For an arbitrary in nite-dimensional Banach space X, we construct examples of strongly-measurable X-valued Pettis integrable functions whose indefinite Pettis integrals are nowhere weakly di erentiable; thus, for these functions the Lebesgue Di erentiation Theorem fails rather spectacularly. We also relate the degree of nondi erentiability of the inde nite Pettis integral to the cotype of X, fr...

Let Gm resp. Gf be the minimal resp. formal Kac-Moody group, associated to a symmetrizable generalized Cartan matrix, over a field F of characteristic 0. Let F [Gm] be the algebra of strongly regular functions on Gm. We denote by Ĝm resp. Ĝf certain monoid completions of Gm resp. Gf , build by using the faces of the Tits cone. We show that there is an action of Ĝf × Ĝf on the spectrum of F-valu...

We discuss local convergence of Newton’s method to a singular solution x∗ of the nonlinear equations F (x) = 0, for F : IR → IR. It is shown that an existing proof of Griewank, concerning linear convergence to a singular solution x∗ from a starlike domain around x∗ for F twice Lipschitz continuously differentiable and x∗ satisfying a particular regularity condition, can be adapted to the case i...

Let {T (t) : t > 0} be a strongly continuous semigroup of linear contractions in Lp, 1 < p < ∞, of a σ-finite measure space. In this paper we prove that if there corresponds to each t > 0 a positive linear contraction P (t) in Lp such that |T (t)f | ≤ P (t)|f | for all f ∈ Lp, then there exists a strongly continuous semigroup {S(t) : t > 0} of positive linear contractions in Lp such that |T (t)...

For a nonnegative n× n matrix A, we find that there is a polynomial f(x) ∈ R[x] such that f(A) is a positive matrix of rank one if and only if A is irreducible. Furthermore, we show that the lowest degree such polynomial f(x) with tr f(A) = n is unique. Thus, generalizing the well-known definition of the Hoffman polynomial of a strongly connected regular digraph, for any irreducible nonnegative...

We discuss local convergence of Newton’s method to a singular solution x∗ of the nonlinear equations F (x) = 0, for F : IR → IR. It is shown that an existing proof of Griewank, concerning linear convergence to a singular solution x∗ from a starlike domain around x∗ for F twice Lipschitz continuously differentiable and x∗ satisfying a particular regularity condition, can be adapted to the case i...

A strongly polynomial sequence of graphs (Gn) is a sequence (Gn)n∈N of finite graphs such that, for every graph F , the number of homomorphisms from F to Gn is a fixed polynomial function of n (depending on F ). For example, (Kn) is strongly polynomial since the number of homomorphisms from F to Kn is the chromatic polynomial of F evaluated at n. In earlier work of de la Harpe and Jaeger, and m...

Let f be a density on the real line and let f,~ be the kernel estimate of f in which the smoothing factor is obtained by maximizing the cross-validated likelihood product according to the method of Duin and Habbema, Hermans and Vandenbroek . Under mild regularity conditions on the kernel and f, we show, among other things that f Jf,~ f ( --~ 0 almost surely if and only if the sample extremes of...

The restricted arc-connectivity λ′ of a strongly connected digraph G is the minimum cardinality of an arc cut F in G such that every strongly connected component of G−F contains at least two vertices. This paper shows that for a d-regular strongly connected digraph with order n and diameter k ≥ 4, if λ′ exists, then λ′(G) ≥ min { (n − dk−1)(d− 1) dk−1 + d− 2 , 2d− 2 } As consequences, the restr...

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