Fully symmetric interpolatory integration rules are constructed for multidimensional integrals over infinite integration regions with a Gaussian weight function. The points for these rules are ...determined by successive extensions of the one-dimensional three-point Gauss-Hermite rule. The new rules are shown to be efficient and only moderately unstable.
In this paper the multiple integration formulas of Gaussian methods with positive coefficient for fuzzy integrations are discussed and then are followed by convergence theorem. The proposed ...algorithms are illustrated by solving some numerical examples.
Education Today 2013: The OECD Perspective Development, Organisation for Economic Co-operation and; Innovation, Centre for Educational Research and
Education Today 2013: The OECD Perspective,
2014.
eBook, Book, Journal Article
What does the OECD have to say about the state of education today? What are the main OECD messages on early childhood education, teacher policies and tertiary education? What about student ...performance, educational spending and equity in education? OECD work on these important education topics and others have been brought together in a single accessible source updating the first edition of Education Today which came out in March 2009. Organised into eight chapters, this report examines early childhood education, schooling, transitions beyond initial education, higher education, adult learning, outcomes and returns, equity, and innovation. The chapters are structured around key findings and policy directions emerging from recent OECD educational analyses. Each entry highlights the main message in a concise and accessible way, with a brief explanation and reference to the original OECD source.
When multiple integrals are approximately evaluated using Korobov cubature formulas, it is necessary to introduce a parameter characterizing the uniform distribution of the grid nodes. A new ...parameter for Korobov parallelepipedal grids is proposed, and an algorithm for its computation is described.
How can we escape Thomae's relations? KRATTENTHALER, Christian; RIVOAL, Tanguy
Journal of the Mathematical Society of Japan,
1/2006, Letnik:
58, Številka:
1
Journal Article
Recenzirano
Odprti dostop
In 1879, Thomae discussed the relations between two generic hypergeometric _3F_2 -series with argument 1. It is well-known since then that, in combination with the trivial ones which come from ...permutations of the parameters of the hypergeometric series, Thomae had found a set of 120 relations. More recently, Rhin and Viola asked the following question (in a different, but equivalent language of integrals): If there exists a linear dependence relation over \bm{Q} between two convergent _3F_2 -series with argument 1, with integral parameters, and whose values are irrational numbers, is this relation a specialisation of one of the 120 Thomae relations? A few years later, Sato answered this question in the negative, by giving six examples of relations which cannot be explained by Thomae's relations. We show that Sato's counter-examples can be naturally embedded into two families of infinitely many _3F_2 -relations, both parametrised by three independent parameters. Moreover, we find two more infinite families of the same nature. The families, which do not seem to have been recorded before, come from certain _3F_2 -transformation formulae and contiguous relations. We also explain in detail the relationship between the integrals of Rhin and Viola and _3F_2 -series.
We point out that a proper use of the Hoeffding-ANOVA decomposition for symmetric statistics of finite urn sequences, previously introduced by the author, yields a decomposition of the space of ...square-integrable functionals of a Dirichlet-Ferguson process, written L²(D), into orthogonal subspaces of multiple integrals of increasing order. This gives an isomorphism between L²(D) and an appropriate Fock space over a class of deterministic functions. By means of a well-known result due to Blackwell and MacQueen, we show that each element of the nth orthogonal space of multiple integrals can be represented as the L² limit of U-statistics with degenerate kernel of degree n. General formulae for the decomposition of a given functional are provided in terms of linear combinations of conditioned expectations whose coefficients are explicitly computed. We show that, in simple cases, multiple integrals have a natural representation in terms of Jacobi polynomials. Several connections are established, in particular with Bayesian decision problems, and with some classic formulae concerning the transition densities of multiallele diffusion models, due to Littler and Fackerell, and Griffiths. Our results may also be used to calculate the best approximation of elements of L²(D) by means of U-statistics of finite vectors of exchangeable observations.
Small gaps between primes Sivak-Fischler, J.
Acta mathematica Hungarica,
9/2007, Letnik:
116, Številka:
4
Journal Article
Recenzirano
Combining Goldston-Yildirim's method on k-correlations of the truncated von Mangoldt function with Maier's matrix method, we show that Equation for all r . 1 where p(n) denotes the nth prime number ...and g is Euler's constant. This is the best known result for any r . 11.
Relaxation of Nonlocal Singular Integrals Grasmair, M.; Scherzer, O.
Numerical functional analysis and optimization,
06/2005, Letnik:
26, Številka:
4-5
Journal Article
Recenzirano
In this paper we study well-posedness of a class of nonconvex variational principles arising in regularization theory for denoising of data with sampling errors and level set regularization methods ...for inverse problems. These models result in minimization of nonconvex, singular functionals involving (possibly) non-local operators.
Romberg’s method, which is used to improve the accuracy of one-dimensional integral evaluation, is extended to multiple integrals if they are evaluated using the product of composite quadrature ...formulas. Under certain conditions, the coefficients of the Romberg formula are independent of the integral’s multiplicity, which makes it possible to use a simple evaluation algorithm developed for one-dimensional integrals. As examples, integrals of multiplicity two to six are evaluated by Romberg’s method and the results are compared with other methods.
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