In the research of the natural ecological environment, environmental scattering is an important research content whose results are widely used. In order to fully study the relevant characteristics of the natural grassland environment, this paper established a grassland broadband clutter model, summarized the dominant and recessive laws of the grassland environment broadband clutter, and provided a new perspective for related environmental monitoring which would be helpful to solve the problem of target detection in complex environments. The main work of this paper is as follows: The dielectric constant model of the grass was established, and the curve of the dielectric constant with frequency was obtained. The Monte Carlo method combined with the Gaussian spectrum function was used to generate a two-dimensional Gaussian surface to simulate the actual grass surface. A broadband clutter model was established, and, considering the calculation efficiency and accuracy, the Sparse Matrix/Canonical Grid (SMCG) was used to calculate the surface scattering coefficient. Then, the environmental clutter amplitude with different radar bandwidth (10, 40, 80 MHz), surface roughness (h = 0.1 m, h = 0.2 m, h = 0.4 m) and grazing angle (30°, 60°) were calculated and the probability density function (PDF) was obtained. The results show that the higher the radar resolution, the larger the incident angle, and the rougher the interface, all of which will cause the tail of the PDF to become longer which will not only reduce the detection probability of the radar, but also the tracking accuracy. The research results can be used for civilian and military field.

Optimal grasping points for a robotic gripper were derived, based on object and hand geometry, using deep neural networks (DNNs). The optimal grasping cost functions were derived using probability density functions for each local cost function of the normal distribution. Using the DNN, the optimum height and width were set for the robot hand to grasp objects, whose geometric and mass centre points were also considered in obtaining the optimum grasping positions for the robot fingers and the object. The proposed algorithm was tested on 10 differently shaped objects and showed improved grip performance compared to conventional methods.

We establish some weighted integral inequalities of Ostrowski, Čebyšev and Lupaş type and give applications for continuous probability density functions supported on infinite intervals.

A stochastic simulation modeling framework was developed for measuring the impact of weed management technologies in terms of their risk and efficacy. The framework explicitly accounted for the variability of environmental conditions, which underpins risk, and its effect upon the weed population dynamics and crop yields. It was applied to wild oat and wild radish in spring wheat as case studies. Technologies considered included a postemergence herbicide, preseeding tillage, increased crop density, and a selective spray-topping (seed-head sterilizing) herbicide. All stages of the weed life cycle were specified as random variables on the basis of triangular probability distributions, which either were derived from daily environmental conditions or specified as a subjective probability distribution. By using probability density functions the study identified the risks to changes in the weed seed bank and weed density associated with various integrated weed management strategies. This approach not only quantified the probabilities associated with the different outcomes, but also identified how the probability distributions of outcomes were changed as a result of different technology combinations used in weed management strategies. For instance, strategies involving a selective spray-topping herbicide to control seed rain not only resulted in lower seed banks, but the range in possible values was considerably reduced, implying lower likelihood of a population increase. The use of such a risk framework not only benefits weed scientists in terms of designing more effective weed management technologies, but can also assist farmer adoption by being able to quantify the probabilities of success and failure of a technology.

Many perennial bioenergy grasses have the potential to escape cultivation and invade natural areas. We quantify dispersal, a key component in invasion, for two bioenergy candidates:Miscanthus sinensis and M. × giganteus. For each species, approximately 1 × 106 caryopses dispersed anemochorously from a point source into traps placed in annuli near the source (0.5 to 5 m; 1.6 to 16.4 ft) and in arcs (10 to 400 m) in the prevailing wind direction. For both species, most caryopses (95% for M. sinensis and 77% for M. × giganteus) were captured within 50 m of the source, but a small percentage (0.2 to 3%) were captured at 300 m and 400 m. Using a maximum-likelihood approach, we evaluated the degree of support in our empirical dispersal data for competing functions to describe seed-dispersal kernels. Fat-tailed functions (lognormal, Weibull, and gamma (Γ)) fit dispersal patterns best for both species overall, but because M. sinensis dispersal distances were significantly affected by wind speed, curves were also fit separately for dispersal distances in low, moderate, and high wind events. Wind speeds shifted the M. sinensis dispersal curve from a thin-tailed exponential function at low speeds to fat-tailed lognormal functions at moderate and high wind speeds. M. sinensis caryopses traveled farther in higher wind speeds (low, 30 m; moderate, 150 m; high, 400 m). Our results demonstrate the ability of Miscanthus caryopses to travel long distances and raise important implications for potential escape and invasion of fertile Miscanthus varieties from bioenergy cultivation.

Some better estimates for the difference between the integral mean of a function and its mean over a subinterval are established. Various applications for special means and probability density functions are also given.

Given a rectangle containing uniformly distributed random points, how far are the points from the rectangle’s centre? In this paper we provide closed-form expressions for the cumulative distribution function and probability density function that characterise the distance. An expression for the average distance to the centre of the rectangle is also provided.

The probabilistic solutions of the nonlinear stochastic dynamic (NSD) systems with polynomial type of nonlinearity are investigated with the subspace-EPC method. The space of the state variables of large-scale nonlinear stochastic dynamic system excited by white noises is separated into two subspaces. Both sides of the Fokker-Planck-Kolmogorov (FPK) equation corresponding to the NSD system is then integrated over one of the subspaces. The FPK equation for the joint probability density function of the state variables in another subspace is formulated. Therefore, the FPK equation in low dimensions is obtained from the original FPK equation in high dimensions and it makes the problem of obtaining the probabilistic solutions of large-scale NSD systems solvable with the exponential polynomial closure method. Examples about the NSD systems with polynomial type of nonlinearity are given to show the effectiveness of the subspace-EPC method in these cases.

In this paper we present formulae for contact distributions of a Voronoi tessellation generated by a homogeneous Poisson point process in the d-dimensional Euclidean space. Expressions are given for the probability density functions and moments of the linear and spherical contact distributions. They are double and simple integral formulae, which are tractable for numerical evaluation and for large d. The special cases d = 2 and d = 3 are investigated in detail, while, for d = 3, the moments of the spherical contact distribution function are expressed by standard functions. Also, the closely related chord length distribution functions are considered.

The total claim amount for a fixed period of time is, by definition, a sum of a random number of claims of random size. In this paper we explore the probabilistic distribution of the total claim amount for claims that follow a Weibull distribution, which can serve as a satisfactory model for both small and large claims. As models for the number of claims we use the geometric, Poisson, logarithmic and negative binomial distributions. In all these cases, the densities of the total claim amount are obtained via Laplace transform of a density function, an expansion in Bell polynomials of a convolution and a subsequent Laplace inversion.

Les méthodes mécano-fiabilistes ont démontré leur efficacité dans la prise en compte des incertitudes lors de la conception de systèmes mécaniques. Elles reposent sur une modélisation mécanique du comportement du système vis-à-vis des modes de défaillance et sur une description probabiliste des paramètres incertainsintervenant dans les modèles de comportement. La perte de la fonctionque doit satisfaire un système mécanique résulte de l'occurrencede situations potentielles telles que celles mises en évidence au coursd'une analyse par arbre de défaillance (Perfect FaultTree Analysis – P-FTA), par exemple. L'intérêt d'un tel arbre estd'identifier chaque scénario de défaillance au niveau le plus élémentaire possible. À cette démarche qualitative, nous associons unedémarche quantitative basée sur les méthodes probabilistes defiabilité afin de mesurer l'importance de chaque variable et de chaquecombinaison de variables dans la perte de fonction. Nous proposons donc une méthode d'analyse fiabiliste de systèmes mécaniques au moyend'arbres de défaillance physiques et probabilistes. La méthodologieproposée est illustrée au moyen d'une application concernant larupture d'un arbre de boîte de vitesses.

In a unified approach, this paper presents distributional properties of a Voronoi tessellation generated by a homogeneous Poisson point process in the Euclidean space of arbitrary dimension. Probability density functions and moments are given for characteristics of the ‘typical’ edge in lower-dimensional section hyperplanes (edge lengths, adjacent angles). We investigate relationships between edges and their neighbours, called Poisson points or centres; namely angular distributions, distances, and positions of neighbours relative to the edge. The approach is analytical, and the results are given partly explicitly and partly as integral expressions, which are suitable for the numerical calculations also presented.

The joint probability density function of the occupation time of a three-state stochastic process with constant transition matrix: aij ≠ 0, i ≠ j, i, j = 1, 2, 3 but αii = 0, i = 1, 2, 3 is studied.