Geometric Bounds for Buckling-Induced Auxetic Metamaterials Undergoing Large Deformation

The performance of a metamaterial is dominated by the geometric features and deformation mechanisms of its microstructure. For a certain mechanism, the geometric features have bounds in which the performance of a metamaterial such as negative Poisson’s ratio (NPR) can be designed. Previous investigation on buckling-induced auxetic metamaterial revealed that there is a geometric limit for its microstructure to exhibit auxetic behaviour in infinitesimal deformation. However, the limit for auxetic metamaterials undergoing large deformation is different from that under small deformation and has not been reported yet. In this paper, the geometric limit was investigated in an elastic and infinitesimal deformation range using linear buckling analysis. Furthermore, experimentally validated finite element models were used to identify the geometric limits for auxetic metamaterials undergoing large deformation. Depending on the control parameters of the topology, the bounds were represented by a line strip for one control parameter, an area for two control parameters and spatial domain surrounded by a 3D surface for three parameters. The limit was determined by the shape and size of the void of the metamaterials and it was identified through the large deformation analysis as well as the linear buckling analysis. We found that there was a significant difference in the geometric bounds obtained through those two methods. The results from this study can be used to design an auxetic metamaterial for different applications and to control the auxetic performance.

Geometric Limits of Julia Sets of Maps zn + exp(2πiθ) as n → ∞

We show that the geometric limit as n → ∞ of the Julia sets J(Pn,c) for the maps Pn,c(z) = zn + c does not exist for almost every c on the unit circle. Furthermore, we show that there is always a subsequence along which the limit does exist and equals the unit circle.

Perpetuities in Fair Leader Election Algorithms

We consider a broad class of fair leader election algorithms, and study the duration of contestants (the number of rounds a randomly selected contestant stays in the competition) and the overall cost of the algorithm. We give sufficient conditions for the duration to have a geometric limit distribution (a perpetuity built from Bernoulli random variables), and for the limiting distribution of the total cost (after suitable normalization) to be a perpetuity. For the duration, the proof is established via convergence (to 0) of the first-order Wasserstein distance from the geometric limit. For the normalized overall cost, the method of proof is also convergence of the first-order Wasserstein distance, augmented with an argument based on a contraction mapping in the first-order Wasserstein metric space to show that the limit approaches a unique fixed-point solution of a perpetuity distributional equation. The use of these two steps is commonly referred to as the contraction method.

Perpetuities in Fair Leader Election Algorithms

We consider a broad class of fair leader election algorithms, and study the duration of contestants (the number of rounds a randomly selected contestant stays in the competition) and the overall cost of the algorithm. We give sufficient conditions for the duration to have a geometric limit distribution (a perpetuity built from Bernoulli random variables), and for the limiting distribution of the total cost (after suitable normalization) to be a perpetuity. For the duration, the proof is established via convergence (to 0) of the first-order Wasserstein distance from the geometric limit. For the normalized overall cost, the method of proof is also convergence of the first-order Wasserstein distance, augmented with an argument based on a contraction mapping in the first-order Wasserstein metric space to show that the limit approaches a unique fixed-point solution of a perpetuity distributional equation. The use of these two steps is commonly referred to as the contraction method.

Simulation Study of a Novel Controllable Metamorphic Palletizing Robot Mechanism

The palletizing robot mechanism was investigated in this paper. The concepts of the metamorphic mechanism and controllable mechanism were introduced into the design of palletizing robot mechanism, a novel controllable metamorphic palletizing robot mechanism with geometric limit, which can achieve a variety of flexible action, was designed. The working space of output of the controllable metamorphic palletizing robot mechanism was obtained through simulation study, and the kinematics characteristics of the mechanism were analyzed. The work of this paper provides some references for the practical application of such kind of mechanism in the palletizing robot.

GEOMETRIC LIMITS OF MANDELBROT AND JULIA SETS UNDER DEGREE GROWTH

First, for the family Pn,c(z) = zn + c, we show that the geometric limit of the Mandelbrot sets Mn(P) as n → ∞ exists and is the closed unit disk, and that the geometric limit of the Julia sets J(Pn,c) as n tends to infinity is the unit circle, at least when |c| ≠ 1. Then, we establish similar results for some generalizations of this family; namely, the maps z ↦ zt + c for real t ≥ 2 and the rational maps z ↦ zn + c + a/zn.

The Limit Behavior of Dual Markov Branching Processes

A dual Markov branching process (DMBP) is by definition a Siegmund's predual of some Markov branching process (MBP). Such a process does exist and is uniquely determined by the so-called dual-branching property. Its q-matrix Q is derived and proved to be regular and monotone. Several equivalent definitions for a DMBP are given. The criteria for transience, positive recurrence, strong ergodicity, and the Feller property are established. The invariant distributions are given by a clear formulation with a geometric limit law.