Strange Attractors in Dissipative Nambu Mechanics : Classical and Quantum Aspects

We extend the framework of Nambu-Hamiltonian Mechanics to include dissipation in $R^{3}$ phase space. We demonstrate that it accommodates the phase space dynamics of low dimensional dissipative systems such as the much studied Lorenz and R\"{o}ssler Strange attractors, as well as the more recent constructions of Chen and Leipnik-Newton. The rotational, volume preserving part of the flow preserves in time a family of two intersecting surfaces, the so called {\em Nambu Hamiltonians}. They foliate the entire phase space and are, in turn, deformed in time by Dissipation which represents their irrotational part of the flow. It is given by the gradient of a scalar function and is responsible for the emergence of the Strange Attractors. Based on our recent work on Quantum Nambu Mechanics, we provide an explicit quantization of the Lorenz attractor through the introduction of Non-commutative phase space coordinates as Hermitian $N \times N$ matrices in $R^{3}$. They satisfy the commutation relations induced by one of the two Nambu Hamiltonians, the second one generating a unique time evolution. Dissipation is incorporated quantum mechanically in a self-consistent way having the correct classical limit without the introduction of external degrees of freedom. Due to its volume phase space contraction it violates the quantum commutation relations. We demonstrate that the Heisenberg-Nambu evolution equations for the Quantum Lorenz system give rise to an attracting ellipsoid in the $3 N^{2}$ dimensional phase space.Comment: 35 pages, 4 figures, LaTe

Copper-mediated coupling reactions and their applications in natural products and designed biomolecules synthesis

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Long-term safety and efficacy of patisiran for hereditary transthyretin-mediated amyloidosis with polyneuropathy: 12-month results of an open-label extension study

Background: Hereditary transthyretin-mediated amyloidosis is a rare, inherited, progressive disease caused by mutations in the transthyretin (TTR) gene. We assessed the safety and efficacy of long-term treatment with patisiran, an RNA interference therapeutic that inhibits TTR production, in patients with hereditary transthyretin-mediated amyloidosis with polyneuropathy. Methods: This multicentre, open-label extension (OLE) trial enrolled patients at 43 hospitals or clinical centres in 19 countries as of Sept 24, 2018. Patients were eligible if they had completed the phase 3 APOLLO or phase 2 OLE parent studies and tolerated the study drug. Eligible patients from APOLLO (patisiran and placebo groups) and the phase 2 OLE (patisiran group) studies enrolled in this global OLE trial and received patisiran 0·3 mg/kg by intravenous infusion every 3 weeks with plans to continue to do so for up to 5 years. Efficacy assessments included measures of polyneuropathy (modified Neuropathy Impairment Score +7 [mNIS+7]), quality of life, autonomic symptoms, nutritional status, disability, ambulation status, motor function, and cardiac stress, with analysis by study groups (APOLLO-placebo, APOLLO-patisiran, phase 2 OLE patisiran) based on allocation in the parent trial. The global OLE is ongoing with no new enrolment, and current findings are based on the interim analysis of the patients who had completed 12-month efficacy assessments as of the data cutoff. Safety analyses included all patients who received one or more dose of patisiran up to the data cutoff. This study is registered with ClinicalTrials.gov, NCT02510261. Findings: Between July 13, 2015, and Aug 21, 2017, of 212 eligible patients, 211 were enrolled: 137 patients from the APOLLO-patisiran group, 49 from the APOLLO-placebo group, and 25 from the phase 2 OLE patisiran group. At the data cutoff on Sept 24, 2018, 126 (92%) of 137 patients from the APOLLO-patisiran group, 38 (78%) of 49 from the APOLLO-placebo group, and 25 (100%) of 25 from the phase 2 OLE patisiran group had completed 12-month assessments. At 12 months, improvements in mNIS+7 with patisiran were sustained from parent study baseline with treatment in the global OLE (APOLLO-patisiran mean change –4·0, 95 % CI –7·7 to −0·3; phase 2 OLE patisiran –4·7, –11·9 to 2·4). Mean mNIS+7 score improved from global OLE enrolment in the APOLLO-placebo group (mean change from global OLE enrolment −1·4, 95% CI –6·2 to 3·5). Overall, 204 (97%) of 211 patients reported adverse events, 82 (39%) reported serious adverse events, and there were 23 (11%) deaths. Serious adverse events were more frequent in the APOLLO-placebo group (28 [57%] of 49) than in the APOLLO-patisiran (48 [35%] of 137) or phase 2 OLE patisiran (six [24%] of 25) groups. The most common treatment-related adverse event was mild or moderate infusion-related reactions. The frequency of deaths in the global OLE was higher in the APOLLO-placebo group (13 [27%] of 49), who had a higher disease burden than the APOLLO-patisiran (ten [7%] of 137) and phase 2 OLE patisiran (0 of 25) groups. Interpretation: In this interim 12-month analysis of the ongoing global OLE study, patisiran appeared to maintain efficacy with an acceptable safety profile in patients with hereditary transthyretin-mediated amyloidosis with polyneuropathy. Continued long-term follow-up will be important for the overall assessment of safety and efficacy with patisiran. Funding: Alnylam Pharmaceuticals