Resolvent Positive Linear Operators Exhibit the Reduction Phenomenon

The spectral bound, s(a A + b V), of a combination of a resolvent positive linear operator A and an operator of multiplication V, was shown by Kato to be convex in b \in R. This is shown here, through an elementary lemma, to imply that s(a A + b V) is also convex in a > 0, and notably, \partial s(a A + b V) / \partial a <= s(A) when it exists. Diffusions typically have s(A) <= 0, so that for diffusions with spatially heterogeneous growth or decay rates, greater mixing reduces growth. Models of the evolution of dispersal in particular have found this result when A is a Laplacian or second-order elliptic operator, or a nonlocal diffusion operator, implying selection for reduced dispersal. These cases are shown here to be part of a single, broadly general, `reduction' phenomenon.Comment: 7 pages, 53 citations. v.3: added citations, corrections in introductory definitions. v.2: Revised abstract, more text, and details in new proof of Lindqvist's inequalit

Localization of Denaturation Bubbles in Random DNA Sequences

We study the thermodynamic and dynamic behaviors of twist-induced denaturation bubbles in a long, stretched random sequence of DNA. The small bubbles associated with weak twist are delocalized. Above a threshold torque, the bubbles of several tens of bases or larger become preferentially localized to \AT-rich segments. In the localized regime, the bubbles exhibit ``aging'' and move around sub-diffusively with continuously varying dynamic exponents. These properties are derived using results of large-deviation theory together with scaling arguments, and are verified by Monte-Carlo simulations.Comment: TeX file with postscript figure

Stochastic Chemical Reactions in Micro-domains

Traditional chemical kinetics may be inappropriate to describe chemical reactions in micro-domains involving only a small number of substrate and reactant molecules. Starting with the stochastic dynamics of the molecules, we derive a master-diffusion equation for the joint probability density of a mobile reactant and the number of bound substrate in a confined domain. We use the equation to calculate the fluctuations in the number of bound substrate molecules as a function of initial reactant distribution. A second model is presented based on a Markov description of the binding and unbinding and on the mean first passage time of a molecule to a small portion of the boundary. These models can be used for the description of noise due to gating of ionic channels by random binding and unbinding of ligands in biological sensor cells, such as olfactory cilia, photo-receptors, hair cells in the cochlea.Comment: 33 pages, Journal Chemical Physic

Generalized Rayleigh and Jacobi processes and exceptional orthogonal polynomials

We present four types of infinitely many exactly solvable Fokker-Planck equations, which are related to the newly discovered exceptional orthogonal polynomials. They represent the deformed versions of the Rayleigh process and the Jacobi process.Comment: 17 pages, 4 figure

Complexity in the Classroom Workshop: Teaching and Learning the Cynefin Framework by Applying it to the Classroom

Complex adaptive systems are both an important fundamental principle in systems engineering education and a reality of all engineering education. The Cynefin framework, as created by Snowden and Boone (2007), is a decision-making tool that helps the engineer recognize the type of system within which they are operating and then respond in a manner that is appropriate for the cause-and-effect relationships associated with that system type. The types of system, or the domains, fall into five categories and their liminal spaces: obvious, where the cause-and-effect relationships are clear to everyone involved; complicated, where the cause-and-effect relationships are clear to those who have appropriate expertise; complex, where the cause-and-effect relationships are not predictable or necessarily even visible; chaos, where there are no cause-and-effect relationships; and disorder, where it is unclear what system context should be the focus. In this hands-on SPECIAL SESSION, participants will explore a new way to teach complex adaptive systems by experiencing it. The new pedagogical application is to use a variation on the collaborative inquiry technique where learners move through one or more cycles of delving into a system (collecting evidence), experience discussion guided through the Cynefin framework, and shared reflection on the meaning of the systems domain knowledge to operating and thriving in the system. The system we will use in the special session is a multi-institutional course wherein participants will be able to explore how additional layers of complexity and their changing cause-and-effect relationships impact pedagogical decisions to create different learning experiences. The course, cardio-vascular engineering, is an example of systems engineering topics taught in a biomedical engineering environment. The facilitators of this special session include two faculty who have experience in both teaching systems engineering and in collaborative inquiry, as well as two faculty who are part of the creation and delivery of the cardio-vascular engineering course. The course is offered simultaneously over multiple institutions with a unified syllabus that accounts for learning needs and contexts of all the students. Learning objectives for the special session include: • Increase knowledge of the Cynefin framework of complex systems; • Practice a pedagogical technique for teaching systems engineering concepts; • Reflect on using systems engineering fundamental knowledge to create learning environments in different ways, particularly as the context needs of learners and industry continue to change; and • Gain exposure to a successful course taught simultaneously across multiple institutions and student levels

Consequences of an incorrect model specification on population growth

We consider stochastic differential equations to model the growth of a population ina randomly varying environment. These growth models are usually based on classical deterministic models, such as the logistic or the Gompertz models, taken as approximate models of the "true" (usually unknown) growth rate. We study the effect of the gap between the approximate and the "true" model on model predictions, particularly on asymptotiv behavior and mean and variance of the time to extinction of the population

Statistical mechanics of ecosystem assembly

We introduce a toy model of ecosystem assembly for which we are able to map out all assembly pathways generated by external invasions. The model allows to display the whole phase space in the form of an assembly graph whose nodes are communities of species and whose directed links are transitions between them induced by invasions. We characterize the process as a finite Markov chain and prove that it exhibits a unique set of recurrent states (the endstate of the process), which is therefore resistant to invasions. This also shows that the endstate is independent on the assembly history. The model shares all features with standard assembly models reported in the literature, with the advantage that all observables can be computed in an exact manner.Comment: Accepted for publication in Physical Review Letter