Cardiometabolic disorders, including cardiovascular disease, obesity and type 2 diabetes are an extreme burden on the health of individuals and societies worldwide. The combination of genetic susceptibility, environmental, and lifestyle factors drive disease development. Therefore, understanding the exact molecular changes occuring in their pathogenesis brings not only insights into mechanism, but also potential novel therapeutic angles. Epigenetic changes -the mitotically heritable chemical marks or packaging of DNA that influence gene expression without changing the genetic code itself -have been identified as highly compelling targets. Therapeutic success modulating the epigenome has initially come in cancer, but is now expanding into non-malignant diseases. Here, we discuss the different epigenetic mechanisms implicated in cardiometabolic disease and the advancing science of epigenetic therapeutics for these disorders.
The interesting background and historical development of KdV equations were discussed widely. These equations describe the propagation of water waves in weakly non linear and weakly dispersive medium. Referring to physical derivation of KdV equations, scientists used to impose shallow water equations, thus the formal or physical derivation of KdV equations. However, these equations have rarely been derived rigorously. The aim of this paper is to giving insight into their rigorous mathematical derivation, instead of only referring to. Thereby, a rigorous derivation of two extended KdV equations: one on the velocity, other on the surface elevation. With this aim in mind, the primary research method for this paper will depend on the definition of consistency. Hence, a rigorous justification of new extended KdV equations will be provided thanks to this definition. This result provides a precise mathematical answer to a question raised by several authors in the last years, that is the verification of the extended KdV equations, derived previously, using formal methods.
Discovered experimentally by Russell and described theoretically by Korteweg and de Vries, KdV equation has been a nonlinear evolution equation describing the propagation of weakly dispersive and weakly nonlinear waves. This equation received a lot of attention from mathematical and physical communities as an integrable equation. The objectives of this paper are: first, providing a rigorous mathematical derivation of an extended KdV equations, one on the velocity, other on the surface elevation, next, solving explicitly the one on the velocity. In order to derive rigorously these equations, we will refer to the definition of consistency, and to find an explicit solution for this equation, we will use the sine-cosine method. As a result of this work, a rigorous justification of the extended Kdv equation of fifth order will be done, and an explicit solution of this equation will be derived.
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