In the paper [2] Hsia noted that the forms x2+xy+y2+9z2 and x2+3y2+3yz+3z2 constitute a genus and that both forms are regular; he asked whether there exist any other genera containing two or more regular forms. In this note it is proved that the forms

are regular. They constitute a genus with discriminant 27 (in the normalization used by Brandt and Intrau in [1]). It is noteworthy that Hsia's genus has the same discriminant.

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.

The two centred expansion of the Coulomb Green's function arises naturally in discussing the static interaction energy of two charge distributions ρ1, and ρ2. This is given by the well-known expression

A slow steady motion of incompressible viscous liquid in the space between two fixed concentric spherical boundaries is considered. The motion arises from two point-sources of strengths ±2m at arbitrary points A, B on the outer sphere r = a. The velocity is calculated as the vector sum of the velocities in two simpler motions in each of which there is an axis of symmetry so that a stream function can be used. The force exerted by the liquid on the inner boundary r = b is similarly the resultant of two forces, each passing through the common centre of the spheres; it can be simply expressed in terms of a, b, m and the vector .

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.

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.

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.

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”.

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.

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†:

A monic polynomial in ${\mathbf{F} }_{q} [t] $ of degree $n$ over a finite field ${\mathbf{F} }_{q} $ of odd characteristic can be written as the sum of two irreducible monic elements in ${\mathbf{F} }_{q} [t] $ of degrees $n$ and $n- 1$ if $q$ is larger than a bound depending only on $n$. The main tool is a sufficient condition for simultaneous primality of two polynomials in one variable $x$ with coefficients in ${\mathbf{F} }_{q} [t] $.

Let L/K be an extension of number fields and let be the subgroup of the unit group consisting of the elements that are roots of units of . Denote by (L/K, B) the number of points in with relative height in the sense of Bergé-Martinet at most B. Here ℙ1(L) stands for the one-dimensional projective space over L. In this paper is proved the formula (L/K, B) = CB2 + O(B2−1/[L:Q]), where C is a constant given in terms of invariants of L/K such as the regulators, class number and discriminant.

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

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.

We prove a generalization of a theorem of Ryshkov relating the Voronoï vectors of lattices to the defining conditions for the Minkowski fundamental domain . This is then used to prove that a Minkowski reduced basis of a lattice of dimension n < 7 consists of strict Voronoï vectors.

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 .

We consider an elliptic self-adjoint first-order differential operator $L$ acting on pairs (2-columns) of complex-valued half-densities over a connected compact three-dimensional manifold without boundary. The principal symbol of the operator $L$ is assumed to be trace-free and the subprincipal symbol is assumed to be zero. Given a positive scalar weight function, we study the weighted eigenvalue problem for the operator $L$ . The corresponding counting function (number of eigenvalues between zero and a positive $\unicode[STIX]{x1D706}$ ) is known to admit, under appropriate assumptions on periodic trajectories, a two-term asymptotic expansion as $\unicode[STIX]{x1D706}\rightarrow +\infty$ and we have recently derived an explicit formula for the second asymptotic coefficient. The purpose of this paper is to establish the geometric meaning of the second asymptotic coefficient. To this end, we identify the geometric objects encoded within our eigenvalue problem—metric, non-vanishing spinor field and topological charge—and express our asymptotic coefficients in terms of these geometric objects. We prove that the second asymptotic coefficient of the counting function has the geometric meaning of the massless Dirac action.

Abstract. We show that the set of T-numbers in Mahler's classification of transcendental numbers supports a measure whose Fourier transform vanishes at infinity. A similar argument shows that U-numbers also support such a measure.

Let ˜ be an equivalence relation on a topological space X. A point x ε X i s stable with respect to ˜ if it is in the interior of an equivalence class. We may also add, if ambiguity arises, that x is stable under perturbations in X. Let E be a Banach space, and let L(E) be the Banach space of continuous linear endomorphisms of E, with norm given by |T| = sup{ |T(x) | : |x| = 1}. In this paper we discuss stability of elements of L(E) with respect to some natural equivalence relations.