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1. Fock–Goncharov dual cluster varieties and Gross–Siebert mirrors

   https://doi.org/10.1515/crelle-2023-0043  

   Argüz, Hülya; Bousseau, Pierrick (July 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Cluster varieties come in pairs: for any $\mathcal{𝒳}${\mathcal{X}}cluster variety there is an associated Fock–Goncharov dual $\mathcal{𝒜}${\mathcal{A}}cluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi–Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross–Siebert program. In this paper we bridge the theory of cluster varieties with the algebro-geometric framework of Gross–Siebert mirror symmetry. Particularly, we show that the mirror to the $\mathcal{𝒳}${\mathcal{X}}cluster variety is a degeneration of the Fock–Goncharov dual $\mathcal{𝒜}${\mathcal{A}}cluster varietyand vice versa. To do this, we investigate how the cluster scattering diagram of Gross, Hacking, Keel and Kontsevich compares with the canonical scattering diagram defined by Gross and Siebert to construct mirror duals in arbitrary dimensions. Consequently, we derive an enumerative interpretation of the cluster scattering diagram. Along the way, we prove the Frobenius structure conjecture for a class of log Calabi–Yau varieties obtained as blow-ups of toric varieties. 

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2. Nitrogen isotopic constraints on nutrient transport to the upper ocean

   https://doi.org/10.1038/s41561-021-00836-8  

   Fripiat, François; Martínez-García, Alfredo; Marconi, Dario; Fawcett, Sarah E.; Kopf, Sebastian H.; Luu, Victoria H.; Rafter, Patrick A.; Zhang, Run; Sigman, Daniel M.; Haug, Gerald H. (November 2021, Nature Geoscience)

   Abstract Ocean circulation supplies the surface ocean with the nutrients that fuel global ocean productivity. However, the mechanisms and rates of water and nutrient transport from the deep ocean to the upper ocean are poorly known. Here, we use the nitrogen isotopic composition of nitrate to place observational constraints on nutrient transport from the Southern Ocean surface into the global pycnocline (roughly the upper 1.2 km), as opposed to directly from the deep ocean. We estimate that 62 ± 5% of the pycnocline nitrate and phosphate originate from the Southern Ocean. Mixing, as opposed to advection, accounts for most of the gross nutrient input to the pycnocline. However, in net, mixing carries nutrients away from the pycnocline. Despite the quantitative dominance of mixing in the gross nutrient transport, the nutrient richness of the pycnocline relies on the large-scale advective flow, through which nutrient-rich water is converted to nutrient-poor surface water that eventually flows to the North Atlantic. 

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3. A Tropical View on Landau–Ginzburg Models

   https://doi.org/10.1007/s10114-024-2295-y  

   Carl, Michael; Pumperla, Max; Siebert, Bernd (January 2024, Acta Mathematica Sinica, English Series)

   This paper, largely written in 2009/2010, fits Landau–Ginzburg models into the mirror symmetry program pursued by the last author jointly with Mark Gross since 2001. This point of view transparently brings in tropical disks of Maslov index 2 via the notion of broken lines, previously introduced in two dimensions by Mark Gross in his study of mirror symmetry for P2 . A major insight is the equivalence of properness of the Landau–Ginzburg potential with smoothness of the anticanonical divisor on the mirror side. We obtain proper superpotentials which agree on an open part with those classically known for toric varieties. Examples include mirror LG models for non-singular and singular del Pezzo surfaces, Hirzebruch surfaces and some Fano threefolds. 

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4. Assessing the Financial Sustainability of a Virtual Clinic Providing Comprehensive Diabetes Care

   https://doi.org/10.1177/19322968251340664  

   Dupenloup, Paul; Guan, Grace; Aleppo, Grazia; Bergenstal, Richard M; Hood, Korey; Kruger, Davida; McArthur, Teresa; Olson, Beth; Oser, Sean; Oser, Tamara; et al (May 2025, Journal of Diabetes Science and Technology)

   Introduction:The Virtual Diabetes Specialty Clinic (VDiSC) study demonstrated the feasibility of providing comprehensive diabetes care entirely virtually by combining virtual visits with continuous glucose monitoring support and remote patient monitoring (RPM). However, the financial sustainability of this model remains uncertain. Methods:We developed a financial model to estimate the variable costs and revenues of virtual diabetes care, using visit data from the 234 VDiSC participants with type 1 or type 2 diabetes. Data included virtual visits with certified diabetes care and education specialists (CDCES), endocrinologists, and behavioral health services (BHS). The model estimated care utilization, variable costs, reimbursement revenue, gross profit, and gross profit margin per member, per month (PMPM) for privately insured, publicly insured, and overall clinic populations (75% privately insured). We performed two-way sensitivity analyses on key parameters. Results:Gross profit and gross profit margin PMPM (95% confidence interval) were estimated at $−4 ($−14.00 to $5.68) and −4% (−3% to −6%) for publicly insured patients; $267.26 ($256.59-$277.93) and 73% (58%-88%) for privately insured patients; and $199.41 ($58.43-$340.39) and 67% (32%-102%) for the overall clinic. Profits were primarily driven by CDCES visits and RPM. Results were sensitive to insurance mix, cost-to-charge ratio, and commercial-to-Medicare price ratio. Conclusions:Virtual diabetes care can be financially viable, although profitability relies on privately insured patients. The analysis excluded fixed costs of clinic infrastructure, and securing reimbursement may be challenging in practice. The financial model is adaptable to various care settings and can serve as a planning tool for virtual diabetes clinics. 

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5. Nonlinear speed-ups in ultracold quantum gases

   https://doi.org/10.1209/0295-5075/ac9fed  

   Deffner, Sebastian (November 2022, Europhysics Letters)

   Abstract Quantum mechanics is an inherently linear theory. However, collective effects in many body quantum systems can give rise to effectively nonlinear dynamics. In the present work, we analyze whether and to what extent such nonlinear effects can be exploited to enhance the rate of quantum evolution. To this end, we compute a suitable version of the quantum speed limit for numerical and analytical examples. We find that the quantum speed limit grows with the strength of the nonlinearity, yet it does not trivially scale with the “degree” of nonlinearity. This is numerically demonstrated for the parametric harmonic oscillator obeying Gross-Pitaevskii and Kolomeisky dynamics, and analytically for expanding boxes under Gross-Pitaevskii dynamics. 

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