Lp-bounds on curvature,elliptic estimates and rectifiability of singular setsBornes Lp sur la courbure,estimées elliptiques et rectifiabilité d'ensembles singuliers

We announce results on rectifiability of singular sets of pointed metric spaces which are pointed Gromov–Hausdorff limits on sequences of Riemannian manifolds, satisfying uniform lower bounds on Ricci curvature and volume, and uniform L_p-bounds on curvature. The rectifiability theorems depend on estimates for |Hess_h|_L2p, (|?Hess_h·|Hess_h|^p?2)_L2, where Δh=c, for some constant c. We also observe that (absent any integral bound on curvature) in the Kähler case, given a uniform 2-sided bound on Ricci curvature, the singular set has complex codimension 2. To cite this article: J. Cheeger, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 195–198.     

Interpolatory model reduction of parameterized bilinear dynamical systems

Interpolatory projection methods for model reduction of nonparametric linear dynamical systems have been successfully extended to nonparametric bilinear dynamical systems. However, this has not yet occurred for parametric bilinear systems. In this work, we aim to close this gap by providing a natural extension of interpolatory projections to model reduction of parametric bilinear dynamical systems. We introduce necessary conditions that the projection subspaces must satisfy to obtain parametric tangential interpolation of each subsystem transfer function. These conditions also guarantee that the parameter sensitivities (Jacobian) of each subsystem transfer function are matched tangentially by those of the corresponding reduced-order model transfer function. Similarly, we obtain conditions for interpolating the parameter Hessian of the transfer function by including additional vectors in the projection subspaces. As in the parametric linear case, the basis construction for two-sided projections does not require computing the Jacobian or the Hessian.     

List coloring triangle‐free hypergraphs

A triangle in a hypergraph is a collection of distinct vertices u, v, w and distinct edges e, f, g with , and . Johansson [Tech. report (1996)] proved that every triangle‐free graph with maximum degree Δ has list chromatic number . Frieze and Mubayi (Electron J Comb 15 (2008), 27) proved that every linear (meaning that every two edges share at most one vertex) triangle‐free triple system with maximum degree Δ has chromatic number . The restriction to linear triple systems was crucial to their proof. We provide a common generalization of both these results for rank 3 hypergraphs (edges have size 2 or 3). Our result removes the linear restriction from 8 , while reducing to the (best possible) result [Johansson, Tech. report (1996)] for graphs. In addition, our result provides a positive answer to a restricted version of a question of Ajtai Erd?s, Komlós, and Szemerédi (combinatorica 1 (1981), 313–317) concerning sparse 3‐uniform hypergraphs. As an application, we prove that if is the collection of 3‐uniform triangles, then the Ramsey number satisfies for some positive constants a and b. The upper bound makes progress towards the recent conjecture of Kostochka, Mubayi, and Verstraëte (J Comb Theory Ser A 120 (2013), 1491–1507) that where C₃ is the linear triangle. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 487–519, 2015     

Log-Concave Functions And Poset Probabilities

   elements of some (finite) poset , write for the probability that precedes in a random (uniform) linear extension of . For define

The role of informal learning spaces in enhancing student engagement with mathematical sciences

By helping create a shared, supportive, learning community, the creative use of custom-designed spaces outside the classroom has a major impact on student engagement. The intention is to create spaces that promote peer interaction within and across year groups, encourage closer working relationships between staff and students and support specific coursework activities – particularly group work. Such spaces make better use of time since students are motivated to stay and work during long gaps in their timetable, can provide a sense of ‘home’ within the institution and can lead to a cohesive community of practice. In this paper, we describe how this has been achieved and currently delivered in Mathematics at Sheffield Hallam University and provide some detailed analysis of the student usage of the space.     

Online Prediction with History-Dependent Experts: The General Case

We study the problem of prediction of binary sequences with expert advice in the online setting, which is a classic example of online machine learning. We interpret the binary sequence as the price history of a stock, and view the predictor as an investor, which converts the problem into a stock prediction problem. In this framework, an investor, who predicts the daily movements of a stock, and an adversarial market, who controls the stock, play against each other over N turns. The investor combines the predictions of $n\ge 2$ experts in order to make a decision about how much to invest at each turn, and aims to minimize their regret with respect to the best-performing expert at the end of the game. We consider the problem with history-dependent experts, in which each expert uses the previous d days of history of the market in making their predictions. We prove that the value function for this game, rescaled appropriately, converges as $N\to \mathrm{\infty }$ at a rate of $O\left(N^{-1/6}\right)$ to the viscosity solution of a nonlinear degenerate elliptic PDE, which can be understood as the Hamilton-Jacobi-Issacs equation for the two-person game. As a result, we are able to deduce asymptotically optimal strategies for the investor. Our results extend those established by the first author and R.V. Kohn [14] for $n=2$ experts and $d\le 4$ days of history. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.     

Pd‐Catalysed Direct Arylation Polymerisation for Synthesis of Low‐Bandgap Conjugated Polymers and Photovoltaic Performance

Low‐bandgap conjugated copolymers based on a donor–acceptor structure have been synthesised via palladium‐complex catalysed direct arylation polymerisation. Initially, we report the optimisation of the synthesis of poly(cyclopentadithiophene‐alt‐benzothiadiazole) (PCPDTBT) formed between cyclopentadithiophene and dibromobenzothiadiazole units. The polymerisation condition has been optimised, which affords high‐molecular‐weight polymers of up to M _n = 70 k using N‐methylpyrrolidone as a solvent. The polymers are used to fabricate organic photovoltaic devices and the best performing PCPDTBT device exhibits a moderate improvement over devices fabricated using the related polymer via Suzuki coupling. Similar polymerisation conditions have also been applied for other monomer units.