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In earlier work we identified a global, non-axisymmetric instability associated with the presence of an extreme in the radial profile of the key function L(r) ≡ (ΣΩ/κ 2 )S 2/Γ in a thin, inviscid, nonmagnetized accretion disk. Here, Σ(r) is the surface mass density of the disk, Ω(r) the angular rotation rate, S(r) the specific entropy, Γ the adiabatic index, and κ(r) the radial epicyclic frequency. The dispersion relation of the instability was shown to be similar to that of Rossby waves in planetary atmospheres. In this paper, we present the detailed linear theory of this Rossby wave instability and show that it exists for a wider range of conditions, specifically, for the case where there is a "jump" over some range of r in Σ(r) or in the pressure P (r). We elucidate the physical mechanism of this instability and its dependence on various parameters, including the magnitude of the "bump" or "jump," the azimuthal mode number, and the sound speed in the disk. We find large parameter range where the disk is stable to axisymmetric perturbations, but unstable to the non-axisymmetric Rossby waves. We find that growth rates of the Rossby wave instability can be high, ∼ 0.2Ω K for relative small "jumps" or "bumps". We discuss possible conditions which can lead to this instability and the consequences of the instability.

Farnesyl pyrophosphate synthetase (FPPS) synthesizes farnesyl pyrophosphate through successive condensations of isopentyl pyrophosphate with dimethylallyl pyrophosphate and geranyl pyrophosphate. Nitrogen-containing bisphosphonate drugs used to treat osteoclast-mediated bone resorption and tumor-induced hypercalcemia are potent inhibitors of the enzyme. Here we present crystal structures of substrate and bisphosphonate complexes of FPPS. The structures reveal how enzyme conformational changes organize conserved active site residues to exploit metal-induced ionization and substrate positioning for catalysis. The structures further demonstrate how nitrogen-containing bisphosphonates mimic a carbocation intermediate to inhibit the enzyme. Together, these FPPS complexes provide a structural template for the design of novel inhibitors that may prove useful for the treatment of osteoporosis and other clinical indications including cancer.Post-translational modification of C-terminal CAAX sequences by covalent attachment of isoprenyl chains is crucial for intracellular localization and proper function of small GTPases such as Ras, Rac, Rho, and CDC42 (1, 2). The substrates for these modifications are the 15-carbon isoprenoid farnesyl pyrophosphate (FPP) 1 or the 20-carbon isoprenoid geranyl-geranyl pyrophosphate synthesized by enzymes of the mevalonate pathway (3) (Fig. 1A). A key branch point enzyme of the mevalonate pathway is farnesyl pyrophosphate synthetase (FPPS), a ϳ30-kDa Mg 2ϩ -dependent homodimeric enzyme that synthesizes (E,E)-FPP from isopentyl pyrophosphate (IPP) and dimethylallyl pyrophosphate (DMAPP) (4, 5) (Fig. 1B). Interest in understanding FPPS activity stems from the recent discovery that FPPS is the molecular target of nitrogencontaining bisphosphonates (6,7,31,32). Bisphophonates are non-cleavable pyrophosphate (P-O-P) analogues in which the central oxygen is replaced by a carbon (P-C-P) with various side chains (Fig. 1C). Against parasitic organisms (8, 9) these agents have been shown in vitro to disrupt cell growth through FPPS inhibition. In people, bisphosphonates are targeted to bone tissue (10) where FPPS inhibition in bone-resorbing osteoclasts is a current therapeutic approach for treating postmenopausal osteoporosis (11,12). Because of their bone-targeting properties, bisphosphonates have also found use as agents to treat tumor-induced hypercalcemia (13), Paget's disease (14), and osteolytic metastases (15).Although structures of apo-and ligand-bound avian FPPS have been solved (16,17), the active sites are unassembled and do not provide substantial information concerning catalysis. Thus, to resolve the molecular basis of catalysis, and also to understand the structural features governing bisphosphonate recognition, we determined the structures of unliganded Staphylococcus aureus FPPS (FPPS-Sa), as well as two Escherichia coli FPPS (FPPS-Ec) ternary complexes. These ternary complexes include a 2.4-Å "substrate-bound" structure containing IPP and the noncleavable DMAPP analogue dimethyla...

Symplectic maps (canonical transformations) are treated from the Lie algebraic point of view using Lie series and Lie algebraic techniques. It is shown that under very general conditions an analytic symplectic map can be written as a product of Lie transformations. Under certain conditions this product of Lie transformations can be combined to form a single Lie transformation by means of the Campbell–Baker–Hausdorff theorem. This result leads to invariant functions and generalizes to several variables a classic result of Birkhoff for the case of two variables. It also provides a new approach since the connection between symplectic maps, Lie algebras, invariant functions, and Birkhoff’s work has not been previously recognized and exploited. It is expected that the results obtained will be applicable to the normal form problem in Hamiltonian mechanics, the use of the Poincaré section map in stability analysis, and the behavior of magnetic field lines in a toroidal plasma device.

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We suggest a new approach that could be used for modeling both the large-scale behavior of astrophysical jets and the magnetically dominated explosions in astrophysics. We describe a method for modeling the injection of magnetic fields and their subsequent evolution in a regime where the free energy is magnetically dominated. The injected magnetic fields, along with their associated currents, have both poloidal and toroidal components, and they are not force free. The dynamic expansion driven by the Lorentz force of the injected fields is studied using threedimensional ideal magnetohydrodynamic simulations. The generic behavior of magnetic field expansion, the interactions with the background medium, and the dependence on various parameters are investigated.

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