BACKGROUND Goblet cell hyperplasia is a classic but variable pathologic finding in COPD. Current literature shows that smoking is a risk factor for chronic bronchitis but the relationship of these clinical features to the presence and magnitude of large airway goblet cell hyperplasia has not been well described. We hypothesized that current smokers and chronic bronchitics would have more goblet...

An interesting problem of algebraic computation is the computation of the greatest common divisor (GCD) of a set of polynomials. The GCD is usually linked with the characterisation of zeros of a polynomial matrix description of a system. The problem of finding the GCD of a set of n polynomials on R[x] of a maximal degree q is a classical problem that has been considered before (Karcanias et al....

In this article, we prove the existence and uniqueness of a certain distribution function on the unit interval. This distribution appears in Brent's model of the analysis of the binary gcd algorithm. The existence and uniqueness of such a function has been conjectured by Richard Brent in his original paper [1]. Donald Knuth also supposes its existence in [4] where developments of its properties...

1 SMP-based parallel algorithms and implementations for polynomial factoring and GCD are overviewed. Topics include polynomial factoring modulo small primes, univariate and multivariate p-adic lifting, and reformulation of lift basis. Sparse polynomial GCD is also covered.

We consider a generalization of the gcd-sum function, and obtain its average order with a quasi-optimal error term. We also study the reciprocals of the gcd-sum and lcm-sum functions.

A family of graphs possesses the common gcd property if the greatest common divisor of the degree sequence of each graph in the family is the same. In particular, any family of trees has the common gcd property. Let F = {H1, . . . ,Hr} be a family of graphs having the common gcd property, and let d be the common gcd. It is proved that there exists a constant N = N(F ) such that for every n > N ...

A 3-equitable prime cordial labeling of a graph G with vertex set V is a bijection f from V to {1, 2, ..., |V |} such that if an edge uv is assigned the label 1 if gcd(f(u), f(v)) = 1 and gcd(f(u) + f(v), f(u)−f(v)) = 1, the label 2 if gcd(f(u), f(v)) = 1 and gcd(f(u) + f(v), f(u)− f(v)) = 2 and 0 otherwise, then the number of edges labeled with i and the number of edges labeled with j differ b...

Richard Zippel’s sparse modular GCD algorithm is widely used to compute the monic greatest common divisor (GCD) of two multivariate polynomials over Z. In this report, we present how this algorithm can be modified to solve the GCD problem for polynomials over finite fields of small cardinality. When the GCD is not monic, Zippel’s algorithm cannot be applied unless the normalization problem is r...