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1. Relative Dynamics and Stability of Point Vortices on the Sphere

   Tomoki Ohsawa (December 2022, IEICE proceeding series)

   We exploit the SO(3)-symmetry of the Hamiltonian dynamics of N point vortices on the sphere to derive a Hamiltonian system for the relative dynamics of the vortices. The resulting system combined with the energy--Casimir method helps us prove the stability of the tetrahedron relative equilibria when all of their circulations have the same sign---a generalization of some existing results on tetrahedron relative equilibria of identical vortices. 

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2. A new canonical reduction of three-vortex motion and its application to vortex-dipole scattering

   https://doi.org/10.1063/5.0208538  

   Anurag, Atul; Goodman, Roy H; O'Grady, Ellison K (June 2024, Physics of Fluids)

   We introduce a new reduction of the motion of three point vortices in a two-dimensional ideal fluid. This proceeds in two stages: a change of variables to Jacobi coordinates and then a Nambu reduction. The new coordinates demonstrate that the dynamics evolve on a two-dimensional manifold whose topology depends on the sign of a parameter κ2 that arises in the reduction. For κ2>0, the phase space is spherical, while for κ2<0, the dynamics are confined to the upper sheet of a two-sheeted hyperboloid. We contrast this reduction with earlier reduced systems derived by Gröbli, Aref, and others in which the dynamics are determined from the pairwise distances between the vortices. The new coordinate system overcomes two related shortcomings of Gröbli's reduction that have made understanding the dynamics difficult: their lack of a standard phase plane and their singularity at all configurations in which the vortices are collinear. We apply this to two canonical problems. We first discuss the dynamics of three identical vortices and then consider the scattering of a propagating dipole by a stationary vortex. We show that the points dividing direct and exchange scattering solutions correspond to the locations of the invariant manifolds of equilibria of the reduced equations and relate changes in the scattering diagram as the circulation of one vortex is varied to bifurcations of these equilibria. 

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3. A Plethora of Cluster Structures on 𝐺𝐿_{𝑛}

   https://doi.org/10.1090/memo/1486  

   Gekhtman, M; Shapiro, M; Vainshtein, A (May 2024, Memoirs of the American Mathematical Society)

   We continue the study of multiple cluster structures in the rings of regular functions on $G{L}_n$, $S{L}_n$and $Ma{t}_n$that are compatible with Poisson–Lie and Poisson-homogeneous structures. According to our initial conjecture, each class in the Belavin–Drinfeld classification of Poisson–Lie structures on a semisimple complex group $\mathcal{G}$corresponds to a cluster structure in $\mathcal{O}\left(\mathcal{G}\right)$. Here we prove this conjecture for a large subset of Belavin–Drinfeld (BD) data of $A_n$type, which includes all the previously known examples. Namely, we subdivide all possible $A_n$type BD data into oriented and non-oriented kinds. We further single out BD data satisfying a certain combinatorial condition that we call aperiodicity and prove that for any oriented BD data of this kind there exists a regular cluster structure compatible with the corresponding Poisson–Lie bracket. In fact, we extend the aperiodicity condition to pairs of oriented BD data and prove a more general result that establishes an existence of a regular cluster structure on $S{L}_n$compatible with a Poisson bracket homogeneous with respect to the right and left action of two copies of $S{L}_n$equipped with two different Poisson-Lie brackets. Similar results hold for aperiodic non-oriented BD data, but the analysis of the corresponding regular cluster structure is more involved and not given here. If the aperiodicity condition is not satisfied, a compatible cluster structure has to be replaced with a generalized cluster structure. We will address these situations in future publications. 

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4. On Mixing of Markov Chains: Coupling, Spectral Independence, and Entropy Factorization.

   Antonio Blanca, Pietro Caputo (January 2022, SODA 2022)

   For general spin systems, we prove that a contractive coupling for an arbitrary local Markov chain implies optimal bounds on the mixing time and the modified log-Sobolev constant for a large class of Markov chains including the Glauber dynamics, arbitrary heat-bath block dynamics, and the Swendsen-Wang dynamics. This reveals a novel connection between probabilistic techniques for bounding the convergence to stationarity and analytic tools for analyzing the decay of relative entropy. As a corollary of our general results, we obtain O(n log n) mixing time and Ω(1/n) modified log-Sobolev constant of the Glauber dynamics for sampling random q-colorings of an n-vertex graph with constant maximum degree Δ when q > (11/6–∊0)Δ for some fixed ∊0 > 0. We also obtain O(log n) mixing time and Ω(1) modified log-Sobolev constant of the Swendsen-Wang dynamics for the ferromagnetic Ising model on an n-vertex graph of constant maximum degree when the parameters of the system lie in the tree uniqueness region. At the heart of our results are new techniques for establishing spectral independence of the spin system and block factorization of the relative entropy. On one hand we prove that a contractive coupling of any local Markov chain implies spectral independence of the Gibbs distribution. On the other hand we show that spectral independence implies factorization of entropy for arbitrary blocks, establishing optimal bounds on the modified log-Sobolev constant of the corresponding block dynamics. 

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5. Crystal structure of a tris(2-aminoethyl)methane capped carbamoylmethylphosphine oxide compound

   https://doi.org/10.1107/S2056989024008478  

   Wackerle, Brandon G; Werner, Eric J; Staples, Richard J; Biros, Shannon M (September 2024, Acta Crystallographica Section E Crystallographic Communications)

   The molecular structure of the tripodal carbamoylmethylphosphine oxide compound diethyl {[(5-[2-(diethoxyphosphoryl)acetamido]-3-{2-[2-(diethoxyphosphoryl)acetamido]ethyl}pentyl)carbamoyl]methyl}phosphonate, C₂₅H₅₂N₃O₁₂P₃, features six intramolecular hydrogen-bonding interactions. The phosphonate groups have key bond lengths ranging from 1.4696 (12) to 1.4729 (12) Å (P=O), 1.5681 (11) to 1.5811 (12) Å (P—O) and 1.7881 (16) to 1.7936 (16) Å (P—C). Each amide group adopts a nearly perfecttransgeometry, and the geometry around each phophorus atom resembles a slightly distorted tetrahedron. 

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