Sequential escapes: onset of slow domino regime via a saddle connection

We explore sequential escape behaviour of coupled bistable systems under the influence of stochastic perturbations. We consider transient escapes from a marginally stable “quiescent” equilibrium to a more stable “active” equilibrium. The presence of coupling introduces dependence between the escape processes: for diffusive coupling there is a strongly coupled limit (fast domino regime) where the escapes are strongly synchronised while for intermediate coupling (slow domino regime) without partially escaped stable states, there is still a delayed effect. These regimes can be associated with bifurcations of equilibria in the low-noise limit. In this paper, we consider a localized form of non-diffusive (i.e. pulse-like) coupling and find similar changes in the distribution of escape times with coupling strength. However, we find transition to a slow domino regime that is not associated with any bifurcations of equilibria. We show that this transition can be understood as a codimension-one saddle connection bifurcation for the low-noise limit. At transition, the most likely escape path from one attractor hits the escape saddle from the basin of another partially escaped attractor. After this bifurcation, we find increasing coefficient of variation of the subsequent escape times.     

On the affordances of the MaxEP principle

Optimality principles have long been popular in the natural sciences and enjoyed much successes in various applications. However these principles seem to be disparate, each applied in limited contexts and there are far too many of them causing some consternation among scientists and philosophers of science regarding the ad-hoc nature of the optimality arguments. In this paper, we discuss the Maximum entropy production (MaxEP) as a plausible over-arching principle to understand stable configurations in fluid mechanics and related problems. The MaxEP being based upon sound physical arguments and in the immutable laws of thermodynamics along with the fact that it has been successfully co-opted across disciplines makes it worthy of attention. We discuss various physical and metaphysical aspects of this principle and use it to analyze some model problems regarding patterns in particle sedimentation such as sedimentation of a particle in Newtonian and non-Newtonian fluids and stable deformation of a falling droplet.     

Polygonal invariant curves for a planar piecewise isometry

We investigate a remarkable new planar piecewise isometry whose generating map is a permutation of four cones. For this system we prove the coexistence of an infinite number of periodic components and an uncountable number of transitive components. The union of all periodic components is an invariant pentagon with unequal sides. Transitive components are invariant curves on which the dynamics are conjugate to a transitive interval exchange. The restriction of the map to the invariant pentagonal region is the first known piecewise isometric system for which there exist an infinite number of periodic components but the only aperiodic points are on the boundary of the region. The proofs are based on exact calculations in a rational cyclotomic field. We use the system to shed some light on a conjecture that PWIs can possess transitive invariant curves that are not smooth.

     

Unfolding the torus: Oscillator geometry from time delays

Summary We present a simple method of plotting the trajectories of systems of weakly coupled oscillators. Our algorithm uses the time delays between the “firings” of the oscillators. For any system ofn weakly coupled oscillators there is an attracting invariantn-dimensional torus, and the attractor is a subset of this invariant torus. The invariant torus intersects a suitable codimension-1 surface of section at an (n−1)-dimensional torus. The dynamics ofn coupled oscillators can thus be reduced,in principle, to the study of Poincaré maps of the (n−1)-dimensional torus. This paper gives apractical algorithm for measuring then−1 angles on the torus. Since visualization of 3 (or higher) dimensional data is difficult we concentrate onn=3 oscillators. For three oscillators, a standard projection of the Poincaré map onto the plane yields a projection of the 2-torus which is 4-to-1 over most of the torus, making it difficult to observe the structure of the attractor. Our algorithm allows a direct measurement of the 2 angles on the torus, so we can plot a 1-to-1 map from the invariant torus to the “unfolded torus” where opposite edges of a square are identified. In the cases where the attractor is a torus knot, the knot type of the attractor is obvious in our projection.     

Designing tubular conducting polymer actuators for wireless electropumping

Rational design and shaping of soft smart materials offer potential applications that cannot be addressed with rigid systems. In particular, electroresponsive elastic materials are well-suited for developing original active devices, such as pumps and actuators. However, applying the electric stimulus requires usually a physical connection between the active part and a power supply. Here we report about the design of an electromechanical system based on conducting polymers, enabling the actuation of a wireless microfluidic pump. Using the electric field-induced asymmetric polarization of miniaturized polypyrrole tubes, it is possible to trigger simultaneously site-specific chemical reactions, leading to shrinking and swelling in aqueous solution without any physical connection to a power source. The complementary electrochemical reactions occurring at the opposite extremities of the tube result in a differential change of its diameter. In turn, this electromechanical deformation allows inducing highly controlled fluid dynamics. The performance of such a remotely triggered electrochemically active soft pump can be fine-tuned by optimizing the wall thickness, length and inner diameter of the material. The efficient and fast actuation of the polymer pump opens up new opportunities for actuators in the field of fluidic or microfluidic devices, such as controlled drug release, artificial organs and bioinspired actuators.

Tubular conducting polymer actuators are used for developing a wireless electropumping device. Bipolar electrochemistry, allowing symmetry breaking in terms of polarization and electrochemical reactions, is the key ingredient for efficient pumping.     

Interpolation of Sparse Multivariate Polynomials over Large Finite Fields with Applications

We develop a Las Vegas-randomized algorithm which performs interpolation of sparse multivariate polynomials over finite fields. Our algorithm can be viewed as the first successful adaptation of the sparse polynomial interpolation algorithm for the complex field developed by M. Ben-Or and P. Tiwari (1988, in “Proceedings of the 20th ACM Symposium on the Theory of Computing,” pp. 301–309, Assoc. Comput. Mach., New York) to the case of finite fields. It improves upon a previous result by D. Y. Grigoriev, M. Karpinski, and M. F. Singer (1990, SIAM J. Comput.19, 1059–1063) and is by far the most time efficient algorithm (time and processor efficient parallel algorithm) for the problem when the finite field is large. As applications, we obtain Monte Carlo-randomized parallel algorithms for sparse multivariate polynomial factorization and GCD over finite fields. The efficiency of these algorithms improves upon that of the previously known algorithms for the respective problems.     

Stereoselective Lewis acid-catalyzed alpha-acylvinyl additions

Silyloxyallenes derived from alpha-hydroxypropargylsilanes undergo efficient addition to aldehydes with catalytic amounts of Lewis acids. The allenes are accessed from the corresponding propargylsilanes in a base-catalyzed 1,2-Brook rearrangement/SE2' process. Enantioenriched propargylsilanes are synthesized by a new zinc-promoted addition of alkynes to acylsilanes in up to 74% ee. This work demonstrates that conversion to the silyloxyallenes occurs with minimal erosion in optical activity. The use of a chiral chromium(III) Lewis acid effects the catalytic asymmetric addition of racemic silyloxyallenes to aromatic aldehydes in up to 92% ee. The overall reaction is broad in scope and accommodates a wide variety of aromatic and aliphatic substituents on both the propargylsilane and aldehyde.     

On the discrepancy between two Zagreb indices

We examine the quantity

$S\left(G\right)=\sum _{uv\in E\left(G\right)}min(degu,degv)$

over sets of graphs with a fixed number of edges. The main result shows the maximum possible value of $S\left(G\right)$ is achieved by three different classes of constructions, depending on the distance between the number of edges and the nearest triangular number. Furthermore we determine the maximum possible value when the set of graphs is restricted to be bipartite, a forest, or to be planar given sufficiently many edges. The quantity $S\left(G\right)$ corresponds to the difference between two well studied indices, the irregularity of a graph and the sum of the squares of the degrees in a graph. These are known as the first and third Zagreb indices in the area of mathematical chemistry.