It is shown that an abelian group admits a non-discrete locally compact group topology if and only if it has a subgroup algebraically isomorphic to the group of p-adic integers or to an infinite product of non-trivial finite cyclic groups. It is also proved that an abelian group admits a non-totally-disconnected locally compact group topology if and only if it has a subgroup algebraically isomorphic to the group of real numbers. Further, if an abelian group admits one non-totally-disconnected locally compact group topology then it admits a continuum of such topologies, no two of which yield topologically isomorphic topological groups.

Recently, through a repeated application of one-dimensional finite integral transforms, Cinelli(1) gave a solution for the temperature distribution in a hollow circular cylinder of finite length. Since no new ideas or techniques are introduced, the extension claimed in (1) with regard to the finite Hankel transform technique employed in the transformation of the radial space variable in the hollow cylinder problem is trivial, in view of well-known works by Sneddon(2) and Tranter (3), to mention a few. The list of the finite Hankel transforms given in (1) for a variety of boundary conditions at r = a and r = b is the result of routine, algebraic manipulations well known from the general theory of eigenvalue problems specialized for the hollow cylinder. In this list a set of seemingly different series expansions is given for the inverse Hankel transform for each combination of boundary conditions at the two radial surfaces. In each case, the two expressions for inversion can readily be shown to be identical to each other when use is made of the frequency equation. One of the inversion forms is therefore unnecessary once the other is given. Furthermore, the general solution as given by equation (54) of Cinelli(1)does not satisfy his boundary conditions (27), (28), (29) and (30), unless these latter are homogeneous.

Consider a long, finite sequence xl, x2, …, xN of repeated measurements on the same physical quantity. The usual model is that the data consists of observed values on a sequence X1, X2, …, XN of independent and identically distributed random variables. It is customary to require that the Xi have a finite mean and variance: µ = E(Xi) is the ‘true value’ of the quantity being measured; and

measures the variability in the measuring process.

In an earlier paper we studied the impact of equivalences between derived categories of group rings on their cohomology rings. Especially the group of auto-equivalences TrPic(RG) of the derived category of a group ring RG as introduced by Raphaël Rouquier and the author defines an action on the cohomology ring of this group. We study this action with respect to the restriction map, transfer, conjugation and the local structure of the group G.

We identify the support of a tempered distribution by evaluation of a sequence of test functions against the Fourier transform of the distribution. This improves previous results by removing the restriction that the distribution's Fourier transform be in and be of polynomial growth. We use an apparently new technical lemma that implies that certain bounded approximate identities for are also topological approximate identities for elements of the space of Schwartz functions.

A generalized Euler–Lagrange equation is presented. It provides a unified approach to boundary value problems in potential theory, diffusion, magnetostatics, and integral equations.

Suppose we have a sample of N independent random variables X1, …, XN where Xi has the distribution F(X|θ, øi). θ is a k-dimensional ‘structural’ parameter (θ(1), …, θ(k)), and the øi are scalar or vector ‘incidental’ parameters in some given space. The Xi may be scalar or vector random variables which are either discrete in which case we write f(X|θ, øi) for the probability associated with a given point, or else continuous random variables with a probability density f(X|θ, øi). In either case we sup pose the support of the probability distribution to be fixed. We aim to estimate the true value of θ by maximum-likelihood methods.

It is well established by observation that in the ordinary flow of a stream there is a slow transverse flow outwards from the middle near the bottom, and inwards near the top; and such a circulation seems to be required to explain the fact that the filament of maximum onward velocity is depressed some distance below the centre of the free surface. Qualitative explanations of the phenomenon have been offered, depending on the extra friction near the sides, but it seems that the true explanation must depend on something more fundamental. For any explanation to be satisfactory it must show uniform linear flow to be impossible; and actually it can at least be rendered highly probable that such flow can exist over a very wide range of conditions.

The hysteresis losses and demagnetizing coefficients of rectangular Fe-Si strips of various dimensions have been investigated. The hysteresis loss is approximately independent of the value of N, in agreement with a simple theoretical argument. The value of the demagnetizing coefficient is in fair agreement with the value calculated for the inscribed ellipsoid.

1. I proved in 1908(1) that if A = Σam and B = Σbn are convergent, and

for large m and n, then

is convergent (necessarily to AB); and this theorem has been extended in a number of directions both by other writers and by myself. Thus we may replace (1), when am and bn are real, by

we may use conditions unsymmetrical in am and bn; we may put the same problem for the product of any number of series; and we may consider modes of ‘Dirichlet multiplication’ based on sequences λm and μn, reducing to Cauchy's when λm = m and μn = n. The appropriate references will be found in(2).

The classical Adams spectral sequence [1] has been an important tool in the computation of the stable homotopy groups of spheres . In this paper we make another contribution to this computation.

In the classical Poincaré–Bendixson theory the object of study are the limit sets of a continuous flow on the 2-sphere S2 and the behaviour of the orbits near them (see [7, 9]). In [2] the second author proved that an assertion similar to the Poincaré–Bendixson theorem is true in the wider class of the 1-dimensional invariant (internally) chain recurrent continua of flows on S2. On the other hand, it is known that among the closed 2-manifolds, the 2-sphere S2, the projective plane RP2 and the Klein bottle K2 are the only ones for which the Poincaré–Bendixson theorem is true (see [1, 8, 11]).

The motivation of the present paper was to examine to what extent the main results of [2] carry over to flows on RP2 and K2. A first attempt to study chain recurrent sets of flows on closed 2-manifolds other than the 2-sphere was [3]. As one expects, the results of [2] carry over easily to RP2, since chain recurrence behaves well with respect to regular covering maps of compact manifolds, as we show in Section 3. The situation with K2 is quite different, since it is doubly covered by the 2-torus T2, where we have no Poincaré–Bendixson theorem. Actually, the Poincaré–Bendixson theorem for 1-dimensional invariant chain recurrent continua of flows on K2 is not true. For example, identifying suitably the boundary periodic orbits of a 2-dimensional Reeb flow on a closed annulus (see [7, chapter III, 2·6]) we get a flow on K2 with a 1-dimensional invariant chain recurrent continuum consisting of the unique periodic orbit and another orbit, which spirals against it in positive and negative time. As we prove in Theorem 4·4, this situation, or concatenations of it, is the only one where the Poincaré–Bendixson theorem for 1-dimensional invariant chain recurrent continua of flows on K2 is not true. Then, we are concerned with the topological structure of the 1-dimensional chain components of a flow on K2 with finitely many singularities. In Proposition 4·6 we find when such a set consists of finitely many orbits and is homeomorphic to a finite graph. An example shows that the hypothesis of Proposition 4·6 is essential. Finally, in Theorem 4·9 we give a description of the structure of the 1-dimensional chain components of a flow on K2 with finitely many singular points.

Many investigations bave sbown the presence of a “continuous” β-ray spectrum accompanying tbe β-ray lines in tbe naturai spec-trum of Ra B and Ra C. The existence of sucb a spectrum is not easily accounted for if one is to regard the nucleus as existing in sharply quantised states. It has been pointed out that this spectrum may be a spurious effect, arising from scattering from the walls of the apparatus and not from within the atom itself. Therefore it is of importance to obtain accurate information regarding the continuous spectrum and its relation to the line spectrum.

Let R be the field of rational numbers, {x} = {x1, z2, …}, {y} = {y1, y2, …} be two countably infinite sets of variables and t an indeterminate. Let (λ) = (λ1, λ2, …, λm) be a partition of n. Then Littlewood (5) has shown that

can be expressed in the form

where Qλ(x, t) and Qλ(y, t) denote certain symmetric functions on the sets {x} and {y} respectively. In addition

where is the partition of n conjugate to (λ). In fact, Littlewood (5) showed that

where the summation is over all terms obtained by permutations of the variables xi (i = 1, 2, …) and

.

An unknotting disc is the ‘trace’ in ℝ4 of a homotopy of a diagram of a knot in ℝ3, which shrinks the diagram to a point. In this paper we study unknotting discs which have as singularities only ordinary triple points. It turns out that the Arf invariant of the knot is determined by the number of triple points in which all three branches of the disc intersect pairwise with the same index. We call such a triple point coherent. This interpretation of the Arf invariant has a surprising consequence:

Let S ⊂ ℝ4 be a taut immersed sphere which has as singularities only ordinary triple points. Then the number of coherent triple points in S is even. For example, it is easy to show that there is a taut immersed sphere S with Euler number six of the normal bundle and which has exactly three ordinary double points and no other singularities. So, our result implies that the three double points of S can not be pushed together to create an ordinary triple point without the appearance of new singularities.

Here ‘taut’ means that the restriction of one of the coordinate functions on S has exactly two (non-degenerate) critical points, i.e. is a perfect Morse function.

Introduction. For applications in differential topology it is desirable to be able to find restrictions on the possible real and complex vector bundles over a given manifold. These restrictions usually state that the top component of some specific rational multinomial in the various characteristic classes is an integral multiple of the fundamental cocycle. For example, in proving non-embedding or non-immersion theorems (compare (1); (10)) one could test whether the normal bundle is consistent with these integrality conditions.

For a compact irreducible 3-manifold containing 2-sided projective planes and admitting a ℤ2-action, we consider the problem of when there exists a ℤ2-equivariant hierarchy. We show that a ℤ2-equivariant hierarchy always exists if either every irreducible summand of the orientable double cover has infinite first homology or every 2-sided projective plane is boundary parallel.

If A = (am, n) is a regular matrix, then for any sequence x = {xn}, Am(x) will denote the transform and A-lim x denotes if that limit exists. We shall denote by the set of bounded sequences which are summed by A. If B is another regular matrix with then we say that B is b-stronger than A. In that case B must be b-consistent with A (see (4) and (6)), i.e. if then

If {μn} is a sequence of positive real numbers with we say that A and B are (μn)-consistent if every sequencer x = {xn} satisfying xn = 0(μn) which is summed by both A and B is summed to the same value by both matrices. A finite set of matrices A1, A2, …, AN is said to be simultaneously (μn)-inconsistent (b-consistent) if whenever is summed by Ai with (i = l, 2, …, N) then implies that The set of sequences, is denoted by

Let f{θ, φ) be a function defined for the range 0 ≤ φ π, 0 ≤ φ ≤ 2π on a sphere S. The ultraspherical series associated with this function is

where

and the ultraspherical polynomials are defined by the following expansion

.