Catálogo de publicaciones - libros

A simple differential equation and its solution. () is a given function and () is the unknown function. A differential equation. If () = 0, () ≡ is a solution. A differential equation. The substitution = leads to a separable equation for .

A useful characterization of closed sets, and a definition of the closure of a set. Definition of a neighborhood. Convergence of a sequence in ℝ. If the sequence does not converge, it . Definition of a Cauchy sequence.

Definition of a convex set. The empty set is, by definition, convex. The first set is convex, while the second is not convex. Properties of convex sets. ( and are real numbers.) Definition of a convex combination of vectors. co() is the of a set in ℝ.

Definition of (global) maximum (minimum) of a function of variables. As collective names, we use points and values, or points and values. Used to convert minimization problems to maximization problems.

If either of the problems (15.1) and (15.2) has a finite optimal solution, so has the other, and the corresponding values of the objective functions are equal. If either problem has an “unbounded optimum”, then the other problem has no admissible solutions.

The . A necessary condition for the solution of (16.1). An alternative form of the Euler equation. The . A necessary condition for the solution of (16.1). Sufficient conditions for the solution of (16.1). Adding condition (16.5) gives sufficient conditions.

Definition of an sequence. Boundedness conditions. and are given numbers. The obtained from period and onwards, given that the state vector is at = . The of problem (17.8). Properties of the value function, assuming that at least one of the boundedness conditions in (17.10) is satisfied.

Definition of a linear combination of vectors. Definition of linear dependence and independence. A characterization of linear independence for vectors in ℝ. (See (19.23) for the definition of rank.) A characterization of linear independence for vectors in ℝ. (A special case of (18.4).)

Notation for a , where is the element in the th row and the th column. The matrix has . If = , the matrix is of order . An matrix. (All elements below the diagonal are 0.) The transpose of (see (19.11)) is called .

The general definition of a determinant of order , by along the th row. The value of the determinant is independent of the choice of . Expanding a determinant by a row or a column in terms of the cofactors of the same row or column, yields the determinant. Expanding by a row or a column in terms of the cofactors of a different row or column, yields 0.