Exact Boundary Conditions at an Artificial Boundary for Partial Differential Equations in Cylinders

The numerical solution of partial differential equations in unbounded domains requires a finite computational domain. Often one obtains a finite domain by introducing an artificial boundary and imposing boundary conditions there. This paper derives exact boundary conditions at an artificial boundary for partial differential equations in cylinders. An abstract theory is developed to analyze the general linear problem. Solvability requirements and estimates of the solution of the resulting finite problem are obtained by use of the notions of exponential and ordinary dichotomies. Useful representations of the boundary conditions are derived using separation of variables for problems with constant tails. The constant tail results are extended to problems whose coefficients obtain limits at infinity by use of an abstract perturbation theory. The perturbation theory approach is also applied to a class of nonlinear problems. General asymptotic formulas for the boundary conditions are derived and displayed in detail

The Numerical Calculation of Traveling Wave Solutions of Nonlinear Parabolic Equations

Traveling wave solutions have been studied for a variety of nonlinear parabolic problems. In the initial value approach to such problems the initial data at infinity determines the wave that propagates. The numerical simulation of such problems is thus quite difficult. If the domain is replaced by a finite one, to facilitate numerical computations, then appropriate boundary conditions on the "artificial" boundaries must depend upon the initial data in the discarded region. In this work we derive such boundary conditions, based on the Laplace transform of the linearized problems at ±∞, and illustrate their utility by presenting a numerical solution of Fisher’s equation which has been proposed as a model in genetics

Using Avida to test the effects of natural selection on phylogenetic reconstruction methods

Phylogenetic trees group organisms by their ancestral relationships. There are a number of distinct algorithms used to reconstruct these trees from molecular sequence data, but different methods sometimes give conflicting results. Since there are few precisely known phylogenies, simulations are typically used to test the quality of reconstruction algorithms. These simulations randomly evolve strings of symbols to produce a tree, and then the algorithms are run with the tree leaves as inputs. Here we use Avida to test two widely used reconstruction methods, which gives us the chance to observe the effect of natural selection on tree reconstruction. We find that if the organisms undergo natural selection between branch points, the methods will be successful even on very large time scales. However, these algorithms often falter when selection is absent

Hyperboloidal evolution of test fields in three spatial dimensions

We present the numerical implementation of a clean solution to the outer boundary and radiation extraction problems within the 3+1 formalism for hyperbolic partial differential equations on a given background. Our approach is based on compactification at null infinity in hyperboloidal scri fixing coordinates. We report numerical tests for the particular example of a scalar wave equation on Minkowski and Schwarzschild backgrounds. We address issues related to the implementation of the hyperboloidal approach for the Einstein equations, such as nonlinear source functions, matching, and evaluation of formally singular terms at null infinity.Comment: 10 pages, 8 figure

Multidomain Spectral Method for the Helically Reduced Wave Equation

We consider the 2+1 and 3+1 scalar wave equations reduced via a helical Killing field, respectively referred to as the 2-dimensional and 3-dimensional helically reduced wave equation (HRWE). The HRWE serves as the fundamental model for the mixed-type PDE arising in the periodic standing wave (PSW) approximation to binary inspiral. We present a method for solving the equation based on domain decomposition and spectral approximation. Beyond describing such a numerical method for solving strictly linear HRWE, we also present results for a nonlinear scalar model of binary inspiral. The PSW approximation has already been theoretically and numerically studied in the context of the post-Minkowskian gravitational field, with numerical simulations carried out via the "eigenspectral method." Despite its name, the eigenspectral technique does feature a finite-difference component, and is lower-order accurate. We intend to apply the numerical method described here to the theoretically well-developed post-Minkowski PSW formalism with the twin goals of spectral accuracy and the coordinate flexibility afforded by global spectral interpolation.Comment: 57 pages, 11 figures, uses elsart.cls. Final version includes revisions based on referee reports and has two extra figure

Expression of toll-like receptors in non-endemic nasopharyngeal carcinoma

BackgroundNasopharyngeal carcinoma (NPC) is a malignant disease with an enigmatic etiology. NPC associates with Epstein-Barr virus (EBV) and human papillomaviruses (HPVs), while immunological factors also play a role in carcinogenesis. Toll-like receptors (TLRs) are pattern recognition receptors that participate in the immunological defence against pathogens, but their functions are also linked to cancer.MethodsIn our whole population-based study, we retrieved 150 Finnish NPC cases and studied their tumour samples for TLR1, TLR2, TLR4, TLR5, TLR7, and TLR9 expressions by immunohistochemistry, and for the presence of EBV and high-risk HPVs with EBV RNA and HPV E6/E7 mRNA in situ hybridizations. In addition, we analyzed the TLR expression patterns according to age, tumour histology, EBV/HPV status, and outcome.ResultsWe found that all TLRs studied were highly expressed in NPC. Viral status of the tumours varied, and 62% of them were EBV-positive, 14% HPV-positive, and 24% virus-negative. The tumours with strong TLR2(nucl) or TLR5 expression were mostly virus-negative or HPV-positive keratinizing squamous cell carcinomas, and the patients with these tumours were significantly older than those with mild or negative TLR2(nucl)/TLR5 expression. In Kaplan-Meier analysis, the patients with strong TLR5 expression had worse survival compared to the patients with negative or mild TLR5 expression, but the results were linked to other patient and tumour characteristics. In multivariable-adjusted Cox regression analysis, the patients with positive TLR7 tumour expression had better overall survival than those with no TLR7 expression. The 5-year overall survival rates according to TLR7 expression were 66% (mild), 52% (moderate or strong), and 22% (negative).ConclusionsTLRs are highly expressed in non-endemic NPC. Intensity of TLR2 and TLR5 expressions correlate with viral status, and TLR7 seems to be an independent prognostic factor of non-endemic NPC.</p