Explicit Coefficients Research Articles

Estimating competition coefficients: strong competition among three species of frugivorous flies.

Despite the abundance of studies on competitive interactions, relatively few experiments have been used to fit explicit competition models and estimate competition coefficients. Such estimates are valuable for making contact between theoretical and empirical studies, which tend to measure competition in different units. To quantify the strength of competitive interactions among the larvae of three species of frugivorous flies, I manipulated the densities of each species to investigate all three pairwise interactions. The densities of each species were changed independently (i.e., using a response surface experimental design), which allowed maximum likelihood estimation of the competition coefficients for each species, based on the Hassell and Comins competition model. The effects of competitor density on larval survival, time to emergence, and the weight of emerging adults were also analyzed to investigate the responses of individual species to density. The estimates of the competition coefficients suggest that the larvae of these flies experience strong asymmetric competition for resources, and raise questions as to how these species coexist. For each pair, one of the species was largely unaffected by interspecific competition, but decreased the performance of the other.

From canonical tensor operators of SU(3) and (3) to bi-orthogonal coupling coefficients: explicit expansion

Expansion of the matrix elements of SU(3) and canonical tensor operators in terms of the bi-orthogonal coupling coefficients and overlaps of the Draayer-Akiyama construction are considered. Special bi-orthogonal extremal isoscalar factors (with subscripts as multiplicity labels and proportional to q-Racah (q-6j) coefficients or the generalized Wilson-Rahman rational bi-orthogonal functions in terms of balanced or hypergeometric series) are used as a natural basis for extremal matrix elements of the highest weight component of the canonical tensor operators of SU(3) in the generating function approach of Biedenharn, Lohe and Louck. The expansion that is obtained (triple sum), together with previously derived asymmetric seed isofactors and elementary overlaps, gives the explicit overlap coefficients and can be used to derive SU(3) and canonical tensor operators as well as new explicit normalized seed isofactors with Regge-type symmetry, specified for the minimal null space case.

Singularity spectrum of multifractal functions involving oscillating singularities

We give general mathematical results concerning oscillating singularities and we study examples of functions composed only of oscillating singularities. These functions are defined by explicit coefficients on an orthonormal wavelet basis. We compute their Holder regularity and oscillation at every point and we deduce their spectrum of oscillating singularities.

Orthogonalization of the projected states and isofactors

The explicit direct and inverse orthogonalization coefficients (OCs) are derived for the projected basis states of the two parametric covariant irreducible representations of (including Elliott's n = 3 case) and (Draayer), with the alternative choices of the Gram - Schmidt processes, began, respectively, from the lowest (Vergados) or highest absolute value of the intrinsic parameter K. The different direct and inverse OCs (the latter equivalent to the definite boundary isofactors, or the boundary resubducing coefficients) are found to be connected by the definite analytical continuation - contraction procedures with mutually related OCs of the biorthogonal states and the boundary orthonormal pseudocanonical and paracanonical isofactors of . The orthogonalization coefficients considered are presented as definite algebraic-polynomial structures under the square root, with the linear factors and numerator - denominator polynomials in terms of the partition-dependent functions of Biedenharn and Louck, .

Infinite systems of equations. The effect of truncation

We consider an infinite system of equations of the form x m + Σ∞ n=1 k mn x n = a m , 1 ≤ m ≤ ∞, or in matrix notation (I + K)x = a, where I denotes the infinite unit matrix, and where the elements k mn of the matrix K and the components a n of the column vector a are given. With the solution {x n } of this system we associate the sum S = Σ∞ n=1 b n X n = b'x. In practice this system is replaced by the finite truncated system involving N unknowns, ξ m (N) + Σ N n=1 k mn ξ n (N) = a m , 1 ≤ m ≤ N, or in matrix notation [(I + K)] N [ξ(N)] N = [a] N , with the associated sum S(N) = Σ N n=1 b n ξ n (N) = [b]' N [ξ(N)] N , where the symbol [a] N , denotes a column vector of dimension N and the symbol [A] N denotes a matrix of dimension N x N. We wish to find how S(N) approaches the limit S, that is, we wish to find the truncation error Δ(N) = S - S(N) = b'(I + K) -1 a - [b]' N ([I + K] N ) -1 [a] N as a function of the large parameter N. The elements of the inverse matrices (I+K) -1 and ([I+K] N ) -1 cannot in general be found explicitly even when the elements of (I + K) are known explicitly. It will be seen (see Theorem 2.1 below) that an explicit bound can always be found for Δ(N) but this bound may not give the true order of magnitude. There are infinite systems for which the form of the asymptotic expansion of the components {x M } of the infinite-dimensional solution vector x can readily be found for large suffixes M. It is shown that for such systems this expansion for x M can be used to improve the bound for Δ(M), and in suitable cases to find the form of an asymptotic expression for Δ(N), through not necessarily the explicit coefficients. The theory is illustrated by means of the example x m - Σ∞ n=1 x n /mn(m + n + 1) = 1/m 2 , 1 ≤ m < ∞, with the associated sum S = Σ x n /n 2 . It is shown that Δ(N) ∼ α/N 3 + β/N 4 + O(1/N 9/2 ), where α and β are certain constants ; the bound of Theorem 2.1 gives Δ(N) = O(1/N 3/2 ). The methods used in the present paper are elementary, involving at most Schwarz's Inequality (Σ X m Y m ) 2 ≤ (Σ X m 2 )(Σ Y m 2 ).

Application of an eigenfunction representation to the scattering of a plane wave by an anisotropically coated circular cylinder

It is well known that in an unbounded anisotropic homogeneous medium there are eigen plane wave solutions, and the fields inside a circle can be expressed as an integral of eigen plane waves with respect to the polar angle. In this paper the fields inside a circle are expanded as a finite summation of eigen plane waves with different polar angles. Each eigen plane wave is a wavefunction of the first kind, which can be represented by a series. From this series form of wavefunctions of the first kind for anisotropic media, series representation with explicit coefficients of the circular cylindrical wavefunctions of the second, third, fourth kind for homogeneous anisotropic media are obtained. This paper further deals with the numerical implementation of the above eigenfunction representation for homogeneous anisotropic media. Applications of the theory are made to the case of two-dimensional scattering of a plane wave by an anisotropically coated cylinder and comparisons are made with publishing data. The computational efficiency of the method expounded in this paper is compared with previous work.

Coupling Coefficients of Quasispin Group SO(5)t11The project supported by Doctoral Program Foundation of Institution of Higher Education of China and National Natural Science Foundation of China.

A recurrent method for coupling coefficient calculations of five-dimensional quasispin group SO(5)t is proposed. The explicit coupling coefficients such as , and with are given.

Explicit weighting coefficients for predicting ARMA time series from the finite past

Explicit formulas are given for the weighting coefficients in the linear minimum variance predictor of a wide sense stationary autoregressive-moving average time series k steps ahead, given n + 1 successive observations of a realization of the process. The formulas involve determinants whose entries are the values of certain polynomials related to the autoregressive part of the process at the zeros of the polynomial that defines the moving average part. The number of observations n + 1 enters into the formulas as parameter, and not so as to increase their complexity as n grows large. Formulas are also given for the variance of the prediction.

Integral expansion often reducing to the density-gradient expansion, extended to non-Markov stochastic processes: Consequent non-Markovian stochastic equation whose leading terms coincide with Schrödinger's.

An integral expansion is obtained that reduces under explicitly given conditions to the density-gradient expansion for the number density of stochastic particles. Explicit coefficients in terms of moments are calculated up to and including the fourth order, corresponding to the super-Burnett approximation. The expansion is proven to be valid for both Markov and non-Markov processes, including those with infinite memory, typical of a stochastic motion with inertia. An application of this expansion is the relation between local average velocities in Markovian stochastic processes and the drift velocity of a probability cloud. As a further application we obtain a non-Markovian stochastic equation whose leading terms coincide with the Schr\"odinger equation. The additional terms could be interpreted as corrections that produce contributions to atomic spectra of the fine-structure type. In the case of scattering, where the density gradients can be locally high, the correction terms become relevant and they could perhaps explain the differences between the calculated and measured rotational and/or vibrational cross sections for hydrogen molecules.

Normal forms for generalised Hopf bifurcations

Explicit coefficients of normal forms for generalised Hopf bifurcations are calculated to the seventh (but, to save space, they are given only to the fifth) order near the critical point. The real part of the third-order coefficient reproduces, at the critical point, the third-order focal value of Andronov et al (1973).

A family of 2 × 2 extreme matrices and a generalization of Clarkson's inequalities

With a view to generalize the classical Clarkson inequalities for 1 < p < ∞, we construct and study a family of 2 × 2 extreme matrices on lp (2), two-dimensional lp -spaces (2 < p < ∞). The scalar (skew-) field may be the reals, the complex numbers or the quaternions. The generalization sought for, in two versions, is then obtained with simple, explicit coefficients.

An algorithm for fast polynomial approximation

An algorithm is presented for the computation of the coefficients of least square fitting polynomials. The algorithm is such that multiplication is not required with respect to the data points. Since the only multiplication required is by system constants, and that only for the final computation, this algorithm is suitable for real time computation on small microprocessors. Explicit coefficients are given for the quadratic case.

Robust prediction of band-limited signals from past samples (Corresp.)

For a complex-valued deterministic signal of finite energy band-limited to the normalized frequency band <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|w| \leq \pi</tex> explicit coefficients <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{a_{kn}\}</tex> are found such that for any <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</tex> satisfying <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0 &lt; T \leq 1/2</tex> , <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\left| f(t)-\sum^{2n}_{k=1}a_{kn}f(t - kT)\right| \leq E_{f}\cdot \beta^{n}</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E_{f}</tex> is the signal energy and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\beta \doteq 0.6863</tex> . Thus the estimate of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(t)</tex> in terms of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2n</tex> past samples taken at a rate equal to or in excess of twice the Nyquist rate converges uniformly at a geometric rate to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(t)</tex> on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(- \infty , \infty)</tex> . The suboptimal coefficients <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{a_{kn}\}</tex> have the desirable property of being pure numbers independent of both the particular band-limited signal and of the selected sampling rate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/T</tex> . It is also shown that these same coefficients can be used to estimate the value of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(t)</tex> of a wide-sense stationary random process in terms of past samples.

Perturbation series for the double cnoidal wave of the Korteweg–de Vries equation

By means of the theorems proved earlier by the author, the problem of the double cnoidal wave of the Korteweg–de Vries equation is reduced to four algebraic equations in four unknowns. Two of the unknowns are the nonlinear phase speeds c1 and c2. Another is a physically irrelevant integration constant. The fourth unknown is the off-diagonal element of the symmetric, 2×2 theta matrix, which in turn gives the explicit coefficients of the Riemann theta function. The double cnoidal wave u(x,t) is then obtained by taking the second x-derivative of the logarithm of the theta function. Two separate forms of these four nonlinear ‘‘residual’’ equations are given. One is obtained from the Fourier series of the theta function and is useful for small wave amplitude. The other is based on the Gaussian series of the theta function and is highly efficient in the large amplitude regime where the double cnoidal wave is the sum of two solitary waves. Both sets of residual equations can be solved via perturbation theory and results are given to fourth order in the Fourier case and second order in the Gaussian case. The Gaussian-based perturbation series has the remarkable property that it converges more and more rapidly as the wave amplitude increases; the zeroth-order solution is the familiar double solitary wave. Numerical comparisons show that the two complementary perturbation series give accurate results in all the important regions of parameter space. (The ‘‘unimportant’’ regions are those in which the double cnoidal wave is an ordinary cnoidal wave subject to a very weak perturbation.) This is turn implies that even for moderate wave amplitude where the nonlinear interactions are not weak, and yet the solitary wave peaks are not well separated, at least to the eye, it is still qualitatively legitimate to describe the double cnoidal wave as either the sum of two sine waves or of two solitary waves of different heights.

Tensor formalism in anharmonic calculations

A new method is presented to compute cartesian tensors in the expansion of curvilinear internal coordinates. Second- and higher-order coefficients are related to the metrics of the space of displacements. Components of the metric tensor are taken from existing tables of inverse kinetic energy matrix elements or, when rotations are involved, derived from general invariance conditions of scalars within a molecule. This leads to a tensor formalism particularly convenient in dealing with curvilinear coordinates in anharmonic calculations of vibrational frequencies. Formulae are given for elements of the potential energy matrix, related to quadratic and cubic force constants in terms of Christoffel symbols. The latter quantities are also used in the expansion of redundancy relations, with explicit coefficients given up to the third order.

Series expansions of means

A mean M( u, v) is defined to be a homogeneous symmetric function of two positive real variables satisfying min( u, v) ⩽ M( u, v) ⩽ max( u, v) for all u and v. Setting M( u, v) = uM(1, vu −1) = uM(1, 1 − t), 0 ⩽ t < 1, we determine power series expansions in t of various generalized means, including μ p(1, 1 − t) = [ 1 2 + (1 − t) p 2 ] 1 p , m p(u, v) = [ (v p + 1 − u p + 1) (v − u)(p + 1) ] 1 p (Stolarsky's mean), M p(u, v) = (u p + v p) (u p− 1 + v p − 1) (Lehmer's mean), E(r, s; u, v) = [ r(u s − v s) s(u r − v r) ] 1 (s − r) (Leach and Sholander's mean), and G(r, s; u, v) = [ (u s + v s) (u r + v r) ] 1 (s − r) (Gini's mean). The explicit power series coefficients and recurrence relations for these coefficients are found. Finally, applications are shown by proving a theorem that generalizes one due to Lehmer.

Three‐dimensional tracer model study of atmospheric CO2: Response to seasonal exchanges with the terrestrial biosphere

A three‐dimensional tracer transport model is used to investigate the annual cycle of atmospheric CO2 concentration produced by seasonal exchanges with the terrestrial biosphere. The tracer model uses winds generated by a global general circulation model to advect and convect CO2; no explicit diffusion coefficients are employed. A biospheric exchange function constructed from a map of net primary productivity, and Azevedo's (1982) seasonality of CO2 uptake and release closely simulates the annual cycles at coastal stations. The results show that zonal homogeneity in surface CO2 concentrations can never be achieved at mid‐latitudes where the time scale for zonal mixing is longer than the time scale for biospheric exchange. Analysis of the zonal mean balance in the lower troposphere reveals that atmospheric transport processes may alter the CO2 response to local biospheric exchanges by 50% or more. Hence year‐to‐year variation of the annual CO2 cycle may result from the natural variability of the atmospheric circulation as well as from changes in the sources and sinks.

Bumpy torus transport in the low collision frequency limit

A variational formulation of transport for a closed field line, large aspect ratio bumpy torus in the low collision frequency limit is presented. The relatively large radial excursions made by those particles with canceling magnetic and electric poloidal drifts requires generalization of previous neoclassical techniques. General expressions for the transport coefficients of both charge species are given. Explicit numerical coefficients are evaluated under certain simplifying assumptions for both the banana and nonbanana species.

A practical method of inverting the metric matrices of crystals

When orthogonalising the localised basis functions in crystals with Lowdin's symmetrical method (1956), the inverse of the metric matrix plays a crucial role. A practical method of inverting the general metric matrices of crystals is evaluated. The procedure is based on the cluster method, introduced for spherically symmetric states by Lundqvist and Froman (1950), and it exploits completely the rotational symmetry of the crystal. The general equations for solving the elements of the inverse metric matrix are derived, and explicit coefficients are given for s and p states. Calculations have been performed for some alkali halides and magnesium oxide, and numerical results for sodium fluoride are given and considered in some detail. The series expansion approximation for the inverse metric matrix elements is found to be reasonable for alkali halides but not at all suitable for magnesium oxide. However, 'exact' values for inverse metric matrix elements should be used in all accurate theoretical calculations.

A Numerical Coning Model

Abstract The numerical simulation of coning behavior bas been one of the most difficult applications of numerical analysis techniques. Coning simulations have generally exhibited severe saturation instabilities in the vicinity of the well unless time-step sizes were severely restricted. The instabilities were a result of using mobilities based on saturations existing at the beginning of the time step. The time-step size limitation, usually the order of a few minutes, resulted in an excessive amount of computer time required to simulate coning behavior. This paper presents a numerical coning model that exhibits stable saturation and production behavior during cone formation and after breakthrough. Time-step sizes a factor of 100 to 1,000 times as large as those previously possible may be used in the simulation. To ensure stability, both production rates and mobilities are extrapolated production rates and mobilities are extrapolated implicitly to the new time level. The finite-difference equations used in the model are presented together with the technique for incorporating the updated mobilities and rates. Example calculations which indicate the magnitude of the time-truncation errors are included. Various factors which affect coning behavior are discussed. Introduction The usual formulation of numerical simulation models for multiphase flow involves the evaluation of flow coefficient terms at the beginning of a time step and assumes that these terms do not change over the time step. These assumptions are valid only if the values of pressure and saturation in the system do not change significantly over the time step. The design of a finite-difference model to evaluate coning behavior of gas or water in a single well usually results in a model which uses radial coordinates. A two-dimensional single-well model is illustrated in Fig. 1. This type of model will often produce finite-difference blocks with pore volumes less than 1 bbl near the wellbore while producing large blocks with pore volumes greater producing large blocks with pore volumes greater than 1 million bbl near the external radius. If one chooses to use a reasonable time-step size of, say, 1 to 10 days, then normal well rates would result in a flow of several hundred pore volumes per time step through blocks near the wellbore. Therefore the assumption that saturations remain constant, for the purpose of coefficient evaluation, is not valid. Welge and Weber presented a paper on water coning which recognized the limitation of using explicit coefficients and applied an arbitrary limitation on the maximum saturation change over a time step. While this method is workable for a certain class of problems, it is not rigorous and is not generally applicable. In 1968, Coats proposed a method to solve the gas percolation problem which is similar in that it also results from explicit mobilities. This proposal involved adjusting the relative permeability to gas at the beginning of the time step so that an individual block would not be over-depleted of gas during a time step. This method is not conveniently extended to two dimensions nor to coning problems where a block is voided many times during a time step. Blair and Weinaug explored the problems resulting from explicitly determined coefficients and formulated a coning model with implicit mobilities and a solution technique utilizing Newtonian iteration. While this method is rigorous, achieving convergence on certain problems is difficult and, in many cases, time-step size is still severely restricted. In addition to the problems resulting from explicit flow-equation coefficients in coning models, the specification of rates requires attention to ensure that the saturations remain stable in the vicinity of the producing block. SPEJ P. 418