Sentence examples for be the principal solution from inspiring English sources

Definition 3.6 Let Y i ( M ) = [ X i T U i T ] T be the principal solution of (1.5) at M = 0.

Corollary 2.2 Let Y i ( 0 ) and Y ˆ i ( N + 1 ) be the principal solutions of (2.1) and (2.2) associated with symplectic fundamental matrices Z i ( 0 ), Z ˆ i ( N + 1 ) such that conditions (2.4) hold.

and the conjoined bases Y i ( M ), Y ˆ i ( M ) of (2.1), (2.2) with the initial conditions Y M ( M ) = [ 0 I ] T, Y ˆ M ( M ) = [ 0 I ] T at i = M are said to be the principal solutions at M. Note that for conjoined bases Y i, Y ˆ i of (2.1), (2.2) there exist symplectic fundamental matrices Z i, Z ˆ i such that Y i = Z i [ 0 I ] T, Y ˆ i = Z ˆ i [ 0 I ] T (2.4).

then is the principal solution of (1.3).

If (1.4) then is the principal solution of (1.3).

In the case of TPS, U is the principal solution of the 3D biharmonic equation (Bookstein, 1989; Harder and Desmarais, 1972).

For the particular case when Y i, Y ˆ i are the principal solutions of (2.1), (2.2), we have the following corollary from Theorem 2.1 (see [[13], Corollary 2.4]).

Remark 2.4 (i) Assume that Y i, Y ˆ i are conjoined bases and Y i ( M ), Y ˆ i ( M ) are the principal solutions of (2.1), (2.2) at M ∈ Z.

Moreover, when (S A,mathbf{b})) is nonempty, the principal solution (hat{mathbf{x}}) becomes the maximum solution and (S A,mathbf{b})) can be determined by the maximum solution and a finite number of minimal solutions.

(ii) If and, then is not the principal solution of (1.3).  .

But Herman said the judge had not approved the use of the bunks, and he believed the tents were "now the principal solution" to temporarily easing the jail problem.