Dynamics of Boolean networks: an exact solution

The dynamics of Boolean networks (BN) with quenched disorder and thermal noise is studied via the generating functional method. A general formulation, suitable for BN with any distribution of Boolean functions, is developed. It provides exact solutions and insight into the evolution of order parameters and properties of the stationary states, which are inaccessible via existing methodology. We identify cases where the commonly used annealed approximation is valid and others where it breaks down. Broader links between BN and general Boolean formulas are highlighted.

A method for systematic artifact selection decision making

Artifact selection decisions typically involve the selection of one from a number of possible/candidate options (decision alternatives). In order to support such decisions, it is important to identify and recognize relevant key issues of problem solving and decision making (Albers, 1996; Harris, 1998a, 1998b; Jacobs & Holten, 1995; Loch & Conger, 1996; Rumble, 1991; Sauter, 1999; Simon, 1986). Sauter classifies four problem solving/decision making styles: (1) left-brain style, (2) right-brain style, (3) accommodating, and (4) integrated (Sauter, 1999). The left-brain style employs analytical and quantitative techniques and relies on rational and logical reasoning. In an effort to achieve predictability and minimize uncertainty, problems are explicitly defined, solution methods are determined, orderly information searches are conducted, and analysis is increasingly refined. Left-brain style decision making works best when it is possible to predict/control, measure, and quantify all relevant variables, and when information is complete. In direct contrast, right-brain style decision making is based on intuitive techniques—it places more emphasis on feelings than facts. Accommodating decision makers use their non-dominant style when they realize that it will work best in a given situation. Lastly, integrated style decision makers are able to combine the left- and right-brain styles—they use analytical processes to filter information and intuition to contend with uncertainty and complexity.

Passive mode-locked lasing by injecting a carbon nanotube-solution in the core of an optical fiber

In this paper, we propose a saturable absorber (SA) device consisting on an in-fiber micro-slot inscribed by femtosecond laser micro fabrication, filled by a dispersion of Carbon Nanotubes (CNT). Due to the flexibility of the fabrication method, efficient and simple integration of the mode-locking device directly into the optical fiber is achieved. Furthermore, the fabrication process offers a high level of control over the dimensions and location of the micro-slots. We apply this fabrication flexibility to extend the interaction length between the CNT and the propagating optical field along the optical fiber, hence enhancing the nonlinearity of the device. Furthermore, the method allows the fabrication of devices that operate by either a direct field interaction (when the central peak of the propagating optical mode passes through the nonlinear media) or an evanescent field interaction (only a fraction of the optical mode interacts with the CNT). In this paper, several devices with different interaction lengths and interaction regimes are investigated. Self-starting passively modelocked laser operation with an enhanced nonlinear interaction is observed using CNT-based SAs in both interaction regimes. This method constitutes a simple and suitable approach to integrate the CNT into the optical system as well as enhancing the optical nonlinearity of CNT-based photonic devices.

SQUID magnetometry as a tool for following a clock reaction in solution

SQUID magnetometry, normally used to characterise the properties of solids, was used to follow a clock reaction in solution, namely the auto-catalytic oxidation of [Co(ii)EDTA] by HO, in real time and it was shown that, in combination with other methods (e.g., magnetic resonance proton relaxation studies and UV-vis absorption analysis), SQUID magnetometry can be a powerful method in elucidating and interpreting the time-profile of chemical reactions so as long as reactants, intermediates and products have suitably large differences in their respective magnetic susceptibilities. © 2009 The Royal Society of Chemistry.

On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach

We investigate the problem of determining the stationary temperature field on an inclusion from given Cauchy data on an accessible exterior boundary. On this accessible part the temperature (or the heat flux) is known, and, additionally, on a portion of this exterior boundary the heat flux (or temperature) is also given. We propose a direct boundary integral approach in combination with Tikhonov regularization for the stable determination of the temperature and flux on the inclusion. To determine these quantities on the inclusion, boundary integral equations are derived using Green’s functions, and properties of these equations are shown in an L2-setting. An effective way of discretizing these boundary integral equations based on the Nystr¨om method and trigonometric approximations, is outlined. Numerical examples are included, both with exact and noisy data, showing that accurate approximations can be obtained with small computational effort, and the accuracy is increasing with the length of the portion of the boundary where the additionally data is given.

An alternating boundary integral based method for a Cauchy problem for the Laplace equation in a quadrant

We study the Cauchy problem for the Laplace equation in a quadrant (quarter-plane) containing a bounded inclusion. Given the values of the solution and its derivative on the edges of the quadrant the solution is reconstructed on the boundary of the inclusion. This is achieved using an alternating iterative method where at each iteration step mixed boundary value problems are being solved. A numerical method is also proposed and investigated for the direct mixed problems reducing these to integral equations over the inclusion. Numerical examples verify the efficiency of the proposed scheme.

A variational method and approximations of a Cauchy problem for elliptic equations

A Cauchy problem for general elliptic second-order linear partial differential equations in which the Dirichlet data in H½(?1 ? ?3) is assumed available on a larger part of the boundary ? of the bounded domain O than the boundary portion ?1 on which the Neumann data is prescribed, is investigated using a conjugate gradient method. We obtain an approximation to the solution of the Cauchy problem by minimizing a certain discrete functional and interpolating using the finite diference or boundary element method. The minimization involves solving equations obtained by discretising mixed boundary value problems for the same operator and its adjoint. It is proved that the solution of the discretised optimization problem converges to the continuous one, as the mesh size tends to zero. Numerical results are presented and discussed.

An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions

We consider a Cauchy problem for the Laplace equation in a two-dimensional semi-infinite region with a bounded inclusion, i.e. the region is the intersection between a half-plane and the exterior of a bounded closed curve contained in the half-plane. The Cauchy data are given on the unbounded part of the boundary of the region and the aim is to construct the solution on the boundary of the inclusion. In 1989, Kozlov and Maz'ya [10] proposed an alternating iterative method for solving Cauchy problems for general strongly elliptic and formally self-adjoint systems in bounded domains. We extend their approach to our setting and in each iteration step mixed boundary value problems for the Laplace equation in the semi-infinite region are solved. Well-posedness of these mixed problems are investigated and convergence of the alternating procedure is examined. For the numerical implementation an efficient boundary integral equation method is proposed, based on the indirect variant of the boundary integral equation approach. The mixed problems are reduced to integral equations over the (bounded) boundary of the inclusion. Numerical examples are included showing the feasibility of the proposed method.

An integral equation method for a mixed initial boundary value problem for unsteady Stokes system in a doubly-connected domain

We present a novel numerical method for a mixed initial boundary value problem for the unsteady Stokes system in a planar doubly-connected domain. Using a Laguerre transformation the unsteady problem is reduced to a system of boundary value problems for the Stokes resolvent equations. Employing a modied potential approach we obtain a system of boundary integral equations with various singularities and we use a trigonometric quadrature method for their numerical solution. Numerical examples are presented showing that accurate approximations can be obtained with low computational cost.

Recovering boundary data in planar heat conduction using boundary integral equation method

We consider a Cauchy problem for the heat equation, where the temperature field is to be reconstructed from the temperature and heat flux given on a part of the boundary of the solution domain. We employ a Landweber type method proposed in [2], where a sequence of mixed well-posed problems are solved at each iteration step to obtain a stable approximation to the original Cauchy problem. We develop an efficient boundary integral equation method for the numerical solution of these mixed problems, based on the method of Rothe. Numerical examples are presented both with exact and noisy data, showing the efficiency and stability of the proposed procedure and approximations.

A method for systematic artifact selection decision making

Artifact selection decisions typically involve the selection of one from a number of possible/candidate options (decision alternatives). In order to support such decisions, it is important to identify and recognize relevant key issues of problem solving and decision making (Albers, 1996; Harris, 1998a, 1998b; Jacobs & Holten, 1995; Loch & Conger, 1996; Rumble, 1991; Sauter, 1999; Simon, 1986). Sauter classifies four problem solving/decision making styles: (1) left-brain style, (2) right-brain style, (3) accommodating, and (4) integrated (Sauter, 1999). The left-brain style employs analytical and quantitative techniques and relies on rational and logical reasoning. In an effort to achieve predictability and minimize uncertainty, problems are explicitly defined, solution methods are determined, orderly information searches are conducted, and analysis is increasingly refined. Left-brain style decision making works best when it is possible to predict/control, measure, and quantify all relevant variables, and when information is complete. In direct contrast, right-brain style decision making is based on intuitive techniques—it places more emphasis on feelings than facts. Accommodating decision makers use their non-dominant style when they realize that it will work best in a given situation. Lastly, integrated style decision makers are able to combine the left- and right-brain styles—they use analytical processes to filter information and intuition to contend with uncertainty and complexity.

Exact solution of the Bragg-diffraction problem in sillenites

A method for the exact solution of the Bragg-difrraction problem for a photorefractive grating in sillenite crystals based on Pauli matrices is proposed. For the two main optical configurations explicit analytical expressions are found for the diffraction efficiency and the polarization of the scattered wave. The exact solution is applied to a detailed analysis of a number of particular cases. For the known limiting cases there is agreement with the published results.

A variational method for identifying a spacewise-dependent heat source

The inverse problem of determining a spacewise-dependent heat source for the parabolic heat equation using the usual conditions of the direct problem and information from one supplementary temperature measurement at a given instant of time is studied. This spacewise-dependent temperature measurement ensures that this inverse problem has a unique solution, but the solution is unstable and hence the problem is ill-posed. We propose a variational conjugate gradient-type iterative algorithm for the stable reconstruction of the heat source based on a sequence of well-posed direct problems for the parabolic heat equation which are solved at each iteration step using the boundary element method. The instability is overcome by stopping the iterative procedure at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented which have the input measured data perturbed by increasing amounts of random noise. The numerical results show that the proposed procedure yields stable and accurate numerical approximations after only a few iterations.

An iterative method for the reconstruction of a stationary flow

In this article, an iterative algorithm based on the Landweber-Fridman method in combination with the boundary element method is developed for solving a Cauchy problem in linear hydrostatics Stokes flow of a slow viscous fluid. This is an iteration scheme where mixed well-posed problems for the stationary generalized Stokes system and its adjoint are solved in an alternating way. A convergence proof of this procedure is included and an efficient stopping criterion is employed. The numerical results confirm that the iterative method produces a convergent and stable numerical solution. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007

A variational conjugate gradient method for determining the fluid velocity of a slow viscous flow

The problem considered is that of determining the fluid velocity for linear hydrostatics Stokes flow of slow viscous fluids from measured velocity and fluid stress force on a part of the boundary of a bounded domain. A variational conjugate gradient iterative procedure is proposed based on solving a series of mixed well-posed boundary value problems for the Stokes operator and its adjoint. In order to stabilize the Cauchy problem, the iterations are ceased according to an optimal order discrepancy principle stopping criterion. Numerical results obtained using the boundary element method confirm that the procedure produces a convergent and stable numerical solution.