This second part follows on directly from the first part that appears on pages 125–156 of this volume. The references for this part are included amongst the references appearing at the end of the first part.

In [1], [2] Besicovitch showed that it is possible to translate each straight line in the plane so that the union of all the translates has zero plane measure. More recently Besicovitch and Rado [3] and independently Kinney [12] showed that the same can be done with arcs of circles instead of straight lines (see also Davies [6]). Allowing rotations as well as translations, Ward [18] showed that all plane polygonal curves can be “packed” thus (allowing overlapping) into zero plane measure, and then Davies [7], making use of Besicovitch's construction, showed translations alone to be sufficient, although these papers in fact contained stronger results concerning Hausdorff measure; the results were further generalized in [16]. The question has naturally been asked whether the class of all plane rectifiable curves can be packed by isometries (translations and rotations) into zero plane measure, but a special case of the main theorem of the present paper shows that this is impossible. The corresponding question remains open for the much smaller class of algebraic curves, or even conies.

We show that a complete metric space X has an essentially unique “nice” zero-dimensional dense Gδ subset, and derive from this a complete algebraic description of the “category algebra” (of Borel modulo first category sets) of X.

We denote by S the unit sphere in ℝ3, and µ is the rotationally invariant measure, generalizing surface area on S; thus µS = 4π. We identify directions (or unit vectors) in ℝ3 with points on S, and prove the following:

Theorem 1. If E is a subset of ℝ3 of Lebesgue measure zero, then for µ almost all directions α, every plane normal to α intersects E in a set of plane measure zero.

Let X be a compact metric space. By an invariant measure on X we shall mean a finitely additive non-negative real-valued function μ. on the Borel σ-algebra in X, such that μ(X) = 1 and Borel subsets of X have equal measure if there exists an isometry from one onto the other (not necessarily extendable to the whole of X). We note in passing that an isometric copy of a Borel set is necessarily itself Borel, by a well-known theorem of Souslin (see [3], §39. V). In Problem 2 of the Scottish Book (17. VII. 1935; see [2]), Banach and Ulam asked whether every non-empty compact metric space admits an invariant measure. This problem remains open; we give a positive answer in a veryspecial case.

A generalization and simplification of F. John's theorem on ellipsoids of minimum volume is proven. An application shows that for 1 ≤ p < 2, there is a subspace E of Lp and a λ > 1 such that 1E has no λ-unconditional decomposition in terms of rank one operators.

If C is a convex open set of a Hausdorff locally convex space, then for every compact set K ⊂ C, the closed convex hull of K is contained in C. It follows that the same is true if C is a countable intersection of convex open sets of E. However, as it is well known (and also follows from §7(a)) a convex of Gδ is not always a countable intersection of open convex sets. This paper is devoted to showing that the Gδ convex sets have some remarkable properties.

The temporal and spatial linear instability of Poiseuille flow through pipes of arbitrary cross-section is discussed for large Reynolds numbers (R). For a pipe whose aspect ratio is finite, neutral stability (lower branch) is found to be governed by disturbance modes of large axial wavelength (of order hR, where h is a characteristic cross-sectional dimension). By contrast, spatial instability for finite aspect ratios is governed by length scales between O(h) and O(hR). When the aspect ratio is increased to O(R1/7), however, these two characteristic length scales both become O(R1/7h) and a match with plane channel flow instability is achieved. Thus the general cross-section produces temporal and spatial instability if the aspect ratio is O(R1/7). Further, in the flow in a rectangular pipe neutral stability (lower branch) exists for some finite aspect ratios, while for the flow in any non-circular elliptical pipe spatial instability is possible. It is suggested that both temporal and spatial instability occur for a wide range of pipe cross-sections of finite aspect ratio. Part 2 (Smith 1979a), which studies the upper branch neutral stability, confirms the importance of the O(hR) scale modes in neutral stability for finite aspect ratios.

We are concerned with invertible transformations of the unit n-dimensional cube In, 2 ≤ n ≤ ∞, which preserve n-dimensional Lebesgue measure μ. Following Halmos [4], we denote the space of all such transformations by G = G(In), and the subset of G consisting of homeomorphisms by M = M(In). We ask to what extent, and in what sense, can we approximate an arbitrary transformation g in G by a homeomorphism h in M. New results are obtained in the course of presenting a new proof of the theorem of J. Oxtoby and H. E. White, Jr., stated below.

A study complementary to Part 1 (Smith 1979) is made of the linear stability characteristics, at high Reynolds number (R), of Poiseuille How through tubes with closed cross-sections. The first significant deviation of the upper branch of the neutral stability curve (Part 1 having described the lower bilanch) from that of plane Poiseuille flow arises when the aspect ratio is decreased from infinity to O(R1/11). The axial wavenumber α on the upper branch is then O(R-1/11). A further decrease of the aspect ratio, to a finite value, forces this α to fall sharply to O(R-1). A similar phenomenon occurs for the lower branch (Part 1). Thus the two branches are likely to meet only when the aspect ratio becomes finite, with the neutrally stable disturbances then having very large axial length scales.

A higher-order, double boundary-layer theory is employed to investigate the mass transport velocity due to two-dimensional standing waves in a system consisting of two semi-infinite, homogeneous fluids of different densities and viscosities. For moderately large wave amplitudes, the leading correction to the tangential mass transport velocity near the interface is extremely significant and may typically contribute about 20% of the total velocity.

In this note we remark on certain “universal fixed messages” for the Rivest-Shamir-Adleman cryptosystem [1], and we describe how in certain cases these universal fixed messages can play a role in an attempt to break the cryptosystem. This use of the universal fixed messages in an attempt to break the RSA cryptosystem† involves a certain prime factorization technique using the fixed messages. We characterize the situation in which such an attempt would be successful in practice, and we show that a user of the RSA cryptosystem can easily arrange to avoid this situation. Hence the use of the present technique does not pose a threat to the security of the RSA cryptosystem.

Suppose that we have a system of congruences ai (mod ni) 1 < n1 < … < ni < … < nk such that every integer is congruent to at least one ai (mod ni), then we say that it is a covering system of congruences. If ni | m, 1 ≤ i ≤ k, we say that m is a covering number. We shall use the symbol ℕ to denote the natural numbers together with zero, then m is a covering number if, for each q there is an aq such that

Before stating the main theorem, we would like to recall the basic properties of the “zeta-functions” attached to cusp forms on SL (2, ℤ). Let k be an even integer ≥ 12 and f a cusp form of weight k on SL (2, ℤ) with q-expansion We shall assume that c1 = 1, and that f is an eigenfunction of the Hecke operators. Define φ(s) as the Dirichlet series The series and the product

over the primes are equal and absolutely convergent for Re (S) > ½(k + 1).

Let be a finite sequence of positive integers. If we want to show that contains primes, or at least almost-primes (i.e. numbers with few prime factors), sieve methods give better results if weights of a certain kind are attached to the elements of .

Barban and Vehov posed in [1] the problem of minimizing the quadratic form

under the conditions

where 1 ≤ z1 < z2 and μ(n) is the Möbius function. They also commented on the connections of this problem with zero-density estimates of L-functions near the lines δ = 1 and ½, and with Linnik's prime number theorem. This program was carried out some years later by other authors, in the first place by Selberg [10], and later by Motohashi ([7]–[9]), Graham [2] and the present author ([4], [5]).

Let f = f(x, y) be a quadratic form with real coefficients in two integer variables x, y. Let V(f) be the set of values taken by f(x, y) at points (x, y) ≠ (0,0). Impose the same conditions on a second form f′. Trivially, f equivalent to f′ implies V(f) = V(f′). It will be shown that the converse implication holds in general for definite forms; the obvious exception f = x2 + xy + y2, f′ = x2 + 3y2 will be shown to be essentially the only one.

The first problem in diophantine approximation is for a given real number ξ, and positive number x to find a fraction s/t with t ≤ x which is close to ξ. This problem can be rephrased in geometric terms. Given a vector v in ℝ2, find a vector a = (a1, a2) with integer coordinates and 1 1 ≤ al ≥ x such that the vectors a and v are nearly parallel. Simultaneous approximation of d − 1 real numbers can be recast in terms of approximation of the angle between two vectors in d-dimensional Euclidean space.

This paper originated with the observation that while all of the known stable lattice packings of spheres are highly symmetric, it is futile to try to prove a converse statement: the ordinary integer-lattice provides a distinctly unstable packing of spheres, but admits a large group of orthogonal symmetries nonetheless. The integerlattice is in fact very unstable—the slightest perturbation places the spheres in a more efficient configuration. We will call such a lattice fragile. The purpose of this note is to prove that a highly symmetric lattice must be either stable or fragile.

Recently the notion of elementary symmetrization attracted new attention in the field of convex sets (see [L; W]), and it was proved that Minkowski's “Quermassintegrale” are decreased by elementary symmetrization. On the other hand, the concept of Schwarz symmetrization for Borel functions gained new interest from its possible applications in the field of elliptic partial differential equations (see [S1, S2, HI, T1, T2, HY1, HY2, GL1, GL2]).