The gradual disengagement of mathematics from logic. Beginning with a historical review of the development of mathematical thought, we have to consider successively (cf. 10, pp. 139-140):

(1) The observational period. For some familiar regularities of (outer or inner) experience of time and space, which, to any attainable degree of approximation, seemed invariable, absolute and sure invariability was postulated. These regularities were called axioms and were put into language. Thereupon extensive systems of properties were developed from the linguistic substratum of the axioms by means of reasoning, guided by experience but linguistically following and using the principles of classical logic. This logic was considered autonomous, and mathematics was considered more or less dependent on logic.

More than sixty years ago, Sylvester (13) proposed the following problem: Let n given points have the property that the straight line joining any two of them passes through a third point of the set. Must the n points all lie on one line?

An alleged solution (not by Sylvester) advanced at the time proved to be fallacious and the problem remained unsolved until about 1933 when it was revived by Erdös (7) and others.

Let us compare two properties of sets of non-negative integers: (1) the set a has property Γ, if there exists an effective procedure which when applied to any element of a different from its maximum (which α does not necessarily possess) yields the next larger element of α; (2) the set α has property Δ, if there exists an effective procedure which when applied to any element of α different from its minimum yields the next smaller element of α. It is readily seen that every recursive set has both properties.

A super-reflexive Banach space is defined to be a Banach space B which has the property that no non-reflexive Banach space is finitely representable in B. Super-reflexivity is invariant under isomorphisms; a Banach space B is super-reflexive if and only if B* is super-reflexive. This concept has many equivalent formulations, some of which have been studied previously.

Let $G$ be an algebraic group and let $X$ be a generically free $G$-variety. We show that $X$ can be transformed, by a sequence of blowups with smooth $G$-equivariant centers, into a $G$-variety ${{X}^{'}}$ with the following property: the stabilizer of every point of ${{X}^{'}}$ is isomorphic to a semidirect product $U\rtimes A$ of a unipotent group $U$ and a diagonalizable group $A$.

As an application of this result, we prove new lower bounds on essential dimensions of some algebraic groups. We also show that certain polynomials in one variable cannot be simplified by a Tschirnhaus transformation.

A useful general strategy for the construction of interesting families of vertex-transitive graphs is to begin with some family of transitive permutation groups and to construct for each group Γ in the family all graphs G whose vertex–set is the orbit V of Γ and for which Γ ≦ Aut (G), where Aut (G) denotes the automorphism group of G. For example, if we consider the family of cyclic groups 〈(0 1 … n – 1)〉 generated by cycles (0, 1 … n – 1) of length n, then the corresponding graphs are the n-vertex circulant graphs.

In this paper we consider transitive permutation groups of degree mn generated by a “rotation” ρ which is a product of m disjoint cycles of length n and by a “twisted translation” t; such that τρτ–l = ρα for some α.

In recent years Boolean Algebra has come to play a prominent role in the analysis and synthesis of switching circuits [1; 4]. One general synthesis problem in which this algebra has proved useful is the following. Let there be given n input leads each of which can assume one of two possible states. It is desired to construct a network with these n input leads and a single output lead also capable of assuming either of two states. Furthermore, the state of the output lead for each of the 2n states of the input leads is prescribed.

Matrices with real quaternion elements have been dealt with in earlier papers by Wolf (10) and Lee (4). In the former, an elementary divisor theory was developed for such matrices by using an isomorphism between n×n real quaternion matrices and 2n×2n matrices with complex elements. In the latter, further results were obtained (including, mainly, the transforming of a quaternion matrix into a triangular form under a unitary similarity transformation) by using a different isomorphism.

The infinitely divisible distributions on ${{\mathbb{R}}^{+}}$ of random variables ${{C}_{t}},\,{{S}_{t}}\,\text{and}\,{{T}_{t}}$ with Laplace transforms

$${{\left( \frac{1}{\cosh \sqrt{2\lambda }} \right)}^{t}},{{\left( \frac{\sqrt{2\lambda }}{\sinh \sqrt{2\lambda }} \right)}^{t}},\,\text{and}\,{{\left( \frac{\tanh \sqrt{2\lambda }}{\sqrt{2\lambda }} \right)}^{t}}$$

respectively are characterized for various $t\,>\,0$ in a number of different ways: by simple relations between their moments and cumulants, by corresponding relations between the distributions and their Lévy measures, by recursions for their Mellin transforms, and by differential equations satisfied by their Laplace transforms. Some of these results are interpreted probabilistically via known appearances of these distributions for $t\,=\,1\,\,\text{or}\,2$ in the description of the laws of various functionals of Brownian motion and Bessel processes, such as the heights and lengths of excursions of a one-dimensional Brownian motion. The distributions of ${{C}_{1}}\,\text{and}\,{{S}_{2}}$ are also known to appear in the Mellin representations of two important functions in analytic number theory, the Riemann zeta function and the Dirichlet $L$-function associated with the quadratic character modulo 4. Related families of infinitely divisible laws, including the gamma, logistic and generalized hyperbolic secant distributions, are derived from ${{S}_{t}}\,\text{and}\,{{C}_{t}}$ by operations such as Brownian subordination, exponential tilting, and weak limits, and characterized in various ways.

In this paper we use number-theoretic properties to classify ordinary graphs that are finite and have no isolated vertices. The classification depends on whether there is an assignment of real values, usually rational integer values, to the edges of the graph, such that the set of assigned values and the set of vertex sums of these values, summed at each vertex over all the edges incident to the vertex, will be a pair of sets with prescribed properties. Then we seek corresponding graph-theoretic properties.

It is possible to describe the problem in terms of a symmetric matrix having specified properties for its row sums, but in this paper we make no use of this interpretation; however, see (3).

In [6], by means of convex functions Φ :R→R, Hardy, Littlewood and Pólya proved a theorem characterizing the strong spectral order relation for any two measurable functions which are defined on a finite interval and which they implicitly assumed to be essentially bounded (cf. [6, the approximation lemma on p. 150 and Theorem 9 on p. 151 of their paper]; see also L. Mirsky [10, pp. 328-329] and H. D. Brunk [1,Theorem A, p. 820]).

Section 1 of this paper contains all the work which deals exclusively with generalizations of Hadamard matrices. The non-existence theorem proven here (Theorem 1.10) generalizes a theorem of Hall and Paige [15] on the non-existence of complete mappings in certain groups.

In Sections 2 and 3, we consider the duals of (Hanani) transversal designs; these dual structures, which we call (s, r, µ)-nets, are a natural generalization of the much studied (Bruck) nets which in turn are equivalent to sets of mutually orthogonal Latin squares. An (s, r, µ)-net is a set of s2µ points together with r parallel classes of blocks. Each class consists of s blocks of equal cardinality. Two non-parallel blocks meet in precisely µ points. It has been proven that r is always less than or equal to (s2µ – l) / (s – 1).

1. General remarks. The heuristic strength of the general principle of relativity lies in the fact that it considerably reduces the number of imaginable sets of field equations; the field equations must be covariant with respect to all continuous transformations of the four coordinates. But the problem becomes mathematically well-defined only if we have postulated the dependent variables which are to occur in the equations, and their transformation properties (field-structure). But even if we have chosen the field-structure (in such a way that there exist sufficiently strong relativistic field-equations), the principle of relativity does not determine the field-equations uniquely. The principle of “logical simplicity” must be added (which, however, cannot be formulated in a non-arbitrary way). Only then do we have a definite theory whose physical validity can be tested a posteriori.

Let Pn(α,β) be the Jacobi polynomial of degree n, order (α,β), α,β > – 1, defined by

[9, p. 67], and let Rn(α,β)(x) = Pn(α,β)(x)/Pn(αβ)(1). Then for n ≧ m,

where

Since Rn(α, β)(l) = 1, it follows that

(1)

It is known that if (the ultraspherical case) or if α = β + 1, then α = β + 1, then g(k, n, m) ≧ 0.

We begin with some definitions and a formulation of the problem treated in subsequent sections. Also included in this section is a brief indication of some of the known results dealing with the n-omino enumeration problem. Some of what follows together with more details may be found in [3] or [4].

We present “reiteration theorems” with limiting values $\theta =0$ and $\theta =1$ for a real interpolation method involving broken-logarithmic functors. The resulting spaces lie outside of the original scale of spaces and to describe them new interpolation functors are introduced. For an ordered couple of (quasi-) Banach spaces similar results were presented without proofs by Doktorskii in $[\text{D}]$.

Our method using CH is a blend of two earlier constructions (Hajnal-Juhász [2] and Ostaszewski [4]) of hereditarily separable (HS), regular, non-Lindelöf, first countable spaces. [4] produces a much better space than ours in § 1 ; it has all of our properties except that it is not realcompact (which is probably more interesting), and it is countably compact as well; however, the construction works only under ◇, which implies the continuum hypothesis (CH) but is not equivalent to it.

This paper makes a unified development of what the authors know about the existence of nearest points to closed subsets of (real) Banach spaces. Our work is made simpler by the methodical use of subderivatives. The results of Section 3 and Section 7 in particular are, to the best of our knowledge, new. In Section 5 and Section 6 we provide refined proofs of the Lau-Konjagin nearest point characterizations of reflexive Kadec spaces (Theorem 5.11, Theorem 6.6) and give a substantial extension (Theorem 5.12). The main open question is: are nearest points dense in the boundary of every closed subset of every reflexive space? Indeed can a proper closed set in a reflexive space fail to have any nearest points? In Section 7 we show that there are some non-Kadec reflexive spaces in which nearest points are dense in the boundary of every closed set.

In the preceding paper Bose, Shrikhande, and Parker give their important discovery of the disproof of Euler's conjecture on Latin squares. In this paper we show that their results can be strengthened to imply that N(n), the maximal number of pairwise orthogonal Latin squares of order n, tends to infinity with n. In fact there exists a positive constant c, such that N(n) > nc for all sufficiently large n.

Our proof involves no new combinatorial insights, but is based entirely on a number-theoretical investigation of the following inequality due to Bose and Shrikhande.

A Steiner triple system (briefly STS) is a pair where S is a finite set and is a collection of 3-subsets of S (called triples) such that every pair of distinct elements of S belongs to exactly one triple of . The number |S| is called the order of . It is well-known that there is an STS of order if and only if or 3 (mod 6). Therefore in saying that a certain property concerning STS is true for all it is understood that or 3 (mod 6). An STS of order v will sometimes be denoted by .