Functional Equations: History, Applications and Theory

Approach your problems from It isn't that they can't see the right end and begin with the solution. It is that they the answers. Then one day, can't see the problem. perhaps you will find the G.K. Chesterton. The Scandal of final question. Father Brown 'The Point of a Pin' . 'The Hermit Clad ~n Crane Feathers' in R. van Gulik's The Chinese Haze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathe­ matics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) ~n re­ gional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical pro­ gramming profit from homotopy theory; Lie algebras are rele­ vant to filtering; and prediction and electrical en~ineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existinf, classifi~ation schemes. They draw upon widely different sections of mathematics.

Contenu

Essays.- 1. On history, applications and theory of functional equations (Introduction).- 2. The esthetics and usefulness of functional equations.- 3. On the historical role of functional equations.- 4. Functional equation methods applied to economic problems: some examples.- 5. On multiple uses of some classical equations.- 6. On some unexpected theoretical and practical applications of functional equations.- 7. Functional equations on groupoids and related structures.- Papers.- 1. Some recent applications of functional equations.- 2. Groupoids with ?-kernels.- 3. The regularity of solutions of functional equations and hypoellipticity.- 4. An improvement of the Nevanlinna-Pólya theorem.- 5. On polynomials in additive and multiplicative functions.- 6. Truncations of distribution functions.- 7. Kurepa's functional equation on Gaussian semigroups.- 8. Some recent results on information measures, a new generalization and some 'real life' interpretations of the 'old' and new measures.- 9. On a characterization of the Shannon concentration measure.- 10. Closed invariant curves of a noncontinuously differentiate recurrence.- 11. Invariant curves as solutions of functional equations.- 12. The cycle theorem for flows.