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https://mathoverflow.net/questions/430864 | 2 | At the Wikipedia there are the differential formulation for [Euler-Bernoulli Beam](https://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory) \eqref{1} and [Timoshenko Beam](https://en.wikipedia.org/wiki/Timoshenko%E2%80%93Ehrenfest_beam_theory) \eqref{2}
$$
\begin{align}
&\dfrac{d^2}{dx^2}\left(EI\dfrac{d^2w}{... | https://mathoverflow.net/users/173662 | Mechanics: Model beam using differential vectorial formulation | The beam equations \eqref{1} and \eqref{2} "admit a vector formulation" in the sense that they can be rigorously deduced from the 3D theory of (nonlinear) elasticity. I'm not aware of a precise reference dealing specifically with those two examples, but the *Encyclopedia of Physics* entry [1] by Antman (which also allu... | 3 | https://mathoverflow.net/users/113756 | 430955 | 174,521 |
https://mathoverflow.net/questions/429953 | 5 | We are given a convex shape $S$ in the $d$-dimensional Euclidean space, whose boundary is formed by portions of $2d$ different spheres, one portion per sphere. The radius of each sphere is the same, $a+1$, and the spheres centers are $\pm a\textbf{e}\_j$ for $1\le j\le d$, where $a\ge 0$. Hence, for $a>0$ each sphere p... | https://mathoverflow.net/users/115803 | Volume of a shape whose boundary consists of portions of spheres symmetrically placed about the origin in $d\gg 1$ dimensions | If we set $a = \beta d^2$ for $\beta > 0$ then $$\frac{V(S)}{2^d} \to \exp(-1/(6\beta)).$$
The idea is to choose $x \in [-1,1]^d$ uniformly at random. Then $\frac{V(S)}{2^d}$ is just the probability that $x \in S$.
Let $S\_j^+$ (respectively $S\_j^-$) denote the ball centered at $e\_j a$ (respectively $-e\_j a$) of... | 4 | https://mathoverflow.net/users/69870 | 430958 | 174,522 |
https://mathoverflow.net/questions/430474 | 0 | Let $R$ be a non commutative ring. We will say that an element of $R$ is isolated if it is zero divisor and nothing nonzero annihilates it at the same time on both sides.
Note that there are many classes of rings that do not contain isolated elements.
$\bullet$ If $R$ is an integral ring, it does not contain isolat... | https://mathoverflow.net/users/168671 | Do you know of any indecomposable ring that has no isolated elements and is neither reversible, nor integral, nor nilpotent, nor unitary? | Let $F$ be a field, and let
$$
R:=F\langle x,y\, :\, x^2=xy=y^2=0\rangle.
$$
Notice that $\{1,x,y,yx\}$ is an $F$-basis for $R$ as an $F$-vector space.
This ring is not reversible since $xy=0$ but $yx\neq 0$.
If by "integral" you mean a domain, then this ring is clearly not a domain (since it isn't reversible).
I... | 4 | https://mathoverflow.net/users/3199 | 430959 | 174,523 |
https://mathoverflow.net/questions/430961 | 2 | Let $X := \mathbb{P}^1$, $S\subset X$ a finite set of points, $U := X - S$, and $j : U\rightarrow X$ the inclusion.
Let $F$ be a complex local system on $U$ of rank $r$, and let $F\_0$ be a typical fiber, so $F\_0$ is a complex vector space. On p14 of Katz's book *Rigid local systems*, he says that
$$\chi(X,j\_\*F) =... | https://mathoverflow.net/users/88840 | Formula for the Euler characteristic of a local system on $\mathbb{P}^1$ | The answer to your question at the end is negative. In fact, $h^2(D, j\_\*F)= h^2(D, j\_\* F)=0$. In fact, the cohomology of a sufficiently small disc around a point in any complex variety, with coefficients in any fixed constructible sheaf, vanishes in all positive degrees.
This is because taking global sections on ... | 5 | https://mathoverflow.net/users/18060 | 430962 | 174,524 |
https://mathoverflow.net/questions/430609 | 7 | Let $a\_n$ is a *binomial sum*, for example
$$
a\_n := \sum\_{k} \binom{n-k}{k} \quad \text{or} \quad \sum\_{k=0}^n\binom{n+k}{n-k}\binom{2k}{k}
\quad \text{or} \quad \sum\_{k=0}^n\sum\_{\ell=0}^k\binom{n}{k}\binom{n+k}{k}\binom{k}{\ell}^3
$$
These are [Fibonacci](https://oeis.org/A000045), [Delannoy](https://oeis.org/... | https://mathoverflow.net/users/4040 | Congruences of binomial sums | I found the reference: Theorem 9.1 in Boris Adamczewski and Jason P. Bell, [Diagonalization and Rationalization of algebraic Laurent series](https://arxiv.org/abs/1205.4090), *Ann. Sci. Éc. Norm. Supér.* **46** (2013), 963–1004.
The authors even discuss the *Apéry sequence* modulo 5, right after the theorem.
It's s... | 4 | https://mathoverflow.net/users/4040 | 430966 | 174,527 |
https://mathoverflow.net/questions/430971 | -1 | If $0<\beta<1$ and $0<x<1,$ how to prove that $$h(x)-2x+(4-2^{1+\beta})x^{1+\beta}<0,$$ where $$h(x)=(1+x)^{1+\beta}-x^{1+\beta}-1.$$The numerical simulation shows that it is true.
| https://mathoverflow.net/users/484832 | A proof of an interesting inequality | $\newcommand\be\beta$We have to show that
$$g(x):=(1+x)^{1+\be}+(3-2^{1+\be}) x^{1+\be}-2 x-1<0; \tag{1}\label{1}$$
here and in what follows, $\be$ and $x$ are in $(0,1)$.
Let $r(x):=\dfrac x{1+x}$, so that $r$ is increasing on $(0,1)$.
Hence, $$g''(x)=(1+\be)\be x^{\be-1}(r(x)^{1-\be}+3-2^{1+\be})$$
can only switch ... | 1 | https://mathoverflow.net/users/36721 | 430973 | 174,529 |
https://mathoverflow.net/questions/430997 | 5 | If $G$ is a finite group acting on a finite set $X$, we have Burnside's formula that counts the number of orbits $|X/G|$ as:
$$ |X/G| = \frac1{|G|} \sum\_{g\in G} |X^g|,
$$
with $X^g$ being the set of elements in $X$ fixed by $g$.
Now, consider the finite set $X$ of $\mathbb F\_p$-points of $\mathbb P^1\_{\mathbb F\_... | https://mathoverflow.net/users/409881 | How to make Burnside's formula compatible with point counting for varieties over finite fields? | The issue is that $X(\mathbb{F}\_p)/G$ is not the same thing as $(X/G)(\mathbb{F}\_p)$. A simpler example is to take $p$ odd, $X = \mathbb{A}^1$ and let $S\_2$ act by $\pm 1$. There are $\tfrac{p+1}{2}$ orbits, but the quotient space is $\mathbb{A}^1$ with the quotient map $x \mapsto x^2$. The quotient space has $p$ $\... | 8 | https://mathoverflow.net/users/297 | 430999 | 174,538 |
https://mathoverflow.net/questions/430724 | 3 | $\DeclareMathOperator\ex{ex}$We write $K\_{2,\dots,2}^{(r)}$ to denote the $r$-uniform hypergraph with vertex set $\{1,2\}\times\{1,\dots,r\}$ and hyperedge set $\{(1,1),(1,2)\}\times \{(2,1),(2,2)\} \dots \times \{(r,1),(r,2)\}$.
For integer $n$, the Turan number $\ex(n,K\_{2,\dots,2}^{(r)})$ denotes the maximum num... | https://mathoverflow.net/users/130484 | Bounds for $\mathrm{ex}(n,K_{2,\dots,2}^{(r)})$ | A recent paper by Conlon, Pohoata and Zakharov provides new lower bounds and a survey on the history of the problem.
It seems that the upper bound is the state of the art.
Now we know that for every $r \geq 2$ there are lower bounds better than the one obtained from probabilistic deletion.
In particular for $r =3$ it h... | 2 | https://mathoverflow.net/users/45545 | 431001 | 174,539 |
https://mathoverflow.net/questions/430741 | 16 | Let $F\_i$ denote the $i$th Fibonacci number (with $F\_1=F\_2=1$). Define
$$ P\_n(x) = \prod\_{i=1}^n (1+x^{F\_{i+1}}). $$
Let $\nu\_k(n)$ denote the number of coefficients of the polynomial $P\_n(x)$
that are equal to the positive integer $k$. Evidence suggests that for
sufficiently large $n$ (depending on $k$), $\nu\... | https://mathoverflow.net/users/2807 | Number of coefficients equal to $k$ in certain "Fibonacci polynomials" | I will use the set up of [my answer](https://mathoverflow.net/q/266215) to a previous question about "Fibonacci polynomials".
The key observation is that the coefficient of $x^m$ in $P(x)$ equals the number of "unrollings" of the Zeckendorf representation $Z\_m$ (viewed as a $01$-string) of $m$, where any substring $... | 5 | https://mathoverflow.net/users/7076 | 431002 | 174,540 |
https://mathoverflow.net/questions/430929 | 3 | Let $X$ be a connected and compact $d$-dimensional smooth manifold; where $d$ is a positive integer. Does *(or rather, when does)* there exist a metric $\rho$ on $X$ generating $X$'s topology and a countable number of sets $\{X\_n\}\_{n}$ such that
* $\bigcup\_n\,X\_n = X$,
* Each $X\_n$ is $\rho$-geodesically convex... | https://mathoverflow.net/users/36886 | Partitioning a smooth manifold into geodesically convex sets | Choose a triangulation of $X$.
Let us equip $X$ with a length metric such that each simplex is standard.
We may think that $X$ subcomplex of a standard simplex $S$ of large dimension.
Since each face of $S$ is convex; it follows that each simplex is a convex set in $X$.
Therefore the covering of $X$ by the simplexes ... | 4 | https://mathoverflow.net/users/1441 | 431007 | 174,541 |
https://mathoverflow.net/questions/430953 | 1 | Let $M$ be an $n-m$ dimensional sub-manifold of $\mathbf R^n$ defined by the following set of equations:
\begin{equation}
f\_1(\vec x)=0, \\
\vdots \\
f\_m(\vec x)=0,
\end{equation}
where $\vec x$ are coordinates in $\mathbf R^n$.
Given a point $\vec x\_0$ in $M$, how can the Ricci scalar (calculated from the induced... | https://mathoverflow.net/users/485792 | Ricci scalar of submanifold of $\mathbf R^n$ | *This is an extended comment answering [Ricci scalar of sub-manifold of $\mathbf R^n$](https://mathoverflow.net/questions/430953/ricci-scalar-of-sub-manifold-of-mathbf-rn?noredirect=1#comment1109048_430953)*
Assuming the $f\_i$ are independent, at every point $x$ their gradients span the orthogonal complement to the ... | 2 | https://mathoverflow.net/users/3948 | 431013 | 174,542 |
https://mathoverflow.net/questions/431015 | 3 | Let $Z$ be an object in a stable (or triangulated/whatever) category $\mathcal C$. I believe it follows from Thomason's theorem (see [The classification of triangulated subcategories](https://doi.org/10.1023/A:1017932514274)) that the triangulated categories generated by
$$Y = Z \oplus \Sigma Z$$
and
$$X = Y \oplus \Si... | https://mathoverflow.net/users/2362 | How to construct $X \oplus \Sigma X$ from $X \oplus \Sigma X \oplus \Sigma X \oplus \Sigma^2 X$ without splitting an idempotent? | The cofibre of $0\oplus 1\oplus 1\oplus 1$ on $X=Z\oplus \Sigma Z\oplus\Sigma Z\oplus\Sigma^2Z$ is $Z\oplus\Sigma Z=Y$.
| 10 | https://mathoverflow.net/users/10366 | 431018 | 174,544 |
https://mathoverflow.net/questions/410332 | 8 | Let $r(n):=r\_3(\mathbb{F}\_3^n)=\max\{|A|: A \subset \mathbb{F}\_3^n, \ A \text{ is 3-AP-free}\}$.
[Edel](https://link.springer.com/article/10.1023%2FA%3A1027365901231) proved that $r(n)\geq 2.217^n$ for sufficiently large $n$. His proof is by giving a construction of a cap-set $A$ in $\mathbb{F}\_3^{480}$. Then obs... | https://mathoverflow.net/users/225950 | Known approaches for the lower bound on cap-set problem | I've just proved a new lower bound of $2.218^n$, in my paper 'New lower bounds for cap sets': <https://arxiv.org/abs/2209.10045>.
My new bound comes from extending Edel's ideas, with better computational methods (including a SAT solver) and introducing a new theoretical construction. I also conjecture that a lower bo... | 8 | https://mathoverflow.net/users/490000 | 431023 | 174,547 |
https://mathoverflow.net/questions/430957 | 1 | $\DeclareMathOperator\Ext{Ext}$Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $F,G$ be finitely generated free $R$-modules and $f:F\to G$ be an $R$-linear map such that $f(F)\subseteq \mathfrak m G$. Let $X$ be a finitely generated $R$-module, and let $\alpha : 0\to F \to A\_{\alpha} \to X \to 0$ be a short exac... | https://mathoverflow.net/users/135389 | On image of map $\text{Ext}^1_R(X,F)\to \text{Ext}^1_R(X,G)$ induced by $R$-linear map of free modules $F\to G$ with entries in the maximal ideal | This is true for any $\mathrm{Ext}$-degree (and, in fact, without many hypotheses except that $\mathfrak m$ is finitely generated and $F$ is free).
Let $(x\_1,\dots,x\_n)$ be generators of the maximal ideal $\mathfrak m$. Then there is a surjection $G^n \to \mathfrak m G$ given by
$$(g\_1,\dots,g\_n) \mapsto \sum x\_... | 1 | https://mathoverflow.net/users/360 | 431026 | 174,549 |
https://mathoverflow.net/questions/430807 | 3 | Let $G=(V,E)$ be a finite undirected graph which we equip with its usual graph *geodesic distance* $d\_G$ making $(G,d\_G)$ into a metric space; let $1<\#V<\infty$. For a given $1<N< \#V$ *what conditions do I need* on $G$ so that does there exist *disjoint* subsets $V\_1,\dots,V\_N\subseteq V$ such that
* $\biguplus... | https://mathoverflow.net/users/36886 | "Geodesic coherent" partition of a graph | [Pilipczuk and Siebertz](https://arxiv.org/abs/1807.03683) proved that every planar graph has such a partition with an even stronger property. Namely, each part $V\_i$ is a geodesic path, and the graph obtained by contracting each part has [treewidth](https://en.wikipedia.org/wiki/Treewidth) at most 8. This result was ... | 4 | https://mathoverflow.net/users/2233 | 431028 | 174,551 |
https://mathoverflow.net/questions/431031 | 3 |
>
> Does the process of 'constructing the category of presheaves' always/never stabilize? Does it stabilize for some special class of categories?
>
>
>
That is, work in a foundation that allows for multiple levels of 'categorical largeness' and let $\mathcal{C}$ be a category. Recursively define
1. $\mathcal{C... | https://mathoverflow.net/users/92164 | Stabilization of taking presheaf categories | The question is extremely dependent on how size issues are handled and many choice that can be made, so that it is very hard to give a general answer.
If you really work with general presheaves categories then I cannot think of any interesting functors $C\_i \to C\_{i+1}$ that would define an interesting value at lim... | 11 | https://mathoverflow.net/users/22131 | 431033 | 174,553 |
https://mathoverflow.net/questions/431030 | 6 | The proof that $AC$ is independent of $\sf ZF$ axioms is done by forcing and constructibility, and these don't beg any consistency strength more than that of $\sf ZF$.
>
> Is there a known similar proof of independence of $AC$ from $\sf Z$ that is done at the consistency level of $\sf Z$ itself?
>
>
>
>
> Mo... | https://mathoverflow.net/users/95347 | Is there a proof of independence of AC from Z that is done in Z? | Towards a partial answer:
I have not read it myself yet, but it seems that the relative consistency of $\mathsf{AC}$ with $\mathsf{Z}$ was proved by Mathias in his paper [*The strength of MacLane set theory*](https://www.dpmms.cam.ac.uk/%7Eardm/maclane.pdf). To quote Mathias' summary (available here):
>
> The pap... | 9 | https://mathoverflow.net/users/8133 | 431035 | 174,554 |
https://mathoverflow.net/questions/431038 | 4 | An element $g\in G$ in a group $G$ is called **infinitely divisible** if $b=y^n$ for infinitely many different $n\in {\Bbb Z}$. It is not hard to find a finite CW-complex (or even a compact manifold) with a fundamental group containing an infinitely divisible element. For example, consider a group generated by $x$ and ... | https://mathoverflow.net/users/3377 | Infinitely divisible elements in Gromov hyperbolic groups | YCor has answered your question in the comments.
But here is another proof anyway. The "asymptotic translation lengths" of elements (in a fixed word hyperbolic group) are uniformly rational. Also, the asymptotic translation length of an element is zero if and only if the element has finite order. Thus if an element i... | 6 | https://mathoverflow.net/users/1650 | 431043 | 174,557 |
https://mathoverflow.net/questions/420989 | 4 | As you may know, there has been very recently a big breakthrough concerning upper bounds for the capset problem over $\mathbb{F}\_3^n$ (and further generalizations to $\mathbb{F}\_q^n$). I was wondering which other configurations have been studied so far in this context. For instance, the corner problem has also been s... | https://mathoverflow.net/users/46573 | Extremal problems in additive combinatorics (over finite fields) | The methods of Edel's lower bound, which I have improved in [this paper](https://arxiv.org/abs/2209.10045), are not specific to the setting of $\mathbb{F}\_3^n$ as far as I can see. My result and Edel's both come from considering 3 cap sets $A\_0, A\_1, A\_2 \subseteq \mathbb{F}\_3^6$ and then extending them, but these... | 6 | https://mathoverflow.net/users/490000 | 431049 | 174,559 |
https://mathoverflow.net/questions/431045 | 1 | Let $f:X\to Y$ be a birational morphism of smooth projective variety. We assume that $f(V)\simeq U$ isomorphism induced by $f$, where $V\subset X$ and $U\subset Y$ are two Zariski open sets. Let $x\in V$, $C$ be a curve passing through $x$ in $X$ and $L$ be a line bundle over $Y$. Then is the following true?
$$f^{\*}... | https://mathoverflow.net/users/211682 | Intersection pairing and birational morphisms | First note $\overline{f(C \cap V)} = f(C)$, since $f$ is closed and $C$ (and hence also $f(C)$) is irreducible. Also $f$ induces a birational map $C \to f(C)$, so $f\_\* [C] = [f(C)]$ where $[\cdot]$ denotes rational equivalence classes.
Then by the projection formula¹
$$f\_\* (f^\* L \cdot [C]) = L \cdot f\_\* [C] =... | 2 | https://mathoverflow.net/users/111897 | 431061 | 174,564 |
https://mathoverflow.net/questions/430655 | 3 | I ask the same question on MathStackExchange but receive no answer.
I'm reading Kollár Mori Chapter 2.3. And they state the following lemma:
>
> Lemma 2.29. Let $X$ be a smooth variety and $\Delta = \sum a\_iD\_i$ a sum of distinct prime divisors. Let $Z\subseteq X$ be a closed subvariety of codimension $k$. Let ... | https://mathoverflow.net/users/167083 | Blowing up of a singular subvariety | Possibly the simplest example is to consider the blowup of a reduced and irreducible curve $C$ in a smooth 3-fold $X$ with a point $P\in C\subset X$ which is locally analytically isomorphic to
$$ 0\in \Gamma=\mathbb{V}(xy,yz,zx)\subset \mathbb{A}^3\_{x,y,z} $$
(i.e. in a small neighbourhood of $P\in X$, the curve $C$ l... | 3 | https://mathoverflow.net/users/104695 | 431068 | 174,566 |
https://mathoverflow.net/questions/431062 | 3 | Let $h \in C^2\_{\mathrm{ub}}(\mathbb{R}^{2n})$, where $C\_{\mathrm{ub}}^k$ consists of $C^k$-functions that are bounded and uniformly continuous along with their derivatives up to $k$th-order.
It is clear that the Hamiltonian vector field $X$ is $C^1$ and globally Lipschitz, hence the Hamiltonian flow $\Phi\_t$ exis... | https://mathoverflow.net/users/485160 | Uniform continuity of Hamiltonian flow | Denote by $L$ the Lipschitz constant of the Hamiltonian vector field $\mathfrak{X}$ and by $\varphi\_t$ the flow generated by $\mathfrak{X}$. Then for any $x, y \in \mathbb{R}^{2n}$, by the chain rule, \begin{align\*}
& |\varphi\_t(x) - \varphi\_t(y)|^2 = \\
& \quad |x - y|^2 + 2 \int\_0^t \langle \mathfrak{X}(\varphi\... | 3 | https://mathoverflow.net/users/64449 | 431069 | 174,567 |
https://mathoverflow.net/questions/431091 | 4 | The Wikipedia article on Bass-Serre theory claims that graphs of groups (in the context of Bass-Serre theory) "can be viewed as one dimensional versions of orbifolds." I hazily see a connection between a graph of groups and the notion of an orbifold, but I have no concrete sense of what the connection is nor how it is ... | https://mathoverflow.net/users/163850 | In what sense is Bass-Serre theory the one-dimensional version of orbifold theory | A graph is a $1$-dimensional manifold with singularities and a graph of groups is, roughly, a $1$-dimensional orbifold with singularities. Every graph of groups has a [Bass-Serre covering tree](https://en.wikipedia.org/wiki/Bass%E2%80%93Serre_theory#Bass%E2%80%93Serre_covering_trees) which is its universal cover as an ... | 4 | https://mathoverflow.net/users/290 | 431095 | 174,573 |
https://mathoverflow.net/questions/431047 | 10 | Let $\mathcal P\_n$ be the set of trigonometric polynomials of degree less than or equal to $n$ and let $\lVert\cdot\rVert\_\infty$ be the supremum norm. The error of the best approximation of $f$ of degree $n$ is defined as
$$e\_n(f)=\inf\_{p\in\mathcal P\_n}\lVert f-p\rVert\_\infty.$$
A theorem of Bernstein says ... | https://mathoverflow.net/users/210944 | A function is of bounded variation if and only if the errors of its best approximation by trigonometric polynomials satisfy $\sum\frac{e_n}n<\infty$? | This is not true. If $f(x)=x \sin( \frac 1x)$ near the origin, then $f$ is $1/2$ Holder continuos and $e\_n(f) \approx \frac{1}{\sqrt n}$ by Jackson theorem, so that the series $\sum\_n \frac{e\_n(f)}{n}$ converges. But is not of bounded variation, since $f' \not \in L^1$.
EDIT
Also the converse does not hold. Take... | 5 | https://mathoverflow.net/users/150653 | 431098 | 174,574 |
https://mathoverflow.net/questions/431083 | 31 | Given $\ell\ge 1$, we say a graph $G$ is $\ell$-good if for each $u,v\in G$ (not necessarily distinct), the number of walks of length $\ell$ from $u$ to $v$ is odd. We say a graph $G$ is good if it is $\ell$-good for some $\ell\ge 1$.
Do good graphs exist? For clarity, I am only talking about simple graphs (which lac... | https://mathoverflow.net/users/130484 | Do graphs with an odd number of walks of length $\ell$ between any two vertices exist? | A graph without loops cannot be good.
Assume the contrary, let $G$ have $n$ vertices and be good.
Let $A$ be the adjacency matrix of $G$, let $\lambda\_1,\ldots,\lambda\_n$ be its eigenvalues over some extension of $\mathbb{F}\_2$. We have $\sum\_{i=1}^n \lambda\_i=\mathrm{tr} A=0$.
That $A$ is good means that $A... | 37 | https://mathoverflow.net/users/3106 | 431102 | 174,576 |
https://mathoverflow.net/questions/428023 | 3 | My question concerns the proof of Proposition 4.2 in Bhatt-Mathew’s paper on the arc-topology, but my confusion is completely general and anyone familiar with limits in $\infty$-categories would know what to do. The situation is that they have a map of $R$-algebras $V\to \tilde V:=V\_{\mathfrak p}\times V/\mathfrak p$ ... | https://mathoverflow.net/users/37110 | How to simplify this homotopy totalization coming from an arc-cover into a pullback? | Let $X$, $Y$, and $Z$ be Kan complexes. We wish to show that
$X\times\_YZ$ in $\mathrm{Spc}$ can be computed as the limit of the diagram
$$(\*)\qquad X\times Z\rightrightarrows X\times Y\times Y\times Z\substack{\rightarrow\\[-1em] \rightarrow \\[-1em] \rightarrow}X\times Y^{\times 6}\times Z\substack{\rightarrow\\[-1e... | 0 | https://mathoverflow.net/users/37110 | 431107 | 174,578 |
https://mathoverflow.net/questions/430220 | 2 | A Markov Chain $M$ has only one stationary distribution $q$.
For distribution $p$, with $D\_{TV}(p,Mp)=x$, can we bound $D\_{TV}(p,q)$?
Clearly, $x=0$ implies $D\_{TV}(p,q)=0$. Does general bound hold?
We may write $q=\lim\_{n\mapsto \infty} \frac{p+Mp+M^2p+\cdots}{n}$.
| https://mathoverflow.net/users/4987 | On the distance to the stationary distribution | It is impossible to bound $D\_{TV}(p,q)$ in terms of $x=D\_{TV}(p,Mp)$ without further assumptions on the chain, like expansion. This is due to the phenomenon known as **metastability**.
Rich examples are discussed in [1], [2] and [3], for instance.
The simplest example is a chain on two states $a,b$ with transitio... | 4 | https://mathoverflow.net/users/7691 | 431108 | 174,579 |
https://mathoverflow.net/questions/431113 | 3 | I'm looking into matrix representations of Hurwitz groups, beginning with 2x2 matrices. There are many representations with finite characteristic, namely the ${\rm PSL}(2,p^n)$ groups, where $n=1$ if $p=7$ or $p=1(\operatorname{mod} 7)$, and $n=3$ otherwise. However, what about in characteristic zero?
If we define ma... | https://mathoverflow.net/users/38744 | Relators of the "most general" 2x2 matrix Hurwitz group | I think the answer to your question, is yes, this is a faithful projective representation of the triangle group $G = \langle a,b \mid a^2,b^3,(ab)^7 \rangle$, although I have only a hazy understanding of the details.
In Corollary 3.2 of [this paper by Plesken](https://www.sciencedirect.com/science/article/pii/S002186... | 3 | https://mathoverflow.net/users/35840 | 431125 | 174,584 |
https://mathoverflow.net/questions/428243 | 7 | This post is based on <https://math.stackexchange.com/questions/2822589/dissect-square-into-triangles-of-same-perimeter>, [On a possible variant of Monsky's theorem](https://mathoverflow.net/questions/394721/on-a-possible-variant-of-monskys-theorem) and [Cutting convex polygons into triangles of same diameter](https://... | https://mathoverflow.net/users/142600 | Partitioning convex polygons into triangles of equal area and perimeter | Question 1 indeed has a negative answer, as previous responders have speculated. See e.g. the 2008 paper by Monsky and Jepsen:
[Constructing Equidissections for Certain Classes of Trapezoids"](https://doi.org/10.1016/j.disc.2007.10.031), Discrete Mathematics, 308 (23): 5672–5681, doi:10.1016/j.disc.2007.10.031, Zbl 1... | 2 | https://mathoverflow.net/users/167341 | 431126 | 174,585 |
https://mathoverflow.net/questions/431104 | 2 | Recall a set of integers $S$ is said to be an additive basis for the natural numbers if there is a $k$ such that every positive integer is expressible as a sum of at most $k$ elements of $S$. Similarly, a set $S$ is said to be an asymptotic additive basis for the natural numbers if there is a $k$ such that every suffic... | https://mathoverflow.net/users/127690 | Additive basis of a set union the square of the set | In the paper "On additive bases. II", Deshouillers and Fouvry prove a conjecture (made in part I, by a different set of authors) that for each sequence $K$ of positive integers, there is a set $A$ such that $A^k$ is a basis precisely when $k$ belongs to $K$. See
J. London Math. Soc. (2) 14 (1976), no. 3, 413–422.
In ... | 6 | https://mathoverflow.net/users/16510 | 431145 | 174,590 |
https://mathoverflow.net/questions/431155 | 3 | Let $A\subset \mathbb R$. Is it true that
$$
\dim(A+A)\le 2\dim A
$$
for some dimensions – say, lower box for the LHS and upper box for the RHS.
| https://mathoverflow.net/users/8131 | Dimension of sumset vs sum of dimensions | $A+A$ is a Lipschitz image of the set $A\times A\subset \mathbb{R}^2$ under the map $(x,y)\to x+y$. If $A$ is covered by $N$ balls of radius $\varepsilon$, then $A\times A$ is covered by $N^2$ balls of radius, say, $\sqrt{2}\varepsilon$, thus the box dimension (lower or upper) of $A\times A$ does not exceed twice that ... | 5 | https://mathoverflow.net/users/4312 | 431156 | 174,591 |
https://mathoverflow.net/questions/431065 | 2 | Let $A\_n$ be a sequence of $d \times d$ matrices converging to a matrix $A$, all invertible and diagonalizable. We can define the Lyapunov spectrum of the corresponding dynamical system:
$$ \chi = \big\{ \lim\_{n \to \infty} \frac1n \log\|A^{(n)}v\| :\ v \in \Bbb R^d \big\}$$
where $A^{(n)} = A\_n A\_{n-1} A\_{n-2} ... | https://mathoverflow.net/users/380456 | Lyapunov exponents of convergent sequence of matrices | So the answer is yes. A brief version is that it suffices to check this for the top exponent and then to use exterior powers to deduce the result for subsequent exponents.
(**Slightly modified to make $\lambda$ the leading Lyapunov exponent**)
Next, if $E$ is the sum of the generalized eigenspaces of all the eigenv... | 1 | https://mathoverflow.net/users/11054 | 431161 | 174,592 |
https://mathoverflow.net/questions/431167 | 10 | Let $K$ be a number field. Let $\mathcal{O}\_K^{\*}$ be the units in the ring of integers of this field.
I am interested in knowing how many units $u,v \in \mathcal{O}\_K^{\*}$ exist such that $u + v$ is also a unit. This is equivalent to knowing units $u$ such that $1+u$ is also a unit.
In particular, I'm interest... | https://mathoverflow.net/users/94546 | When are sums of units in a number field also units? | I only want to add to Gerry's comment that it is a well-known result by Siegel that the number of exceptional units for a fixed number field is finite. There are even bounds on this number by Evertse, which only depend on the degree of the field extension. There also exist examples of number fields (by Triantafillou) w... | 11 | https://mathoverflow.net/users/155336 | 431169 | 174,593 |
https://mathoverflow.net/questions/431172 | 2 | Could anyone please recommend a known website where I can find a database/library that has systems of polynomial equations with $n$ variables and $m$ parameters?
I need some real examples to test my elimination algorithm.
Systems with more variables than equations will also do.
Thanks
| https://mathoverflow.net/users/491824 | Library/Database of parametric polynomial systems | There is a [database of polynomial systems](http://homepages.math.uic.edu/%7Ejan/demo.html), which comes with [PHCpack](http://homepages.math.uic.edu/%7Ejan/PHCpack/phcpack.html) by Jan Verschelde.
| 3 | https://mathoverflow.net/users/7076 | 431187 | 174,599 |
https://mathoverflow.net/questions/431188 | 6 | Let
$$
H\_\lambda=-\frac{d^2}{dx^2}+\lambda^2 x^2,\quad\lambda>0.
$$ It is known that the spectrum of $H\_\lambda$ is the set $\{(2n-1)\lambda,n\in \Bbb N^\*\}$. Now put
$$
(U\_\mu \phi)(x)= e^{\mu\over 2}\phi (e^{\mu}x)\mu \in \Bbb R.
$$
It is easy to check that $\{U\_\mu,\mu\in\Bbb R\}$ forms a one-parameter unitary ... | https://mathoverflow.net/users/172078 | Spectrum of the complex harmonic oscilllator | Indeed, this is the result of [Davies - Pseudo-Spectra, the Harmonic Oscillator and Complex Resonances](https://www.jstor.org/stable/53393) (1982): The resolvent operator $(H-zI)^{-1}$ of
$$H=-d^2/dx^2+cx^2,\;\;\operatorname{Re}c>0,\;\; \operatorname{Im}c>0,$$
is compact for all $z$ not in the spectrum consisting of th... | 8 | https://mathoverflow.net/users/11260 | 431191 | 174,600 |
https://mathoverflow.net/questions/431117 | 19 | Is there a good generalisation of Laurent series for several complex variables?
I am interested in generalised power series that have some terms with negative powers, but not too many. In single variable complex analysis, "not too many" means that the (Laurent) series has only a finite number of terms with a negative... | https://mathoverflow.net/users/175280 | Laurent series in several complex variables | The easiest way to deal with series like $\sum\_{n=0}^\infty z^n w^{-n}$ is with iterated Laurent series. This series is an element of the ring $\mathbb{Z}((w))[[z]]$: power series in $z$ whose coefficients are Laurent series in $w$. (In this case Laurent polynomials in $w$ would suffice.)
A much more general, though... | 16 | https://mathoverflow.net/users/10744 | 431192 | 174,601 |
https://mathoverflow.net/questions/431197 | 2 | Suppose $x\in SG(\sigma^2)$ is a sub-Gaussian random vector, i.e.
$\left<u,x\right>\quad \forall u\in \mathbb{S}^{n-1}$ is a sub-Gaussian random variable.
My question is : what condition on the random matrix $A$ can guarantee that $Ax$ is again a sub-Gaussian random vector?
I know that $\|A\|\in L^{\infty}$ is on... | https://mathoverflow.net/users/491840 | What condition on random matrix can preserve sub-Gaussian property? | If you only have the hypothesis of sub-Gaussianity, this is the best you can do. Work in dimension $n=1$ for simplicity, let $X\sim N(0,1)$, and let $A$ be independent of $X$. If $AX$ is to be sub-Gaussian, the [Laplace transform condition](https://en.wikipedia.org/wiki/Sub-Gaussian_distribution) will demand
$$
\math... | 1 | https://mathoverflow.net/users/99418 | 431203 | 174,605 |
https://mathoverflow.net/questions/431202 | 13 | As far as I understand, the cobordism hypothesis provides a construction of all (appropriately defined) fully-extended TQFTs. In particular, given a fully-dualizable object in a certain category, one can in principle construct the entire TQFT (e.g. the partition function on an arbitrary closed manifold of full dimensio... | https://mathoverflow.net/users/491848 | Practical consequences of the geometric cobordism hypothesis |
>
> My question is: does this lead to a more-or-less explicit construction of any non-trivial quantum field theories? If so, this would be extremely interesting since only a handful of interacting quantum field theories have been constructed in more than two dimensions, and none in more than three dimensions.
>
>
>... | 8 | https://mathoverflow.net/users/402 | 431206 | 174,606 |
https://mathoverflow.net/questions/431205 | 2 | Let $T$ be a ring with involution $s:T\rightarrow T$. And let
$$h:T\otimes T^\text{op} \rightarrow T\otimes T^\text{op}$$ be the ring automorphism given by $h(a\otimes b)=s(b)\otimes s(a)$.
**suppose that $$ K\_{0}(h): K\_{0}(T\otimes T^\text{op})\rightarrow K\_{0}(T\otimes T^\text{op}) $$ is the identity map.**
I ... | https://mathoverflow.net/users/165456 | Induced map in k-theory by an involution | I don't think so: Let us take $T=\mathbb{Z}[x^{\pm 1}]$, and let us take $s$ the identity. Then $T\otimes T^{op}= \mathbb{Z}[x^{\pm 1},y^{\pm 1}]$ has $K\_0 = \mathbb{Z}$ detected by rank (by Grothendieck-Serre), so $K\_0(h)$ is the identity. But I think in $K\_1$ the elements given by the units $x$ and $y$ are differe... | 1 | https://mathoverflow.net/users/39747 | 431210 | 174,607 |
https://mathoverflow.net/questions/430761 | 2 | $\DeclareMathOperator\SL{SL}$Fix an integer $p\geq 1$ and a cocompact lattice $\Gamma\subset \SL(p+1,\mathbb{R})$. Consider the manifold
$$
M\_{\Gamma}:=\SL(p+1,\mathbb{R})/\Gamma.
$$
Let $A\subset \SL(p+1,\mathbb{R})$ be the subgroup of diagonal matrices with positive entries. The action
$$
A\curvearrowright M\_{\Gamm... | https://mathoverflow.net/users/150945 | Is the orbit foliation of the Weyl chamber flow Riemannian? | These foliations are very far from being Riemannian.
Consider the case $p=1$, and take $PSL(2,\mathbb R)$ instead of $SL(2,\mathbb R)$ (just to simplify a bit). I'll explain how to construct an example where one non-compact leaf converges to a compact one (a circle). Such behavior implies that the foliation is not Ri... | 2 | https://mathoverflow.net/users/943 | 431213 | 174,610 |
https://mathoverflow.net/questions/431209 | 3 | Let $V$ be an elementary abelian $p$-group of size $p^n$. Let $G$ be a finite group with $V\unlhd G$ such that $G/V=H$ is simple (like $\operatorname{PSL}(m,q)$ with $q$ a power of $p$ or any other finite simple group of Lie type in characteristic $p$). Moreover, $V$ is a faithful irreducible $H$-module (equivalently, ... | https://mathoverflow.net/users/64643 | Extensions of a simple group by an elementary abelian $p$-group | I think this can often fail even for the non-trivial representation $V$ of smallest possible dimension.
For a reductive group $G$ over $\mathbf{Z}\_p$, the group $G(\mathbf{Z}/p^2 \mathbf{Z})$ will typically be a non-split extension of $G(\mathbf{F}\_p)$ by the adjoint representation $V$, and for various $G$ the adjo... | 7 | https://mathoverflow.net/users/491858 | 431217 | 174,611 |
https://mathoverflow.net/questions/425117 | 4 | $\DeclareMathOperator\SU{SU}\DeclareMathOperator\SL{SL}$What are the maximal closed subgroups of $ \SU\_3 $?
This question is inspired by [Lie subgroups of SU(3)](https://mathoverflow.net/questions/65522/lie-subgroups-of-su3). Interesting partial answers to that question, treating only the case of connected subgroups... | https://mathoverflow.net/users/387190 | What are the maximal closed subgroups of $ \operatorname{SU}_3 $? | $\DeclareMathOperator\SU{SU}\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\Cl{Cl}$Yes the the above is the correct list of maximal closed subgroups of $ \SU\_3 $.
[Antoneli, Forger, and Gaviria - Maximal Subgroups of Compact Lie Groups](https://arxiv.org/abs/math/0605784)
classifies all ma... | 5 | https://mathoverflow.net/users/387190 | 431221 | 174,613 |
https://mathoverflow.net/questions/431177 | 5 | Consider a complete smooth Riemannian manifold $(M,g)$.
I think that it is not difficult to prove that the $k$ Hodge Laplacian is essentially selfadjoint in the relevant $L^2$ space of $k$ forms, when defining it on the smooth compactly supported $k$ forms.
The standard argument by Chernoff should apply to this case,... | https://mathoverflow.net/users/45539 | Properties of non-integer powers of the Hodge Laplacian | This is a nice question. I think there is a trick that makes it obvious enough not to need a special reference. First, if both sides are to be applied to $k+1$-forms, to be consistent your desired identity needs to be
$$
\delta\_{k+1} (\Delta\_{k+1})^{\alpha} = (\Delta\_k)^{\alpha} \delta\_{k+1} . \tag{1}
$$
The tri... | 2 | https://mathoverflow.net/users/2622 | 431222 | 174,614 |
https://mathoverflow.net/questions/430745 | 7 | What is the origin of the abacus bijection (aka the rim hook bijection, aka the Stanton-White bijection, aka James's bijection)?
Igor Pak, in his 2000 article "Ribbon tile invariants" (Transactions of the American Mathematical Society, volume 352 (2000), pages 5525-5561), summarizes the situation thus: "The theorem g... | https://mathoverflow.net/users/3621 | Origin of the abacus bijection | After looking at Robinson's book, my coauthors and I settled upon the following wording:
The version of the bijection that we use is due to Gordon James (see \cite{JamesKerber}) but different forms of it seem to have been discovered independently by various people working in the field of modular representation theory... | 4 | https://mathoverflow.net/users/3621 | 431225 | 174,615 |
https://mathoverflow.net/questions/431219 | 7 | Let $\*$ be a binary operation on a set $M$, with an identity element $e\in M$.
A *monoid representation* of $(M,\*,e)$ is a map $\delta:M\to (S\to S)$ for some set $S$, such that $\delta(e)=\mathrm{id}\_S$, and $\delta(a\*b)=\delta(a)\circ\delta(b)$ for all $a,b\in M$. (A representation could also be called an *acti... | https://mathoverflow.net/users/4600 | Which monoids have a faithful irreducible representation? | **Cleaner rewrite:**
I have a bit more time, so here is a cleaner rewrite. This notion is usually called transitive rather than irreducible, although the terms irreducible and minimal are both used.
If $M$ has a minimal left ideal $L$,then $L$ is a transitive $M$-set and every transitive action is a quotient of $L$... | 6 | https://mathoverflow.net/users/15934 | 431227 | 174,616 |
https://mathoverflow.net/questions/431224 | -3 | I am reading
* imenez, J., Echevarria, J.I., Sousa, T. and Gutierrez, D. (2012), *SMAA: Enhanced Subpixel Morphological Antialiasing* Computer Graphics Forum, 31: 355-364. <https://doi.org/10.1111/j.1467-8659.2012.03014.x>
where I have encountered these three equations
\begin{gather\*}
e\_l=\lvert L-L\_l\rvert>T ... | https://mathoverflow.net/users/491860 | Can you do boolean and of 1 and a number less than 1? | The paragraph before says
>
> We calculate the maximum contrast $c\_{max}$ for all these edges and compare it with the contrast for the left edge [this is $c\_l$]. If the latter is above a threshold of $0.5\cdot c\_{max}$ the edge is preserved; otherwise, it is ignored.
>
>
>
So this reads to me like a conditi... | 2 | https://mathoverflow.net/users/4177 | 431230 | 174,618 |
https://mathoverflow.net/questions/431215 | 3 | Let $d \ge 2$ be a positive integer. For $x=(x\_1,\dotsc,x\_{d-1},x\_d)$, we write $x'=(x\_1,\dotsc,x\_{d-1})$. Let $\mathbb{H}^d=\{x=(x',x\_d) \mid x\_d>0\}$ denote the $d$-dimensional upper half-space.
Then, we obtain the following result.
>
> Let $u \in C^2\_c(\mathbb{R}^{d-1})$, $v \in C\_c^1(\mathbb{R}^{d-1}... | https://mathoverflow.net/users/68463 | Boundedness of an extension operator | Let me give an estimate for the first derivatives of $v:=P\_1 h$ in terms of the sup-norm of $h$.
Setting $z=x'+x\_d \xi$ we get
$$v(x)=\int\_{\mathbb {R}^{d-1}}\phi \left (\frac{z-x'}{x\_d}\right )h(z) x\_d^{2-d} \, dz.
$$
Then
$$v\_{x\_d}(x)=(2-d)\int\_{\mathbb {R}^{d-1}}\phi \left (\frac{z-x'}{x\_d}\right )h(z) x\_d... | 2 | https://mathoverflow.net/users/150653 | 431266 | 174,625 |
https://mathoverflow.net/questions/431251 | 1 | Let $G$ be a profinite topological group with two closed subgroup $G\_1$ and $G\_2$. Suppose $G\_1$ is normal in $G$ and $G=G\_1G\_2$. Let $H\_i$ be an open subgroup in $G\_i$ for $i=1,2$.
**Question:** Is $ H\_1H\_2:=\{h\_1h\_2\mid h\_1\in H\_1 \text{ and } h\_2\in H\_2\}$ also open in $G$?
| https://mathoverflow.net/users/11750 | Openness of product of two open subgroups | Yes (even without assuming $G\_1$ normal).
Indeed, first, since $G\_1G\_2=G$, for some finite subset $F$ we have $FH\_1H\_2F=G$, i.e., finitely many left-right translates of the compact subset $H\_1H\_2$ cover $G$. Hence (Baire) $H\_1H\_2$ has nonempty interior.
Next, $H\_1\times H\_2$ left-right acts on $G$ and $H... | 5 | https://mathoverflow.net/users/14094 | 431267 | 174,626 |
https://mathoverflow.net/questions/431275 | 6 | Consider the surface $S\_{\epsilon}$ defined as:
\begin{align}
%S&=\{\vec x \in \mathbf{R}^3: x=0\}, \\
S\_{\epsilon}&=\{\vec x \in \mathbf{R}^3:f\_{\epsilon}(\vec x)\equiv\epsilon ((x^2 + y^2 - 4)^2 + z^2 - 1) + x=0\}.
\end{align}
The topology of $S\_{\epsilon}$ is $\mathbf S^2$ when $\epsilon$ is small but non-zero, ... | https://mathoverflow.net/users/485792 | Topology change induced by small perturbation | The formal statement you are thinking of when you assert "The topology changes only when..." is Ehresmann's theorem: a proper smooth submersion is a fiber bundle, and hence all fibers are diffeomorphic. Here "proper" means that the inverse image of any compact set is compact. It is a useful fact that proper maps are cl... | 10 | https://mathoverflow.net/users/40804 | 431276 | 174,627 |
https://mathoverflow.net/questions/431248 | 3 | Consider a long cylinder $C = D \times (-L,L) \subset \mathbf{R}^3$, with heat applied to its horizontal boundary according to $\varphi$ and perfectly insulated ends. The steady state $u: C \to \mathbf{R}$ representing the temperature is the solution of the system
\begin{equation}
\begin{cases}
\Delta u = 0 \quad \text... | https://mathoverflow.net/users/103792 | Heating a long cylinder: steady states | $\newcommand{\R}{\mathbb R}$Sorry for being too sketchy in the following answer, time permitting, I'll try to expand.
---
*Step 0.* Some more-or-less classical potential theory. Let $D$ be an open set in $\R^d$ (with $d \geqslant 3$ for simplicity), and assume that $D$ is sufficiently regular (for example, Lipsch... | 5 | https://mathoverflow.net/users/108637 | 431285 | 174,630 |
https://mathoverflow.net/questions/429291 | 8 | *Throughout, I work in $\mathsf{MK}$ in order to be able to conveniently quantify over logics; if one prefers, we can restrict attention to (say) $\Sigma\_{17}$-definable logics and work in $\mathsf{ZFC}$. By "logic" I mean "regular logic containing $\mathsf{FOL}$ and having countably many formulas in a finite language... | https://mathoverflow.net/users/8133 | On the strength of higher-logic analogues of $\mathsf{ZFC}$ + Montague's Reflection Principle | The first chromatic cardinal is the first Mahlo cardinal. (Per the connection with the reflection in the question, I assume that in the definition, $α$ and the first argument of $c\_i$ need not be inaccessible.) If we allowed $c\_i$ for $i≤δ$, then the first chromatic cardinal would be the first Mahlo above $δ$.
If $... | 3 | https://mathoverflow.net/users/113213 | 431287 | 174,632 |
https://mathoverflow.net/questions/431277 | 5 | I need a general method for solving systems of logical equations like:
$$
\begin{equation\*}
\begin{cases}
c\_{0} = a\_{0} \land b\_{0}\\\\
c\_{1} = a\_{0} \land b\_{1} ⊕ a\_{1} \land b\_{0}\\\\
c\_{2} = a\_{0} \land b\_{2} ⊕ a\_{1} \land b\_{1} ⊕ a\_{2} \land b\_{0}\\\\
c\_{3} = a\_{1} \land b\_{2} ⊕ a\_{2} \l... | https://mathoverflow.net/users/491919 | Solve system of logical equations | Let $A=a\_0+a\_1x+a\_2x^2\cdots+a\_nx^n$, $B=b\_0+b\_1x+\cdots+b\_mx^m$, and $C=AB=c\_0+\cdots+c\_{n+m}x^{n+m}$, where arithmetic occurs over $\mathbb{F}\_2$. Then your problem is exactly equivalent to recovering $\{A,B\}$ from $C$. This is the problem of [factorization of polynomials over finite fields](https://en.wik... | 9 | https://mathoverflow.net/users/8938 | 431289 | 174,633 |
https://mathoverflow.net/questions/431299 | 0 | Consider a metric space $(X,d)$ and let $\kappa$ be a cardinal. We say that $(X,d)$ is $\kappa$-**homogenous**, if every (surjective) isometry $h:X\_1 \to X\_2$ between subspaces of $(X,d)$ of size $< \kappa$ extends into an automorphism of $(X,d)$, i.e., into an isometry $f:(X,d) \to (X,d)$ such that $f\upharpoonright... | https://mathoverflow.net/users/322046 | $\omega$-homogenous space which is not $\omega_1$-homogenous | No: if $X$ compact metric is $\omega$-homogeneous then it is $\omega\_1$-homogeneous.
Indeed let $f:X\_1\to X\_2$ be a bijective isometry, with $X\_1$ countable. Write $X\_1$ as increasing union of finite subsets $F\_n$. So $f\_{|F\_n}$ extends to a bijective isometry $g\_n$ of $X$. Since the isometry group of $X$ is... | 1 | https://mathoverflow.net/users/14094 | 431309 | 174,638 |
https://mathoverflow.net/questions/431318 | 6 | I've been trying to understand the (4 line!) proof of Lemma 2.3 of [Limits of small functors](https://arxiv.org/abs/math/0610439), on small functors into copresheaf categories $\mathbf{Set}^\mathcal C$. To me it seems to be using that the functor $\mathbf{Set}^\mathcal C \to \mathbf{Set}^{\text{ob } \mathcal C}$, defin... | https://mathoverflow.net/users/133974 | Day and Lack's "Limits of small functors": Lemma 2.3 | Small and nicely small are indeed equivalent. I would consider this a fairly classical observation in the topic of small functor, at least when $M = Set$, so I wouldn't be surprised that Day and Lack are using it implicitly - maybe without realizing it - but I haven't had time to go re-read their paper in details to an... | 8 | https://mathoverflow.net/users/22131 | 431323 | 174,642 |
https://mathoverflow.net/questions/431320 | 4 | Suppose we are in the following situation: $(X,d)$ is a metric space and $Y$ is a subspace of $X$. Furthermore we have a different metric $\delta$ defined on $Y$ such that $\delta$ is bi Lipschitz equivalent to $d|\_Y$. Is it possible to extend $\delta$ to e metric $\bar{\delta}$ on the whole $X$ such that $\bar{\delta... | https://mathoverflow.net/users/153260 | Extending a metric in a bi-Lipschitz way | Yes. Up to multiply $d$ with a scalar, we can suppose that for some $0<c\le 1$ we have $cd \le\delta\le d$ on $Y\times Y$.
Define $$d'(x,x')=\min(d(x,x'),D(x,x'));\quad \text{where}$$ $$D(x,x')=\inf\_{y,y'\in Y} d(x,y)+\delta(y,y')+d(y',x').$$
Clearly $cd\le d'\le d$ on $X\times X$. It remains to prove the triangle... | 4 | https://mathoverflow.net/users/14094 | 431324 | 174,643 |
https://mathoverflow.net/questions/431185 | 0 | **Setup:$\quad$**
Suppose that $(X\_n)$ is a stationary ergodic process with $E|X\_1|<\infty$.
Given $X^{(n)}=(X\_1, \dots, X\_n)$, select a standard Efron bootstrap subsample $(X\_{n,1}^\*, \dots, X\_{n,m(n)}^\*)$ by pulling $m(n)$ times with replacement from a uniform distribution $U(\{X\_1, \dots, X\_n\})$, i.e.... | https://mathoverflow.net/users/104268 | WLLN for bootstrap means of stationary ergodic processes? | **Answered in comments above**
It seems as though the answer should be yes. I would suggest writing $X\_n$ as $Y\_n+Z\_n$ where $Y\_n$ is $X\_n$ if $|X\_n|\le m(n)^{1/3}$ and 0 otherwise; similarly $Z\_n$ is $X\_n$ if $|X\_n|>m(n)^{1/3}$ and 0 otherwise. Then the Einmal and Rosalsky result applies to the Bootstrap av... | 0 | https://mathoverflow.net/users/11054 | 431335 | 174,644 |
https://mathoverflow.net/questions/431313 | 2 | Let $S$ be a hyperbolic surface of genus $g \geq 2$.
A *discrete* geodesic lamination on $S$ is a set of disjoint, simple, closed geodesics.
Let $L\_{1}$ and $L\_{2}$ be two discrete geodesic laminations on $S$. We say that $L\_1$ and $L\_2$ fill $S$, if $S \setminus (L\_{1} \cup L\_{2})$ is a disjoint union of open ... | https://mathoverflow.net/users/169634 | Pair of laminations that fill on a closed surface | The answer is "no". For consider the case where $L\_1 = L\_2$ is a single simple closed geodesic.
---
Now you've added the hypothesis that $L\_1$ and $L\_2$ have no common leaf, the answer becomes "yes". Here is a sketch.
Let $X\_i = S - L\_i$. If $X\_1$ (say) is a union of pants then $L\_1$ is a pants decompos... | 2 | https://mathoverflow.net/users/1650 | 431340 | 174,645 |
https://mathoverflow.net/questions/405935 | 3 | The paper 'Trisections, intersection forms and the Torelli group' by Peter Lambert-Cole quotes the following formula for the Casson invariant of a knot $K$ in a homology $3$-sphere in terms of the linking form $l$ on a Seifert surface $\Sigma$ for $K$ of genus $g$:
$$
\lambda'(K)=\sum\_{i=1}^g \big(l(a\_i,a\_i)l(b\_i,b... | https://mathoverflow.net/users/156392 | Formula for the Casson invariant in terms of the linking form | Maybe you've already found a source or this is irrelevant by now, but I believe I took it from Morita's paper "Casson's invariant for homology 3-spheres and characteristic classes of surface bundles I". Specifically Proposition 3.2.
| 3 | https://mathoverflow.net/users/126853 | 431345 | 174,647 |
https://mathoverflow.net/questions/430869 | 2 | Who first used the corner quotes, $\ulcorner$ and $\urcorner$, for the notion of Gödel number? They can also be written as`\Godelnum` with Sam Buss's macro.
They were used by Joseph R. Shoenfield, in *Mathematical Logic*, 1967, as from page 122.
The corner quotes are used prevalently in provability logic, and in ot... | https://mathoverflow.net/users/37385 | First use of corner quotes for Gödel numbers | While this is not a full answer, I hope the following observations can still be of some use.
Kreisel and Lévy were using corner quotes, explicitly for Gödel numbers, around the same time as Schoenfield (G. Kreisel and A. Lévy, [Reflection Principles and their Use for Establishing the Complexity of Axiomatic Systems](... | 4 | https://mathoverflow.net/users/103227 | 431347 | 174,648 |
https://mathoverflow.net/questions/431331 | 5 | Let $(P,\leq)$ be a directed set with uncountable cofinality. For every element $p\in P$, we are given a finite set $c\_p\subset P\smallsetminus \{p\}$ of "incompatible elements". We say that a subset $Q\subseteq P$ is *compatible* if for every $q\in Q$ we have $c\_q\cap Q = \emptyset$.
**Question:** is it always pos... | https://mathoverflow.net/users/54309 | Searching for cofinal subsets of directed sets subject to finite constraints | No, it is not always possible. For a counterexample, let $P$ be the set of all finite subsets of some uncountable set $X$, ordered by inclusion (that is, $a \leq b \Leftrightarrow a \subseteq b$).
In this poset, a subset $D$ of $P$ is cofinal if and only if for every finite subset $a$ of $X$, there is some $b \in D$ wi... | 4 | https://mathoverflow.net/users/70618 | 431353 | 174,649 |
https://mathoverflow.net/questions/431152 | 6 | A characteristic feature of [Berkovich spaces](https://en.wikipedia.org/wiki/Berkovich_space) is that they are locally connected (in fact, locally contractible). I'd like to understand the proof. The key ingredient seems to be Corollary 2.2.8 in Berkovich's [book](https://www.google.com/books/edition/Spectral_Theory_an... | https://mathoverflow.net/users/2362 | Why are Berkovich spaces locally connected? | Let $X = \mathcal{M}(A)$ be an affinoid space. There is a map $\mathcal{M}(A) \to \operatorname{Spec}(A)$ sending a seminorm to its kernel. It is continuous, surjective and induces a bijection between the connected components of $\mathcal{M}(A)$ and $\operatorname{Spec}(A)$. Let me prove the last point.
First assume ... | 2 | https://mathoverflow.net/users/4069 | 431354 | 174,650 |
https://mathoverflow.net/questions/431338 | 5 | Consider, $M$, a smooth $m$ dimensional submanifold of $\mathbf R^n$. Does there exist a smooth map $X: \mathbf{R}^m\to\mathbf R^n$ such that $M=X(\mathbf R^m)$?
$X$ may have points at which the Jacobian is singular, which means that $M$ doesn't have to be diffeomorphic to $\mathbf{R}^m$. Furthermore, the stereograph... | https://mathoverflow.net/users/485792 | Can a smooth manifold be realised as the image of a smooth function? | My comment turned answer:
Any smooth $m$-manifold $M$ admits a complete Riemannian metric (for example, as [this answer](https://mathoverflow.net/a/18851/172802) says, any manifold embeds into some Euclidean space as a closed subset by Whitney embedding theorem).
So if we endow any connected smooth manifold $M$ wit... | 13 | https://mathoverflow.net/users/172802 | 431358 | 174,651 |
https://mathoverflow.net/questions/431343 | 2 | Suppose $\mathfrak{g}$ is a real form of a semisimple Lie algebra $\mathfrak{g}\_\mathbb{C} = \mathfrak{g} \otimes\_\mathbb{R} \mathbb{C}$. Then we have the following:
* There is an equivalence of monoidal categories between the category of finite-dimensional *complex* representations of $\mathfrak{g}$ and the catego... | https://mathoverflow.net/users/29738 | Are finite-dimensional real representations of semisimple real Lie algebras completely reducible? | As pointed out by YCor in the comments, the answer is Yes. A reference has been pointed out to me: Chapter III, Section 7, Theorem 8 in Jacobson's book *Lie algebras*.
| 1 | https://mathoverflow.net/users/29738 | 431359 | 174,652 |
https://mathoverflow.net/questions/431363 | 2 | I have seen an equivalence claimed in a few places, but I do not know of a reference that actually proves it with details and it has been a while since I took graduate courses on all this. Apologies if this is something standard. A reference would be appreciated, particularly if it illuminates the general picture.
Th... | https://mathoverflow.net/users/43134 | Equivalence of Hilbert space norm associated to the harmonic oscillator and a sum of Sobolev and weighted $L^2$ norms | Chapter 9 of my book [https://www-users.cse.umn.edu/~garrett/m/v/current\_version.pdf](https://www-users.cse.umn.edu/%7Egarrett/m/v/current_version.pdf)
(that is mostly aimed at applications to automorphic forms) considers this. See section 9.8, in particular.
I really don't know of other references for this sort of ... | 3 | https://mathoverflow.net/users/15629 | 431364 | 174,654 |
https://mathoverflow.net/questions/431368 | 4 | $\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\PSp{PSp}\DeclareMathOperator\USp{USp}\DeclareMathOperator\BSp{BSp}\DeclareMathOperator\BUSp{BUSp}\DeclareMathOperator\BPSp{BPSp}$Let $\USp(n,\mathbb{C})$ and $\Sp(n,\mathbb{C})$ be the compact symplectic group and the symplectic group, respectively. This is,
$$
\Sp(n,\ma... | https://mathoverflow.net/users/121001 | CW structure for $\mathrm{BSp}(n,\mathbb{C})$ and $\mathrm{BPSp}(n,\mathbb{C})$ in degrees $4i$ | I'll assume that by $BG$ you mean ``any space of the form $E/G$, where $E$ is a contractible space with free $G$-action''. (The alternative would be to define $BG$ as the geometric realisation of a specific simplicial space, but then $BU(n)$ would not have even cells.) But then you can take $G=Sp(n,\mathbb{C})=\text{Au... | 4 | https://mathoverflow.net/users/10366 | 431385 | 174,659 |
https://mathoverflow.net/questions/430915 | 4 | Let $\Sigma$ be a compact orientable connected $2$-manifold with a non-empty boundary. Let $\widehat \pi(\Sigma)$ denote the set of free homotopy classes of
curves in $\Sigma$. We say $x\in \widehat \pi(\Sigma)$ is peripheral, if there is representative $\alpha\colon \Bbb S^1\to \Sigma$ of $x$ with $\text{im}(\alpha)\s... | https://mathoverflow.net/users/363264 | Characterization of a non-trivial non-peripheral element of the free homotopy classes of a compact bordered surface | This is true for all (compact, connected, oriented) surfaces that admit essential non-peripheral simple closed curves.
The sphere, disk, annulus and pants do not admit essential non-peripheral curves. The sphere and disk have trivial fundamental group. So the answer is "yes" vacuously for the sphere and disk, and "no... | 3 | https://mathoverflow.net/users/1650 | 431391 | 174,661 |
https://mathoverflow.net/questions/431375 | 1 | I'm reading through the paper [Poincaré type and spectral gap inequalities with fractional Laplacians on Hamming cube](https://arxiv.org/abs/1802.04411).
However, I'm having a difficult time understanding the following proof: [Lemma 2.1 page 3](https://i.stack.imgur.com/epHpF.png)).
I understand the general goal of... | https://mathoverflow.net/users/491992 | Proof of lower bound on variance | It looks to me like the authors just wanted to put in some ridiculous values which "are clearly sufficient" so that they don't have to work out the details.
The choice $\eta = c\_1^{1/10}$ should ensure that the term $\int\_{|g-1|\leq \eta} |g|^\beta \operatorname{sgn}(g) \, d\mu$ converges to 1 for $c\_1 \rightarrow... | 0 | https://mathoverflow.net/users/106046 | 431409 | 174,665 |
https://mathoverflow.net/questions/431407 | 2 | Let $f: \mathbb R \to \mathbb R$ be a nonnegative measurable function, and $\{q\_n\}$ some enumeration of the rational numbers. Suppose for every $0 < r < 1$ it holds that
$$\sum\_{n = 0}^\infty r^n f(x + q\_n)$$
converges for almost every $x \in \mathbb R$.
**Question:** Does this imply that for every compact se... | https://mathoverflow.net/users/173490 | Does this condition on $f$ imply essential boundedness on compacts? | No, this is not true. Let $f \in L^1(\mathbb R)$ and $g\_r(x)=\sum\_{n=0}^\infty r^n f(x+q\_n) \leq \infty$. Then $\|g\_r\|\_1=\frac{\|f\|\_1}{1-r}$ and hence $g\_r(x)<\infty$ a.e. If $E\_r$ is the null set where $g\_r$ may not converge then the series converge for all $x$ outside $\cup\_n E\_{1-\frac 1n}$, for every $... | 6 | https://mathoverflow.net/users/150653 | 431416 | 174,667 |
https://mathoverflow.net/questions/430194 | 4 | Let $M$ be a countable transitive model of $\sf ZF$, Let $V\_\alpha, V\_{\alpha+1}$ be two stages of the cumulative hierarchy in $M$, let $f: V\_\alpha \to V\_{\alpha+1}$ be a bijection such that for any set $S \in V\_{\alpha+2}$ we have both $f[S] \in M ; f^{-1}[S] \in M$ . Let $F: V\_{\alpha+1} \to V\_{\alpha+2}; F(x... | https://mathoverflow.net/users/95347 | Is it possible to have such nesting of functions? | I think I have an answer to this question of mine. if $M$ in the question satisfy $AC$, then there must exist some set $S$ in $V\_{\alpha+3}$ that doesn't have both $F[S]; F^{-1}[S]$ being elements of $M$, otherwise we can interpret $\sf NF+AC$ which is inconsistent. However, the proof of that result is long.
| 0 | https://mathoverflow.net/users/95347 | 431417 | 174,668 |
https://mathoverflow.net/questions/431383 | 3 | Let $f:X\to S$ be a morphism between algebraic varieties which are smooth over a field of characteristic zero. We define the (derived) direct image functor $f\_+:\mathsf{D}^b(\mathcal{D}\_X)\to \mathsf{D}^b(\mathcal{D}\_S)$ as in basically every textbook. (That's what [HTT] denotes by $\int\_f$ on page 40, for example.... | https://mathoverflow.net/users/131975 | Artin vanishing for D-modules (i.e., when is $f_+$ t-exact?) | The functor $f\_+$ by definition is the composite of a right t-exact functor (tensor product with the transfer bimodule) and a left t-exact functor (the derived pushforward of sheaves). In the case $f$ is affine, the latter functor is t-exact, so $f\_+$ is indeed right-exact.
On the other hand, D-module pushforward u... | 3 | https://mathoverflow.net/users/7762 | 431424 | 174,669 |
https://mathoverflow.net/questions/431425 | 1 | ### I would like to know if this equation is solvable for $a$ and $\alpha$:
\begin{equation}
\Sigma = \Gamma + a \left( \alpha 1^\top + 1\alpha^\top \right) +a^2 b
\end{equation}
---
1. $\Sigma$ & $\Gamma$ are known. Both are $D\times D$ matrices.
2. $\Sigma$ is symmetric positive definite.
3. $\Gamma$ is symme... | https://mathoverflow.net/users/492060 | Two unknowns: one vector, one scalar, one equation | A small rearrangement yields
$$
\Sigma-\Gamma = a(\alpha + a 1 )1^T + a1 \alpha^T.
$$
So for solvability the rank of $\Sigma-\Gamma$ can at most be $2$.
If $\mathrm{rank}\ \Sigma-\Gamma = 0$, then $\Sigma = \Gamma$. Then a solution is $a=0$ and any $\alpha$ with $\alpha^T1=0$. If we assume $a\neq 0$ we equivalently n... | 3 | https://mathoverflow.net/users/85570 | 431437 | 174,671 |
https://mathoverflow.net/questions/431448 | 3 | What is the consistency strength of the following prinicple?
Every set is a member of a [supertransitive](https://en.wikipedia.org/wiki/Supertransitive_class) model of $\sf ZFC$.
Formally this means adding the following sentence to axioms of $\sf ZFC$:
$\forall x \exists M :( M \models {\sf ZFC}) \land \operatorn... | https://mathoverflow.net/users/95347 | What is the strength of having all sets being elements of supertransitive models of ZFC? | Every level of the cumulative hierarchy is supertransitive, so "There is a proper class of worldly cardinals" clearly implies the principle in question.
Conversely, if $M\models\mathsf{ZFC}$ is supertransitive then $M$ computes powersets correctly: if $x\in M$ then $\mathcal{P}(x)\subseteq M$ so $\mathcal{P}(x)^M=\ma... | 6 | https://mathoverflow.net/users/8133 | 431468 | 174,678 |
https://mathoverflow.net/questions/431461 | 2 | Let $M$ be a smooth (or just topological) closed manifold. Let $C(M)$ denote the cone over $M$, i.e.
$C(M)$ equals to $M\times [0,\infty)$ with $M\times \{0\}$ contracted to a point. The image of $M\times \{0\}$ in $C(M)$ is called the origin.
**What is the dualizing complex of $C(M)$? In particular what is its stalk... | https://mathoverflow.net/users/16183 | Dualizing complex of the cone over a manifold | The stalk of the dualizing complex at a point is the shift of reduced homology of the link at that point. In this case, the link is $M$ and so the homology of the stalk in degree $i$ is $\tilde H\_{i-1}(M)$.
| 6 | https://mathoverflow.net/users/52918 | 431469 | 174,679 |
https://mathoverflow.net/questions/431398 | 1 | On the nLab, given a local $S$-topos $E$, a concrete sheaf is defined as an object that is separated with respect to the local isomorphisms (the morphisms that are inverted by the global sections functor $\Gamma:E\rightarrow S$): <https://ncatlab.org/nlab/show/concrete+sheaf#in_a_local_topos>
However, separated means... | https://mathoverflow.net/users/122987 | Concrete sheaves | To summarize the discussion in the comments:
* There are two nontrivial Grothendieck topologies used in the definition of a concrete sheaf: the Grothendieck topology T used to define sheaves, and the Grothendieck topology C used to define concrete presheaves.
* The Grothendieck topology T (and its site S) are given t... | 2 | https://mathoverflow.net/users/402 | 431475 | 174,681 |
https://mathoverflow.net/questions/431474 | 4 | **Question**: How to calculate this summation $S=\sum\_{k=0}^m a^k b^{m-k} {n\_1\choose k} {n\_2\choose m-k} $? Where $m<n\_1,m<n\_2$
**Remark1**: When $a=b$, I know the above summation $S=a^m\sum\_{k=0}^m {n\_1\choose k} {n\_2\choose m-k} =a^m {n\_1+n\_2\choose m} $.
**Remark2**: This summation looks somewhat simi... | https://mathoverflow.net/users/492126 | How to calculate this summation $\sum_{k=0}^m a^k b^{m-k} {n_1\choose k} {n_2\choose m-k} $? | In terms of a hypergeometric function you would have
$$S=\sum\_{k=0}^m a^k b^{m-k} {n\_1\choose k} {n\_2\choose m-k}=b^m \binom{n\_2}{m} \, \_2F\_1\left(-m,-n\_1;-m+n\_2+1;a/b\right).$$
I don't see a simpler closed-form expression for arbitrary parameters, but if you fix $n\_1$ this does simplify to a simple rational f... | 5 | https://mathoverflow.net/users/11260 | 431476 | 174,682 |
https://mathoverflow.net/questions/431128 | 3 | Let $\Sigma$ be a compact oriented connected bordered surface other than the pair of pants. Let $\Gamma:=\{\gamma\_i\}$ be a finite collection of simple closed curves on $\Sigma$ such that each component of $\Sigma\setminus\cup\_i\gamma\_i$ is homeomorphic to either $\Bbb S^1\times (0,1)$ or $\{z\in \Bbb R^2:|z|<1\}$. ... | https://mathoverflow.net/users/363264 | A closed curve can be homotopic to remove all intersections with a filling $\Gamma$ if it has zero geometric intersection numbers with $\Gamma$ | This is true, here's a proof, by a kind of "Whitney trick".
Perturb the set of curves $\{\gamma\_i\} \cup \{c\}$ to put it into general position, so they are pairwise transverse and there is no triple point. Let $|c| = \sum\_i |c \cap \gamma\_i|$.
For each $i$ such that $|c \cap \gamma\_i| > 1$, since $GI(c,\gamma\... | 3 | https://mathoverflow.net/users/20787 | 431494 | 174,687 |
https://mathoverflow.net/questions/431466 | 1 | $\DeclareMathOperator\SL{SL}$Let $ G $ be a noncompact simple Lie group. For example $ \SL\_n $. Let $ \Gamma $ be a lattice in $ G $. Consider the action of $ \Gamma $ on the Lie algebra of $ G $ by conjugation. Is this representation of $ \Gamma $ always irreducible?
For example, I think it is true that all lattice... | https://mathoverflow.net/users/387190 | A lattice in $ \operatorname{SL}_n $ is Ad-irreducible | Per the request to post it as an answer.
Notice that the Ad representation is a polynomial representation into $\operatorname{GL}(\operatorname{Lie}(G))$.
We do know that $\operatorname{Ad}(G)$ acts irreducibly, and $\Gamma$ is Zariski dense by Borel's density theorem. Hence $\operatorname{Ad}\rvert\_{\Gamma}$ is als... | 4 | https://mathoverflow.net/users/8857 | 431496 | 174,688 |
https://mathoverflow.net/questions/430944 | 14 | I need to find the determinant of matrix defined by
\begin{align\*}
& a\_{i,1}=a\_{1,j}=1,\quad \forall 1\leq i,j\leq n,\\ & a\_{i,j}=a\_{i-1,j}+a\_{i,j-1}+i-j, \quad \forall 1< i,j\leq n.
\end{align\*}
Numerically, for $n=1$ to $12$; I found that $\det(A)=F\_n$ where $F\_n$ is the Fibonacci sequence.
How to prove ... | https://mathoverflow.net/users/126827 | Determinant equal to Fibonacci sequence | The idea of Jandri can also be applied to a slightly more general case:
Let $$ a(i,j,x,y,z,t)= x\binom{i+j-1}{j}-y \binom{i+j-2}{i}+z(j-i)+t,$$
$A\_n(x,y,z,t)$ be the matrix with entries $a(i,j,x,y,z,t)$ and $D\_n(x,y,z,t)=\det(A\_n(x,y,z,t).$
We apply the following operations on the matrix $A\_n(x,y,z,t):$ We chan... | 2 | https://mathoverflow.net/users/5585 | 431503 | 174,691 |
https://mathoverflow.net/questions/431411 | 5 | $\newcommand{\H}{\mathcal{H}}$
$\newcommand{\A}{\mathcal{A}}$
Recall that a coideal $\H$ over $\omega$ is **selective** if for every $\{A\_n : n < \omega\} \subseteq \H$, where $i < j \implies A\_i \supseteq A\_j$, there exists some $B \in \H$ such that $B/n := \{m \in B : m > n\} \subseteq A\_n$ for all $n \in B$. The... | https://mathoverflow.net/users/146831 | Non-trivial examples of selective coideals of $\omega$ | See Section 12 in [S. Todorcevic, Topics in Topology, Lecture notes in Mathematics Vol. 1652, Springer-Verlag Berlin Heidelberg 1997](https://doi.org/10.1007/BFb0096295)
| 3 | https://mathoverflow.net/users/492151 | 431506 | 174,693 |
https://mathoverflow.net/questions/431509 | 0 | Fix $m\in \mathbb{N}.$
For each $n\in \mathbb{N},$ let $A\_n\in \mathbb{M}\_{m}(\mathbb{C}),$ $X\_n\in \mathbb{C}^m,$ and $B\_n\in \mathbb{C}^m.$ Suppose that
$$X\_{n+1}=A\_n X\_n+B\_n,$$
$$\lim\_{n\rightarrow \infty} A\_n=A\_0,$$
$$|\det(A\_0)|>1,$$
the moduli of all entries of $A\_0$ are greater than 1,
and
$$\limsup... | https://mathoverflow.net/users/492150 | Vector recurrences (asymptotic property) | $\newcommand\1{\mathbf1}\newcommand\R{\mathbb R}$Yes, this is of course possible. E.g., this will be so if for all $n$ the matrix $A\_n$ is the diagonal matrix with $1,2\dots,2$ on the diagonal, $B\_n=0$, and $X\_n=[1,0,\dots,0]^\top$.
---
The OP has changed the question, thereby invalidating the above answer. Af... | 2 | https://mathoverflow.net/users/36721 | 431516 | 174,694 |
https://mathoverflow.net/questions/431422 | 16 | Context
=======
In a [recent paper](https://arxiv.org/abs/2209.06838) involving entanglement in linear optics, we came across some summations involving [Catalan numbers](https://en.wikipedia.org/wiki/Catalan_number) and permutations. In particular, these sums arise when doing integration over the unitary group $\math... | https://mathoverflow.net/users/492058 | Conjecture on sum over permutations of products of Catalan numbers | The problem naturally fits in the framework of breakpoint graphs (per Peter Taylor's observation), which makes it possible to obtain a differential equation for the generating function
$$H(x,u,s\_1,s\_2,\dots) := \sum\_{\ell\geq 0} x^{2\ell} \sum\_{\tau\in S\_{2\ell}} u^{\xi^{(\ell)}(\tau)} \prod\_{i=1}^{\#(\tau)} s\_{... | 6 | https://mathoverflow.net/users/7076 | 431527 | 174,699 |
https://mathoverflow.net/questions/431510 | 12 | In differential geometry it is often natural to speak of infinite-dimensional manifolds (e.g., the manifold of mappings between two smooth manifolds). Different versions of [generalized smooth spaces](https://ncatlab.org/nlab/show/generalized+smooth+space) are proposed for this purpose. I find the most natural and pref... | https://mathoverflow.net/users/148161 | Are there textbooks on differential geometry in the language of smooth sets or smooth derived stacks? | “[Diffeology](http://math.huji.ac.il/%7Epiz/Site/TheBook-rep.html)” by Patrick Iglesias-Zemmour is probably the closest match.
He develops differential forms and de Rham cohomology, fiber bundles, connections, and symplectic geometry in the language of diffeological spaces, i.e., concrete sheaves of sets on the site ... | 12 | https://mathoverflow.net/users/402 | 431529 | 174,700 |
https://mathoverflow.net/questions/431292 | 4 | A space $X$ is said to be Menger if for each sequence $(\mathcal{U}\_n)$ of open covers of $X$, there is a sequence $(\mathcal{V}\_n)$ such that $\mathcal{V}\_n$ is a finite subcollection of $\mathcal{U}\_n$, $n\in\omega$, and $\{\bigcup\mathcal{V}\_n:n\in\omega\}$ is an open cover of $X$. A space $X$ is Lindelöf if ea... | https://mathoverflow.net/users/476373 | Is there any example of a Lindelöf space that has no Menger dense subspaces? | Every space $X$ with a dense Menger subspace must be *weakly Menger*, that is, for each sequence $\{\mathcal{U}\_n: n<\omega \}$ of open covers there is a finite sub-collection $\mathcal{V}\_n \subset \mathcal{U}\_n$ such that $\bigcup \{\mathcal{V}\_n: n <\omega \}$ is dense in $X$. Therefore it's enough to find a Lin... | 5 | https://mathoverflow.net/users/11647 | 431530 | 174,701 |
https://mathoverflow.net/questions/430032 | 3 |
>
> Suppose we have $N$ lines in general position (any two lines, but no three lines, meet at a point) ($N\geq 3$). Let the smallest bounded region have area $1$. Determine the minimum (or possibly infimum) of the total bounded area.
>
>
>
For example, $3$ lines create one triangular region while $4$ create one ... | https://mathoverflow.net/users/490865 | Minimize total area bounded by $N$ lines in general position | I wound up contacting Dr. János Pach (an expert in relevant fields) regarding the problem on the recommendation of my first professor. Apparently the problem (originally in a slightly different form) is a minor unsolved problem first posed by Fejes Tóth in 1987:
>
> L. Fejes Tóth, On Spherical Tilings Generated by ... | 1 | https://mathoverflow.net/users/490865 | 431537 | 174,704 |
https://mathoverflow.net/questions/431460 | 2 | Let $I$ be the Dynkin diagram vertex set and $K$ be a proper nonempty subset of it. Let $w\_0^K$ be the longest word of the Dynkin subdiagram $K$, which might be a disjoint union of connected Dynkin diagrams.
I saw in the paper "[Preprojective algebras and partial flag varieties](http://aif.cedram.org/item?id=AIF_200... | https://mathoverflow.net/users/169192 | Particular reduced expression of the longest element of Weyl group | Whenever $\ell(wv)=\ell(w)+\ell(v)$, you can construct a reduced word for $wv$ by producing one for $w$ and one for $v$ and then concatenating them. So, if you know an algorithm for producing reduced words, you can just do that.
On the other hand, there’s actually a particularly nice interpretation in this case. Redu... | 5 | https://mathoverflow.net/users/66 | 431540 | 174,705 |
https://mathoverflow.net/questions/431512 | 1 | I have just asked the calculation of the following summation [see here](https://mathoverflow.net/questions/431474/how-to-calculate-this-summation-sum-k-0m-ak-bm-k-n-1-choose-k-n-2-ch) $$S(a,b,m,n\_1,n\_2)=\sum\_{k=0}^m a^k b^{m-k} {n\_1\choose k} {n\_2\choose m-k}, $$
which is motivated by the calculation of the follow... | https://mathoverflow.net/users/492126 | How to calculate this limit (if exist)? | $\newcommand\ka\kappa\newcommand{\de}{\delta}\newcommand\ep\varepsilon$The problem obviously reduces to this one: find
\begin{equation}
L:=\lim\_{m\to\infty}\frac{S(c,m,n\_1,n\_2)}{S(c,m-1,n\_1-1,n\_2-1)},
\end{equation}
where $c:=a/b\ne1$ and
\begin{equation}
S(c,m,n\_1,n\_2):=\sum\_{k=0}^m c^k A\_k(m,n\_1,n\_2),
\... | 4 | https://mathoverflow.net/users/36721 | 431551 | 174,712 |
https://mathoverflow.net/questions/431565 | 6 | The density theorem in the ordinary category theory asserts that every presheaf on a small category is a colimit of representables in a canonical way. In Lemma 5.1.5.3 of *Higher Topos Theory*, Lurie proves an $\infty$-categorical version of this. There is a part of the proof which is not clear to me, and I hope someon... | https://mathoverflow.net/users/144250 | Density Theorem for $\infty$-Categories (HTT, Lemma 5.1.5.3) | I'm going to deal with the case of $\mathcal E^1\subset \mathcal E^0$. Note that by composition of pullbacks, $\mathcal E^0 = \mathcal C\_{/X}\times\_{\mathcal P(S)}\mathcal P(S)\_{j(s)/}$, while $\mathcal E^1= \mathcal C\_{/X}\times\_\mathcal C\{j(s)\}$.
Further, because $\mathcal C\to \mathcal P(S)$ is a full subca... | 4 | https://mathoverflow.net/users/102343 | 431572 | 174,717 |
https://mathoverflow.net/questions/431564 | 12 | As the title suggest it seems standard conjectures mean different things depending on the context. I had the impression that in characteristic 0 they are a list of conjectures about varieties over an arbitrary field of characteristic zero and any Weil cohomology theory (not necessarily the usual suspects) just like the... | https://mathoverflow.net/users/127776 | What exactly do the standard conjectures in characteristic zero refer to? | To prove that the standard conjectures are true for any Weil cohomology over a given field $k$, it suffices to prove them for an arbitrary chosen Weil cohomology over each subfield of finite type of $k$.
Here is a way to summarize the principle of the proof.
There is, for each field $k$ the category $M(k)$ of pure ... | 17 | https://mathoverflow.net/users/1017 | 431573 | 174,718 |
https://mathoverflow.net/questions/431539 | 2 | Meusnier's theorem states that all curves on a surface $S$ embedded in $\Bbb R^3$ passing through a given point $P$ and having the same tangent $v\in T\_PS$ also have the same normal curvature.
I think the same is true for [geodesics torsion.](https://encyclopediaofmath.org/wiki/Geodesic_torsion) See [here](https://e... | https://mathoverflow.net/users/54507 | Is the Meusnier's theorem true for geodesics torsion? | I believe that your $\tau\_g$ is what É. Cartan calls the 'geodesic torsion' in his 1945 book *Les systèmes différentiels extérieurs et leurs applications géométriques*. He denotes his geodesic torsion by $1/T\_g$. He gives your formula for $1/T\_g$ in Chapter 7 as part of equation $(17)$.
The point is that what the ... | 5 | https://mathoverflow.net/users/13972 | 431574 | 174,719 |
https://mathoverflow.net/questions/431521 | 5 | Let $\mathcal{F}$ denote a family of countable subsets of $\mathbb{R}$, such that for each $U, V\in\mathcal{F}$ we have that $U\subseteq V$, or $V\subseteq U$. Let $(\mathcal{F}, \preceq)$ denote the inclusion partial order of $\mathcal{F}$.
1. Is it true that $(\mathcal{F}, \preceq)$ is isomorphic to $(S, \leq)$ whe... | https://mathoverflow.net/users/100140 | Question about a family of nested countable subsets of $\mathbb{R}$ | $\let\sset\subseteq\def\cF{\mathcal F}\def\R{\mathbb R}\def\Q{\mathbb Q}$The answer to both questions is negative: let $\preceq$ be a well order of type $\omega\_1$ on a subset of $\R$, and let $\cF$ consist of all proper initial segments of $\preceq$. Then $\cF$ is a family of countable sets totally ordered by $\subse... | 7 | https://mathoverflow.net/users/12705 | 431575 | 174,720 |
https://mathoverflow.net/questions/431568 | 14 | In his "Théorie de chaleur" Fourier proves that the zeros of Bessel function $J\_0(x)$ are all real.
I want to ask if there is a modern version of this proof exist in literature?
If someone can provide me with an elegant compact proof or reference for it , it would be helpful.
| https://mathoverflow.net/users/145223 | Fourier's proof of reality of all roots of Bessel function $J_0(x)$ | Fourier proof was incomplete. Fourier used the following
**Statement.** A real entire function has only real zeros if
its derivatives have the following property:
If $x$ is a real root of $f^{(n)}$ then $f^{(n-1)}(x)f^{(n+1)}(x)<0.$
But he verified this only for polynomials. Fourier's proof was criticized by Cauchy... | 24 | https://mathoverflow.net/users/25510 | 431579 | 174,721 |
https://mathoverflow.net/questions/431543 | 1 | A functional Hilbert space $\mathscr H=\mathscr H(\Omega)$ is a Hilbert space of complex valued functions on a (nonempty) set $\Omega$, which has the property that point evaluations are continuous i.e. for each $\lambda\in \Omega$ the map $f\mapsto f(\lambda)$ is a continuous linear functional on $\mathscr H$. The Ries... | https://mathoverflow.net/users/113054 | Is $\sup\{\|A\widehat{k}_{\lambda}\|: \lambda\in\Omega\}=\sup\{\|A^*\widehat{k}_{\lambda}\|: \lambda\in\Omega\}$? where $A$ is an operator on RKHS | No, this fails already when $\Omega=\{ 1, 2\}$, so $H=\mathbb C^2$ and $\widehat{k}\_j=e\_j$. We can take
$$
A= \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix} .
$$
Then $N(A)=\sqrt{2}$, $N(A^\*)=1$.
| 2 | https://mathoverflow.net/users/48839 | 431593 | 174,724 |
https://mathoverflow.net/questions/431592 | 1 | Suppose that $f: X \to Y$ is a smooth proper map between two smooth manifolds. Is it always possible to represent $f$ as a composition of a closed embedding $g: X \to Z$ with a proper submersion $h: Z \to Y$?
**Motivation:**
Firstly, closed embeddings and proper submersions are in a sense the simplest kinds of prop... | https://mathoverflow.net/users/157863 | Decomposing proper map into closed embedding and proper submersion | Yes. Choose a smooth (but not necessarily closed) embedding $i:X\to W$ where the manifold $W$ is compact, for example a sphere. Together $i$ and $f$ give a smooth map $X\to W\times Y$ that is both proper and an embedding.
| 3 | https://mathoverflow.net/users/6666 | 431595 | 174,726 |
https://mathoverflow.net/questions/431585 | 3 | Suppose that $\tau \in \mathbf{H}$ belongs to the complex upper half plane. The quotient $\mathbf{C}/(\mathbf{Z}+\mathbf{Z}\tau)$ gives an elliptic curve over $\mathbf{C}$. Write this elliptic curve as $E\_{\tau}$. We can write $E\_{\tau}$ as follows:
$$E\_{\tau}: y^2 = 4x^3 - \left( \dfrac{27 j(\tau)}{j(\tau)-1728} ... | https://mathoverflow.net/users/394740 | Hecke operators on universal elliptic curves | The universal elliptic curve can be written $y'^2 = 4x'^3- g\_2 x - g\_3$ where $g\_2$ and $g\_3$ are Eisenstein series. Given any differential on $E$, simply change coordinates to this family, then divide by $dx'/y'$ to obtain a modular form of weight $1$.
The point is that $g\_2$ is modular of weight $4$ and $g\_3$... | 4 | https://mathoverflow.net/users/18060 | 431605 | 174,729 |
https://mathoverflow.net/questions/431602 | 2 | Let $X$ be a compact complex manifold with canonical bundle $K\_X$. Assume the Kodaira dimension $\kappa(X)$ is positive (but *not* maximal, i.e., $X$ is *not* of general type). Let $\varphi\_m : X \dashrightarrow Y\_m \subset \mathbb{P}^{N\_m}$ denote the $m$th pluricanonical map given by the sections of the $m$th ten... | https://mathoverflow.net/users/105103 | Rational curves on the image of the pluricanonical maps | Not only could $Y\_m$ contain a rational curve for all $m$, $Y\_m$ could *be* a rational curve for all $m$.
Take $C$ a hyperelliptic curve, $E$ an elliptic curve, $\tau$ the hyperelliptic involution on $C$, $\sigma$ the translation on $E$ by a point of order $2$.
Let $X = (E \times C)/ (\sigma \times \tau)$.
Sinc... | 6 | https://mathoverflow.net/users/18060 | 431607 | 174,731 |
https://mathoverflow.net/questions/431450 | 1 | Quick context: This is a transposition of exercise 6.3 of Remco Van der Hofstad (2016) and it is relevant to some problem i encoutered in my research.
For each $n \in \mathbb{N}$, define a series of positive numbers $w\_i^n$ with $1\leq i\leq n$. Also denote an uniformly chosen numbers by $w\_U = W\_n$ where $U$ is p... | https://mathoverflow.net/users/164762 | Maximum of a sequence is $o(\sqrt{n})$ | First suppose that hypotheses (a) and (b) hold. I will write $w\_{i,n}$ instead of $w\_i^n$ for clarity.
Given $\epsilon>0$, the Monotone convergence theorem implies that there exists $M$ such that $E(W)-E(W \wedge M) <\epsilon$, where $\wedge$ indicates minimum. By (a) and the bounded convergence theorem,
$$E(W\wedg... | 2 | https://mathoverflow.net/users/7691 | 431608 | 174,732 |
https://mathoverflow.net/questions/302516 | 19 | Classically, we can explicitly construct the free Abelian group $\newcommand{\Z}{\mathbb{Z}}\Z[X]$ on a set $X$ as the set of finitely-supported functions $X \to \Z$, and so easily see that the unit map $X \to \Z[X]$ (inclusion of generators) is injective.
Constructively, this construction of $\Z[X]$ doesn’t work for... | https://mathoverflow.net/users/2273 | Constructively, is the unit of the “free abelian group” monad on sets injective? | I'd like to present a more modular proof. The key lemma (which I learnt from Christian Sattler) is that any set $X$ is a filtered colimit of finite sets. Indeed, let $I$ be the category of pairs $(n, f)$ with $n \in \mathbb N$ and $f : [n] \to X$, where a morphism $(n, f) \to (m, g)$ is a function $p : [n] \to [m]$ suc... | 4 | https://mathoverflow.net/users/113846 | 431619 | 174,735 |
https://mathoverflow.net/questions/431627 | 0 | Let $$A\_n=\{(x\_1,x\_2,x\_3,\cdots,x\_n):x\_i \in [q] \text{ for } i \in [n], x\_1 < x\_2, x\_2 > x\_3, x\_3 < x\_4, \cdots , (-1)^{n}x\_{n-1} < (-1)^{n} x\_n\}.$$ What is the cardinality of $A\_n$?
I have 2 observations. First, let $$B\_n=\{(x\_1,x\_2,x\_3,\cdots,x\_n):x\_i \in [q] \text{ for } i \in [n], x\_1 \leq... | https://mathoverflow.net/users/492228 | Upper bound of the number of oscillatory sequences | In the terminology of Chapter 3 of *Enumerative Combinatorics*,
vol. 1, second ed., you are asking for the value of the strict order
polynomial of the zigzag poset $Z\_n$ at $q$ (see Exercise
3.66). Equivalently, it is equal to $(-1)^n\Omega\_{Z\_n}(-q)$, where
$\Omega\_{Z\_n}$ is the ordinary order polynomial. The num... | 7 | https://mathoverflow.net/users/2807 | 431629 | 174,737 |
https://mathoverflow.net/questions/417965 | 3 | Suppose $X,Y,Z$ are real-valued random variables on some probability space. Suppose I know everything there is to know about the moments
$$
M\_{p,q,r}=\mathbb{E}(X^pY^qZ^r)
$$
for $p,q,r\in\mathbb{N}\_0$. Suppose also that these moments do not grow too fast so the moment problem is determinate, i.e., the moments comple... | https://mathoverflow.net/users/7410 | Moment criterion for conditional independence | If the moment problem is determinate and in addition $Z$ is bounded, then $X\perp Y\mid Z$ if and only if
$$\mathbb E\left[X^pY^qZ^r\right]=\lim\_{n\to\infty}\mathbb E\left[X^p\mathbf Z\_n^{\mathsf T}\right]\mathbb E\left[\mathbf Z\_n\mathbf Z\_n^{\mathsf T}\right]^+\mathbb E\left[\mathbf Z\_nZ^r\mathbf Z\_n^{\mathsf T... | 2 | https://mathoverflow.net/users/492208 | 431637 | 174,741 |
https://mathoverflow.net/questions/431449 | 2 | $\DeclareMathOperator\Tr{Tr}$I have a problem with understanding the proof of Proposition 6.8 in the book ,,Elements of Noncommutative Geometry''. One can find the formulation of this proposition [here](https://ibb.co/V2BHGkY) and the proof [here](https://ibb.co/dDhDHxS). The context is the following: $V$ is a *real* i... | https://mathoverflow.net/users/24078 | Another formula for the Schwinger term — problems with a calculation | I combine the two questions in the OP in a single question: find a proof of proposition 6.8,
$$
\alpha(A,B)=\tfrac{1}{8}i\,{\rm Tr}\,J[J,A][J,B].\qquad\qquad(6.8)
$$
---
I note that the Schwinger term $\alpha(A,B)$ is defined as the trace of antilinear operators on a complex vector space, while the trace in (6.8)... | 2 | https://mathoverflow.net/users/11260 | 431640 | 174,743 |
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