We have n points randomly distributed in a d-dimensional unit hypercube. We randomly sample k of those points and center a ball with radius r on each of those k points. Does there exist an estimate of the radius r such that r is the smallest radius expected to cover all the points with those k possibly overlapping balls?

I am looking for a good accessible reference that would summarize properties of zeros of complex analytic functions.

For my purpose, it would be interesting to see a discussion on the following topics:

- Zeros of single variable analytic function are discrete (isolated)
- Zeros of multivariable analytic functions are not isolated
- The set of zeros of an analytic function of $n$ variables roughly speaking lives in $n-1$ dimensional space.

I have been reading "Functional Theorem of Several Complex Variables" by Krantz. While I found the book interesting to read, I don't think that it is very accessible to a non-mathematician. Right now looking for a reference that would be more accessible to a senior Ph.D. student in engineering with a good background in single variable complex analysis. In any case, any reference on this subject that you can provide would be of great help to me.

A countable structure A is strongly reducible to a structure B if there is a uniform turing functional which, given a copy of the atomic diagram of B, computes a copy of the atomic diagram of A.

A is said to be effectively interpretable in B if A is strongly reducible to B via a computable functor from the category of copies of B to the category of copies of A.

- See this paper of Harrison-Trainor/Melnikov/Miller/Montalban for a precise definition
*(and note that this isn't the starting definition; that definition is given on page $3$, and its characterization by functors is Theorem 5)*. Roughly speaking, let $\mathcal{A}$ and $\mathcal{B}$ be respectively the categories whose objects are copies of $A$ and $B$ and whose morphisms are isomorphisms (in the usual sense); then a computable functor reducing $\mathcal{A}$ to $\mathcal{B}$ is a functor from $\mathcal{B}$ to $\mathcal{A}$ given by a pair of Turing functionals, one sending objects in $\mathcal{B}$ to objects in $\mathcal{A}$ and the other sending morphisms in $\mathcal{B}$ to morphisms in $\mathcal{A}$. The claim that this yields a strict strengthening of strong (= Medvedev) reducibility is made without proof on page $5$ of the linked paper.

What is an example of two countable structures A,B such that A is strongly reducible to B but A is not effectively interpretable in B?

i.e. does there exist an integer $C > 0$ such that $11, 11 + C, ..., 11 + 10C$ are all prime?

I have an application where I need to work with the following idea. I need to work with the shwartz space on $\cup_{n=1}^CR^n$ and its dual by defining an inner product as

$\langle\nu,\phi\rangle=\sum_{n=1}^C\int_{R^n}\phi(x_1,\cdots,x_n)\,d\nu(x_1,\cdots,x_n)$. Does all the properties of schwartz space and its dual on just $R^n$ holds even for the case of the space $\cup_{n=1}^CR^n$? If it is true, can we extend this to the case when $C=\infty$? Please give some suggestions and references for this.

I have recently started to read a bit about geometry and topology. Hopf fibration, Lense spaces, CW complexes, stuff that are discussed in Hatcher's Algebraic Geometry and other things that require good visualization. What is apparent to me is that the further I go, the less I understand what is going on. I have searched on YouTube and found some really nice animations for some of these topics but good animations are rare like gems.

Advanced stuff in mathematics are less discussed and available on the internet. I have realized that if I want to understand math one day, at some point I should be able to create my own animations. Now, my question is rather directed at people with experience in teaching advanced mathematics or currently doing research in mathematics in areas where geometric intuition is absolutely necessary. What kind of tools do you use? Do you develop them on your own in your research team/group? Can an independent person have access to them? Is it possible for an independent person to develop this kind of tools on their own?

Can you think of a situation where you couldn't understand a geometric concept visually but you created an animation that demystified it for you?

I hope this is research level. Suppose $E$ is the direct limit of finite spectra, say $E=\mathrm{colim }\ E_i$, which itself is not finite. I wonder how much and under which conditions the inverse limit $\mathrm{lim}\ D(E_i)$ is a good candidate for playing role of $D(E)$? Here, I write $D$ for the $S$-duality functor. I feel there is problem with the possibility of existence of phantom maps here. I will be very grateful for any advise on this, or possibly some references on the topic.

As to what one might expect. For instance, if $f:X\to Y$ is a map between finite spectra then $f_*=0$ if and only if $D(f)_*=0$. Or if we know about dimension of bottom cell of $E$ then that would tell about the dimension of the top cell of the dual spectrum. Here, $f_*$ is the map induced in ordinary homology and I consider CW spectra. And if $D(E)$ can always be identified with the function spectrum $F(E,S^0)$?!

I also would like to know what features of duality for finite spectra fail to hold in general. For the moment, homological behavior is very interesting for me. I also wonder if my worry about phantom maps has any point?!?

Disclaimer

Sorry in advance for vagueness. I'm still trying to get my ideas right on this one.

SetupSo, let $P$ be a distribution on a Euclidean space $X$ with an $\ell_p$ metric, and let $P_\epsilon$ be another distribution on $X$ whose Wasserstein distance (or KL diverence, for simplicity and as a starting point) from $P$ is at most $\epsilon$.

QuestionWhat is an effective way for sampling from $P_\epsilon$ given that sampling from $P$ is easy ? Alternative, how to compute expectations w.r.t $P_\epsilon$, i.e computing things like $\mathbb E_{x \sim P_\epsilon}[f(x)]$, or $\max_{P_\epsilon} \mathbb E_{x \sim P_\epsilon}[f(x)]$ for a smooth function $f$ ?

Is this problem linked to another well-studied problem ?

The singular Cauchy operator is defined by $$S_\Gamma :f \to \int_\Gamma \frac{f(\xi)}{\xi-z} d\xi , z\in \Gamma.$$ Is this operator bounded in Morrey spaces and weighted Morrey spaces? i.e. is there a constant $c$ such for any $f$ belongs to Morrey space the following inequality is true $$|| S_\Gamma f|| < c ||f\|$$

Let $U^{m} \subset \mathbb{R}^{m}$ be an open set. Suppose $\varphi$ is an immersion of $U^{m}$ into $\mathbb{R}^{m+n}$ satisfying the following condition:

For each point $p \in \varphi(U^{m})$, the nullity of the second fundamental form of $\varphi(U^{m})$ is equal to $m-1$.

Then, it is well-known that

- $\varphi(U^{m})$ is foliated by $(m-1)$-planes, along which the tangent space of $\varphi(U^{m})$ is constant (i.e., isomorphic to the same linear subspace of $\mathbb{R}^{m}$);
- With the induced metric, $\varphi(U^{m})$ is flat.

Recall that a submanifold $M$ of a Riemannian manifold $\overline{M}$ is said to be a *full submanifold* if it is not contained in any totally geodesic submanifold $N$ of $\overline{M}$ with $\dim N < \dim \overline{M}$.

In general, can $\varphi(U^{m})$ be a full submanifold?

Let $P(z)=\sum_{k=1}^na_kz^k $ be a polynomial of degree $n$ having all its zeros in $|z|\leq 1.$ Then what is the best value for 'L' $$\Re\left(\sum_{k=1}^nP(zw_k)\frac{w_k}{(w_k-1)^2}\right)\geq L$$ on $|z|=1$ where $w_k,\; 1\leq k\leq n$ are the roots of the polynomial $z^n+|a_0|=0?$

$ \def\ri{\mathop{\rm ri}} $

If a convex set $C\subseteq\mathbb{R}^n$ contains the origin, $0\in C$, then clearly $0\in\ri C'$ for some $C'\subseteq C$ (where $\ri$ denotes the relative interior). I am interested in the greatest such $C'$. Precisely: Given a convex set $C$, find the greatest one of all subsets of $C$ that contain $0$ as a relative interior point. I believe the following statements are true:

Such greatest subset exists for every convex set $C$. In other words, if $R(C)$ denotes the union of all subsets of $C$ that contain $0$ as a relative interior point, then $0\in\ri R(C)$.

If $C$ is convex, then $R(C)$ is convex.

$R(C)$ is the (unique) face of $C$ that contains $0$ as a relative interior point.

$C\neq\emptyset$ and $0\notin\ri C$ imply $\dim R(C)<\dim C$.

Do these statements hold? If so, how to prove them? If not, what are counter-examples?

This question is from the paper, The Analysis of Elliptic Families
II. Dirac Operators, Eta Invariants, and the Holonomy Theorem, Commun. Math. Phys. 107, 103-163 (1986) --- *Proposition 2.8*.

Suppose that $Tr[D^uexp(-tD^2)]=\frac{C_{-n/2}}{t^{n/2}}+\cdots+ \frac{C_{-1/2}}{t^{1/2}}+O(t^{1/2})$ as $t\to0$. Here $D$ denotes the Dirac operator and $D^u$ denotes the $u$-derivative of $D$. (I think one could ignore the exact definition here.)

And, we have that $$\Gamma(\frac{s+1}2)\eta(s)=-s\int^\infty_0t^{\frac{s-1}2}Tr[D^uexp(-tD^2)] dt.$$

**Q** By the asymptotic formula of the "heat kernel", how could we deduce that
$$\eta(0)=-2C_{-1/2}/\sqrt\pi.$$

PS: What I do not understand:

- Why the other terms vanish under the integral?

- Since the integral is from zero to infinity, is it safe to use the asymptotic formula near $0$?

Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded domain with a smooth boundary. Fix $1<p<n$.

Let $A \in W^{1,p}(\Omega;\text{End}(\mathbb{R}^n)) \cap C(\Omega;\text{End}(\mathbb{R}^n))$ with $\det A > 0$ a.e.

Do there exist $u_n \in C^{\infty}\big(\Omega,\text{GL}(\mathbb{R}^n)\big)$ such that $u_n \to A$ in $W^{1,p}_{loc}$?

I am also interested in a weaker result: Are there $u_n \in C^{\infty}\big(\Omega,\text{End}( \mathbb{R}^n)\big)$, such that $u_n(x) \in \text{GL}(\mathbb{R}^n)$ a.e. **and**
$u_n \to A$ in $W^{1,p}_{loc}$?

I don't really need the $u_n$ to be defined on all $\Omega$. It suffices that for **every** arbitrarily small ball in $\Omega$, there would be a neighbourhood where such a sequence $u_n$ would be defined.

The problem is that *it is not always true* that $A_x \in \text{GL}( \mathbb{R}^n)$ for *every* $x \in \Omega$. The rank can fall on a subset of measure zero.

**If we knew $A(x) \in \text{GL}(\mathbb{R}^n)$ everywhere then the answer would be positive.** This follows from the facts that "being invertible" is an open condition, and that continuous Sobolev maps can be *approximated uniformly* by smooth maps over compact subsets.

In more detail, let $K \subseteq \Omega$ be compact. Since we assumed $A \in C\big(\Omega, \text{GL}(\mathbb{R}^n) \big)$, the map $\psi:x \to A_x$, considered as a map $K \to \text{GL}( \mathbb{R}^n)$, is continuous. Thus $\psi(K) $ is compact and $\text{dist}\big(\psi(K),\partial \text{GL}(\mathbb{R}^n)\big)>0$.

Now consider each component of $\psi(x)=A_x \in \text{End}(\mathbb{R}^n) $. We can approximate each component of $\psi$ using mollification on an open subset of $\Omega$ containing $K$. Since each component is a continuous function, the mollifications *converge uniformly* on $K$. This implies that from a certain point in the mollified sequence, $\text{dist}(u_n,A)<\text{dist}\big(\psi(K),\partial \text{GL}(\mathbb{R}^n)\big)$, so the $u_n$ are invertible.

I'm looking for a gentle an concise introduction to complex-variable differential equations. Eventually, I need to look at complex PDEs, but I assume one starts with complex ODEs.

Mostly, I'm just looking for definitions and basic existence and uniqueness results. Online results, from what I've found, just "leaped" to advanced or specialized topics.

Hi maybe someone on here can help me. I have been stuck on showing this fact for several months. I asked this question in the stack exchange and it has floated around for a while but to no avail.

The question I would like to answer is: If my filtration $\{\mathcal{F}_t\}_{t \geq 0}$ satisfies the usual conditions, and a cadlag process is adapted to that filtration, then that process is progressively measurable. The reason I would like to know this is because I am trying to figure out the proof of Theorem 6 in Protter's Stochastic Integration and Differential Equations. I have a hunch that what may have been used is the statement in the title. I've seen it used a couple of places already, but I cannot figure out why it is true. For example in this question https://math.stackexchange.com/questions/1682370/measurability-of-a-stopped-random-variable, both answers use the statement.

Here is Protter's theorem 6: Let $\{\mathcal{F}_t\}_{t \geq 0}$ be a filtration satisfying the usual conditions. Let $T$ be a finite stopping time for the filtration $\{\mathcal{F}_t\}_{t \geq 0}$. The $\{A \in \mathcal{F}: \text{ for each }t \geq 0, A \cap T^{-1}([0,t]) \in \mathcal{F}_t\} = \sigma( \{X_T: X \text{ is a cadlag process adapted to } \{\mathcal{F}_t\}_{t \geq 0} \})$

In Protter's theorem, he claims that

As far as I'm aware, cadlag means that ALMOST every path is cadlag. I showed that if a process $X$ has each sample path cadlag and is adapted, then it is progressively measurable by using the the functions $X^n(s,\omega) = X(0,\omega)\mathbf{1}_{\{0\}}(s)+\sum\limits_{k = 1}^{2^n}X(tk/2^n,\omega)\mathbf{1}_{(t(k-1)/2^n, tk/2^n]}(s)$ on $[0,t]\times \Omega$.

$X^n$ is $\mathcal{B}([0,t]) \bigotimes \mathcal{F}_t$ measurable and by right continuity, $X^n$ converges everywhere to $X$ on $[0,t]\times \Omega$, so X is progressively measurable.

However, if we drop the assumption that $X$ has every path cadlag and instead just almost every path is cadlag (ie $X$ is a cadlag process) we can let $N = \{\omega \in \Omega: X(\cdot,\omega) \text{ is not cadlag}\}$. Then N is a null set. Even if $\mathcal{F}_t$ is complete, $N$ is a measurable null set in $\mathcal{F}_t$ and therefore $[0,t]\times N$ is null in $\mathcal{B}([0,t]) \bigotimes \mathcal{F}_t$. Since $X^n$ converges to $X$ pointwise on $([0,t]\times N)^c$ we have that $X^n$ converges to $X$ almost everywhere. Now, if we don't know that the product measure space ($[0,t] \times \Omega, \mathcal{B}([0,t]) \bigotimes \mathcal{F}_t) $ is complete, then we can't say that the limit $X$ is $\mathcal{B}([0,t]) \bigotimes \mathcal{F}_t$ measurable.

So if anyone can clear up my confusion I would be very appreciative. I've been stumped for weeks now. I think that there must be another way to show the result than how I did with the simpler case above using sequences.

In theorem 6 of Protter, I was able to show that regardless of whether $\mathcal{F}_t$ is complete or not, $\mathcal{G}^* = \sigma(\{X_T: X \text{ is everywhere cadlag and adapted to } \{\mathcal{F}_t\}\}) = \{A \in \mathcal{F}: \text{ for each } t \geq 0, A \cap T^{-1}([0,t]) \in \mathcal{F_t}\} = \mathcal{F}_T$

However, I still cannot show the analogue for $\mathcal{F_t}$ satisfying the usual conditions and $X_t$ cadlag almost everywhere.

**EditI could show Protter's theorem 6 by following along in Bass' book using simpler stopping times that approximate $T$, but still have not solved the question I posted here. So now, I do not need a solution to Protter's theorem, but to the statement in the title.

Let $\mathcal F$ be a weakly compact subset of $L^1(\Omega)$. Dunford–Pettis theorem says that $\mathcal F$ is uniformly integrable. Also, by de la Vallée-Poussin theorem we can find an increasing convex function $\Phi:[0,\infty)\to[0,\infty)$ that is super linear, i.e. $$ \frac {\Phi(t)} t\to \infty \quad\text{ as $t\to\infty$ } $$ such that $\{\Phi(u):u\in\mathcal F \}$ is bounded in $L^1(\Omega)$.

Is it possible to assert further that we can find such $\Phi$ so that the family $\{\Phi(u):u\in\mathcal F \}$ is also equi-integrable?

Ideally I would also want $\Phi$ to be $C^1$ but I guess we can do that by smoothing out $\Phi$, if we manage to find one.

Consider the Cauchy problem: $$ \frac{\partial u}{\partial t} + \mathrm{i}\mkern1mu A(x,D_x) u = f \quad 0< t < T; \qquad u = u_0 \quad \text{when}\; t = 0, $$ where $A$ has real principal symbol $a(x,\xi)$. This problem is discussed in a number of sources (Hormander v.iii, Taylor's $\Psi$DO, etc. ).

Let $S(t,s)$ be the propagator from $s$ to time $t$, then when $f=0$ one has $$ \operatorname{WF}(S(t,0)u_0) = \chi_t \operatorname{WF}(u_0) $$ where $\chi_t$ is the flow generated by the Hamiltonian $H_a$.

Now take $f \in C^0([0,T], H^s(\mathbb{R}^n))$, $u_0 = 0$ (for simplicity), then the equation is well-posed and the solution is given by: $$ u(t,x) = \int_0^t S(t,s) f(s) d s$$

**My questions:**

What is the wavefront set of $u$ as a distribution in $(0,T)\times\mathbb{R}^n$?

I'm missing references on the question.

My feeling is that maybe seeing integration as the push-forward of the projection $\pi(t,s,x) = (t,x)$ might work. But then I guess I should see $S(t,s)$ as an FIO in $(t,s,x)$ which sounds strange and probably there is a more elementary derivation

This question is a continuation of a question I asked a couple weeks ago.

Let $(\Omega, \mathcal{C})$ be the Cantor space of binary sequences equipped with the usual product topology, and let $(\Omega, \mathcal{F})$ be the associated Borel measurable space. Let $P$ be a finitely additive probability measure on $(\Omega, \mathcal{F})$ that assigns clopen sets positive measure. That is, letting $\omega \in \Omega$ and $\omega^n$ denote the first $n$ bits of $\omega$, we assume that $P(\omega^n)>0$. So for all $F \in \mathcal{F}$, the conditional probability $P(F \mid \omega^n)$ is well-defined. Let $\mathcal{A}$ denote the smallest algebra containing the clopen sets. Then, $\sigma(\mathcal{A}) = \mathcal{F}$.

If $P$ is countably additive, then by martingale convergence $P(F \mid \omega^n) \to 1_F(\omega)$ for almost all $\omega$, and therefore, by Egorov's theorem, almost uniformly as well.

In the linked question, one can observe that almost uniform convergence holds without the assumption that $P$ is countably additive provided $P$ satisfies the following two properties.

(1) For all $\epsilon > 0$ and all $F \in \mathcal{F}$, there exists $A \in \mathcal{A}$ such that $P(F \triangle A)<\epsilon.$

(2) $P$ is countably additive on $\mathcal{G}$, the smallest algebra containing the open sets.

Even in the finitely additive setting, almost uniform convergence implies convergence in probability, and it is not too difficult to show that $$P(\omega: |P(F \mid \omega^n) - 1_F(\omega)|>\epsilon) \to 0 \ \ \text{for all} \ \ \epsilon>0, F \in \mathcal{F}$$ is equivalent to (1). Hence, (1) and (2) are sufficient for the almost uniform convergence of conditional probabilities and (1) is also necessary.

Is (2) necessary? I.e., does $P(F \mid \omega^n) \to 1_F(\omega)$ almost uniformly for all $F \in \mathcal{F}$ imply that $P$ is countably additive on $\mathcal{G}$?

I would also like to see an example of a finitely additive probability that satisfies (1) and (2) but is not countably additive. I know that there are merely finitely additive probabilities that satisfy (1).

Relatedly (and this is getting a bit far afield from the main questions, but I think it's worth asking here anyway), is it true that for every subalgebra $\mathcal{H}$ of $\mathcal{F}$ there is a merely finitely additive probability on $\mathcal{F}$ that is countably additive on $\mathcal{H}$? Or does countable additivity on some subalgebra force countable additivity on all of $\mathcal{F}$?

In the following $k$ is an algebraically closed field of characteristic $0$.

Consider the category $SH(k)$ (the Morel-Voevodsky stable motivic homotopy category).

By the work of Voevodsky (see for instance here), there is a tower (the slice tower): $$f_{q+1}E\to f_{q}E\to s_{q}E\to f_{q+1}E[1]$$ for $E$ an spectrum and $q\in \mathbb{Z}$.

Now consider the category $DM(k)$ (the Voevodsky´s big triangulated tensor category of motives).

There is a functor $$EM:DM(k)\to SH(k)$$

induced by the equivalence of triangulated categories Ho(Mod-$M\mathbb{Z})\cong DM(k)$ and the forgetful functor Ho(Mod-$M\mathbb{Z})\to SH(k)$.

Let $M(\mathbb{P}^{n})\in DM(k)$ be the motive associated to the projective space. We know that $M(\mathbb{P}^{n})$ has the cell decomposition $$M(\mathbb{P}^{n})=\bigoplus_{i=0}^{n}\mathbb{Z}(i)[2i].$$

I have the following questions:

Is it true that $s_{q}EM\left(\bigoplus_{i=0}^{n}\mathbb{Z}(i)[2i]\right)=\bigoplus_{i=0}^{n}s_{q}EM(\mathbb{Z}(i)[2i])$?

What is, for example, $EM(\mathbb{Z}(1)[2])$?

How can I compute $s_{q}EM(M(\mathbb{P}^{n}))\in DM(k)$?