We give sufficient conditions for the spectra and essential spectra of certain classes of operators to be contained in or coincide with an interval of the form [μ, ∞).

Let K be a field and * an involution of K. Let V and V′ be vector spaces over K of dimensions greater than or equal to 3, and let P(V) and P(F′) denote the projective spaces associated with V and V′ respectively. The fundamental theorem of projective geometry states that a bijection σ between P(V) and P(V′), which preserves the collinearity of points in P(V) and P(V′), is induced by a semi-linear bijection ø between V and V′ with respect to an automorphism α of K. We now consider the following additional structure on V and V′. Let f and f′ be hermitian forms with respect to * on V and V′ respectively; they define twisted polarities in P(V) and P(V′). We prove the following theorem. If there are no self-polar points in P(V) and P(V′) and σ preserves the twisted polarities, then ø is “almost” an isomorphism of the hermitian forms in the sense that

for some non-zero A in K such that A = A*. We give examples of forms f and f′ satisfying the above condition and we illustrate our theorem.

The stability of a two-dimensional vortex sheet against small disturbances in the plane of flow is examined. An integro-differential equation for the disturbances is derived and the possibility of solving it approximately is discussed. The approximation is equivalent to saying that short waves grow in a fashion determined by the local strength of the vortex sheet and it is shown that this need not be true throughout the evolution of the disturbance unless the growth rate is the same everywhere. It is possible for the disturbance on a distant part of the vortex sheet to control what happens locally, if the disturbance on the distant part is growing more rapidly. The approximate theory is applied to the tightly-wound spirals of aerodynamic interest and these are shown to be stable.

The problem of finding an axiomatic characterisation of dimension was first tackled by Menger, who gave a set of five independent axioms characterising the dimension (in the sense of dim, ind, or Ind since they are all equal on separable metric spaces) of subsets of the plane [7, p. 156]. The question of whether Menger's axioms characterise the dimension of more general spaces is still unsettled. Recently, Nishiura “11” obtained a set of seven independent axioms characterising the dimension of separable metric spaces. By modifying one of Nishiura's axioms, Aarts [1] then obtained an axiomatic characterisation of the strong inductive dimension (Ind) of metric spaces. Also, Ščepin [12] and Lokucievskiĭ [9] have obtained different axioms for dim on the class of compact metric and compact spaces, respectively. We present here four sets of independent axioms that characterise the dimension function u-dim, which is defined on the class of all uniform spaces.

Let M be a finite extension field of the rational numbers Q, and let CM denote the ideal class group of M. Let l be a rational prime, and let AM denote the l-class group of M (i.e., the Sylow l-subgroup of CM). If G is any finite abelian l-group, we define

where ℤl is the ring of l-adic integers and Fl is the finite field of l elements. One of the classical results of algebraic number theory is the specification of rank AM when M is a quadratic extension of Q and l = 2. This result was obtained by means of Gauss's theory of genera. A generalization of this result can be found in [1], where A. Fröhlich has obtained upper and lower bounds for rank AM when M is a cyclic extension of Q of degree l. His methods also show how to compute rank AM exactly when l = 3. In [4], G. Gras has described a procedure for analyzing the l-class groups of relatively cyclic extensions of degree l. However when l > 3, the computations can be very difficult.

Let V(x) denote the number of distinct values not exceeding x taken by Euler's ø-function, so that we have π(x) ≤ V(x) ≤ x. It was shown by Erdős and Hall [1] that for each fixed B > 2√(2/log 2), the estimate

holds; moreover we stated that the ratio V(x)/π(x) tends to infinity with x, faster than any fixed power of log log x. Our aim in the present paper is to prove the following result.

The object of this paper is to give a new characterization of the set of Pisot-Vijayaraghavan numbers (P.V.-numbers for short). As usual, [x] and [x] represent respectively the integer part and the fractional part of the real number x, and

Schaefer's fixed-point theorem is used to obtain sufficient conditions for the existence of a periodic solution of the non-linear differential equation f(D)x+BMg(D)x = p. The most significant feature of these conditions is a geometrical restriction on the range of the matrix M which is the same as the elliptic ball criterion encountered in stability theory. The extension of the results to delay-differential equations with constant time lags is also discussed.

Let L denote the ordinary differential operator given by Lf = (pf″)″ + (qf′)′ + rf, with p″, q′ and r continuous functions on [0,∞), and with p>0, q ≦ 0, and r ≧ 0. It is proved that if the equation Lg = 0 possesses a non-oscillatory solution, then any non-trivial solution f to Lf = 0 such that f(0) = f′(0) = 0 is eventually bounded away from zero.

This theorem is used to prove that, for a general class of functions q and r containing the polynomials as a very special case, the equation Lg = 0 has at most two linearly independent square integrable solutions, when p is identically one, q ≦ 0 and r ≧ 0.

Finally, the main theorem is applied to show that certain sixth-order operators are limit-3.