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https://mathoverflow.net/questions/270419 | 4 | Is there already name for the generalization of Clothoids to curves on smooth manifolds, i.e. where the curve's curvature depends linearly on the curve's length-parameter?
In the euclidean plane Clothoids are a suitable idealized model for the trajectory of vehicles moving at constant speed while the steering wheel ... | https://mathoverflow.net/users/31310 | Name for Curves from Driving on Smooth Manifolds | Since this is a reference request: in [this article](http://www.ams.org/mathscinet-getitem?mr=1464047), Arroyo, Barros, and Garay just call what you are describing a *Cornu spiral* without much fanfare. ("Cornu spiral" or "Euler spiral" are other common names for the clothoid.)
In [this report/blog post](http://www.t... | 2 | https://mathoverflow.net/users/3948 | 270424 | 121,121 |
https://mathoverflow.net/questions/270386 | 47 | Let $\det\_d = \det((x\_{i,j})\_{1 \leq i,j\leq d})$ be the determinant of a generic $d \times d$ matrix. Suppose $k \mid d$, $1 < k < d$. Can $\det\_d$ be written as the determinant of a $k \times k$ matrix of forms of degree $d/k$?
Even writing $\det\_4$ as the determinant of a $2 \times 2$ matrix of quadratic form... | https://mathoverflow.net/users/88133 | Is the determinant equal to a determinant? | I think a result of Hochster allows to get a quick proof that it is not possible to express the determinant of the generic $d \times d$ matrix as the determinant of a $k \times k$ matrices with entries being homogeneous forms of degree $\dfrac{d}{k}$, provided that $1 <k <d$.
I will work over an algebraically closed ... | 30 | https://mathoverflow.net/users/37214 | 270446 | 121,128 |
https://mathoverflow.net/questions/270416 | 5 | Let $f:\Omega\to\mathbb{R}$ be of $C^k$ class, let $0\in\Omega\subset\mathbb{R}^n$ and let $\Omega$ be star shaped at $0.$
From [Hadamard's Lemma](https://en.wikipedia.org/wiki/Hadamard%27s_lemma) we know that we can write function $f$ as
$$f(x)=f(0)+\sum\_{i=1}^n \overbrace{x\_i\int\_0^1\frac{\partial f}{\partial ... | https://mathoverflow.net/users/62635 | $C^k$ version of Hadamard's Lemma. Differentiability of the remainder | No, not in general. Consider $f(x,y)=(x+y)|x+y|$. This is $C^1$ with partial derivatives $f\_x=f\_y=2|x+y|$, but you lose one derivative when you form
$$
R= x \int\_0^1 2|tx+ty|\, dt = x|x+y| .
$$
| 9 | https://mathoverflow.net/users/48839 | 270452 | 121,130 |
https://mathoverflow.net/questions/270450 | 3 | I am wondering if there is a uniform bound $C$ (independent of $\lambda>10$):
$$\sum\_{k=-\infty}^{-1}\Big|\int\_{2^k}^{2^{1+k}}\frac{\sin(\lambda t^3)}{t}dt\Big|\le C.$$
Remark: (1) An easy upper bound is $C\log\lambda$, since $$\Big|\int\_{2^k}^{2^{1+k}}\frac{\sin(\lambda t^3)}{t}dt\Big|\lesssim \min\{1,\lambda 2^{... | https://mathoverflow.net/users/89123 | Uniform bound for an oscillatory sum | You are not exploiting that the integrands are oscillatory for $\lambda t^3\gtrsim 1$, and the usual way to do this is by integration by parts. Let me change notations slightly and use $n=-k$.
$$
\int\_{2^{-n-1}}^{2^{-n}} e^{i\lambda t^3} \, \frac{dt}{t} =\int\_{2^{-3n-3}}^{2^{-3n}} e^{i\lambda s}\, \frac{ds}{3s} = O(2... | 5 | https://mathoverflow.net/users/48839 | 270453 | 121,131 |
https://mathoverflow.net/questions/270445 | 2 | Let $X$ be a scheme and consider any big site structure on $X$. Is it still true that the inclusion $Qcoh(X) \hookrightarrow Mod(X)$ is exact? In particular, is it left exact? For the usual Zariski topology this is clearly true. But I doubt when the morphisms in a site have no conditions (specifically no flatness) if t... | https://mathoverflow.net/users/56870 | Exactness of the inclusion of quasi-coherent sheaves into the sheaves of modules on a big site | You are probably thinking about the following type of phenomenon:
**Example.** Let $X = \mathbb A\_k^1$, and consider the ringed site $((\operatorname{\underline{Sch}}/X)\_{\text{fppf}},\mathcal O)$, where $\mathcal O$ means the sheaf whose value on $T \in \operatorname{\underline{Sch}}/X$ is $\Gamma(T,\mathcal O\_T)... | 3 | https://mathoverflow.net/users/82179 | 270456 | 121,133 |
https://mathoverflow.net/questions/254651 | 2 | $\newcommand{\M}{M}$
$\newcommand{\N}{N}$
$\newcommand{\TM}{TM}$
$\newcommand{\TN}{TN}$
$\newcommand{\TstarM}{T^\*M}$
$\newcommand{\Ga}{\Gamma}$
Let $\M,\N$ be smooth manifolds, $\phi:\M \to \N$ be a smooth map. Let $\nabla$ be a **symmetric** connection on $\TN$, and let $X,Y \in \Ga(\TM)$.
Then the following hol... | https://mathoverflow.net/users/46290 | Does "symmetry" of a pullback connection should be obvious? | $\newcommand{\id}{\operatorname{Id}}$
Well, there is a natural way to view this "pullback-symmetry":
**Exterior derivative commutes with pullbacks:**
Let $f:M \to N$ be a smooth map, $E$ a vector bundle over $N$ with a connection $\nabla$. Then, there is a pullback operation: $ \Omega^k(N,E) \stackrel{f^\*}{\to} ... | 0 | https://mathoverflow.net/users/46290 | 270464 | 121,137 |
https://mathoverflow.net/questions/270441 | 3 | I am interested in a random graph $G\sim G(n,p)$. I know that if $p<<1/n$, then $G$ will be a forest. I happen to be interested in the boundary case where $p=c/n$, where $c<1$ is a constant. Does $G$ *fail* to be a forest with high probability in this case? Or is it still a forest some positive proportion of the time? ... | https://mathoverflow.net/users/4 | Threshold for appearance of a cycle | In this case, assuming $c\lt 1$ is independent of $n$, the number of cycles is asymptotically Poisson with constant mean $f(c)$. So asymptotically there is a constant nonzero probability $e^{-f(c)}$ that there are no cycles. With a rough calculation that needs checking, I got
$$ e^{-f(c)} = \sqrt{1-c}\,\exp\bigl({{\tex... | 7 | https://mathoverflow.net/users/9025 | 270465 | 121,138 |
https://mathoverflow.net/questions/270448 | 1 | Let $R$ be a Discrete Valutation Ring and let $A$ be a $n \times n$ matrix taking values in $R$. Suppose $A$ has invariant factors $(\alpha\_1, \alpha\_2, \cdots, \alpha\_n)$ (i.e. performing elementary rows and columns moves we get $diag(\alpha\_1, \cdots, \alpha\_n$)). Suppose $U$ is invertible:
>
>
> >
> > Doe... | https://mathoverflow.net/users/110074 | Invariant Factors of a product $AU$ with $U$ an invertible matrix | I'll assume that elementary row and column transformations have determinant one. Otherwise the question is trivial, as darij grinberg pointed out in his comment above. My answer expands on Mohan's comment in this case.
The answer is **yes** and follows from
>
>
> >
> > **Claim**. Let $R$ be a local ring and let... | 2 | https://mathoverflow.net/users/84349 | 270480 | 121,145 |
https://mathoverflow.net/questions/270484 | 3 | I am totally new to microlocal analysis, and have been studying [Jared Wunsch's notes](http://www.math.northwestern.edu/~jwunsch/micronotes.pdf). I have been puzzling over the properties of the wavefront set.
Let $X$ be a compact Riemannian manifold, and $\Psi^m(X)$ denote the space of pseudodifferential operators of... | https://mathoverflow.net/users/104213 | Characterisation of the wavefront set | Your intuitive characterization does not make sense: the function $u$ is defined on some neighborhood of $x\_0$ and $(x\_0,\xi\_0)$ belongs to the sphere bundle. On the other hand, you may salvage part of your statements by DEFINING smoothness at $(x\_0,\xi\_0)$ by your intuitive hunch.
In particular, you have $π\_1(... | 4 | https://mathoverflow.net/users/21907 | 270488 | 121,147 |
https://mathoverflow.net/questions/270487 | 3 | I've encountered the following sequences
$$
a\_k=2^{k+1}\sum\_{j=0}^{k-1}a\_{k-1-j}a\_j,\;a\_0=1
$$
$$
b\_k=(k+1)\sum\_{j=0}^{k-1}b\_{k-1-j}b\_j,\;b\_0=1.
$$
I would like to have an estimate of the growth of these sequences as $k$ grows.
After looking here and there, I found the Catalan's numbers defined by
$$
... | https://mathoverflow.net/users/33135 | Determining the asymptotic behavior of a sequence | One type of Catalan's $q$-analogue is due to Carlitz ([see the paper for this and more](http://homepage.univie.ac.at/josef.hofbauer/85jct_catalan.pdf))
$$C\_{n+1}(q)=\sum\_{k=0}^nC\_k(q)\,C\_{n-k}(q)\,q^{(k+1)(n-k)}, \qquad C\_0:=1.$$
[Blieberger and Kirschenhofer](http://institute.unileoben.ac.at/mathstat/personal/(20... | 5 | https://mathoverflow.net/users/66131 | 270499 | 121,153 |
https://mathoverflow.net/questions/257712 | 15 | Let $R$ be a regular algebra over a field $k$ of char 0. Let $D$ be its corresponding algebra of differential operators.
As in the general setting of non-commutative algebra we can tensor right $D$-modules with left $D$-modules to get $R$-modules. However in this case we have more operations available to us.
Let ... | https://mathoverflow.net/users/22810 | Why should the tensor product of $\mathcal{D}_X$-modules over $\mathcal{O}_X$ be a $\mathcal{D}_X$-module? | OK, I'll give it a shot. The bi-algebra structure on $D$ is something that I found very confusing too, so I will try to spell it out as best I understand. These ideas were explained to me by Pavel Safronov, and I found these notes by Gabriella Bohm be helpful <https://arxiv.org/abs/0805.3806> (though they deal with a m... | 13 | https://mathoverflow.net/users/7762 | 270503 | 121,156 |
https://mathoverflow.net/questions/269993 | 6 | Let $L$ be a finite relational language. Let $T$ be a complete theory with infinite models. If $T$ is an almost sure theory then it has the finite model property (in the sense that any model $M$ of $T$ has the property that if $M\models \varphi$, where $\varphi$ is an $L$ sentence, then there is some finite $L$ structu... | https://mathoverflow.net/users/nan | Sufficient conditions for the finite model property | In my paper [Disjoint $n$-amalgamation and pseudofinite countably categorical theories](https://arxiv.org/abs/1510.03539), I did some work on this question for the restricted class of countably categorical theories. (The question is already very hard for these theories! For example, the finite model property for the th... | 2 | https://mathoverflow.net/users/2126 | 270505 | 121,158 |
https://mathoverflow.net/questions/270519 | 3 | I know that if we assume zero sharp we obtain for a class of standard ordinals
$L\_\alpha\prec L\_\beta$ , $\alpha <\beta$ and obviously $L\_\alpha\in L\_\beta$.
My question is :
If we only assume the existence of a standard transitive model ,and from it we have a countable standard ordinal $\alpha$, $L\_\alpha\model... | https://mathoverflow.net/users/104010 | Elementary extension $L_\alpha\prec N$ such that $L_\alpha \in N$ | The answer is no, there can be no such elementary end-extension $N$ of the minimal model $L\_\alpha$ with $L\_\alpha\in N$. The reason is that $N$ would have to be $\omega$-standard, since it has the same $\omega$ as $L\_\alpha$, and thus $N$ would have the correct understanding of the theory ZFC, and it would think th... | 8 | https://mathoverflow.net/users/1946 | 270522 | 121,166 |
https://mathoverflow.net/questions/270542 | 0 | Let $f$ be a compactly supported function in $\Omega \subset \mathbb{R}^3$ and
$\Delta u=f$ in $\Omega$
such that $D^{\alpha}u=0$ on $\partial \Omega$ for every multi-index $\alpha$ with $|\alpha| \geq 0$. Is $f$ necessarily zero?
| https://mathoverflow.net/users/42326 | Solution of Poisson equation vanishing at the boundary of any order | No. Take any $u$ which is not zero, but compactly supported in $\Omega$. Then
define $f=\Delta u$; it will be also compactly supported, and non-zero.
| 4 | https://mathoverflow.net/users/25510 | 270544 | 121,170 |
https://mathoverflow.net/questions/270534 | 6 | I have quite a practical question motivated by physics.
Consider the Riccati equation whose solution gives a quantum-mechanical (QM) analogue of the classical momentum:
$$
(p(x))^2 + \dfrac{\hbar}{i}p'(x)= 2 (E-V(x)) \quad.
$$
Clearly, in the $\hbar\to0$ limit one obtains the definition of the classical momentum:
$... | https://mathoverflow.net/users/100168 | Riemann surface from Riccati equation | The answer to the highlighted question is "no". When $V$ is a polynomial, the
general solution of the Riccati equation is single valued, it is a meromorphic function in the complex plane. To prove the statement, make the coefficient at
$p'$ equal to $1$ by scaling of $x$, and then reduce your Riccati equation to a line... | 9 | https://mathoverflow.net/users/25510 | 270546 | 121,171 |
https://mathoverflow.net/questions/270526 | 14 | Is it possible to give an example of $n$ dimensional manifold with the property that the tangent bundle $TM$ cannot be expressed as Whitney sum of two subbundles? It is certain true for two sphere; it is certainly not true for three dimensional manifold since every three manifold is parallelizable. You can always split... | https://mathoverflow.net/users/24078 | Splitting of tangent bundle | To expand on my comment above, suppose that $TS^{2k}\cong\xi\oplus\eta$ for some non-trivial vector bundles $\xi$ and $\eta$ over $S^{2k}$ of dimensions $m$ and $\ell$, respectively. Hence $0<m,\ell<2k$. Since $S^{2k}$ is simply-connected, both $\xi$ and $\eta$ are oriented, hence possess Euler classes
$$
e(\xi)\in H^m... | 17 | https://mathoverflow.net/users/8103 | 270549 | 121,173 |
https://mathoverflow.net/questions/270537 | 2 | For varieties $X,Y$ over an algebraically closed field, and a surjective morphism $f:X\rightarrow Y$, $\dim f^{-1}(y)\geq\dim X-\dim Y$ for all closed $y\in Y$, and $\dim f^{-1}(y)=\dim X-\dim Y$ for all closed $y$ in a nonempty open subset of $Y$. If the requirement that $X$ and $Y$ are of finite type is dropped, and ... | https://mathoverflow.net/users/83073 | Fiber Dimension Theorem for infinite-dimensional schemes | (1) is wrong in this generality since the fiber dimensions can be different on dense subsets. Let, e.g., $R:=\mathbb C[x]$ and $S$ the localization of $R$ in $E$ where $E$ is the multipliciatively closed subset of all polynomials which do not have a zero in $\mathbb Q$. Put $X=\text{Spec }S$ and $Y=\text{Spec }R$. Then... | 3 | https://mathoverflow.net/users/89948 | 270560 | 121,178 |
https://mathoverflow.net/questions/270547 | 3 | Suppose that we have a Markov process $\{Z\_t\}\_{t=0}^\infty$, where $Z\_t \geq 0$ for any $t$. Assume that, conditioning on $Z\_t = z\_t$, we have
$
\mathbb{E}\{Z\_{t+1}|Z\_t = z\_t\} \leq \kappa z\_t^2
$. Here $\kappa > 0$ is a constant.
Question: Conditioning on that the realization of $Z\_0$ is sufficiently sm... | https://mathoverflow.net/users/82358 | Superlinear Convergence of a Markov Chain | It is not true in general.
If $Z\_t$ is not bounded, the expectation can diverge to infinity. For example, define:
$$ Z\_t = \left\{\begin{array}{lc}2^{2^{t}} & \text{with probability $\epsilon/2$}\\0 &\text{otherwise}\end{array}\right.$$
Then $E[Z\_0]=\epsilon$ and $Z\_{t+1} = (Z\_t)^2$ but $E[Z\_t]$ diverges to... | 3 | https://mathoverflow.net/users/90045 | 270567 | 121,179 |
https://mathoverflow.net/questions/270543 | 9 | The question is a very little more than what's in the title. It is easy (for some values of ‘easy’) to produce examples of endoscopic groups that are not subgroups. When I asked a colleague, he mentioned $\mathrm{PGL}\_3$ as an endoscopic group of $\mathrm G\_2$. However, in this example, $\mathrm{PGL}\_3$ is isogenous... | https://mathoverflow.net/users/2383 | Endoscopic group that is not a subgroup | If I understand the definition correctly, a connected reductive group $H$ is an endoscopic group for a connected reductive group $G$ if its Langlands dual $H^\vee$ is a connected centralizer in $G^\vee$.
So $H=SO(2p+1)\times SO(2q+1)$ is endoscopic in $G=SO(2p+2q+1)$ since $H^\vee=Sp(2p)\times Sp(2q)$ is a centralize... | 12 | https://mathoverflow.net/users/89948 | 270573 | 121,181 |
https://mathoverflow.net/questions/270565 | 8 | Let $\text{FinVec}$ denote the category of finite dimensional vector spaces over some field $k$, and let $F:\text{FinVec}\to \text{FinVec}$ be a contravariant functor such that $F^2$ is naturally isomorphic to the identity. Is $F$ naturally isomorphic to the canonical duality functor $V\mapsto V^\*=\text{Hom}(V,k)$?
... | https://mathoverflow.net/users/745 | Are there other dualities on finite vector spaces besides the canonical one? | $FinVec$ and its opposite are enriched in finite dimensional $k$-vector spaces. Assume that $F$ is an enriched functor. Then consider the covariant functor:
$$F(-)^\*: FinVec \to FinVec$$
It is a $Vec$-enriched functor. Finite direct sums are absolute limits and so are preserved by this functor ([see here](https:/... | 10 | https://mathoverflow.net/users/184 | 270574 | 121,182 |
https://mathoverflow.net/questions/270523 | 4 | I am currently investigating some finite speed propagation property for nonlinear wave equations, and I am asking myself if there is a way of proving such a property just using energy estimates like in [the linear wave equation](https://mathoverflow.net/questions/172172/finite-speed-of-propagation-of-wave-equation), wh... | https://mathoverflow.net/users/94414 | Finite speed propagation by finite energy method | For an equation that is actually hyperbolic, this is well-known. Here are some classical references:
* Lars Garding, *Cauchy's Problem for Hyperbolic Equations* (1958)
* Jean Leray, *Hyperbolic Differential Equations* (Institute of Advanced Study, 1953): see especially Chapter VI, section 4 as well as the extensions ... | 6 | https://mathoverflow.net/users/3948 | 270576 | 121,183 |
https://mathoverflow.net/questions/270590 | 2 | In the 1994 paper [On the Maxwell-Klein-Gordon Equation with Finite Energy](https://projecteuclid.org/download/pdf_1/euclid.dmj/1077288008) of Klainerman and Machedon, the proof of Proposition 1.1 contains the following statement. For $\phi$ (the scalar field of) a classical solution of the Maxwell-Klein-Gordon system,... | https://mathoverflow.net/users/104213 | Maxwell-Klein-Gordon energy estimates in Klainerman and Machedon's 1994 paper | There is a typo in the paper. Look at the bottom two lines of page 22 which I transcribe here
\begin{align}
\|\phi\|\_{L^3} & \leq \ldots \\
& \lesssim \mathscr{I}\_0^{1/2} (1+t)^{1/2} ( \mathscr{I}\_0 + \|A\|\_{L^6} \|\phi\|\_{L^3} )^{1/2}\\
& \lesssim \mathscr{I}\_0 \color{red}{(1+t)} (1 + \|\phi\|\_{L^3})^{1/2}
... | 2 | https://mathoverflow.net/users/3948 | 270591 | 121,188 |
https://mathoverflow.net/questions/270414 | 0 | Let $ n $ denote a square free positive composite integer, $ \omega(n) $ its number of prime factors, $ P\_{i}(n) $ its $ i $ -th prime factor.
Can we determine an asymptotics for the number of $ n $ below $ x $ such that $ P\_{\omega(n)}-P\_{1}(n)<\left(\dfrac{n}{\omega(n)}\right)^{1/\omega(n)} $?
| https://mathoverflow.net/users/13625 | Asymptotics for a peculiar kind of squarefree numbers | Please ignore the previous version of this answer.
Motivated by Lucia's comment, we use smooth numbers to show that the number in question is $o(x)$. First note that for $100\%$ of all integers one has
$P\_1(n)\leq \log \log \log n$, as a typical application of the naive Eratosthenes' sieve. Also by Hardy-Ramanujan ... | 2 | https://mathoverflow.net/users/9232 | 270601 | 121,191 |
https://mathoverflow.net/questions/270603 | 14 | I am preparing a presentation discussing the structure of the cut locus on Riemannian and Finsler manifolds. Since my audience will consist mostly of applied mathematicians, I was hoping to include some nice applications to other (non-geometric) areas of mathematics which might appeal to the majority of the audience. W... | https://mathoverflow.net/users/100615 | Applications of cut locus structure theorems | The cut locus of a point on a compact manifold has zero measure. So this allows to use a single chart centered at the point e.g. to compute integrals (typically radial coordinates in a chart given by the exponential).
You can exemplify this with the sphere (easy, the cut locus is a single point) and tori. The tori ar... | 9 | https://mathoverflow.net/users/6129 | 270613 | 121,195 |
https://mathoverflow.net/questions/270458 | 6 | In Wilson's paper [*"The structure of the level surfaces of a Lyapunov function,"*](http://www.sciencedirect.com/science/article/pii/0022039667900356) he states in Corollary 1.3 that the level sets of a smooth Lyapunov function are diffeomorphic to a standard sphere. (The Lyapunov function is for a globally asymptotica... | https://mathoverflow.net/users/89166 | On Wilson's claim that Lyapunov function level sets are not exotic spheres | Matthew. I had a look at Wilson's paper; he is of course rigourous; he says that
* $V^{-1}(c)$ is homotopy-equivalent to $S^{n-1}$ for every $n$;
* $V^{-1}(c)$ is diffeomorphic to $S^{n-1}$ for every $n\neq 4, 5$
which corresponds to what was known in 1967. Nowadays, one can say a little more:
* $V^{-1}(c)$ is al... | 6 | https://mathoverflow.net/users/105095 | 270618 | 121,198 |
https://mathoverflow.net/questions/270617 | 11 | I am learning étale cohomology, and we defined the étale cohomological dimension of a scheme $X$ as the minimum of $n$ where $H\_{ét} ^n (X,F)$ vanishes for all the torsion sheaves $F$, yet I don't get why we only restrict our view to torsion sheaves.
This seems to be a very stupid question, but I really have no idea... | https://mathoverflow.net/users/110342 | Why define étale cohomological dimension as it is defined? | I think it's a combination of two things:
First, we do etale cohomology with torsion sheaves only, because of pathologies with non-torsion sheaves, and so we care about the cohomological dimension with torsion sheaves only.
Second, the cohomological dimension is actually different if we include nontorsion sheaves. ... | 11 | https://mathoverflow.net/users/18060 | 270623 | 121,200 |
https://mathoverflow.net/questions/270628 | 2 | Suppose $\mathcal{A}$ is a sub-algebra of $C([0,1],\mathbb{R})$.
If $\mathcal{A}$ separates points in $[0,1]$, does it follow $\dim\mathcal{A}=\infty$?
| https://mathoverflow.net/users/110332 | points separation and dimensions | Without loss of generality, we may assume that $\mathcal{A}$ contains the constant functions (otherwise, just add them to $\mathcal{A}$, it will increase the dimension by $1$). Hence, $\mathcal{A}$ satisfies the conditions of Stone-Weierstrass theorem, and so is dense in $\mathcal{C}([0,1],\mathbb{R})$.
Now if $\dim\... | 4 | https://mathoverflow.net/users/53155 | 270631 | 121,203 |
https://mathoverflow.net/questions/270636 | 1 | Let $X$, $Y$ be Banach spaces. The set of all bounded linear operators from $X$ to $Y$ is denoted by $L(X,Y)$. $L(X,Y)$ becomes a Banach space with the operator norm.
The set of all compact linear operators from $X$ to $Y$ is denoted by $K(X,Y)$.
It is well known that $K(X,Y)$ is a closed subspace of $L(X,Y)$.
*... | https://mathoverflow.net/users/68463 | Additional conditions for uniform convergence | Hard to know what you are looking for but a necessary and sufficient cndition would be that the sequence be relatively compact in the norm topology.
| 1 | https://mathoverflow.net/users/106519 | 270647 | 121,209 |
https://mathoverflow.net/questions/270657 | 4 | What is the difference of Bass and Quillen K-theory groups of a ring $R$.
More concretely, what is $K^{Bass}\_{i}(R)$? does it equal to $K^{Quillen}\_{i}(R)$ if $R$ is a regular ring ?
Does it make sense to say that for some rings, the Bass and the Quillen K-theory agree ?
I read that Bass K-theory is sometime... | https://mathoverflow.net/users/82229 | Bass and Quillen K-theory |
>
> I read that Bass K-theory is sometimes called the non-connective K-theory
>
>
>
Basically, Quillen $K$-theory can be viewed as a spectrum (in the sense of topology), instead of just as a space (this is an important enrichment, not just a generalization for sake of generalization!). A spectrum $X$ also has ho... | 6 | https://mathoverflow.net/users/48362 | 270664 | 121,216 |
https://mathoverflow.net/questions/270530 | 15 | I am trying to prove or disprove
$$\sum\_{k=1}^{\infty}e^{-\lambda\_{k}t}c\_{k} \xrightarrow{t\to 0} \sum\_{k=1}^{\infty}c\_{k} ,$$
where $\sum c\_{k}<\infty, \sum c\_{k}^{2}<\infty\text{ and }\frac{\lambda\_{k}}{k}\to c$(Weyl's law). The $c\_{k},\lambda\_{k},t\in \mathbb{R}$ and
**Update**: the $\lambda\_{k}$ a... | https://mathoverflow.net/users/99863 | Tauberian theorem $\sum_{k=1}^{\infty}e^{-\lambda_{k}t}c_{k} \xrightarrow{t\to 0} \sum_{k=1}^{\infty}c_{k} $ | The $\lambda\_k$ correspond to the Laplacian eigenvalues of a domain in $\mathbb{R}^2$ implies that
\begin{equation}
0<\lambda\_1\le \lambda\_2\le\cdot\cdot\cdot\le \lambda\_n\le\cdot\cdot\cdot
\end{equation}
and $\lambda\_n\rightarrow\infty $ as $n\rightarrow\infty$. Hence for any $t>0$
\begin{align}
\sum\_{n=1}^{\inf... | 2 | https://mathoverflow.net/users/110367 | 270671 | 121,218 |
https://mathoverflow.net/questions/270595 | 5 | Let $S$ be a closed surface of genus $g > 0$ and $[S,S] = Hom(\pi\_{1}(S),\pi\_{1}(S))$ be the monoid of (homotopy classes of) continuous maps from $S$ to itself. Consider the semi-group $A$ of elements that induce the $0$ map on $H\_{2}(S,\mathbb{Z})$.
**Question.** Is there a "nice" set of generators for $A$? (Mayb... | https://mathoverflow.net/users/99732 | Monoid of continuous self-maps of (real) surfaces | Here's some sort of geometric description of maps of degree 0. I don't know how realistically you can find a generators-and-relations description of the semigroup of such maps.
It is a theorem due to Kneser that I learned from [this answer](https://mathoverflow.net/a/20759/40804) that any map of degree 0 between clos... | 2 | https://mathoverflow.net/users/40804 | 270684 | 121,225 |
https://mathoverflow.net/questions/270434 | 9 | Dear Mathoverflow Community,
Suppose that $\Omega$ is a domain in the Riemann Sphere $\widehat{\mathbb{C}}$ with $\infty \in \Omega$, and assume that every connected component of $\partial \Omega$ is a round circle.
Let $\Gamma(\Omega)$ denote the Schottky group of $\Omega$, that is, the free group of Möbius and an... | https://mathoverflow.net/users/1162 | Can the limit set of an infinitely generated Schottky group have positive area? | Two relevant references:
1. W. Abikoff, Some remarks on Kleinian groups. 1971 Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969) pp. 1–5. Ann. of Math. Studies, No. 66. Princeton Univ. Press, Princeton, N.J.
Among other things, he constructs an infinitely generated free Kleinian subgr... | 5 | https://mathoverflow.net/users/21684 | 270687 | 121,226 |
https://mathoverflow.net/questions/270686 | 6 | Is it consistent to have a $(\kappa,\kappa,2)$-saturated ideal $I$ on $\kappa$ that is $\kappa$-complete and $\kappa$ is not weakly compact? Here $\kappa$ is inaccessible. An ideal is $(\kappa,\kappa, 2)$-saturated, if for any collection $\{A\_i: i<\kappa\}\subset I^+$, there exists a sub collection of size $\kappa$ su... | https://mathoverflow.net/users/23835 | $(\kappa, \kappa, 2)$-saturated ideals? | Suppose $\kappa$ is regular uncountable and $I$ is a $\kappa$-aditive ideal on $\kappa$ such that forcing with $I$ is $\kappa$-Knaster. Force with $I$: Let $G$ be a generic filter and $j:V \to M \subseteq V[G]$ be the generic embedding with critical point $\kappa$ ($M$ is the well founded generic ultrapower). Let $T$ b... | 5 | https://mathoverflow.net/users/110372 | 270691 | 121,230 |
https://mathoverflow.net/questions/270666 | 2 | Denote by $\text{SO}\_0(1,4)$ the identity component of the special linear isometry group $\mathrm{SO}(1,4)$ of the Lorentz-Minkowski space $\mathbb{R}\_1^5$, that is, of
$$\text{SO}(1,4)=\left\{X\in\text{SL}(5,\mathbb{R})\mid X^tI\_{1,4}X=I\_{1,4}\right\},\quad\text{where}\;I\_{1,4}:=\text{diag}(-1,1,1,1,1).$$
Is ther... | https://mathoverflow.net/users/84866 | Matrix expression for elements of $\text{SO}_0(1,4)$ | The [Cartan decomposition](https://en.wikipedia.org/wiki/Cartan_decomposition#Cartan_decomposition_on_the_Lie_group_level):
$$
\mathrm{SO}\_0(1,4)=\left\{\left(
\begin{array}{c|c}
1&0\\\hline
0&A
\end{array}\right)
\exp\left(\begin{array}{c|c}
0&{}^tb\\\hline
b&0
\end{array}\right): A\in\mathrm{SO}(4), b\in\mathbf R^{4... | 1 | https://mathoverflow.net/users/19276 | 270708 | 121,238 |
https://mathoverflow.net/questions/270705 | 2 | Let $f\in k[x\_0,...,x\_n]\_d$ be a degree $d$ homogeneous polynomial in $n+1$ variables.
Is there a way to associate to $f$ a form $g(y\_1,...,y\_m)$ which is symmetric in the sets of binary variables $y\_i = [y\_0^i:y\_1^i]$ of degree $d\_i$ in each set for a suitable choice of $m$ and $d\_i$?
For instance we co... | https://mathoverflow.net/users/14514 | Homogeneous polynomials and symmetric binary forms | Not yet an answer but you can see $f$ as an element of the symmetric power $S^d(V^{\vee})$ for $V$ a vector space of dimension $n+1$. You can take $V=S^n(W)$ with $W$ of dimension 2. That should give something like your representation for $g$. The indices labeling the $x$ basis of $V^{\vee}$ become labels for monomials... | 2 | https://mathoverflow.net/users/7410 | 270710 | 121,239 |
https://mathoverflow.net/questions/270624 | 13 | For a polynomial $f(x) = \sum\_{i=0}^dc\_ix^i \in \mathbb Z[x]$ of degree $d$, let
$$
H(f):=\max\limits\_{i=0,1,\ldots, d}\{|c\_i|\}
$$
denote the naive height. Further, define
$$
R(M, r, d) := \#\{f(x) \colon \text{$H(f) \leq M$, $\deg f = d$ and $f(x)$ has extactly $r$ real roots}\}.
$$
I wonder if anything i... | https://mathoverflow.net/users/22733 | Proportion of polynomials of a fixed degree with a certain number of real roots | The quadratic case can be dealt with as follows. A quadratic polynomial $f(x) = ax^2 + bx + c \in \mathbb{Z}[x]$ has two distinct real roots if and only if $\Delta(f) = b^2 - 4ac > 0$, and a pair of complex conjugate roots if and only if $\Delta(f) < 0$.
We now let $a,b,c$ vary in the box $[-X,X]^3$. We first pick a... | 8 | https://mathoverflow.net/users/10898 | 270714 | 121,240 |
https://mathoverflow.net/questions/270712 | 0 | Here is a question I have asked on Math Stack Exchange <https://math.stackexchange.com/questions/2290917/prime-numbers-property-mertens-theorem-related-sequence> , that I would like this community to address. Yes someone responded there, but it seems to have some errors. So: Merten's third theorem states that $$ \lim\_... | https://mathoverflow.net/users/110322 | Prime numbers property. A Merten's third theorem like sequence | I don't think this question is of research level, but let me answer it. Using that the natural logarithm is a concave function, we have
$$ \log T\_\alpha(p\_n)=\alpha\log p\_n+\sum\_{i=1}^n\log\left(1-\frac{1}{p\_i^\alpha}\right)<\alpha\log p\_n-\sum\_{i=1}^n\frac{1}{p\_i^\alpha}.\tag{$\*$}$$
Using the Lebesgue-Stieltj... | 6 | https://mathoverflow.net/users/11919 | 270715 | 121,241 |
https://mathoverflow.net/questions/270728 | 2 | t's probably common knowledge that there are Diophantine equations which do not admit any solutions in the integers, but which admit solutions modulo nn for every nn. This fact is stated, for example,
**Conjecture** Let $m,n $be integer,find all
$$m^4+n^4=10m^2n^2+1$$
I this equality have only foursolution$(m,n)=(0... | https://mathoverflow.net/users/38620 | Solve this diophantine equation: $m^4+n^4=10m^2n^2+1$ | You have already listed all the possible solutions $(m,n)$ in which either $m=0$ or $n=0$.
Let us suppose that $(m,n) \in \mathbb{N} \times \mathbb{N}$ is a solution of your equation. Then, the discriminant $\Delta$ of the polynomial
$$p(z)=z^{2} - 10 \, z \,n^{2} +(n^{4}-1)$$
is necessarily a perfect square. Sin... | 9 | https://mathoverflow.net/users/1593 | 270729 | 121,244 |
https://mathoverflow.net/questions/270638 | 2 | Let $X$ be a smooth projective surface such that $\chi(O\_X)=1$ and $K\_X^2\geq 3$(or just $K\_X^2>0$). Let $D$ be $(-1)$-class, i.e: $D^2=-1,D^2+D.K\_X=-2$(equivalently, $D^2=-1, \chi(-D)=0$). I wonder whether we can show that $D$ is effective.
If $X$ is rational surface and $K\_X^2>0$, then such $D$ is effective. ... | https://mathoverflow.net/users/41650 | Effectivity of $(-1)$-class on smooth projective surface | It is easy to find counterexamples with $X$ of general type. Take $S$ of general type with $\chi(S)=1$, $K^2\_S>1$ and nonzero torsion in $Pic(X)$ (there are plenty of such surfaces, even with $h^1(\mathcal O)=h^1(\mathcal O)=0$). Now blow up $S$ to get $X$ and an effective $-1$ curve $E$ and set $D=E+L$, where $L$ is ... | 3 | https://mathoverflow.net/users/10610 | 270737 | 121,248 |
https://mathoverflow.net/questions/270734 | 0 | It is well-known that a *connected* topological group can be generated by any neighborhood of the identity. There are non-connected topological groups for which this is still true, such as $\mathbb{Q}$. My question is: is there any characterization of those non-connected topological groups for which the result is true?... | https://mathoverflow.net/users/110393 | system of generators for non-connected topological groups | Immediate restatements: let $G$ be a topological group. Then $G$ is generated by any of its neighborhoods of 1 $\Leftrightarrow$ the only open subgroup of $G$ is $G$ $\Leftrightarrow$ $G$ has no nontrivial continuous action on any discrete set. All these properties hold if $G$ is connected and you're asking about when ... | 2 | https://mathoverflow.net/users/14094 | 270741 | 121,249 |
https://mathoverflow.net/questions/270748 | 14 | Let's say I have a smooth irreducible subvariety $X$ of $\mathbb{CP}^n$ with some fixed Hilbert polynomial. What are the best bounds known for the sum of the Betti numbers of $X$? That such a bound exists follows from the boundedness of (a component of) the Hilbert scheme.
I imagine that one could get a reasonable bo... | https://mathoverflow.net/users/51424 | Bounds on Betti numbers of subvarieties? | There is a very recent paper by Zak : <http://mathecon.cemi.rssi.ru/zak/files/Castelnuovo%20Bounds%20for%20Higher%20Dimensional%20Varieties.pdf> which deals with this issue. He has found many new bounds on the total Betti numbers. For instance, he proves that if $X \subset \mathbb{P}^n$ is smooth of degree $d$ and of c... | 17 | https://mathoverflow.net/users/37214 | 270751 | 121,253 |
https://mathoverflow.net/questions/270740 | 3 | A real entire function
$$\psi(x)=\sum\_{k=0}^{\infty} \gamma\_k\frac{x^k}{k!}$$
is said to be in the Laguerre-Polya class, denoted $\psi(x) \in \mathcal{LP}$, if it can be represented in the form
\begin{eqnarray\*}
\psi(x)=c x^m e^{-\alpha x^{2}+\beta x} \prod\_{k=1}^{\infty}\left(1+x/x\_{k}\right)e^{- x/x\_k},
\en... | https://mathoverflow.net/users/110052 | Turan Inequalities | These were first proved by Laguerre himself (They are called sometimes Laguerre's inequalities). The LP class can be characterized as the closure of real polynomials with real zeros, so it is enough to prove the inequalities for real polynomials whose all zeros are real, and this is sort of elementary.
First notice t... | 4 | https://mathoverflow.net/users/25510 | 270753 | 121,254 |
https://mathoverflow.net/questions/270702 | 4 | Suppose $f(z)=c\_1z+c\_2z^2+\cdots+c\_nz^n$ is a univalent map on the unit disk.
You may assume $c\_1=1$. All coefficients are complex.
Is there a sharp bound on the modulus of the last coefficient $c\_n$?
How about the other coefficients?
| https://mathoverflow.net/users/110332 | Bounds on coefficients: univalent maps | There is something to say about $c\_n$, if not much about the lower order coefficients.
Since $f(z)$ is a *schlicht* function, $f'(z)\neq0$ throughout the disk. That means, each root of the polynomial $f'(z)$ has modulus $\vert z\vert\geq1$. From
$$f'(z)=1+2c\_2z+\cdots+nc\_nz^{n-1},$$
we know that the product of the... | 6 | https://mathoverflow.net/users/66131 | 270760 | 121,257 |
https://mathoverflow.net/questions/270762 | 6 | Let $A\_1\leftarrow A\_2\leftarrow A\_3\leftarrow\dotsb$ be a projective system of abelian groups with the projection maps $p\_{ij}\colon A\_j\to A\_i$, $j\ge i$. The derived functor of projective limit $\varprojlim\_n^1 A\_n$ is constructed as the cokernel of the map
$$
\mathrm{id}-\mathit{shift}\colon\prod\nolimits\... | https://mathoverflow.net/users/2106 | The Mittag-Leffler condition as necessary and sufficient | This is due to Emmanouil as far as I know. See [this](http://dx.doi.org/10.1016/0040-9383(94)00056-5).
| 8 | https://mathoverflow.net/users/110414 | 270765 | 121,258 |
https://mathoverflow.net/questions/270735 | 8 | A result of Sole, Planat and Omar's paper, ''Quantum mechanics and the Riemann Hypothesis'', (Theorem 2 with $b=2$), says the RH is equivalent to the statement that for every large enough integer $k$, one has
$$\dfrac{\sigma(N\_k)}{N\_{k}\log\log N\_{k}} > \dfrac{6e^{\gamma}}{\pi^2} $$
where $\sigma(u)$ is the sum ... | https://mathoverflow.net/users/110396 | Ambiguity in Nicolas' criterion for the Riemann Hypotheis? | From the paper you mention and the result of Nicolas, the following three statements are known to be logically equivalent:
1. The Riemann hypothesis.
2. The inequality $\frac{N\_k}{\phi(N\_k) \log\log N\_k} > e^\gamma$ holding for all $k$. (Nicolas's criterion)
3. The inequality $\frac{N\_k}{\phi(N\_k) \log\log N\_k}... | 25 | https://mathoverflow.net/users/766 | 270768 | 121,259 |
https://mathoverflow.net/questions/270782 | 19 | Suppose one wants to use a theorem that was published quite a long time ago (+80 years) in a paper that is using a terminology and notations that are very much out-dated (making the paper very hard to read). Is it okay if we want to reformulate the result as well as the proof in an article using a more modern language ... | https://mathoverflow.net/users/103312 | Reproving a known theorem in an article | I don't think academic math comes up with general policies for things like this. The dreaded "common sense" should be applied. How well known is the old result, for example? Some things are very old but everyone knows them.
Or..to give an extreme example: Suppose your result was little more than a corollary of an ol... | 14 | https://mathoverflow.net/users/93613 | 270784 | 121,265 |
https://mathoverflow.net/questions/270806 | 5 | Let $f(x,y,t):[-1,1]^3\to \mathbb{R}$ be a real-analytic function. Assume that for any fixed $x,y$, $f(x,y;t)$ is not a constant function $[-1,1]\to \mathbb{R}$. Since the zeros of a non-constant real-analytic function of one variable are isolated, we denote the number of the zeros of $f(x,y;t)$ on $[-1,1]$ by $N(x,y)$... | https://mathoverflow.net/users/89123 | Number of zeros of a real analytic function | Let's do this for two variables $x,t$ rather than three for ease of notation. Also, I assume that by real analytic on a compact set, you mean real analytic on some open neighborhood of this set.
Then your claim follows because $N(x)$, with the zeros counted according to multiplicity, is upper semicontinuous, so if $N... | 2 | https://mathoverflow.net/users/48839 | 270808 | 121,275 |
https://mathoverflow.net/questions/270804 | 7 | If I have an entire function give as a power series $f(z)=\sum\_{i=0}^{\infty}a\_iz^i$, is there a way/technique to check if the function is surjective? Weierstarss factorization theorem gives that $f$ is not surjective if and only if $f=\exp(g)+c$ for some $\require{cancel}\cancel{surjective}$ entire function $g$. How... | https://mathoverflow.net/users/69275 | Surjective entire functions | Your statement that if $f$ is not surjective, then $f=e^g+c$ where $g$ is surjective is wrong: $g$ does not have to be surjective. Example: $f(z)=e^{e^z}$. By the way, this example permits an infinite iteration: there
is an infinite sequence of entire functions $f\_n$ such that all of them are zero-free, and $e^{f\_{n+... | 18 | https://mathoverflow.net/users/25510 | 270818 | 121,279 |
https://mathoverflow.net/questions/270756 | 3 | An ideal $\mathcal{I}$ on the positive integers $\mathbf{N}$ is a *P-ideal* if for every sequence $(A\_n)$ of sets in $\mathcal{I}$ there exists $A \in \mathcal{I}$ such that $A\_n\setminus A$ is finite for all $n$.
Moreover, an ideal $\mathcal{I}$ is said to be *analytic* if (equipping $\mathcal{P}(\mathbf{N})$ with... | https://mathoverflow.net/users/32898 | Existence of maximal analytic P-ideal | As already mentioned in the comments, a free ultrafilter considered as a subset of Cantor space (or Cantor set) cannot be analytic, so the answer to the Question 1 is No. (Even without the assumption that the given ideal is P-ideal; we get that no maximal ideal can be analytic.)
This follows immediately if we show th... | 4 | https://mathoverflow.net/users/8250 | 270822 | 121,281 |
https://mathoverflow.net/questions/270776 | 1 | Let $(\Omega,\mathcal{F},P)$ be a probability space, $\{\mathcal{F}\_n \subseteq \mathcal{F}\}\_{n \in \mathbb{N}}$ an increasing filtration, $S$ a finite set and $X: \Omega \rightarrow S$ a random variable. Denote $\mathcal{P}(\Omega)$ the space of probability measures on $(\Omega, \mathcal{F})$ and assume that $\{Q\_... | https://mathoverflow.net/users/11146 | Sum of information gains is almost surely convergent? | Let me know if I'm talking nonsense but it looks like if you just consider the usual entropy $H(P)=\sum\_k P(k)\log P(k)$ for probability measures $P$ on $\mathbb N$, then
$$
E[H(X\_\*Q\_{n+1})]-E[H(X\_\*Q\_{n})]= E I\_n
$$
simply because of the identity
$$
p\log p-q\log q=p\log \frac pq+(p-q)\log q
$$
(the linear in ... | 2 | https://mathoverflow.net/users/1131 | 270830 | 121,284 |
https://mathoverflow.net/questions/270829 | 14 | This is a followup from [a question](https://math.stackexchange.com/questions/2284672/smoothness-of-on-equivariant-maps-of-positive-definite-matrices) I asked on math.SE, which received a helpful answer but unfortunately not a complete one. $\def\Sym{\mathrm{Sym}\_{n\times n}}$
$\def\s{\mathrm{Sym}}\def\sp{\s^+}$Let $\... | https://mathoverflow.net/users/33510 | If an equivariant map is smooth on diagonal matrices, is it smooth everywhere? | I think, one can argue as follows.
1. Let $D\subseteq\text{Sym}$ be the diagonal matrices. Since $\exp:D\to D^+$ and $\exp:\text{Sym}\to\text{Sym}^+$ are compatible diffeomorphisms it suffices to answer the analogous problem for $D\subseteq\text{Sym}$.
2. For $m=0,\ldots,n-1$ let $c\_m:D\to D:(x\_i)\mapsto(x\_i^m)$. ... | 9 | https://mathoverflow.net/users/89948 | 270847 | 121,290 |
https://mathoverflow.net/questions/270852 | 1 | I am looking for an example of an infinite dimensional $C^{\*}$-algebra whose second dual is amenable. Can anyone supply a suggested reference? Many thanks in advance.
**Edit:** If this is inappropriate for overflow, please forgive my rudeness.
| https://mathoverflow.net/users/48568 | Example of an amenable enveloping von-Neumann algebra | It was a silly question. Any subhomogeneous $C^{\*}$-algebra will suffice.
| 1 | https://mathoverflow.net/users/48568 | 270854 | 121,293 |
https://mathoverflow.net/questions/270779 | 2 | An operation on a category $C$ is a functor $$F: C^n \to C.$$
I'd like to detect those construction that are *universal.* Easiest example is coproduct, which is a colimit.
In this sense to be universal means that there is a small category $I$ and a functor $$ Q : C^n \times I \to C $$ such that $$\text{colim} Q(c\_... | https://mathoverflow.net/users/104432 | Detecting Universals | I think you are not asking the right question. A distinguishing feature of the coproduct as an operation $C^n \to C$ (on categories with coproducts) is that it is (lax) natural *in $C$*: given a functor $F : C \to D$ where $D$ also has coproducts we get an induced diagram which lax (oplax?) commutes in the sense that t... | 7 | https://mathoverflow.net/users/290 | 270859 | 121,295 |
https://mathoverflow.net/questions/270855 | 10 | Recall that an inaccessible cardinal $\kappa$ is a Woodin cardinal if for every $A\subseteq V\_\kappa$ there is an unbounded set in $\kappa$ of $\lambda$ such that $V\_\kappa\models\lambda$ is $A$-strong.
This raises the question, of whether or not intermediate notions have been defined in the literature, and what so... | https://mathoverflow.net/users/7206 | "Weakly" Woodin cardinals | (As I pointed out in a comment) yes, partial Woodinness is common in arguments in inner model theory. Accordingly, you obtain determinacy results addressing specific pointclasses (typically, well beyond projective). To illustrate this, let me "randomly" highlight two examples:
* See [here](https://andrescaicedo.wordp... | 8 | https://mathoverflow.net/users/6085 | 270866 | 121,299 |
https://mathoverflow.net/questions/270867 | 5 | **Background**
--------------
We can define Miller Forcing as the poset of nonempty perfect rational trees. That is, we define:
* $p\subset 2^{<\omega}$ is a perfect tree iff it is closed downwards (for all $s, n$, if $s \in p$ and $n \in \omega$, then $s|n \in p$) and every branch splits (for all $s \in p$ there e... | https://mathoverflow.net/users/nan | Miller real is not in the closure of sets under some conditions | For any closed set $F\in V$, one can show $D\_F=\{p\in \mathbb{P}: \exists \text{open }U \ [p]\subset U\& U\cap F=\emptyset\}$ is dense. Just a comment though, I believe when people are talking about Miller forcing, another form (essentially the same) is more common, i.e subtrees of $\omega^{<\omega}$ such that each no... | 2 | https://mathoverflow.net/users/23835 | 270872 | 121,301 |
https://mathoverflow.net/questions/269582 | 12 | This is a theoretical question about poker-type games. I'm not going to specify the rules. You can consider No Limit Texas Hold'em or some simple theoretical model, where each player holds a number from the $(0,1)$ interval. We only consider playing heads-up, i.e., when two players play.
Playing against Darth Vader i... | https://mathoverflow.net/users/955 | Should you bet in poker against Darth Vader? | Here is an answer to the updated question:
Suppose that there are two betting rounds. Darth Vader has three types of hands. Type 1 wins with probability 1. Type 2 is a draw that hits (becomes a winning hand) with probability $1/10$ between the betting rounds. You can't see whether type 2 hands hit. Vader has Type 1 $... | 3 | https://mathoverflow.net/users/2954 | 270873 | 121,302 |
https://mathoverflow.net/questions/270875 | 8 | Foundational uniqueness and representability results on (co)homology theories in algebraic topology frequently make a point of assuming additivity, indicating at least some people think it's worthwhile considering theories not satisfying the wedge axiom despite the inapplicability of familiar results.
1. **What are ... | https://mathoverflow.net/users/5792 | (Co)homology theories not satisfying the wedge axiom | Here's an example. (Rather than calling it a stupid example, I'll call it an example for which I know no applications.)
Let $A$ be a torsion-free abelian group and $B$ be an injective abelian group. Then we can define new versions of cohomology and homology:
$$
\begin{align\*}
F^n(X,U) &= H^n(X,U) \otimes A \\
G\_n(X... | 12 | https://mathoverflow.net/users/360 | 270880 | 121,305 |
https://mathoverflow.net/questions/270817 | 3 | In a meanwhile deleted question I had mentioned my observation, that $$H\left(2^m-1\right) = H\left(3\*(2^m-1)\right) = H\left(3\*(2^m-1)\ +\ 1\right)$$
where $H()$ denotes the Hamming weight.
>
> **Question:**
>
>
> are there other examples of combinations of a set $\Sigma\subset\mathbb{N}$ with a functio... | https://mathoverflow.net/users/31310 | Certain Integer Sets of Elements with Common Hamming-weight Preserving Integer Function | The following sequences work as $\Sigma$ for $f(x)=x^2$
* $x\_n=2^n-1$ has Hamming weight $n$ as does $x\_n^2=2^{2n}-2^{n+1}+1$ (the number $2^{2n}-2^{n+1}$ has a run of $n-1$ ones followed by $n+1$ zeros in binary).
* Slightly more complicated is $x\_n=2^n+2^{n-2}-1$. Here $H(x\_n)=n-1$ and
$$x\_n^2=2^{2n}+2^{n-1}\l... | 3 | https://mathoverflow.net/users/15503 | 270881 | 121,306 |
https://mathoverflow.net/questions/270892 | 1 | In the following link <http://www.math.jhu.edu/~eriehl/ssets.pdf> on page 8 there is the following diagram
$$\begin{array}
XX\_m .\mathcal{F}[n] & \stackrel{f\_{\*}} {\longrightarrow} & X\_m.\mathcal{F}[m] \\
\downarrow{f^{\*}} & & \downarrow{\gamma\_m} \\
X\_n. \mathcal{F}[n] & \stackrel{\gamma\_n}{\longrightarrow} & ... | https://mathoverflow.net/users/94297 | Coend as a universal edge | Try to have a look to section 1.2 of [this](https://arxiv.org/pdf/1501.02503.pdf), and feel free to comment with help requests!
| 1 | https://mathoverflow.net/users/7952 | 270895 | 121,311 |
https://mathoverflow.net/questions/268426 | 5 | I have a question about the Morgan's paper ``The algebraic topology of smooth complex varieties''. Let $\Bbbk=\mathbb{C}$. Given a smooth non singular variety $X$ with normal crossing divisor $D$, the sheaf logarithmic forms $\mathcal{A}\_{DR}^{\bullet}(\log(D))$ is defined as follows: for an open set $U$, a form in $\... | https://mathoverflow.net/users/41970 | A question about the logarithmic complex and Morgan's paper | For the first part of the question: what is well-known is that the cohomology of the open variety $X\setminus D$ can be computed as the hypercohomology of the holomorphic logarithmic complex $\Omega\_X^\bullet(\log D)$. This is well explained in Voisin's book *Hodge Theory and Complex Algebraic Geometry*, corollary 8.1... | 5 | https://mathoverflow.net/users/79968 | 270910 | 121,318 |
https://mathoverflow.net/questions/270891 | 6 | Let $b\_{n,j}\in \mathbb{C}$ for each $n,j\in \mathbb{N}$. I was wondering if there is some characterization of those $b\_{n,j}$ such that for **all** *bounded* sequences $s\_j\in \mathbb{C}, j\in \mathbb{N}$, we have that
$$\sigma\_n := \sum\_{j=1}^{+\infty} b\_{n,j} s\_j$$
converges. Thus we want that the sequence $(... | https://mathoverflow.net/users/62246 | Summation of bounded sequences | Your condition is equivalent to the convergence in $\ell^1$ of $b\_n=(b\_{nj})\_{j\ge 1}$.
First of all, we indeed must insist that $b\_n=(b\_{nj})\_{j\ge 1}\in\ell^1$ for each fixed $n$, or otherwise we couldn't even define $\sum b\_{nj}s\_j$ for general bounded $s$.
Recall that $(\ell^1)^\*=\ell^{\infty}$, and ob... | 9 | https://mathoverflow.net/users/48839 | 270915 | 121,320 |
https://mathoverflow.net/questions/270919 | 24 | Elementary toposes form an elementary class in that they are axiomatizable by (finitary) first-order sentences in the "language of categories" (consisting of a sort for objects, a sort for morphisms, function symbols for domain and codomain, and a ternary relation symbol on morphisms for composition). Grothendieck topo... | https://mathoverflow.net/users/nan | Precise relationship between elementary and Grothendieck toposes? | There are known statements that are true in any Grothendieck topos, but not in every elementary topos with NNO. For instance:
1. Freyd's theorem that a complete small category is a preorder is not constructively provable and can fail in elementary toposes with NNO, such as the effective topos; but can be shown to hol... | 19 | https://mathoverflow.net/users/49 | 270923 | 121,322 |
https://mathoverflow.net/questions/270942 | 2 | Is it possible to generalise Zellweger’s [logic alphabet](https://en.wikipedia.org/wiki/Logic_alphabet) for more than two Boolean variables?
Can it be done by only using the 16 binary connectives?
Thanks.
| https://mathoverflow.net/users/29611 | Logic Alphabet for more than Two Variables | Well, when the number of Boolean variables is $n=3$, since the number of connectives is $2^{2^n}=(2^{2^2})^{n-1}$, we can use infix notation like $pxqyr$, where $p$, $q$, $r$ are Boolean variables and $x$, $y$ are among the 16 symbols. The interpretation is that if $p$ is true then the value is $qxr$, otherwise $qyr$.
... | 1 | https://mathoverflow.net/users/4600 | 270948 | 121,333 |
https://mathoverflow.net/questions/270911 | 2 | Let $\mathfrak{g}$ be a complex simple Lie algebra, and $W$ its Weyl group. We make a choice of simple roots, and note $\Phi\_+$ the corresponding set of positive roots, while $\Phi$ is the set of all roots. Finally, I denote $z^{\lambda}$ the character associated to a weight $\lambda$.
I claim that $$\sum\limits\_{... | https://mathoverflow.net/users/61018 | A corollary to Weyl's formula on roots of Lie algebras | I don't have a reference but one can deduce it from the denominator identity
$$
\prod\_{\alpha>0}(1-z^\alpha)=\sum\_w(-1)^{\ell(w)}z^{\rho-w\rho}.
$$
With $\beta=w\alpha$ write
$$
\prod\_{\alpha>0}(1-z^{w\alpha})=\prod\_{\beta>0,w^{-1}\beta>0}(1-z^\beta)
\prod\_{\beta<0,w^{-1}\beta>0}(1-z^\beta)
$$
The second factor eq... | 3 | https://mathoverflow.net/users/89948 | 270957 | 121,340 |
https://mathoverflow.net/questions/270963 | 5 | Let $ f:X \rightarrow Y$ be finite covering map of simplicial sets of finite degree, say $d$. Let $\varSigma^{\infty}$ denote the functor from the category of simplicial sets to spectra which is an additive category, where
$\varSigma^{\infty}Y = \{n \mapsto S^{n} \wedge Y \}$.
I want to know the sketch or referenc... | https://mathoverflow.net/users/110510 | Transfer map of simplicial sets | In the more general version with compact (finitely dominated) fibers, this is called the **Becker-Gottlieb transfer**. You can find a long list of references on the [nlab](https://ncatlab.org/nlab/show/Becker-Gottlieb+transfer). Here are a few of them:
* *Becker, James C.; Gottlieb, Daniel H.*, [**The transfer map an... | 2 | https://mathoverflow.net/users/43054 | 270971 | 121,347 |
https://mathoverflow.net/questions/270571 | 12 | There has been a lots of approaches to the notion of n-categorical diagram and n-categorical pasting diagram:
[Street : "Parity complexes"](http://archive.numdam.org/article/CTGDC_1991__32_4_315_0.pdf)
[Power : " An n-categorical pasting theorem"](https://www.researchgate.net/profile/John_Power2/publication/2256091... | https://mathoverflow.net/users/22131 | n-categorical pasting diagram overview | I can provide some pieces of knowledge for Street and Johnson, and, to a lesser extent, for Steiner. Since my reputation on mathoverflow is not important enough, I am not able to include too many links and I can only cite figures in a master thesis I made for an internship on this subject : [Parity complexes and pastin... | 14 | https://mathoverflow.net/users/110515 | 270989 | 121,355 |
https://mathoverflow.net/questions/270988 | 1 | Let $V$ be a compact Riemannian manifold, $G$ the set of diffeomophisms of $V$, let $\nu$ be a probability measure in $G$. Suppose that $\exp\_{x}$ is diffeomorphism in $\mathcal{B}\_{2I}(x)\subset V$ ball of radius $2I$ for all $x\in V$, then we define
$$
\begin{array}{rl}
\delta\_{1}(T) & ={\displaystyle \sup\left\{\... | https://mathoverflow.net/users/109184 | About the ergodic theorem of Birkhoff in the context of a compact Riemannian manifold | What this boils down to is the following:
Let $\tau$ be a measure-preserving transformation of a probability space $(\Omega,\mathbb P)$. Suppose that $f$ is a measurable function on $\Omega$ such that $\int f\,d\mathbb P(\omega)<\infty$. Then $(1/n)f(\tau^n \omega)\to 0$ a.e.
The cheapest way to see this is from t... | 1 | https://mathoverflow.net/users/11054 | 270995 | 121,358 |
https://mathoverflow.net/questions/269786 | 76 | What computational mathematics problems that could be used as proof-of-work problems for cryptocurrencies? To make this question easier to answer, I want proof-of-work systems that work in cryptocurrencies that contain many different kinds of proof-of-work problems (the use of many different kinds of proof-of-work prob... | https://mathoverflow.net/users/22277 | What computational problems would be good proof-of-work problems for cryptocurrency mining? | [Here is a recent paper](https://boolesrings.org/jvanname/2017/07/22/nebula-the-cryptocurrency-that-will-produce-the-reversible-computer/), which has been received positively in the cryptocurrency community. I will expand on this paper here.
While conventional hash functions do not allow one to construct very useful... | 11 | https://mathoverflow.net/users/22277 | 270997 | 121,360 |
https://mathoverflow.net/questions/270983 | 2 | A **commutative semiring-like** structure is a structure $(R, {+}, {\cdot})$ where $+$ and $\cdot$ are associative and commutative, and $\cdot$ distributes over $+$.
Is there a commutative semiring-like $R$ such that we can embed the two element lattice $\{\bot, \top\}$ into $R$ so that meet and join are affine mappi... | https://mathoverflow.net/users/1176 | Affine embedding of the two-point lattice into a semiring-like structure | There is no commutative semiring with the desired
properties.
To see this, suppose instead that the semiring
$R$ contains a subset $\{F, T\}$ supporting affine operations
$\vee, \wedge$ that are lattice operations on $\{F, T\}$.
$R$ then has
an affine operation
$m(x,y,z) := (x\wedge y)\vee (x\wedge z)\vee (y\wedge z... | 5 | https://mathoverflow.net/users/75735 | 270999 | 121,362 |
https://mathoverflow.net/questions/245342 | 5 | A *torsionless* abelian group $A$ is one for which any element $a\neq 0$ can be sent to a nonzero element of $Z$ by some homomorphism $A\rightarrow Z$ (integers). Equivalently, $A$ can be embedded as a subgroup of a Cartesian power of $Z$, $Z^I$.
A *separable* abelian group $A$ is one for which any element $a$ such t... | https://mathoverflow.net/users/17764 | Torsionless not separable abelian groups | The answer is no.
Consider your (renamed) example $V = \mathbf{Z}^{(\omega)} + 2\mathbf{Z}^\omega$. It is quotient of the direct sum $$A=\mathbf{Z}^{(\omega)} \oplus 2\mathbf{Z}^\omega\simeq \mathbf{Z}^{(\omega)} \oplus \mathbf{Z}^\omega$$ by a subgroup $B$ isomorphic to the intersection, namely the countable subgrou... | 2 | https://mathoverflow.net/users/14094 | 271004 | 121,363 |
https://mathoverflow.net/questions/270711 | 3 | Define H(a) to be the space of holomorphic functions $f(z)$ on $S\_a:=\{z:|\Re z| < a\}$ with
$$ ||f||^2\_a:=\int\_{S\_a} |f(x+iy)|^2 (1+|x+iy|)^{100} dx \, dy < \infty. $$
Two questions:
1. Is H(2) dense in H(1)?
2. Are the entire functions dense in H(1)?
Is there a nice kernel to convolve things with?
This occu... | https://mathoverflow.net/users/95002 | density of holomorphic functions in vertical strips | It's possible I'm over-thinking this:
If $f \in H(1)$, let
$$ f\_\delta(z) = \frac{1}{\sqrt{\pi\delta}} \int\_{(0)} f(w) \exp ((w-z)^2/\delta) dw, $$
then
$$ f\_\delta(x+iy) = \frac{i\exp (x^2/\delta)}{\sqrt{\pi\delta}} \int\_{-\infty}^\infty f(i(t+y)) e^{-2itx/\delta} \exp (-t^2/\delta) dt, $$
so $f\_\delta$ is entire... | 0 | https://mathoverflow.net/users/95002 | 271007 | 121,365 |
https://mathoverflow.net/questions/270985 | -3 | A graph $G=(V,E)$ is said to be *vertex-critical* if removing a vertex $v\in V$ reduces the chromatic number $\chi(\cdot)$. *Edge-criticality* is defined in a similar manner. Moreover, $G$ is called *contraction-critical* if contracting any edge reduces the chromatic number.
*Questions.*
1) Are edge- and vertex-cri... | https://mathoverflow.net/users/8628 | Concepts of criticality in graph theory | Concerning your first question, every edge-critical graph without isolated vertices must be vertex-critical, but not vice versa. For instance, the complement of a $7$-cycle is vertex-critical but not edge-critical.
Concerning your second question, every vertex-critical graph must be contraction-critical as well. Supp... | 4 | https://mathoverflow.net/users/90417 | 271034 | 121,377 |
https://mathoverflow.net/questions/270976 | 5 | Let $f$ be a modular newform of weight $k \geq 2$, level $N$ (square free) and trivial nebentypus. Let $V\_{f}$ be the $p$-adic (p odd) Galois representation associated $f$. We denote by $V\_{f,l}:= V\_{f}|\_{G\_{l}}$. Let $\chi$ be a quadratic character of conductor $l$. Suppose that $(N,l)=1$. Then $f \otimes \chi$ i... | https://mathoverflow.net/users/110519 | Local Galois representation associated to twist of modular form | I think it helps to put things in a larger perspective.
To an eigencuspform $f$ and a prime number $\ell$ is attached on the one hand an irreducible, admissible representation $\pi(f)\_{\ell}$ of $\operatorname{GL}\_{2}(\mathbb Q\_{\ell})$ which is either
1) An irreducible principal series $\pi(\chi,\psi)$,
2) A ... | 4 | https://mathoverflow.net/users/2284 | 271041 | 121,378 |
https://mathoverflow.net/questions/271039 | 12 | A matrix of real numbers is called totally positive if all its minors are non-negative. A well-known example is the Pascal matrix $(\binom{i}{j})$.
Is it true that the minors of the $q$-Pascal matrix $({\binom{i}{j}}\_q)$ are polynomials with non-negative coefficients?
| https://mathoverflow.net/users/5585 | Total positivity of $q$-Pascal matrix? | Yes. Consider the set $V$ of points with integer coordinates as vertices of a weighted directed acyclic graph. Namely, for any $(i,j)\in V$, the edge from $(i,j)$ to $(i+1,j)$ has weight 1, the edge from $(i,j)$ to $(i,j+1)$ has weight $q^{i}$. Then $\binom{n+k}{k}\_q$ is a weighted sum of paths from the origin to $(n,... | 12 | https://mathoverflow.net/users/4312 | 271042 | 121,379 |
https://mathoverflow.net/questions/271016 | 6 | It is known that, given an abelian category $\mathcal B$, the derived category $\mathrm D (\mathcal B)$ is a triangulated category. In particular, given a distinguished triangle $E\to F\to G \to E[1]$ in $\mathrm D ( \mathcal B)$, taking cohomology of complexes gives a long exact sequence
$$
\cdots \to H^\*(E) \to H^\*... | https://mathoverflow.net/users/69190 | the cohomology objects in the t-structure and long exact sequence | (1) The functor $H^0$ is cohomological [BBD, Théorème 1.3.6].
(2) A t-structure is, basically, a way to determine an abelian subcategory inside a triangulated category *together with* a cohomological functor with values in this subcategory. This permits a formulation within the derived category of sheaves of *pervers... | 4 | https://mathoverflow.net/users/6348 | 271044 | 121,380 |
https://mathoverflow.net/questions/271057 | 1 | Let us look at the abelian group $V$ of integer valued sequences modulo sequence which are zero almost everywhere.
Let me fix one constant $k\in \mathbb{N}$, which I will omit from the notation.
One specific sequence is given by $a\_n = \binom{n+k}{n}$. Let $\Sigma^m a$ be the shift of $a$ by $m$, i.e. $(\Sigma^ma)\_... | https://mathoverflow.net/users/3969 | Zero combinations of certain sequences | Consider the Laurent polynomial $f(t)=\sum \lambda\_m t^m$. It is easy to prove that your conditions are equivalent to a system of $k+1$ equations $f^{(i)}(1)=0$ for $i=0,\dots,k$. In other words, $f(t)=(t-1)^{k+1}h(t)$ for some Laurent polynomial $h$ with integer coefficients. If $h(-1)\ne 0$, we get $|f(-1)|\geqslant... | 1 | https://mathoverflow.net/users/4312 | 271062 | 121,385 |
https://mathoverflow.net/questions/271056 | 3 | I am looking for examples of locally finitely presentable categories which admit a symmetric monoidal structure, such that the tensor product preserves colimits in each variable, but the unit is not finitely presentable, and/or there is a tensor product of finitely presentable objects which is not finitely presentable.... | https://mathoverflow.net/users/2841 | locally finitely presentable tensor categories | One can take the category of modules over a Laurent polynomial ring in one variable $\textrm{Mod}\;k[t,t^{-1}]$ and think of $k[t,t^{-1}]$ as the group algebra of $\mathbb{Z}$. The corresponding cocommutative Hopf algebra structure provides a symmetric monoidal structure $(\otimes\_k, k)$ on $\textrm{Mod}\;k[t,t^{-1}]$... | 7 | https://mathoverflow.net/users/310 | 271063 | 121,386 |
https://mathoverflow.net/questions/271047 | 5 | Consider a special case of the Hilbert's 10th problem:
$f(\vec{x})=g(\vec{y})$, where $\vec{x}$ and $\vec{y}$ are **disjoint** ( i.e, the LHS and RHS do not have any common variables), moreover, $f$ and $g$ are polynomials with **positive** coefficients.
The question is, with the restrictions, whether the undecida... | https://mathoverflow.net/users/110554 | Undecidability of Diophantine equations with disjoint variables? | It is undecidable.
The only integral point on $x^3+x=y^2$ is $(0,0)$.
Let $F(\vec{y})=0$ be undecidable diophantine equation with
positive coefficients and not depending on $x$.
Take $f(x)=x^3+x$ and $g(\vec{y})=F^2$ leading to $x^3+x=F^2(\vec{y})$.
To get $F$ from $F'$ with negative coefficients use sum of squ... | 9 | https://mathoverflow.net/users/12481 | 271066 | 121,388 |
https://mathoverflow.net/questions/270096 | 5 | Are there formulas similar to the Riemann-Weil formula for other arithmetical functions like $ \mu (n) $ or $ \lambda (n) $, for example a sum of the form $ \sum\_{n=1}^{\infty}a(n) f(n) $ with this sum being related to the sum over the imaginary part of the Riemann zeros involving the Fourier transform of $ f(x)$?
T... | https://mathoverflow.net/users/23964 | Do Riemann-Weil formulas exist for functions other than the Mangoldt function $ \Lambda (n) $ | The reason that $\mu(n)$ and $\lambda(n)$ have such expressions is that the corresponding Dirichlet generating functions can be expressed in terms of the Riemann zeta function:
$$
\sum\_n \frac{\mu(n)}{n^s}=\frac{1}{\zeta(s)}
$$
and
$$
\sum\_n \frac{\lambda(n)}{n^s}=\frac{\zeta(2s)}{\zeta(s)}.
$$
In general, if the Dir... | 8 | https://mathoverflow.net/users/6756 | 271082 | 121,393 |
https://mathoverflow.net/questions/271083 | 2 | This comes from the paragraph following equation (27) on page 6 of [this paper](https://arxiv.org/pdf/1506.02438.pdf). It's not crucial to the argument — any such bound will do — but it's not clear to me why this particular bound is appropriate.
>
> Using a discount $\gamma < 1$ corresponds to dropping the terms wi... | https://mathoverflow.net/users/32620 | Given ∈ (0, 1), why is ˡ negligible for l ≫ 1/(1-)? | For $\gamma$ approaching 0, $\gamma^{1/(1-\gamma)}$ approaches $\gamma$.
For $\gamma$ approaching 1, $\gamma^{1/(1-\gamma)}$ approaches $1/e$. If $l$ is much larger than $1/(1-\gamma)$, then $\gamma^l$ is much smaller than $1/e$.
| 2 | https://mathoverflow.net/users/756 | 271084 | 121,394 |
https://mathoverflow.net/questions/271059 | 14 | For a sequence $(x\_{\alpha})$ of surreal numbers indexed by the set of all ordinal numbers, we say that $\lim x\_{\alpha}=l$ ($l$ is a surreal number) if for each surreal $\epsilon>0$, there exists an ordinal $\beta$ such that $|x\_{\alpha}-l|<\epsilon$ for each ordinal $\alpha>\beta$.
Consider the sequence $x\_{\al... | https://mathoverflow.net/users/74664 | The surreal version of $e$ | $\DeclareMathOperator{\ee}{e}$If $\varepsilon$ is an infinitesimal surreal, the quantity $\log(1+\varepsilon)$ is actually equal to the formal sum *à la Hahn series* $\sum \limits\_{n \in \mathbb{N}} \frac{(-1)^n\varepsilon^{n+1}}{n+1}$, and $\log(1+\varepsilon) - \varepsilon$ is negligeable with respect to $\varepsilo... | 7 | https://mathoverflow.net/users/45005 | 271092 | 121,398 |
https://mathoverflow.net/questions/271100 | 8 | I first posed this question when I was a first year student. I came up with some ad hoc arguments as to why the result is true (a bit of numerical experimentation), but never had a proof. I forgot about it until the other day, and thought it was still an interesting question. First, some motivation.
Take the expressi... | https://mathoverflow.net/users/nan | The continuous Taylor series; are they just Taylor series? | No. In fact I will show the map from $1$-periodic signed measures $\mu$ to $$F\_\mu(x) = \int\_{-\infty}^\infty \mu(y) e^{-y^2}\frac{x^y}{\Gamma(y+1)}dy$$ is injective for "reasonable" signed measures $\mu$.
Indeed, consider what happens to the following expression as $y\_0$ goes to $\infty$ while the residue of $y\_... | 8 | https://mathoverflow.net/users/18060 | 271104 | 121,404 |
https://mathoverflow.net/questions/271078 | 5 | This came out of some work on the *digamma function*.
Let $(x)\_k=x(x+1)\cdots(x+k-1)$ denote the Pochhammer symbol. Then,
>
> **Question.** Can you prove/disprove this identity?
> $$\pmb{\frac{(\frac12)\_j^2}{j!^2}}\sum\_{i=0}^{j-1}\frac4{2i+1}
> =\sum\_{i=0}^{j-1}\pmb{\frac{(\frac12)\_i^2}{i!^2}}\frac1{j-i}.$... | https://mathoverflow.net/users/66131 | Identity with Pochhammer and harmonic numbers | Here's a sketch of a proof using "creative telescoping."
Let
$$T(i,j) = \frac{j!^2}{(\tfrac12)\_j^2}\cdot \frac{(\tfrac12)\_i^2}{i!^2}\frac1{j-i}.$$
Since the identity holds for $j=1$, it suffices to show that
$$\sum\_{i=0}^{j} T(i,j+1) -\sum\_{i=0}^{j-1}T(i,j)=\frac{4}{2j+1};$$
i.e., that
$$T(j,j+1) +\sum\_{i=0}^{... | 10 | https://mathoverflow.net/users/10744 | 271108 | 121,406 |
https://mathoverflow.net/questions/271071 | 11 | The classical Catalan numbers are given by $C\_n=\frac1{n+1}\binom{2n}n$, for which there is a plethora of generalizations of interpretations in the literature. Still, let us consider one more such:
$$C\_n(k):=\binom{kn}{n,\dots,n}\prod\_{j=0}^{k-1}\frac1{jn+1};$$
where the multinomial coefficient is simply $\frac{(kn)... | https://mathoverflow.net/users/66131 | Another extension of Catalan: divisibility | Here's a slightly different way to look at this. We have
$$C\_n(k)=\prod\_{j=1}^{k}\left(\frac{1}{(j-1)n+1}\binom{jn}{n}\right).$$
And each factor
$$\frac{1}{(j-1)n+1}\binom{jn}{n}=\binom{jn}{n}-(j-1)\binom{jn}{n-1}$$
is an integer. For $C\_n(p)$ let us look at the last factor $\frac{1}{(p-1)n+1}\binom{pn}{n}$. Since ... | 10 | https://mathoverflow.net/users/2384 | 271113 | 121,407 |
https://mathoverflow.net/questions/271109 | 1 | Suppose I have $k$ tasks that I can run independently on $n$ machines where $k\geq n$. Let $t\_i\in\mathbb{N}$ be the number of seconds that task $i$ takes to be done on any machine (for any $i\in[k] :=\{0,\ldots,k-1\}$). I want to assign the tasks to the $n$ machines such that the parallel run-time gets minimized. Thi... | https://mathoverflow.net/users/8628 | $k$ tasks on $n$ machines | Let me quote a really great computer science professor (<http://www.cs.ucsb.edu/~teo/>), who taught this bound and others in one of his lectures: "Two times optimum".
In fact, we always have
$$D\_n = 2 - \frac{1}{n}$$
as a bound.
To see this, let's first look at the worst thing that can happen with $M(f\_s)$: While s... | 2 | https://mathoverflow.net/users/109932 | 271118 | 121,409 |
https://mathoverflow.net/questions/208440 | 10 | Consider a good enough scheme $X$ (e.g. an algebraic variety over a field). Let $X\_i$ be the set of points of dimension $i$ in $X$. Then we have the Gersten complex in Quillen's K-theory:
$$
\oplus\_{x\in X\_{i+1}}K\_{n+1}(\kappa(x))\to \oplus\_{x\in X\_i}K\_n(\kappa(x))\to \oplus\_{x\in X\_{i-1}}K\_{n-1}(\kappa(x))... | https://mathoverflow.net/users/11599 | Gersten complexes in Quillen's and Milnor's K-theories | Yes, the natural multiplication morphisms induce a morphism of Gersten complexes from Milnor to Quillen K-theory. The basic points are made in the paper
* M. Rost. "Chow groups with coefficients", Doc. Math. 1 (1996), pp. 319-393, [link to DocMath page](http://www.emis.de/journals/DMJDMV/vol-01/16.html).
That pape... | 5 | https://mathoverflow.net/users/50846 | 271120 | 121,410 |
https://mathoverflow.net/questions/268941 | 7 | Let $\mathbb{Z}\_p$ be the ring of $p$-adic integers, $\mathbb{Q}\_p$ the field of fractions of $\mathbb{Z}\_p$, and $\mathbb{C}\_p$ the completion of the algebraic closure of $\mathbb{Q}\_p$. Let $v\_p$ be the $p$-adic valuation on $\mathbb{C}\_p$ with $v\_p(p)=1$. For each $m\ge1$ define the set
$$
A\_m=\mathbb{C}\_p... | https://mathoverflow.net/users/109085 | Characterization of Krasner analytic functions on the complement of $p$-adic integers | If you impose some growth conditions on the boundary of ${\mathbb C}\_p\setminus {\mathbb Z\_p}$ to the Krasner-analytic function $F$, plus some invariance under the absolute Galois group of ${\mathbb Q}$, you have a caracterization in the spirit of Amice-Fresnel theorem. I use $p$-adic absolute value instead of $p$-ad... | 3 | https://mathoverflow.net/users/45381 | 271147 | 121,420 |
https://mathoverflow.net/questions/271150 | 4 | The modular curve $Y(n)$ which over $\mathbb{C}$ is $\mathcal{H}/\Gamma(n)$ is often viewed as a curve defined over $\mathbb{Q}(\zeta\_n)$. However, if one twists the moduli problem to be given by pairs $(E,\alpha)$ where $E$ is an elliptic curve over some $\mathbb{Q}$-scheme $S$, and $\alpha$ is an isomorphism:
$$\alp... | https://mathoverflow.net/users/15242 | Are there "primitive" modular functions for $\Gamma(n)$ with Fourier coefficients in $\mathbb{Q}$? | I think this is in fact always true. Let $f$ be a rational function on this moduli space which is defined over $\mathbb Q$.
Recall that the Tate curve over is an elliptic curve $E / \mathbb Q((q))$ which is $q$-adically analytically and holomorphically isomorphic to the quotient $\mathbb G\_m / \langle q \rangle$. So... | 4 | https://mathoverflow.net/users/18060 | 271152 | 121,423 |
https://mathoverflow.net/questions/271107 | 6 | Let $f$ be a real-valued continuous function on the interval $[0,1]$ and satisfy the following estimate
$$
\left|\int\_0^1 f(t) e^{st}dt\right|\le Cs^{\frac12},\quad s>1,
$$
where the constant $C$ is independent of $s$.
Can we assert that $f$ is identically zero on $[0,1]$?
| https://mathoverflow.net/users/33232 | integral depending on a parameter | Michael has essentially answered this in his comment, but let me make this more explicit.
In fact, a stronger statement is true: If $F(z)=\int\_0^1 f(t)e^{tz}\, dt$ satisfies $|F(s)|\lesssim e^{(a+\epsilon)s}$ for $s>1$ and all $\epsilon>0$ (but with possibly $\epsilon$ dependent implied constants), then $f=0$ on $[a... | 4 | https://mathoverflow.net/users/48839 | 271161 | 121,426 |
https://mathoverflow.net/questions/271158 | 3 | Suppose I have a function $f \in \mathcal C^{\alpha, L}([0,1])$, where
$\mathcal C^{\alpha, L}([0,1])$ is the space of $\alpha$-smooth Hölder
functions with norm $L$. I am interested in efficiently approximating $f$ using
a kernel-based method; we might have
\begin{align\*}
\widetilde f(x) = \sum\_{j = 1}^k \mu\_j \ps... | https://mathoverflow.net/users/46385 | Constructive approximation of Hölder functions using kernel functions | If you are just concerned with logistic kernels, and you are willing to put a mild assumption on the $\beta$-Hölder function $f$ to be estimated, then [Rousseau] assumes mild condition $\boldsymbol{A}\_0$ and proved the $k^{-\beta}$ decay in Theorem 3.1, which is also cited by [Kruijer&Rousseau]. The good thing is that... | 2 | https://mathoverflow.net/users/25437 | 271162 | 121,427 |
https://mathoverflow.net/questions/271167 | 4 | It could be related to my previous question [here](https://mathoverflow.net/questions/271163/different-definitions-of-derived-functors).
Let $\mathcal F$ be a sheaf on a topological space $X$. Hartshorne in his book on Algebraic geometry defines the sheaf cohomology by
$$
H^i(X, \mathcal F)= R^i\Gamma(X,-)(\mathcal F... | https://mathoverflow.net/users/69190 | Different definition of sheaf cohomology | Taking global sections is the same thing as computing the hom from $O\_X$. In other words, there is an isomorphism of functors $\Gamma(X,-)\cong\hom(O\_X,-)$, so both functors have the same derived functors.
As for using the derived category to define cohomology: yes, simply because that is what the derived category ... | 13 | https://mathoverflow.net/users/1409 | 271168 | 121,430 |
https://mathoverflow.net/questions/271142 | 7 | I want to determine $\mathrm D\_{\mathrm{cris}}$ of certain twists of the Galois representations attached to modular forms. For one particular twist it is not clear to me how $\mathrm D\_{\mathrm{cris}}$ looks like.
Let $f\in\mathrm S\_k(\Gamma\_1(N),\psi)$ be a newform of weight $k\ge2$, level $N$, nebentype $\psi$,... | https://mathoverflow.net/users/33820 | How large is Dcris of certain twists of modular forms? | The isomorphism class of the $G\_{\mathbf{Q}\_p}$-representation $V\_f$ determines (up to scaling) a class in $H^1(\mathbf{Q}\_p, \delta \epsilon^{-1})$. The condition that $\mathbf{D}\_{\mathrm{cris}}(V\_f(\psi)(n))$ is 1-dimensional is exactly requiring that this extension is in Bloch--Kato's $H^1\_{\mathrm{f}}$.
N... | 4 | https://mathoverflow.net/users/2481 | 271175 | 121,433 |
https://mathoverflow.net/questions/271156 | 10 | Given a spherical fusion category $\mathcal C$, the [Levin-Wen model](https://arxiv.org/pdf/cond-mat/0404617.pdf) constructs a lattice field theory: to each oriented surface with a triangulation, it assigns a state space $\mathcal H$ and a Hamiltonian $H$, whose space of ground states is independent of the choice of tr... | https://mathoverflow.net/users/97265 | Is there a 1-dimensional analogue of the correspondence between the Levin-Wen and Turaev-Viro models? | Yes, there is an analogous 1d result, but it's not very interesting (which probably accounts for you not finding it in the literature).
In order for there to be a corresponding lattice model, you want the TQFT to be fully extended. This means you should start with a noncommutative Frobenius algebra (the Hilbert space... | 6 | https://mathoverflow.net/users/284 | 271188 | 121,438 |
https://mathoverflow.net/questions/270736 | 3 | I am interested in the following variation of decomposing a graph into a tree. It is related to both the standard tree-decomposition of a graph and tree-cut decompositions.
Given a graph $G$, we want to find a tree $T$ and a partition $\{X\_v \subseteq V(G): v \in V(T)\}$ of the vertices of $G$ indexed by the vertice... | https://mathoverflow.net/users/20940 | A variant of tree or tree-cut decompositions | I mentioned your type of decomposition to my colleague Konstantinos and he pointed out that it had appeared in the literature under the name "strong tree-decomposition"! D. Seese introduced it in 1985 ('Tree-partite graphs and the complexity of algorithms'). Bodlaender and Engelfriet then picked it up in 1994 in an art... | 2 | https://mathoverflow.net/users/37432 | 271196 | 121,439 |
https://mathoverflow.net/questions/238978 | 15 | The harmonic numbers are given by $$H\_n=\sum\_{k=1}^n\frac{1}{k}.$$
Numerical calculation suggests
$$
\sum\_{k=1}^{n}(-1)^k{n\choose k}{n+k\choose k}\sum\_{i=1}^{k}\frac{1}{n+i}=(-1)^nH\_n.
$$
I can not give a proof of this identity. How to prove it?
Hints, references or proof are all welcome.
| https://mathoverflow.net/users/6104 | A combinatorial identity involving harmonic numbers | First we prove the formula
$$\sum\_{k=0}^n (-1)^k\binom{n}{k}\binom {x+k}{k} = (-1)^n\binom xn,\tag{1}$$
which is special case of Vandermonde's theorem:
$$\begin{aligned}
\sum\_{k=0}^n (-1)^k\binom{n}{k}\binom {x+k}{k}
&= \sum\_{k=0}^n \binom n{n-k} \binom{-x-1}{k}\\
&= \binom{n-x-1}{n} = (-1)^n\binom xn.
\end{align... | 12 | https://mathoverflow.net/users/10744 | 271200 | 121,441 |
https://mathoverflow.net/questions/271201 | 6 | Let $R$ be a ring (assumed associative and unital) whose additive group is a finitely generated abelian group. As a reduction step in a paper I'm working on, we need to know that $R$ is a quotient of another ring $S$ whose additive group is a finite rank free abelian group. We believe we have a proof, but it is rather ... | https://mathoverflow.net/users/321 | Quotients of rings with finite free additive group | For a finite ring $R$ you can write it as a quotient of the monoid ring $\mathbb ZR$ (with respect to the multiplicative structure) which has a finitely generated free additive group.
| 2 | https://mathoverflow.net/users/15934 | 271209 | 121,444 |
https://mathoverflow.net/questions/271211 | 0 | Suppose $f(x)=\prod\_{k=1}^n(x-a\_k)$ where all $a\_k>0$.
Expand the function $\frac1f$ at $\infty$ so that
$$\frac1{f(x)}=\frac{b\_n}{x^n}+\frac{b\_{n+1}}{x^{n+1}}+\cdots.$$
**Does it follow that each $b\_m$ is positive, for $m\geq n$?**
| https://mathoverflow.net/users/110332 | A polynomial and its reciprocal expansion | Yes, it's pretty easy.
$$
\frac{1}{f(x)}
= \prod\_{k=1}^n \frac{1}{x-a\_k}
= x^n \prod\_{k=1}^n \frac{1}{1-a\_k/x}
= x^n \prod\_{k=1}^n \sum\_{i=0}^\infty \left(\frac{a\_k}{x}\right)^i.
$$
From this is it clear that your $b\_n$ coefficients are positive.
| 2 | https://mathoverflow.net/users/11926 | 271212 | 121,445 |
https://mathoverflow.net/questions/271163 | 5 | In principle one uses the notion of derived category, and the other doesn't.
Suppose $F: \mathcal A \to \mathcal B$ is a left exact (additive) functor between abelian categories, and suppose the category $\mathcal A$ has enough injective objects. Then we have two kinds of terminology of derived functor:
(1) The sta... | https://mathoverflow.net/users/69190 | Different definitions of derived functors | The total right derived functor ${\bf R}F(-)$ contains a bit more information than just its individual cohomologies ${\bf R}^iF(-) = H^i({\bf R}F(-))$. This information can indeed be described as a kind of gluing data, and can be encoded as suitable $k$-invariants. For example, if $C\_{\bullet}$ is a cochain complex in... | 9 | https://mathoverflow.net/users/51164 | 271222 | 121,449 |
https://mathoverflow.net/questions/271197 | 1 | Let $\mu(n)$ be the Mobius function.
Let us define $\mu^+(n)$ to be $\mu(n)$ if $\mu(n)>0$ and $0$ otherwise. Is there a known asymptotic formula for
$$
\sum\_{n \leq N} \mu^+(n),
$$
and similarly for
$$
\sum\_{ \substack{n \leq N \\ n \equiv a (\mod q)}} \mu^+(n).
$$
I would greatly any appreciate references or c... | https://mathoverflow.net/users/84272 | Distribution of Mobius function | I'll have a stab at this. The relation
$$
\sum\_{n\leq X:n\equiv a~(mod~q)} \mu^2(n)=\frac{6}{\pi^2} \prod\_{p|q}
\left(1-\frac{1}{p^2} \right)\frac{X}{q}+E(X,q,a)
$$
where the error term $E$ is $O\_{\varepsilon}\left(\sqrt{X/q}
+q^{\frac{1}{2}+\varepsilon}\right) $ provided $q\leq X^{\frac{2}{3}-\varepsilon},$
togeth... | 4 | https://mathoverflow.net/users/17773 | 271233 | 121,455 |
https://mathoverflow.net/questions/271230 | 2 | Let $R$ be a strictly henselian local ring of dimension 2, satisfying Serre's condition $S\_2$. Let $X = \text{Spec }R$, and let $f : Y\rightarrow X$ be a finite morphism inducing an isomorphism over the complement of the unique closed point of $X$. Is $f$ an isomorphism?
Is this true in higher dimensions as well (po... | https://mathoverflow.net/users/15242 | Finite morphism inducing an isomorphism away from codimension 2 to a strict henselian ring of dimension 2 satisfying $S_2$ is an isomorphism? | This is not quite true as stated, even for normal rings rings $R$. For example let $R = k[[x,y]]$ and $S = k[[x,y]] \times k$ and the map $R$ to $S$ sends $x,y$ to themselves in the first coordinate and sends $x, y$ to zero in the second.
But I think this is basically the only thing that can go wrong. Say now that $... | 3 | https://mathoverflow.net/users/3521 | 271234 | 121,456 |
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