It is shown that Lipschitz quotient mappings between finite dimensional spaces behave nicely (e.g., are bijective in the case of equal dimensions) if the Lipschitz and co-Lipschitz constants are close to each other. For Lipschitz quotient mappings of the plane, a bound for the cardinality of the pre-image of a point in terms of the ratio of the constants is obtained.

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.

There are many convenient ways in which a plane triangle can be defined and given projective coordinates. It can most simply be treated as an ordered triad of points (A, B, C) or dually as an ordered triad of lines (a, b, c), but it may seem more natural to regard it as a triad of points and an associated triad of lines which together satisfy the familiar incidence conditions. Again, the triangle for which Schubert [1] developed a calculus was a septuple, but Semple [2] has shown the advantages of a calculus for a triangle defined as an octuple.

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 J = (s1, s2, … ) be a collection of relatively prime integers, and suppose that π(n) = |J∩{1,2,…, n}| is a regularly varying function with index a satisfying 0 < α < l. We investigate the “stationary random sieve” generated by J, proving that the number of integers less than k which escape the action of the sieve has a probability mass function with approximate order k-α/2 in the limit as k → ∞. This result may be used to deduce certain asymptotic properties of the set of integers which are divisible by no s є J, in that it gives new information about the usual deterministic (that is, non-random) sieve. This work extends previous results valid when si=pi2, the square of the ith prime.

In a series of papers we have analysed the embedding of certain groups H as normal subgroups of absolutely irreducible skew linear groups G, see [7], [8], [9] and [10]. Here we drop the absolutely irreducibility assumption on G. If H is locally finite we derive relatively strong conditions on G, although not as strong as when G is absolutely irreducible. If H is abelian, very much in contrast to the absolutely irreducible case, we show that nothing can be said. The phrase “bounded by an integer-valued function of n only” we abbreviate to “n-bounded”.

As usual in Waring's problem, we let G(k) be the least number s such that all sufficiently large natural numbers can be written as the sum of s or fewer k-th powers of positive integers. Hardy and Littlewood gave the first general upper bound, G(k)≤(k − 2)2k − 1 + 5. Later [4], they reduced this to order k2 by first constructing an auxiliary set of natural numbers below x which are sums of sk-th powers, of cardinality with for large k. The argument, which is brief and elementary (see [15, Chapter 6]), is now referred to by R. C. Vaughan's term “diminishing ranges”, since the k-th powers are taken from intervals of decreasing lengths. This idea of choosing the variables from restricted ranges was refined by Vinogradov, whose application to exponential sum estimates gave for large k (see [18]). The recent iterative method of Vaughan and Wooley [16, 19, 21], which halves Vinogradov's bound, may be viewed as an evolved diminishing ranges argument, producing an auxiliary set with .

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].

For a natural number n, let λ(n) denote the order of the largest cyclic subgroup of (ℤ/nℤ)*. For a given integer a, let Na(x) denote the number of n ≤ x coprime to a for which a has order λ(n) in (ℤ/nℤ)*. Let R(n) denote the number of elements of (ℤ/nℤ)* with order λ(n). It is natural to compare Na(x) with ∑n≤xR(n)/n. In this paper we show that the average of Na(x) for 1 ≤ a ≤ y is indeed asymptotic to this sum, provided y ≥ exp((2 + ε)(log x log log x)1/2), thus improving a theorem of the first author who had this for y ≥ exp((log x)3/4;). The result is to be compared with a similar theorem of Stephens who considered the case of prime numbers n.

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.

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 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.

Assuming the abc-conjecture, it is shown that there are only finitely many powerful binomial coefficients with 3≤k≤n/2 in fact, if q2 divides , then . Unconditionally, it is shown that there are N1/2+σ(1) powerful binomial coefficients in the top N rows of Pascal's Triangle.

Consider a uniform body whose behaviour is described by the linear field equations of quasi-static, coupled thermoelasticity†:

Let be a topological property. We say that the locally finite sum theorem holds for the property if the following is true:

“If {Fα : α ∈ Λ} be a locally finite closed covering of X such that each Fα possesses the property , then X possesses .” The above property is known as the locally finite sum theorem (referred to as LFST in the present note). The LFST has been of interest to many people for it holds for many interesting properties such as metrizability, paracompactness, normality, collectionwise normality, local compactness, stratifiability, the property of being a normal M-space etc. etc. In [10], a large number of properties for which the LFST holds have been noted. In the same paper, a general method for proving this has been obtained. It has been shown that if a property is such that it is preserved under finite-to-one, closed continuous maps and also preserved under disjoint topological sums, then the LFST holds for and the same has been used to establish the LFST for a large number of properties. In [9,10], several interesting consequences of the LFST have been obtained. In the present note, some more interesting consequences of it have been obtained in regular, normal, collectionwise normal and countably paracompact spaces. Also, the LFST has been established for some other topological properties

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.

Erdös, Kestelman and Rogers [1[ showed that, if A1, A2,… is any sequence of Lebesgue measurable subsets of the unit interval [0, 1] each of Lebesgue measure at least η > 0, then there is a subsequence {Ani} (i = 1, 2,…) such that the intersection contains a perfect subset (and is therefore of power ). They asked for what Hausdorff measure functions φ(t) is it possible to choose the subsequence to make the intersection set ∩Ani of positive φ-measure. In the present note we show that the strongest possible result in this direction is true. This is given by the following theorem.

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.

An asymptotic theory is developed for a class of fourth-order differential equations. Under a general conditions on the coefficients of the differential equation we obtained the forms of the asymptotic solutions such that the solutions have different orders of magnitude for large x.

It is shown here that given a discrete (and infinite) set of points in the plane, it is possible to arrange them on a polygonal path so that every angle on the polygonal path is at least 9∘. This has been known to hold for finite sets (with 20∘). The main result holds for discrete sets in higher dimensions as well, with a smaller bound on the angle.