Optimal order results for a class of regularization methods using unbounded operators

A class of regularization methods using unbounded regularizing operators is considered for obtaining stable approximate solutions for ill-posed operator equations. With an a posteriori as well as an a priori parameter choice strategy, it is shown that the method yields the optimal order. Error estimates have also been obtained under stronger assumptions on the generalized solution. The results of the paper unify and simplify many of the results available in the literature. For example, the optimal results of the paper include, as particular cases for Tikhonov regularization, the main result of Mair (1994) with an a priori parameter choice, and a result of Nair (1999) with an a posteriori parameter choice. Thus the observations of Mair (1994) on Tikhonov regularization of ill-posed problems involving finitely and infinitely smoothing operators is applicable to various other regularization procedures as well. Subsequent results on error estimates include, as special cases, an optimal result of Vainikko (1987) and also some recent results of Tautenhahn (1996) in the setting of Hilbert scales.     

On Anastassiou's Generalizations of the Ostrowski Inequality and Related Results

We provide a generalization of a recent result of Anastassiou related to thewell-known Ostrowski inequality, as well as some related results. Ourresults subsume, extend, and consolidate a number of known results.     

On the moments and distribution of discrete Choquet integrals from continuous distributions

We study the moments and the distribution of the discrete Choquet integral when regarded as a real function of a random sample drawn from a continuous distribution. Since the discrete Choquet integral includes weighted arithmetic means, ordered weighted averaging functions, and lattice polynomial functions as particular cases, our results encompass the corresponding results for these aggregation functions. After detailing the results obtained in [J.-L. Marichal, I. Kojadinovic, Distribution functions of linear combinations of lattice polynomials from the uniform distribution, Statistics & Probability Letters 78 (2008) 985–991] in the uniform case, we present results for the standard exponential case, show how approximations of the moments can be obtained for other continuous distributions such as the standard normal, and elaborate on the asymptotic distribution of the Choquet integral. The results presented in this work can be used to improve the interpretation of discrete Choquet integrals when employed as aggregation functions.     

Difference analogue of the Lemma on the Logarithmic Derivative with applications to difference equations

The Lemma on the Logarithmic Derivative of a meromorphic function has many applications in the study of meromorphic functions and ordinary differential equations. In this paper, a difference analogue of the Logarithmic Derivative Lemma is presented and then applied to prove a number of results on meromorphic solutions of complex difference equations. These results include a difference analogue of the Clunie lemma, as well as other results on the value distribution of solutions.     

Projection-proximal methods for general variational inequalities

In this paper, we consider and analyze some new projection-proximal methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is a weaker condition than monotonicity. The proposed methods include several new and known methods as special cases. Our results can be considered as a novel and important extension of the previously known results. Since the general variational inequalities include the quasi-variational inequalities and implicit complementarity problems as special cases, results proved in this paper continue to hold for these problems.     

The Hazard Rate and the Reversed Hazard Rate Orders, with Applications to Order Statistics

In this paper we first point out a simple observation that can be used successfully in order to translate results about the hazard rate order into results about the reversed hazard rate order. Using it, we derive some interesting new results which compare order statistics in the hazard and in the reversed hazard rate orders; as well as in the usual stochastic order. We also simplify proofs of some known results involving the reversed hazard rate order. Finally, a few further applications of the observation are given.     

Random Latin square graphs

In this paper we introduce new models of random graphs, arising from Latin squares which include random Cayley graphs as a special case. We investigate some properties of these graphs including their clique, independence and chromatic numbers, their expansion properties as well as their connectivity and Hamiltonicity. The results obtained are compared with other models of random graphs and several similarities and differences are pointed out. For many properties our results for the general case are as strong as the known results for random Cayley graphs and sometimes improve the previously best results for the Cayley case. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Common fixed points Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces

In this paper, we introduce a general iteration scheme for a finite family of asymptotically quasi-nonexpansive mappings. The new iterative scheme includes the modified Mann and Ishikawa iterations, three-step iterative scheme of Xu and Noor and Khan and Takahashi scheme as special cases. Our results are generalizations as well as refinement of several known results in the current literature.     

Predicting daily exchange rate with singular spectrum analysis

This paper uses univariate and multivariate singular spectrum analysis for predicting the value and the direction of changes in the daily pound/dollar exchange rate. In prediction of daily pound/dollar rate, we use the rescaled and bootstrapped daily euro/dollar rate as a guidepost for the singular spectrum analysis method. We use the random walk model as a benchmark to evaluate performances of the singular spectrum analysis as a prediction method. Empirical results show that the forecast based on the multivariate singular spectrum analysis compares favorably to the forecast of the random walk model both for predicting the value and the direction of changes in the daily pound/dollar exchange rate. We compared the prediction results based on an error correction model in the context of a restricted vector autoregressive model and compared them with the prediction results by a random walk as well as by those of singular spectrum and multiple singular spectrum models and found that the VEC results are inferior.     

On truncated incomplete decompositions

In the present paper we introduce truncated incomplete decompositions (TrILU) for constant coefficient matrices. This new ILU variant saves most of the memory and work usually needed to compute and store the factorization. Further it improves the smoothing and preconditioning properties of standard ILU-decompositions. Besides describing the algorithm, we give theoretical results concerning stability and convergence as well as the smoothing property and robustness for TrILU smoothing in a multi-grid method. Further, we add numerical results of TrILU as smoother in a multi-grid method and as preconditioner in a pcg-method fully confirming the theoretical results.This work was supported by Deutsche Forschungsgemeinschaft.     

Sequential Lagrangian Conditions for Convex Programs with Applications to Semidefinite Programming

In this paper it is shown that, in the absence of any regularity condition, sequential Lagrangian optimality conditions as well as a sequential duality results hold for abstract convex programs. The significance of the results is that they yield the standard optimality and duality results for convex programs under a simple closed-cone condition that is much weaker than the well-known constraint qualifications. As an application, a sequential Lagrangian duality, saddle-point conditions, and stability results are derived for convex semidefinite programs.The authors are grateful to the referee and Professor Franco Giannessi for valuable comments and constructive suggestions which have contributed to the final preparation of the paper.     

Auxiliary Principle Technique for Equilibrium Problems

In this paper, we use the auxiliary principle technique to suggest and analyze a number of iterative methods for solving mixed quasiequilibrium problems. We prove that the convergence of these new methods requires either partially relaxed strongly monotonicity or peudomonotonicity, which is a weaker condition than monotonicity. Our proof of convergence is very simple as compared with others. These new results include several new and known results as special cases. Our results represent refinement and improvement of the previous known results for equilibrium and variational inequalities problems.     

Solutions with internal jump for an autonomous elliptic system of FitzHugh–Nagumo type

Systems of elliptic partial differential equations which are coupled in a noncooperative way, such as the FitzHugh–Nagumo type studied in this paper, in general do not satisfy order preserving properties. This not only results in technical complications but also yields a richer solution structure. We prove the existence of multiple nontrivial solutions. In particular we show that there exists a solution with boundary layer type behaviour, and we will give evidence that this autonomous system for a certain range of parameters has a solution with both a boundary and an internal layer. The analysis uses results from bifurcation theory, variational methods, as well as some pointwise a priori estimates. The final section contains some numerically obtained results.     

变系数再生散度线性模型和它的估计

本文研究了一类变系数再生散度线性模型.利用局部最大似然的方法,得到了兴趣参数的估计,同时也研究了局部权和光滑参数的确定以及统计推断,结果推广了文献[11]的工作.     

Applications of time scale symplectic systems without normality

Recently, the authors obtained new characterizations of the positivity and nonnegativity of a time scale quadratic functional F with separable endpoints related to a time scale symplectic system (S). In these results, the assumption of normality is absent. In this paper we present applications of such results. Namely, without assuming normality we derive Sturmian comparison theorems, results for general jointly varying endpoints, and characterizations of the positivity of F via the corresponding time scale Riccati equation, a certain perturbed quadratic functional, and a time scale Riccati inequality. These results generalize and unify many recent as well as classical ones.     

Uniform recovery from subgaussian multi-sensor measurements

Parallel acquisition systems are employed successfully in a variety of different sensing applications when a single sensor cannot provide enough measurements for a high-quality reconstruction. In this paper, we consider compressed sensing (CS) for parallel acquisition systems when the individual sensors use subgaussian random sampling. Our main results are a series of uniform recovery guarantees which relate the number of measurements required to the basis in which the solution is sparse and certain characteristics of the multi-sensor system, known as sensor profile matrices. In particular, we derive sufficient conditions for optimal recovery, in the sense that the number of measurements required per sensor decreases linearly with the total number of sensors, and demonstrate explicit examples of multi-sensor systems for which this holds. We establish these results by proving the so-called Asymmetric Restricted Isometry Property (ARIP) for the sensing system and use this to derive both nonuniversal and universal recovery guarantees. Compared to existing work, our results not only lead to better stability and robustness estimates but also provide simpler and sharper constants in the measurement conditions. Finally, we show how the problem of CS with block-diagonal sensing matrices can be viewed as a particular case of our multi-sensor framework. Specializing our results to this setting leads to a recovery guarantee that is at least as good as existing results.     

When every multilinear mapping is multiple summing

In this paper we give a systematized treatment to some coincidence situations for multiple summing multilinear mappings which extend, generalize and simplify the methods and results obtained thus far. The application of our general results to the pertinent particular cases gives several new coincidences as well as easier proofs of some known results (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Positive solutions of nonlinear problems involving the square root of the Laplacian

We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of Gidas-Spruck type. In addition, among other results, we prove a symmetry theorem of Gidas-Ni-Nirenberg type.