Weak Results Research Articles

Differentiable semigroups are Lie groups

We present here a modern, detailed proof to the following theorem which was introduced by Garrett Birkhoff [1] in 1938. If S is a local semigroup with neighborhood of 1 homeomorphic to a Banach space and with multiplication strongly differentiable at 1, then S is a local Lie Group. Although this theorem is more than 50 years old and remains the strongest result relating to Hilbert′s fifth problem in the infinite dimensional setting, it is frequently overlooked in favor of weaker results. Therefore, it is the goal of the authors here to clarify its importance and to demonstrate a proofwhich is more accessible to contemporary readers than the one offered by Birkhoff.

On the stability of a gaseous sphere against non-radial perturbations

A simplified proof of the Antonov-Lebovitz theorem, asserting that any spherical barotropic star having a mass density decreasing monotonically outwards and vanishing at its surface is stable to all non-radial perturbations, is presented. A simple argument showing in a straightforward way a related but somewhat weaker result is also developed, according to which any such star is stable if and only if it is stable to radial perturbations. Extension of these results to a star with a non-decreasing specific entropy distribution is also briefly discussed

REGULARITY OF SOLUTIONS FOR SINGULAR SCHRÖDINGER EQUATIONS

Local and global regularity properties of weak solutions of the Schrödinger equation −Δu+qu=λu play an important role in the spectral theory of the corresponding operator [Formula: see text]. Central among these properties is local boundedness of the solutions u, which is derived in an elementary way for potentials q whose negative parts q− lie in the local Kato class K loc . The method also provides mean value inequalities for and, in case q+ is in K loc too, continuity of u. To employ these mean value inequalities for bounds on eigenfunctions of T in a fixed direction, classes Kρ are introduced which reflect the behavior of q at infinity. A couple of examples allow to compare these classes with more conservative ones like the Stummel class Q and the global Kato class K. The fundamental property of local boundedness of solutions also serves as a base for a very short proof of the self-adjointness of T if the operator is bounded from below and q−∈K loc . If q(x) is permitted to go to −∞, as |x|→∞, a large class K ρ which guarantees self-adjointness of T is derived and contains the case q−(x)= O (|x|2). The Spectral Theorem then allows to deduce rapidly decaying bounds on eigenfunctions for discrete eigenvalues, at least if q−(x)= o (|x|2). This is also the condition under which the existence of a bounded solution is sufficient to guarantee λ∈σ(T). Here q−(x)= O (|x|2) appears as a borderline case and is discussed at some length by means of an explicit example. The class of admissible operators extending to these borderline cases with potentials singular locally and at infinity, the regularity results for solutions being mostly optimal, as demonstrated by numerous examples, yet the proofs being shorter and more straightforward than those to be found in literature for smaller classes and weaker results, the sets Kρ under consideration and the methods employed appear to be quite natural.

Uniform distribution of integral points on multidimensional ellipsoids

Let Q(X), XT=(x1,...,xl), be a positive definite, integral-valued, primitive, quadratic form of l⩾4 variables, let τ(η) be the number of solutions of Eq. Q(X)=n, let τ(η,Ω) be the number of the solution of the equation Q(X)=n such that X/√ɛΩ, where Ω is an arbitrary convex domain on Q(X)=1 with a piecewise smooth boundary. One investigates the asymptotic behavior of the quantity τ(η,Ω) (n→∞). In the case of an even l⩾4 the result is formulated in the following manner: for (n,N)=1 and n→∞ one has , e>o, whereμ(Ω) is the measure of the domain Ω, normalized by the conditionμ(E)=1, where E is the ellipsoid Q(X)=1. Weaker results have been obtained earlier by various authors. In the case of the simplest domains (“belt,” “cap”) the remainder in (1) can be brought to the form . The last estimate for large l is close to an unimprovable one. The proof makes use of the theory of modular forms and of Deligne's estimates.

The generation and propagation of internal layers

4 = c2u, f(u, x)7 (1.1) is studied, where f is of bistable type for each x. Here “bistable” means that f vanishes for exactly three values of U, and fU is negative at the two outer ones; the prototype is f = u u3. When f is independent of x, the equation has a dynamical theory which is pretty well understood. Its most striking feature in that-case is the existence of travelling wave solutions, which are globally stable. This is also true for the heterogeneous equation (1. l), although such travelling wave solutions have up to now only been studied by formal asymptotics. They are not travelling waves in the strictest sense, but are rather frontal structures (confined to thin zones of width O(E)) which move with variable velocity. Starting with smooth arbitrary initial data Us which do not depend on E, the formation of these fronts at certain locations can be predicted, and their approximate subsequent motion can be determined by solving an ordinary differential equation. This heuristic analysis is explained in Section 2. The purpose of this paper is to provide the heuristics with a rigorous footing. In fact, we obtain detailed estimates for the error made by approximating the exact solution by the function constructed in Section 2. For simplicity, we treat the case when there is only a single front, although multiple fronts may clearly occur. The process under consideration consists of two parts, proceeding under two different time scales: which are the initial generation of a layer, followed by its propagation. The analysis describing the first part is the more involved. (If comparison functions based on the ordinary differential equation in Section 2 are used, the analysis simplifies, but weaker results are obtained.)

Modelling properties of spanning trees: Diameter and distance sum

The diameters of graphs and their spanning trees have been extensively studied and used in a wide variety of applications. Another important invariant of a spanning tree is its distance sum—that is, the sum of the distances between each pair of nodes in the tree. First, several results on the possible values for diameters of spanning trees are reviewed. Then it is shown that although the corresponding properties do not hold for distance sums of spanning trees, some weaker results of a similar type are true.

Quasi-bounded and singular thermal potentials

In his monograph [2], Doob presented the theories of quasi-bounded and singular functions associated with the Laplace and heat equations. In particular, on p. 712, he proved in the classical case that a potential is quasi-bounded if and only if its associated measure vanishes on polar sets. However, on p. 714, he proved only weaker results for the thermic case, namely that a potential is quasi-bounded (i) if its associated measure vanishes on semi-polar sets, and (ii) only if its measure vanishes on polar sets. The main purpose of this note is to establish that the exact analogue of the result for the classical case holds for the thermic case. We also use a related theorem on thermal potentials, to prove a second characterization of those which are quasi-bounded, analogous to one due to Arsove and Leutwiler for the classical case. We also give the corresponding characterizations of singular thermal potentials.

The relationship between the rate of inflation and its unpredictability in high inflation Latin American countries

Pourgerami and Maskus (PM) (1987) find three “difficulties” in our analysis of the relationship between the rate of inflation (P) and its unpredictability (Pf) in Latin American countries.’ The first of these is a point very well taken, namely that the positive relationship we hypothesized might well be offset by a negative one arising because higher rates of inflation could be expected to induce the relevant economic agents to invest more in generating the relevant information. in utilizing it efficiently and in developing the instruments capable of hedging against inflation. However, since any such adjustments would be institutional in nature and take time to be carried out. it is obvious that this consideratipn would not be relevant for short-run analysis involving year-to-year or especially quarter-to-quarter changes in inflation rates as in our analysis. The other “difficulties.” namely that quarterly observations are more relevant than annual ones in testing such relationships and that rational expectations (RE) models of the type employed by PM are preferable to adaptive expectations models of the type we used, are criticisms more relevant to the short-run relation between inflation and its predictability, but, as we shall presently demonstrate. are not persuasive. Nevertheless. since the results presented by PM (as they have noted) show the relationship between PI and P to be considerably weaker than was indicated in earlier results (including our own). the substantial difference in results deserves explanation. Since their time coverage. frequent!. of adjustment of expectations. model of esprct;ltions and to some extent countr! coverage were all different than ours, it is not possible to learn from their results the reason for the difference. By employing the same time and country coverage as in PM, and also an estimation procedure (Hildreth-Lu) which corrects for autocorrelation, and continuing to use a modified version of our adaptive expectations model (MAE): but varying the duration of the periods, the results presented in Table 1 shed light on the source of the difference. As can be seen, the use of quarterly instead of annual data does not have a large effect on the results, thereby suggesting that PM’s use of quarterly data is not the source of their weaker results. Indeed, note that in six of the seven high inflation countries the slope coefficient is larger and more significant when quarterly data are employed than when annual data are employed. The largest differences are between the results obtained from our own MAE model estimated with quarterly data and those obtained by PM with their RE model with the same data. In all seven cases the PM estimate of @, is significantly lower than our own corresponding estimate. Which estimate is more reliable? Even if one were to agree in principle with PM that RE models are superior to AE models of price expectations, the particular RE model employed by PM can hardly be considered acceptable, let alone superior to our MAE model. With their 20 right hand side variables, i.e.. four lags of each of five different explanatory variables. relative to our four right hand side variables, it is not surprising that PM havq been able to increase the explained variation of P. correspondingly reduc-

Prime models and almost decidability

This paper introduces and investigates a notion that approximates decidability with respect to countable structures. The paper demonstrates that there exists a decidable first order theory with a prime model that is not almost decidable. On the other hand it is proved that if a decidable complete first order theory has only countably many complete types, then it has a prime model that is almost decidable. It is not true that every decidable complete theory with only countably many complete types has a decidable prime model. It is not known whether a complete decidable theory with only countably many countable models up to isomorphism must have a decidable prime model. In [1] a weaker result was proven—if every complete extension, in finitely many additional constant symbols, of a theory T fails to have a decidable prime model, then T has 2ω nonisomorphic countable models. The corresponding statement for saturated models is false, even if all the complete types are recursive, as was shown in [2]. This paper investigates a variation of the open question via a different notion of effectiveness—almost decidable.A tree Tr will be a subset of ω<ω that is closed under predecessor. For elements f, g in ω<ω ∪ ωω, ƒ ⊲ g iffdf ∀i < lh(ƒ)[ƒ(i) = g(i)].

On consecutivek'th power residues

It is known that for fixedk≽2 and every sufficiently large primep there exists an integerr such thatr andr+1 are bothk'th power residues modulop. It has been conjectured that the least positiver with this property is bounded in terms ofk only. We prove this conjecture in the casek is a prime, and a weaker result in the general case.

Prime models and almost decidability

This paper introduces and investigates a notion that approximates decidability with respect to countable structures. The paper demonstrates that there exists a decidable first order theory with a prime model that is not almost decidable. On the other hand it is proved that if a decidable complete first order theory has only countably many complete types, then it has a prime model that is almost decidable. It is not true that every decidable complete theory with only countably many complete types has a decidable prime model. It is not known whether a complete decidable theory with only countably many countable models up to isomorphism must have a decidable prime model. In [1] a weaker result was proven—if every complete extension, in finitely many additional constant symbols, of a theory T fails to have a decidable prime model, then T has 2ω nonisomorphic countable models. The corresponding statement for saturated models is false, even if all the complete types are recursive, as was shown in [2]. This paper investigates a variation of the open question via a different notion of effectiveness—almost decidable.A tree Tr will be a subset of ω<ω that is closed under predecessor. For elements f, g in ω<ω ∪ ωω, ƒ ⊲ g iffdf ∀i < lh(ƒ)[ƒ(i) = g(i)].

On the Unimodality of High Convolutions of Discrete Distributions

It is shown that if $\{p_j\}$ is a discrete density function on the integers with support contained in $\{0, 1, \cdots, d\}$, and $p_0 > 0, p_1 > 0, p_{d - 1} > 0, p_d > 0$, then there is an $n_0$ such that the $n$-fold convolution $\{p_j\}^{\ast_n}$ is unimodal for all $n \geq n_0$. Examples show that this result is nearly best possible, but weaker results are proved under less restrictive assumptions.

Bound on the maximum negative ionization of atoms and molecules

It is proved that ${N}_{c}$, the number of negative particles that can be bound to an atom of nuclear charge $z$, satisfies ${N}_{c}<2z+1$. For a molecule of $K$ atoms, ${N}_{c}<2Z+K$ where $Z$ is the total nuclear charge. As an example, for hydrogen ${N}_{c}=2$, and thus ${\mathrm{H}}^{\ensuremath{-}\ensuremath{-}}$ is not stable, which is a result not proved before. The bound particles can be a mixture of different species, e.g., electrons and $\ensuremath{\pi}$ mesons; statistics plays no role. The theorem is proved in the static-nucleus approximation, but if the nuclei are dynamical, a related, weaker result is obtained. The kinetic energy operator for the particles can be either $\frac{{[\frac{p\ensuremath{-}eA(x)}{c}]}^{2}}{2m}$ (nonrelativistic with magnetic field) or ${{{[pc\ensuremath{-}eA(x)]}^{2}+{m}^{2}{c}^{4}}}^{\frac{1}{2}}\ensuremath{-}m{c}^{2}$ (relativistic with magnetic field). This result is not only stronger than that obtained before, but the proof (at least in the atomic case) is simple enough to be given in an elementary quantum-mechanics course.

Tail behavior of birth-and-death and stochastically monotone processes

The possible types of tail σ-field of non-homogeneous continuous-time birth-and-death processes are described. Some weaker results are obtained for the tail behavior of stochastically monotone processes.

On the survival probability of a slightly advantageous mutant gene with a general distribution of progeny size--a branching process model.

A branching process method is employed to study the survival probability of a slightly advantageous mutant gene with a general distribution of progeny size in a large population. A counter-example to a classic proposition is given. A somewhat weaker result is proved.

Consistency of Single Linkage for High-Density Clusters

Abstract High-density clusters are defined on a population with density f in r dimensions to be the maximal connected sets of form {x | f(x) ≥ c}. Single-linkage clustering is evaluated for consistency in detecting such high-density clusters—other standard hierarchical techniques, such as average and complete linkage, are hopelessly inconsistent for these clusters. The asymptotic consistency of single linkage closely depends on the percolation problem of Broadbent and Hammersley—if small spheres are removed at random from a solid, at which density of spheres will water begin to flow through the solid? If there is a single critical density such that no flow takes place below a certain density, and flow occurs through a single connected set above that density, then single linkage is consistent in separating high-density clusters (by disjoint single-linkage clusters that include a positive fraction of sample points in the respective clusters and pass arbitrarily close to all points in the respective clusters). The existence of a single critical point remains a conjecture. A weaker result is proved that shows that single-linkage clusters detect high-density clusters if there is a low enough valley separating them.

On bahadur's representation of quantiles in nonregular cases

The paper considers Bahadur’s representation of quantiles (see Bahadur (1966), Ann. Math. Statist., 37, 577-580) in cases where the usual assumptions regarding the existence and boundedness of the derivatives of the distribution function in a neighbourhood of the population quantile(s) of interest are not met. Following Kiefer((1967), Ann. Math. Statist., 38, 1323-1342) we provide an exact order of the remainder term in Bahadur’s representation. A weaker result regarding the order of the remainder term is also provided under weaker regularity assumptions.

Approximationsfehler der Ableitungen von interpolationspolynomen

Let p n ∈ π n ( n = 1, 2, 3,…) denote the polynomial interpolating a given function u∈ C 1|−1, 1| at the extremum points t j = cos jπ n . 0 ⩽ j ⩽ n. of the nth C̆ebys̆ev polynomial. Denote the C̆ebys̆ev norm on C¦−1, 1¦ by ∥∥ . Then the following estimate holds: ∥ u′- p′ n ∥⩽ (2 + 2 log n) inf{∥ u′- q∥: q∈ Π n-1} . This improves earlier results by Schönhage ( Math. Z., 94 (1966), 79–83) and the author ( J. Approx. Theory, 23 (1978). 261–266). The new result is based on the use of a generalized Lebesgue function D n . For the given interpolation points the estimate is sharp. Similar but weaker results are obtained for higher derivatives.

The number of generators of a module over a fully bounded ring

We prove a theorem giving an upper bound for the number of generators of a module in terms of local data, for a class of rings which includes Noetherian rings which are right fully bounded. Forster and Swan have proved such results for commutative Noetherian rings and finite algebras over these rings. Eisenbud and Evans have shown, further, that from any given set of generators for a finitely generated module over such an algebra, the number predicted by the theorem can be obtained by a certain kind of elementary transformation. This result we also obtain for our wider class of rings. Regarding these results as results about the existence and structure of epimorphisms F -+ A, where F is free and A the module, we obtain also corresponding results were F is a direct sum of copies of some projective module other than R. The chief difficulties throughout are those involved in finding substitutes for arguments involving localization, which is not an available tool in our context. For prime rings of Krull dimension one, we can remove the boundedness condition, and also obtain for hereditary Noetherian prime (HNP) rings the analogue of a result of Roiter’s for lattices over orders. We also obtain a weaker result for epimorphisms F + A, where F is a direct sum of copies of some finitely presented module which is not projective, and a uniqueness result (up to right equivalence) for epimorphisms F + A, where F is a large enough free module. In the following, terms like “Noetherian” and “Goldie”-&e applied on both sides unless specified otherwise. If A is a module, we let the direct sum of n copies of A be denoted wi (rather than A”), in or&r to avoid confusion when referring to ideals. If A is a module, we let g(A) be the smallest number which is the cardinality of a set of generators for A. A prime ring is right bounded if every essential right ideal contains a nonzero two-sided ideal, and a ring R is right fully bounded if for every prime ideal P, R/P is right

On the Occurrence of large gaps between prime numbers

Let pn denote the nth prime number. Erdös asked whetherfor some constant c<1. Moreno [7] obtained a somewhat weaker result and subsequently Wolke [10] proved that