Türkiye Mathematical Sciences

Decay of solutions for a singular parabolic m-biharmonic equation with logarithmic nonlinearity

Erhan Pişkin — 2024-12-31

In this work, we obtain a singular parabolic m-biharmonic equation with logarithmic nonlinearity. We establish the decay of the solutions and then derive it using the potential well method and the Hardy-Sobolev inequality.

Sequences Generated by the Lattice Path Problem with Various Vector Sets

CEMİL KARAÇAM — 2024-12-31

The lattice path problem involves finding a path between two specific points in space using only certain predefined vectors. The goal is to establish a relationship between the number of lattice paths to a point and the emergence of specific number sequences. This was achieved by analyzing lattice paths in a table within a Cartesian coordinate system. The number of paths to a particular cell, starting from the first column of the table, was computed, and the results were analyzed through a computer program. This method led to the discovery of Fibonacci, Pell and Pell-Lucas and Tribonacci sequences. Upon examining tables in dimensions higher than two, it was observed that the numbers found corresponded to the products of these special number sequences. Recursive relations were derived for the vector sets used in this process, and through these relations, identities among the number sequences were established

On Some Weighted Inequalities on Time Scale with Nabla Calculus

Ayşe Sena Abalı — 2024-12-31

The theory of time scales is one of the important cornerstones of functional analysis and operator theory. Recently, it has been the subject of many studies from different disciplines. For example, it has become the field of study of many researchers working in mathematics, economics, physics, optics, engineering, and other fields. This study first mentions the basic features of the time scale and nabla calculus. Then a new approach to the weighted Ostrowski-type inequality is presented using nabla calculus on time scales.

Three-Key 12-Note Polyhybrid Encryption

CEMİL KARAÇAM — 2024-12-31

In this paper, a dynamic encryption algorithm rich in music and mathematics has been developed. The foundation of this algorithm is the Pythagorean diatonic note series, combined with the alteration tools (Accidentals) in Turkish music. A 12-note, three-key system with equivalent notes we defined was used to create three different and two-notation encryption databases. The corresponding values of characters' ASCII codes in base 12 were structured to match the 12-tone note system in accordance with three different key structures. Formulas necessary for the integration of the three-key system were derived. From a transmission perspective, a symmetric encryption algorithm supporting addition, subtraction, and symmetry operations in base 12 was created. Fixed and dynamic parameter versions of the encryption key were designed. A musical composite function operation, based on the infrastructure of flat, sharp, and the 12-tone note system, was designed and implemented, benefiting from the properties of flat and sharp accidentals. A dynamic musical encryption design was developed with time and serial numbers, three different musical keys, and a staff-supported mathematical composite function structure, ensuring data transmission and character set. Through the integration of mathematical operations with musical terms, an original algorithm was designed, making it highly resistant to breaking, while minimizing the similarity between characters. In terms of data diversity, mathematical operations, and transmission variety, it stands out from similar studies due to its richness.

A New Approach to Ostrowski Inequalities on Time Scale with Nabla Calculus

Lütfi Akın — 2024-12-31

The theory of time scales has recently become prominent in many disciplines of the scientific world. It has become the field of study of many researchers working in mathematics and economics, physics, optics, engineering, and other fields. Inequalities also have an important place in this theory. In this study, we bring a new approach to the generalized Ostrowski-type inequality using the nabla calculus on time scales.