The fourth moment of Ramanujan ?-function

1983 ◽  
Vol 266 (2) ◽  
pp. 233-239 ◽  
Author(s):  
Carlos J. Moreno ◽  
Freydoon Shahidi
Keyword(s):  
Fourth Moment

scholarly journals Exact tensor closures for the three-dimensional Jeffery's equation

2011 ◽  
Vol 680 ◽  
pp. 321-335 ◽  
Author(s):  
STEPHEN MONTGOMERY-SMITH ◽  
WEI HE ◽  
DAVID A. JACK ◽  
DOUGLAS E. SMITH
Keyword(s):  
Fibre Orientation ◽  
Elliptic Integral ◽  
Moment Tensor ◽  
Three Dimensional ◽  
Exact Formula ◽  
Fourth Moment ◽  
Newtonian Flow ◽  
Curve Fit ◽  
Explicit Formulae ◽  
Jeffery’S Equation

This paper presents an exact formula for calculating the fourth-moment tensor from the second-moment tensor for the three-dimensional Jeffery's equation. Although this approach falls within the category of a moment tensor closure, it does not rely upon an approximation, either analytic or curve fit, of the fourth-moment tensor as do previous closures. This closure is orthotropic in the sense of Cintra & Tucker (J. Rheol., vol. 39, 1995, p. 1095), or equivalently, a natural closure in the sense of Verleye & Dupret (Developments in Non-Newtonian Flow, 1993, p. 139). The existence of these explicit formulae has been asserted previously, but as far as the authors know, the explicit forms have yet to be published. The formulae involve elliptic integrals, and are valid whenever fibre orientation was isotropic at some point in time. Finally, this paper presents the fast exact closure, a fast and in principle exact method for solving Jeffery's equation, which does not require approximate closures nor the elliptic integral computation.

Fourth Moment Structure of Markov Switching Multivariate GARCH Models

2019 ◽  
Author(s):  
Maddalena Cavicchioli
Keyword(s):  
Closed Form ◽  
Moving Average ◽  
Markov Switching ◽  
Computational Cost ◽  
Sufficient Conditions ◽  
Impulse Response Function ◽  
Real Data ◽  
Multivariate Garch ◽  
Fourth Moment ◽  
Spectral Density Matrix

Abstract We derive sufficient conditions for the existence of second and fourth moments of Markov switching multivariate generalized autoregressive conditional heteroscedastic processes in the general vector specification. We provide matrix expressions in closed form for such moments, which are obtained by using a Markov switching vector autoregressive moving-average representation of the initial process. These expressions are shown to be readily programmable in addition of greatly reducing the computational cost. As theoretical applications of the results, we derive the spectral density matrix of the squares and cross products, propose a new definition of multivariate kurtosis measure to recognize heavy-tailed features in financial real data, and provide a matrix expression in closed form of the impulse-response function for the volatility. An empirical example illustrates the results.

An inverse scattering method in time-domain electromagnetics

1981 ◽  
Vol 59 (10) ◽  
pp. 1348-1353
Author(s):  
Sujeet K. Chaudhuri
Keyword(s):  
Inverse Scattering ◽  
Time Domain ◽  
Random Noise ◽  
Power Series Expansion ◽  
Inverse Scattering Method ◽  
Wave Number ◽  
Moment Condition ◽  
Scattering Method ◽  
Fourth Moment ◽  
Rayleigh Coefficient

An inverse scattering model, based on the time-domain concepts of electromagnetic theory is developed. Using the first five (zeroth to fourth) moment condition integrals, the Rayleigh coefficient and the next higher order nonzero coefficient of the power series expansion in k (wave number) of the object backscattering response are recovered. The Rayleigh coefficient and the other coefficient thus recovered are used (with the ellipsoidal assumption for the object shape) to determine the dimensions and orientation of the object.Some numerical results of the application of this coefficient recovery technique to conducting ellipsoidal scatterers are presented. The performance of the software system in the presence of normally distributed random noise is also studied.

scholarly journals THE FOURTH HISTORICAL MOMENT IN SOCIAL RESEARCH ‘THE CRISIS OF REPRESENTATION’: A CHAPTER REVIEW

2021 ◽  
Vol 4 (17) ◽  
pp. 138-144
Author(s):  
Asma Shughail Aqib Al Hashimi ◽  
Adi Anuar Azmin
Keyword(s):  
Qualitative Research ◽  
Current Literature ◽  
Quantitative Research ◽  
Social Research ◽  
Fourth Moment ◽  
Historical Overview ◽  
Social Scientists ◽  
Socially Constructed ◽  
Crisis Of Representation ◽  
Historical Moment

The historical moments of qualitative research reflect socially constructed quasi-historic conventions that remain crosscut and overlapping till the present. This progressive narrative is well represented and assessed in a historical overview by Denzin & Lincoln (2018) in their book “The Sage handbook of qualitative research” in the introduction “The discipline and practice of qualitative research”. Through a chapter review, this article particularly discusses the fourth moment of Quantitative research coined as “The crisis of representation”, which is believed to be the crossroads where social scientists remain entangled between the science and humanity perspective while conducting social research in order to forward social realities. This period of confusion simultaneously forwarded the multi-paradigm (positivism, postpositivism, and interpretivism), all of which have unique characteristics that are suitable for specific research. Thus, this paper sheds light on the overview of the crisis of representation and further explains the types of crises that occurred during this historical moment, including the crisis of representation, the crisis of legitimation, and a crisis of praxis. It is expected that apart from extending current literature this paper would support social scientists for selecting appropriate methods and paradigms as well as to justify their selection.

scholarly journals Consistency of an Estimator for Change Point in Volatility of Financial Returns

2021 ◽  
Vol 13 (1) ◽  
pp. 56
Author(s):  
Josephine Njeri Ngure ◽  
Anthony Gichuhi Waititu
Keyword(s):  
Exchange Rate ◽  
Change Point ◽  
Point Estimation ◽  
Fourth Moment ◽  
Financial Returns ◽  
Test Statistic ◽  
Data Set ◽  
United States Dollar ◽  
Binary Segmentation ◽  
Non Parametric

A non parametric Auto-Regressive Conditional Heteroscedastic model for financial returns series is considered in which the conditional mean and volatility functions are estimated non-parametrically using Nadaraya Watson kernel. A test statistic for unknown abrupt change point in volatility which takes into consideration conditional heteroskedasticity, dependence, heterogeneity and the fourth moment of financial returns, since kurtosis is a function of the fourth moment is considered. The test is based on L2norm of the conditional variance functions of the squared residuals. A non-parametric change point estimator in volatility of financial returns is further obtained. The consistency of the estimator is shown theoretically and through simulation. An application of the estimator in change point estimation in volatility of United States Dollar/Kenya Shilling exchange rate returns data set is made. Through binary segmentation procedure, three change points in volatility of the exchange rate returns are estimated and further accounted for.

Reliability Analysis of Uncertain Nonlinear System Subject to Correlated Failure

2008 ◽  
Vol 44-46 ◽  
pp. 515-522
Author(s):  
X.F. Zhang ◽  
Yi Min Zhang ◽  
Xian Zhen Huang
Keyword(s):  
Reliability Analysis ◽  
System Reliability ◽  
Failure Modes ◽  
Fourth Moment ◽  
Strongly Correlated ◽  
Hysteretic Model ◽  
Analysis Method ◽  
Random Parameters ◽  
Uncertain Nonlinear System ◽  
Probability Information

On the basis of the Bouc-Wen hysteretic model, a numerical method for the reliability analysis of stochastic multi-degree-of-freedom hysteretic system with correlated failure modes is presented. Under the first passage model, considering the random caused by hysteretic loop itself, the theory of incomplete probability information and the fourth-moment technique and Gram Charlier series are employed to develop a numerical reliability analysis method systematically. The numerical example reveals that in most of cases, though system is characterized by a set of independent random parameters, the responses are strongly correlated, and correlation coefficient between the responses is fluctuated with time. The system reliability with correlated failure modes is evaluated with proposed method, and the result obtained by this method is compared well with the Monte-Carlo simulations.

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