Semidefinite approximation for mixed binary quadratically constrained quadratic programs

Motivated by applications in wireless communications, this paper develops semidefinite programming (SDP) relaxation techniques for some mixed binary quadratically constrained quadratic programs (MBQCQP) and analyzes their approximation performance. We consider both a minimization and a maximization model of this problem. For the minimization model, the objective is to find a minimum norm vector in $N$-dimensional real or complex Euclidean space, such that $M$ concave quadratic constraints and a cardinality constraint are satisfied with both binary and continuous variables. {\color{blue}By employing a special randomized rounding procedure, we show that the ratio between the norm of the optimal solution of the minimization model and its SDP relaxation is upper bounded by \cO(Q^2(M-Q+1)+M^2) in the real case and by \cO(M(M-Q+1)) in the complex case.} For the maximization model, the goal is to find a maximum norm vector subject to a set of quadratic constraints and a cardinality constraint with both binary and continuous variables. We show that in this case the approximation ratio is bounded from below by \cO(\epsilon/\ln(M)) for both the real and the complex cases. Moreover, this ratio is tight up to a constant factor

Psychological crisis intervention for the family members of patients in a vegetative state

OBJECTIVES: Family members of patients in a vegetative state have relatively high rates of anxiety and distress. It is important to recognize the problems faced by this population and apply psychological interventions to help them. This exploratory study describes the psychological stress experienced by family members of patients in a vegetative state. We discuss the effectiveness of a psychological crisis intervention directed at this population and offer suggestions for future clinical work. METHODS: A total of 107 family members of patients in a vegetative state were included in the study. The intervention included four steps: acquisition of facts about each family, sharing their first thoughts concerning the event, assessment of their emotional reactions and developing their coping abilities. The Symptom Check List-90 was used to evaluate the psychological distress of the participants at baseline and one month after the psychological intervention. Differences between the Symptom Check List-90 scores at the baseline and follow-up evaluations were analyzed. RESULTS: All participants in the study had significantly higher Symptom Check List-90 factor scores than the national norms at baseline. There were no significant differences between the intervention group and the control group at baseline. Most of the Symptom Check List-90 factor scores at the one-month follow-up evaluation were significantly lower than those at baseline for both groups; however, the intervention group improved significantly more than the control group on most subscales, including somatization, obsessive-compulsive behavior, depression, and anxiety. CONCLUSION: The results of this study indicate that the four-step intervention method effectively improves the mental health of the family members who received this treatment and lessens the psychological symptoms of somatization, obsessive-compulsive behavior, depression and anxiety

7-[(5,5-Dimethyl-2-oxido-1,3,2-dioxaphosphinan-2-yl)­oxy]-4-methyl-2H-chromen-2-one

The title compound, C15H17O6P, was obtained from a reaction of 4-methyl-7-hy­droxy­coumarin and 2-chloro-5,5-dimethyl-1,3,2-dioxaphosphinane 2-oxide. There are two mol­ecules in the asymmetric unit in which the benzopyran ring system is almost planar [r.m.s. deviation for each molecule = 0.003 Å]. In the crystal, C—H⋯O hydrogen bonds and π–π stacking inter­actions [with centroid–centroid distances of 3.743 (3) and 3.727 (3) Å] link the two mol­ecules. The dioxaphospho­rinane ring adopts a chair conformation in both asymmetric molecules

3′,6′-Bis(ethyl­amino)-2′,7′-dimethyl-2-{[2-[(E)-3,4-methyl­enedioxy­benzyl­idene­amino]eth­yl}spiro­[isoindoline-1,9′-xanthen]-3-one

The title compound, C36H36N4O4, was prepared as a spiro­lactam ring formation of the rhodamine dye for comparison with a ring-opened form. The xanthene ring system is approximately planar [r.m.s. deviations from planarity = 0.023 (9) Å for the xanthene ring]. The dihedral angles formed by the spiro­lactam and 1,3-benzodioxole rings with the xanthene ring system are 86.8 (1) and 74.3 (1)°, respectively