Commutant Lifting Theorem for the Bergman Space

The central question of this paper is the one of finding the right analogue of the Commutant Lifting Theorem for the Bergman space L_a². We also analyze the analogous problem for weighted Bergman spaces L_a,α², − 1 < α < ∞.     

Commutant Lifting for Optimal Control of Time-varying Systems

This paper uses the commutant lifting theorem for representations of the nest algebra to deal with the optimal control of infinite dimensional linear timevarying systems.We solve the model matching pro...     

A Local Lifting Theorem for Jointly Subnormal Families of Unbounded Operators

A local lifting theorem for bounded operators that intertwine a pair of jointly subnormal families of unbounded operators is proved. Each family in question is assumed to be composed of operators defined on a common invariant domain consisting of “joint” analytic vectors. This result can be viewed as a generalization of the local lifting theorem proved by Sebestyén, Thomson and the present authors for pairs of bounded subnormal operators.     

A Geometric Approach to Global Optimization

In this paper we consider the problem of optimizing a piecewise-linear objective function over a non-convex domain. In particular we do not allow the solution to lie in the interior of a prespecified region R. We discuss the geometrical properties of this problems and present algorithms based on combinatorial arguments. In addition we show how we can construct quite complicated shaped sets R while maintaining the combinatorial properties.     

A Geometric Approach to Acyclic Orientations

The set of acyclic orientations of a connected graph with a given sink has a natural poset structure. We give a geometric proof of a result of Jim Propp: this poset is the disjoint union of distributive lattices.     

Pseudo-Rigid Bodies: A Geometric Lagrangian Approach

The pseudo-rigid body model is viewed within the context of continuum mechanics and elasticity theory. A Lagrangian reduction, based on variational principles, is developed for both anisotropic and isotropic pseudo-rigid bodies. For isotropic Lagrangians, the reduced equations of motion for the pseudo-rigid body are a system of two (coupled) Lax equations on so(3)×so(3) and a second-order differential equation on the set of diagonal matrices with a positive determinant. Several examples of pseudo-rigid bodies such as stretching bodies, spinning gas cloud and Riemann ellipsoids are presented.     

Dilations and Completions for Gabor Systems

Let be a full rank time-frequency lattice in ℝ ^d ×ℝ ^d . In this note we first prove that any dual Gabor frame pair for a Λ-shift invariant subspace M can be dilated to a dual Gabor frame pair for the whole space L ²(ℝ ^d ) when the volume v(Λ) of the lattice Λ satisfies the condition v(Λ)≤1, and to a dual Gabor Riesz basis pair for a Λ-shift invariant subspace containing M when v(Λ)>1. This generalizes the dilation result in Gabardo and Han (J. Fourier Anal. Appl. 7:419–433, [2001]) to both higher dimensions and dual subspace Gabor frame pairs. Secondly, for any fixed positive integer N, we investigate the problem whether any Bessel–Gabor family G(g,Λ) can be completed to a tight Gabor (multi-)frame G(g,Λ)∪(∪ _j=1 ^N G(g _j ,Λ)) for L ²(ℝ ^d ). We show that this is true whenever v(Λ)≤N. In particular, when v(Λ)≤1, any Bessel–Gabor system is a subset of a tight Gabor frame G(g,Λ)∪G(h,Λ) for L ²(ℝ ^d ). Related results for affine systems are also discussed. Communicated by Chris Heil.     

Time-Varying Linear Control Systems: A Geometric Approach

The concept of [A, B]-invariance for time-varying (not necessarilycontinuous) linear systems is introduced. A natural approachvia time-varying subspaces is given. A disturbance decouplingand a noninteracting problem is studied.     

A Network Approach to a Geometric Packing Problem

We investigate a geometric packing problem (derived from an industrial setting) that involves fitting patterns of regularly spaced disks without overlap. We first derive conditions for achieving a feasible placement of a given set of patterns and then construct a network formulation that facilitates the calculation of such a placement. A heuristic utilizing this network representation is also outlined. Additionally, we show a connection to the well-known Periodic Scheduling Problem.     

A Sort of Gray-scale Morphological Dilations and Erosions

We introduce first a sort of gray-scale morphological dilations and erosions, which might have some further applications in image analysis. Then we show that the dilation and the erosion defined here form adjunctive pairs. The duality between the dilation and the erosion and some other properties, such as the commuting property with translation and homothety, of these operators are discussed as well.     

A New Geometric Approach to Lie Systems and Physical Applications

The characterization of systems of differential equations admitting a superposition function allowing us to write the general solution in terms of any fundamental set of particular solutions is discussed. These systems are shown to be related with equations on a Lie group and with some connections in fiber bundles. We develop two methods for dealing with such systems: the generalized Wei–Norman method and the reduction method, which are very useful when particular solutions of the original problem are known. The theory is illustrated with some applications in both classical and quantum mechanics.     

A Geometric Approach to Actuator Failure in Robotic Manipulators

The use of robotic manipulators in remote and sensitive areas calls for more robust solutions when handling joint failure, and the industry demands mathematically robust approaches to handle even the worst case scenarios. Thus, a systematic analysis of the effects of external forces on manipulators with passive joints is presented. In parallel manipulators passive joints can appear as a design choice or as a result of torque failure. In both cases a good understanding of the effects that passive joints have on the mobility and motion of the parallel manipulator is crucial. We first look at the effect that passive joints have on the mobility of the mechanism. Then, if the mobility, considering passive joints only, is not zero we find a condition for which the parallel manipulator is conditionally equilibrated with respect to a specific external force. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Geometric Integration Approach to Nonsmooth,Nonconvex Optimisation

The optimisation of nonsmooth, nonconvex functions without access to gradients is a particularly challenging problem that is frequently encountered, for example in model parameter optimisation problems. Bilevel optimisation of parameters is a standard setting in areas such as variational regularisation problems and supervised machine learning. We present efficient and robust derivative-free methods called randomised Itoh–Abe methods. These are generalisations of the Itoh–Abe discrete gradient method, a well-known scheme from geometric integration, which has previously only been considered in the smooth setting. We demonstrate that the method and its favourable energy dissipation properties are well defined in the nonsmooth setting. Furthermore, we prove that whenever the objective function is locally Lipschitz continuous, the iterates almost surely converge to a connected set of Clarke stationary points. We present an implementation of the methods, and apply it to various test problems. The numerical results indicate that the randomised Itoh–Abe methods can be superior to state-of-the-art derivative-free optimisation methods in solving nonsmooth problems while still remaining competitive in terms of efficiency.

     

Hardy-Sobolev空间上某类乘子的换位以及相似性

证明了Hardy-Sobolev空间H_β~2上的乘子T_z~n~β相似于⊕_1~n T_z~β,讨论了HardySobolev空间H_β~2(β≤0)上以恒不为零的单叶解析函数的幂为符号的乘子的换位.     

Lipschitz Continuity Results for a Class of Variational Inequalities and Applications: A Geometric Approach

We consider a class of parametric variational inequalities which, under suitable assumptions, admit an equivalent integral formulation. We study the Lipschitz continuity of the solution and treat in detail the case in which the parametric constraint set is a polytope. Finally, the results obtained are applied to the time-dependent traffic equilibrium problem.