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 data analysis, introducing a myriad of different often confusing notations. The Euclidean 3-space, natural generalization both well-known Gibbs-Heaviside vector Hamilton's quaternions, is used he...

The movement of water and chemicals in soils is generally better described with multidimensional nonequilibrium models than with more commonly used one-dimensional and/or equilibrium models. This paper presents analytical solutions for non-equilibrium solute transport in semi-infinite porous media during steady unidirectional flow. The solutions can be used to model transport in porous media wh...

Quantitative bounds on rates of approximation by linear combinations of Heaviside plane waves are obtained for sufficiently differentiable functions f which vanish rapidly enough at infinity: for d odd and f ∈ Cd(R), with lower-order partials vanishing at infinity and dth-order partials vanishing as ‖x‖−(d+1+ε), ε > 0, on any domain ⊂ R with unit Lebesgue measure, the L2( )-error in approximati...

In this paper we show how to construct the Evans function for traveling wave solutions of integral neural field equations when the firing rate function is a Heaviside. This allows a discussion of wave stability and bifurcation as a function of system parameters, including the speed and strength of synaptic coupling and the speed of axonal signals. The theory is illustrated with the construction...

A truly Meshless Local Petrov-Galerkin (MLPG) method is developed for solving 3D elasto-static problems. Using the general MLPG concept, this method is derived through the local weak forms of the equilibrium equations, by using a test function, namely, the Heaviside step function. The Moving Least Squares (MLS) are chosen to construct the shape functions. The penalty approach is used to impose ...

We present a robust second-order accurate method for discretizing the multidimensional Heaviside and the Dirac delta functions on irregular domains. The method is robust in the following ways: (1) Integrations of source terms on a co-dimension one surface are independent of the underlying grid and therefore stable under perturbations of the domain’s boundary; (2) The method depends only on the ...

Abstract. In this paper we propose a variant of the level set formulation for identifying curves separating regions into different phases. In classical level set approaches, the sign of n level set functions are utilized to identify up to 2n phases. The novelty in our approach is to introduce a piecewise constant level set function and use each constant value to represent a unique phase. If 2n ...