The n-ary subdivision
schemes contrast favorably with their binary analogues because they are capable
to produce limit functions with the same (or higher) smoothness but smaller
support. We present an algorithm to generate the 4-point n-ary non-stationary scheme for trigonometric, hyperbolic and
polynomial case with the parameter for describing curves. The performance,
analysis and comparison of the 4-point ternary scheme are also presented.

Abstract:
We present a general formula to generate the family of odd-point ternary approximating subdivision schemes with a shape parameter for describing curves. The influence of parameter to the limit curves and the sufficient conditions of the continuities from C^{0} to C^{5} of 3- and 5-point schemes are discussed. Our family of 3-point and 5-point ternary schemes has higher order of derivative continuity than the family of 3-point and 5-point schemes presented by [Jian-ao Lian, On a-ary subdivision for curve design: II. 3-point and 5-point
interpolatory schemes, Applications and Applied Mathematics: An International Journal, 3(2), 2008, 176-187]. Moreover, a 3-point ternary cubic B-spline is special case of our family of 3-point ternary scheme. The visual quality of schemes with examples is also demonstrated.

A general formula for 4-point α-Ary approximating subdivision scheme for curve designing is introduced
for any arity α≥2. The new scheme is extension of B-spline of degree 6.
Laurent polynomial method is used to investigate the continuity of the scheme.
The variety of effects can be achieved in correspondence for different values
of parameter. The applications of the proposed scheme are illustrated in
comparison with the established subdivision schemes.

In this paper, we propose and analyze a tensor product subdivision scheme which is the extension of three point scheme for curve modeling. The usefulness of the scheme is illustrated by considering different examples along with its application in surface modeling.

Abstract:
In this paper, we propose and analyze a subdivision scheme which unifies 3-point approximating subdivision schemes of any arity in its compact form and has less support, computational cost and error bounds.？ The usefulness of the scheme is illustrated by considering different examples along with its comparison with the established subdivision schemes. Moreover, B-splines of degree 4and well known 3-point schemes [1, 2, 3, 4, 6, 11, 12, 14, 15] are special cases of our proposed scheme.

Abstract:
A new 4-point 3 quaternary approximating subdivision scheme with one shape parameter is proposed and analyzed. Its smoothness and approximation order are higher but support is smaller in comparison with the existing binary and ternary 4-point subdivision schemes.

Abstract:
Helicoverpa armigera (Hubner) (Lepidoptera: Noctuidae) completed its larval stage in 17.325±0.326 days passing through six instars under laboratory protocol, 26±1 °C, 60-70% RH and 16 hours` daylight. The larvae moulted for 2nd instar, two days after hatching from eggs. Average stadiel periods for 2nd, 3rd, 4th, 5th, and 6th instars were 2.07, 2.15, 2.48, 3.12, 3.55 and 3.95 days respectively. The last larval stage did not moult but was contracted and shortened into grub like pre-pupal stage. The average length measured for each instar (first to sixth) was 3.4, 4.6, 9.7, 17, 28.35, 36.85 mm respectively. The average pupal period was 13.2 days for female and 15.4 days for male. Fecundity of moths fed on sucrose solution was significantly higher than water fed females. The unfed females laid few eggs none was viable.

Abstract:
We present an efficient and simple algorithm to generate 4-point n-ary interpolating schemes. Our algorithm is based on three simple steps: second divided differences, determination of position of vertices by using second divided differences, and computation of new vertices. It is observed that 4-point n-ary interpolating schemes generated by completely different frameworks (i.e., Lagrange interpolant and wavelet theory) can also be generated by the proposed algorithm. Furthermore, we have discussed continuity, H？lder regularity, degree of polynomial generation, polynomial reproduction, and approximation order of the schemes. 1. Introduction In general, subdivision schemes can be divided into two categories: approximating and interpolating. For interpolating curve subdivision, new vertices are computed and added to the old polygons for each time of subdivision and the limit curve passes through all the vertices of the original control polygon. Interpolating subdivision schemes are more attractive than approximating schemes in computer aided geometric designs because of their interpolation property. In addition, the interpolation subdivisions are more intuitive to the users. Initial work on interpolating subdivision schemes was started by Dubuc [1]. Later on, Deslauriers and Dubuc [2] have introduced a family of schemes by using Lagrange polynomials indexed by the size of the mask and the arity. In [3], Dyn et al. have studied a family of interpolating schemes with mask size of four. Consequent to this, the research communities are interested in introducing higher arity schemes (i.e., ternary, quaternary -ary) which give better results and less computational cost. Lian [4] has constructed both the -point -ary for any and -point -ary for any odd interpolatory subdivision schemes for curve design by using wavelet theory. Mustafa and Rehman [5] have presented general formulae for the mask of -point -ary interpolating and approximating schemes for any integer and . These formulae provide mask of higher arity schemes and generalize lower arity schemes. Mustafa et al. [6] have presented an explicit formula for the mask of odd points -ary, for any odd , interpolating subdivision schemes. In [7], it has been proved that the large support and higher arity schemes may outperform than small support and lower arity schemes. Even though these schemes are not in practice. It has been suggested that the research on large support and higher arity schemes may continue. The multistage approach is very handy to construct subdivision schemes. This idea is variously used by

Abstract:
We presented a general formula to generate the family of even-point ternary approximating subdivision schemes with a shape parameter for describing curves. Some sufficient conditions for to continuity and approximation order for certain ranges of parameter are discussed. The proposed even-point ternary schemes compare remarkably with existing even-point ternary schemes because they are able to generate limit functions with higher smoothness and approximation order. In addition, we measured curvature and torsion that assist the quality of subdivided curves. 1. Introduction and Preliminaries There are numbers of binary subdivision schemes in the literature. The interest in investigating arities higher than two has been started by Hassan et al. [1, 2]. Nowadays, we have numbers of ternary schemes introduced by [3–7] and so forth, But the research communities are still gaining interest in introducing schemes higher than three arities (i.e., quaternary, quinary, senary, , -ary). Mustafa and Khan [8] introduced a new 4-point quaternary approximating subdivision scheme. Lian [9, 10] introduced 3-, 4-, 5-, and 6-point -ary schemes. Lian [11] also offered -point and -point interpolating -ary schemes for curve design. The 2-, 3-,…, 6-point binary and ternary schemes are very common in the literature. The schemes involving convex combination of more or less than six points at coarse refinement level to insert a new point at next refinement level is introduced by Ko et al. [7]. They introduced - and -point binary schemes. Zheng et al. [12] investigated ternary interpolatory schemes with an odd number of control points, namely, -point ternary interpolatory subdivision scheme. They also investigated ternary even symmetric -point [13] and -ary [14] approximating subdivision scheme and presented the general ternary even symmetric -point approximating subdivision rule and design alternative smooth ternary subdivision scheme of higher order. Mustafa and Rehman [15] presented general formulae for the mask of -point -ary approximating as well as interpolating subdivision schemes for any integers and . This motivates us to present the family of even-point ternary schemes with high smoothness and more degree of freedom for curve design. Proposed schemes not only provide the mask of even-point schemes but also generalize and unify several well-known schemes. Moreover, we measured curvature and torsion that can be used to describe the quality of curve. Also we compared plot of curvature and torsion, obtained by proposed schemes with the other existing schemes. A general