**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**1701

# World Academy of Science, Engineering and Technology

## [Mathematical and Computational Sciences]

### Online ISSN : 1307-6892

##### 1701 Application of Legendre Transformation to Portfolio Optimization

**Authors:**
Peter Benneth,
Tsaroh N. Theophilus,
Prince Benjamin

**Abstract:**

This research work aims at studying the application of Legendre Transformation Method (LTM) to Hamilton Jacobi Bellman (HJB) equation which is an example of optimal control problem. We discuss the steps involved in modelling the HJB equation as it relates to mathematical finance by applying the Ito’s lemma and maximum principle theorem. By applying the LTM and dual theory, the resultant HJB equation is transformed to a linear Partial Differential Equation (PDE). Also, the Optimal Investment Strategy (OIS) and the optimal value function were obtained under the exponential utility function. Furthermore, some numerical results were also presented with observations that the OIS under exponential utility is directly proportional to the appreciation rate of the risky asset and inversely proportional to the instantaneous volatility, predetermined interest rate, risk averse coefficient. Finally, it was observed that the optimal fund size is an increasing function of the risk free interest rate. This result is consistent with some existing results.

**Keywords:**
Legendre transformation method,
Optimal investment strategy,
Ito’s lemma,
Hamilton Jacobi Bellman equation,
Geometric Brownian motion,
financial market.

##### 1700 On Decomposition of Maximal Prefix Codes

**Authors:**
Nikolai Krainiukov,
Boris Melnikov

**Abstract:**

We study the properties of maximal prefix codes. The codes have many applications in computer science, theory of formal languages, data processing and data classification. Our approaches to study use finite state automata (so-called flower automata) for the representation of prefix codes. An important task is the decomposition of prefix codes into prime prefix codes (factors). We discuss properties of such prefix code decompositions. A linear time algorithm is designed to find the prime decomposition. We used the GAP computer algebra system, which allows us to perform algebraic operations for free semigroups, monoids and automata.

**Keywords:**
Maximal prefix code,
regular languages,
flower automata,
prefix code decomposing.

##### 1699 Labyrinth Fractal on a Convex Quadrilateral

**Authors:**
Harsha Gopalakrishnan,
Srijanani Anurag Prasad

**Abstract:**

Quadrilateral Labyrinth Fractals are a type of fractals presented in this paper. They belong to a unique class of fractals on any plane quadrilateral. The previously researched labyrinth fractals on the unit square and triangle inspire this form of fractal. This work describes how to construct a quadrilateral labyrinth fractal and looks at the circumstances in which it can be understood as the attractor of an iterated function system. Furthermore, some of its topological properties and the Hausdorff and box-counting dimensions of the quadrilateral labyrinth fractals are studied.

**Keywords:**
Fractals,
labyrinth fractals,
dendrites,
iterated function
system,
non-self similar,
non-self affine,
connected,
path connected.

##### 1698 Approximation to the Hardy Operator on Topological Measure Spaces

**Authors:**
Kairat T. Mynbaev,
Elena N. Lomakina

**Abstract:**

We consider a Hardy type operator generated by a family of open subsets of a Hausdorff topological space. The family is indexed with non-negative real numbers and is totally ordered. For this operator, we obtain two-sided bounds of its norm, a compactness criterion and bounds for its approximation numbers. Previously bounds for its approximation numbers have been established only in the one-dimensional case, while we do not impose any restrictions on the dimension of the Hausdorff space. The bounds for the norm and conditions for compactness have been found earlier but our approach is different in that we use domain partitions for all problems under consideration.

**Keywords:**
Approximation numbers,
boundedness and
compactness,
multidimensional Hardy operator,
Hausdorff
topological space.

##### 1697 Unveiling the Mathematical Essence of Machine Learning: A Comprehensive Exploration

**Authors:**
Randhir Singh Baghel

**Abstract:**

In this study, the fundamental ideas guiding the dynamic area of machine learning—where models thrive and algorithms change over time—are rooted in an innate mathematical link. This study explores the fundamental ideas that drive the development of intelligent systems, providing light on the mutually beneficial link between mathematics and machine learning.

**Keywords:**
Machine Learning,
deep learning,
Neural Network,
optimization.

##### 1696 Localized Meshfree Methods for Solving 3D-Helmholtz Equation

**Authors:**
Reza Mollapourasl,
Majid Haghi

**Abstract:**

In this study, we develop local meshfree methods known as radial basis function-generated finite difference (RBF-FD) method and Hermite finite difference (RBF-HFD) method to design stencil weights and spatial discretization for Helmholtz equation. The convergence and stability of schemes are investigated numerically in three dimensions with irregular shaped domain. These localized meshless methods incorporate the advantages of the RBF method, finite difference and Hermite finite difference methods to handle the ill-conditioning issue that often destroys the convergence rate of global RBF methods. Moreover, numerical illustrations show that the proposed localized RBF type methods are efficient and applicable for problems with complex geometries. The convergence and accuracy of both schemes are compared by solving a test problem.

**Keywords:**
Radial basis functions,
Hermite finite difference,
Helmholtz equation,
stability.

##### 1695 Identifying Factors Contributing to the Spread of Lyme Disease: A Regression Analysis of Virginia’s Data

**Authors:**
Fatemeh Valizadeh Gamchi,
Edward L. Boone

**Abstract:**

This research focuses on Lyme disease, a widespread infectious condition in the United States caused by the bacterium Borrelia burgdorferi sensu stricto. It is critical to identify environmental and economic elements that are contributing to the spread of the disease. This study examined data from Virginia to identify a subset of explanatory variables significant for Lyme disease case numbers. To identify relevant variables and avoid overfitting, linear poisson, and regularization regression methods such as ridge, lasso, and elastic net penalty were employed. Cross-validation was performed to acquire tuning parameters. The methods proposed can automatically identify relevant disease count covariates. The efficacy of the techniques was assessed using four criteria on three simulated datasets. Finally, using the Virginia Department of Health’s Lyme disease dataset, the study successfully identified key factors, and the results were consistent with previous studies.

**Keywords:**
Lyme disease,
Poisson generalized linear model,
Ridge
regression,
Lasso Regression,
elastic net regression.

##### 1694 Analysis of the Impact of Rainfall Change on the Seasonal Monsoon over the Jaipur District

**Authors:**
Randhir Singh Baghel

**Abstract:**

In this work, long-term spatiotemporal changes in rainfall are investigated and assessed at the meteorological divisional level using whole-year data from Rajasthan, India. Data from each of the district's eight tehsils are studied to see how the rainfall pattern has altered over the last 10 years. We primarily compare information from the Jaipur district in Rajasthan, India, at the tehsil level. We looked at the full year, and from January to December, there was constantly more rain than any other month. Furthermore, we compare the research of annual and monthly rainfall. Havey rainfall is also shown for two months, July and August.

**Keywords:**
Climate change,
temperature,
seasonal monsoons,
rainfall variability.

##### 1693 The Analogue of a Property of Pisot Numbers in Fields of Formal Power Series

**Authors:**
Wiem Gadri

**Abstract:**

This study delves into the intriguing properties of Pisot and Salem numbers within the framework of formal Laurent series over finite fields, a domain where these numbers’ spectral characteristics, Λm(β) and lm(β), have yet to be fully explored. Utilizing a methodological approach that combines algebraic number theory with the analysis of power series, we extend the foundational work of Erdos, Joo, and Komornik to this setting. Our research uncovers bounds for lm(β), revealing how these depend on the degree of the minimal polynomial of β and thus offering a characterization of Pisot and Salem formal power series. The findings significantly contribute to our understanding of these numbers, highlighting their distribution and properties in the context of formal power series. This investigation not only bridges number theory with formal power series analysis but also sets the stage for further interdisciplinary research in these areas.

**Keywords:**
Pisot numbers,
Salem numbers,
Formal power series,
Minimal polynomial degree.

##### 1692 The Algorithm to Solve the Extend General Malfatti’s Problem in a Convex Circular Triangle

**Authors:**
Ching-Shoei Chiang

**Abstract:**

The Malfatti’s problem solves the problem of fitting three circles into a right triangle such that these three circles are tangent to each other, and each circle is also tangent to a pair of the triangle’s sides. This problem has been extended to any triangle (called general Malfatti’s problem). Furthermore, the problem has been extended to have 1 + 2 + … + n circles inside the triangle with special tangency properties among circles and triangle sides; it is called the extended general Malfatti’s problem. In the extended general Malfatti’s problem, call it Tri(Tn), where Tn is the triangle number, there are closed-form solutions for the Tri(T₁) (inscribed circle) problem and Tri(T₂) (3 Malfatti’s circles) problem. These problems become more complex when n is greater than 2. In solving the Tri(Tn) problem, n > 2, algorithms have been proposed to solve these problems numerically. With a similar idea, this paper proposed an algorithm to find the radii of circles with the same tangency properties. Instead of the boundary of the triangle being a straight line, we use a convex circular arc as the boundary and try to find Tn circles inside this convex circular triangle with the same tangency properties among circles and boundary as in Tri(Tn) problems. We call these problems the Carc(Tn) problems. The algorithm is a mO(Tn) algorithm, where m is the number of iterations in the loop. It takes less than 1000 iterations and less than 1 second for the Carc(T16) problem, which finds 136 circles inside a convex circular triangle with specified tangency properties. This algorithm gives a solution for circle packing problem inside convex circular triangle with arbitrarily-sized circles. Many applications concerning circle packing may come from the result of the algorithm, such as logo design, architecture design, etc.

**Keywords:**
Circle packing,
computer-aided geometric design,
geometric constraint solver,
Malfatti’s problem.

##### 1691 Artificial Neural Network Approach for Inventory Management Problem

**Authors:**
Govind Shay Sharma,
Randhir Singh Baghel

**Abstract:**

The stock management of raw materials and finished goods is a significant issue for industries in fulfilling customer demand. Optimization of inventory strategies is crucial to enhancing customer service, reducing lead times and costs, and meeting market demand. This paper suggests finding an approach to predict the optimum stock level by utilizing past stocks and forecasting the required quantities. In this paper, we utilized Artificial Neural Network (ANN) to determine the optimal value. The objective of this paper is to discuss the optimized ANN that can find the best solution for the inventory model. In the context of the paper, we mentioned that the k-means algorithm is employed to create homogeneous groups of items. These groups likely exhibit similar characteristics or attributes that make them suitable for being managed using uniform inventory control policies. The paper proposes a method that uses the neural fit algorithm to control the cost of inventory.

**Keywords:**
Artificial Neural Network,
inventory management,
optimization,
distributor center.

##### 1690 The Impact of Seasonality on Rainfall Patterns: A Case Study

**Authors:**
Priti Kaushik,
Randhir Singh Baghel,
Somil Khandelwal

**Abstract:**

This study uses whole-year data from Rajasthan, India, at the meteorological divisional level to analyze and evaluate long-term spatiotemporal trends in rainfall and looked at the data from each of the thirteen tehsils in the Jaipur district to see how the rainfall pattern has altered over the last 10 years. Data on daily rainfall from the Indian Meteorological Department (IMD) in Jaipur are available for the years 2012 through 2021. We mainly focus on comparing data of tehsil wise in the Jaipur district, Rajasthan, India. Also analyzed is the fact that July and August always see higher rainfall than any other month. Rainfall usually starts to rise around week 25th and peaks in weeks 32nd or 33rd. They showed that on several occasions, 2017 saw the least amount of rainfall during a long span of 10 years. The greatest rain fell between 2012 and 2021 in 2013, 2019, and 2020.

**Keywords:**
Data analysis,
extreme events,
rainfall,
descriptive case studies,
precipitation temperature.

##### 1689 An Ecological Model for Three Species with Crowley–Martin Functional Response

**Authors:**
Randhir Singh Baghel,
Govind Shay Sharma

**Abstract:**

In this paper, we explore an ecosystem that contains a three-species food chain. The first and second species are in competition with one another for resources. However, the third species plays an important role in providing non-linear Crowley-Martin functional support for the first species. Additionally, the third species consumes the second species in a linear fashion, taking advantage of the available resources. This intricate balance ensures the survival of all three species in the ecosystem. A set of non-linear isolated first-order differential equations establish this model. We examine the system's stability at all potential equilibrium locations using the perturbed technique. Furthermore, by spending a lot of time observing the species in their natural habitat, the numerical illustrations at suitable parameter values for the model are shown.

**Keywords:**
Competition,
predator,
response function,
local stability,
numerical simulations.

##### 1688 Optimal Relaxation Parameters for Obtaining Efficient Iterative Methods for the Solution of Electromagnetic Scattering Problems

**Authors:**
Nadaniela Egidi,
Pierluigi Maponi

**Abstract:**

The approximate solution of a time-harmonic electromagnetic scattering problem for inhomogeneous media is required in several application contexts and its two-dimensional formulation is a Fredholm integral equation of second kind. This integral equation provides a formulation for the direct scattering problem but has to be solved several times in the numerical solution of the corresponding inverse scattering problem. The discretization of this Fredholm equation produces large and dense linear systems that are usually solved by iterative methods. To improve the efficiency of these iterative methods, we use the Symmetric SOR preconditioning and propose an algorithm to evaluate the associated relaxation parameter. We show the efficiency of the proposed algorithm by several numerical experiments, where we use two Krylov subspace methods, i.e. Bi-CGSTAB and GMRES.

**Keywords:**
Fredholm integral equation,
iterative method,
preconditioning,
scattering problem.

##### 1687 Comparison of Bayesian and Regression Schemes to Model Public Health Services

**Authors:**
Sotirios Raptis

**Abstract:**

Bayesian reasoning (BR) or Linear (Auto) Regression (AR/LR) can predict different sources of data using priors or other data, and can link social service demands in cohorts, while their consideration in isolation (self-prediction) may lead to service misuse ignoring the context. The paper advocates that BR with Binomial (BD), or Normal (ND) models or raw data (.D) as probabilistic updates can be compared to AR/LR to link services in Scotland and reduce cost by sharing healthcare (HC) resources. Clustering, cross-correlation, along with BR, LR, AR can better predict demand. Insurance companies and policymakers can link such services, and examples include those offered to the elderly, and low-income people, smoking-related services linked to mental health services, or epidemiological weight in children. 22 service packs are used that are published by Public Health Services (PHS) Scotland and Scottish Government (SG) from 1981 to 2019, broken into 110 year series (factors), joined using LR, AR, BR. The Primary component analysis found 11 significant factors, while C-Means (CM) clustering gave five major clusters.

**Keywords:**
Bayesian probability,
cohorts,
data frames,
regression,
services,
prediction.

##### 1686 Pure Scalar Equilibria for Normal-Form Games

**Authors:**
H. W. Corley

**Abstract:**

A scalar equilibrium (SE) is an alternative type of equilibrium in pure strategies for an n-person normal-form game G. It is defined using optimization techniques to obtain a pure strategy for each player of G by maximizing an appropriate utility function over the acceptable joint actions. The players’ actions are determined by the choice of the utility function. Such a utility function could be agreed upon by the players or chosen by an arbitrator. An SE is an equilibrium since no players of G can increase the value of this utility function by changing their strategies. SEs are formally defined, and examples are given. In a greedy SE, the goal is to assign actions to the players giving them the largest individual payoffs jointly possible. In a weighted SE, each player is assigned weights modeling the degree to which he helps every player, including himself, achieve as large a payoff as jointly possible. In a compromise SE, each player wants a fair payoff for a reasonable interpretation of fairness. In a parity SE, the players want their payoffs to be as nearly equal as jointly possible. Finally, a satisficing SE achieves a personal target payoff value for each player. The vector payoffs associated with each of these SEs are shown to be Pareto optimal among all such acceptable vectors, as well as computationally tractable.

**Keywords:**
Compromise equilibrium,
greedy equilibrium,
normal-form game,
parity equilibrium,
pure strategies,
satisficing equilibrium,
scalar equilibria,
utility function,
weighted equilibrium.

##### 1685 Mathematical Analysis of Stock Prices Prediction in a Financial Market Using Geometric Brownian Motion Model

**Authors:**
Edikan E. Akpanibah,
Ogunmodimu Dupe Catherine

**Abstract:**

The relevance of geometric Brownian motion (GBM) in modelling the behaviour of stock market prices (SMP) cannot be over emphasized taking into consideration the volatility of the SMP. Consequently, there is need to investigate how GBM models are being estimated and used in financial market to predict SMP. To achieve this, the GBM estimation and its application to the SMP of some selected companies are studied. The normal and log-normal distributions were used to determine the expected value, variance and co-variance. Furthermore, the GBM model was used to predict the SMP of some selected companies over a period of time and the mean absolute percentage error (MAPE) were calculated and used to determine the accuracy of the GBM model in predicting the SMP of the four companies under consideration. It was observed that for all the four companies, their MAPE values were within the region of acceptance. Also, the MAPE values of our data were compared to an existing literature to test the accuracy of our prediction with respect to time of investment. Finally, some numerical simulations of the graphs of the SMP, expectations and variance of the four companies over a period of time were presented using MATLAB programming software.

**Keywords:**
Stock Market,
Geometric Brownian Motion,
normal and log-normal distribution,
mean absolute percentage error.

##### 1684 Improved Triple Integral Inequalities of Hermite-Hadamard Type

**Authors:**
Leila Nasiri

**Abstract:**

In this paper, we present the concept of preinvex functions on the co-ordinates on an invex set and establish some triple integral inequalities of Hermite-Hadamard type for functions whose third order partial derivatives in absolute value are preinvex on the co-ordinates. The results presented here generalize the obtained results in earlier works for functions whose triple order partial derivatives in absolute value are convex on the co-ordinates on a rectangular box in R^{3}.

**Keywords:**
Co-ordinated preinvex functions,
Hermite-Hadamard
type inequalities,
partial derivatives,
triple integral.

##### 1683 A Review on Higher Order Spline Techniques for Solving Burgers Equation Using B-Spline Methods and Variation of B-Spline Techniques

**Authors:**
Maryam Khazaei Pool,
Lori Lewis

**Abstract:**

This is a summary of articles based on higher order B-splines methods and the variation of B-spline methods such as Quadratic B-spline Finite Elements Method, Exponential Cubic B-Spline Method Septic B-spline Technique, Quintic B-spline Galerkin Method, and B-spline Galerkin Method based on the Quadratic B-spline Galerkin method (QBGM) and Cubic B-spline Galerkin method (CBGM). In this paper we study the B-spline methods and variations of B-spline techniques to find a numerical solution to the Burgers’ equation. A set of fundamental definitions including Burgers equation, spline functions, and B-spline functions are provided. For each method, the main technique is discussed as well as the discretization and stability analysis. A summary of the numerical results is provided and the efficiency of each method presented is discussed. A general conclusion is provided where we look at a comparison between the computational results of all the presented schemes. We describe the effectiveness and advantages of these methods.

**Keywords:**
Burgers’ Equation,
Septic B-spline,
Modified Cubic
B-Spline Differential Quadrature Method,
Exponential Cubic
B-Spline Technique,
B-Spline Galerkin Method,
and Quintic B-Spline
Galerkin Method.

##### 1682 On a Conjecture Regarding the Adam Optimizer

**Authors:**
Mohamed Akrout,
Douglas Tweed

**Abstract:**

The great success of deep learning relies on efficient optimizers, which are the algorithms that decide how to adjust network weights and biases based on gradient information. One of the most effective and widely used optimizers in recent years has been the method of adaptive moments, or Adam, but the mathematical reasons behind its effectiveness are still unclear. Attempts to analyse its behaviour have remained incomplete, in part because they hinge on a conjecture which has never been proven, regarding ratios of powers of the first and second moments of the gradient. Here we show that this conjecture is in fact false, but that a modified version of it is true, and can take its place in analyses of Adam.

**Keywords:**
Adam optimizer,
Bock’s conjecture,
stochastic
optimization,
average regret.

##### 1681 Node Insertion in Coalescence Hidden-Variable Fractal Interpolation Surface

**Authors:**
Srijanani Anurag Prasad

**Abstract:**

The Coalescence Hidden-variable Fractal Interpolation Surface (CHFIS) was built by combining interpolation data from the Iterated Function System (IFS). The interpolation data in a CHFIS comprise a row and/or column of uncertain values when a single point is entered. Alternatively, a row and/or column of additional points are placed in the given interpolation data to demonstrate the node added CHFIS. There are three techniques for inserting new points that correspond to the row and/or column of nodes inserted, and each method is further classified into four types based on the values of the inserted nodes. As a result, numerous forms of node insertion can be found in a CHFIS.

**Keywords:**
Fractal,
interpolation,
iterated function system,
coalescence,
node insertion,
knot insertion.

##### 1680 Forecasting the Fluctuation of Currency Exchange Rate Using Random Forest

**Abstract:**

The exchange rate is one of the most important economic variables, especially for a small, open economy such as Albania. Its effect is noticeable on one country's competitiveness, trade and current account, inflation, wages, domestic economic activity and bank stability. This study investigates the fluctuation of Albania’s exchange rates using monthly average foreign currency, Euro (Eur) to Albanian Lek (ALL) exchange rate with a time span from January 2008 to June 2021 and the macroeconomic factors that have a significant effect on the exchange rate. Initially, the Random Forest Regression algorithm is constructed to understand the impact of economic variables in the behavior of monthly average foreign currencies exchange rates. Then the forecast of macro-economic indicators for 12 months was performed using time series models. The predicted values received are placed in the random forest model in order to obtain the average monthly forecast of Euro to Albanian Lek (ALL) exchange rate for the period July 2021 to June 2022.

**Keywords:**
Exchange rate,
Random Forest,
time series,
Machine Learning,
forecasting.

##### 1679 An Insurer’s Investment Model with Reinsurance Strategy under the Modified Constant Elasticity of Variance Process

**Authors:**
K. N. C. Njoku,
Chinwendu Best Eleje,
Christian Chukwuemeka Nwandu

**Abstract:**

One of the problems facing most insurance companies is how best the burden of paying claims to its policy holders can be managed whenever need arises. Hence there is need for the insurer to buy a reinsurance contract in order to reduce risk which will enable the insurer to share the financial burden with the reinsurer. In this paper, the insurer’s and reinsurer’s strategy is investigated under the modified constant elasticity of variance (M-CEV) process and proportional administrative charges. The insurer considered investment in one risky asset and one risk free asset where the risky asset is modeled based on the M-CEV process which is an extension of constant elasticity of variance (CEV) process. Next, a nonlinear partial differential equation in the form of Hamilton Jacobi Bellman equation is obtained by dynamic programming approach. Using power transformation technique and variable change, the explicit solutions of the optimal investment strategy and optimal reinsurance strategy are obtained. Finally, some numerical simulations of some sensitive parameters were obtained and discussed in details where we observed that the modification factor only affects the optimal investment strategy and not the reinsurance strategy for an insurer with exponential utility function.

**Keywords:**
Reinsurance strategy,
Hamilton Jacobi Bellman equation,
power transformation,
M-CEV process,
exponential utility.

##### 1678 Designing Social Care Policies in the Long Term: A Study Using Regression, Clustering and Backpropagation Neural Nets

**Authors:**
Sotirios Raptis

**Abstract:**

Linking social needs to social classes using different criteria may lead to social services misuse. The paper discusses using ML and Neural Networks (NNs) in linking public services in Scotland in the long term and advocates, this can result in a reduction of the services cost connecting resources needed in groups for similar services. The paper combines typical regression models with clustering and cross-correlation as complementary constituents to predict the demand. Insurance companies and public policymakers can pack linked services such as those offered to the elderly or to low-income people in the longer term. The work is based on public data from 22 services offered by Public Health Services (PHS) Scotland and from the Scottish Government (SG) from 1981 to 2019 that are broken into 110 years series called factors and uses Linear Regression (LR), Autoregression (ARMA) and 3 types of back-propagation (BP) Neural Networks (BPNN) to link them under specific conditions. Relationships found were between smoking related healthcare provision, mental health-related health services, and epidemiological weight in Primary 1(Education) Body Mass Index (BMI) in children. Primary component analysis (PCA) found 11 significant factors while C-Means (CM) clustering gave 5 major factors clusters.

**Keywords:**
Probability,
cohorts,
data frames,
services,
prediction.

##### 1677 Base Change for Fisher Metrics: Case of the q−Gaussian Inverse Distribution

**Authors:**
Gabriel I. Loaiza O.,
Carlos A. Cadavid M.,
Juan C. Arango P.

**Abstract:**

It is known that the Riemannian manifold determined by the family of inverse Gaussian distributions endowed with the Fisher metric has negative constant curvature κ = −1/2 , as does the family of usual Gaussian distributions. In the present paper, firstly we arrive at this result by following a different path, much simpler than the previous ones. We first put the family in exponential form, thus endowing the family with a new set of parameters, or coordinates, θ1, θ2; then we determine the matrix of the Fisher metric in terms of these parameters; and finally we compute this matrix in the original parameters. Secondly, we define the Inverse q−Gaussian distribution family (q < 3), as the family obtained by replacing the usual exponential function by the Tsallis q−exponential function in the expression for the Inverse Gaussian distribution, and observe that it supports two possible geometries, the Fisher and the q−Fisher geometry. And finally, we apply our strategy to obtain results about the Fisher and q−Fisher geometry of the Inverse q−Gaussian distribution family, similar to the ones obtained in the case of the Inverse Gaussian distribution family.

**Keywords:**
Base of Changes,
Information Geometry,
Inverse
Gaussian distribution,
Inverse q-Gaussian distribution,
Statistical
Manifolds.

##### 1676 Solution of Two-Point Nonlinear Boundary Problems Using Taylor Series Approximation and the Ying Buzu Shu Algorithm

**Authors:**
U. C. Amadi,
N. A. Udoh

**Abstract:**

One of the major challenges faced in solving initial and boundary problems is how to find approximate solutions with minimal deviation from the exact solution without so much rigor and complications. The Taylor series method provides a simple way of obtaining an infinite series which converges to the exact solution for initial value problems and this method of solution is somewhat limited for a two point boundary problem since the infinite series has to be truncated to include the boundary conditions. In this paper, the Ying Buzu Shu algorithm is used to solve a two point boundary nonlinear diffusion problem for the fourth and sixth order solution and compare their relative error and rate of convergence to the exact solution.

**Keywords:**
Ying Buzu Shu,
nonlinear boundary problem,
Taylor series algorithm,
infinite series.

##### 1675 Optimizing Data Evaluation Metrics for Fraud Detection Using Machine Learning

**Authors:**
Jennifer Leach,
Umashanger Thayasivam

**Abstract:**

The use of technology has benefited society in more ways than one ever thought possible. Unfortunately, as society’s knowledge of technology has advanced, so has its knowledge of ways to use technology to manipulate others. This has led to a simultaneous advancement in the world of fraud. Machine learning techniques can offer a possible solution to help decrease these advancements. This research explores how the use of various machine learning techniques can aid in detecting fraudulent activity across two different types of fraudulent datasets, and the accuracy, precision, recall, and F1 were recorded for each method. Each machine learning model was also tested across five different training and testing splits in order to discover which split and technique would lead to the most optimal results.

**Keywords:**
Data science,
fraud detection,
machine learning,
supervised learning.

##### 1674 The Analysis of Different Classes of Weighted Fuzzy Petri Nets and Their Features

**Authors:**
Yurii Bloshko,
Oksana Olar

**Abstract:**

This paper presents the analysis of six different classes of Petri nets: fuzzy Petri nets (FPN), generalized fuzzy Petri nets (GFPN), parameterized fuzzy Petri nets (PFPN), T2GFPN, flexible generalized fuzzy Petri nets (FGFPN), binary Petri nets (BPN). These classes were simulated in the special software PNeS® for the analysis of its pros and cons on the example of models which are dedicated to the decision-making process of passenger transport logistics. The paper includes the analysis of two approaches: when input values are filled with the experts’ knowledge; when fuzzy expectations represented by output values are added to the point. These approaches fulfill the possibilities of triples of functions which are replaced with different combinations of t-/s-norms.

**Keywords:**
Fuzzy petri net,
intelligent computational techniques,
knowledge representation,
triangular norms.

##### 1673 The Possibility of Solving a 3x3 Rubik’s Cube under 3 Seconds

**Authors:**
Chung To Kong,
Siu Ming Yiu

**Abstract:**

Rubik's cube was invented in 1974. Since then, speedcubers all over the world try their best to break the world record again and again. The newest record is 3.47 seconds. There are many factors that affect the timing including turns per second (tps), algorithm, finger trick, and hardware of the cube. In this paper, the lower bound of the cube solving time will be discussed using convex optimization. Extended analysis of the world records will be used to understand how to improve the timing. With the understanding of each part of the solving step, the paper suggests a list of speed improvement technique. Based on the analysis of the world record, there is a high possibility that the 3 seconds mark will be broken soon.

**Keywords:**
Rubik’s cube,
convex optimization,
speed cubing,
CFOP.

##### 1672 Matrix Completion with Heterogeneous Observation Cost Using Sparsity-Number of Column-Space

**Authors:**
Ilqar Ramazanli

**Abstract:**

The matrix completion problem has been studied broadly under many underlying conditions. In many real-life scenarios, we could expect elements from distinct columns or distinct positions to have a different cost. In this paper, we explore this generalization under adaptive conditions. We approach the problem under two different cost models. The first one is that entries from different columns have different observation costs, but, within the same column, each entry has a uniform cost. The second one is any two entry has different observation cost, despite being the same or different columns. We provide complexity analysis of our algorithms and provide tightness guarantees.

**Keywords:**
Matrix completion,
adaptive learning,
heterogeneous
cost,
Matroid optimization.