In this note we introduce a natural generalization of the ordinary convolution of arithmetic functions: If f and g are arithmetic functions,

defines the K-product of f and g. If the kernel K(n) ≡ E(n) = 1, the K-product is the ordinary convolution Σd|nf(d)g(n/d);, if K(n) ≡ ∊(n) = [1/n], then the K-product is the unitary product Σf(d)g(n/d), summed over d|n, (d, n/d) = 1 (1, 2). We give in Theorem 1 a characterization of all associative kernels, i.e., kernels for which the corresponding K-product is associative.

Tchebycheff proposed the problem of finding n + 1 constants A, x1, x2, . . , xn ( — 1 ≤ x1 < x2 < . . . < xn ≤ +1) such that the formula

(1)

is exact for all algebraic polynomials of degree ≤n. In this case it is clear that A = 2/n. Later S. Bernstein (1) proved that for n ≥ 10 not all the xi's can be real. For a history of the problem and for more references see Natanson (4).

A subset G of a non-empty partially ordered set C is called normal if it coincides with the set of all upper bounds of the set of lower bounds of G. This is equivalent to stipulating that G be the set of all upper bounds of some subset of C called a set of generators for G. When ordered by inclusion, the family of all normal subsets of C forms a complete lattice with maximum C and minimum empty or singleton. The meet operation is simply point set intersection; whence, the meet of a family Gi of normal subsets is the set of upper bounds of ∪ Fi where Fi generates Gi for each i. A normal subset is called proper if it is neither void nor C, and the proper normal subsets. of C form a boundedly complete lattice.

If f(z) is a non-constant, entire function, then Hadamard's three-circles theorem asserts that

is a convex, increasing function of log r. Hence, by well-known properties of logarithmically convex functions,

Using the methods developed in (2 and 3), in this paper we study some properties of the configuration of generators and points of a cone in an w-dimensional finite projective space. The configuration of secants and external points of a quadric in a finite plane of even characteristic is also studied. I t is shown that these configurations lead to several series of partially balanced incomplete block (PBIB) designs. PBIB designs are defined in Bose and Shimamoto (1). A PBIB design with m associate classes is an arrangement of v treatments in b blocks such that.

The purpose of this paper is to establish the following theorem :

Theorem 1. Let be a finite group with abelian Sylow 2-subgroups in which the centralizer of every involution is solvable. Then either is solvable or else /O is isomorphic to a subgroup of PΓL(2, q) containing PSL(2, q), where either q = 3 or 5 (mod 8), q ≥ 5, or q = 2n, n ≥ 2.

R. E. Johnson (10), Utumi (18), and Findlayand Lambek (7) have defined for each ring R a unique maximal "ring of right quotients" Q. When R is a commutative integral domain (in this paper an integral domain need not be commutative) or an Ore domain, then Q is the usual division ring of quotients of R. Moreover, it is well known that in these special cases, if R is totally ordered, then so is Q.

The main purpose of this paper is to study the ring of quotients Q, and in particular its order properties, for certain lattice-ordered rings R.

In this paper we shall investigate the singular behaviour of the solutions to the elliptic equation

(1.1)

where A (r2), C(r2) are entire functions of the complex variable

A graph G for purposes here is a finite set of elements called vertices and a finite set of elements called edges such that each edge meets exactly two vertices, called the end-points of the edge. An edge is said to join its end-points.

A matching in G is a subset of its edges such that no two meet the same vertex. We describe an efficient algorithm for finding in a given graph a matching of maximum cardinality. This problem was posed and partly solved by C. Berge; see Sections 3.7 and 3.8.

Let Md and be Riemannian manifolds. We shall say that an isometric immersion ϕ: Md —> is isotropic provided that all its normal curvature vectors have the same length. The class of such immersions is closed under compositions and Cartesian products. Umbilic immersions (e.g. Sd ⊂ Rd+1) are isotropic, but the converse does not hold. If M and are Kähler manifolds of constant holomorphic curvature, then any Kähler immersion of M in is automatically isotropic (Lemma 6). We shall find the smallest co-dimension for which there exist non-trivial immersions of this type, and obtain similar results in the real constant-curvature case.

In the first part of this paper I shall demonstrate that one irrational root of the algebraic equation

creates an algebraic number field, out of which n — 1 irrationals can be chosen so that they yield a periodic Jacobi-Perron algorithm. The coefficients in (1) are subject to certain restrictions which will be elaborated below.

Much work has been done in the investigation of the properties of solutions of linear elliptic systems of partial differential equations. Among these systems, the class of Beltrami systems has been studied for many years and has been shown to be of fundamental importance. Another class, perhaps of equal importance, is the class defined by Bers (1), which the author has taken the liberty of calling Bers systems. Solutions of these systems will be called Beltrami and Bers functions respectively.

Characterizations for prime and semi-prime rings satisfying the right quotient conditions (see § 1) have been determined by A. W. Goldie in (4 and 5). A ring R is prime if and only if the right annihilator of every non-zero right ideal is zero. A natural generalization leads one to consider right R-modules having the properties that the annihilator in R of every non-zero submodule is zero and regular elements in R annihilate no non-zero elements of the module. This is the motivation for the definition of prime module in § 1.

In this work we obtain a simultaneous extension of Theorems 1.6 and 1.7 in Agmon and Nirenberg (1), together with a partial extension of the result on backward unicity for parabolic equations by Lions and Malgrange (4).

Let H be a Hilbert space. (·) and | | are the notations for the scalar product and the norm in this space. Consider in H a family B(t), 0 ≤ t ≤ T, of closed linear operators with dense domain DB(t) (varying) with t. Let L2(0, T, H) be the space of Bochner square-integrable vector-valued functions with values in H. Our main result is the following

Let n be a finite affine plane with n points on a line. We denote by G(II) the graph whose vertices are all points and lines of II, with two vertices adjacent if and only if one is a point, the other is a line, and the point and line are incident. Let L(11) denote the line graph of G(II), i.e., the vertices of L(II) are the edges of G(II), and two vertices of L(II) are adjacent if the corresponding edges of G(II) are adjacent. It is clear that L(II) is a regular, connected graph with n2(n + 1) vertices and valence 2n — 1.

A family of circuits of a graph G is said to be independent if no two of the circuits have a common vertex; it is called edge-independent if no two of them have an edge in common. A set of vertices will be called a representing set for the circuits (for the sake of brevity we shall call it a representing set), if every circuit of G passes through at least one vertex of the representing set. Denote by I(G) = k the maximum number of circuits in an independent family and by R(G) the minimum number of vertices of a representing set. Dirac and Gallai asked whether there is any relation between I(G) and R(G) (trivially R(G) ≥ I(G)).

Let f(z) be a non-constant entire function and let {dn} be a sequence of complex numbers such that

The set of equations

(1.1)

Let A be a complex Banach algebra and Lr (Ll) be the lattice of all closed right (left) ideals in A. Following Tomiuk (5), we say that A is a right complemented algebra if there exists a mapping I —> IP of Lτ into Lr such that if I ∊ Lr, then I ∩ Ip = (0), (Ip)p = I, I ⴲ Ip = A and if I1, I2 ∊ Lr with I1 ⊆ I2 then .

If in a Banach algebra A every proper closed right ideal has a non-zero left annihilator, then A is called a left annihilator algebra. If, in addition, the corresponding statement holds for every proper closed left ideal and r(A) = (0) = l(A), A is called an annihilator algebra (1).

The theorems of the present paper deal with conditions which are necessary and sufficient in order that a solvable or nilpotent infinite group should have a completely reducible matrix representation over a given algebraically closed field.

It is known (17) that a locally nilpotent group of matrices is always solvable. Thus the first theorem of the present paper is a partial generalization of Theorem 1 of (16), which states:

If G is a locally nilpotent subgroup of the full linear group GL(n, P) over a perfect field P, then G is completely reducible if and only if each matrix of G is diagonizable (by a similarity transformation over some extension field of P).