parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/366177 | 8 | In its simplest version, the recursion theorem states that for any $m\in\mathbb{N}$ and any function $g:\mathbb{N}\rightarrow\mathbb{N}$, there exists a function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $f(0)=m$ and $f(n+1) = g(f(n))$. There are many more complicated versions, with multiple variables and parameter... | https://mathoverflow.net/users/5017 | What subsystem of second-order arithmetic is needed for the recursion theorem? | As Wojowu already pointed out $\mathsf{RCA}\_0$ proves recursion theorem. You could find a proof in Simpson's book [1], Section II.3.
In fact primitive recursion theorem is equivalent to $\Sigma^0\_1\textsf{-Ind}$ over $\mathsf{RCA}\_0^{\star}$. Here $\mathsf{RCA}\_0^{\star}$ is $\mathsf{EA}+\Delta^0\_1\text{-}\maths... | 11 | https://mathoverflow.net/users/36385 | 366182 | 153,795 |
https://mathoverflow.net/questions/365861 | 2 | I would like to compute the following integral:
$$
I\_\ell(\alpha) := \int\_{-1}^1 dx \, |x| J\_0(\alpha \sqrt{1 - x^2}) P\_\ell(x)
\tag{1}
\label{1}
$$
where $\alpha \geq 0$, $J\_0$ is the zeroth-order Bessel function of the first kind, $P\_\ell(x)$ is the Legendre polynomial of order $\ell$, and $\ell$ is an arbi... | https://mathoverflow.net/users/161234 | Computing the integral $\int_{-1}^1 dx \, |x| J_0(\alpha \sqrt{1 - x^2}) P_\ell(x)$ | Thanks to the [comment](https://mathoverflow.net/questions/365861/computing-the-integral-int-11-dx-x-j-0-alpha-sqrt1-x2-p-ell#comment924242_365861) by Johannes, the solution can indeed be obtained by using the following identities:
\begin{equation}
P\_\ell(z)
=
\frac{1}{2^\ell}
\sum\limits\_{k=0}^{\left\lfloor \frac{... | 4 | https://mathoverflow.net/users/161234 | 366191 | 153,800 |
https://mathoverflow.net/questions/366196 | 7 | Is there any characterization on the set of integers $n$ such that there is a 3-connected 5-regular simple $n$-vertex planar graph?
| https://mathoverflow.net/users/148974 | There is a 3-connected 5-regular simple $n$-vertex planar graph iff $n$ satisfies....? | There is a 3-connected 5-regular simple $n$-vertex planar graph if and only if $n=12$ or $n \ge 16$ is even. See [Recursive generation of 5-regular graphs](https://link.springer.com/chapter/10.1007%2F978-3-642-00202-1_12) by Mahdieh Hasheminezhad, Brendan D. McKay, Tristan Reeves in *WALCOM: Algorithms and Computation*... | 10 | https://mathoverflow.net/users/14807 | 366199 | 153,803 |
https://mathoverflow.net/questions/366171 | 6 | Let $S$ be a geometrically connected smooth projective surface over $\mathbb{Q}\_p$. Can it be put in a proper flat $\mathbb{Z}\_p$-scheme with a geometrically integral special fiber?
| https://mathoverflow.net/users/nan | Smooth projective surface with geometrically integral reduction | That is not true. There are probably shorter answers than the following. Let $K$ be a field, and denote a separable closure by $K^{\text{sep}}$. Let $n>1$ be an integer.
**Definition**. A **Severi-Brauer variety** over $K$ of relative dimension $n-1$ is a proper, smooth $K$-scheme whose base change to $K^{\text{sep}}... | 5 | https://mathoverflow.net/users/13265 | 366201 | 153,805 |
https://mathoverflow.net/questions/366192 | 2 | Does anyone know of a English translation of "Une inégalité pour martingales à indices multiples
et ses applications" by Renzo Cairoli. Or could translate the statement of the martingale convergence theoerm and his definition of multiindex martingale.
Link; <http://archive.numdam.org/article/SPS_1970__4__1_0.pdf>
R... | https://mathoverflow.net/users/83682 | English translation of "Une inégalité pour martingales à indices multiples et ses applications" | $\newcommand\Om\Omega$ $\newcommand\F{\mathcal F}$ $\newcommand\M{\mathcal M}$
With the help from Google Translate:
>
> Throughout the work $m$ is a fixed integer $\ge2$ and $j$ runs through the
> integers from $1$ to $m$. For each $j$, $(\Om\_j,\F\_j,P\_j)$ is a probability space. Set $\Om=\prod\limits\_j\Om\_j$, ... | 1 | https://mathoverflow.net/users/36721 | 366211 | 153,807 |
https://mathoverflow.net/questions/366147 | 6 | Let $f: X \to S$ be a proper morphism ($S$ locally noetherian), and $X \to S' \to S$ its Stein factorisation. By Zariski's Main Theorem the number of geometric connected components of the fibers of $f$ can be read from the cardinal of the fibers of the finite $S' \to S$. In particular if all fibers of $f$ are geometric... | https://mathoverflow.net/users/161405 | Proper morphisms with geometrically reduced and connected fibers | Here is a standard example. Take $\mathbb{P}^1\subset\mathbb{P}^3$ of large degree and let $S$ be the cone, with the vertex $p$, only singular point. Let $f:X\to S$ the blow up of $p$. One can check that $X$ is smooth and thus the Stein factorization $S'$ is the normalization of $S$. The fiber over $p$ in $X$ is smooth... | 6 | https://mathoverflow.net/users/9502 | 366214 | 153,808 |
https://mathoverflow.net/questions/366215 | -2 | Per the title, what are some of the oldest abstract algebra books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am already aware of the books of Dickson and van der Waerden.
| https://mathoverflow.net/users/126532 | Oldest abstract algebra book with exercises? | For a pre-20th century textbook: [Modern Higher Algebra](https://archive.org/details/3rdedlessonintro00salmuoft) by George Salmon (1876) has exercises (with solutions).
---
If I may broaden the query from "abstract algebra" to more general "algebra", I note that [Elements of Algebra](https://en.wikipedia.org/wiki... | 6 | https://mathoverflow.net/users/11260 | 366221 | 153,811 |
https://mathoverflow.net/questions/366202 | 5 | I was recently learning Furstenberg's theorem on random products of $SL(2,R)$ matrices, and came across with a simple example that confused me:
Considering random products of two matrices $A=\begin{pmatrix}
2 &0\\
0 &1/2
\end{pmatrix}$ and $B=\begin{pmatrix}
0 &1\\
-1 &0
\end{pmatrix}$ with probability $1/2$ and $1/2... | https://mathoverflow.net/users/144686 | Random products of $SL(2,R)$ matrices and Furstenberg's theorem | Because its Lyapunov exponent is zero, but that is not what you are computing.
Instead of looking at random walks on $SL(2, \mathbb{R})$, let me focus on random walks on $\mathbb{R}\_+^\*$, as there is the same issue. Let $(X\_n)$ be i.i.d. in $\mathbb{R}\_+^\*$, and to make things simple, assume that there are only ... | 6 | https://mathoverflow.net/users/75670 | 366223 | 153,812 |
https://mathoverflow.net/questions/366216 | 1 | Assume that the algebraically independent polynomials $f, g\in\mathbb{C}[x, y]$ are such that the Jacobian matrix $\text{Jac}\_{x, y}^{f, g}\in\mathbb{C}\setminus\{0\}$.
Is it true that $\mathbb{C}[x, y] = \mathbb{C}[f, g]+g\cdot\mathbb{C}[x, y]$?
| https://mathoverflow.net/users/100359 | Question about Jacobian subalgebra | This is still is equivalent to JC.
Your equality says, $\mathbb{C}[x,y]=\mathbb{C}[f,g]+g\mathbb{C}[x,y]$, the last term is equal to $\mathbb{C}[f]+g\mathbb{C}[f,g]+g\mathbb{C}[x,y]=\mathbb{C}[f]+g\mathbb{C}[x,y]$, since $g\mathbb{C}[f,g]\subset g\mathbb{C}[x,y]$. This says, the map $\mathbb{C}[f]\to \mathbb{C}[x,y]/... | 1 | https://mathoverflow.net/users/9502 | 366232 | 153,815 |
https://mathoverflow.net/questions/366222 | 1 | Statistical mechanics is all about taking thermodynamic limits and, as far as I know, there are more than one way to define such limits. Consider the following theorem:
**Theorem:** In the thermodynamic limit, the pressure:
$$\psi(\beta,h) := \lim\_{\Lambda \uparrow \mathbb{Z}^{d}}\psi\_{\Lambda}^{\#}(\beta, h) $$
is... | https://mathoverflow.net/users/150264 | What is the definition of the thermodynamic limit of a thermodynamic quantity? | It means that if you consider *any* sequence of sets $(\Lambda\_n)\_{n\in\mathbb{N}}$ converging to $\mathbb{Z}^d$ in the sense of van Hove, then the sequence of numbers $(\psi\_{\Lambda\_n}^\#(\beta,h))\_{n\in\mathbb{N}}$ is convergent. (Moreover, the theorem claims that the limit is independent of the sequence chosen... | 5 | https://mathoverflow.net/users/5709 | 366238 | 153,817 |
https://mathoverflow.net/questions/366241 | 6 | Let $\mathbb{C} \ni z \mapsto M(z)$ be a square matrix depending holomorphically on a parameter $z$ with the property that $\operatorname{dim}\ker(M(z)))=1$ for $z $ away from a discrete set $D \subset \mathbb{C}$ and $\operatorname{dim}\ker(M(z)))\ge 1$ for $z \in D.$
I ask: Is it always possible to choose a continu... | https://mathoverflow.net/users/119875 | Continuity of eigenvectors | Yes. Let the size of your matrix be $n$.
Your condition implies that there is an $n-1\times n-1$ submatrix whose determinant is not identically equal to $0$.
Assume without loss of generality that this is the submatrix formed by the first $n-1$ rows and columns. Then we can set $u\_n=1$ and find a vector $u(z)$ such th... | 7 | https://mathoverflow.net/users/25510 | 366247 | 153,821 |
https://mathoverflow.net/questions/366153 | 4 | I need literature about zeroes of polynomials and equation resolution in associative algebraic structures with zero-divisors, but I am having difficulties to find it.
For example, there is literature in the resolution of quadratic(and remarks on other degrees) in quaternions and split-quaternions.
It seems that sea... | https://mathoverflow.net/users/161411 | Literature on the polynomials and equations, in structures with zero-divisors | For **associative algebras**, as your required, see Plotkin, *Algebras with the same (algebraic) geometry*, Israel J. Math., 96 (2) (1996), 511–522.
This is, being more precise, part of this nice relatively new field of Universal Algebraic Geometry which discuss such things.
For a survey I recommend A. Shevlyakov, ... | 3 | https://mathoverflow.net/users/160378 | 366249 | 153,822 |
https://mathoverflow.net/questions/366234 | 6 | This could be a soft question. I am trying to show that the $n$-th Taylor series coefficient of a function is $O(n^{-5/2})$. However, because the function is a function composition of another function with itself, it seems intractable to compute high-order derivatives. I was wondering if there are methods that can boun... | https://mathoverflow.net/users/33278 | Analyzing the decay rate of Taylor series coefficients when high-order derivatives are intractable | Complex analysis can help. The rate of Taylor coefficients is determined by:
a) the radius of convergence, which is equal to the radius of the largest disk $|z|<r$
where your function is analytic. This radius is responsible for the exponential asymptotics, and
b) the nature of singularities on the circle $|z|=r$.
... | 9 | https://mathoverflow.net/users/25510 | 366250 | 153,823 |
https://mathoverflow.net/questions/366239 | 1 | Let $n,q$ be positive integers. We are interested to the cases where $n>q$.
Let $F:\mathbb B^n\to\mathbb S^{q-1}$ be a continuous (differentiable, if needed) map, such that $F(1,0^{n-1})=(1,0^{q-1})$, $F(-1,0^{n-1})=(-1,0^{q-1})$ and $F(\{0\}^1\times\mathbb S^{n-2})\subset\{0\}^1\times\mathbb S^{q-2}\simeq\mathbb S^{... | https://mathoverflow.net/users/118469 | Can a restriction of a null-homotopic spherical map be null-homopotic? | There exists nullhomotopic maps preserving codimension one equators such that the restriction to the equators is not nullhomotopic. Very easy examples come from taking non nullhomotopic maps of spheres and then extending them to a sphere of dimension higher by a nonsurjective map.
However, one can in fact come up wit... | 7 | https://mathoverflow.net/users/134512 | 366253 | 153,825 |
https://mathoverflow.net/questions/366168 | 2 | I have a research work concerning the equation: $$x '(t) + g (x (t)) = f (t),\quad \forall t\in \mathbb R$$
f and g are defined and continuous in $\mathbb R$ and with values in $\mathbb R$.
Furthermore f is assumed to be T periodic (There is no initial condition)
**First question:**
Assume that there is a periodi... | https://mathoverflow.net/users/126827 | $x '(t) + g (x (t)) = f (t),\quad \forall t\in \mathbb R$ have periodic solution $\iff\; \frac 1T \int_0 ^ T f (t) dt \in g (\mathbb R) $ | The condition is not sufficient. Take for example the equation
$$x'=x^2+\sin t$$ with $g(x)=-x^2$ in your notation. Then
$$\int\_0^{2\pi} \sin t\, dt=0=g(0).$$ If $x$ were a $2\pi$-periodic solution, then
$$
0=x(2\pi)-x(0)=\int\_0^{2\pi} x'(t)\, dt=\int\_0^{2\pi}(x^2(t)+\sin t)\, dt= \int\_0^{2\pi}x^2(t)\, dt$$
would i... | 3 | https://mathoverflow.net/users/150653 | 366265 | 153,827 |
https://mathoverflow.net/questions/366260 | 0 | Let $a\_0,\cdots,a\_n$ be algebraic integers. Is $h(a\_0,\cdots,a\_n)\le\max\_{0\le i\le n}\log(\max(1,|a\_i|))$ where $h(a\_0,\cdots,a\_n)$ denotes the logarithmic Weil height?
Thanks in advance.
| https://mathoverflow.net/users/33128 | Logarithmic Weil height | It depends on what you mean by $|\cdot|$, but probably no.
If by $|\cdot|$ you mean the absolute value on $\mathbb C$, and your algebraic integers are elements of $\mathbb C$, then the answer is no. The logarithmic Weil height of $1-\sqrt3\in\mathbb A^1(\overline{\mathbb Q})$ is $\frac12\log(2)$, which is strictly bi... | 3 | https://mathoverflow.net/users/126183 | 366268 | 153,829 |
https://mathoverflow.net/questions/366267 | 7 | I always had the dream to design a course for my graduate students like "mathematical models of the continuum". This course should cover history of real numbers, the Measure Problem, the Banach-Tarski Paradox, but also Baire/Cantor/Polish spaces, and it should look at infinite games and determinacy as well.
Obviously... | https://mathoverflow.net/users/161420 | Reference for graduate-level text or monograph with focus on "the continuum" | I like your idea of such a course a lot! If it is appropriate to recommend a book in German language, I think this one could be the perfect match:
Oliver Deiser (2007): *Reelle Zahlen: Das klassische Kontinuum und die natürlichen Folgen*
I own this book and can say it covers all the topics that you mentioned, and i... | 6 | https://mathoverflow.net/users/161374 | 366274 | 153,830 |
https://mathoverflow.net/questions/366275 | 3 | How to find the general solution of a differential equation with a shift, in the following form?
$$\frac{\partial}{\partial t}g(x,t)=g(x-\Delta,t)+\frac{\partial^2}{\partial x^2} g(x,t)$$
where $\Delta > 0$. And what about the following?
$$\frac{\partial}{\partial t}g(x,t)=g(x,t-\Delta)+\frac{\partial^2}{\partial... | https://mathoverflow.net/users/37545 | How to solve a differential equation in the form $\frac{\partial}{\partial t}g(x,t)=g(x-\Delta,t)+\frac{\partial^2}{\partial x^2} g(x,t)$? | Fourier transform $G(k,t)=\int\_{-\infty}^\infty e^{ikx} g(x,t)dx$ with respect to $x$, then
$$\frac{\partial}{\partial t}G(k,t)=e^{ik\Delta}G(k,t)-k^2 G(k,t),$$
hence
$$G(k,t)=\exp\left(te^{ik\Delta}-tk^2\right)G(k,0).$$
For the second differential equation you would similarly Fourier transform with respect to $t$.
| 3 | https://mathoverflow.net/users/11260 | 366277 | 153,831 |
https://mathoverflow.net/questions/366282 | 1 | Given a connected (undirected) graph with an even number of vertices, consider how many ways are there to pair up vertices so that each pair is connected by an edge. Is there a known classification of all such graphs into those with $0$ ways, $1$ way, and more than $1$ way?
For example, a star belongs to the first ca... | https://mathoverflow.net/users/83212 | Pairing up vertices in a graph | In general, finding the number of perfect matchings is #P-complete. But using the Edmonds "blossom" algorithm, one can decide effectively if the number is zero or not. And it's also easy using the same algorithm to figure out if the number is exactly 1 (in principle we can try removing the edges one by one and check in... | 5 | https://mathoverflow.net/users/14302 | 366284 | 153,833 |
https://mathoverflow.net/questions/366278 | 0 | Define the metric $d(f,g)\triangleq \sup\_{x \in [0,1]} \|f(x)-g(x)\|$ on the set $\operatorname{B}$ of uniformly bounded functions from the interval $[0,1]$ to $\mathbb{R}$, fix $g \in \operatorname{B}$, and define the map $F:\operatorname{B}\rightarrow [0,\infty)$ by $F(f):=d(g,f)$. Is the map $F$ continuous? It cert... | https://mathoverflow.net/users/36886 | Uniform distance from a discontinuous function is continuous | If $d$ is a metric on $B$ then the mapping $F(f) := d(g,f)$ is certainly continuous with respect to the topology induced by the metric $d$:
Let $(f\_n)\_{n\in\mathbb{N}}$ a sequence in $B$ that converges to $f \in B$ w.r.t. $d$. This is equivalent to
$$
d(f,f\_n) \to 0.
$$
Hence, by the triangular inequality
$$
F(f\_... | 5 | https://mathoverflow.net/users/114751 | 366289 | 153,834 |
https://mathoverflow.net/questions/366288 | 1 | Consider a system of ordinary differential equations of the form
$$
\dot{x}(t) + \frac{1}{t}Ax(t) = Q(x(t))
$$
where $x(t) \in \mathbb{C}^n$, $A \in \mathrm{Mat}\_{n\times n}(\mathbb{C})$ is a constant matrix, and $Q: \mathbb{C}^n \to \mathbb{C}^n$ is homogeneous of degree $2$, i.e. $Q(\lambda x) = \lambda^2 Q(x)$ for ... | https://mathoverflow.net/users/123207 | Regular singular point of non-linear ODE: $\dot{x}(t) + t^{-1}Ax(t) = Q(x(t))$ | There's nothing inherently linear about constructing power series solutions à la Frobenius. The existence and uniqueness theory for a class of singular non-linear ODEs, of which yours is a special case, is treated for instance in Ch.IX of
>
> Wasow, W., *Asymptotic expansions for ordinary differential equations*, (... | 2 | https://mathoverflow.net/users/2622 | 366295 | 153,835 |
https://mathoverflow.net/questions/366226 | 0 | The Lie derivative of a general covariant $4$-tensor is given by
$$\mathcal{L}\_{K}R\_{abcd} = X^{e}\nabla\_{e}R\_{abcd} + R\_{ebcd}\nabla\_{a}X^{e} + R\_{aecd}\nabla\_{b}X^{e} + R\_{abed}\nabla\_{c}X^{e} + R\_{abce}\nabla\_{d}X^{e},$$
where $X^{a}$ is a y smooth vector field. If the $(0,4)$ covariant tensor $R$ is t... | https://mathoverflow.net/users/99716 | Curvature collineation and the Killing identity | Using the identity in [here](https://mathoverflow.net/questions/334060/on-equation-delta-circ-partial-partial-x-partial-partial-x-circ-delta/334696#334696), we have
$$ [\nabla\_{[a}, \mathcal{L}\_X] R\_{bc]de} = \pi\_{d[a}{}^f R\_{bc]fe} + \pi\_{e[a}{}^f R\_{bc]df} \tag{1}$$
where we used that the first order deformati... | 2 | https://mathoverflow.net/users/3948 | 366298 | 153,838 |
https://mathoverflow.net/questions/366310 | 26 | One of the major results in graph theory is the graph structure theorem from Robertson and Seymour
<https://en.wikipedia.org/wiki/Graph_structure_theorem>. It gives a deep and fundamental connection between the theory of graph minors and topological embeddings, and is frequently applied for algorithms.
I was working ... | https://mathoverflow.net/users/161328 | Why did Robertson and Seymour call their breakthrough result a "red herring"? | Seymour and Robertson have indeed said that, and in fact they wrote that in their 2003 article in which they published the graph structure theorem.
Here is the quote from Robertson and Seymour „Graph Minors. XVI. Excluding a non-planar graph“ (Journal of Combinatorial Theory, Series B, Vol. 89, Issue 1, Sept. 2003, p... | 30 | https://mathoverflow.net/users/156936 | 366314 | 153,844 |
https://mathoverflow.net/questions/366316 | 2 | Fix $m,n \in \mathbb{N}$ with $m \ge n+1$. Take $m$ points in general position in $\mathbb{R}^n$ and let $P$ be their convex hull. What is the maximal number of (external, codimension-one) faces that $P$ can have, in terms of $m$ and $n$?
(Apologies if this is a well-known quantity.)
| https://mathoverflow.net/users/159965 | Polytope with most faces | The upper bound conjecture of Motzkin, made a theorem by McMullen in 1970, states that the highest number of facets among all polytopes with $m$ vertices in $\mathbb R^n$ is the number of facets of the cyclic polytope $\Delta(m,n)$.
| 5 | https://mathoverflow.net/users/125523 | 366319 | 153,846 |
https://mathoverflow.net/questions/366306 | 3 | I'm having trouble finding references for in-depth examples of perverse sheaves, so answers in the form of such a reference would be most helpful.
I want to construct an example of an intersection complex not concentrated in a single (natural) cohomology degree. Reading BBD, it seems the definition of intermediate ex... | https://mathoverflow.net/users/149523 | Example of an intersection complex not concentrated in a single degree | Sorry I haven't read your entire question, which is a bit long. This is really just an extended comment to address the "where I should look next?" part. Suppose $X$ has an isolated singularity $x$, and $j:U\to X$ is the smooth complement. Then the formula on top of page 60 of BBD would simplify to
$$j\_{!\*}\overline{\... | 4 | https://mathoverflow.net/users/4144 | 366322 | 153,847 |
https://mathoverflow.net/questions/366327 | 0 | I'm studying this classical paper in nonlinear dynamics in biophysics and I want to understand it properly. I'm stuck at this two-variable problem which I've struggled with for too long now.
$$ \dot M = \frac{1}{1+E^m} - a M $$
$$ \dot E = M - b E $$
Where $ \dot z = \frac{dz}{dt} $. It turns out later that $ m \ge... | https://mathoverflow.net/users/161594 | On proving the absence of limit cycles in a dynamical system | $$
\dot{X} = \dot{M} = \frac{1}{1+(E\_0 + Y)^m } -a(M\_0 +X) =
$$
$$
\frac{1}{1+E\_0^m + mE\_0^{m-1} Y + O(Y^2) } -aM\_0 -aX =
$$
$$
\frac{1}{1+E\_0^m} \left( 1-\frac{mE\_0^{m-1} }{1+E\_0^m } Y + O(Y^2) \right) -aM\_0 -aX
$$
Now use $1/(1+E\_0^m) =abE\_0 $ as well as $aM\_0 = abE\_0 $, yielding
$$
\dot{X} =-a^2 b^2 m ... | 1 | https://mathoverflow.net/users/134299 | 366332 | 153,850 |
https://mathoverflow.net/questions/366157 | 5 | Let $F=F(H,H)$ be the space of bounded Fredholm operators in a Hilbert space $H$ with topology inherited from the norm operator topology, and let $X$ be a compact topological space.
For a continuous map $T\colon X\to F$, there exists a closed subspace $W\subseteq H$ with $\dim H/W<\infty$ such that $W\cap\ker T\_x=0$... | https://mathoverflow.net/users/101270 | Equivalence of families indexes of Fredholm operators | Notice that $V^\perp\cap\ker T\_x^\*=0$ for every $x$, because for every $w\in V^\perp\cap\ker T\_x^\*$, $u\in H$, and $v\in V$, one has
$$\langle w,T\_x(u)+v \rangle = \langle w,T\_x(u) \rangle = 0.$$
By composing two isomorphisms $\ker P\_{V^\perp}T \ni u \mapsto u\oplus T(-u) \in \ker T^V$ and $\ker P\_{V^\perp}T\co... | 2 | https://mathoverflow.net/users/7591 | 366336 | 153,851 |
https://mathoverflow.net/questions/366342 | 2 | Let $f(r)=\frac{1}{(-log r)^{\alpha}}$ and let $\xi\_r$ be the unique value in $(0,r)$ such that$f(\xi\_r)=\frac{1}{r}\int\_{0}^rf(t)dt$, where $\alpha\in (0,1)$ and $r>0$. My question is about the order of $\xi\_r$?
Can we have $\xi\_r=o(r)$ as $r$ tends to $0$? moreover, can we get $\xi\_r=o(\frac{r}{log^2(1/r)})$?... | https://mathoverflow.net/users/99411 | Small-$r$ asymptotics of an integral of $1/\log ^\alpha r$ | Starting from the integral (in terms of an incomplete Gamma function)
$$g(r)=\frac{1}{r}\int\_0^r (-\ln t)^{-\alpha}\,dt=\frac{1}{r}\Gamma(1-\alpha,-\ln r),$$
we expand for $r\rightarrow 0$, or $y\equiv -\ln r\rightarrow\infty$,
$$g(r)=y^{-1-\alpha}\bigl(y-\alpha+{\cal O}(1/y)\bigr).$$
We then wish to solve $g(r)=(-\ln... | 5 | https://mathoverflow.net/users/11260 | 366352 | 153,854 |
https://mathoverflow.net/questions/366343 | 2 | Assume that $\mathcal{A}$ is a Lie $\mathbb{C}$-algebra. Also denote the Lie $\mathbb{C}$-algebra $\mathcal{D}=\mathbb{C}[x\_1,\ldots, x\_n]\partial\_{x\_1}\oplus\ldots\oplus\mathbb{C}[x\_1,\ldots, x\_n]\partial\_{x\_n}$. My questions are as follows
Let $\mathcal{A}\otimes\_{\mathbb{C}}\mathbb{C}(t)$, $\mathcal{D}\ot... | https://mathoverflow.net/users/100140 | Does $\mathcal{A}\otimes\mathbb{C}(t)\cong\mathcal{D}\otimes\mathbb{C}(t)$ imply an isomorphism of Lie algebras? | Put $\mathcal{A}'=\mathcal{A}\otimes\_{\mathbb{C}}\mathbb{C}(t)$ and similarly for $\mathcal{D}'$. Choose an isomorphism $f\colon\mathcal{D}'\to\mathcal{A'}$.
Choose a (countable) basis $\mathcal{D}\_0=\{d\_i:i\in\mathbb{N}\}$ for $\mathcal{D}$ over $\mathbb{C}$. Then $f(\mathcal{D}\_0)$ is a basis for $\mathcal{A}'... | 2 | https://mathoverflow.net/users/10366 | 366354 | 153,855 |
https://mathoverflow.net/questions/366339 | 3 | $$ A := \begin{pmatrix}
1 & \frac{1}{2} & \frac{1}{3} & \cdots & \frac{1}{n}\\
\frac{1}{2} & \frac{1}{2} & \frac{1}{3} & \cdots & \frac{1}{n}\\
\frac{1}{3} & \frac{1}{3} & \frac{1}{3} & \cdots & \frac{1}{n}\\
\vdots & \vdots & \vdots & \ddots & \frac{1}{n}\\
\frac{1}{n} & \frac{1}{n} & \frac{1}{n} & \frac{1}{n} & \frac... | https://mathoverflow.net/users/161607 | Maximum eigenvalue of a covariance matrix of Brownian motion | In this answer I show that the largest eigenvalue is bounded by $5< 3 + 2\sqrt{2}$. I will first use the interpretation of this matrix as the covariance matrix of the Brownian motion at times $(\frac{1}{n},\dots, 1)$ (I reversed the order so that the sequence of times is increasing, which is more natural for me).
We ... | 2 | https://mathoverflow.net/users/24953 | 366369 | 153,862 |
https://mathoverflow.net/questions/366307 | 1 | The colorful Carathéodory theorem (Bárány, 1982) considers $d+1$ "colors" $X\_1,\ldots,X\_{d+1}\subseteq \mathbb{R}^d$, and a point $x$ in the convex hull of each color ($x\in \text{conv}(X\_i)$ for each $i\in[d+1]$). It says that there exists a set of $d+1$ points of distinct colors, such that $x$ is in their convex h... | https://mathoverflow.net/users/34461 | Is the following generalization of the Caratheodory theorem true? | There is a counterexample in the plane, for example the following four points in convex position: $X\_1=(0,0), X\_2=(0,1), X\_3=\{(10,0),(10,10)\}$.
(This was originally posted as a comment to a previous version of the question).
| 2 | https://mathoverflow.net/users/24076 | 366370 | 153,863 |
https://mathoverflow.net/questions/366350 | 3 | Let $G$ be a pro-algebraic group, that is, a projective limit of algebraic groups $G\_i$. Let $V\_i$ be an inductive system of finite dimensional rational $G\_i$-representations, so that the colimit $V$ is a rational $G$ representation.
What conditions are required on $G$ to ensure that
$$
H^n(G,V)\simeq \mathrm{coli... | https://mathoverflow.net/users/121425 | When does cohomology of a pro-algebraic group commute with filtered colimits of coefficients? | This isomorphism always holds, and no conditions are needed. Here I presume that the pro-algebraic group $G$ is pro-affine (as the context of the question seems to suggest).
Let $G$ be a pro-affine pro-algebraic group over a field $k$. Denote by $C=O(G)$ the ring of regular functions on $G$. What is important for us ... | 3 | https://mathoverflow.net/users/2106 | 366386 | 153,867 |
https://mathoverflow.net/questions/366371 | 2 | I've always been interested in having a good understanding of Gentzen's proof of the consistency of arithmetic.
Being more precise, he showed that $PRA + WF(\epsilon\_0) \vdash Con(PA)$.
I am interested in an exposition of his work that
**1)** Is transparent on which parts of the consistency proof uses the well-f... | https://mathoverflow.net/users/160378 | Reference request on Gentzen's proof of the consistency of PA | Pohlers's 1989 book 'Proof Theory, An Introduction' gives a very clean, streamlined approach (based on work by Tait.)
Takeuti's presentation in his 'Proof Theory' is closer to Gentzen's original proof, but is much less readable than Pohlers.
| 4 | https://mathoverflow.net/users/24734 | 366387 | 153,868 |
https://mathoverflow.net/questions/366375 | 6 | Is $\arcsin(1/4) / \pi$ rational? An approximation given by a calculator seem to suggest that it isn't, but I found no proof. Thanks in advance!
| https://mathoverflow.net/users/16559 | Is $\arcsin(1/4) / \pi$ irrational? | This is a partial case of the classical result.
<https://en.wikipedia.org/wiki/Niven%27s_theorem>
| 14 | https://mathoverflow.net/users/4312 | 366388 | 153,869 |
https://mathoverflow.net/questions/366378 | 3 | I am trying to find all finite subgroups of $SL\_2(\mathbb{C})$ that arise as a semi-direct product of two finite (non-trivial) subgroups of $SL\_2(\mathbb{C})$. Is there any such characterization? Any hint/reference will be most welcome.
| https://mathoverflow.net/users/45397 | Finite subgroups of $SL_2(\mathbb{C})$ arising as a semi-direct product | Reducible finite subgroups $G$ of ${\rm SL}(2,\mathbb{C})$ are Abelian (and even cyclic), so these can be dealt with easily.
Every finite cyclic group can occur, and only the finite cyclic $p$-groups are not direct products of two non-trivial subgroups.
Irreducible, but imprimitive, subgroups $G$ of ${\rm SL}(2,\ma... | 10 | https://mathoverflow.net/users/14450 | 366391 | 153,870 |
https://mathoverflow.net/questions/366377 | 6 | Consider a compact closed category, i.e., a symmetric monoidal category with a unit $\eta$ and co-unit $\epsilon$. It seems natural to demand that the tensor product of two units (for different objects) is a unit again (for the tensor product of the objects). That is, we have an equality between the morphisms
$$
1\xrig... | https://mathoverflow.net/users/115363 | Tensor product of unit and co-unit in a closed compact category | The definition of compact closed category merely says that for every object $A$ there exists an object $A^\star$ and a unit $\eta\_A$ and counit $\varepsilon\_A$ satisfying the snake equations. That is, it is a property, not structure. Since this data is unique up to isomorphism, you can choose different units and coun... | 5 | https://mathoverflow.net/users/10368 | 366397 | 153,871 |
https://mathoverflow.net/questions/366340 | 4 | If one wants to do $p$-adic analysis and geometry, it is often bad so adapt "naively" complex analytic ideas, basically because $\mathbb{Q}\_p$ is disconnected. The modern approach to this is, to my knowledge, the theory of rigid analytic spaces and Berkovich spaces. For instance in the theory of Berkovich spaces, the ... | https://mathoverflow.net/users/152554 | Identity theorem in $p$-adic geometry/analysis | Let me assume that you have an analytic function $f$ defined on a closed one-dimensional unit disc, which is to say the spectrum of the Tate algebra $k\{T\}$. Then, Weierstrass preparation theorem tells you that $f$ may be written as a product of a polynomial $P$ and a nowhere vanishing function. If $P$ is non-zero, it... | 2 | https://mathoverflow.net/users/4069 | 366400 | 153,872 |
https://mathoverflow.net/questions/366231 | 9 | A theorem of Fontaine says that if a geometrically connected smooth proper variety $X$ over $\mathbb{Q}$ has good reduction everywhere then $h^{i, j}(X)=0$ for $i\neq j$, $i+j\leq 3$.
This means that the variety's Hodge numbers can "disqualify" it from having good reduction everywhere. For example, there do exist var... | https://mathoverflow.net/users/nan | Hodge numbers rule out good reduction | Because the inverse Hodge problem is a very difficult question, I think it's unlikely that there will be an answer for this question for *any* given set of Hodge numbers. For most given Hodge diamonds, we don't even know whether it can be realised by any smooth projective variety (or Kähler manifold); and when it can, ... | 8 | https://mathoverflow.net/users/82179 | 366408 | 153,875 |
https://mathoverflow.net/questions/364576 | 11 | In his 2008 paper *Hyperlinear and Sofic Groups: A Brief Guide*, Pestov asked (Open Question 9.5) whether every group with the Haagerup property is hyperlinear (or sofic). Has this question been answered in the meanwhile?
A short recap of the relevant definitions: A group is hyperlinear (sofic) if it embeds into the ... | https://mathoverflow.net/users/95776 | Are groups with the Haagerup property hyperlinear? | Thompson's group $F$ has the Haagerup property [1], but it is not known if it is hyperlinear according to Narutaka Ozawa's comment.
[1] Farley. *Finiteness and CAT(0) properties of diagram groups*. Topology, 2003.
| 4 | https://mathoverflow.net/users/95776 | 366409 | 153,876 |
https://mathoverflow.net/questions/366390 | 3 | Let $X$ be a proper geodesic space which is uniquely geodesic. Let $\phi:[0,1]\times[0,1] \to X$ be a function which satisfies the following:
The maps $\phi(0,\cdot)$, $\phi(\cdot,0)$, $\phi(1,\cdot)$, and $\phi(\cdot,1)$ are all (linearly parametrized) geodesics. Furthermore, for each fixed $s$, the map $\phi(s,\cdo... | https://mathoverflow.net/users/160011 | On a geodesic mapping of a square | This is not true. Let $X$ be the unit sphere, or some hemisphere thereof, which we describe first in spherical coordinates.
Let $f(s,0)$ go east along the equator, $(\theta,\phi)=(2s\pi/3,\pi/2)$.
Let $f(s,1)$ go south from the North Pole, $(\theta,\phi)=(\pi,s\pi/3)$
Let $f(s,t)$ be $t$ of the way from $f(s,0)$ ... | 3 | https://mathoverflow.net/users/nan | 366412 | 153,878 |
https://mathoverflow.net/questions/366416 | 3 | Let $D$ be a domain in $\mathbb{C}$ with $n$ boundary components. From the work of Koebe, we know that $D$ can be conformally mapped to a parallel slit domain of a specified angle of inclination (relative to the real axis).
**Question:** What boundary regularity does this map possess?
**Further remarks:** This map,... | https://mathoverflow.net/users/105103 | Regularity of a conformal map | EDIT. I just realized that my previous answer was incorrect.
The slits in the Kobe theorem that you consider are bounded,
since the image under the mapping function with your normalization
has $\infty$ inside the domain. Therefore the map is smooth if you assume
that the boundary of $D$ is smooth.
The smoothness depe... | 6 | https://mathoverflow.net/users/25510 | 366419 | 153,880 |
https://mathoverflow.net/questions/366420 | 12 | Let $M$ denote a smooth manifold, and $\omega \in \Omega^2(M, \mathbb{R})$ a symplectic form. The classical version of Darboux's theorem states that for any $x \in M$, there exists an open neighborhood $U$ of $x$ together with local coordinates $p\_1,\dots, p\_n, q\_1, \dots, q\_n$ on $U$ such that
\begin{equation}
\om... | https://mathoverflow.net/users/127878 | Is there an algebraic version of Darboux's theorem? | In fact, the opposite is true. For $X$ a smooth proper variety over $\mathbb C$ and $\omega$ a nonzero symplectic form on $X$. Then there is no nonempty Zariski open set on which $\omega$ is the pullback (under any map) of the standard symplectic form. The same should work for etale open sets, and even for smooth morph... | 18 | https://mathoverflow.net/users/18060 | 366422 | 153,881 |
https://mathoverflow.net/questions/366418 | 2 | Comments on the question [Are those distributional solutions that are functions, the same as weak solutions?](https://mathoverflow.net/questions/366411/are-those-distributional-solutions-that-are-functions-the-same-as-weak-solution?noredirect=1#comment925399_366411) suggest there might not be a standard definition of t... | https://mathoverflow.net/users/38783 | Is there a standard definition of weak form of a nonlinear PDE? | The study of nonlinear PDEs is almost always done in an ad hoc way. This is in sharp contrast to how research is done in almost every other area of modern mathematics. Although there are commonly used techniques, you usually have to customize them for each specific PDE.
| 4 | https://mathoverflow.net/users/613 | 366432 | 153,886 |
https://mathoverflow.net/questions/366439 | 4 | Let ${\cal U}$ be a non-principal ultrafilter on $\omega$, and for each $n\in\omega$, let $p\_n$ denote the $n$th prime, that is $p\_0 = 2, p\_1=3, \ldots$
Next we introduce the following standard equivalence relation on $\big(\prod\_{n\in\omega}\mathbb{Z}/p\_n\mathbb{Z}\big)$: we say $a \simeq\_{\cal U} b$ for $a,b ... | https://mathoverflow.net/users/8628 | Multiplicative and additive groups of the field $(\prod_{n\in\omega}\mathbb{Z}/p_n\mathbb{Z})/\simeq_{\cal U}$ | **1. $(K,+)$ and $(\mathbb R,+)$ are isomorphic.**
The additive group of any field $K$ is a vector space over its prime field ($\mathbb F\_p$ or $\mathbb Q$), hence it is determined up to isomorphism by the characteristic of $K$ and its degree over the prime field (which is just $|K|$ for uncountable $K$). Here, $K$ ... | 9 | https://mathoverflow.net/users/12705 | 366445 | 153,891 |
https://mathoverflow.net/questions/366458 | 1 | Any unitary matrix $U$ can be diagonalized by another unitary matrix $V$,
$$U=VDV^\dagger,$$
where $D={\rm diag}(z\_1,z\_2,...,z\_N)$ is diagonal.
If $U$ is taken at random uniformly with respect to Haar measure, then $V$ and $D$ are independent and $D$ has the Weyl distribution, $P(D)\propto \prod\_{j<k}|z\_k-z\_j|^... | https://mathoverflow.net/users/83671 | Eigenvectors of random unitary matrices | The invariance of the Haar measure implies that the probability to draw the matrix $U$ from the unitary group is unchanged if you replace $U$ by $U\_0 U U\_0^\dagger$, with $U\_0$ an arbitrary unitary matrix. Since this conjugation changes the unitary matrix of eigenvectors from $V$ into $U\_0V$, it means that $V$ and ... | 1 | https://mathoverflow.net/users/11260 | 366469 | 153,900 |
https://mathoverflow.net/questions/366442 | 1 | Let $u\_m = \ln ^2 m$.
Does there exist a non-increasing sequence of positive numbers $\{g\_n\}\_{n \in \mathbb{N}}$, $g\_n \to 0$, such that
$$\sum\limits\_{n \in \mathbb{N} } g\_n = \infty, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$
$$\sum\li... | https://mathoverflow.net/users/41071 | Convergence properties of related series | Call the two series $S\_1, S\_2$. Start out by letting $g\_1=1$. Whatever we do afterwards, this makes sure that $S\_1\ge 1$. Next, fix an $M$ such that $u\_M e^{-1\cdot u\_1}\ge 2$, and then give $g\_2, \ldots, g\_M$ a common small value that will give us
$$
e^{-\sum\_{j=2}^M g\_j u\_j}\ge \frac{1}{2} .
$$
This guaran... | 2 | https://mathoverflow.net/users/48839 | 366473 | 153,902 |
https://mathoverflow.net/questions/366414 | 0 | [I asked a [version](https://math.stackexchange.com/questions/3747401/definition-of-a-system-of-recurrent-events) of this question on MSE a few weeks ago and didn't get any useful feedback. Apologies if I am just being stupid.]
I am reading the paper [*A note on the Borel-Cantelli lemma* by Kochen and Stone](http://p... | https://mathoverflow.net/users/8187 | Definition of a system of recurrent events | Assume the random walk starts at the origin. Let $T\_0=0$ and for $i\geq 1$ let $T\_i$ be the time of the $i$th visit of the walk to the origin after time $0$. Let $X\_i=T\_i-T\_{i-1}$. Then the $X\_i$ are i.i.d. (you may like to think of this in terms of the strong Markov property, for example), and they take even pos... | 1 | https://mathoverflow.net/users/5784 | 366477 | 153,903 |
https://mathoverflow.net/questions/366476 | 3 | In the lecture [Notions of Scalar Curvature - IAS](https://youtu.be/uBrrKaBrPKU) around 8:00, Gromov states the following result, which he claims he does "slightly uncarefully":
>
> Suppose $(X,g\_X)$ and $(Y,g\_Y)$ are Riemannian manifolds, their sectional curvature satisfy $\sec(Y,g\_Y)\leq \kappa\leq \sec(X,g\_X... | https://mathoverflow.net/users/106283 | Reference request: extendability of Lipschitz maps as a synthetic notion of curvature bounds | I can give a partial answer. The theorem you quote is a generalization of [Kirszbraun's theorem](https://en.wikipedia.org/wiki/Kirszbraun_theorem) (which covers the case where $X$ and $Y$ are Hilbert spaces), and a special case of a beautiful theorem of [Lang and Schroeder](https://link.springer.com/article/10.1007/s00... | 3 | https://mathoverflow.net/users/142382 | 366496 | 153,909 |
https://mathoverflow.net/questions/366507 | 2 | I have recently been getting into origami and reading Robert J. Lang's (a physicist and one of the leading modern origami artist) books. In the book *Origami Design Secrets* he showed a sequence of more and more complicated origami bases (the starting points for many origami creations).
[Here are the first 5 elements... | https://mathoverflow.net/users/161460 | Solution for the sequence of the number of "major flaps" in origami bases and its relation to other sequences | What's up, Subhasish! (I'm actually the friend Subhasish mentioned to @AmirSagiv)
To start off with, the first couple terms of the sequence I found with my solution were: $1, 2, 4, 5, 9, 13, 25, 41, 81, 145, 289$.
I found a solution that starts working from the third term. However, it uses a different function for ... | 1 | https://mathoverflow.net/users/161735 | 366527 | 153,916 |
https://mathoverflow.net/questions/366421 | 12 | For $\Gamma$ a set of second-order sentences in the empty language, say that a set $X$ is **$\Gamma$-pseudofinite** if $X$ is infinite but for every sentence $\varphi\in\Gamma$ which is satisfied in every finite pure set we have $X\models\varphi$. For example, $\mathsf{ZF}$ proves that the sentence "I can be linearly o... | https://mathoverflow.net/users/8133 | The strength of "There are no $\Pi^1_1$-pseudofinite sets" | My "hunch" in the comments to the question appears to be correct! This model comes from *Howard, Paul E.; Yorke, Mary F.*, [**Definitions of finite**](http://dx.doi.org/10.4064/fm-133-3-169-177), Fundam. Math. 133, No. 3, 169-177 (1989). [ZBL0704.03033](https://zbmath.org/?q=an:0704.03033). The paper has a few confusin... | 6 | https://mathoverflow.net/users/2000 | 366529 | 153,917 |
https://mathoverflow.net/questions/332243 | 6 | Is there a good reference on how RSK [(and the 3 other variants)](https://arxiv.org/pdf/math/0510676.pdf)
interact with crystal operators on the semi-standard tableaux $(P,Q)$ in the image?
That is, we have biwords, $W$ which are in bijection with pairs of semi-standard tableaux $(P,Q)$ under RSK. Now, we act on $P$ ... | https://mathoverflow.net/users/1056 | RSK and crystal operators | I figured out the details, and [wrote it up here](https://www.math.upenn.edu/%7Epeal/polynomials/tableauOperators.htm#crystals-RSK). I did not manage to find a good reference. There are a few nice surveys on RSK and on crystals, but a survey covering how different tableau operators interact would be nice to see someone... | 2 | https://mathoverflow.net/users/1056 | 366533 | 153,921 |
https://mathoverflow.net/questions/366535 | 4 | Fix an algebraically closed field $k$. Let $X$ and $Y$ be proper varieties over $k$. If there is a connected scheme $B$ of finite type over $k$ such that $X$ and $Y$ embed in a proper flat family over $B$ is there also an irreducible such scheme?
| https://mathoverflow.net/users/nan | Deformation equivalent varieties over an irreducible base | Not in general. Perhaps the simplest example is given by the Hilbert scheme of curves $C$ of degree 3 in $\mathbb{P}^3$ with $\chi (\mathscr{O}\_C)=1$. This has 2 components, one (of dimension 12) corresponding to twisted cubics and the other (of dimension 15) parametrizing the union of a plane cubic and a point in $\m... | 7 | https://mathoverflow.net/users/40297 | 366536 | 153,922 |
https://mathoverflow.net/questions/366549 | 7 | Let $V = (\mathbb{R}^n, g)$, where $g$ is the Euclidean inner product on $V$. Denote by $G$ the orthogonal group $O(V) = O(n)$ and by $\mathfrak{g}$ the Lie algebra of $G$.
Let $W \subset \Lambda^2V^\* \odot \Lambda^2V^\*$ be the subset satisfying the algebraic Bianchi identity. More precisely, let $R(v\_1,v\_2,v\_3,... | https://mathoverflow.net/users/81645 | What are all invariant polynomials on the space of algebraic curvature tensors? | I am not sure that this has a "nice" answer. Your question can be reformulated as follows. Let $\mathcal{A}\_n$ be the space of algebraic curvature tensors on $\mathbb{R}^n$. A homogenous polynomial $P$ on $\mathcal{A}\_n$ is the same as an element of $S^k\mathcal{A}\_n$, the $k$-th symmetric tensor power of $\mathcal{... | 5 | https://mathoverflow.net/users/16702 | 366558 | 153,931 |
https://mathoverflow.net/questions/231325 | 15 | I am interested in the Hausdorff dimension of the Apollonian circle packing.
There seem to be two numerical calculations of the value:
```
1.305686729(10)
```
from [P.B Thomas and D.Dhar, *The Hausdorf*[sic!] *dimension of the Apollonian packing of circles* Journal of Physics A: Mathematical and General, Volume 27... | https://mathoverflow.net/users/39495 | Hausdorff dimension of Apollonian circle packing, 1.305686729, 1.305688 or yet something else? | It seems in the meantime, there are new results:
[Bai, Zai-Qiao; Finch, Steven R. *Precise calculation of Hausdorff dimension of Apollonian gasket*. Fractals 26 (2018), no. 4, 9 pp.](https://www.worldscientific.com/doi/pdf/10.1142/S0218348X18500500)
claims a better approximation is
$$1.30568672804987718464598620685... | 3 | https://mathoverflow.net/users/39495 | 366565 | 153,934 |
https://mathoverflow.net/questions/366553 | 6 | Let $X$ be a configuration space, a finite-dimensional manifold. By "quantum mechanics on $X$" I mean a linear evolution equation on complex-valued functions on $X$, determined by a Hamiltonian $H\in \text{End} [L^2(X,\mathbb{C})]$, with endomorphisms defined in an appropriate densely defined sense. (I am not requiring... | https://mathoverflow.net/users/7108 | Path integral as quantum mechanics on the tangent bundle | Path integrals over simultaneous position/velocity, or more commonly position/momentum, degrees of freedom are known as *phase space path integrals*. I don't know very much about the rigorous construction of path integral measures, and even less so about their phase space version. However, there does appear to be at le... | 4 | https://mathoverflow.net/users/2622 | 366569 | 153,935 |
https://mathoverflow.net/questions/366578 | 4 | Let $k$ be an algebraically closed field and $X,Y$ two $k$-schemes. We fix a $k$-point in $X$ and in $Y$ each, which we denote by abuse of notation by $P$. Since the pushout of schemes along closed embeddings exists, we know that there exists a scheme $X\coprod\_{P} Y$. We know that the cocartesian diagram is also cart... | https://mathoverflow.net/users/152554 | Pushout of schemes and étale cohomology | Probably it's better to take a sheaf-theoretic approach. We have maps $z\_X: X \to X\coprod\_P Y$, $z\_Y: Y \to X\coprod\_P Y$, $q: \operatorname{Spec} k \to X\coprod\_P Y$.
There is an exact sequence of sheaves on $X\coprod\_P Y$ $$0 \to \Lambda \to z\_{X\*} \Lambda \oplus z\_{Y\*} \Lambda \to q\_\* \Lambda \to 0 $$... | 6 | https://mathoverflow.net/users/18060 | 366582 | 153,939 |
https://mathoverflow.net/questions/366491 | 3 | (Edited)
**I need a reference to the following result:**
-----------------------------------------------
If $u \in H^2(B\_1^+) \cap {\rm Lip}(B\_1^+)$ satisfies
\begin{cases}
{\rm div}(F(x,u,\nabla u)) = F\_0(x,u,\nabla u) \quad & {\rm in} \ B\_1^+ \\
u = 0 & {\rm on} \ B\_1'
\end{cases}
where
$$F \in C^{1,\b... | https://mathoverflow.net/users/131957 | Reference to a Classical Regularity Theorem | The discussion from Section 13.1 in the book of Gilbarg and Trudinger shows that $u \in C^{1,\,\alpha}\left(B\_{3/4}^+\right)$. From here one can apply Schauder estimates for linear equations. For example, one can pass the divergence on the left hand side and view $u$ as a solution to a non-divergence form linear equat... | 2 | https://mathoverflow.net/users/16659 | 366583 | 153,940 |
https://mathoverflow.net/questions/366561 | 12 | Let $K\_0$ and $ K\_1$ be knots in $S^3$. They are called *smoothly concordant* if there is a smoothly properly embedded cylinder $S^1 \times [0,1]$ in $S^3 \times [0,1]$ such that $\partial (S^1 \times [0,1]) = -(K\_0) \cup K\_1$.
Let $Y\_0$ and $ Y\_1$ be integral homology spheres, i.e., $H\_\*(Y\_i; \mathbb Z) = H... | https://mathoverflow.net/users/nan | Relating smooth concordance and homology cobordism via integral surgeries | I will call $X\_n(K)$ the trace of $n$-surgery along $K$, that is a 4-manifold diffeomorphic to the union of $B^4$ and an $n$-framed 2-handle attached along $K \subset S^3 = \partial B^4$.
Call $A \subset S^3 \times I$ the concordance from $K\_0$ to $K\_1$.
Consider $X\_1 := X\_n(K\_1)$, viewed as $B^4 \cup S^3\times... | 5 | https://mathoverflow.net/users/13119 | 366585 | 153,941 |
https://mathoverflow.net/questions/366407 | 15 | This is a cross-posted on MSE [here](https://math.stackexchange.com/questions/3764930/in-infinite-dimensions-is-it-possible-that-convergence-of-distances-to-a-sequen).
Let $(X,d)$ be a metric space. Say that $x\_n\in X$ is a P-sequence if $\lim\_{n\rightarrow\infty}d(x\_n,y)$ converges for every $y\in X.$ Say that $(... | https://mathoverflow.net/users/147463 | In infinite dimensions, is it possible that convergence of distances to a sequence always implies convergence of that sequence? | That every Banach space is contained in a $P$-complete Banach space follows immediately from the following
Theorem.
Let $X$ be a Banach space. Then there exists a Banach space $Y$ containing $X$ in which no separated sequence is a $P$-sequence.
Modulo "abstract nonsense", which I will explain later, the theorem fol... | 10 | https://mathoverflow.net/users/2554 | 366587 | 153,942 |
https://mathoverflow.net/questions/366551 | 3 | If $G\left(A\cup B,\ E=\lbrace\lbrace a, b\rbrace\,|\, a\in A,\, b\in B\rbrace\right)$ is a weighted bipartite graph and $M\_0$ an initial perfect matching, then the optimality of $M\_0$ can be verified by the absence of negative cycles in the associated residual network $N\left(V=A\cup B,\,F=\lbrace(a\_i,b\_j)\,|\,e\_... | https://mathoverflow.net/users/31310 | Is it possible to improve the weight of perfect bipartite matchings faster than with Bellman-Ford? | If I understand notation correctly $e\_{ij}$ is the edge $\{a\_i, b\_j\}$ in $G$. I'll let $w\_{ij}$ be the weight $e\_{ij}$. I'll give an example showing the alternative method can fail to detect a negative cycle in $N$. Consider
$$w\_{11} = \epsilon$$
$$w\_{12} = B$$
$$w\_{13} = B$$
$$w\_{21} = B$$
$$w\_{22} = A$$
... | 3 | https://mathoverflow.net/users/51668 | 366588 | 153,943 |
https://mathoverflow.net/questions/364923 | 5 | Does the small étale topos of a smooth proper variety over a perfect field of positive characteristic determine its Hodge numbers? We consider it as a Grothendieck topos over the étale topos of the field.
| https://mathoverflow.net/users/nan | Does the étale topos determine the Hodge numbers? | No. See Proposition 2.14 in *[Canonical models of surfaces of general type in positive characteristic](http://www.numdam.org/item/PMIHES_1988__67__97_0/)*
| 2 | https://mathoverflow.net/users/nan | 366606 | 153,950 |
https://mathoverflow.net/questions/366601 | 14 | Assume $\sf ZF+AD$. Is there some inner model $M$ containing all the ordinals such that $M\models\sf ZF+AD$ as well?
What if we require $\omega\_1$ and/or $\omega\_2$ to be computed correctly?
Can we say anything about these models (e.g. $M\models V=L(\Bbb R)$)?
Is it at all consistent?
There's no real reason t... | https://mathoverflow.net/users/7206 | Is there a minimal inner model for determinacy? | Assuming AD, Woodin showed that no inner model that is missing a real correctly computes $\omega\_1$. (This almost follows from Theorem 9 of Velickovic-Woodin's "[Complexity of the set of reals of inner models of set theory](https://arxiv.org/pdf/math/9501203.pdf).")
Therefore if you take an inner model $M$ of AD + $V ... | 17 | https://mathoverflow.net/users/102684 | 366620 | 153,954 |
https://mathoverflow.net/questions/366512 | 3 | In the book "Harmonic Measure" by Garnett and Marshall, we have the following result:
**Lemma I.2.3** Let $\mu$ be a positive Borel measure on $\partial{\mathbb{D}}$ and let $\{I\_{j}\}$ be a finite sequence of open intervals in $\partial{\mathbb{D}}$. Then $\{I\_{j}\}$ contains a pairwise disjoint subfamily $\{J\_{k... | https://mathoverflow.net/users/80052 | A Covering Lemma for Arbitrary Measures | The answer is no for $n \ge 2$. Consider open cubes $Q(c,1)=\{c\_i <x\_i <c\_i+1\}$ of side $1$. Starting from $Q(0,1)$ and moving $c$ along the diagonal joining $0$ with $(1,\dotsc,1)$ one constructs $N$ cubes $Q(c\_i,1)$ such that two of them always intersect and no cube is contained in the union of the others. Take ... | 1 | https://mathoverflow.net/users/150653 | 366624 | 153,955 |
https://mathoverflow.net/questions/366622 | 9 | Let $G$ be the supergroup $\text{GL}(m|n)$. It has a tautological representation $V= \mathbb{C}^{m|n}$.
For every natural number $d$ we have a natural map $$\Phi\_d:\mathbb{C} S\_d\to \text{End}\_G(V^{\otimes d})$$
where $\sigma\in S\_d$ is sent to the linear transformation given by tensor permuting $V^{\otimes d}$ acc... | https://mathoverflow.net/users/41644 | Schur Weyl duality for the supergroup $\text{GL}(m|n)$ | Schur Weyl duality holds in the super case, as well. There is the double centralizer property, thus a positive answer to Q1, and also a characterization of the kernel as those ideals of $\mathbb C[S\_d]$ which correspond to partitions that don't fit inside the (m,n)-hook.
See the paper ["Hook Young diagrams with appl... | 6 | https://mathoverflow.net/users/2384 | 366632 | 153,959 |
https://mathoverflow.net/questions/366635 | 4 | Let $X\to \mathrm{Spec}\:\mathbb{Z}\_{(p)}$ be a smooth proper morphism with a geometrically connected generic fiber. Assume that the special fiber has an $\mathbb{F}\_p$-point.
Via the isomorphism $H^{\*}\_{\mathrm{cris}}(X\_{\mathbb{F}\_p}/\mathbb{Z}\_p)\approx H^{\*}\_{\mathrm{dR}}(X\_{\mathbb{Z}\_p})$ the absolut... | https://mathoverflow.net/users/nan | Is the action of the absolute Frobenius on de Rham cohomology induced by an algebraic map? | The answer is no. For a generic elliptic curve E, every algebraic map $E\to E$ is, up to shift by elements of $E$, a multiplication map $[k]\colon E\to E$. In particular, it acts on $H^1\_{\mathrm{dR}}(E)$ as multiplication by $k$. But $\mathrm{Fr}$ acts on the 2-dimensional space $H^1(E)$ as a linear map with determin... | 7 | https://mathoverflow.net/users/115052 | 366637 | 153,961 |
https://mathoverflow.net/questions/366611 | 3 | I am looking for a bit of orientation with regards to computational topology resources, as I am personally totally ignorant on the subject. I have lots of different links in $S^3$ (hundreds of millions) lying around in some boxes my garage and I would like to start sorting through them and getting rid of the ones that ... | https://mathoverflow.net/users/99414 | Looking through a bunch of links for unlinks? | It will of course depend on where your examples are coming from. But here are some lightweight approaches.
1. Randomize the triangulation of the link complement a few times and then simplify. Do you get a standard triangulation of the unlink complement? (Regina and Snappy will both do something like this.)
2. Compute... | 3 | https://mathoverflow.net/users/1650 | 366642 | 153,962 |
https://mathoverflow.net/questions/361835 | 3 | [Sierpiński number](https://en.wikipedia.org/wiki/Sierpinski_number) is an odd integer $k$ such that $2^nk+1$ is composite for all $n\in{\mathbb N}$. In the paper [Sur un probleme concernant les nombres $k\cdot2^n+1$](https://www.e-periodica.ch/digbib/view?pid=edm-001:1960:15#133), zbl:0093.04602 (1960), Sierpiński pro... | https://mathoverflow.net/users/106742 | Do Sierpiński numbers of Izotov type have a covering set? |
>
> So, for Sierpiński numbers of the Izotov type $(\*)$, was
> a bigger covering set found between 1995 and 2015?
>
>
>
No, it was not. And it is conjectured that none exists. In the Math Stack Exchange thread linked by Gerry Myerson in a comment to the question, I give other examples of Sierpiński numbers for ... | 1 | https://mathoverflow.net/users/66308 | 366647 | 153,965 |
https://mathoverflow.net/questions/366641 | 1 | Let $(X,d)$ be a (separable, complete) metric space with uniformly strictly non-positive curvature in the sense of Alexandrov, i.e. $(X,d)$ satisfies a $CAT(K)$ inequality for some $K<0$.
Does it hold that the 2-Wasserstein space on $(X,d)$ has strictly non-positive curvature?
I suspect this is false, but haven't f... | https://mathoverflow.net/users/100163 | Wasserstein space with strictly non-positive sectional curvature | Yes, it is false. Note that the space $W\_2(X)$ contains the space $\tfrac1{\sqrt{n}}\cdot X^{\times n}/S\_n$ where the group $S\_n$ permutes the $X$-factors. While space $X^{\times n}$ is $\mathrm{CAT}(0)$, the quotient $X^{\times n}/S\_n$ is not --- for example, take $X=$ Lobachevsky plane.
| 3 | https://mathoverflow.net/users/1441 | 366651 | 153,966 |
https://mathoverflow.net/questions/366663 | 12 | Define $a\_n$ as follows:
$$
a\_1=1,\ \ a\_{n+1}=na\_n+1\
$$
At this time, the sequence $a\_n$ is as follows:
$$
a\_n=\sum\_{k=1}^{n}\frac{(n-1)!}{(k-1)!}
$$
I made some discoveries about this sequence.
The first:$$a\_k\equiv 0\pmod{m}\Rightarrow a\_{k+Nm}\equiv 0\pmod{m}~~~~\forall k,m,N\in\mathbb{N}$$
The secon... | https://mathoverflow.net/users/161811 | Is $\sum_{k=1}^{n}\frac{(n-1)!}{(k-1)!}$ composite for $n\geq 4$? | $a\_n$ is composite for $4 \le n \le 2016$.
$a\_{2017}$ appears to be prime (it passes a strong pseudoprime test). I have not tried to certify that it is prime (this would take a while as the number has 5789 digits).
| 9 | https://mathoverflow.net/users/4854 | 366665 | 153,970 |
https://mathoverflow.net/questions/366674 | 2 | If $H\_i = (V\_i, E\_i)$ are [hypergraphs](https://en.wikipedia.org/wiki/Hypergraph) for $i = 1,2$ , we say that they are *isomorphic* if there is a bijection $f:V\_1 \to V\_2$ such that for all $e\subseteq V\_1$ we have $e\in E\_1$ if and only if $f(e)\in E\_2$.
If $(X,\tau)$ is a topological space, we let the *dens... | https://mathoverflow.net/users/8628 | Isomorphic hypergraphs of dense sets of Hausdorff spaces | Counterexamples abound. Here are a few of them.
**Theorem.** If $\emptyset\ne X\subseteq\mathbb R$ and $X\subseteq\operatorname{cl}(\operatorname{int}(X))$, then $\mathcal D(X)\cong\mathcal D(\mathbb R)$.
**Proof.** Construct an infinite sequence of pairwise disjoint open intervals $I\_n$ so that $\bigcup\_{n=1}^\i... | 3 | https://mathoverflow.net/users/43266 | 366691 | 153,976 |
https://mathoverflow.net/questions/366687 | 5 | I am interested in the status of the conjecture about the minimum number of edge crossings $cr(K\_{m,n})$ in a drawing of the complete bipartite graph $K\_{m,n}$.
The Wikipedia article <https://en.wikipedia.org/wiki/Tur%C3%A1n%27s_brick_factory_problem> led me to study the original papers of Zarankiewicz (*On a probl... | https://mathoverflow.net/users/161420 | Conjecture about minimal number of edge crossings in complete bipartite graphs | The Electronic Journal of Combinatorics has many [Dynamic Surveys](https://www.combinatorics.org/ojs/index.php/eljc/issue/view/Surveys) one of which is [The Graph Crossing Number and its Variants: A Survey](https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS21/pdf) by Schaefer which first appeared in 2013 ... | 7 | https://mathoverflow.net/users/51668 | 366692 | 153,977 |
https://mathoverflow.net/questions/366680 | 5 | I would be interested in recommendations for easily accessible/somehow simply readable texts to Cochran-Orr-Teichner's filtrations on knot concordance:
>
> Tim D. Cochran, Kent E. Orr, and Peter Teichner. "Knot concordance, Whitney towers and L2-signatures." Annals of Mathematics (2003): 433-519.
>
>
>
Any sug... | https://mathoverflow.net/users/nan | Reference for Cochran-Orr-Teichner's filtrations on knot concordance | There are summarys of parts of Cochran, Teichner and Orr's paper in:
* [These lecture notes](https://www.maths.ed.ac.uk/%7Ev1ranick/surgery/sliceknots2.pdf) of Peter Teichner, typed up by Julia Collins and Mark Powell;
* [Mark Powell's 2011 Edinburgh PhD thesis;](https://www.maths.ed.ac.uk/%7Ev1ranick/papers/powellth... | 5 | https://mathoverflow.net/users/8103 | 366694 | 153,979 |
https://mathoverflow.net/questions/366703 | 3 | Let $V$ be a smooth complex projective variety. Choose a very ample class $H\in H^2(V, \mathbb{Q})$. Can there exist finite étale morphisms $\phi\_k:V\to V$ for each $k\geq 1$ such that $\phi^\*\_kH=kH$?
| https://mathoverflow.net/users/nan | Étale covers pulling back a very ample class to any integer multiple | Yes, this is possible.
Let $E$ be the elliptic curve with equation $y^2 =x^3 -x$, and let $V = E^2$. Then endomorphisms of the abelian surface $V$ are given by two-by-two matrices over $\mathbb Z[i]$.
Let $H$ be the sum of the pullbacks of the class of $O(1)$ (in other words, three times the identity) from both ell... | 5 | https://mathoverflow.net/users/18060 | 366706 | 153,982 |
https://mathoverflow.net/questions/366708 | 5 | It is a standard result that for a CW complex $X$, the chern character
$$\text{ch}: K^\*(X)\otimes\_{\mathbb{Z}} \mathbb{Q}\to H^\*(X,\mathbb{Q})$$
induces an isomorphism. Suppose now that $X$ is an open manifold and consider the chern character with compact support
$$\text{ch}\_{\text{cs}}: K^\*\_{\text{cs}}(X)\... | https://mathoverflow.net/users/109370 | Compactly supported chern character | Yes, this is true. For any generalized cohomology theory $E$, the compactly supported $E$-cohomology of a space $X$ is
$$E\_{\mathit{cs}}^\*(X) := \varinjlim\limits\_{K\subseteq X:\text{ $K$ compact}} E^\*(X, X\setminus K).$$
The Chern character is a natural isomorphism of cohomology theories, so is compatible with... | 6 | https://mathoverflow.net/users/97265 | 366710 | 153,984 |
https://mathoverflow.net/questions/366711 | 1 | I have perhaps a very simple question where I lack some inutition at the moment: Is the expression
$$\sup\_{\alpha < 0, \lambda \in \mathbb N}\int\_{-\infty}^{\alpha} e^{-\lambda t^4} \ dt \int\_{\alpha}^0 e^{\lambda t^4} \ dt $$
finite or not?-My main concern is $\alpha$ close to zero since $t^4$ is not strongly con... | https://mathoverflow.net/users/119875 | Uniform boundedness of integral? | Let $a:=\alpha$ and $x:=\lambda$. Using now the substitution $t=x^{-1/4}s$ and introducing $b:=x^{1/4}a$, we see that the problem is to show that $\sup\_{b\le0,x\ge1}x^{-1/2}I(b)J(b)=\sup\_{b\le0}I(b)J(b)<\infty$, where
$$I(b):=\int\_b^0 e^{s^4}\,ds=\int\_0^{-b}e^{s^4}\,ds,\quad J(b):=\int\_{-\infty}^b e^{-s^4}\,ds$$
f... | 4 | https://mathoverflow.net/users/36721 | 366717 | 153,986 |
https://mathoverflow.net/questions/366712 | 6 | Can 17 positive integers *in arithmetic progression* be found such that that no four of them have, pairwise, a common divisor greater than 1, but, likewise, no four of them are, pairwise, relatively prime?
Because R(4,4)=18, 18 such numbers are impossible.
At <https://puzzling.stackexchange.com/questions/100391/sev... | https://mathoverflow.net/users/60732 | Ramsey's number R(4,4) with arithmetic progressions | No, there cannot be 17 such numbers in arithmetic progression (and there cannot be 5 such numbers with the corresponding property for triples).
Suppose we have such an arithmetic progression of length $k$, say $x,x+d,\ldots,x+(k-1)d$. I claim that if a prime $p$ divides any two of them then either it divides all of t... | 5 | https://mathoverflow.net/users/385 | 366724 | 153,991 |
https://mathoverflow.net/questions/366404 | 2 | Let $X$ and $Y$ be two $\rho$ correlated Gaussian vectors, such that $X,Y\sim N(0,1)^n$ and $E[X\_iY\_i]=\rho$.
Let $M\_X = f(X)$ and $M\_Y = f(Y)$ be $k$-bit functions of $X$ and $Y$, that is $H(X)=H(Y)=k$.
We then have that $M\_X - X - Y - M\_Y$ is a Markov Chain.
By the standard Strong Data Processing Inequality... | https://mathoverflow.net/users/5429 | Strong Data Processing Inequality for capped channels | What happens if $n = k = 1$ and $X = Y$? In this case $\rho = 1$. Let $M\_X = 0$ if $X < 0$ and 1 otherwise. Let $M\_Y = 0$ if $Y < 0$ and 1 otherwise. Then $I(M\_X;M\_Y) = H(M\_X) - H(M\_X|M\_Y) = 1$. This seems to contradict your wanted inequality.
| 1 | https://mathoverflow.net/users/82838 | 366744 | 153,997 |
https://mathoverflow.net/questions/366741 | 1 | $\DeclareMathOperator\At{At}\DeclareMathOperator\Obj{Obj}\DeclareMathOperator\Mor{Mor}$According to <https://ncatlab.org/nlab/show/Atiyah+Lie+groupoid#idea> the *Atiyah Lie groupoid* $\At(P)$ of a principal $G$ bundle $\pi:P \rightarrow X$ is a category for which $$\Obj(\At(P))=\lbrace \pi^{-1}(x): x \in X \rbrace$$ an... | https://mathoverflow.net/users/86313 | What is the natural Lie groupoid structure on the Atiyah Lie groupoid of a principal $G$-bundle? | Contrary to what is claimed in the comments, I would argue
that the definition given in nLab's Idea section is rigorous
enough to be an actual definition in a research-level paper,
possibly with an additional phrase thrown in like
“The sets of objects and morphisms are equipped with the obvious
smooth structures that t... | 6 | https://mathoverflow.net/users/402 | 366747 | 153,998 |
https://mathoverflow.net/questions/366746 | 5 | I am searching for (two) presentations of the group $\mathbf{PGL}\_3(\mathbb{F}\_2)$ for which the generators are involutions $a, b, c$, and such that the following relations are present [among extra relations, of course], in two separate cases:
REP. 1: $(ab)^4 = 1, (bc)^4 = 1, (ac)^2 = 1$;
REP. 2: $(ab)^3 = (bc)^3... | https://mathoverflow.net/users/12884 | Presentations of $\mathbf{PGL}_3(\mathbb{F}_2)$ by three involutions | This cannot be done. Let $G\_1$ and $G\_2$ be the groups
$$G\_1 = \langle a,b,c | a^2 = b^2 = c^2 = (ab)^4 = (bc)^4 = (ac)^2 \rangle$$
$$G\_2 = \langle a,b,c | a^2 = b^2 = c^2 = (ab)^3 = (bc)^3 = (ac)^3 \rangle.$$
You are looking for surjections from the $G\_j$ onto $PGL\_3(\mathbb{F}\_2)$, which is the simple group ... | 13 | https://mathoverflow.net/users/297 | 366750 | 154,000 |
https://mathoverflow.net/questions/366733 | 22 | Let $p(n)$ be the number of partitions of $n\geq 0$. We can let $n$ be
any complex number in Rademacher's convergent infinite series for
$p(n)$. (See e.g. equation (24) [here](https://mathworld.wolfram.com/PartitionFunctionP.html).)
For what $n$ does it converge? Does it define an analytic function for
such $n$? If so,... | https://mathoverflow.net/users/2807 | Does Rademacher's convergent series for p(n) define an analytic function? | Edit. We can write the series in the form
$$p(n)=\sum A\_k(z)\frac{d}{dz}f(z/k^2),$$
where $|A\_k(z)|\leq Ck^{1/2}e^{C\_1(\Im z)^+},$ where $y^+=\max\{ y,0\},$ and $C\_j$ are various positive absolute contants, $f(z)=(\sinh\sqrt{z})/\sqrt{z},\; z=C\_2(n-1/24)$.
Notice that $f$ is an even entire function of order $1/2$,... | 19 | https://mathoverflow.net/users/25510 | 366751 | 154,001 |
https://mathoverflow.net/questions/366748 | 0 | Let $P=A\_1\times A\_2,$ where $A\_1,A\_2\subset \mathbb{R}$ are set of positive Lebesgue measure, and $Z\subset \mathbb{R}^2,$ be a set of zero Lebesgue measure. Can we always find positive Lebesgue measure sets $B\_1,B\_2\subset \mathbb{R}$ such that
$$B\_1\times B\_2 \subset \overline{P\setminus Z}?$$
What extra con... | https://mathoverflow.net/users/161865 | Problem regarding Lebesgue measure in $\mathbb{R}^2$ | For $i=1,2$, let
$$Q:=C\_1\times C\_2,$$
where
$$C\_i:=\{x\in A\_i\colon\forall r>0\ |B(x,r)\cap A\_i|>0\},$$
$B(x,r):=(x-r,x+r)$, and $|\cdot|$ denotes the Lebesgue measure in $\mathbb R^d$, for any $d\ge1$. Then $|C\_i|=|A\_i|>0$, by (say) the Lebesgue density theorem.
For all $(x\_1,x\_2)\in Q$, all real $r>0$, an... | 2 | https://mathoverflow.net/users/36721 | 366753 | 154,002 |
https://mathoverflow.net/questions/366681 | 5 | Given $n$ binary sequences $s\_i$ ($1\le i\le n$) with common period $T$. Let $s\_i^{t\_i}$ denote the sequence obtained by cyclically shifting $s\_i$ for $t\_i$ bits. The $n$ sequences form a **good** system if under any combination of $\{t\_i\}\_{i=1}^n$, for each sequence $s\_i$ there always exists $\tau\_i$ such th... | https://mathoverflow.net/users/52871 | NP-hardness of a sequence problem | I assume you mean "there always exist $\tau\_i$ such that $s\_i^{t\_i}(\tau\_i) = 1$ and $s\_j^{t\_j}(\tau\_i) = 0$ for $i \neq j$", i.e. you want that no matter how the sequences are shifted, each sequence has at least one bit which is zero in the other shifted sequences, and that's the slot when it manages to send it... | 5 | https://mathoverflow.net/users/123634 | 366762 | 154,004 |
https://mathoverflow.net/questions/366765 | 185 |
>
> **QUICK FINAL UPDATE**: Just wanted to thank you MO users for all your support. Special thanks for the fast answers, I've accepted first one, appreciated the clarity it gave me. I've updated my torus algorithm with ${\rm cr}(G)$. Works fine on my full test set, i.e. evidence for ${\rm cr}(G)={\rm pcr}(G)$ on toru... | https://mathoverflow.net/users/161819 | Issue UPDATE: in graph theory, different definitions of edge crossing numbers - impact on applications? | $\DeclareMathOperator\cr{cr}\DeclareMathOperator\pcr{pcr}$For the pair crossing number $\pcr(G)$, the short answer is **yes** the crossing lemma holds for drawings on the sphere, but it is **not known** whether it also holds on the torus.
The best and most current reference for you could be the survey article from Sc... | 148 | https://mathoverflow.net/users/156936 | 366766 | 154,006 |
https://mathoverflow.net/questions/366683 | 4 | It is known that the space $H^{\infty}$ of bounded holomorphic functions on the unit disk $D$ is non-separable with respect to the supremum norm $\|\cdot\|\_{\infty}^{D}$. Let $E\subset D$ be connected and not a singleton. Then, $\|\cdot\|\_{\infty}^{E}$ (the supremum norm over $E$) is in fact a norm.
>
> What are ... | https://mathoverflow.net/users/53155 | Supremum over which sets makes $H^{\infty}$ non-separable? | Jochen Wengenroth suggested to look at Carleson's interpolation theorem, and it seems like it completely answers my question. Namely, the following is true.
>
> Let $E$ be a subset of $D$. Then $H^\infty$ is non-separable with respect to $\|\cdot\|\_\infty^E$ if and only if $\overline{E}$ intersects the unit circle... | 6 | https://mathoverflow.net/users/53155 | 366767 | 154,007 |
https://mathoverflow.net/questions/366726 | 8 | This is reposted from MSE, but perhaps it is more appropriate to post it here. Let me know if I'm wrong.
Among the many unsuccessful attempts at solving the Poincaré conjecture, I'm wondering if there was an approach that went along the lines of showing that a closed simply connected $3$-manifold could be endowed wit... | https://mathoverflow.net/users/143629 | Learning from unsuccessful attempts at the Poincaré conjecture | Thurston approaches 3-manifolds by cutting them up along various surfaces (one first cuts along spheres [Kneser-Milnor] and then along tori [Jaco-Shalen-Johannson]) into pieces which each admit a locally homogeneous geometric structure, modelled on a homogeneous space with an invariant Riemannian metric. A compact, sim... | 15 | https://mathoverflow.net/users/13268 | 366769 | 154,008 |
https://mathoverflow.net/questions/366761 | 4 | See for example <https://www.sciencedirect.com/science/article/pii/0021869387901542> for the definition of the Lusztig a-function.
>
> Question 1: Is there a table for the values of Lusztig's a-function for a given Dynkin type?
>
>
>
Can one at least find those values when the corresponding simple Lie-algebra ... | https://mathoverflow.net/users/61949 | Computation of the Lusztig a-function | In Addition to Geck-Pfeiffer: Small values of the **a**-function are also contained in Geck, Jacon - Representations of Hecke algebras at roots of unity. In particular, for $G\_2$ it's in Table 1.3.; for $F\_4$ it's table 1.2.; other values are available through combinatorially formulas (for example type $A$ is complet... | 4 | https://mathoverflow.net/users/3041 | 366782 | 154,011 |
https://mathoverflow.net/questions/366768 | 4 | **Some introduction:**
Given a homogeneous structure called "dilation" in $R^n$: For $t\geq 0$
$$D\_t: R^n\rightarrow R^n$$
$$D\_t(x)=(t^{a\_1}x\_1,...,t^{a\_n}x\_n)$$
where $1=a\_1\leq...\leq a\_n$, and $a\_i$ are all integers. And we call $Q=a\_1+...+a\_n$ the homogeneous dimension. In our problem, we only consider w... | https://mathoverflow.net/users/145357 | How to estimate the order of this integral with parameter | It looks like you care only about the order of magnitude (i.e., an answer up to a constant factor), in which case it is fairly easy.
First, ignore all coefficients. Setting them to $1$ just changes the answer at most constant number of times. Now, suppose we have the denominator of the form $\sum\_{(\alpha,\beta)} x^... | 4 | https://mathoverflow.net/users/1131 | 366794 | 154,014 |
https://mathoverflow.net/questions/366777 | 3 | $\DeclareMathOperator\Hom{Hom}$I'm trying to understand morphism of Verma modules and consider the following example.
**PART 1:**
Consider $\mathfrak{g}=\mathfrak{gl}\_3$ over $\mathbb{C}$ with positive roots
\begin{equation\*}\Phi\_+=\{\alpha\_1=(1,-1,0),\alpha\_2=(1,0,-1),\alpha\_3=(0,1,-1)\},\end{equation\*}
which... | https://mathoverflow.net/users/135674 | Morphism of Verma modules | **PART 1:**
The element $u$ must have weight $-\alpha\_2$, since $\mu = \lambda - \alpha\_2.$
In $U(\mathfrak{n^-})$ there are only two linearly independent elements that have such weight (assuming PBW basis with respect to fixed order of generators based on positive roots): $y\_{\alpha\_2}$ and $y\_{\alpha\_1}y\_{... | 2 | https://mathoverflow.net/users/6818 | 366800 | 154,016 |
https://mathoverflow.net/questions/366796 | 11 | In the paper [Hall's theorem for hypergraphs (Aharoni and Haxell, 2000)](https://onlinelibrary.wiley.com/doi/abs/10.1002/1097-0118(200010)35:2%3C83::AID-JGT2%3E3.0.CO;2-V),
the authors prove a theorem on the existence of perfect matchings in bipartite hypergraphs, using Sperner's lemma. At the last page (6), they say t... | https://mathoverflow.net/users/34461 | Proving Hall's marriage theorem using Sperner's lemma | Penny Haxell's 2011 paper [On Forming Committees](https://www.jstor.org/stable/10.4169/amer.math.monthly.118.09.777) in the American Mathematical Monthly explicitly uses Sperner's lemma to prove Hall's theorem for bipartite graphs (see theorem 4.1 and 4.2).
| 14 | https://mathoverflow.net/users/11260 | 366801 | 154,017 |
https://mathoverflow.net/questions/366776 | 6 | Let $\Gamma,\Sigma\subset \mathrm{SL}\_2({\mathbb R})$ be cocompact arithmetic subgroups. They are called *commensurable in the wider sense*, if there exists
$g\in \mathrm{SL}\_2({\mathbb R})$, such that the intersection of $\Gamma$ and $g\Sigma g^{-1}$ has finite index in both.
The *trace field* of $\Gamma$, denoted $... | https://mathoverflow.net/users/nan | Number of Fuchsian groups with same trace field | No, this follows from a [result of Bogwang Jeon](https://www.ams.org/journals/tran/2019-371-01/S0002-9947-2018-07271-3/home.html). He showed that given a number field $K$ and quaternion algebra $A$ over $K$ with $A\otimes\_K \mathbb{R} \cong M\_2(\mathbb{R})$, one can find a fuchsian surface of genus $g$ having $K$ as ... | 5 | https://mathoverflow.net/users/1345 | 366803 | 154,018 |
https://mathoverflow.net/questions/366792 | 2 | I asked [this question](https://math.stackexchange.com/questions/3766785/show-that-a-linear-map-commutes-with-a-symmetric-group-action) on MSE but I want to ask it again here with some more context sine it received no answers. In Chapter 3 (Algebra) of the book *Operads in Algebra, Topology and Physics* by Markl, Shnid... | https://mathoverflow.net/users/144957 | Detailed proof of $\mathfrak{s}^{-1}\mathrm{End}_V\cong \mathrm{End}_{\Sigma V}$ | Your actual Question has nothing to do with operads. Perhaps it is clarifying to consider the following more general setting: let $G$ be a group, $X$ and $Y$ be right $G$-sets, and $f : X \to Y$ be a function. If $g, h \in G$ and $f$ commutes with the actions of $g$ and of $h$ then it commutes with the action of $gh$:
... | 7 | https://mathoverflow.net/users/318 | 366809 | 154,022 |
https://mathoverflow.net/questions/366245 | 6 | I have observed some similar things between a reformulation of the [Sunflower conjecture](https://en.wikipedia.org/wiki/Sunflower_(mathematics)#Sunflower_lemma_and_conjecture) (see also conjecture 1.3 in *[Improved bounds for the sunflower lemma](https://arxiv.org/abs/1908.08483)*) and [Szemerédi's theorem](https://en.... | https://mathoverflow.net/users/51189 | Is there any relationship between Szemerédi's theorem and Sunflower conjecture? | I do not know of a direct connection to Roth or Szemerédi over the integers. However, the paper
>
> N. Alon, A. Shpilka and C. Umans, *On Sunflowers and Matrix Multiplication*, 2012 IEEE 27th Conference on Computational Complexity, Porto, 2012, pp. 214-223, doi:[10.1109/CCC.2012.26](https://doi.org/10.1109/CCC.2012... | 8 | https://mathoverflow.net/users/119533 | 366814 | 154,024 |
https://mathoverflow.net/questions/366812 | 5 | Let $ \Bbb S^{d-1}=\{(x\_1,\cdots ,x\_d): x\_1^2+ \cdots +x\_d^2=1\}\subset \Bbb R^d$ be the unit
sphere. Let $\nabla u= (\partial\_{x\_1}u,\cdots, \partial\_{x\_d}u)$ be the gradient of a function $u\in C\_c^\infty(\Omega)$ with $\Omega \subset \Bbb R^d$ open. For $e\in \Bbb S^{d-1}$, we write $$\nabla u(x)\cdot e = \... | https://mathoverflow.net/users/112207 | Optimizing the gradient norm on the unit sphere | I do not believe there is a simple formula to express $I(u)$, but for sure for most of the functions the inequality
$$
I(u)<\int\_\Omega |\nabla u|^p\, dx
$$
is sharp. For example if $\Omega=B$ is a ball and $u(x)=f(|x|)$ is a radial function, then $\nabla u$ is a vector field orthogonal to the sphere $\mathbb{S}^{d-1}... | 4 | https://mathoverflow.net/users/121665 | 366815 | 154,025 |
https://mathoverflow.net/questions/366821 | 6 | Let $K$ a local field ($K$ finit extension of $\mathbb{Q}\_p$), $\mathcal{O}\_K$ the integer of $K$ and $k$ the residue field of $\mathcal{O}\_K$.
Let $\psi:\mathbb{P}^1\_K\to\mathbb{P}^1\_K$ a finit separable morphism, $\widetilde{\psi}=\Psi:\mathbb{P}^1\_{\mathcal{O}\_K}\to\mathbb{P}^1\_{\mathcal{O}\_K}$ a model of... | https://mathoverflow.net/users/34066 | Ramification and reduction | In your setting, you can just do everything concretely using the derivative.
The correct statement is for $\overline{Q}$ in $\mathbb P^1\_k$,
$$e(\overline{Q}) + \operatorname{swan}(\overline{Q}) = 1 + \sum\_{\substack{ i \in \{1,\dots n \} \\ \overline{P}\_i = \overline{Q} }} (e\_i - 1).$$
This is under your assum... | 6 | https://mathoverflow.net/users/18060 | 366822 | 154,026 |
https://mathoverflow.net/questions/366824 | 6 | In the Elephant, Peter Johnston remarks that internal categories may be regarded as simplicial objects that “preserve all limits that happen to exist in $\Delta^{op}$“ (I guess you might call this a flat functor). This is because the join in $\Delta$ is a limit.
Does a similar statement exist for the symmetric simpli... | https://mathoverflow.net/users/75783 | Groupoids as models of symmetric simplicial sets | You can definitely characterize groupoids as presheaves on $Fin\_+$ preserving some colimtis (i.e. sending some colimits in $Fin\_+$ to limits in Set). In fact Groupoids are the presheaf on $Fin\_+$ that preserve the colimits comming from $\Delta$.
However, the Category $Fin\_+$ has much more colimits than $\Delta$, ... | 7 | https://mathoverflow.net/users/22131 | 366825 | 154,027 |
https://mathoverflow.net/questions/366823 | 4 | Under the null hypothesis, if we have
$$\sqrt{n} \vec{x} \, \rightarrow\_d \, N(0, I\_p),$$
the test statistic can be construct as:
$$\hat{\Psi} = n \vec{x}^{\top} \vec{x} \, \rightarrow\_d \,\chi^2\_p.$$
And we reject the null hypothesis if $\hat{\Psi} > \chi^2\_{p, 1 - \alpha}$ under level $\alpha$.
Now, if under t... | https://mathoverflow.net/users/153595 | The power of chi-square test | If $\Sigma=I\_p$, then the distribution of
$\sum\_{j=1}^p\xi\_j^2$ for $(\xi\_1,\cdots,\xi\_p)^\top\sim N(\vec{\mu},\Sigma)$ is the [non-central chi-square distribution](https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution#Derivation_of_the_pdf) with $p$ degrees of freedom and non-centrality parameter $\ve... | 3 | https://mathoverflow.net/users/36721 | 366827 | 154,028 |
https://mathoverflow.net/questions/366848 | 7 | Given matrices $A, B \in \Bbb R^{3 \times 3}$ whose ranks satisfy $\mbox{rank} (A), \mbox{rank} (B) \geq 2$, I would like to prove that for large (or small) enough scalar $\alpha \in \mathbb{R} \setminus \{0\}$ the following does hold.
$$\mbox{rank} (A+\alpha B) \geq 2$$
This seems to be true by hand waving argumen... | https://mathoverflow.net/users/161920 | Rank of sum of two matrices | Since rank$(A)\geq 2$, the matrix $A$ has a $2\times 2$ submatrix $a$ with nonzero determinant; the determinant is a continuous function of the matrix elements, so adding a sufficiently small perturbation $\alpha B$ to $A$ will leave $\det a\neq 0$ and hence the rank of $A+\alpha B$ remains $\geq 2$.
If instead of sm... | 12 | https://mathoverflow.net/users/11260 | 366850 | 154,032 |
https://mathoverflow.net/questions/366847 | 1 | Let $C(\mathbb{R})$ be equipped with the topology of compact convergence (or equivalently the compact-open topology). Then, is the subset $\left\{f\in C(\mathbb{R}):
\text{$f$ injective}
\right\}$ an open subset therein?
| https://mathoverflow.net/users/36886 | Openness of the set of injective functions in $C(\mathbb{R})$? | Based on Matthew's post here we go:
Let $f\_n(x)\triangleq \left|\frac{x}{n+1}\cos(\frac{x-1}{n})\right| + \left(1-\frac1{n+1}\right) x$ and $f(x)=x$ and $\sup\_{x \in [0,1] }\|f\_n(x)-f(x)\| \in \mathscr{O}(n^{-1})$.
This provides a counter example on $C((0,1))$ and then just use the homeomorphism:
$$
\begin{align... | 3 | https://mathoverflow.net/users/36886 | 366851 | 154,033 |
https://mathoverflow.net/questions/366855 | 4 | Two questions, the first: What is the smallest non negative integer that we do not know yet is the Tarski number of a group?
The second question is the same as in the title: What is the latest progress on Tarski numbers?
| https://mathoverflow.net/users/84700 | Latest progress on Tarski numbers | **Q1:** The state of the art as reported in 2014, see [arXiv:1406.2097](https://arxiv.org/abs/1406.2097), is that the only numbers which are known to be Tarski numbers of some groups are 4,5,6. Tarski numbers $<4$ are forbidden, which suggests that 7 is the answer to the question *"What is the smallest non negative int... | 4 | https://mathoverflow.net/users/11260 | 366857 | 154,035 |
https://mathoverflow.net/questions/366856 | 12 | This is a research question in the history of math, I suppose.
As a non-native english speaker I became used to mathematical expressions like 'dynamical' and 'tangential'. When using them in daily conversation as substitutes for 'dynamic' and 'tangent' I got frowned upon by native english speakers who claimed to have... | https://mathoverflow.net/users/150549 | Why is a dynamical system not a dynamic system? | Here are the two entries from Anthony Lo Bello's *Origins of Mathematical Words* (John Hopkins, 2013) which is very informative, entertaining, and perhaps curmudgeonly. In his parlance, following the lexicographer Samuel Johnson, a "low word" is one with an "irregular combination" of roots that has "little or no etymol... | 6 | https://mathoverflow.net/users/14807 | 366863 | 154,038 |
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