Sufficient Geometric Conditions Research Articles

Planar Poincaré domains: Geometry and Steiner symmetrization

We determine geometric necessary and sufficient conditions on a class of strip-like planar domains in order for them to satisfy the Poincare inequality with exponentp, where 1≤p<∞. The characterization uses hyperbolic geodesics in the domain and a metric which depends onp and generalizes the quasi-hyperbolic metric in the casep=2. As an application, we show that the Poincare inequality is preserved under Steiner symmetrization of these domains but not in general.

Finite solvable multiprimitive groups

The investigation of the structure of finite solvable multiprimitive groups through an induced geometry leads to a classification of finite solvable multiprimitive groups of up to derived length 3 and to a necessary and sufficient geometric condition for when a finite solvable nC-group is multiprimitive. An application is made to finite groups which have all their splitting systems equivalent.

Feedback linearization and driftless systems

The problem of dynamic feedback linearization is recast using the notion of dynamic immersion. We investigate here a “generic” property which holds at every point of a dense open subset, but may fail at some points of interest, such as equilibrium points. Linearizable systems are then systems that can be immersed into linear controllable ones. This setting is used to study the linearization of driftless systems: a geometric sufficient condition in terms of Lie brackets is given; this condition is also shown to be necessary when the number of inputs equals two. Though noninvertible feedbacks are nota priori excluded, it turns out that linearizable driftless systems with two inputs can be linearized using only invertible feedbacks, and can also be put into a chained form by (invertible) static feedback. Most of the developments are done within the framework of differential forms and Pfaffian systems.

Lyapunov-Schmidt Reduction and Melnikov Integrals for Bifurcation of Periodic Solutions in Coupled Oscillators

We consider a system of autonomous ordinary differential equations depending on a small parameter such that the unperturbed system has an invariant manifold of periodic solutions. The problem addressed here is the determination of sufficient geometric conditions for some of the periodic solutions on this invariant manifold to survive after perturbation. The main idea is to use a Lyapunov-Schmidt reduction for an appropriate displacement function in order to obtain the bifurcation function for the problem in a form which can be recognized as a generalization of the subharmonic Melnikov function. Thus, the multidimensional bifurcation problem can be cast in a form where the geometry of the problem is clearly incorporated. An important application can be made in case the uncoupled system of differential equations is a system of oscillators in resonance. In this case the invariant manifold of periodic solutions is just the product of the uncoupled oscillations. When each of the oscillators has one degree of freedom, the bifurcation function is computed by quadrature along the unperturbed oscillations. Additional applications include the computation of entrainment domains for a sinusoidally forced van der Pol oscillator and the computation of mutual synchronization domains for a system of inductively coupled van der Pol oscillators.

Self-Similar Lattice Tilings

We study the general question of the existence of self-similar lattice tilings of Euclidean space. A necessary and sufficient geometric condition on the growth of the boundary of approximate tiles is reduced to a problem in Fourier analysis that is shown to have an elegant simple solution in dimension one. In dimension two we further prove the existence of connected self-similar lattice tilings for parabolic and elliptic dilations. These results apply to produce Haar wavelet bases and certain canonical number systems.

Equivalence of Nonlinear Systems to Input-Output Prime Forms

The problem of transforming nonlinear control systems into input-output prime forms is dealt with, using state space, static state feedback, and also output space transformations. Necessary and sufficient geometric conditions for the solvability of this problem are obtained. The results obtained generalize well-known results both on feedback linearization as well as input-output decoupling of nonlinear systems. It turns out that, from a computational point of view, the output space transformation is the crucial step, that is performed by constructing rectifying coordinates for a nested sequence of distributions on the output manifold.

Performance of the Gibbs, Hit-and-Run, and Metropolis Samplers

Abstract We consider the performance of three Monte Carlo Markov-chain samplers—the Gibbs sampler, which cycles through coordinate directions; the Hit-and-Run (HR and the Metropolis sampler, which moves with a probability that is a ratio of likelihoods. We obtain several analytical results. We provide a sufficient condition of the geometric convergence on a bounded region S for the H&R sampler. For a general region S, we review the Schervish and Carlin sufficient geometric convergence condition for the Gibbs sampler. We show that for a multivariate normal distribution this Gibbs sufficient condition holds and for a bivariate normal distribution the Gibbs marginal sample paths are each an AR(1) process, and we obtain the standard errors of sample means and sample variances, which we later use to verify empirical Monte Carlo results. We empirically compare the Gibbs and H&R samplers on bivariate normal examples. For zero correlation, the Gibbs sampler provid...

A sufficient condition for non-linear non-interacting control with stability via dynamic state-feedback: block-partitioned outputs

A geometric sufficient condition is given for the first time to solve the problem of non-linear block non-interacting control with stability by means of dynamic state-feedback. This result extends a previous sufficient condition given by the same author (Battilotti 1991).

Nests of subspaces in Banach space and their order types

This paper addresses some questions which arise naturally in the theory of nests of subspaces in Banach space. The order topology on the index set of a nest is discussed, as well as the method of spatial indexing by a vector; sufficient geometric conditions for the existence of such a vector are found. It is then shown that a continuous nest exists in any Banach space. Applications and examples follow; in particular, an extension of the Volterra nest in L ∞ [ 0 , 1 ] {L^\infty }[ {0,1} ] to a continuous one, a continuous nest in a Banach space having no two elements isomorphic to one another, and a characterization of separable L p {\mathcal {L}_p} -spaces in terms of nests.

Uniform, Sobolev extension and quasiconformal circle domains

This paper contributes to the theory of uniform domains and Sobolev extension domains. We present new features of these domains and exhibit numerous relations among them. We examine two types of Sobolev extension domains, demonstrate their equivalence for bounded domains and generalize known sufficient geometric conditions for them. We observe that in the plane essentially all of these domains possess the trait that there is a quasiconformal self-homeomorphism of the extended plane which maps a given domain conformally onto a circle domain. We establish a geometric condition enjoyed by these plane domains which characterizes them among all quasicircle domains having no large and no small boundary components.

Robust Disturbance Decoupling Problem for Parameter Dependent Families of Linear Systems

Abstract We consider the Problem of Disturbance Decoupling by means of a single state feedback for a whole family of systems depending on a parameter. We give a geometric necessary and sufficient condition for the existence of solution to such problem under the hypothesis that the systems of the family are generically injective and the dependence on p is polynomial or, respectively, the set of possible values for p is finite

Motion and structure from point correspondences with error estimation: planar surfaces

The determination of motion and structure of a planar scene from monocular image sequences, is studied. A new and simpler linear algorithm, that gives closed-form solutions for motion and structure parameters using point correspondences between two images, assuming that the coplanar points undergo a rigid motion in 3-D space, is presented. A series of analytical results is established. From two perspective views of a planar scene which is undergoing a rigid motion, there are generally two (normalized) interpretations for motion parameters and the positions of the object plane. These two interpretations, one vertical and the other illusive, are both valid in the sense that they render the same pair of images. The authors identify all the special cases in which the number of interpretations is not two, and derive necessary and sufficient geometrical conditions for those special cases to occur. The approach to error estimation is based on the first-order perturbation. The estimated errors provide quantitative assessment for the accuracy of the solutions. They also indicate degenerate or nearly degenerate configurations in the presence of noise.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Bounded solutions of the Schr�dinger equation on noncompact Riemannian manifolds

Necessary and sufficient geometric conditions are proved for the equation Δu−Q(x)u=0, Q(x)≥0, to have a bounded nontrivial solution on a noncompact Riemannian manifold. The results imply as corollaries conditions for parabolicity and stochastic completeness of a manifold, previously established by other methods.

Deterministic feedback linearization, Girsanov transformations and finite-dimensional filters

The problem of extending state-feedback linearization methods of deterministic control theory to stochastic systems is addressed. For Stratonovitch stochastic differential equations with smooth vector fields, necessary and geffucient geometric conditions for local and global linearization by diffeomorphism and absolutely continuous change of probability law are obtained, using the interpretation of Girsanov transformations as state-feedback on Brownian motions. For stochastic systems with single-with single-input (or onde-dimensional Brownian motion) and single-output (or one-dimensional observation process), necessary and sufficient geometric conditions to transform the Duncan-Mortensen-Zakai (DMZ) equation of filtering into that of an affine prime system are obtained, as well as interpretation of gauge transformation as Girsanov change of probability law.

Zero-input limit cycles due to rounding in digital filters

A necessary condition for the existence of zero-input limit cycles of period T due to rounding in general digital filters is established. Also, geometric sufficient conditions are presented for the existence of limit cycles of period one, of period two and of period T due to rounding. Some illustrative examples are included. >

Functions not constant on fractal quasi-arcs of critical points

This paper provides geometric sufficient conditions for an arc to be a critical set for some function not constant along that arc—an example of which was first discovered by Whitney in 1935. In particular, any fractal subarc of a quasi-circle has this property. The maximum degree of differentiability of the function is closely connected to the arc’s geometry.

Controllability in codimension one

Local and global controlability of analytic affine control systems Σ with an arbitrary number of controls are studied, assuming strong accessibility of Σ and codimension one of the Lie algebra T′ generated by the input vector fields, i.e., dim T′ ( p) = n − 1 at every p ϵ M. Controls are assumed to have no a priori bound. From the study of the set H where a necessary condition for local controllability at a point is verified and assuming some transversality relations, an easy to verify geometric condition is proved: H is a submanifold and Σ is locally controllable at every point of H outside a codimension one submanifold. A geometric sufficient condition for global controllability on simply connected manifolds is then obtained: if every leaf of T′ intersects the manifold H and some transversality relations (including those involved in the local condition) are verified, Σ is globally controllable.

A sharp sufficient geometric condition for the existenee of global real analytic solutions on a bounded domain

A sharp sufficient geometric condition for the existenee of global real analytic solutions on a bounded domain

Local controllability of nonlinear systems

The local controllability of control systems with an arbitrary number of controls is considered, first on an open set and then at a given point; necessary conditions are derived concerning the Lie algebra T′ generated by the input vector fields, and applied to gradient systems. The main results are a geometric sufficient condition for local controllability at a point, and an equivalent condition based on the computation of Lie brackets, assuming T′ to be ( n − 1)-dimensional.

Strong and Weak Convexity of Sets and Functions

In this paper we study two classes of sets, strongly and weakly convex sets. For each class we derive a series of properties which involve either the concept of supporting ball, an obvious extension of the concept of supporting hyperplane, or the normal cone to the set. We also study a class of functions, denoted ρ-convex, which satisfy for arbitrary points x1 and x2 and any value λ ∈ [0, 1] the classical inequality of convex functions up to a term ρ(1 − λ) λ‖x1 − x2‖2. Depending on the sign of the constant ρ the function is said to be strongly or weakly convex. We provide characteristic properties of this class of sets and we relate it to strongly and weakly convex sets via the epigraph and the level sets. Finally, we give three applications: a separation theorem, a sufficient condition for global optimum of a nonconvex programming problem, and a sufficient geometrical condition for a set to be a manifold.