Central limit theorem for multiplicative class functions on the symmetric group

Hambly, Keevash, O'Connell and Stark have proven a central limit theorem for the characteristic polynomial of a permutation matrix with respect to the uniform measure on the symmetric group. We generalize this result in several ways. We prove here a central limit theorem for multiplicative class functions on symmetric group with respect to the Ewens measure and compute the covariance of the real and the imaginary part in the limit. We also estimate the rate of convergence with the Wasserstein distance.Comment: 23 pages; the mathematics is the same as in the previous version, but there are several improvments in the presentation, including a more intuitve name for the considered function

Shift in critical temperature for random spatial permutations with cycle weights

We examine a phase transition in a model of random spatial permutations which originates in a study of the interacting Bose gas. Permutations are weighted according to point positions; the low-temperature onset of the appearance of arbitrarily long cycles is connected to the phase transition of Bose-Einstein condensates. In our simplified model, point positions are held fixed on the fully occupied cubic lattice and interactions are expressed as Ewens-type weights on cycle lengths of permutations. The critical temperature of the transition to long cycles depends on an interaction-strength parameter $\alpha$. For weak interactions, the shift in critical temperature is expected to be linear in $\alpha$ with constant of linearity $c$. Using Markov chain Monte Carlo methods and finite-size scaling, we find $c = 0.618 \pm 0.086$. This finding matches a similar analytical result of Ueltschi and Betz. We also examine the mean longest cycle length as a fraction of the number of sites in long cycles, recovering an earlier result of Shepp and Lloyd for non-spatial permutations.Comment: v2 incorporated reviewer comments. v3 removed two extraneous figures which appeared at the end of the PDF

Advanced layout of a fibre Bragg grating strain gauge rosette

A temperature-compensated strain-sensing scheme based on fiber Bragg gratings (FBGs) that is suitable for strain mapping applications is described. FBGs are bonded to a backing patch, together with an extra grating that is used for temperature compensation/measurement. The patch provides a simpler and more robust way of attaching the FBGs to a structure than directly mounting a bare fiber, though it was necessary to design it in such a way that there was minimal reduction in the strain transferred from the structure to the sensing fibers. Finite element (FE) analysis was used to help design the patch, which was then constructed accordingly. The authors have demonstrated experimentally that the use of the backing patch produces a reduction in strain sensitivity of only around 4%, which is slightly better than theoretically predicted values. The temperature measuring FBG had to be bonded in such a way that it experienced the changes in temperature, but not the strain, to which the structure was subjected. A design for doing this was developed and proven. The use of a backing patch to develop a rosette configuration of Bragg gratings, each having a different peak reflective wavelength, is described. The rosette configuration is one that is frequently used with electrical strain gauges and allows here to determine both the magnitude and the direction of strain

Structural damage location with fiber bragg grating rosettes and lamb waves

The aim of this study is to present the results of testing a damage detection and damage localization system based on fiber Bragg grating sensors. The objective of the system is to detect and locate damage in structures such as those found in aerospace applications. The damage identification system involves Bragg gratings for sensing ultrasound by detecting the linear strain component produced by Lamb waves. A tuneable laser is used for the interrogation of the Bragg gratings to achieve high sensitivity detection of ultrasound. The interaction of Lamb waves with damage, e.g., the reflection of the waves at defects, allows the detection of damage in structures by monitoring the Lamb wave propagation characteristics. As the reflected waves produce additional components within the original signal, most of the information about the damage can be found in the differential signal of the reference and the damage signal. Making use of the directional properties of the Bragg grating the direction of the reflected acoustic waves can be determined by mounting three of the gratings in a rosette configuration. Two suitably spaced rosettes are used to locate the source of the reflection, i.e., the damage, by taking the intersection of the directions given by each rosette. A genetic algorithm (GA) can be used to calculate that intersection and to account for any ambiguities from the Lamb wave measurements. The performance of the GA has been studied and optimized with respect to the localization task. Initial experiments are carried out on an aluminum structure, where holes were drilled to simulate the presence of damage. The results show very good agreement between the calculated and actual positions of the damage

Spatial random permutations with small cycle weights

We consider the distribution of cycles in two models of random permutations, that are related to one another. In the first model, cycles receive a weight that depends on their length. The second model deals with permutations of points in the space and there is an additional weight that involves the length of permutation jumps. We prove the occurrence of infinite macroscopic cycles above a certain critical density