In this paper, we study the Dirichlet problem for systems of mean value equations on a regular tree. We deal both with the directed case (the equations verified by the components of the system at a node in the tree only involve values of the unknowns at the successors of the node in the tree) and the undirected case (now the equations also involve the predecessor in the tree). We find necessary and sufficient conditions on the coefficients in order to have existence and uniqueness of solutions for continuous boundary data. In a particular case, we also include an interpretation of such solutions as a limit of value functions of suitable two-players zero-sum games.

The notion of indicator of an analytic function, that describes the function’s growth along rays, was introduced by Phragmen and Lindelöf. Trigonometric convexity is a defining property of the indicator. For multivariate cases, an analogous property of trigonometric convexity was not known so far. We prove the property of trigonometric convexity for the indicator of multivariate analytic functions, introduced by Ivanov. The results that we obtain are sharp. Derivation of a multidimensional analogue of the inverse Fourier transform in a sector and obtaining estimates on its decay is an important step of our proof.

This paper goes beyond Katz–Sarnak theory on the distribution of curves over finite fields according to their number of rational points, theoretically, experimentally, and conjecturally. In particular, we give a formula for the limits of the moments measuring the asymmetry of this distribution for (non-hyperelliptic) curves of genus $g\geq 3$. The experiments point to a stronger notion of convergence than the one provided by the Katz–Sarnak framework for all curves of genus $\geq 3$. However, for elliptic curves and for hyperelliptic curves of every genus, we prove that this stronger convergence cannot occur.

To understand the p-adic étale cohomology of a proper smooth variety over a p-adic field, Faltings compared it to the cohomology of his ringed topos, by the so-called Faltings’ main p-adic comparison theorem, and then deduced various comparisons with p-adic cohomologies originating from differential forms. In this article, we generalize the former to any proper and finitely presented morphism of coherent schemes over an absolute integral closure of $\mathbb {Z}_p$ (without any smoothness assumption) for torsion abelian étale sheaves (not necessarily finite locally constant). Our proof relies on our cohomological descent for Faltings’ ringed topos, using a variant of de Jong’s alteration theorem for morphisms of schemes due to Gabber–Illusie–Temkin to reduce to the relative case of proper log-smooth morphisms of log-smooth schemes over a complete discrete valuation ring proved by Abbes–Gros. A by-product of our cohomological descent is a new construction of Faltings’ comparison morphism, which does not use Achinger’s results on $K(\pi ,1)$-schemes.

We continue our study from Peterzil et al. (2022, Preprint, arXiv:2208.08293) of finite-dimensional definable groups in models of the theory $T_{\partial }$, the model companion of an o-minimal ${\mathcal {L}}$-theory T expanded by a generic derivation $\partial $ as in Fornasiero and Kaplan (2021, Journal of Mathematical Logic 21, 2150007).

We generalize Buium’s notion of an algebraic D-group to ${\mathcal {L}}$-definable D-groups, namely $(G,s)$, where G is an ${\mathcal {L}}$-definable group in a model of T, and $s:G\to \tau (G)$ is an ${\mathcal {L}}$-definable group section. Our main theorem says that every definable group of finite dimension in a model of $T_\partial $ is definably isomorphic to a group of the form

$$ \begin{align*}(G,s)^\partial=\{g\in G:s(g)=\nabla g\},\end{align*} $$

for some ${\mathcal {L}}$-definable D-group $(G,s)$ (where $\nabla (g)=(g,\partial g)$).

We obtain analogous results when T is either the theory of p-adically closed fields or the theory of pseudo-finite fields of characteristic $0$.

In this paper, we investigate the theory of heights in a family of stacky curves following recent work of Ellenberg, Satriano, and Zureick-Brown. We first give an elementary construction of a height which is seen to be dual to theirs. We count rational points having bounded ESZ-B height on a particular stacky curve, answering a question of Ellenberg, Satriano, and Zureick-Brown. We also show that when the Euler characteristic of stacky curves is non-positive, the ESZ-B height coming from the anti-canonical divisor class fails to have the Northcott property. We prove that a stacky version of a conjecture of Vojta is equivalent to the $abc$-conjecture.

We prove a weak version of the cross-product conjecture: $\textrm {F}(k+1,\ell ) \hskip .06cm \textrm {F}(k,\ell +1) \ge (\frac 12+\varepsilon ) \hskip .06cm \textrm {F}(k,\ell ) \hskip .06cm \textrm {F}(k+1,\ell +1)$, where $\textrm {F}(k,\ell )$ is the number of linear extensions for which the values at fixed elements $x,y,z$ are k and $\ell $ apart, respectively, and where $\varepsilon>0$ depends on the poset. We also prove the converse inequality and disprove the generalized cross-product conjecture. The proofs use geometric inequalities for mixed volumes and combinatorics of words.

Let A be a separable (not necessarily unital) simple $C^*$-algebra with strict comparison. We show that if A has tracial approximate oscillation zero, then A has stable rank one and the canonical map $\Gamma $ from the Cuntz semigroup of A to the corresponding lower-semicontinuous affine function space is surjective. The converse also holds. As a by-product, we find that a separable simple $C^*$-algebra which has almost stable rank one must have stable rank one, provided it has strict comparison and the canonical map $\Gamma $ is surjective.

We study the ring of regular functions on the space of planar electrical networks, which we coin the grove algebra. This algebra is an electrical analog of the Plücker ring studied classically in invariant theory. We develop the combinatorics of double groves to study the grove algebra, and find a quadratic Gröbner basis for the grove ideal.

We introduce a class of Falconer distance problems, which we call of restricted type, lying between the classical version and its pinned variant. Prototypical restricted distance sets are the diagonal distance sets, k-point configuration sets given by

$$ \begin{align*}\Delta^{\mathrm{diag}}(E)= \{ \,|(x,x,\dots,x)-(y_1,y_2,\dots,y_{k-1})| : x, y_1, \dots,y_{k-1} \in E\, \}\end{align*} $$

$E\subset \mathbb {R}^d$

$k\ge 3$

$\Delta ^{\mathrm{diag}}(E)$

(0.1)

$$ \begin{align} \dim(E)> \begin{cases} \frac{2d+1}3, & k=3, \\ \frac{(k-1)d}k,& k\ge 4. \end{cases} \end{align} $$

$C^\omega $

Given an affine Coxeter group W, the corresponding Shi arrangement is a refinement of the corresponding Coxeter hyperplane arrangements that was introduced by Shi to study Kazhdan–Lusztig cells for W. Shi showed that each region of the Shi arrangement contains exactly one element of minimal length in W. Low elements in W were introduced to study the word problem of the corresponding Artin–Tits (braid) group and turns out to produce automata to study the combinatorics of reduced words in W. In this article, we show, in the case of an affine Coxeter group, that the set of minimal length elements of the regions in the Shi arrangement is precisely the set of low elements, settling a conjecture of Dyer and the second author in this case. As a by-product of our proof, we show that the descent walls – the walls that separate a region from the fundamental alcove – of any region in the Shi arrangement are precisely the descent walls of the alcove of its corresponding low element.