matrices in $\mathbb{R}^3$ that generate the free group in two generators is a usually the biggest part of the ... | 14 | https://mathoverflow.net/users/1898 | 290994 | 128,257 |
https://mathoverflow.net/questions/291005 | 4 | Define $d(n)=\sum\_{t|n}1$ (it is equal to $\sigma\_0(n)=\tau(n)$). There are some estimates for upper bound of $$\sum\_{x\leq N}d(f(x))$$ in the terms of $N$, where $f(x)$ is a polynomial, even from Erdos. So far I know for its application in Diophantine sets ($m$-tuples) when $f(x)=x^2-r^2$ for some integer $r$. Are ... | https://mathoverflow.net/users/41466 | Sum of number of divisors function | Estimates on these quantities are used in Elsholtz and Tao's work on the [Erdos-Straus conjecture](https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Straus_conjecture). See their paper "[Counting the number of solutions to the Erdos-Straus equation on unit fractions](https://arxiv.org/abs/1107.1010)" and Tao's [blog pos... | 5 | https://mathoverflow.net/users/630 | 291006 | 128,262 |
https://mathoverflow.net/questions/290956 | 7 | Let $H$ be a complex Hilbert space and $\mathcal{L}(H)$ be the algebra of all bounded linear operators on $E$.
>
> If $A,B\in \mathcal{L}(H)$, It is true that $\overline{\text{Im}(A)}\otimes \overline{\text{Im}(B)}\subset \overline{\text{Im}(A\otimes B)}$?
>
>
>
I try as follows:
Let $z\in \overline{\text{Im... | https://mathoverflow.net/users/113054 | It is true that $\overline{\text{Im}(A)}\otimes \overline{\text{Im}(B)}\subset \overline{\text{Im}(A\otimes B)}$? | The proof is okay. You did not specify the norm on the tensor product $H⊗H$
in which you take the closure. I do not know which one you want. Most naturally this would be the Hilbert-space or $\ell^2$ tensor product (which also describes the space of Hilbert-Schmidt operators). But the proof works for all reasonable nor... | 7 | https://mathoverflow.net/users/26935 | 291026 | 128,268 |
https://mathoverflow.net/questions/291014 | 9 | We are interested in understanding how a higher genus curve in a smooth surface of general type can degenerate into the non-normal locus of the limit of a degeneration of surfaces. More precisely:
Let $\mathscr{X} \to B$ be a family of surfaces over a smooth curve $B$ so that the general fiber $X\_\eta$ is a smooth s... | https://mathoverflow.net/users/14339 | Degeneration of curves inside a family of surfaces | Let $S$ be the "usual" pinch point surface defined by $x^2t=y^2z$ in $\mathbb P^3$ and $T\subseteq \mathbb P^3$ an arbitrary general surface of degree $d-3\geq 2$. Let $X\_0=S\cup T$. Note that then $\deg X\_0=d$. Let $\ell\subseteq S$ be the double line and $H\subseteq \mathbb P^3$ a general surface of sufficiently hi... | 5 | https://mathoverflow.net/users/10076 | 291035 | 128,273 |
https://mathoverflow.net/questions/291003 | 5 | The Riemann hypothesis is equivalent to the assertion that the De Bruijn-Newman constant $ \Lambda $ , as defined in <https://www.sciencedirect.com/science/article/pii/S0001870809001133/pdf?md5=d2b0cbb38f79b80de06d8b9c99836fab&pid=1-s2.0-S0001870809001133-main.pdf&_valck=1>, fulfills $ \Lambda\leq 0 $. On the other han... | https://mathoverflow.net/users/13625 | Does $ M(x)=O(\sqrt{x}) $ if and only if the De Bruijn-Newman constant is negative? | The de Bruijn-Newman constant is nonnegative, as proved in this brand new [preprint](https://arxiv.org/abs/1801.05914) by Rodgers and Tao. It is also conjectured, but not proven yet, that $M(x)=O(\sqrt{x})$ is false, in which case it is actually equivalent to $\Lambda<0$. Time will tell.
| 14 | https://mathoverflow.net/users/11919 | 291043 | 128,278 |
https://mathoverflow.net/questions/291020 | 2 | For a homogeneous degree $d$ polynomial $P$, the symmetric or Waring rank $W(P)$ is the minimum $r$ such that $P = \sum\_{j=1}^r l\_j^d$, where $l\_j$s are linear forms. Now, is the Waring rank sub-multiplicative, i.e. for two homogeneous degree $d$ polynomials $P$ and $Q$, $W(P Q) \leq W(P) \, W(Q)$?
| https://mathoverflow.net/users/98093 | Is the Waring rank homogeneous polynomials sub-multiplicative? | You asked:
>
> Is $W(P \otimes Q) \leq W(P) W(Q)$ for two homogeneous polynomials $P$ and $Q$?
>
>
>
First, I think you have to be a little bit careful about the difference between tensor product and multiplication of polynomials. If you are just using the ordinary tensor product, then the resulting $P \otimes... | 7 | https://mathoverflow.net/users/88133 | 291044 | 128,279 |
https://mathoverflow.net/questions/291060 | 4 | For two measures $\mu, \nu$ on the same space say that $\mu$ is *absolutely continuous* with respect to $\nu$ ($\mu \ll \nu$) whenever $\nu(A)=0$ implies that $\mu(A)=0$ too.
Let $(\Omega, \mathsf P$) be a probability space and let $S$ be a separable metric space. For an $S$-valued random variable $W$ on $\Omega$, de... | https://mathoverflow.net/users/15129 | Absolute continuity of measures - reference sought | This statement is false. Example: $Y=X=X\_1=U$ and $Y\_1=V$, where (say) $U$ and $V$ are independent standard normal r.v.'s.
| 5 | https://mathoverflow.net/users/36721 | 291062 | 128,284 |
https://mathoverflow.net/questions/136794 | 8 | I am trying to compute the group
$H\_1(\mathrm{SL}\_2(\mathbb{Z}\_2),M)$, where $\mathbb{Z}\_2$ are $2$-adic integers and M is a module $\mathbb{Z}\_2 \oplus \mathbb{Z}\_2$. I suppose that the group acts on $M$ by matrix multiplication.
I found a similar-looking computation in the paper of Dupont and Sah
"Homology o... | https://mathoverflow.net/users/21620 | Homology of special linear group over local field | First, some general remarks on the situation for $R$ an arbitrary commutative ring. Since ${\rm diag}(-1,-1)$ acts by multiplication by $-1$ on $R^{\oplus 2}$, the homology groups ${\rm H}\_i({\rm SL}\_2(R),R^{\oplus 2})$ are $2$-torsion for $i\geq 1$. (This is called the center-kills-argument.) This doesn't happen in ... | 5 | https://mathoverflow.net/users/50846 | 291070 | 128,287 |
https://mathoverflow.net/questions/290766 | 3 | * This question is about $U\_q ( \hat{\mathfrak{sl}}\_2 )$ representation theory. There is a notion of vertex operators $\Phi\_{\pm }(z)$ of first and $\Psi\_{\pm}(z)$ of the second type. They are defined to be intertwiners
$$\Phi(z): V(\Lambda\_i) \rightarrow V(\Lambda\_{1-i}) \otimes V\_z $$
$$\Psi(z): V(\Lambda\_i) ... | https://mathoverflow.net/users/62601 | Does the Leclerc-Thibon involution exchange vertex operators of the first and second type? | I found the answer to the first question. Second and third questions remain.
\begin{align}
\Phi\_+ (z) \rightarrow K^{-1/2} \Psi\_+ (q^{-1} z) \\
\Phi\_- (z) \rightarrow K^{1/2} \Psi\_- (q^{-1} z)
\end{align}
Equivalently
\begin{align}
\Psi\_+ (z) \rightarrow K^{-1/2} \Phi\_+ (q^{-1} z) \\
\Psi\_- (z) \rightarr... | 1 | https://mathoverflow.net/users/62601 | 291086 | 128,292 |
https://mathoverflow.net/questions/291097 | 2 | Let $f:X\dashrightarrow Y$ be the flip of a small contraction $\phi:X\rightarrow Z$, and let $\psi:Y\rightarrow Z$ be the small contraction such that $\psi\circ f = \phi$. Let $Exc(\phi), Exc(\psi)$ be the exceptional loci of $\phi$ and $\psi$ respectively.
Do we always have $\dim(Exc(\psi)) = \operatorname{codim}\_... | https://mathoverflow.net/users/nan | Flipping and flipped loci | I don't think so. In general we always have
$$
\dim \text{Exc}\phi+\dim \text{Exc}\psi\geq \dim X-1.
$$
This is proved in Lemma 5-1-7 [Kawamata, Matsuda, Matsuki, Introduction to Minimal Model Program].
But in general the equality may not hold. I did not come up with any example according to my knowledge, but I reme... | 3 | https://mathoverflow.net/users/42636 | 291112 | 128,300 |
https://mathoverflow.net/questions/291106 | 3 | Let $G$ be a countable (that is edit) residually-$p$ group and let $\hat{G}\_p$ be its pro-$p$ completion.
1. If $\hat{G}\_p$ is finitely generated does it mean that $G$ is finitely generated?
2. If $\hat{G}\_p$ is finitely presented does it mean that $G$ is finitely presented? (I think the Grigorchuk group is not fi... | https://mathoverflow.net/users/5034 | Finitely generated and finitely presented groups and their pro-$p$ completions | As I said in a comment, the answer to 1 is trivially no.
The answer to 2 (as edited) is also no. Platonov and Tavgen produced a finitely generated, infinitely presented subgroup $H$ in the square $F\times F$ of a free groups such that the inclusion induces an isomorphism of profinite completions (and hence of pro-$p$... | 5 | https://mathoverflow.net/users/14094 | 291113 | 128,301 |
https://mathoverflow.net/questions/291116 | 3 | Let $D$ be a base-point-free divisor on a normal projective variety $X$, and let $Y$ be the image of the morphism $f\_{D}:X\rightarrow Y$ induced by $D$. Assume that $f\_D$ is birational.
Now, let $X(D)=Proj\left(\bigoplus\_{k\geq 0}H^{0}(X,kD)\right)$. Is $X(D)$ the normalization of $Y$ ?
| https://mathoverflow.net/users/nan | Ring of sections and normalization | The modified assertion is true. For a field $k$, for every proper $k$-scheme $X$, for every $k$-morphism $$f:X\to \mathbb{P}^n,$$ the irreducible curves in $X$ that are contracted by $f$ are precisely the irreducible curves having degree $0$ with respect to the invertible sheaf $\mathcal{L}:=f^\*\mathcal{O}(1)$. These ... | 4 | https://mathoverflow.net/users/13265 | 291139 | 128,312 |
https://mathoverflow.net/questions/291104 | 9 | **Definition 1:**
A [clutter](https://en.wikipedia.org/wiki/Sperner_family#Clutters) $C$ is said to have the *packing property* if $C$ and all of its minors satisfy the König property.
where,
vertex cover of $C$ is a set of vertices that have non-empty intersection with all of the edges. The minimum cardinality of ... | https://mathoverflow.net/users/68302 | Definition of packing property | That Def 1 and Def 2 are equivalent is a well-known [Conjecture](http://www.dtic.mil/dtic/tr/fulltext/u2/a277340.pdf), still open as far as I know. Curiously, you can translate the whole conjecture to the language of commutative algebra, see for example page 26 of this [survey](https://arxiv.org/pdf/1708.03010.pdf). It... | 8 | https://mathoverflow.net/users/2083 | 291149 | 128,316 |
https://mathoverflow.net/questions/291160 | 2 | Theorem 2.1 in the book [‘Theory of Hp spaces by Peter. L Duren](https://books.google.co.in/books/about/Theory_of_Hp_Spaces.html?id=fs4rPPcJ7HUC&redir_esc=yhttp://) states that : Any function $f$ analytic on the unit disc belongs to the Nevanlinna class iff it is of the form $\frac{g}{h}$ where $g$ and $h$ are bounded ... | https://mathoverflow.net/users/119639 | Regarding representation of an outer function | No, g and h are the exp of the Poisson integral of the parts of log|f| that are less than 1 and bigger than 1 (last one taken with a minus and put in the denominator) respectively
For example, f(z)=(1-z)/(1+z) is outer and is the ratio of the bounded functions 1-z and 1+z as obviously 1/f is not bounded either!
| 0 | https://mathoverflow.net/users/119864 | 291162 | 128,318 |
https://mathoverflow.net/questions/291154 | 3 | Suppose $X/\mathbb C$ is a projective $\mathbb Q$-factorial variety with wild singularities. Let $N$ be a nef **$\mathbb R$-Cartier** divisor. Then is it possible that there are infinitely many curves $C\_i \subset X$, such that $C\_i \cdot N>0$ are infinitely close to $0$?
Notice that if $N$ is a $\mathbb Q$-Cartier... | https://mathoverflow.net/users/29730 | Infinitely small intersections with nef $\mathbb R$-Cartier divisors | This is possible. Basically, if $N$ is a point on the boundary of the nef cone that is not in the span of the rational points on the boundary, then we can approximate $N$ arbitrarily closely by rational points in the interior. For this, we need a variety whose nef cone is not a rational polyhedral cone:
**Example.** ... | 7 | https://mathoverflow.net/users/82179 | 291169 | 128,320 |
https://mathoverflow.net/questions/290975 | 7 | Let $G$ be a connected algebraic group over an algebraically closed field $\overline{k}$ acting on an irreducible variety $X$. A geometric quotient is a morphism of varieties $\pi: X \rightarrow X/\sim$ which on closed points (that is, as a morphism of classical varieties) satisfy the following:
(i): $\pi$ is a surj... | https://mathoverflow.net/users/38145 | Rosenlicht's theorem and rationality questions | Jim Humphreys was right: No generalization is necessary. In Theorem 2 of
[Rosenlicht, Maxwell: Some basic theorems on algebraic groups, Amer. J. Math. 78 (1956) 401--443](https://www.jstor.org/stable/2372523?seq=1#page_scan_tab_contents),
the result is already stated over arbitrary base fields.
| 6 | https://mathoverflow.net/users/89948 | 291177 | 128,321 |
https://mathoverflow.net/questions/291157 | 2 | I'll write the two lemmas I have questions about, and then ask my questions. For reference, I'm using the following definition of Gorenstein:
$\mathbf{Definition\ 1.15}$ A local noetherian ring $A$ is called Gorenstein if $A$ as a module over itself has a finite injective resolution.
Moreover, I am assuming, as sta... | https://mathoverflow.net/users/119460 | A question on some lemmas in Orlov's "Triangulated Categories of Singularities and D-Branes in Landau-Ginzburg Models" (Exts vanishing) | The first question. Denote $Q^{-1} = \mathscr{F}$ and $Q^{k+1} = \mathscr{G}$, so that we have an exact sequence
$$
0 \to Q^{-1} \to Q^0 \to \dots \to Q^k \to Q^{k+1} \to 0.
$$
Consider the spectral sequence whose first term is $\underline{Ext}^i(Q^j,O\_X)$ and which converges to $0$. Then the required comparison is gi... | 2 | https://mathoverflow.net/users/4428 | 291181 | 128,324 |
https://mathoverflow.net/questions/291173 | 21 | There are few results that I am aware of where one can prove something stronger by assuming the existence of Siegel zeros than by assuming the GRH. For example Heath-Brown proved the existence of Siegel zeros imply the twin prime conjecture while it is unknown under GRH. Also there is: Let $P(a,q)$ be the least prime $... | https://mathoverflow.net/users/84272 | Reasons behind assuming the existence of Siegel zeros can be used to prove something stronger than assuming GRH? | Roughly speaking, GRH asserts that the Möbius function $\mu$ is "orthogonal" to all Dirichlet characters $\chi$, in the sense that correlations such as $\sum\_{n \leq x} \mu(n) \overline{\chi(n)}$ are very small. This is the expected behaviour of the Möbius function, and through various standard analytic number theory ... | 47 | https://mathoverflow.net/users/766 | 291182 | 128,325 |
https://mathoverflow.net/questions/291171 | 9 | Let $M$ be a smooth manifold of dimension $\ge 3$, equipped with a conformal structure (or a Riemannian metric). Then, the group of conformal diffeomorphisms is a finite dimensional Lie group.
A proof of this theorem can be found in "Transformation groups in differential geometry" by Kobayashi (Theorem 6.1, pg 143).
... | https://mathoverflow.net/users/46290 | Proofs that the conformal group in dimension $\ge 3$ is a Lie group | Answers to the OP's question depend on where the OP is willing to start. To prove that an abstractly defined group is (i.e., has the structure of) a Lie group, one will have to use *something* nontrivial, as this is not a trivial task, in general.
For example, É. Cartan's statement that the set of (smooth?, $C^k$?, a... | 24 | https://mathoverflow.net/users/13972 | 291183 | 128,326 |
https://mathoverflow.net/questions/291166 | 2 | Consider the Hardy space $H^p, 0<p\leq\infty$ ([defined here](http://math.stackexchange.com/questions/2586370/if-a-function-belongs-to-the-hardy-space)).
It is said that given any two outer functions $x\_1$ and $x\_2$ in $H^p$, there exists $a\_1$ and $a\_2$ in $H^\infty$ such that $a\_1x\_1=a\_2x\_2$ . And $a\_1s$ a... | https://mathoverflow.net/users/119639 | Regarding outer functions again | If $f$ is outer, $1/f$ is outer. (More generally $f$ raised to any real - say non-zero to avoid constants - power is outer.) Any outer function has no zeros as those are factored out with Blaschke products, so we can talk about $\log f$ and any complex power of $f$ in the disk.
$1/f$ always belongs to the Nevanlinna ... | 0 | https://mathoverflow.net/users/119876 | 291190 | 128,327 |
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