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Mathematics and Statistics
Number of Followers: 5 Open Access journal ISSN (Print) 2332-2071 - ISSN (Online) 2332-2144 Published by Horizon Research Publishing [54 journals] |
- Tensor Multivariate Trace Inequalities and Their Applications
Abstract: Publication date: May 2021
Source:Mathematics and Statistics Volume 9 Number 3 Shih Yu Chang and Hsiao-Chun Wu In linear algebra, the trace of a square matrix is defined as the sum of elements on the main diagonal. The trace of a matrix is the sum of its eigenvalues (counted with multiplicities), and it is invariant under the change of basis. This characterization can be used to define the trace of a tensor in general. Trace inequalities are mathematical relations between different multivariate trace functionals involving linear operators. These relations are straightforward equalities if the involved linear operators commute, however, they can be difficult to prove when the non-commuting linear operators are involved. Given two Hermitian tensors H1 and H2 that do not commute. Does there exist a method to transform one of the two tensors such that they commute without completely destroying the structure of the original tensor' The spectral pinching method is a tool to resolve this problem. In this work, we will apply such spectral pinching method to prove several trace inequalities that extend the Araki–Lieb–Thirring (ALT) inequality, Golden–Thompson(GT) inequality and logarithmic trace inequality to arbitrary many tensors. Our approaches rely on complex interpolation theory as well as asymptotic spectral pinching, providing a transparent mechanism to treat generic tensor multivariate trace inequalities. As an example application of our tensor extension of the Golden–Thompson inequality, we give the tail bound for the independent sum of tensors. Such bound will play a fundamental role in high-dimensional probability and statistical data analysis.
PubDate: May 2021
- No Finite Time Blowup for 3D Incompressible Navier Stokes Equations via
Scaling Invariance
Abstract: Publication date: May 2021
Source:Mathematics and Statistics Volume 9 Number 3 Terry E. Moschandreou The problem to The Clay Math Institute "Navier-Stokes, breakdown of smooth solutions here on an arbitrary cube subset of three dimensional space with periodic boundary conditions is examined. The incompressible Navier-Stokes Equations are presented in a new and conventionally different way here, by naturally reducing them to an operator form which is then further analyzed. It is shown that a reduction to a general 2D N-S system decoupled from a 1D non-linear partial differential equation is possible to obtain. This is executed using integration over n-dimensional compact intervals which allows decoupling. The operator form is considered in a physical geometric vorticity case, and a more general case. In the general case, the solution is revealed to have smooth solutions which exhibit finite-time blowup on a fine measure zero set and using the Prékopa-Leindler and Gagliardo-Nirenberg inequalities it is shown that for any non zero measure set in the form of cube subset of 3D there is no finite time blowup for the starred velocity for large dimension of cube and small d. In particular vortices are shown to exist and it is shown that zero is in the attractor of the 3D Navier-Stokes equations.
PubDate: May 2021
- Comparing the Performance of AdaBoost, XGBoost, and Logistic Regression
for Imbalanced Data
Abstract: Publication date: May 2021
Source:Mathematics and Statistics Volume 9 Number 3 Sharmeen Binti Syazwan Lai Nur Huda Nabihan Binti Md Shahri Mazni Binti Mohamad Hezlin Aryani Binti Abdul Rahman and Adzhar Bin Rambli An imbalanced data problem occurs in the absence of a good class distribution between classes. Imbalanced data will cause the classifier to be biased to the majority class as the standard classification algorithms are based on the belief that the training set is balanced. Therefore, it is crucial to find a classifier that can deal with imbalanced data for any given classification task. The aim of this research is to find the best method among AdaBoost, XGBoost, and Logistic Regression to deal with imbalanced simulated datasets and real datasets. The performances of these three methods in both simulated and real imbalanced datasets are compared using five performance measures, namely sensitivity, specificity, precision, F1-score, and g-mean. The results of the simulated datasets show that logistic regression performs better than AdaBoost and XGBoost in highly imbalanced datasets, whereas in the real imbalanced datasets, AdaBoost and logistic regression demonstrated similarly good performance. All methods seem to perform well in datasets that are not severely imbalanced. Compared to AdaBoost and XGBoost, logistic regression is found to predict better for datasets with severe imbalanced ratios. However, all three methods perform poorly for data with a 5% minority, with a sample size of n = 100. In this study, it is found that different methods perform the best for data with different minority percentages.
PubDate: May 2021
- Block Method for the Solution of First Order Nonlinear ODEs and Its
Application to HIV Infection of CD4+T Cells Model
Abstract: Publication date: May 2021
Source:Mathematics and Statistics Volume 9 Number 3 Adeyeye Oluwaseun and Omar Zurni Some of the issues relating to the human immunodeficiency virus (HIV) epidemic can be expressed as a system of nonlinear first order ordinary differential equations. This includes modelling the spread of the HIV virus in infecting CD4+T cells that help the human immune system to fight diseases. However, real life differential equation models usually fail to have an exact solution, which is also the case with the nonlinear model considered in this article. Thus, an approximate method, known as the block method, is developed to solve the system of first order nonlinear differential equation. To develop the block method, a linear block approach was adopted, and the basic properties required to classify the method as convergent were investigated. The block method was found to be convergent, which ascertained its usability for the solution of the model. The solution obtained from the newly developed method in this article was compared to previous methods that have been adopted to solve same model. In order to have a justifiable basis of comparison, two-step length values were substituted to obtain a one-step and two-step block method. The results show the newly developed block method obtaining accurate results in comparison to previous studies. Hence, this article has introduced a new method suitable for the direct solution of first order differential equation models without the need to simplify to a system of linear algebraic equations. Likewise, its convergent properties and accuracy also give the block method an edge over existing methods.
PubDate: May 2021
- Stationary and Non-Stationary Models of Extreme Ground-Level Ozone in
Peninsular Malaysia
Abstract: Publication date: May 2021
Source:Mathematics and Statistics Volume 9 Number 3 Siti Aisyah Zakaria Nor Azrita Mohd Amin Noor Fadhilah Ahmad Radi and Nasrul Hamidin High ground-level ozone (GLO) concentrations will adversely affect human health, vegetations as well as the ecosystem. Therefore, continuous monitoring for GLO trends is a good practice to address issues related to air quality based on high concentrations of GLO. The purpose of this study is to introduce stationary and non-stationary model of extreme GLO. The method is applied to 25 selected stations in Peninsular Malaysia. The maximum daily GLO concentration data over 8 hours from year 2000 to 2016 are used. The factors of this distribution are anticipated using maximum likelihood estimation. A comparison between stationary (constant model) and non-stationary (linear and cyclic model) is performed using the likelihood ratio test (LRT). The LRT is based on the larger value of deviance statistics compared to a chi-square distribution providing the significance evidence to non-stationary model either there is linear trend or cyclic trend. The best fit model between selected models is tested by Akaike's Information Criterion. The results show that 25 stations conform to the non-stationary model either linear or cyclic model, with 14 stations showing significant improvement over the linear model in location parameter while 11 stations follow the cyclic model. This study is important to identify the trends of ozone phenomenon for better quality risk management.
PubDate: May 2021
- Numerical Treatment for Solving Fuzzy Volterra Integral Equation by Sixth
Order Runge-Kutta Method
Abstract: Publication date: May 2021
Source:Mathematics and Statistics Volume 9 Number 3 Rawaa Ibrahim Esa Rasha H Ibraheem and Al.i F Jameel There has recently been considerable focus on finding reliable and more effective numerical methods for solving different mathematical problems with integral equations. The Runge–Kutta methods in numerical analysis are a family of iterative methods, both implicit and explicit, with different orders of accuracy, used in temporal and modification for the numerical solutions of integral equations. Fuzzy Integral equations (known as FIEs) make extensive use of many scientific analysis and engineering applications. They appear because of the incomplete information from their mathematical models and their parameters under fuzzy domain. In this paper, the sixth order Runge-Kutta is used to solve second-kind fuzzy Volterra integral equations numerically. The proposed method is reformulated and updated for solving fuzzy second-kind Volterra integral equations in general form by using properties and descriptions of fuzzy set theory. Furthermore a Volterra fuzzy integral equation, based on the parametric form of a fuzzy numbers, transforms into two integral equations of the second kind in the crisp case under fuzzy properties. We apply our modified method using the specific example with a linear fuzzy integral Volterra equation to illustrate the strengths and accurateness of this process. A comparison of evaluated numerical results with the exact solution for each fuzzy level set is displayed in the form of table and figures. Such results indicate that the proposed approach is remarkably feasible and easy to use.
PubDate: May 2021
- Approximation treatment for Linear Fuzzy HIV Infection Model by
Variational Iteration Method
Abstract: Publication date: May 2021
Source:Mathematics and Statistics Volume 9 Number 3 Hafed H Saleh Azmi A. and Ali. F. Jameel There has recently been considerable focus on finding reliable and more effective approximate methods for solving biological mathematical models in the form of differential equations. One of the well-known approximate or semi-analytical methods for solving linear, nonlinear differential well as partial differential equations within various fields of mathematics is the Variational Iteration Method (VIM). This paper looks at the use of fuzzy differential equations in human immunodeficiency virus (HIV) infection modeling. The main advantage of the method lies in its flexibility and ability to solve nonlinear equations easily. VIM is introduced to provide approximate solutions for linear ordinary differential equation system including the fuzzy HIV infection model. The model explains the amount of undefined immune cells, and the immune system viral load intensity intrinsic that will trigger fuzziness in patients infected by HIV. CD4+T-cells and cytototoxic T-lymphocytes (CTLs) are known for the immune cells concerned. The dynamics of the immune cell level and viral burden are analyzed and compared across three classes of patients with low, moderate and high immune systems. A modification and formulation of the VIM in the fuzzy domain based on the use of the properties of fuzzy set theory are presented. A model was established in this regard, accompanied by plots that demonstrate the reliability and simplicity of the methods. The numerical results of the model indicate that this approach is effective and easily used in fuzzy domain.
PubDate: May 2021
- On the Up-to-date Course of Mathematical Logic for the Future Math
Teachers
Abstract: Publication date: May 2021
Source:Mathematics and Statistics Volume 9 Number 3 E. N. Sinyukova S. V. Drahanyuk and O. O. Chepok All-round development of the everyday logic of students should be considered as one of the most important tasks of general secondary education on the whole and general secondary mathematics education in particular. We discuss the problem of organization in teachers' training institutions of higher education and the expedient training of the future math teachers at institutions of general secondary education. The main goal is to ensure their ability to realize all their future professional activities and the necessary participation in forming the everyday logic of their pupils. The authors think that vocational educational program of training is that the future secondary school math teachers must contain a separate course of mathematical logic including at least 90 training hours (3 credits ECTS). Although the content filling of the course cannot be irrespective of the general level of arrangement of mathematics education in the corresponding country, it ought to be a subject of discussion of the international mathematics community and managers in the sphere of higher mathematics education. Simultaneously, the role, the place, and the expedient structure of such a course in the corresponding training programs should be under discussion. The article represents the authors' point of view on the problems indicated above. The research has a qualitative characteristic as a whole. Only some of its conclusions have statistical corroboration.
PubDate: May 2021
- A Goal Programming Approach for Multivariate Calibration Weights
Estimation in Stratified Random Sampling
Abstract: Publication date: May 2021
Source:Mathematics and Statistics Volume 9 Number 3 Siham Rabee Ramadan Hamed Ragaa Kassem and Mahmoud Rashwaan Calibration estimation is one of the most important ways to improve the precision of the survey estimates. It is a method in which the designs weights are modified as little as possible by minimizing a given distance measure to the calibrated weights respecting a set of constraints related to suitable auxiliary information. This paper proposes a new approach for Multivariate Calibration Estimation (MCE) of the population mean of a study variable under stratified random sampling scheme using two auxiliary variables. Almost all literature on calibration estimation used Lagrange multiplier technique in order to estimate the calibrated weights. While Lagrange multiplier technique requires all equations included in the model to be differentiable functions, some un- differentiable functions may be faced in some cases. Hence, it is essential to look for using another technique that can provide more flexibility in dealing with the problem. Accordingly, in this paper, using goal programming approach is newly suggested as a different approach for MCE. The theory of the proposed calibration estimation is presented and the calibrated weights are estimated. A comparison study is conducted using actual and generated data to evaluate the performance of the proposed approach for multivariate calibration estimator with other existing calibration estimators. The results of this study prove that using the proposed GP approach for MCE is more flexible and efficient compared to other calibration estimation methods of the population mean.
PubDate: May 2021
- Per Capita Expenditure Modeling Using Spatial EBLUP Approach – SAE
Abstract: Publication date: May 2021
Source:Mathematics and Statistics Volume 9 Number 3 Luthfatul Amaliana Ani Budi Astuti and Nur Silviyah Rahmi Per capita expenditure of an area is a welfare indicator of the community. It is also a reflection of the economic capacity in meeting basic needs. Bali is the second richest province in Indonesia. This study aims to model the per capita expenditure of Bali at the sub-district level using Spatial-EBLUP (SEBLUP) approach in SAE. Small area estimation (SAE) modeling is an indirect estimation approach capable of increasing the effectiveness of sample sizes and minimizing variance. The heterogeneity of an area is influenced by other areas around. Everything is related to one another, but something closer will be more influential than something far away. Therefore, the spatial effect can be included in the random effect of a model small area, which is called as SEBLUP model. The selection of a spatial weights matrix is very important in spatial data modeling. It represents the neighborhood relationship of each spatial observation unit. A SEBLUP model needs a spatial weights matrix, which can be based on distance (radial distance and power distance), contiguity (queen), and a combination of distance and contiguity (radial distance and queen contiguity). The result of the implementation of the SEBLUP approach in per capita expenditure of Bali shows that the SEBLUP model with radial distance spatial weights matrix is the best model with the smallest ARMSE. South Denpasar Sub-district is the most prosperous sub-district with the highest per capita expenditure in Bali. Meanwhile, Abang Sub-district is the smallest per capita expenditure.
PubDate: May 2021
- An Approximation to Zeros of the Riemann Zeta Function Using Fractional
Calculus
Abstract: Publication date: May 2021
Source:Mathematics and Statistics Volume 9 Number 3 A. Torres-Hernandez and F. Brambila-Paz In this paper an approximation to the zeros of the Riemann zeta function has been obtained for the first time using a fractional iterative method which originates from a unique feature of the fractional calculus. This iterative method, valid for one and several variables, uses the property that the fractional derivative of constants are not always zero. This allows us to construct a fractional iterative method to find the zeros of functions in which it is possible to avoid expressions that involve hypergeometric functions, Mittag-Leffler functions or infinite series. Furthermore, we can find multiple zeros of a function using a singe initial condition. This partially solves the intrinsic problem of iterative methods, which in general is necessary to provide N initial conditions to find N solutions. Consequently the method is suitable for approximating nontrivial zeros of the Riemann zeta function when the absolute value of its imaginary part tends to infinity. Some examples of its implementation are presented, and finally 53 different values near to the zeros of the Riemann zeta function are shown.
PubDate: May 2021
- Almost Interior Gamma-ideals and Fuzzy Almost Interior Gamma-ideals in
Gamma-semigroups
Abstract: Publication date: May 2021
Source:Mathematics and Statistics Volume 9 Number 3 Wichayaporn Jantanan Anusorn Simuen Winita Yonthanthum and Ronnason Chinram Ideal theory plays an important role in studying in many algebraic structures, for example, rings, semigroups, semirings, etc. The algebraic structure Г-semigroup is a generalization of the classical semigroup. Many results in semigroups were extended to results in Г-semigroups. Many results in ideal theory of Г-semigroups were widely investigated. In this paper, we first focus to study some novel ideals of Г-semigroups. In Section 2, we define almost interior Г-ideals and weakly almost interior Г-ideals of Г-semigroups by using the concept ideas of interior Г-ideals and almost Г-ideals of Г-semigroups. Every almost interior Г-ideal of a Г-semigroup S is clearly a weakly almost interior Г-ideal of S but the converse is not true in general. The notions of both almost interior Г-ideals and weakly almost interior Г-ideals of Г-semigroups are generalizations of the notion of interior Г-ideal of a Г-semigroup S. We investigate basic properties of both almost interior Г-ideals and weakly almost interior Г-ideals of Г-semigroups. The notion of fuzzy sets was introduced by Zadeh in 1965. Fuzzy set is an extension of the classical notion of sets. Fuzzy sets are somewhat like sets whose elements have degrees of membership. In the remainder of this paper, we focus on studying some novelties of fuzzy ideals in Г-semigroups. In Section 3, we introduce fuzzy almost interior Г-ideals and fuzzy weakly almost interior Г-ideals of Г-semigroups. We investigate their properties. Finally, we give some relationship between almost interior Г-ideals [weakly almost interior Г-ideals] and fuzzy almost interior Г-ideals [fuzzy weakly almost interior Г-ideals] of Г-semigroups.
PubDate: May 2021
- On the Stochastic Processes on 7-Dimensional Spheres
Abstract: Publication date: May 2021
Source:Mathematics and Statistics Volume 9 Number 3 Nurfa Risha and Muhammad Farchani Rosyid We studied isometric stochastic flows of a Stratonovich stochastic differential equation on spheres, i.e., on the standard sphere and Gromoll-Meyer exotic sphere . In this case, and are homeomorphic but not diffeomorphic. The standard sphere can be constructed as the quotient manifold with the so-called -action of S3, whereas the Gromoll-Meyer exotic sphere as the quotient manifold with respect to the so-called -action of S3. The corresponding continuous-time stochastic process and its properties on the Gromoll-Meyer exotic sphere can be obtained by constructing a homeomorphism . The stochastic flow can be regarded as the same stochastic flow on S7, but viewed in Gromoll-Meyer differential structure. The flow on and the corresponding flow on constructed in this paper have the same regularities. There is no difference between the stochastic flow's appearance on S7 viewed in standard differential structure and the appearance of the same stochastic flow viewed in the Gromoll-Meyer differential structure. Furthermore, since the inverse mapping h-1 is differentiable on , the Riemannian metric tensor on , i.e., the pull-back of the Riemannian metric tensor G on the standard sphere , is also differentiable. This fact implies, for instance, the fact that the Fokker-Planck equation associated with the stochastic flow and the Fokker-Planck equation associated with the stochastic differential equation have the same regularities provided that the function β is C1-differentiable. Therefore both differential structures on S7 give the same description of the dynamics of the distribution function of the stochastic process understudy on seven spheres.
PubDate: May 2021
- Reducing Approximation Error with Rapid Convergence Rate for Non-Negative
Matrix Factorization (NMF)
Abstract: Publication date: May 2021
Source:Mathematics and Statistics Volume 9 Number 3 Jayanta Biswas Pritam Kayal and Debabrata Samanta Non-Negative Matrix Factorization (NMF) is utilized in many important applications. This paper presents development of an efficient low rank approximate NMF algorithm for feature extraction related to text mining and spectral data analysis. NMF can be used for clustering. NMF factorizes a positive matrix A to two positive matrices W and H matrices where A=WH. The proposal uses k-means clustering algorithm to determine the centroid of each cluster and assigns the centroid coordinates of each cluster as one column for W matrix. The initial choice of W matrix is positive. The H matrix is determined with gradient descent algorithm based on thin QR optimization. The performance comparison of the proposed NMF algorithm is illustrated with results. The accurate choice of initial positive W matrix reduces approximation error and the use of thin QR algorithm in combination with gradient descent approach provides rapid convergence rate for NMF. The proposed algorithm is implemented with the randomly generated matrix in MATLAB environment. The number of significant singular values of the generated matrix is selected as the number of clusters. The error and convergence rate comparison of the proposed algorithm with the current algorithms are demonstrated in this research. The accurate measurement of execution time for individual program is not possible in MATLAB. The average time execution over 200 iterations is therefore calculated with an increasing iteration count of the proposed algorithm and the comparative results are presented.
PubDate: May 2021
- Application of Supersaturated Design to Study the Spread of Electronic
Games
Abstract: Publication date: May 2021
Source:Mathematics and Statistics Volume 9 Number 3 Alanazi Talal Abdulrahman Randa Alharbi Osama Alamri Dalia Alnagar and Bader Alruwaili A supersaturated design is an important method that relies on factorial designs whose number of factors is greater than experiments' number. The analysis of supersaturated designs is challenging due to the complexity of the design matrix. This problem is challenging due to the fact that the design matrix has a complicated structure. Identification of the variable including the active factor plays an essential role when supersaturated design is used to analyse the data. A variable selection technique to screen active effects in the SSDs and regression analysis are applied to our case study. This study set out to examine the actual reasons for the spread of electronic games statistically such as Saudi society. An online survey provided quantitative data from 200 participants. Respondents were randomly divided into two conditions (Yes+, No-) and asked to respond to one of two sets of the causes of electronic games. The responses was analysed using contrast method with supersaturated designs and regression methods using the SPSS computer software to determine the actual causes that led to the spread of electronic games. The findings indicated that because of their constant preoccupation, some parents resort to such games in order to get rid of the child's inconvenience and insufficient awareness among parents of the dangers of these games, and excessive pampering is the factor that led to the spread of electronic games in Saudi society statistically. On this basis, it is recommended that Saudi government professionals develop an operational plan to study these causes to take actions. In future investigations, no recent studies address the external environmental aspects that could influence gaming among individuals, and hence further research is required in this field.
PubDate: May 2021
- Volume Minimization of a Closed Coil Helical Spring Using ALO, GWO, DA,
FA, FPA, WOA, CSO. BA, PSO and GSA
Abstract: Publication date: May 2021
Source:Mathematics and Statistics Volume 9 Number 3 Rejula Mercy. J and S. Elizabeth Amudhini Stephen Springs are important members often used in machines to exert force, absorb energy and provide flexibility. In mechanical systems, wherever flexibility or relatively a large load under the given circumstances is required, some form of spring is used. In this paper, non-traditional optimization algorithms, namely, Ant Lion Optimizer, Grey Wolf Optimizer, Dragonfly optimization algorithm, Firefly algorithm, Flower Pollination Algorithm, Whale Optimization Algorithm, Cat Swarm Optimization, Bat Algorithm, Particle Swarm Optimization, Gravitational Search Algorithm are proposed to get the global optimal solution for the closed coil helical spring design problem. The problem has three design variables and eight inequality constraints and three bounds. The mathematical formulation of the objective function U is to minimize the volume of closed coil helical spring subject to constraints. The design variables considered are Wire diameter d, Mean coil diameter D, Number of active coils N of the spring. The proposed methods are tested and the performance is evaluated. Ten non-traditional optimization methods are used to find the minimum volume. The problem is computed in the MATLAB environment. The experimental results show that Particle Swarm Optimization outperforms other methods. The results show that PSO gives better results in terms of consistency and minimum value in terms of time and volume of a closed coil helical spring compared to other methods. When compared to other Optimization methods, PSO has few advantages like simplicity and efficiency. In the future, PSO could be extended to solve other mechanical element problems.
PubDate: May 2021
- A New Three-Parameter Weibull Inverse Rayleigh Distribution: Theoretical
Development and Applications
Abstract: Publication date: May 2021
Source:Mathematics and Statistics Volume 9 Number 3 Adeyinka Solomon Ogunsanya Waheed Babatunde Yahya Taiwo Mobolaji Adegoke Christiana Iluno Oluwaseun R. Aderele and Matthew Iwada Ekum In this work, a three-parameter Weibull Inverse Rayleigh (WIR) distribution is proposed. The new WIR distribution is an extension of a one-parameter Inverse Rayleigh distribution that incorporated a transformation of the Weibull distribution and Log-logistic as quantile function. The statistical properties such as quantile function, order statistic, monotone likelihood ratio property, hazard, reverse hazard functions, moments, skewness, kurtosis, and linear representation of the new proposed distribution were studied theoretically. The maximum likelihood estimators cannot be derived in an explicit form. So we employed the iterative procedure called Newton Raphson method to obtain the maximum likelihood estimators. The Bayes estimators for the scale and shape parameters for the WIR distribution under squared error, Linex, and Entropy loss functions are provided. The Bayes estimators cannot be obtained explicitly. Hence we adopted a numerical approximation method known as Lindley's approximation in other to obtain the Bayes estimators. Simulation procedures were adopted to see the effectiveness of different estimators. The applications of the new WIR distribution were demonstrated on three real-life data sets. Further results showed that the new WIR distribution performed credibly well when compared with five of the related existing skewed distributions. It was observed that the Bayesian estimates derived performs better than the classical method.
PubDate: May 2021
- Statistical Analyses on Factors Affecting Retirement Savings Decision in
Malaysia
Abstract: Publication date: May 2021
Source:Mathematics and Statistics Volume 9 Number 3 Nurul Sima Mohamad Shariff and Waznatul Widad Mohamad Ishak Retirement savings decision is related to the individual judgment on savings planning, and preparation for the retirement. Several factors may affect this decision towards retirement savings. Some of them are demographic factors and other determinants, such as financial knowledge and management, future expectation, social influences and risk tolerance. Due to this interest, this study aims to impact of such factors on retirement savings decision. Furthermore, this study will also discuss the retirement savings decision among Malaysians at different age groups. The data were collected through a survey strategy by using a set of questionnaires. The questions were divided into several sections on the demographic profile, Likert-scale questions on the factors, and the retirement savings decisions. The technique sampling used in this study is a random sampling with 385 respondents. As such, several statistical procedures will be utilized such as the reliability test, Kruskal-Wallis H test, and the ordered probit model. The results of this study found that age, financial knowledge and management, future expectation, and social influences were the significant determinants towards retirement savings decision in Malaysia.
PubDate: May 2021
- Relative Complexity Index for Decision-Making Method
Abstract: Publication date: May 2021
Source:Mathematics and Statistics Volume 9 Number 3 Harliza Mohd Hanif Daud Mohamad and Rosma Mohd Dom The complexity of a method has been discussed in the decision-making area since complexity may impose some disadvantages such as loss of information and a high degree of uncertainty. However, there is no empirical justification to determine the complexity level of a method. This paper focuses on introducing a method of measuring the complexity of the decision-making method. In the computational area, there is an established method of measuring complexity named Big-O Notation. This paper adopts the method for determining the complexity level of the decision-making method. However, there is a lack of applying Big-O in the decision-making method. Applying Big-O in decision-making may not be able to differentiate the complexity level of two different decision-making methods. Hence, this paper introduces a Relative Complexity Index (RCI) to cater to this problem. The basic properties of the Relative Complexity Index are also discussed. After the introduction of the Relative Complexity Index, the method is implemented in Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method.
PubDate: May 2021
- Z-Score Functions of Dual Hesitant Fuzzy Set and Its Applications in
Multi-Criteria Decision Making
Abstract: Publication date: May 2021
Source:Mathematics and Statistics Volume 9 Number 3 Zahari Md Rodzi Abd Ghafur Ahmad Nur Sa’aidah Ismail Wan Normila Mohamad and Sarahiza Mohmad Dual hesitant fuzzy set (DHFS) consists of two parts: membership hesitant function and non-membership hesitant function. This set supports more exemplary and flexible access to set degrees for each element in the domain and can address two types of hesitant in this situation. It can be considered a powerful tool for expressing uncertain information in the decision-making process. The function of z-score, namely z-arithmetic mean, z-geometric mean, and z-harmonic mean, has been proposed with five important bases, these bases are hesitant degree for dual hesitant fuzzy element (DHFE), DHFE deviation degree, parameter α (the importance of the hesitant degree), parameter β (the importance of the deviation degree) and parameter ϑ (the importance of membership (positive view) or non-membership (negative view). A comparison of the z-score with the existing score function was made to show some of their drawbacks. Next, the z-score function is then applied to solve multi-criteria decision making (MCDM) problems. To illustrate the proposed method's effectiveness, an example of MCDM specifically in pattern recognition has been shown.
PubDate: May 2021
- Two Observations in the Application of Logarithm Theory and their
Implications for Economic Modeling and Analysis
Abstract: Publication date: May 2021
Source:Mathematics and Statistics Volume 9 Number 3 Oluremi Davies Ogun The contents of this paper apply to researches in the fields of economics, statistics – physical or life sciences, other social sciences, accounting and finance, business management and mathematics – core and applied. First, I discussed the misconception and the implications thereof, inherent in the conventional practice of entering interest rates as natural or untransformed series in data analysis most especially, regression models. The trends and variabilities of both transformed and untransformed interest rate series were shown to be similar thereby enhancing the likelihood of similar performances in regressions. By extension therefore, the indicated conventional practice unnecessarily and unjustifiably precluded elasticity inference on the coefficients of interest rates and summing up to procedural inefficiency as an independent computation of elasticity became the only available option. Percentages were not the equivalence of percentage changes and thus only series in growth terms hence, percentage changes should be spared log transformation. Secondly, the paper stressed the imperative to avoid unwieldy and theory incongruent expressions in post preliminary data analysis, by flagging the idea that regression models, in particular, of the growth varieties, should as much as practicable, sync with the dictates of modern time series econometrics in the specification of final equations.
PubDate: May 2021
- On Some Properties of Leibniz's Triangle
Abstract: Publication date: May 2021
Source:Mathematics and Statistics Volume 9 Number 3 R. Sivaraman One of the Greatest mathematicians of all time, Gotfried Leibniz, introduced amusing triangular array of numbers called Leibniz's Harmonic triangle similar to that of Pascal's triangle but with different properties. I had introduced entries of Leibniz's triangle through Beta Integrals. In this paper, I have proved that the Beta Integral assumption is exactly same as that of entries obtained through Pascal's triangle. The Beta Integral formulation leads us to establish several significant properties related to Leibniz's triangle in quite elegant way. I have shown that the sum of alternating terms in any row of Leibniz's triangle is either zero or a Harmonic number. A separate section is devoted in this paper to prove interesting results regarding centralized Leibniz's triangle numbers including obtaining a closed expression, the asymptotic behavior of successive centralized Leibniz's triangle numbers, connection between centralized Leibniz's triangle numbers and Catalan numbers as well as centralized binomial coefficients, convergence of series whose terms are centralized Leibniz's triangle numbers. All the results discussed in this section are new and proved for the first time. Finally, I have proved two exceedingly important theorems namely Infinite Hockey Stick theorem and Infinite Triangle Sum theorem. Though these two theorems were known in literature, the way of proving them using Beta Integral formulation is quite new and makes the proof short and elegant. Thus, by simple re-formulation of entries of Leibniz's triangle through Beta Integrals, I have proved existing as well as new theorems in much compact way. These ideas will throw a new light upon understanding the fabulous Leibniz's number triangle.
PubDate: May 2021
- The Seasonal Reproduction Number of p.vivax Malaria Dynamics in Korea
Abstract: Publication date: Mar 2021
Source:Mathematics and Statistics Volume 9 Number 2 Anne M. Fernando Ana Vivas Barber and Sunmi Lee Understanding the dynamics of Malaria can help in reducing the impact of the disease. Previous research proved that including animals in the human transmission model, or 'zooprophylaxis', is effective in reducing transmission of malaria in the human population. This model studies plasmodium vivax malaria and has variables for animal population and mosquito attraction to animals. The existing time-independent Malaria population ODE model is extended to time-dependent model with the differences explored. We introduce the seasonal mosquito population, a Gaussian profile based on data, as a variant for the previous models. The seasonal reproduction number is found using the next generation matrix, endemic and stability analysis is carried out using dynamical systems theory. The model includes short and long term human incubation periods and sensitivity analysis on parameters and all simulations are over three year period. Simulations show for each year larger peaks in the infected populations and seasonal reproduction number during the summer months and we analyze which parameters have more sensitivity in the model and in the seasonal reproduction number. Analysis provides conditions for disease free equilibrium (DFE) and the system is found to be locally asymptotically stable around the DFE when R0
PubDate: Mar 2021
- A New Solution for The Enzymatic Glucose Fuel Cell Model with Morrison
Equation via Haar Wavelet Collocation Method
Abstract: Publication date: Mar 2021
Source:Mathematics and Statistics Volume 9 Number 2 Kuntida Kawinwit Akapak Charoenloedmongkhon and Sanoe Koonprasert Integral equations are essential tools in various areas of applied mathematics. A computational approach to solving an integral equation is important in scientific research. The Haar wavelet collocation method (HWCM) with operational matrices of integration is one famous method which has been applied to solve systems of linear integral equations. In this paper, an approximated analytical method based on the Haar wavelet collocation method is applied to the system of diffusion convection partial differential equations with initial and boundary conditions. This system determines the enzymatic glucose fuel cell with the chemical reaction rate of the Morrison equation. The enzymatic glucose fuel cell model describes the concentration of glucose and hydrogen ion that can be converted into energy. During the process, the model reduces to the linear integral equation system including computational Haar matrices. The computational Haar matrices can be computed by HWCM coding in the Maple program. Illustrated examples are provided to demonstrate the preciseness and effectiveness of the proposed method. The results are shown as numerical solutions of glucose and hydrogen ion.
PubDate: Mar 2021
- A Dirac Delta Operator
Abstract: Publication date: Mar 2021
Source:Mathematics and Statistics Volume 9 Number 2 Juan Carlos Ferrando If T is a (densely defined) self-adjoint operator acting on a complex Hilbert space H and I stands for the identity operator, we introduce the delta function operator at T. When T is a bounded operator, then is an operator-valued distribution. If T is unbounded, is a more general object that still retains some properties of distributions. We provide an explicit representation of in some particular cases, derive various operative formulas involving and give several applications of its usage in Spectral Theory as well as in Quantum Mechanics.
PubDate: Mar 2021
- On Non-Associative Rings
Abstract: Publication date: Mar 2021
Source:Mathematics and Statistics Volume 9 Number 2 Ida Kurnia Waliyanti Indah Emilia Wijayanti and M. Farchani Rosyid Jordan ring is one example of the non-associative rings. We can construct a Jordan ring from an associative ring by defining the Jordan product. In this paper, we discuss the properties of non-associative rings by studying the properties of the Jordan rings. All of the ideals of a non-associative ring R are non-associative, except the ideal generated by the associator in R. Hence, a quotient ring can be constructed, where is the ideal generated by associators in R. The fundamental theorem of the homomorphism ring can be applied to the non-associative rings. By a little modification, we can find that is isomorphic to . Furthermore, we define a module over a non-associative ring and investigate its properties. We also give some examples of such modules. We show if M is a module over a non-associative ring R, then M is also a module over if is contained in the annihilator of R. Moreover, we define the tensor product of modules over a non-associative ring. The tensor product of the modules over a non-associative ring is commutative and associative up to isomorphism but not element by element.
PubDate: Mar 2021
- Solving One-Dimensional Porous Medium Equation Using Unconditionally
Stable Half-Sweep Finite Difference and SOR Method
Abstract: Publication date: Mar 2021
Source:Mathematics and Statistics Volume 9 Number 2 Jackel Vui Lung Chew Jumat Sulaiman and Andang Sunarto A porous medium equation is a nonlinear parabolic partial differential equation that presents many physical occurrences. The solutions of the porous medium equation are important to facilitate the investigation on nonlinear processes involving fluid flow, heat transfer, diffusion of gas-particles or population dynamics. As part of the development of a family of efficient iterative methods to solve the porous medium equation, the Half-Sweep technique has been adopted. Prior works in the existing literature on the application of Half-Sweep to successfully approximate the solutions of several types of mathematical problems are the underlying motivation of this research. This work aims to solve the one-dimensional porous medium equation efficiently by incorporating the Half-Sweep technique in the formulation of an unconditionally-stable implicit finite difference scheme. The noticeable unique property of Half-Sweep is its ability to secure a low computational complexity in computing numerical solutions. This work involves the application of the Half-Sweep finite difference scheme on the general porous medium equation, until the formulation of a nonlinear approximation function. The Newton method is used to linearize the formulated Half-Sweep finite difference approximation, so that the linear system in the form of a matrix can be constructed. Next, the Successive Over Relaxation method with a single parameter was applied to efficiently solve the generated linear system per time step. Next, to evaluate the efficiency of the developed method, deemed as the Half-Sweep Newton Successive Over Relaxation (HSNSOR) method, the criteria such as the number of iterations, the program execution time and the magnitude of absolute errors were investigated. According to the numerical results, the numerical solutions obtained by the HSNSOR are as accurate as those of the Half-Sweep Newton Gauss-Seidel (HSNGS), which is under the same family of Half-Sweep iterations, and the benchmark, Newton-Gauss-Seidel (NGS) method. The improvement in the numerical results produced by the HSNSOR is significant, and requires a lesser number of iterations and a shorter program execution time, as compared to the HSNGS and NGS methods.
PubDate: Mar 2021
- Some Remarks and Propositions on Riemann Hypothesis
Abstract: Publication date: Mar 2021
Source:Mathematics and Statistics Volume 9 Number 2 Jamal Salah In 1859, Bernhard Riemann, a German mathematician, published a paper to the Berlin Academy that would change mathematics forever. The mystery of prime numbers was the focus. At the core of the presentation was indeed a concept that had not yet been proven by Riemann, one that to this day baffles mathematicians. The way we do business could have been changed if the Riemann hypothesis holds true, which is because prime numbers are the key element for banking and e-commerce security. It will also have a significant influence, impacting quantum mechanics, chaos theory, and the future of computation, on the cutting edge of science. In this article, we look at some well-known results of Riemann Zeta function in a different light. We explore the proofs of Zeta integral Representation, Analytic continuity and the first functional equation. Initially, we observe omitting a logical undefined term in the integral representation of Zeta function by the means of Gamma function. For that we propound some modifications in order to reasonably justify the location of the non-trivial zeros on the critical line: s= 1/2 by assuming that ζ(s) and ζ(1-s) simultaneously equal zero. Consequently, we conditionally prove Riemann Hypothesis.
PubDate: Mar 2021
- On Three-Dimensional Mixing Geometric Quadratic Stochastic Operators
Abstract: Publication date: Mar 2021
Source:Mathematics and Statistics Volume 9 Number 2 Ftameh Khaled and Pah Chin Hee It is widely recognized that the theory of quadratic stochastic operator frequently arises due to its enormous contribution as a source of analysis for the investigation of dynamical properties and modeling in diverse domains. In this paper, we are motivated to construct a class of quadratic stochastic operators called mixing quadratic stochastic operators generated by geometric distribution on infinite state space . We also study regularity of such operators by investigating of the limit behavior for each case of the parameter. Some of non-regular cases proved for a new definition of mixing operators by using the shifting definition, where the new parameters satisfy the shifted conditions. A mixing quadratic stochastic operator was established on 3-partitions of the state space and considered for a special case of the parameter Ɛ. We found that the mixing quadratic stochastic operator is a regular transformation for and is a non-regular for . Also, the trajectories converge to one of the fixed points. Stability and instability of the fixed points were investigated by finding of the eigenvalues of Jacobian matrix at these fixed points. We approximate the parameter Ɛ by the parameter , where we established the regularity of the quadratic stochastic operators for some inequalities that satisfy . We conclude this paper by comparing with previous studies where we found some of such quadratic stochastic operators will be non-regular.
PubDate: Mar 2021
- Formulation of a New Implicit Method for Group Implicit BBDF in Solving
Related Stiff Ordinary Differential Equations
Abstract: Publication date: Mar 2021
Source:Mathematics and Statistics Volume 9 Number 2 Norshakila Abd Rasid Zarina Bibi Ibrahim Zanariah Abdul Majid and Fudziah Ismail This paper proposed a new alternative approach of the implicit diagonal block backward differentiation formula (BBDF) to solve linear and nonlinear first-order stiff ordinary differential equations (ODEs). We generate the solver by manipulating the numbers of back values to achieve a higher-order possible using the interpolation procedure. The algorithm is developed and implemented in C ++ medium. The numerical integrator approximates few solution points concurrently with off-step points in a block scheme over a non-overlapping solution interval at a single iteration. The lower triangular matrix form of the implicit diagonal causes fewer differentiation coefficients and ultimately reduces the execution time during running codes. We choose two intermediate points as off-step points appropriately, which are proven to guarantee the method's zero stability. The off-step points help to increase the accuracy by optimizing the local truncation error. The proposed solver satisfied theoretical consistency and zero-stable requirements, leading to a convergent multistep method with third algebraic order. We used the well-known and standard linear and nonlinear stiff IVP problems used in literature for validation to measure the algorithm's accuracy and processor time efficiency. The performance metrics are validated by comparing them with a proven solver, and the output shows that the alternative method is better than the existing one.
PubDate: Mar 2021
- The Varying Threshold Values of Logistic Regression and Linear
Discriminant for Classifying Fraudulent Firm
Abstract: Publication date: Mar 2021
Source:Mathematics and Statistics Volume 9 Number 2 Samingun Handoyo Ying-Ping Chen Gugus Irianto and Agus Widodo The aim of the research is to find the best performance both of logistic regression and linear discriminant which their threshold uses some various values. The performance tools used for evaluating classifier model are confusion matrix, precision-recall, F1 score and receiver operation characteristic (ROC) curve. The Audit-risk data set are used for the implementation of the proposed method. The screening data and dimension reduction by using principal component analysis (PCA) are the first step that must be conducted before the data are divided into the training and testing set. After the training process for obtaining the classifier model parameters has been completed, the calculation of performance measures is done only on the testing set where the various constants are added to the threshold value of both classifier models. The logistic regression classifier has the best performance of 94% on the precision-recall, 91.7% on the F1-score, and 0.906 on the area under curve (AUC) where the threshold values are on the interval between 0.002 and 0.018. On the other hand, the linear discriminant classifier has the best performance when the threshold value is 0.035 and its performance value is respectively the precision-recall of 94%, the F1-score of 91.7%, and the AUC of 0.846.
PubDate: Mar 2021
- Polya's Problem Solving Strategy in Trigonometry: An Analysis of Students'
Difficulties in Problem Solving
Abstract: Publication date: Mar 2021
Source:Mathematics and Statistics Volume 9 Number 2 Dwi Sulistyaningsih Eko Andy Purnomo and Purnomo This study is focused on investigating errors made by students and the various causal factors in working on trigonometry problems by applying sine and cosine rules. Samples were taken randomly from high school students. Data were collected in two ways, namely a written test that was referred to Polya's strategy and interviews with students who made mistakes. Students' errors were analyzed with the Newman concept. The results show that all types of errors occurred with a distribution of 3.83, 19.15, 24.74, 24.89 and 27.39% for reading errors (RE), comprehension error (CE), transformation errors (TE), process skill errors (PSE), and encoding errors (EE), respectively. The RE, CE, TE, PSE, and EE are marked by errors in reading symbols or important information, misunderstanding information and not understanding what is known and questioned, cannot change problems into mathematical models and also incorrectly use signs in arithmetic operations, student inaccuracies in the process of answering and also their lack of understanding in fraction operations, and the inability to deduce answers, respectively. An anomaly occurs because it turns out students who have medium trigonometry achievements make more mistakes than students who have low achievement.
PubDate: Mar 2021
- Instrument Test Development of Mathematics Skill on Elementary School
Abstract: Publication date: Mar 2021
Source:Mathematics and Statistics Volume 9 Number 2 Viktor Pandra Badrun Kartowagiran and Sugiman The aims of this research are: 1) producing the test instrument of mathematics skill on elementary school which is valid and reliable, 2) finding out the characteristics of the test instrument of mathematics skill on elementary school. The instrument test development in this research uses the development model of Wilson, Oriondo and Antonio which is modified. The number of testing sample in this research is 160 students in each class. This research results: 1) the validity index of aiken v is 0.979 in grade IV and 0.988 in grade V. The coefficient of instrument skill in class IV and V are 0.883 and 0.954. 2) the compatibility model in this research is it is suitable for 1PL model or parameter b (difficulty level). The result of parameter analysis of test item in class IV and V, shows that the overall item is in good category which is between -2 to 2. The case indicates that the overall item is accepted and reliable to be used for measuring the development of mathematics skill of elementary school students.
PubDate: Mar 2021
- Numerical Solution for Fuzzy Diffusion Problem via Two Parameter
Alternating Group Explicit Technique
Abstract: Publication date: Mar 2021
Source:Mathematics and Statistics Volume 9 Number 2 A. A. Dahalan and J. Sulaiman The computational technique has become a significant area of study in physics and engineering. The first method to evaluate the problems numerically was a finite difference. In 2002, a computational approach, an explicit finite difference technique, was used to overcome the fuzzy partial differential equation (FPDE) based on the Seikkala derivative. The application of the iterative technique, in particular the Two Parameter Alternating Group Explicit (TAGE) method, is employed to resolve the finite difference approximation resulting after the fuzzy heat equation is investigated in this article. This article broadens the use of the TAGE iterative technique to solve fuzzy problems due to the reliability of the approaches. The development and execution of the TAGE technique towards the full-sweep (FS) and half-sweep (HS) techniques are also presented. The idea of using the HS scheme is to reduce the computational complexity of the iterative methods by nearly/more than half. Additionally, numerical outcomes from the solution of two experimental problems are included and compared with the Alternating Group Explicit (AGE) approaches to clarify their feasibility. In conclusion, the families of the TAGE technique have been used to overcome the linear system structure through a one-dimensional fuzzy diffusion (1D-FD) discretization using a finite difference scheme. The findings suggest that the HSTAGE approach is surpassing in terms of iteration counts, time taken, and Hausdorff distance relative to the FSTAGE and AGE approaches. It demonstrates that the number of iterations for HSTAGE approach has decreased by approximately 71.60-72.95%, whereas for the execution time, the implementation of HSTAGE method is between 74.05-86.42% better. Since TAGE is ideal for concurrent processing, this method has been seen as the key benefit as it consumes sets of independent tasks that can be performed at the same time. The ability of the suggested technique is projected to be useful for the advanced exploration in solving any multi-dimensional FPDEs.
PubDate: Mar 2021
- Prospective Filipino Teachers' Disposition to Mathematics
Abstract: Publication date: Mar 2021
Source:Mathematics and Statistics Volume 9 Number 2 Restituto M. Llagas Jr. Studying mathematics comprises acquiring a positive disposition toward mathematics and seeing mathematics as an effective way of looking at real-life situations. This study aimed to correlate the disposition to Mathematics of prospective Filipino teachers to some teacher-related variables. The participants were the prospective Filipino teachers at the University of Northern Philippines (UNP) and at the Divine Word College of Vigan (DWCV). Two sets of instruments were utilized in the study – the self-report questionnaire and the Mathematics Dispositional Functioning Inventory developed by Beyers [1]. Frequency and percentage, weighted mean, and chi-square were utilized for data analysis. Results show that the overall disposition to mathematics of the participants is "Positive". The cognitive, affective, and conative aspects received a positive disposition. However, some items show an uncertain disposition to mathematics. The participants' profile variables have no significant relationship with their cognitive and conative disposition to mathematics. A training plan was conceptualized to provide information on the results of the study, to enhance the awareness and understanding of dispositions, to equip appropriate methods in solving mathematical problems, and to provide enrichment activities that will foster a positive disposition to mathematics and consequently will improve prospective teachers' and students' performance. Teachers are influential to the development of the students of effective ways of learning, doing, and thinking about mathematics. Understanding how attitudes are learned to establish an association between the teacher's disposition and students' attitude and performance. Thus, fostering dispositions to mathematics through training improves prospective Filipino teachers' and students' performance.
PubDate: Mar 2021
- On Application of Max-Plus Algebra to Synchronized Discrete Event System
Abstract: Publication date: Mar 2021
Source:Mathematics and Statistics Volume 9 Number 2 A. A. Aminu S. E. Olowo I. M. Sulaiman N. Abu Bakar and M. Mamat Max-plus algebra is a discrete algebraic system developed on the operations max () and plus (), where the max and plus operations are defined as addition and multiplication in conventional algebra. This algebraic structure is a semi-ring with its elements being real numbers along with ε=-∞ and e=0. On the other hand, the synchronized discrete event problem is a problem in which an event is scheduled to meet a deadline. There are two aspects of this problem. They include the events running simultaneously and the completion of the lengthiest event at the deadline. A recent survey on max-plus linear algebra shows that the operations max () and plus () play a significant role in modeling of human activities. However, numerous studies have shown that there are very limited literatures on the application of the max-plus algebra to real-life problems. This idea motivates the basic algebraic results and techniques of this research. This paper proposed the discrepancy method of max-plus for solving n×n system of linear equations with n≤n, and further show that an nxn linear system of equations will have either a unique solution, an infinitely many solutions or no solution whiles nxn linear system of equations has either an infinitely many solutions or no solution in (). Also, the proposed concept was extended to the job-shop problem in a synchronized event. The results obtained have shown that the method is very efficient for solving n×n system of linear equations and is also applicable to job-shop problems.
PubDate: Mar 2021
- On the Gaussian Approximation to Bayesian Posterior Distributions
Abstract: Publication date: Jul 2021
Source:Mathematics and Statistics Volume 9 Number 4 Christoph Fuhrmann Hanns-Ludwig Harney Klaus Harney and Andreas M¨uller The present article derives the minimal number N of observations needed to approximate a Bayesian posterior distribution by a Gaussian. The derivation is based on an invariance requirement for the likelihood . This requirement is defined by a Lie group that leaves the unchanged, when applied both to the observation(s) and to the parameter to be estimated. It leads, in turn, to a class of specific priors. In general, the criterion for the Gaussian approximation is found to depend on (i) the Fisher information related to the likelihood , and (ii) on the lowest non-vanishing order in the Taylor expansion of the Kullback-Leibler distance between and , where is the maximum-likelihood estimator of , given by the observations . Two examples are presented, widespread in various statistical analyses. In the first one, a chi-squared distribution, both the observations and the parameter are defined all over the real axis. In the other one, the binomial distribution, the observation is a binary number, while the parameter is defined on a finite interval of the real axis. Analytic expressions for the required minimal N are given in both cases. The necessary N is an order of magnitude larger for the chi-squared model (continuous ) than for the binomial model (binary ). The difference is traced back to symmetry properties of the likelihood function . We see considerable practical interest in our results since the normal distribution is the basis of parametric methods of applied statistics widely used in diverse areas of research (education, medicine, physics, astronomy etc.). To have an analytical criterion whether the normal distribution is applicable or not, appears relevant for practitioners in these fields.
PubDate: Jul 2021
- Inference on P[Y < X] for Geometric Extreme Exponential Distribution
Abstract: Publication date: Jul 2021
Source:Mathematics and Statistics Volume 9 Number 4 Reza Pakyari Geometric Extreme Exponential Distribution (GEE) is one of the statistical models that can be useful in fitting and describing lifetime data. In this paper, the problem of estimation of the reliability R = P(Y < X) when X and Y are independent GEE random variables with common scale parameter but different shape parameters has been considered. The probability R = P(Y < X) is also known as stress-strength reliability parameter and demonstrates the case where a component has stress X and is subjected to strength Y. The reliability R = P(Y < X) has applications in engineering, finance and biomedical sciences. We present the maximum likelihood estimator of R and study its asymptotic behavior. We first study the asymptotic distribution of the maximum likelihood estimators of the GEE parameters. We prove that the maximum likelihood estimators and so the reliability R have asymptotic normal distribution. A bootstrap confidence interval for R is also presented. Monte Carlo simulations are performed to assess he performance of the proposed estimation method and validity of the confidence interval. We found that the performance of the maximum likelihood estimator and also the bootstrap confidence interval is satisfactory even for small sample sizes. Analysis of a dataset has been given for illustrative purposes.
PubDate: Jul 2021
- Finitely Generated Modules's Uniserial Dimensions Over a Discrete
Valuation Domain
Abstract: Publication date: Jul 2021
Source:Mathematics and Statistics Volume 9 Number 4 Samsul Arifin Hanni Garminia and Pudji Astuti We present some methods for calculating the module's uniserial dimension that finitely generated over a DVD in this article. The idea of a module's uniserial dimension over a commutative ring, which defines how far the module deviates from being uniserial, was recently proposed by Nazemian etc. They show that if R is Noetherian commutative ring, which implies that every finitely generated module over R has uniserial dimension. Ghorbani and Nazemians have shown that R is Noetherian (resp. Artinian) ring if only the ring R X R has (resp. finite) valuation dimension. The finitely generated modules over valuation domain are further examined from here. However, since the region remains too broad, further research into the module's uniserial dimensions that finitely generated over a DVD is needed. In the case of a DVD R, a finitely generated module over R can, as is well-known, be divided into a direct sum of torsion and a free module. Therefore, first, we present methods for determining the primary module's uniserial dimension, and then followed by methods for the general finitely generated module. As can be observed, the module's uniserial dimension is a function of the elementary divisors and the rank of the non torsion module item, which is the major finding of this work.
PubDate: Jul 2021
- Time Series Forecasting with Trend and Seasonal Patterns using NARX
Network Ensembles
Abstract: Publication date: Jul 2021
Source:Mathematics and Statistics Volume 9 Number 4 Hermansah Dedi Rosadi Abdurakhman and Herni Utami In this research, we propose a Nonlinear Auto-Regressive network with exogenous inputs (NARX) model with a different approach, namely the determination of the main input variables using a stepwise regression and exogenous input using a deterministic seasonal dummy. There are two approaches in making a deterministic seasonal dummy, namely the binary and the sine-cosine dummy variables. Approximately half the number of input variables plus one is contained in the neurons of the hidden layer. Furthermore, the resilient backpropagation learning algorithm and the tangent hyperbolic activation function were used to train each network. Three ensemble operators are used, namely mean, median, and mode, to solve the overfitting problem and the single NARX model's weakness. Furthermore, we provide an empirical study using actual data, where forecasting accuracy is determined by Mean Absolute Percent Error (MAPE). The empirical study results show that the NARX model with binary dummy exogenous is the most accurate for trend and seasonal with multiplicative properties data patterns. For trend and seasonal with additive properties data patterns, the NARX model with sine-cosine dummy exogenous is more accurate, except the fact that the NARX model uses the mean ensemble operator. Besides, for trend and non-seasonal data patterns, the most accurate NARX model is obtained using the mean ensemble operator. This research also shows that the median and mode ensemble operators, which are rarely used, are more accurate than the mean ensemble operator for data that have trend and seasonal patterns. The median ensemble operator requires the least average computation time, followed by the mode ensemble operator. On the other hand, all of our proposed NARX models' accuracy consistently outperforms the exponential smoothing method and the ARIMA method.
PubDate: Jul 2021
- An Analysis about Fourier Series Estimator in Nonparametric Regression for
Longitudinal Data
Abstract: Publication date: Jul 2021
Source:Mathematics and Statistics Volume 9 Number 4 M. Fariz Fadillah Mardianto Gunardi and Herni Utami Fourier series is a function that is often used Mathematically and Statistically especially for modeling. Here, Fourier series can be constructed as an estimator in nonparametric regression. Nonparametric regression is not only using cross section data, but also longitudinal data. Some of nonparametric regression estimators have been developed for longitudinal data case, such as kernel, and spline. In this study, we concentrate to develop an inference analysis that related to Fourier series estimator in nonparametric regression for longitudinal data. Nonparametric regression based on Fourier series is capable to model data relationship with fluctuation or oscillation pattern that represents with sine and cosine functions. For point estimation analysis, Penalized Weighted Least Square (PWLS) is used to determine an estimator for parameter vector in nonparametric regression. Different with previous studies, PWLS is used to get smooth estimator. The result is an estimator for nonparametric regression curve for longitudinal data based on Fourier series approach. In addition, this study also investigated the asymptotic properties of the nonparametric regression curve estimators using the Fourier series approach for longitudinal data, especially linearity and consistency. Some study cases based on previous research and a new study case is given to make sure that Fourier series estimator in nonparametric regression has good performance in longitudinal data modeling. This study is important in order to develop further inferences Statistics, such as interval estimation and test hypothesis that related nonparametric regression with Fourier series estimator for longitudinal data.
PubDate: Jul 2021
- Time Sensitive Analysis of Antagonistic Stochastic Processes and
Applications to Finance and Queueing
Abstract: Publication date: Jul 2021
Source:Mathematics and Statistics Volume 9 Number 4 Jewgeni H. Dshalalow Kizza Nandyose and Ryan T. White This paper deals with a class of antagonistic stochastic games of three players A, B, and C, of whom the first two are active players and the third is a passive player. The active players exchange hostile attacks at random times of random magnitudes with each other and also with player C. Player C does not respond to any attacks (that are regarded as a collateral damage). There are two sustainability thresholds M and T are set so that when the total damages to players A and B cross M and T, respectively, the underlying player is ruined. At some point (ruin time), one of the two active players will be ruined. Player C's damages are sustainable and some rebuilt. Of interest are the ruin time and the status of all three players upon as well as at any time t prior to . We obtain an analytic formula for the joint distribution of the named processes and demonstrate its closed form in various analytic and computational examples. In some situations pertaining to stock option trading, stock prices (player C) can fluctuate. So in this case, it is of interest to predict the first time when an underlying stock price drops or significantly drops so that the trader can exercise the call option prior to the drop and before maturity T. Player A monitors the prices upon times assigning 0 damage to itself if the stock price appreciates or does not change and assumes a positive integer if the price drops. The times are themselves damages to player B with threshold T. The "ruin" time is when threshold M is crossed (i.e., there is a big price drop or a series of drops) or when the maturity T expires whichever comes first. Thus a prior action is needed and its time is predicted. We illustrate the applicability of the game on a number of other practical models, including queueing systems with vacations and (N,T)-policy.
PubDate: Jul 2021
- Three Dimensional Fractional Fourier-Mellin Transform, and its
Applications
Abstract: Publication date: Jul 2021
Source:Mathematics and Statistics Volume 9 Number 4 Arvind Kumar Sinha and Srikumar Panda The main objective of the paper is to study the three-dimensional fractional Fourier Mellin transforms (3DFRFMT), their basic properties and applicability due to mainly use in the radar system, reconstruction of grayscale images, in the detection of the human face, etc. Only the fractional Fourier transform is based on time-frequency distribution, whereas only the fractional Mellin transform is on scale covariant transformation. Both transforms can discover action in the definite assortment. The fractional Fourier transform is applicable for controlling the range of shift, whereas the fractional Mellin transform is accustomed to managing the range of rotation and scaling of the function. So, combining both transformations, we get an elegant expression for 3DFRFMT, which can be used in several fields. The paper introduces the concept of three-dimensional fractional Fourier Mellin transforms and their applications. Modulation property is the most useful concept in the signal system, radar technology, pattern reorganization, and many more in the integral transform. Parseval's identity applies to the conservation of energy in the universe. Thus we establish the modulation theorem, Parseval's theorem, scaling theorem, analytic theorem for three-dimensional fractional Fourier Mellin transform. We also give some examples of three-dimensional fractional Fourier-Mellin transform on some functions. Finally, we provide three-dimensional fractional Fourier-Mellin transform applications for solving homogeneous and non-homogeneous Mboctara partial differential equations that we can apply with advantages to solve the different types of problems in signal processing systems. The transform is beneficial in a maritime strategy as a co-realtor to control moments in any specific three-dimensional space. The concept is the most powerful tool to deal with any information system problems. After obtaining the generalization, we can explore many more ideas in applying three-dimensional fractional Fourier-Mellin transformations in many real word problems.
PubDate: Jul 2021
- Modified Variational Iteration Method for Solving Nonlinear Partial
Differential Equation Using Adomian Polynomials
Abstract: Publication date: Jul 2021
Source:Mathematics and Statistics Volume 9 Number 4 S. A. Ojobor and A. Obihia The aim of this paper is to solve numerically the Cauchy problems of nonlinear partial differential equation (PDE) in a modified variational iteration approach. The standard variational iteration method (VIM) is first studied before modifying it using the standard Adomian polynomials in decomposing the nonlinear terms of the PDE to attain the new iterative scheme modified variational iteration method (MVIM). The VIM was used to iteratively determine the nonlinear parabolic partial differential equation to obtain some results. Also, the modified VIM was used to solve the nonlinear PDEs with the aid of Maple 18 software. The results show that the new scheme MVIM encourages rapid convergence for the problem under consideration. From the results, it is observed that for the values the MVIM converges faster to exact result than the VIM though both of them attained a maximum error of order 10-9. The resulting numerical evidences were competing with the standard VIM as to the convergence, accuracy and effectiveness. The results obtained show that the modified VIM is a better approximant of the above nonlinear equation than the traditional VIM. On the basis of the analysis and computation we strongly advocate that the modified with finite Adomian polynomials as decomposer of nonlinear terms in partial differential equations and any other mathematical equation be encouraged as a numerical method.
PubDate: Jul 2021
- Z-Score Functions of Hesitant Fuzzy Sets
Abstract: Publication date: Jul 2021
Source:Mathematics and Statistics Volume 9 Number 4 Zahari Md Rodzi Abd Ghafur Ahmad Norul Fadhilah Ismail and Nur Lina Abdullah The hesitant fuzzy set (HFS) concept as an extension of fuzzy set (FS) in which the membership degree of a given element, called the hesitant fuzzy element (HFE), is defined as a set of possible values. A large number of studies are concentrating on HFE and HFS measurements. It is not just because of their crucial importance in theoretical studies, but also because they are required for almost any application field. The score function of HFE is a useful method for converting data into a single value. Moreover, the scoring function provides a much easier way to determine each alternative's ranking order for multi-criteria decision-making (MCDM). This study introduces a new hesitant degree of HFE and the z-score function of HFE, which consists of z-arithmetic mean, z-geometric mean, and z-harmonic mean. The z-score function is developed with four main bases: a hesitant degree of HFE, deviation value of HFE, the importance of the hesitant degree of HFE, α, and importance of the deviation value of HFE, β. These three proposed scores are compared with the existing scores functions to identify the proposed z-score function's flexibility. An algorithm based on the z-score function was developed to create an algorithm solution to MCDM. Example of secondary data on supplier selection for automated companies is used to prove the algorithms' capability in ranking order for MCDM.
PubDate: Jul 2021
- Two-Sided Group Chain Sampling Plans Based on Truncated Life Test for
Generalized Exponential Distribution
Abstract: Publication date: Jul 2021
Source:Mathematics and Statistics Volume 9 Number 4 Nazrina Aziz Zahirah Hasim and Zakiyah Zain Acceptance sampling is an important technique in quality assurance; its main goal is to achieve the most accurate decision in accepting lot using minimum resources. In practice, this often translates into minimizing the required sample sizes for the inspection, while satisfying the maximum allowable risks by consumer and producer. Numerous sampling plans have been developed over the past decades, the most recent being the incorporation of grouping to enable simultaneous inspection in the two-sided chain sampling which considers information from preceding and succeeding samples. This combination offers improved decision accuracy with reduced inspection resources. To-date, two-sided group chain sampling plan (TSGCh) for characteristic based on truncated lifetime has only been explored for Pareto distribution of the 2nd kind. This article introduces TSGCh sampling plan for products with lifetime that follows generalized exponential distribution. It focuses on minimizing consumer's risk and operates with three acceptance criteria. The equations that derived from the set conditions involving generalized exponential and binomial distributions are mathematically solved to develop this sampling plan. Its performance is measured on the probability of lot acceptance and number of minimum groups. A comparison with the established new two-sided group chain (NTSGCh) indicates that the proposed TSGCh sampling plan performs better in terms of sample size requirement and consumers' protection. Thus, this new acceptance sampling plan can reduce the inspection time, resources, and costs via smaller sample size (number of groups), while providing the desired consumers' protection.
PubDate: Jul 2021
- Approximate Solution of Higher Order Fuzzy Initial Value Problems of
Ordinary Differential Equations Using Bezier Curve Representation
Abstract: Publication date: Jul 2021
Source:Mathematics and Statistics Volume 9 Number 4 Sardar G Amen Ali F Jameel and Abdul Malek Yaakob The Bezier curve is a parametric curve used in the graphics of a computer and related areas. This curve, connected to the polynomials of Bernstein, is named after the design curves of Renault's cars by Pierre Bézier in the 1960s. There has recently been considerable focus on finding reliable and more effective approximate methods for solving different mathematical problems with differential equations. Fuzzy differential equations (known as FDEs) make extensive use of various scientific analysis and engineering applications. They appear because of the incomplete information from their mathematical models and their parameters under uncertainty. This article discusses the use of Bezier curves for solving elevated order fuzzy initial value problems (FIVPs) in the form of ordinary differential equation. A Bezier curve approach is analyzed and updated with concepts and properties of the fuzzy set theory for solving fuzzy linear problems. The control points on Bezier curve are obtained by minimizing the residual function based on the least square method. Numerical examples involving the second and third order linear FIVPs are presented and compared with the exact solution to show the capability of the method in the form of tables and two dimensional shapes. Such findings show that the proposed method is exceptionally viable and is straightforward to apply.
PubDate: Jul 2021
- The Effect of Independent Parameter on Accuracy of Direct Block Method
Abstract: Publication date: Jul 2021
Source:Mathematics and Statistics Volume 9 Number 4 Iskandar Shah Mohd Zawawi Zarina Bibi Ibrahim and Khairil Iskandar Othman Block methods that approximate the solution at several points in block form are commonly used to solve higher order differential equations. Inspired by the literature and ongoing research in this field, this paper intends to explore a new derivation of block backward differentiation formula that employs independent parameter to provide sufficient accuracy when solving second order ordinary differential equations directly. The use of three backward steps and five independent parameters are considered adequately in generating the variable coefficients of the formulas. To ascertain only one parameter exists in the derived formula, the order of the method is determined. Such independent parameter retains the favorable convergence properties although the values of parameter will affect the zero stability and truncation error. An ability of the method to compute the approximated solutions at two points concurrently is undeniable. Another advantage of the method is being able to solve the second order problems directly without recourse to the technique of reducing it to a system of first order equations. The essential of the error analysis is to observe the effect of independent parameter on the accuracy, in the sense that with certain appropriate values of parameter, the accuracy is improved. The performance of the method is tested with some initial value problems and the numerical results confirm that the maximum error and average error obtained by the proposed method are smaller at certain step size compared to the other conventional direct methods.
PubDate: Jul 2021
- Applications of the Differential Transformation Method and Multi-Step
Differential Transformation Method to Solve a Rotavirus Epidemic Model
Abstract: Publication date: Jan 2021
Source:Mathematics and Statistics Volume 9 Number 1 Pakwan Riyapan Sherif Eneye Shuaib Arthit Intarasit and Khanchit Chuarkham Epidemic models are essential in understanding the transmission dynamics of diseases. These models are often formulated using differential equations. A variety of methods, which includes approximate, exact and purely numerical, are often used to find the solutions of the differential equations. However, most of these methods are computationally intensive or require symbolic computations. This article presents the Differential Transformation Method (DTM) and Multi-Step Differential Transformation Method (MSDTM) to find the approximate series solutions of an SVIR rotavirus epidemic model. The SVIR model is formulated using the nonlinear first-order ordinary differential equations, where S; V; I and R are the susceptible, vaccinated, infected and recovered compartments. We begin by discussing the theoretical background and the mathematical operations of the DTM and MSDTM. Next, the DTM and MSDTM are applied to compute the solutions of the SVIR rotavirus epidemic model. Lastly, to investigate the efficiency and reliability of both methods, solutions obtained from the DTM and MSDTM are compared with the solutions from the Runge-Kutta Order 4 (RK4) method. The solutions from the DTM and MSDTM are in good agreement with the solutions from the RK4 method. However, the comparison results show that the MSDTM is more efficient and converges to the RK4 method than the DTM. The advantage of the DTM and MSDTM over other methods is that it does not require a perturbation parameter to work and does not generate secular terms. Therefore the application of both methods
PubDate: Jan 2021
- On One Mathematical Model of Cooling Living Biological Tissue
Abstract: Publication date: Jan 2021
Source:Mathematics and Statistics Volume 9 Number 1 B. K. Buzdov When cooling living biological tissue (active, non-inert medium), cryomedicine uses cryo-instruments with various forms of cooling surface. Cryoinstruments are located on the surface of biological tissue or completely penetrate into it. With a decrease in the temperature of the cooling surface, an unsteady temperature field appears in the tissue, which in the general case depends on three spatial coordinates and time. To date, there are a large number of scientific publications that consider mathematical models of cryodestruction of biological tissue. However, in the overwhelming majority of them, the Pennes equation (or some of its modifications) is taken as the basis of the mathematical model, from which the linear nature of the dependence of heat sources of biological tissue on the desired temperature field is visible. This character of the dependence does not allow one to describe the actually observed spatial localization of heat. In addition, Pennes' model does not take into account the fact that the freezing of the intercellular fluid occurs much earlier than the freezing of the intracellular fluid and the heat corresponding to these two processes is released at different times. In the proposed work, a new mathematical model of cooling and freezing of living biological tissue are built with a flat rectangular applicator located on its surface. The model takes into account the above features and is a three-dimensional boundary-value problem of the Stefan type with nonlinear heat sources of a special type and has applications in cryosurgery. A method is proposed for the numerical study of the problem posed, based on the use of locally one-dimensional difference schemes without explicitly separating the boundary of the influence of cold and the boundaries of the phase transition. The method was previously successfully tested by the author in solving other two-dimensional problems arising in cryomedicine.
PubDate: Jan 2021
- Fixed Point Theorems in Complex Valued Quasi b-Metric Spaces for
Satisfying Rational Type Contraction
Abstract: Publication date: Jan 2021
Source:Mathematics and Statistics Volume 9 Number 1 J. Uma Maheswari A. Anbarasan and M. Ravichandran The notion of complex valued metric spaces proved the common fixed point theorem that satisfies rational mapping of contraction. In the contraction mapping theory, several researchers demonstrated many fixed-point theorems, common fixed-point theorems and coupled fixed-point theorems by using complex valued metric spaces. The idea of b-metric spaces proved the fixed point theorem by the principle of contraction mapping. The notion of complex valued b-metric spaces, and this metric space was the generalization of complex valued metric spaces. They explained the fixed point theorem by using the rational contraction. In the metric spaces, we refer to this metric space as a quasi-metric space, the symmetric condition d(x, y) = d(y, x) is ignored. Metric space is a special kind of space that is quasi-metric. The Quasi metric spaces were discussed by many researchers. Banach introduced the theory of contraction mapping and proved the theorem of fixed points in metric spaces. We are now introducing the new notion of complex quasi b-metric spaces involving rational type contraction which proved the unique fixed point theorems with continuous as well as non-continuous functions. Illustrate this with example.
PubDate: Jan 2021
- Generalized Relation between the Roots of Polynomial and Term of
Recurrence Relation Sequence
Abstract: Publication date: Jan 2021
Source:Mathematics and Statistics Volume 9 Number 1 Vipin Verma and Mannu Arya Many researchers have been working on recurrence relation which is an important topic not only in mathematics but also in physics, economics and various applications in computer science. There are many useful results on recurrence relation sequence but there main problem to find any term of recurrence relation sequence we need to find all previous terms of recurrence relation sequence. There were many important theorems obtained on recurrence relations. In this paper we have given special identity for generalized kth order recurrence relation. These identities are very useful for finding any term of any order of recurrence relation sequence.
Authors define a special formula in this paper by this we can find direct any term of a recurrence relation sequence. In this recurrence relation sequence to find any terms we need to find all previous terms so this result is very important. There is important property of a relation between coefficients of recurrence relation terms and roots of a polynomial for second order relation but in this paper, we gave this same property of recurrence relation of all higher order recurrence relation. So finally, we can say that this theorem is valid all order of recurrence relation only condition that roots are distinct. So, we can say that this paper is generalization of property of a relation between coefficients of recurrence relation terms and roots of a polynomial. Theorem: - Let C1 and C2 are arbitrary real numbers and suppose the equation (1) Has X1 and X2 are distinct roots. Then the sequence is a solution of the recurrence relation (2) . For n= 0, 1, 2 …where β1 and β2 are arbitrary constants. Proof: - First suppose that of type we shall prove is a solution of recurrence relation (2). Since X1, X2 and X3 are roots of equation (1) so all are satisfied equation (1) so we have, . Consider . This implies . So the sequence is a solution of the recurrence relation. Now we will prove the second part of theorem. Let is a sequence with three . Let . So (3). (4). Multiply by X1 to (3) and subtracts from (4). We have similarly we can find . So we can say that values of β1 and β2 are defined as roots are distinct. So non- trivial values ofβ1 and β2 can find and we can say that result is valid. Example: Let be any sequence such that n≥3 and a0=0, a1=1, a2=2. Then find a10 for above sequence. Solution: The polynomial of above sequence is . Solving this equation we have roots are 1, 2, and 3 using above theorem we have (7). Using a0=0, a1=1, a2=2 in (7) we have β1+β2+β3=0 (8). β1+2β2+3β2=1 (9).β1+4β2+9β3=2 (10) Solving (8), (9) and (10) we have , , . This implies . Now put n=10 we have a10=-27478. Recurrence relation is a very useful topic of mathematics, many problems of real life may be solved by recurrence relations, but in recurrence relation there is a major difficulty in the recurrence relation. If we want to find 100th term of sequence, then we need to find all previous 99 terms of given sequence, then we can get 100th term of sequence but above theorem is very useful if coefficients of recurrence relation of given sequence satisfies the condition of the above theorem, then we can apply above theorem and we can find direct any term of sequence without finding all previous terms.
PubDate: Jan 2021
- Fuzzy Time Series Forecasting Model Based on Intuitionistic Fuzzy Sets via
Delegation of Hesitancy Degree to the Major Grade De-i-fuzzification
Method
Abstract: Publication date: Jan 2021
Source:Mathematics and Statistics Volume 9 Number 1 Nik Muhammad Farhan Hakim Nik Badrul Alam Nazirah Ramli and Norhuda Mohammed Fuzzy time series is a powerful tool to forecast the time series data under uncertainty. Fuzzy time series was first initiated with fuzzy sets and then generalized by intuitionistic fuzzy sets. The intuitionistic fuzzy sets consider the degree of hesitation in which the degree of non-membership is incorporated. In this paper, a fuzzy set time series forecasting model based on intuitionistic fuzzy sets via delegation of hesitancy degree to the major grade de-i-fuzzification approach was developed. The proposed model was implemented on the data of student enrollments at the University of Alabama. The forecasted output was obtained using the fuzzy logical relationships of the output, and the performance of the forecasted output was compared with the fuzzy time series forecasting model based on fuzzy sets using the mean square error, root mean square error, mean absolute error, and mean absolute percentage error. The results showed that the forecasting model based on induced fuzzy sets from intuitionistic fuzzy sets performs better compared to the fuzzy time series forecasting model based on fuzzy sets.
PubDate: Jan 2021
- A Note on Lienard-Chipart Criteria and its Application to Epidemic Models
Abstract: Publication date: Jan 2021
Source:Mathematics and Statistics Volume 9 Number 1 Auni Aslah Mat Daud An important part of the study of epidemic models is the local stability analysis of the equilibrium points. The linear algebra method which is commonly employed is the well-known Routh-Hurwitz criteria. The criteria give necessary and sufficient conditions for all of the roots of the characteristic polynomial to be negative or have negative real parts. To date, there are no epidemic models in the literature which employ Lienard-Chipart criteria. This note recommends an alternative linear algebra method namely Lienard-Chipart criteria, to significantly simplify the local stability analysis of epidemic models. Although Routh-Hurwitz criteria are a correct method for local stability analysis, Lienard-Chipart criteria have advantages over Routh-Hurwitz criteria. Using Lienard-Chipart criteria, only about half of the Hurwitz determinants inequalities are required, with the remaining conditions of each set concern with only the sign of the alternate coefficients of the characteristic polynomial. The Lienard-Chipart criteria are especially useful for polynomials with symbolic coefficients, as the determinants are usually significantly more complicated than original coefficients as degree of the polynomial increases. Lienard-Chipart criteria and Routh-Hurwitz criteria have similar performance for systems of dimension five or less. Theoretically, for systems of dimension higher than five, verifying Lienard-Chipart criteria should be much easier than verifying Routh-Hurwitz criteria and the advantage of Lienard-Chipart criteria may become clear. Examples of local stability analysis using Lienard-Chipart criteria for two recently proposed models are demonstrated to show the advantages of simplified Lienard-Chipart criteria over Routh-Hurwitz criteria.
PubDate: Jan 2021
- Application of Fuzzy Linear Regression with Symmetric Parameter for
Predicting Tumor Size of Colorectal Cancer
Abstract: Publication date: Jan 2021
Source:Mathematics and Statistics Volume 9 Number 1 Muhammad Ammar Shafi Mohd Saifullah Rusiman and Siti Nabilah Syuhada Abdullah The colon and rectum is the final portion of the digestive tube in the human body. Colorectal cancer (CRC) occurs due to bacteria produced from undigested food in the body. However, factors and symptoms needed to predict tumor size of colorectal cancer are still ambiguous. The problem of using linear regression arises with the use of uncertain and imprecise data. Since the fuzzy set theory's concept can deal with data not to a precise point value (uncertainty data), this study applied the latest fuzzy linear regression to predict tumor size of CRC. Other than that, the parameter, error and explanation for the both models were included. Furthermore, secondary data of 180 colorectal cancer patients who received treatment in general hospital with twenty five independent variables with different combination of variable types were considered to find the best models to predict the tumor size of CRC. Two models; fuzzy linear regression (FLR) and fuzzy linear regression with symmetric parameter (FLRWSP) were compared to get the best model in predicting tumor size of colorectal cancer using two measurement statistical errors. FLRWSP was found to be the best model with least value of mean square error (MSE) and root mean square error (RMSE) followed by the methodology stated.
PubDate: Jan 2021
- Impact of Sleep on Usage of the Smart Phone at the Bedtime– A Case
Study
Abstract: Publication date: Jan 2021
Source:Mathematics and Statistics Volume 9 Number 1 Navya Pratyusha M Rajyalakshmi K Apparao B V and Charankumar G Pittsburgh Sleep Quality Index (PSQI) Scoring (Buysse et al. 1989) is a powerful method to measure the sleep quality index based on the scores of various factors namely duration of sleep, sleep disturbance, sleep latency, day dysfunction due to sleepiness, sleep efficiency, need meds to sleep and overall sleep quality. Mainly we focused on the smart phones' usage and its impact on the quality of sleep at the bed time. Many studies have proved that the usage of smart phones at bed time affects the sleep quality, health and productivity. In the present study, we have collected data randomly from the middle-aged adults and observed the relation between gender and the quality of sleep using phi coefficient. It is clearly observed that as we move from males to females, we move negatively from good sleep quality to poor sleep quality. It indicates that males have poor sleep quality than females. We also performed an analysis of variance to test the hypothesis that there is any association between the smart phones' usage and its impact on quality of sleep at bed time.
PubDate: Jan 2021
- Fourier Method in Initial Boundary Value Problems for Regions with
Curvilinear Boundaries
Abstract: Publication date: Jan 2021
Source:Mathematics and Statistics Volume 9 Number 1 Leontiev V. L. The algorithm of the generalized Fourier method associated with the use of orthogonal splines is presented on the example of an initial boundary value problem for a region with a curvilinear boundary. It is shown that the sequence of finite Fourier series formed by the method algorithm converges at each moment to the exact solution of the problem – an infinite Fourier series. The structure of these finite Fourier series is similar to that of partial sums of an infinite Fourier series. As the number of grid nodes increases in the area under consideration with a curvilinear boundary, the approximate eigenvalues and eigenfunctions of the boundary value problem converge to the exact eigenvalues and eigenfunctions, and the finite Fourier series approach the exact solution of the initial boundary value problem. The method provides arbitrarily accurate approximate analytical solutions to the problem, similar in structure to the exact solution, and therefore belongs to the group of analytical methods for constructing solutions in the form of orthogonal series. The obtained theoretical results are confirmed by the results of solving a test problem for which both the exact solution and analytical solutions of discrete problems for any number of grid nodes are known. The solution of test problem confirm the findings of the theoretical study of the convergence of the proposed method and the proposed algorithm of the method of separation of variables associated with orthogonal splines, yields the approximate analytical solutions of initial boundary value problem in the form of a finite Fourier series with any desired accuracy. For any number of grid nodes, the method leads to a generalized finite Fourier series which corresponds with high accuracy to the partial sum of the Fourier series of the exact solution of the problem.
PubDate: Jan 2021
- The Performance Analysis of a New Modification of Conjugate Gradient
Parameter for Unconstrained Optimization Models
Abstract: Publication date: Jan 2021
Source:Mathematics and Statistics Volume 9 Number 1 I M Sulaiman M Mamat M Y Waziri U A Yakubu and M Malik Conjugate Gradient (CG) method is the most prominent iterative mathematical technique that can be useful for the optimization of both linear and non-linear systems due to its simplicity, low memory requirement, computational cost, and global convergence properties. However, some of the classical CG methods have some drawbacks which include weak global convergence, poor numerical performance both in terms of number of iterations and the CPU time. To overcome these drawbacks, researchers proposed new variants of the CG parameters with efficient numerical results and nice convergence properties. Some of the variants of the CG method include the scale CG method, hybrid CG method, spectral CG method, three-term CG method, and many more. The hybrid conjugate gradient (CG) algorithm is among the efficient variant in the class of the conjugate gradient methods mentioned above. Some interesting features of the hybrid modifications include inherenting the nice convergence properties and efficient numerical performance of the existing CG methods. In this paper, we proposed a new hybrid CG algorithm that inherits the features of the Rivaie et al. (RMIL*) and Dai (RMIL+) conjugate gradient methods. The proposed algorithm generates a descent direction under the strong Wolfe line search conditions. Preliminary results on some benchmark problems show that the proposed method efficient and promising.
PubDate: Jan 2021
- Some Properties on Fréchet-Weibull Distribution with Application to
Real Life Data
Abstract: Publication date: Jan 2021
Source:Mathematics and Statistics Volume 9 Number 1 Deepshikha Deka Bhanita Das Bhupen K Baruah and Bhupen Baruah Research, development and extensive use of generalized form of distributions in order to analyze and modeling of applied sciences research data has been growing tremendously. Weibull and Fréchet distribution are widely discussed for reliability and survival analysis using experimental data from physical, chemical, environmental and engineering sciences. Both the distributions are applicable to extreme value theory as well as small and large data sets. Recently researchers develop several probability distributions to model experimental data as these parent models are not adequate to fit in some experiments. Modified forms of the Weibull distribution and Fréchet distribution are more flexible distributions for modeling experimental data. This article aims to introduce a generalize form of Weibull distribution known as Fréchet-Weibull Distribution (FWD) by using the T-X family which extends a more flexible distribution for modeling experimental data. Here the pdf and cdf with survival function [S(t)], hazard rate function [h(t)] and asymptotic behaviour of pdf and survival function and the possible shapes of pdf, cdf, S(t) and h(t) of FWD have been studied and the parameters are estimated using maximum livelihood method (MLM). Some statistical properties of FWD such as mode, moments, skewness, kurtosis, variation, quantile function, moment generating function, characteristic function and entropies are investigated. Finally the FWD has been applied to two sets of observations from mechanical engineering and shows the superiority of FWD over other related distributions. This study will provide a useful tool to analyze and modeling of datasets in Mechanical Engineering sciences and other related field.
PubDate: Jan 2021
- Corporate Domination Number of the Cartesian Product of Cycle and Path
Abstract: Publication date: Jan 2021
Source:Mathematics and Statistics Volume 9 Number 1 S. Padmashini and S. Pethanachi Selvam Domination in graphs is to dominate the graph G by a set of vertices , vertex set of G) when each vertex in G is either in D or adjoining to a vertex in D. D is called a perfect dominating set if for each vertex v is not in D, which is adjacent to exactly one vertex of D. We consider the subset C which consists of both vertices and edges. Let denote the set of all vertices V and the edges E of the graph G. Then is said to be a corporate dominating set if every vertex v not in is adjacent to exactly one vertex of , where the set P consists of all vertices in the vertex set of an edge induced sub graph , (E1 a subset of E) such that there should be maximum one vertex common to any two open neighborhood of different vertices in V(G[E1]) and Q, the set consists of all vertices in the vertex set V1, a subset of V such that there exists no vertex common to any two open neighborhood of different vertices in V1. The corporate domination number of G, denoted by , is the minimum cardinality of elements in C. In this paper, we intend to determine the exact value of corporate domination number for the Cartesian product of the Cycle and Path .
PubDate: Jan 2021