Incorporating the field border effect to reduce the predicted uncertainty of pollen dispersal model in Asia

The presence of the field border (FB), such as roadways or unplanted areas, between two fields is common in Asian farming system. This study evaluated the effect of the FB on the cross-pollination (CP) and predicted the CP rate in the field considering and not considering FB. Three experiments including 0, 6.75, and 7.5 m width of the FB respectively were conducted to investigate the effect of distance and the FB on the CP rate. The dispersal models combined kernel and observation model by calculating the parameter of observation model from the output of kernel. These models were employed to predict the CP rate at different distances. The Bayesian method was used to estimate parameters and provided a good prediction with uncertainty. The highest average CP rates in the field with and without FB were 74.29% and 36.12%, respectively. It was found that two dispersal models with the FB effect displayed a higher ability to predict average CP rates. The correlation coefficients between actual CP rates and CP rates predicted by the dispersal model combined zero-inflated Poisson observation model with compound exponential kernel and modified Cauchy kernel were 0.834 and 0.833, respectively. Furthermore, the predictive uncertainty was reducing using the dispersal models with the FB effect.

Spheroidal Domains and Geometric Analysis in Euclidean Space

Clifford's geometric algebra has enjoyed phenomenal development over the last 60 years by mathematicians, theoretical physicists, engineers, and computer scientists in robotics, artificial intelligence and data analysis, introducing a myriad of different and often confusing notations. The geometric algebra of Euclidean 3-space, the natural generalization of both the well-known Gibbs-Heaviside vector algebra and Hamilton's quaternions, is used here to study spheroidal domains, spheroidal-graphic projections, the Laplace equation, and its Lie algebra of symmetries. The Cauchy-Kovalevska extension and the Cauchy kernel function are treated in a unified way. The concept of a quasi-monogenic family of functions is introduced and studied.

REGULARIZATION OF SOME PERTURBED INTEGRAL OPERATORS IN THE SPACES Lp

The article presents some generalizations and refinements of the article [1]: examples of integral (non-compact) operators with point wise singularities which are admissible perturbations of the Noetherian operators are constructed; a connection between the regularizes of the perturbed and original operators is established and the equality between the indices of the perturbed and the original operators is proved. The presented results are based on the formulas obtained in this paper for the composition of an operator with the Cauchy kernel and the operators with point wise singularities.

An approximate solution of one singular integro-differential equation using the method of orthogonal polynomials

Two computational schemes for solving boundary value problems for a singular integro-differential equation, which describes the scattering of H-polarized electromagnetic waves by a screen with a curved boundary, are constructed. This equation contains three types of integrals: a singular integral with the Cauchy kernel, integrals with a logarithmic singularity and with the Helder type kernel. The integrands, along with the solution function, contain its first derivative. The proposed schemes for an approximate solution of the problem are based on the representation of the solution function in the form of a linear combination of the Chebyshev orthogonal polynomials and spectral relations that allows to obtain simple analytical expressions for the singular component of the equation. The expansion coefficients of the solution in terms of the Chebyshev polynomial basis are calculated by solving a system of linear algebraic equations. The results of numerical experiments show that on a grid of 20 –30 points, the error of the approximate solution reaches the minimum limit due to the error in representing real floating-point numbers.