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1. Signed permutohedra, delta‐matroids, and beyond

   https://doi.org/10.1112/plms.12592  

   Eur, Christopher; Fink, Alex; Larson, Matt; Spink, Hunter (March 2024, Proceedings of the London Mathematical Society)

   Abstract We establish a connection between the algebraic geometry of the type permutohedral toric variety and the combinatorics of delta‐matroids. Using this connection, we compute the volume and lattice point counts of type generalized permutohedra. Applying tropical Hodge theory to a new framework of “tautological classes of delta‐matroids,” modeled after certain vector bundles associated to realizable delta‐matroids, we establish the log‐concavity of a Tutte‐like invariant for a broad family of delta‐matroids that includes all realizable delta‐matroids. Our results include new log‐concavity statements for all (ordinary) matroids as special cases. 

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2. Covariate-adjusted log-rank test: guaranteed efficiency gain and universal applicability

   https://doi.org/10.1093/biomet/asad045  

   Ye, Ting; Shao, Jun; Yi, Yanyao (July 2023, Biometrika)

   Summary Nonparametric covariate adjustment is considered for log-rank-type tests of the treatment effect with right-censored time-to-event data from clinical trials applying covariate-adaptive randomization. Our proposed covariate-adjusted log-rank test has a simple explicit formula and a guaranteed efficiency gain over the unadjusted test. We also show that our proposed test achieves universal applicability in the sense that the same formula of test can be universally applied to simple randomization and all commonly used covariate-adaptive randomization schemes such as the stratified permuted block and the Pocock–Simon minimization, which is not a property enjoyed by the unadjusted log-rank test. Our method is supported by novel asymptotic theory and empirical results for Type-I error and power of tests. 

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3. Temperature, resources and predation interact to shape phytoplankton size–abundance relationships at a continental scale

   https://doi.org/10.1111/geb.13748  

   Gjoni, Vojsava; Glazier, Douglas S; Wesner, Jeff S; Ibelings, Bastiaan W; Thomas, Mridul K (November 2023, Global Ecology and Biogeography)

   Abstract AimCommunities contain more individuals of small species and fewer individuals of large species. According to the ‘metabolic theory of ecology’, the relationship of log mean abundance with log mean body size across communities should exhibit a slope of −3/4 that is invariant across environmental conditions. Here, we investigate whether this slope is indeed invariant or changes systematically across gradients in temperature, resource availability and predation pressure. Location1048 lakes across the USA. Time Period2012. Major Taxa StudiedPhytoplankton. ResultsWe found that the size–abundance relationship across all sampled phytoplankton communities was significantly lower than −3/4 and near −1 overall. More importantly, we found strong evidence that the environment affects the slope: it varies between −0.33 and −0.93 across interacting gradients of temperature, resource (phosphorus) supply and zooplankton predation pressure. Therefore, phytoplankton communities have orders of magnitude more small or large cells depending on environmental conditions across geographical locations. ConclusionOur results emphasise the importance of the environmental factors' effect on macroecological patterns that arise through physiological and ecological processes. An investigation of the mechanisms underlying the link between individual energetics constrain and macroecological patterns would allow to predict how global warming and changes in nutrients will alter large‐scale ecological patterns in the future. 

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4. Testing the Predictions of Surprisal Theory in 11 Languages

   https://doi.org/10.1162/tacl_a_00612  

   Wilcox, Ethan G.; Pimentel, Tiago; Meister, Clara; Cotterell, Ryan; Levy, Roger P. (January 2023, Transactions of the Association for Computational Linguistics)

   Abstract Surprisal theory posits that less-predictable words should take more time to process, with word predictability quantified as surprisal, i.e., negative log probability in context. While evidence supporting the predictions of surprisal theory has been replicated widely, much of it has focused on a very narrow slice of data: native English speakers reading English texts. Indeed, no comprehensive multilingual analysis exists. We address this gap in the current literature by investigating the relationship between surprisal and reading times in eleven different languages, distributed across five language families. Deriving estimates from language models trained on monolingual and multilingual corpora, we test three predictions associated with surprisal theory: (i) whether surprisal is predictive of reading times, (ii) whether expected surprisal, i.e., contextual entropy, is predictive of reading times, and (iii) whether the linking function between surprisal and reading times is linear. We find that all three predictions are borne out crosslinguistically. By focusing on a more diverse set of languages, we argue that these results offer the most robust link to date between information theory and incremental language processing across languages. 

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5. A Refutation of the Pach-Tardos Conjecture for 0–1 Matrices

   https://doi.org/10.1007/s00493-025-00187-7  

   Pettie, Seth; Tardos, Gábor (December 2025, Combinatorica)

   Abstract The theory of forbidden 0–1 matrices generalizes Turán-style (bipartite) subgraph avoidance, Davenport-Schinzel theory, and Zarankiewicz-type problems, and has been influential in many areas, such as discrete and computational geometry, the analysis of self-adjusting data structures, and the development of the graph parametertwin width. The foremost open problem in this area is to resolve thePach-Tardos conjecturefrom 2005, which states that if a forbidden pattern$$P\in \{0,1\}^{k\times l}$$ $P\in {\{0,1\}}^{k\times l}$isacyclic, meaning it is the bipartite incidence matrix of a forest, then$$\operatorname {Ex}(P,n) = O(n\log ^{C_P} n)$$ $Ex(P,n)=O\left(n{log}^{C_P}n\right)$, where$$\operatorname {Ex}(P,n)$$ $Ex(P,n)$is the maximum number of 1s in aP-free$$n\times n$$ $n\times n$0–1 matrix and$$C_P$$ $C_P$is a constant depending only onP. This conjecture has been confirmed on many small patterns, specifically allPwith weight at most 5, and all but two with weight 6. The main result of this paper is a clean refutation of the Pach-Tardos conjecture. Specifically, we prove that$$\operatorname {Ex}(S_0,n),\operatorname {Ex}(S_1,n) \ge n2^{\Omega (\sqrt{\log n})}$$ $Ex(S_0,n),Ex(S_1,n)\ge n{2}^{\Omega \left(\sqrt{logn}\right)}$, where$$S_0,S_1$$ $S_0,S_1$are the outstanding weight-6 patterns. We also prove sharp bounds on the entire class ofalternatingpatterns$$(P_t)$$ $\left(P_t\right)$, specifically that for every$$t\ge 2$$ $t\ge 2$,$$\operatorname {Ex}(P_t,n)=\Theta (n(\log n/\log \log n)^t)$$ $Ex(P_t,n)=\Theta \left(n{(logn/loglogn)}^t\right)$. This is the first proof of an asymptotically sharp bound that is$$\omega (n\log n)$$ $\omega (nlogn)$. 

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