Examples of refinable componentwise polynomials

This short note presents four examples of compactly supported symmetric refinable componentwise polynomial functions: (i) a componentwise constant interpolatory continuous refinable function and its derived symmetric tight wavelet frame; (ii) a componentwise constant continuous orthonormal and interpolatory refinable function and its associated symmetric orthonormal wavelet basis; (iii) a differentiable symmetric componentwise linear polynomial orthonormal refinable function; (iv) a symmetric refinable componentwise linear polynomial which is interpolatory and differentiable.     

The higher-order derivatives of spectral functions

We are interested in higher-order derivatives of functions of the eigenvalues of real symmetric matrices with respect to the matrix argument. We describe a formula for the k-th derivative of such functions in two general cases.The first case concerns the derivatives of the composition of an arbitrary (not necessarily symmetric) k-times differentiable function with the eigenvalues of symmetric matrices at a symmetric matrix with distinct eigenvalues.The second case describes the derivatives of the composition of a k-times differentiable separable symmetric function with the eigenvalues of symmetric matrices at an arbitrary symmetric matrix. We show that the formula significantly simplifies when the separable symmetric function is k-times continuously differentiable.As an application of the developed techniques, we re-derive the formula for the Hessian of a general spectral function at an arbitrary symmetric matrix. The new tools lead to a shorter, cleaner derivation than the original one.To make the exposition as self contained as possible, we have included the necessary background results and definitions. Proofs of the intermediate technical results are collected in the appendices.     

Symmetric spaces of higher rank do not admit differentiable compactifications

Any nonpositively curved symmetric space admits a topological compactification, namely the Hadamard compactification. For rank 1 spaces, this topological compactification can be endowed with a differentiable structure such that the action of the isometry group is differentiable. Moreover, the restriction of the action on the boundary leads to a flat model for some geometry (conformal, CR or quaternionic CR depending of the space). One can ask whether such a differentiable compactification exists for higher rank spaces, hopefully leading to some knew geometry to explore. In this paper we answer negatively.     

Birkhoff''s theorem on manifolds

Brikhoff's theorem states that if the geometry of a given region of space-time is first spherically symmetric and secondly a solution to the Einstein empty space equations, that then that geometry is a piece of the Schwarzschild geometry. Here we show that Birkhoff's theorem holds on differentiable manifolds in general.     

Characteristic roots and vectors of a diifferentiable family of symmetric matrices

LetA(x) be a differentiable family of k × k symmetric matrices where x runs through a domain D in RⁿWe prove that if λ is a continuous function onDsuch that, for every x εD,λ(x) is a characteristic root of A(x) of constant multiplicity m, then λ is a differentiable function and there exists, locally, a differentiable family of ortho-normal bases for the eigenspace. The case n = 1 has been known in the standard treatises on the perturbation theory for linear operators.     

Bol loops as sections in semi-simple Lie groups of small dimension

Using the relations between the theory of differentiable Bol loops and the theory of affine symmetric spaces we classify all connected differentiable Bol loops having an at most nine-dimensional semi-simple Lie group as the group topologically generated by their left translations. We show that all these Bol loops are isotopic to direct products of Bruck loops of hyperbolic type or to Scheerer extensions of Lie groups by Bruck loops of hyperbolic type.This paper was supported by DAAD.     

Differentiable structure of the set of coaxial stress–strain tensors

In order to study stress–strain tensors, we consider their representations as pairs of symmetric 3 × 3‐matrices and the space of such pairs of matrices partitioned into equivalence classes corresponding to change of bases. We see that these equivalence classes are differentiable submanifolds; in fact, orbits under the action of a Lie group. We compute their dimension and obtain miniversal deformations. Finally, we prove that the space of coaxial stress–strain tensors is a finite union of differentiable submanifolds. Copyright © 2008 John Wiley & Sons, Ltd.     

Decomposition theorems for groups of diffeomorphisms in the sphere

We study the algebraic structure of several groups of differentiable diffeomorphisms in . We show that any given sufficiently smooth diffeomorphism can be written as the composition of a finite number of diffeomorphisms which are symmetric under reflection, essentially one-dimensional and about as differentiable as the given one.

     

拟线性正对称组具非线性特征的边值问题

谷超豪在[1]中就报告了拟线性正对称方程组边值问题解的存在和唯一性定理,详细的证明发表于[2],在[7]中对此有所改进,所建立的理论适用于非特征的齐次边值的情况。[3]参考了[2],[4]的做法,把[2]的结果推广到拟线性正对称组具某类特征的边值问题中去,但要求边界条件仍是齐次的。本文讨论边界条件是非齐次的,且是非线性的特征边值问题,在这种情况下建立了可微解的存在和唯一性定理。     

EXTENSION OF SMOOTHING FUNCTIONS TO SYMMETRIC CONE COMPLEMENTARITY PROBLEMS

The paper uses Euclidean Jordan algebras as a basic tool to extend smoothing functions, which include the Chen-Mangasarian class and the Fischer-Burmeister smoothing functions, to symmetric cone complementarity problems. Computable formulas for these functions and their Jacobians are derived. In addition, it is shown that these functions are Lipschitz continuous with respect to parameter # and continuously differentiable on J × J for any μ 〉 0.     

New smooth C-functions for symmetric cone complementarity problems

In this paper, with the help of the Jordan-algebraic technique we introduce two new complementarity functions (C-functions) for symmetric cone complementary problems, and show that they are continuously differentiable and strongly semismooth everywhere. The work was partly supported by the National Natural Science Foundation of China (10671010, 70471002).     

Dogleg路径信赖域方法

本文提出一类新的解无约束最优化问题的信整域方法。这类方法是通过对一般对称矩阵的Bunch-Parlett分解来产生搜索路径。它们既可以解目标函数是二次可微的也可以解目标函数是非二次可微的最优化问题，并且在由算法得到点列的任意聚点上，二次连续可微的目标函数的Hesse阵都是正定或半正定的。我们证明在一些较弱的条件下，算法是整体收敛的；对一致凸函数，是二次收敛的。一些数值结果表明这种新的方法是非常有效的。     

The symmetric eigenvalue complementarity problem

In this paper the Eigenvalue Complementarity Problem (EiCP) with real symmetric matrices is addressed. It is shown that the symmetric (EiCP) is equivalent to finding an equilibrium solution of a differentiable optimization problem in a compact set. A necessary and sufficient condition for solvability is obtained which, when verified, gives a convenient starting point for any gradient-ascent local optimization method to converge to a solution of the (EiCP). It is further shown that similar results apply to the Symmetric Generalized Eigenvalue Complementarity Problem (GEiCP). Computational tests show that these reformulations improve the speed and robustness of the solution methods.

     

On the solution of the symmetric eigenvalue complementarity problem by the spectral projected gradient algorithm

This paper is devoted to the eigenvalue complementarity problem (EiCP) with symmetric real matrices. This problem is equivalent to finding a stationary point of a differentiable optimization program involving the Rayleigh quotient on a simplex (Queiroz et al., Math. Comput. 73, 1849–1863, 2004). We discuss a logarithmic function and a quadratic programming formulation to find a complementarity eigenvalue by computing a stationary point of an appropriate merit function on a special convex set. A variant of the spectral projected gradient algorithm with a specially designed line search is introduced to solve the EiCP. Computational experience shows that the application of this algorithm to the logarithmic function formulation is a quite efficient way to find a solution to the symmetric EiCP.     

一类不可微规划的高阶对称对偶性

本文我们利用一个可微函数给出了一对高阶对称规划问题 ,其中目标函数包含了Rn 中一紧凸集的支撑函数 .在引入高阶F 凸性 (F 伪凸性 ,F 拟凸性 )后 ,证明了高阶弱、高阶强及高阶逆对称对偶性质 .     

A Measure Of Asymmetry For Positive Semidefinite Matrices

It is known that a continuous map is the gradient of a convex function if and only if it is cyclically monotone. Also, a differentiable map F is the gradient of a function if and only if the matrices F ′(x) are symmetric for all x in the domain. Based on this connection between symmetry and monotonicity, we define a measure of asymmetry for positive semidefinite matrices.     

Fej\'{e}r type inequalities for Harmonically-convex functions with applications

In this paper, a new weighted identity involving harmonically symmetric functions and differentiable functions is established. By using the notion of harmonic symmetricity, harmonic convexity and some auxiliary results, some new Fej\''{e}r type integral inequalities are presented. Applications to special means of positive real numbers are given as well.     

Longtime behavior of the hyperbolic equations with an arbitrary internal damping

This paper is devoted to a study of the longtime behavior of the hyperbolic equations with an arbitrary internal damping, under sharp regularity assumptions that both the principal part coefficients and the boundary of the space domain (in which the system evolves) are continuously differentiable. For this purpose, we derive a new point-wise inequality for second differential operators with symmetric coefficients. Then, based on a global Carleman estimate, we establish an estimate on the underlying resolvent operator of the equation, via which, we show the logarithmic decay rate for solutions of the hyperbolic equations.     

A modified homogeneous potential reduction algorithm for solving the monotone semidefinite linear complementarity problem

In this paper, we study the performance of the Semidefinite Linear Complementarity Problem (SDLCP) for symmetric matrices that is equipped with a continuously differentiable potential function. A practical homogeneous self-dual potential reduction algorithm based on this potential function is prescribed, and we establish a computational basis for interior point methods with the use of HKM directions towards the central trajectory for the monotone SDLCP. Our computational implementation maintains a global linear polynomial time convergence, while achieving strong practical performance in comparison with existing solution methods.