Cooperative effects in random sequential polymer reactions

We consider the kinetics of a reaction in a polymer system where each reacted unit protects its (r—1)-nearest neighbours completely against reaction and modifies the reactivity of the r-nearest neighbours. The number of sequences containing a specified numberof unreacted units, together with the over-all extent of reaction, is determined as function of time. In a continuum limit the model also describes the time evolution of a sequential filling of hard rods on a line     

PROPERTIES OF TENSOR COMPLEMENTARITY PROBLEM AND SOME CLASSES OF STRUCTURED TENSORS

This paper deals with the class of Q-tensors, that is, a Q-tensor is a real tensor A such that the tensor complementarity problem(q, A):finding an x ∈ R~n such that x ≥ 0, q+Ax~(m-1)≥ 0, and x~T(q+Ax~(m-1)) = 0,has a solution for each vector q ∈ R~n. Several subclasses of Q-tensors are given: P-tensors, R-tensors, strictly semi-positive tensors and semi-positive R_0-tensors. We prove that a nonnegative tensor is a Q-tensor if and only if all of its principal diagonal entries are positive, and so the equivalence of Q-tensor,R-tensors, strictly semi-positive tensors was showed if they are nonnegative tensors. We also show that a tensor is an R_0-tensor if and only if the tensor complementarity problem(0, A) has no non-zero vector solution, and a tensor is a R-tensor if and only if it is an R_0-tensor and the tensor complementarity problem(e, A) has no non-zero vector solution, where e =(1, 1 ···, 1)~T.     

Structural Properties of Tensors and Complementarity Problems

In this paper, one of our main purposes is to prove the boundedness of the solution set of tensor complementarity problems such that the specific bounds depend only on the structural properties of such a tensor. To achieve this purpose, firstly, we prove that this class of structured tensors is strictly semi-positive. Subsequently, the strictly lower and upper bounds of operator norms are given for two positively homogeneous operators. Finally, with the help of the above upper bounds, we show that the solution set of tensor complementarity problems has the strictly lower bound. Furthermore, the upper bounds of spectral radius are obtained, which depends only on the principal diagonal entries of tensors.     

Strong convergence of a regularization method for Rockafellar’s proximal point algorithm

In this paper, for a monotone operator T, we shall show strong convergence of the regularization method for Rockafellar’s proximal point algorithm under more relaxed conditions on the sequences {r _k } and {t _k }, $$\lim\limits_{k\to\infty}t_k = 0;\quad \sum\limits_{k=0}^{+\infty}t_k = \infty;\quad\ \liminf\limits_{k\to\infty}r_k > 0.$$ Our results unify and improve some existing results.     

Polymer‐free,low tension graphene mechanical resonators

Graphene resonators are fabricated using a polymer‐free, direct transfer method onto metal reinforced holey carbon grids. The resonators are distinguished by the absence of organic residues and excellent crystallinity. The normal mode frequencies are measured using a Fabry–Perot technique; resonance curves indicate highly linear behaviour but very little built‐in strain, which is consistent with device geometry examined by atomic force microscopy. We conclude that the oscillators' restoring force is due instead to graphene's intrinsic bending rigidity; our measurements indicate a value of approximately 1.0 eV, consistent with previous theoretical and experimental work. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Robust trajectory tracking control for a laboratory helicopter

This paper presents a robust nonlinear control strategy to deal with the trajectory tracking control problem for a laboratory helicopter. The helicopter model is considered as a nominal one with uncertainties such as unmodeled nonlinear dynamics, parametric uncertainties, and external disturbances. The proposed control approach incorporates the feedback linearization technique (FLT) and the signal compensation technique. The FLT is first applied to achieve the linearization of the nominal nonlinear model for reducing the conservation of the robust compensator design. A nominal controller based on the linear quadratic regulation method is designed for the linearized nominal system, whereas a robust compensator is introduced to restrain the influences of the uncertainties. It is shown that the trajectory tracking errors of the closed-loop system are ultimately bounded, and the boundaries can be specified by choosing the controller parameters. Simulation and experimental results on the lab helicopter verify the effectiveness of the proposed method.     

Positive eigenvalue-eigenvector of nonlinear positive mappings

We show that an （eventually） strongly increasing and positively homogeneous mapping T defined on a Banach space can be turned into an Edelstein contraction with respect to Hilbert＇s projective metric. By applying the Edelstein contraction theorem, a nonlinear version of the famous Krein- Rutman theorem is presented, and a simple iteration process {T^kx/｜｜T^kx｜｜} （ x ∈ P^＋） is given for finding a positive eigenvector with positive eigenvalue of T. In particular, the eigenvalue problem of a nonnegative tensor A can be viewed as the fixed point problem of the Edelstein contraction with respect to Hilbert＇s projective metric. As a result, the nonlinear Perron-Frobenius property of a nonnegative tensor A is reached easily.     

Admissible consensus and consensualization of high-order linear time-invariant singular swarm systems

Admissible consensus analysis and consensualizing controller design problems for high-order linear time-invariant singular swarm systems are investigated. Firstly, by projecting the state of a singular swarm system onto a consensus subspace and a complement consensus subspace, a necessary and sufficient condition for admissible consensus is presented in terms of linear matrix inequalities (LMIs). An approach to decrease the calculation complexity is proposed, by which only three LMIs independent of the number of agents need to be checked. Then, by using the changing variable method, sufficient conditions for admissible consensualization are shown. An explicit expression of the consensus function is given, and it is shown that the modes of the consensus function can be arbitrarily placed if each agent is R$R$-controllable and impulse controllable and the interaction topology has a spanning tree. Finally, theoretical results are applied to deal with cooperative control problems of multi-agent supporting systems.