Open Access Original Research Article

Detection of Fracture Bones in X-ray Images Categorization

Oyeranmi Adigun, Babatunde Ronke, Rufai Mohammed, Aigbokhan Edwin

Journal of Advances in Mathematics and Computer Science, Page 1-11
DOI: 10.9734/jamcs/2020/v35i430265

Fractured bone detection and categorization is currently receiving research attention in computer aided diagnosis system because of the ease it has brought to doctors in classification and interpretation of X-ray images.  The choice of an efficient algorithm or combination of algorithms is paramount to accurately detect and categorize fractures in X-ray images, which is the first stage of diagnosis in treatment and correction of damaged bones for patients. This is what this research seeks to address. The research design involves data collection, preprocessing, segmentation, feature extraction, classification and evaluation of the proposed method. The sample dataset were x-ray images collected from the Department of Radiology, National Orthopedic Hospital, Igbobi-Lagos, Nigeria as well as Open Access Medical Image Repositories. The image preprocessing involves the conversion of images in RGB format to grayscale, sharpening and smoothing using Unsharp Masking Tool.  The segmentation of the preprocessed image was carried out by adopting the Entropy method in the first stage and Canny edge method in the second stage while feature extraction was performed using Hough Transformation. Detection and classification of fracture image employed a combination of two algorithms;  K-Nearest Neighbor (KNN) and Support Vector Machine (SVM) for detecting fracture locations based on four classification types: (normal, comminute, oblique and transverse).Two performance assessment methods were employed to evaluate the developed system. The first evaluation was based on confusion matrix which evaluates fracture and non-fracture on the basis of TP (True Positive), TN (True negative), FP (False Positive) and FN (False Negative). The second appraisal was based on Kappa Statistics which evaluates the type of fracture by determining the accuracy of the categorized fracture bone type. The result of first assessment for fracture detection shows that 26 out of 40 preprocessed images were fractured, resulting to the following three values of performance metrics: accuracy value of 90%, sensitivity of 87% and specificity of 100%. The Kappa coefficient error assessment produced accuracy of 83% during classification. The proposed method can find suitable use in categorization of fracture types on different bone images based on the results obtained from the experiment.

Open Access Original Research Article

Adaptive Scheme for ANOVA Models

Gilbert Biney, Gabriel Asare Okyere, Abukari Alhassan

Journal of Advances in Mathematics and Computer Science, Page 12-23
DOI: 10.9734/jamcs/2020/v35i430266

This paper deals with the concept of adaptive scheme and with an application to the Oneway ANOVA model under uncorrelated errors. Oneway ANOVA model is sensitive to nonnormality as well as variance heterogeneity. To overcome these problems, an adaptive scheme is proposed. The adaptive test is a two step procedure. The given data is first examined and classified based on measures of skewness and tailweight. Secondly, a selector statistic is used for selecting a test to be conducted. A 10,000 simulations were conducted to compare the performance of the two models from different continuous distributions. Analysis of real data sets on equal and unequal sample sizes were performed to evaluate the efficiency of the two models. The findings showed that our adaptive scheme outperformed the parametric F-test in symmetric or skewed distributions with varying tailweights except for symmetric and medium-tailed distributions.

Open Access Original Research Article

Fast and Effective Region-based Depth Map Upsampling with Application to Location Map-Free Reversible Data Hiding

Kuo-Liang Chung, Yu-Ling Tseng, Tzu-Hsien Chan, Ching-Sheng Wang

Journal of Advances in Mathematics and Computer Science, Page 24-45
DOI: 10.9734/jamcs/2020/v35i430268

In this paper, we rst propose a fast and eective region-based depth map upsampling method, and then propose a joint upsampling and location map-free reversible data hiding method, simpled called the JUR method. In the proposed upsampling method, all the missing depth pixels are partitioned into three disjoint regions: the homogeneous, semi-homogeneous, and non- homogeneous regions. Then, we propose the depth copying, mean value, and bicubic interpolation approaches to reconstruct the three kinds of missing depth pixels quickly, respectively. In the proposed JUR method, without any location map overhead, using the neighboring ground truth depth pixels of each missing depth pixel, achieving substantial quality, and embedding capacity merits. The comprehensive experiments have been carried out to not only justify the execution-time and quality merits of the upsampled depth maps by our upsampling method relative to the state-of-the-art methods, but also justify the embedding capacity and quality merits of our JUR method when compared with the state-of-the-art methods.

Open Access Original Research Article

Mathematical Modeling and Analysis of Mathematics Anxiety Behavior on Mathematics Performance in Kenya

O. M. Nathan, K. O. Jackob

Journal of Advances in Mathematics and Computer Science, Page 46-62
DOI: 10.9734/jamcs/2020/v35i430269

We propose a deterministic model that describes the dynamics of students who have the capabilityWe propose a deterministic model that describes the dynamics of students who have the capabilityto perform well in mathematics examinations and engage in careers that demand its applicationand the negative inuence of individuals with mathematics anxiety on the potential students.Our model is based on SIR classical infectious model classes with Susceptible (S) and Infected (I)taken as Math anxious students (Ax) and Removed (R) adopted as achievers students (Aa) . Themodel is shown to be both epidemiologically and mathematically well posed. In particular, weprove that all solutions of the model are positive and bounded; and that every solution with initialconditions in remains in the set for all time. The existence of unique math anxious-freeand endemic equilibrium points is proved and the basic reproduction number R0 computed usingnext generation matrix approach. A global stability of anxious-free and the endemic equilibria areperformed using Lasselles invariance principle of Lyapunov functions. Sensitivity analysis showsthat achievement rate of potential achievers and achievement rate of math anxious students are the most sensitive parameters. This indicates that effort should be directed towards theseparameters, by having well trained mathematics staff and the best printed and technological resources so as to control the spread of mathematics anxiety. Furthermore, scaling up the understanding level of mathematics algorithms, lowers the mathematics anxiety level and consequently, the spread of mathematics anxiety amongst students reduces. Lastly, some numerical simulations are performed to verify the theoretical analysis result using Matlab software.

Open Access Original Research Article

An Operational Matrix of Hermite Polynomials for Solving Nonlinear Fractional Dierential Equations

Hatice Yalman Kosunalp, Mustafa Gulsu

Journal of Advances in Mathematics and Computer Science, Page 63-71
DOI: 10.9734/jamcs/2020/v35i430270

In this paper, an effective technique known as the operational matrix method is utilised to solve nonlinear form of fractional dierential equations (FDEs). An explicit effort is placed on the derivation of Hermite polynomials operational matrix with the Caputo sense. The main motivation behind this work is to convert a nonlinear type of FDE into a set of algebraic equations with the consideration of initial conditions. The problem is therefore simplied by these equations to be solved with the proposed method. In order to conrm the effectiveness of the proposed approach, numerical and analytical solutions for a number of nonlinear FDEs are presented. Due to the high simplicity of the proposed approach in practice, it can be comfortably implemented in various aspects of applied science domain.

Open Access Original Research Article

Solitary Wave Solutions for the Shallow Water Wave Equations and the Generalized Klein-Gordon Equation Using Exp(-ϕ(η) )-Expansion Method

Md. Mamunur Rashid, Whida Khatun

Journal of Advances in Mathematics and Computer Science, Page 72-86
DOI: 10.9734/jamcs/2020/v35i430272

In this article, we investigate the exact and solitary wave solutions for the shallow water wave equations and the generalized Klein-Gordon equation using the exp -expansion method. A wave transformation is applied to convert the problem into the form of an ordinary differential equation. By using this method, we found the explicit solitary wave solutions in terms of the hyperbolic functions, trigonometric functions, exponential functions and rational functions. The extracted solution plays a significant role in many physical phenomena such as electromagnetic waves, nonlinear lattice waves, ion sound waves in plasma, nuclear physics, shallow water waves and so on. It is noted that the method is reliable, straightforward and an effective mathematical tool for analytic treatment of nonlinear systems of partial differential equation in mathematical physics and engineering.

Open Access Original Research Article

A New Common Fixed-Point Theorem for Two Pairs of Mappings in Parametric Metric Space

Anil Bakhru, Manoj Ughade, Richa Gupta

Journal of Advances in Mathematics and Computer Science, Page 87-105
DOI: 10.9734/jamcs/2020/v35i430273

Our aim of this paper is to prove a new general common fixed point theorem for two pair of mappings under a different set of conditions using the idea of weakly compatible mappings satisfying a general class of contractions defined by an implicit relation in the frame work of parametric metric space, which unify, extend and generalize most of the existing relevant common fixed point theorems from the literature. Some related results and illustrative an example to highlight the realized improvements is also furnished.

Open Access Original Research Article

On Generalized Opial's Integral Inequalities in q-Calculus

B. Abubakari, M. M. Iddrisu, K. Nantomah

Journal of Advances in Mathematics and Computer Science, Page 106-114
DOI: 10.9734/jamcs/2020/v35i430274

In this paper, we establish results for q-analogues of generalized Opial integral inequalities and also present some extensions of the analogues. Using the concepts of q-differentiability and continuity of functions and the application of the Holder's integral inequality we establish the results.

Open Access Original Research Article

Discrete Artificial Dragonflies Algorithm in Agent Based Modelling for Exact Boolean kSatisfiability Problem

Hamza Abubakar, Sagir Abdu M., Surajo Yusuf, Yusuf Abdurrahman

Journal of Advances in Mathematics and Computer Science, Page 115-134
DOI: 10.9734/jamcs/2020/v35i430275

The development of metaheuristics and Boolean Satisfiability representation plays an important part in a neural network (NN) and Artificial Intelligence (AI) communities. In this paper, a new hybrid discrete version of the artificial dragonfly algorithm (DADA) applying a minimum objective function in agent-based modelling (ABM) obeying a specified procedure to optimize the states of neurons, for optimal Boolean Exact Satisfiability representation on NETLOGO as a dynamic platform. We combined the artificial dragonfly algorithm for its random searching ability that encourages diverse solutions and formation of static swarm’s mechanism to stimulus computational problems to converge to the best global optimal search space. The global performance of the proposed DADA was compared with genetic algorithm (GA)  that are available in the literature based on the global minimum ratio (gM), Local Minimum Ratio (yM), Computational time (CPU) and Hamming distance (HD).  The final results showed good agreement between the proposed DADA and discrete version of GA to efficiently optimize the Exact-kSAT problem.  It found that DADA-ABM has high potentiality for optimizing or modelling a network that is very hard or often impossible to capture by exact or traditional optimization modelling techniques such as Boolean satisfiability problem is better than existing methods in the literature.

Open Access Original Research Article

Modeling the Diffraction of Electromagnetic Waves over Underwater Objects; the Wiener-Hopf Integral Equation

Yajni Warnapala, Cole Foster

Journal of Advances in Mathematics and Computer Science, Page 135-144
DOI: 10.9734/jamcs/2020/v35i430276

This research, inspired by the loss of Malaysian Airline Flight 370, investigates the feasibility of obtaining good convergence results for a model of the interaction of electromagnetic waves over the surface of the Spherical Biconcave Disc. The Galerkin Method is used to numerically solve the Dirichlet and Neumann exterior boundary value problems for the Wiener-Hopf Integral Equation over the half-plane of the Spherical Biconcave Disc. This modeling accounts for the attenuation losses of the propagating electromagnetic wave as a result of absorption and scattering in lossy media with comparison to lossless propagation. The numerical results of this research nds good convergence for this model as well as limitations in the transmission of electromagnetic waves underwater.