İsabeyoǧlu, Zeynep; Erişir, Tülay; Azak, Ayşe Zeynep

This study examines the spinor representations of (Formula presented.) (tangent and normal), (Formula presented.) (normal and binormal), (Formula presented.) (tangent and binormal) and (Formula presented.) (tangent, normal and binormal)–Smarandache curves in three-dimensional Euclidean space (Formula presented.). Spinors are complex column vectors and move on Pauli spin matrices. Isotropic vectors in the (Formula presented.) complex vector space form a two-dimensional surface in the (Formula presented.) complex space. Additionally, each isotropic vector in (Formula presented.) space corresponds to two vectors in (Formula presented.) space, called spinors. Based on this information, our goal is to establish a relationship between curve theory in differential geometry and spinor space by matching a spinor with an isotropic vector and a real vector generated from the vectors of the Frenet–Serret frame of a curve in three-dimensional Euclidean space. Accordingly, we initially assume two spinors corresponding to the Frenet–Serret frames of the main curve and its ( (Formula presented.), (Formula presented.), (Formula presented.) and (Formula presented.) )–Smarandache curves. Then, we utilize the relationships between the Frenet frames of these curves to examine the connections between the two spinors corresponding to these curves. Thus, we give the relationships between spinors corresponding to these Smarandache curves. For this reason, this study creates a bridge between mathematics and physics. This study can also serve as a reference for new studies in geometry and physics as a geometric interpretation of a physical expression.

Hu, Yi-Teng; Şat, Murat

This paper deals with second-order differential pencils with a fixed frozen argument on a finite interval. We obtain the trace formulae under four boundary conditions: Dirichlet–Dirichlet, Neumann–Neumann, Dirichlet–Neumann, Neumann–Dirichlet. Although the boundary conditions and the corresponding asymptotic behaviour of the eigenvalues are different, the trace formulae have the same form which reveals the impact of the frozen argument.

Manav Tatar, Nesrin; Agarwal, Ravi P.

In the realm of generalized modular metric spaces, we substantiate the validity of fixed-point theorems with Branciari contractions. This paper expands and broadens the original theorems in this context. Subsequently, by building upon this foundation, we explore various integral contractions to identify and characterize fixed points within this context. To highlight the practical implications of our work, we introduce the concept of the best proximity pair, thereby culminating in the best approximation theorem. We apply this theoretical construct to a specific example—one that is guided by the best approximation method described in prior research.

Turkoglu, Duran; Manav, Nesrin

We present the concept of multivalued mappings in generalized modular metric spaces (GMMS). In addition, we give Caristi and Feng-Liu fixed point results for this type of mappings in GMMS. Then, we obtain an application for final outcomes in the sense of Jleli and Samet.