We prove the existence of a solution u(.,.;α,β) of the Darboux problem uxy∈F(x, y, u), u(x,0) = α(x), u(0, y) = β(y), which is continuous with respect to (α,β). We assume that F is Lipschitzean with respect to u but not necessarily convex valued.

The three families of classical groups of linear transformations (complex, orthogonal, symplectic) give rise to the three great branches of differential geometry (complex analytic, Riemannian and symplectic). Complex analytic geometry derives most of its interest from complex algebraic geometry, while symplectic geometry provides the general framework for Hamiltonian mechanics.

These three classical groups “intersect” in the unitary group and the three branches of differential geometry correspondingly “intersect” in Kähler geometry, which includes the study of algebraic varieties in projective space. This is the basic reason why Hodge was successful in applying Riemannian methods to algebraic geometry in his theory of harmonic forms.

In various problems in harmonic analysis (4, 5, 6), I have required an estimate for the integral for the hypergeometric function

in the case where a>c−b>0, b>0, and 0 ≦ x < 1 (the integral is then unbounded as x→l−). Although there are innumerable identities for hypergeometric functions, few inequalities for these functions seem to be known, and in estimating the integral (1) I employed ad hoc arguments that made no use of the theory of hypergeometric functions. The estimates obtained were adequate for my purposes, but were far from sharp, and the object of this note is to show that, by assembling a few known facts concerning hypergeometric functions, we can obtain sharp inequalities for the integral (1). We also give (in §4) several related inequalities that can be obtained by transforming the integral.

The theory of *-representations of Banach *-algebras on Hilbert space is one of the most useful and most successful parts of the theory of Banach algebras. However, there are only scattered results concerning the representations of general Banach algebras on Banach spaces. It may be that a comprehensive representation theory is impossible. Nevertheless, for some special algebras interesting and worthwhile results can be proved. This is true for (Y), the algebra of all bounded operators on a Banach space Y, and for (Y), the subalgebra of (Y) consisting of operators with finite dimensional range. The representations of (Y) are studied in a recent paper by H. Porta and E. Berkson (6), and in another recent paper (8), P. Chernoff determines the structure of the representations of (Y) (and also of some more general algebras of operators). In both these papers, (Y), which is the socle of the algebras under consideration, plays an important role in the theory. This suggests the possibility that a more general representation theory can be formulated in the case of a normed algebra with a nontrivial socle. This we attempt to do in this paper.

This paper included an analytical proof of the following theorem:

If five spheres 1, 2, 3, 4, 5, pass through the same point, and if the four points in which the four spheres 1, 2, 3, 4, intersect in sets of three be coplanar; and if the same be true for the sets 1, 2, 3, 5; 1, 2, 4, 5; 1, 3, 4, 5; it will also be true for the remaining set 2, 3, 4, 5.

The same theorem is true for similar and similarly situated quadric surfaces.

The theory of graphical integration is founded on the graphical integration of parabolic arcs. The details and the different technical applications of these problems are developed in the fundamental work of J. Massau; we find there very simple pure constructive methods for the case of parabolas till the 3rd order.

1. The notation for a rectangular array can be extended so as to admit of arrays in which the number of rows exceeds the number of columns.

Let

denote the aggregate of all determinants of the mth order which can be formed from the rectangular array of pq elements by deleting p – m columns and q – m rows.

Let H be a finite or infinite dimensional Lie algebra. Barnes [2] and Towers [5] considered the case when H is a finite-dimensional Lie algebra over an arbitrary field, and all maximal subalgebras of H have codimension 1. Barnes, using the cohomology theory of Lie algebras, investigated solvable algebras, and Towers extended Barnes's results to include all Lie algebras. In [4] complex finite-dimensional Lie algebras were considered for the case when all the maximal subalgebras of H are not necessarily of codimension 1 but when

where S(H) is the set of all Lie subalgebras in H of codimension 1. Amayo [1]investigated the finite-dimensional Lie algebras with core-free subalgebras of codimension 1 and also obtained some interesting results about the structure of infinite dimensional Lie algebras with subalgebras of codimension 1.

A solution of a triad of integral equations involving Bessel functions is given. This, like earlier ones, is in the form of a pair of Fredholm integral equations, which may be solved by iteration in certain cases. In spite of a slightly more general formulation of the problem, the kernels of these equations are simpler than those given in earlier solutions. Certain extensions are considered and a formal solution given. Application is made to the problem of incompressible inviscid flow normal to an annular disc, and to the flow due to the slow rotation of such a disc in a viscous fluid.

According to Mason [1] a right near-ring N is called (i) left (right) strongly regular if for every a there is an x in N such that a = xa2 (a = a2x) and (ii) left (right) regular if for every a there is an x in N such that a = xa2 (a = a2x) and a = axa. He proved that for a zerosymmetric near-ring with identity, the notions of left regularity, right regularity and left strong regularity are equivalent. The aim of this note is to prove that these three notions are equivalent for arbitrary near-rings. We also show that if N satisfies dec on iV-subgroups, then all the above four notions are equivalent.

For bounded operators, the theory of the joint numerical range has been developed by various authors [1,2,3,4,5]. Especially, the properties of commuting normal n-tuples are discussed in detail. Our purpose here is to show that many results in the above references still hold in the case of unbounded normal operators (see Theorem 2.3, Corrollary 3.5, Theorem 4.1, Theorem 4.2). Besides, the operator algebras are closely related to the theory of joint spectrum and joint numerical ranges in the boundedcase (cf. [1,3]). How about unbounded operators? It seems that one must consider unbounded operator algebras. Some work has been done in this direction for the joint spectrum of unbounded normal operators [9]. In the last section of this paper, we provide some intimate relations between the joint numerical range and the unbounded operator algebras for unbounded normal operators.

Following the papers by H. W. Turnbull and J. Williamson, I have verified that the 122 forms, of the system of two quaternary quadrics and are actually irreducible. The original 1917 system contained 125 forms, which Williamson reduced by three. The present verification shews that no further reduction is possible. The proof was carried out as follows. I first constructed the whole system in canonical form with and for the two quadrics, and then listed the degrees in the coefficients and variables u, p, x of these concomitants. I next made Diophantine equations between these degrees for testing the supposed reducibility and found them to be impossible, except for Williamson's reduced forms.