Angel Tocino

Runge-Kutta methods for numerical solution of stochastic differential equations

The way to obtain deterministic Runge-Kutta methods from Taylor approximations is generalized for stochastic differential equations, now by means of stochastic truncated expansions about a point for sufficiently smooth functions of an Ito process. A class of explicit Runge-Kutta schemes of second order in the weak sense for systems of stochastic differential equations with multiplicative noise is developed. Also two Runge-Kutta schemes of third order have been obtained for scalar equations with constant diffusion coefficients. Numerical examples that compare the proposed schemes to standard ones are presented.

Weak Second Order Conditions for Stochastic Runge--Kutta Methods

A general procedure to construct weak methods for the numerical solution of stochastic differential systems is presented. As in the deterministic case, the procedure consists of comparing the stochastic expansion of the approximation with the corresponding Taylor scheme. In this way the authors obtain the order conditions that a stochastic Runge--Kutta method must satisfy to have weak order two. Explicit examples of generalizations of the classical family of second order two-stage explicit Runge--Kutta methods are shown. Also numerical examples are presented.

Symplectic conditions for exponential fitting Runge-Kutta-Nyström methods

In this note, simplecticity conditions easy to handle for constructing symplectic Runge-Kutta-Nystrom methods fitted to trigonometric functions are given. These conditions generalize that of [1] when the frequencies tends to zero.

Modeling the Arabidopsis seed shape by a cardioid: Efficacy of the adjustment with a scale change with factor equal to the Golden Ratio and analysis of seed shape in ethylene mutants

A new model for the description of Arabidopsis seed shape based on the comparison of the outline of its longitudinal section with a transformed cardioid is presented. The transformation consists of scaling the horizontal axis by a factor equal to the Golden Ratio. The elongated cardioid approximates the shape of the Arabidopsis seed with more accuracy than other figures. The length to width ratio in wild-type Columbia Arabidopsis dry seeds is close to the Golden Ratio and decreases over the course of imbibition. Dry seeds of etr1-1 mutants presented a reduced length to width ratio. Application of the new model based on the cardioid allows for comparison of shape between wild-type and mutant genotypes, revealing other general alterations in the seeds in ethylene signaling pathway mutants (etr1-1).

Multiple stochastic integrals with Mathematica

In the construction of numerical methods for solving stochastic differential equations it becomes necessary to calculate the expectation of products of multiple stochastic integrals. Well-known recursive relationships between these multiple integrals make it possible to express any product of them as a linear combination of integrals of the same type. This article describes how, exploiting the symbolic character of Mathematica, main recursive properties and rules of Ito and Stratonovich multiple integrals can be implemented. From here, a routine that calculates the expectation of any polynomial in multiple stochastic integrals is obtained. In addition, some new relations between integrals, found with the aid of the program, are shown and proved.

Mean-square stability analysis of numerical schemes for stochastic differential systems

For ordinary differential systems, the study of A-stability for a numerical method reduces to the scalar case by means of a transformation that uncouples the linear test system as well as the difference system provided by the method. For stochastic differential equations (SDEs), mean-square stability (MS-stability) has been successfully proposed as the generalization of A-stability, and numerical MS-stability has been analyzed for one-dimensional equations. However, unlike the deterministic case, the extension of this analysis to multi-dimensional systems is not straightforward. In this paper we give necessary and sufficient conditions for the MS-stability of multi-dimensional systems with one Wiener noise. The criterion presented does not depend on any norm. Based on the Routh-Hurwitz theorem, we offer a particular criterion of MS-stability for two-dimensional systems in terms of their coefficients. In addition, a counterpart criterion of MS-stability is given for numerical schemes applied to multi-dimensional systems. The MS-stability behavior of a stochastic numerical method is determined by the comparison of its stability region with the stability region of the system. As an application, the numerical MS-stability of @q-methods applied to bi-dimensional systems is investigated.

New Itô--Taylor expansions

A new expression for weak truncated Ito-Taylor expansions of functionals of Ito processes is proposed. The new truncated expansion is expressed, as in the ordinary case, in terms of powers of the increments of the variables. A systematic procedure to obtain such expansions and general results in order to avoid some parts of the calculation are presented. As an application, expansions of order up to 6 are derived.

Two-step Milstein schemes for stochastic differential equations

A set of conditions on the parameters of stochastic linear two-step schemes for their order 1.0 mean-square convergence are established. Then two-step Milstein schemes are defined and necessary and sufficient conditions for their MS-stability are given. Regions of MS-stability are determined and plotted for Adams-Bashforth, Adams-Moulton and BDF Milstein schemes. Numerical experiments confirming the theoretical results are shown.

Seed shape in model legumes: Approximation by a cardioid reveals differences in ethylene insensitive mutants of Lotus japonicus and Medicago truncatula

Seed shape in the model legumes Lotus japonicus and Medicago truncatula is described. Based in previous work with Arabidopsis, the outline of the longitudinal sections of seeds is compared with a cardioid curve. L. japonicus seeds adjust well to an unmodified cardioid, whereas accurate adjustment in M. truncatula is obtained by the simple transformation of scaling the vertical axis by a factor equal to the Golden Ratio. Adjustments of seed shape measurements with simple geometrical forms are essential tools for the statistical analysis of variations in seed shape under different conditions or in mutants. The efficiency of the adjustment to a cardioid in the model plants suggests that seed morphology may be related to genome complexity. Seeds of ethylene insensitive mutants present differences in size and shape as well as altered responses to imbibition. The biological implication and meaning of these relationships are discussed.

Ethylene, free radicals and the transition between stable states in plant morphology

Treatment with hydrogen peroxide has notable effects in the morphology of the root apex in Arabidopsis seedlings. The result was described as consisting in two aspects: first, a reduction in curvature values in the root profile. Second, alterations in size and shape of the cells in the root cap. Cells of the root cap were smaller and had higher circularity index.1 The results of peroxide treatment were similar to alterations in the root apex of ethylene insensitive mutants and wild-type seedlings treated with ethylene inhibitors. This brings new evidence in favour of the association between ethylene and hydrogen peroxide signalling that was recently demonstrated in stomatal cells.2 Notable changes in morphology under peroxide treatment were previously reported3 in other biological systems. In the following paragraphs we make emphasis on the need of an accurate analysis of morphology. This aspect that has not received the attention required in biology, a discipline dominated by functional analysis. We suggest that the observed morphological characteristics in the root apex treated with peroxide may be the manifestation of global processes of adaptation in the organism. Alternative forms of roots grown in water or in peroxide are stable situations representing different global configurations that may have other (genomic, physiological) traits associated. Each form represents a different mode of adaptation to environmental change. The accurate description of morphology in organisms, with particular emphasis in model systems, and their variations under stress, is needed to identify and understand the basis of genomic organization and plasticity.