PROBLEMS AND SOLUTIONS 101

Given the special positions,

Yl=Y2=81, Y3=Y4=82, Ys=Y6=a3,

find the best ui and the resulting maximum Idet (A

REFERENCES

[1] J. S. LEW, Kinematic theory of signature verification measurements, Math. Bioscience, 48 (1980) pp.25-51.

A Constrained Matrix Optimization Problem

Problem 81-4, by H. WOLKOWlCZ (University of Alberta).Given a real symmetric n n matrix B and three subspaces L1, L2 and L3 of R n,

determine the (unique) real symmetric n n matrix A which is closest to B in theEuclidean norm (Hilbert-Schmidt norm) and which is negative semi-definite (nsd) onL1, positive semi-definite (psd) on L2 and 0 on L3.

Principal Value of an Integral

Problem 81-5*, by H. E. FETTIS (Mountain View, California).It is known that

PVcos A0 dO r sin A

cos 4 -cos 0 sin

when A is an integer (PV denotes the Cauchy principal value of the integral). Evaluatethe integral for nonintegral A.

SOLUTIONS

The Rogers-Ramanujan Identities

Problem 74-12, by G. E. AOREWS (Pennsylvania State University).It is well known that the identity

On(x)= y,. (_l),x,(3x_l/2 2n n+l)=- n+ =(1-x ...(1-x),

where

I-[ (1-x=1

A-/+I)(1--X’)-I for 0-<B =<A,

[]=0 otherwise,

may be used to prove the Rogers-Ramanujan identities [G. H. Hardy, Ramanufan, pp.95-98]. This suggests the importance of the polynomials

and

g(x)= E (-1)Xx x(x-1)/2x=- n +A

hn(x) (-1)Xx x(Sx+3)/2 2n +1x=-o n+l+a

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