Newton’s method for generalized equations: a sequential implicit function theorem

In an extension of Newton’s method to generalized equations, we carry further the implicit function theorem paradigm and place it in the framework of a mapping acting from the parameter and the starting point to the set of all associated sequences of Newton’s iterates as elements of a sequence space. An inverse function version of this result shows that the strong regularity of the mapping associated with the Newton sequences is equivalent to the strong regularity of the generalized equation mapping.     

On the Teichmüller theorem and the heights theorem for quadratic differentials

By using the Marden-Strebel heights theorem for quadratic differentials, we provide a concrete method for finding the Teichmüller differential associated with the Teichmüller mapping between compact or finitely punctured Riemann surfaces.

     

A cocycle construction for groups

In the spirit of Wielandt’s method of introducing the transfer mapping, a cocycle construction with several applications is given. In particular, we derive constructions of certain group automorphisms under fairly general conditions.     

A hybrid extragradient method for general variational inequalities

In this paper, we introduce and study a hybrid extragradient method for finding solutions of a general variational inequality problem with inverse-strongly monotone mapping in a real Hilbert space. An iterative algorithm is proposed by virtue of the hybrid extragradient method. Under two sets of quite mild conditions, we prove the strong convergence of this iterative algorithm to the unique common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general variational inequality problem, respectively. L. C. Zeng’s research was partially supported by the National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality grant (075105118). J. C. Yao’s research was partially supported by a grant from the National Science Council of Taiwan.     

Sensitivity Analysis of Wave-equation Tomography: A Multi-scale Approach

Earthquakes, viewed as passive sources, or controlled sources, like explosions, excite seismic body waves in the earth. One detects these waves at seismic stations distributed over the earth’s surface. Wave-equation tomography is derived from cross correlating, at each station, data simulated in a reference model with the observed data, for a (large) set of seismic events. The times corresponding with the maxima of these cross correlations replace the notion of residual travel times used as data in traditional tomography. Using first-order perturbation, we develop an analysis of the mapping from a wavespeed contrast (between the “true” and reference models) to these maxima. We develop a construction using curvelets, while establishing a connection with the geodesic X-ray transform. We then introduce the adjoint mapping, which defines the imaging of wavespeed variations from “finite-frequency travel time” residuals. The key underlying component is the construction of the Fréchet derivative of the solution to the seismic Cauchy initial value problem in wavespeed models of limited smoothness. The construction developed in this paper essentially clarifies how a wavespeed model is probed by the method of wave-equation tomography.     

A generalization of D. A. Grave's method for plane boundary-value problems in harmonic potential theory

The article describes and proves D. A. Grave's method that solves classical plane boundary-value problems for the Green's function of the Laplace equation in regions whose boundaries are smooth analytical curves defined by finite-order irreducible polynomials. The proposed method has certain advantages compared with the method that constructs the Green's function by conformally mapping the original region onto the unit disk. A class of regions are identified for which Grave's methods produces an explicit analytical solution in convergent-series form. This is a natural generalization of the conformal mapping method for simplest regions. Translated from Prikladnaya Matematika i Informatika, No. 1, pp. 5–19, 1999.     

Generate gene expression profile from high-throughput sequencing data

This work presents two methods, the Least-square and Bayesian method, to solve the multiple mapping problem in extracting gene expression profiles through the next-generation sequencing. We parallel the tag sequences to genome, and partition them to improving the methods’ efficiency. The essential feature of these methods is that they can solve the multiple mapping problem between genes and short-reads, while generating almost the same estimation in single-mapping situation as the traditional approaches. These two methods are compared by simulation and a real example, which was generated from radiation-induced lung cancer cells (A549), through mapping short-reads to human ncRNA database. The results show that the Bayesian method, as realized by Gibbs sampler, is more efficient and robust than the Least-square method.     

A regularization method for the numerical solution of elliptic boundary control problems with pointwise state constraints

A Lavrentiev type regularization technique for solving elliptic boundary control problems with pointwise state constraints is considered. The main concept behind this regularization is to look for controls in the range of the adjoint control-to-state mapping. After investigating the analysis of the method, a semismooth Newton method based on the optimality conditions is presented. The theoretical results are confirmed by numerical tests. Moreover, they are validated by comparing the regularization technique with standard numerical codes based on the discretize-then-optimize concept. The authors acknowledge support through DFG Research Center “Mathematics for Key Technologies” (FZT 86) in Berlin.     

On some properties and examples of Nevanlinna domains

The properties of Nevanlinna domains are considered. These domains arise in the problems of approximation by polyanalytic functions. Several analytic and geometric properties (both new and earlier known) of Nevanlinna domains are described. In particular, a new method for constructing Nevanlinna domains with boundaries belonging to the class C¹ is proposed, and new examples of such domains whose boundaries do not belong to the class C^1,α for α ∈ (0, 1) are presented. This method is based on the property of pseudocontinuation of a conformal mapping from the unit disk onto a Nevanlinna domain. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2006, Vol. 253, pp. 204–213.     

On the conformal, concircular, and spin mappings of gravitational fields

A mapping ρ: of two Riemannian or pseudo-Riemannian spaces is called aspin mapping if for each geodesic curve γ in M_n its image ρ^oγ is a spin-curve in the space . In gravitational fields spin-curves describe the trajectories of uniformly accelerated particles of constant mass with simultaneous self-rotation. We prove: 1) a conformal mapping is a spin mapping only when it is concircular; 2) every conformal mapping of Einstein space is a spin mapping. The latter makes it possible to give a local representation of the metrics of all gravitational fields that admit spin mappings. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 44–47. Original article submitted March 17, 1993.     

Jordan Homomorphisms and Harmonic Mappings

 We show that each Jordan homomorphism R → R′ of rings gives rise to a harmonic mapping of one connected component of the projective line over R into the projective line over R′. If there is more than one connected component then this mapping can be extended in various ways to a harmonic mapping which is defined on the entire projective line over R. Received December 7, 2001; in revised form April 28, 2002 Published online January 7, 2003     

Nonholonomous geodesics as solutions to Euler-Lagrange integral equations and the differential of the exponential mapping

Equations of horizontal geodesics on a Riemannian (or pseudo-Riemannian) manifold with nonholonomous distribution are obtained using the Euler-Lagrange method in Pontryagin’s formulation. It is shown that if the distribution and the metric tensor of the distribution are C ^k -smooth, k ≥ 1, then any regular solution to the variational problem is C ^{k + 1}-smooth. The differential of the exponential mapping is obtained for nonholonomous distribution with the condition of cyclicity with respect to “vertical” coordinates. This differential is nonsingular provided that the distribution is strongly bracket generating.     

Resonance elliptic variational inequalities with discontinuous nonlinearities of linear growth

We consider resonance elliptic variational inequalities with second-order differential operators and discontinuous nonlinearities of linear growth. The theorem on existence of a strong solution is proved. The initial-value problem is reduced to the problem of existence of a fixed point for a compact multivalued mapping and then the existence of this point is established by the Leray–Schauder method.     

A smoothing operator polynomial in the Hilbert spaceH λ

A mapping and an operator polynomial are considered as elements of the space H(λ). The polynomial is proved to be smoothing for the mapping in the metric of the space H(λ), that is, to be a solution to the corresponding extremum problem. Bibliography:4 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 77, 1993, pp. 27–34.     

Functor of weakly additive τ-smooth functionals and mappings

We present a criterion for the τ-smoothness of weakly additive order-preserving functionals and establish that the functor of τ-smooth weakly additive functionals preserves the perfectness of mappings. We prove that if a continuous mapping between Tikhonov spaces is such that the mapping between the corresponding spaces of weakly additive functionals generated by it is open, then the indicated mapping is also open. It is shown that the converse statement is true under certain additional conditions.     

(K, K′)-quasiconformal Harmonic Mappings

We prove that a harmonic diffeomorphism between two Jordan domains with C ² boundaries is a (K, K′) quasiconformal mapping for some constants K ≥ 1 and K′ ≥ 0 if and only if it is Lipschitz continuous. In this setting, if the domain is the unit disk and the mapping is normalized by three boundary points condition we give an explicit Lipschitz constant in terms of simple geometric quantities of the Jordan curve which surrounds the codomain and (K, K′). The results in this paper generalize and extend several recently obtained results.     

Stability Analysis for Minty Mixed Variational Inequality in Reflexive Banach Spaces

This paper is devoted to the stability analysis for a class of Minty mixed variational inequalities in reflexive Banach spaces, when both the mapping and the constraint set are perturbed. Several equivalent characterizations are given for the Minty mixed variational inequality to have nonempty and bounded solution set. A stability result is presented for the Minty mixed variational inequality with Φ-pseudomonotone mapping in reflexive Banach space, when both the mapping and the constraint set are perturbed by different parameters. As an application, a stability result for a generalized mixed variational inequality is also obtained. The results presented in this paper generalize and extend some known results in Fan and Zhong (Nonlinear Anal., Theory Methods Appl. 69:2566–2574, 2008) and He (J. Math. Anal. Appl. 330:352–363, 2007).     

A characterization theorem for the nearest point mappings of linearC-sets

Summary Suppose U is a set, F is a field of subsets of U, P_AB is the set of all bounded finitely additive functions from F into ℝ, and P _A ⁺ is the set of non-negative valued elements of P_AB. In this paper it is proved that if β is a mapping from P _A ⁺ into P _A ⁺ then β is the nearest point mapping for a linear C-set iff β commutes with a certain collection of nonlinear mappings. Entrata in Redazione il 6 ottobre 1976.     

Optimization of the Local Search in the Training for SAMANN Neural Network

In this paper, we discuss the visualization of multidimensional data. A well-known procedure for mapping data from a high-dimensional space onto a lower-dimensional one is Sammon’s mapping. This algorithm preserves as well as possible all interpattern distances. We investigate an unsupervised backpropagation algorithm to train a multilayer feed-forward neural network (SAMANN) to perform the Sammon’s nonlinear projection. Sammon mapping has a disadvantage. It lacks generalization, which means that new points cannot be added to the obtained map without recalculating it. The SAMANN network offers the generalization ability of projecting new data, which is not present in the original Sammon’s projection algorithm. To save computation time without losing the mapping quality, we need to select optimal values of control parameters. In our research the emphasis is put on the optimization of the learning rate. The experiments are carried out both on artificial and real data. Two cases have been analyzed: (1) training of the SAMANN network with full data set, (2) retraining of the network when the new data points appear.