Tridiagonal canonical matrices of bilinear or sesquilinear forms and of pairs of symmetric, skew-symmetric, or Hermitian forms

Tridiagonal canonical forms of square matrices under congruence or *congruence, pairs of symmetric or skew-symmetric matrices under congruence, and pairs of Hermitian matrices under *congruence are given over an algebraically closed field of characteristic different from 2. (C) 2008 Elsevier Inc. All rights reserved.

A canonical form for nonderogatory matrices under unitary similarity

A square matrix is nonderogatory if its Jordan blocks have distinct eigenvalues. We give canonical forms for (1) nonderogatory complex matrices up to unitary similarity, and (2) pairs of complex matrices up to similarity, in which one matrix has distinct eigenvalues. The types of these canonical forms are given by undirected and, respectively, directed graphs with no undirected cycles. (C) 2011 Elsevier Inc. All rights reserved.

Topics in randomized numerical linear algebra

This thesis studies three classes of randomized numerical linear algebra algorithms, namely: (i) randomized matrix sparsification algorithms, (ii) low-rank approximation algorithms that use randomized unitary transformations, and (iii) low-rank approximation algorithms for positive-semidefinite (PSD) matrices.

Randomized matrix sparsification algorithms set randomly chosen entries of the input matrix to zero. When the approximant is substituted for the original matrix in computations, its sparsity allows one to employ faster sparsity-exploiting algorithms. This thesis contributes bounds on the approximation error of nonuniform randomized sparsification schemes, measured in the spectral norm and two NP-hard norms that are of interest in computational graph theory and subset selection applications.

Low-rank approximations based on randomized unitary transformations have several desirable properties: they have low communication costs, are amenable to parallel implementation, and exploit the existence of fast transform algorithms. This thesis investigates the tradeoff between the accuracy and cost of generating such approximations. State-of-the-art spectral and Frobenius-norm error bounds are provided.

The last class of algorithms considered are SPSD "sketching" algorithms. Such sketches can be computed faster than approximations based on projecting onto mixtures of the columns of the matrix. The performance of several such sketching schemes is empirically evaluated using a suite of canonical matrices drawn from machine learning and data analysis applications, and a framework is developed for establishing theoretical error bounds.

In addition to studying these algorithms, this thesis extends the Matrix Laplace Transform framework to derive Chernoff and Bernstein inequalities that apply to all the eigenvalues of certain classes of random matrices. These inequalities are used to investigate the behavior of the singular values of a matrix under random sampling, and to derive convergence rates for each individual eigenvalue of a sample covariance matrix.

Remarks on the classification of a pair of commuting semilinear operators

Gelfand and Ponomarev [I.M. Gelfand, V.A. Ponomarev, Remarks on the classification of a pair of commuting linear transformations in a finite dimensional vector space, Funct. Anal. Appl. 3 (1969) 325-326] proved that the problem of classifying pairs of commuting linear operators contains the problem of classifying k-tuples of linear operators for any k. We prove an analogous statement for semilinear operators. (C) 2011 Elsevier Inc. All rights reserved.

The Jordan canonical form for a class of weighted directed graphs

Recently, Cardon and Tuckfield (2011) [1] have described the Jordan canonical form for a class of zero-one matrices, in terms of its associated directed graph. In this paper, we generalize this result to describe the Jordan canonical form of a weighted adjacency matrix A in terms of its weighted directed graph.

Perturbations on the antidiagonals of Hankel matrices

Given a strongly regular Hankel matrix, and its associated sequence of moments which defines a quasi-definite moment linear functional, we study the perturbation of a fixed moment, i.e., a perturbation of one antidiagonal of the Hankel matrix. We define a linear functional whose action results in such a perturbation and establish necessary and sufficient conditions in order to preserve the quasi-definite character. A relation between the corresponding sequences of orthogonal polynomials is obtained, as well as the asymptotic behavior of their zeros. We also study the invariance of the Laguerre-Hahn class of linear functionals under such perturbation, and determine its relation with the so-called canonical linear spectral transformations. © 2013 Elsevier Ltd. All rights reserved.

SpliceDB: database of canonical and non-canonical mammalian splice sites

A database (SpliceDB) of known mammalian splice site sequences has been developed. We extracted 43 337 splice pairs from mammalian divisions of the gene-centered Infogene database, including sites from incomplete or alternatively spliced genes. Known EST sequences supported 22 815 of them. After discarding sequences with putative errors and ambiguous location of splice junctions the verified dataset includes 22 489 entries. Of these, 98.71% contain canonical GT–AG junctions (22 199 entries) and 0.56% have non-canonical GC–AG splice site pairs. The remainder (0.73%) occurs in a lot of small groups (with a maximum size of 0.05%). We especially studied non-canonical splice sites, which comprise 3.73% of GenBank annotated splice pairs. EST alignments allowed us to verify only the exonic part of splice sites. To check the conservative dinucleotides we compared sequences of human non-canonical splice sites with sequences from the high throughput genome sequencing project (HTG). Out of 171 human non-canonical and EST-supported splice pairs, 156 (91.23%) had a clear match in the human HTG. They can be classified after sequence analysis as: 79 GC–AG pairs (of which one was an error that corrected to GC–AG), 61 errors corrected to GT–AG canonical pairs, six AT–AC pairs (of which two were errors corrected to AT–AC), one case was produced from a non-existent intron, seven cases were found in HTG that were deposited to GenBank and finally there were only two other cases left of supported non-canonical splice pairs. The information about verified splice site sequences for canonical and non-canonical sites is presented in SpliceDB with the supporting evidence. We also built weight matrices for the major splice groups, which can be incorporated into gene prediction programs. SpliceDB is available at the computational genomic Web server of the Sanger Centre: http://genomic.sanger.ac.uk/spldb/SpliceDB.html and at http://www.softberry.com/spldb/SpliceDB.html.

Constructing a Canonical form of a Matrix in Several Problems about Combinatorial Designs

Partially supported by the Bulgarian Science Fund contract with TU Varna, No 487.

Krylov subspace methods for approximating functions of symmetric positive definite matrices with applications to applied statistics and anomalous diffusion

Matrix function approximation is a current focus of worldwide interest and finds application in a variety of areas of applied mathematics and statistics. In this thesis we focus on the approximation of A^(-α/2)b, where A ∈ ℝ^(n×n) is a large, sparse symmetric positive definite matrix and b ∈ ℝ^n is a vector. In particular, we will focus on matrix function techniques for sampling from Gaussian Markov random fields in applied statistics and the solution of fractional-in-space partial differential equations. Gaussian Markov random fields (GMRFs) are multivariate normal random variables characterised by a sparse precision (inverse covariance) matrix. GMRFs are popular models in computational spatial statistics as the sparse structure can be exploited, typically through the use of the sparse Cholesky decomposition, to construct fast sampling methods. It is well known, however, that for sufficiently large problems, iterative methods for solving linear systems outperform direct methods. Fractional-in-space partial differential equations arise in models of processes undergoing anomalous diffusion. Unfortunately, as the fractional Laplacian is a non-local operator, numerical methods based on the direct discretisation of these equations typically requires the solution of dense linear systems, which is impractical for fine discretisations. In this thesis, novel applications of Krylov subspace approximations to matrix functions for both of these problems are investigated. Matrix functions arise when sampling from a GMRF by noting that the Cholesky decomposition A = LL^T is, essentially, a `square root' of the precision matrix A. Therefore, we can replace the usual sampling method, which forms x = L^(-T)z, with x = A^(-1/2)z, where z is a vector of independent and identically distributed standard normal random variables. Similarly, the matrix transfer technique can be used to build solutions to the fractional Poisson equation of the form ϕn = A^(-α/2)b, where A is the finite difference approximation to the Laplacian. Hence both applications require the approximation of f(A)b, where f(t) = t^(-α/2) and A is sparse. In this thesis we will compare the Lanczos approximation, the shift-and-invert Lanczos approximation, the extended Krylov subspace method, rational approximations and the restarted Lanczos approximation for approximating matrix functions of this form. A number of new and novel results are presented in this thesis. Firstly, we prove the convergence of the matrix transfer technique for the solution of the fractional Poisson equation and we give conditions by which the finite difference discretisation can be replaced by other methods for discretising the Laplacian. We then investigate a number of methods for approximating matrix functions of the form A^(-α/2)b and investigate stopping criteria for these methods. In particular, we derive a new method for restarting the Lanczos approximation to f(A)b. We then apply these techniques to the problem of sampling from a GMRF and construct a full suite of methods for sampling conditioned on linear constraints and approximating the likelihood. Finally, we consider the problem of sampling from a generalised Matern random field, which combines our techniques for solving fractional-in-space partial differential equations with our method for sampling from GMRFs.

Crosstalk between developmental and tumour-specific signalling pathways : kallikrein-related serine peptidases and nodal in prostrate cancer

Prostate cancer is an important male health issue. The strategies used to diagnose and treat prostate cancer underscore the cell and molecular interactions that promote disease progression. Prostate cancer is histologically defined by increasingly undifferentiated tumour cells and therapeutically targeted by androgen ablation. Even as the normal glandular architecture of the adult prostate is lost, prostate cancer cells remain dependent on the androgen receptor (AR) for growth and survival. This project focused on androgen-regulated gene expression, altered cellular differentiation, and the nexus between these two concepts. The AR controls prostate development, homeostasis and cancer progression by regulating the expression of downstream genes. Kallikrein-related serine peptidases are prominent transcriptional targets of AR in the adult prostate. Kallikrein 3 (KLK3), which is commonly referred to as prostate-specific antigen, is the current serum biomarker for prostate cancer. Other kallikreins are potential adjunct biomarkers. As secreted proteases, kallikreins act through enzyme cascades that may modulate the prostate cancer microenvironment. Both as a panel of biomarkers and cascade of proteases, the roles of kallikreins are interconnected. Yet the expression and regulation of different kallikreins in prostate cancer has not been compared. In this study, a spectrum of prostate cell lines was used to evaluate the expression profile of all 15 members of the kallikrein family. A cluster of genes was co-ordinately expressed in androgenresponsive cell lines. This group of kallikreins included KLK2, 3, 4 and 15, which are located adjacent to one another at the centromeric end of the kallikrein locus. KLK14 was also of interest, because it was ubiquitously expressed among the prostate cell lines. Immunohistochemistry showed that these 5 kallikreins are co-expressed in benign and malignant prostate tissue. The androgen-regulated expression of KLK2 and KLK3 is well-characterised, but has not been compared with other kallikreins. Therefore, KLK2, 3, 4, 14 and 15 expression were all measured in time course and dose response experiments with androgens, AR-antagonist treatments, hormone deprivation experiments and cells transfected with AR siRNA. Collectively, these experiments demonstrated that prostatic kallikreins are specifically and directly regulated by the AR. The data also revealed that kallikrein genes are differentially regulated by androgens; KLK2 and KLK3 were strongly up-regulated, KLK4 and KLK15 were modestly up-regulated, and KLK14 was repressed. Notably, KLK14 is located at the telomeric end of the kallikrein locus, far away from the centromeric cluster of kallikreins that are stimulated by androgens. These results show that the expression of KLK2, 3, 4, 14 and 15 is maintained in prostate cancer, but that these genes exhibit different responses to androgens. This makes the kallikrein locus an ideal model to investigate AR signalling. The increasingly dedifferentiated phenotype of aggressive prostate cancer cells is accompanied by the re-expression of signalling molecules that are usually expressed during embryogenesis and foetal tissue development. The Wnt pathway is one developmental cascade that is reactivated in prostate cancer. The canonical Wnt cascade regulates the intracellular levels of β-catenin, a potent transcriptional co-activator of T-cell factor (TCF) transcription factors. Notably, β-catenin can also bind to the AR and synergistically stimulate androgen-mediated gene expression. This is at the expense of typical Wnt/TCF target genes, because the AR:β-catenin and TCF:β-catenin interactions are mutually exclusive. The effect of β-catenin on kallikrein expression was examined to further investigate the role of β-catenin in prostate cancer. Stable knockdown of β-catenin in LNCaP prostate cancer cells attenuated the androgen-regulated expression of KLK2, 3, 4 and 15, but not KLK14. To test whether KLK14 is instead a TCF:β-catenin target gene, the endogenous levels of β-catenin were increased by inhibiting its degradation. Although KLK14 expression was up-regulated by these treatments, siRNA knockdown of β-catenin demonstrated that this effect was independent of β-catenin. These results show that β-catenin is required for maximal expression of KLK2, 3, 4 and 15, but not KLK14. Developmental cells and tumour cells express a similar repertoire of signalling molecules, which means that these different cell types are responsive to one another. Previous reports have shown that stem cells and foetal tissues can reprogram aggressive cancer cells to less aggressive phenotypes by restoring the balance to developmental signalling pathways that are highly dysregulated in cancer. To investigate this phenomenon in prostate cancer, DU145 and PC-3 prostate cancer cells were cultured on matrices pre-conditioned with human embryonic stem cells (hESCs). Soft agar assays showed that prostate cancer cells exposed to hESC conditioned matrices had reduced clonogenicity compared with cells harvested from control matrices. A recent study demonstrated that this effect was partially due to hESC-derived Lefty, an antagonist of Nodal. A member of the transforming growth factor β (TGFβ) superfamily, Nodal regulates embryogenesis and is re-expressed in cancer. The role of Nodal in prostate cancer has not previously been reported. Therefore, the expression and function of the Nodal signalling pathway in prostate cancer was investigated. Western blots confirmed that Nodal is expressed in DU145 and PC-3 cells. Immunohistochemistry revealed greater expression of Nodal in malignant versus benign glands. Notably, the Nodal inhibitor, Lefty, was not expressed at the mRNA level in any prostate cell lines tested. The Nodal signalling pathway is functionally active in prostate cancer cells. Recombinant Nodal treatments triggered downstream phosphorylation of Smad2 in DU145 and LNCaP cells, and stably-transfected Nodal increased the clonogencity of LNCaP cells. Nodal was also found to modulate AR signalling. Nodal reduced the activity of an androgen-regulated KLK3 promoter construct in luciferase assays and attenuated the endogenous expression of AR target genes including prostatic kallikreins. These results demonstrate that Nodal is a novel example of a developmental signalling molecule that is reexpressed in prostate cancer and may have a functional role in prostate cancer progression. In summary, this project clarifies the role of androgens and changing cellular differentiation in prostate cancer by characterising the expression and function of the downstream genes encoding kallikrein-related serine proteases and Nodal. Furthermore, this study emphasises the similarities between prostate cancer and early development, and the crosstalk between developmental signalling pathways and the AR axis. The outcomes of this project also affirm the utility of the kallikrein locus as a model system to monitor tumour progression and the phenotype of prostate cancer cells.

Urban dynamic origin-destination matrices estimation

The aim of this paper is to explore a new approach to obtain better traffic demand (Origin-Destination, OD matrices) for dense urban networks. From reviewing existing methods, from static to dynamic OD matrix evaluation, possible deficiencies in the approach could be identified: traffic assignment details for complex urban network and lacks in dynamic approach. To improve the global process of traffic demand estimation, this paper is focussing on a new methodology to determine dynamic OD matrices for urban areas characterized by complex route choice situation and high level of traffic controls. An iterative bi-level approach will be used, the Lower level (traffic assignment) problem will determine, dynamically, the utilisation of the network by vehicles using heuristic data from mesoscopic traffic simulator and the Upper level (matrix adjustment) problem will proceed to an OD estimation using optimization Kalman filtering technique. In this way, a full dynamic and continuous estimation of the final OD matrix could be obtained. First results of the proposed approach and remarks are presented.

Mineralized human primary osteoblast matrices as a model system to analyse interactions of prostate cancer cells with the bone microenvironment

Prostate cancer metastasis is reliant on the reciprocal interactions between cancer cells and the bone niche/micro-environment. The production of suitable matrices to study metastasis, carcinogenesis and in particular prostate cancer/bone micro-environment interaction has been limited to specific protein matrices or matrix secreted by immortalised cell lines that may have undergone transformation processes altering signaling pathways and modifying gene or receptor expression. We hypothesize that matrices produced by primary human osteoblasts are a suitable means to develop an in vitro model system for bone metastasis research mimicking in vivo conditions. We have used a decellularized matrix secreted from primary human osteoblasts as a model for prostate cancer function in the bone micro-environment. We show that this collagen I rich matrix is of fibrillar appearance, highly mineralized, and contains proteins, such as osteocalcin, osteonectin and osteopontin, and growth factors characteristic of bone extracellular matrix (ECM). LNCaP and PC3 cells grown on this matrix, adhere strongly, proliferate, and express markers consistent with a loss of epithelial phenotype. Moreover, growth of these cells on the matrix is accompanied by the induction of genes associated with attachment, migration, increased invasive potential, Ca2+ signaling and osteolysis. In summary, we show that growth of prostate cancer cells on matrices produced by primary human osteoblasts mimics key features of prostate cancer bone metastases and thus is a suitable model system to study the tumor/bone micro-environment interaction in this disease.

Osteogenic induction of human bone marrow-derived mesenchymal progenitor cells in novel synthetic polymer-hydrogel matrices

The aim of this project was to investigate the in vitro osteogenic potential of human mesenchymal progenitor cells in novel matrix architectures built by means of a three-dimensional bioresorbable synthetic framework in combination with a hydrogel. Human mesenchymal progenitor cells (hMPCs) were isolated from a human bone marrow aspirate by gradient centrifugation. Before in vitro engineering of scaffold-hMPC constructs, the adipogenic and osteogenic differentiation potential was demonstrated by staining of neutral lipids and induction of bone-specific proteins, respectively. After expansion in monolayer cultures, the cells were enzymatically detached and then seeded in combination with a hydrogel into polycaprolactone (PCL) and polycaprolactone-hydroxyapatite (PCL-HA) frameworks. This scaffold design concept is characterized by novel matrix architecture, good mechanical properties, and slow degradation kinetics of the framework and a biomimetic milieu for cell delivery and proliferation. To induce osteogenic differentiation, the specimens were cultured in an osteogenic cell culture medium and were maintained in vitro for 6 weeks. Cellular distribution and viability within three-dimensional hMPC bone grafts were documented by scanning electron microscopy, cell metabolism assays, and confocal laser microscopy. Secretion of the osteogenic marker molecules type I procollagen and osteocalcin was analyzed by semiquantitative immunocytochemistry assays. Alkaline phosphatase activity was visualized by p-nitrophenyl phosphate substrate reaction. During osteogenic stimulation, hMPCs proliferated toward and onto the PCL and PCL-HA scaffold surfaces and metabolic activity increased, reaching a plateau by day 15. The temporal pattern of bone-related marker molecules produced by in vitro tissue-engineered scaffold-cell constructs revealed that hMPCs differentiated better within the biomimetic matrix architecture along the osteogenic lineage.

In vitro characterization of natural and synthetic dermal matrices cultured with human dermal fibroblasts

The ideal dermal matrix should be able to provide the right biological and physical environment to ensure homogenous cell and extracellular matrix (ECM) distribution, as well as the right size and morphology of the neo-tissue required. Four natural and synthetic 3D matrices were evaluated in vitro as dermal matrices, namely (1) equine collagen foam, TissuFleece®, (2) acellular dermal replacement, Alloderm®, (3) knitted poly(lactic-co-glycolic acid) (10:90)–poly(-caprolactone) (PLGA–PCL) mesh, (4) chitosan scaffold. Human dermal fibroblasts were cultured on the specimens over 3 weeks. Cell morphology, distribution and viability were assessed by electron microscopy, histology and confocal laser microscopy. Metabolic activity and DNA synthesis were analysed via MTS metabolic assay and [3H]-thymidine uptake, while ECM protein expression was determined by immunohistochemistry. TissuFleece®, Alloderm® and PLGA–PCL mesh supported cell attachment, proliferation and neo-tissue formation. However, TissuFleece® contracted to 10% of the original size while Alloderm® supported cell proliferation predominantly on the surface of the material. PLGA–PCL mesh promoted more homogenous cell distribution and tissue formation. Chitosan scaffolds did not support cell attachment and proliferation. These results demonstrated that physical characteristics including porosity and mechanical stability to withstand cell contraction forces are important in determining the success of a dermal matrix material.

Engineered silk fibroin protein 3D matrices for in vitro tumor model

3D in vitro model systems that are able to mimic the in vivo microenvironment are now highly sought after in cancer research. Antheraea mylitta silk fibroin protein matrices were investigated as potential biomaterial for in vitro tumor modeling. We compared the characteristics of MDA-MB-231 cells on A. mylitta, Bombyx mori silk matrices, Matrigel, and tissue culture plates. The attachment and morphology of the MDA-MB-231 cell line on A. mylitta silk matrices was found to be better than on B. mori matrices and comparable to Matrigel and tissue culture plates. The cells grown in all 3D cultures showed more MMP-9 activity, indicating a more invasive potential. In comparison to B. mori fibroin, A. mylitta fibroin not only provided better cell adhesion, but also improved cell viability and proliferation. Yield coefficient of glucose consumed to lactate produced by cells on 3D A. mylitta fibroin was found to be similar to that of cancer cells in vivo. LNCaP prostate cancer cells were also cultured on 3D A. mylitta fibroin and they grew as clumps in long term culture. The results indicate that A. mylitta fibroin scaffold can provide an easily manipulated microenvironment system to investigate individual factors such as growth factors and signaling peptides, as well as evaluation of anticancer drugs.