This paper analyzes a stochastic Allen--Cahn equation for the dynamics of biomolecular damage and repair. The system is driven by two distinct noise processes: a multiplicative cylindrical Wiener process, modeling continuous background stochastic fluctuations, and a jump-type noise, modeling the abrupt, localized damage induced by external shocks. The drift of the equation is singular and covers the typical logarithmic Flory-Huggins potential required in phase-separation dynamics. We prove well-posedness of the model in a strong probabilistic sense, and analyze its long-time behavior in terms of existence and uniqueness of invariant measures, ergodicity, and mixing properties. Eventually, we present an Euler--Maruyama scheme to simulate the model and illustrate how it captures fundamental biological phenomena, such as damage clustering, stress-induced topology perturbations, and damage dynamics.

The Cahn-Hilliard equation and extensions, notably the Cahn-Hilliard-Darcy and Cahn-Hilliard-Navier-Stokes systems, provide widely used frameworks for coupling interfacial thermodynamics with flow. This review surveys the thermodynamic structures underlying these models, focusing on the formulation of free energy functionals, dissipation mechanisms, and variational principles. We compare structural properties, emphasizing how these models encode conservation laws and energy dissipation. A central theme is the translation of these thermodynamic structures into numerical practice by providing representative discretisation strategies that aim to preserve mass conservation, stability, and energy decay. Particular attention is paid to the trade-offs between accuracy, efficiency, and structure preservation in large-scale simulations.

In this paper, we develop and numerically implement a novel approach for solving the inverse source problem of the acoustic wave equation in three dimensions. By injecting a small high-contrast droplet into the medium, we exploit the resulting wave field perturbation measured at a single external point over time. The method enables stable source reconstructions where conventional approaches fail due to ill-posedness, with potential applications in medical imaging and non-destructive testing. Key contributions include: 1. Implementation of a theoretically justified asymptotic expansion, from [33], using the eigensystem of the Newtonian operator, with error analysis for the spectral truncation. 2. Novel numerical schemes for solving the time-domain Lippmann-Schwinger equation and reconstructing the source via Riesz basis expansions and mollification-based numerical differentiations. 3. Reconstruction requiring only single-point measurements, overcoming traditional spatial data limitations. 4. 3D numerical experiments demonstrating accurate source recovery under noise (SNR of the order $1/a$), with error analysis for the droplet size (of the order $a$) and the number of spectral modes $N$.

Several complexity measures such as degree, sparsity and multiplicative index for cryptographic functions including the Diffie-Hellman mapping and the discrete logarithm in a finite field have been studied in the literature. In 2022, Reis and Wang introduced another complexity measure, the additive index, of a self-mapping of a finite field. In this paper, under certain conditions, we determine lower bounds on the additive index of the univariate Diffie-Hellman mapping and a self-mapping of $mathbb{F}_q$ which can be identified with the discrete logarithm in a finite field.

We study an elastic Calderon-type inverse problem: recover the mass density $ρ(x)$ in a bounded domain $Ωsubsetmathbb{R}^3$ from the Neumann-to-Dirichlet map associated with the isotropic Lamé system $mathcal{L}_{λ,μ}u+ω^2ρ(x)u=0$. We introduce a constructive strategy that embeds a subwavelength periodic array of resonant high-density (hard) inclusions to create an effective medium with a uniform negative density shift. Specifically, we place a periodic cluster of inclusions of size $a$ and density $ρ_1asymp a^{-2}$ strictly inside $Ω$. For frequencies $ω$ tuned to an eigenvalue of the elastic Newton (Kelvin) operator of a single inclusion, we show that as $ato0$ and the number of inclusions $Mtoinfty$, the Neumann-to-Dirichlet map $Λ_D$ converges to an effective map $Λ_{mathcal{P}}$ corresponding to a background density shift $-mathcal{P}^2$, with the operator norm estimate $|Λ_D-Λ_{mathcal{P}}|le Ca^αmathcal{P}^6$ for some $α>0$ determined by the geometric scaling. Around this negative background we derive a first-order linearization of $Λ_{mathcal{P}}$ in terms of $ρ$ and the Newton volume potential for the shifted Lamé operator. Testing the linearized relation with complex geometric optics solutions yields an explicit reconstruction formula for the Fourier transform of $ρ$, and hence a global density recovery scheme. The results provide a metamaterial-inspired analytic framework for inverse coefficient problems in linear elasticity and a concrete paradigm for leveraging nanoscale resonators in reconstruction algorithms.

We consider the Lamé transmission problem in $mathbb{R}^3$ with a bounded isotropic elastic inclusion in a high-contrast setting, where the interior-to-exterior Lamé moduli and densities scale like $1/τ$ as $τto0$. We study the scattering resonances of the associated self-adjoint Hamiltonian, defined as the poles of the meromorphic continuation of its resolvent. We obtain a sharp asymptotic description of resonances near the real axis as $τto0$. Near each nonzero Neumann eigenvalue of the interior Lamé operator there is a cluster of resonances lying just below it in the complex plane; in this wavelength-scale regime the imaginary parts are of order $τ$ with non-vanishing leading coefficients. In addition, near zero (a subwavelength regime), we identify resonances with real parts of order $sqrtτ$ and prove a lifetime dichotomy: their imaginary parts are of order $τ$ generically, but of order $τ^2$ for an explicit admissible set $mathcal E$. This yields a classification of long-lived elastic resonances in the high-contrast limit. We also establish resolvent asymptotics for both fixed-size resonators and microresonators. We derive explicit expansions with a finite-rank leading term and quantitative remainder bounds, valid near both wavelength-scale and subwavelength resonances. For microresonators, at the wavelength scale the dominant contribution is an anisotropic elastic point scatterer. Near the zero eigenvalue, the leading-order behaviour is of monopole or dipole type, and we give a rigorous criterion distinguishing the two cases.

We consider time-harmonic electromagnetic scattering by a cluster of hybrid dielectric-plasmonic dimers in $mathbb{R}^3$. Each dimer consists of a high-contrast dielectric nanoparticle and a moderately contrasting plasmonic nanoparticle separated by a subwavelength distance. The cluster is assumed to contain many such dimers whose size $a$ is small compared to the wavelength, with intra-dimer and inter-dimer distances scaling like $a^{t_1}$ and $a^{t_2}$, and the frequency is tuned near suitable electric and magnetic resonances of the associated Newtonian and magnetization operators on the reference shapes. Under these geometric, contrast and spectral assumptions, we derive a Foldy--Lax type approximation for the Maxwell system. We show that the scattered field and its far field admit asymptotic expansions in terms of four moments attached to each dimer, which solve an explicit finite-dimensional linear system. We prove invertibility of this system under quantitative smallness conditions on the contrast and the dimer density, and we obtain error estimates uniform in the number of dimers. By extracting the dominant components, we further show that each hybrid dimer behaves, at leading order, as a co-located electric and magnetic dipole driven by the local fields, and we identify the corresponding $6times 6$ polarizability matrix. This provides a discrete model for clusters of hybrid dimers that is suitable for fast forward simulations, inverse schemes, and as input for effective-medium descriptions. In particular, it suggests parameter regimes where clusters of hybrid dimers can generate (double) negative effective permittivity and permeability and bi-anisotropic constitutive laws and eventually hyperbolic media.

We aim to solve a topology optimization problem where the distribution of material in the design domain is represented by a density function. To obtain candidates for local minima, we want to solve the first order optimality system via Newton's method. This requires the initial guess to be sufficiently close to the a priori unknown solution. Introducing a stepsize rule often allows for less restrictions on the initial guess while still preserving convergence. In topology optimization one typically encounters nonconvex problems where this approach might fail. We therefore opt for a homotopy (continuation) approach which is based on solving a sequence of parametrized problems to approach the solution of the original problem. In the density based framework the values of the design variable are constrained by 0 from below and 1 from above. Coupling the homotopy method with a barrier strategy enforces these constraints to be satisified. The numerical results for a PDE-constrained compliance minimization problem demonstrate that this combined approach maintains feasibility of the density function and converges to a (candidate for a) locally optimal design without a priori knowledge of the solution.

In this paper we investigate the influence of the discretization of PDE constraints on shape and topological derivatives. To this end, we study a tracking-type functional and a two-material Poisson problem in one spatial dimension. We consider the discretization by a standard method and an enriched method. In the standard method we use splines of degree $p$ such that we can control the smoothness of the basis functions easily, but do not take any interface location into consideration. This includes for p=1 the usual hat basis functions. In the enriched method we additionally capture the interface locations in the ansatz space by enrichment functions. For both discretization methods shape and topological sensitivity analysis is performed. It turns out that the regularity of the shape derivative depends on the regularity of the basis functions. Furthermore, for point-wise convergence of the shape derivative the interface has to be considered in the ansatz space. For the topological derivative we show that only the enriched method converges.