Spectral density estimates with partial symmetries and an application to Bahri–Lions-type results

We are concerned with the existence of infinitely many solutions for the problem \(-\Delta u=|u|^{p-2}u+f\) in \(\Omega \), \(u=u_0\) on \(\partial \Omega \), where \(\Omega \) is a bounded domain in \(\mathbb {R}^N\), \(N\ge 3\). This can be seen as a perturbation of the problem with \(f=0\) and \(u_0=0\), which is odd in u. If \(\Omega \) is invariant with respect to a closed strict subgroup of O(N), then we prove infinite existence for all functions f and \(u_0\) in certain spaces of invariant functions for a larger range of exponents p than known before. In order to achieve this, we prove Lieb–Cwikel–Rosenbljum-type bounds for invariant potentials on \(\Omega \), employing improved Sobolev embeddings for spaces of invariant functions.     

Harmonic maps from compact Kähler manifolds to exceptional hyperbolic spaces

It has been conjectured that a lattice in a noncompact group of real rank one, other than SU(1,n), cannot be isomorphic to the fundamental group of a compact Kähler manifold; moreover, it is known to be true for SO(1,n). In this note it is shown that this conjecture also holds for the case of uniform lattices in F_4(?20), the group of isometries of the Cayley hyperbolic plane. The result is a consequence of a classification theorem for harmonic maps between Kähler and Cayley hyperbolic manifolds.     

Approximation of inverse operators by a new family of high‐order iterative methods

The main goal of this paper is to approximate inverse operators by high‐order Newton‐type methods with the important feature of not using inverse operators. We analyse the semilocal convergence, the speed of convergence, and the efficiency of these methods. We determine that Chebyshev's method is the most efficient method and test it on two problems: one associated to the heat equation and the other one to a boundary value problem. We consider examples with matrices that are close to be singular and/or are badly conditioned. We check the robustness and the stability of the methods by considering situations with many steps and noised data. Copyright © 2013 John Wiley & Sons, Ltd.     

How to Improve the Domain of Starting Points for Steffensen's Method

We analyze the semilocal convergence of Steffensen's method, using a novel technique, which is based on recurrence relations, for solving systems of nonlinear equations. This technique allows analyzing the convergence of Steffensen's method to solutions of equations, where the function involved can be both differentiable and nondifferentiable. Moreover, this technique also allows enlarging the domain of starting points for Steffensen's method from certain predictions with the simplified Steffensen method.     

Second-Derivative-Free Variant of the Chebyshev Method for Nonlinear Equations

In this paper, we introduce a numerical method for nonlinear equations, based on the Chebyshev third-order method, in which the second-derivative operator is replaced by a finite difference between first derivatives. We prove a semilocal convergence theorem which guarantees local convergence with R-order three under conditions similar to those of the Newton-Kantorovich theorem, assuming the Lipschitz continuity of the second derivative. In a subsequent theorem, the latter condition is replaced by the weaker assumption of Lipschitz continuity of the first derivative.     

A Lorentz force flow meter for application at blast furnaces: design and calibration

A reference Lorentz force flow meter (LFF) has been developed to measure molten steel mass flow at the end of the runner of an experimental blast furnace. It works according to the principles of Lorentz force velocimetry [1] in which a static magnetic field interacts with a liquid metal stream. The magnetic field lines are generated by an arrangement of permanent magnets and penetrate the entire cross-section of the flow generating eddy currents and a total Lorentz force inside the melt. This force is proportional to the mass flow of the liquid metal and owing Newton's third law, there is a counter force of the same magnitude acting on the magnet system which is connected to a load cell. For accurate flow rate measurements, a “dry and wet calibration” of the LFF needs to be performed [2]. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Kramer Sampling Theorem Revisited

The classical Kramer sampling theorem provides a method for obtaining orthogonal sampling formulas. Besides, it has been the cornerstone for a significant mathematical literature on the topic of sampling theorems associated with differential and difference problems. In this work we provide, in an unified way, new and old generalizations of this result corresponding to various different settings; all these generalizations are illustrated with examples. All the different situations along the paper share a basic approach: the functions to be sampled are obtaining by duality in a separable Hilbert space \(\mathcal {H}\) through an \(\mathcal {H}\) -valued kernel K defined on an appropriate domain.     

Residual Julia set for functions with the Ahlfors' Property

We consider two classes of functions studied by Epstein [A.L. Epstein, Towers of finite type complex analytic maps, Ph.D. thesis, City University of New York, 1993] and by Herring [M.E. Herring, An extension of the Julia–Fatou theory of iteration, Ph.D. thesis, Imperial College, London, 1994], which have the Ahlfors' Property. We prove under some conditions on the Fatou and Julia sets that the singleton buried components are dense in the Julia set for these classes of functions.     

Lagrangian Duality in Set-Valued Optimization

In this paper, we study optimization problems where the objective function and the binding constraints are set-valued maps and the solutions are defined by means of set-relations among all the images sets (Kuroiwa, D. in Takahashi, W., Tanaka, T. (eds.) Nonlinear analysis and convex analysis, pp. 221–228, 1999). We introduce a new dual problem, establish some duality theorems and obtain a Lagrangian multiplier rule of nonlinear type under convexity assumptions. A necessary condition and a sufficient condition for the existence of saddle points are given. The authors thank the two referees for valuable comments and suggestions on early versions of the paper. The research of the first author was partially supported by Ministerio de Educación y Ciencia (Spain) Project MTM2006-02629 and by Junta de Castilla y León (Spain) Project VA027B06.