Inherent size constraints on prokaryote gene networks due to "accelerating" growth

Networks exhibiting "accelerating" growth have total link numbers growing faster than linearly with network size and can exhibit transitions from stationary to nonstationary statistics and from random to scale-free to regular statistics at particular critical network sizes. However, if for any reason the network cannot tolerate such gross structural changes then accelerating networks are constrained to have sizes below some critical value. This is of interest as the regulatory gene networks of single celled prokaryotes are characterized by an accelerating quadratic growth and are size constrained to be less than about 10,000 genes encoded in DNA sequence of less than about 10 megabases. This paper presents a probabilistic accelerating network model for prokaryotic gene regulation which closely matches observed statistics by employing two classes of network nodes (regulatory and non-regulatory) and directed links whose inbound heads are exponentially distributed over all nodes and whose outbound tails are preferentially attached to regulatory nodes and described by a scale free distribution. This model explains the observed quadratic growth in regulator number with gene number and predicts an upper prokaryote size limit closely approximating the observed value.Comment: Corrected error in biological input parameter: 15 pages, 10 figure

Finding a point in the relative interior of a polyhedron

A new initialization or `Phase I' strategy for feasible interior point methods for linear programming is proposed that computes a point on the primal-dual central path associated with the linear program. Provided there exist primal-dual strictly feasible points - an all-pervasive assumption in interior point method theory that implies the existence of the central path - our initial method (Algorithm 1) is globally Q-linearly and asymptotically Q-quadratically convergent, with a provable worst-case iteration complexity bound. When this assumption is not met, the numerical behaviour of Algorithm 1 is highly disappointing, even when the problem is primal-dual feasible. This is due to the presence of implicit equalities, inequality constraints that hold as equalities at all the feasible points. Controlled perturbations of the inequality constraints of the primal-dual problems are introduced - geometrically equivalent to enlarging the primal-dual feasible region and then systematically contracting it back to its initial shape - in order for the perturbed problems to satisfy the assumption. Thus Algorithm 1 can successfully be employed to solve each of the perturbed problems.\ud We show that, when there exist primal-dual strictly feasible points of the original problems, the resulting method, Algorithm 2, finds such a point in a finite number of changes to the perturbation parameters. When implicit equalities are present, but the original problem and its dual are feasible, Algorithm 2 asymptotically detects all the primal-dual implicit equalities and generates a point in the relative interior of the primal-dual feasible set. Algorithm 2 can also asymptotically detect primal-dual infeasibility. Successful numerical experience with Algorithm 2 on linear programs from NETLIB and CUTEr, both with and without any significant preprocessing of the problems, indicates that Algorithm 2 may be used as an algorithmic preprocessor for removing implicit equalities, with theoretical guarantees of convergence

Genetically modified galaxies: performing controlled experiments in cosmological galaxy formation simulations

This thesis develops and applies a novel approach to studying the formation of galaxies in our Universe. Galaxies grow through gravitational amplification of early-Universe overdensities, within which gas reaches sufficient densities to trigger star formation. A galaxy's mass growth is therefore seeded randomly, originating from quantum inflationary perturbations. Understanding how this intrinsic stochasticity in histories couples with strongly non-linear astrophysics is key to interpreting the observed diversity of the galaxy population. To provide new insights to this issue, we clarify and extend the "genetic modification'' framework in Chapter 2. This approach generates alternative versions of a simulation’s initial conditions, each version with a carefully engineered change to the galaxy’s history. This in turn creates controlled experiments allowing us to construct a causal account of the galaxy's response to modifying its merger history. We introduce a new class of variance modifications aiming at improving control over several mergers. We then evolve these variance-modified initial conditions using the simulation code RAMSES, first studying dark matter halo formation (Chapter 3). We causally recover the known correlation between halo formation time and concentration when modifying the merger histories of two haloes, and further establish how late major mergers determine concentrations at fixed formation time. We then turn to the formation of ultra-faint dwarf galaxies with high-resolution hydrodynamical simulations. Scanning through histories, we demonstrate that earlier forming ultra-faints have higher stellar mass today and predict a new class of highly diffuse ultra-faint galaxies which assemble through late mergers (Chapter 4). We finally use a larger suite of objects (Chapter 5) to show how ultra-faints growing sufficiently in dynamical mass after reionization can accrete gas and re-ignite star formation. We conclude that, by transforming cosmological histories into tuneable parameters, "genetically modified'' experiments generate new insights on the complexity of dark matter halo and galaxy formation

A Review on Distribution Network Reconfiguration

In distribution systems, network reconfiguration is comprehended by changing the best possible status of placing the sectionalizing switches. It is generally performed for reducing the losses or for balancing the load in the system. Generally for the distribution systems, meshed networks are configured in a feeble manner with manifold supply points, but they are controlled with radial configurations by unbolting the redundant branches. The system should be recompensed, otherwise it creates disturbance which lead to variation in supply voltages. In this paper, a study related to network reconfiguration techniques is presented in order to avoid network congestion showing the importance of FACTS devices along with an overview of issues related to voltage collapses in distribution system

Projection methods in conic optimization

There exist efficient algorithms to project a point onto the intersection of a convex cone and an affine subspace. Those conic projections are in turn the work-horse of a range of algorithms in conic optimization, having a variety of applications in science, finance and engineering. This chapter reviews some of these algorithms, emphasizing the so-called regularization algorithms for linear conic optimization, and applications in polynomial optimization. This is a presentation of the material of several recent research articles; we aim here at clarifying the ideas, presenting them in a general framework, and pointing out important techniques

On the relationship between control barrier functions and projected dynamical systems

In this paper, we study the relationship between systems controlled via Control Barrier Function (CBF) approaches and a class of discontinuous dynamical systems, called Projected Dynamical Systems (PDSs). In particular, under appropriate assumptions, we show that the vector field of CBF-controlled systems is a Krasovskii-like perturbation of the set-valued map of a differential inclusion, that abstracts PDSs. This result provides a novel perspective to analyze and design CBF-based controllers. Specifically, we show how, in certain cases, it can be employed for designing CBF-based controllers that, while imposing safety, preserve asymptotic stability and do not introduce undesired equilibria or limit cycles. Finally, we briefly discuss about how it enables continuous implementations of certain projection-based controllers, that are gaining increasing popularity.Comment: To be presented at the 62nd IEEE Conference on Decision and Control, Dec. 13-15, 2023, Singapor