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Being driven by the goal of finding edge modes and of explaining the occurrence of edge modes in the case of time-modulated metamaterials in the high-contrast and subwavelength regime, we analyse the topological properties of Floquet normal forms of periodically parameterized time-periodic linear ordinary differential equations d dt X = A α ( t ) X α ∈ T d . In fact, our main goal being the question whether an analogous principle as the bulk-boundary correspondence of solid-state physics is possible in the case of Floquet metamaterials, i.e., subwavelength high-contrast time-modulated metamaterials. This paper is a first step in that direction. Since the bulk-boundary correspondence states that topological properties of the bulk materials characterize the occurrence of edge modes, we dedicate this paper to the topological analysis of subwavelength solutions in Floquet metamaterials. This work should thus be considered as a basis for further investigation on whether topological properties of the bulk materials are linked to the occurrence of edge modes. The subwavelength solutions being described by a periodically parameterized time-periodic linear ordinary differential equation d dt X = A α ( t ) X α ∈ T d , we put ourselves in the general setting of periodically parameterized time-periodic linear ordinary differential equations and introduce a way to (topologically) classify a Floquet normal form F ,   P of the associated fundamental solution X α ( t ) = P ( α , t ) exp ( t F α ) α ∈ T d . This is achieved by analysing the topological properties of the eigenvalues and eigenvectors of the monodromy matrix X α ( T ) and the Lyapunov transformation P ( α , t ) . The corresponding topological invariants can then be applied to the setting of Floquet metamaterials. In this paper these general results are considered in the case of a hexagonal structure. We provide two interesting examples of topologically non-trivial time-modulated hexagonal structures., Competing Interests: Conflict of interestOn behalf of all authors, the corresponding author states that there is no conflict of interest., (© The Author(s) 2023.)

The distribution of voltage in sub-micron cellular domains remains poorly understood. In neurons, the voltage results from the difference in ionic concentrations which are continuously maintained by pumps and exchangers. However, it not clear how electro-neutrality could be maintained by an excess of fast moving positive ions that should be counter balanced by slow diffusing negatively charged proteins. Using the theory of electro-diffusion, we study here the voltage distribution in a generic domain, which consists of two concentric disks (resp. ball) in two (resp. three) dimensions, where a negative charge is fixed in the inner domain. When global but not local electro-neutrality is maintained, we solve the Poisson-Nernst-Planck equation both analytically and numerically in dimension 1 (flat) and 2 (cylindrical) and found that the voltage changes considerably on a spatial scale which is much larger than the Debye screening length, which assumes electro-neutrality. The present result suggests that long-range voltage drop changes are expected in neuronal microcompartments, probably relevant to explain the activation of far away voltage-gated channels located on the surface membrane.

We perform global and local analysis of oscillatory and damped spherically symmetric fundamental solutions for Helmholtz operators (-Δ± β 2 ) in d -dimensional, R -radius hyperbolic H R d and hyperspherical S R d geometry, which represent Riemannian manifolds with positive constant and negative constant sectional curvature respectively. In particular, we compute closed-form expressions for fundamental solutions of (-Δ± β 2 ) on H R d , (-Δ± β 2 ) on S R d , and present two candidate fundamental solutions for (-Δ - β 2 ) on S R d . Flat-space limits, with their corresponding asymptotic representations, are used to restrict proportionality constants for these fundamental solutions. In order to accomplish this, we summarize and derive new large degree asymptotics for associated Legendre and Ferrers functions of the first and second kind. Furthermore, we prove that our fundamental solutions on the hyperboloid are unique due to their decay at infinity. To derive Gegenbauer polynomial expansions of our fundamental solutions for Helmholtz operators on hyperspheres and hyperboloids, we derive a collection of infinite series addition theorems for Ferrers and associated Legendre functions which are generalizations and extensions of the addition theorem for Gegenbauer polynomials. Using these addition theorems, in geodesic polar coordinates for dimensions greater than or equal to three, we compute Gegenbauer polynomial expansions for these fundamental solutions, and azimuthal Fourier expansions in two-dimensions.

We present a numerical method to efficiently and accurately re-compute the Coulomb potential of a large ensemble of charged particles after a subset of the particles undergoes a change of position. Errors are bounded even after a large number of such shifts, making it practical for use in Monte Carlo Markov chain methods in molecular dynamics, computational astrophysics, computational chemistry, and other applications. The method uses truncated multipole expansions of the potential energy functional and a tree decomposition of the computational domain to reduce the computational complexity. Computational costs scale logarithmically in the size of the problem. Scaling, accuracy, and efficiency are confirmed with numerical experiments. The new method outperforms a direct calculation for moderate problem sizes.