Abstract: The Introduction explains the issues raised by constructive logic for different traditions; principally, for mainstream constructive mathematics treated here, and for the foundational tradition, the subject of Part II. The material is arranged under the following headings. 1. Flashy metatheorems and some neglected implications . For example, classical proofs of Π 0 2 theorems are essentially no harder to unwind than corresponding constructive, more precisely, intuitionistic proofs. In short, intuitionistic restrictions tend to be too weak in practice, incidentally despite the (little known) elegance of their metatheory documented in a digression. 2. Weakness of the constructive meaning of Π 0 3 → Π 0 3 . This is illustrated by reference to various elementary implications of this form between familiar finiteness theorems in arithmetic. A proviso is formulated which corrects that weakness, and is often satisfied in practice. 3. Metamathematics of classical systems : modern variants of Hubert's program without detour via constructive logic. For example, some recalcitrant proofs of Π 0 2 theorems have in fact been unwound by suitable functional interpretations such as the n.c.i.; cut elimination is generally less efficient. 4. Applying the n.c.i.: reassurances (about additional skills needed for successful applications, beyond knowing the metatheory of the n.c.i.). Attention is drawn to proof strategies used in ordinary mathematics, of comparable generality to logical metatheorems, where similar additional skills are needed, and have proved to be available. Much the same applies to successful applications of model theory which, contrary to superficial impressions, are shown to involve operations on proof's, and not only on the theorem proved. 5. Algebraization: reminders and surprises (principal section). By reference mainly to recent literature, the potential of algebraization and a couple of other general mathematical strategies in the sense of 4. above is documented, and contrasted with logical contributions to the constructive tradition in mathematics. Remarkably many significant and general points are illustrated by work on sums of squares and by the Appendix . The latter supplements section 2. by reference to the most sophisticated implication between finiteness theorems in the number-theoretic literature. Particular attention is given to realizing the 'proviso' mentioned in 2., or, equivalently, to making a proper choice between various meanings of the popular phrase: from a counterexample to the conclusion we get one to the premise. - The Appendix can be read independently of the main text.

Abstract: In many branches of pure mathematics it can be surprisingly hard to recognize when a question has, in fact, been answered. A clearcut proof of a theorem or the discovery of a counterexample leaves no doubt in the reader's mind that a solution has been found. But when an "explicit solution" to a problem is given, it may happen that more work is needed to evaluate that " solution," in a particular case, than exhaustively to examine all of the possibilities directly from the original formulation of the problem. In such a situation, other things being equal, we may justifiably question whether the problem has in fact been solved. Examples of this sort can turn up anywhere, but here we will concentrate on problems in combinatorial mathematics, specifically those of the type "how manyare there?" Such enumeration problems lie at the heart of the subject, and it is important to be able to recognize solutions when they appear. The point, of course, is that sometimes the "answer" is presented as a formula that is so messy and long, and so full of factorials and sign alternations and whatnot, that we may feel that the disease was preferable to the cure. An answer to such an enumeration question may be given by means of a generating function, a recurrence relation, or by an explicit formula. Each of these is, in essence, just an algorithm for the computation of the counting sequence that is to be determined. How do we judge the usefulness of such answers? Obviously we might be able to do many things with the answer, such as to make asymptotic estimates, to discover congruence relations, to delight in its elegance, and so forth. We're going to restrict attention here to the appraisal of solutions from the point of view of how easily they allow us to calculate the number of objects in the set that is being studied. The quality of such -formulas should therefore be judged by the usual combination of esthetic and quantitative benchmarks that are used on algorithms. In particular, the quantitative criterion is the computational complexity: the amount of work required to get an answer. We suggest here that the same criterion should be applied to enumeration formulas. We will see that a corollary of this attitude is that our decision as to what constitutes an answer may be time-dependent: as faster algorithms for listing the objects become available, a proposed formula for counting the objects will have to be comparably faster to evaluate. For concreteness, suppose that for each integer n > 0 there is a set Sn that we want to count. Let f(n) = I Sn I (the cardinality of Sn), for each n. Suppose further that a certain formula has been found, say

Abstract: As counterexamples to a conjecture of Randic, pairs of nonisomorphic trees with the same collections of distance degree sequences are presented.

Abstract: In this paper, we give a counterexample to the "hard" polyhedra game, proposed by Hironaka some four years ago. Hironaka has proved that an affirmative solution of this game would imply the local uniformization theorem for an algebraic variety over an algebraically closed field of any characteristic. Although the answer turns out to be negative, we hope that this example may be useful for the problem of resolution of singularities: either for constructing a counterexample to the original problem or for modifying the rules of the game so that there will exist a winning strategy for the first player. The original, simpler version of the game, proposed by Hironaka in 1970 (see [2]) does have such a winning strategy (see [4] for a proof) which gives local uniformization in some special cases. I am very grateful to Professor Hironaka for his patient advice and discussions, as well as his warm hospitality during my stay in Kyoto.

Abstract: In a recent paper Lovasz, Neumann-Lara, and Plummer studied Mengerian theorems for paths of bounded length. Their study led to a conjecture concerning the extent to which Menger's theorem can fail when restricted to paths of bounded length. In this paper we offer counterexamples to this conjecture.

Abstract: In this paper we report the results of experimental studies of zero-level, one-level, and two-level search rearrangement backtracking. We establish upper and lower limits for the size problem for which one-level backtracking is preferred over zero-level and two-level methods, thereby showing that the zero-level method is best for very small problems. The one-level method is best for moderate size problems, and the two-level method is best for extremely large problems. Together with our theoretical asymptotic formulas, these measurements provide a useful guide for selecting the best search rearrangement method for a particular problem.

Abstract: A counterexample is given to show the partial correctness of a matrix decomposition algorithm proposed by Ramanujam in 1973. It is pointed out that the edge coloring approach results in the most efficient algorithms for the decomposition problem.

Abstract: : The authors give an example of a curved exponential where the maximum likelihood estimate is not third order efficient either in the sense of Fisher-Rao or Rao.

Abstract: Local perturbations of the dynamics of infinite quantum systems are considered. It is known that, if the Mo/ller morphisms associated to the dynamics and its perturbation are invertible, the perturbed evolution is isomorphic to the unperturbed one, and thereby shares its ergodic properties. It was claimed by V. Ya. Golodets [Theor. Math. Phys. 23, 525 (1975)] that the above condition holds whenever the observable algebra is asymptotically abelian for the unperturbed evolution, and the perturbed evolution has a KMS state. The present paper contains a counterexample to this statement, and a construction of a spatial representation of the Mo/ller morphisms.

Abstract: For every positive integer c, we construct a pair Gc, Hc of infinite, nonisomorphic graphs both having exactly c components such that Gc and Hc are hypomorphic, i.e., Gc and Hc have the same families of vertex-deleted subgraphs. This solves a problem of Bondy and Hemminger. Furthermore, the pair G1, H1 is an example for a pair of non-isomorphic, hypomorphic, connected graphs also having connected complements—a property not shared by any of the previously known counterexamples to the reconstruction conjecture for infinite graphs.

Abstract: Let L denote the class of functions g(z) = z + bo + bIz' + analytic and univalent in I z I> I except for a simple pole at oo. A well-known conjecture asserts that I bn I 3 and n = 4. In ?2, we generalize a variational method of Goluzin and develop second-variational techniques. This enables us in ?3 to construct explicit counterexamples to the conjecture for all n > 4. In fact, the conjectured extremal function does not even provide a local maximum for Re{bn}, n > 4.

Abstract: In the infinite dimensional case it is known from a Dineen-Nachbin counterexample that the natural generalization of the Paley-Wiener-Schwartz theorem is no longer valid: The image through the Fourier transform F of the space ℰ′(E) is made of entire functions on E′ that, besides the usual inequality, satisfy a more technical condition. We define and study here a dense subspace, denoted by E(E), of ℰ(E) with a proper topology such that F (E′(E)) is made of the entire functions on E′ that satisfy only the classical inequality. For this reason this space E(E) is suited for the study of convolution and partial differential equations in spaces of C∞ functions on locally convex spaces.

Abstract: Miller and Farr's algorithm for the index of redundancy is shown to be incorrect by means of a counterexample. It is also argued that their interpretation of the proportion of the total variance in one multivariate response explained by another multivariate response with respect to a given component of the first response is misleading.

Abstract: In the letter we state a theorem modifying the Shanks conjecture on the stability of the planar least-square inverse of a 2-D polynomial based on our observation of various counterexamples.

Abstract: In this paper an example is presented showing that Rosen's decomposition method can be convergent to some nonoptimal solution. In this an example is presented showing that Rosen's decomposition method can be convergent to some nonoptimal solution.

Abstract: It is erroneously stated, in the note added in proof on p. 514, that the support of a Radon measure on a quasi-F-space is Stonian. Frederick K. Dashiell gives a counterexample, Example 3.8 on p. 412 of his paper Nonweakly compact operators from order-Cauchy complete C(S) lattices, with application to Baire classes, Trans. Amer. Math. Soc. 266 (1981), 397-413. This counterexample measure is in fact nonatomic. It is not known whether Liapounoff's convexity theorem is valid for quasi-F-algebras. It is not known whether it is necessary for the validity of Liapounoff's convexity theorem on a Boolean algebra J1 that every nonatomic Radon measure on the Stone space Xs must have Stonian support. A characterization of those compact Hausdorff spaces X (or just the totally disconnected ones) so that every nonatomic measure has Stonian support is not known.

Abstract: It is the purpose of this note to investigate the relationships among four concepts relating to symmetry in graphs: point-symmetry, line-symmetry, arc-symmetry and reversibility; especially which of the first three properties do not imply reversibility. Holt has found a counterexample to one such question and we construct a counterexample to another using a Cayley graph. Both examples are nowhere reversible, a property which is stronger than nonreversibility.

Abstract: Actually, one can get u to be the orthogonal measure corresponding to the vector state co defined by £ and a maximal abelian subalgebra of the commutant £&' of j/. (Such a measure is maximal with respect to the Choquet ordering also in the nonseparable case, see [7], for a short proof). It may be surprising that no separability condition on jaf is assumed. The example given by J. L. Taylor in [15] shows that every state cp in the support of such a measure may fail to be a pure state, contradicting an assertion in an earlier paper of M. Tomita. Therefore in our theorem n^ will not be the GNS representation n^ corresponding to