Raz Kupferman

Geometry and Mechanics in the Opening of Chiral Seed Pods

Two joined latex strips show complex twisting behavior similar to that of seed pods. We studied the mechanical process of seed pods opening in Bauhinia variegate and found a chirality-creating mechanism, which turns an initially flat pod valve into a helix. We studied configurations of strips cut from pod valve tissue and from composite elastic materials that mimic its structure. The experiments reveal various helical configurations with sharp morphological transitions between them. Using the mathematical framework of “incompatible elasticity,” we modeled the pod as a thin strip with a flat intrinsic metric and a saddle-like intrinsic curvature. Our theoretical analysis quantitatively predicts all observed configurations, thus linking the pod’s microscopic structure and macroscopic conformation. We suggest that this type of incompatible strip is likely to play a role in the self-assembly of chiral macromolecules and could be used for the engineering of synthetic self-shaping devices.

Extracting macroscopic dynamics: model problems and algorithms

In many applications, the primary objective of numerical simulation of timeevolving systems is the prediction of coarse-grained, or macroscopic, quantities. The purpose of this review is twofold: first, to describe a number of simple model systems where the coarse-grained or macroscopic behaviour of a system can be explicitly determined from the full, or microscopic, description; and second, to overview some of the emerging algorithmic approaches that have been introduced to extract effective, lower-dimensional, macroscopic dynamics. The model problems we describe may be either stochastic or deterministic in both their microscopic and macroscopic behaviour, leading to four possibilities in the transition from microscopic to macroscopic descriptions. Model problems are given which illustrate all four situations, and mathematical tools for their study are introduced. These model problems are useful in the evaluation of algorithms. We use specific instances of the model problems to illustrate these algorithms. As the subject of algorithm development and analysis is, in many cases, in its infancy, the primary purpose here is to attempt to unify some of the emerging ideas so that individuals new to the field have a structured access to the literature. Furthermore, by discussing the algorithms in the context of the model problems, a platform for understanding existing algorithms and developing new ones is built.

Optimal prediction with memory

Abstract Optimal prediction methods estimate the solution of nonlinear time-dependent problems when that solution is too complex to be fully resolved or when data are missing. The initial conditions for the unresolved components of the solution are drawn from a probability distribution, and their effect on a small set of variables that are actually computed is evaluated via statistical projection. The formalism resembles the projection methods of irreversible statistical mechanics, supplemented by the systematic use of conditional expectations and new methods of solution for an auxiliary equation, the orthogonal dynamics equation, needed to evaluate a non-Markovian memory term. The result of the computations is close to the best possible estimate that can be obtained given the partial data. We present the constructions in detail together with several useful variants, provide simple examples, and point out the relation to the fluctuation–dissipation formulas of statistical physics.

Elastic theory of unconstrained non-Euclidean plates

Non-Euclidean plates are a subset of the class of elastic bodies having no stress-free configuration. Such bodies exhibit residual stress when relaxed from all external constraints, and may assume complicated equilibrium shapes even in the absence of external forces. In this work we present a mathematical framework for such bodies in terms of a covariant theory of linear elasticity, valid for large displacements. We propose the concept of non-Euclidean plates to approximate many naturally formed thin elastic structures. We derive a thin plate theory, which is a generalization of existing linear plate theories, valid for large displacements but small strains, and arbitrary intrinsic geometry. We study a particular example of a hemispherical plate. We show the occurrence of a spontaneous buckling transition from a stretching dominated configuration to bending dominated configurations, under variation of the plate thickness.

No justified complaints: on fair sharing of multiple resources

Fair allocation has been studied intensively in both economics and computer science. This subject has aroused renewed interest with the advent of virtualization and cloud computing. Prior work has typically focused on mechanisms for fair sharing of a single resource. We consider a variant where each user is entitled to a certain fraction of the systems resources, and has a fixed usage profile describing how much he would want from each resource. We provide a new definition for the simultaneous fair allocation of multiple continuously-divisible resources that we call bottleneck-based fairness (BBF). Roughly speaking, an allocation of resources is considered fair if every user either gets all the resources he wishes for, or else gets at least his entitlement on some bottleneck resource, and therefore cannot complain about not receiving more. We show that BBF has several desirable properties such as providing an incentive for sharing, and also promotes high overall utilization of resources; we also compare BBF carefully to another notion of fairness proposed recently, dominant resource fairness. Our main technical result is that a fair allocation can be found for every combination of user requests and entitlements. The allocation profile of each user is proportionate to the users profile of requests. The main problem is that the bottleneck resources are not known in advance, and indeed one can find instances that allow different solutions with different sets of bottlenecks. Therefore known techniques such as linear programming do not seem to work. Our proof uses tools from the theory of ordinary differential equations, showing the existence of a sequence of points that converge upon a solution. It is constructive and provides a practical method to compute the allocations numerically.

A Central-Difference Scheme for a Pure Stream Function Formulation of Incompressible Viscous Flow

We present a numerical scheme for incompressible viscous flow, formulated as an equation for the stream function. The pure stream function formulation obviates the difficulty associated with vorticity boundary conditions. The resulting biharmonic equation is discretized with a compact scheme and solved with an algebraic multigrid solver. The advection of vorticity is implemented with a high-resolution central scheme that remains stable and accurate in the presence of large gradients. The accuracy and robustness of the method are demonstrated for high Reynolds number flows in a lid-driven cavity.

Emergence of structure in a model of liquid crystalline polymers with elastic coupling

Abstract We have solved the equations of transient shear flow for a model of a liquid crystalline polymer that contains a long-range interaction term in the nematic potential. The model exhibits a rich set of dynamics when the structure and momentum equations are coupled, including a periodic vorticity ‘burst’ near the shearing surface and very large gradients in velocity at discrete planes. Textures that are independent of the macroscopic scale were not observed.

Unresolved computation and optimal predictions

We present methods for predicting the solution of time-dependent partial differential equations when that solution is so complex that it cannot be properly resolved numerically, but when prior statistical information can be found. The sparse numerical data are viewed as constraints on the solution, and the gist of our proposal is a set of methods for advancing the constraints in time so that regression methods can be used to reconstruct the mean future. For linear equations we offer general recipes for advancing the constraints; the methods are generalized to certain classes of nonlinear problems, and the conditions under which strongly nonlinear problems and partial statistical information can be handled are briefly discussed. Our methods are related to certain data acquisition schemes in oceanography and meteorology. c 1999 John Wiley & Sons, Inc.

LONG-TERM BEHAVIOUR OF LARGE MECHANICAL SYSTEMS WITH RANDOM INITIAL DATA

We study the long-time behaviour of large systems of ordinary differential equations with random data. Our main focus is a Hamiltonian system which describes a distinguished particle attached to a large collection of heat bath particles by springs. In the limit where the size of the heat bath tends to infinity, the trajectory of the distinguished particle can be weakly approximated, on finite time intervals, by a Langevin stochastic differential equation. We examine the long-term behaviour of these trajectories, both analytically and numerically. We find ergodic behaviour manifest in both the long-time empirical measures and in the resulting auto-correlation functions.

Optimal Prediction for Hamiltonian Partial Differential Equations

Optimal prediction methods compensate for a lack of resolution in the numerical solution of time-dependent differential equations through the use of prior statistical information. We present a new derivation of the basic methodology, show that field-theoretical perturbation theory provides a useful device for dealing with quasi-linear problems, and provide a nonlinear example that illuminates the difference between a pseudo-spectral method and an optimal prediction method with Fourier kernels. Along the way, we explain the differences and similarities between optimal prediction, the representer method in data assimilation, and duality methods for finding weak solutions. We also discuss the conditions under which a simple implementation of the optimal prediction method can be expected to perform well.