Intervals of 1-Lipschitz aggregation operators, quasi-copulas, and copulas with given affine section

Best lower and upper bounds for 1-Lipschitz aggregation operators with a given affine section are given. These are used to determine best bounds for quasi-copulas and copulas with a given affine section. However, in general there is no greatest copula with a given non-decreasing affine section. These results are used to study (quasi-)copulas with arbitrary affine sections.     

On product affine hyperspheres in R~(n+1)

In this paper, we study locally strongly convex affine hyperspheres in the unimodular affine space R~(n+1) which, as Riemannian manifolds, are locally isometric to the Riemannian product of two Riemannian manifolds both possessing constant sectional curvature. As the main result, a complete classification of such affine hyperspheres is established. Moreover, as direct consequences, 3-and 4-dimensional affine hyperspheres with parallel Ricci tensor are also classified.     

Intervals of 1-Lipschitz aggregation operators,quasi-copulas,and copulas with given affine section

Best lower and upper bounds for 1-Lipschitz aggregation operators with a given affine section are given. These are used to determine best bounds for quasi-copulas and copulas with a given affine section. However, in general there is no greatest copula with a given non-decreasing affine section. These results are used to study (quasi-)copulas with arbitrary affine sections. A significant part of this work was done during a visit of the second author at the Johannes Kepler University, Linz (Austria). The second author was supported by the grant VEGA 1/3012/06 and by the Science and Technology Assistance Agency (Contract No. APVT-20-003204). Both authors would like to thank the anonymous referee whose comments (including the two copulas C ₁ and C ₂ given in the Conclusion) not only solved the originally stated open problem in the negative, but also allowed them to formulate two more interesting open problems.     

Reductive group actions on affine quadrics with 1-dimensional quotient: Linearization when a linear model exists

We study reductive group actions on complex affine quadrics. Such an action is called linearizable if it is equivalent to the restriction of a linear orthogonal action in the ambient affine space of the quadric. A linear model for a given action is a linear orthogonal action with the same orbit types and equivalent slice representations. We prove that if a reductive group action on an affine quadric with a 1-dimensional quotient has a linear model, then the action is linearizable. As a consequence, the action is linearizable if certain topological conditions are satisfied.     

Affine immersion ofn-dimensional manifold intoR n+n(n+1)/2 and affine minimality

We formulate an affine theory of immersions of ann-dimensional manifold into the Euclidean space of dimensionn+n(n+1)/2 and give a characterization of critical immersions relative to the induced volume functional in terms of the affine shape operator.     

The weighted euclidean 1-center problem

We present a polynomial algorithm for the weighted 1-center problem (indeed minimization of the ratio of convex quadratic and an affine function over a polyhedral set). In the location problem, the complexity is polynomial in the dimension of the space.     

Nonconforming rotated Q_1 element on non-tensor product anisotropic meshes

In this paper, we consider the nonconforming rotated Q1 element for the second order elliptic problem on the non-tensor product anisotropic meshes, i.e. the anisotropic affine quadrilateral meshes. Though the interpolation error is divergent on the anisotropic meshes, we overcome this difficulty by constructing another proper operator. Then we give the optimal approximation error and the consistency error estimates under the anisotropic affine quadrilateral meshes. The results of this paper provide some hints to derive the anisotropic error of some finite elements whose interpolations do not satisfy the anisotropic interpolation properties. Lastly, a numerical test is carried out, which coincides with our theoretical analysis.     

Fourier transforms related to a root system of rank 1

We introduce an algebra consisting of difference-reflection operators and multiplication operators that can be considered as a q = 1 analogue of Sahi's double affine Hecke algebra related to the affine root system of type . We study eigenfunctions of a Dunkl-Cherednik-type operator in the algebra , and the corresponding Fourier transforms. These eigenfunctions are nonsymmetric versions of the Wilson polynomials and the Wilson functions.     

Nonconforming rotated Q 1 element on non-tensor product anisotropic meshes

In this paper, we consider the nonconforming rotated Q ₁ element for the second order elliptic problem on the non-tensor product anisotropic meshes, i.e. the anisotropic affine quadrilateral meshes. Though the interpolation error is divergent on the anisotropic meshes, we overcome this difficulty by constructing another proper operator. Then we give the optimal approximation error and the consistency error estimates under the anisotropic affine quadrilateral meshes. The results of this paper provide some hints to derive the anisotropic error of some finite elements whose interpolations do not satisfy the anisotropic interpolation properties. Lastly, a numerical test is carried out, which coincides with our theoretical analysis.     

Superlinearly convergent affine scaling interior trust-region method for linear constrained LC<Superscript>1</Superscript> minimization

We extend the classical affine scaling interior trust region algorithm for the linear constrained smooth minimization problem to the nonsmooth case where the gradient of objective function is only locally Lipschitzian. We propose and analyze a new affine scaling trust-region method in association with nonmonotonic interior backtracking line search technique for solving the linear constrained LC1 optimization where the second-order derivative of the objective function is explicitly required to be locally Lipschitzian. The general trust region subproblem in the proposed algorithm is defined by minimizing an augmented affine scaling quadratic model which requires both first and second order information of the objective function subject only to an affine scaling ellipsoidal constraint in a null subspace of the augmented equality constraints. The global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions where twice smoothness of the objective function is not required. Applications of the algorithm to some nonsmooth optimization problems are discussed.     

SOME THEOREMS ON CONVEX HYPERSURFACES IN AN AFFINE SPACE

Let M and M~* be hypersurfaces in an affine space A~(n+1) of dimension n+1. The main results of this paper are the following:(1) Two types of integral formulas for M and M~*.(2) Some conditions for an affine convex hypersurface to be an affine sphere.(3) Some conditions for M and M~* to be different only by an affine transformation or a translation.     

Hall Polynomials for Affine Quivers of Type Am（m≥1）

It is well known that Hall polynomials as structural coefficients play an important role in the structure of Lie algebras and quantum groups. By using the properties of representation categories of affine quivers, the task of computing Hall polynomials for affine quivers can be reduced to counting the numbers of solutions of some matrix equations. This method has been applied to obtain Hall polynomials for indecomposable representations of quivers of type Am（m≥1）     

Riemann extensions and affine differential geometry

In order to define an affine immersion of manifolds in affine differential geometry, it is necessary to choose a set of normal planes to the immersed manifold. The theory is then developed after this choice has been made. However, it was shown by A.G.Walker [WA1] that a torsion-free affine connexion on a manifold determines canonically a pseudo-Riemannian metric on the cotangent bundle, called the Riemann-extension of the affine connexion. By making use of this pseudo-Riemannian metric it is possible to define an affine immersion without making a suitable choice of normal planes.     

伪仿射对偶框架

对于给定冗余的仿射框架X（Ф）={Ф1,…,ФL）,存在对偶Y（ψ）={ψ1,…,ψL）,这个对偶可能不是仿射框架,因为它不满足Bessel条件．这样的对偶我们称为伪仿射对偶,并给出伪仿射对偶框架的一种构造公式．     

About C 1-minimality of the hyperbolic Cantor sets

In this work we prove that a C ^1+α -hyperbolic Cantor set contained in S ¹ that is close to an affine Cantor set is not C ¹-minimal.     

Some theorems in affine differential geometry

In this paper we prove that an affine hypersphere with scalar curvature zero in a unimodular affine space of dimensionn+1 must be contained either in an elliptic paraboloid or in an affine image of the hypersurfacex ₁ x ₂...x _n+1=const. We prove also that an affine complete, affine maximal surface is an elliptic paraboloid if its affine normals omit 4 or more directions in general position.The Project Supported by National Natural Science Foundation of China     

Hausdorff dimension bounds for smoothness of holonomies for codimension 1 hyperbolic dynamics

We prove that the stable holonomies of a proper codimension 1 attractor Λ, for a C^r diffeomorphism f of a surface, are not C^1+θ for θ greater than the Hausdorff dimension of the stable leaves of f intersected with Λ. To prove this result we show that there are no diffeomorphisms of surfaces, with a proper codimension 1 attractor, that are affine on a neighbourhood of the attractor and have affine stable holonomies on the attractor.     

Nonconforming rotated Q 1 element on non-tensor product anisotropic meshes

In this paper, we consider the nonconforming rotated Q ₁ element for the second order elliptic problem on the non-tensor product anisotropic meshes, i.e. the anisotropic affine quadrilateral meshes. Though the interpolation error is divergent on the anisotropic meshes, we overcome this difficulty by constructing another proper operator. Then we give the optimal approximation error and the consistency error estimates under the anisotropic affine quadrilateral meshes. The results of this paper provide some hints to derive the anisotropic error of some finite elements whose interpolations do not satisfy the anisotropic interpolation properties. Lastly, a numerical test is carried out, which coincides with our theoretical analysis.