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1. Uniform stability for local discontinuous Galerkin methods with implicit-explicit Runge-Kutta time discretizations for linear convection-diffusion equation

   https://doi.org/10.1090/mcom/3842  

   Wang, Haijin ; Li, Fengyan ; Shu, Chi-Wang ; Zhang, Qiang ( November 2023 , Mathematics of Computation)

   In this paper, we consider the linear convection-diffusion equation in one dimension with periodic boundary conditions, and analyze the stability of fully discrete methods that are defined with local discontinuous Galerkin (LDG) methods in space and several implicit-explicit (IMEX) Runge-Kutta methods in time. By using the forward temporal differences and backward temporal differences, respectively, we establish two general frameworks of the energy-method based stability analysis. From here, the fully discrete schemes being considered are shown to have monotonicity stability, i.e. the$L^2$norm of the numerical solution does not increase in time, under the time step condition$\tau \le \mathcal {F}(h/c, d/c^2)$, with the convection coefficient$c$, the diffusion coefficient$d$, and the mesh size$h$. The function$\mathcal {F}$depends on the specific IMEX temporal method, the polynomial degree$k$of the discrete space, and the mesh regularity parameter. Moreover, the time step condition becomes$\tau \lesssim h/c$in the convection-dominated regime and it becomes$\tau \lesssim d/c^2$in the diffusion-dominated regime. The result is improved for a first order IMEX-LDG method. To complement the theoretical analysis, numerical experiments are further carried out, leading to slightly stricter time step conditions that can be used by practitioners. Uniform stability with respect to the strength of the convection and diffusion effects can especially be relevant to guide the choice of time step sizes in practice, e.g. when the convection-diffusion equations are convection-dominated in some sub-regions.

    

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2. The stable exotic Cuntz algebras are higher-rank graph algebras

   https://doi.org/10.1090/bproc/180  

   Boersema, Jeffrey ; Browne, Sarah ; Gillaspy, Elizabeth ( March 2024 , Proceedings of the American Mathematical Society, Series B)

   For each odd integer$n \geq 3$, we construct a rank-3 graph$\Lambda _n$with involution$\gamma _n$whose real$C^*$-algebra$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda _n, \gamma _n)$is stably isomorphic to the exotic Cuntz algebra$\mathcal E_n$. This construction is optimal, as we prove that a rank-2 graph with involution$(\Lambda ,\gamma )$can never satisfy$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma )\sim _{ME} \mathcal E_n$, and Boersema reached the same conclusion for rank-1 graphs (directed graphs) in [Münster J. Math.10(2017), pp. 485–521, Corollary 4.3]. Our construction relies on a rank-1 graph with involution$(\Lambda , \gamma )$whose real$C^*$-algebra$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma )$is stably isomorphic to the suspension$S \mathbb {R}$. In the Appendix, we show that the$i$-fold suspension$S^i \mathbb {R}$is stably isomorphic to a graph algebra iff$-2 \leq i \leq 1$.

    

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3. Planar Algebras in Braided Tensor Categories

   https://doi.org/10.1090/memo/1392  

   Henriques, André ; Penneys, David ; Tener, James ( February 2023 , Memoirs of the American Mathematical Society)

   We generalize Jones’ planar algebras by internalising the notion to a pivotal braided tensor category$\mathcal {C}$. To formulate the notion, the planar tangles are now equipped with additional ‘anchor lines’ which connect the inner circles to the outer circle. We call the resulting notion ananchored planar algebra. If we restrict to the case when$\mathcal {C}$is the category of vector spaces, then we recover the usual notion of a planar algebra.

   Building on our previous work on categorified traces, we prove that there is an equivalence of categories between anchored planar algebras in$\mathcal {C}$and pivotal module tensor categories over$\mathcal {C}$equipped with a chosen self-dual generator. Even in the case of usual planar algebras, the precise formulation of this theorem, as an equivalence of categories, has not appeared in the literature. Using our theorem, we describe many examples of anchored planar algebras.

    

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4. On-chip spin-orbit locking of quantum emitters in 2D materials for chiral emission

   https://doi.org/10.1364/OPTICA.463481  

   Ma, Yichen ; Zhao, Haoqi ; Liu, Na ; Gao, Zihe ; Mohajerani, Seyed Sepehr ; Xiao, Licheng ; Hone, James ; Feng, Liang ; Strauf, Stefan ( August 2022 , Optica)

   Light carries both spin angular momentum (SAM) and orbital angular momentum (OAM), which can be used as potential degrees of freedom for quantum information processing. Quantum emitters are ideal candidates towards on-chip control and manipulation of the full SAM–OAM state space. Here, we show coupling of a spin-polarized quantum emitter in a monolayer${\mathrm{W}\mathrm{S}\mathrm{e}}_2$with the whispering gallery mode of a${\mathrm{S}\mathrm{i}}_3{\mathrm{N}}_4$ring resonator. The cavity mode carries a transverse SAM of$\mathrm{\sigma <\#comment/>}=\pm <\#comment/>1$in the evanescent regions, with the sign depending on the orbital power flow direction of the light. By tailoring the cavity–emitter interaction, we couple the intrinsic spin state of the quantum emitter to the SAM and propagation direction of the cavity mode, which leads to spin–orbit locking and subsequent chiral single-photon emission. Furthermore, by engineering how light is scattered from the WGM, we create a high-order Bessel beam which opens up the possibility to generate optical vortex carrying OAM states.

    

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5. From African “tam-tam” to nonlinear optics [Invited]

   https://doi.org/10.1364/JOSAB.399177  

   Kofané, Timoléon C. ; Tabi, Conrad B. ; Moubissi, Alain B. ; Tchawoua, Clément ( September 2020 , Journal of the Optical Society of America B)

   The field of nonlinear optics has been investigated extensively since its beginning in 1961. This development is in both the theory of nonlinear effects and the theory of nonlinear interactions in nonlinear media, and in the applications of nonlinear devices. The mathematical basis of nonlinear optics is Maxwell’s system of equations governing propagation of electromagnetic waves in a material medium, combined with relations accounting for the nonlinear response of the medium to the electromagnetic field. This review paper presents the contribution of African researchers to recent theoretical advances in the study of nonlinear interactions between electromagnetic waves with different types of nonlinear media, such as optical fibers, metamaterials, and Bose–Einstein condensates, that constitute a fascinating source of temporal, spatial, and spatiotemporal phenomena in the physics of light, and that lead to modulational instability, optical pulse compression, rogue waves,$\mathcal{P}\mathcal{T}$-symmetric phenomena, supercontinuum generation, and dissipative solitons.

    

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