Differential equations for the cuspoid canonical integrals

Differential equations satisfied by the cuspoid canonical integrals I_n(a) are obtained for arbitrary values of n≥2, where n−1 is the codimension of the singularity and a=(ɑ_1,ɑ_2,...,ɑ_(n−1)). A set of linear coupled ordinary differential equations is derived for each step in the sequence I_n(0,0,...,0,0) →I_n(0,0,...,0,ɑ_(n−1)) →I_n(0,0,...,ɑ_(n−2),ɑ_(n−1)) →...→I_n(0,ɑ_2,...,ɑ_(n−2),ɑ_(n−1)) →I_n(ɑ_1,ɑ_2,...,ɑ_n−2,ɑ_(n−1)). The initial conditions for a given step are obtained from the solutions of the previous step. As examples of the formalism, the differential equations for n=2 (fold), n=3 (cusp), n=4 (swallowtail), and n=5 (butterfly) are given explicitly. In addition, iterative and algebraic methods are described for determining the parameters a that are required in the uniform asymptotic cuspoid approximation for oscillating integrals with many coalescing saddle points. The results in this paper unify and generalize previous researches on the properties of the cuspoid canonical integrals and their partial derivatives

High-m Kink/Tearing Modes in Cylindrical Geometry

The global ideal kink equation, for cylindrical geometry and zero beta, is simplified in the high poloidal mode number limit and used to determine the tearing stability parameter, $\Delta^\prime$. In the presence of a steep monotonic current gradient, $\Delta^\prime$ becomes a function of a parameter, $\sigma_0$, characterising the ratio of the maximum current gradient to magnetic shear, and $x_s$, characterising the separation of the resonant surface from the maximum of the current gradient. In equilibria containing a current "spike", so that there is a non-monotonic current profile, $\Delta^\prime$ also depends on two parameters: $\kappa$, related to the ratio of the curvature of the current density at its maximum to the magnetic shear, and $x_s$, which now represents the separation of the resonance from the point of maximum current density. The relation of our results to earlier studies of tearing modes and to recent gyro-kinetic calculations of current driven instabilities, is discussed, together with potential implications for the stability of the tokamak pedestal.Comment: To appear in Plasma Physics and Controlled Fusio

Entanglement consumption of instantaneous nonlocal quantum measurements

Relativistic causality has dramatic consequences on the measurability of nonlocal variables and poses the fundamental question of whether it is physically meaningful to speak about the value of nonlocal variables at a particular time. Recent work has shown that by weakening the role of the measurement in preparing eigenstates of the variable it is in fact possible to measure all nonlocal observables instantaneously by exploiting entanglement. However, for these measurement schemes to succeed with certainty an infinite amount of entanglement must be distributed initially and all this entanglement is necessarily consumed. In this work we sharpen the characterisation of instantaneous nonlocal measurements by explicitly devising schemes in which only a finite amount of the initially distributed entanglement is ever utilised. This enables us to determine an upper bound to the average consumption for the most general cases of nonlocal measurements. This includes the tasks of state verification, where the measurement verifies if the system is in a given state, and verification measurements of a general set of eigenstates of an observable. Despite its finiteness the growth of entanglement consumption is found to display an extremely unfavourable exponential of an exponential scaling with either the number of qubits needed to contain the Schmidt rank of the target state or total number of qubits in the system for an operator measurement. This scaling is seen to be a consequence of the combination of the generic exponential scaling of unitary decompositions combined with the highly recursive structure of our scheme required to overcome the no-signalling constraint of relativistic causality.Comment: 32 pages and 14 figures. Updated to published versio

Impact reactivity of materials at very high oxygen pressure

The requirements for impact testing of materials in an oxygen atmosphere at pressures from 82.7 MPa (12,000 psi) to 172 MPa (25,000 psi) were evaluated. The impact tester system was evaluated for potential pressure increases from 69 MPa (10,000 psi) to 82.7 MPa (12,000 psi). The low pressure oxygen and nitrogen systems, the impact tower, the impact test cell, and the high pressure oxygen system were evaluated individually. Although the structural integrity of the impact test cell and the compressor were sufficient for operation at 82.7 MPa (12,000 psi), studies revealed possible material incompatibility at that pressure and above. It was recommended that if a component should be replaced for 82.7 MPa (12,000 psi) operation the replacement should meet the final objectives of 172 MPa (25,000 psi). Recommended changes in the system include; use of Monel 400 for pressures above 82.7 MPa (12,000 psi), use of bellows to replace the seal in the impact tester, use of a sapphire window attached to a fiber optic for event sensing, and use of a three diaphragm compressor