On the eigenvalues of quaternion matrices

This article is a continuation of the article [F. Zhang, Ger?gorin type theorems for quaternionic matrices, Linear Algebra Appl. 424 (2007), pp. 139–153] on the study of the eigenvalues of quaternion matrices. Profound differences in the eigenvalue problems for complex and quaternion matrices are discussed. We show that Brauer's theorem for the inclusion of the eigenvalues of complex matrices cannot be extended to the right eigenvalues of quaternion matrices. We also provide necessary and sufficient conditions for a complex square matrix to have infinitely many left eigenvalues, and analyse the roots of the characteristic polynomials for 2?×?2 matrices. We establish a characterisation for the set of left eigenvalues to intersect or be part of the boundary of the quaternion balls of Ger?gorin.     

古斯—汉欣位移的发现与发展

目前古斯-汉欣位移泛指有限束宽光束在界面处反射或透射时,在入射平面内会出现偏离几何光学预言的一小段横向移动量。古斯—汉欣位移的研究可追溯至牛顿时代,但直到1947年才在实验上发现了该位移,此后人们开始对其逐步深入研究。作为一种基本的物理现象,它已延伸到声学、量子力学和物质波等领域。文章作者结合自身研究经历与认识,从古斯—汉欣位移的发现、理论解释及最新的研究进展等方面对其进行介绍。     

Classes of stable complex matrices defined via the theorems of Geršgorin and Lyapunov

We consider (and characterize) mainly classes of (positively) stable complex matrices defined via methods of Ger?gorin and Lyapunov. Although the real matrices in most of these classes have already been studied, we sometimes improve upon (and even correct) what has been previously published. Many of the classes turn out quite naturally to be the products of common sets of matrices. A Venn diagram shows how the classes are related.     

An alternative to the Brauer set

We derive an inclusion region for the eigenvalues of a matrix that can be considered an alternative to the Brauer set. It is accompanied by non-singularity conditions.     

New regions including eigenvalues of Toeplitz matrices

Two new eigenvalue inclusion regions for matrices with a constant main diagonal are given. We then apply these results to Toeplitz matrices, and obtain two regions including all eigenvalues of Toeplitz matrices. Furthermore, it is proved that the new regions are tighter than those in [Melman A. Ovals of Cassini for Toeplitz matrices, Linear and Multilinear Algebra. 2012;60:189–199].     

BSB5-Bestimmung durch direkte coulometrische Bestimmung des verbrauchten Sauerstoffs

Zusammenfassung In einem geschlossenen System werden 250 ml der Probe zur Bestimmung des Sauerstoffbedarfs thermostatisiert bei 20° C angesetzt. Zu diesem System geh?rt der Anodenraum einer Wasserelektrolyse, die elektrisch gespeist wird mit einem Coulometer, das durch einen Quecksilberschalter, der aufΔp dieses Systems anspricht, gesteuert wird. Im gleichfalls angeschlossenen Natronkalkrohr wird gebildetes CO₂ gebunden. Die Einschaltzeiten des Coulometers werden mit einer mechanisch-elektrischen übertragung und einem mV-Schreiber festgehalten. Die Aufzeichnung stellt die Intensit?t des biologischen Wachstums im Verh?ltnis zur abgelaufenen Zeit dar. Aus dem Gesamtausschlag (mV) kann der Sauerstoffverbrauch nach 5 Tagen errechnet werden.

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Determination of the biochemical oxygen demand (BOD₅) by direct coulometric determination of the oxygen consumed

Summary 250 ml of the sample are given into a closed system thermostatically adjusted to 20° C. This system includes the anodic compartment of a water electrolysis cell connected to a coulometer, which is controlled by a mercury switch responding toΔp of the system. CO₂ is bound in a soda lime tube. The switching times of the coulometer are transmitted to a mV-recorder. Thus the diagram shows the intensity of biological growth in relation to time. The oxygen demand after 5 days can be calculated from the total response.

     

Suitability of Tedlar gas sampling bags for siloxane quantification in landfill gas

Landfill or digester gas can contain man-made volatile methylsiloxanes (VMS), usually in the range of a few milligrams per normal cubic metre (Nm³). Until now, no standard method for siloxane quantification exists and there is controversy with respect to which sampling procedure is most suitable. This paper presents an analytical and a sampling procedure for the quantification of common VMS in biogas via GC-MS and polyvinyl fluoride (Tedlar^®) bags. Two commercially available Tedlar bag models are studied. One is equipped with a polypropylene valve with integrated septum, the other with a dual port fitting made from stainless steel. Siloxane recovery in landfill gas samples is investigated as a function of storage time, temperature, surface-to-volume ratio and background gas. Recovery was found to depend on the type of fitting employed. The siloxanes sampled in the bag with the polypropylene valve show high and stable recovery, even after more than 30 days. Sufficiently low detection limits below 10 μg Nm⁻³ and good reproducibility can be achieved. The method is therefore well applicable to biogas, greatly facilitating sampling in comparison with other common techniques involving siloxane enrichment using sorption media.     

黑体辐射公式的多种推导及其在近代物理构建中的意义(Ⅰ)

黑体辐射是近代物理史上一只会下金蛋的鹅,是近代物理的摇篮。黑体辐射研究的意义还在于这是唯一一个涉及c,k,h三个普适常数的物理情景。黑体辐射谱抗测量误差的特性带来了辐射标准和绝对温度参照,谱分布公式对模型的不敏感则使得黑体辐射成为独特的物理研究母题。黑体辐射谱分布公式,普朗克多角度推导过,德拜推导过,艾伦菲斯特推导过,劳厄推导过,洛伦兹和庞加莱深入讨论过,泡利推导过,玻色推导过,爱因斯坦在20多年的时间里多角度推导过且产出最为丰硕,近代还有从相对论角度的推导,每一个角度的推导都带来了物理学的新内容,这包括量子力学、固体量子论、受激辐射、量子统计、相对论统计,等等。认真回顾黑体辐射研究的历史细节,考察其中的思想概念演化,不啻于体验一次教科书式的学(做)物理之旅,比如也可以尝试给出能量局域分立化的简单新证明。