Fast matrix computations for pair-wise and column-wise commute times and Katz scores

We first explore methods for approximating the commute time and Katz score between a pair of nodes. These methods are based on the approach of matrices, moments, and quadrature developed in the numerical linear algebra community. They rely on the Lanczos process and provide upper and lower bounds on an estimate of the pair-wise scores. We also explore methods to approximate the commute times and Katz scores from a node to all other nodes in the graph. Here, our approach for the commute times is based on a variation of the conjugate gradient algorithm, and it provides an estimate of all the diagonals of the inverse of a matrix. Our technique for the Katz scores is based on exploiting an empirical localization property of the Katz matrix. We adopt algorithms used for personalized PageRank computing to these Katz scores and theoretically show that this approach is convergent. We evaluate these methods on 17 real world graphs ranging in size from 1000 to 1,000,000 nodes. Our results show that our pair-wise commute time method and column-wise Katz algorithm both have attractive theoretical properties and empirical performance.Comment: 35 pages, journal version of http://dx.doi.org/10.1007/978-3-642-18009-5_13 which has been submitted for publication. Please see http://www.cs.purdue.edu/homes/dgleich/publications/2011/codes/fast-katz/ for supplemental code

Numerical Solutions of Matrix Differential Models using Cubic Matrix Splines II

This paper presents the non-linear generalization of a previous work on matrix differential models. It focusses on the construction of approximate solutions of first-order matrix differential equations Y'(x)=f(x,Y(x)) using matrix-cubic splines. An estimation of the approximation error, an algorithm for its implementation and illustrative examples for Sylvester and Riccati matrix differential equations are given.Comment: 14 pages; submitted to Math. Comp. Modellin

Efficient mixed rational and polynomial approximation of matrix functions

This paper presents an efficient method for computing approximations for general matrix functions based on mixed rational and polynomial approximations. A method to obtain this kind of approximation from rational approximations is given, reaching the highest efficiency when transforming nondiagonal rational approximations with a higher numerator degree than the denominator degree. Then, the proposed mixed rational and polynomial approximation can be successfully applied for matrix functions which have any type of rational approximation, such as Pade, Chebyshev, etc., with maximum efficiency for higher numerator degrees than the denominator degrees. The efficiency of the mixed rational and polynomial approximation is compared with the best existing evaluating schemes for general polynomial and rational approximations, providing greater theoretical accuracy with the same cost in terms of matrix multiplications. It is well known that diagonal rational approximants are generally more accurate than the corresponding nondiagonal rational approximants which have the same computational cost. Using the proposed mixed approximation we show that the above statement is no longer true, and nondiagonal rational approximants are in fact generally more accurate than the corresponding diagonal rational approximants with the same cost. (C) 2012 Elsevier Inc. All rights reserved.This work has been supported by Universitat Politecnica de Valencia grant PAID-06-011-2020.Sastre, J. (2012). Efficient mixed rational and polynomial approximation of matrix functions. Applied Mathematics and Computation. 218(24):11938-11946. https://doi.org/10.1016/j.amc.2012.05.064S11938119462182

Single-marker identification of head and neck squamous cell carcinoma cancer stem cells with aldehyde dehydrogenase

Background In accord with the cancer stem cell (CSC) theory, only a small subset of cancer cells are capable of forming tumors. We previously reported that CD44 isolates tumorigenic cells from head and neck squamous cell cancer (HNSCC). Recent studies indicate that aldehyde dehydrogenase (ALDH) activity may represent a more specific marker of CSCs. Methods Six primary HNSCCs were collected. Cells with high and low ALDH activity (ALDH high /ALDH low ) were isolated. ALDH high and ALDH low populations were implanted into NOD/SCID mice and monitored for tumor development. Results ALDH high cells represented a small percentage of the tumor cells (1% to 7.8%). ALDH high cells formed tumors from as few as 500 cells in 24/45 implantations, whereas only 3/37 implantations of ALDH low cells formed tumors. Conclusions ALDH high cells comprise a subpopulation cells in HNSCCs that are tumorigenic and capable of producing tumors at very low numbers. This finding indicates that ALDH activity on its own is a highly selective marker for CSCs in HNSCC. © 2010 Wiley Periodicals, Inc. Head Neck, 2010Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/77970/1/21315_ftp.pd

Incentives, Information, and Emergent Collective Accuracy

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/93557/1/mde2560.pd

Magnetic energy-level diagrams of high-spin (Mn12_{12}-acetate) and low-spin (V15_{15}) molecules

The magnetic energy-level diagrams for models of the Mn12 and V15 molecule are calculated using the Lanczos method with full orthogonalization and a Chebyshev-polynomial-based projector method. The effect of the Dzyaloshinskii-Moriya interaction on the appearance of energy-level repulsions and its relevance to the observation of steps in the time-dependent magnetization data is studied. We assess the usefulness of simplified models for the description of the zero-temperature magnetization dynamics

Efficient computation of the matrix cosine

Trigonometric matrix functions play a fundamental role in second order differential equation systems. This work presents an algorithm for computing the cosine matrix function based on Taylor series and the cosine double angle formula. It uses a forward absolute error analysis providing sharper bounds than existing methods. The proposed algorithm had lower cost than state-of-the-art algorithms based on Hermite matrix polynomial series and Padé approximants with higher accuracy in the majority of test matrices.This work has been supported by Universitat Politecnica de Valencia Grant PAID-06-011-2020.Sastre, J.; Ibáñez González, JJ.; Ruiz Martínez, PA.; Defez Candel, E. (2013). Efficient computation of the matrix cosine. Applied Mathematics and Computation. 219:7575-7585. https://doi.org/10.1016/j.amc.2013.01.043S7575758521

Linearly scaling direct method for accurately inverting sparse banded matrices

In many problems in Computational Physics and Chemistry, one finds a special kind of sparse matrices, termed "banded matrices". These matrices, which are defined as having non-zero entries only within a given distance from the main diagonal, need often to be inverted in order to solve the associated linear system of equations. In this work, we introduce a new O(n) algorithm for solving such a system, being n X n the size of the matrix. We produce the analytical recursive expressions that allow to directly obtain the solution, as well as the pseudocode for its computer implementation. Moreover, we review the different options for possibly parallelizing the method, we describe the extension to deal with matrices that are banded plus a small number of non-zero entries outside the band, and we use the same ideas to produce a method for obtaining the full inverse matrix. Finally, we show that the New Algorithm is competitive, both in accuracy and in numerical efficiency, when compared to a standard method based in Gaussian elimination. We do this using sets of large random banded matrices, as well as the ones that appear when one tries to solve the 1D Poisson equation by finite differences.Comment: 24 pages, 5 figures, submitted to J. Comp. Phy

Tuberculosis suspicion and knowledge among private and public general practitioners: Questionnaire Based Study in Oman

<p>Abstract</p> <p>Background</p> <p>Early detection of smear positive TB cases by smear microscopy requires high level of suspicion of TB among primary care physicians. The objective of this study is to measure TB suspicion and knowledge among private and public sector general practitioners using clinical vignette-based survey and structured questionnaire.</p> <p>Methods</p> <p>Two questionnaires were distributed to both private and public GPs in Muscat Governorate. One questionnaire assessed demographic information of the respondent and had 10 short clinical vignettes of TB and non-TB cases. The second questionnaire had questions on knowledge of TB, its diagnosis, treatment, follow up and contact screening based on Ministry of Health policy. TB suspicion score and TB Knowledge score were computed and analyzed.</p> <p>Results</p> <p>A total of 257 GPs participated in the study of which 154 were private GPs. There was a significant difference between private and public GPs in terms of age, sex, duration of practice and nationality. Among all GPs, 37.7% considered TB as one of the three most likely diagnoses in all 5 TB clinical vignettes. Private GPs had statistically significantly lower TB suspicion and TB knowledge scores than public GPs.</p> <p>Conclusion</p> <p>In Oman, GPs appear to have low suspicion and poor knowledge of TB, particularly private GPs. To strengthen TB control program, there is a need to train GPs on TB identification and adopt a Private Public Mix (PPM) strategy for TB control.</p

Optimized arrays for 2D cross-borehole electrical tomography surveys

The use of optimized arrays generated using the ‘Compare R’ method for cross-borehole resistivity measurements is examined in this paper. We compare the performances of two array optimization algorithms, one that maximizes the model resolution and another that minimizes the point spread value. Although both algorithms give similar results, the model resolution maximization algorithm is several times faster. A study of the point spread function plots for a cross-borehole survey shows that the model resolution within the central zone surrounded by the borehole electrodes is much higher than near the bottom end of the boreholes. Tests with synthetic and experimental data show that the optimized arrays generated by the ‘Compare R’ method have significantly better resolution than a ‘standard’ measurement sequence used in previous surveys. The resolution of the optimized arrays is less if arrays with both current (or both potential) electrodes in the same borehole are excluded. However, they are still better than the ‘standard’ arrays