The Evans wave equation [1] of general relativity is expressed in spinor form, thus producing the Dirac equation in general relativity. The Dirac equation in special relativity is recovered in the limit of Euclidean or flat spacetime. By deriving the Dirac equation from the Evans equation it is demonstrated that the former originates in a novel metric compatibility condition, a geometrical cons...

A second-order stochastic parabolic equation with zero Dirichlet boundary conditions is considered in a sufficiently smooth bounded domain. Existence, uniqueness, and regularity of the solution are established without assuming any compatibility relations. To control the solution near the boundary of the region, special Sobolev-type spaces with weights are introduced. To illustrate the results, ...

Many crystalline materials exhibit twinned microstructures, where well-defined orientation relationships define the special symmetry between different, elastically anisotropic twin variants. When such twins are subjected to external loading, additional internal stresses necessarily occur at the twin boundaries in order to maintain compatibility. These compatibility stresses are constant inside ...

We consider the simultaneous reconstruction of the absorption coefficient and the source term in a linear transport equation from available boundary measurements. This problem finds applications in SPECT, a medical imaging modality. When the absorption coefficient is known, recently derived inversion formulas for the attenuated Radon transform can be used to reconstruct the source term. Moreove...

The growing use of information technology and the Internet has led to the emergence of professional virtual communities (PVCs). Why are PVC members willing to spend time and energy to provide valuable knowledge to others? This study extended Icek Ajzen’s(1991) theory of planned behavior (TPB) to analyze what determines PVC members’ willingness to share knowledge. Empirical data were collected f...

In this paper we propose the notion of dynamic deviation measure, as a dynamic timeconsistent extension of the (static) notion of deviation measure. To achieve time-consistency we require that a dynamic deviation measures satisfies a generalised conditional variance formula. We show that, under a domination condition, dynamic deviation measures are characterised as the solutions to a certain cl...

Discretization processes leading to numerical schemes sometimes produce undesirable effects. One potentially serious problem is that a discretization may produce the loss of validity of some of the physical principles or mathematical properties originally present in the continuous equation. Such loss may lead to uncertain results such as numerical instabilities or unexpected non-physical soluti...

3. The Implicit Function Theorem and its special case, the Inverse Function Theorem. Relation to the Contraction Mapping Fixed Point Theorem. Computation of several iterates of the approximating sequence to the implicit function. Continuity and differentiability of the implicit function. Application to differentiable variation of the fixed point in the Contraction Mapping Fixed Point Theorem. 4...

This paper presents a class of parallel numerical integration methods for stii systems of ordinary diierential equations which can be partitioned into loosely coupled subsystems. The formulas are called decoupled backward diierentiation formulas, and they are derived from the classical formulas by restricting the implicit part to the diagonal subsystem. With one or several subsystems allocated ...