This article studies the generalized Nash equilibrium (GNE) seeking problem of second-integrator multiplayer systems. In particular, each player is endowed with an individual payoff function with respect to collective decision variables, and simultaneously, a coupling inequality constraint and a set constraint are imposed to each player. The players communicate with their local neighbors over a directed topology. To begin with, a distributed-observer-based seeking strategy is synthesized by leveraging a proper composite variable. It is first demonstrated using nonsmooth analysis that the established distributed observer enables each player to accurately estimate the decision variables of others in terms of a strongly connected topology condition. Upon this basis, all the decision variables are then shown to converge to the expected GNE asymptotically borrowing from convex theory. In addition, three extension results are also given under the built GNE seeking framework. First, under the postulation that the velocity information is unavailable, a velocity-free distributed GNE seeking strategy is synthesized for second-integrator systems by implementing a proper auxiliary dynamics. Second, we consider nonlinear Euler-Lagrange systems with unknown inertia parameters and synthesize an improved distributed GNE seeking strategy resorting to an adaptation technique. Third, we focus on integrator chain systems and synthesize a modified distributed GNE seeking strategy using a new composite variable based on a proper coordinate transformation. For three extension cases, we all show in detail the achievement of the GNE seeking objective. Finally, a practical example is simulated to confirm the built GNE seeking results.

Label-specific features serve as an effective strategy to learn from multi-label data with tailored features accounting for the distinct discriminative properties of each class label. Existing prototype-based label-specific feature transformation approaches work in a three-stage framework, where prototype acquisition, label-specific feature generation and classification model induction are performed independently. Intuitively, this separate framework is suboptimal due to its decoupling nature. In this paper, we make a first attempt towards a unified framework for prototype-based label-specific feature transformation, where the prototypes and the label-specific features are directly optimized for classification. To instantiate it, we propose modelling the prototypes probabilistically by the normalizing flows, which possess adaptive prototypical complexity to fully capture the underlying properties of each class label and allow for scalable stochastic optimization. Then, a label correlation regularized probabilistic latent metric space is constructed via jointly learning the prototypes and the metric-based label-specific features for classification. Comprehensive experiments on 14 benchmark data sets show that our approach outperforms the state-of-the-art counterparts.

ABSTRACT The fractional Fourier transform (FRFT)—a generalization of the well‐known Fourier transform (FT)—is a comparatively new and powerful mathematical tool for signal processing. Many results in Fourier analysis have currently been extended to the FRFT, including the ordinary convolution theorem. However, the extension of the ordinary convolution theorem associated with the FRFT has been developed differently and is still not having a widely accepted closed‐form expression. In this paper, a generalized convolution theorem for the FRFT is proposed, and the dual of it is also presented. The ordinary convolution theorem and some of its existing extensions related to the FRFT are shown to be special cases of the derived results. Moreover, some applications of the derived results are presented. Copyright © 2012 John Wiley & Sons, Ltd.