Let $\Phi$ be the unique solution of $$\begin{cases} \frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\ \Phi(x,0) = x \quad x \in \mathbb{R}^N \end{cases}$$ where we have assumed $f$ smooth.

How do you prove that $\Phi(\cdot, t)$ is smooth and in particular that the following holds?

$$\begin{cases} \frac{d}{dt} \nabla \Phi(x,t) = \nabla_1 f(\Phi(x,t),t)\nabla \Phi(x,t) \quad t>0 \\ \nabla\Phi(x,0) = 1 \quad x \in \mathbb{R}^N \end{cases}$$ and $$\begin{cases} \frac{d}{dt} J \Phi(x,t) = \mathrm{div} f(\Phi(x,t),t)J \Phi(x,t) \quad t>0 \\ \nabla\Phi(x,0) = 1 \quad x \in \mathbb{R}^N \end{cases}$$ where $Jf = det \nabla f$.

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Suppose I have an allergy to non-Hausdorff spaces but I really want to do, say, arithmetic geometry. I wonder if there is some perverse way I could develop scheme theory that would accomodate for my needs.

I am not sure what is the best way to formalize this question but here is an attempt. Does there exist a functor from schemes to the category of Hausdorff topological spaces/continuous maps that is one of the following

- full and faithful
- faithful and admits a left adjoint
- full and admits a left adjoint
- faithful and admits a right adjoint
- full and admits a right adjoint?

P.S. I believe that there is a faithful functor from schemes to sets (take the union of the underlying set and all the stalks and do some power-set acrobatics) and there is a faithful functor from sets to Hausdorff spaces (discrete topology) so at least a faithful functor should exist.

I am trying to understand the essential image of the following functor. Given a scheme $X$, we consider the corresponding small Zariski site $X_{zar}$. For a commutative ring $\Lambda$, let $\mathcal D(\Lambda)$ denote the derived $\infty$-category of $\Lambda$, and let $Fun(X_{zar}^{op}, \mathcal D(\Lambda))$ denote the $\infty$-category of the $\mathcal D(\Lambda)$-valued presheaves on $X_{zar}$. Then for any $F$ in $\mathcal D(X, \Lambda)$, the derived $\infty$-category of sheaves of $\Lambda$-modules, we can define $i(F)\in Fun(X_{zar}^{op}, \mathcal D(\Lambda))$ by the formula:$$i(F)(U)=R\Gamma_{zar}(U, F)$$ Here is my question: is $i(F)$ already an $\infty$-categorical sheaf?

An $\infty$-categorical sheaf $G$ here is a presheaf satisfying Čech descent (I am even not sure this "definition" is correct in practice...), i.e. for any covering $U=\cup U_i$ $$G(U)\simeq \lim_{n}G(ČU)$$ where $ČU$ is the corresponding Čech complex. A naive idea is that at least in the bounded below case if we take an injective resolution $I$ of $F$, then the homotopy limit here is just the limit and this becomes the definition of sheaf.

G is a graph on n vertices. Each vertex has degree at least 3, and G is not bipartite. Let k be the size of the smallest odd cycle in G. What is the largest k can be, as a function of n?

Vertices of degree 0 and 1 are clearly irrelevant. If we allowed vertices of degree 2 and n was odd, we could have G just be an n-cycle, and get k=n. So these cases kind of break down trivially. But when all degrees are at least 3, we can't have an n-cycle without having lots of chords. If the large cycle is odd and you start adding approximate diameters, you make a good construction until you've gone most of the way around, then you start getting small odd cycles again.

Per the title, what are some of the oldest number theory books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am already aware of the books of Dickson and Hardy.

Motivation for this question. Some person came up to me before class and asked, are you the person asking the ridiculous "oldest book with exercises" questions on MO? I said yes, and he asked I could do one on number theory. So here we are.

Suppose we have a Coxeter group whose diagram is given by a simplex. In other words, $G=\langle g_1,\ldots ,g_k|(g_i)^2=e,\,(g_ig_j)^3=e \rangle$. How many words of length N simplify to the identity? What is the recursion/generating function? The case k=2 is easy, because the group is finite; the corresponding generating function is $E(x)=\frac{1}{3}[2/(1−x^2)+1/(1−4x^2)]$. I expect k=3 is likewise readily doable. Is there a general solution? What if we change the relations to $(g_ig_j)^m=e \,\,\forall i,j$?

More generally, if I give a group element in this group whose shortest word is $g_{i_1}...g_{i_p}$, how many words of length N are equivalent to it?

Keeping in mind the word problem is solvable for Coxeter groups.

Given a complete graph $(V,E)$ with $n$ vertices $V$ and walks $p \in V^{l+1}$ of length $l$. We say the edges of walks $p$ are the multiset

$$ e_p = \{ (p_i,p_{i+1}) \mid 1 \leq i \leq l \}. $$ Also the frequency of vertex $v \in V$ in walk $p$ is $p[v] = |\{ i \mid p_i = v \}|$. The frequency of all vertices $f_p \in \mathbb{N}^n$ in walk $p$ is then given by $$ f_p = [ p[v_1], p[v_2], \dots, p[v_n] ]. $$

Now, two walks $p$ and $q$ are called edge-invariant if $f_p = f_q$ and $e_p = e_q$. How many different walks of length $l$, which are not edge-invariant, exist? More precisely, i.e., $$ max_{P \subset V^{l+1}} | \{ P \mid \forall p, q \in P: e_p \neq e_q \textrm{ or } f_p \neq f_q \textrm{ for } p \neq q \}|. $$

For example, $n = 2$, it's $4, 8, 14, 22, 32, 44, 58, 74, 92, 112, 134, 158$ for $l = 1,\dots,12$. Specifically for $l = 3$, out of the 16 possible walks, two are edge-invariant: $$ e_{[1,1,2,1]} = e_{[1,2,1,1]} \textrm{ and } e_{[2,2,1,2]} = e_{[2,1,2,2]}. $$

For $n = 3$, it's given here as $9, 27, 75, 186, 414, 840, 1578, 2784, 4662, 7476, 11556$ for $l = 1,\dots,11$.

It seems to be related to k-abelian equivalence classes, where the general problem seems to be rather difficult, but maybe the 2-abelian cases is simpler?

This question already has an answer here:

Let $X$ be a smooth projective complex variety, if every algebraic morphism from $X(\mathbb C)$ to $X(\mathbb C)$ has a fixed point, must every continuous map from $X(\mathbb C)$ to $X(\mathbb C)$ have a fixed point? I don't know whether the two conditions are the same.

How about the story over real numbers？

This question was asked recently on MO and then deleted by the owner, user Aalon. I think the question deserves to be answered, which is what I will try to do here. Aalon was reading this paper, where it is claimed (on page 32, in line 5) that \begin{equation} P\Big(\int_0^1(W_t)_-\,dt\le\varepsilon\Big)\le\varepsilon \tag{1} \end{equation} for $\varepsilon>0$, where $W_\cdot$ is the standard Wiener process. The "reason" for this claim given in the mentioned paper (on page 32, in line 4) was that \begin{equation} \int_0^1(W_t)_-\,dt\ge\Big|\int_0^1 W_t\,dt\Big|. \tag{2} \end{equation} Aalon had doubts about both (1) and (2).

The metric in sub-Riemannian geometry is often called the Carnot-Carathéodory metric.

**Question 1.** What is the origin of this name? Who was the first to introduce it?

I believe that the "Carathéodory" part of the name could be related to his work in theoretical thermodynamics [1], but I do not really know how it is related to his work.

**Question 2.** How is the notion of Carnot-Carathéodory metric related to the work of Carathéodory?

I know that Carnot groups are special examples of sub-Riemannian manifolds, but is it the reason for "Carnot" part in the name of the metric?

**Question 3.** What does the "Carnot" part of the name of the metric stand for?

**[1] C. Carathéodory,** *Untersuchungen uber die Grundlagen der Thermodynamik*.
*Math. Ann.* 67 (1909), 355–386.

Maybe someone familiar with Willard's textbook can help me out. Problem section 23G on pg. 174 is titled **piecewise metrizability**. The first problem is:

- If a Tychonoff space $X$ is the union of a locally finite collection of closed, metrizable subspaces, then X is metrizable.

This fact is well-known. It is 4.4.19 in Engelking's book. It also appears in Nagata's seminal metrization paper from 1950. However, the conclusion is true without the assumption that $X$ is Tychonoff. This leads me to think there is a simpler proof under this assumption. If you know of one, please give me an outline or hint.

More interesting is the next problem from Willard:

- If a $T_4$ space $X$ is the union of any locally finite collection of metrizable subspaces, then $X$ is metrizable. [Use 15.10]

This is less well-known, because it is clearly not true. Consider the one point compactification of an uncountable discrete space. It is normal and non-metrizable, but is the union of just two metrizable subspaces: one discrete and the other a singleton. The hint refers to the standard "shrinking" theorem for point-countable **open** covers of normal spaces. So, it seems likely that the word "open" was just accidentally omitted. The following fact follows easily from the first exercise and 15.10:

Fact: If a $T_4$ space $X$ is the union of a locally finite collection of open, metrizable subspaces, then $X$ is metrizable.

Certainly, someone must have noticed this before now. However, I haven't been able to find any reference to, or use of, the above fact in print. Does anyone know of a better citation for this than "Corrected version of exercise 2 in 23G of [W]"?

Thanks in advance,

-Jeff

Refs:

- Willard, Stephen
*General topology*Originally printed 1970 by AddisonWesley; currently in print by Dover Publications (2004) - Engelking, Ryszard
*General topology*Heldermann Verlag (1989) - Nagata, Jun-iti
*On a necessary and sufficient condition of metrizability*J. Inst. Polytech. Osaka City Univ. (1950)

PS: Unfortunately, exercises 1 and 2 also appear (without correction) in Patty's more recent textbook, *Foundations of Topology,* Jones and Bartlett Publishers (2009). They appear to have been copied verbatim from Willard.

Let's view $\mathbb{R}$ as a vector space over $\mathbb{Q}$, and pick some basis $(v_\alpha)_{0 \leq \alpha < \mathfrak{c}}$ of it. We can then consider the subspace $L$ spanned by $(v_\alpha)_{0 < \alpha < \mathfrak{c}}$ (ie leaving out one vector from the basis). Given the horrible way we have built $L$, I don't suppose there is much a priory reason for $L$ to be measurable. However, I am wondering whether we can say something about the outer measure of $L$.

Let $M^2 \subset \mathbb{S}^3$ be a closed and orientable embedded (and minimal, if important) surface. Choose a unit normal vector field $\eta: M \to \mathbb{S}^3$ along $M$ and a point $p_0 \in \mathbb{S}^3$ such that $-p_0 \not \in M$, and define a function $c_{p_0} : M \to \mathbb{R}$ by $$c_{p_0}(p) = \frac{\langle \eta(p), p_0 \rangle}{1 + \langle p, p_0 \rangle}.$$

An alternative expression for $c_{p_0}$ is

$$c_{p_0}(p) = \frac{\langle \eta(p), p_0 - \langle p, p_0 \rangle p \rangle}{1 + \langle p, p_0 \rangle} = \left\langle \eta(p), -\tan\left( \frac{d_{p_0}(p)}{2} \right) \nabla d_{p_0}(p) \right\rangle,$$

where $d_{p_0}(p)$ is the distance between $p$ and $p_0$ and $\nabla d_{p_0}$ is the gradient (in the sphere) of this function.

Here is my question: is it possible to choose $p_0 \in \mathbb{S}^3$ and $\eta$ such that:

- $-p_0 \not \in M$,
- $\int_M c_{p_0} \, \mathrm{d} A \leq 0$,
- if $\Omega$ denotes the connected component of $\mathbb{S}^3 \setminus M$ such that $\eta$ points outside $\Omega$, then $-p_0 \not \in \Omega$?

I am interested in this setup because I want to apply the divergence theorem on $\Omega$, but $-p_0$ cannot belong to this region since it is a singularity of $c_{p_0}$.

If we have quenched decay of correlation, can we transfer it to annealed decay of correlation? To be precise, let us consider following setting:

Given transformations $T_{\omega}: (S^1, dm) \to (S^1, dm)$ indexed by $\omega \in (\Omega, \sigma, \mathbb{P})$ where $\sigma $ is ergodic probability on $\Omega$ and $dm $ is Leb measure on circle $S^1$. define random composition $T_{\omega}^n:=T_{\sigma^{n-1}\omega} \circ \dots \circ T_{\omega}$. Assume we have quasi-invariant absolutely continuous probability $\mu_{\omega}:= h_{\omega} dm$ such that $(T_{\omega})_{*} \mu_{\omega}=\mu_{\sigma \omega}, h_{\omega} \in Lip(S^1)$.

So we have invariant transformation $\tau $ defines on skew product space $(\Omega \times S^1, \mu)$:

$ \tau (\omega, x): =(\sigma \omega, T_{\omega}x) \text{ where } \mu = d\mu_{\omega}d\mathbb{P} \text{ is invariant probability on } \Omega \times S^1 $.

we study two decay of correlations:

$\forall \phi, \psi \in L^{\infty}(\Omega \times S^1)$ and $\psi_{\omega} \in Lip(S^1)$ for all $\omega \in \Omega$

Quenched decay of correlation:

$|\int \phi_{\sigma^n \omega} \circ T^n_{ \omega} \cdot \psi_{ \omega}d\mu_{\omega}-\int \phi_{\sigma^n \omega} d\mu_{\sigma^n \omega} \int \psi_{\omega} d\mu_{\omega}| \le C_{\omega} \cdot ||\phi||_{\infty} \cdot C_{\psi}\cdot e^{-n}$, where $\mathbb{P}(C_{\omega} >m) \le \frac{1}{m^2}$

Anneal decay of correlation:

$|\int \phi \circ \tau^n \cdot \psi d\mu-\int \phi d\mu \cdot \int \psi d\mu| \le ||\phi||_{\infty} \cdot C_{\psi} \cdot e^{-n} $

I tried to prove Quenched decay of correlation $\implies $ Annealed decay of correlation, but got stuck because $\psi-\int \psi d\mu$ is no longer fiber-wise mean zero.

is it really true or has counter-example? Thanks in advanced.

A semifinite trace $\tau$ on $M_{+}$ (for a von Neumann algebra $M$) is said to be normal if $\tau(\sup x_i ) = \sup \tau(x_i)$ for an bounded increasing net of positive operators $(x_i)_{i \in I}$.

Is it true that $\tau(\inf x_i) = \inf \tau(x_i)$ for a bounded decreasing net of positive operators $(x_i)_{i \in I}$?

If the trace were finite, it is easy enough to see that the above holds using the increasing net $(x_0 - x_i)_{i \in I}$ and the fact that $\tau(x_0)$ is finite. In the semifinite case, in order for the same strategy to work we must have the following result: if $\tau(\inf x_i) < \infty$, then there is an index $j \in I$ such that for all $i \ge j$, we have $\tau(x_i) < \infty$. This seems like something that should be true. But I am unable to come up with a quick proof.

Thank you.

The standard practice when doing a recursive construction seems to be to list all of the desired properties for the construction first:

We construct $(x_n)$, $(y_n)$, $(z_n)$ such that

(1)

(2)

(3)

... (a bunch of properties about the individual items $x_n,z_n,y_n$ and their relationships to $x_k, y_k,z_k$'s when $k<n$)

and then explicitly write out the construction for $n=0$, and then do the ``recursive step''.

My question is, if the list of items is sufficiently short and the construction is not too complex, is it necessary to write the construction like this? Would it be acceptable to simply begin the construction, do a few cases at the beginning, $n=0,1,2$ until you get the feel for things, and then just say "continue in this manner"? And then when proving things about the construction, just say "By construction,...".

When I say acceptable, I mean acceptable for publication in a good journal.

Are there any mathematical advantages to actually writing the conditions (1),(2),(3),... beforehand?

So I am looking for examples of the following phenomenon.

Suppose that $V$ is a variety with a computable equational theory which is not locally finite. Suppose that $G$ is an infinite finitely presented group generated by elements $s_{1},\dots,s_{k}$. Suppose furthermore that $G$ has a polynomial time computable normal form. In other words, there is a function $f:\{w_{1},\dots,w_{k},w^{-1}_{1},\dots,w_{k}^{-1}\}^{*}\rightarrow\mathbb{N}$ computable in polynomial time such that if $d_{i},e_{i}\in\{-1,1\},a_{i},b_{i}\in\{1,\dots,k\}$ for each $i$, then $f(w_{a_{1}}^{d_{1}}\dots w_{a_{u}}^{d_{u}})=f(w_{b_{1}}^{e_{1}}\dots w_{b_{v}}^{e_{v}})$ if and only if $s_{a_{1}}^{d_{1}}\dots s_{a_{u}}^{d_{u}}=s_{b_{1}}^{e_{1}}\dots s_{b_{v}}^{e_{v}}$. Suppose that there is some $n$ along with two matrices of terms $$(t_{i,j}(x_{1},...,x_{n}))_{1\leq i\leq n,1\leq j\leq k},(t^{*}_{i,j}(x_{1},...,x_{n}))_{1\leq i\leq n,1\leq j\leq k}.$$ These terms come from the variety $V$. Then whenever $X\in V,$ define actions of $\{w_{1},\dots,w_{k},w^{-1}_{1},\dots,w_{k}^{-1}\}^{*}$ on $X^{n}$ $$(x_{1},\dots,x_{n})\cdot w_{j}=(t_{1,j}(x_{1},\dots,x_{n}),\dots,t_{n,j}(x_{1},\dots,x_{n}))$$ and $$(x_{1},\dots,x_{n})\cdot w_{j}^{-1}=(t_{1,j}^{*}(x_{1},\dots,x_{n}),\dots,t_{n,j}^{*}(x_{1},\dots,x_{n})).$$

Then the variety $V$ satisfies the identities $$(x_{1},\dots,x_{n})\cdot w_{a_{1}}^{d_{1}}\dots w_{a_{u}}^{d_{u}}=(x_{1},\dots,x_{n})\cdot w_{b_{1}}^{e_{1}}\dots w_{b_{v}}^{e_{v}}$$ if and only if $s_{a_{1}}^{d_{1}}\dots s_{a_{u}}^{d_{u}}=s_{b_{1}}^{e_{1}}\dots s_{b_{v}}^{e_{v}}$, and the equational theory of the variety $V$ is axiomatized by the identities of the form $$(x_{1},\dots,x_{n})\cdot w_{a_{1}}^{d_{1}}\dots w_{a_{u}}^{d_{u}}=(x_{1},\dots,x_{n})\cdot w_{b_{1}}^{e_{1}}\dots w_{b_{v}}^{e_{v}}$$ where $d_{i},e_{i}\in\{-1,1\},a_{i},b_{i}\in\{1,\dots,k\}$ for each $i.$

In this case, the group $G$ acts on $X^{n}$ for each $X\in V$ by letting $$(x_{1},...,x_{n})\cdot s_{a_{1}}^{d_{1}}\dots s_{a_{u}}^{d_{u}}=(x_{1},...,x_{n})\cdot w_{a_{1}}^{d_{1}}\dots w_{a_{u}}^{d_{u}}$$ where $d_{i},e_{i}\in\{-1,1\},a_{i},b_{i}\in\{1,\dots,k\}$ for each $i.$

**Examples**

Example 1: The symmetry group $S_{k}$. Suppose that $\tau_{i}=(i,i+1)$. Then we set $$(x_{1},\dots,x_{k})\tau_{i}=(x_{1},\dots,x_{i-1},x_{i+1},x_{i},x_{i+2},\dots,x_{k}).$$

Example 2: The braid group $B_{k}$ acting on racks and quandles. Suppose that $(X,*)$ is a rack or a quandle (Let us use left-distributivity $x*(y*z)=(x*y)*(x*z)$). Then define $$(x_{1},\dots,x_{n})\cdot\sigma_{i}=(x_{1},\dots,x_{i-1},x_{i}*x_{i+1},x_{i},x_{i+2},\dots,x_{n}).$$

Example 3: $\mathrm{Aut}(F_{k})$ acting on tuples from groups.

Example 4: Finite direct products of groups.

In the case where the group $G$ is finite or though the conditions I have listed out are otherwise not technically satisfied (for example by the infinite strand braid group $B_{\infty}$ on $X^{\mathbb{N}}$ for each rack $X$), feel free to give an answer anyways if you feel the answer is still interesting yet does not stray from the spirit of the question too much.

What is the procedure to formally prove that no norm exists in $\mathbb{R}^n$, that induces a metric $d$?

My first instinctive idea would be to show that $d$ is a metric in $\mathbb{R}^n$, but after this I don't know any further. What could I achieve by following this road?

The specific problem I am working on is to prove that no norm $||\cdot||$ in $\mathbb{R}^2$ exists, that induces the metric $d$:

$d((x_\text{1},y_\text{1}),(x_\text{2},y_\text{2})) := \begin{cases} |y_\text{1}-y_\text{2}|, & \text{if }x_\text{1} = x_\text{2},\\ |y_\text{1}| + |x_\text{1}-x_\text{2}| + |y_\text{2}|, & \text{if }x_\text{1} \not= x_\text{2}. \end{cases}$

I do not intend to get solutions for my specific problem, but maybe a similar problem exists elsewhere, which I could study. Of course, I have searched MathOverflow, and other sources.

What is the reason for over counting when you start by inserting one of the four balls into different four boxes? (There may be boxes that do not insert anything)

Please explain for me....