Random Operator Approach for Word Enumeration in Braid Groups

We investigate analytically the problem of enumeration of nonequivalent primitive words in the braid group B_n for n >> 1 by analysing the random word statistics and the target space on the basis of the locally free group approximation. We develop a "symbolic dynamics" method for exact word enumeration in locally free groups and bring arguments in support of the conjecture that the number of very long primitive words in the braid group is not sensitive to the precise local commutation relations. We consider the connection of these problems with the conventional random operator theory, localization phenomena and statistics of systems with quenched disorder. Also we discuss the relation of the particular problems of random operator theory to the theory of modular functionsComment: 36 pages, LaTeX, 4 separated Postscript figures, submitted to Nucl. Phys. B [PM

Single-molecule imaging of Wnt3A protein diffusion on living cell membranes

Wnt proteins are secreted, hydrophobic, lipidated proteins found in all animals that play essential roles in development and disease. Lipid modification is thought to facilitate the interaction of the protein with its receptor, Frizzled, but may also regulate the transport of Wnt protein and its localisation at the cell membrane. Here, by employing single-molecule fluorescence techniques, we show that Wnt proteins associate with and diffuse on the plasma membranes of living cells in the absence of any receptor binding. We find that labelled Wnt3A transiently and dynamically associates with the membranes of Drosophila S2 cells, diffuses with Brownian kinetics on flattened membranes and on cellular protrusions, and does not transfer between cells in close contact. In S2R+ cells, which express Frizzled receptors, membrane diffusion rate is reduced and membrane residency time is increased. These results provide direct evidence of Wnt3A interaction with living cell membranes and represent a new system for investigating the dynamics of Wnt transport.We thank the Royal Society for the University Research Fellowship of S.F.L. (UF120277). We thank the MRC for funding A.A.J’s studentship (MR/J004103/1). We also thank the Bone Research Society, UK for a Barbara Mawer travel award and the Tissue and Cell Engineering Society for Short Scientific Mission award to A.A.J. This work was supported in part by the National Institute of General Medical Sciences, Grant No. R01GM086196 (to W.E.M)

Disorder-assisted error correction in Majorana chains

It was recently realized that quenched disorder may enhance the reliability of topological qubits by reducing the mobility of anyons at zero temperature. Here we compute storage times with and without disorder for quantum chains with unpaired Majorana fermions - the simplest toy model of a quantum memory. Disorder takes the form of a random site-dependent chemical potential. The corresponding one-particle problem is a one-dimensional Anderson model with disorder in the hopping amplitudes. We focus on the zero-temperature storage of a qubit encoded in the ground state of the Majorana chain. Storage and retrieval are modeled by a unitary evolution under the memory Hamiltonian with an unknown weak perturbation followed by an error-correction step. Assuming dynamical localization of the one-particle problem, we show that the storage time grows exponentially with the system size. We give supporting evidence for the required localization property by estimating Lyapunov exponents of the one-particle eigenfunctions. We also simulate the storage process for chains with a few hundred sites. Our numerical results indicate that in the absence of disorder, the storage time grows only as a logarithm of the system size. We provide numerical evidence for the beneficial effect of disorder on storage times and show that suitably chosen pseudorandom potentials can outperform random ones.Comment: 50 pages, 7 figure

Some Aspects of Multifractal analysis

The aim of this survey is to present some aspects of multifractal analysis around the recently developed subject of multiple ergodic averages. Related topics include dimensions of measures, oriented walks, Riesz products etc

2023 Roadmap on ammonia as a carbon-free fuel

The 15 short chapters that form this 2023 ammonia-for-energy roadmap provide a comprehensive assessment of the current worldwide ammonia landscape and the future opportunities and associated challenges facing the use of ammonia, not only in the part that it can play in terms of the future displacement of fossil-fuel reserves towards massive, long-term, carbon-free energy storage and heat and power provision, but also in its broader holistic impacts that touch all three components of the future global food-water-energy nexus

Random ballistic growth and diffusion in symmetric spaces

Sequential ballistic deposition (BD) with next-nearest-neighbor (NNN) interactions in a N-column box is viewed a time-ordered product of N\times N-matrices consisting of a single sl_2-block which has a random position along the diagonal. We relate the uniform BD growth with the diffusion in the symmetric space H_N=SL(N,R)/SO(N). In particular, the distribution of the maximal height of a growing heap is connected with the distribution of the maximal distance for the diffusion process in H_N. The coordinates of H_N are interpreted as the coordinates of particles of the one--dimensional Toda chain. The group-theoretic structure of the system and links to some random matrix models are also discussed.Comment: 29 pages, 7 figures. Revised and published version. To appear in Nuclear Physics

Fractures in myelomeningocele

BACKGROUND: In patients with myelomeningocele (MMC), a high number of fractures occur in the paralyzed extremities, affecting mobility and independence. The aims of this retrospective cross-sectional study are to determine the frequency of fractures in our patient cohort and to identify trends and risk factors relevant for such fractures. MATERIALS AND METHODS: Between March 1988 and June 2005, 862 patients with MMC were treated at our hospital. The medical records, surgery reports, and X-rays from these patients were evaluated. RESULTS: During the study period, 11% of the patients (n = 92) suffered one or more fractures. Risk analysis showed that patients with MMC and thoracic-level paralysis had a sixfold higher risk of fracture compared with those with sacral-level paralysis. Femoral-neck z-scores measured by dual-energy X-ray absorptiometry (DEXA) differed significantly according to the level of neurological impairment, with lower z-scores in children with a higher level of lesion. Furthermore, the rate of epiphyseal separation increased noticeably after cast immobilization. Mainly patients who could walk relatively well were affected. CONCLUSIONS: Patients with thoracic-level paralysis represent a group with high fracture risk. According to these results, fracture and epiphyseal injury in patients with MMC should be treated by plaster immobilization. The duration of immobilization should be kept to a minimum (<4 weeks) because of increased risk of secondary fractures. Alternatively, patients with refractures can be treated by surgery, when nonoperative treatment has failed