A number of authors, including Apostol [1], Carlitz [2], Mikolás [5] and Rademacher [9] have obtained linear relations for the Dedekind sums

(the inhomogeneous sum) and the homogeneous sum

Here denotes the periodic extension into ℝ of the Bernoulli polynomial Bm(X) on [0, 1] given by the relation

with the exception that we define

1. Saks calls an integral complete if it satisfies two conditions, of which one, the condition (C), is as follows.

If f(x) is defined in (a, b) and is integrable in each interval (a + ε, b − η), where a < a + ε < b − η < b, and

exists, then f(x) is integrable in (a, b) and the integral over (a, b) is equal to the above limit.

In this paper the classical Hardy transformation (5) has been extended to distributions. The celebrated Hankel transformation, Y-transformation and H-transformation are particular cases of the Hardy transformation. The Hankel transformation has been extended to distributions by Zemanian(12). The distributional theory has been explored by Zemanian(13), Koh and Zemanian(6), and Dubey and Pandey(2). The famous Y- and its reciprocal H-transformations have not been extended to distributions. The inversion and some of the properties of these distributional transformations can be deduced as particular cases of the results contained in this paper.

During the course of a cloud chamber investigation of the occurrence of rare events accompanying the passage of fast β-particles through matter, it was found* that existing methods of introducing high-energy electrons into the chamber were unsatisfactory. Electrons of sufficiently high energy could only be obtained from sources which emit, in addition, a large amount of γ-radiation. Absorption of this γ-radiation in the walls and gas of the chamber resulted in the ejection of numbers of stray electrons, whose presence rendered the observation of the fast β-ray tracks a matter of some difficulty.

In a previous paper the present writer gave a solution of the problem of determining the circulation around a thin elliptic cylinder in a steady stream of slightly viscous fluid when the major axis of the cylinder is inclined at a small angle to the direction of the flow at infinity. The present note gives a largely qualitative analysis of the manner in which this circulation is built up when the motion of the fluid or cylinder is started instantaneously from rest with a given velocity.

1. It often happens that we have a series of observed data for different values of the argument and with known standard errors, and wish to remove the random errors as far as possible before interpolation. In many cases previous considerations suggest a form for the true value of the function; then the best method is to determine the adjustable parameters in this function by least squares. If the number required is not initially known, as for a polynomial where we do not know how many terms to retain, the number can be determined by finding out at what stage the introduction of a new parameter is not supported by the observations*. In many other cases, again, existing theory does not suggest a form for the solution, but the observations themselves suggest one when the departures from some simple function are found to be much less than the whole range of variation and to be consistent with the standard errors. The same method can then be used. There are, however, further cases where no simple function is suggested either by previous theory or by the data themselves. Even in these the presence of errors in the data is expected. If ε is the actual error of any observed value and σ the standard error, the expectation of Σε2/σ2 is equal to the number of observed values. Part, at least, of any irregularity in the data, such as is revealed by the divided differences, can therefore be attributed to random error, and we are entitled to try to reduce it.

New methods are presented for the evaluation of lattice Green's functions for the body-centred cubic lattice.

This paper is essentially concerned with the following problem. Let G be a topological group. AG-set is a set S equipped with a continuous action (where S is given the discrete topology). These G-sets form a category BG. It is well known that if G and H are discrete groups, BG is equivalent to BH iff G is isomorphic to H. However, this result cannot be extended much beyond the discrete case. The problem is: when is BG equivalent to BH, for topological groups G and H?

The annihilator $J^{\perp}$ of a weak*-closed inner ideal $J$ in the JBW*-triple $A$ consists of elements $b$ of $A$ for which $\{J\,b\,A\}$ is equal to zero, and the kernel $\mathrm{Ker}(J)$ of $J$ consists of those elements $b$ in $A$ for which $\{J\,b\,J\}$ is equal to zero. The annihilator $J^{\perp}$ is also a weak*-closed inner ideal in $A$, and $A$ enjoys the Peirce decomposition \begin{eqnarray*} A &=& J \oplus_M J^{\perp} \oplus J_1\\ &=& J_2 \oplus_M J_0 \oplus J_1, \end{eqnarray*} where $J_1$ is the intersection of the kernels of $J$ and $J^{\perp}$. When all of the usual Peirce arithmetical relations hold, $J$ is said to be a Peirce inner ideal. A pair $(J,K)$ of weak*-closed inner ideals is said to be compatible when \[A = \bigoplus_{j,k = 0}^2 J_j \cap K_k.\] By analysing the biannihilator $J^{\perp\perp}$ of a Peirce inner ideal $J$ it is shown that, if $J$ is a Peirce inner ideal, then $J^{\perp}$ is a Peirce inner ideal. Furthermore, if $K$ is a weak*-closed inner ideal in $A$ such that $(J,K)$ is a compatible pair, then $(J^{\perp},K)$ is a compatible pair.

A line source whose strength varies sinusoidally with time and also with the co-ordinate measured along its length is situated parallel to the shoreline of a beach of angle ¼π0. Both long-and short-wave solutions are found. It is shown that for certain positions of the source, long waves are not radiated to infinity, while in the short-wave regime, the solutions take the form of edge-waves, with resonances occurring at certain wavenumbers. Computations of the free-surface contours are presented for a range of wavenumbers.

Complex hyperbolic packs are hypersurfaces of complex hyperbolic plane H2ℂ which may be considered as dual to the well known bisectors. In this paper we study the geometric aspects associated to packs.

We show how the techniques introduced in [1] and [2] can be employed to derive endpoint Lp → Lq bounds for the X-ray transform associated to the line complex generated by the curve t → (t, t2, . . ., td−1). Almost-sharp Lorentz space estimates are produced as well.

In a previous paper we discussed a uniform algebraic method of solution of problems in which a prescribed real rational function (or polynomial) g(x) was to be approximated in a given finite interval by a real rational function (or polynomial)f(x) with prescribed numerator and denominator degrees, the approximation being Tchebysheffian, i.e. such as to make the ‘deviation’ of f, max |f − g| in the interval, as small as possible.

An arithmetic Fuchsian group is necessarily of finite covolume and so of the first kind. From the structure theorem for finitely generated Fuchsian groups those of the first kind which can be generated by two elements are triangle groups, groups of signature (1;q;0) or (1; ; 1) or groups of signature (0;2,2,2,e;0) where e is odd 6. It is known that there are finitely many conjugacy classes of arithmetic groups with these signatures or indeed with any fixed signature 3, 15. In the case of non-cocompact groups, the arithmetic groups are conjugate to groups commensurable with the classical modular group and are easily determined. For the other groups described above the space of conjugacy classes of all such Fuchsian groups of fixed signature can be described in terms of the traces of a pair of generating elements and their product. In the case of triangle groups this space is a single point. This description has been utilized to determine all classes of triangle groups 13 and groups of signature (1;q;0) which are arithmetic 15. In this paper we determine all classes of groups of signature (0;2,2,2,e;0) with e odd which are arithmetic. The techniques involving traces used so profitably in 15 are not so fruitful in this case. Consequently we have in the main resorted to a quite different method which does not rely on having a precise description of the space of conjugacy classes and hence could be applicable to groups other than those which have rank 2. Extensive use is made of results of Borell on the structure of arithmetic Fuchsian groups which require detailed information on the number fields defining the quaternion algebras. Consequently, the results are given in terms of the quaternion algebra which is determined by its defining field and ramification set, and maximal orders in that quaternion algebra.

This paper studies the compact p∞-Selmer Iwasawa module X(E/F∞) of an elliptic curve E over a False Tate curve extension F∞, where E is defined over ℚ, having multiplicative reduction at the odd prime p. We investigate the p∞-Selmer rank of E over intermediate fields and give the best lower bound of its growth under certain parity assumption on X(E/F∞), assuming this Iwasawa module satisfies the H(G)-Conjecture proposed by Coates–Fukaya–Kato–Sujatha–Venjakob.

Let I be a set with |I| > 1, let Hi(i ∈ I) be nontrivial groups and let H = *i ∈ IHi be their free product. Let R be a cyclically reduced element of Hi and write

if G is (isomorphic to) H/{R}H.

It is an important result in frame theory that the coproduct of a family of regular Lindelöf frames is Lindelöf [3]. We show that this ‘Lindelöf Tychonoff Theorem’ or ‘LTT’ is independent of ZF and indeed lies close in logical strength to the Axiom of Countable Choice, quite unlike the case with the usual (frame) Tychonoff Theorem. Along the way we construct the regular Lindelöf coreflection and obtain a simple proof of the LTT as a corollary.

Let G be a locally compact topological group, and let μ be the left Haar measure on G, with μ the corresponding outer measure. If R' denotes the non-negative extended real numbers, B (G) the Borel subsets of G, and V = {μ(C):C ∈ B(G)}, then we can define ΦG: V × V → R' by

where AB denotes the product set of A and B in G. Then clearly

so that a knowledge of ΦG will give us some idea of how the outer measure of the product set AB compares with the measures of the sets A and B.

Parametrized equations of state of the form

where s = (s1, … sr) ∈ Rr, x ∈ R, and F is smooth, are common in applied science. Often there exist multiple states x for given s, which may bifurcate as s varies.

May the triforce be the 3-uniform hypergraph on six vertices with edges {123′, 12′3, 1′23}. We show that the minimum triforce density in a 3-uniform hypergraph of edge density δ is δ4–o(1) but not O(δ4).

Let M(δ) be the maximum number such that the following holds: for every ∊ > 0 and $G = {\mathbb{F}}_2^n$ with n sufficiently large, if A ⊆ G × G with A ≥ δ|G|2, then there exists a nonzero “popular difference” d ∈ G such that the number of “corners” (x, y), (x + d, y), (x, y + d) ∈ A is at least (M(δ)–∊)|G|2. As a corollary via a recent result of Mandache, we conclude that M(δ) = δ4–o(1) and M(δ) = ω(δ4).

On the other hand, for 0 < δ < 1/2 and sufficiently large N, there exists A ⊆ [N]3 with |A| ≥ δN3 such that for every d ≠ 0, the number of corners (x, y, z), (x + d, y, z), (x, y + d, z), (x, y, z + d) ∈ A is at most δc log(1/δ)N3. A similar bound holds in higher dimensions, or for any configuration with at least 5 points or affine dimension at least 3.