Let $A/\mathbf{F}_{p^n}$ be an Abelian variety with $\bar{A} = A \times_{\mathbf{F}_{p^n}} \bar{\mathbf{F}}_{p^n}$.

If $\ell \neq p$ is prime, there is a natural isomorphism $$\mathrm{H}^2_{\mathrm{et}}(\bar{A},\mu_{\ell^m}) = \mathrm{Hom}(\Lambda^2A[\ell^m],\mu_{\ell^m}).$$

What is the correct analogue (with fppf cohomology?) of this and of $T_\ell A = \varprojlim_mA[\ell^m]$ and of the Weil pairing $T_\ell A \times T_\ell A^t \to \mathbf{Z}_\ell(1)$ for $\ell = p$?

Edit: On the Weil pairing: See Oda, Tadao: *The first De Rham cohomology group and Dieudonne modules*. In: Ann. Sci. Éc. Norm. Supér. (4), 2 (1969), 63–135, p. 66 f., Theorem 1.1:

Let $f: \mathscr{A} \to \mathscr{A}'$ be an $X$-isogeny of Abelian schemes. The Weil pairing $$ \langle\cdot,\cdot\rangle_f: \ker(f) \times_X \ker(f^t) \to \mathbf{G}_m $$ is a non-degenerate and biadditive pairing of finite flat $X$-group schemes, i. e. it defines a canonical $X$-isomorphism $$ \ker(f^t) = (\ker(f))^t. $$ Moreover, it is functorial in $f$.

It remains the question on $$\mathrm{H}^2_{\mathrm{et}}(\bar{A},\mu_{\ell^m}) = \mathrm{Hom}(\Lambda^2A[\ell^m],\mu_{\ell^m}).$$

Edit 2: For $\ell \neq \mathrm{char}\,k$, this can be proved as follows: $$H^q(\bar{A},\mu_{\ell^n}) = \Lambda^qH^1(\bar{A},\mu_{\ell^n}) \otimes \mu_{\ell^n}^{\otimes(-q+1)} = \Lambda^q\mathrm{Hom}(A[\ell^n],\mu_{\ell^n}) \otimes \mu_{\ell^n}^{\otimes(-q+1)} = \Lambda^q\mathrm{Hom}(A[\ell^n],\mathbf{Z}/\ell^n) \otimes \mu_{\ell^n} = \mathrm{Hom}(\Lambda^qA[\ell^n],\mu_{\ell^n}).$$ But the calculation of the $q$-th étale cohomology of $\bar{A}$ in http://jmilne.org/math/articles/1986b.pdf, Theorem 15.1(b) requires $\ell \neq \mathrm{char}\,k$.

I start with some known preliminaries on the problem:

**Classical result.** The one-dimensional Cauchy functional equation
$$
\forall x,y \in \mathbb{R}, \,\,\,f(x+y)=f(x)+f(y)
$$
with $f:\mathbb{R}\to \mathbb{R}$ is only solved by the trivial solutions $f(x)=cx$, for some $c \in \mathbb{R}$, if $f$ satisfies for some additional conditions, e.g., continuity.

**Classical result with restricted domain.** Now let $\mathbb{R}^+:=(0,\infty)$. It is clear from the proof of the above classical result that if $f:\mathbb{R}^+\to \mathbb{R}^+$ is a continuous function such that
$$
\forall x,y \in \mathbb{R}^+, \,\,\,f(x+y)=f(x)+f(y) \, ,
$$
then there exists $c \in \mathbb{R}^+$ such that $f(x)=cx$ for all $x$.

**Multidimensional Cauchy functional equation.** It is also well known that if $f:\mathbb{R}^2\to \mathbb{R}$ is a continuous function such that
$$
\forall x,y \in \mathbb{R}^2, \,\,\,f(x+y)=f(x)+f(y),
$$
then there exist $A,B \in \mathbb{R}$ such that $f(x,y)=Ax+By$ for all $(x,y) \in \mathbb{R}^2$.

I know that the following generalization holds true as well. In particular, I already know how to prove it, by using a variant of the classical proof. In the following, a cone $C\subseteq \mathbb{R}^2$ is a set for which $\alpha x+\beta y \in C$ whenever $\alpha,\beta \in \mathbb{R}^+$ and $x,y \in C$.

**Fact.** Let $C\subseteq \mathbb{R}^2$ be a non-empty cone and $f:C \to \mathbb{R}$ be a continuous function such that
$$
\forall x,y \in C, \,\,\,f(x+y)=f(x)+f(y).
$$
Then there exist $A,B \in \mathbb{R}$ such that $f(x,y)=Ax+By$ for all $(x,y) \in C$.

Is it a known result? In such case, does anyone have a reference for this result?

Let $\mathcal{S}_x$ and $\mathcal{S}_y$ be a finite discrete sets, such that

$$ 0 < |\mathcal{S}_x| < \infty, \qquad 0 < |\mathcal{S}_y| < \infty, \qquad \mathcal{S}_x \cap \mathcal{S}_y = \emptyset $$

Let $\mathcal{T}_x$ and $\mathcal{T}_y$ be non-empty sets such that

$$ \mathcal{T}_x \subseteq \mathcal{S}_x, \quad \mathcal{T}_x \neq \emptyset; \qquad \mathcal{T}_x \subseteq \mathcal{S}_y, \quad \mathcal{T}_y \neq \emptyset; $$

Let $\mathcal{X} = \left\{ x_i \right\}_{i=1}^m$, and $\mathcal{Y} = \left\{ y_i \right\}_{j=1}^n$ be partitions of $\mathcal{S}_x$ and $\mathcal{S}_y$ respectively, i.e.

$$ \sqcup_{i=1}^m x_i = \mathcal{S}_x; \qquad \sqcup_{j=1}^n y_j = \mathcal{S}_y $$

where $\sqcup \cdot$ is a disjoint union.

Define:

- measures $\mu_x: \mathcal{X} \to \left[0,\, |\mathcal{T}_x|\right]\;$ and $\;\mu_y: \mathcal{Y} \to \left[0,\, |\mathcal{T}_y|\right]$ as

$$ \mu_x(A) = |A \cap \mathcal{T}_x|, \qquad \forall\, A \subseteq \mathcal{X} $$

$$ \mu_y(B) = |B \cap \mathcal{T}_y|, \qquad \forall\, B \subseteq \mathcal{Y} $$

- probability measures $P_x : \mathcal{X} \to [0, 1]\;$ and $\;P_y : \mathcal{Y} \to [0, 1]$ as

$$ P_x(A) = \frac{\mu_x(A)}{|\mathcal{T}_x|}, \qquad \forall\, A \subseteq \mathcal{X} $$

$$ P_y(B) = \frac{\mu_y(B)}{|\mathcal{T}_y|}, \qquad \forall\, B \subseteq \mathcal{Y} $$

Now I want to come up with joint probability mass function $P_{xy}$ of random variables $X$ and $Y$ such that

- Marginal distribution of $X$ is $P_x$ and marginal distribution of $Y$ is $P_y$, i.e.

$$ \sum_{y \in \mathcal{Y}} P_{xy}(x,y) = P_x(x), \qquad x \in \mathcal{X} $$

$$ \sum_{x \in \mathcal{X}} P_{xy}(x,y) = P_y(y), \qquad y \in \mathcal{Y} $$

- $X$ and $Y$ are
**not**independent, i.e.

$$ P_{xy} \neq P_x P_y $$

So far I had 2 ideas:

$$ P_{xy}(A,B) = \frac{\mu_x(A) + \mu_y(B)}{|\mathcal{T}_x| + |\mathcal{T}_y|}, \qquad \forall\, A \subseteq \mathcal{X},\, B \subseteq \mathcal{Y} $$ but I'm not sure whether condition 1 is not satisfied.

$$ P_{xy}(A,B) = \frac{\mu_x(A) \mu_y(B)}{|\mathcal{T}_x| |\mathcal{T}_y|}, \qquad \forall\, A \subseteq \mathcal{X},\, B \subseteq \mathcal{Y} $$ but now looks like they are independent.

Let $\mathcal{S}_x$ and $\mathcal{S}_y$ be a finite discrete sets, such that

$$ 0 < |\mathcal{S}_x| < \infty, \qquad 0 < |\mathcal{S}_y| < \infty, \qquad \mathcal{S}_x \cap \mathcal{S}_y = \emptyset $$

Let $\mathcal{T}_x$ and $\mathcal{T}_y$ be non-empty sets such that

$$ \mathcal{T}_x \subseteq \mathcal{S}_x, \quad \mathcal{T}_x \neq \emptyset; \qquad \mathcal{T}_x \subseteq \mathcal{S}_y, \quad \mathcal{T}_y \neq \emptyset; $$

Let $\mathcal{X} = \left\{ x_i \right\}_{i=1}^m$, and $\mathcal{Y} = \left\{ y_i \right\}_{j=1}^n$ be partitions of $\mathcal{S}_x$ and $\mathcal{S}_y$ respectively, i.e.

$$ \sqcup_{i=1}^m x_i = \mathcal{S}_x; \qquad \sqcup_{j=1}^n y_j = \mathcal{S}_y $$

where $\sqcup \cdot$ is a disjoint union.

Define:

- measures $\mu_x: \mathcal{X} \to \left[0,\, |\mathcal{T}_x|\right]\;$ and $\;\mu_y: \mathcal{Y} \to \left[0,\, |\mathcal{T}_y|\right]$ as

$$ \mu_x(A) = |A \cap \mathcal{T}_x|, \qquad \forall\, A \subseteq \mathcal{X} $$

$$ \mu_y(B) = |B \cap \mathcal{T}_y|, \qquad \forall\, B \subseteq \mathcal{Y} $$

- probability measures $P_x : \mathcal{X} \to [0, 1]\;$ and $\;P_y : \mathcal{Y} \to [0, 1]$ as

$$ P_x(A) = \frac{\mu_x(A)}{|\mathcal{T}_x|}, \qquad \forall\, A \subseteq \mathcal{X} $$

$$ P_y(B) = \frac{\mu_y(B)}{|\mathcal{T}_y|}, \qquad \forall\, B \subseteq \mathcal{Y} $$

Now I want to come up with joint probability mass function $P_{xy}$ of random variables $X$ and $Y$ such that

- Marginal distribution of $X$ is $P_x$ and marginal distribution of $Y$ is $P_y$, i.e.

$$ \sum_{y \in \mathcal{Y}} P_{xy}(x,y) = P_x(x), \qquad x \in \mathcal{X} $$

$$ \sum_{x \in \mathcal{X}} P_{xy}(x,y) = P_y(y), \qquad y \in \mathcal{Y} $$

- $X$ and $Y$ are
**not**independent, i.e.

$$ P_{xy} \neq P_x P_y $$

So far I had 2 ideas:

$$ P_{xy}(A,B) = \frac{\mu_x(A) + \mu_y(B)}{|\mathcal{T}_x| + |\mathcal{T}_y|}, \qquad \forall\, A \subseteq \mathcal{X},\, B \subseteq \mathcal{Y} $$ but I'm not sure whether condition 1 is not satisfied.

$$ P_{xy}(A,B) = \frac{\mu_x(A) \mu_y(B)}{|\mathcal{T}_x| |\mathcal{T}_y|}, \qquad \forall\, A \subseteq \mathcal{X},\, B \subseteq \mathcal{Y} $$ but now looks like they are independent.

It is conjectured that for a discrete, finitely presented group $G$ such that $BG$ satisfies Poincaré duality, there actually exists a closed smooth manifold $M$ which is homotopy equivalent to $BG$.

This is somehow pointing in the opposite direction as Borel's conjecture, which implies that the homeomorphism type of such a manifold $M$ is uniquely determined.

Who conjectured this first? Is it also due to Borel, or was it Wall, or somebody else?

On pp. 152-3 of Hydon's *Symmetry Methods for Differential Equations* (2000 ed.), he lists some computer packages for symmetry-finding. This related Mathematica StackExchange question mentions the SYM Mathematica package and Maple's DEtools/symgen. Does SAGE have anything similar for doing symmetry-finding?

The proof of Gödel's incompleteness theorem can be streamlined by means of the Carnap-Gödel diagonal lemma and the ensuing fixed point theorem $\vdash_S G\leftrightarrow\lnot\Pi\ulcorner G\urcorner$ together with adequacy conditions upon the provability predicate $\Pi$. Formal system S is incomplete if S is consistent and $\vdash_S A \Leftrightarrow \ \vdash_S\Pi\ulcorner A\urcorner$.

Is there a similar streamlining of the proof of the undecidability of the Halting Problem that appeals to the Carnap-Gödel diagonal lemma and fundamental adequacy conditions?

Let $G$ be a weakly amenable group, in the sense that it has a net of finitely supported functions $\varphi:G\to \mathbb{C}$ which converge point wise to 1 and their cb norm is bounded uniformly by some constant $C$.

It is known that this is equivalent to completely bounded approximation property (CBAP) for the reduced group $C^*$-algebra of $G$.

**Question:** If $G$ is weakly amenable, does the full group $C^*$-algebra have CBAP? Namely as a multipliers, do the functions $\varphi$ extend to completely bounded multipliers of the full group $C^*$-algebra, with uniformly bounded norms?

If the answer is negative, then what other $C^*$-completions of the group ring $\mathbb{C}G$ are known to have the CBAP under the assumption that $G$ is weakly amenable?

I apologise in advance if my question is too basic.

Some notation:

$(X,\cal{X})$ denotes a measurable metric space where $X$ is a metric space and $\cal{X}$ is the associated Borel sigma algebra.

$B(X)$ is the space of all bounded continuous functions defined on $X$.

Let $\{\mu_n\}$ and $\{\nu_n\}$ be sequences of probability measures on the above measurable space $(X, \mathcal{X})$. Assume that each $\mu_n$ is absolutely continuous with respect to $\nu_n$, with an density $h_n\in B(X)$. Suppose that $\mu_n\to \mu$, $\nu_n\to \nu$ in the weak star topology and $h_n$ converges to a bounded continuous function $h$.

**Question:**
I would like to know if $\mu\ll\nu$. If so, is $h$ the density? If not, is there some condition in order
to have $\mu\ll\nu$?

Other information that can be useful is that each $\nu_n$ and $\mu_n$ has support in a compact subset $K_n\subset X$ which increase to $X$, i.e, $X=\bigcup K_n$.

... and really *without even the possibility of having objects*, so it's not a matter of just finding the "correct" flavour of Fukaya category to use.

**Question:** Does there exist interesting symplectic manifolds $(M,\omega)$ without:

- Any unobstructed Lagrangians (and therefore, with a $Ob(Fuk(M,\omega)) = \varnothing$)?
- Any (smooth) Lagrangians?
- Any (immersed) Lagrangians?

Let $\Omega\subset R^n$ be bounded, $0\leq u\in L^\infty((0,T)\times\Omega)$, $0\leq v\in L^1((0,T)\times\Omega)$. If $u(0,x)=0$ for all $x\in \Omega$ and $$\frac{d}{dt}\int_\Omega u(t,x)dx\leq \int_\Omega v(t,x)u(t,x)dx.$$ Whether we have $u(t,x)=0$ for all $(t,x)\in(0,T)\times\Omega$? If not, whether we can add some other conditions.

What kind of hyper-surfaces, functions I am dealing with here?

A smooth hyper-surface $S^{n-1}$ in $R^{n}$ is defined by an equation $(x\frac{\partial f(x)}{\partial x})=\rho>0$ with a smooth positive definite (non-homogeneous) function $f(x)>0$ for $\forall x\in R^{n}\setminus0$.

Looks like that it is homeomorphic to a (n-1)-sphere in $R^{n}$ with $0$ as its volume internal point. It seems to be kind of convex, but not in terms of any strict notion of convexity I am aware of. What are those types of surfaces in general? Or these types of functions?

Let us now include the possibility of (limit to) $\rho=0$. Assuming that $f(0)=0$, how to make sure that the function $f(x)$ is such, that the equation $(x\frac{\partial f(x)}{\partial x})=0$ has the only solution $x=0$? Will positive Hessian suffice? Positive semi-definite in a vicinity of $0$ only? Will it be necessary, or sufficient, or both? Are there any theorems on this?

I'll be really grateful for a help in clarifying the concepts and for useful specific references.

Thank you.

For the second part, prove the same identity using the technique called “Double counting” or “Combinatorial argument”. It’s a combinatorial proof technique for showing that two expressions are equal by demonstrating that they are two ways of counting the size of one set. In other words, we need to come up with a “story” for what both, the left-hand side and the right-hand side, are counting.

If anybody wants to have fun solving trigonometrics systems, I am highy interested :)

I've got the following system to solve, which includes trigonometric functions. $x_{0}$, $x_{1}$, $y_{0}$, and $y_{1}$, are known.

I'm looking for $\theta_{0}$ and $\theta_{1}$ and I am stuck... Any idea ?

$ [x_{1}cos(\theta_{1}) - y_{1}sin(\theta_{1}) ] - [x_{0}cos(\theta_{0}) - y_{0}sin(\theta_{0})] = 1 $ $ [x_{1}sin(\theta_{1}) + y_{1}cos(\theta_{1}) ] - [x_{0}sin(\theta_{0}) + y_{0}cos(\theta_{0})] = 1 $

Any help will be gratefully accepted !

Thank you in advance :)

The Quillen-Suslin theorem asserts that there are no nontrivial vector bundles over the affine space $\mathbb{A}^{n+1}$, $n\geq 0$.
Let's work over the complex numbers. What can be said about vector bundles on the *punctured* affine space $X_n=\mathbb{A}^{n+1}\smallsetminus\{0\}$?
According to this paper, there seem to be room for nontrivial vector bundles.

Let $\mathbb{C}^{*}$ act on $X_n$ by the action $\lambda.(x_0,\dots,x_n):=(\lambda x_0,\lambda x_1,\dots, \lambda x_n)$ whose quotient is $\mathbb{P}^n$. Notice that equivariant v.b. on $X_n$ are in bijection -via pullback- with v.b. on $\mathbb{P}^n$, and the latter form already a rich moduli problem on its own. In this question we concentrate on the specificity of $X_n$

**1.** Is there some sort of classification of v.b. on $X_n$, taking as a starting base -say- the "classification" of stable v.b. on $\mathbb{P}^n$ given by the corresponding moduli spaces?

What about particular ranks, for example the case of **line bundles**?

**2.** Are there vector bundles on $X_n$ that are not pullbacks of v.b. on $\mathbb{P}^n$, that is, v.b. on $X_n$ that do not admit an equivariant structure?

Let $\bar{X}$ be a compact Riemann surface of genus $g>0$. Let $X$ be $\bar{X}$ minus a finite set of points $\{a_1,\ldots,a_n\}$ ($n\geq 1$). Let $X^{(r)}$ be the configuration space of $r$ ordered distinct points on $X$: $$X^{(r)}=X^r-\bigcup_{i\neq j}\{(x_1,\ldots,x_r): x_i=x_j\}$$

(1) Is there an explicit description of the Lie algebra associated to the lower central series of the fundamental group of $X^{(r)}$ (=pure braid group of $r$ strands on $X$) in terms of generators and relations?

(2) Are there cases (i.e. specific curves) for which an explicit description of $H^1_{dR}(X^{(r)})$ is known?

Any relevant references will be appreciated. For (1), a paper of Bezrukavnikov addresses $X=\bar{X}$ case. A paper of Nakamura, Takao and Ueno studies the non-compact situation but works with the weight central series (as opposed to the lower central series). In (2), by an explicit description I mean an explicit basis of $H^1_{dR}(X^{(r)})$.