We prove that the coefficients of the mock theta functions

\begin{eqnarray*} f(q) = \sum_{n=1}^{\infty} \frac{ q^{n^2}}{(1+q)^2 (1+q^2)^2 \cdots (1+q^n)^2 } \end{eqnarray*}

\begin{eqnarray*} \omega(q)=1+\sum_{n=1}^\infty \frac{q^{2n^2+2n}}{(1+q)^2(1+q^3)^2\cdots (1+q^{2n+1})^2} \end{eqnarray*}

It has been shown that univalent circle packings filling the complex plane C are unique up to similarities of C. Here we prove that bounded degree branched circle packings properly covering C are uniquely determined, up to similarities of C, by their branch sets. In particular, when branch sets of the packings considered are empty we obtain the earlier result.

We also establish a circle packing analogue of Liouville's theorem: if f is a circle packing map whose domain packing is infinite, univalent, and has recurrent tangency graph, then the ratio map associated with f is either unbounded or constant.

1. The only double binary forms for which the complete systems of invariants and covariants are known appear to be the (1, 1), (2, 1) and (2, 2) forms, for which the complete systems were determined sixty years ago by Peano. In the present paper we determine the complete system of the binary (3, 1) form and establish its irreducibility. The system proves to contain twenty forms, and is thus scarcely more complicated than that of the (2, 2) form which includes eighteen concomitants, and much simpler than that of the (2, 1, 1) form derived by Gilham. It is hoped in a subsequent note to interpret the system geometrically.

The conjugate g of a periodic and integrable function f is not necessarily integrable, even when f is monotone inside its fundamental interval. Paley and Wiener, however, proved that g is integrable if f is monotone and odd. A simpler proof was given later by Zygmund‡.

Let D(R) be the Schwartz space of C∞ functions with compact support on R and let H(D) be the space of all C∞ functions defined on R for which every element is the Hilbert transform of an element in D(R), i.e.

where the integral is defined in the Cauchy principal-value sense. Introducing an appropriate topology in H(D), Pandey [3] defined the Hilbert transform Hf of f ∈ (D(R))′ as an element of (H(D))′ by the relation

and then with an appropriate interpretation he proved that

.

In this paper we give an intrinsic description of the space H(D) and its topology, thereby providing a solution to an open problem posed by Pandey ([4], p. 90).

The solution for a simply supported many-sided polygonal plate does not agree with that for the corresponding circular plate. This paper describes the earlier work of Rao and Rajaiah on polygonal plates and then explains why best convergence of series solutions occurs when the boundary conditions are defined as

Previous investigations by the author into the Stokes phenomenon pertaining to solutions of the differential equation dnu/dzn = (–1)nzmu are extended in order to find when different equations have the same set of Stokes multipliers, with perhaps a series of zeros being additionally allowed. The reason for periodic cycles to exist (with n fixed and m varying), with the same Stokes multipliers regained after a complete cycle, is traced to certain transformation properties of the equations. Within the first cycle (with n fixed and m varying) further remarkable identities exist between the Stokes multipliers, and this also is traced to special transformations between the equations. Relations are found toexist when values of m are chosen so that the highest common factors of the two integers n and n/(n + m) are identical. Finally, a transformation of the independent variable is deduced whereby the set of Stokes multipliers for an equation of order n is identical (apart from the additional zeros) to that for an equation of lower order. A hierarchy of equations is thrown up, whereby certain basic equations are transformed to yield more advanced equations of higher order and different m.

Conditional on the extended Riemann hypothesis, we show that with high probability, the characteristic polynomial of a random symmetric $\{\pm 1\}$ -matrix is irreducible. This addresses a question raised by Eberhard in recent work. The main innovation in our work is establishing sharp estimates regarding the rank distribution of symmetric random $\{\pm 1\}$ -matrices over $\mathbb{F}_p$ for primes $2 < p \leq \exp(O(n^{1/4}))$ . Previously, such estimates were available only for $p = o(n^{1/8})$ . At the heart of our proof is a way to combine multiple inverse Littlewood–Offord-type results to control the contribution to singularity-type events of vectors in $\mathbb{F}_p^{n}$ with anticoncentration at least $1/p + \Omega(1/p^2)$ . Previously, inverse Littlewood–Offord-type results only allowed control over vectors with anticoncentration at least $C/p$ for some large constant $C > 1$ .

1. Introduction. This Department has made extensive observations with a proton magnetometer towed behind a ship (Hill (2)). The magnetometer consists of a bottle of water surrounded by a coil of wire through which a current flows and produces a polarizing field of about 100 oersted. A small proportion of the protons in the water are polarized along the resultant field. When the current is turned off they precess about the Earth's field and produce in the coil an e.m.f. whose frequency is a measure of the field.

An approximation is found to the solution of the partial differential equation

in the region −1 ≤ x ≤ 1, t > 0, where u satisfies a general linear boundary condition on x = ± 1. This approximation is a polynomial in x, and is an exact solution of a perturbed form of the differential equation. By choosing the perturbation appropriately, this approach is mathematically equivalent to some recent methods for solving the differential equation in the form of a Chebyshev series. Better approximations to the required solution (and particularly to the eigenvalues) are obtained by choosing the perturbation to satisfy a least squares criterion.

In his well-known treatise, Banach mentions that there exist complete normed bi-orthogonal systems in any finite-dimensional (real) Banach space—a result he attributes to H. Auerbach ((1), p. 238). The purpose of this note is to present a proof of this result (valid for complex as well as real Banach spaces), and to give some applications to the theory of tensor products of Banach spaces.

The examples are unconnected.

According to some published footnotes, examples are known of Serre fibrations which are not Hurewicz fibrations. However, as far as I know no such example has been published. The following example, which was stimulated by a more complicated example of G. Horrocks, seems worth mentioning, since it is extremely simple and does suggest some further questions.

In this paper we study the homology of groups with coefficients in metabelian Lie powers, and apply the results to obtain information about elements of finite order in certain free central extensions of groups. Perhaps the most prominent example to which our results apply is the relatively free group

where Fd is the (absolutely) free group of rank d. Thus Fd(Bc) is the free group of rank d in the variety Bc of all groups which are both centre-by-(nilpotent of class ≤ c − 1)-by-abelian and soluble of derived length ≤ 3. It was pointed out in [1] that the order of any torsion element in Fd(Bc) divides c if c is odd and 2c if c is even. This, however, is a conditional result as it does not answer the question of whether or not there are any torsion elements in (1·1). Up to now, this question had only been answered in case when c is a prime number [1] or c = 4 [8]. In these cases Fd (Bc) is torsion-free if d ≤ 3, and elements of finite order do occur in Fd(Bc) if d ≥ 4. Moreover, the torsion elements in Fd(Bc) form a subgroup, and the precise structure of this torsion subgroup was exhibited in [1] in the case when c is a prime and in [8] for c = 4. In the present paper we add to this knowledge. On the one hand, we show that for any prime p dividing c the group Fd(Bc) has no elements of order p for all d up to a certain upper bound, which takes arbitrarily large values as c varies over all multiples of p. On the other hand, we show that for prime powers does contain elements of order p whenever d ≥ 4. Finally, we exhibit the precise structure of the p-torsion subgroup of when p ≠ 2. Precise statements are given below (Corollaries 1 and 2). Our results on (1·1) are a special case of more general results (Theorems 1′−3′) which refer to a much wider class of groups, and which are, in their turn, a consequence of our main results on the homology of metabelian Lie powers (Theorems 1–3).

Let f(t) be defined and measurable for t > 0, and suppose that it has non-vanishing moment constants

where the integrals are Cauchy-Lebesgue integrals, or possibly if f(t) is null for t > t0 but not for t < t0 (a convention to which we adhere throughout).

The differential equation

defining the function of the amplitude sL is integrated by an iterative procedure. An expression for the phase

is obtained in terms of integrals of ωL and its derivatives. The method is applied to the numerical calculation of Coulomb functions and the accuracy of the results is discussed. Recurrence relations are used to obtain øL, L ≽ 1.

Throughout this paper k will be an algebraically closed field, R the ring k[[X1, …, XN]] and (kN, 0) the k-scheme Spec(R).

Given an infinite system of linear equations

where the aij depend on a parameter λ, the eigenvalue problem is to determine values of λ for which xj (j = 1, 2, …) are not all zero. This problem (Taylor (3) and Vaughan (4)) can arise in the vibration of rectangular plates. Little theoretical work, however, appears to have been done concerning the existence and determination of the eigenvalues. The usual procedure (see (3) and (4)) is to consider a truncated or reduced system of N equations and find the values of λ for which the determinant of the N × N matrix [aij] vanishes. If a particular λ tends to a constant value as N is increased then this value is assumed to be an eigenvalue. The question therefore arises as to what happens if no limit exists. Can we assert that there are no eigenvalues? By constructing an appropriate example we show that the non-existence of a limit does not imply the non-existence of eigenvalues. In order to construct our example we first establish a result concerning the Legendre polynomials.

A Haken sphere $S$ in a 3-manifold $M$ with a Heegaard splitting $V\cup_F W$ is a separating 2-sphere in $M$ which intersects the Heegaard surface $F$ only in an essential circle in $F$. We describe how any two Haken spheres in the genus 2 Heegaard splitting of connected sum of two lens spaces are related, which extends Scharlemann–Thompson's corresponding result on Haken spheres in genus 2 Heegaard splitting for $S^3$.

The Atiyah–Singer equivariant signature formula implies that the products of isometrically inequivalent classical spherical space forms with the circle are not homeomorphic, and in fact the same conclusion holds if the circle is replaced by a torus of arbitrary dimension. These results are important in the study of group actions on manifolds. Algebraic $K$-theory yields standard classes of counterexamples for topological and smooth analogs of spherical spaceforms. The results of this paper characterize pairs of nonhomeomorphic topological spherical space forms whose products with a given torus of arbitrary dimension are homeomorphic, and the main result is that the known counterexamples are the only ones that exist. In particular, this and basic results in lower algebraic $K$-theory show that if such products are homeomorphic, then the products are already homeomorphic if one uses a 3-dimensional torus. Sharper results are established for important special cases such as fake lens spaces. The methods are basically surgery-theoretic with some input from homotopy theory. One consequence is the existence of new infinite families of manifolds in all dimensions greater than three such that the squares of the manifolds are homeomorphic although the manifolds themselves are not. Analogous results are obtained in the smooth category.