Francis C. Motta

Highly ordered nanoscale surface ripples produced by ion bombardment of binary compounds

Nanoscale surface ripples generated by oblique-incidence ion bombardment of a solid are generally full of defects, and this has prevented the widespread adoption of ion bombardment as a nanofabrication tool. We advance a theory that predicts that remarkably defect-free ripples can be produced by ion bombardment of a binary material if the ion species, energy and angle of incidence are appropriately chosen. This high degree of order results from the coupling between the surface height and composition, and cannot be achieved by bombarding an elemental material.

Random cyclic dynamical systems

For X a finite subset of the circle and for 0 X which maps each point to the clockwise furthest element of X within angular distance less than 2 pi r. We study the discrete dynamical system on X generated by f_r, and especially its expected behavior when X is a large random set. We show that, as |X| -> infinity, the expected fraction of periodic points of f_r tends to 0 if r is irrational and to 1/q if r = p/q is rational with p and q coprime. These results are obtained via more refined statistics of f_r which we compute explicitly in terms of (generalized) Catalan numbers. The motivation for studying f_r comes from Vietoris-Rips complexes, a geometric construction used in computational topology. Our results determine how much one can expect to simplify the Vietoris-Rips complex of a random sample of the circle by removing dominated vertices.

Topological Data Analysis: Developments and Applications

Topological Data Analysis (TDA) and its mainstay computational device, persistent homology (PH), has established a strong track record of providing researchers across the data-driven sciences with new insights and methodologies by characterizing low-dimensional geometric structures in high-dimensional data. When combined with machine learning (ML) methods, PH is valued as a discriminating-feature extraction tool. This work highlights many of the recent successes at the intersection of TDA and ML, introduces some of the foundational mathematics underpinning TDA, and summarizes the efforts to strengthen the bridge between TDA and ML. Thus, this document is a launching point for experimentalists and theoreticians to consider what can be learned from the shape of their data.

Optimally Topologically Transitive Orbits in Discrete Dynamical Systems

Abstract Every orbit of a rigid rotation of a circle by a fixed irrational angle is dense. However, the apparent uniformity of the distribution of iterates after a finite number of iterations appears strikingly different for various choices of a rotation angle. Motivated by this observation, we introduce a scalar function on the orbits of a discrete dynamical system defined on a bounded metric space, called the linear limit density, which we interpret as a measure of an orbits approach to density. Utilizing the three-distance theorem, we compute the exact value of the linear limit density of orbits of rigid rotations by irrational rotation angles with period-1 continued fraction expansions. We further show that any discrete dynamical system defined by an orientation-preserving diffeomorphism of the circle has an orbit with a larger linear limit density than any orbit of the rigid rotation by the golden number. Bernoulli shift maps acting on sequences over a finite alphabet provide another illustrative class of dynamical systems with dense orbits. Our study of the efficiency of an orbits approach to density leads us to demonstrate the existence of a class of infinite sequences with finite linear limit density constructed by recursively extending finite de Bruijn sequences.

Reconciling conflicting models for global control of cell-cycle transcription

ABSTRACT Models for the control of global cell-cycle transcription have advanced from a CDK-APC/C oscillator, a transcription factor (TF) network, to coupled CDK-APC/C and TF networks. Nonetheless, current models were challenged by a recent study that concluded that the cell-cycle transcriptional program is primarily controlled by a CDK-APC/C oscillator in budding yeast. Here we report an analysis of the transcriptome dynamics in cyclin mutant cells that were not queried in the previous study. We find that B-cyclin oscillation is not essential for control of phase-specific transcription. Using a mathematical model, we demonstrate that the function of network TFs can be retained in the face of significant reductions in transcript levels. Finally, we show that cells arrested at mitotic exit with non-oscillating levels of B-cyclins continue to cycle transcriptionally. Taken together, these findings support a critical role of a TF network and a requirement for CDK activities that need not be periodic.