Gamma Law Research Articles

Subordination on ultrametric spaces

Using the formalism of continuous-time random walks (CTRW) the authors discuss particle diffusion on ultrametric spaces (UMS). For broad waiting time distributions psi (t) approximately t- alpha -1 the temporal ( alpha ) and the energetic ( gamma ) parameters combine multiplicatively (subordinate): i.e. S(t), the mean number of UMS sites visited, follows a S(t) approximately talpha gamma law. This mirrors previous findings for CTRW on fractals.

On the Generalized Noncentral Chi-Squared Distribution Induced by an Elliptical Gamma Law

On the Generalized Noncentral Chi-Squared Distribution Induced by an Elliptical Gamma Law

On the generalized noncentral chi-squared distribution induced by an elliptical gamma law

SUMMARY Certain important statistical inference and classification problems lead to the consideration of quadratic forms in rotationally invariant random variables. In this paper noncentral probabilities under a multivariate elliptical gamma distribution are studied. The density and cumulative distribution function of the squared noncentral radii are expressed in a Laguerre series form and certain recurrences are derived. Finally, two numerical approximations are discussed.

On a characterization of gamma distributions via exponential mixtures

A new and simpler proof of Morrison's result that within exponential mixtures only IFR gamma mixing produces linearly increasing mean residual life functions is given. A parallel and new characterization of the DFR gamma laws follows as a consequence. The method of proof used suggests a general result on the infinite divisibility of the mixing distributions in exponential mixtures.

On a characterization of gamma distributions via exponential mixtures

A new and simpler proof of Morrison's result that within exponential mixtures only IFR gamma mixing produces linearly increasing mean residual life functions is given. A parallel and new characterization of the DFR gamma laws follows as a consequence. The method of proof used suggests a general result on the infinite divisibility of the mixing distributions in exponential mixtures.

Relation between the Gruneisen ratio and the pressure dependence of poisson's ratio for metals

From the material above and reading of the literature, one should remember the following:The study of materials under shock loading requires a change in experimental configuration from rod geometry to plate geometry. In rod geometry (uniaxial stress) the stress intensity is limited by plasticity. In plate geometry (uniaxial strain), the stress intensity is governed by bulk properties—material failure determines the limiting pressure.Shock waves occur when a material is stressed far beyond its elastic limit by a pressure disturbance.Because wave velocity increases with pressure above the elastic limit, a smooth pressure disturbance “shocks up.”Because the rarefaction wave moving into the shocked region travels faster than the shock front, the shock is attenuated from behind.The peak pressure for propagating shocks is given by P=ρUu.(3.47) where ρ is the material's density; U, the shock velocity and u, the particle velocity.Because the pressure generated by shock wave propagation can exceed material strength by several orders of magnitude, we can view the early stages of materials response as hydrodynamic. Inertia governs the process. Strength effects appear in the late stages of the event.The conservation of mass, momentum and energy equations for transition of unshocked material to a shocked state are known collectively as the Rankine-Hugoniot jump conditions.The Hugoniot curve is determined from a number of plate impact experiments varying the velocity of the flyer plate for each point on the curve. The loading path is not the curve but the Rayleigh line. The unloading path is the isentrope of the material. For all practical purposes, we take the Hugoniot curve to be the unloading path.The Hugoniot equation is a fit to data used to generate the Hugoniot curve. Its form depends on the plane in which we choose to view the data. The Hugoniot in the U-u plane for many materials is a straight line given by U=C0+su,(3.48) where the parameters C0 and s are determined from experimental fits to the data. There is no theoretical support for the linearity of this equation. It just works out that way for many materials. For the exceptions to the rule, a higher-order polynomial is used to fit the data.The Hugoniot is by itself not enough to fully characterize a material. It only describes states attainable by a shock transition. For a complete description we need to add an EOS and initial conditions.An EOS is an attempt to describe material behavior at the continuum level beginning by considering the interatomic forces and their effects on the lattice structure to a given set of initial conditions. A full description of material behavior of this type cannot at present be obtained from first principles. Thus, EOS used in practice combine experimental data such as the Hugoniot with an underlying theoretical structure. A great number of EOS formulations are available, but only a few are used in wave propagation codes. For impact velocities below 2 km/s where the material remains a solid, the Mie—Gruneisen form is very popular, due mainly to the simplicity of the formulation and the availability of data in many compilations. For impacts and explosive loading at higher velocities, where melting and vaporization can occur, the Tillotson Equation has been used extensively. For explosive detonation products, the BKW, JWL and gamma law EOS are popular.Because experimental data are introduced into the EOS, mainly through the Hugoniot, it exhibits the same characteristics as any empirical equation. Extrapolation very far from the underlying data base generates nonsensical results.EOS data is generated almost always in compression. The EOS is then used to describe tensile behavior in geometries far removed from those for which the data were generated.If the material in this and the preceding chapter is a complete mystery to you, then you need to spend a year or two studying stress and shock wave propagation in solids subjected to intense, impulsive loading. At this point in your development you should not be using a hydrocode except under the close supervision of an experienced user. You are a danger to yourself and to those who believe the results you generate.

Decision analysis of a gamma hydrologic variate

Hydrologists have studied the statistical nature of summer‐type storms and the decision problems associated with containing their runoff. In this tradition, winter storms are examined here. In contrast to summer storms the gamma family rather than the exponential fits the Tucson rainfall record. Use of the gamma law entails greatly increased computational complexity, and a major effort is made to achieve efficiency of calculation. This study begins by reporting statistical tests indicating that the gamma family describes the precipitation process. A natural conjugate family for the gamma law is derived and discussed. Numerical methods for Bayesian analysis of the (somewhat recalcitrant) gamma variate are explained and tested on simulated data. The concluding sections give examples (return time and levee height) of decision making when the gamma family is the underlying law. The method of decision analysis is applicable to other hydrologic problems in which the gamma distribution (also called the Pearson type 3) is a plausible model. Moreover, it is believed that the bulk of the numerical methods derived herein will prove useful in significantly many other Bayesian decision theory contexts.

Stochastic Properties of the Foraging Behavior of Six Species of Migratory Shorebirds

AbstractA statistical description of the temporal and sequential organization of foraging behavior in six species of migrant shorebirds is presented. Intervals between successive pr.edation attempts produce distributions approximating gamma or exponential probability laws. These distributions undergo seasonal changes in the direction of more shorter intervals in winter. The events constituting the intervals between predation attempts exhibit ecologically meaningful relationships and their analysis has provided some insight into the mechanisms of behavioral control. Testable hypotheses may be suggested from these results. Sequential patterns of foraging events show strong second order redundancy (Markovian) but have weaker dependencies extending over higher orders. As a result of this analysis it was found that stereotypy in foraging behavior was higher in winter. The findings of this detailed analysis of hunting strategy correspond to the results of a coarse analysis of behavior reported previously (BAKER, I97I). The methods used here may be generally useful in studying the evolution of ecological behavior.

A Critical Age-Dependent Branching Process with Immigration

It is shown that the number of cells alive at time $t$, denoted by $Z_0(t)$, of a critical age dependent branching process with immigration satisfies $t^{-1}Z_0(t)$ approaches a gamma law with specified constants in distribution.