The notion of quantum field remains at this time still rather elusive from a rigorous standpoint. In conventional physical theory such a field is defined in essentially the same way as in the original work of Heisenberg and Pauli (1) by a function ϕ(x, y, z, t) on space-time whose values are operators. It was recognized very early, however, by Bohr and Rosenfeld (2) that, even in the case of a free field, no physical meaning could be attached to the values of the field at a particular point—only the suitably smoothed averages over finite space-time regions had such a meaning. This physical result has a mathematical counterpart in the impossibility of formulating ϕ(x, y, z, t) as a bona fide operator for even the simplest fields (in any fashion satisfying the most elementary non-trivial theoretical desiderata), while on the other hand for suitable functions f, the integral ∫ϕ(x, y, zy t)f(x, y, z, t)dxdydzdt could be so formulated.

The classical result of Plancherel for Fourier cosine transforms of functions f(x) of the class L2(0, ∞) states that (see (7) for references)

converges in mean square to a function g(x) which also belongs to L2(0, ∞), and furthermore

Some years ago in a series of papers (1; 2; 3) on summation formulae I showed that a similar symmetrical theory for narrower classes of functions and ordinary convergence of the integrals can also be developed.

In 1949 I published in the Acta Mathematica (vol. 81) a rather long paper: “L'intégrale de Riemann-Liouville et le problème de Cauchy.” This work will be quoted in the sequel as Acta paper. Only minor local references to this paper will be made here, and knowledge of it is not required for the reading of the present article. The notations used here are slightly different from those used in my former paper.

In the Acta paper I introduce multiple integrals and of the Riemann- Liouville type depending on a parameter α and converging for sufficiently large values of α. I give the solution of the Cauchy problem for the wave equation in a unique formula, the same for space-time of odd or even dimensions, implying an analytic continuation with respect to the parameter α.

This paper is an extension of the ideas discussed in (3, §§ 14-16); the extension consisting of the use of the third and fourth symmetric Riemann derivative instead of the Schwarz or second symmetric Riemann derivative.

The J2integral, due to James (1), is defined in (3) as follows. Let f(x) be measurable on [a, b] and finite at each point; if there exists a continuous function F(x) such that D2F = f everywhere on (a, b),

then

1.1

The definition is unique since if F(x) and G(x) are continuous and D2F =D2G everywhere then

This note can be regarded as an addendum to the paper (4). On the complex Hilbert space of vectors x = (x1, x2, … ,) a matrix A is said to be bounded if there exists a constant M such that ||Ax|| ≦ M||x|| whenever ||x||2 = Σ|xk|2 < ∞ ; the least such M is denoted by ||A||. Only bounded matrices A and vectors x satisfying ||x|| < ∞ will be considered in the sequel. The spectrum of A, denoted by sp(A), is the set of values for which the resolvent R(λ) = (A — λI)-1 fails to be bounded. The notation A ≥ 0 or A > 0, where A = (aij), means that, for all i and j , aij ≥ 0 or aij 0 respectively.

The matrix spaces under consideration are the four main types of irreducible bounded symmetric domains given by Cartan (5). Let z = (zjk) be a matrix of complex numbers, z' its transpose, z* its conjugate transpose and I = I(n) the identity matrix of order n. Then the first three types are defined by

(1)

where z is an n by m matrix (n ≤ m), a symmetric or a skew-symmetric matrix of order n (16). The fourth type is the set of complex spheres satisfying

(2)

where z is an n by 1 matrix. It is known that each of these domains possesses a distinguished boundary B which in the first three cases is given by

(3)

(In the case of skew symmetric matrices the distinguished boundary is given by (2) only if n is even.)

In this paper the authors prove Theorem 1 on maps of partially ordered sets into themselves, and derive some fixed point theorems as corollaries.

Here, for any partially ordered set P, and any mapping f : P → P and any point a ∈ P, a well ordered subset W(a) ⊂ P is constructed. Except when W(a) has a last element ε greater than or not comparable to f(ε), W(a), although constructed differently, is identical with the set A of Bourbaki (3) determined by a, f , and P1: {x|x ∈ P, x ≤ f(x)}.

Theorem 1 and the fixed point Theorems 2 and 4, as well as Corollaries 2 and 4, are believed to be new.

Corollaries 1 and 3 are respectively the well-known theorems given in (1, p. 54, Theorem 8, and Example 4).

The fixed point Theorem 3 is that of (1, p. 44, Example 4); and has as a corollary the theorem given in (2) and (3).

The proofs are based entirely on the definitions of partially and well ordered sets and, except in the cases of Theorem 4 and Corollary 4, make no use of any form of the axiom of choice.

Shapiro and Forman have presented in (4) an abstract formulation of prime number theorems which includes the various prime number theorems; for primes in arithmetic progressions, for prime ideals in ideal classes etc. The methods of proofs are “elementary” and follow closely Shapiro's proof for the primes in arithmetic progression (for reference see bibliography in (4)).

The author has followed in (1) some ideas of Yamamoto (5) on arithmetic linear transformations to introduce a symbolic calculus in dealing with arithmetic functions. This calculus proved to be very useful in unifying many of the “elementary” proofs in the behaviour of arithmetic functions. In (6) Yamamoto has extended his theory to ideals in algebraic number fields, and with this extension the symbolic calculus of (1) can be extended to cover the abstract case of prime number theorem in countable free abelian groups as discussed in (4).

In this paper, we define and begin the study of an extensive family of simple n-person games based in a natural way on block designs, and hitherto for the most part unexplored except for the finite projective games (13). They should serve at least as a proving ground for conjectures about simple games. It is shown that many of these games are not strong and that many do not possess main simple solutions. In other cases, it is shown that they have no equitable main simple solution, that is, one in which the main simple vector has equal components. On the other hand, the even-dimensional finite projective games PG(2s, pn) with s > 1 possess equitable main simple solutions, although they are not strong either. These results are obtained by means of the study of the possible blocking coalitions. Interpretations in terms of graph theory, network flows, and linear programming are discusssed, as well as k-stability, automorphism groups, and some unsolved problems.

In the classification of almost alternative algebras relative quasiequivalence by Albert two new classes of algebras of type (γ, δ) were introduced, namely those of type (1, 1) and ( —1, 0) (1, equations (34), (35), and Theorem 6). Since rings of type (1, 1) and ( — 1, 0) are anti-isomorphic it suffices to consider those of type (1, 1). They may be defined as rings satisfying

1

and

2

for all elements x, y, and z of the ring, where the associator (a, b, c) is given by (a, b, c) = (ab)c — a (bc).

Actually the identity

2'

together with (1) implies (2) whenever the characteristic of the ring is different from 2. This may readily be verified by linearizing (2’). Consequently we may use (1) and (2’) as the defining relations for a ring of type (1, 1).

The two-dimensional unitary space, U2, is a complex vector space of points (x, y) = (x1 + ix2, y1 + iy2), for which the distance between (x, y) and (x', y') is defined by . A unitary transformation is a linear transformation which preserves distance. A line is the set of points (x, y) satisfying some complex equation ax + by = c. A unitary transformation is a (unitary) reflection if it is of finite period n > 1 and leaves a line pointwise invariant. Thus à unitary matrix represents a reflection if its two characteristic roots are 1 and a complex nth root (n > 1) of 1.

If m, n are integers, “m = n” will mean “m = n (mod 2 ).“ The cardinal number of a set A will be denoted by |A|. The set whose elements are a1 a2, … , an will be denoted by {a1, a2, … , an}. The empty set will be denoted by Ø. If A, B, C are sets, A — B will denote the set of those elements of A which do not belong to B, and A — B — C will denote (A — B)— C. The expression Σε∈Af(ε) will be denoted by f. A. The statements “f = g on A,” “f ≡ g on A” will mean that f(ε) = g(ε) or f(ε) ≡ g(ε) respectively for every ε ∈ A.

The most important notion in topology is that of a homeomorphism f: X → Y from a topological space X onto a topological space Y. If a homeomorphism f: X → Y exists, then the topological spaces X and F are said to be homeomorphic (or topologically equivalent), in symbols,

X ≡ Y.

The relation ≡ among topological spaces is obviously reflexive, symmetric, and transitive; hence it is an equivalence relation. For an arbitrary family F of topological spaces, this equivalence relation ≡ divides /Mnto disjoint equivalence classes called the topology types of the family F. Then, the main problem in topology is the topological classification problem formulated as follows.

The topological classification problem: Given a family F of topological spaces, find an effective enumeration of the topology types of the family F and exhibit a representative space in each of these topology types.

Let ∧ be a lattice in Euclidean n-space, that is, ∧ is a set of points ε1a1 + … + εnan where a1… , an are linearly independent vectors and the ε run over all integers. Let μ denote the Lebesgue measure. A closed convex set F is called a fundamental region for ∧ if the sets F + x (x ∈∧) cover the whole space without overlapping; that is, if F0 is the interior of F, and 0 ≠ x ∈ ∧, then F0 ∩ (F0 + x) = ϕ.

This paper deals with finite sequences of integers. Typical of the problems we shall treat is the determination of the number of sequences of length n, consisting of the integers 1, 2, … , m, which have a longest increasing subsequence of length α. Throughout the first part of the paper we will deal only with sequences in which no numbers are repeated. In the second part we will extend the results to include the possibility of repetition. Our results will be stated in terms of standard Young tableaux.

In this paper we consider again the group-theoretic configuration studied in (1) and (2). Let G be an additive group (not necessarily abelian), let M be a system of operators for G, and let ϕ be a family of admissible subgroups which form a complete lattice relative to intersection and compositum. Under these circumstances we call G an M — ϕ group. In (1) we studied the normal chains for an M — ϕ group and the relation between certain normal chains. In (2) we considered the possibility of representing an M — ϕ group as the direct sum of certain of its subgroups, and proved that with suitable restrictions on the M — ϕ group the analogue of the following theorem for finite groups holds: A group is the direct product of its Sylow subgroups if and only if it is nilpotent. Here we show that under suitable hypotheses (hypotheses (I), (II), and (III) stated at the beginning of §3) it is possible to generalize to M — ϕ groups many of the Sylow theorems of classical group theorem.

Let L be a Lie algebra over a field of characteristic 0 and let D(L) be the derivation algebra of L, that is, the Lie algebra of all derivations of L. Then it is natural to ask the following questions: What is the structure of D(L)? What are the relations of the structures of D(L) and L? It is the main purpose of this paper to present some results on D(L) as the answers to these questions in simple cases.

Concerning the questions above, we give an example showing that there exist non-isomorphic Lie algebras whose derivation algebras are isomorphic (Example 3 in § 5). Therefore the structure of a Lie algebra L is not completely determined by the structure of D(L). However, there is still some intimate connection between the structure of D(L) and that of L.

We will consider the following enumeration problem. Let A and B be finite sets with α and β elements in each set respectively. Let n be some positive integer such that n ≦ αβ. A subset S of the product set A × B of exactly n distinct ordered pairs (ai, bj) is said to be admissible if given any a ∈ A and b ∈ B, there exist elements (ai, bj) and (ak, bl) (they may be the same) in S such that ai = a and bl = b. We shall find here a generating function for the number N(α, β n) of distinct admissible subsets of A × B and from this generating function, an explicit expression for N(α, β n). In obtaining this result, the idea of a cut probability is used. This approach in a problem of enumeration may be of interest.

The basic problem of linear programming is to minimize (or maximize) a linear function of a finite number of variables constrained by a finite number of linear inequalities. From a mathematical point of view the subject may be regarded as being divided into two areas. One is primarily analytical and deals with certain questions of duality and consistency. The other is algorithmic and is concerned with computational questions and methods.

A number of combinatorial problems may be regarded as particular instances of the following rather general situation. Given a set X composed of n elements x1, x2, ..., xn, and m subsets X1 X2, … , Xm of X, find a minimal system of representatives for X1, X2, … , Xm. That is, single out a subset X* of X such that Xi ∩ X* is non-empty for i = 1,2, … ,m, and no subset of X containing fewer elements than X* has this property. To illustrate, each of the following can be thought of in these terms.