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Let Λ be an ordered abelian group. It is shown that groups in a certain class can have no non-trivial action of end type on a Λ-tree. A similar result is obtained for irreducible actions.
Erdős (see [4]) asked if there are infinitely many integers k which are not a difference or a sum of two powers, i.e., if there are infinitely many positive integers k with k≠|um ± vn| for u, v, m, n ε ℤ. This is certainly a very difficult problem. For example, it is known that the Catalan equation, i.e., the equation um − vn = 1 with uv≠0 and min (m, n)≥2 has only finitely many solutions (u, v, m, n), but there is no other positive integer k≥1 for which it is known that the equation
has only finitely many solution (u, v, m, n) with min (m, n)≥2.
This paper concerns the spectrum of the r-dimensional Sturm–Liouville equation y″ + (λI − Q(x))y = 0 with the Dirichlet boundary conditions, where Q is an r × r symmetric matrix. It is proved that, under certain conditions on Q, this problem can only have a finite number of eigenvalues with multiplity r. Further discussion is given for the multiplicities of eigenvalues when Q is an r × r Jacobian matrix.
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).
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].
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 .
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.
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.
The volume of the Lp-centroid body of a convex body K ⊂ ℝd is a convex function of a time-like parameter when each chord of K parallel to a fixed direction moves with constant speed. This fact is used to study extrema of some affine invariant functionals involving the volume of the Lp-centroid body and related to classical open problems like the slicing problem. Some variants of the Lp-Busemann-Petty centroid inequality are established. The reverse form of these inequalities is proved in the two-dimensional case.
Let v be a Henselian valuation of any rank of a field K and its unique prolongation to a fixed algebraic closure of K having value group . For any subfield L of , let R(L) denote the residue field of the valuation obtained by restricting to L. Using the canonical homomorphism from the valuation ring of v onto its residue field R(K), one can lift any monic irreducible polynomial with coefficients in R(K) to yield a monic irreducible polynomial with coefficients in K. In an attempt to generalize this concept, Popescu and Zaharescu introduced the notion of lifting with respect to a (K, v)-minimal pair (α, δ) belonging to × . As in the case of usual lifting, a given monic irreducible polynomial Q(y) belonging to R(K(α))[y] gives rise to several monic irreducible polynomials over K which are obtained by lifting with respect to a fixed (K, v)-minimal pair (α, δ). If F, F1 are two such lifted polynomials with coefficients in K having roots θ, θ1, respectively, then it is proved in the present paper that in case (K, v) is a tame field, it is shown that K(θ) and K(θ1) are indeed K-isomorphic.
Absolute curvature measures for locally finite unions of sets with positive reach are introduced, extending the definition of Zähle [13] by taking into account the absolute value of the index function. It is shown that this definition differs from that of Matheron [5] and Schneider [12]. An intersection formula of Crofton type for absolute curvature measures is proved. The role of absolute curvature measures in geometric statistics is illustrated by an example.