A descent hybrid conjugate gradient method based on the memoryless BFGS update

In this work, we present a new hybrid conjugate gradient method based on the approach of the convex hybridization of the conjugate gradient update parameters of DY and HS+, adapting a quasi-Newton philosophy. The computation of the hybrization parameter is obtained by minimizing the distance between the hybrid conjugate gradient direction and the self-scaling memoryless BFGS direction. Furthermore, a significant property of our proposed method is that it ensures sufficient descent independent of the accuracy of the line search. The global convergence of the proposed method is established provided that the line search satisfies the Wolfe conditions. Our numerical experiments on a set of unconstrained optimization test problems from the CUTEr collection indicate that our proposed method is preferable and in general superior to classic conjugate gradient methods in terms of efficiency and robustness.     

On Newton-like methods of ��bounded deterioration�� using recurrent functions

We provide a semilocal convergence analysis for Newton-like methods of ??bounded deterioration?? in a Banach space setting. We establish tighter error bounds on the distances involved, and a more precise information on the location of the solution, under the same or weaker hypotheses than before (Argyros, Acta Math. Sin. (Engl. Ser.), 23:2087?C2096, 2007; Deuflhard, Newton methods for nonlinear problems. Affine invariance and adaptive algorithms, Springer Series in Computational Mathematics, vol. 35. Springer, Berlin, 2004; Ezquerro and Hern??ndez, IMA J. Numer. Anal., 22:187?C205, 2002) using recurrent functions. Numerical examples are also provided involving polynomial, integral, and differential equations.     

Evaluation of credit risk based on firm performance

This paper investigates whether productive inefficiency measured as the distance from the industry’s ‘best practice’ frontier is an important ex-ante predictor of business failure. We use samples of French textiles, wood and paper products, computers and R&D companies to obtain efficiency estimates for individual firms in each industry. These efficiency measures are derived from a directional technology distance function constructed empirically using non-parametric data envelopment analysis (DEA) methods. Estimating binary and ordered logit regression models we find that productive efficiency has significant explanatory power in predicting the likelihood of default over and above the effect of standard financial indicators.     

Systematic approaches to experimentation: The case of Pick's theorem

In this paper two 10th graders having an accumulated experience on problem-solving ancillary to the concept of area confronted the task to find Pick's formula for a lattice polygon's area. The formula was omitted from the theorem in order for the students to read the theorem as a problem to be solved. Their working is examined and emphasis is given to highlighting the students’ range of systematic approaches to experimentation in the context of problem solving and aspects of control that are reflected in these approaches.     

Logarithmic Dimension Bounds for the Maximal Function Along a Polynomial Curve

Let ℳ denote the maximal function along the polynomial curve (γ ₁ t,…,γ _d t ^d ):

A Kantorovich-type analysis of Broyden’s method using recurrent functions

We provide a semilocal convergence analysis for Broyden’s method for approximating locally unique solutions of nonlinear operator equations. Using the majorant principle we show that under the same or weaker hypotheses, in combination with our new idea of recurrent functions, we can find weaker sufficient conditions for the convergence of Broyden’s method as well as finer error bounds on the distances involved, and a more precise information on the location of the solution than before (Broyden, Math. Comput. 19:577–593, 1965; Chen, Ann. Inst. Stat. Math. 42:387–401, 1990; Dennis, Nonlinear Functional Analysis and Applications, pp. 425–472, Academic Press, San Diego, 1971; Li and Fukushima, Ann. Oper. Res. 103:71–97, 2001). Numerical examples are also provided involving polynomial, integral, and differential equations.     

On the midpoint method for solving equations

Using our idea of recurrent functions, we provide a new semilocal convergence analysis for the midpoint method (MPM) introduced by Argyros and Chen. We show that this way the error estimates are tighter, and the sufficient convergence conditions can be weaker. Moreover, we also show that the Newton-type method (NTM) introduced by Wu and Zhao (using the same information as (MPM)) can always be replaced by the (MPM).Numerical results where our results apply to solve nonlinear equations, but others cannot are also provided in this study.     

Complete description of invariant Einstein metrics on the generalized flag manifold SO(2n)/U(p)?× U(n ? p)

We find the precise number of non-K?hler SO(2n)-invariant Einstein metrics on the generalized flag manifold M = SO(2n)/U(p)×U(n−p) with n ≥ 4 and 2 ≤ p ≤ n−2. We use an analysis on parametric systems of polynomial equations and we give some insight towards the study of such systems. We also examine the isometric problem for these Einstein metrics.     

Investigating brain tumor differentiation with diffusion and perfusion metrics at 3T MRI using pattern recognition techniques

The aim of this study was to evaluate the contribution of diffusion and perfusion MR metrics in the discrimination of intracranial brain lesions at 3T MRI, and to investigate the potential diagnostic and predictive value that pattern recognition techniques may provide in tumor characterization using these metrics as classification features. Conventional MRI, diffusion weighted imaging (DWI), diffusion tensor imaging (DTI) and dynamic-susceptibility contrast imaging (DSCI) were performed on 115 patients with newly diagnosed intracranial tumors (low-and- high grade gliomas, meningiomas, solitary metastases). The Mann–Whitney U test was employed in order to identify statistical differences of the diffusion and perfusion parameters for different tumor comparisons in the intra-and peritumoral region. To assess the diagnostic contribution of these parameters, two different methods were used; the commonly used receiver operating characteristic (ROC) analysis and the more sophisticated SVM classification, and accuracy, sensitivity and specificity levels were obtained for both cases. The combination of all metrics provided the optimum diagnostic outcome. The highest predictive outcome was obtained using the SVM classification, although ROC analysis yielded high accuracies as well. It is evident that DWI/DTI and DSCI are useful techniques for tumor grading. Nevertheless, cellularity and vascularity are factors closely correlated in a non-linear way and thus difficult to evaluate and interpret through conventional methods of analysis. Hence, the combination of diffusion and perfusion metrics into a sophisticated classification scheme may provide the optimum diagnostic outcome. In conclusion, machine learning techniques may be used as an adjunctive diagnostic tool, which can be implemented into the clinical routine to optimize decision making.