Define Ψ(x, y) to be the number of positive integers n ≤ x such that n has no prime divisor larger than y. We present a simple algorithm that approximates Ψ(x, y) in O(y{ log log x log y + 1 log log y }) floating point operations. This algorithm is based directly on a theorem of Hildebrand and Tenenbaum. We also present data which indicate that this algorithm is more accurate in practice than o...

The grain yield (Y) of crops is determined by several Y components that reflect positive or negative effects. Conventionally, ordinary Y components are screened for the highest direct effect on Y. Increasing one component tends to be somewhat counterbalanced by a concomitant reduction in other component (s) due to competition for assimilates. Therefore, it has been suggested that components...

and Applied Analysis 3 To complete the following algorithm, we need the following elementary property. Lemma 7. Let F(x, y) ∈ R[x, y] and p(x) be an irreducible polynomial in R[x]. Consider the module equation F (x, y) = 0 (mod p (x)) . (20) The number of all solutions y = y(x) (modp(x)) is thus at most deg y F. Assumption. Throughout this algorithm, for any F(x, y) ∈ R[x, y], one can solve all...

Deene (x;y) to be the number of positive integers n x such that n has no prime divisor larger than y. We present a simple algorithm that approximates (x; y) in O(yf log log x log y + 1 log log y g) oating point operations. This algorithm is based directly on a theorem of Hildebrand and Tenenbaum. We also present data which indicate that this algorithm is more accurate in practice than other kno...

Ψ(x, y) denotes the number of positive integers ≤ x and free of prime factors > y. Hildebrand and Tenenbaum gave a smooth approximation formula for Ψ(x, y) in the range (log x)1+ < y ≤ x, where is a fixed positive number ≤ 1/2. In this paper, by modifying their approximation formula, we provide a fast algorithm to approximate Ψ(x, y). The computational complexity of this algorithm is O( √ (log ...

An algorithm is presented for generating a succinct encoding of all pairs shortest path information in an n-vertex directed graph G with O(n) edges. The edges have real-valued costs, but the graph contains no negative cycles. The algorithm runs in O('Y(G)n + ('Y(G) )'log 'Y(G)) time, where 'Y(G) is a topological emhedding measure that we define. The algorithm uses a decomposition of the graph i...

The size |Y | of the output space Y is exponential and optimization over the entire space Y is computationally expensive. Hence in the sequential dual optimization method, the optimization of (A.6) is restricted to the set Yi = {y : αiy > 0} maintained for each example. For clarity, we present the sequential dual optimization method to solve (A.2) in Algorithm 3. The algorithm starts with Yi = ...

This paper develops for the first time the mathematical foundations of an efficient stochastic algorithm for solving steady-state population balance equations, which is suitable for coupling to computational fluid dynamics codes. The properties and performance of the algorithm are charted using a systematic comparison with an existing stochastic algorithm. The algorithm is extended to 1D geomet...

We consider the deterministic complexity of the problem of polynomial factorization over finite fields given a finite field Fq and a polynomial h(x, y) ∈ Fq[x, y] compute the unique factorization of h(x, y) as a product of irreducible polynomials. This problem admits a randomized polynomial-time algorithm and no deterministic polynomial-time algorithm is known. In this chapter, we give a determ...