Rank-preserving geometric means of positive semi-definite matrices

The generalization of the geometric mean of positive scalars to positive definite matrices has attracted considerable attention since the seminal work of Ando. The paper generalizes this framework of matrix means by proposing the definition of a rank-preserving mean for two or an arbitrary number of positive semi-definite matrices of fixed rank. The proposed mean is shown to be geometric in that it satisfies all the expected properties of a rank-preserving geometric mean. The work is motivated by operations on low-rank approximations of positive definite matrices in high-dimensional spaces.© 2012 Elsevier Inc. All rights reserved.

Regression on fixed-rank positive semidefinite matrices: A Riemannian approach

The paper addresses the problem of learning a regression model parameterized by a fixed-rank positive semidefinite matrix. The focus is on the nonlinear nature of the search space and on scalability to high-dimensional problems. The mathematical developments rely on the theory of gradient descent algorithms adapted to the Riemannian geometry that underlies the set of fixedrank positive semidefinite matrices. In contrast with previous contributions in the literature, no restrictions are imposed on the range space of the learned matrix. The resulting algorithms maintain a linear complexity in the problem size and enjoy important invariance properties. We apply the proposed algorithms to the problem of learning a distance function parameterized by a positive semidefinite matrix. Good performance is observed on classical benchmarks. © 2011 Gilles Meyer, Silvere Bonnabel and Rodolphe Sepulchre.

Low-rank optimization on the cone of positive semidefinite matrices

We propose an algorithm for solving optimization problems defined on a subset of the cone of symmetric positive semidefinite matrices. This algorithm relies on the factorization X = Y Y T , where the number of columns of Y fixes an upper bound on the rank of the positive semidefinite matrix X. It is thus very effective for solving problems that have a low-rank solution. The factorization X = Y Y T leads to a reformulation of the original problem as an optimization on a particular quotient manifold. The present paper discusses the geometry of that manifold and derives a second-order optimization method with guaranteed quadratic convergence. It furthermore provides some conditions on the rank of the factorization to ensure equivalence with the original problem. In contrast to existing methods, the proposed algorithm converges monotonically to the sought solution. Its numerical efficiency is evaluated on two applications: the maximal cut of a graph and the problem of sparse principal component analysis. © 2010 Society for Industrial and Applied Mathematics.

Riemannian metric and geometric mean for positive semidefinite matrices of fixed rank

This paper introduces a new metric and mean on the set of positive semidefinite matrices of fixed-rank. The proposed metric is derived from a well-chosen Riemannian quotient geometry that generalizes the reductive geometry of the positive cone and the associated natural metric. The resulting Riemannian space has strong geometrical properties: it is geodesically complete, and the metric is invariant with respect to all transformations that preserve angles (orthogonal transformations, scalings, and pseudoinversion). A meaningful approximation of the associated Riemannian distance is proposed, that can be efficiently numerically computed via a simple algorithm based on SVD. The induced mean preserves the rank, possesses the most desirable characteristics of a geometric mean, and is easy to compute. © 2009 Society for Industrial and Applied Mathematics.

A comparison of silicon oxide and nitride as host matrices on the photoluminescence from Er3+ ions

This paper compares the properties of silicon oxide and nitride as host matrices for Er ions. Erbium-doped silicon nitride films were deposited by a plasma-enhanced chemical-vapour deposition system. After deposition, the films were implanted with Er3+ at different doses. Er-doped thermal grown silicon oxide films were prepared at the same time as references. Photoluminescence features of Er3+ were inspected systematically. It is found that silicon nitride films are suitable for high concentration doping and the thermal quenching effect is not severe. However, a very high annealing temperature up to 1200 degrees C is needed to optically activate Er3+ which may be the main obstacle to impede the application of Er-doped silicon nitride.

Calculation of the band structure of semiconductor quantum wells using scattering matrices

The transfer-matrix method widely used in the calculation of the band structure of semiconductor quantum wells is found to have limitations due to its intrinsic numerical instability. It is pointed out that the numerical instability arises from free-propagating transfer matrices. A new scattering-matrix method is developed for the multiple-band Kane model within the envelope-function approximation. Compared with the transfer-matrix method, the proposed algorithm is found to be more efficient and stable. A four-band Kane model is used to check the validity of the method and the results are found to be in good agreement with earlier calculations.

Substitution Matrices of Residue Triplets Derived from Protein Blocks.

In protein sequence alignment, residue similarity is usually evaluated by substitution matrix, which scores all possible exchanges of one amino acid with another. Several matrices are widely used in sequence alignment, including PAM matrices derived from homologous sequence and BLOSUM matrices derived from aligned segments of BLOCKS. However, most matrices have not addressed the high-order residue-residue interactions that are vital to the bioproperties of protein.With consideration for the inherent correlation in residue triplet, we present a new scoring scheme for sequence alignment. Protein sequence is treated as overlapping and successive 3-residue segments. Two edge residues of a triplet are clustered into hydrophobic or polar categories, respectively. Protein sequence is then rewritten into triplet sequence with 2 · 20 · 2 = 80 alphabets. Using a traditional approach, we construct a new scoring scheme named TLESUMhp (TripLEt SUbstitution Matrices with hydropobic and polar information) for pairwise substitution of triplets, which characterizes the similarity of residue triplets. The applications of this matrix led to marked improvements in multiple sequence alignment and in searching structurally alike residue segments. The reason for the occurrence of the ‘‘twilight zone,’’ i.e., structure explosion of lowidentity sequences, is also discussed.

Dendrimer films as matrices for electrochemical fabrication of novel gold/palladium bimetallic nanostructures

Electrodeposition of novel Au/Pd bimetallic nanostructures with dendrimer films as matrices has been reported. The dendrimers exhibited highly open structures arising from protonation of amines and this made them have good penetrability for solvent molecules. The unique properties of dendrimers obviously affected the morphologies and compositions of deposited bimetallic nanostructures compared with those from unmodified surfaces. Field-emitted scanning electron microscopy, energy dispersive spectroscopy, X-ray photoelectron spectroscopy and UV-vis spectroscopy were used to characterize these nanostructures.

Electrochemical biosensors based on advanced bioimmobilization matrices

Biosensors have experienced rapid, extensive development. To maintain the bioactivity of biomolecules and to give the electrochemical output signal required, appropriate bioimmobilization matrices for biomolecules are critical.In this review, we describe some advanced membrane materials (including hydrogels, sol-gel-derived organic-inorganic composites and lipid membranes), introduce electrochemical biosensors based on bioimmobilization materials and describe their performance.Biosensors operating in extreme conditions and displaying direct electron transfer with electrodes based on these advanced membrane materials are attractive. Recent developments in nanomaterials include biosensors, so we emphasize the intersection of nanomaterials with advanced membrane materials in biosensors.

Relationship between luminescence and matrices in the system Al2O3-B2O3-Ce2O3

The system Al2O3-B2O3-Ce2O3, with Al/B ratio varying from 4.5 to 2 and Ce/(Al + B) = 0.02, has been prepared at the temperature from 1 200 to 1 400 degrees C. Relationship between luminescence and matrices in the system Al2O3-B2O3-Ce2O3 was investigated. It was found that some changes of the matrices occurred with Al/B ratio varying from 4.5 to 2. These results lead to a great change in luminescence properties, indicating variation of Ce-surroundings of crystal field. With the decrease of the ratio from 3 to 2, the excitation and emission peaks shift to shorter wavelengths. it was also found that a new type of rare earth luminescent materials was obtained with appropriate Al/B ratio.

Boundary triplets and M-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices

B.M. Brown, M. Marletta, S. Naboko, I. Wood: Boundary triplets and M-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices, J. London Math. Soc., June 2008; 77: 700-718. The full text of this article will be made available in this repository in June 2009 Sponsorship: EPSRC,INTAS

Diseño y desarrollo de herramientas en cerámica, La Calera. Asistencia técnica para el mejoramiento de la pasta cerámica y proceso de secado de piezas de alfarería San Sebastián. Mejoramiento de la pasta cerámica y proceso de secado de piezas de alfarería. Acondicionamiento y manejo de hornos, Momil. Mejoramiento de la pasta cerámica y proceso de secado de piezas de alfarería. Asistencia para la elaboración de matrices y moldes para piezas de cerámica acondicionamiento y manejo de hornos, Ciénaga De Oro y Planeta Rica.

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