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Deriving abelian KdV and NLS hierarchies, we describe non-abelian symmetries and “pre-Lax” elementary approach to Lax pairs. Discrete symmetries of spectral problems are considered in Sect. 4.2. Here we prove Darboux classical theorem and discuss a modern theory of dressing chains.

Recent work in constructive mathematics show that Hilbert's program works for a large part of abstract algebra. Furthermore the arguments we get are not only elementary but also mathematically simpler. We present an example where the simplification was significant enough to suggest an improved version of a classical theorem. For this we use a general method to transform some logically complex first-order formulae in a geometrical form which may be interesting in itself.

In [1], partial words were defined to be partial mappings of a set {1, ..., n} to a finite alphabet. We continue the research of periodic partial words started in that paper. The main goal is to clarify the interaction of different periods of a word. In particular, we exhibit some propeties of periodic partial words, which are quite similar to classical Theorem of Fine and Wilf.

The objective of this chapter is to prove the fundamental theorems of Eilen-berg [75] and Debreu ([65], [67]). Of the several approaches that may be used to prove these classical theorems, we have chosen the one based on Debreu1 s Open Gap Lemma.

We propose an approach to approximate classical reasoning via well-known theorem-proving techniques. Unlike other approaches, our approach takes into acconnt the interplay of knowledge bases and queries and thus allows for query-sensitive approximate reasoning. We demonstrate that our approach deals extremely well with the examples found in the literature. This reveals that conventional theorem-proving techniques can account for approximate reasoning.

We prove several lattice theoretical fixpoint theorems based on the classical theorem of Knaster and Tarski. These theorems give sufficient conditions for a system of generally nonmonotonic functions on a complete lattice to define a unique minimal fixpoint. The primary objective of this paper is to develop a domain theoretic framework to study the semantics of logic programs as well as various rule-based systems where the rules define generally nonmonotonic functions on lattices.

The doctrine stated above (in 1927) became, in the late forties and the fifties, the foundation of a fairly extensive body of mathematics, in which one replaces the vague intuitive concept of “law” or “rule” by the precisely-defined concept of “recursive function”. This chapter presents the main results. Of course, every positive result of constructive mathematics translates into a corresponding theorem of recursive mathematics. A large number of such translations of positive results has been spelled out explicitly in Aberth [1980]. There also exists a variety of strange and delightful, sometimes counter-intuitive, negative results in recursive mathematics. Negative, that is, in the sense that they show that some classical theorem fails in recursive mathematics; although many of these results are positive in the sense that they actually exhibit a counterexample to the recursive version of the theorem. One value of such results is the following: if we show that a theorem fails in recursive mathematics, we can stop wasting our time trying to prove it in constructive mathematics — at least unless we are willing to use some axiom which contradicts Church’s thesis. Just to give one example: a number of mathematicians tried to prove that if ƒ: [0,1]→R is uniformly continuous and everywhere positive, then inf ƒ(x) is positive too — a problem which was mentioned in Bishop [1967] and Bridges [1979]. But this fails in recursive mathematics, as we shall see below.

In this note we wish to show how the idea of strong balancedness clarifies the properties of separable abelian groups and helps to (i) reprove some of the classical theorems, (ii) extend well-known results and (iii) construct new separable abelian groups. Specifically, we show that strongly balanced subgroups of separable abelian groups are again separable. This not only extends but also gives a simpler proof of the classical theorem of L. Fuchs on summands of separable groups. Thus separable abelian groups may be characterized as the strongly balanced images of completely decomposable groups. Our point of view enables us to formulate our results for the class of m-separable groups, where m is an arbitrary cardinal.

In this chapter we shall prove the main theorems of the L2-harmonic analysis on G//K. We shall obtain an explicit Plancherel formula for the Harish-Chandra transform f→ℋf as well as an inversion formula that expresses f in terms of ℋ f. We shall also obtain an analogue of the classical theorem of Paley and Wiener, characterizing the Harish-Chandra transforms of elements of Cc∞ (G//K).

The classical theorem of Riemann asserts that if the two-way infinite sequence of numbers c n tends to 0 as n→± ∞, then the series $$ \sum\limits_{{ - \infty }}^{{ + \infty }} {{c_{n}}{e^{{inx}}}} $$ (S) integrated termwise twice has sum $$F(x) = \tfrac{1}{2}{c_0}{x^2} - \sum\limits_{ - \infty }^{ + \infty } {'{c_n}{n^{ - 2}}{e^{inx}}} $$ which is smooth at each x, that is $$ \frac{{F\left( {x + h} \right) + F\left( {x - h} \right) - 2F\left(x \right)}}{h} \to 0\left( {h \to 0} \right) $$ (1.1) (1.1) As a matter of fact, the last relation holds uniformly in x.