In the present paper we obtain first approximation formulae for the regions of dynamical instability of linear canonical systems. These formulae are analogous to the formulae for Hamiltonian systems stated by Krein and Jakubovič [5] and proved by Pittel' and Juzefovič [8]. Special cases were considered by Malkin [7] and Jakubovič [3]. Related papers are Hale [2] and Jakubovič [4]. However, our method differs from the methods used by these authors and seems to us to be both simpler and more general.

Let Σn−0∞an, be a given infinite series and {sn} the sequence of its partial sums. Let {pn} be a sequence of constants, real or complex, and let us write

If

as n → ∞, then we say that the series is summable by the Nörlund method (N, pn) to σ And the series a,Σan, is said to be absolutely summable (N, pn) or summable |N, Pn| if {σn} is of bounded variation, i.e.,

This paper discusses the relationship between two equivalence relations on the class of finite nilpotent groups. Two finite groups are conformal if they have the same number of elements of all orders. (Notation: G ≈ H.) This relation is discussed in [4] pp 107–109 where it is shown that conformality does not necessarily imply isomorphism, even if one of the groups is abelian. However, if both groups are abelian the position is much simpler. Finite conformal abelian groups are isomorphic.

In 1923 Nörlund [1] considered the difference equation

and showed that the formal solution of (1)

obtained by iteration, although in general divergent, is in fact Abel summable to a solution of (1). He writes the arbitary constant c as

and thus the “principal solution” of the difference equation is

Salem [1] gave the following criterion for Fourier-Lebesgue sequences .

Denote by E the set of continuously differentiate periodic functions u such that u' has an absolutely convergent Fourier series, and let E1 denote the set of uσE satisfying ∥u∥∞≦1.

Let E be a real Banach space. A mapping f of E into E is said to be bounded if f maps every bounded set into a bounded set.

A group G is called a c-group if each of its subnormal subgroups is characteristic in G. It is the object of this note to give a characterization of finite solvable c-groups.

Let E be a real Banach space. The set of all continuous linear mappings of E into E is a Banach algebra under the usual algebraic operations and the operator bound as norm. We denote this Banach algebra by ℒ, if E is a separate Hilbert space.

Let f = f(x) = f(x1, x2,…, xn) be an indefinite n-ary quadratic form of determinant det (f); that is, f(x) = x' Ax where A is a real symmetric matrix with determinant det (f). Such a form is said to take the value v if there exists integral x ≠ 0 such that f(x) = v.

Let G be a group and let Ω(G) denote the semigroup of all mappings of G into G with the usual composition of mappings as multiplication, namely g(θ1θ22) = (gθ1)θ2.

It is well known that the problem of determining the most economical covering of n-dimensional Eucidean space, by equal spheres whose centres form a lattice, may be formulated in terms of positive definite quadratic forms, as follows: Let

be positive definite, and

For real α, set

(the minimum being taken over integral x),

Let {f(n)} be a linear recurrence of order two, i.e.

for all positive integers n.

Let G be a locally compact Abelian Hausdorff group (abbreviated LCA group); let X be its character group and dx, dx be the elements of the normalised Haar measures on G and X respectively. If 1 < p, q < ∞, and Lp(G) and Lq(G) are the usual Lebesgue spaces, of index p and q respectively, with respect to dx, a multiplier of type (p, q) is defined as a bounded linear operator T from Lp(G) to Lq(G) which commutes with translations, i.e. τxT = Tτx for all x ∈ G, where τxf(y) = f(x+y). The space of multipliers of type (p, q) will be denoted by Lqp. Already, much attention has been devoted to this important class of operators (see, for example, [3], [4], [7]).

In 1959, Professor N. A. Court [2] generated synthetically a twisted cubic C circumscribing a tetrahedron T as the poles for T of the planes of a coaxal family whose axis is called the Lemoine axis of C for T. Here is an analytic attempt to relate a normal rational curve rn of order n, whose natural home is an n-space [n], with its Lemoine [n—2] L such that the first polars of points in L for a simplex S inscribed to rn pass through rn anf the last polars of points on rn for S pass through L. Incidently we come across a pair of mutually inscribed or Moebius simplexes but as a privilege of odd spaces only. In contrast, what happens in even spaces also presents a case, not less interesting, as considered here.

This is the first of several papers which grew out of an attempt to provide C (X, Y), the family of all continuous functions mapping a topological space X into a topological space Y, with an algebraic structure. In the event Y has an algebraic structure with which the topological structure is compatible, pointwise operations can be defined on C (X, Y). Indeed, this has been done and has proved extremely fruitful, especially in the case of the ring C (X, R) of all continuous, real-valued functions defined on X [3]. Now, one can provide C(X, Y) with an algebraic structure even in the absence of an algebraic structure on Y. In fact, each continuous function from Y into X determines, in a natural way, a semigroup structure for C(X, Y). To see this, let ƒ be any continuous function from Y into X and for ƒ and g in C(X, Y), define ƒg by each x in X.

A sequence of integers 0 <a1 < a2 <… no term of which divides any other will be called a primitive sequence. Throughout this paper c1, c2,… will denote suitable positive absolute constants. Behrend [1] proved that for every primitive sequence

Although varieties of groups can in theory be determined as well by the identical relations which the groups all satisfy as by some structural property inherited by subgroups, factor groups and cartesian products which the groups have in common, it seems in practice just as hard to answer questions about properties of a group from knowledge of identical relations as it is from, say, a presentation. Many of the important questions connected with Burnside's problems exemplify this difficulty: we still do not know if there is a bound on the derived length of finite groups of exponent 4, nor whether there is a bound on the nilpotency class of finite groups of exponent p (p ≧ 5, a fixed prime).