Important Mathematical Properties Research Articles

Analysis of leukemia and forest fires data using new Poisson Quasi-Shanker distribution

In this study, a new two-parameter discrete distribution is introduced by mixing Poisson and Quasi-Shanker distributions. The new model is named Poisson Quasi-Shanker-(PQS) distribution. Some important mathematical properties of the proposed distribution are derived including factorial moments, moment-generating function, variance, and dispersion index. The parameters of PQS distribution are estimated using the maximum likelihood estimation approach. A Monte Carlo simulation is carried out to illustrate the behavior of derived estimators. The applicability of the new model is assessed using two datasets from different areas. It is concluded the proposed distribution is more flexible as compared to the considered competitive distributions.

Different Estimation Methods for A New Extension Three Parameter Log-Logistic Distribution with Applications

This article presents a new extension of the three-parameter log-logistic distribution, the so-called heavy-tailed log-logistic (HTLL) distribution. Some important mathematical properties of the new HTLL distribution are calculated. In addition, some numerical results of moments for the HTLL distribution are calculated. Extensive simulations were performed to investigate the estimation of the model parameters using many established approaches, including maximum likelihood (ML), least squares (LS), weighted least squares (WLS), Cramer-von Mises (CVM), Anderson-Darling (AD), and right-tail Anderson-Darling (RTAD). The simulation results show that the AD approach has the highest efficiency among these approaches. The usefulness of the newly proposed model is demonstrated by analyzing two real data sets.

An improved one-parameter weighted distribution with mathematical properties, estimation, and applications

In this study, an improved extension of the Haq distribution was introduced. The new model is named “size-biased Haq distribution (SBHD)”. Its two special cases length-biased and area-biased Haq distribution are discussed. The important mathematical properties such as moments, moment generating function, variance, skewness, kurtosis, Rényi entropy, survival function, hazard function, and mean residual life are derived. The model parameters are estimated using five different estimation methods. A comprehensive simulation study was utilized to illustrate the behavior of derived estimators. Two datasets from different fields are utilized to illustrate the flexibility of the new distributions. It is concluded that the new model is more flexible and efficiently analyzed when the datasets are compared to the considered competitive distributions.

Using Hydrodynamic Similarity as a Verification Method for Impact Cratering Simulations in the FLAG Hydrocode

Hydrodynamic codes (hydrocodes) are common tools for modeling hypervelocity impacts to provide insight into the physical phenomenon. Hydrocodes can simulate impacts from micrometer to kilometer spatial scales and reach impact velocities difficult to achieve in experimental settings. However, numerical models are approximations, and demonstrating that a numerical method is capable of providing physical results for these models is essential. In this work, we employ a hydrocode verification technique that leverages hydrodynamic similarity, a mathematical property of the conservation equations of fluid mechanics that form the basis for hydrocode models. Using the FLAG hydrocode, we simulate aluminum (Al) and basalt projectiles and targets at spatial scales spanning 7 orders of magnitude (hundreds of micrometers to kilometers). These materials were chosen because Al-6061 is a common material in spacecraft and satellites and basalt is a useful approximation of rocky astronomical bodies. Our results show that hydrodynamic similarity holds for each material model used and across spatial scales. We show that under certain conditions hydrodynamic similarity can apply in the presence of gravity and that similarity does not hold in the presence of strength models. We conclude that the FLAG hydrocode preserves important mathematical properties of fluid dynamics in hypervelocity impacts of Al-6061 and basalt.

A Study of Elliptic Curve Cryptography and Its Applications

This paper aims to provide a comprehensive review on Elliptic Curve Cryptography (ECC), a public key cryptographic system and its applications. The paper discusses important mathematical properties and operations of elliptic curves, like point addition and multiplication operations and its implementation in cryptographic methods such as encryption and decryption. This paper provides a detailed workout on important mathematical problems on elliptic curves and ECC which provides insight into working of essential cryptographic techniques in ECC. And the paper also provides a literature review of research works based on ECC in various fields such as Internet of Things (IoT), Cloud computing, Blockchain and Image Security. And the paper further provides insight into the recent applications of ECC in fields like IoT and Blockchain by comprehensively discussing the proposed mechanism for each of the recent applications and also briefly discussing the security of the proposed mechanism.

The Remkan Distribution and Its Applications

Due to the ever growing demand for the development of new lifetime distributions to meet the goodness of fit demand of complex datasets, two-parameter distributions has been proposed in recent times. This study therefore aims to contribute to this demand. We propose a new two-parameter lifetime distribution known as the Remkan distribution. Important mathematical properties of the Remkan distribution such as the moments and other related measures, and moment generating function were derived and the model parameters estimated using the maximum likelihood estimate technique. Finally, the flexibility of the new Remkan distribution was illustrated using a real life dataset and the results showed that the new Remkan distribution was the best amongst other competing two parameter distributions.

Assessment of fabric characteristics with the development of sand liquefaction in cyclic triaxial tests: A DEM study

This paper studies fabric properties with the evolution of sand liquefaction by performing a series of 3D constant-volume cyclic triaxial DEM tests. With calibrated parameters for HN31 sand, the consistency between the obtained DEM results and the counterpart experiments is presented. The evolution of fabric characteristics is then assessed with the coordination number and mechanical coordination number, respectively. For liquefaction issues, the conventional coordination number outperforms the mechanical one since the floating particles with the number of contacts less than or equal to 1 can be fully considered. To analyse the complex inter-particle contacts in a granular assembly, the second-order contact normal fabric tensor is briefly introduced with its important mathematical properties. The second invariant of its deviatoric part can prove to be a proper index describing the degree of anisotropy in the principal fabric space spanned by the three eigenvectors belonging to this tensor. Following the direction of the applied loading, the fabric anisotropy accumulates in a gradual manner and finally reaches a threshold value corresponding to liquefaction triggering on the macro scale. This value is fabric-dependent and thus independent of loading intensity. For gaining a complete insight of sand liquefaction on both the micro and macro scales, the shear strain can act as a bridge to describe the evolution of coordination number and the anisotropy degree, respectively.

The Copoun Distribution and Its Mathematical Properties

Due to the ever growing demand for the development of new lifetime distributions to meet the goodness of fit demand of complex datasets, two-parameter distributions has been proposed in recent times. This study therefore aims to contribute to this demand. We propose a new two-parameter lifetime distribution known as the Copoun distribution. Important mathematical properties of the new distribution such as the moments and other related measures, and moment generating function were derived. Finally, the values of the mean, standard deviation, coefficient of variation, skewness, and kurtosis of the Copoun distribution shows that the distribution has the tendency to shift to higher values overall (increasing mean) and narrow around this increased central tendency (decreasing spread, variation and increasing peakedness).

Large-scale closed and generalized networks of ribosome flow model with different site sizes

Dynamical systems play a key role in understanding the different biophysical aspects of many complex transport phenomena. The ribosome flow model with different site sizes (RFMD) is a dynamical system relevant to model various biological and physical processes such as phosphorelay, motion of vehicles along an isolated road, and more. However, many transport processes take place concurrently which give rise to networks comprising multiple tracks e.g., vehicular motion, cargo transport along the cytoskeletal highway, etc. Here, we present two large-scale network models: RFMDs network with a pool (RFMDNP), and the generalized network of RFMDs (RFMDN). The RFMDNP is a closed system that can model the simultaneous movement of particles along tracks having sites of different capacities in a resource-limited environment. We prove that the RFMDNP admits a continuum of linearly ordered steady-state points and it always phase-locks with the periodic excitations in the parameters. Furthermore, we also show that increasing any of the parameters in an RFMD leads to an increase in the output rate of this RFMD, and the output rate of other RFMDs all increase or decrease. Next, we analyze a generalized network of RFMDs that models static connections between the RFMDs and prove the important mathematical properties: (a) the network admits a unique steady-state, and (b) the problem of determining the connecting weights to maximize the steady-state network output rate is a convex optimization problem. We believe that these networks can have wide applicability to model various non-linear phenomena since their behavior is predictable and ordered.

Shifts in student attention on algorithmic and creative practice tasks

AbstractIn mathematics classrooms, it is common practice to work through a series of comparable tasks provided in a textbook. A central question in mathematics education is if tasks should be accompanied with solution methods, or if students should construct the solutions themselves. To explore the impact of these two task designs on student behavior during repetitive practice, an eye-tracking study was conducted with 50 upper secondary and university students. Their eye movements were analyzed to study how the two groups shifted their gaze both within and across 10 task sets. The results show that when a solution method was present, the students reread this every time they solved the task, while only giving minute attention to the illustration that carried information supporting mathematical understanding. Students who practiced with tasks without a solution method seemed to construct a solution method by observing the illustration, which later could be retrieved from memory, making this method more efficient in the long run. We discuss the implications for teaching and how tasks without solution methods can increase student focus on important mathematical properties.

The Cubic Rank Transmuted Inverse Exponential Distribution

This paper uses the recently proposed cubic rank transmutation technique to develop and propose a new distribution in the transmutation family called the Cubic Rank Transmuted Inverse Exponential distribution (CRTIE). Important mathematical properties of the new distribution including Quantile function, and Reliability analysis were derived. The estimates of the parameters of the new distribution were also derived.

On the control of density-dependent stochastic population processes with time-varying behavior

The study of density-dependent stochastic population processes (DDSPPs) is important from a historical perspective as well as from the perspective of a number of existing and emerging applications today. In more recent applications of these processes, it can be especially important to include time-varying parameters for the rates that impact the density-dependent population structures and behaviors. Under a mean-field scaling, we show that such time-inhomogeneous DDSPPs converge to a corresponding nonautonomous dynamical system. We then analogously establish that the optimal control of such time-inhomogeneous DDSPPs converges to the optimal control of the limiting dynamical system. An analysis of both the dynamical system and its optimal control renders various important mathematical properties of interest.

Properties and Applications of the Type I Half-logistic Nadarajah-Haghighi Distribution

A new three-parameter distribution called the type I half-logistic Nadarajah-Haghighi (T IHL N H ) is proposed. We discussed some important mathematical and statistical properties of the new model such as an explicit form of its r th moment, mean deviations,quantile function, Bonferroni and Lorenz curves. The Shannon entropy and Renyi entropy are computed, the expression for the Kullback-Leibler divergence measure is provided. The model parameters estimation was approached by the maximum likelihoodestimation (MLE), and the information matrix is obtained. The finite sample properties of the MLEs are investigated numerically by simulation studies; by examining the bias and mean square error of the estimators, and the results was satisfactory. We used tworeal data applications to demonstrate the superior performance of the T IHL N H in terms of fit over some other existing lifetime models.

An online learning algorithm for a neuro-fuzzy classifier with mixed-attribute data

General fuzzy min–max neural network (GFMMNN) is one of the efficient neuro-fuzzy systems for data classification. However, one of the downsides of its original learning algorithms is the inability to handle and learn directly from mixed-attribute data without using encoding techniques. While categorical feature encoding methods can be used with the GFMMNN learning algorithms, they exhibit many shortcomings. Other improved approaches proposed in the literature are not suitable for online learning algorithms working in the dynamically changing environments without ability to retrain or access full historical data, which are usually required for many real world applications. This paper proposes an extended online learning algorithm for the GFMMNN. The proposed method can handle the datasets with both continuous and categorical features. It uses the change in the entropy values of categorical features of the samples contained in a hyperbox to determine if the current hyperbox can be expanded to include the categorical values of a new training instance. An extended architecture of the original GFMMNN and its new membership function are introduced for mixed-attribute data. Important mathematical properties of the proposed learning algorithms are also presented and proved in this paper. The extensive experiments confirmed superior and stable classification performance of the proposed approach in comparison to other relevant learning algorithms for the GFMM model.

On the mixture of Shanker and gamma distributions with applications to engineering data

There are several reliability and life testing challenges that may be solved using the gamma and Shanker distributions. In this paper, a new lifetime model of two parameters is suggested, referred to as Shanker-Gamma distribution (SGD). The SGD is developed by mixing two well-known distributions, namely; Shanker distribution of one parameter and gamma distribution of two parameters to propose a new continuous more flexibility distribution for modeling real data in different areas. Some important statistical and mathematical properties of the distribution are derived such as the distribution function, moment generating function, rth moment, the coefficients of variation, skewness and kurtosis, and the distributions of order statistics. The maximum likelihood estimation (MLE) method is adopted to estimate the model parameters and a simulation study is conducted investigate the efficiency of the MLE. Also, the Rényi entropy, stochastic ordering, Lorenz and Bonferroni curves and the Gini index are obtained. Additionally, some reliability functions are presented and it is noted that the model have decreasing hazard rate function based on its parameters. Numerical calculations are provided to study the various statistical properties of the SG distribution for various values of the distribution parameters. Finally, two real data sets include tensile strength of polyester fibers and failure time explore the potentiality of the suggested model as compared with some alternatives models. For both data sets, it is revealed that the SGD is more flexible than its competitors considered in this paper. Also, the MLE values decrease as the sample size increases.

Analytical modelling of a slotless tubular interior permanent magnet machine considering iron poles permeability

AbstractAn analytical method is proposed to model the magnetic field distribution across a slotless tubular interior permanent magnet machine. In order to consider the permeability of both permanent magnets and iron poles of the slider, in the proposed approach, instead of choosing each one as an individual sub‐domain, they are taken as an integrated sub‐domain. Therefore, the governing partial differential equation alters from Poisson equation to the governing magnetostatics equation for inhomogeneous (space‐dependent permeability) materials. The solution is derived through separation of variables method in which a Sturm‐Liouville problem is encountered. The solution and its important mathematical properties, such as the orthogonality and generalised Fourier series, are fully addressed. These properties are very helpful for creating linear interface equations in terms of the integral coefficients. In addition, since the number of sub‐domains is decreased, the total number of the unknowns of the final matrix of equations is reduced effectively. To validate the proposed method, a case study is simulated by the aid of computer software and its results are compared with those of finite element method. It has been observed that the results are in good agreement.

Exponentiated Generalized Exponential Geometric Distribution: Model, Properties and Applications

In this article, a new distribution called Exponentiated Generalized Exponential Geometric Distribution is formulated. We have derived some important mathematical properties like hazard function, probability density function, survival function, quantiles, the measures of skewness based on quartiles and coefficient of kurtosis based on octiles. To estimate the parameters of the new distribution, we have applied the three commonly used estimation methods namely maximum likelihood estimation (MLE), least-square (LSE) method and Cramer-Von-Mises (CVM) method. We have used R programming as well as analytical methods for data analysis. For model validation, we have used different information criteria as Akaike’s information criteria, and Bayesian information criteria (BIC) etc. For the assessment of potentiality of the new distribution, we have considered a real dataset and the goodness-of-fit attained by proposed distribution is compared with some competing distributions. It is found that the proposed model fits the data very well and more flexible as compared to some other models.

The Generalized Odd Log-Logistic Fréchet Distribution for Modeling Extreme Values

We introduce a new extension of the Fréchet distribution for modeling the extreme values. The new model generalizes eleven distributions at least, five of them are quite new. Some important mathematical properties of the new model are derived. We assess the performance of the maximum likelihood estimators (MLEs) via a simulation study. The new model is better than some other important competitive models in modeling the breaking stress data, the glass fibers data and the relief time data.

Bi‐objective optimization of the tactical allocation of job types to machines: mathematical modeling, theoretical analysis, and numerical tests

AbstractWe introduce a tactical resource allocation model for a large aerospace engine system manufacturer aimed at long‐term production planning. Our model identifies the routings a product takes through the factory, and which machines should be qualified for a balanced resource loading, to reduce product lead times. We prove some important mathematical properties of the model that are used to develop a heuristic providing a good initial feasible solution. We propose a tailored approach for our class of problems combining two well‐known criterion space search algorithms, the bi‐directional ε‐constraint method and the augmented weighted Tchebycheff method. A computational investigation comparing solution times for several solution methods is presented for 60 numerical instances.

Half Cauchy Exponential Geometric Distribution: Model, Properties and Application

In this article, a new distribution called Half Cauchy Exponential Geometric Distribution is introduced. We have derived some important mathematical properties like hazard function, probability density function, survival function, quantiles, the measures of skewness based on quartiles and coefficient of kurtosis based on octiles of the new distribution. To estimate the parameters of the new distribution, we have applied the three commonly used estimation methods namely maximum likelihood estimation (MLE), least-square (LSE) method and Cramer-Von-Mises (CVM) method. We have used R programming as well as analytical methods for data analysis. For the assessment of potentiality of the new distribution, we have considered a real dataset and the goodness-of-fit attained by proposed distribution is compared with some competing distributions.It is found that the proposed model fits the data very well and more flexible as compared to some other models.