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We have investigated the single and double diffractive production of dileptons and photons in ultra-peripheral collisions at the Large Hadron Collider (LHC). Utilizing advanced theoretical models that integrate quantum electrodynamics (QED) and Quantum Chromodynamics (QCD) frameworks, we analyze the differential cross sections of these processes, with particular emphasis on the role of the Pomeron and resolved Pomeron structures, as well as resolved photon structures. Our research employs diffractive production mechanisms to predict dilepton and photon production rates under various LHC energy scenarios. Our results demonstrate distinct production patterns for single and double diffractive processes, highlighting their potential as probes for studying the electromagnetic structure of heavy ions and the dynamics of soft interactions in high-energy collisions. This paper provides new insights into the photon-mediated and Pomeron-mediated production mechanisms and sets the stage for future experimental investigations at collider facilities., Comment: 30 pages, 12 figures

Geometric objects are primarily represented using curves and surfaces and the subdivision schemes are the basic tools for these representations. This study is based on a new thought that there is a special relation between the binary and some kinds of the quaternary subdivision schemes. Due to the defined relation the quaternary subdivision schemes can also be formulated by the binary subdivision schemes. This study presents the generalized formula in compact form that contains the subdivision rules of (3n-1)-point quaternary approximating subdivision scheme which are based on the predefined 2n-points binary approximating subdivision scheme. Firstly, we derive a relation between the quaternary approximating subdivision scheme and the even-point binary approximating subdivision scheme. By using this relation, we next derive two types of generalized quaternary approximating subdivision scheme that are based on the even and odd values of n. Then we apply these generalized formulas on the known binary schemes for specific values of $n$. This gives us the corresponding quaternary approximating subdivision schemes. We also analyze some of the well-known features of binary and its corresponding quaternary approximating subdivision schemes. These results are equally applicable on parametric and non-parametric subdivision schemes., Comment: 36 pages, 28 figures

A new family of combined subdivision schemes with one tension parameter is proposed by the interpolatory and approximating subdivision schemes. The displacement vectors between the points of interpolatory and approximating subdivision schemes provide the flexibility in designing the limit curves and surfaces. Therefore, the limit curves generated by the proposed subdivision schemes variate in between or around the approximating and interpolatory curves. We also design few analytical algorithms to study the properties of the proposed schemes theoretically. The efficiency of these algorithms are analyzed by calculating their time complexity. The graphical representations and graphical properties of the proposed schemes are also analyzed., Comment: 28 Pages, 42 Figures

In this article, we present a new method to construct a family of (2N+2)-point binary subdivision schemes with one tension parameter where N is a non-negative integer. The construction of the family of schemes is based on repeated local translation of points by certain displacement vectors. Therefore, the refinement rules of a (2N+2)-point scheme for N=M are recursively obtained from the refinement rules of the (2N+2)-point schemes for N=0,1,2,...,M-1. The complexity, polynomial reproduction and polynomial generation of these schemes are increased by two for the successive values of $N$. Furthermore, we modify this family of schemes to a family of (2N+3)-point schemes with two tension parameters. Moreover, a family of interproximate subdivision schemes with tension parameters is also introduced, which allows a different tension value for each edge and vertex of the initial control polygon. Interproximate schemes generate curves and surfaces such that some initial control points are interpolated and others are approximated., Comment: 102 figures, 27 pages

In this article, families of non-linear subdivision schemes are presented that are based on univariate polynomials up to degree three. Theses families of schemes are constructed by using dynamic iterative re-weighed least squares method. These schemes are suitable for fitting scattered data with noise and outliers. Although these schemes are non-interpolatory, but have the ability to preserve the shape of the initial polygon in case of non-noisy initial data. The numerical examples illustrate that the schemes constructed by non-linear polynomials give better performance than the schemes that are constructed by linear polynomials (Computer-Aided Design, 58, 189-199). Moreover, the numerical examples show that these schemes have the ability to reproduce polynomials and do not cause over and under fitting of the data. Furthermore, families of non-linear bivariate subdivision schemes are also presented that are based on linear and non-linear bivariate polynomials., Comment: There are 98 figures and 41 pages in this paper