Continuous Symmetries of Difference Equations

Lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations. In this article we review the results of a much more recent program: the use of Lie groups to study difference equations. We show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners. One is to use generalized symmetries acting on solutions of difference equations, but leaving the lattice invariant. The other is to restrict to point symmetries, but to allow them to also transform the lattice.Comment: Review articl

Ebola: translational science considerations

We are currently in the midst of the most aggressive and fulminating outbreak of Ebola-related disease, commonly referred to as “Ebola”, ever recorded. In less than a year, the Ebola virus (EBOV, Zaire ebolavirus species) has infected over 10,000 people, indiscriminately of gender or age, with a fatality rate of about 50%. Whereas at its onset this Ebola outbreak was limited to three countries in West Africa (Guinea, where it was first reported in late March 2014, Liberia, where it has been most rampant in its capital city, Monrovia and other metropolitan cities, and Sierra Leone), cases were later reported in Nigeria, Mali and Senegal, as well as in Western Europe (i.e., Madrid, Spain) and the US (i.e., Dallas, Texas; New York City) by late October 2014. World and US health agencies declared that the current Ebola virus disease (EVD) outbreak has a strong likelihood of growing exponentially across the world before an effective vaccine, treatment or cure can be developed, tested, validated and distributed widely. In the meantime, the spread of the disease may rapidly evolve from an epidemics to a full-blown pandemic. The scientific and healthcare communities actively research and define an emerging kaleidoscope of knowledge about critical translational research parameters, including the virology of EBOV, the molecular biomarkers of the pathological manifestations of EVD, putative central nervous system involvement in EVD, and the cellular immune surveillance to EBOV, patient-centered anthropological and societal parameters of EVD, as well as translational effectiveness about novel putative patient-targeted vaccine and pharmaceutical interventions, which hold strong promise, if not hope, to curb this and future Ebola outbreaks. This work reviews and discusses the principal known facts about EBOV and EVD, and certain among the most interesting ongoing or future avenues of research in the field, including vaccination programs for the wild animal vectors of the virus and the disease from global translational science perspective