Recall that a ring homomorphism from a field 𝐿1 to a field 𝐿2 is a monomorphism, and that if 𝐿 : 𝐾 is an algebraic field extension and that τ : 𝐿 → 𝐿 is a homomorphism which is fixed on 𝐾, then τ is an automorphism of 𝐿 (Theorem 4.9).

In this chapter, we discuss the fundamental limit – the Shannon limit – on data rate for a communications link. We motivate this limit by providing a sketch of the derivation. To construct this sketch, we discuss the idea of the ratio of hypersphere volumes and how the radius of Gaussian vectors converges to a known radius as the dimensionality goes to infinity. We provide a second approach to think about the Shannon limit, or equivalently channel capacity, of a data link by considering mutual information and entropy. By using a simple line-of-sight channel, we discuss the resulting link theoretical capacity and an approach to estimating the practical limit on data rate. Finally, we provide an overview of source coding approaches.

Political entities are not immutable. They are subject to change. New states appear and old states disappear. Federations, mergers, dissolutions and secessions take place. International law has to incorporate such events into its general framework with the minimum of disruption and instability. Such changes have come to the fore since the end of the Second World War and the establishment of over 100 new, independent countries.

In the long march of mankind from the cave to the computer a central role has always been played by the idea of law – the idea that order is necessary and chaos inimical to a just and stable existence. Every society, whether it be large or small, powerful or weak, has created for itself a framework of principles within which to develop. What can be done, what cannot be done, permissible acts, forbidden acts, have all been spelt out within the consciousness of that community. Progress, with its inexplicable leaps and bounds, has always been based upon the group as men and women combine to pursue commonly accepted goals, whether these be hunting animals, growing food or simply making money.

Recent years have seen an appreciable growth in the level of understanding of the dangers facing the international environment and an extensive range of environmental problems is now the subject of serious international concern. These include atmospheric pollution, marine pollution, global warming and ozone depletion, the dangers of nuclear and other extra-hazardous substances and threatened wildlife species. Such problems have an international dimension in two obvious respects. First, pollution generated from within a particular state often has a serious impact upon other countries. The prime example would be acid rain, whereby chemicals emitted from factories rise in the atmosphere and react with water and sunlight to form acids. These are carried in the wind and fall eventually to earth in the rain, often thousands of miles away from the initial polluting event. Secondly, it is now apparent that environmental problems cannot be resolved by states acting individually. Accordingly, cooperation between the polluting and the polluted state is necessitated. However, the issue becomes more complicated in those cases where it is quite impossible to determine from which country a particular form of environmental pollution has emanated. This would be the case, for example, with ozone depletion. In other words, the international nature of pollution, both with regard to its creation and the damage caused, is now accepted as requiring an international response.

In this chapter, we discuss the ideas of signal acquisition and radio-to-radio synchronization in both time and frequency. We address the critical question: Is anyone out there? We discuss the uncertainty in time and frequency alignment between radios. We introduce and analyze the performance of multiple signal acquisition techniques: energy, cross-correlation, normalized inner product, and autocovariance detectors. We develop the maximum likelihood estimators for temporal and spectral synchronization for single-carrier approaches. Finally, we also introduce a temporal synchronization approach for an OFDM symbol.

In this chapter we leave the study of algebraic extensions, and consider problems concerning transcendence.