Several new quadrature formulas for polynomial integration in the triangle

Working paper submitted to arxiv.org by authorsWe present several new quadrature formulas in the triangle for exact integration of polynomials. The points were computed numerically with a cardinal function algorithm which imposes that the number of quadrature points $N$ be equal to the dimension of a lower dimensional polynomial space. Quadrature forumulas are presented for up to degree $d=25$, all which have positive weights and contain no points outside the triangle. Seven of these quadrature formulas improve on previously known results

Quadrature Strategies for Constructing Polynomial Approximations

Finding suitable points for multivariate polynomial interpolation and approximation is a challenging task. Yet, despite this challenge, there has been tremendous research dedicated to this singular cause. In this paper, we begin by reviewing classical methods for finding suitable quadrature points for polynomial approximation in both the univariate and multivariate setting. Then, we categorize recent advances into those that propose a new sampling approach and those centered on an optimization strategy. The sampling approaches yield a favorable discretization of the domain, while the optimization methods pick a subset of the discretized samples that minimize certain objectives. While not all strategies follow this two-stage approach, most do. Sampling techniques covered include subsampling quadratures, Christoffel, induced and Monte Carlo methods. Optimization methods discussed range from linear programming ideas and Newton's method to greedy procedures from numerical linear algebra. Our exposition is aided by examples that implement some of the aforementioned strategies

Quadrature Strategies for Constructing Polynomial Approximations

Finding suitable points for multivariate polynomial interpolation and approximation is a challenging task. Yet, despite this challenge, there has been tremendous research dedicated to this singular cause. In this paper, we begin by reviewing classical methods for finding suitable quadrature points for polynomial approximation in both the univariate and multivariate setting. Then, we categorize recent advances into those that propose a new sampling approach, and those centered on an optimization strategy. The sampling approaches yield a favorable discretization of the domain, while the optimization methods pick a subset of the discretized samples that minimize certain objectives. While not all strategies follow this two-stage approach, most do. Sampling techniques covered include subsampling quadratures, Christoffel, induced and Monte Carlo methods. Optimization methods discussed range from linear programming ideas and Newton's method to greedy procedures from numerical linear algebra. Our exposition is aided by examples that implement some of the aforementioned strategies

Gain-of-function RAF1 mutations cause Noonan and LEOPARD syndromes with hypertrophic cardiomyopathy

Noonan and LEOPARD syndromes are developmental disorders with overlapping features, including cardiac abnormalities, short stature and facial dysmorphia. Increased RAS signaling owing to PTPN11, SOS1 and KRAS mutations causes similar to 60% of Noonan syndrome cases(1-6), and PTPN11 mutations cause 90% of LEOPARD syndrome cases7. Here, we report that 18 of 231 individuals with Noonan syndrome without known mutations (corresponding to 3% of all affected individuals) and two of six individuals with LEOPARD syndrome without PTPN11 mutations have missense mutations in RAF1, which encodes a serine- threonine kinase that activates MEK1 and MEK2. Most mutations altered a motif flanking Ser259, a residue critical for autoinhibition of RAF1 through 14-3-3 binding. Of 19 subjects with a RAF1 mutation in two hotspots, 18 (or 95%) showed hypertrophic cardiomyopathy (HCM), compared with the 18% prevalence of HCM among individuals with Noonan syndrome in general. Ectopically expressed RAF1 mutants from the two HCM hotspots had increased kinase activity and enhanced ERK activation, whereas non-HCM-associated mutants were kinase impaired. Our findings further implicate increased RAS signaling in pathological cardiomyocyte hypertrophy