Given a p-subgroup P of a finite group G we express the number of p-blocks of G with defect group P as the p-rank of a symmetric integer matrix indexed by the N(P)/P-conjugacy classes in PC(P)/P. We obtain a combinatorial criterion for P to be a defect group in G.

Let X be a Hausdorff topological space and let ρ be a metric on it, not necessarily related to the topology. The space X is said to be fragmented by the metric ρ if each nonempty set in X has nonempty relatively open subsets of arbitrary small ρ-diameter. This concept was introduced by Jayne and Rogers (see [2]) while they studied the existence of Borel selections for upper semicontinuous set-valued maps.

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.

In this paper we generalize and extend work by Languasco, Perreli and Zaccagnini on the Montgomery–Hooley theorem in short intervals. We are thus able to obtain asymptotic formulae for the mean-square error in the distribution of general sequences in arithmetic progressions in short intervals. As applications we consider lower and upper bound approximations to the characteristic function of the primes in a short interval and an average over short intervals for the von Mangoldt function.

We prove a dimension-free strong uniformity theorem, and apply it in the configuration space of a large system of non-interacting particles, to describe the fast approach to equilibrium starting from off-equilibrium, and its long-term stability.

In the so-called Bernoulli model of the kinetic theory of gases, where (1) the particles are dimensionless points, (2) they are contained in a cube container, (3) no attractive or exterior force is acting on them, (4) there is no collision between the particles, (5) the collisions against the walls of the container are according to the law of elastic collision, we deduce from Newtonian mechanics two basic probabilistic limit laws about particle-counting. The first one is a global central limit theorem, and the second one is a local Poisson law. The key fact is that we can prove the “realistic limit”: first the time interval is fixed and the number of particles tends to infinity (where the density particle/volume remains a fixed constant), and then, in the second step, the time tends to infinity.

A Fourier restriction estimate is obtained for a broad class of conic surfaces by adding a weight to the usual underlying measure. The new restriction estimate exhibits a certain affine-invariance and implies the sharp ${L}^{p} - {L}^{q} $ restriction theorem for compact subsets of a type $k$ conical surface, up to an endpoint. Furthermore, the chosen weight is shown to be, in some quantitative sense, optimal. Appended is a discussion of type $k$ conical restriction theorems which addresses some anomalies present in the existing literature.

In this paper I prove the following result:

Theorem. Let f(d) be a multiplicative function such that |f(d)| ≤ 1 and ∑{ l/p : p ε P} = ∞, where P denotes the set of primes p for which f{p) = −1, and let v(n) denote the number of distinct prime factors ofn. Then for almost all n,

where A is an arbitrary constant > 3/e.

We apply Newton’s method to stochastic functional evolution equations in Hilbert spaces using semigroup methods. The first-order convergence is based on our generalization of the Gronwall-type inequality. We also establish a second-order convergence in a probabilistic sense.

We shall prove the following

THEOREM 1. Let α1, …, αn be any positive algebraic numbers and let u1…, un, ν be positive integers, relatively prime in pairs, such that ν ≥ 2 and ui > v for at least one i (1 ≤ i ≤ n). Then for every ε > 0 there are only a finite number of positive integers v such that the inequality

is satisfied, where for real α we understand by ‖α‖ the distance of α from the nearest integer.

The problem of finding necessary and sufficient condi-tions for the existence of trapped modes in waveguides has been known since 1943. [10]. The problem is the following: consider an infinite strip M in ℝ2(or an infinite cylinder with the smooth boundary in ℝn). The spectrum of the(positive) Laplacian, with either Dirichlet or Neumann boundary conditions, acting on this strip is easily computable via the separation of variables; the spectrum is absolutely continuous and equals [v0,+∞). Here, v0 is the first threshold, i.e., eigenvalue of the cross-section of the cylinder (so v0 = 0 in the case of Neumann conditions). Let us now consider the domain (the waveguide) which is a smooth compact perturbation of M (for example, weinsert an obstacle inside M). The essential spectrum of the Laplacian acting on still equals [v0, +ℝ), but there may be additional eigenvalues, which are often called trapped modes; the number of these trapped modes can be quite large, see examples in [11] and [8].

The equation

where ℱ is a certain complex-valued function of the given real periodic function λ, is studied analytically and numerically. The equation is motivated physically by a boundary-layer stability problem in which λ represents the skin-friction of the undisturbed basic flow profile. It is proved that no periodic neutral solutions exist for any attached basic flow and the implications of this result for certain vortex-wave interactions are discussed.

In our paper [12[ we made extensive use of the details of the proofs given in our earlier paper [11], and, in particular, we claimed that Lemma 3 of [11] holds, not just when Y is a metric space, but also when Y is a Hausdorff space, provided X × Y is a Fréchet space. In a corrigendum to [11], we give a corrected version of this Lemma 3, but it seems to depend, in an essential way, on the assumption that Y is a metric space, or at least a perfectly normal space. In this note we show that a modified version of this Lemma 3 enables us to justify all the theorems in [12] by use of a modified method of selection.

In this paper time-harmonic surface wave motion for progressive waves incident normally on and scattered by a partially immersed fixed vertical barrier in water of infinite depth is considered in the presence of surface tension. The problem for the velocity potential is solved, as others have been previously, by first supposing that the free-surface slopes at the barrier are prescribed and the formal solution in terms of these is obtained explicitly by complex-variable methods. To simplify the calculation the known solution corresponding to zero free-surface slopes at the barrier is subtracted out first and emphasis is placed on determining the residual potential. Finally, an appropriate dynamical edge condition is imposed on the formal solution to determine the required values of the edge-slope constants and hence fully solve the transmission problem. The problem was first examined some time ago using a complex-variable reduction procedure before the advent of this condition, although an explicit formal solution was not obtained, that earlier work forms a basis for the present investigation. It is noted in conclusion how the solution of the problem for waves generated by a partially immersed non-uniform heaving vertical plate may easily be obtained in a similar manner, since the formal solution required is just the residual potential determined in our main problem.

Oler's lattice-point theorem gives a sharp upper bound for the lattice-point enumerator GΛ of a certain class of lattices in the plane. We give a sharp lower bound for GΛ of the corresponding class of lattices in all dimensions. This result is closely related to the Blichfeldt-van der Corputgeneralization of Minkowski's fundamental lattice-point theorem.

Let S be a compact set in some euclidean space, such that every homo-thetic copy λS of S, with 0 < λ < 1, can be expressed as the intersection of some family of translates of S. It is shown that S has this property precisely when it is star-shaped, and is such that every point in the complement of S is visible from some point (necessarily on the boundary) of the kernel of S. Alternatively, S can be characterized as a compact star-shaped set, whose maximal convex subsets are cap-bodies of its kernel.

We consider an arithmetic function defined independently by John G. Thompson and Greg Simay, with particular attention to its mean value, its maximal size, and the analytic nature of its Dirichlet series generating function.

This note is concerned with those linear viscoelastic materials which are known to be compatible with thermodynamics in a sense to be denned below. It is proved that equality of its instantaneous and equilibrium elastic moduli is a necessary and sufficient condition for such a material to be elastic.