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Hungarian Problem Book 1 (Number 11) (Bk. 1) by E. Rapaport

By E. Rapaport

Publication via Rapaport, E.

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Extra info for Hungarian Problem Book 1 (Number 11) (Bk. 1)

Sample text

Hence, I and II cannot hold simultaneously. Thus, i=>n. • We remark that if either A or 5 is vacuous, then we revert to Tucker's theorem 3 or Motzkin's theorem 2. We remark further that in all of the above theorems of the alternative the systems I are all homogeneous. 4 The above theorems of the alternative subsume in essence all other theorems of this type. We derive below some of these theorems directly from the above ones. Gordan's theorem [Gordan 73] For each given matrix A, either I Ax > 0 has a solution x or II A'y = 0, y > 0 has a solution y but never both.

Each/< is a numerical function on X. A vector function / has a certain property (for example continuity) whenever each of its components/, has that property. EXAMPLE If X = R, then d is the familiar real single-valued func of Rn. The m components /,, i' = 1, . . , m, of / are numerical functions on Rn. Linear vector functions on Rn An m-dimensional vector function / defined on Rn is said to be linear if f(x) = Ax + b where A is some fixed m X n matrix and b is some fixed vector in Rm. It follows that if / is a linear function on Rn then (Conversely, the last two relations could be used to define a linear vector function on Rn, from which it could be shown that/(a;) = Ax -f 6 [Berge 63, p.

The m components of the vector f(x) are denoted by fi(x), . . , fm(x). Each/< is a numerical function on X. A vector function / has a certain property (for example continuity) whenever each of its components/, has that property. EXAMPLE If X = R, then d is the familiar real single-valued func of Rn. The m components /,, i' = 1, . . , m, of / are numerical functions on Rn. Linear vector functions on Rn An m-dimensional vector function / defined on Rn is said to be linear if f(x) = Ax + b where A is some fixed m X n matrix and b is some fixed vector in Rm.

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