Mathematics and Computer Science: Contemporary Developments Vol. 5

Dynamics of HIV Infection of CD4\(^+\)T Cells: A Fractional Approach

A. R. Meshram — 2024-10-09

The dynamics of Mathematical model of Human Immunodeficiency Virus with three non overlapping classes has been taken into consideration in this chapter. A mathematical model that calculates susceptible CD4⁺T cells, infected CD4⁺T cells and virus particles has been examined here using the fractional differential transform method(FDTM) with stability analysis. A stability of the fractional nonlinear model with Hurwitz state matrix is examined using the Lyapunov direct method. A brief review of literature for integer order as well as fractional order on mathematical biological modeling has been collected to solidify our mathematical approach to solve the proposed HIV model. A nonlinear mathematical model of differential equations has been put forward and analyzed by applying FDTM. The proposed technique gives a solution in the form convergent series as a linear combination in the form of polynomial. An infinite series solution of the system of differential equation is computed by defining fixed components with different time intervals. Furthermore, the solution calculated through FDTM ( integer order) is correlated with the solution calculated using DTM and Laplace Adomain Decomposition Method. Additionally, the graphical representation of the model is given using the fourth order Runge Kutta Method. The solution is analyzed numerically and graphically by using the software Python.

Oscillating Dumbbell Dynamics in Viscous Fluids: Analytical Insights

Mohammed Mattar Al-Hatmi — 2024-10-09

The aim of this chapter is to investigate analytically the motion of oscillating dumbbell, two micro-spheres connected by a spring, in a viscous incompressible fluid at low Reynolds number. The analytical approach used in the study offers a detailed and precise understanding of the motion characteristics, which can help improve the design and optimization of systems involving oscillatory motion in viscous environments. The oscillating dumbbell consists of one conducting sphere and assumed to be actively in motion under the action of an external oscillator field while the other is non-conducting sphere. As result, the oscillating dumbbell moves due to the induced ow oscillation of the surrounding fluid. The fluid ow past the spheres is described by the Stokes equation and the governing equation in the vector form for the oscillating dumbbell is solved asymptotically using the two-timing method. For illustrations, applying a simple oscillatory external field, a systematic description of the average velocity of the oscillating dumbbell is formulated. The trajectory of the oscillating dumbbell was found to be inversely proportional to the frequency of the external field, and the results demonstrated that the oscillating dumbbell moves in a circular path with a speed that decreases inversely with the length of the spring. It is worth noting that the average velocities of the oscillating dumbbell is given in the most general form which could be applicable to study a full three-dimensional problem.

Optimizing University Course Timetabling Using Graph Coloring Techniques

Ogunkan Stella Kehinde — 2024-10-09

This research presents an innovative approach to the University Course Timetabling Problem (UCTP) through the application of graph coloring techniques aimed at achieving optimal scheduling accuracy. By partitioning the conflict graph into independent color classes, time slots are assigned to create a conflict-free timetable. The study utilizes data from the Ladoke Akintola University of Technology (LAUTECH) to construct a course conflict graph, where courses are represented as vertices and conflicts as edges. Venue allocation corresponding to the assigned time slots is accomplished using a first fit packing algorithm. The proposed model is implemented in Python and evaluated using Halstead complexity metrics, yielding results of Program Volume (PV) at 18.45 kbits, Program Length (PL) at 0.51, Program Effort (PE) at 1,037,684, Program Difficulty (PD) at 1.97, and Execution Time (ET) at 20.45 seconds. The findings demonstrate significant improvements over existing models, resulting in a more efficient conflict-free course timetable. This work contributes valuable insights for addressing various scheduling, optimization, and NP-hard computational challenges.

Existence of Moments in Distributions of the Form Tan(X)

Peter Kopanov — 2024-10-09

In this work, we consider the existence of the moments of functions of random variables supported on a bounded interval. Our approach begins by working with an arbitrary diffeomorphism, but later we restrict attention to the tan function–the corresponding distribution is a generalization of the Cauchy distribution, which is derived when one applies tan to a uniformly distributed variable. For a continuous random variable X, we derive a necessary and sufficient condition for the existence of a moment of a given order of the distribution of tan(X) in terms of the behaviour of the probability density of X near the points ± \(\frac{\pi}{2}\). As a consequence, we obtain classes of examples, somewhere the moments exist and somewhere they do not at all.

The n-th Smallest Term for any Finite Sequence of Real Numbers

Josimar da Silva Rocha — 2024-10-09

This paper presents various methods for determining the formula that gives the n-th smallest term in a given finite sequence of real numbers. We introduce recursive methods, MAX-MIN methods, and MIN-MAX methods, and compare their computational effciency. As an aplication, we find a formula that gives the median of any finite sequence of real numbers.

Mathematical Modelling of Forestry Biomass Conservation and Wildlife Population Dynamics

Rachana Pathak — 2024-10-09

Researchers have proposed various mathematical models to study forest depletion caused by resource-independent industrialization (population) by considering the spatial distribution of both forest biomass and the density of industrialization within a single homogeneous habitat. In this study, a non-linear mathematical model for the conservation of forestry biomass and wildlife population is developed and examined. The model assumes that the growth rate of wildlife conservation is directly proportional to the reduction of forestry biomass caused by the wildlife population. Additionally, in modelling process, the study investigate the impact of illegal trade on both forestry biomass and wildlife populations. The model equations are studied to understand the nature and stability of equilibrium points using the theory of nonlinear ordinary differential equations and numerical simulations. Moreover, sufficient conditions for the persistence of the system are derived through differential inequalities. The analysis reveals that the depletion of forestry biomass and wildlife populations can be mitigated through effective wildlife conservation. To achieve optimal results in wildlife conservation, it is crucial to minimize the rate of governance failure.

Building Logic Gate Circuits in Python

Pankaj Dumka — 2024-10-09

This paper presents the development of a Python module designed for various Logic Gates. Three circuits were built using the functions defined in Python for the logic gates, and the outcomes were found to align completely with those reported in the literature. The circuits were tested with different inputs obtaining the expected outputs.

Geogebra Applets and Gemini Artificial Intelligence in Separable Variable Differential Equations in Engineering Students of Antofagasta Chile

Jorge Olivares Funes — 2024-10-09

This paper unifies the advantages of combining GeoGebra applets and Gemini AI to enhance mathematics learning for university students, specifically in the differential equations course at the University of Antofagasta, Chile. This combination of technological tools offers an innovative and effective approach to solving separable differential equations, providing students with a motivating learning experience. GeoGebra, a free dynamic geometry software, enables the creation of interactive applets that facilitate the visual and conceptual understanding of mathematics. In turn, Gemini AI complements GeoGebra applets by guiding students and adapting to their level of understanding and learning style. This integration not only improves comprehension compared to traditional methods but also fosters a more personalized and engaging learning experience. Various studies have demonstrated the benefits of using GeoGebra in mathematics teaching, and now, the inclusion of Gemini AI amplifies these advantages by offering a more adaptive learning experience. Together, this combination represents a significant advancement in the use of educational technologies to innovate mathematics learning in higher education.

A Proof of the Twin Prime Conjecture

A. W. Draut — 2024-10-09

The traditional definition of the twin prime conjecture is that there is an infinite number of twin primes. The traditional definition of a twin prime is a pair of odd primes separated by one even number, e.g., 29 and 31.

The twin prime conjecture is a long standing problem. In many introductory textbooks on number theory, the author has a section (usually in the first chapter) on open problems. The Twin Prime Conjecture is usually listed¹.. Britannica has a comment on it. Go to "britannica.com" and search on "twin prime conjecture (number theory)". From the article: "The first statement of the twin prime conjecture was given in 1846 by French mathematician Alphonse de Polignac, ..." Some mathematicians have suggested that Euclid (circa 300 BC) hinted at the twin prime conjecture when he did his work on the infinitude of the primes.

We give a distinctly new approach to the twin prime conjecture. We don't use the methods of analytic number theory. Instead, we use Eratosthenes' sieve2 . We are not interested in the primes that are uncovered. Instead, we find in the sparse sequences of natural numbers that remain after each implementation of the sieve, useful structures that we call Eratosthenes' Patterns. These structures reveal a number of twin primes, and with each implementation of the sieve, these structures reveal increasing numbers of twin primes.

The essence of our proof is to show that the number of twin primes between p_nand \(p^2_n\) approaches infinity as n approaches infinity.

Moreover, we have found a second constellation of primes, two primes that differ by four. In each of the Eratosthenes Patterns these primes are equal in number to the twin primes. We suggest that these are as significant as the twin primes. They are also infinite in number.

In order, we describe Eratosthenes' Sieve, Eratosthenes' Patterns, and the twins. Then we give the proof.