Showing papers on "Uniform boundedness published in 1986"
TL;DR: In this article, a deep analogy of certain aspects of harmonic analysis on a free group and harmonic analysis in SL(2,R) has been found, which has attracted considerable attention in the last ten years or so.
Abstract: Harmonic analysis on a free group F has attracted considerable attention in the last ten years or so. There seem to be two reason for that: one is the discovery of deep analogy of certain aspects of harmonic analysis on a free group and harmonic analysis on SL(2,R), cf. e.g. fundamental works of P. Cartier [4], A. Figa-Talamanca and M. A. Picardello [8], the other being the interest in the C∗-algebra generated by the regular representation of F , cf. A. Connes [5], J. Cuntz [6], U. Haagerup [10], M. Pimsner and D. Voiculescu [14].
86 citations
TL;DR: In this article, weak continuity and Frechet differentiability of M-functionals were shown to be a necessary condition for the existence of the Frechet derivative of the M-estimator.
Abstract: : A necessary requirement for existence of the Frechet derivative is that the defining psi function is uniformly bounded, and this naturally excludes those nonrobust estimators such as the maximum likelihood estimator in normal parametric models. In this paper the methods of nonsmooth analysis, described in the book by F.H. Clarke (1983), are introduced to the theory of statistical expansions, and are used here in the proofs of weak continuity and Frechet differentiability of M-functionals. Subsequently the conditions for Frechet differentiability given in Clarke (1983) can be relaxed to include most popular M-functionals. Additional keywords: distribution functions; M-estimators; robustness; gross error sensitivity; weak continuity; asymptotic expansions; asymptotic normality; selection functional; local uniqueness. (Author).
69 citations
TL;DR: In this article, a unified theory of periodicity of dissipative ordinary and functional differential equations in terms of uniform boundedness is discussed, and sufficient conditions for the uniform boundedess are given by means of Liapunov functionals having a weighted norm as an upper bound.
Abstract: We discuss a unified theory of periodicity of dissipative ordinary and functional differential equations in terms of uniform boundedness. Sufficient conditions for the uniform boundedness are given by means of Liapunov functionals having a weighted norm as an upper bound. The theory is developed for ordinary differential equations, equations with bounded delay, and equations with infinite delay.
59 citations
TL;DR: In this article, the authors studied the uniform behavior of the empirical brownian bridge over families of functions bounded by a function F (the observations are independent with common distribution P) under some suitable entropy conditions which were already used by Kolcinskii and Pollard.
Abstract: In this paper we study we uniform behavior of the empirical brownian bridge over families of functions F bounded by a function F (the observations are independent with common distribution P). Under some suitable entropy conditions which were already used by Kolcinskii and Pollard, we prove exponential inequalities in the uniformly bounded case where F is a constant (the classical Kiefer's inequality (1961) is improved), as well as weak and strong invariance principles with rates of convergence in the case where F belongs to L2+δ(P) with δe]0,1] (our results improve on Dudley, Philipp's results (1983) whenever F is a Vapnik-Cervonenkis class in the uniformly bounded case and are new in the unbounded case).
54 citations
TL;DR: In this article, a memoryless feedback controller is developed for a model reference control system which is linear, uncertain and has retarded arguments, under certain assumptions guarantees that all solutions are uniformly bounded and also that the system trajectories are always attracted towards some stable hypersurface in the space, no matter what the uncertainties and initial conditions are.
Abstract: A memoryless feedback controller is developed for a model reference control system which is linear, uncertain and has retarded arguments. The controller under certain assumptions guarantees that all solutions are uniformly bounded and also that the system trajectories are always attracted towards some stable hypersurface in the space, no matter what the uncertainties and initial conditions are. Along the hypersurface the motion of the trajectories are insensitive to system parameter variations.
42 citations
31 citations
TL;DR: In this paper, it was shown that the best L2 approximant of a continuous function defined on some interval [−a,b]⊆(−1,1) of the real axis is a finite Blaschke product.
Abstract: It is shown that, if h(x) is any continuous function defined on some interval [−a,b]⊆(−1,1) of the real axis, then, in general, its best L2 approximant, in the class of functions holomorphic and bounded by unity in the unit disk of the complex plane, is a finite Blaschke product. An upper bound is placed on the number of factors of the latter and a method for its construction is given. The paper contains a discussion of the use of these results in performing a stable analytic continuation of a set of data points under a condition of uniform boundedness, as well as some numerical examples.
26 citations
18 Jun 1986
TL;DR: In this article, the stability analysis of composite hybrid dynamical feedback systems of the type depicted in Figure 1, consisting of a block (usually the plant) which is described by an operator L and of a finite dimensional block described by a system of difference equations (usually a digital controller), is addressed.
Abstract: We address the stability analysis of composite hybrid dynamical feedback systems of the type depicted in Figure 1, consisting of a block (usually the plant) which is described by an operator L and of a finite dimensional block described by a system of difference equations (usually a digital controller). We establish results for the well-posedness, attractivity, asymptotic stability, uniform boundedness, asymptotic stability in the large, and exponential stability in the large for such systems. The hypotheses of our results are phrased in terms of the I/O properties of L and in terms of the Lyapunov stability properties of the subsystem described by the indicated difference equations. The applicability of our results is demonstrated by two specific examples.
21 citations
15 citations
TL;DR: In this paper, the authors discuss the asymptotic behavior of (C 0 ) semigroups on Hilbert space and the application of the mean ergodic theorem in big Hilbert spaces.
Abstract: Publisher Summary This chapter discusses the asymptotic behavior of ( C 0 ) semigroups on Hilbert space The semigroups of interest discussed in this chapter are the uniformly bounded and the contraction semigroups The chapter discusses the application of the mean ergodic theorem in “big” Hilbert spaces The first such space is the set of Hilbert–Schmidt operators on H The second is the space of pseudo-periodic locally square integrable H -valued function on R In the chapter, bound states and decaying and scattering states of a contraction semigroup are described The existence and characterization of periodic and pseudo-periodic solutions of time dependent evolution equations are discussed in the chapter An asymptotic equipartition of energy, scattering theory for abstract wave equations, and other equations of higher order in time are described in the chapter Wiener's Theory is discussed and a result of this theorem is also described in the chapter
14 citations
TL;DR: In this article, a new discretization technique for weakly nonlinear boundary value problems, especially for two point boundary value problem, is proposed, and the exact solution can be uniformly bounded from below and from above, respectively.
Abstract: In the present paper a new discretization technique for weakly nonlinear boundary value problems, especially for two point boundary value problems, is proposed. By means of the upper and lower discretizations the exact solution can be uniformly bounded from below and from above, respectively.
TL;DR: In this article, a generalization of the almost disturbance decoupling problem by state feedback is considered, where a second output is bounded with respect to the accuracy of decoupled output.
Abstract: This paper is concerned with a generalization of the almost disturbance decoupling problem by state feedback. Apart from approximate decoupling from the external disturbances to a first to-be-controlled output, we require a second output to be uniformly bounded with respect to the accuracy of decoupling. The problem is studied using the geometric approach to linear systems. We introduce some new almost controlled invariant subspaces and study their geometric structure. Necessary and sufficient conditions for the solvability of the above problem are formulated in terms of these controlled invariant subspaces. A conceptual algorithm is introduced to calculate the feedback laws needed to achieve the design purpose.
TL;DR: In this paper, the authors consider vector-valued functions with components which are entire functions for growth problems and show that all such functions satisfy a class of differential equations, and their components are also of bounded index and exponential type.
TL;DR: For non-uniform Lp-versions of the Berry-Esseen theorem, convergence of absolute moments and probabilities of moderate deviation (in some restricted cases) follow as consequences as discussed by the authors.
18 Jun 1986
TL;DR: In this article, a deterministic treatment for the control of a class of systems with time-varying uncertainties is presented, based on the knowledge of the present state, which guarantees desired performance of the system, including uniform boundedness and uniform ultimate boundedness.
Abstract: We present a deterministic treatment for the control of a class of systems with time-varying uncertainties. For the controller design we assume only that the uncertainties lie within some prescribed sets. The controller, based on the knowledge of the present state, guarantees desired performance of the system, including uniform boundedness and uniform ultimate boundedness. If, however, only output instead of state is available for the controller, the construction of a suitable observer is introduced. The resulting plant-observer-controller system is shown to preserve the above-mentioned desired performance. The theoretical result is illustrated by an application to the suspension control of a Maglev (magnetically levitated) Vehicle.
01 Jan 1986
TL;DR: In this paper, a general result concerning the local uniform boundedness of the set of approximate optimal solutions of a continuously parametrized family of optimization problems is given, which extends Hogan's Theorem and contains the already classical local and global boundedness results for the level sets of the quasiconvex and convex functions, respectively.
Abstract: A general result concerning the local uniform boundedness of the set of approximate optimal solutions of a continuously parametrized family of optimization problems is given. This result extends Hogan's Theorem and contains, in particular, the already classical local and global boundedness results for the level sets of the quasiconvex and convex functions, respectively. Also, a result concerning the existence of minimizing points for partially inf-bounded functions is obtained as a straightforward consequence. Finally, some general stability properties for a continuously parametrized family of convex optimization problems and their corresponding duals are obtained.
15 Jul 1986
TL;DR: A recursion-theoretic result telling when a family of reductions to a class can be replaced by a single oracle Turing machine is derived.
Abstract: We derive a recursion-theoretic result telling when a family of reductions to a class
can be replaced by a single oracle Turing machine. The theorem is a close analogue of the Uniform Boundedness Theorem of functional analysis, specializing it to the Cantor-set topology on ℙ(Σ*). This generalizes one of the main theorems of J. Grollmann and A. Selman [FOCS '84], namely that NP-hardness implies uniform NP-hardness for ‘promise problems’. We investigate other consequences and problems arising from the theorem.
TL;DR: One-leg methods for general differential-algebraic equations are analyzed in this article, where sufficient stability conditions are formulated for transferable DAL equations and a natural scaling is proposed to obtain matrices with uniformly bounded condition in the linear systems to be solved.
Abstract: One-leg methods for general differential-algebraic equations are analyzed. Sufficient stability conditions are formulated for transferable differential-algebraic equations. A natural scaling is proposed to obtain matrices with uniformly bounded condition in the linear systems to be solved.
TL;DR: In this article, it was shown that in an infinite prism the linearization at a natural state of a family of semi-inverse solutions is a combination of elementary St.-Venant solutions, namely, extension bending and torsion.
Abstract: It is shown that in an infinite prism the linearization at a natural state of a family of Ericksen’s semi-inverse solutions is a combination of elementary St.-Venant solutions, namely, extension bending and torsion. Moreover, the span of these St. Venant solutions is precisely the linear manifold of solutions having locally uniformly bounded strain energy. This implies that any solution of a linearized problem in a finite prism continuable to a solution in the infinite prism in a manner that its energy on any portion of fixed length remains bounded, is an elementary St.-Venant solution.
TL;DR: In this article, the authors investigated the topological character of the Lemma of Hayman in the study of the asymptotic behavior near x of meromorphic functions and showed that the local analyticity of the functions can be replaced by continuity.
Abstract: We investigate the topological character of a Lemma of Hayman in his study of the asymptoticbehaviour near x of meromorphic functions: it turns out that the local analyticity of the functionscan be replaced by continuity. The main result is that a function f continuous in C either has x as an asymptotic value or is bounded on a pathgoing to x or is uniformly bounded on a sequence uf closed curves surrounding the origin and receding to x.
TL;DR: In this paper, the authors showed that the mechanism of diffusion in the equation of heat conduction is not sufficiently strong to smooth out highly nonuniform, albeit bounded, distributions of initial data.
Abstract: with initial condition in L”(R), which did not stabilize at x = 0. This indicated that the mechanism of diffusion in the equation of heat conduction is not sufficiently strong to smooth out highly nonuniform, albeit bounded, distributions of initial data. Later on, Repnikov and Eidelman [20], [21] gave necessary and sufficient conditions for stabilization of solutions of certain parabolic equations, including (1.3). Their result stated that within the class of uniformly bounded solutions one has
TL;DR: In this article, the lower semi-equicontinuity of a family of convex multivalued mappings from a normed space into another one and their norms is shown.
Abstract: This paper will present some relations between the lower semi-equicontinuity of some family of multivalued mappings from a normed space into another one and their norms. Further, the BANACH-STEINHAUS Principle of the uniform boundedness will be proved for the family of convex multivalued mappings. This result yields some applications on stability of solutions of nonlinear multivalued equations.
TL;DR: In this paper, mild solutions on the real line of the differential equation in a Banach space, where A is the infinitesimal generator of a Co semi-group Tt, were considered.
Abstract: In this work we consider mild solutions on the whole real line of the differential equation in a Banach space, u'(t) = Au(t) , where A is the infinitesimal generator of a Co semi-group Tt, which is uniformly bounded. We prove that solutions with precompact range possess a generalized normality property (in Bochner's sense).
01 Jan 1986
TL;DR: The computer-aided execution of a verification algorithm ensures the uniform boundedness of all perturbed solutions with initial values and parameters from finite neighborhoods in a system of ordinary differential equations.
Abstract: Summary The computer-aided execution of a verification algorithm ensures the uniform boundedness of all perturbed solutions with initial values and parameters from finite neighborhoods. This test is applied with respect to the system of ordinary differential equations as following from a Fourier-approximation and a collocation method. Applications are concerned with a system of three PDEs (of first order each) from continuum mechanics. The decisive importance of the computer-aided execution of suitable enclosure methods is shown.
TL;DR: In this paper, the scalar Volterra integral equation u(t) + ∝ 0 t g(t, s, u(s)) ds = ǫ(t), whereǫ is the half line.
01 Mar 1986
TL;DR: In this article, it is shown that a real-valued, bounded, Baire class one function of a real variable is the derivative of its indefinite integral at every point except possibly those in a set which is both of measure zero and of first category.
Abstract: It is well known that a real-valued, bounded, Baire class one function of a real variable is the derivative of its indefinite integral at every point except possibly those in a set which is both of measure zero and of first category. In the present paper, a bounded, Darboux, Baire class one function is constructed to have the property that its indefinite integral fails to be differentiable at non-cr-porous set of points. Such functions are then shown to be "typical" in the sense of category in several standard function spaces. We shall denote by B1, DB1, bB1, and bDB1 the spaces of Baire class 1 func- tions, Baire 1 Darboux functions, bounded Baire 1 functions, and bounded Baire 1 Darboux functions, respectively, all defined on the interval (0,1) and all equipped with the topology of uniform convergence. The word "typical" in the title refers to any property which holds for most elements in a space of functions in the sense of category; i.e., the collection of functions not possessing the property is of first category in the space. Perhaps a better title for the present paper would be, "Another typical property of DB1 functions" for many such properties are known. A virtually complete catalog of such results can be found in the survey article (2) by Ceder and Pearson. In the process of writing (4) the present authors became curious about the "size" of the set of points at which bounded functions in certain classes can fail to be the derivatives of their indefinite integrals. Clearly, for functions in bBx this exception must be both of measure zero and of first category. In (4) we were especially concerned with finding circumstances under which this set would be a-porous. (The concept of a er-porous set was introduced in (3) by E. P. Dolzenko. The porosity of a set E of real numbers at the point x on the real line is the value where l(x,r,E) denotes the length of the largest open interval contained in the intersection of the complement of E with the interval (x - r, x + r). The set E is porous if it has positive porosity at each of its points, and it is cr-porous if it is a countable union of porous sets. Thus, cr-porous sets are of both measure zero and first category. Dolzenko showed that cr-porous sets are the natural exceptional sets for certain types of boundary behavior for complex functions defined, for example, in the upper half plane. More recently, cr-porous sets have been found to play a useful role in describing behavior of real functions.) In (4) we showed that any function in the subclass of bB1 consisting of bounded approximately symmetric functions is the derivative of its indefinite integral except at a cr-porous set of points. As noted in (4), it is easy to see that an arbitrary function in bB1 need
TL;DR: In this article, the reproduction of derivatives is based on the construction of functions which are the upper and lower boundaries of a set of derivatives, and certain linear programming problems of constructing these functions are solved.
Abstract: Reproduction algorithms that are of optimal accuracy for the derivatives of functions which are specified, with an error, by their values, the second derivative of these functions being bounded by a known constant, are discussed. The reproduction of derivatives is based on the construction of functions which are the upper and lower boundaries of a set of derivatives. Certain linear programming problems of constructing these functions are solved. A special form of the objective functions and of the constraints makes it possible to obtain explicit formulae for the boundary functions.
01 Jan 1986
TL;DR: In this article, the stability analysis of composite hybrid dynamical feedback systems of the type depicted in Figure 1, consisting of a block (usually the plant) which is described by an operator L and of a finite dimensional block described by a system of ODEs, is presented.
Abstract: We address the stability analysis of composite hybrid dynamical feedback systems of the type depicted in Figure 1, consisting of a block (usually the plant) which is described by an operator L and of a finite dimensional block described by a system of ordinary differential equations (usually the controller) We establish results for the well-posedness, attractlvity, asymptotic stability, uniform boundedness, asymptotic stability in the large, and exponential stability in the large for such systems The hypotheses of these results are phrased in terms of the I/O properties of L and in terms of the Lyapunov stability properties of the subsystem described by the Indicated ordinary differential equations The applicability of our results is demonstrated by means of general specific examples (involving Cg-semlgroups, partial differential equations or Integral equations which determine L)