The modified particle filter, which we call MTU-PF, also allows the application of an adaptive stepsize choice in the time-continuous case to avoid degeneracy problems.
If the boundary value conditions are non-zero valued on the edges of cylinder Q, then the solution of this problem is continuous on the line of degeneracy (r=0) only if (alpha>1), i.e., only in the case of the strong degeneracy of equation (2). 2.
It follows directly from (49) that u_{0}(0,varphi,z)=0quadtext{for } alphageq1 and u_{0}(0,varphi,z =sum _{n=1}^{infty} frac{alpha_{0n}^{0}}{I _{0} ( 2mu_{n}R^{1-alpha} ) }singamma_{n}zquadtext{for }alpha< 1, for each ((varphi,z inoverline{S}), i.e., the solution (u_{0}) is continuous on the line of degeneracy (r=0).
end{aligned} In the opposite case, (g_{i}(0,varphi=1,20), (i=1,2), i.e., the functions (g_{i}), (i=1,2), do not satisfy all conditions of Theorem 1 under which there exists the solution of Problem D1 (64), (64) continuous at the line of degeneracy (r=0).
Here, ν is a degeneracy index, and we tacitly imply that continuous components in the indices (n,ν) are integrated.
Here, ν is a degeneracy index and the notation implies that continuous components in the indices (n, ν) are integrated.
In the case of the weak degeneracy ((alpha<1)), the solution of Problem D1 is continuous on the line of degeneracy if only the boundary function is equal to zero at the points (P_{0}) and (P_{H}). .
In addition to the degeneracy and self-sharpening of the GPME with continuous coefficients, detailed in [1], increased numerical challenges occur in the discontinuous coefficients case.
In other words, there is a degeneracy (infinite or finite depending on whether the symmetry is continuous or discrete) of distinct asymmetric solutions of identical (lowest) energy, the whole set of which maintains the symmetry of the theory.
As well as the degeneracy associated with the constants of motion of the WS ansatz, we also find continuous families of solutions for instantaneously coupled neurons, resulting from the reversibility of the reduced model and the form of the synaptic input.
