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...

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...

Let p > 3 be a prime. We show that p−1 k=0 −1/(p + 1) k p+1 ≡ 0 (mod p 5) and p−1 k=0 1/(p − 1) k p−1 ≡ 0 (mod p 4). For any positive integer m ≡ 0 (mod p), we prove that p−1 k=0 (−1) km p/m − 1 k m ≡ 0 (mod p 4), and p−1 k=1 (−1) km k 2 p/m − 1 k m ≡ 1 p p−1 k=1 1 k (mod p 3) if p > 5. The paper also contains some open conjectures.

We introduce “mod n cryptanalysis,” a form of partitioning attack that is effective against ciphers which rely on modular addition and bit rotations for their security. We demonstrate this attack with a mod 3 attack against RC5P, an RC5 variant that uses addition instead of xor. We also show mod 5 and mod 257 attacks against some versions of a family of ciphers used in the FireWire standard. We...

Gabriel showed that for a unital ring R, there exists bijective correspondence between the set of filters R and Giraud subcategories Mod(R). In this paper we prove result analogous to one given by Gabriel: small preadditive category C, left C Mod(C).