It has been observed previously that Random 3SAT exhibits a phase transition at a critical ratio of constraints to variables, where the average frequency of satissability falls abruptly from near 1 to near 0. In this paper we look beyond satissability to implicates and prime implicates of non-zero length and show experimentally that, for any given length, these exhibit their own phase transitio...

Leptoquarks are hypothetical new particles, which couple quarks directly to leptons. They experienced a renaissance in recent years as they prime candidates explain the so-called \textit{flavor anomalies}, i.e. deviations between Standard Model predictions and measurements $b\to s\ell^{+}\ell^{-}$ c\tau\nu$ processes anomalous magnetic moment of muon. At one-loop level these particles unavoidab...

It is known that any square matrix A of size n over a field of prime characteristic p that has rank less than n/(p− 1) has a permanent that is zero. We give a new proof involving the invariant Xp. There are always matrices of any larger rank with non-zero permanents. It is shown that when the rank of A is exactly n/(p − 1), its permanent may be factorized into two functions involving Xp. Let n ...

The Paley graph Pq, where q ≡ 1 (mod 4) is a prime power, is the graph with vertices the elements of the finite field Fq and an edge between x and y if and only if x − y is a non-zero square in Fq. This paper gives new results on some colouring problems for Paley graphs and related discussion.

Let p be an odd prime number and let G be an extraspecial pgroup. The purpose of the paper is to show that G has no non-zero essential mod-p cohomology (and in fact that H∗(G, Fp) is Cohen-Macaulay) if and only if |G| = 27 and exp(G) = 3.

The purpose of these notes is to illustrate the descent technique used in [Maz72] and [Maz77] to bound the rank of abelian varieties A/Q; this method is used in [Maz77] to prove that the Eisenstein quotient of J0(N)/Q (where N is a prime number) has rank 0. We begin with a brief review of the standard method of descent. We then explain how fppf cohomology can be used following this model to pro...

Resolution improvements in dipolar resonant excitation have been examined in a round-rod quadrupolar collision cell for values of the Mathieu characteristic exponent beta equal to n/p, where n and m are small integers (prime beta values) versus other beta values where n and p are not small (ordinary beta values). The trajectories of ions moving in the time-varying electric fields of a quadrupol...

2 DDH Problem Definition : {x← Z∗ p , y ← Z∗ p : (gx, gy, gxy)} ≈c {x← Z∗ p , y ← Z∗ p , z ← Z∗ p : (gx, gy, gz)} DDH problem states that the above two tuples are completely indistinguishable. Zp = {0, 1, ..., p− 1} where p is a very large prime and Z∗ p contains all non-zero elements in above set such that gcd(x, p) = 1. gxy which is dependant of x and y looks totally indistinguisable from a c...

In this paper we look at the properties of modules and prime ideals in finite dimensional noetherian rings. This paper is divided into four sections. The first section deals with noetherian one-dimensional rings. Section Two deals with what we define a “zero minimum rings” and explores necessary and sufficient conditions for the property to hold. In Section Three, we come to the minimal prime i...