Lessons learned from infertility investigations in the public sector

Objectives. To determine the main factors causing infertility in an urban, tertiary hospital population. To establish if any such major causal factor could be used to rationalise and improve the service for infertile couples in the public sector.Design. A retrospective analysis of the hospital records of 206 women who had a tubal patency test (hysterosalpingogram) performed and the results of the investigations performed in the couples with infertility.Results. Of the 206 women 38 (18.5%) had normal fallopian tubes on hysterosalpingogram; 33 (16%) had unilateral obstruction and 135 (65.5%) had bilateral tubal obstruction. Of the latter group 81 (60%) had significant hydrosalpinges. Semen analysis results in 148 partners (71.8%) demonstrated a normal count in 85 (62%), normal motility in 70 (51%) and normal morphology in only 25 (18%). Testing for ovulation (mid-luteal phase progesterone) was positive in 91 of 124 women tested (73%). Compliance, technical and logistical problems were encountered with both semen analysis and mid-luteal phase progesterone tests.Conclusions. Infertility is a major problem in South Africa, with limited resources for investigation and treatment in the public sector. Tubal factor infertility was the most common cause of infertility demonstrated in this study. In the presence of bilateral tubal obstruction with hydrosalpinges the prognosis is so poor that unless assisted reproductive techniques are available and affordable, further infertility investigations do not seem justified. Recommendations on an approach to the infertile couple in the public sector is outlined

On the Convergence of Ritz Pairs and Refined Ritz Vectors for Quadratic Eigenvalue Problems

For a given subspace, the Rayleigh-Ritz method projects the large quadratic eigenvalue problem (QEP) onto it and produces a small sized dense QEP. Similar to the Rayleigh-Ritz method for the linear eigenvalue problem, the Rayleigh-Ritz method defines the Ritz values and the Ritz vectors of the QEP with respect to the projection subspace. We analyze the convergence of the method when the angle between the subspace and the desired eigenvector converges to zero. We prove that there is a Ritz value that converges to the desired eigenvalue unconditionally but the Ritz vector converges conditionally and may fail to converge. To remedy the drawback of possible non-convergence of the Ritz vector, we propose a refined Ritz vector that is mathematically different from the Ritz vector and is proved to converge unconditionally. We construct examples to illustrate our theory.Comment: 20 page

Purifying and Reversible Physical Processes

Starting from the observation that reversible processes cannot increase the purity of any input state, we study deterministic physical processes, which map a set of states to a set of pure states. Such a process must map any state to the same pure output, if purity is demanded for the input set of all states. But otherwise, when the input set is restricted, it is possible to find non-trivial purifying processes. For the most restricted case of only two input states, we completely characterize the output of any such map. We furthermore consider maps, which combine the property of purity and reversibility on a set of states, and we derive necessary and sufficient conditions on sets, which permit such processes.Comment: 5 pages, no figures, v2: only minimal change

Fast linear algebra is stable

In an earlier paper, we showed that a large class of fast recursive matrix multiplication algorithms is stable in a normwise sense, and that in fact if multiplication of $n$-by-$n$ matrices can be done by any algorithm in $O(n^{\omega + \eta})$ operations for any $\eta > 0$, then it can be done stably in $O(n^{\omega + \eta})$ operations for any $\eta > 0$. Here we extend this result to show that essentially all standard linear algebra operations, including LU decomposition, QR decomposition, linear equation solving, matrix inversion, solving least squares problems, (generalized) eigenvalue problems and the singular value decomposition can also be done stably (in a normwise sense) in $O(n^{\omega + \eta})$ operations.Comment: 26 pages; final version; to appear in Numerische Mathemati

Two-sided Grassmann-Rayleigh quotient iteration

The two-sided Rayleigh quotient iteration proposed by Ostrowski computes a pair of corresponding left-right eigenvectors of a matrix $C$. We propose a Grassmannian version of this iteration, i.e., its iterates are pairs of $p$-dimensional subspaces instead of one-dimensional subspaces in the classical case. The new iteration generically converges locally cubically to the pairs of left-right $p$-dimensional invariant subspaces of $C$. Moreover, Grassmannian versions of the Rayleigh quotient iteration are given for the generalized Hermitian eigenproblem, the Hamiltonian eigenproblem and the skew-Hamiltonian eigenproblem.Comment: The text is identical to a manuscript that was submitted for publication on 19 April 200

A Matrix Hyperbolic Cosine Algorithm and Applications

In this paper, we generalize Spencer's hyperbolic cosine algorithm to the matrix-valued setting. We apply the proposed algorithm to several problems by analyzing its computational efficiency under two special cases of matrices; one in which the matrices have a group structure and an other in which they have rank-one. As an application of the former case, we present a deterministic algorithm that, given the multiplication table of a finite group of size $n$, it constructs an expanding Cayley graph of logarithmic degree in near-optimal O(n^2 log^3 n) time. For the latter case, we present a fast deterministic algorithm for spectral sparsification of positive semi-definite matrices, which implies an improved deterministic algorithm for spectral graph sparsification of dense graphs. In addition, we give an elementary connection between spectral sparsification of positive semi-definite matrices and element-wise matrix sparsification. As a consequence, we obtain improved element-wise sparsification algorithms for diagonally dominant-like matrices.Comment: 16 pages, simplified proof and corrected acknowledging of prior work in (current) Section

Further results on non-diagonal Bianchi type III vacuum metrics

We present the derivation, for these vacuum metrics, of the Painlev\'e VI equation first obtained by Christodoulakis and Terzis, from the field equations for both minkowskian and euclidean signatures. This allows a complete discussion and the precise connection with some old results due to Kinnersley. The hyperk\"ahler metrics are shown to belong to the Multi-Centre class and for the cases exhibiting an integrable geodesic flow the relevant Killing tensors are given. We conclude by the proof that for the Bianchi B family, excluding type III, there are no hyperk\"ahler metrics.Comment: 21 pages, no figure

Read-It: A Multi-modal Tangible Interface for Children Who Learn to Read

Multi-modal tabletop applications offer excellent opportunities for enriching the education of young children. Read-It is an example of an interactive game with a multi-modal tangible interface that was designed to combine the advantages of current physical games and computer exercises. It is a novel approach for supporting children who learn to read. The first experimental evaluation has demonstrated that the Read-It approach is indeed promising and meets a priori expectations

A weakly stable algorithm for general Toeplitz systems

We show that a fast algorithm for the QR factorization of a Toeplitz or Hankel matrix A is weakly stable in the sense that R^T.R is close to A^T.A. Thus, when the algorithm is used to solve the semi-normal equations R^T.Rx = A^Tb, we obtain a weakly stable method for the solution of a nonsingular Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the solution of the full-rank Toeplitz or Hankel least squares problem.Comment: 17 pages. An old Technical Report with postscript added. For further details, see http://wwwmaths.anu.edu.au/~brent/pub/pub143.htm