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

Development of a synthetic aperture radar design approach for wide-swath implementation

The first phase of a study program to develop an advanced synthetic aperture radar design concept is presented. Attributes of particular importance for the system design include wide swath coverage, reduced power requirements, and versatility in the selection of frequency, polarization and incident angle. The multiple beam configuration provides imaging at a nearly constant angle of incidence and offers the potential of realizing a wide range of the attributes desired for an orbital imaging radar for Earth resources applications

Multilingual clients’ experience of psychotherapy

The present study focuses on the experiences of 182 multilingual clients who had been exposed to various therapeutic approaches in various countries. An on-line questionnaire was used to collect quantitative and qualitative data. The analysis of feedback from clients with multilingual therapists showed that clients use or initiate significantly more code-switching (CS) than their therapists, and that it typically occurs when the emotional tone is raised. Gender was unrelated to CS frequency. CS is used strategically when discussing episodes of trauma and shame, creating proximity or distance. CS allows clients to express themselves more fully to the therapist, adding depth and nuance to the therapy. The therapist’s multilingualism promotes empathy and clients’ own multilingualism constitutes an important aspect of their sense of self. Multilingual clients benefit from a therapeutic environment where multilingualism is appreciated, and where they can use CS