Prime and Semiprime Inner Functions

This paper studies the structure of inner functions under the operation of composition, and in particular the notions or primeness and semiprimeness. Results proved include the density of prime finite Blaschke products in the set of finite Blaschke products, the semiprimeness of finite products of thin Blaschke products and their approximability by prime Blaschke products. An example of a nonsemiprime Blaschke product that is a Frostman Blaschke product is also provided.

Prime and semiprime inner functions

Abstract This paper studies the structure of inner functions under the operation of composition, and in particular the notions or primeness and semiprimeness. Results proved include the density of prime finite Blaschke products in the set of finite Blaschke products, the semiprimeness of finite products of thin Blaschke products and their approximability by prime Blaschke products. An example of a nonsemiprime Blaschke product that is a Frostman Blaschke product is also provided.

Measuring Correlation Between HIV Sequence Variation and Disease Stage of HIV-Infected Individuals

With a virus such as Human Immunodeficiency Virus (HIV) that has infected millions of people worldwide, and with many unaware that they are infected, it becomes vital to understand how the virus works and how it functions at the molecular level. Because there currently is no vaccine and no way to eradicate the virus from an infected person, any information about how the virus interacts with its host greatly increases the chances of understanding how HIV works and brings scientists one step closer to being able to combat such a destructive virus. Thousands of HIV viruses have been sequenced and are available in many online databases for public use. Attributes that are linked to each sequence include the viral load within the host and how sick the patient is currently. Being able to predict the stage of infection for someone is a valuable resource, as it could potentially aid in treatment options and proper medication use. Our approach of analyzing region-specific amino acid composition for select genes has been able to predict patient disease state up to an accuracy of 85.4%. Moreover, we output a set of classification rules based on the sequence that may prove useful for diagnosing the expected clinical outcome of the infected patient.

Ultrabithorax Confers Spatial Identity in a Context-Specific Manner in the Drosophila Postembryonic Ventral Nervous System

In holometabolous insects such as Drosophila melanogaster, neuroblasts produce an initial population of diverse neurons during embryogenesis and a much larger set of adult-specific neurons during larval life. In the ventral CNS, many of these secondary neuronal lineages differ significantly from one body segment to another, suggesting a role for anteroposterior patterning genes. Here we systematically characterize the expression pattern and function of the Hox gene Ultrabithorax (Ubx) in all 25 postembryonic lineages. We find that Ubx is expressed in a segment-, lineage-, and hemilineage-specific manner in the thoracic and anterior abdominal segments. When Ubx is removed from neuroblasts via mitotic recombination, neurons in these segments exhibit the morphologies and survival patterns of their anterior thoracic counterparts. Conversely, when Ubx is ectopically expressed in anterior thoracic segments, neurons exhibit complementary posterior transformation phenotypes. Our findings demonstrate that Ubx plays a critical role in conferring segment-appropriate morphology and survival on individual neurons in the adult-specific ventral CNS. Moreover, while always conferring spatial identity in some sense, Ubx has been co-opted during evolution for distinct and even opposite functions in different neuronal hemilineages.