For the interval system of equations defined by [x] = [A][x] + [b] we derive necessary and sufficient criteria for the existence of solutions [x]. Furthermore we give necessary and sufficient criteria for the convergence of powers of [A]. In contrast to former results we treat complex interval arithmetics.

Purpose Brain death/death by neurologic criteria (BD/DNC) may be determined in many countries by a clinical examination that shows coma, brainstem areflexia, and apnea, provided the conditions causing reversible loss of brain function are excluded a priori . To date, accounts of recovery from BD/DNC in adults have been limited to noncompliance with guidelines. Clinical features We report the case of a 72-yr-old man with a combined primary infratentorial (hemorrhagic) and secondary global (anoxic) brain lesion in whom decompressive craniectomy of the posterior fossa and six-hour therapeutic hypothermia (33–34°C) followed by 8-hour rewarming to ≥ 36°C were conducted. Thirteen hours later, clinical findings of brain function loss were documented in addition to guideline-compliant exclusion of reversible causes (arterial hypotension, intoxication, depressant drug effects, relevant metabolic or endocrine disequilibrium, chronic hypercapnia, neuromuscular disorders, and administration of a muscle relaxant). Since a primary infratentorial brain lesion was present, German guidelines required further ancillary testing. Doppler ultrasonography revealed some preserved cerebral circulation, and BD/DNC was not diagnosed. Approximately 24 hr after rewarming to ≥ 36°C, the patient exhibited respiratory efforts. He continued with assisted respiration until final asystole/apnea, without regaining additional brain function other than mild signs of hemispasticity. Follow-up computed tomography showed partial herniation of the cerebellum through the craniectomy gap of the posterior fossa, alleviating caudal brain stem compression. Conclusions Therapeutic decompressive craniectomy of the posterior fossa may allow for delayed reversal of apnea. In these patients, proof of cerebral circulatory arrest should be mandatory for diagnosing BD/DNC. Supplementary Information The online version contains supplementary material available at 10.1007/s12630-022-02265-6.

A previously healthy 39-year-old man was admitted for treatment of COVID-19 six days after the onset of symptoms. Imaging revealed bilateral extended, patchy confluent infiltrates. On oxygen 4 L/min the saturation was consistently >90%. Six days later the patient was no longer febrile, and after a further four days his oxygenation was satisfactory breathing ambient air. Two follow-up swabs at this time were negative for SARS-CoV-2. Enoxaparin 4000 IU s.c. was given daily to prevent thrombosis. One day before planned discharge, the patient reported the sudden onset of severe breathing-related pain. The D-dimer level was elevated (5.0 mg/L; normal value <0.5 mg/L). Thoracic CT a day later showed pulmonary artery embolism in both lungs with multiple bilateral filling defects in the pulmonary arteries of the lower lobes and wedge-shaped infiltrates in the dorsobasal segments of the lower lobes (a). There were also bilateral peripheral ground-glass infiltrates compatible with COVID-19 (b). The infarct pneumonia was treated with antibiotics, and oxygen 2 L/min had to be given for two days. Deep vein thrombosis in the legs or pelvis was ruled out by duplex sonography. The patient was discharged in stable condition on oral anticoagulants.

For the interval system of equations defined by [x] = [A][x]+[b] with ρ(|[A]|) ≥ 1 we derive a necessary and sufficient criterion for the existence and uniqueness of solutions [x]. Generalizing former results we allow the absolute value |[A]| of [A] to be reducible.

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