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1. Uniform congruence counting for Schottky semigroups in SL2(𝐙)

   https://doi.org/10.1515/crelle-2016-0072  

   Magee, Michael; Oh, Hee; Winter, Dale (August 2019, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let Γ be a Schottky semigroup in {\mathrm{SL}_{2}(\mathbf{Z})} ,and for {q\in\mathbf{N}} , let {\Gamma(q):=\{\gamma\in\Gamma:\gamma=e~{}(\mathrm{mod}~{}q)\}} be its congruence subsemigroupof level q . Let δ denote the Hausdorff dimension of the limit set of Γ.We prove the following uniform congruence counting theoremwith respect to the family of Euclidean norm balls {B_{R}} in {M_{2}(\mathbf{R})} of radius R :for all positive integer q with no small prime factors, \#(\Gamma(q)\cap B_{R})=c_{\Gamma}\frac{R^{2\delta}}{\#(\mathrm{SL}_{2}(%\mathbf{Z}/q\mathbf{Z}))}+O(q^{C}R^{2\delta-\epsilon}) as {R\to\infty} for some {c_{\Gamma}>0,C>0,\epsilon>0} which are independent of q .Our technique also applies to give a similar counting result for the continued fractions semigroup of {\mathrm{SL}_{2}(\mathbf{Z})} ,which arises in the study of Zaremba’s conjecture on continued fractions. 

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2. Equitable colourings of Borel graphs

   https://doi.org/10.1017/fmp.2021.12  

   Bernshteyn, Anton; Conley, Clinton T. (January 2021, Forum of Mathematics, Pi)

   Abstract Hajnal and Szemerédi proved that if G is a finite graph with maximum degree $$\Delta $$ , then for every integer $$k \geq \Delta +1$$ , G has a proper colouring with k colours in which every two colour classes differ in size at most by $$1$$ ; such colourings are called equitable. We obtain an analogue of this result for infinite graphs in the Borel setting. Specifically, we show that if G is an aperiodic Borel graph of finite maximum degree $$\Delta $$ , then for each $$k \geq \Delta + 1$$ , G has a Borel proper k -colouring in which every two colour classes are related by an element of the Borel full semigroup of G . In particular, such colourings are equitable with respect to every G -invariant probability measure. We also establish a measurable version of a result of Kostochka and Nakprasit on equitable $$\Delta $$ -colourings of graphs with small average degree. Namely, we prove that if $$\Delta \geq 3$$ , G does not contain a clique on $$\Delta + 1$$ vertices and $$\mu $$ is an atomless G -invariant probability measure such that the average degree of G with respect to $$\mu $$ is at most $$\Delta /5$$ , then G has a $$\mu $$ -equitable $$\Delta $$ -colouring. As steps toward the proof of this result, we establish measurable and list-colouring extensions of a strengthening of Brooks’ theorem due to Kostochka and Nakprasit. 

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3. Momentum dependent optical lattice induced by an artificial gauge potential

   https://doi.org/10.1103/PhysRevResearch.4.013124  

   Chen, Zekai; Yao, Hepeng; Haber, Elisha; Bigelow, Nicholas P. (February 2022, Physical Review Research)

   We propose an experimentally feasible method to generate a one-dimensional optical lattice potential in an ultracold Bose gas system that depends on the transverse momentum of the atoms. The optical lattice is induced by the artificial gauge potential generated by a periodically driven multilaser Raman process. We study the many-body Bose-Hubbard model in an effective 1D case and show that the superfluid–Mott-insulator transition can be controlled via tuning the transverse momentum of the atomic gas. Such a feature enables us to control the phase of the quantum gas in the longitudinal direction by changing its transverse motional state.We examine our prediction via a strong-coupling expansion to an effective 1D Bose-Hubbard model and a quantum Monte Carlo calculation and discuss possible applications. 

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4. A reaction network approach to the convergence to equilibrium of quantum Boltzmann equations for Bose gases

   https://doi.org/10.1051/cocv/2021079  

   Craciun, Gheorghe; Tran, Minh-Binh (January 2021, ESAIM: Control, Optimisation and Calculus of Variations)

   Buttazzo, G.; Casas, E.; de Teresa, L.; Glowinski, R.; Leugering, G.; Trélat, E.; Zhang, X. (Ed.)

   When the temperature of a trapped Bose gas is below the Bose-Einstein transition temperature and above absolute zero, the gas is composed of two distinct components: the Bose-Einstein condensate and the cloud of thermal excitations. The dynamics of the excitations can be described by quantum Boltzmann models. We establish a connection between quantum Boltzmann models and chemical reaction networks. We prove that the discrete differential equations for these quantum Boltzmann models converge to an equilibrium point. Moreover, this point is unique for all initial conditions that satisfy the same conservation laws. In the proof, we then employ a toric dynamical system approach, similar to the one used to prove the global attractor conjecture, to study the convergence to equilibrium of quantum kinetic equations. 

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5. Boosting engine performance with Bose–Einstein condensation

   https://doi.org/10.1088/1367-2630/ac47cc  

   Myers, Nathan M.; Peña, Francisco J.; Negrete, Oscar; Vargas, Patricio; De Chiara, Gabriele; Deffner, Sebastian (February 2022, New Journal of Physics)

   Abstract At low-temperatures a gas of bosons will undergo a phase transition into a quantum state of matter known as a Bose–Einstein condensate (BEC), in which a large fraction of the particles will occupy the ground state simultaneously. Here we explore the performance of an endoreversible Otto cycle operating with a harmonically confined Bose gas as the working medium. We analyze the engine operation in three regimes, with the working medium in the BEC phase, in the gas phase, and driven across the BEC transition during each cycle. We find that the unique properties of the BEC phase allow for enhanced engine performance, including increased power output and higher efficiency at maximum power. 

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