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1. Elliptic bihamiltonian structures from relative shifted Poisson structures

   https://doi.org/10.1112/topo.12315  

   Hua, Zheng; Polishchuk, Alexander (December 2023, Journal of Topology)

   Abstract In this paper, generalizing our previous construction, we equip the relative moduli stack of complexes over a Calabi–Yau fibration (possibly with singular fibers) with a shifted Poisson structure. Applying this construction to the anticanonical linear systems on surfaces, we get examples of compatible Poisson brackets on projective spaces extending Feigin–Odesskii Poisson brackets. Computing explicitly the corresponding compatible brackets coming from Hirzebruch surfaces, we recover the brackets defined by Odesskii–Wolf. 

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2. Fluid data structures

   https://doi.org/10.1145/3315507.3330197  

   Balakrishnan, Darshana; Ziarek, Lukasz; Kennedy, Oliver (June 2019, DBPL 2019: Proceedings of the 17th ACM SIGPLAN International Symposium on Database Programming Languages)

   Functional (aka immutable) data structures are used extensively in data management systems. From distributed systems to data persistence, immutability makes complex programs significantly easier to reason about and implement. However, immutability also makes many runtime optimizations like tree rebalancing, or adaptive organizations, unreasonably expensive. In this paper, we propose Fluid data structures, an approach to data structure design that allows limited physical changes that preserve logical equivalence. As we will show, this approach retains many of the desirable properties of functional data structures, while also allowing runtime adaptation. To illustrate Fluid data structures, we work through the design of a lazy-loading map that we call a Fluid Cog. A Fluid Cog is a lock-free data structure that incrementally organizes itself in the background by applying equivalence-preserving structural transformations. Our experimental analysis shows that the resulting map structure is flexible enough to adapt to a variety of performance goals, while remaining competitive with existing structures like the C++ standard template library map. 

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3. Densely computable structures

   https://doi.org/10.1093/logcom/exab057  

   Calvert, Wesley; Cenzer, Douglas; Harizanov, Valentina (September 2021, Journal of Logic and Computation)

   Abstract In recent years, computability theorists have extensively studied generically and coarsely computable sets. This study of approximate computability was originally motivated by asymptotic density problems in combinatorial group theory. We generalize the notions of generic and coarse computability of sets, introduced by Jockusch and Schupp, to arbitrary structures by defining generically and coarsely computable and computably enumerable structures. There are two directions in which these notions could potentially trivialize: either all structures could have a densely computable copy or only those having a computable (or computably enumerable) copy. We show that some particular classes of structures realize each of these extremal conditions, while other classes realize neither of them. To further explore these concepts, we introduce a graded family of elementarity conditions for substructures, in which we require that the dense sets under consideration be ‘strong’ substructures of the original structure. Here, again, for a given class, the notion could trivialize in the same two directions and we show that both are possible. For each class that we investigate, there is some natural number $$n$$ such that requiring $$\varSigma _{n}$$ elementarity of substructures is enough to trivialize the class of generically or densely computable structures, witnessing the essentially structural character of these notions. 

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4. Abstracted quantitative structures

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5. Dynamic DNA Structures

   https://doi.org/10.1002/smll.201900228  

   Zhang, Yingwei; Pan, Victor; Li, Xue; Yang, Xueqin; Li, Haofei; Wang, Pengfei; Ke, Yonggang (April 2019, Small)

   Abstract Dynamic DNA structures, a type of DNA construct built using programmable DNA self‐assembly, have the capability to reconfigure their conformations in response to environmental stimulation. A general strategy to design dynamic DNA structures is to integrate reconfigurable elements into conventional static DNA structures that may be assembled from a variety of methods including DNA origami and DNA tiles. Commonly used reconfigurable elements range from strand displacement reactions, special structural motifs, target‐binding DNA aptamers, and base stacking components, to DNA conformational change domains, etc. Morphological changes of dynamic DNA structures may be visualized by imaging techniques or may be translated to other detectable readout signals (e.g., fluorescence). Owing to their programmable capability of recognizing environmental cues with high specificity, dynamic DNA structures embody the epitome of robust and versatile systems that hold great promise in sensing and imaging biological analytes, in delivering molecular cargos, and in building programmable systems that are able to conduct sophisticated tasks. 

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