Design and fabrication of highly dispersive semiconductor double-chirped mirrors

We report on the development of semiconductor double-chirped mirrors with the group delay dispersion of ?3,800 ± 100 fs² in the wavelength range between 1,058 ÷ 1,064 nm and reflectivity of 99.1 %. The simplified plane-wave reflection transfer method was used to design the mirror multilayer stack. The mirror contains an epitaxial AlAs/GaAs structure topped with a SiNx antireflective layer.     

Watched guards in art galleries

The art gallery problem asks how many guards are sufficient to see every point of the interior of a polygon. A set of guards is called watched if each guard itself is seen by at least one of its colleagues. In 1994, Hernández-Pe?alver wrote that watched guards always suffice to guard any polygon with n vertices. However in 2001, Michael and Pinciu, and independently Żyliński, presented a class of polygons that required more than watched guards – which disproved the Hernández-Pe?alver’s result – and they established a new tight bound for watched guards: . Combinatorial bounds for watched guards in orthogonal polygons were independently given by Hernández-Pe?alver , and by Michael and Pinciu, who proved the -bound to be tight. In this paper, tight bounds for polygons of miscellaneous shapes are presented: watched guards for monotone and spiral polygons, and vertex watched guards for star-shaped polygons.     

Polynomial approximation of rational Bézier curves with constraints

We present an efficient method to solve the problem of the constrained least squares approximation of the rational Bézier curve by the polynomial Bézier curve. The presented algorithm uses the dual constrained Bernstein basis polynomials, and exploits their recursive properties. Examples are given, showing the effectiveness of the algorithm.     

Harmonic Functions on Metric Measure Spaces: Convergence and Compactness

We investigate the properties of harmonic functions defined on a metric measure space. Especially, sequences of harmonic functions are examined, i.e. their convergence and compactness. Moreover, Harnack‘s inequality is shown.     

On Nonhomogeneous Slip Boundary Conditions for 2D Incompressible Exterior Fluid Flows

The paper examines the issue of existence of solutions to the steady Navier-Stokes equations in an exterior domain in ℝ². The system is studied with nonhomogeneous slip boundary conditions. The main results proves the existence of weak solutions for arbitrary data.      

Pontryagin space structure in reproducing kernel Hilbert spaces over *-semigroups

The geometry of spaces with indefinite inner product, known also as Krein spaces, is a basic tool for developing Operator Theory therein. In the present paper we establish a link between this geometry and the algebraic theory of *-semigroups. It goes via the positive definite functions and related to them reproducing kernel Hilbert spaces. Our concern is in describing properties of elements of the semigroup which determine shift operators which serve as Pontryagin fundamental symmetries.     

On functions convex in the direction of the real axis with a fixed second coefficient

In the paper we consider the class Γ of analytic and univalent functions f in the unit disk Δ, normalized by f(0) = f′(0) − 1 = 0, having real coefficients and such that f(Δ) is convex in the direction of the real axis. We are especially interested in some subclasses of Γ. The most important of them is Γ(c) consisting of those functions which have the second coefficients of the Taylor expansion fixed and equal to c. We obtain the Koebe set for this class as well as for the classes Γ⁺(c) and Γ⁻(c) of functions which are in some sense convex in the direction of positive and negative axes respectively.     

Optimal sampling design for global approximation of jump diffusion stochastic differential equations

The paper deals with strong global approximation of stochastic differential equations (SDEs) driven by two independent processes: a nonhomogeneous Poisson process and a Wiener process. We assume that the jump and diffusion coefficients of the underlying SDE satisfy jump commutativity condition (see Chapter 6.3 in [21]). We establish the exact convergence rate of minimal errors that can be achieved by arbitrary algorithms based on a finite number of observations of the Poisson and Wiener processes. We consider classes of methods that use equidistant or nonequidistant sampling of the Poisson and Wiener processes. We provide a construction of optimal methods, based on the classical Milstein scheme, which asymptotically attain the established minimal errors. The analysis implies that methods based on nonequidistant mesh are more efficient, with respect to asymptotic constants, than those based on the equidistant mesh.     

Chasing robbers on random graphs: Zigzag theorem

In this paper, we study the vertex pursuit game of Cops and Robbers where cops try to capture a robber on the vertices of the graph. The minimum number of cops required to win on a given graph G is the cop number of G. We present asymptotic results for the game of Cops and Robber played on a random graph G(n,p) for a wide range of p = p(n). It has been shown that the cop number as a function of an average degree forms an intriguing zigzag shape. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010