The Weyl calculus for hermitian matrices

The Weyl calculus is a means of constructing functions of a system of hermitian operators which do not necessarily commute with each other. This note gives a new proof of a formula, due to E. Nelson, for the Weyl calculus associated with a system of hermitian matrices.

     

Functional calculus under Kreiss type conditions

It is shown that for an algebraic Banach space operator T , the Kreiss condition, ‖(zI – T )^–1‖ ≤ , |z | > 1, implies the following functional calculus estimate $$ \Vert f (T) \Vert \le {16 \over {\pi} }\, C \cdot {\rm deg} (T) \, \Vert f \Vert _{\infty}\, , $$ where deg(T ) is the degree of the minimal polynomial annihilating T . This result extends the known estimates of the powers of T for Kreiss operators on finite dimensional spaces. In the case of a general Kreiss operator, an estimate of the rational calculus is proved: $$ \Vert r(T) \Vert \le {16 \over {\pi} }\, C ( {\rm deg}(r) + 1) \, \Vert r \Vert _{\infty} \, . $$ Similar estimates hold for the polynomial calculus under generalized Kreiss conditions. A link is also established between the sharp constant in the first estimate and the norm of the best solution for a Nevanlinna–Pick type interpolation problem in analytic Besov classes. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Functional calculus of Hermitian elements and Bernstein inequalities

An element a of a unital Banach algebra A is said to be Hermitian if ‖ exp(ita)‖ = 1 for t ∈ ?. We consider some problems concerned with the functional calculus of Hermitian elements and related to estimates for the norm of ?(a), where ? is an admissible function (symbol). Let K be a compact set in ?, and let a be a Hermitian element whose spectrum coincides with K. Then ‖? (a)‖_A ≤ ‖?(D)‖_K, where D is the differential operator ?id/dx and ‖?(D)‖_K is the norm of ?(D) in the Bernstein space B _K of L ^∞(?)-functions whose Fourier transforms are supported in K. We find a differential equation for the extremals of ?(D) and describe them explicitly in the case of an arbitrary complex polynomial ?.     

稀疏函数型回归模型变量选择及实证分析

传统函数型回归模型变量选择方法,忽略了对稀疏函数型数据的讨论.提出了稀疏函数型数据情形下函数型回归模型的变量选择方法,基于条件期望对稀疏函数型自变量进行函数型主成分分析,并以估计的正交特征函数作为基函数对模型进行展开.这种方法可以有效解决对稀疏函数型变量的选择.作为实证分析,选取2002年到2011年全国34个气象观测站的年降水量,月度平均气温,光照时长,湿度,最高气温和最低气温数据,分别比较讨论了密集和稀疏情形下,原始样本和Bootstrap样本的函数型回归模型变量选择的结果,结果显示新方法具有较好的选择效果.     

A note on the identity operators of fractional calculus

The application of an identity operator for Saigo’s fractional calculus operators is shown by evaluating the limit of an indeterminate form. Its special case yields the result which has been used as an infinitesimal generator in the semigroup theory. Also, an identity operator for the recently introduced multi-dimensional fractional operators (due to Srivastava and Raina [8]) is discussed.     

Direct constructions for cyclically relative difference matrices with five rows

In this article, we construct a (6p, 5, 1) cyclic difference matrix with one hole of size 6 for any prime p > 5 and a (2p, 5, 1) cyclic difference matrix with one hole of size 2 for any prime p ≡ 1, 13 or 17 (mod 24). © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 391–399, 2006     

非线性泛函差分方程解的性质

In this paper, some sufficient and necessary conditions are established for the oscillatory of solutions for nonlinear functional difference equations, which extend and improve some corresponding results obtained and are discrete analogues of the corresponding results for the continuous version.     

一类缺项算子矩阵正补的谱刻画

对 于 具 有 离 散 谱 的 正 算 子 Ａ， Ｂ， 我 们 给 出 了 一 个 闭 集 成 为 ２ × ２ 缺 项 算 子 矩 阵 Ａ ？？ Ｂ 的某个正 补的谱的一些 判别条件     

Balancedly splittable Hadamard matrices

Balancedly splittable Hadamard matrices are introduced and studied. A connection is made to the Hadamard diagonalizable strongly regular graphs, maximal equiangular lines set, and unbiased Hadamard matrices. Several construction methods are presented. As an application, commutative association schemes of 4, 5, and 6 classes are constructed.     

Functional calculus for semigroup generators via transference

In this article we apply a recently established transference principle in order to obtain the boundedness of certain functional calculi for semigroup generators. In particular, it is proved that if −A   generates a _C0$C_0$-semigroup on a Hilbert space, then for each τ>0$\tau >0$ the operator A   has a bounded calculus for the closed ideal of bounded holomorphic functions on a (sufficiently large) right half-plane that satisfy f(z)=O(e^−τRe(z))$f(z)=O(e^{-\tau \mathrm{Re}(z)})$ as |z|→∞$|z|\to \infty$. The bound of this calculus grows at most logarithmically as τ↘0$\tau \searrow 0$. As a consequence, f(A)$f(A)$ is a bounded operator for each holomorphic function f (on a right half-plane) with polynomial decay at ∞. Then we show that each semigroup generator has a so-called (strong) m  -bounded calculus for all m∈N$m\in \mathbb{N}$, and that this property characterizes semigroup generators. Similar results are obtained if the underlying Banach space is a UMD space. Upon restriction to so-called γ-bounded semigroups, the Hilbert space results actually hold in general Banach spaces.     

Effective algorithms for computing triangular operator in Schubert calculus

We develop two parallel algorithms progressively based on C++ to compute a triangle operator problem, which plays an important role in the study of Schubert calculus. We also analyse the computational complexity of each algorithm by using combinatorial quantities, such as the Catalan number, the Motzkin number, and the central binomial coefficients. The accuracy and efficiency of our algorithms have been justified by numerical experiments.     

p-Laplace型非线性泛函差分方程正解的存在性

作者研究下述的p-Laplace型非线性泛函差分方程边值问题正解和多重正解的存在性,得到相应的充分条件,其中Φp(u)=|u|p-2u,P>1,(?)(0)= 0,C+={(?):(?)∈ C,(?)(k)≥0,k∈[-r,0])．     

Matrix calculus and related hypergeometric functions

Using only simple tools of a version of matrix calculus we describe a way how to generate series expansions of hypergeometric functions of one (or two) matrix argument(s) in a straightforward manner without the need of computing the underlying zonal polynomials.     

Norm equalities and inequalities for operator matrices

Several norm equalities and inequalities for operator matrices are proved in this paper. These results, which depend on the structure of circulant and skew circulant operator matrices, include pinching type inequalities for weakly unitarily invariant norms.     

具p-Laplace算子型泛函差分方程的正解

In this paper,the author studies the boundary value problems for a p-Laplacian functional difference equation.By using a fixed point theorem in cones,sufficient conditions are established for the existence of the positive solutions.     

Functional calculus for non-commuting operators with real spectra via an iterated Cauchy formula

We define a smooth functional calculus for a non-commuting tuple of (unbounded) operators A_j on a Banach space with real spectra and resolvents with temperate growth, by means of an iterated Cauchy formula. The construction is also extended to tuples of more general operators allowing smooth functional calculii. We also discuss the relation to the case with commuting operators.