The singular fields in the vicinity of crack front or junction have been accurately represented by the singular elements by taking account the variable order of singularities and the angular profiles of field variables.
In this paper we extend this equation to the concept of fractional variable-order derivative by replacing the Laplace operator in Equation (2.1) by the variable-order Laplace operator to obtain: i h 2 π ∂ Ψ ( r, t ) ∂ t = [ − [ h 2 π ] 2 2 m ∇ α ( x, t ) + V ( r, t ) ] Ψ ( r, t ), 0 < α ( x, t ) ≤ 2. (2.3).
To validate this idea, we firstly extract Vehicular Mobility Pattern (VMP) by employing the Variable-order Markov (VOM) models[5] from real trace data collected from over 4,000 taxis over several months in Shanghai.
In addition, a single PSG design may also have some advantages because serial PSGs can be affected by individually variable order-effects [43].
We introduced the concept of nonisotropic spherical Riesz potential operators generated by the -distance of variable order on -sphere and its -boundedness were investigated.
In another study by Vernikos et al. [ 21], authors used variable order motifs and relative entropy for the detection of compositionally deviating regions.
By transforming the variable, they found a first-order Hamiltonian system equivalent to the second-order p-Laplacian system (1).
In our case, the observed sequences were best fitted by a variable-order Markov model of maximum order 2 (Figure 7).
Performing differences between these pairs of variates allows then to consider a change detection analysis based on linear combinations of the original variables, ordered by correlation (similarity) between pairs [12].
We also tested a variable ordering obtained by a depth first search (DFS) (Sedgewick, 2002), but found that the ordering obtained by a breadth first search resulted in a much better performance on average.
