Instead of saying that propositional constituents are Fregean senses, Russell could say that 'Phosphorus is bright' expresses a proposition of the form '∃x{[M x) & S x)] & ∀y{[M y) & S y)] → y = x]} & B x)'; where 'E x)' and 'M x)' indicate different functions, specified (respectively) in terms of evenings and mornings.
We begin by considering the particular case where A k and f k in problem (5) are specified, respectively, by the integral averages Ā k ( t j + 1 ) z : = 1 k ∫ t j t j + 1 A ( s ) z d s and f ̄ k ( t j + 1 ) : = 1 k ∫ t j t j + 1 f ( s ) d s, (39).
Denote by u h (t) the unique solution of problem (18) in Theorem 1. Assume that A hk and f hk in problem (19) are specified, respectively, by Ā hk and f ̄ h k in (44) and denote by vhk,j, with j = 0,1,..., n, the unique solution of problem (19).
Denote by u h (t) the unique solution of problem (18) in Theorem 3. Assume that A hk and f hk in problem (19) are specified, respectively, by à hk and f ̃ h k in (45), (46) and denote by v hk,j, j = 0, 1,n..,n, the unique solution of problem (19).
We begin by considering the particular case, where A hk and f hk in problem (19) are specified, respectively, by the integral averages Ā h k ( t j ) z : = 1 k ∫ t j t j + 1 A h ( s ) z d s and f ̄ h k ( t j ) : = 1 k ∫ t j t j + 1 f h ( s ) d s, (44).
Theorem 7. Let Assumption 1 be satisfied and assume that constant K satisfies: 2Kk < 1. Denote by u(t) the unique solution of problem (3) in Theorem 1. Assume that A k and f k in problem (5) are specified, respectively, by Ā k and f ^ k in (39) and denote by vk,j, j = 0, 1,..., n, the unique solution of problem (5) in Theorem 2. Let Assumption 3 be satisfied.
where c c k and CTR G,k) were defined in "Problem definition" section and specify respectively the CPC for campaign k and the CTR for campaign k and user profile G.
The exact solution involves the eigenvalues of the problem that are generally real or complex (not integers), and coincide with the degree of the associated Legendre functions and order of the spherical Bessel functions which specify respectively the latitudinal and radial dependencies of the wave field.
If any, please specify', respectively.
The EC code numbers consist of four elements which specify, respectively, the enzyme class, subclass (the bond broken or hydrolyzed, the group transferred, etc)., sub-subclass (usually the nature of the substrate), and the serial number of the enzyme in the respective sub-subclass [ 47].
The process noise and measurement noise are unknown, bounded and confined to a specified ellipsoidal set respectively, which is handled by transformation to the inequality constraints.
