Finite element approximation of the Neumann eigenvalue problem in domains with multiple cracks

^** Email: belhach{at}poncelet.univ-metz.fr^*** Email: bucur{at}math.univ-metz.fr^**** Email: jmse{at}math.univ-metz.fr We study the Neumann–Laplacian eigenvalue problem in domainswith multiple cracks. We derive a mixed variational formulationwhich holds on the whole geometric domain (including the cracks)and implements efficient finite-element discretizations forthe computation of eigenvalues. Optimal error estimates aregiven and several numerical examples are presented, confirmingthe efficiency of the method. As applications, we numericallyinvestigate the behaviour of the low eigenvalues in domainswith a large number of cracks.     

Real-space observation of spectral degeneracy breaking in a waveguide-coupled disk microresonator

We report on the real-space observation of resonant frequency splitting in a high-Q waveguide-coupled silicon-on-insulator microdisk resonator. Phase sensitive near-field analysis reveals the stationary nature of the two resonant states, and spectral investigations clearly show their orthogonality. These measurements emphasize the role of the coupling waveguide in this splitting phenomenon. The symmetry of the two stationary whispering gallery modes is clearly observed and is found to follow the axial symmetry of the waveguide-coupled microdisk as it has been reported by earlier theoretical predictions.     

Maximal surfaces and the universal Teichmüller space

We show that any element of the universal Teichmüller space is realized by a unique minimal Lagrangian diffeomorphism from the hyperbolic plane to itself. The proof uses maximal surfaces in the 3-dimensional anti-de Sitter space. We show that, in AdS _n+1, any subset E of the boundary at infinity which is the boundary at infinity of a space-like hypersurface bounds a maximal space-like hypersurface. In AdS₃, if E is the graph of a quasi-symmetric homeomorphism, then this maximal surface is unique, and it has negative sectional curvature. As a by-product, we find a simple characterization of quasi-symmetric homeomorphisms of the circle in terms of 3-dimensional projective geometry.     

A general ultrasound-assisted access to room-temperature ionic liquids

The replacement of common organic solvents by room-temperature ionic liquids (RTILs) is a topical subject in both academia and industry. In the last decades, the number of applications for RTILs has followed an exponential curve and spilled over the boundaries of chemistry. Still, one of the main drawbacks of these compounds is their difficult access. The present ultrasound-assisted method affords a general and easy access to a large variety of room-temperature ionic liquids.     

Nonparametric estimation of the local Hurst function of multifractional Gaussian processes

A new nonparametric estimator of the local Hurst function of a multifractional Gaussian process based on the increment ratio (IR) statistic is defined. In a general frame, the point-wise and uniform weak and strong consistency and a multidimensional central limit theorem for this estimator are established. Similar results are obtained for a refinement of the generalized quadratic variations (QV) estimator. The example of the multifractional Brownian motion is studied in detail. A simulation study is included showing that the IR-estimator is more accurate than the QV-estimator.     

Linearly repetitive Delone sets are rectifiable

We show that every linearly repetitive Delone set in the Euclidean d  -space R^d${\mathbb{R}}^d$, with d?2$d?2$, is equivalent, up to a bi-Lipschitz homeomorphism, to the integer lattice Z^d${\mathbb{Z}}^d$. In the particular case when the Delone set X   in R^d${\mathbb{R}}^d$ comes from a primitive substitution tiling of R^d${\mathbb{R}}^d$, we give a condition on the eigenvalues of the substitution matrix which ensures the existence of a homeomorphism with bounded displacement from X   to the lattice βZ^d$\beta {\mathbb{Z}}^d$ for some positive β. This condition includes primitive Pisot substitution tilings but also concerns a much broader set of substitution tilings.     

Carrier-injection-based electro-optic modulator on silicon-on-insulator with a heterogeneously integrated III-V microdisk cavity

A compact electro-optic modulator on silicon-on-insulator is presented. The structure consists of a III-V microdisk cavity heterogeneously integrated on a silicon-on-insulator wire waveguide. By modulating the loss of the active layer included in the cavity through carrier injection, the power of the transmitted light at the resonant wavelength is modulated; approximately 10 dB extinction ratio and 2.73 Gbps dynamic operation are demonstrated without using any special driving techniques.     

Dynamic mechanical relaxation and thermal creep of high-entropy La30Ce30Ni10Al20Co10 bulk metallic glass

Spatial frequency shift(SFS) microscopy with evanescent wave illumination shows intriguing advantages, including large field of view(FOV), high speed, and good modularity. However, a missing band in the spatial frequency domain hampers the SFS superresolution microscopy from achieving resolution better than 3 folds of the Abbe diffraction limit. Here, we propose a novel tunable large-SFS microscopy, making the resolution improvement of a linear system no longer restricted by the detection numerical aperture(NA). The complete wide-range detection in the spatial frequency domain is realized by tuning the illumination spatial frequency actively and broadly through an angle modulation between the azimuthal propagating directions of two evanescent waves. The vertical spatial frequency is tuned via a sectional saturation effect, and the reconstructed depth information can be added to the lateral superresolution mask for 3D imaging. A lateral resolution of λ/9, and a vertical localization precision of ~λ/200(detection objective NA = 0.9) are realized with a gallium phosphide(GaP) waveguide. Its unlimited resolution enhancing capability is demonstrated by introducing a designed metamaterial chip with an unusual large refractive index. Besides the great resolution enhancement, this method shows better anti-noise capability than classical structured illumination microscopy without SFS tunability. This method is chip-compatible and can potentially provide a massproducible illumination chip module achieving the fast, large-FOV, and deep-subwavelength 3D nanoscopy.     

Pd–H and Ni–H phase diagrams using cluster variation method and Monte Carlo simulation

This work focuses on interstitial solid solutions of hydrogen in the face-centred cubic (fcc) host lattice of palladium and nickel, using a first-principles based approach. Cluster Variation Method (CVM) and Monte Carlo simulation algorithms were especially designed, allowing a coupled use of both techniques, to study hydrogen–vacancy interactions inside an fcc metallic host lattice. First-principles calculations provided the H–Vac interaction energies by structure inversion method. The phase diagrams and thermodynamic properties were computed using only theoretical inputs. The mechanisms leading to the formation of the miscibility gaps observed for both Pd–H and Ni–H systems and the hydrogen ordering on palladium interstitial lattice were reproduced without any empirical term.