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1. THE DYNAMICAL MORDELL–LANG CONJECTURE FOR ENDOMORPHISMS OF SEMIABELIAN VARIETIES DEFINED OVER FIELDS OF POSITIVE CHARACTERISTIC

   https://doi.org/10.1017/S1474748019000318  

   Corvaja, Pietro; Ghioca, Dragos; Scanlon, Thomas; Zannier, Umberto (March 2021, Journal of the Institute of Mathematics of Jussieu)

   null (Ed.)

   Abstract Let $$K$$ be an algebraically closed field of prime characteristic $$p$$ , let $$X$$ be a semiabelian variety defined over a finite subfield of $$K$$ , let $$\unicode[STIX]{x1D6F7}:X\longrightarrow X$$ be a regular self-map defined over $$K$$ , let $$V\subset X$$ be a subvariety defined over $$K$$ , and let $$\unicode[STIX]{x1D6FC}\in X(K)$$ . The dynamical Mordell–Lang conjecture in characteristic $$p$$ predicts that the set $$S=\{n\in \mathbb{N}:\unicode[STIX]{x1D6F7}^{n}(\unicode[STIX]{x1D6FC})\in V\}$$ is a union of finitely many arithmetic progressions, along with finitely many $$p$$ -sets, which are sets of the form $$\{\sum _{i=1}^{m}c_{i}p^{k_{i}n_{i}}:n_{i}\in \mathbb{N}\}$$ for some $$m\in \mathbb{N}$$ , some rational numbers $$c_{i}$$ and some non-negative integers $$k_{i}$$ . We prove that this conjecture is equivalent with some difficult diophantine problem in characteristic 0. In the case $$X$$ is an algebraic torus, we can prove the conjecture in two cases: either when $$\dim (V)\leqslant 2$$ , or when no iterate of $$\unicode[STIX]{x1D6F7}$$ is a group endomorphism which induces the action of a power of the Frobenius on a positive dimensional algebraic subgroup of $$X$$ . We end by proving that Vojta’s conjecture implies the dynamical Mordell–Lang conjecture for tori with no restriction. 

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2. CUTTING A PART FROM MANY MEASURES

   https://doi.org/10.1017/fms.2019.33  

   BLAGOJEVIĆ, PAVLE V.; PALIĆ, NEVENA; SOBERÓN, PABLO; ZIEGLER, GÜNTER M. (January 2019, Forum of Mathematics, Sigma)

   Holmsen, Kynčl and Valculescu recently conjectured that if a finite set $$X$$ with $$\ell n$$ points in $$\mathbb{R}^{d}$$ that is colored by $$m$$ different colors can be partitioned into $$n$$ subsets of $$\ell$$ points each, such that each subset contains points of at least $$d$$ different colors, then there exists such a partition of $$X$$ with the additional property that the convex hulls of the $$n$$ subsets are pairwise disjoint. We prove a continuous analogue of this conjecture, generalized so that each subset contains points of at least $$c$$ different colors, where we also allow $$c$$ to be greater than $$d$$ . Furthermore, we give lower bounds on the fraction of the points each of the subsets contains from $$c$$ different colors. For example, when $$n\geqslant 2$$ , $$d\geqslant 2$$ , $$c\geqslant d$$ with $$m\geqslant n(c-d)+d$$ are integers, and $$\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{m}$$ are $$m$$ positive finite absolutely continuous measures on $$\mathbb{R}^{d}$$ , we prove that there exists a partition of $$\mathbb{R}^{d}$$ into $$n$$ convex pieces which equiparts the measures $$\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{d-1}$$ , and in addition every piece of the partition has positive measure with respect to at least $$c$$ of the measures $$\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{m}$$ . 

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3. Constitutive relations for compressible granular flow in the inertial regime

   https://doi.org/10.1017/jfm.2019.476  

   Schaeffer, D. G.; Barker, T.; Tsuji, D.; Gremaud, P.; Shearer, M.; Gray, J. M. (September 2019, Journal of Fluid Mechanics)

   Granular flows occur in a wide range of situations of practical interest to industry, in our natural environment and in our everyday lives. This paper focuses on granular flow in the so-called inertial regime, when the rheology is independent of the very large particle stiffness. Such flows have been modelled with the $$\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$$ -rheology, which postulates that the bulk friction coefficient $$\unicode[STIX]{x1D707}$$ (i.e. the ratio of the shear stress to the pressure) and the solids volume fraction $$\unicode[STIX]{x1D719}$$ are functions of the inertial number $$I$$ only. Although the $$\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$$ -rheology has been validated in steady state against both experiments and discrete particle simulations in several different geometries, it has recently been shown that this theory is mathematically ill-posed in time-dependent problems. As a direct result, computations using this rheology may blow up exponentially, with a growth rate that tends to infinity as the discretization length tends to zero, as explicitly demonstrated in this paper for the first time. Such catastrophic instability due to ill-posedness is a common issue when developing new mathematical models and implies that either some important physics is missing or the model has not been properly formulated. In this paper an alternative to the $$\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$$ -rheology that does not suffer from such defects is proposed. In the framework of compressible $$I$$ -dependent rheology (CIDR), new constitutive laws for the inertial regime are introduced; these match the well-established $$\unicode[STIX]{x1D707}(I)$$ and $$\unicode[STIX]{x1D6F7}(I)$$ relations in the steady-state limit and at the same time are well-posed for all deformations and all packing densities. Time-dependent numerical solutions of the resultant equations are performed to demonstrate that the new inertial CIDR model leads to numerical convergence towards physically realistic solutions that are supported by discrete element method simulations. 

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4. The Furstenberg–Poisson boundary and CAT(0) cube complexes

   https://doi.org/10.1017/etds.2016.124  

   FERNÓS, TALIA (September 2018, Ergodic Theory and Dynamical Systems)

   We show under weak hypotheses that$$\unicode[STIX]{x2202}X$$, the Roller boundary of a finite-dimensional CAT(0) cube complex$$X$$is the Furstenberg–Poisson boundary of a sufficiently nice random walk on an acting group$$\unicode[STIX]{x1D6E4}$$. In particular, we show that if$$\unicode[STIX]{x1D6E4}$$admits a non-elementary proper action on$$X$$, and$$\unicode[STIX]{x1D707}$$is a generating probability measure of finite entropy and finite first logarithmic moment, then there is a$$\unicode[STIX]{x1D707}$$-stationary measure on$$\unicode[STIX]{x2202}X$$making it the Furstenberg–Poisson boundary for the$$\unicode[STIX]{x1D707}$$-random walk on$$\unicode[STIX]{x1D6E4}$$. We also show that the support is contained in the closure of the regular points. Regular points exhibit strong contracting properties. 

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5. -DISPLAYS AND RAPOPORT–ZINK SPACES

   https://doi.org/10.1017/S1474748018000373  

   Bültel, O.; Pappas, G. (July 2020, Journal of the Institute of Mathematics of Jussieu)

   null (Ed.)

   Let $$(G,\unicode[STIX]{x1D707})$$ be a pair of a reductive group $$G$$ over the $$p$$ -adic integers and a minuscule cocharacter $$\unicode[STIX]{x1D707}$$ of $$G$$ defined over an unramified extension. We introduce and study ‘ $$(G,\unicode[STIX]{x1D707})$$ -displays’ which generalize Zink’s Witt vector displays. We use these to define certain Rapoport–Zink formal schemes purely group theoretically, i.e. without $$p$$ -divisible groups. 

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