For some c > 0.

$\xi''(s) = O(\log^3(t))$ for $|t| \geq 2$ and $\sigma > 1 - \frac{c}{\log(|t|)}$.

Here David Hansen says

By hard results of Deligne, Grothendieck, and Gabber, $Ri^! \mathcal{E}$ is still bounded and constructible...

What are these results of Deligne, Grothendieck and Gabber? What is the reference where I can find them?

Let $\mathbf{H}P^\infty$ denote the infinite-dimensional quaternionic projective space. The inclusion of its bottom cell defines a map $S^4 \to \mathbf{H}P^\infty$. Does this extend to a map $\Omega S^5 \to \mathbf{H}P^\infty = BSU(2)$?

Since $\Omega S^5$ is the James construction on $S^4$, this question would be very easy to answer (in the positive) if $\mathbf{H}P^\infty$ was a homotopy associative H-space --- but it's known that this is not true. (If $Y$ is a homotopy associative H-space, then any map $X\to Y$ from a path-connected space $X$ admits a unique extension to a H-space map $\Omega \Sigma X \to Y$.) However, the composite $S^4\to BSU(2) \to BSU$ does extend to a map $f_\xi:\Omega S^5\to BSU$ classifying a bundle $\xi$ over $\Omega S^5$; it is easy to see that the Chern classes $c_i(\xi)$ vanish for $i\geq 3$, so the map $f_\xi$ factors, at least on cohomology, through $BSU(2)$.

One natural expectation for the desired map is that it gives a map of fiber sequences from the EHP fiber sequence $S^2 \to \Omega S^3 \to \Omega S^5$ to the Hopf invariant fiber sequence $$S^2 \to \mathbf{C}P^\infty = BS^1 \to \mathbf{H}P^\infty = BS^3$$ via the map $\Omega S^3 \to \mathbf{C}P^\infty$ extending the inclusion of the bottom cell of the target. In fact, thinking along these lines shows that we'd get the desired map if $S^2$ was a loop space, which it isn't.

An approach to constructing the desired map comes from equivariant considerations. Namely, the bottom $C_2$-equivariant cell of $\mathbf{C}P^\infty$ under the complex conjugation action is the one-point compactification $S^\rho$ of the regular representation $\rho$ of $C_2$. This gives a map $\Omega S^{\rho+1} \to \mathbf{C}P^\infty$, and hence a map $(\Omega S^{\rho+1})_{hC_2} \to (\mathbf{C}P^\infty)_{hC_2} = \mathbf{H}P^\infty$. To get the desired map, it therefore suffices to construct a nonequivariant map $\Omega S^5 \to (\Omega S^{\rho+1})_{hC_2}$, but it's not clear to me how/whether such a map exists.

I'd like to remark that looping the map $\Omega S^5\to \mathbf{H}P^\infty$ defines a map $\Omega^2 S^5\to S^3$. This is already known to exist: it is the map appearing in work of Cohen-Moore-Neisendorfer.

I have know this argument for decades. I have no idea of its source. If anyone knows (not guesses) its origin, then I would be very appreciative. My guesses are among Ralph Fox, JHC Whitehead, RH Bing, GT Whyburn, RL Moore. But a reference as well as a name would be the best outcome. Even a reference to a textbook with the argument (no guesses please, I can do that myself) would be good.

Here is the argument.

Let K be a continuum (compact connected set) in the plane. Let K+r represent the hirozontal translate of K by r units parallel to the x-axis.

To prove: if r>0, s>0, t=r+s, then if K and K+r are disjoint and if k and K+s are disjoint, then K and K+t are disjoint.

Proof:

Consider A=K, B=K+r, C=K+t=K+r+s. Assume A and B are disjoint, and B and C are disjoint. This matches the hypothesis. We must show that A and C are disjoint.

Let M be a point of greatest height in B and let m be a point of least height in B. Let d be 1/3 of the smaller of the two distances d(A,B) and d(B,C). The N_d(B) neighborhood of all points no more than d from B is open and connected. It is also disjoint from N_d(A) and N_d(C). There is a polygonal embedded path P from M to m in N_d(B) that never goes over hight M or below height m. So P union the vertical ray to + infinty from M union the vertical ray to - infinity from m is an embedding T of the real line missing N_d(A) and N_d(C). The point M-r is in A and the point M+s is in C. So A is "to the left" of T and C is "to the right" of T. So T separates A from C in the plane and A and C are disjoint.

Application 1:

Let K be a continuum in the plane and let L be a line segment of length t with both endpoints in K. Let r+s=t (all numbers positive). Then either there is a line segment parallel to L of length r with both endpoints in K or a line segment parallel to L if length s with both endpoints in K.

Proof: Rotate everything until L is parallel to the x-axis and reinterpret "with both endpoints in K" as "K intersects its traslate by the length of the segment."

Application 2:

Under the assumption of App. 1, and n is a positive integer, then there is a line segment of length d/n with both endpoints in K.

Proof: It holds for n=1, and if it holds for n-1, then it holds for n by App. 1 since App. 1 gives line segments of either length d(n-1)/n or d/n.

Application 3:

If a simple closed curve represents an element (m,n) in H_1 of a torus, then gcd(m,n)=1.

Proof: Lift the simple closed curve to a path in the plane. Let d be the lenght of the straight line segment from the beginning to the end of the path. Let g=gcd(m,n) and assume g>1. By App. 3, there is a straight line segment of length d/g with both endpoints in the path. But the endpoints project to the same point in the torus and the original curve must have been singular.

Application 4:

If a simple closed curve lives in an annulus, then it represents a generator of pi_1.

Proof: similar to App. 3.

Let $G$ be a Lie group and $\pi:E_G\rightarrow M $ be a principal $G$-bundle.

I have seen in many places that a connection on $(E_G,M,G)$ is a splitting of the Atiyah sequence

$$ 0\rightarrow \text{ad}(E_G)\rightarrow \text{At}(E_G)\rightarrow T M\rightarrow 0$$

where $\text{ad}(E_G)$ is the adjoint vector bundle for $E_G$ and $\text{At}(E_G)$ is the Atiyah bundle for $E_G$.

Reference given for this is Atiyah's paper. This paper is slightly difficult to read.

Can some one give an outline of this construction or point out some exposition where this is written in detail.

p V ( p Λ q ) =p

I am confusing in this question

Let $\gamma \geq 0$ and consider the fractional derivative operator defined in Fourier domain by $$\mathcal{F} \{\mathrm{D}^{\gamma} \varphi \} (\omega) = (\mathrm{i} \omega)^{\gamma} \mathcal{F}\{\varphi\} (\omega),$$ where $\varphi \in \mathcal{S}(\mathbb{R})$ is a smooth and rapidly decaying function.

Of course, the definition can be extended to much more functions than $\varphi \in \mathcal{S}(\mathbb{R})$, including some, but not all, tempered distributions. It is for instance possible to extend $\mathrm{D}^{\gamma}$ to any compactly supported distribution (as for any convolution operator from $\mathcal{S}(\mathbb{R})$ to $\mathcal{S}'(\mathbb{R})$).

My question is the following: Is there a good notion of the "domain of definition" of the operator $\mathrm{D}^{\gamma}$, understood as the largest topological vector space $\mathcal{S}(\mathbb{R}) \subseteq \mathcal{X} \subseteq \mathcal{S}'(\mathbb{R})$ such that $\mathrm{D}^{\gamma} : \mathcal{X} \rightarrow \mathcal{S}'(\mathbb{R})$ is well-defined and continuous? Or at least, if the question is somehow meaningless, any natural construction that will include many tempered distributions in a satisfactory* manner?

*To give a bit of context, I am especially interested by the fractional case where $\gamma \notin \mathbb{N}$. The question is obvious for $\gamma = n \in \mathbb{N}$, since one can select $\mathcal{X} = \mathcal{S}'(\mathbb{R})$. However. when $\gamma$ is purely fractional, there is no hope to define the product $(\mathrm{i} \omega)^{\gamma} \mathcal{F}\{u\} (\omega)$ when $u \in \mathcal{S}'(\mathbb{R})$ is too irregular around the origin, which means morally that $u$ growth too fast at infinity. "In a satisfactory manner" would be a way of specifying properly a good "growth property" of $u \in \mathcal{X}$.

This question consists of 5 related sub-questions.

- Does there exist a non-empty scheme $X$ such that any reduced scheme whose underlying space is homeomorphic to that of $X$ (possibly via a homeomorphism not induced by any morphism of schemes) is isomorphic to $X$? Spectra of fields do not work, for example.
- Does there exist a non-empty scheme $X$ such that any morphism of schemes $Y\rightarrow X$ with a reduced source that induces a homeomorphism on underlying spaces is an isomorphism?
- Does there exist a non-empty scheme $X$ such that any morphism of schemes $Y\rightarrow X$ with a reduced source that induces a homeomorphism on underlying spaces has isomorphic source and target?
- Does there exist a non-empty scheme $X$ such that any morphism of schemes $X\rightarrow Y$ with a reduced target that induces a homeomorphism on underlying spaces is an isomorphism?
- Does there exist a non-empty scheme $X$ such that any morphism of schemes $X\rightarrow Y$ with a reduced target that induces a homeomorphism on underlying spaces has isomorphic source and target?

If the answer to any of these questions is positive, is there some classification or references studying such questions, at least?

Let $L$ be a very ample line bundle on a variety $X$ of dimension $n$. The slope of a torsion free sheaf $E$ on $X$ is given by $$\mu_{L}(E) : = \dfrac{c_{1}(E).L^{n-1}}{rank(E)}$$

Let $\pi : \widetilde{\mathbb{P}^{3}} \rightarrow \mathbb{P}^{3}$ be the blowup de $\mathbb{P}^{3}$ along a regular curve $C$. What is an ample sheaf on $\widetilde{\mathbb{P}^{3}}$? Does it make sense to talk about the stability of tangent sheaf of $\widetilde{\mathbb{P}^{3}}$? I can't find any references in this matter.

Thanks in advance.

Let $S$ be a closed surface and $G$ be a reductive Lie groups. Goldman (see here) proved that for a fairly general class of groups $G$, $M=Hom(\pi_1(S),G)/G$ admits a symplectic structure where the quotient is by conjugation action. Hence the space of all "real valued" functions $C^{\infty}(M,\mathbb{R})$ admit a Lie bracket $\{\,,\}$.

Suppose $G$ is one of the following Lie groups: $GL_n(\mathbb{R}), GL_n(\mathbb{C}), SL_n(\mathbb{R}), SL_n(\mathbb{C}).$ Given any $x\in\pi_1(S)$, we define a function $f_x:M\rightarrow \mathbb{R}$ by $f_x(\rho)=\Re(\mathrm{tr}(\rho(x))),$ where $\Re$ is the real-part of a complex number. Given $x,y\in\pi_1(S)$, Goldman gave explicit formulas for $\{f_x,g_y\}$.

In some papers the authors consider the Lie bracket between two complex valued function on $M$ and used Goldman's formula and paper as a reference. For example Section 4 of this, Page 542 of this and this, considered the Lie bracket of the trace functions (not just the real part) defined similarly as above.

My question is: what is the Lie bracket in $C^\infty(M,\mathbb{C})$ and how is it related to the Lie bracket of $C^\infty(M,\mathbb{R})$?

Any kind of suggestion/reference/comment will be extremely helpful. Thanks in advance.

Define $\text{rad}_{23}(2^m3^nr)=2^{\text{sign}(m)}3^{\text{sign}(n)}r$, where $m,n\ge0$ and $2,3\nmid r\in\mathbb{N}$.

For a triple $a+b=c$ define the quality $q_{23}(a,b,c)=\frac{\log(c)}{\log(\text{rad}_{23}(abc))}$.

Has anyone attempted to *specifically* prove that only finitely many primitive triples have quality $q_{23}>1+\epsilon$ for any given $\epsilon>0$? Is there reason to hope this may be within reach of well established methods?

Based on this table of $abc$-triples, only 2 mini-$abc$-triples: are known with quality $\ge1.4$:

$37 + 2^{15} = 3^8 \times 5,\ \ \ \ \ \ \ q_{23}=1.48291$

$5 + 3^{11} = 2^{10} \times 173,\ \ \ \ \ q_{23}=1.41268$

I have been trapped in solving the following ODE for a long time. I wonder if it has unique analytical solution \begin{equation} [b+c_B(\bar{\beta}^H-\bar{\beta}^L)]\frac{dF(x)}{dx}+c_BF(x)-c_BF(x+\bar{\beta}^H-\bar{\beta}^L-b/c_B)-c_B=0. \end{equation}

I could try to assume $F(x)$ is linear. But I was wondering if there is a way to prove or disprove the uniqueness of the solution of this ODE? Is there a systematic way to find all the solutions to this equation? Thanks for the comments. Now I know that this is a delay differential equation. In my problem, I have $x\geq b/c_B+\bar{\beta}^L-\bar{\beta}^H$. And in fact, $F(x)$ is a cumulative distribution function in my case.

Let $f: X \rightarrow Y$ be a morphism between smooth projective varieties over an algebraic closed field. The graph of $f$ namely $\Gamma_f$ is a cycle inside $X \times Y$. When is $\Gamma_f$ nef? When is it a complete intersection of ample divisors? Are there some useful criterions or restrictions?

If $X=Y$ and $f=id$ then this is a problem about the diagonal cycle, for example in the curve case the diagonal is ample iff genus of $X$ is zero. How about the surface or threefold case?

In the general morphism case with $Y$ being a curve (or just $\mathbb P^1$), the graph is a divisor, maybe the property can be described in a simple way.

Suppose that $X,Y$ are scalar random variables supported on some standard Lebesgue probability space $(\Omega, \mathrm{P})$, such that $X \overset{\mathrm{d}}{=} Y$ in the sense that their pushforward measures are equal, $X_*(\mathrm{P}) = Y_*(\mathrm{P})$. Does there exist a nondegenerate random variable $Z$ on $(\Omega, \mathrm{P})$ satisfying $X + Z \overset{\mathrm{d}}{=} Y + Z$?

In the case that $Y = X\circ T$ for some measure preserving automorphism $T: \Omega \rightarrow \Omega$ (this appears to be often the case by: Random variables with same distribution), and we have in addition the factorization $T = S^2$ for some automorphism $S: \Omega \rightarrow \Omega$, then clearly we can set $Z = X \circ S$, whence

$X + Z = X + X\circ S \overset{\mathrm{d}}{=} X \circ S + X \circ S^2 = Z + Y$.

Are there more general conditions than this, or perhaps conditions under which the answer to the question is negative?

I'm considering a ratio of incomplete Selberg integral: $$f_n(a,b)=\frac{\int_{\Delta_a}\prod_{i=1}^nx_i^{\alpha-\frac{n+1}{2}}\prod_{i=1}^n(1-x_i)^{-1/2}\prod_{i<j}|x_i-x_j|}{\int_{\Delta_b}\prod_{i=1}^nx_i^{\alpha-\frac{n+1}{2}}\prod_{i=1}^n(1-x_i)^{-1/2}\prod_{i<j}|x_i-x_j|}$$ where $$\Delta_a=\{(x_1,...,x_n):0<x_1<a,0<x_2<a,...,0<x_n<a\}$$ and $1>a>b>0$, $\alpha>\frac{n+1}{2}$.

My question is, how can we upper bound $f_n(a,b)$? Is there a bound like some powers of $Ca/b$ for some constant $C>0$?

A non-empty topological space without isolated points is called *maximal* if every finer topology on that space has at least an isolated point. The existence of a (Hausdorff) maximal space is a simple consequence of Zorn's Lemma.

Note that in a maximal space $(X, \tau)$, nowhere dense sets are closed (and discrete). Indeed, if there were a nowhere dense set $N$ that is not closed then $\tau \cup \{X \setminus N \}$ would be a subbase for a finer topology without isolated points on $X$. In particular, a maximal space can't contain any non-trivial convergent sequences, because a discrete set is nowhere dense in a space without isolated points.

Let $\mathfrak{max}$ be the minimal weight of a Hausdorff maximal space. Clearly $\aleph_1 \leq \mathfrak{max} \leq \mathfrak{c}$, but we can prove a better lower bound, namely: $\mathfrak{d} \leq \mathfrak{max}$.

If $X$ is a countable Hausdorff maximal space then $w(X) \geq \mathfrak{d}$.

Proof: Fix a point $x \in X$ and let $\{U_n: n < \omega \}$ be a maximal pairwise disjoint family of non-empty open sets with the property that $x \notin \overline{U_n}$, for every $n< \omega$: Clearly $x \in \overline{\bigcup \{U_n: n < \omega \}}$. Let $\{x^n_k: k < \omega \}$ be an enumeration of $U_n$.

Suppose by contradiction that $w(X)=\kappa < \mathfrak{d}$ and let $\{B_\alpha: \alpha < \kappa \}$ enumerate a local base at $x$. For every $\alpha < \kappa$, the set $B_\alpha$ intersects infinitely many $U_n$'s, so we can find an integer-valued function $f_\alpha$ with infinite domain $\subseteq \omega$ such that $x^n_{f_\alpha(n)} \in B_\alpha \cap U_n$, for every $n \in dom(f_\alpha)$.

Since $\kappa < \mathfrak{d}$, we can find a function $f: \omega \to \omega$ such that, for every $\alpha < \kappa$, there is $n \in dom(f_\alpha)$ with $f_\alpha(n) < f(n)$. Let $D=\{x^n_k: k \leq f(n), n < \omega \}$. Then $D$ is discrete and $x \in \overline{D} \setminus D$, so $D$ is a nowhere dense set in $X$ which is not closed and that contradicts maximality.

QUESTION: Is $\mathfrak{max}=\mathfrak{d}$ in ZFC?

A housewife is waiting for guests and has prepared a cake. She doesn't know how many guests will come, but it will be $n-1$, $n$, or $n+1$.
*What is the minimal number $f(n)$ of pieces the cake should be cut to make it possible to divide between guests equally?*

For $n=2$, $f(n)=f(2)=4$:

The problem was posed 16.10.2018 by Oleksandr Maksymets on page 76 of Volume 2 of the Lviv Scottish Book.

**The prize:** Cooked duck or lunch + beer!

This comes out of a series of transformations, so I'll just get to the main focus here.

Define the functions $$F_n(x)=\frac12\left(x+2+2\sqrt{x+1}\right)^n+\frac12\left(x+2-2\sqrt{x+1}\right)^n. \tag1$$ It's straightforward to obtain $$F_n(x)=\sum_{r=0}^n\binom{2n}{2r}(x+1)^r. \tag2$$

**QUESTION 1.** On the other hand, how does one obtain (3) by manipulating (1)?
$$F_n(x)=\sum_{r=0}^n\frac{n}{n+r}\binom{n+r}{2r}4^rx^{n-r}. \tag3$$

**QUESTION 2.** As an aside, I'd also be pleased with a combinatorial proof of the sums (2)=(3).

Let

- $f\in C^3(\mathbb R)$ with $f>0$ and $$\int f(x)\:{\rm d}x=1$$
- $g:=\ln f$ (and assume $g'$ is Lipschitz continuous)
- $n\in\mathbb N$, $$s(x,y):=\sum_{i=1}^n\left(g(y_i)-g(x_i)\right)$$ and $$h(x,y):=\min\left(1,e^{s(x,\:y)}\right)$$ for $x,y\in\mathbb R^n$
- $x\in\mathbb R^n$ and $Y$ be a $\mathbb R^n$-valued normally distributed random variable on a probability space $(\Omega,\mathcal A,\operatorname P)$ with mean vector $x$ and covariance matrix $\sigma I_n$ for some $\sigma>0$ ($I_n$ denoting the $n\times n$ identity matrix)

I want to make the following argumentation rigorous: By Taylor's theorem, \begin{equation}\begin{split}h(x,Y)-h(x,(x_1,Y_2,\ldots,Y_n))&=\frac{\partial h}{\partial y_1}(x,(x_1,Y_2,\ldots,Y_n))(Y_1-x_1)\\&+\frac12\frac{\partial^2h}{\partial y_1^2}(x,(Z_1,Y_2,\ldots,Y_n))(Y_1-x_1)^2\end{split}\tag1\end{equation} for some real-valued random variable $Z_1$ with $Z_1\in[\min(x_1,Y_1),\max(x_1,Y_1)]$. Thus, \begin{equation}\begin{split}\left.\operatorname E\left[h(x,(y_1,Y_2,\ldots,Y_n))\right]\right|_{y_1\:=\:Y_1}&=\operatorname E\left[\min\left(1,e^A\right)\right]+g'(x_1)\operatorname E\left[1_{\left\{\:A\:<\:0\:\right\}}e^A\right](Y_1-x_1)\\&+\frac12(g''(Z_1)+\left|g'(Z_1)\right|^2)\left.\operatorname E\left[1_{\left\{\:B\:<\:0\:\right\}}e^B\right]\right|_{z_1\:=\:Z_1}(Y_1-x_1)^2.\end{split}\tag2\end{equation} Above, I wrote $$A:=\sum_{i=2}^n(g(Y_i)-g(x_i))$$ and $$B:=g(z_1)-g(x_1)+\sum_{i=2}^n(g(Y_i)-g(x_i))$$ in order to make the equation more readable (you need to replace them where they occur).

**Question 1**: There are two issues: The first one is that $(x,y)\mapsto\min(x,y)$ is partially differentiable in both arguments except on the diagonal $\Delta_2:=\left\{(x,y)\in\mathbb R^2:x=y\right\}$. Are we able to conclude the existence of $Z_1$ anyway? Note that $$\frac{\partial h}{\partial y_1}(x,y)=\begin{cases}\displaystyle g'(y_1)e^{s(x,\:y)}&\text{, if }s(x,y)<0\\0&\text{, if }s(x,y)>0\end{cases}\tag3$$ and $$\frac{\partial^2h}{\partial y_1^2}(x,y)=\begin{cases}\displaystyle(g''(y_1)+|g'(y_1)|^2)e^{s(x,\:y)}&\text{, if }s(x,y)<0\\0&\text{, if }s(x,y)>0\end{cases}\tag4$$ for all $y\in\mathbb R^n$.

**Question 2**: The second issue is the case $s(x,y)=0$. In order for $(3)$ to hold, we need to show that the probability of the corresponding event is $0$ (this seems to be related to the question whether the set on which the occurring function is not differentiable has Lebesgue measure $0$; and it's clear that $\Delta$ has Lebesgue measure $0$). How can we do that?

While it's clear that $h$ is partially differentiable with respect to the second variable except on a countable set, it is not clear to me why $h$ is even twice differentiable with respect to the second variable except on a set (at least) of Lebesgue measure $0$ (see this related question).

**EDIT**: Please take note of this related question which might yield a solution for question 2.

It was proven by Colin and Honda in Stabilizing the monodromy of an open book decomposition that any diffeomorphism can be made pseudo-Anosov and right-veering after a series of positive stabilizations.

Also we have the following result by A. Fathi in Dehn twists and pseudo-Anosov diffeomorphisms that says that if $\phi$ is pseudo-Anosov and $\gamma \subset S$ is a simple closed curve, then $t_\gamma^n \circ \phi$ is pseudo-Anosov except for at most seven consecutive values of $n$.

My question relaxes the conditions on $\phi$ and changes "composing with Dehn twists" to "stabilizing". More concretely:

Let $\tilde\phi:S \to S$ be a diffeomorphism of an oriented compact surface with boundary such that $\phi|_{\partial S} = id$ and such that $\phi$ contains a pseudo-Anosov piece in its Nielsen-Thurson decomposition. Let $c \subset S$ be a properly embedded arc. Attach a $1$-handle to $S$ along $\partial c$ and stabilize $\tilde\phi$ using this handle and the arc $c$. Call $\phi$ the stabilized diffeomorphism. Can we be sure that $\phi$ contains a pseudo-Anosov piece as well? Do there exist any necessary and sufficient conditions on $c$ so that the above is true?