The component-by-component construction algorithm for rank-1 lattices can be formulated elegantly as a repeated matrix-vector product. As was shown in an earlier paper, this matrix-vector product can be done in time O(n log(n)) for n prime. Here we extend this result to general n using facts from algebra to obtain a construction cost of O(sn log(n)) for a rank-1 lattice in s dimensions with n p...

for the arithmetic mean of the distinct prime factors of n, then pa(n) is uniformly distributed modulo 1. Furthermore, it was also shown that for large x the number of n ≤ x such that pa(n) is an integer of order of magnitude O(x/ log log x). This was made more precise by the first author in [4], who showed that the counting function of the above n is asymptotically equal to (c+o(1))x/ log log ...

Suppose a 1 < a 2 < : : : < a Z are distinct integers in f1; : : :; N g. If a 0 +" 1 a 1 +" 2 a 2 + + " Z a Z is prime for all choices of " 1 ; " 2 ; : : : ; " Z 2 f0; 1g, then Z (9=2 + o(1)) log N= log log N. The argument uses a modiied form of Gallagher's Larger Sieve, and shows that if Z > (9=2 + o(1)) log N= log log N , then for some prime p < Z 2 , the subset sums " 1 a 1 + " 2 a 2 + + " Z...

where GQ = Gal ( Q/Q ) is the absolute Galois group of Q and ` is a fixed rational prime. For example, ρ = ρE,` may be the `-adic representation of an elliptic curve E over Q, or ρ = ρf may be the `-adic representation associated to a modular form. The continuity of such Galois representations implies the image lies in GL2(O) for some ring of integers O with maximal ideal λ in a finite extensio...

Schrag and Crawford ( 1996) present strong experimental evidence that the occurrence of prime implicates of varying lengths in random instances of 3-SAT exhibits behaviour similar to the wellknown phase transition phenomenon associated with satisfiability. Thus, as the ratio of number of clauses (m) to number of propositional variables (n) increases, random instances of 3-SAT progress from form...

We present a fast derandomization scheme for the PROFIT/COST problem. Through the applications of this scheme we show the time complexity of O(log 2 n loglog n) for the + 1 vertex coloring problem using O((m + n)= log logn) processors on the CREW PRAM, the time complexity of O(log 2:5 n) for the maximal independent set problem using O((m+n)= log 1:5 n) processors on the CREW PRAM and the time c...

We announce some new results about systems of residue sets. A residue set R C Z is an arithmetic progression R = {a,a±n,a± 2n,...}. The positive integer n is referred to as the modulus of R. Following Znam [21] we denote this set by a(n). We need several number-theoretic functions. p(ra)-the least prime divisor of a natural number ra, P(m)-the greatest prime divisor of ra, A(m)-the greatest div...

let $n$ be a submodule of a module $m$ and a minimal primary decomposition of $n$ is known‎. ‎a formula to compute baer's lower nilradical of $n$ is given‎. ‎the relations between classical prime submodules and their nilradicals are investigated‎. ‎some situations in which semiprime submodules can be written as finite intersection of classical prime submodule are stated‎.

The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs, which can be characterized by their vertex count n and a set D of divisors of n in such a way that they have vertex set Zn and edge set {{a, b} : a, b ∈ Zn, gcd(a− b, n) ∈ D}. Using tools from convex optimization, we study the maxim...

For a positive integer n that is not a power of 2, precisely the same family of convex polygons with n sides is optimal in three different geometric problems. These polygons have maximal perimeter relative to their diameter, maximal width relative to their diameter, and maximal width relative to their perimeter. We study the number of different convex n-gons E(n) that are extremal in these thre...