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Recently Littlewood and Offord1 proved the following lemma Let x1, x2, .-. , x, be complex numbers with I x1I ? 1. Consider the sums Fr_, erxr , where the ek are ± 1. Then the number of the sums ~r_,etx, which fall into a circle of radius r is not greater than cr2n(log n)n-1-1. In the present paper we are going to improve this to cr2 7 zii 1 / 2. The case(More)
A phylogenetic tree, also called an ''evolutionary tree,'' is a leaf-labeled tree which represents the evolutionary history for a set of species, and the construction of such trees is a fundamental problem in biology. Here we address the issue of how many sequence sites are required in order to recover the tree with high probability when the sites evolve(More)
Introduction. If the numbers of vertices and edges of a (linear) graph are suitably restricted, it is to be expected that something can be said about the configurations which the graph contains. As far as we know the first result in this direction is due to Turin .' He proved that a graph with kn vertices and Ck,2n 2 +1 edges always contains a complete(More)
THEOREM. Each set of 2n-1 integers contains some subset of n elements the sum of which is a multiple of n. PROOF. Assume first n = p (p prime). Our theorem is trivial for p = 2, thus henceforth p > 2. We need the following LEMMA. Let p > 2 be a prime and A = {a,, a 2 ,. . ., a,} 2 5 s < p a s tegers each prime to p satisfying ca, a 2 (mod p). Then the set a(More)
We describe how deletion-correcting codes may be enhanced to yield codes with double-strand DNA-sequence codewords. This enhancement involves abstractions of the pertinent aspects of DNA; it nevertheless ensures specificity of binding for all pairs of single strands derived from its codewords—the key desideratum of DNA codes– i.e. with binding feasible only(More)
The construction of evolutionary trees is a fundamental problem in biology, and yet methods for reconstructing evolutionary trees are not reliable when it comes to inferring accurate topologies of large divergent evolutionary trees from realistic length sequences. We address this problem and present a new polynomial time algorithm for reconstructing(More)
A set of points some of which are connected by an edge will be called a graph G. Two vertices are connected by at most one edge, and loops (i .e ., edges whose endpoints coincide) will be excluded. Vertices will be denoted by a, 0,-• • , edges will be denoted by e l , e 2 ,. .. or by (a, /3) where the edge (a, ,0) connects the vertices a and 0. G-e 1-• •(More)
  • P Erdös, M Kac
  • 1939
In the present note we state without proofs some results concerning additive functions, the proofs of which depend partially on statistical methods. A function f(m) is called additive if for (ml, m) = 1 one has f(mr*ms) = f(mr) + f(mz). We assume furthermore that fw) = f(ip) and 1 f&) 1 < 1 for every prime p. None of these assumptions is essential but they(More)