The solutions to some operator equations with corresponding operators not necessarily having closed ranges

In this paper, we study the solvability of the operator equations in the general setting of infinite-dimensional Hilbert space with corresponding operators no necessarily having closed range. We get the necessary and sufficient conditions for the existences of the solutions and obtain formulae in each case for the general selfadjoint, positive and real positive solutions to these operator systems.     

Some equivalent definitions of Besov spaces of generalized smoothness

In this paper, we present some alternative definitions of Besov spaces of generalized smoothness, defined via Littlewood–Paley‐type decomposition, involving weak derivatives, polynomials, convolutions and generalized interpolation spaces.     

Enterprise security economics: A self‐defense versus cyber‐insurance dilemma

We propose a model that optimizes enterprise investments in cybersecurity using expected utility theory. The model allows computing (a) investment in self‐defense to reduce the risk of security breaches, (b) investment in cyber insurance to transfer the residual risk to insurance companies, and (c) investment in forensic readiness to make the insured firms capable of generating provable insurance claims about security breaches. A three‐phase–based model of vulnerability rate evolution over time is proposed and used to estimate the different planned security expenditures throughout the investment horizon. At the starting time of investment, a decision maker invests to cover the existing risk of breach and periodically spends to cover the additional risk observed due to the release of new vulnerabilities. In this work, the intermediate tranches are determined while considering three different attitudes of decision makers, namely, optimistic, pessimistic, and realistic. An analysis is conducted to assess the performance of the proposed models.     

Real zeros of the zero-dimensional parametric piecewise algebraic variety

The piecewise algebraic variety is the set of all common zeros of multivariate splines. We show that solving a parametric piecewise algebraic variety amounts to solve a finite number of parametric polynomial systems containing strict inequalities. With the regular decomposition of semi-algebraic systems and the partial cylindrical algebraic decomposition method, we give a method to compute the supremum of the number of torsion-free real zeros of a given zero-dimensional parametric piecewise algebraic variety, and to get distributions of the number of real zeros in every n-dimensional cell when the number reaches the supremum. This method also produces corresponding necessary and sufficient conditions for reaching the supremum and its distributions. We also present an algorithm to produce a necessary and sufficient condition for a given zero-dimensional parametric piecewise algebraic variety to have a given number of distinct torsion-free real zeros in every n-cell in the n-complex. This work was supported by National Natural Science Foundation of China (Grant Nos. 10271022, 60373093, 60533060), the Natural Science Foundation of Zhejiang Province (Grant No. Y7080068) and the Foundation of Department of Education of Zhejiang Province (Grant Nos. 20070628 and Y200802999)     

跳-扩散风险模型的最优投资策略和破产概率研究

对盈余投资于金融市场的跳-扩散风险模型的最优投资策略和破产概率进行了研究,得到最优投资策略和最小破产概率的显示解,发现破产概率满足Lundberg等式．最后通过数值计算,得到最小破产概率与无风险利率,投资和相关系数之间的关系,以及无风险利率和相关系数对最优投资策略的影响．     

Rapid determination of ginkgolic acids in Ginkgo biloba kernels and leaves by direct analysis in real time‐mass spectrometry

A novel method based on direct analysis in real time integrated with mass spectrometry was established and applied into rapid determination of ginkgolic acids in Ginkgo biloba kernels and leaves. Instrument parameter settings were optimized to obtain the sensitive and accurate determination of ginkgolic acids. At the sample introduction speed of 0.2 mm/s, high intensity of [M–H]^– ions for ginkgolic acids were observed in the negative ion mode by utilization of high‐purity helium gas at 450°C. Two microliters of methanol extract of G. biloba kernels or leaves dropped on the surface of Quick‐Strip module was analyzed after solvent evaporated to dryness. A series of standard solutions of ginkgolic acid 13:0 in the range of 2–50 mg/L were analyzed with a correlation coefficient r  = 0.9981 and relative standard deviation (n  = 5) from 12.5 to 13.7%. The limit of detection was 0.5 mg/L. The results of direct analysis in real time‐mass spectrometry were in agreement with those observed by thermochemolysis gas chromatography. The proposed method demonstrated significant potential in the application of the high‐throughput screening and rapid analysis for ginkgolic acids in dietary supplements.     

超实数系统及其一致连续问题

基于模型论的观点与方法介绍超实数系统的构造及其公理体系,以说明非标准分析.由于超实数系统是普通实数系统的一个真扩张,所以,实数系统中的一致连续问题可推广至超实数系统.     

薄膜层状外延生长的光学原位实时监测方法

综述了几种用于监控薄膜外延生长的光学原位实时监测方法的进展。其中光反射差法／光反射各向异性谱（ＲＤＳ／ＲＡＳ）和ｐ偏振反射谱（ＰＲＳ），表面光吸收（ＳＰＡ），椭偏仪（ＳＥ）等。在外延过程中已观测到了薄膜层状外延周期振荡。     

带交易费用的可数无限态两期经济的均衡存在性

To some two-period economies with countable infinite state spaces,the existence of expectation equilibrium of real asset economies with transaction costs is given. This work extends the researches of Zame in 1993.     

The continuum as a formal space

A constructive definition of the continuum based on formal topology is given and its basic properties studied. A natural notion of Cauchy sequence is introduced and Cauchy completeness is proved. Other results include elementary proofs of the Baire and Cantor theorems. From a classical standpoint, formal reals are seen to be equivalent to the usual reals. Lastly, the relation of real numbers as a formal space to other approaches to constructive real numbers is determined. Received: 11 November 1996