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Let L be a set of n lines in R d , for d ≥ 3. A joint of L is a point incident to at least d lines of L, not all in a common hyperplane. Using a very simple algebraic proof technique, we show that the maximum possible number of joints of L is Θ(n d/(d−1)). For d = 3, this is a considerable simplification of the orignal algebraic proof of Guth and Katz [8],… (More)

— We present sufficient conditions for robust relay delayed semiglobal stabilization of second order systems, which relate the upper bound to an uncertain time delay and the parameters of the plant. We also suggest an algorithm of delayed relay control gain adaptation for semiglobal stabilization, which is based on delayed information about the sign of the… (More)

Let A be a nonsingular n by n matrix over the finite field GF q , k = n 2 , q = p a , a ≥ 1, where p is prime. Let P (A, q) denote the number of vectors x in (GF q) n such that both x and Ax have no zero component. We prove that for n ≥ 2, and q > 2 2n 3 , P (A, q) ≥ [(q − 1)(q − 3)] k (q − 2) n−2k and describe all matrices A for which the equality holds.… (More)

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