The deterministic analog of the Markov property of a time-homogeneous Markov process is the semigroup property of solutions of an autonomous differential equation. The semigroup property arises naturally when the solutions of a differential equation are unique, and leads to a semiflow. We prove an abstract result on measurable selection of a semiflow for the situations without uniqueness. We outline applications to ODEs, PDEs, differential inclusions, etc. Our proof of the semiflow selection theorem is motivated by N. V. Krylov's Markov selection theorem. To accentuate this connection, we include a new version of the Markov selection theorem related to more recent papers of Flandoli & Romito and Goldys et al.
Conventional preparations of amphotericin B (AmB) at established therapeutic doses are known to increase nonspecific immune responses. It remains to be established whether higher doses of the less toxic liposomal preparation of AmB maintains a beneficial effect on the immune response to fungal infections. Examination of the effect of treatment of human peripheral blood mononuclear cells from healthy subjects with various doses of both liposomal AmB (L-AmB) and deoxycholate AmB (d-AmB) on proliferation, cell viability, and percentage of apoptosis demonstrated that, although both L-AmB and d-AmB at low doses significantly increased nonspecific proliferative responses, L-AmB, but not d-AmB, treatment maintained this beneficial effect at higher doses. High doses of d-AmB, but not L-AmB, resulted in significantly decreased cell viability and increased apoptosis. This study provides further evidence in healthy human subjects for choosing L-AmB over conventional preparations in the clinical treatment of fungal infections requiring systemic high-dose treatment with AmB.
A semi-process is an analog of the semi-flow for non-autonomous differential equations or inclusions. We prove an abstract result on the existence of measurable semiprocesses in the situations where there is no uniqueness. Also, we allow solutions to blow up in finite time and then obtain local semi-processes.
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