A graph in which each line is designated as either positive or negative is called a signed graph S. The sign of a cycle in S is a product of the signs of its lines. A signed graph in which every cycle is positive is called balanced. This concept was introduced by Harary in (3) and the following characterisation of balanced signed graphs was given.

It is well known that the real, skew-symmetric, non-singular, bilinear forms of n + n variables have no invariants. In fact, each of these forms may be transformed into one and the same form, for instance into the one which occurs in the usual representation of the complex group. The standard proofs of this theorem break down in case of infinite forms which are bounded in the sense of Hilbert, one of the impediments being the possibility of a continuous spectrum. The object of this note is to show that, while the usual proofs break down, the theorem itself is true in Hubert's case also.

The concept of “root vectors” is investigated for a class of multiparameter eigenvalue problems

where operate in Hilbert spaces Hm and . Previous work on this “uniformly elliptic” class has demonstrated completeness of the decomposable tensors x1 ⊗…⊗ xk in a subspace G of finite codimension in H=H1 ⊗…⊗ Hk, but questions remain about extending this to a basis of H. In this work, bases of elements ym, in general nondecomposable but satisfying recursive equations of the type are constructed for the “root subspaces” corresponding to λ∈ℝk.

Rings of invariants are identified for some automorphisms θ of certain iterated skew polynomial rings R, including the enveloping algebra of sl2(k), the Weyl algebra A1 and their quantizations. We investigate how finite-dimensional simple R-modules split over the ring of invariants Rθ and how finite-dimensional simple Rθ-modules extend to R.

G denotes a locally compact abelian group and M(G) the convolution algebra of regular bounded Borel measures on G. An ideal I of M(G) closed in the usual (total variation) norm topology is called an L-ideal if μ ∈ I, ν≪ μ (ν absolutely continuous with respect to μ) implies that ν ∈ I. Here we are concerned with the L-idealsL1(G), , and M0(G) where, as usual, L1(G) denotes the set of measures absolutely continuous with respect to Haar measure, denotes the radical of L1(G) in M(G) and M0(G) denotes the set of measures whose Fourier-Stieltjes transforms vanish at infinity.

Throughout this note A denotes a ring with identity, and “ module ” means “ left unitary module ”. In (2), C. Yohe studied elemental annihilator rings (e.a.r. for brevity). An e.a.r. is defined as a ring in which every ideal is the annihilator of an element of the ring. For example, a semi-simple, Artinian ring is an e.a.r. A is a l.e.a.r. (left elemental annihilator ring) if every left ideal is the left annihilator of an element of the ring. A r.e.a.r. (right elemental annihilator ring) is denned similarly.

In the theory of ordinary linear differential equations with three regular singularities and in the theory of their special and limiting cases, integral representations of the solutions are known to be very important. It seems that there is no corresponding simple integral representation of the solutions of ordinary linear differential equations with four regular singularities (Heun's equation) or of particular (e.g. Lamé's equation) or limiting (e.g. Mathieu's equation) cases of such equations. It has been suggested (Whittaker 1915 c) that the theorems corresponding in these latter cases to integral representations of the hypergeometric functions involve integral equations of the second kind. Such integral equations have been discovered for Mathieu functions (Whittaker 1912, cf. also Whittaker and Watson 1927 pp. 407–409 and 426) as well as for Lame functions (Whittaker 1915 a and b, cf. also Whittaker and Watson 1927 pp. 564–567) and polynomial or “quasi-algebraic” solutions of Heun's equation (Lambe and Ward 1934). Ince (1921–22) investigated general integral equations connected with periodic solutions of linear differential equations.

Eddington has considered equations of the gravitational field in empty space which are of the fourth differential order, viz. the sets of equations which express the vanishing of the Hamiltonian derivatives of certain fundamental invariants. The author has shown that a wide class of such equations are satisfied by any solution of the equations

where Gμν and gμν are the components of the Ricci tensor and the metrical tensor respectively, whilst λ is an arbitrary constant. For a V4 this applies in particular when the invariant referred to above is chosen from the set

where Bμνσρ is the covariant curvature tensor. K3 has been included since, according to a result due to Lanczos3, its Hamiltonian derivative is a linear combination of and , i.e. of the Hamiltonian derivatives of K1 and K2. In fact

In 1964 Green and Rivlin (1) introduced a theory of simple force and stress multipoles founded on conventional kinematics. Using a work formula, the force and stress multipoles were defined with the help of the velocity field and its spatial derivatives. More recently, within the framework of this general study, Bleustein and Green (2) examined the theory of the simplest multipolar fluid, the dipolar fluid, and formulated constitutive equations for a homogeneous incompressible linear dipolar fluid.

The differential operation known as Lie derivation was introduced by W. Slebodzinski in 1931, and since then it has been used by numerous investigators in applications in pure and applied mathematics and also in physics. A recent monograph by Kentaro Yano (2) devoted to the theory and application of Lie derivatives gives some idea of the wide range of its uses. However, in this monograph, as indeed in other treatments of the subject, the Lie derivative of a tensor field is defined by means of a formula involving partial derivatives of the given tensor field. It is then proved that the Lie derivative is a differential invariant, i.e. it is independent of a transformation from one allowable coordinate system to another. Sometimes some geometrical motivation is given in explanation of the formula, but this is seldom very satisfying.

Etant donnés une conique K, dont l'équation est K = 0, et un point P (α, β), l'équation générale des coniques qui passant par les points d'intersection de la conique K et du cercle P de rayon nul, qui a le point (α, β) pour centre, est

Comme les quatre points d'intersection du cercle P et de la conique K sont imaginaires, le système (1) comprend un seul couple de droites réelles Δ et Δ. Ces droites seront dites, par analogie avec une expression proposée par Chasles, les conjointes du point P et de la conique K.

In (2) a finite p-group G is said to be nearly regular if it has the following two properties:

(i) There exists a central subgroup Z of order p and G/Z is regular.

(ii) If x ∈ G and y ∈ γ2(G), then gp{x, y} is regular.

Let pn>0 be such that pn diverges, and the radius of convergence of the power series

is 1. Given any series σan with partial sums sn, we shall use the notation

and

Let G be any finite group, and p any prime number. (All groups to be considered here are finite, and we assume this without further comment.) We denote by Kp(G) the unique smallest normal subgroup of G for which the quotient G/Kp(G) is a p-group. G/Kp(G) is called the p-residual of G. W. Gaschütz (2, Satz 7) has proved the following

Theorem. Set K = Kp(G). If the Sylow p-subgroups of K are abelian, then G splits over K.