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Let be a relation from to and a relation from to . Then we define the ○ of and as the set of all (, ) in × such that there is an element in with and . ○ is a relation from to .

Let be the number of changes of sign in the sequence of coefficients , , … , , in (2.8). The number of positive real roots of () = 0, counting the multiplicities of the roots, is or minus a positive even number. If = 1, the equation has exactly one positive real root.

Any function built from continuous functions by additions, subtractions, multiplications, divisions, and compositions, is continuous where defined.

Let () be a continuous, homothetic function defined in a connected cone . Assume that is strictly increasing along each ray in , i.e. for each ≠ in , () is a strictly increasing function of . Then there exist a homogeneous function and a strictly increasing function such that () = (()) for all in

• When is a utility function, and and are goods, is called the (abbreviated MRS).

• When is a production function and and are inputs, is called the (abbreviated MRTS).

• When (, ) = 0 is a production function in implicit form (for given factor inputs), and and are two products, is called the (abbreviated MRPT).

System (6.4) has if there is a set of of the variables that can be freely chosen such that the remaining − variables are uniquely determined when the variables have been assigned specific values. If the variables are restricted to vary in a set in ℝ, the system has .

If is convex on the interval and is a random variable with finite expectation, then ([]) ≤ [()] If is strictly convex, the inequality is strict unless is a constant with probability 1.

If a series is absolutely convergent, then the sum is independent of the order in which terms are summed. A conditionally convergent series can be made to converge to any number (or even diverge) by suitable rearranging the order of the terms.

Definition of the of the integral. and are constants. Special integration results.

The solutions of a homogeneous, linear secondorder difference equation with constant coefficients and . , , and are arbitrary constants. If the function is itself a solution of the homogeneous equation, multiply the trial solution by . If this new trial function also satisfies the homogeneous equation, multiply the trial function by again. (See Hildebrand (1968), Sec. 1.8 for the general procedure.)