We discuss the influence of positive and negative feedback on the stability of a system, which is not clear-cut, and involves complex, mathematical problems. We show in particular that positive feedback can have a stabilising effect on some systems. We also point out the role that positive feedback plays in the digital treatment of signals required by cellular signalling, drawing on analogies f...

Positive feedback is a ubiquitous signal transduction motif that allows systems to convert graded inputs into decisive, all-or-none outputs. Here we investigate why the positive feedback switches that regulate polarization of budding yeast, calcium signaling, Xenopus oocyte maturation, and various other processes use multiple interlinked loops rather than single positive feedback loops. Mathema...

Mathematical analysis of positive feedback loops in proteolytic systems has previously suggested that when the active enzymes are subject to inhibitory control these systems will exhibit threshold behavior. This is demonstrated in the present study, for the autolytic activation of factor XII in the presence of a contact activator and an irreversible inhibitor of factor XIIa. The threshold betwe...

“Hybrid” climate policy simulations have sought to bridge the gap between “bottom-up” engineering and “top-down” macroeconomic models by integrating the former's energy technology detail into the latter's macroeconomic framework. Construction of hybrid models is complicated by the need to numerically calibrate them to multiple, incommensurate sources of economic and engineering data. I develop ...

A state in quantum mechanics is defined as a positive operator of norm 1. For finite systems, this may be thought of as a positive matrix of trace 1. This constraint of positivity imposes severe restrictions on the allowed evolution of such a state. From the mathematical viewpoint, we describe the two forms of standard dynamical equations global (Kraus) and local (Lindblad) and show how each of...

We show a new and constructive proof of the two-squares theorem, based on a somewhat unusual, but very effective, way of rewriting the so-called extended Euclid’s algorithm. Rather than simply verifying the result — as it is usually done in the mathematical community — we use Euclid’s algorithm as an interface to investigate which numbers can be written as sums of two positive squares. The prec...

This is not a proof. (No good proof uses “and so forth” to gloss over a key point!) But the idea is as follows: we can keep iterating the above argument as many times as we want, deducing at each stage that since S contains the natural number which is one greater than the last natural number we showed that it contained. Now it is a fundamental part of the structure of the positive integers that...

Mathematical cognition, paragon of abstraction, is marked by systematic associations between number and space, often described as “mental number lines.” In three experiments, we demonstrate a novel effect: a sagittal (back-to-front) number line for negative and positive integers. In a paradigm requiring full-body movements, participants judged numerical magnitude (Exp. 1-2) and parity (Exp. 3)....

Viruses can amplify their genomes following different replication modes (RMs) ranging from the stamping machine replication (SMR) model to the geometric replication (GR) model. Different RMs are expected to produce different evolutionary and dynamical outcomes in viral quasispecies due to differences in the mutations accumulation rate. Theoretical and computational models revealed that while SM...