Let R be a Noetherian local ring with maximal ideal m and lull ring of fractions Q. In this paper we consider a numerical function EHI: ℤ → ℤ, where I is an m-primary ideal of R, that coincides with the Hilbert function HI for positive values and that takes account of the fractional powers of I for negative values. We focus our attention on the one-dimensional case. Among other results we characterize one-dimensional Gorenstein local rings by means of the symmetry of EHR in Theorem 2.1, we show that the extended Hilbert function is not determined by the Hilbert function in Example 2.2. and we generalize to m-primary ideals the upper bound for e1(m) given by Matlis for the maximal ideal.

We continue our study of the different representations of an integer n as a sum of two squares initiated in our final paper written with Paul Erdős [1]. We let r(n) denote the number of representations n = a2 + b2 counted in the usual way, that is, with regard to both the order and sign of a and b. We have

where x is the non-principal Dirichlet character (mod 4); moreover, r(n)≥0 if and only if n has no prime factor p ≡ 3 (mod 4) with odd exponent. We define the function b(n) on the sequence of representable numbers as the least possible value of |b|—for example, we have b(13) = 2,b(25) = 0, b(65) = 1—and we write

here and throughout the paper the star denotes that the sum is restricted to the representable integers. The problem considered here is to find an asymptotic formula for this sum, or, less ambitiously, to determine the order of magnitude of the function B(x).

We characterize injective continuous maps on the space of real or complex rectangular matrices preserving adjacent pairs of matrices. We also extend Hua's fundamental theorem of the geometry of rectangular matrices to the infinite-dimensional case. An application in the theory of local automorphisms is presented.

This paper treats rational homology groups of one-point compactifications of spaces of complex monic polynomials with multiple roots. These spaces are indexed by number partitions. A standard reformulation in terms of quotients of orbit arrangements reduces the problem to studying certain triangulated spaces Xλ,μ.

A diophantine system is studied from which is deduced an analogue of van der Corput's A5-process in order to bound analytic exponential sums of the form . The saving has now to be taken to the exponent 1/20 instead of 1/32. Our main application is a “ninth derivative test” for exponential sums which is essential for giving new exponent pairs in [3].

This paper considers the dynamics of a three-dimensional nonlinear autonomous system that models the behaviour of an electrical circuit. Results on the existence of stable periodic oscillations and the behaviour of Poincaré–Bendixon types are obtained. The work is based on a variation of classical monotone systems theory.

For an indefinite quadratic form f(x1,…,xn) of discriminant d. let P(f) denote the greatest lower bound of the positive values assumed by f for integers x1, …, xn. This paper investigates the values of P3/|d| for nonzero ternary forms of signature −1, and finds the only remaining class of forms with .

Let L be a Galois extension of the number field K. Set n = nK = deg K/ℚ, nL = deg L/ℚ and nL/K = deg L/K. Let I = IL/K denote the group of fractional ideals of K whose prime decomposition contains no prime ideals that ramify in L, and let P = {(α)ΣI: αΣK*, α>0}. Following Hecke [9}, let (λ1, λ2, …, λn − 1) be a basis for the torsion-free characters on P that satisfy λi(α) = 1 (1≤i≤n − 1) for all units α>0 in , the ring of integers of K. Fixing an extension of each λi to a character on I, then λi,(α) (1 ≤i≤n − 1) are defined for all ideals α of K that do not ramify in L. So, for such ideals, we can define . Then the small region of K referred to above is

for 0<l<½ and with the notation that, for any α ∈ ℝ, we set β. where β is the unique real satisyfing .

This paper establishes the main features of the spectral theory for the singular two-point boundary-value problemandwhich models the buckling of a rod whose cross-sectional area decays to zero at one end. The degree of tapering is related to the rate at which the coefficient A tends to zero as s approaches 0. We say that there is tapering of order p ≥ 0 when A ∈ C([0, 1]) with A(s) > 0 for s ∈ (0, 1] and there is a constant L ∈ (0, ∞) such that lims→0A(s)/sp = L. A rigorous spectral theory involves relating (1)−(3) to the spectrum of a linear operator in a function space and then investigating the spectrum of that operator. We do this in two different (but, as we show, equivalent) settings, each of which is natural from a certain point of view. The main conclusion is that the spectral properties of the problem for tapering of order p = 2 are very different from what occurs for p < 2. For p = 2, there is a non-trivial essential spectrum and possibly no eigenvalues, whereas for p < 2, the whole spectrum consists of a sequence of simple eigenvalues. Establishing the details of this spectral theory is an important step in the study of the corresponding nonlinear model. The first function space that we choose is the one best suited to the mechanical interpretation of the problem and the one that is used for treating the nonlinear problem. However, we relate this formulation in a precise way to the usual L2 setting that is most common when dealing with boundary-value problems.

In this paper we present a result of existence of infinitely many arbitrarily small positive solutions to the following Dirichlet problem involving the p-Laplacian,where Ω ∈ RN is a bounded open set with sufficiently smooth boundary ∂Ω, p > 1, λ > 0, and f: Ω × R → R is a Carathéodory function satisfying the following condition: there exists t̄ > 0 such thatPrecisely, our result ensures the existence of a sequence of a.e. positive weak solutions to the above problem, converging to zero in L∞(Ω).

The main goal of this paper is to prove that the classical theorem of local inversion for functions extends in finite dimension to everywhere differentiable functions. As usual, a theorem of implicit functions can be deduced from this “Local Inversion Theorem”. The deepest part of the local inversion theorem consists of showing that a differentiable function with non-vanishing Jacobian determinant is locally one-to-one. In turn, this fact allows one to extend the Darboux property of derivative functions on ℝ (the range of the derivative is an interval) to the Jacobian function Df of a differentiable function, under the condition that this Jacobian function does not vanish. It is also proved that these results are no longer true in infinite dimension. These results should be known in whole or part, but references to a complete proof could not be found.

We consider the Dirichlet problem for the equation -Δpu = q(|x|)f(u) in an annulus Ω ⊂ Rn, n ≥ 1, where Δpu = div(|∇u|p−2∇u) is the p-Laplacian operator, p > 1. With no assumption on the behaviour of the nonlinearity f either at zero or at infinity, we prove existence and localization of positive radial solutions for this problem by applying Schauder's fixed-point theorem. Precisely, we show the existence of at least one such solution each time the graph of f passes through an appropriate tunnel. So, it is easy to exhibit multiple, or even infinitely many, positive solutions. Moreover, upper and lower bounds for the maximum value of the solution are obtained. Our results are easily extended to the exterior of a ball, when n > p.

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.

Fermat gave the first example of a set of four positive integers {a1, a2, a3, a4} with the property that aiaj+1 is a square for 1≤i<j≤4. His example was {1, 3, 8, 120}. Baker and Davenport [1] proved that the example could not be extended to a set of 5 positive integers such that the product of any two of them plus one is a square. Kangasabapathy and Ponnudurai [6], Sansone [9] and Grinstead [4] gave alternative proofs. The construction of such sets originated with Diophantus who studied the problem when the ai are rational numbers. It is conjectured that there do not exist five positive integers whose pairwise products are all one less than the square of an integer. Recently Dujella [3] proved that there do not exist nine such integers. In this note we address the following related problem. Let V denote the set of pure powers, that is, the set of positive integers of the form xk with x and k positive integers and k>1. How large can a set of positive integers A be if aa′ + 1 is in V whenever a and a′ are distinct integers from A? We expect that there is an absolute bound for |A|, the cardinality of A. While we have not been able to establish this result, we have been able to prove that such sets cannot be very dense.

The notion of generalized X-ray for star sets in a Riemannian manifold is introduced to prove uniqueness theorems for convex bodies contained in a simply convex neighbourhood of a two-manifold. These results extend to the whole space and to arbitrary dimension when spaces of constant curvature are considered. As a consequence, a characterization of centrally symmetric convex bodies is obtained.

Given a sequence of linear formsin m ≥ 2 complex or p-adic numbers α1, …,αm ∈ Kv with appropriate growth conditions, Nesterenko proved a lower bound for the dimension d of the vector space Kα1 + ··· + Kαm over K, when K = Q and v is the infinite place. We shall generalize Nesterenko's dimension estimate over number fields K with appropriate places v, if the lower bound condition for |Rn| is replaced by the determinant condition. For the q-series approximations also a linear independence measure is given for the d linearly independent numbers. As an application we prove that the initial values F(t), F(qt), …, F(qm−1t) of the linear homogeneous q-functional equationwhere N = N(q, t), Pi = Pi(q, t) ∈ K[q, t] (i = 1, …, m), generate a vector space of dimension d ≥ 2 over K under some conditions for the coefficient polynomials, the solution F(t) and t, q ∈ K*.

General upper bounds for lattice kissing numbers are derived using Hurwitz zeta functions and new inequalities for Mellin transforms.