This paper introduces a new technique for the construction of an 8-point subdivision scheme (SS) with a shape parameter mu. Some well-known SSs are particular cases of our proposed 8point binary SS. We find the parametric range in which the proposed SS generates C1-continuous limit curves. It is observed that our proposed SS is C3-continuous for mu = 5 2048 . Some important properties of the proposed SS like the symmetry of basic limit function, exactness, approximation order, and convexity preservation are discussed. We also discuss error bounds and curvature of the limit curves of the proposed SS. Further, we illustrate the effectiveness of the shape parameter of the proposed SS through various applications of the SS. It is noted that the proposed SS has the potential to generate fractal curves for suitable choices of the shape parameter.

This paper introduces a new class of stationary refinement algorithms that combine the advantages of the four-point interpolatory algorithm and the cubic B-spline. This algorithm includes a tension parameter. This analysis identifies the range of this parameter and specifies conditions under which the proposed algorithm maintains shape-preserving properties such as monotonicity and convexity. The algorithm achieves improved smoothness, reaching C3 and fourth-order accuracy, while maintaining the same support length as both parent algorithms. Unlike many high-order algorithms that are nonlinear and complex, this method remains simple and efficient. Numerical examples are provided to demonstrate its practical performance, and the proposed algorithm is particularly well-suited for a discontinuous type of data set.

In this paper, we introduce a new combined four-point ternary refinementscheme capable of generatingboth interpolatingand approx-imating curves. The scheme is constructedby translatinga quintic B-Spline refinement scheme to new positions using displacementvectors that incorporatefour shape control parameters.We demonstratethat this combined scheme produces limit curves with up to C4 continuityand al-so provides significantflexibility in crafting smooth curves with reducedsupportsize. Furthermore,we establishthatvarious well-known inter-polatory and approximating refinement schemes are special casesof our proposedapproach.Somekey properties,including polynomial generation, reproduction,and supportwidth, are examined.Additionally, we investigate the fractal generation characteristicsof the refinement scheme, revealing that specificparameterchoicesenable effective constructionof fractalcurves.Through a series of numericalexperiments, we validate thatthe limit curves producedby our scheme accurately reflect the associ-ated control polygons, yielding more realisticresults comparedto existingmethods.

In this paper, we introduce a novel combined 5-point quinary refinement scheme that incorporates five parameters to enhance its adapt-ability and resilience. The development of the scheme stands on translat-ing a 3-point approximating quinary refinement scheme to the new loca-tion by introducing the displacement vectors. We also demonstrate that the combined scheme generates numerous sub-schemes within a given set of parameters. The combined scheme has the modest support with C3 degree of smoothness that attains better shape control and flexibil-ity. Additionally, we discuss the important characteristic like polynomial generation, reproduction and approximation order. Moreover, we analyze that the proposed scheme exhibits complex behavior under specific pa-rameter configuration. Further, we present some visual depiction based on polygons to verify the outcomes of the scheme. Lastly, to showcase the efficiency of the scheme the comparison with the classical and recently proposed schemes is presented.

Subdivision schemes are a crucial component of geometric modeling, widely applied in curve and surface design. Classical schemes such as the six-point interpolatory scheme and the quintic B-spline are well known for their efficiency and smoothness. Yet, both have inherent drawbacks, particularly with respect to approximation order and smoothness. This study introduces a novel subdivision scheme that blends the strengths of the six-point interpolatory and the quintic B-spline schemes. The proposed scheme achieves sixth-order approximation and C5 smoothness while preserving the support size of the six-point scheme. A key feature of the scheme is the introduction of a tension parameter, which provides flexibility to control the trade-off betweensmoothness and approximation order. Moreover, the scheme preserves essential properties such as monotonicity and convexity under mild conditions. Despite the other higher-order shape-preserving schemes, which are non-linear and computationally complex, this scheme is linear and stationary. Experimental results confirm that the proposed scheme consistently produces accurate and visually elegant curves, outperforming existing schemes in both approximation order and smoothness.

The interpolating and approximating refinement schemes are well-studied algorithms to generate smooth curves and surfaces. In this paper, we propose a novel non-symmetric refinement scheme that combines the strengths of interpolating and approximating refinement algorithms. The construction of the proposed scheme is derived from a classical 5-point approximating refinement scheme, which is systematically translated to new positions using displacement vectors. These vectors are parameterized by three independent shape control parameters, allowing for adjustable curve behavior and enhanced flexibility in design. By appropriately selecting these parameters, we demonstrate that the resulting limit curves can achieve up to [Formula: see text] continuity, which is significant for applications requiring high smoothness. One of the key advantages of our scheme is its ability to generate smooth curves with reduced support size, leading to improved computational efficiency and locality of influence. The proposed scheme is applicable in areas such as geometric modeling, and curve design in graphics and textile. Furthermore, we rigorously analyze essential mathematical properties of the scheme, including polynomial generation, linear reproduction, and the compactness of the support. We also investigate the limit stencil of the generated curves. A comprehensive comparison with several existing refinement schemes is presented. To support our theoretical findings, we include some graphical illustrations that showcase the performance, flexibility, and visual quality of the curves generated by our scheme.

In this study, we proposed a family of $ m $-point quaternary approximating subdivision schemes, characterized by an explicit formula involving three parameters. One of these parameters served as a shape control parameter, allowing for flexible curve design, while the other two parameters identify different members of the family and determined the smoothness of the resulting limit curves. We conducted a thorough analysis of the proposed schemes, covering their smoothness properties, polynomial generation, and reproduction capabilities. Additionally, we examined the behavior of the Gibbs phenomenon within the family both theoretically and graphically, highlighting the advantages of the proposed schemes in eliminating undesirable oscillations. A comparative study with existing subdivision schemes demonstrated the effectiveness and versatility of our approach. The results indicated that the proposed family offered enhanced smoothness and control, making it suitable for a wide range of applications in computer graphics and geometric modeling.

In this article, we define several fundamental characteristics and put forward some basic fixed point results in the context of hyperbolic space for generalized (alpha, /3 )-nonexpansive type-1 mappings. Additionally, we present triangle convergence and strong convergence results within the framework of hyperbolic space for these types of mappings. Lastly, we provide some numerical examples to highlight our main result and comparison the iterative procedures that we use in our study with various iterative techniques from the literature. The results in this study enhance, broaden and unite corresponding results in the literature.

The main objective of this research is to analyze some geometrical properties of a quaternary four-point interpolatory subdivision scheme ([Formula: see text]-scheme) with a shape parameter and then find the precise range of the shape parameter of the [Formula: see text]-scheme to generate fractal and convexity of limit curves. That allows the curve to change easily and be more flexible without altering its control points. Therefore, by taking the various values of tension parameters, the curve still preserves its characteristics and geometrical configuration. These geometric modeling examples show that our techniques can be easily performed, and they can also provide us with an alternative strong strategy for the modeling of complex figures.

Tensor product has been at the heart of continuous geometry almost from its origin, and many applications for the notion of tensor product have been discussed. Subdivision schemes (SSs) are widely used in Computer Aided Geometric Design (CAGD) and several attempts have been made to link surface models obtained by tensor product schemes to the limit surfaces generated by stationary and non-stationary SSs. In this paper, we introduce a bi-variate relaxed four-point approximating subdivision by taking tensor product of a relaxed four-point subdivision scheme. We discuss some important properties like continuity, Holder regularity, joint spectral radius and local analysis with invariant neighborhood of the proposed schemes. Some graphical examples are also given to illustrate the impact of the proposed scheme.

•The higher dimensional Burger’s equations are considered.•The reduction method is used.•The exponential, hyperbolic and rational functions are retrieved.•The obtained results are given by figures.

The structure of q-rung orthopair fuzzy sets (q-ROFSs) is a generalization of fuzzy sets (FSs), intuitionistic FSs (IFSs), and Pythagorean FSs (PFSs). The notion of q-ROFSs has the proficiency of coping with uncertainty without any restrictions. In addition, the structure of q-ROFSs can effectively cope with the situations involving dual opinions without any restrictions, instead of dealing with only single opinion or dual opinions under certain restrictions. In clustering problems, the correlation coefficients are worthwhile because they provide the degree of similarity or correlation between two elements or sets. The theme of this study is to formulate the correlation coefficients for q-ROFSs that are basically the generalization of correlation coefficients of IFSs and PFSs. Moreover, an application of these correlation coefficients to a clustering problem is proposed. Also, an analysis of the outcomes is carried out. Furthermore, a comparison is carried out among the correlation coefficients for q-ROFSs and the existing ones. Finally, the downsides of the existing works and benefits of the correlation coefficients for q-ROFSs are discussed.

This paper introduces a family of shape-preserving binary approximating subdivision schemes by applying a shape-preserving variant on the Lane-Riesenfeld algorithm. Using the symbols of subdivision schemes, we determine convergence and smoothness, Hölder continuity, and support size of the limit curves. Furthermore, these schemes produce monotonic and convex curves under the certain conditions imposed on the initial data.

In this paper, we analyze shape-preserving behavior of a relaxed four-point binary interpolating subdivision scheme. These shape-preserving properties include positivity-preserving, monotonicity-preserving and convexity-preserving. We establish the conditions on the initial control points that allow the generation of shape-preserving limit curves by the four-point scheme. Some numerical examples are given to illustrate the graphical representation of shape-preserving properties of the relaxed scheme.

Shape preservation has been the heart of subdivision schemes (SSs) almost from its origin, and several analyses of SSs have been established. Shape preservation properties are commonly used in SSs and various ways have been discovered to connect smooth curves/surfaces generated by SSs to applied geometry. With an eye on connecting the link between SSs and applied geometry, this paper analyzes the geometric properties of a ternary four-point rational interpolating subdivision scheme. These geometric properties include monotonicity-preservation, convexity-preservation, and curvature of the limit curve. Necessary conditions are derived on parameter and initial control points to ensure monotonicity and convexity preservation of the limit curve of the scheme. Furthermore, we analyze the curvature of the limit curve of the scheme for various choices of the parameter. To support our findings, we also present some examples and their graphical representation.

In this paper, a generalized algorithm to develop a class of approximating binary subdivision schemes is presented. The proposed algorithm is based on three-point approximating binary and four-point interpolating binary subdivision schemes. It contains a parameter which classifies members of the new class of subdivision schemes. A set of efficient properties, for instance, polynomial generation and reproduction, support, continuity, and Hölder continuity, is discussed. Moreover, applications of the proposed subdivision schemes are given in order to demonstrate their variety, flexibility, and visual performance.

Closed-loop supply chain networks are gaining research popularity due to environmental, economic and social concerns. Such networks are primarily designed to overcome carbon footprints and to retrieve end of life products from customers. This study considers a multi echelon closed-loop supply chain in the presence of machine disruption. A multi-objective model is presented to optimize the total cost, the total time and emissions in a closed-loop supply chain network. The aim is to analyze the trade-off between the objectives of cost, time, and emissions and how these decisions are impacted by the selection of different available machines. A number of solution approaches are tested on a case study from the tire industry. The results suggest the improved performance of the hybrid heuristic and the importance of controlling disruption in a closed-loop supply chain network. Furthermore, there is a trade-off between the different objective functions which can help the decision maker to choose a particular solution according to the preference of an organization. Finally, conclusion and future research avenues are provided.

In this paper, we present the shape-preserving properties of the four-point ternary non-stationary interpolating subdivision scheme (the four-point scheme). This scheme involves a tension parameter. We derive the conditions on the tension parameter and initial control polygon that permit the creation of positivity- and monotonicity-preserving curves after a finite number of subdivision steps. In addition, the outcomes are generalized to determine conditions for positivity- and monotonicity-preservation of the limit curves. Convexity-preservation of the limit curve of the four-point scheme is also analyzed. The shape-preserving behavior of the four-point scheme is also shown through several numerical examples.

In this article, we present a family of 4-point odd-ary interpolating non-stationary schemes. This family of schemes is based on Lagrange trigonometric polynomial. These non-stationary schemes reproduce functions spanned by \(\{ 1, \cos \alpha (x), \sin \alpha (x) \}\). Some examples are also given to show visual performance of the schemes.

In this paper, we propose an elegant strategy for constructing a family of binary univariate subdivision schemes, starting with two binary schemes. The members of the proposed family of schemes are categorized by a parameter, for even and odd values of this parameter resulting schemes are primal and dual in nature respectively. It is shown that the new resulting schemes have higher smoothness and Hölder exponents while less magnitude of artifacts as compared to their parent binary schemes. It is further noticed that resulting schemes have cubic polynomial reproducing property. Support of basic limit function of proposed schemes is discussed. Limit stencil and artifact analysis are also carried out. Numerical study shows that proposed family of schemes is free from Gibbs phenomenon.