By V. A. Tkachenko

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Corollary 7. (i) (iii) There exist orthogonal designs of types (i,I,2,1,3,1,3,1,3), (1,1,2,2,2,2,2,2,2) and (ii) (i,i,2,1,2,1,2,1,2) (iv) (I,I,2,1,i,i,I,i,i) in order 16. Proof. Use S = XlA 1 +x2A 2+XsA 3 and (i) R= BI, P= B2+B3, (ii) R= BI,P = B3, 51 (iii) R = BI+B2, P = B3, (iv) R = BI, P = B2, respectively in the theorem, where the A i and B i are defined in (i) of Section 2. Corollary 8. There exist orthogonal designs of types (ii) (2,2,4,l,a,l,a,l,a), or 6, a=1,2,3,4,5,6 or 7, (i) (2,2,4,3,5,3,5,3,5), (iii) (l,l,2,4,2,a,2,a,2,a), a=2,4 (iv) (1,7,1,7,1,7,1,7) in order 32.

Robinson, private communication, 1975. [5] D. Shapiro, private communication, 1975. [s] Jennifer Wallis, Combinatorial Theory. , 1967). A note on amicable Hadamard matrices, Util~tas Math. 3 (1973), 119-125. [7] Jennifer Seberry Wallis, [8] Jennifer Wallis, Soc. [9] Warren W. Wolfe, Warren W. Wolfe, Math. Preprint [ii] (v,k,l) configurations and Hadamard matrices, (to appear). J. Austral. Math. ii (1970), 297-309. Math. Preprint [i0] On the existence of Hadamard matrices, Warren W. Wolfe, Rational quadratic forms and orthogonal designs, No.

Proof. W(60,53). We have the enunciation from [6, Lemma 16] except for the W(60,51) and It remains then to show these two weighing matrices exist. ] respectively. AE The W(60,53) is found by replacing the variables of the orthogonal design of type (2,5,5,8) in order 20 by the matrices B, J, B and A respectively. 4. CONSTRUCTIONS USING CIRCULANT WEIGHING MATRICES It was shown in [14] that orthogonal designs of types (l,l,l,q2), (l,l,q2,q2), (l,q2,q2 q2), (q2,q2 q2 q2), (2q2 2(q2+2q+2)l exist in order 4(q2+q+l) when q is a prime power.

### A problem in the spectral theory of an ordinary differential operator in a complex domain by V. A. Tkachenko

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