On differentiability and analyticity of positive definite functions

We derive a set of differential inequalities for positive definite functions based on previous results derived for positive definite kernels by purely algebraic methods. Our main results show that the global behavior of a smooth positive definite function is, to a large extent, determined solely by the sequence of even-order derivatives at the origin: if a single one of these vanishes then the function is constant; if they are all non-zero and satisfy a natural growth condition, the function is real-analytic and consequently extends holomorphically to a maximal horizontal strip of the complex plane.

Paleomagnetic study of the Great Foum Zguid dyke (southern Morocco): A positive contact test related to metasomatic processes

When a paleomagnetic pole is sought for in an igneous body, the host rocks should be subjected to a contact test to assure that the determined paleopole has the age of the intrusion. If the contact test is positive, it precludes the possibility that the measured magnetization is a later effect. Therefore, we investigated the variations of the remanent magnetization along cross-sections of rocks hosting the Foum Zguid dyke (southern Morocco) and the dyke itself. A positive contact test was obtained, but it is mainly related with Chemical/Crystalline Remanent Magnetization due to metasomatic processes in the host-rocks during magma intrusion and cooling, and not only with Thermo-Remanent Magnetization as ordinarily assumed in standard studies. Paleomagnetic data obtained within the dyke then reflect the Earth magnetic field during emplacement of this well-dated (196.9 +/- 1.8 Ma) intrusion.

Positive Solutions of the Dirichlet Problem for the One-dimensional Minkowski-Curvature Equation

We discuss existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear ordinary differential equation-(u' / root 1 - u'(2))' = f(t, u). Depending on the behaviour of f = f(t, s) near s = 0, we prove the existence of either one, or two, or three, or infinitely many positive solutions. In general, the positivity of f is not required. All results are obtained by reduction to an equivalent non-singular problem to which variational or topological methods apply in a classical fashion.

Positive solutions of fourth order problems with clamped beam boundary conditions

n this paper we make an exhaustive study of the fourth order linear operator u((4)) + M u coupled with the clamped beam conditions u(0) = u(1) = u'(0) = u'(1) = 0. We obtain the exact values on the real parameter M for which this operator satisfies an anti-maximum principle. Such a property is equivalent to the fact that the related Green's function is nonnegative in [0, 1] x [0, 1]. When M < 0 we obtain the best estimate by means of the spectral theory and for M > 0 we attain the optimal value by studying the oscillation properties of the solutions of the homogeneous equation u((4)) + M u = 0. By using the method of lower and upper solutions we deduce the existence of solutions for nonlinear problems coupled with this boundary conditions. (C) 2011 Elsevier Ltd. All rights reserved.

Complex variable positive definite functions

In this paper we develop an appropriate theory of positive definite functions on the complex plane from first principles and show some consequences of positive definiteness for meromorphic functions.

Positive radial solutions of the Dirichlet problem for the Minkowski-Curvature equation in a ball

We study the existence and multiplicity of positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation { -div(del upsilon/root 1-vertical bar del upsilon vertical bar(2)) in B-R, upsilon=0 on partial derivative B-R,B- where B-R is a ball in R-N (N >= 2). According to the behaviour off = f (r, s) near s = 0, we prove the existence of either one, two or three positive solutions. All results are obtained by reduction to an equivalent non-singular one-dimensional problem, to which variational methods can be applied in a standard way.

Predictors of infant positive, negative and self-direct coping during face to face still-face in a Portuguese preterm sample

Past studies found three types of infant coping behaviour during Face-to-Face Still-Face paradigm (FFSF): a Positive Other-Directed Coping; a Negative Other-Directed Coping and a Self-Directed Coping. In the present study, we investigated whether those types of coping styles are predicted by: infants’ physiological responses; maternal representations of their infant’s temperament; maternal interactive behaviour in free play; and infant birth and medical status. The sample consisted of 46, healthy, prematurely born infants and their mothers. At one month, infant heart rate was collected in basal. At three months old (corrected age), infant heart-rate was registered during FFSF episodes. Mothers described their infants’ temperament using a validated Portuguese temperament scale, at infants three months of corrected age. As well, maternal interactive behaviour was evaluated during a free play situation using CARE-Index. Our findings indicate that positive coping behaviours were correlated with gestational birth weight, heart rate (HR), gestational age, and maternal sensitivity in free play. Gestational age and maternal sensitivity predicted Positive Other-Direct Coping behaviours. Moreover, Positive Other-Direct coping was negatively correlated with HR during Still-Face Episode. Self-directed behaviours were correlated with HR during Still-Face Episode and Recover Episode and with maternal controlling/intrusive behaviour. However, only maternal behaviour predicted Self-direct coping. Early social responses seem to be affected by infants’ birth status and by maternal interactive behaviour. Therefore, internal and external factors together contribute to infant ability to cope and to re-engage after stressful social events.