Spline collocation for singular integro-differential equations over (0.1)

Summary This paper analyses the convergence of spline collocation methods for singular integro-differential equations over the interval (0.1). As trial functions we utilize smooth polynomial splines the degree of which coincides with the order of the equation. Depending on the choice of collocation points we obtain sufficient and even necessary conditions for the convergence in sobolev norms. We give asymptotic error estimates and some numerical results.     

Continuous functions with impermeable graphs

We construct a Hölder continuous function on the unit interval which coincides in uncountably (in fact continuum) many points with every function of total variation smaller than 1 passing through the origin. We conclude that this function has impermeable graph—one of the key concepts introduced in this paper—and we present further examples of functions both with permeable and impermeable graphs. Moreover, we show that typical (in the sense of Baire category) continuous functions have permeable graphs. The first example function is subsequently used to construct an example of a continuous function on the plane which is intrinsically Lipschitz continuous on the complement of the graph of a Hölder continuous function with impermeable graph, but which is not Lipschitz continuous on the plane. As another main result, we construct a continuous function on the unit interval which coincides in a set of Hausdorff dimension 1 with every function of total variation smaller than 1 which passes through the origin.     

Effect of the aliphatic chain length on electrical double layer formation at the liquid/vacuum interface in the [C(n)mim][BF4] ionic liquid series

Neutral impact collision ion scattering spectroscopy (NICISS) was used to determine the surface structure of three ionic liquids, 1-hexyl-3-methylimidazolium [C(6)mim], 1-octyl-3-methylimidazolium [C(8)mim], 1-decyl-3-methylimidazolium [C(10)mim] tetrafluoroborates [BF(4)]. Concentration depth profiles of the elements in an ionic liquid (IL) homologous series with a common anion were obtained. We show that separation between the oppositely charged ions is seen for all three ionic liquids, resulting in an electrical double layer formation. The surface charge shifts from more positive to more negative with increasing aliphatic chain length.     

Multiplexed inkjet functionalization of silicon photonic biosensors

The transformative potential of silicon photonics for chip-scale biosensing is limited primarily by the inability to selectively functionalize and exploit the extraordinary density of integrated optical devices on this platform. Silicon biosensors, such as the microring resonator, can be routinely fabricated to occupy a footprint of less than 50 × 50 μm; however, chemically addressing individual devices has proven to be a significant challenge due to their small size and alignment requirements. Herein, we describe a non-contact piezoelectric (inkjet) method for the rapid and efficient printing of bioactive proteins, glycoproteins and neoglycoconjugates onto a high-density silicon microring resonator biosensor array. This approach demonstrates the scalable fabrication of multiplexed silicon photonic biosensors for lab-on-a-chip applications, and is further applicable to the functionalization of any semiconductor-based biosensor chip.     

Nonlinear responses from the interaction of two progressing waves at an interface

For scalar semilinear wave equations, we analyze the interaction of two (distorted) plane waves at an interface between media of different nonlinear properties. We show that new waves are generated from the nonlinear interactions, which might be responsible for the observed nonlinear effects in applications. Also, we show that the incident waves and the nonlinear responses determine the location of the interface and some information of the nonlinear properties of the media. In particular, for the case of a jump discontinuity at the interface, we can determine the magnitude of the jump.     

Travel Time Tomography

We survey some results on travel time tomography. The question is whether we can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as geometry problems, the boundary rigidity problem and the lens rigidity problem. The boundary rigidity problem is whether we can determine a Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function between boundary points. The lens rigidity problem problem is to determine a Riemannian metric of a Riemannian manifold with boundary by measuring for every point and direction of entrance of a geodesic the point of exit and direction of exit and its length. The linearization of these two problems is tensor tomography. The question is whether one can determine a symmetric two-tensor from its integrals along geodesics. We emphasize recent results on boundary and lens rigidity and in tensor tomography in the partial data case, with further applications.     

Relèvements des revêtements de courbes faiblement ramifiés

Let X be a smooth projective curve over a perfect field of characteristic p>0 and G a finite group of automorphism of X. Let ν(X,G) be the characteristic of the versal equivariant deformation ring R(X,G) of (X,G). When the ramification is weak (i.e., all second ramification groups are trivial), we prove that ν(X,G) ∈ {0,p} and we compute R(X,G).

---

Résumé Soit X une courbe projective lisse sur un corps parfait de caractéristique p>0 et G un groupe fini d'automorphismes de X. Nous considérons la caractéristique ν(X,G) de l'anneau versel R(X,G) de déformations équivariantes de (X,G). Dans le cas d'une ramification faible (où tous les seconds groupes de ramification sont triviaux), nous démontrons que ν(X,G) ∈ {0,p} et nous calculons R(X,G).