Accessing large global charge via the ϵ-expansion

We compute the lowest operator dimension ∆(J; D) at large global charge J in the O(2) Wilson-Fisher model in D = 4 − ϵ dimensions, to leading order in both 1/J and ϵ. While the effective field theory approach of [1] could only determine ∆(J; 3) as a series expansion in 1/J up to an undetermined constant in front of each term, this time we try to determine the coefficient in front of J3/2 in the ϵ-expansion. The final result for ∆(J; D) in the (resummed) ϵ-expansion, valid when J ≫ 1/ϵ ≫ 1, turns out to be$$ \Delta \left(J;D\right)=\left[\frac{2\left(D-1\right)}{3\left(D-2\right)}{\left(\frac{9\left(D-2\right)\pi }{5D}\right)}^{\frac{D}{2\left(D-1\right)}}{\left[\frac{5\Gamma \left(\frac{D}{2}\right)}{24{\pi}^2}\right]}^{\frac{1}{D-1}}{\epsilon}^{\frac{D-1}{2\left(D-1\right)}}\right]\times {J}^{\frac{D}{D-1}}+O\left({J}^{\frac{D-2}{D-1}}\right) $$ Δ J D = 2 D − 1 3 D − 2 9 D − 2 π 5 D D 2 D − 1 5 Γ D 2 24 π 2 1 D − 1 ϵ D − 1 2 D − 1 × J D D − 1 + O J D − 2 D − 1 where next-to-leading order onwards were not computed here due to technical cumbersomeness, despite there are no fundamental difficulties. We also compare the result at ϵ = 1,$$ \Delta (J)=0.293\times {J}^{3/2}+\cdots $$ Δ J = 0.293 × J 3 / 2 + ⋯ to the actual data from the Monte-Carlo simulation in three dimensions [2], and the discrepancy of the coefficient 0.293 from the numerics turned out to be 13%. Additionally, we also find a crossover of ∆(J; D) from ∆(J) ∝ $$ {J}^{\frac{D}{D-1}} $$ J D D − 1 to ∆(J) ∝ J, at around J ∼ 1/ϵ, as one decreases J while fixing ϵ (or vice versa), reflecting the fact that there are no interacting fixed-point at ϵ = 0. Based on this behaviour, we propose an interesting double-scaling limit which fixes λ ≡ Jϵ, suitable for probing the region of the crossover. I will give ∆(J; D) to next-to-leading order in perturbation theory, either in 1/λ or in λ, valid when λ ≫ 1 and λ ≪ 1, respectively.

Streamer compass validation and verification

During the acquisition of seismic data in a marine environment, the positions of the hydrophone groups are derived from data obtained using compass units deployed at intervals along the length of a streamer. Interpretation of the compass data to derive the streamer position has always proved difficult. The data is a set of measurements of spatial gradients at known distances along the streamer. The integration method used to derive the streamer shape requires additional data to resolve the undetermined constant. Thus the derived positions of the hydrophone groups have a translational indeterminacy. The compass measurements are independent and all measured azimuths can he expected to have similar error estimates. However streamer shapes, derived using a least‐squares polynomial fitting algorithm are critically dependent on the accuracy of data obtained from the first and last units. These features have lead to innovative schemes to position the two ends of the streamers. Lasers, acoustics, and radio positioning, have all been used to locate accessible points closest to the streamer ends viz the towpoints and the tailbuoy. While this additional data has improved the quality of streamer positioning, these methods still leave the compasses as the only means of generating position information in the region where the hydrophone groups are actually situated.

Pulsation of Ventilated Cavities

The problem of pulsating supercavities under artificial ventilation is analytically treated as a resonance problem of a two-dimensional gas-liquid system using a linearized method. A simple kinematical consideration and a dynamical model of the flow lead to solutions for frequency and amplitude of pulsations. The criteria of pulsation are given in terms of a formula relating σv. and σ Maximum air-carrying capacities of pulsating cavities are also estimated. Most of the formulas involve an undetermined constant which must be estimated by using experimental data. The analytical results are compared with the experimental data obtained at the St. Anthony Falls Hydraulic Laboratory, and, in general, good agreement is obtained. It is found that pulsation is possible only for a two-dimensional cavity or a cavity in which a substantial portion of the span can be regarded as two-dimensional. The existence of a free surface is also essential to pulsation. The strong effect of the free surface suggests that pulsation may become an important problem in the open sea only when submergence is relatively small.

Correlation of Creep Properties by a Diffusion Analogy

The necessity of atom movements for both creep and self-diffusion suggests a method of correlating the constant-stress creep properties of pure metals. The concept of steady-state creep is discarded, and two empirical approximations for the strain-time behavior of pure polycrystalline metals lead to a creep equation defining an activation energy as the only undetermined constant. For cases where the orientation of individual crystals is important, a second constant is required. Application of this equation to published creep data effects a correlation which indicates that the apparent activation energies observed for creep and self-diffusion show a similar temperature dependence. The effect of stress upon the activation energy for several metals is described approximately, but the need for further experiments encompassing lower values of stress is revealed. The qualitative effects of impurities, grain size, cold-working, and surface conditions upon creep as predicted by a diffusion analogy are found to be in agreement with experimental results, but it is noted that the analogy does not hold if creep deformation is obtained as a result of slip.

The Flow of an Incompressible Fluid through an Axial Turbo-Machine with any Number of Rows

SummaryA method is described whereby, at any point in an infinite parallel annulus, the approximate axial velocity due to a single row of high aspect ratio blades may be calculated from a knowledge of the conditions of flow adjacent to the blades. The method is based on the assumption of a simplified expression for the radial velocity, being the product of an unknown function of the radius and an exponential term independent of the radius containing an undetermined constant; the function and the undetermined constant are calculated by reference to the conditions of flow in the plane of the row considered. The flow due to any number of rows is then obtained by summing the radial velocity fields due to each row and obtaining the axial velocities by integration of the equation of continuity.The solution of the problem with infinitely many rows is shown to have a simple form by virtue of the fact that the flow (provided that the velocities remain finite) settles down to a pattern which is periodic by one stage pitch.