A new 2-point hybrid block method for the numerical solution of first-order stiff systems of ordinary differential equations in initial value problems with optimal stability property is presented. The necessary and sufficient conditions for the convergence of the proposed implicit block numerical scheme for solving stiff ODEs are established. The stability and convergence analysis of the method show that it is consistent, zero-stable, and convergent. The absolute stability region of the method is plotted, indicating that the method is A-stable. The method is implemented in Microsoft Dev C++ environment using the C programming language and Newton’s iteration, and some selected first-order stiff initial value problems are solved. The numerical results obtained for the proposed method are compared with the existing fifth order fully implicit 2-point block backward differentiation formula and 2-point block backward differentiation formula with two off-step points methods. The comparison reveals that the new method outperforms both methods in terms of accuracy but are competing in terms of computation time as we reduce the step size. It is evident that the method converges faster.

A new complex derivative is defined for the first time. Some theorems linked to the definitions are given first. Applications of the new derivative to calculus and dynamics are discussed. The analysis may open newhorizons to undergraduate students and lecturers

Bernoulli bases are developed to approximate the solutions of two-point BVP in modelling viscoelastic flows in which the shifted Chebyshev collocation points are used as collocation nodes. Properties of Bernoulli bases are then used to reduce the two-point BVP in modelling viscoelastic flows to systems of nonlinear algebraic equations. The results show the agreement between the exact solutions and the approximate solutions. Form the numerical results we see that the proposed method gives accurate results.

A new definition of the Fibonacci Matrix is given. The elements of the matrix consist of the Fibonacci numbers. First a preliminary knowledge on Fibonacci sequences and their properties are given. Then the new definition of the matrix is given together with some properties. The difference from the common definition is also discussed. The determinant of the matrix and its properties are posed and proven. Applications to systems of algebraic equations are also outlined.

In light of the remarkable advancements in communication information technology and multimedia internet technology, there arises an increasingly demanding need for the proficient handling of image data pertaining to passenger luggage. This encompasses the critical aspects of image transmission, storage, and compression. The contemporary landscape of passenger figure images presents a formidable data magnitude, which poses considerable trials upon the constraints of limited bandwidth. Consequently, the implementation of digital image compression strives to capture the overarching characteristics of the images utilizing a diminished bit count, while concurrently minimizing the potential degradation during the subsequent image restoration process. The present study incorporates principal component analysis (PCA) in the context of the security inspection industry, specifically focusing on passenger figure images. The application of PCA is integrated within the research framework of image compression and reconstruction systems. In this regard, MATLAB software is utilized to simulate the experiments, followed by an in-depth analysis of the obtained results. The findings demonstrate that by ensuring a cumulative contribution rate of more than 85% from a select few principal components following variable dimension reduction, it is feasible to achieve clear and distortion-free passenger figure images.

Soft set theory, introduced by Molodtsov, is as an important mathematical tool to deal with uncertainty and it has been applied to many fields both as theoretical and application aspects. Since 1999, different kinds of soft set operations has been defined and used in various types. In this paper, we define a new kind of soft set operation called, complementary soft binary piecewise plus operation and investigate its basic algebraic properties. Moreover, by examining the distribution rules, we contribute to the soft set literature by obtaining the relationships between this new soft set operation and some other types of soft set operations such as soft restrcited, extended, soft binary piecewise, and complementary soft binary piecewise operations. As proposing new soft set operations and obtaining their algebraic properties and implementations opens up new avenues for handling parametric data challenges in terms of decision-making methods and new cryptography approaches, and analyzing the algebraic structure of soft sets from the standpoint of new soft set operations offers a thorough understanding of the algebraic structure of soft sets, this paper can be regarded as both theoretical and application study.

The intersection problem is one of the basic problems in combinatorial design theorys. In this paper, two cases of resolvable balanced incomplete block designs of size four are investigated. As the basic step of the study on the intersection problem for resolvable balanced incomplete block designs, we have determined some values of the intersection numbers. It will play an important role on the further recursive constructions.

This study establishes mathematically a comprehensive approach for evaluating the solutions of Bessel’s function of the first kind for different values of the order n. The study revealed that the knowledge of gamma function, Maclaurin’s series and basic laws of indices plays a significant role in evaluating the Bessel’s function of order n. The results showed that the solutions, J-1(x) = J-2(x) = J-3(x) = 0 for x = 0, 1, 2, 3, 4…….12; the pattern of variation for the various graphs of J0(x), J1/2(x), J-1/2(x), J1(x), J3/2(x), J-3/2(x), J2(x), J5/2(x), J-5/2(x) and J3(x) were investigated and the results showed that there is a slight decrease in the values of J0(x) from x = 0 to 3 which then increase steadily from x = 4 to 12, for J1/2(x), J3/2(x), J2(x) and J3(x), the values increases from x = 0 to 12. The values of J-1/2(x) increases steadily from x = 0 to 12 in the form of a parabola. For J1(x), the graph increase from x = 0 to 2 then decreases at x = 3 and increases continuously from x = 4 to 12. The figure depicting J-3/2(x) showed that when x = 0, J-3/2(0) = ∞ , negative values were obtained at x = 1 and 2, the values then increases steadily from x = 4 to 12. The figure for J5/2(x) shows that the values increases from x = 0 to 2 and decreases negatively from x = 3 to 8 and increases steadily from x = 9 to 12. The figure for J-5/2(x) depicts that when x = 0, J-5/2(0) = ∞, the values decrease from x = 1 to 2 and decreases further but negatively from x = 3 to 12.

Set theory is considered as the foundation of all mathematics since many mathematical concepts cannot be defined precisely without using set-theoretical concepts. In this study, we define new complemental binary operations, called union complements, intersection left complement, and union right complement and investigate their properties in detail. We contribute to the literature of sets by illuminating the relationships between these complemental binary operations and inclusive\exclusive complements via researching the distribution rules. Moreover, we show that the set of all the sets together with these new complemental binary operations form some algebraic structures. Finally, with the inspiration of these novel concepts, we give an application to group theory as regards subgroups by defining new type of subgroups in order to prompt the reader to think via interesting questions. Since the concept of operations of soft set theory, one of the most popular theory for uncertainty modeling in the past twenty four years, is the crucial notion for developing the theory and since all the types of soft set operations are based on the classical set operations, generation of new complemental binary operations on sets, and thus on soft sets and derivation of their algebraic properties will provide new perspectives for solving problems related to parametric data.

Soft set theory, introduced by Molodtsov, is as an efficacious mathematical tool to deal with uncertainty and it has been applied to many fields both as theoretical and application aspect. Since its inception, different kinds of soft set operations are defined and used in various types. In this paper, we define a new kind of soft set operation called, complementary soft binary piecewise gamma operation and we investigate its basic algebraic properties. Moreover by examing the distribution rules, it is aimed to contribute to the soft set literature by obtaining the relationships between this new soft set operation and other types of soft set operations such as extended soft set operations, complementary extended soft set operations, soft binary piecewise operations, complementary soft binary piecewise operations and restricted soft set operations. This paper can be regarded as a theoritical study of soft set theory.