This study presents a generalization of the DD Stancu operator reported in the literature by combining it with the lambda Bernstein operator. Using the basis functions of the proposed operator, geometric modeling applications related to vascular visualization, botanical form representation, and entomological form modeling are presented; thus, the role of approximation theory in mathematical modeling and the aforementioned fields is emphasized. The biological modeling and botanical illustration generated using the basis functions of the operator demonstrate the effectiveness of the method in representing organic shapes by enabling the λ-parameter to capture with high precision the natural forms of leaves and butterflies, their internal curves, and venation structures. The model, developed to represent structural changes such as narrowing and widening in vascular geometry, was designed solely as a geometric visualization tool rather than as a validated medical simulation or diagnostic method, and is made available for further studies.

With the advent of quantum computing, traditional cryptography algorithms are at risk of being broken. Post-quantum encryption algorithms, developed to include mathematical challenges to make it impossible for quantum computers to solve problems, are constantly being updated to ensure that sensitive information is protected from potential threats. In this study, a hybrid examination of a (p,q)-Bernstein-type polynomial, which is an argument that can be used for encryption algorithms with a post-quantum approach, was made from a mathematical and cryptography perspective. In addition, we have aimed to present a new useful operator that approximates functions and can be used in cases where it is not possible to work with functions in the fields of technology, medicine, and engineering. Based on this idea, a new version of the (p,q)-Bernstein-Schurer operator was introduced in our study on a variable interval and the convergence rate was calculated with two different methods. At the same time, the applications of the theoretical situation in the study were presented with the help of visual illustrations and tables related to the approach. Additionally, our operator satisfied the statistical-type Korovkin theorem and is suitable for variable interval approximation. This is the first paper to study the statistical convergence properties of (p,q)-Bernstein-Schurer operators defined on a variable bounded interval, to obtain special matrices with the help of (p,q)-basis functions, and to give an application of (p,q)-type operators for encrypted image transmission.

The existence of natural gas and rare mineral reserves, energy transmission lines, and sacred places within its borders makes that geography a target for other countries, whether neighboring or not. These countries spend most of their budgets on war technologies and good defense. There are many factors to consider when choosing the location of a military base, which is vital in terms of both defense and logistic support. This study aimed to determine the criteria that should be taken into account in determining the borderline security and selecting the location of military bases of great strategic importance by getting rid of the disadvantages of classical decision-making processes. For this purpose, a solution to the problem was sought with the method obtained by combining the AHP method, one of the latest approaches in the decision-making process, with neutrosophic logic. In order to enable the experts to cope with uncertain information and to prevent errors in preference values due to differences in individual approaches, three expert opinions were obtained and the Delphi method was used to increase the advantages of the neutrosophic analytic hierarchy process (N-AHP) method by utilizing the degree of consensus. Expert opinions were received to determine, prioritize, and group the criteria using the Delphi method, and after these criteria were analyzed, their importance levels were determined by weighting the criteria using the N-AHP method. Thus, an important study in which these two compatible methods were used together for the establishment of a military base was presented to researchers. When the criteria weights of the 12 sub-criteria are analyzed, it was concluded that ease of logistics access is the most important criterion for base location selection.