Uniform Boundedness of Iterates of Analytic Mappings Implies Linearization: a Simple Proof and Extensions

A well-known result in complex dynamics shows that if the iterates of an analytic map are uniformly bounded in a complex domain, then the map is analytically conjugate to a linear map. We present a simple proof of this result in any dimension. We also present several generalizations and relations to other results in the literature.     

Analysis of direct searches for discontinuous functions

It is known that the Clarke generalized directional derivative is nonnegative along the limit directions generated by directional direct-search methods at a limit point of certain subsequences of unsuccessful iterates, if the function being minimized is Lipschitz continuous near the limit point. In this paper we generalize this result for discontinuous functions using Rockafellar generalized directional derivatives (upper subderivatives). We show that Rockafellar derivatives are also nonnegative along the limit directions of those subsequences of unsuccessful iterates when the function values converge to the function value at the limit point. This result is obtained assuming that the function is directionally Lipschitz with respect to the limit direction. It is also possible under appropriate conditions to establish more insightful results by showing that the sequence of points generated by these methods eventually approaches the limit point along the locally best branch or step function (when the number of steps is equal to two). The results of this paper are presented for constrained optimization and illustrated numerically.     

A new proof of the stable manifold theorem

We give a new proof of the stable manifold theorem for hyperbolic fixed points of smooth maps. This proof shows that the local stable and unstable manifolds are projections of a relation obtained as a limit of the graphs of the iterates of the map. The same proof generalizes to the setting of stable and unstable manifolds for smooth relations.     

Slow escaping points of quasiregular mappings

This article concerns the iteration of quasiregular mappings on \(\mathbb {R}^d\) and entire functions on \(\mathbb {C}\). It is shown that there are always points at which the iterates of a quasiregular map tend to infinity at a controlled rate. Moreover, an asymptotic rate of escape result is proved that is new even for transcendental entire functions. Let \(f:\mathbb {R}^d\rightarrow \mathbb {R}^d\) be quasiregular of transcendental type. Using novel methods of proof, we generalise results of Rippon and Stallard in complex dynamics to show that the Julia set of f contains points at which the iterates \(f^n\) tend to infinity arbitrarily slowly. We also prove that, for any large R, there is a point x with modulus approximately R such that the growth of \(|f^n(x)|\) is asymptotic to the iterated maximum modulus \(M^{n}(R,f)\).     

Convergence of Implementable Descent Algorithms for Unconstrained Optimization

Descent algorithms use sufficient descent directions combined with stepsize rules, such as the Armijo rule, to produce sequences of iterates whose cluster points satisfy some necessary optimality conditions. In this note, we present a proof that the whole sequence of iterates converges for quasiconvex objective functions.     

Approximation with the output of linear control systems

In this paper we give a new proof that for controllable and observable linear systems every L²[0,T] function can be approximated in the L²[0,T] sense with an output function generated by an L²[0,T] input function. We also give a new characterization of how continuous functions on [0,T] are uniformly approximated by an output generated by a continuous input function. The relative degree of the transfer function of the system determines those functions that can be approximated. We further show that if the initial data is allowed to vary then every continuous function is uniformly approximated by outputs generated by continuous functions.     

Snapback repellers and semiconjugacy for iterated entire mappings

A theorem of Poincaré guarantees existence of the local conjugacy of an entire analytic mapping with an hyperbolically unstable fixed point to the linearized mapping. Since the local conjugacy can be extended to a global conjugacy, it is a valuable tool for the global study of dynamics. Especially we focus on snapback repellers which are defined as entire orbits which tend to an unstable fixed point in the past and snap back to the same fixed point. Snapback repellers correspond to the zeros of the semiconjugacy. It turns out that in general there exist infinitely many snapback points and for each one of them there exist infinitely many snapback repellers. The exceptional classes of functions with a different behavior are characterized. The proof exploits the Theorem of Picard about the range of values that an analytic function assumes near an essential singularity. Furtheron, we related the multiplicity of the zeros of the semiconjugacy to the occurrence of critical points in the corresponding snapback repeller. For quadratic mappings and their iterates, the zeros of the semiconjugacy have at most multiplicity two.     

Trust region affine scaling algorithms for linearly constrained convex and concave programs

We study a trust region affine scaling algorithm for solving the linearly constrained convex or concave programming problem. Under primal nondegeneracy assumption, we prove that every accumulation point of the sequence generated by the algorithm satisfies the first order necessary condition for optimality of the problem. For a special class of convex or concave functions satisfying a certain invariance condition on their Hessians, it is shown that the sequences of iterates and objective function values generated by the algorithm convergeR-linearly andQ-linearly, respectively. Moreover, under primal nondegeneracy and for this class of objective functions, it is shown that the limit point of the sequence of iterates satisfies the first and second order necessary conditions for optimality of the problem. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.The work of these authors was based on research supported by the National Science Foundation under grant INT-9600343 and the Office of Naval Research under grants N00014-93-1-0234 and N00014-94-1-0340.     

On the iterates of positive linear operators preserving the affine functions

In this note we study the limit behavior of the iterates of a large class of positive linear operators preserving the affine functions and, as a byproduct of our result, we obtain the limit of the iterates of Meyer-König and Zeller operators.     

Double scaling limit in the random matrix model: The Riemann‐Hilbert approach

We prove the existence of the double scaling limit in the unitary matrix model with quartic interaction, and we show that the correlation functions in the double scaling limit are expressed in terms of the integrable kernel determined by the ψ function for the Hastings‐McLeod solution to the Painlevé II equation. The proof is based on the Riemann‐Hilbert approach, and the central point of the proof is an analysis of analytic semiclassical asymptotics for the ψ function at the critical point in the presence of four coalescing turning points. © 2003 Wiley Periodicals, Inc.     

A simple justification of the singular limit for equatorial shallow‐water dynamics

The equatorial shallow‐water equations at low Froude number form a symmetric hyperbolic system with large variable‐coefficient terms. Although such systems are not covered by the classical Klainerman‐Majda theory of singular limits, the first two authors recently proved that solutions exist uniformly and converge to the solutions of the long‐wave equations as the height and Froude number tend to 0. Their proof exploits the special structure of the equations by expanding solutions in series of parabolic cylinder functions. A simpler proof of a slight generalization is presented here in the spirit of the classical theory. © 2008 Wiley Periodicals, Inc.     

On the Uniformity of Distribution of the Naor–Reingold Pseudo-Random Function

We show that the new pseudo-random number function, introduced recently by M. Naor and O. Reingold, possesses one more attractive and useful property. Namely, it is proved that for almost all values of parameters it produces a uniformly distributed sequence. The proof is based on some recent bounds of character sums with exponential functions.     

Sur les automorphismes analytiques des variétés hyperboliques

Results of H. Cartan about holomorphic automorphisms on bounded domains are generalized to the case of hyperbolic manifolds in the sense of Kobayashi. In this setting, we give an identity theorem together with its topological version. We show also that a sequence of automorphisms which converges uniformly on some nonempty open set, has a limit on the whole space which is an automorphism. At the end of the paper, conditions are given for the sequence of iterates of a self holomorphic map in order to be an automorphism.     

Polynomial operators and local smoothness classes on the unit interval

We obtain a characterization of local Besov spaces of functions on [-1,1] in terms of algebraic polynomial operators. These operators are constructed using the coefficients in the orthogonal polynomial expansions of the functions involved. The example of Jacobi polynomials is studied in further detail. A by-product of our proofs is an apparently simple proof of the fact that the Cesàro means of a sufficiently high integer order of the Jacobi expansion of a continuous function are uniformly bounded.     

Implementing Pure Adaptive Search with Grover's Quantum Algorithm

Pure adaptive search (PAS) is an idealized stochastic algorithm for unconstrained global optimization. The number of PAS iterations required to solve a problem increases only linearly in the domain dimension. However, each iteration requires the generation of a random domain point uniformly distributed in the current improving region. If no regularity conditions are known to hold for the objective function, then this task requires a number of classical function evaluations varying inversely with the proportion of the domain constituted by the improving region, entirely counteracting the PAS apparent speedup. The Grover quantum computational search algorithm provides a way to generate the PAS iterates. We show that the resulting implementation, which we call the Grover adaptive search (GAS), realizes PAS for functions satisfying certain conditions, and we believe that, when quantum computers will be available, GAS will be a practical algorithm.     

A forward–backward penalty scheme with inertial effects for monotone inclusions. Applications to convex bilevel programming

ABSTRACT

We investigate a forward–backward splitting algorithm of penalty type with inertial effects for finding the zeros of the sum of a maximally monotone operator and a cocoercive one and the convex normal cone to the set of zeroes of an another cocoercive operator. Weak ergodic convergence is obtained for the iterates, provided that a condition expressed via the Fitzpatrick function of the operator describing the underlying set of the normal cone is verified. Under strong monotonicity assumptions, strong convergence for the sequence of generated iterates is proved. As a particular instance we consider a convex bilevel minimization problem including the sum of a non-smooth and a smooth function in the upper level and another smooth function in the lower level. We show that in this context weak non-ergodic and strong convergence can be also achieved under inf-compactness assumptions for the involved functions.     

Approximation of semigroups and cosine functions in spaces of periodic functions

In this article we furnish a representation of the solutions of some classes of first-order and second-order evolution problems as limit of iterates of classical sequences of approximating operators. The method is based on Trotter's theorem on the approximation of semigroups which is applied here also for the approximation of groups and cosine functions. We apply this method in spaces of continuous periodic functions and using some classical sequences of trigonometric polynomials.     

Über Die Konvergenz von Einzelschrittverfahren zur Minimierung konvexer Funktionen

A first convergence proof for free steering minimization methods was given by SCHECHTER [3]. The function f to be minimized was assumed by him to be twice continously differentiable and strictly convex. These conditions were weakened by ELKIN [1], who considered continously differentiable and uniformly convex functions and who employed a slight modification of SCHECHTER's convergence proof. The following article gives a new proof for convergence of free steering methods. We assume f to be strictly convex and continously differentiable.

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Aus Kap. I 1. der Dissertation des Verfassers [2].     

A nonlinear programming algorithm based on non-coercive penalty functions

 We consider first the differentiable nonlinear programming problem and study the asymptotic behavior of methods based on a family of penalty functions that approximate asymptotically the usual exact penalty function. We associate two parameters to these functions: one is used to control the slope and the other controls the deviation from the exact penalty. We propose a method that does not change the slope for feasible iterates and show that for problems satisfying the Mangasarian-Fromovitz constraint qualification all iterates will remain feasible after a finite number of iterations. The same results are obtained for non-smooth convex problems under a Slater qualification condition. Received: September 2000 / Accepted: June 2002 Published online: March 21, 2003 Research partially supported by CAPES, Brazil Research partially supported by CNPq, Brazil, and CONICIT, Venezuela. Mathematics Subject Classification (1991): 20E28, 20G40, 20C20     

An empirical central limit theorem for intermittent maps

We prove an empirical central limit theorem for the distribution function of a stationary sequence, under a dependence condition involving only indicators of half line. We show that the result applies to the empirical distribution function of iterates of expanding maps with a neutral fixed point at zero as soon as the correlations are summable.