Polynomial Moments Research Articles

Search for theZ(4430)−atBABAR

We report the results of a search for $Z(4430)^-$ decay to $J/\psi\pi^-$ or $\psi(2S)\pi^-$ in $B^{-,0}\to \jpsi\pi^- K^{0,+}$ and $B^{-,0}\to \psitwos \pi^- K^{0,+}$ decays. The data were collected with the BABAR detector at the SLAC PEP-II asymmetric-energy $e^+e^-$ collider operating at center of mass energy 10.58 \gev, and the sample corresponds to an integrated luminosity of 413 fb$^{-1}$. Each $K\pi$ mass distribution exhibits clear $K^{\ast}(892)$ and $K^{\ast}_2(1430)$ signals, and the efficiency-corrected spectrum is well-described by a superposition of the associated Breit-Wigner intensity distributions, together with an $S$-wave contribution obtained from the LASS $I=1/2$ $K\pi$ scattering amplitude measurements. Each $K\pi$ angular distribution varies significantly in structure with $K\pi$ mass, and is represented in terms of low-order Legendre polynomial moments. We find that each $J/\psi\pi$ or $\psi(2S)\pi$ mass distribution is well-described by the reflection of the measured $K\pi$ mass and angular distribution structures. We see no significant evidence for a $Z(4430)^-$ signal for any of the processes investigated, neither in the total $J/\psi\pi$ or $\psi(2S)\pi$ mass distribution, nor in the corresponding distributions for the regions of $K\pi$ mass for which observation of the $Z(4430)^-$ signal was reported. We obtain branching fraction upper limits ${\cal{B}}(B^-\to Z^-\bar{K^0}, Z^-\to J/\psi\pi^-)<1.5\times 10^{-5}$, ${\cal{B}}(B^0\to Z^-K^+, Z^-\to J/\psi\pi^-)<0.4\times 10^{-5}$, ${\cal{B}}(B^-\to Z^-\bar{K^0}, Z^-\to \psi(2S)\pi^-)<4.7\times 10^{-5}$, and ${\cal{B}}(B^0\to Z^-K^+, Z^-\to \psi(2S)\pi^-)<3.1\times 10^{-5}$ at 95% confidence level, where the $Z(4430)^-$ mass and width have been fixed to the reported central values.

Queueing analysis for the OFDMA downlink: Throughput regions, delay and exponential backlog bounds

While multiuser scheduling strategies have been intensively studied in the context of OFDMA downlink recently, the resulting throughput and delay performance affecting strongly end-to-end performance of wireless communication systems can still be evaluated by simulations only. In this paper we approach this problem and consider bounds for delay and queue backlog for a large class of scheduling policies. Adopting a general state space Markov chain model the concept of policy-specific throughput regions is introduced. Then, under the regime of the policy, a recursive formula for calculating all polynomial moments of the queue backlog is derived. Moreover, it is shown that even exponential decay of the tail distribution can be obtained under proper circumstances. Based on these results, upper bounds on the buffer overflow probability are derived giving insights for practical buffer dimensioning problems in UMTS LTE systems.

A New Approach to Recovery of Discontinuous Galerkin

A new recovery operator P : Q disc (T ) ! Q disc+1(M) for discontinuous Galerkin is derived. It is based on the idea of projecting a discontinuous, piecewise polynomial solution on a given mesh T into a higher order polynomial space on a macro mesh M. In order to do so, we deflne local degrees of freedom using polynomial moments and provide global degrees of freedom on the macro mesh. We prove consistency with respect to the local L2-projection, stability results in several norms and optimal anisotropic error estimates. As an example, we apply this new recovery technique to a stabilized solution of a singularly perturbed convection-difiusion problem using bilinear elements.

Discrete polynomial moments for real-time geometric surface inspection

A thorough analysis of discrete polynomial moments and their suitability for application to geometric surface inspection is presented. A new approach is taken to the analysis based on matrix algebra, revealing some formerly unknown fundamental properties. It is proven that there is one and only one unitary polynomial basis that is complete, i.e., the polynomial basis for a Chebychev system. Furthermore, it is proven that the errors in the computation of moments are almost exclusively associated with the application of the recurrence relationship, and it is shown that QR decomposition can be used to eliminate the systematic propagation of errors. It is also shown that QR decomposition produces a truly orthogonal basis set despite the presence of stochastic errors. Fourier analysis is applied to the polynomial bases to determine the spectral distribution of the numerical errors. The new unitary basis offers almost perfect numerical behavior, enabling the modeling of larger images with higher-degree polynomials for the first time. The application of a unitary polynomial basis eliminates the need to compute pseudo-inverses. This improvement in numerical efficiency enables real-time modeling of surfaces in industrial surface inspection. Two applications in industrial quality control via artificial vision are demonstrated.

Aggregation in a one-dimensional stochastic gas model with finite polynomial moments of particle speeds

We consider a one-dimensional system of auto-gravitating sticky particles with random initial speeds and describe the process of aggregation in terms of the largest cluster size Ln at any fixed time prior to the critical time. We study the asymptotic behavior of Ln for the warm gas, i.e., for a system of particles with nonzero initial speeds vi(0) = ui, where (ui) is a family of i.i.d. random variables with mean zero, unit variance, and finite pth moment E(|ui|p) < ∞, p ≥ 2, and obtain sharp lower and upper bounds for Ln(t). Bibliography: 17 titles.

Approximate moment matrix decomposition in wavelet Galerkin BEM

We present an improvement to the wavelet Galerkin BEM in [J. Tausch, A variable order wavelet method for the sparse representation of layer potentials in the non-standard form, J. Numer. Math. 12 (3) (2004) 233–254]. In the non-standard form representation of integral operators the number of wavelets in every level determines the efficiency of the underlying Galerkin scheme. In order to increase this number, the partial singular value decomposition (PSVD) of the moment matrix is employed. Since the resulting wavelets do not exactly satisfy the conventional vanishing moment condition, i.e. a certain number of polynomial moments of the wavelets are zero, we introduce the concept of “quasi-vanishing moment”. For integral equations of the second kind we prove that the asymptotic convergence of the full Galerkin scheme can be retained by controlling the cut-off for singular values in the PSVD appropriately. Numerical results show that our improvement can largely decrease the time and memory usage while preserving the optimal convergence rate of the method.

Cross sections and beam asymmetries fore→p→enπ+in the nucleon resonance region for1.7⩽Q2⩽4.5GeV2

The exclusive electroproduction process e→p→e'nπ+ was measured in the range of the photon virtuality Q2=1.7−4.5 GeV2, and the invariant mass range for the nπ+ system of W=1.15−1.7 GeV using the CEBAF Large Acceptance Spectrometer. For the first time, these kinematics are probed in exclusive π+ production from protons with nearly full coverage in the azimuthal and polar angles of the nπ+ center-of-mass system. The nπ+ channel has particular sensitivity to the isospin ½ excited nucleon states, and together with the pπ0 final state will serve to determine the transition form factors of a large number of resonances. The largest discrepancy between these results and present modes was seen in the σLT' structure function. In this experiment, 31,295 cross section and 4,184 asymmetry data points were measured. Because of the large volume of data, only a reduced set of structure functions and Legendre polynomial moments can be presented that are obtained in model-independent fits to the differential cross sections.18 MoreReceived 5 September 2007DOI:https://doi.org/10.1103/PhysRevC.77.015208©2008 American Physical Society

Legendre modified moments for Euler's constant

Polynomial moments are often used for the computation of Gauss quadrature to stabilize the numerical calculation of the orthogonal polynomials, see [W. Gautschi, Computational aspects of orthogonal polynomials, in: P. Nevai (Ed.), Orthogonal Polynomials-Theory and Practice, NATO ASI Series, Series C: Mathematical and Physical Sciences, vol. 294. Kluwer, Dordrecht, 1990, pp. 181–216 [6]; W. Gautschi, On the sensitivity of orthogonal polynomials to perturbations in the moments, Numer. Math. 48(4) (1986) 369–382 [5]; W. Gautschi, On generating orthogonal polynomials, SIAM J. Sci. Statist. Comput. 3(3) (1982) 289–317 [4]] or numerical resolution of linear systems [C. Brezinski, Padé-type approximation and general orthogonal polynomials, ISNM, vol. 50, Basel, Boston, Stuttgart, Birkhäuser, 1980 [3]]. These modified moments can also be used to accelerate the convergence of sequences to a real or complex numbers if the error satisfies some properties as done in [C. Brezinski, Accélération de la convergence en analyse numérique, Lecture Notes in Mathematics, vol. 584. Springer, Berlin, New York, 1977; M. Prévost, Padé-type approximants with orthogonal generating polynomials, J. Comput. Appl. Math. 9(4) (1983) 333–346]. In this paper, we use Legendre modified moments to accelerate the convergence of the sequence H n - log ( n + 1 ) to the Euler's constant γ . A formula for the error is given. It is proved that it is a totally monotonic sequence. At last, we give applications to the arithmetic property of γ .

Higher-order macroscopic measures

We present generalizations of Hill's classical results concerned with the macroscopic strain and stress measures. Generalizations involve polynomial boundary conditions and polynomial moments of the microscopic fields. It is shown that for higher-order polynomials certain boundary conditions and moments should be excluded from considerations in order to guarantee unique relationships between boundary data and macroscopic measures. Particularly simple relationships are obtained for spherical specimens, for which higher-order macroscopic measures are defined in terms of spherical harmonics. Also it is demonstrated that higher-order macroscopic measures and constitutive equations can be useful in multi-scale analysis of problems formulated in terms of integral equations.

A Markov chain model of a polling system with parameter regeneration

We study a model of a polling system, that is, a collection of d queues with a single server that switches from queue to queue. The service time distribution and arrival rates change randomly every time a queue is emptied. This model is mapped to a mathematically equivalent model of a random walk with random choice of transition probabilities, a model which is of independent interest. All our results are obtained using methods from the constructive theory of Markov chains. We determine conditions for the existence of polynomial moments of hitting times for the random walk. An unusual phenomenon of thickness of the region of null recurrence for both the random walk and the queueing model is also proved.

Long time asymptotics for constrained diffusions in polyhedral domains

We study long time asymptotic properties of constrained diffusions that arise in the heavy traffic analysis of multiclass queueing networks. We first consider the classical diffusion model with constant coefficients, namely a semimartingale reflecting Brownian motion (SRBM) in a d -dimensional positive orthant. Under a natural stability condition on a related deterministic dynamical system [P. Dupuis, R.J. Williams, Lyapunov functions for semimartingale reflecting brownian motions, Annals of Probability 22 (2) (1994) 680–702] showed that an SRBM is ergodic. We strengthen this result by establishing geometric ergodicity for the process. As consequences of geometric ergodicity we obtain finiteness of the moment generating function of the invariant measure in a neighborhood of zero, uniform time estimates for polynomial moments of all orders, and functional central limit results. Similar long time properties are obtained for a broad family of constrained diffusion models with state dependent coefficients under a natural condition on the drift vector field. Such models arise from heavy traffic analysis of queueing networks with state dependent arrival and service rates.

Finite-Moment Tensor of the 3 September 2002 Yorba Linda Earthquake

We have developed procedures for inverting broadband waveforms for the finite-moment tensors (fmts) of regional earthquakes. The fmt is defined in terms of second-order polynomial moments of the source space–time function; it removes the fault-plane ambiguity of the centroid moment tensor (cmt) and yields several additional parameters of seismological interest: the characteristic length Lc , width Wc , and duration Tc of the faulting, as well as the directivity vector v d of the fault slip. Following McGuire et al. (2001), we represent the observed waveform relative to the synthetic in terms of two frequency-dependent differential times, a phase delay δτp ( ω ) and an amplitude-reduction time δτq ( ω ), which we measure using the generalized seismological data functional (gsdf) method (Gee and Jordan, 1992). We numerically calculate the fmt partial derivatives, which allows us to use synthetics computed by using any forward-modeling tools. We have tested our methodology on Southern California Seismic Network (scsn) recordings of the 03 September 2002 Yorba Linda earthquake ( M W 4.3). Using 1D synthetic Green’s functions, we determined the cmt and resolved fault-plane ambiguity. To resolve the details of source finiteness, we employed a joint-inversion technique that recovers the cmt parameters of the aftershocks and the cmt and fmt parameters of the mainshock. The joint system of equations relating the data to the source parameters of the mainshock–aftershock cluster is denuisanced for path anomalies in both observables; this projection operation effectively corrects the mainshock data for path-related anomalies in a way similar to, but more flexible than, empirical Green’s function (egf) techniques. Our results indicate that the Yorba Linda rupture occurred as left-lateral slip on a fault-patch conjugate to the nearby, right-lateral Whittier fault. We obtained source dimensions of Lc = 0.7 ± 0.1 km, Wc = 0.4 ± 0.1 km, and Tc = 0.2 ± 0.05 sec, implying a stress drop of about 3.2 MPa, and we found a directivity of 0.8 ± 0.2, oriented up and to the northeast. The inferred fault plane is consistent with the mainshock–aftershock distribution relocated by Hauksson et al. (2002).

Necessary conditions for geometric and polynomial ergodicity of random-walk-type

We give necessary conditions for geometric and polynomial convergence rates of randomwalk- type Markov chains to stationarity in terms of existence of exponential and polynomial moments of the invariant distribution and the Markov transition kernel. These results complement the use of Foster-Lyapunov drift conditions for establishing geometric and polynomial ergodicity. For polynomially ergodic Markov chains, the results allow us to derive exact rates of convergence and exact relations between the moments of the invariant distribution and the Markov transition kernel. In an application to Markov chain Monte Carlo we derive tight rates of convergence for symmetric random walk Metropolis.

Zonal function network frames on the sphere

We introduce a class of zonal function network frames suitable for analyzing data collected at scattered sites on the surface of the unit sphere of a Euclidean space. Our frames consist of zonal function networks and are well localized. The frames belonging to higher and higher scale wavelet spaces have more and more vanishing polynomial moments. The main technique is applicable in the general setting of separable Hilbert spaces, in which context, we study the construction of new frames by perturbing an orthonormal basis.

Convergence rates and moments of Markov chains associated with the mean of Dirichlet processes

We give necessary and sufficient conditions for geometric and polynomial ergodicity of a Markov chain on the real line with invariant distribution M α equal to the distribution of the mean of a Dirichlet process with parameter α. This extends the applicability of a recent MCMC method for sampling from M α . We show that the existence of polynomial moments of α is necessary and sufficient for geometric ergodicity, while logarithmic moments of α are necessary and sufficient for polynomial ergodicity. As corollaries it is shown that α and M α have polynomial moments of the same order, while the order of the logarithmic moments differ by one.

Teleseismic inversion for the second-degree moments of earthquake space-time distributions

Determining detailed descriptions of the space–time distribution of an earthquake's rupture using only teleseismic data is often a difficult and non unique process. However, a description of an earthquake that goes beyond a point source is often needed for many seismotectonic problems. We have developed a method that inverts measurements of the shifts in traveltime and amplitude of various seismic phases at global stations for the second degree polynomial moments of an event's space–time distribution. Our values for the second moments of the 1995 Mw 8.0 Jalisco, Mexico, thrust event correspond to a characteristic rupture length of 121±10 km along strike, a characteristic rupture width of 76±11 km downdip, a characteristic duration of 29±2 s, and a direction of rupture propagation along the strike of the subduction zone to the NW. For the 1995 Mw 7.2 Gulf of Aqaba earthquake, our estimates of the second moments correspond to a characteristic rupture length of 53±2 km, a characteristic duration of 12±1 s, and rupture propagation to the NE along the strike of the Dead Sea Transform. Our estimates of the first and second moments agree well with values from local seismic and geodetic networks. For both the Jalisco and Aqaba events, we are able to resolve the fault plane ambiguity associated with the events' moment tensors using our estimates of the second spatial moment.

Rupture dimensions of the 1998 Antarctic Earthquake from low‐frequency waves

We inverted frequency dependent phase and amplitude measurements from 1st orbit Rayleigh waves at global stations for the 1st and 2nd degree polynomial moments of the stress‐glut rate tensor. The higher moments of the slip‐rate distribution determine the fault plane and approximate rupture dimensions. The results show strong rupture propagation to the west with an average velocity of the instantaneous centroid of 3.6±.1 km/s. The rupture had a characteristic length of 178±46 km in the east‐west direction and a characteristic duration of 48±2 s. The results are consistent with unilateral rupture on the east‐west fault plane of the focal mechanism and rule out significant rupture on the north‐south nodal plane.

Toward Objective Selection of Representative Microscope Images

Scientists wishing to communicate the essential characteristics of a pattern (such as an immunofluorescence distribution) currently must make a subjective choice of one or two images to publish. We therefore developed methods for objectively choosing a typical image from a set, with emphasis on images from cell biology. The methods involve calculation of numerical features to describe each image, calculation of similarity between images as a distance in feature space, and ranking of images by distance from the center of the feature distribution. Two types of features were explored, image texture measures and Zernike polynomial moments, and various distance measures were utilized. Criteria for evaluating methods for assigning typicality were proposed and applied to sets of images containing more than one pattern. The results indicate the importance of using distance measures that are insensitive to the presence of outliers. For collections of images of the distributions of a lysosomal protein, a Golgi protein, and nuclear DNA, the images chosen as most typical were in good agreement with the conventional understanding of organelle morphologies. The methods described here have been implemented in a web server ( http://murphylab.web.cmu.edu/services/TypIC).

Determination of the polynomial moments of the seismic moment rate density distribution with positivity constraints

SUMMARY We present a new formulation of the inverse problem of determining the temporal and spatial power moments of the seismic moment rate density distribution, in which its positivity is enforced through a set of linear conditions. To test and demonstrate the method, we apply it to artificial data for the great 1994 deep Bolivian earthquake. We use two different kinds of faulting models to generate the artificial data. One is the Haskell-type of faulting model. The other consists of a collection of a few isolated points releasing moment on a fault, as was proposed in recent studies of this earthquake. The positions of 13 teleseismic stations for which P- and SH-wave data are actually available for this earthquake are used. The numerical experiments illustrate the importance of the positivity constraints without which incorrect solutions are obtained. We also show that the Green functions associated with the problem must be approximated with a low approximation error to obtain reliable solutions. This is achieved by using a more uniform approximation than Taylor's series. We also find that it is necessary to use relatively long-period data first to obtain the low- (0th and 1st) degree moments. Using the insight obtained into the size and duration of the process from the first-degree moments, we can decrease the integration region, substitute these low-degree moments into the problem and use higher-frequency data to find the higher-power moments, so as to obtain more reliable estimates of the spatial and temporal source dimensions. At the higher frequencies, it is necessary to divide the region in which we approximate the Green functions into small pieces and approximate the Green functions separately in each piece to achieve a low approximation error. A derivation showing that the mixed spatio-temporal moments of second degree represent the average speeds of the centroids in the corresponding direction is given.

Linear-scaling tight binding from a truncated-moment approach.

We present an approximation to the total-energy tight-binding (TB) method based on use of the kernel polynomial method within a truncated subspace. Chebyshev polynomial moments of the Hamiltonian matrix are generated in a stable and efficient manner through recursive matrix-vector multiples. To compute the energy, either the electronic density of states (DOS) or the zero-temperature Fermi function is smeared by convolution with the kernel polynomial, with Jackson damping to minimize Gibbs oscillations while maintaining the positivity of the DOS. These are shown to give approximate lower and upper bounds, respectively, on the exact TB energy, and are averaged to obtain an improved energy estimate. The scaling of the computational work is made linear in the number of atoms by truncating the moment computation at a certain range about each atom. Energy derivatives necessary for molecular dynamics are obtained via a matrix-polynomial derivative relation. The method converges to exact TB as the number of moments and the truncation range are increased. We demonstrate the convergence properties and viability of the method for materials simulations in an examination of defects in silicon. We also discuss the relative importance of truncation range versus number of moments. {copyright} {ital 1996 The American Physical Society.}