scholarly journals Univariate approximating schemes and their non-tensor product generalization

2018 ◽  
Vol 16 (1) ◽  
pp. 1501-1518 ◽  
Author(s):  
Ghulam Mustafa ◽  
Robina Bashir
Keyword(s):  
Initial Data ◽  
Tensor Product ◽  
Limit Curve ◽  
Subdivision Schemes ◽  
Tension Parameter ◽  
The Family

AbstractThis article deals with univariate binary approximating subdivision schemes and their generalization to non-tensor product bivariate subdivision schemes. The two algorithms are presented with one tension and two integer parameters which generate families of univariate and bivariate schemes. The tension parameter controls the shape of the limit curve and surface while integer parameters identify the members of the family. It is demonstrated that the proposed schemes preserve monotonicity of initial data. Moreover, continuity, polynomial reproduction and generation of the schemes are also discussed. Comparison with existing schemes is also given.

scholarly journals Recursive Process for Constructing the Refinement Rules of New Combined Subdivision Schemes and Its Extended Form

2021 ◽  
Vol 2021 ◽  
pp. 1-23
Author(s):  
Rabia Hameed ◽  
Ghulam Mustafa ◽  
Jiansong Deng ◽  
Shafqat Ali
Keyword(s):  
New Method ◽  
Subdivision Scheme ◽  
Control Points ◽  
Local Translation ◽  
Extended Form ◽  
Subdivision Schemes ◽  
Tension Parameter ◽  
Displacement Vectors ◽  
The Family ◽  
Initial Control

In this article, we present a new method to construct a family of 2 N + 2 -point binary subdivision schemes with one tension parameter. The construction of the family of schemes is based on repeated local translation of points by certain displacement vectors. Therefore, refinement rules of the 2 N + 2 -point schemes are recursively obtained from refinement rules of the 2 N -point schemes. Thus, we get a new subdivision scheme at each iteration. Moreover, the complexity, polynomial reproduction, and polynomial generation of the schemes are increased by two at each iteration. Furthermore, a family of interproximate subdivision schemes with tension parameters is also introduced which is the extended form of the proposed family of schemes. This family of schemes allows a different tension value for each edge and vertex of the initial control polygon. These schemes generate curves and surfaces such that some initial control points are interpolated and others are approximated.

scholarly journals Odd-Ary Approximating Subdivision Schemes and RS Strategy for Irregular Dense Initial Data

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Muhammad Aslam ◽  
W. P. Abeysinghe
Keyword(s):  
Initial Data ◽  
Regularization Method ◽  
Noise Removal ◽  
Limit Curve ◽  
Control Data ◽  
Subdivision Schemes ◽  
Initial Control ◽  
Data Points ◽  
Variational Regularization

We investigate the implementation of approximating subdivision schemes on noisy or irregular initial control data. Presence of noise in the initial data generates oscillatory curves by subdivision schemes. To reduce or completely eliminate these oscillations, we combine subdivision schemes with other noise removal schemes such as variational regularization method. This setup will allow us to produce the limit curve with less oscillations and still stay as close as possible to the initial data points.

scholarly journals Nuclear subalgebras of UHF C*-algebras

1986 ◽  
Vol 29 (1) ◽  
pp. 97-100 ◽  
Author(s):  
R. J. Archbold ◽  
Alexander Kumjian
Keyword(s):  
Tensor Product ◽  
Inductive Limit ◽  
Inductive Limits ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
The Family ◽  
Algebraic Tensor Product

A C*-algebra A is said to be approximately finite dimensional (AF) if it is the inductive limit of a sequence of finite dimensional C*-algebras(see [2], [5]). It is said to be nuclear if, for each C*-algebra B, there is a unique C*-norm on the *-algebraic tensor product A ⊗B [11]. Since finite dimensional C*-algebras are nuclear, and inductive limits of nuclear C*-algebras are nuclear [16];,every AF C*-algebra is nuclear. The family of nuclear C*-algebras is a large and well-behaved class (see [12]). The AF C*-algebras for a particularly tractable sub-class which has been completely classified in terms of the invariant K0 [7], [5].

References and Remarks on Rotating Fluids

2006 ◽  
Author(s):  
Jean-Yves Chemin ◽  
Benoit Desjardins ◽  
Isabelle Gallagher ◽  
Emmanuel Grenier
Keyword(s):  
Initial Data ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Incompressible Limit ◽  
Rotating Fluids ◽  
Inviscid Fluids ◽  
First Case ◽  
The Family ◽  
Whole Space

The problem investigated in this part can be seen as a particular case of the study of the asymptotic behavior (when ε tends to 0) of solutions of systems of the type where Δε is a non-negative operator of order 2 possibly depending on ε, and A is a skew-symmetric operator. This framework contains of course a lot of problems including hyperbolic cases when Δε = 0. Let us notice that, formally, any element of the weak closure of the family (uε)ε>0 belongs to the kernel of A. We can distinguish from the beginning two types of problems depending on the nature of the initial data. The first case, known as the well-prepared case, is the case when the initial data belong to the kernel of A. The second case, known as the ill-prepared case, is the general case. In the well-prepared case, let us mention the pioneer paper by S. Klainerman and A. Majda about the incompressible limit for inviscid fluids. A lot of work has been done in this case. In the more specific case of rotating fluids, let us mention the work by T. Beale and A. Bourgeois and T. Colin and P. Fabrie. In the case of ill-prepared data, the nature of the domain plays a crucial role. The first result in this case was established in 1994 in the pioneering work by S. Schochet for periodic boundary conditions. In the context of general hyperbolic problems, he introduced the key concept of limiting system (see the definition on page 125). In the more specific case of viscous rotating fluids, E. Grenier proved in 1997 in Theorem 6.3, page 125, of this book. At this point, it is impossible not to mention the role of the inspiration played by the papers by J.-L. Joly, G. Métivier and J. Rauch (see for instance and). In spite of the fact that the corresponding theorems have been proved afterwards, the case of the whole space, the purpose of Chapter 5 of this book, appears to be simpler because of the dispersion phenomena.

scholarly journals A Family of Binary Approximating Subdivision Schemes based on Binomial Distribution

2019 ◽  
Vol 38 (4) ◽  
pp. 1087-1100 ◽  
Author(s):  
Muhammad Asghar ◽  
Ghulam Mustafa
Keyword(s):  
Tensor Product ◽  
Binomial Distribution ◽  
Probability Generating Function ◽  
Visual Performance ◽  
Complete Analysis ◽  
Subdivision Schemes ◽  
B Splines ◽  
Surface Subdivision ◽  
Product Surface ◽  
Tensor Product Surface

A simplest way is introduced to generate a generalized algorithm of univariate and bivariate subdivision schemes. This generalized algorithm is based on the symbol of uniform B-splines subdivision schemes and probability generating function of Binomial distribution. We present a family of binary approximating subdivision schemes which has maximum continuity and less support size. Our proposed family members P4, P5, P6, and P7, have C7, C9, C11 and C13 continuities respectively. In fact, we use Binomial probability distribution to increase the continuity of uniform B-splines subdivision schemes up to more than double. We present the complete analysis of one family member of proposed schemes and give a visual performance to check smoothness graphically. In our analysis, we present continuity, Holder regularity, degree of generation, degree of reproduction and limit stencils analysis of proposed family of subdivision schemes. We also present a survey of high continuity subdivision schemes. Comparison shows that our proposed family of subdivision schemes gives high continuity of subdivision schemes comparative to existing subdivision schemes. We have found that as complexity increases the continuity also increases. In the last, we give the general formula for tensor product surface subdivision schemes and also present the visual performance of proposed tensor product surface subdivision schemes.

scholarly journals A New Class of 2q-Point Nonstationary Subdivision Schemes and Their Applications

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 639 ◽  
Author(s):  
Abdul Ghaffar ◽  
Mehwish Bari ◽  
Zafar Ullah ◽  
Mudassar Iqbal ◽  
Kottakkaran Sooppy Nisar ◽  
...  
Keyword(s):  
Curve Surface ◽  
Asymptotic Equivalence ◽  
Shape Parameters ◽  
Limit Curve ◽  
Subdivision Schemes ◽  
Local Shape ◽  
New Class ◽  
Stationary Subdivision

The main objective of this study is to introduce a new class of 2 q -point approximating nonstationary subdivision schemes (ANSSs) by applying Lagrange-like interpolant. The theory of asymptotic equivalence is applied to find the continuity of the ANSSs. These schemes can be nicely generalized to contain local shape parameters that allow the user to locally adjust the shape of the limit curve/surface. Moreover, many existing approximating stationary subdivision schemes (ASSSs) can be obtained as nonstationary counterparts of the proposed ANSSs.

11.—Equations d'Evolution du Second Ordre Associées à des Operateurs Maximaux Monotones

1976 ◽  
Vol 75 (2) ◽  
pp. 131-147 ◽  
Author(s):  
Laurent Véron
Keyword(s):  
Hilbert Space ◽  
Initial Data ◽  
Maximal Monotone Operator ◽  
Existence And Uniqueness ◽  
Real Hilbert Space ◽  
Maximal Monotone ◽  
Uniqueness Theorems ◽  
Bounded Solutions ◽  
The Family ◽  
Measure Spaces

SYNOPSISThis paper extends some recent results of V. Barbu and H. Brézis. It is concerned with bounded solutions of the problem pu″+qu′ ∈ Au, u(0) = a, where A is a maximal monotone operator in a real Hilbert space H and p and q are real functions. Existence and uniqueness theorems are proved, with results on integrability of solutions in various measure spaces on R+. T(t) denotes the family of contractions of D(A) generated by the equation and we obtain a regularising effect on the initial data. Some properties of this family of contractions are studied.

scholarly journals A Family of Even-Point Ternary Approximating Schemes

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Abdul Ghaffar ◽  
Ghulam Mustafa
Keyword(s):  
General Formula ◽  
Shape Parameter ◽  
Sufficient Conditions ◽  
Approximation Order ◽  
Subdivision Schemes ◽  
The Family ◽  
Curvature And Torsion

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 C0 to C7 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.

scholarly journals A Family of High Continuity Subdivision Schemes Based on Probability Distribution

2019 ◽  
Vol 38 (2) ◽  
pp. 389-398 ◽  
Author(s):  
Muhammad Asghar ◽  
Muhammad Javed Iqbal ◽  
Ghulam Mustafa
Keyword(s):  
Probability Distribution ◽  
Geometric Modeling ◽  
Probability Generating Function ◽  
Hölder Regularity ◽  
Binomial Probability ◽  
Shape Preserving ◽  
Subdivision Schemes ◽  
Smooth Curves ◽  
The Family ◽  
Holder Regularity

Subdivision schemes are famous for the generation of smooth curves and surfaces in CAGD (Computer Aided Geometric Design). The continuity is an important property of subdivision schemes. Subdivision schemes having high continuity are always required for geometric modeling. Probability distribution is the branch of statistics which is used to find the probability of an event. We use probability distribution in the field of subdivision schemes. In this paper, a simplest way is introduced to increase the continuity of subdivision schemes. A family of binary approximating subdivision schemes with probability parameter p is constructed by using binomial probability generating function. We have derived some family members and analyzed the important properties such as continuity, Holder regularity, degree of generation, degree of reproduction and limit stencils. It is observed that, when the probability parameter p = 1/2, the family of subdivision schemes have maximum continuity, generation degree and Holder regularity. Comparison shows that our proposed family has high continuity as compare to the existing subdivision schemes. The proposed family also preserves the shape preserving property such as convexity preservation. Subdivision schemes give negatively skewed, normal and positively skewed behavior on convex data due to the probability parameter. Visual performances of the family are also presented.

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