Main Theorem. Suppose f, k, and g are real functions on R which are locally Lipschitz and p is a nonconstant, continuous, real function on [0, T ], T > 0. Also suppose all solutions of the initial value problem (1.1) can be extended to [0, T ]. If there exist real numbers a1 and a2 for which g(a1) ≤ p(t) ≤ g(a2) holds for each 0 ≤ t ≤ T , then there exists T0, 0 < T0 < T , such that if p is T0-...

For a bounded operator T acting on a complex Banach space, we show that if T −λ is not surjective, then λ is an isolated point of the surjective spectrum σsu(T ) of T if and only if X = H0(T−λ)+K(T−λ), where H0(T ) is the quasinilpotent part of T and K(T ) is the analytic core for T . Moreover, we study the operators for which dimK(T ) < ∞. We show that for each of these operators T , there exi...

Using a probabilistic approach, we extend for general Q-linearly independent sequences a result of Túran concerning the sequence (log p ℓ), p ℓ being the ℓ-th prime. For instance let λ 1 , λ 2 ,. .. be linearly independent over Q. We prove that there exists a constant C 0 such that for any positive integers N and ω, if T > 4ω

We initiate the study of reducts of relational structures up to primitive positive interdefinability: After providing the tools for such a study, we apply these tools in order to obtain a classification of the reducts of the logic of equality. It turns out that there exists a continuum of such reducts. Equivalently, expressed in the language of universal algebra, we classify those locally close...

Two problems related to the analysis of noun phrases in which demonstratives occur are discussed: (1) it is shown that there exists an infinite number of syntactically complex demonstrative noun phrases and thus an infinite number of noun phrases which are neither purely referential not purely quantificational, (2) some problems concerning the semantic role of the common noun to which demonstra...

The so-called first selection lemma states the following: given any set P of n points in R, there exists a point in R contained in at least cdn − O(n) simplices spanned by P , where the constant cd depends on d. We present improved bounds on the first selection lemma in R. In particular, we prove that c3 ≥ 0.00227, improving the previous best result of c3 ≥ 0.00162 by Wagner [Wag03]. This makes...

Three components of determiner meanings – truth conditions, implicatures, and presuppositions – have been identified. One of the major findings in acquisition, related to the truth conditions of the quantifiers, has been that children go through at least two stages of non-adult interpretation of the quantifier every (Philip, 1995). More recently, researchers (Noveck, 2001; Gualmini et al., 2001...

We study Pisot numbers β ∈ (1, 2) which are univoque, i.e., such that there exists only one representation of 1 as 1 = P n≥1 snβ −n, with sn ∈ {0, 1}. We prove in particular that there exists a smallest univoque Pisot number, which has degree 14. Furthermore we give the smallest limit point of the set of univoque Pisot numbers.

Let 0<ai<a2< ■ ■ -be any infinite sequence of integers. Denote by N(di, n) the number of a^n. I conjectured that to every sequence atthere corresponds a sequence bj of density 0 (i.e., such that lim„=0O (l/n)N(bj, n) = 0) so that every sufficiently large integer is of the form a,-+6yLorentz2 in a recent paper proved this conjecture; in fact, he showed that there exists a sequence bj with the re...