**Mathematics Research Project (Paper) Topics and Ideas**

Mathematics includes the study of such topics as quantity, structure, space, and change. It has no generally accepted definition. Mathematicians seek and use patterns to formulate new conjectures; they resolve the truth or falsity of conjectures by mathematical proof. We have developed a list of interesting mathematics research project (paper) topics as follows:

**1. Evaluating the Validity of Perron’s Paradox**

**Discuss:** Perron’s paradox, highlights the danger of considering a solution to a mathematical problem if the solution is actually nonexistent.

For example, if we assume the largest natural number exists and it is N,

Therefore if

N>1,

then

N^2>N,

and this contradicts our hypothesis that N is the largest natural number, hence the largest natural number is

N=1,

and this would seem to be absurd, hence it emphasizes the danger of assuming a solution exists while it is actually nonexistent.

Let’s reexamine this again.

**2. Evaluating the Validity of Video Education for Mathematical Studies: Case Study of videos of mathematical concepts made by students**

**Discuss:** This study presents a research scenario of a teaching experience in which students use ICT to develop video tutorials of the mathematical concepts from the Mathematics for Business Studies and evaluates the effectiveness of the process on their understanding of the subject matter.

**3. A Critical Analysis of the Henstock Integral and Riemann Integral**

**Discuss:** The Riemann integral, which is familiar to the reader from calculus, can be loosely defined as the limit of the sum of narrow “strips” of the area under a function taken as the width of each strip goes to 0. This Study will apply the Henstock-Kurzweil integrals to deal with linear differential equations.

**4. What is Musical Mathematics? Evaluating the Music Composers who have used Mathematical Formulae in Their Compositions**

**Discuss:** Many Music composers have used mathematical formulas in their compositions. This study will evaluate the best in history and develop a mathematically perfect tune for a specific audience.

**5. Examining the Relationship between Horizontal/Vertical asymptote of reciprocal functions and exponential function.**

**Discuss:** Using the Maclaurin power expansion, for

|x| < 1

x/(1-x) = x{1+ x + x² +..+ xⁿ+..}

= x + x² + .. + xⁿ + ..

The exponential function has an estimate

ex = 1 + x + (1/2)x² + (1/6)x³+

**Note**

|ex – 1 – (x/(1-x))| = O(x²)

therefore, for small values for x, the given rational function is a good approximation of an exponential function.

This Study will examine the connections and possible alternation of formula utilization in real-life calculations.

**6. Mathematical Solutions for Dementia Prevention and Medication**

**Discuss:** This Study will examine the effects of using Math Protocols and Math’s games for the elderly on Dementia Progression.

**7. The Founder of Algebra; Thales or Pythagoras or Diophantus or Al-Khwarizmi or any other?**

**Discuss:** Algebra is essential and is taught to every student in high school, but who is responsible for inventing algebra? It was discovered and developed at different times and in different locations, and these discoveries and new ideas eventually came together to give us what we collectively call algebra today.

**8. Principles and Processes of Developing Mathematically Generated Brain Signals**

**Discuss:** This Study will examine the difference between natural human signals and an artificially generated signal. Evaluating what we need to do to improvise and minimize the difference between the two. The result takes us closer to a better analytical and predictive study of Human brains.

**9. The Mathematics Behind Beauty: Can Beauty be calculated? Let’s Talk about the Golden Ratio.**

**Discuss:** If the amount of beauty is proportional to the quantity and variety of identity relations, then the smallest quantity of beauty would be one identity relation.

**10. Developing a Formula for calculating the Impact of Automation Techniques on Cost of Quality in Nigeria**

**Discuss:** Nigerian Companies are largely manual systems, and development is coming to Nigerian Companies sooner rather than later, the main development is improved automation, this study will calculate the effect of automation on Nigerian Companies Quality, Cost Efficiency and Employment landscape.

**11. Evaluating the Validity of Meta-Heuristic and Heuristic Algorithms?**

**Discuss:** When tackling, High-Quality Algorithm Solutions, Meta-Heuristic and Heuristic Algorithms are often valid methods of approach. However, a large body of mathematicians do not rate Meta-Heuristic and Heuristic Algorithms are valid. This study will examine why.

**12. Evaluating the Utilization of the Dice CAPTCHA by Advanced Statistical Analysis**

**Discuss:** This Study will analyse the utilization of Dice CAPTCHA by advanced statistical analysis. An experiment will be performed on a population of 200 Internet users, characterised by age and Internet experiences, to which the solution to the Dice CAPTCHA is required on a system.

**13. Evaluating the Validity of the “DMAS” rule in Mathematical Education**

**Discuss:** Division, Multiplication, Addition and Subtraction (DMAS) is the elementary rule for the order of operation of the Binary operations. What is the scientific and technical reason behind this mathematical myth through Multiplication before Division also gives the same result mostly? DMAS, a nice tool but has less convincing/appealing to admit its order of operation.

**14. The Effect of Complex Mathematical Problem Solving on the Human Brain (A Case Study of Chess)**

**Discuss:** Want to transform your brain and make better decisions? Learn chess, a game synonymous with intelligence and brainpower. This Study will aim to prove that chess improves the cognitive structure of a person’s brain.

**15. A Critical Review of the Magnetostrsiction and Magneto-elasticity Theory**

**Discuss:** This Study will be a deep dive into the current modern status of magnetostrsiction (MS), magneto-elastic (MEL) coupling and magneto-elasticity (ME) theory and models, particularly the standard theory for magnetostrsiction.

**16. Understanding Conditional law of a branching process observing a subpopulation (Industrial Mathematics)**

**Discuss:** This Study is concerned with filtering the cardinality of a branching process observing the cardinality of a subpopulation. In this model, both the processes, state and observation are pure jump processes and may have common jump times. Preliminary properties are studied in the tree framework.

**17. Understanding Algorithms for Machine Optimization (Industrial Mathematics)**

**Discuss:** This Study will seek to understand how the machine ‘thinks’: Understanding opacity in machine learning algorithms.

**18. A Critical Review of Maxwell’s Equations: Boundary Element Methods**

**Discuss:** What are the essential needs of modern life? One can say without any hesitation, electric lighting, heating and cooling, telephones, electric motors and generators, radio, television, X-ray machine, ECG machine, radars and weather forecasting system equipment, etc.

**19. A Critical Review of Models of Hysteresis and Applications (Industrial Mathematics)**

**Discuss:** The phenomenon of hysteresis occurs in a large number of practical situations and plays a very important role in areas like ferromagnetism, ferroelectricity, phase transitions from solid to liquid or liquid to solid.

**20. A Critical Review of Stochastic Differential Delay Equations**

**Discuss:** In Mathematics, delay differential equation is one type of differential equation in which the derivative of the unknown function at a certain time and stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution.

**21. Measuring and Modelling Soil Depth Functions in Farms in Northern Nigeria**

**Discuss:** The changes of soil particle size distribution in a soil profile can be used as an indicator of soil formation and processes and has been used as a proxy for soil age or degree of development.

**22. Developing a Mathematics Tutoring System with Metacognitive Strategies**

**Discuss:** This Study will investigate the metacognitive mathematics tutor impact on students learning and metacognition. Metacognitive Mathematics tutor is an intelligent tutoring system which supports three metacognitive learning strategies: self-explanation, self-questioning and self-monitoring.

**23. Understanding the Cross-Entropy Method for Continuous Multi-Extremal Optimization**

**Discuss:** In recent years, the cross-entropy method has been successfully applied to a wide range of discrete optimization tasks. In this paper we consider the cross-entropy method in the context of continuous optimization

**24. Comparison of a linear and non-linear method for determination of optimum equilibrium isotherm for adsorption of copper (II) onto treated Martial sand**

**Discuss:** A comparison of linear and non-linear regression method for predicting the optimum isotherm was made using the experimental adsorption equilibrium data of Cu (II) onto treated Martil sand (TMS). It is well known that in curve fitting of a linear and non-linear form of adsorption isotherms on a set of experimental data, different values are achieved for isotherm parameters and R2. What “mathematical” explanation can be given for this discrepancy? e.g. Langmuir adsorption isotherm can be written in different forms and for a special set of data there are different R2 as followings:

Non-linear form:

**qe=qm.K.Ce/(1+K.Ce) ,**

**R2=0.9991**

Linear form 1:

**1/qe=1/qm+(1/(qm.K).1/Ce ,**

**R2=0.9715**

Linear form 2:

**qe=qm-(1/K).qe/Ce , **

**R2=0.4542**

**25. Understanding Mathematical Intuition and its real-world applications**

**Discuss:** The notion of ‘intuition’ can depend on whether a philosophical or psychological perspective is exerted. From a perspective of the philosophy of mathematics, mathematics develops through a particular mathematical intuition shared by all humans, but linguistic representations of mathematics have departed from the intuitive mathematical constructions.

**26. A Comparative Analysis of Curve-linear and Cartesian Coordinate Systems**

**Discuss:** Coordinate systems provide a way to define a point in space in either one, two, or three dimensions. The most prevalent coordinate system used in linear motion applications is the Cartesian system.

**27. High Efficiency Computation of Monte-Carlo Simulations with Variance Reduction**

Discuss: Companies perform Monte-Carlo transport simulations on large and complex tokamak models such as ITER. These simulations are challenging as streaming and deep penetration effects are equally important. In order to make such simulations tractable, both variance reduction (VR) techniques and parallel computing are used. This study will development techniques that improve the efficiency of Monte-Carlo Simulations.

**28. Evaluating the effects of hot-carrier energy relaxation on collapsing field domains**

**Discuss:** Multiple “collapsing” field domains are a physical reason for superfast switching and sub-terahertz (sub-THz) emission experimentally observed in powerfully avalanching GaAs structures. This phenomenon, however, has been studied so far without considering carrier energy relaxation and that essentially has restricted the possibility of correct interpretation of experimental results.

**29. Intrusive Teaching Methods for Mathematics**

Discuss: This study analyzes the teaching techniques utilized by elementary, secondary and college teachers in teaching mathematics and the effects the teacher’s mathematical knowledge plas on its effectiveness.

**30. A Deep dive into Mathematical Descriptors for Spatial Phenomenon**

**Discuss:** This Study will analyze what other types of spatial descriptors exist. The spatial descriptors symbols based on the nature of the given data. There are curves to describe the solution set of relations y=f(x), G(x,y)=0 as st. lines parabolas, circles, etc. Also, surfaces describe the solution set of relations G(x,y,z)=0 as st. planes, ellipsoid, elliptic cones, general manifolds. Where (x,y,z) represent the coordinates of a point in the real space. Also, intervals, areas, and volumes describe regions. Sequences are described as dots or points in the coordinates system.

**31. A Comparative Analysis of Linear-Elastic & Isotropic Elasticity Tensor**

**Discuss:** This Study analyzes the structure of both transport and elastic properties of Linear-Elasticity and Isotropic-Elasticity of porous materials.

**32. Understanding Pell’s Equation: Proof of Euclid’s formula**

**Discuss:** Euclid’s formula by a, b, c is sufficient for the triangle to be Pythagorean is apparent from the fact that for positive integers m and n, m > n, the a, b, and c given by the formula are all positive integers, and from the fact that

**a=m2−n2, b=2mn, c=m2+n2 where m>n>0**

for any 2 values of m and n the above formula will give a Pythagorean Triple.

To get a Primitive Pythagorean triple, m and n have to co-prime and not both odd.

This Study will Seek to understand this formula

**33. Developing 3D Mapping System: Understanding the points required to map a 3D shape**

**Discuss:** A shape is fixed by three points in 3D. In terms of translations and rotations, you can think of it this way. If point A maps to A’, that takes care of three translations, but the shape can freely rotate attached to A’. If a second point B maps to B’, the shape can only rotate around line A’B’. A third point C’ fixates that last degree of freedom.

**34. Gender Roles in Instrumentation in Mathematics Education**

**Discuss:** Contrary to the common perception that mathematics is a pure mental activity, the importance of tools in mathematical activity has been largely acknowledged: Ç many of the actions of doing mathematics involve selecting, using and creating tools.

**35. Understanding Image Processing and Fourier Wavelet Methods**

**Discuss:** Image processing based on the continuous or discrete image transforms are classic techniques. The image transforms are widely used in image filtering, data description, etc.

**36. A Comparative Analysis of Flipped Classroom and Traditional Classroom in a Given School**

**Discuss:** This study aims to determine the perception of the lecturers’/students’ and their achievement between two learning cultures, the traditional classroom and flipped classroom in a Given School.

**37. Solving Inverse of Large Matrix: Utilizing Arnoldi Method to Solve Quadratic Eigenvalue Problems**

**Discuss:** First a large size matrix requires sufficient memory to inverse the matrix. Secondly, there are several mathematical techniques available to solve the inverse of a matrix.

**38. Children are the Building Blocks of Economic Complexity: The Economic Complexity Index Formula**

**Discuss:** This Study examines the question of economic complexity is equivalent to asking whether we can infer properties such as the diversity and exclusivity of the Lego pieces inside a child’s bucket by looking only at the models of a group of children.

**39. Formation, Coordination, Development and Adaptation of Small Groups of Complex Systems**

**Discuss:** A conceptually elegant analysis of groups as systems. Although the systems approach has been growing more influential in various fields of social psychology in the last ten years, no one has put forward a definitive analysis that applies with fidelity the general systems approach to group processes.

**40. A Critical Review of Quantum Evolutionary Algorithm: Analysing Invasive Weed Optimization**

**Discuss:** This Study will analyse a new optimization algorithm based on developing the invasive weed optimization algorithm by quantum computing.

**41. Developing a Detection Algorithm for Melanoma using a Hybrid Neural Network**

**Discuss:** Melanoma is among the most dangerous forms of cancers. Early detection is the best defense against melanoma. This Study will analyse a new efficient method for detecting melanoma via images.

**42. Meta-Heuristics Algorithm and Global Solution: Sampling Distribution for the Probability of Finding a Global Max/Min Levels**

**Discuss:** Meta-heuristics are the most recent development in approximate search methods for solving complex optimization problems that arise in business, commerce, engineering, industry, and many other areas.

**43. Validating the impact of Doppler Shift on the rate of information transmission?**

**Discuss:** When a modulated signal lasting a

time =T

is subjected to an optical Doppler shift

K (K=shifted frequency/original frequency)

the Doppler-shifted signal will last

T/K.

This because all shifted harmonics of its Fourier series with a (fundamental period of T) will last

1/K times

the period of the original harmonic.

**44. A Critical Review of Mathematics Anxiety: Why Primary School Studies are afraid of Mathematics**

**Discuss:** Thousands of studies have been published that touch on the topic of “math anxiety.” The overwhelming fear of math, regardless of one’s actual aptitude, affects students of all ages, from kindergarten to grad school. This study will analyze the reasons for this and best teaching practices to reduce it.

**45. Understanding Monte Carlo Methods (Industrial Mathematics)**

**Discuss:** Monte Carlo simulations are used to model the probability of different outcomes in a process that cannot easily be predicted due to the intervention of random variables. It is a technique used to understand the impact of risk and uncertainty in prediction and forecasting models.

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