On the geometry of the domain of the solution of nonlinear Cauchy problem

We consider the Cauchy problem for a second order quasi-linear partial differential equation with an admissible parabolic degeneration such that the given functions described the initial conditions are defined on a closed interval. We study also a variant of the inverse problem of the Cauchy problem and prove that the considered inverse problem has a solution under certain regularity condition. We illustrate the Cauchy and the inverse problems in some interesting examples such that the families of the characteristic curves have either common envelopes or singular points. In these cases the definition domain of the solution of the differential equation contains a gap.Comment: accepted for publication in the book Lie groups, differential equations and geometry in Springer Unip

Suicide risk in schizophrenia: learning from the past to change the future

Suicide is a major cause of death among patients with schizophrenia. Research indicates that at least 5–13% of schizophrenic patients die by suicide, and it is likely that the higher end of range is the most accurate estimate. There is almost total agreement that the schizophrenic patient who is more likely to commit suicide is young, male, white and never married, with good premorbid function, post-psychotic depression and a history of substance abuse and suicide attempts. Hopelessness, social isolation, hospitalization, deteriorating health after a high level of premorbid functioning, recent loss or rejection, limited external support, and family stress or instability are risk factors for suicide in patients with schizophrenia. Suicidal schizophrenics usually fear further mental deterioration, and they experience either excessive treatment dependence or loss of faith in treatment. Awareness of illness has been reported as a major issue among suicidal schizophrenic patients, yet some researchers argue that insight into the illness does not increase suicide risk. Protective factors play also an important role in assessing suicide risk and should also be carefully evaluated. The neurobiological perspective offers a new approach for understanding self-destructive behavior among patients with schizophrenia and may improve the accuracy of screening schizophrenics for suicide. Although, there is general consensus on the risk factors, accurate knowledge as well as early recognition of patients at risk is still lacking in everyday clinical practice. Better knowledge may help clinicians and caretakers to implement preventive measures. This review paper is the results of a joint effort between researchers in the field of suicide in schizophrenia. Each expert provided a brief essay on one specific aspect of the problem. This is the first attempt to present a consensus report as well as the development of a set of guidelines for reducing suicide risk among schizophenia patients