Study on cyanobiphenyl nematic doped by silver nanoparticles

We study the effects of doped (1%wt and 2%wt) silver nanoparticles on material properties of nematic liquid crystal: 4-pentyl-4′-cyanobiphenyl. Using differential scanning calorimetry, electrical and dielectric measurements methods, we show that the doped NPs do not affect the nematic’s phase clearing point, lower the dielectric anisotropy, viscosity, switching-off time, and increase the threshold voltage and elasticity of the nematic. We report that the doped materials temperature behaviour of Frederick’s transition threshold voltage and switching-off time, deviates from the expected behaviour for pristine nematics. To explain this anomalous behaviour, we perform data analysis of the governing Frederick’s transition material parameters of studied samples. We show that the elastic parameter of doped samples is not following the predictions of Maier–Saupe theory, which is valid for conventional nematics. We report that the doped samples temperature behaviour of the elastic parameter follows the predictions of the Gelbart and Ben-Shaul theory.     

Mean Curvature,Threshold Dynamics,and Phase Field Theory on Finite Graphs

In the continuum, close connections exist between mean curvature flow, the Allen-Cahn (AC) partial differential equation, and the Merriman-Bence-Osher (MBO) threshold dynamics scheme. Graph analogues of these processes have recently seen a rise in popularity as relaxations of NP-complete combinatorial problems, which demands deeper theoretical underpinnings of the graph processes. The aim of this paper is to introduce these graph processes in the light of their continuum counterparts, provide some background, prove the first results connecting them, illustrate these processes with examples and identify open questions for future study. We derive a graph curvature from the graph cut function, the natural graph counterpart of total variation (perimeter). This derivation and the resulting curvature definition differ from those in earlier literature, where the continuum mean curvature is simply discretized, and bears many similarities to the continuum nonlocal curvature or nonlocal means formulation. This new graph curvature is not only relevant for graph MBO dynamics, but also appears in the variational formulation of a discrete time graph mean curvature flow. We prove estimates showing that the dynamics are trivial for both MBO and AC evolutions if the parameters (the time-step and diffuse interface scale, respectively) are sufficiently small (a phenomenon known as “freezing” or “pinning”) and also that the dynamics for MBO are nontrivial if the time step is large enough. These bounds are in terms of graph quantities such as the spectrum of the graph Laplacian and the graph curvature. Adapting a Lyapunov functional for the continuum MBO scheme to graphs, we prove that the graph MBO scheme converges to a stationary state in a finite number of iterations. Variations on this scheme have recently become popular in the literature as ways to minimize (continuum) nonlocal total variation.     

Bragg structure and the first spectral gap

Periodic media are routinely used in optical devices and, in particular, photonic crystals to create spectral gaps, prohibiting the propagation of waves with certain temporal frequencies. In one dimension, Bragg structures, also called quarter-wave stacks, are frequently used because they are relatively easy to manufacture and the spectrum exhibits large spectral gaps at explicitly computable frequencies. In this short work, we use variational methods to demonstrate that within an admissible class of pointwise-bounded, periodic media, the Bragg structure uniquely maximizes the first spectral gap-to-midgap ratio.     

Optimization of spectral functions of Dirichlet–Laplacian eigenvalues

We consider the shape optimization of spectral functions of Dirichlet–Laplacian eigenvalues over the set of star-shaped, symmetric, bounded planar regions with smooth boundary. The regions are represented using Fourier-cosine coefficients and the optimization problem is solved numerically using a quasi-Newton method. The method is applied to maximizing two particular nonsmooth spectral functions: the ratio of the nth to first eigenvalues and the ratio of the nth eigenvalue gap to first eigenvalue, both of which are generalizations of the Payne–Pólya–Weinberger ratio. The optimal values and attaining regions for n ? 13 are presented and interpreted as a study of the range of the Dirichlet–Laplacian eigenvalues. For both spectral functions and each n, the optimal attaining region has multiplicity two nth eigenvalue.     

Minimal Convex Combinations of Three Sequential Laplace-Dirichlet Eigenvalues

We study the shape optimization problem where the objective function is a convex combination of three sequential Laplace-Dirichlet eigenvalues. That is, for α≥0, β≥0, and α+β≤1, we consider $\inf\{ \alpha\lambda_{k}(\varOmega)+\beta\lambda _{k+1}(\varOmega)+(1-\alpha-\beta) \lambda_{k+2}(\varOmega)\colon\varOmega\mbox { open set in } \mathbb{R}^{2} \mbox{ and } |\varOmega|\leq1\} $ . Here λ _k (Ω) denotes the k-th Laplace-Dirichlet eigenvalue and |?| denotes the Lebesgue measure. For k=1,2, the minimal values and minimizers are computed explicitly when the set of admissible domains is restricted to the disjoint union of balls. For star-shaped domains, we show that for k=1 and α+2β≤1, the ball is a local minimum. For k=1,2, several properties of minimizers are studied computationally, including uniqueness, connectivity, symmetry, and eigenvalue multiplicity.