Finiteness conditions in the stable module category

We study groups whose cohomology functors commute with filtered colimits in high dimensions. We relate this condition to the existence of projective resolutions which exhibit some finiteness properties in high dimensions, and to the existence of Eilenberg–Mac Lane spaces with finitely many n-cells for all sufficiently large n. To that end, we determine the structure of completely finitary Gorenstein projective modules over group rings. The methods are inspired by representation theory and make use of the stable module category, in which morphisms are defined through complete cohomology. In order to carry out these methods, we need to restrict ourselves to certain classes of hierarchically decomposable groups.     

Spaces with Dual Order-Isomorphic to ℓ1

We establish that an ordered Banach space is order-isomorphic to c₀ if and only if it is a -Dedekind complete vector lattice and its norm dual is order-isomorphic to ₁.     

A Convergence Analysis of Newton-Like Method for Singular Equations Using Recurrent Functions

We provide new semilocal convergence results for Newton-like method involving outer or generalized inverses in a Banach space setting. Using our new idea of recurrent functions and the same or weaker conditions than before [5-19 A. Ben-Israel and N.E. Greville ( 1974 ). Generalized Inverses: Theory and Applications, Pure and Applied Mathematics . Wiley-Interscience , New York . X. Chen and T. Yamamoto ( 1989 ). Convergence domains of certain iterative methods for solving nonlinear equations . Numer. Funct. Anal. Optimiz. 10 : 37 – 48 . J.E. Dennis , Jr. ( 1968 ). On Newton-like methods . Numer. Math. 11 : 324 – 330 . P. Deuflhard and C. Heindl ( 1979 ). Convergence theorems for Newton's method and extensions to related methods . SIAM J. Numer. Anal. 16 : 1 – 10 . J.M. Gutiérrez ( 1997 ). A new semilocal convergence theorem for Newton's method . J. Comp. Appl. Math. 79 : 131 – 145 . J.M. Gutiérrez , M.A. Hernández , and M.A. Salanova ( 1995 ). Accessibility of solutions by Newton's method . Internat. J. Comput. Math. 57 : 239 – 247 . W.M. Häubler ( 1986 ). A Kantorovich-type convergence analysis for the Gauss–Newton methods . Numer. Math. 48 : 119 – 125 . L.V. Kantorovich and G.P. Akilov ( 1964 ). Functional Analysis . Pergamon Press , Oxford . M.Z. Nashed and X. Chen ( 1993 ). Convergence of Newton-like methods for singular operator equations using outer inverses . Numer. Math. 66 : 235 – 257 . F.A. Potra and V. Ptàk ( 1980 ). Sharp error bounds for Newton's process . Numer. Math. 34 : 67 – 72 . W.C. Rheinboldt ( 1968 ). A unified convergence theory for a class of iterative processes . SIAM J. Numer. Anal. 5 : 42 – 63 . W.C. Rheinboldt ( 1977 ). An adaptive continuation process for solving systems of nonlinear equations . Polish Academy of Sciences, Banach Ctr. Publ. 3 : 129 – 142 . T. Yamamoto ( 1987 ). A method for finding sharp error bounds for Newton's method under the Kantorovich assumptions . Numer. Math. 49 : 203 – 230 . T. Yamamoto ( 1987 ). A convergence theorem for Newton-like methods in Banach spaces . Numer. Math. 51 : 545 – 557 . T. Yamamoto ( 1989 ). Uniqueness of the solution in a Kantorovich-type theorem of Haubler for the Gauss–Newton method . Japan J. Appl. Math. 6 : 77 – 81 . ], we provide more precise information on the location of the solution and finer bounds on the distances involved. Moreover, since our Newton–Kantorovich-type hypothesis is weaker than before, we can now cover cases not previously possible.

Applications and numerical examples involving a nonlinear integral equation of Chandrasekhar-type and a differential equation with Green's function are also provided in this study.     

Collective coordinates and the mechanism for conformational transitions of complex molecules

Reduction of dimensionality is crucial for the deeper understanding of the mechanism for large-amplitude conformational transitions of complex molecules. By taking up a six-atomcluster as an illustrative example, we present a general methodology to understand conformational transitions of molecules in terms of the low-dimensional dynamics of molecular gyration radii. The dynamics of gyration radii is generally governed by the interplay between the ordinary potential force and a dynamical force called the internal centrifugal force. We show that the internal centrifugal force can be more important than the original potential barrier and gives rise to a new dynamical barrier that truly dominates the conformational transitions of the system. This kind of dynamical effect should be crucially important in a wide class of molecular reaction dynamics. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pressure anisotropy effects on nonlinear electrostatic excitations in magnetized electron-positron-ion plasmas

The propagation of linear and nonlinear electrostatic waves is investigated in a magnetized anisotropic electron-positron-ion (e-p-i) plasma with superthermal electrons and positrons. A two-dimensional plasma geometry is assumed. The ions are assumed to be warm and anisotropic due to an external magnetic field. The anisotropic ion pressure is defined using the double adiabatic Chew-Golberger-Low (CGL) theory. In the linear regime, two normal modes are predicted, whose characteristics are investigated parametrically, focusing on the effect of superthermality of electrons and positrons, ion pressure anisotropy, positron concentration and magnetic field strength. A Zakharov-Kuznetsov (ZK) type equation is derived for the electrostatic potential (disturbance) via a reductive perturbation method. The parametric role of superthermality, positron content, ion pressure anisotropy and magnetic field strength on the characteristics of solitary wave structures is investigated. Following Allen and Rowlands [J. Plasma Phys. 53, 63 (1995)], we have shown that the pulse soliton solution of the ZK equation is unstable to oblique perturbations, and have analytically traced the dependence of the instability growth rate on superthermality and ion pressure anisotropy.     

Purkinje imaging system to measure anterior segment scattering in the human eye

We present an instrument based on Purkinje imaging that permits the objective measurement of the amount of scattering associated with the eye's anterior segment, avoiding the contribution from the retina. The experimental system records the fourth Purkinje image, and adequate processing is used to compute a parameter that quantifies the scattering. The method was first tested in an artificial eye and later in normal young eyes wearing customized contact lenses that induced different amounts of scatter. We were able to detect scattering increments, which indicates that this technique may be used as an objective tool to quantify the level of scattering in the anterior segment of the living human eye. The future use of this technique in clinical environments might help to estimate the level of corneal haze in eyes undergoing refractive surgery or/and scattering within the lens during cataract development.     

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.