We show that necessary and sufficient conditions for the existence of a semi-evenly partite star factorization of the symmetric complete tripartite digraph K~I,n2,n3 are (i) k is even, k 2 4 and (ii) nl = n2 = n3 == 0 (mod k(k -1)/3) for k == 0 (mod 6) and nl = n2 = n3 == 0 (mod k(k 1)) for k == 2,4 (mod 6).

Let h(m) = m mod 2. Let m = 5 i.e. 101 binary string, then h(m) = 5 = 25 ≡ 1 mod 2. Now if we change m = 5 to m = 6 i.e. binary string 101 to 110 then h(m) = 6 = 36 ≡ 0 mod 2. So if the message m has changed from even to odd or odd to even then we can detect it. However if m = 5 is modified to m = 7, then h(m) = 7 = 49 ≡ 1 mod 2. This change cannot be detected! Hence we cannot use h(m) = m mod ...

The notion of modular arithmetic is related to that of the remainder in Euclidean division. The operation of finding the remainder is sometimes referred to as the modulo operation, and denoted with ”mod” used as an infix operator. For example, the remainder of the division of 16 by 12 is denoted by 16 mod 12; as this remainder is 4, we have 16 mod 12 = 4. The congruence, indicated by ”≡” follow...

Abstract Aim To evaluate the effect of using different short fiber-reinforced resin composites on fracture resistance and flexural strength endodontically treated tooth compare it with nano-filled composite. Methods Sixty human premolars were used for test. Root canals sequentially enlarged a Pro-taper system from SX to F3 obturated Gutta-percha. Mesio-Occluso-Distal (MOD) cavities prepared in ...

(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...