Abstract. Fix a field F. A zero-nonzero pattern A is said to be potentially nilpotent over F if there exists a matrix with entries in F with zero-nonzero pattern A that allows nilpotence. In this paper we initiate an investigation into which zero-nonzero patterns are potentially nilpotent over F, with a special emphasis on the case that F = Zp is a finite field. As part of this investigation, w...

We study the modification of Altarelli-Feruglio $A_4$ flavor symmetry model by adding three singlet flavons $\xi'$, $\xi''$ and $\rho$ is augmented with extra $Z_2\times Z_2^ \prime$ to prevent unwanted terms in our study. The addition these lead two higher order corrections form perturbation parameters $\epsilon$ $\epsilon^\prime$. These yield deviation from exact tri-bimaximal (TBM) neutrino ...

Much of the wonderful invariant theory of the 19 century worked over fields of characteristic zero, while the theory for prime characteristic lagged behind. However, the Frobenius (p power) map in characteristic p > 0 leads to a rich theory of invariants in prime characteristic. This theory is closely bound up with the Steenrod Algebra, which allows us to derive new invariants from known invari...

We consider properties of the CKM phase in the heterotic orbifold models. We find that at the renormalizable level the CKM phase vanishes identically for the prime orbifolds, whereas it can be non-zero for some non-prime orbifolds. In particular, we study in detail the Z6-I orbifold which allows for a non-trivial CKM phase and analyze the modular properties of the corresponding Jarlskog invaria...

Surely the Howard Government is ‘tough on drugs’! The Prime Minister has used this very phrase repeatedly, unambiguously and emphatically to describe his government’s approach to illicit drugs. Also, Mr Howard has often described his personal attitude to illegal drugs as ‘zero tolerance’. After a 6:3 majority of the Ministerial Council on Drug Strategy supported a scientific trial of prescripti...

For a prime p ≡ 5 (mod 8) satisfying certain conditions, we show that there exist an infinitude of K3 surfaces parameterized by certain solutions to Pell’s equation X2 − pY 2 = 4 in the projective 5-space that are counterexamples to the Hasse principle explained by the Brauer-Manin obstruction. Further, these surfaces contain no zero-cycle of odd degree over Q. As an illustration for the main r...

Let R be the Z-module generated by the irreducible characters of the symmetric group Sd . We determine bases for the kernel of the decomposition map. It is known that R ⊗Z F is isomorphic to the radical quotient of the Solomon descent algebra when F is a field of characteristic zero. We show that when F has prime characteristic and I d br is the kernel of the decomposition map for prime p then ...

Let R be a commutative ring with identity and M an R–module. If M is either locally cyclic projective or faithful multiplication then M is locally either zero or isomorphic to R. We investigate locally cyclic projective modules and the properties they have in common with faithful multiplication modules. Our main tool is the trace ideal. We see that the module structure of a locally cyclic proje...

We present eecient zero-knowledge proof systems for quasi-safe prime products and other related languages. Quasi-safe primes are a relaxation of safe primes, a class of prime numbers useful in many cryptographic applications. Our proof systems achieve higher security and better eeciency than all previously known ones. In particular, all our proof systems are perfect or statistical zero-knowledg...

Let M be a matroid on a set E and let w : E −→ G be a weight function, where G is a cyclic group. Assuming that w(E) satisfies the Pollard’s Condition (i.e. Every non-zero element of w(E)−w(E) generates G), we obtain a formulae for the number of distinct base weights. If |G| is a prime, our result coincides with a result Schrijver and Seymour. We also describe Equality cases in this formulae. I...