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Models of various physical, biological, and artificial phenomena follow a power law over multiple magnitudes. This article presents a modified Pareto model with an upside-down shape and an adjustable right tail. The moments, quantiles, failure rate, mean residual life, and quantile residual life functions are examined. In addition, some stochastic ordering characteristics of the proposed model are investigated. The estimation of the parameters using the maximum likelihood estimator, the mean square error, and the Anderson–Darling estimator is explored, and a simulation study is conducted to analyze their behavior. Finally, we compare the proposed model with alternative methods for describing a dataset on the strength of carbon fiber and a dataset on customer waiting times in a bank.

The recent advancements in using micro-organisms effectively for biofuels are the key reason for conducting this study. Therefore, the current theoretical work addresses the Oswald – DeWaele power-law models of bioconvection nanofluid flow containing both nanoparticles and gyrotactic microorganisms. This study has examined two major elements of gyrotactic microorganisms: swimming behavior and the development of bioconvection structures. In addition, the Lie-group approach of transformations is used to establish similarity representations of the governing equations rather than the existing transformations. To achieve the model’s aims, the Legendre-based collocation method (LBCM) was adopted. The findings are analyzed using graphs to examine the patterns of the relevant parameters in the problem. Compared with existing results, the present analysis is effective, fast-converging, and accurate. The bioconvection parameters have a significant impact on the density of motile microorganisms, nanoparticle volume fraction, flow momentum, and temperature distribution. Furthermore, the current work finds application in nanofluid fuel. [ABSTRACT FROM AUTHOR]

The temperature effect on viscosity of liquid and semi-liquid foods has been traditionally described by the Arrhenius equation, a few other mathematical models, and more recently by the WLF and VTF (or VFT) equations. The essence of the Arrhenius equation is that the viscosity is proportional to the absolute temperature's reciprocal and governed by a single parameter, namely, the energy of activation. However, if the absolute temperature in K in the Arrhenius equation is replaced by T + b where both T and the adjustable b are in °C, the result is a two-parameter model, which has superior fit to experimental viscosity-temperature data. This modified version of the Arrhenius equation is also mathematically equal to the WLF and VTF equations, which are known to be equal to each other. Thus, despite their dissimilar appearances all three equations are essentially the same model, and when used to fit experimental temperature-viscosity data render exactly the same very high regression coefficient. It is shown that three new hybrid two-parameter mathematical models, whose formulation bears little resemblance to any of the conventional models, can also have excellent fit with r2 ∼ 1. This is demonstrated by comparing the various models' regression coefficients to published viscosity-temperature relationships of 40% sucrose solution, soybean oil, and 70°Bx pear juice concentrate at different temperature ranges. Also compared are reconstructed temperature-viscosity curves using parameters calculated directly from 2 or 3 data points and fitted curves obtained by nonlinear regression using a larger number of experimental viscosity measurements. [ABSTRACT FROM AUTHOR]

Human performance in selection tasks is frequently described by Fitts' law, which states that the average time needed to move to a target and select it is linearly related to the index of difficulty for the task, which is the logarithm of the task characteristic C = A / W , where A is the target distance and W is the target width. The coefficients in this linear relationship can vary across interaction conditions, such as when using distinct interaction devices, and can for instance be used to define throughput, which is a standardized measure to compare interaction conditions. Although Fitts' law has proven very useful in the field of human–computer interaction (HCI) over the past 50 years, there are several issues with Fitts' law that argue in favor of more advanced statistical modeling of experimental data. More specifically, we propose two generalizations of Fitts' law. The first generalization is not to limit the data analysis to average movement times, but to consider the distributions of the observed times instead. The second generalization is to extend Fitts' law to a more general relationship between task characteristics and index of difficulty and to use this generalized model to come up with additional measures that can be used alongside throughput. We use data from an experiment with both selection and tracing tasks to illustrate the proposed analysis method. The primary goal of the experiment is to compare the task performance in four experimental conditions, corresponding to all combinations of interaction device (mouse or pen) and target orientation (horzizontal/vertical versus oblique movements). First, we show that non-linear transformations on the measured task completion times are indeed advised to resolve problems with the normality and homoscedasticity of the data, especially in case of the tracing task. Second, we show that in case of the selection task the data supports a linear relationship between the logarithm of the task characteristic C and the logarithm of the movement time, which corresponds to a power-law between movement time and task characteristic, an alternative to Fitts' law that has previously been proposed by several authors. In case of the tracing task, the data supports a power-law function in between a linear and a logarithmic function. We conclude by demonstrating how multiple performance measures can be used simultaneously when comparing interaction conditions. [ABSTRACT FROM AUTHOR]

There is a lack of accurate protocol for predicting aboveground biomass (AGB) and carbon pools in Iran's Hyrcanian forests. This study aimed to figure out the most accurate model for the site-specific prediction of AGB. Site-specific allometric equations on the basis of power-law function and artificial neural network (ANN) type of multi-layer perception based on back-propagation training algorithm were developed to model engineering application for predicting the AGB. The AGB was measured by destructively sampling and weighing 174 fallen trees in the field and breast height diameter ( D ), total height ( H ) and basic wood density ( ρ ) were recorded as explanatory variables for developing allometric equations and ANN models. The findings showed that the simple allometry model including the geometrical variable ( D 2 Hρ ) was a highly accurate predictor with the highest certainty among the allometric equations (Adj. R 2 = 0.91; CF = 1.04; AIC = −430). Furthermore, the product of D 2 Hρ and the primary variables were the effective input nodes for training algorithm in the ANN. Statistical issues such as collinearity among the parameters, reliability of parameters and application of dubious empirical equations (…) were the main problems for developing allometric equations; however, there was no limiting factor for designing the models in ANN. According to training and testing data set, the best architecture designed in the ANN model was composed of two hidden layers and 20 neurons in each layer including function of tangent sigmoid. The results showed that the best designed model in ANN predicted the AGB with the higher accuracy (RMSE% = 7.3) than allometric equations. Thus, ANN is offered rather than traditional protocols for predicting the AGB in natural forest ecosystems. [ABSTRACT FROM AUTHOR]

The species-area relationship (SAR) is known to overestimate species extinction but the underlying mechanisms remain unclear to a great extent. Here, I show that when total species number in an area is unknown, the SAR model exaggerates the estimation of species extinction. It is proposed that to accurately estimate species extinction caused by habitat destruction, one of the principal prerequisites is to accurately total the species numbers presented in the whole study area. One can better evaluate and compare alternative theoretical SAR models on the accurate estimation of species loss only when the exact total species number for the whole area is clear. This presents an opportunity for ecologists to simulate more research on accurately estimating Whittaker's gamma diversity for the purpose of better predicting species loss. [ABSTRACT FROM AUTHOR]

In this study melt rheological behavior of lactose-filled polyethylene glycol (PEG) composites as a low melting polymeric carrier for controlled release drugs was investigated using a capillary rheometer. The effect of lactose concentration and process variables such as temperature and ram speed on the flow behavior of PEG has been studied. The composites were found to be shear thinning in behavior when extruded, and the results were well described by power-law model in each case. Stronger shear thinning behavior was observed by raising the filler concentration and decreasing the temperature, while the flow index has been decreased. In all compositions a significant increase in shear viscosity was found by an increase in the filler content. In fact, shear viscosity increased linearly by weight fraction of filler, but there was a dramatic increase after the filler content raised above 20 wt% of lactose which might be the result of the strong interaction among filler particles. Furthermore, decreasing the process temperature resulted in an increase in shear viscosity, and the temperature dependence of shear viscosity decreased as the shear rate increased. The extensional viscosity of composites was calculated in each case. The results showed that the ratio of the extensional viscosity to shear viscosity was in the range of 500-1200. [ABSTRACT FROM AUTHOR]

Abstract: Mathematical modeling has become an integral component in biotechnology, in which these models are frequently used to design and optimize bioprocesses. Canonical models, like power-laws within the Biochemical Systems Theory, offer numerous mathematical and numerical advantages, including built-in flexibility to simulate general nonlinear behavior. The construction of such models relies on the estimation of unknown case-specific model parameters by way of experimental data fitting, also known as inverse modeling. Despite the large number of publications on this topic, this task remains the bottleneck in canonical modeling of biochemical systems. The focus of this paper concerns with the question of identifiability of power-law models from dynamic data, that is, whether the parameter values can be uniquely and accurately identified from time-series data. Existing and newly developed parameter identifiability methods were applied to two power-law models of biochemical systems, and the results pointed to the lack of parametric identifiability as the root cause of the difficulty faced in the inverse modeling. Despite the focus on power-law models, the analyses and conclusions are extendable to other canonical models, and the issue of parameter identifiability is expected to be a common problem in biochemical system modeling. [ABSTRACT FROM AUTHOR]

Abstract: The mathematical modelling of signal transduction pathways has become a valuable aid to understanding the complex interactions involved in intracellular signalling mechanisms. An important aspect of the mathematical modelling process is the selection of the model type and structure. Until recently, the convention has been to use a standard kinetic model, often with the Michaelis–Menten steady state assumption. However this model form, although valuable, is only one of a number of choices, and the aim of this article is to consider the mathematical structure and essential features of an alternative model form — the power-law model. Specifically, we analyse how power-law models can be applied to increase our understanding of signal transduction pathways when there may be limited prior information. We distinguish between two kinds of power law models: a) Detailed power-law models, as a tool for investigating pathways when the structure of protein–protein interactions is completely known, and; b) Simplified power-law models, for the analysis of systems with incomplete structural information or insufficient quantitative data for generating detailed models. If sufficient data of high quality are available, the advantage of detailed power-law models is that they are more realistic representations of non-homogenous or crowded cellular environments. The advantages of the simplified power-law model formulation are illustrated using some case studies in cell signalling. In particular, the investigation on the effects of signal inhibition and feedback loops and the validation of structural hypotheses are discussed. [Copyright &y& Elsevier]

The Human Genome Project, DNA microarrays, proteome chips and metabolic profiles, among others, are generating unprecedented amounts of valuable information about the genetic and metabolic responses of organisms to stimuli. This wealth of information poses a great scientific challenge, namely the development of novel, effective methods for functional analysis and interpretation. It is proposed here that biomathematical systems models must accompany data generation and management. A particularly effective framework for this purpose is canonical modeling based on Biochemical Systems Theory. The key concept of this theory is the formulation of biological phenomena as systems of differential equations, in which all processes are represented as products of power-law functions. [Copyright &y& Elsevier]

• The paper shows that the expansion of agricultural land can be described by a power-law model. • Additionally, it shows that Government spending per unit of area to foster agriculture also follows a power-law model. • These results challenge common perceptions that these variables result from complex interactions. • Deviations from Power-law trajectories are due to the relative importance of centroid localities. • Some applications of the scale-invariant model for land use policy are discussed. The expansion dynamics of the agricultural surface is a complex process, since it requires decisions among different actors under a multitude of socioeconomic and natural conditions, expectations, and risks. This paper shows that despite this complexity, and at odds with the intuition, the density of agricultural lands and the government spending to foster agriculture around a human settlement displays a simple power-law relationship regarding the distance to such a settlement. The theoretical proof of this relationship is based on theoretical and empirical findings made by several authors on the expansion of population centers, river networks, species and ecosystem's distribution. The empirical proof is made by fitting a simple power-law model having as response variables: density of agricultural land; government spending coverage, and government spending per unit of agricultural surface, while the distance to the centroid settlement is used as independent variable. Results show that the scaling exponents of the power-law relationship estimated from those models, with data from rural settlements in Mexico, are similar statistically to the expected values from theoretical models for the three expansion processes tested. Three applications of the scale-invariant attribute characterizing the expansion process of agricultural lands are presented: i) monitoring agricultural activities at the territories; ii) a proxy to estimate the expected growth of the agricultural surface; and iii) a typology to characterize government assistance at local level. [ABSTRACT FROM AUTHOR]