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1. -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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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. On the computability of rotation sets and their entropies

   https://doi.org/10.1017/etds.2018.45  

   BURR, MICHAEL A.; SCHMOLL, MARTIN; WOLF, CHRISTIAN (February 2020, Ergodic Theory and Dynamical Systems)

   null (Ed.)

   Let $$f:X\rightarrow X$$ be a continuous dynamical system on a compact metric space $$X$$ and let $$\unicode[STIX]{x1D6F7}:X\rightarrow \mathbb{R}^{m}$$ be an $$m$$ -dimensional continuous potential. The (generalized) rotation set $$\text{Rot}(\unicode[STIX]{x1D6F7})$$ is defined as the set of all $$\unicode[STIX]{x1D707}$$ -integrals of $$\unicode[STIX]{x1D6F7}$$ , where $$\unicode[STIX]{x1D707}$$ runs over all invariant probability measures. Analogous to the classical topological entropy, one can associate the localized entropy $$\unicode[STIX]{x210B}(w)$$ to each $$w\in \text{Rot}(\unicode[STIX]{x1D6F7})$$ . In this paper, we study the computability of rotation sets and localized entropy functions by deriving conditions that imply their computability. Then we apply our results to study the case where $$f$$ is a subshift of finite type. We prove that $$\text{Rot}(\unicode[STIX]{x1D6F7})$$ is computable and that $$\unicode[STIX]{x210B}(w)$$ is computable in the interior of the rotation set. Finally, we construct an explicit example that shows that, in general, $$\unicode[STIX]{x210B}$$ is not continuous on the boundary of the rotation set when considered as a function of $$\unicode[STIX]{x1D6F7}$$ and $$w$$ . In particular, $$\unicode[STIX]{x210B}$$ is, in general, not computable at the boundary of $$\text{Rot}(\unicode[STIX]{x1D6F7})$$ . 

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4. LINEAR AND QUADRATIC UNIFORMITY OF THE MÖBIUS FUNCTION OVER

   https://doi.org/10.1112/S0025579319000032  

   Bienvenu, Pierre-Yves; Lê, Thái Hoàng (January 2019, Mathematika)

   We examine correlations of the Möbius function over $$\mathbb{F}_{q}[t]$$ with linear or quadratic phases, that is, averages of the form 1 $$\begin{eqnarray}\frac{1}{q^{n}}\mathop{\sum }_{\deg f0$$ if $$Q$$ is linear and $$O(q^{-n^{c}})$$ for some absolute constant $c>0$ if $$Q$$ is quadratic. The latter bound may be reduced to $$O(q^{-c^{\prime }n})$$ for some $$c^{\prime }>0$$ when $Q(f)$ is a linear form in the coefficients of $$f^{2}$$ , that is, a Hankel quadratic form, whereas, for general quadratic forms, it relies on a bilinear version of the additive-combinatorial Bogolyubov theorem. 

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5. COVER TIME FOR THE FROG MODEL ON TREES

   https://doi.org/10.1017/fms.2019.37  

   HOFFMAN, CHRISTOPHER; JOHNSON, TOBIAS; JUNGE, MATTHEW (January 2019, Forum of Mathematics, Sigma)

   The frog model is a branching random walk on a graph in which particles branch only at unvisited sites. Consider an initial particle density of $$\unicode[STIX]{x1D707}$$ on the full $$d$$ -ary tree of height $$n$$ . If $$\unicode[STIX]{x1D707}=\unicode[STIX]{x1D6FA}(d^{2})$$ , all of the vertices are visited in time $$\unicode[STIX]{x1D6E9}(n\log n)$$ with high probability. Conversely, if $$\unicode[STIX]{x1D707}=O(d)$$ the cover time is $$\exp (\unicode[STIX]{x1D6E9}(\sqrt{n}))$$ with high probability. 

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