(1) g 2 = g + 1 (mod p). It i s obvious that if (1) holds then so do (2) g 3 = g 2 + g (mod p) , (3) g 4 = g 3 + g 2 (mod p) , e t c .

We give necessary and sufficient conditions for the decomposition of complete bipartite multigraph Km,n(λ) into paths and cycles having k edges. In particular, we show that such decomposition exists in Km,n(λ), when λ ≡ 0 (mod 2), m, n ≥ k2 , m + n > k, and k(p + q) = 2mn for k ≡ 0(mod 2) and also when λ ≥ 3, λm ≡ λn ≡ 0(mod 2), k(p + q) = λmn, m, n ≥ k, (resp., m,n ≥ 3k/2) for k ≡ 0(mod 4) (re...

Well-known necessary conditions for the existence of a generalized Bhaskar Rao design, GBRD(v, 3, λ;G) with v ≥ 4 are: (i) λ ≡ 0 (mod |G|), (ii) λ(v − 1) ≡ 0 (mod 2), (iii) λv(v − 1) ≡ 0 (mod 3), (iv ) if |G| ≡ 0 (mod 2) then λv(v − 1) ≡ 0 (mod 8). In this paper we show that these conditions are sufficient whenever (i) the group G has odd order or (ii) the order is of the form 2q for q = 3 or q...

1 Dirichlet’s theorem — set up Dirichlet’s famous theorem from 1837 asserts that if k is a positive integer and a is an integer coprime to k, then there are infinitely many primes p with p ≡ a (mod k). In this unit we will prove the stronger assertion that the sum of the reciprocals of the primes p ≡ a (mod k) is divergent. Important in the proof is the use of the orthogonality relations for th...

We have investigated, with and without the influence of X-inactivation, the relationship between autosomal gene-dosage and gene-product in a mammalian system, the mouse. The gene was mitochondrial malic enzyme (Mod-2), shown to lie on Chromosome 7 between the albino (c) and shaker-1 (sh-1) loci, and the enzyme was its product, mitochondrial malic enzyme (MOD-2). Gene duplication, with and witho...

The fetal modified myocardial performance index (Mod-MPI) is a noninvasive, pulsed-wave Doppler-derived measure of global myocardial function. This review assesses the progress in technical refinements of its measurement and the potential for automation to be the crucial next step. The Mod-MPI is a ratio of isovolumetric to ejection time cardiac time intervals, and the potential for the left ve...

The finite ring Zk = Z(+, .) mod p k of residue arithmetic with odd prime power modulus is analysed. The cyclic group of units Gk in Zk(.) has order (p − 1).p , implying product structure Gk ≡ Ak.Bk with |Ak| = p− 1 and |Bk| = p , the ”core” and ”extension subgroup” of Gk respectively. It is shown that each subgroup S ⊃ 1 of core Ak has zero sum, and p+1 generates subgroup Bk of all n ≡ 1 mod p...

The suppressor of Hairy wing (su(Hw)) protein inhibits the function of transcriptional enhancers located distally from the promoter with respect to the location of su(Hw)-binding sites. This polarity is due to the ability of the su(Hw)-binding region to form a chromatin insulator. Mutations in modifier of mdg4 (mod(mdg4)) enhance the effect of su(Hw) by inhibiting the function of enhancers loca...

As seen from the International Technology Roadmap for Semiconductors (ITRS) [1], the source/drain junction depth is being reduced each year in order to continue MOS transistor scaling. This year, leading-edge foundries and integrated device manufacturers (IDMs) are scheduled to move 65nm processes into pre-production, while simultaneously developing 45nm processes in R&D. In designing and manuf...

The framework of dependency pairs allows automated termination and innermost termination proofs for many TRSs where such proofs were not possible before. In this paper we present a reenement of this framework in order to prove termination in a modular way. Our mod-ularity results signiicantly increase the class of term rewriting systems where termination resp. innermost termination can be prove...