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1. Asymptotic Behaviour of Time Stepping Methods for Phase Field Models

   https://doi.org/10.1007/s10915-020-01391-x  

   Cheng, Xinyu ; Li, Dong ; Promislow, Keith ; Wetton, Brian ( March 2021 , Journal of Scientific Computing)

   null (Ed.)

   Abstract Adaptive time stepping methods for metastable dynamics of the Allen–Cahn and Cahn–Hilliard equations are investigated in the spatially continuous, semi-discrete setting. We analyse the performance of a number of first and second order methods, formally predicting step sizes required to satisfy specified local truncation error $$\sigma $$ σ in the limit of small length scale parameter $$\epsilon \rightarrow 0$$ ϵ → 0 during meta-stable dynamics. The formal predictions are made under stability assumptions that include the preservation of the asymptotic structure of the diffuse interface, a concept we call profile fidelity. In this setting, definite statements about the relative behaviour of time stepping methods can be made. Some methods, including all so-called energy stable methods but also some fully implicit methods, require asymptotically more time steps than others. The formal analysis is confirmed in computational studies. We observe that some provably energy stable methods popular in the literature perform worse than some more standard schemes. We show further that when Backward Euler is applied to meta-stable Allen–Cahn dynamics, the energy decay and profile fidelity properties for these discretizations are preserved for much larger time steps than previous analysis would suggest. The results are established asymptotically for general interfaces, with a rigorous proof for radial interfaces. It is shown analytically and computationally that for most reaction terms, Eyre type time stepping performs asymptotically worse due to loss of profile fidelity. 

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2. The emergence of a virus variant: dynamics of a competition model with cross-immunity time-delay validated by wastewater surveillance data for COVID-19

   https://doi.org/10.1007/s00285-023-01900-0  

   Pell, Bruce ; Brozak, Samantha ; Phan, Tin ; Wu, Fuqing ; Kuang, Yang ( March 2023 , Journal of Mathematical Biology)

   We consider the dynamics of a virus spreading through a population that produces a mutant strain with the ability to infect individuals that were infected with the established strain. Temporary cross-immunity is included using a time delay, but is found to be a harmless delay. We provide some sufficient conditions that guarantee local and global asymptotic stability of the disease-free equilibrium and the two boundary equilibria when the two strains outcompete one another. It is shown that, due to the immune evasion of the emerging strain, the reproduction number of the emerging strain must be significantly lower than that of the established strain for the local stability of the established-strain-only boundary equilibrium. To analyze the unique coexistence equilibrium we apply a quasi steady-state argument to reduce the full model to a two-dimensional one that exhibits a global asymptotically stable established-strain-only equilibrium or global asymptotically stable coexistence equilibrium. Our results indicate that the basic reproduction numbers of both strains govern the overall dynamics, but in nontrivial ways due to the inclusion of cross-immunity. The model is applied to study the emergence of the SARS-CoV-2 Delta variant in the presence of the Alpha variant using wastewater surveillance data from the Deer Island Treatment Plant in Massachusetts, USA.

    

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3. Stability of the centers of group algebras of $GL_n(q)$

   https://doi.org/https://doi.org/10.1016/j.aim.2019.04.026  

   Wan, Jinkui ; Wang, Weiqiang ( April 2019 , Advances in mathematics)

   The center $Z_n(q)$ of the integral group algebra of the general linear group $GL_n(q)$ over a finite field admits a filtration with respect to the reflection length. We show that the structure constants of the associated graded algebras $\mathscr{G}_n(q)$ are independent of $n$, and this stability leads to a universal stable center with positive integer structure constants which governs the algebras $\mathscr{G}_n(q)$ for all $n$. Various structure constants of the stable center are computed and several conjectures are formulated. Analogous stability properties for symmetric groups and wreath products were established earlier by Farahat-Higman and the second author. 

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4. Regular Supercuspidal Representations

   Tasho Kaletha ( July 2019 , Journal of the American Mathematical Society)

   We show that, in good residual characteristic, most supercuspidal representations of a tamely ramified reductive p-adic group G arise from pairs (S,\theta), where S is a tame elliptic maximal torus of G, and \theta is a character of S satisfying a simple root-theoretic property. We then give a new expression for the roots of unity that appear in the Adler-DeBacker-Spice character formula for these supercuspidal representations and use it to show that this formula bears a striking resemblance to the character formula for discrete series representations of real reductive groups. Led by this, we explicitly construct the local Langlands correspondence for these supercuspidal representations and prove stability and endoscopic transfer in the case of toral representations. In large residual characteristic this gives a construction of the local Langlands correspondence for almost all supercuspidal representations of reductive p-adic groups. 

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5. On codimension one stability of the soliton for the 1D focusing cubic Klein-Gordon equation

   https://doi.org/10.1090/cams/32  

   Lührmann, Jonas ; Schlag, Wilhelm ( February 2024 , Communications of the American Mathematical Society)

   We consider the codimension one asymptotic stability problem for the soliton of the focusing cubic Klein-Gordon equation on the line under even perturbations. The main obstruction to full asymptotic stability on the center-stable manifold is a small divisor in a quadratic source term of the perturbation equation. This singularity is due to the threshold resonance of the linearized operator and the absence of null structure in the nonlinearity. The threshold resonance of the linearized operator produces a one-dimensional space of slowly decaying Klein-Gordon waves, relative to local norms. In contrast, the closely related perturbation equation for the sine-Gordon kink does exhibit null structure, which makes the corresponding quadratic source term amenable to normal forms (see Lührmann and Schlag [Duke Math. J. 172 (2023), pp. 2715–2820]).

   The main result of this work establishes decay estimates up to exponential time scales for small “codimension one type” perturbations of the soliton of the focusing cubic Klein-Gordon equation. The proof is based upon a super-symmetric approach to the study of modified scattering for 1D nonlinear Klein-Gordon equations with Pöschl-Teller potentials from Lührmann and Schlag [Duke Math. J. 172 (2023), pp. 2715–2820], and an implementation of a version of an adapted functional framework introduced by Germain and Pusateri [Forum Math. Pi 10 (2022), p. 172].

    

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