We consider a multitype branching process with immigration in a random environment introduced by Key in [12]. It was shown by Key that the branching process is subcritical in the sense that it converges to a proper limit law. We complement this result by a strong law of large numbers and a central limit theorem for the partial sums of the process. In addition, we study the asymptotic behavior o...

Recent experimental work in lung morphogenesis has described an elegant pattern of branching phenomena. Two primary forms of branching have been identified: side branching and tip splitting. In our previous study of lung branching morphogenesis, we used a 4 variable partial differential equation (PDE), due to Meinhardt, as our mathematical model to describe the reaction and diffusion of morphog...

Axon branching plays a critical role in establishing the accurate patterning of neuronal circuits in the brain. However, the mechanisms that control axon branching remain poorly understood. Here we report that knockdown of the brain-enriched signaling protein JNK-interacting protein 3 (JIP3) triggers exuberant axon branching and self-contact in primary granule neurons of the rat cerebellar cort...

During mammary gland development, an epithelial bud undergoes branching morphogenesis to expand into a continuous tree-like network of branched ducts [1]. The process involves multiple cell types that are coordinated by hormones and growth factors coupled with signaling events including Wnt and Hedgehog [2-5]. Primary cilia play key roles in the development of many organs by coordinating extrac...

This paper constructs a cirquent calculus system and proves its soundness and completeness with respect to the semantics of computability logic. The logical vocabulary of the system consists of negation ¬, parallel conjunction ∧, parallel disjunction ∨, branching recurrence ◦ | , and branching corecurrence ◦| . The article is published in two parts, with (the previous) Part I containing prelimi...

We investigate the x log x condition for a general (Crump–Mode– Jagers) multi-type branching process with arbitrary type space by constructing a size-biased population measure that relates to the ordinary population measure via the intrinsic martingale Wt. Sufficiency of the x log x condition for a non-degenerate limit of Wt is proved and conditions for necessity are investigated. GENERAL BRANC...

This paper constructs a cirquent calculus system and proves its soundness and completeness with respect to the semantics of computability logic. The logical vocabulary of the system consists of negation ¬, parallel conjunction ∧, parallel disjunction ∨, branching recurrence ◦ | , and branching corecurrence ◦| . The article is published in two parts, with (the present) Part I containing prelimin...

This paper surveys the recent results on the complexity of Boolean functions in terms of Boolean circuits, formulas, and branching programs. The primary aim is to give accessible proofs of the more di cult theorems proving lower bounds on the complexity of speci c functions in restricted computational models. These include bounded depth circuits, monotone circuits, and bounded width branching p...

We survey multivariate limit theorems in the framework of the contraction method for recursive sequences as arising in the analysis of algorithms, random trees or branching processes. We compare and improve various general conditions under which limit laws can be obtained, state related open problems and given applications to the analysis of algorithms and branching recurrences. AMS subject cla...