where summation with respect to repeated indices is implied,, is a strictly positive definite matrix, uniformly in, for and are smooth functions defined in and satisfies the condition:,.
One is that a new and simple rule to update the penalty parameter ρ in (P ρ ) is derived, the other is that, same as in [18], the uniformly positive definite restriction on the Lagrangian Hessian estimate is relaxed.
We regularize the variable coefficient Helmholtz equations arising from implicit time discretizations for resistive MHD, in a way that leads to a symmetric positive-definite system uniformly in the time step.
N is uniformly positive definite i.e. N t ≥δ I m,0≤t≤T, a.s.s
If the coefficient (D mathbf {x},t)) is uniformly positive definite, then there exists a unique solution of (2.4).
for some δ> 0; (ii) P or Q is uniformly positive definite, and F is uniformly nondegenerate, i.e. |F t |≥δ,0≤t≤T, a.s., for some δ> 0. .
For any ϵ > 0 the function ( 1 - | ϵ x | ) + is compactly supported, positive definite and approximates 1 uniformly on compact sets as ϵ → 0. Hence the function f ϵ ( x ) = ( 1 - | ϵ x | ) + f ( x ) is positive definite, approximates f uniformly on R and supp f ϵ ⊆ Ω ¯ ∩ - 1 ϵ, 1 ϵ.
Several robust stabilities of time-varying systems with parametric uncertainties, such as general robust stability, robustly asymptotical stability and exponential stability, are studied using uniformly positive definite matrix functions and the Lyapunov method.
for some δ> 0; P or Q is uniformly positive definite, and F is uniformly nondegenerate, i.e. |F t |≥δ,0≤t≤T, a.s., for some δ> 0. Let us define the dynamic formulation of the stochastic McKean-Vlasov control problem.
(Basic Estimate) Let (Omega ) be a domain in ({mathbb C}^n) with pseudoconvex boundary, and let (varphi ) be a function on (Omega ) such that the Hermitian matrix (7) is uniformly positive definite at each point of (Omega ).
