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Lower and upper bounds for solutions of the congruence xm ≡ a(mod n)

Zakarczemny M.

Technical Transactions

2017

Vol. 11(114)

161--168

Let n, m be natural numbers with n ≥ 2. We say that an integer a, (a, n) = 1, is the m-th power residue modulo n if there exists an integer x such that xm ≡ a(mod n). Let C(n) denote the multiplicative group consisting of the residues modulo n which are relatively prime to n. Let s(n, m, a) be the smallest solution of the congruence xm ≡ a(mod n) in the set C(n). Let t(n, m, a) be the largest solution of the congruence xm ≡ a(mod n) in the set C(n). We will give an upper bound for s(n, m, a) and a lower bound for t(n, m, a).

Niech n, m będą liczbami naturalnymi, takimi że n ≥ 2. Powiemy, że liczba całkowita a, (a, n) = 1, jest m-tą resztą kwadratową modulo n, jeśli istnieje liczba całkowita x, taka że xm ≡ a(mod n). Niech C(n) będzie grupą multiplikatywną zawierającą reszty modulo n, względnie pierwsze z n. Oznaczmy przez s(n, m, a) najmniejsze rozwiązanie równania xm ≡ a(mod n) w zbiorze C(n). Oznaczmy przez t(n, m, a) największe rozwiązanie równania xm ≡ a(mod n) w zbiorze C(n). Podamy górne oszacowanie na s(n, m, a) oraz dolne na t(n, m, a).

Proper feedback compensators for a strictly proper plant by polynomial equations

Callier F. M., Kraffer F.

International Journal of Applied Mathematics and Computer Science

2005

Vol. 15, no 4

493-507

We review the polynomial matrix compensator equation XlDr + YlNr = Dk (COMP), e.g. (Callier and Desoer, 1982, Kučera, 1979; 1991), where (a) the right-coprime polynomial matrix pair (Nr,Dr) is given by the strictly proper rational plant right matrix-fraction P = NrD-1 r , (b) Dk is a given nonsingular stable closed-loop characteristic polynomial matrix, and (c) (Xl, Yl) is a polynomial matrix solution pair resulting possibly in a (stabilizing) rational compensator given by the left fraction C = X-1 l Yl. We recall first the class of all polynomial matrix pairs (Xl, Yl) solving (COMP) and then single out those pairs which result in a proper rational compensator. An important role is hereby played by the assumptions that (a) the plant denominator Dr is column-reduced, and (b) the closed-loop characteristic matrix Dk is row-column-reduced, e.g., monically diagonally degree-dominant. This allows us to get all solution pairs (Xl, Yl) giving a proper compensator with a row-reduced denominator Xl having (sufficiently large) row degrees prescribed a priori. Two examples enhance the tutorial value of the paper, revealing also a novel computational method.