We present a novel approach for dealing with optimal approximate merging of two adjacent Bezier eurves with G^(2)-continuity.Instead of moving the control points,we minimize the distance between the original curves an...We present a novel approach for dealing with optimal approximate merging of two adjacent Bezier eurves with G^(2)-continuity.Instead of moving the control points,we minimize the distance between the original curves and the merged curve by taking advantage of matrix representation of Bezier curve's discrete structure,where the approximation error is measured by L_(2)-norm.We use geometric information about the curves to generate the merged curve,and the approximation error is smaller.We can obtain control points of the merged curve regardless of the degrees of the two original curves.We also discuss the merged curve with point constraints.Numerical examples are provided to demonstrate the effectiveness of our algorithms.展开更多
This paper presents a quadratic programming method for optimal multi-degree reduction of B6zier curves with G^1-continuity. The L2 and I2 measures of distances between the two curves are used as the objective function...This paper presents a quadratic programming method for optimal multi-degree reduction of B6zier curves with G^1-continuity. The L2 and I2 measures of distances between the two curves are used as the objective functions. The two additional parameters, available from the coincidence of the oriented tangents, are constrained to be positive so as to satisfy the solvability condition. Finally, degree reduction is changed to solve a quadratic problem of two parameters with linear constraints. Applications of degree reduction of Bezier curves with their parameterizations close to arc-length parameterizations are also discussed.展开更多
The concept of a fuzzy topology on a fuzzy set has been introduced in [1]. The aim of this work is to introduce fuzzy δ*-continuity and fuzzy δ**-continuity in this in new situation and to show the relationships bet...The concept of a fuzzy topology on a fuzzy set has been introduced in [1]. The aim of this work is to introduce fuzzy δ*-continuity and fuzzy δ**-continuity in this in new situation and to show the relationships between fuzzy continuous functions where we confine our study to some of their types such as, fuzzy δ-continuity, fuzzy continuity, after presenting the definition of a fuzzy topology on a fuzzy set and giving some properties related to it.展开更多
ωB-splines have many optimal properties and can reproduce plentiful commonly-used analytical curves.In this paper,we further propose a non-stationary subdivision method of hierarchically and efficiently generatingωB...ωB-splines have many optimal properties and can reproduce plentiful commonly-used analytical curves.In this paper,we further propose a non-stationary subdivision method of hierarchically and efficiently generatingωB-spline curves of arbitrary order ofωB-spline curves and prove its C^k?2-continuity by two kinds of methods.The first method directly prove that the sequence of control polygons of subdivision of order k converges to a C^k?2-continuousωB-spline curve of order k.The second one is based on the theories upon subdivision masks and asymptotic equivalence etc.,which is more convenient to be further extended to the case of surface subdivision.And the problem of approximation order of this non-stationary subdivision scheme is also discussed.Then a uniform ωB-spline curve has both perfect mathematical representation and efficient generation method,which will benefit the application ofωB-splines.展开更多
In the paper,we define weakly δ-continuous correspondences on super-space,On the basis of δ-open(closed) sets,θ-open(closed) sets and regular open(closed) sets in topological space,some equivalent conditions of thi...In the paper,we define weakly δ-continuous correspondences on super-space,On the basis of δ-open(closed) sets,θ-open(closed) sets and regular open(closed) sets in topological space,some equivalent conditions of this kind of correspondences are obtained,and some applications of subset nets and convergence nets are given.展开更多
Extension of a B-spline curve or surface is a useful function in a CAD system. This paper presents an algorithm for extending cubic B-spline curves or surfaces to one or more target points. To keep the extension curve...Extension of a B-spline curve or surface is a useful function in a CAD system. This paper presents an algorithm for extending cubic B-spline curves or surfaces to one or more target points. To keep the extension curve segment GC^2-continuous with the original one, a family of cubic polynomial interpolation curves can be constructed. One curve is chosen as the solution from a sub-class of such a family by setting one GC^2 parameter to be zero and determining the second GC^2 parameter by minimizing the strain energy. To simplify the final curve representation, the extension segment is reparameterized to achieve C-continuity with the given B-spline curve, and then knot removal from the curve is done. As a result, a sub-optimized solution subject to the given constraints and criteria is obtained. Additionally, new control points of the extension B-spline segment can be determined by solving lower triangular linear equations. Some computing examples for comparing our method and other methods are given.展开更多
Although widely used in permeation reaction barrier(PRB)for strengthening the removal of various heavy metals,zero-valent iron(ZVI)is limited by various inherent drawbacks,such as easy passivation and poor electron tr...Although widely used in permeation reaction barrier(PRB)for strengthening the removal of various heavy metals,zero-valent iron(ZVI)is limited by various inherent drawbacks,such as easy passivation and poor electron transfer.As a solution,a synergistic system with PRB and electrokinetics(PRB-EK)was established and applied for the efficient removal of Cr(Ⅵ)-contaminated groundwater.As the filling material of PRB,ZVI/Fe_(3)O_(4)/activated carbon(ZVI/Fe_(3)O_(4)/AC)composites were synthesized by ball milling and thermal treatment.A series of continuous flow column experiments and batch tests was conducted to evaluate the removal efficiency of Cr(Ⅵ).Results showed that the removal efficiency of Cr(Ⅵ)remained above 93%even when the bed volume(BV)reached 2000 under the operational parameters(iron/AC mass ratio,2:1;current,5 m A).The mechanism of Cr(Ⅵ)removal by the PRB-EK system was revealed through field emission scanning electron microscopy images,X-ray diffraction,X-ray photoelectron spectroscopy,Fe^(2+) concentration,and redox potential(E h)values.The key in Cr(Ⅵ)reduction was the Fe^(2+)/Fe^(3+) cycle driven by the surface microelectrolysis of the composites.The application of an externally supplied weak direct current maintained the redox process by enhancing the electron transfer capability of the system,thereby prolonging the column lifetime.Cr(Ⅵ)chemical speciation was determined through sequential extraction,verifying the stability and safety of the system.These findings provide a scientific basis for PRB design and the in-situ remediation of Cr(Ⅵ)-contaminated groundwater.展开更多
Recently, Wardowski [Fixed Point Theory Appl., 2012: 94, 2012] introduced and studied a new contraction called F-contraction to prove a fixed point result as a generalization of the Banach contraction principle. In th...Recently, Wardowski [Fixed Point Theory Appl., 2012: 94, 2012] introduced and studied a new contraction called F-contraction to prove a fixed point result as a generalization of the Banach contraction principle. In this paper, we introduce an α-β-FG-contraction and generalize the Wardowski fixed point result in b-metric and ordered b-metric spaces. As an application of our results we deduce Suzuki type fixed point results for β-FG-contractions.Moreover, we discuss some illustrative examples to highlight the realized improvements.展开更多
In this article,we adopt the C-type spline of degree 2 to model and blend basic shapes including conics and circle arcs.The C-type spline belongs to theωB-spline category of splines that are capable of blending polyn...In this article,we adopt the C-type spline of degree 2 to model and blend basic shapes including conics and circle arcs.The C-type spline belongs to theωB-spline category of splines that are capable of blending polynomial,trigonometric and hyperbolic functions.Commonly used basic shapes can be exactly represented by these types of splines.We derive explicit formulas for the convenience of modeling the basic curves.The entire blending curve is C^1-continuous.In comparison with the existing best blending method by rational G^2 splines,which are rational splines of degree 3,the proposed method allows simpler representation and blending of the basic curves,and it can represent numerous basic shapes including the hyperbolic types.We also design a subdivision method to generate blending curves;this method is precise for the basic curves and approximate for the blending sections.The subdivision process is efficient for modeling and rendering.It has also proven to be C^1-continuous by the asymptotically equivalent theory and the continuity of stationary subdivision method.In addition,we extend the proposed methods to cases involving the modeling and blending of basic surfaces.We provide many examples that illustrate the merits of our methods.展开更多
In this paper, we introduce the notion of αδ-US spaces. Also we study the concepts of αδ-convergence, sequentially αδ-compactness, sequentially αδ-continunity and sequentially αδ-sub-continuity and derive so...In this paper, we introduce the notion of αδ-US spaces. Also we study the concepts of αδ-convergence, sequentially αδ-compactness, sequentially αδ-continunity and sequentially αδ-sub-continuity and derive some of their properties.展开更多
The aim of this work is to introduce some weak forms of continuity in bitopological spaces. Then we use these new forms of weak continuity to give many decompositions of?i-continuity and pairwise continuity.
In this paper. the concepts of θ-continuous functions and θ-compactness in fyzzifying topology characterized in terms of θ-open sets is given.some properties of θ-continuous functions and θ-compactness are discus...In this paper. the concepts of θ-continuous functions and θ-compactness in fyzzifying topology characterized in terms of θ-open sets is given.some properties of θ-continuous functions and θ-compactness are discussed.展开更多
This paper presents a fast algorithm （BS2 Algorithm） for fitting C 1 surfaces to scat- tered data points. By using energy minimization, the bivariate spline space S 2 1（△ m,n （2） ） is introduced to construct a ...This paper presents a fast algorithm （BS2 Algorithm） for fitting C 1 surfaces to scat- tered data points. By using energy minimization, the bivariate spline space S 2 1（△ m,n （2） ） is introduced to construct a Cl-continuous piecewise quadratic surface through a set of irregularly 3D points. Moreover, a multilevel method is also presented. Some experimental results show that the accuracy is satisfactory. Furthermore, the BS2 Algorithm is more suitable for fitting surfaces if the given data points have some measurement errors.展开更多
基金supported by the National Natural Science Foundation of China(No.60773179)the National Basic Research Program(973)of China(No.G2004CB318000)
文摘We present a novel approach for dealing with optimal approximate merging of two adjacent Bezier eurves with G^(2)-continuity.Instead of moving the control points,we minimize the distance between the original curves and the merged curve by taking advantage of matrix representation of Bezier curve's discrete structure,where the approximation error is measured by L_(2)-norm.We use geometric information about the curves to generate the merged curve,and the approximation error is smaller.We can obtain control points of the merged curve regardless of the degrees of the two original curves.We also discuss the merged curve with point constraints.Numerical examples are provided to demonstrate the effectiveness of our algorithms.
基金Project supported by the National Natural Science Foundation ofChina (No. 60473130)the National Basic Research Program(973) of China (No. G2004CB318000)
文摘This paper presents a quadratic programming method for optimal multi-degree reduction of B6zier curves with G^1-continuity. The L2 and I2 measures of distances between the two curves are used as the objective functions. The two additional parameters, available from the coincidence of the oriented tangents, are constrained to be positive so as to satisfy the solvability condition. Finally, degree reduction is changed to solve a quadratic problem of two parameters with linear constraints. Applications of degree reduction of Bezier curves with their parameterizations close to arc-length parameterizations are also discussed.
文摘The concept of a fuzzy topology on a fuzzy set has been introduced in [1]. The aim of this work is to introduce fuzzy δ*-continuity and fuzzy δ**-continuity in this in new situation and to show the relationships between fuzzy continuous functions where we confine our study to some of their types such as, fuzzy δ-continuity, fuzzy continuity, after presenting the definition of a fuzzy topology on a fuzzy set and giving some properties related to it.
基金the National Natural Science Foundation of China(61772164,61761136010)the Natural Science Foundation of Zhejiang Province(LY17F020025).
文摘ωB-splines have many optimal properties and can reproduce plentiful commonly-used analytical curves.In this paper,we further propose a non-stationary subdivision method of hierarchically and efficiently generatingωB-spline curves of arbitrary order ofωB-spline curves and prove its C^k?2-continuity by two kinds of methods.The first method directly prove that the sequence of control polygons of subdivision of order k converges to a C^k?2-continuousωB-spline curve of order k.The second one is based on the theories upon subdivision masks and asymptotic equivalence etc.,which is more convenient to be further extended to the case of surface subdivision.And the problem of approximation order of this non-stationary subdivision scheme is also discussed.Then a uniform ωB-spline curve has both perfect mathematical representation and efficient generation method,which will benefit the application ofωB-splines.
文摘In the paper,we define weakly δ-continuous correspondences on super-space,On the basis of δ-open(closed) sets,θ-open(closed) sets and regular open(closed) sets in topological space,some equivalent conditions of this kind of correspondences are obtained,and some applications of subset nets and convergence nets are given.
文摘Extension of a B-spline curve or surface is a useful function in a CAD system. This paper presents an algorithm for extending cubic B-spline curves or surfaces to one or more target points. To keep the extension curve segment GC^2-continuous with the original one, a family of cubic polynomial interpolation curves can be constructed. One curve is chosen as the solution from a sub-class of such a family by setting one GC^2 parameter to be zero and determining the second GC^2 parameter by minimizing the strain energy. To simplify the final curve representation, the extension segment is reparameterized to achieve C-continuity with the given B-spline curve, and then knot removal from the curve is done. As a result, a sub-optimized solution subject to the given constraints and criteria is obtained. Additionally, new control points of the extension B-spline segment can be determined by solving lower triangular linear equations. Some computing examples for comparing our method and other methods are given.
基金financial support from the National Natural Science Foundation of China(Nos.21906044 and 21477034)the Key Science and Technology Program of Henan Province,China(No.132102210129)+3 种基金the Basic Scientific and Technological Frontier Project of Henan Province(No.162300410046)the Innovation Scientists and Technicians Troop Construction Projects of Henan Province,the Scientific Research Foundation from Soochow University(No.Q416000117)the Technology Department of the Henan Science and Technology Fund Project(No.202102310603)the Cultivating National Scientific Research Project Funds,Henan Normal University(No.5101219170804)。
文摘Although widely used in permeation reaction barrier(PRB)for strengthening the removal of various heavy metals,zero-valent iron(ZVI)is limited by various inherent drawbacks,such as easy passivation and poor electron transfer.As a solution,a synergistic system with PRB and electrokinetics(PRB-EK)was established and applied for the efficient removal of Cr(Ⅵ)-contaminated groundwater.As the filling material of PRB,ZVI/Fe_(3)O_(4)/activated carbon(ZVI/Fe_(3)O_(4)/AC)composites were synthesized by ball milling and thermal treatment.A series of continuous flow column experiments and batch tests was conducted to evaluate the removal efficiency of Cr(Ⅵ).Results showed that the removal efficiency of Cr(Ⅵ)remained above 93%even when the bed volume(BV)reached 2000 under the operational parameters(iron/AC mass ratio,2:1;current,5 m A).The mechanism of Cr(Ⅵ)removal by the PRB-EK system was revealed through field emission scanning electron microscopy images,X-ray diffraction,X-ray photoelectron spectroscopy,Fe^(2+) concentration,and redox potential(E h)values.The key in Cr(Ⅵ)reduction was the Fe^(2+)/Fe^(3+) cycle driven by the surface microelectrolysis of the composites.The application of an externally supplied weak direct current maintained the redox process by enhancing the electron transfer capability of the system,thereby prolonging the column lifetime.Cr(Ⅵ)chemical speciation was determined through sequential extraction,verifying the stability and safety of the system.These findings provide a scientific basis for PRB design and the in-situ remediation of Cr(Ⅵ)-contaminated groundwater.
基金funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, JeddahDSR, KAU for financial supportthe Ministry of Education, Science and Technological Development of Serbia, Grant No. 174002
文摘Recently, Wardowski [Fixed Point Theory Appl., 2012: 94, 2012] introduced and studied a new contraction called F-contraction to prove a fixed point result as a generalization of the Banach contraction principle. In this paper, we introduce an α-β-FG-contraction and generalize the Wardowski fixed point result in b-metric and ordered b-metric spaces. As an application of our results we deduce Suzuki type fixed point results for β-FG-contractions.Moreover, we discuss some illustrative examples to highlight the realized improvements.
基金This work described in this article was supported by the National Science Foundation of China(61772164,61272032)Provincial Key Platforms and Major Scientific Research Projects in Universities and Colleges of Guangdong(2017KTSCX143)the Natural Science Foundation of Zhejiang Province(LY17F020025).
文摘In this article,we adopt the C-type spline of degree 2 to model and blend basic shapes including conics and circle arcs.The C-type spline belongs to theωB-spline category of splines that are capable of blending polynomial,trigonometric and hyperbolic functions.Commonly used basic shapes can be exactly represented by these types of splines.We derive explicit formulas for the convenience of modeling the basic curves.The entire blending curve is C^1-continuous.In comparison with the existing best blending method by rational G^2 splines,which are rational splines of degree 3,the proposed method allows simpler representation and blending of the basic curves,and it can represent numerous basic shapes including the hyperbolic types.We also design a subdivision method to generate blending curves;this method is precise for the basic curves and approximate for the blending sections.The subdivision process is efficient for modeling and rendering.It has also proven to be C^1-continuous by the asymptotically equivalent theory and the continuity of stationary subdivision method.In addition,we extend the proposed methods to cases involving the modeling and blending of basic surfaces.We provide many examples that illustrate the merits of our methods.
文摘In this paper, we introduce the notion of αδ-US spaces. Also we study the concepts of αδ-convergence, sequentially αδ-compactness, sequentially αδ-continunity and sequentially αδ-sub-continuity and derive some of their properties.
文摘The aim of this work is to introduce some weak forms of continuity in bitopological spaces. Then we use these new forms of weak continuity to give many decompositions of?i-continuity and pairwise continuity.
文摘In this paper. the concepts of θ-continuous functions and θ-compactness in fyzzifying topology characterized in terms of θ-open sets is given.some properties of θ-continuous functions and θ-compactness are discussed.
基金Acknowledgments. This work was supported by the National Natural Science Foundation of China （Nos. U0935004,11071031,11071037,10801024）, and the Fundamental Funds for the Central Universities. should be changed to Acknowledgments. This work is partly supported by the National Natural Science Foundation of China （Nos. U0935004,11071031,10801024）, the Fundamental Funds for the Central Universities （DUT10ZD112, DUT11LK34）, and National Engineering Research Center of Digital Life, Guangzhou 510006, China.
文摘This paper presents a fast algorithm （BS2 Algorithm） for fitting C 1 surfaces to scat- tered data points. By using energy minimization, the bivariate spline space S 2 1（△ m,n （2） ） is introduced to construct a Cl-continuous piecewise quadratic surface through a set of irregularly 3D points. Moreover, a multilevel method is also presented. Some experimental results show that the accuracy is satisfactory. Furthermore, the BS2 Algorithm is more suitable for fitting surfaces if the given data points have some measurement errors.
