By Bouyukliev I., Fack V., Winne J.
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Extra info for 2-(31,15,7), 2-(35,17,8) and 2-(36,15,6) designs with automorphisms of odd prime order, and their related Hadamard matrices and codes
We assume that matrix A0 (composed of the mean values of the corresponding elements of A(t)) has no eigenvalues with positive real parts, but has eigenvalues with zero real parts. In this case the study of the stability of the trivial solution of a system dx = εA(t)x dt becomes a more complicated task. Z. Shtokalo [1946, 1961] for the investigation of the stability of systems with almost periodic coeﬃcients that are close to constants. 1) k=1 where ε > 0 is a small parameter, A is a constant n × n matrix, Ak (t), k = 1, .
31) has the form t tA2 y2 (t) = e e(t−s)A2 f2 (s)ds. 33) 0 Therefore, a solution that is bounded on [0, ∞) should have the representation ∞ e(t−s)A2 f2 (s)ds. 31). 27). 27) are the solutions of systems of integral equations t tA1 y1 (t) = e e(t−s)A1 (D(s, ε)y(s))1 ds, y1 (0) + 0 ∞ e(t−s)A2 (D(s, ε)y(s))2 ds. 25) are the solutions of a system of integral equations y(t) = t etA1 0 0 0 y(0) − 0 ∞ 0 0 0 e(t−s)A2 + e(t−s)A1 0 0 0 D(s, ε)y(s)ds+ D(s, ε)y(s)ds. 25), such that y11 (0) = y12 (0), |y 1 (t)|, |y 2 (t)| ≤ r0 for t ≥ 0.
By substituting the last expression into the matrix equation we get the equations for determining the matrices Dl : (iλl I − A)Dl + Dl A = C1l . 5) Since the matrix A has only real eigenvalues the intersection of the spectral of matrices (iλl I − A) and A is empty. 5) has a unique solution (see, for example, Gantmacher , Daleckii and Krein ). All subsequent matrix equations have the same structure. Matrices Bi (i = 2, . . , m) can be determined as mean values of the right-hand side of the corresponding matrix equations.
2-(31,15,7), 2-(35,17,8) and 2-(36,15,6) designs with automorphisms of odd prime order, and their related Hadamard matrices and codes by Bouyukliev I., Fack V., Winne J.