代数不等式的分拆降维方法与机器证明

利用双变元对称型所构成实线性空间的特点,设计了一种特殊形式的基,基中元素是非负的.如果一个元在此基下的坐标非负,则该元自身也是非负的.于是要证明某个元非负将被归结为证明其在指定基下的坐标非负.通常坐标中的变元数,少于原对称型的变元数,从而起到了降低维数的作用.对非对称型,可通过对称化转换为对称型来处理.根据该方法编制了Maple通用程序Bidecomp.虽此方法并非完备的,但大量的应用实例表明了此种方法证明多项式型不等式的有效性.     

On the pointwise limits of bivariate lagrange projectors

A linear algebra proof is given of the fact that the nullspace of a finite-rank linear projector, on polynomials in two complex variables, is an ideal if and only if the projector is the bounded pointwise limit of Lagrange projectors, i.e., projectors whose nullspace is a radical ideal, i.e., the set of all polynomials that vanish on a certain given finite set. A characterization of such projectors is also given in the real case. More generally, a characterization is given of those finite-rank linear projectors, on polynomials in d complex variables, with nullspace an ideal that are the bounded pointwise limit of Lagrange projectors. The characterization is in terms of a certain sequence of d commuting linear maps and so focuses attention on the algebra generated by such sequences.     

Restricting statistical convergence

We first introduce a new notion called statistical convergence of order α and primarily show that it gives rise to a decreasing chain of closed linear subspaces of the space of all bounded real sequences with sup norm which never coincides with the class of convergent sequences and in fact their intersection properly contains the class of convergent sequences. We then show that the same method can be applied for double sequences also and introduce the notion of statistical convergence of order (α,β).     

赋范空间有界序列收敛的等价性

Two kinds of convergent sequences on the real vector space m of all bounded sequences in a real normed space X were discussed in this paper, and we prove that they are equivalent, which improved the results of [1].     

Convergent regular splittings for nonnegative matrices

MP matrices are those real matrices which possess a nonnegative, nonsingular l-inverse. This paper characterizes the nonnegative MP matrices and hence, determines when a nonnegative matrix A has a convergent regular splitting M—Q which induces the linear stationary iterative scheme x_k+1=M^-1Qx_k+M^-1b to solve Ax=b.     

A structure-preserving doubling algorithm for nonsymmetric algebraic Riccati equation

In this paper, we propose a structure-preserving doubling algorithm (SDA) for the computation of the minimal nonnegative solution to the nonsymmetric algebraic Riccati equation (NARE), based on the techniques developed for the symmetric cases. This method allows the simultaneous approximation to the minimal nonnegative solutions of the NARE and its dual equation, requiring only the solutions to two linear systems and several matrix multiplications per iteration. Similar to Newton's method and the fixed-point iteration methods for solving NAREs, we also establish global convergence for SDA under suitable conditions, using only elementary matrix theory. We show that sequences of matrices generated by SDA are monotonically increasing and quadratically convergent to the minimal nonnegative solutions of the NARE and its dual equation. Numerical experiments show that the SDA algorithm is feasible and effective, and outperforms Newton's iteration and the fixed-point iteration methods. This research was supported in part by RFDP (20030001103) & NSFC (10571007) of China and the National Center for Theoretical Sciences in Taiwan. This author's research was supported by NSFC grant 1057 1007 and RFDP grant 200300001103 of China.     

Two-Sided Closed Ideals of Certain Algebras of Unbounded Operators

This paper investigates the two-sided uniformly closed ideals of the maximal Op*-algebra L⁺(D) of (bounded or unbounded) operators on a dense domain D in a HILBERT space. It is assumed that D is a FRECHET space with respect to the graph topology. The set of all non-trivial two-sided closed ideals of L⁺(D) is well-ordered by inclusion and the α-th closed ideal ??_α is generated by the orthogonal projections onto HILBERTian subspaces of D of dimension less then ??_α. An element A in L⁺(D) belongs to the minimal closed ideal ??₀ if and only if the following two equivalent conditions are satisfied: a) A maps bounded subsets of D into relatively compact sets. b) A maps weakly convergent sequences in D into convergent sequences.     

Statistical convergence in intuitionistic fuzzy n-normed linear spaces

The aim in our article is to introduce the notion of statistical convergence and statistically Cauchy sequences in intuitionistic fuzzy n-normed linear spaces. The paper shows that some properties of statistical convergence of real sequences also hold for sequences in this space. Characterization for statistically convergent and statistically Cauchy sequences is also given. Further, the concept of statistical limit points and statistical cluster points are introduced and their relation with limit points of sequences have been investigated.     

Global exponential stability of nonresident computer virus models

This paper is concerned with nonresident computer virus models which are defined on the nonnegative real vector space. By using differential inequality technique, we employ a novel argument to show that the virus-free equilibrium is globally exponentially stable, and the exponential convergent rate can be unveiled. Moreover, a numerical simulation is given to demonstrate our theoretical results.     

Efficient Recognition of Totally Nonnegative Matrix Cells

The space of m×p totally nonnegative real matrices has a stratification into totally nonnegative cells. The largest such cell is the space of totally positive matrices. There is a well-known criterion due to Gasca and Peña for testing a real matrix for total positivity. This criterion involves testing mp minors. In contrast, there is no known small set of minors for testing for total nonnegativity. In this paper, we show that for each of the totally nonnegative cells there is a test for membership which only involves mp minors, thus extending the Gasca and Peña result to all totally nonnegative cells.     

On Asymptotics of Toeplitz Determinants with Symbols of Nonstandard Smoothness

We prove Szegös strong limit theorem for Toeplitz determinants with a symbol having a nonstandard smoothness. We assume that the symbol belongs to the Wiener algebra and, moreover, the sequences of Fourier coefficients of the symbol with negative and nonnegative indices belong to weighted Orlicz classes generated by complementary N-functions both satisfying the ⁰ ₂-condition and by weight sequences satisfying some regularity and compatibility conditions.     

Global convergence of modified multiplicative updates for nonnegative matrix factorization

Nonnegative matrix factorization (NMF) is the problem of approximating a given nonnegative matrix by the product of two nonnegative matrices. The multiplicative updates proposed by Lee and Seung are widely used as efficient computational methods for NMF. However, the global convergence of these updates is not formally guaranteed because they are not defined for all pairs of nonnegative matrices. In this paper, we consider slightly modified versions of the original multiplicative updates and study their global convergence properties. The only difference between the modified updates and the original ones is that the former do not allow variables to take values less than a user-specified positive constant. Using Zangwill’s global convergence theorem, we prove that any sequence of solutions generated by either of those modified updates has at least one convergent subsequence and the limit of any convergent subsequence is a stationary point of the corresponding optimization problem. Furthermore, we propose algorithms based on the modified updates that always stop within a finite number of iterations.     

On a Conjecture of Phadke and Thakare

We prove the connectedness of the set of all nonzero bounded linear operators on a complex Hilbert space having a generalized inverse.     

Fatou's identity and Lebesgue's convergence theorem

The classical Fatou lemma for bounded sequences of nonnegative integrable functions is represented as an equality. A similar result is stated for measure convergent sequences. Neither result requires a uniform integrability assumption. For the latter a converse is proven. Two extensions of Lebesgue's convergence theorem are presented.

     

Weak bases of vector measures

We solve the problem of representation of measures with values in a Banach space as the limits of weakly convergent sequences of vector measures whose basis is a given nonnegative measure. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 10, pp. 1436–1440, October, 2007.     

On inverse spectrum problems for normal nonnegative matrices

For a set σ with n complex numbers, some sufficient conditions are found for σ to be the spectrum of an n ×n normal (entrywise) nonnegative (positive) matrix. After proving a fundamental theorem and introducing the companion set σ' of σ which consists of real numbers, we prove that if σ' satisfies any known sufficient conditions for a real set to be the spectrum of a nonnegative matrix introduced by Suleimanova, Perfect, Salzmann and Kellogg respectively, then σ is the spectrum of an n×n normal nonnegative matrix.     

On inverse spectrum problems for normal nonnegative matrices

For a set σ with n complex numbers, some sufficient conditions are found for σ to be the spectrum of an n ×n normal (entrywise) nonnegative (positive) matrix. After proving a fundamental theorem and introducing the companion set σ′ of σ which consists of real numbers, we prove that if σ′ satisfies any known sufficient conditions for a real set to be the spectrum of a nonnegative matrix introduced by Suleimanova, Perfect, Salzmann and Kellogg respectively, then σ is the spectrum of an n×n normal nonnegative matrix.     

谱表示

<正> Stone M.对Hilbert空间中一个具有简单谱的自伴算子建立了谱表示定理,即有实轴上的有限Borel测度μ,使得同构于L~2(μ),同时变A为乘以自变量λ的算子.Jauch等([2])讨论了一列交换的自伴算子完全集谱表示定理,但要求一个关于测度绝对连续性的假定.此外,依据约化理论([3])可知,如果A是可分Hilbert空间的自伴     

A generalization of the Helly theorem for functions with values in a uniform space

In this paper we consider sequences of functions that are defined on a subset of the real line and take on values in a uniform Hausdorff space. For such sequences we obtain a sufficient condition for the existence of pointwise convergent subsequences. We prove that this generalization of the Helly theorem includes many results of the recent research. In addition, we prove that the sufficient condition is also necessary for uniformly convergent sequences of functions. We also obtain a representation for regular functions whose values belong to the uniform space.     

ECT-B-splines defined by generalized divided differences

ECT-spline curves are generated from different local ECT-systems via connection matrices. If they are nonsingular, lower triangular and totally positive there is a basis of the space of ECT-splines consisting of functions having minimal compact supports, normalized either to form a nonnegative partition of unity or to have integral one. In this paper such ECT-B-splines are defined by generalized divided differences. This definition reduces to the classical one in case of a Schoenberg space. Under suitable assumptions it leads to a recursive method for computing the ECT-B-splines that reduces to the de Boor–Mansion–Cox recursion in case of ordinary polynomial splines and to Lyche's recursion in case of Tchebycheff splines [Mühlbach and Tang, Calculation of ECT-B-splines and of ECT-spline curves recursively, in preparation].There is an ECT-spline space naturally adjoint to every ECT-spline space. We also construct B-splines via generalized divided differences for this space and study relations between the two adjoint spaces.