Hypercomplex Algebras and their application to the mathematical formulation of Quantum Theory

Quantum theory (QT), namely in terms of Schr\"odinger's 1926 wave functions in general requires complex numbers to be formulated. However, it soon turned out to even require some hypercomplex algebra. Incorporating Special Relativity leads to an equation (Dirac 1928) requiring pairwise anti-commuting coefficients, usually $4\times 4$ matrices. A unitary ring of square matrices is an associative hypercomplex algebra by definition. Since only the algebraic properties and relations of the elements matter, we replace the matrices by biquaternions. In this paper, we first consider the basics of non-relativistic and relativistic QT. Then we introduce general hypercomplex algebras and also show how a relativistic quantum equation like Dirac's one can be formulated using biquaternions. Subsequently, some algebraic preconditions for operations within hypercomplex algebras and their subalgebras will be examined. For our purpose equations akin to Schr\"odinger's should be able to be set up and solved. Functions of complementary variables should be Fourier transforms of each other. This should hold within a purely non-real subspace which must hence be a subalgebra. Furthermore, it is an ideal denoted by $\mathcal{J}$. It must be isomorphic to $\mathbb{C}$, hence containing an internal identity element. The bicomplex numbers will turn out to fulfil these preconditions, and therefore, the formalism of QT can be developed within its subalgebras. We also show that bicomplex numbers encourage the definition of several different kinds of conjugates. One of these treats the elements of $\mathcal{J}$ like the usual conjugate treats complex numbers. This defines a quantity what we call a modulus which, in contrast to the complex absolute square, remains non-real (but may be called `pseudo-real'). However, we do not conduct an explicit physical interpretation here but we leave this to future examinations.Comment: 21 pages (without titlepage), 14 without titlepage and appendi

Basel II and Fostering the Disclosure of Banks' Internal Credit Ratings

Advances in risk management capabilities should make it profitable for major banks to rely on internal credit ratings to calculate Basel II capital requirements (IRB approach). Firms and, more generally, market participants would benefit from the disclosure of these ratings. Banks, however, have no incentives to make their rating data publicly available. This paper proposes a regulatory framework to efficiently solve this incentive issue. It first shows that there are benefits in fostering the disclosure of internal ratings. In particular, it would reduce the cost of capital for both large and small borrowers while facilitating investor diversification. To be sure, internal rating disclosure would also have its costs. For example, market volatility may increase, whereas lender-borrower relationships may suffer. This paper argues that it is possible to design a regulatory framework under which the benefits of rating disclosure clearly outweigh its costs. Banks opting for the IRB approach would have to provide their internal ratings to one of several regional entities. The latter would consolidate the data collected and give each borrower a rating equal to the average of the ratings it gets from its lenders. The average rating would be disclosed to the public unless the borrower has opted for non-disclosure. Relying upon multiple regional entities may cause some uncertainty (a given firm may get diverging average ratings), but it would also reduce moral hazard effects and foster competition in the rating industr

Removal of metabolic heat from man working in a protective suit

A water cooled garment was constructed and used to study the characteristics of independent regional cooling of the body in contrast to the current practice of uniform cooling. The cooling pads in the garment were grouped to provide independent control of water inlet temperatures and flow rates to six regions: head, upper torso, lower torso, arms, thighs, and lower legs. Experiments with and without the cooling suit were conducted with five test subjects standing and walking on a treadmill on selected schedules. Steady state and, to a lesser extent, transient characteristics were obtained

A study of the thermal behavior of living biological tissue with application to thermal control of protective suits

Investigating biothermal model of living tissue for application to thermal control of protective clothin

On-Going Board Reforms: One Size Fits All and Regulatory Capture

Proponents of board reforms assume that corporate structures and director-specific provisions matter. This paper argues that reformers have set minimum standards, but failed to take into account various trade-offs and regulatory capture effects. It is thus suggested that the flexibility of existing provisions be increased and that this new approach be used to improve shareholder protection against board failures in general and failures of institutional investor boards in particula

Codetermination as a (Partial) Substitute for Mandatory Disclosure?

Employee board representation can serve information dissemination purposes. This comment argues that in Germany, which (still) has a bank-oriented financial system, transparency may be better served by using codetermination (under which up to half of the supervisory board is composed of employees) as an information channel, rather than by importing the mandatory disclosure requirements that are typical of Anglo-Saxon jurisdictions. There are two reasons for this. One is that the evidence about the efficiency of mandatory disclosure requirements is mixed. The other is that mandatory disclosure is an intrinsic component of market-oriented financial systems such as the US financial system. Transplanting a market-oriented component into a bank-oriented financial system brings the risk of inconsistencies, as it affects the complementarities that exist among the intrinsic components of a bank-oriented system. This comment thus concludes that before suggesting ways to transplant further mandatory disclosure requirements into the German financial system, or improve existing ones, one should consider how to use or improve the use of codetermination as an information dissemination channe