For each finite field F of prime order there is a constant c such that every 4-connected matroid has at most c inequivalent representations over F. We had hoped that this would extend to all finite fields, however, it was not to be. The (m,n)-mace is the matroid obtained by adding a point freely to M(Km,n). For all n ≥ 3, the (3, n)-mace is 4-connected and has at least 2 representations over an...

Counting rational points on Jacobian varieties of hyperelliptic curves over finite fields is very important for constructing hyperelliptic curve cryptosystems (HCC), but known algorithms for general curves over given large prime fields need very long running times. In this article, we propose an extremely fast point counting algorithm for hyperelliptic curves of type y = x + ax over given large...

Counting the order of the Jacobian group of a hyperelliptic curve over a nite eld is very important for constructing a hyperelliptic curve cryptosystem (HECC), but known algorithms to compute the order of a Jacobian group over a given large prime eld need very long running times. In this note, we propose a practical polynomial-time algorithm to compute the order of the Jacobian group for a hype...

Let G be a (p, q) graph. Let f : V (G) → {1, 2, . . . , k} be a map. For each edge uv, assign the label gcd (f(u), f(v)). f is called k-prime cordial labeling of G if |vf (i) − vf (j)| ≤ 1, i, j ∈ {1, 2, . . . , k} and |ef (0) − ef (1)| ≤ 1 where vf (x) denotes the number of vertices labeled with x, ef (1) and ef (0) respectively denote the number of edges labeled with 1 and not labeled with 1....

Let $G= (V,E)$ be a $(p,q)$-graph. A bijection $f: Eto{1,2,3,ldots,q }$ is called an edge-prime labeling if for each edge $uv$ in $E$, we have $GCD(f^+(u),f^+(v))=1$ where $f^+(u) = sum_{uwin E} f(uw)$. Moreover, a bijection $f: Eto{1,2,3,ldots,q }$ is called a semi-edge-prime labeling if for each edge $uv$ in $E$, we have $GCD(f^+(u),f^+(v))=1$ or $f^+(u)=f^+(v)$. A graph that admits an  ...

To get a matching lower bound, we showed that as long as we can construct a finite projective plane of order d, then we have a bipartite graph that is C4-free and has Θ(n ) edges where the number of vertices n = d + d + 1. In this lecture we will construct such objects when d is a prime power. This construction is also due to Reiman 1958. Let q be a prime power, i.e. q = p where p is a prime an...

(a) Prove that for each integer n > 1 there are infinitely many (Z/nZ)-extensions of Q ramified at only one prime. (b) Prove that for each integer n > 2 there are no (Z/nZ)2-extensions of Q ramified at only one prime. Why does this not contradict the fact that (Z/pZ)2-extensions of Qp exists for every p? (c) Given an explicit example of a (Z/2Z)2-extension of Q ramified at only one prime (with ...

1 . Throughout this paper, we use the following notation : c•1 , c2 , . . ., X0 , X1 , . . . denote positive absolute constants. We denote the number of elements of the finite set S by BSI . We write ex =exp (x) . We denote the least prime factor of n by p(n) . We write pall n if pain but pa+1 f n . v(n) denotes the number of the distinct prime factors of n, while the number of all the prime fa...

Let K be an infinite compact space, let C(K) be the algebra of continuous complex-valued functions of K, let F be a well-ordered chain of nonmaximal prime ideals of C(K), let IF be the smallest element of F and let MF be the unique maximal ideal of C(K) containing the elements of F . Assuming the continuum hypothesis, we show that if |C(K)/IF | = 20 , and if there exists a sequence (Gn)n≥1 of s...

The objective of this study was to develop a palatable, precooked goat roast. Goat legs (n = 64) were fabricated into 220 roasts and assigned to one of four spice treatments: control (CON), Italian, Mexican, prime rib. After being injected with a 15-percent brine mixture, roasts were smoked to an internal temperature of 63°C, vacuum packaged, and frozen at -10°C. Roasts were thawed and reheated...