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Backward compact dynamics is deduced for a non-autonomous Benjamin-Bona-Mahony (BBM) equation on an unbounded 3D-channel. A backward compact attractor is defined by a time-dependent family of backward compact, invariant and pullback attracting sets. The theoretical existence result for such an attractor is derived from the backward flattening property, and this property is proved to be equivalent to the backward asymptotic compactness in a uniformly convex Banach space. Finally, it is shown that the BBM equation has a backward compact attractor in a Sobolev space under some suitable assumptions, such as, backward translation boundedness and backward small-tail. Both spectrum decomposition and cut-off technique are used to give all required backward uniform estimates.

*fmincon*in Matlab, LOQO and MINOS and, as a comparison, the piecewise constant contracts are also designed by solving the single-level nonconvex programming which is equivalent to the principal-agent bilevel programming model with corresponding discrete distributions. Numerical tests show that: to design a piecewise linear contract, which is much better than a piecewise constant contract, it needs only to solve a much lower dimensional optimization problem and hence needs much less computing time. Numerical experiences also show that the modified constraint shifting homotopy method is feasible and robust.

Semi-supervised learning is an attractive method in classification problems when insufficient training information is available. In this investigation, a new semi-supervised classifier is proposed based on the concept of maximum vector-angular margin, (called S$^3$MAMC), the main goal of which is to find an optimal vector $c$ as close as possible to the center of the dataset consisting of both labeled samples and unlabeled samples. This makes S$^3$MAMC better generalization with smaller VC (Vapnik-Chervonenkis) dimension. However, S$^3$MAMC formulation is a non-convex model and therefore it is difficult to solve. Following that we present two optimization algorithms, mixed integer quadratic program (MIQP) and DC (difference of convex functions) program algorithms, to solve the S$^3$MAMC. Compared with the supervised learning methods, numerical experiments on real and synthetic databases demonstrate that the S$^3$MAMC can improve generalization when the labelled samples are relatively few. In addition, the S$^3$MAMC has competitive experiment results in generalization compared to the traditional semi-supervised classification methods.

The circadian clock is a self-sustaining oscillator that has a period of about 24 hours at the molecular level. The oscillator is a transcription-translation feedback loop system composed of several genes. In this paper, a scalar nonlinear differential equation with two delays, modeling the transcriptional co-regulation in mammalian circadian clock, is proposed and analyzed. Sufficient conditions are established for the asymptotic stability of the unique nontrivial positive equilibrium point of the model by studying an exponential polynomial characteristic equation with delay-dependent coefficients. The existence of the Hopf bifurcations can be also obtained. Numerical simulations of the model with proper parameter values coincide with the theoretical result.

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