The effects of a thin film dopant precursor on the structure and properties of femtosecond-laser irradiated silicon

Femtosecond (fs) laser irradiation of a silicon substrate coated with a thin film is a flexible approach to producing metastable alloys with unique properties, including near-unity sub-band gap absorptance extending into the infrared. However, dopant incorporation from a thin film during fs-laser irradiation is not well understood. We study the thin film femtosecond-laser doping process through optical and structural characterization of silicon fs-laser doped using a selenium thin film, and compare the resulting microstructure and dopant distribution to fs-laser doping with sulfur from a gaseous precursor. We show that a thin film dopant precursor significantly changes the laser-material interactions, modifying both the surface structuring and dopant incorporation processes and in turn affecting p–n diode behavior.     

A stochastic approach to the harmonic map heat flow on manifolds with time-dependent Riemannian metric

We first prove stochastic representation formulae for space–time harmonic mappings defined on manifolds with evolving Riemannian metric. We then apply these formulae to derive Liouville type theorems under appropriate curvature conditions. Space–time harmonic mappings which are defined globally in time correspond to ancient solutions to the harmonic map heat flow. As corollaries, we establish triviality of such ancient solutions in a variety of different situations.     

A Strong Maximum Principle for parabolic equations with the p-Laplacian

We prove the Strong Maximum Principle (SMP) under suitable assumptions for a class of quasilinear parabolic problems with the p  -Laplacian, p>1$p>1$, on bounded cylindrical domains of R^N+1${\mathbb{R}}^{N+1}$,

_∂tu−_Δpu−λ|u|^p−2u≥0,${\partial }_{t}u-{\mathrm{\Delta }}_{p}u-\lambda {|u|}^{p-2}u\ge 0,$

with nonnegative initial–boundary conditions and λ≤0$\lambda \le 0$, and we give some counterexamples to the SMP if some of our assumptions are violated. We show that the Hopf Maximum Principle holds for 1<p<2$1<p<2$, and give a counterexample to it for p>2$p>2$. Also the Weak Maximum Principle for λ≤_λ1$\lambda \le {\lambda }_1$ is established.     

Quasi-static Maxwell's equations with a dissipative non-linear boundary condition: Full discretization

We study a time dependent eddy current equation for the magnetic field H accompanied with a nonlinear boundary condition, which is a generalization of the classical Silver–Müller condition for a non-perfect conductor. More exactly, the relation between the normal components of electric (E) and magnetic (H  ) fields obeys the following power law (linearized for small and large values) ν×E=ν×(|H×ν|^α−1H×ν)$\mathit{\nu }\times \mathit{E}=\mathit{\nu }\times ({|\mathit{H}\times \mathit{\nu }|}^{\alpha -1}\mathit{H}\times \mathit{\nu })$ for some α∈(0,1]$\alpha \in (0,1]$. We design a linear fully discrete approximation scheme to solve this nonlinear problem. The convergence of the approximations to a weak solution is proved, error estimates describing the dependence of the error on discretization parameters are derived as well. The efficiency of the proposed method is supported by numerical experiments.     

Inverse Sturm-Liouville problems with homogeneous delay

We study the inverse Sturm-Liouville problem with a delay $$- y''(x) + q(x)y(\alpha x), q \in L_2 [0,\pi ], \alpha \in (0,1],$$ and the boundary conditions y(0) = y(π) = 0.     

Variational analysis of the coupling between a geometrically exact Cosserat rod and an elastic continuum

We formulate the static mechanical coupling of a geometrically exact Cosserat rod to a nonlinearly elastic continuum. In this setting, appropriate coupling conditions have to connect a one-dimensional model with director variables to a three-dimensional model without directors. Two alternative coupling conditions are proposed, which correspond to two different configuration trace spaces. For both, we show existence of solutions of the coupled problems, using the direct method of the calculus of variations. From the first-order optimality conditions, we also derive the corresponding conditions for the dual variables. These are then interpreted in mechanical terms.     

On the Kluvánek construction of the Lebesgue integral with respect to a vector measure

The Kluvánek construction of the Lebesgue integral is extended in two directions. First, instead of a compact interval [a, b] in the real line an abstract non-empty set X is considered, instead of the ring generated by subintervals of [a, b] an arbitrary ring A of subsets of X. Secondly, instead of the length of intervals (λ([c, d]) = d?c) any vector measure λ: A→V is considered, where V is a Riesz space.     

On the guaranteed convergence of new two-point root-finding methods for polynomial zeros

We presented new two-point methods for the simultaneous approximation of all n simple (real or complex) zeros of a polynomial of degree n. We proved that the R-order of convergence of the total-step version is three. Moreover, computationally verifiable initial conditions that guarantee the convergence of one of the proposed methods are stated. These conditions are stated in the spirit of Smale’s point estimation theory; they depend only on available data, the polynomial coefficients, polynomial degree n and initial approximations \(x_{1}^{(0)},\ldots ,x_{n}^{(0)}\) , which is of practical importance. Using the Gauss-Seidel approach we state the corresponding single-step version and consequently its prove that the lower bound of its R-order of convergence is at least 2 + y _n > 3, where y _n ∈ (1, 2) is the unique positive root of the equation y ⁿ ? y ? 2 = 0. Two numerical examples are given to demonstrate the convergence behavior of the considered methods, including global convergence.     

On the Parameterization of Rational Ringed Surfaces and Rational Canal Surfaces

Ringed surfaces and canal surfaces are surfaces that contain a one-parameter family of circles. Ringed surfaces can be described by a radius function, a directrix curve and vector field along the directrix curve, which specifies the normals of the planes that contain the circles. In particular, the class of ringed surfaces includes canal surfaces, which can be obtained as the envelopes of a one-parameter family of spheres. Consequently, canal surfaces can be described by a spine curve and a radius function. We present parameterization algorithms for rational ringed surfaces and rational canal surfaces. It is shown that these algorithms may generate any rational parameterization of a ringed (or canal) surface with the property that one family of parameter lines consists of circles. These algorithms are used to obtain rational parameterizations for Darboux cyclides and to construct blends between pairs of canal surfaces and pairs of ringed surfaces.