Let G be a finite group and C(G) be the family of representative conjugacy classes of subgroups of G. The matrix whose H;K-entry is the number of fixed points of the set G=K under the action of H is called the table of marks of G where H;K run through all elements in C(G). In this paper, we compute the table of marks and the markaracter table of groups of order pqr where p, q, r are prime numbers.

We refine the techniques of our previous paper [KM1] to prove that the average order of vanishing of L-functions of primitive automorphic forms of weight 2 and prime level q satisfies ∑ f∈S2(q)∗ ords= 1 2 L(f, s) ≤ C|S2(q) ∗| with C = 6.5, for all q large enough. On the Birch and Swinnerton-Dyer conjecture, this implies rank J0(q) ≤ C dim J0(q) for q prime large enough.

Problem 3-1. Prime Triangles. TA Notes: It is a little-known fact (to us, anyway) that the Java 1.4.1 class library contains a bug in part of its primality tester. Specifically, the Lucas-Lehmer test is implemented incorrectly, and sometimes reports that certain large numbers are composite when they are actually prime. This slowed down our own trianglefinding algorithm for large values of k, an...

In 1876, E. Lucas showed that a quick proof of primality for a prime p could be attained through the prime factorization of p − 1 and a primitive root for p. V. Pratt’s proof that PRIMES is in NP, done via Lucas’s theorem, showed that a certificate of primality for a prime p could be obtained in O(log p) modular multiplications with integers at most p. We show that for all constants C ∈ R, the ...

Objective(s): Infection with tuberculosis (TB) is regarded as a major health issue. Due to the emergence of antibiotic resistance during TB treatment, prevention via vaccination is one of the most effective ways of controlling the infection. DNA vaccines are developed at a greater pace due to their ability in generating a long-lasting immune response, higher safety com...

The $J/{\psi}$ weak decays are rare but possible within the standard model of elementary particles. Inspired by potential prospects at future intensity frontier, $C$ parity violating ${\to}$ ${\pi}{\eta}^{({\prime})}$, ${\eta}{\eta}^{\prime}$ and strangeness changing ${\pi}K$, $K{\eta}^{({\prime})}$ studied with perturbative QCD approach. It is found that have relatively large branching ratios,...

• M. Monge, Generation of the symmetric field by Newton polynomials in prime characteristic, published in Rocky Mountains Journal (2012). It was proved by Dvornicich-Zannier that the polynomials xa + ya, xb + yb, xc+yc generate the field of symmetric function in x, y over Q, when a,b,c have no common factor. We extend the result in prime characteristic p, under the additional hypothesis that a,...

The purpose of this paper is to complete the classification of binary self-dual [48, 24, 10] codes with an automorphism of odd prime order. We prove that if there is a self-dual [48, 24, 10] code with an automorphism of type p-(c, f) with p being an odd prime, then p = 3, c = 16, f = 0. By considering only an automorphism of type 3-(16, 0), we prove that there are exactly 264 inequivalent self-...

Local distinguishability of sets generalized Bell states (GBSs) is investigated. We first clarify the conditions such that a set GBSs can be locally transformed to certain type GBS easily distinguishable within local operations and one-way classical communication. then show that, if space dimension $d$ prime, these are necessary sufficient for in $\mathbb{C}^d \otimes \mathbb{C}^d$ distinguisha...

BACKGROUND We explored the concept of heterologous prime/boost vaccination using 2 therapeutic vaccines currently in clinical development aimed at treating chronically infected hepatitis C virus (HCV) patients: prime with a DNA-based vaccine expressing HCV genotype 1a NS3/4A proteins (ChronVac-C) and boost with a modified vaccinia virus Ankara vaccine expressing genotype 1b NS3/4/5B proteins (M...