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1. Computing Optimal Persistent Cycles: Polynomial and Hard Cases

   Dey, Tamal K; Hou, Tao; Mandal, Sayan (January 2020, ACM-SIAM Symposium on Discrete Algorithms)

   Persistent cycles, especially the minimal ones, are useful geometric features functioning as augmentations for the intervals in the purely topological persistence diagrams (also termed as barcodes). In our earlier work, we showed that computing minimal 1-dimensional persistent cycles (persistent 1-cycles) for finite intervals is NP-hard while the same for infinite intervals is polynomially tractable. In this paper, we address this problem for general dimensions with Z2 coefficients. In addition to proving that it is NP-hard to compute minimal persistent d-cycles (d>1) for both types of intervals given arbitrary simplicial complexes, we identify two interesting cases which are polynomially tractable. These two cases assume the complex to be a certain generalization of manifolds which we term as weak pseudomanifolds. For finite intervals from the d-th persistence diagram of a weak (d+1)-pseudomanifold, we utilize the fact that persistent cycles of such intervals are null-homologous and reduce the problem to a minimal cut problem. Since the same problem for infinite intervals is NP-hard, we further assume the weak (d+1)-pseudomanifold to be embedded in R^{d+1}Rd+1 so that the complex has a natural dual graph structure and the problem reduces to a minimal cut problem. Experiments with both algorithms on scientific data indicate that the minimal persistent cycles capture various significant features of the data. 

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2. Interval Signal Temporal Logic From Natural Inclusion Functions

   https://doi.org/10.1109/LCSYS.2023.3337744  

   Baird, Luke; Harapanahalli, Akash; Coogan, Samuel (January 2023, IEEE Control Systems Letters)

   We propose an interval extension of Signal Temporal Logic (STL) called Interval Signal Temporal Logic (I-STL). Given an STL formula, we consider an interval inclusion function for each of its predicates. Then, we use minimal inclusion functions for the min and max functions to recursively build an interval robustness that is a natural inclusion function for the robustness of the original STL formula. The resulting interval semantics accommodate, for example, uncertain signals modeled as a signal of intervals and uncertain predicates modeled with appropriate inclusion functions. In many cases, verification or synthesis algorithms developed for STL apply to I-STL with minimal theoretic and algorithmic changes, and existing code can be readily extended using interval arithmetic packages at negligible computational expense. To demonstrate I-STL, we present an example of offline monitoring from an uncertain signal trace obtained from a hardware experiment and an example of robust online control synthesis enforcing an STL formula with uncertain predicates. 

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3. Watershed and fire severity are stronger determinants of soil chemistry and microbiomes than within-watershed woody encroachment in a tallgrass prairie system

   https://doi.org/10.1093/femsec/fiab154  

   Mino, Laura; Kolp, Matthew R.; Fox, Sam; Reazin, Chris; Zeglin, Lydia; Jumpponen, Ari (November 2021, FEMS Microbiology Ecology)

   ABSTRACT Fire can impact terrestrial ecosystems by changing abiotic and biotic conditions. Short fire intervals maintain grasslands and communities adapted to frequent, low-severity fires. Shrub encroachment that follows longer fire intervals accumulates fuel and can increase fire severity. This patchily distributed biomass creates mosaics of burn severities in the landscape—pyrodiversity. Afforded by a scheduled burn of a watershed protected from fires for 27 years, we investigated effects of woody encroachment and burn severity on soil chemistry and soil-inhabiting bacteria and fungi. We compared soils before and after fire within the fire-protected, shrub-encroached watershed and soils in an adjacent, annually burned and non-encroached watershed. Organic matter and nutrients accumulated in the fire-protected watershed but responded less to woody encroachment within the encroached watershed. Bioavailable nitrogen and phosphorus and fungal and bacterial communities responded to high-severity burn regardless of encroachment. Low-severity fire effects on soil nutrients differed, increased bacterial but decreased fungal diversity and effects of woody encroachment within the encroached watershed were minimal. High-severity burns in the fire-protected watershed led to a novel soil system state distinct from non-encroached and encroached soil systems. We conclude that severe fires may open grassland restoration opportunities to manipulate soil chemistry and microbial communities in shrub-encroached habitats. 

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4. A Bias Bound Approach to Non-parametric Inference

   https://doi.org/10.1093/restud/rdz065  

   Schennach, Susanne M (January 2020, The Review of Economic Studies)

   Abstract The traditional approach to obtain valid confidence intervals for non-parametric quantities is to select a smoothing parameter such that the bias of the estimator is negligible relative to its standard deviation. While this approach is apparently simple, it has two drawbacks: first, the question of optimal bandwidth selection is no longer well-defined, as it is not clear what ratio of bias to standard deviation should be considered negligible. Second, since the bandwidth choice necessarily deviates from the optimal (mean squares-minimizing) bandwidth, such a confidence interval is very inefficient. To address these issues, we construct valid confidence intervals that account for the presence of a non-negligible bias and thus make it possible to perform inference with optimal mean squared error minimizing bandwidths. The key difficulty in achieving this involves finding a strict, yet feasible, bound on the bias of a non-parametric estimator. It is well-known that it is not possible to consistently estimate the pointwise bias of an optimal non-parametric estimator (for otherwise, one could subtract it and obtain a faster convergence rate violating Stone’s bounds on the optimal convergence rates). Nevertheless, we find that, under minimal primitive assumptions, it is possible to consistently estimate an upper bound on the magnitude of the bias, which is sufficient to deliver a valid confidence interval whose length decreases at the optimal rate and which does not contradict Stone’s results. 

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5. Analytic reparametrizations of translation toral flows with countable Lebesgue spectrum

   https://doi.org/10.3934/dcds.2023063  

   Abdedou, Fatna; Fayad, Bassam; Kessi, Arezki (January 2023, Discrete and Continuous Dynamical Systems)

   We give an example of a real analytic reparametrization of a minimal translation flow on $$\mathbb{T}^{5}$$ that has a Lebesgue spectrum with infinite multiplicity. As a consequence, we see that the dynamics on a non-Diophantine invariant torus of an almost integrable Hamiltonian system can be spectrally equivalent to a Bernoulli flow. 

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