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Recent works on three-planet mean motion resonances (MMRs) have highlighted their importance for understanding the details of the dynamics of planet formation and evolution. While the dynamics of two-planet MMRs are well understood and approximately described by a one-degree-of-freedom Hamiltonian, little is known of the exact dynamics of three-body resonances besides the cases of zeroth-order MMRs or when one of the bodies is a test particle. In this work, I propose the first general integrable model for first-order three-planet mean motion resonances. I show that one can generalize the strategy proposed in the two-planet case to obtain a one-degree-of-freedom Hamiltonian. The dynamics of these resonances are governed by the second fundamental model of resonance. The model is valid for any mass ratio between the planets and for every first-order resonance. I show the agreement of the analytical model with numerical simulations. As examples of application, I show how this model could improve our understanding of the capture into MMRs as well as their role in the stability of planetary systems. [ABSTRACT FROM AUTHOR]

Background: Livedoid vasculopathy (LV) is a chronic dermatosis associated with micro-thrombosis of the vessels of the dermis, leading to ischemic lesions and painful skin ulcerations of the lower limbs. This thrombosing occlusive vasculopathy, clearly distinct from 'classical vasculitis' (not related to alteration of vessel walls), may lead to peripheral neuropathy. Objective: To clarify the main clinical, electrophysiological and pathological characteristics of peripheral neuropathy linked to LV. Method: We presented a series of personal cases of peripheral neuropathy due to LV. We also conducted a review of the literature (since the first description of LV in 1974) using multiple combinations of keywords from 'PubMed', 'Google Scholar' and 'ScienceDirect' databases according to the 'Preferred Reporting Items for Systematic reviews and Meta-Analyses' guidelines. Results: We identified 16 patients (6 personal cases and 10 cases from the medical literature). Our personal cases were five females and one male, with a median age (at the onset of cutaneous signs of LV) of 38 (range 25–62). Several types of skin lesions of the lower limbs were observed. Median age at the onset of peripheral neuropathy symptoms was 48 years (range 29–66), with a main clinical and electrophysiological pattern of mononeuropathy multiplex. Discussion: We observed a typical pattern of peripheral neuropathy, mostly mononeuropathy multiplex, whose pathophysiology might be related to occlusions of the small vessels of the nerves, as seen in the dermis. Moreover, LV may also be associated with other types of peripheral neuropathies (sometimes of autoimmune etiology) not directly related to the skin lesions. Conclusion: The 'ischemic form' of peripheral neuropathy linked to LV is mainly responsible for sensory disturbances (with multifocal distribution), sometimes for motor disturbances. This type of peripheral neuropathy has to be distinguished from 'classical vasculitic neuropathies' which are usually treated with antithrombotic therapies. [ABSTRACT FROM AUTHOR]