We briefly survey a number of important recent achievements in Theoretical Computer Science (TCS), especially Computational Complexity Theory. We will discuss the PCP Theorem, its implications to inapproximability on combinatorial optimization problems; space bounded computations, especially deterministic logspace algorithm for undirected graph connectivity problem; deterministic polynomial-time primality test; lattice complexity, worst-case to averagecase reductions; pseudorandomness and extractor constructions; and Valiant’s new theory of holographic algorithms and reductions.

Abstract. Let m> 0 and q> 1 be relatively prime integers. We find an explicit period νm(q) such that for any integers n � 0 and r we have n + νm(q) r

Abstract. Let q> 1 and m> 0 be relatively prime integers. We find an explicit period νm(q) such that for any integers n> 0 and r we have h n + νm(q) i r

Abstract. Let m> 0 and q> 1 be relatively prime integers. We find an explicit period νm(q) such that for any integers n � 0 and r we have n + νm(q) r