A complexity calculus for recursive tree algorithms

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Effective Bounds for P-Recursive Sequences

We describe an algorithm that takes as input a complex sequence $(u_n)$ given by a linear recurrence relation with polynomial coefficients along with initial values, and outputs a simple explicit upper bound $(v_n)$ such that $|u_n| \leq v_n$ for all $n$. Generically, the bound is tight, in the sense that its asymptotic behaviour matches that of $u_n$. We discuss applications to the evaluation of power series with guaranteed precision.Comment: 26 page

Transition probabilities for general birth-death processes with applications in ecology, genetics, and evolution

A birth-death process is a continuous-time Markov chain that counts the number of particles in a system over time. In the general process with $n$ current particles, a new particle is born with instantaneous rate $\lambda_n$ and a particle dies with instantaneous rate $\mu_n$. Currently no robust and efficient method exists to evaluate the finite-time transition probabilities in a general birth-death process with arbitrary birth and death rates. In this paper, we first revisit the theory of continued fractions to obtain expressions for the Laplace transforms of these transition probabilities and make explicit an important derivation connecting transition probabilities and continued fractions. We then develop an efficient algorithm for computing these probabilities that analyzes the error associated with approximations in the method. We demonstrate that this error-controlled method agrees with known solutions and outperforms previous approaches to computing these probabilities. Finally, we apply our novel method to several important problems in ecology, evolution, and genetics

Lowering β-Amyloid Levels Rescues Learning and Memory in a Down Syndrome Mouse Model

β-amyloid levels are elevated in Down syndrome (DS) patients throughout life and are believed to cause Alzheimer's disease (AD) in adult members of this population. However, it is not known if β-amyloid contributes to intellectual disability in younger individuals. We used a γ-secretase inhibitor to lower β-amyloid levels in young mice that model DS. This treatment corrected learning deficits characteristic of these mice, suggesting that β-amyloid-lowering therapies might improve cognitive function in young DS patients

Analytic combinatorics of non-crossing configurations

AbstractThis paper describes a systematic approach to the enumeration of ‘non-crossing’ geometric configurations built on vertices of a convex n-gon in the plane. It relies on generating functions, symbolic methods, singularity analysis, and singularity perturbation. Consequences are both exact and asymptotic counting results for trees, forests, graphs, connected graphs, dissections, and partitions. Limit laws of the Gaussian type are also established in this framework; they concern a variety of parameters like number of leaves in trees, number of components or edges in graphs, etc