2n Terms Research Articles

Estimating covariance functions of multivariate skew-Gaussian random fields on the sphere

This paper considers a multivariate spatial random field, with each component having univariate marginal distributions of the skew-Gaussian type. We assume that the field is defined spatially on the unit sphere embedded in R3, allowing for modeling data available over large portions of planet Earth. This model admits explicit expressions for the marginal and cross covariances. However, the n-dimensional distributions of the field are difficult to evaluate, because it requires the sum of 2n terms involving the cumulative and probability density functions of a n-dimensional Gaussian distribution. Since in this case inference based on the full likelihood is computationally unfeasible, we propose a composite likelihood approach based on pairs of spatial observations. This last being possible thanks to the fact that we have a closed form expression for the bivariate distribution. We illustrate the effectiveness of the method through simulation experiments and the analysis of a real data set of minimum and maximum surface air temperatures.

Extending BEAMS to incorporate correlated systematic uncertainties

New supernova surveys such as the Dark Energy Survey, Pan-STARRS and the LSST will produce an unprecedented number of photometric supernova candidates, most with no spectroscopic data. Avoiding biases in cosmological parameters due to the resulting inevitable contamination from non-Ia supernovae can be achieved with the BEAMS formalism, allowing for fully photometric supernova cosmology studies. Here we extend BEAMS to deal with the case in which the supernovae are correlated by systematic uncertainties. The analytical form of the full BEAMS posterior requires evaluating 2N terms, where N is the number of supernova candidates. This `exponential catastrophe' is computationally unfeasible even for N of order 100. We circumvent the exponential catastrophe by marginalising numerically instead of analytically over the possible supernova types: we augment the cosmological parameters with nuisance parameters describing the covariance matrix and the types of all the supernovae, τi, that we include in our MCMC analysis. We show that this method deals well even with large, unknown systematic uncertainties without a major increase in computational time, whereas ignoring the correlations can lead to significant biases and incorrect credible contours. We then compare the numerical marginalisation technique with a perturbative expansion of the posterior based on the insight that future surveys will have exquisite light curves and hence the probability that a given candidate is a Type Ia will be close to unity or zero, for most objects. Although this perturbative approach changes computation of the posterior from a 2N problem into an N2 or N3 one, we show that it leads to biases in general through a small number of misclassifications, implying that numerical marginalisation is superior.

New Techniques for the Computation of Linear Recurrence Coefficients

The n coefficients of a fixed linear recurrence can be expressed through its m≤2n terms or, equivalently, the coefficients of a polynomial of a degree n can be expressed via the power sums of its zeros—by means of a polynomial equation known as the key equation for decoding the BCH error-correcting codes. The same problem arises in sparse multivariate polynomial interpolation and in various fundamental computations with sparse matrices in finite fields. Berlekamp's algorithm of 1968 solves the key equation by using order of n2 operations in a fixed field. Several algorithms of 1975–1980 rely on the extended Euclidean algorithm and computing Padé approximation, which yields a solution in O(n(log n)2 log log n) operations, though a considerable overhead constant is hidden in the “O” notation. We show algorithms (depending on the characteristic c of the ground field of the allowed constants) that simplify the solution and lead to further improvements of the latter bound, by factors ranging from order of log n, for c=0 and c>n (in which case the overhead constant drops dramatically), to order of min (c, log n), for 2≤c≤n; the algorithms use Las Vegas type randomization in the case of 2<c≤n.

Propagation of SH-waves in a layered medium

Summary. A set of recurrence relations similar to that of Kennett suitable for SH-wave generation in an (n+ l)-layered half-space is presented. The recurrence relations contain no growing terms, thus providing a stable and efficient algorithm for computing complete SH synthetic seismograms. The complete expansion of these recurrence relations gives the explicit form of the transfer function for SH-waves. The transfer function for a point source in layer s of the stratification is a series of 2n terms in the denominator and a series of 2n−s+1 terms in the numerator. The result of Wang from ray summation is shown to be a special case of our general result. Numerical comparison of the algorithm of this paper with the propagator matrix method is also made.

Spatial arrays generated by two signal waves and one pump wave

Two signal waves combine with a pump wave to form two-dimensional gratings in nonlinear medium by means of |E|(2n) terms in the electric polarization. |E|(4) terms can be used to add and subtract the phases of the two signal waves.

Investigations of the pion-nucleus optical potential from pionic atoms

The explicit dependence of the zero-energy π −-nucleus optical potential on its various parameters, as well as its implicit dependence on neutron radii, is investigated for a selected set of particularly accurate level shift and width data. Among the various effects studied are some possible variations in the form of the Lorentz-Lorenz term, terms induced by the so-called “angle-transformation”, and the smearing of the π 2N terms which allows for a finite range in the absorption process. The pionic atom data are equally well fitted by any of the several forms used for the optical potential, with the persistent failure to fit the recently reported widths of 3d levels in 181Ta, Re and 209Bi and the width of the 1s level in 23Na.

Response Function Method for Solving Dirichlet's Problems. II. Fast Calculation of the Solutions

A method for rapid calculation of the response functions φN is presented for a Dirichlet problem of the Laplace equation in a rectangular domain. The series for φN consists of N terms (N is the total number of sampling points for the boundary value). If N is set equal to an integral power of two and the series is modified so that it contains 2N terms, the algorithm of fast Fourier transform is applied to this series. Further, a convenient method is described, by which a set of φN can produce other sets of φN. It is shown that the computing time for obtaining the solutions is greatly reduced.

Length of prime disjunctive normal forms

The author presents an example of a Logical function of n variables that has a prime disjunctive normal form that asymptotically contains 2n terms. Bibliography of 2 references.

Convergence acceleration from the point of view of linear programming

The best possible bounds for the sum of an alternating series with completely monotonic terms, when 2N terms have been computed, are determined. It is shown that their difference decreases exponentially withN. Various generalizations are indicated. The optimal application of Euler's transformation is also discussed. The error of that method also decreases exponentially, though the logarithmic decrement is only about 2/3 compared with the best possible error bounds.

Solution of Linear Integral Equations Using Padé Approximants

It is shown that the exact solution of a nonhomogeneous linear integral equation with a kernel K of rank n is given by forming the Padé approximant P(n, n) from the first 2n terms of the perturbation series solution. It follows that for a compact kernel K, the solution is limn→∞ P(n, n); this gives meaning to a large class of perturbation series when the perturbation is large. The possible extension of this result to wider classes of equations is discussed.

Inverse Overlap Matrix for Periodic Arrays of Atoms

The calculation of the inverse overlap matrix for an infinite chain of single orbital atoms is reduced to the problem of calculating the roots of a polynomial of degree n constructed from the overlap integrals between n neighbors. The familiar method of diagonalizing the overlap matrix, inverting it, and then transforming back to the original representation is used. The final transformation leads to contour integrals which can be evaluated by the method of residues. This diagonalization method is shown to be useful also for finite chains of atoms with Born von Kárman boundary conditions, for chains of atoms with more than one orbital per atom, and for calculating the inverse root of the overlap matrix of single-orbital atoms when the only overlap is between nearest neighbors. Application of the method to two- and three-dimensional arrays of atoms leads to contour integrals containing branch points which cannot be reduced to simple analytic expressions. However, it is shown that an iterative procedure can be devised which permits the calculation of 2n terms of the Löwdin expansion for the inverse with only 2n matrix multiplications.