General Moment Research Articles

Undrained failure envelope for skirted foundations under general loading

Three-dimensional failure envelopes can be used to define the bearing capacity and proximity to failure of shallow foundations under general vertical, horizontal and moment loading (V, H, M/B). Different structures, and different load cases for the same structure, cover varying domains of (±V, ±H, ±M/B) load space; therefore, a fully encompassing failure envelope in (V, H, M/B) load space is a useful tool to define ultimate limit states for design. In this technical note, a closed-form expression is proposed that enables prediction of undrained bearing capacity of skirted foundations under general in-plane loading, valid for a range of embedment ratios and soil shear strength heterogeneities.

Moment-Tensor Determination by Nonlinear Inversion of Amplitudes

We propose an inversion method of the direct wave amplitudes P, SV, and SH to determine the general moment tensor (MT) and estimate the associated uncertainties. This method is a generalization of the one we developed to determine the double-couple (DC) or fault-plane solution from sparse observations in reservoir contexts (Godano et al., 2009). Like the previous one, it is based on a simulated annealing inversion algorithm. First, we test the reliability of this new method on synthetic data. The inversion allows retrieval of the expected moment tensor, but the analysis of the uncertainty associated with the solution shows that the resolution of the inversion is sensitive not only to the direct wave amplitudes but also to uncertainties in the velocity model and the event location. Next, we test the method on four induced microearthquakes recorded by the Soultz-sous-Forets hot-dry-rock (HDR) reservoir network. The inversion correctly converges for three events, and the obtained MTs display a dominant DC part (70 to 90%). The uncertainty associated with the non-DC part is relatively high. This prevents us from determining if the small non-DC part is significant or an artifact related to noise in the data and/or uncertainties in the velocity model and at the event location.

General Moments of the Inverse Real Wishart Distribution and Orthogonal Weingarten Functions

We study a random positive definite symmetric matrix distributed according to a real Wishart distribution. We compute general moments of the random matrix and of its inverse explicitly. To do so, we employ the orthogonal Weingarten function, which was recently introduced in the study of Haar-distributed orthogonal matrices. As applications, we give formulas for moments of traces of a Wishart matrix and its inverse.

Modelling deformation due to a pressurized ellipsoidal cavity, with reference to the Campi Flegrei caldera, Italy

[1] Magmatic unrest can be successfully monitored and studied from modeling of the induced surface deformation; one limiting factor however is the small number of available magmatic source models. Here we have obtained expressions (quadrupole approximation) for displacements and stresses from the inflation of any pressurized triaxial ellipsoid in an infinite elastic medium. The expressions can be evaluated by combining the effects of seven suitable point sources and are approximately valid also for a heterogeneous half-space. Till now, the only available (approximate or exact) expressions for finite expansion sources referred to spheres, prolate spheroids, and horizontal circular cracks embedded in a homogeneous half-space. Our approach allows to model also oblate spheroids and non-axisymmetric sources, whose effects were previously estimable only in the far field through a moment tensor representation. We also show that when the deformation source is a vertically flattened ellipsoid, the inversion of superficial displacements using a general moment tensor may lead to wrong physical models of the deformation source itself. This may be the case for the Campi Flegrei caldera, as well as for other calderas.

A MULTIVARIATE WEIBULL DISTRIBUTION

A multivariate survival function of Weibull Distribution is developed by expanding the theorem by Lu and Bhattacharyya. From the survival function, the probability density function, the cumulative probability function, the determinant of the Jacobian Matrix, and the general moment are derived.

The Smallest Eigenvalue of Hankel Matrices

Let H_N=(s_{n+m}),n,m\le N denote the Hankel matrix of moments of a positive measure with moments of any order. We study the large N behaviour of the smallest eigenvalue lambda_N of H_N. It is proved that lambda_N has exponential decay to zero for any measure with compact support. For general determinate moment problems the decay to 0 of lambda_N can be arbitrarily slow or arbitrarily fast. In the indeterminate case, where lambda_N is known to be bounded below by a positive constant, we prove that the limit of the n'th smallest eigenvalue of H_N for N tending to infinity tends rapidly to infinity with n. The special case of the Stieltjes-Wigert polynomials is discussed.

Critical Evaluation of Equivalent Moment Factor Procedures for Laterally Unsupported Beams

This paper compares the numerous approaches to determining equivalent moment factors used in evaluating the elastic critical moment of laterally unsupported beams for a wide variety of moment distributions. The investigation revealed that the procedure used currently in the Canadian design standard produces unacceptable results for the majority of the common bending moment distributions considered. Large abrupt changes in Cb values with only slight changes in the shape of the moment diagram were observed in six out of the 12 moment distribution comparisons, which contributes to the overall poor performance of the procedure.The study also revealed drawbacks inherent in other methods. Overall, the quarter-point moment equations developed for general moment distributions capture the trends of the numerical data reasonably well. However, for example, the evaluations show that the 2005 AISC equation produces non-conservative results in some situations, while the British equation, although generally conservative, produces comparatively less accurate results. Other equations examined capture the trends of the numerical data more consistently by implementing a square root format in the quarter-point moment method. However, they produce results that exceed the numerical data in several cases, implying that they are too aggressive for design purposes. To capture the best features of the various methods investigated, yet improve the overall suitability for general design purposes, a modified quarter-point moment equation using the square root format is proposed. Not only does it simulate the trends of the numerical solutions closely, but it also produces reasonable and conservative equivalent moment factors, even in cases where other methods do not. Like all quarter-point moment methods, the proposed equation does not produce good results in some situations where concentrated moments are applied. Nevertheless, it is believed to be appropriate for the vast majority of typical design cases.

Square-Full Divisors of Square-Full Integers

AbstractWe consider some general real moments associated with square-full divisors of square-full numbers and compute the contribution to these moments given by the square divisors.

Portfolio selection in multidimensional general and partial moment space

This paper develops a general approach for the single period portfolio optimization problem in a multidimensional general and partial moment space. A shortage function is defined that looks for possible increases in odd moments and decreases in even moments. A main result is that this shortage function ensures sufficient conditions for global optimality. It also forms a natural basis for developing tests on the influence of additional moments. Furthermore, a link is made with an approximation of an arbitrary order of a general indirect utility function. This non-parametric efficiency measurement framework permits to differentiate mainly between portfolio efficiency and allocative efficiency. Finally, information can, in principle, be inferred about the revealed risk aversion, prudence, temperance and other higher-order risk characteristics of investors.

A Stochastic Controller Design Using an Approximate Solution of FPK Equation

The Fokker-Planck-Kolmogorov (FPK) equation is a powerful tool with Markovian response for analyzing dynamic systems. In this study we consider the design of a stochastic controller using FPK for a system subjected to a wide band random disturbance with constant power spectral density. The time-domain dynamic system equation is transformed into a general dynamic statistical moment equation in the stochastic domain via the FPK equation. The system transformation cannot be realized in real time. In order to improve stochastic controller design in real time, a new method of designing a stochastic controller and its implementation are proposed. The practical efficiency of the proposed method using the approximate probability density function is verified via numerical simulation.

General moment theorems for nondistinct unrestricted partitions

A well-known result from Hardy and Ramanujan [“Aysmptotic formulae in combinatory analysis,” Proc. Lond. Math. Soc. 17, 75 (1918)] gives an asymptotic expression for the number of possible ways to express an integer as the sum of smaller integers. In this vein, we consider the general partitioning problem of writing an integer n as a sum of summands from a given sequence Λ of nondecreasing integers. Under suitable assumptions on the sequence Λ, we obtain results using associated zeta functions and saddle-point techniques. We also calculate higher moments of the sequence Λ as well as the expected number of summands. Applications are made to various sequences, including those of Barnes and Epstein types. These results are of potential interest in statistical mechanics in the context of Bose–Einstein condensation.

Landau collision operators and general moment equations for an electron-ion plasma

The general moment equations for an electron-ion plasma are established. The distribution functions for electrons and ions are expanded in terms of orthogonal polynomials of random velocity variables in contrast to the total velocity variables [J.-Y. Ji and E. D. Held, Phys. Plasmas 13, 102103 (2006)]. The moments of the streaming part of the kinetic equation are explicitly written with simple formulas. A simple version of the exact linearized Coulomb collision integrals is presented for like species. The electron-ion and ion-electron operators that conserve momentum and energy are also calculated with a small mass-ratio approximation. It is shown in the relaxation theory that the Lorentz operator, as a replacement of the like-species operator, is acceptable only for the high-order harmonic moments.

Razumikhin-Type Theorems on Stability of Neutral Stochastic Functional Differential Equations

Recently, Mao (1995) initiated the study of exponential stability of neutral stochastic functional differential equations (NSFDEs) and employed the Razumikhin techniques to investigated the stability problem of NSFDEs. However, the importance of general asymptotic stability has not been considered. This paper studies Razumikhin-type theorems on general pth moment asymptotic stability of NSFDEs. The theorems established in this paper work for some cases to which the existing results cannot be applied.

Tables of crystallographic properties of magnetic space groups

Tables of crystallographic properties of the reduced magnetic superfamilies of space groups, i.e. the 7 one-dimensional, 80 two-dimensional and 1651 three-dimensional group types, commonly referred to as magnetic space groups, are presented. The content and format are similar to that of non-magnetic space groups and subperiodic groups given in International Tables for Crystallography. Additional content for each representative group of each magnetic space-group type includes a diagram of general positions with corresponding general magnetic moments, Seitz notation used as a second notation for symmetry operations, and general and special positions listed with the components of the corresponding magnetic moments allowed by symmetry.

Semiparametric efficiency in GMM models with auxiliary data

We study semiparametric efficiency bounds and efficient estimation of parameters defined through general moment restrictions with missing data. Identification relies on auxiliary data containing information about the distribution of the missing variables conditional on proxy variables that are observed in both the primary and the auxiliary database, when such distribution is common to the two data sets. The auxiliary sample can be independent of the primary sample, or can be a subset of it. For both cases, we derive bounds when the probability of missing data given the proxy variables is unknown, or known, or belongs to a correctly specified parametric family. We find that the conditional probability is not ancillary when the two samples are independent. For all cases, we discuss efficient semiparametric estimators. An estimator based on a conditional expectation projection is shown to require milder regularity conditions than one based on inverse probability weighting.

Strong convergence of pairwise NQD random sequences

Strong limit theory is one of the most important problems in probability theory. Some results on the convergence of pairwise NQD random sequences have been presented. This paper further analyzes the strong convergence of pairwise NQD sequences and generalizes partial results of Wu [Q.Y. Wu, Convergence properties of pairwise NQD random sequences, Acta Math. Sinica 45 (3) (2002) 617–624 (in Chinese)]. Since no general moment inequalities are given as so far, we avoid this problem and obtain a class of strong limit theorem for NQD sequences and some corresponding conclusions by use of truncation methods and generalized three series theorem, which are the supplements to the previous fruits.

Moment inequalities for spatial processes

This paper establishes a general moment inequality for spatial processes satisfying the α -mixing condition [cf., Tran, 1990. Kernel density estimation on random fields. J. Multivariate Analy. 34, 37–53]. Such a general moment inequality is a nontrivial extension of the corresponding result established in Cox and Kim [1995. Moment bounds for mixing random variables useful in nonparametric function estimation. Stochastic Process. Appl. 56, 151–158] for the time series case. As is the case for the Cox–Kim inequality for nonparametric estimation of time series, the new inequality is useful in nonparametric kernel estimation of spatial processes.

On the general relativistic symmetric hyperbolic moment system for matter and radiation: local rotational symmetry

For the one-dimensional, locally rotationally symmetric case, we employ the relativistic Boltzmann equations and the moment methods to derive a 6D causal evolution system of first-order symmetric hyperbolic form for a fluid-radiation mixture, i.e., a relativistic gas composed of material particles and photons. The material medium is assumed to be a nonbarotropic perfect gas and the evolution equations for this medium are obtained by using the local Jüttner equilibrium distribution function as seen by an observer with arbitrary 4-velocity. In order to derive the evolution equations for the photon gas, we develop the modified Grad-type method which consists of expanding the phase density (i.e., the number density of photons) about a local Planck equilibrium distribution function and including the trace-free anisotropic pressure in the expansion. In our approach, it is significant that the matter fluid's peculiar velocity and the radiative heat flux (relative to an arbitrary 4-velocity vector field) are incorporated into the model in a non-perturbative manner, thereby allowing arbitrarily large values for the two nonvanishing components of these dynamical variables. The additional difference about the treatment here as opposed to other treatments is that we do not neglect terms in the evolution equations which involve small first-order quantities multiplying the spatial frame derivatives of hydrodynamical variables.

Spheroidal oscillations of the Earth stimulated by the Sumatra-Andaman earthquake with CDSN data

The 0S2∼0S54 spheroidal modes of Earth’s free oscillations, triggered by the great Sumatra-Andaman earthquake of 26 December 2004 are retrieved from VHZ data recorded by seven upgraded stations of China Digital Seismograph Network (CDSN). We compare these spheroidal modes with theoretical free oscillation spectra calculated from the Preliminary Reference Earth Model (PREM) and find a coincidence between their periods. Spectral splitting phenomenon is observed obviously in 0S2, 0S3, 0S4, 2S1 and 1S2 free oscillation modes. What is most noticeable is that the oscillation mode 2S1 is reported for the second time (the first time by Rosat et al) without any data stacking. We simulated the split singlet of 0S2 mode on seven CDSN stations based on general focal mechanism and seismic moment of the earthquake. The result shows that seismic moment of the earthquake can reach 1023 N·m. We also find that the recording of Earth’s free oscillations carries abundant information of source mechanism and earthquake location, which is applicable to the detailed study of source rupture parameters.

General Order Selection Combining for Nakagami and Weibull Fading Channels

In this paper, some analytical results for general order selection (GOS) over independent but not necessarily identically distributed (i.n.d.) Weibull and Nakagami fading channels are presented. The Weibull fading parameters are assumed to be equal whereas the Nakagami fading parameters are assumed to be integer-valued. It is shown that the pdf of the q-th order statistic can be expressed as a linear combination of Weibull and Nakagami pdf's respectively. Closed-form expressions for the moment generating functions and general moments are derived. In addition, exact closed-form expressions for the symbol error rate are obtained for a number of modulation schemes. Numerical results show that for the same average channel gains, the performance on i.n.d. channels may be better or worse than on i.i.d. channels.