Correction Bounds on measures satisfying moment conditions

The Annals of Applied Probability (2002) 12 1114-113

A new look at nonnegativity on closed sets and polynomial optimization

We first show that a continuous function f is nonnegative on a closed set $K\subseteq R^n$ if and only if (countably many) moment matrices of some signed measure $d\nu =fd\mu$ with support equal to K, are all positive semidefinite (if $K$ is compact $\mu$ is an arbitrary finite Borel measure with support equal to K. In particular, we obtain a convergent explicit hierarchy of semidefinite (outer) approximations with {\it no} lifting, of the cone of nonnegative polynomials of degree at most $d$. Wen used in polynomial optimization on certain simple closed sets \K (like e.g., the whole space $\R^n$, the positive orthant, a box, a simplex, or the vertices of the hypercube), it provides a nonincreasing sequence of upper bounds which converges to the global minimum by solving a hierarchy of semidefinite programs with only one variable. This convergent sequence of upper bounds complements the convergent sequence of lower bounds obtained by solving a hierarchy of semidefinite relaxations

A cross-disciplinary and multi-method approach of multilingualism in psychotherapy

In this chapter Jean-Marc and Beverley will share their experiences of working with mixed methods in an under-researched area. As we shall see, her interest in larger sampling groups introduced her to some of the advantages of quantitative research. Together with Jean-Marc, who expands on the methods in detail in this chapter, Beverley was able to research multilingual therapy from several angles

Psychotherapy across languages: beliefs, attitudes and practices of monolingual and multilingual therapists with their multilingual patients

The present study investigates beliefs, attitudes and practices of 101 monolingual and multilingual therapists in their interactions with multilingual patients. We adopted a mixed-method approach, using an on-line questionnaire with 27 closed questions which were analysed quantitatively and informed questions in interviews with one monolingual and two multilingual therapists. A principal component analysis yielded a four-factor solution accounting for 41% of the variance. The first dimension, which explained 17% of variance, reflects therapists’ attunement towards their bilingual patients (i.e., attunement versus collusion). Further analysis showed that the 18 monolingual therapists differed significantly from their 83 bi- or multilingual peers on this dimension. The follow up interviews confirmed this result. Recommendations based on these findings are made for psychotherapy training and supervision to attend to a range of issues including: the psychological and therapeutic functions of multi/bilingualism; practice in making formulations in different languages; the creative therapeutic potential of the language gap