scholarly journals Local Convexity-PreservingC2Rational Cubic Spline for Convex Data

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Muhammad Abbas ◽  
Ahmad Abd Majid ◽  
Jamaludin Md. Ali
Keyword(s):  
Shape Parameter ◽  
Cubic Spline ◽  
Convex Curve ◽  
Shape Feature ◽  
Local Convexity ◽  
Shape Parameters ◽  
Numerical Examples ◽  
2D Data ◽  
Industrial Demand ◽  
Made In

We present the smooth and visually pleasant display of 2D data when it is convex, which is contribution towards the improvements over existing methods. This improvement can be used to get the more accurate results. An attempt has been made in order to develop the local convexity-preserving interpolant for convex data usingC2rational cubic spline. It involves three families of shape parameters in its representation. Data dependent sufficient constraints are imposed on single shape parameter to conserve the inherited shape feature of data. Remaining two of these shape parameters are used for the modification of convex curve to get a visually pleasing curve according to industrial demand. The scheme is tested through several numerical examples, showing that the scheme is local, computationally economical, and visually pleasing.

scholarly journals ULTRAMETRIC AND NON-LOCALLY CONVEX ANALOGUES OF THE GENERAL CURVE LEMMA OF CONVENIENT DIFFERENTIAL CALCULUS

2008 ◽  
Vol 50 (2) ◽  
pp. 271-288
Author(s):  
HELGE GLÖCKNER
Keyword(s):  
Differential Calculus ◽  
Vector Spaces ◽  
Local Convexity ◽  
Topological Vector Spaces ◽  
Infinite Dimensional ◽  
General Curve ◽  
Infinite Dimensional Analysis ◽  
Locally Convex ◽  
Differentiability Properties ◽  
Made In

AbstractThe General Curve Lemma is a tool of Infinite-Dimensional Analysis that enables refined studies of differentiability properties of maps between real locally convex spaces to be made. In this article, we generalize the General Curve Lemma in two ways. First, we remove the condition of local convexity in the real case. Second, we adapt the lemma to the case of curves in topological vector spaces over ultrametric fields.

scholarly journals B-splines Algorithms for Solving Fredholm Linear Integro-Differential Equations

2004 ◽  
Vol 1 (2) ◽  
pp. 340-346
Author(s):  
Baghdad Science Journal
Keyword(s):  
Differential Equations ◽  
Spline Function ◽  
Cubic Spline ◽  
Second Order ◽  
Third Order ◽  
B Spline ◽  
Numerical Examples ◽  
B Splines ◽  
The Third ◽  
New Procedures

Algorithms using the second order of B -splines [B (x)] and the third order of B -splines [B,3(x)] are derived to solve 1' , 2nd and 3rd linear Fredholm integro-differential equations (F1DEs). These new procedures have all the useful properties of B -spline function and can be used comparatively greater computational ease and efficiency.The results of these algorithms are compared with the cubic spline function.Two numerical examples are given for conciliated the results of this method.

scholarly journals Geometric Modeling Using New Cubic Trigonometric B-Spline Functions with Shape Parameter

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2102
Author(s):  
Abdul Majeed ◽  
Muhammad Abbas ◽  
Faiza Qayyum ◽  
Kenjiro T. Miura ◽  
Md Yushalify Misro ◽  
...  
Keyword(s):  
Geometric Modeling ◽  
Shape Parameter ◽  
3D Models ◽  
Basis Functions ◽  
Shape Parameters ◽  
B Spline ◽  
Spline Curves ◽  
Spline Basis ◽  
Cubic Polynomial ◽  
2D And 3D

Trigonometric B-spline curves with shape parameters are equally important and useful for modeling in Computer-Aided Geometric Design (CAGD) like classical B-spline curves. This paper introduces the cubic polynomial and rational cubic B-spline curves using new cubic basis functions with shape parameter ξ∈[0,4]. All geometric characteristics of the proposed Trigonometric B-spline curves are similar to the classical B-spline, but the shape-adjustable is additional quality that the classical B-spline curves does not hold. The properties of these bases are similar to classical B-spline basis and have been delineated. Furthermore, uniform and non-uniform rational B-spline basis are also presented. C3 and C5 continuities for trigonometric B-spline basis and C3 continuities for rational basis are derived. In order to legitimize our proposed scheme for both basis, floating and periodic curves are constructed. 2D and 3D models are also constructed using proposed curves.

scholarly journals Data Visualization Using Rational Trigonometric Spline

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Uzma Bashir ◽  
Jamaludin Md. Ali
Keyword(s):  
Data Visualization ◽  
The Other ◽  
Shape Features ◽  
Shape Parameters ◽  
Shape Preserving ◽  
Numerical Examples ◽  
Data Points ◽  
Trigonometric Spline ◽  
The Given

This paper describes the use of trigonometric spline to visualize the given planar data. The goal of this work is to determine the smoothest possible curve that passes through its data points while simultaneously satisfying the shape preserving features of the data. Positive, monotone, and constrained curve interpolating schemes, by using aC1piecewise rational cubic trigonometric spline with four shape parameters, are developed. Two of these shape parameters are constrained and the other two are set free to preserve the inherited shape features of the data as well as to control the shape of the curve. Numerical examples are given to illustrate the worth of the work.

Shape Control Learning Algorithm for Neural Networks

2013 ◽  
Vol 347-350 ◽  
pp. 2270-2274
Author(s):  
Dai Yuan Zhang
Keyword(s):  
Neural Networks ◽  
Artificial Neural Networks ◽  
Numerical Experiments ◽  
Shape Control ◽  
Learning Algorithm ◽  
Neural System ◽  
Cubic Spline ◽  
New Method ◽  
Shape Parameters ◽  
Control Learning

A new kind of shape control learning algorithm (SCLA) for training neural networks is proposed. We use the rational cubic spline (with quadratic denominator) to implement a new neural system for shape control, and construct a new kind of artificial neural networks based on given patterns. The shape can be controlled by some shape parameters, which is much different from the known algorithms for training neural networks. The numerical experiments indicate that the new method proposed in this paper demonstrates good results.

scholarly journals Some Remarks on Estimate of Mittag-Leffler Function

2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Jia Jia ◽  
Zhen Wang ◽  
Xia Huang ◽  
Yunliang Wei
Keyword(s):  
Dynamic Analysis ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Fractional Order Systems ◽  
Numerical Examples ◽  
Estimation Process ◽  
Made In

The estimate of Mittag-Leffler function has been widely applied in the dynamic analysis of fractional-order systems in some recently published papers. In this paper, we show that the estimate for Mittag-Leffler function is not correct. First, we point out the mistakes made in the estimation process of Mittag-Leffler function and provide a counterexample. Then, we propose some sufficient conditions to guarantee that part of the estimate for Mittag-Leffler function is correct. Meanwhile, numerical examples are given to illustrate the validity of the two newly established estimates.

A Study of Kansei Engineering in PET Bottle Silhouette

2011 ◽  
Author(s):  
Khusnun Widiyati ◽  
Hideki Aoyama
Keyword(s):  
Shape Parameter ◽  
Taguchi Methods ◽  
Emotional Attachment ◽  
Kansei Engineering ◽  
Shape Parameters ◽  
Pet Bottles ◽  
Emotional Evaluation ◽  
Physical Attributes ◽  
Artificial Neural Network Ann ◽  
The Many

Today’s customers see the product not only based on its functionality and its value, but also based on its aesthetic view. And from day to day, the level of consumers’ aesthetic satisfaction is advanced. Companies have to struggle in recognizing and feeding the customers with higher and higher level of aesthetics design, in order to win the competition. Kansei Engineering is a powerful tool during product designs that analyze the design in relation with consumers’ feeling toward the product. In this saturated market, moreover, the application of Kansei Engineering might be a way out to provide a product emotionally attached to the customers’ feeling. PET bottles, particularly bottles distributed in Japan, is an example of “everyday-beverage-container” which has many variations in the shape. Among the many product attributes, product form/shape is one of the product attribute that can attract emotional attachment to the customer. In this paper, physical attributes of PET bottle which evoke consumers to have certain emotional attachment were evaluated using Kansei Engineering. In order to do this, 18 models of PET bottle generated using Taguchi Methods, and 9 emotional evaluation words were applied in a questionnaire. By using Taguchi Methods, important shape parameters that evoke customer to certain emotional feeling were identified. Validation to the Taguchi Methods’ finding was validated using Artificial Neural Network (ANN). The validation was performed by mapping the Kansei/emotional space to shape parameter space. Evaluation towards the result from Taguchi Methods and ANN was performed. Comparison between Taguchi Methods’ and ANN’s result showed that both result were correlated.

Entropies of Mixing (EOM) and the Lorenz Order

2006 ◽  
Vol 13 (01) ◽  
pp. 75-90 ◽  
Author(s):  
B. H. Lavenda
Keyword(s):  
Shannon Entropy ◽  
Power Distribution ◽  
Shape Parameter ◽  
Sufficient Conditions ◽  
Fractional Part ◽  
Concave Function ◽  
Value Theory ◽  
Invariant Distribution ◽  
Shape Parameters ◽  
Single Class

Polynomial nonadditive, or pseudo-additive (PAE), entropies are related to the Shannon entropy in that both are derived from two classes of parent distributions of extreme-value theory, the Pareto and power distributions. The third class is the exponential distribution, corresponding to the Shannon entropy, to which the other two tend as their shape parameters increase without limit. These entropies all belong to a single class of entropies referred to as EOM. EOM is defined as the normalized difference between the dual of the Lorentz function and the Lorenz function. Sufficient conditions for majorization involve finding a separable, Schur-concave function, like the EOM, which increases as the distribution becomes more uniform or less spread out. Lorenz ordering has been associated to the degree in which the Lorenz curve is bent. This criterion is valid for tail distributions, and fails in the case where the distribution is limited on the right. EOM provide criteria for inequality in the Lorenz ordering sense: In the Pareto case, an increase in the shape parameter implies a decrease in inequality and the EOM decreases, whereas for the power distribution an increase in the shape parameter corresponds to an increase in inequality leading to an increase in the EOM. An analogy is drawn between Gauss' invariant distribution for the probability of the fractional part of a continued fraction and the area criterion in Lorenz ordering, analogous to the Gini index criterion. The tendency to approach the invariant distribution, as the number of partial quotients increases without limit, is shown to be analogous to the tendency to approach the invariant area, as the shape parameters increase without limit.

scholarly journals Generalized Developable Cubic Trigonometric Bézier Surfaces

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 283
Author(s):  
Muhammad Ammad ◽  
Md Yushalify Misro ◽  
Muhammad Abbas ◽  
Abdul Majeed
Keyword(s):  
Shape Parameter ◽  
Surface Characteristics ◽  
Surface Shape ◽  
Shape Parameters ◽  
New Approach ◽  
Bézier Surface ◽  
Bezier Surface ◽  
Bézier Surfaces ◽  
Bezier Surfaces ◽  
Shape Adjustment

This paper introduces a new approach for the fabrication of generalized developable cubic trigonometric Bézier (GDCT-Bézier) surfaces with shape parameters to address the fundamental issue of local surface shape adjustment. The GDCT-Bézier surfaces are made by means of GDCT-Bézier-basis-function-based control planes and alter their shape by modifying the shape parameter value. The GDCT-Bézier surfaces are designed by maintaining the classic Bézier surface characteristics when the shape parameters take on different values. In addition, the terms are defined for creating a geodesic interpolating surface for the GDCT-Bézier surface. The conditions appropriate and suitable for G1, Farin–Boehm G2, and G2 Beta continuity in two adjacent GDCT-Bézier surfaces are also created. Finally, a few important aspects of the newly formed surfaces and the influence of the shape parameters are discussed. The modeling example shows that the proposed approach succeeds and can also significantly improve the capability of solving problems in design engineering.

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