By Clifford Truesdell

ISBN-10: 0691095779

ISBN-13: 9780691095776

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**Extra info for An essay toward a unified theory of special functions**

**Example text**

Lovdsz 36 yA 3 w v20 min y . x where 1 denotes the vector It is well-known that if x, y run through non-negative real vectors, (10) and (11) have a common optimum. But now we are interested in integer vector solutions. Let B be a (0, 1)-matrix such that (i) any column u of B satisfies Au C 1, (ii) every maximal (0, 1)-vector with this property is a column of B. Consider two further programs: yB 3 w y 30 min y * 1 Bx s 1 x 20 rnaxw-x Theorem 5. Assume that the optimum of (10) ( = the optimum of (11)) is an integer for any (0, 1)-vector w.

A hypergraph H is balanced if and only if v ( H s ; l , q ) = ~ * ( H s ; l , q () S C X , q E N " ' ) . Proof. 1. 12. A hypergraph H is balanced i f and only i f every partial hypergraph H' has the Menger property. Proof. Let H be a balanced hypergraph on X = {xl,x2,. . } and let p E N " . 6, H'p' is balanced. 9, v ( H ;p. , So every partial hypergraph H' C H also satisfies v ( H ' ;p, 1 ) = T ( H ' ;p, 1). Hence H' has the Menger property. Conversely, let H be a hypergraph whose partial hypergraphs have the Menger property.

We have thus to prove that T, n (&\ To)= 0. This is clear for p odd since Eo C B and T, C R in this case. Suppose p is even. We show that E,, fl E = 0 for all edges E E gP. Let E, E 8, and suppose for a contradiction that there is xo E E, f l ( E o \ { z } ) . Let x, E Tp-lf l E,. By the minimality of T,-, there is Ep-lE such that E,-, fl T,-, = {x,}. Since is red in c,-~ by ( 2 . p) we have n E, Tp-I,hence E,-, n E, C EP-,n Tn-,= { x p } , and thus E,-, n E, = { x , } . Repeating this procedure with instead of E,, and inductively, we get a sequence E P , ~ P , E p -,,.