Lyapunov equation for infinite-dimensional discrete bilinear systems

Mean-square stability for discrete systems requires that uniform convergence is preserved between input and state correlation sequences. Such a convergence preserving property holds for an infinite-dimensional bilinear system if and only if the associate Lyapunov equation has a unique strictly positive solution.     

Banach空间Fuzzy多值隐拟变分包含

在Banach空间中并在不具紧性的条件下，引入和研究了一类fuzzy多值隐拟变分包含及相应的fuzzy隐预解算子方程，借助预解算子技巧，建立了Banach空间中fuzzy多值隐拟变分包含与预解算子方程的等价性，得到了该类fuzzy多值隐拟变分包含的迭代算法与某些解的存在性定理及解的迭代逼近。     

Multiplicative Operator Splittings in Nonlinear Diffusion: From Spatial Splitting to Multiple Timesteps

Operator splitting is a powerful concept used in many diversed fields of applied mathematics for the design of effective numerical schemes. Following the success of the additive operator splitting (AOS) in performing an efficient nonlinear diffusion filtering on digital images, we analyze the possibility of using multiplicative operator splittings to process images from different perspectives.We start by examining the potential of using fractional step methods to design a multiplicative operator splitting as an alternative to AOS schemes. By means of a Strang splitting, we attempt to use numerical schemes that are known to be more accurate in linear diffusion processes and apply them on images. Initially we implement the Crank-Nicolson and DuFort-Frankel schemes to diffuse noisy signals in one dimension and devise a simple extrapolation that enables the Crank-Nicolson to be used with high accuracy on these signals. We then combine the Crank-Nicolson in 1D with various multiplicative operator splittings to process images. Based on these ideas we obtain some interesting results. However, from the practical standpoint, due to the computational expenses associated with these schemes and the questionable benefits in applying them to perform nonlinear diffusion filtering when using long timesteps, we conclude that AOS schemes are simple and efficient compared to these alternatives.We then examine the potential utility of using multiple timestep methods combined with AOS schemes, as means to expedite the diffusion process. These methods were developed for molecular dynamics applications and are used efficiently in biomolecular simulations. The idea is to split the forces exerted on atoms into different classes according to their behavior in time, and assign longer timesteps to nonlocal, slowly-varying forces such as the Coulomb and van der Waals interactions, whereas the local forces like bond and angle are treated with smaller timesteps. Multiple timestep integrators can be derived from the Trotter factorization, a decomposition that bears a strong resemblance to a Strang splitting. Both formulations decompose the time propagator into trilateral products to construct multiplicative operator splittings which are second order in time, with the possibility of extending the factorization to higher order expansions. While a Strang splitting is a decomposition across spatial dimensions, where each dimension is subsequently treated with a fractional step, the multiple timestep method is a decomposition across scales. Thus, multiple timestep methods are a realization of the multiplicative operator splitting idea. For certain nonlinear diffusion coefficients with favorable properties, we show that a simple multiple timestep method can improve the diffusion process.     

An extended minimax disparity to determine the OWA operator weights

This paper contributes to extend the minimax disparity to determine the ordered weighted averaging (OWA) model based on linear programming. It introduces the minimax disparity approach between any distinct pairs of the weights and uses the duality of linear programming to prove the feasibility of the extended OWA operator weights model. The paper finishes with an open problem.     

Mean-square stability for discrete bilinear systems in Hilbert space

A discrete system is uniformly mean-square stable if uniform convergence is preserved between input and state correlation sequences, and if nuclearity is preserved between the input and state correlation limits. This paper supplies a contraction-free condition ensuring mean-square stability for infinite-dimensional bilinear systems evolving in a separable Hilbert space.     

Choosing OWA operator weights in the field of Social Choice

One of the most important issues in the theory of OWA operators is the determination of associated weights. This matter is essential in order to use the best-suited OWA operator in each aggregation process. Given that some aggregation processes can be seen as extensions of majority rules to the field of gradual preferences, it is possible to determine the OWA operator weights by taking into account the class of majority rule that we want to obtain when individuals do not grade their pairwise preferences. However, a difficulty with this approach is that the same majority rule can be obtained through a wide variety of OWA operators. For this reason, a model for selecting the best-suited OWA operators is proposed in this paper.     

Laplace spectra as fingerprints for image recognition

In the area of image retrieval from data bases and for copyright protection of large image collections there is a growing demand for unique but easily computable fingerprints for images. These fingerprints can be used to quickly identify every image within a larger set of possibly similar images. This paper introduces a novel method to automatically obtain such fingerprints from an image. It is based on a reinterpretation of an image as a Riemannian manifold. This representation is feasible for gray value images and color images. We discuss the use of the spectrum of eigenvalues of different variants of the Laplace operator as a fingerprint and show the usability of this approach in several use cases. Contrary to existing works in this area we do not only use the discrete Laplacian, but also with a particular emphasis the underlying continuous operator. This allows better results in comparing the resulting spectra and deeper insights in the problems arising. We show how the well known discrete Laplacian is related to the continuous Laplace-Beltrami operator. Furthermore, we introduce the new concept of solid height functions to overcome some potential limitations of the method.     

Discrete quadratic curvature energies

We present a family of discrete isometric bending models (IBMs) for triangulated surfaces in 3-space. These models are derived from an axiomatic treatment of discrete Laplace operators, using these operators to obtain linear models for discrete mean curvature from which bending energies are assembled. Under the assumption of isometric surface deformations we show that these energies are quadratic in surface positions. The corresponding linear energy gradients and constant energy Hessians constitute an efficient model for computing bending forces and their derivatives, enabling fast time-integration of cloth dynamics with a two- to three-fold net speedup over existing nonlinear methods, and near-interactive rates for Willmore smoothing of large meshes.