Editor(s)

Dr. Belkacem Chaouchi
Professor,
Department of Mathematics, Faculty of Sciences, Khemis Miliana University, Algeria.

ISBN 978-93-91473-37-2 (Print)
ISBN 978-93-91473-38-9 (eBook)
DOI: 10.9734/bpi/ctmcs/v5

This book covers key areas of mathematics and computer science. The contributions by the authors include duration of illness, Length of stay, Mardia test, mental disorder, multivariate normal distribution, Psychiatric rating scale, multivariate modelling, human face recognition, fourth - order Biharmonic equation, finite difference, cognitive behaviour therapy, mini mental status examination, simulation, zero one truncated binomial distribution, history of mathematics, education of mathematics, edge detection, flatfoot, medical image processing, GS lattice, soft valuation, soft pseudo metric lattice, soft metric lattice, soft topological lattice, neighborhood prime labeling, star graph, flower graph, comb graph, duplication, fusion, vertex switching, comparative fit index, eysenck personality questionnaire, parental mental health, root mean square error, structural equation models, mean graphs, line graphs, bimean graphs, conceptual design, management system architecture, real-time systems, system design methodology, ontological knowledge bases, Quasi-differential expressions, product operators, regular and singular end-points, singular differential operators, deficiency indices, direct sum spaces. This book contains various materials suitable for students, researchers and academicians in the field of mathematics and computer science.

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### Chapters

##### Determining the Length of Stay and Duration of Illness for Psychiatric Inpatients Using Multivariate Modelling

Alka Sabharwal, Sakshi Kaushik

Current Topics on Mathematics and Computer Science Vol. 5, 15 July 2021, Page 1-22
https://doi.org/10.9734/bpi/ctmcs/v5/3214F

Mental and behavioural disorders is a significant contributor of global burden of disease. As per WHO estimates, this burden is likely to increase by 15 percent by 2020, significantly impacting health and major social, human rights and economic consequences in all countries of the world. This paper provides a procedure for estimation of length of stay (LOS) in the hospital, total duration of illness (TDI) and recent duration of illness (RDI) using multivariate normal (MVN) and bivariate (BVN) normal distributions. To accomplish this, a retrospective data of 146 patients with complete record history, diagnosed with mental and behavioural disorders (as per APA’s DSM-V as well as the WHO’s ICD-10) is collected from the Department of Psychiatry, Lady Hardinge Medical College & Smt. S.K, Hospital, New Delhi, India for the calendar year 2013-2014. The estimated values of the above mentioned variables are found to be consistent with the observed values. Finally MVN distribution is applied to estimate the variables LOS and TDI for the patients for whom the information on these variables is not known. The model derived in this paper will facilitate the medical fraternity to not only guide the patients about their approximate length of stay in the hospital at the time of admission, but also assist them in development and management of appropriate interventions for patients.

##### The Application of the 25-Grid Point 3D Biharmonic Equation in Human Face Recognition

Abdulaziz B. M. Hamed

Current Topics on Mathematics and Computer Science Vol. 5, 15 July 2021, Page 23-33
https://doi.org/10.9734/bpi/ctmcs/v5/3134F

The paper used mathematical methods to create a model of human face images. Human face recognition has emerged as a major issue, occupying an active area in a variety of fields. It has strong ties to the broader field of pattern recognition and is in high demand in commercial and security applications that use biometric systems. The fundamental issue in face recognition is the difficulty in numerically solving the fourth-order Biharmonic Equation to generate an Elliptic Surface. Combining Elliptic Surface discrete numbers with human face images is similarly difficult. The research looked at the effectiveness of solving the three-dimensional Biharmonic Equation in the field of human face identification, as well as the modelling of face images in order to identify people based on their facial information. The procedure included dividing the surface using the forward finite differences approach to obtain the coefficient matrices of the grid points that form the elliptic surface, and then extracting human face photos from a database using a 3D camera. MATLAB was then used to extract the statistic data information and plot the curves of these data from the images. It has been found that the 25-grid point's surface is conservative field.

##### Estimating the Effect of Interactive Workshop on Cognitive Abilities of Elderly People Staying in Old Age Homes through Truncated Binomial Model

Alka Sabharwal, Babita Goyal

Current Topics on Mathematics and Computer Science Vol. 5, 15 July 2021, Page 34-45
https://doi.org/10.9734/bpi/ctmcs/v5/3125F

Cognitive decline is prevalent in older adults and is associated with decline in performance of instrumental activities which adversely affects the wellbeing of older adults. The aim of this study is to build a model that would estimate the benefit from the score based 21 days’ workshop in terms of their cognitive abilities of old adults, suffering from mild cognitive disorders. A truncated binomial distribution with missing zero and one classes, having parameters (ni, p), ni known, is fitted to estimate the unknown parameter p, by the methods of maximum likelihood and simulation. The data was collected on 71 subjects residing in 5 old age homes in Delhi, India, using Mini Mental Status Examination tool before and after a score-based 21 days cognitive behaviour therapy workshop. The probability of benefiting from the workshop is estimated only for those test attempts which were more than ten percent completed. The maximum likelihood estimate and simulation estimate of p were 0.838552 and 0.835967 respectively. Our Zero One Truncated Binomial Distribution model computed the probability of benefit (more than or equal to 0.5) from the workshop and found that by using the estimated values of p from the methods of maximum likelihood estimation and simulation, out of 71 subjects 51 (71.83%) and 53 (74.65%) were benefitted respectively. Our study concluded that there are positive effects of psycho-social interventions on quality of life and mental health.

##### Co-existence of History of Mathematics and Education of Mathematics: A Comparative Study

Saied Seyed Agha Banihashemi

Current Topics on Mathematics and Computer Science Vol. 5, 15 July 2021, Page 46-51
https://doi.org/10.9734/bpi/ctmcs/v5/3081F

Education of Mathematics and History of Mathematics are both old subjects.The question is whether or not these two important subjects can help each other. Unfortunately, this idea divided mathematics society into two groups: those who believe that history of mathematics can help with Education of Mathematics and those who believe that history of mathematics not only cannot help with mathematics education but also causes some confusion. In this paper, I'll present some comparisons and draw some conclusions on how history of mathematics can make mathematics education more active and interesting.

##### Study on Cloud Image Processing and Analysis Based Flatfoot Classification Approach

Ming-Shen Jian, Jun-Hong Shen, Yu-Chih Chen, Chao-Chun Chang, Yi-Chi Fang, Ci-Cheng Chen, Wei-Han Chen

Current Topics on Mathematics and Computer Science Vol. 5, 15 July 2021, Page 52-72
https://doi.org/10.9734/bpi/ctmcs/v5/3396F

The Cloud Image Processing and Analysis based Flatfoot Classification approach is proposed in this study to assist doctors in determining flat feet. The proposed technique can eliminate noise and shape images of the foot based on X-ray images by using image processing and analysis set up on different virtual machines on the cloud. After image processing, the individual X-ray image is divided into four blocks using the proposed division approach, which takes into account the percentage of each foot. Each divided image can be given to distinct analysis algorithms for key-point discovery by dividing the original image into four individual sub-partitions. Each image can be processed based on individual virtual machine on cloud. The system can identify four decision points of each block using the proposed techniques implemented on cloud for individual sub-partitions of the original image. The system can automatically detect flat feet based on the integration of processing data from various algorithms. The information and identification results might also be given to the doctor for manual identification. Additionally, the decision point can be manually chosen. To put it another way, the system can produce more accurate and objective findings based on the doctor's selection. Based on the dpi of the X-ray image, the simulation shows that accuracy can be improved. Furthermore, the performance of different methods for determining decision points varies.

##### Study on Generalized Soft Lattice and Soft Valuation

Manju John, D. Susha

Current Topics on Mathematics and Computer Science Vol. 5, 15 July 2021, Page 73-88
https://doi.org/10.9734/bpi/ctmcs/v5/3203F

A soft set is a generalization of a fuzzy set and is an efficient tool for modelling imprecise and vague data. The mathematical structures partially ordered sets and lattices play an important role in mathematics as well as in other disciplines such as computer science, engineering, cryptography, etc. Here we apply soft set theory to lattice theory and introduce the concept of a soft partial ordering and more concepts related to it. Mainly we introduce the concept of a generalized soft lattice(gs lattice) and investigate some of its fundamental properties. Further, we define a soft real valued function called soft valuation on a gs lattice and study its major properties. Moreover, we discuss the notion of a soft distance function in terms of soft valuation and using that function we discuss the conditions under which a gs lattice becomes a soft metric lattice. We conclude this study by defining a soft topological lattice and describe that when a gs lattice with a soft valuation becomes a soft topological lattice.

##### Study on Neighborhood Prime Labeling

T. J. Rajesh Kumar

Current Topics on Mathematics and Computer Science Vol. 5, 15 July 2021, Page 89-96
https://doi.org/10.9734/bpi/ctmcs/v5/11408D

A labeling or numbering of a graph G is an assignment of labels to the vertices of G that induces for each edge uv a labeling depending on the vertex labels f(u) and f(v). In this paper, neighborhood prime labeling of Star graph, Complete bipartite graph, Fan graph, Friendship graph, Flower graph, Bistar graph, Comb graph, splitting graph of Pn and Splitting graph of K1,n are discussed. Also discuss the neighborhood prime labeling in Ring sum of cycle Cn and K1,n and some graph operations on the Fan graph and Friendship graph.

##### Effect of Parental Mental Health in Estimation of One Personality Trait from Others with Structural Equation Models

Alka Sabharwal, Babita Goyal, Nidhi Arora Dhingra

Current Topics on Mathematics and Computer Science Vol. 5, 15 July 2021, Page 97-111
https://doi.org/10.9734/bpi/ctmcs/v5/3183F

Introduction: Adolescence is a transformational but vulnerable age. A favorable atmosphere at home and surroundings plays a pivotal role in development in different dimensions of personality. Lack of a support system may result in some psychiatric disorders among adolescents which might needs to be addressed. Eysenck Personality Questionnaire is a psychiatric tool to assess various dimensions of personality.

Objective: Adolescence is the beginning of the biological, social and psychological maturity. Personality starts taking shape at this age and the personality traits developed during this phase are likely to remain with a person for the whole life. Role of parents at this phase of life becomes very important to shape the personality. With this background we wish to examine the role of parents’ mental health which actually affects home environment, on adolescent personality.

Methods: Structural equation models with two and three layers were applied to estimate personality traits as identified by Eysenck Personality Questionnaire on two groups of adolescents: (i) The ‘Control’ group whose parents were not reported any psychiatric disorder; and (ii) the ‘Case’ group whose parent(s) were suffering from some psychiatric disorder as diagnosed by DSM-IV.

Results: The conventional three-layer model for ‘Control’ group estimated ‘Psychoticism’ directly by ‘Lie-scale’ and indirectly by ‘Extraversion’ with p-values < 0.05. The two-layered model yielded no significant difference between the base and the proposed model for ‘Control’ and ‘Case’ group as p-values were more than 0.05. The variability of Neuroticism was estimated up to 63.6% by a linear combination of Psychoticism, Lie-scale and Gender in Layer 1 of the model for the ‘Control’ group.

Conclusion: In this study either through the conventional three-layer model or the proposed two-layer models, we concluded that some information about one dimension can be extracted from the other dimension with EPQ even if the correlation is not significant. Also, the proposed two-layer model established the effect of gender at 10% level of significance in case of ‘Control’ group. Psychoticism and Neuroticism mean scores were higher for the ‘Case’ groups than those for the ‘Control’ group thus establishing the effect of parental mental health on adolescents’ personality.

##### The Definition of Bimean Graphs

Teena Liza John, T. K. Mathew Varkey

Current Topics on Mathematics and Computer Science Vol. 5, 15 July 2021, Page 112-119
https://doi.org/10.9734/bpi/ctmcs/v5/3636F

Let G be a(p,q) graph and let: V(G) $$\rightarrow$${0,1,2,…..,q} be an injection. The graph G is said to be a mean graph if for each edge there exists an induced map $$f^*:E(G)\rightarrow{1,2,…,q}$$ defined by $$f^*(uv) = {f(u)+f(v) \over 2}$$, if  $$f(u)+f(v)$$ is even or $${f(u)+f(v)+1 \over 2}$$, if $$f(u)+f(v)$$ is odd. The line graph of G is a graph in which the vertices are the edges (lines) of G and the two points of L(G) are adjacent whenever the corresponding lines of G are adjacent. In this chapter, we investigate the meanness of both G and L(G) and thus introduce the definition of bimean graphs.

##### A Conceptual Framework for the Design of a Complex Real-time Management System (CoDeCS)

Kosolapov Anatolii

Current Topics on Mathematics and Computer Science Vol. 5, 15 July 2021, Page 120-129
https://doi.org/10.9734/bpi/ctmcs/v5/1722C

Modern information management systems are complicated systems that integrate territorial and functionally scattered complexes using network technology and real-time operations. The importance of their conceptual design grows when designing and developing such systems.  The huge number of used parameters, as well as the incompleteness of data, need the usage of particular system design methodologies and frameworks. The work is based on the national standard 34 and addresses the notion of modern information management system architecture as well as the main stages of system design. The methodology for creating complex real-time management (information) systems is proposed. Ontological models of knowledge about the problem domain provide the foundation of the methodology.  The proposed CoDeCS methodology is complemented by a set of tool programs for solving practical engineering tasks that can be used autonomously.

##### Studies on the Deficiency Indices of Product Differential Operators in Direct Sum Spaces

Sobhy El-Sayed Ibrahim

Current Topics on Mathematics and Computer Science Vol. 5, 15 July 2021, Page 130-147
https://doi.org/10.9734/bpi/ctmcs/v5/1817C

In this chapter, we explained the deficiency index problem for the product differential operators which are generated by a general ordinary quasi-differential expressions $$\tau_1$$, $$\tau_2$$, ..., $$\tau_n$$ each of order n with complex coefficients in the direct sum $$\oplus^N_{P=1}$$$$L^2_w (I_p)$$ of spaces of functions defined on each of the separate intervals in the circumstances of regular and singular end-points. The domains of these operators are described in terms of boundary conditions featuring $$L^2_w$$-solutions of the differential equations. These findings are more extensive of those of formally symmetric expression $$\tau$$ studied in [1 - 8], and those of general quasi-differential expressions $$\tau$$ in [9, 10, 11].