The best recovery of a linear functional Lf, f=f(x,y), on the basis of given linear functionals L jf,j=1,2,...,N in a sense of Sard has been investigated, using analogy of Peano's theorem. The best recovery of a ...The best recovery of a linear functional Lf, f=f(x,y), on the basis of given linear functionals L jf,j=1,2,...,N in a sense of Sard has been investigated, using analogy of Peano's theorem. The best recovery of a bivariate function by given scattered data has been obtained in a simple analytical form as a special case.展开更多
There are several examples of spaces of univariate functions for which we have a characterization of all sets of knots which are poised for the interpolation problem. For the standard spaces of univariate polynomials,...There are several examples of spaces of univariate functions for which we have a characterization of all sets of knots which are poised for the interpolation problem. For the standard spaces of univariate polynomials, or spline functions the mentioned results are well-known. In contrast with this, there are no such results in the bivariate case. As an exception, one may consider only the Pascal classic theorem, in the interpolation theory interpretation. In this paper, we consider a space of bivariate piecewise linear functions, for which we can readily find out whether the given node set is poised or not. The main tool we use for this purpose is the reduction by a basic subproblem, introduced in this paper.展开更多
In this paper,we consider the clustering of bivariate functional data where each random surface consists of a set of curves recorded repeatedly for each subject.The k-centres surface clustering method based on margina...In this paper,we consider the clustering of bivariate functional data where each random surface consists of a set of curves recorded repeatedly for each subject.The k-centres surface clustering method based on marginal functional principal component analysis is proposed for the bivariate functional data,and a novel clustering criterion is presented where both the random surface and its partial derivative function in two directions are considered.In addition,we also consider two other clustering methods,k-centres surface clustering methods based on product functional principal component analysis or double functional principal component analysis.Simulation results indicate that the proposed methods have a nice performance in terms of both the correct classification rate and the adjusted rand index.The approaches are further illustrated through empirical analysis of human mortality data.展开更多
This paper analyses the local behavior of the simple off-diagonal bivariate quadratic function approximation to a bivariate function which has a given power series expansion about the origin.It is shown that the simpl...This paper analyses the local behavior of the simple off-diagonal bivariate quadratic function approximation to a bivariate function which has a given power series expansion about the origin.It is shown that the simple off-diagonal bivariate quadratic Hermite-Padé form always defines a bivariate quadratic function and that this function is analytic in a neighbourhood of the origin.Numerical examples compare the obtained results with the approximation power of diagonal Chisholm approximant and Taylor polynomial approximant.展开更多
This paper analysis the local behavior of the bivariate quadratic function approximation to a bivariate function which has a given power series expansion about the origin. It is shown that the bivariate quadratic Herm...This paper analysis the local behavior of the bivariate quadratic function approximation to a bivariate function which has a given power series expansion about the origin. It is shown that the bivariate quadratic Hermite-Padé form always defines a bivariate quadratic function and that this function is analytic in a neighborhood of the origin.展开更多
This paper proposes a universal framework for constructing bivariate stochastic processes,going beyond the limitations of copulas and offering a potentially simpler alternative.The achieved generality of the construct...This paper proposes a universal framework for constructing bivariate stochastic processes,going beyond the limitations of copulas and offering a potentially simpler alternative.The achieved generality of the construction methods extends its applicability to diverse stochastic processes also including discrete as well as continuous time cases.The initially given two arbitrary univariate stochastic processes{Y_(t)},{Z_(t)},are only assumed to share the same time t.When considered as describing(time dependent)random quantities that are physically separated(the baseline case),the processes are independent.From this trivial case we move to the case when physical interactions between the quantities make them stochastically dependent random variables at any moment t.For each time epoch t,we impose stochastic dependence on two“initially independent”random variables Y_(t),Z_(t) by multiplying the product of their survival functions by a proper“dependence factor”φ_(t)(y_(t), z_(t)),obtaining in this way a universal(“canonical”)form valid for any(!)bivariate distribution.In some known cases,however,this form may become complicated thou it always exists and is unique.The dependence factor,basically,but not always,has the form φ_(t)(y, z)=exp[-∫^(y)_(0)∫^(z)_(0)Ψ_(t)(s ,u )dsdu]whenever such a continuous function Ψ_(t)(s ,u ) exists,for each t.That representation of stochastic dependence by the functions Ψ_(t)(s ,u ) leads,in turn,to the phenomenon of change of the original(baseline)hazard rates of the marginals,similar to those analyzed by Cox and,especially Aalen for single pairs(or sets)of,time independent,random variables.That is why,until Section 4,we consider only single random vectors(Y,Z)'joint survival functions,mostly as a preparation to the theory of bivariate stochastic processes{(Y_(t),Z_(t))}constructions as initiated in Section 4.The bivariate constructions are illustrated by examples of some applications in biomedical and econometric areas.展开更多
Non-tensor product bivariate fractal interpolation functions defined on gridded rectangular domains are constructed. Linear spaces consisting of these functions are introduced. The relevant Lagrange interpolation prob...Non-tensor product bivariate fractal interpolation functions defined on gridded rectangular domains are constructed. Linear spaces consisting of these functions are introduced. The relevant Lagrange interpolation problem is discussed. A negative result about the existence of affine fractal interpolation functions defined on such domains is obtained.展开更多
In this paper,we enumerate the set of Motzkin trees with n edges according to the number of leaves,the number of vertices adjacent to a leaf,the number of protected nodes,the number of(protected)branch nodes,and the n...In this paper,we enumerate the set of Motzkin trees with n edges according to the number of leaves,the number of vertices adjacent to a leaf,the number of protected nodes,the number of(protected)branch nodes,and the number of(protected)lonely nodes.Explicit formulae as well as generating functions are obtained.We also find that,as n goes to infinity,the proportion of protected branch nodes and protected lonely nodes among all vertices of Motzkin trees with n edges approaches 4/27 and 2/9.展开更多
To better describe and understand the time dynamics in functional data analysis,it is often desirable to recover the partial derivatives of the random surface.A novel approach is proposed based on marginal functional ...To better describe and understand the time dynamics in functional data analysis,it is often desirable to recover the partial derivatives of the random surface.A novel approach is proposed based on marginal functional principal component analysis to derive the representation for partial derivatives.To obtain the Karhunen-Lo`eve expansion of the partial derivatives,an adaptive estimation is explored.Asymptotic results of the proposed estimates are established.Simulation studies show that the proposed methods perform well in finite samples.Application to the human mortality data reveals informative time dynamics in mortality rates.展开更多
A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, we propose the Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangu...A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, we propose the Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations. We show that, if two continuous piecewise algebraic curves of degrees m and n respectively meet at ranT distinct points over a cross-cut triangulation, where T denotes the number of cells of the triangulation, then any continuous piecewise algebraic curve of degree m ＋ n - 2 containing all but one point of them also contains the last point.展开更多
文摘The best recovery of a linear functional Lf, f=f(x,y), on the basis of given linear functionals L jf,j=1,2,...,N in a sense of Sard has been investigated, using analogy of Peano's theorem. The best recovery of a bivariate function by given scattered data has been obtained in a simple analytical form as a special case.
文摘There are several examples of spaces of univariate functions for which we have a characterization of all sets of knots which are poised for the interpolation problem. For the standard spaces of univariate polynomials, or spline functions the mentioned results are well-known. In contrast with this, there are no such results in the bivariate case. As an exception, one may consider only the Pascal classic theorem, in the interpolation theory interpretation. In this paper, we consider a space of bivariate piecewise linear functions, for which we can readily find out whether the given node set is poised or not. The main tool we use for this purpose is the reduction by a basic subproblem, introduced in this paper.
基金supported by National Natural Science Foundation of China (Grant Nos.12261007)Natural Science Foundation of Guangxi Province (Grant No.2020GXNSFAA297225)。
文摘In this paper,we consider the clustering of bivariate functional data where each random surface consists of a set of curves recorded repeatedly for each subject.The k-centres surface clustering method based on marginal functional principal component analysis is proposed for the bivariate functional data,and a novel clustering criterion is presented where both the random surface and its partial derivative function in two directions are considered.In addition,we also consider two other clustering methods,k-centres surface clustering methods based on product functional principal component analysis or double functional principal component analysis.Simulation results indicate that the proposed methods have a nice performance in terms of both the correct classification rate and the adjusted rand index.The approaches are further illustrated through empirical analysis of human mortality data.
基金Supported by National Natural Science Foundation of China( 699730 1 0,1 0 2 71 0 2 2 )
文摘This paper analyses the local behavior of the simple off-diagonal bivariate quadratic function approximation to a bivariate function which has a given power series expansion about the origin.It is shown that the simple off-diagonal bivariate quadratic Hermite-Padé form always defines a bivariate quadratic function and that this function is analytic in a neighbourhood of the origin.Numerical examples compare the obtained results with the approximation power of diagonal Chisholm approximant and Taylor polynomial approximant.
基金Supported by the NNSF of China(10271022, 60373093)Supported by the Science and Technology Development Foundation of Education Department of Liaoning Province(2004C060)
文摘This paper analysis the local behavior of the bivariate quadratic function approximation to a bivariate function which has a given power series expansion about the origin. It is shown that the bivariate quadratic Hermite-Padé form always defines a bivariate quadratic function and that this function is analytic in a neighborhood of the origin.
文摘This paper proposes a universal framework for constructing bivariate stochastic processes,going beyond the limitations of copulas and offering a potentially simpler alternative.The achieved generality of the construction methods extends its applicability to diverse stochastic processes also including discrete as well as continuous time cases.The initially given two arbitrary univariate stochastic processes{Y_(t)},{Z_(t)},are only assumed to share the same time t.When considered as describing(time dependent)random quantities that are physically separated(the baseline case),the processes are independent.From this trivial case we move to the case when physical interactions between the quantities make them stochastically dependent random variables at any moment t.For each time epoch t,we impose stochastic dependence on two“initially independent”random variables Y_(t),Z_(t) by multiplying the product of their survival functions by a proper“dependence factor”φ_(t)(y_(t), z_(t)),obtaining in this way a universal(“canonical”)form valid for any(!)bivariate distribution.In some known cases,however,this form may become complicated thou it always exists and is unique.The dependence factor,basically,but not always,has the form φ_(t)(y, z)=exp[-∫^(y)_(0)∫^(z)_(0)Ψ_(t)(s ,u )dsdu]whenever such a continuous function Ψ_(t)(s ,u ) exists,for each t.That representation of stochastic dependence by the functions Ψ_(t)(s ,u ) leads,in turn,to the phenomenon of change of the original(baseline)hazard rates of the marginals,similar to those analyzed by Cox and,especially Aalen for single pairs(or sets)of,time independent,random variables.That is why,until Section 4,we consider only single random vectors(Y,Z)'joint survival functions,mostly as a preparation to the theory of bivariate stochastic processes{(Y_(t),Z_(t))}constructions as initiated in Section 4.The bivariate constructions are illustrated by examples of some applications in biomedical and econometric areas.
基金Supported in part by the NKBRSF(G1998030600) in part by the Doctoral Program Foundation of Educational Department of China(1999014115).
文摘Non-tensor product bivariate fractal interpolation functions defined on gridded rectangular domains are constructed. Linear spaces consisting of these functions are introduced. The relevant Lagrange interpolation problem is discussed. A negative result about the existence of affine fractal interpolation functions defined on such domains is obtained.
基金Supported by the National Natural Science Foundation of China(Grant No.11861045)Gansu Province Science Foundation for Youths(Grant No.20JR10RA187)the Hongliu Foundation of First-Class Disciplines of Lanzhou University of Technology,China。
文摘In this paper,we enumerate the set of Motzkin trees with n edges according to the number of leaves,the number of vertices adjacent to a leaf,the number of protected nodes,the number of(protected)branch nodes,and the number of(protected)lonely nodes.Explicit formulae as well as generating functions are obtained.We also find that,as n goes to infinity,the proportion of protected branch nodes and protected lonely nodes among all vertices of Motzkin trees with n edges approaches 4/27 and 2/9.
基金supported by National Natural Science Foundation of China(Grant Nos.11861014,11561006 and 11971404)Natural Science Foundation of Guangxi Province(Grant No.2018GXNSFAA281145)+1 种基金Humanity and Social Science Youth Foundation of Ministry of Education of China(Grant No.19YJC910010)the Intramural Research Program of the Eunice Kennedy Shriver National Institute of Child Health and Human Development,National Institutes of Health,USA。
文摘To better describe and understand the time dynamics in functional data analysis,it is often desirable to recover the partial derivatives of the random surface.A novel approach is proposed based on marginal functional principal component analysis to derive the representation for partial derivatives.To obtain the Karhunen-Lo`eve expansion of the partial derivatives,an adaptive estimation is explored.Asymptotic results of the proposed estimates are established.Simulation studies show that the proposed methods perform well in finite samples.Application to the human mortality data reveals informative time dynamics in mortality rates.
基金The first author is supported by National Natural Science Foundation of China （Grant Nos. U0935004, 11071031, 11001037, 10801024） and the Fundamental Research Funds for the Central Universities （Grant Nos. DUT10ZDll2, DUT10JS02） the second author is supported by the 973 Program （2011CB302703）, National Natural Science Foundation of China （Grant Nos. U0935004, 60825203, 61033004, 60973056, 60973057, 61003182）, and Beijing Natural Science Foundation （4102009） We thank the referees for valuable suggestions which improve the presentation of this paper.
文摘A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, we propose the Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations. We show that, if two continuous piecewise algebraic curves of degrees m and n respectively meet at ranT distinct points over a cross-cut triangulation, where T denotes the number of cells of the triangulation, then any continuous piecewise algebraic curve of degree m ＋ n - 2 containing all but one point of them also contains the last point.
