Using the Cayley-Hamilton theorem with N-partitioned matrices

A definition is given for the characteristic equation of anN-partitioned matrix. It is then proved that this matrix satisfies its own characteristic equation. This can then be regarded as a version of the Cayley-Hamilton theorem, of use withN-dimensional systems.

Transnational urbanism in Port of Spain: returning middle-class urban elites

Port of Spain, Trinidad offers an ideal context in which to analyze pre-retirement return migration to a Global South urban realm, expanding transnational urban research beyond the conventional focus on Global North metropolitan destinations. In this article, we draw on the transnational narratives of a selected sample of relatively youthful Trinidadians, who have spent many years abroad acquiring education and professional experience, but who have then decided to return in mid-career to the capital region of the island nation of their birth, or of their parent(s). Theoretically, we position these returning professionals as members of a "middling" transnational urban class whose return is at least partly motivated by a desire to "make a difference." Our results contribute to a growing literature that documents the role of transnational middle-class urban elites returning elsewhere in the Carribbean: "middling" transnational urbanism is reshaping key facets of urbanization in the Global South.

One-port de-embedding technique for the quasi-optical characterization of integrated components

We describe a one-port de-embedding technique suitable for the quasi-optical characterization of terahertz integrated components at frequencies beyond the operational range of most vector network analyzers. This technique is also suitable when the manufacturing of precision terminations to sufficiently fine tolerances for the application of a TRL de-embedding technique is not possible. The technique is based on vector reflection measurements of a series of easily realizable test pieces. A theoretical analysis is presented for the precision of the technique when implemented using a quasi-optical null-balanced bridge reflectometer. The analysis takes into account quantization effects in the linear and angular encoders associated with the balancing procedure, as well as source power and detector noise equivalent power. The precision in measuring waveguide characteristic impedance and attenuation using this de-embedding technique is further analyzed after taking into account changes in the power coupled due to axial, rotational, and lateral alignment errors between the device under test and the instruments' test port. The analysis is based on the propagation of errors after assuming imperfect coupling of two fundamental Gaussian beams. The required precision in repositioning the samples at the instruments' test-port is discussed. Quasi-optical measurements using the de-embedding process for a WR-8 adjustable precision short at 125 GHz are presented. The de-embedding methodology may be extended to allow the determination of S-parameters of arbitrary two-port junctions. The measurement technique proposed should prove most useful above 325 GHz where there is a lack of measurement standards.