Computing the common zeros of two bivariate functions via Bézout resultants

The common zeros of two bivariate functions can be computed by finding the common zeros of their polynomial interpolants expressed in a tensor Chebyshev basis. From here we develop a bivariate rootfinding algorithm based on the hidden variable resultant method and Bézout matrices with polynomial entries. Using techniques including domain subdivision, Bézoutian regularization, and local refinement we are able to reliably and accurately compute the simple common zeros of two smooth functions with polynomial interpolants of very high degree (≥ 1000). We analyze the resultant method and its conditioning by noting that the Bézout matrices are matrix polynomials. Two implementations are available: one on the Matlab Central File Exchange and another in the roots command in Chebfun2 that is adapted to suit Chebfun’s methodology

Pan-cancer analysis of whole genomes

Cancer is driven by genetic change, and the advent of massively parallel sequencing has enabled systematic documentation of this variation at the whole-genome scale(1-3). Here we report the integrative analysis of 2,658 whole-cancer genomes and their matching normal tissues across 38 tumour types from the Pan-Cancer Analysis of Whole Genomes (PCAWG) Consortium of the International Cancer Genome Consortium (ICGC) and The Cancer Genome Atlas (TCGA). We describe the generation of the PCAWG resource, facilitated by international data sharing using compute clouds. On average, cancer genomes contained 4-5 driver mutations when combining coding and non-coding genomic elements; however, in around 5% of cases no drivers were identified, suggesting that cancer driver discovery is not yet complete. Chromothripsis, in which many clustered structural variants arise in a single catastrophic event, is frequently an early event in tumour evolution; in acral melanoma, for example, these events precede most somatic point mutations and affect several cancer-associated genes simultaneously. Cancers with abnormal telomere maintenance often originate from tissues with low replicative activity and show several mechanisms of preventing telomere attrition to critical levels. Common and rare germline variants affect patterns of somatic mutation, including point mutations, structural variants and somatic retrotransposition. A collection of papers from the PCAWG Consortium describes non-coding mutations that drive cancer beyond those in the TERT promoter(4); identifies new signatures of mutational processes that cause base substitutions, small insertions and deletions and structural variation(5,6); analyses timings and patterns of tumour evolution(7); describes the diverse transcriptional consequences of somatic mutation on splicing, expression levels, fusion genes and promoter activity(8,9); and evaluates a range of more-specialized features of cancer genomes(8,10-18).Peer reviewe

Steepest Descent Adaptation of Min-Max Fuzzy If-Then Rules

A new technique for adaptation of fuzzy membership functions in a fuzzy inference system is proposed. The technique relies upon the isolation of the specific membership function that contributed to the final decision, followed by the updating of this function's parameters using steepest descent. The error measure used is thus back propagated from output to input, through the min and max operators used during the inference stage. This is feasible because the operations of min and max are continuous differentiable functions and therefore can be placed in a chain of partial derivatives for steepest descent backpropagation adaptation. More interestingly, the partials of min and max (or any other order statistic, for that matter) act as `pointers' with the result that only the function that gave rise to the min or max is adapted; the others are not. To illustrate, let a # max#b 1 # b 2 # ####b N #. Then a#b n # 1whenb n is the maximum and is otherwise zero. We apply this property to the f..

Multi-variate Hardy-type lattice point summation and Shannon-type sampling

The famous Shannon sampling theorem gives an answer to the question of how a one-dimensional time-dependent bandlimited signal can be reconstructed from discrete values in lattice points. In this work, we are concerned with multi-variate Hardy-type lattice point identities from which space-dependent Shannon-type sampling theorems can be obtained by straightforward integration over certain regular regions. An answer is given to the problem of how a signal bandlimited to a regular region in q-dimensional Euclidean space allows a reconstruction from discrete values in the lattice points of a (general) q-dimensional lattice. Weighted Hardy-type lattice point formulas are derived to allow explicit characterizations of over- and undersampling, thereby specifying not only the occurrence, but also the type of aliasing in a thorough mathematical description. An essential tool for the proof of Hardy-type identities in lattice point theory is the extension of the Euler summation formula to second order Helmholtz-type operators involving associated Green functions with respect to the “boundary condition” of periodicity. In order to circumvent convergence difficulties and/or slow convergence in multi-variate Hardy-type lattice point summation, some summability methods are necessary, namely lattice ball and Gauß–Weierstraß averaging. As a consequence, multi-variate Shannon-type lattice sampling becomes available in a proposed summability context to accelerate the summation of the associated cardinal-type series. Finally, some aspects of constructive approximation in a resulting Paley–Wiener framework are indicated, such as the recovery of a finite set of lost samples, the reproducing Hilbert space context of spline interpolation