Residual methods for the large-scale matrix pth root and some related problems

The problem of finding the pth root of a matrix has received special attention in the last few years. Standard approaches for this problem include and combine some variations of Newton’s method, which in turn involve matrix factorizations that, in general, are not suitable for large-scale problems. Motivated by some recently developed low-cost iterative schemes for nonlinear problems, we consider and analyze specialized residual methods that only require a few matrix-matrix products per iteration, and hence are suitable for the large-scale case. As a by-product we also discuss the advantages of residual methods for general nonlinear problems whose variables separate. Preliminary and encouraging numerical results are presented for computing pth roots of large-scale symmetric and positive definite matrices, for different values of p.     

A new inversion-free method for a rational matrix equation

Motivated by the classical Newton-Schulz method for finding the inverse of a nonsingular matrix, we develop a new inversion-free method for obtaining the minimal Hermitian positive definite solution of the matrix rational equation X+A^∗X^-1A=I, where I is the identity matrix and A is a given nonsingular matrix. We present convergence results and discuss stability properties when the method starts from the available matrix AA^∗. We also present numerical results to compare our proposal with some previously developed inversion-free techniques for solving the same rational matrix equation.     

Mild solutions for impulsive neutral functional differential equations with state-dependent delay

In this work, we establish the existence of mild solutions for a class of impulsive neutral functional differential equations with state-dependent delay.     

Desingularization of singular Riemannian foliation

Let F{\mathcal{F}} be a singular Riemannian foliation on a compact Riemannian manifold M. By successive blow-ups along the strata of F{\mathcal{F}} we construct a regular Riemannian foliation [^(F)]{\hat{\mathcal{F}}} on a compact Riemannian manifold [^(M)]{\hat{M}} and a desingularization map [^(r)]:[^(M)]? M{\hat{\rho}:\hat{M}\rightarrow M} that projects leaves of [^(F)]{\hat{\mathcal{F}}} into leaves of F{\mathcal{F}}. This result generalizes a previous result due to Molino for the particular case of a singular Riemannian foliation whose leaves were the closure of leaves of a regular Riemannian foliation. We also prove that, if the leaves of F{\mathcal{F}} are compact, then, for each small ${\epsilon >0 }${\epsilon >0 }, we can find [^(M)]{\hat{M}} and [^(F)]{\hat{\mathcal{F}}} so that the desingularization map induces an e{\epsilon}-isometry between M/F{M/\mathcal{F}} and [^(M)]/[^(F)]{\hat{M}/\hat{\mathcal{F}}}. This implies in particular that the space of leaves M/F{M/\mathcal{F}} is a Gromov-Hausdorff limit of a sequence of Riemannian orbifolds {([^(M)]_n/[^(F)]_n)}{\{(\hat{M}_{n}/\hat{\mathcal{F}}_{n})\}}.     

Residual algorithm for large-scale positive definite generalized eigenvalue problems

In the positive definite case, the extreme generalized eigenvalues can be obtained by solving a suitable nonlinear system of equations. In this work, we adapt and study the use of recently developed low-cost derivative-free residual schemes for nonlinear systems, to solve large-scale generalized eigenvalue problems. We demonstrate the effectiveness of our approach on some standard test problems, and also on a problem associated with the vibration analysis of large structures. In our numerical results we use preconditioning strategies based on incomplete factorizations, and we compare with and without preconditioning with a well-known available package.     

Schrödinger equations with critical nonlinearity, singular potential and a ground state

We study semilinear elliptic equations in a generally unbounded domain Ω⊂RN when the pertinent quadratic form is nonnegative and the potential is generally singular, typically a homogeneous function of degree −2. We prove solvability results based on the asymptotic behavior of the potential with respect to unbounded translations and dilations, while the nonlinearity is a perturbation of a self-similar, possibly oscillating, term f_∞ of critical growth satisfying , j∈Z, s∈R. This paper focuses on two qualitatively different cases of this problem, one when the quadratic form has a generalized ground state and another where the presence of potential does not change the energy space. In the latter case we allow nonlinearities with oscillatory critical growth. An important example of such quadratic form is the one on RN with the radial Hardy potential −μ|x|⁻² with μ=μ_∗ in the first case, μ<μ_∗ in the second case, where is the largest constant for which the energy form remains nonnegative.     

Improved thermogravimetric system for processing of oil sludge using HY zeolite catalyst

The oil sludge residue presents an aggregate of hydrocarbons, organic and inorganic impurities, and water. The microporous and mesoporous zeolites are considered promising catalysts for processing of petroleum residues generated in refining processes. The aim of this work was to study the degradation of petroleum sludge obtained from primary processing, with applications of an improved thermogravimetry system and HY zeolite, at specific temperature ranges and degradation times, in order to obtain light gases and distillate fuels. The NaY zeolite was synthesized under hydrothermal treatment of a gel containing sodium silicate, sodium aluminate, and water. The obtained solid material was filtered, dried and calcined, and then ion-exchanged with ammonium chloride and calcined in order to obtain its protonic acid form (HY). The samples’ characterization by TG/DTG, XRD, and SEM proved that the crystalline structure of the faujasite zeolite was obtained. The thermal and catalytic degradation of the petroleum sludge was performed with 1.0 g of sample containing 10% of HY zeolite in the temperatures of 100, 200, 300, 400, and 500 °C, varying the time from 0 to 60 min to each temperature, using an oven with temperature program system, adapted with a Shimadzu precision balance. The curves obtained with this system evidenced that the presence of HY zeolite improves the degradation of the residue, with decreasing of the activation energy for the processes, as determined using the Arrhenius model.

     

Optimization of Antenna Current Feeding for the Alfvén Eigenmodes Active Diagnostic System of JET

The possibility of exploring proper phasing of the feeding currents in the existing antenna of the Alfvén Eigenmodes Active Diagnostic system of JET, to excite pure toroidal spectra of Toroidal Alfvén Eigenmodes, is numerically investigated. Special attention is given to the actual perturbed fields excited in the plasma, which are calculated self-consistently using the antenna version of the CASTOR code. It is found that due to the close spacing of the JET antenna modules and quasi degeneracy of modes with medium to high values of the toroidal mode number n, although a proper choice of the phasing of the feeding currents of the antenna modules indeed leads to an increase of the perturbed fields of the selected mode, modes with nearby values of n are also excited with large amplitudes, so that a scheme to proper select the detected modes remains necessary. A scheme using different antenna position distribution is proposed to achieve successful optimization.     

One-pot synthesis of a chitosan-based hydrogel as a potential device for magnetic biomaterial

This describes the cross-linking/co-polymerization reaction of chitosan (CS), acrylic acid (AAc), and N, N′-methylenebisacrylamide (MBA) in the presence of citrate-covered-γ-Fe₂O₃ nanoparticules. A gelling process was verified by means of spectroscopic methods; Fourier transform infrared (FT-IR) and solid-state ¹³C-CP/MAS nuclear magnetic resonance (NMR). The corresponding signals of the gelling process, in the ¹³C NMR spectra, for the magnetic hydrogel were shifted to lower values due to embedding of the citrate-covered-γ-Fe₂O₃ nanoparticules. The X-ray diffraction (XRD) confirmed that the crystallinity of the magnetic hydrogel exhibited a different crystalline structure to that without magnetic properties. The Mössbauer and magnetization analysis revealed that the magnetic hydrogel displays a high lattice strain, due to bonded iron atom covalence and superparamagnetism. From scanning electronic microscope (SEM) micrographs, no separation phase coexists between the magnetic nanoparticules and cross-linked hydrogel, indicating an excellent dispersion throughout the hydrogel. The swelling rate was dependent on the cross-linking degree of the hydrogel and ionic strength of the aqueous solution.     

Ultrastructure and cytochemistry of lipid granules in the many-celled magnetotactic prokaryote, 'Candidatus Magnetoglobus multicellularis'

Conspicuous cytoplasmic granules are reported in a magnetotactic multicellular prokaryote named ‘Candidatus Magnetoglobus multicellularis’. Unfortunately, this microorganism, which consists of an assembly of gram-negative bacterial cells, cannot yet be cultivated, limiting the biochemical analysis of the granules and preventing in vitro studies with starvation/excess of nutrients. In this scenario, light and electron microscopy techniques were used to partially address the nature of the granules. Besides magnetosomes, three types of inclusions were observed: small (mean diameter = 124 nm) polyhydroxyalkanoate-like (PHA) granules, large (diameters ranging from 0.11 to 2.5 μm) non-PHA lipid granules, and rare phosphorus-rich granules, which probably correspond to polyphosphate bodies. The PHA granules were rounded in projection, non-reactive with OsO₄, and suffered the typical plastic deformation of PHAs after freeze fracturing. The nature of the large granules, consisting of round globular structures (mean diameter = 0.76 μm), was classified as non-PHA based on the following data: (a) multilayered structure in freeze-fracture electron microscopy, typical of non-PHA lipids; (b) Nile blue fluorescence imaging detected non-PHA lipids; (c) imidazole buffered osmium tetroxide and ruthenium red cytochemistry stained the globules, which appeared as electron-dense granules instead of electron lucent as PHAs do. Most likely, ‘Candidatus Magnetoglobus multicellularis’ stores carbon mainly as unusual lipid granules, together with smaller amounts of PHAs.