A number n is said to be ordinary if the smallest number with exactly n divisors is p q1−1 1 · · · p qa−1 a where q1 · · · qa is the prime factorization of n and q1 ≥ . . . ≥ qa (and where pk denotes the k-th prime). We show here that all square–free numbers are ordinary and that the set of ordinary numbers has natural density one.

and Applied Analysis 3 From the ideas of Theorem D to Theorem F, it is natural to ask whether the values a, b in Theorem C can be replaced by two polynomials Q1, Q2. The main purpose of this paper is to investigate this problem. We prove the following result. Theorem 1.2. Let Q1 z a1z a1,p−1zp−1 · · · a1,0 and Q2 z a2z a2,p−1zp−1 · · · a2,0 be two polynomials such that degQ1 z degQ2 z p (where ...

We have used time-lapse video microscopy to study the behavior of a neuron, Q1, that pioneers the posterior commissure of the embryonic grasshopper. Our goal is to use time-lapse video as a tool to acquire a precise picture of normal development over time, and thereby identify stereotypic activities that might indicate important interactions necessary for proper formation of the commissure. We ...

Astrid Oberson, Else K. Bünemann, Dennis K. Friesen, I.M. Rao, Paul C. Smithson, Benjamin L. Turner, and Emmanuel Frossard, Institute of Plant Sciences, Swiss Federal Institute of Technology, Zurich, Switzerland School of Earth and Environmental Sciences, University of Adelaide, Adelaide, Australia International Center for the Improvement of Maize and Wheat (CIMMYT) and International Q1 Fertili...

MOTIVATION Gene class testing (GCT) is a statistical approach to determine whether some functionally predefined classes of genes express differently under two experimental conditions. GCT computes the P-value of each gene class based on the null distribution and the gene classes are ranked for importance in accordance with their P-values. Currently, two null hypotheses have been considered: the...

Let the functions q1 be analytic and let q2 be analytic univalent in the unit disk. Using the methods of differential subordination and superordination, sufficient conditions involving the Schwarzian derivative of a normalized analytic function f are obtained so that either q1 z ≺ zf ′ z /f z ≺ q2 z or q1 z ≺ 1 zf ′′ z /f ′ z ≺ q2 z . As applications, sufficient conditions are determined relati...