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.

Trevor Evans in (8) introduced postulates for a non-associative number theory similar to, but less general than, those of A. Robinson (9). Evans' number theory is also non-commutative under addition and multiplication, but an alternative equality axiom also suggested by Robinson leads to a number theory which is commutative under addition and still non-associative except in the special case:

We answer the following questions negatively: Does there exist a simple locally finite barely transitive group (LFBT-group)? More precisely we have: There exists no simple LFBT -group. We also deal with the question, whether there exists a LFBT-group G acting on an infinite set Ω so that G is a group of finitary permutations on Ω. Along this direction we prove: If there exists a finitary LFBT-group G, then G is a minimal non-FC p-group. Moreover we prove that: If a stabilizer of a point in a LFBT-group G is abelian, then G is metabelian. Furthermore G is a p-group for some prime p, G/G′ ≅ Cp∞, and G′ is an abelian group of finite exponent.

Asymptotic Expressions for the Bessel Functions.

From the asymptotic expansion for Ku(z) it follows that, if − π < amp z < π,

This theorem is also true if amp z = ± π; to prove this consider the formula

Fourier's Theorem, as usually stated, is

Most writers give this without limitation, but De Morgan (Diff. and Int. Calc. pp. 618 &c.) directs attention to what he calls the apparent neglect by previous writers of the limitation of the theorem to functions which satisfy the condition

In many biological diffusion-reaction studies, it was found that one should include the effect of density dependent rates, drift terms and spatially varying growth rates, in order to obtain more accurate results. (See e.g. [7],[10], [8] , [3]). On the other hand, many recent mathematical results on reaction-diffusion systems do not include such general setting. This article investigates the behaviour of competing-species reaction-diffusion model under this more general situation. Efforts are made to obtain results concerning coexistence, survival and extinction, by methods similar to that in [5], [6].

Let F = GF(q). To any polynomial G ∈ F[x] there is associated a mapping Ĝ on the set IF of monic irreducible polynomials over F. We present a natural and effective theory of the dynamics of Ĝ for the case in which G is a monic q-linearized polynomial. The main outcome is the following theorem.

Assume that G is not of the form , where l ≥ 0 (in which event the dynamics is trivial). Then, for every integer n ≥ 1 and for every integer k ≥ 0, there exist infinitely many μ ∈ IF. having preperiod k and primitive period n with respect to Ĝ.

Previously, Morton, by somewhat different means, had studied the primitive periods of Ĝ when G = xq – ax, α a non-zero element of F. Our theorem extends and generalizes Morton's result. Moreover, it establishes a conjecture of Morton for the class of q-linearized polynomials.