Characterizations of Abelian varieties (3-folds) in positive characteristic

From this question Characterizations of complex Abelian varieties (especially 3-folds) among projective nonsingular varieties? I learned that if $X$ is a smooth complex projective variety of dimension $g$, then $X$ is a torsor over an Abelian variety (its Albanese variety) if and only if $\omega_X \cong \mathcal{O}_X$ and ${\rm h^1}(X; \mathcal{O}_X) = g$.

I would like to know the corresponding statement over an algebraically closed field of positive characteristic. If necessary, we could exclude small primes:

By the Bombieri-Mumford classification of surfaces in positive characteristic, I know that the above statement holds for $g=2$ and characteristics different from 2 and 3. In those small characteristics, one also gets examples of "quasi-hyperelliptic surfaces" (essentially because the Albanese could be a non-reduced group scheme), and they distinguish the two classes via étale cohomology (which as far as I can tell, I cannot calculate for my examples of interest).

For my own purposes, I want the answer for $g=3$, but the answer in general is also welcome. If it helps, I also know that ${\rm h}^i(X; \mathcal{O}_X) = \binom{g}{i}$ for all $i$.

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Enumerate spanning trees

I am using Pawel Winter's algorithm to enumerate all spanning trees. What I need to do now is enumerate all spanning trees where one edge say e1 remains in the tree and the edge e2 is in e1's fundamental cutset. I'm not sure where to start so any information is appreciated.

Edit: is there a graph operation that can be performed to guarantee the edge e2 is in e1's cutset? …Kind of like how contracting edges helps find all spanning trees with that edge in it.

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Is a non-singular invertible ergodic transformation on a measure space isomorphic to its inverse?

A non-singular, invertible, ergodic transformation is a quadruple $(X,\mathcal B, \mu, T)$ where $(X,\mathcal B, \mu)$ is a measure space and $T$ is an invertible, measurable automorphism where $\mu$ and $\mu\circ T$ are equivalent measures.

Two such systems $(X,\mathcal B, \mu, T)$ and $(Y,\mathcal C, \nu, S)$ are isomorphic when there exists an isomorphism $\phi: X \mapsto Y$ where $$ S\phi x = \phi T x.$$

It seems it should be obvious that $(X,\mathcal B, \mu, T)$ and $(X,\mathcal B, \mu, T^{-1})$ are isomorphic. If I assume $X$ is a product space and $T$ is the $+1$ odometer action then I can prove it myself. But I would rather quote someone and be done with it.

Do you know of a source to quote for this?

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Ranks of higher incidence matrices of designs

In 1978 Doyen, Hubaut and Vandensavel proved that if $S$ is a Steiner triple system $S(2,3,v)$ then the $GF(2)$ rank of its incidence matrix $N$ is $$ Rk_{2}(N)=v-(d_{p}+1), $$ where $d_{p}$ is the projective dimension of $S$.

I would like to know if there are known generalizations of this result in one (or better both!) of the following ways:

1. Replacing $S(2,3,v)$ with a more general design $S_{\lambda}(t,k,v)$.

2. Replacing the incidence matrix $N$ with a higher incidence matrix $N_{s}$.

$N_{s}$ has $\binom{v}{s}$ rows indexed by all $s$-subsets of $\{1,2,\ldots,v\}$ and its columns are indexed by the blocks of $S$, with $N_{s}(A,B)=1$ if $A \subseteq B$ and $0$ otherwise. The usual incidence matrix $N$ is equal to $N_{1}$.

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Canonical divisor of varieties of maximal Albanese dimension

Let $X$ be a smooth projective variety over an algebraically closed field. Suppose that $X$ is of maximal Albanese dimension, i.e., $X$ admits a generically finite morphism to an abelian variety. Then does $K_X$ have to be effective?

If the characteristic is 0, then this seems to follow from this criterion of maximal Albanese dimension. So I mainly interest in the positive characteristic case. The assertion is easy for curves, and for surfaces it can be verified by using the classification of surfaces. But I can't handle it in the general case.

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Benchmark datasets for 0/1 multidimensional knapsack problem, with the constraint that each bin can only hold a subset of all available items [closed]

I am working on an algorithm to assign observable events to smart-cameras in a smart-camera network. Each smart-camera can only observe dedicated groups of events due to the limitations of their field of view. Each camera can only communicate with its immediate neighbors. Each camera's "knowledge" is limited to its neighborhood (no routing of information). The algorithm shall be able to find the "optimal" solution when running instances of it on each smart-camera can only access the local knowledge of each camera.

This problem is similar to the 0/1 multidimensional knapsack problem, with the constraint that each bin can only hold a subset of all available items. In my case, a subset means the events that are within e.g. 10 meters of the cameras viewing range. So does anyone know benchmark test-data-sets for the 0/1 multidimensional knapsack problem, with the constraint that only subsets of items can be assigned to each bin? Thanks in advance

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binary intersection property in finite dimension

I have recently discovered a survey article called "The Hahn-Banach Theorem: The Life and Times" by Narici and Beckenstein and I have read it with great amusement:

I was particularly fascinated by Nachbin's theorem on extendibility, which I was not aware of. To put it shortly: Given two normed spaces $X,Y$ and a subspace $U$ of $X$, it is true that any bounded linear operator from $U$ to $Y$ can be extended to a bounded linear operator from $X$ to $Y$ with same norm if and only if $Y$ has the binary intersection property (BIP), i.e., if and only if every collection of pairwise intersecting closed balls has nonempty intersection.

At a certain point the authors claim that $R^n$, endowed with the $\ell^p$-norm, has the binary intersection property if and only if $n=1$ or $p=\infty$. It is clear that the Euclidean $R^2$-space does not have the binary intersection property cf. here, but I was not able to find a counterexample for $p=1$.

Thus, I started checking for some reference and eventually found another article by Narici, where he indeed quickly proves that $R^2$ with the 1-norm does have the binary intersection property (p. 13, third-to-last remark) and justify the fact that all $R^2$ with the $p$-norm in the reflexive range do not by the smoothness (in the sense of Fréchet differentiability) of their balls; and refer to Goodner's paper from 1950 for the proof of the fact that $R^n$, endowed with the $p$-norm, does not have the binary intersection property for $n>1$ and $1<p<\infty$.

So, the only open question is whether $R^n$ with the 1-norm has the binary intersection property. My geometrical intuition suggests me that this should be true, but Corollary 4.10 in Goodner's paper seems to state that this is not the case for any $n>2$. It surprises me that a counterexample is not immediately found, and I cannot quite follow Goodner's proof.

Does anybody know of a more direct proof and/or a simple pictorial counterexample?

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A question regarding a family of one-relator groups

Are the family of groups $\langle a,b| [a,b]^n\rangle$ non-isomorphic? If yes, how to show that. (Here, $[a,b]= aba^{-1}b^{-1}$)

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Must diffeomorphisms on $S^n$ have fixed points?

Question.

Let $N_1 \simeq S^k$ and $N_2 \simeq S^l$ be disjoint smoothly embedded spheres in $S^n$ with $k + l = n$. Suppose a diffeomorphism $\psi: S^n \to S^n $ preserves $N_1$ and $N_2$, and that each restriction $\psi|_{N_i}: N_i \to N_i$ is fixed-point-free. Is it necessarily true that $\psi$ has at least one fixed point on all of $S^n$?

I think when one of the sphere has dimension 1, the claim can be proven by the Brouwer's fixed-point. If two spheres have dimension $\geq 2$, the Lefschetz's fixed point theorem might be relevant, I think.

Are there known results or counterexamples for this situation?

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Explicit homotopy equivalence $\operatorname{BDiff} \rightarrow \operatorname{BGL}$

Let $\operatorname{BGL}(d)$ and $\operatorname{BDiff}(\mathbb{R}^d)$ be the simplicial Spaces defined as the nerves of the obvious topological groupoids.

I am looking for an explicit weak homotopy equivalence $f:\operatorname{BDiff}(\mathbb{R}^d) \rightarrow \operatorname{BGL}(d)$ between them (or between their fat/thin realizations).

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Question about formula for Siegel-Eisenstein series with character

What's the definition for a Siegel-Eisenstein series with 2 characters?

I know in the $\mathrm{SL}_2$ case it's something like this: let $\psi, \chi$ be two characters with modulus $u,v$ respectively, with $uv=N$, with $\chi$ primitive and we consider

$$ G_k^{\psi,\chi}(z) = \sum_{c=0}^{u-1} \sum_{d=0}^{v-1} \sum_{e=0}^{u-1} \psi(c) \bar{\chi}(d) \sum_{(a,b) \equiv (cv,d+ev) (\bmod\; N)} (az+b)^{-k}$$

here we have the additional technical condition $(\psi\chi)(-1) \cdot (-1)^k=1$ as well.

I know how to attach one character to Siegel Eisenstein series of type $\Gamma_0^{(n)}$ (I'm primarily interested in the case of degree=genus 2). It is given by

$$ E_n^{k,\chi}(Z) := \sum_{M \in \Gamma_\infty^{(n)} \backslash \Gamma_0^{(n)}(N)} \bar{\chi}(\det D) \det(CZ+D)^{-k}$$

One can see that $\Phi E_2^{k,\chi} = E_{k,\chi}$ which leads to a nice formulation of the $L$-function of $ E_2^{k,\chi}$ (at least away from prime factors away from $N$)

My question is: is there a way to attach two characters to the Siegel-Eisenstein series?

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A finiteness result for Galois cohomology of a reductive group over a global field

Let $K$ be a global field and let $V_K$ denote the set of places of $K$. Let $G$ be a connected reductive group over $K$, and let $\xi\in H^1(K,G)$ be a Galois cohomology class. Let $S(\xi)\subset V_K$ denote the set of places $v\in V_K$ such that the localization of $\xi$ at $v$ is non-trivial: $${\rm loc}_v(\xi)\neq 1\in H^1(K_v,G).$$ It is well known (and easy to prove) that the set $S(\xi)$ is finite.

Question. What is a reference for this well-known fact?

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Lie algebra of a real/complex algebraic matrix group in terms of power series?

In real or complex algebraic matrix groups, one can define the Lie algebra using the ordinary matrix exponential as the set of matrices $M$ such that $\exp(tM) \in G$ for all sufficiently small $t \in \mathbb{R}$ (here, according to Hall’s book, we consider real 1-parameter-subgroups even if $G$ is a complex group). I am wondering about an algebraic version of this result using power series.

Q: Is the following a correct (equivalent) definition of the Lie algebra for…what class of groups? (For example, semisimple? Linearly reductive? Matrix?) Is there a good reference for this, or whatever the correct result along these lines is, if there is one?

Def (attempt): If $G \subseteq GL_n(\mathbb{F})$ is an $\mathbb{F}$-algebraic matrix group ($\mathbb{F} \in \{\mathbb{R}, \mathbb{C}\}$), then an $n \times n$ matrix $M \in Mat_n(\mathbb{F})$ is in its (matrix) Lie algebra iff the power series $\exp(\varepsilon M)$, viewed as a matrix over the field of Laurent series $\mathbb{F}(\!(\varepsilon)\!)$, is an element of $G(\mathbb{F}(\!(\varepsilon)\!))$.

(Or maybe we always consider it over $\mathbb{R}(\!(\varepsilon)\!)$ even in the complex case, by analogy with using real 1-parameter subgroups even when $G$ is a complex group?)

Some additional context/background:

If $G$ is a real or complex algebraic group, which is thus also a Lie group, then if I'm remembering & understanding correctly, one can define the Lie algebra in this setting either algebraically or analytically, as:

1. The algebraic tangent space at the identity. I believe this is equivalently the set of $v$ such that $I + \varepsilon v \in G(\mathbb{F}[\varepsilon]/\varepsilon^2)$, the vector space $(\mathfrak{m} / \mathfrak{m}^2)^*$ where $\mathfrak{m}$ is the ideal of regular functions vanishing at the identity, or the set of derivations on $\mathbb{F}[G]$ that are invariant under all left translations by $G$.

2. The analytic tangent space at the identity. I believe this is equivalently the set of vectors $v$ such that $v = \frac{d}{dt} g(t)|_{t=0}$ where $g(t)$ is a real-1-parameter subgroup, or where $g(t)$ is just a (differentiable, I guess) path passing through the identity at $t=0$, or – in the case of matrix groups – the set of matrices $M$ such that $\exp(tM) \in G$ for all sufficiently small $t \in \mathbb{R}$.

In the cases of GL, SL, SU, SO, Sp (admittedly, all reductive), in their defining representations I checked and the above attempted definition seems to work just fine, but I don’t see why it should hold in general. Holding in general is like saying “If I have a point up to first order ($\exp(\varepsilon M)$ is in $G(\mathbb{F}[\varepsilon]/\varepsilon^2)$), then we get a point up to infinite order ($\exp(\varepsilon M)$ is in $G(\mathbb{F}(\!(\varepsilon)\!))$)", and the latter sounds a little too strong to me to hold in general (say, by analogy with stuff in deformation theory, where you can have first-order deformations that don’t extend to deformations over the power series ring). But even if it (or something like it) does hold, it'd be great to have a reference for it, which I've been unable to find.

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Concentration of very dependent Markov chains

Consider the following simple Markov chain $ X_1\to X_2\to\cdots\to X_n $ where each $X_i$ is $\{-1,1\}$-valued and $X_1\sim\mathrm{Unif}(\{-1,1\})$ (such that the chain is stationary). The flip probability is $\delta$, i.e., $\Pr[X_{i+1} = -x\,|\,X_i = x] = \delta$ for $x\in\{-1,1\}$. I'm interested in exponentially tight bounds on the tail of $ |\frac{1}{n}\sum_{i=1}^nX_i| $.

If $ \delta $ is very close to $1/2$, then $X_i$'s are weakly dependent and we would expect that $|\frac{1}{n}\sum_{i=1}^nX_i|$ is very close to $0$. Therefore, the rare event is $|\frac{1}{n}\sum_{i=1}^nX_i|\ge\epsilon$. There are many bounds in the literature on the tail $\Pr[|\frac{1}{n}\sum_{i=1}^nX_i|\ge\epsilon]$.

If $\delta$ is very close to $0$, then $X_i$'s are strongly dependent and we would expect that $|\frac{1}{n}\sum_{i=1}^nX_i|$ is very close to $1$. Therefore, the rare event is $|\frac{1}{n}\sum_{i=1}^nX_i|\le1-\epsilon$. I'm interested in obtaining exponentially tight upper bounds on the tail $\Pr[|\frac{1}{n}\sum_{i=1}^nX_i|\le1-\epsilon]$. (Note that this looks like anti-concentration on the face of it, but I'm not sure if it's appropriate to call it so, since we're not lower bounding the probability of rare events.)

I looked into the literature and it seems that most existing bounds (e.g., Theorem 1.2 here) behave well in the weakly dependent regime but behave poorly in the strongly dependent regime. In particular, for the specific parameter regime in my original context, I couldn't even find a bound that beats a trivial union bound: $$\Pr\left[\left|\frac{1}{n}\sum_{i=1}^nX_i\right|\le1-\epsilon\right]\le\Pr\left[\left|\frac{1}{n}\sum_{i=1}^nX_i\right|<1\right] = \Pr\left[\exists i,\,X_i\ne X_{i+1}\right]\le n\cdot\delta.$$ Here I think of $n$ as something like $\frac{1}{\delta}(\log\frac{1}{\delta})^{-3}$.

My questions are:

1. What is the actual scaling of the above tail as a function of $n$ and $\delta$?
2. More generally, how much is known about large deviation of strongly dependent Markov chains?

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On the integral points of simple algebraic group over non-archimedean local field of positive characteristic

Let $K$ be a non-archimedean local field of characteristic $p>0$ and $\mathcal{O}_{K}$ its valuation ring with maximal ideal $\mathfrak{m}$. Let $\mathbf{G}$ be a connected, simply connected $K$-simple subgroup of ${\rm GL}_{r}$. Let $\Delta:=\mathbf{G}(\mathcal{O}_{K})=\mathbf{G}\cap {\rm GL}_{r}(\mathcal{O}_{K})$. Then it's claimed on page 111 of the book Subgroup Growth that

$\Delta$ contains an open $\mathcal{O}_{K}$-perfect pro-$p$ group $G$ in the sense of Section 4.4, i.e. $G$ is a $\mathcal{O}_{K}$-standard group such that $[G,G]=G_{2}$. (We refer to Section 13.5 of the book Analytic Pro-P Groups for more details.)

There is no proof there, and it just cites Exercise 13.11 in Analytic Pro-P Groups, which states that

If $\mathcal{G}$ is any simple Chevalley group scheme, then the first congruence subgroup $\mathcal{G}^{1}(\mathcal{O}_{K}):=\ker(\mathcal{G}(\mathcal{O}_{K})\to > \mathcal{G}(\mathcal{O}_{K}/\mathfrak{m}))$ is an $\mathcal{O}_{K}$-perfect pro-$p$ group.

I am not familiar with the theory of algebraic groups. Could someone please explain whether this is correct and give a more specific and detailed proof?

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Number of increasing (monotone) surjections [closed]

Let $f \colon A \to B$ be an increasing (or monotone) surjection from a finite completely ordered set $A$ to a finite completely ordered set $B$ of, respectively, $|A|=m$ and $|B|=n$ elements. By increasing $f$ I mean $f(a') \geq f(a)$ for all $a'>a$. Question: how many increasing surjections are there? It is well known (relatively easy to understand and derive) that the number of unrestricted surjections is given in terms of the Stirling number of the second kind by $S_{mn} = {m \brace n}n!$, but I can't think of a simple argument that could give the number $S_{mn}^{+}$ of increasing surjections. Does it have to do with ordered partitions of $A$, in which case maybe $S_{mn}^{+} = S_{mn}\,/n!$?

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Eigenvector quadratic form inequality

Given matrix $B\in\mathbb{R}_+^{n\times n}$ and scalar $\alpha \in \mathbb{R}_{+}$, let $A:=\alpha B+B^T/\alpha$. Note that $B$ and $A$ have nonnegative entries and that $\alpha$ controls degree of asymmetry of $A$. Let $v \in \mathbb{R}^n_+, \Vert v\Vert =1$ be the Perron eigenvector of $A$ that achieves largest (real-valued) eigenvalue. Numerical explorations suggest the following simple inequality: $$\vert u^* B u \vert \le v^T B v$$ whenever $u\in\mathbb{C}^n, \Vert u \Vert =1$ is an eigenvector of $A$. This is easy to prove for $\alpha=1$ (symmetric $A$) using Courant-Fischer. Any ideas for how to prove or disprove this for other $\alpha$?

[Note: This builds off another one of my MO question's. However, here it is formulated in a more general and self-contained setting.]

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Some simplicial invariants associated to a finite group

I participated in a talk about abstract combinatorial simplicial complex. During the talk the following question came to my mind.

Let $G$ be a finite group. We construct a simplicial complex $\Delta$ whose set of vertices is $G$. A face $F \in \Delta$ is a subset $F \subset G$ whose elements mutually commute. I am curious about the topological realization $X_G=|\Delta|$ of $\Delta$. The invariant I associate to a group $G$ is the Euler characteristic $\chi(X_G)$ of $X_G$. Is this invariant introduced already? What is the group theoretical interpretation of homological dimensions(Betti numbers) of this simplicial complex $X_G$?Are there some interpretation of these quantities in combinatorial or geometric group theory?

I think that the following can be found in the book Hatcher Algebraic topology Recall that an abstract simplicial complex $\Delta$ is a finite set $V=\{v_1,v_2,\ldots,v_n\}$ as vertices. This structure is equipped with a collection $\mathfrak{F}$ of subsets of $V$, called collection of faces, which is closed under inclusion. Assume that $\Delta $ is an abstract simplicial complex with vertices $v_i, i=1,\ldots, n$, the topological realization $X_{\Delta}$ of this complex is the simplicial complex generated by the standard base $e_i, i=1 \ldots n$ of $\mathbb{R}^n$ whose face is determined by $\Delta$

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Matrix formula for point of intersection of three circles through six points

Suppose we have the six points in the cartesian plane $(x_i, y_i)$ for $1 \leq i \leq 6$. Further suppose that we draw three circles through them, so that each circle passes through three of the points - say circles 123, 345, and 561.
Is there a matrix determinant formula to determine whether these three circles are concurrent? The long way of solving this problem (explicit formulas for the circles and their intersection points) is not what I need - it's too untenable.

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Is the solvable closure of $\mathbb{F}_p(t)$ PAC?

It is a famous open problem in field arithmetic whether $\mathbb{Q}^{\mathrm{solv}}$, the solvable closure of $\mathbb{Q}$, is pseudo algebraically closed (PAC). That is, whether every absolutely irreducible $\mathbb{Q}$-variety admits a rational point over some solvable extension of $\mathbb{Q}$.

My question is whether anything is known about the analogous situation over function fields. In particular, is it known whether the solvable closure of $\mathbb{F}_p(t)$ is PAC?

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