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We consider a system of two balance laws of Keyfitz–Kranzer type with varying generalized Chaplygin gas, which exhibits negative pressure and is a product of a function of time and the inverse of a power of the density. The Chaplygin gas is a fluid designed to accommodate measurements for the early universe and late-time universal expansion while obeying the pressure–density–time relation. We produce an explanation and description of the non-self-similar Riemann solutions, including the non-classical singular solutions. We also find that due to a direct dependence on time, a change in the regions allowing for combinations of classical and non-classical singular solutions occurs; therefore, a Riemann solution can have different solutions over several time intervals. Our findings are confirmed numerically using the Local Lax–Friedrichs scheme. [ABSTRACT FROM AUTHOR]

Interferon-beta is a current treatment for multiple sclerosis (MS). Interferon-beta is thought to exert its therapeutic effects on MS by down-modulating the immune response by multiple potential pathways. Here, we document that treatment of MS patients with interferon beta-1a (Rebif) results in a significant increase in the levels and function of the protein tyrosine phosphatase SHP-1 in PBMCs. SHP-1 is a crucial negative regulator of cytokine signaling, inflammatory gene expression, and CNS demyelination as evidenced in mice deficient in SHP-1. In order to examine the functional significance of SHP-1 induction in MS PBMCs, we analyzed the activity of proinflammatory signaling molecules STAT1, STAT6, and NF-kappaB, which are known SHP-1 targets. Interferon-beta treatment in vivo resulted in decreased NF-kappaB and STAT6 activation and increased STAT1 activation. Further analysis in vitro showed that cultured PBMCs of MS patients and normal subjects had a significant SHP-1 induction following interferon-beta treatment that correlated with decreased NF-kappaB and STAT6 activation. Most importantly, experimental depletion of SHP-1 in cultured PBMCs abolished the anti-inflammatory effects of interferon-beta treatment, indicating that SHP-1 is a predominant mediator of interferon-beta activity. In conclusion, interferon-beta treatment upregulates SHP-1 expression resulting in decreased transcription factor activation and inflammatory gene expression important in MS pathogenesis.