Taking the Fibonacci sequence G0 = 1; G1 = b 2 f1; 3; 5g and Gn+1 = 3 Gn + Gn 1 (n 1) with an integer 2 m 2 N , we get a purely periodic sequence fGn( mod m)g. Consider any shortest full period and form a frequency block Bm 2 N to consist of the frequency values of the residue d when d runs through the complete residue system modulo m. The purpose of this paper is to show that such frequency bl...

Twenty-five years ago, W. M. Snyder extended the notion of a repunit Rn to one in which for some positive integer b, Rn(b) has a b-adic expansion consisting of only ones. He then applied algebraic number theory in order to determine the pairs of integers under which Rn(b) has a prime divisor congruent to 1 modulo n. In this paper, we show how Snyder’s theorem follows from existing theory pertai...

We study how to construct a number greater than one whose powers are uniformly distributed modulo 1. Also we prove that for every λ > 0 there is a dense set of computable numbers α > 1 such that the discrepancy of {λα}n=1 is O(N − 1 4+ε ).

We present a new program for predicting protein subcellular localization from amino acid sequence. WoLF PSORT is a major update to the PSORTII program, based on new sequence data and incorporating new features with a feature selection procedure. Following SWISS-PROT, we divided eukaryotes into three groups: fungi, plant, and animal. For the 2113 fungi proteins divided into 14 categories; we fou...

Abstract. A seminal theorem due to Weyl [14] states that if (an) is any sequence of distinct integers, then, for almost every x ∈ R, the sequence (anx) is uniformly distributed modulo one. In particular, for almost every x in the unit interval, the sequence (anx) is uniformly distributed modulo one for every computable sequence (an) of distinct integers. Call such an x UD random. Here it is sho...

A set B ⊆ N is called low for Martin-Löf random if every Martin-Löf random set is also Martin-Löf random relative to B. We show that a ∆2 set B is low for Martin-Löf random iff the class of oracles which compress less efficiently than B, namely the class C = {A | ∀n K(n) ≤ K(n)} is countable (where K denotes the prefix free complexity and ≤ denotes inequality modulo a constant). It follows that...

Algorithm development for finding quasiperiodic regions in sequences is at the core of many problems arising in biological sequence analysis. We solve an important problem in this area. Let A be an alphabet of size n and A’ denote the set of sequences of length 1 over A. Given a sequence S = ~1.52 . . .sl E A’, a positive integer p is called a period of S if s; = s;+~ for 1 5 i 5 1 p. S is call...

For a triple (n, t,m) of positive integers, we attach to each t-subset S = {a1, . . . , at} ⊆ {1, . . . , n} the sum f(S) = a1 + · · · + at (modulo m). We ask: for which triples (n, t,m) are the ( n t ) values of f(S) uniformly distributed in the residue classes mod m? The obvious necessary condition, that m divides ( n t ) , is not sufficient, but a q-analogue of that condition is both necessa...