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We show that a large class of maximally degenerating families of We anticipate that wall structures can be constructed quite generally from maximal degenerations. The construction given here then provides the homogeneous coordinate ring of the mirror degeneration along with a canonical basis. The appearance of a canonical basis of sections for certain degenerations points towards a good compactification of moduli of certain polarized varieties via stable pairs, generalizing the picture for K3 surfaces [Gross, Hacking, Keel, and Siebert,

In an earlier work (Publ. Inst. Hautes Études Sci., 122 (2015), 65–168) the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi–Yau varieties (those with maximal boundary) has a canonical vector space basis parameterized by the integral tropical points of the mirror. Further, the structure constants for the multiplication rule in this basis should be given by counting broken lines (certain combinatorial objects, morally the tropicalizations of holomorphic discs). Here we prove the conjecture in the case of cluster varieties, where the statement is a more precise form of the Fock–Goncharov dual basis conjecture (Publ. Inst. Hautes Études Sci., 103 (2006), 1–211). In particular, under suitable hypotheses, for each Y the partial compactification of an affine cluster variety U given by allowing some frozen variables to vanish, we obtain canonical bases for H^0(Y,\\mathcal{O}_Y) extending to a basis of H^0(U,\\mathcal{O}_U). Each choice of seed canonically identifies the parameterizing sets of these bases with integral points in a polyhedral cone. These results specialize to basis results of combinatorial representation theory. For example, by considering the open double Bruhat cell U in the basic affine space Y, we obtain a canonical basis of each irreducible representation of \\operatorname {SL}_r, parameterized by a set which each choice of seed identifies with the integral points of a lattice polytope. These bases and polytopes are all constructed essentially without representation-theoretic considerations. Along the way, our methods prove a number of conjectures in cluster theory, including positivity of the Laurent phenomenon for cluster algebras of geometric type.

The goal of this paper is to give a general theory of logarithmic Gromov-Witten invariants. This gives a vast generalization of the theory of relative Gromov-Witten invariants introduced by Li-Ruan, Ionel-Parker, and Jun Li and completes a program first proposed by the second named author in 2002. One considers target spaces XX carrying a log structure. Domains of stable log curves are log smooth curves. Algebraicity of the stack of such stable log maps is proven, subject only to the hypothesis that the log structure on XX is fine, saturated, and Zariski. A notion of basic stable log map is introduced; all stable log maps are pull-backs of basic stable log maps via base-change. With certain additional hypotheses, the stack of basic stable log maps is proven to be proper. Under these hypotheses and the additional hypothesis that XX is log smooth, one obtains a theory of log Gromov-Witten invariants.

We construct from a real affine manifold with singularities (a tropical manifold) a degeneration of Calabi-Yau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a striking feature of our approach is that it yields an explicit and canonical order-by-order description of the degeneration via families of tropical trees. This gives complete control of the B-model side of mirror symmetry in terms of tropical geometry. For example, we expect that our deformation parameter is a canonical coordinate, and expect period calculations to be expressible in terms of tropical curves. We anticipate this will lead to a proof of mirror symmetry via tropical methods.

The rechargeable lithium metal battery has attracted wide attention as a next-generation energy storage technology. However, simultaneously achieving high cell-level energy density and long cycle life in realistic batteries is still a great challenge. Here we investigate the degradation mechanisms of Li || LiNi 0.6 Mn 0.2 Co 0.2 O 2 pouch cells and present fundamental linkages among Li thickness, electrolyte depletion and the structure evolution of solid–electrolyte interphase layers. Different cell failure processes are discovered when tuning the anode to cathode capacity ratio in compatible electrolytes. An optimal anode to cathode capacity ratio of 1:1 emerges because it balances well the rates of Li consumption, electrolyte depletion and solid–electrolyte interphase construction, thus decelerating the increase of cell polarization and extending cycle life. Contrary to conventional wisdom, long cycle life is observed by using ultra-thin Li (20 µm) in balanced cells. A prototype 350 Wh kg −1 pouch cell (2.0 Ah) achieves over 600 long stable cycles with 76% capacity retention without a sudden cell death. The development of Li metal batteries requires understanding of cell-level electrochemical processes. Here the authors investigate the interplay between electrode thickness, electrolyte depletion and solid–electrolyte interphase in practical pouch cells and demonstrate the construction of high-energy long-cycle Li metal batteries.

Background In South Africa, human geographic mobility is high as people engage in both permanent and temporary relocation, predominantly from rural to urban areas. Such mobility can compromise healthcare access and utilisation. The objective of this paper is to explore healthcare utilisation and its determinants in a cohort of internal migrants and permanent residents (non-migrants) originating from the Agincourt sub-district in South Africa’s rural northeast. Methods A 5-year cohort study of 3800 individuals aged 18 to 40 commenced in 2017. Baseline data have been collected from 1764 Agincourt residents and 1334 temporary, mostly urban-based, migrants, and are analysed using bivariate analyses, logistic and multinomial regression models, and propensity score matching analysis. Results Health service utilisation differs sharply by migrant status and sex. Among those with a chronic condition, migrants had 0.33 times the odds of non-migrants to have consulted a health service in the preceding year, and males had 0.32 times the odds of females of having used health services. Of those who utilised services, migration status was further associated with the type of healthcare utilised, with 97% of non-migrant rural residents having accessed government facilities, while large proportions of migrants (31%) utilised private health services or consulted traditional healers (25%) in migrant destinations. The multinomial logistic regression analysis indicated that, in the presence of controls, migrants had 8.12 the relative risk of non-migrants for utilising private healthcare (versus the government-services-only reference category), and 2.40 the relative risk of non-migrants for using a combination of public and private sector facilities. These findings of differential utilisation hold under statistical adjustment for relevant controls and for underlying propensity to migrate. Conclusions Migrants and non-migrants in the study population in South Africa were found to utilise health services differently, both in overall use and in the type of healthcare consulted. The study helps improve upon the limited stock of knowledge on how migrants interface with healthcare systems in low and middle-income country settings. Findings can assist in guiding policies and programmes to be directed more effectively to the populations most in need, and to drive locally adapted approaches to universal health coverage.

As announced in Gross and Siebert (in Algebraic geometry: Salt Lake City 2015, Proceedings of Symposia in Pure Mathematics, vol 97, no 2. AMS, Providence, pp 199–230, 2018) in 2016, we construct and prove consistency of the canonical wall structure . This construction starts with a log Calabi–Yau pair ( X ,  D ) and produces a wall structure, as defined in Gross et al. (Mem. Amer. Math. Soc. 278(1376), 1376, 1–103, 2022). Roughly put, the canonical wall structure is a data structure which encodes an algebro-geometric analogue of counts of Maslov index zero disks. These enumerative invariants are defined in terms of the punctured invariants of Abramovich et al. (Punctured Gromov–Witten invariants, 2020. arXiv:2009.07720v2 [math.AG]). There are then two main theorems of the paper. First, we prove consistency of the canonical wall structure, so that, using the setup of Gross et al. (Mem. Amer. Math. Soc. 278(1376), 1376, 1–103, 2022), the canonical wall structure gives rise to a mirror family. Second, we prove that this mirror family coincides with the intrinsic mirror constructed in Gross and Siebert (Intrinsic mirror symmetry, 2019. arXiv:1909.07649v2 [math.AG]). While the setup of this paper is narrower than that of Gross and Siebert (Intrinsic mirror symmetry, 2019. arXiv:1909.07649v2 [math.AG]), it gives a more detailed description of the mirror.

Vigilante violence tends to take place in areas or situations in which the state is unable or unwilling to provide for the safety of certain groups. Crime control vigilantism can be understood as an alternative means of controlling crime and providing security where the state does not. The violent punishment inherent in vigilante activity is generally with the ultimate goal of providing safety and security, and thus should theoretically \"fit the crime\" and not be excessive. However, in many acts of vigilante violence this is not the case, and vigilantism takes on an extraordinarily violent character. This article examines vigilante violence in three South African townships through the micro-sociological perspective of violence developed by Randall Collins (2008), \"forward panic.\" Forward panic is a process whereby the tension and fear marking most potentially violent conflict situations is suddenly released, bringing about extraordinary acts of violence. Based on data from eighteen interviews gathered from the Johannesburg townships of Diepsloot, Freedom Park, and Protea South, I analyze respondents' accounts and experiences with vigilante violence using the framework of forward panic. The data confirm that many acts of vigilante violence in South Africa's townships can be clearly categorized as episodes of forward panic and that although Collins's conception of forward panic focuses on the individual, the conditions that create the emotional potential for forward panic in an individual can be structural and thus create the potential for forward panic in entire groups or parts of communities.

The Endangered Species Act requires actions that improve the passage and survival rates for migrating salmonoids and other fish species that sustain injury and mortality when passing through hydroelectric dams. To develop a low-cost revolutionary acoustic transmitter that may be injected instead of surgically implanted into the fish, one major challenge that needs to be addressed is the micro-battery power source. This work focuses on the design and fabrication of micro-batteries for injectable fish tags. High pulse current and required service life have both been achieved as well as doubling the gravimetric energy density of the battery. The newly designed micro-batteries have intrinsically low impedance, leading to significantly improved electrochemical performances at low temperatures as compared with commercial SR416 batteries. Successful field trial by using the micro-battery powered transmitters injected into fish has been demonstrated, providing an exemplary model of transferring fundamental research into practical devices with controlled qualities.

We prove a decomposition formula of logarithmic Gromov–Witten invariants in a degeneration setting. A one-parameter log smooth family $X \\longrightarrow B$ with singular fibre over $b_0\\in B$ yields a family $\\mathscr {M}(X/B,\\beta ) \\longrightarrow B$ of moduli stacks of stable logarithmic maps. We give a virtual decomposition of the fibre of this family over $b_0$ in terms of rigid tropical maps to the tropicalization of $X/B$. This generalizes one aspect of known results in the case that the fibre $X_{b_0}$ is a normal crossings union of two divisors. We exhibit our formulas in explicit examples.