Rectangular Probability Distribution Research Articles

From a single slit to periodic, modulated, and quasiperiodic crystals – a physical space diffraction analysis of aperiodic systems

Using a statistical approach to diffraction analysis, it was shown that the scale of difficulty in the analysis of the diffraction image of a one-dimensional quasicrystal is not significantly different from the analysis of the diffraction image for a single slit. In both cases we get a rectangular probability distribution, for which the Fourier transform leads to analytical expressions on the envelopes of diffraction peaks expressed by means of simple trigonometric functions (sinc). The generalization to 2d (decagonal quasicrystals) and 3d (icosahedral quasicrystals) requires more complex calculations, but conceptually, it does not significantly differ from the calculations for model one-dimensional quasicrystals.

Brittle to quasibrittle transition in a compound fiber bundle.

The brittle to quasibrittle transition has been studied for a compound of two different kinds of fibrous materials, having distinct difference in their breaking strengths under the framework of the fiber bundle model. A random fiber bundle model has been devised with a bimodal distribution of the breaking strengths of the individual fibers. The bimodal distribution is assumed to consist of two symmetrically placed rectangular probability distributions of strengths p and 1-p, each of width d, and separated by a gap 2s. Different properties of the transition have been studied varying these three parameters and using the well-known equal load-sharing dynamics. Our study exhibits a brittle to quasibrittle transition at the critical width d_{c}(s,p)=p(1/2-s)/(1+p) confirmed by our numerical results.

The equilibrium liquidus temperatures of rhenium–carbon, platinum–carbon and cobalt–carbon eutectic alloys

The eutectic alloys rhenium–carbon, platinum–carbon and cobalt–carbon have been proposed as reference standards for thermometry, with temperature and uncertainty values specified within the mise en pratique of the definition of the kelvin. These alloys have been investigated in a collaboration of eleven national measurement institutes and laboratories. Published results reported the point-of-inflection in the melting curve with extremely low uncertainties. However, to be considered as standards it is necessary to stipulate what phenomenon a temperature value has been ascribed to; specifically, this should be a thermodynamic state. Therefore, the data have been further evaluated and the equilibrium liquidus temperatures determined based on a consideration of limits and assuming a rectangular probability distribution. The values are: for rhenium–carbon 2747.91 ± 0.44 K, for platinum–carbon 2011.50 ± 0.22 K and for cobalt–carbon 1597.48 ± 0.14 K, with uncertainties at approximately a 95% coverage probability. It is proposed that these values could be used as the basis of thermodynamic temperature measurement at high temperatures (above 1300 K).

Estimated of the associated uncertaintes with the linearity test of activimeters

Activimeters determine the activity of radioactive samples and are validated by performance tests. This research determined the expanded uncertainties associated with the linearity test. We used two activimeters and a source of 99Tcm for testing recommended by the IAEA protocol, which consider the decay of radioactive samples. The expanded uncertainties evaluated were not correlated with each other and our analysis considered a rectangular probability distribution. The results are also presented in graphical form by the function of normalized activity measured in terms of conventional true value.

Rectangular distribution whose end points are not exactly known: curvilinear trapezoidal distribution

Metrologists often represent the state of knowledge concerning a quantity about which scant specific information is available by a rectangular probability distribution. The end points are frequently specified by subjective judgment; therefore, they are inexactly known. If the states of knowledge about the end points may be represented by other (narrower) rectangular distributions, then the resulting probability distribution looks like a trapezoid whose sloping sides are curved. We refer to such a probability distribution as curvilinear trapezoid. Depending on the limits of rectangular distributions for the end points, the curvilinear trapezoidal distribution may be asymmetric. In a previous paper we had shown that if the mid-point of a rectangular distribution is known and the state of knowledge about the half-width may be represented by a rectangular distribution then the resulting distribution is symmetric curvilinear trapezoid. In this paper, we describe the probability density function of a curvilinear trapezoidal distribution which arises from inexactly known end points. Then we give compact analytic expressions for all moments including the expected value and the variance. Next we discuss how random numbers from such a distribution may be generated. We compare the curvilinear trapezoid which arises from inexactly known end points with the corresponding trapezoid whose sloping sides are straight. We also compare the curvilinear trapezoid which arises from inexactly known end points with the curvilinear trapezoid which arises when the mid-point is known and the state of knowledge about the half-width may be represented by a rectangular distribution. The results presented in this paper are useful in evaluating uncertainty according to the Guide to the Expression of Uncertainty in Measurement (GUM) as well as Supplement 1 to the GUM (GUM-S1).

The effect of the different uncertainty models in dose expected volume histogram computation

Dose expected volume histograms are a useful alternative to dose volume histograms in order to take into account computation uncertainty when prescribing, planning and reporting external beam radiation therapy. Due to the type B nature of computation uncertainties, a rectangular probability distribution was assumed in its definition. In the present work, the changes of dose expected volume histograms when using other recommended point dose uncertainty models are investigated. Results show that the choice of probability distribution, among the ones recommended for uncertainty modelling, has a small effect on the result. Thus, the standard uncertainty of point dose computations is the main parameter affecting dose expected volume histograms.

Multiple nearest-neighbor exchange model for the frustrated magnetic molecules{Mo72Fe30}and{Mo72Cr30}

Our measurements of the differential susceptibility $\ensuremath{\partial}M/\ensuremath{\partial}H$ of the frustrated magnetic molecules ${{\text{Mo}}_{72}{\text{Fe}}_{30}}$ and ${{\text{Mo}}_{72}{\text{Cr}}_{30}}$ reveal a pronounced dependence on magnetic field $(H)$ and temperature $(T)$ in the low $H$--low $T$ regime, contrary to the predictions of the existing models. Excellent agreement with experiment is achieved upon formulating a nearest-neighbor classical Heisenberg model where the 60 nearest-neighbor exchange interactions in each molecule, rather than being identical as has been assumed heretofore, are described by a two-parameter rectangular probability distribution of values of the exchange constant. We suggest that the probability distribution provides a convenient phenomenological platform for summarizing the combined effects of multiple microscopic mechanisms that disrupt the idealized picture of a Heisenberg model based on a single value of the nearest-neighbor exchange constant.

A Novel Method for the Evaluation of Uncertainty in Dose–Volume Histogram Computation

Dose-volume histograms (DVHs) are a useful tool in state-of-the-art radiotherapy treatment planning, and it is essential to recognize their limitations. Even after a specific dose-calculation model is optimized, dose distributions computed by using treatment-planning systems are affected by several sources of uncertainty, such as algorithm limitations, measurement uncertainty in the data used to model the beam, and residual differences between measured and computed dose. This report presents a novel method to take them into account. To take into account the effect of associated uncertainties, a probabilistic approach using a new kind of histogram, a dose-expected volume histogram, is introduced. The expected value of the volume in the region of interest receiving an absorbed dose equal to or greater than a certain value is found by using the probability distribution of the dose at each point. A rectangular probability distribution is assumed for this point dose, and a formulation that accounts for uncertainties associated with point dose is presented for practical computations. This method is applied to a set of DVHs for different regions of interest, including 6 brain patients, 8 lung patients, 8 pelvis patients, and 6 prostate patients planned for intensity-modulated radiation therapy. Results show a greater effect on planning target volume coverage than in organs at risk. In cases of steep DVH gradients, such as planning target volumes, this new method shows the largest differences with the corresponding DVH; thus, the effect of the uncertainty is larger.

Random and systematic measurement uncertainties considered in the evaluation of parameters derived from the course of the tabletting process

The uncertainty of some characteristic parameters describing the course of the tabletting process, namely the area quotient according to Emschermann and Müller, the apparent net work, the slope of the Heckel-plot, and the parameters of the modified Weibull function, was calculated according to the German norm DIN 1319-4 1999. The method allows to consider random and systematic uncertainties in a consistent way as variances of normal and rectangular probability distributions, respectively, or other suitable probability distributions based on Bayesian statistics and the principle of maximum entropy. So, random and systematic uncertainties known from a calibration and validation study of the measurement of force and displacement, and the uncertainty of the true density were included meaningfully into the uncertainty of the resulting tabletting parameters using the propagation of uncertainties according to the Gauss method. The standard uncertainty for the results calculated this way seems to be suitable for a critical evaluation of the tabletting data.

Effect of Systematic and Random Errors in Thermodynamic Models on Chemical Process Design and Simulation: A Monte Carlo Approach

A Monte Carlo method is presented to separate and study the effects of systematic and random errors present in thermodynamic data on chemical process design and simulation. From analysis of thermodynamic data found in the literature, there is clear evidence of the presence of systematic errors, in particular, for liquid−liquid equilibria data. For systematic errors, the data are perturbed systematically with a rectangular probability distribution, and to analyze random errors, the perturbation is carried out randomly with normal probability distributions. Thermodynamic parameters are obtained from appropriate regression methods and used to simulate a given unit operation and to obtain cumulative frequency distributions, providing a quantitative risk assessment and a better understanding of the role of uncertainty in process design and simulation. The results show that the proposed method can clearly distinguish when one type of error is more significant. Potential applications are safety factors, process modeling, and experimental design.

What motion distributions yield global transparency and spatial segmentation?

We have examined the ability of observers to parse bimodal local-motion distributions into two global motion surfaces, either overlapping (yielding transparent motion) or spatially segregated (yielding a motion boundary). The stimuli were random dot kinematograms in which the direction of motion of each dot was drawn from one of two rectangular probability distributions. A wide range of direction distribution widths and separations was tested. The ability to discriminate the direction of motion of one of the two motion surfaces from the direction of a comparison stimulus was used as an objective test of the perception of two discrete surfaces. Performance for both transparent and spatially segregated motion was remarkably good, being only slightly inferior to that achieved with a single global motion surface. Performance was consistently better for segregated motion than for transparency. Whereas transparent motion was only perceived with direction distributions which were separated by a significant gap, segregated motion could be seen with abutting or even partially overlapping direction distributions. For transparency, the critical gap increased with the range of directions in the distribution. This result does not support models in which transparency depends on detection of a minimum size of gap defining a bimodal direction distribution. We suggest, instead, that the operations which detect bimodality are scaled (in the direction domain) with the overall range of distributions. This yields a flexible, adaptive system that determines whether a gap in the direction distribution serves as a segmentation cue or is smoothed as part of a unitary computation of global motion.

Magnetic properties of a disordered Ising lattice with a rectangular probability distribution of the exchange integrals.

The rectangular probability distribution of the exchange integrals is applied for the description of a disordered Ising ferromagnet. In particular, the numerical results for the simple quadratic lattice are presented and compared with the results obtained for other probability distributions

Random walks and diffusion

The usual solution of the diffusion equation gives inaccurate results for short diffusion times for displacements which are relatively large compared to the maximum range of the diffusion. A method is proposed for solving the diffusion equation by a random-walk approximation technique which utilizes the appropriate generating functions and the complex integration method of steepest descent. The power of the method is illustrated for a one-dimensional random walk (SRW) and for physical situations in which the elementary diffusion process is governed by a normal or a rectangular probability distribution. Formulas for higher approximations applicable to any probability distribution are also derived. The method, which can also be applied to two and three dimensions, is capable of yielding high accuracy for the diffusion solution over the entire diffusion domain.

Snap-Through Buckling of a Viscoelastic von Mises Truss in a Random Temperature Field

The reduction of the expected lifetime of a temperature-sensitive viscoelastic von Mises truss caused by small random temperature fluctuations is calculated, assuming the probability density of the temperature process to be a known (symmetric) function. Results are derived for a normal and a rectangular probability distribution of the temperature fluctuations.

Linear Control of Non-Linear Systems

ABSTRACT For some simple relay control systems it is shown that if the control signal input to the relay is made an appropriate linear combination of the state variables, nearly optimum performance can be achieved. The performance index used is integral -square-error; its value when the switching function is linear is calculated and compared with the value obtained when the switching function is the optimal (usually non-linear) one. E.g. for a second-order plant and a command signal which is a step of random magnitude having a rectangular probability distribution, linear switching can give a mean integral-square-error which exceeds the optimal value by less than 1%.