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/290491 | 23 | In a comment to a [recent question](https://mathoverflow.net/questions/290459/is-it-consistent-with-zf-that-v-to-v-ast-ast-is-always-an-isomorphism), Jeremy Rickard asked whether it is consistent with ZF that the map $V \to V^{\*\*}$ from a vector space to its double dual is always surjective. We know that "always inje... | https://mathoverflow.net/users/3106 | Is it consistent with ZF that $V\to V^{\ast \ast}$ is always surjective? | $\def\fin{\text{finite}}\def\count{\text{countable}}$This is a CW answer to write up the proof which Harry West gave on [another answer](https://mathoverflow.net/a/401202/297), as pointed out by Gro-Tsen. Thanks to Prof. West for the clever solution and Gro-Tsen for pointing it out. I'm just rewriting it to record deta... | 15 | https://mathoverflow.net/users/297 | 415774 | 169,477 |
https://mathoverflow.net/questions/415773 | 4 | Here is a link for the definition of Wick product
<https://encyclopediaofmath.org/wiki/Wick_product>, which defines the Wick product recursively. My question is where do these two equations come from? I mean the equations
$$
\left\langle: f\_{1}^{k\_{1}} \cdots f\_{n}^{k\_{n}}:\right\rangle=0
$$
and
$$
\frac{\partial}{... | https://mathoverflow.net/users/69279 | Motivation for the axioms in Wick product | The Wick product :$A\_1A\_2A\_3$: is a specific way to order noncommuting operators $A\_1,A\_2,A\_3$. The concept was introduced by Gian-Carlo Wick in 1950 to avoid "infinite expectation values" that arise from the zero-point-motion of harmonic oscillators. Basically you reorder the operators in such a way that the exp... | 6 | https://mathoverflow.net/users/11260 | 415786 | 169,481 |
https://mathoverflow.net/questions/415743 | 0 | Given iid random variables $\xi\_1,\dots,\xi\_n,\dots$ with probability $P(\xi\_1=-\log 3)=\frac{1}{2}=P(\xi\_1=\log 5-\log 3)$. Let $S\_n=\sum\_{i=1}^n \xi\_i$. This is a random walk on $R$. If let a stopping time $T:=\inf\{n: S\_n\le -4\log 3\}$. I want to get $\mathbb{P}(\tau<\infty)$.
I try to follow the solution... | https://mathoverflow.net/users/168083 | Expectation value of random walk on $\mathbb{R}$ | 1. In the [MSE setting](https://math.stackexchange.com/questions/1108134/unbounded-stopping-time), there could be no overshoot: there, $X\_1+ \dots+X\_\tau$ is exactly $1$ (on the event $\{\tau<\infty\}$). In contrast, in your setting there will be, with a nonzero probability, an undershoot: $P(S\_T<-4\ln3)>0$. So, the... | 2 | https://mathoverflow.net/users/36721 | 415795 | 169,482 |
https://mathoverflow.net/questions/415801 | 0 | An informal investigation of a sum.
Consider this sum:
$$S =\sum\_{k=2}^{\infty}(k^{1/k} -1)$$
Does this converge? How does it behave as it diverges, if it diverges? If $k$ equaled $1$ we would get $0$ so we start at $k=2$. Very generally we can find that: $k^{1/k}$ will always be $1$ + a remainder. After subtracting... | https://mathoverflow.net/users/nan | The exploration of the asymptotic behavior of a simple sum. $\sum_{k=1}^{\infty} (k^{1/k} - 1)$ | The OP asks the question whether the series $S =\sum\_{k=2}^{\infty}(k^{1/k} -1)$ diverges.
The answer is that the series diverges, because $$k^{\frac{1}{k}}-1 = \exp\left(\frac{\ln k}{k}\right)-1>\frac{\ln k}{k} > \frac{1}{k}$$
and $\sum\_{k=2}^\infty 1/k=\infty$.
| 10 | https://mathoverflow.net/users/11260 | 415802 | 169,486 |
https://mathoverflow.net/questions/415682 | 2 | Say I have $N$ random variables $X\_1,\cdots,X\_i,\cdots,X\_N$, with zero mean and finite variance. $X\_i$ and $X\_j$ are independent iif $|i-j|>m$, and positively correlated otherwise (say the covariance is of $\mathcal{O}(1)$). It is well-known that the sums of $N$ of these random variables are distributed as a norma... | https://mathoverflow.net/users/173974 | Weakly dependent central limit theorem | The answer is yes. Say we have a stationary process, but we observe samples at random times $\{t\_n\}$ which itself is a stochastic point process (e.g. Poisson process). The resulting sample is also a stationary process ([Ref](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.851.9334&rep=rep1&type=pdf)).
I a... | 0 | https://mathoverflow.net/users/173974 | 415804 | 169,487 |
https://mathoverflow.net/questions/415782 | 3 | Let $V$ be a $\mathbb{C}$-vector space of dimension $N$, let $d$ be a positive integer, let $l \leq N$ be a positive integer, and let $U \subseteq S^d(V)$ be a linear subspace of codimension $k=\binom{l+d-1}{d}$. Does there exist a linear subspace $W \subseteq V$ of dimension $l$ for which $S^d(V)=U \oplus S^d(W)$?
I... | https://mathoverflow.net/users/150898 | Given a subspace $U \subseteq S^d(V)$, does there always exist a complement of the form $S^d(W)$? | Without any further hypothesis on $U$, the answer is no. Take $V = \mathbb{C}^3$, $d=2$, $l=2$ and $U = S^2(W\_1)$ where $W\_1$ is a $\mathbb{C}^2$ inside $V$. Then for any other $W\_2 \subset V$ of dimension $2$, we have:
$$S^2(W\_1 \cap W\_2) \subset U \cap S^2(W\_2)$$
and obiously $\dim S^2(W\_1 \cap W\_2) \geq ... | 6 | https://mathoverflow.net/users/37214 | 415807 | 169,488 |
https://mathoverflow.net/questions/415709 | 3 | Suppose $A$ is an abelian variety over a number field $K$ and call $M$ the maximal torsion free quotient of $A(\overline{K})$ equipped with its Galois action.
>
> For the first Galois cohomology of $M$, do we have $H^1(\text{Gal}\_K, M) = 0$? Is $H^1(\text{Gal}\_K, M)$ at least finite?
>
>
>
If $H^1(\text{Gal}... | https://mathoverflow.net/users/nan | Galois cohomology of abelian varieties | I think that all Galois cohomology groups $\mathrm{H}^i(\mathrm{Gal}\_K,M)$ vanish for $i>0$. As you observed, it suffices to prove that
$\mathrm{H}^i(\mathrm{Gal}\_K/U,M^U)$ vanishes for all open normal subgroups $U$ of $\mathrm{Gal}\_K$. Since $\mathrm{Gal}\_K/U$ is a finite
group, the Galois cohomology group $\mathr... | 4 | https://mathoverflow.net/users/85592 | 415809 | 169,490 |
https://mathoverflow.net/questions/415805 | 4 | Suppose $L/F$ is a finite Galois extension of number fields. Let $E$ be an elliptic curve over $F$ and $E\_L$ its base change to $L$.
>
> Do we have that $\text{Sha}(E/F)$ is finite if and only if $\text{Sha}(E\_L/L)^{\text{Gal}(L/F)}$ is finite?
>
>
>
Analogously, write $\text{Sha}(E\_{\overline{F}}/\overline... | https://mathoverflow.net/users/nan | Tate-Shafarevich groups under finite Galois field extensions | The remark added to the question shows that the kernel of $Ш(E/F) \to Ш(E/L)^G$ is finite where $G$ is the finite Galois group of $L/F$.
$\DeclareMathOperator{\coker}{coker}$
Here is an argument why the cokernel is finite. It may be too complicated.
Let $p$ be a prime and let us show that the cokernel on the $p$-prim... | 5 | https://mathoverflow.net/users/5015 | 415811 | 169,491 |
https://mathoverflow.net/questions/413714 | 0 | I would like to define standard Gram matrices, and use them to help me understand the symmetries of lattices.
I define "standard Gram matrix" as the Gram matrix g that minimizes the deviation from Toeplitz and that satisfies abs(inv(g)) = abs(g).
I have obtained Gram matrices for D4, E8, A15+ with small deviations ... | https://mathoverflow.net/users/385881 | Standard Gram matrices for lattices |
>
> How do I uniquely determine a standard Gram matrix for a lattice? Are there any other definitions of "standard Gram matrix" for lattices? Do you have any literature references in which "standard Gram matrix" is defined for any reason?
>
>
>
Yes, this has been done in [A Canonical Form for Positive Definite M... | 2 | https://mathoverflow.net/users/101207 | 415814 | 169,492 |
https://mathoverflow.net/questions/415806 | 5 | I'm trying to calculate the weak limit of $\mathcal{E}\_N(x)=\sum\_{k=1}^{2^N}\delta\_{x-Z\_k}$ , with $Z\_k=X\_k-\max\_{k\leq 2^N}X\_k$, $\{X\_k\}$ being $2^N$ copies of i.i.d. Gaussians with mean zero and variance $N$(so it can be seen as discretization of Brownian motion).
I'm considering the Laplace transform, si... | https://mathoverflow.net/users/174600 | Limit of the extremal process of i.i.d. Gaussians see from the tip | $\newcommand\R{\mathbb R}\newcommand{\ep}{\varepsilon}\newcommand{\de}{\delta}\newcommand{\vpi}{\varphi}$For $\phi=c\,1\_{[-A,\infty)}$ with $c\ge0$ and $A\ge0$, the expectation in question converges to
\begin{equation\*}
\frac1{1+(e^c-1)e^{A\sqrt{\ln4}}}. \tag{1}\label{1}
\end{equation\*}
---
Indeed, let $n:=2^... | 6 | https://mathoverflow.net/users/36721 | 415825 | 169,496 |
https://mathoverflow.net/questions/414238 | 3 | It seems to be a commonplace in harmonic analysis that if some operator (say, Fourier multiplier) is bounded on $L^p(\mathbb{R}^n)$ then by transference the similar operator is also bounded on $L^p(\mathbb{T}^n)$ (the most common example of application of this general principle is Hilbert transform for $n=1$). There ex... | https://mathoverflow.net/users/69086 | Fourier multipliers and transference on cyclic groups | Ok, I found the answer myself, so I'll post it here. It turned out that the results for multipliers should be transferred not from $L^p(\mathbb{T})$ but from $\ell^p(\mathbb{Z})$. That is, the boundedness of operators that figure in the question follows from the boundedness of the following operator on $\ell^p(\mathbb{... | 1 | https://mathoverflow.net/users/69086 | 415827 | 169,498 |
https://mathoverflow.net/questions/415820 | 1 | Suppose $X$ is a Hausdorff (I'm happy to also assume "non compact") topological space that can be written as the topological direct limit
$$\varinjlim\_{a\in J} K\_a$$
for $J$ a directed set and $K\_a$ compact Hausdorff subspaces of $X$, where all bonding maps are continuous non-surjective immersions (I'm happy to as... | https://mathoverflow.net/users/nan | Approximations by compact sub-spaces | Q1. Consider the space of rationals $\mathbb{Q}$.
It is the direct limit of the family of finite unions of convergent sequences (including their limits), ordered by inclusion, as a set is closed iff its intersection with every convergent sequence is closed.
Now choose for every convergent sequence $\mathbf{q}=\langle... | 2 | https://mathoverflow.net/users/5903 | 415828 | 169,499 |
https://mathoverflow.net/questions/415832 | 0 | I'm looking for a closed solution or an approximation to $$\int x^{-a} \text{erf}\left( b - c x^{-d} \right) dx,$$
where $a, b, c, d > 0$.
| https://mathoverflow.net/users/103291 | Solution or approximation to $\int x^{-a} \text{erf}\left( b - c x^{-d} \right) dx$? | From the comments I understand that the OP seeks an approximation of
$$I=\int\_{x\_1}^{x\_2} x^{-a} \text{erf}\left( b - c x^{-d} \right)\, dx$$
for $x\_2\gg x\_1\gg 1$. A complication which will limit the accuracy of the approximation is that $d\ll 1$. If I ignore that for a moment, and assume all coefficients $a,b,c,... | 3 | https://mathoverflow.net/users/11260 | 415843 | 169,504 |
https://mathoverflow.net/questions/415815 | 6 | Consider $\mathcal{X}$ a projective and flat scheme over $\text{Spec}(\mathcal{O}\_K)$, with $\mathcal{O}\_K$ the ring of integers of a number field $K$.
Let $F/K$ vary over all finite Galois number field extensions and define $\mathbf{A} := \mathbf{A}\_{\overline{K}}$ as the direct limit of the topological rings $\m... | https://mathoverflow.net/users/nan | Adèlic points and algebraic closure | Since $\mathcal X$ is projective, a section is given by finitely many coordinates. If the $i$'th coordinate lies in $\mathbf A\_{F\_i}$ for some extension $F\_i$ of $K$, then all the coordinates lie in $\mathbf A\_{F}$ for $F$ the composition of the $F\_i$.
So $\mathcal X(\mathbf A\_{\overline {K}})$ agrees with the ... | 6 | https://mathoverflow.net/users/18060 | 415850 | 169,507 |
https://mathoverflow.net/questions/415846 | 2 | $\DeclareMathOperator\Vol{Vol}$In 1.7 on p.224 of the following paper, there is a rigidity result for compact manifolds whose sectional curvature is almost $-1$.
[Gromov, M.. Manifolds of negative curvature. J. Differential Geometry 13 (1978), no. 2, 223-230.](https://projecteuclid.org/journals/journal-of-differentia... | https://mathoverflow.net/users/12904 | On a closed manifold whose curvature is close to "hyperbolic" | I know two ways to complete the argument.
1. The lower curvature bound is preserved under GH convergence, and so is the upper curvature bound provided there is a lower bound on convexity radius. In the universal cover the convexity radius is infinite. So the universal cover converges to an Alexandrov space of curvatu... | 4 | https://mathoverflow.net/users/1573 | 415853 | 169,508 |
https://mathoverflow.net/questions/415794 | 8 | In the smooth setting, Whitney's approximation theorem says the following: If $M,N$ are smooth manifolds and $f,g:M\to N$ are smooth functions that are continuously homotopic (ie there is a continuous homotopy $H:M\times [0,1]\to N$ between them) then they are also smoothly homotopic (ie there exists a smooth homotopy ... | https://mathoverflow.net/users/351083 | Whitney's approximation theorem for Lipschitz manifolds | For compact Lipschitz manifolds this follows from the main result of the paper
by Liu, Luofei, Yu, Hanfu,
Liu, Ye
"[Converting uniform homotopies into Lipschitz homotopies via moduli of
continuity.](https://www.sciencedirect.com/science/article/abs/pii/S0166864120303205)" Topology Appl. 285 (2020)
But the above paper... | 5 | https://mathoverflow.net/users/18050 | 415856 | 169,509 |
https://mathoverflow.net/questions/415863 | 12 | Given a matrix $A\in \operatorname{SL}\_d(\mathbb{Z})$ ([the special linear group](https://en.wikipedia.org/wiki/Special_linear_group)) satisfying the two conditions: (1) no eigenvalue of $A$ is a root of unity, (2) the characteristic polynomial of $A$ is irreducible over $\mathbb{Q}$.
>
> **QUESTION.** Does it fol... | https://mathoverflow.net/users/66131 | Is there an eigenvalue of modulus larger than 1? | Since the characteristic polynomial is irreducible, eigenvalues are simple and hence the matrix is $\mathbf{C}$-diagonalizable. If all were on the unit cercle, it would follow that $\{ A^n:n\in\mathbf{Z}\}$ is bounded. But since it is contained in the set of integral points, this would force it to be finite, and hence ... | 23 | https://mathoverflow.net/users/14094 | 415864 | 169,511 |
https://mathoverflow.net/questions/415871 | 2 | If $\{X\_i, i\in I\}$ is a directed system of abelian groups such that we have
$$\varinjlim\_{i\in I}X\_i = 0$$
is it true that for every $i$ and large enough $j\ge i$ the transition map $f\_{i,j} : X\_i\to X\_j$ is the zero map?
If the transition maps satisfy this, then of course the direct limit is zero. If the $X\... | https://mathoverflow.net/users/nan | Transition maps in trivial direct limit | No. Take each $G\_n$ free abelian with basis $e\_k$ with $k\in \mathbb N$. Here $n$ runs over the natural number. Let the map from $G\_n$ to $G\_{n+1}$ kill $e\_0$ and send $e\_k$ to $e\_{k-1}$ for $k>0$. Then no map is zero but each element in each group eventually maps to $0$ far enough down. So the direct limit is 0... | 3 | https://mathoverflow.net/users/15934 | 415872 | 169,513 |
https://mathoverflow.net/questions/415867 | 2 | Let $X$ and $Y$ be compact Hausdorff spaces and let $\varphi:X\to Y$ be continuous with a property that if $A$ is a nowhere dense zero-set in $Y$, then $\varphi^{-1}(A)$ is nowhere dense in $X$. Let $Z=\varphi(X)$.
>
> Does $\varphi$ still have the analogous property as a map into $Z$?
>
>
>
Note that the cond... | https://mathoverflow.net/users/53155 | Is a certain property of a continuous map preserved under "surjectification"? | The answer here is negative: for $Y$ take the remainder $\beta\omega\setminus\omega$ of the Stone-Cech remander of the discrete space $\omega$ of finite ordinals. In the space $Y$ take any countable discrete subspace $D$ and let $Z$ be the closure of $D$. Since $Y$ has no isolated points, the space $D$ is nowhere dense... | 2 | https://mathoverflow.net/users/61536 | 415873 | 169,514 |
https://mathoverflow.net/questions/415877 | 2 | Consider a number field $K$ and a finite Galois field extension $L/K$. Let $E$ be an elliptic curve over $K$ and consider the abelian group
$$E(L)\otimes L^{\times}.$$
Every element $g$ in $\text{Gal}(L/K)$ acts on it by $g(P\otimes \alpha) = g(P)\otimes g(\alpha)$ for $P\in E(L)$ and $\alpha\in L^{\times}$.
>
... | https://mathoverflow.net/users/nan | Galois invariants and tensor products | Consider $E:y^2=x^3-3$ over $\mathbb Q$. Then $E(\mathbb Q)$ is trivial, hence so is $E(\mathbb Q)\otimes\mathbb Q^\times$. On the other hand, over $L=\mathbb Q(\sqrt{-3})$ we have a point $P=(0,\sqrt{-3})\in E(L)$ which is mapped to its opposite by the complex conjugation. The same is true of $\sqrt{-3}\in L$, therefo... | 6 | https://mathoverflow.net/users/30186 | 415879 | 169,517 |
https://mathoverflow.net/questions/415885 | 3 | The generating function of the product of Legendre polynomials for the same $n$ is given by
\begin{aligned}
\sum\_{n=0}^{\infty} z^{n} \mathrm{P}\_{n}(\cos \alpha) \mathrm{P}\_{n}(\cos \beta)&=\frac{\mathrm{F}\left(\frac{1}{2} ,\frac{1}{2} ;1;\frac{4z\sin \alpha \sin \beta }{1-2z\cos (\alpha +\beta )+z^{2}}\right)}{\... | https://mathoverflow.net/users/476902 | Generating function of the product of Legendre polynomials | Using the recursion relation
$$
P\_{n-1} (x) = x P\_n (x) - \frac{x^2 -1}{n} \frac{d}{dx} P\_n (x) \ ,
$$
you can reduce your expression to a sum of the generating function you quote and a combined derivative (the $d/dx$) and integral w.r.t. $z$ (to generate the $1/n$) thereof.
| 5 | https://mathoverflow.net/users/134299 | 415888 | 169,520 |
https://mathoverflow.net/questions/415886 | 2 | Consider the nonlinear map $F\_i:\mathbb R^2 \to \mathbb R$
$F\_i(x):=\varepsilon^2\langle x, A\_i x\rangle +\varepsilon\langle b\_i,x \rangle + x\_i,$
where $A\_i$ is some matrix and $b\_i$ some vector
Can we uniquely solve the equation $(F\_1(x),F\_2(x))=z\_0 \in \mathbb R^2$ for $\varepsilon>0$ small enough su... | https://mathoverflow.net/users/150549 | Analytic solution of low-dimensional Riccati equation | This is a particular instance of the analytic implicit function theorem; see e.g. [this MO page](https://mathoverflow.net/questions/73388/analytic-implicit-function-theorem) and further references there.
| 2 | https://mathoverflow.net/users/36721 | 415898 | 169,522 |
https://mathoverflow.net/questions/415897 | 4 | Let $\mathbb{B}$ be an infinite $\sigma$-complete Boolean algebra. By $\mathbb{B}^\omega$ we denote the countable product of $\mathbb{B}$ with the coordinate-wise operations. Let us call a homomorphism $\varphi\colon\mathbb{B}^\omega\to\mathbb{B}$ *generalized limit* if for every sequence $(A\_n)\in\mathbb{B}^\omega$ w... | https://mathoverflow.net/users/15860 | Generalized limits in Boolean algebras | If the Boolean algebra $B$ is complete, then I claim that there exists generalized limits. I am going to generalize this answer from limits indexed by $\omega$ to a limit of a net indexed by an arbitrary directed set.
Let $B$ be a complete Boolean algebra. Suppose that $D$ is a directed set. Then
let $I$ be the ideal... | 2 | https://mathoverflow.net/users/22277 | 415920 | 169,532 |
https://mathoverflow.net/questions/415922 | 3 | Let $ X $ be an affine, normal, $ \mathbb{G}\_{a} $-variety with action $ \beta $ over a field $ k $ of characteristic zero. The action of $ \mathbb{G}\_{a} $ is obtained from $ \delta \in \operatorname{Der}\_{k}(\mathcal{O}(X)) $ via
\begin{align\*}
\beta^{\sharp}(f) =\exp(t\delta)(f)\\
= \sum\_{j=0}^{\infty} \frac{\d... | https://mathoverflow.net/users/470753 | Does anyone know the first times the term plinth ideal of a $ \mathbb{G}_{a} $-representation was used? | G. Freudenburg, in [Algebraic theory of locally nilpotent derivations](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.470.10&rep=rep1&type=pdf) introduces the term "plinth ideal" in a way that suggests it was not used before, on page 10, with the footnote:
*The term plinth commonly refers to the base of a c... | 2 | https://mathoverflow.net/users/11260 | 415923 | 169,533 |
https://mathoverflow.net/questions/415890 | 1 | Given a simple complex Lie group $G$ (I might say upfront that I am mostly interested with exceptional Lie algebras) and a nilpotent orbit $\mathcal{O}\subset G$ I would like to describe the intersection of $\mathcal{O}$ with Levi subgroups (or probably, their derived groups) and Pseudo-Levi subgroups.
I guess that the... | https://mathoverflow.net/users/64702 | Unipotent orbits and intersection with Levi and pseudo-Levi subgroups | You can't always find a pseudo-Levi such that the intersection is a principal = regular orbit, but you can always find a **Levi** subalgebra such that the intersection is distinguished, i.e. the connected centraliser in the derived subgroup is unipotent. This is the Bala-Carter theorem.
Now, there is an improvement t... | 1 | https://mathoverflow.net/users/26635 | 415926 | 169,535 |
https://mathoverflow.net/questions/415903 | 0 | Let $(X,d)$ be a metric space and $m$ be a Borel measure on $(X,d)$. The measure $m$ is called Ahlors regular if $m(B(x,r))\asymp r^q$ for some $q>0$ and each $x\in X$. Is there a name for measures satisfying the following relaxation:
There is some strictly monotone continuous increasing function $\omega:[0,\infty]\r... | https://mathoverflow.net/users/36886 | Terminology "upper" Ahlfors regular measure | In the case $\omega(r) = Cr^q$:
* the terminology "upper Ahlfors ($q$-)regular" is certainly in use. See, e.g., <https://arxiv.org/pdf/1803.04819.pdf> .
* the terminology "($q$-)Frostman measure" is also used, referring to [Frostman's lemma](https://en.wikipedia.org/wiki/Frostman_lemma). See, e.g., [https://www.maths... | 0 | https://mathoverflow.net/users/476947 | 415929 | 169,537 |
https://mathoverflow.net/questions/415904 | 2 | Let $A$ be a finite abelian group, and $\sigma$ a non-trivial action of $A$ on $\mathbb{C}$ by real algebra automorphisms. In particular, using $\sigma$, we can view $\mathbb{C}^{\times}$ as a module over $A$. Let $\xi$ be a 2-cocycle representing a class in $H^2(A;\mathbb{C}^{\times}\_{\sigma})$. We can consider the r... | https://mathoverflow.net/users/105094 | Real representations of twisted group algebras | The algebra $A:=\mathbb{C}\_\sigma^\xi[G]$ is the universal $\mathbb{R}$-algebra that
1. contains $\mathbb{C}$ and symbols $\{u\_g \mid g\in G\}$ such that
2. $u\_g u\_h = \xi(g,h) u\_{gh}$ and $u\_g z u\_g^{-1} = {^g z}$ holds, where I use ${^g z}$ as a shorthand for $\sigma(g)(z)$.
This is an example of a "crosse... | 4 | https://mathoverflow.net/users/3041 | 415931 | 169,539 |
https://mathoverflow.net/questions/415935 | 4 | Consider all Pythagorean triangles $a^2 + b^2 = p^2$ in which the hypotenuse $p$ is a prime number. Let $h(x) = \sum\_{p \le x}p^2$, $a(x) = \sum\_{p \le x}ab$ and $r(x) = \sum\_{p \le x}(a+b)^2$. Is it true that:
$$
\lim\_{x \to \infty}\frac{h(x)}{r(x)} = \frac{\pi}{2+\pi}
$$
$$
\lim\_{x \to \infty}\frac{a(x)}{r(x... | https://mathoverflow.net/users/23388 | Relation between $\pi$, area and the sides of Pythagorean triangles whose hypotenuse is a prime number | **Q:** Is $\lim\_{x \to \infty}\frac{h(x)}{r(x)} = \frac{\pi}{2+\pi} $ ?
**A:** use that $r(x)=h(x)+2a(x)$, hence
$$\frac{h(x)}{r(x)} = \frac{h(x)/a(x)}{2+h(x)/a(x)}$$
and
$$\lim\_{x\rightarrow\infty}\frac{h(x)}{a(x)}=\pi$$
in view of <https://math.stackexchange.com/a/3481801/87355>
| 6 | https://mathoverflow.net/users/11260 | 415948 | 169,542 |
https://mathoverflow.net/questions/415895 | 2 | **Definition.** A semigroup $X$ is called *$E$-separated* if for any distinct idempotents $x,y\in X$ there exists a homomorphism $h:X\to Y$ to a semilattice $Y$ such that $h(x)\ne h(y)$.
Observe that $X$ is $E$-separated if and only if the smallest semilattice congruence on $X$ is idempotent-separating. This seems to... | https://mathoverflow.net/users/61536 | $E$-separated semigroups | I finally found an answer to my own question: by an old (nontrivial) result of [Putcha and Weissglass](https://msp.org/pjm/1971/39-1/p21.xhtml), a semigroup $X$ is $E$-separated if and only if it is viable.
A semigroup $X$ is *viable* if for any elements $x,y\in X$ with $\{xy,yx\}\subseteq E(X)$ we have $xy=yx$.
Ho... | 1 | https://mathoverflow.net/users/61536 | 415962 | 169,545 |
https://mathoverflow.net/questions/415970 | 17 | For $n\in \mathbb{N}$ let $S\_n$ denote the set of permutations (bijections) $\pi: \{0,\ldots,n-1\}\to \{0,\ldots,n-1\}$. A *transposition* swaps exactly $2$ elements and is often denoted by $(i \; k)$ if $i\neq k\in\{0,\ldots,n-1\}$ are the elements being swapped.
For $n\in\mathbb{N}$ let $E\_n$ be the number of ele... | https://mathoverflow.net/users/8628 | Fraction of $S_n$ reachable by using every transposition once as $n\to\infty$? | The value is $1/2$.
The problem can be reformulated as follows: How to sort a non-sorted list $(a\_1,...,a\_n)$ that is a permutation of $\{1 .. .n\}$ with the [parity](https://en.wikipedia.org/wiki/Parity_of_a_permutation) the same as $n \choose 2$ by exactly one transposition between every pair of indices?
In thi... | 25 | https://mathoverflow.net/users/125498 | 415978 | 169,551 |
https://mathoverflow.net/questions/415982 | 4 | [Szőkefalvi-Nagy's theorem](https://en.wikipedia.org/wiki/Sz.-Nagy%27s_dilation_theorem) says the following: if $A$ is a contraction on a Hilbert space $H$, then there exists a unitary $U$ on a Hilbert space $H'\supset H$ for which $A^n=P\_HU^nP\_H$ for all $0\le n$. ($P\_H$ denotes projection onto $H$.)
If $H$ is fi... | https://mathoverflow.net/users/159965 | Status of finite-dimensional Ando's theorem | In their 2013 paper "[Unitary $N$-dilations for tuples of commuting matrices](https://doi.org/10.1090/S0002-9939-2012-11714-9)" McCarthy and Shalit showed that there does indeed exist an $N$-dilation to a finite-dimensional space (see Theorem 1.2 in the paper).
By the way, I found this paper by taking a look at the r... | 4 | https://mathoverflow.net/users/102946 | 415985 | 169,552 |
https://mathoverflow.net/questions/392467 | 4 | I am currently looking at a few simple properties of the Witt ring of a field $K$ (by which I mean the ring of Witt classes of quadratic forms, not the ring of Witt vectors), which are clearly true when the Pythagoras number of $K$ is $1$, and clearly false when it is $3$, but I am not sure what can happen when it is $... | https://mathoverflow.net/users/68479 | Witt ring of a field with Pythagoras number $2$ | Let $p(K)$ be the Pythagoras number of $K$. Suppose that $p(K) = 2$.
If $K$ is a formally real field, then the torsion subgroup of $W(K)$ has exponent 2. If $K$ is a nonreal field, then this is false.
For example, let $K$ be the finite field with $q$ elements where $q \equiv 3 \bmod 4$. Then $p(K) = 2$, but $W(K)$ ha... | 1 | https://mathoverflow.net/users/476997 | 415999 | 169,557 |
https://mathoverflow.net/questions/415934 | 1 | It is well-known that if $O$ is an orthogonal map, then $\Delta u(Ox) = \Delta u$ where $\Delta$ is the Laplacian.
Now, let $A$ be a constant invertible matrix, then we define the weighted Laplacian
$$\Delta\_A = \langle \nabla, A \nabla \rangle.$$
My question is: Does there exist a function $\varphi\_O$ such tha... | https://mathoverflow.net/users/150564 | Orthogonal invariance of (weighted) Laplacian | $\newcommand{\De}{\Delta}$The answer is: not in general.
Indeed, suppose the contrary. Let
\begin{equation}
A:=\begin{pmatrix}
0&1/2\\
1/2&0
\end{pmatrix},
\end{equation}
so that
\begin{equation}
(\De\_A f)(s,t)=\frac{\partial^2 f(s,t)}{\partial s\,\partial t}=f^{(1,1)}(s,t).
\end{equation}
Let
\begin{equation}
... | 1 | https://mathoverflow.net/users/36721 | 416005 | 169,559 |
https://mathoverflow.net/questions/416010 | 3 | Let say I have two $n$ x $n$ matrices $A$ and $B$ where all elements are real positive values. I want to find some $n$ x $n$ permutation matrix $P$ such that $\operatorname{tr}(P A P ^T B)$ is minimized. Does there exist such an algorithm or technique?
| https://mathoverflow.net/users/477004 | Algorithm to minimize $\operatorname{tr}(PAP^TB)$? | This problem is NP-hard.
Let $A$ be the adjacency matrix of an $n$-cycle plus the all-ones matrix and $B$ the adjacency matrix of a graph $G$ plus the all-ones matrix.
Then, if $G$ has a Hamiltonian cycle, the maximum trace is achieved when the elements of $PAP^T$ with value $2$ corresponds to a Hamiltonian cycle i... | 5 | https://mathoverflow.net/users/125498 | 416020 | 169,562 |
https://mathoverflow.net/questions/415987 | 4 | Usually, I like working with determinants related to the Vandermonde matrix, i.e.
$$\det(x\_j^{i-1})=\prod\_{i<j}(x\_j-x\_i).$$
However, I run into some unusual matrix and its determinant. Define the $(2n)\times (2n)$ matrix $\mathbb{M}\_{2n}$ with entries
$$\mathbb{M}\_{2n}(i,j)=
\begin{cases}
(x\_j-x\_i)(x\_i-x\_{2n}... | https://mathoverflow.net/users/66131 | A determinant of perfect square polynomials | The answer is **yes** for any $n \geq 2$. (The $n=1$ case should be treated separately.)
The determinant, as a polynomial, contains $(x\_m−x\_{m+1})$ $(m=1,2,...,2n-1)$ and $(x\_{2n}-x\_1)$ as factors, because setting $x\_m=x\_{m+1}$ will make the $m$th and $(m+1)$th column of $\mathbb{M}\_{2n}$ equal, and setting $x... | 6 | https://mathoverflow.net/users/125498 | 416028 | 169,565 |
https://mathoverflow.net/questions/416032 | 5 | This question is inspired by [Maximal compact subgroup of $\mathrm{SL}(2,\mathbb{H})$](https://mathoverflow.net/questions/195060/maximal-compact-subgroup-of-mathrmsl2-mathbbh). Consider the embedding $\operatorname{U}(n,\mathbb{H})\subset \operatorname{GL}(n,\mathbb{H}) $. Since $\operatorname{U}(n,\mathbb{H})\cong \op... | https://mathoverflow.net/users/40297 | Why is $\operatorname{U}(n,\mathbb{H})\subset \operatorname{SL}(n,\mathbb{H}) $? | Yes, relying on the fact that elements of the symplectic group (over $\mathbf{C}$, and hence over $\mathbf{R}$) have determinant 1.
Indeed, an element $g$ of $\mathrm{U}(n,\mathbf{H})$ preserves the canonical Hermitian form $b$. Let $b'$ be any imaginary component of $b$. Then $b'$ is a $g$-invariant real-valued symp... | 4 | https://mathoverflow.net/users/14094 | 416036 | 169,569 |
https://mathoverflow.net/questions/416007 | 4 | In his book *Introduction to Infinity-Categories,* Land in his Theorem 3.3.16 asserts an equivalence of $\infty$-categories where one of the categories $\mathrm{LFib}(\mathcal C)$ is the full subcategory of $(\mathrm{Cat}\_\infty)\_{/\mathcal C}$ on vertices that are left fibrations $?\to\mathcal C$ ($\mathcal C$ some ... | https://mathoverflow.net/users/37110 | How to define the $\infty$-category of left fibrations? | First let me point out a small typo : it should be functors over $\mathcal C$ *which send cocartesian edges to cocartesian edges* (this doesn't matter for left fibrations, but for cocartesian fibrations it does).
For your main point, you are right that they are different quasi-categories: the simplicial sets are not ... | 3 | https://mathoverflow.net/users/102343 | 416040 | 169,573 |
https://mathoverflow.net/questions/415793 | 6 | A famous conjecture in topology asserts:
*The Euler characteristic of a closed aspherical $2n$-manifold $M$ satisfies $(-1)^n\chi(M) \geq 0$*.
This was conjectured by Hopf for manifolds with non-positive sectional curvature and (much later) by Thurston for all aspherical manifolds.
By the classification of surfac... | https://mathoverflow.net/users/14233 | Status of the Hopf-Thurston sign conjecture in dimension 4 | There has been a lot of work on cases of this conjecture connected to Coxeter groups. M. Davis and R. Charney made a conjecture that comes from these cases in 1995 in The Euler characteristic of a non-positively curved piecewise Euclidean manifold. Also
have a look at the article Vanishing theorems and conjectures for ... | 8 | https://mathoverflow.net/users/124004 | 416043 | 169,575 |
https://mathoverflow.net/questions/416049 | 4 | In my recent [MO question](https://mathoverflow.net/questions/415987/a-determinant-of-perfect-square-polynomials), Darij Grinberg mentioned a closely related (structure-wise) determinant, that is,
$$\det\left(x\_{\min\{i,j\}}\right)\_{i,j}^{1,m}=x\_1(x\_2-x\_1)(x\_3-x\_2)\cdots(x\_m-x\_{m-1}).$$
In particular, $\det(\m... | https://mathoverflow.net/users/66131 | Characteristic polynomial of a simple matrix: Chebyshev? | Yes, the characteristic polynomial is given by $(-1)^m U\_{2m}(1/2\lambda ) \lambda^{2m} $.
The inverse matrix is given by $$\begin{pmatrix} 2 & -1 & 0 & 0 & \dots \\ -1 & 2 & -1 & 0 & \dots \\ 0 & -1 & 2 & -1 & \dots \\ 0 & 0 & -1 & 2 & \dots % \\ 0 & 0 & 0 & -1 & \dots
\\ \dots & \dots & \dots & \dots & 1\end{pmat... | 8 | https://mathoverflow.net/users/18060 | 416053 | 169,578 |
https://mathoverflow.net/questions/416022 | 1 | **Background:**
I am reading the paper: *Best constant in Sobolev inequality* by Talenti (see [here](https://math.jhu.edu/%7Ejs/Math646/talenti.sobolev.pdf)) and I am trying to understand the following step.
On p. 365, the author is arguing that the solutions to the following equation
$$\left(r^{m-1}\left|u^{\pri... | https://mathoverflow.net/users/68232 | Extremizers of the Sobolev inequality | Take any extremizer $u$ as in the [paper you linked](https://math.jhu.edu/%7Ejs/Math646/talenti.sobolev.pdf).
From the conditions $u(r)=r^{1-m/2}v(r)$, $u(r)=o(r^{1-m/2})$, and $u'(r)=o(r^{-m/2})$ (as $r\to\infty$), we deduce $v(r)=r^{m/2-1}u(r)$, $v(r)=o(1)$, and $v'(r)=(m/2-1)r^{m/2-2}u(r)+r^{m/2-1}u'(r)=o(1/r)$.
... | 3 | https://mathoverflow.net/users/36721 | 416058 | 169,580 |
https://mathoverflow.net/questions/416054 | 5 | Let $E$ be an elliptic curve over $\mathbb Q$. Let $F$ be the rational function field of $E$.
The $K\_2$ group of $F$ [may be described](https://en.wikipedia.org/wiki/Algebraic_K-theory#Matsumoto%27s_theorem) by elements in $F^\times ⊗\_\mathbb{Z} F^\times$ quotiented by the relations $\langle f ⊗ (1 − f)\rangle (f ∈... | https://mathoverflow.net/users/125498 | Computation of the torsion of K-groups related to elliptic curves | The most efficient way I know to detect whether an element of $K\_2(E)$ is torsion is to use the elliptic dilogarithm. This relies however on a conjecture, and it is not an exact method, in the sense that it uses floating-point arithmetic.
Consider the map
\begin{align\*}
\beta : F^\times \otimes F^\times & \to \math... | 5 | https://mathoverflow.net/users/6506 | 416069 | 169,582 |
https://mathoverflow.net/questions/415784 | 4 | Let $\mathbf{F}$ be a discrete Fourier transform (DFT) matrix such that
\begin{align}
F\_{m,n}=e^{-j2\pi(m-1)(n-1)/N},\quad m,n=1,\ldots,N.
\end{align}
What we can say about the singular value decomposition (SVD) of truncated DFT matrix (the following matrix)?
\begin{align}
\tilde{\mathbf{F}} =
\begin{bmatrix}
\mathbf... | https://mathoverflow.net/users/68835 | Singular value decomposition of truncated discrete Fourier transform matrix | Let me insert a factor $N^{-1/2}$, so that the Fourier transform is unitary:
$$U\_{mn}=N^{-1/2}e^{-2\pi i(m-1)(n-1)/N},\quad m,n=1,\ldots,N.$$
We truncate the $N\times N$ matrix $U$ to the $k\times k$ upper left corner,
$$U^{(k)}\_{mn}= N^{-1/2}e^{-2\pi i(m-1)(n-1)/N},\quad m,n=1,\ldots,k\leq N.$$
A characterisation ... | 3 | https://mathoverflow.net/users/11260 | 416071 | 169,583 |
https://mathoverflow.net/questions/416047 | 3 | I encountered the following problem in one of my research projects which can be encapsulated as follows. Let's say we have a set $\mathcal{C}$ of functions $f$ defined from $\mathbb R\_+$ to $\mathbb R$, and we have two functionals $A = A(f) \,\colon\, \mathcal{C} \to \mathbb{R}\_+$ and $B = B(f) \,\colon\, \mathcal{C}... | https://mathoverflow.net/users/163454 | Obtaining the "best possible" inequality by tuning hyper-parameters | $\newcommand{\ga}{\gamma}$Letting $\ga\to\infty$ (with $A,B,\lambda,f(0)$ fixed), we see that the left-hand side of your inequality
\begin{equation}
\begin{aligned}
&A + (\lambda + \gamma)^2 + \gamma^2 \\
&\leq 2\,\sqrt{B + 2f(0)(\lambda - \gamma) + \lambda^2 + \gamma^2}\,\sqrt{A + (\lambda + \gamma)^2 + \gamma^2} \... | 3 | https://mathoverflow.net/users/36721 | 416073 | 169,584 |
https://mathoverflow.net/questions/416056 | 4 | This question is motivated by a real-world application related to an art project that involves displaying images, but my search hit a dead end after finding the wikipage about [**Kirkman systems**](https://en.wikipedia.org/wiki/Kirkman%27s_schoolgirl_problem) (other related terms include [**Steiner systems**](https://e... | https://mathoverflow.net/users/22971 | Enumerating subsets with no triple appearing together more than once | In fact, Wikipedia article on [Steiner systems](https://en.wikipedia.org/wiki/Steiner_system) that you linked already provides an answer to your question:
>
> An $S(3,4,n)$ is called a **Steiner quadruple system**. A necessary and sufficient condition for the existence of an $S(3,4,n)$ is that $n \equiv 2\ \text{or... | 9 | https://mathoverflow.net/users/7076 | 416074 | 169,585 |
https://mathoverflow.net/questions/370332 | 4 | Murthy numbers, in a given base, are positive integers, such as 2009 in base 10, which are not relatively prime to their reversal, that is, the number written backwards (in base 10 such numbers are [AO71249](http://oeis.org/A071249) in the OEIS).
In base 10, numbers from 8432 to 8440 are all Murthy numbers. Are there... | https://mathoverflow.net/users/60732 | Runs of consecutive numbers all of which are Murthy numbers | Let $m+1, \dots, m+n$ be a sequence of $n$ consecutive Murthy numbers such that each $m+i$ shares with its reversal $\overline{m+i}$ a prime factor $p\_i\equiv 3\pmod4$ such that 10 is a quadratic nonresidue modulo $p\_i$. We will show how to construct such a sequence of $n+1$ consecutive Murthy numbers.
Define $t :=... | 3 | https://mathoverflow.net/users/7076 | 416079 | 169,586 |
https://mathoverflow.net/questions/416090 | 8 | In [this recent](https://arxiv.org/abs/2004.04550) preprint, the authors construct a certain uncountable family of non-finitely presented FP groups. Recall that group is an FP group if the trivial $\mathbb Z[G]$-module $\mathbb Z$ has a finite projective resolution by finitely generated $\mathbb Z[G]$-projectives.
In... | https://mathoverflow.net/users/102343 | Non-finitely presented FP groups with cohomological dimension $2$ | The Bestvina-Brady construction of non-finitely presented groups of type FP produces groups of cohomological dimension two. Bestvina-Brady groups are parametrized by finite flag simplicial complexes. The Bestvina-Brady group is of type FP iff the flag complex is acyclic (=has the same ordinary homology as a point), is ... | 11 | https://mathoverflow.net/users/124004 | 416092 | 169,587 |
https://mathoverflow.net/questions/416104 | 0 | Let us consider the measure algebra $M(\mathbb{R})$ consisting of all Radon measures on reals.
Let $\mu$ be a Radon measure in $M(\mathbb{R})$ and $\delta\_0$ be the point mass measure concentrated on 0, which is also the multiplicative identity of $M(\mathbb{R})$.
>
> Q. Does there exist any strictly positive real... | https://mathoverflow.net/users/84390 | An equation in the convolution measure algebra on reals | The answer is no in general. Indeed, let $\hat\mu$ denote the Fourier transform of $\mu$, so that $\hat\mu(t)=\int e^{itx}\mu(dx)$ for real $t$. Then the equality
$$\mu^\*\*\mu=r\delta\_0+\sum\_1^n\mu\_i^\*\*\mu\_i$$
would imply
$$|\hat\mu|^2=r+\sum\_1^n|\hat\mu\_i|^2\ge r>0.$$
Taking now any $\mu$ with $\hat\mu(t)\to0... | 4 | https://mathoverflow.net/users/36721 | 416112 | 169,592 |
https://mathoverflow.net/questions/416077 | 8 | Suppose that $M$ is a *compact, real*
algebraic subset of $\mathbb R^n$ and $f:\mathbb R^n \to \mathbb R^m$ is the projection to the first $m$ coordinates. If $f$ maps $M$ *bijectively* unto its image $f(M)$, is it true that $f(M)$ is algebraic?
| https://mathoverflow.net/users/13842 | Projections of compact real algebraic sets | The answer is no. (Although the previous example I gave was bad.)
Let $C$ be the curve $y^2 = x^2 (x-1)(2-x)$, so $C$ has a smooth component with $1 \leq x \leq 2$, and also a node at $(0,0)$. Let $M$ be the normalization of $C$; explicitly,
$$M = \{ (x,y,z) : z^2 = (x-1)(2-x),\ y=xz \}.$$
Then the projection $M \to ... | 9 | https://mathoverflow.net/users/297 | 416123 | 169,594 |
https://mathoverflow.net/questions/416089 | 10 | Let $P(n)$ of a sequence $s(1),s(2),s(3),...$ be obtained by leaving $s(1),...,s(n)$ fixed and reverse-cyclically permuting every $n$ consecutive terms thereafter; apply $P(2)$ to $1,2,3,...$ to get $PS(2)$, then apply $P(3)$ to $PS(2)$ to get $PS(3)$, then apply $P(4)$ to $PS(3)$, etc. The limit of $PS(n)$ is $a(n)$ (... | https://mathoverflow.net/users/231922 | Prime numbers from permutation | **1. Answer to the question.**
**Claim 1.** $a(n)=2(n-1)$ iff the number $d=a(n)$ moves only leftwards (while it moves).
*Proof.* At each move, every moving number moves either to the right or $1$ left.
The number $d=a(n)$ came to the $n$th position during $P(n-1)$, moving $1$ left. If $d$ moved leftwards through... | 9 | https://mathoverflow.net/users/17581 | 416125 | 169,595 |
https://mathoverflow.net/questions/415108 | 3 | The Schur multipliers of finite simple groups are known and easily accessible:
<https://en.wikipedia.org/wiki/List_of_finite_simple_groups>
Moreover, as a consequence of the second Whitehead's Lemma, if $L$ is a finite-dimensional simple Lie algebra over a field $\mathbb{F}$ of characteristic zero, then its Schur mul... | https://mathoverflow.net/users/17582 | Schur multiplier of finite-dimensional simple Lie algebras in positive characteristic | As $L$ is simple, it has trivial abelianization, so one has $H\_1(L,\mathbb{F})=0$. Therefore, it follows from the Universal Coefficient Theorem for Lie algebras that $H\_2(L, \mathbb{F})\cong H^2(L, \mathbb{F})$. For a finite-dimensional graded Lie algebra of Cartan type over an algebraically closed field of character... | 2 | https://mathoverflow.net/users/14653 | 416133 | 169,599 |
https://mathoverflow.net/questions/416106 | 9 | I'm really interested in Topological Quantum Field Theory (TQFT) and am currently planning to focus on it in my undergraduate thesis. My university, unfortunately, does not allow double majors in mathematics and physics or even a minor in physics, hence I fear that I do not have enough background in physics. I have tak... | https://mathoverflow.net/users/476977 | Undergraduate research in Topological Quantum Field Theory | I am not sure if this is an answer or a request for clarification.
There are a lot of different topics in math (and in physics) that go by the name 'topological quantum field theory'. Beyond the initial hand-waving about '(some) observables not depending on the spacetime manifold's geometric structure', they aren't a... | 10 | https://mathoverflow.net/users/35508 | 416136 | 169,601 |
https://mathoverflow.net/questions/416147 | 6 | Let $C$ be an irreducible curve of arithmetic genus $1$ over a field $k$ and with a double $k$-point $p\in C$.
Is $C$ rational over $k$?
If $C$ is a plane cubic the answer is positive since we can parametrize $C$ with the lines through $p$.
In general I would embed $C$ in some projective space $\mathbb{P}^n$ and ... | https://mathoverflow.net/users/14514 | Singular curves of genus 1 | There's no problem over finite fields, but there is a problem over fields that have a nontrivial Brauer class. If you take a genus $0$ curve that's not rational (say a plane quadric), it will always have points over a degree $2$ extension (intersect with a line), and then you can glue two of them together to get a noda... | 10 | https://mathoverflow.net/users/18060 | 416150 | 169,603 |
https://mathoverflow.net/questions/416145 | 6 | We can consider the generalized Harmonic numbers $$H\_{n,m} := \sum\_{k=1}^{n} \frac{1}{k^{m}} $$ as a partial version of the Riemann zeta function, because $$\lim\_{n \to \infty} H\_{n,m} = \zeta(m). $$
We could also define the "partial Sophomore's Dream function" $$S\_{r} := \sum\_{q=1}^{r}\frac{1}{q^{q}} ,$$
as we... | https://mathoverflow.net/users/93724 | Has the "partial Sophomore's Dream function" been studied before? | We have that $S=\sum\_{n=1}^\infty \frac 1 {n^n}$ and $S\_r=\sum\_{n=1}^r \frac 1 {n^n}$. Thus,
$A=\sum\_{r=1}^\infty \sum\_{n=r+1}^\infty \frac 1 {n^n} = \sum\_{n=2}^\infty \sum\_{r=1}^{n-1} \frac 1 {n^n}$
But then, $A=\sum\_{n=2}^\infty \frac {n-1} {n^n}$. To evaluate this, consider the integral $\int\_0^1 t^{-tx... | 6 | https://mathoverflow.net/users/114143 | 416154 | 169,605 |
https://mathoverflow.net/questions/416131 | 2 | Let $\mathsf{T}$ be a rigid abelian tensor category and suppose that we're given fiber functors $\omega\_M:\langle M\rangle \to \mathsf{vect}\_k$ for every object $M$ of $\mathsf{T}$. Is there a canonical way to obtain a fiber functor on $\mathsf{T}$?
| https://mathoverflow.net/users/131975 | Given fiber functors on all the subcategories of the form $\langle M\rangle$, can we obtain a fiber functor on the whole category? | I think the answer is no in general.
Consider, for example, a tannakian category $M$ over a field $k$, and
suppose for simplicity that it is a
union $M=\bigcup M\_{n}$, $M\_{n}\subset M\_{n+1}$, of neutral tannakian
categories. Is $M$ neutral? In other words, if we assume that each $M\_{n}$ has
a $k$-valued fibre fun... | 2 | https://mathoverflow.net/users/nan | 416161 | 169,606 |
https://mathoverflow.net/questions/416121 | 0 | Let $h$ be a random variable and $g(h)$ be a real-valued function of $h$.
We know that if h is a real-random variable then:
$E\_h[g(h)] = \int\_{-\infty}^{\infty} f(h) g(h) dh$ where f(h) is the PDF of h.
I want to learn if there is an integral to express the expectation over $h$ when $h$ is a complex-random variab... | https://mathoverflow.net/users/477072 | Integral form of expectation with respect to complex random variables | There is an unfortunate mismatch between the notations and concepts in probability used by mathematicians and by practitioners of many engineering fields. In the latter fields, it is conventional to refer to an expectation "with respect to" a particular random variable; in the former one may refer to $\operatorname E(g... | 4 | https://mathoverflow.net/users/6316 | 416166 | 169,607 |
https://mathoverflow.net/questions/416183 | 1 | Given an undirected graph $G = (V, E)$, a *vertex k-cut* of $G$ is a vertex subset of $V$ the removing of which disconnects the graph in at least $k$ connected components
(from <https://cris.unibo.it/handle/11585/713744>).
Given a biconnected graph, the minimal cardinality of a vertex 2-cut is 2, by definition.
My ... | https://mathoverflow.net/users/68336 | Minimal cardinality of a vertex $k$-cut of a biconnected graph | Question (if I got it right):
>
> can one get $k$ connected components by removing less than $k$ vertices from a biconnected graph?
>
>
>
Answer:
Yes. Consider the graph $G=(V,E)$ with $V=\{a,b,v\_1,...,v\_n\}$ and $E=\{av\_i,bv\_i : i=1,...,n\}$. It is 2-connected. Removing nodes $a$ and $b$ you get $n$ com... | 3 | https://mathoverflow.net/users/477125 | 416189 | 169,612 |
https://mathoverflow.net/questions/416187 | 3 | Let $(M^2,g)$ be a noncompact orientable Riemannian surface without boundary. Let $A \in \Gamma(\operatorname{Sym}(TM))$ be a section of the bundle of symmetric endomorphisms of $TM$, that is, for each $p \in M$, the linear map $A\_p : T\_p M \to T\_p M$ is symmetric with respect to the inner product $g\_p$. Assume tha... | https://mathoverflow.net/users/85934 | Global choice of eigenvectors on an open surface | Not necessarily. To construct a counter-example, start from the other direction. Suppose that the tangent bundle of $M$ can be split as the direct sum $TM = L\_1\oplus L\_2$ where $L\_1$ and $L\_2$ are non-trivial smooth line bundles. Then you can easily construct a metric $g$ on $M$ such that $L\_1$ and $L\_2$ are $g$... | 7 | https://mathoverflow.net/users/13972 | 416192 | 169,614 |
https://mathoverflow.net/questions/416201 | 8 | The question is in the title:
>
> **Question:** Which inscribed $n$-dimensional polytope (inscribed in the unit sphere) with $2^n$ vertices has the largest possible volume?
>
>
>
Is it the $n$-dimensional cube? If not, how much larger can its volume be?
| https://mathoverflow.net/users/156219 | Inscribed $n$-polytope with $2^n$ vertices of maximal volume | For $n=3$, the maximal volume polytope with 8 vertices is described in that paper.
*Berman, J. D.; Hanes, K.*, [**Volumes of polyhedra inscribed in the unit sphere in (E^3)**](http://dx.doi.org/10.1007/BF01435416), Math. Ann. 188, 78-84 (1970). [ZBL0187.19604](https://zbmath.org/?q=an:0187.19604).The link is paywalle... | 17 | https://mathoverflow.net/users/908 | 416203 | 169,618 |
https://mathoverflow.net/questions/416168 | 6 | The equation $z^2=\overline{z}$ has four zeros and this example motivates us to generalize the problem to this form;
How many zeros does the equation $P(z)=\overline{z}$ have if $P(z)$ is a polynomial of degree $n>1?$ Can we find the bound for the number of zeros of this problem? The example motivate us to conjecture t... | https://mathoverflow.net/users/128472 | Fixed points of a function $z\mapsto\overline{P(z)}$ of a complex variable | Function $z\mapsto\overline{P(z)}$ has at most $3d-2$ fixed points, where $d\geq 2$
is the degree of $P$, and this is best possible. This remarkable result is due to Khavinson and Świa̧tek,
[MR1933331](https://mathscinet.ams.org/mathscinet-getitem?mr=1933331)
Khavinson, Dmitry, Świa̧tek, Grzegorz ,
[On the number of ... | 8 | https://mathoverflow.net/users/25510 | 416204 | 169,619 |
https://mathoverflow.net/questions/416023 | 1 | For which properties (P) [of groups] does the following hold:
given a group $G$ which has a finite presentation with at most $n$ relations of length at most $\ell$, there is a $R(n,\ell)$ so that, if the ball of radius $R$ in the [labelled/unlabelled] Cayley graph of $G$ [w.r.t. to the generating set of said presenta... | https://mathoverflow.net/users/18974 | Which properties can be read off the balls of a Cayley graph? | This question gestures in a few different directions. Since I think they may all be of some interest, I'll attempt to summarise what's known in a few of these directions here. Hopefully some of this helps!
**Finitely generated groups**
For a finitely generated group $G$ with a given generating set $S$, we can ask w... | 3 | https://mathoverflow.net/users/1463 | 416206 | 169,621 |
https://mathoverflow.net/questions/416198 | 4 | Let $\varphi(x)=\frac{1}{\sqrt{2\pi}}\exp(-x^2/2)$ be the Gaussian density and
$f:\mathbb{R}\to\mathbb{R}$ another measurable function.
Under what conditions can $f$ be recovered from its convolution with $\varphi$? In other words, under what conditions does $f\ast\varphi=0$ imply that $f$ is zero a.e?
If $f\in L^1... | https://mathoverflow.net/users/477138 | Recovering a function from its Gaussian convolution | $\newcommand{\vpi}{\varphi}\newcommand\R{\mathbb R}\newcommand\C{\mathbb C}\newcommand{\ep}{\varepsilon} $The minimal condition
\begin{equation\*}
|f|\*\vpi<\infty \tag{1}\label{1}
\end{equation\*}
is already enough for the recovery of $f$.
Indeed, since $\vpi(x-u)=\vpi(u)e^{xu}e^{-x^2/2}$, condition \eqref{1} can b... | 4 | https://mathoverflow.net/users/36721 | 416212 | 169,622 |
https://mathoverflow.net/questions/412037 | 7 | Let $(A, \Delta)$ be a compact quantum group in the sense of Woronowicz. Is it true that the comultiplication $\Delta : A \to A \otimes A$ always injective?
This is true for both the universal (because one has a counit) and the reduced (because the Haar state is faithful) version, but the general case seems more deli... | https://mathoverflow.net/users/128540 | Is the comultiplication of a compact quantum group always injective? | No, the comultiplication need not be injective. When $\Gamma$ is a countable group and $\pi : \Gamma \to \mathcal{U}(H)$ is a faithful unitary representation with the property that $\pi \otimes \pi$ is weakly contained in $\pi$, we write $A = C^\*\_\pi(\Gamma)$ and there is a unique unital $\*$-homomorphism $\Delta : A... | 6 | https://mathoverflow.net/users/159170 | 416221 | 169,623 |
https://mathoverflow.net/questions/416119 | 4 | $\DeclareMathOperator\SL{SL}$I am currently looking at the paper titled "$\SL(2,\mathbb{C})$ quotients de $(\mathbb{P^1})^n$" by Marzia Polito. The author has considered diagonal action of $\SL(2,\mathbb{C})$ over $(\mathbb{P^1})^n$ given by $(g,(x\_1, x\_2,\cdots,x\_n))=(gx\_1, gx\_2,\cdots,gx\_n)$ for $g\in \SL(2,\ma... | https://mathoverflow.net/users/211682 | Question regarding semistability of a point of GIT quotient | The details of this proof can be found in introductory texts on geometric invariant theory (GIT). I recommend Dolgachev's textbook Lectures on Invariant Theory, Chapter 11, where one can find a proof via the Hilbert-Mumford numerical criterion. The technical overhead to learning the Hilbert-Mumford numerical criterion ... | 2 | https://mathoverflow.net/users/477081 | 416224 | 169,624 |
https://mathoverflow.net/questions/415991 | 6 | The Computational Diffie Hellman (CDH) problem is to compute $g^{XY}$ given $g^X$ and $g^Y$ where $g$ generates the group. The Discrete Logarithm (DLOG) problem is to compute $X$ given $g^X$. The latter in $P$ implies the former in $P$.
>
> Are there groups where CDH problem is easy and in $P$ while DLOG is difficu... | https://mathoverflow.net/users/10035 | Groups in which Computational Diffie Hellman is in $P$ but Discrete Logarithm is not known to be in $P$ | Some background for this answer:
1. A computation is *non-uniform* if, in addition to an input $x$, one gets an "advice string" that depends solely on the *length* of $x$. One can view this as precomputation, although this advice string need not be computable (this should not be relevant here though).
In this answer,... | 2 | https://mathoverflow.net/users/101207 | 416238 | 169,625 |
https://mathoverflow.net/questions/416099 | 0 | Mathematica gives me the following solution to $\int{x^{-a} (b -cx^{-d})^e }$:
$$\int{x^{-a} (b -cx^{-d})^e dx} = -\frac{b^{e}x^{1-a} \, \_2F\_1\left(\frac{a-1}{d},-e;\frac{a+d-1}{d};\frac{c x^{-d}}{b}\right)}{a-1},$$
where $a,b,c,d,e > 0$.
I'd like to figure out what are the steps involved in this solution.
I st... | https://mathoverflow.net/users/103291 | What are the steps involved in the solution to $\int{x^{-a} (b -cx^{-d})^e }dx$? | The proof starts by applying the change of variable $z=(c/b)x^{-d}$ and then using the following identity
$$\int (1-z)^{u} z^{v} dz = \frac{z^{v+1}}{v+1} \, \_2F\_1(-u, v+1; v+2; z), |x|<1,$$
the proof is concluded.
| 1 | https://mathoverflow.net/users/103291 | 416243 | 169,627 |
https://mathoverflow.net/questions/416241 | 9 | Let $B$ be the closed unit ball in $\mathbb R^3$ centered at the origin and let $U= \{x\in \mathbb R^3\,:\, \frac{1}{2}\leq |x| \leq 1\}.$ Let
$$ S\_U= \{u \in C^{\infty}(U)\,:\, \Delta u =0 \quad\text{on $U^{\textrm{int}}$}\},$$
and
$$ S\_B= \{u \in C^{\infty}(B)\,:\, \Delta u =0 \quad\text{on $B^{\textrm{int}}$}\}.$$... | https://mathoverflow.net/users/50438 | Density of restrictions of harmonic functions inside a ball | No. If $\varphi \in C^\infty\_c(B)$ is a bump function equal to $1$ in $\lvert x\rvert \leq 1/2$ then from Green's theorem we have
$$ \int\_B u \Delta \varphi = 0$$
for all $u \in S\_B$, but the same is not true in general for typical $u \in S\_U$, which by the Cauchy–Schwarz inequality implies that $u$ is a positive d... | 20 | https://mathoverflow.net/users/766 | 416245 | 169,628 |
https://mathoverflow.net/questions/416230 | 2 | The Fourier transform of the (indicator function of the) unit ball is well known to be given by the Bessel functions (see [Fourier transform of the unit sphere](https://mathoverflow.net/questions/149692/fourier-transform-of-the-unit-sphere)).
What can be said about the $\ell\_1$ ball, that is the set of points in $\m... | https://mathoverflow.net/users/7581 | Fourier transform of the unit ball in L1 metric | The Fourier transform $F(\mathbf{k})$ of the unit ball in L1 metric can be evaluated as follows$^\ast$.
*Notation:* $\theta(x)$ is the unit step function, $\delta(x)$ is the delta function, and ${\cal P}$ denotes the principal value of the integral.
$$F(k\_1,\ldots k\_n)=\int \cdots\int e^{i\mathbf{k\cdot x}}\thet... | 4 | https://mathoverflow.net/users/11260 | 416254 | 169,630 |
https://mathoverflow.net/questions/171703 | 14 | My question is about a variant of the usual notion of relative constructibility, $\le\_c$ (which an earlier version of this question confusingly denoted "$\le\_L$"), in set theory.
Fix a countable transitive model $W\models\mathsf{ZFC+V\not=L}$. By a theorem of Barwise, given any set $A\in W$ there is a (possibly-ill... | https://mathoverflow.net/users/8133 | When is $A$ "$L$-ish" whenever $B$ is "$L$-ish"? | Remarks:
(i) I'm interpreting the definition of $\leq\_{L,\mathrm{end}}$ as quantifying over set models $W'$, not proper classes.
(ii) I'm considering the main question (comparing the two orders), particularly in the case that $V$ has large cardinals, but mainly not in the case of "particular interest", i.e. where ... | 3 | https://mathoverflow.net/users/160347 | 416259 | 169,632 |
https://mathoverflow.net/questions/416249 | 1 | Let $f(x)$ be a strictly increasing function such that $\lim\limits\_{x\to\pm\infty}f(x)=\pm\infty$ and $\lim\limits\_{x\to+\infty}f'(x)e^{f(x)-x}=+\infty$. If $f(a)=0$ for some prescribed $a\in\Bbb R$, does a minimum exist for $$\int\_{\Bbb R}xe^{f(x)-x^2}\,dx$$ and if so, what would be the minimiser $f^\*$? The prese... | https://mathoverflow.net/users/113397 | Does a minimiser exist for this Gaussian-like functional? | $\newcommand\R{\mathbb R}$For any $a\in\R$, there is no minimizer of
\begin{equation\*}
I(f):=\int\_\R xe^{f(x)-x^2}\,dx \tag{-1}\label{-1}
\end{equation\*}
over all $f\in F\_a$, where $F\_a$ is the set of all strictly increasing functions $f\colon\R\to\R$ such that $\lim\_{x\to\pm\infty}f(x)=\pm\infty$, $\lim\_{x\to\... | 1 | https://mathoverflow.net/users/36721 | 416260 | 169,633 |
https://mathoverflow.net/questions/416271 | 4 | I am interested in extensions $A\leq B$ of commutative rings with the property that for all ideals $I\leq A$ we have $IB\cap A=I$. Is there a standard name for this property, or a standard reference for results about it?
| https://mathoverflow.net/users/10366 | Terminology/literature for $\forall I\leq A,\; IB\cap A=I$ | Such extension is called "cyclically pure". An extension is called pure if the induced map $A\otimes\_A M\to B\otimes\_A M$ is injective for any $A$ module $M$. If the map $A\to B$ splits as map of $A$-modules, then it is pure, and the converse holds if $B$ is finitely presented as an $A$-module.
Also clearly, purity... | 3 | https://mathoverflow.net/users/2083 | 416283 | 169,643 |
https://mathoverflow.net/questions/416269 | 8 | Let $d$ be a positive number.
There is a two dimensional recurrence relation as follow:
$$R(n,m) = R(n-1,m-1) + R(n,m-d)$$
where
$R(0,m) = 1$ and $R(n,0) = R(n,1) = \cdots = R(n, d-1) = 1$ for all $n,m>0$.
How to analyze the asymptotics of $R(n, kn)$ for fixed $k$?
It is easy to see that
$$R(n, kn) = O\left( c\_{k,d}... | https://mathoverflow.net/users/115910 | How to find the asymptotics of a linear two-dimensional recurrence relation | Your generating function $f(x,y)$ is convergent on a polydisk $|x|<\epsilon$, $|y|< \delta$, for some $\epsilon, \delta < 1$. We can reduce $\epsilon$ so that $\epsilon \ll \delta^k$ and the domain of $f(x,y)$ to the product of the polydisk $|x|<\epsilon$ and the annulus $\delta' < |y| < \delta$, with $\delta' \sim \de... | 3 | https://mathoverflow.net/users/2622 | 416295 | 169,647 |
https://mathoverflow.net/questions/416289 | 0 | This question is an "outgrowth" of <https://math.stackexchange.com/questions/4380919/> which led to a numerically-generated two-parameter function $f\_b(n)$, where $b$ is the number base $2,3,4,\ldots$, and $n$ is as follows, paraphrasing the original question...
What is the number of $n$-character words consisting o... | https://mathoverflow.net/users/476477 | What is this numerically-generated function? | First, we notice that the first character must be 1. Let's take it off and focus on the remaining $n-1$ characters, where the number of 1's in each prefix must be at least as many as the number of the other $b-1$ digits.
Replacing each 1 with `[` and each other digit with `]`, we get a truncated [Dyck word](https://e... | 0 | https://mathoverflow.net/users/7076 | 416299 | 169,649 |
https://mathoverflow.net/questions/320814 | 4 | A functor $N\colon\mathrm{Cat}\_{A\_\infty}\longrightarrow\mathrm{Cat}\_\infty$ is constructed in a paper [1] by Faonte. This gives a way to get an $\infty$-category by starting with an $A\_\infty$-category.
Going the other way, is it possible to *define* linear $A\_\infty$-categories as special $\infty$-categories?
... | https://mathoverflow.net/users/130058 | Is it possible to define linear $A_\infty$-categories as special $\infty$-categories? | An affirmative to the "conjecture" above is fully recorded in this work:
<https://arxiv.org/abs/2003.05806>
In Remark 1.2, for example, we comment how Gepner-Haugseng's results imply that k-linear A-infinity categories are precisely k-chain-complex-enriched infinity-categories.
This passes through the infinity-cate... | 4 | https://mathoverflow.net/users/3593 | 416305 | 169,651 |
https://mathoverflow.net/questions/416302 | 15 | First of all, I am interested in the general case of a non-orientable manifold but let's for now consider the projective plane $\mathbb{R}P^2.$ In short, I am curious if there is any relation between the diffeomorphism group $\text{Diff}(\mathbb{R}P^2)$ of the projective plane and the diffeomorphism group $\text{Diff}(... | https://mathoverflow.net/users/131858 | Diffeomorphism group of the projective plane | Two different answers using almost identical techniques! Allen's response got me to think through my response more carefully. Let me edit in a comment to point out my sloppiness, as it points out a useful detail in the machinery we are using.
$\DeclareMathOperator\Diff{Diff}\DeclareMathOperator\Emb{Emb}\DeclareMathOp... | 15 | https://mathoverflow.net/users/1465 | 416310 | 169,653 |
https://mathoverflow.net/questions/416220 | 2 | I originally asked this on MSE, but did not get an answer there.
Let $M$ be a von Neumann algebra. Let $\varphi: M\_+ \to [0, \infty]$ be a weight on $M$. Consider
\begin{align\*}&\mathfrak{p}\_\varphi:= \{x\in M\_+: \varphi(x) < \infty\}\\
&\mathfrak{n}\_\varphi:= \{x \in M: \varphi(x^\*x) < \infty\}\\
& \mathfrak{m... | https://mathoverflow.net/users/216007 | Decomposition of an element as a difference of positive elements in the definition subalgebra of a weight (Takesaki) | I don't know what Takesaki had in mind for the proof, but what you're asking is incorrect. Here is a counterexample where $\phi$ in the counterexample is a normal faithful semifinite (n.f.s.) weight.
Let $H$ be a separable infinite dimensional Hilbert space, and $M = B(H\oplus H)$. Fix a faithful normal state $\psi$ ... | 4 | https://mathoverflow.net/users/126109 | 416318 | 169,656 |
https://mathoverflow.net/questions/416246 | 1 | I am interested in the following situation: Given any $n>1$ suppose I have a codimension-1 foliation of $R^n\_{++}$ (i.e. the subset of strictly positive $n$-vectors) arising from an $(n-1)$-dimensional, involutive, plane field $F$. Further, suppose that the normal to the subspace spanned by the vectors of each $F\_p$ ... | https://mathoverflow.net/users/92328 | Codimension-1 foliations of Euclidean space with strictly positive normal bundle | I believe the answer is yes. It's enough to assume that the last coordinate of each normal is positive. Then each leaf is a graph $x\_n=x\_n(x\_1,\ldots, x\_{n-1})$ on some domain (depending on the leaf) in $\mathbb R^{n-1}$. This function must go to infinity at the boundary of the domain (if the boundary is nonempty) ... | 1 | https://mathoverflow.net/users/18050 | 416327 | 169,660 |
https://mathoverflow.net/questions/416258 | 14 | Erdős and Selfridge (1971) state that the following is "implied by an unpublished result of Rosser" which they claim appears in a forthcoming book on sieve methods by Halberstam and Richert.
I guess I could try to find something implying it in that 400-page book (called "Sieve Methods", available online in unsearchable... | https://mathoverflow.net/users/4600 | Unpublished result of Rosser in Sieve Methods book | This is part of the theory of the "linear sieve." Chapter 8 of the book of Halberstam and Richert deals with this topic. Alternatively you could look at Iwaniec's paper [On the error term in the linear sieve](http://matwbn.icm.edu.pl/ksiazki/aa/aa19/aa1911.pdf). The result you want can be extracted from the lower bound... | 10 | https://mathoverflow.net/users/38624 | 416328 | 169,661 |
https://mathoverflow.net/questions/416333 | 2 | Let $A$ be real matrix with $M > 1$ rows and $K > 2$ columns, and each entry $a\_{m,k} \in (0,1)$, with each row summing to $1$. For all $m$
$$
\sum\_{k=1}^{K} a\_{m,k} = 1
$$
I want to find out if for any row $m$
$$
\sum\_{k=1}^{K} \frac{(a\_{m,k})^2}{\sum\_{i=1}^{M} a\_{i,k}} \geq \frac{1}{M}
$$
I believe that I have... | https://mathoverflow.net/users/477213 | Inequality for matrix with rows summing to 1 | If I am not missing something, this seems a direct application of [Titu's lemma](https://brilliant.org/wiki/titus-lemma/)
$$
\sum\_{k=1}^K \frac{x\_k^2}{y\_k} \geq \frac{\left(\sum\_{k=1}^K x\_k \right)^2}{\sum\_{k=1}^K y\_k}, \quad x\_k \geq 0, y\_k > 0,
$$
which is quickly proved using Cauchy-Schwarz's inequality (as... | 1 | https://mathoverflow.net/users/1898 | 416337 | 169,665 |
https://mathoverflow.net/questions/407857 | 4 | It is known that [fully faithful functors are closed under pushouts in Cat](https://ncatlab.org/nlab/show/full+and+faithful+functor) (e.g. Lemma 4.9 of [this paper](https://lmcs.episciences.org/742/pdf)). Are locally fully faithful 2-functors closed under (strict) 2-pushouts in the 2-category 2-Cat of 2-categories, (st... | https://mathoverflow.net/users/152679 | Are locally fully faithful 2-functors closed under 2-pushout in 2-Cat? | No. I'm not sure whether you're asking about pushout along an arbitrary 2-functor, or just about pushing out a locally fully faithful functor along *another* locally fully faithful functor. Either way the answer is no.
Consider the inclusion of 1-categories into 2-categories as the locally discrete ones. This functor... | 1 | https://mathoverflow.net/users/2362 | 416341 | 169,667 |
https://mathoverflow.net/questions/416339 | 6 | Suppose that $\boldsymbol{t}\sim \mathcal{N}(\boldsymbol{u};\boldsymbol{0},\boldsymbol{M})=f\_{\boldsymbol{t}}(\boldsymbol{u})$, where $\boldsymbol{t}$ is a $N$-dimensional gaussian random vector, and
\begin{equation}
\boldsymbol{M}=
\left[
\begin{matrix}
1&m\_{12}&\cdots& m\_{1N}\\
m\_{21}&1&\cdots& m\_{2N}\\
\vdots&\... | https://mathoverflow.net/users/163557 | Is this function monotonically increasing? | Yes, this is a special case of [Slepian's inequality](https://en.wikipedia.org/wiki/Slepian%27s_lemma).
| 5 | https://mathoverflow.net/users/36721 | 416343 | 169,668 |
https://mathoverflow.net/questions/416242 | 1 | How to prove that
$$\int (1-z)^{u} z^{v} dz = \frac{z^{v+1}}{v+1} \, \_2F\_1(-u, v+1; v+2; z)?$$
| https://mathoverflow.net/users/103291 | How to prove that $\int (1-z)^{u} z^{v} dz$ is equal to $\frac{z^{v+1}}{v+1}_2F_1(-u, v+1; v+2; z)$? | The generalized binomial coefficients are
$$
\binom{u}{n} := \frac{(-1)^n(-u)\_n}{n!} ,
$$
here we used the notation $(-u)\_n$ for the rising factorial: $(-u)\_n := \prod\_{k=0}^{n-1}(-u+k)$.
Taylor's expansion of $(1-z)^u$ around $z=0$ is
$$
(1-z)^u = \sum\_{n=0}^\infty \binom{u}{v}(-z)^n = \sum\_{n=0}^\infty \frac{... | 0 | https://mathoverflow.net/users/103291 | 416364 | 169,670 |
https://mathoverflow.net/questions/416346 | 1 | Define
$$
F(\lambda,x):=g(x)+\lambda\int\limits\_{\partial\_e Y}f(y)d\mu\_x(y)
$$
where
* $Y$ is a convex space and $w^\*$-compact, moreover $Y$ forms a Bauer simplex (in particular $Y$ corresponds to some subset of some algebraic state space),
* its extreme boundary $\partial\_e Y$ is $w^\*$-compact,
* the points $x... | https://mathoverflow.net/users/145631 | Derivatives of infimum in variational problem | Let $t:=\lambda$, so that
$$R(t):=\inf\_{x\in Y}F(t,x).$$
Suppose that $\int\_{\partial\_e Y}f\,d\mu\_x$ is lower-semicontinuous in $x$ (with respect to the appropriate topology, which you appear to assume to be the $w^\*$-topology with respect to some unspecified duality). Then the set
$$Y\_t:=\{x\in Y\colon F(t,x)=R(... | 0 | https://mathoverflow.net/users/36721 | 416373 | 169,676 |
https://mathoverflow.net/questions/416311 | 16 | A Schwartz function on $\mathbb R^d$ is a $C^\infty$ function, such that all differentials of order $k \ge 0$ decay faster than any polynomial. They include the class $C^\infty\_c(\mathbb R^d)$ of compactly supported, smooth functions.
I would like two know, if for every Schwartz function $f$, there are Schwartz func... | https://mathoverflow.net/users/123409 | Is every Schwartz function the product of two Schwartz functions? | Yes, such a decomposition exists. More general given a compact set $B\subset\mathcal{S}(\mathbb{R}^{n})$ there is a function $\varphi\in\mathcal{S}(\mathbb{R}^{n})$ and a compact set $C\subset\mathcal{S}(\mathbb{R}^{n})$ with $B=\varphi C$. This property is called the *compact strong factorisation property* by J. Voigt... | 16 | https://mathoverflow.net/users/83700 | 416374 | 169,677 |
https://mathoverflow.net/questions/416100 | 2 | This question was motivated by [Singular value decomposition of truncated discrete Fourier transform matrix](https://mathoverflow.net/q/415784/11260)
Consider for integers $1\leq k\leq N$, $1\leq n\_0\leq N-k+1$ the $k\times k$ sub-unitary matrix $W(N,k,n\_0)$ with elements
$$[W(N,k,n\_0)]\_{nm}=N^{-1/2}e^{2\pi i(n-1... | https://mathoverflow.net/users/11260 | Unit singular value conjecture for discrete Fourier transform submatrix | The conjecture is true. Moreover we show that
if $W(N,N,1)$ is replaced by any unitary $N \times N$ matrix
then $\max(2k-N,0)$ is still a lower bound on the multiplicity,
and is usually sharp (though for $k<N$ there are easy exceptions
such as the $N \times N$ identity matrix).
First observe that if $T$ is any "sub-u... | 3 | https://mathoverflow.net/users/14830 | 416382 | 169,679 |
https://mathoverflow.net/questions/416146 | 3 | Let $V$ be a $\mathbb{C}$-vector space of dimension $N \geq 2$, let $d$ be a positive integer, let $l < N$ be a positive integer, and let $U \subseteq S^d(V)$ be a linear subspace of codimension $k=\binom{l+d-1}{d}$. Suppose that there exists a basis of $U$ of the form $\{u\_1^{\vee d},\dotsc, u\_r^{\vee d}\}$ such tha... | https://mathoverflow.net/users/150898 | Given a subspace $U \subseteq S^d(V)$ of a particular form, does there always exist a complement of the form $S^d(W)$? | The answer is still no and a variant of [my previous counter-example](https://mathoverflow.net/a/415807/37214) will provide a counter-example for this new question.
Let $V = \mathbb{C}^4$ with basis $\{x,y,z,t\}$ and $\mathbb{C}^3 \subset V$ with basis $\{x,y,z\}$. Put $U = S^2 \mathbb{C}^3 \oplus \mathbb{C}\cdot t^2... | 1 | https://mathoverflow.net/users/37214 | 416401 | 169,683 |
https://mathoverflow.net/questions/416331 | 9 | This might be an easy question, maybe the example I'm looking for is common knowledge. As always, recall that a topological space $X$ is *scattered* if and only if every non-empty subset $Y$ of $X$ contains at least one point which is isolated in $Y$. It is known that any first countable, $T\_3$, Lindelöf and scattered... | https://mathoverflow.net/users/146942 | Example of an uncountable scattered space with some properties | There exists an example for this question under Continuum Hypothesis, more precisely, under the assumption $\mathfrak b=\omega\_1$. In this case by Theorem 10.2 in the van Douwen's survey paper ``[The integers and Topology](https://www.sciencedirect.com/science/article/pii/B9780444865809500069)'', there exists an uncou... | 10 | https://mathoverflow.net/users/61536 | 416404 | 169,685 |
https://mathoverflow.net/questions/416405 | 2 | If $p : X \rightarrow Y$ is the dominant surjective finite morphism of varieties between smooth projective varieties over of same dimension over $\mathbb{C}$, then do we know about the properties of the pull-back maps $p^{\star} : H^{k,k}(Y) \rightarrow H^{k,k}(X)$ is surjective/injective? If someone could provide refe... | https://mathoverflow.net/users/476114 | When does a morphism between varieties induce surjection/injection morphism on cohomology? | For a finite map between smooth varieties, the induced pullback map on cohomology is injective (hence it is somewhat rarely surjective - only when it's an isomorphism).
This is because there a is a trace map $H^{k,k}(X) \to H^{k,k}(Y)$, and the composition of the two $H^{k,k}(Y) \to H^{k,k}(X) \to H^{k,k}(Y)$ is mult... | 8 | https://mathoverflow.net/users/18060 | 416408 | 169,686 |
https://mathoverflow.net/questions/416411 | 6 | Let $X$ be a compact complex manifold. What are obstructions to the existence of a real subbundle $V$ of $TX$ such that $TX = V \otimes \mathbb{C}$?
For example, does $\mathbb{CP}^n$ have such a structure?
| https://mathoverflow.net/users/392184 | Existence of a real structure on the tangent bundle of a complex manifold | For a vector bundle to have real structure, it must be isomorphic to its complex conjugate, hence it's $i$th Chern class must be equal to its own negation (i.e. $2$-torsion) for all odd $i$. So $2c\_1, 2c\_3, 2 c\_5, \dots$ are all obstructions.
Since $2c\_1$ is nonzero for the tangent bundle of projective space, $\m... | 7 | https://mathoverflow.net/users/18060 | 416412 | 169,688 |
https://mathoverflow.net/questions/416344 | 2 | It is well known that each $n\in\mathbb N=\{0,1,2,\ldots\}$ can be written as $2w^2+x^2+y^2+z^2$ with $w,x,y,z\in\mathbb N$. Furthermore,
$$\{2w^2+x^2+y^2:\ w,x,y\in\mathbb N\}=\mathbb N\setminus\{4^k(16m+14):\ k,m\in\mathbb N\}.$$
Motivated by this, here I pose the following novel question.
**Question 1.** Can each ... | https://mathoverflow.net/users/124654 | Can each natural number be represented by $2w^2+x^2+y^2+z^2+xyz$ with $x,y,z\in\mathbb N$? | The answer to question 1 is yes - the other questions seem to me to be more difficult.
If $n$ is odd, then there are non-negative integers $w$, $x$ and $y$ so that $n = 2w^{2} + x^{2} + y^{2}$. One way to see this is that the class number of this quadratic form $Q\_{1} = 2w^{2} + x^{2} + y^{2}$ is $1$, and so every l... | 5 | https://mathoverflow.net/users/48142 | 416426 | 169,693 |
https://mathoverflow.net/questions/416437 | 0 | If I choose a principal bundle, let us say $G\rightarrow P \rightarrow B$, with $G=U(1)$, $P=S^1 \times S^1$ and $B=S^1$. Can I follow the identity element of the group over a curve at the base. How isn't this equivalent to a preferred Horizontal subspace (and hence a canonical connection) ?
| https://mathoverflow.net/users/477214 | Non existence of preferred Horizontal subspace on a bundle | 1. A trivial principal bundle, $P= G\times B\to B$ has a preferred connection (which is also flat). There is an obvious map $B\to \{1\}\times B\subset P$ and you can take as horizontal bundle the image of its differential.
2. If your principal bundle $P\to B$ is not trivial, but is trivializable, i.e. exists a section ... | 2 | https://mathoverflow.net/users/99042 | 416442 | 169,697 |
https://mathoverflow.net/questions/416430 | 4 | I can deduce some results about this from the prime number theorem or from results about the [primorial function](https://en.wikipedia.org/wiki/Primorial#Characteristics) $p\#$ but I'm wondering what the state of the art is:
Given integers $0\le a\_1<a\_2<\dots <a\_c$, what bound can we put on the least modulus $m$ s... | https://mathoverflow.net/users/4600 | Least modulus distinguishing some integers | I'm not sure about the state of art, but here is a rough estimate in terms of $c$ and $\ell:=a\_c-a\_1$ in the case $c\ll \ell$.
First, we notice that for an integer $M$ satisfying
$$M\# ~>~ \big(\frac{c}{2(c-1)}\ell\big)^{c(c-1)/2} ~\geq~ \prod\_{1\leq i<j\leq c} (a\_j-a\_i),$$
where the second inequality follows fr... | 3 | https://mathoverflow.net/users/7076 | 416457 | 169,703 |
https://mathoverflow.net/questions/416301 | 3 | **Update:**
Now I know why my method fails. But I still wanna know how to work out the original question, that is to show the exactness of the chain complex $C\_\*(X)$ except for two positions $n=0,N-1$.
---
Here is a little problem about the normalization theorem in the theory of simplicial objects.
Given a fi... | https://mathoverflow.net/users/471160 | The exactness of the associated chain complex of a simplicial free abelian group over a finite set and the normalization theorem | Leaving an answer so that this question can be marked resolved.
Your chain complex is called the "complex of injective words" and proving that it is highly connected is maybe not so direct. Here is a paper by an author I respect <https://arxiv.org/pdf/1608.04496.pdf> with a "simple" proof that looks pretty involved t... | 2 | https://mathoverflow.net/users/9068 | 416459 | 169,704 |
https://mathoverflow.net/questions/416466 | 0 | Let $V$ be a $\mathbb{C}$-vector space, and let $f\_1,\dots,f\_n \in S^d(V^\*)$ be homogeneous polynomials of degree $d$ for which $V(f\_1,\dots, f\_n)=\{0\}$.
Must there exist a positive integer $k\geq d$ such that for all $v \in S^k(V^\*)$ there exists $g\_1,\dots, g\_n \in S^{k-d}(V^\*)$ for which $v=\sum\_{i=1}^n... | https://mathoverflow.net/users/150898 | A variation on the projective Nullstellensatz | Since $V(f\_1,\dots,f\_n)=0$, by the Nullestellensatz
$$ \sqrt{ (f\_1,\dots, f\_n)} = I ( V( f\_1,\dots, f\_n)) = I(0) = (x\_1,\dots, x\_r)$$
where $x\_1,\dots, x\_r$ are a basis for $V^\*$.
So for each $i$ from $1$ to $r$, $x\_i^{e\_i} \in (f\_1,\dots, f\_n)$ for some $e\_i \in \mathbb N$.
It follows that $$(x\_1,... | 4 | https://mathoverflow.net/users/18060 | 416469 | 169,708 |
https://mathoverflow.net/questions/416468 | 7 | Theorem 2 in [these notes](https://www.math.ias.edu/%7Elurie/278xnotes/Lecture16-Enumerations.pdf)[1] states that, roughly, that each Grothendieck topos can be built (using limits and colimits) from localic topoi. To what extent is that related to the [theorem](https://mathoverflow.net/questions/412504/an-extension-of-... | https://mathoverflow.net/users/477332 | Every Grothendieck topos can be built from localic topoi | The groupoid representation of Joyal and Tierney may be identified with the truncated simplicial diagram appearing in the statement of Theorem 2 of the Lurie notes mentioned in the question. That is, a groupoid object is precisely a diagram of the shape indicated in Theorem 2 satisfying some axioms, which are automatic... | 10 | https://mathoverflow.net/users/2362 | 416470 | 169,709 |
https://mathoverflow.net/questions/416467 | 2 | I am reading a paper about the curve shortening flow which make use of one inequality but I don't know where does it come from where f(x,t) is a smooth function and C is a constant depending on time t. Since the curve $\gamma$ is closed and bounded, it also compact,$$\underset{\gamma}{\max}|f|^2\le C\int\_\gamma|f'|^2+... | https://mathoverflow.net/users/470164 | Question of an inequality from curve-shortening flow | $\newcommand{\thh}{\theta}$The inequality in the proof of Corollary 4.4.4 in the linked paper is stated there (without proof) literally as follows:
\begin{equation}
\max|f|^2\le C\int|f'|^2+f^2. \tag{1}\label{1}
\end{equation}
The paper does not appear to specify the meaning of $C$ or $\int$ or $f$. The proof only say... | 4 | https://mathoverflow.net/users/36721 | 416478 | 169,712 |
https://mathoverflow.net/questions/416362 | 1 | Consider in 1D the operator given by
$$
\mathcal{L} = \frac{d^2}{dx^2} - V'(x)\frac{d}{dx},
$$
where $V(x)$ is a convex, sufficiently quickly growing potential, so that $\mathcal{L}$ has a complete set of eigenfunctions $\psi\_n(x)$ (orthogonal with respect to a stationary distribution $\rho(x)dx$).
Let $\psi\_1$ be ... | https://mathoverflow.net/users/2192 | Monotonicity of the top eigenfunction of the generator of a diffusion | This is rather a way to get a (positive) answer than a complete one, some missing details will be clear in a moment.
Let me use $A$ for $\mathcal L$ and $B=A-V''$. The symmetrizing measure (for both) is $e^{-V}\, dx$. Both operators are negative and self-adjoint in $L^2(e^{-V}\, dx)$ and the first eigenvalue of $A$ i... | 1 | https://mathoverflow.net/users/150653 | 416480 | 169,714 |
https://mathoverflow.net/questions/409095 | 8 | Say that a logic $\mathcal{L}$ satisfies the **weak test property** iff for all $\mathfrak{A}\subseteq\mathfrak{B}$ we have $(1)\implies(2)$ below:
1. For each $\mathcal{L}$-formula $\varphi$ with parameters from $\mathfrak{A}$ we have $$\vert\varphi^\mathfrak{B}\cap\mathfrak{A}^{arity(\varphi)}\vert=\vert\varphi^\ma... | https://mathoverflow.net/users/8133 | Does "agreement on cardinalities" imply second-order elementary substructurehood? | No, second order logic does not have the weak test property: let $\mathfrak{B}=(\mathbb{R},{<})$ (that is, the real numbers with the only predicate being the usual "less than" order) and let $\mathfrak{A}=(\mathbb{R}\backslash\{0\},{<})$. Then $\mathfrak{B}\models$"I am a complete linear order" (completeness as in "for... | 5 | https://mathoverflow.net/users/160347 | 416485 | 169,715 |
https://mathoverflow.net/questions/416348 | 3 | Assuming that $i \geq 0$, let $p\_i$ denote an $i$-th prime: $p\_0 = 2, p\_1 = 3, \ldots$ Then $b\_i$ denotes the second-to-last bit of $p\_i$, i.e. $b\_i = \left\lfloor p\_i/2 \right\rfloor \bmod 2.$
The sequences $B\_0$ and $B\_1$ are constructed by the following algorithm: if $b\_i$ is equal to $i \bmod 2$, append... | https://mathoverflow.net/users/122796 | How to explain a particular property of the second-to-last bits of primes? | I think the phenomena you observe are explained more by the way you construct $B\_0$ and $B\_1$ than by the properties of primes (though properties of primes seem to play a role). If we suppose instead that $b\_i$ were random, obtained by a coin flip, it's easy to see that for each bit of $B\_0$ or $B\_1$, the probabil... | 5 | https://mathoverflow.net/users/18060 | 416487 | 169,716 |
https://mathoverflow.net/questions/416481 | 6 | It can be found, at Hartshorne exercise 4.2.7 for example, that in the case where $\operatorname{char}(k) \neq 2$ we have a nice correspondence between etale degree 2 covers of a curve $C$ and 2-torsion in $JC$. This correspondence, as constructed in Hartshorne, breaks down in the case where $\operatorname{char}(k) = 2... | https://mathoverflow.net/users/160814 | Degree-2 étale covers of curves in characteristic 2 vs torsion points on the Jacobian | There is a duality between degree 2 coverings and two-torsion points on the Jacobian — i.e. both form elementary abelian 2-groups, and these groups are naturally dual.
This is the Artin–Milne Poincaré duality pairing in fppf cohomology (Corollary 4.9 of [Duality in the Flat Cohomology of Curves](https://eudml.org/doc... | 6 | https://mathoverflow.net/users/18060 | 416504 | 169,721 |
https://mathoverflow.net/questions/414531 | 3 | **Does every three dimensional compact solvmanifold admit either Euclidean, nil, or sol geometry?**
definitions/motivation/background:
A solvmanifold is a manifold $ M $ admitting a transitive action by a solvable Lie group $ G $. In other words
$$
M \cong G/H
$$
where $ G $ is solvable and $ H $ is a closed subgro... | https://mathoverflow.net/users/387190 | 3 dimensional solvmanifolds and Thurston geometries | Yes.
First observe that every compact 3d solvmanifold admits a Thurston geometry. To see why: Let $ M $ be a compact solvmanifold. Then by theorem of Auslander $ M $ is finitely covered by a compact special solvmanifold $ M' $ (a compact special solvmanifold is the quotient of a simply connected solvable Lie group $ ... | 2 | https://mathoverflow.net/users/387190 | 416509 | 169,724 |
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