The impact of a natural time change on the convergence of the Crank-Nicolson scheme

We first analyse the effect of a square root transformation to the time variable on the convergence of the Crank-Nicolson scheme when applied to the solution of the heat equation with Dirac delta function initial conditions. In the original variables, the scheme is known to diverge as the time step is reduced with the ratio of the time step to space step held constant and the value of this ratio controls how fast the divergence occurs. After introducing the square root of time variable we prove that the numerical scheme for the transformed partial differential equation now always converges and that the ratio of the time step to space step controls the order of convergence, quadratic convergence being achieved for this ratio below a critical value. Numerical results indicate that the time change used with an appropriate value of this ratio also results in quadratic convergence for the calculation of the price, delta and gamma for standard European and American options without the need for Rannacher start-up steps

A Penalty Method for the Numerical Solution of Hamilton-Jacobi-Bellman (HJB) Equations in Finance

We present a simple and easy to implement method for the numerical solution of a rather general class of Hamilton-Jacobi-Bellman (HJB) equations. In many cases, the considered problems have only a viscosity solution, to which, fortunately, many intuitive (e.g. finite difference based) discretisations can be shown to converge. However, especially when using fully implicit time stepping schemes with their desirable stability properties, one is still faced with the considerable task of solving the resulting nonlinear discrete system. In this paper, we introduce a penalty method which approximates the nonlinear discrete system to first order in the penalty parameter, and we show that an iterative scheme can be used to solve the penalised discrete problem in finitely many steps. We include a number of examples from mathematical finance for which the described approach yields a rigorous numerical scheme and present numerical results.Comment: 18 Pages, 4 Figures. This updated version has a slightly more detailed introduction. In the current form, the paper will appear in SIAM Journal on Numerical Analysi

Coherence of a room-temperature CW GaAs/GaAlAs injection laser

The temporal coherence of a stripe-geometry double-heterojunction GaAs/GaAlAs laser operating CW at room temperature was determined. A heterodyne detection scheme was used involving the mixing of the laser field with a frequency-shifted and time-delayed image of itself in an interferometer. Because the laser device oscillated in several longitudinal modes, the autocorrelation function of its output exhibited resonances for specific time delays. The rate at which the amplitude of these resonances decreased with increasing time delays provided a measure of an apparent coherence length associated with individual longitudinal modes. The coherence length, so defined, was found to increase linearly with drive current in excess of threshold. This observation is interpreted as evidence that the intrinsic linewidth of a longitudinal mode is inversely proportional to the coherent optical power in that mode. Apparent coherence lengths were a few centimeters for a few milliwatts of total optical power emitted per facet. For a perfectly balanced interferometer, a sharp heterodyne beat signal was also observed when the laser device was operated considerably below threshold, i.e., in the LED mode

Invasive pulmonary aspergillosis complicating COVID-19 in the ICU - A case report.

It is not yet known, if critically ill COVID-19 patients are prone to fungal infections. We report a 69-year-old patient without typical risk factors for invasive pulmonary aspergillosis (IPA), who developed IPA two weeks after onset of symptoms. Our report shows that IPA may occur in critically ill COVID-19 patients

Evaluating Terrain for Harvesting Equipment Selection

A terrain evaluation model, utilizing a geographic information system, has been developed as a tool for planning large-scale industrial timber harvesting operations. The model combines terrain descriptions with machine operating criteria to produce maps delineating operable areas. The integration of the model with a harvest planning decision support system is discussed and an example is presented

Penalty Methods for the Solution of Discrete HJB Equations -- Continuous Control and Obstacle Problems

In this paper, we present a novel penalty approach for the numerical solution of continuously controlled HJB equations and HJB obstacle problems. Our results include estimates of the penalisation error for a class of penalty terms, and we show that variations of Newton's method can be used to obtain globally convergent iterative solvers for the penalised equations. Furthermore, we discuss under what conditions local quadratic convergence of the iterative solvers can be expected. We include numerical results demonstrating the competitiveness of our methods.Comment: 31 Pages, 7 Figure

What are the triggers of Asian visitor satisfaction and loyalty in the Korean heritage site?

Based on complexity theory, this study examines a configurational model that uses motivation antecedents and demographic configurations to explore the causal recipes that lead to high and low levels of Asian visitor satisfaction and loyalty. Data were collected from 183 Chinese and Japanese visitors to the Hanok heritage site in Seoul, South Korea. Asymmetrical modeling using a fuzzy-set qualitative comparative analysis was applied and a combination of desired behavioral outcomes identified. Hanok experience from the motivation configuration and gender from the demographic configuration appeared as necessary conditions to make visitors satisfied and loyal. Key tenets of complexity theory are supported by the study's findings

The Cinchona Primary Amine-Catalyzed Asymmetric Epoxidation and Hydroperoxidation of α,β-Unsaturated Carbonyl Compounds with Hydrogen Peroxide

Using cinchona alkaloid-derived primary amines as catalysts and aqueous hydrogen peroxide as the oxidant, we have developed highly enantioselective Weitz–Scheffer-type epoxidation and hydroperoxidation reactions of α,β-unsaturated carbonyl compounds (up to 99.5:0.5 er). In this article, we present our full studies on this family of reactions, employing acyclic enones, 5–15-membered cyclic enones, and α-branched enals as substrates. In addition to an expanded scope, synthetic applications of the products are presented. We also report detailed mechanistic investigations of the catalytic intermediates, structure–activity relationships of the cinchona amine catalyst, and rationalization of the absolute stereoselectivity by NMR spectroscopic studies and DFT calculations