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https://mathoverflow.net/questions/406308 | 1 | Let $W$ be a one dimensional standard Brownian motion, and $\sigma: [0, \infty) \to \mathbb R$ a Borel function with $c < \sigma < C$ for some constants $c, C > 0$.
Does there exist some $M > 0$ such that, conditional on $W\_t > -1$ for all $0 \leq t \leq 1$ and $W\_1 > M$,
$\int\_0^1 \sigma(t) \, dW\_t \geq 0$, al... | https://mathoverflow.net/users/173490 | Conditioning the stochastic integral on the driving Brownian motion being large | No such $M$ exists for the following $\sigma$. Partition $[0,1]$ into countably many intervals, with endpoints $t\_0=0,t\_1,t\_2,...$ and let $\sigma$ take value 1 on the odd ones and value 2 on the even ones. Given any $M$, the following event $A\_M$ has positive probability:
$A\_M$ requires that the increments $W\_... | 1 | https://mathoverflow.net/users/7691 | 406377 | 166,562 |
https://mathoverflow.net/questions/406373 | 0 | I fairly new in the field of Stochastic Processes and Markov Chains so excuse my ignorance.
My question is: If we have a sequence of Markov chains such that each one has a stationary distribution $\pi^{(n)}$ and the chains converge in some way to another Markov chain that has stationary distribution $\pi$, can we say... | https://mathoverflow.net/users/416334 | Convergence of stationary distributions of a sequence of Markov Chains | We assume that the Markov chains are on a finite state space, that $P\_n \to P$ pointwise, and the limit matrix $P$ is irreducible, so its stationary measure $\pi$ is unique. Let $\pi^{(n\_k)} \to \mu$ be a convergent subsequence of $\pi^{(n)}$. Then $\pi^{(n\_k)}P\_{n\_k}=\pi^{(n\_k)}$, so continuity of multiplication... | 5 | https://mathoverflow.net/users/7691 | 406382 | 166,565 |
https://mathoverflow.net/questions/406381 | 3 | A **normed division algebra over $\mathbb{R}$** is a pair $(A,\lVert{-}\rVert)$ with
* $A$ an $\mathbb{R}$-algebra with a unit $1\_A$;
* $\lVert{-}\rVert\colon A\to\mathbb{R}\_{\geq0}$ a norm on $A$;
such that:
* For each $a\in A$, there exists a unique $a^{-1}\in A$ such that $a^{-1}a=1\_A=aa^{-1}$;
* For each $... | https://mathoverflow.net/users/130058 | Is there a classification of the $p$-adic normed division algebras? | Clearly you are assuming some kind of finite-dimensionality over the center.
To classify the finite-dimensional associative division algebras over $\mathbf Q\_p$, or more generally over a local field, it's standard to fix the center. A *$K$-central algebra* here will mean a $K$-algebra whose center is $K$, so $\mathb... | 9 | https://mathoverflow.net/users/3272 | 406384 | 166,566 |
https://mathoverflow.net/questions/406293 | 12 | Let $\mathbb{C}P^n$ be the $n$-dimensional complex projective space and denote $[z\_0:\dots:z\_n]$ its points. If we glue $[z\_0:\dots:z\_n]$ and $[\overline{z\_0}:\dots:\overline{z\_n}]$ for any $[z\_0:\dots:z\_n]\in\mathbb{C}P^n$, where $\overline{z}$ denotes the complex conjugation of $z$, then we obtain a quotient ... | https://mathoverflow.net/users/392187 | A quotient space of complex projective space | This answer gives information about the cohomology of $\overline{\mathbb CP^n}$. Perhaps someone will recognize this as the cohomology of a familiar space.
The conjugation is an action of $\Sigma\_2$ on $\mathbb CP^n$. We are interested in the orbit space of this action. Recall that the fixed point of this action is ... | 12 | https://mathoverflow.net/users/6668 | 406389 | 166,569 |
https://mathoverflow.net/questions/406288 | 1 | \begin{gather\*}
M\_g=(x\_1\times x\_2\times\dotsb\times x\_n)^{1/n} \\
M\_a=\frac1 n\times (x\_1+x\_2+\dotsb+x\_n).
\end{gather\*}
Is it true that $$\lvert M\_g-M\_a\rvert \leq (\max(x\_i) /\min(x\_i)) \times(\max(x\_i) - \min(x\_i))?\label{1}\tag{1}$$
And is it true that
$$\lvert M\_g-M\_a\rvert\leq (\max(x\_i)... | https://mathoverflow.net/users/110301 | Compare AM and GM | The inequality $(2)$ (even with factor $\frac12$ in the r.h.s.) follows from the inequality quoted in [this answer](https://mathoverflow.net/q/215094):
$$M\_a - M\_g \leq \frac1{2n\min\_k x\_k} \sum\_{i=1}^n (x\_i - M\_a)^2.$$
First, we notice that
\begin{split}
(x\_i - M\_a)^2 &\leq \max\_k x\_k\cdot |x\_i-M\_a|\\
&= ... | 3 | https://mathoverflow.net/users/7076 | 406390 | 166,570 |
https://mathoverflow.net/questions/406368 | 9 | Let $G$ be a finitely presented group and let $L(G)$ be the Magnus Lie algebra associated to the lower central series of $G$. This $L(G)$ is a graded Lie ring generated by its degree 1 piece $L\_1(G) = G^{ab}$. Must $L(G)$ be a finitely presented Lie ring? If it makes it easier, I would be happy to tensor with the rati... | https://mathoverflow.net/users/416222 | Is the Magnus Lie algebra of a finitely presented group finitely presented | $\newcommand{\Z}{\mathbf{Z}}$No.
Take the Baumslag-Solitar group $$G=\mathrm{BS}(1,3)=\langle t,x\mid txt^{-1}=x^3\rangle=\mathbf{Z}\ltimes\_3\mathbf{Z}[1/3]$$
then the lower central series satisfies $G^1=G$, $G^i=2^{i-1}\mathbf{Z}[1/3]$ for all $i\ge 2$, which has index $2^{i-1}$ in $\mathbf{Z}[1/3]$. So $G^1/G^2\... | 11 | https://mathoverflow.net/users/14094 | 406397 | 166,574 |
https://mathoverflow.net/questions/406385 | 3 | In Grunbaum's paper, [A result on graph coloring](https://projecteuclid.org/journals/michigan-mathematical-journal/volume-15/issue-3/A-result-on-graph-coloring/10.1307/mmj/1029000043.full?tab=ArticleLinkCited), the following conjecture was posed:
>
> Let $G$ be a graph with $n$ nodes with $\Delta(G) < k$. There exi... | https://mathoverflow.net/users/130484 | Proper graph colorings with similar sized color classes | As mentioned in Grunbaum's paper, this is a conjecture of Erdős from 1964. It was solved completely by Hajnal and Szemerédi in 1970 and is now known as the Hajnal-Szemerédi theorem. A simpler proof was given by Kierstead and Kostochka in 2008. See the wikipedia page on [equitable coloring](https://en.wikipedia.org/wiki... | 5 | https://mathoverflow.net/users/2233 | 406398 | 166,575 |
https://mathoverflow.net/questions/406414 | 1 | In the question ([$C(X)$ as finitely generated $C^\*$-algebra](https://mathoverflow.net/questions/233512/cx-as-finitely-generated-c-algebra)), the answer show that spectrum of an abelian unital finitely generated C\*-algebra is homeomorphic to compact subset of $\mathbb{C}^{n}$. I would like to know more about the deta... | https://mathoverflow.net/users/172458 | finitely generated C*-algebra as $C(X)$ | Let $X$ be a compact Hausdorff space. Let $C(X)$ be the set of all continuous functions $f:X\rightarrow\mathbb{C}$. I claim that $X$ embeds into $\mathbb{C}^{n}$ if and only if the $C^{\*}$-algebra $C(X)$ is generated by $n$ functions. To prove this fact, we need to the complex Stone Weierstrass theorem.
We say that ... | 3 | https://mathoverflow.net/users/22277 | 406425 | 166,584 |
https://mathoverflow.net/questions/406427 | 13 | Is the consistency of classical third-order arithmetic provable in the logic of a topos with natural numbers?
(My guess would be yes, but I haven't seen this anywhere.)
Edit: in the original version I used the name PA$\_3$ as an abbreviation for classical third-order arithmetic, and comments have followed suit, but... | https://mathoverflow.net/users/170446 | Consistency proof in topos logic | Let $\Omega\_{\neg\neg} = \{p \in \Omega \mid \neg\neg p \Rightarrow p\}$ be the object of $\neg\neg$-stable truth values, and let us write $P\_{\neg\neg}(A) = {\Omega\_{\neg\neg}}^A$ for the object of $\neg\neg$-stable subobjects of $A$. Observe that $\Omega\_{\neg\neg}$ is a complete Boolean algebra, and this fact ca... | 17 | https://mathoverflow.net/users/1176 | 406445 | 166,595 |
https://mathoverflow.net/questions/406367 | 25 | Is there a result showing that something along the lines of the three body problem is undecidable? Or are they known to be decidable or neither?
I mean problems along the lines of the following formulated in some suitable system:
Given masses, velocities and positions in 3 dimensions and a distance d (assume all ex... | https://mathoverflow.net/users/23648 | Decidability of 3 body problem | The paper [Undecidability in $\mathbb{R}^n$: Riddled Basins,
the KAM Tori, and the Stability of the Solar System](http://philsci-archive.pitt.edu/13175/1/parker2003.pdf) by Matthew W. Parker (*Philosophy of Science* **70** (April 2003), 359–382) comes close to answering your question. A classical problem in the same sp... | 14 | https://mathoverflow.net/users/3106 | 406449 | 166,596 |
https://mathoverflow.net/questions/406424 | 1 | Is there a solution to this integral?
$$\int\_{0}^{y} x^{-a} \exp \left[- \frac{(b - cx^{-d})^2}{2} \right] dx,$$
where $a > 0$ and $d > 0$.
| https://mathoverflow.net/users/103291 | Solution to $\int_{0}^{y} x^{-a} \exp \left[- \frac{(b - cx^{-d})^2}{2} \right] dx$ | By using series expansion, change of variable and Eq. (3.381.9) of the book: "I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 8th ed. Burlington, MA, USA: Academic Press, 2015", I was able to find this solution:
$$\int\_{0}^{y} x^{-a} \exp \left[- \frac{(b - cx^{-d})^2}{2} \right] dx
\\=... | 3 | https://mathoverflow.net/users/103291 | 406452 | 166,597 |
https://mathoverflow.net/questions/406451 | 11 | $\newcommand{\Grp}{\mathrm{Grp}}$Consider the category of groups $\Grp$, and within it we have the solvable groups $S$. Then any group $G$ determines the functor from solvable groups: $$h\_G:=\text{hom}\_{\Grp}(\\_,G):S^{\mathrm{op}}\rightarrow \text{Set}$$
Does this functor determine the group $G$? More concretely, ... | https://mathoverflow.net/users/128502 | Are groups determined by their morphisms from solvable groups? | Let $G, G'$ be two non-isomorphic Tarski monsters of prime exponent $p$ or two non-isomorphic torsion-free Tarski monsters. Then for every solvable group $A$, $\mathbb{hom}(A,G)\cong \mathbb{hom}(A,G')$.
| 24 | https://mathoverflow.net/users/157261 | 406454 | 166,598 |
https://mathoverflow.net/questions/406458 | 2 | Consider [*General Set Theory*](https://en.wikipedia.org/wiki/General_set_theory) ($ \mathsf { GST } $) axiomatized by the following.
1. **Axiom of Extensionality:** The sets $ x $ and $ y $ are the same set if they have the same members:
$$ \forall x \forall y \bigl ( \forall z ( z \in x \leftrightarrow z \in y ) \r... | https://mathoverflow.net/users/76416 | Representation of the equality relation between hereditarily finite sets in weak set theories | This question is based on a misunderstanding of the situation already in $\mathsf{PA}$: equality is representable in $\mathsf{PA}$ only for *terms*, or at the very least relatively simple formulas. To see this consider, given an arbitrary sentence $\varphi$, the formula (modulo obvious abbreviations) $$\psi(x)\equiv(x=... | 4 | https://mathoverflow.net/users/8133 | 406459 | 166,601 |
https://mathoverflow.net/questions/406407 | 1 | I recently saw a question here on mathoverflow: «For what n and t can a square be partitioned into n similar rectangles in t congruence classes?», where Joseph Gordon gave a proof that, indeed, a square can be partitioned into n non-congruent similar rectangles for any $n\ge3$. His method involves the use of Fibonacci-... | https://mathoverflow.net/users/415477 | Tiling a square with similar non-congruent rectangles. What is the aspect ratio of the rectangles as n grows large? | You are correct.
Let $\varphi$ be the golden ratio and $r\_n$ be the ratio you want (long side to short).
According to the answer you mention, for a certain positive constant $A\_n:$ $$r\_n=\frac{F\_{n-1} A\_n + F\_{n-2}}{F\_{n-2} A\_n + F\_{n-3}}. \tag{\*}$$ It follows that (for odd $n$) $$\frac{F\_{n-1}}{F\_{n-2}... | 1 | https://mathoverflow.net/users/8008 | 406466 | 166,604 |
https://mathoverflow.net/questions/406485 | 2 | A famous theorem by Sumihiro states that, given a normal quasi-projective variety $X$ with a regular $G$-action (where $G$ is a connected linear algebraic group), there is a G-equivariant
projective embedding $$X\hookrightarrow \mathbb{P}^n,$$ where $G$ acts on $\mathbb{P}^n$ linearly.
The closure of the image of this ... | https://mathoverflow.net/users/114985 | $\mathbb{C}^*$-equivariant smooth completion of a quasiprojective variety | Since you seem to work in characteristic zero (even over $\mathbb{C}$), you can first take any $\mathbb{C}^\times$-equivariant completion, and then take its **canonical** resolution of singularities (see, e.g., Bierstone, E. and P. Milman, “Functoriality in resolution of singularities”, Research Institute
for Mathemati... | 3 | https://mathoverflow.net/users/4428 | 406486 | 166,608 |
https://mathoverflow.net/questions/406343 | 9 | We say a complex manifold $X$ has the *resolution property* if every coherent sheaf $\mathcal{M}$ on $X$ admits a surjection $\mathcal{E}\twoheadrightarrow \mathcal{M}$ by some finite rank locally free sheaf $\mathcal{E}$.
It is well-known that any projective manifold has the resolution property ([SGA 6, Expose II, 2... | https://mathoverflow.net/users/24965 | Does $X\times Y$ have the resolution property if both $X$ and $Y$ have? | Please find below a short argument in the case of schemes. The answer is positive for algebraic spaces too; in that case it can be proven using the characterization: $X$ has the resolution property $\Leftrightarrow$ $X = X'/GL\_n$ with $X'$ quasi-affine $+$ products of quasi-affines are quasi-affine $+$ a bit more work... | 7 | https://mathoverflow.net/users/152991 | 406487 | 166,609 |
https://mathoverflow.net/questions/406426 | 7 | Let $K$ be a $p$-adic field (a finite extension of the field of $p$-adic numbers ${\mathbb Q}\_p$).
Let $T$ be a $K$-torus with character group $X={\sf X}^\*(T)$ and cocharacter group $Y={\sf X}\_\*(T)=X^\vee$.
I would like to have *explicit cocycles* for all Galois cohomology classes in $H^1(K,T)$.
We can compute $H... | https://mathoverflow.net/users/4149 | Explicit cocycles for the first Galois cohomology of a $p$-adic torus | Answer of James S. Milne: Most probably, this homomorphism
$$\lambda\colon\, H^{-1}(\Gamma\_{L/K}, Y)\overset\sim\longrightarrow H^1(\Gamma\_{L/K}, Y\otimes\_{\Bbb Z} L^\times)$$
is just the cup-product with the fundamental class in $H^2(\Gamma\_{L/K}, L^\times)$.
| 1 | https://mathoverflow.net/users/4149 | 406509 | 166,614 |
https://mathoverflow.net/questions/406417 | 5 | The $\mathbf{A}^1$-invariance of vector bundles have been discussed in, for example, [this paper](https://projecteuclid.org/journals/duke-mathematical-journal/volume-166/issue-10/Affine-representability-results-in-A1-homotopy-theory-I--Vector/10.1215/00127094-0000014X.short) by Asok, Hoyois and Wendt. This of course im... | https://mathoverflow.net/users/100553 | $\mathbf{A}^1$-invariance of Brauer groups and $H^2_{\mathrm{et}}(-;\mathbb{G}_m)$ | (For $i=0$, the map $H\_{\mathrm{et}}^{0}(\operatorname{Spec} A,\mathbb{G}\_{m}) \to H\_{\mathrm{et}}^{0}(\operatorname{Spec} A[t],\mathbb{G}\_{m})$ is an isomorphism if and only if $A$ is reduced.)
For $i=1$, it is a theorem of Traverso that $H\_{\mathrm{et}}^{1}(\operatorname{Spec} A,\mathbb{G}\_{m}) \to H\_{\mathr... | 6 | https://mathoverflow.net/users/15505 | 406516 | 166,616 |
https://mathoverflow.net/questions/406475 | 10 | Let $R$ be a ring (possibly noncommutative with zero-divisors). A non-unit and non-zero-divisor element $r \in R$ will be called *irreducible* if for all $a,b \in R$ such that $r=ab$, then $a$ or $b$ is a unit.
Many extensions $R$ of $\mathbb{Z}$ do not keep its set of irreducible elements, for example $2 = (1+i)(1-i... | https://mathoverflow.net/users/34538 | Existence of a finite extension of ℤ providing a finite extension of the primes | This post started as some minor observations, but I believe it now contains a full proof that there is no such ring. Throughout, we let $R$ be an example of the kind wanted.
**Observation 1**: $R$ is indecomposable.
*Proof of observation 1*: Write $R=S\_1\times S\_2$, with $S\_1,S\_2\neq 0$. Suppose, by way of cont... | 5 | https://mathoverflow.net/users/3199 | 406520 | 166,617 |
https://mathoverflow.net/questions/406436 | 1 | Let $f(n)$ be [A007814](https://oeis.org/A007814), the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$.
Then we have an integer sequence given by
\begin{align}
a(0)=a(1)&=1\\
a(2n)& = a(n)+a(n-2^{f(n)})+a(2n-2^{f(n)})\\
a(2n+1) ... | https://mathoverflow.net/users/231922 | Formula from the recurrence relation | The conjectured formula can be proved by induction on $\mathrm{wt}(n)$.
For $\mathrm{wt}(n)=0$, we have $n=0$ and the conjectured formula trivially holds.
Now, for a positive integer $\ell$, suppose that the conjectured formula holds for all $\mathrm{wt}(n)<\ell$. Let us prove it for $\mathrm{wt}(n)=\ell$. So, let ... | 2 | https://mathoverflow.net/users/7076 | 406527 | 166,618 |
https://mathoverflow.net/questions/406531 | 1 | Let $G$ be a Lie group and let $M \rightarrow B$ be a $G$-equivariant vector bundle with typical fiber $E$.
Suppose that $B$ and $E$ are both symplectic manifolds with a $G$-Hamiltonian action. Can we conclude from this that $M$ is also a symplectic manifold with a Hamiltonian action?
I have been told that given a ... | https://mathoverflow.net/users/172459 | Symplectic structure on a vector bundle | Let $f:M\rightarrow B$ be the symplectic fibration. A fibration whose typical fibre $F$ is a symplectic manifold, and the coordinate change are elements of the group of symplectomorphisms of $(F,\omega\_F)$. For the first question, a natural idea is to have the symplectic form $\omega\_M$ of $M$ compatible with the sym... | 2 | https://mathoverflow.net/users/80891 | 406536 | 166,619 |
https://mathoverflow.net/questions/406539 | 5 | Let $k\in\mathbb N$. Given a finite graph with two subsets of vertices $X$ and $Y$, Menger's Theorem gives a criterion for when there are $k$ pairwise disjoint paths starting in $X$ and ending in $Y$.
Now let $X\_1$, $Y\_1$, $\dots$,$X\_k$, $Y\_k$ be subsets of vertices. Is there a "similar" condition for when there ... | https://mathoverflow.net/users/51389 | Menger's theorem with restrictions on where the paths can begin and end | There is no known necessary and sufficient condition like in Menger's theorem.
However, there is a polynomial-time algorithm that decides if the paths exist. This is one of the main results of the Graph Minors Project of Robertson and Seymour (see [Graph Minors XIII](https://www.sciencedirect.com/science/article/pii/... | 10 | https://mathoverflow.net/users/2233 | 406540 | 166,620 |
https://mathoverflow.net/questions/404004 | 8 | Given a continuum $X$ (compact metrizable connected $X$) let $K(X)$ denote the hyperspace of nonempty compact subspaces of $X$ with the Vietoris topology and let $C(X)$ denote the (closed) subspace of $K(X)$ consisting of connected sets.
It is well known (and has been discussed on [MO before](https://mathoverflow.net... | https://mathoverflow.net/users/49381 | When does $C(X)$, $X$ a continuum, admit a continuous choice function? | I will post a CW answer to my own question to remove it from the unanswered list since Benjamin doesn't seem interested in turning his comment into an answer, but this is all informations taken from the section of *Hyperspaces* that he suggested.
To begin with let's fix the terminology *selectible* to refer to a cont... | 2 | https://mathoverflow.net/users/49381 | 406551 | 166,623 |
https://mathoverflow.net/questions/406548 | 4 | Let $C\_p$, $1<p<\infty$, be the Schatten-$p$-class.
Let $x\in C\_p$ be positive and $\|x\|\_p =1$.
Let $E$ be the conditional expectation onto the diagonal part.
If $\|E(x)\|\_p \ge 1-\delta $ for some small $\delta>0$, then can we get an estimate for
$\|x - E(x)\|\_p$ in terms of $\delta$? In particular, if $\delta\t... | https://mathoverflow.net/users/91769 | Norm estimate for the difference between a positive operator and its expectation | Yes. This follows from the uniform convexity of the Schatten $p$ classes.
Indeed, for every unitary diagonal operator $u$, we have $\| (x + u x u^\*)/2\|\_p \geq \| E( x+uxu^\*)/2\|\_p = \|E(x)\|p \geq 1-\delta$ ($E$ is a contraction). Therefore, we have $\|x - uxu^\*\|\_p\leq\varepsilon$ (where $\varepsilon = \varep... | 3 | https://mathoverflow.net/users/10265 | 406560 | 166,625 |
https://mathoverflow.net/questions/406544 | 2 | It is known that if a real continuous function $f(x)$ satisfies a local $\alpha$-Hölder condition on a closed interval $[a,b]$, the box dimension of the graph of $f(x)$ on $[a,b]$ will be not greater than $2-\alpha$.
But if the proposition above is taken inversely, that is, if the box dimension of the graph of a real... | https://mathoverflow.net/users/152618 | A question about box dimension and Hölder condition | The box dimension is not countably stable, so even one exceptional point could drive it up. For example, given $\epsilon>0$, the function $f(x)=x^{1/2} \sin(1/x)$ on $[0,1]$, with $f(0)=0$, is locally Lipschitz on $(0,1]$, yet the graph of $f$ has box dimension at least $1.25$. Indeed, for $j<k$, to cover the graph abo... | 2 | https://mathoverflow.net/users/7691 | 406576 | 166,632 |
https://mathoverflow.net/questions/406557 | 11 | A few months ago, I read a nice elementary proof of Lang's theorem:
**Theorem:** Let $G$ be a connected linear algebraic group over $\overline{\mathbb{F}}\_p$ and let $F : G \to G$ be a Frobenius map. Then the Lang map $L : G \to G$ defined by $L(g) = g^{-1}F(g)$ is surjective.
The proof showed that $L$ was both fi... | https://mathoverflow.net/users/175051 | Reference request: Elementary proof of Lang's theorem | The following is a rewrite of the proof of Lemma G from Steinberg's [On theorems of Lie-Kolchin, Borel, and Lang](https://doi.org/10.1016/B978-0-12-080550-1.50032-9). Let $G$ be an algebraic group over a finite field $k=\mathbb{F}\_q$ and let $A$ be the coordinate ring of $G$.
**Background from algebraic groups:** We... | 10 | https://mathoverflow.net/users/297 | 406580 | 166,634 |
https://mathoverflow.net/questions/406588 | 4 | I have precedently posted the same question on Math.Stackexchange (<https://math.stackexchange.com/questions/4277856/quasi-compact-surjective-morphism-of-smooth-k-schemes-is-flat>), but to no avail; I hope this is not too low-level for this site.
In the article "The Greenberg functor revisited'' (<https://doi.org/10.... | https://mathoverflow.net/users/171983 | Quasi-compact surjective morphism of smooth k-schemes is flat | The simplest blowup morphism $\mathrm{Bl}\_0(\mathbb{A}^2) \to \mathbb{A}^2$ (with center at a point) is not flat.
EDIT. Here is an example with affine morphism. Let
$$
X = \{ x\_1y\_1 + x\_2y\_2 + x\_3y\_3 = 0 \} \subset \mathbb{A}^4\_{x\_1,x\_2,x\_3,x\_4} \times \mathbb{A}^4\_{y\_1,y\_2,y\_3}
$$
and let $f \colon X... | 6 | https://mathoverflow.net/users/4428 | 406591 | 166,639 |
https://mathoverflow.net/questions/406592 | 0 | Consider the following block matrix:
$$
A = \begin{bmatrix}
0 & I\\
-M & -I
\end{bmatrix}
$$
Suppose matrix $M$ is positive definite and satisfies $M\succeq \alpha I$, where $\alpha>0$ is a constant. When will matrix $A$ be Hurwitz stable, i.e., all of the eigenvalues have negative real parts?
| https://mathoverflow.net/users/172027 | Conditions for a block matrix to be Hurwitz stable | The eigenvalues of $A$ are $(-1 \pm \sqrt{1-4s})/2$ where $s$ is an eigenvalue of $M$. If $s \ge 1/4$, these have real part $-1/2$, while if $0 < s < 1/4$, they are both real and negative. So it's always true when $M$ is positive definite.
| 1 | https://mathoverflow.net/users/13650 | 406595 | 166,641 |
https://mathoverflow.net/questions/406559 | 0 | For each $n\ge 1$, consider $X^i\_t=1-\beta t + W^i\_t$ for $i=1,\ldots n$ and $t\ge 0$, where $\beta>0$ and $(W^i\_t)\_{t\ge 0}$ are independent Brownian motions. $\phi\equiv \big((\phi^1\_t)\_{t\ge 0},\ldots, (\phi^N\_t)\_{t\ge 0}\big)$ is said to be an allocation strategy if every $(\phi^i\_t)\_{t\ge 0}$ is progress... | https://mathoverflow.net/users/261243 | Number of drifted Brownian motions that never hit zero under allocation | I think its O(1).
$$\lbrace \tau^{\phi}\_i > t \rbrace = \lbrace \tau^{\phi}\_i > t , \int\_0^t \phi^i\_sds > \frac {\beta t} 2 \rbrace \cup \lbrace \tau^{\phi}\_i > t,\int\_0^t \phi^i\_sds < \frac {\beta t} 2 \rbrace = $$,
so $$1\_{\lbrace \tau^{\phi}\_i > t \rbrace} \le 1\_{ \lbrace \tau^{\phi}\_i > t , \int\_0^t \ph... | 1 | https://mathoverflow.net/users/143907 | 406607 | 166,643 |
https://mathoverflow.net/questions/406601 | 5 | I am specifically referencing the property that, given a braided monoidal category with a braiding $c$ and left and right unitors $\lambda, \rho$,
$$
\lambda\_A \circ c\_{A,I}=\rho\_{A},
$$
for any object $A$. This equation is stated in almost every reference (first done so by [Joyal and Street](https://www.sciencedire... | https://mathoverflow.net/users/419447 | What is the proof of the compatibility of a braiding with the unitors? | Using the notation of [Joyal and Street](https://www.sciencedirect.com/science/article/pii/S0001870883710558) §2, here’s a proof of $\newcommand{\r}{\rho}\newcommand{\l}{\lambda}\newcommand{\x}{\otimes}\newcommand{\comp}{\!\!\cdot\!}\r\_A = \l\_A \comp c\_{A,I}$. Since $\l$ and $c$ are invertible, it suffices to prove ... | 5 | https://mathoverflow.net/users/2273 | 406623 | 166,646 |
https://mathoverflow.net/questions/406608 | 4 | Let $U(n)$ be the compact manifold of unitary $(n \times n)$-matrices and let $\mu\_n$ denote the Haar-probability measure on $U(n)$. For $m < n$ does there exists a measurable (maybe even continuous or smooth) map
$$
F: \ U(n) \rightarrow U(m)
$$
with the property, that
$$
\mu\_m(A) = \mu\_n(F^{-1}(A))
$$
for every $A... | https://mathoverflow.net/users/409412 | A 'projective' property of the Haar U(n) measure | For any $m<n$ the $n\times n$ unitary matrix $\Omega$ has the block decomposition
$$\Omega=\begin{pmatrix} A&B\\ C&D\end{pmatrix},$$
where $A$ has dimensions $m\times m$, $D$ has dimensions $(n-m)\times(n-m)$, $B$ has dimensions $m\times(n-m)$ and $C$ has dimensions $(n-m)\times m$. Up to a set of measure zero, the mat... | 3 | https://mathoverflow.net/users/11260 | 406640 | 166,649 |
https://mathoverflow.net/questions/406631 | 0 | Let $p$ be a point in the $2$-dimensional hyperbolic space $H\_2$. Consider a normal coordinate system $(x,y)$ at centered at $p$. Let $o\_x\in H\_2$ be the point of coordinate $(x,0)$ and let $C\_x$ be the hyperbolic circle centered in $o$ passing through $p$.
I have difficulties with the $2$ following questions:
... | https://mathoverflow.net/users/176470 | Half space vs growing balls in the hyperbolic plane | Your question is actually best reformulated in terms of horospheres and horoballs. Indeed, the set of points described in normal coordinates as $\{(x,0), x\ge 0\}$ is precisely a geodesic ray $\gamma$ in the hyperbolic plane issued from the point $p$. Then the union of the interiors of all your $C\_x$ is precisely the ... | 1 | https://mathoverflow.net/users/8588 | 406642 | 166,650 |
https://mathoverflow.net/questions/406618 | 7 | I'm confused by a subtle point in the definition of analytic sets. Suppose I have a Polish space $X$. Now I start with the collection of Borel sets in $X$ and take all their continuous images in $X$. Do I get the entire family of analytic sets in this way? In other words, can I say in good conscience that the **analyti... | https://mathoverflow.net/users/106622 | In a Polish space, is every analytic set the continuous image of a Borel set from the same Polish space? | The answer is positive.
If $X$ is countable all subsets of $X$ are Borel, so they are their own continuous image through the identity function.
If $X$ is uncountable then it contains a copy of the Cantor space, but the Cantor space contains a copy of the Baire space $\mathcal N$ (which is necessarily $G\_\delta$ in... | 6 | https://mathoverflow.net/users/49381 | 406643 | 166,651 |
https://mathoverflow.net/questions/406546 | 4 | Consider Motzkin paths with the following weight:
All up-steps and the horizontal steps on height $0$ have weight $1$, all down-steps have weight $t$ and the horizontal steps on even heights have weight $1-t$ while those on odd heights have weight $t-1.$
Computations suggest that the weight $c\_n(t)$ of all paths o... | https://mathoverflow.net/users/5585 | A special class of weighted Motzkin paths | Let $F\_k(x)$ be a generating function for the total weight of Motzkin paths of length $n$ starting and ending at height $k$ and not going below that height. Let $G\_k(x)$ be a similar generating function with an additional restriction that paths do not come to height $k$, except at the beginning and at the end. Then $... | 3 | https://mathoverflow.net/users/7076 | 406646 | 166,652 |
https://mathoverflow.net/questions/406653 | 1 | Assume that $X\sim \mathcal N(\sigma\_1,\mu\_1)$ and $Y\sim \mathcal N(\sigma\_2,\mu\_2)$.
I want to estimate $\frac{\mu\_1+\mu\_2}{2}$ after observing $X,Y$.
In my setting, $\sigma\_1,\sigma\_2$ are known and we want to estimate the average of the means (e.g., what is the MLE of it?).
---
In the special case... | https://mathoverflow.net/users/47499 | Estimating the average of two gaussians' mean | The maximum likelihood estimator (MLE) for $(\mu\_1,\mu\_2)$ is $(X,Y)$. So, by the [functional invariance of the MLE](https://en.wikipedia.org/wiki/Maximum_likelihood_estimation#Functional_invariance) (that is, simply by definition), the MLE of $g(\mu\_1,\mu\_2):=(\mu\_1+\mu\_2)/2$ is $g(X,Y):=(X+Y)/2$, which also, ob... | 3 | https://mathoverflow.net/users/36721 | 406655 | 166,656 |
https://mathoverflow.net/questions/406651 | 5 | Let $G$ be a simple graph with vertex set $V$, such that for any two vertices $u,v\in V$, we have at least $k$ edge-disjoint paths of length $2$ (i.e., formed by $2$ edges) connecting $u$ with $v$. Let $n=|V|$ be the total number of vertices of $G$.
---
**Question:** What is the minimum value of $k$, expressed as... | https://mathoverflow.net/users/115803 | Graph combinatorial optimization problem | The answer is $k=n-2$. To see this, first note that $k \geq n-2$, since the complete graph on $n$ vertices minus an edge has the desired property for $k=n-3$. For the other inequality suppose that $G$ is an $n$-vertex graph such that for all distinct $u, v \in V(G)$, there are at least $n-2$ edge-disjoint paths between... | 6 | https://mathoverflow.net/users/2233 | 406656 | 166,657 |
https://mathoverflow.net/questions/406614 | 2 | Let $M$ be an almost surely continuous martingale that is not almost surely constant in time - that is, it is not the case that almost surely, $M\_t = M\_0$ for all $t$.
Assume further that $M$ is a time homogeneous Markov process, and that it is transitive, in the sense that for any measurable subset $U$ of $\mathbb... | https://mathoverflow.net/users/173490 | If a continuous function of a Markov martingale is a martingale, does the function have to be affine linear? | If you allow for an arbitrary starting point, then just use the optional stopping theorem for $f(M\_t)$: $$\begin{aligned} f(x) & = \mathbb E^x f(M(\tau\_{(a,b)})) \\ & = \mathbb P^x(M(\tau\_{(a,b)}) = b) f(b) + P^x(M(\tau\_{(a,b)}) = a) f(a) \\ & = \frac{x-a}{b-a} f(b) + \frac{b-x}{b-a} f(a) , \end{aligned}$$
as desir... | 4 | https://mathoverflow.net/users/108637 | 406657 | 166,658 |
https://mathoverflow.net/questions/403397 | 25 | Consider an integer of the form $$N = 1 + \sum\_{i=1}^r d\_i^2$$ where $d\_i \in \mathbb{N}\_{\ge 3}$ and $d\_i^2$ divides $N$.
**Question**: Must $r$ be greater than or equal to $9$?
*Checking* (with SageMath): It is true for $N \le 500000$.
*Remark*: If it is true in general then it is *optimal* because $$144 =... | https://mathoverflow.net/users/34538 | Sum of squares and divisibility | I hope so. But please double check (or, better, simplify) the argument below.
Denote $N=qs^2$ for $q$ squarefree. Then each $d\_i$ divides $s$, say $d\_i=s/m\_i$ and we get $$q=1/s^2+\sum\_{i=1}^r 1/m\_i^2, \quad\quad\quad (\heartsuit)$$
and the sum of $r+1$ square reciprocals is integer. Since $d\_i\geqslant 3$, we ... | 8 | https://mathoverflow.net/users/4312 | 406674 | 166,662 |
https://mathoverflow.net/questions/406638 | 6 | $\newcommand{\U}{\mathcal{U}}$
$\newcommand{\P}{\mathbb{P}}$
$\newcommand{\Q}{\mathbb{Q}}$
$\newcommand{\F}{\mathcal{F}}$
Recall the following equivalent definitions of a Ramsey ultrafilter over $\omega$:
>
> **Theorem (Ramsey Ultrafilter).** Let $\U$ be a non-principal ultrafilter over $\omega$. TFAE:
>
>
> 1. F... | https://mathoverflow.net/users/146831 | Ramsey ultrafilters on partial order | Presumably in (2) you meant to assume the sets $A\_p$ belong to $\mathcal U$.
The nontrivial thing is to show that (1) implies (2). The main point is that the Ramsey property of $\mathcal U$ implies that $\mathbb P$ has a $\mathcal U$-large suborder isomorphic either to $\omega$, $\omega^\*$, or an infinite discrete or... | 4 | https://mathoverflow.net/users/102684 | 406676 | 166,663 |
https://mathoverflow.net/questions/406663 | 7 | $\newcommand{\Cats}{\mathsf{Cats}}\newcommand{\MonCats}{\mathsf{MonCats}}\newcommand{\BrMonCats}{\mathsf{BrMonCats}}\newcommand{\SymMonCats}{\mathsf{SymMonCats}}\newcommand{\CMon}{\mathsf{CMon}}\newcommand{\Mon}{\mathsf{Mon}}\newcommand{\Z}{\mathbb{Z}}\newcommand{\Ab}{\mathsf{Ab}}\newcommand{\Grp}{\mathsf{Grp}}\newcomm... | https://mathoverflow.net/users/130058 | This is not a tensor: tensoring abelian groups over groups | The one line answer is that the category $\mathsf{Ab}$ of abelian groups is enriched over the **skew-monoidal category** $\mathsf{Gp}$ of groups, and that this "faux-tensor" defines a **skew-action** of the skew-monoidal category $\mathsf{Gp}$ on $\mathsf{Ab}$.
A *skew-monoidal structure* on a category $\mathcal{C}$ ... | 8 | https://mathoverflow.net/users/57405 | 406678 | 166,664 |
https://mathoverflow.net/questions/406680 | 1 | I'm trying to follow an argument in C. Giraud's "**High Dimensional Statistics**" (2nd Ed, p. 11 / $\S$ 1.2.3). The specific page is accessible via Google Books [here](https://www.google.com/books/edition/Introduction_to_High_Dimensional_Statist/YiM_EAAAQBAJ?hl=en&gbpv=1&dq=high%20dimensional%20statistics&pg=PR19&print... | https://mathoverflow.net/users/423486 | tail probability of max of Gaussians | The correct version of this formula is
$$P(\max\_j |\epsilon\_j| \ge x) = 1-(1-P(|\epsilon\_1| \ge x))^p
\underset{x \to\infty}\sim p \,P(|\epsilon\_1| \ge x)$$
for each real $p>0$,
which follows because $(1-u)^p=1-(p+o(1))u$ as $u\to0$.
(The reproduction quality of the preview of the book is indeed terrible. It is ... | 1 | https://mathoverflow.net/users/36721 | 406681 | 166,665 |
https://mathoverflow.net/questions/406689 | 1 | I posed [this question on math.stackexchange.com](https://math.stackexchange.com/q/4279365/64809) but have gotten no answer. So I post the question here in order to obtain an answer.
---
$\forall x\in \mathbf R^{n+1}$, let $x\_{(0)}\le x\_{(1)}\le\,\cdots\le x\_{(n)}$ denote the non-decreasing rearrangement of $x... | https://mathoverflow.net/users/32660 | A sufficient condition for weak majorization from below | We should prove that the sum of any $k$ $y$'s is not less than the sum of certain $k$ $x$'s (indeed, this property is equivalent to the condition that the sum of $k$ smallest $y$'s is not less than the sum of $k$ smallest $x$'s.) Let our $k$ $y$'s contain $y\_i$'s for $i=0,1,\ldots,p$, but do not contain $y\_{p+1}$ (th... | 1 | https://mathoverflow.net/users/4312 | 406691 | 166,667 |
https://mathoverflow.net/questions/406604 | 3 | **Let $\Omega$ be domain in $\mathbb{C}^n$. Suppose we have taken two distinct points from $\Omega$. Does there exist a domain $U$ in $\mathbb{C}$ such that there is a holomorphic function from $U$ to $\Omega$ whose range contains these two points?**
I tried to prove the identity theorem in several complex variables.... | https://mathoverflow.net/users/422133 | Holomorphic connectedness in several complex variables | Let $\Omega$ be a domain in $\mathbb C^n$. Fix two points $z\_0$, $z\_1$ in $\Omega$. Then
there exists a curve $\alpha : [0, 1] \to \Omega$ connecting these points. Using the Weierstrass approximation theorem there is a polynomial map $P : [0, 1] \to\Omega$ with
$P(0) = z\_0$ and $P(1) = z\_1$. Then it is easy to choo... | 2 | https://mathoverflow.net/users/343739 | 406697 | 166,669 |
https://mathoverflow.net/questions/406699 | 6 | Is there an example of a birational morphism of smooth complex projective varieties $f\colon X\to Y$, that cannot be factored into a chain $X\to X\_1\to\cdots\to X\_n\to Y$ of blow-down along smooth centers?
(By weak factorization theorem, we know in general that $f$ can be factorized into a zig-zag of blow-ups and b... | https://mathoverflow.net/users/nan | Birational morphism that is not successive blow-down along smooth centers? | Let $X \subset \mathbb{P}^2\_{x\_i} \times \mathbb{P}^6\_{y\_j}$ be given by the equations
$$
x\_1y\_1 + x\_2y\_2 + x\_3y\_3 = x\_1y\_4 + x\_2y\_5 + x\_3y\_6 = 0.
$$
It is smooth because its projection to $\mathbb{P}^2$ is a $\mathbb{P}^4$-fibration. This also implies that the rank of the Picard group of $X$ is 2. Now ... | 11 | https://mathoverflow.net/users/4428 | 406702 | 166,670 |
https://mathoverflow.net/questions/406705 | 7 | Suppose that $T \in TC(l^2( \mathbb{Z}))$ is trace class.
Consider its kernel $ T(i,j) = \langle e\_i, T e\_j \rangle $ where $ \{e\_i\}\_{i \in \mathbb{Z}}$ is an ONB for $l^2( \mathbb{Z})$. Now, consider the operator given by the kernel $T(i,j) K(i,j) $ for some numbers $K(i,j)$ such that $\sup\_{i,j} \vert K(i,j)\ve... | https://mathoverflow.net/users/143779 | If I multiply the coefficients of a trace-class operator with bounded complex numbers is it still trace class? | It's a little more complicated than I thought! Frederik Ravn Klausen pointed out an error. Still, I maintain that the product needn't even be bounded.
As the answer to [this question](https://math.stackexchange.com/questions/2907991/hadamard-product-optimal-bound-on-operator-norm) shows, in $M\_n$ you can find a unit... | 10 | https://mathoverflow.net/users/23141 | 406706 | 166,672 |
https://mathoverflow.net/questions/406713 | 1 | Let $G=(V\_1,E\_1)$ be a simple graph with vertex set $\{v\_1,v\_2,\ldots,v\_n\}$ and let $G'=(V\_2,E\_2)$ be another copy of $G$ with vertex set $\{u\_1,u\_2,\ldots,u\_n\}$. Assume $V\_1\cap V\_2= \emptyset$.
Let $H=(V,E)$ be a graph with $V=V\_1 \cup V\_2$ and $E=E\_1\cup E\_2\cup \{u\_1v\_1，u\_2v\_2, \ldots, v\_nv... | https://mathoverflow.net/users/375270 | What's the name of the graph operation of connecting two copies of a graph with a perfect matching? | I dont't know of a standard name, but it is $G \square K\_2$, where $\square$ denotes the [Cartesian product](https://en.wikipedia.org/wiki/Cartesian_product_of_graphs).
| 3 | https://mathoverflow.net/users/2233 | 406714 | 166,674 |
https://mathoverflow.net/questions/406619 | 2 | Let $H(q)$ be the set of reduced residues $mod(q)$ and $\Phi(a)$ Euler totient function. How can I evaluate
$\min\_{q\leq x}\frac{1}{q}\sum\_{a\epsilon\ H(q)}\frac{\Phi (a)}{a}$
| https://mathoverflow.net/users/169583 | Minimal value for the specific summatory Euler Phi function | Let us find a good approximation for your sum for a given large $q$. I will use the notation $\varphi(a)$ for the Euler function. First of all, by Möbius inversion,
$$
\sum\_{a\in H(q)}\frac{\varphi(a)}{a}=\sum\_{a\leq q}\left(\sum\_{d\mid (a,q)}\mu(d)\right)\frac{\varphi(a)}{a}=\sum\_{d\mid q}\mu(d)S\_d(q),
$$
where
$... | 2 | https://mathoverflow.net/users/101078 | 406715 | 166,675 |
https://mathoverflow.net/questions/406683 | 8 | $\def\CC{\mathbb{C}}\def\ZZ{\mathbb{Z}}\def\GL{\operatorname{GL}}$This question is about an assertion in [Mixed Hodge polynomials of character varieties](https://arxiv.org/abs/math/0612668), by Hausel and Rodriguez-Villegas. Fix positive integers $g$ and $n$ and let $\zeta$ be a primitive $n$-th root of unity.
Let
$$... | https://mathoverflow.net/users/297 | Why is the $\operatorname{GL}_n$ character variety "cohomologically" the product of the $\operatorname{PGL}_n$ character variety and a torus? | You need to keep reading to the proof of Theorem 2.2.12.
The main point is that the $PGL\_n$-character variety $\tilde{\mathcal{M}}\_n/\mathbb{C}$ is both a quotient by a torus $\mathcal{M}\_1/\mathbb{C}\cong (\mathbb{C}^\*)^{2g}$ of the $GL\_n$-character $\mathcal{M}\_n/\mathbb{C}$ and also a quotient of a finite gr... | 7 | https://mathoverflow.net/users/12218 | 406720 | 166,678 |
https://mathoverflow.net/questions/406671 | 2 | Consider the compact Lie groups $U(l)$ (the unitary group) and $U(1) \times SU(l)$ for some natural number $l$. Both the groups have the same Lie algebra $\frak{gl}\_l$. Which means that they both have the "same" dense Peter-Weyl subalgebra $PW$ of their $C^\*$-algebra of continuous functions.
The Gelfand-Naimark the... | https://mathoverflow.net/users/153228 | Gelfand-Naimark and Peter-Weyl for the unitary group | There were silly errors in my comments, so let me start again. You have a canonical surjection $U(1) \times SU(k) \to U(k)$ given by scalar multiplication, so that any representation of $U(k)$ lifts to a representation of $U(1) \times SU(k)$; the question is whether or not every representation of $U(1) \times SU(k)$ de... | 1 | https://mathoverflow.net/users/6999 | 406722 | 166,679 |
https://mathoverflow.net/questions/405736 | 4 | (This is a spin-off of [Determine the minimal elements of a Dynkin system generated by a finite set of finite sets](https://mathoverflow.net/q/405305))
Let $\Omega$ be a finite set. A [Dynkin system](https://en.wikipedia.org/wiki/Dynkin_system) on $\Omega$ is a subset of the power set of $\Omega$ containing $\Omega$,... | https://mathoverflow.net/users/3032 | What is the number of finite Dynkin systems? | We can compute the number of Dynkin systems for small $n$ using an almost-brute-force method. For efficient computation we represent a set ${a\_i}$ as a binary number $\sum\_i 2^{a\_i}$; then $\Omega = \{0, 1, 2, \ldots, n-1\}$ is represented by $2^n - 1$ and we can compute the complement with bitwise exclusive or by $... | 4 | https://mathoverflow.net/users/46140 | 406723 | 166,680 |
https://mathoverflow.net/questions/406712 | 3 | Let $(f\_n)\_n:X \to \mathbb R$ be a sequence of measurable functions on a measurable space $X$ converging pointwise to a function $f:X \to \mathbb R$, and let $(\mu\_n)\_n$ be a sequence of finite measures (e.g probability measures) on $X$ such that each $f\_n$ is integrable w.r.t $\mu\_n$.
>
> **Question.** *Unde... | https://mathoverflow.net/users/78539 | Dominated convergence theorem when the measure space also varies with $n$ | $\newcommand\ep\varepsilon$The conjunction of the following conditions is enough:
1. The $f\_n$'s are uniformly bounded: $|f\_n|\le M$ for some real $M>0$ and all $n$;
2. $X$ is Polish;
3. $f\_n\to f$ uniformly on every compact $K\subseteq X$;
4. $\mu\_n\to\mu$ weakly for some $\mu$.
Indeed, take any real $\ep>0$. ... | 6 | https://mathoverflow.net/users/36721 | 406728 | 166,682 |
https://mathoverflow.net/questions/406727 | 1 | This is a follow-up to my [previous question](https://mathoverflow.net/questions/406653/estimating-the-average-of-two-gaussians-mean).
Assume that $X\sim \mathcal N(\mu\_1,\sigma\_1^2)$ and $Y\sim \mathcal N(\mu\_2,\sigma\_2^2)$.
I want to estimate $\frac{\mu\_1+\mu\_2}{2}$ after observing $X,Y$.
In my setting, $... | https://mathoverflow.net/users/47499 | Estimating the average of two gaussians' mean with minimal squared error | $\newcommand\si\sigma$Clearly, the best estimator of $\mu\_1$ is $X$, no matter what $\si\_1$ and $\si\_2$ are. Similarly, the best estimator of $\mu\_2$ is $Y$, no matter what $\si\_1$ and $\si\_2$ are. So, one may argue, a good estimator of $(\mu\_1+\mu\_2)/2$ is the substitution estimator $(X+Y)/2$.
As was shown i... | 1 | https://mathoverflow.net/users/36721 | 406732 | 166,685 |
https://mathoverflow.net/questions/406735 | 2 | Is there a name for an adjunction between two categories such that
i) the unit of the adjunction is a natural isomorphism,
ii) the counit of the adjunction is a natural isomorphism?
| https://mathoverflow.net/users/153228 | Name for a categorical adjunction that is a "semi-equivalence" | An adjunction for which the unit is a natural isomorphism is called a [coreflective adjunction](https://ncatlab.org/nlab/show/coreflective+subcategory). An adjunction for which the counit is a natural isomorphism is called a [reflective adjunction](https://ncatlab.org/nlab/show/reflective+subcategory).
| 8 | https://mathoverflow.net/users/152679 | 406739 | 166,687 |
https://mathoverflow.net/questions/406644 | 1 | We are given a $n\times n$ symmetric matrix $M$ whose entries are positive integers.
Let $z\_{i,j}:=\frac{M\_{i,j}}{M\_{i,j}+\sum\_{k\neq i,j}\min(M\_{i,k},M\_{k,j})}$ for all $1\le i<j \le n$, and
$z:=\sum\_{i<j}z\_{i,j}$.
---
**Question:** Can we prove that the maximum value $z^\*$ of $z$ over *all* matrices ... | https://mathoverflow.net/users/115803 | Combinatorial graph optimization problem on integer adjacency matrices | Consider $$M\_{i,j} = \begin{cases} 1 & \textrm{if } i \equiv j \pmod 2 \\ N & \textrm{if } i \not\equiv j \pmod 2\end{cases}$$ where $N > 1$. Then
$$\min(M\_{i,k}, M\_{k,j}) = \begin{cases} 1 & \textrm{if } i \equiv k \pmod 2 \;\vee\; k \equiv j \pmod 2 \\ N & \textrm{if } i \not\equiv k \pmod 2 \;\wedge\; k \not\equi... | 2 | https://mathoverflow.net/users/46140 | 406744 | 166,688 |
https://mathoverflow.net/questions/406626 | 4 | I am looking for a triangulation of an $n$-dimensional simplex such that all sub-simplices are of comparable size, and are "as close as possible" to a regular simplex : the latter property could be formalized as "the minimum $n$-dimensional angle is bounded away from $0$, uniformly in the size of the triangulation".
I ... | https://mathoverflow.net/users/422680 | Triangulation of a simplex | You are looking for the edgewise subdivision:
*Edelsbrunner, H.; Grayson, D. R.*, [**Edgewise subdivision of a simplex**](http://dx.doi.org/10.1007/s004540010063), Discrete Comput. Geom. 24, No. 4, 707-719 (2000). [ZBL0968.51016](https://zbmath.org/?q=an:0968.51016).
The basic idea is to slice your simplex k times... | 6 | https://mathoverflow.net/users/59302 | 406748 | 166,690 |
https://mathoverflow.net/questions/370967 | 1 | Let $H$ be a Hilbert space. A vector subspace $W\subset B(H)$ is called a Fredholm subspace if there is an upper bound for the absolute value of Fredholm index of all Fredholm operators $T$ in $W$.
>
> Is there a classification of all $C^\*$-algebras $A$ which admit an irreducible representation $\phi:A \to B(H)$ i... | https://mathoverflow.net/users/36688 | Fredholm $C^*$-algebras | Any unital $C^\*$-algebra $A$ has an irreducible representation $\phi$ such that every Fredholm operator in $\phi(A)$ has index 0.
To see it, let me first repeat something from Nik Weaver's previous answer: if $\pi$ is a representation of $A$ such that $\pi(A)$ intersects the compact operators trivially, then any Fre... | 1 | https://mathoverflow.net/users/14497 | 406749 | 166,691 |
https://mathoverflow.net/questions/406760 | 4 | Let $P\_i=V\_{i}V\_{i}^{\top}\in\mathbb{R}^{m\times m}$ where $\forall i\in[T]: V\_{i}\in\mathbb{R}^{m\times n}$ is a “tall” matrix (i.e., $m \ge n$) with orthonormal columns. Note that these matrices are symmetric PSD.
Is the product of all these matrices, i.e., $P\_T P\_{T-1}\cdots P\_1$, necessarily diagonalizable... | https://mathoverflow.net/users/100796 | Is the product of projection matrices diagonalizable? | No. Take
$$v\_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \qquad v\_2 = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \qquad v\_3 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}.$$
Then
$$P\_1=v\_1 v\_1^T = \begin{bmatrix} 1&0 \\ 0&0 \end{bmatrix} \qquad P\_2=v\_2 v\_2^T = \begin{bmatrix} 1&1 \\ 1&1 \end{bmatrix} \qquad P\_3=v\_3 v\_3^T = ... | 7 | https://mathoverflow.net/users/297 | 406761 | 166,693 |
https://mathoverflow.net/questions/406773 | 6 | I was looking at two sequences of integers, both with prominent place is combinatorics. The first one appears, for instance, in [Stieltjes moment sequences for pattern-avoiding permutations](https://arxiv.org/pdf/2001.00393.pdf) (see page 23)
$$a\_n=\sum\_{k=0}^n\frac{\binom{2k}k\binom{n+1}{k+1}\binom{n+2}{k+1}}{(n+1)^... | https://mathoverflow.net/users/66131 | A moment sequence and Motzkin numbers. Modular coincidence? | Let's start with $b\_n$. Since Catalan number $C\_k$ is odd iff $k=2^m-1$, from [Lucas theorem](https://en.wikipedia.org/wiki/Lucas%27s_theorem) it follows that
$$b\_n=\sum\_{k=0}^n \binom{n}{2k}C\_k \equiv\sum\_{m\geq 0}\binom{n}{2(2^m-1)}\equiv 1+\nu\_2(\lfloor n/2\rfloor+1)\pmod2,$$
where $\nu\_2(\cdot)$ is the 2-ad... | 8 | https://mathoverflow.net/users/7076 | 406786 | 166,699 |
https://mathoverflow.net/questions/406785 | 10 | Let $A$ and $B$ be skew lines in $\mathbb{R}^3$. Choose four points $a\_1, a\_2, a\_3, a\_4$ on $A$ and four points $b\_1, b\_2, b\_3, b\_4$. For all $i,j \in [4]$ draw a line segment from $a\_i$ to $b\_j$. Since $A$ and $B$ are skew, none of these line segments intersect each other. Thus, this is a straight line drawi... | https://mathoverflow.net/users/2233 | Is this drawing of $K_{4,4}$ knotted? | Here is a proof that all such embeddings are knotless. Consider the four half-planes bounded by $A$ which each contain one of the points $b\_i$. Then these planes give an open-book decomposition which contains $K\_{4,4}$. Each plane contains four edges, but if $L$ is a cycle of $K\_{4,4}$, then it can only contain at m... | 5 | https://mathoverflow.net/users/126206 | 406788 | 166,701 |
https://mathoverflow.net/questions/406782 | 1 | Suppose it is given that there exists a ‘strictly positive’ vector $\vec x \in (0,1)^k$, which lies on the probability simplex $\sum\_i x\_i = 1$. What is the least number of inequalities of the form $\vec g\_i^T \vec x \geq 0$ required to ensure that the solution set is a singleton vector? Note that there is no restri... | https://mathoverflow.net/users/424786 | How many inequalities do I need to ensure a unique solution? | $\newcommand\v\vec\newcommand\R{\mathbb R}$For any natural $k\ge2$, $k$ inequalities (but not fewer than $k$) will suffice.
Indeed, take any vector $\v a\in(0,\infty)^k$ such that $\v1\cdot\v a=1$, where $\v1:=(1,\dots,1)\in\R^k$ and $\cdot$ denotes the dot product. Take any linearly independent vectors $\v g\_1,\dot... | 4 | https://mathoverflow.net/users/36721 | 406792 | 166,703 |
https://mathoverflow.net/questions/406468 | 1 | $\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\GL{GL}
$Let $F$ be local field of characteristic zero and $(W,\langle,\rangle)$ be a $2n$-dimensional symplectic space over $F$.
Let $X,X^\*$ be maximal totally isotropic subspaces of $W$, which are dual with respect to $\langle,\rangle$.
L... | https://mathoverflow.net/users/35898 | Part of some generic representation is also generic? |
>
> Let $\pi$ be the irreducible generic unramified representation of $Sp(W) $ that is a subquotient of $Ind(\chi\_1, \dots, \chi\_n)$.
>
>
>
I think the key here is to realise that this *does not exist* for all values of $\chi$. If the induction is irreducible (which is true for "sufficiently general" tuples of... | 2 | https://mathoverflow.net/users/2481 | 406797 | 166,706 |
https://mathoverflow.net/questions/406803 | 3 | Let $\Lambda$ be a simply laced root lattice and $w$ a Coxeter element of the Weyl group of $\Lambda$.
**Question:** Is it true that the action of $w$ on the $\mathbb{F}\_2$-vector space $\Lambda/2\Lambda$ is semisimple?
Motivation: I want to show that, if $F$ denotes the subset of $w$-fixed points on $\Lambda/2\La... | https://mathoverflow.net/users/110362 | Action of Coxeter element on mod $2$ root lattice is semisimple | The answer seems to be negative.
According to [this](https://mathoverflow.net/q/52100) answer, an operator $X\in\operatorname{M}(n,\mathbb{F}\_q)$ is semisimple if and only if $X^{q^m}=X$, where $m=\operatorname{lcm}(2,\ldots,n)$.
Now consider the standard realization of the root system of type $\mathsf{A}\_n$ insi... | 2 | https://mathoverflow.net/users/5018 | 406810 | 166,709 |
https://mathoverflow.net/questions/406738 | 3 | Let $wt(n)$ be [A000120](https://oeis.org/A000120), number of $1$'s in binary expansion of $n$ (or the binary weight of $n$)
and
$$n=2^{t\_1}(1+2^{t\_2+1}(1+\dots(1+2^{t\_{wt(n)}+1}))\dots)$$
Then we have an integer sequence given by
$$a(n)=\sum\limits\_{j=0}^{2^{wt(n)}-1}m^{wt(n)-wt(j)}\prod\limits\_{k=0}^{wt(n)-1}(1+... | https://mathoverflow.net/users/231922 | Sum with Stirling numbers of the second kind | The [same idea](https://mathoverflow.net/q/405210) of grouping terms by the number of unit bits, as well as grouping by the value of $\mathrm{wt}(j)$ (representing $j$ via individual bits) works here:
\begin{split}
s(n,m) &= \sum\_{\ell=0}^n \sum\_{t\_1 + \dots + t\_\ell \leq n-\ell} \sum\_{j\_1,\dots,j\_\ell\in\{0,1\}... | 2 | https://mathoverflow.net/users/7076 | 406817 | 166,713 |
https://mathoverflow.net/questions/406769 | 4 | A (real) normed space $(V, \lVert \cdot \rVert\_V)$ is called strictly convex if for all $x, y \in V \setminus \{ 0 \}$ we have
\begin{equation}
\lVert x + y \rVert\_V = \lVert x \rVert\_V + \lVert y \rVert\_V \implies \exists c >0 \enspace x = cy.
\end{equation}
A completion of a normed space $(V, \lVert \cdot \rVert... | https://mathoverflow.net/users/170491 | Is a completion of strictly convex normed space strictly convex? | No, the completion of a strictly convex normed space can fail to be strictly convex.
To put it differently, there are non strictly convex Banach spaces with a dense strictly convex subspace.
Here is a possible construction. To make things easier, it is tempting to start with a space where there is a good control on... | 6 | https://mathoverflow.net/users/10265 | 406818 | 166,714 |
https://mathoverflow.net/questions/204724 | 12 | (Split off from [Does every CAT(0) space embed in a measurable integral of $\mathbb{R}$-trees?](https://mathoverflow.net/questions/204710/does-every-cat0-space-embed-in-a-product-of-trees/204723) )
Fix an integer $k \ge 2$, and let
$MC0\_k \subset \mathbb{R}^{\binom{k}{2}}$ be the set of possible squared-distances be... | https://mathoverflow.net/users/5010 | What are the extremal CAT(0) metrics? | Let me describe a 6-point counterexample.
Let $K$ be a 2-dimensional cone with total angle $\theta=2{\cdot}\pi+\varepsilon$, where $\varepsilon$ is small and positive (any $0<\varepsilon<\tfrac\pi2$ will do).
Note that $K$ is CAT(0).
Consider the following 6 points in $K$: the tip $p$ + and an orbit $\{x\_1,x\_2,x\... | 6 | https://mathoverflow.net/users/1441 | 406833 | 166,720 |
https://mathoverflow.net/questions/406789 | 3 | Let $u$ be an harmonic function in a cylindrical domain $B\_2^{n-1}\times(-1,1)\subset\mathbb{R}^n$, and suppose its level sets $\Gamma\_t=\{u=t\}$ are graphs of functions on $B\_2^{n-1}$.
Consider a linear parametrization of $u$:
$$u\_t:=u-t.$$
Then the nodal set of $u\_t$ is $t$-level set of $u$:
$$\{u\_t=0\}=\{u=t\}... | https://mathoverflow.net/users/151368 | Geometric flow by the level sets of a harmonic function | There is a lot of current study of the level sets of harmonic functions in this exact way.
See these papers: <https://arxiv.org/abs/1209.4669> <https://arxiv.org/abs/2108.08402> and <https://arxiv.org/abs/1911.06754> .
Some may not look directly related to what you're asking, but note that anytime you are using the... | 3 | https://mathoverflow.net/users/1540 | 406836 | 166,721 |
https://mathoverflow.net/questions/406609 | 0 | This is a continuation of [Number of drifted Brownian motions that never hit zero under allocation](https://mathoverflow.net/questions/406559/number-of-drifted-brownian-motions-that-never-hit-zero-under-allocation)
For each $n\ge 1$, consider $X^i\_t=1+\beta t + W^i\_t$ for $i=1,\ldots n$ and $t\ge 0$, where $\beta>0... | https://mathoverflow.net/users/261243 | Does fixed allocation increase the proportion of positively drifted Brownian motions surviving forever? | This is not an answer to your question, but a similar result related to the case $\beta=0$ can be found in [Optimal surviving strategy for drifted Brownian motions with absorption](https://projecteuclid.org/journals/annals-of-probability/volume-46/issue-3/Optimal-surviving-strategy-for-drifted-Brownian-motions-with-abs... | 0 | https://mathoverflow.net/users/nan | 406837 | 166,722 |
https://mathoverflow.net/questions/406752 | 9 | Factorization in the ring $\mathbb{Z}[x]/(x^2+1)\mathbb{Z}[x]\cong \mathbb{Z}[i]$ is well known. For instance, $5$ and $13$ (and any prime $\equiv 1\pmod{4}$) are no longer prime.
The factorization of $5$ lifts to $\mathbb{Z}[x]/((x^2+1)^2)\mathbb{Z}[x]$, but it isn't a simple consequence of Hensel's lemma. The troub... | https://mathoverflow.net/users/3199 | Hensel's lemma, Bezout's identity, and the integers | Assume that $\mathcal{O} = \mathbb{Z}[x]/(q(x))$ is the ring of integers of a number field $K$. One can search for a prime $p$ which is not inert in $\mathcal{O}$ and such that there is a principal prime ideal above $p$. I think such primes exist because the density of principal prime ideals is $1/h$, where $h$ is the ... | 6 | https://mathoverflow.net/users/6506 | 406841 | 166,725 |
https://mathoverflow.net/questions/405634 | 3 | Define
$$RT(n,K\_l,f(n))=ex\_l(n,f(n))=\max\_G\{e(G): K\_l \not\subset G, v(G)=n, \alpha(G)\leq f(n)\}$$
and the Ramsey-Turán density function $f\_l:(0,1] \to \mathbb{R}$ as
$$f\_l(\alpha)=\lim\_{n\to \infty}\frac{ex\_l(n,\alpha n)}{{n\choose 2}}.$$
By Turán Theorem we have that $f\_l(\alpha)=1-\frac{1}{l-1}$ for eve... | https://mathoverflow.net/users/225950 | Ramsey-Turán density function is well defined | I answered the question in detail for $RT(n, K\_l, \alpha n)$ [here](https://math.stackexchange.com/a/4282372/383078) on Math StackExchange.
| 1 | https://mathoverflow.net/users/106323 | 406842 | 166,726 |
https://mathoverflow.net/questions/406775 | 9 | It seems to me that a considerably simpler proof [see below] of Birkhoff's ergodic theorem can be obtained for bounded observables than for more general $L^1$ observables. Therefore, I feel like it would be nicer to prove Birkhoff's ergodic theorem by starting off with the bounded case and then extending to the general... | https://mathoverflow.net/users/15570 | Can Birkhoff's ergodic theorem for integrable functions easily be deduced from Birkhoff's ergodic theorem for bounded functions? | There is a simple reduction of the Birkhoff ergodic Theorem for $L^1$ functions to the bounded case using Kakutani-Rokhlin towers, that I learned from H. Furstenberg and B. Weiss decades ago. We use the notation of the post, and assume that $T$ is ergodic. Given $f \in L^1(X)$ we may assume it is nonnegative, and then ... | 9 | https://mathoverflow.net/users/7691 | 406846 | 166,727 |
https://mathoverflow.net/questions/406849 | 0 | Let $X\_1, \ldots, X\_n$ be independent and identically distributed random variables. Let $f:\mathbb{R}^n \to \mathbb{R}$ be a bounded difference function, i.e., for any $x,y \in \mathbb{R}^n$ that differ only in coordinate $i$, we have $|f(x) - f(y)| \leq c\_i$. Under this setting, [McDiarmid’s inequality](https://en.... | https://mathoverflow.net/users/16976 | CLT for bounded difference functions | In the general setting you propose, the answer is negative. Suppose that $X\_i$ take the values $\pm 1$ with equal probability. Let $h: {\mathbb R} \to {\mathbb R}$ be a Lipschitz function with constant 1, e.g., $$ (\*) \quad h(x)=\max\{x,0\} \, ,$$ and define
$$f\_n(x\_1,\dots,x\_n):=\sqrt{n} \cdot h \left(\frac{x\_1+... | 2 | https://mathoverflow.net/users/7691 | 406851 | 166,730 |
https://mathoverflow.net/questions/406859 | 1 | In [Tka] the author writes:
>
> "Every topological space $X$ can be represented as an open continuous image of a completely regular submetrizable space $Y$ (in other words, $Y$ admits a continuous one-to-one mapping onto a metrizable space) — the corresponding construction is given on p. 331 of [Eng]".
>
>
>
B... | https://mathoverflow.net/users/142738 | Open images of submetrizable spaces | It seems quite likely that it was intended to be page 331 of the Engelking's book published in 1977 in PWN, Warszava1 - rather then the later edition by Heldermann (revised and completed edition, 1989).
I will quote in full exercise 4.2.D (page 331 in the older edition, page 264 in the newer edition):
>
> 4.2.D. ... | 1 | https://mathoverflow.net/users/8250 | 406862 | 166,731 |
https://mathoverflow.net/questions/395720 | 11 | **Remark 1:** On p.384 of volume 3 of Gauss's *Werke*, which is a part of an unpublished treatise on the arithmetic geometric mean, Gauss makes the following remark:
>
> **On the theory of the division of numbers into four squares**:
> The theorem that the product of two sums of four squares is itself a sum of four... | https://mathoverflow.net/users/118562 | Explanation of several unpublished remarks of Gauss on representations of a given number as sums of two, three and four squares | Let me add a few remarks concerning 2. If $p \equiv 3 \bmod 4$, then ${\mathbb F}\_p(i) = {\mathbb F}\_{p^2}$. The relative norm of $x+iy$ is the product of $x+iy$ and its conjugate $x-iy$, but the latter is the image of the Frobenius automorphism, i.e., $x-iy = (x+iy)^p$. This shows that $x^2 + y^2 = (x+iy)(x-iy) = (x... | 3 | https://mathoverflow.net/users/3503 | 406866 | 166,732 |
https://mathoverflow.net/questions/406864 | 2 | I'm working with the notion of direct integrals as in Dixmier. Briefly: Given a measurable space $X$ and a family of separable Hilbert spaces $(H\_x)\_{x\in X}$, a measurable structure is a subspace $Y$ of the sections $\Pi\_x H\_x$ which satisfies the following axioms:
(1) for all $u\in Y$, $x\mapsto \|u(x)\|\_{H\_x... | https://mathoverflow.net/users/70434 | Measurable structures for direct integrals | Up to isomorphism, there are no other examples. But literally speaking, there are other examples. You could choose a non-measurable map $U$ from $X$ to the unitary group $\mathcal{U}(H)$, for instance by fixing a nontrivial unitary $U\_0 \in \mathcal{U}(H)$ and putting $U(x) = U\_0$ when $x$ belongs to a nonmeasurable ... | 4 | https://mathoverflow.net/users/159170 | 406874 | 166,735 |
https://mathoverflow.net/questions/406877 | 4 | Let $f(x),g(x)$ be polynomials in $\mathbb{Q}[x]$. If $\mathrm{deg}(f)\geq2$ and $f$ irreducible, is the composition $f(g(x))$ always reduced (has no repeated irreducible factors)?
(If we do not ask $\mathrm{deg}(f)\geq2$ we can take $f(x)=x-1, g(x)=x^2+1$; if we do not ask $f$ be reducible, we can take $f(x)=(x-1)x$... | https://mathoverflow.net/users/nan | Does irreducible polynomial remain reduced by pre-composition? | It may have repeated irreducible factor. Take $f(x)=x^2+1$ and $g(x)=x+f(x) h(x)$ so that $g'(i)=0$. Then $f^2$ divides $f(g) $.
| 12 | https://mathoverflow.net/users/4312 | 406883 | 166,738 |
https://mathoverflow.net/questions/406884 | 2 | I am trying to find smooth functions $f : \mathbb{R}\_+ \to \mathbb{R}\_+$ such that the quantity $$\Delta\_f(x) := 2f(x)-f(2x)$$
is positive for $x$ large enough and has the **greatest asymptotic growth**.
---
It seems clear that one cannot go beyond a linear growth, since $\Delta\_f$ vanishes for linear functio... | https://mathoverflow.net/users/50777 | Largest asymptotic growth for $2f(x)-f(2x)$ | Let us discretise the problem by setting $a\_n=2^{-n}f(2^n)$, $b\_n=2^{-n-1}\Delta\_f(2^n)$. Then your relation becomes,
$$b\_n=a\_n-a\_{n+1}.$$
since $a\_n,b\_n$ are non-negative, we conclude that
$$\sum\_{n=1}^\infty b\_n<\infty.$$
This is a necessary and sufficient condition. Indeed, take any summable sequence $b\_n... | 3 | https://mathoverflow.net/users/25510 | 406891 | 166,741 |
https://mathoverflow.net/questions/406886 | 4 | $\DeclareMathOperator{\Inv}{Inv}\DeclareMathOperator{\Erg}{Erg}$This is mostly curiosity on my part and I hope that the MO community might be able to help.
For $c\in (0,4]$ consider the logistic map
$$
T\_c:[0,1]\to[0,1],\;\;T\_c(x)=cx(1-x).
$$
Denote by $\Inv\_c$ the collection of Borel probability measures on $[0,1... | https://mathoverflow.net/users/20302 | Ergodic measures for the logistic map | When $c=4$, the map $T\_4(x)=4x(1-x)$ on the unit interval is semi-conjugate to the transformation $z\mapsto z^2$ of the unit circle via $z\mapsto\frac{1}{2}-\frac{1}{4}\left(z+\frac{1}{z}\right)$:
$$
T\_4\left(\frac{1}{2}-\frac{1}{4}\left(z+\frac{1}{z}\right)\right)=\frac{1}{2}-\frac{1}{4}\left(z^2+\frac{1}{z^2}\right... | 10 | https://mathoverflow.net/users/128556 | 406892 | 166,742 |
https://mathoverflow.net/questions/406872 | 2 | Say $\ell^2-\ell-\delta\equiv0\bmod p$ is a polynomial $x^2-x-\delta$ with root $\ell$ and we lift to $\ell'^2-\ell'-\delta\equiv0\bmod p^2$ by Hensel lifting my question is following: if $g^y\equiv \ell^2\bmod p$ held at a generator $g$ then is there an explicit generator $g'$, which can be found in polynomial time, f... | https://mathoverflow.net/users/10035 | A problem on generators and Hensel lifting | Write $g'=g(1+ap)$ so that
$$g'^y\equiv g^y(1 + pya)\pmod{p^2}.$$
It follows that we can take
$$a = \frac{(\ell^2/g^{y}\bmod p^2)-1}{py},$$
which can be computed in polynomial time.
| 2 | https://mathoverflow.net/users/7076 | 406898 | 166,744 |
https://mathoverflow.net/questions/406857 | 5 | Let $K$ be a finite extension of $\mathbb{Q}\_p$. Let $F$ be a Lubin–Tate formal group law defined over $K$ with endomorphism $f(T)$ corresponding to $\pi$ (a uniformizer of $K$). Then one can define the logarithm of $F$ to be $\lambda\_F(T) = \lim\_{n\rightarrow\infty}\pi^{-n}f^n(T)$. Then $\lambda\_F(T) = T + \text{h... | https://mathoverflow.net/users/378621 | Question about log and exp of a formal group law | The radius of convergence of any formal group $F$ (one-dimensional, finite height) is $1$, in other words $L\_F$ will converge at all $z\in\Bbb C\_p$ with $v(z)>0$.
In particular, the logarithm is convergent at all torsion points of the formal group. What goes wrong, goes wrong with the exponential, whose series is n... | 13 | https://mathoverflow.net/users/11417 | 406912 | 166,748 |
https://mathoverflow.net/questions/406905 | 9 | Suppose $A\_1,\dots,A\_n$ are measurable subsets of the plane that are all related by rigid motions such that $|(A\_1 \cup \dots \cup A\_n)^c| = 0$ and $|A\_i \cap A\_j| = 0$ for all $1 \leq i < j \leq n$, where $|S|$ denotes the Lebesgue measure of $S$.
Must each $A\_i$ have the property that $|A\_i \cap B(r)|/|B(r)... | https://mathoverflow.net/users/3621 | Tiling the plane with finitely many congruent pieces | Write $A\_i=T\_i(A)$ for $A=A\_1$, where each $T\_i$ is a rigid motion. For each $i$, we have $|A\_i\cap B(r)|=|T\_i(A\cap T\_i^{-1}(B(r))|=|A\cap T\_i^{-1}(B(r))|$. The symmetric difference between this set and $A\cap B(r)$ is contained in the symmetric difference of $T\_i^{-1}(B(r))$ and $B(r)$, whose measure is $O(r... | 11 | https://mathoverflow.net/users/30186 | 406913 | 166,749 |
https://mathoverflow.net/questions/406921 | 1 | Let $X = G/\Gamma$ denote the Iwasawa threefold, where
$$G = \left\{\begin{pmatrix} 1 & z\_1 & z\_3\\ 0 & 1 & z\_2\\ 0 & 0 & 1\end{pmatrix} : z\_1, z\_2, z\_3 \in \mathbb{C} \right\},$$
and $\Gamma$ is the discrete subgroup $$\Gamma=\left\{\begin{pmatrix} 1 & z\_1 & z\_3\\ 0 & 1 & z\_2\\ 0 & 0 & 1\end{pmatrix} : z\... | https://mathoverflow.net/users/105103 | Reference for the Hodge diamond of the Iwasawa threefold | Should have waited a few minutes, it's on page 49 of Danielle Angela's *Cohomological aspects of non-Kähler manifolds*: <https://arxiv.org/pdf/1302.0524.pdf>
| 2 | https://mathoverflow.net/users/105103 | 406922 | 166,751 |
https://mathoverflow.net/questions/406920 | 1 | Let $A\_{n}(F) $ denote the $n \times n$ skew symmetric matrices over a finite field $F$. Suppose $n$ be even and $N$ be a subspace of $A\_{n}(F) $. Now if all the non-zero matrices in $N$ are invertible, then the maximum the dimension of $N$ will be $n/2$. The upper bound of this maximum dimension follows from Chevall... | https://mathoverflow.net/users/215016 | Problem concerning about an $n$-subspace of $ A_{n}(F) $ | If you take $n/2 \times n/2$ matrix $A$, with an irreducible minimal polynomial (as can be constructed with a [companion matrix](https://en.wikipedia.org/wiki/Companion_matrix)), the subspace $W = \text{span}\left(I, A, \ldots, A^{n/2-1}\right)$ will be a subspace of matrices of dimension $n/2$ where all non-zero matri... | 2 | https://mathoverflow.net/users/7838 | 406925 | 166,753 |
https://mathoverflow.net/questions/406863 | 1 | Let $f: \mathbb R \to \mathbb R$ be a Lipschitz strictly monotone (so, in particular, invertible) function. Let $u: \mathbb R \to \mathbb R$. If $f \circ u \in BV$ can we conclude that $u \in BV$?
| https://mathoverflow.net/users/157076 | If $f \circ u \in BV$ and $f$ is strictly monotone, then is $u \in BV$? | The answer is no.
E.g., let $f(x):=\min(1,|x|)x$ for real $x$. Then $f$ is Lipschitz and strictly monotone.
Let then $g$ be any function in $BV$ such that $g(1/n)=(-1)^n/n^2$ for all natural $n$; it is easy to see that such a function exists. (For instance, let $g=0$ on $(-\infty,0]\cup(1,\infty)$ and let $g$ be mo... | 6 | https://mathoverflow.net/users/36721 | 406930 | 166,755 |
https://mathoverflow.net/questions/406901 | 1 | Let $(X(t))\_{t \in [-1,1]}$ be a centered non-stationary smooth gaussian process with covariation function $\rho(t,s) = \mathbb E[X(t)X(s)]$. For $t\_0 \in (-1,1)$ and $\epsilon \in (-1-t\_0,1-t\_0)$, define
$$
p\_X(t\_0,\epsilon) : = \mathbb P(X(t) = 0\,\text{ for some } t \in [t\_0-\epsilon,t\_0+\epsilon])
$$
>
... | https://mathoverflow.net/users/78539 | Lower-bound on zero-crossing probability of the nonstationary gaussian process $X(t) = tU+(1-t^2)^{1/2}V$, with $(U,V) \sim N(0,I_2)$ | $\newcommand\ep\epsilon\newcommand\si\sigma\newcommand\th\theta$In your concrete example,
$$p\_X(t\_0,\ep)=P\Big(m\_1<\frac VU<m\_2\Big),$$
where
$$m\_1:=\min\_{t\in[t\_0-\ep,t\_0+\ep]}r(t)
=r(t\_0+\ep),\quad
m\_2:=\max\_{t\in[t\_0-\ep,t\_0+\ep]}r(t)
=r(t\_0-\ep),\quad r(t):=-\frac t{(1-t^2)^{1/2}};$$
this follows bec... | 1 | https://mathoverflow.net/users/36721 | 406932 | 166,756 |
https://mathoverflow.net/questions/406543 | 3 | The following theorem and proof are in **Applications of the proper forcing axiom**, the Baumgartner's paper in the book *Handbook of Set-theoretic topology*.
$3.6$
***THEOREM***. Assume PFA. Suppose that for each $\alpha < \omega\_1$ a set $S\_\alpha \subseteq \omega\_1$ is given such that, for every limit ordinal $... | https://mathoverflow.net/users/169636 | Adding a closed unbounded set containing of only limit ordinals with a special property | Maybe the following idea works: Given a condition as above, also require the following:
(1) for each $\alpha, f(\alpha)$ is indecomposable,
(2) suppose $dom(p)=\{\beta\_0 < \beta\_1 < \cdots < \beta\_n\}$. Then there exists a closed subset $C$ of $f(\beta\_n)$ of order type $\beta\_n$ such that $C \cap \bigcup\_{\g... | 3 | https://mathoverflow.net/users/11115 | 406945 | 166,764 |
https://mathoverflow.net/questions/406946 | 1 | This statement is proved by [Vizing](https://books.google.co.kr/books?id=leL0Y5N0bFoC&pg=PA30&lpg=PA30&dq=V.%20G.%20Vizing,%20Vertex%20colorings%20with%20given%20colors%20(Russian).%20Diskret.%20Analiz.%2029%20(1976),%203%E2%80%9310.&source=bl&ots=zKFBHSs3nD&sig=ACfU3U0UAZFYC02b2_YFOK1NUSg8KP91BA&hl=ko&sa=X&ved=2ahUKEw... | https://mathoverflow.net/users/384338 | $K_{k,m}$ is $k$-choosable if and only if $m<k^k$ | Assume that $K\_{k,m}$ is not $k$-choosable with some lists of admissible colors. Let $A\_1$, $A\_2$, $\ldots$, $A\_k$ be sets of admissible colors in the small part (that with $k$ vertices). Choose arbitrarily colors $a\_i\in A\_i$ for all $i=1,\ldots,k$. The large part must contain a vertex with admissible colors $a\... | 1 | https://mathoverflow.net/users/4312 | 406947 | 166,765 |
https://mathoverflow.net/questions/406734 | 4 | Recall the construction of the reduced crossed product:
>
> Let $\Gamma$ be a discrete group and $A$ be a $C^\*$-algebra with an action $\alpha: \Gamma\to \operatorname{Aut}(A)$. Consider the $\*$-algebra $C\_c(\Gamma,A)$ of finitely supported functions $\Gamma \to A$ with the $\alpha$-twisted multiplication and in... | https://mathoverflow.net/users/nan | A completely positive equivariant map $\varphi: A \to B$ induces a map $A \rtimes_r \Gamma \to B \rtimes_r \Gamma.$ | This is Exercise 4.1.4 in the book by Brown+Ozawa (is that where this question ultimately comes from?) The "Hint" is "The reduced case is easy." Hmm.
Well, it might perhaps help to look at this section of the book. Indeed, Proposition 4.1.5, *and its proof*, shows that if $F\subseteq\Gamma$ is a finite-set, and $P:\e... | 7 | https://mathoverflow.net/users/406 | 406976 | 166,774 |
https://mathoverflow.net/questions/406959 | 0 | When I first saw the definition of general Sobolev spaces with real exponent I immediately got interested in the following problem: pick several of your favourite irregular functions/distributions and now ask how singular are they, i.e. what is the largest exponent $s$ such that this function/distribution belongs to $H... | https://mathoverflow.net/users/24078 | What is the critical exponent for irregular function in the Sobolev scale? | Consider the distribution with Fourier series $$\sum\_{n \ge 1} \frac{\cos(nt)}{n^{1/2+\alpha} \log n} \,.$$
This will be in the Sobolev space $H^s=H^{s,2}$ for $s \le \alpha$ but not for $s>\alpha$.
| 1 | https://mathoverflow.net/users/7691 | 406980 | 166,775 |
https://mathoverflow.net/questions/406985 | -2 | Is there a exact formula for number of decompositions of $2n-1$ into a difference of two squares?
Examples:
```
3: 1 | 21: 1
4: 1 | 22: 1
5: 2 | 23: 3
6: 1 | 24: 1
7: 1 | 25: 2
8: 2 | 26: 2
9: 1 | 27: 1
10: 1 | ... | https://mathoverflow.net/users/147145 | The number of decompositions of $2n-1$ into a difference of two squares? | Any odd composite integer $m$ can be written as $m=pq$. We can switch it to a difference of squares:
>
> $$m=pq$$
> $$4m=4pq$$
> $$4m=2pq+2pq$$
> $$4m=2pq+2pq+q^{2}-q^{2}+p^{2}-p^{2}$$
> $$4m=\left(p+q\right)^{2}-\left(q-p\right)^{2}$$
> $$m=\frac{\left(p+q\right)^{2}}{4}-\frac{\left(q-p\right)}{4}^{2}$$
> $$m=\lef... | 0 | https://mathoverflow.net/users/147145 | 406986 | 166,779 |
https://mathoverflow.net/questions/406978 | 7 | $\DeclareMathOperator\SL{SL}\DeclareMathOperator\trace{trace}$Let $A \in \SL(2,\mathbb{R})$ and $\trace(A)>2$. Is it true that $$\lVert A\rVert \leq \lVert A^2\rVert,$$
where $\lVert \rVert$ is the operator norm that is the first singular value?
$$\lVert A \rVert =\sqrt{\lambda\_{\text{max}}(A^\*A)}=\sigma\_{\text{max}... | https://mathoverflow.net/users/127839 | Is it true that $\lVert A\rVert \leq \lVert A^2\rVert$ for $A\in \operatorname{SL}(2, \mathbb{R})$ when $\operatorname{trace}(A)>2$? | We can do this by a calculation. The assumptions on the determinant and trace are equivalent to having eigenvalues $\lambda,1/\lambda$, with $\lambda>1$. We can rotate the first eigenvector to the $e\_1$ position, and then
$$
A=\begin{pmatrix} \lambda & b \\ 0 & 1/\lambda \end{pmatrix} ,
$$
so
$$
A^\*A=\begin{pmatrix} ... | 11 | https://mathoverflow.net/users/48839 | 406987 | 166,780 |
https://mathoverflow.net/questions/407002 | 4 | Let $E$ be an elliptic curve over $\mathbf{Q}$. Then we can base-change $E$ to $\mathbf{C}$ and apply the uniformization theorem to obtain:
$$E(\mathbf{C}) \cong \mathbf{C}/(\mathbf{Z} + \mathbf{Z} \tau ) $$
for some complex number $\tau$ in the upper half plane. I've done a few numerical tests on Sage, and I've found ... | https://mathoverflow.net/users/394740 | The real part of the period of an elliptic curve | However, if $E$ is defined over $\mathbb R$, then it's always possible to find a $\tau$ of the form either $\tau=ti$ or $\tau=\frac12+ti$ so that $E(\mathbb C)$ is analytically isomorphic (over $\mathbb R$, even) to $\mathbb C/(\mathbb Z+\mathbb Z\tau)$. So possibly the examples you were looking at are defined over $\m... | 8 | https://mathoverflow.net/users/11926 | 407005 | 166,787 |
https://mathoverflow.net/questions/406902 | 2 | Let $f(n)$ be [A053645](https://oeis.org/A053645), distance to largest power of $2$ less than or equal to $n$; write $n$ in binary, change the first digit to zero, and convert back to decimal.
Let $g(n)$ be [A007814](https://oeis.org/A007814), the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary c... | https://mathoverflow.net/users/231922 | Pair of recurrence relations with $a(2n+1)=a(2f(n))$ | As proved in [this answer](https://mathoverflow.net/q/405996), for $n=2^tk$ with $2\nmid k$, we have
$$a\_1(n)=\sum\_{i=0}^t \binom{t}{i} a\_1(2^i(k-1)+1).$$
Then for $n=2^{t\_1}(1+2^{t\_2}(1+\dots(1+2^{t\_m}))\dots)$ with $t\_1\geq 0$ and $t\_j\geq 1$ for $j\geq 2$, we have
\begin{split}
a\_1(n) &= \sum\_{i\_1=0}^{t... | 1 | https://mathoverflow.net/users/7076 | 407010 | 166,788 |
https://mathoverflow.net/questions/406990 | 4 | Can a Vopenka cardinal be supercompact?
I asked a [weaker question](https://mathoverflow.net/questions/400732/is-vopenkas-principle-ord-has-the-tree-property-consistent) on here before. Unfortunately, I don't know enough set theory to see whether the positive answer there generalizes to a positive answer here.
| https://mathoverflow.net/users/2362 | Can a Vopenka cardinal be supercompact? | If $\kappa$ is almost huge with target $\lambda,$ then $V\_{\lambda}$ thinks that $\kappa$ is a supercompact Vopenka cardinal.
I'll take for granted the standard facts about almost huge cardinals listed here: <https://neugierde.github.io/cantors-attic/Huge>
In particular, $\kappa$ is Vopenka (in $V$), and this is j... | 9 | https://mathoverflow.net/users/109573 | 407024 | 166,792 |
https://mathoverflow.net/questions/406795 | 2 | Let $\Delta\_n(x\_1, \ldots, x\_n)$ denote the Vandermonde determinant $\displaystyle \prod\_{1 \leq i < j \leq n}(x\_j - x\_i)$. Let $c\_1, \ldots, c\_n$ and $K$ be nonnegative integers satisfying
$$c\_1 + \cdots +c\_n = K + m\frac{n(n-1)}{2},$$
where $m$ is a positive integer. Then the coefficient of $x\_1^{c\_1}\cdo... | https://mathoverflow.net/users/93753 | Coefficient of a term in a several variable polynomial multipled with Vandermonde determinant | Note that $\Delta\_n(x\_1^m,\ldots,x\_n^m)=\det(x\_i^{k\_j})$, where $k\_j:=(j-1)m$. Two key observations are the following (both work for arbitrary non-negative integers $k\_1<k\_2<\ldots<k\_n$, if $\sum c\_j=\sum k\_j+K$):
1. (algebraic) $$\left[x\_1^{c\_1}x\_2^{c\_2}\ldots x\_n^{c\_n}\right](x\_1+\ldots+x\_n)^K\de... | 4 | https://mathoverflow.net/users/4312 | 407039 | 166,797 |
https://mathoverflow.net/questions/406941 | 11 | Have there been any successful mathematicians that also happen to be mathematical fictionalists? Let's say success is defined by at least one article published in a non-pay journal.
I ask because this seems like a very extreme position for a working mathematician to have. Also, fictionalism is a very recent position.... | https://mathoverflow.net/users/38066 | Mathematical fictionalism | As I suggested in response to a [related MO question](https://mathoverflow.net/q/298749), one difficulty with answering this type of question is that most mathematicians outside of logic and set theory lack well-developed "positions" on the types of questions that occupy much of the attention of philosophers of mathema... | 26 | https://mathoverflow.net/users/3106 | 407040 | 166,798 |
https://mathoverflow.net/questions/406511 | 4 | I have been doing research on the Niemeier lattices with root systems of type, $A\_{k}^n$ and I am particularly interested in the finite groups permuting the constituent root systems. These groups seemed random at first, and they appeared to have a connection to certain projective linear groups over Galois fields with ... | https://mathoverflow.net/users/178390 | Structure of the permutation groups acting on the root systems of Niemeier lattices of type $A_{k}^n$ | The information required for answering this question is contained in [1]. There it is scattered over many chapters; so I will give a summary here.
Let $N$ be a Niemeier lattice, $R$ be the sublattice of $N$ generated by
its roots and and $R^\*$ be the dual lattice of $R$. Then $G = R^\*/R$ is
an Abelian group, the *g... | 2 | https://mathoverflow.net/users/105705 | 407047 | 166,799 |
https://mathoverflow.net/questions/407046 | 5 | Fix a (countable) set $\mathcal{P}$ of atomic propositional variables. Recall a *Kripke model $\mathcal{K}$ for intuitionistic propositional logic (IPL)* consists of:
* A preorder $(W,\leq)$
* For each $w \in W$, a (classical) valuation $\varphi\_w\colon \mathcal{P} \to 2$
such that for all $w \leq v$ and $x \in \m... | https://mathoverflow.net/users/136473 | Possible values of "Kripke rank" for formulae in IPL | The finite model property of intuitionistic logic implies that every unprovable formula has finite rank. On the other hand, all positive integers are ranks of some formulas; there are many families of formulas one could use to show this, but for example, the formulas
$$\bigvee\_{i=0}^n\Bigl(\bigwedge\_{j<i}p\_j\to p\_i... | 8 | https://mathoverflow.net/users/12705 | 407056 | 166,802 |
https://mathoverflow.net/questions/407062 | 3 | Let $X$ be a Polish space and let $(\mu\_i)\_{i=1}^{\infty}$ be a sequence of probability measures in the Wasserstein space $\mathcal{P}(X)$ on $X$. Let $(\beta\_i)\_{i=1}^{\infty}$ be a summable sequence in $(0,\infty)$. For every positive integer $k$, define the probability measures
$$
\nu\_k = (\sum\_{1\leq i\leq k}... | https://mathoverflow.net/users/36886 | Wasserstein convergence of "series expansion'' of probability measure | It is true and clear if the metric space $X$ has a finite diameter, but false in general: Take $\beta\_i=2^{-i}$ and $\mu\_i$ the point mass at $3^i$.
Details: In the case $D=$diam$(X)<\infty$, write $s\_k=\sum\_{i \le k} \beta\_i$
and $s=\sum\_{i<\infty} \beta\_i$. Then $$\nu\_\infty=\frac{s\_k}{s} \nu\_k+\frac{s-s\... | 6 | https://mathoverflow.net/users/7691 | 407063 | 166,804 |
https://mathoverflow.net/questions/407065 | 1 | Following
[About uniform continuity](https://mathoverflow.net/questions/402661/about-uniform-continuity)
Let $E$ be a topological space, for all $a \in E$, we associate an open set of $E$, $U(a)$ containing $a$.
We will say that $\{U (a), a \in E\}$ is a uniform covering of $E$ if
For any dense $B$ in $E$ the c... | https://mathoverflow.net/users/110301 | Uniform covering and uniform continuity | Let $(E',d')$ be the Euclidean plane $\mathbb{R}^2$ with the usual metric.
Let $(E,d)$ be the interval $[-1,1]$ with the usual metric.
Let $f$ be the inclusion map of $[-1,1]$ to the interval $[-1,1]\times \{0\}$; it is an isometry to its image and is uniformly continuous.
Consider the following cover of $E'$:
$$ U... | 1 | https://mathoverflow.net/users/3948 | 407074 | 166,808 |
https://mathoverflow.net/questions/407057 | 7 | Let $k$ be a field, $X$ an algebraic variety, and $G$ a smooth algebraic group, acting on $X$ via $(g,x)\mapsto g\cdot x$.
Fixing $x$ in $X$ a $k$-point, there is a map $f\_x:G\rightarrow X$ sending $g\mapsto g\cdot x$.
Now, assuming that $df\_x:T\_eG\rightarrow T\_xX$ is surjective, how can one show that the orbit... | https://mathoverflow.net/users/166993 | Open orbits under the action of an algebraic group | $\DeclareMathOperator\Im{Im}$I don't think that smoothness of $X$ is necessary, just the fact that it is geometrically integral (I assume your varieties are geometrically integral). (As noted by Jason Starr, smoothness of $G$ is necessary.)
Here is a possible argument, please let me know if there are any mistakes.
... | 5 | https://mathoverflow.net/users/339730 | 407081 | 166,810 |
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